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abstract: 'Convection over a wavy heated bottom wall in the air flow has been studied in the experiments with the Rayleigh number $\sim 10^8$. It is shown that the mean temperature gradient in the flow core inside the large-scale circulation is directed upward, that corresponds to the stably stratified flow. In this experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequencies of the observed large-scale waves vary from 0.01 Hz to 0.1 Hz, while the turbulent time in the integral scale is about 0.5 s. The measured spectrum of these waves contains several localized maxima, that implies an existence of waveguide resonators for the internal gravity waves. For the same imposed mean temperature differences between bottom and upper walls, but in the experiments with a smooth plane bottom wall, there are much less locations with a stably stratified flow, and the internal gravity waves are not observed.'
author:
- 'L. Barel'
- 'A. Eidelman'
-
- 'G. Fleurov'
- 'N. Kleeorin'
- 'A. Levy'
- 'I. Rogachevskii'
title: Detection of standing internal gravity waves in experiments with convection over a wavy heated wall
---
Introduction
============
Temperature stratified turbulence in convective and stably stratified flows has been investigated theoretically, experimentally and in numerical simulations due to numerous applications in geophysical, astrophysical and industrials flows (see, e.g., [@T73; @MY75; @KF84; @Z91; @G92; @N11; @C14; @AGL09; @LX10; @CS12]). One of the key ingredients of stably stratified flows are internal gravity waves. In atmospheric and oceanic turbulence they have been a subject of intense research (see, e.g., [@B74; @GH75; @M81; @N02; @S10; @GM79; @FA03; @SS02; @FE81; @FEF84; @EF84; @F88; @EF93; @C99; @Z02; @JSG03; @BP04; @JRF05; @SN15]). In the atmosphere, internal gravity waves exist at scales ranging from meters to kilometers, and are measured by direct probing or remote sensing using radars and lidars [@FA03; @C99]. The sources of internal gravity waves can be flows over complex terrain, strong wind shears, convective and other local-scale motions underlying the stably stratified layer, and wave-wave interactions [@FA03; @SS02]. The internal gravity waves propagation is complicated by variable wind and density profiles causing refraction, reflection, focusing, and ducting.
The internal gravity waves can strongly affect the small-scale turbulence. In particular, these waves create additional productions of turbulent energy and additional vertical turbulent fluxes of momentum and heat. In particular, the waves emitted at a certain level, propagate upward, and the losses of wave energy cause the production of turbulence energy. These effects have been studied theoretically in [@ZKR09; @KRZ18], where the energy- and flux-budget (EFB) turbulence closure theory which accounts for large-scale internal gravity waves (IGW) for stably stratified atmospheric flows has been developed. For the stationary (in statistical sense) and homogeneous turbulence, the EFB theory without large-scale IGW [@ZKR07; @ZKR08; @ZKR10; @ZKR13] yields universal dependencies of the main turbulence parameters on the flux Richardson number (defined as the ratio of the consumption of turbulent kinetic energy (TKE) needed for overtaking buoyancy forces to the TKE production by the velocity shear). Due to the large-scale IGW, these dependencies lose their universality [@ZKR09; @KRZ18]. The maximum value of the flux Richardson number (universal constant $\approx 0.25$ in the absence of the large-scale IGW) becomes strongly variable. In the vertically homogeneous stratification, the flux Richardson number increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy and even can reach very small values [@ZKR09; @KRZ18].
Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction of the wave propagation (downward or upward), the internal gravity waves either strengthen or weaken the total vertical flux of momentum [@ZKR09; @KRZ18].
Even in a convective turbulence, stably stratified regions can be formed, where the mean temperature gradient in the flow core inside the large-scale circulation is directed upward [@TSW17; @NSS2000]. In these regions internal gravity waves are generated, which affect the turbulence. In spite of many studies of stratified turbulence and internal gravity waves, a mechanism of formation of the stably stratified regions in convective turbulence, and generation of internal gravity waves in these regions is not comprehensively studied and understood.
In various flows a complex terrain (i.e., canopy and various topography) strongly affects the stratified turbulence [@B95]. It changes local temperature gradients, heat and mass fluxes, and affects a local structure of fluid flows. Therefore, one of the important questions — what is the effect of complex terrain on convective and stably stratified turbulence?
To model the effect of complex terrain, the Rayleigh-Bénard convection (RBC) with modulated boundaries has been investigated in [@KP78; @SWB08; @WSB12; @WSB14; @FPZ11; @TSW17; @ZSVL17; @RCC01]. In particular, the influence of a modulated boundary on RBC by a lithographically fabricated periodic texture on the bottom plate has been studied experimentally in [@WSB14]. The different convection patterns have been obtained by varying the Rayleigh number and the wave number of the modulated boundary. For small Rayleigh numbers, convection takes the form of straight parallel rolls. With increasing Rayleigh number, a secondary instability is excited and the convection has more complex patterns [@WSB14]. This secondary instability has been studied theoretically and numerically in [@FPZ11].
The roughness effect on the heat transport in RBC has been investigated in two-dimensional numerical simulations in [@TSW17; @ZSVL17] by varying the height and wavelength of the roughness elements. The sinusoidal roughness profile has been chosen in [@TSW17; @ZSVL17]. The ultimate regime of thermal convection (when the boundary layers undergo a transition leading to the generation of smaller scales near the boundaries that increase the system’s efficiency in transporting the heat) has been found in [@TSW17]. This regime in which the heat flux becomes independent of the molecular properties of the fluid, has been predicted in [@K62] (see also [@GL11]). The first experiment designed to use roughness to reach the ultimate regime at accessible Rayleigh numbers has been made in [@RCC01]. One of the key role of the roughness elements is the production and release of the plumes from the roughness elements, resulting formation of larger plumes. This can cause an increase in the efficiency of the heat transfer [@TSW17; @SPV06]. The existence of two universal regimes in RBC, namely the ultimate regime and the classical boundary-layer-controlled regime with increased Rayleigh number, have been demonstrated in [@ZSVL17]. The transition from the first to the second regime is determined by the competition between bulk and boundary layer flow. The bulk-dominated regime corresponds to the ultimate regime.
In the present study, we investigate another aspects related to effects of complex terrain on turbulent convection. In particular, we study formation of the stably stratified regions and excitation of internal gravity waves in laboratory experiments with turbulent convection over a wavy heated bottom wall in the air as the working fluid. An interactions of the large-scale convective circulation with a wavy heated bottom wall strongly modifies the mean temperature spatial distribution producing stably stratified regions in the flow core, and causes generation of large-scale standing internal gravity waves. The measured spectrum of the waves contains several localized maxima, that indicates an existence of the waveguide resonators of the internal gravity waves in the stably stratified regions. We have compared the results of these experiments for the same imposed mean temperature differences between bottom and upper walls with those obtained in the experiments with a smooth plane bottom wall.
This paper is organized as follows. In Section II we determine the frequencies of the standing internal gravity waves in stably stratified flow. In Section III we describe the experimental set-up and instrumentation. The results of laboratory study and comparison with the theoretical predictions are presented in Section IV. Finally, conclusions are drawn in Section V.
Internal gravity waves
======================
Let us first consider the internal gravity waves propagating in a stably stratified fluid in the absence of turbulence and neglecting dissipations. These waves are described by the following momentum and entropy equations written in the Boussinesq approximation: $$\begin{aligned}
{\partial {\bm V}^{\rm W} \over \partial t} + ({\bm V}^{\rm W} \cdot {\bm \nabla}) {\bm V}^{\rm W} &=& - {{\bm \nabla} P^{\rm W} \over \rho} + g \, S^{\rm W} {\bm e} ,
\label{B1}\\
{\partial S^{\rm W} \over \partial t} + ({\bm V}^{\rm W} \cdot {\bm \nabla}) S^{\rm W} &=& - g^{-1} N^2 \, V^{\rm W}_j e_j,
\label{B2}\end{aligned}$$ (see, e.g., [@T73; @M81; @N02; @S10]), where ${\bm V}^{\rm W}$, $S^{\rm W}$ and $P^{\rm W}$ are the velocity, entropy and pressure, respectively, which are caused by the internal gravity waves, ${\bm e}$ is the vertical unit vector, ${\bm g}$ is the acceleration due to gravity, $N(z)= (g \nabla_z S)^{1/2}$ is the Brunt-Väisälä frequency, $\rho$ and $S$ are the density and entropy of fluid, respectively. The entropy is defined as $S=c_{\rm v} \, \ln(P \, \rho^{-\gamma})$ and we use the equation of state for the perfect gas, where $T$ and $P$ are the fluid temperature and pressure, and $\gamma=c_{\rm p}/c_{\rm v}$ is the ratio of the specific heats, $c_{\rm p}$ and $c_{\rm v}$, at constant pressure and volume, respectively. The Boussinesq approximation with div ${\bm V}^{\rm W} =0$ is applied here.
Linearised Eqs. (\[B1\]) and (\[B2\]) yield the frequency $\omega$ of the internal gravity waves: $$\begin{aligned}
\omega = N(z) {k_h \over k} ,
\label{B11}\end{aligned}$$ where ${\bm k}={\bm k}_h + {\bm e} k_z$ is the wave vector and ${\bm k}_h=(k_x, k_y)$ is the wave vector in the horizontal direction. Propagation of the internal gravity waves in the stably stratified flow in the approximation of geometrical optics is determined by the following Hamiltonian equations: $$\begin{aligned}
{\partial {\bm r} \over \partial t} &=& {\partial \omega \over \partial {\bm k}},
\label{B16}\\
{\partial {\bm k} \over \partial t} &=& - {\partial \omega \over \partial {\bm r}},
\label{B10}\end{aligned}$$ (see, e.g., [@W62]), where ${\bm r}$ is the radius-vector of the centre of the wave packet. Since the Brunt-Väisälä frequency $N=N(z)$, the only non-zero spatial derivative, $\nabla_z \omega \not=~0$, is in the vertical direction. Therefore, Eq. (\[B10\]) yields ${\bm k}_h=$ const. The vertical component of the wave vector $k_z=k_z(z)$ is determined from Eq. (\[B11\]): $$\begin{aligned}
k_z(z)=k_h \left({N^2(z) \over \omega^2} -1\right)^{1/2} .
\label{B12}\end{aligned}$$
Taking twice [**curl**]{} to exclude the pressure term in linearised Eq. (\[B1\]), calculating the time derivative of the obtained equation and using linearised Eq. (\[B2\]), we arrive at the following equation: $$\begin{aligned}
{\partial^2 \over \partial t^2} \, \Delta {\bm V}^{\rm W} &=& \left[{\bm \nabla} ({\bm e} \cdot {\bm \nabla}) - {\bm e} \Delta \right] V^{\rm W}_z N^2(z) ,
\label{B3}\end{aligned}$$ which is equivalent to the system of equations for the vertical and horizontal velocity components: $$\begin{aligned}
{\partial^2 \over \partial t^2} \, \Delta V_z^{\rm W} &=& - N^2(z) \Delta_\perp V^{\rm W}_z ,
\label{B4}\\
{\partial^2 \over \partial t^2} \, \Delta {\bm V}^{\rm W}_\perp &=& {\bm \nabla}_\perp \Delta_z \left[V^{\rm W}_z N^2(z) \right] .
\label{B5}\end{aligned}$$ Here ${\bm V}^{\rm W}={\bm V}^{\rm W}_\perp + {\bm e} V_z^{\rm W}$, ${\bm V}^{\rm W}_\perp=(V^{\rm W}_x, V^{\rm W}_y)$ is the horizontal velocity for IGW and $\Delta=\Delta_\perp + \Delta_z$. Solution of Eq. (\[B4\]) for $V_z^{\rm W}(t,{\bm r})$ we seek for in the form of standing internal gravity waves existing in the range $z_{\rm min} \leq z \leq z_{\rm max}$: $$\begin{aligned}
V_z^{\rm W}(t,{\bm r}) &=& V_\ast \cos (\omega t) \cos ({\bm k}_h \cdot {\bm r})
\nonumber\\
&& \times \sin \left(\int_{z_{\rm min}}^z k_z(z') \,dz' + \varphi\right) .
\label{B6}\end{aligned}$$ Substituting Eq. (\[B6\]) into Eq. (\[B5\]), we obtain solution for the horizontal velocity ${\bm V}^{\rm W}_\perp(t,{\bm r})$: $$\begin{aligned}
{\bm V}^{\rm W}_\perp(t,{\bm r}) &=& - {\bm k}_h {k_z(z) \over k_h^2} \,
V_\ast \cos (\omega t) \sin ({\bm k}_h \cdot {\bm r})
\nonumber\\
&& \times \cos \left(\int_{z_{\rm min}}^z k_z(z') \,dz'+ \varphi\right) .
\label{B8}\end{aligned}$$ Using Eqs. (\[B2\]) and (\[B6\]), we obtain solution for $S^{\rm W}(t,{\bm r})$: $$\begin{aligned}
S^{\rm W}(t,{\bm r}) &=& {N^2(z) \over g \, \omega} \,
V_\ast \sin (\omega t) \cos ({\bm k}_h \cdot {\bm r})
\nonumber\\
&& \times \sin \left(\int_{z_{\rm min}}^z k_z(z') \,dz'+ \varphi\right) .
\label{B9}\end{aligned}$$
The solution (\[B6\]) should satisfy the following boundary conditions: $V_z^{\rm W}(z=z_{\rm min})=0$ and $V_z^{\rm W}(z \approx z_{\rm max})=0$. The boundary condition, $V_z^{\rm W}(z \approx z_{\rm max})=0$, at the vicinity $z \approx z_{\rm max}$ implies that $$\begin{aligned}
\int_{z_{\rm min}}^{z_{\rm max}} k_z(z') \,dz' = \pi \left(m + {1 \over 4} \right) ,
\label{B7}\end{aligned}$$ where $z_{\rm max}$ is the reflection (or ”turning") point in which $k_z(z=z_{\rm max})=0$. To get the condition (\[B7\]), we use an analogy between this problem and the quantum mechanics problem that is related to the behavior of the wave function near the turning points in the semi-classical limit [@LL13]. The wave functions are described in terms of the Airy functions, and Eq. (\[B7\]) is analogous to the Bohr-Sommerfeld quantization condition. In particular, the phase $\varphi$ in Eq. (\[B8\]) is determined using asymptotics \[$\propto \sin (\int_{z_{\rm min}}^{z} k_z(z') \,dz'+ \pi/4)$\] for the Airy function in the vicinity of $z \approx z_{\rm max}$.
Equation (\[B7\]) yields the frequencies of the standing internal gravity waves for the quadratic profile of the Brunt-Väisälä frequency $N^2(z) = N^2_0(1-z^2/L_N^2)$: $$\begin{aligned}
\omega_m &=& {N_0 \over k_h L_N} \biggl\{\biggl[\left(m + {1 \over 4}\right)^2 + \left(k_h L_N\right)^2 \biggr]^{1/2}
\nonumber\\
&&- \left(m + {1 \over 4}\right)\biggr\} ,
\label{B14}\end{aligned}$$ where $m=0, 1, 2, ...$. Equation (\[B14\]) implies the existence of a discrete spectrum of the standing internal gravity waves. In the long wavelength limit, $k_h L_N \ll 1$, Eq. (\[B14\]) yields $$\begin{aligned}
\omega_m = {2 N_0 \over 4 m +1} k_h L_N .
\label{B15}\end{aligned}$$ We will use Eq. (\[B14\]) in the experimental study of convection over a wavy heated bottom wall, where the large-scale internal gravity waves are excited in the stably stratified regions formed inside the flow core of the large-scale circulation.
Experimental set-up
===================
In this section we describe the experimental set-up. The experiments have been conducted in air as the working fluid in rectangular chamber with dimensions $L_x \times L_y \times L_z$, where $L_x = L_z = 26$ cm, $L_y=58$ cm and the axis $z$ is in the vertical direction. The side walls of the chamber are made of transparent Perspex with the thickness of $1$ cm.
A vertical mean temperature gradient in the turbulent air flow is formed by attaching two aluminium heat exchangers to the bottom and top walls of the test section (a heated bottom and a cooled top wall of the chamber). The temperature difference between the top and bottom plates, $\Delta T$, in the experiments are from 40 K to 70 K. The top plate is a bottom wall of the tank with cooling water. Temperature of water circulating through the tank and the chiller is kept constant within $0.1$ K. Cold water is pumped into the cooling system through two inlets and flows out through two outlets located at the side wall of the cooling system.
We study the effects of complex terrain on the structure of the velocity and temperature fields in stratified turbulence. In the laboratory experiments complex terrain is modelled by a wavy bottom wall of the chamber which is manufactured from aluminium. The wavy bottom wall has a sinusoidal modulation containing 7 periods with wavelength 7.8 cm and amplitude 1 cm. A sketch of the experimental set-up is shown in Fig. \[Fig1\]. The results of these experiments for the same imposed mean temperature differences between bottom and upper walls are compared with those obtained in the experimental set-up with a smooth plane bottom wall.
![\[Fig1\] A sketch of the experimental set-up: heat exchangers at the bottom (1) and top (2) walls; temperature probe (3) equipped with 12 E-thermocouples; a wavy bottom wall (4) with a sinusoidal modulation.](Fig1.eps){width="9.5cm"}
The temperature field is measured with a temperature probe equipped with 12 E-thermocouples (with the diameter of 0.13 mm and the sensitivity of $\approx 65 \, \mu$V/K) attached to a rod with a diameter 4 mm. The spacing between thermocouples along the rod is 22 mm. Each thermocouple is inserted into a 1 mm diameter and 45 mm long case. A tip of a thermocouple protruded at the length of 15 mm out of the case. The temperature is measured for 11 rod positions with 22 mm intervals in the horizontal direction, i.e., at 121 locations in a flow. Due to space constraints, the probe is positioned at a 45 degree angle. The exact position of each thermocouple is measured using images captured with the optical system employed in Particle Image Velocimetry (PIV) measurements. A sequence of 500 temperature readings for every thermocouple at every rod position is recorded and processed.
The velocity field is measured using a Stereoscopic Particle Image Velocimetry, see Refs. [@AD91; @RWK07; @W00]. In the experiments we use LaVision Flow Master III system. A double-pulsed light sheet is provided by a Nd-YAG laser (Continuum Surelite $ 2 \times
170$ mJ). The light sheet optics includes spherical and cylindrical Galilei telescopes with tuneable divergence and adjustable focus length. We use the progressive-scan 12 bit digital CCD camera (with pixel size $6.7 \, \mu$m $\times \,
6.7 \, \mu$m and $1280 \times 1024$ pixels) with a dual-frame-technique for cross-correlation processing of captured images. A programmable Timing Unit (PC interface card) generated sequences of pulses to control the laser, camera and data acquisition rate.
An incense smoke with sub-micron particles is used as a tracer for the PIV measurements. Smoke is produced by high temperature sublimation of solid incense grains. Analysis of smoke particles using a microscope (Nikon, Epiphot with an amplification of 560) and a PM-300 portable laser particulate analyzer shows that these particles have an approximately spherical shape and that their mean diameter is of the order of $0.7
\mu$m. The maximum tracer particle displacement in the experiment is of the order of $1/4$ of the interrogation window. The average displacement of tracer particles is of the order of $2.5$ pixels. The average accuracy of the velocity measurements is of the order of $4 \%$ for the accuracy of the correlation peak detection in the interrogation window of the order of $0.1$ pixel (see, e.g., Refs. [@AD91; @RWK07; @W00]).
We determine the mean and the r.m.s. velocities, two-point correlation functions and an integral scale of turbulence from the measured velocity fields. Series of 520 pairs of images acquired with a frequency of 2 Hz, are stored for calculating velocity maps and for ensemble and spatial averaging of turbulence characteristics. The center of the measurement region coincides with the center of the chamber. We measure velocity in a flow domain $256 \times 503$ mm$^2$ with a spatial resolution of 393 $\mu$m / pixel. The velocity field in the probed region is analyzed with interrogation windows of $32
\times 32$ pixels. In every interrogation window a velocity vector is determined from which velocity maps comprising $27 \times 53$ vectors are constructed. The mean and r.m.s. velocities for every point of a velocity map are calculated by averaging over 520 independent velocity maps, and the obtained averaged velocity map is averaged also over the central flow region.
The two-point correlation functions of the velocity field are determined for every point of the central part of the velocity map (with $16
\times 16$ vectors) by averaging over 520 independent velocity maps, which yields 16 correlation functions in horizontal and vertical directions. The two-point correlation function is obtained by averaging over the ensemble of these correlation functions. An integral scale of turbulence, $\ell$, is determined from the two-point correlation functions of the velocity field. The turbulence time scale at the integral scale is $\tau = \ell / \sqrt{\langle {\bf u}^2 \rangle}$, where ${\bf u}$ are the velocity fluctuations and $\sqrt{\langle {\bf u}^2 \rangle}$ is the r.m.s. of the velocity fluctuations. In the experiments we evaluated the variability between the first and the last 20 velocity maps of the series of the measured velocity field. Since very small variability is found, these tests show that 520 velocity maps contain enough data to obtain reliable statistical estimates. The size of the probed region does not affect our results.
In our study we employ a triple decomposition whereby the instantaneous temperature $T^{\rm tot}=T
+ \theta$, where $\theta$ are the temperature fluctuations and $T$ is the temperature determined by sliding averaging of the instantaneous temperature field over the time that is by one order of magnitude larger than the characteristic turbulence time. This temperature $T$ is given by a sum, $T
=\overline{T} + T^{\rm W}$, where $T^{\rm W}$ are the long-term variations of the temperature $T$ due to the large-scale standing internal gravity waves around the mean value $\overline{T}$. The mean temperature $\overline{T}$ is obtained by the additional averaging of the temperature $T$ over the time 400 s.
The time interval during which temperature field is measured at every point, is $400$ s, which corresponds to 500 data points of the temperature field (over which we perform averaging). In the temperature measurements, the acquisition frequency of the temperature is $1.25$ Hz, and the corresponding acquisition time is $0.8$ s. It is larger than the characteristic turbulence time (see below), and is much smaller than the period of the long-term oscillations of the mean temperature caused by internal gravity waves. Therefore, the acquisition frequency of temperature is high enough to provide sufficiently long time series for statistical estimation of the mean temperature $\overline{T}$ and the long-term variations $T^{\rm W}$ of the temperature due to the large-scale internal gravity waves. Similar experimental set-ups, measurement techniques for temperature and velocity fields and data processing procedures have been used previously in our experimental study of different aspects of turbulent convection [@EEKR06; @BEKR09] and stably stratified turbulence [@BEKR11; @EEKR13].
Experimental results
====================
In this section we describe experimental results related to formation of stably stratified regions in turbulent convection and excitation of internal gravity waves. We perform two sets of experiments with (i) a wavy bottom wall and (ii) a smooth plane bottom wall for the same imposed mean temperature differences $\Delta T$ between bottom and upper walls. The velocity measurements show that in the both sets of experiments, a one large-scale circulation is observed for the temperature differences $\Delta T= 40$ K and $\Delta T= 50$ K. In this section we present all results only for these values of $\Delta T$. In Fig. \[Fig2\] we show the mean velocity patterns obtained in the experiments with the wavy bottom wall (upper panel) and smooth plane bottom wall (bottom panel). A difference in the mean velocity patterns is clearly seen at the vicinity of the wavy bottom wall where the flows with the sinusoidal modulation of the mean velocity field are observed (see upper panel in Fig. \[Fig2\]). Other comparison of the results obtained in these two sets of experiments is discussed in the second part of this section.
![\[Fig2\] The mean velocity patterns for the experiments with the wavy bottom wall (upper panel) and the smooth plane bottom wall (bottom panel) obtained at $\Delta T= 50$ K. Here the $z$ and $y$ are measured in mm. ](Fig2a.eps "fig:"){width="7.4cm"} ![\[Fig2\] The mean velocity patterns for the experiments with the wavy bottom wall (upper panel) and the smooth plane bottom wall (bottom panel) obtained at $\Delta T= 50$ K. Here the $z$ and $y$ are measured in mm. ](Fig2b.eps "fig:"){width="6.8cm"}
Now we discuss parameters in the experiments with the wavy bottom wall. The characteristic turbulence time is $\tau = 0.28 - 0.62$ seconds, while the characteristic period for the large-scale circulatory flow is about $10$ s, which is by order of magnitude larger than $\tau$. These two characteristic times are much smaller than the time during which the velocity fields are measured $(\sim 260$ s). The maximum Rayleigh number, ${\rm Ra}=
\alpha \, g \, L_z^3 \, \Delta T /(\nu \, \kappa)$, is about $10^8$, where $\alpha$ is the thermal expansion coefficient, $\nu$ is the kinematic viscosity, $\kappa$ is the thermal diffusivity, and $L_z$ in the experiments with a wavy bottom wall is the hight measured from the lower point of a wavy wall with sinusoidal modulation to the upper wall of the chamber.
The temperature measurements in the experiments with a wavy bottom wall, show that the mean temperature gradient in the flow core (193 mm $\leq y \leq$ 333 mm, 40 mm $\leq z \leq$ 180 mm) inside the large-scale circulation is directed upward, that corresponds to the stably stratified flow. We have measured the temperature field in this region in 80 locations. In Fig. \[Fig3\] we show profiles of the vertical mean temperature gradient $\nabla_z \overline{T}$ in the experiments for different temperature differences $\Delta T$ between the bottom and top walls. These profiles of the vertical mean temperature gradient are also averaged over $y$-coordinate in the core flow (193 mm $\leq y \leq$ 333 mm) of the large-scale circulation. In particular, they are averaged over 8 vertical profiles measured at 8 equidistant positions located in the $y$ direction with the interval $\Delta y = 20$ mm between the temperature probes.
There are two maxima of the mean temperature gradient separated by one minimum in the center of the core flow (see Fig. \[Fig3\]). At the minimum of $\nabla_z \overline{T}$, the mean temperature gradient is positive for the dashed curve (corresponding to the experiments with $\Delta T= 40$ K), and the minimum of $\nabla_z \overline{T}$ tends to very small negative value for the solid curve (for the experiments with $\Delta T= 50$ K). In the case of negative vertical mean temperature gradient $\nabla_z \overline{T}$, the value of $N^2 <0$ and the internal gravity waves cannot exist in the vicinity of this region. This means that for $\Delta T= 50$ K, there are two waveguides for the internal gravity waves, while in the first case ($\Delta T= 40$ K) there is only one waveguide. In these experiments the maximum gradient Richardson number, Ri $=N^2/{\rm Sh}^2$, based on the mean velocity shear, ${\rm Sh}$, of the large-scale circulation is about 1. Here $N$ is the Brunt-Väisälä frequency.
![\[Fig3\] Profiles of the vertical mean temperature gradient $\nabla_z \overline{T}$ in the experiments for different temperature differences $\Delta T$ between the bottom and top walls: 40 K (dashed) and 50 K (solid). Here the hight $z$ is measured in mm, and the mean temperature gradient is measured in K/cm. ](Fig3.eps){width="8.5cm"}
To study internal gravity waves, we determine spectrum of the long-term variations of the temperature field characterising the large-scale IGW. From this spectral analysis we obtain main frequencies of IGW. The spectrum function $E_{T}(f) = \tilde T(f) \, \tilde T^\ast(f)$ for the temperature field (containing 250 frequency data points) has been determined at 80 locations of the stably stratified region, where the temperature $\tilde T(t)=T^{\rm tot} - \overline{T}$ and $\tilde T(f)$ is its Fourier component. For every frequency $f$, the obtained spectrum functions $E_{T}(f)$ have been averaged over 80 locations. In Figs. \[Fig4\] and \[Fig5\] we show the averaged spectrum function $E_{\tilde T}(f)$ of the temperature field in the experiments for the temperature differences $\Delta T= 50$ K and $\Delta T= 40$ K, respectively.
Comparing the main frequencies of internal gravity waves obtained in the temperature measurements with the profile of the Brunt-Väsäla frequency (based on the measured profile of the mean temperature gradient), we determine the horizontal wave-numbers for the observed internal gravity waves. Figures \[Fig6\] and \[Fig7\] show the theoretical dependencies for the frequencies of the internal gravity waves versus the normalized horizontal wavelength $\lambda_h/L_N$ given by Eq. (\[B14\]), where $L_N$ is the characteristic vertical scale of the Brunt-Väsäla frequency variations and $\lambda_h=2 \pi/k_h$ is the horizontal wavelength. The theoretical curves are plotted for the first three main modes of the large-scale IGW.
![\[Fig4\] The averaged spectrum function $E_{\tilde T}(f)$ of the temperature field obtained in the experiments for the temperature differences $\Delta T= 50$ K between the bottom and top walls. The main frequencies of the large-scale IGW measured in Hz are indicated above the maxima of $\tilde T$. The function $\tilde T(f)$ is measured in $K^2$/Hz.](Fig4.eps){width="8.5cm"}
![\[Fig5\] The averaged spectrum function $E_{\tilde T}(f)$ of the temperature field obtained in the experiments for the temperature differences $\Delta T= 40$ K between the bottom and top walls. The main frequencies of the large-scale IGW measured in Hz are indicated above the maxima of $\tilde T$. The function $\tilde T(f)$ is measured in $K^2$/Hz.](Fig5.eps){width="9.5cm"}
![\[Fig6\] The frequencies of the internal gravity waves (measured in Hz) versus the normalized horizontal wavelength $\lambda_h/L_N$: theoretical curves for different modes $m=1$ (solid), $m=2$ (dashed) and $m=3$ (dashed-dotted) and measured frequencies associated with different modes: $m=1$ (diamonds), $m=2$ (snowflakes) and $m=3$ (circles) in the experiments for the temperature differences $\Delta T= 50$ K between the bottom and top walls. ](Fig6.eps){width="9.5cm"}
![\[Fig7\] The frequencies of the internal gravity waves (measured in Hz) versus the normalized horizontal wavelength $\lambda_h/L_N$: theoretical curves for different modes $m=1$ (solid), $m=2$ (dashed) and $m=3$ (dashed-dotted) and measured frequencies associated with different modes: $m=1$ (diamonds), $m=2$ (snowflakes) and $m=3$ (circles) in the experiments for the temperature differences $\Delta T= 40$ K between the bottom and top walls. ](Fig7.eps){width="9.0cm"}
Using the theoretical curves and the measured frequencies of the long-term variations of the temperature fields in the experiments with the temperature differences $\Delta T= 50$ K (Fig. \[Fig6\]) and $\Delta T= 40$ K (Fig. \[Fig7\]), we determine the horizontal wavelengthes associated with different modes of the large-scale IGW. Note that in the experiments where there is only a one waveguide, it is possible to observe two dominant modes. For instance, in the case of two waveguides for the large-scale IGW, we observe four wave frequencies (two in the each waveguide), see Fig. \[Fig6\]. On the other hand, in the experiments with $\Delta T= 40$ K where there is only one waveguide, we observe two frequencies of large-scale standing internal gravity waves (see Fig. \[Fig7\]).
![\[Fig8\] The vertical profiles of the rms velocities $u_y^{\rm rms}$ (stars) and $u_z^{\rm rms}$ (circles) in the experiments with the wavy bottom wall (upper panel) and the smooth plane bottom wall (bottom panel) obtained at $\Delta T= 50$ K. Here the $z$ is measured in mm, and the rms velocities $u_y^{\rm rms}$ and $u_z^{\rm rms}$ are measured in cm/s. ](Fig8a.eps "fig:"){width="7.5cm"} ![\[Fig8\] The vertical profiles of the rms velocities $u_y^{\rm rms}$ (stars) and $u_z^{\rm rms}$ (circles) in the experiments with the wavy bottom wall (upper panel) and the smooth plane bottom wall (bottom panel) obtained at $\Delta T= 50$ K. Here the $z$ is measured in mm, and the rms velocities $u_y^{\rm rms}$ and $u_z^{\rm rms}$ are measured in cm/s. ](Fig8b.eps "fig:"){width="7.5cm"}
To compare the results for the two sets of experiments with the wavy and smooth plane bottom walls, conducted at the same imposed mean temperature differences between bottom and upper walls, we show in Fig. \[Fig8\] the vertical profiles of the rms velocity components $u_y^{\rm rms}=\sqrt{\langle u_y^2 \rangle}$ and $u_z^{\rm rms}=\sqrt{\langle u_z^2 \rangle}$. In particular, in the experiments with the wavy bottom wall, there is a minimum in the vertical profiles of the rms velocities in the center part of the chamber. This minimum may be caused by the presence of stably stratified turbulence in the center part of the chamber where the intensity of the velocity fluctuations are partially depleted by the production of the temperature fluctuations. On the other hand, in the experiments with the smooth plane bottom wall, the vertical distribution of the rms velocities is essentially less inhomogeneous.
In the mean temperature field we observe much more differences for these two kind of experiments. In particular, in the experiments with a smooth plane bottom wall there are much less locations with positive mean temperature gradient (corresponding to the stably stratified flows) and internal gravity waves are not observed there.
Conclusions
===========
In the present study we perform laboratory experiments in convection over a wavy heated bottom wall. An interaction of the large-scale circulation with the wavy heated bottom wall strongly affects the spatial structure of the mean temperature field in the flow core. In particular, we have found that there are many locations with stably stratified regions in the flow core of the large-scale circulation, and the large-scale standing internal gravity waves are excited in these regions. The spectrum of these waves contains several localized maxima, that is an indication of existence of the waveguide resonators for the internal gravity waves. The theoretical predictions are in an agreement with the obtained experimental results. On the other hand, in the experiments with a smooth plane bottom wall but the same imposed mean temperature differences between bottom and upper walls, there are much less locations with a stably stratified turbulence and the internal gravity waves are not detected. The turbulence in the region with stably stratified flows in the experiments with the wavy bottom wall is inhomogeneous, e.g., there is a minimum in the vertical profiles of the rms velocities in the center part of the chamber, where the intensity of the velocity fluctuations are partially depleted by the production of the temperature fluctuations. On the other hand, in the experiments with the smooth plane bottom wall, the vertical distribution of the rms velocities is nearly homogeneous.
We thank A. Krein for his assistance in construction of the experimental set-up. The detailed comments on our manuscript by the two anonymous referees which essentially improved the presentation of our results, are very much appreciated. This research was supported in part by the Israel Science Foundation governed by the Israel Academy of Sciences (grant No. 1210/15).
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Introduction
============
The equilibrium crystal shape (ECS) is that shape which, in the limit of infinitely large volume, yields the minimum free energy of a crystal. For a given arbitrary surface orientation and unit cell the atomic reconstruction that yields the lowest surface free energy can be determined. However, it is well known that in general this will not result in a thermodynamically stable situation, because the surface can further lower its energy by faceting on a macroscopic scale. The ECS provides a set of surface orientations that exist in thermodynamic equilibrium. Except for some situations with degenerate surface energies surfaces of any other orientations will facet.
The faceting of GaAs surfaces has been studied experimentally. Whereas Weiss [*et al*]{}. [@weiss:89] studied the different surface orientations exposed on a round shaped crystal with low-energy electron-diffraction (LEED), Nötzel [*et al*]{}. [@noetzel:92] investigated various planar high-index surfaces with reflection high-energy electron-diffraction (RHEED). Both groups observed for different high-index surface orientations faceting into low-index surfaces. Moreover, surface energies play a major role in the formation of islands during heteroepitaxy. For example, InAs grows on GaAs in the Stranski-Krastanov mode. [@leonard:93] The surface energy of InAs being lower than that of GaAs, first a uniform wetting-layer forms. During further deposition of InAs three-dimensional islands are formed due to strain relaxation. Recently, these quantum dots have attracted great interest. [@leonard:93; @moison:94; @grundmann:95; @shchukin:95] Besides other quantities like the elastic relaxation energy of the islands, the absolute InAs surface energies of the involved facets which we assume to be similar to those of GaAs enter into the theory of the shape and size of the islands.
Both experimental as well as calculated [*absolute*]{} values of the surface energy as a function of orientation are quite scarce. The surface energy has been measured for the GaAs (110) surface in a fracture experiment.[@messmer:81] Relative surface energies and the ECS of Si have been determined [@eaglesham:93; @bermond:95], but to our knowledge no such measurements have been carried through for GaAs. Moreover, it is often difficult to establish whether an observed surface really represents thermodynamic equilibrium. At low temperatures faceting and therefore thermodynamic equilibration may be hindered by insufficient material transport. At high temperatures, kinetics may govern the surface morphologies due to evaporation.
The purpose of this work is to present the [*absolute*]{} values for the surface energy of the GaAs (110), (100), (111), and (111) surfaces calculated from first principles, and the ECS constructed from these data. Empirical potentials do not produce reliable surface properties. [*Ab initio*]{} calculations have been carried out by various groups for different surface orientations of GaAs. Qian [*et al*]{}. [@qian:88a] used an [*ab initio*]{} pseudopotential method to calculate the absolute surface energy of the GaAs (110) surface. They found very good agreement with the experimental cleavage energy. Northrup and Froyen [@northrup:93], Qian [*et al*]{}.[@qian:88], and Ohno [@ohno:93] determined the (100) reconstruction with lowest energy. The absolute surface energies for these reconstructions were not given, however. Kaxiras [*et al*]{}.[@kaxiras:87] calculated energies for GaAs (111) reconstructions relative to the surface energy of the ideal (111) surface. For the (111) surface Kaxiras [*et al*]{}. [@kaxiras:87a] and Northrup [*et al.*]{} [@biegelsen:90] calculated relative surface energies for different $(2 \times 2)$ reconstructions. Based on their results they predicted the (111) equilibrium reconstruction.
However, for geometrical reasons it is impossible to derive absolute surface energies for the (111) and (111) orientations of GaAs from such total-energy calculations. Chetty and Martin [@chetty:92; @chetty:92a] solved this problem by introducing an energy density, which enables the computation of the energies of the top and the bottom surfaces of the slab separately. Having calculated the absolute surface energies for the ideal reference surfaces they transformed the relative surface energies of Kaxiras [*et al*]{}.[@kaxiras:87; @kaxiras:87a] and Northrup [*et al.*]{}[@biegelsen:90] to absolute surface energies. A comparison of these absolute values, however, shows that the two results differ significantly. This difference is not yet understood, and we will come back to it in Section IV below.
We have calculated absolute surface energies for the different orientations directly (i.e., without introducing a reference surface) and consistently with one and the same set of parameters and pseudo-potentials. Before we will detail our results and the ECS of GaAs in Section IV, we will first give an overview of GaAs surface properties in Section II and describe the computational details in Section III.
Chemical Potential and Surface Reconstruction
=============================================
The stable surface reconstruction is the one with the lowest surface free energy. In our case the substrate consists of two elements and thus the difference of the number of atoms of the two species enters as another degree of freedom in addition to the atomic geometry. Non-stoichiometric surfaces are considered by allowing the surface to exchange atoms with a reservoir, which is characterized by a chemical potential. The equilibrium is determined by the minimum of the free energy $$\gamma_{\rm surface} A = E_{\rm surface} - \sum_i \mu_i N_i .$$ The surface free energy $\gamma_{\rm surface}A$ of the surface area $A$ has been calculated for zero temperature and pressure and neglecting zero point vibrations. The chemical potential $\mu_i$ is the free energy per particle in the reservoir for the species $i$, and $N_i$ denotes the number of particles of the species $i$. The temperature dependence is ignored because the contributions tend to cancel for free energy differences.
In experiment the value of the chemical potential can be varied over a certain interval. This interval is limited by the bulk chemical potentials of the condensed phases of Ga and As [@qian:88; @biegelsen:90], corresponding to the two following situations: On the one hand the surface can be in equilibrium with excess Ga-metal, which has the chemical potential $\mu_{\rm Ga
(bulk)}$, and the GaAs bulk with chemical potential $\mu_{\rm GaAs}$. On the other hand the surface can be in equilibrium with bulk As and, again, the GaAs bulk. Both reservoirs can act as sinks and sources of surface atoms. The upper limit of each chemical potential is determined by the condensed phase of the respective element, $$\mu_i < \mu_{i \rm (bulk)},
\label{muup}$$ because otherwise the elemental phase would form on the GaAs surface. Furthermore, in thermodynamic equilibrium the sum of chemical potentials of Ga and As must be equal to the bulk energy per GaAs pair, $$\begin{aligned}
\mu_{\rm Ga} + \mu_{\rm As} & = & \mu_{\rm GaAs} \nonumber\\
& = & \mu_{\rm Ga (bulk)} + \mu_{\rm As (bulk)} - \Delta H_f.
\label{heat}\end{aligned}$$ For the heat of formation we have calculated a value of 0.64 eV using a plane-wave cutoff of 10 Ry which is in good agreement with the experimental value [@hcp:86] of 0.74 eV. For the bulk calculations we computed the bulk energy of Ga in an orthorhombic structure [@qian:88] and the bulk energy of As in a trigonal structure [@needs:86].
In this work we give the surface energies in dependence of the As chemical potential. Therefore, we write equations (\[muup\]) and (\[heat\]) in the following form $$\mu_{\rm As (bulk)} - \Delta H_f < \mu_{\rm As} < \mu_{\rm As (bulk)}.$$ The surface energy is calculated from the total energy $E_{\rm tot}$, $$\gamma_{\rm surface} A = E_{\rm tot} - \mu_{\rm GaAs} N_{\rm Ga}
- \mu_{\rm As}(N_{\rm As} - N_{\rm Ga}).
\label{esurface}$$ The stoichiometry of the surface, $\Delta N = N_{\rm As} - N_{\rm
Ga}$, determines the slope of the surface energy versus the chemical potential. A consistent counting method for $\Delta N$ has to be applied to all orientations. We apply the method of Chetty and Martin [@chetty:91] which utilizes the bulk symmetries of the crystal. For example, following their counting method the ideal (110) cleavage surface is stoichiometric, i.e. the difference $\Delta N$ is equal to zero. Thus the surface energy of the (110) cleavage surface is independent of the chemical potential.
When the chemical potential is varied, different reconstructions with different surface stoichiometries become thermodynamically stable. All experimentally observed reconstructions, however, fulfill certain conditions. First of all, GaAs surfaces favor to be semiconducting, as this leads to a low surface energy. Surface bands in the bulk gap and especially surface bands crossing the Fermi-level will lead to a higher surface energy. The electron counting model [@harrison:79; @pashley:89] gives a simple criterion whether a surface can be semiconducting or not. In the bulk the $sp^3$ hybridized orbitals of GaAs form bonding and antibonding states. At the surface there are partially filled dangling bonds. Their energies are shown schematically in Fig. \[energy\_levels\], they are estimated from the atomic $s$ and $p$ eigenenergies of either species.
=8.7cm
Compared to the dispersion of the conduction and the valence bands, the dangling bond energy of the cation (Ga) falls into the conduction band and therefore it should be empty. The dangling bond energy of the anion (As) lies in the valence band and therefore it should be filled. Thus there has to occur an electron transfer from the Ga to the As dangling bonds. For a low-energy semiconducting surface the dangling bonds in the conduction band have to be empty, exactly filling all the dangling bonds in the valence band. Otherwise the surface becomes metallic and has a higher surface energy. Ga and As surface atoms are added to, or removed from, the ideal bulk-truncated polar surfaces to obtain a low-energy semiconducting surface.
Secondly, the electron transfer from the Ga dangling bonds to the As dangling bonds has consequences for the geometry of the surface reconstructions. The surface Ga atom which has lost an electron favors a $sp^2$ like hybridization. Therefore the Ga atom relaxes inwards and forms a more planar configuration. The dangling bond of arsenic is completely filled and the As atom prefers to form bonds with its three $p$ orbitals. Therefore the bond angle of the surface As atom is close to 90$^\circ$, and the As atom relaxes outwards. These configurations resemble the bond geometry of small molecules like GaH$_3$ and AsH$_3$ and are a general result for surfaces of III-V semiconductors. [@alves:91]
Computational Details
=====================
To determine the surface energies we carried out total-energy calculations using density-functional theory.[@hohenberg:64; @kohn:65] We applied the local-density approximation to the exchange-correlation functional, choosing the parameterization by Perdew and Zunger [@perdew:81] of Ceperley and Alder’s [@ceperley:80] data for the correlation energy of the homogeneous electron gas. The surfaces were described by periodically repeated slabs. All computations were done with an extended version of the computer code [*fhi93cp*]{}. [@stumpf:94] The program employs [*ab initio*]{} pseudopotentials and a plane-wave basis-set. It was generalized to additionally compute the energy density according to Chetty and Martin. [@chetty:92]
The slab geometry leads to serious problems when surface energies of zinc-blende structures are to be calculated for arbitrary orientations. To derive the surface energy from a total energy calculation both surfaces of the slab have to be equivalent. Though such slabs can be constructed for the (110) and the (100) orientation, this is impossible for the (111) orientation: The (111) and the (111) surfaces of GaAs are inequivalent. This follows from the simple geometric property of the zinc-blende structure that the Ga-As double layers are Ga and As terminated on the top and bottom side of the slab, respectively. Chetty and Martin[@chetty:92] solved this problem by introducing an energy density. The energy density itself, however, does not bear any physical significance, only the integrals of the energy density over suitable parts of the supercell (e.g., volumes bounded by bulk mirror planes) lead to well-defined, physically meaningful energies. [@chetty:92] We have checked the accuracy of this approach for our GaAs slabs: Variation of the surface reconstruction on the bottom side of the (100) and (111) slabs results in a negligible change of the surface energy of the surface on the top ($<0.7$ meV/[Å]{}$^2$).
[*Ab initio*]{} norm-conserving pseudopotentials were generated with Hamann’s scheme. [@hamann:89] The cutoff radii for pseudoization have been chosen equal to 0.58 [Å]{}, 0.77 [Å]{}, and 1.16 [Å]{} for the $s$, $p$, and $d$ wave-functions of Ga, and equal to 0.61 [Å]{}, 0.60 [Å]{}, and 1.07 [Å]{} for $s$, $p$, and $d$ wave-functions of As. The semi-local pseudopotentials were further transformed into fully separable Kleinman-Bylander pseudopotentials [@kleinman:82], with the $d$ potential chosen as the local potential. The logarithmic derivatives of the different potentials were examined and various transferability tests [@stumpf:91], e.g. “hardness” tests, were performed. All together the potentials showed good transferability. The structures of the bulk phases of Ga and As are well described by these potentials, the theoretical lattice constants being only slightly smaller than the experimental ones with a relative deviation below 3.5%.
The wave functions were expanded into plane waves [@ihm:79] with a kinetic energy up to 10 Ry. This leads to a convergence error in the surface energies of less than 3 meV/[Å]{}$^2$. The electron density was calculated from special [**k**]{}-point sets [@monkhorst:76], their density in reciprocal space being equivalent to 64 [**k**]{}-points in the whole (100) ($1 \times 1$) surface Brillouin-zone.
For the (100), (111), and (111) surfaces “pseudo-hydrogen” was used to saturate the bottom surfaces of the slabs. [@shiraishi:90] Pseudo-hydrogen denotes a Coulomb-potential with a non-integer core-charge $Z$, together with $Z$ electrons. The Ga and As atoms of these surfaces were fixed at their ideal bulk positions. The Ga terminated surface was saturated with pseudo-hydrogen with an atomic number of $Z=1.25$. On each dangling bond of a Ga surface atom one pseudo-hydrogen was placed. Similarly, the As terminated surface was saturated with pseudo-hydrogen with an atomic number of 0.75. The saturated surfaces are semiconducting without any surface states in the bulk band-gap. There are two main advantages using this pseudo-hydrogen. First of all, the interaction of both surfaces with each other is in this way minimal. Secondly, the surface atoms which are saturated with the pseudo-hydrogen can be kept fixed at ideal bulk positions. Thus thinner slabs can be used and charge sloshing is suppressed.
For polar surfaces, such as the ideal (111) surface, a difficulty arises due to charge transfer from one side of the slab to the opposite side. This charge transfer is hindered by a semiconducting surface, e.g. the pseudo-hydrogen saturated surface at the bottom of the slab. We estimate the uncertainty due to charge transfer to be smaller than 1.4 meV/[Å]{}$^2$ for a polar surface, comparing the surface energies of the pseudo-hydrogen saturated surface derived from two calculations. One is carried through with a semiconducting surface on the top of the slab, the other one with a metallic surface.
We have carried out computations for a large variety of reconstructions of the GaAs (110), (100), (111), and (111) surfaces, which have previously been suggested in literature. Starting from some initial geometry, the atom positions in the topmost layers of the slab were relaxed until the forces on the atoms were smaller than 50 meV/[Å]{}. The other layers were kept fixed at their ideal bulk positions with a bulk lattice-constant of 5.56 Åwhich had been determined theoretically at the same cutoff energy as the slab calculations and using 384 [**k**]{}-points in the whole Brillouin-zone. This value is 1.4% smaller than the experimental lattice constant[@landolt:82] neglecting zero point vibrations.
Results and Discussion
======================
(110) Surface
-------------
The (110) surface is one of the most extensively studied GaAs surfaces (Ref. and references therein). The (110) plane is the cleavage plane of III-V semiconductors. Containing the same number of cations (Ga) and anions (As) it is intrinsically neutral. The cleavage surface does not reconstruct, only a relaxation of surface atomic positions within the $(1 \times 1)$ surface unit cell is observed. The charge from the Ga dangling-bond is transfered into the As dangling-bond, which becomes completely filled. The orbitals of both surface atoms rehybridize, and the zigzag chains of Ga and As surface atoms tilt, with the As atom being raised and the Ga atom being lowered. Thereby the Ga surface atom acquires a nearly planar bonding configuration, while the As surface atom relaxes towards a pyramidal configuration with orthogonal bonds.
We have calculated the surface energy of the relaxed cleavage surface shown in Fig. \[110\_geo\](a). It is stoichiometric ($\Delta N = 0$) and semiconducting.
=7.8cm
In addition, we considered two other surface structures: The Ga terminated (110) surface is shown in Fig. \[110\_geo\](b). Formally it can be constructed from the cleavage surface by substituting all top-layer As atoms by Ga atoms. This surface has a stoichiometry of $\Delta N = -2$ per $(1 \times 1)$ cell, and it fulfills the electron counting criterion. Nevertheless, it is not semiconducting, because the bands of the Ga-Ga surface bonds cross the Fermi-level. The Ga surface atoms do not relax in the same way as the respective Ga and As atoms in the cleavage surface, instead they almost stay in the same plane. Finally, we have calculated the surface energy of the As terminated (110) surface (see Fig. \[110\_geo\](c)). Here the Ga surface atoms have been replaced by As atoms, which yields a surface with a stoichiometry of $\Delta N = 2$ per $(1 \times 1)$ surface unit cell. Also this surface fulfills the electron counting criterion, and it is semiconducting. Both As dangling-bonds are completely filled and lie beneath the Fermi-level. Similar to the Ga terminated surface, the As surface atoms do not relax significantly, but stay in the same plane.
For all three (110) surface reconstructions we used the same super cell, with slabs composed of nine atomic layers and a vacuum region with a thickness equivalent to seven atomic layers. The whole surface Brillouin zone was sampled with 48 special [**k**]{}-points. [@monkhorst:76]
The calculated surface energies are shown in Fig. \[110\] for the three surface structures we have considered.
=8.7cm
For a large range of the chemical potential the cleavage surface is energetically most favorable. Our result for the surface energy of 52 meV/[Å]{}$^2$ is in good agreement with the value of 57eV/[Å]{}$^2$ meV which was calculated by Qian [*et al*]{}. [@qian:88a] using essentially the same [*ab initio*]{} method. Both results compare very well with the experimental surface energy of 54 $ \pm $ 9 meV/[Å]{}$^2$ from fracture experiments by Messmer and Bilello. [@messmer:81] In As-rich environments we find the As terminated surface to exist in thermodynamical equilibrium, in agreement with Northrup’s calculation.[@northrup:91] We obtain a value of 45 meV/[Å]{}$^2$ for the surface energy in an As-rich environment. Kübler [*et al.*]{} [@kuebler:80] provided experimental evidence for the existence of this structure. Using LEED they observed that the surface relaxation was removed as the As coverage was increased. In contrast to the As terminated surface, we find the Ga terminated surface to be unstable even under the most extreme Ga rich conditions.
(100) Surface
-------------
Among the different orientations the (100) surface is the one used most widely for the growth of opto-electronic devices. The (100) surface is polar, i.e. the planes parallel to the surface consist of either only Ga or only As atoms. As a consequence, the stable surface structure [@pashley:89] displays various reconstructions which distinctly differ from those found on the (100) faces of the covalent group IV semiconductors. Däweritz [*et al*]{}. [@daeweritz:90] have derived a steady state “phase” diagram for the surface reconstruction as a function of growth conditions. In their diagram they point out 14 different reconstructions. To our knowledge, the [*equilibrium*]{} phase diagram of the (100) surface has not yet been determined. However, there are certain reconstructions which are generally observed during and after growth. While heating the surface Biegelsen [*et al*]{}. [@biegelsen:90a] observed a sequence of phases from the As-rich $c(4 \times 4)$, $(2 \times 4)$ to the Ga-rich $(4 \times 2)$ reconstructions. For each of these surface unit-cells there exists a large variety of possible atomic configurations.
Chadi [@chadi:87] performed tight binding based total energy minimizations to examine the structure of the $(2 \times 1)$ and $(2
\times 4)$ reconstructed surface. For the $(2 \times 4)$ he suggested two possible atomic configurations with three and two As-dimers ($\beta$ and $\beta2$, notation according to Northrup [*et al*]{}.[@northrup:94]) per surface unit-cell. Moreover, he determined the energy difference between the $(2 \times 4)$ and the related $c(2
\times 8)$ reconstruction to be less than 1 meV/[Å]{}$^2$. As the $(2
\times 4)$ and the $c(2 \times 8)$ are very similar and have only small difference in surface energy, we have not calculated the centered reconstructions $c(2 \times 8)$ and $c(8 \times 2)$. Ohno[@ohno:93] and Northrup[@northrup:93] carried through [*ab initio*]{} calculations of the surface energies. Ohno could exclude various configurations of the $(2 \times 1)$ and $(3 \times 1)$ surface unit-cell from being equilibrium structures. Moreover, he concluded that for the $(2 \times 4)$ reconstruction the phase $\beta$ with three surface dimers is stable, which appeared to be in agreement with the STM observations of Biegelsen [*et al*]{}.[@biegelsen:90a] However, calculations by Northrup [*et al.*]{} [@northrup:94] showed that the most stable $(2 \times 4)$ reconstruction contains two As-dimers in the top layer, which has been confirmed by recent high resolution STM observations. [@hashizume:94] Northrup [*et al*]{}. also investigated the energetics of the $(4 \times 2)$ and $c(4 \times 4)$ reconstructions. For the $(4 \times 2)$ reconstruction they found a two-dimer phase to be energetically favorable in agreement with STM investigations.[@xue:95] However, a recent analysis of LEED intensities by Cerd’a [*et al.*]{}[@cerda:95] suggests that the top layer consist of three Ga dimers per $(4 \times 2)$ unit cell. For the $c(4 \times 4)$ reconstruction Northrup [*et al*]{}. considered a three-dimer phase[@biegelsen:90a] which they found to be stable in certain conditions with respect to the $(2 \times 4)$ and $(4 \times
2)$ reconstructions. On the other hand a two-dimer phase was suggested by Sauvage-Simkin [*et al*]{}. [@simkin:89] on the basis of X-ray scattering experiments, and by Larsen [*et al*]{}. [@larsen:83] who studied the surface with a number of different experimental techniques.
In our calculations we have considered all atomic configurations with a $(2 \times 4)$ and a $(4 \times 2)$ surface unit-cell that were previously investigated by Northrup [*et al*]{}. [@northrup:93; @northrup:94] For the $c(4 \times 4)$ reconstruction we took into account the three-dimer phase [@biegelsen:90a] and a structure which has two instead of three As-dimers in the top layer. [@simkin:89; @larsen:83] The total energy calculations were performed using supercells containing seven layers of GaAs. The thickness of the vacuum corresponded to five layers GaAs.
In Fig. \[100\_geo\] the geometries of those surface structures are shown that have minimum surface energy within some range of the chemical potential and therefore exist in thermodynamic equilibrium.
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All four structures fulfill the electron counting criterion and are semiconducting, i.e., the anion dangling bonds are filled and the cation dangling bonds are empty. Furthermore the surfaces display Ga-Ga bonds and As-As bonds, both having filled bonding and empty antibonding states. The $\alpha(2 \times 4)$ reconstruction (Fig.\[100\_geo\](a)) is stoichiometric ($\Delta N = 0 $). In the top layer four As atoms are missing per $(2 \times 4)$ cell. The surface As atoms form two dimers. The Ga-layer underneath is complete, but differs from the bulk geometry by two Ga-Ga bonds which are formed between the Ga atoms in the region of the missing As dimers. Removing the Ga atoms in the missing dimer region one obtains the $\beta2(2
\times 4)$ structure in Fig. \[100\_geo\](b) with a stoichiometry of $\Delta N = \frac{1}{4}$ per $(1 \times 1)$ unit cell. The completely As-terminated $c(4 \times 4)$ surface shown in Fig. \[100\_geo\](c) has a stoichiometry of $\Delta N = \frac{5}{4}$ per $(1 \times 1)$ unit cell. It consists of three As-dimers which are bonded to a complete As-layer beneath. The $\beta2(4 \times 2)$ structure shown in Fig. \[100\_geo\](d) represents the Ga-terminated counterpart of the $\beta2(2 \times 4)$ reconstruction, with Ga atoms exchanged for As atoms and vice versa. Thus the top layer consists of two Ga dimers per $(4 \times 2)$ cell, and the second layer lacks two As atoms. This results in a stoichiometry of $\Delta N = -\frac{1}{4}$ per $(1 \times 1)$ cell.
Our calculated surface energies of these four phases are shown in Fig. \[100\] as a function of the chemical potential.
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We predict the same sequence of equilibrium surface structures as Northrup and Froyen [@northrup:93; @northrup:94] as a function of increasing As coverage: $\beta2$(4$\times$2), $\alpha$(2$\times$4), $\beta2$(2$\times$4), and c(4$\times$4). The $c(4 \times 4)$ structure with only two surface As-dimers per unit cell, which we considered in addition to the structures investigated by Northrup and Froyen, turned out to be unstable. Though this structure is more Ga-rich than the c(4$\times$4) three As-dimer structure shown in Fig.\[100\_geo\](c), even in the Ga-rich environment the two-dimer phase has a surface energy which is 5 meV/[Å]{}$^2$ higher than for the three-dimer phase. Due to the lack of absolute values in previous calculations, quantitatively we can only compare energy differences between surfaces with the same stoichiometry. Further comparison is made difficult by the different range of the chemical potential in our versus Northrup and Froyen’s calculation [@northrup:93; @northrup:94]: Their value for the heat of formation is $\Delta H_{f}$ = 0.92 eV, as opposed to our smaller value of $\Delta H_{f}$ = 0.64 eV. Comparing the three dimer phase $\beta$ with the two dimer phase $\beta2$, which both have the same stoichiometry, we find that the two dimer phase has a surface energy lower by 2 meV/[Å]{}$^2$. This agrees with the result of Northrup and Froyen who report an energy difference of 3 meV/[Å]{}$^2$, and it further confirms the conclusion that the three dimer phase $\beta$ does not exist in equilibrium. On the whole, the agreement with the relative surface energies calculated by Northrup [*et al.*]{} is good. They can be converted to absolute surface energies by shifting them by $\approx$ 65 meV/[Å]{}$^2$, which results in a diagram similar to Fig. \[100\].
All investigated (100) surfaces display similar atomic relaxations which are characterized by the creation of dimers and the rehybridization of threefold coordinated surface atoms. The creation of surface dimers decreases the number of partially occupied dangling bonds, and by rehybridization the surface gains band structure energy. The calculated bond lengths in bulk Ga and As, 2.32 [Å]{} and 2.50 [Å]{}, respectively, can serve as a first estimate for the respective dimer bond lengths on the GaAs surface. Our calculations yield As-dimer lengths between 2.45 and 2.50 [Å]{} for the $\alpha$ and $\beta2$ surface reconstruction. This is within the range of experimentally deduced values which scatter between 2.2 and 2.9 [Å]{} [@chambers:92; @xu:93; @li:93] and it is similar to the dimer lengths of 2.53 and 2.55 [Å]{} which were determined by Northrup [*et al*]{}. [@northrup:95] On the $c(4 \times 4)$ reconstructed surface the calculated As-dimer lengths are 2.57 [Å]{} for the central dimer and 2.53 [Å]{} for the two outer dimers of the three-dimer strings in the surface unit-cell. Using X-ray scattering Sauvage-Simkin [*et al*]{}.[@simkin:89] determined these bond-lengths as $2.63 \pm 0.06$ [Å]{} and $2.59 \pm 0.06$ [Å]{}. Very recently Xu [*et al.*]{}[@xu:95] suggested that the dimers on the c(4$\times$4) structure should be tilted by $4.3^\circ$. However, as for the $(2 \times 4)$ reconstructions we find the dimers to be parallel to the surface, in agreement with several previous experiments[@biegelsen:90a; @simkin:89]. Even when starting with an initial configuration with surface dimers tilted by $8^\circ$ we find the dimers to relax back to the symmetric positions with a residual tilt angle less than $0.1^\circ$. The Ga-Ga dimer bond length is calculated to be 2.4 [Å]{} on the $\beta2(4 \times 2)$ reconstruction and 2.5 [Å]{} on the $\alpha(2 \times 4)$ structure which agrees with previous [*ab initio*]{} calculations. [@northrup:95] From a recent LEED investigation of the Ga rich (100) surface Cerd’a [*et al*]{}.[@cerda:95] deduced that the stable $(4 \times 2)$ reconstructed surface displays three Ga-dimers per unit cell with unusual dimer lengths of 2.13 [Å]{} and 3.45 [Å]{}. In our calculation, however, this three dimer phase is energetically slightly less favorable than the two dimer phase $\beta2(4 \times 2)$ by 0.8 meV/[Å]{}$^2$. Therefore, it should not be stable at least at low temperatures. Furthermore, we found the Ga dimer length to be 2.4 [Å]{} and no local minimum for Cerd’a’s unusually large dimer length.
The rehybridization of the $sp^3$ orbitals located at the threefold coordinated Ga-atoms drives the relaxation towards a preferentially flat Ga-bond configuration. On the Ga terminated $\beta2 (4 \times
2)$ structure this leads to a decreased spacing between the Ga top layer and the neighboring As layer which amounts to roughly half of the bulk interlayer spacing. Also on the $\alpha (2 \times 4)$ and $\beta2(2 \times 4)$ surfaces the threefold coordinated Ga atoms which bond to As relax towards the plane of their neighboring As atoms. Together with a slight upward shift of the top layer As atoms this leads to a steepening[@ohno:93] of the As dimer block. The change of the angle between the bonds of the threefold coordinated As atoms is less pronounced. However, the trend is obvious: except for the $c(4
\times 4)$ structure, we find the As bond-angles to be always smaller than 109.5$^\circ$, which is the angle of the ideal tetrahedral coordination. The As bonds on the $c(4 \times 4)$ surface behave differently from those on the other three surfaces because the top-layer As atoms are bonded to a second layer which consist of As instead of Ga. A decrease of the angle between the bonds of all threefold coordinated As atoms would require a change in the As-As bond lengths, which probably costs more energy than would be gained from rehybridization.
(111) Surface
-------------
The polar (111) orientation of GaAs has been studied within density-functional theory by Kaxiras [*et al*]{}.[@kaxiras:86; @kaxiras:86a; @kaxiras:87], who computed surface energies relative to the ideal unreconstructed surface for various atomic geometries. They found that under As-rich conditions an As trimer geometry yields the lowest surface energy, whereas a Ga vacancy reconstruction is preferred under Ga-rich conditions. Haberern and Pashley [@haberern:90] and Thornton [*et al*]{}.[@thornton:95] confirmed this experimentally. Haberern and Pashley interpreted their STM images to show an array of Ga vacancies with a (2$\times$2) periodicity. Thornton [*et al*]{}. observed both the As triangle model and the Ga vacancy model in STM. Here we concentrate on the following reconstructions of the Ga terminated (111) surface: the As adatom, the As trimer, the Ga vacancy model, and, for comparison but not as a reference system as in previous work, the truncated-bulk geometry.
The ideal (111) surface (see Fig. \[111a\_geo\](a)) has a stoichiometry $\Delta N = -\frac{1}{4}/(1 \times 1)$.
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It does not fulfill the electron counting criterion. Each Ga dangling-bond is filled with 3/4 of an electron and therefore the ideal surface has to be metallic. To create a neutral semiconducting surface, following the electron counting criterion one can either add an As surface atom to, or remove a Ga surface atom from, every $(2
\times 2)$ surface unit cell. Therefore we consider three different $(2 \times 2)$ reconstructions. First of all, the As adatom model is shown in Fig. \[111a\_geo\](b). This reconstruction is stoichiometric. The As adatom binds to the Ga surface atoms. It exhibits a nearly orthogonal bond configuration, while the Ga atom with the empty dangling bond relaxes towards the plane of the As atoms. Secondly, we consider the As trimer model shown in Fig.\[111a\_geo\](c). This model has a stoichiometry $\Delta N =
\frac{1}{2} /(1 \times 1)$, it also fulfills the electron counting criterion and it is semiconducting. The three extra As atoms form a trimer with each As atom binding to one Ga atom. The dangling-bonds of the As atoms are completely filled and the dangling-bond of the the Ga atom which is not bonded to As trimer atoms is completely empty. This Ga atom relaxes into the plane of the As atoms of the layer below. Finally, we calculated the Ga vacancy model (see Fig.\[111a\_geo\](d)). The removal of one Ga surface atom causes the surface to be stoichiometric. The Ga surface atoms have completely empty dangling-bonds and relax into the plane of the As atoms. The three As atoms surrounding the vacancy have completely filled dangling-bonds.
We used the same super cell for the calculations of the (111) and the (111) surfaces. Only the bulk-truncated surface was calculated within a $(1 \times 1)$ surface unit cell, else always a $(2 \times
2)$ unit cell was used. The slab consisted of five (111) double layers. The vacuum region had a thickness equivalent to four (111) double layers. The whole Brillouin zone of the $(2 \times 2)$ surface unit cell was sampled with 16 special [**k**]{}-points, corresponding to 64 [**k**]{}-points in the Brillouin zone of the $(1 \times 1)$ cell. Absolute surface energies of the (111) reconstructions were determined using the energy density formalism. The results are shown in Fig.\[111a\].
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The Ga vacancy model is the most favorable reconstruction for a large range of the chemical potential from a Ga-rich to an As-rich environment. Only in very As-rich environments the As trimer model has a lower energy. The Ga vacancy model has a surface energy of 54 meV/[Å]{}$^2$, whereas, the As trimer model has a surface energy of 51 meV/[Å]{}$^2$ in As-rich environment at $\mu_{\rm As} = \mu_{\rm As
(bulk)}$. The Ga vacancy reconstruction was observed experimentally by Haberern and Pashley [@haberern:90] and Tong [*et al*]{}. [@tong:84] Thornton [*et al*]{}. additionally observed the As trimer reconstruction.
Two other groups have performed similar [*ab initio*]{} calculations. Using their energy density formalism, Chetty and Martin[@chetty:92a] derived a value of 131 meV/[Å]{}$^2$ for the surface energy of the ideal (111) surface in a Ga-rich environment, which is much larger than our value of 93 meV/[Å]{}$^2$. Secondly, we can compare our results to the relative surface energies of Kaxiras [*et al*]{}.[@kaxiras:86a; @kaxiras:87a] They arrived at the same qualitative conclusions. However, quantitatively their relative surface energies are not easily comparable to ours because they used As$_4$ gas to define the As-rich environment. Therefore they obtained a larger interval for the chemical potential. We derive for the surface energy difference of the As adatom and Ga vacancy structure a value of 13 meV/[Å]{}$^2$, whereas Kaxiras [*et al*]{}. calculate a much larger difference of 47 meV/[Å]{}$^2$. Using their own result for the ideal surface, Chetty and Martin transformed the relative surface energies of Kaxiras [*et al*]{}. to absolute surface energies. In comparison to our results, all these surface energies contain the same shift towards higher energy as the ideal surface mentioned above. We will discuss this difference below and explain, why we believe our results to be accurate.
Tong [*et al*]{}. [@tong:84] performed a LEED analysis for the geometry of the Ga vacancy reconstruction. Their geometry data compare very well with the theoretical data of Chadi [@chadi:84], Kaxiras [*et al*]{}. [@kaxiras:86a] and ours. For the Ga vacancy reconstruction we find an average bond angle of the $sp^2$-bonded Ga surface atom of 119.8$^\circ$ in agreement with Tong [*et al*]{}. The bond angles of the $p^3$-bonded As atom of 87.0$^\circ$ and 100.6$^\circ$ average to 91.5 $^\circ$ which again compare very well with the value of 92.9$^\circ$ by Tong [*et al*]{}. The bonds of the $p^3$-bonded As atom are strained by -1.6% and 2.6% with respect to the GaAs bulk bonds. Tong [*et al*]{}. measured a value -1.3% and 1.9%, respectively.
Furthermore, for the As trimer reconstruction we compare our geometry data to theoretical data of Kaxiras [*et al*]{}. [@kaxiras:86a] The threefold-coordinated As adatoms form bond angles to the neighboring As adatoms of 60$^\circ$ due to symmetry reasons. The bond angle of the As adatom to the next Ga atom is 106.2$^\circ$. Therefore we get an average bond angle of 90.8$^\circ$ which is in good agreement with the 91.7$^\circ$ of Kaxiras [*et al*]{}. The surface Ga-As bonds are strained by 1.4 %, whereas Kaxiras [*et al*]{}. find the same bond length as in the bulk. The As-As bonds have a bond length of 2.44 [Å]{}, 2.4 % shorter than that in As bulk. The Ga surface atom which is not bond to an As adatom relaxes into the plane of the As atoms with a bond angle of 118.4$^\circ$ and a bond length which is 2.6 % shorter than in GaAs bulk. These values are slightly larger than the 114.7$^\circ$ and 1.0 % reported by Kaxiras [*et al*]{}.
(111) Surface
-------------
The polar GaAs (111) surface differs from the (111) surface, as the bulk-truncated (111) surface is terminated by As atoms, while the (111) surface is Ga terminated. At first sight the (111) surfaces might seem to be still analogous to the (111) surfaces, only that the Ga and As atoms have to be exchanged. However, this analogy is not useful, because As and Ga have different electronic properties, and therefore the (111) and (111) surfaces do not exhibit equivalent reconstructions. Stoichiometric (111) surfaces are gained by adding a Ga atom per $(2 \times 2)$ surface unit cell to the bulk-truncated surface or by removing an As surface atom.
Kaxiras [*et al*]{}. [@kaxiras:87a] calculated the relative surface energy for various $(2 \times 2)$ reconstructions. Biegelsen [*et al*]{}. [@biegelsen:90] studied the (111) surface both experimentally and theoretically. Using STM they observed an As trimer $(2 \times 2)$ reconstruction for As-rich environments. A $(\sqrt{19}
\times \sqrt{19})$ reconstruction which is dominated by two-layer hexagonal rings was identified for Ga-rich environments.
Due to the large unit cell the $(\sqrt{19} \times \sqrt{19})$ reconstruction is computationally quite expensive, and in this work we thus only consider $(2 \times 2)$ reconstructions. First of all, for comparison, we calculate the surface energy of the ideal (i.e., relaxed bulk-truncated) surface shown in Fig. \[111b\_geo\](a).
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This surface is not stoichiometric $(\Delta N = \frac{1}{4} /(1 \times
1))$. The dangling-bond of each As surface atom is filled with 5/4 of an electron. Therefore the surface is metallic. Secondly, the Ga adatom model shown in Fig. \[111b\_geo\](b) was considered. Through adding of an additional Ga surface atom the surface has become stoichiometric and semiconducting. The dangling-bond of the Ga adatom is completely empty, whereas the dangling-bond of the As atom which is not bond to the Ga adatom is completely filled. Furthermore, we also consider an As trimer model (see Fig. \[111b\_geo\](c)). In contrast to the (111) surface the As trimer is bond to As surface atoms. This reconstruction has a stoichiometry of $\Delta N = 1$ per $(1 \times
1)$ surface unit cell. Each As surface atom has a completely filled dangling-bond. Therefore, the surface is semiconducting. Furthermore, we calculate the surface energy for the As vacancy model which is shown in Fig. \[111b\_geo\](d). The removal of the As surface atom causes the surface to be stoichiometric. The three neighboring Ga atoms have completely empty dangling-bonds. The surface fulfills the electron counting criterion and is semiconducting. Finally, we calculate the Ga trimer model (see Fig.\[111b\_geo\](e)) to compare with Kaxiras [*et al*]{}.[@kaxiras:87a] and Northrup [*et al*]{}. [@biegelsen:90] This surface model has a stoichiometry of $\Delta N = -1/2$ per $(1 \times
1)$ surface unit cell and also fulfills the electron counting criterion. However, it is metallic for the same reason as the Ga terminated (110) surface.
The calculations for the (111) surface were carried out with the same parameters and supercell as those for the (111) surface outlined in the previous section. The results are shown in Fig. \[111b\].
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For As-rich environments we find that the As trimer model is the most favorable reconstruction, as observed experimentally by STM and confirmed by previous [*ab initio*]{} calculations.[@biegelsen:90] This reconstruction has a very low surface energy of 43 meV/[Å]{}$^2$. In a Ga-rich environment the Ga adatom reconstruction has the lowest energy (69 meV/[Å]{}$^2$) among all the structures we calculated. The $(\sqrt{19} \times \sqrt{19})$ reconstruction found experimentally was not included in our approach. However, as suggested by Biegelsen [*et al*]{}. our present data can be used to restrict the range of possible values for the surface energy of the $(\sqrt{19}
\times \sqrt{19})$ reconstruction consistent with observation: It has to be smaller than the surface-energy of the Ga adatom model on the one hand, and it has to be larger than the minimum energy of the As-trimer surface (plus a small correction of $-3$ meV/Å$^2$ to account for the non-stoichiometry of the $\sqrt{19}\times\sqrt{19}$ reconstruction) on the other hand. Therefore, we conclude that energy of the $(\sqrt{19} \times \sqrt{19})$ reconstruction is in the range between 40 and 69 meV/[Å]{}$^2$. Considering also the energetical competition with facets of other orientations, even a slightly more stringent condition can be deduced: For the (111) $(\sqrt{19}
\times \sqrt{19})$ surface in a Ga-rich environment to be stable against faceting into {110} surfaces, its surface energy has to be less than 63 meV/[Å]{}$^2$.
In comparison to the relative surface energies calculated by Kaxiras [*et al*]{}. [@kaxiras:87a] our energy difference between the As vacancy and the Ga adatom structure of 2 meV/[Å]{}$^2$ is only slightly smaller than their value of 6 meV/[Å]{}$^2$. However, they state that for the Ga-rich environment the Ga trimer structure is 24 meV/[Å]{}$^2$ more favorable than the Ga adatom structure. In contrast, we agree with Northrup [*et al*]{}. [@biegelsen:90] that the Ga trimer is energetically quite unfavorable. It has a 29 meV/[Å]{}$^2$ higher surface energy than the Ga adatom. Also the other relative surface energies compare quite well with the already mentioned calculations of Northrup [*et al*]{}. [@biegelsen:90] although they derived a larger heat of formation (0.92 eV as opposed to our value of 0.64 eV). Relative to the Ga adatom our surface energies of the As trimer are about 10 meV/[Å]{}$^2$ larger than theirs. Also, they find a slightly larger energetic separation between the As vacancy and Ga adatom structures. Their value for this energy difference is 6 meV/[Å]{}$^2$, whereas our value is 2 meV/[Å]{}$^2$. However, these differences are small and do not affect the physical conclusions. Chetty and Martin derived the absolute surface energies using their result for the ideal (111) surface and the relative surface energies of Kaxiras [*et al*]{}. and Northrup [*et al*]{}. In contrast to the (111) their value of 69 meV/[Å]{}$^2$ for the ideal (111) surface in the Ga-rich environment is much smaller than ours of 97 meV/[Å]{}$^2$. Therefore, this time in comparison to our data the results all contain the same shift to lower surface energies as the ideal surface. However, the sum of the (111) and (111) surface energies from Chetty and Martin is close to ours. Therefore it is the splitting of the slab total energy into contributions from the (111) and the (111) side that comes out differently. In our calculations both sides are energetically similar which seems to be plausible in view of the fact that the flat (i.e., not faceted) surfaces have been observed experimentally.
With respect to the calculated geometry we find that the As-As bond length in the trimer is 2.46 [Å]{}, 1.6 % shorter than in bulk As. The As trimer atoms each bind to an As atom 2.30 [Å]{} beneath the As trimer plane in agreement with Northrup [*et al*]{}. [@biegelsen:90] The remaining As atom which is not bond to the trimer relaxes outwards and is 1.74 [Å]{} below the trimer plane. This compares reasonably well with the slightly larger value of 1.89 [Å]{} by Northrup [*et al*]{}. For the two Ga surface models the separation of the adatom or trimer plane and the closest As (rest atom) plane amounts to 0.98 [Å]{} for the Ga adatom model, and 1.98 [Å]{} for the Ga trimer model. Northrup [*et al*]{}. derived values of 0.98 [Å]{} and 1.90 [Å]{}.
Equilibrium Crystal Shape (ECS)
-------------------------------
As opposed to liquids, crystals have non-trivial equilibrium shapes because the surface energy $\gamma(\hat{\bf m})$ depends on the orientation $\hat{\bf m}$ of the surface relative to the crystallographic axes of the bulk. Once $\gamma(\hat{\bf m})$ is known, the ECS is determined by the Wulff construction, [@wulff:01; @wortis:88] which is equivalent to solving $$r(\hat{\bf h}) = \min_{\hat{\bf m}}
\left(\frac{\gamma({\hat{\bf m}})} {\hat{\bf m} \cdot \hat{\bf h}} \right).$$ Here $r(\hat{\bf h})$ denotes the radius of the crystal shape in the direction $\hat{\bf h}$. When the surface energy $\gamma(\hat{\bf m})$ is drawn as polar plot, the ECS is given by the interior envelope of the family of planes perpendicular to $\hat{\bf m}$ passing through the ends of the vectors $\gamma(\hat{\bf m})\,\hat{\bf m}$. Under the assumption that only the (110), (100), (111), and (111) facets exist, we construct the ECS from the calculated surface energies of these facets. Thus there may exist additional thermodynamically stable facets that are missing on our ECS. To be sure to construct the complete shape one would have to calculate the surface energy for every orientation. However, from experiments it is known that the low Miller-indices surface orientations we consider are likely to be the energetically most favorable ones.
As the GaAs surface energies depend on the chemical environment, the ECS becomes a function of the chemical potential. In Fig. \[3d\_ecs\] the ECS is shown for an As-rich environment and zero temperature.
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The different facets have been marked in the Figure and the ECS reflects the symmetry of bulk GaAs. To investigate the dependence of the ECS on the chemical potential we will focus on the cross-section of the ECS with a (110) plane through the origin. This cross-section includes the complete information from all four calculated surfaces, because they all possess surface normals within this plane. The ECS is shown for three different chemical environments in Fig. \[ecs\].
Note that in a Ga-rich environment the (111)$(\sqrt{19} \times
\sqrt{19})$ reconstruction would be energetically more favorable than the (111)(2$\times$2) Ga-adatom reconstruction used for the construction of the ECS at this chemical potential, i.e., the experimental (111) facet appears somewhat closer to the origin. For an As-rich environment we find that all four considered surface orientations exist in thermodynamic equilibrium. Furthermore, the (111) surface exists within the full range of accessible chemical potentials. This is in contrast to the result Chetty and Martin [@chetty:92a] derived from the work of Kaxiras [*et al*]{}.[@kaxiras:87]: They stated that the (111) surface has a high energy and thus it should not exist as a thermodynamic equilibrium facet. However, experimental work of Weiss [*et al*]{}.[@weiss:89] using a cylindrical shaped sample indicates that between the (110) and (111) orientation all surfaces facet into (110) and (111) orientations. The $(2 \times 2)$ superstructure of the (111) surface has been observed on these faceted surfaces. If the (111) orientation of GaAs were instable, the appearance of facets other than (111) on the cylindrical crystal were to be expected.
In Fig. \[ecs\] one can see that the ECS becomes smaller for As-rich environments. The As terminated reconstructions have surface energies about 20 % smaller than those found in Ga-rich environments, which are mostly stoichiometric like the Ga vacancy. In contrast to the surface reconstructions found for As-rich environments no similar Ga terminated reconstructions are observed. Another remarkable feature of the ECS is that the surface energies do not vary very much with the orientation. For Ga-rich environments they vary by about $\pm 10 \%$, whereas for As-rich environments they vary only by $\pm 5 \%$.
Our calculated ECS imposes restrictions on the surface energies of other surface orientations: When it has been proven experimentally that a facet exists in thermodynamic equilibrium, one can derive a lower and an upper limit for its surface energy. The limits are given by the surface energy of the neighboring facets on our ECS together with appropriate geometry factors. They follow from the conditions that (a) the surface energy has to be sufficiently small, so that the surface does not facet into {110}, {100}, {111}, and {111} orientations, and (b) that the surface energy is not so small that neighboring facets are cut off by this plane and thus vanish from the ECS. In a similar way the Wulff construction yields a lower limit for the surface energy of any facet that does not exist in thermodynamic equilibrium.
Recently the shape of large three-dimensional InAs islands (diameter $\sim$ 2000 Å) grown by MOVPE on a GaAs(100) substrate has been observed by E. Steimetz [*et al.*]{} [@steimetz:96] These islands are presumably relaxed, the misfit of the lattice constants being compensated by a dislocation network at the InAs-GaAs interface. Thus the facets displayed on these islands should be identical to the facets on the ECS of InAs. In fact, the observed shapes are compatible with an ECS like that of GaAs shown in Fig. \[3d\_ecs\], with {110}, {100}, {111}, and {111} facets being clearly discernible. Due to the similarity between InAs and GaAs we take this as another confirmation of our results as opposed to those of Chetty and Martin. [@chetty:92a]
Summary and Conclusion
======================
The GaAs surface energies of different orientations have been calculated consistently with one and the same parameters and pseudo-potentials. The surface energies of the (110), (100), (111) and (111) surfaces are given in dependence of the chemical potentials.
For the (111) and (111) surfaces we find a large difference to previous results of Chetty and Martin. [@chetty:92a] They derived a difference of about 62 meV/[Å]{}$^2$ between the surface energies of the ideal (111) and (111) surfaces, whereas we calculate a difference of about $-4$ meV/[Å]{}$^2$. Consequently the absolute surface energies calculated by Chetty and Martin using data of Kaxiras [*et al*]{}. [@kaxiras:87; @kaxiras:87a] and Northrup [*et al*]{}.[@biegelsen:90] contain the above difference of 66 meV/[Å]{}$^2$. This is due to a different splitting of the slab energy into contributions from the (111) and (111) surfaces, as Chetty and Martin’s and our sum of the (111) and (111) surface energies are essentially equal. Obtaining high surface energies for the (111) surfaces Chetty and Martin have to conclude that the (111) facet should be unfavorable and not exist in thermodynamic equilibrium. In contrast our surface energies for the (111) surface are lower and therefore we conclude that it exists in thermodynamic equilibrium which appears to be in agreement with experimental observations.
As already stated by Chetty and Martin [@chetty:92a] there are significant differences between the results of Kaxiras [*et al*]{}.and Northrup [*et al*]{}. for the (111) surface: Kaxiras [*et al*]{}. find the Ga trimer structure to be energetically favorable in Ga-rich environments, whereas we agree with Northrup [*et al*]{}. and find it energetically unfavorable. This is also confirmed by experiment.
Having calculated the absolute surface energies for different orientations we are in the position to construct the ECS of GaAs. We have to keep in mind, however, that it is implicitly assumed that only the (110), (100), (111) and (111) surfaces exist in equilibrium. For a more refined discussion of faceting further calculations also for higher-index surfaces would have to be performed. From our ECS we conclude that in As-rich environment all four orientations exist in thermodynamic equilibrium. For a given chemical potential the variation of the surface energy with orientation is small and less than $\pm 10 \%$. Our ECS of GaAs gives indication for the ECS of InAs or other III-V semiconductors which show similar surface reconstructions.
Acknowledgments
===============
We thank E. Steimetz for helpful discussion and a copy of Ref. prior to publication. This work was supported in part by the Sfb 296 of the Deutsche Forschungsgemeinschaft.
Present Address: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.\
Electronic Address: moll@mit.edu
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|
---
author:
- 'B. W. Holwerda[^1], R. J. Allen, W. J. G. de Blok, A. Bouchard, R. A. González-Lópezlira, P. C. van der Kruit, and A. Leroy'
subtitle: Dust and Gas Surface Densities
title: 'The Opacity of Spiral Galaxy Disks IX;'
---
Introduction
============
The radio 21-cm emission of atomic hydrogen () observed in the disks of spiral galaxies is a powerful tracer of the presence and dynamics of the interstellar medium (ISM), extending to well outside the typical scale of the stellar disk. Its origin is likely a mix of “primordial" [@Fall80], or recently accreted material [@Sancisi08], recycled matter [ejecta raining back onto the disk; e.g., @Oosterloo07], and skins of photo-dissociated material surrounding molecular clouds [@Allen04]. The other components of the ISM, ionised and molecular hydrogen, metals and dust, are all more difficult to trace, because their emission strengths depend on the local degree of excitation which in turn is affected by particle densities and temperatures, photon densities, and stellar and AGN illumination.
Molecular hydrogen is usually traced with CO(J=1-0 or 2-1) line emission, and from it we have derived our knowledge of the molecular clouds in nearby spirals [e.g, @Rosolowsky05a; @Leroy08]. However, it remains an open question how sensitive the CO brightness is to the local volume density and temperature of the ISM, and what is the accuracy with which observations of CO surface brightness can be converted into 2 column densities and ultimately into molecular cloud masses. This conversion is also likely to depend on metallicity and hence galactocentric radius [@Madden97; @Israel97; @Leroy07; @Pohlen10; @Leroy11; @Foyle12].
Nevertheless, a successful and extensive description of the atomic and molecular ISM in spirals and their relation to the star-formation rate is currently being developed, using a multi-wavelength approach to estimate the star-formation rate, and high-resolution and CO observations to characterize the ISM in individual galaxies [@Calzetti05; @Kennicutt07; @Thilker07a; @Bendo10b; @Foyle12], in detail in small samples of galaxies [@Cortese06a; @Boissier07; @Bigiel08; @Leroy08; @Schruba11], or in a generalized way over a population of galaxies [e.g., @Kennicutt98; @Buat02; @Bell03c; @Kannappan04; @West10a; @Catinella10; @Fabello11a]. Star-formation occurs when the combined ISM exceeds a threshold surface density [although the exact threshold is still debated, see e.g., @Bigiel08; @Pflamm-Altenburg08]. The ratio between molecular and neutral ISM is set by the hydrostatic pressure [@Bigiel08; @Leroy08; @Obreschkow09c]. Also, observational models of the role of photo-dissociation in the balance between atomic and molecular hydrogen have made steady progress [@Allen97; @Allen04; @Smith00; @Heiner08a; @Heiner08b; @Heiner09; @Heiner10].
As an alternative to CO, one could use interstellar dust as a tracer of the molecular component in spiral galaxies, since it is linked mechanically to the molecular phase [@Allen86; @Weingartner01b], by mutual shielding from photo-dissociation, and the formation of molecular hydrogen on the surface of dustgrains [e.g., @Cazaux04b]. Interstellar dust can be traced by its emission or its extinction of starlight.
Surface densities of dust in spirals have been obtained from spectral energy distribution models of multi-wavelength data [e.g., @Popescu00; @Popescu02; @Draine07; @Boselli10], from simple (modified) blackbody fits of far-infrared and sub-mm data [@Bendo08; @Bendo10b; @Gordon08; @Gordon10] or FUV/FIR ratios [@Boissier04; @Boissier05; @Boissier07; @Munoz-Mateos11]. The aim is to estimate the typical temperature, mass, composition and emissivity of the dust, and the implied gas-to-dust ratio [@Boissier04; @Boselli10; @Munoz-Mateos09a; @Munoz-Mateos09b; @Pohlen10; @Smith10b; @Roman-Duval10; @Magrini11b; @Galliano11; @Foyle12; @Galametz12].
The most recent [*Herschel*]{} results include a resolved temperature gradient in the disks of spirals [@Bendo10b; @Smith10b; @Engelbracht10; @Pohlen10; @Foyle12], linked to increased illumination of the grains, notably in the spiral arms [@Bendo10b] and bulge [@Engelbracht10]. With sufficient spatial sampling, one can extract the ISM power spectrum but this is only possible with [*Herschel*]{} for local group galaxies [@Combes12]. Based on [*Herschel*]{} data of the Virgo cluster, [@Smith10b], [@Cortese10] and [@Magrini11b] show the spatial coincidence and efficiency of stripping the dust together with the from the disks of spirals in a cluster environment.
In the comparison between the [*Herschel*]{} cold grain emission, and and CO observations, the mass-opacity coefficient of dust grains appears to be too low in M33 [the inner disk, @Braine10], and M99 and M100 [@Eales10]. This is either because (1) its value is not well understood, (2) the conversion factor between CO and molecular hydrogen, , is different in M99 and M100, or (3) the emissivity ($\beta$) is different at sub-mm wavelengths. [@Roman-Duval10] compare CO, and dust in the Large Magellanic Cloud (LMC), and argue that the cause of the discrepancy cannot be a different emissivity, nor a different gas-to-dust ratio, but that CO clouds have 2 envelopes, hence changes with different density environments [an explanation also favored by @Wolfire10]. Other recent results seem to back variations in ; [@Leroy11] find a link between and metallicity based on SED models of a few local group galaxies and the HERACLES CO survey. A solid result from the first [*Herschel*]{} observations is that the gas-to-dust ratio increases with galactocentric radius [@Pohlen10; @Smith10b] as do [@Munoz-Mateos09b; @Bendo10a based on Spitzer data alone]. [@Magrini11b] find a much lower than Galactic CO-to-2 conversion factor based on the relation between metallicity and gas-to-dust ratio radial profiles of several Virgo cluster spirals.
Even with the excellent wavelength coverage of [*Herschel*]{}, the SED fit results remain degenerate between dust mass, temperature and emissivity [see the reviews in @Calzetti01; @Draine03]. It is still especially difficult to distinguish between a mass of very cold (poorly illuminated) dust from dust with much different emissivity characteristics (the emissivity efficiency depends on wavelength as $\lambda^{-\beta}$ in the sub-mm regime with $\beta \ne 2$, which may be typical for very large grains).
While large masses of extremely cold dust can be ruled out with increasing confidence, the level of illumination of the grains by the interstellar radiation field remains a fully free parameter in the SED models. The main uncertainty is complex relative geometry between the dusty filamentary structures and the illuminating stars. Both the grain emissivity and dust/star geometry can be expected to change significantly throughout the disk, i.e., with galactocentric radius or in a spiral arm.
Alternatively to models of dust emission, one can use the absorption of stellar light to trace dust densities. The advantages are higher spatial resolution of optical wavelengths and an independence of dust temperature. However, one needs a known background source of stellar light to measure the transparency of a spiral disk[^2]. Two observational techniques have been developed to measure the opacity of spirals and consequently their dust content. The first one uses occulting galaxy pairs [@Andredakis92; @Berlind97; @kw99a; @kw00a; @kw00b; @kw01a; @kw01b; @Elmegreen01; @Holwerda07c; @Holwerda09; @Keel11 Holwerda et al. [*submitted.*]{}], of which an increasing number are now known thanks to the Sloan Digital Sky Survey and the GalaxyZOO citizen science project [@Lintott08].
The second method uses the number of distant galaxies seen through the disk of a nearby face-on spiral, preferably in Hubble Space Telescope ([*HST*]{}) images. The latter technique is the focus of our “Opacity of Spiral Galaxies” series of papers [@Gonzalez98; @Gonzalez03; @Holwerda05a; @Holwerda05b; @Holwerda05c; @Holwerda05e; @Holwerda05d; @Holwerda07a].[^3] The benefit of using distant galaxies as the background light source is their ubiquity in HST images of nearby galaxies. Now that uniform maps are available from the THINGS project [The Nearby Galaxy Survey, @Walter08], as well as public [*Herschel*]{} data from the KINGFISH [Key Insights on Nearby Galaxies: a Far-Infrared Survey with Herschel, @Skibba11; @Dale12; @Kennicutt11; @Galametz12], and CO(J=2-1) maps from the HERACLES survey [The HERA CO Line Extragalactic Survey, @heracles] for a sub-sample of the galaxies analysed in our “Opacity of Spiral Galaxies” project, we are taking the opportunity to compare our disk opacities to and 2 surface densities to see how they relate.
Our method of determining dust surface densities is certainly not without its own uncertainties (notably cosmic variance, see §\[s:sfm\]) but these are not the ones of sub-mm emission suffers from (grain emissivity, level of stellar illumination, variance within the disk or these). Hence, our motivation for our comparison between the disk opacity and the other tracers of the cold ISM is to serve as an independent check to the new [*Herschel*]{} results.
In section 2, we discuss the origin of our sample and data. Section 3 explains how we derive a disk opacity from the number of distant galaxies. In section 4, we discuss the distant galaxy number as a function of column density and in section 5, we compare the and 2 column densities, dust extinction, averaged over whole WFPC2 fields, and per contour, respectively. Sections 6 and 7 contain our discussion and conclusions.
Galaxy Sample and Data
======================
Our present sample is the overlap between the [@Holwerda05b], the THINGS [@Walter08], and the HERACLES [@heracles] projects. The common 10 disk galaxies are listed in Table \[t:info\]. We use the public THINGS data and early science release data from HERACLES.
Figure \[f:himap\] shows the HST/WFPC2 “footprints” overlaid on the VLA HI maps. In the case of NGC 3621 and NGC 5194, there are two HST/WFPC2 fields available for each galaxy.
VLA 21-cm Line Observations {#s:hi}
---------------------------
For this study we use the THINGS [The Nearby Galaxy Survey, @Walter08] robustly-weighted (RO) integrated total intensity maps (available from <http://www.mpia-hd.mpg.de/THINGS/>). The maps were obtained with the VLA, and converted to surface density using the prescription from [@Walter08], equations 1 and 5, and Table 3. Although the naturally-weighted maps are markedly more sensitive to the largest scale distribution, the robust maps have the highest angular resolution.
The robust maps are better suited for a direct comparison with the number of background galaxies, as we are interested in the column density at the position of each background galaxy and hence at scales smaller than the FOV of the HST/WFPC2 FOV (3 CCDs of $1\farcm3 \times 1\farcm3$). Additionally, we use the WFPC2 footprint as an aperture on the maps (Figure \[f:himap\]).[^4] The column densities averaged over the WFPC2 footprints (an angular scale of $2\farcm3$) on the sample galaxies, and expressed in units of ${\rm M_\odot/pc^2}$ are listed in Table \[t:info\]. These mean column densities include a correction factor (1.36) for Helium contribution to the atomic gas phase.
![The THINGS robustly weighted integrated column density maps. The HST/WFPC2 footprint is overlaid (black outline). NGC 3621 and NGC 5194 have two HST pointings each. A 3 arcminute ruler is shown for scale comparison. Most of the WFPC2 fields in [@Holwerda05] were originally taken for the Cepheid Distance Scale Key Project ([@KeyProject]); they were positioned on spiral arms in the outer, less crowded, parts of the disks to aid in the identification of Cepheid variables. NGC 3031 and NGC 3621 do not have CO observations.[]{data-label="f:himap"}](./holwerda_f1.pdf){width="\textwidth"}
\[t:info\]
----------- ------- ---------- ------------ ------------------- ------------ ---------- ------------------- ------------ ------------------------ ----------------- ----------- --------- ------ ------- ------- -------
Galaxy Dist. $R_{25}$ $R/R_{25}$ $\rm \Sigma_{HI}$ $\rm FWHM$ $L_{CO}$ $\rm \Sigma_{H2}$ $\rm FWHM$ $\rm A_{SFM}$ $\rm \Sigma_d$ $\rm FOV$ log(OH) + 12
() (CO)
(Mpc) (kpc) ($\rm M_\odot$ (kpc) (K ($\rm M_\odot$ (kpc) (mag.) ($\rm M_\odot $ (kpc) KK04 PT05
$\rm / pc^2$) km/s) $\rm / pc^2$) $\rm / pc^2$)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)
NGC925 11.20 5.61 0.50 17.92 0.33 0.23 1.83 0.60 $ -0.4^{ 0.3}_{ 0.3}$ -0.44 4.22 8.20 8.70 -3.49 -3.50 -3.54
NGC2841 14.10 4.07 0.61 8.98 0.41 1.00 7.85 0.75 $ 0.8^{ 0.4}_{ 0.5}$ 0.88 5.32 8.40 9.20 -3.17 -3.27 -3.39
NGC3031 3.60 13.46 0.25 4.47 0.10 … … 0.19 $ 0.8^{ 0.6}_{ 0.5}$ 0.88 1.36 8.50 9.10 -4.05 -4.16 -4.22
NGC3198 13.80 4.26 0.62 18.15 0.40 0.31 2.43 0.74 $ 0.8^{ 0.3}_{ 0.3}$ 0.88 5.20 8.30 8.80 -3.60 -3.63 -3.69
NGC3351 10.00 3.71 0.51 3.52 0.29 1.04 8.19 0.53 $ 1.2^{ 0.5}_{ 0.6}$ 1.32 3.77 8.60 9.20 -3.05 -3.18 -3.33
NGC3621-1 6.64 6.15 0.37 18.80 0.19 … … 0.35 $ 2.2^{ 0.6}_{ 0.6}$ 2.42 2.50 8.30 8.90 -3.25 -3.34 -3.41
NGC3621-2 6.64 6.15 0.38 12.72 0.19 … … 0.35 $ 1.0^{ 0.3}_{ 0.4}$ 1.10 2.50 8.30 8.90 -3.25 -3.54 -3.63
NGC3627 10.05 4.56 0.52 9.15 0.29 3.93 30.98 0.54 $ 2.1^{ 0.7}_{ 0.7}$ 2.31 3.79 … … -2.67 -2.86 -3.01
NGC5194-1 8.40 5.61 0.62 8.97 0.24 3.38 26.66 0.45 $ -0.4^{ 0.4}_{ 0.4}$ -0.44 3.17 8.50 9.10 -2.88 -3.05 -3.21
NGC5194-2 8.41 5.61 0.63 10.11 0.24 4.47 35.26 0.45 $ 1.4^{ 0.6}_{ 0.6}$ 1.54 3.17 8.50 9.10 -2.84 -3.01 -3.14
NGC6946 11.48 5.74 0.75 9.44 0.33 2.12 16.69 0.61 $ 1.1^{ 0.5}_{ 0.6}$ 1.21 4.33 8.30 8.90 -3.28 -3.44 -3.60
NGC7331 14.72 5.24 0.76 21.53 0.43 0.26 2.06 0.78 $ 0.3^{ 0.3}_{ 0.3}$ 0.33 5.55 8.30 8.80 -3.57 -3.61 -3.69
----------- ------- ---------- ------------ ------------------- ------------ ---------- ------------------- ------------ ------------------------ ----------------- ----------- --------- ------ ------- ------- -------
CO(J = 2 $\rightarrow$ 1) Line Observations {#s:co}
-------------------------------------------
The HERACLES project [The HERA CO Line Extragalactic Survey, @heracles; @Walter09] is a project on the IRAM 30m telescope to map the molecular gas over the entire optical disks ($R_{25}$) of 40 nearby galaxies via the CO(J=2-1) emission line. The HERA instrument has comparable spatial (11") and velocity (2.6 km/s) resolutions to the THINGS survey, and good sensitivity ($3 \sigma \approx 3 \rm M_\odot / pc^2$) as well. The HERACLES sample overlaps by design with the THINGS and SINGS [[*Spitzer*]{} Infrared Nearby Galaxy Survey, @SINGS] samples and it also has 8 galaxies in common with our previous work (Table \[t:info\]).
To convert the CO (J=2-1) maps to molecular hydrogen surface density maps, we need the conversion factor (alternatively denoted as $\rm \alpha_{CO}$). For the CO (J=1-0) line, this is commonly assumed to be 4.4. The ratio between the CO(J=1-0) and CO(J=2-1) line is 0.7 according to the HERACLES observations. To convert the CO(J=2-1) map (in K km/s) into molecular surface density, $\rm X_{CO (2-1)} = 4.4/0.7 = 6.3 M_\odot/pc^2$ [@Leroy08]. The mean values of the CO(J=2-1) surface brightness and the molecular hydrogen surface density are listed in Table \[t:info\].
HST/WFPC2 Images {#s:hst}
----------------
The background galaxy counts are based on HST/WFPC2 data, as presented in [@Holwerda05] and [@Holwerda05b]. The footprints of the 12 HST/WFPC2 fields on the integrated maps of 10 THINGS galaxies are shown in Figure \[f:himap\] and we only consider these areas of the disks. The HST fields are predominantly from the Distance Scale Key Project [@KeyProject], and are therefore usually aimed at spiral arms in the outer parts of the main disks, in order to facilitate the identification of Cepheids. The final drizzled WFPC2 images in $F814W$ and $F555W$, from [@Holwerda05], can be obtained at <http://archive.stsci.edu/prepds/sgal/> and the NASA Extragalactic Database.[^5]
Disk Opacity from the Number of Background Galaxies. {#s:sfm}
====================================================
The central premise of our method to measure disk opacity, is that the reduction in the number of distant galaxies seen though a foreground spiral galaxy is a reasonable indication of the transparency of the disk. The number of distant galaxies that can be identified is a function of several factors: the real number of galaxies behind the disk; the crowding by objects in the foreground disk and consequently the confusion in the identification of the distant galaxies, and, finally, absorption of the light from the background galaxies by the interstellar dust in the foreground disk. Since we are only interested in the last one –the dust extinction–, all the other factors need to be mitigated and accounted for. [*HST*]{} provides the superb resolution to identify many distant galaxies, even in the quite crowded fields of nearby spiral galaxies. But to fully calibrate for crowding and confusion, we developed the “Synthetic Field Method” (SFM), in essence a series of artificial galaxy counts under the same conditions as the science field [@Gonzalez98; @Holwerda05a].
If we identify $N$ galaxies in a field, we need to know two quantities to convert this number into a disk opacity measurement: (1) the number ($N_0$) of galaxies we would have identified in this field, without any dust extinction but under the same crowding and confusion conditions, and (2) the dependence ($C$) of the number of galaxies on any increase of dust extinction. The disk’s opacity in $F814W$ is then expressed as: $$\label{eq1}
A_I = -2.5~ C~ \log \left({N \over N_0}\right).$$
If the number of identified galaxies behaved exactly as photons, the parameter $C$ would be unity. We have found it to be close to 1.2 for a typical field, and $N_0$ to depend the surface brightness and granularity of the foreground disk [@Gonzalez03; @Holwerda05e]. From our artificial distant galaxy counts in the WFPC2 fields, we can obtain both $N_0$ and $C$; the first from an artificial count of seeded, undimmed, distant galaxies, and the second from a series of artificial distant galaxy counts with progressive dimming of the seeded galaxies.
Since we cannot know the intrinsic number of distant galaxies behind the foreground disk, we treat the cosmic variance as a source of uncertainty in $N_0$ that can be estimated from the observed 2-point correlation function. This typically is of the same order as the Poisson error in the opacity measurement.[^6] Because the cosmic variance uncertainty is substantial, improvements in the identification of distant galaxies barely improve our errors [see also @mythesis].
To test the general SFM results, we have done several checks against other techniques. The results are consistent with those obtained from occulting galaxy pairs [@Holwerda05b], both the results in [@kw00a; @kw00b] as well as the later opacities found in [@Holwerda07c]. The SFM results are also consistent with the amount of dust reddening observed for the Cepheids in these fields [the majority of which is from the Cepheid Distance Scale Key Project @KeyProject], the dust surface densities inferred from the far-infrared SED [@Holwerda07a discussed below], and the sub-mm fluxes from KINGFISH observations (§\[s:herschel\], below). Even with HST, the number of identifiable galaxies in a given WFPC2 field is relatively small, a fact that results in large uncertainties if the field is further segmented for its analysis, e.g., sub-divided into arm and inter-arm regions. To combat the large uncertainties, we combined the numbers of background galaxies found in different fields, based on certain characteristics of the foreground disks, like galactocentric radius, location in the arm or inter-arm regions [@Holwerda05b], surface brightness [@Holwerda05d], or NIR colour [@Holwerda07b]. Because no uniform and CO maps were available until now, we compared radial profiles to our radial opacity profile in [@Holwerda05c], but this is far from ideal. Now that the THINGS and HERACLES maps are available, we can compare the average opacity of an [*HST*]{} field to its mean and 2 surface densities or, alternatively, rank the distant galaxies based on the foreground disk’s column density at their position.
Dust Surface Densities {#s:sigmad}
----------------------
To convert the above opacity of the spiral disk to a dust surface density, we assume a smooth surface density distribution of the dust (no clumps or fine structure). The dust surface density is then: $$\Sigma_{\rm d} = {1.086 A \over \kappa_{\rm abs}},$$ with $\kappa_{\rm abs}$ for Johnson $I$ from [@Draine03], Table 4; $4.73 \times 10^3$ cm$^2$ g$^{-1}$. The mean opacity ($A_{\rm SFM}$) and implied mean dust surface densities are listed in Table \[t:info\]. The value for $\kappa_{\rm abs}$ changes with the types of grain (and hence with environment in the disk) and the Draine et al. is a value typical for large grains. Variance in $\kappa_{\rm abs}$ is not unusual depending on the prevailing composition of the dust.
The screen approximation to estimate the surface density is common but in fact the dusty ISM is clumped and filamentary in nature with a wide range of densities and temperatures. Typically, the distant galaxies are seen in gaps between the dusty clouds [@Holwerda07b]. The typical value of $A_I \sim 1$ (Figure \[f:sighi-A\]) corresponds to a surface covering factor of 60%, if the clouds were completely opaque. In reality, the disk opacity is a mix of covering factor and the mean extinction of the clouds [on average $\tau_{\rm cloud} = 0.4$ and cloud size 60 pc, @Holwerda07a]. We note that our mass estimates agree with those from a fit to the [*Spitzer*]{} fluxes with the [@Draine01b] model [to within a factor of two @Holwerda07a Figure \[f:draine\]]. [@Draine07] note that the addition of sub-mm information to such a fit may modify the dust mass estimate by a factor of 1.5 or less. Thus, while there is certainly a range of dust densities in each field, we are confident that the estimate from the above expression is a reasonable [*mean*]{} surface density.
Herschel-SPIRE Surface Brightness {#s:herschel}
---------------------------------
Sub-mm data for all our galaxies are available at the [*Herschel*]{} Science Archive[^7], the majority taken for the KINGFISH project[^8]. We therefore check the reliability of the SFM as a tracer of the dust surface density by directly comparing the surface brightness measured by the Spectral and Photometric Imaging REceiver [SPIRE, @Griffin10], onboard [*Herschel*]{} to the opacity as measured by the SFM.
We used the WFPC2 field-of-view as the aperture to measure the fluxes at 250, 350, and 500 $\mu$m (listed in Table \[t:info\]), similar to our measurements of the surface density in the and CO data (§\[s:hi\] and \[s:co\]). These were not aperture corrected because of the unique shape of the aperture. Figure \[f:herschel\] shows the [*Herschel*]{} surface brightnesses versus the SFM opacities for all three wavebands (the 250 and 500 $\mu$m. values are the end points of the horizontal bars). To convert the flux in a [*Herschel-SPIRE*]{} waveband into a dust surface density, one would need both a typical dust temperature or a temperature distribution and the dust’s emissivity. The horizontal bars indicate there is a range of mean temperatures in these disks.
There is a linear relation between the [*Herschel-SPIRE*]{} surface brightnesses and the SFM opacities. The scatter is much less for this relation than between the SFM dust surface density values and those inferred from far-infrared SED models [@Holwerda07a Figure \[f:draine\]]. Hence, we conclude that the SFM opacities are a reasonable indicator for mean dust surface density.
As a qualitative check, we compare the dust surface densities derived for a subset of the KINGFISH sample by [@Galametz12], their Figure A1, to those derived above. Typical values mid-disk for the overlap (NGC 3351, NGC 3521 and NGC 3627), where the WFPC2 images are located, are $\sim0.3 M_\odot/pc^2$, which appear to be typical [i.e., similar to those in @Foyle12]. These values lie a factor two below the ones implied by the SFM (Table \[t:info\]), regardless of the dust emissivity used in the [@Galametz12] fits but the difference is greater for fits where the emissivity is a free parameter. We found similar dust surface densities from the SED model in [@Holwerda07a], based on the [*Spitzer*]{} fluxes alone (Figure \[f:draine\]). Because all these models are based on the [@Draine07] model, we made a second check using the [magphys]{} SED model ([magphys]{}). These dust surface density are to a factor ten below the SFM or Draine et al. estimates. These fits illustrate the importance of the choice of model compared to the inclusion of sub-mm data.
![The Herschel/SPIRE 350 $\mu$m mean surface brightnesses in the WFPC2 field-of-view. Horizontal bars mark the 250 (left) and 500 (right) $\mu$m fluxes in the same field. The width of the horizontal bar is indicative of the mean temperature of the dust in each disk (a wide bar points to higher mean temperature). Variance around the mean surface brightness in each band is substantial due to both Poisson noise and structure in the galaxy disk. The lowest surface brightness point is NGC 3031, the closest galaxy in our sample. This field is right on the edge of the ISM disk (Figure \[f:himap\]) and therefore suffers the most from uncertainties due to internal structure and aperture correction. []{data-label="f:herschel"}](./holwerda_f2.pdf){width="50.00000%"}
![The dust surface density inferred by the SED model from [@Draine07] based on Spitzer fluxes [presented earlier in @Holwerda07a] compared to those from the SFM. There is at most a factor two difference between these, consistent with the lack of sub-mm information in these initial fits. Dashed line is the line of equality. []{data-label="f:draine"}](./holwerda_f2a.pdf){width="50.00000%"}
Column Density and the Number of Distant Galaxies {#s:hi}
==================================================
To improve statistics, one our tactics has been to stack the numbers of galaxies in our fields according to a local characteristic (surface brightness, galactocentric radius etc.). Here we combine the number of background galaxies, both real and artificial, based on the column density at their respective positions. If there is a relation between disk opacity and column density resolved in the THINGS RO maps, it should show as a preference of the real distant galaxies for a specific column density, for example for lower values of $\rm \Sigma_{HI}$. The artificial galaxies would not prefer any column density value in particular.
![[**Top:**]{} histogram of real (hatched) and artificial galaxies, $N$ and $N_0$ respectively, as a function of surface density, $\rm \Sigma_{HI}$. Because all the WFPC2 fields were chosen on spiral arms at the edge of the optical disks, the range of $\rm \Sigma_{HI}$ is limited. [**Bottom:**]{} inferred opacity ($A_I$) as a function of surface density. The dashed line is the relation from [[@Bohlin78]]{} for the Galactic total (+2) gas-to-dust ratio. []{data-label="f:na"}](./holwerda_f3.pdf){width="50.00000%"}
The top panel in Figure \[f:na\] shows the distribution histogram of real (hatched) and artificial (solid) galaxies observed, as a function of foreground galaxy column density. The bottom panel converts the ratio of real and artificial galaxies found at an column density into an opacity, using equation \[eq1\] with C equal to 1.2. The real distant galaxies identified in the HST images do not show a clear preference for a certain column density. Their distribution is very similar to that of the artificial distant background galaxies. As a result, the inferred opacity is constant with column density. In our opinion, this lack of a relation can either be: (1) real, pointing to a break-down in the spatial relation between and dust on scales of 6$^{\prime\prime}$ (corresponding to $\sim0.5$ kpc in our galaxies); or (2) an artifact of stacking results from different fields at various galactocentric radii in different foreground galaxies at diverse distances. We note, however, that the deviation from the [@Bohlin78] relation between column density and extinction (dashed line in bottom panel) is strongest for the lowest column densities, where our statistics are the most robust. In our opinion, this points to that one needs to compare to the total hydrogen column density, including the molecular component[^9].
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
Average Column Densities and Opacity per WFPC2 field {#s:wfpc2}
====================================================
Our second approach is to compare and 2 column densities to disk opacity averaged over each WFPC2 field. Table \[t:info\] lists the average opacity value for each HST/WFPC2 field, and the and 2 column densities averaged over the WFPC2 field-of-view (the footprints in Figure \[f:himap\]). The beams of the and 2 observations are much smaller than the WFPC2 apertures and we expect any aperture correction to the surface densities to be small (Table \[t:info\])[^10]. Figure \[f:sighi-A\], left, plots the opacity versus surface density; there is no clear relation between the two, when averaged over the size of a WFPC2 field. There are two negative values in our present sample \[and the entire [@Holwerda05] sample\], that are probably due to cosmic variance in the number of background galaxies (a background cluster). The opacity values and surface densities span a reasonable range for spiral galaxies. [@Cuillandre01] similarly find little relation between reddening and number of distant galaxies, on one side, and column density, on the other. Figure \[f:sighi-A\], middle panel, shows the relation between disk opacity and mean surface density of 2, inferred from the CO observations. There are fewer useful points, as there are no CO data for three of our WFPC2 fields, and two of the WFPC2 fields show the aforementioned negative opacity. There could be a relation between CO inferred molecular surface density and opacity. Figure \[f:sighi-A\], right panel, shows the relation between disk opacity and mean surface density of total gas (+2). For those galaxies where no CO information as available, we use the mean surface density (open diamonds). For comparison, we show the canonical Galactic relation from Bohlin et al. Opacity appears mostly independent from total gas surface density but with the majority of our points lie above the Bohlin et al. relation. There is surprisingly little of a relation between the gas, total, molecular or atomic, and disk opacity. In part this may be due in part to the different dust clumpiness in each disk, which is observed at a different distance. Alternatively, the metallicity and implicitly the average galactocentric radius of each field is the missing factor in the gas-dust relation in these fields.
![The ratio of dust surface density ($\rm \Sigma_{D}$) to either molecular (2 $\rm \Sigma_{H_2}$, top panel) or atomic hydrogen ( $\rm \Sigma_{HI}$) as a function of galactocentric radius, normalized to the 25 mag/arcsec$^2$ B-band isophotal radius [$R_{25}$ from @RC3]. Open circles are the negative disk opacities from NGC 925 and NGC5194-1. The innermost three points from NGC 3031 and NGC 3621 do not have CO information.[]{data-label="f:sigsR"}](./holwerda_f56.pdf){width="50.00000%"}
One explanation for the lack of a relation in Figure \[f:sighi-A\] is that the measurements were taken at various galactocentric radii (and hence metallicity) in each disk. Figure \[f:sigsR\] plots the ratio between the dust surface density (to facilitate direct comparison) and the two phases of the hydrogen,atomic and molecular in $\rm M_\odot/pc^2$, as a function of radius, scaled to the 25 mag/arcsec$^2$ B-band isophotal radius ($R_{25}$) from [@RC3]. The relation with atomic phase is consistent with a constant fraction of $\rm \Sigma_{D}/\Sigma_{HI} \sim 0.1$ with two exceptions at $R \sim0.5 R_{25}$; NGC 3351 and NGC 3627. Both of these are small disks, of which the WFPC2 field covers a large fraction (see Figure \[f:himap\]), both with prominent spiral arms. NGC 3627 is a member of the Leo triplet and as such may also be a victim of atomic gas stripping or a tidally induced strong spiral pattern. The top panel in \[f:sigsR\] shows the ratio between dust and molecular surface density, consistent with a constant fraction of $\rm \Sigma_{D}/\Sigma_{H2} \sim 0.75$ with one exception; NGC 3198, which is a very flocculant spiral, which a much lower surface density. There is little relation between the dust-to-atomic or dust-to-molecular ratio and radius. Exceptions seem to be either strong spiral arm structure, in the case of , or very flocculant, in the case of 2.
![ The ratio between implied average dust surface density ($\rm \Sigma_d$), and the total hydrogen surface density ($\rm \Sigma_{HI+H_2}$) as a function of radius. The inner three points (open diamonds) do not have CO data. The dashed-dotted line is the ratio from Bohlin et al. (1978).[]{data-label="f:ratio"}](./holwerda_f7.pdf){width="50.00000%"}
By combining the surface densities of and 2 into a single hydrogen surface density ($\rm \Sigma_{HI+H2}$), we can now directly compare the total dust-to-gas surface density ratio. In the cases, where no CO observations are available (NGC 3031 and NGC 3621), we use the ratio with only. Figure \[f:ratio\] shows the dust-to-total-gas ratio as a function of radius. The anomalous ratios of NGC 3351, NGC 3627 and NGC 3198 in Figure \[f:sigsR\] now fall into line.
If we take the points without CO information (open diamond symbols) at face value (assume no molecular gas), Figure \[f:ratio\] suggests an exponential decline of dust-to-total gas; ${\rm \Sigma_d / \Sigma_{HI+H2}} = 0.52 \times e^{- 4.0 {R / R_{25} }}$. A decline of the dust-to-total-gas ratio would be consistent with the relation with metallicity shown in [@Leroy11; @Sandstrom11], and with the trends with radius in the recent [*Spitzer*]{} [e.g., @Munoz-Mateos09b; @Bendo10a] and [*Herschel*]{} results [@Pohlen10; @Smith10b].
However, if we exclude those points without CO information (open diamond symbols) and those with negative SFM measurements (open circles), Figure \[f:ratio\] is in agreement with a [*constant*]{} gas-to-dust ratio of $0.043 \pm 0.02$ (weighted mean). One can reasonably expect a much more substantial contribution by the molecular component in the inner disk, which would bring the three points without CO information into line with this constant fraction. This dust-to-total-gas fraction is approximately a factor two above the typical value in the literature [$\sim$0.01-0.03, @Smith10b; @Leroy11] or the one from [@Bohlin78]. The fact that the ratio between dust and total gas surface density is nearly constant points to dust in both the diffuse disk as well as in the denser molecular clouds.
Discussion
==========
When compared to either phase of hydrogen in these disks, atomic or molecular, the dust density implied by the disk opacity mostly point to a constant ratio. Exceptions seem to point to a change in gas phase due to the strength of spiral arms in the WFPC2 field-of-view; a strong spiral density wave moves gas into the molecular phase and a flocculant structure into the atomic one. A scenario consistent with the density wave origin of spiral structure. In our opinion it illustrates the need for a constraint on both gas phases for a comparison with dust surface density.
Our dust-to-total-gas ratio of 0.043 (Figure \[f:ratio\]) is higher than the values found, for example, in the Local Group spiral galaxies [the Milky Way, M31, and M33 in the case of @Leroy11], or in a single Virgo spiral galaxy with [*Herschel*]{} [NGC 4501, @Smith10b] or the values found by [@Magrini11b]. These studies find the values closest to ours in the outskirts of the respective galaxy disks. There are several explanations for the high dust-to-gas ratio in our measurements: (1) we overestimated the dust surface density, (2) a substantial aperture correction of the CO and surface densities is needed, (3) for large portions of the disk, a different CO-to-2 conversion factor () is appropriate, and (4) a different absorption factor ($\kappa_{abs}$) for a disk average is appropriate. First, we are confident that our dust surface densities are unbiased and reasonably accurate because we checked them agains several other observational techniques (Cepheid reddening, occulting galaxy results, [*Spitzer*]{} FIR SED fits). Our main assumption is that the dust is in a screen, which is a very rough approximation, especially when the probe used is the number of distant objects [i.e, the opacity is also a function of cloud cover @Holwerda07b]. However, our comparison between dust surface densities from an SED fit and the number count of distant galaxies showed good agreement [@Holwerda07a] (Figure \[f:draine\]) to within a factor two. We note that these SED fits were done without sub-mm information but [@Draine07] point out that dust masses can vary with a factor less than 1.5 if the SED of the large grains is done with or without sub-mm information (their Figure 12). The dust surface densities are therefore not likely to be overestimated by more than a factor two. Our comparison with sub-mm fluxes (Figure \[f:herschel\], §\[s:herschel\]) seems to confirm this. The SFM estimate of the dust surface density may well be the upper limit of dust in these disks.\
Secondly, no aperture correction was applied to the CO and surface densities. Because the aperture we use to measure the CO and fluxes is the odd shape of the WFPC2 camera’s field-of-view (Figure \[f:himap\]), an aperture correction is not straightforward. Yet, we estimate that the aperture correction cannot change the reported average surface brightnesses sufficiently, as the resolution of the observations is substantially smaller than the WFPC2 aperture (Table \[t:info\]).\
Thirdly, when averaged over a large portion of the disk, which spans a range in density environments, the CO conversion factor () may well underestimate the total molecular hydrogen surface density, since some molecular clouds the observed CO may be from the “skin” of the GMC and there is not straightforward conversion from CO to 2 volume [@Wolfire10; @Glover11a; @Planck-Collaboration11a; @Shetty11; @Madden11; @Feldmann11a; @Feldmann12b; @Mac-Low12].\
A fourth option is that the dust absorption factor ($\kappa_{abs}$) is different when averaged over different environments and therefore dust grain properties [e.g., @Narayanan11c; @Narayanan12], although this is likely a secondary effect.
If [*all*]{} our dust surface densities would are all [*over*]{}-estimated by a factor $\sim$2, or the aperture correction increased the gas surface densities substantially, one may not need to change the factor to bring our dust-to-gas ratio in line with recent results from [*Herschel*]{}. We suspect, however, that the explanation includes a different , when averaged of a large section of the spiral disk and at different galactocentric radii, extending the range in values found in Local Group spiral galaxies by [@Leroy11].
![The logarithm of the ratio between the total hydrogen surface density ($\rm \Sigma_{HI+H_2}$) and the implied average dust surface density ($\rm \Sigma_d$) as a function of the metallicity ($\rm 12+log(O/H)$), estimated from Figure 7 in [@Moustakas10]. The circles and diamonds are for the calibration from [@KK04] and [@PT05] respectively. Open symbols are those points without CO information. There is no metallicity estimate for NGC 3627. We use the gas-to-dust ratio here in order to compare to the relation from [@Leroy11].[]{data-label="f:metal"}](./holwerda_f8.pdf){width="50.00000%"}
Comparison to Metallicities {#s:metal}
---------------------------
The present consensus is that the dust-to-total-gas ratio depends linearly on the metallicity [see for instance @Leroy11]. Fortunately, uniformly determined metallicity gradients for the SINGS[^11], and hence THINGS and HERACLES, galaxies are presented in [@Moustakas10]. Only NGC 3627 does not have metallicity information. Starting from their linear relation for the radial dependence of metallicity in each galaxy (their Figure 7), we can obtain an estimate of the metallicity for each of our WFPC2 fields. They present two different estimates of metallicity ($\rm log(O/H)$), with either the theoretical calibration from [@KK04] or the empirical one from [@PT05] (see Table \[t:info\]). [@Moustakas10] note that, until the calibration issues are resolved, one should either average the metallicity estimates based on either calibration or use both separately. We will use both calibrations separately for comparison and the total-gas-to-dust ratio to facilitate a direct comparison to Figure 6 in [@Leroy11]. We note that since our WFPC2 fields were placed with crowding issues in mind, our coverage of galactocentric radii (and hence metallicities) is not very large.
Figure \[f:metal\] shows the logarithm of the total-gas-to-dust ratio as a function of metallicity, using either of the two calibrations. Our points lie lower than the linear relation from [@Leroy11] for the gas-to-dust ratio with metallicity, not unexpectedly as we already established that our dust-to-gas values are higher than those previously reported.
However, using the calibration from [@PT05], and discarding those points that lack CO information, there is a reasonable agreement with the relation from [@Leroy11].
Conclusions {#s:concl}
===========
To conclude our “Opacity of spiral disks" project, we have compared the opacity of spiral galaxies, and the hence dust surface density to the surface densities of hydrogen, both atomic and molecular, the original goal of our project. We conclude from this comparison:
1. The disk opacity scales with the [*Herschel-SPIRE*]{} 250 $\mu$m surface brightness (Figure \[f:herschel\]), confirming our assertion that opacity scales with dust surface density to first order.
2. There is little relation between the column density and where a distant galaxy was identified in these fields (Figure \[f:na\]).
3. Averaged over a WFPC2 field, there is only a weak link between disk opacity (or dust surface density) and gas surface density, either atomic, molecular or total (Figure \[f:sighi-A\]), pointing to third factor; radius or metallicity.
4. The dust-to- or dust-to-2 relations with galactocentric radius are both relatively constant (Figure \[f:sigsR\]), but the exceptions point to the role of spiral structure in the dominant gas phase of the ISM.
5. The dust-to-[*total*]{}-gas ratio is close to constant for all our fields $\rm \Sigma_{HI+H_2} = 0.043 \pm 0.024$ (Figure \[f:ratio\]). This higher value can, in our opinion, be attributed to a different conversion to dust surface density or the CO-to-2 conversion factor () for such large sections of disks.
6. Compared to the relation between total-gas-to-dust and metallicity from [@Leroy11], our results are reasonably consistent, provided one uses the [@PT05] calibration of the metallicities of [@Moustakas10] (Figure \[f:metal\]).
Future use of the number of distant galaxies identified through a foreground spiral disk as a probe of dust is critically limited by cosmic variance [@Gonzalez03; @Holwerda05e] but its optimal application will be on a [*single*]{} large [*HST*]{} mosaic of a nearby face-on spiral (e.g., M81 or M101), which will most likely the last contribution of this unique approach to the issue of the dust content of spiral disks.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge the THINGS collaboration for the publication of their surface density maps, based on their Very Large Array radio observations and would like to thank the HERACLES collaboration for making their surface density maps available early. The authors would like to thank F. Walther and S-L. Blyth for useful discussions and feedback. We thank the anonymous referee for his or her excellent report and extraordinary effort.
We acknowledge support from HST Archive grants AR-10662 and AR-10663 and from the National Research Foundation of South Africa. The work of W.J.G. de Blok is based upon research supported by the South African Research Chairs Initiative of the Department of Science and Technology and the National Research Foundation. Antoine Bouchard acknowledges the financial support from the South African Square Kilometre Array Project. R. A. González-Lópezlira acknowledges support from DGAPA (UNAM) grant IN118110.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Based on observations made with the NASA/ESA Hubble Space Telescope, which is a collaboration between the Space Telescope Science Institute (STScI/ NASA), the Space Telescope European Coordinating Facility (ST-ECF/ ESA), and the Canadian Astronomy Data Centre (CADC/NRC/CSA). The Hubble data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584, and by other grants and contracts.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of NASA’s Astrophysics Data System.
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MAGPHYS SED Model {#s:magphys}
=================
As an alternative check of the inferred dust masses, we ran the Multi-wavelength Analysis of Galaxy Physical Properties ([magphys]{}) package on the [*Spitzer*]{} and [*Herschel/SPIRE*]{} surface brightnesses. This is a self-contained, user-friendly model package to interpret observed spectral energy distributions of galaxies in terms of galaxy-wide physical parameters pertaining to the stars and the interstellar medium, following the approach described in [@da-Cunha08]. Figure \[f:magphys\] summarizes the result: dust surface density derived from the [magphys]{} fit compared to those inferred from the number of distant galaxies. In [@Holwerda07a], we found that the [@Draine07] model inferred similar dust optical depths for these disks as the SFM as well as similar (to within a factor two) dust masses. The discrepancy with [magphys]{} illustrates, in our view, the importance of modeling sections of spiral disks in resolved observations with more physical models that include a range of stellar heating parameters [e.g. the models by @Draine07; @Galliano11].
![The dust surface densities from the [magphys]{} fit and inferred from the number of identified background galaxies (SFM) for each WFPC2 aperture. The dashed line denotes a factor ten ratio. [magphys]{} SED models do not take internal structure and differential stellar heating into account.[]{data-label="f:magphys"}](./holwerda_f2b.pdf){width="50.00000%"}
SED of each WFPC2 field with the [magphys]{} fit. {#sed-of-each-wfpc2-field-with-the-magphys-fit. .unnumbered}
=================================================
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}\
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}\
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}\
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
\[f:seds\]
[^1]: E-mail: benne.holwerda@esa.int
[^2]: We used the term “opacity" throughout our project and its publications for historical reasons.
[^3]: Other authors have used distant galaxy counts or colours to estimate extinction in the Magellanic Clouds [@Shapley51; @Wesselink61b; @Hodge74; @Hodge75; @McGillivray75; @Gurwell90; @Dutra01] and other galaxies [@Zaritsky94; @Cuillandre01].
[^4]: In this case it does not matter whether the maps are robustly weighted or naturally weighted.
[^5]: Similar quality products are now also available from the archives at STSCI, the High-Level Archive; [www.hla.stsci.edu](www.hla.stsci.edu).
[^6]: It depends to a degree on the depth of the data. Conservatively, for this kind of fields, the total error is about 3.5 times Poisson [@Gonzalez03].
[^7]: <http://herschel.esac.esa.int/Science_Archive.shtml>
[^8]: Key Insights on Nearby Galaxies: a Far-Infrared Survey with Herschel, PI. R. Kennicutt, [see also @Skibba11a; @Dale12; @Kennicutt11; @Galametz12]
[^9]: Our fields are usually centered on a spiral arm (to observe Cepheids) and this increases the contribution from molecular phase.
[^10]: We chose not to correct the surface densities because of the odd shape of the aperture. Depending on how one treats the edges of the aperture, the average surface density varies with $\sim$10%.
[^11]: Spitzer Infrared Nearby Galaxy Survey [@SINGS].
|
---
abstract: |
Distributed programs are often formulated in popular functional frameworks like [*MapReduce*]{}, *Spark* and *Thrill*, but writing efficient algorithms for such frameworks is usually a non-trivial task. As the costs of running faulty algorithms at scale can be severe, it is highly desirable to verify their correctness.
We propose to employ existing imperative reference implementations as specifications for [*MapReduce*]{} implementations. To this end, we present a novel verification approach in which equivalence between an imperative and a [*MapReduce*]{} implementation is established by a series of program transformations.
In this paper, we present how the equivalence framework can be used to prove equivalence between an imperative implementation of the [*PageRank*]{} algorithm and its [*MapReduce*]{} variant. The eight individual transformation steps are individually presented and explained.
author:
- |
Bernhard Beckert Timo Bingmann Moritz Kiefer\
Peter Sanders Mattias Ulbrich Alexander Weigl
bibliography:
- 'references.bib'
title: Proving Equivalence Between Imperative and MapReduce Implementations Using Program Transformations
---
Introduction
============
Today requirements on the efficiency and scale of computations grow faster than the capabilities of the hardware on which they are to run. Frameworks such as [*MapReduce*]{} [@mapreduce], Spark [@spark] and Thrill [@Thrill] that distribute the computation workload amongst many nodes in a cluster, address these challenges by providing a limited set of operations whose execution is automatically and transparently parallelised and distributed among the available nodes.
In this paper, we use the term “[*MapReduce*]{}” as a placeholder for a wider range of frameworks. While some frameworks such as Hadoop’s MapReduce [@white2012hadoop] strictly adhere to the two functions “map” and “reduce”, the more recent and widely used distribution frameworks provide many additional primitives – for performance reasons and to make programming more comfortable.
Formulating efficient implementations in such frameworks is a challenge in itself. The original algorithmic structure of a corresponding imperative algorithm is often lost during that translation since imperative constructs do not translate directly to the provided primitives. Significant algorithmic design effort must be invested to come up with good [*MapReduce*]{} implementations, and flaws are easily introduced during the process.
The approach we proposed in our previous work [@arxiv] and will refine and apply in this paper supports algorithm engineers in the correct design of [*MapReduce*]{} implementations by providing a transformation framework with which it is possible to interactively and iteratively translate a given imperative implementation into an efficient one that operates within a [*MapReduce*]{} framework.
The framework is thus a verification framework to prove the *behavioural equivalence* between an imperative algorithm and its [*MapReduce*]{} counterpart. Due to often considerable structural differences between the two programming paradigms, our approach is interactive: It requires the specification of intermediate programs to guide the translation process. While it is not in the focus of this publication, our approach is designed to have a high potential for automation: The required interaction is designed to be as high-level as possible. The rules are designed such that their side conditions can be proved automatically, and pattern matching can be used to allow for a more flexible specification of intermediate steps.
We present an approach based on program transformation rules with which a [*MapReduce*]{} implementation of an algorithm can be proved equivalent to an imperative implementation. From an extensive analysis of the example set of the framework Thrill, we were able to identify a set of [13]{} transformation rules such that a chain of rule applications from this set is likely to succeed in showing the equivalence between an imperative and a functional implementation.
We describe a workflow for integrating this approach with existing interactive theorem provers. We have successfully implemented the approach as a prototype within the interactive theorem prover [*Coq*]{} [@Coq].
The main contribution of this paper is the demonstration of the abilities of the framework to establish equivalence between imperative and [*MapReduce*]{} implementations. We do this (1) by motivating and thoroughly explaining the rationales and the nature of the transformation rules and (2) by reporting on the successful application of the framework to a relevant non-trivial case study. We have chosen the [*PageRank*]{} algorithm as the demonstrative example since it is one of the original and best known [*MapReduce*]{}application cases.
#### Overview of the approach.
![Chain of equivalent programs is translated into formalised functional language[]{data-label="fig:approach-overview"}](fig-approach)
The main challenge in proving the equivalence of an imperative and a [*MapReduce*]{} algorithm lies in the potentially large structural difference between the two algorithms. To deal with this, the equivalence of imperative and [*MapReduce*]{} algorithms is not shown in one step, but as a succession of equivalence proofs for structurally closer program versions.
To this end, we require that the translation of the algorithm is broken down (by the user) into a chain of intermediate programs. For each pair of neighbouring programs in this chain, the difference is comparatively small such that the pair can be used as start and end point of a single program transformation step induced by a rule application.
The approach uses two programming languages: One is the imperative input language () in which the imperative algorithm, the intermediate programs, as well as the target [*MapReduce*]{} implementation are stated. Besides imperative language constructs, supports [*MapReduce*]{} primitives.
Each program specified in the high-level imperative language is then automatically translated into the formalised functional language (). The program transformations and equivalence proofs operate on programs in this functional language. The translation of the original, the intermediate and the [*MapReduce*]{} programs form a chain of programs. For each pair of neighbouring programs in the chain, a proof obligation is generated that requires proving their equivalence. These proof obligations are then discharged independently of each other. Since, by construction, the semantics of programs is the same as that of corresponding programs, the equivalence of two programs follows from the equivalence of their translations to . An overview of this process can be seen in [Fig. \[fig:approach-overview\]]{}. Figure \[fig:nonlocal-transformation\] shows two example programs for calculating the element-wise sum of two arrays. The implementation of our approach based on the [*Coq*]{} theorem prover has only limited proof automation and still requires a significant amount of interactive proofs. We are convinced, however, that our approach can be extended such that it becomes highly automatised and only few user interactions or none at all are required – besides providing the intermediate programs. To enable this automation, one of the design goals of the approach was to make the matching of the rewrite rules as flexible as possible. Further challenges include the extension of our approach to features such as references and aliasing which are commonly found in imperative languages.
#### Structure of the paper.
The remainder of the paper is structured as follows: After introducing the supported programming languages and in Sect. \[sec:foundations\], a description of the two different kinds of transformation rules applied in the framework follows in Sect. \[sec:rules\]. The case study on the [*PageRank*]{} example is conducted in Sect. \[sec:example:pagerank\]. After a report on related work in Sect. \[sec:relatedwork\], the paper is wrapped up with conclusions in Sect. \[sec:conclusion\].
Foundations {#sec:foundations}
===========
This section introduces the programming language used to formulate the programs, and briefly describes the language used for the proofs.
The high-level imperative programming language is based on a while language. Besides the usual `while` loop constructor, it possesses a [`for`]{} loop constructor which allows traversal of the entries of an array. The supported types are integers (`Int`), rationals[^1] (`Rat`), Booleans (`Bool`), fixed length arrays (`[T]`) and sum (`T_1 + T_2`) and product types (`T_1 * T_2`). Since arrays are an important basic data type for [*MapReduce*]{}, a number of functions are provided to operate on this data type. Table \[tab:il-funcs\] lists the -functions relevant for this paper. Besides the imperative language constructs, supports a number of [*MapReduce*]{} primitives. In particular, a lambda abstraction for a variable $v$ of type $T$ over an expression $e$ can be used and is written as does not support recursion.
Given that allegedly [*MapReduce*]{} programs tend to be more of a functional than an imperative nature, it might seem odd that we use also for specifying the [*MapReduce*]{} algorithm and not a functional language. However, most existing [*MapReduce*]{} frameworks are not implemented as separate languages, but as frameworks built on top of imperative languages. This implies that the sequential imperative constructs of the host language can also be found in [*MapReduce*]{}programs. Sequential parts of [*MapReduce*]{} algorithms are realised using imperative programming features, while the computational, distributed parts are composed using the [*MapReduce*]{}primitives. Figure \[fig:nonlocal-transformation\] shows two behaviourally equivalent implementations of a routine that computes the sum of the entries of two `Int`-arrays.
fn SumArrays(xs: [Int], ys: [Int]) {
var sum := replicate(n, 0);
for(i : range(0, length(xs))) {
sum[i] := xs[i] + ys[i];
}
return sum;
}
fn SumArraysZipped(xs: [Int], ys: [Int]) {
var sum := replicate(n, 0);
zipped := zip(xs, ys);
for(i : range(0, length(xs))) {
sum[i] := fst(zipped[i]) + snd(zipped[i]);
}
return sum;
}
The programs specified in are then automatically translated into , the functional language based on $\lambda$-calculus in which the equivalence proofs by program transformation are conducted. We follow the work by Radoi et al. [@translatingimperative] and use a simply typed lambda calculus extended by the theories of sums, products, and arrays. Moreover, to allow the translation of both imperative and [*MapReduce*]{}code into , the language also contains constructs for loop iteration and the programming primitives usually found in [*MapReduce*]{} frameworks. is formally defined as a theory in the theorem prover [*Coq*]{} which allowed us to conduct correctness proofs on the rewrite rules.
Without going into the intricacies of the details of the translation between the formalisms, the idea is as follows: Any statement becomes a (higher order) term in in which the currently available local variables ${\mathit{acc}}$ make up the $\lambda$-abstracted variables. The two primitives ${\mathsf{iter}}$ and ${\mathsf{fold}}$ serve as direct translation targets for imperative loops. The ${\mathsf{fold}}$ function is used to translate bounded [`for`]{} loops into . The iterator loop `for(x:xs) ‘{ f ‘}` in is translated into as the expression $\lambda {\mathit{acc}}.\ {\mathsf{fold}}(\hat f, {\mathit{acc}}, \hat{{\mathit{xs}}})$ in which $\hat f$ and $\hat xs$ are the -translations of `f` and `xs`. This starts with the initial loop state ${\mathit{acc}}$ and iterates over each value of the array $\hat{\mathit{xs}}$ updating the loop state by applying $\hat f$. The more general `while` loops cannot be translated using `fold` since that always has bounded number of iterations. Instead, `while` is translated using the ${\mathsf{iter}}$ fixed point operator. The loop `while(c) ‘{ f ‘}` translates as ${\mathsf{iter}}(\lambda {\mathit{acc}}.\ \mathsf{if}\ c({\mathit{acc}})\ \mathsf{then\ inr} \hat
f({\mathit{acc}})\ \mathsf{else\ inl}\ {\mathsf{unit}})$ and is evaluated by repeatedly applying $\hat f$ to the loop state until $\hat f$ returns ${\mathsf{unit}}$ to indicate termination. ${\mathsf{iter}}$ is a partial function; if the loop does not terminate it does not yield a value. The semantics of is defined as a bigstep semantics. A program together with its input parameters are reduced to a value. Ill-typed or non-terminating programs do not reduce to a value. Details on the design of and can be found elsewhere [@arxiv].
The design of ’s core follows the ideas of Chen et al. [@sparkspecification] who describe how to reduce the large number of primitives provided by [*MapReduce*]{} frameworks.
Transformation Rules {#sec:rules}
====================
For the examples shipped with the [*MapReduce*]{} framework Thrill, we analysed how the steps of a transformation of an algorithm into [*MapReduce*]{} would look and detected typical recurring patterns for steps in a transformation. We were able to identify two different categories of transformation rules that are needed for the process:
1. Local, *context-independent rewrite rules* with which a statement of the program can be replaced with a semantically equivalent statement. Such rules may have side conditions on the parts on which they apply, but they cannot restrict the application context (the statements enclosing the part to be exchanged).
2. *Context-dependent equivalence rules* that cannot replace a statement without considering the context. Some transformations are only valid equivalence replacements within their surrounding context. These are not pattern-driven rewrite rules, but follow a deductive reasoning pattern that proves equivalence of a piece of code locally.
Context-independent rules are good for changing the control flow of the program. In Sect. \[sec:trans1\], we will encounter an example of a rule which replaces a loop by an equivalent `map` expression. The data flow, while differently processed, remains the same. The context-independent rules are a powerful tool to bridge larger differences in the control structure between the two programming paradigms. These changes must be anticipated beforehand and cannot be detected and proved on the spot. We identified a total of [13]{} rules that allow us to transform imperative constructs into [*MapReduce*]{} primitives. (See App. B of [@arxiv] for a complete list.) Their rigid search patterns make context-independent rules less flexible in their application.
Context-dependent rules, on the other hand, are suited for transforming a program into a structurally similar program (they do not/little change the control flow); the data flow may be altered however using such rules. It is comprehensible that a change in the data representation is an aspect which cannot be considered locally, but requires analysing the whole program.
The collection of rewrite rules for context-independent replacements comprises various different patterns. The context-dependent case is different. There exists one single rule which can be instantiated for arbitrary coupling predicates and is thereby highly adaptable. We employ relational reasoning using product programs [@productprograms] to show connecting properties. The rule bases on the observation that loops in the two compared programs need not be considered separately but can be treated simultaneously. If $x_1$ ($x_2$) are the variables of the first (second) program, and if the conditions $c_1$ and $c_2$, as well as the loop bodies $b_1$ and $b_2$ refer only to variables of the respective program, then the validity of the Hoare triple $$\label{eq:hoare-1}
\{ x_1 = x_2 \} \texttt{ while($c_1$) \char`\{{} $b_1$ \char`\}{} ;
while($c_2$) \char`\{{} $b_2$ \char`\}{} } \{ x_1 = x_2 \}$$ is implied by the validity of $$\label{eq:hoare-2}
\{ x_1 = x_2 \} \texttt{ while($c_1$) \char`\{{} $b_1$; $b_2$; assert $c_1$=$c_2$; \char`\}{} } \{ x_1 = x_2 \}\rlap{ \enspace.}$$
Condition expresses that given equal inputs, the two loop programs terminate in the same final state. Condition manages to express[^2] this with a single loop. This gives us the chance to prove equivalence using a single *coupling invariant* that relates to both program states. To show equivalence for context-dependent cases, the specification of a relational coupling invariant with which the consistency of the data can be shown is required.
An example of two programs which are equivalent with similar control structure, yet different data representation, has already been presented in Fig. \[fig:nonlocal-transformation\]. Both programs can be shown equivalent by means of the coupling invariant where the subscripts indicate which program a variable belongs to. Sect. \[sec:trans2\] demonstrates the application of the rule within the [*PageRank*]{} example.
In the formal approach (also outlined in [@arxiv]) and the [*Coq*]{} implementation, all rules are formulated on and operate on the level of (which has been designed for exactly this purpose). For the sake of better readability we show the corresponding rules here on the level of and notate the context independent rewrite rules as $$\mathtt{program_1} \quad \leadsto \quad \mathtt{program_2} \qquad (\textsl{rulename})$$ meaning that under the specified side conditions any statement in a program matching `program_1` can be replaced by the corresponding instantiation of `program_2`, yielding a program behaviourally equivalent to original program.
The application of rules are transformations only in a broader sense. The approach is targeted as interacting with a user who provides the machine with the intermediate steps and the rule scheme to be applied. It would likewise be possible to instead allow specifying which rules must be applied and have the user specify the instantiations instead of the resulting intermediate programs.
In particular the context-dependent rule is hardly a rewrite rule due to the deductive nature of the equivalence proof. It is not astonishing though that the transformation into [*MapReduce*]{} does not work completely by means of a set of rewrite patterns since transforming imperative programs to [*MapReduce*]{} programs is more than simple pattern matching process but requires some amount of ingenious input to come up with an efficient implementation.
Example: PageRank {#sec:example:pagerank}
=================
In this section, we demonstrate our approach by applying it to the [*PageRank*]{} algorithm. We present all intermediate programs in and explain the transformations and techniques used in the individual steps. While the implementation executes the actual equivalence proofs on terms, we only present the programs here since these are the intermediate steps specified by the user. Where the translation of a transformation to is not straightforward, we also give an explanation of the transformation expressed on terms.
The algorithm
-------------
[*PageRank*]{}is the algorithm that Google originally successfully employed to compute the rank of web pages in their search engine. This renowned algorithm is actually a special case of sparse matrix-vector multiplication, which has much broader applications in scientific computing. [*PageRank*]{} is particularly well suited as an example for a map reduce implementation and is included in the examples of Thrill and Spark. While more highly optimized [*PageRank*]{}algorithms and implementations exist, we present here a simplified version. The idea behind [*PageRank*]{} is the propagation of reputation of a web page amongst the pages to which it links. If a page with a high reputation links to another page, the latter page’s reputation thus increases.
The algorithm operates as follows: [*PageRank*]{} operates on a graph in which the nodes are the pages of the considered network, and (directed) edges represent links between pages. Pages are represented as integers, and the 2-dimensional array `links` holds the edges of the graph in the form of an adjacency list: the $i$-th component of `links` is an array containing all indices of the pages to which page $i$ links. The result is an array `ranks` of rationals that holds the pagerank value of every page. The initial rank is set to ${\mathit{Rank}}_0(p) = \frac{1}{|\mathtt{links}|}$ for all pages $p$.
In the course of the $k$-th iteration ($k>0$) of the algorithm, the rank of each link target is updated depending on the pages that link *to* the page, i.e., the incoming edges in the graph: $$\begin{aligned}
\Delta_k(p) &= \sum_{(o,p) \in \mathtt{links}} \mathit{{\mathit{Rank}}}_{k-1}(o) &
\mathit{{\mathit{Rank}}}_k(p) &= \delta * \Delta_k(p) + \frac{1-\delta}{|\mathtt{links}|}
\label{eq:pr}\end{aligned}$$ The factor $\delta \in (0,1)$ is a dampening factor to limit the effects of an iteration by weighting the influence of the result of the iteration $\Delta_k(p)$ against the original value ${\mathit{Rank}}_0(p) = \frac{1}{|\mathtt{links}|}$. Our implementation iterates this scheme for a fixed number of times (`iterations`).
[Listing \[lst:pr-1\] on page ]{} shows a straightforward imperative implementation of this algorithm that realises the iterative laws of directly. It marks the starting point of the translation from imperative to [*MapReduce*]{}algorithm. To allow a better comparison between the programs, the programs are listed next to each other at the end of this section.
A context-independent rule application {#sec:trans1}
--------------------------------------
The first step in the chain of transformations from imperative to distributed replaces the [`for`]{} loop used to calculate the weighted new ranks with a call to `map`. This is possible since the values can be computed independently. The `map` expression allows computing the dampened values in parallel and can thereby significantly improve performance. The rewrite rule used here can be applied to all [`for`]{} loops that iterate over the index range of an array where each iteration reads a value from one array at the respective index, applies a function to it and then writes the result back to another array at the same index.
for (i : range(0, length(xs))) {
ys[i] := f(xs[i]);
}
$\leadsto$
ys := map(f, xs);
(*map-introduce*)
Sufficient conditions for the validity of this context-independent transformation are that `f` does not access the index `i` directly and that `xs` and `ys` have the same length. The first condition can be checked syntactically while matching the rule while the second requires a (simple) proof in the context of the rule application. In our implementation, these proofs are executed in [*Coq*]{}.
As mentioned before, [`for`]{} loops in correspond to `fold` operations in . The rewrite rule expressed on thus transforms ${\mathsf{fold}}(\lambda {\mathit{acc}}\ i.\ {\mathit{acc}}[i := f(\mathit{xs}[i])], \mathit{ys},
{\mathsf{range}}(0, {\mathsf{length}}(xs)))$ into ${\mathsf{map}}(f, \mathit{xs})$.
The result of the transformation is shown in [Listing \[lst:pr-2\] on page ]{}. For convenience, in this and the following listings, the modified part of the program is highlighted in colour.
A context-dependent rule application {#sec:trans2}
------------------------------------
In this step, the body of the main [`while`]{} loop is changed to first combine the `links` and `ranks` arrays to an array `outRanks` of tuples using the `zip` primitive. In the remaining loop body, all references to `links` and `ranks` point to this new array and retrieve the original values using the pair projection functions `fst` and `snd`. The process of rewriting all references does not fit easily into the rigid structure of the rewrite rules employed in our approach. We thus resort to using a context-dependent rule using a coupling predicates to prove equivalence of the last and the new loop body. Using the coupling predicate $$\mathtt{newRanks}_{1} = \mathtt{newRanks}_{2} ~~\wedge~~
\mathtt{outRanks}_{2} = \mathtt{zip}(\mathtt{links}_{1}, \mathtt{ranks}_{1})$$ that relates the values in the states of the two programs (we use the subscript indices $1$ and $2$ to refer to variables in the original and the transformed program) we obtain that the loop bodies have equivalent effects, and hence, that the programs are equal.
The result of the transformation is shown in [Listing \[lst:pr-3\] on page ]{}.
Rule *range-remove*
-------------------
In the next transformation the [`for`]{} loop which iterates over all pages as the index range of the array `links` is replaced by a [`for`]{} loop that iterates directly over the elements in `outRanks`. The rewrite rule *range-remove* applied here can be applied to all [`for`]{} loops that iterate over the index range of an array and only use the index to access these array elements. Again this is a side condition for the rule which can be checked syntactically during rule matching.
acc := acc0;
for (i : range(0, length xs)) {
acc := f(acc, xs[i]);
}
$\leadsto$
acc := acc0;
for (x : xs) {
acc := f(acc, x);
}
(*range-remove*)
The result of the application of rewrite rule is shown in [Listing \[lst:pr-4\] on page ]{}.
Expressed on the level of , this rewrite rule transforms terms of the form\
$ \tfold(\lambda \acc\ i.\ f(\acc,\xs[i]), \acc_0, \trange(0,\tlength(\xs))$ into $ \tfold(\lambda \acc\ x .\ f(\acc,x), \acc_0, \xs)
$.
Aggregating link information
----------------------------
The next step is a typical step that can be observed when migrating algorithms into the [*MapReduce*]{} programming model. A computation is split into two consecutive steps: one step processing data locally on individual data points and one step aggregating the results. It can be anticipated already now that these two steps will become the *map* and the *reduce* part of the algorithm.
The newly introduced variable `linksAndContrib` stores the (locally for each node) computed rank contribution as a list of tuples. Assume $({\langles_1, \ldots, s_n\rangle}, r)$ is the $i$-th entry in the array `outRanks`. This means that page $i$ links to page $s_j$ for $j < n$ and has a current rank of $r$. After the newly introduced local computation, the entry becomes the list of pairs ${\langle(s_1, \frac{r}{|\mathtt{links}|}), \ldots, (s_n,
\frac{r}{|\mathtt{links}|})\rangle}$, i.e., the rank is distributed to all successor pages and the data is rearranged with the focus now on the receiving pages.
As in the transformation in Sect. \[sec:trans2\], a context-dependent transformation is employed to prove equivalence using the following relational coupling loop invariant: $$\begin{aligned}
& \mathtt{newRanks}_{1} = \mathtt{newRanks}_{2} \ \wedge \\
& \begin{aligned}
\forall i j.\ & \mathtt{fst}\ \mathtt{linksAndContrib}_{2}[i][j] =
(\mathtt{fst}\ \mathtt{outRanks}_{1}[i])[j] \ \wedge \\
& \mathtt{snd}\ \mathtt{linksAndContrib}_{2}[i][j] = \mathtt{snd}\ \mathtt{outRanks}_{1}[i] / \mathtt{length} (\mathtt{fst}\ \mathtt{outRanks}_{1}[i])
\end{aligned}
\end{aligned}$$
The result of the transformation is shown in [Listing \[lst:pr-5\] on page ]{}. Note that the nested loops in the highlighted block no longer perform the computation of the rank updates (`snd links_rank / length(fst links_rank)`), but only aggregate the contribution updates into new ranks. This transformation is a preparation for collapsing the nested loops in the next step.
Collapsing nested loops {#sec:trans4}
-----------------------
Since the computation of `contribution` has been moved outside in the previous step, the iteration variable `link_contribs` is now only used as the iterated array in the inner [`for`]{} loop. This allows collapsing the nested loops into a single loop using `concat`. This rule can always be applied if the iterated value in the inner [`for`]{} is the only reference to the values the outer [`for`]{} iterates over.
acc := acc0;
for (xs : xss) {
for (x : xs) {
acc := f(acc, x);
}
}
$\leadsto$
``` {mathescape=""}
acc := acc0;
for (x : concat(xss)) {
acc := f(acc, x);
}
$\ $
```
(*concat-intro*)
The program with the two loops collapsed is shown in [Listing \[lst:pr-6\] on page ]{}.
This transformation is succeeded by a step that combines the call to `concat` in the [`for`]{} loop and the `map` operation before the loop into a single call to `flatMap`. Its result is shown in [Listing \[lst:pr-7\] on page ]{}. In , `flatMap` is not a builtin primitive but a synonym for successive calls to `concat` and `map`. This step is thus one which has visible effects on the level of , but no impact on the level of .
Towards [*MapReduce*]{}
-----------------------
Now we are getting closer to a program that adheres to the [*MapReduce*]{} programming model. The penultimate transformation step restructures the processed data by grouping all rank updates that affect the same page. It operates on the array `newRanks` using the function `group`. The updated result is calculated using a combination of `map` and `fold`. The results are then written back to `newRanks`. The effects of the rule on the program structure are more severe than for the other applied transformation rules, yet this grouping pattern is one that is typically observed in the [*MapReduce*]{} transformation process and is implemented as a single rule for that reason.
The corresponding rewrite rule can be applied to all [`for`]{} loops that iterate over an array containing index-value tuples and update an accumulator based on the old value stored for that index and the current value:[^3]
``` {mathescape=""}
acc := acc0;
for ((i,v) : xs) {
acc[i] := f(acc[i], v);
}
$\ $
```
$\leadsto$
``` {mathescape=""}
acc := acc0;
var upd := map((i,vs) => fold(f, acc[i], vs),
group(acc));
for (x : concat(xss)) {
acc := f(acc, x);
}
```
(*group-intro*)
Note that since `acc0` could store values for indices for which there are no corresponding tuples in `xs`, it is necessary to write the results back to that array instead of simply using the result from the `group` operation which would be missing those entries.
The resulting program is shown in [Listing \[lst:pr-8\] on page ]{}.
The final [*MapReduce*]{}implementation
---------------------------------------
In the last step, the expression that groups contributions by index and then sums them up is replaced by the -function `reduceByKey` which is also provided by many [*MapReduce*]{} frameworks. In the lower level language , however, `reduceByKey` is not a primitive function, but a composed expression using ${\mathsf{map}}$, ${\mathsf{fold}}$ and ${\mathsf{group}}$, such that this step changes the program, but has no impact on the ${\textsf{FFL}\xspace}$ level. The resulting implementation using map reduce constructs is shown in [Listing \[lst:pr-9\] on page ]{}. It is very close to the [*MapReduce*]{}implementation of [*PageRank*]{} that is delivered in the example collection of the Thrill framework.
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1. / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
for (pageId : range(0, length(links))) {
var contribution : Rat := ranks[pageId] / length(links[pageId]);
for (outgoingId : links[pageId]) {
newRanks[outgoingId] := newRanks[outgoingId] + contribution;
}
}
for (pageId : range(0, length(links))) {
ranks[pageId] :=
dampening * newRanks[pageId] + (1 - dampening) / length(links);
}
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1. / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
for (pageId : range(0, length(links))) {
var contribution : Rat := ranks[pageId] / length(links[pageId]);
for (outgoingId : links[pageId]) {
newRanks[outgoingId] := newRanks[outgoingId] + contribution;
}
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
var outRanks : [[Int] * Rat] := zip(links, ranks);
for (pageId : range(0, length(links))) {
var contribution : Rat := snd outRanks[pageId] / length(fst outRanks[pageId]);
for (outgoingId : fst outRanks[pageId]) {
newRanks[outgoingId] := newRanks[outgoingId] + contribution;
}
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
var outRanks : [[Int] * Rat] := zip(links, ranks);
for (links_rank : outRanks) {
var contribution : Rat := snd links_rank / length(fst links_rank);
for (outgoingId : fst links_rank) {
newRanks[outgoingId] := newRanks[outgoingId] + contribution;
}
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
var outRanks : [[Int] * Rat] := zip(links, ranks);
var linksAndContrib : [[Int * Rat]] :=
map((links_rank : [Int] * Rat) =>
map((link : Int) =>
(link, snd links_rank / length(fst links_rank)),
fst links_rank),
outRanks);
for (link_contribs : linksAndContrib) {
for (link_contrib : link_contribs) {
newRanks[fst link_contrib] :=
newRanks[fst link_contrib] + snd link_contrib;
}
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
var outRanks : [[Int] * Rat] := zip(links, ranks);
var linksAndContrib : [[Int * Rat]] :=
map((links_rank : [Int] * Rat) =>
map((link : Int) =>
(link, snd links_rank / length(fst links_rank)),
fst links_rank),
outRanks);
for (link_contrib : concat(linksAndContrib)) {
newRanks[fst link_contrib] :=
newRanks[fst link_contrib] + snd link_contrib;
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var newRanks : [Rat] := replicate(length(links), 0);
var outRanks : [[Int] * Rat] := zip(links, ranks);
var linksAndContrib : [Int * Rat] :=
flatMap((links_rank : [Int] * Rat) =>
map((link : Int) =>
(link, snd links_rank / length(fst links_rank)),
fst links_rank),
outRanks);
for (link_contrib : linksAndContrib) {
newRanks[fst link_contrib] :=
newRanks[fst link_contrib] + snd link_contrib;
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var outRanks : [[Int] * Rat] := zip(links, ranks);
var contribs : [Int * Rat] :=
flatMap((links_rank : [Int] * Rat) =>
map((link : Int) => (link,
snd links_rank / length(fst links_rank)),
fst links_rank),
outRanks);
var rankUpdates : [Int * Rat] :=
map((link : Int) (contribs : [Rat]) =>
(link, fold((x: Rat) (y : Rat) => x + y, 0, contribs)),
group(contribs));
var newRanks : [Rat] := replicate(length(links), 0);
for (link_rank : rankUpdates) {
newRanks[fst link_rank] := snd link_rank;
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
------------------------------------------------------------------------
fn pageRank(links : [[Int]], dampening : Rat, iterations : Int) -> [Rat] {
var iter : Int := 0;
var ranks : [Rat] := replicate(length(links), 1 / length(links));
while (iter < iterations) {
var outRanks : [[Int] * Rat] := zip(links, ranks);
var contribs : [Int * Rat] =
flatMap((links_rank : [Int] * Rat) =>
map((link : Int) => (link,
snd links_rank / length(fst links_rank)),
fst links_rank),
outRanks);
var rankUpdates : [Int * Rat] := reduceByKey((x : Rat) (y : Rat) => x + y, 0, contribs);
var newRanks : [Rat] := replicate(length(links), 0);
for (link_rank : rankUpdates) {
newRanks[fst link_rank] := snd link_rank;
}
ranks :=
map((rank : Rat) => dampening * rank + (1 - dampening) / length(links),
newRanks);
iter := iter + 1;
}
return ranks;
}
Related Work {#sec:relatedwork}
============
A common approach to relational verification and program equivalence is the use of product programs [@productprograms]. Product programs combine the states of two programs and interleave their behavior in a single program. *RVT* [@rvt] proves the equivalence of C programs by combining them in a product program. By assuming that the program states are equal after each loop iteration, *RVT* avoids the need for user-specified or inferred loop invariants and coupling predicates. Hawblitzel et al. [@mutualsummaries] use a similar technique for handling recursive function calls. Felsing et al. [@automatingregver] demonstrate that coupling predicates for proving the equivalence of two programs can often be inferred automatically. While the structure of imperative and [*MapReduce*]{} algorithms tends to be quite different, splitting the translation into intermediate steps yields programs which are often structurally similar. We have found that in this case, techniques such as coupling predicates arise naturally and are useful for selected parts of an equivalence proof. De Angelis et al. [@DeAngelis16] present a further generalised approach.
Radoi et al. [@translatingimperative] describe an automatic translation of imperative algorithms to [*MapReduce*]{} algorithms based on rewrite rules. While the rewrite rules are very similar to the ones used in our approach, we complement rewrite rules by coupling predicates. Furthermore we are able to prove equivalence for algorithms for which the automatic translation from Radoi et al. is not capable of producing efficient [*MapReduce*]{} algorithms. The objective of verification imposes different constraints than the automated translation – in particular both programs are provided by the user, so there is less flexibility needed in the formulation of rewrite rules.
Chen et al. [@sparkspecification] and Radoi et al. [@translatingimperative] describe languages and sequential semantics for [*MapReduce*]{} algorithms. Chen et al.describe an executable sequential specification in the Haskell programming language focusing on capturing non-determinism correctly. Radoi et al. use a language based on a lambda calculus as the common representation for the previously described translation from imperative to [*MapReduce*]{} algorithms. While this language closely resembles the language used in our approach, it lacks support for representing some imperative constructs such as arbitrary *while*-loops.
Grossman et al. [@equivalencespark] verify the equivalence of a restricted subset of Spark programs by reducing the problem of checking program equivalence to the validity of formulas in a decidable fragment of first-order logic. While this approach is fully automatic, it limits programs to Presburger arithmetic and requires that they are synchronized in some way.
To the best of our knowledge, we are the first to propose a framework for proving equivalence of [*MapReduce*]{} and imperative programs.
Conclusion {#sec:conclusion}
==========
In this paper we demonstrated how an imperative implementation of a relevant, non-trivial algorithm can be iteratively transformed into an equivalent efficient [*MapReduce*]{} implementation. The presentation bases on the formal framework described in [@arxiv]. Equivalence within this framework is guaranteed since the individual applied transformations are either behaviour-preserving rewrite rules or equivalence proofs using coupling predicates. The example that has been used as a case study in this paper is the [*PageRank*]{} algorithm, a prototypical application case of the [*MapReduce*]{} programming model. The transformation comprises eight transformation steps.
Future work for proving equivalence between imperative and [*MapReduce*]{} implementations includes further automation of the transformation process.
[^1]: In the current implementation, rationals are implemented using integers with the operators being uninterpreted function symbols.
[^2]: is stronger than in general, but is equivalent in case both loops are guaranteed to have the same number of iterations.
[^3]: The actually implemented version of the rule allows `f` to access not only the values `vs`, but also the index `i` it operates on. See [@arxiv] for details.
|
---
abstract: |
We define and study a generalization of Sobol sensitivity indices for the case of a vector output. [*To cite this article: F. Gamboa, A. Janon, T. Klein, A. Lagnoux, C. R. Acad. Sci. Paris, Ser. xx xxx (2013).*]{}
0.5
[**Résumé**]{} 0.5[**Indices de sensibilité pour sorties multivariées.** ]{} Nous définissons et étudions une généralisation des indices de Sobol pour des sorties vectorielles. [*Pour citer cet article : F. Gamboa, A. Janon, T. Klein, A. Lagnoux, C. R. Acad. Sci. Paris, Ser. xx xxx (2013).* ]{}
address:
- 'Institut Mathématique de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex.'
- 'Laboratoire SAF, ISFA Université Lyon 1, 50 avenue Tony Garnier, 69007 Lyon'
author:
- Fabrice Gamboa
- Alexandre Janon
- Thierry Klein
- Agnès Lagnoux
bibliography:
- 'biblio.bib'
title: Sensitivity indices for multivariate outputs
---
,
[Received \*\*\*\*\*; accepted after revision +++++\
Presented by £££££]{}
Introduction
============
Many mathematical models encountered in applied sciences involve a large number of poorly-known parameters as inputs. It is important for the practitioner to assess the impact of this uncertainty on the model output. An aspect of this assessment is sensitivity analysis, which aims to identify the most sensitive parameters. In other words, parameters that have the largest influence on the output. In global stochastic sensitivity analysis, the input variables are assumed to be independent random variables. Their probability distributions account for the practitioner’s belief about the input uncertainty. This turns the model output into a random variable.
When the output is scalar, using the so-called Hoeffding decomposition [@van2000asymptotic], its total variance can be split down into different partial variances. Each of these partial variances measures the uncertainty on the output induced by the corresponding input variable. By considering the ratio of each partial variance to the total variance, we obtain a measure of importance for each input variable called the *Sobol index* or *sensitivity index* of the variable [@sobol1993]; the most sensitive parameters can then be identified and ranked as the parameters with the largest Sobol indices.
Generalization of the Sobol index for multivariate (vector) outputs has been considered in [@lamboni2011multivariate] in an empirical way. In this note, we consider and study a new generalization of Sobol indices for vector outputs. These indices stem from an Hoeffding decomposition and satisfy natural invariance properties. In this note, we define the new sensitivity indices, examine some of their properties and show why they are natural. We also study a Monte-Carlo estimator of these indices, as in practice the exact values are not explicitly computable.
Definitions and Properties
==========================
Preliminaries
-------------
We denote by $X_1, \ldots, X_p$ the input random variables defined on some probability space $(\Omega,\mathbb P)$, and by $Y$ the output: $ Y = f(X_1, \ldots, X_p) $, where $f: {\mathbb{R}}^p \rightarrow {\mathbb{R}}^k$ ($p,k$ are integers). We suppose that $X_1,\ldots,X_p$ are independent, that $Y \in L^2(\Omega, {\mathbb{R}}^k)$, and that the covariance matrix of $Y$ is positive definite.
For any non-empty $r$-subset $\bf u$ of $\{1, \ldots, p\}$, we set $X_{\bf u}=(X_i, i \in \bf u)$ and $X_{\sim \bf u}=(X_i, i \in \{1,\ldots,p\}\setminus \bf u)$.
Definition of the generalized Sobol indices
-------------------------------------------
We recall the Hoeffding decomposition of $f$ [@van2000asymptotic]: $$\label{e:hoeff}
f(X_1, \ldots, X_p) = c + f_{\bf u}({X_{\bf u}}) + f_{\sim\bf u}({X_{\sim\bf u}}) + f_{\bf u,\sim\bf u}({X_{\bf u}},{X_{\sim\bf u}}),$$ where $c\in{\mathbb{R}}^k$, $f_{\bf u}: {\mathbb{R}}^r\rightarrow{\mathbb{R}}^k$, $f_{\sim\bf u}: {\mathbb{R}}^{p-r}\rightarrow{\mathbb{R}}^k$ and $f_{\bf u,\sim\bf u}: {\mathbb{R}}^p\rightarrow{\mathbb{R}}^k$ are given by: $$c = {\mathbb{E}}(Y), \; f_{\bf u}={\mathbb{E}}(Y|{X_{\bf u}})-c, \; f_{\sim\bf u}={\mathbb{E}}(Y|{X_{\sim\bf u}})-c, \; f_{u,\sim\bf u}=Y-f_{\bf u}-f_{\sim\bf u}-c.$$ Taking the covariance matrices of both sides of gives (thanks to $L^2$-orthogonality): $$\label{e:hoeffvar}
\Sigma = C_{\bf u} + C_{\sim\bf u} + C_{\bf u,\sim\bf u},$$ where $\Sigma$, $C_{\bf u}$, $C_{\sim\bf u}$ and $C_{\bf u,\sim\bf u}$ are, respectively, the covariance matrices of $Y$, $f_{\bf u}({X_{\bf u}})$, $f_{\sim\bf u}({X_{\sim\bf u}})$ and $f_{\bf u,\sim\bf u}({X_{\bf u}},{X_{\sim\bf u}})$.
For scalar outputs (ie., when $k=1$), the covariance matrices are scalar (variances), and is interpreted as the decomposition of the total variance of $Y$ as a sum of the variance caused by the variation of the input factors $X_i$ for $i\in\bf u$, the variance caused by the input factors not in $\bf u$, and the variance caused by the interactions of the factors in $\bf u$ and those not in $\bf u$. The (univariate) closed Sobol index $ S^{\bf u, \textrm{Scal}}(f) = \frac{C_{\bf u}}{\Sigma} $ is then interpreted as the sensibility of $Y$ to the inputs in $\bf u$. Due to noncommutativity of the matrix product, a direct generalization of this index is not straightforward.
We now go back to the general case. For any $k \times k$ matrix $M$, can be projected on a scalar by multiplying by $M$ and taking the trace: $${\mathrm{Tr}}(M\Sigma) = {\mathrm{Tr}}(M {C_{\bf u}}) + {\mathrm{Tr}}(M {C_{\sim\bf u}}) + {\mathrm{Tr}}(M {C_{\bf u,\sim\bf u}}).$$ This equation is the natural scalarization of the matricial identity (as, for a symmetric matrix $V$, we have $\sum_{i,j} M_{i,j} V_{i,j} = {\mathrm{Tr}}(MV)$). This suggests to define, when ${\mathrm{Tr}}(M \Sigma)\neq 0$: $$S^{\bf u}(M; f) = \frac{{\mathrm{Tr}}(M {C_{\bf u}})}{{\mathrm{Tr}}(M \Sigma)}$$ as the $M$-sensitivity measure (sensitivity index, or generalized Sobol index) of $Y$ to the inputs in $\bf u$. We can also analogously define: $ S^{\sim\bf u}(M; f) = \frac{{\mathrm{Tr}}(M {C_{\sim\bf u}})}{{\mathrm{Tr}}(M \Sigma)},\;\; S^{\bf u,\sim\bf u}(M; f) = \frac{{\mathrm{Tr}}(M {C_{\bf u,\sim\bf u}})}{{\mathrm{Tr}}(M \Sigma)}$, which measures the sensitivity to, respectively, the inputs not in $\bf u$, and to the interaction between inputs of $\bf u$ and inputs of $\{1,\ldots,p\}\setminus\bf u$. The following lemma is obvious:
1. The generalized sensitivity measures sum up to 1: $$\label{e:somsobol}
S^{\bf u}(M; f) + S^{\sim \bf u}(M; f) + S^{\bf u, \sim \bf u}(M; f) = 1.$$
2. Left-composing $f$ by a linear operator $O$ of ${\mathbb{R}}^k$ changes the sensitivity measure according to: $$\label{e:changesobol}
S^{\bf u}(M; Of) = \frac{{\mathrm{Tr}}(M O {C_{\bf u}}O^t)}{{\mathrm{Tr}}(M O \Sigma O^t)} =
\frac{{\mathrm{Tr}}(O^t M O {C_{\bf u}})}{{\mathrm{Tr}}(O^t M O \Sigma)} =
S^{\bf u}(O^t M O; f).$$
3. For $k=1$, and for any $M \neq 0$, we have $S^{\bf u}(M; f) = S^{\bf u,\textrm{Scal}}(f)$.
The case $M=\textrm{Id}_k$
--------------------------
We now consider the special case $M = \textrm{Id}_k$ (the identity matrix of dimension $k$). We set $S_{\bf u}(f)=S^{\bf u}(\textrm{Id}_k;f)$. The index $S_{\bf u}(f)$ has the following properties:
\[prop:properties\] Suppose that $Y\in L^2(\Omega,{\mathbb{R}}^k)$ and that $\Sigma$ is positive-definite:
1. $0 \leq S_{\bf u}(f) \leq 1$;
2. $S_{\bf u}(f)$ is invariant by left-composition of $f$ by any isometry of ${\mathbb{R}}^k$, i.e. $$\forall O\, k\times k \text{ matrix s.t. } O^tO=\textrm{Id}_k, \;\; S_{\bf u}(Of)=S_{\bf u}(f);$$
3. $S_{\bf u}(f)$ is invariant by left-composition of $f$ by any nonzero homothety of ${\mathbb{R}}^k$.
*Proof:* Point (i): positivity is clear, as ${C_{\bf u}}$ and $\Sigma$ are positive; $S_{\bf u}(f)\leq1$ follows from positivity and . For (ii), we use . Point (iii) is immediate.
The properties in the Proposition above are natural requirements for a sensitivity measure (the isometry invariance property ensures that the resulting indices are “intrinsic” and does not depend on the parametrization of the output). Note that $S_{\bf u}(f)$ is the sum of the partial variances divided by the sum of the total variances of each output coordinate, and the covariances between coordinates are not involved. In the next section, we will show that these requirements can be fulfilled by $S^{\bf u}(M;\cdot)$ iff $M=\lambda \textrm{Id}_k$ for $\lambda\in{\mathbb{R}}^*$. Hence the only “canonical” sensitivity measure is $S_{\bf u}$.
$M=\textrm{Id}_k$ is the only good choice
=========================================
The following Proposition can be seen as a kind of converse of Proposition \[prop:properties\].
Let $M$ be a square $k\times k$ matrix such that ${\mathrm{Tr}}(M V) \neq 0$ for any symmetric positive-definite matrix $V$. Now if for all $f: {\mathbb{R}}^p \rightarrow {\mathbb{R}}^k$, and all subsets $u\subset\{1,\ldots,p\}$, we have that $S^{\bf u}(M;f)$ is invariant by left-composition of $f$ by any isometry of ${\mathbb{R}}^k$, then $S^{\bf u}(M;\cdot) = S_{\bf u}(\cdot)$.
*Proof:* Let $M$ as in the Proposition. We can write $M = M_{Sym} + M_{Antisym}$ where $M_{Sym}^t=M_{Sym}$ and $M_{Antisym}^t=-M_{Antisym}$. Since, for any symmetric matrix $V$, we have ${\mathrm{Tr}}(M_{Antisym} V)=0$, we have $S^{\bf u}(M;f)=S^{\bf u}(M_{Sym};f)$ and we can assume, without loss of generality, that $M$ is symmetric.
We diagonalize $M$ in an orthonormed basis: $M=P D P^t$, where $P^t P = \textrm{Id}_k$ and $D$ diagonal. We have: $$S^{\bf u}(M;f) = \frac{{\mathrm{Tr}}(P D P^t {C_{\bf u}})}{{\mathrm{Tr}}(P D P^t \Sigma)} = \frac{{\mathrm{Tr}}(D P^t {C_{\bf u}}P)}{{\mathrm{Tr}}(D P^t \Sigma P)} = S^{\bf u}(D; P^tf).$$ This shows that $M$ can in fact be assumed diagonal.
Now we want to show that $M = \lambda \textrm{Id}_k$ for some $\lambda\in{\mathbb{R}}^*$. Suppose, by contradiction, that $M$ has two different diagonal coefficients $\lambda_1 \neq \lambda_2$. It is clearly sufficient to consider the case $k=2$. Choose $f=\textrm{Id}_2$ (hence, $p=2$), and ${\bf u}=\{1\}$. We have $\Sigma=\textrm{Id}_2$, and ${C_{\bf u}}=\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)$, hence on the one hand $S^{\bf u}(M;f)=\frac{\lambda_1}{\lambda_1+\lambda_2}$. On the other hand, let $O$ be the isometry which exchanges the two vectors of the canonical basis of ${\mathbb{R}}^2$. We have $S^{\bf u}(M;Of)=\frac{\lambda_2}{\lambda_1+\lambda_2}$, and invariance by isometry is contradicted if $\lambda_1\neq\lambda_2$. We also have $\lambda \neq 0$ since ${\mathrm{Tr}}(M)\neq 0$. Finally, it is easy to check that, for any $\lambda\in{\mathbb{R}}^*$, $S^{\bf u}(\lambda \textrm{Id}_k;\cdot)=S^{\bf u}(\textrm{Id}_k;\cdot)=S_{\bf u}$.
Estimation of $S^{\bf u}(f)$
============================
In general, the covariance matrices ${C_{\bf u}}$ and $\Sigma$ are not analytically available. In the scalar case ($k=1$), it is customary to estimate $ S^{\bf u, \textrm{Scal}}(f) $ by using a Monte-Carlo pick-freeze method [@sobol1993; @janon2012asymptotic], which uses a finite sample of evaluations of $f$. In this Section, we propose a pick-freeze estimator for the vector case which generalizes the $T_N$ estimator studied in [@janon2012asymptotic]. We set: $Y^{\bf u}=f(X_{\bf u}, X_{\sim\bf u}')$ where $X_{\sim\bf u}'$ is an independent copy of $X_{\sim\bf u}$. Let $N$ be an integer. We take $N$ independent copies $Y_1, \ldots, Y_N$ (resp. $Y_1^{\bf u},\ldots,Y_N^{\bf u}$) of $Y$ (resp. $Y^{\bf u}$). For $l=1,\ldots,k$, and $i=1,\ldots,N$, we also denote by $Y_{i, l}$ (resp. $Y_{i, l}^{\bf u}$) the $l^\text{th}$ component of $Y_i$ (resp. $Y_i^{\bf u}$). We define the following estimator of $S_{\bf u}(f)$: $$S_{{\bf u}, N}(f) = \frac{\sum_{l=1}^k \left( \sum_{i=1}^N Y_{i,l} Y_{i,l}^{\bf u} - \frac1N \left( \sum_{i=1}^N \frac{Y_{i,l}+Y_{i,l}^{\bf u}}{2} \right)^2 \right) }{ \sum_{l=1}^k \left( \sum_{i=1}^N \frac{ Y_{i,l}^2+(Y_{i,l}^{\bf u})^2 }{2} - \frac1N \left( \sum_{i=1}^N \frac{ Y_{i,l}+Y_{i,l}^{\bf u} }{2} \right)^2 \right) }.$$ Thanks to the simple form of this estimator, the following Proposition can be proved in a way similar to the one used to prove Proposition 2.2 and Proposition 2.5 of [@janon2012asymptotic] (ie., by an application of the so-called Delta method).
Suppose $Y \in L^4(\Omega,{\mathbb{R}}^k)$, and that $\Sigma$ is positive-definite. Then:
1. $\left(S_{{\bf u},N}(f)\right)_{N}$ is asymptotically normal: there exists $\sigma=\sigma(f)$ so that $ \sqrt N ( S_{{\bf u},N}(f) - S_{\bf u}(f) ) $ converges (for $N \rightarrow +\infty$) in distribution to a centered Gaussian distribution with variance $\sigma^2$.
2. $\left(S_{{\bf u},N}(f)\right)_{N}$ is asymptotically efficient for estimating $S_{\bf u}(f)$ among regular estimator sequences that are function of exchangeable pairs $(Y, Y^{\bf u})$.
[**Acknowledgements. ** This work has been partially supported by the French National Research Agency (ANR) through COSINUS program (project COSTA-BRAVA n°ANR-09-COSI-015). ]{}
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abstract: 'We consider the effect of electron-phonon interactions on edge states in quantum Hall systems with a single edge branch. The presence of electron-phonon interactions modifies the single-particle propagator for general quantum Hall edges, and, in particular, destroys the Fermi liquid even at integer filling. The effect of the electron-phonon interactions may be detected experimentally in the AC conductance or in the tunneling conductance between integer quantum Hall edges.'
address: |
$^*$Institute of Theoretical Physics and $^{\dagger}$Department of Applied Physics\
Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
author:
- 'O. Heinonen$^{*,\dagger,}$[@UCF] and Sebastian Eggert$^*$'
title: 'Electron-phonon interactions on a single-branch quantum Hall edge'
---
A two-dimensional electron gas subjected to a strong perpendicular magnetic field may exhibit the quantum Hall (QH) effect[@QHE_book]. The effect occurs because the electron gas is incompressible at certain densities, which is due to an energy gap to bulk excitations. In the integer QH effect this gap is the kinetic energy gap (the cyclotron energy $\hbar\omega_c$) in a magnetic field, and in the fractional QH effect the gap arises because of the electron-electron interactions. As a result of the bulk energy gap, gapless excitations in the system can only exist at the edges. Such edge excitations are density modulations localized at the edges of the system, and all the low-energy properties of a QH system are determined by the edge excitations.
As was first demonstrated by Wen[@Wen1], the density operators for edge excitations obey a Kac-Moody algebra, similar in structure to that obeyed by Luttinger liquids[@Luttinger1]. Because time-reversal invariance is broken by the magnetic field, the excitations on one edge can only propagate in one direction[@composite] corresponding to ‘chiral Luttinger liquids’[@Wen1]. Wen calculated the single-particle propagator for the electron on the edge of a quantum Hall system at filling factor $\nu=1/(2m+1)$ and showed that it is given by[@Wen1] $G(x,t)\propto(x-v_Ft)^{1/\nu}$. Here, $x$ is a coordinate along the edge, and $v_F$ is the edge velocity determined by the details of the potential confining the electron gas at the edge. It is worth noting the contrast between chiral Luttinger liquids and regular Luttinger liquids in one-dimensional (1D) interacting electron systems without magnetic fields. In the regular Luttinger liquid, the Fermi surface is destroyed by repeated backscattering, and the exponent of the single-particle propagator depends on the details of the electron-electron interactions. On the other hand, a chiral Luttinger liquid at a QH edge is caused by the strong correlations in the bulk, and the exponent of the single-particle propagator is fixed by the topological order in the bulk of the system.
The appearance of the anomalous exponent in the single-particle propagator has important experimental implications. It was shown by Wen[@Wen2] and by Kane and Fisher[@KF1] that the tunneling conductance between two edges of an FQHE system at bulk filling factor $\nu=1/(2m+1)$ depends on temperature as $T^{2(1/\nu-1)}$. The resonant tunneling conductance was calculated by Moon [*et al.*]{}[@Moon1], and Fendley [*et al.*]{}[@Fendley], and measured by Milliken [*et al.*]{}[@Milliken]. The experimentally measured tunneling conductance does indeed exhibit a $T^{2(1/\nu-1)}$-dependence, except for at the very lowest temperatures, where Coulomb interactions between the edges may modify the conductance[@Moon2].
In this letter, we consider the effect of electron-phonon interactions on the QH edge states of spin-polarized QH systems with bulk filling factor $\nu=1/m$, with $m$ an odd integer. Such systems have a single branch of edge excitations on each edge. Electron-phonon interactions in regular Luttinger liquids have been considered previously[@Voit; @Martin]. Martin and Loss[@Martin] showed that coupling the electron system to acoustic phonons destroys the Fermi surface, even in the absence of electron-electron interactions. This effect is only appreciable if the Fermi velocity $v_F$ is of the same order as the sound velocity $v_s$, which may be achieved in strongly correlated electron systems where the role of the Fermi velocity $v_F$ is played by the charge velocity of the Luttinger liquid. The effect may also be appreciable in QH systems, where $v_F$ is determined by the stiffness of the confining potential and the electron density, both of which may be tuned electrostatically by gates. We will show here that the electron-phonon interactions will modify the single-particle propagator even in a [*chiral*]{} Luttinger Liquid. In particular, the electron propagator of QH edge state will be shown to have the form $$G(x,t)\propto
{1\over (x- v_\alpha t)^{T_{11}^2/\nu}}
{1\over (x-v_\beta t)^{T_{12}^2/\nu}}
{1\over (x+v_\gamma t)^{T_{13}^2/\nu}},
\label{prop2}$$ where $v_\alpha$ is a renormalized edge velocity, and $v_{\beta,\gamma}$ are renormalized sound velocities. As a consequence, even the integer quantum Hall edge states will not be Fermi liquids in the presence of electron-phonon interactions.
The modification of the single-particle propagator by the electron-phonon interactions may be detected experimentally, and we will discuss two possibilities. First, the AC conductance will have resonances at (longitudinal) wave-vectors $q$ and frequencies $\omega$ related by $q=\omega/v_\alpha, \ q=\omega/v_\beta,$ and $q=-\omega/v_\gamma$ which may in principle be resolved and detected in an experiment. On the other hand, the DC Hall conductance is [*not*]{} modified by the electron-phonon interactions. Second, the anomalous exponents in the single-particle propagator will modify the tunneling conductance and can in principle be measured, for example in tunneling between two $\nu=1$ edges in a bilayer system[@bilayers] with an overall filling of $\nu_{\rm tot}=2$. In the absence of electron-phonon interactions the edges of such a system are (chiral) Fermi liquids which propagate in the same direction. Coulomb interactions alone cannot change the temperature dependence of the tunneling conductance from $T^0$. Therefore, [*any*]{} temperature behavior of the tunneling conductance at sufficiently low temperatures must be due to the electron-phonon interactions. The electron-phonon interaction also modifies the temperature dependence of the tunneling conductance between counterpropagating edge states at very low temperatures which has been measured by Milliken [*et al.*]{}[@Milliken]
The edge excitations in a quantum Hall system with filling fraction $\nu=1/(2m+1)$ can be described by density modulations of an effectively one-dimensional system[@Wen1] $$H_e \equiv \frac{2 \pi}{L}\frac{v_F}{\nu} \sum_{k>0}
\ J_k J_{-k}, \label{H_e}$$ where the densities $J_k$ obey commutation relations depending on the filling fraction $\nu$: $[J_k, J_{k'}] = - \frac{L\nu k}{2\pi} \delta_{k,-k'}$. Here, $k$ is the wave-vector along the edge. This theory represents a single $U(1)$ Kac-Moody algebra and it is well known how to “bosonize” it in terms of a chiral boson $\phi_R(x)$. The current density $j(x)$ is written as $$j(x) = \sqrt{\frac{\nu}{\pi}} \
\frac{\partial \phi_R}{\partial x}. \label{bos2}$$ Using $J_k = \int dx \ e^{ikx} j(x)$, we can immediately express the Hamiltonian in terms of the annihilation and creation operators of the boson mode expansion: $$a^\dagger_k = \sqrt{\frac{2 \pi}{L\nu k}} J_k,
\ \ \ a_k = \sqrt{\frac{2 \pi}{L\nu k}} J_{-k}$$ (the zero modes are omitted). The full (spin-polarized) electron field can also be written in terms of the chiral boson and has been identified as[@Wen1] $$\psi(x) \propto \exp\left(i \sqrt{1/4 \nu \pi}\phi_R\right)\label{bos1}$$ (this is not to be confused with the quasi-particle field $\chi$ which can also be defined in terms of the chiral boson $\chi \propto e^{i \sqrt{\nu/4 \pi} \phi_R}$, but carries fractional charge).
We now consider the interaction of such a system with phonons $$H_{ph} = v_s \sum_{\vec{k}}
\ |\vec{k}| b_{\vec{k}}^\dagger b_{\vec{k}}^{}. \label{H_ph}$$ One normal mode of the phonons is assumed to be along the quantum Hall edge, so that the crystal displacement $d(x)$ along this edge can be expressed as $$d(x) = \sum_k i (2 L \rho v_s k)^{-1/2} e^{i k x} \ (b_k + b_{-k}^\dagger).
\label{d_x}$$ where $\rho$ is the linear mass density of the crystal. The electron-phonon interaction then becomes $$H_{e-p} = D \int dx \rho(x) \partial_x d(x) =
v_c \int dk \ k (a_k^{} b_k^\dagger + a_k^\dagger b^{}_k), \label{H_e-p}$$ where $D$ is the deformation potential constant and the coupling $v_c = D\sqrt{{\nu/\pi \rho v_s}}$ is independent of $k$. For specific values of these parameters, we consider a QH edge in a GaAs heterojunction. We assume that the edge is along one of the cubic axes of GaAs so that the piezoelectric coupling vanishes and can be ignored. For electrostatic confinement by an electrode with potential $V_g$, $v_F$ is of the order of $\omega_c\ell_B^2/\ell$, where $\ell_B=\left[\hbar c/(eB)\right]^2$ is the magnetic length and $\ell=V_g\epsilon/(4\pi^2 n_0 e)$ is the length scale of the electrostatic confining potential[@Chklovskii] ($\epsilon$ is the static dielectric constant and $n_0$ is the two-dimensional electron density). For a magnetic field strength of about 5 T and a density of $n_0=10^{15}$ m$^{-2}$, this gives a Fermi velocity approximately equal to the average sound velocity $v_s\approx5\times10^3\rm m/s$ in GaAs. Thus, it should be possible to optimize the effects of electron-phonon interactions in GaAs heterojunctions under ordinary conditions. The deformation potential constant $D$ is approximately 7.4 eV [@Levinson]. Assuming an effective cross-sectional area of $10^{-14}$ m$^{-2}$ of the GaAs phonon system in the direction perpendicular to the electron propagation, we then arrive at a coupling velocity $v_c/v_s\sim 0.1$.
At this point it is straightforward to diagonalize the complete Hamiltonian $$H = \sum_{k>0} k \left( v_F a^\dagger_k a_k^{} +
v_s[b^\dagger_k b^{}_k + b^\dagger_{-k} b^{}_{-k}]
+ v_c [ a^\dagger_k (b_k+b^\dagger_{-k}) + h.c.]\right) \label{H}$$ by using a generalized Bogliubov transformation $T$ $$(a_k, b_k, b^\dagger_{-k}) = T\cdot \left(\begin{array}{c}
\alpha_k \\ \beta_k \\ \gamma^\dagger_{-k}\\ \end{array}\right)
\label{bogliubov}$$ where $T$ is given in terms of three variables $\phi, \theta, \eta$: $$T = \left(\begin{array}{ccc}
\cos\phi \cosh\theta & \sin\phi \cosh\theta\cosh\eta + \sinh\theta\sinh\eta
& -\sin\phi \cosh\theta \sinh\eta -\sinh\theta \cosh\eta \\
-\sin\phi & \cos\phi\cosh\eta & -\cos\phi \sinh\eta \\
-\cos\phi \sinh\theta & -\sin\phi \sinh\theta\cosh\eta - \cosh\theta\sinh\eta
& \sin\phi \sinh\theta \sinh\eta -\cosh\theta \cosh\eta \\
\end{array}\right).$$ The Hamiltonian is now written as $$H = \sum_{k>0} k \left(
\alpha_k , \beta_k , \gamma^\dagger_{-k}\right)\cdot A
\cdot \left(\begin{array}{c}
\alpha_k^\dagger \\ \beta_k^\dagger \\ \gamma^{}_{-k}\\ \end{array}\right)
\label{rotated_H}$$ where the coupling matrix $A$ is given by $$A\equiv T^\dagger\cdot \left(\begin{array}{ccc} v_F & v_c & v_c \\
v_c & v_s & 0 \\ v_c & 0 & v_s \\ \end{array} \right)\cdot T$$ For the Hamiltonian to become diagonal, the off-diagonal elements of $A$ are required to vanish, which determines the three angles $\phi, \theta, \eta$ and in turn also the diagonal elements of $A$ (i.e. the renormalized Fermi and sound velocities). Moreover, the boson representing the electron density now becomes according to Eq. (\[bogliubov\]) $$a_k = T_{11} \alpha_k + T_{12} \beta_k + T_{13} \gamma_{-k}^\dagger
\label{rescale}$$ Therefore, the electron field in Eq. (\[bos1\]) must now be expressed in terms of three independent boson fields, namely a right moving “charge” boson $\alpha_k$ and one right and one left moving “sound” boson $\beta_k, \ \gamma_k$. The electron propagator then becomes a product of three factors according to Eqs. (\[bos1\]) and (\[rescale\]) $$G(x,t) \equiv <\psi^\dagger(x,t)\psi(0,0)>
\propto {1\over (x- v_\alpha t)^{T_{11}^2/\nu}}
{1\over (x-v_\beta t)^{T_{12}^2/\nu}}
{1\over (x+v_\gamma t)^{T_{13}^2/\nu}},
\label{propagator}$$ where $v_\alpha=A_{11}$ is the renormalized Fermi velocity and $v_\beta =A_{22}, \ v_\gamma = A_{33}$ are the renormalized sound velocities. In Fig. \[v\] we have plotted the renormalized velocities $v_i \ vs. \ v_F \ (i=\alpha,\beta,\gamma)$ for a coupling of $v_c/v_s=0.1$. The total ‘momentum’ $v_\alpha
+v_\beta-v_\gamma = v_F$ of the electron is conserved by the transformation $T$. The right propagating velocities $v_\alpha$ and $v_\beta$ show a discontinuity at the resonance $v_F=v_s$ of equal magnitude which appears to be quadratic in the coupling $v_c/v_s$. The left propagating velocity $v_\gamma$ is only slightly modified of the order of 1% from its original value $v_s$.
The fact that the electron propagator breaks up into three pieces, corresponding to the normal modes of the Hamiltonian Eq. (\[rotated\_H\]), will have experimental consequences for transport properties. We first calculate the linear response to a scalar potential $\phi(x,y)=-Ey\cos(qx-\omega t),$ where we take $q,\omega>0$. This potential gives an electric field ${\bf E}(x,y)=
E\left[-qy\sin(qx-\omega t)\hat{\bf x}+\cos(qx-\omega t)\hat{\bf y}\right],$ where $y$ can be taken to be constant. The perturbed charge density in response to the potential is then obtained as $$\delta\rho(x,t) =
\frac{e^2Ey}{h}\nu q \cos(qx-\omega t)
\left[{T_{11}^2\over(v_\alpha q-\omega)}
+{T_{12}^2\over (v_\beta q-\omega)}+{T_{13}^2\over(v_\gamma q+\omega)}\right].$$ By using the continuity equation $\partial\rho/(\partial t)+
\partial j(x)/(\partial x)=0$, we can then obtain the current response function to the applied potential as $$\widetilde\sigma(q,\omega)=\frac{e^2\nu}{h}\omega
\left[{T_{11}^2\over(v_\alpha q-\omega)}+{T_{12}^2\over(v_\beta q-\omega)}
+{T_{13}^2\over(v_\gamma q+\omega)}\right].
\label{sigma}$$ Experiments dictate that the DC Hall conductance $\sigma_H=\lim_{\omega\to0}\lim_{q\to0}\widetilde\sigma(q,\omega)$ must not be altered from its quantized value $e^2\nu/h$, which is indeed the case according to Eq. (\[sigma\]) since the matrix elements obey the sum rule $T_{11}^2 + T_{12}^2 - T_{13}^2 = 1$. On the other hand, the AC conductance will exhibit resonance structures when $\omega=v_\alpha q, \ \omega=v_\beta q,\ $ and $\omega=-v_\gamma q$ in response to a potential $\phi(x,y)$. Provided at least two of the ‘spectral weights’ $T_{11}^2$, $T_{12}^2$, and $T_{13}^3$ are not too small, and the corresponding renormalized velocities are not too close, these resonances can then in principle be resolved and detected. In Fig. \[T\] we see that near $v_F/v_s \sim 1$, both $T_{11}^2$ and $T_{12}^2$ are close to 0.5, while $v_\alpha$ and $v_\beta$ are on opposite sides of $v_s$ (Fig. \[v\]). With an experimentally reasonable value of $q\sim10^5$ m$^{-1}$ and $v_s\sim v_F \sim10^3\rm m/s$, this gives a resonance at about $10^8$ Hz, well within experimentally accessible range. Figure \[T\] also shows that the ‘total spectral weight’ of the electron consists mostly of the forward propagating modes, which contribute almost equally at resonance $v_F = v_s$. This means that only very little charge is transported in the counterpropagating direction which was to be expected.
Since the single-particle propagator is changed by the electron-phonon interaction according to Eq. (\[propagator\]), the single-particle density-of-states and properties depending on it such as the tunneling conductance will also be affected by the coupling to the phonons. In particular, we consider the inter-layer tunneling conductance between two edges of a bilayer system with integer filling in each layer, and a total filling factor of $\nu_{\rm tot}=2$. This can in principle be measured by attaching probes to the different layers separately in a system with large enough separation between the layers that the bulk tunneling probability vanishes. A gate at the edge can then be used to adjust the tunneling probability between the edges. At low enough voltage across the tunneling junction, tunneling through the bulk will be suppressed, and only the edge tunneling current appreciable. Note that in this arrangement, the two edges propagate in the same direction.
The tunneling current is determined by the retarded response function[@Wen2] $$X_{\rm ret}(t)=-i\theta(t)\langle\left[
A(t),{A}^\dagger(0)\right]\rangle,
\label{Xret}$$ where $\Gamma A=\Gamma\psi_1(x=0){\psi_2}^\dagger(x=0) + h.c.$ is the tunneling operator, with $\Gamma$ the tunneling amplitude, and $\psi_i(x)$, $i=1,2$, the electron field operator on the two edges. From Eq. (\[propagator\]) and following Wen[@Wen2] it is a straightforward exercise to determine the temperature dependence of the tunneling differential conductance, with the result that $$\left.{dI_t\over dV_t}\right|_{V_t=0}\ \ \propto \ \
T^{2(T_{11}^2+T_{12}^2+T_{13}^2)-2} \ \ = \ \ T^{-4 T_{13}^2}.
\label{tunnel}$$ Due to the fact that $T_{13}^2\not=0$ in the presence of electron-phonon interactions, the differential tunneling conductance will now depend on temperature, in fundamental contrast to the tunneling between two chiral Fermi liquids propagating in the same direction, which is temperature [*independent*]{} at low temperatures, even in the presence of electron-electron interactions between the two edges. From Fig. \[T\] we see that the magnitude of the exponent $-4 T_{13}^2$ for the parameters chosen here is of the order of $10^{-2}$, which is small (the exponent appears to be quadratic in the coupling constant $v_c/v_s$). However, the main point is that any measured temperature dependence at all will be due to the electron-phonon interactions.
In conclusion we have shown that the electron-phonon interaction modifies the chiral Luttinger Liquid on the quantum Hall edge. The single particle propagator becomes a product of three separate modes, one of which is always counterpropagating. From this we have predicted direct experimental consequences for the AC conductivity of the quantum Hall bar and the temperature dependence of the tunneling conductance.
The authors would like to thank Henrik Johannesson and Micheal Johnson for helpful discussions. O.H. expresses his gratitude to Stellan Östlund and Mats Jonson at Chalmers University of Technology and Göteborg University for their hospitality and support during a great sabbatical stay, and to the National Science Foundation for support through grant DMR93-01433. This research was supported in part by the Swedish Natural Science Research Council.
Permanent address: Department of Physics, University of Central Florida, Orlando, FL 32816-2385 See, for example [*The Quantum Hall effect*]{}, R.E. Prange and S.M. Girvin (eds.) (Springer Verlag, New York, 1986). X.-G. Wen, Phys. Rev. B [**41**]{}, 12 838 (1990). J.M. Luttinger, J. Math. Phys. [**15**]{}, 609 (1963); D.C. Mattis and E.H. Lieb, [*ibid.*]{} [**6**]{}, 304) (1965); F.D.M. Haldane, J. Phys. C [**14**]{}, 2585 (1981). For a recent review, see J. Voit SISSA-preprint 9510014 (to be published, Rep. Prog. Phys.). We are here not considering systems with composite edges, which may have counterpropagating modes. X.G. Wen, Phys. Rev. B [**44**]{}, 5708 (1991). C.L. Kane and M.P.A. Fisher, Phys. Lett. [**68**]{}, 1220 (1992); Phys. Rev. B [**46**]{}, 15 233 (1992). K. Moon, H. Yi, C.L. Kane, S.M. Girvin, and M.P.A. Fisher, Phys. Rev. Lett. [**71**]{}, 4381 (1993). P. Fendley, A.W.W. Ludwig, and H. Saleur, Phys. Rev. Lett. [**74**]{}, 3005 (1995). F.P. Milliken, C.P. Umbach, and R.A. Webb, Solid State Comm. (in press); B.G. Levi, Physics Today [**47**]{}, 21 (1994). K. Moon and S.M. Girvin, SISSA-preprint 9511013. J. Voit and H.J. Schulz, Phys. Rev. B [**34**]{}, 7429 (1986); [**36**]{}, 968 (1987); [**37**]{} 10 068 (1988). T. Martin and D. Loss, Int. J. Mod. Phys. [**9**]{}, 495 (1995). For an excellent recent review on bilayer systems, see S.M. Girvin and A.H. MacDonald in [*Novel Quantum Liquids in Low-Dimensional Semiconductor Structures*]{}, S. Das Sarma and A. Pinczuk (eds.) (Wiley, New York, 1995). D.B. Chklovskii, B.I. Shklovskii, and L.I. Glazman, Phys. Rev. B. [**46**]{}, 4026 (1992). , by V.F. Gantmakher and Y.B. Levinson (North-Holland, Amsterdam 1987).
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abstract: 'Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. Entanglement is central to many quantum communication protocols, including dense coding, teleportation and quantum protocols for cryptography. For quantum algorithms, multipartite (many-qubit) entanglement is necessary to achieve an exponential speedup over classical computation. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number $n_q$ of qubits, thus becoming rapidly unfeasible when increasing $n_q$. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in $n_q$. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.'
author:
- 'G. Benenti'
title: 'Entanglement, randomness and chaos'
---
\[1999/12/01 v1.4c Il Nuovo Cimento\]
Introduction
============
Entanglement [@bruss; @pleniovirmani; @horodecki], arguably the most spectacular and counterintuitive manifestation of quantum mechanics, is observed in composite quantum systems. It signifies the existence of non-local correlations between measurements performed on particles that have interacted in the past, but now are located arbitrarily far away. We say that a two-particle state $|\psi\rangle$ is entangled, or non-separable, if it cannot be written as a simple tensor product $|k_1\rangle|k_2\rangle\equiv|k_1\rangle\otimes |k_2\rangle$ of two states which describe the first and the second subsystems, respectively, but only as a superposition of such states: $|\psi\rangle=\sum_{k_1,k_2}c_{k_1k_2}|k_1\rangle|k_2\rangle$. When two systems are entangled, it is not possible to assign them individual state vectors.
The intriguing non-classical properties of entangled states were clearly illustrated by Einstein, Podolsky and Rosen (EPR) in 1935 [@EPR]. These authors showed that quantum theory leads to a contradiction, provided that we accept (i) the reality principle: If we can predict with certainty the value of a physical quantity, then this value has physical reality, independently of our observation; [^1] (ii) the locality principle: If two systems are causally disconnected, the result of any measurement performed on one system cannot influence the result of a measurement performed on the second system. [^2] The EPR conclusion was that quantum mechanics is an incomplete theory. The suggestion was that measurement is in reality a deterministic process, which merely appears probabilistic since some degrees of freedom (hidden variables) are not precisely known. Of course, according to the standard interpretation of quantum mechanics there is no contradiction, since the wave function is not seen as a physical object, but just as a mathematical tool, useful to predict probabilities for the outcome of experiments.
The debate on the physical reality of quantum systems became the subject of experimental investigation after the formulation, in 1964, of Bell’s inequalities [@bell]. These inequalities are obtained assuming the principles of realism and locality. Since it is possible to devise situations in which quantum mechanics predicts a violation of these inequalities, any experimental observation of such a violation excludes the possibility of a local and realistic description of natural phenomena. In short, Bell showed that the principles of realism and locality lead to experimentally testable inequality relations in disagreement with the predictions of quantum mechanics.
Many experiments have been performed in order to check Bell’s inequalities; the most famous involved EPR pairs of photons and was performed by Aspect and coworkers in 1982 [@aspect]. This experiment displayed an unambiguous violation of a Bell’s inequality by tens of standard deviations and an excellent agreement with quantum mechanics. More recently, other experiments have come closer to the requirements of the ideal EPR scheme and again impressive agreement with the predictions of quantum mechanics has always been found. Nonetheless, there is no general consensus as to whether or not these experiments may be considered conclusive, owing to the limited efficiency of detectors. If, for the sake of argument, we assume that the present results will not be contradicted by future experiments with high-efficiency detectors, we must conclude that Nature does not experimentally support the EPR point of view. In summary, the World is not locally realistic.
I should stress that there is more to learn from Bell’s inequalities and Aspect’s experiments than merely a consistency test of quantum mechanics. These profound results show us that *entanglement is a fundamentally new resource*, beyond the realm of classical physics, and that it is possible to experimentally manipulate entangled states. A major goal of quantum information science [@qcbook; @nielsen] is to exploit this resource to perform computation and communication tasks beyond classical capabilities.
Entanglement is central to many quantum communication protocols, including quantum dense coding [@densecoding], which permits transmission of two bits of classical information through the manipulation of only one of two entangled qubits, and quantum teleportation [@teleportation], which allows the transfer of the state of one quantum system to another over an arbitrary distance. Moreover, entanglement is a tool for secure communication [@ekert]. Finally, in the field of quantum computation entanglement allows algorithms exponentially faster than any known classical computation [@shor]. For any quantum algorithm operating on pure states, the presence of multipartite (many-qubit) entanglement is necessary to achieve an exponential speedup over classical computation [@jozsa]. Therefore the ability to control high-dimensional entangled states is one of the basic requirements for constructing quantum computers.
Random numbers are important in classical computation, as probabilistic algorithms can be far more efficient than deterministic ones in solving many problems [@papadimitriou]. Randomness may also be useful in quantum computation. *Random pure states* of dimension $N$ are drawn from the uniform (Haar) measure on pure states [^3] The entanglement content of random pure quantum states is almost maximal [@random-states; @zyczkowski; @hayden] and such states find applications in various quantum protocols, like superdense coding of quantum states [@harrow; @hayden], remote state preparation [@bennett2005], and the construction of efficient data-hiding schemes [@hayden2]. Moreover, it has been argued that random evolutions may be used to characterize the main aspects of noise sources affecting a quantum processor [@emerson1]. Finally, random states may form the basis for a statistical theory of entanglement. While it is very difficult to characterize the entanglement properties of a many-qubit state, a simplified theory of entanglement might be possible for random states [@hayden].
The preparation of a random state or, equivalently, the implementation of a random unitary operator mapping a fiducial $n_q$-qubit initial state, say $|0\rangle\equiv |0\ldots 0\rangle
\equiv|0\rangle \otimes
\ldots \otimes |0\rangle$, onto a typical (random) state, requires a number of elementary one- and two-qubit gates exponential in the number $n_q$ of qubits, thus becoming rapidly unfeasible when increasing $n_q$. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in $n_q$ [@emerson1; @emerson2; @weinstein; @plenio1; @plenio2; @znidaric]. In a sense, pseudo-random states play in quantum information protocols a role analogous to pseudo-random numbers in classical information theory.
Random states can be efficiently approximated by means of random one- and two-qubit unitaries [@emerson1; @emerson2; @weinstein; @plenio1; @plenio2; @znidaric; @gennaro] or by deterministic dynamical systems (maps) in the regime of quantum chaos [@haake; @stoeckmann; @saraceno; @schack; @georgeot; @simone2002; @caves; @weinstein]. These maps are known to exhibit certain statistical properties of random matrices [@haake; @stoeckmann] and are efficient generators of multipartite entanglement among the qubits, close to that expected for random states [@caves; @weinstein]. Note that in this case deterministic instead of random one- and two-qubit gates are implemented, the required randomness being provided by deterministic chaotic dynamics. A related crucial question, which I shall discuss in this review, is whether the generated entanglement is robust when taking into account unavoidable noise sources affecting a quantum computer. That is, decoherence or imperfections in the quantum computer hardware [@qcbook; @varenna05], that in general turn pure states into mixtures, with a corresponding loss of quantum coherence and entanglement content.
This paper reviews previous work concerning several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer (Sec. \[sec:stabilitymultipartite\]); (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics (Sec. \[sec:detect\]); (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment (Sec. \[sec:chaoticenvironments\]). In order to make this paper accessible also to readers without a background in quantum information science and/or in quantum chaos, basic concepts and tools concerning bipartite and multipartite entanglement, random and pseudo-random quantum states and quantum chaos maps are discussed in the remaining sections and appendixes.
Bipartite entanglement {#sec:bipartite}
======================
The von Neumann entropy
-----------------------
In this section, we show that *for pure states* $|\psi\rangle$ a good measure of bipartite entanglement exists: the von Neumann entropy of the reduced density matrices. Given a state described by the density matrix $\rho$, its von Neumann entropy is defined as $$S(\rho)=-{\rm Tr}(\rho\log\rho).$$ Note that, here as in the rest of the paper, all logarithms are base $2$ unless otherwise indicated.
First of all, a few definitions are needed:
*Entanglement cost*: Let us assume that two communicating parties, Alice and Bob, share many Einstein-Podolsky-Rosen (EPR) pairs [^4], say $$|\phi^+\rangle=\frac{1}{\sqrt{2}}\,\big(|00\rangle+|11\rangle\big),
\label{EPRstate}$$ and that they wish to prepare a large number $n$ of copies of a given bipartite pure state $|\psi\rangle$, using only local operations and classical communication. If we call $k_{\min}$ the minimum number of EPR pairs necessary to accomplish this task, we define the entanglement cost as the limiting ratio $k_{\min}/n$, for $n\to\infty$.
*Distillable entanglement*: Let us consider the reverse process; that is, Alice and Bob share a large number $n$ of copies of a pure state $|\psi\rangle$ and they wish to concentrate entanglement, again using only local operations supplemented by classical communication. If $k_{\max}'$ denotes the maximum number of EPR pairs that can be obtained in this manner, we define the distillable entanglement as the ratio $k_{\max}'/n$ in the limit $n\to\infty$.
It is clear that $k_{\max}' \leq k_{\min}$. Otherwise, we could employ local operations and classical communication to create entanglement, which is a non-local, purely quantum resource (it would be sufficient to prepare $n$ states $|\psi\rangle$ from $k_{\min}$ EPR pairs and then distill $k_{\max}'>k_{\min}$ EPR states). Furthermore, it is possible to show that, asymptotically in $n$, the entanglement cost and the distillable entanglement coincide and that the ratios $k_{\min}/n$ and $k_{\max}'/n$ are given by the reduced single-qubit von Neumann entropies. Indeed, we have $$\lim_{n\to\infty} \frac{k_{\min}}{n}
=
\lim_{n\to\infty} \frac{k_{\max}'}{n}
= S(\rho_A) = S(\rho_B),$$ where $S(\rho_A)$ and $S(\rho_B)$ are the von Neumann entropies of the reduced density matrices $\rho_A={\rm Tr}_{B}\big(|\psi\rangle\langle\psi|\big)$ and $\rho_B={\rm Tr}_{A}\big(|\psi\rangle\langle\psi|\big)$, respectively. Therefore, the process that changes $n$ copies of $|\psi\rangle$ into $k$ copies of $|\phi^+\rangle$ is asymptotically reversible. Moreover, it is possible to show that it is faithful; namely, the change takes place with unit fidelity when $n\to\infty$.[^5] The proof of this result can be found in [@bennett]. We can therefore quantify the entanglement of a bipartite pure state $|\psi\rangle$ as $$E_{AB}(|\psi\rangle) = S(\rho_A) = S(\rho_B).
\label{entbipure}$$ It ranges from $0$ for a separable state to $1$ for maximally entangled two-qubit states (the EPR states). Hence, it is common practice to say that the entanglement of an EPR pair is $1$ *ebit*.
The Schmidt decomposition {#sec:schmidt}
-------------------------
The fact that $S(\rho_A)=S(\rho_B)$ is easily derived from the Schmidt decomposition.
*The Schmidt decomposition theorem*: Given a pure state $|\psi\rangle\in\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$ of a bipartite quantum system, there exist orthonormal states $\{|i\rangle_A\}$ for $\mathcal{H}_A$ and $\{|i'\rangle_B\}$ for $\mathcal{H}_B$ such that $$|\psi\rangle =
\sum_{i=1}^k
\sqrt{p_i} \, |i\rangle_A |i'\rangle_B
=
\sqrt{p_1} \, |1\rangle_A |1'\rangle_B +
\cdots +
\sqrt{p_k} \, |k\rangle_A |k'\rangle_B ,
\label{schmidtdec}$$ with $p_i$ positive real numbers satisfying the condition $\sum_{i=1}^k p_i=1$ (for a proof of this theorem see, e.g., [@qcbook; @nielsen]).
It is important to stress that the states $\{|i\rangle_A\}$ and $\{|i'\rangle_B\}$ depend on the particular state $|\psi\rangle$ that we wish to expand. The reduced density matrices $\rho_A=\sum_i p_i |i\rangle_A\,{}_A\langle i|$ and $\rho_B=\sum_i p_i |i'\rangle_B\,{}_B\langle i'|$ have the same non-zero eigenvalues. Their number is also the number $k$ of terms in the Schmidt decomposition (\[schmidtdec\]) and is known as the *Schmidt number* (or the *Schmidt rank*) of the state $|\psi\rangle$. A separable pure state, which by definition can be written as $$|\psi\rangle = |\phi\rangle_A |\xi\rangle_B,$$ has Schmidt number equal to one. Thus, we have the following entanglement criterion: a bipartite pure state is entangled if and only if its Schmidt number is greater than one. For instance, the Schmidt number of the EPR state (\[EPRstate\]) is $2$.
It is clear from the Schmidt decomposition (\[schmidtdec\]) that $$S(\rho_A)=S(\rho_B)=-\sum_i p_i\log p_i.$$ If $N$, $N_A$ and $N_B$ denote the dimensions of the Hilbert spaces $\mathcal{H}$, $\mathcal{H}_A$ and $\mathcal{H}_B$, with $N=N_AN_B$ and $N_A\le N_B$, we have $$0\le E_{AB}(|\psi\rangle)=S(\rho_A)=S(\rho_B)\le \log N_A.$$ A maximally entangled state of two subsystems has $N_A$ equally weighted terms in its Schmidt decomposition and therefore its entanglement content is $\log{N_A}$ ebits. For instance, the EPR state (\[EPRstate\]) is a maximally entangled two-qubit state. Note that a maximally entangled state $|\psi\rangle$ leads to a maximally mixed state $\rho_A$.
The purity
----------
The purity of state described by the density matrix $\rho$ is defined as $$P(\rho)={\rm Tr}(\rho^2).$$ We have $$P(\rho_A)=P(\rho_B)=\sum_{i}p_i^2.$$ The purity is much easier to investigate analytically than the von Neumann entropy. Moreover, it provides the first non-trivial term in a Taylor series expansion of the von Neumann entropy about its maximum.[^6] The purity ranges from $1/N_A$ for maximally entangled states to $1$ for separable states. One can also consider the *participation ratio* $\xi=\frac{1}{\sum_i p_i^2}$, which is the inverse of the purity. This quantity is bounded between $1$ and $N_A$ and is close to $1$ if a single term dominates the Schmidt decomposition (\[schmidtdec\]), whereas $\xi=N_A$ if all terms in the decomposition have the same weight ($p_1=\cdots=p_{N_A}=1/N_A$). The participation ratio $\xi$ represents the effective number of terms in the Schmidt decomposition.
A natural extension of the discussion of this section is to consider *bipartite mixed states*, $\rho=\sum_i p_i|\psi\rangle\langle\psi|$ $(\sum_i p_i=1$), instead of pure states. However, mixed-state entanglement is not as well understood as pure-state bipartite entanglement and is the focus of ongoing research (for a review, see, e.g., Refs. [@bruss; @pleniovirmani; @horodecki]).
By definition, a (generally mixed) state is said to be separable if it can be prepared by two parties (Alice and Bob) in a “classical” manner; that is, by means of local operations and classical communication. This means that Alice and Bob agree over the phone on the local preparation of the two subsystems $A$ and $B$. Therefore, a mixed state is separable if and only if it can be written as $$\rho_{AB} = \sum_k p_k \, \rho_{Ak} \otimes \rho_{Bk}, \quad
\hbox{with } p_k \ge 0 \hbox{ and }
\sum_k p_k = 1,
\label{sepdecomposition}$$ where $\rho_{Ak}$ and $\rho_{Bk}$ are density matrices for the two subsystems. A separable system always satisfies Bell’s inequalities; that is, it only contains classical correlations. Given a density matrix $\rho_{AB}$, it is in general a non-trivial task to prove whether a decomposition as in (\[sepdecomposition\]) exists or not [@pleniovirmani; @horodecki]. We therefore need separability criteria that are easier to test. Two useful tools for the detection of entanglement, the Peres criterion and entanglement witnesses, are reviewed in Appendix \[app:separability\].
Entanglement of random states {#sec:entrandom}
=============================
A simple argument helps understanding why the bipartite entanglement content of a pure random state $|\psi\rangle$ is almost maximal. In a given basis $\{|i\rangle\}$ the density matrix for the state $|\psi\rangle=\sum_i c_i |i\rangle$ is written as follows: $$\rho_{ij} = \langle i | \psi\rangle
\langle \psi | j\rangle= c_i c_j^\star,$$ where $c_i=\langle i | \psi \rangle$ are the components of the state $|\psi\rangle$ in the $\{|i\rangle\}$ basis. In the case of a random state the components are uniformly distributed, with amplitudes $c_i\approx 1 /\sqrt{N}$ and random phases. Here $N$ is the Hilbert space dimension and the value $1/\sqrt{N}$ of the amplitudes ensures that the wave vector $|\psi\rangle$ is normalized. The density matrix can therefore be written as $$\rho \approx
\mathrm{diag}
\left(\frac{1}{{N}},\frac{1}{{N}},\ldots,
\frac{1}{{N}}\right) +
\Omega,
\label{eq:rhototrandom}$$ where $\Omega$ is a ${N}\times{N}$ zero diagonal matrix with random complex matrix elements of amplitude $\approx
1/{N}$. Suppose now that we partition the Hilbert space of the system into two parts, $A$ and $B$, with dimensions ${N}_A$ and ${N}_B$, where ${N}_A{N}_B={N}$. Without loss of generality, we take the first subsystem, $A$, to be the one with the not larger dimension: ${N}_A\le {N}_B$. The reduced density matrix $\rho_A$ is defined as follows: $$\rho_A=\mathrm{Tr}_B \rho=
\sum_{i_B} c_{i_A i_B} c^\star_{i_A^\prime i_B}
|i_A\rangle \langle i_A^\prime|,$$ where $|i\rangle = |i_A i_B\rangle$. Using Eq. (\[eq:rhototrandom\]), we obtain $$\rho_A \approx \mathrm{diag}
\left(\frac{1}{{N}_A},\frac{1}{{N}_A},\ldots,
\frac{1}{{N}_A}\right) +
\Omega_A,
\label{eq:rhoA}$$ where $\Omega_A$ is a zero diagonal matrix with matrix elements of $O\left(\sqrt{{N}_B}/{N}\ll
1/{N}_A\right)$ (sum of ${N}_B\gg 1$ terms of order $1/{N}$ with random phases). Neglecting $\Omega_A$ in (\[eq:rhoA\]), the reduced von Neumann entropy of subsystem $A$ is given by $S(\rho_A)=\log ({N}_A)$, the maximum entropy that the subsystem $A$ can have.
Page’s formula
--------------
The exact mean value $\langle E_{AB}(|\psi\rangle) \rangle$ of the bipartite entanglement is given by Page’s formula, obtained [@random-states] by considering the ensemble of random pure states drawn according to the Haar measure on $U(N)$: $$S_P\equiv \langle E_{AB}(|\psi\rangle) \rangle =
\langle S(\rho_A) \rangle=\langle S(\rho_B)\rangle=
\log{N}_A-\frac{{N}_A}{2{N}_B\ln 2},
\label{Epage}$$ where $\langle\, \cdot\, \rangle$ denotes the (ensemble) average over the uniform Haar measure. For $\log{N}_A\gg 1$, $\langle E_{AB} \rangle$ is close to its maximum value $E_{AB}^{\hbox{max}}(|\psi\rangle)=\log{N}_A\gg 1$. Note that, if we fix $N_A$ and let $N_B\to\infty$, then $\langle E_{AB}\rangle $ tends to $E_{AB}^{\hbox{max}}$.
Remarkably, if we consider the *thermodynamic limit*, that is, we fix $N_A/N_B$ and let $N_A\to\infty$, then the reduced von Neumann entropy concentrates around its average value (\[Epage\]). This is a consequence of the so-called *concentration of measure* phenomenon: the uniform measure on the $k$-sphere $\mathbb{S}^k$ in $\mathbb{R}^{k+1}$ (parametrized, for instance, by $k=N^2-2$ angles in the Hurwitz parametrization [@pozniak; @weinstein]) concentrates very strongly around the equator when $k$ is large: Any polar cap smaller than a hemisphere has relative volume exponentially small in $k$. This observation implies, in particular, the concentration of the entropy of the reduced density matrix $\rho_A$ around its average value [@sommers; @hayden]. This in turn implies that when the dimension $N$ of the quantum system is large it is meaningful to apply statistical methods and discuss typical (entanglement) behavior or random states, in the sense that almost all random states behave in essentially the same way.
Lubkin’s formula
----------------
For random states, the average value of the purity of the reduced density matrices $\rho_A$ and $\rho_B$ is giben by Lubkin’s formula [@lubkin]: $$P_L\equiv \langle P(\rho_A)\rangle =
\langle P(\rho_B)\rangle =
\frac{N_A+N_B}{N_AN_B+1}.
\label{eq:lubkin}$$ Nothe that, if we fix $N_A$ and let $N_B\to\infty$, then $P_L$ tends to its minimum value $1/N_A$. If we fix $N_A/N_B$ and let $N_A\to\infty$, then $P_L\to 0$. For large $N$, the variance $$\sigma_{P}^2=\langle P^2 \rangle - P_L^2 \approx \frac{2}{N^2},
\label{eq:variancelubkin}$$ so that the relative standard deviation $$\frac{\sigma_{P}}{P_L} \approx \frac{\sqrt{2}}{N_A+N_B}$$ tends to zero in the thermodynamic limit $N_A\to \infty$ (at fixed $N_A/N_B$). For *balanced bipartition*, corresponding to $N_A=N_B=\sqrt{N}$, we have $$\frac{\sigma_{P}}{P_L} =O\left(\frac{1}{\sqrt{N}}\right).$$ Note that the fact that ${\sigma_{P}}/{P_L}\to 0$ when $N\to\infty$ is again a consequence of the concentration of measure phenomenon [@sommers; @hayden]. A derivation of Eqs. (\[eq:lubkin\]) and (\[eq:variancelubkin\]) is presented in Appendix \[app:lubkin\].
Pseudo-random states {#sec:pseudo-random}
====================
The generation of a random state $|\psi\rangle$ is exponentially hard. Indeed, starting from a fiducial $n_q$-qubit state $|0\rangle=|0\ldots 0\rangle$ one needs to implement a typical (random) unitary operator $U$ (drawn from the Haar measure on $U({N}=2^{n_q})$) to obtain $|\psi\rangle=U|0\rangle$. Since $U$ is determined by $4^{n_q}-2$ real parameters (for instance, the angles of the Hurwitz parametrization [@pozniak; @weinstein]), its generation requires a sequence of elementary one- and two-qubit gates whose length grows exponentially in the number of qubits. Thus, the generation of random states is unphysical for a large number of qubits. On the other hand, one can consider the generation of pseudo-random states that could reproduce the entanglement properties of truly random states [@emerson1; @emerson2; @weinstein; @plenio1; @plenio2; @znidaric]. In Refs. [@plenio1; @plenio2] it has been proven that the average entanglement of a typical state can be reached to a fixed accuracy within $O(n_q^3)$ elementary quantum gate. This proof holds for a random circuit such that $U$ is the product, $$U=W_{t}W_{t-1}\cdots W_2W_1,
\label{UCNOT}$$ of a sequence of $t=O(n_q^3)$ two-qubit gates $W_k$ independently chosen at each step as follows:
- a pair of integers $(i,j)$, with $i\ne j$ is chosen uniformly at random from $\{1,...,n_q\}$;
- single-qubit gates (unitary transformations) $V_k^{(i)}$ and $V_k^{(j)}$, drawn independently from the Haar measure on $U(2)$, are applied;
- a ${\rm{CNOT}}^{(i,j)}$ gate with control qubit $i$ and target qubit $j$ is applied.[^7]
Therefore, $$W_k={\rm{CNOT}}^{(i,j)}V_k^{(i)}V_k^{(j)}.
\label{WCNOT}$$ The proof [@plenio1; @plenio2] that (\[UCNOT\]) generates to within any desired accuracy the entanglement of a random state in a polynomial number of gates is based on the fact that the evolution of the purity of the two subsystems $A$ and $B$ can be described following a Markov chain approach, with gap in the Markov chain given by $\Delta(n_q)\geq p(n_q)$, where $p(n_q)=O(n_q^{-2})$. If the density matrix $\rho_t$ of the overall system is expanded in terms of Pauli matrices, $$\rho_t=\sum_{\alpha_0,...,\alpha_{n_q-1}=0}^3
c_t^{(\alpha_0,...,\alpha_{n_q-1})} \sigma_0^{(\alpha_0)}\otimes...
\otimes \sigma_{n_q-1}^{(\alpha_{n_q-1})},$$ where $$c_t^{(\alpha_0,...,\alpha_{n_q-1})}=
\frac{1}{N} {\rm Tr}\left(\sigma_0^{(\alpha_0)}\otimes...
\otimes \sigma_{n_q-1}^{(\alpha_{n_q-1})}\rho_t\right),$$ with $\sigma_0\equiv I$, $\sigma_1\equiv\sigma_x$, $\sigma_2\equiv\sigma_y$, $\sigma_3\equiv\sigma_z$, we obtain $$P(\rho_{A,t})=P(\rho_{B,t})=N_AN_B^2
\sum_{\alpha_0,...,\alpha_{n_A-1}=0}^3
[c_t^{(\alpha_0,...,\alpha_{n_A-1},0,...,0)}]^2.
\label{puritypauli}$$ As usual in this paper, we consider $n_A+n_B=n_q$ and $N_AN_B=N$. In the case of model (\[WCNOT\]), the evolution of the column vector $${\bf c_t^2}\equiv {}^t[(c_t^{(0,...,0)})^2,(c_t^{(0,...,0,1)})^2,...,
(c_t^{(3,...,3)})^2]$$ is described [@plenio1; @plenio2] by a Markov chain: $${\bf c_{t+1}^2}=M {\bf c_{t}^2}.$$ Therefore, the asymptotic decay of purity is determined by the second largest eigenvalue $1-\Delta(n_q)$ of the matrix $M$ (the largest eigenvalue is the unit eigenvalue): $$|\langle P(\rho_{A,t})\rangle -P_L|\asymp (1-\Delta(n_q))^t
=\exp\{[\ln(1-\Delta(n_q))]t\}.$$ For model (\[WCNOT\]), one obtains [@plenio1; @plenio2] $\Delta(n_q)\geq p(n_q)$, with $p(n_q)=O(n_q^{-2})$.
Alternatively, a two-qubit gate different from ${\rm{CNOT}}$ [@znidaric] or $W_t$ chosen from the $U(4)$ Haar measure and acting on a a pair $i,j$ of qubits ($i\ne j$) randomly chosen at each step, can be used [@znidaric; @gennaro]. Numerical results for these models [@plenio1; @plenio2; @znidaric] indicate that the above analytic bound is not optimal: there is numerical evidence that $\Delta(n_q)=O(n_q^{-1})$ and therefore $O(n_q^2)$ steps are sufficient to generate the average entanglement of a random state. A different strategy to efficiently approximate the entanglement content of random states is based on quantum chaotic maps [@weinstein; @rossini] and will be discussed in Sec. \[sec:qcent\].
Finally, we note that the Markov chain approach allows an easy derivation of Lubkin’s formula (\[eq:lubkin\]). Let us consider $W_k$ chosen from the $U(4)$ Haar measure. In this case, the Markov matrix $$M=\frac{2}{n_q(n_q-1)}\sum_{i,j} M_{ij}^{(2)},$$ with $M^{(2)}_{ij}$ acting non trivially (differently from identity) only on the subspace spanned by qubits $i$ and $j$ ($i,j=1,...,n_q$ and $i\ne j$). After averaging over the uniform Haar measure on $U(4)$ one can see [@znidaric] that $M^{(2)}_{ij}$ preserves identity ($\sigma_i^{(0)}\otimes \sigma_j^{(0)}
\to \sigma_i^{(0)}\otimes \sigma_j^{(0)}$) and uniformly mixes the other $15$ products $\sigma_i^{(\alpha_i)}\otimes
\sigma_j^{(\alpha_j)}$ [@znidaric]. Matrix elements are therefore $[M^{(2)}_{ij}]_{0,0}=1$, $[M^{(2)}_{ij}]_{0,x}=[M^{(2)}_{ij}]_{x,0}=0$, and $[M^{(2)}_{ij}]_{x,x'}=1/15$ for $x,x'\in \{1,...,15\}$.
The matrix $M$ has an eigenvalue equal to $1$ (with multiplicity $2$) and all the other eigenvalues smaller than $1$. The eigenspace corresponding to the unit eigenvalue of matrix $M$ is spanned by the column vectors $$v_0={}^t(1,0,...,0),\;\;
v_1={}^t(0,1,...,1).$$ The asymptotic equilibrium state $c^2(\infty)=\lim_{t\to \infty}c^2(t)$ is however uniquely determined by the constraints ${\rm Tr}(\rho_t)=1$ and ${\rm Tr}(\rho_t^2)=1$, which impose $$x_0\equiv [c_t^{(0,...,0)}]^2=\frac{1}{N^2},\;\;
x_1\equiv \sum_{\alpha_0,...,\alpha_{n_q-1}\ne (0,...,0)}
[c_t^{(\alpha_0,...,\alpha_{n_q-1})}]^2=\frac{N-1}{N^2}.$$ Finally, we obtain $$c^2(\infty)=x_0 v_0+x_1\frac{1}{N^2-1}v_1=
\frac{1}{N^2}\,{}^{t}(1,\frac{N-1}{N^2-1},...,
\frac{N-1}{N^2-1})$$ and, after substitution of the components of this vector into Eq. (\[puritypauli\]), $$P(\rho_{A,t})=\frac{N_AN_B^2}{N^2}\left[1+
(N_A^2-1)\frac{N-1}{N^2-1}\right],$$ which immediately leads to Lubkin’s formula (\[eq:lubkin\]).
Multipartite entanglement {#sec:multipartite}
=========================
The characterization and quantification of multipartite entanglement is a challenging open problem in quantum information science and many different measures have been proposed [@pleniovirmani; @horodecki]. To grasp the difficulty of the problem, let us suppose to have $n$ parties composing the system we wish to analyze. In order to obtain a complete characterization of multipartite entanglement, we should take into account all possible non-local correlations among all parties. It is therefore clear that the number of measures needed to fully quantify multipartite entanglement grows exponentially with the number of qubits. Therefore, in Ref. [@facchi] it has been proposed to characterize multipartite entanglement by means of a function rather than with a single measure. The idea is to look at the probability density function of bipartite entanglement between all possible bipartitions of the system. For pure states the bipartite entanglement is the von Neumann entropy of the reduced density matrix of one of the two subsystems: $E_{AB}(|\psi\rangle)=S(\rho_A)=S(\rho_B)$.
It is instructive to consider the smallest non-trivial instance where multipartite entanglement can arise: the three-qubit case. Here we have three possible bipartitions, with $n_A=1$ qubit and $n_B=2$ qubits. For a GHZ state [@GHZstate], $$|{\rm GHZ}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle),$$ we obtain $\rho_A=\frac{I}{2}$ for all bipartitions, and therefore $$p(E_{AB})=\delta_{E_{AB},1},$$ namely there is maximum multipartite entanglement, fully distributed among the three qubits. [^8] Note that in this case $\rho_B=\frac{1}{2}(|00\rangle\langle 00|+
|11\rangle\langle 11|)$ is separable and therefore the pairwise entanglement between any two qubits is equal to zero.
For a W state [@Wstate], $$|{\rm W}\rangle=\frac{1}{\sqrt{2}}(|100\rangle+
|010\rangle+|001\rangle),$$ we obtain $\rho_A=\frac{2}{3}|0\rangle\langle 0|+
\frac{1}{3}|1\rangle\langle 1|$ for all bipartitions, and therefore $$p(E_{AB})=\delta_{E_{AB},\bar{E}},$$ where $\bar{E}=-\frac{2}{3}\log\left(\frac{2}{3}\right)
-\frac{1}{3}\log\left(\frac{1}{3}\right)\approx 0.92$, namely the distribution $p(E_{AB})$ is peaked but the amount of multipartite entanglement is not maximal.
As a last three-qubit example, let us consider the state $$|\psi\rangle=\frac{1}{\sqrt{2}}(|000\rangle+
|110\rangle)=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)
|0\rangle,$$ where the first two qubits are in a maximally entangled (Bell) state, while the third one is factorized. In this case, $\rho_A=\frac{I}{2}$ if subsystem $A$ is one of the first two qubits, $\rho_A=|0\rangle\langle 0|$ otherwise. Hence, $$p(E_{AB})= \frac{2}{3}\delta_{E_{AB},1}+\frac{1}{3}\delta_{E_{AB},0},$$ namely the entanglement can be large but the variance of the distribution $p(E_{AB})$ is also large.
For sufficiently large systems (${N} = 2^{n_q} \gg 1$), it is reasonable to consider only balanced bipartitions, i.e., with $n_A = n_B$ ($n_A+n_B=n_q$), since the statistical weight of unbalanced ones ($n_A\ll n_B$) becomes negligible [@facchi]. If the probability density has a large mean value $\langle E_{AB} \rangle \sim n_q$ ($\langle \, \cdot \, \rangle$ denotes the average over balanced bipartitions) and small relative standard deviation $\sigma_{AB}/\langle E_{AB}\rangle \ll 1$, we can conclude that genuine multipartite entanglement is almost maximal (note that $E_{AB}$ is bounded within the interval $[0,n_q]$). As I shall discuss in Sec. \[sec:qcent\], this is the case for random states [@facchi] (see also Ref. [@kendon]).[^9]
Quantum chaos map and entanglement {#sec:qcent}
==================================
The relation between chaos and entanglement is discussed, for instance, in [@furuya; @miller; @lakshminarayan1; @lakshminarayan2; @lakshminarayan3]. More specifically, the use of quantum chaos for efficient and robust generation of pseudo-random states carrying large multipartite entanglement is nicely illustrated by the example of the quantum sawtooth map [@benenti01; @varenna05; @qcbook]. This map is described by the unitary Floquet operator $\hat{U}$: $$|\psi_{t+1}\rangle = \hat{U}|\psi_t\rangle =
e^{-iT\hat{n}^{2}/2} \, e^{ik(\hat{\theta} -\pi)^{2}/2}
|\psi_t\rangle ,
\label{eq:quantmap}$$ where $\hat{n} = -i \, \partial/\partial \theta$, $[\hat{\theta},\hat{n}]=i$ (we set $\hbar=1$) and the discrete time $t$ measures the number of map iterations. In the following I will always consider map (\[eq:quantmap\]) on the torus $0 \leq \theta < 2 \pi$, $- \pi \leq p < \pi$, where $p = T n$. With an $n_q$-qubit quantum computer we are able to simulate the quantum sawtooth map with ${N} = 2^{n_q}$ levels; as a consequence, $\theta$ takes $N$ equidistant values in the interval $0 \leq \theta < 2 \pi$, while $n$ ranges from $-{N}/2$ to ${N}/2 -1$ (thus setting $T=2\pi/{N}$). We are in the quantum chaos regime for map (\[eq:quantmap\]) when $K\equiv kT >0$ or $K<-4$; in particular, in the following I will focus on the case $K=1.5$.
There exists an efficient quantum algorithm for simulating the quantum sawtooth map [@benenti01; @qcbook]. The crucial observation is that the operator $\hat{U}$ in Eq. \[eq:quantmap\] can be written as the product of two operators: $\hat{U}_{k}= e^{ik(\hat{\theta}-\pi)^{2}/2}$ and $\hat{U}_{T}=e^{-iT\hat{n}^{2}/2}$, that are diagonal in the $\theta$- and in the $n$-representation, respectively. Therefore, the most convenient way to classically simulate the map is based on the forward-backward fast Fourier transform between $\theta$ and $n$ representations, and requires $O({N}\log {N})$ operations per map iteration. On the other hand, quantum computation exploits its capacity of vastly parallelize the Fourier transform, thus requiring only $O((\log {N})^2)$ one- and two-qubit gates to accomplish the same task [@benenti01; @qcbook]. In brief, the resources required by the quantum computer to simulate the sawtooth map are only logarithmic in the system size ${N}$, thus admitting an exponential speedup, as compared to any known classical computation. The sawtooth map and the quantum algorithm for its simulation are discussed in details in Appendix \[sec:sawmap\].
Let us first compute the average bipartite entanglement $\langle E_{AB} \rangle$ as a function of the number $t$ of iterations of map (\[eq:quantmap\]). Numerical data in Fig. \[fig:EntGen\] exhibit a fast convergence, within a few kicks, of this quantity to the value $$\langle E^{\, {\rm rand}}_{AB} \rangle=
\frac{n_q}{2} - \frac{1}{2 \ln 2}
\label{eq:entpure}$$ expected for a random state according to Page’s formula [@random-states] (note that this result is obtained from Eq. (\[Epage\]) in the special case $n_A=n_B$). Precisely, as shown in the inset of Fig. \[fig:EntGen\], $\langle E_{AB} \rangle$ converges exponentially fast to $\langle E^{\,{\rm rand}}_{AB} \rangle$, with the time scale for convergence $\propto n_q$. Therefore, the average entanglement content of a true random state is reached to a fixed accuracy within $O(n_q)$ map iterations, namely $O(n_q^3)$ quantum gates. I stress that in our case a deterministic map, instead of random one- and two-qubit gates as in Ref. [@plenio1; @plenio2; @znidaric; @gennaro], is implemented. Of course, since the overall Hilbert space is finite, the above exponential decay in a deterministic map is possible only up to a finite time and the maximal accuracy drops exponentially with the number of qubits. I also note that, due to the quantum chaos regime, properties of the generated pseudo-random state do not depend on initial conditions, whose characteristics may even be very far from randomness (e.g., simulations of Fig. \[fig:EntGen\], start from a completely disentangled state).
![Time evolution of the average bipartite entanglement of a quantum state, starting from a state of the computational basis (eigenstate of the momentum operator $\hat{n}$), and recursively applying the quantum sawtooth map (\[eq:quantmap\]) at $K=1.5$ and, from bottom to top, $n_q= 4, 6, 8, 10, 12$. Dashed lines show the theoretical values of Eq. (\[eq:entpure\]). Inset: convergence of $\langle E_{AB} \rangle (t)$ to the asymptotic value $\langle E^{\, {\rm rand}}_{AB} \rangle$ in Eq. (\[eq:entpure\]); time axis is rescaled with $1/n_q$. This figure is taken from Ref. [@rossini].[]{data-label="fig:EntGen"}](EntGen){width="10.0cm"}
As discussed above, multipartite entanglement should generally be described in terms of a function, rather than by a single number. I therefore show in Fig \[fig:Isto\_eps0\] the probability density function $p(E_{AB})$ for the entanglement of all possible balanced bipartitions of the state $|\psi_{t=30}\rangle$. This function is sharply peaked around $\langle E^{\,{\rm rand}}_{AB}\rangle$, with a relative standard deviation $\sigma_{AB} / \left< E_{AB} \right>$ that drops exponentially with $n_q$ (see the inset of Fig. \[fig:Isto\_eps0\]) and is small ($\sim 0.1$) already at $n_q=4$. For this reason, we can conclude that multipartite entanglement is large and that it is reasonable to use the first moment $\langle E_{AB} \rangle$ of $p(E_{AB})$ for its characterization. The corresponding probability densities for random states is also calculated (dashed curves in Fig. \[fig:Isto\_eps0\]); their average values and variances are in agreement with the values obtained from states generated by the sawtooth map. As we have remarked in Sec. \[sec:entrandom\], the fact that for random states the distribution $p(E_{AB})$ is peaked around a mean value close to the maximum achievable value $E_{AB}^{\rm max}=n_q/2$ is a manifestation of the concentration of measure phenomenon in a multi-dimensional Hilbert space [@sommers; @hayden].
![Probability density function of the bipartite von Neumann entropy over all balanced bipartitions for the state $|\psi_t\rangle$, after $30$ iterations of map (\[eq:quantmap\]) at $K=1.5$. Various histograms are for different numbers of qubits: from left to right $n_q = 8, 10, 12$; dashed curves show the corresponding probabilities for random states. Inset: relative standard deviation $\sigma_{AB} / \langle E_{AB} \rangle$ as a function of $n_q$ (full circles) and best exponential fit $\sigma_{AB} / \langle E_{AB} \rangle \sim e^{- 0.48 \, n_q}$ (continuous line); data and best exponential fit $\sigma_{AB} / \langle E_{AB} \rangle \sim e^{- n_q / 2}$ for random states are also shown (empty triangles, dashed line). This figure is taken from Ref. [@rossini].[]{data-label="fig:Isto_eps0"}](Isto_Lowbound){width="10.0cm"}
Stability of multipartite entanglement {#sec:stabilitymultipartite}
======================================
In order to assess the physical significance of the generated multipartite entanglement, it is crucial to study its stability when realistic noise is taken into account. Hereafter I model quantum noise by means of unitary noisy gates, that result from an imperfect control of the quantum computer hardware [@cirac95]. The noise model of Ref. [@rossini04] is followed. One-qubit gates can be seen as rotations of the Bloch sphere about some fixed axis; I assume that unitary errors slightly tilt the direction of this axis by a random amount. Two-qubit controlled phase-shift gates are diagonal in the computational basis; I consider unitary perturbations by adding random small extra phases on all the computational basis states. Hereafter I assume that each noise parameter $\varepsilon_i$ is randomly and uniformly distributed in the interval $[-\varepsilon, +\varepsilon]$; errors affecting different quantum gates are also supposed to be completely uncorrelated: every time we apply a noisy gate, noise parameters randomly fluctuate in the (fixed) interval $[-\varepsilon, +\varepsilon]$.
Starting from a given initial state $|\psi_0\rangle$, the quantum algorithm for simulating the sawtooth map in presence of unitary noise gives an output state $|\psi_{\varepsilon_I,t}\rangle$ that differs from the ideal output $|\psi_t\rangle$. Here $\varepsilon_I=(\varepsilon_1,\varepsilon_2,...,\varepsilon_{n_d})$ stands for all the $n_d$ noise parameters $\varepsilon_i$, that vary upon the specific noise configuration ($n_d$ is proportional to the number of gates). Since we do not have any a priori knowledge of the particular values taken by the parameters $\varepsilon_i$, the expectation value of any observable $A$ for our $n_q$-qubit system will be given by ${\rm Tr} [\rho_{\varepsilon,t} A]$, where the density matrix $\rho_{\varepsilon,t}$ is obtained after averaging over noise: $$\rho_{\varepsilon,t} =
\left(\frac{1}{2\varepsilon}\right)^{n_d}
\int d \varepsilon_I
|\psi_{\varepsilon_I,t}\rangle
\langle\psi_{\varepsilon_I,t}| \, .
\label{eq:rhomatr}$$ The integration over $\varepsilon_I$ is estimated numerically by summing over $\mathcal{N}$ random realizations of noise, with a statistical error vanishing in the limit $\mathcal{N}\to \infty$. The mixed state $\rho_{\varepsilon}$ may also arise as a consequence of non-unitary noise; in this case Eq. (\[eq:rhomatr\]) can also be seen as an unraveling of $\rho_{\varepsilon}$ into stochastically evolving pure states $|\psi_{\varepsilon_I}\rangle$, each evolution being known as a quantum trajectory [@plenioknight; @brun; @carlo1; @carlo2].
I now focus on the entanglement content of $\rho_{\varepsilon,t}$. Unfortunately, for a generic mixed state of $n_q$ qubits, a quantitative characterization of entanglement is not known, neither unambiguous [@pleniovirmani; @horodecki]. Anyway, it is possible to give numerically accessible lower and upper bounds for the bipartite [*distillable entanglement*]{} $E_{AB}^{(D)} (\rho_{\varepsilon})$: $$\max \left\{ S(\rho_{\varepsilon,A}) -
S(\rho_\varepsilon), 0 \right\} \leq
E_{AB}^{(D)} (\rho_{\varepsilon})
\leq \log \| \rho_{\varepsilon}^{T_B} \| \, ,
\label{eq:entbounds}$$ where $\rho_{\varepsilon,A} =
{\rm Tr}_B (\rho_\varepsilon)$ and $\| \rho_{\varepsilon}^{T_B} \| \equiv {\rm Tr}
\sqrt{(\rho_\varepsilon^{T_B})^\dagger \,
\rho_{\varepsilon}^{T_B}}$ denotes the trace norm of the partial transpose of $\rho_{\varepsilon}$ with respect to party $B$ (see Appendix \[app:peres\] for the definition of the partial transposition operation).
In practice, the quantum algorithm for the quantum sawtooth map is simulated in the chaotic regime with noisy gates and the two bounds in Eq. (\[eq:entbounds\]) for the bipartite distillable entanglement of the mixed state $\rho_{\varepsilon,t}$, obtained after averaging over $\mathcal{N}$ noise realizations, are evaluated. A satisfactory convergence for the lower and the upper bound is obtained after $\mathcal{N}\sim \sqrt{N}$ and $\mathcal{N}\sim N$ noise realizations, respectively. The first moment of the lower ($E_m$) and the upper ($E_M$) bound for the bipartite distillable entanglement is shown as a function of the imperfection strength in Fig. \[fig:Ent\_dest\_nqvar\], upper panels. The various curves are for different numbers $n_q$ of qubits; $\mathcal{N}$ depends on $n_q$ and is large enough to obtain negligible statistical errors (smaller than the size of the symbols). The relative standard deviation of the probability density function (over all balanced bipartitions) for the bipartite distillable entanglement is shown in the lower panels of Fig. \[fig:Ent\_dest\_nqvar\]. Like for pure states, we notice an exponential drop with $n_q$; the distribution width slightly broadens when increasing imperfection strength $\varepsilon$. We can therefore conclude that an average value of the bipartite bipartite distillable entanglement close to the ideal case $\varepsilon=0$ implies that multipartite entanglement is stable.
![Upper graphs: lower $\langle E_m \rangle$ (left panel) and upper bound $\langle E_M \rangle$ (right panel) for the bipartite distillable entanglement as a function of the noise strength at time $t=30$. Various curves stand for different numbers of qubits: $n_q=$ 4 (circles), 6 (squares), 8 (diamonds), 10 (triangles up), and 12 (triangles down). Lower graphs: relative standard deviation of the probability density function for distillable entanglement over all balanced bipartitions as a function of $n_q$, for different noise strengths $\varepsilon$. Dashed lines show a behavior $\sigma / \left< E \right> \sim e^{-n_q/2}$ and are plotted as guidelines. This figure is taken from Ref. [@rossini].[]{data-label="fig:Ent_dest_nqvar"}](Ent_dest_nqvar){width="12.0cm"}
In order to quantify the robustness of multipartite entanglement with the system size, let us define a perturbation strength threshold $\varepsilon^{(R)}$ at which the distillable entanglement bounds drop by a given fraction, for instance to $1/2$, of their $\varepsilon=0$ value, and analyze the behavior of $\varepsilon^{(R)}$ as a function of the number of qubits. Numerical results are plotted in Fig. \[fig:EScal\_eps\_nq\]; both for lower and upper bounds we obtain a power-law scaling close to $$\varepsilon^{(R)} \sim 1/n_q \, .
\label{eq:epsscaling}$$
![Perturbation strength at which the bounds of multipartite entanglement halve (lower bound on the left panel, upper bound on the right panel), as a function of the number of qubits. Dashed lines are best power-law fits of numerical data: $\varepsilon^{(R)} \sim n_q^{-0.79 \pm 0.01}$ at $t=15$, $\varepsilon^{(R)} \sim n_q^{-0.9 \pm 0.01}$ at $t=30$, for both lower and upper bounds. This figure is taken from Ref. [@rossini].[]{data-label="fig:EScal_eps_nq"}](Scal_eps_nq){width="11.0cm"}
It is possible to give a semi-analytical proof of the scaling (\[eq:epsscaling\]) for the lower bound measure, that is based on the quantum Fano inequality [@schumacher96], which relates the entropy $S(\rho_{\varepsilon})$ to the fidelity $F = \langle \psi_t \vert \rho_{\varepsilon,t}
\vert \psi_t \rangle$: $$S(\rho_{\varepsilon})\,\lesssim\, h(F) + (1 - F) \, \log (N^2 - 1),
\label{eq:qfano}$$ where $h(x) = -x \log (x) - (1-x) \log (1-x)$ is the binary Shannon entropy. Since $F \simeq e^{- \gamma \varepsilon^2 n_g t}$ [@rossini04; @bettelli04], with $\gamma \sim 0.28$ and $n_g = 3 n_q^2 + n_q$ being the number of gates required for each map step, we obtain, for $\varepsilon^{2} n_g t \ll 1$, $$S (\rho_{\varepsilon}) \, \le \, \gamma \varepsilon^2 n_g t \left[
- \log ( \gamma \varepsilon^2 n_g t) + 2 n_q + \frac{1}{\ln 2} \right].
\label{eq:entrofano}$$ For sufficiently large systems the second term dominates (for $n_q =12$ qubits, $t=30$ and $\varepsilon \sim 5 \times 10^{-3}$ the other terms are suppressed by a factor $\sim 1/ 10$) and, to a first approximation, we can only retain it. On the other hand, an estimate of the reduced entropy $S(\rho_A)$ is given by the bipartite entropy (\[eq:entpure\]) of a pure random state [@random-states]. Therefore, from Eq. (\[eq:entbounds\]) we obtain the following expression for the lower bound of the distillable entanglement: $$E_{AB}^{(D)} (\rho_{\varepsilon}) \, \ge \,
\frac{n_q}{2} - \frac{1}{2 \ln 2} -
6 \gamma n_q^3 \varepsilon^2 t \,.
\label{eq:fanoscaling}$$ From the threshold definition $E_{AB}^{(D)}
(\rho_{\varepsilon^{(R)}}) = \frac{1}{2} E_{AB}^{(D)} (\rho_{0})
=\frac{1}{2}S(\rho_A)$ we get the scaling (\[eq:epsscaling\]), that is valid when $n_q \gg 1$: $$\varepsilon^{(R)}_m \sim 1/\sqrt{24 \, \gamma \, n_q^2 \, t}.$$ Notice that, for small systems as the ones that can be numerically simulated (see data in Fig. \[fig:EScal\_eps\_nq\]), the first term of Eq. (\[eq:entrofano\]) may introduce remarkable logarithmic deviations from the asymptotic power-law behavior. At any rate, the scaling derived from Eq. (\[eq:fanoscaling\]) is in good agreement with the above shown numerical data, and also reproduces the prefactor in front of the power-law decay (\[eq:epsscaling\]) up to a factor of two.
Detecting entanglement of random states {#sec:detect}
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The entanglement content of high-dimensional random pure states is almost maximal: nevertheless, in this section I will demonstrate that, due to the complexity of such states, the detection of their entanglement is rather difficult [@slovenia].
Random states and the quantum to classical transition
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A motivation to the study of the detection of random states comes from considerations of the quantum to classical transition. Since random states carry a lot of entanglement and entanglement has no analogue in classical mechanics, one can immediately conclude that random states are highly non-classical. On the other hand, as we have seen in Sec. \[sec:qcent\], pseudo-random states with properties close to those of true random states can be efficiently generated by dynamical systems (maps) in the regime of quantum chaos. In such chaotic maps the classical limit is recovered when the number of levels $N\to\infty$. Therefore, one can argue that for large random states, i.e., in the limit $N\to \infty$, the quantum expectation value of an operator with a well defined classical limit will be close to its classical microcanonical average. According to this picture random states in a way “mimic” classical microcanonical density. Expectation values are therefore close to the classical ones.
At first sight this is in striking contrast with the almost maximal entanglement of such states. However, as I shall discuss in the following, the contradiction is only apparent [@slovenia]. The detection of entanglement for a random state appears very difficult at large $N$, as it would demand the control of very finely interwoven degrees of freedom and a measurement resolution inversely proportional to $N$, which seems hardly feasible experimentally. Therefore, as far as the detection of entanglement is concerned, high dimensional random states are effectively classical. [^10]
Moreover, coarse graining naturally appears. For instance, one could repeat several times the measurement of an entanglement witness (the definition of entanglement witness is provided in Appendix \[sec:witness\]) for a random state and the prepared random state would be different from time to time due to unavoidable experimental imperfections. Let us model this problem by considering mixtures of $m$ pure random states, namely $$\rho=\sum_{i=1}^m \frac{1}{m} \ket{\psi_i}\bra{\psi_i},
\label{eq:rho}$$ where the $\ket{\psi_i}$ are mutually independent random pure states, but in general they are not orthogonal. I am going to show that the detection of entanglement is even more difficult for these mixed states, as it requires a number of measurements growing exponentially with $m$.
It is interesting to remark that there are other physical contexts in which formally the same kind of coarse graining naturally appears:
- \(i) [*Time averaging*]{} - For example, if a state $\ket{\psi}$ undergoes a time evolution $\ket{\psi(t)} = U(t)\ket{\psi}$ given in terms of some unitary dynamics $U(t)$, then the time average of a physical observable $A$ over an interval $T$ is given by the expectation value ${\rm Tr}(A \rho)$ for the mixed state $$\rho=\frac{1}{T}\int_0^T {d} t \ket{\psi(t)}\bra{\psi(t)},
\label{eq:rhot}$$ which has an effective rank $m \approx T/t_{\rm corr}$, where $t_{\rm corr}$ is a dynamical correlation time of the dynamics $U(t)$. For a quantum chaotic evolution $U(t)$, the state $\ket{\psi(t)}$ can be, after some time, arguably well described by a random state and the correlation time $t_{\rm corr}$ is expected to be short, so $\rho$ in Eq. (\[eq:rhot\]) may be well approximated by a mixture of $m$ uncorrelated random states analogous to Eq. (\[eq:rho\]).
- \(ii) [*Phase space averaging*]{} - Sometimes it is useful to represent quantum states in terms of distribution functions in the classical phase space, like the Husimi function (see, e.g., [@saraceno]), which can be understood as a convolution of the Wigner function or its coarse graining over a phase space volume $2\pi\hbar$ (to simplify writing, let us consider systems with one degree of freedom). In fact, the Husimi function of a pure state can be understood as a Wigner function of the following mixed state: $$\rho = \frac{1}{2\pi\hbar} \int {d}{q}{d}{p}
\exp\left[-\frac{1}{2\hbar}(\alpha q^2+\alpha^{-1} p^2)\right]
T({q},{p})\ket{\psi}\bra{\psi}T^\dagger({q},{p}),
\label{eq:rhop}$$ where $T({q},{p})$ are unitary phase space translation operators, and $\alpha$ is an arbitrary (squeezing) parameter. A random pure state $\ket{\psi}$ has a Wigner function with random sub-Planck structures with phase space correlation length $l_{\rm corr} \sim \hbar$ which is semi-classically smaller than the coarse-graining width $\sim \hbar^{1/2}$, so $\rho$ in Eq. (\[eq:rhop\]) can be again considered as a mixture (Eq. (\[eq:rho\])) of $m$ random pure states with $m \sim \hbar^{-1/2}$.
Unknown random states
---------------------
In this section is is assumed that the random state $\ket{\psi}$ whose entanglement we would like to detect is unknown so that we are not able to use an optimal entanglement witness $W$ for a particular $\ket{\psi}$. The best one can do is to choose some fixed witness $W$ in advance, independently of the state. Since I am interested in the average behavior over unitary invariant ensemble of pure random states, $W$ can be chosen to be random as well. That is, in the present section I am going to study detection of entanglement with a random entanglement witness, whose precise definition will be given later. What I want to calculate is the distribution of the expectation values $\bracket{\psi}{W}{\psi}$ for a fixed $W$ and an ensemble of random pure states $\ket{\psi}$. Averaging over random states $\ket{\psi}$ we see that the average expectation value $\overline{\bracket{\psi}{W}{\psi}}$ is $$\int d{\cal P}
\bracket{\psi}{W}{\psi}={\rm Tr}(W)/N,$$ where $\overline{\bullet} = \int d{\cal P}\bullet$ denotes an integration over the $U(N)$-invariant (Haar) distribution of pure states $\ket{\psi}$, and I used the fact that for a random state $\ket{\psi}=\sum_i c_i \ket{i}$ we have $\overline{c_i c_j^*}=\delta_{ij}/N$. Let us fix normalization of the entanglement witness $W$ such that ${\rm Tr}(W)=1$. Therefore, the average expectation value $\overline{\bracket{\psi}{W}{\psi}}$ scales $\propto 1/N$. It is therefore convenient to define the rescaled quantity $w=N \bracket{\psi}{W}{\psi}$ such that $\overline{w}=1$, independently of the dimension $N$.
Here and in the following, the investigation is limited to decomposable entanglement witnesses (see Appendix \[sec:witness\]) of the form $W=Q^{\rm T_B}$, with $Q$ positive semidefinite operator.
I first consider the case when $Q$ is a simple rank one projector, that is, $W$ is given by $W=(\ket{\phi}\bra{\phi})^{\rm T_B}$. If $\ket{\phi}$ is a state with a large Schmidt number $r\sim \sqrt{N}$, as it is typical for random $\ket{\phi}$ (I consider equal size subsystems, $N_A=N_B=\sqrt{N}$), then one can show [@slovenia] that the probability density $p(w)=d{\cal P}/dw$ converges to a Gaussian in the limit $N\to \infty$, $$p(w)=\frac{1}{\sqrt{2\pi}}\exp{(-(w-1)^2/2)}.
\label{eq:gauss}$$ Numerical results for finite $N=2^{10}$ are shown in Fig. \[fig:gauss\] (top). The probability of measuring negative $w$, i.e., of detecting entanglement, is $\int_{-\infty}^0{p(w)dw}$ and therefore $${\cal P}(w<0)=(1-{\rm erf}(1/\sqrt{2}))/2 \approx 0.159.$$ Note that this entanglement detection probability is independent of the details of $\ket{\phi}$, provided that its Schmidt number $r$ is large, more precisely $r\propto N$.
\
Since it appears difficult to measures witness operators corresponding to states $\ket{\phi}$ with large Schmidt number $r$, it is interesting to consider the opposite limit of small $r$. In particular, let us consider the extreme case of rank $r=2$. We therefore have only two nonzero terms in the Schmidt decomposition (\[schmidtdec\]), corresponding to the nonzero eigenvalues, $p_1=\lambda$ and $p_2=1-\lambda$, of the reduced density matrix $\rho_A={\rm Tr}_{B}\big(|\phi\rangle\langle\phi|\big)$. In this case, we obtain [@slovenia] $$p(w)=
\cases{
\frac{1}{(1-2\lambda)^2} \left\{\lambda {\rm e}^{-\frac{w}{\lambda}}
+ (1-\lambda){\rm e}^{-\frac{w}{1-\lambda}} \right\}
+ \frac{1}{4\sqrt{\lambda(1-\lambda)}-2}{\rm e}^{-\frac{w}{
\sqrt{\lambda(1-\lambda)}}} & : $w>0$, \cr
\frac{1}{4\sqrt{\lambda(1-\lambda)}+2}{\rm e}^{\frac{w}{
\sqrt{\lambda(1-\lambda)}}} & : $w<0$.
}
\label{eq:pw_sch}$$ Results of numerical simulation for $p(w)$ for two cases, $\lambda=1/2$ and $\lambda=1/26$, are compared in in Fig. \[fig:gauss\] (bottom). The probability of detecting entanglement, i.e., of measuring negative values of $w$ is $${\cal P}(w<0)=1/(4+2/\sqrt{\lambda(1-\lambda)}).$$ For instance, ${\cal P}(w<0)=1/8=0.125$ when $\lambda=1/2$ and ${\cal P}(w<0)=5/72\approx 0.07$ when $\lambda=1/26$. Note that in the limit $\lambda \to 0$, i.e., of a pure separable state for $\ket{\phi}$, $w$ is always positive with an exponential distribution.
I emphasize that, neglecting problems related to finite measurement resolution (an issue that will be discussed in Sec. \[sec:knownrandomstates\]), the entanglement detection probability is, for pure states, independent of the system size $N$.
So far I have discussed only the case when $Q$ is of rank one, $Q=\ket{\phi}\bra{\phi}$. In general, one can consider $Q$ of rank $k$: $Q=\sum_i^k d_i \ket{\phi_i}\bra{\phi_i}$. Since I have fixed ${\rm Tr}(W)=1$, then $\sum_i d_i=1$. Assuming for simplicity that all $d_i$ are the same, $d_i=1/k$, we obtain [@slovenia], in the limit $N\to\infty$, $$p(w)=\sqrt{\frac{k}{2\pi}} {\rm e}^{-k(w-1)^2/2}.
\label{eq:pwr}$$ Therefore, the entanglement detection for pure states is more efficient if one consider $W=Q^{T_B}$, with $Q$ rank-one projector.
It is also interesting to consider the case in which $Q$ is of rank $1$ but we wish to detect the entanglement of mixed states, for instance of a mixture of $m$ pure random states, as given in Eq. (\[eq:rho\]). In this case, the resulting distribution $p(w)$ is, in the limit $N\to\infty$, a Gaussian of variance $1/m$ [@slovenia]. Because the distribution $p(w)$ becomes narrowly peaked about its mean $\overline{w}=1$ with increasing $m$, the probability of measuring negative values decreases with $m$, that is, we obtain $${\cal P}(w<0)=\frac{1-{\rm erf}(\sqrt{m/2})}{2}\asymp
\frac{1}{\sqrt{2\pi m}} {\rm e}^{-m/2}.
\label{eq:pm}$$ This probability decays to zero exponentially with $m$. Therefore, the detection of entanglement for a mixture of random states is very hard. This result outlines the importance of coarse graining to explain the emergence of classicality. For random pure states, a finite success probability in the detection of entanglement exists also in the limit in which the Hilbert space dimension $N\to\infty$. This implies that chaotic dynamics alone is not sufficient to erase any trace of entanglement when going to the classical limit, provided that ideal measurements are possible. On the other hand such erasure becomes very efficient when coarse graining is taken into account, for instance when mixtures instead of pure states are considered.
Known random states {#sec:knownrandomstates}
-------------------
In this section it is assumed that the random state $\ket{\psi}$ whose entanglement we want to measure is known in advance and furthermore, that we are able to prepare an arbitrary decomposable entanglement witness. In addition, we have to assume that our state $\ket{\psi}$ is neither separable, nor bound entangled (see Appendix \[app:separability\]), which is true with probability which converges to one exponentially in $N$. Therefore, for each $\ket{\psi}$ we can prepare an optimal entanglement witness, such that its expectation value will be minimal. As far as decomposable entanglement witnesses are concerned, the optimal choice of $W=W_{\rm opt}$ is to take for $Q$ a projector to the eigenspace corresponding to the minimal (negative) eigenvalue $\lambda_{\rm min}$ of $\rho^{\rm T_B}$, $W_{\rm opt}=(\ket{\phi_{\rm min}}\bra{\phi_{\rm min}})^{\rm T_B}$. The maximal violation of positivity is therefore $${\rm Tr}{(W_{\rm opt}\rho)}=-|\lambda_{\rm min}(\rho^{\rm T_B})|.
\label{eq:Wopt}$$ If we are able to measure the entanglement witness $W_{\rm opt}$ with a given precision it is the size of $\lambda_{\rm min}$ which determines the difficulty of detecting entanglement in $\ket{\psi}$. Note that the optimal entanglement witness $W_{\rm opt}$ depends on the state $\ket{\psi}$. For each state $\ket{\psi}$ we have to pick a different $W_{\rm opt}$.
The expectation value of the minimal eigenvalue equals $\overline{\lambda}_{\rm min}=-4/\sqrt{N}$ [@slovenia]. In fact, the distribution of $\lambda_{\rm min}$ becomes strongly peaked around $-4/\sqrt{N}$ with diminishing relative fluctuations as $N\to\infty$. When we mix several independent (in general non-orthogonal) random vectors, $\rho=\sum_i^m \ket{\psi_i}\bra{\psi_i}/m$, the minimal eigenvalue $\lambda_{\rm min}$ increases and the distribution becomes increasingly sharply peaked (for $m \to \infty$ we get $\rho \to \mathbbm{1}/N$ with all eigenvalues being equal to $1/N$). Note that the average minimal eigenvalue $\bar{\lambda}_{\rm min}$ is positive for $m > m^*$, with $m^* \approx 4 N$.
Although von Neumann entropy of a random state is large all eigenvalues of $\rho^{\rm T_B}$ are very small and will therefore be hard to detect. If we assume that we are able to measure values of $|{\rm Tr}{(W\rho)}|
< \epsilon$ then we can, depending on the scaling of $\epsilon$ with $N$, tell for which values of $m$ the detection of entanglement is possible. If $\epsilon$ does not depend on $N$, i.e., precision does not increase with $N$, then for sufficiently large $N$, such that $4/\sqrt{N} <\epsilon$, detection of entanglement will be impossible. Already a single random state becomes from the viewpoint of entanglement detection “classical”, since measuring a negative expectation value of its optimal entanglement witness is below the detection limit. If on the other hand we are able to measure $\epsilon$ which decreases as $1/\sqrt{N}$, the critical $m_{\rm crit}$, beyond which the entanglement detection is impossible, will be independent of $N$, i.e., in the limit $N\to\infty$ the ratio $m_{\rm crit}/\sqrt{N} \to 0$. If however we are able to detect very small expectation values of order $1/N$, then $m_{\rm crit}$ will be proportional to $N$. Furthermore, even with arbitrary accuracy, detection of entanglement with decomposable entanglement witnesses is in practice impossible beyond $m=m^* \propto N$ (that is, when $\bar{\lambda}_{\rm min}$ becomes positive).
I would like to stress once more than the detection difficulties are a consequence of the complexity of random states. If instead one considers “regular states” such as the GHZ state [@GHZstate] $|{\rm GHZ}\rangle=\frac{1}{\sqrt{2}}
(|0...0\rangle+|1...1\rangle)$, then the optimal witness is $W_{\rm opt}=(|\phi_{\rm min}\rangle\langle\phi_{\rm min}|)^{\rm T_B}$, with $|\phi_{\rm min}\rangle=
\frac{1}{\sqrt{2}}(|0...0\rangle|1...1\rangle
-|1...1\rangle|0...0\rangle)$ which corresponds to the minimal eigenvalue $\lambda_{\rm min}=-1/2$ of $(|{\rm GHZ}\rangle
\langle{\rm GHZ}|)^{\rm T_B}$. Since the value of $\lambda_{\rm min}$ is $-1/2$ instead of $-4/\sqrt{N}$ as typical for a random state, it turns out that it will be much easier to detect entanglement in a “regular” rather than in a random state. This happens in spite of the fact that the entanglement content is much larger in a random than in such a regular state.
Chaotic environments {#sec:chaoticenvironments}
====================
Real physical systems are never isolated and the coupling of the system to the environment leads to decoherence. This process can be understood as the loss of quantum information, initially present in the state of the system, when non-classical correlations (entanglement) establish between the system and the environment. On the other hand, when tracing over the environmental degrees of freedom, we expect that the entanglement between internal degrees of freedom of the system is reduced or even destroyed. Decoherence theory has a fundamental interest, since it provides explanations of the emergence of classicality in a world governed by the laws of quantum mechanics [@zurek]. Moreover, it is a threat to the actual implementation of any quantum computation and communication protocol [@qcbook; @nielsen]. Indeed, decoherence invalidates the quantum superposition principle, which is at the heart of the power of quantum algorithms. A deeper understanding of the decoherence phenomenon is essential to develop quantum technologies.
The environment is usually described as a many-body quantum system. The best-known model is the Caldeira-Leggett model [@caldeiraleggett; @ingold; @weiss], in which the environment is a bosonic bath consisting of infinitely many harmonic oscillators at thermal equilibrium. More recently, first studies of the role played by chaotic dynamics [@park; @blume03; @lee; @saraceno06; @rossini06; @seligman] or random environments [@Pineda2007; @Petruccione2007] in the decoherence process have been carried out.
In the following, it is shown that the many-body environment may be substituted with a closed deterministic system with a small number of degrees of freedom, but chaotic [@rossini06]. In other words, the complexity of the environment arise not from being many-body but from having chaotic dynamics. I consider two qubits coupled to a *single particle, fully deterministic, conservative chaotic “environment”*, described by the kicked rotator model. It is shown that, due to the system-environment interaction, the entropy of the system increases. At the same time, the entanglement between the two qubits decays, thus illustrating the loss of quantum coherence. The evolution in time of the two-qubit entanglement is in good agreement with the evolution obtained in a pure dephasing stochastic model. Since this pure dephasing decoherence mechanism can be derived in the framework of the Caldeira-Leggett model [@palma], a direct link between the effects of a many-body environment and of a chaotic single-particle environment is established.
Let us consider two qubits coupled to a quantum kicked rotator. The overall system is governed by the Hamiltonian $$\hat{H} = \hat{H}^{(1)} + \hat{H}^{(2)} +
\hat{H}^{(\mathrm{kr})} + \hat{H}^{(\mathrm{int})} \, ,
\label{eq:hammodel}$$ where $\hat{H}^{(i)} = \omega_i \, \hat{\sigma}_x^{(i)}$ ($i=1,2$) describes the free evolution of the two qubits, $$\hat{H}^{(\mathrm{kr})} = \frac{\hat{n}^2}{2} + k \cos(\hat{\theta})
\sum_j \delta(\tau-jT)
\label{eq:krotator}$$ the quantum kicked rotator, and $$\hat{H}^{(\mathrm{int})} = \epsilon \: ( \hat{\sigma}_z^{(1)} +
\hat{\sigma}_z^{(2)}) \cos(\hat{\theta}) \sum_j \delta(\tau-jT)$$ the interaction between the qubits and the kicked rotator; as usual, $\hat{\sigma}_\alpha^{(i)}$ ($\alpha = x,y,z$) denote the Pauli operators for the $i$-th qubit. Both the cosine potential in $\hat{H}^{(\mathrm{kr})}$ and the interaction $\hat{H}^{(\mathrm{int})}$ are switched on and off instantaneously (kicks) at regular time intervals $T$. Let us consider the two qubits as an open quantum system and the kicked rotator as their *common* environment. Note that I chose non-interacting qubits as I want their entanglement to be affected exclusively by the coupling to the environment [^11]
The kicked rotator, as the sawtooth map described in detail in Appendix \[sec:sawmap\], belongs to the class of periodically driven systems of Eq. (\[sawham\]), with the external driving described by the potential $V(\theta)=k \cos\theta$, switched on and off instantaneously at time intervals $T$. The evolution from time $tT^-$ (prior to the $t$-th kick) to time $(t+1)T^-$ (prior to the $(t+1)$-th kick) of the kicked rotator in the classical limit is described by the Chirikov standard map: $$\left\{
\begin{array}{l}
n_{t+1} = n_t + k \sin \theta_t, \\
\theta_{t+1} = \theta_t + T n_{t+1},
\end{array}
\right.
\label{eq:chirikov}$$ where $(n,\theta)$ are conjugated momentum-angle variables and $t=\tau/T$ denotes the discrete time, measured in number of kicks. As for the sawtooth map, by rescaling $n \to p = T n$, the dynamics of Eq. (\[eq:chirikov\]) is seen to depend only on the parameter $K = kT$. For $K=0$ the motion is integrable; when $K$ increases, a transition to chaos of the Kolmogorov-Arnold-Moser (KAM) type is observed [@lichtenberg; @arnold]. [^12] The last invariant KAM torus is broken for $K \approx 0.97$. If $K \sim 1$ the phase space is mixed (simultaneous presence of integrable and chaotic components). If $K$ increases further, the stability islands progressively reduce their size; for $K \gg 1$ they are not visible any more. In what follows, I always consider map (\[eq:chirikov\]) on the torus $0 \leq \theta < 2 \pi$, $- \pi \leq p < \pi$. In this case, the Chirikov standard map describes the stroboscopic dynamics of a *conservative* dynamical system with two degrees of freedom which, in the fully chaotic regime $K\gg 1$, relaxes, apart from quantum fluctuations, to the uniform distribution on the torus.
The Hilbert space of the global system is given by $$\mathcal{H} = \mathcal{H}^{(1)} \otimes \mathcal{H}^{(2)} \otimes
\mathcal{H}^{(kr)} \, ,$$ where $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ are the two-dimensional Hilbert spaces associated to the two qubits, and $\mathcal{H}^{(kr)}$ is the Hilbert space for the kicked rotator with $N$ quantum levels.
The time evolution generated by Hamiltonian (\[eq:hammodel\]) in one kick is described by the operator $$\begin{array}{rl}
\hat{U} = & \exp \big[ -i \big( k + \epsilon (\hat{\sigma}_z^{(1)} +
\hat{\sigma}_z^{(2)}) \big) \cos(\hat{\theta}) \big] \\
& \times\, \exp \big[ -i T \frac{\hat{n}^2}{2} \big]
\exp \big( -i \, \delta_1 \, \hat{\sigma}_x^{(1)} \big)
\exp \big( -i \, \delta_2 \, \hat{\sigma}_x^{(2)} \big).
\end{array} \label{eq:kickedevol}$$ The effective Planck constant is $\hbar_{\rm eff}=T = 2 \pi /N$; $\delta_1 = \omega_1 T , \: \delta_2 = \omega_2 T$; $\epsilon$ is the coupling strength between the qubits and the environment. The classical limit $\hbar_{\rm eff} \to 0$ is obtained by taking $T \to 0$ and $k \to \infty$, in such a way that $K = kT$ is kept fixed.
I am interested in the case in which the environment (the kicked rotator) is chaotic (that is, with $K \gg 1$). The two qubits are initially prepared in a maximally entangled state, so that they are disentangled from the environment. Namely, I suppose that at $t=0$ the system is in the state $$\ket{\Psi_0} = \ket{\phi^+} \otimes \ket{\psi_0} \, ,
\label{eq:initial}$$ where $\ket{\phi^+} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right)$ is a Bell state (the particular choice of the initial maximally entangled state is not crucial for what follows), and $\ket{\psi_0} = \sum_n c_n \ket{n}$ is a generic state of the kicked rotator, with $c_n$ random coefficients such that $\sum_n |c_n|^2 =1$, and $\ket{n}$ eigenstates of the momentum operator. The evolution in time of the global system (kicked rotator plus qubits) is described by the unitary operator $\hat{U}$ defined in Eq. (\[eq:kickedevol\]). Therefore, any initial pure state $\ket{\Psi_0}$ evolves into another pure state $\ket{\Psi (t)} = \hat{U}^t \ket{\Psi_0}$. The reduced density matrix $\rho_{12} (t)$ describing the two qubits at time $t$ is then obtained after tracing $\ket{\Psi (t)} \bra{\Psi (t)}$ over the kicked rotator’s degrees of freedom.
In the following I will focus my attention on the time evolution of the *entanglement of formation* (see Appendix \[sec:concurrence\]) $E_{12}$ between the two qubits and that between them and the kicked rotator, measured by the reduced von Neumann entropy $S_{12}=-\mathrm{Tr} \, [ \rho_{12}\log_{2} \rho_{12}]$ of the reduced density matrix $\rho_{12}$. Clearly, for states like the one in Eq. (\[eq:initial\]), we have $E_{12} (0) = 1$, $S_{12} (0) = 0$. As the total system evolves, we expect that $E_{12}$ decreases, while $S_{12}$ grows up, thus meaning that the two-qubit system is progressively losing coherence.
If the kicked rotator is in the chaotic regime and in the semiclassical region $\hbar_{\rm eff}\ll 1$, it is possible to drastically simplify the description of the system in Eq. (\[eq:hammodel\]) by using the *random phase-kick* approximation, in the framework of the Kraus representation formalism. Since, to a first approximation, the phases between two consecutive kicks in the chaotic regime can be considered as uncorrelated, the interaction with the environment can be simply modeled as a phase-kick rotating both qubits through the same random angle about the $z$-axis of the Bloch sphere. This rotation is described in the $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ basis by the unitary matrix $$\label{eq:rotation}
{R} (\theta) = \left[
\begin{array}{cc} e^{- i \epsilon \cos \theta} & 0 \\
0 & e^{i \epsilon \cos \theta} \end{array} \right] \otimes
\left[ \begin{array}{cc} e^{- i \epsilon \cos \theta} & 0 \\
0 & e^{i \epsilon \cos \theta} \end{array} \right],$$ where the angle $\theta$ is drawn from a uniform random distribution in $[ 0, 2\pi )$. The one-kick evolution of the reduced density matrix $\rho_{12}$ is then obtained after averaging over $\theta$: $$\bar{\rho}_{12} = \frac{1}{2\pi} \int_{0}^{2 \pi} \hspace{-2mm} d \theta \,
{R} (\theta) \, e^{-i \delta_2 {\sigma}_x^{(2)}}
e^{-i \delta_1 {\sigma}_x^{(1)}}
\,\rho_{12} \,
e^{i \delta_1 {\sigma}_x^{(1)}} e^{i \delta_2 {\sigma}_x^{(2)}}
{R}^{\dagger} (\theta). \label{eq:randomphase}$$
In order to assess the validity of the random phase-kick approximation, model (\[eq:hammodel\]) is numerically investigated in the classically chaotic regime $K\gg 1$ and in the region $\hbar_{\rm eff}\ll 1$ in which the environment is a semiclassical object. Under these conditions, we expect that the time evolution of the entanglement can be accurately predicted by the random phase model. Such expectation is confirmed by the numerical data shown in Fig. \[fig:confr\_ES\]. Even though differences between the two models remain at long times due to the finite number $N$ of levels in the kicked rotator, such differences appear at later and later times when $N\to \infty$ ($\hbar_{\rm eff}\to 0$). The parameter $K$ has been chosen much greater than one, so that the classical phase space of the kicked rotator can be considered as completely chaotic. Note that the value $K \approx 99.72676$ is chosen to completely wipe off memory effects between consecutive and next-consecutive kicks (see Ref. [@rossini06] for details).
![Reduced von Neumann entropy $S_{12}$ (main figure) and entanglement $E_{12}$ (inset) as a function of time at $K \approx 99.73$, $\delta_1=10^{-2}$, $\delta_2=\sqrt{2}\delta_1$, $\epsilon=8\times 10^{-3}$. The thin curves correspond to different number of levels for the environment (the kicked rotator) ($N=2^9,2^{10},2^{11},2^{12},2^{13},2^{14}$ from bottom to top in the main figure and vice versa in the inset). The thick curves give the numerical results from the random phase model (\[eq:randomphase\]).[]{data-label="fig:confr_ES"}](krotenv.eps){width="10.cm"}
I point out that the random phase model can be derived from the Caldeira-Leggett model with a pure dephasing coupling $\propto ( \hat{\sigma}_z^{(1)} + \hat{\sigma}_z^{(2)}) \sum_k g_k \hat{q}_k$, with $g_k$ coupling constant to the $k$-th oscillator of the environment, whose coordinate operator is $\hat{q}_k$ [@palma; @braun]. This establishes a direct link between the chaotic single-particle environment considered in this paper and a standard many-body environment.
Final remarks
=============
The role of entanglement as a resource in quantum information has stimulated intensive research aimed at unveiling both its qualitative and quantitative aspects. The interest is first of all motivated by experimental implementations of quantum information protocols. Decoherence, which can be considered as the ultimate obstacle in the way of actual implementation of any quantum computation or communication protocol, is due to the entanglement between the quantum hardware and the environment. The decoherence-control issue is expected to be particularly relevant when the state of the quantum system is *complex*, namely when it is characterized by a large amount of multipartite entanglement. It is therefore important, for applications but also in its own right, to scrutinize the robustness and the multipartite features of relevant classes of entangled states. In this context, random states play an important role, both for applications in quantum protocols and in view of a, highly desirable, statistical theory of entanglement.
Such studies have deep links with the physics of complex systems. In classical physics, a well defined notion of complexity, based on the exponential instability of chaos, exists, and has profound links with the notion of algorithmic complexity [@ford; @alekseev]: in terms of the symbolic dynamical description, almost all orbits are random and unpredictable. On the other hand, in spite of many efforts (see [@prosen2007] and references therein) the transfer of these concepts to quantum mechanics still remains elusive. However, there is strong numerical evidence that quantum motion is characterized by a greater degree of stability than classical motion (see [@arrow; @varenna05; @qcbook; @sokolov]). This has important consequences on the stability of quantum algorithms; for instance, the robustness of the multipartite entanglement generated by chaotic maps and discussed in Sec. \[sec:stabilitymultipartite\] is related to the power-law decay of the fidelity time scales for quantum algorithms [@georgeot2000; @thermalization; @benenti01; @frahm] which, in turn, is a consequence of the discreteness of the phase space in quantum mechanics [@arrow; @varenna05; @qcbook; @sokolov]. If we consider the chaotic classical motion (governed by the Liouville equation) of some phase-space density, smaller and smaller scales are explored exponentially fast. These fine details of the density distribution are rapidly lost under small perturbations. In quantum mechanics, there is a lower limit to this process, set by the size of the Planck cell, and this reduces the complexity of quantum motion as compared to classical motion.
Finally, the fundamental, purely quantum notion of entanglement is expected to play a crucial role in characterizing the complexity of a quantum system [@briegel2005]. I believe that studies of complexity and multipartite entanglement will shed some light on series of very important issues in quantum computation and in critical phenomena of quantum many-body condensed matter physics.
Separability criteria {#app:separability}
=====================
The Peres criterion {#app:peres}
-------------------
The Peres criterion [@peres96] provides a necessary condition for the existence of decomposition (\[sepdecomposition\]), in other words, a violation of this criterion is a sufficient condition for entanglement. This criterion is based on the *partial transpose* operation. Introducing an orthonormal basis $\{|i\rangle_{A}|\alpha\rangle_{B}\}$ in the Hilbert space $\mathcal{H}_{AB}$ associated with the bipartite system $A+B$, the density matrix $\rho_{AB}$ has matrix elements $(\rho_{AB})_{i\alpha;j\beta}
= {}_{A}\langle i|{}_B\langle\alpha|
\rho_{AB}|j\rangle_{A}|\beta\rangle_{B}$. The partial transpose density matrix is constructed by only taking the transpose in either the Latin or Greek indices (here Latin indices refer to Alice’s subsystem and Greek indices to Bob’s). For instance, the partial transpose with respect to Alice is given by $$\big(\rho^{T_A}_{AB}\big)_{i\alpha;j\beta} =
\big(\rho_{AB}\big)_{j\alpha;i\beta}.$$ Since a separable state $\rho_{AB}$ can always be written in the form (\[sepdecomposition\]) and the density matrices $\rho_{Ak}$ and $\rho_{Bk}$ have non-negative eigenvalues, then the overall density matrix $\rho_{AB}$ also has non-negative eigenvalues. The partial transpose of a separable state reads $$\rho^{T_A}_{AB} = \sum_k p_k \, \rho^T_{Ak} \otimes \rho_{Bk}.$$ Since the transpose matrices $\rho^T_{Ak}=\rho^\star_{Ak}$ are Hermitian non-negative matrices with unit trace, they are also legitimate density matrices for Alice. It follows that none of the eigenvalues of $\rho^{T_A}_{AB}$ is non-negative. This is a necessary condition for decomposition (\[sepdecomposition\]) to hold. It is then sufficient to have at least one negative eigenvalue of $\rho^{T_A}_{AB}$ to conclude that the state $\rho_{AB}$ is entangled.
It can be shown [@horodecki96] that for composite states of dimension $2\times2$ and $2\times3$, the Peres criterion provides a necessary and sufficient condition for separability; that is, the state $\rho_{AB}$ is separable if and only if $\rho^{T_A}_{AB}$ is non-negative. However, for higher dimensional systems, states exist for which all eigenvalues of the partial transpose are non-negative, but that are non-separable [@horodecki97]. These states are known as *bound entangled states* since they cannot be distilled by means of local operations and classical communication to form a maximally entangled state [@horodecki98].
I stress that the Peres criterion is more sensitive than Bell’s inequality for detecting quantum entanglement; that is, there are states detected as entangled by the Peres criterion that do not violate Bell’s inequalities ([@peres96]).
Entanglement witnesses {#sec:witness}
----------------------
A convenient way to detect entanglement is to use the so-called entanglement witnesses [@horodecki96; @terhal00]. By definition, an entanglement witness is a Hermitian operator $W$ such that ${\rm Tr}\big(W\rho^{(\rm sep)}_{AB}\big)\geq 0$ for all separable states $\rho^{(\rm sep)}_{AB}$ while there exists at least one state $\rho^{(\rm ent)}_{AB}$ such that ${\rm Tr}\big(W\rho^{(\rm ent)}_{AB})<0$. Therefore, the negative expectation value of $W$ is a signature of entanglement and the state $\rho^{(\rm ent)}_{AB}$ is said to be detected as entangled by the witness $W$.
The existence of entanglement witnesses is a consequence of the *Hahn-Banach theorem:* Given a convex and compact set $S$ and $\rho_{AB}\notin S$, there exists a hyperplane that separates $\rho_{AB}$ from $S$.
This fact is illustrated in Fig. \[fig:witness\]. The set $S$ of separable states is a subset of the set of all possible density matrices for a given system. The dashed line represents a hyperplane separating an entangle state $\rho_{AB}$ from $S$. The optimized witness $W_{\rm opt}$ (represented by a full line) is obtained after performing a parallel transport of the above hyperplane, so that it becomes tangent to the set of separable states. Therefore, the optimized witness $W_{\rm opt}$ detects more entangled states than before parallel transport. Note that, in order to fully characterize the set $S$ of separable states one should find all the witnesses tangent to $S$.
![Schematic drawing of entanglement witnesses.[]{data-label="fig:witness"}](witness){width="10.0cm"}
The concept of entanglement witness is close to experimental implementations and detection of entanglement by means of entanglement witnesses has been realized in several experiments [@Bour:04; @wineland; @Haffner:05]. The more negative expectation value of entanglement witness we find, the easier it is to detect entanglement of such a state. The expectation value of $W$ also provides lower bounds to various entanglement measures [@eisert:07; @Guhne:07]. Finally, it is interesting to note that violation of Bell’s inequalities can be rewritten in terms of non-optimal entanglement witnesses [@terhal00; @Hyllus:05].
In general, classification of entanglement witnesses is a hard problem. However, much simpler is the issue with the so-called decomposable entanglement witnesses [@Lewenstein:00]. By definition, a witness is called decomposable if $$W=P+Q^{\rm T_B}, \qquad P,Q\ge 0,
\label{eq:DEW}$$ that is, with positive semidefinite operators $P,Q$. Decomposable entanglement witnesses can only detect entangled states with at least one negative eigenvalues of $\rho^{\rm T_B}$. Therefore, similarly to the Peres criterion, decomposable witnesses do not detect bound entangled states. Note, however, that entanglement witnesses are closer to experimental implementations than the Peres criterion, as full tomographic knowledge about the state is not needed.
Purity of random states {#app:lubkin}
=======================
Let us write a $N$-level random state in the form $$|\psi\rangle =
%\sum_{k=0}^{N-1} c_k |k\rangle =
\sum_{k=0}^{N-1} r_k e^{i\phi_k} |k\rangle,$$ where $\phi_k$ are independent random variables uniformly distributed in $[0,2\pi)$ and ${\bf r}=(r_0,...,r_{N-1})$ is a random point uniformly distributed on the unit hypersphere $\mathbb{S}^{N-1}=
\{{\bf r}\in \mathbb{R}^N \vert {\bf r}^2=1\}$, with distribution function $$p({\bf r})=C_N \prod_{k=0}^{N-1} r_k \delta({\bf r}^2-1),$$ with $C_N$ normalization constant to be determined later.
Given a bipartition of the Hilbert space of the system into two parts, $A$ and $B$, with dimensions $N_A$ and $N_B$, the purity reads $$P=\sum_{j,j'=0}^{N_A-1} \sum_{l,l'=0}^{N_B-1}
r_{jl}r_{j'l}r_{j'l'}r_{jl'}
\exp[i(\phi_{jl}-\phi_{j'l}+\phi_{j'l'}-\phi_{jl'})],$$ where $|k\rangle=|jl\rangle=|j\rangle_A\otimes |l\rangle_B$. Following [@facchi], I split $P$ in two parts: $$P=X+M,$$ where $$X=\sum_{j,j'}{}^\prime \sum_{l,l'}{}^\prime
r_{jl}r_{j'l}r_{j'l'}r_{jl'}
\exp[i(\phi_{jl}-\phi_{j'l}+\phi_{j'l'}-\phi_{jl'})],
\label{eq:X}$$ $$M=\sum_{j,j'}{}^\prime\sum_l r_{jl}^2r_{j'l}^2
+\sum_j \sum_{l,l'}{}^\prime r_{jl}^2r_{jl'}^2
+\sum_{j,l} r_{jl}^4,
\label{eq:M}$$ where $\sum'$ means that equal indexes are banned in the sum. Since $\langle e^{i\phi_k} \rangle =0$ we obtain $\langle X \rangle =0$. Therefore, $$P=\langle M \rangle =
N(N_A+N_B-2) \langle r_0^2 r_1^2 \rangle +
N \langle r_0^4 \rangle,
\label{eq:PM}$$ where I have used $\langle r_k^4 \rangle =\langle r_0^4 \rangle$ for all $k$ and $\langle r_k^2 r_{k'}^2 \rangle =
\langle r_0^2 r_1^2 \rangle$ for all $k, k'$ with $k\ne k'$.
I now evaluate the marginal distribution $$p(r_0,...,r_{m-1})=C_N
r_0\cdots r_{m-1}
\int_0^1 dr_m r_m
\int_0^1 dr_{m+1} r_{m+1}
\cdots
\int_0^1 dr_{N-1} r_{N-1} \delta (r^2-1)$$ $$=\frac{C_N}{2}\,
r_0\cdots r_{m-1}
\int_0^{\sqrt{1-r_0^2-...-r_{m-1}^2}} dr_{m}r_{m}
\cdots
\int_0^{\sqrt{1-r_0^2-...-r_{N-3}^2}} dr_{N-2}r_{N-2}$$ $$=\frac{C_N}{2^{N-m}}\frac{1}{(N-m-1)!}\,
r_0\cdots r_{m-1}
\left(1-\sum_{j=0}^{m-1} r_j^2\right)^{N-m-1}.$$ In particular, $$p(r_0)=\frac{C_N}{2^{N-1}}\,\frac{1}{(N-2)!}\,
r_0(1-r_0^2)^{N-2}.$$ The normalization condition $\int_0^1 dr_0 p(r_0)=1$ allows us to determine $C_N=2^N(N-1)!$. Thus, we obtain $$p(r_0)=2(N-1)r_0(1-r_0^2)^{N-2},$$ $$\langle r_0^4 \rangle
=\int_0^1 dr_0 r_0^4 p(r_0) =\frac{2}{N(N+1)},
\label{eq:pr0}$$ $$p(r_0,r_1)=4(N-1)(N-2)r_0r_1(1-r_0^2-r_1^2)^{N-3},$$ $$\langle r_0^2 r_1^2 \rangle
=\int_0^1 dr_0 r_0^2 \int_0^1 dr_1 r_1^2
p(r_0,r_1) =\frac{1}{N(N+1)}.
\label{eq:pr0pr1}$$ After substitution of Eqs. (\[eq:pr0\]) and (\[eq:pr0pr1\]) into (\[eq:PM\]) we readily obtain Lubkin’s formula (\[eq:lubkin\]).
The variance $\sigma_P^2$ can be computed with the same technique as above [@facchi; @caves]. However, to obtain the variance (\[eq:variancelubkin\]) for large $N$ it is suffcient to replace in Eqs. (\[eq:X\]) and (\[eq:M\]) $r_k$ with its mean value $1/\sqrt{N}$: $$\sigma_P^2=\langle P^2 \rangle -P_L^2=
\langle X^2 \rangle +\langle M^2 \rangle -P_L^2
\approx \langle X^2 \rangle \approx \frac{2}{N^2}.$$ We can see from Eqs. (\[eq:X\]) and (\[eq:M\]) that $X$ and $M$ are sums of $O(N^2)$ terms of order $1/N^2$. Therefore, the central limit theorem implies that, for large $N$, the purity tends to a Gaussian distribution with mean $P_L$ and variance $\sigma_P$. Finally, I note that all moments of the purity have been recently computed [@giraud].
The sawtooth map {#sec:sawmap}
================
The sawtooth map is a prototype model in the studies of classical and quantum dynamical systems and exhibits a rich variety of interesting physical phenomena, from complete chaos to complete integrability, normal and anomalous diffusion, dynamical localization, and cantori localization. Furthermore, the sawtooth map gives a good approximation to the motion of a particle bouncing inside a stadium billiard (which is a well-known model of classical and quantum chaos).
Classical dynamics
------------------
The sawtooth map belongs to the class of periodically driven dynamical systems, governed by the Hamiltonian $$H(\theta,n;\tau) =
\frac{n^2}{2} + V(\theta)
\sum_{j=-\infty}^{+\infty} \delta(\tau-jT) \,,
\label{sawham}$$ where $(n,\theta)$ are conjugate action-angle variables ($0\leq\theta<2\pi$). This Hamiltonian is the sum of two terms, $H(\theta,n;\tau)=H_0(n)+U(\theta;\tau)$, where $H_0(n)=n^2\!/2$ is just the kinetic energy of a free rotator (a particle moving on a circle parametrized by the coordinate $\theta$), while $$U(\theta;\tau) =
V(\theta) \sum_j \delta(\tau-jT)
% - \frac{k \, (\theta - \pi)^2}{2} \, \sum_j \delta(\tau-jT)$$ represents a force acting on the particle that is switched on and off instantaneously at time intervals $T$. Therefore, we say that the dynamics described by Hamiltonian (\[sawham\]) is *kicked*. The corresponding Hamiltonian equations of motion are $$\left\{
\begin{array}{l}
\displaystyle
\dot{n} =
-\frac{\partial{H}}{\partial\theta} =
% k \, (\theta - \pi)
-\frac{d V(\theta)}{d \theta}
\sum_{j=-\infty}^{+\infty} \delta(\tau-jT) \,,
\\[2ex]
\displaystyle
\dot\theta =
\frac{\partial{H}}{\partial{n}} = n \,.
\end{array}
\right.$$ These equations can be easily integrated and one finds that the evolution from time $tT^-$ (prior to the $t$-th kick) to time $(t+1)T^-$ (prior to the $(t+1)$-th kick) is described by the map $$\left\{
\begin{array}{l}
\displaystyle
% {n}_{t+1} = n_t + k \, (\theta - \pi) \,,
{n}_{t+1} = n_t + F (\theta) \,,
\\[2ex]
\displaystyle
\theta_{t+1}= \theta_t + T{n}_{t+1} \,,
\end{array}
\right.
\label{sawmap}$$ where the discrete time $t=\tau/T$ measures the number of map iterations and $F(\theta)=-dV(\theta)/d\theta$ is the force acting on the particle.
In the following, we focus on the special case $V(\theta)=-k(\theta-\pi)^2/2$. This map is called the *sawtooth map*, since the force $F(\theta)=-dV(\theta)/d\theta=k(\theta-\pi)$ has a sawtooth shape, with a discontinuity at $\theta=0$. For such a discontinuous map the conditions of the Kolmogorov-Arnold-Moser (KAM) theorem are not satisfied and, for any $k\ne 0$, the motion is not bounded by KAM tori. By rescaling $n\to{I=Tn}$, the classical dynamics is seen to depend only on the parameter $K=kT$. Indeed, in terms of the variables $(I,\theta)$ map (\[sawmap\]) becomes $$\left\{
\begin{array}{l}
\displaystyle
{I}_{t+1} = I_t + K (\theta - \pi) \,,
\\[2ex]
\displaystyle
{\theta}_{t+1} = \theta_t + {I}_{t+1} \,.
\end{array}
\right.
\label{sawmap2}$$ The sawtooth map exhibits sensitive dependence on initial conditions, which is the distinctive feature of classical chaos: any small error is amplified exponentially in time. In other words, two nearby trajectories separate exponentially, with a rate given by the maximum Lyapunov exponent $\lambda$, defined as $$\lambda =
\lim_{|t|\to\infty} \frac1{t}
\ln\!\left( \frac{\delta_t}{\delta_0} \right) ,$$ where $\delta_t=\sqrt{[\delta I_t]^2+[\delta \theta_t]^2}$. To compute $\delta I_t$ and $\delta \theta_t$, we differentiate map (\[sawmap2\]), obtaining $$\left[
\begin{array}{c}
\delta {I}_{t+1} \\
\delta \theta_{t+1}
\end{array}
\right]
= M
\left[
\begin{array}{c}
\delta I_t \\
\delta\theta_t
\end{array}
\right]
=
\left[
\begin{array}{c@{\quad}c}
1 & K \\
1 & 1+K
\end{array}
\right]
\left[
\begin{array}{c}
\delta I_t \\
\delta\theta_t
\end{array}
\right] .
\label{tangmap}$$ The iteration of map (\[tangmap\]) gives $\delta I_t$ and $\delta \theta_t$ as a function of $\delta I_0$ and $\delta \theta_0$ ($\delta I_0$ and $\delta \theta_0$ represent a change of the initial conditions). The stability matrix $M$ has eigenvalues $\mu_{\pm}=\frac{1}{2}(2+K\pm\sqrt{K^2+4K})$, which do not depend on the coordinates $I$ and $\theta$ and are complex conjugate for $-4\leq{K}\leq0$ and real for $K<-4$ and $K>0$. Thus, the classical motion is stable for $-4\leq{K}\leq0$ and completely chaotic for $K<-4$ and $K>0$. For $K>0$, $\delta_t \propto (\mu_+)^t$ asymptotycally in $t$, and therefore the maximum Lyapunov exponent is $\lambda=\ln\mu_+$. Similarly, we obtain $\lambda=\ln|\mu_-|$ for $K<-4$. In the stable region $-4\le{K}\le0$, $\lambda=0$.
The sawtooth map can be studied on the cylinder \[$I\in(-\infty,+\infty)$\], or on a torus of sinite size ($-{\pi}L\le I < \pi L$, where $L$ is an integer, to assure that no discontinuities are introduced in the second equation of (\[sawmap2\]) when $I$ is taken modulus $2{\pi}L$). Although the sawtooth map is a deterministic system, for $K>0$ and $K<-4$ the motion along the momentum direction is in practice indistinguishable from a random walk. Thus, one has normal diffusion in the action (momentum) variable and the evolution of the distribution function $f(I,t)$ is governed by a Fokker–Planck equation: $$\frac{\partial{f}}{\partial{t}} =
\frac{\partial}{\partial{I}}
\left(\frac12 D \frac{\partial{f}}{\partial{I}} \right) .
\label{fokkerplanck}$$ The diffusion coefficient $D$ is defined by $$D = \lim_{t\to\infty} \frac{\langle(\Delta{I}_t)^2\rangle}{t} \,,$$ where $\Delta{I}\equiv{I}-\langle{I}\rangle$, and $\langle\dots\rangle$ denotes the average over an ensemble of trajectories. If at time $t=0$ we take a phase space distribution with initial momentum $I_0$ and random phases $0\leq\theta<2\pi$, then the solution of the Fokker–Planck equation (\[fokkerplanck\]) is given by $$f(I,t) =
\frac1{\sqrt{2 \pi D t}} \, \exp\!\left[ -\frac{(I-I_0)^2}{2Dt} \right] .$$ The width $\sqrt{\langle(\Delta I_t)^2\rangle}$ of this Gaussian distribution grows in time, according to $$\langle (\Delta{I}_t)^2 \rangle \approx
D(K) \, t \,.$$ For $K>1$, the diffusion coefficient is well approximated by the random phase approximation, in which we assume that there are no correlations between the angles (phases) $\theta$ at different times. Hence, we have $$D(K) \approx
\langle (\Delta{I}^{(1})^2 \rangle =
\frac1{2\pi}\int_0^{2\pi} d\theta (\Delta{I}^{(1)})^2 =
\frac1{2\pi}\int_0^{2\pi} d\theta K^2(\theta-\pi)^2 =
\frac{\pi^2}{3} K^2 ,$$ where $\Delta{I}^{(1)}=
{I}_{t+1}-I_t$ is the change in action after a single map step. For $0<K<1$ diffusion is slowed, due to the sticking of trajectories close to broken tori (known as cantori), and we have $D(K)\approx3.3\,K^{5/2}$ (this regime is discussed in Ref. [@percival]). For $-4<K<0$ the motion is stable, the phase space has a complex structure of elliptic islands down to smaller and smaller scales, and one can observe anomalous diffusion, that is, $\langle(\Delta{J})^2\rangle\propto{t}^\alpha$, with $\alpha\ne1$ (see Ref. [@benenti01]). The cases $K=-1,-2,-3$ are integrable.
Quantum dynamics
----------------
The quantum version of the sawtooth map is obtained by means of the usual quantization rules, $\theta\to\hat{\theta}$ and $n\to{}\hat{n}=-i\partial/\partial\theta$ (we set $\hbar=1$). The quantum evolution in one map iteration is described by a unitary operator $\hat{U}$, called the Floquet operator, acting on the wave vector $|\psi\rangle$: $$|\psi\rangle_{t+1} =
\hat{U}\,|\psi\rangle_t =
\exp\left[
-i \int_{lT^-}^{(l+1)T^-} d\tau H(\hat{\theta},\hat{I};\tau)
\right]
|\psi\rangle_t \,,
\label{sawq}$$ where $H$ is Hamiltonian (\[sawham\]). Since the potential $V(\theta)$ is switched on only at discrete times $lT$, it is straightforward to obtain $$|\psi\rangle_{t+1} =
e^{-i T \hat{n}^2\!/2} \, e^{-iV(\hat{\theta})} \, |\psi\rangle_t,
% e^{-i T \hat{n}^2\!/2} \, e^{ik(\hat{\theta} -\pi)^2\!/2}
%\, |\psi_t\rangle, \,.
\label{sawquantum}$$ which for the sawtooth map is just Eq. (\[eq:quantmap\]). It is important to emphasize that, while the classical sawtooth map depends only on the rescaled parameter $K=kT$, the corresponding quantum evolution (\[sawquantum\]) depends on $k$ and $T$ separately. The effective Planck constant is given by $\hbar_{\rm eff}=T$. Indeed, if we consider the operator $\hat{I}=T\hat{n}$ ($\hat{I}$ is the quantization of the classical rescaled action $I$), we have $$[\hat{\theta},\hat{I}]=T[\hat{\theta},\hat{n}]=i T =i \hbar_{\rm eff}.$$ The classical limit $\hbar_{\rm eff}\to 0$ is obtained by taking $k\to\infty$ and $T\to0$, while keeping $K=kT$ constant.
In the quantum sawtooth map model one can observe important physical phenomena like dynamical localization [@benenti03]. Indeed, due to quantum interference effects, the chaotic diffusion in momentum is suppressed, leading to exponentially localized wave functions. This phenomenon was first found and studied in the quantum kicked-rotator model [@izrailev] and has profound analogies with Anderson localization of electronic transport in disordered materials [@fishman]. Dynamical localization has been observed experimentally in the microwave ionization of Rydberg atoms [@koch] and in experiments with cold atoms [@raizen]. In the quantum sawtooth map also cantori localization takes place: In the vicinity of a broken KAM torus, a cantorus starts to act as a perfect barrier to quantum wave packet evolution, if the flux through it becomes less than $\hbar$ [@geisel; @mackay; @fausto; @prosen; @prange].
Quantum algorithm
-----------------
In the following, we describe an exponentially efficient quantum algorithm for simulation of the map (\[eq:quantmap\]) [@benenti01]. It is based on the forward/ backward quantum Fourier transform between momentum and angle bases. Such an approach is convenient since the Floquet operator $\hat{U}$, introduced in Eq. (\[eq:quantmap\]), is the product of two operators, $\hat{U}_k=e^{ik(\hat{\theta}-\pi)^2\!/2}$ and $\hat{U}_T=e^{-iT\hat{n}^2\!/2}$, diagonal in the $\theta$ and $n$ representations, respectively. This quantum algorithm requires the following steps for one map iteration:
- Apply $\hat{U}_k$ to the wave function $\psi(\theta)$. In order to decompose the operator $\hat{U}_k$ into one- and two-qubit gates, we first of all write $\theta$ in binary notation: $$\theta=2\pi\sum_{j=1}^{n_q} \alpha_j 2^{-j} \,,$$ with $\alpha_i\in \{ 0,1 \}$. Here $n_q$ is the number of qubits, so that the total number of levels in the quantum sawtooth map is $N=2^{n_q}$. From this expansion, we obtain $$(\theta - \pi)^2 =
4\pi^2 \sum_{j_1,j_2=1}^{n_q}
\left( \frac{\alpha_{j_1}}{2^{j_1}} -\frac1{2n} \right)
\left( \frac{\alpha_{j_2}}{2^{j_2}} -\frac1{2n} \right) .$$ This term can be put into the unitary operator $\hat{U}_k$, giving the decomposition $$e^{ik(\theta -\pi)^2\!/2} =
\prod_{j_1,j_2=1}^n
\exp\!\left[
i 2 \pi^2 k
\left(\frac{\alpha_{j_1}}{2^{j_1}} - \frac1{2n}\right)
\left(\frac{\alpha_{j_2}}{2^{j_2}} - \frac1{2n}\right)
\right] ,
\label{ukdec}$$ which is the product of $n_q^2$ two-qubit gates (controlled phase-shift gates), each acting non-trivially only on the qubits $j_1$ and $j_2$. In the computational basis $\{|\alpha_{j_1}\alpha_{j_2}\rangle =
|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ each two-qubit gate can be written as $\exp(i2\pi^2kD_{j_1,j_2})$, where $D_{j_1,j_2}$ is a diagonal matrix: $$D_{j_1,j_2} =
\left[
\begin{array}{cccc}
\frac1{4n^2} & 0 & 0 & 0 \\
0 & -\frac1{2n}\big(\frac1{2^{j_2}}-\frac1{2n}\big) & 0 & 0 \\
0 & 0 & -\frac1{2n}\big(\frac1{2^{j_1}}-\frac1{2n}\big) & 0 \\
0 & 0 & 0 & \big(\frac1{2^{j_1}}-\frac1{2n}\big)
\big(\frac1{2^{j_2}}-\frac1{2n}\big)
\end{array}
\right] .$$
- The change from the $\theta$ to the $n$ representation is obtained by means of the quantum Fourier transform, which requires $n_q$ (single-qubit) Hadamard gates and $\frac12 n_q(n_q-1)$ (two-qubit) controlled phase-shift gates (see, e.g., [@qcbook; @nielsen]).
- In the $n$ representation, the operator $\hat{U}_T$ has essentially the same form as the operator $\hat{U}_k$ in the $\theta$ representation and can therefore be decomposed into $n_q^2$ controlled phase-shift gates, similarly to Eq. (\[ukdec\]).
- Return to the initial $\theta$ representation by application of the inverse quantum Fourier transform.
Thus, overall, this quantum algorithm requires $3n_q^2+n_q$ gates per map iteration ($3n_q^2-n_q$ controlled phase-shifts and $2n_q$ Hadamard gates). This number is to be compared with the $O(n_q2^{n_q})$ operations required by a classical computer to simulate one map iteration by means of a fast Fourier transform. Thus, the quantum simulation of the quantum sawtooth map dynamics is exponentially faster than any known classical algorithm. Note that the resources required to the quantum computer to simulate the evolution of the sawtooth map are only logarithmic in the system size $N$. Of course, there remains the problem of extracting useful information from the quantum computer wave function. For a discussion of this problem, see Refs. [@qcbook; @georgeotvarenna]. Finally, I point out that the quantum sawtooth map has been recently implemented on a three-qubit nuclear magnetic resonance (NMR)-based quantum processor [@corysawtooth].
Entanglement of formation and concurrence {#sec:concurrence}
=========================================
Any state $\rho$ can be decomposed as a convex combination of projectors onto pure states: $$\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|.$$ The entanglement of formation $E$ is defined as the mean entanglement of the pure states forming $\rho$, minimized over all possible decompositions: $$E(\rho_S) = \inf_{\mathrm{dec}} \sum_j p_j E(|\psi_j\rangle),$$ where the (bipartite) entanglement of the pure states $|\psi_j\rangle$ is measured according to Eq. (\[entbipure\]).
The entanglement of formation $E_{12}$ of a generic two-qubit state $\rho_{12}$ can be evaluated in a closed form following Ref. [@wootters98]. First of all we compute the *concurrence*, defined as $C=\max ( \lambda_{1} - \lambda_{2} -
\lambda_{3} - \lambda_{4} , 0 )$, where the $\lambda_{i}$’s are the square roots of the eigenvalues of the matrix $R=\rho_{12} \tilde{\rho}_{12}$, in decreasing order. Here $\tilde{\rho}_{12}$ is the spin flipped matrix of $\rho_{12}$, and it is defined by $\tilde{\rho}_{12}=
(\sigma_{y} \otimes \sigma_{y}) \, \rho_{12}^{\star} \,
(\sigma_{y} \otimes \sigma_{y})$ (note that the complex conjugate is taken in the computational basis $\{ \vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle,
\vert 11 \rangle \}$). Once the concurrence has been computed, the entanglement of formation is obtained as $E= h((1+\sqrt{1-C^{2}})/2)$, where $h$ is the binary entropy function: $h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$, with $x=(1+\sqrt{1-C^{2}})/2$.
The concurrence is widely investigated in condensed matter physics, in relation to the general problem of the behavior of entanglement across quantum phase transitions [@fazioreview]. For studies of the relation between entanglement and integrability to chaos crossover in quantum spin chain, see [@simone; @monasterio; @viola1; @viola2; @viola3; @prosen2007], and references therein.
While working on the topics discussed in this review paper, I had the pleasure to collaborate with Dima Averin, Gabriel Carlo, Giulio Casati, Rosario Fazio, Giuseppe Gennaro, Jae Weon Lee, Carlos Mejía-Monasterio, Simone Montangero, Massimo Palma, Tomaž Prosen, Alessandro Romito, Davide Rossini, Dima Shepelyansky, Valentin Sokolov, Oleg Zhirov and Marko Žnidarič. I would like to express my gratitude to all of them.
[0]{} . . preprint arXiv:quant-ph/0702225v2. . . . , Vol. I: Basic concepts (World Scientific, Singapore, 2004); Vol. II: Basic tools and special topics (World Scientific, Singapore, 2007). (Cambridge University Press, Cambridge, 2000). . , . . . (Addison-Wesley, Reading, Massachusetts, 1994). (Cambridge University Press, Cambridge, 2006). ; ; ; . . . . . . . . . . . . . (2nd Ed.) (Springer-Verlag, 2000). (Cambridge University Press, Cambridge, 1999). . . . . . Proceedings of the “E. Fermi” Varenna School on , Varenna, Italy, 5-15 July 2005, edited by Casati G., Shepelyansky D.L., Zoller P. Benenti G. (IOS Press and SIF, Bologna, 2006); reprinted in . . . . . . . . . , . . . . . . . . . . A noisy gates model close to experimental implementations is discussed in , . . . . . . . . . . . (Wiley-VCH, Weinheim, 1998). (2nd Ed.) (World Scientific, Singapore, 1999). . . . . . . . . . . , . (2nd Ed.) (Springer-Verlag, 1992). (2nd Ed.) (Springer-Verlag, 1997). . , Phys. Today. April 1983, pag. 40. . . ; . preprint arXiv:0807.2902v1 \[nlin.CD\]. ; . . . . . . . . . . . . . . IN[Phys. Rev. A]{}[72]{}[2005]{}[012321]{}. . ; . . . ; for a review see, e.g., . . , , and references therein. ; ; . . . . . . Proceedings of the “E. Fermi” Varenna School on , Varenna, Italy, 5-15 July 2005, edited by Casati G., Shepelyansky D.L., Zoller P. Benenti G. (IOS Press and SIF, Bologna, 2006). . . , and references therein. . . . . .
[^1]: For example, if a system’s wave function $|\psi\rangle$ is an eigenstate of an operator $\hat{A}$, namely, $\hat{A} |\psi\rangle = a |\psi\rangle$, then the value $a$ of the observable $A$ is, using the EPR language, an element of physical reality.
[^2]: Following the theory of relativity, we say that two measurement events are causally disconnected if $(\Delta{x})^2>c^2(\Delta{t})^2$, where $\Delta{x}$ and $\Delta{t}$ are the space and time separations of the two events in some inertial reference frame and $c$ is the speed of light (the two events take place at space-time coordinates $(x_1,t_1)$ and $(x_2,t_2)$, respectively, and $\Delta{x}=x_2-x_1$, $\Delta{t}=t_2-t_1$).
[^3]: The Haar measure on the unitary group $U(N)$ is the unique measure on pure $N$-level states invariant under unitary transformations. It is a uniform, unbiased measure on pure states. For a single qubit ($N=2$), it can be simply visualized as a uniform distribution on the *Bloch sphere* [@qcbook; @nielsen]. The generic state of a qubit may be written as $$|\psi\rangle=\cos\frac{\theta}{2}|0\rangle +
e^{i\phi}\sin\frac{\theta}{2}|1\rangle,
\quad
0\leq\theta\leq\pi, \ 0\leq\phi<2\pi,
\label{sphere}$$ where the states of the *computational basis* $\{|0\rangle,|1\rangle\}$ are eigenstates of the Pauli operator $\hat{\sigma}_z$. The qubit’s state can be represented by a point on a sphere of unit radius, called the Bloch sphere. This sphere is parametrized by the angles $\theta,\phi$ and can be embedded in a three-dimensional space of Cartesian coordinates $(x=\cos\phi\sin\theta,y=\sin\phi\sin\theta,z=\cos\theta)$.\
We also point out that ensembles of *random mixed states* are reviewed in [@karol].
[^4]: A basis of (maximally) entangled states for the two-qubit Hilbert space is provided by the four Bell states (EPR pairs) $$|\phi^\pm\rangle=\frac{1}{\sqrt{2}}\,\big(|00\rangle\pm|11\rangle\big),
\quad
|\psi^\pm\rangle=\frac{1}{\sqrt{2}}\,\big(|01\rangle\pm|10\rangle\big).$$
[^5]: The fidelity $F$ provides a measure of the distance between two, generally mixed, quantum states $\rho$ and $\sigma$: $$F(\rho,\sigma) = \left({\rm Tr} \sqrt{\rho^{1/2}\sigma\rho^{1/2}}\right)^2.$$ The fidelity of a pure state $|\psi\rangle$ and an arbitrary state $\sigma$ is given by $$F(|\psi\rangle,\sigma)={\langle \psi | \sigma | \psi \rangle},$$ which is the square root of the overlap between $|\psi\rangle$ and $\sigma$. Finally, the fidelity of two pure quantum states $|\psi_1\rangle$ and $|\psi_2\rangle$ is defined by $$F(|\psi_1\rangle,|\psi_2\rangle)=
|\langle\psi_1|\psi_2\rangle|^2.$$ We have $0\leq{}F\leq{}1$, with $F=1$ when $|\psi_1\rangle$ coincides with $|\psi_2\rangle$ and $F=0$ when $|\psi_1\rangle$ and $|\psi_2\rangle$ are orthogonal. For further discussions on this quantity see, e.g., [@qcbook; @nielsen]. The average fidelity between random states is studied in [@karolsommers].
[^6]: If we write $p_i=\frac{1+\epsilon_i}{N_A}$, with $\epsilon_i\ll 1$ and $\sum_i \epsilon_i=0$, we obtain $$S(\rho_A)\approx \log N_A -[N_A/(2\ln 2)]P(\rho_A).$$
[^7]: By definition, the ${\rm{CNOT}}^{(i,j)}$ gate acts on the states $\{|xy\rangle\equiv |x\rangle_i\otimes|y\rangle_j=
|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ of the two-qubit computational basis as follows: ${\rm{CNOT}}^{(i,j)}$ turns $|00\rangle$ into $|00\rangle$, $|01\rangle$ into $|01\rangle$, $|10\rangle$ into $|11\rangle$, and $|11\rangle$ into $|10\rangle$. The CNOT (controlled-NOT) gate flips the state of the second (target) qubit if the first (control) qubit is in the state $|1\rangle$ and does nothing if the first qubit is in the state $|0\rangle$. In short, ${\rm{CNOT}^{(i,j)}}|x\rangle_i |y\rangle_j=
|x\rangle_i |x\oplus y\rangle_j$, with $x,y\in\{0,1\}$ and $\oplus$ indicating addition modulo $2$. By definition, given input bits $x,y$, the ${\rm XOR}$ gate outputs $i\oplus j$. Therefore, the (quantum) ${\rm CNOT}$ gate acts on the states of the computational basis as the (classical) ${\rm XOR}$ gate. However, the ${\rm CNOT}$ gate, in contrast to the ${\rm XOR}$ gate, can also be applied to any superposition of the computational basis states. The CNOT gate is the prototypical two-qubit gate that is able to generate entanglement. For instance, CNOT maps the separable state $|\psi\rangle=\frac{1}{\sqrt{2}}
(|0\rangle+|1\rangle)|0\rangle$ onto the maximally entangled (Bell) state $|\phi^+\rangle={\rm CNOT}|\psi\rangle=
\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. Any unitary operation in the Hilbert space of $n_q$ qubits can be decomposed into (elementary) one-qubit and two-qubit CNOT gates.
[^8]: The problem of finding, for a generic number $n_q$ of qubits, maximally multipartite entangled states, that is, pure states for which the entanglement is maximal for each bipartition, is discussed in [@parisi].
[^9]: For a $n_q$-qubit GHZ state, $|{\rm GHZ}\rangle=\frac{1}{\sqrt{2}}
(|0...0\rangle+|1...1\rangle)$, the distribution $p(E_{AB})$ is peaked but at a small and essentially $n_q$-independent value, while for the cluster states [@briegel] the average entanglement $\langle E_{AB} \rangle$ is large and increases with $n_q$ but also the variance is large [@facchi].
[^10]: Of course, this remark does not call into question the utility of high dimensional random states for quantum information processing.
[^11]: As discussed in [@lee], the chaotic environment model discussed in this section could be implemented, at least in principle, using cold atoms in a pulsed optical lattice created by laser fields [@darcy] or superconducting nanocircuits [@romito].
[^12]: The property of complete integrability is very delicate and atypical as it is, in general, destroyed by an arbitrarily weak perturbation that converts a completely integrable system into a KAM-integrable system. The structure of KAM motion is very intricate: the motion is confined to invariant tori for most initial conditions yet a single, connected, chaotic motion component (for more than two degrees of freedom) of exponentially small measure (with respect to the perturbation) arises, which is nevertheless everywhere dense.
|
---
abstract: 'We present results of the updated `SuperChic 3` Monte Carlo event generator for central exclusive production. This extends the previous treatment of proton–proton collisions to include heavy ion (pA and AA) beams, for both photon and QCD–initiated production, the first time such a unified treatment of exclusive processes has been presented in a single generator. To achieve this we have developed a theory of the gap survival factor in heavy ion collisions, which allows us to derive some straightforward results about the $A$ scaling of the corresponding cross sections. We compare against the recent ATLAS and CMS measurements of light–by–light scattering at the LHC, in lead–lead collisions. We find that the background from QCD–initiated production is expected to be very small, in contrast to some earlier estimates. We also present results from new photon–initiated processes that can now be generated, namely the production of axion–like particles, monopole pairs and monopolium, top quark pair production, and the inclusion of $W$ loops in light–by–light scattering.'
bibliography:
- 'references.bib'
---
IPPP/18/90
[**Exclusive LHC physics with heavy ions: SuperChic 3**]{}\
L.A. Harland–Lang$^{1}$, V.A. Khoze$^{2,3}$, M.G. Ryskin$^{3}$\
${}^1$Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom.\
${}^2$Institute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE\
${}^3$Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg, 188300, Russia
Introduction
============
Central Exclusive Production (CEP) is the reaction
hh h+ X+h where ‘+’ signs are used to denote the presence of large rapidity gaps, separating the system $X$ from the intact outgoing hadrons $h$. This simple signal is associated with a broad and varied phenomenology, from low energy QCD to high energy BSM physics, see [@Albrow:2008pn; @Albrow:2010yb; @Tasevsky:2014cpa; @Harland-Lang:2014lxa; @Harland-Lang:2014dta; @N.Cartiglia:2015gve] for reviews. Consequently an extensive experimental programme is planned and ongoing at the LHC, with dedicated proton tagging detectors installed and collecting data in association with both ATLAS and CMS [@AFP1; @Albrow:1753795], while multiple measurements using rapidity gap vetoes have been made by LHCb and ALICE.
CEP may proceed via either QCD or photon–induced interactions, see Fig. \[fig:pCp\], as well as through a combination of both, namely via photoproduction. Although producing the same basic exclusive signal, each mechanism is distinct in terms of the theoretical framework underpinning it and the phenomenology resulting from it. The QCD–initiated mechanism benefits from a ‘$J^{PC}=0^{++}$’ selection rule, permitting the production of a range of strongly interacting states in a precisely defined gluon–rich environment, while also providing a non–trivial test of QCD in a distinct regime from standard inclusive production. The framework for describing photon–initiated production is under very good theoretical control, such that one can in effect use the LHC as a photon–photon collider; this well understood QED initial state provides unique sensitivity to beyond the Standard Model (BSM) effects. Photoproduction can for example provide a probe of low $x$ QCD effects such as gluon saturation in both proton and nuclear targets. For further information and reviews, see [@Khoze:2001xm; @Albrow:2010yb; @Tasevsky:2014cpa; @Harland-Lang:2014lxa; @Harland-Lang:2014dta].
As mentioned above, a range of measurements have been made and are ongoing at the LHC. To support this experimental programme, it is essential to provide Monte Carlo (MC) tools to connect the theoretical predictions for CEP with the experimental measurements. For this reason the authors have previously produced the publicly available `SuperChic` MC [@HarlandLang:2009qe; @HarlandLang:2010ep], subsequently upgraded to version 2 in [@Harland-Lang:2015cta]. This generates a wide range of QCD and photon–initiated processes in $pp$ collisions, with the former calculated using the perturbative ‘Durham’ approach. In addition, this includes a fully differential treatment of the soft survival factor, that is the probability of no additional soft particle production, which would spoil the exclusivity of the event.
Other available MC implementations include: FPMC [@Boonekamp:2011ky], which generates a smaller selection of final–states and does not include a differential treatment of survival effects, although it also generates more inclusive diffractive processes, beyond pure CEP; an implementation of CEP in `Pythia` described in [@Lonnblad:2016hun], which provides a full treatment of initial–state showering effects for a small selection of processes, allowing both pure CEP and semi–exclusive production to be treated on the same footing, while the survival factor is included via the standard `Pythia` treatment of multi-particle interactions (MPI); the `Starlight` MC [@Klein:2016yzr] generates a range of photon–initiated and photoproduction processes in heavy ion collisions; `ExHuME` [@Monk:2005ji], for QCD–initiated production of a small selection of processes; `CepGen` [@Forthomme:2018ecc], which considers photon–initiated production but aims to allow the user to add in arbitrary processes; for lower mass QCD–initiated production, the `Dime` [@Harland-Lang:2013dia], `ExDiff` [@Ryutin:2017qii] and `GenEx` [@Kycia:2017ota] MCs.
As discussed above, the `SuperChic` MC aims to provide a treatment of all mechanisms for CEP, both QCD and photon initiated, within a unified framework. However, so far it has only considered the case of proton–proton (or proton–antiproton) collisions; CEP with heavy ion (pA and AA) beams, so–called ‘ultra–peripheral’ collisions (UPCs), have not been included at all. Such processes are of much interest, with in particular the large photon flux $\sim Z^2$ per ion enhancing the signal for various photon–initiated processes. In this paper we therefore extend the MC framework to include both proton–ion and ion–ion collisions, for arbitrary beams and in both QCD and photon–initiated production.
Indeed, a particularly topical example of this is the case of light–by–light (LbyL) scattering, $\gamma\gamma \to \gamma\gamma$, evidence for which was found by ATLAS [@Aaboud:2017bwk] and more recently CMS [@dEnterria:2018uly]. These represent the first direct observations of this process, and these data already show sensitivity to various BSM scenarios [@Knapen:2016moh; @Ellis:2017edi]. However, one so–far unresolved question is the size of the potential background from QCD–initiated production, $gg \to \gamma\gamma$, which in both analyses was simply taken from the `SuperChic` prediction in $pp$ collisions and scaled by $A^2 R^4$, where the factor $R\sim 0.7$ accounted for gluon shadowing effects, that is assuming that all $A$ nucleons in each ion can undergo CEP. While the normalization of this baseline prediction was in fact left free and set by data–driven methods, it is nonetheless important to address whether such a prediction is indeed reliable, by performing for the first time a full calculation of QCD–initiated production in heavy ion collisions. We achieve this here, and as we will see, predict that this background is much lower than previously anticipated.
A further topical CEP application is the case of high mass production of electroweakly coupled BSM states, for which photon–initiated production will be dominant at sufficiently high mass [@Khoze:2001xm]. Events may be selected with tagged protons in association with central production observed by ATLAS and CMS, during nominal LHC running. There are possibilities, for example, to probe anomalous gauge couplings (see [@Baldenegro:2017aen] and references therein) and search for high mass pseudoscalar states [@Baldenegro:2018hng] in these channels, accessing regions of parameters space that are difficult or impossible to reach using standard inclusive methods. With this in mind, we also present various updates to the photon–initiated production channels. Namely, we provide a refined calculated of Standard Model (SM) LbyL scattering, including the $W$ loops that are particularly important at high mass, as well as generating axion–like particle (ALP), monopole pair and monopolium production. We also include photon–initiated top quark pair production. We label the MC including these updates `SuperChic 3`.
The outline of this paper is as follows. In Section \[sec:heavyion\] we present details of the implementation of CEP in pA and AA collisions, for both photon and QCD–initiated cases. In Section \[sec:newproc\] we discuss the new photon–initiated processes that are included in the MC. In Section \[sec:LbyL\] we take a closer look at LbyL scattering, comparing in detail to the ATLAS and CMS data, and considering both the photon–initiated signal and QCD-initiated background. In Section \[sec:superchic\] we summarise the processes generated by `SuperChic 3` and provide information on its availability. In Section \[sec:conc\] we conclude, and in Appendix \[ap:incoh\] we present some analytic estimates of the expected scaling with $A$ of the QCD–initiated production process in pA and AA collisions, supporting our numerical findings.
![Schematic diagrams for (left) QCD and (right) photon initiated CEP.[]{data-label="fig:pCp"}](CEPgg.pdf "fig:") ![Schematic diagrams for (left) QCD and (right) photon initiated CEP.[]{data-label="fig:pCp"}](CEPgam.pdf "fig:")
Heavy Ion Collisions {#sec:heavyion}
====================
We first consider the photon–initiated production, before moving on to consider the QCD–initiated case.
$\gamma\gamma$ collisions – unscreened case {#sec:gamunsc}
-------------------------------------------
For photon–initiated production in heavy ion collisions, ignoring for now the possibility of additional ion–ion interactions, we can apply the usual equivalent photon approximation [@Budnev:1974de]. The cross section for the production of a system of mass $M_X$ and rapidity $Y_X$ is given by $$\begin{aligned}
\sigma_{N_1 N_2 \to N_1 X N_2} &= \int {\rm d} x_1 {\rm d}x_2 \,n(x_1) n(x_2)\hat{\sigma}_{\gamma \gamma \to X}\;,\\
&= \int {\rm d}M_X {\rm d}Y_X \frac{2 M_X}{s} \,n(x_1) n(x_2) \hat{\sigma}_{\gamma \gamma \to X}\;,\end{aligned}$$ where the photon flux is n(x\_i)=((1-x\_i)F\_E(Q\_i\^2)+F\_M(Q\_i\^2)), in terms of the transverse momentum $q_{i\perp}$ and longitudinal momentum fraction $x_i$ of the parent nucleus carried by the photon[^1]. The modulus of the photon virtuality, $Q^2_i$, is given by $$\label{eq:qi}
Q^2_i=\frac{q_{i_\perp}^2+x_i^2 m_{N_i}^2}{1-x_i}\;,$$ For the proton, we have $m_{N_i}=m_p$ and the form factors are given by $$F_M(Q^2_i)=G_M^2(Q^2_i)\qquad F_E(Q^2_i)=\frac{4m_p^2 G_E^2(Q_i^2)+Q^2_i G_M^2(Q_i^2)}{4m_p^2+Q^2_i}\;,$$ with $$G_E^2(Q_i^2)=\frac{G_M^2(Q_i^2)}{7.78}=\frac{1}{\left(1+Q^2_i/0.71 {\rm GeV}^2\right)^4}\;,$$ in the dipole approximation, where $G_E$ and $G_M$ are the ‘Sachs’ form factors. For the heavy ion case the magnetic form factor is only enhanced by $Z$, and so can be safely dropped. We then have $$\label{eq:wwion}
F_M(Q^2_i)=0\qquad F_E(Q^2_i)=F_p^2(Q_i^2)G_E^2(Q_i^2)\;,$$ where $F_p(Q^2)^2$ is the squared charge form factor of the ion. Here, we have factored off the $G_E^2$ term, due to the form factor of the protons within the ion; numerically this has a negligible impact, as the ion form factor falls much more steeply, however we include this for completeness. The ion form factor is given in terms of the proton density in the ion, $\rho_p(r)$, which is well described by the Woods–Saxon distribution [@Woods:1954zz] \[eq:rhop\] \_p(r)= , where the skin thickness $d \sim 0.5-0.6$ fm, depending on the ion, and the radius $R \sim A^{1/3}$. The density $\rho_0$ is set by requiring that \^3 r\_p(r) = Z. The total nucleon density $\rho_A$ can be defined in a similar way, and is normalised to the mass number $A$. The charge form factor is then simply given by the Fourier transform F\_p(||) = \^3r e\^[i ]{} \_p(r), in the rest frame of the ion; in this case we have $\vec{q}^2 = Q^2$, so that written covariantly this corresponds to the $F(Q^2)$ which appears in . In impact parameter space, the coherent amplitude is given by a convolution of the transverse proton density within the ion, and the amplitude for photon emission from individual protons; hence in transverse momentum space we simply multiply by the corresponding form factor. This is shown in Fig. \[fig:ff\] for the case of ${}^{63}{\rm Cu}$ and ${}^{208}{\rm Pb}$, for which we take [@Chamon:2002mx] \[eq:rdep\] R = (1.31 A\^[1/3]{} - 0.84) [fm]{}, d=0.55 [fm]{}, for concreteness. The sharp fall off with $Q^2$ is clear, with the form factors falling to roughly zero by $\sqrt{Q^2} \sim 3/R\sim 0.1$ GeV; for the smaller Cu ion this extends to somewhat larger $Q^2$ values.
![Normalized charge form factor due to lead and copper ions.[]{data-label="fig:ff"}](tfermi)
The above results, which are written at the cross section level, completely define the situation in the absence of screening corrections. However for the purpose of future discussion we can also write this in terms of the amplitude \[eq:tq1q2\] T(q\_[1]{},q\_[2]{}) = \_1 \_2 q\_[1]{}\^q\_[2]{}\^V\_, where $V_{\mu\nu}$ is the $\gamma\gamma \to X$ vertex, and the normalization factors are given by \_i = ((1-x\_i))\^[1/2]{}. Indeed, the derivation of the equivalent photon approximation at the amplitude level has precisely this Lorentz structure[^2]. This then reduces to the usual cross section level result after noting that we can write $$\begin{aligned}
q_{1_\perp}^i q_{2_\perp}^j V_{ij} =\begin{cases} &-\frac{1}{2} ({\bf q}_{1_\perp}\cdot {\bf q}_{2_\perp})(\mathcal{M}_{++}+\mathcal{M}_{--})\;\;(J^P_z=0^+)\\
&-\frac{i}{2} |({\bf q}_{1_\perp}\times {\bf q}_{2_\perp})|(\mathcal{M}_{++}-\mathcal{M}_{--})\;\;(J^P_z=0^-)\\
&+\frac{1}{2}((q_{1_\perp}^x q_{2_\perp}^x-q_{1_\perp}^y q_{2_\perp}^y)+i(q_{1_\perp}^x q_{2_\perp}^y+q_{1_\perp}^y q_{2_\perp}^x))\mathcal{M}_{-+}\;\;(J^P_z=+2^+)\\
&+\frac{1}{2}((q_{1_\perp}^x q_{2_\perp}^x-q_{1_\perp}^y q_{2_\perp}^y)-i(q_{1_\perp}^x q_{2_\perp}^y+q_{1_\perp}^y q_{2_\perp}^x))\mathcal{M}_{+-}\;\;(J^P_z=-2^+)
\end{cases}\label{Agen}\end{aligned}$$ where $\mathcal{M}_{\pm \pm}$ corresponds to the $\gamma(\pm) \gamma(\pm) \to X$ helicity amplitude. We then have \^2 q\_[1]{}[d]{}\^2 q\_[2]{} |T(q\_[1]{},q\_[2]{}) |\^2 = n(x\_1)n(x\_2) \_[\_1 \_2]{} |\_[\_1 \_2]{}|\^2, after performing the azimuthal angular integration on the left hand side.
The cross section is then given by \[eq:csn\] \_[N\_1 N\_2 N\_1 X N\_2]{} = x\_1 [d]{}x\_2 [d]{}\^2 q\_[1]{}[d]{}\^2 q\_[2]{} \_i |T(q\_[1]{},q\_[2]{}) |\^2, where $\mathcal{PS}_i$ is defined for the $2\to i$ process to reproduce the corresponding cross section $\hat{\sigma}$, i.e. explicitly \_1 = (-M\^2),\_2 = . It is then straightforward to see that this reduces to the usual equivalent photon result. However, as we will see below, we must work at the amplitude level to give a proper account of screening corrections.
$\gamma\gamma$ collisions – screened case {#sec:gamsc}
-----------------------------------------
The inclusion of screening corrections follows in essentially straightforward analogy to the $pp$ case considered in e.g. [@Gotsman:2014pwa; @Harland-Lang:2015cta; @Khoze:2017sdd]. This is most easily discussed in impact parameter space, for which the average eikonal survival factor is given by $$\label{eq:S2}
\langle S^2_{\rm eik} \rangle=\frac{\int {\rm d}^2 b_{1\perp}\,{\rm d}^2 b_{2 \perp}\, |\tilde{T}(s, b_{1\perp}, b_{2\perp})|^2\,{\rm exp}(-\Omega_{A_1 A_2}(s,b_\perp))}{\int {\rm d}^2\, b_{1\perp}{\rm d}^2 b
_{2\perp}\, |\tilde{T}(s,b_{1\perp},b_{2\perp})|^2}\;,$$ where $b_{i\perp}$ is the impact parameter vector of ion $i$, so that $b_\perp=b_{1\perp}+ b_{2\perp}$ corresponds to the transverse separation between the colliding ions. $\tilde{T}(s, b_{1\perp},b_{2\perp})$ is the amplitude in impact parameter space, i.e. (s, b\_[1]{},b\_[2]{})= \^2 q\_[1]{} [d]{}\^2 q\_[2]{} e\^[-i \_[1]{}\_[1]{}]{}e\^[i \_[2]{}\_[2]{}]{}T(s, q\_[1]{},q\_[2]{}), while $\Omega_{A_1 A_2}(s,b_\perp)$ is the ion–ion opacity; physically $\exp(-\Omega_{A_1 A_2}(s,b_\perp))$ represents the probability that no inelastic scattering occurs at impact parameter $b_\perp$. Its calculation is described in the following section. For our purposes it is simpler to work in $q_\perp$ space, for which we introduce the screening amplitude via \[eq:tres\] T\_[res]{}(q\_[1]{},q\_[2]{}) = T\_[el]{}(k\^2\_) T(q\_[1]{}’,q\_[2]{}’), where $q_{1\perp}' =q_{\perp} - k_\perp$ and $q_{2\perp}' = q_{2\perp} + k_\perp$ and $T_{\rm el}$ is the elastic ion–ion amplitude, given by T\_[el]{}(k\^2\_) = 2is \^2 b\_ e\^[i \_\_]{}(1-e\^[-\_[A\_1A\_2]{}(b\_)/2]{}). Then it is straightforward to show that S\^2\_[eik]{} = , and thus we should simply replace $T(q_{1\perp},q_{2\perp}) \to T(q_{1\perp},q_{2\perp}) + T_{\rm res}(q_{1\perp},q_{2\perp})$ for the corresponding amplitude in .
The ion–ion opacity {#sec:opac}
-------------------
![Elastic proton–proton cross section $d\sigma/dt$ at 5,02, 8.16, 39 and 63 TeV (from top to bottom). The predictions calculated within the two–channel model [@Khoze:2018kna] and the one channel eikonal model described in the text are shown by the red and dashed black lines, respectively. In both cases only the $|{\rm Im} A_{el}|^2$ contribution to $d\sigma/dt$ is shown.[]{data-label="fig:sigel"}](sigel)
Having introduced the ion–ion opacity above, which encodes the probability for no additional ion–ion rescattering at different impact parameters, we must describe how we calculate this. The ion–ion opacity is given in terms of the opacity due to nucleon–nucleon interactions, $\Omega_{nn}$, which is in turn given by a convolution of the nucleon–nucleon scattering amplitude $A_{nn}$ and the transverse nucleon densities $T_n$. In particular we have \[eq:omega\] \_[A\_1A\_2]{}(b\_) = \^2 b\_[1]{} [d]{}\^2 b\_[2]{} T\_[A\_1]{}(b\_[1]{})T\_[A\_2]{}(b\_[2]{})A\_[nn]{}(b\_-b\_[1]{}+b\_[2]{}), with $T_A$ given in terms of the nucleon density \[eq:tpn\] T\_A(b\_)= z \_A(r) = z (\_n(r) + \_p(r)), of the corresponding ion. For the case of $pA$ collisions, we simply take T\_A(b\_) \^[(2)]{}(\_), for the $A \to p$ replacement. The nucleon–nucleon scattering amplitude is given in terms of the nucleon opacity $\Omega_{nn}(b_\perp)$ via \[eq:anncoh\] A\_[nn]{}(b\_) = 2 ( 1-e\^[-\_[nn]{}(b\_)/2]{}). Note that this corresponds to the total scattering cross section, as \_[tot]{}\^[nn]{} = \^2 b\_A\_[nn]{}(b\_), see e.g. [@Ryskin:2009tj]. This is the appropriate choice the momentum transfers involved even in purely elastic nucleon–nucleon rescattering will as a rule lead to ion break up. On the other hand for the case of QCD–initiated semi–exclusive production discussed further below, where the ion breaks up, we should take \[eq:annincoh\] A\_[nn]{}(b\_) = 1-e\^[-(b\_)]{}, so that \^[nn]{}\_[inel]{} = \^2 b\_A\_[nn]{}(b\_), which corresponds to a somewhat smaller suppression. To calculate the nucleon opacity we can then apply precisely the same procedure as for $pp$ collisions, see e.g. [@Khoze:2013jsa; @Khoze:2014aca]. This in general requires the introduction of so–called Good–Walker eigenstates [@Good:1960ba] to account for the internal structure of the proton. However, in order to avoid unfeasibly complicated combinatorics we instead apply a simpler one–channel approach here. The parameters of this model are tuned in order to closely reproduce the more complete result of the two–channel model of [@Khoze:2018kna] for the elastic $pp$ cross section in the relevant lower $t$ region, in particular before the first diffractive dip. The result is shown in Fig. \[fig:sigel\].
In more detail, the nucleon opacity is given by (b\_) = -\^2 q\_ e\^[i \_\_]{} A\_[I P]{}(-q\^2), where $A_{I\!\! P}$ is the elastic amplitude due to single Pomeron exchange, given by A\_[I P]{}=is \_0 \^2(t). For the form factors $\beta$ we take \[eq:beta\] (t)=, with the precise numerical values given in Table \[tab:op\] (for other values of $\sqrt{s}$ we use a simple interpolation). We note that in the above, we have the same scattering amplitude in the neutron and protons cases, due the high energy nature of the interaction and dominance of Pomeron exchange in this region.
$\sqrt s$ \[TeV\] $\sigma_0$ \[mb\] $a$ \[GeV$^2$ \] $b$ \[GeV$^{-2}$\] c
------------------- ------------------- ------------------ -------------------- -------
5.02 146 0.180 20.8 0.414
8.16 159 0.190 26.3 0.402
39 228 0.144 23.3 0.397
63 245 0.150 28.0 0.390
: The parameters of the one channel eikonal description of nucleon–nucleon amplitude, described in the text.[]{data-label="tab:op"}
The opacity and probability for no inelastic scattering, $e^{-\Omega_{A_1 A_2}(b_\perp)}$, in lead–lead collisions are shown in Fig. \[fig:opacb\]. For the neutron and proton densities we take as before the Wood–Saxons distribution , with the experimentally determined values [@Tarbert:2013jze] $$\begin{aligned}
\nonumber
R_p &= 6.680\, {\rm fm}\;, &d_p &= 0.447 \, {\rm fm}\;,\\ \label{eq:pbpar}
R_n &= (6.67\pm 0.03)\, {\rm fm}\;, &d_n &= (0.55 \pm 0.01) \, {\rm fm}\;.\end{aligned}$$ The solid curve corresponds to the central values, while for the dashed curves we take values for the neutron density at the lower and upper end of the 1$\sigma$ uncertainties, for illustration. For lower values of $b_\perp \lesssim 2 R$ (here we define $R\sim R_{p,n}$ for simplicity), where the colliding ions are overlapping in impact parameter space, we can see that the probability is close to zero, while for larger $b_\perp \gtrsim 2 R$ this approaches unity, as expected. However we can see that this transition is not discrete, with the probability being small somewhat beyond $2 R$, due both to the non–zero skin thickness of the ion densities and range of the QCD single–Pomeron exchange interaction. This will be missed by an approach that is often taken in the literature, namely to simply to cutoff the cross in impact parameter space when $b_\perp<2R$. Comparing to , we can see that this corresponds to taking instead e\^[-(b)/2]{} = (b-2 R). The value at which this would turn on is indicated in Fig. \[fig:opacb\]. As our more realistic result turns on smoothly above $2R$, this will correspond to somewhat suppressed exclusive cross sections in comparison. For ultra–peripheral photon-initiated interactions, where the dominant contribution to the cross section comes from $b_\perp \gg 2 R$, this will have a relatively mild impact, but for QCD–initiated production a complete treatment is essential.
![Ion–ion opacity (left) and probability for no inelastic scattering (right) for lead–lead collisions, as a function of the lead impact parameter $b_\perp$.[]{data-label="fig:opacb"}](opacpbln "fig:") ![Ion–ion opacity (left) and probability for no inelastic scattering (right) for lead–lead collisions, as a function of the lead impact parameter $b_\perp$.[]{data-label="fig:opacb"}](opacpbn "fig:")
QCD–induced production {#sec:qcdind}
----------------------
We can also apply the above formalism to the case of QCD–initiated diffractive production in heavy ions. We will discuss two categories for this, namely semi–exclusive and fully exclusive production, below.
### Semi–exclusive production {#sec:semiex}
We first consider the case of incoherent QCD–induced CEP. Here, while the individual nucleons remain intact due to the diffractive nature of the interaction, the ion will in general break up. This can therefore lead to an exclusive–like signal in the central detector, with large rapidity gaps between the produced state and ion decay products. If zero degree calorimeter (ZDC) detectors are not used to veto on events where additional forward neutrons are produced, this will contribute to the overall signal.
Nonetheless, as we will see such interactions are strongly suppressed by the requirement that the ions themselves do not interact in addition, producing secondary particles in the central detector, that is due the ion–ion survival factor. The incoherent cross section, prior to the inclusion of survival effects, is simply given by integrating the CEP cross section in $pp$ collisions over the nucleon densities \[eq:sigincoh0\] \_[incoh]{} = \^2 b\_[1]{} [d]{}\^2 b\_[2]{} T\_[A\_1]{}(b\_[1]{}) T\_[A\_2]{}(b\_[2]{}) \_[CEP]{}\^[pp]{} , where $\sigma_{\rm CEP}^{pp}$ is the usual QCD–induced $pp$ cross section as implemented in previous versions of `SuperChic` [@Harland-Lang:2015cta]. Note that here we make the approximation that the nucleon–nucleon CEP interaction is effectively point–like in comparison to the ion radius $R$. Strictly speaking, we should instead convolute the transverse densities, $T_A$, with the form factor due to the range of the nucleon–nucleon CEP interaction, however we have checked that numerically this is a relatively small effect, and omit this in what follows. We can see that then scales like $\sim A_1 A_2$, i.e. with the total number of nucleon pairings. However, this exclude survival effects. To account for these, we simply multiply by the probability for no additional inelastic ion–ion interactions, so that \[eq:sigincoh\] \_[incoh]{} = \^2 b\_[1]{} [d]{}\^2 b\_[2]{} T\_[A\_1]{}(b\_[1]{}) T\_[A\_2]{}(b\_[2]{}) \^[CEP]{} e\^[-\_[A\_1 A\_2]{}( b\_[1]{} - b\_[2]{})]{}, where the opacity $\Omega_{A_1 A_2}$ is calculated as described in Section \[sec:opac\], in particular via and . In fact the MC, we calculate the effective survival factor S\^2\_[incoh]{} = , and multiply the usual $pp$ cross section (calculated differentially in $k_\perp$ space) by this.
Crucially, we have seen in Section \[sec:opac\], see in particular Fig. \[fig:opacb\], that the ion–ion opacity is very large for $b_\perp \lesssim 2 R$, and hence the probability of no additional inelastic interactions is exponentially suppressed. This has the result that in the region of significant nucleon density (where the $T_A$ are not suppressed) we will almost inevitably have additional inelastic interactions, and the corresponding survival factor will be very small. Thus, we will only be left with a non–negligible CEP cross section in the case that the interacting nucleons are situated close to the ion periphery, where the nucleon density and hence inelastic interaction probability is lower. In other words, we do not have $A_1 A_2$ possible nucleon–nucleon interactions, but rather expect a much gentler increase with the ion mass number, with only those nucleons on the surface (or more precisely, the edge of the ion ‘disc’ in the transverse plane) playing a role. We find in particular that to good approximation \[eq:incohsc\] \_[incoh]{} A\^[1/3]{}, for both $AA$ and $pA$ collisions, where in the former case we assume the ions are the same for simplicity. The detailed derivation is given in Appendix \[ap:incoh\]. Comparing to the $Z_1^2 Z_2^2$ scaling of the photon–initiated process, we may therefore expect QCD–initiated CEP to be strongly suppressed; we will see that this is indeed the case below.
### Exclusive production {#sec:exprod}
Alternatively, we can consider the case of coherent ion–ion QCD–induced CEP, which leaves the ions intact. To achieve this, we proceed in a similar way to Sections \[sec:gamunsc\] and \[sec:gamsc\]. In particular we simply have \[eq:tqcdex\] T\_[QCD]{}\^[A\_1 A\_2]{}(q\_[1]{},q\_[2]{}) = T\_[QCD]{}\^[pp]{}(q\_[1]{},q\_[2]{}) F\_[A\_1]{}(Q\_1\^2) F\_[A\_2]{}(Q\_2\^2), where $Q^2$ is given as in (\[eq:qi\]) and $T_{\rm QCD}$ is the QCD–induced CEP amplitude as calculated within the usual Durham model approach, see [@Harland-Lang:2015cta]. Here $F_A$ is the ion form factor, given in terms of the nucleon density $\rho_A$, see . In impact parameter space this corresponds to \_[QCD]{}\^[A\_1 A\_2 ]{}(b\_[1]{},b\_[2]{}) = \^2 b\_[1]{}’[d]{}\^2 b\_[2]{}’ \_[QCD]{}\^[pp]{}(b\_[1]{}’,b\_[2]{}’) T\_[A\_1]{}(b\_[1]{}-b\_[1]{}’) T\_[A\_2]{}(b\_[2]{}-b\_[2]{}’) . Now, the range of the nucleon–nucleon CEP amplitude $T^{pp}$ (which is $\lesssim 1$ fm) is significantly less than the extent of the ion transverse density (i.e. $\sim 7$ fm for a Pb ion). This allows us to take $T_A(b_{\perp}-b_{\perp}') \sim T_A(b_{\perp})$ above, so that $$\begin{aligned}
\tilde{T}_{\rm QCD}^{A_1 A_2 }(b_{1\perp},b_{2\perp}) &\approx T_{A_1}(b_{1\perp})T_{A_2}(b_{2\perp})\int {\rm d}^2 b_{1\perp}'{\rm d}^2 b_{2\perp}' \tilde{T}_{\rm QCD}^{pp}(b_{1\perp}',b_{2\perp}')\;,\\
& = T_{\rm QCD}^{pp}(q_{1\perp}=0,q_{2\perp}=0)\cdot T_{A_1}(b_{1\perp})T_{A_2}(b_{2\perp})\;.\end{aligned}$$ The cross section then becomes $$\begin{aligned}
\nonumber
\sigma_{\rm coh} &= (4\pi^2)^2 |T_{\rm QCD}^{pp}(q_{1\perp}=0,q_{2\perp}=0)|^2 \int {\rm d}^2 b_{1\perp}{\rm d}^2 b_{1\perp}|T_{A_1}(b_{1\perp})|^2 |T_{A_2}(b_{2\perp})|^2 e^{-\Omega_{A_1 A_2}( b_{1\perp} - b_{2\perp})}\;,\\ \label{eq:sigbt}
&\approx (4\pi^2)^2 \frac{\sigma_{\rm CEP}^{pp}}{\pi^2 \left\langle q_{1\perp}^2 \right \rangle \left\langle q_{2\perp}^2 \right \rangle }\int {\rm d}^2 b_{1\perp}{\rm d}^2 b_{1\perp}|T_{A_1}(b_{1\perp})|^2 |T_{A_2}(b_{2\perp})|^2 e^{-\Omega_{A_1 A_2}( b_{1\perp} - b_{2\perp})}\;,\end{aligned}$$ where we define $\left\langle q_{\perp}^2 \right \rangle$ in the second line. This is of the order of the average squared transverse momentum transfer in the $pp$ cross section, i.e. q\_[1]{}\^2 \~, and similarly for $q_{2\perp}$, where we assume the $q_{1\perp}$ and $q_{2\perp}$ dependencies factorise; such an expression is exactly true if we assume a purely exponential form factor in $q_\perp^2$, for example.
We emphasise that in the MC we make use of the general result, with the formalism of Section \[sec:gamsc\] applied to to include survival effects. However, the above result holds to good approximation, and allows us to derive some straightforward expectations for the scaling and size of the coherent contribution. As discussed further in Appendix \[ap:incoh\], under these approximations, for $pA$ collisions we expect a similar $\sim A^{1/3}$ to the incoherent case , but with a parametric suppression \[eq:cohsupp\] \_[coh]{}\^[pA]{} \~ \_[incoh]{}\^[pA]{} . For $AA$ collisions the expected scaling is in fact somewhat gentler in comparison to the incoherent case, with a (squared) parametric suppression \[eq:cohsup\] \_[coh]{}\^[AA]{} \~()\^2A\^[-1/6]{}\_[incoh]{}\^[AA]{} A\^[1/6]{}, where we write $\left \langle q_{i\perp}^2 \right \rangle = \left\langle q_\perp^2 \right \rangle$ and $\sigma_{\rm tot}^{nn}$ is the total $pp$ cross section. We therefore expect some numerical parametric suppression by the ratio of the cross sectional extent of the CEP interaction with each ion ($\sim 4\pi/\left\langle q_\perp^2 \right \rangle$) to total $pp$ cross section. Taking some representative vales for these, numerically we have \[eq:qtsub\] \~ \~0.5. Hence we may expect some suppression in the coherent cross section, although given the relatively mild effect predicted by this approximate result, a precise calculation is clearly necesssary. Note that we here take a rather small value of $\left\langle q_{\perp}^2 \right \rangle \sim 0.1$ ${\rm GeV}^2$, corresponding to a quite steep slope in $q_\perp^2$. This is as expected when $pp$ rescattering effects are included, see e.g. [@HarlandLang:2010ep], which tend to prefer small values of the proton transverse momenta, where the survival factor is larger. A consequence of this is that the observed ratio of the cross section with heavy ions to the proton–proton cross section will depend on the precise process considered and in particular the quantum numbers of the produced state, through the effect this has on the survival factor. Finally, we recall that in the case of ion–ion collisions there is a reasonable probability to excite a ‘giant dipole resonance’ (GDR) via multi–photon exchange between the ions. This effect, not currently included in the MC, will lead to an excited final state, decaying via the emission of additional neutrons. From [@Baltz:2002pp], the probability for this to occur at the relative low impact parameters $b_\perp \sim 2R$ relevant to QCD–initiated CEP is found to be rather large, see Fig. 2 of this reference. We can estimate from this a probability of $\sim$ 50% for GDR excitation in each ion in this region. This will reduce the exclusive and increase the semi–exclusive cross sections predicted here accordingly. If one does not tag neutron emission experimentally via ZDCs this is not an issue, as we simply sum the two contributions, however when comparing to data with such tagging performed a corresponding correction to our predictions should be made.
### Including the participating nucleons {#sec:part}
In principle our calculation of the survival factor in proton–ion and ion–ion collisions, as in for example , i.e. S\^2\_[A\_1 A\_2]{} (b\_) = e\^[-\_[A\_1 A\_2]{}( b\_ )]{}, gives the probability of no inelastic interactions between all nucleons within the overlap in impact parameter of the colliding ions. In particular, this corresponds to a simple Poissonian no interaction probability, with the mean number of inelastic nucleon–nucleon interactions given as in , in terms of the total ion transverse densities $T_A$ integrated over the appropriate impact parameter regions. These therefore in principle take care of all possible nucleon–nucleon interactions, including the particular nucleon–nucleon pairing that undergoes CEP.
However, the survival factor due to this active pair would be better treated separately and included explicitly, as its precise value will depend on the underlying CEP process. More significantly, the exclusive production process must take place close to the peripherary of the ions, where the corresponding nucleon density is low and the average number of nucleon–nucleon interactions contained in the above expression can be below one. Applying the above factor alone will therefore overestimate the corresponding survival factor, giving a value higher than that due to the active pair, and so such a separate treatment is essential.
We therefore include the (process dependent) nucleon–nucleon survival factor explicitly, i.e. the CEP cross section in and amplitude in correspond to those including survival effects in the nucleon–nucleon interaction. On the other hand, having done this we must take care to avoid double counting the possibility for inelastic interactions due to this active pair. Unfortunately this in general requires a careful treatment of the ion structure, moving beyond the opacity above, which is simply given in terms of the total average nucleon density. Here, we base our calculation on the nuclear shell model, and recall that for CEP we are dominated by interactions which occur close to the ion peripherary, which is mainly populated by $N_{\rm shell}$ nucleons with the largest principal and orbital quantum numbers. Each of these contributes to the total average nucleon density T\_A(b\_) = \_[i=1]{}\^[N\_[shell]{}]{} T\_A\^i(b\_) = N\_[shell]{} T\_A\^i(b\_) , where $T_A^i(b_\perp)$ is the contribution from each individual nucleon, which in the last step we assume to be the same for each nucleon. To remove the contribution from the active nucleon that undergoes CEP we therefore simply replace \_[pA]{} \_[pA]{} (1-),\_[AA]{} \_[AA]{} (1-)\^2, in the corresponding opacities. In the case of $^{208}Pb$ the highest shell has $l=3$ for neutrons and $l=2$ for protons, corresponding to 14 neutrons and 12 protons. At the peripherary the proton density is roughly three times smaller than the neutron, and therefore as a rough estimate then we can take $N_{\rm shell} \approx 20$. Hence this correction is rather small, at the $5-10\%$ level.
However, this is not the end of the story. In particular the position of the nucleons in the ion shell are not completely independent, and we can expect some repulsion between them due to $\omega$ meson exchange [@Reid:1968sq]. In the ion peripherary the nucleon density is rather small, and hence it is reasonable to describe this repulsion in the same way as the repulsive ‘core’ in the deuteron wave function [@Reid:1968sq]. Here, the separation between the nearest nucleons cannot be less than $r_{\rm core}=0.6-0.8$ fm. To account for this, we can subtract an interval of length $2r_{\rm core}$ in the $z$ direction from the nucleon density which enters the calculation of the opacity[^3].
In the results which follow we will take $N_{\rm shell}=20$ and $r_{\rm core}=0.8$ fm. The latter gives roughly a $50\%$ increase in the cross section, while as discussed above the former correction is significantly smaller. While this provides our best estimate of the CEP cross section, there is clearly some uncertainty in the precise predictions due to the effects above, conservatively at the $50\%$ level, with the result omitting these two corrections representing a lower bound on the cross section.
### Numerical results
![Ratio of cross sections at $\sqrt{s}=5.02$ TeV in proton–ion ($pA$) and ion–ion ($AA$) collisions to the proton–proton result. The QCD (photon) initiated cases are shown in the left (right) plots. Results with and without survival effects are shown by the solid (dashed) lines. Note that for the QCD–initiated production the survival factor due to the participating nucleon pair is included in all cases.[]{data-label="fig:qcdcomp"}](qcdcompnn "fig:") ![Ratio of cross sections at $\sqrt{s}=5.02$ TeV in proton–ion ($pA$) and ion–ion ($AA$) collisions to the proton–proton result. The QCD (photon) initiated cases are shown in the left (right) plots. Results with and without survival effects are shown by the solid (dashed) lines. Note that for the QCD–initiated production the survival factor due to the participating nucleon pair is included in all cases.[]{data-label="fig:qcdcomp"}](gamcompn "fig:")
In Fig. \[fig:qcdcomp\] (left) we show numerical predictions for the ratio of QCD–initiated cross sections at $\sqrt{s}=5.02$ TeV in proton–ion ($pA$) and ion–ion ($AA$) collisions to the proton–proton result. In all cases we include the survival factor due to the active nucleon pair, but in the solid curves we include the effect due to the additional nucleons present in the ion(s) as well. To be concrete, we show results for $\gamma\gamma$ production within the ATLAS event selection [@Aaboud:2017bwk]. We take for the dependence of the ion radius on $A$, while we show results for $d=0.5, 0.55$ and 0.6 fm (dotted, solid and dashed lines, respectively), including survival effects, in Fig. \[fig:qcdcomp1\] to give an indication of the sensitivity of the cross section to the value of the ion skin thickness. This also provides a clearer demonstration of the trends for the full cross section (i.e. including survival effects): the solids curves in the two plots correspond to the same results.
In all cases, the impact of survival effects is found to be sizeable. Already for proton–ion collisions these reduce the corresponding cross sections by up to two order of magnitude, while in ion–ion collisions the effect is larger still, leading to a reduction of up to four and six orders of magnitude in the semi–exclusive and exclusive cases, respectively. As discussed earlier, this is to be expected: as the range of the QCD–initiated CEP interaction is much smaller than the ion radius, the majority of potential nucleon–nucleon CEP interactions (in the absence of survival effects due to the non–interacting nucleons) would take place in a region of high nucleon density, where additional particle production is essentially inevitable. This is in strong contrast to the case of photon–initiated production, where the long range QED interaction allows all protons in the ion to contribute coherently in an ultra–peripheral process.
![Ratio of QCD–initiated cross sections at $\sqrt{s}=5.02$ TeV in proton–ion ($pA$) and ion–ion ($AA$) collisions to the proton–proton result. Results are shown with different values of the ion skin thickness, as described in the text.[]{data-label="fig:qcdcomp1"}](qcdcomp1)
Considering in more detail the cross sections including survival effects, in the proton–ion case, the relatively gentle scaling of the exclusive and semi–exclusive cross sections with $A$ is clear, which upon inspection are indeed found to follow a rough $\sim A^{1/3}$ trend, consistent with and . As expected from the discussion in Section \[sec:exprod\], the exclusive and semi–exclusive cross sections are of similar sizes. Interestingly, we can see that the precise calculation predicts that the exclusive cross section is in fact somewhat enhanced relative to the semi–exclusive. For the ion–ion case we can see that the semi–exclusive cross section again increases only very gently with $A$, again as expected. Upon inspection, we observe that the trend is consistent with a flatter $A$ dependence then the simple $\sim A^{1/3}$ scaling predicted using the analytic calculation of Appendix \[ap:incoh\]; on closer investigation, we find that this is due to the correct inclusion of the impact parameter dependence of the elastic nucleon–nucleon scattering amplitude in the definition of the opacity , which is omitted in the simplified analytic approach. In the exclusive case, interestingly the cross section in fact decreases with $A$, albeit with a relatively flat behaviour at larger $A$. This is again found to be due to the full calculation of the opacity. Again, numerically the exclusive and semi–exclusive cross sections enter at roughly the same order, with some suppression in the former case, as expected from the discussion in Section \[sec:exprod\].
In Fig. \[fig:qcdcomp\] (right) we show the corresponding cross section ratios for the photon–initiated cross sections. For concreteness, we calculate $Z$ by maximising the binding energy according to the semi–empirical mass formula [@Weizscker:1935zz], i.e. 2+A\^[2/3]{}, with $a_C=0.711$, $a_A=23.7$. The impact of survival effects is in this case found to be significantly more moderate, at the $10 -20\%$ level, due to the well–known result that the photon–initiated interaction takes places at large impact parameters, i.e. ultra–peripherally, where the impact of further ion–ion or proton–ion interactions is relatively small. The dramatic cross section scaling with $A$ in the ion–ion case is also clear, leading to a relative enhancement by many orders of magnitude in comparison to the QCD–initiated case. For proton–ion collisions a milder enhancement is also observed. We note that in both cases the steeply falling $Q^2$ dependence of the ion form factors leads to some suppression relative the naïve $\sim Z^2$ and $Z^4$ scaling in the proton–ion and ion–ion cases.
New processes {#sec:newproc}
=============
In this section we briefly describe the new processes and refinements that have been included in `SuperChic` since the version described in [@Harland-Lang:2015cta].
Light–by–light scattering: $W$ loop contributions
-------------------------------------------------
In previous versions of `SuperChic`, expressions for the fermion loop contributions to the $\gamma \gamma \to \gamma \gamma$ light–by–light scattering process in the $\hat{s} \gg m_f^2$ limit were applied. We now move beyond this approximation, applying the `SANC` implementation [@Bardin:2009gq] of this process, which includes the full dependence on the fermion mass in the loop. This in addition includes the contribution from $W$ bosons, which was not included previously, again with the full mass dependence. We also implement a modified version of the `SANC` implementation for the $gg \to \gamma\gamma$ process, which has the same form as the quark–loop contributions to the light–by–light scattering process, after accounting for the different colour factors and charge weighting.
In Fig. \[fig:lbyl\] (left) we show the diphoton invariant mass distribution due to QCD and photon–initiated CEP in $pp$ collisions at $\sqrt{s}=14$ TeV. The photons are required to have transverse momentum $p_\perp^\gamma > 10$ GeV and pseudorapidity $|\eta^\gamma|<2.4$. We can see that while the former dominates for $M_{\gamma\gamma} \lesssim 150$ GeV, above this the latter is more significant. This is due to the well–known impact of the Sudakov factor in the QCD–initiated cross section [@Khoze:2001xm] which suppresses higher mass production, due to the increasing phase space for additional gluon radiation, so that at high enough mass this compensates the suppression in the photon–initiated cross section due to the additional powers of the QED coupling $\alpha$. We also show the relative contributions of fermion and $W$ boson loops to the photon–initiated cross section. While for $M_{\gamma\gamma} \lesssim 2 M_W$ the latter is as expected negligible, at sufficiently high invariant mass it comes to dominate. In Fig. \[fig:lbyl\] (right) we show the impact of excluding the fermion masses for the QCD–initiated case. The photons are required to have transverse momentum $p_\perp^\gamma > 16$ GeV and pseudorapidity $|\eta^\gamma|<2.4$. We can see that at lower $M_X$ the difference is at the $\sim 30\%$ level, decreasing to below $10\%$ at higher mass, in the considered region. Thus the previous `SuperChic` predictions will have overestimated the cross section by this amount. It should be noted however, that for the $gg\to \gamma\gamma$ case this is below the level of other theoretical uncertainties, due in particular to the gluon PDF and soft survival factor. Moreover, this is a purely LO result, and we may expect higher order corrections to increase the cross section by a correction of this order.
Finally, we note that the MC prediction for QCD–initiated CEP processes such as diphoton production does not include the impact of so–called ‘enhanced’ screening effects. These may be expected to reduce the corresponding cross section by as much as a factor of $\sim 2$ [@Ryskin:2009tk; @Ostapchenko:2017prv], but we leave a detailed study of this to future work. Note that such effects are entirely absent in the case of photon–initiated CEP.
![Diphoton invariant mass distribution due to QCD and photon–initiated CEP in $pp$ collisions at $\sqrt{s}=14$ TeV. The left plot in addition shows the individual contributions from fermion and $W$ loops to the $\gamma\gamma$–initiated process, while the right plot shows the impact of including finite fermion masses.[]{data-label="fig:lbyl"}](lbyl "fig:") ![Diphoton invariant mass distribution due to QCD and photon–initiated CEP in $pp$ collisions at $\sqrt{s}=14$ TeV. The left plot in addition shows the individual contributions from fermion and $W$ loops to the $\gamma\gamma$–initiated process, while the right plot shows the impact of including finite fermion masses.[]{data-label="fig:lbyl"}](diphotcomp "fig:")
ALP production
--------------
New light pseudoscalar ‘axion–like’ particles (ALPs), with dimension–5 couplings to two gauge bosons or derivative interactions to fermions occur in a wide range of BSM models, often resulting from the breaking of some approximate symmetry (for a list of popular references, see e.g. [@Baldenegro:2018hng]). For example, in the context of dark matter, these are often considered as mediators between dark matter and SM particles, while from an observational point of view the coupling to the SM may be sufficiently small so as to evade current constraints. The production of ALPs in ultra–peripheral heavy ion collisions was discussed in [@Knapen:2016moh], and more recently in [@Baldenegro:2018hng] for the case of larger ALP masses, in $pp$ collisions, while the ATLAS evidence for light–by–light scattering [@Aaboud:2017bwk] was used in [@Knapen:2017ebd] and in the recent CMS analysis [@dEnterria:2018uly] to set the most stringent constraints yet on the ALP mass and couplings in certain regions of parameter space.
We implement ALP production according to the Lagrangian =\^a \_a -m\_a\^2 a\^2 -g\_a a F\^\_, where $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu \alpha\beta} F_{\alpha\beta}$. That is, we only consider $\gamma\gamma$ coupling with strength $g_a$, through which the ALP is both produced and decays. We in addition include the possibility of a scalar ALP, through the replacement $\tilde{F} \to F$. For the $\gamma_{\lambda_1}\gamma_{\lambda_1} \to a$ amplitudes these give: $$\begin{aligned}
{\rm Pseudoscalar}&: & & \mathcal{M}_{+-}=\mathcal{M}_{-+}=0\;,& & \mathcal{M}_{++}=-\mathcal{M}_{--}=\frac{g_a M_{\gamma\gamma}^2}{2}\;,\\
{\rm Scalar}&: & & \mathcal{M}_{+-}=\mathcal{M}_{-+}=0\;,& & \mathcal{M}_{++}=\mathcal{M}_{--}=\frac{g_a M_{\gamma\gamma}^2}{2}\;.\end{aligned}$$ As an example, the expected signals due to a 10 and 30 GeV pseudoscalar ALP, with coupling $g_a=5\times 10^{-5}$ ${\rm GeV}^{-1}$, are shown in Fig. \[fig:alp\], overlaid on the continuum light–by–light background. The expected number of events (ignoring any further experimental efficiencies) with $L=10\,{\rm nb}^{-1}$ of $\sqrt{s}=5.02$ TeV Pb–Pb collision data are shown. We note that in both cases these are not excluded by current experimental constraints [@Knapen:2017ebd; @dEnterria:2018uly].
![Diphoton invariant mass distribution at $\sqrt{s}=5.02$ TeV in Pb–Pb collisions, for integrated luminosity $L=10\,{\rm nb}^{-1}$. The result due to the production of an ALP of mass 10 and 30 GeV is shown, with coupling $g_a=5\times 10^{-5}$ ${\rm GeV}^{-1}$, in both cases with a width of $0.5$ GeV included to roughly mimic the effect of experimental resolution. The continuum light–by–light background is also shown. The photons are required to have transverse momentum $p_\perp^\gamma>3$ GeV and pseudorapidity $|\eta^\gamma|<2.4$. The ALP is assumed here only to couple to photons.[]{data-label="fig:alp"}](alp)
Monopole and monopolium production
----------------------------------
Magnetic monopoles complete the symmetry of Maxwell’s equations and explain charge quantization [@Dirac:1931kp]. As such states would be expected to have large electromagnetic couplings, one possibility is to search for the production of monopole pairs, or bound states of monopole pairs (so called ‘monopolium’) through exclusive photon–initiated production at the LHC [@Epele:2012jn]. In the MC we have implemented the CEP of both monopoles pairs, and monopolium, in the latter case followed by the decay to two photons.
For the production of monopole pairs, we simply apply the known results for lepton pair production $\gamma\gamma \to l^+l^-$, but with the replacement $\alpha \to 1/4\alpha$, as required by the Dirac quantisation condition g=N, where we take $N=1$, and $g$ is the monopole charge. We also allow for the so–called $\beta g$ coupling scenario [@Epele:2012jn], for which we simply replace $g \to g \beta$, where $\beta$ is the monopole velocity. In the monopolium case we apply the cross section of [@Epele:2012jn], with the wave function of [@Epele:2007ic], and include the decay to two photons.
$\gamma\gamma \to t\overline{t}$
--------------------------------
We include photon–initiated top quark production. This is implemented using the same matrix elements as the lepton pair production process, with the mass, electric charge and colour factors suitably modified. We find a total photon–initiated cross section of $0.25$ fb in $pp$ collisions at $\sqrt{s}=14$ TeV, and 36 fb in Pb–Pb collisions at $\sqrt{s}=5.02$ TeV. Note that the QCD–initiated cross section in $pp$ collisions is about $0.02$ fb, and so is an order of magnitude smaller, while in Pb–Pb this will be smaller still.
Light–by–light scattering: a closer look {#sec:LbyL}
========================================
Evidence for light–by–light scattering in ultra–peripheral Pb–Pb collisions has been found by ATLAS [@Aaboud:2017bwk] and more recently by CMS [@dEnterria:2018uly]. In both cases, the production of a diphoton system accompanied by no additional particle production is measured, while in the ATLAS case ZDCs are in addition used to measure additional neutral particle production in the forward direction, which would be a signal of semi–exclusive production accompanied by ion break–up.
LbyL QCD (coh.) QCD (incoh.) $A^2 R^4$
------------------------------------------------------ ------ ------------ -------------- -----------
ATLAS 50 0.008 0.05 50
ATLAS (aco $<$ 0.01, $p_\perp^{\gamma\gamma}<2$ GeV) 50 0.007 0.01 10
CMS 103 0.03 0.2 180
CMS (aco $<$ 0.01, $p_\perp^{\gamma\gamma}<1$ GeV) 102 0.02 0.03 30
: Predicted cross sections, in nb, for diphoton final states within the ATLAS [@Aaboud:2017bwk] and CMS [@dEnterria:2018uly] event selections, in Pb–Pb collisions at $\sqrt{s}=5.02$ TeV. That is, the photons are required to have transverse energy $E_\perp^\gamma >2$ (3) GeV and pseudorapidity $|\eta^\gamma|<2.4$, while in the CMS case an additional cut of $m_{\gamma\gamma} > 5$ GeV is imposed. Results with and without an additional acoplanarity cut aco $<$ 0.01, and cut on the combined transverse momentum $p_\perp^{\gamma\gamma}<1$ (2) GeV in the CMS (ATLAS) case are shown. The cross sections for the light–by–light scattering (LbyL) and QCD–initiated photon pair production, in both the coherent and incoherent cases, are given. The result of simply scaling the $pp$ cross section (including the $pp$ survival factor) by $A^2 R^4$ with $R=0.7$ is also shown.[]{data-label="tab:lbylcs"}
However, in addition to the desired photon–initiated signal, there is the possibility that QCD–initiated diphoton production may contribute as a background. We are now in a position for the first time to calculate this, using the results of Section \[sec:qcdind\]. The results for the QCD–initiated background (both coherent and incoherent), as well as the prediction for the light–by–light signal, are shown in Table \[tab:lbylcs\]. We consider both the ATLAS and CMS event selection in the central detectors. Namely, the produced photons are required to have transverse energy $E_\perp^\gamma >2$ (3) GeV and pseudorapidity $|\eta^\gamma|<2.4$ in the case of CMS (ATLAS), while for CMS an addition cut of $m_{\gamma\gamma} > 5$ GeV is imposed. We show results before and after further cuts on the diphoton system $p_\perp^{\gamma\gamma}<1 (2) $ GeV for CMS (ATLAS) and acoplanarity ($1-\Delta \phi_{\gamma\gamma}/\pi<0.01$) are imposed, which are designed to suppress the non–exclusive background.
For the light–by–light signal the predicted cross sections are fully consistent with the ATLAS and CMS results: $$\begin{aligned}
\sigma^{\rm ATLAS} &=70 \pm 24\, ({\rm stat.}) \pm 17 \,({\rm syst.}) \,{\rm nb} \;,\\
\sigma^{\rm CMS} &=120 \pm 46 \,({\rm stat.}) \pm 28\, ({\rm syst.})\pm 4\, ({\rm th.}) \,{\rm nb} \;.\end{aligned}$$ On the other hand, we find that the QCD–initiated background is expected to be very small. In particular, both the incoherent and coherent contributions are expected to be negligible, even before imposing additional acoplanarity cuts.
We can see that incoherent background, which we recall corresponds to the case that the colliding ions do not remain intact, is further suppressed by the additional acoplanarity and $p_\perp^{\gamma\gamma}$ cuts; as we would expect, due to the broader $p_\perp$ spectrum of the incoherent cross section. This is seen more clearly in Fig. \[fig:aco\], which shows the (normalized) acoplanarity distributions in the three cases. We can see that the QED–initiated process is strongly peaked at low acoplanarity ($<$ 0.01), as is the coherent QCD–initiated process, albeit with a somewhat broader distribution due to the broader QCD form factor in this case. On the other hand, for incoherent QCD–initiated production we can see that the spectrum is spread quite evenly over the considered acoplanarity region.
It was suggested in [@d'Enterria:2013yra] that to calculate the QCD–initiated background, understood to be the dominant incoherent part, we can simply scale the corresponding $pp$ cross section by a factor of $A^2 R^4$, where $R\approx 0.7$ accounts for nuclear shadowing effects. As discussed in Section \[sec:qcdind\], this $\sim A^2$ scaling is certainly far too extreme, due to the short–range nature of the QCD interaction and corresponding requirement that only peripheral interactions can lead to exclusive or semi–exclusive production. In addition, we note that as the dominant contribution in this case will come from nucleons situated close to the ion peripherary, where the nucleon number density is relatively low, we can expect shadowing effects to be minimal, and hence we are justified in using the standard proton PDF in the calculation of the CEP cross section. Nonetheless, for the sake of comparison we also show the predictions from this $\sim A^2 R^4$ scaling in Table \[tab:lbylcs\], where we include the $pp$ survival factor. We can see that the cross section prediction in this case is, as expected, much larger, by many orders of magnitude. Such an approach will therefore dramatically overestimate the expected background. On the other hand, the relative reduction with the application of the acoplanarity and $p_\perp^{\gamma\gamma}$ cuts is similar to the semi–exclusive case, being driven by the same QCD form factor which enters in both cases.
![Normalized differential cross sections for exclusive and semi–exclusive diphoton production with respect to the diphoton acoplanarity. The QED–initiated and QCD–initiated (both coherent and incoherent) processes are shown.[]{data-label="fig:aco"}](aco)
It should be emphasised that in both the ATLAS and CMS analyses the normalization of the QCD–initiated background is in fact determined by the data. In particular, the predicted QCD background from this $A^2 R^4$ scaling is allowed to be shifted by a free parameter $f^{\rm norm}$, which is fit to the observed cross section in the aco $>0.01$ region, where the LbyL signal is very low. Interestingly in both analyses a value of $f^{\rm norm} \approx 1$ is preferred, which is significantly larger than our prediction; from Table \[tab:lbylcs\] we can roughly expect $f^{\rm norm} \sim \sigma^{\rm incoh}/\sigma^{A^2 R^4} \sim 10^{-3}$. However, great care is needed in interpreting these results: as is discussed in [@dEnterria:2018uly] this normalization effectively account for [*all*]{} backgrounds that result in large acoplanarity photons, not just those due to QCD–initiated production. Indeed, in this analysis it is explicitly demonstrated that the MC for the background for $e^+ e^-$ production significantly undershoots the data in the large acoplanarity region, and it is suggested that this could be due to events where extra soft photons are radiated. Our results clearly predict that the contribution to the observed events in the large acoplanarity region should not be due to QCD–initiated production, suggesting that a closer investigation of other backgrounds, such as the case of $e^+ e^- + \gamma$ discussed in [@dEnterria:2018uly], would be worthwhile.
Finally, we note that in the ATLAS analysis [@Aaboud:2017bwk] the number of events in the region with diphoton acoplanarity $> 0.01$, where the QED–initiated CEP signal will be strongly suppressed, with and without neutrons detected in the ZDCs is observed. They find 4 events with a ZDC signal, that is with ion dissociation, and 4 without, which roughly corresponds to a $O(10\,{\rm fb})$ cross section in both cases. However from Table \[tab:lbylcs\] we predict a much smaller cross sections of roughly $0.04$ (0.01) fb with (without) ZDC signals, i.e. 0 events in both cases. While some care is needed, in particular as the predictions in Table \[tab:lbylcs\] have not been corrected for detector effects, this predicted QCD contribution is clearly far too low to explain these observed events. We note that the probability of excitation of a GDR in each ion can be rather large (in [@Baltz:2002pp] a probability of $\sim 30\%$ for the related vector meson photoproduction process is predicted), however these should generally lead to events in the acoplanarity $<0.01$ region. Inelastic photon emission can lead to ion break up at larger acoplanarity, but is predicted in [@Hencken:1995me] to be at the % level. Again, clearly further investigation of these issues is required.
`SuperChic 3`: generated processes and availability {#sec:superchic}
===================================================
`SuperChic 3` is a Fortran based Monte Carlo that can generate the processes described above and in [@Harland-Lang:2015cta], with and without soft survival effects. User–defined distributions may be output, as well as unweighted events in the HEPEVT, Les Houches and HEPMC formats. The code and a user manual can be found at http://projects.hepforge.org/superchic.
Here we briefly summarise the processes that are currently generated, referring the reader to the user manual for further details. The QCD–initiated production processes are: SM Higgs boson via the $b\overline{b}$ decay, $\gamma\gamma$, 2 and 3–jets, light meson pairs ($\pi,K,\rho,\eta('),\phi$), quarkonium pairs ($J/\psi$ and $\psi(2S)$) and single quarkonium ($\chi_{c,b}$ and $\eta_{c,b}$). Photoproduction processes are: $\rho$, $\phi$, $J/\psi$, $\psi(2S)$ and $\Upsilon(1S)$. Photon–initiated processes are: $W$ pairs, lepton pairs, $\gamma\gamma$, SM Higgs boson via the $b\overline{b}$ decay, ALPs, monopole pairs and monopolium. $pp$, $pA$ and $AA$ collisions are available for arbitrary ion beams, for QCD and photon–initated processes. For photoproduction, currently only $pp$ and $pA$ beams are included. Electron beams are also included for photon–initiated production.
Conclusions and outlook {#sec:conc}
=======================
In this paper we have presented the updated `SuperChic 3` Monte Carlo generator for central exclusive production. In such a CEP process, an object $X$ is produced, separated by two large rapidity gaps from intact outgoing protons, with no additional hadronic activity. This simple signal is associated with a broad and varied phenomenology, from low energy QCD to high energy BSM physics, and is the basis of an extensive experimental programme that is planned and ongoing at the LHC.
`SuperChic 3` generates a wide range of final–states, via QCD and photon–initiated production and with pp, pA and AA beams. The addition of heavy ion beams is a completely new update, and we have included a complete description of both photon and QCD–initiated production. In the latter case this is to the best of our knowledge the first time such a calculation has been attempted. We have accounted for the probability that the ions do not interact inelastically, and spoil the exclusivity of the final state. While this is known to be a relatively small effect in the photon–initiated case, in the less peripheral QCD–initiated case the impact has been found to be dramatic.
These issues are particularly topical in light of the recent ATLAS and CMS observations of exclusive light–by–light scattering in heavy ion collisions. We have presented a detailed comparison to these results, and have shown that the signal cross section can be well produced by our SM predictions, and any background from QCD–initiated production is expected to be essentially negligible, in contrast to some estimates presented elsewhere in the literature. We find that the presence of additional events outside the signal region, with and without neutrons observed in the ZDCs (indicating ion break up) cannot be explained by the predicted QCD–initiated background. Addressing this open question therefore remains an experimental and/or theoretical challenge for the future.
Finally, there are very promising possibilities to use the CEP channel at high system masses to probe electroweakly coupled BSM states with tagged protons during nominal LHC running, accessing regions of parameters space that are difficult or impossible to reach using standard inclusive search channels. With this in mind, we have presented updates for photon–initiated production in pp collisions, including axion–like particle, monopole pairs and monopolium, as well as an updated calculated of SM light–by–light scattering including $W$ boson loops. These represent only a small selection of possible additions to the MC, and indeed as the programme of CEP measurements at the LHC continues to progress, we can expect further updates to come.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank David d’Enterria and Marek Tasevsky for useful discussions, Vadim Isakov for useful clarifications on questions related to nuclear structure, and Radek [Ž]{}leb[č]{}[' a]{}k for identifying various bugs and mistakes in the previous MC version. LHL thanks the Science and Technology Facilities Council (STFC) for support via grant awards ST/P004547/1. MGR thanks the IPPP at the University of Durham for hospitality. VAK acknowledges support from a Royal Society of Edinburgh Auber award.
$A$ scaling in QCD–induced production {#ap:incoh}
=====================================
In this appendix we derive the scaling behaviour for QCD–initiated production in heavy ion collisions. As discussed in Section \[sec:semiex\], we are interested in the peripheral region, $r \gtrsim R$. We denote the direction of the ion–ion impact parameter $b_\perp$ as $x$ and the orthogonal transverse direction as $y$. We can write the $x$ position for each ion as $x_i = R + \delta x_i$, with $i=1,2$. As we have $R\gg d$ we can expand in $\delta x/R$, to give r\_i-R\_i +x\_i, where we have used that $y_1=y_2=y$ and $z_1=z_2=z$. We then have \[eq:apptb\] T(b\_[i]{}) = z (r) \_0 z e\^[-]{}\_0 z e\^[-]{}e\^[-]{}=\_0 e\^[-]{}e\^[-]{}. In what follows, we will consider for simplicity a point–like QCD interaction. In other words, in the case of exclusive production for the ion–ion opacity we have $$\begin{aligned}
\Omega_{A_1A_2}(b_\perp) &= \int {\rm d}^2 b_{1\perp} {\rm d}^2 b_{2\perp} T_{A_1}(b_{1\perp})T_{A_2}(b_{2\perp})A_{nn}(b_\perp-b_{1\perp}+b_{2\perp})\;,\\
&\approx \sigma_{\rm tot}^{nn} \int {\rm d}^2 b_{1\perp} T_{A_1}(b_{1\perp})T_{A_2}(b_{1\perp}-b_\perp)\;,\end{aligned}$$ which is valid when the $nn$ interaction radius is much smaller than the extent of the ion transverse densities. In setting the normalization we have used . For the case of semi–exclusive production, we simply replace $\sigma_{\rm tot}^{nn} \to \sigma_{\rm inel}^{nn}$, see . As we are only interested in the overall scaling with $A$, we will for simplicity assume $\sigma_{\rm inel}^{nn} \sim \sigma_{\rm tot}^{nn}$, and work with the latter variable in what follows; however, strictly speaking this replacement should be made when considering semi–exclusive production. We now consider the proton–ion and ion–ion cases in turn.
Proton–ion collisions
---------------------
In this case, we take $T_{A_2} (b_\perp) = \delta^{(2)}(\vec{b}_{\perp})$, so that the opacity simply becomes \_[pA]{}(b\_) = \_[tot]{}\^[nn]{} T\_A(b\_) \_[tot]{}\^[nn]{}\_0 (2Rd)\^[1/2]{} e\^[-]{}e\^[-]{}, which defines the constant $\omega$. Here, we have used the fact that for proton–ion collisions, the coordinate choice we have taken above corresponds to setting $y=0$, and we drop the subscript on the $\delta x$ for simplicity. Note that the integration is explicitly only performed over the peripheral region, i.e. over a ring of radius $\sim R$ and thickness $\delta x$, where we will expect a non–negligible contribution to the CEP cross section. Recalling , the incoherent cross section is given by \_[incoh]{}\^[pA]{} = \_[CEP]{}\^[pp]{} \^2 b\_ T\_A(b\_) e\^[-(b\_)]{} = 2R \_[CEP]{}\^[pp]{} x . The exponent falls sharply with increasing $\delta x$, and has a maximum at $\delta x = d \ln \omega$. We can therefore apply the saddle point approximation to evaluate the integral, giving \_[incoh]{}\^[pA]{} 2R \_[CEP]{}\^[pp]{} d \^[-1]{} = \_[CEP]{}\^[pp]{}. Taking $R \approx (4\pi/3)^{1/3} A^{1/3} \rho_0^{-1/3}$, we then have \_[incoh]{}\^[pA]{} ()\^[1/3]{} A\^[1/3]{} \_[CEP]{}\^[pp]{}\~1.0 A\^[1/3]{} \_[CEP]{}\^[pp]{}, where for concreteness we have substituted the values $\sigma_{\rm tot}^{nn}=90$ mb, $\rho_0=0.15 \,{\rm fm}^{-3}$ and $d=0.5$ fm.
For coherent production, we have instead $$\begin{aligned}
\sigma_{\rm coh}^{pA} &= \frac{4\pi}{ \left\langle q_{\perp}^2\right \rangle} \sigma_{\rm CEP}^{pp} \int {\rm d}^2 b_{\perp} T_A(b_{\perp})^2 e^{-\Omega(b_\perp)}\;,\\
&= \frac{4\pi}{ \left\langle q_{\perp}^2\right \rangle} \sigma_{\rm CEP}^{pp} \frac{\omega^2}{(\sigma_{\rm tot}^{nn})^2}2\pi R \int {\rm d} \delta x \exp\left[ -2\frac{\delta x}{d} -\omega e^{-\frac{\delta x}{d}}\right]\;.\end{aligned}$$ The exponent now has a maximum at $\delta x = d \ln \left(\omega/2\right)$, and we find $$\begin{aligned}
\sigma_{\rm coh}^{pA} &= \frac{4\pi}{ \left\langle q_{\perp}^2\right \rangle} 2\pi R \frac{\omega^2}{(\sigma_{\rm tot}^{nn})^2} \sigma_{\rm CEP}^{pp}\cdot \frac{4\pi^{1/2} d}{e^2} \omega^{-2} =\frac{4\pi}{ \left\langle q_{\perp}^2\right \rangle \sigma_{\rm tot}^{nn}} \frac{2^{3/2}}{e} \sigma_{\rm incoh}^{pA}\;,\\
& \sim \frac{4\pi}{ \left\langle q_{\perp}^2\right \rangle \sigma_{\rm tot}^{nn}} \sigma_{\rm coh}\sim 0.2 \cdot A^{1/3} \cdot \sigma_{\rm CEP}^{pp} \;,\end{aligned}$$ where we have substituted numerically as in .
Ion–ion collisions
------------------
For simplicity we will assume that $R_1=R_2=R$ in what follows, although the results can be readily be generalised. In this case, the opacity takes the form \_[AA]{}(b\_) = \_[tot]{}\^[nn]{} \^2 b\_[1]{} T\_A(b\_[1]{}) T\_A(b\_-b\_[1]{})= \_[[tot]{}]{}2R d\_0\^2 x [d]{} y e\^[-]{}e\^[-]{}, where we have imposed the constraint that $\delta x_1 + \delta x_2 = |b_\perp| - 2R \equiv \Delta$, which defines $\Delta$. Performing the integrals we have \[eq:pcalc\] \_[AA]{}(b\_) =\_[tot]{}\^[nn]{}2 (R d)\^[3/2]{}\_0\^2 e\^[-]{} D e\^[-]{}, where we integrate $x$ over the interval $\Delta$. Considering first the incoherent cross section, we have \_[incoh]{}\^[AA]{} = \^2 b\_P e\^[-P]{} where $P=D \Delta e^{-\Delta/d}$.
As before we only integrate over the peripheral region, with a ring of thickness $\Delta$ and radius $2R$. We have $$\begin{aligned}
\label{eq:incohaa}
\sigma_{\rm incoh}^{AA} &= 4\pi R \frac{\sigma_{\rm CEP}^{pp}}{\sigma_{\rm tot}^{nn}} \int {\rm d}\Delta \,P e^{-P}\;,\\
& = 4\pi R \frac{\sigma_{\rm CEP}^{pp}}{\sigma_{\rm tot}^{nn}}\int {\rm d}P\, \frac{e^{-P}}{|{\rm d}\ln P/{\rm d}\Delta|}\;,\\
& = 4\pi R \frac{\sigma_{\rm CEP}^{pp}}{\sigma_{\rm tot}^{nn}}\int {\rm d}P\, \frac{e^{-P}}{\left|\frac{1}{\Delta} - \frac{1}{d}\right|}\;.\end{aligned}$$ The dominant contribution to this last integral comes from the region of $P\sim 1$. As an example, for the case of colliding lead ions, with $R=6.68$ fm, d=0.5 fm and $\sigma_{\rm tot}^{nn}=100$ mb for $\sqrt{s}=5.02$, we find $D\sim 15$ ${\rm fm}^{-1}$ in . Thus $P \sim 1$ implies a rather large value of $\Delta \sim 1.5$ fm, i.e. $\Delta \sim 3 d$. This gives ${\rm d}\ln P/{\rm d}\Delta \sim -2/3d$ and hence the exclusive contribution comes from a ring in $b$ space of radius $R_1+R_2$ and thickness $\delta b \sim 1.5 d$. The $A$–dependence of the cross section is simply \_[incoh]{} \~3R d A\^[1/3]{}, Thus we expect a $\sim A^{1/3}$ scaling, with no additional numerical suppression in the prefactors.
For the coherent case a similar approach can be taken, however instead of we find $$\begin{aligned}
\sigma_{\rm coh}^{AA} &=4\pi R \,\sigma_{\rm CEP}^{pp}\left(\frac{4\pi}{\sigma_{\rm tot}^{nn} \left \langle q_{\perp}^2 \right \rangle}\right)^2 \frac{1}{(2\pi R d)^{1/2}}\int {\rm d}\Delta \,\frac{P^2 e^{-P} }{\Delta} \;,\\
&=4\pi R \,\sigma_{\rm CEP}^{pp}\left(\frac{4\pi}{\sigma_{\rm tot}^{nn} \left \langle q_{\perp}^2 \right \rangle}\right)^2 \frac{1}{(2\pi R d)^{1/2}}\int {\rm d}P \,\frac{P^2 e^{-P} }{\left|1 - \frac{\Delta}{d}\right|} \;,\\
&\sim \left(\frac{4\pi}{\sigma_{\rm tot}^{nn} \left \langle q_{\perp}^2 \right \rangle}\right)^2 \cdot \sigma_{\rm CEP}^{pp} \cdot A^{1/6}\;,\end{aligned}$$ where in the second line we again use that the dominant part of the integral comes from the $P\sim 1$ region.
[^1]: Correspondingly, we have $s= A_1 A_2 s_{nn}$, where $s_{nn}$ is the squared c.m.s. energy per nucleon and $A_i$ is the ion mass number.
[^2]: Strictly speaking this is only true for the contribution proportional to the electric form factors, see [@Harland-Lang:2015cta] for further discussion; however here we indeed take $F_M=0$.
[^3]: To be precise, we omit the region $(-r_{\rm core},r_{\rm core})$, that is we take mean value of $z=0$ for the active nucleon. In the case of the ion–ion opacity, for which additional nucleon–nucleon interactions can take place at different impact parameters to the active nucleon, such a simple replacement will in general overestimate the cross section, but for peripheral collisions this remains a good approximation.
|
---
abstract: 'We address the question of which quantum states can be inter-converted under the action of a time-dependent Hamiltonian. In particular, we consider the problem as applied to mixed states, and investigate the difference between pure and mixed-state controllability introduced in previous work. We provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group. We also address the problem of which states can be prepared if the dynamical Lie group is not sufficiently large to allow the system to be controllable.'
address:
- |
Quantum Processes Group and Dept of Applied Maths, The Open University,\
Milton Keynes, MK7 6AA, United Kingdom
- 'Department of Mathematics and Institute of Theoretical Science, University of Oregon, Eugene, Oregon, 97403, USA'
author:
- S G Schirmer and A I Solomon
- J V Leahy
bibliography:
- 'books.bib'
- 'papers2000.bib'
- 'papers9599.bib'
- 'papers9094.bib'
- 'papers8089.bib'
- 'papers--79.bib'
title: Criteria for reachability of quantum states
---
Introduction
============
The subject of control of quantum systems has been a fruitful area of investigation lately. The growing interest in the subject can be attributed both to theoretical and experimental breakthroughs that have made control of quantum phenomena an increasingly realistic objective, as well as the prospect of many exciting new applications such as quantum computers [@IJMPA16p3335] or quantum chemistry [@SCI288p0824], which attracts researchers from various fields.
Among the theoretical problems that have received considerable attention lately is the issue of controllability of quantum systems. Various aspects such as the controllability of quantum systems with continuous spectra [@IEEE39CDC2803; @JMP24p2608], wavefunction controllability for bilinear quantum systems [@CRAS330p327; @CP267p1; @IEEE39CDC3003], controllability of distributed systems [@PNAS94p4828], controllability of molecular systems [@PRA51p960], controllability of spin systems [@qph0106115], controllability of quantum evolution in NMR spectroscopy [@MP96p1739], and controllability of quantum systems on compact Lie groups [@IEEE39CDC1086; @JPA34p1679; @PRA63n063410; @qph0110147] have been addressed, and related problems such as the dynamical realizability of kinematical bounds on the optimization of observables [@IEEE39CDC3002; @PRA63n025403], and the relation between controllability and universality of quantum gates [@PRA54p1715], as well as the information-theoretic limits of control [@PRL84p1156] have been studied.
In this process, various notions of controllability have been introduced. Recent work on controllability of quantum systems on compact Lie groups has finally shown that the degree of controllability of a quantum system depends on its dynamical Lie group, and that many different notions of controllability are in fact equivalent. In particular, it has been proved that quantum systems evolving on a compact Lie group, such as closed quantum systems with a discrete energy spectrum, are either density matrix / operator controllable, pure-state / wavefunction controllable, or not controllable [@qph0106128; @qph0108114]. For density matrix, operator or completely controllable quantum systems, every kinematically admissible target state or operator can be dynamically realized, and the kinematical bounds on the expectation values (ensemble averages) of observables are always dynamically attainable [@PRA63n025403]. Fortunately, many quantum systems have been shown to be completely controllable [@JPA34p1679; @PRA63n063410; @qph0108114].
Nevertheless, there are quantum systems that are either only pure-state controllable or not controllable at all. For instance, it has been shown that the dynamical Lie group of certain atomic systems with degenerate energy levels is the (unitary) symplectic group, which corresponds to pure-state controllability [@qph0108114]. Other systems with certain symmetries may be either pure-state controllable or non-controllable depending on the symmetry. For instance, given a system with $N$ equally spaced energy levels and uniform dipole moments for transitions between adjacent levels, the dynamical Lie group is the symplectic group if the dimension of its Hilbert space $N$ is even, but it is the orthogonal group if $N$ is odd [@JPA35p2327]. For these systems, the question of dynamical reachability of target states, which is important in many applications, remains. In this paper, we address this problem by studying the action of the dynamical Lie group of pure-state-only and non-controllable quantum systems on the kinematical equivalence classes of states. Explicit criteria for dynamical reachability of states are derived for systems whose dynamical Lie group is the (unitary) symplectic group or the orthogonal group.
Quantum states and kinematical/dynamical equivalence classes
============================================================
We consider a quantum system whose state is represented by a density matrix acting on a Hilbert space $\H$ of dimension $N$. A density matrix always has a discrete spectrum with non-negative eigenvalues $w_n$ that sum to one, $\sum_n w_n =1$, and a spectral resolution of the form $$\label{eq:DM}
\op{\rho} = \sum_{n=1}^N w_n \ket{\Psi_n}\bra{\Psi_n},$$ where $\ket{\Psi_n}$ are the eigenstates of $\op{\rho}$. The $\ket{\Psi_n}$ for $1\le
n\le N$ are elements of the Hilbert space $\H$ and can always be chosen so as to form a complete orthonormal set for $\H$. The $\bra{\Psi_n}$ are the corresponding dual states defined by $$\ip{\Psi_n}{\Psi_m} = \delta_{mn} \qquad \forall m,n.$$
Conservation laws such as conservation of energy and probability require the time evolution of any (closed) quantum system to be unitary. Thus, given a Hilbert space vector $\ket{\Psi_0}$, its time evolution is determined by $\ket{\Psi(t)}=\op{U}(t)
\ket{\Psi_0}$ where $\op{U}(t)$ a unitary operator for all $t$ and $\op{U}(0)=\op{I}$. Hence, a density matrix $\op{\rho}_0$ must evolve according to $$\op{\rho}(t)=\op{U}(t)\op{\rho}_0\op{U}(t)^\dagger,$$ where $\op{U}(t)$ is unitary for all times. This constraint of unitary evolution induces kinematical restrictions on the set of target states that are physically admissible from any given initial state.
Two quantum states represented by density matrices $\op{\rho}_0$ and $\op{\rho}_1$ are *kinematically equivalent* if there exists a unitary operator $\op{U}$ such that $\op{\rho}_1 = \op{U} \op{\rho}_0 \op{U}^\dagger$.
Thus, the constraint of unitary evolution partitions the set of density matrices on $\H$ into (infinitely many) kinematical equivalence classes. It is well known that two density matrices $\op{\rho}_0$ and $\op{\rho}_1$ are kinematically equivalent if and only if they have the same eigenvalues. The kinematical equivalence classes are therefore determined by the eigenvalues of $\op{\rho}$. Furthermore, we introduce the following classification of density matrices according to their eigenvalues, which we shall relate to the degree of controllability of the system.
Every density matrix is of one of the following types.
1. \[type1\] *Completely random ensembles:* Density matrices whose spectrum consists of a single eigenvalue $w_1=\frac{1}{N}$ that occurs with multiplicity $N$.
2. \[type2\] *Pure-state-like ensembles:* Density matrices whose spectrum consists of two distinct eigenvalues, one of which occurs with multiplicity one and the other with multiplicity $N-1$.
3. \[type3\] *General ensembles:* Density matrices whose spectrum consists of at least two distinct eigenvalues, at least one of which occurs with multiplicity $N_1$ where $2\le N_1\le N-2$; or density matrices whose spectrum consists of $N$ distinct eigenvalues ($N\ge 2$).
Note that type (\[type2\]) (pure-state-like ensembles) includes density matrices representing pure states such as $\op{\rho}=\mbox{diag}(1,0,0,0)$ but not every density matrix in this class represents a pure state. For instance, $\op{\rho}=\mbox{diag}
(0.7,0.1,0.1,0.1)$ is of type (\[type2\]) but does *not* represent a pure state.
Given a specific quantum system with a control-dependent Hamiltonian of the form $$\label{eq:H}
\op{H}[f_1(t),\ldots,f_M(t)] = \op{H}_0 + \sum_{m=1}^M f_m(t) \op{H}_m,$$ where the $f_m$, $1\le m\le M$, are (independent) bounded measurable control functions, the question arises which states are dynamically reachable from a given initial state. Clearly, the set of potentially dynamically reachable states is restricted to states within the same kinematical equivalence class as the initial state. However, not every kinematically admissible target state is necessarily dynamically reachable. Since the time-evolution operator $\op{U}(t)$ has to satisfy the Schrodinger equation $$\label{eq:SE}
\rmi\hbar \frac{d}{d t} \op{U}(t) = \op{H}[f_1(t),\ldots,f_M(t)] \op{U}(t),$$ where $\op{H}$ is the Hamiltonian defined above, only unitary operators of the form $$\op{U}(t)=\exp_+\left\{-\frac{\rmi}{\hbar}\op{H}
\left[f_1(t),\ldots,f_M(t)\right]\right\},$$ where $\exp_+$ denotes the time-ordered exponential, qualify as evolution operators. Using, for instance, the Magnus expansion of the time-ordered exponential, it can be seen that only unitary operators of the form $\exp(\op{x})$, where $\op{x}$ is an element in the dynamical Lie algebra $\L$ generated by the skew-Hermitian operators $\rmi\op{H}_0,\ldots,\rmi\op{H}_M$, are dynamically realizable. These operators form the dynamical Lie group $S$ of the system.
Two kinematically equivalent states $\op{\rho}_0$ and $\op{\rho}_1$ are *dynamically equivalent* if there exists a unitary operator $\op{U}$ in the dynamical Lie group $S$ such that $\op{\rho}_1=\op{U}\op{\rho}_0\op{U}^\dagger$.
This dynamical equivalence relation subdivides the kinematical equivalence classes.
In the following, we shall be particularly concerned with the unitary group $U(N)$, the special unitary group $SU(N)$, the (unitary) symplectic group $Sp(\frac{N}{2})$ and the (unitary) orthogonal group $SO(N)$. As usual, the unitary group $U(N)$ is the compact Lie group consisting of all regular $N\times N$ matrices $\op{U}$ that satisfy $\op{U}^\dagger\op{U}=\op{U}\op{U}^\dagger =\op{I}$. The special unitary group $SU(N)$ is the subgroup of $U(N)$ consisting of all unitary matrices $\op{U}\in U(N)$ whose determinant is $+1$. For our purposes in this paper, we define the symplectic group and the special orthogonal group as follows.
The (unitary) *symplectic group* $Sp(\ell)$ is the subgroup of $SU(2\ell)$ consisting of all unitary operators of dimension $2\ell$ that satisfy $\op{U}^T\op{J}\op{U}=\op{J}$ for $$\label{eq:Jsp}
\op{J}= \left( \begin{array}{cc} 0 & \op{I}_\ell \\
-\op{I}_\ell & 0
\end{array} \right),$$ where $\op{I}_\ell$ is the identity matrix of dimension $\ell$.
The (unitary) *special orthogonal group* $SO(N)$ is the subgroup of $SU(N)$ consisting of all unitary operators of dimension $N$ that satisfy $\op{U}^T\op{J}\op{U}=\op{J}$ for $$\label{eq:Jso}
\op{J}= \left( \begin{array}{cc} 0 & \op{I}_\ell \\
\op{I}_\ell & 0
\end{array} \right), \quad N=2\ell, \quad
\op{J}= \left( \begin{array}{ccc} 1 & 0 & 0 \\
0 & 0 & \op{I}_\ell \\
0 & \op{I}_\ell & 0
\end{array} \right), \quad N=2\ell+1.$$
Dynamical Lie group action on the kinematical equivalence classes
=================================================================
The set of quantum states that is dynamically accessible from a given initial state $\op{\rho}_0$ depends on the action of the dynamical Lie group $S$ on the kinematical equivalence classes of density operators.
The dynamical Lie group $S$ of a quantum system is said to *act transitively* on a kinematical equivalence class $\C$ of density matrices if any two states in $\C$ are *dynamically equivalent*.
Since the equivalence class of completely random ensembles \[type (\[type1\]) above\] consists only of a single state $\op{\rho}=\frac{1}{N}\op{I}_N$, it follows immediately that *every* group acts transitively on this equivalence class.
Any dynamical Lie group $S$ that does not act transitively on the kinematical equivalence class of pure states, acts transitively only on the trivial kinematical equivalence class of completely random ensembles. Furthermore, from classical results by Montgomery and Samelson [@43Montgomery], it follows that $U(N)$, $SU(N)$, $Sp(\frac{1}{2}N)$ and $Sp(\frac{1}{2}N)\times U(1)$ are the *only* dynamical Lie groups (up to isomorphism) that act transitively on the equivalence class of pure states. Therefore, any dynamical Lie group $S$ that is not isomorphic to either $U(N)$, $SU(N)$, $Sp(\frac{1}{2}N)$ or $Sp(\frac{1}{2}N)\times U(1)$ acts transitively only on type (\[type1\]) states, i.e., completely random ensembles. $U(N)$ and $SU(N)$ clearly act transitively on *every* kinematical equivalence class of states, which leaves only $Sp(\frac{1}{2}N)$ and $Sp(\frac{1}{2}N)\times U(1)$, whose action on the kinematical equivalence classes of states we shall now address.
We begin by showing that transitive action of $Sp(\frac{1}{2}N)$ on pure states implies transitive action on all equivalence classes of type (\[type2\]). We shall prove this result for the standard representation of $Sp(\frac{1}{2}N)$ as defined above. To see that this is sufficient, note that lemma 4.2 in [@qph0106128] shows that whenever the dynamical Lie algebra of a quantum system of the type considered in this paper is *isomorphic* to $sp(\frac{1}{2}N)$, then it is *conjugate* to $sp(\frac{1}{2}N)$ via an element in $U(N)$. Thus, if the dynamical Lie group $S$ of the system is of type $Sp(\frac{1}{2}N)$ then it is not only isomorphic to the standard representation of $Sp(\frac{1}{2}N)$, but there exists a unitary transformation (basis change) $\op{B}$ that maps any unitary operator in $\op{U}\in S$ to a unitary operator $\tilde{U}=\op{B}
\op{U}\op{B}^\dagger$ in the standard representation of $Sp(\frac{1}{2}N)$, i.e., $S$ is unitarily equivalent to the standard representation of $Sp(\frac{1}{2}N)$.
Note that theorem 6 in [@qph0106128] gives a general condition for transitive action of a dynamical Lie group $S\subset U(N)$ on a kinematical equivalence class of states represented by a density matrix $\op{\rho}$: the action is transitive if and only if $$\label{eq:dimFormula}
\dim U(N) - \dim S = \dim \C_\op{\rho} - \dim (\C_\op{\rho} \cap S),$$ where $\C_\op{\rho}$ is the centralizer of $\op{\rho}$ and $\C_\op{\rho}\cap S$ is the intersection of the centralizer with $S$. However, since determination of the dimension of $\C_\op{\rho}$, and especially $\C_\op{\rho}\cap S$, tends to be very difficult in practice (see \[appendix:B\] for an example) we shall *not* use this result but pursue an alternative approach instead.
\[lemma1\] $Sp(\frac{1}{2}N)$ acts transitively on all kinematical equivalence classes of density matrices whose eigenvalues satisfy $w_1 \neq w_2=w_3=\ldots=w_N$.
Any $\op{\rho}$ with eigenvalues $w_1\neq w_2=w_3=\ldots=w_N$ can be written as $$\op{\rho} = w_1 \ket{\Psi}\bra{\Psi} + w_2 \op{P}(\ket{\Psi}^\perp),$$ where $\op{P}(\ket{\Psi}^\perp)$ is the projector onto the orthogonal complement of the subspace spanned by $\ket{\Psi}$. Hence, any pair of kinematically equivalent states of this type is of the form $$\begin{aligned}
\op{\rho}_0 &=& w_1\ket{\Psi^{(0)}}\bra{\Psi^{(0)}}+w_2\op{P}(\ket{\Psi^{(0)}}^\perp)\\
\op{\rho}_1 &=& w_1\ket{\Psi^{(1)}}\bra{\Psi^{(1)}}+w_2\op{P}(\ket{\Psi^{(1)}}^\perp).\end{aligned}$$ Since $Sp(\frac{1}{2}N)$ acts transitively on the equivalence class of pure states, there exists a unitary operator $\op{U}\in Sp(\frac{1}{2}N)$ such that $\op{U}\ket{\Psi^{(0)}}=
\ket{\Psi^{(1)}}$. Furthermore, $\op{U}$ automatically maps the orthogonal complement of $\ket{\Psi^{(0)}}$ onto the orthogonal complement of $\ket{\Psi^{(1)}}$ since it is unitary and thus we have $$\op{U}\op{\rho}^{(0)}\op{U}^\dagger
= w_1 \ket{\Psi^{(1)}}\bra{\Psi^{(1)}} + w_2 \op{P}(\ket{\Psi^{(1)}}^\perp)
= \op{\rho}^{(1)}.$$ Hence, $Sp(\frac{1}{2}N)$ acts transitively on all equivalence classes of density matrices whose eigenvalues satisfy $w_1\neq w_2=w_3=\ldots=w_N$.
However, the action of $Sp(\frac{1}{2}N)$ on the class of pure states is not two-point transitive as the following example shows.
\[example:one\] Let $N=2\ell$ and $\vec{a}$ and $\vec{b}$ be two unit vectors in $\CC^N$. Since $N=2\ell$, we can partition the vectors as follows $$\vec{a}=\left(\begin{array}{c}\vec{a}_1 \\ \vec{a}_2 \end{array}\right), \quad
\vec{b}=\left(\begin{array}{c}\vec{b}_1 \\ \vec{b}_2 \end{array}\right),$$ where $\vec{a}_j$, $\vec{b}_j$ for $j=1,2$ are vectors in $\CC^\ell$. Since $Sp(\ell)$ acts transitively on the unit sphere in $\CC^N$ it follows that there exists a $\op{U}
\in Sp(\ell)$ such that $\op{U}\vec{a}=\vec{b}$. However, since any unitary operator in $Sp(\ell)$ satisfies $\op{U}^T\op{J}\op{U}=\op{J}$ with $\op{J}$ as in (\[eq:Jsp\]), we have $\op{U}=\op{J}^\dagger\op{U}^*\op{J}$ and thus $\op{J}^\dagger\op{U}^*\op{J}
\vec{a}=\vec{b}$ or equivalently $\op{U}\op{J}\vec{a}^* = \op{J}\vec{b}^*$. Noting that $$\op{J}\vec{a}^*=\left(\begin{array}{c} -\vec{a}_2^* \\
\vec{a}_1^*
\end{array}\right), \quad
\op{J}\vec{b}^*=\left(\begin{array}{c} -\vec{b}_2^* \\
\vec{b}_1^*
\end{array}\right),$$ it thus follows that $\op{U}$ maps $\vec{c}\equiv \op{J}\vec{a}^*$ onto $\vec{d}\equiv
\op{J}\vec{b}^*$. Therefore, given two (orthogonal) unit vectors of the form $\vec{a}$ and $\vec{c}$, it is not possible to find a unitary transformation in $Sp(\ell)$ that maps these two vectors onto two arbitrary (orthogonal) unit vectors. Rather, once we have chosen the image of $\vec{a}$, the image of $\vec{c}$ is fixed.
This lack of two-point transitivity has serious implications for the action of $Sp(\frac{1}{2}N)$, in particular it implies non-transitive action on all kinematical equivalence classes of type (\[type3\]).
\[lemma2\] $Sp(\frac{1}{2}N)$ does not act transitively on kinematical equivalence classes of density matrices with at least three distinct eigenvalues, two of which having multiplicity one.
Any two kinematically equivalent density matrices can be written as $$\op{\rho}_0=\sum_{n=1}^N w_n \ket{\Psi_n}\bra{\Psi_n}, \quad
\op{\rho}_1=\sum_{n=1}^N w_n \ket{\Phi_n}\bra{\Phi_n}.$$ Since there are at least three distinct eigenvalues and two of them have multiplicity one, we may assume $w_1\neq w_n$ for all $n\neq 1$ and $w_2\neq w_n$ for all $n\neq 2$. Thus, $\ket{\Psi_n}$ and $\ket{\Phi_n}$ for $n=1,2$ are unique up to phase factors and any $\op{U}$ such that $\op{\rho}_1=\op{U}\op{\rho}_0\op{U}^\dagger$ must map $\ket{\Psi_n}$ onto $\ket{\Phi_n}$ (modulo phase factors) for $n=1,2$, i.e., $$\op{U}\ket{\Psi_1}=e^{i\phi_1}\ket{\Phi_1}, \quad
\op{U}\ket{\Psi_2}=e^{i\phi_2}\ket{\Phi_2},$$ However, suppose $\ket{\Psi_1}\doteq\vec{a}$, $\ket{\Psi_2}\doteq\vec{c}$ and $\ket{\Phi_1}\doteq \vec{b}$ but $\ket{\Phi_2}\neq e^{i\phi}\vec{d}$, where $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$ are as defined in example \[example:one\]. This example then shows that it is impossible to find a $\op{U}\in Sp(\frac{1}{2}N)$ that simultaneously maps $\ket{\Psi_1}$ onto $\ket{\Phi_1}$ and $\ket{\Psi_2}$ onto $\ket{\Phi_2}$. Therefore, there does not exist a unitary operator in $Sp(\frac{1}{2}N)$ such that $\op{\rho}_1=\op{U}\op{\rho}_0\op{U}^\dagger$.
\[lemma3\] $Sp(\frac{1}{2}N)$ does not act transitively on equivalence classes of density matrices that have at least one non-zero eigenvalue that occurs with multiplicity greater than one but less than $N-1$.
Suppose $w_1$ has multiplicity $N_1$ where $2\le N_1\le N-2$. If $Sp(\frac{1}{2}N)$ acts transitively on the selected equivalence class of states then we must be able to map the $N_1$-dimensional eigenspace $E^{(0)}(w_1)$ for $\op{\rho}_0$ onto the corresponding eigenspace $E^{(1)}(w_1)$ for $\op{\rho}_1$ by a unitary operator in $Sp(\frac{1}{2}N)$ for any $\op{\rho}_0$ and $\op{\rho}_1$ in the same equivalence class. However, it is easy to see that this is not always possible. Suppose $E^{(0)}(w_1)$ contains a pair of vectors of the form $\vec{a}$, $\vec{c}$ as defined above and $E^{(1)}(w_1)$ contains a vector $\vec{b}$ but the related vector $\vec{d}$ is in the orthogonal complement of $E^{(1)}(w_1)$. Then it is impossible to map $E^{(0)}(w_1)$ onto $E^{(1)}(w_1)$ by a $\op{U}\in Sp(\frac{1}{2}N)$. Since the orthogonal complement of $E^{(1)}(w_1)$ has at least dimension two, we can always choose $E^{(1)}(w_1)$ such that $\vec{d}\in E^{(1)}
(w_1)^\perp$. Hence, $Sp(\frac{1}{2}N)$ does not act transitively on the selected equivalence class of states.
Given any two mixed states $\op{\rho}_0$ and $\op{\rho}_1$ related by $\op{\rho}_1=
\op{U}\op{\rho}_0\op{U}^\dagger$ for some $\op{U}\in Sp(\frac{1}{2}N)\times U(1)$, we can find a $\tilde{U}\in Sp(\frac{1}{2}N)$ such that $\op{\rho}_1=\tilde{U}\op{\rho}_0
\tilde{U}^\dagger$. For instance, if $\det\op{U}=e^{i\alpha}$, setting $\tilde{U}=
e^{-i\alpha/N}\op{U}$ produces an operator with $\det(\tilde{U})=1$ that obviously satisfies $$\tilde{U}\op{\rho}_0\tilde{U}^\dagger=\op{U}\op{\rho}_0\op{U}^\dagger=\op{\rho}_1.$$ Thus, $Sp(\frac{1}{2}N)\times U(1)$ acts transitively on a kinematical equivalence class $\C$ of density matrices if and only if $Sp(\frac{1}{2}N)$ does. Combining this observation with the previous lemmas yields the following theorem.
- $U(N)$ and $SU(N)$ act transitively on all kinematical equivalence classes.
- $Sp(\frac{1}{2}N)$ and $Sp(\frac{1}{2}N)\times U(1)$ act transitively on all kinematical equivalence classes of density matrices of type (\[type1\]) or (\[type2\]) and *only* those.
- Any other dynamical Lie group acts transitively only on the trivial kinematical equivalence class of completely random ensembles.
Criteria for reachability of target states
==========================================
Having established that the action of the dynamical Lie groups $Sp(\frac{1}{2}N)$ and $Sp(\frac{1}{2}N)\times U(1)$ is *not* transitive on any kinematical equivalence class of density matrices of type (\[type3\]), and that all other dynamical Lie groups except $U(N)$ and $SU(N)$ act transitively only on the trivial kinematical equivalence class of completely random ensembles, the question of identifying states that are kinematically but not dynamically equivalent arises.
Since dynamical Lie groups can be very complicated, it would be unrealistic to expect that simple criteria for dynamical equivalence of states can be derived for arbitrary dynamical Lie groups. However, for certain types of dynamical Lie groups of special interest, such as $Sp(\frac{1}{2}N)$ \[or $Sp(\frac{1}{2}N)\times U(1)$\] and $SO(N)$ \[or $SO(N)\times U(1)$\], this is possible, as will be shown in the following.
Systems with dynamical Lie group $Sp(\frac{1}{2}N)$ or $Sp(\frac{1}{2}N)\times U(1)$
------------------------------------------------------------------------------------
To address the problem of finding criteria for dynamical equivalence of states for systems whose dynamical Lie group $S$ is isomorphic (unitarily equivalent) to $Sp(\frac{1}{2}N)$, we recall that any unitary operator $\op{U}\in Sp(\ell)$ satisfies $\op{U}^T\op{J}\op{U}=
\op{J}$ for $\op{J}$ as defined in (\[eq:Jsp\]). Thus, any dynamical evolution operator $\op{U}$ for a system of dimension $N=2\ell$ with dynamical Lie group of type $Sp(\ell)$ must satisfy $$\op{U}^T\tilde{J}\op{U}=\tilde{J}$$ for a matrix $\tilde{J}$, which is unitarily equivalent to (\[eq:Jsp\]).[^1] Therefore, we must have $$\op{U} = \tilde{J}^\dagger \op{U}^* \tilde{J}, \quad
\op{U}^\dagger = \tilde{J}^\dagger \op{U}^T \tilde{J}.$$ Two kinematically equivalent states $\op{\rho}_0$ and $\op{\rho}_1$ are thus dynamically equivalent if and only if there exists a unitary operator $\op{U}$ such that $$\op{\rho}_1=\op{U}\op{\rho}_0\op{U}^\dagger \quad \mbox{and} \quad
\op{\rho}_1=\tilde{J}^\dagger\op{U}^*\tilde{J}\op{\rho}_0\tilde{J}^\dagger\tilde{U}^T\tilde{J},$$ or equivalently, $$\label{eq:DE}
\op{\rho}_1 = \op{U}\op{\rho}_0\op{U}^\dagger \mbox{ and }
\underbrace{(\tilde{J}\op{\rho}_1\tilde{J}^\dagger)^*}_{\tilde{\rho}_1}
=\op{U}\underbrace{(\tilde{J}\op{\rho}_0\tilde{J}^\dagger)^*}_{\tilde{\rho}_0}\op{U}^\dagger.$$
Let $N=4$ and $S=Sp(2)$ with $\tilde{J}=\op{J}$ as in (\[eq:Jsp\]).
1. Then $\op{\rho}_0=\mbox{diag}(a,a,b,b)$ ($0\le a,b \le \frac{1}{2}$, $a+b=
\frac{1}{2}$) and $\op{\rho}_1=\mbox{diag}(a,b,b,a)$ are dynamically equivalent since there exists a unitary operator $\op{U}$ such that $\op{\rho}_1=\op{U}
\op{\rho}_0\op{U}^\dagger$ and any such $\op{U}$ clearly maps $\tilde{\rho}_0=
\mbox{diag}(b,b,a,a)$ to $\tilde{\rho}_1 =\mbox{diag}(b,a,a,b)$.
2. $\op{\rho}_0$ and $\op{\rho}_2=\mbox{diag}(a,b,a,b)$, on the other hand, are not dynamically equivalent (unless $b=a$) since $\tilde{\rho}_2=\op{\rho}_2$ but $\tilde{\rho}_0\neq\op{\rho}_0$ and there cannot be a unitary operator such that $\op{\rho}_1=\op{U}\op{\rho}_0\op{U}^\dagger=\op{U}\tilde{\rho}_0\op{U}^\dagger$ if $\op{\rho}_0\neq \tilde{\rho}_0$.
This shows that $S=Sp(2)$ divides any kinematical equivalence class of states with two distinct eigenvalues of multiplicity $\ell=2$ into at least two disjoint subsets of dynamically equivalent states.
Sometimes the condition $\op{U}^T\tilde{J}\op{U}=\tilde{J}$ can also be used directly to show that two states are not dynamically equivalent.
Consider again $N=4$ and $S=Sp(2)$ with $\tilde{J}=\op{J}$ as in (\[eq:Jsp\]) as well as the initial state $\op{\rho}_0=\mbox{diag}(a,b,c,d)$ where $0\le a,b,c,d\le 1$, $a+b+c+d=1$ and $a,b,c,d$ mutually different. We can conclude that the state $\op{\rho}_1=\diag(b,a,c,d)$ is not dynamically equivalent to $\op{\rho}_0$ since we would require a unitary operator of the form $$\op{U} = \left(\begin{array}{cccc} 0 & e^{i\phi_1} & 0 & 0 \\
e^{i\phi_2} & 0 & 0 & 0 \\
0 & 0 & e^{i\phi_3} & 0 \\
0 & 0 & 0 & e^{i\phi_4}
\end{array} \right)$$ which does not satisfy $\op{U}^T\op{J}\op{U}=\op{J}$.
Another way of showing that two (kinematically equivalent) density matrices are not dynamically equivalent is to prove that (\[eq:DE\]) cannot have a solution by showing that the related linear system $$\label{eq:DE2}
\op{\rho}_1\op{U} - \op{U}\op{\rho}_0 = 0, \quad
\tilde{\rho}_1 \op{U} - \op{U}\tilde{\rho}_0 = 0$$ does not have a solution. To verify this, we note that the linear system above can be rewritten in the form $\A\vec{U}=0$ where $\A$ is a matrix with $2N^2$ rows and $N^2$ columns and $\vec{U}$ is a column vector of length $N^2$. If the null space of $\A$ is empty then there is no $\vec{U}$ such that $\A\vec{U}=0$ and hence there is no $N\times
N$ matrix $\op{U}$ that satisfies (\[eq:DE2\]). However, note that if the linear system above *does* have a solution, this does not imply that the states in question are dynamically equivalent since the solution to the linear equation is in general not unitary.
Systems with dynamical Lie group $SO(N)$ or $SO(N)\times U(1)$
--------------------------------------------------------------
From the previous discussion, we know that $SO(N)$ does not act transitively on any kinematical equivalence class other than the trivial one. However, we can establish criteria for dynamical equivalence of states similar to those for $Sp(\frac{1}{2}N)$ by noting that any unitary operator $\op{U}\in SO(N)$ must satisfy $\op{U}^T\op{J}\op{U}
=\op{J}$ for $\op{J}$ as in (\[eq:Jso\]). Therefore, two kinematically equivalent states $\op{\rho}_0$ and $\op{\rho}_1$ are dynamically equivalent under the action of a dynamical Lie group $S$ which is unitarily equivalent to $SO(N)$, if there exists a unitary operator $\op{U}$ such that $$\op{\rho}_1 = \op{U}\op{\rho}_0\op{U}^\dagger \mbox{ and }
\underbrace{(\tilde{J}\op{\rho}_1\tilde{J}^\dagger)^*}_{\tilde{\rho}_1}
=\op{U}\underbrace{(\tilde{J}\op{\rho}_0\tilde{J}^\dagger)^*}_{\tilde{\rho}_0}\op{U}^\dagger.$$ with $\tilde{J}$ unitarily equivalent to (\[eq:Jso\]), and determined as described in \[appendix:A\].
Consider a system with $N=5$ and Hamiltonian $\op{H}=\op{H}_0+f(t)\op{H}_1$ where $$\op{H}_0=\left(\begin{array}{ccccc} -2 & 0 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 2
\end{array} \right), \qquad
\op{H}_1=\left(\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{array} \right).$$ It can be verified using the algorithm described in [@PRA63n063410] that the Lie algebra of this system has dimension $10$, which is equal to the dimension of $so(5)$. Using the technique described in \[appendix:A\], we find that both of the generators $\rmi\op{H}_0$ and $\rmi\op{H}_1$ of the Lie algebra satisfy $\op{x}^T\tilde{J}+\tilde{J}
\op{x}=0$ for $$\tilde{J} = \left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 &-1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 &-1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0
\end{array} \right),$$ which is unitarily equivalent to the standard $\op{J}$ for $so(5)$. We can thus conclude that its dynamical Lie algebra is $so(5)$ and its dynamical Lie group is $SO(5)$. Furthermore, note that the two pure states $$\op{\rho}_0=\left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array} \right), \qquad
\op{\rho}_1=\left(\begin{array}{ccccc} 0.5 & 0 & 0 & 0 & 0.5 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0.5 & 0 & 0 & 0 & 0.5
\end{array} \right)$$ are not dynamically equivalent since $(\tilde{J}\op{\rho}_1\tilde{J}^\dagger)^*=\op{\rho}_1$ but $(\tilde{J}\op{\rho}_0\tilde{J}^\dagger)^*\neq\op{\rho}_0$ and it is thus impossible to find a unitary transformation such that $\op{U}\op{\rho}_0\op{U}^\dagger=\op{\rho}_1
=\op{U}(\tilde{J}\op{\rho}_0\tilde{J}^\dagger)^*\op{U}^\dagger$.
Conclusion
==========
The question of dynamical equivalence of kinematically equivalent quantum states has been been addressed by studying the action of the dynamical Lie group of the system on the kinematical equivalence classes. For systems whose dynamical Lie group is unitarily equivalent to either $Sp(\frac{1}{2}N)$ or $SO(N)$, explicit criteria for dynamical reachability / equivalence of states have been given, and their application illustrated with several examples.
Furthermore, we have provided a classification of density matrices according to their eigenvalues, which divides mixed quantum states into three main types: (i) completely random ensembles, (ii) pure-state-like ensembles, and (iii) general ensembles. We have also proved that the dynamical Lie group $Sp(\frac{1}{2}N)$ acts transitively on all equivalence classes of quantum states of type (\[type1\]) and (\[type2\]), but *only* those.
Although it is known that a pure-state controllable system whose dynamical Lie group $S$ is isomorphic to $Sp(\frac{1}{2}N)$ is *not* density matrix controllable in general [@qph0106128], this result shows that there are more than just a few examples of kinematically equivalent density matrices that are not dynamically reachable from one another in this case. In fact, the action of $S$ is *not* transitive on *almost all* kinematical equivalence classes. This is in marked contrast to the action of $S$ for a density matrix controllable system, which is transitive on *all* kinematical equivalence classes, as well as the action of $S$ for a non-controllable system, which is transitive only on the trivial kinematical equivalence class of completely random ensembles.
SGS acknowledges the hospitality and financial support of the Department of Mathematics and the Institute of Theoretical Science at the University of Oregon, where most of this work was completed. AIS acknowledges the hospitality of the Laboratoire de Physique Théorique des Liquides, University of Paris IV, where he is currently a visiting faculty member.
Finding $\op{J}$ for dynamical Lie groups of type $Sp(\frac{1}{2}N)$ or $SO(N)$ {#appendix:A}
===============================================================================
For the results of the previous sections to be truly useful, we must also address the question of how to determine the $\tilde{J}$ matrix of a given system. To this end, note that the elements of the dynamical Lie algebra $L$ associated with the dynamical groups $Sp(\frac{1}{2}N)$ and $SO(N)$ must satisfy a relation similar to the one satisfied by the elements of the group, namely any $\op{x}\in L$ must satisfy $$\label{eq:Jcond}
\op{x}^T \tilde{J}+ \tilde{J}\op{x} = 0,$$ where $\tilde{J}$ is the same as for the related group. Thus, given a system with total Hamiltonian (\[eq:H\]), this implies in particular that the generators $\rmi\op{H}_m$ of the dynamical Lie algebra must satisfy (\[eq:Jcond\]).
Equation (\[eq:Jcond\]) can be written as a system of linear equations of the form $$\L_m \vec{J} = 0, \quad 0\le m\le M,$$ where $\L_m$ is a square matrix of dimension $N^2$ determined by the generators $\rmi\op{H}_m$ and $\vec{J}$ is a column vector of length $N^2$. The solutions $\vec{J}$ of the above matrix equation can be found by computing the null space of the operator $$\left( \begin{array}{c} \tilde{\L}_0 \\ \vdots \\ \tilde{\L}_M \end{array} \right).$$ If the dynamical Lie group is of type $Sp(\frac{1}{2}N)$ or $SO(N)$ then the nullspace contains a single element $\vec{J}$, which can be rearranged into a square matrix whose eigenvalues agree with whose of the standard $\op{J}$ for the group defined above. That is, concretely,
- if $N=2\ell$ and $\tilde{J}$ has two distinct eigenvalues $+\rmi$ and $-\rmi$, both of which occur with multiplicity $\ell$ then the dynamical Lie group is $Sp(\ell)$;
- if $N=2\ell$ and $\tilde{J}$ has two distinct eigenvalues $+1$ and $-1$, both of which occur with multiplicity $\ell$ then the dynamical Lie group is $SO(2\ell)$;
- if $N=2\ell+1$ and $\tilde{J}$ has two distinct eigenvalues $+1$ and $-1$, occurring with multiplicity $\ell+1$ and $\ell$, respectively, then the dynamical Lie group is $SO(2\ell+1)$;
Hence, the algorithm not only determines $\tilde{J}$ but it also allows us to decide whether the dynamical Lie group is of type $Sp(\frac{1}{2}N)$ or $SO(N)$.
Note that the dynamical Lie group $S$ can only be $Sp(\frac{1}{2}N)$ or $SO(N)$ if all the partial Hamiltonians $\op{H}_m$ of the system have zero trace. However, if any of the partial Hamiltonians $\op{H}_m$ has non-zero trace then the dynamical Lie group of the system can still be $Sp(\frac{1}{2}N)\times U(1)$ or $SO(N)\times U(1)$. To deal with this situation, we note that $S \simeq Sp(\frac{1}{2}N)\times U(1)$ or $S\simeq
SO(N)\times U(1)$ is possible only if the generators $$\op{x}_m=\rmi\op{H}_m -\frac{\rmi}{N}\Tr(\op{H}_m)\op{I}_N, \quad 0\le m\le M$$ of the related trace-zero Lie algebra $L'$ satisfy (\[eq:Jcond\]) for $0\le m\le M$ and we can thus proceed as above to determine $\tilde{J}$.
Comparison of Theorem 1 with Theorem 6 in [@qph0106128] {#appendix:B}
=======================================================
To demonstrate the difficulty in using theorem 6 in [@qph0106128] to verify whether the dynamical Lie group $S$ of a system acts transitively on an equivalence class of density operators, we shall consider a simple example.
Assume the dynamical Lie group of the system is $Sp(2) \subset U(4)$. According to theorem 1 above, $Sp(2)$ does *not* act transitively on the kinematical equivalence class represented by $\op{\rho}=\mbox{diag}(a,a,b,b)$ with $0\le a,b\le \frac{1}{2}$ and $a+b=\frac{1}{2}$ since $\op{\rho}$ is of type (\[type3\]).
To show that the action is not transitive using theorem 6 in [@qph0106128], we note first that $\dim U(4)=16$ and $\dim Sp(2)=10$. Thus, the left hand side in (\[eq:dimFormula\]) is $\dim U(4)-\dim Sp(2)=6$.
To compute the right hand side, we need to determine the centralizer $\C_\op{\rho}$ of $\op{\rho}$. Noting that $$\op{\rho}= \left( \begin{array}{cc} a \op{I}_2 & 0 \\ 0 & b \op{I}_2 \end{array}\right),$$ where $\op{I}_2$ is the indentity matrix in dimension 2, we see that $\op{\rho}$ commutes with every unitary matrix of the form $$\op{U}_C= \left( \begin{array}{cc} \op{A}_1 & 0 \\ 0 & \op{A}_2 \end{array}\right),$$ where $\op{A}_1$ and $\op{A}_2$ are arbitrary unitary matrices in $U(2)$. Thus, the centralizer of $\op{\rho}$ is $U(2)\times U(2)$ and its dimension is $4+4=8$.
To compute the intersection of $\C_\op{\rho}$ with $S=Sp(2)$, we recall that any matrix in $Sp(2)$ must preserve $\op{J}$ as defined in (\[eq:Jsp\]). Concretely, this means $\op{U}_C^T\op{J}\op{U}_C =\op{J}$, i.e., $$\op{U}_C^T \op{J} \op{U}_C =
\left( \begin{array}{cc}
0 & \op{A}_1^T\op{A}_2 \\
-\op{A}_2^T\op{A}_1 & 0
\end{array}\right)
= \left(\begin{array}{cc}
0 & \op{I}_2 \\
-\op{I}_2 & 0
\end{array}\right).$$ Thus, we must have $\op{A}_1^T\op{A}_2 = \op{I}_2$. Noting that $\op{A}_1$ and $\op{A}_2$ are unitary, this is only possible if $\op{A}_1=\op{A}_2^*$, i.e., if $\op{A}_1$ is the complex conjugate of $\op{A}_2$, since $(\op{A}_2^*)^T\op{A}_2=\op{A}_2^\dagger\op{A}_2
=\op{I}$. Hence, the intersection of the centralizer $\C_\op{\rho}$ with $S=Sp(2)$ is $U(2)$, which has dimension $4$. Hence, $\dim\C_\op{\rho}-\dim (\C_\op{\rho}\cap Sp(2))
=8-4=4\neq 6$, i.e., the left and right hand side in (\[eq:dimFormula\]) are *not* equal. Thus we have shown using theorem 6 that the action of $Sp(2)$ on the kinematical equivalence class of $\op{\rho}$ is not transitive.
References {#references .unnumbered}
==========
[^1]: See \[appendix:A\] for details about how to determine $\tilde{J}$.
|
---
abstract: 'We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.'
address:
- 'Felix Schulze: Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany'
- 'Miles Simon: Universität Magdeburg , Universitätsplatz 2, 39106 Magdeburg, Germany'
author:
- Felix Schulze
- Miles Simon
title: |
Expanding solitons with non-negative curvature\
operator coming out of cones
---
0.3 true in
Introduction and statement of results
=====================================
0.1 true in
Let $(M^n,h)$ be a smooth $n$-dimensional, complete, non-compact Riemannian manifold without boundary, with non-negative curvature operator and bounded curvature. In particular $(M,h)$ has non-negative sectional curvature and non-negative Ricci curvature. Any rescaling of this space also has non-negative sectional curvatures, and hence for every sequence of scalings $(M,c_i h,p_i)$, $c_i \in {\mathbb{R}}^+$ , $p_i \in M$, $i \in {\mathbb{N}}$, there exists a subsequence which converges in the pointed Gromov Hausdorff sense to a metric space $(X,d_X)$ (see Proposition 10.7.1 in [@BuBuIv]) which is a [*metric space with curvature $\geq 0$*]{} (see Definition 4.6.2 in [@BuBuIv]). In the case that $c_i \rightarrow 0$ and $p_i
= p$ for all $i \in {\mathbb{N}}$, the limit is known as $(X,d_X,0)$ [*the asymptotic cone at infinity*]{} and it is unique: see Lemma 3.4 of [@GK]. It is the Euclidean cone over a metric space $(V,d_V)$ where $(V,d_V)$ is an Alexandrov space of curvature bounded from below by one, and $0$ is the tip of the cone: see Corollary 3.5 of [@GK] for example.
The Euclidean cone $CV$ over a metric space $V$ is homeomorphic to the space ${\mathbb{R}}^+_0 \times V/
\sim $ with the quotient topology, where $(r,y) \sim
(s,x)$ if and only if ($r= s= 0$) or ($r=
s$ and $x= y$). The metric is given by: $$d_{CV}((r,x),(s,y)):= r^2 + s^2
-2rs\cos(\min(d_V(x,y),\pi)).$$ In the case that $CV$ arises as a Gromov Hausdorff limit in the setting described above, and the sequence $(M,c_i h,p_i)$ is [*non-collapsing*]{}, then $V$ is homeomorphic to $S^{n-1}$. This follows from (unpublished) results of G. Perelman [@PeUn] as explained and simplified by V. Kapovitch in the paper [@Kap1]. See Appendix \[B\].
[**Notation:**]{} A pointed sequence $(X_i,d_i,p_i)$ of metric spaces is [*non-collapsing*]{} if ${{\rm vol}}(B_1(p)) \geq {\delta}>0$ for all $p \in X_i$ and some ${\delta}>0$ independent of $i$.
Our aim is to flow such cones $(CV,d_{CV})$ by Hamilton’s Ricci flow (introduced in [@Ha82]). We will show that a solution to Ricci flow with initial value given by the cone exists, that the solution is immediately smooth, and that it is an expanding Ricci soliton with non-negative curvature operator. For an interval $I$ (open, closed, half open, finite or infinite length) and a smooth manifold $M$ without boundary, a smooth (in space and time) family of complete Riemannian metrics $\{g(\cdot,t) \}_{t \in I}$ solves [**Ricci flow**]{} if $ { {\partial {\over}{\partial t}} }g(t) = - 2\, {\rm Ric}(g(t))$ for all $t \in I$. If $I =(0,T)$, we say $(M,g(t))_{t \in (0,T)}$ is a solution to Ricci flow with initial value $(M,d_0)$ ($(M,d_0)$ a metric space), if $\lim_{t \searrow 0} (M,d(g(t)))= (M,d_0) $ in the Gromov-Hausdorff sense ($(M,d(g))$ is the metric space associated to $(M,g)$).
Let $(M,h)$ be a smooth, complete Riemannian manifold without boundary, with non-negative Ricci curvature. The asymptotic volume ratio ${{\rm AVR}}(M,h)$ is $${{\rm AVR}}(M,h):=
\lim_{r \to \infty} \frac{{{\rm vol}}(B_r(x))} { r^n},$$ where $x$ is an arbitrary point in $M$.
Due to the Bishop Gromov volume comparison principle, ${{\rm AVR}}(M,h)$ is well defined for such manifolds, and does not depend on the point $x$. It also easily follows that $$\begin{aligned}
\frac {{{\rm vol}}(B_r(x))} { r^n} \geq
{{\rm AVR}}(M,h)\ \ \ \ \forall\ r >0. \end{aligned}$$ In particular if $V_0:= {{\rm AVR}}(M,h) >0$, then we have $$\begin{aligned}
\frac {{{\rm vol}}(B_r(x))} { r^n} \geq V_0 >0
\ \ \ \ \forall\ r >0. \end{aligned}$$
\[thm:mainthm\] Let $(M,h)$ be a smooth, complete Riemannian manifold without boundary, with non-negative, bounded curvature operator and positive asymptotic volume ratio $V_0 := {{\rm AVR}}(M,h)>0$. Let $(X,d_X,0)$ be the asymptotic cone at infinity, i.e. the unique Gromov-Hausdorff limit of $(M,c_i
h,p_0)$ for any sequence $c_i \to 0$ of positive numbers and any base point $p_0 \in M$. Then:
- There exists a smooth solution $(M,g(t))_{t \in
[0,\infty)}$ to Ricci flow with $g(0)= h$.
- Let $g^i(t):= c_i g(t/c_i)$, $i \in {\mathbb{N}}$, be the solutions to Ricci flow obtained by rescaling the flows obtained in (i). The pointed solutions $(M,g_i(t),p_0)_{ t \in (0,\infty)}$ converge smoothly, sub-sequentially (in the Hamilton-Cheeger-Gromov sense: see [@Ha95a]) as $i \to \infty$ to a limit solution $(\tilde X,\tilde{g}(t)_{t \in (0,\infty)},{\tilde}x_0)$. This solution $(\tilde X,\tilde{g}(t)_{t \in (0,\infty)},\tilde x_0)$ satisfies $(\tilde{X},d(\tilde{g}(t)),\tilde x_0) \to (X,d_X,0)$ in the Gromov-Hausdorff sense as $t \searrow 0$, $\text{AVR}({\tilde}X, {\tilde}g(t)) = \text{AVR}(X,d_X) = \text{AVR}(M,h) \
\ \forall\ t >0$, and $X$ is homeomorphic to ${\tilde}X$. Hence such a limit solution may be thought of as a solution to Ricci flow with initial value $(X,d_X,0)$. Furthermore, $(\tilde{X},\tilde{g}(t))_{t \in (0,\infty)}$ is an expanding gradient soliton with non-negative curvature operator. That is $\tilde{g}(t)= t (\phi_t)^*\tilde{g}(1)$ and $\tilde{g}(1)$ satisfies $${\rm Ric}(\tilde{g}(1))
-(1/2)\tilde{g}(1) + {}^{{\tilde}g(1)}{\nabla}^2 f$$ for some smooth function $f:M \to {\mathbb{R}}$.
As explained before, in this case $X$ is the Euclidean cone over a metric space $V$, and $V$ is homeomorphic to $S^{n-1}$, see Appendix B. Thus $X$ is homeomorphic to ${\mathbb{R}}^n$ .
In this paper (and in particular the above theorem) $
{}^g {\nabla}^j$ refers to the $j$th covariant derivative with respect to $g$. In order to prove this theorem we require a priori estimates for non-collapsed solutions with non-negative bounded curvature operator. This involves proving a refined version of Lemma 4.3 of [@Si6], suited to the current setting, which we now state.
\[thm:longexist\] Let $(M^n,g_0)$ be a smooth, complete Riemannian manifold without boundary, with bounded non-negative curvature operator and $V_0:= {{\rm AVR}}(M,g_0) >0$. Then there exists a constant $c= c(n,V_0)>0$ and a solution to Ricci-flow $(M,g(t))_{ t \in [0,\infty)}$ with $g(0)=
g_0$ such that $$\sup_M |{{\rm Riem}}(g(t))|
\leq \frac c t\ ,$$ for all $t \in (0,\infty)$.
It is known that any smooth, open solution to Ricci flow with non-negative, bounded curvature operator has constant asymptotic volume ratio, see [@Yokota08 Theorem 7]. That is, in the above theorem we have $$V_t:= \textup{AVR}(M,g(t)) = V_0\, ,$$ for all $t\in[0,\infty)$.
After completing this paper a pre-print of E. Cabezas-Rivas and B. Wilking appeared [@CabezasRivasWilking11] where the necessary a priori estimates are shown to extend our results to open Riemannian manifolds with non-negative (possibly unbounded) complex sectional curvature .
Previous results and structure of the paper
===========================================
The literature that exists on expanding, shrinking and steady solitons is vast. For a very good and current overview of the field, we refer the reader to the survey paper of H-D.Cao [@Cao2]. Here we mention some of the results on expanding solitons relevant to the current setting.
In the paper [@Cao3], the author constructs families of examples of Kähler gradient expanding solitons on ${ \mathbb C}^n$. He also shows that any solution to Kähler Ricci flow with bounded curvature and which
- exists for $t \in (0, \infty)$, and
- has non-negative holomorphic bi-sectional and positive Ricci curvature, and
- has $t { { \rm R }}(\cdot,t) \leq K$ for all $t>0$, and
- $\sup_{(x,t) \in M \times (0,\infty)} t { { \rm R }}(x,t)$ is attained
must itself be an expanding Kähler gradient soliton. This result was generalised to the case of Ricci flow with non-negative curvature operator by B-L.Chen and X-P.Zhu in [@ChZh] (see Proposition 4.2 there). That is, if we have a solution to Ricci flow which satisfies the above with condition $(ii)$ replaced by $({\tilde}{ii})$ [*has non-negative curvature operator and positive Ricci curvature*]{}, then the conclusion is, [*the solution must be an expanding gradient soliton*]{}.
Hence, it is natural to look for solutions satisfying all or some of these conditions, when trying to construct expanding solitons with non-negative curvature operator.
Both of these theorems use the linear trace Harnack inequality of B.Chow and R.Hamilton (see [@ChHa]).
A pre-print of L.Ma [@Ma] appeared after we had completed this paper. The pre-print contains a generalisation of the above result of B-L.Chen and X-P.Zhu [@ChZh]. In the paper [@Ma], the condition $({\tilde}{ii})$ above is replaced by the new $({\tilde}{ii})$ [*has non-negative Ricci curvature and is non-collapsed*]{}. The paper of L.Ma uses the $W_+$ functional, which is a generalisation of Perleman’s $W$ energy (see [@Pe]).
In the papers [@FeIlNi] and [@NiKae] the $W_+$ functional is studied globally and locally. $W_+$ is monotone non-decreasing, and constant precisely on expanders (up to a shift in time). See for example Theorem 1.1 in [@FeIlNi] and also Corollary 6.9 of [@NiKae], where a similar result to that of [@ChZh] is proved using the $W_+$ functional.
It is known that if an expanding soliton has bounded curvature and ${\rm Ric}> 0$, then $0< {{\rm AVR}}< \infty$. This was first proved by R.Hamilton for ${\rm Ric}>0$ (see Proposition 9.46 in the book [@ChLuNi]). In the paper [@CaNi] this result was generalised to the case that the curvature is not necessarily bounded: see proposition 5.1 therein. Sharper estimates for expanding gradient solitons under weaker assumptions are also proved there. We refer the reader to that paper for more details. Similar estimates may also be found in Proposition 4.1 of the paper [@ChTa].
Note that in our setting, we may assume that ${\rm Ric}>0$ after isometrically splitting off a factor ${\mathbb{R}}^m$ (see Section \[Rescaling\] for more details). Hence, the assumption that the manifolds we consider have $ {{\rm AVR}}>0$ is natural.
Further examples of and estimates on expanding, steady and shrinking solitons on ${ \mathbb C}^m$ are given in [@FeIlKn]. In particular, they construct an example of a Ricci flow which starts as a shrinking soliton (for time less than zero), flows into a cone at time zero and then into a smooth expanding soliton (for time bigger than zero). They also include a discussion (with justification) on the desirable properties of a [*weak Ricci flow*]{}.
The splitting result that we prove in Appendix A is essentially derived from that of Hamilton in [@Ha86] (see also [@Cao]).\
[**Structure of the paper:**]{} In chapter three we fix some notation. In chapter four we prove a short time existence result for smooth, complete, non-collapsed Riemannian manifolds with non-negative and bounded curvature operator. In chapter five we show by a scaling argument that these conditions actually imply longtime existence if the asymptotic volume ratio is positive. Furthermore the asymptotic volume ratio for the so obtained solutions remains constant. By blowing down such a flow parabolically we prove in chapter six that we obtain a smooth limiting solution, which evolves out of the asymptotic cone at infinity of the initial manifold. We furthermore show that this solution actually is an expanding soliton. In Appendix A we give a proof of a splitting result, which has its origin in the de Rham Splitting Theorem. In Appendix B we recall an approximation result from V.Kapovitch/G.Perelman.
Notation
==========
For a smooth Riemannian manifold $(M,g)$, and a family $(M,g(t))_{t \in [0,T)}$ of smooth Riemannian metrics, we use the notation
- $(M,d(g))$ is the metric space associated to the Riemannian manifold $(M,g)$,
- $d \mu_g$ is the volume form of the Riemannian manifold $(M,g)$,
- $d(x,y,t)= {\mbox{\rm dist}}_{g(t)}(x,y) = d(g(t))(x,y)$ is the distance between $x$ and $y$ in $M$ with respect to the metric $g(t)$,
- $ {}^g B_r(x)$ is the ball of radius $r$ and centre $x$ measured with respect to $d(g)$,
- $B_r(x,t)$ is the ball of radius $r$ and centre $x$ measured with respect to $d(g(t))$,
- ${{\rm vol}}(\Omega,g)$ is the volume of $\Omega$ with respect to the metric $g$,
- ${{\rm vol}}( {}^g B_r(x))= {{\rm vol}}( {}^g B_r(x),g )$,
- ${{\rm vol}}( B_r(x,t))= {{\rm vol}}(B_r(x,t),g(t))$,
- ${ {\mathcal R} }(g)$ is the curvature operator of $g$,
- ${{\rm Riem}}(g)$ is the curvature tensor of $g$,
- ${\rm Ric}(g)$ is the Ricci curvature tensor of $g$,
- ${ { \rm R }}(g)$ is the scalar curvature of $g$,
- ${ { \rm R }}(p,g)$ is the scalar curvature of the metric $g$ at the point $p$.
- If we write $B_r(x)$ resp. ${{\rm vol}}(B_r(x))$ then we mean ${}^g B_r(x)$ resp. ${{\rm vol}}( {}^g B_r(x))$, where $g$ is a metric which will be clear from the context.
Short time existence {#chapbound}
======================
Let $(M^n,g_0)$ be any smooth, complete manifold with bounded curvature and without boundary. From the results of R.Hamilton [@Ha82] and W-X.Shi [@Sh], we know that there exists a solution $(M,g(t))_{t \in [0,T)}$ to Ricci flow with $g(0)= g_0$ and $T \geq S(n,k_0)>0$ where $k_0 := \sup_M |{{\rm Riem}}(g_0)|$. That is: we can find a solution for a positive amount of time $T$ and $T$ is bounded from below by a constant depending on $k_0$ and $n$. The results of the paper [@Si6] show that if the initial manifold is smooth, complete, without boundary and has non-negative bounded curvature operator and ${{\rm vol}}(B_1(x,0)) \geq v_0 >0$ for all $x \in M$, then the there exists a solution for a time interval $[0,T)$ where $T\geq S(n,v_0)$. Note the difference to the results of Hamilton and Shi: the lower bound on the length of the time interval of existence does not depend on the constant $k_0 := \sup_M |{{\rm Riem}}(g_0)| < \infty$. Some estimates on the evolving curvature were also proved in that paper. We state this result here, and give a proof using the results of [@Si6].
\[Shorttime\] Let $(M,g_0)$ be smooth, complete Riemannian manifold without boundary, with non-negative and bounded curvature operator. Assume also that the manifold is non-collapsed, that is $$\begin{aligned}
{{\rm vol}}(B_1(x,0)) \geq
v_0 > 0 { \ \ \forall \ \ }x \in M. \end{aligned}$$ Then there exist constants $T= T(n,v_0) >0$ and $K(n,v_0)$ and a solution to Ricci flow $(M,g(t))_{t \in [0,T)}$ which satisfies $$\label{diogo}
\begin{split}
&(a_t) \ \ { {\mathcal R} }(g(t)) \geq 0, \\
&(b_t) \ \ {{\rm vol}}(B_1(x,t)) \geq {v_0 /2}, \\
&(c_t) \ \ \sup_{M} |{{\rm Riem}}(g(t))| \leq K^2/t, \\
&(d_t) \ \ d(p,q,s) \geq d(p,q,t) \geq
d(p,q,s) -K (\sqrt t - \sqrt s)
\end{split}$$ for all $ x,p,q \in M$, $ 0
< s \leq t \in [0,T)$, where ${ {\mathcal R} }(g)$ is the curvature operator of $g$.
The proof follows from the results contained in the paper [@Si6] and some other well known facts about Ricci flow. Using the result of [@Sh], we obtain a maximal solution $(M,g(t))_{t \in [0,T_\text{max})}$ to Ricci flow with $g(0)= g_0$, where $T_\text{max}>0$ and $\sup_M
|{{\rm Riem}}(g(t))| < \infty$ for all $t \in [0,T_\text{max})$ and $\lim_{t \nearrow T_\text{max}} \sup_M |{{\rm Riem}}(g(t))|= \infty$ if $T_\text{max} < \infty$. Also the curvature operator of the solution is non-negative at each time, since non-negative curvature operator is preserved for solutions with bounded curvature due to the maximum principle: see [@Ha95] and for example the argument in Lemma 5.1 of [@Si6] (the argument there shows that the maximum principle is applicable to this non-compact setting, in view of the fact that the curvature is bounded. Maximum principles of this sort are well known: see for example [@EH] or [@NiTa2]). Now the result follows essentially by following the proof of Theorem 6.1 in [@Si4]. For convenience we sketch the argument here, and refer the reader to the proof there for more details. Let $[0,T_M)$ be the maximal time interval for which the flow exists and $$\begin{aligned}
\inf_{x \in M} {{\rm vol}}(B_1(x,t)) & > & {v_0 {\over}2}, \label{maximal1}
\end{aligned}$$ for all $t \in [0,T_M)$. Using the maximum principle and standard ODE estimates, one shows easily that $T_M>0$ (see the proof of Theorem 7.1 in [@Si6] for details). The aim is now to show that $T_M \geq S$ for some $S= S(n,V_0)>0$. From Lemma 4.3 of [@Si6] we see that if $T_M \geq 1$ then the estimates $(a_t)$, $(b_t)$ and $(c_t)$ are satisfied for all $t \leq 1$, and $(d_t)$ would then follow from Lemma 6.1 of [@Si6], and hence we would be finished. So w.l.o.g. $T_M \leq 1$. From Lemma 4.3 of [@Si6] once again, $$\label{maximal2}
|{{\rm Riem}}(g(t)| \leq {c_0(n,v_0)
{\over}t}$$ for all $t \in (0,T_M)$ for some $c_0= c_0(n,v_0) < \infty$. First note that $(d_t)$ holds on the interval $(0,T_M)$ in view of Lemma 6.1 in [@Si6], and the fact that ${\rm Ric}(g(t)) \geq
0$. Using Corollary 6.2 of [@Si6], we see that there exists an $S=
S(v_0,c_0(v_0,n))= S(n,v_0)>0$, such that ${{\rm vol}}(B_1(x,t))
> {2v_0/3}$ for all $t \in [0,T_M) \cap
[0,S)$. If $T_M < S$, then we obtain a contradiction to the definition of $T_M$ ($T_M$ is the first time where the condition (\[maximal1\]) is violated). Hence $T_M \geq S$. But then we may use Lemma 4.3, Lemma 6.1 of [@Si6] to show that $(a_t) , (b_t), (c_t)$ and $(d_t)$ are satisfied on $(0,S)$, as required.
\[helpful\] Note that $T_\text{max} \geq (U(n) / k_0)$ and $\sup_{M} |{{\rm Riem}}(g(t))| \leq k_0 {\tilde}k(n)$ for all $t \leq (U(n) / k_0)$ for our solution, where $k_0 := \sup_{x \in M} |{{\rm Riem}}(g_0)| < \infty$ and ${\tilde}k(n),U(n)> 0$ are constants. This is due to the fact that our solution is constructed by extending a Shi solution, and the solutions of Shi satisfy such estimates by scaling.
Long time existence and estimates
=================================
The long time existence result follows essentially from scaling.
\[thm:longtimeexist\] Let $(M,g_0)$ be smooth, complete, without boundary, with non-negative bounded curvature operator. Assume also that ${{\rm AVR}}(M,g_0) =: V_0 > 0$. Then there exists a solution to Ricci flow $(M,g(t))_{t \in [0,\infty)}$ with $g(0) = g_0$. Furthermore, the solution satisfies the following estimates. $$\begin{aligned}
(a_t) & { {\mathcal R} }(g(t)) \geq 0 \cr
(b^\prime_t)& {{\rm AVR}}(M,g(t)) = V_0 \cr \nonumber
(c_t)&\sup_{M} |{{\rm Riem}}(g(t))| \leq {K^2 {\over}t}, \cr \nonumber
(d_t)& d(p,q,0) \geq d(p,q,t) \geq
d(p,q,s) -K (\sqrt t - \sqrt s)
\end{aligned}$$ for all $ t \in
[0,\infty)$ and $p,q \in M$, where $K= K(n,V_0)>0$ is a positive constant and ${ {\mathcal R} }(g)$ is the curvature operator of $g$.
Let $c \in (0,\infty)$ and ${\tilde}g_0 := c g_0$. Then we still have $\sup_M |{{\rm Riem}}({\tilde}g_0)| < \infty$ and ${{\rm AVR}}(M,{\tilde}g_0)=
V_0 >0$ as ${{\rm AVR}}(M,g)$ is a scale invariant quantity. From the Bishop-Gromov comparison principle, we have ${{\rm vol}}({\tilde}B_1(x)) \geq
V_0 >0$ for all $x \in M$. Using the result above (Theorem \[Shorttime\]), we obtain a solution $(M,{\tilde}g(t))_{t \in
[0,T(n,V_0))}$ satisfying ${\tilde}g(0)= {{{\tilde}g}}_0 $ and the estimates $(a_{t}), (b_{t}), (c_{t}), (d_{t})$ for all $ t \in [0,T(n,V_0))$. Setting $g(t):= (1/c) {{\tilde}g}(ct)$, for $t \in [0,(T/c))$ we also obtain a solution to Ricci flow with bounded non-negative curvature operator, satisfying $g(0)= g_0$ and the estimates $(a_t), (c_t) $ and $(d_t)$ for all $t \in [0,(T/c))$, as $(a_t),(c_t),(d_t)$ are invariant under this scaling. Furthermore, by [@Yokota08 Theorem 7] we have ${{\rm AVR}}(M,g(t)) = V_0$, and thus ${{\rm vol}}(B_r(x,t))
\geq V_0 r^n$ for all $r>0$. Now taking a sequence $c_i \to 0$ (in place of $c$ in the argument above), we obtain the result, in view of this estimate, $(a_t)$ and $(c_t)$, and the estimates of Shi and the compactness Theorem of Hamilton [@Ha95a], see [@Ha95]. Note that in fact $\sup_M |{{\rm Riem}}(g(t))| \leq k_0 {\tilde}k(n) $ for all $t
\leq U(n)/k_0$, where $k_0:= \sup_M |{{\rm Riem}}(g_0)|$, in view of Remark \[helpful\]. Additionally $\sup_M |{{\rm Riem}}(g(t))| \leq \frac{K^2
k_0}{U(n)}$ for all $t\geq U(n)/k_0 $ in view of the scale invariant estimate $(c_t)$ and hence the results of Shi (see [@Ha95]) apply.
Rescaling {#Rescaling}
=========
In this chapter we show that it is possible to scale down solutions of the type obtained in Theorem \[thm:longtimeexist\] to obtain an expanding soliton coming out of the asymptotic cone $(X,d_X)$ at infinity of $(M,h)$.
We assume that $(M,h)$ is a smooth manifold with non-negative, bounded curvature operator and positive asymptotic volume ratio $V_0:= \text{AVR}(M,h)>0$. Now let $c_i \rightarrow 0$ be a sequence of positive numbers, converging to zero. Then $(M,c_ih, p_0)$ converges in the pointed Gromov-Hausdorff sense to the metric cone $(X,d_X,0)$. By Theorem \[thm:longtimeexist\] there exists a Ricci flow $(M,g(t))_{t\in[0,\infty)}$ with $g(0)= h$ which satisfies $(a_t),(c_t)$ and $\text{AVR}(M,g(t))= \text{AVR}(M,g(0))$. By Hamilton’s Harnack estimate, see for example equation 10.46 of Chapter 10, §4 of [@ChLuNi] , we have $$\label{eq:harnack}
{ {\partial {\over}{\partial t}} }\big( \, t \, { { \rm R }}(p,g(t))\big)
\geq 0$$ for all $t \in
[0,\infty)$ for all $p \in M$. We define the scaled Ricci flows $(M,g^i(t))_{t\in[0,\infty)}$ by $$g^i(t):= c_ig(t/c_i)\ .$$ Note that these flows still satisfy $(a_t), (c_t)$ and $\text{AVR}(M,g^i(t)) =
\text{AVR}(M,h)$ and we have a uniform lower bound for the injectivity radius for times $t \in [\delta, \infty)$, $\delta >0$, since the curvature is uniformly bounded and the volume of balls is uniformly bounded from below on such time intervals. Hence, we may take a pointed limit of the flows $(M,g^i(t),p_0)_{t \in (0,
\infty)}$ to obtain a smooth Ricci flow $({\tilde}X,\tilde{g}(t),{\tilde}x_0)_{ t \in (0,\infty)}$ with $$\begin{aligned}
{\tilde}V_0 := \text{AVR}({\tilde}{X},{\tilde}{g}(t))
\geq \text{AVR}(M,h)= V_0 > 0 \label{tvz}
\end{aligned}$$ for all $t \in (0,\infty)$. ${\tilde}V_0 $ is a constant by [@Yokota08 Theorem 7]. Note that we also have the following estimates: $d_X(\cdot,\cdot) \geq d({\tilde}g(t))(\cdot,\cdot) \geq d_X(\cdot,\cdot)
- K\sqrt t$ where $(X,d_X,0)$ is the asymptotic cone at infinity of $(M,h)$. These estimates follow after taking a limit of the estimates $(d_t)$ $d(c_ih)(\cdot,\cdot) \geq d(g^i(t))(\cdot,\cdot) \geq d(c_ih)(\cdot,\cdot)
- K\sqrt t$, which hold by construction of our solution. In particular we see that $({\tilde}X, {\tilde}g(t), {\tilde}x_0)$ converges in the pointed Gromov-Hausdorff sense to $(X,d_X,0)$ as $t \rightarrow 0$.
A result of Cheeger and Colding (see Theorem 5.4 of [@ChCo]) gives that volume is continuous under the Gromov-Hausdorff limit of non-collapsing spaces with Ricci curvature bounded below. Thus since $(\tilde{X},
\tilde{g}(t), \tilde{x}_0)$ converges to the asymptotic cone at infinity $(X,d_X,0)$ of $(M,h,p_0)$ as $t\rightarrow 0$, and the Bishop-Gromov volume comparison principle holds, we have $${\tilde}V_0\leq\text{AVR}(X,d_X)= \text{AVR}(M,h) = V_0\ ,$$ and thus ${\tilde}V_0 = V_0$. By , we also have ${ {\partial {\over}{\partial t}} }(t { { \rm R }}(p,g^i(t)) ) \geq 0$ for all $t \in [0,\infty)$ and all $p \in M$. For $p \in M$ define $S(p) := \lim_{t\rightarrow \infty} t { { \rm R }}(p,g(t))$, which is a well defined and positive real (non-infinite) number by and $(c_t)$. Since this quantity is scale-invariant it follows that $$\lim_{i\rightarrow \infty} t_0 { { \rm R }}(p,g^i (t_0))= S(p)$$ for any fixed $t_0>0$. Note that the convergence of $(M,g^i(t),p_0) \to ({\tilde}X,\tilde{g}(t),{\tilde}x_0)$ is smooth on compact sets contained in $(0,\infty) \times {\tilde}X$, and $p_0$ is mapped by the diffeomorphisms involved in the pointed Hamilton-Cheeger-Gromov convergence onto ${\tilde}x_0$, thus we have $$t{ { \rm R }}({\tilde}x_0,\tilde{g}(t))= S(p_0)$$ for all $t>0$. Recall that the evolution equation for the Ricci curvature is given by $$\frac{\partial}{\partial t}{\rm Ric}^{i}_{\ j} = \Delta {\rm Ric}^{i}_{\ j}
+ 2\, {\rm Ric}^{r}_{\ s}\,{{\rm Riem}}^{i\ \ s}_{\ rj}\, .$$ Now note, that if ${\rm Ric}(y,\tilde{g}(t))(Y,Y)= 0$ for some $Y \in T_y {\tilde}X$ then we have in view of the de Rham Decomposition Theorem (see Appendix \[A\] with $h$ of the Decomposition Theorem equal to ${\rm Ric}$) a splitting, $({\tilde}X,\tilde{g}(t)) = (L \times \Omega, h \oplus l(t))$ where $(L,h)$ has zero curvature operator and $(\Omega,l(t))$ has positive Ricci curvature (here we use that ${\rm Ric}(Y,Y) = 0$ implies $\sec(Y,V)=0$ for all $V$ in view of the fact that ${ {\mathcal R} }\geq 0$). In fact $(L,h) = ({\mathbb{R}}^k,h)$ where $h$ is the standard metric. This may be seen as follows. If $(L,h)$ is not $ ({\mathbb{R}}^k,h)$, then the first fundamental group of $(L,h)$ is non-trivial. This would imply in particular that the first fundamental group of $(L \times \Omega, h
\oplus l(t)) = ({\tilde}X,\tilde{g}(t))$ is also non-trivial. Using the same argument given in Theorem 9.1 of [@Si6] (see Lemma \[homeo\] in this paper for some comments thereon), we see that $({\tilde}X,\tilde{g}(t),{\tilde}x_0)$ is homeomorphic to $(X,d_X,0)$, and hence $(X,d_X)$ has non-trivial first fundamental group. But as explained at the beginning of this paper $(X,d_X)$ is a cone over a standard sphere. In particular $(X,d_X)$ is homeomorphic to ${\mathbb{R}}^n$. Hence $(X,d_X)$ has trivial first fundamental group which leads to a contradiction.
Hence, we may write $({\tilde}X,\tilde{g}(t))= ({\mathbb{R}}^k \times \Omega, h
\oplus l(t))$ where $(\Omega,l(t))$ is a solution to the Ricci flow satisfying $(a_t)$, $(c_t)$ and ${\rm Ric}(l(t)) >0$ for all $t>0$. Using Fubini’s theorem, it is easy to see that the asymptotic volume ratio of $l(t)$ is given by $(\omega_{n-k}/\omega_{n}) {\tilde}V_0 > 0$, where $\omega_m$ is the volume of the $m$-dimensional Euclidean unit ball.
We show in the following that $(\Omega,l(t))_{t \in (0,\infty)}$ is a gradient expanding soliton, generated by some smooth function $f$. Hence $({\tilde}X,\tilde{g}(t))= ({\mathbb{R}}^k \times \Omega, h \oplus l(t))$ is a gradient expanding soliton: $ {\nabla}^2 f(t) - {\rm Ric}(l(t)) -
(1/(2t)) l(t) = 0$ on $\Omega$ and $ {\nabla}^2 v(t) -{\rm Ric}(h) - (1/(2t))h
= 0$ on ${\mathbb{R}}^k$ where $v(x,t) = \frac{|x|^2}{4t}$, and hence $ {\nabla}^2
{\tilde}f(t) - {\rm Ric}({\tilde}g(t)) - (1/(2t)){\tilde}g(t) = 0$ for all $t>0$ on ${\tilde}X$ with ${\tilde}f(x,y,t)= f(y,t) + v(x,t)$ for $(x,y) \in ({\mathbb{R}}^k \times \Omega)$.\
For simplicity let us denote $l(t)$ again by $\tilde{g}(t)$, i.e. we assume that $k= 0$.
$(i)$ After completing this paper we noticed that we could use the proof of Proposition 12 of the paper of S. Brendle [@SB] at this point to show that $(X,{\tilde}g(t))$ is an expanding gradient soliton. We include here our original proof which follows the lines of that given in [@ChLuNi].\
$(ii)$ In [@ChLuNi] and [@ChZh] it is assumed that $t { { \rm R }}(\cdot,t)$ achieves its maximum somewhere in order to conclude that the solution is a soliton. We make no such assumption. We show that ${\nabla}R ({\tilde}x_0) = 0$ in view of the fact that ${ {\partial {\over}{\partial t}} }(t
R(t,{\tilde}x_0))= 0$, and then argue as in [@ChLuNi] and [@ChZh].
For the rest of this argument we work with the Riemannian metrics ${\tilde}{g}(t)$. For ease of reading we introduce the notation ${ { \rm R }}(x,t):= { { \rm R }}(x, \tilde{g}(t))$, ${\rm Ric}(x,t):= {\rm Ric}({\tilde}g(t))(x)$ and so on. All metrics and covariant derivatives are taken with respect to the metrics ${\tilde}{g}(t)$. We assume that ${\tilde}g_{ij} = {\delta}_{ij}$ at points where we calculate, and indices that appear twice are summed. We saw before that we may assume that ${ {\partial {\over}{\partial t}} }(t { { \rm R }}({\tilde}x_0,
t ) )= 0$. But then $$\begin{aligned}
\label{testy}
0= { {\partial {\over}{\partial t}} }(t { { \rm R }}({\tilde}x_0,t))_{t=1}= {\Delta}{ { \rm R }}({\tilde}x_0,1) +
2|{\rm Ric}|^2({\tilde}x_0,1) + { { \rm R }}({\tilde}x_0,1). \cr \end{aligned}$$ By Theorem 10.46 in [@ChLuNi], with $v_{ij}= {\rm Ric}_{ij}$, we have $$\label{eq:lintraceharnack} Z(Y):=
\nabla_i\nabla_j{\rm Ric}_{ij}= |{\rm Ric}|^2 + 2(\nabla_j{\rm Ric}_{ij})Y_i
+{\rm Ric}_{ij}Y_iY_j + \frac{{{\rm R}}}{2t} \geq 0\ ,$$ for any tangent vector $Y$. In particular for $$Y=
-({\rm Ric}^{-1})^{ji}\text{div}({\rm Ric})_j {\frac{\partial } {\partial x^i} }=
-(1/2)({\rm Ric}^{-1})^{ji} {\nabla}_j { { \rm R }}{\frac{\partial } {\partial x^i} },$$ we see that $$\label{eq:lintraceharnackv2}
\begin{split}
Z(Y)= &\ (1/2){\Delta}{ { \rm R }}+ |{\rm Ric}|^2 -
(1/2)({\rm Ric}^{-1})^{ij} (\nabla_i { { \rm R }}) (\nabla_j { { \rm R }})\\
&\ + (1/4){\rm Ric}_{ij} ({\rm Ric}^{-1})^{si}{\nabla}_s { { \rm R }}({\rm Ric}^{-1})^{jk}{\nabla}_k { { \rm R }}+ \frac{{{\rm R}}}{2t}\\ = &\ (1/2)\Big(
{\Delta}{ { \rm R }}+2 |{\rm Ric}|^2 - (1/2)({\rm Ric}^{-1})^{ij}(\nabla_i
{ { \rm R }})(\nabla_j { { \rm R }}) + \frac{{{\rm R}}}{t}\Big) \geq 0.
\end{split}$$ Using , we have $$0 \leq Z(Y)({\tilde}x_0,1)= - (1/4)({\rm Ric}^{-1})^{ij}(\nabla_i
{ { \rm R }})(\nabla_j { { \rm R }})({\tilde}x_0,1)$$ and hence $\nabla { { \rm R }}({\tilde}x_0,1)= 0$. This implies that $$Z(Y)({\tilde}x_0,1)= 0$$ which is a global minimum for $Z(Y)$. Now we use the evolution equation for $Z(Y)$, which is given by equation (10.73) in [@ChLuNi]. It implies in particular that $$\begin{aligned}
{ {\partial {\over}{\partial t}} }Z(Y) \geq {\Delta}Z(Y), \end{aligned}$$ and hence by the strong maximum principle, we must have $Z(Y)= 0$ everywhere. By looking once again at the equation (10.73) in [@ChLuNi] and using the matrix Harnack inequality and the fact that $Z(Y)
= 0$, we get $${ {\partial {\over}{\partial t}} }Z(Y) \geq 2v_{ij}({\nabla}_k Y_i -{\rm Ric}_{ik} - (1/(2t)){\delta}_{ik})(
{\nabla}_k Y_j -{\rm Ric}_{jk} - (1/(2t)){\delta}_{jk})\ .$$ If at some point in space and time we have ${\nabla}Y - {\rm Ric}-
(1/(2t)){\tilde}g \neq 0$ as a tensor, then we get $ 2v_{ij}({\nabla}_k Y_i
-{ { \rm R }}_{ik} - (1/(2t)){\delta}_{ik})( {\nabla}_k Y_j -{ { \rm R }}_{jk} -
(1/(2t)){\delta}_{jk}) >0$ which would imply that ${ {\partial {\over}{\partial t}} }Z(Y) >0$ at this point in space and time. This in turn would imply that there are points in space and time with $ Z(Y) > 0$, which is a contradiction. Hence ${\nabla}Y - {\rm Ric}- (1/(2t)){\tilde}g = 0$, which yields that ${\tilde}g(t)$ is an expanding gradient soliton.
For completeness we include the following Lemma, whose statement and proof appeared in the proof we just gave.
\[homeo\] Let $({\tilde}X,{\tilde}g(t),{\tilde}x_0)_{t \in (0,\infty)}:= \lim_{i \to
\infty}(M,g_i(t),p_0)_{t \in (0,\infty)}$ be the solution obtained above, and $(X,d_X,0)$ be the pointed Gromov-Hausdorff limit of $(M,c_ih,p_0)=
(M,g_i(0),p_0)$, ($c_i \to 0$). Then $({\tilde}X,d({\tilde}g(t)),{\tilde}x_0) \to (X,d_X,0)$ in the pointed Gromov-Hausdorff sense as $t \to 0$. That is, the solution flows out of the cone $(X,d_X,0)$. Furthermore, ${\tilde}X$ is homeomorphic to $X$ which is homeomorphic to ${\mathbb{R}}^n$.
(We repeat the proof given above). Using the same argument given in Theorem 9.1 of [@Si6] , we see that $({\tilde}X,\tilde{g}(t),{\tilde}x_0)$ is homeomorphic to $(X,d_X,0)$ ( note that in the argument there, $U$ and $V$ should be [**bounded**]{} open sets: this is sufficient to conclude that the topologies are the same, since any open set can be written as the union of bounded open sets in a metric space). But as explained at the beginning of this paper $(X,d_X)$ is a cone over a sphere, where the topology of the sphere is the same as that of the standard sphere. In particular $(X,d_X)$ is homeomorphic to ${\mathbb{R}}^n$.
[de Rham splitting]{}\[A\] In this appendix, we explain some known splitting results, which follow from the de Rham Splitting Theorem. We give proofs for the reader’s convenience. We follow essentially the argument given in the proof of Theorem 2.1 of [@Cao] which follows closely that of Lemma 8.2/Theorem 8.3 of [@Ha86]. For a two tensor $\beta_{ij}$ we let $\beta^{i}_{\ j}:= g^{il}\beta_{lj}$, i.e. this defines an endomorphism from each tangent space into itself.
Let ${h(t)}_{t \in [0,T)}$ be a smooth (in space and time) bounded family of symmetric two tensors defined on a simply connected complete manifold $M^n$ without boundary, satisfying the evolution equation $$\begin{aligned}
{ {\partial {\over}{\partial t}} }h^i_{\ j}= {}^{g(t)}{\Delta}h^i_{\ j} + \phi^i_{\ j}
\end{aligned}$$ where $h(x,t), \phi(x,t) \geq 0$ for all $(x,t) \in M \times [0,T)$ in the sense of matrices. We also assume $g$ is a smooth family of metrics (in space and time) satisfying $^{g_0}|D^{(i,j)} g| +
{}^{g_0}|D^{(i,j)} h| \leq k(i,j)< \infty $ everywhere, where $i,j
\in {\mathbb{N}}$ and $D^{(i,j)}$ refers to taking $i$ time derivatives and $j$ covariant derivatives with respect to $g_0$, and $k(i,j) \in {\mathbb{R}}$ are constants. Then for all $x \in M, t>0$, the null space of $h(x,t)$ is invariant under parallel translation and constant in time. There is a splitting, $(M,g(t))= (N \times P, r(t) \oplus
l(t))$, where $r,l$ are smooth families of Riemannian metrics such that $h>0$ on $N$ (as a two tensor), and $h = 0$ on $P$.
Let $0 \leq {\sigma}_1(x,t) \leq{\sigma}_2(x,t) \leq \ldots \leq {\sigma}_n(x,t)$ be the eigenvalues of $h(x,t)$ and $\{ e_1(x,t) \}$ orthonormal eigenvectors. Assume that ${\sigma}_1(x_0,t_0) + {\sigma}_2(x_0,t_0) + \ldots
+ {\sigma}_k(x_0,t_0) >0$ at some point $x_0$ and some time $t_0$. Define a smooth function $\eta_{t_0}:M \to {\mathbb{R}}^+_0$ which is positive at $x_0$ and zero outside of $B_1(x_0,0)$ (the ball in $M$ of radius one with respect to $g_0$), and satisfies ${\sigma}_1(\cdot,t_0) +
{\sigma}_2(\cdot,t_0) + \ldots + {\sigma}_k(\cdot,t_0) > \eta_{t_0}(\cdot).$ Solve the Dirichlet problem: $$\begin{split} &{ {\partial {\over}{\partial t}} }\eta_i = {} ^{g(t)} {\Delta}\eta_i \\
&\eta_i(\cdot,t) |_{{\partial}B_i(x_0,0)}= 0
\ \ \forall \ t \in [t_0,T) \\
&\eta_i(\cdot,t_0)= \eta_{t_0}(\cdot)
\end{split}$$ Using the estimates for $g$ we see that the solutions exist for all time and satisfy interior estimates independent of $i$ (see for example Theorem 10.1, chapter IV, §10 in [@LSU]). Thus we may take a subsequence to obtain a smooth solution $\eta: M \times
[t_0,T) \to {\mathbb{R}}$ of the equation $$\begin{split}
&{ {\partial {\over}{\partial t}} }\eta= {}^{g(t)}
{\Delta}\eta \\
&\eta(\cdot,t_0)= \eta_{t_0}(\cdot)\ .
\end{split}$$ From the strong maximum principle, $\eta(\cdot,t) >0$ for all $t>
t_0$. Also, the construction and the estimates on $g$ guarantee that $\sup_{(M \backslash B_i(x_0,0)) \times [t_0,S]} |\eta(\cdot,t)| \to
0 $ as $i \to \infty$. for all $S < T$. We claim that ${\sigma}_1(\cdot,t) + \ldots + {\sigma}_k(\cdot,t) \geq e^{-at}\eta(\cdot,t)
$ for all $t\geq t_0$. One proves first, that ${\sigma}_1(\cdot,t) +
\ldots + {\sigma}_k(\cdot,t) - e^{-at}\eta(\cdot,t) + {\varepsilon}e^{\rho^2(\cdot,t)(1+at) + at} \geq 0 $ for arbitrary small ${\varepsilon}>0$ and an appropriately chosen constant $a$, where here $\rho(x,t)=
{\mbox{\rm dist}}(x,x_0,t)$ ($a>0 $ does not depend on ${\varepsilon}$: $a$ depends on the constants in the statement of the Theorem). This is done by using the maximum principle. See for example the argument in the proof of Lemma 5.1 in [@Si6] for details. Now let ${\varepsilon}$ go to zero. This implies ${\sigma}_1(\cdot,t) + \ldots + {\sigma}_k(\cdot,t) \geq \eta(\cdot,t)
$ for all $t \geq t_0$ and hence ${\sigma}_1(\cdot,t) + \ldots +
{\sigma}_k(\cdot,t) >0$ for all $t>t_0$. Thus $$dim (null (h(x,t)))=
\max \{ i \in \{0, \ldots, n\} | {\sigma}_1(x,t) + \ldots + {\sigma}_i(x,t)= 0
\}$$ is independent of $x \in M$ for all $t>t_0$ and a decreasing function in time. Hence $rank(h(x,t))$ is constant in space and time for some short time interval $ t_0 < t < t_0 + {\delta}$ for any $t_0 \in [0,T)$. Now we let $v$ be a smooth vector field in space and time lying in the null space of $h$ (at each point in space and time). We can always construct such sections which have length one in a small neighbourhood, by defining it locally smoothly, and then multiplying by a cut-off function. We follow closely the proof of Lemma 8.2 of Hamilton ([@Ha86]) and Theorem 2.1 of [@Cao]. In the following we use the notation ${\nabla}$ and ${\Delta}$ to refer to $ {}^{g(t)}{\nabla}$ and $ {}^{g(t)}{\Delta}$. Using $ h(v,v)
\equiv 0$ we get $$\label{ddth}
\begin{split}
0={ {\partial {\over}{\partial t}} }\Big(h(v,v)\Big) &= \Big({ {\partial {\over}{\partial t}} }h^i_{\ j}\Big)v_iv^j +
h^i_{\ j}\Big(\Big ({ {\partial {\over}{\partial t}} }v_i\Big) v^j
+ v_i\Big({ {\partial {\over}{\partial t}} }v^j\Big)\Big) \\
&= \Big({ {\partial {\over}{\partial t}} }h^i_{\ j}\Big)v^iv^j,
\end{split}$$ since $h^i_{\ j}v_i= 0$ and $h^i_{\ j}v^j = 0$. Furthermore, since $
h^i_{\ j} v_i v^j \equiv 0$ we get $$\label{laph}
\begin{split}
0 &= {\Delta}( h^i_{\ j} v_i v^j) \\
&= ({\Delta}h)^i_{\ j} v_i v^j + 2 {\Delta}(v_i)
h^i_{\ j} v^j \\
&\ \ + 4 g^{kl} v^j{\nabla}_k h^i_{\ j}
{\nabla}_l v_i + 2 g^{kl} h^i_{\ j} {\nabla}_k v^j
{\nabla}_l v_i
\end{split}$$ The term $2 {\Delta}(v_i) h^i_{\ j} v^j$ is once again zero, since $h^i_{\ j} v^j= 0$. Using this, , and the evolution equation for $h$ we get $$\begin{split}
0 &= \Big( { {\partial {\over}{\partial t}} }(h^i_{\ j}) - ({\Delta}h)^i_{\ j} -
\phi^i_{\ j}\Big)(v_iv^j)\\
&= 4 g^{kl} v^j {\nabla}_k h^i_{\ j} {\nabla}_l v_i + 2 g^{kl}
h^i_{\ j}
{\nabla}_k v^j {\nabla}_l v_i - \phi^i_{\ j}v_iv^j
\end{split}$$ Now use $$v^j{\nabla}_k h^i_{\ j}= {\nabla}_k(v^j
h^i_{\ j}) - h^i_{\ j} {\nabla}_k v^j = - h^i_{\ j} {\nabla}_k v^j$$ to conclude $$\begin{aligned}
2 g^{kl} h^i_{\ j}
{\nabla}_k v^j {\nabla}_l v_i + \phi^i_{\ j}v_i v^j= 0
\label{maineqapp} \end{aligned}$$ Since $\phi(v,v) \geq 0$ (and $h\geq 0$) we see that $\phi^i_{\ j}v_i v^j= 0$. That is, $v$ is also in the null space of $\phi$. But then, shows that $X_R(x,t): = {\nabla}_{R} v(x,t)$ is in the null space of $h$ for any vector $R \in T_x M $ (choose orthonormal coordinates at $x$ at time $t$, so that ${\frac{\partial } {\partial x^1} }(x):= R/{ \|}R { \|}_{g(x,t)}$ and use this in equation ). This shows that the null space of $h$ is invariant under parallel transport for each fixed time, as explained in the following for the readers convenience:
- Let $v_1(x), \ldots, v_k(x)$ be a smooth o.n. basis for $null(h(x,t))$ in a small spatial neighbourhood of $x_0$, and extend this to a smooth family $v_1, \ldots, v_n$ of vectors which is an o.n. basis everywhere in a small spatial neighbourhood of $x_0$. Let $X_0\in T_{x_0}M$ satisfy $g(X_0,v_i(x_0)) = 0$ for all $i \in \{k+1, \ldots, n\}$ and let $\gamma:[0,1]\rightarrow M$ be any smooth curve, starting in $x_0$ and whose image is contained in the neighbourhood of $x_0$ in question. Then parallel transport $X_0$ along $\gamma$. Call this vector field $X$. Write $X(\tau) = \sum_{i = 1}^n X^i(\tau)v_i(\gamma(\tau))$. We claim $ X(\tau) = \sum_{i = 0}^{k} X^i(\tau) v_i(\gamma(\tau))$. Let $X^{\top}(\tau) = \sum_{i = 1}^k X^i(\tau) v_i(\gamma(\tau)) $, and $X^{\perp}(\tau) = \sum_{i = k+1}^n X^i(\tau) v_i(\gamma(\tau)) $. First note that for $i \in \{1, \ldots, k\}$, and $V$ the tangent vector field along $\gamma$: $$g({\nabla}_V (X^{\perp}), v_i)
= V(g(X^{\perp}, v_i)) - g(X^{\perp}, {\nabla}_V v_i)
= 0$$ in view of $ {\nabla}_V v_i \in span\{ v_1, \ldots, v_k\}$ and $X^{\perp} \in span\{ v_{k +1}, \ldots, v_n\}$. Furthermore, for $j \in \{ k+1, \ldots, n\}$ we have $$\begin{split}
g({\nabla}_V (X^{\perp}), v_j) &= g({\nabla}_V (X- X^{\top}), v_j)\\
&= - g({\nabla}_V (X^{\top}), v_j)\\
&= -g (\sum_{i=1}^k V(X^i)v_i,v_j) - g(\sum_{i=1}^k X^i {\nabla}_V v_i,v_j)\\
&= -\sum_{i=1}^k X^i g({\nabla}_V v_i,v_j)\\
&= 0
\end{split}$$ in view of the fact that ${\nabla}_V v_i \in span\{ v_1, \ldots, v_k\}$. Hence $X^{\perp}$ is also parallel along $\gamma$. Since $X^\perp(0)=0$ we have $X^\perp \equiv 0$.
We have also shown that $ null(h) \subseteq null(\phi)$. Let $v(x_0,s)$ for $s \in(t, t+{\delta})$ be smoothly dependent on time, and $ v(x_0,s) \in null(h(x_0,s))$ for each $s \in(t, t+{\delta})$. Extend this vector at each time $s \in (t,t + {\delta})$ by parallel transport along geodesics emanating from $x_0$ to obtain a local smooth vector field $v(\cdot,\cdot)$ which satisfies $ v(x,s) \in null(h(x,s))$ for all $x$ (in a small ball) and all $s \in (t,t+ {\delta})$. In particular, $${\nabla}_iv \in null(h(x,s)) \ \ \text{and}\ \
{\Delta}v(x,s)= (g^{kl}{\nabla}_k {\nabla}_l v)(x,s) \in null(h(x,s)).$$ Since ${\nabla}_iv(x_0,s)=0$ we can compute $$\begin{split}
0={ {\partial {\over}{\partial t}} }\Big(h^i_{\ j} v_i\Big) &= \Big(h^i_{\ j} { {\partial {\over}{\partial t}} }v_i\Big)
+ \big(v_i ({\Delta}h)^i_{\ j}\big) + v_i \phi^i_{\ j} \\
&= h^i_{\ j} { {\partial {\over}{\partial t}} }v_i + {\Delta}(v_i h^i_{\ j}) +v_i\phi^i_{\ j} \\
&= h^i_{\ j} { {\partial {\over}{\partial t}} }v_i
\end{split}$$ where we have used that $ v \in
null(\phi)$. Hence ${ {\partial {\over}{\partial t}} }v(x_0,s) \in null(h(x_0,s))$. Assume that at time $s_0$ we have $null(h(x_0,s_0))=
{\mathbb{R}}^k \subseteq {\mathbb{R}}^n= T_{x_0} M$ and let $\{ e_1(t),
\ldots, e_n(t) \}$ be a smooth (in time) o.n. basis of vectors with $\{ e_1(t), \ldots, e_k(t) \}$ a smooth (in time) o.n. basis of vectors of $null(h(x_0,t))$. Let $e_i^l(t):= \langle e_i(t), e^l(0)\rangle$, where $\{ e^1(0), \ldots, e^n(0) \}$ refer to the standard basis vectors of ${\mathbb{R}}^n$ and $\langle \cdot , \cdot \rangle$ is the standard inner product on ${\mathbb{R}}^n$. ${ {\partial {\over}{\partial t}} }e_i(t) \in null(h(t))$ for all $i \in \{ 1, \ldots, k\}$ implies ${ {\partial {\over}{\partial t}} }e_i(t) = \sum_{j= 1}^k a_i^j(t)e_j(t)$ for some smooth functions $a_i^j: [0, \infty) \to {\mathbb{R}}$, $i,j
\in \{1, \ldots, k\}$. Then we have a system of ODEs ($l
\in \{1, \ldots, n\}$, $i \in \{ 1, \ldots, k\}$) $$\begin{split}
& { {\partial {\over}{\partial t}} }e_{i}^l (t)= \sum_{j= 1}^k a_{i}^j(t)e_j^{l} (t) \\
& e_{i}^l (0)= {\delta}_i^l.
\end{split}$$ By assuming $e_j^{l}(t)= 0$ for all $l \geq k +1$ we still have a solvable system, and hence the solution satisfies (by uniqueness) $e_j^{l}(t)= 0$ for all $l \geq k +1$. That is $\{ e_1(t), \ldots, e_k(t) \}$ remains in ${\mathbb{R}}^k$.
$null(h(x_0,t))$ is a space which is invariant under parallel transport (from the argument above). Hence the de Rham splitting theorem (see [@deR]) says, $M $ splits [**isometrically at time $s$** ]{} as $ N(s) \oplus
P(s)$ where $h(\cdot,s) = 0$ on $P(s)$ and $h(\cdot,s)>0$ on $N(s)$. We can do this it every time $s$. But the second part of the argument shows that $N(s)=
N(s_0)$ for all $s$ and $P(s)= P(s_0)$ for all $s$.
[An approximation result by V.Kapovitch/G.Perelman.]{}\[B\] Let $(M_i,d_i,p_0)$ be a non-collapsing sequence of non-negatively curved $n-$dimensional, smooth, complete manifolds without boundary such that $(M^n_i,d_i,p_0) \to (X,d_X,0)$ as $i \to \infty$ (in the GH sense) where $X =CV$ is an Euclidean cone with non-negative curvature over the metric space $(V,d_V)$ (with sectional curvature not less than 1 in the sense of Alexandrov), and $(M_i,d_i,p_0)$ are smooth with $\sec \geq 0$. This is the situation examined in the introduction. It is well known that the space of directions $\Sigma_{0}(X)$ of $(X,d_X)$ at $0$ is $(V,d_V)$: see Theorem 10.9.3 (here we have used that the tangent cone of $X$ at $0$ is equal to $X$, since $X$ is a cone). Now Theorem 5.1 of [@Kap1] says that $\Sigma_{0}(X)$ is homeomorphic to $\Sigma_{p_0} M_i$ (for $i$ big enough) which is isometric to the standard sphere $S^{n-1}$ since the $(M_i,g_i)$ are smooth manifolds. That is $V$ is homeomorphic to $S^{n-1}$.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
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|
---
address:
- |
Astronomy\
Mail Stop 105-24, Robinson Lab\
California Institute of Technology\
Pasadena, CA 91125\
U.S.A.\
E-mail: tjg@astro.caltech.edu
- |
Netherlands Foundation for Research in Astronomy and Kapteyn Astronomical Institute\
E-mail: ger@nfra.nl
author:
- 'T. J. Galama'
- 'A. G. De Bruyn'
title: 'The Unique Potential of SKA Radio Observations of Gamma-Ray Bursts.'
---
Introduction
============
Gamma-ray bursts (GRBs) are the strongest phenomenon seen at $\gamma$-ray wavelengths. Since their discovery in the 1970s these events, which emit the bulk of their energy in the $0.1 - 1.0$ MeV range, and whose durations span milliseconds to tens of minutes, posed one of the great unsolved problems in astrophysics. Until recently, no counterparts (quiescent as well as transient) could be found and observations did not provide a direct measurement of their distance, and thereby the true energy output was unknown by several orders of magnitude. The breakthrough came in early 1997, when the Wide Field Cameras aboard the Italian-Dutch BeppoSAX satellite allowed rapid and accurate localization of GRBs. Follow-up on these positions resulted in the discovery of X-ray [@cfh+97], optical [@vgg+97] and radio afterglows [@fkn+97]. These observations revealed that GRBs come from ‘cosmological’ distances. GRBs are by far the most luminous photon sources in the universe, with (isotropic) peak luminosities in $\gamma$ rays up to $10^{52}$ erg/s, and total energy budgets up to several $10^{53-54}$ erg (e.g., [@kdo+99]). The optical signal from GRB is regularly seen to be 10 magnitudes brighter (absolute) than the brightest supernovae, and once even 18 magnitudes brighter[@abb+99].
Here we discuss the current status of GRB afterglow observations (Sect. \[Fire\] to \[sec:bea\]) and discuss the unique potential of SKA observations of GRBs (Sect. \[sec:ska\]).
Relativistic blast-wave models {#Fire}
==============================
GRB afterglows are in good agreement with, so called, fireball-plus-relativistic blast-wave models (see [@pir99] for an extensive review). The basic model is a point explosion with an energy of order $10^{52}$ ergs, which leads to a ‘fireball’, an optically thick radiation-electron-positron plasma with initial energy much larger than its rest mass that expands ultra-relativistically. The GRB may be due to a series of ‘internal shocks’ that develop in the relativistic ejecta before they collide with the ambient medium. When the fireball runs into the surrounding medium a ‘forward shock’ ploughs into the medium and heats it, and a ‘reverse shock’ does the same to the ejecta. As the forward shock is decelerated by increasing amounts of swept-up material it produces a slowly fading ‘afterglow’ of X rays, then ultraviolet, optical, infrared, millimetre, and radio radiation.
[**Confirmation of the relativistic blast-wave model.**]{} Radio light curves of the afterglow of GRB970508 show variability on time scales of less than a day, but these dampen out after one month [@fkn+97] (see Fig. \[fig:radiofrail\]). Interpreting this as the effect of source expansion on the diffractive interstellar scintillation a source size of roughly 10$^{17}$ cm was derived, corresponding to a mildly relativistic expansion of the shell [@fkn+97].
The first X-ray and optical (but see [@gtv+99]) afterglows show power-law temporal decays, with power-law exponents in the range 1 to 2. These afterglow light curves agree well with the predictions of the relativistic blast-wave model (e.g., [@wrm97]).
The broad-band afterglow spectra are also power laws (in four distinct regions); together with the observed decrease of the cooling break and the peak frequency the observations conform nicely with simple relativistic blast-wave models in which the emission is synchrotron radiation by electrons accelerated in a relativistic shock [@gwb+98; @wg99].
The brightness temperature of the GRB990123 optical flash [@abb+99] exceeds the Compton limit of $10^{12}$K, confirming the highly relativistic nature of the GRB source [@gbw+99].
Progenitors and the cause of the explosion {#sec:sne}
==========================================
The GRB and the afterglow are produced when relativistic ejecta are slowed down; no observable radiation emerges directly from the ‘hidden engine’ that powers the GRB. Thus, in spite of all recent discoveries the origin of GRBs remains unknown (although an important link may be provided by the possible connection of GRBs to SNe). Currently popular models for the origin of GRBs are the neutron star-neutron star and neutron star-black hole mergers, white dwarf collapse, and core collapses of very massive stars (‘failed’ supernovae or hypernovae). These models can in principle provide the required energies.
[**SN1998bw/GRB980425.**]{} Galama et al. [@gvv+98] discovered a relatively rare and bright SN of type Ic within the small BeppoSAX localization of GRB980425, and suggested that the two objects are connected. A conservative estimate of the probability of a chance coincidence of the supernova and the GRB is $9 \times
10^{-5}$ [@gvv+98]. In the radio, the SN rapidly brightened and became one of the most luminous radio SNe [@kfw+98]. Kulkarni et al. [@kfw+98] drew attention to the fact that the radio emitting shell in SN1998bw must be expanding at relativistic velocities, $\Gamma \gsim 2$; see Fig. \[fig:radiofrail\]. This relativistic shock could well have produced the GRB at early times. The consequence of accepting such an association is that the $\gamma$-ray peak luminosity of GRB980425 and its total $\gamma$-ray energy budget are much smaller (a factor of $\sim$ 10$^5$) than those of ‘normal’ GRBs. GRB980425 is thus a member of a new class of GRBs – low luminosity GRBs related to nearby SNe (SN1998bw is at $z = 0.0085$). Such GRBs may well be the most most frequently occuring GRBs!
[**GRBs and SNe.**]{} But perhaps most, if not all, GRBs are associated with supernovae. There is growing evidence linking the usual GRBs – the cosmologically located GRBs – to SNe. The most direct evidence comes from the suggestion of an underlying SN in the afterglow of GRB980326 [@bkd+99]. The SN is revealed readily by its distinctive UV-poor spectrum against the broad-band afterglow. Also for GRB970228 there is evidence that a supernova dominated the light curves at late times [@rei99; @gtv+99]. So there may not be a dichotomy between ‘normal’ and supernova GRBs, only a gradual transition. The relation between cosmologically located GRBs like GRBs (980326, 970228) and GRB980425/SN1998bw is as yet unclear.
[**Collapsar model.**]{} All these observational developments support the collapsar model pioneered by Woosley and collaborators (see [@wmh99] and refs therein) in which massive stars core collapse to form black holes. Energy is somehow extracted from the spinning black hole and the jets drill their way out and power the GRBs and their afterglows.
GRBs as potential probes of the high-redshift universe {#sec:sfr}
======================================================
Host galaxies have been seen in most optical afterglow images. The detection of \[O II\] $\lambda$ 3727 and Lyman $\alpha$ emission from some hosts indicates that these are sites of vigorous star formation. The observed connection between some GRBs and star forming regions suggests that GRBs occur at critical phases in the evolution of massive stars. If GRBs are related to the deaths of massive stars (whose total lifetime is very short), their rate is proportional to the star formation rate (SFR). In that case GRBs may very well be at very high redshifts, with $z \sim 6$ or greater, for the faintest bursts (e.g. [@wbbn98]). GRBs may therefore become a powerful tool to probe the far reaches of the universe by guiding us to regions of very early star formation, and the (proto) galaxies and (proto) clusters of which they are part. The redshifts determined so far range between $z=0.41$ and $z = 3.42$.
The early afterglow {#sec:early}
===================
The discovery of a very bright and brief optical flash coincident in time with GRB990123 [@abb+99] shows that the early optical signal from GRB can be some 18 magnitudes brighter than the brightest supernovae. The reverse shock could cause emission that peaks in the optical waveband and is observed only during or just after the GRB. GRB990123 would then be the first burst in which all three emitting regions have been seen: internal shocks causing the GRB, the reverse shock causing the prompt optical flash, and the forward shock causing the afterglow [@mr99; @sp99b; @gbw+99].
Strongly anisotropic outflow (beaming) {#sec:bea}
======================================
An important uncertainty concerns the possible beaming of the $\gamma$-ray and afterglow emissions. This has an immediate impact on the burst energetics, and the nature and number of events needed to account for the observed burst rate [@rho99]. If the afterglow is beamed with opening angle $\theta$, a change of the light curve slope occurs at the time when the Lorentz factor $\Gamma$ of the blast wave equals $1/\theta$. Slightly later the jet begins a lateral expansion, which causes a further steepening of the light curve. Perhaps such a transition has been observed in the optical afterglow light curve of GRB990123 (e.g., [@kdo+99]). A similar transition was better sampled in afterglow data of GRB990510; optical observations of GRB990510, show a clear steepening of the rate of decay of the light between $\sim$ 3 hours and several days [@hbf+99]. Together with radio observations [@hbf+99], which reveal a similar steepening of the decline, it is found that the transition is very much frequency-independent; this virtually excludes explanations in terms of the passage of the cooling or the peak frequency, but is what is expected in case of beaming. Harrison et al. (1999) derive a jet opening angle of $\theta = 0.08$ radians, which for this burst would reduce the total energy in $\gamma$ rays to $\sim 10^{51}$ erg.
The unique potential of SKA observations of GRB afterglows. {#sec:ska}
===========================================================
- [**Interstellar Scintillation.**]{} As discussed in Sect. \[Fire\], the observations of GRB afterglows are in good agreement with the relativistic-blast wave model. However, the expansion rate and size of the blast wave have never been observed directly. Observations of the size, expansion rate and the shape of the GRB remnant would provide a stringent test of the relativistic blast wave model. The size $d$ of the GRB remnant is of the order of
$$d = \gamma c t,
\label{eq:size}$$
where $\gamma$ is the Lorentz factor of the blast wave, $c$ is the speed of light and $t$ the time in the observer’s frame. Hence, after 1 week the source size is $2 \gamma$ light-weeks, which, at a typical redshift of $z \sim1$ corresponds to an angular size of $\sim$ 1 microarcsecond.
Let us take GRB970508 as an example. Radio light curves of the afterglow of this GRB show rapid variability on time scales of less than a day, but these dampen out after one month [@fkn+97]. The reduced flux density modulations, interpreted as diffractive interstellar scintillation (DISS), are caused by the expansion of the source and then imply an angular diameter of at most a few $\mu$arcsec. At the redshift of GRB970508 this corresponds to a linear diameter of roughly 10$^{17}$ cm corresponding to a mildly relativistic expansion of the shell [@fkn+97]. Similar estimates of the source size were derived from the observed flux density for frequencies below the self-absorption frequency [@fkn+97], and from the presence of several breaks in the spectral energy distribution of GRB970508 [@wg99]. It is clear that with the current resolution and sensitivity of earth-bound Very Large Baseline Interferometry it is going to be impossible to ever obtain a direct measurement of the source size, except for the nearest GRBs like GRB980425/SN1998bw.
However, the example of GRB970508 shows that indirect source size estimates of GRB remnants can be obtained by observations of interstellar scintillation (DISS and RISS). The current observations are still severely sensitivity limited but as we discuss below there can be fantastic progress with the sensitivity provided by SKA.
Strong scattering can be observed at frequencies below the transition frequency $\nu_0$ (typically $\nu_0 \sim$ 5 GHz at high galactic latitudes [@wal98]). For $\nu>\nu_0$, the scattering is weak and the modulations scale as $(\nu/\nu_0)^{-17/12}<1$. For $\nu<\nu_0$ we will see strong diffractive scintillation only if the size of the source, $\theta_S<\theta_D=\theta_{F_0}(\nu/\nu_0)^{6/5}$. Depending on the properties of the turbulent plasma screen we may expect to encounter such conditions in the first day(s) after the GRB event.
The other ISS parameters of interest are the decorrelation timescale $t_{\rm
diff}$ (time for significant changes in the detected flux), and the bandwidth over which the diffractive ISS is decorrelated, $\Delta\nu =
\nu_0 (\nu/\nu_0)^{22/5}$. These scale as: $t_{\rm diff} \propto
\nu^{1.2}$ and $\Delta \nu_{\rm d} \propto \nu^{4.4}$. As emphasized by Goodman [@goo97] all these observables carry independent information on the properties of the ISM.
Observations of the modulation index, decorrelation time scale and decorrelation bandwidth as a function of frequency, and the determination of the transition frequency $\nu_0$ between weak and strong scattering, in early GRB afterglows, hence provide a wealth of information on the dynamically changing size and shape of the source. In principle the scintillation tool allows us to infer a crude measure of the morphology of the radio source: is it ring-like, as in spherical blast-wave models, or jet-like, in currently popular models. In the decaying, sub- or non-relativistic phase, the source may develop double structure (approaching and receding jet components) leading to distinct patterns in the dynamically changing spectra.
The scintillation method, however, will need a proper calibration before it will release its potential. This is clearly shown by the inferred sizes of two recently discovered scintillating quasars. For the radio quasar PKS 0405-385 [@kjw+97] an angular size of $\sim$ 5 $\mu$arcsec was estimated. However, for J1819+3845, Dennett-Thorpe and de Bruyn [@dtb00] estimate a size of $\sim$ 25 $\mu$arcsec at 5 GHz (possibly related to relatively nearby plasma turbulence). The calibration of scintillation screen properties can be provided by the observation of angularly nearby pulsars, the perfect point sources, of which SKA could easily detect about 1 per square degree.
SKA will provide the right technical specifications for these exciting and unique observations. The proposed instantaneous bandwidth: $0.5 +
f/5$ GHz and the large number of spectral channels: 10$^4$, allow the recording of dynamic spectra of GRBs, as is now common for pulsars.
- [**Synchrotron self absorption.**]{} The large instantaneous bandwidth, the large number of spectral channels, and sensitiviy of SKA will also allow detailed study of the transition from optically thin to optically thick frequencies and the evolution of the shape and location of this transition. Such high quality observations will provide strong constraints on models of GRB afterglows.
- [**Supernova-GRBs.**]{} As discussed in Sect. \[sec:sne\] the most common GRBs may very well be the low luminosity GRBs like GRB 980425/SN1998bw. The bright radio emission of such sources may easily be detected out to redshifts of $z \sim 1$, with SKA’s sensitivity. From the duration of the radio phase in SN1998bw, its typical brightness of $\sim$ 10 mJy at GHz frequencies, and some simple order-of-magnitude extrapolations to the whole sky, we derive that at any given time there will be several tens of such fading supernova-GRB radio afterglows above a flux density of 1 $\mu$Jy per square degree! Most of these sources will be distant, hence small and scintillating. This is how they could be discerned. What distinguishes them from AGN, however, is that they would appear at places where previously there would have been no radio source. A survey of the sky, carried out with SKA in its first year, could provide the template, against which to pick up these new sources. (Nota bene, this is also how new radio SNe would be discovered, but they are typically orders of magnitude fainter and rarely reach the radio brightness temperatures of GRB afterglows).
- [**Rapid response.**]{} The early radio afterglow emission of GRBs is currently hard to observe: the sources are very faint at ages less than 1 day and the current response time, which is typically a few hours, is too slow. Soon, with the launch of the gamma-ray satellite HETE-II, the response time may be very much improved upon (positions will be available within tens of seconds of the event), but the sensitivity is expected to remain problematic for such early observations. SKA will provide the required sensitivity plus it will have this other unique capability: the possibility of very rapid response (we may, however, have to build a few SKA’s to cover both the northern and southern hemispheres, and provide 24h watch!). We envision that SKA may be triggered directly by future gamma-ray spacecraft, and then rapidly electronically steer to the location on the sky. Such observations may provide insight in the physics of the fireball at very early times, for example we may expect to detect emission from the reverse shock [@sp99a].
- [**Wide Field Surveys.**]{} SKA will be a unique instrument for surveying large areas of sky (as noted above in connection with the GRB980425/SN1988bw association). If the GRB luminosity function is bimodal (i.e. GRBs like 980425 versus the more distant GRBs like 970508) then such surveys may be dominated by supernova-GRBs. However, a substantial number of the more distant GRBs may be discovered too, depending on the amount of collimation into jets. If the GRB luminosity function is not bimodal, but very wide then we may also detect a substantial number of radio afterglows from intermediate luminosity GRBs ($E_\gamma \sim 10^{48-50}$ erg). Thus, thousands of GRBs and supernova-GRBs may be discovered from such surveys by their distinct observational characteristics (for example a self-absorbed synchrotron spectrum, observed ISS and the characteristic damping of ISS fluctuations with time due to expansion of the source, the fact that GRB afterglows are not expected to be recurrent, etc). An important aspect of a radio selected survey is its unbiased-to-dust nature. Afterglows may thus be discovered independent of an optical or even a GRB identification.
The statistics of radio afterglows may be compared with the numbers expected from specific spherical- or jet-fireball models. As the bulk Lorentz factor, $\Gamma$ decreases with time after the event, the observer sees more and more of the emitting surface, $\theta \sim
1/\Gamma$. It follows that if gamma ray bursts are highly collimated, many more radio transients should be observed without associated gamma rays than with them. The ratio of expected (assuming spherical symmetry) to observed number of afterglows is thus a direct measure of the amount of collimation in GRB fireballs.
The number of radio afterglows may also be compared to that of optical afterglows (subject to obscuration by dust) to reveal possibly dusty environments (expected for massive star progenitor models).
- [**The high-$z$ universe and the cosmic star formation/death history.**]{} A GRB at a redshift $z \sim 1$ may easily be a mJy bright. With SKA’s senstivity such radio counterparts can be detected out to redshifts of 10 or greater. SKA will also be sufficiently sensitive that the radio emission of GRB hosts can be studied. Currently this is barely feasible. Such observations may provide information on the progenitors of GRBs (e.g., young or old stellar populations). Also, as discussed in Sect. \[sec:sfr\] GRBs are expected to trace the star formation rate (SFR) in the universe. Vice versa by observing GRB hosts we will learn about the star formation (and star death!) history. Observations in the (sub-)mm band suggest that star formation at high redshift is dominated by dusty star burst galaxies [@bsik99]. Estimates of the SFR of GRB host galaxies can be accurately determined by radio observations. These observations will be insensitive to dust, a crucial fact if one wants to be complete.
References {#references .unnumbered}
==========
[10]{}
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T. Piran. Gamma-ray bursts and the fireball model. , 1999. in press.
T. J. Galama et al. Evidence for a supernova in reanalyzed optical and near-infrared images of GRB970228. submitted, astro-ph/9907264, 1999.
R. A. M. J. Wijers, M. J. Rees, and P. Mészáros. Shocked by [GRB]{} 970228: the afterglow of a cosmological fireball. , 288:L51–L56, 1997.
T. J. [Galama]{} et al. The radio-to-X-ray spectrum of GRB970508 on 1997 May 21.0 UT. , 500:L97–101, 1998.
R. A. M. J. [Wijers]{} and T. J. [Galama]{}. Physical parameters of GRB970508 and GRB971214 from their afterglow synchrotron emission. , 523:177–186, 1999.
T. J. [Galama]{} et al. The effect of magnetic fields on gamma-ray bursts inferred from multi-wavelength observations of the burst of 23 January 1999. , 398:394–399, 1999.
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D. A. Frail, E. Waxman, and S. R. Kulkarni. A 450-day light curve of the radio afterglow of [GRB]{} 970508: Fireball calorimetry. ApJ (Let) submitted, 1999.
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Daniel E. [Reichart]{}. GRB970228 revisited: Evidence for a supernova in the light curve and late spectral energy distribution of the afterglow. , 521:L111–L115, 1999.
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---
abstract: 'We discuss the charge carrier dynamics of the heavy-fermion compound CeCoIn$_5$ in the metallic regime measured by means of quasi-optical THz spectroscopy. The transmittance of electromagnetic radiation through a CeCoIn$_5$ thin film on a dielectric substrate is analyzed in the single-particle Drude framework. We discuss the temperature dependence of the electronic properties, such as the scattering time and dc-conductivity and compare with transport measurements of the sheet resistance. Towards low temperatures, we find an increasing mismatch between the results from transport and Drude-analyzed optical measurements and a growing incapability of the simple single-particle picture describing the charge dynamics, likely caused by the evolving heavy-fermion nature of the correlated electron system.'
address:
- '1. Physikalisches Institut, University of Stuttgart, D-70569 Stuttgart, Germany'
- 'Department of Physics, Kyoto University, Kyoto 606-8502, Japan'
- 'Research Center for Low Temperature and Materials Science, Kyoto University, Kyoto 606-8502, Japan'
- 'Department of Advanced Materials Science, University of Tokyo, Chiba 277-8561, Japan'
- 'Department of Physics, Kyoto University, Kyoto 606-8502, Japan'
- '1. Physikalisches Institut, University of Stuttgart, D-70569 Stuttgart, Germany'
author:
- 'Uwe S. Pracht'
- 'Julian Simmendinger, Martin Dressel'
- 'Ryota Endo, Tatsuya Watashige, Yousuke Hanaoka, Masaaki Shimozawa'
- Takahito Terashima
- Takasada Shibauchi
- Yuji Matsuda
- Marc Scheffler
title: 'Charge Carrier Dynamics of the Heavy-Fermion Metal CeCoIn$_5$ Probed by THz Spectroscopy'
---
CeCoIn$_5$ ,heavy fermions ,Drude model ,charge carrier dynamics ,THz spectroscopy
Introduction {#i}
============
Amongst the different heavy-fermion materials, CeCoIn$_5$ has a prominent role as the Ce-based heavy-fermion superconductor with the highest $T_c$ = 2.3 K and as the most studied material of the 115-family of Ce compounds.[@petrovic2001; @sarrao2007; @thompson2013] Detailed information about electronic excitations and charge dynamics in heavy fermions can be obtained by optical spectroscopy in different spectral ranges.[@basov2011; @scheffler2013] Concerning infrared optics, CeCoIn$_5$ has been examined in great detail.[@singley2002; @mena2005; @burch2007; @okamura2015] These studies focused on signatures of the hybridization of conduction and $f$ electrons whereas the dynamics of the mobile charge carriers in heavy-fermion materials have to be probed at even lower frequencies, in the GHz and THz ranges.[@scheffler2013; @webb1986; @degiorgi1999; @marc_nature; @dressel2006; @scheffler2006; @scheffler10] Previous GHz and THz studies on CeCoIn$_5$ have explicitly addressed the superconducting transition,[@ormeno2002; @nevirkovets2008; @SudhakarRao2009; @truncik2013] while the heavy-electron charge dynamics in the metallic state of CeCoIn$_5$ have hardly been addressed by optics so far. The main reason here is that the sensitivity of broadband GHz and THz experiments is limited and for studies on highly conductive materials requires thin-film samples,[@marc_rsi; @marc_strip; @Pra13] which are difficult to grow in the case of heavy-fermion metals. Here we explicitly study the charge response of a CeCoIn$_5$ thin film.
Methods {#m}
=======
The sample under study is a 70nm thick film of CeCoIn$_5$ deposited via molecular beam epitaxy on a dielectric $5\times5\times0.5\,$mm$^3$ MgF$_2$ substrate. This technique has already been applied for growth of several thin-film systems based on CeCoIn$_5$ and CeIn$_3$.[@shishido2010; @mizukami2011; @shimozawa2012; @goh2012; @shimozawa2014] Other than in previous THz studies [@Sch13] on CeCoIn$_5$ thin films, we do not need additional metallic buffer layers to achieve high-quality samples. THz transmission measurements, however, still remain challenging for several reasons. First, the film needs to be thin enough to allow for a detectable transmission signal, while the sample quality favors thick films. Here we have chosen a thickness of 70nm, which is a compromise between sample quality and suitability for our experimental technique. Second, thin films of CeCoIn$_5$ often rapidly degrade in ambient air conditions so that exposure time must be cut to a minimum.[@Sch13] Third, the aperture, through which the focused THz radiation passes before it is transmitted through the sample, needs to have a diameter $d_a$ smaller than the sample. We have chosen $d_a=$ 3mm which restricts our accessible spectral range to wavelengths shorter than 1mm due to diffraction effects.
After deposition in Kyoto, the film was sealed in a glass tube under vacuum conditions before it was shipped to Stuttgart. Right after removal from the glass tube, the sample was mounted onto the THz sample holder, transferred to the cryostat, and rapidly cooled down in He-gas atmosphere. Here, the overall exposure time to ambient air was less than 5 minutes. The entire optical measurements were subsequently performed during a period of about 36 hours. During this time, the sample was always kept below 150K. Afterwards, it was removed from the cryostat and contacted in standard 4-point geometry in order to measure the dc-sheet resistance, and correspondingly obtain the dc transport resistivity $\rho_{dc}$. Even after a measurement time of $\sim$60 hours including an exposure time of $\sim$30 minutes we could not infer any signs of degradation from the THz spectra.
![\[res\] (Color online) dc transport resistivity $\rho_{dc}$ versus temperature of the CeCoIn$_5$ film studied in this work. Though being rather thin, all characteristics known from single-crystal CeCoIn$_5$ are well recovered. The inset shows a schematic drawing of the bilayer system. ](res-eps-converted-to.pdf)
![\[Tr\_SO34\] (Color online) Raw transmittance of 70nm CeCoIn$_5$ film as function of temperature and frequency. The pronounced oscillation pattern is caused by the dielectric substrate acting as a Fabry-Perot resonator. At high temperatures, the oscillation pattern is constant while it acquires a strong frequency dependence towards low temperatures: the transmittance increases with increasing frequency and decreasing temperature except for the low frequency- and temperature limit, where it is suppressed.](Tr_SO34.pdf)
The transmittance was measured using a set of tunable backward-wave oscillators as sources of coherent and monochromatic THz radiation and a He-cooled bolometer as detector.[@Pra13] Measurements were conducted in a spectral range spanning 13 - 46cm$^{-1}$ (i.e. 0.4 - 1.4THz). Sample temperatures between 6K and 150K were maintained in a home-built cryostat. We omitted measurements at higher and lower temperatures in order to restrain the measurement time and avoid degradation. Since our transmittance data is for a two-layer system (substrate + film), we measured a bare reference substrate in the same run to disentangle the properties of both layers.
Results and Discussion {#r}
======================
The temperature dependence of the dc transport resistivity $\rho_{dc}$ is shown in Fig. \[res\]. The sample exhibits all characteristic regimes well known for CeCoIn$_5$,[@petrovic2001; @malinowski2005] which is consistent with an excellent film quality. Starting at room temperature, the system behaves like a normal metal and $\rho_{dc}$ decreases slightly, passes a minimum at around $T_{min}=165$K and then increases again due to incoherent Kondo scattering. This increase levels off at around $T_{max}=40$K, where the system enters the coherent heavy-fermion state which then goes along with a rapid reduction of $\rho_{dc}$ upon further cooling before the curve bends down to the superconducting transition at presumably $T_c\approx 1.8$K slightly below our lowest measured temperature.
The raw transmittance is displayed in Fig. \[Tr\_SO34\] as function of frequency and temperature. The spectra feature pronounced Fabry-Perot (FP) oscillations that stem from multiple reflections inside the substrate.[@Pra13] At $T=150$K, the highly-conductive metallic film suppresses the overall transmittance (at the FP peaks) to less than 1%. Upon cooling, the spectra acquire a strong frequency dependence beyond the FP pattern. This is most pronounced in the high-frequency limit, where at $T=6$K the transmittance is about 4 times larger than at high temperatures. At intermediate frequencies, this enhancement is less strong and at the lowest frequencies it is even reversed at around 30K. Measurements of the bare substrate reveal only minor losses at 150K, which disappear below $\sim$80K. Thus, the observed frequency and temperature dependence of the transmittance can completely be attributed to the electronic properties of the CeCoIn$_5$ film. Such a behavior at THz frequencies is expected for a good metal, where the electron scattering rate $\Gamma = 1/\tau$, with $\tau$ the time between two scattering events, shifts into the examined spectral range upon cooling. In our case of CeCoIn$_5$ the shift of $\Gamma$ is attributed to the gradual emergence of a coherent heavy-fermion state with a concomitant slowing down of the Drude relaxation rate combined with a reduction of temperature-dependent scattering e.g. due to phonons.[@millis1986]
![\[Drude\] (Color online) Temperature dependence of (a) the dc resistivity obtained from optical and transport probes, (b) the scattering time $\tau$, (c) the scattering rate $\Gamma=1/\tau$, and (d) $\tau/\sigma_0$, which is a measure for the electron-mass enhancement. ](DrudeParametersSO34-eps-converted-to.pdf)
As further data analysis, we fitted the transmittance (see Fig. \[fits\](a) below) to well-established Fresnel equations for multiple reflections [@Dre02], where the complex refractive index $n+ik$ is expressed in terms of the Drude conductivity $$\sigma_1(\omega)+i\sigma_2(\omega)=\sigma_0\left(\frac{1}{1+(\omega \tau)^2}+i\frac{\omega \tau}{1+(\omega \tau)^2}\right)\label{eq:Drude}$$ with angular frequency $\omega=2\pi \nu$ and temperature-dependent parameters dc conductivity $\sigma_0$ and scattering time $\tau=1/\Gamma$. Fig. \[Drude\] (a) displays $\rho_0=1/\sigma_0$, i.e. the dc resistivity from optical Drude analysis, and compares it to $\rho_{dc}$ obtained from the transport measurement. In the regime of incoherent Kondo scattering, i.e. between $T_{min}$ and $T_{max}$, the results of both the optical and transport measurements coincide. Between $T_{min}$ and the inflection point of the $\rho_{dc}(T)$ curve, $\tau$ and $\Gamma$, see Fig. \[Drude\] (b) and inset (d), remain roughly constant. For lower temperatures, $\tau$ and $\Gamma$ tend to increase and decrease, respectively, signaling the emergence of the underlying heavy-fermion state. At around 25K, results from the optical and transport probes in Fig. \[Drude\] (a) start to deviate. We observe a more rapid decrease of $\rho_{dc}$ compared to $\rho_0$ which even levels off at around 13K. Down to this temperature, $\tau$ and $\Gamma$ display a strong temperature dependence in the now well developed heavy-fermion state. At even lower temperatures, $\rho_0$ remains fairly constant and tends to a slight increase which is in clear contrast to the transport result. At the same time, the temperature dependence of $\tau$ and $\Gamma$ becomes less pronounced. In the Drude framework, $\sigma_0$ is given by $$\sigma_0=\frac{Ne^2\tau}{m}\label{eq:mass}$$ where $N,e$ and $m$ are the electron density, charge, and mass. In this free electron picture, the temperature dependence of $\rho_0$ is usually determined by $\tau$ given that the number of carriers is constant. In heavy-fermion metals, however, $m$ is renormalized by electron-electron interactions and the effective mass $m^*$ becomes strongly temperature dependent. While $\rho_0$ remains fairly constant in the heavy-fermion regime, $\tau$ drastically increases. Even without knowledge of $N$ we can infer the temperature dependence of $m^*$ by plotting $\tau/\sigma_0=m^*/(Ne^2)$, see Fig. \[Drude\](c). Upon cooling, the mass enhancement already sets in well before the heavy-fermion state is fully developed, speeds up, and eventually levels off towards the lowest temperature. Assuming a constant value of $N$, this would translate to a mass enhancement of roughly four between the highest and lowest temperature of this study. This value is small when compared to those found for CeCoIn$_5$ from other techniques such as specific heat [@petrovic2001; @kim2001] or quantum oscillations [@settai2001; @mccollam2005], but those probes usually reveal the effective mass only for very low temperatures, whereas our work indicates that the mass enhancement upon cooling is not complete yet at our lowest temperature of 6 K. Closer to our approach are optical studies at higher frequencies, in the infrared regime,[@singley2002; @mena2005] which found a frequency-dependent effective mass enhancement which amounts to approximately 20 and 13, respectively, at their lowest frequencies and temperatures (30 cm$^{-1}$, 10 K and 40 cm$^{-1}$, 8 K, respectively). Since our optical measurements reach lower frequencies and temperatures, more pronounced mass enhancement is expected, which is not found in the data of Fig. \[Drude\](c). This can be explained since the simple Drude analysis performed here does not take into account any possible frequency dependence of scattering rate and effective mass. As we will show below, this description becomes more inaccurate for decreasing temperature. Here, a more detailed analysis based on data of the full optical response (real and imaginary parts) is desired.
![\[fits\] (Color online) (a) Transmittance spectra and Fresnel fits including Drude conductivity for several temperatures. The spectra and fits are shifted for clarity. While at high temperatures the transmittance is well described by the fit, the fidelity gradually decreases towards low temperatures signaling a breakdown of the simple single-particle Drude description. (b) and (c) schematically explain such behavior by comparing the Drude and Fermi-liquid (FL) predictions for the optical response: in FL theory the scattering rate $\Gamma$ (dashed) is frequency dependent, which leads to different spectra in the real part $\sigma_1$ of the conductivity (full) and in the transmittance.](Tr_fitted-eps-converted-to.pdf)
However, the trends that we find for $\rho_0, \tau, \Gamma$ and $m^*$ in the present work are in good agreement with previous results [@Sch13] on CeCoIn$_5$ thin films, where the analysis was less straightforward due to metallic buffer layers beneath the thin film. In [@Sch13] the discrepancy between optical and transport probes was explained by a rapid degradation of the film during the measurement. Here, this seems less likely as we could not observe any signs of degradation in the optical spectra.
Fig. \[fits\] (a) comprises the transmittance together with the Fresnel fits including the single-particle Drude formula, Eq. (\[eq:Drude\]). At high temperatures, where no mismatch between optical and transport probes was visible, the fit captures the transmittance in the entire frequency range very well. Upon cooling down, the agreement between the simple theory and experimental data becomes less good: most severe deviations arise in the low and high frequency limits, where the actual transmittance drops below the theory expectation. Furthermore, a small phase shift arises at intermediate frequencies. By judging the fit fidelity, one can understand the discrepancy between $\rho_0$ and $\rho_{dc}$ as an increasing incapability of the single-particle Drude theory to reproduce the dynamical properties of CeCoIn$_5$. Indeed, in a number of correlated electron systems a frequency dependence of $\tau$ was found resulting from electron-electron interactions, and by Kramers-Kronig relations, one could also expect a frequency-dependent effective mass. This might also explain the failure of the Drude theory describing the many-body heavy-fermion state in CeCoIn$_5$ at sufficiently low temperatures.
In Fig. \[fits\](b) and (c) we qualitatively explain how such deviations from Drude behavior in the transmittance spectra can arise due to electronic correlations. Our reference for an interacting electron system is a Fermi liquid (FL) with optical properties well understood from the theoretical side [@gurzhi1959; @maslov2012; @Berthod2013] and recently studied experimentally.[@scheffler2013; @schneider2014; @stricker2014] For CeCoIn$_5$, non-FL behavior (with linear temperature dependence of dc resistivity compared to the FL quadratic behavior) was found experimentally for many properties and is expected for the THz response, but in lack of an appropriate non-FL prediction, we here refer to the FL case to demonstrate the overall behavior. FL theory predicts a quadratic frequency dependence of the scattering rate $\Gamma(\omega) = \Gamma(\omega=0) + b \omega^2$ (with the prefactor $b$ depending on material) compared to the frequency-independent $\Gamma$ within the Drude response. As a result, $\sigma_1(\omega)$ for FL notably deviates from the Drude case, with the characteristic feature being a higher $\sigma_1$ at higher frequency (non-Drude trail).[@Berthod2013] Such differences in $\sigma_1$ correspond to differences in the transmittance, as is shown in Fig. \[fits\](c): if one compares the transmittance spectra of an interacting electron system (our CeCoIn$_5$ data at low temperature and the FL in the schematic figure) with the spectrum for a Drude metal, one finds that the maxima of the FP oscillations can be modeled properly in a limited (intermediate) frequency range, while the maxima in the Drude case surpass those of the interacting case for both lower and higher frequency. Therefore we interpret the insufficient Drude fits of our low-temperature transmittance data as evidence for electronic correlations in the THz response of CeCoIn$_5$. Whether these can be described as FL optics or whether, as expected, non-FL features govern the THz properties remains to be seen from further studies that should address the phase shift in the THz response in addition to the transmittance.
Summary {#s}
=======
In summary, we discussed the transmittance of THz radiation through a high-quality thin film of CeCoIn$_5$ measured by quasi-optical spectroscopy and compared it to transport measurements of the dc resistivity $\rho_{dc}$. We found a perfect agreement of the dc resistivity $\rho_0=1/\sigma_0$ obtained from Drude optics and $\rho_{dc}$ in the regime of incoherent Kondo scattering. At lower temperatures, the scattering time $\tau$ and effective mass $m^*$ acquire a strong temperature dependence and the agreement between $\rho_{dc}$ and $\rho_0$ becomes worse. We attribute this to an increasing incapability of the simple single-particle picture, i.e. the Drude theory, in favor of a more advanced description that accounts for the electronic correlations associated with the low-temperature heavy-fermion state. With the recent improvements in growing high-quality thin films, optical experiments at THz and GHz frequencies become feasible and we hope that our results motivate further investigations illuminating the unconventional charge carrier dynamics in CeCoIn$_5$.
Acknowledgements {#a}
================
This study was supported by the DFG. The work in Japan was supported by KAKENHI from JSPS. U.S.P. acknowledges financial support from the Studienstiftung des deutschen Volkes.
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---
abstract: 'We present a simple proof of the continuity, in the sense distributions, of the minors of the differential matrices of mappings belonging to grand Sobolev spaces. Such function spaces were introduced in connection with a problem on minimal integrability of the Jacobian and are useful in certain aspects of geometric function theory and partial differential equations.'
address: |
Sobolev Institute of Mathematics\
4 Acad. Koptyug avenue, Novosibirsk 630090, Russia\
Peoples’ Friendship University\
6 Miklukho-Maklaya str., Moscow 117198, Russia
author:
- Anastasia Molchanova
title: |
A note on the continuity of minors in\
grand Lebesgue spaces
---
Introduction {#sec:intro}
============
The academic literature on enlarged function spaces has grown considerably in recent times. Authors are typically concerned with the general theory of function spaces and applications in PDEs. An attractive feature of such function spaces is that they require a minimum of a priori assumptions, while member functions retain specific attractive properties such as continuity or regularity. Particular examples of these spaces are the so-called grand Lebesgue and grand Sobolev spaces. These spaces first appear in a paper by T. Iwaniec and C. Sbordone [@IwaSbo1992] in which they investigate minimal conditions for the integrability of the Jacobian of an orientation-preserving Sobolev mapping. Fundamental properties of these spaces have since been established such as duality and reflexivity [@Fio2000; @FioKar2004], as well as the boundedness of various integral operators [@JaiSinSin2016; @Kok2010; @KokMes2009]. For further discussion of grand spaces, the interested reader is referred to [@CapFioKar2008; @CasRaf2016; @DonSboSch2013; @FioGupJai2008; @FioMerRak2001; @FioMerRak2002; @FioSbo1998; @GreIwaSbo1997; @JaiSinSin2017; @Sbo1996; @Sbo1998].
It is well known that if a sequence of mappings $f_m$ converges weakly in the Sobolev space $W^{1,n}_{\rm loc}$ to a mapping $f_0$, then all $k\times k$-minors, $k=1, \dots n$, of matrices $Df_m$ tend to the corresponding $k\times k$-minors of the matrix $Df_0$, in the sense of distributions (*in* $D'$), see [@Mor1966 Ch. 9] for the particular case $n=2$, [@Resh1982 §4.5] and [@Dac2008 Theorem 8.20] for $n\geq 2$. The weak continuity of such minors plays a key role in the calculus of variations respecting the lower semicontinuity problem, see [@BenKru2017; @Dac2008] and references therein for more information. The related question of the integrability of the Jacobian (which is a particular case of a minor) under minimal assumptions, is partially motivated by applications such as nonlinear elasticity theory [@Dac1982]. Significant results were obtained for mappings with nonnegative Jacobians, which are sometimes called ‘orientation-preserving’ mappings. Specifically, S. Müller proved that if $|Df| \in L^n$ and $J_f(x)\geq 0$, then the Jacobian possesses the higher integrability $J_f(x) \in L \log L$ [@Mul1990]. Further generalizations can be found in [@Gre1998; @KosZho2002] and associated references. Following integrability, continuity theorems for Jacobians in corresponding spaces are the next natural step towards more general approximation results. In this way T. Iwaniec and A. Verde obtained, in [@IwaVer1999], the strong continuity of Jacobians in $L \log L$, while L. D’Onofrio and R. Schiattarella in [@DonSch2013] proved a continuity theorem for orientation preserving mappings $f_k$ belonging to the grand Sobolev space $W^{1,n)}$. Provided that we have the additional requirement of uniformly vanishing $n$-modulus, i.e.$$\lim\limits_{\varepsilon \to 0+} \varepsilon \sup\limits_{k \geq 1}
\int\limits_{\Omega}|Df_k (x)|^{n-n\varepsilon} \,dx =0,$$ the weak continuity of Jacobians is obtained by L. Greco, T. Iwaniec, and U. Subramanian [@GreIwaSub2003].
This paper proves continuity theorems for the minors of the differential matrix of mappings belonging to grand Sobolev spaces (see Section \[sec:preliminaries\] for the definitions). More precisely,
\[th:main\] Let $\Omega \subset \mathbb{R}^n$ and $f_m = (f_m^1, \dots, f_m^k)\colon \Omega \to \mathbb{R}^k$, $1 \leq k \leq n$, $m \in \mathbb{N}$, be a sequence of mappings locally bounded in $W^{1,p),\delta} (\Omega)$ with $p > k$. Assume that $f_m$ converges in $L^{1}_{\rm loc}$ to $f_0 = (f_0^1, \dots, f_0^k)$ as $m \to \infty$, then the sequence of forms $\omega_m = df_{m}^{1}\wedge \dots \wedge df_{m}^{k}$ converges to $\omega_0 = df_{0}^{1}\wedge \dots \wedge df_{0}^{k}$ in $D'$ and is locally bounded in $L^{\frac{p}{k}),\delta} (\Omega)$.
The case $p=k$ requires some additional conditions, since it makes use of the property of the coincidence between the distributional Jacobian and the point-wise Jacobian (Theorem \[lem:weak\_J\] below). The same technique used in obtaining proof of the main result, with minor changes, allows us to prove the following results.
\[thm:main-2\] Let $f_m = (f_{m}^{1}, \dots, f_{m}^{k})$, $1 \leq k \leq n$, $m \in \mathbb{N}$, be a sequence of mappings locally bounded in $W^{1,k)}$ and with $Df_m \in L^{k)}_{b}$. Assume that $f_m$ converges in $L^{1}_{\rm loc}$ to $f_0 = (f_{0}^{1}, \dots, f_{0}^{k})$ as $m \to \infty$ and forms $\omega_m = df_{m}^{1}\wedge \dots \wedge df_{m}^{k}$ and $\omega_0 = df_{0}^{1}\wedge \dots \wedge df_{0}^{k}$ are locally integrable. It follows that $\omega_m$ converges to $\omega_0$ in $D'$ and is locally bounded in $L^{1)}$.
\[thm:main-3\] Let $f_m = (f_{m}^{1}, \dots, f_{m}^{k})$, $1 \leq k \leq n$, $m \in \mathbb{N}$, be a sequence of mappings locally bounded in $W^{1,k)}$ and with $Df_m \in L^{k)}_{b}$. Assume that $f_m$ converges in $L^{1}_{\rm loc}$ to $f_0 = (f_{0}^{1}, \dots, f_{0}^{k})$ as $m \to \infty$ and all $k$-minors of matrix $Df_m$ are nonnegative. It follows that $\omega_m = df_{m}^{1}\wedge \dots \wedge df_{m}^{k}$ converges to $\omega_0 = df_{0}^{1}\wedge \dots \wedge df_{0}^{k}$ in $D'$ and is locally bounded in $L^{1)}$.
The stated results are similar to those of [@GreIwaSub2003] but the proof, based on a technique used by Yu. Reshetnyak [@Resh1982], is comparatively simple and requires us to know only basic properties of the theory of differential forms and Sobolev spaces. Moreover, this method allows us to easily extend the results for grand Sobolev spaces $W^{1,p)}$ to grand Sobolev spaces with respect to measurable functions $W^{1,p),\delta}$, as stated in Theorem \[th:main\].
Preliminaries {#sec:preliminaries}
=============
For a bounded open subset $\Omega$ in $\mathbb{R}^n$, $n\geq 1$, vector functions $f = (f^1, \dots, f^n) \colon \Omega \to \mathbb{R}^n$ are called mappings of the Sobolev class $W^{1,p} (\Omega,\mathbb{R}^n)$, $1\leq p \leq \infty$, if all coordinate functions $f^i$, $i = 1,2,\dots, n$, belong to $W^{1,p} (\Omega,\mathbb{R})$. Throughout this paper the symbol $Df$ stands for the differential matrix and $J_f$ denotes its determinant, the Jacobian.
For $0 < q < \infty$ the *grand Lebesgue space* $L^{q)}(\Omega)$ consists of all measurable functions $f \colon \Omega \to \mathbb{R}$ such that $$\label{def:grand_norm}
\|f\|_{L^{q)}} = \sup\limits_{0 < \varepsilon < \varepsilon_0}
\left(\frac{\varepsilon}{|\Omega|} \int\limits_{\Omega} |f(x)|^{q-\varepsilon} \, dx \right)^{\frac{1}{q-\varepsilon}} < \infty,$$ where $\varepsilon_0 = q-1$ if $q > 1$ and $\varepsilon_0 \in (0,q)$ if $0<q\leq 1$.
Grand Lebesgue spaces have been thoroughly studied by many different authors. We refer the interested reader to the reviews given in articles [@DonSboSch2013; @FioForGog2018; @JaiSinSin2017] and [@CasRaf2016 §7.2]. However, we now state some basic properties of these spaces which will be useful for the results that follow.
For the case $q>1$, the continuous embeddings $$L^{q} \subset L^{q)} \subset L^{q-\varepsilon}, \quad
\text{ for }
0 < \varepsilon < q-1,$$ hold, and are strict. This can be easily seen by considering a unit ball $B(0,1)$ and the function $f(x) = |x|^{-\frac{n}{q}}$. In this case $f$ belongs to $L^{q)}(B(0,1))$ but not $L^{q}(B(0,1))$.
Spaces $L^{q)}$ for $q>1$ are known to be non-reflexive Banach spaces [@Fio2000].
The space $L_b^{q)}$ consists of all functions $f \in L^{q)}$ such that $$\lim\limits_{\varepsilon \to 0+} \varepsilon \int\limits_{\Omega} |f(x)|^{q-\varepsilon} \, dx = 0.$$
The space $L_b^{q)}$ is the closure of $L^q$ in the norm $\|\cdot\|_{L^{q)}}$ and $L_b^{q)} \neq L^{q)}$ see [@CarSbo1997; @Gre1993]. The validity of this latter claim is easy to see by considering once again the function $f(x) = |x|^{-\frac{n}{q}}$ on the unit ball $B(0,1)$, for which $f\not\in L_b^{q)}(B(0,1))$, since $$\varepsilon \int\limits_{\Omega} |f(x)|^{q-\varepsilon} \, dx = \frac{q}{n} |B(0,1)| \not \to 0
\text{ as }
\varepsilon \to 0+.$$
The embeddings $L^{q, p} \subset L^{q, \infty} \subset L^{q)}$ and $L^{q}(\log L)^{-1} \subset L_b^{q)} \subset L^{q)}$ also hold, where $L^{q, p}$ are Lorenz spaces, and $L^{q}(\log L)^{-1}$ are Orlicz spaces. For further discussions of embeddings of these spaces, we refer the reader to [@DonSboSch2013; @GiaGrePas2010; @Gre1993; @IwaSbo1992].
As is seen from , grand Lebesgue spaces can be characterized as controlling the blow-up of the Lebesgue norm by the parameter $\varepsilon$. Indeed, the norm of the function $f$, belonging to $\bigcap\limits_{0<\varepsilon < q-1} L^{q-\varepsilon}$ but not $L^q$, must blow up, i.e., $\|f\|_{L^{q-\varepsilon}} \to \infty$, when $\varepsilon \to 0$. Thus, a natural generalization is to substitute for $\varepsilon$ a measurable function $\delta(\varepsilon)$, which is positive a.e. [@CapForGio2013].
\[def:GLd\] For $0 < q < \infty$ the *grand Lebesgue space* $L^{q), \delta}(\Omega)$ *with respect to $\delta$* consists of all measurable functions $f \colon \Omega \to \mathbb{R}$ such that $$\|f\|_{L^{q),\delta}} = \sup\limits_{0 < \varepsilon < \varepsilon_0}
\left(\frac{\delta(\varepsilon)}{|\Omega|} \int\limits_{\Omega} |f(x)|^{q-\varepsilon} \, dx \right)^{\frac{1}{q-\varepsilon}} < \infty,$$ where $\delta \in L^{\infty}((0,\varepsilon_0), (0,1])$ is a left continuous function such that $\lim\limits_{\varepsilon \to 0+} \delta(\varepsilon) = 0$ and $\delta^{\frac{1}{q-\varepsilon}}(\varepsilon)$ is nondecreasing, $\varepsilon_0 = q-1$ if $q > 1$ and $\varepsilon_0 \in (0,q)$ if $0<q\leq 1$.
If $\delta(\varepsilon) = \varepsilon$, the space $L^{q),\delta}$ is equivalent to $L^{q)}$. If $\delta(\varepsilon) = \varepsilon^{\theta}$ with $\theta >0$, we denote the resulting space by $L^{q),\theta}$. It was first introduced and studied in [@GreIwaSbo1997]. In [@CapForGio2013] it was also shown that for $q>1$ $$L^{q} \subset L^{q),\delta} \subset L^{q-\varepsilon} \quad
\text{ for }
0 < \varepsilon \leq q-1.$$
The definition of convergence in the sense of distributions is standard. We say that the sequence $f_m \in X(\Omega)$ converges *in the sense of distributions* (*in* $D'$) to $f_0$ if, for every function $\varphi \in C_0^{\infty}(\Omega)$, $$\int\limits_\Omega f_m(x) \varphi(x) \, dx \to \int\limits_\Omega f_0(x) \varphi(x) \, dx \quad \text{as } m \to \infty.$$ It is well-known that $f_m$ converges to $f_0$ weakly in $L^{p}$ if and only if the sequence $\{f_m\}_{m\in\mathbb{N}}$ is bounded in $L^{p}$ and $f_m$ converges in the sense of distributions to $f_0$.
We now make some brief comments on exterior algebra that will be useful for the results that follow. Let $\omega$ be differential $k$-forms, where $1 \leq k \leq n$. If $I = (i_1,i_2, \dots, i_k)$ is a $k$-tuple with $1 \leq i_1 < i_2 < \dots < i_k \leq n$, a differential form $\omega$ can be represented as $$\omega = \sum \limits_{I} \omega_{I}(x)\, dx^{i_1} \wedge \dots \wedge dx^{i_k} = \sum \limits_{I} \omega_{I}(x)\, dx^{I}.$$ Note that the sequence of $k$-forms $\omega_m$ converges to $\omega_0$ in $D'$ as $m \to \infty$ if the coefficients of the forms $\omega_m$ converge in $D'$ to the corresponding coefficients of $\omega_0$.
The calculus of differential forms is a powerful tool in the study of the analytical and geometrical properties of mappings. Thus, for mappings $f$ in Sobolev class $W^{1,p}$, with $p\geq n$, the Jacobian can be represented by the $n$-form $$J_f =df_1\wedge \dots \wedge df_n.$$
To deal with the borderline case $p=k$ we need the integration-by-parts formula, $$\label{eq:Det=det}
\int\limits_{\Omega} \varphi(x) J_f(x) \,dx =
- \int\limits_{\Omega} f^n \, df^1 \wedge df^2 \dots \wedge df^{n-1} \wedge d \varphi.$$ It is easy to see that holds for $f\in W^{1,n}(\Omega)$. In general, Sobolev embeddings and the Hölder inequality ensure that for $f\in W^{1,\frac{n^2}{n+1}}_{\rm loc} (\Omega)$, the right-hand-side of can be considered as a distribution, called the *distributional Jacobian* $\mathcal{J}_f$, and defined by the rule $$\mathcal{J}_f[\varphi] = - \int\limits_{\Omega} f^n \, df^1 \wedge df^2 \dots \wedge df^{n-1} \wedge d \varphi$$ for every test function $\varphi \in C^{\infty}_0 (\Omega)$. A function $f = x+ \frac{x}{|x|}$, with $\Omega$ being a unit ball, shows that fails as soon as $f\in W^{1,p}(\Omega)$, $p<n$. The natural question of the coincidence of the distributional and the point-wise Jacobians is thoroughly studied in [@Gre1993; @IwaSbo1992; @Mul1990], as well as in [@IwaMar2001 §7.2] and [@HajIwaMalOnn2008 §6.2]. We need the following results for grand Lebesgue spaces.
\[lem:weak\_J\] Let $f = (f^1, \dots, f^n) \in W^{1,1}_{\rm loc} (\Omega)$ be a function such that $J_f \in L^{1}_{\rm loc} (\Omega)$ and $|Df| \in L^{n)}_{b} (\Omega)$. Then holds for all compactly supported test functions $\varphi \in C^{\infty}_0 (\Omega)$.
\[lem:weak\_J>0\] Let $f = (f^1, \dots, f^n) \in W^{1,1}_{\rm loc} (\Omega)$ be a function such that $J_f (x) \geq 0$ a.e. in $\Omega$ and $|Df| \in L^{n)}_{b} (\Omega)$. Then holds for all compactly supported test functions $\varphi \in C^{\infty}_0 (\Omega)$.
Before we proceed to the proof of the main results, we need the following auxiliary lemma, which can be found in [@Resh1982 §4.5], and for which we now provide a proof for the convenience of the reader.
\[lem:conv\_dif\] Let $\omega_m$ be a sequence of differential $k$-forms, bounded in $L^{1}_{\rm loc} (\Omega)$, that converges in $D'$ to a form $\omega_0$ as $m \to \infty$. Assume that each of the forms $\omega_m$, $m \in \mathbb{N}$, has in $\Omega$ a generalized differential, and that the sequence $d\omega_m$ is bounded in $L^{1}_{\rm loc} (\Omega)$. It follows that the forms $d\omega_m$ converge to $d\omega_0$ in $D'$ as $m \to \infty$.
Consider an arbitrary $C^\infty$-smooth, compactly supported $(n-k-1)$-form $\alpha$. From the definition of a generalized differential we have $$\int\limits_\Omega \omega_m \wedge d\alpha = (-1)^{k-1}\int\limits_\Omega d\omega_m \wedge \alpha.$$ Since $\omega_m \to \omega_0$ in $D'$ and $d\alpha $ is a $(n-k)$-form of the class $C_0^\infty (\Omega)$, we obtain $$\int\limits_\Omega \omega_m \wedge d\alpha \xrightarrow[m \to \infty]{} \int\limits_\Omega \omega_0 \wedge d\alpha = (-1)^{k-1}\int\limits_\Omega d\omega_0 \wedge \alpha.$$ And finally $$\int\limits_\Omega d\omega_m \wedge \alpha \xrightarrow[m \to \infty]{} \int\limits_\Omega d\omega_0 \wedge \alpha$$ for all test $(n-k-1)$-forms $\alpha \in C_0^\infty (\Omega)$.
We now make use of Lemma \[lem:conv\_dif\] for grand Lebesgue spaces.
\[lem:conv\_dif\_GL\] Let $\omega_m$ be a sequence of differential $k$-forms, locally bounded in $L^{p),\delta} (\Omega)$, that converges in $D'$ to a form $\omega_0$ as $m \to \infty$. Assume that each of the forms $\omega_m$, $m \in \mathbb{N}$, has a generalized differential in $\Omega$, and that the sequence $d\omega_m$ is locally bounded in $L^{q),\delta} (\Omega)$. It follows that the forms $d\omega_m$ converge to $d\omega_0$ in $D'$ as $m \to \infty$.
For a mapping $f = (f^1, \dots f^n)\colon \Omega \to \mathbb{R}^n$, we define the $k \times k$-minors of the differential matrix as $$\frac{\partial f^I}{\partial x^J} = \frac{\partial (f^{i_1}, \dots f^{i_k})}{\partial (x^{j_1}, \dots x^{j_k})}$$ for ordered $k$-tuples $I= (i_1,i_2, \dots, i_k)$ and $J= (j_1,j_2, \dots, j_k)$. The representation $$df^{i_1} \wedge \dots \wedge df^{i_k} = \sum \limits_{J}\frac{\partial f^I}{\partial x^J}\, dx^{j_1} \wedge \dots \wedge dx^{j_k}$$ is valid.
Since in the proofs we investigate the properties of a particular $k \times k$ minor, it suffices to consider mappings $f \colon \Omega \to \mathbb{R}^k$ instead of maps into $\mathbb{R}^n$; also, this makes the notation simpler. Moreover, the condition *“$f_m$ converges in $L^{1}_{\rm loc}$ to $f_0$ as $m \to \infty$”* results from the statement *“there exists a subsequence converging weakly in $W^{1,q}_{\rm loc}$ to $f_0$ for all $1\leq q < p$”*. Indeed, by the Sobolev embeddings we can find a subsequence $f_{m_l}$, which converges to $f_0$ in $L^s_{\rm loc}$, for some $1\leq s < \frac{nq}{n-q}$. The Hölder inequality and boundedness of $\Omega$ then guarantee that $f_0$ is also an $L^1_{\rm loc}$-limit of $f_{m_l}$.
Proof of the main results
=========================
We will prove Theorem \[th:main\] by induction on $k$. The case of $k=1$ follows directly from Lemma \[lem:conv\_dif\_GL\]. Assume that the lemma has been proven for some general $k$, and let $f_m\colon \Omega \to \mathbb{R}^{k+1}$ be a sequence of mappings of class $W^{1,p),\delta} (\Omega)$, $p > k+1$. The sequence $f_m$ is locally bounded in $W^{1,p),\delta} (\Omega)$, consequently, also bounded in $W^{1,p-\varepsilon} (\Omega)$ for $0 < \varepsilon < p-1$, and is locally convergent in $L^1$ to $f_0$. From the Sobolev embedding theorem we obtain that $f_m \to f_0$ in $L^s$ for $s < \frac{n(p-\varepsilon)}{n-p + \varepsilon}$.\
<span style="font-variant:small-caps;">Step I.</span> Let us consider the forms $$\label{def:uvw}
\begin{aligned}
& u = dy^1 \wedge d y^2 \wedge \dots \wedge dy^k, \\
& v = (-1)^k y^{k+1} u = (-1)^k y^{k+1} dy^1 \wedge \dots \wedge dy^k, \\
& w = u \wedge d y^{k+1} = dy^1 \wedge d y^2 \wedge \dots \wedge dy^k \wedge d y^{k+1}
\end{aligned}$$ in $\mathbb{R}^{k+1}$.
It is easy to see that $w = d v$.
Consider also the pull-backed forms $$\label{def:pullbacked_forms}
\begin{aligned}
& \tilde \omega_m = f^*_m u = df_{m}^{1} \wedge d f_{m}^{2} \wedge \dots \wedge df_{m}^{k}, \\
& \psi_m = f^*_m v = (-1)^k f_{m}^{k+1} \tilde \omega_m, \\
& \omega_m = f^*_m w = df_{m}^{1} \wedge d f_{m}^{2} \wedge \dots \wedge df_{m}^{k+1}.
\end{aligned}$$ Then $\omega_m = d \psi_m$ for each $m$. In fact $\omega_m$, $\psi_m \in L^1$, since for each of $j$, the functions $f_{m}^{j}$, $df_{m}^{j}$ lie in $L^{\tilde{p}}$, where $p > \tilde{p} \geq k+1$. Thus, for any $(n-k-1)$-form $\eta \in C_0^\infty (\Omega)$, $$\label{eq:weak_J}
\int_\Omega \omega_m \wedge \eta = (-1)^{k-1} \int_\Omega \psi_m \wedge d \eta.$$ By the induction hypothesis $\tilde \omega_m \to \tilde \omega_0$ in $D'$ and $\tilde \omega_m$ is locally bounded in $L^{p/k)}$.\
<span style="font-variant:small-caps;">Step II.</span> Let $\xi$ be an arbitrary $C^\infty$-smooth, compactly supported $(n-k)$-form. Let us show that $$\label{eq:conv_0}
\int_\Omega f_{m}^{k+1} \tilde \omega_m \wedge \xi \to
\int_\Omega f_{0}^{k+1} \tilde \omega_0 \wedge \xi.$$ Indeed, fix $0 < \varepsilon = \frac{k}{n+1} < p-1$. Then, by the Sobolev embedding theorem, $f_{m}^{k+1} \to f_{0}^{k+1}$ in $L^s$, $s < \frac{n(p-\varepsilon)}{n-p + \varepsilon}$. Put $s' = \frac {p- \varepsilon}{k}$ and $s = \frac{p - \varepsilon }{p - k - \varepsilon}$, then $\frac{1}{s'} + \frac{1}{s} = 1$. Hence $$\label{eq:conv_1}
\left| \int_\Omega f_{m}^{k+1} \tilde \omega_m \wedge \xi -
\int_\Omega f_{0}^{k+1} \tilde \omega_m \wedge \xi \right| \\
\leq C \|\tilde \omega_m\|_{L^{s'} (A)} \|f_{m}^{k+1} - f_{0}^{k+1}\|_{L^{s} (A)}
\to 0,$$ where $A = \operatorname{supp} \xi$. Further, for any $\gamma >0$ let $f \in C_0^\infty (\Omega)$ be such that $\|f - f_{0}^{k+1}\|_{L^s(\Omega)} < \gamma$. Then $$\begin{gathered}
\left| \int_\Omega f_{0}^{k+1} \tilde \omega_m \wedge \xi -
\int_\Omega f_{0}^{k+1} \tilde \omega_0 \wedge \xi \right| \leq
\left| \int_\Omega (f_{0}^{k+1} - f) \tilde \omega_m \wedge \xi \right| \\
+
\left| \int_\Omega f (\tilde \omega_m \wedge \xi - \tilde \omega_0 \wedge \xi) \right|
+ \left| \int_\Omega (f - f_{0}^{k+1}) \tilde \omega_0 \wedge \xi \right| \to 0
\end{gathered}$$ as $m \to \infty$. The first and the third terms are less than $C\gamma$ due to the choice of $f$, the second one tends to zero by the induction hypothesis. Since $\gamma$ is arbitrary, this implies that $$\label{eq:conv_2}
\int_\Omega f_{0}^{k+1} \tilde \omega_m \wedge \xi \to
\int_\Omega f_{0}^{k+1} \tilde \omega_0 \wedge \xi.$$ The convergence follows from and . This means that the sequence of forms $\psi_m = f_{m}^{k+1} \tilde \omega_m$ converges to the form $\psi_0 = f_{0}^{k+1} \tilde \omega_m$ in $D'$.
It remains to show that the sequences of forms $\psi_m$ and $d \psi_m$ are bounded in $L^{q),\delta}$ for $q = \frac{p}{k+1}$. Indeed, the Hölder inequality provides
$$\begin{gathered}
\bigg(\int_\Omega |\psi_m|^{q - \varepsilon} \, dx \bigg)^{\frac{1}{q-\varepsilon}} =
\bigg(\int_\Omega |f_{m}^{k+1} \tilde \omega_m|^{q - \varepsilon} \, dx \bigg)^{\frac{1}{q-\varepsilon}} \\
\leq
\bigg(\int_\Omega |f_{m}^{k+1}|^{(q - \varepsilon) \frac{p - \varepsilon}{q - \varepsilon}} \, dx \bigg)^{\frac{1}{p-\varepsilon}}
\bigg(\int_\Omega |\tilde \omega_m|^{(q - \varepsilon) \frac{p - \varepsilon}{p - q}} \, dx \bigg)^{\frac{p-q}{(p-\varepsilon)(q - \varepsilon)}}.
\end{gathered}$$
Here $\frac{p - \varepsilon}{q-\varepsilon} > 1$ as $p - \varepsilon > q - \varepsilon$.
Multiplying by $\delta(\varepsilon)$ and taking the supremum, we obtain $$\begin{gathered}
\label{est:psi}
\|\psi_m\|_{L^{q),\delta}} \\
\leq
\sup\limits_{0 < \varepsilon < q-1}\bigg(\delta(\varepsilon)\int_\Omega |f_{m}^{k+1}|^{(q - \varepsilon) \frac{p - \varepsilon}{q - \varepsilon}} \, dx \bigg)^{\frac{1}{p-\varepsilon}}
\bigg(\delta(\varepsilon)\int_\Omega |\tilde \omega_m|^{(q - \varepsilon) \frac{p - \varepsilon}{p - q}} \, dx \bigg)^{\frac{p-q}{(p-\varepsilon)(q - \varepsilon)}} \\
\leq
\sup\limits_{0 < \varepsilon < p-1}\bigg(\delta(\varepsilon)\int_\Omega |f_{m}^{k+1}|^{p - \varepsilon} \, dx \bigg)^{\frac{1}{p-\varepsilon}}
\sup\limits_{0 < \varepsilon' < \frac{p}{k}-1}\bigg(\delta(\varepsilon')\int_\Omega |\tilde \omega_m|^{\frac{p}{k} - \varepsilon'} \, dx \bigg)^{\frac{1}{p/k-\varepsilon'}}
\\
\leq
\|f_{m}^{k+1}\|_{L^{p),\delta}} \|\tilde \omega_m\|_{L^{p/k),\delta}}.
\end{gathered}$$
The last inequality is valid for $\varepsilon' = \frac{\varepsilon (2p + pk - \varepsilon k - \varepsilon)}{pk}$, which satisfies\
$\frac{(q - \varepsilon)(p - \varepsilon)}{p - q} = \frac{p}{k} - \varepsilon'$. It is easy to check that $\varepsilon < \varepsilon'$, and from Definition \[def:GLd\] we can deduce that $\delta$ is a nondecreasing function, and thus $\delta (\varepsilon)^{\frac{1}{p/k - \varepsilon'}} \leq \delta (\varepsilon')^{\frac{1}{p/k - \varepsilon'}}$.
In order to make sure that $0 < \varepsilon' < \frac{p}{k}-1$, we show that $$h(\varepsilon) = pk \left(\frac{p}{k} -1 - \varepsilon'\right) = (k + 1) \varepsilon^2 - (2p + pk)\varepsilon + p^2 - pk >0.$$ First, note that $h(0) > 0$ and $h\left(\frac{p}{k+1} - 1\right) >0$. Moreover, $h'(\varepsilon) = 2 (k + 1) \varepsilon - (2p + pk) < 0$ if $\varepsilon < \frac{2p + pk}{2 (k + 1)}$ with $\frac{p}{k+1} - 1 < \frac{2p + pk}{2 (k + 1)}$, i.e., $h(\varepsilon)$ decreases for $0< \varepsilon < \frac{p}{k+1} - 1$ and takes positive values at the boundary points. Thus, $h(\varepsilon)>0$ for all $\varepsilon \in (0, \frac{p}{k+1} - 1)$, and so it follows that $0 < \varepsilon' < \frac{p}{k}-1$.
In view of this, we can consider the supremum over all $0 < \varepsilon' < \frac{p}{k} - 1$, and its value is not less than the supremum over all $0 < \varepsilon < q - 1=\frac{p}{k+1} - 1$. This completes the proof of .
The same arguments show that $d \psi_m = \omega_m = \tilde \omega_m \wedge d f_{m}^{k+1}$ is bounded in $L^{\frac{p}{k+1}),\delta}$. By Lemma \[lem:conv\_dif\_GL\], this implies that $\omega_m \to \omega_0$ in $D'$.
Here, we need some modifications of the proof of Theorem \[th:main\]. At Step I we use Lemma \[lem:weak\_J\] to obtain the relation . Note that Lemma \[lem:weak\_J\] can be modified for $k$-forms by considering $f^I = (f^1,f^2, \dots, f^k, x^{i_{k+1}}, \dots, x^{i_n})$, where $x^{i_l}$ is a corresponding coordinate function.\
<span style="font-variant:small-caps;">Step I.</span> Recall that $p=k+1$. Let us consider the forms $u$, $v$, $w$ and their pullbacks $\tilde \omega_m$, $\psi_m$, and $\omega_m$ defined by and , correspondingly. Now we use Lemma \[lem:weak\_J\] to obtain $\omega_m = d \psi_m$ for each $m$.
Indeed, $\omega_m = df_{m}^{1} \wedge d f_{m}^{2} \wedge \dots \wedge df_{m}^{k+1} \in L^1_{\rm loc}$ by the hypothesis of Theorem \[thm:main-2\], the local integrability of $\psi_m = (-1)^k f_{m}^{k+1} df_{m}^{1} \wedge d f_{m}^{2} \wedge \dots \wedge df_{m}^{k}$ follows from $f\in W_{\rm loc}^{1,\frac{n(k+1)}{n+1}}$, as $\frac{n(k+1)}{n+1} < k+1$. Then we have $df_{m}^{1} \wedge d f_{m}^{2} \wedge \dots \wedge df_{m}^{k} \in L_{\rm loc}^{\frac{n(k+1)}{k(n+1)}}$ and, from the Sobolev embedding theorem $f_{m}^{k+1} \in L_{\rm loc}^{\frac{n(k+1)}{n-k}}$. The Hölder inequality provides the required integrability, as $\frac{k(n+1)}{n(k+1)} + \frac{n-k}{n(k+1)} = 1$.
Hence, for any $(n-k-1)$-form $\eta \in C_0^\infty (\Omega)$, $$\int_\Omega \omega_m \wedge \eta = (-1)^{k-1} \int_\Omega \psi_m \wedge d \eta.$$ By the induction hypothesis $\tilde \omega_m \to \tilde \omega_0$ in $D'$ and the sequence $\tilde \omega_m$ is locally bounded in $L^{p/k)}$.\
<span style="font-variant:small-caps;">Step II.</span> All the estimates of Step II in the proof of Theorem \[th:main\] are satisfied if we consider in the definition of the grand Lebesgue norm $\varepsilon_0 = \frac{k+2 - \sqrt{k^2+4k}}{2} < 1$. According to Lemmas \[lem:weak\_J\] and \[lem:weak\_J>0\], we can replace the local integrability condition of $\omega_m$ by non-negativity of all $k$-minors of the matrix $Df_m$.
Let $\xi$ be an arbitrary $C^\infty$-smooth, compactly supported $(n-k)$-form. Let us show that $$\label{eq:conv_0_1}
\int_\Omega f_{m}^{k+1} \tilde \omega_m \wedge \xi \to
\int_\Omega f_{0}^{k+1} \tilde \omega_0 \wedge \xi.$$ To this end, fix $0 < \varepsilon = \frac{k}{n+1} < k = p-1$. From the Sobolev embedding theorem $f_{m}^{k+1} \to f_{0}^{k+1}$ in $L^s$, $s < \frac{n(k+1-\varepsilon)}{n-k-1 + \varepsilon}$. Put $s' = \frac {k+1- \varepsilon}{k}$ and $s = \frac{k+1 - \varepsilon }{1 - \varepsilon}$, then $\frac{1}{s'} + \frac{1}{s} = 1$. Hence $$\label{eq:conv_1_1}
\left| \int_\Omega f_{m}^{k+1} \tilde \omega_m \wedge \xi -
\int_\Omega f_{0}^{k+1} \tilde \omega_m \wedge \xi \right| \\
\leq C \|\tilde \omega_m\|_{L^{s'} (A)} \|f_{m}^{k+1} - f_{0}^{k+1}\|_{L^{s} (A)}
\to 0,$$ where $A = \operatorname{supp} \xi$. Furthermore, for any $\gamma > 0$ let $f \in C_0^\infty (\Omega)$ be such that $\|f - f_{0}^{k+1}\|_{L^s(\Omega)} < \gamma$, then $$\begin{gathered}
\left| \int_\Omega f_{0}^{k+1} \tilde \omega_m \wedge \xi -
\int_\Omega f_{0}^{k+1} \tilde \omega_0 \wedge \xi \right| \leq
\left| \int_\Omega (f_{0}^{k+1} - f) \tilde \omega_m \wedge \xi \right| \\
+
\left| \int_\Omega f (\tilde \omega_m \wedge \xi - \tilde \omega_0 \wedge \xi) \right|
+ \left| \int_\Omega (f - f_{0}^{k+1}) \tilde \omega_0 \wedge \xi \right| \to 0
\end{gathered}$$ as $m \to \infty$. The first and the third terms are less than $C\gamma$ due to the choice of $f$, and the second one tends to zero by the induction hypothesis. Since $\gamma$ is arbitrary, this implies that $$\label{eq:conv_2_1}
\int_\Omega f_{0}^{k+1} \tilde \omega_m \wedge \xi \to
\int_\Omega f_{0}^{k+1} \tilde \omega_0 \wedge \xi.$$ The relation indicated in follows from and . This means that the sequence of forms $\psi_m = f_{m}^{k+1} \tilde \omega_m$ converges to the form $\psi_0 = f_{0}^{k+1} \tilde \omega_m$ in $D'$.
It remains to check that the sequences of forms $\psi_m$ and $d \psi_m$ are bounded in $L^{1)}$. The Hölder inequality provides $$\begin{gathered}
\bigg(\int_\Omega |\psi_m|^{1 - \varepsilon} \, dx \bigg)^{\frac{1}{1-\varepsilon}} =
\bigg(\int_\Omega |f_{m}^{k+1} \tilde \omega_m|^{1 - \varepsilon} \, dx \bigg)^{\frac{1}{1-\varepsilon}} \\
\leq
\bigg(\int_\Omega |f_{m}^{k+1}|^{(1 - \varepsilon) \frac{k+1 - \varepsilon}{1- \varepsilon}} \, dx \bigg)^{\frac{1}{k+1-\varepsilon}}
\bigg(\int_\Omega |\tilde \omega_m|^{(1 - \varepsilon) \frac{k+1 - \varepsilon}{k}} \, dx \bigg)^{\frac{k}{(k+1-\varepsilon)(1 - \varepsilon)}};
\end{gathered}$$ here $\frac{k+1 - \varepsilon}{1-\varepsilon} > 1$.
Multiplying by $\varepsilon$ and taking the supremum, we obtain $$\begin{gathered}
\label{est:psi_1}
\|\psi_m\|_{L^{1)}} \\
\leq
\sup\limits_{0 < \varepsilon < \varepsilon_0}\bigg(\varepsilon\int_\Omega |f_{m}^{k+1}|^{(1 - \varepsilon) \frac{k+1 - \varepsilon}{1 - \varepsilon}} \, dx \bigg)^{\frac{1}{k+1-\varepsilon}}
\bigg(\varepsilon\int_\Omega |\tilde \omega_m|^{(1 - \varepsilon) \frac{k+1 - \varepsilon}{k}} \, dx \bigg)^{\frac{k}{(k+1-\varepsilon)(1 - \varepsilon)}} \\
\leq
\sup\limits_{0 < \varepsilon < k}\bigg(\varepsilon\int_\Omega |f_{m}^{k+1}|^{k+1 - \varepsilon} \, dx \bigg)^{\frac{1}{k+1-\varepsilon}}
\sup\limits_{0 < \varepsilon' < \frac{k+1}{k}-1}\bigg(\varepsilon'\int_\Omega |\tilde \omega_m|^{\frac{k+1}{k} - \varepsilon'} \, dx \bigg)^{\frac{1}{(k+1)/k-\varepsilon'}}
\\
\leq
\|f_{m}^{k+1}\|_{L^{k+1)}} \|\tilde \omega_m\|_{L^{\frac{k+1}{k})}}.
\end{gathered}$$
The last inequality is valid for $\varepsilon' = \frac{\varepsilon (2 + k - \varepsilon )}{k}$, which satisfies $\frac{(1 - \varepsilon)(k+1 - \varepsilon)}{k} = \frac{k+1}{k} - \varepsilon'$. It is easy to check that $\varepsilon < \varepsilon'$. In order to make sure that $0 < \varepsilon' < \frac{k+1}{k}-1 = \frac{1}{k}$, note that the roots of $h(\varepsilon) = \varepsilon^2 - (2 + k)\varepsilon + 1 $, $\varepsilon_{1,2} = \frac{k+2 \pm \sqrt{k^2+4k}}{2}$ are not less than $\varepsilon_0 = \frac{k+2 - \sqrt{k^2+4k}}{2}$, and $h(0) = 1 > 0$.
In view of this, we can consider the supremum over all $0 < \varepsilon' < \frac{1}{k}$, and, by doing so, its value is seen to increase. This completes the proof of the estimate .
The same arguments show that $d \psi_m = \omega_m = \tilde \omega_m \wedge d f_{m}^{k+1}$ is bounded in $L^{\frac{p}{k+1})}$. By Lemma \[lem:conv\_dif\_GL\], this implies that $\omega_m \to \omega_0$ in $D'$.
Acknowledgment {#acknowledgment .unnumbered}
--------------
The author warmly thanks professor Sergey Vodopyanov and my great friend Dr. Ian McGregor for the numerous discussions on, and useful comments about this paper.
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|
[ Deducing rest masses of quarks with ]{}
[a three step quantization ]{}
[Jiao Lin Xu]{}
[The Center for Simulational Physics, The Department of Physics and Astronomy]{}
[University of Georgia, Athens, GA 30602, USA]{}
E- mail: [ jxu@hal.physast.uga.edu]{}
**Abstract.**[ Using a three step quantization and phenomenological formulae, we can deduce the rest masses and intrinsic quantum numbers (I, S, C, B and Q) of quarks from only one unflavored elementary quark family ]{}$\epsilon $[ with S = C = B = 0 in the vacuum. Then using sum laws, we can deduce the rest masses and intrinsic quantum numbers of baryons and meson from the deduced quarks. The deduced quantum numbers match experimental results exactly. The deduced rest masses are consistent with experimental results. This paper predicts some new quarks \[d]{}$_{s}$[(773), d$_{s}$(1933), ]{}$\text{u}_{c}$[(6073) , ]{}$\text{d}_{b}$[(9333)\], baryons \[]{}$\Lambda _{c} $[(6699), ]{}$\Lambda _{b}$[(9959)\] and mesons \[D(6231), B(9502)\].PACS: 12.60.-i; 12.39.-x; 14.65.-q; 14.20.-c Key word: beyond the standard model ]{}
[ ]{}
**1. Introduction**
One hundred years ago, classic physics had already been fully developed. Most physical phenomena could be explained with this physics. Black body spectrum, however, could not be explained by the physics of that time, leading Planck to propose a quantization postulate to solve this problem [@Planck]. The Planck postulate eventually led to quantum mechanics. Physicists already clearly knew that the black body spectrum was a new phenomenon outside the applicable area of classic physics. The development from classic physics to quantum physics depended mainly on new physical ideas (a quantization postulate) rather than complex mathematics and extra dimensions of space.
Today we face a similar situation. The standard model [@Standard] is in excellent accord with almost all current data.... It has been enormously successful in predicting a wide range of phenomena, but it cannot deduce the mass spectra of quarks. So far, no theory has been able to successfully do so. Like black body spectrum, this mass spectrum may need a new theory outside the standard model. M. K. Gaillard, P. D. Grannis, and F. J Sciulli have already pointed out [@Standard] that the standard model is incomplete... We do not expect the standard model to be valid at arbitrarily short distances. However, its remarkable success strongly suggest that the standard model will remain an excellent approximation to nature at distance scales as small as 10$^{-18}$m... high degree of arbitrariness suggests that a more fundamental theory underlies the standard model. The mass spectrum of quarks is outside the applicable area of the standard model. Physicists need to find a new and more fundamental theory that underlies the standard model. The history of quantum physics shows that a new physics theory’s primary need is new physical ideas. A three step quantization is the new physical idea. Using this quantization, we try to deduce the masses of quarks.
Today, physics’ foundation is quantum physics (quantum mechanics and quantum field theory), not classic physics. We work with quantum systems (quarks and hadrons) that are a level deeper than the system (atoms and molecules) faced by Planck and Bohr. Therefore, the quantized systems are quantum systems (not the classic system). If Planck and Bohr got correct quantizations for atoms and molecules using only one simple quantization, we must use more steps and more complex quantizations. It is worth emphasizing that deducing the rest masses and the intrinsic quantum numbers of the quarks using the three step quantization may be one level deeper than the standard model. Hopefully, the three step quantization can help physicists discover a more fundamental theory underlying the standard model [@Standard], just as the Planck-Bohr quantization did.
**2. The elementary quarks and their free excited quarks**
**2.1 The elementary quarks** ****
We assume that there is only one elementary quark family $\epsilon $ with s = I = $\frac{1}{2}$ and two isospin states ($\epsilon _{u}$ has I$_{Z}$ = $\frac{1}{2}$ and Q = +$\frac{2}{3}$, $\epsilon _{d}$ has I$_{Z}$ = $\frac{-1}{2}$ and Q = -$\frac{1}{3}$). For $\epsilon _{u}$ (or $\epsilon _{d}$),there are three colored (red, yellow and blue) quarks. Thus, there are six Fermi elementary quarks in the $\epsilon $ family with S = C = B = 0 in the vacuum. $\epsilon _{u}$ and $\epsilon _{d}$ have SU(2) symmetry.
As a colored (red, yellow or blue) elementary quark $\epsilon _{u}$ (or $\epsilon _{d}$) is excited from the vacuum, its color, electric charge, rest mass and spin do not change, but it will get energy. The free excited state of the elementary quark $\epsilon _{u}$ is the u-quark with either a red, yellow or blue color, Q = $\frac{2}{3}$, rest mass m$_{\epsilon
_{u}}^{\ast }$, I = s = $\frac{1}{2}$ and I$_{z}$ = $\frac{1}{2}$ . The free excited state of the elementary quark $\epsilon _{d}$ is the d-quark with either a red, yellow or blue color, Q = - $\frac{1}{3},$ rest mass m$_{\epsilon _{d}}^{\ast }$, I = s = $\frac{1}{2}$ and I$_{z}$ = -$\frac{1}{2}$. Since $\epsilon _{u}$ and $\epsilon _{d}$ have SU(2) symmetry, the u-quark and the d-quark also have SU(2) symmetry.
**2..2 The free motion of excited quark**
For the excited quark free motion, we generally use the Dirac equation, $$\text{i}\hslash \frac{\partial \psi }{\partial \text{t}}\text{=}\frac{\hslash c}{i}\text{(}\alpha _{1}\frac{\partial \psi }{\partial x^{1}}\text{+}\alpha _{2}\frac{\partial \psi }{\partial x^{2}}\text{+}\alpha _{3}\frac{\partial \psi }{\partial x^{3}}\text{) + }\beta m_{\epsilon }^{\ast }\text{C}^{2}\psi \text{.} \label{Dirac}$$Our purpose, however, is to find rest masses of the excited quarks. The rest masses are the energy of the excited quark at rest. Corresponding to an excited quark at rest, the free motion Dirac equation (\[Dirac\]) reduces [@Bjorken] to $$\text{i}\hslash \frac{\partial \psi }{\partial \text{t}}\text{ = }\beta
m_{\epsilon }^{\ast }\text{c}^{2}\psi . \label{Rest-Dirac}$$The equation has four solutions:
$$\psi ^{1}=e^{-i\frac{\text{mc}^{2}}{\hslash }t}\left[
\begin{array}{c}
1 \\
0 \\
0 \\
0\end{array}\right] ,\psi ^{2}=e^{-i\frac{\text{mc}^{2}}{\hslash }t}\left[
\begin{array}{c}
0 \\
1 \\
0 \\
0\end{array}\right] ,\psi ^{3}=e^{+i\frac{\text{mc}^{2}}{\hslash }t}\left[
\begin{array}{c}
0 \\
0 \\
1 \\
0\end{array}\right] ,\psi ^{1}=e^{+i\frac{\text{mc}^{2}}{\hslash }t}\left[
\begin{array}{c}
0 \\
0 \\
0 \\
1\end{array}\right]$$
$\psi ^{1}$ and $\psi ^{2}$ correspond to positive energy (quark) and $\psi
^{3}$ and $\psi ^{4}$ correspond to negative energy (antiquark). If we only consider quark and omit antiquark, we can get the two-component Pauli equation. Bjorken and Drell wrote [@Bjorken]: In particular, we wish to show that they have a sensible nonrelativistic reduction to the two-component Pauli spin theory. Thus we can use the low energy limit of the Dirac equation– Pauli equation to find the rest masses of quarks.
For single quark free low energy motion, the quark with spin up (s$_{z}$ = $\frac{1}{2}$) will have the same energy as the same quark with spin down (s$_{z}$ = -$\frac{1}{2}$). Thus, for single free low energy limits, we can omit the spin of the quark, and the two-component Pauli equation can be approached by the Schrödinger [@Schrodinger] equation. The Schrödinger equation is not to be looked down on. In fact, it is useful in deducing the rest masses of some baryons [@Daliz] and quarkonium mesons c$\overline{c}$ and b$\overline{b}$ [@Martin] and [@Kwong].
When we use the Schrödinger equation to approach the Pauli equation, we cannot forget the static energy of the excited quark. We will deal with this energy as a constant potential energy (V) at any location. The approximate Schrödinger equation is:
$$\frac{\hslash ^{2}}{\text{2}m_{\epsilon }^{\ast }}\nabla ^{2}\psi \text{ + (}\mathbb{E}\text{-V)}\psi \text{ = 0} \label{Schrodinger}$$
where $m_{\epsilon }^{\ast }$ is the unknown rest mass of the excited quark. V is the static energy (constant), and it is the minimum excited energy of an elementary quark from the vacuum. The solution of (\[Schrodinger\]) is the eigen wave function and the eigen energy of the free u-quark or the free d-quark:
$$\begin{tabular}{l}
eigen function $\psi _{\overrightarrow{k}}\text{(}\overrightarrow{\text{r}}\text{) }\backsim \text{ exp(i}\overrightarrow{k}\cdot \overrightarrow{r}\text{),}$ \\
eigen energy $\mathbb{E}\text{ = V +}\frac{\hslash ^{2}}{\text{2}m_{\epsilon
}^{\ast }}\text{[(k}_{1}\text{)}^{2}\text{+(k}_{2}\text{)}^{2}\text{+(k}_{3}\text{)}^{2}\text{].}$\end{tabular}
\label{Wave+Energy}$$
According to the Quark Model [@Quark; @Model] a proton p = uud and a neutron n = udd. Omitting electromagnetic mass of quarks, from ([Wave+Energy]{}), at $\overrightarrow{k}$ = 0, we have the rest masses
$$\begin{aligned}
\text{M}_{p} &\text{=}&\text{m}_{u}^{\ast }\text{+m}_{u}^{\ast }\text{+m}_{d}^{\ast }\text{ -}\left\vert \text{E}_{bind}\right\vert \approx \text{M}_{n}\text{= m}_{u}^{\ast }\text{+m}_{d}^{\ast }\text{+m}_{d}^{\ast }\text{ -}\left\vert \text{E}_{bind}\right\vert \text{= 939 Mev } \label{939} \\
&\rightarrow &\text{m}_{u}^{\ast }\text{ = m}_{d}^{\ast }\text{ = V =\ }\frac{1}{3}\text{(939 +}\left\vert \text{E}_{bind}\right\vert \text{) = 313 +}\Delta \text{ (Mev)} \label{313}\end{aligned}$$
where E$_{bind}$ is the total binding energy of the three quarks in a baryon. $\Delta $ represents $\frac{1}{3}\left\vert \text{E}_{bind}\right\vert $, and is an unknown large positive constant. Since no free quark has been found, we assume $$\Delta \text{ = }\frac{1}{3}\left\vert \text{E}_{bind}\right\vert \text{
\TEXTsymbol{>}\TEXTsymbol{>} M}_{p}\text{.} \label{Dalta}$$The free excited u(313+$\Delta $)-quark and d(313+$\Delta $)-quark have large rest masses (313+$\Delta $) M$_{p}$ = 938 Mev. This is a reason that the Schrödinger equation of the low energy free quark is a good approximation of the Dirac equation. The large rest masses of excited quarks guarantee that the Schrödinger equation is a very good approximation.
Now we deduce the energy bands (quarks) with a three step quantization method.
**3 Deducing energy bands with a three step Quantization**
Today’s physics must continue to develop from the standard model into a more fundamental physics [@Standard]. The new theory will deduce the rest masses and intrinsic quantum numbers (I, S, C, B and Q) of the quarks. Recall that the development from classic physics to quantum physics began with the Planck-Bohr quantization. We try to use a three step quantization method to start the long time procedure.
**3.1 Recall Planck and Bohr’s works**
Planck’s [@Planck] energy quantization postulate states that any physical entity whose single ‘coordinate’ execute simple harmonic oscillations (i.e., is a sinusoidal function of time) can possess only total energy $\varepsilon $ which satisfy the relation $\varepsilon =nh\nu $, n = 0, 1, 2, 3, .... where $\nu $ is the frequency of the oscillation and h is a universal constant. Planck selects reasonable energy from a continuous energy spectrum.
Bohr’s [@Bohr] orbit quantization tells us that an electron in an atom moves in a circular orbit about the nucleus...obeying the laws of classical mechanics. But instead of the infinity of orbits which would be possible in classical mechanics, it is only possible for an electron to move in an orbit for which its orbital angular momentum L is an integral multiple of Planck’s constant h, divided by 2$\pi $. Using the quantization condition (L = $\frac{nh}{2\pi }$, n = 1, 2, 3, ... ), Bohr selects reasonable orbits from infinite orbits.
Drawing from these great physicists’ works, we find that the most important law is to use quantized conditions and symmetries (as circular orbit) to select reasonable energy levels from a continuous energy spectrum.
**3.2 Three step Quantization**
In order to get the short-lived quarks, we quantize the free motion (quantum plane wave function of the Schrödinger equation) of an excited quark (\[Wave+Energy\]) to select energy bands from the continuous energy. The energy bands will correspond to short-lived quarks.
3.2.1 The first step quantizing condition
For free motion of an excited quark with continuous energy (\[Wave+Energy\]), we assume the wave vector $\overrightarrow{k}$ has the symmetries of the regular rhombic dodecahedron in $\overrightarrow{k}$-space (see Fig. 1).
\[Fig1\]
We assume that the axis $\Gamma $-H in Fig.1 has length $\frac{2\pi }{a}$ with an unknown constant a. The first step quantizing conditions are: $$\begin{tabular}{l}
k$_{1}$ = $\frac{2\pi }{a}$(n$_{1}$-$\xi $), \\
k$_{2}$ = $\frac{2\pi }{a}$(n$_{2}$-$\eta $), \\
k$_{3}$ = $\frac{2\pi }{a}$(n$_{3}$-$\zeta $).\end{tabular}
\label{K(1,2,3)}$$Putting (\[K(1,2,3)\]) into (\[Wave+Energy\]), we get (\[BandW\])$\ $and (\[E(nk)\]):
$$\begin{aligned}
\psi _{\overrightarrow{k}}\text{(}\overrightarrow{\text{r}}\text{) }
&\thickapprox &\text{ exp}\frac{2\pi i}{a}\text{[(n}_{1}\text{-}\xi \text{)x
+ (n}_{2}\text{ - }\eta \text{)y+(n}_{3}\text{ - }\zeta \text{)z],}
\label{BandW} \\
\mathbb{E}\text{(}\vec{k}\text{,}\vec{n}\text{) } &\text{=}&\text{313 + }\Delta \text{ + }\alpha \text{[(n}_{1}\text{-}\xi \text{)}^{2}\text{+(n}_{2}\text{-}\eta \text{)}^{2}\text{+(n}_{3}\text{-}\zeta \text{)}^{2}\text{]}
\notag \\
&\text{=}&\text{313 + }\Delta \text{ + 360 E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{) (Mev).} \label{E(nk)}\end{aligned}$$
where $\alpha $ =$\frac{\text{h}^{2}}{\text{2m}_{\epsilon }^{\ast }\text{a}^{2}}$ = 360 Mev (\[360\]). E($\overrightarrow{\kappa }$,$\overrightarrow{n}$) = \[(n$_{1}$-$\xi $)$^{2}$+(n$_{2}$-$\eta $)$^{2}$+(n$_{3}$-$\zeta $)$^{2} $\]. For $\overrightarrow{n}$ = (n$_{1}$, n$_{2}$, n$_{3}$), n$_{1}$, n$_{2}$ and n$_{3}$ are $\pm \func{integer}$s and zero. The $\overrightarrow{\kappa } $ = ($\xi $, $\eta $, $\zeta $) has the symmetries of a regular rhombic dodecahedron. In order to deduce the short-lived quarks, we must further quantize the $\overrightarrow{n}$ and $\overrightarrow{\kappa }$ as follows.
3.2.2 The second step quantization
We will quantize the $\overrightarrow{n}$ = (n$_{1}$, n$_{2}$, n$_{3}$) values further. If we assume n$_{1}$ = *l*$_{2}$ *+ l*$_{3}$, n$_{2}$ =* l*$_{3}$ *+ l*$_{1}$ and n$_{3}$ =* l*$_{1}$ *+ l*$_{2},$ so that $$\begin{tabular}{l}
\textit{l}$_{1}$ = $\frac{1}{2}$(-n$_{1}$ + n$_{2}$ + n$_{3}$) \\
\textit{l}$_{2}$ = $\frac{1}{2}$(+n$_{1}$ - n$_{2}$ + n$_{3}$) \\
\textit{l}$_{3}$ = $\frac{1}{2}$(+n$_{1}$ + n$_{2}$ - n$_{3}$).\end{tabular}
\label{l-n}$$The second step quantizing condition is that only those values of $\overrightarrow{n}$ = (n$_{1}$, n$_{2}$, n$_{3}$) are allowed that make $\overrightarrow{l}$ = *(l*$_{1}$*, l*$_{2}$*, l*$_{3}$) an integer vector. This is a second step quantization for $\vec{n}$ values. For example, $\vec{n}$ cannot take the values (1, 0, 0) or (1, 1, -1), but can take (0, 0, 2) and (1, -1, 2). From E($\overrightarrow{\kappa }$,$\overrightarrow{n}$) = \[(n$_{1}$-$\xi $)$^{2}$ +(n$_{2}$-$\eta $)$^{2}$ +(n$_{3}$-$\zeta $)$^{2}$\], we can give a definition of the equivalent $\overrightarrow{n}$: for $\overrightarrow{\kappa }$ = ($\xi $, $\eta ,$ $\varsigma )$ = 0, all $\overrightarrow{n}$ values that give the same E($\overrightarrow{\kappa }$,$\overrightarrow{n}$) value are equivalent n-values. We show the low level equivalent $\overrightarrow{n}$-values that satisfy condition (\[l-n\]) in the following list (\[nnn\]) (note $\overline{\text{n}_{i}}$ = - n$_{i}$):
$$\begin{tabular}{|l|}
\hline
{\small E(}$\overrightarrow{n}${\small ,0) = 0\ : (0, 0, 0) \ \ \ \ \ \ \ \
\ \ Notes: }[$\overline{\text{{\small 1}}}${\small 12 }$\equiv $ (-1,1,2)
and $\overline{\text{{\small 1}}}${\small 1}$\overline{\text{{\small 2}}}\equiv $ (-1,1,-2)] \\ \hline
{\small E(}$\overrightarrow{n}${\small ,0) = 2\ :}$\ ${\small (101, }$\overline{\text{{\small 1}}}${\small 01, 011, 0}$\overline{\text{{\small 1}}}
${\small 1, 110, 1}$\overline{\text{{\small 1}}}${\small 0, }$\overline{\text{{\small 1}}}${\small 10, }$\overline{\text{{\small 1}}}\overline{\text{{\small 1}}}${\small 0, 10}$\overline{\text{{\small 1}}}${\small , }$\overline{\text{{\small 1}}}${\small 0}$\overline{\text{{\small 1}}}${\small , 01}$\overline{\text{{\small 1}}}${\small , 0}$\overline{\text{{\small 1}}}\overline{\text{{\small 1}}}${\small )} \\ \hline
{\small E(}$\overrightarrow{n}${\small ,0) = 4\ :\ \ (002, 200}, {\small 200}$\text{, }\overline{\text{{\small 2}}}\text{{\small 00}, }${\small 0}$\overline{\text{{\small 2}}}${\small 0, 00}$\overline{\text{{\small 2}}}${\small )} \\ \hline
{\small E(}$\overrightarrow{n}${\small ,0) = 6:}
\begin{tabular}{l}
{\small 112, 211, 121, }$\overline{\text{{\small 1}}}${\small 21,}$\overline{\text{{\small 1}}}${\small 12, 2}$\overline{\text{{\small 1}}}${\small 1},
{\small 1}$\overline{\text{{\small 1}}}${\small 2, 21}$\overline{\text{{\small 1}}}${\small ,12}$\overline{\text{{\small 1}}}${\small ,}$\overline{\text{{\small 2}}}${\small 11, 1}$\overline{\text{{\small 2}}}${\small 1, 11}$\overline{\text{{\small 2}}}$,{\small \ } \\
$\overline{\text{{\small 11}}}${\small 2, }$\overline{\text{{\small 1}}}${\small 2}$\overline{\text{{\small 1}}}$,{\small \ 2}$\overline{\text{{\small 11}}}${\small , }$\overline{\text{{\small 21}}}${\small 1}, $\overline{\text{{\small 12}}}${\small 1, 1}$\overline{\text{{\small 12}}}${\small , 1}$\overline{\text{{\small 21}}}${\small ,}$\overline{\text{{\small 1}}}${\small 1}$\overline{\text{{\small 2}}}${\small ,}$\overline{\text{{\small 2}}}$1$\overline{\text{{\small 1}}}$,{\small \ }$\overline{\text{{\small 211}}}${\small , }$\overline{\text{{\small 121}}},${\small \ }$\overline{\text{{\small 112}}},$\end{tabular}
\\ \hline
\end{tabular}
\label{nnn}$$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
3.2.3 The third step Quantization
The vector $\overrightarrow{\kappa }$ = ($\xi $, $\eta $, $\zeta $) in ([BandW]{}) and (\[E(nk)\]) has the symmetries of the regular rhombic dodecahedron in k-space (see Fig. 1). From Fig. 1, we can see that there are four kinds of symmetry points ($\Gamma $, H, P and N) and six kinds of symmetry axes ($\Delta $, $\Lambda $, $\Sigma $, D, F and G) in the regular rhombic dodecahedron. The coordinates ($\xi $, $\eta $, $\varsigma $) of the symmetry points and axes are:
$$\overrightarrow{\kappa }_{\Gamma }=(0,0,0),\overrightarrow{\kappa }_{\text{H}}=(0,0,1),\overrightarrow{\kappa }_{\text{p}}\text{ }\text{= (}\frac{\text{1}}{\text{2}}\text{, }\frac{\text{1}}{\text{2}}\text{, }\frac{\text{1}}{\text{2}}\text{), }\overrightarrow{\kappa }_{\text{N}}\text{ }\text{= (}\frac{\text{1}}{\text{2}}\text{, }\frac{\text{1}}{\text{2}}\text{, }0\text{).}
\label{S-Point}$$
$$\begin{tabular}{ll}
$\overrightarrow{\kappa }_{\Delta }\text{ }\text{= (0, 0, }\zeta \text{),\ \
0}\leq \zeta \text{ }\leq \text{1; }$ & $\overrightarrow{\kappa }_{\Lambda }\text{ = (}\xi \text{, }\xi \text{, }\xi \text{), \ 0}\leq \xi \leq \frac{\text{1}}{\text{2}}\text{;}$ \\
$\overrightarrow{\kappa }_{\Sigma }\text{{} }\text{= (}\xi \text{, }\xi
\text{, 0), \ 0}\leq \xi \text{ }\leq \frac{\text{1}}{\text{2}}\text{;}$ & $\overrightarrow{\kappa }_{\text{D}}\text{ }\text{= (}\frac{\text{1}}{\text{2}}\text{, }\frac{\text{1}}{\text{2}}\text{, }\xi \text{), \ 0}\leq \xi \leq
\frac{\text{1}}{\text{2}}\text{;}$ \\
$\overrightarrow{\kappa }_{\text{G}}\text{ }\text{= (}\xi \text{, 1-}\xi
\text{, 0), \ }\frac{\text{1}}{\text{2}}\leq \xi \text{ }\leq \text{1;}$ & $\overrightarrow{\kappa }_{\text{F}}\text{ = (}\xi \text{, }\xi \text{, 1-}\xi \text{), \ 0}\leq \xi \leq \frac{\text{1}}{\text{2}}\text{.}$\end{tabular}
\label{Sym-Axes}$$
The third step quantizing condition is that the coordinates ($\xi $, $\eta $, $\zeta $) of $\ \overrightarrow{\kappa }$ in (\[E(nk)\]) only take the coordinate values of the six symmetry axes (\[Sym-Axes\]).
**3.3 Energy bands**
From the low energy free wave motion of a excited elementary quark $\epsilon
$ with a continuous energy spectrum {$\mathbb{E}$ = V +$\frac{\hslash ^{2}}{2m}$\[(k$_{1}$)$^{2}$+(k$_{2}$)$^{2}$+(k$_{3}$)$^{2}$\] (\[Wave+Energy\])}, using the three step quantization, we obtain a new energy formula {$\mathbb{E}$($\vec{k}$,$\vec{n}$) =313 + $\Delta $ + $\alpha $\[(n$_{1}$-$\xi $)$^{2}$+(n$_{2}$-$\eta $)$^{2}$+(n$_{3}$-$\zeta $)$^{2}$\] (\[E(nk)\])} with quantized $\vec{n}$ values of (\[nnn\]) and $\vec{k}$ values of ([Sym-Axes]{}). The energy (\[E(nk)\]) with a $\overrightarrow{n}$ = (n$_{1}$, n$_{2}$, n$_{3}$) of (\[nnn\]) and a $\vec{k}$ = ($\xi $, $\eta $, $\varsigma $) of (\[Sym-Axes\]) forms an energy band. The formula ([E(nk)]{}) with quantized $\vec{n}$ of (\[nnn\]) and $\vec{k}$ of ([Sym-Axes]{}) is the formula that can deduced all energy bands.
**3.4 Deducing energy bands**
After getting (\[E(nk)\]), (\[nnn\]) and (\[Sym-Axes\]), we can deduce low energy bands of the six symmetry axes. As an example, we will deduce the single energy bands of the $\Delta $-axis. For the $\Delta $-axis, $\overrightarrow{\kappa }_{\Delta }$ = (0, 0, $\zeta $) from (\[Sym-Axes\]). Putting $\overrightarrow{\kappa }_{\Delta }$ = (0, 0, $\zeta $) into (\[E(nk)\]), we get $\mathbb{E}_{\Delta }$[(]{}$\vec{k}$[,]{}$\vec{n}$[) ]{}= 313+$\Delta $+360\[(n$_{1}$)$^{2}$+(n$_{2}$)$^{2}$+(n$_{3}$-$\zeta $)$^{2}$\]. For point$\Gamma ,\overrightarrow{\kappa }_{\Gamma }$ = (0, 0, $0$) from (\[S-Point\]), E$_{\Gamma }$($\overrightarrow{\kappa }
$,$\overrightarrow{n}$) = (n$_{1}$)$^{2}$+(n$_{2}$)$^{2}$+(n$_{3}$)$^{2}$. For point-H$,\overrightarrow{\kappa }_{\text{H}}$ = (0, 0, 1) from ([S-Point]{}), E$_{\text{H}}$($\overrightarrow{\kappa }$,$\overrightarrow{n}$) = (n$_{1}$)$^{2}$+(n$_{2}$)$^{2}$+(n$_{3}$-1)$^{2}$. $$\begin{tabular}{l}
T$\text{he }\Delta \text{-axis, }\mathbb{E}_{\Delta }\text{{\small (}}\vec{k}\text{{\small ,}}\vec{n}\text{{\small )}=313+}\Delta \text{+360[(n}_{1}\text{)}^{2}\text{+(n}_{2}\text{)}^{2}\text{+(n}_{3}\text{-}\zeta \text{)}^{2}\text{]}$ \\
the $\Gamma $-$\text{point, }\overrightarrow{\kappa }_{\Gamma }\text{\
=(0,0,0), E}_{\Gamma }\text{(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{)=(n}_{1}\text{)}^{2}\text{+(n}_{2}\text{)}^{2}\text{+(n}_{3}\text{)}^{2}$ \\
the H-$\text{point, }\overrightarrow{\kappa }_{\text{H}}\text{\ =(0,0,1), E}_{\text{H}}\text{(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{)=(n}_{1}\text{)}^{2}\text{+(n}_{2}\text{)}^{2}\text{+(n}_{3}\text{-1)}^{2}\text{.}$\end{tabular}
\label{D-1}$$Putting $\text{(n}_{1}\text{, n}_{2}\text{, n}_{3}$[) ]{}values of the single bands of the $\Delta $-axis (Table B1 of [@0502091]) into $\mathbb{E}_{\Delta }$($\vec{k}$,$\vec{n}$), E$_{\Gamma }$($\overrightarrow{\kappa }$,$\overrightarrow{n}$) and E$_{\text{H}}$($\overrightarrow{\kappa }$,$\overrightarrow{n}$) (\[D-1\]), we can find energy bands as shown in Table 1:
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Table 1 The Single Energy Bands of the $\Delta $-Axis (the $\Gamma $-H axis)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\begin{tabular}{|l|l|l|l|l|}
\hline
$\_[1]{}\_[2]{}\_[3]{}${\small )} & $\_[Start]{}$ & minim.
E & $${\small (}$${\small ,}$${\small ) (Band)} & $\_[end]{}$
\\ \hline
{\small (0, 0, 0)} & $\_$ & 313+$$ & 313+$$+$\^[2]{}$ & $\_[H]{}$ \\ \hline
{\small (0, 0, 2) } & $\_[H]{}$ & {\small 673}+$$ & 313+$$+(2-$$)$\^[2]{}$ & $\_$ \\ \hline
{\small (0, 0, }$${\small ) } & $\_$ &
{\small 1753}+$$ & 313+$$+(2+$$)$\^[2]{}$ & $\_[H]{}$ \\ \hline
{\small (0, 0, 4)}$$ & $\_[H]{}$ & {\small 3553}+$$ & 313+$$+(4-$$)$\^[2]{}$ & $\_$
\\ \hline
{\small (0, 0, }$${\small )} & $\_$ & {\small 6073}+$$ & 313+$$+(4+$$)$\^[2]{}$ & $\_[H]{}$ \\ \hline
(0, 0, 6) & $\_[H]{}$ & {\small 9313}+$$ & 313+$ $+(6-$$)$\^[2]{}$ & $\_$ \\ \hline
& ... & ... & ... & ... \\ \hline
\end{tabular}\ \ $
$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{)}_{Start}$[is the value of ]{}$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{) at the start point of the energy band}$
$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{)}_{end}$[is the value of ]{}$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{) at the end point of the energy band}$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Similarly, we can deduce the single energy bands of the $\Sigma $-axis. For the $\Sigma $-axis, $\overrightarrow{\kappa }_{\Sigma }$= ($\xi $, $\xi $, $0 $). Putting the $\overrightarrow{\kappa }$ into (\[E(nk)\]), we have [ ]{}$\mathbb{E}_{\Sigma }$[(]{}$\vec{k}$[,]{}$\vec{n}$[)]{} =313+ $\Delta $ + $360$\[(n$_{1}$-$\xi $)$^{2}$+(n$_{2}$-$\xi $)$^{2}$+(n$_{3}$)$^{2}$\] $\text{\ 0}\leq \zeta \leq \frac{\text{1}}{2}$. For point N$,$ $\overrightarrow{\kappa }_{\text{N}}$ = ($\frac{1}{2}$, $\frac{1}{2}$, 0) from (\[S-Point\]), E$_{\text{N}}$($\overrightarrow{\kappa }$,$\overrightarrow{n}$) = (n$_{1}$-$\frac{1}{2}$)$^{2}$ +(n$_{2}$-$\frac{1}{2}$)$^{2}$ +(n$_{3}$)$^{2}$.$$\begin{tabular}{l}
$\mathbb{E}_{\Sigma }${\small (}$\vec{k}${\small ,}$\vec{n}${\small )} =313+
$\Delta $ + $360$[(n$_{1}$-$\xi $)$^{2}$+(n$_{2}$-$\xi $)$^{2}$+(n$_{3}$)$^{2}$] \\
$\overrightarrow{\kappa }_{\text{N}}$\ = ($\frac{1}{2}$, $\frac{1}{2}$, 0), E$_{\text{N}}$($\overrightarrow{\kappa }$,$\overrightarrow{n}$) = (n$_{1}$-$\frac{1}{2}$)$^{2}$+(n$_{2}$-$\frac{1}{2}$)$^{2}$+(n$_{3}$)$^{2}$ \\
$\overrightarrow{\kappa }_{\Gamma }$\ = (0, 0, 0), E$_{\text{N}}$($\overrightarrow{\kappa }$,$\overrightarrow{n}$) = (n$_{1}$)$^{2}$+(n$_{2}$)$^{2}$+(n$_{3}$)$^{2}$ {\small (\ref{D-1})}\end{tabular}
\label{S-1}$$Putting $\text{(n}_{1}\text{,n}_{2}\text{,n}_{3}$[) ]{}values of the single bands of the $\Sigma $-axis (Table B2 of [@0502091]) into $\mathbb{E}$($\vec{k}$,$\vec{n}$), [E]{}$_{\text{N}}$[(]{}$\overrightarrow{\kappa }$[,]{}$\overrightarrow{n}$[) and E]{}$_{\Gamma }$[(]{}$\overrightarrow{\kappa }$[,]{}$\overrightarrow{n}$[) (\[S-1\]), ]{}we can deduce energy bands as shown in Table 2
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Table 2 The Single Energy Bands of the $\Sigma $-Axis (the $\Gamma $-N axis) $\text{ }$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\begin{tabular}{|l|l|l|l|l|}
\hline
$\_[1]{}\_[2]{}\_[3]{}$ & $\_[Start]{}$ & minim.E & $\_ ${\small =313+}+$$+360$$ & $\_[end]{}$ \\ \hline
{\small (0, 0, 0)} & $\_$ & {\small 313}+ $$
& 313+$$+720$\^[2]{}$ & $\_[N]{}$ \\
\hline
$()$ & $\_[N]{}$ & $$+ $$ & 313+$$+720(1-$$)$\^[2]{}$ & $\_$
\\ \hline
$()$ & $\_$ & $$+$$
& 313+$$+720(1+$$)$\^[2]{}$ & $\_[N]{} $ \\ \hline
$()$ & $\_[N]{}$ & $$+$$ & 313+$$+720(2-$$)$\^[2]{}$ & $\_$
\\ \hline
$()$ & $\_$ & $$+$$
& 313+$$+720(2+$$)$\^[2]{}$ & $\_[N]{}$ \\ \hline
$()$ & $\_[N]{}$ & $$+$$ & 313+$$+720(3-$$)$\^[2]{}$ & $\_ $ \\ \hline
{\small (-}3,-3,0{\small )} & $\_$ & $$+$$ & 313+$$+720(3+$$)$\^[2]{}$ & $\_[N]{}$ \\ \hline
(4, 4, 0) & $\_[N]{}$ & $$+$$ & 313+$$+720(4-$$)$\^[2]{}$ & $\_$
\\ \hline
\end{tabular}\ $
$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{)}_{Start}$[is the value of ]{}$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{) at the start point of the energy band}$
$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{)}_{end}$[is the value of ]{}$\text{E(}\overrightarrow{\kappa }\text{,}\overrightarrow{n}\text{) at the end point of the energy band}$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Following the two examples in Tables 1 and 2, we can deduce all low energy bands of the six axes using the formulae (\[E(nk)\]), (\[nnn\]) and ([Sym-Axes]{}) (see Appendix B of [@0502091])
From these energy bands, we can deduce quarks using phenomenological formulae.
**4 The phenomenological formulae**[ ]{}
In order to deduce the** **short-lived quarks from energy bands, we assume the following phenomenological formulae for the rest mass and intrinsic quantum numbers (I, S, C, B and Q) of energy bands:
1). For a group of degenerate energy bands (number = deg) with the same energy and equivalent $\overrightarrow{n}$ values (\[nnn\]), the isospin is
$$\text{2I + 1 = }\deg \rightarrow \text{I = }\frac{\text{deg - 1}}{\text{2}}
\label{IsoSpin}$$
2). The strange number S of an energy band (quark) that lies on an axis with a rotary fold R of the regular rhombic dodecahedron is $$\text{S = R - 4.} \label{S-Number}$$
3). For energy bands with deg R and R - deg $\neq $ 2 (such as the single energy bands on the $\Gamma $-H axis and the $\Gamma $-N axis), the strange number is $$\text{S = S}_{axis}\text{+ }\Delta \text{S, \ }\Delta \text{S = }\delta
\text{(}\widetilde{n}\text{) + [1-2}\delta \text{(S}_{axis}\text{)]Sign(}\widetilde{n}\text{)} \label{S+DS}$$where $\delta $($\widetilde{n}$) and $\delta $(S$_{axis}$) are Dirac functions and S$_{axis}$ is the strange number (\[S-Number\]) of the axis. For an energy band with $\overrightarrow{n}$ = (n$_{1}$, n$_{2}$, n$_{3})$, $\widetilde{n}$ is defined as $$\widetilde{n}\text{ }\equiv \frac{\text{n}_{1}\text{+n}_{2}\text{+n}_{3}}{\left\vert \text{n}_{1}\right\vert \text{+}\left\vert \text{n}_{2}\right\vert \text{+}\left\vert \text{n}_{3}\right\vert }\text{. Sgn(}\widetilde{n}\text{) = }\left[ \text{\begin{tabular}{l}
+1 for $\widetilde{n}$ \TEXTsymbol{>} 0 \\
0 \ \ for $\widetilde{n}$ = 0 \\
-1 \ for $\widetilde{n}$ \TEXTsymbol{<} 0\end{tabular}
}\right] \label{n/n}$$
$$\text{If }\widetilde{n}\text{ = 0 \ \ \ \ }\Delta \text{S = }\delta \text{(0) = +1 \ from (\ref{S+DS}) and (\ref{n/n}).} \label{n=0-DS=+1}$$
$$\text{If \ }\widetilde{n}\text{ = }\frac{0}{0}\text{, we assume }\Delta
\text{S = - S}_{Axis}\text{ .} \label{DaltaS=-Sax}$$
Thus, for $\overrightarrow{n}$ = (0, 0, 0), from (\[DaltaS=-Sax\]), we have $$\text{S = S}_{Axis}\text{+ }\Delta \text{S = S}_{Axis}\text{- S}_{Axis}\text{
= 0.} \label{S=0 of n=0}$$
4). If an energy band with S = +1, we call it has a charmed number C (C = 1): $$\text{if }\Delta \text{S = +1}\rightarrow \text{S = S}_{Ax}\text{+}\Delta
\text{S = +1, }C\text{ }\equiv \text{+1.} \label{Charmed}$$If an energy band with S = -1$,$ which originates from $\Delta S=+1$ (S$_{Ax} $= -2), and there is an energy fluctuation,$\ $we call it has a bottom number B:$\ \ \ \ \ \ \ \ \ \ \ \ \ $$$\text{for single bands, if\ }\Delta \text{S = +1}\rightarrow \text{S = -1
and }\Delta \text{E}\neq \text{0, B}\equiv \text{-1.} \label{Battom}$$
5). The elementary quark $\epsilon _{u}$ (or $\epsilon _{d}$) determines the electric charge Q of an excited quark. For an excited quark of $\epsilon
_{u} $ (or $\epsilon _{d}$), Q = +$\frac{2}{3}$ (or -$\frac{1}{3}$). For an excited quark with isospin I, there are 2I +1 members . For I$_{z}$ 0, Q = +$\frac{2}{3}$; I$_{z}$ 0, Q = -$\frac{1}{3}$ and I$_{z}$ = 0, $$\begin{aligned}
\text{ if S+C+b }\text{\TEXTsymbol{>} 0, Q = Q} &&_{\epsilon _{u}(0)}\text{
= }\frac{2}{3}\text{;} \label{2/3} \\
\text{ if S+C+b }\text{\TEXTsymbol{<} 0, Q = Q} &&_{\epsilon _{d}(0)}\text{
= -}\frac{1}{3}\text{.} \label{- 1/3}\end{aligned}$$There is no quark with I$_{z}$ = 0 and S + C + b = 0.
6). Since the experimental full width of baryons is about 100 Mev order, for simplicity, we assume that a fluctuation energy $\Delta $E of a quark roughly is
$$\Delta \text{E = 100 S[(1+S}_{Ax}\text{)(J}_{S,}\text{+S}_{Ax}\text{)]}\Delta \text{S \ \ \ J}_{S}\text{=}\left\vert \text{S}_{Ax}\right\vert +\text{1,2,3, ....} \label{Dalta-E}$$
Fitting experimental results, we can get $$\alpha \text{ = 360 Mev.} \label{360}$$The rest mass (m$^{\ast }$) of a quark is the minimum energy of the energy band. From (\[E(nk)\]), (\[360\]) and (\[Dalta-E\]), the rest mass (m$^{\ast }$) of the quark is $$\begin{tabular}{l}
$\text{m}^{\ast }\text{ = \{313+ 360 minimum[(n}_{1}\text{-}\xi \text{)}^{2}\text{+(n}_{2}\text{-}\eta \text{)}^{2}\text{+(n}_{3}\text{-}\zeta \text{)}^{2}\text{]+}\Delta \text{E+}\Delta \text{\} (Mev)}$ \\
\ \ \ \ = m + $\Delta $ \ (Mev),\end{tabular}
\label{Rest Mass}$$This formula (\[Rest Mass\]) is the united quark mass formula.
Using above phenomenological formulae, we will show that each energy band corresponds to a short-lived quark and deduce its intrinsic quantum numbers I, S, C, B and Q.** **The minimum energy of the energy band is the rest mass of the short-lived quark corresponding to the energy band.
**5 Deducing quarks using the phenomenological formulae from energybands **
**5.1 Deducing quarks from the energy bands in Tables 1 and 2**
From deduced single energy bands of the $\Delta $-axis in Table 1, we can use the above formulae (\[IsoSpin\])-(\[Rest Mass\]) to deduce the quarks. For the $\Delta $-axis, R = 4, strange number S$_{\Delta }$ = 0 from (\[S-Number\]). For single energy bands, I = 0 from (\[IsoSpin\]); and S = S$_{\Delta }$+ $\Delta $S = $\Delta $S = $\delta $($\widetilde{n}$) - Sign($\widetilde{n}$) from (\[S+DS\]). For $\overrightarrow{n}$ = (0, 0, -2) and (0, 0, -4), $\Delta $S = +1 from (\[n/n\]) and (\[S+DS\]); for n = (0, 0, 2), (0, 0, 4) and (0, 0, 6) $\Delta $S = -1 from (\[n/n\]) and ([S+DS]{}). Using (\[Charmed\]), (\[n/n\]) and (\[S+DS\]), we can find the charmed number C = +1 when n = (0, 0, -2) and (0, 0, -4). From (\[2/3\]), we can find Q = $\frac{2}{3}$ when n = (0, 0, -2) and (0, 0, -4); from (\[- 1/3\]), Q = - $\frac{1}{3}$ when n = (0, 0, 2), (0, 0, 4) and (0, 0, 6). From (\[Dalta-E\]) and (\[Rest Mass\]), we can find the rest masses (minimE + $\Delta $E) . We list all results in Table 3:
[|l|]{} Table 3. The u$_{C}$(m$^{\ast }$)-quarks and the d$_{S}$(m$^{\ast }$)-quarks on the $\Delta $-axis\
S$_{axis}$ = 0, I = 0, S = $\Delta $S = $\delta $($\widetilde{n}$) + \[1-2$\delta $(S$_{axis}$)\]Sign($\widetilde{n}$), $\widetilde{n}\text{ }\equiv
\frac{\text{n}_{1}\text{+n}_{2}\text{+n}_{3}}{\left\vert \text{n}_{1}\right\vert \text{+}\left\vert \text{n}_{2}\right\vert \text{+}\left\vert \text{n}_{3}\right\vert }$\
$\text{n}_{1,}\text{n}_{2,}\text{n}_{3}$ $\text{E}_{Point}$ MinimE $\Delta \text{S}$ J I S C Q $\Delta \text{E}$ $q_{\text{Name}}(m^{\ast })$
------------------------------------------ -------------------------------- --------------- ------------------- --------------------------------- --------------- ---- --- ---------------- ------------------- --------------------------------------------
$\text{{\small 0,\ \ 0, \ 0}}$ $\text{E}_{\Gamma }\text{=0}$ 313 0 J$\text{ = 0}$ $\frac{1}{2}$ 0 0 $\frac{2}{3}$ 0 $\text{u(313+}\Delta \text{)}$
$\text{{\small 0, \ 0, \ 2}}$ $\text{E}_{H}\text{=1}$ 673 -1 J$_{\text{S,H}}\text{ =1}$ 0 -1 0 -$\frac{1}{3}$ 100 $\text{d}_{S}\text{(773+}\Delta \text{)}$
$\text{{\small 0, \ 0, -2}}$ $\text{E}_{\Gamma }\text{=4}$ 1753 +1 J$_{\text{C,}\Gamma }\text{=1}$ 0 0 1 $\frac{2}{3}$ 0 $\text{u}_{C}\text{(1753+}\Delta \text{)}$
$\text{{\small 0, \ 0, \ 4}}$ $\text{E}_{H}\text{=9}$ 3553 -1 J$_{\text{S,H}}\text{ =2}$ 0 -1 0 -$\frac{1}{3}$ 200 $\text{d}_{S}\text{(3753+}\Delta \text{)}$
$\text{{\small 0, \ 0, -4}}$ $\text{E}_{\Gamma }\text{=16}$ 6073 +1 J$_{\text{C,}\Gamma }\text{=2}$ 0 0 1 $\frac{2}{3}$ 0 $\text{u}_{C}\text{(6073+}\Delta \text{)}$
$\text{{\small 0, \ 0, \ 6}}$ $\text{E}_{H}\text{=25}$ $\text{9313}$ -1 J$_{\text{S,H}}\text{ =3}$ 0 -1 0 -$\frac{1}{3}$ 300 $\text{d}_{S}\text{(9613+}\Delta \text{)}$
\
Similarly, for the $\Sigma $-axis, R = 2, strange number S$_{axis}$ = -2 from (\[S-Number\]). For single energy bands, I = 0 from (\[IsoSpin\]). From (\[S+DS\]), S = S$_{axis}$+ $\Delta $S = -2+ $\Delta $S; the $\Delta $S = $\delta $($\widetilde{n}$) + Sign($\widetilde{n}$). For $\overrightarrow{n}$ = (1, 1, 0), (2, 2, 0), (3, 3, 0) and (4, 4, 0), $\Delta $S = +1 from (\[n/n\]) and (\[S+DS\]); for $\overrightarrow{n}$ = (-1, -1, 0), (-2, -2, 0) and (-3, -3, 0), $\Delta $S = -1 from (\[n/n\]) and (\[S+DS\]). Using (\[Battom\]), (\[n/n\]), (\[S+DS\]) and (\[Dalta-E\]), we can find the bottom number B = -1 when $\overrightarrow{n}$ = (3, 3, 0) and (4, 4, 0). From (\[2/3\]) and (\[- 1/3\]),we can find the electric charge Q = -$\frac{1}{3}$ for all quarks. From (\[Dalta-E\]) and (\[Rest Mass\]), we can deduce rest masses (minimE + $\Delta $E) of quarks from the energy bands in Table 2. We list all results in Table 4:
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Table 4. The d$_{b}$(m$^{\ast }$)-Quarks, d$_{S}$(m$^{\ast }$)-Quarks and d$_{\Omega }$(m$^{\ast }$)-Quarks of the $\Sigma $-Axis
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
S$_{axis}$ = -2, I = 0, S = S$_{axis}+\Delta $S = $\delta $($\widetilde{n}$) + \[1-2$\delta $(S$_{axis}$)\]Sign($\widetilde{n}$), $\widetilde{n}\text{ }\equiv \frac{\text{n}_{1}\text{+n}_{2}\text{+n}_{3}}{\left\vert \text{n}_{1}\right\vert \text{+}\left\vert \text{n}_{2}\right\vert \text{+}\left\vert \text{n}_{3}\right\vert }$
$\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
E$\_[Point]{}$ & $\_[1]{}\_[2]{}\_[3]{}$ & $$ &
S & B & Q & $ $ & I & minimE & $$ & $\_[Name]{}\^$ \\ \hline
$\_$ & (0, 0, 0) & +2$\^$ & 0 & 0 & $$ & J$\_$ & $$ & 313 & 0 & d(313$$) \\ \hline
$\_[N]{}$ & $()$ & +1 & -1 & 0
& $$ & J$\_$ & 0 & $$ & 0 & $\_[S]{}$ \\ \hline
$\_$ & $()$ & - 1 & -3 & 0 & $$ & J$\_$ & 0 & $$ & 0 & $\_$ \\ \hline
$\_[N]{}$ & $()$ & +1 & -1 & 0
& $$ & J$\_$ & 0 & $$ & 0 & $\_[S]{}$ \\ \hline
$\_$ & $()$ & - 1 & -3 & 0 & $$ & J$\_$ & 0 & $$ & 0 & $\_$ \\ \hline
$\_[N]{}$ & $()$ & +1 & 0 & -1
& $$ & J$\_$ & 0 & $$ & 100 & $\_[B]{}$ \\ \hline
$\_$ & $()$ & - 1 & -3 & 0 & $$ & J$\_$ & 0 & $$ & -300 & $\_$ \\ \hline
$\_[N]{}$ & $()$ & +1 & 0 & -1
& $$ & J$\_$ & 0 & $$ & 200 & $\_[B]{}$ \\ \hline
\end{tabular}\ $
$^{\ast }$For $\overrightarrow{n}\text{ = (n}_{1}\text{, n}_{2}\text{, n}_{3})$ = (0, 0, 0), $\Delta $S = - S$_{axis}$= +2 from (\[DaltaS=-Sax\])
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**5.2 The five deduced Ground quarks**
From Table 3 and Table 4, we find: The unflavored (S = C = B = 0) ground quarks are u(313+$\Delta $) and d(313+$\Delta $). The strange quarks d$_{s}$(493), d$_{s}$(773), d$_{s}$(1933), d$_{S}$(3753) d$_{s}$(9613), $\text{d}_{\Omega }\text{(1033+}\Delta \text{), d}_{\Omega }\text{(3193+}\Delta \text{) and d}_{\Omega }\text{(6493+}\Delta \text{); The ground strange quark is }$d$_{s}$(493). The charmed quarks u$_{c}$(1753) and u$_{c}$(6073); the charmed ground quark is u$_{c}$(1753). The bottom quarks d$_{b}$(4913) and d$_{b}$(9333); the bottom ground quark is d$_{b}$(4913). (in Table 11 of [0502091]{} we have shown all low energy quarks, the five deduced ground quarks are still the ground quarks of all quarks). For the four flavored quarks, there are five ground quarks \[u(313+$\Delta $), d(313+$\Delta $), d$_{S}$(493+$\Delta $), u$_{C}$(1753+$\Delta $) and d$_{b}$(4913+$\Delta $)\] in the quark spectrum. These five ground quarks correspond to the five quarks [@Quarks] of the current Quark Model: u $\leftrightarrow $ u(313+$\Delta $), d $\leftrightarrow $ d(313+$\Delta $), s $\leftrightarrow $ d$_{S} $(493+$\Delta $), c $\leftrightarrow $ u$_{C}$(1753+$\Delta $) and b $\leftrightarrow $ d$_{b}$(4913+$\Delta $).
We can compare the rest masses and intrinsic quantum numbers (I, S, C, b and Q) [@Quarks] of the current quarks with the deduced values of the five ground quarks. The deduced intrinsic quantum numbers (I, S, C, b and Q ) of the five ground quarks are exactly the same as the five current quarks as shown in Table 5A:
----------------------------------------------------------------------------------------------------------------------------
Table 5A. The Five Deduced Ground Quarks and Current Quarks
$\begin{tabular}{|l|l|l|l|l|l|}
\hline
Dq(m)$\^[\$]{}$,Cq$\^$ & u(313), u & d(313), d & d$\_[s]{}$(493), s & u$\_[c]{}$(1753), c & d$\_[b]{}$(4913), b \\ \hline
Strange S & 0 \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \ \ \ 0 & -1 \ \ \ \ \ \ \
-1 & 0 \ \ \ \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \\ \hline
Charmed C & 0 \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \
\ \ 0 & 1 \ \ \ \ \ \ \ \ \ \ \ \ 1 & 0 \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \\ \hline
Bottom B & 0 \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \ \
\ 0 & 0 \ \ \ \ \ \ \ \ \ \ \ \ 0 & -1 \ \ \ \ \ \ \ \ \ \ \ \ -1 \\ \hline
Isospin I & $$ \ \ \ \ \ \ \ \ $$ & $$ \ \
\ \ \ \ \ \ $$ & 0 \ \ \ \ \ \ \ \ \ 0 & 0 \ \ \ \ \ \ \ \ \ \ \
\ 0 & 0 \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \\ \hline
I$\_[Z]{}$ & $$ \ \ \ \ \ \ \ \ $$ & -$$ \ \ \
\ \ -$$ & 0 \ \ \ \ \ \ \ \ \ 0 & $0$ \ \ \ \ \ \ \ \ \ \ \ \ 0\
& 0 \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \\ \hline
Electric $\_[q]{}$ & $$ \ \ \ \ \ \ \ \ $$ & -$$ \ \ \ \ \ -$$ & -$$ \ \ \ \ \ \ -$$\ & $$ \ \ \ \ \ \ \ \ \ \ \ $$ & -$$
\ \ \ \ \ \ \ \ \ \ -$$\ \\ \hline
\end{tabular}\ $
Dq(m)$^{\$}$= Deduced ground quark; Cq$^{\ast }$= Current quark
----------------------------------------------------------------------------------------------------------------------------
The deduced rest masses of the five ground quarks are roughly a constant (about 390 Mev) larger than the masses of the current quarks, as shown in Table 5B.
‘
-------------------------------------------------------------------------------------------------------
Table 5B Comparing the Rest Masses of Deduced and Current Quarks
$\begin{tabular}{|l|l|l|l|l|l|}
\hline
Current Quark & Up & Down & Strange & Charmed & bottom \\ \hline
Current Quark(m) & u(2.8) & d(6) & s(105) & c(1225) & b(4500) \\ \hline
Current quark mass & {\small 1.5 to 4} & {\small 4 to 8} & {\small 80 to 130}
& {\small 1250 to 1350} &
\begin{tabular}{l}
{\small 4.1 to 4.4 G.} \\
{\small 4.6 to 4.9 G.}\end{tabular}
\\ \hline
Deduced Quark (m) & u(313) & d(313) & d$\_[S]{}$(493) & u(1753) & d$\_[b]{}$(4913)
\\ \hline
$\_[Curr.]{}\_[Deduced.]{}$ & 310 & 307 &
388 & 528 & 413 \\ \hline
\end{tabular}\ $
The rest mass of a deduced quark m$^{\ast }$= m + $\Delta \ \rightarrow $ m = m$^{\ast }$ - $\Delta $
-------------------------------------------------------------------------------------------------------
These mass differences may originate from different energy reference systems. If we use the same energy reference system, the deduced masses of ground quarks will be roughly consistent with the masses of the corresponding current quarks. Of course, the ultimate test is whether or not the baryons and mesons composed of the deduced quarks are consistent with experimental results.
We will deduce the baryons and mesons composed of the quarks in Table 3 and 4 in this paper.
**6 Deducing the baryons of the quarks in Table 3 and 4**
According to the Quark Model [@Quark; @Model], a colorless baryon is composed of three different colored quarks. Using sum laws (\[SumB\]) and the deduced quarks in Tables 3 and 4, we can deduce the baryons as shown in Table 6. From Tables 3 and 4, we can see that there is a term $\Delta \ $of the rest masses of quarks. $\Delta $ is a very large unknown constant. Since the rest masses of the quarks in a baryon are large (from $\Delta $) and the rest mass of the baryon composed by three quarks is not, we infer that there will be a strong binding energy (E$_{Bind}$ = - 3$\Delta $) to cancel 3$\Delta $ from the three quarks: M$_{\text{B}}\text{ = m}_{q_{1}}^{\ast }$ + m$_{q_{_{N}(313)}}^{\ast }$ + m$_{_{q_{_{N}(313)}}}^{\ast }$- $\left\vert \text{E}_{Bind}\right\vert $ = m$_{q_{1}}$+ m$_{q_{_{N}(313)}}$+ m$_{_{q_{_{N}(313)}}}$+ 3$\Delta $ - 3$\Delta $ = m$_{q_{1}}$+ m$_{q_{_{N}(313)}}$+ m$_{_{q_{_{N}(313)}}}$. Thus we will omit the term 3$\Delta \ $in the three quark masses and the term $-3\Delta \ $in the binding energy from now on. For simplicity’s sake, we only deduced baryons composed of at least two free excited quark q$_{N}$(313) \[u(313), d(313)\] since other baryons have much lower productivity. For these baryons, sum laws are:
$$\begin{tabular}{l}
Baryon strange number $\ \text{S}_{\text{B}}\text{ }\text{= S}_{q_{1}}\text{+ S}_{q_{_{N}(313)}}\text{+ S}_{q_{_{N}(313)}}=\text{ S}_{q_{1}}\text{,}$ \\
baryon charmed number $\text{C}_{\text{B\ }}\text{ }\text{= C}_{q_{1}}\text{
+ C}_{q_{_{N}(313)}}\text{ + C}_{q_{_{N}(313)}}\text{= C}_{q_{1}}\text{,}$
\\
baryon bottom number $\ $B$_{\text{B}}\text{=}\text{ B}_{q_{1}}\text{+ B}_{q_{_{N}(313)}}\text{+ B}_{q_{_{N}(313)}}\text{= B}_{q_{1}}\text{,}$ \\
baryon electric charge $\ \text{Q}_{\text{B}}\text{ = Q}_{q_{1}}\text{+ Q}_{q_{_{N}(313)}}\text{+ Q}_{_{q_{_{N}(313)}}}.$ \\
baryon mass M$_{\text{B}}$ = \ $\text{m}_{q_{1}}\text{+ m}_{q_{_{N}(313)}}\text{+ m}_{_{q_{_{N}(313)}}}($except charmed baryons)\end{tabular}
\label{SumB}$$
For charmed baryons, M$_{\text{B}}$ = $\text{m}_{q_{1}}\text{+ m}_{q_{_{N}(313)}}\text{+ m}_{_{q_{_{N}(313)}}}$+ $\Delta e$$$\Delta e\text{ = 100C(2I-1)} \label{Ebin of B}$$where C is the charmed number and I is the isospin of the charmed baryons. Using sum laws (\[SumB\]) and binding energy formula (\[Ebin of B\]), we deduce the baryons shown in Table 6 from the quarks in Tables 3 and 4:
[l]{} Table 6. The Baryons of the Quarks in Table 3 and Table 4\
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[q]{}$_{i}^{I_{z}}$(m) [q]{}$_{j}$ [q]{}$_{k}$ [I]{} [S]{} [C]{} [b]{} [Q]{} [M]{} [Baryon]{} [Exper.]{} $\frac{\Delta M}{M}$[%]{}
--------------------------------- ------------- ------------- --------------- -------- ------- -------- -------- ----------- ------------------------------ ------------------------------------- ---------------------------
[u]{}$^{\frac{1}{2}}$[(313)]{} [u]{} [d]{} $\frac{1}{2}$ [0]{} [0]{} [0]{} [1]{} [939]{} [p(939)]{} [p(938)]{} [0.11]{}
[d]{}$^{\frac{-1}{2}}$[(313)]{} [u]{} [d]{} $\frac{1}{2}$ [0]{} [0]{} [0]{} [0]{} [939]{} [n(939)]{} [n(940)]{} [0.11]{}
[d]{}$_{s}^{0}$[(493)]{} [u]{} [d]{} [0]{} [-1]{} [0]{} [0]{} [0]{} [1119]{} $\Lambda $[(1119)]{} $\Lambda ^{0}$[(1116)]{} [0.27]{}
[u]{}$_{c}^{0}$[(1753)]{} [u]{} [d]{} [0]{} [0]{} [1]{} [0]{} [1]{} [2279]{} $\Lambda $\Lambda _{c}^{+}$[(2285)]{} [0.3.]{}
_{c}$[(2279)]{}
[u]{}$_{c}^{0}$[(1753)]{} [u]{} [u]{} [1]{} [0]{} [1]{} [0]{} [2]{} [2479]{} $\Sigma $\Sigma _{c}^{++}$[(2455)]{} [1.0]{}
_{c}^{++}$[(2479)]{}
[u]{}$_{c}^{0}$[(1753)]{} [u]{} [d]{} [1]{} [0]{} [1]{} [0]{} [1]{} [2479]{} $\Sigma $\Sigma _{c}^{+}$[(2455)]{} [1.0]{}
_{c}^{+}$[(2479)]{}
[u]{}$_{c}^{0}$[(1753)]{} [d]{} [d]{} [1]{} [0]{} [1]{} [0]{} [0]{} [2479]{} $\Sigma $\Sigma _{c}^{-}$[(2455)]{} [1.0]{}
_{c}^{-}$[(2479)]{}
[d]{}$_{b}^{0}$[(4913)]{} [u]{} [d]{} [0]{} [0]{} [0]{} [-1]{} [0]{} [5539]{} $\Lambda _{b}$[(5539)]{} $\Lambda _{b}^{0}$[(5624)]{} [1.5]{}
[d]{}$_{S}^{0}$[(773)]{} [u]{} [d]{} [0]{} [-1]{} [0]{} [0]{} [0]{} [1399]{} $\Lambda $[(1399)]{} $\Lambda $[(1405)]{} [0.4]{}
[d]{}$_{S}^{0}$[(1933)]{} [u]{} [d]{} [0]{} [-1]{} [0]{} [0]{} [0]{} [2559]{} $\Lambda $[(2559)]{} $\Lambda $[(2585)]{}$^{\ast \ast }$ [1.0]{}
[d]{}$_{S}^{0}$[(3753)]{} [u]{} [d]{} [0]{} [-1]{} [0]{} [0]{} [0]{} [4375]{} $\Lambda $[(4375)]{} [Prediction]{}
[d]{}$_{S}^{0}$[(9613)]{} [u]{} [d]{} [0]{} [-1]{} [0]{} [0]{} [0]{} [10239]{} $\Lambda $[(10239)]{} [Prediction]{}
[u]{}$_{c}^{0}$[(6073)]{} [u]{} [d]{} [0]{} [0]{} [1]{} [0]{} [1]{} [6699]{} $\Lambda [Prediction]{}
_{C}^{+}$[(6699)]{}
[d]{}$_{b}^{0}$[(9333)]{} [u]{} [d]{} [0]{} [0]{} [0]{} [-1]{} [9959]{} $\Lambda _{b}^{0}$[(9959)]{} [Prediction]{}
[d]{}$_{\Omega }^{0}$[(1033)]{} [d]{} [d]{} [0]{} [-3]{} [0]{} [0]{} [-1]{} [1659]{} $\Omega ^{-}$[(1659)]{} $\Omega ^{-}$[(1672)]{} [0.8]{}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\
[u ]{}$\equiv $[ u(313) and d ]{}$\equiv $[ d(313). ]{}$\Lambda $[(2585)]{}$^{\ast \ast }$[ Evidence of existence only fair.]{}
Table 6 shows that the deduced intrinsic quantum numbers ( I, S, C, b and Q) of the baryons match experimental results [@Baryon04] exactly and that the deduced rest masses of the baryons are consistent with more than 98.5% of experimental results [@Baryon04]. These are strong supports for the deduced the rest masses and intrinsic quantum numbers of the quarks.
**7 Deducing the mesons of the quarks in Tables 3 and 4**
According to the Quark Model [@Quark; @Model], a colorless meson is composed of a quark q$_{i}$ with a color and an antiquark $\overline{q_{j}}$ with the anticolor of the quark q$_{i}$. For each flavor, the three different colored quarks have the same I, S, C, B, Q and m. Thus we can omit the color when we deduce the rest masses and intrinsic quantum numbers of the mesons. For mesons, the sum laws are
$$\begin{tabular}{l}
Meson strange number $\text{S}_{\text{M}}\text{ }\text{= S}_{q_{i}}\text{+ S}_{\overline{q_{j}}}\text{,}$ \\
meson charmed numbers $\text{C}_{\text{M\ }}\text{ }\text{= C}_{q_{i}}\text{
+ C}_{\overline{q_{j}}}\text{ ,}$ \\
meson bottom number B$_{\text{M}}\text{=}\text{ B}_{q_{i}}\text{+ B}_{\overline{q_{j}}}\text{,}$ \\
meson electric charge $\text{Q}_{\text{M}}\text{ = Q}_{q_{i}}\text{+ Q}_{\overline{q_{j}}}.$\end{tabular}
\label{Sum(SCbQ)}$$
There is a strong interaction between the quark and antiquark (colors), but we do not know how large it is. Since the rest masses of the quarks in mesons are large (from $\Delta $) and the rest mass of the meson composed of the quark and antiquark is not, we infer that there will be a large portion of binding energy (- 2$\Delta $) to cancel 2$\Delta $ from the quark and antiquark and a small amount of binding energy as shown in the following $$\text{E}_{B}\text{(q}_{i}\overline{q_{j}}\text{) = -2}\Delta \text{ - 337 +
100[}\frac{\Delta \text{m}}{\text{m}_{g}}\text{ +DS -}\ \widetilde{m}\text{+}\gamma \text{(i,j) -2I}_{i}\text{I}_{j}\text{]} \label{Eb-Meson}$$where $\Delta $ = $\frac{1}{3}\left\vert \text{E}_{bind}\right\vert $ ([Dalta]{}) is $\frac{1}{3}$ of the binding energy of a baryon (an unknown large constant, $\Delta $ m$_{\text{P}}$= 938 Mev), $\Delta $m = $\left\vert \text{m}_{i}\text{-m}_{j}\right\vert $, DS =$\left\vert \text{(}\Delta \text{S)}_{i}\text{- (}\Delta \text{S)}_{j}\right\vert $. m$_{g}$ = 939 (Mev) unless $$\begin{tabular}{|l|l|l|l|}
\hline
m$_{i}$(or m$_{j}$) equals & m$_{C}\geqslant $ 6073 & m$_{b}\geqslant $ 9333
& m$_{S}\geqslant $ 9613 \\ \hline
$\ \ \ \text{m}_{g}${\small \ will equal to} & 1753(Table 4) & 4913 (Table7)
& 3753(Table 4). \\ \hline
\end{tabular}
\label{m(g)}$$
$\ \widetilde{m}$ = $\frac{m_{i}\times m_{j}}{\text{m}_{g_{i}}\times \text{m}_{g_{j}}}$ m$_{g_{i}}$ = m$_{g_{j}}^{\text{ \ }}$ = 939 (Mev) unless $$\begin{tabular}{|l|l|l|l|l|l|}
\hline
\ m$_{i}$(or m$_{j}$) & m$_{q_{_{N}}}$=313 & m$_{d_{s}}$=493 & m$_{u_{c}}\succeq $1753 & m$_{d_{S}}$\TEXTsymbol{>} 3753, & m$_{d_{b}}\succeq $
4913 \\ \hline
\ m$_{g_{j}}$ (or m$_{g_{j}}^{\text{ \ }})$ & 313 & 493 & 1753 & 3753, &
4913. \\ \hline
\end{tabular}
\label{M(gi)}$$ If q$_{i\text{ }}$and q$_{j}$ are both ground quarks in Table 5, $\gamma $(i, j) = 0. If q$_{i\text{ }}$and q$_{j}$ are not both ground quarks, for q$_{i\text{ }}$= q$_{j}$, $\gamma $(i, j) = -$1$; for q$_{i\text{ }}\neq $ q$_{j}$,$\ \gamma $(i, j) = +1. S$_{i}$ (or S$_{j}$) is the strange number of the quark q$_{i}$ (or q$_{j}$). I$_{i}$ (or I$_{j}$) is the isospin of the quark q$_{i}$ (or q$_{j}$).
From the quarks in Tables 3 and 4, we can use (\[Sum(SCbQ)\]) and ([Eb-Meson]{}) to deduce the rest masses and the intrinsic quantum numbers (I, S, C, b and Q) of mesons as shown in Table 7.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\ \ \ \ \ \ \ \ \ \ \ \text{Table 7.\ \ The Deduced Mesons of the Quarks in
Tables 3 and 4}$
$\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$\_[i]{}\^[S]{}\_[i]{}$ & $$ & {\small DS} & {\small -100}$$ & {\small E}$\_[bind]{}$ &
{\small Deduced} & {\small Experiment} & {\small R} \\ \hline
$\_[N]{}\^[0]{}${\small (313)}$$ &
{\small 0} & $[0]{}$ & {\small -100} & {\small - 487}$\^[\#]{}$ & $${\small (139)} & $${\small (138)} & {\small 0.7} \\ \hline
{\small q}$\_[N]{}\^[0]{}${\small (313)}$$
& {\small 19} & {\small 1} & {\small -100} & {\small - 318} & {\small K(488)}
& {\small K(494)} & {\small 0.2} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (493)}$$
& {\small 0} & {\small 0} & {\small -100} & {\small - 437} & $${\small (549)} & $${\small (548)} & {\small 0.2} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (1753)}$$ & {\small 153} & {\small 1} & {\small -100} & {\small - 184} &
{\small D(1882)} & {\small D(1869)} & {\small 0.7} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (1753)}$$ & {\small 134} & {\small 0} & {\small -100} & {\small - 303} &
{\small D}$\_[S]{}${\small (1943)} & {\small D}$\_[S]{}${\small (1969)} & {\small 0.4} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (1753)}$$ & {\small 0} & {\small 0} & {\small -100} & {\small -437} &
{\small J/}$${\small (3069)} & {\small J/}$${\small (3097)} &
{\small 0.9} \\ \hline
$\_[N]{}\^[0]{}${\small (313)}$$ & {\small 490} & {\small 1} & {\small -100} & {\small \ 153}$\^$ & {\small B(5379)} & {\small B(5279)} & {\small 1.9} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (493)}$$ & {\small 471} & {\small 0} & {\small -100} & {\small \ 34}$\^$ & {\small B}$\_[S]{}${\small (5440)} & {\small B}$\_[S]{}${\small (5370)}
& {\small 1.3} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (1753)}$$ & {\small 337} & {\small 0} & {\small -100} & {\small -100} &
{\small B}$\_[C]{}${\small (6566)} & {\small B}$\_[C]{}${\small (6400)} & {\small 2.6} \\ \hline
{\small d}$\_[b]{}\^[1]{}${\small (4913)}$$ & {\small 0} & {\small 0} & {\small -100} & {\small -\ 437} & $${\small (9389)} & $${\small (9460)} & {\small 0.8} \\
\hline
{\small d}$\_[S]{}\^[-1]{}${\small (773)}$$ & {\small 0} & {\small 0} & {\small -68} & {\small -505} & $${\small (1041)} & $${\small (1020)} & {\small 2.0} \\ \hline
{\small d}$\_[S]{}\^[-1]{}${\small (3753)}$$ & {\small 0} & {\small 0} & {\small -1597} & {\small -2034} & $${\small (5472)} & {\small prediction} & \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (1933)}$$ & {\small 0} & {\small 0} & {\small -424} & {\small -861} & $ ${\small (3005)} & $\_[c]{}${\small (2980)} & {\small 0.8} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (9333)}$$ & {\small 0} & {\small 0} & {\small -361} & {\small -798} & $${\small (17868)} & {\small prediction} & {\small ?} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (6073)}$$ & {\small 0} & {\small 0} & {\small -1200} & {\small -1637} & $${\small (10509)} & $${\small (10355)} & {\small 1.5} \\ \hline
{\small d}$\_[S]{}\^[-1]{}${\small (9613)}$$ & {\small 0} & {\small 0} & {\small -656} & {\small -1093}
& $${\small (18133)} & {\small prediction} & \\ \hline
{\small d}$\_\^[-1]{}${\small (1033)}$$ & {\small 0} & {\small 0} & {\small -121} & {\small -558} & $${\small (1508)} & {\small f}$\_[0]{}${\small (1507)} & {\small 0.7} \\ \hline
{\small q}$\_[N]{}\^[0]{}${\small (313)}$$ & {\small 49} & {\small 1} & {\small -82} & {\small -170} &
{\small K(916)} & {\small K(892)} & {\small 2.7} \\ \hline
{\small q}$\_[N]{}\^[0]{}${\small (313)}$$ & {\small 171} & {\small 1} & {\small -206} & {\small -170} &
{\small K(2076)} & {\small K}$\_[4]{}\^${\small (2045)} & {\small 1.5} \\
\hline
{\small q}$\_[N]{}\^[0]{}${\small (313)}$$
& {\small 347} & {\small 1} & {\small -400} & {\small -190} & {\small K(3876)} & {\small prediction} & {\small ?} \\ \hline
{\small q}$\_[N]{}\^[0]{}${\small (313)}$$ & {\small 248} & {\small 1} & {\small -256} & {\small -145} &
{\small K(9781)} & {\small prediction} & {\small ?} \\ \hline
{\small q}$\_[N]{}\^[0]{}${\small (313)}$$ & {\small 183} & {\small 1} & {\small -190} & {\small -144} &
{\small B(9502)} & {\small prediction} & {\small ?} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (6073)}$$ & {\small 328} & {\small 1} & {\small -346.4} & {\small -155} &
{\small D(6231)} & {\small prediction} & {\small ?} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (6073)}$$ & {\small 318.3} & {\small 0} & {\small -346.4} & {\small -265} &
{\small D}$\_[s]{}${\small (6301)} & {\small prediction} & {\small ?} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (493)}$$ & {\small 30} & {\small 2} & {\small -256.1} & {\small -90} & $${\small (1177)} & $${\small (1170)} & {\small 0.6} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (493)}$$ & {\small 347} & {\small 2} & {\small -339.7} & {\small -90} & $${\small (4156)} & $${\small (4159)} & {\small .07} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (493)}$$ & {\small 243} & {\small 2} & {\small -256.1} & {\small -50} & $${\small (10056)} & $${\small (10023)} & {\small 0.4} \\ \hline
{\small u}$\_[C]{}\^[1]{}${\small (1753)}$$ & {\small 104} & {\small 2} & {\small -82.3} & {\small -15} &
{\small D}$\_[S]{}${\small (2511)} & {\small D}$\_[S\_[1]{}]{}${\small (2535)} &
{\small 1.0} \\ \hline
{\small d}$\_[S]{}\^[1]{}${\small (9613)}$$ & {\small 235} & {\small 0} & {\small -211} & {\small -212} & $${\small (10174)} & $${\small (10232)} & {\small 0.6} \\ \hline
\end{tabular}$
$^{\#}$[For q]{}$_{N}^{0}$[(313)]{}$\overline{_{N}^{0}\text{{\small (313)}}}$[, I]{}$_{i}$[I]{}$_{j}$ [= ]{}$\frac{1}{4}\rightarrow $ [100( -2I]{}$_{i}$[I]{}$_{j}$[) = - 50]{}
$^{\ast }$[The total binding energy (153-2]{}$\Delta $[) and (34-2]{}$\Delta $[) are negative from (\[Eb-Meson\])]{}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Table 7 shows that the deduced intrinsic quantum numbers match experimental results [@Meson04] exactly. The deduced rest masses are consistent with experimental results.
**8 Predictions **[ ]{}
This paper predicts some quarks, baryons and mesons shown in the following list:
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[q]{}$_{i}$[(m)]{} [q]{}$_{j}$ [q]{}$_{k}$ [Baryon]{} [q]{}$_{i}$[(m)]{}$\overline{\text{q}_{j}\text{(m)}}$ [Meson]{} [q]{}$_{i}$[(m)]{}$\overline{\text{{\small q}}_{i}\text{{\small (m)}}}^{\$}$
-------------------------------------- ------------- ------------- ----------------------------- -- ----------------------------------------------------------------------------------------- ------------------- -- ------------------------------------------------------------------------------
$\text{u}_{C}$[(6073)]{} [u]{} [d]{} $\Lambda _{C}$[(6699)]{} $\text{u}_{C}^{1}$[(6073)]{}$\overline{\text{{\small q}}_{N}^{0}\text{{\small (313)}}}$ [D(6231)]{} $\psi $[(10509)]{}$^{{\small \#}}$
$\text{d}_{b}^{1}$[(9333)]{} [u]{} [d]{} [B(9959)]{} [q]{}$_{N}^{0}$[(313)]{}$\overline{\text{d}_{b}^{1}\text{{\small (9333)}}}$ [B(9502)]{} $\Upsilon
$[(17868)]{}
$\text{d}_{S}\text{({\small 773})}$ [u]{} [d]{} $\Lambda $[(1399)]{}$^{\#}$ [q]{}$_{N}^{0}$[(313)]{}$\overline{\text{d}_{S}^{-1}\text{{\small (773)}}}$ [K(916)]{}$^{\#}$ $\phi $[(1041)]{}$^{{\small \#}}$
$\text{d}_{S}\text{({\small 1933})}$ [u]{} [d]{} $\Lambda $[(2559)]{}$^{\#}$ [q]{}$_{N}^{0}$[(313)]{}$\overline{\text{d}_{S}^{1}\text{{\small (1933)}}}$ [K(2076)]{} $\eta $[(3005)]{}$^{{\small \#}}$
$\text{d}_{S}\text{({\small 3753})}$ [u]{} [d]{} $\Lambda $[(4379)]{} [q]{}$_{N}^{0}$[(313)]{}$\overline{\text{d}_{S}^{-1}\text{{\small (3753)}}}$ [K(3876)]{} $\eta $[(5472)]{}
$\text{d}_{S}\text{({\small 9613})}$ [u]{} [d]{} $\Lambda $[(10239)]{} [q]{}$_{N}^{0}$[(313)]{}$\overline{\text{{\small d}}_{S}^{-1}\text{{\small (9613)}}}$ [K(9781)]{} $\eta $[(18133)]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$^{\$}$[The last column shows the mesons of the pair quarks \[q]{}$_{i}$[(m)]{}$\overline{\text{{\small q}}_{i}\text{{\small (m)}}}$[\] that the]{}
[ q]{}$_{i}$[(m) is in the first column.]{}
$\ \ \ \ \ \ \ \ \ \ \Lambda ^{0}$[(1399)]{}$^{\#}$
\[experimental
$\Lambda ^{0}$[(1406) with (]{}$\frac{\Delta M}{M}$[%) = 0.5%\],]{}
[ ]{}$\Lambda $[(2559)]{}$^{\#}$[ \[experimental ]{}$\Lambda $[(2585)]{}$^{\ast \ast }$[with (]{}$\frac{\Delta M}{M}$[%) = 1.0%\],]{}
[ K(916)]{}$^{\#}$
\[experimental K
$^{\ast }$[(892) with (]{}$\frac{\Delta M}{M}$[%) = 2.7%\],]{}
[ ]{}$\ \ \ \ \ \ \eta $[(1041)]{}$^{\#}$[ \[experimental ]{}$\phi $[(1020) with (]{}$\frac{\Delta M}{M}$[%) = 2%\],]{}
[ ]{}$\ \ \ \ \ \ \eta $[(3005)]{}$^{\#}$[ \[experimental ]{}$\eta _{c}$[(2980) with (]{}$\frac{\Delta M}{M}$[%) = 0.8%\].]{}
It is very important to pay attention to the $\Upsilon $(3S)-meson (mass m = 10,355.2 $\pm $ 0.4 Mev, full width $\Gamma $ = 26.3 $\pm $ 3.5 kev). We compare the mesons J/$\psi $(3097), $\Upsilon $(9460) and $\Upsilon $(10355) shown as follow list
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
u$_{C}^{1}$(1753)$\overline{\text{u}_{C}^{1}\text{(1753)}}$ = J/$\psi $(3069) \[J/$\psi $(3096.916$\pm $0.011), $\Gamma $ = 91.0 $\pm $ 3.2kev\]
d$_{b}^{1}$(4913)$\overline{\text{d}_{b}^{1}\text{(4913)}}$ = $\Upsilon $(9389) \[$\Upsilon $(9460.30$\pm $0.26), $\ \ \ \ \ \ \ \Gamma $ = 53.0 $\pm $ 1.5kev\]
u$_{C}^{1}$(6073)$\overline{\text{u}_{C}^{1}\text{(6073)}}$ = $\psi $(10509) \[$\Upsilon $(10.355.2 $\pm $ 0.4), $\ \ \ \ \ \Gamma $ = 26.3 $\pm $ 3.5 kev\]
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\Upsilon $(3S) has more than three times larger of a mass than J/$\psi $(1S) (m = 3096.916 $\pm $ 0.011 Mev) and more than three times longer of a lifetime than J/$\psi $(1S) (full width $\Gamma $ = 91.0 $\pm $ 3.2 kev). It is well known that the discovery of J/$\psi $(1S) is also the discovery of charmed quark c (u$_{c}$(1753)) and that the discovery of $\Upsilon $(9460) is also the discovery of bottom quark b (d$_{b}$(4913)). Similarly the discovery of $\Upsilon $(3S) will be the discovery of a very important new quark—the u$_{C}$(6073)-quark.
**9 Discussion **
1). From the low energy free wave motion of a excited elementary quark $\epsilon $ with a continuous energy spectrum {$\mathbb{E}$ = V +$\frac{\hslash ^{2}}{2m}$\[(k$_{1}$)$^{2}$+(k$_{2}$)$^{2}$+(k$_{3}$)$^{2}$\]}, using the three step quantization, we obtain a new energy formula {$\mathbb{E}$($\vec{k}$,$\vec{n}$) =313 + $\Delta $ + $\alpha $\[(n$_{1}$-$\xi $)$^{2}$+(n$_{2}$-$\eta $)$^{2}$+(n$_{3}$-$\zeta $)$^{2}$\]} with quantized $\vec{n}$ values (\[nnn\]) and $\vec{k}$ values (\[Sym-Axes\]). The energy ([E(nk)]{}) with a $\overrightarrow{n}$ = (n$_{1}$, n$_{2}$, n$_{3}$) of ([nnn]{}) and a $\vec{k}$ = ($\xi $, $\eta $, $\varsigma $) of (\[Sym-Axes\]) forms an energy band. If the free eigen wave function and eigen energy of the Schrödinger equation are first quantization, the three step quantization is the second quantization. This second quantization products new quantum– shout-lived quarks.
2). The rest masses and the intrinsic quantum numbers (I, S, C, B and Q) are necessary for the standard model, but they cannot be deduced by the standard model. Using (\[IsoSpin\]) - (\[Rest Mass\]), we deduce the rest masses and intrinsic quantum numbers of quarks from the energy bands. The deduced rest masses and quantum numbers of baryons and mesons from these masses and numbers of quarks, are** **consistent with experimental results. This is a strong support for the three step quantization.
3). The five quarks of the current Quark Model correspond to the five deduced ground quarks \[u$\leftrightarrow $u(313), d$\leftrightarrow $d(313), s$\leftrightarrow $d$_{s}$(493), c$\leftrightarrow $u$_{c}$(1753) and b$\leftrightarrow $d$_{b}$(4913)\] (see Table 11 of [@0502091])). The current Quark Model uses only these five quarks to explain baryons and mesons. In early times, this was reasonable, natural and useful. Today, however, it is not reasonable that physicists use only these five current quarks since physicists have discovered many high energy baryons and mesons that are composed of more high energy quarks.
4). The energy band excited quarks u(313) with $\overrightarrow{n}$ = (0, 0, 0) in Table 3 and d(313) with $\overrightarrow{n}$ = (0, 0, 0) in Table 4 will be short-lived quarks. They are, however, lowest energy quarks. Since there is no lower energy position that they can decay into, they are not short-lived quarks. Because they have the same rest mass and intrinsic quantum numbers as the free excited quarks u(313) and d(313), they cannot be distinguished from the free excited quarks u(313) and d(313) by experiments. The u(313) and d(313) with $\overrightarrow{n}$ = (0, 0, 0) will be covered up by free excited u(313) and d(313) in experiments. Therefore, we can omit u(313) and d(313) with $\overrightarrow{n}$ = (0, 0, 0) since the probability that they are produced much small than the free excited u(313)-quark and d(313)-quark. There are only long-lived and free excited the u(313)-quark and the d(313)-quark in both theory and experiments.
5). The fact that physicists have not found any free quark shows that the binding energies are very large. The baryon binding energy -3$\Delta $ (meson - 2$\Delta $ ) is a phenomenological approximation of the color’s strong interaction energy in a baryon (a meson). The binding energy -3$\Delta $ (-2$\Delta $) is always cancelled by the corresponding parts 3$\Delta $ of the rest masses of the three quarks in a baryon (2$\Delta $ of the quark and antiquark in a meson). Thus we can omit the binding energy -3$\Delta $ (-2$\Delta $) and the corresponding rest mass parts 3$\Delta $ (2$\Delta $) of the quarks when we deduce rest masses of baryons (mesons). This effect makes it appear as if there is no strong binding energy in baryons (mesons).
**10 Conclusions **
1). There is only one elementary quark family $\epsilon $ with three colors and two isospin states ($\epsilon _{u}$ with I$_{Z}$ = $\frac{1}{2}$ and Q = +$\frac{2}{3}$, $\epsilon _{d}$ with I$_{Z}$ = $\frac{-1}{2}$ and Q = -$\frac{1}{3}$) for each color. Thus there are six Fermi (s = $\frac{1}{2}$) elementary quarks with S = C = B = 0 in the vacuum. $\epsilon _{u}$ and $\epsilon _{d}$ have SU(2) symmetries.
2). All quarks inside hadrons are the excited states of the elementary quark $\epsilon $. There are two types of excited states: free excited states and energy band excited states. The free excited states are only the u-quark and the d-quark. They are long-lived quarks. The energy band excited states are the short-lived quarks, such as d$_{s}$(493), d$_{s}$(773), u$_{c}$(1753) and d$_{b}$(4913).
3). Since all quarks inside hadrons are excited states of the same elementary quark $\epsilon $, all quarks (m 313 Mev) will eventually decay into the q$_{N}$(313)-quark \[(u(313) and d(313)\].
4). The three step quantization is a someway second quantization that products the short-lived quarks.
5). There is a large binding energy -3$\Delta $ (or -2$\Delta $) among three quarks in a baryon (or between the quark and the antiquark in a meson). It may be a possible foundation for the quark confinement.
6). We have deduced the rest masses and intrinsic quantum numbers of quarks (Table 3 and 4), baryons (Table 6) and mesons (Table 7) using the three step quantization method and phenomenological formulae. The deduced intrinsic quantum numbers of baryons and mesons match the experimental results [Baryon04]{} and [@Meson04] exactly, while the deduced rest masses of the baryons and the mesons are consistent with more than 98% of experimental results [@Baryon04] and [@Meson04].
7). The current Quark Model is the five ground quark approximation of a more fundamental model.
8). This paper predict some new quarks \[$\text{u}_{C}$(6073) , $\text{d}_{b}$(9333) and d$_{S}$(773)\], baryons \[$\Lambda _{C}$(6699)[ and ]{}B(9959)\] and mesons \[D(6231), B(9502) and K(3876)\].
**Acknowledgments**
I sincerely thank Professor Robert L. Anderson for his valuable advice. I acknowledge** **my indebtedness to Professor D. P. Landau for his help also. I would like to express my heartfelt gratitude to Dr. Xin Yu for checking the calculations. I sincerely thank Professor Yong-Shi Wu for his important advice and help. I thank Professor Wei-Kun Ge for his support and help. I sincerely thank Professor Kang-Jie Shi for his advice.
[99]{} R. M. Eisberg, *Fundamentals of Mondern Physics*, (John Wiley & Sons, New York, 1961) p.64.
M. K. Gaillard, P. D. Grannis, and F. J Sciulli, Rev. Mod. Phys., **71** No.2 Centenary S96 (1999).
J. D. Bjorken and S. D. Drell, *Relativistic Quantum Mechanics, (Mc*Graw-Hill, New York, 1965) * p.10*.
H. Grosse and A. Martin, *Particle Physics and Schrödinger Equation*. (Cambridge, University Press, 1997).
R. H. Daliz and L. J. *Reinders, in Hardron Structure as Know from Electromagnetic and Strong Interactions, proceedings of the Hadrons ’77Conferece*, p.11 (Veda, 1979).
J. Sttubble and A. Martin, Phys. Lett. B **271,** 208 (1991).
W. Kwong and J. L. Rosner, Phys. Rev. **D38,** 279 (1988).
M. Gell-Mann, Phys. Lett. **8,** 214 (1964); G. Zweig, CERN Preprint CERN-Th-401, CERN-Th-412 (1964); Particle Data Group, Phys. Lett. **B592**, 154 (2004).
R. M. Eisberg, *Fundamentals of Mondern Physics*, (John Wiley & Sons, New York, 1961) p.114.
J. L. Xu, hep-ph/0502091.
Particle Data Group, Phys. Lett. **B592**, 37 (2004).
Particle Data Group, Phys. Lett. **B592**, 66–78 (2004).
Particle Data Group, Phys. Lett. **B592**, 38–65 (2004).
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abstract: 'Let $K$ be a number field, $\overline{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in Gal(\overline{K}/K)$, the Mordell-Weil group $E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of $\overline{K}$ under $\sigma$ has infinite rank.'
address: 'Department of Mathematics, Indiana University, Bloomington, Indiana 47405'
author:
- 'Bo-Hae Im'
date: 'January 20, 2003'
title: ' Mordell-Weil groups and the rank of elliptic curves over large fields '
---
Introduction
============
In [@fj74], G. Frey and M. Jarden showed that if $K$ is an infinite field of finite type and $A$ is an abelian variety of dimension $d\geq 1$ defined over $K$, then for any positive integer $n$, there is a subset of $Gal(\overline{K}/K)^n$ of Haar measure 1 such that for every $n$-tuple $(\sigma_1,\ldots,\sigma_n)$ belonging to the subset, the group of rational points $A(\overline{K}(\sigma_1,\ldots,\sigma_n))$ of $A$ over the fixed subfield of $\overline{K}$ under $(\sigma_1,\ldots,\sigma_n)$ has infinite rank.
In [@larsen], M. Larsen proved that for a number field $K$ and an elliptic curve $E/K$ over $K$, there is a nonempty open subset $\Sigma$ of $Gal(\overline{K}/K)$ such that for any $\sigma\in
\Sigma$, the Mordell-Weil group $E(\overline{K}^{\sigma})$ of $E$ over the fixed field under $\sigma$ has infinite rank.
It is natural to ask if such an open subset can be the whole Galois group $Gal(\overline{K}/K)$. We have a positive answer for elliptic curves defined over ${\mathbb{Q}}$. In [@im], we proved that for any elliptic curve $E/{\mathbb{Q}}$, the rank of $E(\overline{{\mathbb{Q}}}^{\sigma})$ is infinite, for every $\sigma\in
Gal(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$. Our approach in [@im] is arithmetic: taking advantage of the modularity of elliptic curves over ${\mathbb{Q}}$ and the complex multiplication theory and constructing an infinite supply of rational points of $E$ consisting of Heegner points.
This paper is motivated by [@fj74], [@im] and [@larsen] and we prove in section 3 that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma \in
Gal(\overline{K}/K)$, the Mordell-Weil group $E(\overline{K}^{\sigma})$ over the fixed subfield of $\overline{K}$ under $\sigma$ has infinite rank. Here, we approach by using Diophantine geometry which is a completely different method from the one that we use in [@im].
The main strategy for constructing infinitely many linearly independent rational points of $E$ over $\overline{K}^{\sigma}$ for $\sigma\in Gal(\overline{K}/K)$ is approximately as follows: find a finite group $G$, a ${\mathbb{Z}}$-free ${\mathbb{Z}}[G]$-module $M$ and an infinite sequence $\{K_i/K\}_{i=1}^{\infty}$ of linearly disjoint finite Galois extensions of $K$ with $Gal(K_i/K)\cong G$ such that for each $i$, $E(K_i)\otimes{\mathbb{Q}}$ contains a $G$-submodule isomorphic to $M\otimes{\mathbb{Q}}$. If $M^G=0$ but $M^g\neq 0$ for each $g\in G$, then we can find ${\mathbb{Q}}$-independent points of $E(K_i)\cap E(\overline{K}^{\sigma})$ for any $\sigma\in Gal(\overline{K}/K)$. For any pair $(G,M)$, $G$ acts on $E\otimes M\cong E^r$. Suppose we find a projective line ${\mathbb{P}}^1$ in $(E\otimes M)/G$ over $K$. If its preimage $X$ in $E\otimes M$ under the quotient map is an irreducible curve over $K$, then by the Hilbert irreducibility theorem ([@l83], Chapter 9), most points in ${\mathbb{P}}^1(K)$ determine points in $E^r(K_i)$ with $Gal(K_i/K)=G$; the coordinates generate the desired $G$-submodule of $E(K_i)\otimes{\mathbb{Q}}$. In this paper, we take for $G$ the alternating group $A_n$ on $n=2k$ letters and for the module $M$ the irreducible $(n-1)$-dimensional quotient of the permutation representation of $A_n$.
In section 2, first, we show that $S_n$ admits a nontrivial action on the $(n-1)$-fold product $E^{n-1}$ of $E$ and its quotient $E^{n-1}/S_n$ by $S_n$ is isomorphic to the $(n-1)$-dimensional projective space ${\mathbb{P}}^{n-1}$. We also find some properties of transitive subgroups of $S_n$ which contain a transposition and observe properties of subgroups of $A_n$ which occur as branched Galois coverings of a projective line.
In section 3, if $K$ is totally imaginary and $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then we show that for some even integer $n$, there is a projective line over $K$ in $E^{n-1}/S_n$ whose preimage in $E^{n-1}/A_n$ under the double cover is a curve of genus 0, which gives infinitely many linearly independent points of $E$ over the fixed field of each $\sigma\in Gal(\overline{K}/K)$. In section 4, as a special case, by the Hilbert irreducibility theorem ([@l83], Chapter 9) and the density of the Hilbert sets over ${\mathbb{Q}}$ in ${\mathbb{R}}$, we prove that if $K$ is a number field and $K_{ab}$ is the maximal abelian extension of $K$, then for any complex conjugation automorphism $\sigma\in Gal(\overline{K}/K)$, the rank of $E((K_{ab})^{\sigma})$ is infinite. Hence, the rank of $E(\overline{K}^{\sigma})$ is infinite.
Then, in section 5, we show that if $\sigma\in
Gal(\overline{K}/K)$ is not a complex conjugation automorphism, then, there is a totally imaginary finite extension of $K$ which is fixed under $\sigma$. So by applying this to extend the ground field to a totally imaginary extension for such automorphisms in $Gal(\overline{K}/K)$ and combining the result of infinite rank of the case of totally imaginary number fields, and the case of complex conjugation automorphisms, we get a more general result that if $K$ is an arbitrary number field and $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in Gal(\overline{K}/K)$ the rank of $E(\overline{K}^{\sigma})$ is infinite.
I wish to thank my thesis advisor, Michael Larsen for suggesting this problem, his guidance, valuable discussions and helpful comments on this paper.
Action of $S_n$ on $E^{n-1}$ and branched Galois coverings of ${\mathbb{P}}^1$
===============================================================================
Let $n\geq 2$ be an integer. First, let $S_n$ be the symmetric group on $n$ letters and $A_n$ the alternating subgroup of $S_n$. Denote the $n$-fold product of $E$ by $E^n$. Naturally, $S_n$ acts on $E^n$ by permutation, *i.e.* if we denote its action by ‘$\cdot$’, for $\sigma \in S_n$ and an $n$-tuple $(P_1,\ldots,P_n)\in E^n$, $\sigma\cdot(P_1,\ldots,P_n)=(P_{\sigma(1)},\ldots,P_{\sigma(n)})$. So does $A_n$ on $E^n$. Let $\Sigma :E^n \rightarrow E$ be the map defined by the sum of coordinates of an $n$-tuple. Then identify $E^{n-1}$ with $n$-tuples of elements in $E$ which sum to $O$ *i.e.* $Ker(\Sigma)$. $S_n$ still acts on $E^{n-1}\cong
Ker(\Sigma)$ by the nontrivial induced permutation action.
Through the paper, we always consider $E^{n-1}$ as $Ker(\Sigma)$ so that a point in $E^{n-1}$ (or its quotient $E^{n-1}/S_n$ by $S_n$) is a $n$-tuple $(P_1,\ldots,P_n)\in E^{n-1}$ whose coordinates sum to $O$.
The following lemma gives the structure of the quotient space $E^{n-1}/S_n$ of $E^{n-1}$ by $S_n$.
\[lem:action\] For each $n\geq 2$, $S_n$ admits a nontrivial action on $E^{n-1}$. And the quotient space $E^{n-1}/S_n$ of $E^{n-1}$ by $S_n$ is isomorphic to the $(n-1)$-dimensional projective space ${\mathbb{P}}^{n-1}$.
Identify $E^{n-1}$ with $n$-tuples of elements in $E$ which sum to $O$ *i.e.* with the set $Ker(\Sigma)$, where $\Sigma :E^n
\rightarrow E$ is the map defined by the sum of coordinates of an $n$-tuple. Then, for each $(P_1,\ldots,P_n)\in Ker(\Sigma)\cong
E^{n-1}$, there is a rational function $f$ on $E$ such that $\sum\limits _{i=1}^n (P_i)=(f)+n(O)$ as divisors. This gives a map from $E^{n-1}$ to the linear space of all rational functions $f$ on $E$ such that $(f)+n(O) \geq 0$. Denote this linear space by $|n(O)|$. Then, by the Riemann-Roch Theorem ([@h52], Chapter IV, Theorem 1.3), the dimension of this space is $n$ as a vector space so it gives an ($n-1$)-dimensional projective space. We choose a basis $f_0,\ldots, f_{n-1}$ of the space $|n(O)|$ and define a map $\phi
: E^{n-1} \rightarrow {\mathbb{P}}^{n-1}$ in the following way.
For each $(P_1,\ldots,P_n)\in Ker(\Sigma)\cong E^{n-1}$, there is a rational function $f$ on $E$ such that $\sum\limits _{i=1}^n
(P_i)=(f)+n(O)$. Write $f =\sum\limits _{i=0}^{n-1}a_i f_i$ with $a_0,\ldots,a_{n-1}\in{\mathbb{C}}$. Then, define $\phi(P_1,\ldots,P_n)=(a_0:a_1:\cdots:a_{n-1})\in {\mathbb{P}}^{n-1}$.
Then two $n$-tuples which sum to $O$ in $E^{n-1}$ map onto the same point in ${\mathbb{P}}^{n-1}$ under $\phi$ if and only if they are the same up to permutations of $S_n$. This implies that the quotient space $E^{n-1}/S_n$ is isomorphic to the projective space ${\mathbb{P}}^{n-1}$.
Now we find some properties of subgroups of $S_n$ which act transitively on $\{1,2,\ldots,n\}$ and contain a transposition. The following lemma assumes a weaker condition than in ([@jar2], Lemma 1.4).
\[lem:semidirect\] If $H$ is a subgroup of $S_n$ containing a transposition and $H$ acts transitively on $\{1,2,\ldots,n\}$, then there are positive integers $m$ and $k$ such that $mk=n$, where $m>1$, $k\geq 1$ and there are a normal subgroup $M$ of $H$ and a subgroup $K$ of $S_k$ such that $M\cong (S_m)^k$, $H/M \cong K$ and $K$ acts transitively on $\{1,2,\ldots,k\}$, where $(S_m)^k=\underbrace{S_m
\times \cdot\cdot\cdot \times S_m}_{k~ times}$.
Moreover, $M\cap A_n \unlhd H\cap A_n$ and $(H\cap A_n)/(M\cap
A_n)\cong K$ and $H\cap A_n$ acts transitively on $\{1,2,\ldots,n\}$. In particular, if $n$ is a prime $p$, then $H\cong S_p$ and $H\cap A_p\cong A_p$.
Without loss of generality, we may assume that the transposition $(12)\in H$. Define a relation $\sim$ on $\{1,2,\ldots,n\}$ by: for $x,y\in
\{1,2,\ldots,n\}$, $$x\sim y ~~\mbox{if and only if }~~ x=y
\mbox{ or there is a transposition }(xy)\in H.$$ Then, this relation is an equivalence relation. In fact, the transitivity of the relation holds, since if $(xy)\in H$ and $(yz)\in H$, then $(xz)=(xy)(yz)(xy)\in H$.
Since $(12)\in H$, $1\sim 2$, hence there is at least one nonempty equivalence class of $\{1,2,\ldots,n\}$. Suppose $x\sim y$. Then $(xy)\in H$. Let $x'\in \{1,2,\ldots,n\}$. Since $H$ acts transitively on $\{1,2,\ldots,n\}$, there is $h\in H$ such that $h(x)=x'$. Now $h(xy)h^{-1}=(h(x)~h(y))=(x'~h(y))$, which is in $H$. Hence $h(x)=x'\sim h(y)$, *i.e.* $x\sim y$ iff $h(x)\sim h(y)$. Therefore, each equivalence class has the same number of elements.
Let $k$ be the number of equivalence classes and let $C_1,\ldots,
C_k$ be the equivalence classes of $\{1,2,\ldots,n\}$. And let $m=\frac{n}{k}$. Each class $C_i$ has $m$ elements. Note that $m\geq 2$, since $(12)\in H$.
For each $h\in H$, on each class $C_i$, $h(C_i)=C_{h_i}$, for some $h_i\in\{1,2,\ldots,k\}$, since we have showed in the above that $x\sim y$ iff $h(x)\sim h(y)$. And $h$ gives a bijection of $C_i$ and $C_{h_i}$. Hence we have a natural map $\phi_h :
\{C_i\}_{1\leq i\leq k}\rightarrow \{C_i\}_{1\leq i\leq k}$ defined by $\phi_h(C_i)=C_{h_i}$ where $i, h_i=1,2,\ldots,k$. This shows that $\phi_h$ permutes equivalence classes $C_1,\ldots,C_k$. Hence we get a permutation $\sigma_h\in S_k$ such that $\sigma_h(i)=h_i$, where $i_k$ is given by $h(C_i)=C_{h_i}$.
So we can define a map $\pi : H\rightarrow S_k$ given by $\pi(h)=\sigma_h$ defined as above. Then $\pi$ is a group homomorphism, since $hh'(x)=h(h'(x))$, for $h,h'\in H$ and $x\in
\{1,2,\ldots,n\}$.
Let $K=Image(\pi)$ and $M=ker(\pi)$. Then $M\unlhd H$ and $K\leq
S_k$. Moreover, $K$ acts transitively on $\{1,2,\ldots,k\}$, since $C_1\sqcup C_2\sqcup\cdot\cdot\cdot \sqcup C_k=\{1,2,\ldots,n\}$ and $H$ acts on $\{1,2,\ldots,n\}$ transitively.
Now we show that $M \cong \underbrace{S_m \times \cdot\cdot\cdot
\times S_m}_{k~ times}:=(S_m)^k$. Let $S(C_i)$ be the group of all permutations on elements of $C_i$. For any $h\in M$, $h$ has a decomposition, $h=h_1h_2\cdot\cdot\cdot h_k$, where each permutation $h_i$ is a product of disjoint cycles in $S(C_i)$, since $h$ is stable on each class. If $h,g\in M$, let $h=h_1h_2\cdot\cdot\cdot h_k$ and $g=g_1g_2\cdot\cdot\cdot g_k$, where $h_i, g_i\in S(C_i)$, then $hg=h_1g_1h_2g_2\cdot\cdot\cdot
h_kg_k$, since $h_i$ and $g_j$ are disjoint for $i\neq j$. Hence we get an injective homomorphism $f:M\rightarrow S(C_1)\times
\cdot\cdot\cdot \times S(C_k)$ defined by $f(h)=(h_1,\ldots,h_k)$, where $h=h_1h_2\cdot\cdot\cdot h_k$ and $h_i\in S(C_i)$. Since $S(C_i)\cong S_m$ is generated by transpositions and for any $x_i,y_i\in C_i$, there is a transposition $(x_iy_i)\in H$, $f((x_1y_1)\cdot\cdot\cdot (x_ky_k))=((x_1y_1),\ldots ,(x_ky_k))$. Hence $f$ is surjective. Therefore, $M\cong S(C_1)\times
\cdot\cdot\cdot \times S(C_k)\cong (S_m)^k$. By the first isomorphism theorem, $H/M\cong K$.
From the above, we get a short exact sequence of groups, $$1\longrightarrow (S_m)^k \stackrel{f}{\longrightarrow} H
\stackrel{\pi}{\longrightarrow} K \longrightarrow 1.$$ Now we show that the following sequence is exact: $$1\longrightarrow
(S_m)^k\cap A_n \stackrel{f'}{\longrightarrow} H\cap A_n
\stackrel{\pi'}{\longrightarrow} K \longrightarrow 1,$$ where $f'$ is the restriction of the inclusion $f$ to $(S_m)^k\cap A_n$.
First, it is obvious that $f'$ is injective, since $f$ is injective. Moreover, since $ker(\pi)=Image(f)$, we have $ker(\pi')=ker(\pi)\cap A_n=Image(f)\cap A_n=Image(f')$. So this implies the exactness of the middle one. Now we need to show *$\pi'$* is surjective. For any $\sigma\in K$, there is $h\in
H$ such that $\pi(h)=\sigma$, since $\pi$ is surjective. If $h$ is an even permutation, then $h\in H\cap A_n$ and $\pi'(h)=\pi(h)=\sigma$. If $h$ is not even, then consider $\sigma$ as a permutation of $\{C_1,\ldots,C_k\}$ as in the above. Then there are two distinct integers $i$ and $j\in
\{1,2,\ldots,k\}$ such that $\sigma(C_i)=C_j$. Since $C_i$ has at least two elements, there are two elements $a,b\in C_i$. $\emph{i.e.}$ $a\sim b\in C_i$. Hence $(ab)\in H$. Moreover, $(ab)\in Ker(\pi)$ from the construction. Hence $(ab)\circ h$ is an even permutation and $\pi'((ab)\circ h)=\pi((ab)\circ
h)=\pi(h)=\sigma$. Hence $\pi'$ is surjective. Therefore, $M\cap
A_n\cong (S_m)^k\cap A_n=ker(f') \unlhd H\cap A_n$ and $(H\cap
A_n)/((S_m)^k\cap A_n)\cong K$.
Now we show that $H\cap A_n$ acts transitively on $\{1,2,\ldots,n\}$. If $k=1$, then $m=n$, hence $H\cong S_n$. Therefore, $H\cap A_n = A_n$, which acts transitively on $\{1,2,\ldots,n\}$. Assume that $k\geq 2$. Let $a,b\in
\{1,2,\ldots,n\}$. We need to find an even permutation $\sigma \in
H$ such that $\sigma(a)=b$. If both $a$ and $b$ are in the same class $C_i$ for some $i\in \{1,2,\ldots,k\}$, then there is $(ab)\in H$. Since $k\geq 2$, we choose two distinct elements $c$ and $d\in C_j$ for some $j\neq i$. Then if let $\sigma=(ab)(cd)$, then $\sigma\in H\cap A_n$ and $\sigma(a)=b$.
If $a$ and $b$ are in distinct classes, say $a\in C_i$ and $b\in
C_j$ for $i\neq j$, then there is $\tau\in K$ such that $\tau(i)=j$. Since $\tau$ is a bijection between $C_i$ and $C_j$, there are $b'\in C_j$ and $a'\in C_i$ such that $\tau(a)=b'$ and $\tau(a')=b$. If $\tau$ is an even permutation, then let $\sigma=(aa')(bb')\circ \tau$. Then $\sigma(a)=b$ and $\sigma\in
H\cap A_n$. If $\tau$ is odd, then let $\sigma=(bb')\circ\tau$. Then $\sigma(a)=b$ and $\sigma\in H\cap A_n$. This completes the proof.
\[lem:invari\] If $H$ is a transitive subgroup of $S_n$ and $(V,\rho)$ is the permutation representation of $S_n$, then the restriction of $(V,\rho)$ to $H$ has one 1-dimensional invariant subspace.
Let $e_1,\ldots,e_n$ be a basis for the restriction $(V,\rho')$ of the permutation representation of $S_n$ to $H$. Let $H_1=\{h\in H \mid h(1)=1\}$ be the stabilizer of 1 in $H$. Let $W$ be the subspace of $V$ generated by $e_1$. Then $W$ is invariant under $H_1$. Moreover, since $H$ acts transitively on $\{1,2,\ldots,n\}$, we have exactly $n$ left cosets of $H_1$ in $H$. Hence we can identify the permutation representation $(V,\rho')$ with the induced representation $\left(\bigoplus \limits _{i=1}^n \rho'_{g_i}(W),~
\mbox{Ind}_{H_1}^H(1)\right)$ of $H$ by the trivial representation $(W, 1)$ of $H_1$, where $g_i$ are representatives of left cosets of $H_1$ in $H$.
If we denote by $1$ the trivial representation of $H$, then by Frobenius reciprocity, $$\langle ~1,
\mbox{Ind}_{H_1}^{H}(1)~\rangle _{H}
=\langle~\mbox{Res}^{H_1}_{H}(1),1~\rangle _{H_1}=\langle ~1,1~
\rangle _{H_1}=1.$$ Therefore, the restriction $(V,\rho)$ of the permutation representation to $H$ has one 1-dimensional invariant subspace.
\[cor:HA\_n\] If $H$ is a transitive subgroup of $S_n$ and $H$ contains a transposition, then the restriction of the permutation representation of $S_n$ to $H\cap A_n$ has one 1-dimensional invariant subspace.
This follows from Lemma \[lem:invari\], since $H\cap A_n$ acts transitively on $\{1,2,\ldots,n\}$ by Lemma \[lem:semidirect\].
\[lem:inva\] Let $n\geq 1$ be an integer. Let $\sigma \in A_n$ have $k$ disjoint cycles, for some positive integer $k\leq n$. Then there are $k$ fixed vectors under $\sigma $ in the permutation representation of $A_n$.
Let $e_1,\ldots, e_n$ be a basis of the permutation representation of $A_n$. Let $\sigma\in A_n$ have $k$ disjoint cycles. Then, they form $k$ partitions $C_1,\ldots,C_k$ of $\{1,2,\ldots,n\}$.
For $1\leq i\leq k$, let
$$v_i=\sum\limits_{j\in C_i} e_j.$$ Then, these $k$ vectors are fixed under $\sigma$.
\[lem:cycle\] For any even integer $n$, every element in $A_n$ has more than one cycle.
Let $\sigma\in A_n$ have the cycle decomposition $\sigma = (a_{11}\cdot\cdot\cdot a_{1 m_1})(a_{21}\cdot\cdot\cdot
a_{2 m_2})\cdot\cdot\cdot(a_{k1}\cdot\cdot\cdot a_{k m_k})$, where $a_{ij}\in\{1,2,\ldots,n\}$ are distinct and $m_1+m_2+\cdot\cdot\cdot+m_k=n$ for some positive integer $m_i$. If $k=1$, then $m_1=n$ and $\sigma=(a_{11}a_{12}\cdot\cdot\cdot
a_{1n})$ has one cycle of length of the even integer $n$ which is an odd permutation, hence it is not in $A_n$. Thus, we must have that $k\geq 2$. This implies that $\sigma\in A_n$ has at least two cycles.
The following two lemmas show that subgroups of $S_n$ which occur as Galois coverings of a projective line (which is isomorphic to a projective closure of a base-point free linear system of $E$) act transitively on $\{1,2,\ldots,n\}$ and contain a transposition.
\[lem:tran\] Let $K$ be a number field and $E/K$ an elliptic curve over $K$. Suppose that there is a projective line $L$ in $E^{n-1}/S_n \cong
{\mathbb{P}}^{n-1}$ which is a projective closure of a base point-free linear system of $E$. Let $C$ be the preimage of $L$ in $E^{n-1}/A_n$ under the double cover from $E^{n-1}/A_n$ to $E^{n-1}/S_n$ and let the preimage in $E^{n-1}$ of $C$ under the quotient map of $E^{n-1}$ by $A_n$ have a decomposition $X_1\cup
X_2\cup\cdot\cdot\cdot \cup X_k$ into irreducible components $X_i$.
Then, for each $i=1,\ldots,k$, the morphism $X_i\rightarrow C$ is a Galois covering with $H_i=Gal(X_i/C)$ a subgroup of $A_n$ and each $X_i\rightarrow L$ is also a Galois covering with $M_i=Gal(X_i/L)$ such that $M_i\leq S_n$ and $M_i\cap A_n = H_i$. Moreover, the $M_i$ are conjugate to each other and each $M_i$ acts transitively on $\{1,2,\ldots,n\}$.
For each $i=1,\ldots,k$, the morphism $\psi_i:X_i\rightarrow
C$ is the quotient map by the stabilizer $H_i$ of $X_i$ in $A_n$, that is, $H_i=\{\sigma\in A_n \mid \sigma\cdot X_i=X_i\}$. Since $X_i$ is an irreducible component of the preimage of $C$ under the action of $A_n$, for any $\sigma\neq 1\in H_i$, $X_i$ is not contained in the kernel of $1-\sigma$ acting on $E^n$. So $\psi_i$ is a regular Galois covering map with Galois group $Gal(X_i/C)=H_i$ a subgroup of $A_n$. Similarly, each map $\phi_i:X_i \rightarrow L$ is also a Galois covering with Galois group $Gal(X_i/L)$ which is the stabilizer of $X_i$ in $S_n$ and $\phi_i$ is the composite of the Galois covering from $X_i$ to $C$ with $H_i=Gal(X_i/C)$ and the double cover from $C$ to $L$. Hence $M_i
\leq S_n$ and $M_i\cap A_n = H_i$.
First, we show that the $M_i$ are conjugate to each other. Note that $S_n$ acts transitively on $\{X_1,X_2,\ldots,X_k\}$, since $X_1\cup X_2\cup\cdot\cdot\cdot \cup X_k$ is the preimage of the irreducible curve $L$. Hence for each $i$, there is $\tau_i\in
S_n$ such that $\tau_i\cdot X_i=X_1$. Let $\sigma\in M_1$. Then $\tau_i\cdot X_i=X_1=\sigma\cdot X_1=\sigma\tau_i\cdot X_i$. Hence $\tau_i^{-1}\sigma\tau_i\cdot X_i=X_i$. Hence $\tau_i^{-1}\sigma\tau_i\in M_i$. This proves that $\tau_i^{-1}M_1\tau_i=M_i$ for each $i$.
Next, we show that each $M_i$ acts transitively on $\{1,2,\ldots,n\}$. It is enough to show that $M_1$ acts transitively on $\{1,2,\ldots,n\}$, since $M_i$ are conjugate to each other. Note that $E^{n-1}/S_n$ is isomorphic to the projective closure of the linear space $|n(O)|$ of all rational functions $f$ such that $(f)+n(O)\geq 0$ by Lemma \[lem:action\]. Since the curve $L\subset E^{n-1}/S_n$ is a projective closure of a base point-free linear system of $E$, there exists an elliptic function $f$ in $H^0(E,\mathcal{L}(n(O)))$ which has $n$ zeros which sum to $O$ of $E$ such that the base-point free linear system is generated by $f$ and the constant function $1$. Parameterize an open dense subset of the projective line $L$ in ${\mathbb{P}}^{n-1}\cong E^{n-1}/S_n$ by the parameter $\lambda$ such that $f-\lambda$ represents a point of the open subset. Let $g:E^{n-1} \rightarrow E$ be defined by $g(P_1,\ldots,P_n)=P_1$ where the sum of $(P_1,\ldots,P_n)$ is $O$ as a point of $E^{n-1}$. Then, the curve $X_1 \subseteq E^{n-1}$ maps to $E$ through $g$ as well as to the projective line $L\in E^{n-1}/S_n$ through the quotient map by $S_n$. So $X_1$ maps to $E\times L$ and projects onto $L$.
Choose a fundamental domain so that the distinct zeros $z_1,\ldots,z_n$ of $f-\lambda$ are in the interior of the domain. Let $i\in \{2,3,\ldots,n\}$ be fixed. We can take a path from $z_1$ to $z_i$ so that the path doesn’t pass through the other zeros of $f-\lambda$. Moving along this path in $E^{n-1}$, let $f-\lambda '$ be the image in $E^{n-1}/S_n$ of the end point of the path. Then $z_i$ is a common zero of both $f-\lambda$ and $f-\lambda '$, we get $\lambda=\lambda'$. Hence the image of the path in $E^{n-1}/S_n$ is a closed loop starting and ending at $\lambda$, that is, $f-\lambda$. This implies that the path from $z_1$ to $z_i$ stays in the connected component $X_1$. Since $\phi_1$ is a Galois covering with the Galois group $M_1$ which is the stabilizer of $X_1$ in $S_n$ and $X_1$ maps to $E\times L$ through $g$ which is defined as a first coordinate of a point of the preimage of $f-\lambda\in L$ and the morphism $\phi_1$, and the first coordinates of the preimage of $f-\lambda$ are $z_1$ and $z_i$ under the closed loop, this shows that there is an element $\sigma\in M_1$ such that $\sigma(1)=i$. This shows $M_1$ acts transitively on $\{1,2,\ldots,n\}$.
The following lemma has a similar setting as in ([@jar2], Lemma 1.5). But here, we assume that there is a divisor in a given projective line which decomposes into the sum of one ramified divisor of degree 2 and other divisors of odd degree or unramified divisors under a Galois covering, while the lemma in ([@jar2], Lemma 1.5) assumes every divisor decomposes into one ramified divisor of degree 2 and other unramified divisors.
\[lem:(12)\] Suppose there is a curve $L \subset E^{n-1}/S_n\cong {\mathbb{P}}^{n-1}$ which is isomorphic to a projective closure of a base-point free linear system on $E$ and the normalization of its preimage in $E^{n-1}$ under the quotient map is $X_1\cup
X_2\cup\cdot\cdot\cdot \cup X_k$ such that for each $m\in\{1,2,\ldots,k\}$, the Galois covering $M_m:=Gal(X_m/L)$ is a subgroup of $S_n$. Then if $L$ contains a divisor $D=2(P_1)+\sum\limits _{i=2}^{\ell} k_i (P_i) -n(O)$, where $P_i$ are points of $E$ such that $P_i\neq P_j$, $2P_1+\sum\limits _{i=2}^{\ell} k_i P_i=O$ and $k_i$ are odd integers $\geq 1$ with $\sum\limits
_{i=2}^{\ell} k_i=n-2$, then each $M_m$ contains a transposition.
Note that if $k_i=1$, for all $ i=2,\ldots,\ell$, then we apply the proof in ([@jar2], Lemma 1.5) with the given divisor $D$ to get a transposition. Now we assume the general case when $k_i$ are odd integers.
For each $m=1,\ldots,k$, let $\phi_i: X_i\rightarrow L$ be the restriction of the quotient map of $E^{n-1}$ by $S_n$ with $M_m=Gal(X_m/L)$. Let $M_m$ act by permutation of coordinates of each point: for $\sigma\in M_m$, $\sigma\cdot
(P_1,P_2,\ldots,P_n)=(P_{\sigma(1)},P_{\sigma(2)},\ldots,P_{\sigma(n)})$, where $P_n=-(P_1+P_2+\cdot\cdot\cdot+P_{n-1})$.
Suppose $L$ contains a divisor $D=2(P_1)+\sum\limits _{i=2}^{\ell}
k_i (P_i) -n(O)$, where $P_1,\ldots,P_{\ell}$ are distinct points of $E$ and $k_i$ are odd integers such that $\sum\limits_{i=2}^{\ell} k_i=n-2$. Let $f$ be the function whose divisor is equivalent to $D$ and let $z_i$ be the zeros of $f$ corresponding to $P_i$ for each $i=1,\ldots,{\ell}$, *i.e.* $f(z)=(z-z_i)^{k_i}\left(a_{i0}+a_{i1}(z-z_i)+\cdot\cdot\cdot+\right)$ with $a_{i0}\neq 0$. Then by Hensel’s Lemma ([@sil86], Chapter IV, Lemma 1.2), for a number $\lambda$ with small $|\lambda|$, $f-\lambda=(z-z_i)^{k_i}(a_{i0}+a_{i1}(z-z_i)+\cdot\cdot\cdot+\cdot\cdot)-\lambda$ has zeros at $$Q_{1,1}=z_1+\left(\frac{\lambda}{a_{10}}\right)^{\frac{1}{2}}+A_1,
~~~
Q_{1,2}=z_1-\left(\frac{\lambda}{a_{10}}\right)^{\frac{1}{2}}+A_2,$$ $$\mbox{and }
Q_{i,j}=z_i+\left(\frac{\lambda}{a_{i0}}\right)^{\frac{1}{k_i}}\zeta
_{k_i}^{j-1}+B_{i,j}, \mbox{ for each } i=2,\ldots,{\ell}, \mbox{
and } j=1,\ldots,k_i,$$ where $A_1,A_2,$ and $B_{i,j}$ are convergent Puiseux series in $\lambda$ such that each term of $A_1$ and $A_2$ is of higher degree than $\lambda^{\frac{1}{2}}$ and each term of $Q_{i,j}$ is of higher degree than $\lambda^{\frac{1}{k_i}}$, and $\zeta_{k_i}$ is a primitive $k_i$-th root of unity.
Note that each quotient map $\phi_m$ by $M_m$ is surjective. Let $k_1=2$. Choose a small enough number $\lambda$ such that for all $i=1,\ldots,{\ell}$, the circles centered at $z_i$ with radius $\left(\frac{\lambda}{a_{i0}}\right)^{\frac{1}{k_i}}$ do not intersect each other. For each $i=1,\ldots,{\ell}$, $k_i$ points $Q_{i,j}$ for $j=1,\ldots,k_i$ lie in the circle centered at $z_i$.
Since these circles are closed paths and $Q_{i,j}$ are zeros of one fixed function $f-\lambda$, their preimages in $E^{n-1}$ still stay in one component $X_m$ for some $m$. Therefore, for each $i=1,\ldots,{\ell}$, there exists a cycle $\tau_i$ in $S_n$ of length $k_i$ which permutes $Q_{i,j}$ for $j=1,\ldots,k_i$ and the product of all $\tau_i$ is in $M_m$. In particular, $\tau_i$ are disjoint cycles and $\tau_1$ is a transposition permuting $Q_{1,1}$ and $Q_{1,2}$.
Let $c=lcm(k_2,\ldots,k_{\ell})$. Then, $c$ is odd, since all $k_i$ for $i=2,\ldots,{\ell}$ are odd. Since $\tau_1$ is of order $2$, $(\tau_1\tau_2\cdot\cdot\cdot\tau_{\ell})^c=\tau_1\in M_m$. Hence, $M_m$ contains a transposition $\tau_1$. Since $M_m$ are conjugate to each other by Lemma \[lem:tran\], every $M_m$ has a transposition.
$E$ over totally imaginary number fields with a rational point $P$ such that $2P\neq O$ and $3P\neq O$
======================================================================================================
First, we show that if $K$ is a totally imaginary number field and $E/K$ has a $K$-rational point $P$ such that $6P\neq O$, then, for some even integer $n$, there is a projective line over $K$ in $E^{n-1}/A_n \cong {\mathbb{P}}^{n-1}$ whose preimage under the quotient map of $E^{n-1}$ by $A_n$ is a curve of genus 0 in $E^{n-1}$ over $K$. We will need the following lemma to show the existence of such a projective line. We start with the definition of *rank* of quadratic forms that we use in this paper.
The rank of a quadratic form $\Phi$ on the space $V$ is the codimension of the orthogonal complement of $V$ with respect to $\Phi$ in the sense of $($[@ser], Chapter IV, Section 1.2, pp.28$)$.
\[lem:intersec\] Suppose $\Phi_1$ and $\Phi_2$ are two quadratic forms defined over $\overline{K}$ such that for all $r,s\in \overline{K}$, not both zero, the form $r\Phi_1+s\Phi_2$ is of rank $\geq 5$. Then the intersection of the zero loci of $\Phi_1$ and $\Phi_2$ is not entirely contained in a finite union of hyperplanes.
By the abuse of the notation, we denote the intersection of two hypersurfaces defined by $f$ and $g$ by $f\cap g$ and the union of them by $f\cup g$.
Suppose codim$( \Phi_1 \cap\Phi_2\cap L)=2$ for some hyperplane by $L$. If both intersections $\Phi_1\cap L$ and $\Phi_2\cap L$ are irreducible, then, $\Phi_1\cap L=\Phi_2\cap L$. Thus, $\Phi_1
\equiv c\Phi_2 (\bmod~ L)$, for some $c\in \overline{K}$, that is, $\Phi_1-c\Phi_2=LL'$ for some linear form $L'$. Hence, the pencil of $\Phi_1$ and $\Phi_2$ contains some form which has rank $\leq
2$, which leads to a contradiction to the hypothesis.
So we assume that a quadratic form, say $\Phi_1$ intersected with $L$ is reducible into two hyperplanes defined by linear forms $L_2$ and $L_3$ on the original space. Then, $\Phi_1\equiv L_2L_3
~(\bmod~ L)$. Therefore, for some linear form $L_4$, $\Phi_1=LL_4+L_2L_3$ so it has rank $\leq 4$, which is a contradiction to the hypothesis. Hence, we have shown that for every hyperplane by $L$, $$\mbox{codim}( \Phi_1 \cap\Phi_2\cap L)< 2.$$
Now, suppose the intersection of $\Phi_1$ and $\Phi_2$ is entirely contained in the union of hyperplanes by $L_1, \ldots,L_n$. Then, $$\min\limits_{1\leq i\leq n}\mbox{codim}( \Phi_1\cap\Phi_2
\cap L_i)=\mbox{codim}(\Phi_1\cap\Phi_2)=2,$$ which is impossible. This completes the proof.
We will need the following weak approximation of quadrics.
\[prop:appro\] There exists a function $F:\;{\mathbb{N}}\rightarrow{\mathbb{N}}$ with the following property: Given a non-negative integer $n$, a number field $K$, a $K$-vector space $V$, an $n$-dimensional $K$-vector space of quadratic forms $W\subset {\operatorname{Sym}}^2 V$ on $V$, and a finite set of places $S$ of $K$, if for every non-zero $w\in W$, there exists an $F(n)$-dimensional subspace $V_w\subset V$ on which $w$ is non-degenerate, then the intersection of all quadrics in $W$, $X_W(K)$, is dense in $\prod_{v\in S} X_W(K_s)$.
See ([@imlarsen], Theorem 5).
\[prop:An\] Let $K$ be a totally imaginary number field. If $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for some even integer $n$, there is a projective line over $K$ in $E^{n-1}/S_n \cong {\mathbb{P}}^{n-1}$ as a projective closure of a base-point free linear system of $E$ such that the normalization of its preimage under the double cover is a curve of genus 0 over $K$ in $E^{n-1}/A_n$ which contains a divisor of a rational function on $E$ of degree $n$ which has one double zero and all other zeros of odd order (including simple zeros).
By Lemma \[lem:action\], $E^{n-1}/S_n\cong {\mathbb{P}}^{n-1}$ which is isomorphic to the $(n-1)$-dimensional projective space ${\mathbb{P}}(H^0(E,\mathcal{L}(n(O))))$.
If $f$ is an elliptic function of degree $n$, holomorphic except at a unique pole $O$, the vector space spanned by $f$ and $1$ defines a pencil of all divisors $(a+bf)+n(O)$ with $a,b\in{\mathbb{C}}$ on $E$ linearly equivalent to $n(O)$, or equivalently, a line on $E^{n-1}/S_n\cong {\mathbb{P}}^{n-1}$. Note that since $P$ is neither 2-torsion nor 3-torsion, we have that $-2P\notin \{P, O\}$. Now we find an elliptic function $f$ of degree $n=2k$ for some integer $k$, whose derivative is of the form $f'=lh^2$, where $l$ has the divisor $2(P)+(-2P)-3(O)$ and $h$ is in the vector space of elliptic functions defined over $K$ with divisors $\geq (1-k)(O)$. Let $y+ax+b=0$ be the affine tangent line at a $K$-rational point $P$ and let $l:=y+ax+b$. Let $f$ and $f'$ be as follows:
Case I : Suppose $n\equiv 0$ (mod 4). Let $n=4m$ for some integer $m$. For parameters $a_0,\ldots,a_{m-1}$, $b_0,\ldots,b_{m-3}$, $d_0,\ldots,d_{2m-2}$, $c_1,\ldots,c_{2m}$ to be determined and the given tangent line $l=0$, let $$f(z)=
y(d_{2m-2}x^{2m-2}+\cdot\cdot\cdot+d_1x+d_0)+c_{2m}x^{2m}+\cdot\cdot\cdot+c_1x$$$$\mbox{ and } f'(z)=l(h(z))^2,$$ where $ h(z)=
a_{m-1}x^{m-1}+\cdot\cdot\cdot+a_1x+a_0+y(x^{m-2}+\cdot\cdot\cdot+b_1x+b_0).$
Case II : Suppose $n\equiv 2$ (mod 4). Let $n=4m+2$ for some integer $m$. For parameters $a_0,\ldots,a_{m-1}$, $b_0,\ldots,b_{m-2}$, $d_0,\ldots,d_{2m-1}$, $c_1,\ldots,c_{2m+1}$ to be determined and the given tangent line $l=0$, let $$f(z)=
y(d_{2m-1}x^{2m-1}+\cdot\cdot\cdot+d_1x+d_0)+c_{2m+1}x^{2m+1}+\cdot\cdot\cdot+c_1x$$$$\mbox{ and } f'(z)=l(h(z))^2,$$ where $h(z)=x^m+a_{m-1}x^{m-1}+\cdot\cdot\cdot+a_1x+a_0+y(b_{m-2}x^{m-2}+\cdot\cdot\cdot+b_
1x+b_0).$
From the equations obtained by equating the coefficient of each $x^{i}y^j$-term of $f'(z)$ with that of the derivative of $f(z)$ given in the above (equivalently, by equating $f$ with the integral of $f'$ along two periods of $E$), we get two quadratic equations over $K$ in $\frac{n-4}{2}$ variables, namely $a_0,\ldots,a_{m-1}, b_0,\ldots, b_{m-4}$ and $b_{m-3}$ if $n=4m$, and $a_0,\ldots,a_{m-1}, b_0,\ldots,b_{m-3}$ and $b_{m-2}$ if $n=4m+2$. Homogenize these two quadratic equations to get two quadratic forms in $\frac{n-4}{2}+1$ variables with a new variable. We need to find a common isotropic vector over $K$ of two quadratic forms which defines a common solutions of two original quadratic equations, (that is, which is outside the hyperplane at $\infty$) and defines $f'=lh^2$ such that $h(-2P)\neq 0$ in the above notation of cases I and II.
Let $D$ be a fundamental domain of $E$ and $C_1$ and $C_2$ be two line segments dividing the fundamental domain of $E$ into four congruent parallelograms and $I_1 $ and $I_2$ the first half line segments of $C_1$ and $C_2$ respectively as shown below.
(8,8) (-1,3) (-1.7,2.3)[ $O$]{} (-1,3)[(1,0)[6]{}]{} (0,5)[(1,0)[6]{}]{} (1,7)[(1,0)[6]{}]{} (-1,3)[(1,2)[2]{}]{} (2,3)[(1,2)[2]{}]{} (5,3)[(1,2)[2]{}]{} (-3.5,.8)[$\langle$A fundamental domain $D$ of $E$ with two periods $C_1$ and $C_2\rangle$]{} (1.5,5.4)[ $I_1$]{} (0.3,5.1)[(1,0)[2.7]{}]{} (2.8,5.1)[(-1,0)[2.7]{}]{} (6.3,4)[(-1,1)[1]{}]{} (6.5,3.5)[ $C_1$]{} (1.55,4)[ $I_2$]{} (1.9,3.1)[(1,2)[.95]{}]{} (2.8,4.9)[(-1,-2)[.95]{}]{} (5.5,7.5)[(-2,-1)[1.5]{}]{} (5.7,7.8)[ $C_2$]{}
Let $M=max\{F(2), 5\}$, where $F$ is the function given in Proposition \[prop:appro\]. We can choose $2M$ holomorphic functions $f_1,\ldots,f_{2M}$ on $I_1\cup I_2$ such that $$\displaystyle\int_{I_1} lf_i f_j dz = \displaystyle\int_{I_2}
lf_i f_j dz =0, \mbox{ for } i\neq j, \hspace{1.2 cm}$$ $$\hspace{.23 cm} \displaystyle\int_{I_1\cup I_2} lf_i ^2 dz =
\int _{I_1} lf_i^2 dz \neq 0, \mbox{ for } i=1,2,\ldots,M,$$ $$\mbox{ and } \displaystyle\int_{I_1\cup I_2} lf_i ^2 dz = \int
_{I_2} lf_i^2 dz\neq 0, \mbox{ for }i=M+1,,\ldots,2M.$$
Since the Weierstrass $\wp$-function $x=\wp(z): I_1\cup
I_2\rightarrow {\mathbb{C}}$ is injective, its inverse $\wp^{-1}$ is well-defined on the image $\wp(I_1\cup I_2)$ and the image is a compact contractible set in ${\mathbb{C}}$. Hence the complement of $\wp(I_1\cup I_2)$ is connected. So by Mergelyan’s Theorem ([@rudin], pp.390), each holomorphic function $f_i\circ
\wp^{-1} :\wp(I_1\cup I_2)\rightarrow {\mathbb{C}}$ can be approximated by some polynomial $p_i(z)$, for each $i=1,\dots, 2M$. Moreover, since $K$ is a totally imaginary number field, $K$ is dense in $\left(\prod\limits_{v\in S_{\infty}} {\mathbb{C}}\right)$ with respect to the usual topology for any embeddings of $K$ in ${\mathbb{C}}$, where $S_{\infty}$ is the set of all infinite places. Hence we may assume that coefficients of $p_i(z)$ are in $K$. So each $f_i$ can be approximated by the polynomial $p_i(x)$ in terms of $x=\wp(z)$ with coefficients in $K$.
Let $W$ be a space of dimensional $\geq 2M$ generated by elliptic functions including all of $p_i(x)$ for $i=1,\ldots,2M$. Then, any two quadratic forms $\Phi_1$ and $\Phi_2$ over $K$ obtained from the homogenization of the integration on $W$ satisfy the property: $$\mbox{for any } r,s\in \overline{K} \mbox{ not both zero, any form in the pencil }
r\Phi_1+s\Phi_2 \mbox{ is of rank }\geq M.$$ For example, if $r=0$, then the $M$ functions $p_i(x)$ for $i=M+1,\ldots,2M$ generate an M-dimensional non-degenerate subspace of $W$ for the form $r\Phi_1+s\Phi_2$. If $s=0$, then $p_i(x)$ for $i=1,\ldots,M$ generates an M-dimensional non-degenerate subspace for $r\Phi_1+s\Phi_2$. And if neither $r$ nor $s$ is zero, either the $M$ functions $p_i(x)$ for $i=M+1,\ldots,2M$ or for $i=1,\ldots,M$ generate an M-dimensional non-degenerate subspace.
Hence, any pencil of $\Phi_1$ and $\Phi_2$ has rank $\geq F(2)$, since $M\geq F(2)$. Then, by Proposition \[prop:appro\], it has the weak approximation, since $K$ is totally imaginary. Therefore, the set of $K$-rational points in the intersection of $\Phi_1$ and $\Phi_2$ on a non-degenerate subspace of dimension $\geq M$ is Zariski-dense in the variety defined by $\Phi_1$ and $\Phi_2$. Let $L$ be the hyperplane at $\infty$ and $L'$ the hyperplane defined by $h(-2P)$ in the above notation of $f'=lh^2$ in case I or II. By Lemma \[lem:intersec\], the intersection of two forms $\Phi_1$ and $\Phi_2$ is not contained in the union of two hyperplanes defined by $L$ and $L'$. Hence, by the density of $K$-rational points, we can get a nontrivial common zero over $K$ which is a common zero of two original quadratic equations which defines an elliptic function $f$ such that $f'=lh^2$, for some elliptic function $h$ such that $h(-2P)\neq 0$.
Now we take the projective closure $V$ over $K$ of the linear subspace of $
{\mathbb{P}}(H^0(E,\mathcal{L}(n(O))))$ generated by $f$ and the constant function $1$ over $K$. Note that the linear space generated by $f$ and $1$ is a base-point free linear system on $E$ from the construction. Then $V$ is isomorphic to the projective line ${\mathbb{P}}^1(K)$. And the normalization $X \subseteq E^{n-1}/A_n$ of its preimage under the 2-1 map from $E^{n-1}/A_n$ to $E^{n-1}/S_n$ meets the ramification divisor wherever the divisor $f-\lambda$ for some $\lambda$ has a zero or a pole of even multiplicity $\geq 2$, that is, wherever its derivative $(f-\lambda)'=f'$ has a zero or a pole of odd order. And $X$ has only two points which meet the ramification locus at $-2P$ and $O$ to odd contact order by Lemma \[lem:contact\] below. Hence by the Hurwitz formula, the normalization of $X$ in $E^{n-1}/A_n$ is a curve of genus 0 defined over $K$. By subtracting the constant $\lambda_p=f(-2P)$ from $f$, the function $f-\lambda_p$ has one double zero at $-2P$ and other zeros of odd order, since $f'$ has only one simple zero at $-2P$ and other zeros of even order.
\[lem:contact\] Under the same notation as in the proof of Proposition \[prop:An\], if an elliptic function $f$ has a zero (or a pole) at a point $P$ of order $m$, the contact order of $f$ with the ramification locus of the double cover from $E^{n-1}/A_n$ onto $E^{n-1}/S_n$ at $P$ is $m-1$.
Suppose $f$ has a zero $\alpha$ corresponding to the zero $P$ of order $m$. Let $f(z)=(z-\alpha)^m(a_0+a_1(z-\alpha)+\cdot\cdot\cdot +$ higher terms in $(z-\alpha))$, where $a_0\neq 0$.
Note that the ramification locus under the quotient map from $E^{n-1}$ to $E^{n-1}/S_n$ is the zero locus of $\prod\limits_{i<j}(z_i-z_j)$, where $z_i$ are zeros of $f-\lambda$, for a parameter $\lambda$, that is, the quotient map is ramified whenever $f-\lambda$ has a double zero. By considering the ramification index, since the degree of the map from $E^{n-1}/A_n$ onto $E^{n-1}/S_n$ is 2, the ramification locus under the double cover from $E^{n-1}/A_n$ onto $E^{n-1}/S_n$ is the zero locus of the discriminant of $f-\lambda$, that is, $$\prod\limits_{i<j}(z_i-z_j)^2,$$ where $z_i$ are zeros of $f-\lambda$. If we write the discriminant of $f-\lambda$ in terms of $\lambda$ with small $|\lambda|$, then its degree with respect to $\lambda$ is the contact order of $f$ at $P$. We may assume that $\alpha=0$ by translation. Hence we have $$f-\lambda=0 \Leftrightarrow
z^m(a_0+a_1z+a_2z+\cdot\cdot\cdot+\mbox{ higher terms in } z
)-\lambda =0.$$ By Hensel’s Lemma ([@sil86], Chapter IV, Lemma 1.2) on ${\mathbb{C}}[[\lambda^{\frac{1}{m}}]]$, all zeros of $z^m(a_0+a_1z+a_2z+\cdot\cdot\cdot+\mbox{ higher
terms in } z )-\lambda$ are $$z_i=\left(\frac{\lambda}{a}\right)^{\frac{1}{m}} \zeta _m^i+A_i(\lambda), \mbox{for } 0\leq i\leq m-1,$$ where $\zeta_m$ is a primitive $m$th root of unity, and $A_i(\lambda)$ is a convergent Puiseux series in $\lambda$, that is, a convergent power series in $\lambda^{\frac{1}{m}}$. Hence the degree of the discriminant of $f-\lambda$ with respect to $\lambda$ is $\frac{1}{m} \cdot{ m\choose 2}\cdot 2=m-1$, which is the contact order at $\alpha$ with the ramification locus. For a pole, we proceed similarly, replacing $f$ by $1/f$.
Next, we examine the Galois theory of the fixed fields $\overline{K}^{\sigma}$ for automorphisms $\sigma\in
Gal(\overline{K}/K)$. We give some definitions.
\[def:real\] A field $F$ is $($formally$)$ real, if $-1$ is not a sum of squares in $F$. A real field $F$ is real closed, if no algebraic extension of $F$ is real.
\[lem:brauer\] Let $K$ be a number field. Then for any $\sigma\in
Gal(\overline{K}/K)$, $$Gal(\overline{K}/\overline{K}^{\sigma}) \cong
\prod\limits_{p\in S}{\mathbb{Z}}_p \mbox{\hspace{.2 in} or\hspace{.2in}}
{\mathbb{Z}}/2{\mathbb{Z}},$$ where $S$ is a set of prime integers. In particular, if $K$ is totally imaginary, $Gal(\overline{K}/\overline{K}^{\sigma})$ has no torsion element, hence, the Brauer group $Br(\overline{K}^{\sigma})$ of the fixed field under $\sigma$ is trivial.
Let $\sigma\in Gal(\overline{K}/K)$. $Gal(\overline{K}/\overline{K}^{\sigma})$ is isomorphic to the closure of the subgroup generated by $\sigma$ in the sense of the Krull topology by ([@morandi], Theorem 17.7). Hence, $$Gal(\overline{K}/\overline{K}^{\sigma}) \cong
\prod\limits_{p\in S}{\mathbb{Z}}_p\times \prod\limits_{p\in
T}{\mathbb{Z}}/p^{m_p}{\mathbb{Z}}\cong \prod\limits_{p\in S}\langle
\sigma_p\rangle \times \prod\limits_{p\in T}\langle
\tau_p\rangle,$$ where $S$ and $T$ are disjoint sets of primes, $m_p$ are positive integers, and $\tau_p$ has a finite order $p^{m_p}$. But since any element in $Gal(\overline{K}/\overline{K}^{\sigma})$ has the order $1, 2$ or $\infty$ by Artin-Schreier Theorem ([@issacs], Theorem (25.1)), the torsion part of $Gal(\overline{K}/\overline{K}^{\sigma})$ is trivial or ${\mathbb{Z}}/2{\mathbb{Z}}$. Moreover, if there are $q\in T$ and $p\in S\cup
(T-\{q\})$, then, $\tau_q$ is an involution and its fixed field $\overline{K}^{\tau_q}$ is a real closed field by ([@issacs], Theorem (25.13)). Also $\sigma_p^{-1}\tau_q\sigma_p=\tau_q$, so $\sigma_p$ induces a nontrivial automorphism of $\overline{K}^{\tau_q}$. This contradicts the uniqueness of an isomorphism between two real closed fields ([@l93], XI, §2, Theorem 2.9, pp.455). Therefore, $$Gal(\overline{K}/\overline{K}^{\sigma})\cong \prod\limits_{p\in S}{\mathbb{Z}}_p\mbox{\hspace{.2 in}or \hspace{.2 in}}
{\mathbb{Z}}/ 2{\mathbb{Z}}.$$
On the other hand, if $Gal(\overline{K}/\overline{K}^{\sigma})$ is isomorphic to $
{\mathbb{Z}}/ 2{\mathbb{Z}}$ generated by $\tau$, then $[\overline{K}:\overline{K}^{\tau}]=2$, so $\overline{K}^{\tau}$ is real-closed by ([@issacs], Theorem (25.13)), so it has a real embedding by ([@issacs], Theorem (25.18)). Hence if $K\subseteq\overline{K}^{\tau}$ is totally imaginary, then $Gal(\overline{K}/\overline{K}^{\sigma})$ is isomorphic to $\prod\limits_{p\in S}{\mathbb{Z}}_p$. Then, since $\overline{K}^*$ is a divisible topological $\prod\limits_{p\in S}{\mathbb{Z}}_p$-group, $H^2(\prod\limits_{p\in
S}{\mathbb{Z}}_p, \overline{K}^*)$ is trivial by ([@nsw], Chapter 1, §6. Proposition 1.6.13.(ii)). Therefore, $Br(\overline{K}^{\sigma})=0$.
\[lem:conic\] Let $K$ be a totally imaginary number field. Then for any $\sigma\in Gal(\overline{K}/K)$, a conic curve $X$ defined over $K$ has a $\overline{K}^{\sigma}$-rational point.
Let $\sigma\in Gal(\overline{K}/K)$. Since a conic can be identified with an element of $Br(\overline{K}^{\sigma})$ as a Severi-Brauer variety of dimension 1, and $Br(\overline{K}^{\sigma})=0$ by Lemma \[lem:brauer\], a conic is isomorphic to ${\mathbb{P}}^1$ over $\overline{K}^{\sigma}$. Equivalently, it has a $\overline{K}^{\sigma}$-rational point.
Let $f\in K(t_1,\ldots,t_m)[X_1,\ldots,X_n]$ be a polynomial with coefficients in the quotient field $K(t_1,\ldots,t_m)$ of $K[t_1,\ldots,t_m]$ which is irreducible over $K(t_1,\ldots,t_m)$. We define $$H_K(f)=\{(a_1,\ldots,a_m)\in K^m\mid
f(a_1,\ldots,a_m,X_1,\ldots,X_n) \mbox{ is irreducible over }
K\}$$ to be the Hilbert set of $f$ over $K$. We need the following lemma.
\[lem:hilbert\] Let $L$ be a finite separable extension of $K$ and let $f\in
L(t_1,\ldots,t_m)[X_1,\ldots,X_n]$ is an irreducible polynomial over the quotient field $L(t_1,\ldots,t_m)$. Then, there exists a polynomial $p\in K[t_1,\ldots,t_m,X_1,\ldots,X_n]$ such that $p$ is irreducible over $K(t_1,\ldots,t_m)$ and $H_K(p) \subseteq
H_L(f)$.
For a given irreducible polynomial $f \in L(t_1,\ldots,t_m)[X_1,\ldots,X_n]$, by ([@jar], Ch.11, Lemma 11.6), there is an irreducible polynomial $q$ $\in
K(t_1,\ldots,t_m)[X_1,\ldots,X_n]$ such that $H_K(q)\subseteq
H_L(f)$. By ([@jar], Ch.11, Lemma 11.1), there is an irreducible polynomial $p \in K[t_1,\ldots,t_m,X_1,\ldots,X_n]$ which is irreducible over $K(t_1,\ldots,t_m)$ such that $H_K(p)
\subseteq H_K(q)$. Hence the Hilbert set $H_L(f)$ of $f$ over $L$ contains the Hilbert set $H_K(p)$ of $p$ over $K$.
Let $G$ be a finite group and $\Lambda$ an $n$-dimensional $G$-representation. Then, $G$ acts on $E\otimes \Lambda$ through its action on $\Lambda$. Define $E\otimes\Lambda$ to be the abelian variety representing the functor $S \mapsto
E(S)\otimes_{{\mathbb{Z}}}\Lambda$, where $S$ is any scheme over the ground field and $E(S)$ is the functor of points associated to $E$. Then, as an abelian variety, $E\otimes\Lambda$ is just $E^n$, since the action of $G$ on $E\otimes\Lambda$ is only though $\Lambda$. With this background, we prove the following proposition.
\[prop:group\] Let $K$ be a totally imaginary number field and $\sigma\in
Gal(\overline{K}/K)$. Let $G$ be a nontrivial finite group and $\Lambda$ an $n$-dimensional integral $G$-representation for a positive integer $n$. Then $G$ acts on $E^n\cong E\otimes \Lambda$ through $\Lambda$. Suppose that there is a curve $X$ of genus 0 in $E^n/G$ over $K$. Suppose the preimage of $X$ under the quotient map by $G$ is decomposed into $k$ irreducible curves $C_1,\ldots, C_k$ such that each $C_i\rightarrow X$ is the Galois covering with $Gal(C_i/X)
\leq G$, then $X$ cannot be decomposed completely, i.e. $k<|G|$, and $Gal(C_i/X)$ are conjugate to each other in $G$. And for an irreducible component $C\subseteq E^n$ in the preimage of $X$, there exist a finite extension $F$ of $K$ and an infinite sequence $\{L_i/F\}_{i=1}^{\infty}$ of linearly disjoint finite Galois extensions of $F$ such that $F\subseteq
\overline{K}^{\sigma}$ and $Gal(L_i/F)$ is naturally isomorphic to $Gal(C/X)$ as a subgroup of $G$. And for each $i$, there is a submodule $M_i$ of $E(L_i)\otimes {\mathbb{Q}}$ isomorphic to $\Lambda\otimes {\mathbb{Q}}$ as a $Gal(L_i/F)$-module via the inclusion $Gal(L_i/F) \hookrightarrow G$.
In particular, if $K$ is an arbitrary number field and $X$ is isomorphic to ${\mathbb{P}}^1$ over $K$, then this holds with $F=K$. And if the preimage of $X$ in $E^n$ is irreducible, then each $Gal(L_i/F)$ is isomorphic to $G$ itself.
Let $\sigma\in Gal(\overline{K}/K)$. If the curve $X$ of genus 0 has a $K$-rational point, then $X\cong{\mathbb{P}}^1$ over $K$. If not, it is isomorphic to a conic curve. Then, since $K$ is totally imaginary, by Lemma \[lem:conic\], for every $\sigma\in Gal(\overline{K}/K)$, $X$ has a $\overline{K}^{\sigma}$-rational point. Choose a point of $X$ over $\overline{K}^{\sigma}$ and let $F$ be the field of definition of this point. Then $F\subseteq \overline{K}^{\sigma}$, $F$ is a finite extension of $K$, and $X$ is isomorphic to ${\mathbb{P}}^1$ over $F$. Now we consider $X\subseteq E^n/G$ as ${\mathbb{P}}^1$ over $F$. Note that if $X\cong {\mathbb{P}}^1$ over $K$, then we can take $F=K$.
First, suppose that the preimage of the curve $X$ in $E^n$ under the quotient map by $G$ is an irreducible curve $C$ with the function field $F(C)$. Then the restricted quotient map $\phi:C\rightarrow X$ by $G$ realizes $F(C)$ as a Galois extension of the function field $F(x)$ of $X(F)\cong {\mathbb{P}}^1_F$ with the Galois group isomorphic to $G$. By the theorem of the primitive element, there exists $t\in F(C)$ such that $F(C)=F(x,t)$ and $$g_mt^{m}+g_{m-1}t^{m-1}+\cdot\cdot\cdot+g_1t+g_0=0,$$ where $g_i$ are polynomials in $F[x]$. Choose a minimal polynomial of $t$ over $F(x)$ and clear its denominators so that we let $f(x,y)$ be a minimal polynomial of $t$ in $F[x,y]$. Then, $f$ is absolutely irreducible over $F$ so it is irreducible over $F(x)$.
By ([@sil], Lemma), the set $\bigcup\limits_{[L:F]\leq k}
E(L)_{tor}$ is a finite set, where the union runs over all finite extensions $L$ of $F$ whose degree over $F$ is $\leq k$, where $k=|G|$. Let $L'$ be a finite field extension of $F$ over which all points of $\bigcup\limits_{[L:F]\leq k} E(L)_{tor}$ are defined. Applying Lemma \[lem:hilbert\] and ([@jar], Lemma 12.12) to $f$ over $L'$, we can choose $x_1\in H_F(f)\cap K$ such that the specialization $x \mapsto x_1$ preserves the Galois group $G$ and there is a point $Q_1$ of $C\subseteq E^n\cong E\otimes
\Lambda $ in the preimage $\phi^{-1}((1:x_1))$ of $(1:x_1)\in
{\mathbb{P}}^1(F)\cong X$ under $\phi$ is defined over a finite Galois extension $L_1$ of $F$ with $Gal(L_1/F)\cong G$, that is, the preimage of $(1:x_1)$ under $\phi$ consists of a single point Spec $L_1$. Let $\Lambda^*$ be the dual of $\Lambda$ with the action of $G$. Then the morphism from Spec $L_1$ to $E\otimes \Lambda$ induces a ${\mathbb{Z}}[G]$-linear map $g :\Lambda^{*} \rightarrow E(L_1)$ given by $\lambda^*\mapsto \sum\limits_j\lambda^*(\lambda_j)P_j$, where $Q_1=\sum\limits_j P_i\otimes \lambda_i\in E\otimes
\Lambda$.
Since $f(x_1,y)$ is irreducible over $L'$, two extensions $L_1$ and $L'$ are linearly disjoint over $F$. So for $\lambda^*\in
\Lambda^*$, $g(\lambda^*)\in E(L_1)$ is a non-torsion point. So if we let $M_1\subseteq E(L_1)\otimes {\mathbb{C}}$ be the submodule generated by the points of $E(L_1)$ in the image of $\Lambda^*$ under the given map $g$ in the above, it is a submodule of $E(L_1)\otimes {\mathbb{Q}}$ isomorphic to $\Lambda^*\otimes {\mathbb{Q}}$ as a $Gal(L_1/F)$-module via the natural isomorphism $Gal(L_1/F)\cong
G$. Since $\Lambda$ is a finite dimensional integral representation, it is isomorphic to its dual $\Lambda^*$ as $G$-representations. So $M_1$ is isomorphic to $\Lambda\otimes
{\mathbb{Q}}$ as a $Gal(L_1/F)$-module.
Suppose the preimage of $X$ is decomposed into a union of irreducible curves $C_1\cup C_2\cup\cdot\cdot\cdot\cup C_k$. Then, $G$ acts transitively on the set of $k$ curves and each $Gal(C_i/K)$ can be identified with the stabilizer of $C_i$ in $G$ so $Gal(C_i/K)$ are conjugate to each other. So if $k=|G|$, then this implies that $C_i\cong {\mathbb{P}}^1$ in $E^n$ which is impossible, because no abelian variety contains ${\mathbb{P}}^1$ as a subvariety. So $k <|G|$. Let $C$ be one of irreducible components $C_i$. Applying the same argument with the quotient map from the fixed component $C$ to $X$, we get a Galois extension $L_1$ of $F$ with the Galois group $Gal(L_1/F)$ which is isomorphic to the stabilizer of $C$ in $G$ which is $Gal(C/X)\leq G$ and a $Gal(L_1/F)$-submodules $M_1$ of $E(L)\otimes {\mathbb{Q}}$ generated by $n$ non-torsion points of $E(L_1)$ and it is isomorphic to $\Lambda\otimes {\mathbb{Q}}$ as a $Gal(L_1/F)$-module via the natural inclusion $Gal(L_1/F)\hookrightarrow G$.
Inductively, suppose we have found linearly disjoint finite Galois extensions $L_1,L_2,\ldots,L_{k}$ of $F$ and for each $i=1,2,\ldots,k$, there is a submodule $M_i$ of $E(L_i)\otimes
{\mathbb{Q}}$ isomorphic to $\Lambda\otimes{\mathbb{Q}}$ as a $Gal(L_i/F)$-module via the natural inclusion $Gal(L_1/F)\hookrightarrow G$. By applying Lemma \[lem:hilbert\] and ([@jar], Lemma 12.12) to $f$ over the composite field $L'L_1L_2\cdot\cdot\cdot L_k$, there is a point $x_{k+1}\in X(F)$ such that the specialization $x
\mapsto x_{k+1}$ preserves the Galois group $G$ and a point in the preimage of $x_{k+1}$ in $C$ is defined over a Galois extension $L_{k+1}$ of $F$ which is linearly disjoint from $L'L_1L_2\cdot\cdot\cdot L_k$ and has the Galois group isomorphic to a subgroup of $G$. Then similarly, we get a $Gal(L_{k+1}/F)$-submodule $M_{k+1}$ generated by $n$ non-torsion points of $E(L_{k+1})$ isomorphic to $\Lambda\otimes {\mathbb{Q}}$ via $Gal(L_{k+1}/F)\hookrightarrow G$. This completes the proof.
\[cor:group\] Let $K$ be a totally imaginary number field and $E/K$ an elliptic curve over $K$ with a $K$-rational point such that $2P\neq O$ and $3P\neq O$. Let $\Lambda$ be the $(n-1)$-dimensional irreducible quotient representation space of the natural permutation representation of the alternating group $A_n$ by the trivial representation. Let $\sigma\in Gal(\overline{K}/K)$. Then for some even integer $n$, there exist a finite extension $F\subseteq \overline{K}^{\sigma}$ over $K$ and an infinite sequence $\{L_i/F\}_{i=1}^{\infty}$ of linearly disjoint finite Galois extensions of $F$ such that $Gal(L_i/F)$ acts transitively on $\{1,2,\ldots,n\}$ as a subgroup of $A_n$. And for each Galois extension $L_i$ of $K$, there is a submodule $M_i$ of $E(L_i)\otimes {\mathbb{Q}}$ isomorphic to the $(n-1)$-dimensional irreducible quotient representation space $\Lambda\otimes{\mathbb{Q}}$ as a $Gal(L_i/F)$-module via the natural inclusion $Gal(L_i/F)\hookrightarrow A_n$.
By Proposition \[prop:An\], there is a curve $X$ of genus 0 defined over $K$ in $E^{n-1}/A_n$, for some even integer $n$. So by Proposition \[prop:group\], there exist such an infinite sequence of Galois extensions $L_i$ and submodules $M_i$ of $E(L_i)\otimes {\mathbb{Q}}$ isomorphic to the $(n-1)$-dimensional irreducible quotient representation space $\Lambda\otimes{\mathbb{Q}}$ of $A_n$ as a $Gal(L_i/F)$-module via the natural inclusion $Gal(L_i/F)\hookrightarrow A_n$. And by Proposition \[prop:An\], Proposition \[prop:group\], and Lemma \[lem:tran\], for each $Gal(L_i/F)$ as a subgroup of $A_n$, there is a subgroup $H_i\leq S_n$ such that $H_i\cap A_n\cong Gal(L_i/F)$ and it acts transitively on $\{1,2,\ldots,n\}$. Moreover, the image of $X$ given by Proposition \[prop:An\] under the 2-to-1 map from $E^{n-1}/A_n$ onto $E^{n-1}/S_n$ has a divisor which decomposes into one divisor of ramification degree 2 and other divisors of odd degree. So by Lemma \[lem:(12)\], $H_i$ contains a transposition. Therefore, by Lemma \[lem:semidirect\], $Gal(L_i/F)$ also acts transitively on $\{1,2,\ldots,n\}$.
\[lem:indep\] Let $E/K$ be an elliptic curve over a number field $K$. Let $d$ be a positive integer $\geq 2$. Suppose $\{L_i/K\}^{\infty}_{i=1}$ is an infinite sequence of linearly disjoint finite Galois extensions of $K$ whose degrees $[L_i:K]$ are $\leq d$ and $\{P_i\}^{\infty}_{i=1}$ is an infinite sequence of points in $E(\overline{K})$ such that for each $i$, $P_i\in E(L_i)$ but $P_i\notin E(K)$. Then, there is an integer $N$ such that $\{P_i\}_{i\geq N}$ is a sequence of linearly independent non-torsion points of $E$.
By ([@sil], Lemma), the set $S=\bigcup\limits_{[L:K]\leq d}
E(L)_{tor}$ is a finite set, where the union runs all over finite extensions $L$ of $K$ whose degree over $K$ is $\leq d$. So there is a finite extension $F$ of $K$ over which all points of $S$ are defined and there is an integer $n$ such that $nP=O$, for all $P\in S$. Let $n$ be such a fixed integer and let $T$ be the set of all points $P$ of $E(\overline{K})$ such that $n P\in E(K)$. Then, since $E(K)$ is finitely generated by the Mordell-Weil Theorem ([@sil86], Chapter VIII), there is a finite extension $F'$ of $K$ over which all points of $T$ are defined. Then, all but finitely many fields $L_i$ in the given sequence $\{L_i/K\}_{i=1}^{\infty}$ are linearly disjoint from $F$ and $F'$ over $K$. This implies that there is an integer $N$ such that points $P_i$ for all $i\geq N$ are non-torsion points in $E(L_i)$. And by linear disjointness of fields $L_i$, $F$ and $F'$ over $K$, we have that for all $i\geq N$, $$E(L_i)\cap S \subseteq E(K)_{tor} \mbox{~~and ~~} E(L_i)\cap T\subseteq E(K).$$ Note that since each $P_i\notin E(K)$, we have that for any integer $m\geq N$ and for each $i$ such that $N\leq i\leq m$, there is an automorphism $\tau_i\in Gal(\overline{K}/K)$ such that $\tau_i|_{L_j}=id_{L_j}$ for all $N\leq j\neq i\leq m$, but $\tau_i(P_i)\neq P_i$. Moreover, we may choose such a $\tau_i$ that $\tau_i(P_i)-P_i$ is not a torsion point. In fact, otherwise, for every restriction $\tau_i|_{L_i}\in Gal(L_i/K)$ of $\tau_i$, $\tau_i|_{L_i}(P_i)-P_i$ is a torsion point in $E(L_i)$. Hence, $\tau_i|_{L_i}(P_i)-P_i \in E(L_i)\cap S \subseteq E(K)_{tor}.$ Then, $n(\tau_i|_{L_i}(P_i)-P_i)=O$ so $\tau_i|_{L_i}(nP_i)=nP_i$ for all $\tau_i|_{L_i}\in Gal(L_i/K)$. This implies $nP_i\in E(K)$ so $P_i\in T\cap E(L_i)\subseteq E(K)$ which contradicts the assumption that $P_i\notin E(K)$. Hence, we conclude that for each $i$ such that $N\leq i\leq m$, there is an automorphism $\tau_i\in
Gal(\overline{K}/K)$ such that $\tau_i|_{L_j}=id_{L_j}$ for all $N\leq j\neq i\leq m$, but $\tau_i(P_i)-P_i$ is a non-trivial and non-torsion point of $E$.
Let $m\geq N$ be a given positive integer. Suppose that for some integers $a_i$, $$a_NP_{N}+a_{N+1}P_{N+1}+\cdots+a_mP_{m}=0.$$ By the claim above, for each $i=N, N+1, \ldots, m$, there is an automorphism $\tau_i\in Gal(\overline{K}/K)$ such that $\tau_i|_{L_{j}}=id_{L_{j}}$ for all $1\leq j\neq i \leq m$ but $\tau_i(P_i)-P_i$ is a non-trivial and non-torsion point of $E$. Now we apply such $\tau_i$ to get $$a_NP_{N}+a_{N+1}P_{N+1}+\cdots+a_{i-1}P_{i-1}+a_i\tau_i(P_{i})+a_{i+1}P_{i+1}+\cdots+a_mP_{i_m}=0.$$ So by subtracting, we get $a_i(P_{i}-\tau_i(P_{i}))=0$, which implies $a_i=0$. Hence any non-torsion points in $\{P_{i}\}_{i\geq
N}$ are linearly independent.
\[thm:totally\] Let $K$ be a totally imaginary number field. Suppose $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$. Then for each $\sigma \in Gal(\overline{K}/K)$, $E(\overline{K}^{\sigma})$ has infinite rank.
Let $\sigma\in Gal(\overline{K}/K)$. By Proposition \[prop:An\] and Corollary \[cor:group\], there are a finite extension $F\subseteq \overline{K}^{\sigma}$ over $K$ and an infinite sequence $\{L_i/F\}_{i=1}^{\infty}$ of linearly disjoint finite Galois extensions of $F$ such that the Galois group $Gal(L_i/F)$ acts transitively on $\{1,2,\ldots,n\}$ as a subgroup of $A_n$ for some even integer $n$. And for each $i$, there is a $Gal(L_i/F)$-submodule of $E(L_i)\otimes{\mathbb{Q}}$ which is isomorphic to the restriction of the natural ($n-1$)-dimensional quotient of the permutation representation of $A_n$ to $Gal(L_i/F)$.
Let $\sigma _i = \sigma|_{L_i}$ be the restriction of $\sigma$ to $L_i$. Then since $F\subseteq \overline{K}^{\sigma}$, $\sigma_i|_F=id_F$. Therefore, $\sigma_i\in Gal(L_i/F)\leq A_n$. Let $E(L_i^{\sigma _i})$ be the group of fixed points of $E(L_i)$ under $\sigma_i$. Then obviously, $E(L_i^{\sigma _i})\subseteq
E(\overline{K}^{\sigma})$. Since $n$ is even, each $M_i$ of $E(L_i)\otimes{\mathbb{Q}}$ has a fixed element $v_i$ under $\sigma_i$ by Lemma \[lem:cycle\] and Lemma \[lem:inva\].
Note that each $v_i$ is not a torsion point and not defined over $F$. In fact, if $v_i$ is defined over $F$, then $v_i$ is fixed under every element in $Gal(L_i/F)$. But since $Gal(L_i/F)$ acts transitively on $\{1,2,\ldots,n\}$, by Lemma \[lem:invari\], there is no fixed vector of the restriction to $Gal(L_i/F)$ of the $(n-1)$-dimensional quotient of the permutation representation of $A_n\leq S_n$. Then, by Lemma \[lem:indep\], there is an integer $N$ such that $\{v_i\}_{i\geq
N}$ are linearly independent.
Since $v_i\in E(L_i^{\sigma _i})\otimes{\mathbb{Q}}$ for each $i$, the module generated by $\{v_i\}_{i=1}^{\infty}$ over ${\mathbb{Q}}$ is a submodule of $E\left(\prod \limits _{i=1}^{\infty} L_i^{\sigma
_i}\right)\otimes{\mathbb{Q}}$. Hence $$E(\overline{K}^{\sigma})\otimes{\mathbb{Q}}\supseteq E\left(\prod
\limits _{i=1}^{\infty} L_i(\sigma _i)\right)\otimes{\mathbb{Q}}\supseteq
\{v_i\}_{i=1}^{\infty}\supseteq \{v_i\}_{i\geq N}$$ is infinite dimensional.
Infinite rank over the fixed fields under complex conjugation automorphisms
===========================================================================
The only difficulty in proving the rank of $E(\overline{K}^{\sigma})$ is infinite is that $\overline{K}^{\sigma}$ may have a real embedding. Now we consider complex conjugation automorphisms of $\overline{K}$ and prove that without hypothesis on rational points of elliptic curves and the ground field, the rank of an elliptic curve over the fixed field under every complex conjugation automorphism is infinite.
A field $F$ is called an ordered field with the positive set $P$, if $F=P\bigsqcup \{0\}\bigsqcup -P$, a disjoint union, where $P$ is a subset of $F$ closed under addition and multiplication.
Now we prove the following two lemmas by using the relation between real fields (refer Definition \[def:real\]) and ordered fields.
\[lem:real1\] If a field $F$ is ordered (or real) and algebraic over ${\mathbb{Q}}$, then $F$ has a real embedding $\theta$, that is, $\theta(F) \subseteq {\mathbb{R}}\cap \overline{F}$.
By ([@issacs], Chapter 25, Corollary (25.22), pp.411), a field $F$ is ordered if and only if it is real. Hence, $F$ is real. And since $F$ is real and algebraic over ${\mathbb{Q}}$, by ([@issacs], Theorem (25.18), pp. 410), there exists an isomorphism from $F$ into ${\mathbb{R}}\cap
\overline{F}$.
We give some equivalent statements of complex conjugation automorphisms.
\[lem:real2\] The following statements are equivalent: for an automorphism $\sigma\in Gal(\overline{K}/K)$,\
(1) $\overline{K}^{\sigma}$ has a real embedding $\theta$, that is, $\theta(\overline{K}^{\sigma}) \subseteq {\mathbb{R}}\cap\overline{K}$\
(2) $\sigma$ is a complex conjugation automorphism, that is, the order of $\sigma$ in $Gal(\overline{K}/K)$ is 2.\
(3) $\overline{K}^{\sigma} \cong{\mathbb{R}}\cap\overline{K}$
Suppose (1). Then, $$\langle \sigma \rangle \cong
Gal(\overline{K}/\overline{K}^{\sigma})\cong
Gal(\overline{K}/\theta(\overline{K}^{\sigma})) \unrhd
Gal(\overline{K}/{\mathbb{R}}\cap \overline{K}) \cong {\mathbb{Z}}/2{\mathbb{Z}},$$ since $[\overline{K}:{\mathbb{R}}\cap \overline{K}]=2$. Hence, $Gal(\overline{K}/\overline{K}^{\sigma})$ has a torsion subgroup of order 2. Then, by Lemma \[lem:brauer\], $Gal(\overline{K}/\overline{K}^{\sigma})$ itself is isomorphic to ${\mathbb{Z}}/2{\mathbb{Z}}$. Since $\sigma$ is not trivial, we have $Gal(\overline{K}/\overline{K}^{\sigma}) \cong
Gal(\overline{K}/{\mathbb{R}}\cap \overline{K})$, hence, the order of $\sigma$ is $2$, which implies (2). And $\overline{K}^{\sigma}
\cong {\mathbb{R}}\cap \overline{K}$, which implies (3).
Now we suppose (3). Then, the order of $\sigma$ equals the degree $[\overline{K}:\overline{K}^{\sigma}]$ which is equal to $[\overline{K}:{\mathbb{R}}\cap\overline{K})]=2$, and this implies (2).
Suppose (2). Then, $[\overline{K}:\overline{K}^{\sigma}]=2$. By by ([@issacs], Theorem (25.13)), $\overline{K}^{\sigma}$ is real closed. Then, it is real and algebraic over ${\mathbb{Q}}$. So by Lemma \[lem:real1\], it has a real embedding. This implies (1).
The following lemma gives the density of the Hilbert sets over a number field $K$ with respect to any real embeddings of $K$ into ${\mathbb{R}}$.
\[lem:dense\] Let $K$ be a number field and $\tau_1,\ldots,\tau_m$ be a family of real embeddings of $K$. For $i=1,2,\ldots,k$, let $f_i(x,y)\in K[x,y]$ be irreducible polynomials over $K(x)$. Let $H_K(f_i)$ be the Hilbert set of $f_i$ over $K$. Then $$\left(\bigcap\limits _{i=1}^k
H_K(f_i)\right)~\cap~\left(\bigcap\limits
_{j=1}^m\tau_j^{-1}(I)\right)\neq~~ \emptyset,$$ for any open interval $I$ in ${\mathbb{R}}$.
This is a special case of ([@geyer], Lemma 3.4).
\[thm:rank\] Let $K$ be a number field, $K_{ab}$ the maximal abelian extension of $K$, and E/K an elliptic curve over $K$. Then, for any complex conjugation automorphism $\sigma\in
Gal(\overline{K}/K)$, $E((K_{ab})^{\sigma})$ has infinite rank. Hence, $E(\overline{K}^{\sigma})$ has infinite rank.
For a complex conjugation automorphism $\sigma\in
Gal(\overline{K}/K)$, there exists a real embedding $\theta$ such that $\theta(\overline{K}^{\sigma})\subseteq {\mathbb{R}}\cap
\overline{K}$, by Lemma \[lem:real2\]. Note that $\sigma(\sqrt{-1})=-\sqrt{-1}$, since otherwise $\sigma(\sqrt{-1})=\sqrt{-1}$ and then, $\sqrt{-1}\in
\overline{K}^\sigma$ and $0<(\theta(\sqrt{-1}))^2=\theta(-1)=-1$ which is a contradiction. And for any element $\alpha\in K$ such that $\theta(\alpha)> 0$, $\sigma(\sqrt{\alpha})=\sqrt{\alpha}$, since otherwise, $\sigma(\sqrt{\alpha})=-\sqrt{\alpha}$, hence $\sigma(\sqrt{-\alpha})=\sqrt{-\alpha}$, then, $ \sqrt{-\alpha}\in
\overline{K}^\sigma$ and $0<(\theta(\sqrt{-\alpha}))^2=-\theta(\alpha)<0$ which is a contradiction.
Let’s consider a Weierstrass equation of $E/K$, $y^2=x^3+ax+b$, for $a,b\in K$. Then there exists $\alpha\in {\mathbb{R}}$ such that $x^3+\theta(a)x+\theta(b)>0$ for all $x>\alpha$. Let $I=(\alpha,\infty)$ be the open interval of all real numbers $>\alpha$. Let $f(x,y)=y^2-(x^3+ax+b)$. Then $f$ is an absolutely irreducible polynomial in $K[x,y]$, hence irreducible over $K(x)$. Let $H_K(f)$ be the Hilbert set of $f$ over $K$. Note that the restriction $\theta|_K$ of $\theta$ to $K$ is a real embedding of $K$.
By Lemma \[lem:dense\], there is an element $x_1\in H_K(f)\cap \theta|_{K}^{-1}(I)$. Then $x_1^3+ax_1+b\in K $ and since $\theta(x_1)>\alpha$, $\theta(x_1^3+ax_1+b)$ is positive, hence $\sigma(\sqrt{x_1^3+ax_1+b})=
\sqrt{x_1^3+ax_1+b}$. Hence $\sigma$ fixes $\sqrt{x_1^3+ax_1+b}$. Let $K_1=K(\sqrt{x_1^3+ax_1+b})$. Then since $f(x_1,y)$ is irreducible in $K[y]$, $K_1$ is a quadratic extension of $K$ and $K_1\subseteq\overline{K}(\sigma)$.
Inductively, suppose we have constructed linearly disjoint quadratic extensions $K_1,$ $\ldots,K_{n-1}$ of $K$ such that $K_i=K(\sqrt{x_i^3+ax_i+b})$ for $x_i\in H_K(f)\cap
\theta|_{K}^{-1}(I)$. Let $L_{n-1}=K_1\cdots K_{n-1}$ be the composite field extension over $K$. By Lemma \[lem:dense\] again, there is $x_n\in H_{L_{n-1}}(f)\cap \theta|_K^{-1}(I)$. Let $K_n$ $=K(\sqrt{x_n^3+ax_n+b})$. Then similarly, we can show that $K_n$ is a quadratic extension of $K$ and $K_n\subseteq\overline{K}(\sigma)$. Moreover, $K_n$ is linearly disjoint from all $K_1,K_2,\ldots,K_{n-1}$, since $x_n\in
H_{L_{n-1}}(f)$.
Hence we have obtained $\{x_i\}_{i=1}^{\infty}\subseteq K$ and an infinite sequence $\{K_i/K\}_{i=1}^{\infty}$ of linearly disjoint quadratic extensions of $K$ such that $K_i=K(\sqrt{x_i^3+ax_i+b})$. For each $i$, let $P_i$ be a point of $E(K_i)$ whose $x$-coordinate is $x_i$. Note that $P_i\notin
E(K)$, for each $i$. Hence, by Lemma \[lem:indep\], for some $N$, $\{P_i\}_{i\geq N}$ consists of linearly independent non-torsion points of $E(\overline{K})$. In particular, since $K_i$ are abelian extensions of $K$ and are fixed under $\sigma$, $P_i$ are points of $E((K_{ab})^{\sigma})$. Hence $$E\left((K_{ab})^{\sigma}\right)\otimes {\mathbb{Q}}\supseteq
E\left(\prod \limits _{i=N}^{\infty} K_i\right)\otimes {\mathbb{Q}}\supseteq \{P_i\otimes 1\}_{i\geq N}$$ is infinite dimensional. And $E(\overline{K}^{\sigma})$ has infinite rank as well.
More general result : $E$ over arbitrary number fields with a rational point which is neither 2-torsion nor 3-torsion
======================================================================================================================
In this section, we prove a more general result than the result of Theorem \[thm:totally\] for $E/K$ with a rational point $P$ such that $2P\neq O$ and $3P\neq O$ without hypothesis on the ground field $K$. To do so, we need the following lemma and proposition.
\[lem:real\] For a number field $K$, let $\sigma\in
Gal(\overline{K}/K)$. If $\sigma$ does not fix any totally imaginary finite extensions of $K$, then $\sigma$ is a complex conjugation automorphism.
Since $\overline{K}^{\sigma}$ is algebraic over ${\mathbb{Q}}$, by Lemma \[lem:real1\] and Lemma \[lem:real2\], it is enough to show that $\overline{K}^\sigma$ is ordered.
If $L$ is a finite extension of $K$ such that $L\subseteq
\overline{K}^\sigma$, then $L$ is not totally imaginary by the assumption. Let $\tau_1,\ldots,\tau_r$ be all real embeddings of $L$.
For $\alpha\in L^*$ ($=L-\{0\}$), if $\tau_i(\alpha)<0$ for all $i=1,\ldots,r$, then, $L(\sqrt{\alpha})$ is totally imaginary (otherwise, $L(\sqrt{\alpha})$ has a real embedding $\rho$ and $\rho|_L=\tau_i$ for some $i$. But we have $0<
(\rho(\sqrt{\alpha}))^2=\rho(\alpha)=\tau_i(\alpha)$, which contradicts $\tau_i(\alpha)<0$ for all $i$). Hence, $\sigma$ does not fix $\sqrt{\alpha}$ by the assumption, so $\sigma(\sqrt{\alpha})=-\sqrt{\alpha}$. This implies that for $\beta\in L^*$, if $\tau_i(\beta)>0$ for all $i=1,\ldots,r$, then $\sigma(\sqrt{-\beta})=-\sqrt{-\beta}$ and since $\tau_i(-1)=-1<0$ for all $i$, $\sigma(\sqrt{-1})=-\sqrt{-1}$, hence $\sigma(\sqrt{\beta})=\sqrt{\beta}$.
Therefore, there is a homomorphism $h: \prod\limits_{i=1}^r \{\pm
1\}\rightarrow \{\pm 1\}$ such that the action of $\sigma$ on $\sqrt{\alpha}$ for $\alpha\in L^*$ depends only on the image of the vector of signs of $\alpha$ under $h$. In other words, for $\alpha\in L^*$, we let $f: L^*\rightarrow \prod\limits_{i=1}^r
\{\pm 1\}$ be a homomorphism defined by $$f(\alpha)=(\mbox{sign}(\tau_1(\alpha)),\ldots,\mbox{sign}(\tau_r(\alpha))),$$ and $g: L^*\rightarrow \{\pm 1\}$ defined by $$g(\alpha)=\mbox{
the sign of }\frac{\sigma(\sqrt{\alpha})}{\sqrt{\alpha}}_,$$ so $\sigma(\sqrt{\alpha})=g(\alpha)\sqrt{\alpha}$, then there exists a homomorphism $h: \prod\limits_{i=1}^r \{\pm 1\} \rightarrow
\{\pm 1\}$ such that $h\circ f=g$.
Note that from the above explanation on totally positive or totally negative elements of $L^*$, we get $$(*)\hspace{1.5 in} h(-1,\ldots,-1)=-1 \mbox{\hspace{.2in} and
\hspace{.2in}} h(1,\ldots,1)=1.\hspace{1.5 in}$$ In particular, there is always a vector consisting of $-1$ in all but one coordinate and $1$ in the remaining coordinate which lies in the kernel of $h$. In fact, there are $r$ vectors consisting of $-1$ in all but one coordinate and $1$ in the remaining coordinate: $$v_1=(1,-1,\ldots,-1),v_2=(-1,1,-1,\ldots,-1),\ldots,v_r=(-1,\ldots,-1,1).$$ If all $r$ vectors map to $-1$ under $h$, then $$(-1)^r=\prod\limits_{i=1}^r h(v_i)=h\left(\prod\limits_{i=1}^r
v_i\right)=h((-1)^{r-1},\ldots, (-1)^{r-1}).$$ But this contradicts ($*$) by taking an even and odd integer $r$. Therefore, at least one of $v_i$ must map to $1$, so it lies in the kernel of $h$. Without loss of generality, we may assume that $v_1$ maps to $1$ under $h$.
Hence, we can choose $\alpha\in L^*$ such that $\sigma(\sqrt{\alpha})=\sqrt{\alpha}$ and $\tau_1(\alpha)>0$ but $\tau_i(\alpha)<0$ for all $i=2,\ldots, r$ and let $L'=L(\sqrt{\alpha})$. Then, $L'$ is fixed under $\sigma$ so $L'$ is not totally imaginary. Let $\rho$ be a real embedding of $L'$. Then, since $\alpha$ is positive only with respect to $\tau_1$, $$0<
(\rho(\sqrt{\alpha}))^2=\rho(\alpha)=\rho|_L(\alpha)=\tau_1(\alpha).$$ Hence, $$\rho(\sqrt{\alpha})=\pm \sqrt{\tau_1(\alpha)}.$$ This shows that $L'$ has exactly two real embeddings $\rho_1$ and $\rho_2$ such that $$\rho_1(\sqrt{\alpha})=\sqrt{\tau_1(\alpha)}, \hspace{.15 in}
\rho_2(\sqrt{\alpha})=-\sqrt{\tau_1(\alpha)},\mbox{~ and ~}
\rho_{i}|_L=\tau_1, \mbox{ for } i=1,2.$$
We proceed the same argument on $L'$ with two real embeddings $\rho_i$ as before and get a homomorphism $h':\{\pm 1\}\times
\{\pm 1\}\rightarrow \{\pm 1\}$ and $f': L'^*\rightarrow \{\pm
1\}\times \{\pm 1\}$ given by $f(\beta)=($sign$(\rho_1(\beta)),
$sign$(\rho_2(\beta)))$, $g':L'^*\rightarrow \{\pm 1\}$ given by $g'(\beta)=\mbox{ the sign of
}\frac{\sigma(\sqrt{\beta})}{\sqrt{\beta}}$, such that $h'\circ
f'=g'$ on $L'^*$. Again, we have that $$(**)\hspace{1.7 in} h'(-1,-1)=-1 \mbox{\hspace{.2in} and
\hspace{.2in}} h'(1,1)=1.\hspace{1.7 in}$$ This implies that $h'$ cannot send both $(-1,-1)$ and $(1,1)$ to the same value $1$ or $-1$. So either $h'(-1,1)=-1$ and $h'(1,-1)=1$ or $h'(-1,1)=1$ and $h'(1,-1)=-1$. Therefore, $h'$ is the projection onto either the first factor or the second factor. Without loss generality, we assume that $h'$ is the projection onto the first factor, that is, $g'$ is defined by the sign of the first real embedding $\rho_1$. Then, if $\beta, \gamma\in L'^*$ such that $\sqrt{\beta},
\sqrt{\gamma}$ are fixed under $\sigma$, then $\rho_1(\beta)>0$ and $\rho_1(\gamma)>0$ so $\rho_1(\beta+\gamma)>0$. Hence, $$g'(\beta+\gamma)=h'(f'(\beta+\gamma))=h'(1, a)=1, \mbox{ where } a= 1\mbox{ or } -1.$$ So $\sigma(\sqrt{\beta+\gamma})=\sqrt{\beta+\gamma}.$ And obviously, $\sigma(\sqrt{\beta\gamma})=\sqrt{\beta\gamma}.$
We have shown that the set of $\beta\in L'^*$ with $\sigma(\sqrt{\beta})=\sqrt{\beta}$ is closed under addition and multiplication. Therefore, for any two elements $a$ and $b\in
\overline{K}^\sigma-\{0\}$, by applying the preceding argument with taking $L$ as a finite extension of $K$ generated by $a^2$ and $b^2$, the set of squares in $ \overline{K}^\sigma-\{0\}$ are closed under addition and multiplication. Hence, if we let $S$ be the set of squares in $ \overline{K}^\sigma-\{0\}$ and denote the the set of non-squares in $ \overline{K}^\sigma-\{0\}$ by $-S$, then $\overline{K}^\sigma= S \bigsqcup \{0\}\bigsqcup -S$, a disjoint union. Hence, $\overline{K}^\sigma$ is ordered with the positive set $S$. This completes the proof.
\[prop:comptot\] For a number field $K$, let $\sigma\in
Gal(\overline{K}/K)$. If $\sigma$ is not a complex conjugation automorphism, then there is a totally imaginary finite extension $L$ over $K$ such that $L \subseteq \overline{K}^\sigma$.
It follows Lemma \[lem:real\].
The following is our more general theorem without hypothesis on the ground field or the given automorphisms.
\[thm:main\] Let $K$ be a number field and $E/K$ an elliptic curve over $K$ with a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$. Then, for each $\sigma\in
Gal(\overline{K}/K)$, the rank of $E(\overline{K}^\sigma)$ is infinite.
Let $\sigma\in Gal(\overline{K}/K)$. If $\sigma$ is a complex conjugation automorphism, then $E(\overline{K}^\sigma)$ has infinite rank by Theorem \[thm:rank\]. If $\sigma$ is not a complex conjugation automorphism, then by Proposition \[prop:comptot\], there is a totally imaginary finite extension $L$ of $K$ such that $L \subseteq \overline{K}^\sigma$. Hence $\sigma\in Gal(\overline{K}/L)$. Now consider $E/L$ defined over $L$ by replacing the ground field $K$ by $L$. Then, since the given $K$-rational point $P$ is also defined over $L$, we apply Theorem \[thm:totally\] to complete the proof.
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|
---
abstract: 'The supersymmetric extension of a model introduced by Lukierski, Stichel and Zakrewski in the non-commutative plane is studied. The Noether charges associated to the symmetries are determined. Their Poisson algebra is investigated in the Ostrogradski–Dirac formalism for constrained Hamiltonian systems. It is shown to provide a supersymmetric generalization of the Galilei algebra with a two-dimensional central extension.'
address:
- |
Instituto de Matemática y Física, Universidad de Talca,\
Casilla 747, Talca, Chile
- |
Centre de recherches mathématiques, Université de Montréal,\
Montreal, Quebec, Canada, H3C 3J7
- |
Department of Mathematics and Statistics and Department of Physics,\
McGill University, Montreal, Quebec, Canada, H3A 2J5
author:
- Luc Lapointe
- Hideaki Ujino
- Luc Vinet
title: 'Supersymmetry in the Non-Commutative Plane'
---
,
and
non-commutative plane, supersymmetry, Noether charge,Poisson algebra, central extension
11.30.-j; 11.30.Pb; 12.60.Jv; 02.20.Sv
Introduction
============
There has been an increased interest lately in the study of physics in non-commutative space-time. This stems, in particular, from advances in string theory [@SW] and from the Connes program [@Connes1; @Connes2] (see also [@Jackiw]). In this context, two models [@DH1; @LSZ] with interesting features were recently introduced in the non-commutative plane in independent and different ways. In the two cases, which have been shown to be related with each other [@DH2],[^1] the non-commutativity of the space coordinates is intimately related to the invariance of the model under the Galilei group with a two-dimensional central extension. While the dynamics is described using coadjoint orbits and canonical symplectic structures in ref. [@DH1], the Lagrangian picture is used in ref. [@LSZ]. In this paper, we shall focus exclusively on the latter description, which posits a non-relativistic classical model in two dimensions described by the Lagrangian $$L_{\rm b} := \dfrac{1}{2}m\dot{x}_i^2-k\epsilon_{ij}\dot{x}_i
\ddot{x}_j, \quad i,j=1,2,
\label{eq:LSZ}$$ where $\epsilon_{ij}$ is the Levi–Civita antisymmetric metric [@LSZ], and where, as will be the case throughout the paper, the Einstein convention on the summation of repeated indices is employed. The model [(\[eq:LSZ\])]{} was shown to have the (2+1)-Galilean symmetry [@LL] with a two-dimensional central extension parametrized by the mass $m$ and the coupling parameter $k$.
We study in this paper a supersymmetrized version of the model [(\[eq:LSZ\])]{}, which has been also introduced in ref. [@LSZ2].[^2] In addition to the intrinsic interest of the generalized model, an additional motivation is the exploration of the supersymmetric enlargement of the Galilei algebra with a two-dimensional central extension. We shall also identify the presence of the higher conformal and superconformal symmetries in the original and in the supersymmetric models.
The paper is organized as follows. In section \[sec:model\], we introduce the supersymmetric model. In section \[sec:eom\], the equations of motion and the canonical structure of the supersymmetric model are presented through the Ostrogradski–Dirac formalism. In section \[sec:lpa\], we obtain the Noether charges associated with the symmetries and investigate the Poisson algebra that they generate. The final section includes a summary and concluding remarks.
The supersymmetric model and its symmetries {#sec:model}
===========================================
We shall consider a generalization of the Lagrangian [(\[eq:LSZ\])]{} involving a two-dimensional free-fermion term: $$L
=L_{\rm b}+\dfrac{\I}{2}\xi_i\dot{\xi}_i+\cdots,
\quad \xi_i,\ i=1,2:\ \text{Grassmannian}$$ and supplemented by additional terms so that $L$ be invariant under the infinitesimal supersymmetric transformation, $$\begin{split}
& \delta_{Q}x_i:=\I\alpha\xi_i,\ \delta_{Q}\xi_i:=-m\alpha \dot{x}_i \\
& \alpha:\ \text{an infinitesimal Grassmannian parameter}
\end{split}
\label{eq:supert}$$ up to a total time-derivative, $\delta_Q L=\frac{\D \Lambda_Q}{\D t}$. It is straightforward to check that the following Lagrangian $$L=L_{\rm b}+L_{\rm f}
:=\dfrac{1}{2}m\dot{x}_i^2-k\epsilon_{ij}\dot{x}_i\ddot{x}_j
+\dfrac{\I}{2}\xi_i\dot{\xi}_i+\dfrac{\I k}{m}\epsilon_{ij}
\dot{\xi}_i\dot{\xi}_j
\label{eq:SLSZ}$$ remains invariant under the infinitesimal supersymmetric transformation [(\[eq:supert\])]{} up to a total time-derivative.[^3] To be more specific, we have $$\delta_Q L
=\dfrac{\D \Lambda_Q}{\D t},
\quad \Lambda_{Q}:=\alpha\Bigl(\dfrac{1}{2}\I m\dot{x}_i\xi_i-\I k
\epsilon_{ij}\dot{x}_i\dot{\xi}_j\Bigr).
\label{eq:LambdaQ}$$ We shall refer to the system described by the Lagrangian [(\[eq:SLSZ\])]{} as the sLSZ model.
Extending the symmetry analysis of $L_{\rm b}$ [@LSZ; @LL] to the system containing the Grassmannian variables [@C1; @C2], we observe that the sLSZ model exhibits Galilean supersymmetry. The corresponding transformations take the form, $$\begin{split}
& \delta_{\rm r}x_i:= -\epsilon_{ij}x_j\phi_{\rm b},\
\delta_{\rm r}\xi_i:=-\epsilon_{ij}\xi_j\phi_{\rm f},\
\delta_{\rm G}x_i:=v_i t \\
& \delta_{\rm t}x_i:=d_i, \
\delta_{\rm t}\xi_i:=\delta_i, \
\delta_{\tau}t:=\tau, \quad i=1,2
\end{split}
\label{eq:Galileit}$$ where the infinitesimal parameters $\phi_{\rm b,f}$, $v_i$, $d_i$, $\delta_i$ and $\tau$ are respectively the rotation angles of the bosonic and fermionic variables, the velocity of the Galilei boost of the bosonic variables and the translation shifts of the bosonic, fermionic and time variables. Among the parameters, only the $\delta_i$’s are Grassmannian, or fermionic. The sLSZ model is shown to be strictly invariant under the time and space translations as well as under the rotations, $$\delta_{\tau}L=0,\ \delta_{\rm t}L=0,\ \delta_{\rm r}L=0,
\label{eq:Galileii}$$ and furthermore, to be invariant under the Galilei boosts for the bosonic coordinates up to a total time-derivative, $$\delta_{\rm G}L=\dfrac{\D\Lambda_{{\rm G}}}{\D t},\quad
\Lambda_{\rm G}=v_i\bigl(mx_i-k\epsilon_{ij}\dot{x}_j\bigr).
\label{eq:LambdaG}$$ We should note that the Grassmannian variables do not transform under Galilei boosts: $\delta_{\rm G}\xi_i=0$.
The sLSZ model is also observed to have conformal and superconformal symmetries [@dHV]. Consider the infinitesimal dilations, conformal and superconformal transformations given by $$\begin{split}
& \delta_{\rm d}x_i:=g_{\rm b}\Bigl(t\dot{x}_i-\dfrac{1}{2}x_i
-\dfrac{2k}{m}\epsilon_{ij}\bigl(t\ddot{x}_j-\dfrac{1}{2}\dot{x}_j
\bigr)\Bigr), \\
& \delta_{\rm d}\xi_i:=g_{\rm f}\Bigl(t\dot{\xi}_i
-\dfrac{2k}{m}\epsilon_{ij}t\ddot{\xi}_j\Bigr) , \\
& \delta_{\rm c}x_i:=h_{\rm b}\Bigl(t^2\dot{x}_i-tx_i
-\dfrac{2k}{m}\epsilon_{ij}\bigl(t^2\ddot{x}_j-t\dot{x}_j\bigr)
\Bigr), \\
& \delta_{\rm c}\xi_i:=h_{\rm f}\Bigl(
t^2\dot{\xi}_i-\dfrac{2k}{m}\epsilon_{ij}t^2\ddot{\xi}_j
\Bigr), \\
& \delta_{\rm s}x_i:=\I\beta\Bigl(t\xi_i-\dfrac{2k}{m}\epsilon_{ij}
\bigl(t\dot{\xi}_j-\dfrac{1}{2}\xi_j\bigr)\Bigr) , \\
& \delta_{\rm s}\xi_i:=-m\beta\Bigl(t\dot{x}_i-x_i
-\dfrac{2k}{m}\epsilon_{ij}\bigl(t\ddot{x}_j
-\dfrac{1}{2}\dot{x}_j\bigr)\Bigr),
\end{split}
\label{eq:conformalt}$$ where $g_{\rm b,f}$, $h_{\rm b,f}$ and the Grassmannian variable $\beta$ are infinitesimal parameters. In each of these cases, the Lagrangian remains invariant up to a total time-derivative: $$\begin{aligned}
& \delta_{\rm d}L=\dfrac{\D \Lambda_{\rm d}}{\D t},
& \Lambda_{\rm d}:= & g_{\rm b}
\Bigl(\dfrac{m}{2}t\dot{x}_i^2-3k\epsilon_{ij}t\dot{x}_i\ddot{x}_j
+\dfrac{2k^2}{m}\bigl(t\ddot{x}_i^2-t\dot{x}_i\dddot{x}_i
-\dfrac{1}{2}\dot{x}_i\ddot{x}_i\bigr)\Bigr) \nonumber\\
& & & + \I g_{\rm f}\Bigl(\dfrac{1}{2}t\xi_i\dot{\xi}_i
+\dfrac{2k}{m}\epsilon_{ij}t\bigl(\dot{\xi}_i\dot{\xi}_j
-\dfrac{1}{2}\xi_i\ddot{\xi}_j\bigr)
+\dfrac{4k^2}{m^2}t\dot{\xi}_i\ddot{\xi}_i\Bigr), \nonumber\\
& \delta_{\rm c}L=\dfrac{\D \Lambda_{\rm c}}{\D t},
& \Lambda_{\rm c} := & h_{\rm b}
\Bigl(\dfrac{m}{2}\bigl(t^2\dot{x}_i^2-x_i^2\bigr)
-k\epsilon_{ij}\bigl(3t^2\dot{x}_i\ddot{x}_j-x_i\dot{x}_j\bigr) \nonumber\\
& & & +\dfrac{2k^2}{m}\bigl(t^2\ddot{x}_i^2-t^2\dot{x}_i\dddot{x}_i
-t\dot{x}_i\ddot{x}_i\bigr)\Bigr) \label{eq:conformali}\\
& & & + \I h_{\rm f}\Bigl(\dfrac{1}{2}t^2\xi_i\dot{\xi}_i
+\dfrac{2k}{m}\epsilon_{ij}t^2\bigl(\dot{\xi}_i\dot{\xi}_j
-\dfrac{1}{2}\xi_i\ddot{\xi}_j\bigr)
+\dfrac{4k^2}{m^2}t^2\dot{\xi}_i\ddot{\xi}_i\Bigr), \nonumber\\
& \delta_{\rm s}L=\dfrac{\D \Lambda_{\rm s}}{\D t},
& \Lambda_{\rm s} := & \I\beta
\Bigl(\dfrac{m}{2}\bigl(t\dot{x}_i\xi_i+x_i\xi_i\bigr)
+k\epsilon_{ij}\bigl(t\ddot{x}_i\xi_j-3t\dot{x}_i\dot{\xi}_j
+\dfrac{1}{2}\dot{x}_i\xi_j\bigr) \nonumber\\
& & & +\dfrac{2k^2}{m}\bigl(2t\ddot{x}_i\dot{\xi}_i-t\dot{x}_i\ddot{\xi}_i
-\dfrac{1}{2}\dot{x}_i\dot{\xi}_i\bigr)\Bigr). \nonumber\end{aligned}$$
The equations of motion and the canonical structure {#sec:eom}
===================================================
The Euler–Lagrange equations $$\dfrac{\D}{\D t}\Bigl(\dfrac{\partial L}{\partial \dot{x}_i}
-\dfrac{\D}{\D t}\Bigl(\dfrac{\partial L}{\partial\ddot{x}_i}\Bigr)\Bigr)
-\dfrac{\partial L}{\partial x_i}=0 \text{ and }
\dfrac{\D}{\D t}\Bigl(\dfrac{\partial L}{\partial\dot{\xi}_i}\Bigr)
-\dfrac{\partial L}{\partial \xi_i}=0$$ reduce to the following equations of motion for our model:
$$\begin{aligned}
& \dfrac{\D}{\D t}\Bigl(
m\dot{x}_i-2k\epsilon_{ij}\ddot{x}_j\Bigr)=0,\quad
\dfrac{\D}{\D t}\Bigl(-\dfrac{1}{2}\I\xi_i
+\dfrac{2k}{m}\I\epsilon_{ij}\dot{\xi}_j\Bigr)
-\dfrac{1}{2}\I\dot{\xi}_i = 0,
\label{eq:bELe} \\
& \Leftrightarrow
m\ddot{x}_i-2k\epsilon_{ij}\dddot{x}_j=0,
\quad -\I\dot{\xi}_i+\dfrac{2k}{m}\I\epsilon_{ij}\ddot{\xi}_j=0.
\label{eq:fELe}
\end{aligned}$$
\[eq:ELe\]
\
Note that the right derivative [@C1; @C2] is employed to define the derivative in the fermionic coordinates. This convention will be used throughout the paper.
Due to the presence of second order time-derivatives in the Lagrangian, in order to formulate the sLSZ model in the Hamiltonian description of the Ostrogradski formalism, we need to introduce three kinds of momenta:
$$\begin{aligned}
& p_i:=\dfrac{\partial L}{\partial \dot{x}_i}
-\dfrac{\D}{\D t}\Bigl(\dfrac{\partial L}{\partial\ddot{x}_i}\Bigr)
=m\dot{x}_i-2k\epsilon_{ij}\ddot{x}_j,
\label{eq:p}\\
& \tilde{p}_i:=\dfrac{\partial L}{\partial\ddot{x}_i}
=k\epsilon_{ij}\dot{x}_j,
\label{eq:ptilde}\\
& \pi_i:=\dfrac{\partial L}{\partial \dot{\xi}_i}
=-\dfrac{1}{2}\I\xi_i+\dfrac{2k}{m}\I\epsilon_{ij}\dot{\xi}_j.
\label{eq:pi}
\end{aligned}$$
\[eq:canonical\_momenta\]
\
This suggests that twelve canonical variables $\{x_i,\dot{x}_i,p_i,\tilde{p}_i;\xi_i,\pi_i\}$ should be employed. However, the elements in this set of canonical variables are not independent, as can be seen from eq. [(\[eq:ptilde\])]{}, which leads to two constraints, $$\Phi_i:=\dot{x}_i+\dfrac{1}{k}\epsilon_{ij}\tilde{p}_j=0,
\label{eq:constraint}$$ of the second class [@D]. Therefore, any physical quantity can be described in terms of only ten coordinates. For instance, using the Legendre transformation, $$\begin{split}
H & := \dot{x}_i p_i+\ddot{x}_i\tilde{p}_i+\dot{\xi}_i\pi_i - L \\
& = -\dfrac{m}{2k^2}\tilde{p}_i^2-\dfrac{1}{k}\epsilon_{ij}p_i\tilde{p}_j
-\dfrac{m}{4k}\I\epsilon_{ij}\bigl(\pi_i+\dfrac{1}{2}\I\xi_i\bigr)
\bigl(\pi_j+\dfrac{1}{2}\I\xi_j\bigr) \\
& = H_{\rm b}+H_{\rm f}, \\
H_{\rm b}&:=-\dfrac{m}{2k^2}\tilde{p}_i^2
-\dfrac{1}{k}\epsilon_{ij}p_i\tilde{p}_j,\quad
H_{\rm f}:=-\dfrac{m}{4k}\I\epsilon_{ij}
\bigl(\pi_i+\dfrac{1}{2}\I\xi_i\bigr)
\bigl(\pi_j+\dfrac{1}{2}\I\xi_j\bigr),
\end{split}
\label{eq:Hamiltonian}$$ we obtain the Hamiltonian of the sLSZ model in terms of the ten coordinates $\{x_i,p_i,\tilde{p}_i;\xi_i,\pi_i\}$.
When investigating the canonical equations of motion and the Poisson algebra of the sLSZ model, it is necessary to use the graded Poisson bracket as well as the Dirac bracket. Let $A,B$ be either bosonic or fermionic valued differentiable functions of the canonical variables $\{x_i,\dot{x}_i,p_i,\tilde{p}_i;\xi_i,\pi_i\}$. The graded Poisson bracket $\{A,B\}$ can be defined (in a non-graded form) as $$\{A,B\}:=
\Bigl(
\dfrac{\partial A}{\partial x_i} \dfrac{\partial B}{\partial p_i}
-\dfrac{\partial A}{\partial p_i} \dfrac{\partial B}{\partial x_i}
\Bigr)
+\Bigl(
\dfrac{\partial A}{\partial \dot{x}_i}
\dfrac{\partial B}{\partial \tilde{p}_i}
-\dfrac{\partial A}{\partial \tilde{p}_i}
\dfrac{\partial B}{\partial \dot{x}_i}
\Bigr)
-\Bigl(
\dfrac{\partial B}{\partial \pi_i}
\dfrac{\partial A}{\partial \xi_i}
+\dfrac{\partial B}{\partial \xi_i}
\dfrac{\partial A}{\partial \pi_i}
\Bigr).
\label{eq:gPB}$$ The canonical Poisson brackets among the canonical variables are then $$\{x_i,p_j\}=\delta_{ij},\
\{\dot{x}_i,\tilde{p}_j\}=\delta_{ij},\
\{\xi_i,\pi_j\}=-\delta_{ij},\ \text{others}\ \{\cdot,\cdot\}=0.$$ Due to the constraints $\Phi_i$, we need to use the Poisson bracket defined on the reduced phase space, which is nothing but the Dirac bracket [@D], $$\{A,B\}_{\rm D}:=\{A,B\}-\{A,\Phi_i\}C_{ij}\{\Phi_j,B\},$$ where the matrix $C$ is defined through the relation $C_{ik}\{\Phi_k,\Phi_j\}=\delta_{ij}$. Substitution of the constraints [(\[eq:constraint\])]{} gives the Dirac bracket for the sLSZ model: $$\{A,B\}_{\rm D}:=\{A,B\}-\{A,\Phi_i\}\dfrac{k}{2}\epsilon_{ij}\{\Phi_j,B\}.
\label{eq:DiracB}$$ Choosing the independent variables as $y_{a}:=\{x_i,p_i,\tilde{p}_i;\xi_i,\pi_i\}$, $a=1,\dots,10$, we then have $$\{y_a,y_b\}_{\rm D}=\omega_{ab},
\
\omega:=\left[\begin{array}{ccccc}
0 & E & 0 & 0 & 0 \\
-E & 0 & 0 & 0 & 0 \\
0 & 0 & \dfrac{k}{2}\epsilon & 0 & 0 \\
0 & 0 & 0 & 0 & -E \\
0 & 0 & 0 & -E & 0
\end{array}\right],
\label{eq:CDB}$$ where $$E:=\left[\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right],\
\epsilon:=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0
\end{array}\right],$$ and where $0$ denotes the $2\times 2$ null matrix.
Using the Dirac bracket, the canonical equations of motion read as $$\dot{y}_a=\{y_a,H\}_{\rm D}+\dfrac{\partial y_a}{\partial t} .$$ In the case of the sLSZ model, this leads to $$\begin{split}
& \dot{x}_i=-\dfrac{1}{k}\epsilon_{ij}\tilde{p}_j,\
\dot{p}_i=0,\
\dot{\tilde{p}}_i=-\dfrac{m}{2k}\epsilon_{ij}\tilde{p}_j
-\dfrac{1}{2}p_i,\\
& \dot{\xi}_i=\dfrac{m}{2k}\I\epsilon_{ij}\bigl(\pi_j
+\dfrac{1}{2}\I\xi_j\Bigr),\
\dot{\pi}_i=-\dfrac{m}{4k}\epsilon_{ij}\bigl(\pi_j
+\dfrac{1}{2}\I\xi_j\Bigr),
\end{split}
\label{eq:cem}$$ which is consistent with the Euler–Lagrange equations [(\[eq:fELe\])]{} derived from the Lagrangian. We should note that the equations of motion and the Dirac brackets of the Grassmannian variables can be cast into a simpler form using the variables $$\theta_i^\pm:=\pi_i\pm\dfrac{1}{2}\I\xi_i,
\label{eq:def_theta}$$ as they then read $$\dot{\theta}_i^+=-\dfrac{m}{2k}\epsilon_{ij}\theta_j^+,\
\dot{\theta}_i^-=0,\
\{\theta_i^\pm,\theta_j^\pm\}=\mp\I\delta_{ij},\
\{\theta_i^+,\theta_j^-\}=0.
\label{eq:theta}$$ We shall now investigate the Poisson algebra of the conserved charges of the sLSZ model.
The Noether charges and their Poisson algebra {#sec:lpa}
=============================================
Let the sLSZ Lagrangian with its independent variables $L=L(x_i,\dot{x}_i,\ddot{x}_i,\xi_i,\dot{\xi}_i)$ be denoted for short as $L(x_i,\xi_i)$. According to Noether’s theorem, the invariance, up to a total derivative, of the Lagrangian $L$ with respect to the infinitesimal transformation, $\delta x_i$, $\delta\xi_i$, that is $$\delta L := L(x_i+\delta x_i,\xi_i+\delta\xi_i)-L(x_i,\xi_i)
=\dfrac{\D\Lambda}{\D t} ,$$ implies the conservation of a quantity of the form $$C:=\delta x_i p_i+\delta\dot{x}_i \tilde{p}_i+\delta\xi_i\pi_i-\Lambda.
\label{eq:NC1}$$ Applying the formula [(\[eq:NC1\])]{} to each symmetry transformations [(\[eq:supert\])]{}, [(\[eq:Galileit\])]{} and [(\[eq:conformalt\])]{} (except the time-translation) provides the following 12 conserved quantities: $$\begin{aligned}
&\text{space-translation:} & & \nonumber \\
& C_{\rm t}= d_ip_i+\delta_i\theta_i^-,
& & p_i,\ \theta_i^-,\nonumber\\
&\text{rotation:} & & \nonumber\\
& C_{\rm r}= \phi_{\rm b}\left(\epsilon_{ij}x_i p_j
-\dfrac{1}{k}\tilde{p}_j^2 \right)+\phi_{\rm f}\epsilon_{ij}\xi_i\pi_j,
& & J_{\rm b}:=\epsilon_{ij}x_i p_j-\dfrac{1}{k}\tilde{p}_j^2, \
J_{\rm f}:=\epsilon_{ij}\xi_i\pi_j,\nonumber\\
&\text{Galilei boost:} & & \nonumber \\
& C_{\rm G}= v_i(tp_i-mx_i+2\tilde{p}_i),
& & G_i:=tp_i-mx_i+2\tilde{p}_i, \nonumber\\
& \text{supersymmetric:} & & \nonumber \\
& C_{\rm Q}=\alpha\bigl(p_i(\theta_i^+-\theta_i^-)
-\dfrac{m}{k}\epsilon_{ij}\tilde{p}_i\theta_j^+\bigr),
& & Q:=p_i(\theta_i^+-\theta_i^-)
-\dfrac{m}{k}\epsilon_{ij}\tilde{p}_i\theta_j^+, \nonumber\\
& \text{dilation:} & & \nonumber \\
& C_{\rm d}=g_{\rm b}\Bigl(\dfrac{1}{2m}p_iG_i\Bigr),
& & D:=\dfrac{1}{2m}p_iG_i, \label{eq:NCSLSZ1}\\
& \text{conformal:} & & \nonumber \\
& C_{\rm c}=h_{\rm b}\Bigl(\dfrac{1}{2m}G_i^2\Bigr),
& & K:=\dfrac{1}{2m}G_i^2, \nonumber \\
& \text{superconformal:} & & \nonumber \\
& C_{\rm s}=\beta\Bigl(\dfrac{k}{m}\tilde{Q}-G_i\theta_i^{-}\Bigr),
& & S:=\dfrac{k}{m}\tilde{Q}-G_i\theta_i^{-},\nonumber\\
& & & \tilde{Q}:= \epsilon_{ij}p_i
(\theta_j^+-\theta_j^-)+\dfrac{m}{k}\tilde{p}_i\theta_i^+. \nonumber\end{aligned}$$ In order to replace the time-derivatives of the coordinates with the canonical momenta, we have used the following relations derived from the definitions of the canonical momenta [(\[eq:canonical\_momenta\])]{} and the equations of motion [(\[eq:cem\])]{}: $$\begin{aligned}
& \dot{x}_i=-\dfrac{1}{k}\epsilon_{ij}\tilde{p}_j,\
\ddot{x}_i=\dfrac{1}{2k}\epsilon_{ij}p_j-\dfrac{m}{2k^2}\tilde{p}_i,\
\dddot{x}_i=\dfrac{m^2}{4k^3}\epsilon_{ij}\tilde{p}_j
+\dfrac{m}{4k^2}p_i,\\
& \dot{\xi}_i=\dfrac{m}{2k}\I\epsilon_{ij}\theta_j^+,\
\ddot{\xi}_i=\dfrac{m^2}{4k^2}\I\theta_i^+.\end{aligned}$$ As can be seen in eq. [(\[eq:SLSZ\])]{}, the Lagrangian of the sLSZ model does not explicitly depend on the time $t$, i.e. $\frac{\partial L}{\partial t}=0$. Hence, the conserved quantity corresponding to the time-translation is given by the Hamiltonian [(\[eq:Hamiltonian\])]{}.
We have thus obtained 13 Noether charges: the Hamiltonian [(\[eq:Hamiltonian\])]{} plus the 12 quantities appearing in [(\[eq:NCSLSZ1\])]{}. We now turn to the Poisson algebra that these Noether charges generate.
Since the bosonic and the fermionic coordinates are decoupled in the Hamiltonian [(\[eq:Hamiltonian\])]{}, the bosonic and fermionic parts of the Hamiltonian, $H_{\rm b}$ and $H_{\rm f}$, are independently conserved. Moreover, the canonical momenta $p_i$ are conserved, as is the case in the interaction-free model. We can thus separate the Hamiltonian [(\[eq:Hamiltonian\])]{} into three individually conserved quantities, $H_0$, $H_k$ and $H_{\rm f}$, $$\begin{split}
& H = H_0+H_{k}+H_{\rm f}, \\
& H_0 := \dfrac{1}{2m}p_i^2,\
H_k:=H_{\rm b}-H_0=-\dfrac{m}{2k^2}\tilde{P}_i^2,\
H_{\rm f}=-\dfrac{m}{4k}\I\epsilon_{ij}\theta_i^+\theta_j^+,
\end{split}
\label{eq:separate_Hamiltonian}$$ where the quantities $$\tilde{P}_i:=\dfrac{k}{m}p_i+\epsilon_{ij}\tilde{p}_j
\label{eq:LSZNCP}$$ are the non-commuting modified momenta introduced in ref. [@LSZ]. From the definition of the canonical momenta [(\[eq:canonical\_momenta\])]{}, we have $$\tilde{p}_i=k\epsilon_{ij}\dot{x}_j=O(k),\
\tilde{P}_i=-\dfrac{2k^2}{m}\epsilon_{ij}\ddot{x}_j=O(k^2),\
\theta_i^+=\dfrac{2k}{m}\I\epsilon_{ij}\dot{\xi}_j=O(k),
\label{eq:OE1}$$ and thus observe immediately that $H_k$ and $H_{\rm f}$ vanish in the limit $k\rightarrow 0$.
The Noether charge associated to the superconformal transformation $S$ in eq. [(\[eq:NCSLSZ1\])]{} can also be divided into two independently conserved quantities, $\tilde{Q}$ and $F$, $$S:=\dfrac{k}{m}\tilde{Q}-F,\ F:=G_i\theta_i^-,
\label{eq:separate_S}$$ since $G_i$ and $\theta_i^-$ are themselves conserved.
In addition to $F$, three kinds of “quadratic” conserved quantities, $$\tilde{F}:=\epsilon_{ij}G_i\theta_j^-,\
E:=p_i\theta_i^-,\ \tilde{E}:=\epsilon_{ij}p_i\theta_j^-,
\label{eq:CQSLSZ}$$ arise from the closure of the Dirac brackets. These constructions are similar to that of $F$, in the sense that they are all products of the linear Noether charges of eq. [(\[eq:NCSLSZ1\])]{}.
The Dirac brackets among the conserved charges, $$\{A,B\}_{\rm D}, \quad A, B\in\{p_i,\theta_i^{-},G_i,
J_{\rm b}, J_{\rm f},Q,\tilde{Q},
H_0,
H_k, H_{\rm f},E,\tilde{E},D,F,\tilde{F},K\},
\label{eq:current_algebra}$$ are summarized in table \[tb:current\_algebra\],
[c|m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}]{} $A\backslash B$ & $p_j$ & $\theta_j^-$ & $G_j$ & $J_{\rm b}$ & $J_{\rm f}$ & $Q$ & $\tilde{Q}$ & $H_0$\
$p_i$ & 0 & 0 & $m\delta_{ij}$ & $-\epsilon_{ij}p_j$ & 0 & 0 & 0 & 0\
$\theta_i^-$ & & & 0 & 0 & & & & 0\
$G_i$ & & & $2k\epsilon_{ij}$ & $-\epsilon_{ij}G_j$ & 0 & $m\theta_i^-$ & $m\epsilon_{ij}\theta_j^-$ & $-p_i$\
$J_{\rm b}$ & & & & 0 & 0 & $-\tilde{Q}$ & $Q$ & 0\
$J_{\rm f}$ & & & & & 0 & & & 0\
$Q$ & & & & & & & 0 & 0\
$\tilde{Q}$ & & & & & & & & 0\
$H_0$ & & & & & & & & 0
[c|m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}m[3.8em]{}]{} $A\backslash B$ & $H_k$ & $H_{\rm f}$ & $E$ & $\tilde{E}$ & $D$ & $F$ & $\tilde{F}$ & $K$\
$p_i$ & 0 & 0 & 0 & 0 & $\frac{1}{2}p_i$ & $m\theta_i^-$ & $m\epsilon_{ij}\theta_j^-$ & $G_i$\
$\theta_i^-$ & 0 & 0 & & & 0 & & & 0\
$G_i$ & 0 & 0 & $-m\theta_i^-$ & $-m\epsilon_{ij}\theta_j^-$ & ${\scriptstyle \frac{k\epsilon_{ij}}{m}p_j-\frac{1}{2}G_i}$ & $2k\epsilon_{ij}\theta_j^-$ & $-2k\theta_i^-$ & $\frac{2k}{m}\epsilon_{ij}G_j$\
$J_{\rm b}$ & 0 & 0 & $-\tilde{E}$ & $E$ & 0 & $-\tilde{F}$ & $F$ & 0\
$J_{\rm f}$ & 0 & 0 & & & 0 & & & 0\
$Q$ & $\frac{m}{2k}{\scriptstyle (\tilde{E}+\tilde{Q})}$ & & & 0 & $-\frac{1}{2}E$ & & & $-F$\
$\tilde{Q}$ & $-\frac{m}{2k}{\scriptstyle (E+Q)}$ & & 0 & & $-\frac{1}{2}\tilde{E}$ & $\underline{\scriptscriptstyle 4\I kH_{\rm f}^\dagger-2\I m\tilde{D}}$ & & $-\tilde{F}$\
$H_0$ & 0 & 0 & 0 & 0 & $H_0$ & $E$ & $\tilde{E}$ & $2D$\
$H_k$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\
$H_{\rm f}$ & & 0 & 0 & 0 & 0 & 0 & 0 & 0\
$E$ & & & & 0 & $\frac{1}{2}E$ & & $\underline{\scriptscriptstyle 4\I kH_{\rm f}^\dagger-2\I m\tilde{D}}$ & $F$\
$\tilde{E}$ & & & & & $\frac{1}{2}\tilde{E}$ & $\underline{\scriptscriptstyle 2\I m\tilde{D}-4\I k H_{\rm f}^\dagger}$ & & $\tilde{F}$\
$D$ & & & & & 0 & ${\scriptstyle \frac{1}{2}F+\frac{k}{m}\tilde{E}}$ & ${\scriptstyle \frac{1}{2}\tilde{F}-\frac{k}{m}E}$ & ${\scriptstyle K+\frac{2k}{m}\tilde{D}}$\
$F$ & & & & & & ${\scriptscriptstyle\hspace*{-1em} \underline{2\I mK+\frac{8\I k^2}{m}H_{\rm f}^\dagger}}$ & 0 & $-\frac{2k}{m}\tilde{F}$\
$\tilde{F}$ & & & & & & & ${\scriptscriptstyle\hspace*{-1em}
\underline{2\I mK+\frac{8\I k^2}{m}H_{\rm f}^\dagger}}$ & $\frac{2k}{m}F$\
$K$ & & & & & & & & 0
with $ \tilde{D}$ and $H_{\rm f}^\dagger$ standing in the table for: $$\begin{aligned}
\tilde{D} & :=\dfrac{1}{2m}\epsilon_{ij}p_i G_j
= \dfrac{1}{2}J_{\rm b}-\dfrac{k}{m}\bigl(H_0+H_k\bigr), \\
H_{\rm f}^\dagger & := -\dfrac{m}{4k}\I\epsilon_{ij}\theta_i^- \theta_j^-
= H_{\rm f}-\dfrac{m}{2k}J_{\rm f}.\end{aligned}$$
We note that $H_k+H_{\rm f}$ has a vanishing Dirac bracket with every charge when it is placed to the right in the bracket, $$\{A,H_k+H_{\rm f}\}_{\rm D}=0,\quad A\in\{p_i,\theta_i^{-},G_i,
J_{\rm b}, J_{\rm f},Q,\tilde{Q},H_0,
H_k, H_{\rm f},E,\tilde{E},D,F,\tilde{F},K\}.$$ This result might suggest that $H_k+H_{\rm f}$ belongs, like the charges associated to the mass $m$ and to the parameter $k$, to the center of the Poisson algebra. But this is actually not the case, because it has a non-vanishing Dirac bracket with $Q$ and $\tilde{Q}$ when it is placed to the left of the bracket, $$\{H_k+H_{\rm f},Q\}_{\rm D}=-\dfrac{m}{k}(\tilde{E}+\tilde{Q}),\
\{H_k+H_{\rm f},\tilde{Q}\}_{\rm D}=\dfrac{m}{k}(E+Q),$$ even though $\{H_k+H_{\rm f},\cdot\}_{\rm D}=0$ for every other bracket. In a similar way, we observe that $$\{H_k-H_{\rm f},A\}=0,\
\{Q,H_k-H_{\rm f}\}_{\rm D}=\dfrac{m}{k}(\tilde{E}+\tilde{Q}),\
\{\tilde{Q},H_k-H_{\rm f}\}_{\rm D}=-\dfrac{m}{k}(E+Q),$$ while $\{\cdot,H_k-H_{\rm f}\}_{\rm D}=0$, for all the other ones. Such asymmetries in the Poisson algebra are also seen when charges made out of bosonic and fermionic parts are considered (like for instance $J_{\rm b}\pm J_{\rm f}$).
As we can readily see from eqs. [(\[eq:separate\_Hamiltonian\])]{} and [(\[eq:OE1\])]{}, $H_k$ and $H_{\rm f}$ vanish in the limit $k\rightarrow 0$. Even though the other conserved charges neither vanish nor diverge in this limit, some linear combinations of the charges accidentally vanish. This allows to reconcile our results with those of the well-known $k=0$ situation. For example, $E+Q$ and $\tilde{E}+\tilde{Q}$ that respectively appear as Dirac brackets of $\tilde{Q}$ and $Q$ with $H_k$ and $H_{\rm f}$ in table \[tb:current\_algebra\], $$\{Q,H_k\}_{\rm D}=\dfrac{m}{2k}(\tilde{E}+\tilde{Q})
=-\{Q,H_{\rm f}\}_{\rm D},\
\{\tilde{Q},H_k\}_{\rm D}=-\dfrac{m}{2k}(E+Q)
=-\{\tilde{Q},H_{\rm f}\}_{\rm D},
\label{eq:OE2}$$ should vanish in the limit $k\rightarrow 0$ (even though a factor $1/k$ appears) since $H_k$ and $H_{\rm f}$ vanish in this limit. Using the definitions of the charges [(\[eq:NCSLSZ1\])]{} and [(\[eq:CQSLSZ\])]{}, as well as the expressions for the canonical momenta and the modified second momenta in the original coordinates [(\[eq:OE1\])]{}, one obtains that $$E+Q=-\dfrac{4k^2}{m}\I\ddot{x}_i\dot{\xi}_i=O(k^2), \
\tilde{E}+\tilde{Q}=-\dfrac{4k^2}{m}\I\epsilon_{ij}\ddot{x}_i\dot{\xi}_j
=O(k^2),$$ and thus that all Dirac brackets in eq. [(\[eq:OE2\])]{} indeed vanish when $k\rightarrow 0$. It should be remarked that both $E+Q$ and $\tilde{E}+\tilde{Q}$ have vanishing Dirac brackets with $\{p_i$, $\theta_i^-$, $G_i$, $H_0$, $D$, $F$, $\tilde{F}$, $K\}$.
The above Poisson algebra [(\[eq:current\_algebra\])]{} contains as subalgebras the Galilei algebra, and the two subalgebras obtained by considering only the quantities generated respectively by the bosonic and fermionic variables. The linear generators $\{p_i,\theta_i^+,G_i\}$ and the quadratic generators $\{J_{\rm b}$, $J_{\rm f}$, $Q$, $\tilde{Q}$, $H_0$, $H_{k}$, $H_{\rm f}$, $E$, $\tilde{E}$, $D$, $F$, $\tilde{F}$, $K\}$ also form subalgebras. Besides such obvious subalgebras, the Poisson algebra has a subalgebra generated by $\{p_i$, $\theta_i^-$, $G_i$, $J_{\rm b}$, $J_{\rm f}$, $Q$, $\tilde{Q}$, $H_0$, $H_{k}$, $H_{\rm f}$, $E$, $\tilde{E}\}$, which is the smallest subalgebra that simultaneously contains the Noether charges associated with the Galilean and supersymmetric transformations as well as the full information of the Hamiltonian. We should note that the interaction parts of the bosonic and fermionic Hamiltonians, $H_{k}$ and $H_{\rm f}$, are in the centers of the subalgebras generated by the bosonic and fermionic variables respectively. We finally remark that there obviously exist other nontrivial subalgebras such as, for instance, $\{E+Q$, $\tilde{E}+\tilde{Q}$, $H_k+H_{\rm f}\}$.
Concluding remarks {#sec:remarks}
==================
The sLSZ model [(\[eq:SLSZ\])]{} is a supersymmetric version of the Lagrangian [(\[eq:LSZ\])]{}. As we have shown, the sLSZ model remains invariant under the supersymmetric transformation [(\[eq:supert\])]{}, the (2+1)-dimensional Galilean supersymmetry [(\[eq:Galileit\])]{}, and the conformal and superconformal symmetries [(\[eq:conformalt\])]{}. Using the Ostrogradski–Dirac formalism for constrained Hamiltonian systems, the Poisson algebra associated with the Noether charges of the sLSZ model was investigated in detail, as was summarized in table \[tb:current\_algebra\].
Conformal aspects of the bosonic model [(\[eq:LSZ\])]{} are discussed in ref. [@SZ], where the model is extended to include Coulomb and magnetic vortex interactions. In the context of non-commutative geometry, the non-commutative coordinates introduced in ref. [@LSZ] are modified as $$X_i:=x_i-a\Bigl(\dfrac{2}{m}\tilde{p}_i
-\dfrac{2k}{m^2}\epsilon_{ij}p_j\Bigr),$$ where a dimensionless constant $a\neq 0$ can be chosen arbitrarily [@HP],[^4] in order for the coordinates to behave as a Galilean vector, $$\{G_i,X_j\}_{\rm D}=-t\delta_{ij}, \
\{X_i,X_j\}_{\rm D}=-\dfrac{2k}{m}a^2\epsilon_{ij}.$$ Other non-commutative coordinates introducing an interesting split into “external” and “internal” degrees of freedom in the sLSZ model are discussed in ref. [@LSZ2]. We expect further studies in such directions to be highly relevant in the understanding of the spectrum generating algebra and the representation theory of the sLSZ model as well as its generalizations.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to Professors P. C. Stichel and M. S. Plyushchay for their comments on this work. One of the authors (HU) would like to express his sincere gratitude to the Centre de Recherches Mathématiques, the Université de Montréal and McGill University for their warm hospitality. This author is also supported by a grant for his research activities abroad from the Ministry of Education, Culture, Sports, Science and Technology of Japan. Part of this research was conducted while LL held a NSERC grant. The work of LV is supported in part through a grant from NSERC.
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[^1]: Quite recently, a slight difference between the models in refs. [@DH1; @LSZ] was reported [@HP2].
[^2]: The authors are thankful to Prof. P. C. Stichel for drawing their attention to this reference.
[^3]: Equation (14) in ref. [@LSZ2] reads as $L_{\rm SUSY}^{(0)}\sim L_{\rm b}+mL_{\rm f}$, which is essentially the same as eq. [(\[eq:SLSZ\])]{}. The slight difference in the Lagrangians comes from the non-essential difference in the definitions of the supersymmetric transformation (given in this article by eq. [(\[eq:supert\])]{} and in ref. [@LSZ2] by $\delta_{\rm Q}\xi_i=-\alpha\dot{x}_i$).
[^4]: Actually, the constant $a$ is fixed at unity in ref. [@HP].
|
---
abstract: |
The $D_{4}-D_{5}-E_{6}$ model of gravity and the Standard Model\
with a $130$ $GeV$ truth quark is constructed using $3 \times 3$ matrices of octonions. The model has both continuum and lattice versions. The lattice version uses HyperDiamond lattice structure.
---
\#1\#2
hep-ph/9501252\
THEP-95-1\
January 1995\
revised March 1995
Gravity and the Standard Model\
with $130$ $GeV$ Truth Quark\
from $D_{4}-D_{5}-E_{6}$ Model\
using $3 \times 3$ Octonion Matrices.\
Frank D. (Tony) Smith, Jr.\
e-mail: gt0109e@prism.gatech.edu\
and fsmith@pinet.aip.org\
P. O. Box for snail-mail:\
P. O. Box 430, Cartersville, Georgia 30120 USA\
[WWW URL http://www.gatech.edu/tsmith/home.html](http://www.gatech.edu/tsmith/home.html)\
School of Physics\
Georgia Institute of Technology\
Atlanta, Georgia 30332\
©1995 Frank D. (Tony) Smith, Jr., Atlanta, Georgia USA
Foreword. {#foreword. .unnumbered}
=========
In the $D_{4}-D_{5}-E_{6}$ mode, all tree level force strengths, particle masses, and K-M parameters can be calculated. They can be found at [WWW URL http://www.gatech.edu/tsmith/home.html](http://www.gatech.edu/tsmith/home.html) [@SMI6].
Within 10 % or so, they are consistent with currently accepted experimental values except for the truth quark mass. Acccording to the $D_{4}-D_{5}-E_{6}$ model, it should be $130$ $GeV$ at tree level, whereas CDF at Fermilab has interpreted their data to show that the truth quark mass is about $174 GeV$.
Fermilab’s announcement is at [WWW URL http://fnnews.fnal.gov/](http://fnnews.fnal.gov/) [@FNA].
My opinion is that the CDF interpretation is wrong, and that currently available experimental data indicates a truth quark mass in the range of $120-145$ $GeV$, which is consistent with the $130$ $GeV$ tree level calculation of the $D_{4}-D_{5}-E_{6}$ model.
Details of my opinion about the truth quark mass are at [WWW URL http://www.gatech.edu/tsmith/TCZ.html](http://www.gatech.edu/tsmith/TCZ.html) [@SMI6].
The $D_{4}-D_{5}-E_{6}$ model uses 3 copies of the octonions to represent the 8 first generation fermion particles, the 8 first generation fermion antiparticles, and the 8-dimensional (before dimensional reduction) spacetime.
This paper is an effort to describe the construction of the $D_{4}-D_{5}-E_{6}$ model by starting with elementary structures, and building the model up from them so that the interrelationships of the parts of the model may be more clearly understood.
The starting points I have chosen are the octonions, and $3 \times 3$ matrices of octonions [@FLY].
Since the properties of octonions are not well covered in most textbooks, the first section of the paper is devoted to a brief summary of some of the properties used in this paper.
$3 \times 3$ matrices are chosen because matrices are well covered in conventional math and physics texts, and also because $3 \times 3$ traceless octonion matrices can be used to build up in a concrete way the Lie algebras, Jordan algebras, and symmetric spaces that are needed to construct the $D_{4}-D_{5}-E_{6}$ model.
In particular, $3 \times 3$ traceless octonion matrices can be divided into hermitian and antihermitian parts to form a Jordan algebra $J_{3}^{\bf O}o$ and (when combined with derivations) a Lie algebra $F_{4}$.
Then $J_{3}^{\bf O}o$ and $F_{4}$ can be combined to form the Lie algebra $E_{6}$.
The components of the matrix representation of $E_{6}$ can then be used to construct the Lagrangian of the $D_{4}-D_{5}-E_{6}$ model of Gravity plus the Standard Model for first generation fermions.
By extension to the exceptional Lie algebras $E_{7}$ and $E_{8}$, the model includes the second and third generation fermions.
Since the $E$-series of Lie algebras contains only $E_{6}$, $E_{7}$ and $E_{8}$, there are only three generation of fermions in the model.
To get an overview of what this paper does, represent the $E_{6}$ Lie algebra by
$${\bf R} \otimes
\left(
\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)
\oplus G_{2} \oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)
\right)$$
where the [**R**]{} represents the real scalar field of this representation of $E_{6}$; ${\bf O}$ is octonion; $S^{7}$ represents the imaginary octonions; $a, b, c$ are real numbers (octonion real axis); and $G_{2}$ is the Lie algebra of derivations of ${\bf O}$.
Then, use the components of $E_{6}$ to construct the 8-dimensional Lagrangian of the $D_{4}-D_{5}-E_{6}$ model:
$$\int_{V_{8}} F_{8} \wedge \star F_{8} + \partial_{8}^{2}
\overline{\Phi_{8}} \wedge \star \partial_{8}^{2} \Phi_{8} +
\overline{S_{8\pm}} \not \! \partial_{8} S_{8\pm}$$
where $\star$ is the Hodge dual;
$\partial_{8}$ is the 8-dimensional covariant derivative, $\not \! \partial_{8}$ is the 8-dimensional Dirac operator, and
$F_{8}$ is the 28-dimensional $Spin(8)$ curvature, which come from the $Spin(0,8)$ gauge group subgroup of $E_{6}$, represented here by
$$\left(
\begin{array}{cc}
S^{7}_{1} & 0 \\
& \\
0 & S^{7}_{2}
\end{array}
\right)
\oplus G_{2}$$
$\Phi_{8}$ is the 8-dimensional scalar field, which comes from the ${\bf R}$ scalar part of the representation of $E_{6}$;
$V_{8}$ is 8-dimensional spacetime, and which comes from the ${\bf O}_{v}$ part of the representation of $E_{6}$;
$S_{8\pm}$ are the $+$ and $-$ 8-dimensional half-spinor fermion spaces, which come from the ${\bf O}_{+}$ and ${\bf O}_{-}$ parts of the representation of $E_{6}$;
As a theory with an 8-dimensional spacetime, the $D_{4}-D_{5}-E_{6}$ model is seen to be constructed from the fundamental representations of the $D_{4}$ Lie algebra $Spin(0,8)$:
$\Phi_{8}$ comes from the trivial scalar representation;
$F_{8}$ comes from the 28-dimensional adjoint representation;
$V_{8}$ comes from the 8-dimensional vector representation; and
$S_{8+}$ and $S_{8-}$ come from the two 8-dimensional half-spinor representations.
As all representations of $Spin(0,8)$ can be built by tensor products and sums using the 3 8-dimensional representations and the 28-dimensional adjoint representation, (the 4 of which make up the $D_{4}$ Dynkin diagram, which looks like a Mercedes-Benz 3-pointed star, with the 28-dimensional adjoint representation in the middle) plus the trivial scalar representation, the $D_{4}-D_{5}-E_{6}$ model effectively uses all the ways you can look at $Spin(0,8)$.
If you use exterior wedge products as well as tensor products and sums, you can build all the representations from only the representations on the exterior of the Dynkin diagram, in this case, the 3 points of the star, the 3 8-dimensional vector and half-spinor representations, plus the trivial scalar representation. In this case, it is clear because the adjoint representation is the bivector representation, or the wedge product of two copies of the vector representation. For discussion of more complicated Lie algebras, such as the $E_{8}$ Lie algebra, see Adams [@ADA2].
Our physical spacetime is not 8-dimensional, and the $D_{4}-D_{5}-E_{6}$ model gets to a 4-dimensional spacetime by a process of dimensional reduction.
Dimensional reduction of spacetime to 4 dimensions produces a realistic 4-dimensional Lagrangian of Gravity plus the Standard Model.
Then force strength constants and particle masses are calculated, as are Kobayashi-Maskawa parameters.
(The calculations are at tree level, with quark masses being constituent masses.)
The structures in this paper are exceptional in many senses, and can be studied from many points of view.
This paper is based on the structure of $3 \times 3$ matrices of octonions. For a discussion of $3 \times 3$ matrices of octonions from a somewhat different perspective, see Truini and Biedenharn [@TRU].
I have also looked at the $D_{4}-D_{5}-E_{6}$ model from the Clifford algebra point of view [@SMI3]. Some useful references to Clifford algebras include the books of Gilbert and Murray [@GILB], of Harvey [@HAR], and of Porteous [@POR].
From the point of view of the exceptional Lie algebra $F_{4}$ and the Cayley Moufang plane ${\bf O}P^{2}$, see the paper of Adams [@ADA1].
From the point of view of the 7-sphere, the highest dimensional sphere that is parallelizable, and the only parallelizable manifold that is not a Lie group, see papers by Cederwall and Preitschopf [@CED1], Cederwall [@CED2], Manogue and Schray [@MAN], and Schray and Manogue [@SCH].
From the point of view of Hermitian Jordan Triple systems, see papers by Günaydin [@GUN2; @GUN3] and my paper [@SMI2].
General references for differential geometry, Lie groups, and the symmetric spaces used herein are the books of Besse [@BES], Fulton and Harris [@FUL], Gilmore [@GIL], Helgason [@HEL1], Hua [@HUA], Kobayashi and Nomizu [@KOB], Porteous [@POR], and Postnikov [@POS], and the papers of Ramond [@RAM] and Sudbery [@SUD]. An interesting discussion of symmetries in physics is Saller [@SAL].
A notational comment - this paper uses the same notation for a Lie group as for its Lie algebra. It should be clear from context as to which is being discussed.
I would like to thank Geoffrey Dixon, Sarah Flynn, David Finkelstein, Tang Zhong, Igor Kulikov, Michael Gibbs, John Caputlu-Wilson, Ioannis Raptis, Julian Niles, Heinrich Saller, Ernesto Rodriguez, Wolfgang Mantke, and Marc Kolodner for very helpful discussions (some electronic) over the past year.
Octonions.
==========
There are 480 different ways to write a multiplication table for an octonion product. Since the purpose of this paper is to construct a concrete representation of the $D_{4}-D_{5}-E_{6}$ physics model, one of these is chosen and used throughout.
For a good introduction to octonions, see the books of Geoffrey Dixon [@DIX4] and of Jaak Lohmus, Eugene Paal, and Leo Sorgsepp [@LOH], as well as the paper of Günaydin and Gürsey [@GUN1].
For many more interesting things about octonions, see the book and papers of Geoffrey Dixon [@DIX1; @DIX2; @DIX3; @DIX4; @DIX5; @DIX6; @DIX7].
The following description of octonion products, left and right actions, and automorphisms and derivations, is taken from Dixon’s book and papers cited above, and from a paper by A. Sudbery [@SUD].
Octonion Product.
-----------------
For concreteness, choose one of the 480 multiplications: Let $e_{a}, a=1,...,7$, represent the imaginary units of [**O**]{}, and adopt the cyclic multiplication rule $$e_{a}e_{a+1}=e_{a+5} = e_{a-2},$$ a=1,...,7, all indices modulo 7, from 1 to 7 (another cyclic multiplication rule for [**O**]{}, dual to that above, is $e_{a}e_{a+1}=e_{a+3} = e_{a-4}$). In particular, $$\{q_{1} \rightarrow e_{a}, q_{2} \rightarrow e_{a+1},
q_{3} \rightarrow e_{a+5}\}$$ define injections of [**Q**]{} into [**O**]{} for a=1,...,7. In the multiplication rule Equation (4) the indices range from 1 to 7, and the index 0 representing the octonion real number $1$ is not subject to the rule. This octonion multiplication has some very nice properties.
For example, $$\mbox{if } e_{a}e_{b}=e_{c}, \mbox{ then } e_{(2a)}e_{(2b)}=e_{(2c)}.$$ Equation (6) in combination with Equation (4) immediately implies $$\begin{aligned}
& e_{a}e_{a+2}=e_{a+3}, \nonumber \\
& e_{a}e_{a+4}=e_{a+6}\end{aligned}$$ (so $e_{a}e_{a+2^{n}}=e_{a-2^{n+1}}$, or $e_{a}e_{a+b} = [b^{3} \mbox{ mod } 7]e_{a-2b^{4}}$, $b=1,...,6$, where $b^{3}$ out front provides the sign of the product (modulo 7, $1^{3}=2^{3}=4^{3}=1$, and $3^{3}=5^{3}=6^{3}=-1$ )). Also, 2(7)=7 mod 7, so Equations (6) and (4) imply $$\begin{array}{ccc}
e_{7}e_{1}=e_{5}, & e_{7}e_{2}=e_{3}, & e_{7}e_{4}=e_{6}. \\
\end{array}$$ These modulo 7 periodicity properties are reflected in the full multiplication table: $$\left( \begin{array} {cccccccc}
1 & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7}\\
e_{1}&-1&e_{6}&e_{4}&-e_{3}&e_{7}&-e_{2}&-e_{5}\\
e_{2}&-e_{6}&-1&e_{7}&e_{5}&-e_{4}&e_{1}&-e_{3}\\
e_{3}&-e_{4}&-e_{7}&-1&e_{1}&e_{6}&-e_{5}&e_{2}\\
e_{4}&e_{3}&-e_{5}&-e_{1}&-1&e_{2}&e_{7}&-e_{6}\\
e_{5}&-e_{7}&e_{4}&-e_{6}&-e_{2}&-1&e_{3}&e_{1}\\
e_{6}&e_{2}&-e_{1}&e_{5}&-e_{7}&-e_{3}&-1&e_{4}\\
e_{7}&e_{5}&e_{3}&-e_{2}&e_{6}&-e_{1}&-e_{4}&-1\\
\end{array} \right).$$ Although the octonion product is nonassociative, it is alternative.
An example of nonassociativity from the multiplication table is that $$(e_{1}e_{2})e_{4} = e_{6}e_{2} = -e_{7}$$ is not equal to $$e_{1}(e_{2}e_{4}) = e_{1}e_{5} = e_{7}$$
The alternativity property is the fact that the associator $$[x,y,z] = x(yz) - (xy)z$$ is an alternating function of $x,y,z \in \bf{O}$.
Left and Right Adjoint Algebras of **O.**
-----------------------------------------
The octonion algebra is nonassociative and so is not representable as a matrix algebra.
However, the adjoint algebras of left and right actions of [**O**]{} on itself are associative.
For example, let $u_{1},...,u_{n},x$ be elements of [**O**]{}. Consider the left adjoint map $$x \rightarrow u_{n}(...(u_{2}(u_{1}x))...).$$ The nesting of parentheses forces the products to occur in a certain order, hence this algebra of left-actions is trivially associative, and it is representable by a matrix algebra.
One such representation can be derived immediately from the multiplication table Equation (9). For example, the actions $$x \rightarrow e_{1}x \equiv e_{L1}(x), \mbox{ and
} x \rightarrow xe_{1} \equiv e_{R1}(x)$$ can be identified with the matrices $$e_{L1} \rightarrow \left( \begin{array} {cccccccc}
.&-1&.&.&.&.&.&.\\
1&.&.&.&.&.&.&.\\
.&.&.&.&.&.&-1&.\\
.&.&.&.&-1&.&.&.\\
.&.&.&1&.&.&.&.\\
.&.&.&.&.&.&.&-1\\
.&.&1&.&.&.&.&.\\
.&.&.&.&.&1&.&.\\
\end{array} \right),$$ $$e_{R1} \rightarrow \left( \begin{array} {cccccccc}
.&-1&.&.&.&.&.&.\\
1&.&.&.&.&.&.&.\\
.&.&.&.&.&.&1&.\\
.&.&.&.&1&.&.&.\\
.&.&.&-1&.&.&.&.\\
.&.&.&.&.&.&.&1\\
.&.&-1&.&.&.&.&.\\
.&.&.&.&.&-1&.&.\\
\end{array} \right)$$ (only nonzero entries are indicated). Note that $e_{2}e_{6}=e_{3}e_{4}=e_{5}e_{7}=e_{1}$, but because of the nonassociativity of [**O**]{}, for example, $$e_{1}x = (e_{2}e_{6})x \ne e_{2}(e_{6}x) \equiv e_{L26}(x)$$ in general.
In the octonion algebra, any product from the right can be reproduced as the sum of products from the left, and visa versa.
The left and right adjoint algebras ${\bf O}_{L}$ and ${\bf O}_{R}$ are the same algebra, and this algebra is larger than [**O**]{} itself. In fact, it is isomorphic to [**R**]{}(8).
It is not difficult to prove that $$e_{La...bc...d}= -e_{La...cb...d}$$ if $b \neq c$, all indices from 1 to 7.
So,for example, $e_{1}(e_{2}(e_{3}x)) = -e_{1}(e_{3}(e_{2}x))
= e_{3}(e_{1}(e_{2}x))$.
In addition, $$e_{Lab...pp...c} = -e_{Lab...c}$$ (cancellation of like indices).
Together with $$e_{L7654321} = {\bf 1},$$ Equations (16) and (17) imply that a complete basis for the left=right adjoint algebra of [**O**]{} consists of elements of the form $${\bf 1}, e_{La}, e_{Lab}, e_{Labc},$$ [**1**]{} the identity.
This yields 1+7+21+35=64 as the dimension of the adjoint algebra of [**O**]{}, also the dimension of [**R**]{}(8). $${\bf O}_{L}={\bf O}_{R}={\bf R}(8).$$ The 8-dimensional [**O**]{} itself is the object space of the adjoint algebra.
Aut(O) and Der(O).
------------------
The 14-dimensional exceptional Lie group $G_{2}$ is the automorphism group $Aut(\bf{O})$ of the octonions.
When viewed from the point of view of a linear Lie algebra with bracket product rather than the point of view of a nonlinear global Lie group with group product, the structure that corresponds to the automorphism group $Aut(\bf{O})$ is the algebra of derivations $Der(\bf{O})$ of the octonions.
A derivation is a linear map $D:\bf{O} \rightarrow \bf{O}$ such that for $x,y \in \bf{O}$: $$D(xy) = (Dx)y + x(Dy)$$ Then, from the alternative property of the octonions: $$D(ab)x = [a,b,x] + (1/3)[[a,b],x]$$
Let $C_{d} = L_{d} - R_{d}$, where $L_{d}$ and $R_{d}$ denote left and right multiplication by an imaginary octonion $d$. The imaginary octonions $Im(\bf{O})$ are those octonions in the space orthogonal to the octonion real axis. Then $$D(ab) = (1/6)([C_{a},C_{b}] + C_{[a,b]})$$
The algebra of derivations $Der(\bf{O})$ of the octonions is the Lie algebra $G_{2}$.
A basis for the Lie algebra of $G_{2}$ is represented in ${\bf O}_{L}$ by: $$\{e_{Lab}-e_{Lcd} : e_{a}e_{b} = e_{c}e_{d}\} \rightarrow G_{2}.$$ In ${\bf O}_{R}$ the basis is much the same: $$\{e_{Rab}-e_{Rcd} : e_{a}e_{b} = e_{c}e_{d}\} \rightarrow G_{2}.$$ The stability group of any fixed octonion direction is the 8-dimensional $SU(3) \subset G_{2}$.
A basis for the Lie algebra of the stability group of $e_{7}$ is: $$\{e_{Lab}-e_{Lcd} \in G_{2} : a,b,c,d \ne 7\} \rightarrow su(3).$$ Thus $SU(3)$ is the intersection of $G_{2}$ with $Spin(6)$, and $G_{2} = SU(3) \oplus S^{6}$.
The algebra of derivations does not give the Lie algebra all the antisymmetric maps of a real division algebra $\bf{R, C, H, O}$ unless the algebra is commutative and associative, i.e., $\bf{R, C}$. In the case of $\bf{C}$, the Lie algebra of all antisymmetric maps is $$Der({\bf{C}}) = Spin(2) = U(1)$$.
For $\bf{H}$, which is associative but not commutative, $$Der({\bf{H}}) \subset L_{Im({\bf{H}})} \oplus R_{Im({\bf{H}})}$$ where $L_{Im({\bf{H}})}$ and $R_{Im({\bf{H}})}$ denote left and right multiplication on the quaternions by the imaginary quaternions. $L_{Im({\bf{H}})}$ is isomorphic to and commutes with $R_{Im({\bf{H}})}$. Since $L_{Im({\bf{H}})} = R_{Im({\bf{H}})} =
Spin(3) = SU(2) = Sp(1) = S^{3}$, the Lie algebra $Spin(4)$ of all antisymmetric maps of $\bf{H}$ is given by $$Spin(4) = Spin(3) \oplus Spin(3)$$
For $\bf{O}$, which is neither associative nor commutative, the vector space of all antisymmetric maps of $\bf{O}$ is given by $$Spin(0,8) = Der({\bf{O}}) \oplus L_{Im({\bf{O}})} \oplus
R_{Im({\bf{O}})} =
G_{2} \oplus S^{7} \oplus S^{7}$$ where $S^{7}$ represents the imaginary octonions, notation suggested by the fact that the unit octonions are the 7-sphere $S^{7}$ which, since it is parallelizable, is locally representative of the imaginary octonions.
The 7-sphere $S^{7}$ has a left-handed basis $\{e_{La}\}$ and a right-handed basis $\{e_{Ra}\}$.
Unlike the 3-sphere, which is the Lie group $Spin(3) = SU(2)
= Sp(1)$, $S^{7}$ does not close under the commutator bracket product because $[e_{La},e_{Lb}]/2 = e_{Lab}$ and $[e_{Ra},e_{Rb}]/2 = e_{Rab}$ for $a \neq b$.
To make a Lie algebra out of $S^{7}$, it must be extended to $Spin(0,8)$ by adding the 21 basis elements $\{e_{Lab}\}$ or $\{e_{Rab}\}$.
There are three ways to extend $S^{7}$ to the Lie algebra $Spin(0,8)$.
They result in the left half-spinor, right half-spinor, and vector representations of $Spin(0,8)$: $$\begin{array}{c}
\{e_{La},e_{Lbc}\} \rightarrow Spin(0,8), \mbox{left half-spinor}, \\
\{e_{Ra},-e_{Rbc}\} \rightarrow Spin(0,8), \mbox{right half-spinor}, \\
\{e_{La}+e_{Ra}, e_{Lbc} - e_{Ra}:e_{a}=e_{b}e_{c}\} \rightarrow
Spin(0,8), \mbox{vector}. \\
\end{array}$$ Therefore, the two half-spinor representations and the vector representation of $Spin(0,8)$ all have 8-dimensional [**O**]{} as representation space.
The three representations are isomorphic by triality.
Consider the 28 basis elements $\{e_{La}+e_{Ra}, e_{Lbc} - e_{Ra}:
e_{a}=e_{b}e_{c}\}$ of the vector representation of $Spin(0,8)$: The 21-element subset $\{e_{Lbc} - e_{Ra}:e_{a}=e_{b}e_{c}\}$ is a basis for the Lie algebra $Spin(7) = G_{2} \oplus S^{7}$.
Therefore, the Lie algebra $Spin(0,8) = S^{7} \oplus G_{2}
\oplus S^{7}$.
$3 \times 3$ Octonion Matrices and $E_{6}$.
===========================================
The $D_{4}-D_{5}-E_{6}$ model of physics uses 3 copies of the octonions: $\bf O_{v}$ to represent an 8-dimensional spacetime (prior to dimensional reduction to 4 dimensions); $\bf O_{+}$ to represent the 8 first-generation fermion +half-spinor particles ; and $\bf O_{-}$ to represent the 8 first-generation fermion -half-spinor antiparticles.
Consider the 72-dimensional space of $3 \times 3$ matrices of octonions: $$\left(
\begin{array}{ccc}
{\bf O}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf X} & {\bf O}_{2} & {\bf O}_{-} \\
& & \\
{\bf Z} & {\bf Y} & {\bf O}_{3}
\end{array}
\right)$$ where ${\bf O}_{v}, {\bf O}_{+}, {\bf O}_{-},{\bf O}_{1}, {\bf O}_{2},
{\bf O}_{3}, {\bf X}, {\bf Y}, {\bf Z}$ are octonion, ${\bf O}_{1} = a + S^{7}_{1}, {\bf O}_{2} = b + S^{7}_{2},
{\bf O}_{3} = c + S^{7}_{3}$ $a, b, c$ are real, and $S^{7}_{1}, S^{7}_{2}, S^{7}_{3}$ are imaginary octonion.
Consider the ordinary matrix product $AB$ of two $3 \times 3$ octonion matrices $A$ and $B$.
Now, to construct the Lie algebra $E_{6}$ from $3 \times 3$ octonion matrices, it is useful to split the product $AB$ into antisymmetric and symmetric parts. $$AB = (1/2)(AB - BA) + (1/2)(AB + BA)$$
This will enable us to construct an $F_{4}$ Lie algebra from antiHermitian matrices that will arise from considering the antisymmetric product, and a $J_{3}^{\bf O}o$ Jordan algebra from hermitian matrices that will arise from considering the symmetric product.
Then the Lie algebra $F_{4}$ and the Jordan algebra $J_{3}^{\bf O}o$ will be combined to form the Lie algebra $E_{6}$ whose structure forms the basis of the $D_{4}-D_{5}-E_{6}$ model.
This construction is only possible in this case because of many exceptional structures and symmetries. The $D_{4}-D_{5}-E_{6}$ model therefore inherits remarkable symmetry structures
Antihermitian Matrices and Lie Algebras.
----------------------------------------
Consider the antisymmetric product $(1/2)(AB - BA)$: The 45-dimensional space of antihermitian $3 \times 3$ octonion matrices does not close under the antisymmetric product to form a Lie algebra (here, $\dagger$ denotes octonion conjugation): $$\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & S^{7}_{3}
\end{array}
\right)$$ A product that closes is $$(1/2)(AB - BA - Tr(AB - BA))$$ The form of the product indicates that to get closure, you have to use only the 45-7 = 38-dimensional space of traceless antihermitian $3 \times 3$ octonion matrices: $$\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)$$ However, the Jacobi identity is not satisfied, so you still do not have a Lie algebra.
As was mentioned in Section 1.3, to get the Lie algebra of all antisymmetric maps of the nonassociative, noncommutative real division algebra ${\bf{O}}$ you must include the 14-dimensional Lie algebra of derivations $Der({\bf{O}}) = G_{2}$.
Adding the derivations to the product that closes gives a product that not only closes but also satisfies the Jacobi identity: $$(1/2)(AB - BA - Tr(AB -BA) \oplus D(A,B))$$ where the derivation $D(A,B) = \sum_{ij} D(a_{ij},b_{ij})$ and the derivation $D(x,y)$ acts on the octonion $z$ by using the alternator $[x,y,z] = D(x,y)z$.
The resulting space is the 38+14 = 52-dimensional Lie algebra $F_{4}$. $$\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)
\oplus G_{2}$$ The physical interpretation of this representation of $F_{4}$ in the $D_{4}-D_{5}-E_{6}$ model is:
${\bf O}_{v}$ is 8-dimensional spacetime before dimensional reduction to 4 dimensions;
${\bf O}_{+}$ is the 8-dimensional space representing the first generation fermion particles (the neutrino, the electron, the red, blue and green up quarks, and the red, blue and green down quarks);
${\bf O}_{-}$ is the 8-dimensional space representing the first generation fermion antiparticles before dimensional reduction creates 3 generations; and
$S^{7}_{1} \oplus S^{7}_{2} \oplus G_{2}$ is the $Spin(0,8)$ gauge group before dimensional reduction breaks it down into gravity plus the Standard Model.
Now, consider the symmetric Jordan product and the hermitian matrices that form an algebra under it.
In the next subsection, they will be studied so that their structure can be added to the $F_{4}$ structure of traceless antihermitian matrices plus the derivations $G_{2}$.
Hermitian Matrices and Jordan Algebras.
---------------------------------------
Consider the symmetric product $(1/2)(AB + BA)$:
The 27-dimensional space of hermitian $3 \times 3$ octonion matrices closes under the symmetric product to form the Jordan algebra $J_{3}^{O}$: $$\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & c
\end{array}
\right)$$ Even though the full 27-dimensional space of hermitian matrices forms a Jordan algebra $J_{3}^{O}$, only the 26-dimensional traceless subalgebra $J_{3}^{O}o$ is acted on by $F_{4}$ as its representation space: $$\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ The 52-dimensional $F_{4}$ and the 26-dimensional $J_{3}^{O}o$ combine to form the 78-dimensional Lie algebra $E_{6}$,
$$\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)
\oplus G_{2} \oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$
the $E_{6}$ of the $D_{4}-D_{5}-E_{6}$ model.
$E_{6}$ preserves the cubic determinant pseudoscalar 3-form for $3 \times 3$ octonionic matrices (see [@FUL; @CHE]).
Sudbery [@SUD] has identified $E_{6}$ with $SL(3,{\bf O})$.
Flynn [@FLY], in the context of her physics models, has used such an identification to note the similarity of $E_{6}$ to $SL(3,{\bf C})$, which is also made up of an antisymmetric part, $SU(3)$, plus a symmetric part, an 8-dimensional Jordan algebra, and which preserves a cubic determinant.
Shilov Boundaries of Complex Domains.
=====================================
In the $D_{4}-D_{5}-E_{6}$ model, physical spacetime and the physical spinor fermion representation manifold are Shilov boundaries of bounded complex domains.
The best general reference to Shilov boundaries is Helgason’s 1994 book [@HEL2].
The best set of calculations of volumes, etc., of Shilov boundaries is Hua’s book [@HUA].
That means, for instance, that physical 8-dimensional spacetime is the 8-real dimensional Shilov boundary of of a 16-real dimensional (8-complex-dimensional) bounded complex domain.
To physicists, the most familiar example of bounded complex domains and their Shilov boundaries (other than the unit disk and its Shilov boundary, the circle) probably comes from the twistors of Roger Penrose [@PEN]. Another example, possibly less familiar, is the chronometry theory of I. E. Segal [@SEG] at M.I.T. Still another example, also probably not very familiar, is the use of the geometry of bounded complex domains by Armand Wyler [@WYL] in his effort to calculate the value of the fine structure constant.
To mathematicians, such structures are well known. A standard general reference is the book of Hua [@HUA], in which Shilov boundaries are called characteristic manifolds. Actually, I would prefer the term characteristic boundary, because it would describe it as being part of a boundary that characterizes important structures on the manifold. However, I will use the term Shilov boundary because that seems to be the dominant term in English-language literature.
The simplest example (a mathematical object that Prof. Feller [@FEL] said was the best all-purpose example in mathematics for understanding new concepts) is the unit disk along with its harmonic functions. The unit disk is the bounded complex domain, the unit circle is its Shilov boundary, and the harmonic functions are determined throughout the unit disk by their values on the Shilov boundary.
A more complicated example of such structures, taken from the works of those mentioned above, starts with an 8-real-dimensional 4-complex-dimensional space denoted ${\cal M}^{\bf C}$ with signature $(2,6)$.
[**What are Bounded Complex Homogeneous Domains?**]{}
To see the second example, start with the complexified Minkowski spacetime ${\cal M}_{2,6}^{\bf C}$ of Penrose twistor theory.
${\cal M}^{\bf C}$ is a Hermitian symmetric space that is the coset space $$Spin(2,4) / Spin(1,3) \times U(1)$$. In the mathematical classification notation, it is called a Hermitian symmetric space of type $BDI_{2,4}$.
The Hermitian symmetric coset space is unbounded, but for each Hermitian symmetric space there exists a natural corresponding bounded complex domain.
In this case, $BCI_{2,4}$, the bounded complex domain is called $Type \; IV_{4}$ and consists of the elements of ${\bf C}_{4}$ defined by $$\{ z_{1} ,z_{2}, z_{3} ,z_{4} \mid
|z_{1}|^{2} + |z_{2}|^{2} + |z_{3}|^{2} + |z_{4}|^{2} <
(1 + |z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2}|) / 2 < 1 \}$$
[**What are Shilov boundaries?**]{}
The Shilov boundary (called the characteristic manifold by Hua in [@HUA]) is a subset of the topological boundary of the bounded complex domain.
Following Hua [@HUA], consider the analytic functions on the bounded complex domain. The Shilov boundary is the part of the topological boundary on which every analytic function attains its maximum modulus, and such that for every point on the Shilov boundary, there exists an analytic function on the bounded complex domain that attains its maximum modulus at that point.
The Shilov boundary is closed. Any function which is analytic in the neighborhood of every point of the Shilov boundary is uniquely determined by its values on the Shilov boundary.
In the case of the bounded domain of $Type \; IV_{4}$ Hua [@HUA] shows that the Shilov boundary is $$\{ z = e^{i \theta} x \mid 0 \leq \theta \leq \pi,
x \overline{x} = 1 \} = {\bf R}P^{1} \times S^{3}$$
In the Penrose twistor formalism, it is the 4-dimensional Minkowski spacetime ${\cal M}_{1,3}$ with signature $(1,3)$
There exists a kernel function, the Poisson kernel $P(z, \xi)$ function of a point z in the bounded complex domain and a point $\xi$ in its Shilov boundary, such that, for any analytic $f(z)$, $$f(z) = \int_{Shilov bdy} P(z, \xi) f(\xi)$$ Since all the analytic functions in the bounded complex domain are determinmed by their values on the Shilov boundary,the Shilov boundary should be the proper domain of definition of physically relevant functions. That is why the $D_{4}-D_{5}-E_{6}$ model takes the Shilov boundary to be the relevant manifold for spacetime and for representing fermion particles and antiparticles.
There exists another kernel function, the Bergman kernel function $K(z, \overline{w})$ of two points $z, w$ in the bounded complex domain such that, for any analytic $f(z)$, $$f(z) = \int_{domain} K(z, \overline{w}) f(w)$$
Setting $z = w$ in the Bergman kernel gives a Riemannian metric for the bounded complex domain, which in turn defines invariant differential operators including the Laplacian, which in turn gives harmonic functions.
The Bergman kernel is equal to the ratio of the volume density to the Euclidean volume of the bounded complex domain.
Hua [@HUA] not only gives the above description, he also actually calculates the volumes of the bounded complex domains and their Shilov boundaries.
Suppose that, as in the $D_{4}-D_{5}-E_{6}$ model, different bounded complex domains represent different physical forces whose Green’s functions are determined by their invariant differential operators. Then, since the domain volumes represent the measures of the Bergman kernels of bounded complex domains, and since the physical part of the domain is its Shilov boundary, the ratios of (suitably normalized) volumes of Shilov boundaries should (and do, in the $D_{4}-D_{5}-E_{6}$ model) represent the ratios of force strengths of the corresponding forces.
$E_{6} / (D_{5} \times U(1))$ and $D_{5} / (D_{4} \times U(1))$.
================================================================
2x2 Octonion Matrices and D5.
-----------------------------
The $3 \times 3$ octonion traceless antihermitian checkerboard matrices form a subalgebra of the $3 \times 3$ octonion traceless antihermitian matrices. $$\left(
\begin{array}{ccc}
S^{7}_{1} & 0 & {\bf O}_{v} \\
& & \\
0 & S^{7}_{2} & 0 \\
& & \\
-{\bf O}_{v}^\dagger & 0 & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)$$ It is isomorphic to the algebra of $2 \times 2$ octonion antihermitian matrices: $$\left(
\begin{array}{cc}
S^{7}_{1} & {\bf O}_{v} \\
& \\
-{\bf O}_{v}^\dagger & S^{7}_{2}
\end{array}
\right)$$ When $Der({\bf{O}}) = G_{2}$ is added the result is a subalgebra of the Lie algebra $F_{4}$: $$\left(
\begin{array}{cc}
S^{7}_{} & {\bf O}_{v} \\
& \\
-{\bf O}_{v}^\dagger & S^{7}_{2}
\end{array}
\right)
\oplus G_{2}$$ This Lie algebra is 8+7+7+14 = 36-dimensional $Spin(9)$, also denoted $B_{4}$. $Spin(9)$ is to ${\bf{O}}$ as $SU(2)$ is to ${\bf{C}}$.
The checkerboard $3 \times 3$ octonion traceless hermitian matrices also form a subalgebra of the $3 \times 3$ octonion traceless hermitian matrices: $$\left(
\begin{array}{ccc}
a & 0 & {\bf O}_{v} \\
& & \\
0 & b & 0 \\
& & \\
{\bf O}_{v}^\dagger & 0 & -a-b
\end{array}
\right)$$ It is isomorphic to the 10-dimensional Jordan algebra $J_{2}^{\bf{O}}$ of $2 \times 2$ octonion hermitian matrices: $$\left(
\begin{array}{cc}
a & {\bf O}_{v} \\
& \\
{\bf O}_{v}^\dagger & b
\end{array}
\right)$$ $J_{2}^{\bf{O}}$ has a 9-dimensional traceless Jordan subalgebra $J_{2}^{\bf{O}}o$, and $$J_{2}^{\bf{O}} = J_{2}^{\bf{O}}o \oplus U(1)$$ The 1-dimensional Jordan algebra $J_{1}^{\bf{C}}o$ corresponds to the Lie algebra $U(1)$.
The 36-dimensional $B_{4}$ and the 9-dimensional $J_{2}^{\bf{O}}o$ combine to form the 45-dimensional Lie algebra $D_{5}$, also denoted $Spin(10)$, the $D_{5}$ of the $D_{4}-D_{5}-E_{6}$ model.
Physically, $D_{5}$ contains the gauge group and spacetime parts of $E_{6}$, so the half-spinor fermion particle and antiparticle parts of $E_{6}$ should live in a coset space that is a quotient of $E_{6}$ by $D_{5}$.
Due to complex structure, the quotient must be taken by $D_{5} \times U(1)$ rather than $D_{5}$ alone. The $U(1)$ comes from action on the Jordan algebra $J_{1}^{\bf{C}}o$.
The resulting symmetric space that is the representation space of the first-generation particles and antiparticles in the $D_{4}-D_{5}-E_{6}$ model is the 78-45-1 = 32-real-dimensional hermitian symmetric space $$E_{6} / (D_{5} \times U(1))$$ The 16 particles and antiparticles live on the 16-real-dimensional Shilov boundary of the bounded complex domain that corresponds to the symmetric space. The Shilov boundary is $$(S^{7} \times {\bf{R}}P^{1}) \oplus (S^{7} \times {\bf{R}}P^{1})$$ There is one copy of $S^{7} \times {\bf{R}}P^{1}$ for the 8 first-generation fermion half-spinor particles, and one for the 8 antiparticles.
2x2 Diagonal Octonion Matrices and D4.
--------------------------------------
The $2 \times 2$ octonion antihermitian checkerboard matrices form the diagonal matrix subalgebra of the $2 \times 2$ octonion antihermitian matrices. $$\left(
\begin{array}{cc}
S^{7}_{1} & 0\\
& \\
0 & S^{7}_{2}
\end{array}
\right)$$ When $Der({\bf{O}}) = G_{2}$ is added the result is a subalgebra of the Lie algebras $B_{4}$ and $F_{4}$: $$\left(
\begin{array}{cc}
S^{7}_{1} & 0\\
& \\
0 & S^{7}_{2}
\end{array}
\right)
\oplus G_{2}$$ This Lie algebra is 7+7+14 = 28-dimensional $Spin(0,8)$, also denoted $D_{4}$.
The checkerboard $2 \times 2$ octonion traceless hermitian matrices also form the diagonal subalgebra of the $2 \times 2$ octonion traceless hermitian matrices: $$\left(
\begin{array}{cc}
a & 0\\
& \\
0 & 0
\end{array}
\right)$$ It is isomorphic to the 1-dimensional Jordan algebra $J_{1}^{\bf{C}}o$.
The 1-dimensional Jordan algebra $J_{1}^{\bf{C}}o$ corresponds to the Lie algebra $U(1)$.
The 28-dimensional $D_{4}$ Lie algebra, also denoted $Spin(0,8)$, is (before dimensional reduction) the smallest Lie algebra in the $D_{4}-D_{5}-E_{6}$ model.
Physically, $D_{4}$ contains only the gauge group parts of $D_{5}$ and $E_{6}$, so the 8-dimensional spacetime of $D_{5}$ and $E_{6}$ should live in a coset space that is a quotient of $D_{5}$ by $D_{4}$.
Due to complex structure, the quotient must be taken by $D_{4} \times U(1)$ rather than $D_{4}$ alone. The $U(1)$ comes from action on the Jordan algebra $J_{1}^{\bf{C}}o$.
The resulting symmetric space that is the representation space of the 8-dimensional spacetime in the $D_{4}-D_{5}-E_{6}$ model is the 45-28-1 = 16-real-dimensional hermitian symmetric space $$D_{5} / (D_{4} \times U(1))$$ The physical (before dimensional reduction) spacetime lives on the 8-real-dimensional Shilov boundary of the bounded complex domain that corresponds to the symmetric space. The Shilov boundary is $$S^{7} \times {\bf{R}}P^{1}$$
Global $E_{6}$ Lagrangian.
==========================
Represent the $E_{6}$ Lie algebra by
$${\bf R} \otimes
\left(
\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)
\oplus G_{2} \oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)
\right)$$
where the [**R**]{} represents the real scalar field of this representation of $E_{6}$.
Full $E_{6}$ symmetry of the $D_{4}-D_{5}-E_{6}$ model is a global symmetry, and is useful primarily to: define the spacetime, fermions, and bosons; define the tree-level relative force strengths and particle masses; give generalized supersymmetric relationships among fermions and bosons that may be useful in loop cancellations to produce ultraviolet and infrared finite results; give $CPT$ symmetry, which involves both particle $C$ and spacetime $P$ and $T$ symmetries. Some more discussion of $CPT$, $CP$, and $T$ can be found at [WWW URL http://www.gatech.edu/tsmith/CPT.html](http://www.gatech.edu/tsmith/CPT.html) [@SMI6].
Dynamics of the $D_{4}-D_{5}-E_{6}$ model are given by a Lagrangian action that is the integral over spacetime of a Lagrangian density made up of a gauge boson curvature term, a spinor fermion term (including through a Dirac operator interaction with gauge bosons), and a scalar term.
The 8-dimensional Lagrangian is:
$$\int_{V_{8}} F_{8} \wedge \star F_{8} + \partial_{8}^{2}
\overline{\Phi_{8}} \wedge \star \partial_{8}^{2} \Phi_{8} +
\overline{S_{8\pm}} \not \! \partial_{8} S_{8\pm}$$
where $\star$ is the Hodge dual;
$\partial_{8}$ is the 8-dimensional covariant derivative, $\not \! \partial_{8}$ is the 8-dimensional Dirac operator, and $F_{8}$ is the 28-dimensional $Spin(8)$ curvature, which come from the $Spin(0,8)$ gauge group subgroup of $E_{6}$, represented here by
$$\left(
\begin{array}{cc}
S^{7}_{1} & 0 \\
& \\
0 & S^{7}_{2}
\end{array}
\right)
\oplus G_{2}$$
$\Phi_{8}$ is the 8-dimensional scalar field, which comes from the ${\bf R}$ scalar part of the representation of $E_{6}$;
$V_{8}$ is 8-dimensional spacetime, and which comes from the ${\bf O}_{v}$ part of the representation of $E_{6}$;
$S_{8\pm}$ are the $+$ and $-$ 8-dimensional half-spinor fermion spaces, which come from the ${\bf O}_{+}$ and ${\bf O}_{-}$ parts of the representation of $E_{6}$;
The 8-dimensional Lagrangian is a classical Lagrangian.
To get a quantum action, the $D_{4}-D_{5}-E{6}$ model uses a path integral sum over histories. At its most fundamental level, the $D_{4}-D_{5}-E{6}$ model is a lattice model with 8-dimensional $E_{8}$ lattice spacetime.
The path integral sum over histories is based on a generalized Feynman checkerboard scheme over the $E_{8}$ lattice spacetime.
The original Feynman checkerboard is in 2-dimensional spacetime, in which the speed of light is naturally $c$ = $1$.
In the 4-dimensional spacetime of the $D_{4}$ lattice, the speed of light is naturally $c$ = $\sqrt{3}$.
In 8-dimensional spacetime, the speed of light is naturally $c$ = $\sqrt{7}$, but in the 8-dimensional $E_{8}$ lattice the nearest neighbor vertices have only 4 (not 8) non-zero coordinates and therefore have a natural speed of light of $c$ = $\sqrt{3}$ that is appropriate for 4-dimensional light-cones rather than for 8-dimensional lightcones with $c$ = $\sqrt{7}$.
This means that:
the generalized Feynman checkerboard scheme will not produce lightcone paths in 8-dimensional spacetime;
the dimension of spacetime must be reduced to 4 dimensions, as discussed in Section 7.; and the 8-dimensional Lagrangian is classical, and does not contain gauge fixing and ghost terms.
Gauge-fixing term and ghost terms only appear after dimensional reduction of spacetime permits construction of generalized Feynman checkerboard quantum path integral sums over histories.
The gauge fixing terms are needed to avoid overcounting gauge-equivalent paths, and ghosts then appear.
The $D_{4} = Spin(0,8)$ gauge symmetry will be reduced to gravity plus the Standard Model by dimensional reduction. The gauge symmetries of gravity and the Standard Model will be broken by gauge-fixing and ghost terms, and the resulting Lagrangian will have a BRS symmetry.
The dynamical symmetry that is physically relevant at the initial level is the $D_{4} = Spin(0,8)$ gauge symmetry of the Lagrangian.
It is clear that $Spin(0,8)$ acts through its 8-dimensional vector, +half-spinor, and -half-spinor representations on the ${\bf O}_{v}$, ${\bf O}_{+}$, and ${\bf O}_{-}$ parts of the antihermitian matrix $$\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)$$
It is also clear that $Spin(0,8)$ acts through its 28-dimensional adjoint representation on the $S^{7}_{1}$ and $S^{7}_{2}$ parts of the antihermitian matrix, plus the derivations $G_{2}$, since $$Spin(0,8) = S^{7}_{1} \oplus S^{7}_{2} \oplus G_{2}$$
On the hermitian Jordan algebra matrix $J_{3}^{{\bf{O}}}o$: $$\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ the action of $Spin(0,8)$ (as a subgroup of $E_{6}$ and $F_{4}$) leaves invariant each of the ${\bf O}_{v}$, ${\bf O}_{+}$, and ${\bf O}_{-}$ parts of the matrix, corresponding to the facts that the gauge group $Spin(0,8)$:
does not act as a generalized supersymmetry to interchange spacetime and fermions; and
does not interchange fermion particles and antiparticles.
3 Generations: $E_{6}$, $E_{7}$, and $E_{8}$.
=============================================
In the $D_{4}-D_{5}-E_{6}$ model, the first generation of spinor fermions is represented by the octonions ${\bf O}$, the second by ${\bf O} \oplus {\bf O}$, and the third by ${\bf O} \oplus {\bf O} \oplus {\bf O}$. The global structure of the $D_{4}-D_{5}-E_{6}$ model with 8-dimensional spacetime and first generation fermions is given by $$E_{6} = F_{4} \oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ where $$\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)
= J_{3}^{\bf O}o$$ Here, the ${\bf O}_{+}$ and ${\bf O}_{}$ in $J_{3}^{\bf O}o$ represent the ${\bf O}$ first generation fermion particles and antiparticles. Using the octonion basis $\{ 1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7} \}$ with quaternionic subalgebra basis $\{ 1, e_{1}, e_{2}, e_{6} \}$ and with octonion product Equation (4), the representation is:
$$\begin{array}{|c|c|} \hline
Octonion & Fermion \: Particle \\
basis \: element & \\ \hline
1 & e-neutrino \\ \hline
e_{1} & red \: up \: quark \\ \hline
e_{2} & green \: up \: quark \\ \hline
e_{6} & blue \: up \: quark \\ \hline
e_{4} & electron \\ \hline
e_{3} & red \: down \: quark \\ \hline
e_{5} & green \: down \: quark \\ \hline
e_{7} & blue \: down \: quark \\ \hline
\end{array}$$
The ${\bf O}_{v}$ represents 8-dimensional spacetime.
To represent the ${\bf O} \oplus {\bf O}$ second generation of fermions, you need a structure that generalizes Equation (52) by having two copies of the fermion octonions.
The simplest such generalization is $$?E? =
F_{4} \oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)
\oplus
\left(
\begin{array}{ccc}
0 & {\bf O}_{+} & 0 \\
& & \\
{\bf O}_{+}^\dagger & 0 & {\bf O}_{-} \\
& & \\
0 & {\bf O}_{-}^\dagger & 0
\end{array}
\right)$$ This proposal fails because the ${\bf O}_{+}$ and ${\bf O}_{}$ in $?E?$ are not embedded in $J_{3}^{\bf O}o$ and therefore do not transform like the first generation ${\bf O}_{+}$ and ${\bf O}_{}$ that are embedded in $J_{3}^{\bf O}o$.
Therefore, make the second simplest generalization: $$??E?? =
F_{4} \oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)
\oplus
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ The $??E??$ proposal also fails, because the algebraic structure of the two copies of $J_{3}^{\bf O}o$ is incomplete.
To complete the algebraic structure , a third copy of $J_{3}^{\bf O}o$ must be added, and all three copies must be related algebraically like the imaginary quaternions $\{ i, j, k \}$. This can be done by tensoring $J_{3}^{\bf O}o$ with the imaginary quaternions $S^{3} = SU(2) = Spin(3) = Sp(1)$.
Since the order of the octonions in ${\bf O} \oplus {\bf O}$ should be irrelevant (for example, the octonion pair $\{ e_{i}, 1 \}$ should represent the same fermion as the octonion pair $\{ 1, e_{i} \}$), the structure must include the derivation algebra of the automorphism group of the quaternions, $SU(2)$.
The resulting structure is the 133-dimensional exceptional Lie algebra $E_{7}$: $$E_{7} =
F_{4} \oplus
SU(2)
\oplus
S^{3}
\otimes
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ Therefore $E_{7}$ is the global structure algebra of the second generation fermions.
For the third generation of fermions, note that three algebraically independent copies of $J_{3}^{\bf O}o$ generate seven copies, corresponding to the imaginary octonions $\{ 1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7} \}$; that the imaginary octonions can be represented by $S^{7}$; and that the derivation algebra of the automorphism group of the octonions is $G_{2}$.
Therefore, the 248-dimensional exceptional Lie algebra $E_{8}$: $$E_{8} =
F_{4} \oplus
G_{2}
\oplus
S^{7}
\otimes
\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ is the global structure algebra of the third generation fermions.
Since there are only three Lie algebras in the series $E_{6}, E_{7}, E_{8}$, there are only three generations of fermions.
The first-generation Lie algebra $E_{6}$ has one copy of $J_{3}^{\bf O}o$, corresponding to the complex imaginary $i$.
The second-generation Lie algebra $E_{7}$ adds one more algebraically independent copy of $J_{3}^{\bf O}o$, corresponding to the quaternionic imaginary $j$. Together with the $i$ copy of $J_{3}^{\bf O}o$, the $k$ copy is produced, so $E_{7}$ has in total 3 copies of $J_{3}^{\bf O}o$.
The third-generation Lie algebra $E_{8}$ adds one more algebraically independent copy of $J_{3}^{\bf O}o$, corresponding to the octonionic imaginary $e_{4}$. Together with the $i=e_{1}, j=e_{2}, k=e_{6}$ copies of $J_{3}^{\bf O}o$, the $e_{3}, e_{5}, e_{7}$ copies are produced, so $E_{7}$ has in total 3 copies of $J_{3}^{\bf O}o$.
Therefore, transitions among generation of fermions to a lower one involve elimination of algebraically independent imaginaries.
From 2nd to 1st, it must effectively map $j \rightarrow 1$. This transition can be done in one step.
From 3rd to 2nd, it must effectively map $e_{4} \rightarrow 1$. This transition can also be done in one step.
From 3rd to 1st, it must effectively map $j,e_{4} \rightarrow 1$. This transition cannot be done in one step. The map $e_{4} \rightarrow 1$ only gets you to the 2nd generation. You still need the additional map $j \rightarrow 1$ to get to the 1st.
Since the imaginaries $e_{4}$ and $j$ are orthogonal to each other, there must be an intermediate step that is effectively a phase shift of $\pi / 2$, and the KOBAYASHI-MASKAWA PHASE angle parameter should be
$$\epsilon = \pi / 2$$
Discrete Lattices and Dimensional Reduction.
============================================
The physical manifolds of the $D_{4}-D_{5}-E_{6}$ model should be representable in terms of discrete lattices in order to be formulated as a generalized Feynman checkerboard model.
General references on lattices, polytopes, and related structures are the book of Conway and Sloane [@CON] and the books [@COX1; @COX2] and some papers [@COX3; @COX4] of Coxeter.
Discrete Lattice $D_{4}-D_{5}-E_{6}$ Model.
-------------------------------------------
As seen in section 2, 78-dimensional $E_{6}$ of the $D_{4}-D_{5}-E_{6}$ model is made up of three parts: 38-dimensional space of antihermitian $3 \times 3$ octonion matrices; 14-dimensional space of the Lie algebra $G_{2}$; and 26-dimensional space of the Jordan algebra $J_{3}^{{\bf{O}}}o$.
Antihermitian $3 \times 3$ Octonion Matrices.
---------------------------------------------
Antihermitian $3 \times 3$ octonion matrices: $$\left(
\begin{array}{ccc}
S^{7}_{1} & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
-{\bf O}_{+}^\dagger & S^{7}_{2} & {\bf O}_{-} \\
& & \\
-{\bf O}_{v}^\dagger & -{\bf O}_{-}^\dagger & -S^{7}_{1}-S^{7}_{2}
\end{array}
\right)$$
Individually, ${\bf O}_{v}$, ${\bf O}_{+}$, and ${\bf O}_{-}$ can each be represented by the 8-dimensional $E_{8}$ lattice, as shown by Geoffrey Dixon [@DIX6].
As Geoffrey Dixon [@DIX7] has shown, the two half-spinor spaces ${\bf O}_{+}$, and ${\bf O}_{-}$ taken together can be represented by the 16-dimensional Barnes-Wall lattice $\Lambda_{16}$.
The Barnes-Wall lattices form a series of real dimension $2^{n}$ for $n \geq 2$, and that the Barnes-Wall lattices of real dimension 4 and 8 are the $D_{4}$ and $E_{8}$ lattices.
${\bf O}_{+}$, and ${\bf O}_{-}$ form the Shilov boundary of a 16-complex dimensional bounded complex homogeneous domain, and Conway and Sloane [@CON] note that Quebbemann’s 32-real dimensional lattice is a complex lattice whose 16-real-dimensional real part is the 16-real-dimensional Barnes-Wall $\Lambda_{16}$ lattice.
To represent all three ${\bf O}_{v}$, ${\bf O}_{+}$, and ${\bf O}_{-}$ together, it may be possible to use the 24-dimensional Leech lattice.
Such a Leech lattice representation is likely to be related to the group $$Spin(0,8) = S^{7} \Join S^{7} \Join G_{2}$$ where $\Join$ denotes the fibre product of the fibrations $$Spin(7) \rightarrow Spin(8) \rightarrow S^{7}$$ and $$G_{2} \rightarrow Spin(7) \rightarrow S^{7}$$
In a Leech lattice representation, it is likely that:
the ${\bf O}_{+}$ and ${\bf O}_{-}$ correspond to the two $S^{7}$’s of the $Spin(0,8)$ fibrations, and to the $X$-product of Martin Cederwall [@CED1] and the $XY$-product on which Geoffrey Dixon is working; and
the ${\bf O}_{v}$ corresponds to a 7-dimensional representation of the $G_{2}$ of the $Spin(0,8)$ fibrations.
The entire 26-dimensional space of traceless antihermitian $3 \times 3$ octonion matrices may be represented by the Lorentz Leech lattice $\Pi_{25,1}$, which is closely related to the Monster group.
$S^{7} \oplus S^{7} \oplus G_{2} = D_{4}$.
------------------------------------------
$S^{7}_{1} \oplus S^{7}_{2} \oplus G_{2} = D_{4}$ is the Lie algebra of the $Spin(0,8)$ gauge group of the $D_{4}-D_{5}-E_{6}$ model (before spacetime dimensional reduction).
The gauge group $Spin(0,8)$ acts through its gauge bosons, which can: propagate through spacetime; interact with fermion particles or antiparticles; and interact with each other.
Propagation through spacetime is represented by the 8-dimensional vector representation of $Spin(0,8)$, which in turn is represented by the octonions ${\bf O}_{v}$ and, in discrete lattice version, by the $E_{8}$ lattice.
Interaction with fermion particles is represented by the 8-dimensional +half-spinor representation of $Spin(0,8)$, which in turn is represented by the octonions ${\bf O}_{+}$ and, in discrete lattice version, individually by the $E_{8}$ lattice.
Interaction with fermion antiparticles is represented by the 8-dimensional -half-spinor representation of $Spin(0,8)$, which in turn is represented by the octonions ${\bf O}_{-}$ and, in discrete lattice version, individually by the $E_{8}$ lattice.
Together, the two half-spinor representations are represented by the Barnes-Wall $\Lambda_{16}$ lattice.
Interaction with other gauge bosons is represented by the 28-dimensional adjoint representation of $Spin(0,8)$, which in turn is represented by the exterior wedge bivector product of two copies of the vector octonions ${\bf O}_{v}
\wedge {\bf O}_{v}$ and, in discrete lattice version, by the exterior wedge bivector product of two copies of the $E_{8}$ lattice, $E_{8} \wedge E_{8}$.
Jordan algebra $J_{3}^{{\bf{O}}}o$.
-----------------------------------
26-dimensional space of the Jordan algebra $J_{3}^{{\bf{O}}}o$: $$\left(
\begin{array}{ccc}
a & {\bf O}_{+} & {\bf O}_{v} \\
& & \\
{\bf O}_{+}^\dagger & b & {\bf O}_{-} \\
& & \\
{\bf O}_{v}^\dagger & {\bf O}_{-}^\dagger & -a-b
\end{array}
\right)$$ The 25+1 = 26-dimensional Lorentz Leech lattice $II_{25,1}$ can be used to represent the 26-dimensional Jordan algebra $J_{3}^{{\bf{O}}}o$.
As seen in section 4.,the physically relevant group action on $J_{3}^{{\bf{O}}}o$ for the Lagrangian dynamics of the $D_{4}-D_{5}-E_{6}$ model is not action by the global symmetry group $E_{6}$, but rather action by the gauge group $Spin(0,8)$.
Since action on $J_{3}^{{\bf{O}}}o$ by the gauge group $Spin(0,8)$ leaves invariant each of the ${\bf O}_{v}$, ${\bf O}_{+}$, and ${\bf O}_{-}$ parts of the matrix, the only discrete lattice structure needed for the Lagrangian dynamics of the $D_{4}-D_{5}-E_{6}$ model is the representation of each of ${\bf O}_{v}$, ${\bf O}_{+}$, and ${\bf O}_{-}$ by an $E_{8}$ lattice, and the full Lorentz Leech lattice $II_{25,1}$ is not needed.
However, in case it may be useful to have a discrete lattice description of global symmetries (for example, in looking at generalized supersymmetric relationships among fermions and bosons or at CPT symmetry), here are some characteristics of the Lorentz Leech lattice $II_{25,1}$:
$II_{25,1}$ can be represented (as in Conway and Sloane [@CON]) by the set of vectors $\{ x_{0}, x_{1}, ... , x_{24} | x_{25} \}$ such that all the $x_{i}$ are in ${\bf Z}$ or all in ${\bf Z} + 1/2$ and satisfy $x_{0} + ... + x_{24} - x_{25} \in 2{\bf Z}$
Let $w = (0,1,2,3, ... 23,24|70)$. Since $0^{2}+1^{2}+2^{2}+...+24^{2} = 70^{2}$, $w$ is an isotropic vector in $II_{25,1}$.
Then the Leech roots, or vectors $r$ in $II_{25,1}$ such that $r \cdot r = 2$ and $r \cdot w = -1$ are the vertices of a 24-dimensional Leech lattice.
Also, $(w^{\bot} \cap II_{25,1})/w$ is a copy of the Leech lattice.
The 24-dimensional Leech lattice can be made up of 3 $E_{8}$ lattices, and so corresponds to the off-diagonal part of $J_{3}^{{\bf{O}}}o$.
Each vertex of the Leech lattice has 196,560 nearest neighbors.
A space of 196,560+300+24 = 196,884 dimensions can be used to represent the largest finite simple, group, the Fischer-Greiss Monster of order $$2^{46} \cdot 3^{20} \cdot 5^{9} \cdot 7^{6} \cdot 11^{2}
\cdot 13^{3} \cdot 17 \cdot 19
\cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47
\cdot 59 \cdot 71 =$$ $$= 808,017,424,794,512,875,886,459,904,$$ $$961,710,757,005,754,368,000,000,000$$
Lattice Dimensional Reduction.
------------------------------
### HyperDiamond Lattices.
The lattices of type $D_{n}$ are n-dimensional checkerboard lattices, that is, the alternate vertices of a Zn hypercubic lattice. A general reference on lattices is Conway and Sloane [@CON]. For the n-dimensional HyperDiamond construction from $D_{n}$, Conway and Sloane use an n-dimensional glue vector $[1] = (0.5, ..., 0.5)$ (with n $0.5$’s).
Consider $D_{3}$, the fcc close packing in 3-space. Make a second $D_{3}$ shifted by the glue vector $(0.5, 0.5, 0.5)$.
Then form the union $D_{3} \cup ([1] + D_{3})$.
That is a 3-dimensional Diamond crystal.
When you do the same thing to get a 4-dimensional HyperDiamond, you get $D_{8} \cup ([1] + D_{8})$. The 4-dimensional HyperDiamond $D_{4} \cup ([1] + D_{4})$ is the ${\bf{Z}}^{4}$ hypercubic lattice with null edges.
It is the lattice that Michael Gibbs [@GIB] uses in his Ph. D. thesis advised by David Finkelstein.
When you construct an 8-dimensional HyperDiamond, you get $D_{8} \cup ([1] + D_{8})$ = $E_{8}$, the fundamental lattice of the octonion structures in the $D_{4}-D_{5}-E_{6}$ model described in [hep-ph/9501252](http://xxx.lanl.gov/abs/hep-ph/9501252).
### Dimensional Reduction.
Dimensional reduction of spacetime from 8 to 4 dimensions takes the $E_{8}$ lattice into a $D_{4}$ lattice. The $E_{8}$ lattice can be written in 7 different ways using octonion coordinates with basis $$\{ 1 ,e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7} \}$$ One way is: 16 vertices: $$\pm 1, \pm e_{1}, \pm e_{2}, \pm e_{3}, \pm e_{4},
\pm e_{5}, \pm e_{6}, \pm e_{7}$$ 96 vertices: $$\begin{array} {c}
(\pm 1 \pm e_{1} \pm e_{2} \pm e_{3}) / 2 \\
(\pm 1 \pm e_{2} \pm e_{5} \pm e_{7}) / 2 \\
(\pm 1 \pm e_{2} \pm e_{4} \pm e_{6}) / 2 \\
(\pm e_{4} \pm e_{5} \pm e_{6} \pm e_{7}) / 2 \\
(\pm e_{1} \pm e_{3} \pm e_{4} \pm e_{6}) / 2 \\
(\pm e_{1} \pm e_{3} \pm e_{5} \pm e_{7}) / 2 \\
\end{array}$$ 128 vertices: $$\begin{array} {c}
(\pm 1 \pm e_{3} \pm e_{4} \pm e_{7}) / 2 \\
(\pm 1 \pm e_{1} \pm e_{5} \pm e_{6}) / 2 \\
(\pm 1 \pm e_{3} \pm e_{6} \pm e_{7}) / 2 \\
(\pm 1 \pm e_{1} \pm e_{4} \pm e_{7}) / 2 \\
(\pm e_{1} \pm e_{2} \pm e_{6} \pm e_{7}) / 2 \\
(\pm e_{2} \pm e_{3} \pm e_{4} \pm e_{7}) / 2 \\
(\pm e_{1} \pm e_{2} \pm e_{4} \pm e_{5}) / 2 \\
(\pm e_{2} \pm e_{3} \pm e_{5} \pm e_{6}) / 2 \\
\end{array}$$
Consider the quaternionic subspace of the octonions with basis $\{ 1 ,e_{1},e_{2},e_{6} \}$ and the $D_{4}$ lattice with origin nearest neighbors: 8 vertices: $$\pm 1, \pm e_{1}, \pm e_{2}, \pm e_{6}$$ and 16 vertices: $$(\pm 1 \pm e_{1} \pm e_{2} \pm e_{6}) / 2 \\$$
Dimensional reduction of the $E_{8}$ lattice spacetime to 4-dimensional spacetime reduces each of the $D_{8}$ lattices in the $E_{8}$ lattice to $D_{4}$ lattices.
Therefore, we should get a 4-dimensional HyperDiamond $D_{4} \cup ([1] + D_{4})$.
The 4-dimensional HyperDiamond $D_{4} \cup ([1] + D_{4})$ is the ${\bf{Z}}^{4}$ hypercubic lattice with null edges.
It is the lattice that Michael Gibbs [@GIB] uses in his Ph. D. thesis advised by David Finkelstein.
Here is an explicit construction of the 4-dimensional HyperDiamond.
START WITH THE 24 VERTICES OF A 24-CELL $D_{4}$:
$$\begin{array}{cccc}
+1 & +1 & 0 & 0 \\
+1 & 0 & +1 & 0 \\
+1 & 0 & 0 & +1 \\
+1 & -1 & 0 & 0 \\
+1 & 0 & -1 & 0 \\
+1 & 0 & 0 & -1 \\
-1 & +1 & 0 & 0 \\
-1 & 0 & +1 & 0 \\
-1 & 0 & 0 & +1 \\
-1 & -1 & 0 & 0 \\
-1 & 0 & -1 & 0 \\
-1 & 0 & 0 & -1 \\
0 & +1 & +1 & 0 \\
0 & +1 & 0 & +1 \\
0 & +1 & -1 & 0 \\
0 & +1 & 0 & -1 \\
0 & -1 & +1 & 0 \\
0 & -1 & 0 & +1 \\
0 & -1 & -1 & 0 \\
0 & -1 & 0 & -1 \\
0 & 0 & +1 & +1 \\
0 & 0 & +1 & -1 \\
0 & 0 & -1 & +1 \\
0 & 0 & -1 & -1 \\
\end{array}$$
SHIFT THE LATTICE BY A GLUE VECTOR, BY ADDING
$$\begin{array}{cccc}
0.5 & 0.5 & 0.5 & 0.5 \\
\end{array}$$
TO GET 24 MORE VERTICES $[1] + D_{4}$:
$$\begin{array}{cccc}
+1.5 & +1.5 & 0.5 & 0.5 \\
+1.5 & 0.5 & +1.5 & 0.5 \\
+1.5 & 0.5 & 0.5 & +1.5 \\
+1.5 & -0.5 & 0.5 & 0.5 \\
+1.5 & 0.5 & -0.5 & 0.5 \\
+1.5 & 0.5 & 0.5 & -0.5 \\
-0.5 & +1.5 & 0.5 & 0.5 \\
-0.5 & 0.5 & +1.5 & 0.5 \\
-0.5 & 0.5 & 0.5 & +1.5 \\
-0.5 & -0.5 & 0.5 & 0.5 \\
-0.5 & 0.5 & -0.5 & 0.5 \\
-0.5 & 0.5 & 0.5 & -0.5 \\
0.5 & +1.5 & +1.5 & 0.5 \\
0.5 & +1.5 & 0.5 & +1.5 \\
0.5 & +1.5 & -0.5 & 0.5 \\
0.5 & +1.5 & 0.5 & -0.5 \\
0.5 & -0.5 & +1.5 & 0.5 \\
0.5 & -0.5 & 0.5 & +1.5 \\
0.5 & -0.5 & -0.5 & 0.5 \\
0.5 & -0.5 & 0.5 & -0.5 \\
0.5 & 0.5 & +1.5 & +1.5 \\
0.5 & 0.5 & +1.5 & -0.5 \\
0.5 & 0.5 & -0.5 & +1.5 \\
0.5 & 0.5 & -0.5 & -0.5 \\
\end{array}$$
FOR THE NEW COMBINED LATTICE $D_{4} \cup ([1] + D_{4})$, THESE ARE 6 OF THE NEAREST NEIGHBORS TO THE ORIGIN:
$$\begin{array}{cccc}
-0.5 & -0.5 & 0.5 & 0.5 \\
-0.5 & 0.5 & -0.5 & 0.5 \\
-0.5 & 0.5 & 0.5 & -0.5 \\
0.5 & -0.5 & -0.5 & 0.5 \\
0.5 & -0.5 & 0.5 & -0.5 \\
0.5 & 0.5 & -0.5 & -0.5 \\
\end{array}$$
HERE ARE 2 MORE THAT COME FROM ADDING THE GLUE VECTOR TO LATTICE VECTORS THAT ARE NOT NEAREST NEIGHBORS OF THE ORIGIN:
$$\begin{array}{cccc}
0.5 & 0.5 & 0.5 & 0.5 \\
-0.5 & -0.5 & -0.5 & -0.5 \\
\end{array}$$
THEY COME, RESPECTIVELY, FROM ADDING THE GLUE VECTOR TO:
THE ORIGIN
$$\begin{array}{cccc}
0 & 0 & 0 & 0 \\
\end{array}$$
ITSELF;
AND
THE LATTICE POINT
$$\begin{array}{cccc}
-1 & -1 & -1 & -1 \\
\end{array}$$
WHICH IS SECOND ORDER, FROM
$$\begin{array}{cccc}
-1 & -1 & 0 & 0 \\
plus &&& \\
0 & 0 & 0 & 0 \\
\end{array}$$
FROM
$$\begin{array}{cccc}
-1 & 0 & -1 & 0 \\
plus &&& \\
0 & -1 & 0 & -1 \\
\end{array}$$
OR FROM
$$\begin{array}{cccc}
-1 & 0 & 0 & -1 \\
plus &&& \\
0 & -1 & -1 & 0 \\
\end{array}$$
That the $E_{8}$ lattice is, in a sense, fundamentally 4-dimensional can be seen from several points of view:
the $E_{8}$ lattice nearest neighbor vertices have only 4 non-zero coordinates, like 4-dimensional spacetime with speed of light $c$ = $\sqrt{3}$, rather than 8 non-zero coordinates, like 8-dimensional spacetime with speed of light $c$ = $\sqrt{7}$, so the $E_{8}$ lattice light-cone structure appears to be 4-dimensional rather than 8-dimensional;
the representation of the $E_{8}$ lattice by quaternionic icosians, as described by Conway and Sloane [@CON];
the Golden ratio construction of the $E_{8}$ lattice from the $D_{4}$ lattice, which has a 24-cell nearest neighbor polytope (The construction starts with the 24 vertices of a 24-cell, then adds Golden ratio points on each of the 96 edges of the 24-cell, then extends the space to 8 dimensions by considering the algebraicaly independent $\sqrt{5}$ part of the coordinates to be geometrically independent, and finally doubling the resulting 120 vertices in 8-dimensional space (by considering both the $D_{4}$ lattice and its dual $D_{4}^{\ast}$) to get the 240 vertices of the $E_{8}$ lattice nearest neighbor polytope (the Witting polytope); and
the fact that the 240-vertex Witting polytope, the $E_{8}$ lattice nearest neighbor polytope, most naturally lives in 4 complex dimensions, where it is self-dual, rather than in 8 real dimensions.
Some more material on such things can be found at [WWW URL http://www.gatech.edu/tsmith/home.html](http://www.gatech.edu/tsmith/home.html) [@SMI6].
In referring to Conway and Sloane [@CON], bear in mind that they use the convention (usual in working with lattices) that the norm of a lattice distance is the square of the length of the lattice distance.
It is also noteworthy that the number of vertices in shells of an $E_{8}$ lattice increase monotonically as the radius of the shell increases, while cyclic relationships (see Conway and Sloane [@CON]) appear in the number of vertices in shells of a $D_{4}$ lattice.
Discrete Lattice Effects of $8 \rightarrow 4 \; dim$.
-----------------------------------------------------
$Spin(0,8)$ acts on the octonions ${\bf O}$, the lattice version of which is the $E_{8}$ lattice.
Each vertex in the $E_{8}$ lattice has 240 nearest neighbors, the inner shell of the $E_{8}$ lattice.
Geoffrey Dixon [@DIX7] shows that the 240 vertices in the $E_{8}$ inner shell break down with respect to the two 4-dimensional subspaces of ${\bf O}$, each represented by the inner shell of a $D_{4}$ lattice, as
$$\begin{array}{ccc}
<U,0> & \rightarrow & 24 \; elements \\
<0,V> & \rightarrow & 24 \; elements \\
<W,X> (WX \ast = +/- qm) & \rightarrow & 192 \; elements
\end{array}$$
where $D_{4}^{\ast}$ is the dual lattice to the $D_{4}$ lattice, and where $U,V \in D_{4}$, $W \in D_{4}^{\ast}$, and $X \in \{ \pm 1, \pm i, \pm j, \pm k \} \subset D_{4}^{ \ast }$
The two 24-element sets each have the group structure of the binary tetrahedral group, also the group of 24 quaternion units, and the 24 elements would represent the root vectors of the $Spin0,(8)$ $D_{4}$ Lie algebra in the 4-dimensional space of the $D_{4}$ lattice.
The 192 element set is the Weyl group of the $Spin0,(8)$ $D_{4}$ Lie algebra. The Weyl group is the group of reflections in the hyperplanes (in the $D_{4}$ 4-dimensional space) that are orthogonal to the 24 root vectors.
If the 8-dimensional $E_{8}$ spacetime is reduced to the 4-dimensional $D_{4}$ spacetime, then
$$\begin{array}{ccc}
<U,0> & \rightarrow & 24 \; elements \; of \; D_{4} \; lattice \; \\
&&inner \; shell\\
<0,V> & \rightarrow & 24 \; elements \; of \; binary \; \\
&&tetrahedral \; group\\
<W,X> (WX \ast = +/- qm) & \rightarrow & 192 \;
elements \; of \; Weyl \; group \; \\
&&of \; reduced \; gauge \; group
\end{array}$$
where $U,V \in D_{4}$, $W \in D_{4}^{\ast}$, and $X \in \{\pm 1, \pm i, \pm j, \pm k \} \subset D_{4}^{\ast}$
The 24-element $D_{4}$ lattice inner shell formed by the $U$ elements of Equation (76) form the second shell of the reduced 4-dimensional spacetime $D_{4}$ lattice, as described in the preceding Subsection 7.5. Denote this set by $24U$.
The 24-element $D_{4}$ lattice inner shell formed by the $V$ elements of Equation (76) form the 24-element finite group (binary tetrahedral group of unit quaternions) that is the Weyl group of the internal symmetry gauge group of the 4-dimensional $D_{4}-D_{5}-E_{6}$ model. Denote this group by $24V$.
The 192-element Weyl group of the $D_{4}$ Lie algebra of $Spin(0,8)$ is made up of pairs, the first of which is an element of the 24-element set of $W$ elements. Denote that 24-element set by $24W$. The second part of a pair is an element of the 8-element set of $X$ elements. Denote that 8-element set by $8X$.
After dimensional reduction of spacetime, $Spin(0,8)$ is too big to act as an isotropy group on spacetime, as it acted in 8-dimensional spacetime, which is of the form $$Spin(2,8) / Spin(2,0) \times Spin(0,8)$$.
Therefore, in 4-dimensional spacetime, the 28 infinitesimal generators of $Spin(0,8)$ (which act as 28 gauge bosons in 8-dimensional $D_{4}-D_{5}-E_{6}$ physics) cannot interact according to the commutation relations of $Spin(0,8)$, but must interact according to commutation relations of smaller groups that can act on 4-dimensional spacetime.
What type of action should these smaller groups have on 4-dimensional spacetime?
Isotropy action is sufficient in the case of the 28 $Spin(0,8)$ infinitesimal generators in the 8-dimensional theory, because the gauge boson part of the Lagrangian $\int_{8-dim} F \wedge \star F$ is an integral over 8-dimensional spacetime using a uniform measure that is the same for all 28 gauge boson infinitesimal generators. Physically, the force strength of the $Spin(0,8)$ gauge group can be taken to be $1$ because there is only one gauge group in the 8-dimensional $D_{4}-D_{5}-E_{6}$ model. However, as we shall see now, isotropy action is not sufficient for the gauge groups after dimensional reduction.
After reduction to 4-dimensional spacetime, the 28 infinitesimal generators will have to regroup into more that one smaller groups, each of which will have its own force strength. The 4-dimensional Lagrangian will be the sum of more than one Lagrangians of the form $\int_{4-dim} F \wedge \star F$, each of which will use a different measure in integrating over 4-dimensional spacetime. A factor in determining the relative strengths of the 4-dimensional forces will be the relative magnitude of the measures over 4-dimensional spacetime. Therefore, the measure information should be carried in the $F$ of the 4-dimensional Lagrangian $\int_{4-dim} F \wedge \star F$, which is different for each force, rather than in the overall $\int_{4-dim}$, which should be uniform for all terms in the total 4-dimensional Lagrangian of the $D_{4}-D_{5}-E_{6}$ model.
In the 4-dimensional $D_{4}-D_{5}-E_{6}$ model, the small gauge groups must act transitively on 4-dimensional spacetime, so that they can carry the measure information. Physically, the gauge bosons of the different gauge groups see spacetime differently (see [WWW URL http://www.gatech.edu/tsmith/See.html](http://www.gatech.edu/tsmith/See.html) [@SMI6] ).
Since the 4-dimensional $D_{4}-D_{5}-E_{6}$ model gauge groups are not isotropy groups of 4-dimensional spacetime, but actually act transitively on 4-dimensional manifolds, they are not exactly conventional “local symmetry” gauge groups. In the conventional “local symmetry” picture, you can put a gauge boson infinitesimal generator $x(G)$ of the gauge group $G$ at each point $p$ of the spacetime base manifold, with the choice of $x(G)$ made independently at each point $p$. In the $D_{4}-D_{5}-E_{6}$ model picture, the choice of $x(G)$ at a point not only is the choice of a gauge boson, but also of a “translation” direction in the 4-dimensional spacetime.
When you choose a gauge boson, say a ’red-antiblue gluon“ of the $SU(3)$ color force gauge group, then does your ”choice“ of ”translation direction" fix a physical direction of propagation (say, $(+1-i-j-k)/2$ in quaternionic coordinates for the future lightcone) ?
If it does, the model doesn’t work right. Fortunately for the $D_{4}-D_{5}-E_{6}$ model, there is one more choice to be made independently at each point, and that is the choice to put any given element of the gauge group $G$ in corresponce with with any element of the isotropy subgroup $K$ or with any direction in the 4-dimensional manifold $G / K$ on which $G$ acts transitively.
Therefore, at any given point in the 4-dimensional spacetime, you can choose the “red-antiblue gluon” gauge boson (or any other gauge boson) and the $(+1-i-j-k)/2$ direction (or any other direction), and independent choices can be made at all points in the 4-dimensional spacetime of the $D_{4}-D_{5}-E_{6}$ model.
Since choices of gauge boson and direction of propagation are both made at once, it is natural in the lattice $D_{4}-D_{5}-E_{6}$ model to picture the gauge bosons as living on the links of the spacetime lattice, with the fermion particles and antiparticles living on the vertices.
This type of structure might not work consistently in a model with less symmetry than the $D_{4}-D_{5}-E_{6}$ model. Particularly, the triality symmetry of 8-dimensional spacetime with the 8-dimensional half-spinor representation spaces of the first-generation fermion particles and antiparticles means that the gauge boson symmetry group, which must act on fermion particles and antiparticles in a natural way, also acts transitively on spacetime in a natural way. For a discussion of what types of “generalized supersymmetry” symmetries are useful, and why the $D_{4}-D_{5}-E_{6}$ model probably has the most useful symmetries of any model, see [hep-th/9306011](http://xxx.lanl.gov/abs/hep-th/9306011) [@SMI3] ).
Even after dimensional reduction of spacetime, there is a residual symmetry relationship between the fermion representation spaces and spacetime. Perhaps that residual symmetry might be a way to relate the results of the $D_{4}-D_{5}-E_{6}$ model to the results of the 4-dimensional lattice model of Finkelstein and Gibbs. [@GIB]
What can these smaller groups be? Since 4-dimensional spacetime has quaternionic structure, they must act transitively on 4-dimensional manifolds with quaternionic structure. Such manifolds have been classified by Wolf [@WOL], and they are
$$\begin{array}{|c|c|c|}
\hline
M & Symmetric \: Space & Gauge \: Group \\
\hline
& & \\
S^{4} & Spin(5) \over Spin(4) & Spin(5) \\
& & \\
{\bf C}P^2 & SU(3) \over {SU(2) \times U(1)}
& SU(3) \\
& & \\
S^2 \times S^2 & 2 \: copies \: of \; \left( SU(2) \over U(1) \right)
& SU(2) \\
& & \\
S^1 \times S^1 \times S^1
\times S^1 & 4 \: copies \: of \; U(1) & U(1) \\
& & \\
\hline
\end{array}$$
Therefore the 4 forces have gauge groups $Spin(5)$ (10 infinitesimal generators) $SU(3)$ (8 infinitesimal generators) $SU(2)$ (2 copies, each with 3 infinitesimal generators) and $U(1)$ (4 copies, each with 1 infinitesimal generator) that account for all 28 of the $Spin(0,8)$ gauge bosons.
There are two cases in which 4-dimensional spacetime is made up of multiple copies of lower-dimensional manifolds.
Two copies of the gauge group $SU(2)$ act on 2 copies of $S^{2}$. Each $SU(2)$ has 3 infinitesimal generators, the three weak bosons $\{ W_{-}, W_{0}, W_{+}$. Since a given weak boson cannot carry a different charge in different parts of 4-dimensional spacetime, the 2 copies of $SU(2)$ must be aligned consistently. This means that there is physically only one $SU(2)$ weak force gauge group, and that there are 3 degrees of freedom due to 3 $Spin(0,8)$ infinitesimal generators that are not used. Denote them by $3-SU(2)$.
Four copies of the gauge group $U(1)$ act on 4 copies of $S^{1}$. Each $U(1)$ has 1 infinitesimal generator, the photon. Since a given photon should be the same in all parts of 4-dimensional spacetime, the 4 copies of $U(1)$ must be aligned consistently. This means that there is physically only one $U(1)$ electromagnetic gauge group, and that there are 3 degrees of freedom due to 3 $Spin(0,8)$ infinitesimal generators that are not used. Denote them by $3-U(1)$.
How does all this fit into the structures $24U$, $24V$, $24W$, and $8X$ ?
$24U$ and $24W$ are the $D_{4}$ and $D_{4}^{\ast}$ of the 4-dimensional lattice spacetime as described in the preceding Subsection 7.5 .
The $8X$, being part of a 192-element product with $24W$, should represent a gauge group that is closely connected to spacetime. $8X$ represents the 8-element Weyl group $S_{2}^{3}$ of the gauge group $Spin(5)$. By the MacDowell-Mansouri mechanism [@MAC], the $Spin(5)$ gauge group accounts for Einstein-Hilbert gravity.
$24V$, being entirely from the part of 8-dimensional spacetime that did not survive dimensional reduction, should represent the Weyl groups of gauge groups of internal symmetries. As $24V$ is the 24-element binary tetrahedral group of unit quaternions, the inner shell of the $D_{4}$ lattice, it has a 4-element subgroup $S_{2}^{2}$ made up of unit complex numbers, the inner shell of the Gaussian lattice. $24V / S_{2}^{2}$ is the 6-element group $S_{3}$, which is the Weyl group of the $SU(3)$ gauge group of the color force. The $S_{2}^{2}$ is 2 copies of the Weyl group of the $SU(2)$ gauge group of the weak force. As $U(1)$ is Abelian, and so has the identity for its Weyl group, 4 copies of the $U(1)$ gauge group of electromagnetism can be said to be included in the gauge groups of which $24V$ is the Weyl group.
Therefore, $24V$ gives us the gauge groups of the Standard Model, $SU(3) \times SU(2) \times U(1)$, plus the extra 6 degrees of freedom $3-SU(2)$ and $3-U(1)$ that were discussed above.
It will be seen in Section 8.2 that 5 of the 6 extra degrees of freedom are a link between the Higgs sector of the Standard Model and conformal symmetry related to gravity, and the 6th, a copy of U(1), accounts for the complex phase of propagators in the $D_{4}-D_{5}-E_{6}$ model.
From the continuum limit viewpoint of Section 8.2, those 6 degrees of freedom are combined with the gravity sector to expand the 10-dimensional de Sitter $Spin(5)$ group of the MacDowell-Mansouri mechanism to the 15-dimensional conformal $Spin(2,4)$ group plus the $U(1)$ of the complex propagator phase, which in turn can all be combined into one copy of 16-dimensional $U(4)$.
From the discrete Weyl group point of view of this section, the $3-SU(2)$ and $3-U(1)$ degrees of freedom are represented by the Weyl group $S_{2}$. The $S_{2}$ can be identified with the 3 $S_{2}$’s of the gravity $Spin(5)$ $S_{2}^{3}$ in 3 ways, so the identification effectively expands the gravity sector Weyl group by a factor of 3 from $S_{2}^{3} = S_{2}^{2} \times S_{2}$ to $S_{2}^{2} \times S_{3}$, which is the Weyl group of the conformal group $Spin(2,4)$ and also the Weyl group of $U(4)$.
Continuum Limit Effects of $8 \rightarrow 4 \; dim$.
====================================================
The dimensional reduction breaks the gauge group $Spin(0,8)$ into gravity plus the Standard Model.
The following sections discuss the effects of dimensional reduction on the terms of the 8-dimensional Lagrangian $$\int_{V_{8}} F_{8} \wedge \star F_{8} + \partial_{8}^{2}
\overline{\Phi_{8}} \wedge \star \partial_{8}^{2} \Phi_{8} +
\overline{S_{8\pm}} \not \! \partial_{8} S_{8\pm}$$ of the $D_{4}-D_{5}-E_{6}$ model discussed in Section 5.
and the phenomenological results of calculations based on the 4-dimensional structures.
Scalar part of the Lagrangian
-----------------------------
The scalar part of the 8-dimensional Lagrangian is
$$\int_{V_{8}}
\partial_{8}^{2} \overline{\Phi_{8}} \wedge \star
\partial_{8}^{2} \Phi_{8}$$
As shown in chapter 4 of Göckeler and Schücker [@GOC], $\partial_{8}^{2} \Phi_{8}$ can be represented as an 8-dimensional curvature $F_{H8}$, giving
$$\int_{V_{8}} F_{H8} \wedge \star F_{H8}$$
When spacetime is reduced to 4 dimensions, denote the surviving 4 dimensions by $4$ and the reduced 4 dimensions by $\perp 4$.
Then, $F_{H8} = F_{H44} + F_{H4\perp 4} + F_{H\perp 4 \perp 4}$, where
$F_{H44}$ is the part of $F_{H8}$ entirely in the surviving spacetime;
$F_{H4 \perp 4}$ is the part of $F_{H8}$ partly in the surviving spacetime and partly in the reduced spacetime; and
$F_{H\perp 4 \perp 4}$ is the part of $F_{H8}$ entirely in the reduced spacetime;
The 4-dimensional Higgs Lagrangian is then:
$\int (F_{H44} + F_{H4 \perp 4} + F_{H\perp 4 \perp 4})
\wedge
\star (F_{H44} + F_{H4 \perp 4} + F_{H\perp 4 \perp 4}) =$
$=\int (F_{H44} \wedge \star F_{H44} +
F_{H4 \perp 4} \wedge \star F_{H4 \perp 4} +
F_{H\perp 4 \perp 4} \wedge \star F_{H\perp 4 \perp 4})$.
As all possible paths should be taken into account in the sum over histories path integral picture of quantum field theory, the terms involving the reduced 4 dimensions, $\perp 4$, should be integrated over the reduced 4 dimensions.
Integrating over the reduced 4 dimensions, $\perp 4$, gives
$\int \left( F_{H44} \wedge \star F_{H44} +
\int_{\perp 4} F_{H4 \perp 4} \wedge \star F_{H4 \perp 4} +
\int_{\perp 4} F_{H\perp 4 \perp 4} \wedge
\star F_{H\perp 4 \perp 4} \right)$.
### First term $ F_{H44} \wedge \star F_{H44}$
The first term is just $\int F_{H44} \wedge \star F_{H44}$.
Since they are both $SU(2)$ gauge boson terms, this term, in 4-dimensional spacetime, just merges into the $SU(2)$ weak force term $\int F_{w} \wedge \star F_{w}$.
### Third term $\int_{\perp 4} F_{H\perp 4 \perp 4}
\wedge \star F_{H\perp 4 \perp 4}$
The third term, $ \int \int_{\perp 4} F_{H\perp 4 \perp 4} \wedge
\star F_{H\perp 4 \perp 4}$, after integration over $\perp 4$, produces terms of the form
$ \lambda (\overline{\Phi} \Phi)^{2} - \mu^{2}
\overline{\Phi} \Phi$ by a process similar to the Mayer mechanism (see Mayer’s paper [@MAY] for a description of the Mayer mechanism, a geometric Higgs mechanism).
The Mayer mechanism is based on Proposition 11.4 of
chapter 11 of volume I of Kobayashi and Nomizu [@KOB], stating that:
$2 F_{H\perp 4 \perp 4}(X,Y) = [\Lambda(X), \Lambda(Y)] -
\Lambda([X,Y])$,
where $\Lambda$ takes values in the $SU(2)$ Lie algebra.
If the action of the Hodge dual $\star$ on $\Lambda$ is such that
$\star \Lambda = - \Lambda$ and $\star [\Lambda, \Lambda] =
[\Lambda, \Lambda]$,
then
$F_{H\perp 4 \perp 4}(X,Y) \wedge
\star F_{H\perp 4 \perp 4}(X,Y) =
(1/4)([\Lambda(X), \Lambda(Y)]^{2} - \Lambda([X,Y])^{2} )$.
If integration of $\Lambda$ over $\perp 4$ is $\int_{\perp 4} \Lambda \propto \Phi = (\Phi^{+}, \Phi^{0})$, then
$\int_{\perp 4} F_{H\perp 4 \perp 4} \wedge
\star F_{H\perp 4 \perp 4} = $ $ (1/4) \int_{\perp 4} [\Lambda(X),\Lambda(Y)]^{2} -
\Lambda([X,Y])^{2} = $
$= (1/4) [ \lambda ( \overline{\Phi} \Phi)^{2} - \mu^{2}
\overline{\Phi} \Phi ]$,
where $\lambda$ is the strength of the scalar field self-interaction, $\mu^{2}$ is the other constant in the Higgs potential, and where $\Phi$ is a 0-form taking values in the $SU(2)$ Lie algebra.
The $SU(2)$ values of $\Phi$ are represented by complex
$SU(2) = Spin(3)$ doublets $\Phi = (\Phi^{+}, \Phi^{0})$.
In real terms, $\Phi^{+} = (\Phi_{1} + i \Phi_{2})/ \sqrt{2}$ and $\Phi^{0} = (\Phi_{3} + i \Phi_{4})/ \sqrt{2}$,
so $\Phi$ has 4 real degrees of freedom.
In terms of real components, $\overline{\Phi} \Phi = (\Phi_{1}^{2} + \Phi_{2}^{2} +
\Phi_{3}^{2} + \Phi_{4}^{2})/2 $.
The nonzero vacuum expectation value of the
$ \lambda (\overline{\Phi} \Phi)^{2} - \mu^{2}
\overline{\Phi} \Phi$ term is $v = \mu / \sqrt{\lambda}$, and
$<\Phi^{0}> = <\Phi_{3}> = v / \sqrt{2}$.
In the unitary gauge, $\Phi_{1} = \Phi_{2} = \Phi_{4} = 0$,
and
$\Phi = (\Phi^{+}, \Phi^{0}) = (1/ \sqrt{2})(\Phi_{1} + i
\Phi_{2}, \Phi_{3} + i \Phi_{4}) = (1/ \sqrt{2})
(0, v + H)$,
where $\Phi_{3} = (v + H) / \sqrt{2}$,
$v$ is the Higgs potential vacuum expectation value, and
$H$ is the real surviving Higgs scalar field.
Since $\lambda = \mu^{2} / v^{2}$ and $\Phi = (v + H)
/ \sqrt{2}$,
$(1/4)[ \lambda (\overline{\Phi} \Phi)^{2} - \mu^{2}
\overline{\Phi} \Phi ] = $
$= (1/16) (\mu^{2} / v^{2})(v + H)^{4} -
(1/8) \mu^{2} (v + H)^{2} = $
$= (1/16) [ \mu^{2} v^{2} + 4 \mu^{2} vH +
6 \mu^{2} H^{2} + 4 \mu^{2} H^{3} / v + \mu^{2} H^{4} /
v^{2} - 2 \mu^{2} v^{2} - $
$- 4 \mu^{2} v H - 2 \mu^{2} H^{2} ] = $
$= (1/4) \mu^{2} H^{2} - (1/16) \mu^{2} v^{2}
[ 1 - 4 H^{3} / v^{3} - H^{4} / v^{4} ] $.
### Second term $F_{H4 \perp 4} \wedge
\star F_{H4 \perp 4}$
The second term,
$\int_{\perp 4} F_{H4 \perp 4} \wedge \star F_{H4 \perp 4}$,
gives $\int \partial \overline{\Phi} \partial \Phi$, by a process similar to the Mayer mechanism (see Mayer’s paper [@MAY] for a description of the Mayer mechanism, a geometric Higgs mechanism).
From Proposition 11.4 of chapter 11 of volume I of Kobayashi and Nomizu [@KOB]:
$2 F_{H4 \perp 4}(X,Y) = [\Lambda(X), \Lambda(Y)] -
\Lambda([X,Y])$,
where $\Lambda$ takes values in the $SU(2)$ Lie algebra.
For example, if the $X$ component of $F_{H4 \perp 4}(X,Y)$ is in the surviving $4$ spacetime and the $Y$ component of $F_{H4 \perp 4}(X,Y)$ is in $\perp 4$, then
the Lie bracket product $ [X,Y] = 0$ so that $\Lambda([X,Y]) = 0$ and therefore
$F_{H4 \perp 4}(X,Y) = (1/2) [\Lambda(X),\Lambda(Y)] =
(1/2) \partial_{X} \Lambda(Y) $.
The total value of $F_{H4 \perp 4}(X,Y)$ is then $F_{H4 \perp 4}(X,Y) = \partial_{X}\Lambda(Y) $.
Integration of $\Lambda$ over $\perp 4$ gives
$\int _{Y \epsilon \perp 4} \partial_{X}\Lambda(Y) =
\partial_{X}\Phi$,
where, as above, $\Phi$ is a 0-form taking values in the $SU(2)$ Lie algebra.
As above, the $SU(2)$ values of $\Phi$ are represented by complex $SU(2)=Spin(3)$ doublets $\Phi = (\Phi^{+}, \Phi^{0})$.
In real terms, $\Phi^{+} = (\Phi_{1} + i \Phi_{2}) /
\sqrt{2}$ and $\Phi^{0} = (\Phi_{3} + i \Phi_{4}) / \sqrt{2}$,
so $\Phi$ has 4 real degrees of freedom.
As discussed above, in the unitary gauge, $ \Phi_{1} = \Phi_{2} = \Phi_{4} = 0$, and
$\Phi = (\Phi^{+}, \Phi^{0}) =
(1/ \sqrt{2})(\Phi_{1} + i \Phi_{2},
\Phi_{3} + i \Phi_{4}) = (1 / \sqrt{2})(0, v + H)$,
where $\Phi_{3} = (v + H) / \sqrt{2}$ ,
$v$ is the Higgs potential vacuum expectation value, and
$H$ is the real surviving Higgs scalar field.
The second term is then:
$\int (\int_{\perp 4} - F_{H4 \perp 4} \wedge
\star F_{H4 \perp 4}) =$
$= \int (\int_{\perp 4} (-1/2) [\Lambda(X),\Lambda(Y)] \wedge
\star [\Lambda(X),\Lambda(Y)] ) = \int \partial
\overline{\Phi} \wedge \star \partial \Phi$
where the $SU(2)$ covariant derivative $\partial$ is
$\partial = \partial + \sqrt{\alpha_{w}} (W_{+} +
W_{-}) + \sqrt{\alpha_{w}} \cos{\theta_{w}}^{2} W_{0}$, and $\theta_{w}$ is the Weinberg angle.
Then $\partial \Phi = \partial (v + H) /
\sqrt{2} =$
$= [\partial H + \sqrt{\alpha_{w}} W_{+} (v + H) +
\sqrt{\alpha_{w}} W_{-} (v + H) + \sqrt{\alpha_{w}} W_{0}
(v + H) ] / \sqrt{2}$.
In the $D_{4}-D_{5}-E_{6}$ model the $W_{+}$, $W_{-}$, $W_{0}$, and $H$ terms are considered to be linearly independent.
$v = v_{+} + v_{-} + v_{0}$ has linearly independent components $v_{+}$, $v_{-}$, and $v_{0}$ for $W_{+}$, $W_{-}$, and $W_{0}$.
$H$ is the Higgs component.
$\partial \overline{\Phi} \wedge \star \partial \Phi$ is the sum of the squares of the individual terms.
Integration over $\perp 4$ involving two derivatives $\partial_{X} \partial_{X}$ is taken to change the sign by $i^{2} = -1$.
Then:
$\partial \overline{\Phi} \wedge \star \partial \Phi =
(1/2) (\partial H)^{2} +$
$+ (1/2) [ \alpha_{w} v_{+}^{2} \overline{W_{+}} W_{+} +
\alpha_{w} v_{-}^{2} \overline{W_{-}} W_{-} +
\alpha_{w} v_{0}^{2} \overline{W_{0}} W_{0} ] +$
$+ (1/2) [ \alpha_{w} \overline{W_{+}} W_{+} +
\alpha_{w} \overline{W_{-}} W_{-} +
\alpha_{w} \overline{W_{0}} W_{0} ] [ H^{2} + 2 v H ]$.
Then the full curvature term of the weak-Higgs Lagrangian,
$\int F_{w} \wedge \star F_{w} + \partial
\overline{\Phi} \wedge \star \partial \Phi +
\lambda (\overline{\Phi} \Phi)^{2} -
\mu^{2} \overline{\Phi} \Phi$,
is, by the Higgs mechanism:
$\int [ F_{w} \wedge \star F_{w} +$
$+ (1/2) [ \alpha_{w} v_{+}^{2} \overline{W_{+}} W_{+} +
\alpha_{w} v_{-}^{2} \overline{W_{-}} W_{-} +
\alpha_{w} v_{0}^{2} \overline{W_{0}} W_{0} ] +$
$+ (1/2) [ \alpha_{w} \overline{W_{+}} W_{+} +
\alpha_{w} \overline{W_{-}} W_{-} +
\alpha_{w} \overline{W_{0}} W_{0} ] [ H^{2} + 2 v H ] +$
$+ (1/2) (\partial H)^{2} + (1/4) \mu^{2} H^{2} -$
$- (1/16) \mu^{2} v^{2} [ 1 - 4H^{3} / v^{3} - H^{4} /
v^{4} ] ] $.
The weak boson Higgs mechanism masses, in terms of $v = v_{+} + v_{-} + v_{0}$, are:
$(\alpha_{w} / 2) v_{+}^{2} = m_{W_{+}}^{2}$ ;
$(\alpha_{w} / 2) v_{-}^{2} = m_{W_{-}}^{2}$ ; and
$(\alpha_{w} / 2) v_{0}^{2} = m_{W_{+0}}^{2}$,
with $( v = v_{+} + v_{-} + v_{0} ) = ((\sqrt{2}) /
\sqrt{\alpha_{w}}) ( m_{W_{+}} + m_{W_{-}} + m_{W_{0}} )$.
Then:
$\int [ F_{w} \wedge \star F_{w} +$
$+ m_{W_{+}}^{2} W_{+} W_{+} + m_{W_{-}}^{2} W_{-} W_{-} +
m_{W_{0}}^{2} W_{0} W_{0} +$
$+ (1/2) [ \alpha_{w} \overline{W_{+}} W_{+} +
\alpha_{w} \overline{W_{-}} W_{-} +
\alpha_{w} \overline{W_{0}} W_{0} ] [ H^{2} + 2vH ] +$
$+ (1/2)(\partial H)^{2} + (1/2)(\mu^{2} / 2)H^{2} -$
$- (1/16 \mu^{2} v^{2} [1 - 4H^{3} / v^{3} -
H^{4} / v^{4}]$.
Gauge Boson Part of the Lagrangian.
-----------------------------------
In this subsection, we will look at matrix representations of $Spin(0,8)$ and how dimensional reduction affects them.
For a similar study from the point of view of Clifford algebras, see [hep-th/9402003](http://xxx.lanl.gov/abs/hep-th/9402003) [@SMI3] ). For errata for that paper (and others), see [WWW URL http://www.gatech.edu/tsmith/Errata.html](http://www.gatech.edu/tsmith/Errata.html) [@SMI6] ).
The gauge boson bivector part of the Lagrangian is $$\int_{V_{8}} F_{8} \wedge \star F_{8}$$
It represents the $D_{4}-D_{5}-E_{6}$ model gauge group $Spin(0,8)$ acting in 8-dimensional spacetime.
The $8 \times 8$ matrix representation of the $Spin(0,8)$ Lie algebra with the commutator bracket product \[,\] is
$$\left( \begin{array} {cccccccc}
0 & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} & a_{17} & a_{18}\\
-a_{12} & 0 & a_{23} & a_{24} & a_{25} & a_{26} & a_{27} & a_{28}\\
-a_{13} &-a_{23} & 0 & a_{34} & a_{35} & a_{36} & a_{37} & a_{38}\\
-a_{14} &-a_{24} &-a_{34} & 0 & a_{45} & a_{46} & a_{47} & a_{48}\\
-a_{15} &-a_{25} &-a_{35} &-a_{45} & 0 & a_{56} & a_{57} & a_{58}\\
-a_{16} &-a_{26} &-a_{36} &-a_{46} &-a_{56} & 0 & a_{67} & a_{68}\\
-a_{17} &-a_{27} &-a_{37} &-a_{47} &-a_{57} &-a_{67} & 0 & a_{78}\\
-a_{18} &-a_{28} &-a_{38} &-a_{48} &-a_{58} &-a_{68} &-a_{78} & 0\\
\end{array} \right)$$
To see how the $Spin(0,8)$ is affected by dimensional reduction of spacetime to 4 dimensions, represent $Spin(0,8)$, as in section 3.2, by
$$\left(
\begin{array}{cc}
S^{7}_{1} & 0\\
& \\
0 & S^{7}_{2}
\end{array}
\right)
\oplus G_{2}$$
This Lie algebra is 7+7+14 = 28-dimensional $Spin(0,8)$, also denoted $D_{4}$.
Recall that $S^{7}_{1}$ and $S^{7}_{2}$ represent the imaginary octonions $\{ e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7} \}$. From our present local Lie algebra point of view, they look like linear tangent spaces to 7-spheres, not like global round nonlinear 7-spheres. Therefore, instead of using the Hopf fibration $S^{3} \rightarrow S^{7} \rightarrow S^{4}$, we break the spaces down in accord with dimensional reduction to:
$$S^{7}_{1} \rightarrow {\bf R}^{3}_{1} \oplus {\bf R}^{4}_{1}$$
and
$$S^{7}_{2} \rightarrow {\bf R}^{3}_{2} \oplus {\bf R}^{4}_{2}$$
$G_{2}$ has two fibrations:
$$SU(3) \rightarrow G^{2} \rightarrow S^{6}.$$
$$SU(2) \otimes SU(2) \rightarrow G^{2} \rightarrow
M(G_{2})_{8}$$
where $M(G_{2})_{8}$ is an 8-dimensional homogeneous rank 2 symmetric space.
By choice of which $G_{2}$ fibration to use, $Spin(0,8)$ has two decompositions from octonionic derivations $G_{2}$ to quaternionic derivations $SU(2)$.
First, choose the $SU(3)$ subgroup of $G_{2}$ by choosing the $G_{2}$ fibration $$SU(3) \rightarrow G^{2} \rightarrow S^{6}.$$ Since the $SU(3)$ subgroup of $G_{2}$ is the larger 8-dimensional part of 14-dimensional $G_{2}$, also take the larger ${\bf R}^{4}$ parts of the $S^{7}$’s, to get:
$$\left(
\begin{array}{cc}
{\bf R}^{4}_{1} & 0\\
& \\
0 & {\bf R}^{4}_{2}
\end{array}
\right)
\oplus SU(3)$$
If ${\bf R}^{4}_{1} \oplus {\bf R}^{4}_{2}$ is identified with the local tangent space of the 8-dimensional manifold $$(Spin(0,6) \times U(1)) / SU(3)$$ then we have constructed the $Spin(0,6) \times U(1)$ subgroup of $Spin(0,8)$.
The 8-dimensional manifold $(Spin(0,6) \times U(1)) / SU(3)$ is built up from the 6-dimensional irreducible symmetric space $$Spin(0,6) / U(3)$$ (see the book $Einstein \; Manifolds$ by Besse [@BES]) by adding two $U(1)$’s, one $U(1) = U(3) / SU(3)$ and the other the $U(1)$ in $Spin(0,6) \times U(1)$.
Now that we have built $Spin(0,6) \times U(1)$ from dimensional reduction process acting on $Spin(0,8)$, compare an $8 \times 8$ matrix representation:
$$\left( \begin{array} {cccccccc}
0 & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} & 0 & 0 \\
-a_{12} & 0 & a_{23} & a_{24} & a_{25} & a_{26} & 0 & 0 \\
-a_{13} &-a_{23} & 0 & a_{34} & a_{35} & a_{36} & 0 & 0 \\
-a_{14} &-a_{24} &-a_{34} & 0 & a_{45} & a_{46} & 0 & 0 \\
-a_{15} &-a_{25} &-a_{35} &-a_{45} & 0 & a_{56} & 0 & 0 \\
-a_{16} &-a_{26} &-a_{36} &-a_{46} &-a_{56} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{78}\\
0 & 0 & 0 & 0 & 0 & 0 &-a_{78} & 0\\
\end{array} \right)$$
Now make the second choice, the $SU(2) \times SU(2)$ subgroup of $G_{2}$ by choosing the $G_{2}$ fibration
$$SU(2) \otimes SU(2) \rightarrow G^{2} \rightarrow
M(G_{2})_{8}$$
where $M(G_{2})_{8}$ is an 8-dimensional homogeneous rank 2 symmetric space.
Since the $SU(2) \times SU(2)$ subgroup of $G_{2}$ is the smaller 6-dimensional part of 14-dimensional $G_{2}$, also take the smaller ${\bf R}^{3}$ parts of the $S^{7}$’s, to get:
$$\left(
\begin{array}{cc}
{\bf R}^{3}_{1} & 0\\
& \\
0 & {\bf R}^{3}_{2}
\end{array}
\right)
\oplus SU(2) \oplus SU(2)$$
If ${\bf R}^{3}_{1} \oplus {\bf R}^{3}_{2}$ is identified with the local tangent space of the 6-dimensional manifold $$U(3) / Spin(3)$$ then, since $Spin(3) = SU(2)$, we have constructed a $U(3) \times SU(2)$ subgroup of $Spin(0,8)$.
The 6-dimensional manifold $U(3) / Spin(3)$ is built up from the 5-dimensional irreducible symmetric space $$SU(3) / Spin(3)$$ (see the book $Einstein \; Manifolds$ by Besse [@BES]) by adding the $U(1)$ from $U(1) = U(3) / SU(3)$.
Now that we have built $U(3) \times SU(2)$ from the dimensional reduction process acting on $Spin(0,8)$, note that $U(3)$ = $SU(3) \times U(1)$ so that we have the 12-dimensional Standard Model gauge group $$SU(3) \times SU(2) \times U(1)$$ Actually, it is even nicer to say that it is $$U(3) \times SU(2)$$ by putting the electromagnetic $U(1)$ with the color force $SU(3)$, because, as O’Raifeartaigh says in section 9.4 of his book $Group \; Structure \; of \; Gauge \; Theories$ [@ORA], that is the most natural representation of the Standard Model gauge groups. This is because, when fermion representations are taken into account, the unbroken symmetry is the $U(3)$ of electromagnetism and the color force, while the weak force $SU(2)$ is broken by the Higgs mechanism.
One of the few differences between the Standard Model sector of the 4-dimensional $D_{4}-D_{5}-E_{6}$ model and the Standard Model electroweak structure is that in the $D_{4}-D_{5}-E_{6}$ model the electromagnetic $U(1)$ is most naturally put with the color force $SU(3)$, while in the electroweak Standard Model the electromagnetic $U(1)$ is most naturally put with the weak force $SU(2)$. As O’Raifearteagh [@ORA] points out, that difference is an advantage of the $D_{4}-D_{5}-E_{6}$ model.
Now, look at the Standard Model sector of $D_{4}-D_{5}-E_{6}$ model from the matrix representation point of view:
Consider a $U(4)$ subalgebra of $Spin(0,8)$.
$U(4)$ can be represented (see section 412 G of [@EDM]) as a subalgebra of $Spin(0,8)$ by
$$\begin{array}{|c|c|}
\hline
{\bf{Re}}(U_{4}) &{\bf{Im}}(U_{4}) \\
\hline
{\bf{-Im}}(U_{4}) & {\bf{Re}}(U_{4}) \\
\hline
\end{array}$$
Therefore, the $U(4)$ subalgebra of $Spin(0,8)$ can be represented by
$$\left( \begin{array} {cccccccc}
0 & u_{12} & u_{13} & u_{14} & v_{11} & v_{12} & v_{13} & v_{14} \\
-u_{12} & 0 & u_{23} & u_{24} & v_{12} & v_{22} & v_{23} & v_{24} \\
-u_{13} &-u_{23} & 0 & u_{34} & v_{13} & v_{23} & v_{33} & v_{34} \\
-u_{14} &-u_{24} &-u_{34} & 0 & v_{14} & v_{24} & v_{34} & v_{44} \\
-v_{11} &-v_{12} &-v_{13} &-v_{14} & 0 & u_{12} & u_{13} & u_{14}\\
-v_{12} &-v_{22} &-v_{23} &-v_{24} &-u_{12} & 0 & u_{23} & u_{24}\\
-v_{13} &-v_{23} &-v_{33} &-v_{34} &-u_{13} &-u_{23} & 0 & u_{34}\\
-v_{14} &-v_{24} &-v_{34} &-v_{44} &-u_{14} &-u_{24} &-u_{34} & 0\\
\end{array} \right)$$
(Compare this representation of $U(4)$ with the representation above of the isomorphic algebra $Spin(0,6) \times U(1)$.)
A 12-dimensional subalgebra is
$$\left( \begin{array} {cccccccc}
0 & u_{12} & u_{13} & u_{14} & 0 & v_{12} & v_{13} & v_{14} \\
-u_{12} & 0 & u_{23} & u_{24} & v_{12} & 0 & v_{23} & v_{24} \\
-u_{13} &-u_{23} & 0 & u_{34} & v_{13} & v_{23} & 0 & v_{34} \\
-u_{14} &-u_{24} &-u_{34} & 0 & v_{14} & v_{24} & v_{34} & 0 \\
0 &-v_{12} &-v_{13} &-v_{14} & 0 & u_{12} & u_{13} & u_{14} \\
-v_{12} & 0 &-v_{23} &-v_{24} &-u_{12} & 0 & u_{23} & u_{24} \\
-v_{13} &-v_{23} & 0 &-v_{34} &-u_{13} &-u_{23} & 0 & u_{34} \\
-v_{14} &-v_{24} &-v_{34} & 0 &-u_{14} &-u_{24} &-u_{34} & 0 \\
\end{array} \right)$$
As we have seen in this subsection, the 28 infinitesimal generators of $Spin(8)$ are broken by dimensional reduction into two parts:
16-dimensional $U(4)$ = $Spin(0,6) \times U(1)$, where $Spin(0,6)$ = $SU(4)$ is the conformal group of 4-dimensional spacetime (the conformal group gives gravity by the MacDowell-Mansouri mechanism [@MAC; @MOH], and gauge fixing of the conformal group gives the Higgs scalar symmetry breaking) and the $U(1)$ is the complex phase propagators in the 4-dimensional spacetime; and
Physically, the $U(1)$ is the Dirac complexification and it gives the physical Dirac gammas their complex structure, so that they are ${\bf C}(4)$ instead of ${\bf R}(4)$.
Dirac complexification justifies the physical use of Wick rotation between Euclidean and Minkowski spacetimes, because ${\bf{C}}(4)$ is the Clifford algebra of both the compact Euclidean deSitter Lie group $Spin(0,5)$ and the non-compact Minkowski anti-deSitter Lie group $Spin(2,3)$.
12-dimensional $U(3) \times SU(2)$, where $SU(2)$ is the gauge group of the weak force and $U(3)$ = $SU(3) \times U(1)$ is the $SU(3)$ gauge group of the color force and the $U(1)$ gauge group of electromagnetism.
### Conformal Gravity and Higgs Scalar.
$Spin(0,6)$ is the maximal subgroup of $Spin(0,8)$ that acts on the 4-dimensional reduced spacetime. It acts as the compact version of the conformal group. An $8 \times 8$ matrix representation of $Spin(0,6)$ is
$$\left( \begin{array} {cccccccc}
0 & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} & 0 & 0 \\
-a_{12} & 0 & a_{23} & a_{24} & a_{25} & a_{26} & 0 & 0 \\
-a_{13} &-a_{23} & 0 & a_{34} & a_{35} & a_{36} & 0 & 0 \\
-a_{14} &-a_{24} &-a_{34} & 0 & a_{45} & a_{46} & 0 & 0 \\
-a_{15} &-a_{25} &-a_{35} &-a_{45} & 0 & a_{56} & 0 & 0 \\
-a_{16} &-a_{26} &-a_{36} &-a_{46} &-a_{56} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{array} \right)$$
The MacDowell-Mansouri mechanism [@MAC] produces a classical model of gravity from a $Spin(0,5)$ de Sitter gauge group. As a subalgebra of the $Spin(0,6)$ Lie algebra, the $Spin(0,5)$ Lie algebra can be represented by
$$\left( \begin{array} {cccccccc}
0 & a_{12} & a_{13} & a_{14} & a_{15} & 0 & 0 & 0 \\
-a_{12} & 0 & a_{23} & a_{24} & a_{25} & 0 & 0 & 0 \\
-a_{13} &-a_{23} & 0 & a_{34} & a_{35} & 0 & 0 & 0 \\
-a_{14} &-a_{24} &-a_{34} & 0 & a_{45} & 0 & 0 & 0 \\
-a_{15} &-a_{25} &-a_{35} &-a_{45} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{array} \right)$$
The 5 elements $\{ a_{16}, a_{26},a_{36},a_{46},a_{56} \}$ that are in $Spin(0,6)$ but not in $Spin(0,5)$ are the 1 scale and 4 conformal degrees of freedom that are gauge-fixed by Mohapatra in section 14.6 of [@MOH] to get from the 15-dimensional conformal group $Spin(0,6)$ to the 10-dimensional deSitter group $Spin(0,5)$ so that the MacDowell-Mansouri mechanism [@MAC] can be used to produce Einstein-Hilbert gravity plus a cosmological constant term, an Euler topological term, and a Pontrjagin topological term. As Nieto, Obregon, and Socorro [@NIE] have shown, MacDowell-Mansouri deSitter gravity is equivalent to Ashtekar gravity plus a cosmological constant term, an Euler topological term, and a Pontrjagin topological term.
For further discussion of the MacDowell-Mansouri mechanism, see Freund [@FRE] (chapter 21), or Ne’eman and Regge [@NEE](at pages 25-28), or [Nieto, Obregon, and Socorro](http://xxx.lanl.gov/abs/gr-qc/9402029) [@NIE]
The physical reason for fixing the scale and conformal degrees of freedom lies in the relationship between gravity and the Higgs mechanism.
Since all rest mass comes from the Higgs mechanism, and since rest mass interacts through gravity, it is natural for gravity and Higgs symmetry breaking to be related at a fundamental level.
As remarked by [Sardanashvily](http://xxx.lanl.gov/abs/gr-qc/9410045) [@SAR4], Heisenberg and Ivanenko in the 1960s made the first atttempt to connect gravity with a symmetry breaking mechanism by proposing that the graviton might be a Goldstone boson resulting from breaking Lorentz symmetry in going from flat MInkowski spacetime to curved spacetime.
[Sardanashvily](http://xxx.lanl.gov/abs/gr-qc/9410045) [@SAR4] (see also [gr-qc/9405013](http://xxx.lanl.gov/abs/gr-qc/9405013) [@SAR2] [gr-qc/9407032](http://xxx.lanl.gov/abs/gr-qc/9407032) [@SAR3] [gr-qc/9411013](http://xxx.lanl.gov/abs/gr-qc/9411013) [@SAR5]) proposes that gravity be represented by a gauge theory with group $GL(4)$, that $GL(4)$ symmetry can be broken to either Lorentz $SO(3,1)$ symmetry or $SO(4)$ symmetry, and that the resulting Higgs fields can be interpreted as either the gravitational field (for breaking to $SO(3,1)$ or the Riemannian metric (for breaking to $SO(4)$.
The identification of a pseudo-Riemannian metric with a Higgs field was made by Trautman [@TRA], by Sardanashvily [@SAR1] , and by Ivanenko and Sardanashvily [@IVA]
In the $D_{4}-D_{5}-E_{6}$ model (using here the compact version) the conformal group $Spin(0,6)$ = $SU(4)$ is broken to the de Sitter group $Spin(0,5)$ = $Sp(2)$ by fixing the 1 scale and 4 conformal gauge degrees of freedom.
The resulting Higgs field is interpreted in the $D_{4}-D_{5}-E_{6}$ model as the same Higgs field that gives mass to the $SU(2)$ weak bosons and to the Dirac fermions by the Higgs mechanism.
The Higgs mechanism requires “spontaneous symmetry breaking” of a scalar field potential whose minima are not zero, but which form a 3-sphere $S^{3}$ = $SU(2)$.
In particular, one real component of the complex Higgs scalar doublet is set to $v / \sqrt{2)}$, where $v$ is the modulus of the $S^{3}$ of minima, usually called the vacuum expectation value.
If the $S^{3}$ is taken to be the unit quaternions, then the “spontaneous symmetry breaking” requires choosing a (positive) real axis for the quaternion space.
In the standard model, it is assumed that a random vacuum fluctuation breaks the $SU(2)$ symmetry and in effect chooses a real axis at random.
In the $D_{4}-D_{5}-E_{6}$ model, the symmetry breaking from conformal $Spin(0,6)$ to de Sitter $Spin(0,5)$ by fixing the 1 scale and 4 conformal gauge degrees of freedom is a symmetry breaking mechanism that does not require perturbation by a random vacuum fluctuation.
Gauge-fixing the 1 scale degree of freedom fixes a length scale. It can be chosen to be the magnitude of the vacuum expectation value, or radius of the $S^{3}$.
Gauge-fixing the 4 conformal degrees of freedom fixes the (positive) real axis of the $S^{3}$ consistently throughout 4-dimensional spacetime.
Therefore, the $D_{4}-D_{5}-E_{6}$ model Higgs field comes from the breaking of $Spin(0,6)$ conformal symmetry to $Spin(0,5)$ de Sitter gauge symmetry, from which Einstein-Hilbert gravity can be constructed by the MacDowell-Mansouri mechanism.
Einstein-Hilbert gravity as a spin-2 field theory in flat spacetime: Feynman, in his 1962-63 lectures at Caltech [@FEY], showed how Einstein-Hilbert gravity can be described by starting with a linear spin-2 field theory in flat spacetime, and then adding higher-order terms to get Einstein-Hilbert gravity. The observed curved spacetime is based on an unobservable flat spacetime. (see also Deser [@DES]
The Feynman spin-2 flat spacetime construction of Einstein-Hilbert gravity allows the $D_{4}-D_{5}-E_{6}$ model to be based on a fundamental $D_{4}$ lattice 4-dimensional spacetime.
Quantization in the $D_{4}-D_{5}-E_{6}$ model is fundamentally based on a path integral sum over histories of paths in a $D_{4}$ lattice spacetime using a generalized Feynman checkerboard.
The generalized Feynman checkerboard is discussed at [](http://www.gatech.edu/tsmith/FynCkbd.html) [@SMI6].
Fundamentally, that is nice, but calculations can be very difficult, particularly for quantum gravity.
The work of [Garcia-Compean et. al](http://xxx.lanl.gov/abs/hep-th/9408003) [@GAR] suggests that the most practical approach to quantum gravity may be through BRST symmetry.
Quantization breaks the gauge group invariance of the $D_{4}-D_{5}-E_{6}$ model Lagrangian, because the path integral must not overcount paths by including more than one representative of each gauge-equivalence class of paths. The remaining quantum symmetry is the symmetry of BRST cohomology classes. Knowledge of the BRST symmetry tells you which ghosts must be used in quantum calculations, so the BRST cohomology can be taken to be the basis for the quantum theory.
[Garcia-Compean et. al](http://xxx.lanl.gov/abs/hep-th/9408003) [@GAR] discuss two current approaches to quantum gravity:
string theory, which abandons point particles even at the classical level; and
redefinition of classical general relativity in terms of new variables, the [Ashtekar variables](http://vishnu.nirvana.phys.psu.edu/newvariables.ps) [@ASH] and trying to use the new variables to construct a quantum theory of gravity.
[Nieto, Obregon, and Socorro](http://xxx.lanl.gov/abs/gr-qc/9402029) [@NIE] have shown that the MacDowell-Mansouri $Spin(0,5) = Sp(2)$ de Sitter Lagrangian for gravity used in the $D_{4}-D_{5}-E_{6}$ model is equal to the Lagrangian for gravity in terms of the [Ashtekar variables](http://vishnu.nirvana.phys.psu.edu/newvariables.ps) [@ASH] plus
a cosmological constant term,
an Euler topological term, and
a Pontrjagin topological term.
Therefore, although the quantum gravity methods of string theory cannot be used in the $D_{4}-D_{5}-E_{6}$ model because the $D_{4}-D_{5}-E_{6}$ model uses fundamental point particles at the classical level,
the methods based on [Ashtekar variables](http://vishnu.nirvana.phys.psu.edu/newvariables.ps) [@ASH] are available.
Two such approaches are:
a topological approach based on loop groups; and
an algebraic approach based on getting BRST transformations from Maurer-Cartan horizontality conditions.
The latter approach, which is described in [Blaga, et. al.](http://xxx.lanl.gov/abs/hep-th/9409046) [@BLA] is the approach used for the $D_{4}-D_{5}-E_{6}$ model.
### Chern-Simons Time
An essential part of a quantum theory of gravity is the correct definition of physical time.
[Nieto, Obregon, and Socorro](http://xxx.lanl.gov/abs/gr-qc/9402029) [@NIE] have shown that Lagrangian action of the [Ashtekar variables](http://vishnu.nirvana.phys.psu.edu/newvariables.ps) [@ASH] is a Chern-Simons action if the Killing metric of the de Sitter group is used instead of the Levi-Civita tensor.
[Smolin and Soo](http://xxx.lanl.gov/abs/gr-qc/9405015) [@SMO] have shown that the Chern-Simons invariant of the Ashtekar-Sen connection is a natural candidate for the internal time coordinate for classical and quantum cosmology, so that the $D_{4}-D_{5}-E_{6}$ model uses Chern-Simons time.
### Quantum Gravity plus Standard Model
Another essential part of a quantum theory of gravity is the correct relationship of quantum gravity with the quantum theory of the forces and particles of the Standard Model, to calculate how standard model particles and fields interact in the presence of gravity.
[Moritsch, et. al.](http://xxx.lanl.gov/abs/hep-th/9409081) [@MOR] have done this by using Maurer-Cartan horizontality conditions to get BRST transformations for Yang-Mills gauge fields in the presence of gravity.
### Color, Weak, and Electromagnetic Forces.
12-dimensional $U(3) \times SU(2)$, where $SU(2)$ is the gauge group of the weak force and $U(3)$ = $SU(3) \times U(1)$ is the $SU(3)$ gauge group of the color force and the $U(1)$ gauge group of electromagnetism, can be represented in terms of $8 \times 8$ matrices by
$$\left( \begin{array} {cccccccc}
0 & u_{12} & u_{13} & u_{14} & 0 & v_{12} & v_{13} & v_{14} \\
-u_{12} & 0 & u_{23} & u_{24} & v_{12} & 0 & v_{23} & v_{24} \\
-u_{13} &-u_{23} & 0 & u_{34} & v_{13} & v_{23} & 0 & v_{34} \\
-u_{14} &-u_{24} &-u_{34} & 0 & v_{14} & v_{24} & v_{34} & 0 \\
0 &-v_{12} &-v_{13} &-v_{14} & 0 & u_{12} & u_{13} & u_{14} \\
-v_{12} & 0 &-v_{23} &-v_{24} &-u_{12} & 0 & u_{23} & u_{24} \\
-v_{13} &-v_{23} & 0 &-v_{34} &-u_{13} &-u_{23} & 0 & u_{34} \\
-v_{14} &-v_{24} &-v_{34} & 0 &-u_{14} &-u_{24} &-u_{34} & 0 \\
\end{array} \right)$$
In terms of the fibrations of $S^{7}$ and $G_{2}$, the 12-dimensional subalgebra $U(3) \times SU(2)$ is represented by
$$\left(
\begin{array}{cc}
S^{3}_{a} & 0\\
& \\
0 & S^{3}_{b}
\end{array}
\right)
\oplus SU(2) \oplus SU(2)$$
The result of this decomposition is that the gauge boson bivector part of the 8-dimensional $D_{4}-D_{5}-E_{6}$ Lagrangian $$\int_{V_{8}} F_{Spin(0,8)} \wedge \star F_{Spin(0,8)}$$ breaks down into $$\int_{V_{4}} F_{Spin(0,6)} \wedge \star F_{Spin(0,6)} \oplus
F_{U(1)} \wedge \star F_{U(1)} \oplus
F_{SU(3)} \wedge \star F_{SU(3)} \oplus
F_{SU(2)} \wedge \star F_{SU(2)}$$ The 28 gauge bosons of 8-dimensional $Spin(0,8)$ are broken into four independent (or commuting, from the Lie algebra point of view) sets of gauge bosons:
15 for gravity and Higgs symmetry breaking $Spin(0,6)$, plus 1 for the $U(1)$ propagator phase;
1 for $U(1)$ electromagentism;
8 for color $SU(3)$; and
3 for the $SU(2)$ weak force.
Each of the terms of the form $$\int_{V_{4}} F \wedge \star F$$ contains a force strength constant.
The force strength constants define the relative strengths of the four forces.
One of the factors determining the force strength constants is the relative magnitude of the measures of integration over the 4-dimensional spacetime base manifold in each integral.
The relative magnitude of the measures is proportional to the volume $Vol(M_{force})$ of the irreducible $m_{force}$(real)-dimensional symmetric space on which the gauge group acts naturally as a component of 4(real) dimensional spacetime $M_{force}^{\left( 4 \over m_{force} \right)}$.
The $M_{force}$ manifolds for the gauge groups of the four forces are:
$$\begin{array}{|c|c|c|c|}
\hline
Gauge \: Group & Symmetric \: Space & m_{force}
& M_{force} \\
\hline
& & & \\
Spin(5) & Spin(5) \over Spin(4) & 4 & S^4\\
& & & \\
SU(3) & SU(3) \over {SU(2) \times U(1)}
& 4 & {\bf C}P^2 \\
& & & \\
SU(2) & SU(2) \over U(1) & 2 & S^2 \times S^2 \\
& & & \\
U(1) & U(1) & 1 & S^1 \times S^1 \times S^1
\times S^1 \\
& & & \\
\hline
\end{array}$$
Further discussion of this factor is in [](http://www.gatech.edu/tsmith/See.html) [@SMI6].
The second factor in the force strengths is based on the interaction of the gauge bosons with the spinor fermions through the covariant derivative.
When the spinor fermion term is added to the 4-dimensional $D_{4}-D_{5}-E_{6}$ Lagrangian for each force, you get a Lagrangian of the form $$\int_{V_{4}} F \wedge \star F +
\overline{S_{8\pm}} \not \! \partial_{8} S_{8\pm}$$ The covariant derivative part of the Dirac operator $\not \! \partial_{8}$ gives the interaction between each of the four forces and the spinor fermions.
The strength of each force depends on the magnitude of the interaction of the covariant derivative of the force with the spinor fermions. Since the spinor fermions are defined with respect to a space $Q$ = $S^{7} \times {\bf R}P^{1}$ that is the Shilov boundary of a bounded complex homogeneous domain $D$, the relative strength of each force can be measured by the relative volumes of the part of the manifolds $Q$ and $D$ that are affected by that force.
Let $Vol(Q_{force})$ be the volume of that part of the full compact fermion state space manifold ${\bf R}P^1 \times S^7$ on which a gauge group acts naturally through its charged (color or electromagnetic charge) gauge bosons.
For the forces with charged gauge bosons,
$Spin(5)$ gravity,
$SU(3)$ color force, and
$SU(2)$ weak force,
$Q_{force}$ is the Shilov boundary of the bounded complex homogeneous domain $D_{force}$ that corresponds to the Hermitian symmetric space on which the gauge group acts naturally as a local isotropy (gauge) group.
For $U(1)$ electromagnetism, whose photon carries no charge, the factors $Vol(Q_{U(1)})$ and $Vol(D_{U(1)})$ do not apply and are set equal to $1$.
The volumes $Vol(M_{force})$, $Vol(Q_{force})$, and $Vol(D_{force})$ are calculated with $M_{force}, Q_{force}, D_{force}$ normalized to unit radius.
The factor $1 \over {{Vol(D_{force})}^{\left( 1 \over m_{force} \right)}}$ is a normalization factor to be used if the dimension of $Q_{force}$ is different from the dimension $m_{force}$, in order to normalize the radius of $Q_{force}$ to be consistent with the unit radius of $M_{force}$.
The $Q_{force}$ and $D_{force}$ manifolds for the gauge groups of the four forces are:
$$\begin{array}{|c|c|c|c|c|}
\hline
Gauge & Hermitian & Type & m_{force}
& Q_{force} \\
Group & Symmetric & of & & \\
& Space & D_{force} & & \\
\hline
& & & & \\
Spin(5) & Spin(7) \over {Spin(5) \times U(1)}
& IV_{5} &4 & {\bf R}P^1 \times S^4 \\
& & & & \\
SU(3) & SU(4) \over {SU(3) \times U(1)}
& B^6 \: (ball) &4 & S^5 \\
& & & & \\
SU(2) & Spin(5) \over {SU(2) \times U(1)}
& IV_{3} & 2 & {\bf R}P^1 \times S^2 \\
& & & & \\
U(1) & - & - & 1 & - \\
& & & & \\
\hline
\end{array}$$
The third factor affects only the force of gravity, which has a characteristic mass because the Planck length is the fundamental lattice length in the $D_{4}-D_{5}-E_{6}$ model, so that $\mu_{Spin(0,5)} =
M_{Planck}$ and the weak force, whose gauge bosons acquire mass by the Higgs mechanism, so that $\mu_{Spin(0,5)} =
\sqrt{m_{W+}^{2} + m_{W-}^{2} + m_{W_{0}}^{2}}$.
For the weak force, the relevant factor is $${1 \over \mu_{force}^2} = {1 \over {m_{W+}^{2}
+m_{W-}^{2} + m_{W_{0}}^{2}}}$$
For gravity, it is $${1 \over \mu_{force}^2} = {1 \over {m_{Planck}^{2}}}$$
For the $SU(3)$ color and $U(1)$ electromagnetic forces, $${1 \over \mu_{force}^2} = 1$$
Taking all the factors into account, the calculated strength of a force is taken to be proportional to the product: $$\left(1 \over \mu_{force}^2 \right) \left( Vol(M_{force})
\right)
\left( {Vol(Q_{force})} \over {{Vol(D_{force})}^{ \left( 1
\over m_{force} \right) }} \right)$$
The geometric force strengths, that is, everything but the mass scale factors $1 / \mu_{force}^{2}$, normalized by dividing them by the largest one, the one for gravity.
The geometric volumes needed for the calculations, mostly taken from Hua [@HUA], are
$$\begin{array}{||c||c|c||c|c||c|c||}
\hline
Force & M & Vol(M) & Q
& Vol(Q) & D & Vol(D \\
\hline
& & & & & & \\
gravity & S^4 & 8\pi^{2}/3
& {\bf R}P^1 \times S^4 & 8\pi^{3}/3
& IV_{5} & \pi^{5}/2^{4} 5! \\
\hline
& & & & & &\\
color & {\bf C}P^2 & 8\pi^{2}/3
& S^5 & 4\pi^{3}
& B^6 \: (ball) & \pi^{3}/6 \\
\hline
& & & & & & \\
weak & {S^2} \times {S^2} & 2 \times {4 \pi}
& {\bf R}P^1 \times S^2 & 4 \pi^2
& IV_{3} & \pi^{3} / 24 \\
\hline
& & & & & & \\
e-mag & T^4 & 4 \times {2\pi}
& - & -
& - & - \\
\hline
\end{array}$$
Using these numbers, the results of the calculations are the relative force strengths at the characteristic energy level of the generalized Bohr radius of each force:
$$\begin{array}{|c|c|c|c|c|}
\hline
Gauge \: Group & Force & Characteristic
& Geometric & Total \\
& & Energy & Force & Force \\
& & & Strength & Strength \\
\hline
& & & & \\
Spin(5) & gravity & \approx 10^{19} GeV
& 1 & G_{G}m_{proton}^{2} \\
& & & & \approx 5 \times 10^{-39} \\
\hline
& & & & \\
SU(3) & color & \approx 245 MeV & 0.6286
& 0.6286 \\
\hline
& & & & \\
SU(2) & weak & \approx 100 GeV & 0.2535
& G_{W}m_{proton}^{2} \approx \\
& & & & \approx 1.02 \times 10^{-5} \\
\hline
& & & & \\
U(1) & e-mag & \approx 4 KeV
& 1/137.03608 & 1/137.03608 \\
\hline
\end{array}$$
The force strengths are given at the characteristic energy levels of their forces, because the force strengths run with changing energy levels.
The effect is particularly pronounced with the color force.
In [WWW URL http://www.gatech.edu/tsmith/cweRen.html](http://www.gatech.edu/tsmith/cweRen.html) [@SMI6], the color force strength was calculated at various energies according to renormalization group equations, with the following results:
$$\begin{array}{|c|c|}
\hline
Energy \: Level & Color \: Force \: Strength \\
\hline
& \\
245 MeV & 0.6286 \\
& \\
5.3 GeV & 0.166 \\
& \\
34 GeV & 0.121 \\
& \\
91 GeV & 0.106 \\
& \\
\hline
\end{array}$$
Shifman [WWW URL http://xxx.lan.gov/abs/hep-ph/9501222](http://xxx.lan.gov/abs/hep-ph/9501222) [@SHI] has noted that Standard Model global fits at the $Z$ peak, about $91 \; GeV$, give a color force strength of about 0.125 with $\Lambda_{QCD} \approx 500 \; MeV$, whereas low energy results and lattice calculations give a color force strength at the $Z$ peak of about 0.11 with $\Lambda_{QCD} \approx 200 \; MeV$.
The low energy results and lattice calculations are closer to the tree level $D_{4}-D_{5}-E_{6}$ model value at $91 \; GeV$ of 0.106. Also, the $D_{4}-D_{5}-E_{6}$ model has $\Lambda_{QCD} \approx 245 \; MeV$ (For the pion mass, upon which the $\Lambda_{QCD}$ calculation depends, see [WWW URL http://www.gatech.edu/tsmith/SnGdnPion.html](http://www.gatech.edu/tsmith/SnGdnPion.html) [@SMI6].)
Fermion Part of the Lagrangian
------------------------------
Consider the spinor fermion term $\int {\overline{S_{8\pm}}
\not \! \partial_{8} S_{8\pm}}$
For each of the surviving 4-dimensional $4$ and reduced 4-dimensional $\perp 4$ of 8-dimensional spacetime, the part of $S_{8\pm}$ on which the Higgs $SU(2)$ acts locally is $Q_{3} = {\bf{R}}P^{1} \times S^{2}$.
It is the Shilov boundary of the bounded domain $D_{3}$ that is isomorphic to the symmetric space $\overline{D_{3}} = Spin(5)/SU(2) \times U(1)$.
The Dirac operator $\not \! \partial_{8} $ decomposes as $\not \! \partial = \not \! \partial_{4} +
\not \! \partial_{\perp 4}$, where
$\not \! \partial_{4}$ is the Dirac operator corresponding to the surviving spacetime $4$ and
$\not \! \partial_{\perp 4}$ is the Dirac operator corresponding to the reduced 4 $\perp 4$.
Then the spinor term is $\int {\overline{S_{8\pm}} \not \! \partial_{4} S_{8\pm}} +
\overline{S_{8\pm}} \not \! \partial_{\perp 4} S_{8\pm}$
The Dirac operator term $\not \! \partial_{\perp 4}$ in the reduced $\perp 4$ has dimension of mass.
After integration $\int {\overline{S_{8\pm}} \not \!
\partial_{\perp 4} S_{8\pm}}$ over the reduced $\perp 4$,
$\not \! \partial_{\perp 4}$ becomes the real scalar Higgs scalar field $Y = (v + H)$ that comes
from the complex $SU(2)$ doublet $\Phi$ after action of the Higgs mechanism.
If integration over the reduced $\perp 4$ involving two fermion terms $\overline{S_{8\pm}}$ and $S_{8\pm}$ is taken to change the sign by $i^{2} = -1$, then, by the Higgs mechanism,
$\int \overline{S_{8\pm}} \not \! \partial_{\perp 4} S_{8\pm}
\rightarrow \int(\int_{\perp 4} \overline{S_{8\pm}}
\not \! \partial_{\perp 4} S_{8\pm} ) \rightarrow $
$\rightarrow - \int \overline{S_{8\pm }} YY S_{8\pm } =
- \int \overline{S_{8\pm }} Y(v + H) S_{8\pm }$,
where:
$H$ is the real physical Higgs scalar, $m_{H} = v \sqrt(\lambda / 2)$, and $v$ is the vacuum expectation value of the scalar field $Y$, the free parameter in the theory that sets the mass scale.
Denote the sum of the three weak boson masses by $\Sigma_{m_{W}}$.
$v = \Sigma_{m_{W}}((\sqrt{2}) / \sqrt{\alpha_{w}}) =
260.774 \times \sqrt{2} / 0.5034458 = 732.53 \; GeV$,
a value chosen so that the electron mass will be 0.5110 MeV.
The Higgs vacuum expectation value $v = ( v_{+} + v_{-} + v_{0} )$ is the only particle mass free parameter.
In the $D_{4}-D_{5}-E_{6}$ model, $v$ is set so that the electron mass $m_{e} = 0.5110 MeV$.
In the $D_{4}-D_{5}-E_{6}$ model, $\alpha_{w}$ is calculated to be $\alpha_{w} = 0.2534577$,
so $\sqrt{\alpha_{w}}$ = 0.5034458 and $v$ = 732.53 GeV.
The Higgs mass $m_{H}$ is given by the term $$(1/2)(\partial H)^{2} - (1/2)(\mu^{2} / 2)H^{2} =
(1/2) [ (\partial H)^{2} - (\mu^{2}/2) H^{2} ]$$ to be $$m_{H}^{2} = \mu^{2} / 2 = \lambda v^{2} / 2$$ so that $$m_{H} = \sqrt{(\mu^{2} / 2)} =
\sqrt{\lambda} v^{2} / 2)$$
$\lambda$ is the scalar self-interaction strength. It should be the product of the “weak charges” of two scalars coming from the reduced 4 dimensions in $Spin(4)$, which should be the same as the weak charge of the surviving weak force $SU(2)$ and therefore just the square of the $SU(2)$ weak charge, $\sqrt{(\alpha_{w}^{2})} = \alpha_{w}$, where $\alpha_{w}$ is the $SU(2)$ geometric force strength.
Therefore $\lambda = \alpha_{w} = 0.2534576$, $\sqrt{\lambda} = 0.5034458$, and $v$ = 732.53 GeV,
so that the mass of the Higgs scalar is $$m_{H} = v \sqrt(\lambda / 2) = 260.774 \; GeV.$$
$Y$ is the Yukawa coupling between fermions and the Higgs field.
$Y$ acts on all 28 elements (2 helicity states for each of the 7 Dirac particles and 7 Dirac antiparticles) of the Dirac fermions in a given generation, because all of them are in the same $Spin(0,8)$ spinor representation.
### Calculation of Particle Masses.
Denote the sum of the first generation Dirac fermion masses by $\Sigma_{f_{1}}$.
Then $Y = (\sqrt{2}) \Sigma_{f_{1}} / v$, just as $\sqrt(\alpha_{w}) = (\sqrt{2}) \Sigma_{m_{W}} / v$.
$Y$ should be the product of two factors:
$e^{2}$, the square of the electromagnetic charge $e = \sqrt{\alpha_{E}}$ , because in the term $\int(\int_{\perp 4} \overline{S_{8\pm }} \not \!
\partial _{\perp 4} S_{8\pm } ) \rightarrow
- \int \overline{S_{8\pm }} Y(v + H) S_{8\pm }$ each of the Dirac fermions $S_{8\pm}$ carries electromagnetic charge proportional to $e$ ; and
$1/g_{w}$, the reciprocal of the weak charge $g_{w} = \sqrt{\alpha_{w}}$, because an $SU(2)$ force, the Higgs $SU(2)$, couples the scalar field to the fermions.
Therefore $$\Sigma_{f_{1}} = Y v / \sqrt{2} =
(e^{2} / g_{w}) v / \sqrt{2} = 7.508 \; GeV$$ and
$$\Sigma_{f_{1}} / \Sigma_{m_{W}} = (e^{2} / g_{w}) v /
g_{w} v = e^{2} / g_{w}^{2} = \alpha_{E} / \alpha_{w}$$
The Higgs term $- \int \overline{S_{8\pm}} Y(v + H)$ $S_{8\pm} = - \int \overline{S_{8\pm}}$ $Yv S_{8\pm} - \int \overline{S_{8\pm}} YH S_{8\pm } = $ $= - \int \overline{S_{8\pm}} (\sqrt{2} \Sigma_{f_{1}})
S_{8\pm } - \int \overline{S_{8\pm}} (\sqrt{2}
\Sigma_{f_{1}} / v) S_{8\pm}$.
The resulting spinor term is of the form $\int [ \overline{S_{8\pm}} (\not \! \partial - Yv)
S_{8\pm} - \overline{S_{8\pm}} YH S_{8\pm} ]$ where $(\not \! \partial - Yv)$ is a massive Dirac operator.
How much of the total mass $\Sigma_{f_{1}} =
Y v / \sqrt{2} = 7.5 \; GeV$ is allocated to each of the first generation Dirac fermions is determined by calculating the individual fermion masses in the $D_{4}-D_{5}-E_{6}$ model, and
those calculations also give the values of $$\Sigma_{f_{2}} = 32.9 \; GeV$$ $$\Sigma_{f_{3}} = 1,629 \; GeV$$
as well as second and third generation individual fermion masses, with the result that the individual tree-level lepton masses and quark constituent masses are:
$m_{e}$ = 0.5110 MeV (assumed); $m_{\nu_{e}}$ = $m_{\nu_{\mu}}$ = $m_{\nu_{\tau}}$ = 0; $m_{d}$ = $m_{u}$ = 312.8 MeV (constituent quark mass); $m_{\mu}$ = 104.8 MeV; $m_{s}$ = 625 MeV (constituent quark mass); $m_{c}$ = 2.09 GeV (constituent quark mass); $m_{\tau}$ = 1.88 GeV; $m_{b}$ = 5.63 GeV (constituent quark mass); v $$m_{t} \; = \; 130 \; GeV \; (constituent \; quark \; mass).$$
Here is how the individual fermion mass calculations are done in the $D_{4}-D_{5}-E_{6}$ model.
The Weyl fermion neutrino has at tree level only the left-handed state, whereas the Dirac fermion electron and quarks can have both left-handed and right-handed states, so that the total number of states corresponding to each of the half-spinor $Spin(0,8)$ representations is 15.
Neutrinos are massless at tree level in all generations.
In the $D_{4}-D_{5}-E_{6}$model, the first generation fermions correspond to octonions ${\bf O}$, while second generation fermions correspond to pairs of octonions ${\bf O} \times {\bf O}$ and third generation fermions correspond to triples of octonions ${\bf O} \times {\bf O} \times {\bf O}$.
To calculate the fermion masses in the model, the volume of a compact manifold representing the spinor fermions $S_{8+}$ is used. It is the parallelizable manifold $S^7\times RP^1$.
Also, since gravitation is coupled to mass, the infinitesimal generators of the MacDowell-Mansouri gravitation group, $Spin(0,5)$, are relevant.
The calculated quark masses are constituent masses, not current masses.
In the $D_{4}-D_{5}-E_{6}$ model, fermion masses are calculated as a product of four factors: $$V(Q_{fermion}) \times N(Graviton) \times N(octonion) \times Sym$$
$V(Q_{fermion})$ is the volume of the part of the half-spinor fermion particle manifold $S^7\times RP^1$ that is related to the fermion particle by photon, weak boson, and gluon interactions.
$N(Graviton)$ is the number of types of $Spin(0,5)$ graviton related to the fermion. The 10 gravitons correspond to the 10 infinitesimal generators of $Spin(0,5)$ = $Sp(2)$. 2 of them are in the Cartan subalgebra. 6 of them carry color charge, and may therefore be considered as corresponding to quarks. The remaining 2 carry no color charge, but may carry electric charge and so may be considered as corresponding to electrons. One graviton takes the electron into itself, and the other can only take the first-generation electron into the massless electron neutrino. Therefore only one graviton should correspond to the mass of the first-generation electron. The graviton number ratio of the down quark to the first-generation electron is therefore 6/1 = 6.
$N(octonion)$ is an octonion number factor relating up-type quark masses to down-type quark masses in each generation.
$Sym$ is an internal symmetry factor, relating 2nd and 3rd generation massive leptons to first generation fermions. It is not used in first-generation calculations.
The ratio of the down quark constituent mass to the electron mass is then calculated as follows: Consider the electron, e. By photon, weak boson, and gluon interactions, e can only be taken into 1, the massless neutrino. The electron and neutrino, or their antiparticles, cannot be combined to produce any of the massive up or down quarks. The neutrino, being massless at tree level, does not add anything to the mass formula for the electron. Since the electron cannot be related to any other massive Dirac fermion, its volume $V(Q_{electron})$ is taken to be 1.
Next consider a red down quark $e_{3}$. By gluon interactions, $e_{3}$ can be taken into $e_{5}$ and $e_{7}$, the blue and green down quarks. By weak boson interactions, it can be taken into $e_{1}$, $e_{2}$, and $e_{6}$, the red, blue, and green up quarks. Given the up and down quarks, pions can be formed from quark-antiquark pairs, and the pions can decay to produce electrons and neutrinos. Therefore the red down quark (similarly, any down quark) is related to any part of $S^7\times {\bf R}P^1$, the compact manifold corresponding to
$$\{ 1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7} \}$$
and therefore a down quark should have a spinor manifold volume factor $V(Q_{down quark}$ of the volume of $S^7\times {\bf R}P^1$. The ratio of the down quark spinor manifold volume factor to the electron spinor manifold volume factor is just
$$V(Q_{down quark}) / V(Q_{electron}) =
V(S^7\times {\bf R}P^1)/1 = \pi ^{5} / 3.$$
Since the first generation graviton factor is 6,
$$md/me = 6V(S^7 \times {\bf R}P^1) = 2 {\pi}^5 = 612.03937$$
As the up quarks correspond to $e_{1}$, $e_{2}$, and $e_{6}$, which are isomorphic to $e_{3}$, $e_{5}$, and $e_{7}$ of the down quarks, the up quarks and down quarks have the same constituent mass $m_{u} = m_{d}$.
Antiparticles have the same mass as the corresponding particles.
Since the model only gives ratios of massses, the mass scale is fixed by assuming that the electron mass $m_{e}$ = 0.5110 MeV.
Then, the constituent mass of the down quark is $m_{d}$ = 312.75 MeV, and the constituent mass for the up quark is $m_{u}$ = 312.75 MeV.
As the proton mass is taken to be the sum of the constituent masses of its constituent quarks $$m_{proton} = m_{u} + m_{u} + m_{d} = 938.25 \; MeV$$ The $D_{4}-D_{5}-E_{6}$ model calculation is close to the experimental value of 938.27 MeV.
The third generation fermion particles correspond to triples of octonions. There are $8^3$ = 512 such triples.
The triple $\{ 1,1,1 \}$ corresponds to the tau-neutrino.
The other 7 triples involving only $1$ and $e_{4}$ correspond to the tauon: $$\{ e_{4}, e_{4}, e_{4} \},
\{ e_{4}, e_{4}, 1 \},
\{ e_{4}, 1, e_{4} \},
\{ 1, e_{4}, e_{4} \},
\{ 1, 1, e_{4} \},
\{ 1, e_{4}, 1 \},
\{ e_{4}, 1, 1 \}$$,
The symmetry of the 7 tauon triples is the same as the symmetry of the 3 down quarks, the 3 up quarks, and the electron, so the tauon mass should be the same as the sum of the masses of the first generation massive fermion particles.
Therefore the tauon mass 1.87704 GeV.
Note that all triples corresponding to the tau and the tau-neutrino are colorless.
The beauty quark corresponds to 21 triples. They are triples of the same form as the 7 tauon triples, but for $1$ and $e_{3}$, $1$ and $e_{5}$, and $1$ and$ e_{7}$, which correspond to the red, green, and blue beauty quarks, respectively.
The seven triples of the red beauty quark correspond to the seven triples of the tauon, except that the beauty quark interacts with 6 $Spin(0,5)$ gravitons while the tauon interacts with only two.
The beauty quark constituent mass should be the tauon mass times the third generation graviton factor 6/2 = 3, so the B-quark mass is $m_{b}$ = 5.63111 GeV.
Note particularly that triples of the type $\{ 1, e_{3}, e_{5} \}$, $\{ e_{3}, e_{5}, e_{7} \}$, etc., do not correspond to the beauty quark, but to the truth quark.
The truth quark corresponds to the remaining 483 triples, so the constituent mass of the red truth quark is 161/7 = 23 times the red beauty quark mass, and the red T-quark mass is $$m_{t} = 129.5155 \; GeV$$
The blue and green truth quarks are defined similarly.
The tree level T-quark constituent mass rounds off to 130 GeV.
These results when added up give a total mass of third generation fermions: $$\Sigma_{f_{3}} = 1,629 \; GeV$$
The second generation fermion calculations are:
The second generation fermion particles correspond to pairs of octonions. There are 82 = 64 such pairs. The pair $\{ 1,1 \}$ corresponds to the $\mu$-neutrino. The pairs $\{ 1, e_{4} \}$, $\{ e_{4}, 1 \}$, and $\{ e_{4}, e_{4} \}$ correspond to the muon. Compare the symmetries of the muon pairs to the symmetries of the first generation fermion particles. The pair $\{ e_{4}, e_{4} \}$ should correspond to the $e_{4}$ electron. The other two muon pairs have a symmetry group S2, which is 1/3 the size of the color symmetry group S3 which gives the up and down quarks their mass of 312.75 MeV.
Therefore the mass of the muon should be the sum of the $\{ e_{4}, e_{4} \}$ electron mass and the $\{ 1, e_{4} \}$, $\{ e_{4}, 1 \}$ symmetry mass, which is 1/3 of the up or down quark mass.
Therefore, $m_{\mu}$ = 104.76 MeV.
Note that all pairs corresponding to the muon and the $\mu$-neutrino are colorless.
The red, blue and green strange quark each corresponds to the 3 pairs involving $1$ and $e_{3}$, $e_{5}$, or $e_{7}$.
The red strange quark is defined as the three pairs $1$ and $e_{3}$, because $e_{3}$ is the red down quark. Its mass should be the sum of two parts: the $\{ e_{3}, e_{3} \}$ red down quark mass, 312.75 MeV, and the product of the symmetry part of the muon mass, 104.25 MeV, times the graviton factor.
Unlike the first generation situation, massive second and third generation leptons can be taken, by both of the colorless gravitons that may carry electric charge, into massive particles.
Therefore the graviton factor for the second and third generations is 6/2 = 3.
Therefore the symmetry part of the muon mass times the graviton factor 3 is 312.75 MeV, and the red strange quark constituent mass is $$m_{s} = 312.75 \; MeV + 312.75 \; MeV = 625.5 \; MeV$$
The blue strange quarks correspond to the three pairs involving $e_{5}$, the green strange quarks correspond to the three pairs involving $e_{7}$, and their masses are determined similarly.
The charm quark corresponds to the other 51 pairs. Therefore, the mass of the red charm quark should be the sum of two parts:
the $\{ e_{1}, e_{1} \}$, red up quark mass, 312.75 MeV; and
the product of the symmetry part of the strange quark mass, 312.75 MeV, and
the charm to strange octonion number factor 51/9, which product is 1,772.25 MeV.
Therefore the red charm quark constituent mass is $$m_{c} = 312.75 \; MeV + 1,772.25 \; MeV = 2.085 \; GeV$$
The blue and green charm quarks are defined similarly, and their masses are calculated similarly.
These results when added up give a total mass of second generation fermions: $$\Sigma_{f_{2}} = 32.9 \; GeV$$
### Massless Neutrinos and Parity Violation
It is required (as an ansatz or part of the $D_{4}-D_{5}-E_{6}$ model) that the charged $W_{\pm}$ neutrino-electron interchange must be symmetric with the electron-neutrino interchange, so that the absence of right-handed neutrino particles requires that the charged $W_{\pm}$ $SU(2)$ weak bosons act only on left-handed electrons.
It is also required (as an ansatz or part of the $D_{4}-D_{5}-E_{6}$ model) that each gauge boson must act consistently on the entire Dirac fermion particle sector, so that the charged $W_{\pm}$ $SU(2)$ weak bosons act only on left-handed fermions of all types.
Therefore, for the charged $W_{\pm}$ $SU(2)$ weak bosons, the 4-dimensional spinor fields $S_{8\pm}$ contain only left-handed particles and right-handed antiparticles. So, for the charged $W_{\pm}$ $SU(2)$ weak bosons, $S_{8\pm}$ can be denoted $S_{8 \pm L}$.
The neutral $W_{0}$ weak bosons do not interchange Weyl neutrinos with Dirac fermions, and so may not entirely be restricted to left-handed spinor particle fields $S_{8\pm L}$, but may have a component that acts on the full right-handed and left-handed spinor particle fields $S_{8\pm} = S_{8\pm L} + S_{8\pm R}$.
However, the neutral $W_{0}$ weak bosons are related to the charged $W_{\pm}$ weak bosons by custodial $SU(2)$ symmetry, so that the left-handed component of the neutral $W_{0}$ must be equal to the left-handed (entire) component of the charged $W_{\pm}$.
Since the mass of the $W_{0}$ is greater than the mass of the $W_{\pm}$, there remains for the $W_{0}$ a component acting on the full $S_{8\pm} = S_{8\pm L} + S_{8\pm R}$ spinor particle fields.
Therefore the full $W_{0}$ neutral weak boson interaction is proportional to $(m_{W_{\pm}}^{2} / m_{W_{0}}^{2})$ acting on $S_{8\pm L}$ and $(1 - (m_{W_{\pm}}^{2} / m_{W_{0}}^{2}))$ acting on $S_{8\pm} = S_{8\pm L} + S_{8\pm R}$.
If $(1 - (m_{W_{\pm}}2 / m_{W_{0}}^{2}))$ is defined to be $\sin{\theta_{w}}^{2}$ and denoted by $\xi$, and
if the strength of the $W_{\pm}$ charged weak force (and of the custodial $SU(2)$ symmetry) is denoted by $T$,
then the $W_{0}$ neutral weak interaction can be written as:
$W_{0L} \sim T + \xi$ and $W_{0R} \sim \xi$.
The $D_{4}-D_{5}-E_{6}$ model allows calculation of the Weinberg angle $\theta_{w}$, by $$m_{W_{+}} = m_{W_{-}} = m_{W_{0}} \cos{\theta_{w}}$$
The Hopf fibration of $S^{3}$ as $S^{1} \rightarrow S^{3} \rightarrow S^{2}$ gives a decomposition of the $W$ bosons into the neutral $W_{0}$ corresponding to $S^{1}$ and the charged pair $W_{+}$ and $W_{-}$ corresponding to $S^{2}$.
The mass ratio of the sum of the masses of $W_{+}$ and $W_{-}$ to the mass of $W_{0}$ should be the volume ratio of the $S^{2}$ in $S^{3}$ to the $S^{1}$ in ${S3}$.
The unit sphere $S^{3} \subset R^{4}$ is normalized by $1 / $2.
The unit sphere $S^{2} \subset R^{3}$ is normalized by $1 / \sqrt{3}$.
The unit sphere $S^{1} \subset R^{2}$ is normalized by $1 / \sqrt{2}$.
The ratio of the sum of the $W_{+}$ and $W_{-}$ masses to the $W_{0}$ mass should then be $(2 / \sqrt{3}) V(S^{2}) / (2 / \sqrt{2}) V(S^{1}) =
1.632993$.
The sum $\Sigma_{m_{W}} = m_{W_{+}} + m_{W_{-}} + m_{W_{0}}$ has been calculated to be $v \sqrt{\alpha_{w}} = 517.798 \sqrt{0.2534577} =
260.774 \; GeV$.
Therefore, $\cos{\theta_{w}}^{2} = m_{W_{\pm}}^{2 } /
m_{W_{0}}^{2} = (1.632993/2)^{2} = 0.667$ , and
$\sin{\theta_{w}}^{2} = 0.333$, so $m_{W_{+}} = m_{W_{-}} = 80.9 \; GeV$, and $m_{W_{0}} = 98.9 \; GeV$.
Corrections for $m_{Z}$ and $\theta_{w}$
----------------------------------------
The above values must be corrected for the fact that only part of the $w_{0}$ acts through the parity violating $SU(2)$ weak force and the rest acts through a parity conserving $U(1)$ electromagnetic type force.
In the $D_{4}-D_{5}-E_{6}$ model, the weak parity conserving $U(1)$ electromagnetic type force acts through the $U(1)$ subgroup of $SU(2)$, which is not exactly like the $D_{4}-D_{5}-E_{6}$ electromagnetic $U(1)$ with force strength $\alpha_{E} = 1 / 137.03608 = e^{2}$.
The $W_{0}$ mass $m_{W_{0}}$ has two parts:
the parity violating $SU(2)$ part $m_{W_{0\pm}}$ that is equal to $m_{W_{\pm}}$ ; and
the parity conserving part $m_{W_{00}}$ that acts like a heavy photon.
As $m_{W_{0}}$ = 98.9 GeV = $m_{W_{0\pm}} + m_{W_{00}}$, and as $m_{W_{0\pm}} = m_{W_{\pm}} = 80.9 \; GeV$, we have $m_{W_{00}} = 18 \; GeV$.
Denote by $\tilde{\alpha_{E}} = \tilde{e}^{2}$ the force strength of the weak parity conserving $U(1)$ electromagnetic type force that acts through the $U(1)$ subgroup of $SU(2)$.
The $D_{4}-D_{5}-E_{6}$ electromagnetic force strength $\alpha_{E} = e^{2} = 1 / 137.03608$ was calculated using the volume $V(S^{1})$ of an $S^{1} \subset R^{2}$, normalized by $1 / \sqrt{2}$.
The $\tilde{\alpha_{E}}$ force is part of the $SU(2)$ weak force whose strength $\alpha_{w} = w^{2}$ was calculated using the volume $V(S^{2})$ of an $S^{2} \subset R^{3}$, normalized by $1 / \sqrt{3}$.
Also, the $D_{4}-D_{5}-E_{6}$ electromagnetic force strength $\alpha_{E} = e^{2}$ was calculated using a 4-dimensional spacetime with global structure of the 4-torus $T^{4}$ made up of four $S^{1}$ 1-spheres,
while the $SU(2)$ weak force strength $\alpha_{w} = w^{2}$ was calculated using two 2-spheres $S^{2} \times S^{2}$, each of which contains one 1-sphere of the $\tilde{\alpha_{E}}$ force.
Therefore $\tilde{\alpha_{E}} = \alpha_{E} (\sqrt{2} /
\sqrt{3})(2 / 4) = \alpha_{E} / \sqrt{6}$, $\tilde{e} = e / 4 \sqrt{6} = e / 1.565$ , and
the mass $m_{W_{00}}$ must be reduced to an effective value
$m_{W_{00}eff} = m_{W_{00}} / 1.565$ = 18/1.565 = 11.5 GeV
for the $\tilde{\alpha_{E}}$ force to act like an electromagnetic force in the 4-dimensional spacetime of the $D_{4}-D_{5}-E_{6}$ model:
$\tilde{e} m_{W_{00}} = e (1/5.65) m_{W_{00}} = e m_{Z_{0}}$,
where the physical effective neutral weak boson is denoted by $Z$ rather than $W_{0}$.
Therefore, the correct $D_{4}-D_{5}-E_{6}$ values for weak boson masses and the Weinberg angle are:
$m_{W_{+}} = m_{W_{-}} = 80.9 \; GeV$;
$m_{Z} = 80.9 +11.5 = 92.4 \; GeV$; and
$\sin{\theta_{w}}^{2} = 1 - (m_{W_{\pm}} /
m_{Z})^{2} = 1 - 6544.81/8537.76 = 0.233$.
Radiative corrections are not taken into account here, and may change the $D_{4}-D_{5}-E_{6}$ value somewhat.
K-M Parameters.
---------------
The following formulas use the above masses to calculate Kobayashi-Maskawa parameters:
$$phase \; angle \; \epsilon = \pi / 2$$
$$\sin{\alpha} = [m_{e}+3m_{d}+3m_{u}] /
\sqrt{ [m_{e}^{2}+3m_{d}^{2}+3m_{u}^{2}] +
[m_{\mu}^{2}+3m_{s}^{2}+3m_{c}^{2}] }$$
$$\sin{\beta} = [m_{e}+3m_{d}+3m_{u}] /
\sqrt{ [m_{e}^{2}+3m_{d}^{2}+3m_{u}^{2}] +
[m_{\tau}^{2}+3m_{b}^{2}+3m_{t}^{2}] }$$
$$\sin{\tilde{\gamma}} = [m_{\mu}+3m_{s}+3m_{c}] /
\sqrt{ [m_{\tau}^{2}+3m_{b}^{2}+3m_{t}^{2}] +
[m_{\mu}^{2}+3m_{s}^{2}+3m_{c}^{2}] }$$
$$\sin{\gamma} = \sin{\tilde{\gamma}}
\sqrt{\Sigma_{f_{2}} / \Sigma_{f_{1}}}$$
The resulting Kobayashi-Maskawa parameters are:
$$\begin{array}{|c|c|c|c|}
\hline
& d & s & b
\\
\hline
u & 0.975 & 0.222 & -0.00461 i \\
c & -0.222 -0.000191 i & 0.974 -0.0000434 i & 0.0423 \\
t & 0.00941 -0.00449 i & -0.0413 -0.00102 i & 0.999 \\
\hline
\end{array}$$
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---
abstract: 'The observed number of dwarf galaxies as a function of rotation velocity is significantly smaller than predicted by the standard model of cosmology. This discrepancy cannot be simply solved by assuming strong baryonic feedback processes, since they would violate the observed relation between maximum circular velocity ($v_{\rm max}$) and baryon mass of galaxies. A speculative but tantalising possibility is that the mismatch between observation and theory points towards the existence of non-cold or non-collisionless dark matter (DM). In this paper, we investigate the effects of warm, mixed (i.e warm plus cold), and self-interacting DM scenarios on the abundance of dwarf galaxies and the relation between observed HI line-width and maximum circular velocity. Both effects have the potential to alleviate the apparent mismatch between the observed and theoretical abundance of galaxies as a function of $v_{\rm max}$. For the case of warm and mixed DM, we show that the discrepancy disappears, even for luke-warm models that evade stringent bounds from the Lyman-$\alpha$ forest. Self-interacting DM scenarios can also provide a solution as long as they lead to extended ($\gtrsim 1.5$ kpc) dark matter cores in the density profiles of dwarf galaxies. Only models with velocity-dependent cross sections can yield such cores without violating other observational constraints at larger scales.'
author:
- |
Aurel Schneider$^{1}$, Sebastian Trujillo-Gomez$^{2}$, Emmanouil Papastergis$^{3}$,\
\
[$^1$Institute for Astronomy, Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093, Zurich, Switzerland]{}\
[$^2$Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland]{}\
[$^3$Kapteyn Astronomical Institute, University of Groningen, Landleven 12, Groningen NL-9747AD, The Netherlands]{}\
[$^4$$S^{3}IT$, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland]{}\
[Email: aurel.schneider@phys.ethz.ch]{}
bibliography:
- 'ASbib.bib'
title: Hints against the cold and collisionless nature of dark matter from the galaxy velocity function
---
\[firstpage\]
cosmology: theory – dark matter – Local Group
Introduction
============
Observations of the abundance and structure of dwarf galaxies have the potential to probe the particle nature of dark matter (DM). This is because effects from DM free-streaming or from (self-)interactions have an impact on structure formation at the smallest observable scales.
There are potential inconsistencies between small-scale observations and the standard model of $\Lambda$CDM [based on the observations of the [Planck]{} satellite, @Planck:2015xua]. First of all, haloes predicted by gravity-only simulations greatly outnumber observed galaxies. This long-standing discrepancy has been established for both the Milky-Way satellites [@Klypin:1999uc; @Moore:1999nt] as well as nearby isolated galaxies [e.g. @Tikhonov:2009jq; @Zavala:2009ms] and is usually referred to as the [*over-abundance*]{} (or [*missing satellite*]{}) problem. Second, observations of rotation velocities from stars and gas point towards very shallow inner density profiles of small haloes in strong contrast to predictions from gravity-only simulations. This is generally known as the *cusp-core* [e.g. @deBlok:2009sp] or the *too-big-to-fail* problem [TBTF, @BoylanKolchin:2011de; @Papastergis:2014aba], depending on the context.
The main difficulty with these problems of small-scale structure formation is the fact that they are based on predictions from gravity-only simulations, ignoring any potential effects from baryonic physics. Indeed, it is expected that photo-evaporation from UV sources during reionisation can expel gas from small haloes, effectively preventing star formation and reducing the number of observable dwarf galaxies [e.g. @Gnedin:2000uj; @Okamoto:2008sn]. More recently, it was realised that supernova feedback is energetic enough to reshape the inner parts of halo profiles making them considerably shallower [@Governato:2012fa]. However, the details of how feedback affects the halo profile are still under debate. For example, @Onorbe:2015ija and @Read:2015sta show that the core size depends on the details of the star formation history, while @DiCintio:2013qxa and @Fitts:2016usl connect the core size to the dwarf’s stellar-to-halo mass ratio. Other papers [e.g. @Sawala:2015cdf; @Fattahi:2016nld] point out that the presence of cores depends on the feedback implementation and might not be required to recover the observations.
Despite the ongoing debate about the efficiency of baryon-induced feedback mechanisms, they are generally assumed to be the most likely explanation for the dwarf abundance and structure problems. However, because of our poor understanding of sub-grid effects in hydrodynamical simulations, it has so far been impossible to verify these assumptions from first principles.
A very useful statistic that simultaneously probes both the abundance and the structure of galaxies is the velocity function (VF), i.e. the number density of galaxies as a function of their observed rotation velocity. The VF offers a direct link between observations and theory because the rotation velocities of galaxies act as a tracer of the halo gravitational potential. Since the galaxy velocity function is sensitive to both the abundance and the inner structure of haloes, any model that predicts the observed VF is also likely to solve the overabundance, the cusp-core, and the TBTF problems. This makes the velocity function an ideal probe of structure formation.
In a recent study, @Klypin:2014ira compiled the velocity function of the local 10 Mpc around the Milky Way using rotation velocities ($v_{\rm rot}$) predominantly based on spatially unresolved HI line widths. When compared to the VF from gravity-only simulations, they found a discrepancy in the abundance of galaxies at $v_{\rm rot}<80$ km/s which increases towards lower velocities. However, this comparison is based on the assumption that $v_{\rm rot}$ is a good approximation of the maximum circular velocity ($v_{\rm max}$) of the halo, which is not guaranteed.
In a recent paper, we investigated the relation between the measured rotation velocity of a galaxy and the maximum circular velocity of the halo that hosts the galaxy, and we studied how the resulting $v_{\rm max}$ based velocity function is affected by baryonic processes [@Trujillo-Gomez:2016pix henceforth TG16]. Our main findings were the following: (i) the bias between $v_{\rm rot}$ and $v_{\rm max}$ is not large enough to significantly reduce the mismatch between the observed VF and the one predicted by gravity-only simulations of $\Lambda$CDM; (ii) while baryonic processes are able to reduce the theoretical abundance of galaxies and alleviate the over-abundance problem, they cannot completely solve it without simultaneously violating the observed relation between baryon mass and $v_{\rm max}$ (i.e. the $v_{\rm max}$ baryonic Tully-Fisher relation).
This paper builds upon TG16 and investigates the tantalising possibility that the mismatch between the observed and predicted galaxy velocity function is caused by the underlying particle properties of dark matter. Both non-cold and non-collisionless DM models could provide more natural solutions, as they suppress the amplitude of matter perturbations and/or alter the halo density profiles. As representative examples, we focus on the effective scenarios of warm, mixed (i.e. warm plus cold) and self-interacting DM.
The paper is structured as follows: In Sections \[sec:problem\] and \[sec:theory\] we give a brief summary of the results obtained in TG16, and we present our theoretical model of the DM halo velocity function based on the the extended Press-Schechter approach. Sections \[sec:WDM\], \[sec:MDM\], and \[sec:SIDM\] present the results for warm, mixed, and self-interacting DM. We examine how these DM scenarios affect the halo profiles, the maximum circular velocities, and finally the VF. In Section \[sec:othercandidates\] we discuss qualitatively other potential DM particle scenarios. Our results are summarised in Section \[sec:conclusions\].
Setting up the problem {#sec:problem}
======================
The number density of galaxies as a function of rotational velocities – i.e. the velocity function (VF) – is a very useful observational quantity relating information about galaxy abundance with the underlying halo potentials. This allows to compare theory with observations without detailed knowledge about the galaxy formation efficiency. In this section, we recap the results from TG16, summarising the procedure for obtaining maximum circular velocities ($v_{\rm max}$) from observed HI line widths ($w_{50}$) and how this affects the shape of the VF.
Galaxies in the local universe
------------------------------
In a recent paper, @Klypin:2014ira performed a detailed analysis of the abundance of galaxies within the local volume around the Milky-Way. Their analysis is based on the galaxy catalogue from @Karachentsev:2013ipr [hereafter K13] which they show to be complete down to a limiting magnitude of $M_B=-12$ within 10 Mpc from the Milky-Way. The rotation velocities ($v_{\rm rot}$) of all galaxies in the K13 sample were determined by either relying on inclination corrected unresolved HI line-width measurements or by using the magnitude-velocity relation for the galaxies with no detected HI (a fraction of less than ten percent). @Klypin:2014ira find a slowly rising VF down to $v_{\rm rot}\sim15$ km/s which they claim to be in tension with the $\Lambda$CDM model prediction below $v_{\rm rot}\sim80$ km/s. Similar conclusions have been made previously by @Zavala:2009ms, @TrujilloGomez:2010yh, and @Papastergis:2011xe based on data from the [HIPASS]{} and [ALFALFA]{} HI surveys, respectively[^1].
The statement that there is tension between observations and the $\Lambda$CDM model relies on the assumption that the rotational velocity from HI line-widths, $v_{\rm rot}\equiv w_{50}/(2\sin i)$, can be used as a proxy for the maximum circular velocity ($v_{\rm max}$) of a halo. @Klypin:2014ira showed that this is approximately the case, at least for a subset of selected dwarf galaxies and assuming all haloes to have NFW profiles with a given concentration-mass relation. More recently, several authors have questioned the validity of these assumptions, reporting strong biases between $v_{\rm rot}$ and $v_{\rm max}$ instead. These studies used abundance matching [@Brook:2015ofa; @Brook:2015eva], measurements from zoom-in hydrodynamical simulations of individual galaxies [@Maccio:2016egb], or semi-analytical models [@Yaryura:2016djm; @Obreschkow:2013hka]. In TG16, we used direct observations of galaxies to perform a detailed investigation of possible biases between $v_{\rm rot}$ and $v_{\rm max}$. We showed that $v_{\rm max}$ can be directly recovered from $v_{\rm rot}$, and we argued that the bias between the two is present but not large enough to solve the discrepancy present in the VF. We will now summarise the method developed in TG16.
From $\mathbf{v_{\rm rot}}$ to $\mathbf{v_{\rm max}}$ {#sec:vmax}
-----------------------------------------------------
Most of the galaxies from the K13 sample only have HI line-width measurements (or magnitude-based estimates for the subdominant population of gas-free galaxies) without any spatial information, making it impossible to estimate the corresponding $v_{\rm max}$. There is, however, more information for a sub-sample of galaxies with existing spatially resolved measurements of their kinematics. A sample of 200 galaxies with interferometric HI observations ($v_{\rm out})$ at the outermost HI radius ($r_{\rm out}$) was compiled by @Papastergis:2016aba. We use this catalogue to select all galaxies with $r_{\rm out}>3 r_{1/2}$, where $r_{1/2}$ is the galactic half-light radius (see TG16). This additional selection criterion guarantees that the velocity measurement is not dominated by baryonic effects, including a potential DM core from strong stellar feedback [@Read:2015sta]. The final catalogue consists of 109 galaxies (distributed over the full range of relevant scales) which can be used to estimate the relation between $v_{\rm rot}$ and $v_{\rm max}$.
In a first step, we fit NFW profiles [@Navarro:1995iw] to the observed velocities $v_{\rm out}$ at radius $r_{\rm out}$ in order to determine the corresponding $v_{\rm max}$, which we then compare to $v_{\rm rot}$ from HI line-width measurements[^2]. For the concentrations, we use the relation from @Dutton:2014xda based on [Planck]{} cosmology. In the left panel of Fig. \[fig:intro\], the spatially resolved observed circular velocities $v_{\rm out}$ (black symbols with error bars) are shown together with the fitted velocity profiles based on NFW (grey lines). For most galaxies in the sample, $v_{\rm max}$ is not much larger than $v_{\rm out}$, owing to the fact that $v_{\rm out}$ is observed far out in radius ($r_{\rm out}$). The relation between $v_{\rm rot}$ (i.e. the rotation velocity, obtained via the measurement of unresolved HI line widths) and $v_{\rm max}$ is shown in the middle panel of Fig. \[fig:intro\] (black symbols). The error bars illustrate the sensitivity of the results for varying concentrations within the 1-$\sigma$ scatter given by @Dutton:2014xda. In general, there is a small bias between $v_{\rm rot}$ and $v_{\rm max}$ slowly growing towards very small velocities.
In a second step, we attempt to include effects from baryons on the circular velocity profiles. We therefore repeat the same analysis using the fit from @Read:2015sta [herafter R16] which consists of an NFW profile plus a baryon-induced core proportional to the stellar half-light radius. The mass profile of the R16 fit is given by $$\label{R16profile}
M_{\rm R16}(r)=M_{\rm nfw}(r)f^n,\hspace{0.3cm}f=\left[\tanh\left(\frac{r}{r_c}\right)\right]$$ and is a simple extension of the NFW mass profile ($M_{\rm nfw}$) with two additional free parameters $r_{\rm c}$ and $n$. R16 showed that the sizes of baryon-induced cores are proportional to the half-light radii (i.e. $r_{\rm c}=\eta\, r_{1/2}$) and that $n$ varies between 0 and 1 depending on the individual star formation history of each galaxy. Here we fix $\eta=1.75$ because this was shown by R16 to provide the best match to their simulations. Furthermore, we adopted the value $n=1$ for the second free parameter in order to maximise the effect from baryons.
The effects of the baryon-induced cores are illustrated in the left and middle panel of Fig. \[fig:intro\]. The inner part of the velocity profiles are much steeper (left panel, red lines) reflecting how cores affect the velocity profiles. However, the resulting values for $v_{\rm max}$ (middle panel, red symbols) are nearly indistinguishable from the ones obtained with an NFW fit. This shows that baryon-induced cores do not bias the $v_{\rm max}$-estimates of the selected galaxy sample, which is a direct result of our original selection criteria (i.e. $r_{\rm out}>3r_{1/2}$). Such a selection greatly simplifies the analysis and can be justified as long as the resulting galaxy sample is *representative* for the galaxies that make up the VF. In TG16, we used the baryonic Tully-Fisher (BTF) relation to show that this is indeed the case.
Corrected velocity function
---------------------------
Let us now turn our attention to the velocity function and how it can be corrected to account for the bias between the rotational ($v_{\rm rot}$) and the maximum circular velocity ($v_{\rm max}$). This correction is important, since only $v_{\rm max}$ can be directly related to the halo mass (and therefore to the theory prediction) while $v_{\rm rot}$ depends on the details of the gas distribution within the galaxy.
In the right panel of Fig. \[fig:intro\] we plot both the observed VF based on $v_{\rm rot}$ (green band) and the predicted VF from gravity-only simulations[^3] of a $\Lambda$CDM universe based on $v_{\rm max}$ (black solid line). The two velocity functions agree reasonably well at large velocities but start to diverge below 100 km/s. This apparent discrepancy between observations and theory has been pointed out repeatedly in the past [see e.g. @Tikhonov:2009jq; @Zavala:2009ms; @TrujilloGomez:2010yh; @Papastergis:2011xe].
It is possible to correct the observed VF using the average relation between $v_{\rm max}$ and $v_{\rm rot}$ obtained with the selected galaxy sample (i.e. the fit from the middle panel of Fig. \[fig:intro\]). The resulting $v_{\rm max}$-corrected VF is plotted as a grey band in the right panel of Fig. \[fig:intro\]. Despite being slightly steeper, it remains inconsistent with the prediction from gravity-only simulations. In the following, we use this $v_{\rm max}$-corrected VF and compare it to theoretical predictions including baryon effects as well as modifications induced by the DM model.
Theoretical predictions for the velocity function of haloes {#sec:theory}
===========================================================
The velocity function (VF) is sensitive to both baryonic feedback effects and the particle nature of dark matter, making an accurate modelling both essential and challenging. In this paper we use an analytical approach based on the extended Press-Schechter (EPS) model. This has the advantage of being easily adaptable to different dark matter models, and it can be used to estimate suppression effects from baryons.
Modelling the velocity function {#sec:EPS}
-------------------------------
The calculation of the halo velocity function is based on the EPS approach presented in @Schneider:2013ria [@Schneider:2013wwa]. The first and most important step is to obtain the halo mass function with sharp-$k$ filter $$\begin{aligned}
\label{WDMmassfct}
\frac{dn}{d\ln M}&=&\frac{1}{12\pi^2}\frac{\bar\rho}{M}\nu f(\nu)\frac{P_{\rm lin}(1/R)}{\delta_c^2R^3},\\
\sigma^2(R)&=&\int \frac{d\mathbf{k}^3}{(2\pi)^3} P_{\rm lin}(k)\Theta(1-kR),\end{aligned}$$ where $P_{\rm lin}(k)$ is the linear power spectrum, $\delta_c=1.686$ the collapse threshold, and $\Theta$ the Heaviside step function. The first crossing distribution $f(\nu)$ is obtained from the ellipsoidal collapse model, yielding $$f(\nu)=A\sqrt{2\nu/\pi}(1+\nu^{-p}) {\rm e}^{-\nu/2}$$ with $\nu= (\delta_c/\sigma)^2$, A=0.322, and p=0.3. The halo mass is assigned to the filter scale with the relation $M=4\pi\bar\rho(cR)^3/3$ where $c=2.5$ [see also @Benson:2012su for a similar description].
In order to obtain the maximum circular velocity, we assume all haloes to be described by an NFW profile. This is a good assumption, even for alternative DM models with extended cores, because the radius corresponding to $v_{\rm max}$ lies beyond the scale radius of the halo. For the concentration-mass relation of the $\Lambda$CDM model, we use the power-law relation $$\label{cDutton}
c(M)=10^{1.025}\left(\frac{10^{12}\, {\rm M_{\odot}/h}}{M}\right)^{0.097}$$ from @Dutton:2014xda based on the [Planck]{} cosmology. Alternative DM (ADM) scenarios can have different concentrations and we follow the approach from @Schneider:2014rda which consists of comparing halo formation times between CDM and ADM models and assigning concentrations accordingly. An estimate of the average redshift of halo formation can be obtained by solving the equation $$\begin{aligned}
D(z_c)&=&\left[1+\sqrt{\frac{\pi}{2}}\frac{1}{\delta_c}\sqrt{\sigma^2(F^{1/3}R)-\sigma^2(R)}\right]^{-1}\label{zcoll}\\
&\equiv& \frac{5\Omega_m}{2}H(z_c)\int_{z_c}^{\infty}dz\frac{(1+z)}{H(z)^3}\nonumber\end{aligned}$$ for the collapse redshift $z_c(M)$, where $F=0.05$. Once the function $z_c(M)$ is known for both ADM and CDM, we can link together ADM and CDM haloes with the same collapse redshift and assign concentrations for ADM haloes from Eq. (\[cDutton\]). Although $z_c(M)$ is a rather poor estimate of the *actual* collapse redshift measured in simulations, the resulting concentrations of ADM haloes are surprisingly accurate [see @Schneider:2014rda].
To finally obtain the velocity function, we create a mock sample of haloes drawn from the halo mass function, and we assign concentrations from a log-normal distribution. This allows us to determine a value for the maximum circular velocity according to the relation $$v_{\rm max}=0.465\sqrt{\frac{GM}{r_{\rm vir}}}\left[c^{-1}\ln(1+c)-(1+c)^{-1}\right]^{-1/2}$$ [directly resulting from the NFW profile, see @Sigad:2000cd] and to re-bin the sample in order to obtain the VF $$\Phi(v_{\rm max})\equiv\frac{dn}{d\ln v_{\rm max}}.$$ Similar approaches have been applied by several authors in the past [see e.g. @Zavala:2009ms; @Schneider:2013wwa].
It was shown by @TrujilloGomez:2010yh and @Dutton:2010dw that the maximum circular velocity of small galaxies are not affected by baryonic infall or contraction. Larger galaxies with $v_{\rm max}\gtrsim 100$ km/s have boosted velocities due to their baryonic components. We follow @Klypin:2014ira and correct the maximum circular velocity of massive galaxies by solving the equation $$\label{barcorKlypin}
v_{\rm max}^{\rm dmo}=v_{\rm max}\left[1+\frac{0.35(v_{\rm max}/120\, {\rm kms^{-1}})^6}{1+(v_{\rm max}/120\, {\rm kms^{-1}})^6} \right]^{-1},$$ where $v_{\rm max}^{\rm dmo}$ stands for the maximum circular velocity without baryonic correction.
The EPS approach with sharp-$k$ filter has two distinctive advantages with respect to other methods: First of all, it accurately describes the halo abundance of models with arbitrary power spectra, while the standard EPS model with a real-space tophat filter only works for the CDM scenario [@Schneider:2014rda]. Second, it does not suffer from artificial clumping, which is a serious problem for direct simulations of DM scenarios with suppressed power spectra [see e.g. @Wang:2007he; @Lovell:2013ola; @Hahn:2015sia; @Hobbs:2015dda].
One drawback of the EPS approach is that it does not account for substructures. To correct for this, we multiply the EPS velocity function by a factor of 1.25 so that it matches the predictions from the MultiDark N-body simulations [@Klypin:2014kpa]. This corresponds to adding a constant number of sub-haloes to each velocity bin[^4]. The normalisation is done once and is not changed for different DM models.
Maximising effects from baryons {#sec:maxbaryoneffects}
-------------------------------
The great majority of work on the velocity function has been based on gravity-only $N$-body simulations in the past [e.g. @Gonzalez:1999ek; @Zavala:2009ms; @Zwaan:2009dz; @Papastergis:2011xe; @Obreschkow:2013hka; @Klypin:2014ira]. There are, however, two distinct effects from baryons which should be be accounted for, since they have the potential to significantly alter the VF at dwarf galaxy scales. The first effect is [*baryonic depletion*]{} and consists of a reduction of the maximum circular velocity due to the fact that some of the gas is being pushed out of haloes, reducing the total mass and accretion rate of the halo during its formation. The second effect is [*baryonic suppression*]{}, referring to the fact that feedback can reduce the number of observable galaxies by pushing the luminosity below the sensitivity level of a given survey. The maximum suppression of the VF from both types of baryonic effects were quantified in TG16. Here we summarise these results and show how they can be extended to alternative DM (ADM) scenarios.
The maximum effect from *baryonic depletion* can be obtained by calculating the VF for a cosmology where the entire baryon content is removed. This is achieved by replacing $\sigma_8\rightarrow (1-\Omega_b/\Omega_m)\sigma_8$ as well as $\Omega_m\rightarrow (1-\Omega_b/\Omega_m)\Omega_m$, resulting in a scale-independent decrease of the maximum circular velocities, i.e. $$\label{depl}
v^{\rm depl}_{\rm max}\simeq0.86\, v_{\rm max},$$ independent of the DM model (see TG16 for more details). We want to stress that this corresponds to the *maximum* baryonic depletion, likely to overestimate the true effect.
The second type of effect, the *baryonic suppression*, is more difficult to model as it crucially depends on the details of the suppression mechanism. In TG16 we developed a model-independent approach to quantify the maximum possible suppression of dwarf galaxy numbers. Any decrease of the stellar or gaseous content of galaxies leads to a bend in the relation between $v_{\rm max}$ and $M_{\rm bar}$ – the baryonic Tully-Fisher relation of $v_{\rm max}$. The maximum allowed suppression can therefore be directly constrained by the data without prior knowledge of feedback mechanisms[^5].
In this paper we describe the $v_{\rm max}$-$M_{\rm bar}$ relation with the function $\mathcal{M}(v_{\rm max})$ which provides a good description of the data points. The suppression induced by potential baryonic processes is furthermore parameterised as $$\label{BTFsuppression}
\mathcal{M}_{\rm supp}(v_{\rm max})=[1+(v_s/v_{\rm max})^{4}]^{-5}\mathcal{M}(v_{\rm max})$$ where $v_s$ is a free model parameter. This function leads to a very similar suppression than the one obtained in hydro simulations [see e.g. @Sales:2016dmm]. A more general parametrisation is discussed in TG16.
For the fiducial case of CDM, the function $\mathcal{M}(v_{\rm max})$ is given by a linear least-square fit of the data. It can be shown that this is indeed a very good fit to the data (see TG16). Next, we can use the *likelihood ratio analysis*[^6] to determine the model $\mathcal{M}_{\rm supp}(v_{\rm max})$ with the largest value of $v_c$ that is still in agreement with the data at 3-$\sigma$ confidence level (CL). This function is defined as the model with maximum allowed baryon suppression. For CDM it is given by the parameter $v_s=23$ km/s.
In the right panel of Fig. \[fig:WDMprofiles\] the $v_{\rm max}$-$M_{\rm bar}$ relation of CDM (empty triangles) is plotted together with the linear fit $\mathcal{M}$ (solid black line) and the function of maximum allowed baryon suppression $\mathcal{M}_{\rm supp}$ (dashed black line). While the former is a good fit to the data, the latter is characterised by a strong downturn towards small velocities.
For the alternative DM models discussed in this paper, the values of $v_{\rm max}$ are modified with respect to CDM. As a consequence, the $v_{\rm max}$-$M_{\rm bar}$ relation cannot be described by a linear least-square fit anymore. A more accurate model for $\mathcal{M}$ is obtained when the linear fit from CDM is corrected by accounting for the difference in the average value of $v_{\rm max}$ between ADM and CDM. Based on this corrected function $\mathcal{M}$ for ADM, the maximum baryon suppression model can again be obtained with a likelihood ratio analysis.
Any downturn in the $v_{\rm max}$-$M_{\rm bar}$ relation is expected to have an influence on the velocity function, as it sets the velocity scale below which galaxies become undetectable by a given survey. At the velocity scale $v_c$, where $\mathcal{M}_{\rm supp}(v_{\rm max})$ crosses the survey detectability limit (in terms of baryon mass), half of the galaxies are too faint to be visible in the velocity function. This effect can be modelled as follows $$\begin{aligned}
\label{VFsuppression}
\Phi_{\rm supp}(v_{\rm max}) &=& \mathcal{G}_{\rm supp}(v_{\rm max}) \Phi(v_{\rm max}),\\
\mathcal{G}_{\rm supp}(v_{\rm max})&=&\frac{1}{2}\left[{\rm erf}\left(\frac{\log v_{\rm max}-\log v_c}{\sqrt{2}\log \sigma_c}\right)+1\right],\nonumber\end{aligned}$$ where we assumed a log-normal distribution of galaxies around the mean (see TG16 for more details). For the K13 sample, the detectability limit is at $\sim4.3\times 10^6$ M$_{\odot}$/h resulting in $v_c\sim29.5$ km/s for CDM (as indicated by the black cross in the right panel of Fig. \[fig:WDMprofiles\]). The scatter can be directly measured from the data (ignoring baryon suppression) resulting in $\log\sigma_c\sim0.15$ for most models (including CDM), except for SIDM where the scatter is larger (see Sec. \[sec:SIDM\]).
In the following sections, the theoretical model developed here is applied to cold, warm, mixed, and self-interacting DM models. We always present both the VF without baryon effects as well as the VF with maximum baryon suppression and depletion. These two extreme models quantify the current uncertainty of theory predictions due to unknown feedback effects.
Warm dark matter {#sec:WDM}
================
The first alternative paradigm we consider is the collisionless warm dark matter (WDM) model, which is characterised by a steep suppression of the power spectrum at small scales caused by the free-streaming properties of the DM particles. The scale and exact shape of the suppression depends on both the DM particle mass and its phase-space distribution. In this section, we restrict ourselves to the standard and most studied case of a Fermi-Dirac distributed DM fluid (i.e. the so-called thermal WDM) for which we vary the particle mass. However, other distributions are possible depending on the the exact DM production mechanism [see e.g. @Merle:2014xpa]. These models might lead to somewhat shallower suppressions of the power spectrum, closer to the case of mixed DM (see Sec. \[sec:MDM\]).
There are various constraints of the thermal WDM scenario from the Lyman-$\alpha$ forest [@Seljak:2006qw; @Viel:2005qj; @Viel:2013apy; @Baur:2015jsy], the dwarf galaxy abundance in the local volume [@Polisensky:2010rw; @Kennedy:2013uta; @Horiuchi:2013noa], high-redshift galaxies [@Menci:2016eww; @Menci:2016eui], or stellar ages of Milky-Way satellites [@Chau:2016jzi]. As a rule-of-thumb, the current Lyman-$\alpha$ limits are $m_{\rm TH}\gtrsim 3$ keV while all other limits are around $m_{\rm TH}\gtrsim 2$ keV or weaker[^7].
In this paper we study the representative cases of $m_{\rm TH}=2$, 3, and 4 keV, where the first is in tension with the Lyman-$\alpha$ data but consistent with other limits and the latter two are, roughly speaking, in agreement with observations. The linear power spectra of these models are plotted in Fig. \[fig:powspec\] for illustration. They are indistinguishable from CDM at large scales (low wavenumber $k$) but become strongly suppressed towards smaller scales (high $k$). The suppression scale only depends on the thermal mass ($m_{\rm TH}$) of the WDM model.
Halo profiles {#sec:haloprofilesWDM}
-------------
It has been shown in the past that WDM models with realistic particle masses do not produce halo cores large enough to be observable [@deNaray:2009xj; @VillaescusaNavarro:2010qy; @Maccio:2012qf; @Shao:2012cg]. Instead, the haloes are well described by NFW profiles with a modified concentration-mass relation [@Schneider:2014rda; @Bose:2015mga; @Ludlow:2016ifl]. Rather than monotonically rising towards small masses, as is the case for CDM, the WDM concentrations turn over and decrease again with a maximum around dwarf galaxy scales, with the exact position depending on the DM particle mass [@Eke:2000av; @Schneider:2011yu].
The modified concentrations of the WDM model affect the calculation of the maximum circular velocities. In the left panel of Fig. \[fig:WDMprofiles\] we show the WDM (with $m_{\rm TH}=3$ keV) circular velocity profiles (brown lines) fitted to the data-points of the outermost rotation measurement from the selected sample galaxies (black symbols). The lower concentrations at small scales lead to larger values of $v_{\rm max}$ further out in radius compared to the case of CDM (black lines). This means that in WDM models, a galaxy with a given $v_{\rm rot}$ can be fit into a more massive DM halo compared to CDM.
In the middle panel of Fig. \[fig:WDMprofiles\] we illustrate the difference between WDM and CDM in terms of the relation between $v_{\rm max}$ and $v_{\rm rot}$ (brown and black symbols), where the error-bars indicate the sensitivity of $v_{\rm max}$ to variations in concentration[^8]. For WDM the data is better fitted by a quadratic fit (brown line) compared to a linear fit for CDM (black line).
The right panel of Fig. \[fig:WDMprofiles\] shows the $v_{\rm max}$-$M_{\rm bar}$ relation of the selected galaxy sample for both WDM and CDM (full and empty triangles). While the CDM relation is well described by a linear fit (black solid line), the data bends downwards at small velocities for WDM. We capture this downturn by applying the average shift of $v_{\rm max}$ between WDM and CDM (i.e. the vertical separation between black and brown lines in the middle panel) to the linear fit from CDM, yielding $\mathcal{M}(v_{\rm max})$ for WDM.
Next, we perform a likelihood ratio analysis to find the model $\mathcal{M}_{\rm supp}(v_{\rm max})$ with the maximum baryon suppression. As for CDM, this corresponds to the model with the largest value of $v_s$ allowed by the data at the 3-$\sigma$ CL. This model is shown as dashed brown line in Fig. \[fig:WDMprofiles\]. The brown cross indicates the critical velocity ($v_c$) where the line of maximal suppression crosses the completeness limit of the K13 sample. Below this scale, the observed abundance of galaxies could be reduced due to baryonic processes (see Eq. \[VFsuppression\]).
In Fig. \[fig:WDMprofiles\] we only illustrate the case of WDM with $m_{\rm TH}=3$ keV for brevity. Note, however, that other WDM models show very similar trends with increasing discrepancies between WDM and CDM for decreasing thermal-relic mass $m_{\rm TH}$.
Velocity function
-----------------
Assuming a WDM scenario affects the velocity function (VF) in several non-trivial ways. First of all, the predicted VF is flatter in WDM than in CDM due to a combination of lower halo abundance and lower concentrations. The former reduces the number of observable galaxies while the latter lowers the maximum circular velocity at a given mass scale. Second, the $v_{\rm max}$-corrected VF from observations becomes steeper in WDM compared to CDM, the reason being the modified relation between $v_{\rm rot}$ and $v_{\rm max}$. This is again a direct consequence of the reduced concentrations which cause galaxies of a given $v_{\rm rot}$ to inhabit larger haloes. Both effects are expected to improve the agreement between theory and observations for WDM compared to CDM.
In Fig. \[fig:WDMvelfct\] we show the velocity function of three different WDM models with thermal masses of $m_{\rm TH}=2$ keV (top left), $m_{\rm TH}=3$ keV (top right), and $m_{\rm TH}=4$ keV (bottom left) as well as the VF for CDM (bottom right). Both the flattening of the predicted and the steepening of the observed $v_{\rm max}$-VF are well visible in the plot, the effects becoming more pronounced for decreasing values of $m_{\rm TH}$.
In each panel of Fig. \[fig:WDMvelfct\], the predicted VF with no baryon effects and with maximum baryon effects are shown as solid and dashed lines. While *baryon depletion* induces a horizontal shift towards small velocities, *baryon suppression* leads to a turn-over of the VF below a characteristic velocity ($v_{c}$). The value of $v_c$ depends on the model and becomes larger for a smaller thermal relic mass ($m_{\rm TH}$). In general, the area between these lines (hatched area) illustrates the theoretical uncertainty related to unknown baryon effects.
It is obvious from Fig. \[fig:WDMvelfct\] that the WDM models lead to a better match between theory and observations than CDM. For the models with $m_{\rm TH}=2$ and 3 keV, the $v_{\rm max}$-VF from observations overlaps with the theory prediction (given the uncertainties in baryonic effects). For the coolest model with $m_{\rm TH}=4$ keV, a small tension between observation and theory starts to be visible around $v_{\rm max}\sim40$ km/s, but the discrepancy is still significantly smaller than for the case of CDM (reproduced in the bottom right panel of Fig. \[fig:WDMvelfct\]).
It is remarkable that warm DM models with thermal masses of $m\gtrsim3$ keV are able to solve (or at least significantly alleviate) the problem of the VF. These models are cold enough to agree with the very stringent bounds from the Lyman-$\alpha$ forest, therefore offering a truly viable alternative to standard CDM. Former studies have argued that only extreme WDM scenarios, which are in conflict with constraints from Lyman-$\alpha$ and MW satellite counts, are able to solve the mismatch of the VF [@Zavala:2009ms; @Papastergis:2011xe; @Schneider:2013wwa; @Klypin:2014ira; @Papastergis:2014aba]. These studies, however, did not account for suppression effects from the baryon sector nor for the correction of the observed VF due to larger values of $v_{\rm max}$ for WDM.
The special case of sterile neutrinos
-------------------------------------
The sterile (or right-handed) neutrino is often considered as the prime candidate for WDM. It is a well motivated hypothetical particle based on a straightforward extension of the standard model neutrino sector. Sterile neutrinos can only play the role of DM if their mass is in the keV-range, otherwise they would either not cluster enough or decay too quickly [see e.g. @Adhikari:2016bei].
A popular way to produce sterile neutrino DM in the early universe is via resonant mixing with active neutrinos [@Shi:1998km; @Abazajian:2001nj; @Asaka:2005an]. This production mechanism differs from thermal freeze-out and does not lead to Fermi-Dirac like momentum distributions. As a result, the suppression in the power spectrum can be somewhat shallower than for the case of the standard (thermal relic) WDM, at least for parts of the parameter space [@Ghiglieri:2015jua; @Venumadhav:2015pla].
A similar effect is observed if sterile neutrino DM is produced via the decay of a heavy scalar singlets [@Kusenko:2006rh; @Shaposhnikov:2006xi]. Depending on the coupling of the scalar to the standard model and on the decay width, the resulting sterile neutrino momentum distribution can strongly differ from a Fermi-Dirac function and may lead to shallower suppressions of the power spectra [@Merle:2015oja; @Merle:2015vzu; @Konig:2016dzg].
In terms of the velocity function, sterile neutrino DM is expected to show qualitatively similar effects to the thermal-like WDM models [see e.g. @Lovell:2016fec]. At the quantitative level, some differences are expected due to changes in the shape of the power spectra [@Schneider:2016uqi]. A detailed investigation of the effects of sterile neutrino DM on the VF will be performed in future work.
Mixed dark matter {#sec:MDM}
=================
A straight forward extension to the dark sector is to assume more than one DM species. There is a wealth of possibilities for MDM scenarios including different particle species with different kinds of couplings. These range from two unrelated purely gravitationally interacting DM species to phenomenologically rich scenarios which mirror the baryonic sector. In fact, one could argue that we are already confronted with a MDM universe, since neutrinos are massive, behave exactly like a dark matter fluid, and have a non-negligible effect on structure formation.
In this section we limit ourselves to the simple case of a mixture between cold and (thermal relic) warm DM [see e.g. @Borgani:1996ag; @Palazzo:2007gz; @Boyarsky:2008xj]. This is a hypothetical model with the advantage of yielding a large variety of suppressed power spectra. Similar to WDM, it reduces the number of small galaxies [@Anderhalden:2012jc; @Maccio':2012uh] and produces halo profiles with smaller concentrations [@Schneider:2014rda], but the affected mass range can be much larger.
In addition to the particle mass of the warm component ($m_{\rm TH}$), the MDM model is characterised by the mass fraction of warm to cold species, i.e. $f=\Omega_{\rm WDM}/(\Omega_{\rm WDM}+\Omega_{\rm CDM})$. In this paper we investigate three cases with the same WDM particle mass of $m_{\rm TH}=1.5$ keV and different fractions $f=0.2$, 0.4, and 0.6. Of course, this only covers a very small fraction of the full MDM parameter space, but it serves as an illustration for the kind of corrections that can be expected for other combinations of mixed DM.
The linear power spectra of the three MDM models are shown in Fig. \[fig:powspec\]. They are suppressed with respect to CDM but the shape of the suppression is much shallower than for WDM, spanning many orders of magnitudes in length scale.
Halo profiles {#halo-profiles}
-------------
The haloes of the mixed dark matter scenario are well described by NFW profiles [@Anderhalden:2012qt; @Maccio':2012uh] with reduced concentrations at small scales. The shape of the concentration-mass relation can again be directly obtained from the linear power spectrum of MDM by assigning the same concentrations to haloes with the same collapse redshift [see @Schneider:2014rda].
Fig. \[fig:MDMprofiles\] shows the velocity profiles (left panel) as well as the $v_{\rm max}$-$v_{\rm rot}$ dependence (middle panel) for a mixed DM model with $f=0.4$ and $m_{\rm TH}=1.5$ keV. Similarly to the case of WDM, the values of $v_{\rm max}$ are increased for small galaxies in MDM with respect to CDM. The resulting relation between $v_{\rm max}$ and $v_{\rm rot}$ is well fitted by a curved line which flattens out towards small velocities. The flattening starts at slightly larger scales than for WDM, owing to the smaller mass of the warm component.
The maximum allowed baryon suppression for mixed dark matter is obtained in the same way as for the WDM scenario (see Sec. \[sec:haloprofilesWDM\]). First, we define the function $\mathcal{M}$ describing the $v_{\rm max}$-$M_{\rm bar}$ relation of galaxies in MDM. Then, we determine the maximum allowed baryon suppression using the likelihood ratio analysis. The corresponding function ($\mathcal{M}_{\rm supp}$), is shown as dashed green line in Fig. \[fig:MDMprofiles\]. Finally, we use the completeness limit of the K13 sample to determine the characteristic velocity $v_c$ (green cross). The value of $v_c$ is model dependent (increasing with higher fractions $f$) and sets the largest scale at which the velocity function could be affected by baryon suppression.
For brevity, we illustrate only one MDM scenario in Fig. \[fig:MDMprofiles\]. However, other models show similar trends with growing discrepancies between MDM and CDM for increasing fraction $f$ or decreasing mass $m_{\rm TH}$.
Velocity function
-----------------
Within the mixed dark matter scenario, the velocity function is affected in a similar way to the case of WDM. First of all, the predicted VF is shallower than in the case of CDM, owing to a combination of reduced halo abundance and concentrations. Second, the observed and $v_{\rm max}$-corrected VF becomes steeper because of higher estimates of $v_{\rm max}$ in MDM as opposed to CDM.
Both effects are visible in Fig. \[fig:MDMvelfct\], where we plot the MDM models with $m_{\rm TH}=1.5$ keV and $f=0.6$ (top left), $f=0.4$ (top right), and $f=0.2$ (bottom right). Not surprisingly, all three models provide a much better match between theory and observations than in the case of CDM (bottom right). For the first two models, there is full agreement between $(v_{\rm max}$-corrected) observations (shaded bands) and theory predictions that include the uncertainties of baryon effects (hatched areas, bracketed by the solid and dashed lines). A small tension starts to be visible for the model with $f=0.2$, but the discrepancy between theory and observations remains significantly smaller than for the case of CDM.
The example of MDM illustrates that many alternative DM models have the potential to alleviate the problem of the over-abundance of field galaxies, provided they suppress perturbations at small scales. In Sec. \[sec:othercandidates\] we will briefly discuss some other models with similar characteristics.
Self-Interacting dark matter {#sec:SIDM}
============================
The concept of self-interacting cold dark matter (SIDM) became popular after @Spergel:1999mh showed that it could provide a better match to dwarf galaxy observations than the standard CDM model. However, it was soon realised that strong self-interactions are in conflict with observations at the scale of galaxy clusters, thereby ruling out the most simple SIDM scenarios [@Yoshida:2000uw; @MiraldaEscude:2000qt]. More recently, the SIDM model regained popularity thanks to the realisation that previous limits were set too stringently [@Rocha:2012jg; @Peter:2012jh], and that velocity-dependent SIDM models easily evade limits from clusters while being well motivated by particle physics [@Feng:2009mn; @Feng:2010zp; @Loeb:2010gj]. In addition, some observational studies based on strong-lensing found offsets between the mass centres of the stellar and the DM components in clusters, which could be explained by SIDM models [@Williams:2011pm; @Massey:2015dkw].
Concerning the velocity function (VF), only SIDM with velocity-dependent cross sections has the potential to reduce the discrepancy on small scales without modifying the large scales. All velocity-independent models alter both small and large scales and can therefore be discarded as a solution to the observed discrepancy of the VF. For the velocity-dependent cross section, we follow [@Feng:2010zp; @Loeb:2010gj] and assume a Yukawa force interaction leading to a scattering cross section $$\label{vdCS}
\frac{\sigma}{\sigma_{\rm m}}\simeq \left\{
\begin{array}{ll}
\frac{4\pi}{22.7} \beta^2\ln(1+\beta^{-1}) & \beta< 0.1 \\
\frac{8\pi}{22.7} \beta^2(1+1.5\beta^{1.65})^{-1} & 0.1< \beta< 10^3 \\
\frac{\pi}{22.7} (\ln\beta+1- \frac{1}{2}\ln^{-1}\beta)^2 & \beta> 10^3
\end{array}
\right.$$ where $m$ is the mass of the force carrier and $\beta\equiv\pi v_{\rm m}^2/v^2$. The SIDM model has two free parameters given by $(\sigma_{\rm m}/m)$ and $v_{\rm m}$.
In Fig. \[fig:crosssections\] we show how the SIDM cross sections depend on velocity for the models studied in this paper. All cross sections are largest at low particle velocities and strongly reduced at large velocities, showing their potential to simultaneously produce significant cores for dwarf galaxies while evading galaxy cluster constraints. The light grey band in Fig. \[fig:crosssections\] shows the region of parameter space where the TBTF problem is potentially alleviated due to the reduced central densities of haloes. The dark grey band indicates the region where the largest halo cores are expected. Above this scale, core-collapse starts to dominate, effectively reducing the core sizes despite even larger cross sections [see @Elbert:2014bma].
The cross-sections from Eq. \[vdCS\] have become the standard prescription for velocity-dependent SIDM in the literature. However, there are other models with shallower velocity dependence that are equally justified from a particle physics point-of-view [see @Kaplinghat:2015aga].
From cross sections to halo profiles
------------------------------------
The most striking feature of SIDM models in contrast to the CDM scenario is the flattening of the inner part of DM halo density profiles, which is a result of multiple scattering processes in high density regions. In previous work, SIDM haloes were described by Burkert-profiles [@Burkert:2000di; @Zavala:2012us]. This profile provides a good fit to the inner parts of a halo but slightly deviates towards large radii [@Rocha:2012jg]. In this paper we use the R16 profile [@Read:2015sta] instead, which has the advantage of becoming an NFW profile well beyond the core radius. So far, the R16 profile has only been applied to core transformations induced by baryons, but we show in Appendix \[app:profile\] that it also provides a very accurate fit to simulated SIDM haloes from the literature.
To assign concentrations to SIDM haloes, we use the same relation as for CDM, implicitly assuming that the rare collisions, well beyond the core radius, have a negligible effect on the profile. This assumption seems reasonable but should be tested in the future.
There is a direct relation between the cross section of SIDM and the average core size of haloes. Unfortunately, no simulation-based study has ever investigated this connection systematically. It is, however, possible to determine the approximate core size ($r_c$) analytically, using estimates of the average DM interaction rate ($\Gamma$). Assuming that a fixed number of interactions per Hubble time is required to produce a core, we obtain the relation $$\label{coresizeestimate}
\rho_{\rm nfw}(r_{c}|M)\frac{\langle\sigma v\rangle (r_{c})}{m}\simeq\Gamma.$$ Here, we closely follow the approach of @Dooley:2016ajo, but we assume an NFW profile (instead of a Hernquist profile) and we use an interaction rate of $\Gamma=0.4$ Gyr$^{-1}$ (instead of $\Gamma=1.0$ Gyr$^{-1}$). The latter gives a better match to simulated SIDM haloes in combination with the R16 core definition (see the appendix for more details). Following @Vogelsberger:2012ku, the average velocity-weighted cross sections are given by the integral over the Maxwell-Boltzmann distribution, i.e. $$\label{velweightedcs}
\langle\sigma v\rangle(r)=\frac{1}{\sqrt{4\pi}\sigma_{\rm vel}^3(r)}\int dv v^2 (\sigma v)\exp \left[-\frac{v^2}{4\sigma_{\rm vel}^2(r)}\right].$$ The velocity dispersion $\sigma_{\rm vel}$ depends on the halo profile and can be calculated by solving the isotropic Jeans equation, $d(\rho\sigma_{\rm vel}^2)/dr=-\rho d\phi/dr$ (where $\phi$ is the potential and $\rho$ is the DM density), leading to $$\label{veldisp}
\sigma_{\rm vel}^2 (r)=\frac{G}{\rho_{\rm nfw}(r)}\int_{r}^{\infty}dx\frac{M_{\rm nfw}(x)\rho_{\rm nfw}(x)}{x^2}.$$ Eq. (\[coresizeestimate\]) can be combined with Eqs. (\[velweightedcs\], \[veldisp\]) to find a relation between core size ($r_c$) and halo mass ($M$). Since this relation is calibrated to SIDM simulations from the literature (via the interaction rate parameter $\Gamma$), it is expected to provide reasonably accurate results over a large range of scales. A similar approach to estimate halo cores from SIDM cross sections can be found in @Kaplinghat:2015aga.
In the top-left panel of Fig. \[fig:SIDMvelfct\], we plot the relation between core size ($r_c$) and halo mass ($M$) for the SIDM models studied in this paper. The halo cores only vary by a factor of a few over a large range of mass scales. This is in strong contrast to SIDM models with velocity-independent cross sections, where there is a strong power-law dependence with core size increasing with halo mass [see @Dooley:2016ajo].
Given a halo density profile (i.e. Eq. \[R16profile\]) and a core radius ($r_{c}$) for SIDM, we can perform profile fits to all the galaxies in the selected sample. In the left panel of Fig. \[fig:SIDMprofiles\] we plot the velocity profiles of CDM (grey lines) and SIDM (with $\sigma_m/m=14$ cm$^2$/g, v$_{\rm m}=30$ km/s, dark-blue lines) fitted to the observed circular velocities $v_{\rm out}$ at $r_{\rm out}$ (symbols). The large SIDM cores lead to steep velocity profiles at small radii, requiring that small galaxies inhabit larger haloes. This becomes even more evident in the middle panel of Fig. \[fig:SIDMprofiles\] where the maximum circular velocity ($v_{\rm max}$) is plotted against the observed rotation velocity derived from the HI line width ($v_{\rm rot}$). There is a strong flattening and an increase of scatter visible in $v_{\rm max}$-$v_{\rm rot}$ relation towards small velocity scales (fitted by the solid dark-blue line). The latter is in line with the recently predicted higher variability of SIDM rotation curves with respect to CDM [@Elbert:2016dbb; @Kamada:2016euw; @Creasey:2016aaa][^9]
Finally, we plot the $v_{\rm max}$-$M_{\rm bar}$ relation for SIDM in the right panel of Fig. \[fig:SIDMprofiles\]. The relation is identical to the one of CDM at large velocity scales and shows both a downturn and an increase of scatter towards smaller scales. Following the approach used for the warm and mixed DM models, we define the function $\mathcal{M}(v_{\rm max})$ by modifying the linear fit from CDM (using the average shift between $v_{\rm max}$ from SIDM and CDM, i.e. the vertical separation between the black and dark-blue lines in the middle panel). This leads to the solid blue line in the right panel of Fig. \[fig:SIDMprofiles\]. The model with maximum baryon suppression, $\mathcal{M}_{\rm supp}(v_{\rm max})$, is shown as a dashed blue line. The line crosses the completeness limit of the K13 sample at the characteristic velocity, $v_c$ (dark-blue cross). The value of $v_c$ is model dependent (growing for increasing cross sections) and sets the maximum scale at which the theoretical abundance of galaxies hosted by SIDM haloes could be affected by suppression effects from baryonic processes.
Note that only one particular SIDM model is illustrated in Fig. \[fig:SIDMprofiles\] for brevity. Other scenarios show similar trends with growing differences between SIDM and CDM for larger particle cross-sections.
Velocity function
-----------------
In contrast to warm and mixed dark matter, the self-interacting DM model does not lead to a reduction of the halo abundance but simply modifies their inner structure. Obviously, the disagreement between the observed and predicted velocity function of galaxies can only be solved with substantially larger cores than the ones induced by baryons, which we showed in Sec. \[sec:vmax\] to be not large enough to affect the result.
In Fig. \[fig:SIDMvelfct\] we plot the VF of the four self-interacting DM scenarios introduced above. The two models with large cross sections and cores above $r_c\sim2$ kpc are able to fully reconcile theory with observations (see top panels). The third model is also marginally consistent with observations while the last model only slightly reduces the tension with respect to CDM.
In general, the SIDM models lead to a steepening of the observed $v_{\rm max}$-VF, while the predicted VF does not become shallower (as is the case for WDM and MDM). However, the VF can be more strongly suppressed by baryon processes in SIDM as opposed to CDM. This is due to the shallower $v_{\rm max}$-$v_{\rm rot}$ relation and the increased scatter, allowing for a stronger downturn of the $v_{\rm max}$-BTF relation.
While it is possible to solve the “missing dwarfs" problem with SIDM models, sufficiently strong cross-sections are required, producing cores of $r_c\sim 1-1.5$ kpc or larger (at the relevant dwarf-galaxy scales). It remains to be established whether these models are in agreement with other potential constraints from structure formation. For example, larger cross sections may lead to faster evaporation of substructures (due to high-velocity encounters with particles from the host halo) which could destroy too many satellites in MW sized objects. Detailed simulations are required to refute or confirm these concerns.
Other dark matter candidates {#sec:othercandidates}
============================
In addition to the dark matter models presented in this paper, there are other scenarios with the potential to alleviate the mismatch between the predicted and observed velocity function. We will now mention some of them, however without providing a quantitative analysis.
An obvious DM candidate with the potential to solve the discrepancy are axion-like particles (ALPs). At very low mass scales of $m\sim10^{-23}$ eV, ALPs start to form coherent waves of astrophysical length scales, leading to a suppression of the power spectrum [@Hu:2000ke; @Marsh:2013ywa], and to the formation of soliton cores [@Schive:2014hza]. For this reason, ALPs are sometimes referred to as *fuzzy* or *wave* DM. Whether the soliton cores are sufficiently large to be of relevance for the VF is, however, still unclear [@Schive:2015kza; @Hui:2016ltb].
Another interesting scenario consists of DM particles coupled to some relativistic fluid like photons, neutrinos, or dark radiation [see e.g. @CyrRacine:2012fz; @Boehm:2014vja; @Bringmann:2016ilk]. Such a coupling generates a suppression of the power spectrum usually combined with acoustic oscillations at the suppression scale. In terms of effects on the galaxy abundance, a similar behaviour than for WDM or MDM can be expected. It would furthermore be interesting to establish if the dark acoustic oscillation could still be visible in the galaxy velocity function [see also @Buckley:2014hja].
An interesting new framework to systematically study interactions of the dark sector has recently been presented under the name of *ETHOS* [effective theory of structure formation, @Cyr-Racine:2015ihg; @Vogelsberger:2015gpr]. Some of the example cases investigated by the authors show models with both suppressed power spectra and significant halo cores. The VF of the local volume could provide an ideal testbed to further study such scenarios.
Conclusions {#sec:conclusions}
===========
In the last few years it became evident that the observed *HI velocity width function* of galaxies in the local volume is in tension with predictions from gravity-only simulations based on the standard model of $\Lambda$CDM [e.g. @Zavala:2009ms; @Papastergis:2011xe; @Klypin:2014ira]. The discrepancy cannot be fully solved with baryon effects, such as strong UV photoevaporation or supernova feedback, because these processes induce a downturn in the $v_{\rm max}$-$M_{\rm bar}$ relation, which is not observed [@Trujillo-Gomez:2016pix TG16]. This crucial point is further highlighted in Appendix \[app:comparison\], where we compare to other work based on abundance matching techniques, simulations, and direct mass estimates from observed dwarf galaxies.
In principle, a number of observational effects could be affecting the kinematic analysis performed in this article, and could therefore alter our conclusions regarding the viability of $\Lambda$CDM[^10]. For example, the subsample of selected galaxies (with spatially resolved kinematic measurements) used to determine $v_{\max}$ could be biased with respect to the full sample used for the VF. This possibility was investigated by TG16, who found no systematic differences between the selected and full samples of local volume galaxies. Another possibility is a bias from inaccurate estimates of galaxy inclinations. Indeed, highly inclined galaxies (which give more accurate measurements of $v_{\rm rot}$) tend to have somewhat smaller velocities at a given baryonic mass. This means, however, that reducing inclination errors would shift the observed velocity function of $v_{\rm max}$ further away from the CDM predictions, worsening the discrepancy.
A further potential source of error are dwarf galaxies with an extremely low surface-brightness falling below the survey detection limit. If such objects exist in large numbers, they could potentially explain the difference between the observed and predicted velocity function. A detailed discussion of this possibility is given by @Klypin:2014ira. There it is argued that while some of the smallest galaxies with $v_{\rm rot}<20$ km/s could potentially stay undetected because of extremely low surface-brightness, they are very unlikely to make up a sizeable fraction of the full population[^11].
Finally, the mismatch between the observed and predicted $v_{\rm max}$-VF could originate from errors in the fitting procedure used to determine $v_{\rm max}$, with the main concern being the possibility of inner DM cores induced by stellar feedback. However, existing estimates produced by baryonic feedback effects [@Read:2015sta; @DiCintio:2013qxa] have been shown to be too small to produce a significant effect in galaxies with kinematic measurements at large radii (see Fig. \[fig:intro\] and TG16). Finally, the results of this paper depend on the customary assumption that spatially resolved HI rotation measurements (including corrections for turbulence) can be used to probe the halo potential.
A more speculative but intriguing option is that the mismatch between observed and predicted abundance of isolated galaxies points towards an alternative dark matter sector. In the present paper, we investigated different DM scenarios such as warm, mixed, and self-interacting DM, and we showed that they can be in much better agreement with observations. A more detailed summary of the results follows:
- The warm DM (WDM) scenario is characterised by a steep cutoff in the initial power spectrum, resulting in two important effects: reduced halo abundance, and lower concentrations. The two effects work together to flatten the predicted $v_{\rm max}$-VF. Furthermore and due to the lower concentrations, observed galaxies are expected to reside in more massive haloes, yielding a steeper relation for the observed $v_{\rm max}$-VF. As a result, observations and theory agree for WDM models with (thermal-relic) masses between $m_{\rm TH}\sim 1.8-3.5$ keV (see Fig. \[fig:WDMvelfct\]). This includes lukewarm DM scenarios which are cold enough to avoid the most stringent constraints from the Lyman-$\alpha$ forest. Significantly warmer models with $m_{\rm TH}\lesssim1.5$ keV are disfavoured by the observed $v_{\rm max}$-VF.
- We also considered a simple mixed DM (MDM) scenario with varying fraction of warm to cold dark matter. This model leads to a wide range of shapes for the power spectrum making MDM an ideal testbed for structure formation. The resulting effect on the VF is qualitatively similar to the case of WDM, except that the flattening can affect a wider range of scales. For large parts of the MDM parameter space, the agreement between observations and theory is highly improved with respect to the case of CDM (see Fig. \[fig:MDMvelfct\]). Again, this includes models that are in agreement with Lyman-$\alpha$ limits.
- The self-interacting DM (SIDM) scenario is qualitatively different from warm and mixed DM models in the sense that it does not yield suppressed perturbations at small scales. Instead, the inner parts of halo profiles are flattened due to repeated collisions of DM particles in high-density regions. The flattening of profiles leads to the inevitable conclusion that small galaxies should inhabit more massive haloes compared to CDM. As a result, the observed $v_{\rm max}$-VF becomes steeper for increasing cross sections. The theoretical abundance of galaxies in SIDM models is unchanged with respect to CDM as long as baryonic processes are neglected. The model including the maximum allowed baryonic suppression, on the other hand, is less constrained than for CDM due to the higher values (and the increased scatter) of $v_{\rm max}$ estimated for observed dwarf galaxies. As a result, SIDM models can fully solve the tension between the predicted and observed $v_{\rm max}$-VF as long as they form sufficiently large DM cores of $r_c\gtrsim1.5$ kpc in dwarf galaxies (see Fig. \[fig:SIDMvelfct\]). This is only possible for models with significant cross sections which have to be velocity-dependent to avoid constraints from galaxy clusters. Whether these models are in agreement with other small-scale observables has yet to be established.
In general, all models that either suppress perturbations at dwarf galaxy scales or flatten the inner DM halo density profiles (or a combination of both) can potentially alleviate the mismatch between predicted and observed abundance of galaxies as a function of $v_{\rm max}$. This includes many more scenarios than the ones studied here. Examples are interacting, decaying, late decoupling, or boson dark matter. A more detailed investigation of such models is postponed to future work.
Upcoming large area HI surveys, such as the [APERTIF]{} survey with the [WSRT]{} interferometer and the [WALLABY]{} survey with the [ASKAP]{} interferometer will provide large samples of dwarf galaxies with spatially resolved velocity information and improved HI sensitivity. This will make it possible to track down the remaining potential systematics related to the profile fitting procedure. The new data will furthermore allow to extend the observed velocity function to smaller scales, well below 10 km/s in velocity. This should lead to the discovery of a downturn of both the VF and the $v_{\rm max}$-$M_{\rm bar}$ relation due to the effects of photo-evaporation during reionisation. Once this effect is known to better accuracy, it will be possible to come up with highly improved constraints for the particle nature of dark matter.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Anatoly Klypin for very helpful suggestions on how to improve the present manuscript. AS acknowledges support from the Swiss National Science Foundation (PZ00P2\_161363). EP is supported by a postdoctoral fellowship of the Netherlands Research School for Astronomy (NOVA).
Comparison with other studies {#app:comparison}
=============================
Several recent papers have reported solutions to the overabundance problem of the galactic velocity function within the standard model of $\Lambda$CDM. In this section we discuss how these papers differ from our work and why we think they do not remove the problem.
@Brook:2015ofa performed a detailed investigation of the galactic velocity function for three cases, CDM with baryonic cores, as well as one model of WDM and SIDM both without baryonic cores. Starting with the halo mass function, they use abundance matching (AM) to obtain stellar masses, from which they estimate the extent of stellar and HI discs as well as baryon induced DM cores. This allows them to obtain mock velocity profiles for each halo mass. Based on these velocity profiles, @Brook:2015ofa determine the radius where the circular velocity equals the observed $v_{\rm rot}$ from small galaxies (which leads to an empirical relation between $M_{\rm star}$ and $r_{\rm rot}$). They find this radius to be much smaller than the radius of maximum circular velocity, concluding that $v_{\rm rot}$ is considerably smaller than $v_{\rm max}$. This bias between $v_{\rm rot}$ and $v_{\rm max}$ strongly reduces the initial discrepancy between theory and observation in the VF without, however, completely solving it. @Brook:2015ofa show that the remaining tension can be solved by either assuming baryon-induced cores within $\Lambda$CDM or alternative dark matter (i.e. WDM or SIDM without baryonic cores).
The main weakness of the method applied by @Brook:2015ofa is that it relies on abundance matching. The AM technique matches observations to the $\Lambda$CDM model by *assigning* small galaxies to very large haloes, without *testing* this assignment against observations. As a consequence, @Brook:2015ofa obtain small HI radii (via their empirical $M_{\rm star}$-$r_{\rm rot}$ scaling relation) and therefore small values for $v_{\rm rot}$, which largely alleviates the initial discrepancy between the predicted and observed VF.
Very recently, @Brook:2015eva published a paper, where abundance matching combined with the observed baryonic Tully-Fisher (BTF) relation is used to estimate the bias between $v_{\rm rot}$ and $v_{\rm max}$. They find that the discrepancy between observed and predicted velocity function disappears entirely, provided the adequate Tully-Fisher relation is used. While this paper reveals systematical differences between different observations and highlights the importance of an adequate velocity definition, it does not provide a test for $\Lambda$CDM on its own[^12]. A further necessary requirement for testing the underlying cosmological model is to verify the applied AM relation.
Attempts to verify abundance matching of $\Lambda$CDM by estimating the halo mass of observed galaxies have been performed by several papers in the past. @Papastergis:2014aba used kinematical information from the extended HI content of dwarf galaxies to show that there is a discrepancy between the data and AM relations from the local volume, irrespectively of whether baryon-induced cores are assumed or not. More recently, @Pace:2016oim estimated halo masses based on careful analysis of full HI rotation curves for a few field galaxies, obtaining similar results. @Karukes:2016eiz also find a discrepancy with AM expectations based on $\Lambda$CDM, by analysing the rotation curves of a sample of 36 late-type dwarfs. @Brook:2014hda, on the other hand, used stellar kinematics to estimate halo masses of dwarf galaxies. They find better agreement with AM relations from the local volume, mainly due to their assumption of cored profiles from baryon processes. However, stellar kinematics only probe the very inner region of haloes, which are subject to large uncertainties and potential systematics in the halo mass estimates. Indeed, small differences in the model are amplified leading to large differences in halo mass.
Recently, @Maccio:2016egb published a study on the VF based on the NIHAO[^13] suite of hydrodynamical simulations with full metal cooling and standard recipes for sub-grid effects such as star formation and supernova feedback. They report a very large bias between $w_{50}$ and $v_{\rm max}$ fully solving the discrepancy between the observed and predicted VF. There are two main reasons why the results from @Maccio:2016egb differ from ours. First, the HI content of the NIHAO galaxies is less extended than the one from the selected sample of observed galaxies leading to smaller values of $w_{50}$ (or $v_{\rm rot}$) compared to $v_{\rm max}$ (see TG16 and @Papastergis:2016aaa for a detailed comparison). Second, the strong feedback effects present in the NIHAO simulations make the NIHAO galaxies reside in very massive haloes compared to the mass estimates from the selected sample.
Finally, during the review process of this work, @Brooks:2017rfe [henceforth Brooks17] published a study based on a suite of hydrodynamical simulations that claims to fully solve the apparent discrepancy of the velocity function. Similarly to @Maccio:2016egb, they obtain a more significant bias between $v_{\rm rot}$ and $v_{\rm max}$ compared to what we find in our analysis. A closer look at their results reveals that they are able to completely close the gap between the observed and predicted VF between $v_{\rm rot}=20-50$. However, there is some remaining discrepancy at both larger and smaller velocities, which they correct in the latter case by assuming a cutoff due to reionisation.
As a result of very efficient feedback recipes, both the NIHAO and the Brooks17 simulations obtain larger halo masses and larger maximum circular velocities than what we find by analysing HI kinematics of dwarf galaxies. This is illustrated in Fig. \[fig:appendix1\], where the left and right panels show the $M_{\rm star}-M_{\rm 200}$ and the $v_{\rm rot}-v_{\rm max}$ relations, respectively. For both relations the selected galaxy sample (black symbols with error bars representing the dependence on the concentration parameter) is well described by a power law (black solid lines) while the NIHAO and Brooks17 simulations (red and green circles) exhibit a downturn towards small mass and velocity scales. This is a direct consequence of their strong feedback recipes which reduce the amount of stars and gas in a halo of a given size.
In Fig. \[fig:appendix1\] we also compare our results to independent estimates of halo mass and maximum circular velocity from @Read:2016aaa [henceforth Read16]. They used fully resolved rotation curves of individual field dwarfs accounting for stellar and gaseous components as well as baryon-induced cores. The results of Read16 are shown as purple squares, where bright symbols represent dwarfs with reliable rotation curves while the shaded symbols without error-bars denote data with potential systematics from inclination or signs of disequilibrium (dubbed ‘rogues’ in Read16). It is very encouraging that the results from Read16 agree well with our own mass estimates.
In summary, there is a systematic difference between the halo mass of simulated galaxies (from the NIHAO or the Brooks2017 simulations) and direct mass estimates from local field dwarfs which could point towards a genuine problem of hydrodynamical simulations at dwarf galaxy scales. However, we want to point out that the current observational data is too sparse to support any strong conclusions. Upcoming large area HI surveys with interferometric data will highly improve the observational situation in the next few years.
A new halo density profile for SIDM {#app:profile}
===================================
Here we demonstrate that the R16 profile [@Read:2015sta] is not only suitable to describe cores from baryonic feedback, but also provides very accurate fits to profiles of self-interacting dark matter (SIDM) haloes. In Eq. (\[R16profile\]) we introduced the R16 mass profile which consists of the NFW mass profile multiplied with a simple two-parameter function. The density profile can be obtained from Eq. (\[R16profile\]) by a simple derivative, i.e., $$\rho_{\rm R16}(r) = \rho_{\rm nfw}(r)f^n+\frac{n f^{(n-1)}(1-f^2)}{4\pi r^2r_c}M_{\rm nfw}(r),$$ where $n=1$ and $f(r)$ is given by Eq. (\[R16profile\]). The R16 profile has the advantage of converging to the NFW profile for $r\gg r_c$, where DM self-interactions are negligible.
In Fig. \[fig:appendix2\] we show fits using the R16 profile to a few simulated SIDM profiles found in the literature. The left panel shows data points for two haloes from @Zavala:2012us corresponding to velocity-dependent model with $\sigma_{\rm m}/m=35$ cm$^2$/g and $v_{\rm m}=10$ km/s (blue, dSIDMb) and a velocity-independent model with $\sigma/m=10$ cm$^2$/g (red, SIDM10). The haloes are well fitted with R16 profiles with $r_c=0.8$ kpc and $r_c=1.8$ kpc. The right panel shows two haloes from @Elbert:2014bma out of velocity-independent SIDM simulations with $\sigma/m=1$ cm$^2$/g (yellow, SIDM1) and $\sigma/m=10$ cm$^2$/g (magenta, SIDM10), respectively. Again the R16 profile provides an accurate fit to the simulated halo profiles. The core sizes for these haloes are $r_c=1.2$ kpc and $r_c=2.4$ kpc.
[^1]: Recently, @Bekeraite:2016aaa have shown that there is disagreement between observations and simulations at larger velocities as well (i.e. between 60 km/s and 300 km/s). This tension is, however, not as strong and could be due to the fact that $v_{\rm max}$ of larger galaxies is dominated by the stellar component.
[^2]: All observed values of $v_{\rm out}$ are corrected for pressure support [see Sec. 4.1 in @Papastergis:2016aaa]. For large galaxies (with $v_{\rm out}>120$ km/s) we furthermore subtract the expected contribution from stars and cold gas in the galaxy centres. We have checked that this correction does not affect our final results (see TG16 for more details).
[^3]: The line is based on the MultiDark suite of simulations [@Klypin:2014kpa] and includes a correction for the increase of circular velocities due to the stellar component of galaxies visible beyond 80 km/s [see @TrujilloGomez:2010yh].
[^4]: A constant ratio of subhalo to host-halo numbers is a very good approximation for haloes with $v_{\rm max}<150$ km/s. In @Klypin:2010qw it was shown that this ratio does not change by more than four percent in the range $v_{\rm max}=30-150$ km/s.
[^5]: The argument that baryonic feedback effects suppress the BTF relation is only true if $v_{\rm max}$ is used for the velocity, as it is a direct measure of the halo potential and does not depend on the extent of observable gas (as is the case for $v_{\rm rot}$ for example).
[^6]: The logarithmic likelihood ratio test is based on the measure $D\equiv2\ln(\mathcal{L}_m/\mathcal{L}_0)$, where $\mathcal{L}_m$ is the maximum likelihood and $\mathcal{L}_0$ the likelihood of a constrained model with fixed $v_s$. For a large sample size $D$ is known to be $\chi^2$-distributed and a model with given $v_s$ can therefore be excluded at the confidence level (CL) given by the $p$-value from a $\chi^2$ statistic.
[^7]: The constraints from Lyman-$\alpha$ rely on assumptions about the temperature-evolution of the intergalactic medium. Using very high-redshift quasars and assuming a power-law dependence for the temperature, @Viel:2013apy obtained the limit $m_{\rm TH}\gtrsim 3.3$ keV at 95% CL. This limit is weakened by about 1 keV if the power-law evolution of the temperature is replaced by an abrupt jump in temperature around $z\sim5$ [@Viel:2013apy; @Garzilli:2015iwa]. An even stronger constraint of $m_{\rm TH}\gtrsim 4.1$ keV (at 95% CL) has been found by @Baur:2015jsy using high-redshift quasars from the [BOSS]{} survey. This tight limit is, however, relaxed to $m_{\rm TH}\gtrsim 3.0$ keV if cosmological parameters from [Planck]{} are assumed (instead of the internal parameters from [BOSS]{}).
[^8]: The sizes of the error-bars show the maximum variation in $v_{\rm max}$ if the concentrations are raised or lowered by 1-$\sigma$ with respect to the mean value. They do not include observational uncertainties.
[^9]: The larger scatter of SIDM rotation curves is the result of an interplay between the stellar and the DM components. Depending on the number and distribution of stars in the halo centres, the DM component may or may not experience core collapse, resulting in a large diversity of profiles. In principle, this effect is testable by combining HI rotation curves with the observed stellar density profiles.
[^10]: For a detailed discussion about potential systematics regarding the galaxy sample, see Sec. 4.1 of @Papastergis:2016aaa.
[^11]: Furthermore, the most recent searches for extremely low surface brightness dwarfs around massive spirals (including Local Group dwarfs and MW and M31 satellites) find very few objects with $\mu>27$ mag/arcsec$^2$ brighter than $M_V=-10$ [see e.g. @Merritt:2014rza]. Therefore, most of the dwarfs in the local volume that went undetected due to their low surface brightness should be below the magnitude limit of the catalogue from @Karachentsev:2013ipr, and hence would not affect our conclusions.
[^12]: Indeed, for the ideal case where the BTF, the VF, as well as the AM relation is based on one single set of observations, the discrepancy of the VF has to disappear by construction for a large number of different cosmological models.
[^13]: The Acronym NIHAO stands for Numerical Investigation of a Hundred Astrophysical Objects.
|
---
abstract: 'The photometric characterization of M33 star clusters is far from complete. In this paper, we present homogeneous $UBVRI$ photometry of 708 star clusters and cluster candidates in M33 based on archival images from the Local Group Galaxies Survey, which covers 0.8 deg$^2$ along the galaxy’s major axis. Our photometry includes 387, 563, 616, 580, and 478 objects in the $UBVRI$ bands, respectively, of which 276, 405, 430, 457, and 363 do not have previously published $UBVRI$ photometry. Our photometry is consistent with previous measurements (where available) in all filters. We adopted Sloan Digital Sky Survey $ugriz$ photometry for complementary purposes, as well as Two Micron All-Sky Survey near-infrared $JHK$ photometry where available. We fitted the spectral-energy distributions of 671 star clusters and candidates to derive their ages, metallicities, and masses based on the updated [parsec]{} simple stellar populations synthesis models. The results of our $\chi^2$ minimization routines show that only 205 of the 671 clusters (31%) are older than 2 Gyr, which represents a much smaller fraction of the cluster population than that in M31 (56%), suggesting that M33 is dominated by young star clusters ($<1$ Gyr). We investigate the mass distributions of the star clusters—both open and globular clusters—in M33, M31, the Milky Way, and the Large Magellanic Cloud. Their mean values are $\log(M_{\rm cl}/M_{\odot})=4.25$, 5.43, 2.72, and 4.18, respectively. The fraction of open to globular clusters is highest in the Milky Way and lowest in M31. Our comparisons of the cluster ages, masses, and metallicities show that our results are basically in agreement with previous studies (where objects in common are available); differences can be traced back to differences in the models adopted, the fitting methods used, and stochastic sampling effects.'
author:
- Zhou Fan and Richard de Grijs
title: 'Star clusters in M33: updated $UBVRI$ photometry, ages, metallicities, and masses'
---
Introduction {#s:intro}
============
Since star clusters represent an important component of the galaxies they are associated with, studies of star clusters’ stellar populations and age distributions can provide clues to the formation and evolution of their host galaxies. In addition, since populous star clusters are much more luminous than individual stars, they are usually much easier to observe and study.
At a distance of $847\pm60$ kpc—equivalent to a distance modulus of $(m-M)_0=24.64\pm0.15$ mag [@gal04]—M33 (also known as the Triangulum Galaxy) is the third largest spiral galaxy in the Local Group of galaxies. Since the galaxy is seen relatively face-on, under an inclination of $i=56^{\circ}\pm1^{\circ}$ [@zeh], it is eminently suitable for studies of its star cluster system. At present, the most comprehensive and widely used star cluster catalog is that of @sm07, which combines data on almost all M33 star clusters published in the literature, including information on their photometry, ages, metallicities, and masses. The latest version[^1] (henceforth SM10) includes 595 star clusters and candidates. @pl07 found 104 star clusters in [*Hubble Space Telescope*]{} ([*HST*]{})/Wide Field and Planetary Camera-2 (WFPC2) archival images, including 32 new objects based on new [*HST*]{} observations. Although their observations improved the spatial coverage of the M33 disk, this catalog is still incomplete for the entire disk. These authors found two different star cluster populations on the basis of their sample’s color–magnitude diagram (CMD), including a large number of blue clusters and a smaller number of red objects. They also suggested that relatively more red clusters are found in the galaxy’s outer regions.
Subsequently, @zkh published a list of 4780 extended sources, including 3554 new cluster candidates observed with the MegaCam instrument on the 3.6 m Canada–France–Hawai’i Telescope (CFHT). However, $\sim$60% of these clusters are not considered genuine owing to possible misidentifications [@san09; @san10]. Based on [*HST*]{}/Advanced Camera for Surveys (ACS)–Wide Field Channel (WFC) observations, @san09 presented photometry of 161 M33 star clusters, of which 115 were newly identified. Based on their CMDs, they suggested that these clusters’ ages were between 0.01 and 1 Gyr, whereas their masses range from $5\times10^3 M_\odot$ to $5\times10^4 M_{\odot}$. However these authors also point out that, since their photometry is generally not sufficiently deep to detect the main-sequence turnoff (MSTO), very few of their sample clusters are older than 1 Gyr. Using MegaCam on the CFHT, @san10 identified 2990 extended sources in M33, 599 of which were new cluster candidates and 204 were previously known clusters. Based on CMD analysis, these authors suggested that the majority of the clusters have young to intermediate ages, although their sample also includes some old objects. They suggested that a possible M31–M33 interaction some 3.4 Gyr ago may have triggered an epoch of star (cluster) formation in M33.
Comparison of observational spectral-energy distributions (SEDs) with theoretical stellar population synthesis models by application of $\chi^2$ minimization is a widely used technique to estimate ages, metallicities, reddening values, and masses of extragalactic star clusters. This technique has been applied to the cluster systems in, e.g., M31 [@jiang03; @fan06; @fan10; @ma07; @ma09; @wang10], M33 [@ma01; @ma02a; @ma02b; @ma02c; @ma04a; @ma04b], the Large Magellanic Cloud [LMC; e.g., @da06; @pop12; @grijs13], M82 [@deg03a; @lim13], NGC 3310 and 6745 [@deg03b], as well as for stellar population synthesis model comparisons [@deg05; @fan12].
In this paper, we first obtain photometry for all M33 star clusters in our sample (see Section \[s:samp\] for definition) based on archival images from the Local Group Galaxies Survey [LGGS; @massey]. Using photometry in the $UBVRI, ugriz$ (Sloan Digital Sky Survey; SDSS) bands and Two Micron All-Sky Survey (2MASS) $JHK$ magnitudes [@2mass][^2] when available, the ages and masses of the star clusters in our sample are estimated by comparison of their observed SEDs with updated [ parsec]{} (version 1.1) isochrones [@bre12]. This paper is organized as follows. Section \[s:data\] describes the sample selection and $UBVRI$ photometry. In Section \[s:age\] we describe the simple stellar population (SSP) models used, as well as our method to estimate the cluster ages and metallicities. In Section \[s:mass\] we present the clusters’ mass estimates, and we summarize and conclude the paper in Section \[s:sum\].
Data {#s:data}
====
Sample {#s:samp}
------
Our sample star clusters are mainly selected from @san10, whose database is based on observations with the CFHT/MegaCam camera. Their catalog covering the M33 area contains 2990 objects, including background galaxies, confirmed star clusters, and cluster candidates, as well as unknown objects. The catalog provides the positions and $ugriz$ photometry of all objects. Since our focus is on the star clusters, galaxies and unknown objects were eliminated from the catalog, and we subsequently performed photometry for the 803 star clusters and cluster candidates in their catalog.
We used archival $UBVRI$ images from the LGGS, which covers a region of 0.8 deg$^2$ along the galaxy’s major axis. The images we used consisted of three separate but overlapping fields with a scale from 0.261$''$ pixel$^{-1}$ at the center to 0.258$''$ pixel$^{-1}$ in the corners of each image. The field of view of each mosaic image is $36\times36$ arcmin$^2$. The observations were taken with the Kitt Peak National Observatory 4 m telescope between August 2000 and September 2002. The median seeing of the LGGS images is $\sim$ 1. Although @ma12 inspected the images and obtained $UBVRI$ photometry for all star clusters and unknown objects in @sm07 based on archival LGGS images, there are still hundreds of star clusters from @san10 in this field which do not have published $UBVRI$ photometry. Therefore, here we only perform photometry of the clusters in the LGGS images following the identifications of @san10. We employed the latest version of [ SExtractor]{}[^3] [@ba96] to find the sources in the images and match them to the coordinates of our 803 sample star clusters and candidates. Eventually, we detected 588 clusters and candidates with quality FLAGS = 0, which indicates that there are no problems associated with these objects (i.e., no contamination by nearby sources or saturation effects) in the LGGS images.
To supplement these data, we also include 120 confirmed star clusters from the updated (2010) version of @sm07 [SM10], which were not included in @san10. Thus, the number of clusters in our final sample is 708.
Figure \[fig1\] shows the spatial distribution of all sample clusters and candidates in the M33 field. The three data frames represent the field of view of the @massey data, and the large square outline covers the observed field of @san10. The large green ellipse delineates $D_{25}$, i.e., it corresponds to the $\mu_B = 25$ mag arcsec$^{-2}$ isophote [@boi07]. The yellow solid bullets and green open circles are, respectively, the confirmed star clusters and cluster candidates from @san10. The combined total number of clusters and cluster candidates is 588. These latter clusters have been cross-identified in the @massey image, which we will focus on here; we do not consider clusters outside of the boundaries of the @massey image. The orange symbols represent the 120 star clusters identified by SM10 but not included in @san10. It is clear that most star clusters and candidates are associated with and projected onto the galaxy’s disk (i.e. inside the galaxy’s $D_{25}$).
![Spatial distribution of our star clusters and cluster candidates. The yellow solid bullets (confirmed clusters) and green open circles (candidate clusters) are from @san10; we determined their cross-identifications in the @massey image. The orange symbols represent the clusters identified by SM10 but not included in @san10. The three data frames are the fields of view of @massey, whereas the large square outline covers the observed field of @san10.[]{data-label="fig1"}](fig1.ps)
Integrated photometry {#s:phot}
---------------------
Prior to this work, @massey compiled point-spread-function (PSF) photometry for 146,622 stars (point sources) in the M33 fields, with photometric uncertainties of $<10$% at $UBVRI \sim 23$ mag. However, there are no relevant discussions of the extended sources (e.g., star clusters and galaxies) in the published LGGS papers. Recently, @ma12 derived aperture photometry of 392 star clusters and unknown objects in the catalog of @sm07 in the $UBVRI$ bands. However, there are still several hundred M33 star clusters and cluster candidates identified by @san10 which lack LGGS photometry in the $UBVRI$ bands.
To obtain additional photometric information for the star clusters, we carried out photometric measurements of our sample M33 clusters and candidates. [SExtractor]{} was applied to the LGGS images in all of the $UBVRI$ bands to derive supplementary and homogeneous photometry. The [SExtractor]{} code provides isophotal magnitudes corrected for the flux missed by isophotal-magnitude determination, MAG$\_$ISOCOR. This approach works well for stars but poorly for elliptical (galaxy) profiles with broader wings. [SExtractor]{} also delivers automatic aperture photometry measurements of galaxies based on the first-moment algorithm of @kr80, MAG$\_$AUTO. The MAG$\_$BEST magnitudes can be automatically mapped onto MAG$\_$AUTO if neighbors cannot bias the photometry by more than 10%. In all other cases, MAG$\_$BEST is set to equal MAG$\_$ISOCOR, because the latter measurements are not significantly affected by nearby sources. Thus, we adopted the MAG$\_$BEST magnitudes as our final instrumental magnitudes. As a consequence, we do not need to choose the size of the aperture used. The instrumental magnitudes were calibrated in the standard Johnson–Kron–Cousins $UBVRI$ system by comparing the published magnitudes of stars from @massey, who calibrated their photometry using @lan92 standard stars, with our instrumental magnitudes. Since the magnitudes in @massey are given in the Vega system, our photometry is also tied to that system. The calibration errors range from $\sim0.01$ to $\sim0.03$ mag in the $UBVRI$ bands, with more than 300 secondary standard stars available in each field. Finally, we obtained photometry for 708 objects, with 387, 563, 616, 580, and 478 sources in the individual $UBVRI$ bands, respectively, of which 276, 405, 430, 457, and 363 star clusters and candidates do not have previously published photometry.
Table 1 of @massey shows that the seeing conditions under which the LGGS fields were obtained ranged from $0''.8$ to $1''.2$ in all filters; for most fields the prevailing seeing was around $1''.0$. In their table 3, these authors compared the differences in their calibrated photometry between overlapping fields using well-exposed, isolated stars and found that the median difference was several millimagnitudes. Our photometry has been calibrated relative to that of @massey. We compared the photometric measurements of those clusters that were located in the regions of overlap between different frames and and found differences of only a few $\times 0.01$ mag. While these differences are a little larger than those reported by @massey, this is not unexpected, since star clusters often have more extended and more complicated profiles than stars. For clusters with more than one photometric measurement in overlapping fields, we adopted the magnitude associated with the smallest statistical uncertainty.
Table \[photo\] lists our new broad-band $UBVRI$ magnitudes and the corresponding photometric errors. The latter combine the errors associated with MAG$\_$BEST with those related to the flux calibration, as $$\sigma_i^{2}=\sigma_{{\rm best},i}^{2}+\sigma_{{\rm calib},i}^{2},
\label{eq1}$$ where $i$ represents any of the $UBVRI$ bands, whereas $\sigma_{\rm
best}$ and $\sigma_{\rm calib}$ correspond to the photometric uncertainties associated with the MAG$\_$BEST magnitudes and flux calibration, respectively.
Since [SExtractor]{} applies apertures of different sizes to obtain MAG$\_$BEST magnitudes, depending on the size of the object of interest, we applied aperture growth-curve corrections to all photometric measurements. In fact, although the MAG$\_$BEST values represent the optimum magnitudes in the presence of neighboring sources, they may still systematically underestimate the total flux of extended sources by about 10% [@mc05; @cal08]. Therefore, we corrected for this ‘lost’ flux using the appropriate aperture corrections. We used an approximate aperture radius $r=(a^2+b^2)^{1/2}$ for our photometry by combining half the major axis, $a$, and half the minor axis, $b$. We then calculated the aperture corrections on the basis of template growth curves (derived from the LGGS data) that were most representative of our extended star clusters. The aperture corrections are slightly different for different filters and different images: on average, they are $\sim0.36$ mag for $r \le 3$ pix, $\sim0.10$ mag for $3<r\le5$ pix, $\sim0.05$ mag for $5 < r \le7$ pix, and $\sim0.02$ mag for $r>7$ pix. The maximum aperture used for our photometry is 7.19 pix ($1''.85$). We therefore used $r = 7 \mbox{ pix} \approx1''.8$ as the radius for our full sample’s aperture corrections.
Many previous studies [e.g., @san09; @san10; @pl07] used a fixed aperture of $r=2''.2$ for the photometry of all clusters and a smaller aperture, $r \approx 1''.5$, for color measurements. For comparison, based on the sizes and (elliptical) profiles of the clusters in our sample, we used apertures of $r \le 1''.5$ for the photometry of 95% of our sample clusters. Nevertheless, the apertures adopted in both this article and previous studies result in essentially the same photometry and colors. The object names we use follow the naming convention of @san10 and SM10.
Previously, @pl07 determined integrated $BVI$ aperture photometry for 104 M33 star clusters using (mainly) CFHT images, as well as supplementary [*HST*]{}/WFPC2 archival images. Charge-transfer (in)efficiency (CTE) corrections were applied to the [*HST*]{} magnitudes, and all photometry has been converted to the standard Johnson–Cousins system. Figure \[fig2\] compares our photometry with that of @pl07, which shows that there is little systematic difference: $\Delta V=0.006\pm0.001$ mag [in the sense, this paper minus @pl07], with $\sigma=0.349$ mag. Both our $(B-V)$ and $(V-I)$ colors show good agreement with @pl07 down to the faintest magnitudes. The differences between both sets of photometry are $\Delta (B-V)=0.049\pm0.004$ mag with $\sigma=0.189$ mag, and $\Delta (V-I)=0.025\pm0.003$ mag, $\sigma=0.302$ mag.
![Comparisons of our photometric measurements with those of @pl07 for all star clusters in common. The error bars represent a combination (addition in quadrature) of the uncertainties associated with our photometry and those from the literature.[]{data-label="fig2"}](fig2.ps)
Figure \[fig3\] shows the difference between our photometry and that of @san09, whose database includes integrated photometry of 161 star clusters in M33 based on [*HST*]{}/ACS–WFC observations. The photometric uncertainties associated with the mean offsets are defined as $\sigma/\sqrt{N}$, where $\sigma$ is the standard deviation and $N$ the number of data points. CTE corrections were applied and the photometry has been converted to the standard Johnson–Cousins system. Figure \[fig3\] shows comparisons of the respective sets of magnitudes and colors. Since @san09 do not provide their photometric uncertainties, the error bars in Fig. \[fig3\] reflect our photometric uncertainties only. Our $V$-band magnitudes are in good agreement with those of @san09 down to the faintest magnitudes: $\Delta V=0.162\pm0.016$ mag [this paper minus @san09], with $\sigma=0.304$ mag. However, a few objects have magnitudes that are $>0.5$ mag fainter in our database than in the tables of @san09, i.e., objects 105, 110, and 148 of @san09, which are marked with open circles in Fig. \[fig3\]. We checked the $V$-band images and found that all of these objects are located close to a few fainter neighboring sources, which are treated as independent objects in our photometry but they were regarded as stars belonging to the same cluster by @san09. Both our $(B-V)$ and $(V-I)$ colors show good agreement with @san09. The differences between our measurements and their photometry are $\Delta (B-V)=0.106\pm0.025$ mag with $\sigma=0.233$ mag and $\Delta (V-I)=-0.065\pm0.008$ mag, $\sigma=0.361$ mag.
![As Fig. \[fig2\] but for our photometry and that of @san09. Since @san09 do not provide their photometric uncertainties, the error bars only reflect the uncertainties in our photometry. The open circles are objects 105, 110, and 148 of @san09.[]{data-label="fig3"}](fig3.ps)
We also compared our photometry with the measurements of SM10, who assembled their photometry from the recent literature. Figure \[fig4\] shows the relevant comparisons. Since SM10 do not provide their photometric uncertainties, the error bars only reflect the uncertainties associated with our photometry. Our photometry is generally consistent with that of SM10: $\Delta
V=0.082\pm0.005$ mag (in the sense of this paper minus SM10) with $\sigma=0.322$ mag. Again, both the $(B-V)$ and $(V-I)$ colors show good agreement with SM10 down to the faintest magnitudes. The differences between our colors and their photometry are $\Delta
(B-V)=0.069\pm0.006$ mag with $\sigma=0.267$ mag and $\Delta
(V-I)=0.012\pm0.001$ mag, $\sigma=0.343$ mag.
![As Fig. \[fig3\] but for our photometry and that of SM10. Since SM10 do not provide their photometric uncertainties, the error bars only reflect the uncertainties associated with our photometry.[]{data-label="fig4"}](fig4.ps)
We also compare our newly obtained photometric magnitudes and colors with those published by @ma12 in Fig. \[fig5\]. The error bars shown in this figure are a combination (added in quadrature) of the uncertainties associated with our measurements and those from the literature. For all sources, the $V$-band offset is $\Delta
V=-0.100\pm0.007$ mag [again, in the sense this paper minus @ma12], with $\sigma=0.257$ mag. Our $V$-band magnitudes are in good agreement with the equivalent values of @ma12 for bright sources, $V<18$ mag, while the photometry of @ma12 seems to be somewhat fainter than our photometry for $V>18$ mag. In fact, @ma12 noted that his $V$-band photometry is systematically fainter than the previously published photometric measurements of @sm07, @pl07, and @san09, so this result is in line with our expectations. Our $(U-V)$, $(B-V)$, $(V-R)$, and $(V-I)$ colors show good agreement with the measurements of @ma12. The differences between our and his colors are $\Delta
(U-V)=-0.151\pm0.013$, $\sigma=0.244$ mag; $\Delta
(B-V)=-0.068\pm0.005$, $\sigma=0.154$ mag; $\Delta
(V-R)=0.049\pm0.004$, $\sigma=0.157$ mag; and $\Delta
(V-I)=0.080\pm0.010$, $\sigma=0.274$ mag.
![Comparison of our photometry and colors with the equivalent measurements of @ma12. The error bars are a combination (added in quadrature) of the uncertainties associated with our measurements and those of @ma12.[]{data-label="fig5"}](fig5.ps)
Figure \[fig6\] shows the luminosity function of the M33 star clusters and candidates in our sample, which can be used to estimate the completeness of our photometry. The magnitudes are extinction-corrected absolute $V$-band magnitudes. We adopted a distance modulus to M33 of $(m-M)_0 = 24.64$ mag [@gal04]. Extinction determinations for these star clusters were taken from @pl07 and @san09. For star clusters without published reddening values, we adopted an average reddening of $E(V-I)=0.06$ mag [@sa00; @san09], since for a significant number of clusters deriving individual reddening values is not possible. In fact, @sa00 found that the standard deviation associated with the average reddening value is $\Delta E(V-I)=0.02$ mag, which shows that the scatter in reddening among most M33 star clusters is not significant. Reddening variations may introduce a maximum additional uncertainty of $\sim0.08$ mag in the $U$ band, and much smaller uncertainties in the other, redder filters, particularly in the near-infrared (NIR) bands. This is similar to the uncertainties in our photometry. We thus conclude that variations in the average reddening are unlikely to affect our fit results more significantly for clusters without prior reddening estimates than the individual reddening values determined previously for most of our other sample clusters in M33.
Using a bin size of 0.5 mag, we determined an overall limiting magnitude of $M_V^0= -4.54$ mag, which corresponds to the half-peak height of the distribution; this follows the method adopted in our previous analysis [@fan10] of M31 globular clusters (GCs). In addition, we found that the peak of the distribution occurs at $M_V^0=-6.04\pm0.04$ mag, with $\sigma_{M_V^0}=1.277$ mag, which we adopt as the completeness magnitude threshold. Note that a Gaussian ‘fit’ does not seem to actually fit the data very well. It underpredicts cluster numbers at the bright end and somewhat overpredicts them at the faint end. Therefore, we give more weight to the bright end in our fitting routine. The faint end may not be Gaussian in (true) shape but simply be depleted because of sample incompleteness.
![Reddening-corrected absolute $V$-band magnitude ($M_V^0$) distribution of our M33 star cluster sample. The vertical line at $M_V^0= -4.54$ reflects our estimate of the sample’s 50% completeness limit.[]{data-label="fig6"}](fig6.ps)
To further explore the completeness level of the M33 star cluster sample, we show its spatial distribution in Fig. \[fig7\]. The sample clusters characterized by absolute, extinction-corrected $V$-band magnitudes, $M_V^0 \le -6.04$ mag, which is the peak magnitude of the distribution in Fig. \[fig6\], are shown as green points, while the purple points represent the $M_V^0>-6.04$ mag star clusters in our sample. There is no evidence of any spatial differences between both samples. We fit Gaussian profiles to the distributions in the right ascension (R.A.) and declination (Dec) directions and determined the distribution’s center coordinates: R.A. = 23.449$^\circ$, with $\sigma=0.153^\circ$, and Dec = 30.642$^\circ$, with $\sigma=0.205^\circ$, for the bright sources; R.A. = 23.459$^\circ
(\sigma=0.138^\circ)$ and Dec = 30.623$^\circ (\sigma=0.192^\circ)$ for the faint objects. The two center positions are shown as the plus signs in Fig. \[fig7\].
SED Fits and Results {#s:ana}
====================
We constrain the ages, metallicities, and masses of the star clusters based on SED fitting using $\chi^2$ minimization. We mainly used our photometry (Table \[photo\]) from the LGGS images to do so, supplemented with the SDSS $ugriz$ photometry from @san10. After elimination of those clusters for which photometry is available in too few passbands,[^4] our sample is reduced to 671 star clusters and cluster candidates. We will use this subsample for SED fitting and analysis in the following sections.
Age estimates {#s:age}
-------------
As is common in relation to most ground-based observations, we can only access the [*integrated*]{} spectra and photometry of most extragalactic star clusters. Therefore, the ages, metallicities, and other physical parameters are obtained through analysis of the integrated data. As a matter of fact, a strong age–metallicity degeneracy would likely affect our analysis if only optical photometry were available [@wor94; @ar96; @kaviraj07]. @anders04b recommend to use NIR photometry if available. Inclusion of at least one NIR passband can significantly improve the accuracy of the resulting cluster parameters and partially break this degeneracy. In addition, @deg05 and @wu05 showed that NIR colors can greatly contribute to breaking the age–metallicity and age–extinction degeneracies. Therefore, we will combine our $UBVRI$ photometry with $JHK$ photometry from 2MASS when available to disentangle the degeneracies and obtain more accurate results.
We fit the SEDs with the evolutionary tracks derived from the updated [parsec]{} isochrones, assuming a @chab lognormal stellar initial mass function (IMF).[^5] These [ parsec]{} isochrones are available for metallicities $0.0001 \le Z \le
0.06$ ($-2.2 \le {\rm [M/H]} \le +0.5$ dex) and for stellar masses in the range $0.1 \le M/M_{\odot} \le 12$, with revised diffusion and overshooting for low-mass stars and improvements in the interpolation scheme. Note that, at present, the [parsec]{} isochrones do not include the thermally pulsing asymptotic giant-branch (TP-AGB) phase. We adopted 24 metallicities from $Z= 0.0001$ to 0.06, essentially equally spaced in in $\log Z$ space. The maximum age for a reliable interpolation in metallicity is 13.5 Gyr, or $\log( t \mbox{
yr}^{-1})=10.13$. Therefore, we adopted 71 equally spaced time steps from $\log( t \mbox{ yr}^{-1})=6.6$ (4 Myr) to $\log( t \mbox{
yr}^{-1})=10.1$ (12.6 Gyr) in steps of $\Delta \log( t \mbox{
yr}^{-1}) =0.05$. The models return isochrone tables and integrated SSP magnitudes for a number of photometric systems. We adopted the SDSS $ugriz$, Johnson–Cousins $UBVRI$, and 2MASS $JHK$ systems.
The magnitudes were corrected for reddening (obtained previously; see above) assuming a @ccm extinction curve. Since the wavelength ranges covered by the SDSS $ugriz$ and Johnson–Cousins $UBVRI$ systems are essentially the same, it is not necessary to use both for our SED fitting. We assigned priority to using the broad-band Johnson–Cousins $UBVRI$ photometry, since these filters have wider bandwidths and, hence, the photometry could potentially have higher higher signal-to-noise ratios, all else being equal. Where broad-band Johnson–Cousins magnitudes were not available, the SDSS $ugriz$ photometry was used. Thus, the cluster ages ($t$) could be determined by comparing, in the $\chi^2$ sense, the [parsec]{} SSP synthesis models with the observed SEDs and adopting $Z$ as a free parameter, i.e. $$\chi^2_{\rm min}(t,Z)={\rm
min}\left[\sum_{i=1}^8\left({\frac{m_{\lambda_i}^{\rm
obs}-m_{\lambda_i}^{\rm mod}}{\sigma_i}}\right)^2\right],
\label{eq2}$$ where $m_{\lambda_i}^{\rm mod}(t,Z)$ is the integrated magnitude in the $i^{\rm th}$ filter of a theoretical SSP at age $t$ and for metallicity $Z$, $m_{\lambda_i}^{\rm obs}$ represents the observed, integrated magnitude in the same filter, $m_{\lambda_i}=UBVRI,ugriz,JHK$ when 2MASS data is available or $m_{\lambda_i}=UBVRI,ugriz$ when 2MASS data is not available (all magnitudes were transformed to the AB magnitude system for our SED fits). The errors, which are used as weights ($=1/\sigma^2$) by the fitting routine, are calculated as $$\sigma_i^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}.
\label{eq3}$$ Here, $\sigma_{{\rm obs},i}$ is the observational uncertainty; $\sigma_{{\rm mod},i}$ represents the uncertainty associated with the model itself, for the $i^{\rm th}$ filter. Following @deg05, @wu05, @fan06, @ma07 [@ma09], @fan10, and @wang10, we adopt $\sigma_{{\rm mod},i}=0.05$ mag.
The estimated ages with $1\sigma$ errors of the M33 star clusters are listed in Table \[agemass\]. We estimated the uncertainty associated with a given parameter by fixing all other parameters to their best values, then varied the parameter of interest, and recorded an error corresponding to the $1\sigma~\chi^2_{\rm min}$ value.
Our newly estimated ages, compared with previous results from the literature, are plotted in Fig. \[fig8\]. The top panels are comparisons of SM10 and our results, and the middle panels are those for our estimates and the results of @san09. The latter authors compare MSTO photometry in the observed CMD with @gir00 theoretical isochrones assuming a metal abundance $Z=0.004$, which is the mean of the disk abundance gradient in M33 [cf. @kim; @sa06; @san09]. In the top panels, we note that SM10’s cluster ages and masses exhibit a general one-to-one trend with respect to our results, although the scatter is relatively large. In fact, all cluster ages and masses in SM10 were taken from the Ma et al. series of articles. These latter authors derived these parameters based on SED fits based on the Bruzual & Charlot SSP synthesis models (BC96). This is a similar approach to our method, but based on different theoretical models and photometry. In this article, we use some of the most up-to-date SSP synthesis models currently available, while our photometry is based on (more recent) observations with the Kitt Peak 4 m telescope. In addition, we used NIR photometry here, which can be used successfully to partially break the well-known age–metallicity degeneracy affecting broad-band SED analyses. A comparison of the results of @san09 and the newly derived parameters presented here reveals that the systematic differences, in a logarithmic sense, in the ages, masses, and metallicities between our estimates and those of SM10 (i.e., Ma et al.) are $-0.23\pm0.77$ dex, $-0.32\pm0.73$ dex, and $0.08\pm1.49$ dex, respectively. This scatter is partially caused by the different metallicities adopted and partially because of stochastic sampling effects [see, e.g., @anders13]. However, the ages from @san09 are systematically younger for clusters we determine as $>1$ Gyr old, while they are systematically older than our estimates for objects we return as $<0.1$ Gyr old. This may be partially caused by the different analysis methods or by application of different SSP models with different IMFs.
To check if our method introduces any biases, we also compare the fit results for @san09 and those of SM10. This comparison shows the same trends as seen in the comparisons of @san09 and our results, which indicates that the CMD fitting method of @san09 may be affected by a systematic bias (see the bottom panels of Fig. \[fig8\]). Since @san09 adopted a single metal abundance for their CMD fits, we do not show a comparison of their metallicities with those of SM10. In fact, as @san09 point out, the [*isochrone-derived*]{} ages for clusters younger than $\sim1$ Gyr exhibit very little sensitivity to the assumed metal abundance. This implies that any metallicity difference between our results and those of @san09 will likely have insignificant effects on the final cluster parameters derived, since almost all clusters in the sample of @san09 are younger than 1 Gyr. Even if @san09 had allowed their metal abundances to vary, their fit results would unlikey be affected significantly. Note that this is rather different in comparison with SED fits. In addition, we adopted the extinction values of @san09.
@ppl09 compared cluster ages derived from resolved CMDs with those from integrated photometry and found that ages derived from resolved CMDs cover a relatively smaller range—$7.0 \la \log(t
\mbox{ yr}^{-1}) \la 8.5$—than those estimated based on integrated colors and SEDs [see, e.g., @cha99; @cha02; @ma01; @ma02a; @ma02b; @ma02c; @ma04a; @ma04b]. In addition, @sf97 found that stochastic sampling effects can strongly affect the integrated $VJHK$ magnitudes of star clusters with ages $7.5<\log(t \mbox{ yr}^{-1})<9.25$, in particular for less massive clusters [@buz89; @bj12; @sl11].
In fact, @sl11 discussed the effects of stochastic sampling on CMD fits; such effects are particularly prominent for low-mass star clusters. Since star clusters are composed of finite numbers of stars, stochastic sampling of the stellar IMF can significantly affect the derived integrated cluster parameters, such as their ages, metallicities, and masses. Recently, @anders13 quantified the effects of stochastic sampling of stellar IMFs based on a set of GALEV SSP models for a wide range of (input) masses, metallicities, foreground extinction values, and photometric uncertainties for their model star clusters. They derive the accuracy of the integrated parameter determination in different age ranges based on performing fully sampled integrated-SED fits. For low-mass ($\sim10^3 M_{\odot}$) clusters that are older than 10 Myr, the dispersion in $\log(t \mbox{
yr}^{-1})$ could be as much as $\sim1$–2 dex, while the dispersion in $\log(M_{\rm cl}/M_\odot)$ could be of the same order for \[Fe/H\] = 0.0 dex, foreground $E(B−V) = 0.0$ mag, and assuming photometric uncertainties of 0.1 mag for all $UBVRIJHK$ magnitudes. In addition, using a variety of metallicities and different combinations of photometric passbands, stochastic sampling effects can even lead to differences of $\Delta \log(t \mbox{ yr}^{-1}) >1$ dex. The offset between the estimates of SM10 and @san09 can thus be understood easily. This type of effect could also partially explain the offsets in the derived parameters between SM10 and our determinations, as well as those between SM10 and @san09.
We emphasize that @san09 also point out that, since their photometry is generally not deep enough to detect the MSTO, few of their clusters are returned as older than 1 Gyr. The middle panel in the central row of Fig. \[fig8\] shows a comparison of the mass estimates of @san09 with our new determinations. We note that the masses derived by @san09 range from $5\times10^3 M_{\odot}$ to $5\times10^4 M_{\odot}$, while our estimates cover the range from $\sim10^2 M_{\odot}$ to $\sim10^6 M_{\odot}$. This difference can largely be traced back to the differences in our derived ages.
The left-hand panels in the top and second rows of Fig. \[fig9\] show the age distribution of our sample clusters in M33, as well as the distribution of a representative sample of M31 star clusters [@fan12], for a bin size of 0.3 dex. To allow a reasonable comparison, the parameters, such as age, metallicity, and mass of the M31 star clusters were also all derived based on the [parsec]{} models, using the photometric data from @fan12. We also plot the age distribution of the Milky Way star cluster sample, which is composed of both GCs and open clusters (OCs). In addition, the age estimates of LMC star clusters taken from @bau13 are also plotted in this diagram, for comparison. We note that the M33 star clusters in our sample exhibit two peaks, at ages of $\sim10$ Myr and $\sim1$ Gyr. The mean ages of the cluster samples are $\log(t \mbox{
yr}^{-1})=8.68$, 9.17, 8.46, and 8.48 for M33, M31, MW, and the LMC, respectively. For M33, the clusters with ages in excess of 10 Gyr were most likely created during the epoch when the galaxy formed, while the young star clusters might have been created in a number of mergers during the last few Gyr or by the postulated recent galactic encounter with M31 a few Gyr ago, suggested by @mc09. The age distribution of the star clusters in M31 is dominated by clusters with ages between 1 Gyr and the [*WMAP9*]{} age of $13.77\pm0.06$ Gyr [@ben12]. The age distribution of the Milky Way star clusters is based on the combination of that of the OCs collected in @dias02—the [*New Catalog of Optically Visible Open Clusters and Candidates*]{}; version 3.3—and the GCs, for which we assumed the [*WMAP9*]{} age. The mean age of the Milky Way’s OCs is similar to that of the M33 and LMC clusters. This latter similarity implies that both galaxies have recently gone through one or more periods of active star (cluster) formation. It is clear that there is a much higher fraction of young star clusters in M33 (30.6% are older than 2 Gyr) than in M31, where 55.8% of the clusters are older than 2 Gyr.
The middle panels of Fig. \[fig9\], from the top to the third row, show the metallicity distributions of star clusters in M33, M31, and the Milky Way for a bin size of $\Delta \rm [Fe/H]=0.25$ dex. The mean values of the three distributions are $\rm \overline{[Fe/H]}=-1.01$, $-0.43$, and $-0.19$ dex. The metallicity distribution of the M33 star clusters comes from our SSP fits based on the [parsec]{} models, while the metallicity distribution of the M31 star clusters is from the data of @fan12, also based on the [parsec]{} models. The metallicity distribution of the Milky Way GCs has been plotted based on the GC catalog of @har10 and that of the OCs is based on the [@dias02] catalog, which was most recently updated in 2013 and includes 201 Milky Way OCs with metallicity measurements. We computed the weighted metallicity of the two cluster samples using the numbers of GCs and OCs as weights, which is therefore dominated by the metallicity distribution of the OCs.
Masses of the M33 star clusters {#s:mass}
-------------------------------
We calculated the clusters’ theoretical mass-to-light ratios ($M/L_V$) using the [parsec]{} models, luminosities based on conversion of the $V$-band fluxes, and a distance modulus of $(m-M)_0 = 24.64$ mag. The resulting masses are listed in Table \[agemass\]. Figure \[fig9\] (right column) shows the mass distribution of the M33 star clusters in our sample, as well as the masses of the M31 and Milky Way clusters. The masses of the Galactic GCs were calculated by @go97, while those of 650 OCs with mass estimates are from @pis08. As before, these Galactic cluster mass estimates were combined, using as weights the total numbers of clusters of different types. For comparison, we also include the LMC star cluster data from @bau13. The bin size is $\Delta \log(M_{\rm cl}/M_\odot)=0.3$ dex. The mean mass of the M33 clusters is $\log(M_{\rm
cl}/M_\odot)=4.25$ ($1.78\times10^4M_{\odot}$) while the mean values for the M31 star clusters, the combined sample of Galactic star clusters, and the LMC clusters are $\log(M_{\rm cl}/M_\odot)=5.43$ ($2.69\times10^5M_{\odot}$), $\log(M_{\rm cl}/M_\odot)=2.72$ ($5.24\times10^2M_{\odot}$), and $\log(M_{\rm cl}/M_\odot)=4.18$ ($1.51\times10^4M_{\odot}$), respectively. Figure \[fig9\] shows that the mean mass of the star clusters in M33, which is similar to that of the LMC clusters, is much lower than the equivalent masses in M31 and the Milky Way, suggesting that the M33 cluster population on the whole is dominated by lower-mass clusters.
The mass–metallicity relation (MMR) for star clusters in the ‘blue sequence’ (which is known as the ‘blue tilt’) has been discussed for many external galaxies, e.g., for a sample of six giant elliptical galaxies [@ha09a], M87 [@peng09; @ha09b], the Sombrero galaxy [@ha09c], and M31 [@fan09]. Self-enrichment was considered a reasonable explanation by both @bh09 and @ss08, who suggested that the level of star formation is controlled by supernova feedback, and the efficiency scaling is proportional to the proto-cloud mass. Since we have derived the masses and metallicities of the M33 clusters, we can investigate their MMR. In Fig. \[fig10\], we plot cluster masses as a function of metallicity for our sample star clusters. The filled triangles with associated error bars represent the mean values and $\sigma$’s for each bin. The bin size is 0.5 dex in metallicity. For low metallicities, ${\rm
[Fe/H]}<-0.8$ dex, the cluster masses seem to decrease with metallicity, while for ${\rm [Fe/H]}\ge-0.8$ dex, the cluster masses increase with increasing metallicity.
![Masses versus metallicities for the M33 star clusters. We divided the metallicities into 6 bins, each with a size of 0.5 dex. The filled triangles with error bars represent the mean values and $\sigma$’s for each bin.[]{data-label="fig10"}](fig10.ps)
Figure \[fig11\] shows the extinction-corrected absolute magnitudes as a function of age for our sample star clusters. The solid lines represent theoretical isochrones from the updated [parsec]{} models for masses of $10^3, 10^4, 10^5$, and $10^6M_{\odot}$, for a metallicity of $Z =0.004$. The masses of most star clusters and candidates are between $10^3 M_{\odot}$ and $10^5 M_{\odot}$. The less massive clusters, $M_{\rm cl}<10^3 M_{\odot}$, tend to be young ($<2$ Gyr) while clusters with $M_{\rm cl} > 10^5 M_{\odot}$ are generally old ($>2$ Gyr).
![Extinction-corrected absolute magnitude as a function of cluster age. The solid lines are the expected relations for SSPs from the [parsec]{} model with $Z = 0.004$ and masses of $10^3,
10^4, 10^5$, and $10^6 M_{\odot}$ (from bottom to top).[]{data-label="fig11"}](fig11.ps)
The age–metallicity relations of extragalactic star clusters have been studied by many authors [e.g., @da06; @wang10; @fan10]. Figure \[fig12\] shows the age-versus-mass estimates of our M33 sample clusters, as well as the so-called ‘fading line,’ which is roughly equivalent to the $\sim 50$% completeness limit. The theoretical line is based on the [parsec]{} SSP models for a metallicity of $Z=0.004$. For the detection limit, an absolute magnitude of $M_V\approx-5$ mag is estimated from Fig. \[fig6\]. Indeed, most star clusters lie above the line. The small number of clusters found below the fading line may be due to either a possibly variable photometric completeness level or underestimated extinction. A few faint, young ($<20$ Myr-old) clusters could be young OCs. Note that there are three overdensity regions in this figure, at (i) $\log(t \mbox{ yr}^{-1})\approx7.3$ (20 Myr), (ii) $\log(t \mbox{ yr}^{-1})\approx9$ (1 Gyr), and (iii) the [*WMAP9*]{} age ($\sim13.7$ Gyr); the latter represents the subset of the cluster population that seems to have formed during the time of the galaxy’s formation.
![Mass as a function of age for the M33 star clusters. We also show the fading (‘completeness’) limit for $M_V \approx-5$ mag from Fig. \[fig6\], based on the [parsec]{} SSP models for a metallicity $Z=0.004$.[]{data-label="fig12"}](fig12.ps)
Summary {#s:sum}
=======
We have obtained $UBVRI$ photometry for 708 star clusters and cluster candidates in M33, which were selected from @san10, including 387, 563, 616, 580, and 478 objects in the $UBVRI$ bands, respectively, of which 276, 405, 430, 457, and 363 did not have previously published $UBVRI$ photometry. The [SExtractor]{} code was applied to derive the photometry from LGGS archival images, which cover 0.8 deg$^2$ along the major axis of M33.
We compared our photometry with previous measurements, which showed that our photometry is generally consistent with previous measurements in all filters. The ages, metallicities, and masses of our sample clusters were derived by comparison of their observed SEDs with [ parsec]{} SSP synthesis models. The fits show that only 205 of the 671 clusters (30.55%) are older than 2 Gyr, which is a much smaller fraction than that derived for the M31 globular-like clusters (55.80%), suggesting that the M33 cluster population is dominated by young star clusters ($<1$ Gyr). We also note that the mean mass of the M33 star clusters is $1.78\times10^4M_{\odot}$, which is much less massive than that of the M31 sample ($2.69\times10^5M_{\odot}$) and similar to that of LMC cluster population ($1.51\times10^4M_{\odot}$), but higher than that of Milky Way star clusters (including both GCs and OCs: $5.24\times10^2M_{\odot}$). This may be related to the fact that the mass of M33 is lower that that of M31, and the gravitational potential is not large enough to produce as many GCs as in M31. Instead, the star-formation history of M33 may be similar to that of the LMC. As for the Milky Way, the recent few Gyr have seen the galaxy undergo quiescent evolution (a low star-formation rate).
On the other hand, the mean metallicity of the M33 clusters ($\rm
[Fe/H]=-1.01$ dex) is much lower than that of the M31 star clusters ($\rm [Fe/H]=-0.43$ dex) and also of the Milky Way star clusters ($\rm
[Fe/H]=-0.19$ dex), suggesting that its star-formation history has been rather different from those of either M31 or the Milky Way. Based on the cluster mass distributions we also found that the mean mass of star clusters in M33 is similar to that in the LMC but much lower than that in M31 and higher than that in the Milky Way, indicating that M31 underwent more violent star formation than either M33 or the LMC. We also note that stochastic sampling effects can significantly affect our SED fit results for low-mass clusters (i.e., $M_{\rm cl}
\le 10^3 M_{\odot}$), potentially leading to large differences in integrated cluster ages, metallicities, and masses [@anders13]. The effects of stochasticity in the clusters’ stellar mass functions become weaker with increasing cluster mass. However, even for the highest-mass clusters, $M_{\rm cl}
\simeq 2 \times 10^5 M_{\odot}$, the uncertainties in the derived logarithmic ages are 0.05–0.25 dex; the equivalent uncertainties pertaining to the derived masses are 0.09–0.17 dex. Although these uncertainties are sometimes significant, obtaining accurate, high-resolution spectroscopic observations for statistically large samples of extragalactic star clusters is often prohibitive, particularly if observing time on significantly oversubscribed (large) telescopes is sought. Broad-band imaging and parameter determination based on sophisticated SED fits is the realistic yet a poor man’s alternative approach.
The broad-band SED uncertainties can be further reduced for specific age ranges, e.g., for young clusters that exhibit H$\alpha$ emission, which places additional constraints on the most likely age range. At present, a number of teams are working on quantifying the effects of stochastic sampling; although the underlying message is that such effects may have significant implications in terms of the precision of the derived physical parameters, we argue that a proper understanding of one’s uncertainties is of the utmost importance. One should keep these issues in mind when dealing with physical cluster parameters based on broad-band observations.
This publication makes use of data products 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 U.S. National Science Foundation. This research is supported by the National Natural Science Foundation of China (NFSC) through grants 11003021, 11043006, 11073001, 11373003, and 11373010. ZF also acknowledges a Young Researcher Grant from the National Astronomical Observatories, Chinese Academy of Sciences, while RdG acknowledges research support from the Royal Netherlands Academy of Arts and Sciences (KNAW) under its Visiting Professors Programme.
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[cccc]{}
[^1]: http://www.mancone.net/m33$\_$catalog/, updated in December 2010.
[^2]: http://www.ipac.caltech.edu/2mass/
[^3]: http://www.astromatic.net/software/sextractor; version 2.8.6 was updated on 5 October 2009.
[^4]: We only apply SED fitting to clusters with photometric measurements in $\ge 3$ passbands; measurements in fewer filters lead to highly unreliable results [cf. @anders04b].
[^5]: http://stev.oapd.inaf.it/cgi-bin/cmd
|
---
abstract: |
In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian $p$-groups $E_r$ of rank $r\geq 2$ by exploiting a functor from the module category of a generalized Beilinson algebra $B(n,r)$, $n \leq p$, to $\operatorname{mod}E_r$.\
We define analogs of the above mentioned properties in $\operatorname{mod}B(n,r)$ and give a homological characterization of the resulting subcategories via a $\mathbb{P}^{r-1}$-family of $B(n,r)$-modules of projective dimension one. This enables us to apply general methods from Auslander-Reiten theory and thereby arrive at results that, in particular, contrast the findings for equal images modules of Loewy length two over $E_2$ [@cfs09] with the case $r > 2$. Moreover, we give a generalization of the $W$-modules defined by Carlson, Friedlander and Suslin in [@cfs09].
address: 'Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany'
author:
- Julia Worch
bibliography:
- 'lit.bib'
title: 'Categories of modules for elementary abelian $p$-groups and generalized Beilinson algebras'
---
[^1]
Introduction {#introduction .unnumbered}
============
Addressing representations of finite group schemes over fields of positive characteristic, Carlson, Friedlander and Pevtsova have introduced in [@cafrpe08] the category of modules of constant Jordan type. Their approach involves the theory of $\pi$-points, i.e. certain embeddings $\alpha: k[T]/(T^p) \rightarrow kG$ along which representations of $kG$ can be restricted to the less complicated subalgebra $k \mathbb{Z}_p \cong k[T]/(T^p)$. The representations of $k[T]/(T^p)$ are completely understood in terms of Jordan types, i.e. Jordan block decompositions. Since $kG$ is wild in most cases, it is reasonable to study representations with additional properties. A $kG$-module has constant Jordan type if its Jordan block decomposition does not depend on the chosen $\pi$-point. There is the related notion of the constant $j$-rank property such that a module has constant Jordan type iff it has constant $j$-rank for all $j \geq 1$ (cf. [@fp p. 11]).\
Confining investigations to elementary abelian $p$-groups $E_r=(\mathbb{Z}_p)^{\times r}$ of rank $r \geq 2$, a more restrictive condition has been formulated in [@cfs09] by Carlson, Friedlander and Suslin, where $M \in \operatorname{mod}kE_r$ satisfies the so-called equal images property if there exists a $k$-space $V$ such that $\alpha(t).M=V$ for all $\pi$-points $\alpha$. The dual concept is referred to as the equal kernels property. In [@cfs09], the authors are mainly concerned with the case $r=2$ and they introduce a family of $kE_2$-modules, the so-called $W$-modules, which satisfy the equal images property and are ubiquitous in a sense that every module satisfying the equal images property is a quotient of a $W$-module [@cfs09 4.4]. This relies on the fact that the indecomposable equal images modules of Loewy length two over $kE_2$ are $W$-modules [@cfs09 4.1]. It has been observed that these modules correspond to the preinjective modules over the Kronecker algebra (cf. [@f11 4.2.2]).\
The approach we give in this paper is based on the objective to understand $\operatorname{mod}_n kE_r$, i.e. the full subcategory of $kE_r$-modules with Loewy length bounded by $n \leq p$. The generalized Beilinson algebra $B(n,r)$ provides a faithful exact functor $\mathfrak{F}: \operatorname{mod}B(n,r) \rightarrow \operatorname{mod}_n(kE_r)$. We formulate analogs of the constant Jordan type and constant $j$-rank property as well as the equal images and equal kernels property for $ B(n,r)$-modules and define full subcategories $\operatorname{CJT}(n,r),\ \operatorname{CR}^j(n,r), \ \operatorname{EIP}(n,r), \ \operatorname{EKP}(n,r)\subset \operatorname{mod}B(n,r)$ such that the restrictions of $\mathfrak{F}$ to $\operatorname{EIP}(n,r)$ and $\operatorname{EKP}(n,r)$ reflect isomorphisms and have an essential image consisting of standardly graded modules with the equal images property and costandardly graded modules with the equal kernels property, respectively.\
An immediate advantage of passing over to $B(n,r)$ is that we are able to give a homological characterization of the categories $\operatorname{EIP}(n,r)$ and $\operatorname{EKP}(n,r)$ involving a $\mathbb{P}^{r-1}$-family of $B(n,r)$-modules of projective dimension one which allows us to apply general methods from Auslander-Reiten theory. With this tool in hand, we prove:
The category $\operatorname{EIP}(n,r)$ is the torsion class $\mathcal{T}$ of a torsion pair $(\mathcal{T}, \mathcal{F})$ in $\operatorname{mod}B(n,r)$ such that $\operatorname{EKP}(n,r) \subset \mathcal{F}$ and $\mathcal{T}$ is closed under the Auslander-Reiten translate $\tau$ and contains all preinjective modules.\
Dually, $\operatorname{EKP}(n,r)$ is the torsion-free class $\mathcal{F}'$ of a torsion pair $(\mathcal{T'}, \mathcal{F'})$ in $\operatorname{mod}B(n,r)$ such that $\operatorname{EIP}(n,r) \subset \mathcal{T}'$ and $\mathcal{F}'$ is closed under $\tau^{-1}$ and contains all preprojective modules.\
In particular, there are no non-trivial maps $\operatorname{EIP}(n,r) \rightarrow \operatorname{EKP}(n,r)$.
We can specialize our results to the case $n=2$: The algebra $B(2,r)$ is the path algebra $\mathcal{K}_r$ of the $r$-Kronecker and has wild representation type if and only if $r > 2$. The Auslander-Reiten quiver of the wild hereditary algebra $B(2,r)$ consists of a preprojective component, a preinjective component and infinitely many (regular) components of type $\mathbb{Z}A_{\infty}$ [@ri78]. We summarize our main results for $B(2,r), \ r >2,$ as follows, contrasting the findings for $r=2$:
Let $r >2$, $n \leq p$ and let $\Gamma$ be the Auslander-Reiten quiver of $B(2,r)$.
(i) Let $\mathcal{C}$ be a regular component of $\Gamma$. Then $\operatorname{EIP}(2,r) \cap \mathcal{C}$ and $\operatorname{EKP}(2,r) \cap \mathcal{C}$ are non-empty disjoint cones. The size of the gap $\mathcal{W}(\mathcal{C}) \in \mathbb{N}_0$ between these cones is an invariant of $\mathcal{C}$.
(ii) For each $n \in \mathbb{N}$, there exists a regular component $\mathcal{C}$ of $\Gamma$ such that $\mathcal{W}(\mathcal{C})>n$.
(iii) If $\mathcal{W}(\mathcal{C})=0$, then every object in $\mathcal{C}$ has constant Jordan type.
(iv) If $\mathcal{W}(\mathcal{C})=1$, then either $\mathcal{C} \subset \operatorname{CJT}(2,r)$ or apart from the cones $\operatorname{EIP}(2,r) \cap \mathcal{C}$ and $\operatorname{EKP}(2,r) \cap \mathcal{C}$, there are no other objects of constant Jordan type in $\mathcal{C}$.
Our paper is organized as follows: In Section 1, we recall definitions and basic results and give a generalization of the $W$-modules defined in [@cfs09] to arbitrary rank. We introduce generalized Beilinson algebras and give a homological description of the categories $\operatorname{CJT}(n,r),\ \operatorname{CR}^j(n,r), \ \operatorname{EIP}(n,r)$ and $\operatorname{EKP}(n,r)$ in Section 2 and point out the special role that generalized $W$-modules play in $\operatorname{EIP}(n,r)$. In the final section, we restrict our investigations to modules of Loewy length two and give our more specific results on the $r$-Kronecker together with some examples.
Generalized $W$-modules
=======================
Let us first of all introduce the set up and recall the relevant concepts and some basic results from [@cafrpe08], [@cafr09], [@cfs09] and [@fp]. In doing so, we will present some definitions in a way that is suitable for our purposes.\
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $E_r=(\mathbb{Z}_p)^{\times r}$ be an elementary abelian $p$-group of rank $r\geq2$ with generators $g_1,\ldots,g_r$. Let furthermore $R=k[X_1,\ldots,X_r]$ be the polynomial ring in $r$ variables. Sending $X_i$ to $x_i:=g_i-1$ yields an isomorphism $k[X_1,\ldots,X_r]/(X_1^p,\ldots,X_r^p) \cong kE_r$ between the truncated polynomial ring and the group algebra of $E_r$. Consider furthermore the ideal $I=(X_1,\ldots,X_r)\subseteq R$ generated by all polynomials of degree one as well as the augmentation ideal $J=\operatorname{rad}(kE_r)=(x_1,\ldots,x_r)$ of $kE_r$. We let $\operatorname{mod}kE_r$ be the category of finitely generated $kE_r$-modules and $\operatorname{mod}_n (kE_r) \subset \operatorname{mod}kE_r$ be the full subcategory consisting of modules of Loewy length at most $n$.\
An algebra homomorphism $\alpha: k[T]/(T^p) \rightarrow kE_r$ is called *p-point* if the pullback $\alpha^*(kE_r)$ is a free $k[T]/(T^p)$-module. Note that this is equivalent to saying that $\alpha(t)$ with $t:=T+(T^p)$ is an element in $\operatorname{rad}(kE_r) \backslash \operatorname{rad}^2(kE_r)$ [@cfs09 p. 3]. Given such a $p$-point $\alpha$, for $M \in \operatorname{mod}kE_r$, we consider the linear operator $\alpha(t)_M : M \rightarrow M, \ m \mapsto \alpha(t).m$. The Jordan canonical form of $\alpha(t)_M$ entirely determines the isomorphism type of $M$ as a $k[T]/(T^p)$-module. The sequence of sizes of Jordan blocks is referred to as the [*Jordan type*]{} of $M$ corresponding to $\alpha$ and we write $\operatorname{JType}(\alpha,M)=a_p[p] +\cdots + a_1[1]$, indicating that there are $a_i$ blocks of size $[i]$ for $1 \leq i \leq p$. If this Jordan type does not depend on the $p$-point we choose, we say that $M$ is of [*constant Jordan type*]{} $\operatorname{JType}(M):=\operatorname{JType}(\alpha,M)$.\
We say that $M \in \operatorname{mod}kE_r$ is of *constant $j$-rank* for $j \in \mathbb{N}$ if the rank $\operatorname{rk}\alpha(t)^j_M$ is independent of our choice of $p$-point. Note that $M$ is of constant Jordan type iff $M$ is of constant $j$-rank for all $j \geq 1$ [@fp p. 11]. We denote the subcategories of $\operatorname{mod}kE_r$ consisting of such modules by $\operatorname{CR}^j(kE_r)$ and $\operatorname{CJT}(kE_r)$.\
A module $M \in \operatorname{mod}kE_r$ is said to satisfy the *equal images property* if $\operatorname{im}\alpha(t)_M=\operatorname{rad}M$ for all $p$-points $\alpha$. A module $M \in \operatorname{mod}kE_r$ is said to satisfy the *equal kernels property* if $\ker \alpha(t)_M=\operatorname{soc}M$ for all $p$-points $\alpha$. We denote the corresponding subcategories of $\operatorname{mod}(kE_r)$ by $\operatorname{EIP}(kE_r)$ and $\operatorname{EKP}(kE_r)$, respectively.\
In [@cfs09 1.2, 1.7], it is shown that it suffices to check the above properties for all $p$-points $\alpha$ with $\alpha(t)=\alpha_1x_1+\cdots + \alpha_r x_r$ for a non-trivial element $(\alpha_1,\ldots,\alpha_r) \in k^r \backslash 0$.\
Note that $M$ satisfies the equal images property if and only if its linear dual $M^*$ satisfies the equal kernels property. The category $\operatorname{EIP}(kE_r)$ is image-closed [@cfs09 1.10], and dually $\operatorname{EKP}(kE_r)$ is closed under taking submodules. We have $\operatorname{EIP}(kE_r) \cup \operatorname{EKP}(kE_r) \subseteq \operatorname{CJT}(kE_r)$ [@cfs09 1.9] and furthermore $\operatorname{EIP}(kE_r) \cap \operatorname{EKP}(kE_r)=\operatorname{add}k$ [@f11 4.4.3], where $k$ is the trivial $kE_r$-module and $\operatorname{add}k$ the full subcategory of $\operatorname{mod}kE_r$ whose objects are direct sums of the trivial module $k$.\
We now give a generalization of the $\mathbb{Z}_p \times \mathbb{Z}_p$-modules $W_{n,d}$ defined by Carlson, Friedlander and Suslin in [@cfs09] to elementary abelian $p$-groups of arbitrary rank. The authors show that these so-called $W$-modules are indecomposable equal images modules which play a prominent role in the category $\operatorname{EIP}(k(\mathbb{Z}_p \times \mathbb{Z}_p))$. Whereas in [@cfs09], the modules $W_{n,d}$ are defined via generators and relations, we give an alternative definition that is amenable to generalization to higher rank.\
For all $n \in \mathbb{N},\ d \leq \min\left\{n,p\right\}$, we consider the $R$-module $$M_{n,d}^{(r)}:=I^{n-d}/I^n.$$ By choice of $d$, the canonical action factors through $R/(X_1^p,\ldots,X_r^p)$, so that we can likewise study this module and its linear dual $W_{n,d}^{(r)}:=({M_{n,d}^{(r)}})^*$ over $kE_r$.\
The module ${M_{3,2}^{(3)}}$ can be depicted as follows $$\begin{xy}
\xymatrix@R=2em@C=0.4em{
& & & \stackrel{x_1}{\bullet} \ar@{->}[llld] \ar@{-->}[ld] \ar@{~>}[rrrd] && \stackrel{x_2}{\bullet} \ar@{->}[llld] \ar@{-->}[ld] \ar@{~>}[rrrd]&& \stackrel{x_3}{\bullet} \ar@{~>}[rrrd] \ar@{-->}[rd] \ar@{->}[ld]\\
{\bullet} && {\bullet} && \bullet && \bullet && \bullet && \bullet}
\end{xy}$$ where the dots represent the canonical basis elements given by the monomials in degree one and two and $\rightarrow, \ \dashrightarrow$ and $\rightsquigarrow$ denote the action of $x_1, x_2$ and $x_3$, respectively. It is easy to see that in case $r=2$ we have $W_{n,d}^{(2)}=W_{n,d}$ as defined in [@cfs09]. Modules of the form $M_{n,d}^{(r)}$ will be referred to as [*$M$-modules*]{} and modules of the form $W_{n,d}^{(r)}$ as [*$W$-modules*]{}, respectively. We furthermore set the convention that $M_{n,d}^{(r)}:=M_{n,n}^{(r)}$ and $W_{n,d}^{(r)}:= W_{n,n}^{(r)}$ in case $d>n$.\
The module $M_{n,d}^{(r)}$ satisfies the equal kernels property, since for all $(\alpha_1,\ldots,\alpha_r) \in k^r \backslash 0$ we have $$\ker \left\{\sum_{i=1}^r{\alpha_ix_i}: M_{n,d}^{(r)} \rightarrow M_{n,d}^{(r)} \right\}=I^{n-1}/I^n=\operatorname{soc}(M_{n,d}^{(r)}).$$ Hence $W_{n,d}^{(r)}$ satisfies the equal images property. Some $W$-modules can be recognized as submodules of the group algebra $kE_r$, generalizing [@cfs09 2.2].
There is an isomorphism $$W_{d,d}^{(r)} \cong \operatorname{rad}^{r(p-1)+1-d}(kE_r)$$ for $d\leq p$.
Observe that $kE_r$ is isomorphic to the restricted enveloping algebra of an $r$-dimensional abelian Lie algebra with trivial $p$-map [@fs88 §5] and is equipped with the structure of a Frobenius algebra where the projection $\tau: kE_r \rightarrow k$ onto the coefficient of $x_1^{p-1} \cdots x_r^{p-1}$ defines a non-degenerate associative symmetric bilinear-form $$(.,.): kE_r \times kE_r \rightarrow k, (a,b):=\tau(ab),$$ see [@ber64]. Since there is an isomorphism $kE_r/\operatorname{rad}^d(kE_r) \cong M^{(r)}_{d,d}$, the claimed isomorphism of $kE_r$-modules follows from the associativity of $(.,.)$ together with the isomorphism $$W_{d,d}^{(r)} \cong (kE_r/\operatorname{rad}^d(kE_r))^* \cong (\operatorname{rad}^d(kE_r))^{\perp} =\operatorname{rad}^{r(p-1)+1-d}(kE_r).$$
Observe furthermore that the algebraic group $\operatorname{GL}_r(k)$ acts on the $r$-dimensional vector space $\bigoplus_{i=1}^r kX_i$ and thereby on $R$ and $kE_r$ via automorphisms, leaving $I$ and $J$ invariant. Moreover, consider the action of $\operatorname{GL}_r(k)$ on $\operatorname{mod}kE_r$ sending $M \in \operatorname{mod}kE_r$ to its [*$g$-twist $M^{(g)}$*]{} for $g \in \operatorname{GL}_r(k)$, where $M^{(g)}$ is the $kE_r$-module with underlying vector space $M$ and action given by $x.m:=(g^{-1}.x)m$. We call a module [*$\operatorname{GL}_r(k)$-stable*]{} if there is an isomorphism $M \cong M^{(g)}$ for all $g \in \operatorname{GL}_r(k)$. Since $\operatorname{GL}_r(k)$ acts on $\bigoplus_{i=1}^r kX_i \backslash 0$ with one orbit, $\operatorname{GL}_r(k)$-stable modules are necessarily of constant Jordan type.
\[stable\] Let $n \in \mathbb{N}, d \leq p$. The $kE_r$-modules $M_{n,d}^{(r)}$ and $W_{n,d}^{(r)}$ are $\operatorname{GL}_r(k)$-stable.
Since dualizing and twisting are compatible, it suffices to prove the first claim: The module $I^{n-d}/I^n$ is a subfactor of the $\operatorname{GL}_r(k)$-module $R$ and the map $$\varphi: I^{n-d}/I^n \rightarrow I^{n-d}/I^n, m \mapsto g^{-1}.m$$ defines an isomorphism $M_{n,d}^{(r)} \cong (M_{n,d}^{(r)})^{(g)}$.
In the following, we will make use of the graded structures of the algebras $R$ and $kE_r$ and their module categories, respectively.\
Graded (Artin) algebras and their modules categories were studied by Gordon and Green [@gg82], [@gg82b]. An algebra $\Lambda$ is [*$\mathbb{Z}^n$-graded*]{} for some $n \in \mathbb{N}$ if $\Lambda$ affords a vector space decomposition $\Lambda=\bigoplus_{i \in \mathbb{Z}^n} \Lambda_i$ such that $\Lambda_i \Lambda_j \subseteq \Lambda_{i+j}$ for all $i,j \in \mathbb{Z}^n$. We denote by $|i|:=\sum_{j=1}^n i_j$ the [*value*]{} of $i=(i_1,\ldots,i_n) \in \mathbb{Z}^n$. Graded ideals are defined canonically. A $\Lambda$-module $M$ is [*$\mathbb{Z}^n$-graded*]{} if $M=\bigoplus_{i \in \mathbb{Z}^n} M_i$ such that $\Lambda_i M_j \subseteq M_{i+j}$ for all $i,j \in \mathbb{Z}^n$.\
The category $\operatorname{mod}_{\mathbb{Z}^n}\Lambda$ has the finitely generated $\mathbb{Z}^n$-graded $\Lambda$-modules as objects and the sets of morphisms $\operatorname{Hom}_{\Lambda}^{\mathbb{Z}^n}(M,N)$ are the $\Lambda$-linear maps $\varphi: M \rightarrow N$ with $\varphi(M_i) \subseteq N_i$ for all $i \in \mathbb{Z}^n$. Furthermore, the [*$i$-th shift functor*]{} $[i]: \operatorname{mod}_{\mathbb{Z}^n}\Lambda \rightarrow \operatorname{mod}_{\mathbb{Z}^n}\Lambda$ is defined on objects $M \in \operatorname{mod}_{\mathbb{Z}^n}\Lambda$ to be $M[i]$ where $M[i]_j:=M_{j-i}$. Morphisms are left unchanged.\
If $M, N \in \operatorname{mod}_{\mathbb{Z}^n}\Lambda$ afford gradings $M=\bigoplus_{i \in \mathbb{Z}^n} M_i$ and $N=\bigoplus_{i \in \mathbb{Z}^n} N_i$, then $\operatorname{Hom}_{\Lambda}(M,N)$ affords a grading $\operatorname{Hom}_{\Lambda}(M,N)=\bigoplus_{i \in \mathbb{Z}^n}\operatorname{Hom}_{\Lambda}(M,N)_i$ as a module over the $\mathbb{Z}^n$-graded algebra $\operatorname{End}_{\Lambda}(N)$, where $\operatorname{Hom}_{\Lambda}(M,N)_i=\left\{ \varphi \in \operatorname{Hom}_{\Lambda}(M,N) | \varphi(M_j) \subseteq N_{i+j} \; \forall j \in \mathbb{Z}^n \right\}$.\
Now $R=\oplus_{i \in \mathbb{Z}^r} R_i$ is a $\mathbb{Z}^r$-graded algebra, where $R_i$ is the $k$-span of the polynomial $X_1^{i_1} \cdots X_r^{i_r}$ for all $i=(i_1,\ldots,i_r) \in \mathbb{N}_0^r$ and $R_i=0$ else. Hence all non-trivial homogeneous components are one-dimensional.\
Since the ideal $(X_1^p,\ldots,X_r^p)$ is homogeneous with respect to this grading, $kE_r$ inherits the $\mathbb{Z}^r$-grading from $R$. Furthermore it is $I=\bigoplus_{i \in \mathbb{Z}^r \atop i \neq 0} R_i$ and hence $M_{n,d}^{(r)}=I^{n-d}/I^n$ has both as an $R$- and $kE_r$-module a canonical $\mathbb{Z}^r$-grading $$M_{n,d}^{(r)}=\bigoplus_{i \in \mathbb{Z}^r} M_i$$ where $M_i$ is the vector space spanned by $x_1^{i_1} \cdots x_r^{i_r}=X_1^{i_1} \cdots X_r^{i_r} + I^n$ for all $i \in \mathbb{N}_0^r$ with $n-d \leq |i| \leq n-1$, and $M_i=0$ else. Endowed with this grading, $M_{n,d}^{(r)}$ is generated by its components $M_i$ with $i \in \mathbb{N}_0^r,\ |i|=n-d$. Observe that the $\mathbb{Z}^r$-grading induces a $\mathbb{Z}$-grading both on the algebra and the graded modules in a canonical fashion via $R_i=\bigoplus_{|(j_1,\ldots,j_r)|=i}R_{(j_1,\ldots,j_r)}$ and $M_i=\bigoplus_{|(j_1,\ldots,j_r)|=i}M_{(j_1,\ldots,j_r)}$ for all $i \in \mathbb{Z}$.
For $r\geq2$ and $n \geq d>1$, $d \leq p$, we have an isomorphism of $\mathbb{Z}^r$-graded rings $$\operatorname{End}_{kE_r}(M_{n,d}^{(r)}) \cong kE_r/J^d \oplus \bigoplus_{i \in \mathbb{Z}^r \atop |i|=d-1}k[i]^{s_i}$$ where the right-hand side denotes the trivial extension of $kE_r/J^d$ by a sum of shifts of the trivial $kE_r$-bimodule $k$. In particular, $\operatorname{End}_{kE_r}(M_{n,d}^{(r)})$ is local and commutative.
[By computing $\operatorname{Hom}$-spaces, we will moreover show that the $s_i$ are uniquely determined and that they are all equal to zero iff $n=d$.]{.nodecor}
We claim that for $n \in \mathbb{N}, d \leq \min\left\{n, p \right\}$, there is a monomorphism $$\iota: kE_r/J^d \rightarrow \operatorname{End}_{kE_r}(M_{n,d}^{(r)})$$ of $\mathbb{Z}^r$-graded $k$-algebras. Multiplication by an element of $kE_r$ clearly yields an endomorphism of $M_{n,d}^{(r)}$ and we obtain a homomorphism $kE_r \rightarrow \operatorname{End}_{kE_r}(M_{n,d}^{(r)})$ of $k$-algebras which obviously respects the $\mathbb{Z}^r$-grading. Since $\operatorname{ann}_{kE_r}(M_{n,d}^{(r)})=J^d$, $\iota$ is injective.\
We now show that $\iota$ induces an isomorphism of homogeneous components $$\label{hc}
( kE_r/J^d)_i \cong \operatorname{End}(M_{n,d}^{(r)})_i$$ for $i \in \mathbb{Z}^r$ with $|i| \leq d-2$.\
*Proof of (\[hc\]):* Since $M_{d,d}^{(r)} \cong kE_r/J^d$, the isomorphism in (\[hc\]) is obvious for $n=d$. We thus assume $n >d$. Let $\varphi_i \in \operatorname{End}(M_{n,d}^{(r)})_i$ and $|i| \leq d-2$. Recall that all non-trivial homogeneous components of $M_{n,d}^{(r)}$ are one-dimensional and the module is generated by its homogeneous components $M_j=\left< x_1^{j_1}\cdots x_r^{j_r} \right>_k$ with $|j|=n-d$.\
For all $1 \leq t \leq r$, we denote by $1_t$ the element in $\mathbb{N}_0^r$ with the $t$-th entry being equal to 1 and all others being equal to 0. For all $1 \leq t, t' \leq r$ we denote by $-\mathds{1}_t+ \mathds{1}_{t'}$ the operation on $\mathcal{K}=\left\{ \kappa \in \mathbb{N}_0^r | \ |\kappa|=n-d \right\}$ given by $-\mathds{1}_t+ \mathds{1}_{t'}(\kappa)=\kappa-1_t+1_{t'}$ if $\kappa_t \neq 0$ and $-\mathds{1}_t+ \mathds{1}_{t'}(\kappa)=\kappa$ else. Observe that every non-empty subset of $\mathcal{K}$ that is closed under all such operations is equal to $\mathcal{K}$.\
Let us first of all show that $\varphi_i=0$ if $i$ has a negative entry $i_l$ for some $1 \leq l \leq r$. We know that $\varphi_i$ certainly vanishes on those $M_j$, $|j|=n-d$, with $j_l=0$.\
Now assume $\varphi_i(M_k)=0$, i.e. $\varphi_i(x_1^{k_1}\cdots x_r^{k_r})=0$ for some $k \in \mathbb{N}_0^r,\ |k|=n-d$. Let furthermore $t \in \left\{1,\ldots,r\right\}$ such that $k_t \neq 0$. For all $t' \in \left\{1,\ldots,r \right\}$, we have $$\label{cv}
\ x_t \varphi_i(x_1^{k_1}\cdots x_r^{k_r}\frac{x_{t'}}{x_t})= x_{t'} \varphi_i (x_1^{k_1}\cdots x_r^{k_r}).$$ By our assumption, we have $|i| \leq d-2$ which implies $\varphi_i(x_1^{k_1}\cdots x_r^{k_r}\frac{x_{t'}}{x_t})=0$ and hence $\varphi_i(M_{k-1_t+1_{t'}})=0$. Thus $\left\{\kappa \in \mathcal{K}|\varphi_i(M_{\kappa})=0\right\}$ is non-empty and closed under operations of the form $-\mathds{1}_t+ \mathds{1}_{t'}$ and hence equal to $\mathcal{K}$. We thus obtain $\varphi_i(M_{\kappa})=0$ for all $\kappa \in \mathcal{K}$ and hence $\varphi_i=0$.\
For $i \in \mathbb{N}_0^r$, $|i| \leq d-2$, we use the fact that non-trivial homogeneous components are one-dimensional and obtain $\varphi_i(x_1^{k_1}\cdots x_r^{k_r})=c_k x_1^{k_1+i_1}\ldots x_r^{k_r+i_r}$ for all $k \in \mathbb{N}_0^r, |k|=n-d$ and scalars $c_k$. Comparing coefficents in (\[cv\]) yields that $\varphi_i$ is multiplication by an element of the form $cx_1^{i_1} \cdots x_r^{i_r}$ with $c \in k$. This proves our claim (\[hc\]).\
In case $i \in \mathbb{N}_0^r,\ |i|=d-1$, we have an isomorphism of vector spaces $$\operatorname{End}_{kE_r}(M_{n,d}^{(r)})_{i} \cong \bigoplus_{j \in \mathbb{N}_0^r \atop |j|=n-d} \operatorname{Hom}_k((M_{n,d}^{(r)})_j, (M_{n,d}^{(r)})_{i+j})$$ and hence $\dim_k \operatorname{End}_{kE_r}(M_{n,d}^{(r)})_{i}/\iota((kE_r/J^d)_i) = \dim_k I^{n-d} -1$. The right-hand term is equal to zero if and only if $n=d$. For $i \in \mathbb{Z}^r \backslash \mathbb{N}_0^r,\ |i|=d-1$, we have $$\operatorname{End}_{kE_r}(M_{n,d}^{(r)})_{i} \cong \bigoplus_{j \in \mathbb{N}_0^r \atop {|j|=n-d \atop i+j \in \mathbb{N}_0^r}} \operatorname{Hom}_k((M_{n,d}^{(r)})_j,(M_{n,d}^{(r)})_{i+j})$$ with $( kE_r/J^d)_i=(0)$ and the right-hand term being equal to zero iff $n=d$. Since furthermore $\operatorname{End}(M_{n,d}^{(r)})_i=0$ for $i \in \mathbb{Z}^r, |i| \geq d$, and maps of degree $d-1$ vanish when composed with maps of degree greater than zero, we obtain the above structure of the endomorphism ring.
\[brick\] Let $n \in \mathbb{N}$ and $1<d\leq p$.
(i) The $kE_r$-module $M_{n,d}^{(r)}$ is indecomposable.
(ii) We have $k \cong\operatorname{End}_{kE_r}(M_{n,d}^{(r)})_0=\operatorname{End}_{kE_r}^{\mathbb{Z}}(M_{n,d}^{(r)})$, i.e. $M_{n,d}^{(r)}$ is a brick in $\operatorname{mod}_{\mathbb{Z}} kE_r$.
Jordan types that can be realized via indecomposable modules are of special interest. Counting polynomials, we obtain:
For $n \in \mathbb{N}, d \leq \min\left\{n, p \right\}$, we have $$\operatorname{JType}(M_{n,d}^{(r)} )={{r+n-d-1}\choose{n-d}}[d]+\sum_{i=1}^{d-1}\binom{r+n-2-i}{n-i}[i]=\operatorname{JType}(W_{n,d}^{(r)})$$ and in particular for $n=d \leq p$ $$\operatorname{JType}(M_{n,n}^{(r)})=[n]+\sum_{i=1}^{n-1}{{r+n-i-2} \choose {n-i}}[i]=\operatorname{JType}(W_{n,n}^{(r)}).$$
The indecomposability of $W$- and $M$-modules of Loewy length greater than one over the algebra $kE_2$ follows directly from [@cfs09 4.2], according to which the Jordan type $\sum_{i=1}^p a_i [i]$ of a module with the equal images property is such that $a_{i-1} \neq 0$ whenever $a_i \neq 0$, $i \geq 2$. Taking into account that $\operatorname{EIP}(kE_r)$ and $\operatorname{EKP}(kE_r)$ are closed under direct summands and $\operatorname{JType}(W_{n,d})=(n-d+1)[d] + \sum_{i=1}^{d-1}[i]$, these modules are hence indecomposable if $d \geq 2$. In case $r>2$, this conclusion does not seem to follow from the computation of Jordan types.\
Moreover, for $r=2$, the indecomposable equal images modules of Loewy length 2 are just the modules $W_{n,2}$ [@cfs09 4.1]. We will show in the following sections that in case $r>2$, the situation is completely different and there is no hope to parametrize the indecomposable equal images modules of Loewy length 2 in the same fashion. It seems that $W$-modules are thus not “ubiquitous” in $\operatorname{EIP}(kE_r)$ if $r>2$.
removefromreset[theorem]{}[section]{} addtoreset[theorem]{}[subsection]{}
Equal images modules for generalized Beilinson algebras
=======================================================
In order to understand the subcategories of $\operatorname{mod}kE_r$ introduced in the previous section, we will now consider the category $\operatorname{mod}_{\mathbb{Z}} kE_r$ of $\mathbb{Z}$-graded modules over the $\mathbb{Z}$-graded algebra $kE_r$. When studying objects in $\operatorname{mod}_{\mathbb{Z}} kE_r$ that have a bounded support, the generalized Beilinson algebra $B(n,r)$ comes into play. It turns out that we can define certain analogs of our subcategories of $\operatorname{mod}_n kE_r$ as subcategories of $\operatorname{mod}B(n,r)$ which exhibit interesting properties and behave nicely when it comes to Auslander-Reiten theory. Our main results strongly depend on our homological characterization of these subcategories.
General approach
----------------
We recall from [@gg82] that there is a faithful exact functor $$\mathfrak{F}: \operatorname{mod}_{\mathbb{Z}} kE_r \rightarrow \operatorname{mod}kE_r$$ referred to as the forgetful functor since it “forgets” the grading on objects. $\mathfrak{F}$ preserves indecomposability and has the property that the fibre of an indecomposable object in the essential image of $\mathfrak{F}$ consists of the shifts $M[i],\ i \in \mathbb{Z},$ of a certain indecomposable object $M=\bigoplus_{i \in \mathbb{Z}}M_i \in \operatorname{mod}_{\mathbb{Z}} kE_r$ [@gg82 4.1]. Note furthermore that $\mathfrak{F}$ is not dense.\
If $M=\bigoplus_{i \in \mathbb{Z}} M_i \in \operatorname{mod}_{\mathbb{Z}} kE_r$, then $\operatorname{supp}(M)=\left\{i \in \mathbb{Z}|M_i \neq 0 \right\}$ is called the [*support*]{} of $M$. An object in the essential image of $\mathfrak{F}$ is called *gradable*. We say that $M \in \operatorname{mod}kE_r$ is *$J$-gradable* for $J \subseteq \mathbb{Z}$ if $M \cong \mathfrak{F}(\bigoplus_{i \in \mathbb{Z}} M_i)$ for some $\bigoplus_{i \in \mathbb{Z}} M_i \in \operatorname{mod}_{\mathbb{Z}} kE_r$ such that $\operatorname{supp}(\bigoplus_{i \in \mathbb{Z}} M_i ) \subseteq J$.\
Using the terminology of [@gmmz], the positively graded algebra $\Lambda=kE_r$ is [*standardly graded*]{}, i.e. $\Lambda_0$ is a direct product of $k$, $\Lambda_i$ is finite dimensional for all $i \geq 0$ and $\Lambda_i\Lambda_j=\Lambda_{i+j}$ for all $i,j \geq 0$. We call $M=\bigoplus_{i \in \mathbb{Z}}M_i \in \operatorname{mod}_{\mathbb{Z}} kE_r$ *standardly graded* if $M$ is generated by $M_i$, where $i = \min \operatorname{supp}(M)$. Dually, we say that $M$ is *costandardly graded* if $M$ is cogenerated by $M_i$, where $i = \max \operatorname{supp}(M)$. We refer to their images under $\mathfrak{F}$ as [*standardly gradable*]{} and [*costandardly gradable*]{} objects, respectively.\
For $2\leq n \leq p$, we now consider the full subcategory $\mathcal{C}_{[0,n-1]}$ of $\operatorname{mod}_{\mathbb{Z}} kE_r$ containing those objects $M=\bigoplus_{i \in \mathbb{Z}}M_i \in \operatorname{mod}_{\mathbb{Z}} kE_r$ with $\operatorname{supp}(M) \subseteq [0,n-1]:=\left\{0,\ldots,n-1\right\}$. Hence the essential image of $\mathfrak{F}| _{\mathcal{C}_{[0,n-1]}}$ consists of the $[0,n-1]$-gradable objects in $\operatorname{mod}kE_r$. Observe furthermore that $\mathcal{C}_{[0,n-1]}$ is equivalent to $\operatorname{mod}B(n,r)$, the module category of a generalized Beilinson algebra, where $B(n,r)$ is defined as follows:\
Let $E(n,r)$ be the path algebra of the quiver with $n$ vertices and $r$ arrows between vertices $i$ and $i+1$ for all $0 \leq i \leq n-1$.
$$\begin{xy}
\xymatrix{
0 \ar@/^1pc/[r]^{\gamma_1^{(0)}} \ar@/_1pc/[r]_ {\gamma_r^{(0)}} \ar@{}[r]|{\vdots} & 1 \ar@/^1pc/[r]^{\gamma_1^{(1)}} \ar@/_1pc/[r]_ {\gamma_r^{(1)}} \ar@{}[r]|{\vdots} & 2 \ar@{}[r]|{\cdots} & n-2 \ar@/^1pc/[r]^{\gamma_{1}^{({n-2})}} \ar@/_1pc/[r]_ {\gamma_{r}^{({n-2})}} \ar@{}[r]|{\vdots} & n-1 }
\end{xy}$$
Now let $B(n,r)$ be the factor algebra $E(n,r)/I$ where $I$ is generated by the commutativity relations $\gamma^{(j)}_i \gamma^{(j-1)}_{k}-\gamma^{(j)}_{k} \gamma^{(j-1)}_{i}$ for all $i,k \in \left\{1,\ldots,r\right\}, j \in \left\{1,\ldots,n-2\right\}$. The equivalence between $\mathcal{C}_{[0,n-1]}$ and $\operatorname{mod}B(n,r)$ is such that $M=M_0 \oplus \cdots \oplus M_{n-1} \in \mathcal{C}_{[0,n-1]}$ is a module for $B(n,r)$ where $M_i=e_i M$ for the primitive orthogonal idempotents $e_i \in B(n,r)$ corresponding to the vertex $i$. Hence we use this notation both for objects in $\mathcal{C}_{[0,n-1]}$ and $\operatorname{mod}B(n,r)$. The action of $x_j$ on elements in $M_i$ corresponds to the action of $\gamma_j^{(i)}$ on elements in $e_iM$.\
In the following, we thus regard $\operatorname{mod}B(n,r)$ as a full subcategory of $\operatorname{mod}_{\mathbb{Z}} kE_r$ and we will see in the next section that we gain a lot by viewing $\mathcal{C}_{[0,n-1]}$ as the module category for a bound quiver algebra. A general introduction to representation theory of quivers can be found in [@ass06].\
For all $0 \leq i \leq n-1$, we denote by $S(i)$, $P(i)$ and $I(i)$ the simple, the projective and the injective indecomposable $B(n,r)$-module corresponding to the vertex $i$.\
Restricting $\mathfrak{F}$ to $\operatorname{mod}B(n,r)$ yields a functor $$\mathfrak{F}_{(n,r)}: \operatorname{mod}B(n,r) \rightarrow \operatorname{mod}_n kE_r.$$
We now define subcategories of $\operatorname{mod}B(n,r)$ that correspond to the full subcategories $\operatorname{CR}^j_n(kE_r) \subset \operatorname{CR}^j(kE_r) $, $\operatorname{CJT}_n(kE_r) \subset \operatorname{CJT}(kE_r)$, $\operatorname{EIP}_n(kE_r) \subset \operatorname{EIP}(kE_r)$ as well as $\operatorname{EKP}_n(kE_r) \subset \operatorname{EKP}(kE_r)$ containing modules of Loewy length at most $n$.\
Let therefore $\alpha \in k^r \backslash 0$. Now for the element $\tilde{\alpha}=\sum_{i=0}^{n-2}(\alpha_1 \gamma_1^{(i)} + \cdots + \alpha_r \gamma_r^{(i)}) \in B(n,r)$ and $M=\bigoplus_{i=0}^{n-1}M_i \in \operatorname{mod}B(n,r)$, left-multiplication with $\tilde{\alpha}$ yields a linear operator $$\alpha_M : M \rightarrow M$$ such that for $1 \leq j \leq n-1$, $(\alpha_M)^j$ coincides with the left-multiplication with the element $\sum_{i=0}^{n-j-1}( (\alpha_1 \gamma_1^{(i+j-1)} + \cdots + \alpha_r \gamma_r^{(i+j-1)}) \cdots (\alpha_1 \gamma_1^{(i)} + \cdots + \alpha_r \gamma_r^{(i)}) ) \in B(n,r)$.
For $n \leq p, r \geq 2$, we define full subcategories of $\operatorname{mod}B(n,r)$ as follows:
(a) $\operatorname{EIP}(n,r) := \left\{M \in \operatorname{mod}B(n,r)| \operatorname{im}(\alpha_M)=\bigoplus_{i=1}^{n-1}M_i \ \ \forall \ \alpha \in k^r\backslash 0 \right\}$,
(b) $\operatorname{EKP}(n,r) := \left\{M \in \operatorname{mod}B(n,r)| \ker(\alpha_M)=M_{n-1} \ \ \forall \ \alpha \in k^r\backslash 0 \right\}$,
(c) $\operatorname{CR}^j(n,r) := \left\{ M \in \operatorname{mod}B(n,r)| \exists c_j \in \mathbb{N}_0: \operatorname{rk}(\alpha_{M})^j=c_j \ \ \forall \ \alpha \in k^r\backslash 0 \right\}$,
(d) $\operatorname{CJT}(n,r):= \bigcap_{j=1}^n \operatorname{CR}^j(n,r)$.
[A module $M \in \operatorname{EIP}(n,r)$ is standardly graded and $M_0=0$ implies $M=0$. Dually, a module $M \in \operatorname{EKP}(n,r)$ is costandardly graded and $M_{n-1}=0$ implies $M=0$.]{.nodecor}
Note that the duality $D: \operatorname{mod}B(n,r) \rightarrow \operatorname{mod}B(n,r)$ induced by relabelling the vertices in the reversed order and taking the linear dual is such that $D \operatorname{EIP}(n,r) = \operatorname{EKP}(n,r)$. Moreover observe that we have $\operatorname{EIP}(n,r) \cup \operatorname{EKP}(n,r) \subset \operatorname{CR}^j(n,r)$ for all $j \geq 1$.
\[func\] The restriction of $\mathfrak{F}_{(n,r)}$ to $\mathcal{X} \in \left\{\operatorname{EIP}(n,r), \operatorname{EKP}(n,r), \operatorname{CR}^j(n,r), \operatorname{CJT}(n,r) \right\}$ induces a faithful exact functor $$\mathfrak{F}_{\mathcal{X}}: \mathcal{X} \rightarrow \operatorname{mod}_n kE_r$$ such that
(i) for $\mathcal{X}=\operatorname{EIP}(n,r)$, $\mathfrak{F}_{\mathcal{X}}$ reflects isomorphisms and the essential image consists of the standardly gradable objects in $\operatorname{EIP}_n(kE_r)$.
(ii) for $\mathcal{X}=\operatorname{EKP}(n,r)$, $\mathfrak{F}_{\mathcal{X}}$ reflects isomorphisms and the essential image consists of the costandardly gradable objects in $\operatorname{EKP}_n(kE_r)$.
(iii) for $\mathcal{X}=\operatorname{CR}^j(n,r)$, the essential image of $\mathfrak{F}_{\mathcal{X}}$ consists of the $\left [0,n-1 \right]$-gradable objects in $\operatorname{CR}^j_n(kE_r)$.
Observe that for $M \in \operatorname{mod}B(n,r)$, the linear operator $\alpha_M$ given by $\alpha \in k^r \backslash 0$ corresponds to the linear operator $\alpha(t)_{\mathfrak{F}_{(n,r)}(M)}$ on $\mathfrak{F}_{(n,r)}(M)$ given by the $p$-point $\alpha$ with $\alpha(t)=\alpha_1x_1+ \cdots + \alpha_rx_r$.\
$(i)$: With the preceding observation and in view of the fact that fibres of indecomposables are shifts on an indecomposable object [@gg82 4.1], it is easy to see that for an indecomposable object $M \in \operatorname{mod}_\mathbb{Z} kE_r$ we have $\mathfrak{F}_{(n,r)}(M) \in \operatorname{EIP}_n(kE_r)$ if and only if $M=N[i]$ for some object $N \in \operatorname{EIP}(n,r) \subset \operatorname{mod}_{\mathbb{Z}}kE_r$.\
Given $N \in \operatorname{EIP}(n,r)\backslash 0$, we have $\operatorname{supp}(N)=[0,l]$ for some $0 \leq l \leq n-1$. Thus we have $N[i] \notin \operatorname{EIP}(n,r)$ unless $i=0$ since $\operatorname{supp}N[i] =[i,l+i]$. Since $\mathfrak{F}_{(n,r)}$ commutes with direct sums and $\operatorname{mod}_n kE_r$ is a Krull-Schmidt category, $\mathfrak{F}_{\operatorname{EIP}(n,r)}$ thus reflects isomorphisms. Moreover, $M \in \operatorname{EIP}(n,r)$ is generated by $M_0$ and is hence standardly graded. Thus the essential image of $\mathfrak{F}_{\operatorname{EIP}(n,r)}$ consists of the standardly gradable objects in $\operatorname{EIP}_n(kE_r)$.\
$(ii)$: Dual to $(i)$.\
$(iii)$: Is clear in view of our general observation above.
[A direct consequence of Proposition \[func\] is $\operatorname{EIP}(n,r) \cap \operatorname{EKP}(n,r) = (0)$, since $S(0)$ is the only simple $B(n,r)$-module in $\operatorname{EIP}(n,r)$, $S(n-1)$ the only simple $B(n,r)$-module in $\operatorname{EKP}(n,r)$ and $\operatorname{EIP}(kE_r) \cap \operatorname{EKP}(kE_r) = \operatorname{add}k$ [@f11 4.4.3].]{.nodecor}
Homological characterization
----------------------------
In this section, we will give a new point of view on our subcategories of $\operatorname{mod}B(n,r)$ which enables us to apply general methods from Auslander-Reiten theory. The approach we present is inspired by work of Happel and Unger [@haun] on representations of the generalized Kronecker $\mathcal{K}_r$. The authors construct a representation $X=(X_1,X_2)$ over $\mathcal{K}_r$ corresponding to a given arrow $\gamma$ of $\mathcal{K}_r$ such that the representations $Y=(Y_1,Y_2)$ in the right-perpendicular category $X^{\perp}$ are exactly those for which the operator $\gamma_Y: Y_1 \rightarrow Y_2$ corresponding to $\gamma$ is bijective [@haun 2.1].\
Note that $P(i) \cong kE_r[i]/J^{n-i} kE_r[i]$ in $\operatorname{mod}_{\mathbb{Z}} kE_r$. Let $\alpha \in k^r \backslash 0$. Since $n-i \leq n-1 <p$, the map $$\alpha(i): P(i+1) \rightarrow P(i),\ e_{i+1} \mapsto \alpha_1\gamma^{(i)}_1 + \cdots +\alpha_r\gamma^{(i)}_r,$$ i.e. the right multiplication with $\alpha_1\gamma^{(i)}_1 + \cdots +\alpha_r\gamma^{(i)}_r$, defines an embedding of $B(n,r)$-modules. Composition yields embeddings $$\alpha(i)_j: P(i+j) \rightarrow P(i),\ e_{i+j} \mapsto ( \alpha_1\gamma^{(i+j-1)}_1 + \cdots + \alpha_r\gamma^{(i+j-1)}_r ) \cdots ( \alpha_1\gamma^{(i)}_1 + \cdots + \alpha_r\gamma^{(i)}_r )$$ for all $0 \leq i \leq n-2,\ 1 \leq j \leq n-i-1$. We let $ X_{\alpha}^{i,j}:=\operatorname{coker}\alpha(i)_j= P(i) / \alpha(i)_j(P(i+j))$.\
For $1 \leq j \leq n-1$, $\alpha \in k^r \backslash 0$, we define $$X_{\alpha}^j=\bigoplus_{i=0}^{n-j-1} X_{\alpha}^{i,j}.$$ By definition, we obtain for the projective dimensions of these modules $\operatorname{pd}( X_{\alpha}^{i,j}) = 1$ and hence $\operatorname{pd}(X_{\alpha}^j)=1$. In the following, whenever we write $\operatorname{Hom}$ or $\operatorname{Ext}$, we refer to the category $\operatorname{mod}B(n,r)$.
\[main\] We have
(a) $\operatorname{EIP}(n,r)= \left\{ M \in \operatorname{mod}B(n,r)| \operatorname{Ext}^1(X_{\alpha}^1,M)=0 \; \forall \alpha \in k^r \backslash 0 \right\}$,
(b) $\operatorname{EKP}(n,r)= \left\{ M \in \operatorname{mod}B(n,r)| \operatorname{Hom}(X_{\alpha}^1,M)=0 \; \forall \alpha \in k^r \backslash 0 \right\}$,
(c) $\operatorname{CR}^j(n,r)= \left\{ M \in \operatorname{mod}B(n,r)| \exists c_j \ \dim_k \operatorname{Ext}^1(X_{\alpha}^j,M)=c_j\; \forall \alpha \in k^r \backslash 0 \right\}$.
Consider the projective resolution $0 \rightarrow P(i+j) \stackrel{\alpha(i)_j}{\longrightarrow} P(i) \rightarrow X_{\alpha}^{i,j} \rightarrow 0$ and, for $M \in \operatorname{mod}B(n,r)$, the exact sequence $$0 \rightarrow \operatorname{Hom}(X_{\alpha}^{i,j}, M) \rightarrow \operatorname{Hom}(P(i), M) \rightarrow \operatorname{Hom}(P(i+j), M) \rightarrow \operatorname{Ext}^1(X_{\alpha}^{i,j} ,M) \rightarrow 0.$$ There is a commutative diagram
$$\begin{CD}
\operatorname{Hom}(P(i),M) @> \operatorname{Hom}(\alpha(i)_j,M)>> \operatorname{Hom}(P(i+j),M)\\
@VV \cong V @VV \cong V\\
M_i @> (\alpha_{M})^j|_{M_i}>> M_{i+j}
\end{CD}$$
whence $$(\alpha_{M})^j|_{M_i}: M_{i} \rightarrow M_{i+j}$$ is surjective, resp. injective, if and only if $\operatorname{Ext}^1(X_{\alpha}^{i,j},M)=0$, resp. $\operatorname{Hom}(X_{\alpha}^{i,j},M)=0$. This already yields $(a)$ and $(b)$ and since $$\operatorname{rk}(\alpha_M)^j =\sum_{i=0}^{n-j-1} (\dim_k M_{i+j}-\dim_k\operatorname{Ext}^1(X_{\alpha}^{i,j},M))=\sum_{i=0}^{n-j-1}{\dim_k M_{i+j}}-\dim_k \operatorname{Ext}^1(X_{\alpha}^j,M),$$ we obtain $(c)$.
Hence we have a homological description of the subcategories defined in $\S \ 2.1$ that involves a $\mathbb{P}^{r-1}$-family of $B(n,r)$-modules of projective dimension 1. At this juncture, we exploit fundamental homological properties of $\operatorname{mod}B(n,r)$ that do not hold in $\operatorname{mod}kE_r$.\
Let us list some of the distinctive features of these modules.
\[prop\] Let $\alpha \in k^r \backslash 0$ and let $\iota: B(n,r-1) \rightarrow B(n,r)$ be the embedding defined via $\gamma^{k}_l \mapsto \gamma^{k}_l$ for all $0 \leq k \leq n-2$ and $1 \leq l \leq r-1$.
(i) We have $\operatorname{pd}(X_{\alpha}^j)=1$ for all $1\leq j \leq n-1$.
(ii) The module $X_{\alpha}^{i,j}$ is standardly graded and $\operatorname{supp}(X_{\alpha}^{i,j})=[i,n-1]$.
(iii) We have $\dim_k (X_{\alpha}^{i,j})_i=1$ and the module $X_{\alpha}^{i,j}$ is a brick in $\operatorname{mod}B(n,r)$.
(iv) All proper submodules of $X_{\alpha}^{n-2,1}$ are of the form $P(n-1)^{\oplus m}$ for some $m <r$.
(v) The pullback $\iota^*(X_{(0,\ldots,0,1)}^{i,1})$ is isomorphic to the projective $B(n,r-1)$-module $\tilde{P}(i)$.
In the following, we make use of Auslander-Reiten theory as well as torsion theory. At this point, we will briefly and in a somewhat informal way recall what Auslander-Reiten theory is about. A thorough introduction can be found in [@ass06 IV]. The module category $\operatorname{mod}A$ is described in terms of the [*Auslander-Reiten quiver*]{} $\Gamma(A)$ of the algebra $A$ which is defined as follows:
(i) The vertices of $\Gamma(A)$ correspond to the isomorphism classes $[M]$ of indecomposable $A$-modules.
(ii) The arrows from $[N]$ to $[M]$ correspond to so-called [*irreducible*]{} maps $f: N \rightarrow M$, i.e. $f$ is neither a section nor a retraction and whenever $f=f_1f_2$, then either $f_1$ is a retraction or $f_2$ is a section.
Each non-projective indecomposable module $M$ (non-injective indecomposable module $N$) gives rise to a uniquely determined short exact sequence, an [*Auslander-Reiten*]{} (or [*almost split*]{}) [*sequence*]{}, $$0 \rightarrow N \stackrel{f}{\rightarrow} \bigoplus_{i=1}^t E_i^{n_i} \stackrel{g}{\rightarrow} M \rightarrow 0$$ where $N$ ($M$) is indecomposable, the $E_i$ are pairwise non-isomorphic and indecomposable and the maps $f_{i_1},\ldots,f_{i_{n_i}}: N \rightarrow E_i$, $g_{i_1},\ldots, g_{i_{n_i}}: E_i \rightarrow M$ correspond to bases of the vector spaces of irreducible maps $N \rightarrow E_i$ and $E_i \rightarrow M$, respectively. We write $N=\tau(M)$ ($M=\tau^{-1}(N)$), where $\tau$ is referred to as the [*Auslander-Reiten translation*]{} of $M$ and we denote this in $\Gamma(A)$ by $[N] \dashleftarrow [M]$.\
An indecomposable module $M$ is called [*pre-projective*]{} ([*pre-injective*]{}) if there is $n \in \mathbb{N}_0$ such that $\tau^n(M)$ ($\tau^{-n}(M)$) is projective (injective). A module is referred to as [*regular*]{} if it is neither pre-injective nor pre-projective. Connected components of $\Gamma(A)$ that consist entirely of regular modules are then called regular.\
Furthermore, we say that an indecomposable module $N$ is a [*predecessor*]{} ([*successor*]{}) of $M$ if there is a directed path from $[N]$ to $[M]$ ($[M]$ to $[N]$) in $\Gamma(A)$, i.e. a chain of irreducible maps from $N$ to $M$ ($M$ to $N$). We denote the set of all predecessors and successors by $(\rightarrow M)$ and $(M \rightarrow )$, respectively.\
Given an algebra $A$, a pair $(\mathcal{T},\mathcal{F})$ of full subcategories of $\operatorname{mod}A$ is called [*torsion pair*]{} if the following conditions are satisfied
(a) $\operatorname{Hom}_A(M,N)=0$ for all $M \in \mathcal{T}$, $N \in \mathcal{F}$.
(b) $\operatorname{Hom}_A(M,-)|_{\mathcal{F}}=0$ implies $M \in \mathcal{F}$.
(c) $\operatorname{Hom}_A(-,N)|_{\mathcal{T}}=0$ implies $N \in \mathcal{F}$.
The category $\mathcal{T}\ (\mathcal{F})$ is then called torsion (torsion-free) class of the torsion pair $(\mathcal{T},\mathcal{F})$. According to [@ass06 VI, 1.4], torsion classes correspond to those full subcategories of $\operatorname{mod}A$ that are closed under images and extensions whereas torsion-free classes correspond to the full subcategories of $\operatorname{mod}A$ that are closed under submodules and extensions.
\[preinj\] The category $\operatorname{EIP}(n,r)$ is the torsion class $\mathcal{T}$ of a torsion pair $(\mathcal{T}, \mathcal{F})$ in $\operatorname{mod}B(n,r)$ with $\operatorname{EKP}(n,r) \subset \mathcal{F}$ that is closed under the Auslander-Reiten translate $\tau$ and which contains all preinjective modules.\
Dually, $\operatorname{EKP}(n,r)$ is a torsion-free class $\mathcal{F}'$ of a torsion pair $(\mathcal{T'}, \mathcal{F'})$ in $\operatorname{mod}B(n,r)$ with $\operatorname{EIP}(n,r) \subset \mathcal{T}'$ that is closed under $\tau^{-1}$ and contains all preprojective modules.\
In particular, there are no non-trivial maps $\operatorname{EIP}(n,r) \rightarrow \operatorname{EKP}(n,r)$.
Application of Theorem \[main\] directly yields that $\operatorname{EIP}(n,r)$ is extension closed. Since $\operatorname{pd}(X_{\alpha}^j)=1$ and hence $\operatorname{Ext}^2(X_{\alpha}^j,-)=0$, the class is furthermore image closed. Thus $\operatorname{EIP}(n,r)$ is a torsion class in $\operatorname{mod}B(n,r)$.\
The corresponding torsion-free objects in $\mathcal{F}=\left\{M \in \operatorname{mod}B(n,r)|\operatorname{Hom}(\mathcal{T},M)=0 \right\}$ are those that do not have any non trivial submodules in $\operatorname{EIP}(n,r)$. In particular, all $N \in \operatorname{mod}B(n,r)$ such that $N_0=0$ are torsion-free.\
We now show that for $M \in \operatorname{EIP}(n,r)$, we have $\tau(M) \in \operatorname{EIP}(n,r)$. The Auslander-Reiten formula [@ass06 IV, 2.13] yields an isomorphism $$\operatorname{Ext}^1(X_{\alpha}^1,\tau M) \cong D\underline{\operatorname{Hom}}(M,X_{\alpha}^1),$$ where $$\operatorname{Hom}(M,X_{\alpha}^1) \cong \bigoplus_{i=0}^{n-1} \operatorname{Hom}(M, X_{\alpha}^{i,1}).$$ For $i \geq 1$, we have $(X_{\alpha}^{i,1})_0=0$ (Prop. \[prop\], (ii)) and thus $X_{\alpha}^{i,1} \in \mathcal{F}$. This yields the isomorphism $\operatorname{Hom}(M,X_{\alpha}^1) \cong \operatorname{Hom}(M, X_{\alpha}^{0,1} ).$ Since $[0] \subset [0,n-1]=\operatorname{supp}X_{\alpha}^{0,1}$ (Prop. \[prop\], (ii)) and by definition $\operatorname{im}\alpha_{X_{\alpha}^{0,1}}=0$, we in particular obtain $X_{\alpha}^{0,1} \notin \operatorname{EIP}(n,r)$.\
Since $X_{\alpha}^{0,1}=B(n,r) (X_{\alpha}^{0,1})_0$ as well as $\dim_k (X_{\alpha}^{0,1})_0=1$, this already yields $X_{\alpha}^{0,1} \in \mathcal{F}$ and hence $\operatorname{Hom}(M, X_{\alpha}^{0,1})=0$ which implies $\tau(M) \in \operatorname{EIP}(n,r)$.\
Moreover, Theorem \[main\] directly yields that $\operatorname{EIP}(n,r)$ contains all injective objects in $\operatorname{mod}B(n,r)$ and hence also their $\tau^m$-shifts for all $m \geq0$, i.e. all preinjectives. The dual statement follows using $D$. Hence $\operatorname{EKP}(n,r)$ is closed under taking submodules and $\operatorname{EIP}(n,r) \cap \operatorname{EKP}(n,r) =(0)$ implies $\operatorname{EKP}(n,r) \subset \mathcal{F}$.
Note furthermore that the inclusions $\operatorname{EKP}(n,r) \subset \mathcal{F}$ and $\operatorname{EIP}(n,r) \subset \mathcal{T}'$ are proper. We have $X^{n-2,1}_{\alpha} \in \mathcal{F} \backslash \operatorname{EKP}(n,r)$ for example.
Corollary \[preinj\] implies that a mesh in the Auslander-Reiten quiver of $\operatorname{mod}B(n,r)$
$$\begin{xy}
\xymatrix@R=0.7em@C=0.7em{
& [E_1] \ar[rd] \\
[\tau(M)] \ar[ru] \ar[rd]& \vdots & [M]\\
& [E_t] \ar[ru]
}
\end{xy}$$
with $M$ in $\operatorname{EIP}(n,r)$, is completely contained in $\operatorname{EIP}(n,r)$. We thus obtain
\[pred\] Let $M \in \operatorname{EIP}(n,r)$ be indecomposable. Then $(\rightarrow M) \subseteq \operatorname{EIP}(n,r)$. Dually, for $M \in \operatorname{EKP}(n,r)$, we have $(M \rightarrow) \subseteq \operatorname{EKP}(n,r)$.
We can make more precise statements for $\mathbb{Z}A_{\infty}$-components of $\Gamma(B(n,r))$. These components can be visualized as follows: $$\begin{xy}
\xymatrix@R=0.7em@C=0.7em{
\vdots && \vdots && \vdots &&\vdots && \vdots && \vdots &&\vdots\\
\cdots & \bullet \ar[rd] && \bullet \ar[rd] && \bullet \ar[rd] && \bullet \ar[rd] && \bullet \ar[rd] && \bullet \ar[rd] && \bullet \ar[rd] & \cdots \\
\bullet \ar[rd] \ar[ru] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] &&\bullet \ar@{-->}[ll] \\
\cdots & \bullet \ar[ru] \ar[rd] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[rd] \ar@{-->}[ll] \ar[ru] & \cdots \\
\bullet \ar[ru] \ar[rd] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll]&& \bullet \ar[rd] \ar@{-->}[ll] \ar[ru] && \bullet \ar@{-->}[ll] \\
{ \cdots} & \bullet\ar[ru] \ar[rd] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] & { \cdots} \\
\bullet \ar[rd] \ar[ru] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar@{-->}[ll] \\
{ \cdots} & \bullet \ar[ru] \ar[rd] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[ru] \ar[rd] \ar@{-->}[ll] && \bullet \ar[rd] \ar[ru] \ar@{-->}[ll] & { \cdots} \\
\bullet \ar[ru] && \bullet \ar[ru] \ar@{-->}[ll] && \bullet \ar[ru] \ar@{-->}[ll] && \bullet \ar[ru] \ar@{-->}[ll] && \bullet \ar[ru] \ar@{-->}[ll]&& \bullet \ar[ru] \ar@{-->}[ll] && \bullet \ar[ru] \ar@{-->}[ll]&& \bullet \ar@{-->}[ll]}
\end{xy}$$
Modules in the bottom row of such components are called [*quasi-simple*]{}. Ringel [@ri78] has shown that for each module $M$ in a regular $\mathbb{Z}A_{\infty}$-component $\mathcal{C}$, there exist uniquely determined quasi-simple modules $X$ and $Y$ $\in \mathcal{C}$ and uniquely determined chains of irreducible monomorphisms $X=X_1 \rightarrow \cdots \rightarrow X_{s-1}\rightarrow X_s= M$ and epimorphisms $M=Y_s \rightarrow Y_{s-1} \rightarrow \cdots \rightarrow Y_1=Y$ where $s$ is the so called [*quasi-length*]{} of $M$ and $X$ ($Y$) is referred to as the [*quasi-socle*]{} ([*quasi-top*]{}) of $M$. Moreover, $M$ is uniquely determined by its quasi-length and quasi-socle (quasi-top) whence we write $M=X(s)$ ($M=[s]Y$).\
\[ZA\] Let $\mathcal{C}$ be a regular $\mathbb{Z}A_{\infty}$-component of $\Gamma(B(n,r))$. If $\operatorname{EIP}(n,r) \cap \mathcal{C} \neq \emptyset$, then either $\mathcal{C} \subseteq \operatorname{EIP}(n,r)$ or there exists a quasi-simple module $W_{\mathcal{C}}$ such that $(\rightarrow W_{\mathcal{C}})=\mathcal{C} \cap \operatorname{EIP}(n,r).$ Dually, if $\operatorname{EKP}(n,r) \cap \mathcal{C} \neq \emptyset$, then either $\mathcal{C} \subseteq \operatorname{EKP}(n,r)$ or there exists a quasi-simple module $M_{\mathcal{C}}$ such that $(M_{\mathcal{C}} \rightarrow)=\mathcal{C} \cap \operatorname{EKP}(n,r).$
Since in every regular $\mathbb{Z}A_{\infty}$-component the irreducible maps from top to bottom are surjective, $\operatorname{EIP}(n,r) \cap \mathcal{C} \neq \emptyset$ yields the existence of a quasi-simple module $W$ in $\mathcal{C}$ that belongs to $\operatorname{EIP}(n,r)$. If all quasi-simple modules belong to $\operatorname{EIP}(n,r)$, Corollary \[pred\] yields $\mathcal{C} \subset \operatorname{EIP}(n,r)$. In view of Corollary \[pred\] and the fact that any two quasi-simple modules are successor, resp. predecessor of one another, we can choose $k$ maximal such that $W_{\mathcal{C}}:=\tau^{-k}(W) \in \operatorname{EIP}(n,r)$ and $(\rightarrow W_{\mathcal{C}})=\mathcal{C} \cap \operatorname{EIP}(n,r)$. Dual properties of modules in $\operatorname{EKP}(n,r)$ yield the assertion.
Furthermore, we can extend Corollary \[preinj\] with regard to when the translate of a module satisfies the equal images property.
Let $M=M_0 \oplus M_1 \cdots \oplus M_{n-1} \in \operatorname{mod}B(n,r)$ be indecomposable and generated by $M_0$. If $M_{n-1}=0$, then $\tau M \in \operatorname{EIP}(n,r)$.
Applying Theorem \[main\] again in combination with the Auslander-Reiten formula, it suffices to show that $\operatorname{Hom}(M,X_{\alpha}^1)=0$. Since the module $M$ is generated by $M_0$, we have $\operatorname{Hom}(M,X_{\alpha}^1) \cong \operatorname{Hom}(M,X_{\alpha}^{0,1})$.\
Assume that there is a non-trivial morphism $\varphi: M \rightarrow X_{\alpha}^{0,1}$. Then there exists $m \in M_0$ such that $\varphi(m) \in (X_{\alpha}^{0,1})_0 \backslash 0$. Since $(X_{\alpha}^{0,1})_0$ is one-dimensional (Proposition \[prop\], (iii)) and $X_{\alpha}^{0,1}$ is generated by $(X_{\alpha}^{0,1})_0$ (Proposition \[prop\], (ii)), $\varphi$ is hence surjective. This contradicts the fact that $M_{n-1}=0$ and $(X_{\alpha}^{0,1})_{n-1} \neq 0$.
The next result concerns the special role that $W$- and $M$- modules play as modules for generalized Beilinson algebras.\
Recall that for all $m \in \mathbb{N}, d \leq n$, the $\mathbb{Z}$-graded module $M_{m,d}^{(r)}$ endowed with the grading from $\S \ 1$ satisfies $\operatorname{supp}(M_{m,d}^{(r)})=[m-d,m-1]$. Hence we have $M_{m,d}^{(r)}[n-m] \in \mathcal{C}_{[0,n-1]}$ such that $M_{m,d}^{(r)}[n-m]$ is an object in $\operatorname{EKP}(n,r)$. Likewise, the canonical $\mathbb{Z}$-grading on $W$-modules is such that $\operatorname{supp}(W_{m,d}^{(r)})=[-m+1,-m+d]$ and hence $W_{m,d}^{(r)}[m-1] \in \mathcal{C}_{[0,n-1]}$ is an object in $\operatorname{EIP}(n,r)$. For our duality $D$ on $\operatorname{mod}B(n,r)$, we have $$DM_{m,d}^{(r)}[n-m] \cong W_{m,d}^{(r)}[m-1].$$ Note furthermore that for $1 \leq d \leq n$, we have $M_{d,d}^{(r)}[n-d] \cong P(n-d)$ and $W_{d,d}^{(r)}[d-1] \cong I(d-1)$. Since $M_{m,d}^{(r)}$ is a brick in $\operatorname{mod}_{\mathbb{Z}} kE_r$ by Corollary \[brick\], $M_{m,d}^{(r)}[n-m]$ is a brick in $\operatorname{mod}B(n,r)$.\
In the remainder of this section, we are concerned with $B(n,r)$-modules and hence shorten notation and write $M_{m,d}^{(r)}$ for the $B(n,r)$-module $M_{m,d}^{(r)}[n-m]$ and likewise $W_{m,d}^{(r)}$ for the $B(n,r)$-module $W_{m,d}^{(r)}[m-1]$.\
The following theorem does not hold in case $r=2$. Since modules of the form $M_{m,2}^{(2)}$ are preprojective, we have $\tau(M_{m,2}^{(2)}) \in \operatorname{EKP}(2,2) \backslash 0$ for $m>2$.
\[wmod\] Let $r \geq 3$ and let $n \leq p$, $m > n$. Then $\tau (M_{m,n}^{(r)}) \in \operatorname{EIP}(n,r)$ and dually $\tau^{-1}(W_{m,n}^{(r)}) \in \operatorname{EKP}(n,r)$.
We want to apply Theorem \[main\] again in combination with the Auslander-Reiten formula and thus show that for all $\alpha \in k^r \backslash 0$, there are only trivial maps $M_{m,n}^{(r)} \rightarrow X_{\alpha}^1$. Since $M_{m,n}^{(r)}$ is generated by $(M_{m,n}^{(r)})_0$, we have $\operatorname{Hom}(M_{m,n}^{(r)},X_{\alpha}^1) \cong \operatorname{Hom}(M_{m,n}^{(r)},X_{\alpha}^{0,1})$ and a non-trivial map $\varphi: M_{m,n}^{(r)} \rightarrow X_{\alpha}^{0,1}$ is necessarily surjective (\[prop\], (ii), (iii)). By Proposition \[stable\], $M_{m,n}^{(r)}$ is $\operatorname{GL}_r(k)$-stable. Furthermore, for all $\alpha$, $\beta \in k^r \backslash 0$, there exists $g \in \operatorname{GL}_r(k)$ such that $(X_{\alpha}^{0,1})^{(g)} \cong X_{\beta}^{0,1}$. Since $\operatorname{Hom}(M_{m,n}^{(r)}, X_{\alpha}^{0,1}) \cong \operatorname{Hom}(M_{m,n}^{(r)}, (X_{\alpha}^{0,1})^{(g)})$ we may hence assume that $\alpha=(0,0,\ldots,1)$.\
Now we have $\gamma^{(i)}_r \in \operatorname{ann}_{B(n,r)}X_{\alpha}^{0,1}$ and thus $\gamma^{(i)}_r M_{m,n}^{(r)} \subseteq \ker \varphi$ for all $0 \leq i \leq n-2$. Note that $N:=\sum_{i=0}^{n-2}\gamma^{(i)}_r M_{m,n}^{(r)}$ is a submodule of $M$ such that $\gamma_r^{(i)}$ acts trivially on $\tilde{M}:=M_{m,n}^{(r)}/N$. Observe that for the embedding $\iota$ from Proposition \[prop\], we have $\iota^*(\tilde{M}) \cong M_{m,n}^{(r-1)}$. Moreover, Proposition \[prop\], (v), yields $\iota^*(X_{\alpha}^{0,1}) \cong \tilde{P}(0)$ for the projective indecomposable of $B(n,r-1)$-module corresponding to the vertex $0$.\
Thus there results a split epimorphism $M_{m,n}^{(r-1)} \rightarrow \tilde{P}(0)$ of $B(n,r-1)$-modules which is a contradiction since by Theorem \[brick\] $M_{m,n}^{(r-1)}$ is indecomposable and furthermore non-projective in $\operatorname{mod}B(n,r-1)$ since $m>n$ and $r>2$. Hence we have $\tau (M_{m,n}^{(r)}) \in \operatorname{EIP}(n,r)$. Our duality $D$ on $\operatorname{mod}B(n,r)$ now yields the assertion.
\[cjt\] Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an exact sequence in $\operatorname{mod}B(n,r)$. If $A \in \operatorname{EIP}(n,r)$, then $B \in \operatorname{CR}^j(n,r)$ if and only if $C \in \operatorname{CR}^j(n,r)$.
Since $\operatorname{Ext}^2(X_{\alpha}^j,-)=0$, we get an exact sequence $$\operatorname{Ext}^1(X_{\alpha}^j,A) \rightarrow \operatorname{Ext}^1(X_{\alpha}^j,B) \rightarrow \operatorname{Ext}^1(X_{\alpha}^j,C)\rightarrow 0,$$ where $\operatorname{Ext}^1(X_{\alpha}^j,A)=0$ since $A \in \operatorname{EIP}(n,r)$. Thus the dimension of the rightmost term does not depend on $\alpha$ iff the dimension of the middle term does not.
We close this section on Beilinson algebras with the following statement concerning Auslander-Reiten sequences. In case $n=2$, this is a direct consequence of Theorem $\ref{shape}$ below.
\[indec\] Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an Auslander-Reiten sequence in $\operatorname{mod}B(n,r)$ such that $A$ is in $\operatorname{EIP}(n,r)$ and $C$ is in $\operatorname{EKP}(n,r)$. Then $B$ is an indecomposable module in $\operatorname{CJT}(n,r) \backslash ( \operatorname{EKP}(n,r) \cup \operatorname{EIP}(n,r))$.
Let us first of all show that $B$ is indecomposable. Assume that there exists a decomposition $B=\oplus_{i \in I} B_i$ such that $|I|>2$. Then for reasons of dimension it is not possible that all irreducible maps $A \rightarrow B_i$ are injective and all irreducible maps $B_j \rightarrow C$ are surjective (this would imply $\dim A + \dim C < \dim B_i + \dim B_j \leq \dim B$). Thus there exists an epimorphism $A \rightarrow B_i$ for some $i$ or a monomorphism $B_i \rightarrow C$. This now implies that $B_i$ satisfies the equal images property, respectively the equal kernels property. Now in case $B_i \in \operatorname{EIP}(n,r)$, every morphism $B_i \rightarrow C$ is trivial in view of Corollary \[preinj\]. With the same argument $B_i \in \operatorname{EKP}(n,r)$ yields that every morphism $A \rightarrow B_i$ is trivial, a contradiction. Thus $B$ is indecomposable. Corollary \[preinj\] yields $B \notin \operatorname{EIP}(n,r) \cup \operatorname{EKP}(n,r)$, whereas $B \in \operatorname{CJT}(n,r)$ follows from Lemma \[cjt\].
[Returning to the categories $\operatorname{EIP}(kE_r),\ \operatorname{EKP}(kE_r),\ \operatorname{CR}^j(kE_r)$ and $\operatorname{CJT}(kE_r)$, Lemma \[cjt\] holds in $\operatorname{mod}kE_r$ as well and follows directly from the Snake Lemma [@ass06 I.5, 5.1]. Demanding that $k$ is not a direct summand of the middle-term $B$, Proposition \[indec\] also holds in $\operatorname{mod}kE_r$.]{.nodecor}
The generalized Kronecker quiver
================================
We are now going to confine our investigations to the case $B(2,r)$ where $r \geq 2$. The algebra $B(2,r)$ is isomorphic to $\mathcal{K}_r$, the path algebra of the $r$-Kronecker quiver. Note that $\mathfrak{F}_{(2,r)}$ is dense and so are the functors $\mathfrak{F}_{\mathcal{X}}$ from Proposition \[func\]. Furthermore, we have $\operatorname{CJT}(2,r)=\operatorname{CR}^1(2,r)$. As was mentioned above, the indecomposable equal images modules for $kE_2$ of Loewy length at most 2 have been classified in [@cfs09]: The only indecomposable modules in $\operatorname{EIP}(2,2)$ are the preinjective modules over $\mathcal{K}_2$, i.e. the modules $W_{n,2}$ and the simple injective module $S(1)$ [@f11 4.2.2]. This implies that $\operatorname{EIP}(2,2)$ is the additive closure of the preinjectives. Apart from the preprojective modules, that satisfiy the equal kernels property, there are no other indecomposable modules of constant Jordan type. We show that the situation is completely different for $r \geq 3$.\
The algebra $\mathcal{K}_r$ is wild if $r >2$ and tame if $r=2$. Recall that the Auslander-Reiten translate for the hereditary algebra $\mathcal{K}_r$, $r \geq 2$, is given by $\tau \cong \operatorname{Ext}^1 (-,\mathcal{K}_r)^*$ [@ass06 V.II, 1.9]. The components of the Auslander-Reiten quiver $\Gamma( \mathcal{K}_r)$ of $\operatorname{mod}\mathcal{K}_r$ have the following shape if $r >2$:\
There is exactly one preprojective component $\mathcal{P}$, consisting of the two projective modules and their $\tau^{-1}$-shifts and exactly one preinjective component $\mathcal{I}$ consisting of the two injective modules and their $\tau$-shifts. Ringel has proven in [@ri78], that the remaining (regular) components are of type $\mathbb{Z}A_{\infty}$.\
From now on, we assume that $r > 2$. We write $X_{\alpha}:=X_{\alpha}^1=\operatorname{coker}\alpha(1)$ for $\alpha \in k^r \backslash 0$. The module $X_{\alpha}$ is a brick and has no proper submodules apart from direct sums of $P(1)$ (Proposition \[prop\], (iii), (iv)). Via computing the dimension vectors of the preprojective and preinjective modules, we can conclude that $X_{\alpha}$ is regular and, since it has no proper regular submodules, thus quasi-simple. Moreover, computation yields
\[dual\] Let $\alpha \in k^r \backslash 0$. Then $D X_{\alpha}=\operatorname{Ext}^1(X_{\alpha}, \mathcal{K}_r)^*=\tau X_{\alpha}$.
According to Thereom \[preinj\], we have $\mathcal{I} \subseteq \operatorname{EIP}(2,r)$ and $\mathcal{P} \subseteq \operatorname{EKP}(2,r)$ whereas Theorem \[wmod\] implies that $W_{n,2}^{(r)} \notin \mathcal{I}$ and $M_{n,2}^{(r)} \notin \mathcal{P}$ for $n>2$. Thus these modules are examples of regular modules with the equal images property and with the equal kernels property, respectively.
Regular components
------------------
We will now describe the occurrence of regular equal images and equal kernels modules in the Auslander-Reiten quiver $\Gamma( \mathcal{K}_r)$ of $\operatorname{mod}\mathcal{K}_r$.
In order to show the existence of equal images as well as equal kernels modules in every regular component of $\Gamma( \mathcal{K}_r)$, we record the following dual version of a lemma by Kerner:
\[kerner\] If $X, Y$ are regular modules over a wild hereditary algebra, there exists an integer $N$ with $\operatorname{Hom}(Z,\tau^{-m}(X))=0$ for all $m \geq N$ and all regulars $Z$ with $\dim_k Z \leq \dim_k Y$.
Note that our next result also follows from Corollary \[preinj\] in combination with [@assker96 Theorem (B)], a general result concerning non-splitting torsion pairs for wild hereditary algebras.
\[shape\] Let $\mathcal{C}$ be a regular component of $\Gamma(B(2,r))$. Then $\mathcal{C}$ contains two uniquely determined quasi-simple modules $W_{\mathcal{C}}$ and $M_{\mathcal{C}}$ such that $$(\rightarrow W_{\mathcal{C}})=\mathcal{C} \cap \operatorname{EIP}(2,r) \text{ and }
(M_{\mathcal{C}} \rightarrow)=\mathcal{C} \cap \operatorname{EKP}(2,r).$$
Let $\mathcal{C}$ be a regular component, $X$ be in $\mathcal{C}$. Since we have $\dim_k X_{\alpha} = \dim_k X_{\beta}$ for all $\alpha, \beta \in k^r \backslash 0$ and $\mathcal{K}_r$ is wild, we can apply Lemma \[kerner\] with $Y=X_{\alpha}$ for some $\alpha$ and $Z$ running through all $X_{\beta}$, $\beta \in k^r \backslash 0$. This implies that there exists an $N$ such that $\operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},\tau^{-m}(X))=0$ for all $m \geq N$ and all $\alpha \in k^r \backslash 0$. In view of Theorem \[main\] we thus have $\tau^{-m}(X) \in \operatorname{EKP}(2,r)$ for all $m \geq N$. Dually, $\operatorname{EIP}(2,r) \cap \mathcal{C} \neq \emptyset$. Now apply Proposition \[ZA\].
Thus the regular components of $\Gamma(\mathcal{K}_r)$ have the following shape
$$\begin{xy}
\xymatrix@R=0.5em@C=0.5em{
\vdots && \vdots && \vdots &&\vdots && \vdots && \vdots &&\vdots\\
\cdots & \circ \ar[rd] && \circ \ar[rd] && \circ \ar[rd] && \circ \ar[rd] && \circ \ar[rd] && \circ \ar[rd] && \circ \ar[rd] & \cdots \\
\circ \ar[rd] \ar[ru] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] &&\circ \ar@{-->}[ll] \\
\cdots & \circ \ar[ru] \ar[rd] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[rd] \ar@{-->}[ll] \ar[ru] & \cdots \\
\nabla \ar[ru] \ar[rd] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll]&& \circ \ar[rd] \ar@{-->}[ll] \ar[ru] && \Delta \ar@{-->}[ll] \\
\cdots & \nabla \ar[ru] \ar[rd] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \Delta \ar[ru] \ar[rd] \ar@{-->}[ll] & \cdots \\
\nabla \ar[rd] \ar[ru] && \nabla \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \circ \ar[rd] \ar[ru] \ar@{-->}[ll] && \Delta \ar[ru] \ar[rd] \ar@{-->}[ll] && \Delta \ar@{-->}[ll] \\
\cdots & \nabla \ar[ru] \ar[rd] && \nabla \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \circ \ar[ru] \ar[rd] \ar@{-->}[ll] && \Delta \ar[ru] \ar[rd] \ar@{-->}[ll] && \Delta\ar[rd] \ar[ru] \ar@{-->}[ll] & \cdots \\
\nabla \ar[ru] && \nabla \ar[ru] \ar@{-->}[ll] && \nabla \ar[ru] \ar@{-->}[ll] && \circ \ar[ru] \ar@{-->}[ll] && \circ \ar[ru] \ar@{-->}[ll]&& \Delta \ar[ru] \ar@{-->}[ll] && \Delta \ar[ru] \ar@{-->}[ll]&& \Delta \ar@{-->}[ll]}
\end{xy}$$
where $\nabla$ and $\Delta$ indicate that the corresponding module is an object in $\operatorname{EIP}(2,r)$, resp. in $\operatorname{EKP}(2,r)$. Hence for each regular component $\mathcal{C}$, the [*width*]{} $\mathcal{W}(\mathcal{C})$ of the gap between these two modules, i.e. the natural number $k$ such that $\tau^{k+1}(M_{\mathcal{C}})=W_{\mathcal{C}}$ is an invariant for $\mathcal{C}$.\
[**Examples**]{}
1. Let $\mathcal{C}_n$ be the component containg the module $W_{n,2}^{(r)}$ for $n > 2$. By Theorem \[wmod\] we have $\tau^{-1}(W_{n,2}^{(r)}) \in \operatorname{EKP}(2,r)$ and thus $W_{n,2}^{(r)}=W_{\mathcal{C}_n}$ and $\tau^{-1}(W_{n,2}^{(r)})=M_{\mathcal{C}_n}$. Hence $\mathcal{W}(\mathcal{C}_n)=0$.
2. Let $\mathcal{C}_{\alpha}$ be the component containing the quasi-simple brick $X_{\alpha}$. Recall that by Proposition \[dual\], we have $\tau (X_{\alpha}) \cong D X_{\alpha}$. Since $\mathcal{K}_r$ is wild hereditary, $\tau$ is an equivalence on the full subcategory of regular modules and hence $$\operatorname{Hom}_{\mathcal{K}_r}(X_{\beta},\tau^{-1}(X_{\alpha})) \cong \operatorname{Hom}_{\mathcal{K}_r}(\tau(X_{\beta}),X_{\alpha}) \cong \operatorname{Hom}_{\mathcal{K}_r}(DX_{\beta},X_{\alpha})$$ for all $\beta \in k^r \backslash 0$. Proper submodules of $X_{\alpha}$ are of the form $P(2)^{\oplus m}$ (Proposition \[prop\], (iv)) and, dually, proper factor modules of $DX_{\beta}$ are of the form $I(1)^{\oplus m'}$. Hence the rightmost term is equal to zero. According to Theorem \[main\], we thus obtain $\tau^{-1}(X_{\alpha}) \in \operatorname{EKP}(2,r)$. Using the Auslander-Reiten formula, we can analogously show that $\tau^2(X_{\alpha}) \in \operatorname{EIP}(2,r)$. Hence $\mathcal{W}(C_{\alpha})=2.$
3. Let $\mathcal{C}_{\lambda}$ be the component containing the brick $E^{(\lambda)}$ for $\lambda \in k^r \backslash 0$ with dimension vector $(1,1)$ on which $\gamma_i$ acts via multiplication with $\lambda_i$. Using the Auslander-Reiten formula in combination with Theorem \[main\] we have $$\operatorname{Ext}^1_{\mathcal{K}_r}(X_{\alpha},\tau(E^{(\lambda)})) \cong \operatorname{Hom}_{\mathcal{K}_r} (E^{(\lambda)},X_{\alpha})=0$$ and hence $\tau(E^{(\lambda)}) \in \operatorname{EIP}(n,r)$. Dualizing yields $$\begin{aligned}
\operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},\tau^{-1}(E^{(\lambda)}))& \cong \operatorname{Hom}_{\mathcal{K}_r}(DX_{\alpha},E^{(\lambda)})\\& \cong \operatorname{Hom}_{K_r}(D(E^{(\lambda)}),X_{\alpha})\\
& \cong \operatorname{Hom}_{\mathcal{K}_r} (E^{(\frac{1}{\lambda})},X_{\alpha})=0\end{aligned}$$ where $(\frac{1}{\lambda})_i=\frac{1}{\lambda_i}$ if $\lambda_i \neq 0$ and $(\frac{1}{\lambda})_i=0$ else, for all $1 \leq i \leq r$. Hence $\mathcal{W} (C_{\lambda})=1$.
The examples show that $\mathcal{W}(\mathcal{C})$ indeed varies while running through the different regular components and we will show that there is no upper boundary for this number.
In [@ker90], Kerner has defined an invariant for regular components of a wild hereditary algebra $A$. Let $\mathcal{C}$ be a regular component of $A$ and $X$ some quasi-simple module in $\mathcal{C}$. The quasi-rank of $\mathcal{C}$ is defined via $$\operatorname{rk}\mathcal{C} = \min \left\{m \in \mathbb{Z}| \operatorname{rad}(X,\tau^l X) \neq 0 \; \forall l \geq m \right\},$$ where for two indecomposable modules $X,Y \in \operatorname{mod}\mathcal{K}_r$, $\operatorname{rad}(X,Y)$ is the vector space of all non-isomorphisms from $X$ to $Y$ (cf. [@ass06 A.3, 3.5]). Hence for $l \neq 0$ and $X$ regular, it is $\operatorname{rad}(X,\tau^l X)= \operatorname{Hom}(X,\tau^l X)$. A theorem by Hoshino (cf. [@cb V]) says that for $A=\mathcal{K}_r$, $\operatorname{rk}$ is bounded above by 1. In view of [@ker90 1.6], we can conclude that $\mathcal{C}$ contains a brick if and only if $\operatorname{rk}\mathcal{C}=1$.
\[rkW\]
(i) Let $\mathcal{C}$ be a regular component of $K_r$. If $\mathcal{C}$ does not possess a brick, then we have $|\operatorname{rk}\mathcal{C}| \leq \mathcal{W} (\mathcal{C})$.
(ii) Let $n \in \mathbb{N}$. Then there exists a regular component $\mathcal{C}$ of $\mathcal{K}_r$ such that $\mathcal{W}(\mathcal{C}) > n$.
$(i)$: Choose the quasi-simple module $W_{\mathcal{C}}$ in $\mathcal{C}$ given by Theorem \[shape\]. The module $\tau^{- \mathcal{W} (\mathcal{C}) -1}(W_{\mathcal{C}})=M_C$ satisfies the equal kernels property and hence by Corollary \[preinj\] $\operatorname{Hom}(W_{\mathcal{C}},\tau^{- \mathcal{W} (\mathcal{C}) -1}(W_{\mathcal{C}})) =0$, which implies $\operatorname{rk}{\mathcal{C}}> - \mathcal{W} (\mathcal{C}) -1$. Since $\mathcal{C}$ does not possess a brick and hence $\operatorname{rk}\mathcal{C} \leq 0$ it is $|\operatorname{rk}\mathcal{C}| \leq \mathcal{W}(\mathcal{C})$.\
$(ii)$: In [@kerlu 3.1] it is proven that $$\inf \left\{\operatorname{rk}(\mathcal{C}) |\; \mathcal{C} \in \Omega(\mathcal{K}_r) \right\}=- \infty$$ where $\Omega(\mathcal{K}_r)$ denotes the set of regular components of $\operatorname{mod}\mathcal{K}_r$. Since $\operatorname{rk}\mathcal{C}=1$ iff $\mathcal{C}$ contains a brick, we can conclude $(ii)$ with $(i)$.
For every component $\mathcal{C}$ containg a brick, it is $\operatorname{rk}\mathcal{C}=1$. By contrast, the examples $\mathcal{C}_n$ and $\mathcal{C}_{\alpha}$ show, that some components containing bricks may be distinguished via the invariant $\mathcal{W}$.
The category CJT(2,r)
---------------------
In this subsection, we direct our attention towards the category $\operatorname{CJT}(2,r)$ and make some statemens concerning Auslander-Reiten components of $B(2,r)$. Friedlander and Pevtsova have shown that for the group algebra $kE_r$, the constant $j$-rank property is in fact a property of the components of the stable Auslander-Reiten quiver of $kE_r$ [@fp 4.7]. We will see, that the situation is rather different in our context.\
Unlike $\operatorname{EIP}(2,r)$ and $\operatorname{EKP}(2,r)$, the category $\operatorname{CJT}(2,r)$ is neither closed under images nor under submodules and is hence more difficult to grasp categorically. However, $\operatorname{CJT}(2,r)$ is closed under direct summands [@cafrpe08 3.7]. We are able to make more specific statements about the category $\operatorname{CJT}(2,r)=\operatorname{CR}^1(2,r)$ as opposed to $\operatorname{CR}^j(n,r)$ with $n>2$.
\[cr\] Let $M \in \operatorname{mod}B(2,r)$ be regular and not isomorphic to an $X_{\alpha}$. Let $$0 \rightarrow \tau(M) \rightarrow E \rightarrow M \rightarrow 0$$ be the Auslander-Reiten sequence ending in $M$. If two out of the three modules are of constant rank, then so is the third.
Since $ E \rightarrow M$ is right almost split and $X_{\alpha}$ is indecomposable, any morphism $X_{\alpha} \rightarrow M$ factors through $E$. Hence we get the following exact sequence $$0 \rightarrow \operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},\tau(M)) \rightarrow \operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},E) \rightarrow \operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},M) \rightarrow 0$$ and the assertion follows with Theorem \[main\].
A direct consequence is the following
\[rk0\] Let $\mathcal{C}$ be a regular component in $\Gamma(\mathcal{K}_r)$.
(i) If all quasi-simple modules in $\mathcal{C}$ are of constant rank, then $\mathcal{C} \subseteq \operatorname{CJT}(2,r)$.
(ii) In particular, if $\mathcal{W}(\mathcal{C})=0$, then $\mathcal{C} \subseteq \operatorname{CJT}(2,r)$.
This especially tells us, that there are many indecomposable modules of constant Jordan type in Loewy length two that satisfy neither the equal images property nor the equal kernels property which is not the case if $r=2$. The inclusion $\operatorname{ind}\operatorname{EIP}(2,r) \cup \operatorname{ind}\operatorname{EKP}(2,r) \subseteq \operatorname{ind}\operatorname{CJT}(2,r)$ of indecomposable objects is proper if and only if $r>2$. Since $\mathfrak{F}_{(2,r)}$ is dense, this directly implies the same result for the categories $\operatorname{EIP}_2(kE_r),\ \operatorname{EKP}_2(kE_r)$ and $\operatorname{CJT}_2(kE_r)$.
\[rk1\] Let $\mathcal{C}$ be a regular component with $\mathcal{W}(\mathcal{C})=1$. Then either $\mathcal{C} \subseteq \operatorname{CJT}(2,r)$ or there are no indecomposable modules of constant rank in $\mathcal{C}$ apart from the modules in $\operatorname{EIP}(n,r) \cap \mathcal{C}$ and $\operatorname{EKP}(n,r) \cap \mathcal{C}$.
We first of all show
1. Let $\mathcal{C}$ be a regular component with $\mathcal{W}(\mathcal{C})=n$ and let $W_{\mathcal{C}}$ and $M_{\mathcal{C}}$ be as in Theorem \[shape\]. Then for all $1 \leq k \leq n$ we have the following: If there exists $l\geq k$ such that
(i) $[l]\tau^{-k}(W_{\mathcal{C}})$ is of constant rank, then so is $[l']\tau^{-k}(W_{\mathcal{C}})$ for all $l' \geq k$.
(ii) $\tau^k(M_{\mathcal{C}})(l)$ is of constant rank, then so is $\tau^k(M_{\mathcal{C}})(l')$ for all $l' \geq k$.
[*Proof of (\*)*]{}: We show $(1)$, $(2)$ is dual. Let $l \geq k$ and $[l]\tau^{-k}(W_{\mathcal{C}})$ be of constant rank. Now assume that there is $l' >k$ minimal such that $[l']\tau^{-k}(W_{\mathcal{C}})$ does not have constant rank. The quasi-socle $\tau^{k-l'-1}(W_{\mathcal{C}})$ satisfies the equal images property and we have a short exact sequence (cf. [@ri78 2.2]) $$0 \rightarrow \tau^{k-l'-1}(W_{\mathcal{C}}) \rightarrow [l']\tau^{-k}(W_{\mathcal{C}}) \rightarrow [l'-1]\tau^{-k}(W_{\mathcal{C}}) \rightarrow 0.$$ In view of Lemma \[cjt\], $[l'-1]\tau^{-k}(W_{\mathcal{C}})$ has constant rank, a contradiction to the choice of $l'$.\
Now since $\mathcal{W}(\mathcal{C})=1$, we have $[l]\tau^{-k}(W_{\mathcal{C}})=\tau^{k}(M_{\mathcal{C}})(l)$ for all $k,l \in \mathbb{N}$ and furthermore $$\mathcal{M}=\left\{[l]\tau^{-k}(W_{\mathcal{C}})| l \geq k \geq 1 \right\}=\left\{M \in \mathcal{C}|M \notin \operatorname{EIP}(n,r)\cup \operatorname{EKP}(n,r)\right\}.$$ Now (\*) implies that if the cone $\mathcal{M}$ contains an element of $\operatorname{CJT}(2,r)$, we have $\mathcal{M} \subseteq \operatorname{CJT}(2,r)$.
[**Examples**]{}
1. The component $\mathcal{C}_n$ containing $W_{n,2}^{(r)}$ for $n \geq 3$: It is $\mathcal{W}(\mathcal{C}_n)=0$ and hence Proposition \[rk0\] implies that all modules in $\mathcal{C}_n$ have constant rank.
2. The component $\mathcal{C}_{\alpha}$ containing $X_{\alpha}$: We claim that there are no constant rank modules in $\mathcal{C}_{\alpha}$ apart from the equal images and equal kernels modules. In view of Statement (\*) in the proof of Proposition \[rk1\], we only need to show that $[2]X_{\alpha}$ does not have constant rank. Following [@cb V], it is $$\operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},[j]X_{\alpha})=0$$ for all $j \geq 2$. Hence $\operatorname{Hom}_{\mathcal{K}_r}(X_{\alpha},[2]X_{\alpha})=0$. Since furthermore $[2]X_{\alpha} \notin \operatorname{EKP}(2,r)$, the module can’t be of constant rank.
3. The component $\mathcal{C}_{\lambda}$ containing the module $E^{(\lambda)}$: Since $E^{(\lambda)}$ obviously does not have constant rank and $\mathcal{W}(\mathcal{C}_{\lambda})=1$, Corollary \[rk1\] implies that there are no modules of constant rank in $\mathcal{C}_{\lambda}$ apart from the equal kernels and equal images modules.
Acknowledgements {#acknowledgements .unnumbered}
================
The results of this paper are part of my doctoral thesis which I am currently writing at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for his support as well as helpful remarks and comments. Moreover, I would like to thank Julian Külshammer for proof reading and helpful comments. In particular, I would like to thank Otto Kerner for pointing out literature and results on wild hereditary algebras.
[^1]: Partly supported by the D.F.G. priority programm SPP 1388 “Darstellungstheorie”
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abstract: 'This paper concerns the truly or purely cosmetic surgery conjecture. We give a survey on exceptional surgeries and cosmetic surgeries. We prove that the slope of an exceptional truly cosmetic surgery on a hyperbolic knot in $S^3$ must be $\pm 1$ and the surgery must be toroidal but not Seifert fibred. As consequence we show that there are no exceptional truly cosmetic surgeries on certain types of hyperbolic knot in $S^3$. We also give some properties of Heegaard Floer correction terms and torsion invariants for exceptional cosmetic surgeries on $S^3$.'
author:
- 'Huygens C. Ravelomanana'
title: Exceptional Cosmetic surgeries on $S^3$
---
|
---
abstract: 'Let $\mathbb{N}_0$ be the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective, where $f(u)+ f(v)$ is the sum set of the sets $f(u)$ and $f(v)$. A graph $G$ which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are the sets whose elements are in arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain associated graphs of the given graph $G$, like line graph, total graph, etc.'
author:
- '[**N K Sudev [^1]**]{} and [**K A Germina[^2]**]{}'
title: '**Associated graphs of Certain Arithmetic IASI Graphs**'
---
**Key words**: Integer additive set-indexers, arithmetic integer additive set-indexers, isoarithmetic integer additive set-indexers,biarithmetic integer additive set-indexer, semi-arithmetic set-indexer.\
**AMS Subject Classification : 05C78**
Introduction
============
For all terms and definitions, not defined specifically in this paper, we refer to [@FH] and for more about graph labeling, we refer to [@JAG]. Unless mentioned otherwise, all graphs considered here are simple, finite and have no isolated vertices. All sets mentioned in this paper are finite sets of non-negative integers. We denote the cardinality of a set $A$ by $|A|$.
Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. For all $A, B \subseteq \mathbb{N}_0$, the [*sum set*]{} of $A$ and $B$ is denoted by $A+B$ and is defined as $A + B = \{a+b: a \in A, b \in B\}$.
\[D2\][[@GA] An [*integer additive set-indexer*]{} (IASI, in short) is defined as an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph.]{}
\[D3\][The cardinality of the labeling set of an element (vertex or edge) of a graph $G$ is called the [*set-indexing number*]{} of that element.]{}
In [@GS2], the vertex set $V$ of a graph $G$ is defined to be [*$l$-uniformly set-indexed*]{}, if all the vertices of $G$ have the set-indexing number $l$.
By the term, an arithmetically progressive set, (AP-set, in short), we mean a set whose elements are in arithmetic progression. The common difference of the set-label of an element of $G$ is called the [*deterministic index*]{} of that element.
[@GS7] An [*arithmetic integer additive set-indexer*]{} is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are the sets whose elements are in arithmetic progressions. A graph that admits an arithmetic IASI is called an [*arithmetic IASI graph*]{}.
If all vertices of $G$ are labeled by the sets consisting of arithmetic progressions, but the set-labels of edges are not arithmetic progressions, then the corresponding IASI may be called [*semi-arithmetic IASI*]{}.
\[T-AIASI-g\] [@GS7] A graph $G$ admits an arithmetic IASI if and only if for any two adjacent vertices in $G$, the deterministic index of one vertex is a positive integral multiple of the deterministic index of the other vertex and this positive integer is less than or equal to the cardinality of the set-label of the latter vertex.
\[P-AIASI-1\] If the set-labels of both the end vertices of an edge have the same deterministic indices, say $d$, then the deterministic index of that edge is also $d$.
[[@GS8] If the set-labels of all elements of a graph $G$ consist of arithmetic progressions with the same common difference $d$, then the corresponding IASI is called [*isoarithmetic IASI*]{}. That is, an arithmetic IASI of a graph $G$ is an isoarithmetic IASI if all elements of $G$ have the same deterministic index.]{}
[[@GS8] An arithmetic IASI $f$ of a graph $G$, under which the deterministic indices $d_i$ and $d_j$ of two adjacent vertices $v_i$ and $v_j$ respectively of $G$, holds the conditions $d_j=kd_i$ where $k$ is a non-negative integer such that $1< k \le |f(v_i)|$, is called [*biarithmetic IASI*]{}. If the value of $k$ is unique for all pairs of adjacent vertices of a biarithmetic IASI graph $G$, then that biarithmetic IASI is called [*identical biarithmetic IASI*]{} and $G$ is called an [*identical biarithmetic IASI graph*]{}.]{}
As we study the graphs, the set-labels of whose elements are AP-sets, all sets we consider in this discussion consists of at least three elements which are in ascending order.
In this paper, we investigate the admissibility of arithmetic integer additive set-indexers by certain graphs that are associated to a given graph $G$ and establish some results on arithmetic IASIs.
Isoarithmetic IASIs of Associated Graphs
========================================
In the following discussions, we study admissibility of isoarithmetic IASIs and biarithmetic IASIs by certain graphs associated to a given arithmetic IASI graph.
Throughout this section, we denote the set-label of a vertex $v_i$ of a given graph $G$ by $A_i$, which is a set of non-negative integers.
Let $G$ be an isoarithmetic IASI graph. Then, any non-trivial subgraph of $G$ is also an isoarithmetic IASI Graph.
let $f$ be an arithmetic IASI on $G$ and let $H\subset G$. The proof follows from the fact that the restriction $f|_H$ of $f$ to the subgraph $H$ is an induced isoarithmetic IASI on $H$.
[By [*edge contraction operation*]{} in $G$, we mean an edge, say $e$, is removed and its two incident vertices, $u$ and $v$, are merged into a new vertex $w$, where the edges incident to $w$ each correspond to an edge incident on either $u$ or $v$.]{}
We establish the following theorem for the graphs obtained by contracting the edges of a given graph $G$. The following theorem verifies the admissibility of the graphs obtained by contracting the edges of a given isoarithmetic IASI graph $G$.
Let $G$ be an isoarithmetic IASI graph and let $e$ be an edge of $G$. Then, $G\circ e$ admits an isoarithmetic IASI.
Let $G$ admits an isoarithmetic IASI. Let $e$ be an edge in $E(G)$, the deterministic index of whose end vertices is $d$, where $d$ is a positive integer. Since $G$ is isoarithmetic IASI graph, the set set-label of each edge of $G$ is also an AP-set with the same common difference $d$. $G\circ e$ is the graph obtained from $G$ by deleting the edge $e$ of $G$ and identifying the end vertices of $e$. Label the new vertex thus obtained, say $w$, by the set-label of the deleted edge $e$. Then, each edge incident upon $w$ has a set-label which is also an AP-set with the same common difference $d$. Hence, $G\circ e$ is an isoarithmetic IASI graph.
[[@KDJ] Let $G$ be a connected graph and let $v$ be a vertex of $G$ with $d(v)=2$. Then, $v$ is adjacent to two vertices $u$ and $w$ in $G$. If $u$ and $w$ are non-adjacent vertices in $G$, then delete $v$ from $G$ and add the edge $uw$ to $G-\{v\}$. This operation is known as an [*elementary topological reduction*]{} on $G$.]{}
Let $G$ be a graph which admits an isoarithmetic IASI. Then, any graph $G'$, obtained by applying a finite number of elementary topological reductions on $G$, also admits an isoarithmetic IASI.
Let $G$ be a graph which admits an isoarithmetic IASI, say $f$. Then, all the elements of $G$ are labeled by AP-sets having the same common difference $d$, where $d$ is a positive integer. Let $v$ be a vertex of $G$ with $d(v)=2$. Then, $v$ is adjacent to two non-adjacent vertices $u$ and $w$ in $G$. Now remove the vertex $v$ from $G$ and introduce the edge $uw$ to $G-{v}$. Let $G'=(G-{v})\cup \{uw\}$. Now $V(G')\subset V(G)$. Let $f':V(G')\to \mathcal{P}(\mathbb{N}_0)$ such that $f'(v)=f(v)~ \forall ~v\in V(G')$ (or $V(G)$) and the associated function $f'^+:E(G')\to \mathcal{P}(\mathbb{N}_0)$ defined by $$f'^+(e)= \left\{
\begin{array}{l l}
f^+(e)& \quad \text{if $e\ne uw$}\\
f(u)+f(w)& \quad \text{if $e=uw$}
\end{array} \right.$$ Clearly, $f'$ is an isoarithmetic IASI of $G'$.
Another associated graph of a given graph $G$ is its graph subdivision. The notion of a graph subdivision is given below and its admissibility of arithmetic IASI are established in the following theorem.
[[@RJT] A [*subdivision*]{} of a graph $G$ is the graph obtained by adding vertices of degree two into some or all of its edges.]{}
The graph subdivision $G^{\ast}$ of an isoarithmetic IASI graph $G$ also admits an isoarithmetic IASI.
Let $u$ and $v$ be two adjacent vertices in $G$. Since $G$ admits an isoarithmetic IASI, the set-labels of two vertices $u$, $v$ and the edge $uv$ of $G$ are AP-sets with the common difference $d$, where $d$ is a positive integer. Introduce a new vertex $w$ to the edge $uv$. Now, we have two new edges $uw$ and $vw$ in place of $uv$. Extend the set-labeling of $G$ by labeling the vertex $w$ by the same set-label of the edge $uv$. Then, both the edges $uw$ and $vw$ have the set-labels which are AP-sets with the same common difference $d$. Hence, $G^{\ast}$ admits an isoarithmetic IASI.
Recall the following definition of line graph of a graph.
[[@DBW] For a given graph $G$, its line graph $L(G)$ is a graph such that each vertex of $L(G)$ represents an edge of $G$ and two vertices of $L(G)$ are adjacent if and only if their corresponding edges in $G$ are incident on a common vertex in $G$.]{}
An interesting question we need to address here is whether the line graph of an isoarithmetic IASI graph admits an isoarithmetic IASI. The following theorem answers this question.
If $G$ is an isoarithmetic IASI graph, then its line graph $L(G)$ is also an isoarithmetic IASI graph.
Since $G$ is an isoarithmetic IASI graph, the elements of $G$ have the set-labels whose elements are in arithmetic progression with the same common difference, say $d$, where $d$ is a positive integer. Label each vertex of $L(G)$ by the same set-label of its corresponding edge in $G$. Hence, the set-labels of all vertices in $L(G)$ are AP-sets with the same common difference $d$. Therefore, the set-labels of all edges of $L(G)$ are also AP-sets with the same common difference $d$. That is, $L(G)$ is also an isoarithmetic graph.
[[@MB] The [*total graph*]{} of a graph $G$, denoted by $T(G)$, is the graph having the property that a one-to one correspondence can be defined between its points and the elements (vertices and edges) of $G$ such that two points of $T(G)$ are adjacent if and only if the corresponding elements of $G$ are adjacent (if both elements are edges or if both elements are vertices) or they are incident (if one element is an edge and the other is a vertex). ]{}
If $G$ is an isoarithmetic IASI graph, then its total graph $T(G)$ is also an isoarithmetic IASI graph.
Let the graph $G$ admits an isoarithmetic IASI, say $f$. Then, for any element (a vertex or an edge) $x$ of $G$, the set-label $f(x)$ is an AP-set of non-negative integers with the common difference, say $d$, $d$ being a positive integer. Define a map $f':V(T(G))\to \mathcal{P}(\mathbb{N}_0)$ which assigns the same set-labels of the corresponding elements in $G$ under $f$ to the vertices of $T(G)$. Clearly, $f'$ is injective and for each vertex $u_i$ in $T(G)$, $f'(u_i)$ is an AP-set with the same common difference $d$. Now, define the associated function $f^+:E(T(G))\to \mathcal{P}(\mathbb{N}_0)$ defined by $f'^+(u_iu_j)= f'(u_i)+f'(u_j),~ u_i,u_j\in V(T(G))$. Then, $f'^+$ is injective and each $f'^+(u_iu_j)$ is also an AP-set with the same common difference $d$. Therefore, $f'$ is an isoarithmetic IASI of $T(G)$. This completes the proof.
Biarithmetic IASI of Associated Graphs
======================================
In this section, we discuss the admissibility of biarithmetic IASIs by the associated graphs of a given biarithmetic IASI graph.
\[T-CUIASI2\] [@GS8] A biarithmetic IASI of a graph $G$ is an $l$-uniform IASI if and only if $G$ has $p$ bipartite components, where $p$ is the number of distinct pair $(m_i,n_j)$ of positive integers such that $m_i$ and $n_j$ are the set-indexing numbers of adjacent vertices in $G$ and $l=m_i+n_j-1$.
What are the characteristics of the line graph of a biarithmetic graph? The following results provide a solution to the problem.
Let $G$ be a biarithmetic IASI graph. Then, its line graph $L(G)$ admits an isoarithmetic IASI if and only if $G$ is bipartite.
Let $G$ be a bipartite graph which admits a biarithmetic IASI, with the bipartition $(X,Y)$. Since $G$ admits a biarithmetic IASI, there exists an integer $k>1$ such that the vertices of $X$ are labeled by distinct AP-sets of non-negative integers with common difference $d$ and the vertices of $Y$ are labeled by distinct AP-sets of non-negative integers with common difference $kd$. Then, the set-label of every edge of $G$ is also an AP-set with the common difference $d$. Therefore, the set-labels of all vertices in $L(G)$ are AP-sets with the same common difference $d$. Hence, every edge of $L(G)$ also has a set-label which is an AP-sets with the same common difference $d$. That is, $L(G)$ admits an isoarithmetic IASI.
Conversely, let $L(G)$ is an isoarithmetic IASI graph. Hence, every element of $L(G)$ must be labeled by an AP-set with common difference $d$. Therefore, the all the edges in $G$ must have set labels which are AP-sets with the same common difference $d$. Since, $G$ admits a biarithmetic IASI, the set-label of one end vertex of every edge must be an AP-set with common difference $d$ and the set-label of the other end vertex is an AP-set with the common difference $kd$. Let $X$ be the set of all vertices of $G$ which are labeled by the AP-sets with common difference $d$ and $Y$ be the set of all vertices of $G$ labeled by the AP-sets with common difference $kd$. Since $k>1$, no two vertices in $X$ can be adjacent to each other and no two vertices in $Y$ can be adjacent to each other. Therefore, $(X,Y)$ is a bipartition of $G$. Hence, $G$ is bipartite. This completes the proof.
\[T-kAIASI1\] If the line graph $L(G)$ of a biarithmetic IASI graph $G$ admits a biarithmetic IASI, then $G$ is acyclic.
Assume that $L(G)$ is a biarithmetic IASI graph. If possible, let $G$ contains a cycle $C_n=v_1v_2v_3\ldots v_nv_1$. Let $e_i=v_iv_{i+1}, 1\le i \le n$ and let $u_i$ be the vertex in $L(G)$ corresponding to the edge $e_i$ in $G$. Label each vertex $v_i$ of $G$ by the set whose elements are arithmetic progression with common difference $d_i$ where $d_{i+1}=k.d_i;~ k\ge |f(v_i)|_{min}$. Without loss of generality, let $f(v_1)$ has the minimum cardinality. Since $L(G)$ admits a biarithmetic IASI, adjacent vertices $u_i$ and $u_{i+1}$ in $L(G)$ are labeled by the sets whose elements are in arithmetic progressions whose common differences are $d_i$ and $d_{i+1}=k.d_i$ respectively. Therefore, the corresponding edges $e_i$ and $e_{i+1}$ of $G$ must also have the same set-labeling. Hence, alternate vertices of $G$ can not have the set-labels with the same common difference. Then, $d_i=k^i.d_1, 1<k\le |f(v_1)|$. Here, we notice that the set-label of one end vertex $v_n$ of the edge $v_nv_1$ in the cycle $C_n$ has the common difference $k^n.d_1$ and the set-label of other end vertex $v_1$ has the common difference $d_1$, which is a contradiction to the fact that $G$ is biarithmetic IASI graph. Therefore, G is acyclic.
[The converse of the theorem need not be true. For example, the graph $K_{1,3}$ admits a biarithmetic IASI and is acyclic, but its line graph does not admit a biarithmetic IASI.]{}
What is the condition required for an acyclic graph to admit a biarithmetic IASI? The following theorem establishes the necessary and sufficient condition for a biarithmetic IASI graph to have its line graph, a biarithmetic IASI graph.
\[T-kAIASI2\] The line graph of a biarithmetic IASI graph admits a biarithmetic IASI if and only if $G$ is a path.
The necessary part of the theorem follows from Theorem \[T-kAIASI1\]. Conversely, assume that $G$ is a path. Let $G=v_1v_2v_3\ldots v_n$. Label the vertex $v_i$ by an AP-set with the common difference $d_i$, where $k\le |f(v_i)|_{min}$. Without loss of generality, let $f(v_1)$ has the minimum cardinality. Then, $d_i=k^i.d_1, 1<k\le |f(v_1)|$. Then, the set-label of each edge $e_i$ of $G$ is an AP-set with difference $d_i=k.d_{i-1}$. Hence, the each vertex $u_i$ in $L(G)$ corresponding to the edge $e_i$ has the set-label which is an AP-set with the common difference $d_i=k.d_{i-1}=k^{i-1}.d_1$. Hence, $L(G)$ admits a biarithmetic IASI. This completes the proof.
In the above theorems, the value of $k$ should be within $1$ and $|f(v_i)|min$, the minimum among the set-indexing numbers of the vertices of $G$. A question that arises in this context is about the validity of these results if $k>|f(v_i)|min$. The following result answers this question.
Let $G$ admits a biarithmetic IASI $f$. Let $k=\frac{f(v_i)}{f(v_j)}$ for any two adjacent vertices of $G$. If $k> \min (|f(v_i)|)$, then the line graph of $G$ does not admit an arithmetic IASI.
Let $G$ admits a biarithmetic IASI $f$ and let $V(G)=\{v_1,v_2,v_3,\ldots,\\ v_n\}$ be the vertex set of $G$. If possible, let $k>\min (|f(v_i)|)$. Then, the set-label of the edge $v_iv_{i+1}$ will not be an AP-set. That is, $f$ is a semi-arithmetic IASI. Therefore, the set-label of the vertex $u_l$ of its line graph $L(G)$ corresponding to the edge $v_iv_{i+1}$ in $G$ is not an AP-set. Hence, for $k>|f(v_i)|_{min};~ v_i\in V(G)$, the line graph $L(G)$ of a biarithmetic IASI graph does not admit an arithmetic IASI.
The total graph of an identical biarithmetic IASI graph is an arithmetic IASI graph.
The vertices of $T(G)$ corresponding to the vertices of $G$ have the same set-labels and the edges in $T(G)$ connecting these vertices also preserve the same set-labels of the corresponding edges of $G$. The vertices of $T(G)$ corresponding to the edges of $G$ are given the same set-labels of the corresponding set-labels of the edges of $G$. Hence, all these vertices in $T(G)$ have the same deterministic index, say $d$, and hence the edges in $T(G)$ connecting these vertices also have the same deterministic index $d$. As the deterministic index of an edge and one of its end vertex are the same and the deterministic index of the other end vertex is a positive integral multiple of the deterministic index of the edge, where this integer is less than or equal to the cardinality of the set-label of the other end vertex, the edges corresponding to the incidence relations in $G$ also have the deterministic index $d$. Hence, $T(G)$ admits an arithmetic IASI.
The total graph of a biarithmetic IASI graph is an arithmetic IASI graph.
The vertices of $T(G)$ corresponding to the vertices of $G$ have the same set-labels and the vertices of $T(G)$ corresponding to the edges of $G$ are given the same set-labels of the corresponding set-labels of the edges of $G$. Also, the deterministic index of an edge and one of its end vertex are the same and the deterministic index of the other end vertex is a positive integral multiple of the deterministic index of the other end vertex, where this integer is less than or equal to the cardinality of the set-label of the other end vertex. Hence, for every two adjacent vertices in $T(G)$, the deterministic index of one is a positive integral multiple of of the deterministic index of the other, where this integer is less than or equal to the set-indexing number of the latter. Therefore, by Theorem \[T-AIASI-g\], $T(G)$ is an arithmetic IASI graph.
The following theorem checks whether the total graph corresponding to a biarithmetic IASI graph $G$ admits a biarithmetic IASI.
The total graph of a biarithmetic IASI graph is not a biarithmetic IASI graph.
We observe that every edge in $G$ corresponds to a triangle $K_3$ in its total graph. Since $K_3$ can not admit a biarithmetic IASI, $T(G)$ is not a biarithmetic IASI graph.
If $G$ is an identical biarithmetic IASI graph, will $G\circ e, ~e\in E(G)$ be an identical biarithmetic IASI graph? The cycle $C_4$ is a identical biarithmetic graph, but for any edge $e$ of $C_4$, $C_4\circ e=C_3$, which does not admit an identical biarithmetic IASI. Hence, we observe
A graph obtained from an identical biarithmetic IASI graph by contracting an edge of it, need not a biarithmetic IASI graph.
We also prove a similar for the graphs obtained from a biarithmetic IASI graph by a finite number of topological reductions.
Let $H$ be a graph obtained by finite number of topological reduction on a biarithmetic IASI graph $G$. Then, $H$ is not a biarithmetic IASI graph.
Let $v$ be a vertex of $G$ with degree $2$. Without loss of generality, let the set-label of $v$ be an AP-set with difference $d$. Let $u$ and $w$ be the adjacent vertices of $v$ which are not adjacent to each other. Since $G$ is a biarithmetic graph, both $u$ and $w$ must be labeled by distinct AP-sets with difference $k.d$. Now delete the vertex $v$ and join $u$ and $w$. Let $H=(G-\{v\})\cup \{uw\}$. Then, both the end vertices of the edge $vw$ has the set labels which are AP-sets of the same difference $k.d$. Hence, $H$ does not admit a biarithmetic IASI.
The graph subdivision $G^{\ast}$ of a given biarithmetic IASI graph $G$ does not admit a biarithmetic IASI.
Let $u$ and $v$ be two adjacent vertices in $G$ whose set-labels are AP-sets with common differences $d$ and $k.d$ respectively. Since $G$ admits a biarithmetic IASI, the set-label of the edge $uv$ is AP-set with different difference $d$. If we introduce a new vertex $w$ to the edge $uv$ and extend the set-labeling of $G$ by labeling the vertex $w$ by the same set-label of the edge $uv$, then, the set-labels of both $u$ and $w$ (or $v$ and $w$) are AP-sets with the same difference $d$. Hence, $G^{\ast}$ does not admit a biarithmetic IASI.
Further Points of Discussions
=============================
In this section we make some remarks on semi-arithmetic IASI graphs and their associated graphs. We observe that if the set labels of all vertices of $G$ are AP-sets with distinct differences, then the set-labels of edges will not be AP-sets. Hence, We have the following observations.
The line graph $L(G)$ of a semi-arithmetic IASI graph $G$ does not admit an arithmetic IASI (or a semi-arithmetic IASI).
The Total graph $T(G)$ of a semi-arithmetic IASI graph $G$ does not admit an arithmetic IASI (or a semi-arithmetic IASI).
From the fact that a graph $G$, its subdivision graph, the graph obtained by contracting an edge and the graph obtained by elementary topological reductions have some common edges, we observe the following results.
The graph $G\circ e$, obtained by contracting an edge $e$ of a semi-arithmetic IASI graph $G$,does not admit an arithmetic IASI (or a semi-arithmetic IASI).
The subdivision graph $G^{\ast}$ of a semi-arithmetic IASI graph does not admit an arithmetic IASI (or a semi-arithmetic IASI).
The graph $G'$, obtained by applying elementary topological reduction on a semi-arithmetic IASI graph $G$, does not admit an arithmetic IASI (or a semi-arithmetic IASI).
Conclusion
==========
In this paper, we have discussed some characteristics of certain graphs associated a given graph which admits certain types of arithmetic IASI. We have formulated some conditions for those graph classes to admit arithmetic IASIs. Here, we have discussed about isoarithmetic IASI graphs and biarithmetic IASI graphs only. The existence of similar results for semi-arithmetic IASI graphs are yet to be studied.
The IASIs under which the vertices of a given graph are labeled by different standard sequences of non negative integers, are also worth studying. The problems of establishing the necessary and sufficient conditions for various graphs and graph classes to have certain IASIs still remain unsettled. All these facts highlight a wide scope for further studies in this area.
[15]{} M Behzad, (1969). [*The Connectivity of Total Graphs*]{}, Bull. Austral. Math.Soc.,[**1**]{}, 175-181. J A Bondy and U S R Murty, (2008). [**Graph Theory**]{}, Springer. N Deo, (1974). [**Graph Theory with Applications to Engineering and Computer Science**]{}, PHI Learning. J A Gallian, (2011). [*A Dynamic Survey of Graph Labelling*]{}, The Electronic Journal of Combinatorics (DS 16). K A Germina and T M K Anandavally, (2012). [*Integer Additive Set-Indexers of a Graph:Sum Square Graphs*]{}, Journal of Combinatorics, Information and System Sciences, [**37**]{}(2-4), 345-358. K A Germina, N K Sudev, (2013). [*On Weakly Uniform Integer Additive Set-Indexers of Graphs*]{}, Int. Math. Forum., [**8**]{}(37), 1827-1834. K A Germina, N K Sudev, [*Some New Results on Strong Integer Additive Set-Indexers*]{}, Communicated. F Harary, (1969). [**Graph Theory**]{}, Addison-Wesley Publishing Company Inc. K D Joshi, [**Applied Discrete Structures**]{}, New Age International, (2003). M B Nathanson (1996). [**Additive Number Theory, Inverse Problems and Geometry of Sumsets**]{}, Springer, New York. M B Nathanson (1996). [**Additive Number Theory: The Classical Bases**]{}, Springer-Verlag, New York. N K Sudev and K A Germina, (2014). [*On Integer Additive Set-Indexers of Graphs*]{}, Int. J. Math. Sci.& Engg. Applications, [**8**]{}(2), 11-22. N K Sudev and K A Germina, [*On Arithmetic Integer Additive Set-Indexers of Graphs*]{}, communicated. N K Sudev and K A Germina, [*On Certain Types of Arithmetic Integer Additive Set-Indexers of Graphs*]{}, communicated. N K Sudev and K A Germina, (2014). [*A Study on Semi-Arithmetic Integer Additive Set-Indexers of Graphs*]{}, Int. J Math. Sci. & Engg. Applns, [**8**]{}(3) 140-149 . R J Trudeau, (1993). [**Introduction to Graph Theory**]{}, Dover Pub., New York. D B West, (2001). [**Introduction to Graph Theory**]{}, Pearson Education Inc.
[^1]: Department of Mathematics, Vidya Academy of Science & Technology, Thalakkottukara, Thrissur - 680501, email: [*sudevnk@gmail.com*]{}
[^2]: Department of Mathematics, School of Mathematical & Physical Sciences, Central University of Kerala, Kasaragod, email:[*srgerminaka@gmail.com*]{}
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---
abstract: 'We present aperture synthesis observations in the 21 cm line of pointings centered on the Virgo Cluster region spirals NGC 4307, NGC 4356, NGC 4411B, and NGC 4492 using the Very Large Array (VLA) radiotelescope in its CS configuration. These galaxies were identified in a previous study of the three-dimensional distribution of [[H$\,$I]{}]{} emission in the Virgo region as objects with a substantial dearth of atomic gas and Tully-Fisher (TF) distance estimates that located them well outside the main body of the cluster. We have detected two other galaxies located in two of our fields and observed bands, the spiral NGC 4411A and the dwarf spiral VCC 740. We provide detailed information of the gas morphology and kinematics for all these galaxies. Our new data confirm the strong [[H$\,$I]{}]{} deficiency of all the main targets but NGC 4411B, which is found to have a fairly normal neutral gas content. The VLA observations have also been used to discuss the applicability of TF techniques to the five largest spirals we have observed. We conclude that none of them is actually suitable for a TF distance evaluation, whether due to the radical trimming of their neutral hydrogen disks (NGC 4307, NGC 4356, and NGC 4492) or to their nearly face-on orientation (NGC 4411A and B).'
author:
- 'M. Carmen Toribio and José M. Solanes'
title: '[[H$\,$I]{}]{} Distribution and Tully-Fisher Distances of Gas-Poor Spiral Galaxies in the Virgo Cluster Region'
---
Introduction
============
The regions around rich clusters are the most obvious sites to evidence the transformation of galaxy disks driven by the surrounding intracluster medium (ICM). The increased density of both hot gas and galaxies, as well as the high relative velocities of the latter, set the scene for dramatic effects on their fragile interstellar medium (ISM).
In the local universe, the nearby Virgo Cluster region is an ideal place to quantify these nurturing effects, because its proximity makes it possible to probe the gaseous disks with higher sensitivity and resolution than in any other cluster. Another characteristic that makes this galaxy system very appealing for studies of galaxy evolution is its relative dynamical youth: Virgo has a central region with several substructures in the process of merging, surrounded by suburbia dominated by late-type galaxies that might fall into the cluster during the next Hubble time. As noted by @Vol01 and @Sol01, environmental mechanisms such as ram-pressure stripping may see their effectiveness increased during the built-up of clusters.
There is a long list of studies covering a broad stretch of the electromagnetic spectrum that have investigated the impact of cluster residency on the late-type galaxy population. Plenty of them use data from the 21-cm emission line of the abundant, and easy to strip off, neutral hydrogen ([[H$\,$I]{}]{}) of the disks, as the most direct approach to measure the affectation of the ISM. These investigations generally agree in indicating that Virgo spirals, like those inhabiting other rich clusters, tend to have less neutral gas than their field counterparts, and also in finding evidence for a correlation of the [[H$\,$I]{}]{} deficiency with clustercentric distance, with [[H$\,$I]{}]{}-poor disks typically situated close to the cluster cores and galaxies removed from those regions showing normal gas contents [e.g. @HG86; @Cay94; @Sol01; @Gav05; @Chu08]. The lack of atomic gas, which usually affects the outer disks, is frequently accompanied by an even more severe truncation of the H$\alpha$ emission and the corresponding quenching of star formation beyond that truncation radius [e.g. @KK04; @CK08]. Recently, there are also evidences than it could be associated with H$_{\mathrm{2}}$ reduction too [@Fum09].
Virgo is also the first, and so far the only, cluster region for which the *spatial* distribution of the [[H$\,$I]{}]{} deficiency has been mapped [@Sol02 hereafter Sol02]. By using homogenized Tully-Fisher (TF) distance moduli and 21 cm data from single-dish observations for 161 galaxies, these authors confirmed that the neutral gas deficiency in the Virgo Cluster decreases with increasing 3D barycentric distance. This study, however, also revealed the presence of an unexpectedly large fraction of strongly [[H$\,$I]{}]{}-deficient spirals with TF radial distances pointing to a location well outside the cluster body.
Ensuing studies based on both analytic infall models [@San02] and $N$-body simulations [@Mam04] investigated the possibility that some of the gas-poor spirals in Virgo’s suburbia had lost their gas content in a previous passage through the cluster core and were now lying near the apocenter of their orbits. Both works lead to the identification by @San04 of 13 extremely [[H$\,$I]{}]{}-poor spiral galaxies for which the lack of cold neutral gas could hardly be attributed to ISM-ICM stripping, unless their radial distances were affected by relative errors much larger than the typical uncertainty attributed to TF measurements. Other possibilities for the origin of these [[H$\,$I]{}]{} outliers, such as gas deficiency caused by gravitational interactions (tides or mergers) with companion galaxies, or errors in the [[H$\,$I]{}]{} deficiency estimates arising from morphological misclassifications, were also investigated and considered less probable.
With the aim of shedding more light on this matter, we initiated some time ago a program of dedicated observations of some of the outlying [[H$\,$I]{}]{}-deficient Virgo Cluster spirals found in [@San04]. In this paper, we attempt to improve the results of the aforementioned study, which were based on the analysis of integral galaxy properties retrieved from public databases, by investigating the neutral gas distribution and kinematics, as well as the TF distances, for 4 of these objects by means of deep 21 cm synthesis observations carried out with the Very Large Array (VLA) in its CS configuration[^1]. The paper begins by describing in Section \[selection\] the selection of the targets. The acquisition and reduction of the 21-cm line data, as well as the steps followed in the derivation of the [[H$\,$I]{}]{} synthesis results are discussed in Section \[processing\]. In Section \[analysis\], we analyze case by case the [[H$\,$I]{}]{} properties of all the galaxies showing 21 cm emission in the selected fields of view, while in Section \[tf\] we discuss the applicability of the TF technique to the five large spirals that have been observed. Finally, the results and conclusions of this work are given in Section \[conclusions\].
Galaxy Selection {#selection}
================
We have used the VLA to observe Virgo Cluster galaxies that are faint in the 21 cm line. Three of the targets, NGC 4307, NGC 4356, and NGC 4492, belong to the subset of 13 [[H$\,$I]{}]{}-outliers identified by @San04[^2]. These are galaxies with neutral gas deficiencies deviating by more than $3{\sigma_\mathrm{DEF}}$ from normalcy. A fourth pointing has been centered on NGC 4411B, another spiral with a less extreme [[H$\,$I]{}]{} deficiency. All these objects are among the most gas-poor spiral galaxies lying on the sky between the M49 subcluster and the W’/W cloud region (see Fig. \[location\]) and have TF estimates of their radial distances suggestive of a possible background location far from the cluster core, provided one adopts the currently preferred Virgo mean distance of $d_{\mathrm{Virgo}}\sim 16-17$ Mpc, as suggested by both the measurements of $H_0$ from HST observations of Cepheids and the spatial distribution of the early type galaxy population and X-ray gas [e.g. @Gav99; @Fre02; @San04; @Mei07]. Another characteristic these galaxies have in common is that their systemic velocities do not differ much from the mean cluster velocity. Galaxy properties are compiled in Table \[toribio\_tab1\], including a preliminary estimate of their neutral gas deficiency using the following distance-independent calibrator $$\label{def}
{\mbox{DEF}}=\langle\log{\overline{\Sigma}{_\mathrm{H\mbox{\tiny I}}}}(T)\rangle-\log{\overline{\Sigma}{_\mathrm{H\mbox{\tiny I}}}}\;,$$ which compares the logarithms of the expected and observed values of the *hybrid* [[H$\,$I]{}]{} surface density calculated from the ratio between the intrinsic integrated [[H$\,$I]{}]{} flux and the square of the apparent major optical diameter of a galaxy of morphological type $T$. We have followed @Sol02 and adopted for $\langle\log{\overline{\Sigma}{_\mathrm{H\mbox{\tiny I}}}}(T)\rangle$ the values: 0.24 for Sa, Sab types; 0.38 for Sb; 0.40 for Sbc; 0.34 for Sc; and 0.42 for later types in units of Jy[ km s$^{-1}$]{} per arcmin square. Thus, taking into account that the rms scatter in ${\mbox{DEF}}$ for field galaxies is ${\sigma_\mathrm{DEF}}=0.24$, an object with ${\mbox{DEF}}> 3{\sigma_\mathrm{DEF}}$ has less than $20\%$ of the expected [[H$\,$I]{}]{} mass for a galaxy of its morphology.
We have also included in Table \[toribio\_tab1\] four more galaxies located in some of the fields of view of our main targets. These are: VCC 740, a small spiral in the vicinity of NGC 4356; the first component of the pair NGC 4411A/B, very close to its companion galaxy in both projected position and radial velocity (their centers are separated only by $4\arcmin$ and 11[ km s$^{-1}$]{}, respectively), but whose estimated TF radial distance of $\sim 16$ Mpc (@Sol02) indicates that it is likely a Virgo Cluster member; and two dwarf ellipticals, VCC numbers 933 and 976, lying close to this pair on the sky.
Data Acquisition and Processing {#processing}
===============================
Observations {#acquisition}
------------
The observations published here consist of data obtained at the VLA in its C configuration between July and October 2005. The [[H$\,$I]{}]{} spectral line was observed with the correlator in 4IF mode using on-line Hanning smoothing.
The observational strategy was designed to achieve the best velocity resolution given the bandwidth needed for each galaxy. For observations where the primary target was an edge-on galaxy (NGC 4307 and NGC 4356), we chose to overlap partially the two IFs, each one with a bandwidth of 1.526 MHz and a spectral resolution of 24.4 kHz ($\sim 5.2$[ km s$^{-1}$]{}), whereas for fields with main targets oriented face-on (NGC 4411B and NGC 4492), the two IFs were centered on the heliocentric velocity of the target, the first with a bandwidth of 1.562 MHz and a spectral resolution of 24.4 kHz, and the second using a wider bandwidth of 3.125 MHz, but a lower frequency resolution of 97 kHz ($\sim 20.8$[ km s$^{-1}$]{}). The goal was to search for 21 cm line emission also from possible gaseous tidal tails, extraplanar gas or dwarf companions in the neighborhood of the target objects. The pointing of the field containing NGC 4492 was offset by $3\arcmin$ towards M49, due to its strong, extended radio continuum emission, in order to avoid systematic effects due to the VLA beam squint as well as to pointing uncertainties in individual VLA antennas [@Bhat08; @UC08].
The July and August observations started in the afternoon. An incidence in the electric system of a substation on August 12th led to the partial loss of the observing time initially allocated for NGC 4492, which was compensated by 3 hours of diurnal observation on September 18th. The NGC 4411B field was also observed during the daytime on the 1st and 2nd of October (5 and 4 hrs, respectively). Solar interference was therefore significant only for the September and October observations.
Each galaxy was observed for about 6 hours with an overhead of $\sim
2$ hours for calibration which included two 10-minute scans on each day on the primary calibrator 3C286$\ =\ $1331$+$305 (J2000). The rest of the observing sequences consisted of 30-minute scans of the target fields interspersed with 10-minute observations of the corresponding secondary calibrator. For the data acquired between July and September, 3C273$\ =\ $1229$+$020 (J2000), with a flux of $\sim 32$ Jy, was used as a secondary phase and bandpass calibrator. This calibrator was too close to the Sun during our October observations of the NGC 4411B field. We therefore modified our strategy and observed J1254$+$116 as a secondary calibrator. A summary of the observing parameters is provided in Table \[toribio\_tab2\].
Data Calibration {#calibration}
----------------
The raw UV data were reduced using the Astronomical Image Processing System (AIPS) software package distributed by the National Radio Astronomy Observatory.[^3]
The observations of NGC 4307, NGC 4356, and NGC 4492 were calibrated in a similar way. First, we discarded corrupted data by inspecting a pseudo-continuum database obtained from the vector average of visibilities for channels 4 to 60 at each time stamp —the remaining channels in the low and high velocity ends of the bandpass were discarded from the beginning. The primary flux calibrator 3C286 was used then to determine an initial bandpass as well as zeroth-order amplitude and phase calibration. Next, pseudo-continuum images of 3C273 were obtained and subsequently used to calculate a secondary phase calibration. The same source was the basis to determine a secondary bandpass calibration. This calibration was applied to the data on the observing fields by linear interpolation of bandpass and global phases. Finally, phase self-calibration of the data on the galaxies was applied, and for NGC 4307 and NGC 4356, the two IFs were joined by means of the task UJOIN [channel-by-channel visibility averages for the overlapping channels; see @MU08]. Regarding the NGC 4492 field, we note that during the make-up observation scheduled on September 18th we acquired 3 hours of data that suffered from solar interference. This forced us to reject baselines shorter than 1 k$\lambda$ for that day.
The reduction process just described was unsuitable to the observations of the NGC 4411B field. After discarding corrupted data, we calculated amplitude and phase calibrations from the observations of 3C286 and 1254$+$116. A solution for the shape of the bandpass obtained using 3C286 was applied to the whole data at the same time that the amplitude and phase solution, which was interpolated for NGC 4411B dataset by means of simple linear connection between phases. As for the NGC 4492 September data, we discarded spacings less than 1 k$\lambda$ to avoid the strong contamination by solar RFI.
Continuum Emission Subtraction {#subsec:subtraction}
------------------------------
Continuum emission in our fields was estimated by imaging the vector average of visibilities for line-free channels and subsequently subtracted from the datacube with UVSUB.
We note that comparison of our flux measurements of the brightest sources in each field with the corresponding flux values listed in the NRAO VLA Sky Survey [NVSS, @Con98] shows good agreement, although, on average, our flux measurements are $\sim 5\pm 2\%$ larger than the NVSS values. Given the low signal-to-noise ratio (S/N) of the spectral signal of our targets, this does not affect our total [[H$\,$I]{}]{}flux estimates. These discrepancies, however, were accounted for in the calculation of the corresponding uncertainties.
As mentioned on § \[calibration\], for targets close to the Sun, the images were obtained after excluding the baselines most affected by solar contamination. However, we could not eliminate solar RFI completely, which prompted us to make a second continuum subtraction. The routine UVLIN was used to fit and subtract a first order polynomial to the real and imaginary components of each visibility through the line-free channels. Following the recommendation by @CUH92, we applied UVLIN also to the data not affected by solar contamination in order to subtract any residual continuum emission. Subsequently, we obtained the channel images and examined the statistics of the image cubes to check for artifacts and, in particular, to verify that the distribution of noise in our data cubes was Gaussian-like. Finally, we proceeded to concatenate the datasets for those objects with observations split in two different dates (NGC 4492 and NGC 4411B).
[[H$\,$I]{}]{} Synthesis Results {#results}
--------------------------------
### Channel Maps {#channels}
Image cubes were constructed for the NGC 4307, NGC 4356, and NGC 4411B fields using robustness parameters $\Re=-1$ (closer to uniform weighting) and $\Re=0.7$ (closer to natural weighting). For all galaxies but NGC 4492, we present the results for $\Re=0.7$ as it provides the best compromise between the S/N and resolution when data have full UV-coverage [@Bri95].
The low S/N of the data on NGC 4492 forced us to smooth and taper the observations to improve our sensitivity. Channel maps were obtained from IF 2 (spectral resolution $\sim 20.8$[ km s$^{-1}$]{}) by using a robustness parameter $\Re=1$, which results in lower noise levels and a wider, but still acceptable, synthesized beam size. A Gaussian taper (with a 9 k$\lambda$-width at the $30\%$ level both in the U and V directions) additional weighting was applied on the visibilities to lower the contribution of the long-baseline datapoints.
In all cases, the channel images were CLEANed and the clean components restored with a Gaussian beam similar to the synthesized beam. The characteristics of the deconvolved images are summarized in Table \[toribio\_tab3\].
### Moments
The cleaned image cubes were used to calculate total [[H$\,$I]{}]{} flux images, as well as first and second velocity moments of the 21 cm emission.
We decided whether to keep or not a pixel in the integration by examining a spatially and frequency smoothed version of the datacubes. The spatial smoothing was done by convolving with a Gaussian kernel, whereas a Hanning smoothing was applied in velocity. We selected those pixels from the original datacubes that were above the $3\sigma$ level in the smoothed counterpart, except for NGC 4492, for which the $2\sigma$ level was used (otherwise, almost no signal was left from the galaxy). All flux with absolute value above $0.5\sigma$ was integrated along the velocity axis to obtain the total [[H$\,$I]{}]{} map, and intensity-weighted first and second order moments were computed. Finally, the [[H$\,$I]{}]{} intensity map was corrected for primary beam attenuation and scaled to the column density ${N{_\mathrm{H\mbox{\tiny I}}}}$ of the gas, assuming optically thin [[H$\,$I]{}]{}. When estimating the errors in the latter map, we scaled the rms noise in the channel maps to the square-root of the mean number of adjacent channels that contributed to the total intensity image, and then added a $5\%$ independent uncertainty arising from calibration and correction for primary beam attenuation.
The diameter of the [[H$\,$I]{}]{} disks was measured at a column density of $10^{20}$ atoms cm$^{-2}$ (no correction for beam smearing was applied). The reported uncertainties take into account variations in position angle of major axis as well as the correlation introduced by the synthesized beam (Table \[toribio\_tab4\]).
### Global [[H$\,$I]{}]{} Profiles {#globalprofiles}
In order to determine if we have recovered all of the flux from single-dish measurements, simulated line profiles were derived. The latter were calculated by integrating over the spatial axes for each channel the primary-beam attenuation-corrected emission used in the moment calculation (§ \[moments\]).
We have measured ${W_\mathrm{20}}$ and ${W_\mathrm{50}}$, the profile width at the $20\%$ and $50\%$ of the peak intensity, respectively. For those cases in which a clear double-peaked profile is found (VCC 740, NGC 4411A and NGC 4411B), the peak fluxes on both sides were considered separately when calculating the linewidths. In the remaining cases we used the overall peak flux. The [[H$\,$I]{}]{} linewidths were subsequently corrected for broadening effects due to the finite spectral resolution of the instrument by following the considerations of @VS01. The adopted broadening corrections for the cubes with a spectral resolution of $5.2$[ km s$^{-1}$]{} were $\delta{W_\mathrm{20}}= 0.86$[ km s$^{-1}$]{} and $\delta{W_\mathrm{50}}=0.56$[ km s$^{-1}$]{}, whereas when the resolution was 20.8[ km s$^{-1}$]{}, we adopted $\delta{W_\mathrm{20}}=12.0$[ km s$^{-1}$]{} and $\delta{W_\mathrm{50}}=7.88$[ km s$^{-1}$]{}.
The heliocentric systemic velocity, $v_\mathrm{sys}$, was derived as the average of velocities of channels at 20 and 50% of the peak flux of the profile, and the total [[H$\,$I]{}]{} flux was obtained by integrating the 21 cm line profiles along the velocity axis.
Since some of the profiles have low S/N, we decided to measure the kinematic parameters defined above by running Monte Carlo simulations for each one of the profiles. The values quoted in Table \[toribio\_tab4\] correspond to the mean and 1-$\sigma$ deviation of the distribution of measurements from one thousand random realizations of the profiles by taking into account the flux in each channel and its rms error. The latter was estimated from the rms noise in the channel maps and the correlation introduced by the synthesized beam. The quoted error in the total [[H$\,$I]{}]{} flux has been estimated by adding in quadrature the error estimates from the Monte Carlo technique and a $5\%$ uncertainty arising from the calibration and the correction for primary beam attenuation.
We find, in general, good agreement with single-dish observations (Table \[toribio\_tab4\]). Especially remarkable is the close match between the shapes, linewidths, and total flux densities of our VLA [[H$\,$I]{}]{} line profiles and their counterparts in the ongoing Arecibo Legacy Fast ALFA (ALFALFA) extragalactic [[H$\,$I]{}]{} survey [@Gio05], which recently released the results for the strip of the Virgo Cluster region where our targets are located [@Ken08].
### Position-Velocity Diagrams and Rotation Curves {#rotcurves}
Position-velocity (PV) diagrams along the major axis of each target were also obtained by taking slices of the data cubes through the optical centers and estimating their position angle from the moment maps or from the rotation curves in those cases where this was possible.
Major-axis velocities as a function of angular radius were exclusively derived for the two components of the pair NGC 4411A/B and the dwarf spiral VCC 740, since these were the only targets detected with a high S/N (see next section). To infer the rotation curves we have fitted both a Brandt model and the standard iterative tilted-ring algorithm [@Beg89].
Case by Case Analysis of the [[H$\,$I]{}]{} Synthesis Data {#analysis}
==========================================================
Table \[toribio\_tab4\] summarizes the results inferred from our analysis of the VLA observations for all the galaxies detected. In the printed version of the manuscript we include a complete graphical layout of the NGC 4307 field (Figs. \[channels\_n4307\]–\[synthesis\_n4307\]). Images for the rest of our VLA pointings are available in the electronic version of the manuscript.
We now comment on the properties of the target galaxies from our VLA observations.
NGC 4307
--------
The [[H$\,$I]{}]{} gas disk of NGC 4307 has very small dimensions and is found only deep within the optical disk, in agreement with the strong gas-deficiency estimated for this galaxy. The deficiency parameter ${\mbox{DEF}}$ (eq. \[\[def\]\]) measured from our VLA flux (see Table \[toribio\_tab4\]) shows a pretty good consistency with the value of 1.41 one can infer from the observed [[H$\,$I]{}]{} flux provided by [@Ken08]. We remark that this new value is also in agreement with the extremely [[H$\,$I]{}]{}-deficient status reported in previous works (e.g.@Sol02; @Gav05).
The $0$th-order moment map shows that the maximum of the 21 cm emission is displaced SE with respect to the optical center, in a direction nearly perpendicular to the major axis, with an excess of emission at the approaching side. This asymmetry shows up also in the global [[H$\,$I]{}]{} profile, which has a peaked appearance indicative of centrally concentrated gas. The estimated offset of $\sim 7\arcsec$ is less than half of both the synthesized beam size and the mismatch expected to arise from edge-on ram-pressure stripping in hydrodynamic simulations [@Kro08].
There are no other pieces of evidence susceptible to being interpreted as environmental effects down to the sensitivity limit of the measurements. Moreover, the galaxy is too inclined to allow one to disentangle whether the observed asymmetries can be ascribed to noncircular motions associated with a spiral arm.
The PV diagram along the major axis is still rising at the edge of the measured [[H$\,$I]{}]{} distribution, as if the gas was rotating almost as a pure solid body. Our VLA data is compatible to a large extent with the H$\alpha$ rotation curve derived by @Gav99, which shows a hint of a turn over on both sides of the galaxy. We discuss in Section \[tf\] the risks of using extremely [[H$\,$I]{}]{}-deficient galaxies like this one in a TF analysis.
NGC 4356 and VCC 740
--------------------
The properties of the neutral hydrogen in NGC 4356 resemble those of NGC 4307. This is a nearly edge-on galaxy that like the former one, and even more strongly so, has a very small gas distribution compared with its optical dimensions. The PV diagram along the major axis also rises steeply and shows no signs of turning over, as in the H$\alpha$ rotation curve measured by @Gav99. There is similarly a misalignment between the distribution and motion of the atomic gas and the optical disk. As in the former case, this offset is less than half the synthesized beam size and, hence, relatively small and not necessarily indicative of ram-pressure pushing.
For this galaxy, our VLA linewidth and flux values are somewhat smaller that the most recent estimates by @Ken08. Given the low S/N of this galaxy, the observed difference in total fluxes, which is only of tenths of a mJy[ km s$^{-1}$]{}, does not necessarily imply that some flux has been lost due to the missing short baselines in our synthesis aperture images. Instead, the discrepancies can be the result of low-baseline ripples that seem to affect the single-dish profile.
VCC 740, another highly inclined galaxy detected in this same pointing, has, in contrast, quite a strong S/N and a total [[H$\,$I]{}]{} flux that is about the same that in NGC 4307. Its [[H$\,$I]{}]{} map suggests that the neutral gas has a sharp cutoff at the optical radius of the disk on the approaching side, while the other side has a more gradual falloff and is somewhat more extended. In spite of giving the impression that part of the gas might be missing, both ours and the ALFALFA flux measurements actually result in a negative deficiency parameter ${\mbox{DEF}}$ of $\sim -0.2$, indicative of a perfectly normal [[H$\,$I]{}]{}content.
The gas velocity field of VCC 740, on the other hand, exhibits some warping, as well as inner contours parallel to the minor axis, indicating that $V_{\mathrm{rot}}(R)\propto R$, as the PV diagram shows. Further out on the SE side, the contours show the classical $V$-shape, suggesting that at the sensitivity limit of the measurements the rotation speed is just about to become nearly constant. In the rotation curve model fits, the warp of the velocity field increases the position angle on the external part of the disk. Both Brandt and tilted-ring models yield an estimated inclination of $\sim 70 \degr$ for the internal disk region that drops to $\sim 45 \degr$ on the outside. The fact that the velocity field of this galaxy shows a clear rotation pattern reinforces its morphological classification as a dwarf barred disk instead of the IB type assigned in the LEDA. The angular resolution of our VLA measurements, however, is insufficient to observe the effects of the bar on the gas velocities. No signs of interaction are found between VCC 740 and NGC 4356.
NGC 4411A/B, VCC 933 and VCC 976 {#NGC 4411A/B}
--------------------------------
Contrarily to what has happened with NGC 4307 and NGC 4356, the two low surface brightness spirals NGC 4411A and B have produced integrated [[H$\,$I]{}]{} fluxes that are inconsistent with the values quoted in @Sol02, which were estimated in earliest mapping attempts done with the Arecibo antenna [@Hay81]. Discrepancies with this and other old single-dish measurements [e.g. @Hof89] are ascribed, however, solely to the amplitudes, as the shapes of the global [[H$\,$I]{}]{} profiles look very similar.
Our new data, reinforced by the newest single-dish measurements done at Arecibo by the ALFALFA team [@Ken08], imply that NGC 4411A and NGC 4411B loose their initial status of objects with moderate and strong [[H$\,$I]{}]{} deficiency, respectively (see Table \[toribio\_tab1\]), to be both reclassified as galaxies with quite a normal gas abundance, in accordance with the visual impression obtained from the [[H$\,$I]{}]{}contours overlaid on the Digitized Sky Survey (DSS) image.
These maps show that these two galaxies have [[H$\,$I]{}]{} distributions extending beyond their optical disks, which for NGC 4411A reach up to nearly twice the optical radius, except on the NE side where the [[H$\,$I]{}]{}contours appear compressed. In this latter galaxy the [[H$\,$I]{}]{} is concentrated in a ring with two important regions of emission that emanate perpendicularly from the ends of the bar. The $0$th-order map of NGC 4411B shows an even wider major ring-like structure with a noticeable excess of emission at its northern half near the outer edge. We do not find important displacements from the optical disks, the most remarkable feature being the presence of a depression in the center of both galaxies, as found in other LSB galaxies [@dBMvH96].
We fail in detecting neither gas bridges nor significant intergalactic [[H$\,$I]{}]{} signals between these two targets. Given that both galaxies show normal disk emission with only mild alterations of the symmetry, comparable to those seen in more isolated objects [@Kor00], we feel compelled to classify the system NGC 4411A/B as a visual pair (see also § \[tf\]).
Regarding the kinematics of the neutral gas, we note that the PV diagram of NGC 4411B shows a steep rise well within the stellar disk (of small amplitude given its near face-on orientation) followed by a sharp bend towards a flat part, —with indications of a modest decline on the receding side at the largest radii where the [[H$\,$I]{}]{} is detected—, consistent with the shape of the radial velocity contours and with the well behaved double-horned global profile. In NGC 4411A, the turnover in the rotation velocity is not complete and the 21 cm line profile is affected by a larger asymmetry. The observed behavior of the PV diagrams is typical of objects with a compact distribution of their luminous matter.
For these two galaxies we have gone a step further and inferred also major-axis rotation velocities with the aim of estimating the orientations of the gaseous disks. The apparent inclination inferred from the tilted-ring technique for NGC 4411A is $29^{+5.2}_{-3.7}$ degrees, while for NGC 4411B we get $i=26^{+4.4}_{-4.7}$ degrees. Brandt curve fits to the whole velocity fields result in similar inclinations of $\sim 27\degr$ and $\sim 28\degr$, respectively. Yet it shouldn’t be forgotten that the bumps in the [[H$\,$I]{}]{}line data associated with spiral arms and, especially, the near face-on orientation of these disks, make rotation curve model fits to the velocity fields uncertain and, caveat lector, unable of giving a precise inclination angle. In the next section, we discuss the problems arising from the derivation of this parameter in low-inclination galaxies and provide alternative estimates based on optical images.
Our VLA observations have not detected 21 cm line emission coming from any of the two dwarf elliptical galaxies VCC 933 and VCC 976 also present in this field.
NGC 4492
--------
In spite of the strong image-degrading effects of the Sun for the second half of the observations of NGC 4492, we succeed in detecting its [[H$\,$I]{}]{} signal at a quality level comparable with the previous single-dish observations carried out by @HG86 and @Hof89 —and not too different from the one achieved by the ALFALFA measurements, which have a S/N of 4.6.
In this galaxy, the detected neutral hydrogen is located within the optical radius and shows an important elongation in the SE-NW direction, almost perpendicularly to the galaxy major axis. The [[H$\,$I]{}]{}distribution is strongly asymmetric, with the peak of the [[H$\,$I]{}]{}emission shifted some $30\arcsec$ to the East from the optical center of the galaxy. This gives rise to a synthesized line profile with decreasing flux toward the approaching side. By integrating the latter, we find a total [[H$\,$I]{}]{} flux about a $30\%$ smaller than the ALFALFA’s value (see Table \[toribio\_tab4\]). Again, one may wonder whether possible flux losses in our estimation arising from too a strict rejection of the short baselines could explain this difference. In this respect, we note that during the application of the Gaussian UV-taper weighting to obtain the final datacubes some testing done varying the width of the tapering function showed that the resulting total [[H$\,$I]{}]{} flux was not significantly affected. Therefore, given the low S/R of the detections, the discrepancies in total flux and linewidths with respect to singledish measurements could have been originated by low-level baseline ripples that seem to affect the ALFALFA profile. With this caveat in mind, our VLA measurements assign a new [[H$\,$I]{}]{} deficiency parameter of 1.21 to NGC 4492, which indicates that the gas content for this galaxy could be less than $6\%$ of the expectation value.
The asymmetries on the spatial distribution of the gas for this galaxy are reproduced in the [[H$\,$I]{}]{} dynamics. Thus, the velocity map of NGC 4492 reveals a possible displacement of the dynamical center from the light center (consistent with that observed in the $0$th-order map), while on the PV diagram along the major axis most of the emission is found coming from the receding side. The low S/N of the datacubes, the small size of the [[H$\,$I]{}]{} disk with respect to the beam size, as well as a not too favorable orientation of this galaxy, prevent any attempt to fit a rotation curve model to the velocity data.
TF-based Distance Estimates to our Galaxies {#tf}
===========================================
The most striking aspect of the Virgo’s galaxies selected in the present study is not their high deficiency of neutral gas, but the possibility that this has been attained outside the cluster environment. While in the outskirts of clusters gas stripping can happen by galaxy-galaxy interactions in infalling groups or through collisions with lumps of intergalactic gas [see, for instance, @CK06; @CK08], strong [[H$\,$I]{}]{} deficiencies in the periphery of clusters are expected to be an exception to the rule.
The classification by @San04 of a spiral galaxy as an [[H$\,$I]{}]{}outlier relied on its TF distance. The latter was inferred from the disk’s maximum rotation speed $V_{\mathrm{rot}}$, —which is expected to be a proxy to the total mass of the galaxy and, therefore, also to its intrinsic luminosity—, measured via the width of the [[H$\,$I]{}]{}spectral line. It is then of fundamental importance to assess the feasibility of this technique in galaxies like ours which have gaseous disks deeply altered.
From @Guh88 to @CKH08 the literature is full of TF studies highlighting the risks of using the gas kinematics in the determination of distances to [[H$\,$I]{}]{}-deficient galaxies. The most straight reasoning being that truncated gas disk measurements could underestimate the rotation velocity of a galaxy and, therefore, its mass, biasing low the derived radial distance. However, depending on the interaction mechanism and its geometry, the [[H$\,$I]{}]{} that does not get dislocated may also lost temporarily its equilibrium within the global galactic potential —externally induced disturbances on the kinematics of disks are erased in about 1 Gyr [e.g. @Dal01], a time during which two interacting galaxies can move hundreds of kiloparsecs apart. These effects, as well as induced noncircular or nonplanar gas motions, which may even lead to an overestimation of the true $V_{\mathrm{rot}}$ [@Kro07], possible changes in the observed luminosity resulting from alterations in the star formation rate or, simply, the fact that even for undisturbed galaxies in many cases there is evidence of noncircular motions in their central regions [for instance, due to bars; @Val07], could make the application of the TF technique in strong [[H$\,$I]{}]{}-deficient galaxies totally inefficient.
All this drives us to regard critically the distance estimates of the three galaxies in our sample that exhibit severely truncated gas disks: NGC 4307, NGC 4356, and NGC 4492. Certainly, we have not found evidence of a recent gravitational encounter in our 21 cm line imaging data in the form of gaseous tails and bridges for any of them. Nor the closeness of the systemic velocity of the first two galaxies to the mean cluster velocity supports a recent ram-pressure event inside the core. Yet the fact we do not detect a flat part in their rotation curves, as well as the irregularities in the spatial distribution and kinematics of the [[H$\,$I]{}]{} evidenced in the VLA maps of these objects do not allow us to state confidently that the dynamical equilibrium of the neutral gas has been fully restored after the removal event. Neither can we assert that their luminosities have not been affected. Therefore, we believe that it is not legitimate to use the TF technique to derive the radial distance to any of these three galaxies and that, consequently, their published TF distance measurements might well be largely in error.
Compared to the previous objects, NGC 4411B and its close neighbor NGC 4411A exhibit regular and symmetric gas velocity fields, with flat extended outer parts that appear to satisfy the basic tenet that underlies a TF study. In this case, however, the attempts of estimating the radial distance to these two galaxies are thwarted for an unfavorable viewing angle. At low apparent inclinations ($i\lesssim
40$–$45\degr$), estimates of the orientation of disks become more uncertain, led to deprojected quantities with divergent errors as a face-on orientation is approached, and are skewed towards larger values by nonaxysimmetric features in the images [see @AB03 and references therein]. As a result, the total fractional error in the radial distance for low-inclination galaxies largely exceeds $15\%$, a value usually adopted as representative of the typical uncertainty in the distance arising from good-quality TF data.
This situation is illustrated in Figure \[tf\_distances\], where we show the radial distances to NGC 4411A and B reported in some of the TF catalogs used by @Sol02. Our synthesized [[H$\,$I]{}]{} line profiles have been used to estimate the intrinsic rotational velocities, which we have calculated by exactly following the same prescriptions adopted in the referenced studies that in some cases apply non-null turbulent motion corrections. The uncertainty resulting from inclination measures is shown by a vertical bar whose extent is set by the most extreme values of this angle ever assigned to our galaxies in the literature (which range from $\sim 20\degr$ to $\sim
55\degr$). Open diamonds in the plots indicate radial distances published in the cited references. It is obvious from this figure that the inability to infer accurate inclinations prevents us from establishing the location of both galaxies along the LOS *within the entire Virgo Cluster region*.
We have also depicted in Figure \[tf\_distances\] the [[H$\,$I]{}]{}-based radial distances inferred from our own measurements of the inclination of these galaxies (Sec. \[NGC 4411A/B\]). Double pointed arrows encompass the ranges of distances corresponding to the values of inclinations and errors derived from inspection of the residuals of our tilted-ring model fit to the [[H$\,$I]{}]{} velocity fields. The well-known limitations of the weighting scheme included in the modeling of the [[H$\,$I]{}]{} rotation curves when it comes to measuring inclinations below $40\degr$ [@AB03], have led us to make as well independent fits to the orientation parameters of these disks from Sloan Digital Sky Survey (SDSS) images. We have used the package GALFIT [@Pen02] to decompose the r-band images of both galaxies into several components. The sky contribution apart, we have fitted a bulge plus an exponential disk to the image of NGC 4411B, while for NGC 4411A a bar component has been added too. We have followed @BdV81 and adopted intrinsic axis ratios of 0.18 for NGC 4411A (SBc) and of 0.13 for NGC 4411B (Scd). All this gives estimated inclinations of about $24\degr$ for the stellar disk of NGC 4411A and of $\sim 16\degr$ for the one of NGC 4411B. The uncertainties in these values are difficult to determine, as the analysis of the optical images involves a large number of parameters. Resulting distances are indicated in the plots by asterisks. Comparison with the results from the [[H$\,$I]{}]{} shows that for nearly face-on disks in the Virgo Cluster region inclinations differing by $\sim 10\degr$ can lead, depending on the TF relationship adopted, to differences in distance exceeding 10 Mpc. The very uncertain radial distances and the lack of unmistakable signs of an ongoing interaction do not make it possible to assert that these galaxies are physically connected in spite of their proximity in $z$-space.
Another feature of Figure \[tf\_distances\] that draws one’s attention is the existence of considerable author to author fluctuations in the estimated TF radial distances that cannot be just ascribed to the uncertainties in the observational parameters entering this relationship. For highly inclined galaxies, such discrepancies can be traced back mostly to the adopted TF template, which for a given passband can show systematic variations among different authors, even when similar samples of calibrators are used. We note, for instance, that small datasets are rather sensitive to the trimming of the data and, hence, to the, somewhat subjective, identification of the most deviant measurements. Besides, differences on the fitting methods, or on the adopted data weighting, as well as morphological and incompleteness biases [see, for instance, @Gio97], may give raise to significant variations in the slope and zero-point coefficients of the TF relation. As shown in Figure \[tf\_templates\], where we compare the $B$-band TF template relations defined in the studies of the Virgo Cluster considered by @Sol02, the absolute magnitude assigned to a given galaxy, regardless of its inclination, can vary up to $\sim 1.5$ mag depending on the calibration adopted. This exceeds by far the typical error of $\sim 0.1$–0.2 mag usually assigned to the zero-point calibration of individual TF templates.
Summary and Conclusions {#conclusions}
=======================
This is the first 21 cm synthesis survey of spiral galaxies ever made in which the targets have been specifically chosen on the basis of their expected dearth of cold gas. It has been motivated by the detection, in previous investigations of the neutral gas content on spirals in the Virgo Cluster region, of a significant number of severely [[H$\,$I]{}]{}-deficient disks supposedly located, according to their TF distance estimates, beyond the maximum rebound radius galaxies can bounce after infall. According to this location, these galaxies could not owe their [[H$\,$I]{}]{} deficiency to interactions within the cluster environment.
Our high-sensitivity VLA observations have been aimed at characterizing in detail the spatial distribution and kinematics of the neutral gas in four galaxies suspected of being [[H$\,$I]{}]{}-outliers, in a first effort to gain a better understanding of the origin of this class of objects. At the same time, these synthesis observations in the 21 cm line have provided direct evidence of the risks involved in the application of the TF relationship to disturbed or nearly face-on disks, which can render the derived distances unreliable.
We have detected a total of six galaxies within the four fields initially selected. The main conclusions of the analysis of our VLA data are as follows:
1\) We confirm the strong [[H$\,$I]{}]{} deficiency of three of our four main targets, NGC 4307, NGC 4356, and NGC 4492 (inferred [[H$\,$I]{}]{} contents are a factor $\gtrsim 20$ lower than their corresponding standard values), which is pronounced through the reduced extension of the gaseous disks, a characteristic typical of ram-pressure-stripped galaxies. In contrast, we find that the integrated [[H$\,$I]{}]{} fluxes of our fourth target, NGC 4411B, and its companion, NGC 4411A, are $\sim 2$–$3$ times larger that the old single-dish values used to estimate their [[H$\,$I]{}]{} deficiency in @Sol02. Our measured VLA fluxes, —which for all our targets are compatible with those inferred from the new, sensitive ALFALFA extragalactic [[H$\,$I]{}]{} survey—, indicate that the [[H$\,$I]{}]{} contents of these last two galaxies deviate less than $1\sigma$ from normalcy. This is consistent with our observation that their [[H$\,$I]{}]{} disks extend beyond their optical counterparts, so that only the outermost portions of the cold gas distributions have been affected, if at all. A sixth galaxy with a healthy amount of cold gas, the dwarf spiral VCC 740, has been detected in the field of NGC 4356.
2\) Visual inspection of the images of the most gas-deficient galaxies has revealed signs of asymmetries and lopsidedness, as well as small offsets of the dynamical centers with respect to the optical ones. These are all suggestive, albeit not conclusive, indications of possible gravitational interactions and/or ram-pressure effects, as deviations from flat, axisymmetric disks are also known to prosper in isolated galaxies. This, and the fact that we have not found evidence of gaseous tails or bridges within the limit we have been able to trace the [[H$\,$I]{}]{} ($\sim 3$–$5\times 10^{19}\ \mathrm{cm}^{-2}\
\mathrm{channel}^{-1}$ at the $3\sigma$ level) appear to indicate that none of the galaxies investigated has undergone *recent* gravitational interactions. This means, in particular, that our VLA observations reinforce the classification of NGC 4411A/B as a virtual pair in spite of their closeness on the observational space phase.
3\) Our three targets with highly truncated gas disks exhibit rotation velocities that are still rising at the last measured points. Moreover, the observational evidence gathered do not allow us to assert with complete confidence that the gas remaining tied to the disks has regain dynamical equilibrium, nor the extent to which the luminosity of these [[H$\,$I]{}]{}-deficient galaxies could have been affected. The classification of these objects as [[H$\,$I]{}]{}-outliers could therefore simply obey to the inefficient estimate of their radial distances by means of the TF relationship. The fourth target, NGC 4411B, as well as it space phase neighbor NGC 4411A, show, in contrast, extended gas disks with regular and symmetric velocity fields. In spite of being galaxies presumably in virial equilibrium, their TF-based distances are also problematic because of their nearly face-on apparent orientation, which results in the inability to determine accurate inclinations. This translates into a considerable uncertainty —larger than the resolution necessary to determine unambiguously the region (infall or cluster) where a galaxy belongs— when it comes to placing these two galaxies along the LOS within the Virgo Cluster region.
Aperture synthesis observations in the 21 cm line like the ones presented here are fundamental for probing the impact of cluster residency on the spiral population. Further insight into the identification of the physical processes disturbing the disks can be gained by supplementing this type of data with multifrequency observations.
This work was partly supported by the Dirección General de Investigación Científica y Técnica, under contracts AYA2006–01213 and AYA2007-60366. M.C.T. acknowledges support from a fellowship of the Ministerio de Educación y Ciencia of Spain. We are grateful to all the people and institutions that have made possible the LEDA (`http://leda.univ-lyon1.fr/`), GOLDMine (`http://goldmine.mib.infn.it/`), DSS (`http://archive.stsci.edu/cgi-bin/dss_form`), and SDSS (`http://cas.sdss.org`) databases.
[99]{}
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[^1]: The VLA is a facility of the National Radio Astronomy Observatory.
[^2]: Seven other members of this subset are among the targets of the VIVA (VLA Imaging of Virgo in Atomic gas) survey by J. Kenney, J. van Gorkom, and cols. in which the gas is imaged down to a column-density sensitivity of a few times $10^{19}$ cm$^{-2}$, similar to that of the present observations.
[^3]: Unless otherwise stated, all the quoted routines in capital letters belong to this package.
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abstract: 'Closed analytical expressions for scattering intensity and other global structure factors are derived for a new solvable model of polydisperse sticky hard spheres. The starting point is the exact solution of the “mean spherical approximation” for hard core plus Yukawa potentials, in the limit of infinite amplitude and vanishing range of the attractive tail, with their product remaining constant. The choice of factorizable coupling (stickiness) parameters in the Yukawa term yields a simpler “dyadic structure” in the Fourier transform of the Baxter factor correlation function $q_{ij}(r)$, with a remarkable simplification in all structure functions with respect to previous works. The effect of size and stickiness polydispersity is analyzed and numerical results are presented for two particular versions of the model: i) when all polydisperse particles have a single, size-independent, stickiness parameter, and ii) when the stickiness parameters are proportional to the diameters. The existence of two different regimes for the average structure factor, respectively above and below a generalized Boyle temperature which depends on size polydispersity, is recognized and discussed. Because of its analycity and simplicity, the model may be useful in the interpretation of small-angle scattering experimental data for polydisperse colloidal fluids of neutral particles with surface adhesion.'
address: |
Istituto Nazionale di Fisica della Materia and\
Facoltà di Scienze, Università di Venezia,\
S. Marta DD 2137, I-30123 Venezia, Italy
author:
- Domenico Gazzillo and Achille Giacometti
title: Structure factors for the simplest solvable model of polydisperse colloidal fluids with surface adhesion
---
INTRODUCTION
============
A theoretical determination of scattering intensity and structure factors for fluids with a large number $p$ of components is, in general, a difficult task. In particular, this is true when some kind of polydispersity is present, as occurs in colloidal or micellar solutions [@Pusey91; @Lowen94; @Nagele96]. Size polidispersity means that macroparticles of a same chemical species exhibit several different dimensions within a discrete or continuous set of possible values. Interaction polydispersity then denotes a similar, and usually correlated, dispersion of parameters (charges, etc.) defining the strength of interaction potentials. Even when all macroparticles belong to a unique chemical species, a polydisperse fluid must therefore be treated as a multi-component mixture, with very large $p$ values (of order $10^1\div 10^3$) or, in the infinite-component limit $%
p\rightarrow \infty ,$ with an idealized continuous distribution of some properties [@Stell79].
In structural studies on polydisperse colloidal fluids, a key role can be played by the models for which the Ornstein-Zernike (OZ) integral equations of the liquid state [@Hansen86] admit analytical solutions leading to closed expressions of scattering functions for any finite $p$ and even for $%
p\rightarrow \infty .$ A sufficient condition for this is that the Fourier transforms $\widehat{q}_{ij}\left( k\right) $ of the functions $q_{ij}(r)$, solutions of the Baxter factorized version of the OZ equations [@Baxter70; @Hiroike79], have a peculiar mathematical form, which we refer to as [*dyadic*]{} structure [@Gazzillo97] and will be illustrated in Section II C. Using the dyadicity, the explicit inversion of a related $%
p\times p$ matrix $\widehat{{\bf Q}}\left( k\right) $ is always possible for arbitrary $p$ and closed analytical expressions for the “partial” structure factors $S_{ij}(k)$ [@Ashcroft67] can be obtained. The scattering intensity and other “global” structure factors are then calculated as weighted sums of all partial structure factors.
Usually, the above sums are performed numerically by evaluating $p\left(
p+1\right) /2$ independent contributions $S_{ij}(k)$ at each $k$ [@Stell79; @Griffith86; @Senatore85; @Ginoza99]. This procedure becomes numerically demanding for large $p$, as required in polydisperse mixtures. On the other hand, we stress that the dyadicity property also enables an alternative route (followed in the present work) which avoids the explicit computation of individual $S_{ij}(k)$. The weighted sums can, in fact, be worked out analytically, by a procedure originally proposed by Vrij [@Vrij79] and referred to as “Vrij’s summation” hereafter. The resulting closed analytical expressions of “global” scattering functions hold true for [*any*]{} number $p$ of components, can be easily applied to polydisperse fluids even in the limit $p\rightarrow \infty ,$ and are particularly suitable to fit experimental scattering data. Vrij [@Vrij79] first obtained a closed expression for the scattering intensity of polydisperse hard spheres (HS) within the Percus-Yevick (PY) approximation. Gazzillo [*et al.*]{} [@Gazzillo97] derived similar formulas for polydisperse charged hard spheres (CHS), by using the corresponding analytical solution within the “mean spherical approximation” (MSA).
In the present paper we extend the approach previously exploited for polydisperse HS [@Vrij79] and CHS [@Gazzillo97] to polydisperse “sticky” hard spheres (SHS). This simple model adds to an interparticle hard core repulsion an infinitely strong attraction at contact, and can be applied to real colloidal fluids of neutral spherical particles with a van der Waals (or dispersion) force of attraction, working at very short distances.
Baxter [@Baxter68] proposed the one-component original version of this model, solved the OZ equation with the PY closure and found that such a system presents a liquid-gas phase transition. The adhesive contribution in Baxter’s Hamiltonian is defined by a particular limiting case of a square-well tail in which the depth goes to infinity as the width goes to zero, in such a way that the contribution to the second virial coefficient remains finite but not zero (Baxter limit). Stell [@Stell91] found that SHS of equal diameter in the Baxter limit, when treated exactly rather than in the PY approximation, are not thermodynamically stable. Nevertheless, the PY solution for SHS as a useful colloidal model has received a continuously growing interest in the last two decades [@Stell91; @Regnaut89], partially because of its capability to exhibit a gas-liquid phase transition.
Perram and Smith [@Perram75] and Barboy and Tenne [@Barboy79] extended Baxter’s work to $p$-component fluids, using the same kind of Hamiltonian and the PY approximation. Santos [*et al.*]{} [@Santos98] developed a rational function approximation to go beyond the PY approximation and derived improved expressions for the radial distribution functions and structure factors of SHS mixtures.
Unfortunately, the PY analytical solution for mixtures requires the determination of a set of unknown [*density-dependent*]{} parameters $%
\lambda _{ij}$, related, through $p(p+1)/2$ coupled quadratic equations, to other parameters $\tau _{ij}$ which appear in the potentials as monotonically increasing functions of the temperature $T$ and whose inverses measure the degree of adhesion (“stickiness”) of interacting spheres. In most cases the coefficients $\lambda _{ij}$ for given $\tau _{ij}$ can only be found numerically, and this feature limits the applicability of the SHS-PY model to small $p$ values. As a matter of fact the number of actual applications to polydisperse fluids is very limited. We are aware of a study by Robertus [*et al.* ]{}[@Robertus89] on small angle x-ray scattering from microemulsions, with polydispersity represented by $p=9$ components, a work by Penders and Vrij [@Penders90] on turbidity of silica particles, and an investigation by Duits [*et al.* ]{}[@Duits91] on small angle neutron scattering from sterically stabilized silica particles dispersed in benzene. To simplify the numerical determination of the set $\left\{ \lambda
_{ij}\right\} $, all these papers treat the special case of a single stickiness parameter, $\tau $, independent of particle size.
In general, the SHS-PY solution for mixtures does not have the dyadic structure which allows the [*analytic*]{} inversion of $\widehat{{\bf Q}}%
\left( k\right) $ required to get closed expressions for structure factors of polydisperse systems$.$ To recover the dyadicity and obtain an explicitly solvable model, Herrera and Blum [@Herrera91] used the [*ad hoc* ]{}assumption $\lambda _{ij}=\lambda _i\lambda _j$, in a study on polydisperse CHS with sticky interactions under a MSA/PY closure.
On the other hand, apart from Baxter’s original definition [@Baxter68], there exists a second version of the SHS model, proposed in the one-component case by Brey [*et al.*]{} [@Brey87] and Mier-y-Teran [*et al.*]{} [@Mier89]. Here, the adhesive part of the potential is defined as the limit of a Yukawa tail when both amplitude and inverse range tend to infinity, with their ratio remaining constant. The analytic solution was obtained within the MSA closure [@Mier89], and Ginoza and Yasutomi [@Ginoza96] discussed its relationship to Baxter’s PY solution. Recently, Tutschka and Kahl [@Tutschka98] investigated the multi-component version of this second SHS model and presented MSA expressions of structure functions for both the discrete (finite $p$) and the continuous ($%
p\rightarrow \infty $) polydisperse case. These authors pointed out that the dyadic structure of the SHS-MSA solution can be ensured [*a priori*]{} by imposing from the outset a Berthelot-type rule [@Hansen86] on the coupling (stickiness) parameters $\gamma _{ij\text{ }}$of the Yukawa tail, i.e. $\gamma _{ij\text{ }}=(\gamma _{ii\text{ }}\gamma _{jj\text{ }})^{1/2}.$
Within the same framework of Yukawa-MSA models, the present paper has a threefold aim: i) we shall show that a new choice of Yukawa coupling parameters, $Y_i$, slightly different from Tutschka and Kahl’s ones [@Tutschka98], can produce an even simpler solvable model of SHS, with a remarkable simplification of all analytical results; ii) we shall obtain closed analytical expressions for scattering intensity and other “global” structure factors of SHS by extending the formalism successfully employed for HS and CHS; iii) we shall present numerical applications not only in the case of equal stickiness for all particles, but also when a simple size-dependence of the stickiness parameters $Y_i$ is assumed.
The interplay among stickiness attraction, size polydispersity, and hard core repulsion gives rise to a rather complex behaviour. Nonetheless we shall present a simple unified description of these results, hinging on the introduction of a generalized Boyle temperature.
The paper is organized as follows. In the next section the basic theory on structure factors, integral equations and dyadic matrices will be briefly recalled. The SHS model, its MSA solution under the assumption of factorizable coefficients, expressions for scattering intensity and other “global” structure factors will be given in Sec. III. Numerical results are included in Sec. IV, while the last section is devoted to a summary and some conclusive remarks.
BASIC FORMALISM
===============
Structure factors and scattering intensity
------------------------------------------
All scattering functions of multicomponent or polydisperse fluids with spherically symmetric interparticle potentials can be expressed in terms of “partial” structure factors $S_{ij}(k)$, such as the Ashcroft-Langreth ones [@Ashcroft67]
$$S_{ij}\left( k\right) =\delta _{ij}+H_{ij}\left( k\right) =\delta
_{ij}+\left( \rho _i\rho _j\right) ^{1/2}\widetilde{h}_{ij}\left( k\right) .$$
Here, $k$ is the magnitude of the scattering vector, $\delta _{ij}$ the Kronecker delta$,$ $\rho _i$ the number density of species $i,$ $%
\widetilde{h}_{ij}\left( k\right) $ the three-dimensional Fourier transform of the total correlation function, $h_{ij}\left( r\right) =g_{ij}\left(
r\right) -1,$ with $g_{ij}\left( r\right) $ being the radial distribution function between two particles of species $i$ and $j$ at distance $r$.
The knowledge of the $S_{ij}\left( k\right) $ allows to calculate the scattering intensity as well as some “global” structure factors. The [*coherent scattering intensity* ]{}$I(k)$ for a $p$-component fluid is given by [@Pusey91; @Vrij79]
$$R(k)\equiv I\left( k\right) /V=\rho \sum_{i,j=1}^p\left( x_ix_j\right)
^{1/2}F_i\left( k\right) F_j^{*}\left( k\right) S_{ij}\left( k\right) ,$$
where $V$ is the volume, $\rho =\sum_m\rho _m$ the total number density, while $x_i=\rho _i/\rho $ and $F_i(k)$ denote the molar fraction and form factor of species $i,$ respectively (the asterisk means complex conjugation). The [*measurable average structure factor* ]{}is then defined from the Rayleigh ratio $R(k)$ as [@Pusey91; @Vrij79]
$$S_{{\rm M}}\left( k\right) =R(k)\ /\left[ \ \rho P(k)\right] . \label{s3}$$
with $P(k)=\sum_{m=1}^px_m\left| F_m\left( k\right) \right| ^2.$ As a third useful quantity, we consider the Bathia-Tornton [*number-number structure factor*]{} [@Bhatia70], which is related to number density fluctuations:
$$S_{{\rm NN}}\left( k\right) =\sum_{i,j=1}^p\left( x_ix_j\right)
^{1/2}S_{ij}\left( k\right) .$$
The definition of other global structure factors may be found in Ref. 3. Clearly, $R(k)$, $S_{{\rm M}}\left( k\right) $ and $S_{{\rm NN}%
}\left( k\right) $ involve a unique kind of weighted sum, i.e.,
$$\sum_{i,j=1}^pw_i\left( k\right) w_j^{*}\left( k\right) S_{ij}\left(
k\right) ,$$
with $w_i\left( k\right) $ being equal to $\rho _i^{1/2}F_i\left(
k\right) ,\left[ x_i/P(k)\right] ^{1/2}F_i\left( k\right) ,$ and $%
x_i{}^{1/2},$ respectively.
Integral equations in Baxter form
---------------------------------
Integral equations of the liquid state theory represent a powerful theoretical tool to get the $h_{ij}\left( r\right) $ required to calculate the partial structure factors $S_{ij}\left( k\right) $. The Ornstein-Zernike (OZ) integral equations relate the $h_{ij}\left( r\right) $ functions to the simpler direct correlation functions $c_{ij}\left( r\right) .$ For fluids with spherically symmetric interactions, these equations are [@Hansen86; @Lee88]
$$h_{ij}\left( r\right) =c_{ij}\left( r\right) +\sum_{m=1}^p\rho _m\int d{\bf r%
}^{\prime }\ c_{im}\left( r^{\prime }\right) h_{mj}\left( |{\bf r-r}^{\prime
}|\right)$$
and can be solved only when coupled with an approximate second relationship (a “closure”) among $c_{ij}\left( r\right) $, $h_{ij}\left(
r\right) $ and interparticle potential $u_{ij}\left( r\right) $ [@Hansen86; @Lee88].
By Fourier transformation, the OZ convolution equations become, in $k$-space,
$$\left[ {\bf I}+{\bf H}\left( k\right) \right] \left[ {\bf I}-{\bf C}\left(
k\right) \right] ={\bf I,}$$
where $C_{ij}\left( k\right) \equiv \left( \rho _i\rho _j\right)
^{1/2}\widetilde{c}_{ij}\left( k\right) $ and $\widetilde{c}_{ij}\left(
k\right) $ is the Fourier transform of $c_{ij}\left( r\right) .$ If ${\bf S}%
\left( k\right) $ denotes the symmetric matrix with elements $S_{ij}\left(
k\right) $, then we get
$${\bf S}\left( k\right) ={\bf I}+{\bf H}\left( k\right) =\left[ {\bf I}-{\bf C%
}\left( k\right) \right] ^{-1}, \label{i3}$$
with ${\bf I}$ being the unit matrix of order $p.$ The $%
S_{ij}\left( k\right) $ can therefore be expressed in terms not only of $%
\widetilde{h}_{ij}\left( k\right) $, but also of $\widetilde{c}_{ij}\left(
k\right) .$ However, in this paper we shall use a third representation of $%
S_{ij}\left( k\right) $ based upon the Baxter factor correlation functions $%
q_{ij}(r)$ [@Baxter70]. By means of a Wiener-Hopf factorization of ${\bf %
I}-{\bf C}\left( k\right) ,$ Baxter transformed the OZ equations for HS fluids into an equivalent, but easier to solve, form [@Baxter70]. Later on these equations were extended by Hiroike to any spherically symmetric potentials, without using the Wiener-Hopf factorization [@Hiroike79]. Baxter factorization reads
$${\bf I}-{\bf C}(k)=\widehat{{\bf Q}}^T\left( -k\right) \widehat{{\bf Q}}%
\left( k\right) , \label{i4}$$
where $\widehat{{\bf Q}}\left( k\right) $ has the form $$\widehat{{\bf Q}}\left( k\right) ={\bf I}-\widetilde{{\bf Q}}\left( k\right)
={\bf I}-\int_{-\infty }^{+\infty }dr\ e^{ikr}{\bf Q}\left( r\right) ,$$
with $Q_{ij}(r)=2\pi \left( \rho _i\rho _j\right) ^{1/2}q_{ij}(r)$ ($\ \widehat{{\bf Q}}^T$ is the transpose of $\widehat{{\bf Q}}\ $). Note that $\widehat{Q}_{ij}\left( -k\right) =\widehat{Q}_{ij}^{*}\left( k\right)
. $ For fluids of particles with spherically symmetric interactions including HS repulsions (i.e., $u_{ij}\left( r\right) =+\infty $ when $%
r<\sigma _{ij}\equiv (\sigma _i+\sigma _j)/2,$ with $\sigma _i$ = hard sphere diameter of species $i$), the Baxter equations in $r$-space are
$$\left\{
\begin{array}{l}
rc_{ij}\left( |r|\right) =-q_{ij}^{\ \prime }(r)+2\pi \sum_m\rho
_m\int_{L_{mi}}^\infty dt\ q_{mi}\left( t\right) q_{mj}^{\ \prime }\left(
r+t\right) , \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \\
rh_{ij}\left( |r|\right) =-q_{ij}^{\ \prime }(r)+2\pi \sum_m\rho
_m\int_{L_{im}}^\infty dt\ q_{im}\left( t\right) \left( r-t\right) \\
\qquad \qquad \qquad \qquad \qquad \qquad \times h_{mj}\left( |r-t|\right) ,
\end{array}
\right.$$
where $r>L_{ij}\equiv (\sigma _i-\sigma _j)/2$ and the prime denotes differentiation with respect to $r$.
Using Eqs. ($\ref{i3}$) and ($\ref{i4}$), we get ${\bf S}(k)=\widehat{{\bf Q}%
}^{-1}\left( k\right) \left[ \widehat{{\bf Q}}^{-1}\left( -k\right) \right]
^T$ and the partial structure factors can be written as
$$S_{ij}(k)=\sum_m\widehat{Q}_{im}^{-1}\left( k\right) \widehat{Q}%
_{jm}^{-1}\left( -k\right) =\sum_m\widehat{Q}_{im}^{-1}\left( k\right)
\left[ \widehat{Q}_{jm}^{-1}\left( k\right) \right] ^{*}. \label{i7}$$
On defining
$$s_m(k)=\sum_{i=1}^pw_i(k)\widehat{Q}_{im}^{-1}\left( k\right) , \label{i8}$$
all the “global” structure functions can then be expressed as $$\sum_{i,j=1}^pw_i\left( k\right) w_j^{*}\left( k\right) S_{ij}\left(
k\right) =\sum_{m=1}^ps_m(k)s_m^{*}(k). \label{i9}$$
Dyadic matrices and Vrij’s summation
------------------------------------
The main problem of these analytical calculations hinges on the inversion of the matrix $\widehat{{\bf Q}}\left( k\right) ={\bf I}-\widetilde{{\bf Q}}%
\left( k\right) $, which usually becomes a formidable task with increasing the number $p$ of components. In a particular case, however, the inverse $%
\widehat{{\bf Q}}^{-1}\left( k\right) $ can be easily found for any size of the original matrix. This occurs when $\widehat{Q}_{ij}(k)$ is a [*dyadic*]{} (or Jacobi) matrix, i.e. when it has the peculiar mathematical structure $$\widehat{Q}_{ij}=\delta _{ij}+\sum_{\mu =1}^na_i^{(\mu )}b_j^{(\mu )}\qquad
(i,j=1,\ldots ,p) \label{d1}$$
(the dependence on $k$ was omitted for simplicity). We recall that a matrix $T_{ij}=a_ib_j$ formed by the direct product of two vectors is often referred to as a [*dyad*]{}, ${\bf {ab},}$ while a linear combination of dyads $\sum_\mu \lambda _\mu {\bf {a}^{\left( \mu \right) }b^{\left( \mu
\right) }}$ is called a [*dyadic*]{} [@Mathews65]. Moreover, we shall refer to a sum of $n$ dyads as a [*n-dyadic.*]{}
We caution the reader that, since $\widehat{Q}_{ij}(k)=\delta _{ij}-2\pi
\left( \rho _i\rho _j\right) ^{1/2}\widehat{q}_{ij}\left( k\right) ,$ where $%
\widehat{q}_{ij}\left( k\right) $ is the unidimensional Fourier transform of $q_{ij}\left( r\right) ,$ Eq. (\[d1\]) actually requires that $\widehat{q}%
_{ij}\left( k\right) $ is a $n$-dyadic matrix (of order $p$), but in the following we shall use the same terminology for $\widehat{Q}_{ij}(k)$ as well. The dyadicity is actually present in some solvable models of fluid mixtures: $\widehat{q}_{ij}\left( k\right) $ is $2$-dyadic in the PY solution for neutral HS[@Stell79; @Vrij79], and $3$-dyadic in the MSA solution for CHS [@Blum77].
The dyadic matrices have some special properties, which have already been partially discussed in Ref. 8. Here we recall the main points along with new additional features. Let us associate to the matrix of order $p$ of Eq. (\[d1\]), $\widehat{{\bf Q}}{\bf \ =I}+\sum_{\mu =1}^n{\bf a}^{(\mu )}{\bf b}%
^{(\mu )},$ a matrix ${\bf D}_{{\rm Q}}$ of order $n$ with elements $$d_{\alpha \beta }=\delta _{\alpha \beta }+{\bf a}^{(\alpha )}\cdot {\bf {b}}%
^{(\beta )}\qquad \left( \alpha ,\beta =1,\ldots ,n\right) ,$$ where the dot denotes the usual scalar product of vectors, i.e. $%
{\bf a}^{(\alpha )}\cdot {\bf {b}}^{(\beta )}{\bf =}\sum_{m=1}^pa_m^{(\alpha
)}b_m^{(\beta )}.$ A first property is that any $n$-dyadic matrix $\widehat{%
{\bf Q}}$ of order $p$ has always rank $n$, irrespective of $p.$ Moreover, its determinant $\left| \widehat{{\bf Q}}\right| ,$ which is of order $p,$ turns out to be equal to the determinant $D_{{\rm Q}}\equiv \left| {\bf D}_{%
{\rm Q}}\right| $ of order $n$ (with $n\ll p$ in multi-component fluids). A second property yields the explicit form of the elements of the inverse matrix $\widehat{{\bf Q}}^{-1},$ as
$$\widehat{Q}_{ij}^{-1}=\delta _{ij}-\frac 1{D_{{\rm Q}}}\sum_{\alpha
=1}^n\sum_{\beta =1}^na_i^{(\alpha )}b_j^{(\beta )}\left| {\bf D}_{{\rm Q}%
}\right| ^{\alpha \beta }, \label{d2}$$
where $\left| {\bf D}_{{\rm Q}}\right| ^{\alpha \beta }$ is the cofactor of the ($\alpha ,\beta $)th element in ${\bf D}_{{\rm Q}}$ and the double sum contains $n^2$ determinants of order $n$. Clearly, this expression could also be rewritten as a sum of only $n$ determinants, i.e., $%
\widehat{Q}_{ij}^{-1}=\delta _{ij}-\sum_{\alpha =1}^na_i^{(\alpha )}\widehat{%
D}_j^{(\alpha )}/D_{{\rm Q}},$ where $\widehat{D}_j^{(\alpha )}\equiv
\sum_{\beta =1}^nb_j^{(\beta )}\left| {\bf D}_{{\rm Q}}\right| ^{\alpha
\beta }$ is the determinant obtained from $D_{{\rm Q}}$ by replacing the $%
\alpha $-th [*row*]{} with $b_j^{(1)},...,b_j^{(n)}$ . Alternatively, one could write $\widehat{Q}_{ij}^{-1}=\delta _{ij}-\sum_{\beta =1}^nD_i^{(\beta
)}b_j^{(\beta )}/D_{{\rm Q}},$ with $D_i^{(\beta )}\equiv \sum_{\alpha
=1}^na_i^{(\alpha )}\left| {\bf D}_{{\rm Q}}\right| ^{\alpha \beta }$ being obtained from $D_{{\rm Q}}$ by replacing the $\beta $-th [*column*]{} with $%
a_i^{(1)},...,a_i^{(n)}$ (Cramer rule).
All above expressions reflect the remarkable fact that $\widehat{{\bf Q}}%
^{-1}$ is a $n$-dyadic matrix as well, and it is indeed this property enabling a successful outcome of Vrij’s summation for “global” structure functions. Our starting point is a reformulation of Eq. (\[d2\]) in terms of a determinant of order $n+1:$
$$\widehat{Q}_{ij}^{-1}=\frac 1{D_{{\rm Q}}}\left|
\begin{array}{lllll}
\delta _{ij}\quad & \ \qquad b_j^{(1)} & \qquad b_j^{(2)} & \cdots & \qquad
b_j^{(n)} \\
a_i^{(1)}\quad & ~1+{\bf a}^{(1)}\cdot {\bf {b}}^{(1)} & \qquad {\bf a}%
^{(1)}\cdot {\bf {b}}^{(2)} & \cdots & \qquad {\bf a}^{(1)}\cdot {\bf {b}}%
^{(n)} \\
a_i^{(2)} & \qquad {\bf a}^{(2)}\cdot {\bf {b}}^{(1)}{\bf \quad } & \ 1+{\bf %
a}^{(2)}\cdot {\bf {b}}^{(2)} & \cdots & \qquad {\bf a}^{(2)}\cdot {\bf {b}}%
^{(n)} \\
\ \vdots & \ \qquad \quad \vdots & \ \qquad \quad \vdots & ~\vdots & \
\qquad \quad \vdots \\
a_i^{(n)} & \qquad {\bf a}^{(n)}\cdot {\bf {b}}^{(1)} & \qquad {\bf a}%
^{(n)}\cdot {\bf {b}}^{(2)} & \cdots & 1+{\bf a}^{(n)}\cdot {\bf {b}}^{(n)}
\end{array}
\right| , \label{d3}$$
where ${\bf D}_{{\rm Q}}$ is included as a submatrix. From Eqs. (\[i8\]) and (\[d3\]), one then gets
$$s_m=\frac 1{D_{{\rm Q}}}\left|
\begin{array}{lllll}
\quad w_m & b_m^{(1)} & b_m^{(2)} & \cdots & b_m^{(n)} \\
{\bf w}\cdot {\bf {a}}^{(1)}\quad & d_{11} & d_{12} & \cdots & d_{1n} \\
{\bf w}\cdot {\bf {a}}^{(2)} & d_{21}{\bf \quad } & d_{22} & \cdots & d_{2n}
\\
\ \quad \vdots & ~\vdots & ~\vdots & ~\ \vdots & ~\vdots \\
{\bf w}\cdot {\bf {a}}^{(n)} & d_{n1} & d_{n2} & \ \cdots & d_{nn}
\end{array}
\right| . \label{d4}$$
To perform the sum over $m$ required in Eq. (\[i9\]), we expand this determinant along the first row to get
$$s_m=w_m+\sum_{\alpha =1}^nb_m^{(\alpha )}C_\alpha , \label{d5}$$
where $C_\alpha \equiv T_\alpha /D_{{\rm Q}}$ and $T_\alpha $ $%
\left( \alpha =0,1,\ldots ,n\right) $ is the cofactor of the element $\left(
1,\alpha +1\right) $th in the determinant of Eq. (\[d4\]). Clearly, $%
T_0=D_{{\rm Q}}$. Using Eqs. (\[i9\]) and (\[d5\]), the searched final result is
$$\sum_{i,j=1}^pw_iw_j^{*}S_{ij}={\bf w}\cdot {\bf {w}}^{*}+2\sum_{\alpha =1}^n%
%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
\ \left[ {\bf w}^{*}\cdot {\bf {b}}^{(\alpha )}C_\alpha \right]
+\sum_{\alpha =1}^n\sum_{\beta =1}^n{\bf b}^{(\alpha )}\cdot {\bf {b}}%
^{(\beta )*}\ C_\alpha C_\beta ^{*}, \label{d6}$$
where $%
%TCIMACRO{\func{Re}}
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
\left[ ...\right] $ denotes the real part of a complex number. Vrij [@Vrij79] first performed a similar computation and derived a closed expression for the scattering intensity of HS mixtures. Our expression generalizes Vrij’s one as well as that of Ref. 8 for the scattering intensity of CHS. It is simpler and more compact and can be used to calculate any “global” structure function, for mixtures with [*any*]{} number $p$ of components. It involves a sum of only $(n+1)^2$ terms, and depends on $p$ through some averages represented by scalar products of vectors. Only the number of terms contained in these averages increases with increasing $p,$ and hence the application to polydisperse mixtures is straightforward.
THE MODEL
=========
Sticky hard spheres as a limit of Yukawa particles
--------------------------------------------------
A fluid of SHS can be derived from particles interacting via HS plus Yukawa (HSY) attractive potentials, i.e.
$$-\beta u_{ij}(r)=\left\{
\begin{array}{lll}
+\infty , & & 0<r<\sigma _{ij} \\
\beta A_{ij}e^{-\mu (r-\sigma _{ij})}/r, & & r>\sigma _{ij}
\end{array}
,\right. \qquad \label{y1}$$
in the limit $\mu \rightarrow +\infty $ with $\beta A_{ij}=\mu
K_{ij}$ and $K_{ij}$ independent of $\mu $. Here, $\beta =(k_BT)^{-1},$ where $k_B$ is the Boltzmann constant, and all $A_{ij}$ are positive, with $%
A_{ji}=A_{ij}$ and $K_{ji}=K_{ij}$, as required by the symmetry condition $%
u_{ji}(r)=u_{ij}(r)$. This approach is convenient, since the Baxter equations for HSY mixtures have been solved analytically [@Blum78], for any finite $\mu ,$ within the MSA closure, which adds to the exact hard core condition, $h_{ij}\left( r\right) =-1$ for $r<\sigma _{ij},$ the approximate relationship $c_{ij}\left( r\right) =-\beta u_{ij}(r)$ for $r>\sigma _{ij}.$ The solution is
$$q_{ij}(r)=\left\{
\begin{array}{l}
0,\qquad \qquad r<L_{ij} \\
\frac 12a_i(r-\sigma _{ij})^2+(b_i+a_i\sigma _{ij})(r-\sigma
_{ij})+C_{ij}\left[ e^{-\mu (r-\sigma _{ij})}-1\right] \\
\qquad \qquad \qquad +D_{ij}e^{-\mu (r-\sigma _{ij})},\qquad L_{ij}<r<\sigma
_{ij} \\
D_{ij}e^{-\mu (r-\sigma _{ij})},\qquad r>\sigma _{ij}
\end{array}
\right.$$
where the coefficients are determined by a complicate set of equations [@Blum78]. From these, however, it can be shown that, as $\mu
\rightarrow +\infty ,$ then $C_{ij}\rightarrow -D_{ij}$ and $%
D_{ij}\rightarrow \beta A_{ij}/\mu =K_{ij}.$ The MSA solution for SHS is therefore
$$q_{ij}(r)=\left\{
\begin{array}{l}
\frac 12a_i(r-\sigma _{ij})^2+(b_i+a_i\sigma _{ij})(r-\sigma
_{ij})+K_{ij},\qquad L_{ij}\leq r\leq \sigma _{ij} \\
0,\qquad \qquad \qquad \qquad \text{elsewhere}
\end{array}
\right. \label{y3}$$
$$a_i=\frac 1\Delta +\frac{3\xi _2\sigma _i}{\Delta ^2}-\frac{12\zeta _i}\Delta
,\qquad b_i=\left( \frac 1\Delta -a_i\right) \frac{\sigma _i}2,$$
$$\xi _m=\frac \pi 6\sum_{i=1}^p\rho _i\sigma _i^m,\qquad \zeta _i=\frac \pi 6%
\sum_{j=1}^p\rho _j\sigma _jK_{ij}\ ,\qquad \Delta =1-\xi _3.$$
To calculate $\widehat{Q}_{ij}^{-1}(k),$ we need the unidimensional Fourier transform $\widehat{q}_{ij}\left( k\right) $, i.e.
$$\begin{aligned}
\widehat{q}_{ij}\left( k\right) &=&-e^{{\rm i}X_i}\ \left\{ \left( 1+\frac{%
3\xi _2\sigma _i}\Delta -12\zeta _i\right) \frac 1{4\Delta }\sigma _j^3\
\frac{j_1(X_j)}{X_j}\right. \label{y6} \\
&&\left. +\frac 1{4\Delta }\sigma _i\sigma _j^2\left[ \ j_0(X_j)-{\rm i}%
j_1(X_j)\right] -K_{ij}\sigma _j\ j_0(X_j)\right\} , \nonumber\end{aligned}$$
where $X_m\equiv k\sigma _m/2,$ $\ j_0(x)=\sin x/x$ and $\
j_1(x)=\left( \sin x-x\cos x\right) /x^2$ are spherical Bessel functions, and ${\rm i}$ - when it is not a subscript - is the imaginary unit.
Factorizable coefficients
-------------------------
In general $\widehat{q}_{ij}\left( k\right) ,$ as defined in Eq. (\[y6\]), does not have the required dyadic structure, due to the presence of $K_{ij}$ in the last term. To overcome this difficulty, Tutschka and Kahl [@Tutschka98] proposed the following [*Ansatz:*]{}
$$K_{ij}=\gamma _{ij}\sigma _{ij}^2,\qquad \text{with\qquad }\gamma _{ij\text{
}}=(\gamma _{ii\text{ }}\gamma _{jj\text{ }})^{1/2}\quad \text{%
(Berthelot-rule),} \label{f1}$$
which yields $K_{ij}=\gamma _{ii}^{1/2}\gamma _{jj}^{1/2}(\sigma
_i^2+2\sigma _i\sigma _j+\sigma _j^2)/4.$ Consequently, the last term of Eq. (\[y6\]) splits into three independent contributions, and $\widehat{q}%
_{ij}\left( k\right) $ turns out to be 5-dyadic, in spite of the fact that in the HS limit (no adhesion) it is only 2-dyadic.
We first note that a great simplification occurs with the factorization
$$K_{ij}=Y_iY_j \label{f2}$$
(all $Y_m\geq 0$). In this case, the last term of Eq. (\[y6\]) generates only one contribution, and $\widehat{q}_{ij}\left( k\right) $ becomes simply 3-dyadic and, in a particular case to be discussed later on, even 2-dyadic. The $K_{ij}$ defined by Eq. (\[f2\]) satisfy the Berthelot-rule, i.e. $K_{ij}=(K_{ii}K_{jj})^{1/2}.$ Note that the stickiness parameters $\gamma _{mm\text{ }}$are dimensionless [@Tutschka98], while the $Y_m$ are lengths.
Factorizable adhesive parameters have already been considered by Yasutomi and Ginoza [@Yasutomi96] and Herrera [*et al.* ]{}[@Herrera98] in studies on adhesive-HSY fluids, although no expressions for structure functions were given. Since in those papers $K_{ij}=GG_iG_j$, the relationship with our coefficients is simply given by $Y_m=G^{1/2}G_m.$
Using Eq. (\[f2\]), we get $\zeta _i=\xi _2^Y\ Y_i\ $ with $\xi _2^Y\equiv
\left( \pi /6\right) \sum_{j=1}^p\rho _j\sigma _jY_j$ (dimensionally analogous to $\xi _2$), and therefore
$$\begin{aligned}
\widehat{Q}_{ij}(k) &=&\delta _{ij}-2\pi \left( \rho _i\rho _j\right) ^{1/2}%
\widehat{q}_{ij}\left( k\right) \nonumber \\
&=&\delta _{ij}+\rho _i^{1/2}e^{{\rm i}X_i}\ \left\{ \left( 1+\frac{3\xi
_2\sigma _i}\Delta -{\rm i}\frac{k\sigma _i}2-12\xi _2^Y\ Y_i\ \right) \frac %
\pi {2\Delta }\sigma _j^3\ \frac{j_1(X_j)}{X_j}\right. \label{f3} \\
&&\left. +\frac \pi {2\Delta }\sigma _i\sigma _j^2\ j_0(X_j)-2\pi
Y_iY_j\sigma _j\ j_0(X_j)\right\} \rho _j^{1/2}, \nonumber\end{aligned}$$
which has the required dyadic structure $\widehat{Q}_{ij}=\delta
_{ij}+\sum_{\mu =1}^na_i^{(\mu )}b_j^{(\mu )}.$ We emphasize that the decomposition into $a_i^{(\mu )}$ and $b_j^{(\mu )}$ is not unique. Our choice allows an easy comparison with the corresponding results for polydisperse HS [@Vrij79] and CHS [@Gazzillo97]. After defining
$$\begin{aligned}
\alpha _m(k) &=&\frac \pi {2\Delta }\sigma _m^3\frac{j_1(X_m)}{X_m},
\nonumber \\
\beta _m^{(0)}(k) &=&\frac \pi {2\Delta }\sigma _m^2j_0(X_m),\qquad \beta
_m(k)=\beta _m^{(0)}(k)+\left( \frac{3\xi _2}\Delta -{\rm i}\frac k2\right)
\alpha _m(k), \label{f4} \\
\delta _m^{(0)}(k) &=&-2\pi \sigma _mY_m\ j_0(X_m),\qquad \delta
_m(k)=\delta _m^{(0)}(k)-12\xi _2^Y\ \alpha _m(k), \nonumber\end{aligned}$$
$\widehat{Q}_{ij}(k)$ may be rewritten as
$$\widehat{Q}_{ij}(k)=\delta _{ij}+\rho _i^{1/2}e^{{\rm i}X_i}\left[ \alpha
_j(k)+\sigma _i\beta _j(k)+Y_i\delta _j(k)\right] \rho _j^{1/2}, \label{f5}$$
and the corresponding decomposition is
$$\begin{array}{lll}
a_i^{(1)}=\rho _i^{1/2}e^{{\rm i}X_i}, & \qquad & b_j^{(1)}=\rho
_j^{1/2}\alpha _j, \\
a_i^{(2)}=\rho _i^{1/2}e^{{\rm i}X_i}\sigma _{i\ }, & \qquad &
b_j^{(2)}=\rho _j^{1/2}\beta _j, \\
a_i^{(3)}=\rho _i^{1/2}e^{{\rm i}X_i}Y_i\ , & \qquad & b_j^{(3)}=\rho
_j^{1/2}\delta _j.
\end{array}
\label{f6}$$
If we are interested in the scattering intensity, then Eq. (\[d4\]) with $%
w_m=\rho _m^{1/2}F_m$ yields
$$s_m=\frac{\rho _m^{1/2}}{D_{{\rm Q}}}\left|
\begin{array}{llll}
\ F_m & \qquad \ \alpha _m & \ \qquad \beta _m & \qquad \ \delta _m \\
\left\{ F\right\} & \quad 1+\left\{ \alpha \right\} & \qquad \left\{ \beta
\right\} & \qquad \left\{ \delta \right\} \\
\left\{ \sigma F\right\} & \qquad \left\{ \sigma \alpha \right\} & \quad
1+\left\{ \sigma \beta \right\} & \qquad \left\{ \sigma \delta \right\} \\
\left\{ YF\right\} & \qquad \left\{ Y\alpha \right\} & \qquad \left\{ Y\beta
\right\} & \quad 1+\left\{ Y\delta \right\}
\end{array}
\right| \label{f7}$$
where $D_{{\rm Q}}$ coincides with the cofactor of the (1,1)th element in the $4\times 4$ determinant, and
$$~\left\{ fg\right\} \equiv \sum_{m=1}^p\rho _me^{{\rm i}X_m}f_mg_m=\rho
\left\langle e^{{\rm i}X}fg\right\rangle ,\qquad ~\left\langle
f\right\rangle \equiv \sum_{m=1}^px_mf_m.$$
Here and in the following, angular brackets $\left\langle \cdots
\right\rangle $ denote compositional averages over the distribution of particles (this notation differs from that of Ref. 8, which also contains misprints corrected in Refs. 35,36).
With the chosen decomposition of $\widehat{Q}_{ij}(k)$ the stickiness contributions are confined in the last row and the last column of the determinant of Eq. (\[f7\]). When adhesion is turned off, all these elements vanish apart from their diagonal term, and the $4\times 4$ determinant essentially reduces to the $3\times 3$ HS one. For numerical computation, it is convenient, following Ref. 8, to simplify $s_m$ by using elementary transformations which do not alter the value of the determinant. If we add to the third column the second one multiplied by $-\left( 3\xi
_2/\Delta -{\rm i}k/2\right) $ and to the fourth column the second one multiplied by $12\xi _2^Y$, then $s_m$ becomes
$$s_m=\frac{\rho _m^{1/2}}{D_{{\rm Q}}}\left|
\begin{array}{llll}
\ F_m & \qquad \ \alpha _m & \ \qquad \beta _m^{(0)} & \qquad \ \delta
_m^{(0)} \\
\left\{ F\right\} & \quad 1+\left\{ \alpha \right\} & \qquad \left\{ \beta
^{(0)}\right\} -3\xi _2/\Delta +{\rm i}k/2 & \qquad \left\{ \delta
^{(0)}\right\} +12\xi _2^Y \\
\left\{ \sigma F\right\} & \qquad \left\{ \sigma \alpha \right\} & \quad
1+\left\{ \sigma \beta ^{(0)}\right\} & \qquad \left\{ \sigma \delta
^{(0)}\right\} \\
\left\{ YF\right\} & \qquad \left\{ Y\alpha \right\} & \qquad \left\{ Y\beta
^{(0)}\right\} & \quad 1+\left\{ Y\delta ^{(0)}\right\}
\end{array}
\right| , \label{f9}$$
where $D_{{\rm Q}}$ has been changed accordingly. Expanding the $%
4\times 4$ determinant along the first line and inserting it into Eq. (\[i9\]), the final result for the scattering intensity of SHS is
$$\begin{aligned}
R(k)/\rho &=&\left\langle F^2\right\rangle +\left\langle \alpha
^2\right\rangle \left| C_1\right| ^2+\left\langle \beta ^{(0)2}\right\rangle
\left| C_2\right| ^2+\left\langle \delta ^{(0)2}\right\rangle \left|
C_3\right| ^2 \nonumber \\
&&+2%
%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
\left[ \left\langle F\alpha \right\rangle C_1+\left\langle F\beta
^{(0)}\right\rangle C_2+\left\langle F\delta ^{(0)}\right\rangle C_3\right.
\label{f10} \\
&&\left. +\ \left\langle \alpha \beta ^{(0)}\right\rangle
C_1C_2^{*}+\left\langle \alpha \delta ^{(0)}\right\rangle
C_1C_3^{*}+\left\langle \beta ^{(0)}\delta ^{(0)}\right\rangle
C_2C_3^{*}\right] , \nonumber\end{aligned}$$
where form factors have been assumed to be real quantities, as is indeed the case for spherical homogeneous scattering cores. On the r.h.s. all $k$ arguments have been omitted for simplicity, and the $C_\nu (k)$ have already been defined with reference to Eq. (\[d5\]).
The expression for the average structure factor $S_{{\rm M}}\left( k\right) $ is then obtained after division of the Rayleigh ratio $R(k)$ by $\rho
P(k)=\rho \left\langle F^2(k)\right\rangle .$ Moreover, the Bathia-Thornton number-number structure factor $S_{{\rm NN}}\left( k\right) $ can easily be derived by setting all $F_m=1$ everywhere into the expression of $R(k)/\rho $, with the result
$$\begin{aligned}
S_{{\rm NN}}\left( k\right) &=&1+\rho \left\{ \left\langle \alpha
^2\right\rangle \left| {\cal C}_1\right| ^2+\left\langle \beta
^{(0)2}\right\rangle \left| {\cal C}_2\right| ^2+\left\langle \delta
^{(0)2}\right\rangle \left| {\cal C}_3\right| ^2\right. \nonumber \\
&&+2%
%TCIMACRO{\func{Re} }
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
\left[ \left\langle \alpha \right\rangle {\cal C}_1+\left\langle \beta
^{(0)}\right\rangle {\cal C}_2+\left\langle \delta ^{(0)}\right\rangle {\cal %
C}_3\right. \label{f11} \\
&&\left. \left. +\ \left\langle \alpha \beta ^{(0)}\right\rangle {\cal C}_1%
{\cal C}_2^{*}+\left\langle \alpha \delta ^{(0)}\right\rangle {\cal C}_1%
{\cal C}_3^{*}+\left\langle \beta ^{(0)}\delta ^{(0)}\right\rangle {\cal C}_2%
{\cal C}_3^{*}\right] \right\} , \nonumber\end{aligned}$$
where the ${\cal C}_\nu $ are the analogues of the $C_\nu $ appearing in $R(k)$.
RESULTS FOR POLYDISPERSE FLUIDS
===============================
Size distribution
-----------------
For SHS fluids containing only one chemical species, size polydispersity simply means the presence of a multiplicity of possible diameters. In a “discrete” representation of polydispersity the number $p$ of different diameters is very large but finite, and $x_{i\text{ }}$is the fraction $%
N_i/N $ of particles having diameter $\sigma _i$. On the other hand, a theoretical representation with infinitely many components ($p\rightarrow
\infty )$ and “continuously” distributed diameters is also possible and often used.
Although all formulas of previous Sections refer to a finite number $p$ of components, the polydisperse continuous limit of such expressions can immediately be inferred by the [*replacement rules*]{} $x_\alpha \rightarrow
{\rm d}x=f(\sigma ){\rm d}\sigma $ and $\sum_\alpha x_\alpha ...\rightarrow
\int {\rm d}\sigma f(\sigma )...$, where $f(\sigma ){\rm d}\sigma $ is the fraction $dN/N$ of particles with diameter in the interval $\left( \sigma
,\sigma +{\rm d}\sigma \right) $. As [*molar fraction density function*]{} $%
f(\sigma )$ we choose the Schulz distribution [@Beurten81; @DAguanno91]
$$f(\sigma )=\frac a{\Gamma (a)\left\langle \sigma \right\rangle }\left( a\
\frac \sigma {\left\langle \sigma \right\rangle }\right) ^{a-1}\exp \left(
-a\ \frac \sigma {\left\langle \sigma \right\rangle }\right) \;\;\quad (a>1),$$
where $\Gamma $ is the gamma function[@Abramowitz72], $\left\langle
\sigma \right\rangle $ the average diameter, $a=1/s_\sigma ^2,$ and $%
s_\sigma =\left[ \left\langle \sigma ^2\right\rangle -\left\langle \sigma
\right\rangle ^2\right] ^{1/2}/\left\langle \sigma \right\rangle $ measures the degree of size polydispersity. In the monodisperse limit, $s_\sigma =0,$ the distribution becomes a Dirac delta function centered at $\left\langle
\sigma \right\rangle $. The Schulz function allows an easy analytic evaluation of some averages $\int {\rm d}\sigma f(\sigma )...$, such as the moments $\left\langle \sigma ^m\right\rangle $, which obey a simple relation for $m\geq 1$, i.e. $$\left\langle \sigma ^m\right\rangle =\left[ 1+(m-1)s_\sigma ^2\right]
\left\langle \sigma \right\rangle \left\langle \sigma ^{m-1}\right\rangle
=\left\langle \sigma \right\rangle ^m\prod_{j=1}^{m-1}M_j\ , \label{sd2}$$
with $M_j\equiv 1+js_\sigma ^2\ .$
In most cases, however, analytical integration is hardly feasible, and numerical integration brings back to discrete expressions with large $p$, of order $10^2-10^3$. In practice, the “discrete” representation of polydispersity is the most convenient for numerical purposes, and all formulas of the previous Sections can be employed by assuming $x_\alpha
=f(\sigma _\alpha )\Delta \sigma $, where $\Delta \sigma $ is the grid size of numerical integration.
For fluids with Schulz-distributed diameters the packing fraction, $\eta
\equiv \xi _3=(\pi /6)\rho \left\langle \sigma ^3\right\rangle $, can be written as $\eta =\eta _{{\rm mono}}\left( 1+s_\sigma ^2\right) \left(
1+2s_\sigma ^2\right) ,$ with $\eta _{{\rm mono}}=(\pi /6)\rho \left\langle
\sigma \right\rangle ^3$.
Stickiness distribution
-----------------------
On a dimensional basis, the parameters $Y_i$ must be lengths. Moreover, $%
K_{ij}=Y_iY_j$ must be proportional to $\beta =(k_BT)^{-1}$. If we assume, for simplicity, that stickiness polydispersity and size polydispersity are fully correlated, then the most natural choice for $Y_i$ is
$$Y_i=\gamma _{0\ }\sigma _i\ , \label{sp1}$$
with the dimensionless proportionality factor
$$\gamma _0\ =\left( \frac{\varepsilon _0}{k_BT}\right) ^{1/2}=\frac{%
Y_{\left\langle \sigma \right\rangle }}{\left\langle \sigma \right\rangle },
\label{sp2}$$
where $\varepsilon _0$ denotes an energy and $Y_{\left\langle
\sigma \right\rangle }$ is the stickiness parameter of particles with diameter $\left\langle \sigma \right\rangle .$ This implies that
$$K_{ij}=\gamma _0^2\ \sigma _i\sigma _j=\frac{\varepsilon _0}{k_BT}\ \sigma
_i\sigma _j\ =\frac 1{T^{*}}\ \sigma _i\sigma _j\ ,\ \label{sp3}$$
where we have also introduced a reduced temperature $T^{*}=\left(
k_BT/\varepsilon _0\right) \equiv 1/\gamma _0^2$.
The model of Eq. (\[sp1\]) will be compared with the one of SHS polydisperse in size but not in stickiness (on the analogy of Refs. 20 and 21). In this simpler case all particles have the same $Y_i=Y_{\left\langle
\sigma \right\rangle }=\gamma _0\left\langle \sigma \right\rangle ,$ and the degree of stickiness polydispersity $s_Y$, defined similarly to $s_\sigma ,$ vanishes.
Both these models may be regarded as particular cases (for $\alpha =0$ and $%
\alpha =1$) of a more general size-dependence given by
$$Y_i=Y_{\left\langle \sigma \right\rangle }\left( \frac{\sigma _i}{%
\left\langle \sigma \right\rangle }\right) ^\alpha =\gamma _0\ \frac{\sigma
_i^\alpha }{\left\langle \sigma \right\rangle ^{\alpha -1}}, \label{sp4}$$
with $\alpha \geq 0$. We have examined this generalization for $%
\alpha =2$ and $\alpha =3$, but for the purposes of the present paper we restrict our analysis only to the cases $\alpha =0$ and $\alpha =1.$
The choice $Y_i=\gamma _0\ \sigma _i$ has very interesting properties. First, the corresponding distribution of $Y$-values, related to the size distribution $f_\sigma $ as $f_Y\equiv dN/dY=f_\sigma d\sigma /dY$, is a Schulz function as well, with $\left\langle Y\right\rangle =Y_{\left\langle
\sigma \right\rangle }$ and $s_Y=s_\sigma $. A second more important fact is that only in this special case $\widehat{Q}_{ij}(k)$, in general 3-dyadic for SHS, becomes simply 2-dyadic, i.e.
$$\widehat{Q}_{ij}(k)=\delta _{ij}+\rho _i^{1/2}e^{{\rm i}X_i}\ \left\{
A_i(k)\alpha _j(k)+G_0\sigma _i\beta _j^{(0)}(k)\right\} \rho _j^{1/2},
\label{sp5}$$
with
$$A_i(k)=1+\left( \frac{3\xi _2G_0}\Delta -{\rm i}\frac k2\right) \sigma _i\ ,$$ $$G_0\equiv 1-4\gamma _0^2\Delta =1-\frac{4\varepsilon _0}{k_BT}\left[ 1-\eta
_{{\rm mono}}\left( 1+s_\sigma ^2\right) \left( 1+2s_\sigma ^2\right) \right]
\label{sp6}$$
(now $\xi _2^Y=\gamma _0\ \xi _2$). It is remarkable that this expression for $\widehat{Q}_{ij}(k)$ differs from the HS one only for the presence of $G_0$ ($G_0=1$ for HS). Now the natural dyadic decomposition becomes
$$\begin{array}{lll}
a_i^{(1)}=\rho _i^{1/2}e^{{\rm i}X_i}A_i, & & b_j^{(1)}=\rho _j^{1/2}\alpha
_j, \\
a_i^{(2)}=\rho _i^{1/2}e^{{\rm i}X_i}G_0\sigma _{i\ }, & & b_j^{(2)}=\rho
_j^{1/2}\beta _j^{(0)},
\end{array}$$
while the $4\times 4$ determinant appearing in Eq. (\[f7\]) reduces to a $3\times 3$ one, with a consequent simplification of the formulas for $R(k)/\rho $, $S_{{\rm M}}\left( k\right) $ and $S_{{\rm NN}%
}\left( k\right) $ (all terms depending on subscript 3 vanish in Eqs. (\[f10\]) and (\[f11\])).
Numerical results
-----------------
Because of its importance in the analysis of experimental scattering data, we have focused on the measurable average structure factor $S_{{\rm M}%
}\left( k\right) $. The scattering cores inside the particles have been assumed to be spherical and homogeneous, with form factors $F_m=\ V_m^{{\rm %
scatt}}\Delta n_m3j_1(X_m^{{\rm scatt}})/X_m^{{\rm scatt}},$ where $X_m^{%
{\rm scatt}}=k\sigma _m^{{\rm scatt}}/2,$ $\sigma _m^{{\rm scatt}}\leq
\sigma _m$ is the diameter of a scattering core of species $m$, $V_m^{{\rm %
scatt}}=\left( \pi /6\right) \left( \sigma _m^{{\rm scatt}}\right) ^3$ its volume, and $\Delta n_m\ $its scattering contrast with respect to the solvent. For mixtures with several components belonging to only one chemical species, as in the present paper, $\Delta n_m$ is the same for all particles. For simplicity, we have taken $\sigma _m^{{\rm scatt}}=\sigma _m$.
The polydisperse SHS model depends on the following parameters: the packing fraction $\eta $, the strength $\gamma _0$ of the adhesive interaction, the average diameter $\left\langle \sigma \right\rangle ,$ and the two degrees of polydispersity $s_\sigma $ and $s_Y$. In all numerical calculations we have adopted dimensionless variables, with lengths expressed in units of $%
\left\langle \sigma \right\rangle .$ To understand the influence of each parameter on $S_{{\rm M}}(k)$, it is instructive to first recall the behaviour of a sequence of simpler systems, starting from monodisperse hard spheres and adding in the first two cases either surface attraction or size polydispersity.
### Monodisperse HS and SHS
In pure fluids all particles are equal ($s_\sigma =0=s_Y$), $\eta =\eta _{%
{\rm mono}}$, and $S_{{\rm M}}\left( k\right) =S_{{\rm mono}}\left( k\right)
$ with no form factor involved. Figures 1 and 2 depict the dependence of $S_{%
{\rm mono}}\left( k\right) $ on the parameters ($\eta ,\gamma _0$). Fig. 1 illustrates the evolution of $S_{{\rm mono}}\left( k\right) ,$ as $\eta $ increases from low values up to the freezing one, in the well known case of monodisperse HS of diameter $\sigma $ without stickiness ($\gamma _0=0$). Here we have exploited the PY solution [@Hansen86], which, for HS, coincides with the MSA one. In Fig. 2 the dependence of $S_{{\rm mono}%
}\left( k\right) $ on $\eta $ is displayed for monodisperse SHS, at two fixed $\gamma _0$ values, i.e. 0.5 and 0.7, corresponding to $\gamma
_0^2=\varepsilon _0/\left( k_BT\right) =0.25$ and $0.49$, or to $T^{*}=4$ and $2.\,04$ ($\gamma _0$ and $T^{*}$ have been defined in Eqs. (\[sp2\]) and (\[sp3\]), respectively).
In all these cases (Figs. 1, 2a and 2b), as $\eta $ increases at fixed $%
\gamma _0$, the first peak height and amplitudes of all subsequent oscillations increase, but the behaviour near the origin depends on $\gamma
_0$, as will be discussed in more detail shortly.
On the other hand, the effect of increasing $\gamma _0$ (i.e. increasing the adhesive attraction or decreasing $T$) at fixed $\eta $ can be seen by comparing, for instance, the solid curves ($\eta =0.2$) of Figs. 1 and 2. As $\gamma _0$ increases, the first peak and subsequent maxima are shifted to larger $k$ values, and their amplitudes change as well. However, the most significant effect on $S_{{\rm mono}}\left( k\right) $ occurs near the origin. Here, $S_{{\rm mono}}\left( 0\right) $ substantially increases and becomes the global maximum at large $\gamma _0$. This behaviour can be understood from the explicit expression of $S_{{\rm mono}}\left( 0\right) $, which reads
$$\begin{aligned}
S_{{\rm mono}}\left( 0\right) &=&\left[ \widehat{Q}_{{\rm mono}}\left(
0\right) \right] ^{-2}=\frac{\left( 1-\eta \right) ^4}{\left[ 1+2\eta
-12\gamma _0^2\ \eta \left( 1-\eta \right) \right] ^2} \label{r1} \\
&=&K_T/K_T^{{\rm id}}=\rho k_BTK_T. \nonumber\end{aligned}$$
Since $S_{{\rm mono}}\left( 0\right) $ is related to the isothermal osmotic compressibility $K_T$ and to the density fluctuations [@March76], its drastic increase signals the approach to a gas-liquid phase transition. The critical point can be obtained from the spinodal line, defined by $S_{{\rm mono}}^{-1}\left( 0\right) =0$, and the critical parameters turn out to be: $\eta _{{\rm c}}=\left( \sqrt{3}-1\right)
/2\simeq 0.37$ and $\gamma _{0{\rm c}}^2=\left( \sqrt{3}+2\right) /6\simeq
0.62$ [@Brey87] (corresponding to $\gamma _{0{\rm c}}\simeq 0.79$, or to the reduced [*critical temperature*]{} $T_{{\rm c}}^{*}\simeq 1.61$).
The combined influence of $\eta $ and $\gamma _0$ can be observed going back to Fig. 2. On defining the [*Boyle temperature*]{} $T_{{\rm B}}^{*}$ as the one where the attractive and repulsive forces balance each other in such a way that the second virial coefficient $B_2$ vanishes [@Penders90], we note that the temperatures corresponding to $\gamma _0=0.5$ and $0.7$ lie, respectively, above and below $T_{{\rm B}}^{*}$ (but in both cases above $T_{%
{\rm c}}^{*}$). In fact, for this monodisperse model it is easy to see, from the low-density expansion of $S_{{\rm mono}}\left( 0\right) =1+\rho
\widetilde{h}\left( 0\right) \simeq 1-2B_2\rho +{\cal O}(\rho ^2),$ that $%
B_2=4V_{{\rm HS}}(1-3\gamma _0^2)=4V_{{\rm HS}}\left( 1-3/T^{*}\right) $ with $V_{{\rm HS}}=(\pi /6)\sigma ^3$, and therefore $T_{{\rm B}}^{*}=3$ ($%
\gamma _{0{\rm B}}^2=1/3$ or $\gamma _{0{\rm B}}\simeq 0.58$). Fig. 2 suggests the existence of two different “regimes” for $S_{{\rm mono}%
}\left( k\right) $ above and below the Boyle temperature, respectively. When $T^{*}>T_{{\rm B}}^{*}$ or, equivalently, $\gamma _0<\gamma _{0{\rm B}}$ (“weak-attraction regime”, as in Fig. 2a) the fluid behaves like pure HS without stickiness. Here repulsive forces are dominant, $B_2>0$, and compressibility and density fluctuations, along with the whole $S_{{\rm mono}%
}\left( k\right) $ near the origin, decrease with increasing $\eta $. When $%
T_{{\rm c}}^{*}<T^{*}<T_{{\rm B}}^{*}$ or, equivalently, $\gamma _{0{\rm B}%
}<\gamma _0<\gamma _{0{\rm c}}$ (“strong-attraction regime”), one finds $%
B_2<0,$ while the balance between attractive and repulsive forces becomes more complex. In this case $S_{{\rm mono}}\left( k\right) $ near the origin has a non-monotonic dependence on $\eta $, as in Fig. 2b. Here, compressibility and density fluctuations first increase with $\eta $, in agreement with the low-density expansion of $S_{{\rm mono}}\left( 0\right) $. Then, an inversion occurs at $\eta _0=(6-2T^{*})/(6+T^{*})$ ($\simeq 0.24$ when $T^{*}=2.04$) and afterwards $S_{{\rm mono}}\left( 0\right) $ decreases. In other words, below $T_{{\rm B}}^{*}$ attractive forces seem to be dominant at low packing fraction, whereas repulsion again prevails at higher $\eta .$
### Polydisperse HS without stickiness
Fig. 3 refers to polydisperse HS without surface adhesion ($\gamma _0=0$). Size polydispersities $s_\sigma =0.1,0.3$ have been employed here and in the following, since values in this range are rather common in experimental data from colloidal fluids. The two Schulz distributions have been discretized with a grid size $\Delta \sigma /\left\langle \sigma \right\rangle =0.02,$ and truncated where $f(\sigma )\Delta \sigma \approx 10^{-8}$, i.e. at $%
\sigma _{{\rm cut}}/\left\langle \sigma \right\rangle =1.68$ and $3.48$, respectively. Since each diameter characterizes a different component, these discrete polydisperse mixtures involve $p=85$ and $175$ components. Note that these numbers of components are much larger than those used with the SHS-PY model of Ref. 20.
The effect of size polydispersity is considerable [@Beurten81], as appears from a comparison among Figs. 1, 3a and 3b: with increasing $%
s_\sigma $ at fixed $\eta $, $S_{{\rm M}}\left( k\right) $ slightly increases in the low-$k $ region, its first peak is reduced and shifted to smaller $k$ values, and all subsequent oscillations are progressively dumped, as a result of destructive interference among the several length scales involved.
### SHS polydisperse in size but not in stickiness
At this point we study SHS fluids polydisperse in size but monodisperse in stickiness, with all particles having $Y_i=Y_{\left\langle \sigma
\right\rangle }=\gamma _0\left\langle \sigma \right\rangle $ ($s_\sigma \neq
0$, $\alpha =0\Rightarrow s_Y=0$). This choice will be referred to as Model I and has been prompted by the SHS-PY investigations of Refs. 20-22, where a single, size-independent, stickiness parameter was considered.
Figures 4 and 5 illustrate what happens when a surface adhesion (with $%
\gamma _0=0$.$5,0.7$) is added to size polydispersity. Comparison with Fig. 3 ($\gamma _0=0$) shows that the attractive interaction, in the presence of size polydispersity, produces a further lowering of oscillation amplitudes in the first peak region and beyond. When $\gamma _0=0.7$ and $s_\sigma =0.3$ (Fig. 5b) all curves exhibit an almost complete flattening in the same range.
Near the origin (for $k\left\langle \sigma \right\rangle \lesssim 5$), for both considered cases with $\gamma _0=0.5$ (weak-attraction), only a small increase in $S_{{\rm M}}\left( k\right) $ is found with respect to polydisperse HS without stickiness (Fig. 3), the relative ordering of all curves is unchanged and also coincides with that of the corresponding monodisperse SHS (Fig. 2a). On the other hand, when $\gamma _0=0.7$ and $%
s_\sigma =0.1$ (strong-attraction and low size polydispersity, Fig. 5a) the behaviour of $S_{{\rm M}}\left( k\right) $ close to the origin strongly differs from that of polydisperse HS without stickiness and is similar to the monodisperse SHS case of Fig. 2b. Surface adhesion produces large $S_{%
{\rm M}}\left( 0\right) $ values, which are, however, smaller than the corresponding monodisperse ones. This means that, when $\gamma _0\neq 0$, size polydispersity reduces $S_{{\rm M}}\left( k\right) $ even near the origin (whereas, when $\gamma _0=0$, increasing $s_\sigma $ at fixed $\eta $ determines an increase of $S_{{\rm M}}\left( 0\right) $). This effect of size polydispersity, in the presence of attraction, is amplified when $%
\gamma _0=0.7$ and $s_\sigma =0.3$ (strong-attraction and high size polydispersity, Fig. 5b). Now one observes an interesting return to a “HS-like ordering” of the curves in the low-$k$ region, as in the case $%
\gamma _0=$ 0.5. This behaviour is peculiar of Model I and will be absent in Model II to be presented in the next subsection.
### SHS polydisperse both in size and in stickiness
Next we consider the case of stickiness correlated to the size, according to the linear law $Y_i=\gamma _{0\ }\sigma _i$ ($\alpha =1,$ $s_Y=s_\sigma \neq
0$). This will be referred to as Model II.
The results for $\gamma _0=$ 0.5 , shown in Fig. 6, are qualitatively similar to those of Model I (Fig. 4). When $s_Y=s_\sigma =0.1$ the quantitative differences are very small. However, when $s_Y=s_\sigma =0.3$ the $S_{{\rm M}}(0)$ values lie more clearly above those of Fig. 4b.
For $\gamma _0=$ 0.7 (Fig. 7) the behaviour of $S_{{\rm M}}\left( k\right) $ in the first peak region and beyond is essentially unchanged with respect to Model I, but near the origin differences are larger and significant. Here, when $s_Y=s_\sigma =0.1$ the $S_{{\rm M}}\left( k\right) $ curves are similar to those of Fig. 5a, with larger $S_{{\rm M}}\left( 0\right) $ values (very close to the corresponding monodisperse ones of Fig. 2b), but as $s_Y=s_\sigma =0.3$ there is a qualitative as well as quantitative difference with respect to Model I (Fig. 5b). Indeed in the low-$k$ region the $S_{{\rm M}}\left( k\right) $ curves of Fig. 7b exhibit the same relative ordering present in the previous case with lower polydispersity (Fig. 7a) as well as in the corresponding fully monodisperse fluid (Fig. 2b). This persistence in a “strong-attraction regime” even at high size polydispersity constitutes the main difference between Model I and II. Such a feature can be probably related to the fact that the stickiness distribution of Model II is skewed towards large $Y_{i\text{ }}$ values completely absent in Model I, and this asymmetry implies, on average, stronger attractive forces.
Unfortunately, the behaviour of $S_{{\rm M}}(0)$ in polydisperse models does not admit any simple thermodynamical interpretation. For mixtures, in fact, the average structure factor $S_{{\rm M}}\left( k\right) $ depends on the form factors and $S_{{\rm M}}\left( 0\right) $ is no longer the normalized compressibility [@Nagele96]. Nevertheless, we have been able to account for the aforesaid difference of “regimes” between Model I and II when $%
\gamma _0=0.7$ and $s_\sigma =0.3$ in terms of a single parameter, which generalizes the Boyle temperature of the monodisperse SHS case.
### Generalized Boyle temperature
The Boyle temperature of these polydisperse models can be found by deriving their second virial coefficient $B_2$ from the low-density expansion of $S_{%
{\rm NN}}\left( 0\right) =1-2B_2\rho +{\cal O}(\rho ^2).$ Likewise, to interpret the behaviour of $S_{{\rm M}}\left( k\right) $ previously discussed, we start from the low-density expansion of $S_{{\rm M}}\left(
0\right) .$ A straightforward calculation, employing the dyadic formalism of Sections II and III and not reported here, yields
$$\widehat{Q}_{ij}^{-1}(0)=\delta _{ij}-\rho (x_ix_j)^{1/2}\left[ \frac \pi 6%
\left( \sigma _j^3+3\sigma _i\sigma _j^2\right) -2\pi Y_iY_j\sigma _j\right]
+{\cal O}(\rho ^2),$$
and therefore, from Eq. (\[i7\]),
$$S_{ij}(0)=\delta _{ij}-\rho (x_ix_j)^{1/2}\left[ \frac \pi 6\left( \sigma
_i+\sigma _j\right) ^3-2\pi \left( Y_iY_j\sigma _j+Y_jY_i\sigma _i\right)
\right] +{\cal O}(\rho ^2).$$
Inserting this result into $S_{{\rm M}}\left( 0\right)
=\sum_{i,j}(x_ix_j)^{1/2}\left[ F_i\left( 0\right) F_j^{*}\left( 0\right)
/P(0)\right] \ S_{ij}\left( 0\right) ,$ and using the above mentioned expression for $F_m\left( k\right) $, we obtain $S_{{\rm M}}\left( 0\right)
=1-2B_{2,{\rm F}}\rho +{\cal O}(\rho ^2)$, with
$$B_{2,{\rm F}}=\frac \pi 6\frac 1{\left\langle \sigma ^6\right\rangle }\left[
\left\langle \sigma ^6\right\rangle \left\langle \sigma ^3\right\rangle
+3\left\langle \sigma ^5\right\rangle \left\langle \sigma ^4\right\rangle
-12\ \left\langle \sigma ^3Y\right\rangle \left\langle \sigma
^4Y\right\rangle \right] , \label{r4}$$
which is the analogue of the second virial coefficient, including all form factors. The sign of $B_{2,{\rm F}},$ and therefore the behaviour of $S_{{\rm M}}\left( 0\right) $ at low density (as well as the overall “regime” of $S_{{\rm M}}\left( k\right) ),$ depends on $T^{*}$, which is hidden in the $Y$ terms. On defining a [*generalized Boyle temperature*]{} $%
T_{{\rm B,F}}^{*}$ as the one where $B_{2,{\rm F}}$ vanishes, and employing our assumption $Y_i=\gamma _0\sigma _i^\alpha /\ \left\langle \sigma
\right\rangle ^{\alpha -1}$, we find
$$T_{{\rm B,F}}^{*}=\ \frac{12\left\langle \sigma ^{3+\alpha }\right\rangle \
\left\langle \sigma ^{4+\alpha }\right\rangle }{\left[ \left\langle \sigma
^6\right\rangle \left\langle \sigma ^3\right\rangle +3\left\langle \sigma
^5\right\rangle \left\langle \sigma ^4\right\rangle \right] \ \left\langle
\sigma \right\rangle ^{2\left( \alpha -1\right) }}\ .$$
Exploiting Eq. (\[sd2\]), it follows that
$$\begin{array}{lll}
T_{{\rm B,F}}^{*}=12/\left[ M_4\left( M_5+3M_3\right) \right] & & \text{for
Model I,} \\
T_{{\rm B,F}}^{*}=12M_3/\left( M_5+3M_3\right) & & \text{for Model II,}
\end{array}
\label{r6}$$
where $M_j$ has been defined with reference to Eq. (\[sd2\]). The role of $T_{{\rm B,F}}^{*}$ for $S_{{\rm M}}\left( k\right) $ is the same as that of $T_{{\rm B}}^{*}$ for $S_{{\rm mono}}\left( k\right) $ of monodisperse SHS: above $T_{{\rm B,F}}^{*}$ the behaviour is “HS-like”, whereas a “strong-attraction regime” is found when $T_{{\rm c}%
}^{*}<T^{*}<T_{{\rm B,F}}^{*}$.
Eqs. (\[r6\]) imply that, in both models, $T_{{\rm B,F}}^{*}$ depends on the degree of size polydispersity $s_\sigma $. However, whereas $T_{{\rm B,F}%
}^{*}(s_\sigma )$ of Model I is a rapidly decreasing function, in Model II it exhibits only a very slow decrease asymptotically approaching $18/7\simeq
2.\,57$. Such a difference explains the behaviours displayed in Fig. 5 and 7, which refer to $T^{*}=2.\,04$ (i.e. $\gamma _0=0.7$). In Model I, when $%
s_\sigma =0.1$ one has $T^{*}<T_{{\rm B,F}}^{*}=2.79$, whereas when $%
s_\sigma =0.3$ one finds $T^{*}>T_{{\rm B,F}}^{*}=1.68$. On the other hand, in both cases the temperature of Model II is below the generalized Boyle one, the values of $T_{{\rm B,F}}^{*}$ now being $2.99$ and $2.90$ for $%
s_\sigma =0.1$ and $0.3$, respectively.
Finally, it is instructive to compare $T_{{\rm B,F}}^{*}$ with the true Boyle temperature $T_{{\rm B}}^{*}$ of these polydisperse models$.$ The second virial coefficient, obtained from the low-density expansion of $S_{%
{\rm NN}}\left( 0\right) ,$ turns out to be $B_2=(\pi /6)\left[ \left\langle
\sigma ^3\right\rangle +3\left\langle \sigma ^2\right\rangle \left\langle
\sigma \right\rangle -12\ \left\langle Y\right\rangle \left\langle \sigma
Y\right\rangle \right] ,$ and one obtains $T_{{\rm B}}^{*}=12/\left[
M_1\left( M_2+3\right) \right] $ and $12/\left( M_2+3\right) $ for Model I and II, respectively. Note that for these polydisperse fluids it is always $%
T_{{\rm B}}^{*}<\left( T_{{\rm B}}^{*}\right) _{{\rm mono}}=3$, $T_{{\rm B,F}%
}^{*}<T_{{\rm B}}^{*}$ for Model I, and $T_{{\rm B,F}}^{*}>T_{{\rm B}}^{*}$ for Model II, while in the limit of monodisperse fluids $T_{{\rm B,F}%
}^{*}\rightarrow \left( T_{{\rm B}}^{*}\right) _{{\rm mono}}.$
SUMMARY AND CONCLUSIONS
=======================
In this paper we have presented a new analytically solvable model for multi-component SHS fluids within the MSA closure, using a hard-core-Yukawa potential with factorizable coupling parameters (in the appropriate infinite amplitude and zero range limit). The model is simpler than previous ones available in the literature, since $\widehat{q}_{ij}(k)$ is in general 3-dyadic (Tutschka and Kahl’s model [@Tutschka98] was 5-dyadic), with a consequent great simplification of all analytical formulas.
We have stressed the importance of the “dyadic structure” of $\widehat{q}%
_{ij}(k)$ and recalled the properties of matrices with dyadic elements. Such a feature allows the analytic inversion of $\widehat{Q}_{ij}(k)$ required to get the partial structure factors $S_{ij}(k)$. Through Vrij’s summation, expressions have then been obtained for global structure functions, such as $%
R(k)$, $S_{{\rm M}}\left( k\right) $ and $S_{{\rm NN}}\left( k\right) $. These closed analytical formulas, just as their counterparts for polydisperse HS [@Vrij79] and CHS [@Gazzillo97], allow to “bypass” the computation of the individual $p(p+1)/2$ partial structure factors, which may be a rather difficult task for polydisperse systems with large number of components. Because of their simplicity, our expressions may therefore represent a very useful tool to fit experimental scattering data of real colloidal fluids.
While the presented 3-dyadic expressions hold true for any choice of stickiness parameters $Y_i$, two particular versions of the model have been analyzed numerically. The first one assumes size polydispersity, but a single stickiness parameter for all particles (Model I), while the second one proposes stickiness parameters dependent on the diameters according to a linear law (Model II). Model I is similar to the SHS-PY models for polydisperse colloids known in the literature [@Robertus89; @Penders90; @Duits91], while Model II is the simplest choice for size-dependent stickiness parameters.
The combined influence of hard core repulsion, adhesive attraction and polydispersity can generate a variety of behaviours at the level of measurable average structure factor $S_{{\rm M}}\left( k\right) $. We have recognized the existence of two different “regimes” for $S_{{\rm M}}\left(
k\right) $ both in monodisperse and in polydisperse SHS fluids. Above a temperature $T_{{\rm B,F}}^{*}$, which in the monodisperse case coincides with the Boyle temperature, we have identified a “weak-attraction” behaviour, resembling the HS one. In the range below $T_{{\rm B,F}}^{*}$ but still above the critical temperature, a “strong-attraction” regime sets in, and we have described its features in detail. It is found that $T_{{\rm %
B,F}}^{*}$ is a decreasing function of the degree of size polydispersity $%
s_\sigma .$ It is also worth noting that the behaviour of our SHS-MSA models in the “strong-attraction regime” is in qualitative agreement with that of the SHS-PY model displayed in Fig.s 3 and 4 of Ref. 20, where the existence of two different “regimes” for $S_{{\rm M}}\left( k\right) $ was, however, not recognized.
All our numerical results show that size polydispersity strongly affects the behaviour of $S_{{\rm M}}\left( k\right) $ in the first peak region and beyond, where the influence of stickiness polydispersity is less significant. Models I and II are nearly equivalent in this interval of $k$-values, whereas they may substantially differ near the origin. The present study shows that the use of a single stickiness parameter, instead of more realistic size-dependent ones, may lead to marked differences in the small angle scattering region at sufficiently high $\gamma _0$, i.e. when attraction is strong or temperature is low.
In the small $k$ region the adhesive forces and the specific relationship between stickiness and size parameters have far reaching consequences. Although very little is known experimentally about the correlation between stickiness and size, it is reasonable to expect that larger particles attract each other more strongly. The linear dependence $Y_i=\gamma _{0\
}\sigma _i$ represents the simplest non-trivial choice, but other possibilities could also be taken into account [@Tutschka98]. Duits [*et al.*]{} [@Duits91] found that in some cases the SHS-PY model with a single stickiness parameter, independent of particle size, was unable to fit their experimental scattering data, and these authors already emphasized the possible role of stickiness polydispersity as a cause for the observed deviations. Similar discrepancies between experimental and model $S_{{\rm M}%
}\left( k\right) $ values could be a crucial test for the soundness of our Model II with respect to Model I, as well as of any other choice for the stickiness-size functional relationship.
It would be also instructive to compare our SHS-MSA model with other recent theoretical approaches to polydisperse colloidal fluids. As an example, we mention the “optimized random phase approximation” joined with orthogonal polynomial expansions, proposed by Lado and coworkers [@Leroch99].
However, the most important advantage of the present model lies in its simplicity. In particular, version II has special formal properties, since - only in this case - the expression of $\widehat{q}_{ij}(k)$ becomes 2-dyadic. Therefore version II can indeed be reckoned as the simplest solvable model for polydisperse SHS and it could be a good candidate to tackle the issue of thermodynamics and phase stability of these fluids from a fully analytical point of view. We expect that compact expressions for pressure, chemical potentials, partial structure factors at $k=0$, as well as other quantities required to investigate - for instance - sedimentation [@Bolhuis93], vapor-liquid equilibrium and demixing in the presence of polydispersity, can easily be obtained. We hope to accomplish this task in a forthcoming paper.
Partial financial support by the Italian MURST (Ministero dell’Università e della Ricerca Scientifica through the INFM (Istituto Nazionale di Fisica della Materia) is gratefully acknowledged.
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abstract: 'In this article, we introduce a new family of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.'
address: 'Department of Mathematics, Central University of Rajasthan, Bandarsindri, Kishangarh-305817, Dist.-Ajmer, Rajasthan, India'
author:
- 'Rajbala and Jugal K. Prajapat'
title: 'CERTAIN GEOMETRIC PROPERTIES OF CLOSE-TO-CONVEX HARMONIC MAPPINGS'
---
Introduction
============
Let $\mathcal{H}$ denote the class of complex valued harmonic functions $f$ in $\mathbb{D}$ normalized by $f(0)=f_z(0)-1=0.$ Each such function $f$ can be expressed uniquely as $f=h+\overline{g},$ where $h$ and $g$ have the following power series representations: $$\label{intro1}
h(z) = z + \sum_{n=2}^{\infty} a_nz^n \quad {\rm and } \quad g(z)=\sum_{n=1}^{\infty} b_nz^n.$$ A result of Lewy [@lewy], shows that $f\in \mathcal{H}$ is locally univalent in $\mathbb{D}$ if and only if $J_f(z) = |f_z(z)|^2-|f_{\overline{z}}(z)|^2$ is non-zero in $\mathbb{D},$ and is sense preserving if $J_f(z)>0 \;(z\in \mathbb{D}),$ or equivalently, if the dilatation $w=g' /h'$ is analytic and satisfies $|w|<1$ in $\mathbb{D}.$ Observe that, the class $\mathcal{H}$ reduces to the class $\mathcal{A}$ of normalized analytic functions if the co-analytic part is zero. Let $\mathcal{S}_\mathcal{H}$ be the subclass of $\mathcal{H}$ consisting of univalent and sense-preserving harmonic mappings in $\mathbb{D}.$ The classical family $\mathcal{S}$ of normalized analytic univalent functions is subclass of $\mathcal{S}_{\mathcal{H}}$ as $\mathcal{S}=\{f=h+\overline{g}\in \mathcal{S}_{\mathcal{H}}:g\equiv 0 \quad {\rm in} \quad \mathbb{D} \}.$ Also, we denote by $\mathcal{H}^0=\left\lbrace f\in \mathcal{H}: f_{\overline{z}}(0)=0 \right\rbrace $ and $\mathcal{S}_\mathcal{H}^0=\left\lbrace f\in \mathcal{S} _\mathcal{H}: f_{\overline{z}}(0)=0 \right\rbrace.$ It is well known that the class $\mathcal{S}_\mathcal{H}^0$ is compact and normal, whereas the class $\mathcal{S}_\mathcal{H}$ is normal but not compact. In 1984, Clunie and Sheil-Small [@clunie] investigated the class $\mathcal{S}_\mathcal{H},$ together with some of its geometric subclasses.
A function $h \in \mathcal{A}$ is called close-to-convex in $\mathbb{D},$ if the complement of $h(\mathbb{D})$ can be written as the union of non-intersecting half lines. Let $\mathcal{C}$ denote the class of close-to-convex functions in $\mathbb{D}$. By $\mathcal{C}_{\mathcal{H}},$ we denote the class of close-to-convex harmonic mappings $f=h+\overline{g}$ for which $f(\mathbb{D})$ is close-to-convex in $\mathbb{D}.$ An analytic function $h \in \mathcal{A}$ is close-to-convex in $\mathbb{D},$ if there exists an convex function $\phi$ (not necessarily normalized) in $\mathbb{D}$ such that $$\Re\left( \dfrac{h'(z)}{\phi'(z)}\right)> 0\qquad (z\in \mathbb{D}).$$
If $\phi(z)=z,$ then functions $h\in \mathcal{A}$ which satisfy $\Re( h'(z))>0,$ are close-to-convex in $\mathbb{D}.$ A function $h \in \mathcal{A}$ is said to be close-to-convex function of order $\beta \;(0 \leq \beta <1)$, if it satisfies $\Re(h'(z))>\beta \;(z\in \mathbb{D})$. Let $\mathcal{W}(\alpha,\beta)$ denote a class of functions $h \in \mathcal{A}$ such that $\Re(h'(z)+\alpha zh''(z))>\beta \;\; (\alpha \geq 0, 0 \leq \beta <1).$ The class $\mathcal{W}(\alpha, \beta)$ was studied by Gao and Zohu [@AAaa] for $\beta<1$ and $\alpha>0.$ They determined the extreme points of $\mathcal{W}(\alpha, \beta)$ and obtained a number $\beta(\alpha)$ such that $\mathcal{W}(\alpha, \beta)\subset \mathcal{S}^*$ for fixed $\alpha \in [1,\infty).$ The class $\mathcal{W}(\alpha,\beta)$ is generalization of class $\mathcal{W}(\alpha) \equiv \mathcal{W}(\alpha,0)$, which was studied by Chichra [@chichra77]. In [@singh89], Singh and Singh proved that functions in $\mathcal{W}(1,0)$ are starlike in $\mathbb{D}.$
A harmonic function $f\in \mathcal{H}$ is said to be convex in $\mathbb{D}$, if $f(\mathbb{D})$ is convex in $\mathbb{D}$. We denote by $\mathcal{K}_{\mathcal{H}}\,$ the class of functions in $\mathcal{H}$ which are convex in $\mathbb{D}.$ A sense preserving harmonic mapping $f=h+\overline{g} \in \mathcal{H}$ is known to be convex in $\mathbb{D},$ if $\frac{\partial}{\partial \theta}\left(arg\,\left(\frac{\partial}{\partial \theta}f(re^{i \theta})\right)\right)>0$ for all $z=re^{i \theta}\in \mathbb{D}/\{0\}.$ Hence, $f=h+\overline{g}\in \mathcal{H}$ is convex in $\mathbb{D},$ if $f(z)\neq 0$ for all $z\in \mathbb{D}/\{0\}$ and condition $$\Re\left\{\dfrac{z(h'(z)+zh''(z))+\overline{z(g'(z)+zg''(z))}}{zh'(z)-zg'(z)} \right\}>0$$ is satisfied for all $z \in \mathbb{D}/\{0\}.$
Let $h\in \mathcal{S}$ be given by $h(z)=\sum_{n=0}^{\infty}a_n z^n.$ Then the $n^{th}$ partial sum (or section) of $h(z)$ is defined by $$s_n(h)=\sum_{k=0}^{n}a_kz^k \quad {\rm for}\quad n\in \mathbb{N},$$ where $a_0=0$ and $a_1=1.$ One of the classical results of Szegö [@szego28] shows that if $h \in \mathcal{S},$ then the partial sum $s_n(h)(z)=\sum_{k=0}^{n}a_kz^k$ is univalent in disk $|z|<1/4$ for all $n\geq 2,$ and number $1/4$ can not be replaced by larger one. In [@AAA], Robertson proved that $n^{th}$ partial sum of the Koebe function $k(z)=z/(1-z)^2$ is starlike in the disk $|z|<1-3n^{-1} \log n \quad (n\geq 5),$ and number $3$ can not be replaced by smaller constant. It is known by a result [@Durenp p. 256, 273], that $s_n(h)$ is convex, starlike, or close-to-convex in the disk $|z|<1-3n^{-1} \log n\quad (n\geq 5),$ whenever $h$ is convex, starlike or close-to-convex in $\mathbb{D}.$ The largest radius $r_n$ of univalence of $s_n(h) \,(h \in \mathcal{S})$ is not yet known. However, Jenkins [@jenkins] (see also [@Durenp Section 8.2]) observed that $r_n \geq 1-(4+\varepsilon) n^{-1} \,\mbox{log} \,n$ for each $\epsilon\,(|\epsilon|=1)$ and for large $n$. There exists a considerable amount of results in the literature for partial sums of functions in the class $\mathcal{S}$ and some of its geometric subclasses.
Analogously in the harmonic case, the $(p,q)$-th partial sum of a harmonic mapping $f=h+\overline{g}\in \mathcal{H}$ is defined by $$s_{p,q}(f)=s_p(h)+\overline{s_q(h)},$$ where $s_p(h)=\sum_{k=1}^{p}a_kz^k$ and $s_q(g)=\sum_{k=1}^{q}b_kz^k$, $p,q\geq 1$ with $a_1=1, \, p\geq 1$ and $q\geq2$. In [@li13], Li and Ponnusamy studied the radius of univalency of partial sums of functions in the class $\mathcal{P}_\mathcal{H}^0=\{f=h+\bar{g}\in \mathcal{H}^0: \,\, \Re(h'(z))>|g'(z)| \;(z\in \mathbb{D})\}.$ Further, in [@s13], Li and Ponnusamy studied partial sums of functions in the class $\mathcal{P}_\mathcal{H}^0(\alpha)=\{f=h+\overline{g} \in \mathcal{H}^0: \Re(h'(z)-\alpha) > |g'(z)| \; (\alpha<1, \;z \in \mathbb{D})\}$. Recently, Ghosh and Vasudevarao [@ghosh17] studied a class of harmonic mappings $\mathcal{W}_\mathcal{H}^0(\alpha)=\{f=h+\bar{g} \in \mathcal{H}^0 : \,\Re(h'(z)+\alpha zh''(z))> |g'(z)+\alpha zg''(z)|\; (z \in \mathbb{D})\}$ and gave some results concerning growth, convolution and convex combination for the members of the class $\mathcal{W}_\mathcal{H}^0(\alpha).$
For two analytic functions $\psi_1(z)=\sum_{n=0}^{\infty}a_nz^n$ and $\psi_2(z)=\sum_{n=0}^{\infty}b_nz^n,$ the convolution (or Hardamard product) is defined by $ \left( \psi_1\ast \psi_2 \right) (z)= \sum_{n=0}^{\infty} a_nb_nz^n \; (z\in\mathbb{D}).$ Analogously in the harmonic case, for two harmonic mappings $f_1=h_1+\overline{g_1}$ and $f_2=h_2+\overline{g_2}$ in $\mathcal{H}$ with the power series of the form $$f_1(z)=z+\sum_{n=2}^{\infty}a_nz^n+\overline{\sum_{n=1}^{\infty}b_n\, z^n} \quad {\rm and} \quad f_2(z)=z+\sum_{n=2}^{\infty}A_nz^n+\overline{\sum_{n=1}^{\infty}B_n\, z^n},$$ we define the harmonic convolution as follows: $$\; f_1 \ast f_2 = h_1\ast h_2+\overline{g_1\ast g_2}=z+\sum_{n=2}^{\infty}a_n A_n z^n+\overline{\sum_{n=1}^{\infty}b_n\,B_n\, z^n}.$$ Clearly, the class $\mathcal{H}$ is closed under the convolution, [*i.e.*]{} $\mathcal{H}\ast \mathcal{H}\subset \mathcal{H}.$ In the case of conformal mappings, the literature about convolution theory is exhaustive. Unfortunately, most of these results do not necessarily carry over to the class of univalent harmonic mappings in $\mathbb{D}.$ We refer [@droff1; @kumar; @ELiu], for more information about convolution of harmonic mappings.
We now define a new class of close-to-convex harmonic mappings as follows:
[**Definition 1.1.**]{} For $\alpha \geq 0$ and $0 \leq \beta <1$, let $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ denote the class of harmonic mappings $f=h+\overline{g}$, which is defined by $$\mathcal{W}_\mathcal{H}^0(\alpha,\beta)=\{f=h+\bar{g} \in \mathcal{H}^0: \;\Re(h'(z)+\alpha zh''(z)-\beta)> |g'(z)+\alpha zg''(z)|\quad (z \in \mathbb{D}) \}.$$
We observe that, the class $\mathcal{W}_{\mathcal{H}}^0(\alpha,\beta)$ generalizes several previously studied classes of harmonic mappings, as $\mathcal{W}_{\mathcal{H}}^0(\alpha,0) \equiv \mathcal{W}_{\mathcal{H}}^0(\alpha)$ (see [@ghosh17]), $\mathcal{W}_\mathcal{H}^0(0,\beta) \equiv \mathcal{P}_\mathcal{H}^0(\beta)$ (see [@s13]), $\mathcal{W}_\mathcal{H}^0(1,0) \equiv \mathcal{W}_\mathcal{H}^0$ (see [@nagpal14]), and $\mathcal{W}_\mathcal{H}^0(0,0)\equiv \mathcal{P}_{\mathcal{H}}^0$ (see [@li13]).
In this article, we establish that functions in the class $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ are close-to-convex $\mathbb{D}$. In section $3,$ we obtain certain coefficient inequalities and growth results for the functions in $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$. In section $4,$ we prove that the functions in $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ are closed under convex combinations and establish certain convolution results. In section $5$, we determine the radius of convexity of partial sums $s_{p,q}(f)$ of functions in $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$. Finally, in section $6,$ we consider the harmonic mappings which involve the hypergeometric function and obtain conditions on its parameters such that it belongs to the class $\mathcal{W}_\mathcal{H}^0(\alpha,\beta).$ Further we construct the univalent harmonic polynomials belonging to $\mathcal{W}_\mathcal{H}^0(\alpha,\beta).$ The following results will be needed in our investigation.
\[LEMA\] (see, [@goodman]). Let $p\in \mathcal{P},$ where $\mathcal{P}$ denotes the class of Carathéodory functions in $\mathbb{D}.$ Then $$\left|p'(z)\right|\geq \dfrac{1-|z|}{1+|z|}\qquad {\rm and} \qquad \left|\dfrac{p''(z)}{p'(z)}\right|\leq \dfrac{2}{1-|z|^2} \quad (z\in \mathbb{D}).$$ These inequalities are sharp. Equality occurs for suitable $z\in \mathbb{D}$ if and only if $p(z)=-z-2e^{i \theta}\log (1-z e^{i \theta}) \quad (0\leq \theta\leq 2 \pi).$
\[lm.6\] If the harmonic mapping $f=h+\overline{g}:\mathbb{D}\rightarrow\mathbb{C}$ satisfies $|g'(0)|<|h'(0)|$ and the function $F_\epsilon=h+\epsilon g$ is close-to-convex for every $|\epsilon|=1,$ then $f$ is close-to-convex function.
The Close-to-Convexity
======================
The first result provides a one-to-one correspondence between the classes $\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta)$ of harmonic mappings and the class $\mathcal{W}(\alpha, \beta)$ of analytic functions.
The harmonic mapping $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ if and only if $F_{\epsilon}=h+\epsilon g \in \mathcal{W}(\alpha,\beta)$ for each $|\epsilon|=1$. \[th1\]
Let $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$. Then for each $|\epsilon|=1$, we have $$\begin{aligned}
\Re(F_{\epsilon}'(z)+\alpha zF_{\epsilon}''(z))&=& \Re(h'(z)+\epsilon g'(z)+\alpha z(h''(z)+\epsilon g''(z))\\
&=& \Re(h'(z)+\alpha zh''(z)+\epsilon (g'(z)+\alpha g''(z))\\
&>& \Re(h'(z)+\alpha zh''(z))-|g'(z)+\alpha zg''(z)|> \beta \quad (z \in \mathbb{D}).\end{aligned}$$ Hence $F_{\epsilon} \in \mathcal{W}(\alpha,\beta)$ for each $|\epsilon|=1$. Conversely, let $F_{\epsilon} \in \mathcal{W}(\alpha,\beta).$ Then $$\Re(h'(z)+\alpha zh''(z))> \Re(-\epsilon (g'(z)+\alpha zg''(z)))+\beta \quad (z\in \mathbb{D}).$$ As $\epsilon(|\epsilon|=1)$ is arbitrary, then for an appropriate choice of $\epsilon,$ we obtain $$\Re(h'(z)+\alpha zh''(z)-\beta) >|g'(z)+\alpha zg''(z)| \quad (z\in \mathbb{D}),$$ and hence we conclude that $f \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
To establish the next result, we need to establish that functions in the class $\mathcal{W}(\alpha,\beta)$ are close-to-convex in $\mathbb{D}$, and to prove this, we shall need the following result.
(Jack’s Lemma [@jack71]) Let $\omega(z)$ be analytic in $\mathbb{D}$ with $\omega(0)=0.$ If $|\omega(z)|$ attains its maximum value on the circle $|z|=r<1$ at a point $z_0\in \mathbb{D}$, then we have $z_0 \omega'(z_0)=k\omega(z_0)$ for a real number $k\geq 1.$ \[3\]
If $f \in \mathcal{W}(\alpha,\beta)$, then $\Re(f'(z))>\beta \,\,(0\leq\beta<1)$, and hence $f$ is close-to-convex in $\mathbb{D}$. \[5\]
If $f \in \mathcal{W}(\alpha,\beta),$ then $\Re (\psi(z))>0$, where $\psi(z)=f'(z)+\alpha z f''(z)-\beta$. Let $w$ be an analytic function in $\mathbb{D}$ such that $w(0)=0$ and $$f'(z)=\frac{1+(1-2\beta)w(z)}{1-w(z)}.$$ To prove the result, we need to show that $|w(z)|<1$ for all $z$ in $\mathbb{D}$. If not, then by Lemma \[3\], we could find some $\xi (|\xi|<1)$, such that $|w(\xi)|=1$ and $\xi w'(\xi)=kw(\xi)$, where $k \geq 1$. A computation gives $$\begin{aligned}
\Re \left\{\psi(\xi) \right\}&=&\Re \left\{\frac{1+(1-2\beta)w(\xi)}{1-w(\xi)}+\frac{2\alpha k(1-\beta)w(\xi)}{(1-w(\xi))^2}-\beta \right\} \\
&=& \Re \left\{\frac{ 2\alpha k(1-\beta)w(\xi)}{(1-w(\xi))^2} \right\} = - \frac{4\alpha k(1-\beta) (1-\Re(w(\xi))}{|1-w(\xi)|^4} \leq 0\end{aligned}$$ for $|w(\xi)|=1$. This contradicts the hypotheses. Hence, $|w(z)|<1,$ which lead to $\Re(f'(z))>\beta \,\,(0\leq\beta<1).$
\[newthm\] The functions in the class $\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta)$ are close-to-convex in $\mathbb{D}.$
From Lemma \[5\], we find that functions $F_\epsilon=h+\epsilon g \in \mathcal{W}(\alpha, \beta)$ are close-to-convex in $\mathbb{D}$ for each $\epsilon(|\epsilon|=1).$ Now in view of Lemma \[lm.6\] and Theorem \[th1\], we obtain that functions in $\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta)$ are close-to-convex in $\mathbb{D}.$
Coefficient Inequalities and Growth Estimates
=============================================
The following results provides sharp coefficient bounds for the functions in $\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
\[thm1\] Let $f=h+\overline{g}\in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ be of the form with $b_1=0.$ Then we have $$|b_n|\leq\dfrac{1-\beta}{n(1+\alpha(n-1))}.
\label{eq6}$$ The result is sharp and equality in is obtained by $f(z)=z+\dfrac{1-\beta}{n(1+\alpha(n-1))}\overline{z}^n$.
Since $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$, then using the series representation of $g$, we have $$\begin{aligned}
r^{n-1} n(1+\alpha(n-1))|b_n|&\leq& \frac{1}{2\pi}\int_{0}^{2\pi}|g'(re^{i\theta })+\alpha re^{i\theta}g''(re^{i\theta})|d\theta\\
&<& \frac{1}{2\pi}\int_{0}^{2\pi}\{\Re(h'(re^{i\theta})+\alpha re^{i\theta}h''(re^{i\theta}))-\beta\}d\theta \\
&=& \dfrac{1}{2 \pi} \int_0^{2\pi}\{1-\beta+ n(1+\alpha (n-1))a_n r^{n-1} e^{i(n-1)\theta} \}d \theta =1-\beta. \end{aligned}$$ Now $r\rightarrow 1^{-}$ gives the desired bound. Further, it is easy to see that the equality in is obtained for the function $f(z)=z+\dfrac{1-\beta}{n(1+\alpha(n-1))}\overline{z}^n$.
\[th3\] Let $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ be of the form with $b_1=0.$ Then for $n \geq 2$, we have
- $|a_n|+|b_n|\leq \dfrac{2(1-\beta)}{n(1+\alpha(n-1))},$
- $||a_n|-|b_n||\leq \dfrac{2(1-\beta)}{n(1+\alpha(n-1))},$
- $|a_n|\leq \dfrac{2(1-\beta)}{n(1+\alpha(n-1))}. $
All these results are sharp for the function $f(z)=z+\sum_{n=2}^{\infty}\dfrac{2(1-\beta)}{n(1+\alpha(n-1))}\overline{z}^n$.
\(i) Since $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$, then Theorem \[th1\] implies that $F_{\epsilon}=h+\epsilon g \in \mathcal{W}(\alpha,\beta)$ for each $\epsilon(|\epsilon|=1)$. Thus for each $|\epsilon|=1$, we have $$\Re((h+\epsilon g)'(z)+\alpha z(h+\epsilon g)''(z))>\beta \quad {\rm{for}} \quad z \in \mathbb{D}.$$ This implies that there exists a [*Carathéodory*]{} function of the form $p(z)=1+\sum_{n=1}^{\infty}p_nz^n$, with $\Re(p(z))>0$ in $\mathbb{D}$, such that $$h'(z)+\alpha zh''(z)+\epsilon (g'(z)+\alpha zg''(z))=\beta+(1-\beta)p(z).
\label{eq7}$$ Comparing coefficients on both sides of , we obtain $$n(1+\alpha(n-1))(a_n+\epsilon b_n)=(1-\beta)p_{n-1}\quad {\rm for} \quad n\geq 2.
\label{eq8}$$ Since $|p_n|\leq 2$ for $n\geq 1$ (see [@Durenp p. 41]), and $\epsilon(|\epsilon|=1)$ is arbitrary, therefore the result follows from . Part (ii) and (iii) follows from part (i).
The following result gives a sufficient condition for a function to be in the class $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
\[th5\] Let $f=h+\overline{g} \in \mathcal{H}^0$, where $h$ and $g$ are of the form . If $$\sum_{n=2}^{\infty}n(1+\alpha(n-1))(|a_n|+|b_n|)\leq 1-\beta,
\label{eq13}$$ then $f \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
If $f=h+\bar{g} \in \mathcal{H}^0$, then using , we have $$\begin{aligned}
\Re(h'(z)+\alpha zh''(z))&=& \Re\Big(1+\sum_{n=2}^{\infty} n(1+\alpha(n-1))\,a_n \,z^{n-1}\Big) \\
&\geq& 1-\sum_{n=2}^{\infty}n(1+\alpha(n-1))|a_n| \geq\sum_{n=2}^{\infty}n(1+\alpha(n-1))|b_n|+\beta\\
&\geq& |\sum_{n=2}^{\infty}n(1+\alpha(n-1)) \,b_n|+\beta =|g'(z)+\alpha zg''(z)|+\beta,\end{aligned}$$ and so $f \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
The following theorem gives sharp inequalities in the class $\mathcal{B}_{\mathcal{H}}^0(\alpha, \beta).$
\[th4\] If $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$, then $$|z|-2\sum_{n=2}^{\infty}\dfrac{(-1)^{n-1}(1-\beta)|z|^n}{\alpha n^2+n(1-\alpha)} \leq |f(z)|\leq |z|+2\sum_{n=2}^{\infty}\dfrac{(1-\beta)|z|^n}{\alpha n^2+n(1-\alpha)}.
\label{eq9}$$ Both the inequalities are sharp when $f(z)=z+\sum_{n=2}^{\infty} \dfrac{2(1-\beta)}{\alpha n^2+n(1-\alpha)}\overline{z}^n$, or its rotations.
Let $f=h+\bar{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$. Then $F_{\epsilon}=h+\epsilon g \in \mathcal{W}(\alpha,\beta)$ for each $\epsilon\,(|\epsilon|=1)$. Thus there exists an analytic function $w(z)$ with $w(0)=0$ and $|w(z)|<1$ in $\mathbb{D}$, such that $$F_{\epsilon}'(z)+\alpha zF_{\epsilon}''(z)=\frac{1+(1-2\beta)w(z)}{1-w(z)}.
\label{eq11}$$ Simplifying , we get $$\begin{aligned}
z^{1/\alpha}F_{\epsilon}'(z) =\dfrac{1}{\alpha}\int_{0}^{z}\xi^{\frac{1}{\alpha}-1}\frac{1+(1-2\beta)w(\xi)}{1-w(\xi)}d\xi = \dfrac{1}{\alpha}\int_{0}^{|z|}(te^{i\theta})^{\frac{1}{\alpha}-1}\frac{1+(1-2\beta)w(te^{i\theta})}{1-w(te^{i\theta})}e^{i\theta}dt.\end{aligned}$$ Therefore using Schwarz Lemma, we have $$\begin{aligned}
|z^{1/\alpha}F_{\epsilon}'(z)|=\Big|\dfrac{1}{\alpha}\int_{0}^{|z|}(te^{i\theta})^{\frac{1}{\alpha}-1}\frac{1+(1-2\beta)w(te^{i\theta})}{1-w(te^{i\theta})}e^{i\theta}dt\Big| \leq \frac{1}{\alpha}\int_{0}^{|z|}t^{\frac{1}{\alpha}-1}\frac{1+(1-2\beta)t}{1-t}dt,\end{aligned}$$ and $$\begin{aligned}
|z^{1/\alpha}F_{\epsilon}'(z)|&=&\Big|\dfrac{1}{\alpha}\int_{0}^{|z|}(te^{i\theta})^{\frac{1}{\alpha}-1}\frac{1+(1-2\beta)w(te^{i\theta})}{1-w(te^{i\theta})}e^{i\theta}dt\Big|\\
&\geq &\dfrac{1}{\alpha}\int_{0}^{|z|}t^{\frac{1}{\alpha}-1}\; \Re{\frac{1+(1-2\beta)w(te^{i\theta})}{1-w(te^{i\theta})}}dt\\
&\geq&\frac{1}{\alpha}\int_{0}^{|z|}t^{\frac{1}{\alpha}-1}\; \frac{1+(1-2\beta)t}{1-t}dt.\end{aligned}$$ Further computation gives $$|F'(z)|=|h'(z)+\epsilon g'(z)|\leq 1+2(1-\beta)\sum_{n=1}^{\infty}\frac{|z|^n}{1+\alpha n},
\label{eq12}$$ and $$|F'(z)|=|h'(z)+\epsilon g'(z)|\geq 1+2(1-\beta)\sum_{n=1}^{\infty}\frac{(-1)^n|z|^n}{1+\alpha n}.$$ Since $\epsilon(|\epsilon|=1)$ is arbitrary, it follows from that $$|h'(z)|+| g'(z)|\leq 1+2(1-\beta)\sum_{n=1}^{\infty}\frac{|z|^n}{1+\alpha n},$$ and $$|h'(z)|-| g'(z)|\geq 1-2(1-\beta)\sum_{n=1}^{\infty}\frac{(-1)^n|z|^n}{1+\alpha n}.$$ Let $\Gamma$ be the radial segment from 0 to $z$, then $$\begin{aligned}
|f(z)|&=&\Big|\int_\Gamma \dfrac{\partial f}{\partial \xi}d\xi +\frac{\partial f}{\partial \bar{\xi}}d\bar{\xi}\Big|\leq \int_\Gamma (|h'(\xi)|+| g'(\xi)|)|d\xi|\\
&\leq& \int_{0}^{|z|}\Big( 1+2(1-\beta)\sum_{n=1}^{\infty}\dfrac{|t|^n}{1+\alpha n}\Big)dt=|z|+2(1-\beta)\sum_{n=2}^{\infty}\frac{|z|^n}{\alpha n^2+(1-\alpha)n},\end{aligned}$$ and $$\begin{aligned}
|f(z)|&=&\int_\Gamma \Big|\dfrac{\partial f}{\partial \xi}d\xi +\frac{\partial f}{\partial \bar{\xi}}d\bar{\xi}\Big|\geq \int_\Gamma (|h'(\xi)|-| g'(\xi)|)|d\xi|\\
&\geq& \int_{0}^{|z|}\Big( 1-2(1-\beta
)\sum_{n=1}^{\infty}\frac{(-1)^n|t|^n}{1+\alpha n}\Big)dt=|z|+2(1-\beta)\sum_{n=2}^{\infty}\frac{(-1)^{n-1}|z|^n}{\alpha n^2+(1-\alpha)n}.\end{aligned}$$ Equality in holds for the function $ f(z)=z+\sum_{n=2}^{\infty}\dfrac{2(1-\beta)}{\alpha n^2+(1-\alpha)n}\overline{ z}^n$ or its rotations.
Convex combinations and convolutions
====================================
In this section, we prove that the class $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ is closed under convex combinations and convolutions. A sequence $\{c_n\}_{n=0}^{\infty}$ of non-negative real numbers is said to be a convex null sequence, if $c_n\rightarrow 0$ as $n\rightarrow \infty$, and $c_0-c_1\geq c_1-c_2 \geq c_2-c_3 \geq...\geq c_{n-1}-c_n\geq ...\geq 0.$ To prove results for convolution, we shall need the following Lemma \[7\] and \[8\].
[@LFlf] \[7\] If $\{c_n\}_{n=0}^{\infty}$ be a convex null sequence, then function $q(z)=\dfrac{c_0}{2}+\sum_{n=1}^{\infty}c_nz^n$ is analytic and $\Re(q(z)) >0$ in $\mathbb{D}$.
[@singh89]\[8\] Let the function $p$ be analytic in $\mathbb{D}$ with $p(0)=1$ and $\Re(p(z))>1/2$ in $\mathbb{D}$. Then for any analytic function $f$ in $\mathbb{D}$, the function $p*f$ takes values in the convex hull of the image of $\mathbb{D}$ under $f$.
\[th6\] The class $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ is closed under convex combinations.
Let $f_i=h_i+\overline{g_i} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ for $i=1,2,...n$ and $\sum_{i=1}^{n}t_i=1(0\leq t_i \leq 1)$. Write the convex combination of $f_i's$ as $$f(z)=\sum_{i=1}^{n}t_if_i(z)=h(z)+\overline{g(z)},$$ where $ h(z)=\sum_{i=1}^{n}t_ih_i(z)$ and $ g(z)=\sum_{i=1}^{n}t_ig_i(z)$. Clearly both $h$ and $g$ are analytic in $\mathbb{D}$ with $h(0)=g(0)=h'(0)-1=g'(0)=0.$ A simple computation yields $$\begin{aligned}
\Re(h'(z)+\alpha zh''(z))&=& \Re\Big(\sum_{i=1}^{n}t_i(h'(z)+\alpha zh''(z))\Big) > \sum_{i=1}^{n}t_i(|g_i'(z)+\alpha zg_i''(z)|+\beta)\\
&\geq& |g'(z)+\alpha zg''(z)|+\beta.\end{aligned}$$ This shows that $f \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
\[9\] If $F \in \mathcal{W}(\alpha,\beta)$, then $\Re\Big(\dfrac{F(z)}{z}\Big) > \dfrac{1}{2-\beta}$.
If $F \in \mathcal{W}(\alpha,\beta)$ be given by $F(z)=z+\sum_{n=2}^{\infty}A_nz^n$, then $$\Re \Big(1+\sum_{n=2}^{\infty} n(1+\alpha (n-1))A_n z^{n-1}\Big)>\beta \quad (z \in \mathbb{D}),$$ which is equivalent to $\Re(p(z))>\dfrac{1}{2-\beta}\geq \dfrac{1}{2}\;$ in $\mathbb{D}$, where $p(z)=1+\dfrac{1}{2-\beta}\sum_{n=2}^{\infty}n(1+\alpha (n-1))A_n z^{n-1}.$ Now consider a sequence $\{c_n\}_{n=0}^{\infty}$ defined by $c_0=1$ and $c_{n-1}=\dfrac{2-\beta}{n(1+\alpha(n-1))}$ for $n\geq 2$. We can easily see that the sequence $\{c_n\}_{n=0}^{\infty}$ is convex null sequence and hence in view of Lemma \[7\], the function $q(z)=\frac{1}{2}+\sum_{n=2}^{\infty}\dfrac{2-\beta}{n(1+\alpha(n-1))}z^{n-1}$ is analytic and $\Re(q(z))>0$ in $\mathbb{D}$. Further $$\frac{F(z)}{z}=p(z)*\Big(1+\sum_{n=2}^{\infty}\dfrac{2-\beta}{n(1+\alpha(n-1))}z^{n-1}\Big).$$ Hence an application of Lemma \[8\] gives that $\Re\Big(\dfrac{F(z)}{z}\Big)>\dfrac{1}{2-\beta}$ for $z\in \mathbb{D}$.
\[lm10\] Let $F_1$ and $F_2$ belong to $\mathcal{W}(\alpha,\beta)$, then $F_1*F_2 \in \mathcal{W}(\alpha,\beta)$.
The convolution of $F_1=z+\sum_{n=2}^{\infty}A_nz^n$ and $F_2=z+\sum_{n=2}^{\infty}B_nz^n$ is given by $$F(z)=(F_1*F_2)(z)=z+\sum_{n=2}^{\infty}A_nB_nz^n.$$ Since $zF'(z)=zF_1'(z)*F_2(z)$, therefore a computation shows that $$\frac{1}{1-\beta} \big(F'(z)+z\alpha F''(z)-\beta \big) =\frac{1}{1-\beta}(F_1'(z)+z\alpha F_1''(z)-\beta)*\Big(\frac{F_2(z)}{z}\Big).
\label{eq14}$$ Since $F_1 \in \mathcal{W}(\alpha,\beta)$, hence it satisfy $\Re(F_1'(z)+\alpha zF_1''(z)-\beta)>0.$ Further from Lemma \[9\], we have $\Re(\dfrac{F_2(z)}{z})> \dfrac{1}{2-\beta}\geq\dfrac{1}{2}$ in $\mathbb{D}$. Now applying Lemma \[8\], we get $F=F_1*F_2 \in \mathcal{W}(\alpha,\beta)$.
Now using Lemma \[lm10\], we will show that the class $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ is closed under convolutions.
\[11\] If functions $f_1$ and $f_2$ belong to $\mathcal{W}_\mathcal{H}^0(\alpha,\beta),$ then $f_1*f_2 \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
Let the functions $f_1=h_1+\overline{g_1}$ and $f_2=h_2+\overline{g_2}$ are belongs to $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$. To show $f_1*f_2 \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$, it is sufficient to show that $F_{\epsilon}=h_1*h_2+\epsilon( {g_1*g_2}) \in \mathcal{W}(\alpha,\beta)$ for each $\epsilon(|\epsilon|=1)$. By Lemma \[lm10\], $\mathcal{W}(\alpha,\beta)$ is closed under convolutions. If $h_i+\epsilon g_i \in \mathcal{W}(\alpha,\beta)$ for each $\epsilon(|\epsilon|=1)$ and for $i=1,2$. Then both $F_1$ and $F_2$ given by $$F_1(z)=(h_1-g_1)*(h_2-\epsilon g_2) \quad {\mbox and} \quad F_2(z)=(h_1+g_1)*(h_2+\epsilon g_2),$$ belong to $\mathcal{W}(\alpha,\beta)$. Since $\mathcal{W}(\alpha,\beta)$ is close under convex combinations, then the function $F_{\epsilon}=\dfrac{1}{2}(F_1+F_2)=(h_1*h_2)+\epsilon (g_1*g_2)$ belongs to $\mathcal{W}(\alpha,\beta)$. Hence $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ is closed under convolution.
In [@Goodloe02], Goodloe considered the Hadamard product of a harmonic function with an analytic function defined as follows: $$f\,\widehat{*}\phi=h*\phi+\overline{g*\phi},$$ where $f=h+\overline{g}$ is harmonic function and $\phi$ is an analytic function in $\mathbb{D}.$
\[12\] Let $f \in \mathcal{W}_{\mathcal{H}}^0(\alpha,\beta)$ and $\phi \in \mathcal{A}$ be such that $\Re\Big(\dfrac{\phi(z)}{z}\Big)> \dfrac{1}{2}$ for $z \in \mathbb{D}$, then $f\,\widehat{*}\phi \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
Let $f=h+\overline{g} \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$. To prove that $f\,\widehat{*}\phi$ belongs to $\mathcal{W}_\mathcal{H}^0(\alpha,\beta)$, it suffices to prove that $F_{\epsilon}=h*\phi +\epsilon (g*\phi)$ belongs to $\mathcal{W}(\alpha,\beta)$ for each $\epsilon (|\epsilon|=1)$. Since $f=h+\overline{g}\in \mathcal{W}_\mathcal{H}^0(\alpha,\beta),$ then $F_{\epsilon}=h+\epsilon g$ belongs to $\mathcal{W}(\alpha,\beta)$ for each $\epsilon (|\epsilon|=1)$. Therefore $$\frac{1}{1-\beta} (F_\epsilon'(z)+\alpha zF_{\epsilon}''(z)-\beta)=\frac{1}{1-\beta}(F_{\epsilon}'(z)+\alpha zF_{\epsilon}''(z)-\beta)*\frac{\phi(z)}{z}.$$ Since $\Re\Big(\dfrac{\phi(z)}{z}\Big)> \dfrac{1}{2}$ and $\Re(F_\epsilon'(z)+\alpha zF_\epsilon''(z))>\beta$ in $\mathbb{D}$, then in view of Lemma \[8\], we obtain that $F_\epsilon \in \mathcal{W}(\alpha,\beta)$.
Suppose $f \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$ and $\phi \in \mathcal{K}$, then $f\,\widehat*\phi \in \mathcal{W}_\mathcal{H}^0(\alpha,\beta)$.
It is well known that, if $\phi$ is convex then $\Re\Big(\dfrac{\phi(z)}{z}\Big)> \dfrac{1}{2}$ for $z \in \mathbb{D}$. Hence result follows from Theorem \[12\].
Partial sums
============
In this section, we determine the value of $r$ such that the partial sums of $f \in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta)$ are convex in the disk $|z|<r.$
\[P1\] Let $f=h+\overline{g}\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$ If $p$ and $q$ satisfies one of the following conditions:
- $1=p\,<\,q$
- $3\,\leq \, p\,<\, q$
- $3\leq q<p,$
then $s_{p,q}(f)(z)$ is convex in $|z|<1/4.$
\(i) By assumption, we know that $$s_{1,q}(f)(z)=z+\overline{s_q(g)(z)}=z+\sum_{n=2}^q\overline{b_n z^n}.$$ Since $$\Re\left\{\dfrac{z+\overline{z(zs'_q(g)(z))'}}{z-\overline{zs'_q(g)(z)}} \right\}=\Re \left\{ \dfrac{z+\sum_{n=2}^q \overline{n^2b_n z^n}}{z-\sum_{n=2}^q\overline{n b_n z^n}}\right\}\quad {\rm and} \quad \lim _{z \rightarrow 0} \dfrac{z+\sum_{n=2}^q\overline{n^2b_n z^n}}{z-\sum_{n=2}^q\overline{n b_n z^n}}=1,$$ it suffices to prove $$A=:\Re \left\{\left(z+\sum_{n=2}^q\overline{n^2 b_n z^n}\right) \left(\overline{z}-\sum_{n=2}^qn b_n z^n \right) \right\}>0 \qquad {\rm for}\qquad |z|=1/4.$$ Now, we find that $$\begin{aligned}
\qquad A &=& |z|^2+\Re \left(\sum_{n=2}^q \overline{n^2 b_n z^{n+1}}-\sum_{n=2}^q n b_n z^{n+1}\right)-\Re \left\{\left(\sum_{n=2}^q\overline{n^2 b_n z^n} \right) \left(\sum_{n=2}^q nb_n z^n \right) \right\} \\
&\geq& |z|^2-\sum_{n=2}^q n(n-1)|b_n||z|^{n+1}-\left(\sum_{n=2}^q n^2 |b_n||z|^n \right)\left(\sum_{n=2}^q n|b_n| |z|^n \right).\end{aligned}$$ Further, using Theorem \[thm1\], we obtain $$\begin{aligned}
A &\geq& |z|^2-\sum_{n=2}^q \dfrac{(1-\beta)(n-1)}{1+\alpha (n-1)} |z|^{n+1}-\left(\sum_{n=2}^{\infty}\dfrac{n (1-\beta)}{1+\alpha(n-1)}|z|^n \right) \left(\sum_{n=2}^{\infty}\dfrac{(1-\beta)}{1+\alpha (n-1)}|z|^n \right)\notag \\
&\geq& |z|^2-(1-\beta)\sum_{n=2}^q (n-1) |z|^{n+1}-(1-\beta)^2\left(\sum_{n=2}^{\infty}n |z|^n \right) \left(\sum_{n=2}^{\infty}|z|^n \right) \notag \\
&=& |z|^2-(1-\beta)|z|^3\dfrac{1-q|z|^{q-1}+(q-1)|z|^q}{(1-|z|)^2}\\
&& \qquad -(1-\beta)^2 |z|^4 \dfrac{(2-|z|-(q+1)|z|^{q-1}+q|z|^q)(1-|z|^{q-1})}{(1-|z|)^3}. \notag\end{aligned}$$ Thus, for $|z|=1/4$, we have $$\begin{aligned}
\dfrac{A\,(1-|z|)^3}{|z|^2} &\geq&(1-|z|)^3-(1-\beta)|z|(1-|z|)(1-q|z|^{q-1}+(q-1)|z|^q) \notag \\
&& \;\;\; -(1-\beta)^2|z|^2(2-|z|-(q+3)|z|^{q-1}+(q+1)|z|^q+(q+1)|z|^{2q-2}-q|z|^{2q-1}) \notag \\
&\geq& \dfrac{27}{64}-\dfrac{3}{16}\left(1-\dfrac{q}{4^{q-1}}-\dfrac{q-1}{4^q}\right)-\dfrac{1}{16} \left(\dfrac{7}{4}-\dfrac{q+3}{4^{q-1}}+\dfrac{q+1}{4^q}+\dfrac{q+1}{4^{2(q-1)}}-\dfrac{q}{4^{2q-1}}\right) \notag \\
&=& \dfrac{1}{8}+ \dfrac{12q+14}{4^{q+2}}-\dfrac{3q+4}{4^{2q-1}} = \dfrac{1}{8}+\dfrac{12 q(4^q-1)+14\times 4^q-16}{4^{2q+2}}>0. \notag\end{aligned}$$ Hence the result follows.
\(ii) Let $\sigma_p(h)(z)=\sum_{n=p+1}^{\infty}a_n z^n$ and $\sigma_q(g)(z)=\sum_{n=q+1}^{\infty}b_nz^n,$ so that $h(z)=s_p(h)(z)+\sigma_p(h)(z)$ and $g(z)=s_q(g)(z)+\sigma_q(g)(z).$ Thus for each $|\epsilon|=1$, we may write $$\label{.eq1}
1+z\,\dfrac{s_p''(h)(z)+\epsilon s_q''(g)(z)}{s_p'(h)(z)+\epsilon s'_q(g)(z)}=1+\phi(z)+\psi(z),$$ where $$\phi(z)=\dfrac{z(h''(z)+\epsilon g''(z))}{h'(z)+\epsilon g'(z)}\qquad \rm and$$ $$\psi(z)=\dfrac{\phi(z)(\sigma'_p(h)(z)+\epsilon\sigma'_q(g)(z))-z(\sigma_p''(h)(z)+\epsilon\sigma_q''(g)(z))}{h'(z)+\epsilon g'(z)-(\sigma_p'(h)(z)+\epsilon \sigma'_q(g)(z))}.$$ Since $h+\epsilon g \in \mathcal{P},$ using Lemma \[LEMA\], we have $$\label{.eq2}
|\phi(z)|\leq \dfrac{2|z|}{1-|z|^2}\qquad {\rm and} \qquad |h'(z)+\epsilon g'(z)|\geq \dfrac{1-|z|}{1+|z|}.$$ Now, if $p\leq q,$ then Theorem \[thm1\], yields that $$\begin{aligned}
\label{.eq3}
|\sigma_p'(h)(z)+\epsilon \sigma'_q(g)(z)| &=& \left| \sum_{n=p+1}^qna_nz^{n-1}+\sum_{n=q+1}^\infty n(a_n+\epsilon b_n)z^{n-1} \right| \notag \\
&\leq & \sum_{n=p+1}^{\infty}\dfrac{2(1-\beta)}{1+\alpha(n-1)}|z|^{n-1}\,\leq\, 2(1-\beta)\sum _{n=p+1}^{\infty}|z|^{n-1} \notag \\
&=& 2(1-\beta) \dfrac{|z|^p}{1-|z|}.\end{aligned}$$ Similarly, $$\begin{aligned}
\label{.eq4}
|z(\sigma_p''(h)(z)+\epsilon \sigma _q''(g)(z))| &= &\left|\sum_{n=p+1}^q n(n-1)a_nz^{n-1}+\sum_{n=q+1}^{\infty} n(n-1)(a_n+\epsilon b_n)z^{n-1} \right| \notag \\
&\leq& \sum_{n=p+1}^{\infty}\dfrac{2(1-\beta)(n-1)}{1+\alpha (n-1)}|z|^{n-1}\,\leq \,2(1-\beta)\sum_{n=p+1}^{\infty}(n-1)|z|^{n-1} \notag \\
&= & 2(1-\beta) \left(\dfrac{p|z|^p}{1-|z|}+\dfrac{|z|^{p+1}}{(1-|z|)^2} \right).\end{aligned}$$ Using estimates - , by the triangle inequality we deduce that $$\left| \psi(z)\right| \leq \dfrac{2(1-\beta)|z|^p\{3|z|+|z|^2+p(1-|z|^2)\}}{(1-|z|)\{(1-|z|)^2-2(1-\beta)|z|^p(1+|z|)\}}.$$ Thus $$\begin{aligned}
\Re(1+\phi(z)+\psi(z))&\geq& 1-|\phi(z)|-|\psi(z)|\notag \\
&\geq& 1-\dfrac{2|z|}{1-|z|^2}-\dfrac{2(1-\beta)|z|^p\{3|z|+|z|^2+p(1-|z|^2)\}}{(1-|z|)\{(1-|z|)^2-2(1-\beta)|z|^p(1+|z|)\}}\notag\\
&=& \dfrac{1-|z|^2-2|z|}{1-|z|^2}-\dfrac{2(1-\beta)|z|^p\{3|z|+|z|^2+p(1-|z|^2)\}}{(1-|z|)\{(1-|z|)^2-2(1-\beta)|z|^p(1+|z|)\}}, \notag\end{aligned}$$ which for $|z|=1/4$ gives $$\Re(1+\phi(z)+\psi(z))\geq \dfrac{1}{3}\left\{\dfrac{7}{5}-\dfrac{2(1-\beta)(13+15 p)}{9\times 4^{p-1}-10(1-\beta)} \right\}=B(p,\beta).$$ Since the function $B(p,\beta)$ is monotonically increasing with respect to $p$ for $p\geq3,$ the least estimate shows that $\Re(1+\phi(z)+\psi(z))\geq A(p)\geq A(3)>0.$ Thus implies for each $|\epsilon|=1,$ that the section $s_p(h)+\epsilon s_q(g)$ is convex in $|z|\leq 1/4$ for $3\leq p \leq q.$ As $\epsilon$ is arbitrary, this shows that $s_{p,q}(f)$ is convex in $|z|<1/4,$ for $3\leq p\leq q.$
\(iii) If $p>q,$ then using Theorem \[thm1\], we have $$\begin{aligned}
\label{.eq5}
\left| \sigma_p'(h)(z)+\epsilon \sigma_q'(g)(z)\right| &=& \left|\sum_{n=q+1}^p\epsilon n b_n z^{n-1}+\sum_{n=p+1}^{\infty}n(a_n+\epsilon b_n)z^{n-1} \right| \notag \\
&\leq& \sum_{n=q+1}^p\dfrac{1-\beta}{1+\alpha (n-1)}|z|^{n-1}+\sum_{n=p+1}^{\infty}\dfrac{2(1-\beta)}{1+\alpha(n-1)}|z|^{n-1} \notag \\
&\leq& (1-\beta)\left(\sum_{n=q+1}^p|z|^{n-1} +2\sum _{n=p+1}^{\infty}|z|^{n-1}\right)\,=\, \dfrac{(1-\beta)(|z|^p+|z|^q)}{1-|z|}, \end{aligned}$$ and $$\begin{aligned}
\label{.eq6}
\left| z(\sigma_p''(h)(z)+\epsilon \sigma_q''(g)(z)) \right| &=& \left| \sum_{n=q+1}^p\epsilon n(n-1)b_n z^{n-1}+\sum_{n=p+1}^{\infty}n(n-1)(a_n+\epsilon b_n)z^{n-1}\right| \notag \\
&\leq& \sum_{n=q+1}^p\dfrac{(n-1)(1-\beta)}{1+\alpha(n-1)}|z|^{n-1}+\sum_{n=p+1}^{\infty}\dfrac{2(n-1)(1-\beta)}{1+\alpha(n-1)}|z|^{n-1} \notag \\
&\leq& (1-\beta)\left(\sum_{n=q+1}^{\infty}(n-1)|z|^{n-1}+\sum_{n=p+1}^{\infty}2(n-1)|z|^{n-1} \right)\notag\\
&=& \dfrac{(1-\beta)\{p|z|^p+q|z|^q-(p-1)|z|^{p+1}-(q-1)|z|^{q+1}\}}{(1-|z|)^2}. \end{aligned}$$ Using estimates , and , we obtain that $$|\psi(z)|\leq \dfrac{(1-\beta)}{(1-|z|)}\left(\dfrac{p|z|^p+q|z|^q+3|z|^{p+1}+3|z|^{q+1}-(p-1)|z|^{p+2}-(q-1)|z|^{q+2}}{1-2|z|+|z|^2-(1-\beta)(1+|z|)(|z|^p+|z|^q)} \right).$$ Thus $\Re \left( 1+\phi(z)+\psi(z)\right)\geq 1-|\phi(z)|-|\psi(z)|,$ which for $|z|=1/4$ reduces to $$\begin{aligned}
\Re \left( 1+\phi(z)+\psi(z)\right)&\geq & \dfrac{4}{3}\left(\dfrac{7}{20}-\dfrac{(1-\beta)\{4^p(15 q+13)+4^q(15 p+13)\}}{9\times 4^{p+q}-20(1-\beta)(4^p+4^q)} \right) \notag\\
&>& \dfrac{4}{3}\left(\dfrac{7}{20}-\dfrac{4^p(15 q+13)+4^q(15 p+13)}{9\times 4^{p+q}-20(4^p+4^q)} \right). \notag \end{aligned}$$ Moreover, for $p>q\geq3,$ we have $$\Re \left( 1+\phi(z)+\psi(z)\right)> \dfrac{4}{3}\left(\dfrac{7}{20}-\dfrac{305}{2204}\right)>0, \notag$$ which implies that for each $\epsilon$ with $|\epsilon|=1,$ $s_p(h)+\epsilon s_q(g)$ is convex in $|z|<1/4,$ for $3\leq q\leq p,$ and thus each section $s_{p,q}(f)$ is convex in $|z|<1/4$ for $3\leq q \leq p.$
Let $f=h+\overline{g}\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$ Then
- For $q>2,\, s_{2,q}(f)(z)$ is convex in the disk $|z|<R_1, $ where $R_1$ is smallest positive root of the equation $$\label{.eq7}
1-4r+(6\beta-2)r^2-8(1-\beta)r^3+(1-2\beta)r^4+4(1-\beta)r^5=0$$ in $(0,1).$
- For $p>2,\, s_{p,2}(f)(z)$ is convex in the disk $|z|<R_2,$ where $R_2$ is smallest positive root of the equation $$\label{.eq8}
1-4r+(1-\beta)r^2-(8-3\beta)r^3-(5-2\beta)r^4-(4-3\beta)r^5+3(1-\beta)r^6=0$$ in $(0,1).$
\(i) Let $f=h+\overline{g}\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta),$ and suppose that $p=2<q.$ Then for each $|\epsilon|=1,$ it is sufficient to show that $$X=\Re\left(1+\dfrac{z(s_2''(h)(z)+\epsilon s_q''(g)(z))}{s_2'(h)(z)+\epsilon s'_q(g)(z)} \right)>0$$ in the disk $|z|<R_1.$ For $2=p<q,$ the estimates in - are continue to hold. Therefore, we deduce that $$\begin{aligned}
(1-|z|)X &\geq & \dfrac{1-|z|^2-2|z|}{1+|z|}-\dfrac{2(1-\beta)|z|^2\{3|z|+2(1-|z|^2)+|z|^2\}}{1-2|z|+|z|^2-2(1-\beta)|z|^2(1+|z|)} \notag \\
&=& \dfrac{1-|z|^2-2|z|}{1+|z|}-\dfrac{2(1-\beta)|z|^2\{2+3|z|-|z|^2\}}{1-2|z|+(1-2(1-\beta))|z|^2-2(1-\beta)|z|^3} \notag \\
&=& \dfrac{1-4|z|+\{4-6(1-\beta)\}|z|^2-8(1-\beta)|z|^3+\{2(1-\beta)-1\}|z|^4+4(1-\beta)|z|^5}{(1+|z|)\{1-2|z|+(1-2(1-\beta))|z|^2-2(1-\beta)|z|^3\}} \notag \\
&=& \dfrac{1-4|z|+(6\beta-2)|z|^2-8(1-\beta)|z|^3+(1-2\beta)|z|^4+4(1-\beta)|z|^5}{(1+|z|)\{1-2|z|+(1-2(1-\beta))|z|^2-2(1-\beta)|z|^3\}}, \notag\end{aligned}$$ which is greater then zero in $|z|<R_1,$ where $R_1$ is the smallest positive root of the equation in $(0,1).$
\(ii) Let $f=h+\overline{g}\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta)$ and suppose that $q=2<p.$ Then for each $|\epsilon|=1,$ it is sufficient to show that $$Y=\Re\left(1+\dfrac{z(s_p''(h)(z)+\epsilon s_2''(g)(z))}{s_p'(h)(z)+\epsilon s'_2(g)(z)} \right)>0$$ in the disk $|z|<R_2.$ Since for $2=q<p,$ the estimates in equations , and continue to hold. Therefore we deduce that $$\begin{aligned}
(1-|z|)Y &\geq & \dfrac{1-|z|^2-2|z|}{1+|z|}-\dfrac{(1-\beta)(p|z|^p+2|z|^2+3|z|^{p+1}+3|z|^4-(p-1)|z|^{p+2}-|z|^4)}{1-2|z|+|z|^2-(1-\beta)(|z|^p+|z|^2)(1+|z|)} \notag \\
&=& \dfrac{1-|z|^2-2|z|}{1+|z|}-\dfrac{(1-\beta)|z|^p(p+3|z|-(p-1)|z|^2)+(1-\beta)|z|^2(2+3|z|-|z|^2)}{1-2|z|+|z|^2-(1-\beta)(|z|^p+|z|^2)(1+|z|)} \notag \\
&\geq & \dfrac{1-|z|^2-2|z|}{1+|z|}-\dfrac{(1-\beta)\{|z|^3(3+3|z|-2|z|^2)+|z|^2(2+3|z|-|z|^2)\}}{1-2|z|+|z|^2-(1-\beta)(2|z|^3+|z|^2+|z|^4)} \notag \\
&=& \dfrac{1-|z|^2-2|z|}{1+|z|}-\dfrac{(1-\beta)\{3|z|^3+3|z|^4-2|z|^5+2|z|^2+3|z|^3-|z|^4\}}{1-2|z|+|z|^2-(1-\beta)(2|z|^3+|z|^2+|z|^4)} \notag \\
&=&\dfrac{1-4|z|+(1+\beta)|z|^2-(8-3\beta)|z|^3-(5-2\beta)|z|^4-(4-3\beta)|z|^5+3(1-\beta)|z|^6}{(1+|z|)\{1-2|z|+|z|^2-(1-\beta)(|z|^2+2|z|^3+|z|^4)\}},\notag\end{aligned}$$ which is greater then zero in $|z|<R_2,$ where $R_2$ is the smallest positive root of in $(0,1).$
If $f=h+\overline{g}\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta),$ then $s_{2,2}(f)(z)$ is convex in $|z|<(1+\alpha)/4(1-\beta).$
Let $s_{2,2}(f)(z)\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$ Then for each $|\epsilon|=1$, it is sufficient to show that $$\Re\left(1+\dfrac{z(s_2''(h)(z)+\epsilon s_2''(g)(z)}{s_2'(h)(z)+\epsilon s_2'(g)(z)} \right)>0$$ in the disk $|z|<\dfrac{1+\alpha}{4(1-\beta)}.$ In the view of Theorem \[th3\], we have $$\begin{aligned}
\Re\left(1+\dfrac{z(s_2''(h)(z)+\epsilon s_2''(g)(z)}{s_2'(h)(z)+\epsilon s_2'(g)(z)} \right) &\geq& 1-\left|\dfrac{z(s_2''(h)(z)+\epsilon s_2''(g)(z)}{s_2'(h)(z)+\epsilon s_2'(g)(z)} \right| \notag \\
&=& 1-\left|\dfrac{2(a_2+\epsilon b_2)z}{1+2(a_2+\epsilon b_2)z} \right| \geq 1-\dfrac{2|a_2+\epsilon b_2||z|}{1-2|a_2+\epsilon b_2| |z|} \notag \\
&=& \dfrac{1-\dfrac{4(1-\beta)}{1+\alpha}|z|}{1-\dfrac{2(1-\beta)}{1+\alpha}|z|}>0 \notag.\end{aligned}$$ Hence the result follows.
Applications
============
In this section, we consider the harmonic mappings whose co-analytic part involve the Gaussian hypergeometric function $_2F_1(a,b;c;z)$, which is defined by $$\label{G1}
_2F_1(a,b;c;z)=F(a,b;c;z)=\sum_{n=0}^{\infty}\dfrac{(a)_n \,(b)_n}{(c)_n\, n!}z^n \qquad (z\in \mathbb{D}),$$ where $a,b,c \in \mathbb{C}, c\neq 0, -1, -2, \cdots$ and $(a)_n$ is the Pochhammer symbol defined by $(a)_n=a(a+1)(a+2)\cdots(a+n-1)$ and $(a)_0=1$ for $n\in \mathbb{N}.$ The series is absolutely convergent in $\mathbb{D}.$ Moreover, if $\Re(c-a-b)>0,$ then the series is convergent in $|z|\leq 1.$ Further, for $z=1,$ we have the following well-known Gauss formula [@NMT] $$\label{G2}
F(a,b;c;1)=\dfrac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}<\infty.$$
We shall use the following Lemma to prove our results in this section:
\[lemaG\] [@G21] Let $a,b>0.$ Then the following holds:
- For $c>a+b+1,$ $$\sum_{n=0}^{\infty}\dfrac{(n+1)(a)_n (b)_n}{(c)_n n!}=\dfrac{\Gamma(c) \Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}(ab+c-a-b-1).$$
- For $c>a+b+2,$ $$\sum_{n=0}^{\infty} \dfrac{(n+1)^2(a)_n (b)_n}{(c)_n n!}=\dfrac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)}\left(\dfrac{(a)_2 (b)_2}{(c-a-b-2)_2}+\dfrac{3ab}{c-a-b-1}+1 \right).$$
- For $a,b,c\neq1$ with $c>\,{\rm max}\, \{0, a+b+1\},$ $$\sum_{n=0}^{\infty}\dfrac{(a)_n (b)_n}{(c)_n (n+1)!}=\dfrac{1}{(a-1)(b-1)}\left[\dfrac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}-(c-1) \right].$$
\[thmG\] Let $f_1(z)=z+\overline{ z^2 F(a,b;c;z)},\quad f_2(z)=z+\overline{z(F(a,b;c;z)-1)}$ and $f_3(z)=z+\overline{z\int_0^z F(a,b;c;t)dt},$ where $a,b,c$ are positive real numbers such that $c>a+b+2.$ Then the following holds:
- If $$\label{G3}
\dfrac{\Gamma(c) \Gamma(c-a-b-1)}{\Gamma(c-a) \Gamma(c-b)}\left[ \dfrac{\alpha (a)_2 (b)_2}{c-a-b-2}+(1+4\alpha)ab+2(1+\alpha)(c-a-b-1)\right]\leq 1-\beta,$$ then $f_1\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
- If $$\label{G4}
\dfrac{\Gamma(c)\Gamma(c-a-b-2)}{\Gamma(c-a)(c-b)}\left[\alpha ab(ab+c-1)+(1+\alpha)ab(c-a-b-2)+1 \right] \leq 2-\beta,$$ then $f_2 \in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
- If $a,b,c\neq 1 $ and $c>\,{\rm max}\, \{0, a+b+1\},$ $$\label{G5}
\dfrac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}\left[\alpha ab+(1+2\alpha)(c-a-b-1)+\dfrac{1}{(a-1)(b-1)} \right]$$ $$-\dfrac{(c-1)}{(a-1)(b-1)} \leq 1-\beta,$$ then $f_3 \in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
\(i) Let $f_1(z)=z+\overline{ z^2 F(a,b;c;z)}=z+\overline{\sum_{n=2}^{\infty}C_nz^n,}$ where $$C_n=\dfrac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-2)!} \quad {\rm for}\quad n\geq 2.$$ Therefore, we have $$\begin{aligned}
\sum_{n=2}^{\infty}n(1+\alpha(n-1))|C_n| &=& \sum_{n=2}^{\infty}n(1+\alpha(n-1)) \dfrac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-2)!}\notag \\
&=& (1+\alpha)\sum_{n=0}^{\infty}(n+1)\dfrac{(a)_n(b)_n}{(c)_n n!}+\alpha \sum_{n=0}^{\infty}(n+1)^2\dfrac{(a)_n (b)_n}{(c)_n n!}+\sum_{n=0}^{\infty}\dfrac{(a)_n(b)_n}{(c)_n n!}. \notag\end{aligned}$$ Now, using Lemma \[lemaG\] and Gauss formula , we have\
$\sum_{n=2}^{\infty}n(1+\alpha(n-1))|C_n|=$ $$\dfrac{\Gamma(c) \Gamma(c-a-b-1)}{\Gamma(c-a) \Gamma(c-b)}\left[ \alpha \dfrac{(a)_2 (b)_2}{c-a-b-2}+(1+4\alpha)ab+2(1+\alpha)(c-a-b-1)\right].$$ If holds, then $\sum_{n=2}^{\infty}n(1+\alpha(n-1))|C_n|\leq 1-\beta.$ Hence the result follows.
\(ii) Let $f_2(z)=z+\overline{z(F(a,b;c;z)-1)}=z+\overline{\sum_{n=2}^{\infty}D_nz^n},$ where $$D_n=\dfrac{(a)_{n-1}(b)_{n-1}}{(c)_{n-1}(n-1)!}\quad {\rm for}\quad n\geq2.$$ Therefore, we have $$\begin{aligned}
\label{G7}
\sum_{n=2}^{\infty}n(1+\alpha(n-1))|D_n| &=& \sum_{n=2}^{\infty}n(1+\alpha(n-1))\dfrac{(a)_{n-1}(b)_{n-1}}{(c)_{n-1}(n-1)!} \notag \\
&=& \sum_{n=0}^{\infty}(\alpha(n+1)^2+(1+\alpha)(n+1)+1)\dfrac{(a)_{n+1}(b)_{n+1}}{(c)_{n+1}(n+1)!}. \notag\end{aligned}$$ Now using the identity $(\gamma)_{n+1}=\gamma(\gamma+1)_n$, we have $$\begin{aligned}
\label{G8}
\sum_{n=2}^{\infty}n(1+\alpha(n-1))|D_n|&=& \dfrac{ab}{c} \alpha \sum_{n=0}^{\infty}(n+1)\dfrac{(a+1)_n(b+1)_n}{(c+1)_n n!} \notag \\
&+& \dfrac{ab}{c}\left[(1+\alpha)\sum_{n=0}^{\infty} \dfrac{(a+1)_n(b+1)_n}{(c+1)_n n!}+\sum_{n=0}^{\infty} \dfrac{(a+1)_n(b+1)_n}{(c+1)_n (n+1)!} \right]. \notag \end{aligned}$$ Further, using Lemma \[lemaG\] and Gauss formula , we obtain\
$\sum_{n=2}^{\infty}n(1+\alpha(n-1))|D_n|= $ $$\dfrac{\Gamma(c) \Gamma(c-a-b-2)}{\Gamma(c-a) \Gamma(c-b)}\left[\alpha ab(ab+c-1)+(1+\alpha)ab(c-a-b-2)+1 \right]-1.$$ Now, if holds, then $\sum_{n=2}^{\infty}n(1+\alpha(n-1))|D_n| \leq 1-\beta,$ hence the result follows.
\(iii) Let $f_3(z)=z+\overline{z\int_0^z F(a,b;c;t)dt}=z+\overline{\sum_{n=2}^{\infty}E_nz^n},$ where $$E_n=\dfrac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-1)!}\quad {\rm for}\quad n\geq2.$$ Therefore, $$\begin{aligned}
\label{G9}
\sum_{n=2}^{\infty}n(1+\alpha(n-1))|E_n| &=& \sum_{n=2}^{\infty}n(1+\alpha(n-1))\dfrac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-1)!} \notag \\
&=& \alpha \sum_{n=0} ^{\infty} (n+1) \dfrac{(a)_n(b)_n}{(c)_n n!}+(1+\alpha) \sum_{n=0} ^{\infty}\dfrac{(a)_n(b)_n}{(c)_n n!}+\sum_{n=0} ^{\infty} \dfrac{(a)_n(b)_n}{(c)_n (n+1)!}. \notag\end{aligned}$$ Now using Lemma \[lemaG\] and Gauss formula , we obtain $$\label{10}
\sum_{n=2}^{\infty}n(1+\alpha(n-1))|E_n|=$$ $$\dfrac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}\left[\alpha ab+(1+2\alpha)(c-a-b-1)+\dfrac{1}{(a-1)(b-1)}\right]- \dfrac{(c-1)}{(a-1)(b-1)}.$$ Further, if holds, then the result follows.
Note that for $\eta \in \mathbb{C}/ \{-1, -2, \cdots \}$ and $n\in \mathbb{N}\cup \{0\},$ we have $$\dfrac{(-1)^n(-\eta)_n}{n!}=\binom {\eta}{n} = \dfrac{\Gamma(\eta+1)}{n! \Gamma(\eta-n+1)}.$$ In particular, when $\eta=m (m\in \mathbb{N}, m\geq n), $ we have $$(-m)_n=\dfrac{(-1)^n m!}{(m-n)!}.$$ Using above relations in Theorem \[thmG\], we get harmonic univalent polynomials which belongs to the class $\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$ Setting $a=b=-m\,(m\in \mathbb{N}),$ we get
\[c11\] Let $m\in \mathbb{N},$ $c$ be a positive real number and $$F_1(z)=z+\overline{\sum_{n=0}^m \binom {m}{n} \dfrac{(m-n+1)_n}{(c)_n}z^{n+2}},$$ $$F_2(z)=z+\overline{\sum_{n=0}^m \binom {m}{n} \dfrac{(m-n+1)_n}{(c)_n}z^{n+1}},$$ $$F_3(z)=z+\overline{\sum_{n=0}^m \binom {m}{n} \dfrac{(m-n+1)_n}{(n+1)(c)_n}z^{n+2}}.$$ Then the following holds:
- If $$\dfrac{\Gamma(c) \Gamma(c+2m-1)}{[\Gamma(c+m)]^2}\left[ \dfrac{\alpha m^2 (m-1)^2}{c+2m-2}+(1+4\alpha)m^2+2(1+\alpha)(c+2m-1)\right]\leq 1-\beta,$$ then $F_1\in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
- If $$\dfrac{\Gamma(c)\Gamma(c+2(m-1))}{[\Gamma(c+m)]^2}\left[\alpha m^2(m^2+c-1)+(1+\alpha)m^2(c+2m-2)+1 \right] \leq 2-\beta,$$ then $F_2 \in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
- If $$\dfrac{\Gamma(c)\Gamma(c+2m-1)}{[\Gamma(c+m)]^2}\left[\alpha m^2+(1+2\alpha)(c+2m-1)+\dfrac{1}{(m+1)^2} \right]-\dfrac{(c-1)}{(m+1)^2}\leq 1-\beta,$$ then $F_3 \in \mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
Further setting $m=2$ and $c=1$ in Corollary \[c11\], we get
If $G_1(z)=z+\overline{z^2+4z^3+z^4},\,\,G_2(z)=z+\overline{4z^2+z^3},\,\,$ and $G_3 (z)=z+\overline{z^2+2z^3+\dfrac{1}{3}z^4},$ then the following holds:
- If $2(19\alpha+9)\leq 1-\beta,$ then $G_1(z)\in\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
- If $ 28\alpha+13\leq 2(2-\beta),$ then $G_2(z)\in\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
- If $ 108\alpha+37\leq 6(1-\beta),$ then $G_3(z)\in\mathcal{W}_{\mathcal{H}}^0(\alpha, \beta).$
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abstract: 'The Bochum experimental enhancement of the d+d fusion rate in a deuterated metal matrix at low incident energies is explained by the quantum broadening of the momentum-energy dispersion relation and consequent modification of the high-momentum tail of the distribution function from an exponential to a power-law.'
address:
- 'Dipartimento di Fisica, Università di Cagliari, I-09042 Monserrato, Italy'
- 'Istituto Nazionale di Fisica Nucleare, Cagliari, I-09042 Monserrato, Italy'
- 'Dipartimento di Fisica, Politecnico di Torino, I-10125 Torino, Italy'
- 'State Research Centre of Russian Federation, Troitsk Institute for innovation and fusion research, Centre for Theoretical Physics and Computational Mathematics, Troitsk 142190 Moscow reg., Russia'
author:
- 'M. Coraddu'
- 'M. Lissia'
- 'G. Mezzorani'
- 'Yu.V. Petrushevich'
- 'P. Quarati'
- 'A. N. Starostin'
date: 16 January 2004
title: 'Quantum-tail effect in low energy d+d reaction in deuterated metals'
---
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Introduction
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Anomalous enhancement of the sub-barrier nuclear fusion reaction d(d,p)t in a deuterated metallic matrix has been experimentally observed at energies of the incident beam lower than few keV [@Raiola:02; @Raiola:02a]. Electron screening is not sufficient to explain this enhancement, and other quantitative explanations are missing [@Fiorentini:2002vj].
In fact, the calculated (adiabatic limit) electron screening potential energy $U_{ad}$ is 28 eV, but experiments show an exponential enhancement of the cross section at very low energies that would correspond to a screening energy $U_{ex}=309 \pm12$ eV [@Raiola:02]. Similar behaviors are found in gaseous targets.
These discrepancies between the calculated $U_{ad}$ and the experimentally inferred $U_{ex}$ have not yet been understood [@Fiorentini:2002vj]; these puzzling results could have important consequences also for the study of nuclear fusion in astrophysical environments [@Coraddu:1998yb; @Coraddu:2000nu]. In this paper we discuss a possible explanation of this enhancement which is based on the quantum-uncertainty dispersive effect between energy and momentum that was proposed by Galitskij and Yakimets [@Ga:67] and recently discussed and applied by Starostin et al. [@St:02; @St:00; @Savchenko:1999ap].
Anomalously large electron screening
====================================
Experimental data for the d(d,p)t reaction in a deuterated Tantalium target [@Raiola:02] are reported for beam energies in the range 4-20 keV. The target is cooled with nitrogen at a temperature of 10 $^{\circ}$C, which corresponds to a thermal energy $k T = 2.44\times 10^{-2}$ eV. Data from Ref. [@Raiola:02] are plotted in Fig. \[fig:RaiolaSigmaV\].
In panel (a) of Fig. \[fig:RaiolaSigmaV\] the experimental values of $\sigma(E_{cm}) v_{rel}(E_{cm})$ are plotted versus the center of mass energy $E_{cm}$. Since the incoming particles are not relativistic and the target particles are practically at rest, these data are plotted using $ v_{rel}(E_{cm}) = \sqrt{2 E_{cm}/ \mu} $ and $E_{cm} = E_{beam} /2 $, where $\mu = m_D/2$ is the reduced mass).
The unscreened cross section can be written as: $$\sigma_b(E_{cm}) =
\frac{S(E_{cm})}{E_{cm}}\, \exp{\left( - \pi\,
\sqrt{ \frac{E_G}{E_{cm}}} \right)}
\label{eq:BareCrossSec}$$ with $ E_G = 2 \mu \; Z_1^2 Z_2^2 e^4 / \hbar^2 \, $. The astrophysical factor $S(E_{cm})$ should vary slowly in this energy range and is linearly approximated as $S(E_{cm}) =\, S_0\, +\, S_1 E_{cm} \, $, where $S_0$ and $S_1$ are extrapolated from energies $E_{cm} > 20$ keV (see panel (b) in Fig. \[fig:RaiolaSigmaV\]); the values reported in Ref. [@Raiola:02] are $S_0 = 43$ keV b and $S_1 = 0.53$ b.
The interacting nuclei “feel” a effective potential barrier lower by an amount equal the electron screening potential $U_e$: the resulting screened cross section is: $$\sigma(E_{cm})\, =\, \sigma_b(E_{cm} + U_e) \; .$$ When $U_e \ll E_{cm}$ a correction factor $f_e$ can be factorized: $$\sigma(E_{cm})\, \simeq\, f_e \cdot \sigma_b(E_{cm}) \quad ,
\label{eq:ScreenedCrossSec}$$ where $$f_e(E_{cm} , U_e) =
\exp{\left( \pi\, \sqrt{\frac{E_G}{E_{cm}}} \cdot \frac{U_e}{2 E_{cm}} \right)}
\quad .
\label{eq:EScreenFact}$$ The electron screening potential computed in the adiabatic limit $U_{ad}$ constitutes a theoretical upper limit for $U_e$: $$U_e\, \leq\,U_{ad}\quad .$$ For d+d reactions, Fig. \[fig:RaiolaSigmaV\] compares the “bare curve” of Eq. (\[eq:BareCrossSec\]) and the “screened curve” ($U_e = U_{ad} = 28$ eV) of Eq. (\[eq:ScreenedCrossSec\]) with the experimental data. The screened cross section with $U_e = U_{ad}= 28$ eV underestimates the experimental data by about an order of magnitude in the few keV energy range; the data could be fitted only by using an unphysical electron potential much larger than the adiabatic upper limit: $U_e = 309$ eV $ \gg U_{ad}$. $$S_{exp}=
\sigma(E_{cm})\, E_{cm}\, \exp{\left(\pi\,
\sqrt{\frac{E_G}{E_{cm}}} \right)}
\quad .$$ Equation (\[eq:ScreenedCrossSec\]) implies $$S_{exp}\, =\, f_e \cdot S(E_{cm})$$ and, once again, only the screened curve with $U_e = 309$ eV fits the data.
In summary, there is experimental evidence of anomalously high values of low-energy fusion cross sections that would require electron screening potentials $U_e$ one order of magnitude larger than their adiabatic limit, if explained in terms of screening. In deuterated metal targets, the effect depends strongly on the metal [@Raiola:02a]). Values of $U_e > U_{ad}$ are required also for describing experiments with gas target, but violations of the requirement $U_e\leq U_{ad} $ are less strong than in metal targets.
These anomalous values of $U_e$ are substantially unexplained; the screened potential approach is probably trying to mimic important processes that have been disregarded. One should attempt to find alternative mechanisms that could reproduce the enhancement of the cross section at low energy.
Thermal corrections and quantum uncertainty
============================================
The parameters of the experiments (temperature and beam energy) are such that thermal corrections can be neglected. In fact if we average $\sigma\, v_{rel}$ over a Maxwellian distribution of velocity, the temperature correction factor $f_T$ can be estimated [@Fiorentini:2002vj]: $$\langle \sigma v_{rel}
\rangle_M = f_T \cdot (\sigma v_{rel})_{T=0} \quad ,$$ $$f_T = \exp\left( \frac{ \pi^2 E_G\,
k T}{2\, E_{beam}^2} \right) \quad .
\label{eq:MaxTermCorrFact}$$ For d+d reactions ($\pi^2 E_G = 986$ keV) even at room temperature ($k T \sim 10^{-5}$ keV) and at the lowest energies of the beam ($E_{beam} \sim 1$ keV) the Maxwellian thermal factor $f_T\simeq 1$.
As showed by Galitskij and Yakimets [@Ga:67] many-body collisions broaden the relation between momentum and energy of the particles. Since momentum rather than energy determines the scattering amplitude, the reaction cross section must be averaged over a momentum distribution that may differ from the energy one. The reaction rate for a mono-energetic beam is:
$$\begin{aligned}
\langle \sigma v_{rel} \rangle
& = & \int d^3\mathbf p_a\, v_{rel}\, \sigma(E_{cm}) f(p_a) \\
\label{eq:ReacRate}
f(p_a)
& = & \int_0^\infty d E_a n(E_a)\, \delta_{\gamma}(E_a - {\epsilon_{p_a}}) \quad ,
\label{eq:MomentDistFunc} \end{aligned}$$
where the center of mass energy $E_{cm}$ is function of the beam-particle and target-particle momentum, and the momentum distribution function of the target particles $f(p_a)$ depends on thermal distribution of the target-particle energies $ n(E_a)$, which we take to be Maxwellian, and on the probability that a target particle with energy $E_a$ has momentum $p_a = \sqrt{2 m_a {\epsilon_{p_a}}}$ (dispersion relation), $ \delta_{\gamma}(E_a - {\epsilon_{p_a}})$.
Quantum effects are responsible for the dispersion relation between energy and momentum $ \delta_{\gamma}(E_a - {\epsilon_{p_a}})$ not being a $\delta$-function, but a broader distribution. According to Galitskij and Yakimets [@Ga:67] the relation between energy and momentum is a Lorentzian $$\delta_{\gamma}(E_a - {\epsilon_{p_a}}) =
\frac{1}{\pi}\, \frac{\gamma}{\left( E_a - {\epsilon_{p_a}}\right)^2\, +\, \gamma^2}
\label{eq:QuantDispRel}$$ with $ \gamma = \hbar \nu_{coll} = \hbar\, n\, \sigma_{coll}\, v_{coll} $, where $\nu_{coll}$ and $\sigma_{coll}$ are the collisional frequency and cross section, while $n$ and $v_{coll}$ are the colliding particles density and velocity.
Assuming that the collision cross section could be approximated by the bare Coulomb cross section $\sigma_{coll} = \pi\, e^4 / {\epsilon_{p_a}}^2 $, the resulting $\gamma$ is $$\gamma =
\frac{\pi\, \hbar N_a \rho\, e^4 }{A\, {\epsilon_{p_a}}^2 }\,
\sqrt{\frac{2 E_a}{m_a}}
\quad ,
\label{eq:WidthQuantDispRel}$$ where the relations $v_{coll} = \sqrt{2 E_a / m_a}$ and $n = N_A \rho / A$ have been used with $\rho$ the total density and $A$ the (average) atomic number.
In the asymptotic regime $ {\epsilon_{p_a}}\gg k T $, relevant for the particles that undergo fusion, the Maxwellian contribution $ (2\pi\, m_a\, k T )^{-3/2} \times
\exp\left(-{\epsilon_{p_a}}/(k T) \right) $ is negligible and the distribution of momenta becomes $$f(p_a) \sim \frac{1}{(2\pi\, m_a\, k T )^{3/2}} \times
\frac{\hbar N_A \rho e^4\, k T }{ A\, {\epsilon_{p_a}}^4}\,
\sqrt{\frac{2\pi k T}{m_a}}
\quad .$$ Since only particles in the high-energy tail of the distribution $ {\epsilon_{p_a}}\gg k T $ contribute to the fusion rate, the quantum effect contribution $$\langle \sigma v_{rel} \rangle \sim
\frac{\hbar N_a \rho e^4}{ 2 \pi A\, m_a^2}\,
\int v_{rel}\, \sigma(E_{cm})\, \frac{1}{{\epsilon_{p_a}}^4}\, d^3\mathbf p_a
\label{eq:QeffTerCorr}$$ is the only important contribution to the rate.
More in general, the fact that the relation between energy and momentum is not a $\delta$-function results in a distribution of momentum $ f(p_a)$ with a power-law asymptotic behavior in spite of the energy distribution $n(E_a)$ being exponential. This power-law tail becomes mostly important for reactions whose cross sections select high-momentum particles.
This quantum contribution can be calculated numerically and we are also developing useful analytical approximations: we shall report elsewhere the derivation of these approximations and their comparison with the exact numerical integration.
In the following we give some preliminary results using a parameterization that is motivated by the form of Eq. (\[eq:QeffTerCorr\]) and that can be used to qualitatively estimate the importance of such quantum broadening of the momentum distribution.
If we use $ E_{max} < E_{beam}$ as an upper bound for ${\epsilon_{p_a}}$ in the integral in Eq. (\[eq:QeffTerCorr\]) (the precise value of the low bound is inconsequential) and work in the relevant approximation that the target particles have energies lower than the beam particles, we obtain the following partial parameterization for the dominant contribution $$\langle \sigma v_{rel} \rangle \sim
(\sigma v_{rel})_{T=0}\; \frac{\sqrt{2}}{\pi^2}\,
\frac{\hbar N_a \rho\, e^4}{A\, \sqrt{m_a}}\,
\frac{E_{beam}^2
\exp\left( \frac{\pi \sqrt{2 E_G E_{max}}}{E_{beam}} \right) }
{E_G\, E_{max}^{7/2} }\, \quad .
\label{eq:QeffLowEnApprox}$$
In Fig. \[fig:EnhancFactCompared\] we have plotted the ratio $f_Q = \langle \sigma v_{rel} \rangle / (\sigma v_{rel} )_{T=0}$ with $\langle \sigma v_{rel} \rangle $ given by Eq. (\[eq:QeffLowEnApprox\]) for three different values of $ E_{max}$. The function $f_Q$ shows an evident enhancement at low energy starting from the region of 1-2 keV in qualitative agreement with experiments. The threshold below which this enhancement becomes important depends on $ E_{max} $. We are completing a more detailed calculation that does not require the introduction of the parameter $ E_{max}$ and that, therefore, can better test the relevance of this quantum effect for the experimental results.
Conclusion
==========
The theory of Galitskij and Yakimets predicts that quantum indeterminacy broadens the relation between energy and momentum.
We have performed a preliminary calculation to estimate the effect of this broadening on the momentum distribution of deuteron in metals and, therefore, on the cross section of the reaction d(d,p)t. This calculation shows that this quantum effect should give an important enhancement of the cross section at low energies similar to the one observed in the Bochum experiments.
A more quantitative comparison between theory and experiments requires the use of more sophisticated analytical or numerical analyses [@Starostin:2003next] and the inclusion of the effects of screening both on the fusion cross section and on the ion-collision cross section. This further work is being completed and will be published in the near future.
[00]{} F. Raiola [*et al.*]{}, Eur. Phys. J. [**A 13**]{} (2002) 377. F. Raiola [*et al.*]{}, Phys. Lett. [**B 547**]{} (2002) 193. G. Fiorentini, C. Rolfs, F. L. Villante and B. Ricci, Phys. Rev. C [**67**]{} (2003) 014603 \[arXiv:astro-ph/0210537\]. M. Coraddu, G. Kaniadakis, A. Lavagno, M. Lissia, G. Mezzorani and P. Quarati, Braz. J. Phys. [**29**]{} (1999) 153 \[arXiv:nucl-th/9811081\]. M. Coraddu, M. Lissia, G. Mezzorani and P. Quarati, arXiv:nucl-th/0012002. V. M. Galitskii and V. V. Yakimets, JEPT [**24**]{} (1967) 637. A. N. Starostin, A. B. Mironov, N. L. Aleksandrov, J. N. Fisch, and R. M. Kulsrud, Physica A [**305**]{} (2002) 287. A. N. Starostin, V. I. Savchenko, and N. J. Fisch, Phys. Lett. A [**274**]{} (2000) 64. V. I. Savchenko, arXiv:astro-ph/9904289. M. Coraddu, G. Mezzorani, Yu. V. Petrushevich, P. Quarati, and A. N. Starostin, Physica A, this issue.
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abstract: 'In this letter, we investigate the tradeoff between energy efficiency (EE) and spectral efficiency (SE) in device-to-device (D2D) communications underlaying cellular networks with uplink channel reuse. The resource allocation problem is modeled as a noncooperative game, in which each user equipment (UE) is self-interested and wants to maximize its own EE. Given the SE requirement and maximum transmission power constraints, a distributed energy-efficient resource allocation algorithm is proposed by exploiting the properties of the nonlinear fractional programming. The relationships between the EE and SE tradeoff of the proposed algorithm and system parameters are analyzed and verified through computer simulations.'
author:
- 'Zhenyu Zhou, Mianxiong Dong, Kaoru Ota, Jun Wu, and Takuro Sato, [^1] [^2] [^3] [^4] [^5] [^6] [^7]'
bibliography:
- 'IEEE\_gc\_2014.bib'
title: 'Energy Efficiency and Spectral Efficiency Tradeoff in Device-to-Device (D2D) Communications'
---
[Shell : Bare Demo of IEEEtran.cls for Journals]{}
EE and SE tradeoff, D2D communication, noncooperative game, nonlinear fractional programming.
Introduction
============
communications underlaying cellular networks bring numerous benefits including the proximity gain, the reuse gain, and the hop gain [@D2D_design]. However, the introduction of D2D communications into cellular networks poses many new challenges in the resource allocation design due to the co-channel interference caused by spectrum reuse and limited battery life of user equipments (UEs).
A large number of works have been done in how to optimize the spectral efficiency (SE) through resource allocation in an interference-limited environment (see [@Feiran_WCNC2013; @Doppler_TWC; @Song_JSAC] and references therein). However, most of the previous studies ignore the energy consumption of UEs. In practical implementation, UEs are typically handheld devices with limited battery life and can quickly run out of battery if the energy consumption is ignored in the system design.
A limited amount of works have considered the energy efficiency (EE) optimization problem (see [@Feiran_2012; @Zhou_GC2014v2; @Wu_TVT2014], and references therein). Unfortunately, optimum EE and SE are not always achievable simultaneously and may sometimes even conflict with each other [@EE_SE_tradeoff]. Therefore, it is an urgent task to study the EE and SE tradeoff in D2D communications underlaying cellular networks, which has not been well investigated and analyzed.
In this letter, firstly, we model the resource allocation problem as a noncooperative game, and propose a novel distributed energy-efficient resource allocation algorithm to maximize each UE’s EE subject to the SE requirement and transmission power constraints. Then, we study the EE and SE tradeoff of the proposed algorithm, and analyze and verify the relationships between the tradeoff and system parameters (such as transmission power, channel gain, etc.) through computer simulations.
System Model {#System Model}
============
In this paper, we consider the uplink scenario of a single cellular network. Each cellular UE is allocated with an orthogonal link, and D2D pairs reuse the same channels allocated to cellular UEs in order to improve the SE. The set of UEs is denoted as $\mathcal{S}=\{ \mathcal{N}, \mathcal{K} \}$, where $\mathcal{N}$ and $\mathcal{K}$ denote the sets of D2D UEs and cellular UEs respectively. The total number of D2D links and cellular links are denoted as $N$ and $K$ respectively.
The distributed resource allocation problem is modeled as a noncooperative game. The strategy sets of the $i$-th D2D transmitter and other D2D transmitters in $\mathcal{N} \backslash \{i\}$ are denoted as $\mathbf{p}_i^d$ and $\mathbf{p}_{-i}^d$ respectively. The strategy sets of the $k$-th cellular UE and other cellular UEs in $\mathcal{K} \backslash \{k\}$ are denoted as $\mathbf{p}_k^c$ and $\mathbf{p}_{-k}^c$ respectively. For the $i$-th D2D pair, its EE $U_{i, EE}^d$ (bits/Hz/J) depends not only on $\mathbf{p}_i^d$, but also on the strategies taken by other UEs in $\mathcal{S}\backslash \{i\}$, i.e., $\mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c$, which is defined as $$\begin{aligned}
\label{eq:UE_EED}
&U_{i, EE}^d (\mathbf{p}_i^d, \mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c)=\frac{U_{i, SE}^d}{p_{i, total}^d}\notag\\
&=\frac{\sum_{k=1}^K \log_2 \left( 1+\frac{p_i^k g_{i}^k}{p_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}p_{j}^k g_{j, i}^k+N_0} \right) }{\sum_{k=1}^K \frac{1}{\eta } p_i^k+2p_{cir}},\end{aligned}$$ where $U_{i,SE}^d$ is the SE (bits/s/Hz), and $p_{i, total}^d$ is the total power consumption (W). $p_i^k$, $p_c^k$, and $p_{j}^k$ are the transmission power of the $i$-th D2D transmitter, the $k$-th cellular UE, and the $j$-th D2D transmitter in the $k$-th channel respectively. $g_{i}^k$ is the channel gain of the $i$-th D2D pair, $g^k_{c, i}$ is the interference channel gain between the $k$-th cellular UE and the $i$-th D2D receiver, and $g_{j, i}^k$ is the interference channel gain between the $j$-th D2D transmitter and the $i$-th D2D receiver. $p_c^k g^k_{c, i}$ and $\sum_{j=1, j\neq i}^{N} p_{j}^k g_{j, i}^k$ denote the interference from the cellular UE and the other D2D pairs that reuse the $k$-th channel respectively. $N_0$ is the noise power. $p_{i, total}^d$ is composed of the transmission power over all of the $K$ channels, i.e., $\sum_{k=1}^K \frac{1}{\eta } p_i^k$, and the circuit power of both the D2D transmitter and receiver, i.e., $2p_{cir}$. The circuit power of any UE is assumed as the same and denoted as $p_{cir}$. $\eta$ is the power amplifier (PA) efficiency, i.e., $0 < \eta < 1$.
Similarly, the EE of the $k$-th cellular UE $U_{k, EE}^c$ is defined as $$\begin{aligned}
&U_{k, EE}^c (\mathbf{p}_i^d, \mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c)\notag\\
&=\frac{U_{k, SE}^c}{p_{k, total}^c}=\frac{\log_2 \left( 1+\frac{p_c^k g_c^k}{\sum_{i=1}^{N}p_{i}^k g_{i, c}^k+N_0} \right)}{\frac{1}{\eta} p_c^k+p_{cir}},\end{aligned}$$ where $g_c^k$ is the channel gain between the $k$-th cellular UE and the base station (BS), $g^k_{i, c}$ is the interference channel gain between the $i$-th D2D transmitter and the BS in the $k$-th channel. $\sum_{i=1}^{N}p_{i}^k g_{i, c}^k$ denotes the interference from all of the D2D pairs to the BS in the $k$-th channel. $p_{k, total}^c$ is composed of the transmission power $\frac{1}{\eta} p_c^k$ and the circuit power only at the transmitter side, i.e., $p_{cir}$.
The EE maximization problem for the $i$-th D2D pair is formulated as $$\begin{aligned}
\label{eq:Dproblem}
&\max. \hspace{5mm} U_{i, EE}^d (\mathbf{p}_i^d, \mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c) \\
&\mbox{s.t.} \hspace{5mm} C1: U_{i, SE}^d \geq R_{i, min}^d, \\
&\hspace{9mm} C2: 0 \leq \sum_{k=1}^K p_i^k \leq p_{i, max}^d.\end{aligned}$$
The corresponding EE maximization problem for the $k$-th cellular UE is formulated as $$\begin{aligned}
\label{eq:Cproblem}
&\max. \hspace{5mm} U_{k, EE}^c (\mathbf{p}_i^d, \mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c) \\
&\mbox{s.t.} \hspace{5mm} C3: U_{k, SE}^c \geq R_{k, min}^c,\\
&\hspace{9mm} C4: 0 \leq p_c^k \leq p_{k, max}^c.\end{aligned}$$ The constraints C1 and C3 specify the minimum SE requirements. C2 and C4 are the non-negative constraints on the power allocation variables.
Distributed Energy-Efficient Resource Allocation {#distributed}
================================================
The Objective Function Transformation {#transformation}
-------------------------------------
The objective functions defined in (\[eq:Dproblem\]) and (\[eq:Cproblem\]) are non-convex, but can be transformed into concave functions by using the nonlinear fractional programming developed in [@Dinkelbach]. We define the maximum EE of the $i$-th D2D pair as $q^{d*}_i$, which is given by $$q^{d*}_i=\max. \hspace{1mm} U_{i, EE}^d (\mathbf{p}_i^d, \mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c)=\frac{U_{i, SE}^d(\mathbf{p}_i^{d*})}{p_{i, total}^d(\mathbf{p}_i^{d*})},$$ where $\mathbf{p}_i^{d*}$ is the best response of the $i$-th D2D transmitter given the other UEs’ strategies $\mathbf{p}_{-i}^d$, $\mathbf{p}_k^c$, $\mathbf{p}_{-k}^c$. The following theorem can be proved:
***Theorem 1:*** The maximum EE $q_i^{d*}$ is achieved if and only if $\max. \:\: U_{i, SE}^d (\mathbf{p}_i^d)-q_i^{d*}p_{i, total}^d(\mathbf{p}_i^d)=U_{i,SE}^d (\mathbf{p}_i^{d*})-q_i^{d*}p_{i, total}^d(\mathbf{p}_i^{d*})=0$.
The proof of Theorem 1 is given in Appendix \[theorem1\].
Theorem 1 shows that the transformed problem with an objective function in subtractive form is equivalent to the non-convex problem in fractional form, i.e., they lead to the same optimum solution $\mathbf{p}_i^{d*}$. Similarly, let $q^{c*}_k$ and $\mathbf{p}_k^{c*}$ denote the maximum EE and best response of the $k$-th cellular UE, we have
***Theorem 2:*** The maximum EE $q_k^{c*}$ is achieved if and only if $\max. \:\: U_{k, SE}^c (\mathbf{p}_k^c)-q_k^{c*}p_{k, total}^c(\mathbf{p}_k^c)=U_{k, SE}^c (\mathbf{p}_k^{c*})-q_k^{c*}p_{k, total}^c(\mathbf{p}_k^{c*})=0$.
The Iterative Optimization Algorithm {#algorithm}
------------------------------------
The proposed algorithm is summarized in Algorithm \[offline algorithm\]. $n$ is the iteration index, $L_{max}$ is the maximum number of iterations, and $\Delta$ is the maximum tolerance. $L_{max}$ is set to 10 to ensure that the algorithm converges sufficiently although simulation results in Section \[Simulation Results\] show that the algorithm is able to converge in only 5 iterations. This setting will not increase the computation complexity significantly because the loop will terminate once the algorithm converges sufficiently close to the optimum EE, i.e., when the condition $U_{i, SE}^d(\mathbf{\hat{p}}_i^d)-q_i^d p_{i,total}^d (\mathbf{\hat{p}}_i^d) \leq \Delta$ is satisfied.
At each iteration, for any given $q_i^{d}$ or $q_k^{c}$, the corresponding resource allocation strategies are obtained by solving the following equivalent transformed optimization problems respectively: $$\begin{aligned}
\label{eq:transformed problemD}
&\max . \:\: U_{i, SE}^d (\mathbf{p}_i^d)-q_i^d p_{i, total}^d(\mathbf{p}_i^d) \nonumber\\
&\mbox{s.t.} \:\:\: C1, C2.
\end{aligned}$$ $$\begin{aligned}
\label{eq:transformed problemC}
&\max . \:\: U_{k, SE}^c (\mathbf{p}_k^c)-q_k^c p_{k, total}^c (\mathbf{p}_k^c) \nonumber\\
&\mbox{s.t.} \:\:\: C3, C4.
\end{aligned}$$
Taking the $i$-th D2D pair as an example, the Lagrangian associated with the problem (\[eq:transformed problemD\]) is given by $$\begin{aligned}
&\mathcal{L}_{EE}(\mathbf{p}_i^d, \alpha_i, \beta_i) =U_{i, SE}^d (\mathbf{p}_i^d)-q_i^d p_{i, total}^d (\mathbf{p}_i^d) \notag\\
&+\alpha_i \left( U_{i, SE}^d(\mathbf{p}_i^d)-R_{i, min}^d \right)-\beta_i \left( \sum_{k=1}^K p_i^k-p_{i, max}^d\right),
\end{aligned}$$ where $\alpha_i$, $\beta_i$ are the Lagrange multipliers associated with the constraints C1 and C2 respectively. Since the problem (10) is in a standard concave form with differentiable objective and constraint functions, the Karush-Kuhn-Tucker (KKT) conditions are used to find the optimum solutions and the duality gap is zero (see page 244 in [@convex_optimization]). Another way to prove that the strong duality holds is to prove that the Slater’s condition is satisfied. Define $f_0 (\mathbf{p}_i^d)=-U_{i, SE}^d (\mathbf{p}_i^d)+q_i^d p_{i, total}^d(\mathbf{p}_i^d)$, $f_1 (\mathbf{p}_i^d)=R_{i, min}^d-U_{i, SE}^d(\mathbf{p}_i^d)$, $f_2 (\mathbf{p}_i^d)= - \sum_{k=1}^K p_i^k$, $f_3 (\mathbf{p}_i^d)=\sum_{k=1}^K p_i^k-p_{i, max}^d$, then the EE maximization problem can be written as $$\begin{aligned}
&\min. \hspace{5mm} f_0 (\mathbf{p}_i^d) \\
&\mbox{s.t.} \hspace{5mm} f_1 (\mathbf{p}_i^d) \leq 0 \\
&\hspace{9mm} f_2 (\mathbf{p}_i^d) \leq 0 \\
&\hspace{9mm} f_3(\mathbf{p}_i^d) \leq 0 \end{aligned}$$
Let us define **relint** $\mathcal{D}$ as the relative interior of the feasible domain, and $\mathcal{D}=\cap_{m=1}^{3} \mbox{dom} (f_m)$. We note that $f_0$ and $f_1$ are convex functions, and $f_2$ and $f_3$ are affine functions. If **relint** $\mathcal{D}$ is not empty, there always exists an $\mathbf{p}_i^d \in $ **relint** $\mathcal{D}$ such that $f_1 (\mathbf{p}_i^d) < 0$, which satisfies the Slater’s condition and ensures that the strong duality holds. On the other hand, if **relint** $\mathcal{D}$ is empty, the optimization problem is either infeasible or has only one solution, which is not the interest of this paper.
Alternatively, we can replace $R_{i, min}^d$ by $R_{i, min}^d+\lim_{\xi \to 0^{+}} \xi $ ($\xi >0$) in the constraint C1 so that $U_{i, SE}^d(\mathbf{p}_i^d) \geq R_{i, min}^d+\lim_{\xi \to 0^{+}} \xi$. This always ensures that $$\begin{aligned}
f_1=R_{i, min}^d-U_{i, SE}^d(\mathbf{p}_i^d) \leq R_{i, min}^d-R_{i, min}^d-\lim_{\xi \to 0^{+}} \xi=-\lim_{\xi \to 0^{+}} \xi <0. \end{aligned}$$ This modification of C2 will not affect the stability of the algorithm since the proposed iterative optimization algorithm converges to the optimum EE, which is proved in Theorem 4.
The equivalent dual problem can be decomposed into two subproblems, which is given by $$\label{eq:dual problem}
\displaystyle \min_{\displaystyle (\alpha_i \geq 0, \beta_i \geq 0)}\!\!\!\!. \hspace{5mm} \max_{\displaystyle (\mathbf{p}_i^d)}. \:\:\: \mathcal{L}_{EE}(\mathbf{p}_i^d, \alpha_i, \beta_i)$$ Taking the first-order derivatives of (12) with regard to $p_i^k$, we have $$\begin{aligned}
\frac{\partial \mathcal{L}_{EE}(\mathbf{p}_i^d, \alpha_i, \beta_i)}{\partial p_i^k} \Big |_{p_i^k=\hat{p}_i^k}=0, k=1, \cdots, K\end{aligned}$$ For any given $q_i^d$, the optimum solution is given by $$\label{eq:waterfilling}
\hat{p}_i^{k}=\left[ \frac{\eta(1+\alpha_i) \log_2e }{q_i^d+\eta\beta_i }-\frac{\hat{p}_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}\hat{p}_{j}^k g_{j, i}^k+N_0}{g_{i}^k}\right]^{+},$$ where $[x]^+=\max\{0,x\}$. Equation (\[eq:waterfilling\]) indicates a water-filling algorithm for transmission power allocation, and the interference from the other UEs decreases the water level.
For solving the minimization problem, the Lagrange multipliers can be updated by using the gradient method [@improved_step_size; @subgradient]. The gradient of $\alpha_i$ and $\beta_i$ are given by $$\begin{aligned}
&\frac{\partial \mathcal{L}_{EE}(\mathbf{p}_i^d, \alpha_i, \beta_i)}{\partial \alpha_i}=U_{i, SE}^d(\mathbf{p}_i^d)-R_{i, min}^d, \notag\\
&\frac{\partial \mathcal{L}_{EE}(\mathbf{p}_i^d, \alpha_i, \beta_i)}{\partial \beta_i}=-\left( \sum_{k=1}^K p_i^k-p_{i, max}^d \right).\end{aligned}$$ Then, $\alpha_i$, $\beta_i$ are updated by using the gradient method as $$\begin{aligned}
\alpha_i (\tau +1)&=\left[ \alpha_i(\tau )-\mu_{i, \alpha} (\tau ) \left(U_{i, SE}^d(\mathbf{\hat{p}}_i^d)-R_{i, min}^d \right) \right]^{+},\\
\beta_i (\tau +1)&=\left[ \beta_i(\tau )+\mu_{i, \beta} (\tau ) \left( \sum_{k=1}^K \hat{p}_i^k - p_{i, max}^d \right) \right]^{+},\end{aligned}$$ where $\tau \geq 0$ is the iteration index, $\mu_{i, \alpha} (\tau ) , \mu_{i, \beta} (\tau ) $ are the positive step sizes which are taken in the direction of the negative gradient for the dual variables at iteration $\tau$. The step sizes should be chosen to strike a balance between optimality and convergence speed. Since the Lagrange multiplier updating techniques are beyond the scope of this paper, interested readers may refer to [@improved_step_size; @subgradient] and references therein for details.
Similarly, for any given $q_k^c$, the optimum solution of $k$-th cellular UE is given by $$\label{eq:waterfilling_CE}
\hat{p}_c^k=\left[ \frac{\eta (1+\delta_k) \log_2e }{q_k^c+\eta \theta_k }-\frac{\sum_{i=1}^N \hat{p}_{i}^k g_{i, c}^k+N_0}{g_{c}^k} \right]^+,$$ where $\delta_k, \theta_k$ are the Lagrange multipliers associated with the constraints C3 and C4 respectively.
Complexity Analysis
-------------------
The proposed iterative optimization algorithm is based on the nonlinear fractional programming developed in [@Dinkelbach]. The iterative algorithm solves the convex problem of (\[eq:transformed problemD\]) (or (\[eq:transformed problemC\]) at each iteration. The iterative algorithm produces an increasing sequence of $q_i^d$ (or $q_k^c$) values which are proved to converge to the optimum EE $q_i^{d*}$ at a superlinear convergence rate [@Dinkelbach_superlinear]. Taking the $i$-th D2D pair as an example, in each iteration, (\[eq:transformed problemD\]) is solved by using the Lagrange dual decomposition. The algorithmic complexity of this method is dominated by the calculations given by (\[eq:waterfilling\]), which leads to a total complexity $\mathcal{O} (I_{i, dual}^d I_{i, loop}^{d} K)$ when $K$ is large, where $I_{i, dual}^d$ is the required number of iterations required for reaching convergence, i.e., $I_{i, dual} \leq L_{max}$, and $I_{i, loop}^{d}$ is the required number of iterations for solving the dual problem.
In particular, the dual problem (\[eq:dual problem\]) is decomposed into two subproblems: the inner maximization problem solves the the power allocation problem to find the best strategy and the outer minimization problem solves the master dual problem to find the corresponding Lagrange multipliers. In the inner maximization problem, a total of $I_{i, dual}^d I_{i, loop}^{d}K (N+3)$ real additions, $I_{i, dual}^d I_{i, loop}^{d}K(N+5)$ real multiplications, and $I_{i, dual}^d I_{i, loop}^{d}K$ real comparisons are required. In the outer minimization problems, a total of $I_{i, dual}^d I_{i, loop}^{d}(K+3)$ real additions, $2 I_{i, dual}^d I_{i, loop}^{d}$ real multiplications, and $2 I_{i, dual}^d I_{i, loop}^{d}$ real comparisons are quired. In conclusion, a total of $I_{i, dual}^d I_{i, loop}^{d}(KN+4K+3)$ real additions, $I_{i, dual}^d I_{i, loop}^{d}(KN+5K+2)$ real multiplications, and $I_{i, dual}^d I_{i, loop}^{d}(K+2)$ real comparisons are quired for the $i$-th D2D pair.
$q_i^d \leftarrow 0$, $q_k^c \leftarrow 0$, $L_{max} \leftarrow 10$, $n \leftarrow 1$, $\Delta \leftarrow 10^{-3}$ solve (\[eq:transformed problemD\]) for a given $q_i^d$ and obtain $\mathbf{\hat{p}}_i^d$ $\mathbf{p}_i^{d*}=\mathbf{\hat{p}}_i^d$, and $\displaystyle q_i^{d*}=\frac{U_{i,SE}^d(\mathbf{p}_i^{d*})}{p_{i, total}^d(\mathbf{p}_i^{d*})}$ **break** $\displaystyle q_i^d=\frac{U_{i,SE}^d(\mathbf{\hat{p}}_i^d)}{p_{i, total}^d(\mathbf{\hat{p}}_i^d)}$, and $n=n+1$ solve (\[eq:transformed problemC\]) for a given $q_k^c$ and obtain $\mathbf{\hat{p}}_k^c$ $\mathbf{p}_k^{c*}=\mathbf{\hat{p}}_c$, and $\displaystyle q_k^{c*}=\frac{U_{k,SE}^c (\mathbf{p}_k^{c*})}{p_{k, total}^c(\mathbf{p}_k^{c*})}$ **break** $\displaystyle q_k^c=\frac{U_{k,SE}^c(\mathbf{\hat{p}}_k^c)}{p_{k, total}^c(\mathbf{\hat{p}}_k^c)}$, and $n=n+1$
Distributed Implementation
--------------------------
In the formulated EE maximization problem, the best response of the $i$-th D2D transmitter $\mathbf{p}_i^d$ depends on the strategies of all other UEs, i.e., $\mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c$. In order to obtain this knowledge, each UE has to broadcast its transmission strategy to other UEs. However, we observe that the sufficient information of $\mathbf{p}_{-i}^d, \mathbf{p}_k^c, \mathbf{p}_{-k}^c$ are contained in the form of interference, i.e., $p_c^k g^k_{c, i}$ and $\sum_{j=1, j\neq i}^{N} p_{j}^k g_{j, i}^k$. In this way, each D2D pair has only to estimate the interference on all available channels to determine the power optimization rather than knowing the specific strategies of other UEs. For the $k$-th cellular UE, the BS estimates the interference from D2D pairs on the $k$-th channel and then feeds back this information to the cellular UE. If UEs update their strategies sequentially, player strategies will eventually converge to a Nash equilibrium, which is proved to exist in Theorem 3. The D2D peer discovery techniques and the design of strategy updating mechanism are out of the scope of this paper and will be discussed in future works.
Energy Efficiency and Spectral Efficiency Tradeoff {#tradeoff}
==================================================
For the $i$-th D2D pair, by analyzing the EE and SE relationships, we have the following properties.
***Lemma 1:*** The SE, $U_{i, SE}^{d}$, increases monotonically as $p_i^k$ increases, while the EE, $U_{i, EE}^{d}$, increases firstly and then decreases as $p_i^k$ increases. $U_{i, EE}^d$ is quasiconcave.
The proof of Lemma 1 is given in Appendix \[lemma1\].
***Lemma 2:*** The transformed objective function in subtractive form is a concave function.
The proof of Lemma 2 is given in Appendix \[lemma2\].
***Lemma 3:*** $ \max_{(\mathbf{p}_i^d)} U_{i, SE}^d (\mathbf{p}_i^d)-q_i^d p_{i, total}^d(\mathbf{p}_i^d)$ is monotonically decreasing as $q_i^d$ increases.
The proof of Lemma 3 is given in Appendix \[lemma3\].
***Lemma 4:*** For any feasible $\mathbf{p}_i^{d}$, $\max_{\big(\mathbf{p}_i^{d}\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}\big)-q_i^{d} p_{i, total}^d(\mathbf{p}_i^d ) \geq 0$.
The proof of Lemma 4 is given in Appendix \[lemma4\].
***Theorem 3:*** A Nash equilibrium exists and the optimum strategy set $\{ \mathbf{p}_i^{d*}, \mathbf{p}^{c*}_k \mid i \in \mathcal{N}, k \in \mathcal{K}\}$ obtained by using Algorithm \[offline algorithm\] is the Nash equilibrium.
The proof of Theorem 3 is given in Appendix \[theorem3\].
***Theorem 4:*** The proposed iterative optimization algorithm converges to the optimum EE.
The proof of Theorem 4 is given in Appendix \[theorem4\].
***Corollary 1:*** EE can be increased by a maximum of $\Delta EE=q_i^{d*}-U_{i, EE}^{d}(\mathbf{p}_i^d)$ by either trading off SE with $\Delta SE= U_{i, SE}^{d}(\mathbf{p}_i^{d})-U_{i, SE}^{d}(\mathbf{p}_i^{d*})$ if and only if $p_i^k>p_i^{k*}$, $\forall p_i^{k*} \in \mathbf{p}_i^{d*}$, or by simultaneously increasing SE with $\Delta SE= U_{i, SE}^{d}(\mathbf{p}_i^{d*})-U_{i, SE}^{d}(\mathbf{p}_i^{d})$ if and only if $p_i^k<p_i^{k*}$, $\forall p_i^{k*} \in \mathbf{p}_i^{d*}$.
Corollary 1 can be easily proved by Lemma 1 since that $U_{i,EE}^d$ decreases as $p_i^k$ increases when $p_i^k>p_i^{k*}$, and both $U_{i,EE}^d$ and $U_{i,SE}^d$ increases as $p_i^k$ increases when $p_i^k<p_i^{k*}$.
The EE and SE tradeoffs depend on the specific channel realization in each simulation and a large number of simulations are required to obtain the average result. In order to facilitate analysis and get some insights, we consider a special case that all the signal channels have the same power gain $g$, and all the interference channels have the same power gain $\hat{g}$. The network coupling factor is defined as $ I=\hat{g}/g$ [@Miao_TWC2011]. Assuming that $N_0$ can be ignored comparing to the interference, $U_{i,SE}^d$ and $U_{i,EE}^d$ are given by $$\begin{aligned}
\label{eq:U_SE_D_I}
U_{i,SE}^d &\approx K \log_2 \big(1+ \frac{p_i^k}{p_c^kI+(N-1)p_i^kI} \big),\\
\label{eq:U_EE_D_I}
U_{i,EE}^d &\approx \frac{\eta U_{i,SE}^d \big( 1-(N-1) I (2^{ \frac{U_{i,SE}^d}{K}}-1) \big)}{K p_c^k I (2^{\frac{U_{i,SE}^d}{K}}-1)+2p_{cir}\eta \big( 1-(N-1) I (2^{\frac{U_{i,SE}^d}{K}}-1) \big) }.\end{aligned}$$
Similarly, $U_{k, SE}^c$ and $U_{i, EE}^d$ are given by $$\begin{aligned}
\label{eq:U_SE_C_K}
U_{k,SE}^c &\approx \log_2 \big(1+ \frac{p_c^k}{Np_i^kI} \big),\\
\label{eq:U_EE_C_K}
U_{k,EE}^c &\approx \frac{\eta U_{k,SE}^c }{Np_i^kI(2^U_{k,SE}-1)+p_{cir}\eta}.\end{aligned}$$
***Corollary 2:*** For any given $p_i^k$ and $p_c^k$, both $U_{i,SE}^d$ and $U_{i,EE}^d$ decrease monotonically as $I$ increases. For any finite and positive $I$, $U_{i,EE}^d$ increases firstly and then decreases as $U_{i,SE}^d$ increases. $U_{i,EE}^d \rightarrow 0$ if and only if $U_{i,SE}^d \rightarrow 0$ or $U_{i,SE}^d \rightarrow K\log_2 \big(1+\frac{1}{(N-1)I} \big)$.
The proof of Corollary 2 is given in Appendix \[corollary2\].
***Corollary 3:*** For any given $p_i^k$ and $p_c^k$, both $U_{k,SE}^c$ and $U_{k,EE}^c$ decrease monotonically as $I$ increases. For any finite and positive $I$, $U_{k,EE}^c$ increases firstly and then decreases as $U_{k,SE}^c$ increases. $U_{k,EE}^c \rightarrow 0$ if and only if $U_{k,SE}^c \rightarrow 0$ or $U_{k,SE}^c \rightarrow \infty$.
Similar conclusions hold for cellular links but are omitted here due to space limitation.
Simulation Results {#Simulation Results}
==================
In this section, the EE and SE tradeoff is investigated through computer simulations. There are a total of $N=5$ D2D links and $K=3$ cellular links. For each simulation, the locations of cellular UEs and D2D UEs are generated randomly within a cell with a radius of $500$ m. The maximum D2D transmission distance is $25$ m. The values of simulation parameters and channel gains are inspired by [@Feiran_WCNC2013; @Feiran_2012; @Song_JSAC]. Fig. \[scenario\] shows the locations of D2D UEs and cellular UEs generated in one simulation. The maximum distance between any two D2D UEs that form a D2D pair is $25$ m. The channel gain between the transmitter $i$ and the receiver $j$ is calculated as $d_{i, j}^{-2} |h_{i, j}|^2$ [@Feiran_WCNC2013; @Feiran_2012; @Feiran_ICC2013], where $d_{i, j}$ is the distance between the transmitter $i$ and the receiver $j$, $h_{i, j}$ is the complex Gaussian channel coefficient that satisfies $h_{i, j} \sim \mathcal{CN} (0, 1)$.
Fig. \[EE\_D2D\] shows the normalized average EE of D2D links corresponding to the number of game iterations. We compare the proposed EE maximization algorithm (labeled as “energy-efficient") with the SE maximization algorithm (labeled as “spectral-efficient" ), and the random power allocation algorithm (labeled as “random"). In the spectral-efficient algorithm, each UE is self-interested and wants to maximize its own SE rather than EE, and the power consumption is completely ignored in the optimization process. The results are averaged through a total number of $1000$ simulations and normalized by the maximum value. The normalized average EE of the proposed energy-efficient algorithm converge to $0.429$, while the random algorithm converge to $0.124$ and the spectral-efficient algorithm converge to $0.064$. It is clear that the proposed energy-efficient algorithm significantly outperforms the spectral-efficient algorithm and the random algorithm in terms of EE in an interference-limited environment. The spectral-efficient algorithm has the worst EE performance among the three because power consumption is completely ignored in the optimization process. The random algorithm fluctuates around the equilibrium since that the transmission power strategy is randomly selected.
Fig. \[EE\_SE\_real\] shows the EE and SE tradeoffs for D2D links corresponding to $p_{i,max}^d=\infty, 200$ mW respectively. For each D2D link, the SE requirement is increased from $0$ to $16$ bits/s/Hz with a step of $1$, and the corresponding EE is obtained by Algorithm 1. The average EE of $N$ D2D links is averaged again over a total number of $500$ simulations. For any specified SE requirement ($0 \leq U_{i, EE}^d \leq 16$ bits/s/Hz), there is always a possibility to satisfy the SE requirement if the signal channel gain is large enough compared to the interference channel gain. One simple example is that the $i$-th D2D transmitter and receiver are close to each other but far from the other interference sources. Simulation results show that the maximum achievable EE is limited by $p_{i,max}^d$ (constraint C2), which is particularly obvious in the high SE regime. If the circuit power consumption $p_{cir}$ is taken into consideration, as proved in Lemma 1, the EE, $U_{i, EE}^{d}$, increases firstly and then decreases as $p_i^k$ increases. Since the SE, $U_{i, SE}^{d}$, increases monotonically as $p_i^k$ increases, we can prove that the EE, $U_{i, EE}^{d}$, increases firstly and then decreases as $U_{i, SE}^{d}$ increases, which is in agreement with Fig. \[EE\_SE\_real\]. It is clear that the EE gain achieved by decreasing the transmission power below the power for optimum EE is not able to compensate for the EE loss caused by the circuit power and SE loss.
Fig. \[EE\_SE\_tradeoff\] shows the tradeoff between EE and SE for D2D links in the special case discussed in Section \[tradeoff\]. Cellular UEs are assumed to transmit with $p_c^k=p_{k,max}^c=200$ mW. For each SE, the corresponding EE is obtained by (\[eq:U\_EE\_D\_I\]). Simulation results show that the maximum achievable SE and EE decrease monotonically as $I$ increases, which agrees with Corollary 2. In Fig. \[EE\_SE\_tradeoff\], it is impossible to achieve the corresponding EE for some $U_{i, EE}^d$. The reason is that we consider the special case introduced in Section IV that all the signal channels have the same power gain $g$, and all the interference channels have the same power gain $\hat{g}$. In this special case, the channel gains are fixed and no longer depend on the transmission distance. When $I=-15$ dB, $p_c^k=p_{k, max}^c=200$ mW, $N=5, K=3, p_{i, max}^d=200$ mW, the maximum achievable $U_{i, SE}^d$ calculated by (\[eq:U\_SE\_D\_I\]) is only $8.6182$ bits/s/Hz. Therefore, the solution is infeasible when $U_{i, SE}^d \geq 9$ bits/s/Hz. Both Fig. \[EE\_SE\_real\] and Fig. \[EE\_SE\_tradeoff\] demonstrate that increasing transmission power beyond the power for optimum EE brings little SE improvement but significant EE loss. However, in the case of $I=-10$dB, the EE loss is not so obvious since that the maximum achievable EE is severely limited by the interference.
Fig. \[EE\_SE\_real\_cellular\] shows the EE and SE tradeoffs for cellular links corresponding to $p_{i,max}^d=\infty, 200$ mW respectively. The SE requirement is increased from $0$ to $10$ bits/s/Hz with a step of $0.5$, and the corresponding EE is obtained by Algorithm 1. The average EE of $K$ cellular links is averaged again over a total number of $500$ simulations. Compared with Fig. \[EE\_SE\_real\], the maximum EE is much lower due to the low signal channel gain caused by longer transmission distance in cellular links. In addition, the maximum achievable EE is significantly limited by $p_{k, max}^c$ in low and high SE regimes also due to the long transmission distance.
Fig. \[EE\_SE\_tradeoff\_cellular\] shows the tradeoff between EE and SE for cellular links in the special case discussed in Section \[tradeoff\]. D2D UEs are assumed to transmit with $p_i^k=\frac{p_{i,max}^d}{K}=\frac{200}{3}$ mW. For each SE, the corresponding EE is obtained by (\[eq:U\_EE\_C\_K\]). Simulation results show that the maximum achievable SE and EE decrease monotonically as $I$ increases, which agrees with Corollary 3. Compared with Fig. \[EE\_SE\_tradeoff\], both of the maximum EE and SE are limited due to that a cellular link can only use one channel, while a D2D pair uses $K$ channels.
Conclusion {#Conclusion}
==========
In this paper, we proposed a distributed energy-efficient resource allocation algorithm for D2D communications by exploiting the properties of nonlinear fractional programming. We have analyzed and verified the EE and SE tradeoff of the proposed algorithm through computer simulations. Simulation results demonstrate that increasing transmission power beyond the power for optimum EE brings little SE improvement but significant EE loss. Therefore, the proposed energy-efficient algorithm can bring significant EE improvement subject to little SE loss.
Proof of the Theorem 1 {#theorem1}
======================
The proof of the Theorem 1 is similar to the proof of the Theorem (page 494 in [@Dinkelbach]). Firstly, we prove the necessity proof. For any feasible strategy set $\mathbf{p}_i^d$, $\forall i \in \mathcal{N}$, we have $$\begin{aligned}
\label{eq:sb}
q_i^{d*}=\frac{r_i^d(\mathbf{p}_i^{d*})}{p_{i, total}^d(\mathbf{p}_i^{d*})} \geq \frac{r_i^d(\mathbf{p}_i^d)}{p_{i, total}^d(\mathbf{p}_i^d)}.\end{aligned}$$ By rearranging (\[eq:sb\]), we obtain $$\begin{aligned}
r_i^d(\mathbf{p}_i^{d*})-q_i^{d*}p_{i, total}^d(\mathbf{p}_i^{d*})&=0,\\
r_i^d(\mathbf{p}_i^{d})-q_i^{d*}p_{i, total}^d(\mathbf{p}_i^{d}) &\leq 0.\end{aligned}$$ Hence, the maximum value of $r_i^d(\mathbf{p}_i^{d})-q_i^{d*}p_{i, total}^d(\mathbf{p}_i^{d})$ is $0$, and can only be achieved by $\mathbf{p}_i^{d*}$, which is obtained by solving the EE maximization problem defined in (\[eq:Dproblem\]). This completes the necessity proof.
Now we turn to the sufficiency proof. Assume that $\mathbf{\tilde{p}}_i^{d}$ is the optimum solution which satisfies that $$\begin{aligned}
\label{eq:assumption_theorem1}
r_i^d(\mathbf{p}_i^{d})-q_i^{d*}p_{i, total}^d(\mathbf{p}_i^{d}) &\leq r_i^d(\mathbf{\tilde{p}}_i^{d})-q_i^{d*}p_{i, total}^d(\mathbf{\tilde{p}}_i^{d}) =0.\end{aligned}$$ By rearranging (\[eq:assumption\_theorem1\]), we have $$\begin{aligned}
q_i^{d*}=\frac{r_i^d(\mathbf{\tilde{p}}_i^{d})}{p_{i, total}^d(\mathbf{\tilde{p}}_i^{d})} \geq \frac{r_i^d(\mathbf{p}_i^d)}{p_{i, total}^d(\mathbf{p}_i^d)}.\end{aligned}$$ Hence, $\mathbf{\tilde{p}}_i^{d}$ is also the solution of the EE maximization problem defined in (\[eq:Dproblem\]), i.e., $\mathbf{\tilde{p}}_i^{d}=\mathbf{p}_i^{d*}$. This completes the sufficiency proof.
Proof of the Lemma 1 {#lemma1}
====================
It is easily verified that $\frac{\partial U_{i,SE}^d}{\partial p_i^k}=\frac{g_i^k \log_2e}{p_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}p_{j}^k g_{j, i}^k+N_0+p_i^kg_i^k}>0$. Hence, $U_{i,SE}^d$ increases monotonically with $p_i^k$.
The denominator of $\frac{\partial U_{i,EE}^d}{\partial p_i^k}$ is a positive value, so we only have to consider the numerator, which is defined as $$\begin{aligned}
f(p_i^k)&=\frac{g_i^k \big (\sum_{k=1}^K \frac{1}{\eta } p_i^k+2p_{cir}\big) \log_2e}{p_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}p_{j}^k g_{j, i}^k+N_0+p_i^kg_i^k}\notag\\
&- \frac{1}{\eta }\sum_{k=1}^K \log_2 \left( 1+\frac{p_i^k g_{i}^k}{p_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}p_{j}^k g_{j, i}^k+N_0} \right)\end{aligned}$$ Take the first-order derivative of $f(p_i^k)$, it can be verified that $\frac{\partial f(p_i^k)}{\partial p_i^k}<0$, thus we have $f(\infty) <f(p_i^k)<f(0)$. As $\lim_{p_i^k \to \infty}f(p_i^k)=\frac{1}{\eta } \log_2e - \infty<0$, and $\lim_{p_i^k \to 0}f(p_i^k)=\frac{2 g_i^k p_{cir} \log_2e}{p_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}p_{j}^k g_{j, i}^k+N_0} >0$, we have $\frac{\partial U_{i,EE}^d}{\partial p_i^k}>0$ when $p_i^k<p_i^{k*}$, and $\frac{\partial U_{i,EE}^d}{\partial p_i^k}<0$ when $p_i^k>p_i^{k*}$. Thus, we prove that $U_{i, EE}^d$ increases firstly and then decreases as $p_i^k$ increases.
Since the numerator and denominator of (\[eq:UE\_EED\]) are concave function and affine function of $p_i^k$ respectively, $U_{i, EE}^d$ is quasiconcave (Problem 4.7 in [@convex_optimization]).
Proof of the Lemma 2 {#lemma2}
====================
Taking $U_{i, SE}^d (\mathbf{p}_i^d)-q_i^d p_{i, total}^d(\mathbf{p}_i^d)$ as an example, which is the transformed objective function in subtractive form corresponding to the $i$-th D2D pair. The first part $U_{i, SE}^d (\mathbf{p}_i^d)$ can be rewritten as $$\begin{aligned}
U_{i, SE}^d (\mathbf{p}_i^d)=\sum_{k=1}^K \log_2 \left( 1+\frac{p_i^k g_{i}^k}{p_c^k g^k_{c, i}+\sum_{j=1, j\neq i}^{N}p_{j}^k g_{j, i}^k+N_0} \right),\end{aligned}$$ which is the sum of $K$ concave functions. The second part $-q_i^d p_{i, total}^d(\mathbf{p}_i^d)$ is given by $$\begin{aligned}
-q_i^d p_{i, total}^d(\mathbf{p}_i^d)=-q_i^d \left( \sum_{k=1}^K \frac{1}{\eta } p_i^k+2p_{cir} \right),\end{aligned}$$ which is the sum of $K$ affine functions. Since the sum of a concave function and an affine function is also concave, this completes the proof of Lemma 2.
Proof of the Lemma 3 {#lemma3}
====================
Define $q_i^{d*}<q_i^{d*'}$, and define $\mathbf{p}_i^{d*}$ and $\mathbf{p}_i^{d*'}$ as the corresponding optimum solutions respectively. We have $$\begin{aligned}
& \max_{(\mathbf{p}_i^d)} U_{i, SE}^d (\mathbf{p}_i^d)-q_i^{d*} p_{i, total}^d(\mathbf{p}_i^d) = U_{i, SE}^d (\mathbf{p}_i^{d*})-q_i^{d*} p_{i, total}^d(\mathbf{p}_i^{d*}) \notag\\
& > U_{i, SE}^d (\mathbf{p}_i^{d*'})-q_i^{d*} p_{i, total}^d(\mathbf{p}_i^{d*'})> U_{i, SE}^d (\mathbf{p}_i^{d*'})-q_i^{d*'} p_{i, total}^d(\mathbf{p}_i^{d*'}) \notag\\
& = \max_{(\mathbf{p}_i^d)} U_{i, SE}^d (\mathbf{p}_i^d)-q_i^{d*'} p_{i, total}^d(\mathbf{p}_i^d). \end{aligned}$$
Proof of the Lemma 4 {#lemma4}
====================
Define an feasible solution $\mathbf{\hat{p}}_i^{d}$ such that $q_i^d=\frac{U_{i, SE}^d (\mathbf{\hat{p}}_i^{d})}{p_{i, total}^d (\mathbf{\hat{p}}_i^{d}) }$, we have $$\begin{aligned}
\max_{\big(\mathbf{p}_i^{d}\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}\big)-q_i^{d} p_{i, total}^d(\mathbf{p}_i^d )
\geq U_{i, SE}^d \big(\mathbf{\hat{p}}_i^{d}\big)-q_i^{d} p_{i, total}^d(\mathbf{\hat{p}}_i^d )=0.\end{aligned}$$
Proof of the Theorem 3 {#theorem3}
======================
According to [@game_theory_1994], a Nash equilibrium exists if the utility function is continuous and quasiconcave, and the set of strategies is a nonempty compact convex subset of a Euclidean space. Taking the EE objection function defined in (\[eq:UE\_EED\]) as an example, the numerator $U_{i, SE}^d$ is a concave function of $p_i^k$, $\forall i \in \mathcal{N}, k \in \mathcal{K}$. The denominator $p_{i, total}^d$ is an affine function of $p_i^k$. Therefore, $U_{i, EE}^d$ is quasiconcave (Problem 4.7 in [@convex_optimization]). The set of the strategies $\mathbf{p}_i^d=\{p_i^k \mid 0 \leq \sum_{k=1}^K p_i^k \leq p_{i, max}^d, k \in \mathcal{K} \}$, $\forall i \in \mathcal{N}$, is a nonempty compact convex subset of the Euclidean space $\mathbb{R}^K$. Similarly, it is easily proved that the above conditions also hold for the cellular UE. Therefore, a Nash equilibrium exists in the noncooperaive game.
If the strategy set $\mathbf{p}_i^{d*}$ obtained by using Algorithm \[offline algorithm\] is not the Nash equilibrium, the $i$-th D2D transmitter can choose the Nash equilibrium $\mathbf{\hat{p}}_i^{d}$ ($\mathbf{\hat{p}}_i^{d} \neq \mathbf{p}_i^{d*}$) to obtain the maximum EE $q_i^{d*}$. However, by Theorem 1, $q_i^{d*}$ can only be achieved by choosing $\mathbf{p}_i^{d*}$. Then, we must have $\mathbf{\hat{p}}_i^{d} = \mathbf{p}_i^{d*}$, which contradicts with the assumption. Therefore, $\mathbf{p}_i^{d*}$ is part of the Nash equilibrium. A similar proof holds for $\mathbf{p}^{c*}_k$. It is proved that the set $\{ \mathbf{p}_i^{d*}, \mathbf{p}^{c*}_k \mid i \in \mathcal{N}, k \in \mathcal{K}\}$ obtained by using Algorithm \[offline algorithm\] is the Nash equilibrium.
Proof of the Theorem 4 {#theorem4}
======================
Firstly, we prove that the EE for the $i$-th D2D pair $q_i^d$ increases in each iteration. We denote that $\mathbf{\hat{p}}_i^{d}(n)$ as the optimum resource allocation policies in the $n$-th iteration, and $q_i^{d*}$ as the optimum EE. We denote that $q_i^d (n)$ and $q_i^d (n+1)$ as the EE in the $n$-th iteration and $(n+1)$-th iteration respectively, and we assume that $q_i^d (n) \neq q_i^{d*}$, and $q_i^d (n+1) \neq q_i^{d*}$. $q_i^d (n+1)$ is updated in the $n$-th iteration in the proposed Algorithm 1 as $q_{n+1}=\frac{U_{i, SE}^d \big(\mathbf{\hat{p}}_i^{d}(n)\big)}{p_{i, total}^d \big(\mathbf{\hat{p}}_i^{d}(n)\big) }$. We have $$\begin{aligned}
& \max_{\big(\mathbf{p}_i^{d}(n)\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}(n)\big)-q_i^{d}(n) p_{i, total}^d(\mathbf{p}_i^d (n))\notag\\
&= U_{i, SE}^d \big(\mathbf{\hat{p}}_i^{d}(n)\big)-q_i^d (n) p_{i, total}^d \big( \mathbf{\hat{p}}_i^{d}(n) \big) \notag\\
&=q_i^d (n+1)p_{i, total}^d \big(\mathbf{\hat{p}}_i^{d}(n)\big)-q_i^d (n) p_{i, total}^d \big( \mathbf{\hat{p}}_i^{d}(n)\big) \notag\\
&=p_{i, total}^d \big(\mathbf{\hat{p}}_i^{d}(n)\big) \big( q_i^d (n+1)-q_i^d (n) \big) \stackrel{\mathrm{Theorem 1, Lemma 3, lemma 4}}{>}0 \notag\\
&\stackrel{\mathrm{p_{i, total}^d \big(\mathbf{\hat{p}}_i^{d}(n)\big)>0}}{\Longrightarrow } q_i^d (n+1)>q_i^d (n)
\end{aligned}$$
Secondly, by combining $q_i^d (n+1)>q_i^d (n)$, Lemma 3, and Lemma 4, we can prove that $$\begin{aligned}
& \max_{\big(\mathbf{p}_i^{d}\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}\big)-q_i^{d}(n) p_{i, total}^d(\mathbf{p}_i^d ) \notag\\
&> \max_{\big(\mathbf{p}_i^{d}\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}\big)-q_i^{d}(n+1) p_{i, total}^d(\mathbf{p}_i^d )\notag\\
& > \max_{\big(\mathbf{p}_i^{d}\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}\big)-q_i^{d*} p_{i, total}^d(\mathbf{p}_i^d ) \notag\\
&=U_{i, SE}^d \big(\mathbf{p}_i^{d*}\big)-q_i^{d*} p_{i, total}^d(\mathbf{p}_i^{d*} )=0.\end{aligned}$$ Therefore, $q_i^d (n) $ is increased in each iteration and will eventually approaches $q_i^{d*}$ as long as $L_{max}$ is large enough, and $\max_{\big(\mathbf{p}_i^{d}\big)} U_{i, SE}^d \big(\mathbf{p}_i^{d}\big)-q_i^{d} p_{i, total}^d(\mathbf{p}_i^d )$ will approach zero and satisfy the optimality conditions proved in Theorem 1.
Proof of the Corollary 2 {#corollary2}
========================
Since $\frac{\partial U_{i,SE}^d}{\partial I}= - \frac{kp_i^k\log_2e}{\big( p_c^k+(N-1)p_i^k \big)I^2+p_i^kI}<0$, and $\frac{\partial U_{i,EE}^d}{\partial I}= - \frac{k\eta p_i^k\log_2e}{\bigg( \big( p_c^k+(N-1)p_i^k \big)I^2+p_i^kI \bigg)(kp_i^k+2p_{cir}\eta)}<0$, both $U_{i,SE}^d$ and $U_{i,EE}^d$ decreases monotonically as $I$ increases. The second part is proved by setting the numerator of (\[eq:U\_EE\_D\_I\]) to $0$ and solving the corresponding $U_{i, SE}^d$.
[^1]: Manuscript received May 7, 2014; revised July 4, 2014.
[^2]: This work was partially supported by Fundamental Research Funds for the Central Universities under Grant Number 14MS08, China Mobile Communication Co. Ltd. Research Institute (CMRI), and China Electric Power Research Institute (CEPRI) of State Grid Corporation of China (SGCC).
[^3]: Zhenyu Zhou is with the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, School of Electrical and Electronic Engineering, North China Electric Power University, Beijing, China, 102206.
[^4]: Mianxiong Dong is with the National Institute of Information and Communications Technology, Kyoto, Japan.
[^5]: Kaoru Ota is with the Department of Information and Electric Engineering, Muroran Institute of Technology, Muroran, Hokkaido, Japan (e-mail: ota@csse.muroran-it.ac.jp).
[^6]: Jun Wu is with the School of Information Security Engineering, Shanghai Jiao Tong University, Shanghai, China.
[^7]: Takuro Sato is with the Graduate School of Fundamental Science and Engineering, Waseda University, Tokyo, Japan.
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abstract: 'In a previous paper$^1$, submitted to Journal of Physics A – we presented an infinite class of potentials for which the radial Schrödinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be zero, but for the other part (also infinite), one can have any angular momentum. In the present paper, we study a simple subclass (also infinite) of the whole class for which the solution of the Schrödinger equation is simpler than in the general case. This subclass is obtained by combining another approach together with the general approach of the previous paper. Once this is achieved, one can then see that one can in fact combine the two approaches in full generality, and obtain a much larger class of potentials than the class found in ref. $^1$ We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity.'
---
[****]{} 2 truemm
[**Potentials for which the Radial**]{} 2 truemm [**Schrödinger Equation can be solved**]{}
5 truemm
[**Khosrow Chadan** ]{}\
[*Laboratoire de Physique Théorique*]{}[^1]\
[*Université de Paris XI, Bâtiment 210, 91405 Orsay Cedex, France*]{}\
[*(Khosrow.Chadan@th.u-psud.fr)*]{}
5 truemm [and]{}
5 truemm [**Reido Kobayashi**]{}\
[*Department of Mathematics*]{}\
[*Tokyo University of Sciences, Noda, Chiba 278-8510, Japan*]{}\
[*(reido@ma.noda.tus.ac.jp)*]{}
1 truecm
3 truecm
LPT Orsay 01-2006\
January 2006
\
In a recent paper,$^1$ we showed how, starting from two regular potentials for which the radial Schrödinger equation can be solved explicitly at zero energy –there are many of them$^{1,2}$– one can construct explicitely an infinite number of potentials for which one can solve again, explicitely, the radial equation at zero energy. The solutions for these new potentials are given very simply in terms of the solutions of the two initial potentials. By construction, it is then seen that the method can be applied without any modifications to potentials which are singular (repulsive) at the origin, or are long range (Coulomb, etc), or are even confining, like $\lambda r^2$, $\lambda > 0$, etc. According to the case, one can include, as well, the angular momentum potential $\ell (\ell + 1)/r^2$. Many examples, both for regular and singular potentials, covering all cases, except the confining potentials, are given in ref. $^1$, with explicit solutions of the radial equation.
In the present paper, following a different method, we study a subclass of the general class, which is simpler, and more explicit. As in the general case, one can have, here too, singular (repulsive) potentials at the origin, or long range potentials like Coulomb potential, etc.
For the convenience of the reader, we give here a résumé of ref. $^1$, where all the proofs can be found, together with appropriate references, which are essentially those of the present paper$^{2-8}$.
Consider first the radial Schrödinger equation at zero energy for the $S$-wave
$$\left \{ \begin{array}{l} \varphi ''_0 (r) = V_0 (r) \ \varphi_0 (r) \ , \\ \\ r \in [0, \infty )\quad, \quad \varphi_0(0) = 0\ , \ \varphi ' _0(0) = 1 \ . \end{array} \right .\eqno({\rm A})$$
It is assumed that either $V_0$ is positive (repulsive), or else, if it is negative (attractive), it is weak and sustains no bound states. Moreover, we assume $$\int_0^1 r |V_0(r)|dr < \infty \quad, \quad \int_1^{\infty} r^2 |V_0 (r) |dr < \infty \ . \eqno({\rm B})$$
Under the above conditions on $V_0 (r)$, i.e. absence of bound states and (B), it can be shown that $$\left \{ \begin{array}{l} \varphi_0 (r) = r + o(r) \ {\rm as} \ r \to 0\ , \ \varphi_0 (r) > 0 \ {\rm for \ all}\ r > 0\ , \\ \\ \varphi_0(r) = Ar + B + o(1)\ {\rm as} \ r\to \infty \ , \ 0 < A < \infty\ , \ |B| < \infty \ . \end{array} \right .\eqno({\rm C})$$
A second independent solution of the Schrödinger equation is given by $$\left \{ \begin{array}{l} \chi_0 (r) = \varphi_0 (r) \displaystyle{\int_r^{\infty} {dt \over \varphi_0^2(t)}} \ , \ r > 0 \ ,\\ \\ \chi_0(0) = 1\ , \ W[\varphi_0, \chi_0] = \varphi '_0 \chi_0 - \chi '_0 \varphi_0 = 1 \ . \end{array} \right .\eqno({\rm D})$$
Indeed, it follows from the definition of $\chi_0(r)$ that $\chi ''_0 = V_0 \chi_0$, and $W[\varphi_0, \chi_0] = 1$. Then $\chi_0 (0) = 1$ follows from the first line of (C). One gets then $$\left \{ \begin{array}{l} \chi_0 (r) > 0\ {\rm for}\ r \in [0, \infty )\ , \\ \\ \chi_0(r)= \displaystyle{{1 \over A}} + o(1)\qquad {\rm as } \ r \to \infty \ , \end{array} \right .\eqno({\rm E})$$
where $A$, strictly positive and finite, is defined in (C).
Now, consider the mapping $r \to x(r)$ defined by $$x(r) = {\varphi_0 (r) \over \chi_0 (r)}\quad , \quad {dx \over dr} = {\varphi '_0 \chi_0 - \varphi_0 \chi '_0 \over \chi_0^2(r)} = {1 \over \chi_0^2(r)} > 0\ . \eqno({\rm F})$$
It is obvious that the mapping is one-to-one, and is smooth. It is in fact $C^2$ since $${d^2x \over dr^2} = {-2 \chi '_0 (r) \over \chi_0^3 (r)}\ . \eqno({\rm G})$$
Therefore, we can use $x(r)$ for making a change variable in the Schrödinger equation. Note that one has also, according to (C) and (E), $$\left \{ \begin{array}{l} x(r) = r + o(r) \quad {\rm as}\ r \to 0\ , \\ \\ x(r) = A^2 r + AB + o(1) \qquad {\rm as}\ r \to \infty\ . \end{array} \right .\eqno({\rm F}')$$
We consider now the equation $$\left \{ \begin{array}{l} \varphi '' (r) = V_0 (r) \varphi (r) + [\chi_0 (r)]^{-4} V[x(r)] \varphi (r)\ , \\ \\ \varphi(0) = 0\ , \ \varphi ' (0) = 1 \ .
\end{array} \right .\eqno({\rm H})$$
Note here that, as we saw before, $\chi_0 (r)$ is a smooth, bounded, and strictly positive function for all $r \geq 0$.
We assume again that $$\left \{ \begin{array}{l} \hbox{i) $V_0(r)$ satisfies (B), and sustains no bound states,} \\ \\ \hbox{ii) $V(x)$ satisfies also (B) in the variable $x$,}\\ \\ \hbox{iii) $V(r)$ can have any (finite) number of bound states.}\end{array} \right .\eqno({\rm I})$$
5 truemm
From (B) for $V(x)$, it follows that $$\int_0^{\infty} x |V(x)|dx < \infty\ . \eqno({\rm J})$$
It then follows from the Bargmann bound for the number $n$ of bound states of $V(x)$ that $$n(V) \leq \int_0^{\infty} x |V(x)|dx < \infty\ . \ \sq \eqno({\rm K})$$
5 truemm
If we make now in (H) the change of variable $r \to x(r)$ defined by (F), and the change of function $$\psi (x) = \left [{\varphi (r) \over \chi_0 (r)}\right ]_{r = r(x)} \ , \eqno({\rm L})$$
$r(x)$ being the inverse mapping, well-defined and also $C^2$, mapping $x \in [0, \infty )$ into $r \in [0, \infty )$, we find $$\left \{ \begin{array}{l} \ddot{\psi}(x) = V(x) \psi (x) \\ \\ \psi (0) = 0\ , \ \dot{\psi} (0) = 1 \ .\end{array} \right .\eqno({\rm M})$$
Therefore, if (A) and (M) can be solved explicitely for $V_0(r)$ and $V(x)$, then (H) also can be solved explicitely, and we have, according to (L), $$\varphi (r) = \chi_0 (r)\ \psi (x) = \chi_0 (r)\ \psi \left ( {\varphi_0 (r) \over \chi_0 (r)}\right ) \ . \eqno({\rm N})$$
We have therefore, the following :
5 truemm
Suppose the Schrödinger equation $\varphi '' (r) = v(r) \varphi (r)$, $r \in [0, \infty )$, together with $\varphi (0) = 0$, $\varphi '(0) = 1$, can be solved explicitely for two potentials $V_0(r)$ and $V(r)$, both satisfying the integrability conditions shown in (B). We assume i) $V_0$ sustains no bound states ; ii) $V$ can have bound states, their number $n$ being finite according to (K). Then (H) also can be solved explicitely, and its solution is given by (N). This, of course, can be checked directly by differentiating (N).\
Once we have (N), we can start now with $V_0$ and $V_0 + \chi^{-4} (r) V[x(r)]$, and repeat the operation to get an infinite number of potentials.\
Once the theorem is established, it was then shown in ref. $^1$ that one can generalize it to the case where $V_0(r)$ in (H) can be singular and repulsive at the origin, violating therefore $rV_0(r) \in L^1(0, 1)$, or be long range and repulsive at infinity, violating therefore $r^2V_0(r) \in L^1(1, \infty )$, provided always that it sustains no bound states. Also $V(r)$ can be more general than it was assumed. Many explicit examples, illustrating all these cases, were given. Of course, if $V_0$, and or $V$, violate (B) at $r=0$ or $r= \infty$, one must modify, accordingly, the boundary conditions. Full details are given in ref. $^1$. For each case, one singular example is shown below. One must secure, of course, each time that the corresponding $\chi_0(r)$ does not vanish for $r \geq 0$, i.e. absence of bound states for $V_0$. This is no problem since in all the explicit examples we give for singular potentials, we are dealing with (modified) Bessel functions, and the locations of the zeros of these functions are known.$^7$ One can also introduce angular momentum by adding $\ell (\ell + 1)/r^2$ to $V_0$.$^{1}$\
We have assumed that $V_0$ has no bound states. If $V(x)$ sustains $n$ bound states, then, according to the nodal theorem,$^{5,6}$ $\psi (x)$, solution of (M), has $n$ zeros for $x > 0$. It follows then from (N) that $\varphi (r)$ has the same number of zeros for $r > 0$, and, therefore, that the potential $V_0(r) + \chi_0^{-4} (r)$ $V(x)$ sustains the same number of bound states as $V(r)$.\
Potentials for which (A) and (M) can be solved explicitely, are many. Not only the classical examples$^{1, 2}$ $$\left \{ \begin{array}{l}V(r) = \displaystyle{{\lambda a^2 \over (1 + ar)^4}} \quad , \quad a > 0 \ , \\ \\ \varphi (r) =\left ( \displaystyle{ {1 + ar \over a \sqrt{\lambda}}}\right ) \sinh \left ( \displaystyle{{\sqrt{\lambda} ar \over 1 + ar }} \right ) \ ; \end{array} \right .
\eqno({\rm O})$$
$$\left \{ \begin{array}{l}V(r) = \displaystyle{{\lambda b^2 \over (b^2 + r^2)^2}} \quad , \quad (b > 0) \ , \\ \\ \varphi (r) = \displaystyle{ {(b^2 + r^2)^{1/2} \over \sqrt{\lambda -1}}} \sinh \left ( \sqrt{\lambda -1} \ Arctg \displaystyle{{r \over b}} \right ) \ ; \end{array} \right .
\eqno({\rm P})$$
and $$\left \{ \begin{array}{l}V(r) = \lambda e^{-\mu r} \quad , \quad \mu > 0 \ , \\ \\ \varphi (r) = \alpha I_0 \left ( \displaystyle{ {2\sqrt{\lambda} \over \mu}} \ e^{- \mu r/2} \right ) + \beta K_0 \left ( \displaystyle{{2\sqrt{\lambda} \over \mu}} \ e^{-\mu r/2} \right ) \ , \end{array} \right .
\eqno({\rm Q})$$
where $I_0$ and $K_0$ are modified Bessel and Hankel functions of order zero,$^7$ and $\alpha$ and $\beta$ are determined as to have $\varphi (0) = 0$, $\varphi ' (0) = 1$, but also the full class of infinite other potentials we found in reference 1, as well as all the Bargmann potentials, etc$^{2,4}$. Therefore, one can use them as $V_0$ and $V$ in (H), and obtain many explicitely soluble examples. The above formulas are written for $\lambda > 1$, or for $\lambda >0$. For $\lambda < 0$, sinh goes to sin, and $I_0$ and $K_0$ to $J_0$ and $N_0$. It is known that, for $\lambda > 0$, $I_0$ and $K_0$ have no zeros for $r \geq 0$,$^7$ and for $\lambda < 0$, Bessel functions have, usually, oscillations. Remember that whatever the potential we choose for $V_0$, it should not sustain bound states, i.e. be either repulsive, or, if attractive, be weak.\
One can also include a singular repulsive potential as well (singular at the origin), like $$V_0(r) = {g \over r^n}\quad , \quad g > 0 \quad , \quad n > 2\ .
\eqno({\rm R})$$
And it can be checked easily that everything works as before. We leave the details for the reader. They can be found in $^1$. We just note that here $\chi_0(0) = \infty$, and $\chi_0(\infty ) = 1$. Since $V_0$ is singular now, we must assume $V_0 > 0$, for, we know that, in such a case, i.e. with singular and attractive potentials at the origin, violating (B), we don’t have a unique self-adjoint extension of the Hamiltonian in $L^2 ({I \hskip - 1 truemm R}^3)$.$^{2,3,6,8}$ The simplest case here is to take $n = 4$. Then, the two independent solutions of (A) are given by, up $$n = 4
\left \{ \begin{array}{l} \chi_0 = \displaystyle{{r \over \sqrt{g}}} \sinh (\sqrt{g}/r)\ , \\ \\ \varphi_0 (r) = r \exp (- \sqrt{g}/r) \ . \end{array} \right .
\eqno({\rm S})$$
Note that, in accordance with $^1$, because the potential violates (B) at the origin, we have $\chi_0 (0) = \infty$, and $\chi_0(\infty ) = 1$. As an example of a long range potential, we consider (M) with $$V(x) = {\alpha \over x}\ ,
\eqno({\rm T})$$ and we find, according to $^1$, that the solution is given by $$\psi (x) = \sqrt{{x \over \alpha}} \ I_1\left ( 2 \sqrt{\alpha x}\right ) \ ,
\eqno({\rm U})$$
where $I_1$ is the modified Bessel function.$^7$ If $\alpha > 0$, we would have no bound states. If $\alpha < 0$, $I_1$ goes to $J_1$, the ordinary Bessel function, which has infinitely many oscillations as $x \to \infty$, and, therefore according to the nodal theorem,$^{5,6}$ we would have infinitely many bound states accumulating at energy $E = 0$, as is well-known. It is also known that, for $\alpha > 0$, $I_1$ does not vanish for $x > 0$, and increases exponentially as $x \to \infty$.$^{7}$
Much work has been done, of course, for finding potentials for which the Schrödinger equation can be solved. We refer the reader to the paper of G. Lévai,$^9$ which contains full references to earlier works.\
\
For this purpose, we use a transformation of the Schrödinger equation devised some years ago by one of the authors and Harald Grosse, which has very smoothing effects on the potential.$^4$ It is as follows. Consider the radial Schrödinger equation at zero energy \[1e\] {
[l]{} ” (r) = V\_0 (r) (r) , V\_0 ,\
\
r \[0, ), (0) = 0 .
.
$V_0(r)$ is assumed to be a regular potential, i.e. to satisfy the Bargmann-Jost-Kohn condition$^{2,3}$ \[2e\] rV\_0 (r) L\^1(0, ) .
We introduce now \[3e\] W\_0(r) = - \_r\^ V\_0(t) dt ; U\_0 (r) = \_r\^ W\_0 (t) dt .
Our transformation is now defined by$^4$ \[4e\] x = \_0\^r e\^[2 U\_0(t)]{} dt , (x) = e\^[U\_0 (r)]{} (r) .
5 truemm Since, for large values of $r$, we have $|V_0| < r |V_0| \in L^1(\infty )$, $W_0$ is well-defined and is an absolutely continuous function for all $r > 0$. Then it is easily seen that $W_0(r) \in L^1(0, \infty )$. Indeed, \[5e\] |U\_0(r)| &=&| \_r\^ W\_0(r) dr | \_0\^ | W\_0 (r)| dr \_0\^ dr \_r\^ | V\_0 (t)| dt\
&=&\_0\^ | V\_0(t) | dt \_0\^t dr = \_0\^ t| V\_0(t)| dt < .
It follows that $U_0(r)$ is a well-defined, bounded, and absolutely continuous function for all $r \geq 0$, and is also continuously differentiable for $r > 0$. $\sq$\
From the above results, it is obvious that the transformation (\[4e\]) is a nice smooth transformation, and we have a smooth one to one mapping \[6e\] r \[0, ) x\[0, ) ,
and $(dx/dr ) = \exp (2U_0(r)) > 0$. Obviously, we have also \[7e\] (0) = 0 (0) = 0 .
Making the transformation (\[4e\]) in (\[1e\]), we find, with $\dot{} = {d \over dx}$, \[8e\] {
[l]{} (x) = (x) = (x) (x)\
\
x \[0, ) , (0) = 0 .
.
Now, since $U_0 (r)$ is a very smooth and bounded function, for all $r \geq 0$, and $U( \infty ) = 0$, it is obvious from the definition (\[4e\]) that $x$ and $r$ are very close to each other, and we have \[9e\] {
[ll]{} x e\^[2U\_0(0)]{} r + o(r), & r 0 ,\
\
x r + 0(1), & r .
.
Therefore, as far as integrability at $x = 0$ and $x = \infty$ are concerned, we have \[10e\] \_0\^ x dx \~\_0\^ r W\_0\^2 (r) dr .
To show that the last integral is absolutely convergent is now very easy. Indeed, from the definition of $W_0(r)$, (\[3e\]), \[11e\] | r W\_0 (r) | \_r\^ t | V\_0 (t) | dt < \_0\^ t | V\_0(t) | dt = C < .
Therefore, \[12e\] | r W\_0\^2(r)| C | W\_0 (r)| ,
and since $W_0 (r)$ was shown to be $L^1(0, \infty )$, the same is true for $r W_0^2(r)$. Therefore, in (\[8e\]), $x \widetilde{V}(x) \in L^1(0, \infty )$, and $\widetilde{V}(x)$ is a regular potential.
Consider now the equation \[13e\] ” (r) &=& (r)\
&=& V(r) (r) .
Note here the variable $x = x(r)$ in $V_1(x)$ ! Both $rV_0(r)$ and $xV_1(x)$ are assumed to be $L^1(0, \infty )$, i.e. satisfy (\[2e\]). Since we showed that $r W_0^2(r)$ was also $L^1(0, \infty )$, and $x$ and $r$ are always of the same order by virtue of (\[9e\]), the full potential in (\[13e\]) satisfies the same integrability condition.
After making the transformation (\[4e\]), one finds then, with $\dot{} = (d/dx)$, \[14e\] {
[l]{} (x) = V\_1(x) (x) ,\
\
x \[0, ) , (0) = 0 , xV\_1(x) L\^1(0, ) .
.
Therefore, if the equation (\[14e\]) with $V_1(x)$ can be solved explicitely, we can solve also explicitely (\[13e\]), and its solution is given by \[15e\] (r) = e\^[-U\_0 (r)]{} (x)
where $x$ is defined in (\[4e\]). Therefore, we have completed our programme, i.e. starting from two potentials $V_0$ and $V_1$, and knowing that the Schrödinger equation can be solved explicitely for $V_1$, to find a new potential for which the same holds. One can check again, directly, that (\[15e\]) is, indeed, a solution of (\[13e\]).
The connection with ref. $^1$ is as follows. Consider \[16e\] ” \_0= ( V\_0 + W\_0\^2) \_0 .
A solution of this equation, called $\chi_0 (r)$, with $\chi_0(0) = 1$, is \[17e\] \_0 (r) = [e\^[-U\_0 (r)]{} e\^[-U\_0(0)]{}]{} .
This has no zeros for $r \geq 0$. Therefore, the physical solution of (\[16e\]), called $\varphi_0(r)$, with $\varphi_0(0) = 0$, and given by$^1$ \[18e\] \_0 (r) = \_0 (r) \_0\^r [dt \_à\^2(t)]{} ,
as can be checked easily, has also no zeros for $r > 0$. Therefore, according to the nodal theorem$^{5, 6}$, $V_0 + W_0^2$ cannot have bound states, whatever the sign of $V_0$ is. According to (\[17e\]), the full potential in (\[13e\]) can be written \[19e\] V(r) = ( V\_0(r) + W\_0\^2 (r) ) + \_0\^[-4]{}(r) V\_1(x) .
This is in complete analogy with reference $^1$, where we had \[20e\] V(r) = V\_0(r) + \_0\^[-4]{}(r) V\_1(x) ,
assuming there $V_0(r)$ to be either positive, or negative but weak enough in order not to have bound states in (\[1e\]). In the present paper, $V_0$ is replaced by $V_0 + W_0^2$, with the same properties, but now with simple explicit solutions given by (\[17e\]) and (\[18e\]), and no restrictions on the sign of $V_0$.\
So far, we did not assume anything on the signs of $V_0$ and $V_1$ in (\[13e\]). They can have any sign. As we showed, $V_0 + W_0^2$ alone cannot have bound states. So in order to have bound states, $V_1(x)$ must be negative, and strong enough. Now, again, the nodal theorem$^{5, 6}$, applied to (\[14e\]), shows that, if $V_1(x)$ sustains $n$ bound states, then $\psi (x)$ has $n$ zeros (nodes) for $x > 0$. Therefore, according to (\[15e\]), $\varphi (r)$ also has $n$ nodes for $r > 0$. So, there are $n$ bound states also for the full potential (\[18e\]). In conclusion, whatever the sign of $V_0$ is, $(V_0 + W_0^2)$ sustains no bound states, and $V_0 + W_0^2 + \chi_0^{-4} V(x)$ and $V(r)$ have the same number of bound states, where $x$ is given in (\[4e\]).\
Generalization to include the centrifugal potential $\ell (\ell + 1)/r^2$ in $V_0 + W_0^2$ is straightforward.$^4$. We define now \[24e\] {
[l]{}W\_ (r) = - V\_0(t) t\^[-2]{} dt ,\
\
U\_ (r) = W\_ (t) r\^[2]{} dt .
.
Since \[25e\] | r\^[2 ]{} W\_ (r) | \_r\^ V\_0 (t) dt < , r > 0 ,
it is obvious that $r^{2\ell} W_{\ell} (r)$, like $W_0(r)$, is bounded for $r > 0$, and $\in L^1 (0 , \infty )$. And $U_{\ell}(r)$ is a nice bounded and smooth function for all $r \geq 0$, as was the case for $U_0(r)$. Now, we define, \[26e\] \_ (r) = r\^[-]{} e\^[-U\_ (r)]{} .
It is then easily seen that $\chi_{\ell} (r)$ satisfies \[27e\] ”\_(r) = [(+ 1) r\^2]{} \_(r)+ V\_0 \_ + r\^[4]{} W\_\^2 (r) \_ (r) .
The generalization of (\[13e\]) is now \[28e\] ”\_ (r) = \_ (r) ,
and one finds, after the transformations \[29e\] x = x(r) = \_0\^r t\^[2]{} e\^[2U\_(t)]{} dt , \_ (x) = r\^ e\^[U\_ (r)]{} \_ (r) ,
the differential equation \[30e\] \_ (x) = V\_1(x) \_ (x) .
Therefore, the conclusion is exactly as for the case $\ell = 0$. Remember that we have assumed $xV_1(x) \in L^1(0, \infty )$. The solution of (\[28e\]) is, therefore, given by $$\varphi_{\ell}(r) = r^{-\ell} \ e^{-U_{\ell}(r)} \ \psi_{\ell} (x) \ . \eqno(25')$$
Making $\ell = 0$, we find, of course, (\[13e\])-(\[15e\]).\
With the potentials (O)-(Q) for $V_0$ and $V_1$, and (\[3e\]), (\[13e\]), (\[14e\]) and (\[15e\]), one can construct easily many explicit examples. We leave the details to the reader. With (R) for $V_0$, $n=4$, and (\[17e\]) and (\[18e\]), it is easily found that $$\chi_0 (r) = \exp (g/6r^2)\ , \eqno(17')$$
and $$\varphi_0 (r) = \left \{ \begin{array}{l} r^3 \exp (-g/6r^2) + \cdots\ , \quad r\to 0\\ \\ r - \sqrt{g} + \cdots\ , \quad r \to \infty \ .\end{array} \right . \eqno(18')$$
One can then take for $V_1$ one of the potentials (O)-(Q).\
\
In this section, we are going to combine both transformations, namely, the transformation of section I, and the tranformation of section II. Consider the three potentials \[36e\] {
[l]{}
.
We assume that one can solve explicitely \[37e\] {
[l]{} ” (r) = v(r) (r), v = V\_0 (r) , V(r) ,(0) = 0 , ’ (0) = 1 .
.
Nothing is assumed for the explicit solution for $v = V_1(r)$.
As we saw in section I, the assumptions on $V_0(r)$, entail the existence of the solution $\chi_0(r)$ of (\[37e\]), given by (D), such that \[38e\] {
[l]{} \_0(r) > 0 [for all]{} r 0 , \_0 (0) = 1 , \_0() = , 0 < A < ,
.
that is, a smooth and strictly positive bounded function. We introduce now \[39e\] {
[l]{}W\_1(r) = - V\_1(t) \_0\^2(t) dt , U\_1(r) = W\_1(t) .
.
It follows now that $W_1$ and $U_1$ have the same properties as $W$ and $U$ introduced earlier in (\[3e\]), and shown in [**Remark 1**]{}, (\[5e\]), and (\[12e\]) : \[40e\] {
[l]{} W\_1(r)L\^1(0, ),rW\_1\^2(r) L\^1(0, ).
. Let us now introduce, as in section II, the mapping \[41e\] r x = x(r) = \_0\^r e\^[2U\_1(t)]{} [dt \_0\^2(t)]{} .
It is a smooth, and twice differentiable one-to-one mapping for $r > 0$ : \[42e\] {
[l]{} r\[0, ) x \[0, ) ,r = 0 x= 0, r = x = .
.
Indeed, $U_1(\infty ) = 0$, $\chi_1(\infty ) = {1 \over A} \not= \infty$, as shown in (E), and \[43e\] {
[l]{} = > 0, A\^2 [for]{} r , = - e\^[2U\_1(r)]{} .
.
Remember that $\chi ''_0 = V_0 \chi_0$. $\chi_0$ being continuous and bounded for all $r\geq 0$, and $V_0 \in L^1$ by assymption for all $r > 0$, the same is true for $\chi ''_0$ : $\chi ''_0(r) \in L^1$ for $r > 0$. Therefore, $\chi '_0(r)$ is bounded and continuous for $r > 0$. The inverse mapping $r(x)$ is also, of course, a smooth and twice differentiable mapping for $x > 0$. Also, it is obvious from the first part of (\[43e\]), that \[44e\] {
[l]{}x(r) = e\^[2U\_1(0)]{} r + o(1), r 0 ,x(r) = A\^2r + o(r), r .
.
$x$ and $r$ are, therefore, of the same order as $r \to 0$ or $r\to \infty$.
Consider now the Schrödinger equation at zero energy \[45e\] {
[l]{}” (r) = (r) , (0) = 0 .
.
Note here, again, the appearance of $x$, given by (\[41e\]), in $V(x)$. Remember also that $rW_1^2(r) \in L^1(0, \infty )$, as shown in (\[40e\]). It follows that all the potentials in (\[45e\]) are regular, i.e. $rv(r) \in L^1(0, \infty )$ by assumption. Making the change of function, where $U_1$ is defined by (\[39e\]), \[46e\] {
[l]{}(r) (x) = \_[r=r(x)]{} , (0) = 0(0) = 0 ,
.
differentiating twice $\psi (x)$ with respect to $x$ defined by (\[41e\]), and using the first part of (\[43e\]) and (\[45e\]), we find \[47e\] {
[l]{} (x) = V(x) (x) .(0) = 0 .
.
Since it was assumed that this equation can be solved explicitely, we have achieved our goal, and the solution of (\[45e\]), according to (\[46e\]), is given by \[48e\] (r) = \_0 (r) e\^[-U\_1(r)]{} (x) ,
where $x$ is explicitely defined by (\[41e\]) in terms of $\chi_0 (r)$ and $V_1(r)$ through (\[39e\]). We can summarize our results in the following theorem :\
Given three potentials $V_0$, $V_1$, and $V$ satisfying the assumptions (\[36e\]), and assuming that (\[37e\]) can be solved explicitely for $V_0$ and $V$, the solution of (\[45e\]) is given explicitely by (\[48e\]). Obviously, we have $\varphi (0) = 0$.\
Making $V_0(r) = 0$, i.e. $\chi_0 (r) = 1$, we find the results of section II, and making $V_1(r) = 0$, the results of ref. $^1$, as given in the introduction. Also, having now a new potential with the explicit solution $\varphi$, we can repeat the operation with $V_0$ and the new potential, and continue indefinitely the process. We should note here that all the explicit examples of potentials we have given in ref. $^1$, some of which are reproduced in the present paper, lead to soluble Schrödinger equation for any coupling constant in front of the potential. We have, therefore, a great (infinite) variety of soluble potentials. Also, as we said in the Abstract, once the solutions are exhibited, one can check directly, by differentiation, that they satisfy indeed the appropriate equations. One can, of course, include also here the angular momentum, either by proceeding as in section II, or else by making \[41NEW\] {
[l]{} V\_0 (r) V\_0 (r) + ,\
\
r\^[2+2]{} V\_0 (r) L\^1(1 , )
. in (\[37e\]), and replacing $\varphi_0 (r)$ and $\chi_0 (r)$ by the appropriate solutions of $\varphi_{\ell} (r)$ and $\chi_{\ell} (r)$. Details are given in .$^{1}$ Since $V_0$ is assumed to sustain no bound states, the same is true when one makes (\[41NEW\]). And one shows again that the mapping $r \to x(r)$ is one to one, and twice differentiable. One has $r \in [0, \infty ) \Leftrightarrow x \in [0, \infty )$, and one can proceed as shown before in this section by replacing $\chi_0 (r)$ by $\chi_{\ell} (r)$ in (\[39e\]), (\[41e\]), (\[45e\]), and (\[46e\]). We obtain now another kind of potentials with explicit solutions.\
One of the authors (KC) would like to thank Professors Kenro Furutani and Takao Kobayashi, and the Department of Mathematics of the Science University of Tokyo, for warm hospitality and financial support.\
1. K. Chadan and R. Kobayashi (2005), ArXives math-ph/0510047, 12 Oct. 2005. Submitted to J. Phys. A. In the figure 2, the curves I and II should be interchanged.
2. R. Newton, Scattering Theory of Waves and Particles (Springer, New York, 2nd edition, 1982). See specially chapter 14 for many soluble examples.
3. A. Galindo and P. Pascual, Quantum Mechanics, two volumes (Springer, Berlin, 1990), vol. I.
4. K. Chadan and H. Grosse, J. Phys. A, [**16**]{}, 955 (1983).
5. R. Courant and D. Hilbert, Methods of Mathematical Physics (Intescience, New York, 1952), volume I.
6. E.-A. Coddington and N. Levinson, Theory of Ordinarly Differential Equations (Mc-Graw-Hill, New York, 1955).
7. A. Erdélyi, editor, Higher Transcendal Functions, vol. II (McGraw-Hill, New York, 1953).
8. E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, 1969).
9. G. Lévai, J. Phys. A, [**22**]{}, 689 (1984).
[^1]: Unité Mixte de Recherche UMR 8627 - CNRS
|
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abstract: 'We present Gemini Planet Imager (GPI) adaptive optics near-infrared images of the giant planet-forming regions of the protoplanetary disk orbiting the nearby ($D=54$ pc), pre-main sequence (classical T Tauri) star TW Hydrae. The GPI images, which were obtained in coronagraphic/polarimetric mode, exploit starlight scattered off small dust grains to elucidate the surface density structure of the TW Hya disk from $\sim$80 AU to within $\sim$10 AU of the star at $\sim$1.5 AU resolution. The GPI polarized intensity images unambiguously confirm the presence of a gap in the radial surface brightness distribution of the inner disk. The gap is centered near $\sim$23 AU, with a width of $\sim$5 AU and a depth of $\sim$50%. In the context of recent simulations of giant planet formation in gaseous, dusty disks orbiting pre-main sequence stars, these results indicate that at least one young planet with a mass $\sim$0.2 $M_J$ could be present in the TW Hya disk at an orbital semi-major axis similar to that of Uranus. If this (proto)planet is actively accreting gas from the disk, it may be readily detectable by GPI or a similarly sensitive, high-resolution infrared imaging system.'
author:
- 'Valerie A. Rapson, Joel H. Kastner Maxwell A. Millar-Blanchaer, Ruobing Dong'
title: |
Peering into the Giant Planet Forming Region of the\
TW Hydrae Disk with the Gemini Planet Imager
---
Introduction
============
Recent numerical simulations of planet formation processes in disks orbiting young stars demonstrate that such planet-building activity should leave distinct imprints on disk dust density structures [e.g., @Zhu2014; @Dong_etal2015b; @Dong_etal2015a]. Planet-disk interactions cause material that is co-orbiting with a massive planet to be transported outward via deposition of angular momentum onto the disk, creating pressure gradients that are manifested in disk density structures such as gaps, rings or spirals [e.g., @Bryden1999; @Pinilla2012; @Dong_etal2015b; @Dong_etal2015a]. Many such examples of ring/gap features have now been detected via submm-wave interferometry within disks orbiting young (age $<$10 Myr), low-mass stars [e.g., @Hughes2007; @Andrews2009; @Isella2010; @Andrews2011; @Rosenfeld2013]. With the advent of extreme adaptive optics (EAO) near-infrared polarimetric imaging instruments, such as the Gemini Planet Imager (GPI) on Gemini South and SPHERE on the Very Large Telescope, we can probe as close or closer to the central stars — within a few AU — by imaging light scattered off micron-sized (or smaller) dust grains [e.g., @Garufi2013; @Rapson2015; @Thalmann2015]. Comparison of such near-infrared, polarimetric images with similarly high-resolution ALMA submm imaging of thermal emission from dust [e.g., @ALMA2015] potentially offers a powerful means to characterize the radial and azimuthal dust density distributions within giant planet formation regions of protoplanetary disks and, perhaps, to pinpoint the locations of young planets [@Zhu2014; @Dong_etal2015a].
Thanks to its combination of proximity [D$=$54 pc; @Torres2008], advanced age [$\sim$8 Myr; @Ducourant2014], and nearly face-on viewing geometry [$i\approx7^\circ$; @qi2004], the disk orbiting TW Hya represents particularly fertile ground for such searches for evidence of planet formation. TW Hya is a $\sim$0.8 M$_{\odot}$ star of optical spectral type K6 [@torres2006]. Its relatively gas-rich, massive [$\sim$0.05 $M_\odot$; @Bergin2013] circumstellar disk has been imaged over a broad range of wavelengths, revealing rich disk structure and chemistry. @andrews2012 and [@Rosenfeld2012] imaged TW Hya with the Submillimeter Array and ALMA and found its CO gas disk extends to $\gtrsim$215 AU, similar to the radial extent of scattering of starlight by small (micron- and submicron-sized) dust grains [@Weinberger2002; @Apai2004], while the larger (mm-sized) dust grain population traced by 870 $\mu$m continuum emission displays a sharp outer edge at 60 AU. @debes2013 used HST/STIS coronagraphic imaging to trace starlight scattered off dust grains in the outer disk and find a gap in the disk at $\sim$80 AU that they contend could be opened by a planetary companion of a few hundredths of a Jupiter mass. A ring of C$_2$H emission extending from $\sim$45 AU to $\sim$120 AU overlaps this 80 AU gap and may trace a region of highly efficient photodissociation of hydrocarbons in the surface layers of the disk [@kastner2015]. Emission from N$_2$H$^+$ imaged with the SMA reveals an inner hole at R$\lesssim$30 AU that likely traces the inner edge of the CO “snow line,” where CO molecules freeze out onto dust grains [@qi2013]. @akiyama2015 identified a possible gap in the disk interior to this snow line, at R$\sim$20 AU, via polarimetric near-infrared imaging of light scattered off dust grains with Subaru/HiCIAO. Modeling of the spectral energy distribution of the disk [@calvet2002; @menu2014] further indicates the presence of a cavity interior to R$\lesssim$4 AU. Within this innermost disk region, the disk evidently harbors a few tenths of a lunar mass in submicron-sized dust grains [@Eisner2006].
Here, we present EAO near-IR coronagraphic/polarimetric images of the TW Hya disk that we obtained with the Gemini Planet Imager [GPI; @Macintosh2008; @Macintosh2014] on the Gemini South telescope. These images directly probe the disk dust structure, via the intensity of scattered incident starlight, to within $\sim$10 AU of TW Hya with unprecedented linear spatial resolution of $\sim$1.5 AU. The GPI imaging observations hence allow us to study the disk structure within the giant planet forming region around TW Hya, complementing and elaborating on the aforementioned recent mm-wave interferometric, polarized near-infrared, and optical imaging data obtained for this heavily scrutinized protoplanetary disk.
Observations and Data Reduction
===============================
Gemini/GPI polarimetric images of TW Hya were obtained through $J$ (1.24 $\mu$m) and $K1$ (2.05 $\mu$m) band filters and 0.184$''$ and 0.306$''$ diameter coronagraphic spots on April 22, 2015. Eight (four-image) sets of $J$ band images and 11 sets of $K1$ band images were obtained at waveplate position angles of 0$^\circ$, 22$^\circ$, 45$^\circ$, and 68$^\circ$ with exposure times of 60 s each. The $K1$ band images were obtained first, and over the course of the observations, the airmass and DIMM seeing ranged from 1.018$"$ to 1.034$"$ and 1.66$"$ to 0.8$"$ , respectively. Under these conditions, GPI’s EAO system delivers nearly diffraction-limited imaging, with angular resolution of $\sim$0.032$''$ and $\sim$0.055$''$ at $J$ and $K1$, respectively, on the 8 m of the Gemini South telescope (which has an effective pupil aperture of 7.77 m). Due to the target’s relatively faint I magnitude [$I \sim 9.2$; @barrado2006], the EAO system had difficulties keeping the focal plane mask properly aligned; only 25 of the (44) $K1$ image frames were included in the final reduction.
Polarimetric images in each filter were reduced using the GPI pipeline v1.3.0 [@Maire2010; @Perrin2014_pipe], following methods similar to those described in @Perrin2014 and @Rapson2015. To summarize, the images were cleaned by removal of correlated noise due to detector readout electronics and microphonics. We then performed interpolation over bad pixels, and subtracted background images from the target $K1$ frames. Calibration spot grids, which define the location of each polarization spot pair produced by the lenslet array, were used to extract intensity data from each raw image and produce a pair of orthogonally polarized images. Satellite spots on both the $J$ and $K1$ band images were used to determine the location of the the central star behind the coronagraph. The cubes were cleaned using a double differencing routine, and then instrumental polarization was subtracted as described in @MillarBlanchaer2015. Each image was then smoothed using a Gaussian filter with a FWHM of $\sim$2.3 pixels ($\sigma$ = 1 pixel) to reduce pixel to pixel noise. The cubes were all registered to place the obscured star at the center, and were then rotated to the same sky coordinate alignment (i.e., N up and E to the left). The orthogonally polarized images obtained at the four different waveplate angles were combined via the GPI pipeline to produce a Stokes data cube with slices $I$, $Q$, $U$ and $V$. Lastly, the radial and tangential Stokes parameter images $Q_r$ and $U_r$ were calculated via the pipeline from the Stokes cube [as in @schmid2006 albeit with the opposite sign convention]. Assuming that all the polarized flux is in the tangential component (which is expected for stellar photons single-scattered off circumstellar material), calculation of $Q_r$ avoids the positive bias induced by calculating total polarized intensity as the sum of the squares of the Stokes $Q$ and $U$ images. As absolute flux calibration of GPI polarized intensity images remained uncertain as of the writing of this paper, the $Q_r$ images are presented here in instrumental units.
Results
=======
In Fig. 1 we display total polarized intensity ($Q_r$) and radially scaled polarized intensity ($Q_r \times R^2$) images of the TW Hya disk. The total intensity images ($I$) are dominated by the residual PSF, and the disk was not detected; indeed, due to its azimuthal symmetry, the disk is indistinguishable from the stellar PSF in unpolarized angular differential imaging. The $Q_r$ images trace the intensity of scattered starlight as a function of position, while the radially scaled $Q_r$ images account for the dilution of incident starlight and thereby better represent the dust density distribution across the disk surface [e.g., @Garufi2014; @Rapson2015]. The total polarized intensity images reveal a bright central ring with relatively sharp outer edge near $\sim$0.3$"$ ($R\sim18$ AU), a deficit of scattered light just outside this radius, and a fainter scattering halo extending to at least $\sim$1$"$ ($R\sim55$ AU). All of these features are seen at both $J$ and $K1$, and are especially evident in the scaled polarized intensity images (righthand panels of Fig. 1). The latter images also appear to show an inner dust cavity within $\sim$15 AU, although this region of the images lies very near the inner working angle of the coronagraph (see below). The scattered light from the disk is detected at higher signal-to-noise ratio at $J$ band because of the combination of lower background and brighter incident radiation field, and, possibly, greater dust scattering efficiency at this wavelength.
In Fig. 2, we present radial brightness profiles obtained from the $J$ and $K1$ band polarized intensity and scaled polarized intensity images along directions parallel and perpendicular to disk position angle, i.e., the line marking the intersection of the disk equatorial plane and the plane of the sky, as inferred from CO kinematics [151$^{\circ}$, measured east from north; @Rosenfeld2012]. The radial profiles obtained from the unscaled $J$ and $K1$ polarized intensity images (lefthand panels of Fig. 2) display a sharp dip in surface brightness near $\sim$20–25 AU, as well as a weaker inflection near $\sim$30 AU. The profiles furthermore indicate that the outer scattered light halo drops smoothly out to $\sim$100 AU, a displacement roughly corresponding to the corner of the field of view of GPI at $J$ band. The radial profiles of the scaled polarized intensity at $J$ and $K1$ (righthand panels of Fig. 2) more clearly show the sharp dip feature (hereafter referred to as a gap) to be centered at $R\sim23$ AU; its FWHM of $\sim$5 AU indicates that the gap is well resolved by GPI.
The radial profiles in Fig. 2 also indicate that the polarized intensity is relatively isotropic, with large-scale asymmetries only of order $\sim$10% and no clear systematic differences in surface brightness along directions parallel vs. perpendicular to the disk position angle. This is consistent with the modest disk inclination determined from radio line data [$i \approx 7^\circ$; @Rosenfeld2012 and refs. therein]. In the $K1$ image, the inner disk surface brightness appears somewhat brighter to the southeast. Given that the same asymmetry is not present in the $J$ image, however (compare left panels of Figs. 1 and 2), the apparent inner-disk asymmetry at $K1$ is most likely caused by poor centering of the star+disk under the coronagraph.
In Figure 3, we present surface brightness profiles obtained by averaging the $J$ and $K1$ polarized intensity images over concentric elliptical annuli with minor:major axis ratios of 0.99 (approximating a circular disk with inclination of 7$^\circ$) at single-pixel (0.014$"$) intervals, with the ellipses oriented at a position angle of $151^\circ$. These profiles, which are discussed in detail in §4, decline rapidly from $R\sim12$ AU to the gap feature at $R\sim20$–25 AU, appear nearly flat out to $\sim$40 AU, and then drop precipitously beyond this radius. There is also an inflection in surface brightness at $R\sim80$–90 AU that is more apparent in the (higher S/N ratio) $J$ band radial profile.
The profiles in Figs. 2, 3 furthermore suggest that the bright inner disk has a ring-like structure; i.e., the surface brightness of polarized intensity peaks at $\sim10$–15 AU, with a potential decline interior to this radius. We note, however, that the peak in polarized intensity appears to lie closer to the star at $J$ than at $K1$ (peaks near $\sim$10 AU and $\sim$12 AU, respectively). This suggests that the decrease in polarized intensity near the inner working angle of the coronagraph may be instrumental in origin, although there remains the possibility that there is in fact a deficit of small dust grains in the TW Hya disk within $\sim$10 AU of the star. If real, this inner cavity dimension would be somewhat larger than previously deduced from SED fitting [which yielded an inner cavity size scale of $\sim$3–4 AU; @calvet2002; @menu2014].
Discussion
==========
The structure of the TW Hya dust disk from $\sim$10 AU to $\sim$80 AU
---------------------------------------------------------------------
Previous near-infrared coronagraphic imaging of TW Hya disk was conducted with HiCIAO, in polarimetric mode, on the 8 m Subaru telescope [@akiyama2015]. This imaging revealed a potential gap at a radial position of $R\sim20$ AU in the scattered-light surface brightness distribution of the TW Hya disk. This feature is readily apparent in the radial polarized intensity profiles extracted from our GPI coronagraphic/polarimetric imaging (Figs. 2, 3). The GPI radial profiles furthermore provide refined measurements of the position and width of the inner gap detected by Subaru/HiCIAO, as well as the dependence of scattered light on radius from $\sim$10 AU to $\sim$80 AU. Specifically, we find the scaled polarized intensity (Fig. 2, right panels) displays a clear local minimum at $R\sim23$ AU with a FWHM of $\Delta R\sim5$ AU and depth of $\sim$50%. In the azimuthally averaged radial profiles (Fig. 3), this feature has a bowl-shaped appearance. The GPI data also appear to confirm the presence of a gap near $R\sim80$–90 AU that was inferred from HST imaging [@debes2013], in the form of a weak inflection in the GPI $J$ band surface brightness profiles. We caution, however, that this radius roughly corresponds to the limit of the GPI field of view at $J$.
In characterizing the potential gap at $\sim$20 AU, @akiyama2015 parameterized the azimuthally averaged polarized intensity of the TW Hya disk in terms of a “stair-like” decline in radial power law ($r^\gamma$), with a drop from $\sim$10 to $\sim$20 AU characterized by $\gamma \approx -1.4$, a flattening to $\gamma \approx -0.3$ between $\sim$20 AU and $\sim$40 AU, and then a turnover to $\gamma \approx -2.7$ out to $\sim$80 AU. Adopting the same “three-zone” model (Fig. 3, top panel), we find similar values of $\gamma \approx -1.7$ and $\gamma \approx -0.4$, respectively, in the two zones within $\sim$40 AU, but we measure a considerably steeper decline of $\gamma \approx -3.9$ beyond $\sim$40 AU.
However, the clear appearance of the gap-like feature at $\sim$23 AU in Fig. 2 suggests it is preferable to parameterize the azimuthally averaged polarized intensity profiles (Fig. 3) in terms of a smooth decline in the inner disk (from $\sim$10 AU to $\sim$45 AU) that is interrupted by the gap feature at $\sim$23 AU, and a turnover to a steeper decline beyond $\sim$45 AU (i.e., a “two-zone” model). Adopting this parameterization, we find the radial dependence of polarized intensity is characterized by $\gamma \approx -1.2$ between $\sim$10 AU and $\sim$45 AU and by $\gamma \approx -3.9$ between $\sim$45 AU and $\sim$80 AU (Fig. 3, top). The latter value is rather extreme compared with the value $\gamma \approx -2$ that is predicted for radial scattered-light profiles by simulations that adopt “conventional” surface density and scale height profiles [e.g., @Dong2015 see next section].
TW Hya and V4046 Sgr: similarities and differences
--------------------------------------------------
In the right panel of Fig. 3, we compare the radial near-IR polarized intensity profiles of TW Hya with those obtained from GPI imaging of V4046 Sgr [@Rapson2015], a close binary, classical T Tauri system that, like TW Hya, is relatively nearby and evolved ($D \approx 73$ pc; age $\sim$20 Myr). The TW Hya disk and the V4046 Sgr (circumbinary) disk are also similarly massive and chemically rich [@Kastner2014 and references therein]. The comparison demonstrates that the two disks display similar slopes of azimuthally averaged polarized intensity in the $\sim$12–30 AU range; both show gaps, in the form of depressions in polarized intensity, near $\sim$20 AU; and both show turnovers to much steeper radial polarized intensity profiles in their outer disks. For TW Hya, the turnover to a steeper slope occurs at $\sim$45 AU, as noted above, whereas for V4046 Sgr the turnover lies near $\sim$28 AU and the slope of the outer disk profile is steeper [$\gamma \approx -5.5$; @Rapson2015]. The gap feature near $\sim$20 AU is evidently also much wider and deeper in the TW Hya disk than in the V4046 Sgr disk.
In our tests of Monte Carlo scattering models, we found that the radial polarized intensity gradient is relatively insensitive to the radial gradients in disk surface density and scale height, within reasonable ranges for these parameters (see §4.3). There would hence appear to be two potential explanations for the turnovers in the radial polarized intensity profiles of both disks: shadowing of the outer disk by the inner disk, or a sharp change in the radial gradient of the specific surface density or scale height of small and/or highly-IR-reflective dust grains. With regard to these alternatives, it is intriguing that the radial positions of the turnovers of the polarized intensity profiles in the TW Hya and V4046 Sgr disks at $\sim$45 AU and $\sim$28 AU, respectively, appear to closely coincide with the inner edge of a ring of C$_2$H emission [TW Hya; @kastner2015] and the inner edge of a ring of submm-wave continuum emission [V4046 Sgr; @Rapson2015]. In the case of TW Hya, the ring of C$_2$H may mark a region of enhanced stellar-irradiation-driven desorption of volatiles from grain surfaces, and/or photodestruction of dust grains themselves, in disk surface layers [@kastner2015]. In the case of V4046 Sgr, the ring of submm emission is likely due to a zone of dust grain growth and grain size segregation processes that are potentially related to planet building [@Rosenfeld2013]. The apparent spatial correlations between the turnovers in radial polarized intensity profiles (Fig. 3) and these zones of C$_2$H and large grain production may or may not be significant, but are deserving of further study, alongside further explorations of the radial intensity profiles generated by Monte Carlo disk scattering models (§4.3).
TW Hya disk gap: clearing by a giant planet?
--------------------------------------------
Both the depth ($\sim$0.5 dex) and width ($\sim$5 AU) of the apparent gap at $R\sim23$ AU in the TW Hya disk, as imaged by GPI, appear generally consistent with the predictions of simulations of disk gap clearing by giant planets [@Dong_etal2015a]. In Figs. 4 and 5, we directly compare the GPI polarized intensity images and radial profiles, respectively, with model images and profiles of a protoplanetary disk with an embedded planet of mass 0.16 $M_J$ ($2\times10^{-4}$ $M_\star$) at an orbital semimajor axis of 21 AU. The star/disk/planet model and resulting simulated near-IR images have been generated via the same methodology and Monte Carlo radiative transfer code as is described in detail in @Dong_etal2015a. To investigate the effect of signal-to-noise (S/N) ratio on the comparisons of model and data, we generated low-noise and high-noise model images consisting of runs with $10^9$ and $10^8$ photons, respectively. The latter runs yield images more closely resembling the S/N ratios of the GPI images near the gap region of the disk. The images were then convolved with an approximate instrumental PSF.
The similarity of the ring-gap structure in the model and data is evident from Fig. 4. In the low-noise model images, a spiral feature resulting from planet-induced resonances in the disk is also apparent both within and outside the darker gap near 20 AU. In the high-noise model images, only the in-gap portion of the spiral is detectable, as the surface brightness enhancement of the outer spiral is modest ($\sim$5%) compared with that of the in-gap spiral ($\sim$30%). Such a spiral structure is not apparent in the GPI images, although given the S/N ratio of the GPI images within and around the gap (S/N $\sim$ 10), the outer spiral would not be detectable (middle panels of Fig. 4). Such a direct comparison is further complicated by a modest asymmetry in surface brightness at radii $\stackrel{<}{\sim}$25 AU in the GPI images that is due to imperfect centering of the star behind the coronagraph (§2).
In Fig. 5 we compare the observed and model radial profiles, where the latter have been renormalized in intensity so as to match the GPI data. We find the model intensity renormalizations necessary to match the data in the $J$ and $K1$ bands are the same to within $\sim$20%, indicating that the @Dong_etal2015a model well reproduces the wavelength dependence of scattered light. It is evident from Fig. 5 that the width, depth, and overall shape of the gap feature in the model disk with embedded 0.16 $M_J$ planet provide a good match match to the GPI data, over the range $\sim$15 AU to $\sim$25 AU.
It is also evident that the model slope is steeper than the data for radii $\stackrel{<}{\sim}$15 AU and $\stackrel{>}{\sim}$25 AU. Under the assumptions that the radial profile of the dust surface density ($\Sigma_{\rm dust}$) and the scale height of the dust in the vertical direction ($h_{\rm dust}$) both obey simple power laws and that the dust population is uniform (i.e., the dust scattering properties do not vary) throughout the disk, we have experimented with disk models having various radial dependences of $\Sigma_{\rm dust}$ and $h_{\rm dust}$. We found that changing $\Sigma_{\rm dust}$ and $h_{\rm dust}/r$ from their original [@Dong_etal2015a] radial dependencies of $\Sigma_{\rm dust} \propto 1/r$ and $h_{\rm dust}/r\propto r^{0.25}$ to $\Sigma_{\rm dust} \propto r$ or $h_{\rm dust}/r\propto r^{0.5}$, respectively, results in a change of a factor of a few in absolute disk brightness in the near-IR. However, these power law parameter changes have relatively little impact on the radial dependence of scattered light; specifically, the radial scattered light power law index $\gamma$ changes by at most $\sim$0.3. This is because scattered light emerges from the disk surface with an intensity that is largely determined by the (grazing) incidence angle of starlight on the surface [@Takami2014]; changes to the simple radial power-law dependences of $\Sigma_{\rm dust}$ and $h_{\rm dust}$, while changing the disk flaring angle somewhat, do not yield significant changes to the radial dependence of this grazing incidence angle.
The foregoing suggests that future work aimed at exploring the discrepancy between the observed and model radial profiles should focus on invoking more complex disk radial and vertical dust density dependences (e.g., broken power laws), potential radial and vertical gradients in the grain scattering properties, and/or disk shadowing effects. The first two possibilities are well motivated by observations at other wavelengths (§4.2) as well as by the likelihood that grain growth within disks can produce complicated radial dependencies of $\Sigma_{\rm dust}$ for specific dust grain sizes, superimposed on simple power-law $\Sigma_{\rm dust}$ forms for the disk as a whole [@Birnstiel2010; @Birnstiel2012; @Pinilla2012]. Meanwhile, present models suggest that an outer disk that is completely shadowed by the inner disk can display a very steep scattered light radial power law index [see, e.g., Fig. 6 of @Dong2015].
Conclusions
===========
We have presented GPI polarized intensity images of TW Hya that reveal the surface dust density structure of the disk from $\sim$80 AU to within $\sim$10 AU of the star at $\sim$1.5 AU resolution. The GPI imaging unambiguously confirms the presence of a gap in the disk at $\sim$23 AU, with a width of $\sim$5 AU and a depth of $\sim$50%. The comparison between these GPI data and a Monte Carlo radiative transfer model of a disk with embedded planet (§4.3) appears to provide compelling evidence that a (proto)planet is likely responsible for clearing this gap in the TW Hya disk. We caution, however, that other mechanisms may be responsible for generating gap and ring structure in protoplanetary disks. For example, the gap in the TW Hya disk that we have resolved with GPI could be a natural consequence of the radial dependence of grain fragmentation rates [@Birnstiel2015]. This hypothesis could be tested via ALMA submm continuum imaging of the TW Hya disk at spatial resolution comparable to that of the GPI imaging. Alternatively, @Zhang2015 have proposed that disk gaps may form at radii corresponding to ice condensation fronts, which act as catalysts for rapid, localized grain growth. Although such a model may pertain to the TW Hya disk, we note that the gap at $\sim23$ AU seen in GPI and Subaru/HiCIAO imaging lies well outside the ($\sim$5 AU) radius where water ice is expected to form on dust grains, and well inside the ($\sim$30 AU) inner edge of a ring of N$_2$H$^+$ submm line emission that is hypothesized to mark the radius where the disk midplane drops to the ($\sim$20 K) temperature at which CO freezes out [@qi2013].
Followup coronagraphic imaging with GPI and with SPHERE (on the VLT) is hence now warranted, to test the hypothesis that at least one gas giant planet with a mass of $\sim$0.2 $M_{Jup}$ is actively forming in the TW Hya disk at an orbital semi-major axis similar to that of Uranus. Deeper polarimetric imaging with these instruments could reveal the spiral signatures that result from a massive planet exciting resonances within the disk (Fig. 4). Thermal emission from the planet itself would likely test the limits of GPI’s detection capabilities in its angular and spectral differential imaging mode [e.g., @Macintosh2015]. However, if the planet is actively accreting disk gas, its accretion luminosity may outshine the planet itself by several orders of magnitude in the near-infrared [@Zhu2015]. Such a disk-embedded, accreting protoplanet should be readily detectable in orbit about TW Hya by the present generation of EAO near-infrared imaging systems [e.g., @Sallum2015].
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![Left: azimuthally averaged $J$ (blue) and $K1$ (red) radial surface brightness profiles extracted from the GPI images of TW Hya (solid curves), with measured slopes for “two-radial-zone” and “three-radial-zone” [@akiyama2015] models overlaid (black dashed and dotted lines, respectively; see text). Right: comparison of TW Hya radial surface brightness profiles with those of V4046 Sgr [dashed curves; adapted from @Rapson2015]. ](Fig3aR1.pdf "fig:") ![Left: azimuthally averaged $J$ (blue) and $K1$ (red) radial surface brightness profiles extracted from the GPI images of TW Hya (solid curves), with measured slopes for “two-radial-zone” and “three-radial-zone” [@akiyama2015] models overlaid (black dashed and dotted lines, respectively; see text). Right: comparison of TW Hya radial surface brightness profiles with those of V4046 Sgr [dashed curves; adapted from @Rapson2015]. ](Fig3bR1.pdf "fig:")
![Comparison of simulated $J$ (top row) and $K1$ (bottom row) polarized intensity (scattered light) images obtained from Monte Carlo models of a disk with embedded 0.16 $M_{Jup}$ planet [left and middle panels; adapted from @Dong2015] with GPI images (right panels). In each row, low-noise and high-noise model images (see text) are presented in the left and middle panels, respectively. All images are displayed on a linear intensity scale and have a field of view of 90 AU $\times$ 90 AU ($1.67''\times1.67''$).](Fig4R1.pdf)
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abstract: 'We give an elementary proof for Lewis Bowen’s theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups $K$ and every free product $\Gamma$ of infinite amenable groups, the factor $\Gamma {\curvearrowright}K^{\Gamma}/K$ of the Bernoulli action $\Gamma {\curvearrowright}K^{\Gamma}$ by the diagonal $K$-action, is isomorphic with a Bernoulli action of $\Gamma$.'
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[**Stable orbit equivalence of Bernoulli actions of free\
groups and isomorphism of some of their factor actions**]{}
by Niels Meesschaert, Sven Raum and Stefaan Vaes
Free, ergodic and probability measure preserving (p.m.p.) actions $\Gamma {\curvearrowright}(X,\mu)$ of countable groups give rise to II$_1$ factors ${\mathord{\text{\rm L}}}^\infty(X) \rtimes \Gamma$ through the group measure space construction of Murray and von Neumann. It was shown in [@Si55] that the isomorphism class of the II$_1$ factor ${\mathord{\text{\rm L}}}^\infty(X) \rtimes \Gamma$ only depends on the orbit equivalence relation on $(X,\mu)$ given by $\Gamma {\curvearrowright}(X,\mu)$. This led Dye in [@Dy58] to a systematic study of group actions up to orbit equivalence, where he proved the fundamental result that all free ergodic p.m.p. actions of ${\mathbb{Z}}$ are orbit equivalent. Note that two such actions need not be isomorphic (using entropy, spectral measure, etc). In [@OW79] Ornstein and Weiss showed that actually all orbit equivalence relations of all free ergodic p.m.p. actions of infinite amenable groups are isomorphic with the unique ergodic hyperfinite equivalence relation of type II$_1$.
The nonamenable case is far more complex and many striking rigidity results have been established over the last 20 years, leading to classes of group actions for which the orbit equivalence relation entirely determines the group and its action. We refer to [@Sh05; @Fu09; @Ga10] for a comprehensive overview of measured group theory. On the other hand there have so far only been relatively few orbit equivalence “flexibility” results for nonamenable groups. Two results of this kind have been obtained recently by Lewis Bowen in [@Bo09a; @Bo09b]. In [@Bo09a] Bowen proved that two Bernoulli actions ${\mathbb{F}}_n {\curvearrowright}X_0^{{\mathbb{F}}_n}$ and ${\mathbb{F}}_n {\curvearrowright}X_1^{{\mathbb{F}}_n}$ of the same free group ${\mathbb{F}}_n$, but with different base probability spaces, are always orbit equivalent. Note that this is a nontrivial result because Bowen proved earlier in [@Bo08] that these Bernoulli actions can only be isomorphic if the base probability spaces $(X_0,\mu_0)$ and $(X_1,\mu_1)$ have the same entropy.
Two free ergodic p.m.p. actions $\Gamma_i {\curvearrowright}(X_i,\mu_i)$ are called stably orbit equivalent if their orbit equivalence relations can be restricted to non-negligible measurable subsets ${\mathcal{U}}_i \subset X_i$ such that the resulting equivalence relations on ${\mathcal{U}}_0$ and ${\mathcal{U}}_1$ become isomorphic. The number $\mu_1({\mathcal{U}}_1)/\mu_0({\mathcal{U}}_0)$ is called the compression constant of the stable orbit equivalence. In [@Bo09b] Bowen proved that the Bernoulli actions ${\mathbb{F}}_n {\curvearrowright}X_0^{{\mathbb{F}}_n}$ and ${\mathbb{F}}_m {\curvearrowright}X_1^{{\mathbb{F}}_m}$ of two different free groups are stably orbit equivalent with compression constant $(n-1)/(m-1)$.
The first aim of this article is to give an elementary proof for the above two theorems of Bowen. The concrete stable orbit equivalence that we obtain between ${\mathbb{F}}_n {\curvearrowright}X_0^{{\mathbb{F}}_n}$ and ${\mathbb{F}}_m {\curvearrowright}X_1^{{\mathbb{F}}_m}$ is identical to the one discovered by Bowen. The difference between the two approaches is however the following: rather than writing an explicit formula for the stable orbit equivalence, we construct actions of ${\mathbb{F}}_n$ and ${\mathbb{F}}_m$ on (subsets of) the same space, having the same orbits and satisfying an abstract characterization of the Bernoulli action.
Secondly our simpler methods also yield a new orbit equivalence flexibility (actually isomorphism) result that we explain now. Combining the work of many hands [@GP03; @Io06; @GL07] it was shown in [@Ep07] that every nonamenable group admits uncountably many non orbit equivalent actions (see [@Ho11] for a survey). Nevertheless it is still an open problem to give a concrete construction producing such an uncountable family. For a while it has been speculated that for any given nonamenable group $\Gamma$ the actions $$\label{eq.family}
\Bigl\{\Gamma {\curvearrowright}K^\Gamma / K \; \Big| \; \text{$K$ a compact second countable group acting by diagonal translation on $K^\Gamma$}\; \Bigr\}$$ are non orbit equivalent for nonisomorphic $K$. Indeed, in [@PV06 Proposition 5.6] it was shown that this is indeed the case whenever every $1$-cocycle for the Bernoulli action $\Gamma {\curvearrowright}K^\Gamma$ with values in either a countable or a compact group ${\mathcal{G}}$ is cohomologous to a group homomorphism from $\Gamma$ to ${\mathcal{G}}$. By Popa’s cocycle superrigidity theorems [@Po05; @Po06], this is the case when $\Gamma$ contains an infinite normal subgroup with the relative property (T) or when $\Gamma$ can be written as the direct product of an infinite group and a nonamenable group. Conjecturally the same is true whenever the first $\ell^2$-Betti number of $\Gamma$ vanishes (cf. [@PS09]).
In the last section of this paper we disprove the above speculation whenever $\Gamma = \Lambda_1 * \cdots * \Lambda_n$ is the free product of $n$ infinite amenable groups, in particular when $\Gamma = {\mathbb{F}}_n$. We prove that for these $\Gamma$ and for every compact second countable group $K$ the action $\Gamma {\curvearrowright}K^\Gamma/K$ is isomorphic with a Bernoulli action of $\Gamma$. As we shall see, the special case $\Gamma = {\mathbb{F}}_n$ is a very easy generalization of [@OW86 Appendix C.(b)] where the same result is proven for $K = {\mathbb{Z}}/ 2 {\mathbb{Z}}$ and $\Gamma = {\mathbb{F}}_2$.
More generally, denote by ${\mathcal{G}}$ the class of countably infinite groups $\Gamma$ for which the action $\Gamma {\curvearrowright}K^{\Gamma}/K$ is isomorphic with a Bernoulli action of $\Gamma$. Then by [@OW86] the class ${\mathcal{G}}$ contains all infinite amenable groups. We prove in Theorem \[thm.stability\] that ${\mathcal{G}}$ is stable under taking free products. By the results cited above, ${\mathcal{G}}$ does not contain groups that admit an infinite normal subgroup with the relative property (T) and ${\mathcal{G}}$ does not contain groups that can be written as the direct product of an infinite group and a nonamenable group. So it is a very intriguing problem which groups belong to ${\mathcal{G}}$.
Terminology and notations {#terminology-and-notations .unnumbered}
-------------------------
A measure preserving action $\Gamma {\curvearrowright}(X,\mu)$ of a countable group $\Gamma$ on a standard probability space $(X,\mu)$ is called *essentially free* if a.e. $x \in X$ has a trivial stabilizer and is called *ergodic* if the only $\Gamma$-invariant measurable subsets of $X$ have measure $0$ or $1$. Two free ergodic probability measure preserving (p.m.p.) actions $\Gamma {\curvearrowright}(X,\mu)$ and $\Lambda {\curvearrowright}(Y,\eta)$ are called
- *conjugate,* if there exists an isomorphism of groups $\delta : \Gamma {\rightarrow}\Lambda$ and an isomorphism of probability spaces $\Delta : X {\rightarrow}Y$ such that $\Delta(g \cdot x) = \delta(g) \cdot \Delta(x)$ for all $g \in \Gamma$ and a.e. $x \in X$;
- *orbit equivalent,* if there exists an isomorphism of probability spaces $\Delta : X {\rightarrow}Y$ such that $\Delta( \Gamma \cdot x) = \Lambda \cdot \Delta(x)$ for a.e. $x \in X$;
- *stably orbit equivalent,* if there exists a nonsingular isomorphism $\Delta : {\mathcal{U}}{\rightarrow}{\mathcal{V}}$ between non-negligible measurable subsets ${\mathcal{U}}\subset X$ and ${\mathcal{V}}\subset Y$ such that $\Delta(\Gamma \cdot x \cap {\mathcal{U}}) = \Lambda \cdot \Delta(x) \cap {\mathcal{V}}$ for a.e. $x \in {\mathcal{U}}$. Such a $\Delta$ automatically scales the measure by the constant $\eta({\mathcal{V}})/\mu({\mathcal{U}})$, called the *compression constant* of the stable orbit equivalence.
We say that two p.m.p. actions $\Gamma {\curvearrowright}(X_i,\mu_i)$ of the same group are *isomorphic* if they are conjugate w.r.t. the identity isomorphism ${\mathord{\text{\rm id}}}: \Gamma {\rightarrow}\Gamma$, i.e. if there exists an isomorphism of probability spaces $\Delta : X_0 {\rightarrow}X_1$ such that $\Delta(g \cdot x) = g \cdot \Delta(x)$ for all $g \in \Gamma$ and a.e. $x \in X_0$.
Recall that for every countable group $\Gamma$ and standard probability space $(X_0,\mu_0)$, the Bernoulli action of $\Gamma$ with base space $(X_0,\mu_0)$ is the action $\Gamma {\curvearrowright}X_0^\Gamma$ on the infinite product $X_0^\Gamma$ equipped with the product probability measure, given by $(g \cdot x)_h = x_{hg}$ for all $g,h \in \Gamma$ and $x \in X_0^\Gamma$. If $\Gamma$ is an infinite group and $(X_0,\mu_0)$ is not reduced to a single atom of mass $1$, then $\Gamma {\curvearrowright}X_0^\Gamma$ is essentially free and ergodic.
Statement of the main results {#statement-of-the-main-results .unnumbered}
-----------------------------
We first give an elementary proof for the following theorem of Lewis Bowen.
\[thm.A\] For fixed $n$ and varying base probability space $(X_0,\mu_0)$ the Bernoulli actions ${\mathbb{F}}_n {\curvearrowright}X_0^{{\mathbb{F}}_n}$ are orbit equivalent.
If also $n$ varies, the Bernoulli actions ${\mathbb{F}}_n {\curvearrowright}X_0^{{\mathbb{F}}_n}$ and ${\mathbb{F}}_m {\curvearrowright}Y_0^{{\mathbb{F}}_m}$ are stably orbit equivalent with compression constant $(n-1)/(m-1)$.
Next we study factors of Bernoulli actions and prove the following result.
\[thm.B\] If $\Gamma = \Lambda_1 * \cdots * \Lambda_n$ is the free product of $n$ infinite amenable groups and if $K$ is a nontrivial second countable compact group equipped with its normalized Haar measure, then the factor action $\Gamma {\curvearrowright}K^{\Gamma}/K$ of the Bernoulli action $\Gamma {\curvearrowright}K^\Gamma$ by the diagonal translation action of $K$ is isomorphic with a Bernoulli action of $\Gamma$. In particular, keeping $n$ fixed and varying the $\Lambda_i$ and $K$, all the actions $\Gamma {\curvearrowright}K^\Gamma/K$ are orbit equivalent.
In the particular case where $\Gamma = {\mathbb{F}}_n$, the action $\Gamma {\curvearrowright}K^{\Gamma}/K$ is isomorphic with the Bernoulli action $\Gamma {\curvearrowright}(K \times \cdots \times K)^\Gamma$ whose base space is an $n$-fold direct product of copies of $K$.
Acknowledgment {#acknowledgment .unnumbered}
--------------
We are extremely grateful to Lewis Bowen for his remarks on the first versions of this article. Initially we only proved that the factors of Bernoulli actions of ${\mathbb{F}}_n$ in Theorem \[thm.B\] are orbit equivalent with a Bernoulli action of ${\mathbb{F}}_n$. Lewis Bowen remarked that it was unknown whether these actions are actually isomorphic to Bernoulli actions. Triggered by this remark we proved the above version of Theorem \[thm.B\].
Preliminaries {#sec.prelim}
=============
Let $(X,\mu)$ and $(Y,\eta)$ be standard probability spaces. We call $\Delta$ a *probability space isomorphism* between $(X,\mu)$ and $(Y,\eta)$ if $\Delta$ is a measure preserving Borel bijection between conegligible subsets of $X$ and $Y$. We call $\Delta$ a *nonsingular isomorphism* if $\Delta$ is a null set preserving Borel bijection between conegligible subsets of $X$ and $Y$.
Given a sequence of standard probability spaces $(X_n,\mu_n)$, we consider the *infinite product* $X = \prod_n X_n$ equipped with the infinite product measure $\mu$. Then, $(X,\mu)$ is a standard probability space. The coordinate maps $\pi_n : X {\rightarrow}X_n$ are measure preserving and independent. Moreover, the Borel $\sigma$-algebra on $X$ is the smallest $\sigma$-algebra such that all $\pi_n$ are measurable.
Conversely, assume that $(Y,\eta)$ is a standard probability space and that $\theta_n : Y {\rightarrow}X_n$ is a sequence of Borel maps. Then, the following two statements are equivalent.
1. There exists an isomorphism of probability spaces $\Delta : Y {\rightarrow}X$ such that $\pi_n(\Delta(y)) = \theta_n(y)$ for a.e. $y \in Y$.
2. The maps $\theta_n$ are measure preserving and independent, and the $\sigma$-algebra on $Y$ generated by the maps $\theta_n$ equals the entire Borel $\sigma$-algebra of $Y$ up to null sets.
The proof of this equivalence is standard: if the $\theta_n$ satisfy the conditions in 2, one defines $\Delta(y)_n := \theta_n(y)$.
Assume that $\Gamma {\curvearrowright}(X,\mu)$ and $\Lambda {\curvearrowright}(Y,\eta)$ are essentially free ergodic p.m.p. actions. Assume that $\Delta : X {\rightarrow}Y$ is an orbit equivalence. By essential freeness, we obtain the a.e. well defined Borel map ${\omega}: \Gamma \times X {\rightarrow}\Lambda$ determined by $$\Delta(g \cdot x) = {\omega}(g,x) \cdot \Delta(x) \quad\text{for all}\;\; g \in \Gamma \;\;\text{and a.e.}\;\; x \in X \; .$$ Then, ${\omega}$ is a *$1$-cocycle* for the action $\Gamma {\curvearrowright}(X,\mu)$ with values in the group $\Lambda$. In general, whenever ${\mathcal{G}}$ is a Polish group and $\Gamma {\curvearrowright}(X,\mu)$ is a p.m.p. action, we call a Borel map ${\omega}: \Gamma \times X {\rightarrow}{\mathcal{G}}$ a $1$-cocycle if ${\omega}$ satisfies $${\omega}(gh,x) = {\omega}(g,h \cdot x) \, {\omega}(h,x) \quad\text{for all}\;\; g,h \in \Gamma \;\;\text{and a.e.}\;\; x \in X \; .$$ Two $1$-cocycles ${\omega},{\omega}' : \Gamma \times X {\rightarrow}{\mathcal{G}}$ are called *cohomologous* if there exists a Borel map ${\varphi}: X {\rightarrow}{\mathcal{G}}$ such that $${\omega}'(g,x) = {\varphi}(g \cdot x) \, {\omega}(g,x) \, {\varphi}(x)^{-1} \quad\text{for all}\;\; g \in \Gamma \;\;\text{and a.e.}\;\; x \in X \; .$$
Also a *stable orbit equivalence* gives rise to a $1$-cocycle, as follows. So assume that $\Gamma {\curvearrowright}(X,\mu)$ and $\Lambda {\curvearrowright}(Y,\eta)$ are essentially free ergodic p.m.p. actions and that $\Delta : {\mathcal{U}}{\rightarrow}{\mathcal{V}}$ is a nonsingular isomorphism between the nonnegligible subsets ${\mathcal{U}}\subset X$ and ${\mathcal{V}}\subset Y$, such that $\Delta({\mathcal{U}}\cap \Gamma \cdot x) = {\mathcal{V}}\cap \Lambda \cdot \Delta(x)$ for a.e. $x \in {\mathcal{U}}$. To define the Zimmer $1$-cocycle ${\omega}: \Gamma \times X {\rightarrow}\Lambda$, one first uses the ergodicity of $\Gamma {\curvearrowright}(X,\mu)$ to choose a Borel map $p : X {\rightarrow}{\mathcal{U}}$ satisfying $p(x) \in \Gamma \cdot x$ for a.e. $x \in X$. Then, ${\omega}: \Gamma \times X {\rightarrow}\Lambda$ is uniquely defined such that $$\Delta(p(g \cdot x)) = {\omega}(g,x) \cdot \Delta(p(x)) \quad\text{for all}\;\; g \in \Gamma \;\;\text{and a.e.}\;\; x \in X \; .$$ One checks easily that ${\omega}$ is a $1$-cocycle and that, up to cohomology, ${\omega}$ does not depend on the choice of $p : X {\rightarrow}{\mathcal{U}}$.
In this article, we often use $1$-cocycles for p.m.p. actions $\Gamma {\curvearrowright}(X,\mu)$ of a free product group $\Gamma = \Gamma_1 * \Gamma_2$. Given $1$-cocycles ${\omega}_i : \Gamma_i \times X {\rightarrow}{\mathcal{G}}$, one checks easily that there is a unique $1$-cocycle ${\omega}: \Gamma \times X {\rightarrow}{\mathcal{G}}$, up to equality a.e., satisfying ${\omega}(g,x) = {\omega}_i(g,x)$ for all $g \in \Gamma_i$ and a.e. $x \in X$.
Orbit equivalence of co-induced actions
=======================================
Let $\Lambda {\curvearrowright}(X,\mu)$ be a p.m.p. action. Assume that $\Lambda < G$ is a subgroup. The co-induced action of $\Lambda {\curvearrowright}X$ to $G$ is defined as follows. Choose a map $r : G {\rightarrow}\Lambda$ such that $r(\lambda g) = \lambda r(g)$ for all $g \in G,\lambda \in \Lambda$ and such that $r(e) = e$. Note that the choice of such a map $r$ is equivalent to the choice of a section $\theta : \Lambda \backslash \Gamma {\rightarrow}\Gamma$ satisfying $\theta(\Lambda e) = e$. Indeed, the formula $g = r(g) \, \theta(\Lambda g)$ provides the correspondence between $\theta$ and $r$.
Once we have chosen $r : G {\rightarrow}\Lambda$, we can define a $1$-cocycle ${\Omega}: \Lambda \backslash G \times G {\rightarrow}\Lambda$ for the right action of $G$ on $\Lambda \backslash G$, given by $\Omega(\Lambda k,g) = r(k)^{-1} r(kg)$ for all $g,k \in G$.
Classically, whenever ${\omega}: G \times X {\rightarrow}\Lambda$ is a $1$-cocycle for an action of $G$ on $X$, we can induce an action $\Lambda {\curvearrowright}Y$ to an action $G {\curvearrowright}X \times Y$ given by $g \cdot (x,y) = (g \cdot x, {\omega}(g,x) \cdot y)$.
The co-induced action is defined by a similar formula. So assume that $\Lambda {\curvearrowright}(X,\mu)$ is a p.m.p. action and that $\Lambda < G$ is a subgroup. Choose $r : G {\rightarrow}\Lambda$ with the associated $1$-cocycle ${\Omega}: \Lambda \backslash G \times G {\rightarrow}\Lambda$, as above. Then the formula $$G {\curvearrowright}X^{\Lambda \backslash G} \quad\text{where}\quad (g \cdot y)_{\Lambda k} = \Omega(\Lambda k,g) \cdot y_{\Lambda kg}$$ yields a well defined action of $G$ on the product probability space $X^{\Lambda \backslash G}$. It is easy to check that $G {\curvearrowright}X^{\Lambda \backslash G}$ is a p.m.p. action and that $(\lambda \cdot y)_{\Lambda e} = \lambda \cdot y_{\Lambda e}$ for all $\lambda \in \Lambda$ and $y \in X^{\Lambda \backslash G}$. A different choice of $r : G {\rightarrow}\Lambda$ leads to a cohomologous $1$-cocycle ${\Omega}$ and hence an isomorphic action.
Given a subgroup $\Lambda < G$, a subset $I \subset G$ is called a *right transversal* of $\Lambda < G$ if $I \cap \Lambda g$ is a singleton for every $g \in G$.
Up to isomorphism the co-induced action can be characterized as the unique p.m.p. action $G {\curvearrowright}Y$ for which there exists a measure preserving map $\rho : Y {\rightarrow}X$ with the following properties.
1. $\rho(\lambda \cdot y) = \lambda \cdot \rho(y)$ for all $\lambda \in \Lambda$ and a.e. $y \in Y$.
2. The factor maps $y \mapsto \rho(g \cdot y)$, $g \in G$, generate the Borel $\sigma$-algebra on $Y$, up to null sets.
3. If $I \subset G$ is a right transversal of $\Lambda < G$, then the maps $y \mapsto \rho(g \cdot y)$, $g \in I$, are independent.
To prove this characterization, first observe that the co-induced action satisfies properties 1, 2 and 3 in a canonical way, with $\rho(y) = y_{\Lambda e}$. Conversely assume that $G {\curvearrowright}Y$ satisfies these properties. Fix a right transversal $I \subset G$ for $\Lambda < G$, with $e \in I$. Combining properties 1 and 2, we see that the factor maps $y \mapsto \rho(g \cdot y)$, $g \in I$, generate the Borel $\sigma$-algebra on $Y$, up to null sets. A combination of property 3 and the characterization of product probability spaces in Section \[sec.prelim\] then provides the isomorphism of probability spaces $\Delta : Y {\rightarrow}X^{\Lambda \backslash G}$ given by $\Delta(y)_{\Lambda g} = \rho(g \cdot y)$ for all $y \in Y$, $g \in I$. The right transversal $I \subset G$ for $\Lambda < G$ allows to uniquely define the map $r : G {\rightarrow}\Lambda$ such that $r(\lambda g) = \lambda$ for all $\lambda \in \Lambda$ and $g \in I$. This choice of $r$ provides a formula for the co-induced action $G {\curvearrowright}X^{\Lambda \backslash G}$. It is easy to check that $\Delta(g \cdot y) = g \cdot \Delta(y)$ for all $g \in G$ and a.e. $y \in Y$.
\[rem.bete\]\
[**1.**]{} The above characterization of the co-induced action yields the following result that we use throughout the article: the co-induction of the Bernoulli action $\Lambda {\curvearrowright}(X_0,\mu_0)^\Lambda$ is isomorphic with the Bernoulli action $G {\curvearrowright}(X_0,\mu_0)^G$. Indeed, the Bernoulli action $G {\curvearrowright}(X_0,\mu_0)^G$, together with the canonical factor map $X_0^G {\rightarrow}X_0^\Lambda$, satisfies the above characterization of the co-induced action.
[**2.**]{} In certain cases, for instance if $G = \Gamma * \Lambda$, there exists a group homomorphism $\pi : G {\rightarrow}\Lambda$ satisfying $\pi(\lambda) = \lambda$ for all $\lambda \in \Lambda$. Then $r : G {\rightarrow}\Lambda$ can be taken equal to $\pi$ and the co-induced action $G {\curvearrowright}X^{\Lambda \backslash G}$ is of the form $(g \cdot y)_{\Lambda k} = \pi(g) \cdot y_{\Lambda k g}$ for all $g,k \in G$ and $y \in X^{\Lambda \backslash G}$.
[**3.**]{} We often make use of diagonal actions: if $\Lambda {\curvearrowright}(X,\mu)$ and $\Lambda {\curvearrowright}(Y,\eta)$ are p.m.p. actions, we consider the diagonal action $\Lambda {\curvearrowright}X \times Y$ given by $\lambda \cdot (x,y) = (\lambda \cdot x,\lambda \cdot y)$. We make the following simple observation: if $\Lambda < G$ and if we denote by $G {\curvearrowright}{\widetilde{X}}$, resp. $G {\curvearrowright}{\widetilde{Y}}$, the co-induced actions of $\Lambda {\curvearrowright}X$, resp. $\Lambda {\curvearrowright}Y$, to $G$, then the co-induced action of the diagonal action $\Lambda {\curvearrowright}X \times Y$ to $G$ is precisely the diagonal action $G {\curvearrowright}{\widetilde{X}}\times {\widetilde{Y}}$.
[**4.**]{} Assume that $\Lambda {\curvearrowright}(X,\mu)$ is a p.m.p. action and that $\Lambda < G$ is a subgroup. Denote by $G {\curvearrowright}Y$ the co-induced action and by $\rho : Y {\rightarrow}X$ the canonical $\Lambda$-equivariant factor map. Whenever $\Delta_0 : X {\rightarrow}X$ is a p.m.p. automorphism that commutes with the $\Lambda$-action, there is a unique p.m.p. automorphism $\Delta : Y {\rightarrow}Y$, up to equality a.e., that commutes with the $G$-action and such that $\rho(\Delta(y)) = \Delta_0(\rho(y))$ for a.e. $y \in Y$. Writing $Y = X^{\Lambda \backslash \Gamma}$, the automorphism $\Delta$ is just the diagonal product of copies of $\Delta_0$. Later we use this easy observation to canonically lift a p.m.p. action $K {\curvearrowright}(X,\mu)$ of a compact group $K$, commuting with the $\Lambda$-action, to a p.m.p. action $K {\curvearrowright}Y$ that commutes with the $G$-action. Moreover, $\rho$ becomes $(\Lambda \times K)$-equivariant. Writing $Y = X^{\Lambda \backslash \Gamma}$, the action $K {\curvearrowright}Y$ is the diagonal $K$-action.
We prove that orbit equivalence is preserved under co-induction to a free product. We actually show that the preservation is “$K$-equivariant” in a precise way that will be needed in the proof of Theorem \[thm.B\]. The case where $K = \{e\}$, i.e. co-induction from $\Lambda$ to $\Gamma * \Lambda$, is due to Lewis Bowen [@Bo09a]. Recall that similarly as in the case of countable groups, a p.m.p. action $G {\curvearrowright}(X,\mu)$ of a second countable locally compact group $G$ is called essentially free if a.e. $x \in X$ has a trivial stabilizer (cf. Lemma \[lem.compact-free\] in the appendix).
\[thm.coinduced\] Let $\Lambda_0,\Lambda_1$ and $\Gamma$ be countable groups and $K$ a compact second countable group. Assume that $\Lambda_i \times K {\curvearrowright}(X_i,\mu_i)$ are essentially free p.m.p. actions. Denote $G_i := \Gamma * \Lambda_i$ and denote by $G_i {\curvearrowright}Y_i$ the co-induced action of $\Lambda_i {\curvearrowright}X_i$ to $G_i$, together with the natural actions $K {\curvearrowright}Y_i$ that commute with $G_i {\curvearrowright}Y_i$ (see Remark \[rem.bete\].4).
- If the actions $\Lambda_i {\curvearrowright}X_i/K$ are orbit equivalent, then the actions $G_i {\curvearrowright}Y_i/K$ are orbit equivalent.
- If the actions $\Lambda_i {\curvearrowright}X_i/K$ are conjugate w.r.t. the group isomorphism $\delta : \Lambda_0 {\rightarrow}\Lambda_1$, then the actions $G_i {\curvearrowright}Y_i/K$ are conjugate w.r.t. the group isomorphism ${\mathord{\text{\rm id}}}* \delta : G_0 {\rightarrow}G_1$.
We start by proving the first item of the theorem.
Let $\Delta_0 : X_0/K {\rightarrow}X_1/K$ be an orbit equivalence between the actions $\Lambda_i {\curvearrowright}X_i/K$. Denote by $x \mapsto {\overline{x}}$ the factor map from $X_i$ to $X_i / K$. Since $K$ acts essentially freely on $X_i$ and $K$ is compact, Lemma \[lem.compact-free\] in the appendix provides measurable maps $\theta_i : X_i {\rightarrow}K$ satisfying $\theta_i(k \cdot x) = k \theta_i(x)$ a.e. and such that $$\Theta_i : X_i {\rightarrow}K \times X_i/K : x \mapsto (\theta_i(x) , {\overline{x}})$$ is a measure preserving isomorphism. Defining $\Delta := \Theta_1^{-1} \circ ({\mathord{\text{\rm id}}}\times \Delta_0) \circ \Theta_0$, we have found a measure preserving isomorphism $\Delta : X_0 {\rightarrow}X_1$ that is $K$-equivariant and satisfies $\Delta((\Lambda_0 \times K) \cdot x) = (\Lambda_1 \times K) \cdot \Delta(x)$ for a.e. $x \in X_1$. Using this $\Delta$ we may assume that $\Lambda_0, \Lambda_1$ and $K$ act on the same probability space $(X,\mu)$ such that the $K$-action commutes with both the $\Lambda_i$-actions and such that $(\Lambda_0 \times K) * x = (\Lambda_1 \times K) \cdot x$ for a.e. $x \in X$. Here and in what follows, we denote the action of $\Lambda_0 \times K$ by $*$ and the action of $\Lambda_1 \times K$ by $\cdot$. We have $k * x = k \cdot x$ for all $k \in K$ and a.e. $x \in X$.
Write $Y = X^{\Lambda_1 \backslash \Gamma * \Lambda_1}$ and denote by $\cdot$ the co-induced action $G_1 {\curvearrowright}Y$ of $\Lambda_1 {\curvearrowright}X$ to $G_1$. Also denote by $\cdot$ the diagonal action $K {\curvearrowright}Y$, which commutes with $G_1 {\curvearrowright}Y$. Define the $(\Lambda_1 \times K)$-equivariant factor map $\rho : Y {\rightarrow}X : \rho(y) = y_{\Lambda_1 e}$.
Define the Zimmer $1$-cocycles $$\begin{aligned}
& \eta : \Lambda_0 \times X {\rightarrow}\Lambda_1 \times K : \eta(\lambda_0,x) \cdot x = \lambda_0 * x \quad\text{for a.e.}\;\; x \in X_1, \lambda_0 \in \Lambda_0 \; ,\\
& \eta' : \Lambda_1 \times X {\rightarrow}\Lambda_0 \times K : \eta'(\lambda_1,x) * x = \lambda_1 \cdot x \quad\text{for a.e.}\;\; x \in X_1, \lambda_1 \in \Lambda_1 \; .\end{aligned}$$ Since the $\Lambda_0$-action commutes with the $K$-action on $X$, we have that $$\label{eq.my-formula}
\eta(\lambda_0, k * x) = k \eta(\lambda_0,x) k^{-1} \quad\text{for all}\;\; k \in K, \lambda_0 \in \Lambda_0 \;\;\text{and a.e.}\;\; x \in X \; .$$ We define a new action $G_0 {\curvearrowright}Y$ denoted by $*$ and determined by $$\gamma * y = \gamma \cdot y \;\;\text{for}\;\; \gamma \in \Gamma, y \in Y \quad\text{and}\quad \lambda_0 * y = \eta(\lambda_0, \rho(y)) \cdot y \;\;\text{for}\;\; \lambda_0 \in \Lambda_0, y \in Y \; .$$ Because of , the action $G_0 {\curvearrowright}Y$ commutes with $K {\curvearrowright}Y$.
Define ${\omega}: G_0 \times Y {\rightarrow}G_1 \times K$ as the unique $1$-cocycle for the action $G_0 \overset{*}{{\curvearrowright}} Y$ satisfying ${\omega}(\gamma,y) = \gamma$ for all $\gamma \in \Gamma$ and ${\omega}(\lambda_0,y) = \eta(\lambda_0,\rho(y))$ for all $\lambda_0 \in \Lambda_0$. Then the equality $g * y = {\omega}(g,y) \cdot y$ holds when $g \in \Gamma$ and when $g \in \Lambda_0$. So the same equality holds for all $g \in G_0$ and a.e. $y \in Y$. In particular $G_0 * {\overline{y}}\subset G_1 \cdot {\overline{y}}$ for a.e. ${\overline{y}}\in Y/K$.
Define ${\omega}' : G_1 \times Y {\rightarrow}G_0 \times K$ as the unique $1$-cocycle satisfying ${\omega}'(\gamma,y) = \gamma$ for all $\gamma \in \Gamma$ and ${\omega}'(\lambda_1,y) = \eta'(\lambda_1,\rho(y))$ for all $\lambda_1 \in \Lambda_1$. As above, it follows that $g \cdot y = {\omega}'(g,y) * y$ for all $g \in G_1$ and a.e. $y \in Y$. Hence, $G_1 \cdot {\overline{y}}\subset G_0 * {\overline{y}}$ for a.e. ${\overline{y}}\in Y/K$. We already proved the converse inclusion so that $G_1 \cdot {\overline{y}}= G_0 * {\overline{y}}$ for a.e. $y \in Y/K$.
We prove now that the action $G_0 \overset{*}{{\curvearrowright}} Y$ together with the $\Lambda_0$-equivariant factor map $\rho : Y {\rightarrow}X$ satisfies the abstract characterization for the co-induced action of $\Lambda_0 {\curvearrowright}X$ to $G_0$. Once this is proven, the theorem follows because $\rho$ is moreover $K$-equivariant and the action $G_0 {\curvearrowright}Y$ commutes with the $K {\curvearrowright}Y$ (see Remark \[rem.bete\].4).
We first need to prove that the maps $y \mapsto \rho(g*y)$ are independent and identically distributed when $g$ runs through a right transversal of $\Lambda_0 \subset G_0$. If $g \in G_i = \Gamma * \Lambda_i$, denote by $|g|$ the number of letters from $\Gamma - \{e\}$ that appear in a reduced expression of $g$. By convention, put $|g| = 0$ if $g \in \Lambda_i$. Define the subsets $I_n \subset G_0$ given by $I_0 := \{e\}$ and $$I_n := \bigl\{g \in G_0 \;\big| \; |g|=n \;\;\text{and the leftmost letter of a reduced expression of $g$ belongs to $\Gamma-\{e\}$}\;\bigr\} \; .$$ Similarly define $J_n \subset G_1$ and note that $\bigcup_{n=0}^\infty J_n$ is a right transversal for $\Lambda_1 < \Gamma * \Lambda_1$. So, in the construction of the co-induced action, we can choose the $\Lambda_1$-equivariant map $r : G_1 {\rightarrow}\Lambda_1$ such that $r(g) = e$ for all $g \in J_n$ and all $n \in {\mathbb{N}}$. Hence $(g \cdot y)_{\Lambda_1 e} = y_{\Lambda_1 g}$ for all $g \in J_n$, $n \in {\mathbb{N}}$ and a.e. $y \in Y$. For $j \in \Lambda_1 \backslash G_1$ we put $|j|=n$ if $j = \Lambda_1 g$ with $g \in J_n$.
Denote ${\omega}(g,y) = ({\omega}_1(g,y),{\omega}_K(g,y))$ with ${\omega}_1(g,y) \in G_1$ and ${\omega}_K(g,y) \in K$. Similarly write $\eta(\lambda,x) = (\eta_1(\lambda,x),\eta_K(\lambda,x))$. Note that for $\lambda \in \Lambda_0 - \{e\}$ we have $\eta_1(\lambda,x) \neq e$ for a.e. $x \in X$. Indeed, if $\eta_1(\lambda,x) = e$ for a fixed $\lambda \in \Lambda_0 - \{e\}$, then the element $(\lambda,\eta_K(\lambda,x)^{-1})$ of $\Lambda_0 \times K$ stabilizes $x$ and the essential freeness of $\Lambda_0 \times K {\curvearrowright}X$ implies that this can only happen for $x$ belonging to a negligible subset of $X$. One then proves easily by induction on $n$ that
- for a.e. $y \in Y$ and all $n \in {\mathbb{N}}$, the map $g \mapsto {\omega}_1(g,y)$ is a bijection of $I_n$ onto $J_n$,
- for all $n \in {\mathbb{N}}, g \in I_n$, the map $y \mapsto {\omega}(g,y)$ only depends on the coordinates $y_j$, $|j| \leq n-1$.
Since for all $g \in I_n$ we have ${\omega}_1(g,y) \in J_n$, it follows that $$\label{eq.nuttig}
\rho(g*y) = (g*y)_{\Lambda_1 e} = ({\omega}(g,y) \cdot y)_{\Lambda_1 e} = {\omega}_K(g,y) \cdot y_{\Lambda_1 {\omega}_1(g,y)}$$ for all $n \in {\mathbb{N}}$, $g \in I_n$ and a.e. $y \in Y$. We now use Lemma \[lem.indep\] to prove that for all $n \in {\mathbb{N}}$, the set $\{y \mapsto \rho(g*y) \mid g \in I_n\}$ forms a family of independent random variables that are independent of the coordinates $y_j$, $|j| \leq n-1$, and that only depend on the coordinates $y_j$, $|j| \leq n$. More concretely, we write ${\mathcal{J}}_n = \{ \Lambda_1 g \mid |g| \leq n\}$ and we apply Lemma \[lem.indep\] to the countable set ${\mathcal{J}}_n - {\mathcal{J}}_{n-1}$, the direct product $$Z := X^{{\mathcal{J}}_{n-1}} \times X^{{\mathcal{J}}_n - {\mathcal{J}}_{n-1}}$$ and the family of measurable maps ${\omega}_g : Z {\rightarrow}K \times ({\mathcal{J}}_n - {\mathcal{J}}_{n-1})$ indexed by $g \in I_n$, only depending on the coordinates $y_j$, $j \in {\mathcal{J}}_{n-1}$ and given by $${\omega}_g : y \mapsto ({\omega}_K(g,y), \Lambda_1 {\omega}_1(g,y)) \; .$$ Since $g \mapsto {\omega}_1(g,y)$ is a bijection of $I_n$ onto $J_n$, we have that $g \mapsto \Lambda_1 {\omega}_1(g,y)$ is a bijection of $I_n$ onto ${\mathcal{J}}_{n} - {\mathcal{J}}_{n-1}$. A combination of Lemma \[lem.indep\] and formula then implies that $\{y \mapsto \rho(g*y) \mid g \in I_n\}$ is a family of independent random variables that are independent of the coordinates $y_j$, $j \in {\mathcal{J}}_{n-1}$. By construction, these random variables only depend on the coordinates $y_j$, $|j| \leq n$. Having proven these statements for all $n \in {\mathbb{N}}$, it follows that $\{y \mapsto \rho(g*y) \mid g \in \bigcup_n I_n\}$ is a family of independent random variables.
Denote by ${\mathcal{B}}_0$ the smallest $\sigma$-algebra on $Y$ such that $Y {\rightarrow}X_1 : y \mapsto \rho(g * y)$ is ${\mathcal{B}}_0$-measurable for all $g \in G_0$. It remains to prove that ${\mathcal{B}}_0$ is the entire $\sigma$-algebra of $Y$. Note that by construction, the map $Y {\rightarrow}Y : y \mapsto g * y$ is ${\mathcal{B}}_0$-measurable for all $g \in G_0$. Since $\rho$ is $K$-equivariant and the actions $K {\curvearrowright}Y$ and $G_0 {\curvearrowright}Y$ commute, we also get that the map $y \mapsto k * y$ is ${\mathcal{B}}_0$-measurable for every $k \in K$. We must prove that $y \mapsto y_i$ is ${\mathcal{B}}_0$-measurable for every $n \in {\mathbb{N}}$ and $i \in \Lambda_1 \backslash G_1$ with $|i|=n$. This follows by induction on $n$, because for all $g \in J_n$ we have $$y_{\Lambda_1 g} = \rho(g \cdot y) = \rho({\omega}'(g,y) * y)$$ and because $y \mapsto {\omega}'(g,y)$ only depends on the coordinates $y_j$, $|j| \leq n-1$.
To prove the second item of the theorem, it suffices to make the following observation. If the actions $\Lambda_i {\curvearrowright}X_i /K$ are conjugate w.r.t. the isomorphism $\delta : \Lambda_0 {\rightarrow}\Lambda_1$, then in the proof of the first item, the Zimmer $1$-cocycle $\eta$ is of the form $\eta(\lambda_0,x) = (\delta(\lambda_0),\eta_K(\lambda_0,x))$. So the $1$-cocycle ${\omega}: G_0 \times Y {\rightarrow}G_1 \times K$ is of the form ${\omega}(g,y) = (({\mathord{\text{\rm id}}}* \delta)(g),{\omega}_K(g,y))$. This immediately implies that the actions $G_i {\curvearrowright}Y_i/K$ are conjugate w.r.t. the isomorphism ${\mathord{\text{\rm id}}}* \delta$.
\[cor.indep-base\] For fixed $n$ and varying base probability space $(X_0,\mu_0)$ the Bernoulli actions ${\mathbb{F}}_n {\curvearrowright}X_0^{{\mathbb{F}}_n}$ are orbit equivalent.
By Remark \[rem.bete\].1, the co-induction of a Bernoulli action is again a Bernoulli action over the same base space. Let $X_0$ and $X_1$ be nontrivial base probability spaces. By Dye’s theorem [@Dy58], the Bernoulli actions ${\mathbb{Z}}{\curvearrowright}X_0^{{\mathbb{Z}}}$ and ${\mathbb{Z}}{\curvearrowright}X_1^{\mathbb{Z}}$ are orbit equivalent. By Theorem \[thm.coinduced\] their co-induced actions to ${\mathbb{F}}_n = {\mathbb{F}}_{n-1} * {\mathbb{Z}}$ are orbit equivalent. But these co-induced actions are isomorphic to the Bernoulli actions ${\mathbb{F}}_n {\curvearrowright}X_i^{{\mathbb{F}}_n}$.
We used the following easy independence lemma.
\[lem.indep\] Let $(X,\mu)$ and $(X_0,\mu_0)$ be standard probability spaces and let $H {\curvearrowright}(X_0,\mu_0)$ be a measure preserving action. Let $I$ be a countable set. Consider $Z = X \times X_0^I$ with the product probability measure. Assume that ${\mathcal{F}}$ is a family of measurable maps ${\omega}: Z {\rightarrow}H \times I$. Write ${\omega}(x,y) = ({\omega}_1(x,y),{\omega}_2(x,y))$. Assume that
- for almost every $z \in Z$, the map ${\mathcal{F}}{\rightarrow}I : {\omega}\mapsto {\omega}_2(z)$ is injective,
- for every ${\omega}\in {\mathcal{F}}$, the map $z \mapsto {\omega}(z)$ only depends on the variable $Z {\rightarrow}X : (x,y) \mapsto x$.
Then, $\{(x,y) \mapsto {\omega}_1(x,y) \cdot y_{{\omega}_2(x,y)} \mid {\omega}\in {\mathcal{F}}\}$ is a family of independent identically $(X_0,\mu_0)$-distributed random variables that are independent of $(x,y) \mapsto x$.
Since the maps ${\omega}\in {\mathcal{F}}$ only depend on the variable $(x,y) \mapsto x$, we view ${\omega}\in {\mathcal{F}}$ as a map from $X$ to $H \times I$. We have to prove that $\{(x,y) \mapsto {\omega}_1(x) \cdot y_{{\omega}_2(x)} \mid {\omega}\in {\mathcal{F}}\}$ is a family of independent identically $(X_0,\mu_0)$-distributed random variables that are independent of $(x,y) \mapsto x$. But conditioning on $x \in X$, we get that the variables $$X_0^I {\rightarrow}X_0 : y \mapsto {\omega}_1(x) \cdot y_{{\omega}_2(x)}$$ are independent and $(X_0,\mu_0)$-distributed because the coordinates ${\omega}_2(x)$, for ${\omega}\in {\mathcal{F}}$, are distinct elements of $I$ and because the action $H {\curvearrowright}X_0$ is measure preserving. So the lemma is proven.
Stable orbit equivalence of Bernoulli actions
=============================================
Denote by $a,b$ the standard generators of ${\mathbb{F}}_2$. Denote by $\langle a \rangle$ and $\langle b \rangle$ the subgroups of ${\mathbb{F}}_2$ generated by $a$, resp. $b$. Let $(X_0,\mu_0)$ be a standard probability space and consider the Bernoulli action ${\mathbb{F}}_2 {\curvearrowright}X_0^{{\mathbb{F}}_2}$ given by $(g \cdot x)_h = x_{hg}$.
Whenever $(X_0,\mu_0)$ is a probability space, the Bernoulli action $\Gamma {\curvearrowright}X_0^\Gamma$ can be characterized up to isomorphism as the unique p.m.p. action $\Gamma {\curvearrowright}X$ for which there exists a factor map $\pi : X {\rightarrow}X_0$ such that the maps $x \mapsto \pi(g \cdot x)$, $g \in \Gamma$, are independent and generate, up to null sets, the whole $\sigma$-algebra of $X$.
We prove the stable orbit equivalence of Bernoulli actions as a combination of the following three lemmas. Fix $\kappa \in {\mathbb{N}}$, $\kappa \geq 2$, and denote $X_0 = \{0,\ldots,\kappa-1\}$ equipped with the uniform probability measure. Let $(Y_0,\eta_0)$ be any standard probability space (that is not reduced to a single atom). Denote by $r : {\mathbb{F}}_2 {\rightarrow}{\mathbb{Z}}/\kappa {\mathbb{Z}}$ the group morphism determined by $r(a) = 0$ and $r(b) = 1$. Identify $X_0$ with ${\mathbb{Z}}/\kappa {\mathbb{Z}}$ and denote by $\cdot$ the action of ${\mathbb{Z}}/\kappa {\mathbb{Z}}$ on $X_0$ given by addition in ${\mathbb{Z}}/\kappa {\mathbb{Z}}$.
\[lem.een\] Consider the action ${\mathbb{F}}_2 {\curvearrowright}X := X_0^{\langle b \rangle \backslash {\mathbb{F}}_2}$ given by $(g \cdot x)_{\langle b \rangle h} = r(g) \cdot x_{\langle b \rangle h g}$. Let ${\mathbb{F}}_2 {\curvearrowright}Y_0^{{\mathbb{F}}_2}$ be the Bernoulli action. Then the diagonal action ${\mathbb{F}}_2 {\curvearrowright}X \times Y_0^{{\mathbb{F}}_2}$ given by $g \cdot (x,y) = (g \cdot x,g \cdot y)$ is orbit equivalent with a Bernoulli action of ${\mathbb{F}}_2$.
\[lem.twee\] The action ${\mathbb{F}}_2 {\curvearrowright}X$ defined in Lemma \[lem.een\] is stably orbit equivalent with compression constant $1/\kappa$ with a Bernoulli action of ${\mathbb{F}}_{1 + \kappa}$.
\[lem.drie\] Let $\Gamma {\curvearrowright}(X,\mu)$ be any free ergodic p.m.p. action of an infinite group $\Gamma$. Assume that $\kappa \in {\mathbb{N}}$ and that $\Gamma {\curvearrowright}X$ is stably orbit equivalent with compression constant $1/\kappa$ with a Bernoulli action of some countable group $\Lambda$. Let $(Y_0,\eta_0)$ be any standard probability space and $\Gamma {\curvearrowright}Y_0^\Gamma$ the Bernoulli action. Then also the diagonal action $\Gamma {\curvearrowright}X \times Y_0^\Gamma$ is stably orbit equivalent with compression constant $1/\kappa$ with a Bernoulli action of $\Lambda$.
### Proof of Theorem \[thm.A\] {#proof-of-theorem-thm.a .unnumbered}
We already deduce Theorem \[thm.A\] from the above three lemmas.
We first prove that Lemmas \[lem.een\], \[lem.twee\], \[lem.drie\] yield a Bernoulli action of ${\mathbb{F}}_2$ that is stably orbit equivalent with compression constant $1/\kappa$ with a Bernoulli action of ${\mathbb{F}}_{1+\kappa}$. Indeed, by Lemma \[lem.een\] a Bernoulli action of ${\mathbb{F}}_2$ is orbit equivalent with the diagonal action ${\mathbb{F}}_2 {\curvearrowright}X \times Y_0^{{\mathbb{F}}_2}$. By Lemma \[lem.twee\], the action ${\mathbb{F}}_2 {\curvearrowright}X$ is stably orbit equivalent with compression constant $1/\kappa$ with a Bernoulli action of ${\mathbb{F}}_{1+\kappa}$. But then, Lemma \[lem.drie\] says that the same holds for the diagonal action ${\mathbb{F}}_2 {\curvearrowright}X \times Y_0^{{\mathbb{F}}_2}$.
Combined with Corollary \[cor.indep-base\] it follows that *all* Bernoulli actions of ${\mathbb{F}}_2$ are stably orbit equivalent with all Bernoulli actions of ${\mathbb{F}}_m$, $m \geq 2$, with compression constant $1/(m-1)$. By transitivity of stable orbit equivalence, all Bernoulli actions of ${\mathbb{F}}_n$ and ${\mathbb{F}}_m$ are stably orbit equivalent with compression constant $(n-1)/(m-1)$.
### Proof of Lemma \[lem.een\] {#proof-of-lemma-lem.een .unnumbered}
View ${\mathbb{Z}}$ as the subgroup of ${\mathbb{F}}_2$ generated by $b$. Let ${\mathbb{Z}}{\curvearrowright}Y_0^{\mathbb{Z}}$ be the Bernoulli action. Consider the action ${\mathbb{Z}}{\curvearrowright}X_0 \times Y_0^{\mathbb{Z}}$ given by $g \cdot (x,y) = (r(g) \cdot x, g \cdot y)$. Note that ${\mathbb{Z}}{\curvearrowright}X_0 \times Y_0^{\mathbb{Z}}$ is a free ergodic p.m.p. action. Using Remark \[rem.bete\] (statements 1, 2 and 3), one gets that the action ${\mathbb{F}}_2 {\curvearrowright}X \times Y_0^{{\mathbb{F}}_2}$ given in the formulation of Lemma \[lem.een\] is precisely the co-induction of ${\mathbb{Z}}{\curvearrowright}X_0 \times Y_0^{\mathbb{Z}}$ to ${\mathbb{F}}_2$. By Dye’s theorem [@Dy58], the free ergodic p.m.p. action ${\mathbb{Z}}{\curvearrowright}X_0 \times Y_0^{{\mathbb{Z}}}$ is orbit equivalent with a Bernoulli action of ${\mathbb{Z}}$. By Remark \[rem.bete\].1, the co-induction of the latter is a Bernoulli action of ${\mathbb{F}}_2$. So by Theorem \[thm.coinduced\], the action ${\mathbb{F}}_2 {\curvearrowright}X \times Y_0^{{\mathbb{F}}_2}$ is orbit equivalent with a Bernoulli action of ${\mathbb{F}}_2$.
### Proof of Lemma \[lem.twee\] {#proof-of-lemma-lem.twee .unnumbered}
We have $X = X_0^{\langle b \rangle \backslash {\mathbb{F}}_2}$ and the action ${\mathbb{F}}_2 {\curvearrowright}X$ is given by $(g \cdot x)_{\langle b \rangle h} = r(g) \cdot x_{\langle b \rangle hg}$. Write $Z = X_0^{\mathbb{Z}}$ and denote by $\rho : X {\rightarrow}Z$ the factor map given by $\rho(x)_n = x_{\langle b \rangle a^n}$. Denote by $\cdot$ the Bernoulli action ${\mathbb{Z}}{\curvearrowright}Z$ and note that $\rho(a^n \cdot x) = n \cdot \rho(x)$ for all $x \in X$ and $n \in {\mathbb{Z}}$.
Define the subsets $V_i$, $i=0,\ldots,\kappa-1$, of $Z$ given by $V_i := \{z \in Z \mid z_0 = i\}$. Similarly define $W_i \subset X$ given by $W_i = \rho^{-1}(V_i)$. Note that $W_0$ has measure $1/\kappa$. To prove the lemma we define a p.m.p. action of ${\mathbb{F}}_{1+\kappa}$ on $W_0$ such that ${\mathbb{F}}_{1+\kappa} * x = {\mathbb{F}}_2 \cdot x \cap W_0$ for a.e. $x \in W_0$ and such that ${\mathbb{F}}_{1+\kappa} {\curvearrowright}W_0$ is a Bernoulli action.
By Dye’s theorem [@Dy58], there exists a Bernoulli action ${\mathbb{Z}}\overset{*}{{\curvearrowright}} V_0$ such that ${\mathbb{Z}}* z = {\mathbb{Z}}\cdot z \cap V_0$ for a.e. $z \in V_0$. Denote by $\eta : {\mathbb{Z}}\times V_0 {\rightarrow}{\mathbb{Z}}$ the corresponding $1$-cocycle for the $*$-action determined by $n * z = \eta(n,z) \cdot z$ for $n \in {\mathbb{Z}}$ and a.e. $z \in V_0$.
Since the Bernoulli action ${\mathbb{Z}}\overset{\cdot}{{\curvearrowright}} Z$ is ergodic and since all the subsets $V_i \subset Z$ have the same measure, we can choose measure preserving isomorphisms ${\alpha}_i : V_0 {\rightarrow}V_i$ satisfying ${\alpha}_i(z) \in {\mathbb{Z}}\cdot z$ for a.e. $z \in {\mathbb{Z}}$ and take ${\alpha}_0$ to be the identity isomorphism. Let ${\varphi}^0_i : V_0 {\rightarrow}{\mathbb{Z}}$ and $\psi^0_i : V_i {\rightarrow}{\mathbb{Z}}$ be the maps determined by ${\alpha}_i(z) = {\varphi}^0_i(z) \cdot z$ for a.e. $z \in V_0$ and ${\alpha}_i^{-1}(z) = \psi^0_i(z) \cdot z$ for a.e. $z \in V_i$. Define the corresponding measure preserving isomorphisms $\theta_i : W_0 {\rightarrow}W_i$ given by $\theta_i(x) = {\varphi}_i(x) \cdot x$ and $\theta_i^{-1}(x) = \psi_i(x) \cdot x$ where ${\varphi}_i(x) = a^{{\varphi}^0_i(\rho(x))}$ and $\psi_i(x) = a^{\psi^0_i(\rho(x))}$.
Denote by $a$ and $b_i$, $i = 0,\ldots,\kappa-1$, the generators of ${\mathbb{F}}_{1+\kappa}$. Define the p.m.p. action ${\mathbb{F}}_{1+\kappa} \overset{*}{{\curvearrowright}} W_0$ given by $$a^n * x = a^{\eta(n,\rho(x))} \cdot x \quad\text{and}\quad b_i * x = \theta_{i+1}^{-1} (b \cdot \theta_i(x)) \quad\text{for all}\;\; x \in W_0 \; .$$ Note that the action is well defined: if $x \in W_0$, then $\theta_i(x) \in W_i$ and hence $b \cdot \theta_i(x) \in W_{i+1}$. We use the convention that $W_\kappa = W_0$ and $\theta_\kappa = {\mathord{\text{\rm id}}}$. Observe that $\rho(a^n * x) = n * \rho(x)$ for all $n \in {\mathbb{Z}}$ and a.e. $x \in W_0$.
It remains to prove that ${\mathbb{F}}_{1+\kappa} * x = {\mathbb{F}}_2 \cdot x \cap W_0$ for a.e. $x \in W_0$ and that ${\mathbb{F}}_{1+\kappa} {\curvearrowright}W_0$ is a Bernoulli action.
Denote by ${\omega}: {\mathbb{F}}_{1+\kappa} \times W_0 {\rightarrow}{\mathbb{F}}_2$ the unique $1$-cocycle for the $*$-action determined by $${\omega}(a^n,x) = a^{\eta(n,\rho(x))} \quad\text{and}\quad {\omega}(b_i,x) = \psi_{i+1}(b \cdot \theta_i(x)) \, b \, {\varphi}_i(x) \; .$$ By construction, the formula $g * x = {\omega}(g,x) \cdot x$ holds for all $g \in \{a,b_0,\ldots,b_{\kappa-1}\}$ and a.e. $x \in W_0$. Since ${\omega}$ is a $1$-cocycle for the action ${\mathbb{F}}_{1+\kappa} \overset{*}{{\curvearrowright}} W_0$, the same formula holds for all $g \in {\mathbb{F}}_{1+\kappa}$ and a.e. $x \in W_0$. In particular, ${\mathbb{F}}_{1+\kappa} * x \subset {\mathbb{F}}_2 \cdot x \cap W_0$ for a.e. $x \in W_0$. To prove the converse inclusion we define the inverse $1$-cocycle for ${\omega}$.
Define $q_0 : Z {\rightarrow}V_0$ given by $q_0(z) = {\alpha}_i^{-1}(z)$ when $z \in V_i$. Denote by $\eta' : {\mathbb{Z}}\times Z {\rightarrow}{\mathbb{Z}}$ the $1$-cocycle for the $\cdot$-action determined by $q_0(n \cdot z) = \eta'(n,z) * q_0(z)$. Whenever $z \in V_0$, we have $z = q_0(z)$ and hence $$\label{eq.sven}
\eta'(\eta(n,z),z) * z = \eta'(\eta(n,z),z) * q_0(z) = q_0(\eta(n,z) \cdot z) = q_0(n*z) = n * z \; .$$ Since $*$ is an essentially free action of ${\mathbb{Z}}$, it follows that $\eta'(\eta(n,z),z) = n$ for all $n \in {\mathbb{Z}}$ and a.e. $z \in V_0$.
Denote by ${\omega}' : {\mathbb{F}}_2 \times X {\rightarrow}{\mathbb{F}}_{1+\kappa}$ the unique $1$-cocycle for the $\cdot$-action determined by $${\omega}'(a^n,x) = a^{\eta'(n,\rho(x))} \quad\text{for $n \in {\mathbb{Z}}$ and a.e.\ $x \in X$, and}\quad {\omega}'(b,x) = b_i \quad\text{for a.e.\ $x \in W_i$.}$$ Define $q : X {\rightarrow}W_0$ given by $q(x) = \theta_i^{-1}(x)$ when $x \in W_i$. Note that $\rho(q(x)) = q_0(\rho(x))$ for a.e. $x \in X$. We prove that $q(g \cdot x) = {\omega}'(g,x) * q(x)$ for all $g \in {\mathbb{F}}_2$ and a.e. $x \in X$. If $g = a^n$ for some $n \in {\mathbb{Z}}$, we know that both $q(g \cdot x)$ and ${\omega}'(g,x) * q(x)$ belong to $\langle a \rangle \cdot x$. So to prove that they are equal, it suffices to check that they have the same image under $\rho$. The following computation shows that this is indeed the case. $$\begin{aligned}
& \rho(q(a^n \cdot x)) = q_0(\rho(a^n \cdot x)) = q_0(n \cdot \rho(x)) = \eta'(n,\rho(x)) * q_0(\rho(x)) \quad\text{while,} \\
& \rho({\omega}'(a^n,x) * q(x)) = \rho(a^{\eta'(n,\rho(x))} * q(x)) = \eta'(n,\rho(x)) * \rho(q(x)) = \eta'(n,\rho(x)) * q_0(\rho(x)) \; .\end{aligned}$$ Since by definition of the action $*$ we have that $b_i * \theta_i^{-1}(x) = \theta_{i+1}^{-1}(b \cdot x)$ whenever $x \in W_i$, the formula ${\omega}'(g,x) * q(x) = q(g \cdot x)$ also holds when $g = b$. Hence, the same formula holds for all $g \in {\mathbb{F}}_2$ and a.e. $x \in X$. In particular, ${\mathbb{F}}_2 \cdot x \cap W_0 \subset {\mathbb{F}}_{1+\kappa} * x$ for a.e. $x \in W_0$. The converse inclusion was already proven above. Hence, ${\mathbb{F}}_{1+\kappa} * x = {\mathbb{F}}_2 \cdot x \cap W_0$ for a.e. $x \in W_0$.
Denote by ${\mathcal{J}}\subset {\mathbb{F}}_{1+\kappa}$ the union of $\{e\}$ and all the reduced words that start with one of the letters $b_i^{\pm 1}$, $i = 0,\ldots,\kappa-1$. Note that ${\mathcal{J}}$ is a right transversal for $\langle a \rangle < {\mathbb{F}}_{1+\kappa}$. It remains to prove that $$\{ W_0 {\rightarrow}V_0 : x \mapsto \rho(g * x) \mid g \in {\mathcal{J}}\}$$ is a family of independent random variables that generate, up to null sets, the whole ${\sigma}$-algebra on $W_0$. Indeed, we already know that ${\mathbb{Z}}\overset{*}{{\curvearrowright}} V_0$ is a Bernoulli action so that it will follow that ${\mathbb{F}}_{1+\kappa} {\curvearrowright}W_0$ is the co-induction of a Bernoulli action, hence a Bernoulli action itself (see Remark \[rem.bete\].1).
We equip both ${\mathbb{F}}_2$ and ${\mathbb{F}}_{1+\kappa}$ with a length function. For $g \in {\mathbb{F}}_2$ we denote by $|g|$ the number of letters $b^{\pm 1}$ appearing in the reduced expression of $g$, while for $g \in {\mathbb{F}}_{1+\kappa}$ we denote by $|g|$ the number of letters $b_i^{\pm 1}$, $i = 0,\ldots,\kappa-1$, appearing in the reduced expression of $g$. By induction on the length of $g$, one easily checks that $|{\omega}(g,x)| \leq |g|$ for all $g \in {\mathbb{F}}_{1+\kappa}$ and a.e. $x \in W_0$, and that $|{\omega}'(g,x)| \leq |g|$ for all $g \in {\mathbb{F}}_2$ and a.e. $x \in X$.
We next claim that $$\label{eq.inverse}
{\omega}'({\omega}(g,x),x) = g \quad\text{for all $g \in {\mathbb{F}}_{1+\kappa}$ and a.e.\ $x \in W_0$.}$$ Once this claim is proven, it follows that $|{\omega}(g,x)| = |g|$ for all $g \in {\mathbb{F}}_{1+\kappa}$ and a.e. $x \in W_0$ : indeed, the strict inequality $|{\omega}(g,x)| < |g|$ would lead to the contradiction $$|g| = |{\omega}'({\omega}(g,x),g)| \leq |{\omega}(g,x)| < |g| \; .$$ First note that for $g = a^n$ formula follows immediately from . So it remains to prove when $g = b_i$. First observe that ${\omega}'({\varphi}_i(x),x) = e$ for a.e. $x \in W_0$. Indeed, $${\omega}'({\varphi}_i(x),x) * x = q({\varphi}_i(x) \cdot x) = q(\theta_i(x)) = x$$ and since the $*$-action of $\langle a \rangle$ on $W_0$ is essentially free, it follows that ${\omega}'({\varphi}_i(x),x) = e$. Similarly, ${\omega}'(\psi_i(x),x) = e$ for a.e. $x \in W_i$. Take $x \in W_0$ and write $x' := b {\varphi}_i(x) \cdot x$. Note that $x' = b \cdot \theta_i(x)$ and that $x' \in W_{i+1}$. So, $${\omega}'({\omega}(b_i,x),x) = {\omega}'(\psi_{i+1}(x') \, b \, {\varphi}_i(x) , x) = {\omega}'(\psi_{i+1}(x'), x') \, {\omega}'(b, \theta_i(x)) \, {\omega}'({\varphi}_i(x),x) = e \, b_i \, e = b_i \; .$$ So holds for $g = a^n$ and $g = b_i$. Hence holds for all $g \in {\mathbb{F}}_{1+\kappa}$. Note that implies that the action ${\mathbb{F}}_{1+\kappa} \overset{*}{{\curvearrowright}} W_0$ is essentially free. Indeed, if $g \in {\mathbb{F}}_{1+\kappa}$, $x \in W_0$ and $g * x = x$, it follows that ${\omega}(g,x) \cdot x = x$. Since the $\cdot$-action is essentially free, we conclude that ${\omega}(g,x) = e$. But then by $$g = {\omega}'({\omega}(g,x),x) = {\omega}'(e,x) = e \; .$$
Define the subsets ${\mathcal{C}}(n) \subset \langle b \rangle \backslash {\mathbb{F}}_2$ given by ${\mathcal{C}}(n) := \{\langle b \rangle g \mid g \in {\mathbb{F}}_2, |g| \leq n\}$. Also define ${\mathcal{J}}_n := \{g \in {\mathcal{J}}\mid |g| \leq n\}$. We prove by induction on $n$ that the following two statements hold.
- If $g \in {\mathbb{F}}_{1+\kappa}$ and $|g| \leq n$, then $x \mapsto {\omega}(g,x)$ only depends on the coordinates $x_i$, $i \in {\mathcal{C}}(n)$.
- The set $\{W_0 {\rightarrow}V_0 \mid x \mapsto \rho(g * x) \mid g \in {\mathcal{J}}_n\}$ is a family of independent random variables that only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n)$.
Since $e$ is the only element in ${\mathcal{J}}$ of length $0$, statements $1_0$ and $2_0$ are trivial. Assume that statements $1_n$ and $2_n$ hold for a given $n$.
Any element in ${\mathbb{F}}_{1+\kappa}$ of length $n+1$ can be written as a product $gh$ with $|g| = 1$ and $|h| = n$. By the cocycle equality, we have $${\omega}(gh,x) = {\omega}(g,h*x) \, {\omega}(h,x) = {\omega}(g, {\omega}(h,x) \cdot x) \, {\omega}(h,x) \; .$$ By statement $1_n$, we know that the map $x \mapsto {\omega}(g,x)$ only depends on the coordinates $x_i$, $i \in {\mathcal{C}}(1)$, and that the map $x \mapsto {\omega}(h,x)$ only depends on on the coordinates $x_i$, $i \in {\mathcal{C}}(n)$. So, $x \mapsto {\omega}(gh,x)$ only depends on the coordinates $x_i$, $i \in {\mathcal{C}}(n)$, and the map $$x \mapsto ({\omega}(h,x) \cdot x)_{\langle b \rangle k} = r({\omega}(h,x)) \cdot x_{\langle b \rangle k {\omega}(h,x)} \quad\text{for}\;\; |k| \leq 1 \; .$$ Again by statement $1_n$ these maps only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n+1)$, so that statement $1_{n+1}$ is proven.
Define, for $i = 0,\ldots,\kappa-1$ and ${\varepsilon}= \pm 1$, $${\mathcal{J}}_n^{i,{\varepsilon}} := \bigl\{g \in {\mathbb{F}}_{1 + \kappa} \; \big| \; |g| = n \;\;\text{and}\;\; |b_i^{{\varepsilon}} g| = n+1 \bigr\} \; .$$ It follows that $${\mathcal{J}}_{n+1} = {\mathcal{J}}_n \cup \bigcup_{i \in \{0,\ldots,\kappa-1\}, {\varepsilon}\in \{\pm 1\}} b_i^{\varepsilon}\, {\mathcal{J}}_n^{i,{\varepsilon}} \;.$$ Since we assumed that statement $2_n$ holds, in order to prove statement $2_{n+1}$, it suffices to show that $$\{x \mapsto \rho(b_i^{\varepsilon}g * x) \mid i=0,\ldots,\kappa-1, {\varepsilon}= \pm 1, g \in {\mathcal{J}}_n^{i,{\varepsilon}}\}$$ is a family of independent random variables that only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n+1)$, and that are independent of the coordinates $x_i$, $i \in {\mathcal{C}}(n)$.
Note that $\rho(b_i g * x) = {\alpha}_{i+1}^{-1} (\rho(b \cdot \theta_i(g*x)))$ while $\rho(b_i^{-1} g * x) = {\alpha}_i^{-1}(\rho(b^{-1} \cdot \theta_{i+1}(g*x)))$. The value of $\rho(b \cdot \theta_i(g*x))$ at $0$ is constantly equal to $i+1$, while the value of $\rho(b^{-1} \cdot \theta_{i+1}(g*x))$ at $0$ is constantly equal to $i$. Therefore we have to prove that $$\label{eq.variables}
\begin{split}
\{x \mapsto \rho & (b \cdot \theta_i(g*x))_m \mid i=0,\ldots,\kappa-1 , g \in {\mathcal{J}}_n^{i,+} , m \in {\mathbb{Z}}- \{0\} \} \\ & \cup
\{x \mapsto \rho(b^{-1} \cdot \theta_{i+1}(g*x))_m \mid i=0,\ldots,\kappa-1 , g \in {\mathcal{J}}_n^{i,-} , m \in {\mathbb{Z}}- \{0\} \}
\end{split}$$ is a family of independent random variables that only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n+1)$, and that are independent of the coordinates $x_i$, $i \in {\mathcal{C}}(n)$.
Write $${\omega}_i^{\varepsilon}(g,x) := \begin{cases} b \, {\varphi}_i(g * x) \, {\omega}(g,x) &\quad\text{if ${\varepsilon}= 1$,} \\
b^{-1} \, {\varphi}_{i+1}(g*x) \, {\omega}(g,x) &\quad\text{if ${\varepsilon}= -1$.}\end{cases}$$ The random variables in are precisely equal to $$\label{eq.variablester}
\{x \mapsto r({\omega}_i^{\varepsilon}(g,x)) \cdot x_{\langle b \rangle a^m {\omega}_i^{\varepsilon}(g,x)} \mid i=0,\ldots,\kappa-1 , {\varepsilon}= \pm 1 , g \in {\mathcal{J}}_n^{i,{\varepsilon}} , m \in {\mathbb{Z}}- \{0\} \} \; .$$ So we have to prove that is a family of independent random variables that only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n+1)$, and that are independent of the coordinates $x_i$, $i \in {\mathcal{C}}(n)$. By statement $1_n$, the maps $x \mapsto {\omega}_i^{\varepsilon}(g,x)$, and in particular $x \mapsto r({\omega}_i^{\varepsilon}(g,x))$, only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n)$. So, we have to prove that $$\label{eq.variablesbis}
\{x \mapsto x_{\langle b \rangle a^m {\omega}_i^{\varepsilon}(g,x)} \mid i=0,\ldots,\kappa-1 , {\varepsilon}= \pm 1 , g \in {\mathcal{J}}_n^{i,{\varepsilon}} , m \in {\mathbb{Z}}- \{0\} \} \; .$$ is a family of independent random variables that only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n+1)$, and that are independent of the coordinates $x_i$, $i \in {\mathcal{C}}(n)$.
We apply Lemma \[lem.indep\] to the countable set ${\mathcal{C}}(n+1) - {\mathcal{C}}(n)$ and the direct product $$X_0^{{\mathcal{C}}(n)} \times X_0^{{\mathcal{C}}(n+1)-{\mathcal{C}}(n)} \; .$$ Since the maps $x \mapsto {\omega}_i^{\varepsilon}(g,x)$ only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n)$, it remains to check that the cosets $\langle b \rangle a^m {\omega}_i^{\varepsilon}(g,x)$ belong to ${\mathcal{C}}(n+1) - {\mathcal{C}}(n)$ and that they are distinct for fixed $x \in W_0$ and varying $i \in \{0,\ldots,\kappa-1\}$, ${\varepsilon}\in \{\pm 1\}$ and $g \in {\mathcal{J}}_n^{i,{\varepsilon}}$.
Note that ${\omega}(b_i^{\varepsilon}g,x) \in \langle a \rangle \, {\omega}_i^{\varepsilon}(g,x)$. Hence, $$|{\omega}_i^{\varepsilon}(g,x)| = |{\omega}(b_i^{\varepsilon}g,x)| = |b_i^{\varepsilon}g| = n+1$$ because $g \in {\mathcal{J}}_n^{i,{\varepsilon}}$. Since $|{\omega}(g,x)| = n$ and $|{\omega}_i^{\varepsilon}(g,x)| = n+1$, it follows from the defining formula of ${\omega}_i^{\varepsilon}$ that the first letter of ${\omega}_i^{\varepsilon}(g,x)$ must be $b^{\varepsilon}$. So the first letter of $a^m {\omega}_i^{\varepsilon}(g,x)$, $m \neq 0$, is $a^{\pm 1}$. This implies that $\langle b \rangle a^m {\omega}_i^{\varepsilon}(g,x)$ belongs to ${\mathcal{C}}(n+1) - {\mathcal{C}}(n)$. It also follows that if $$\langle b \rangle a^m {\omega}_i^{{\varepsilon}}(g,x) = \langle b \rangle a^{m'} {\omega}_{i'}^{{\varepsilon}'}(g',x) \; ,$$ then we must have $m = m'$, ${\varepsilon}= {\varepsilon}'$ and ${\omega}_i^{\varepsilon}(g,x) = {\omega}_{i'}^{{\varepsilon}'}(g',x)$. Assume ${\varepsilon}= {\varepsilon}' = 1$, the other case being analogous. So, $${\varphi}_i(g*x) \, {\omega}(g,x) = {\varphi}_{i'}(g',x) \, {\omega}(g',x) \; .$$ Applying these elements to $x$, it follows that $\theta_i(g*x) = \theta_{i'}(g'*x)$. Since the ranges of $\theta_i$ and $\theta_{i'}$ are disjoint for $i \neq i'$, it follows that $i=i'$. So, $g*x = g'*x$. Since we have seen above that the action ${\mathbb{F}}_{1+\kappa} \overset{*}{{\curvearrowright}} W_0$ is essentially free, it follows that $g=g'$.
We have proven that is a family of independent random variables that only depend on the coordinates $x_i$, $i \in {\mathcal{C}}(n+1)$, and that are independent of the coordinates $x_i$, $i \in {\mathcal{C}}(n)$. So, statement $2_{n+1}$ holds.
To conclude the proof of the lemma, it remains to show that the random variables $x \mapsto \rho(g * x)$, $g \in {\mathbb{F}}_{1+\kappa}$, generate up to null sets the whole ${\sigma}$-algebra of $W_0$. Denote by ${\mathcal{B}}_0$ the $\sigma$-algebra on $W_0$ generated by these random variables. By construction, $x \mapsto g*x$ is ${\mathcal{B}}_0$-measurable for every $g \in {\mathbb{F}}_{1+\kappa}$. Since $x \mapsto \rho(x)$ is ${\mathcal{B}}_0$-measurable, the formula $$q(a^n \cdot x) = a^{\eta'(n,\rho(x))} * x$$ shows that $x \mapsto q(a^n \cdot x)$ is ${\mathcal{B}}_0$-measurable for every $n \in {\mathbb{Z}}$. Denote by ${\mathcal{B}}_1$ the smallest $\sigma$-algebra on $X$ containing ${\mathcal{B}}_0$, containing the subsets $W_0,\ldots,W_{\kappa-1} \subset X$ and making $q : X {\rightarrow}W_0$ a ${\mathcal{B}}_1$-measurable map. Note that the restriction of ${\mathcal{B}}_1$ to $W_0$ equals ${\mathcal{B}}_0$ and that ${\mathcal{U}}\subset X$ is ${\mathcal{B}}_1$-measurable if and only if $q({\mathcal{U}}\cap W_i)$ is ${\mathcal{B}}_0$-measurable for every $i=0,\ldots,\kappa-1$. It therefore suffices to prove that ${\mathcal{B}}_1$ is the whole ${\sigma}$-algebra of $X$. By construction, $\rho : X {\rightarrow}Z$ is ${\mathcal{B}}_1$-measurable and by the above, also $x \mapsto a^n \cdot x$ is ${\mathcal{B}}_1$-measurable for every $n \in {\mathbb{Z}}$. If $x \in W_i$, we have that $b \cdot x = \theta_{i+1}^{-1}(b_i * \theta_i(x))$ and it follows that $x \mapsto b \cdot x$ is ${\mathcal{B}}_1$-measurable. Hence, $x \mapsto g \cdot x$ is ${\mathcal{B}}_1$-measurable for every $g \in {\mathbb{F}}_2$. Since $\rho$ is ${\mathcal{B}}_1$-measurable, it follows that $x \mapsto x_{\langle b \rangle g}$ is ${\mathcal{B}}_1$-measurable for every $g \in {\mathbb{F}}_2$. Hence ${\mathcal{B}}_1$ is the entire product ${\sigma}$-algebra.
### Proof of Lemma \[lem.drie\] {#proof-of-lemma-lem.drie .unnumbered}
We denote by a dot $\cdot$ the action of $\Gamma$ on $X$. Let $X_1 \subset X$ be a subset of measure $1/\kappa$. We are given a p.m.p. action $\Lambda \overset{*}{{\curvearrowright}} X_1$ such that $\Lambda * x = \Gamma \cdot x \cap X_1$ for a.e. $x \in X_1$ and such that $\Lambda {\curvearrowright}X_1$ is isomorphic with a $\Lambda$-Bernoulli action. This means that we have a probability space $U$ and a factor map $\pi : X_1 {\rightarrow}U$ such that the random variables $\{x \mapsto \pi(\lambda * x) \mid \lambda \in \Lambda\}$ are independent, identically distributed and generating the Borel $\sigma$-algebra of $X_1$. Denote by ${\omega}: \Lambda \times X_1 {\rightarrow}\Gamma$ the $1$-cocycle determined by ${\omega}(\lambda,x) \cdot x = \lambda * x$ for all $\lambda \in \Lambda$ and a.e. $x \in X_1$. Put $Y = Y_0^\Gamma$ and define the action $\Lambda {\curvearrowright}X_1 \times Y$ given by $$\lambda * (x,y) = {\omega}(\lambda,x) \cdot (x,y) = (\lambda * x, {\omega}(\lambda,x) \cdot y) \; .$$ By construction, $\Lambda * (x,y) \subset \Gamma \cdot (x,y) \cap X_1 \times Y$. But also the converse inclusion holds. Indeed, if we have $\gamma \in \Gamma$, $x \in X_1$ and $y \in Y$ such that $\gamma \cdot x \in X_1$, we can take $\lambda \in \Lambda$ such that $\lambda * x = \gamma \cdot x$. Hence ${\omega}(\lambda,x) = \gamma$ and also $\gamma \cdot (x,y) = \lambda * (x,y)$.
It remains to prove that $\Lambda {\curvearrowright}X_1 \times Y$ is isomorphic with a $\Lambda$-Bernoulli action.
By ergodicity of $\Gamma {\curvearrowright}X$, choose a partition (up to measure zero) $X = X_1 \sqcup \cdots \sqcup X_\kappa$ with $\mu(X_i) = 1/\kappa$ and choose measurable maps ${\varphi}_i : X_1 {\rightarrow}\Gamma$ such that the formulae $\theta_i(x) = {\varphi}_i(x) \cdot x$ define measure space isomorphisms $\theta_i : X_1 {\rightarrow}X_i$. Take ${\varphi}_1(x) = e$ for all $x \in X_1$. Define the measurable map $$\rho : X_1 \times Y {\rightarrow}U \times Y_0^\kappa : \rho(x,y) = (\pi(x),y_{{\varphi}_1(x)},\ldots,y_{{\varphi}_\kappa(x)}) \; .$$ We prove that $\rho$ is measure preserving and that the random variables $\{(x,y) \mapsto \rho(\lambda*(x,y)) \mid \lambda \in \Lambda\}$ are independent, identically distributed and generating the Borel $\sigma$-algebra of $X_1 \times Y$.
We first claim that for a.e. $x \in X_1$ $$\label{eq.F}
{\mathcal{F}}:= \Bigl({\varphi}_i(\lambda * x) {\omega}(\lambda,x)\Bigr)_{\lambda \in \Lambda \;\text{and}\; i=1,\ldots,\kappa}$$ is an enumeration of $\Gamma$ without repetitions. Observe that $${\varphi}_i(\lambda * x) {\omega}(\lambda,x) \cdot x = \theta_i(\lambda * x) \; .$$ It follows that ${\mathcal{F}}\cdot x = \Gamma \cdot x$. Since $\Gamma {\curvearrowright}X$ is essentially free, it follows that ${\mathcal{F}}$ enumerates the whole of $\Gamma$. If ${\varphi}_i(\lambda * x) {\omega}(\lambda,x) = {\varphi}_j(\lambda' * x) {\omega}(\lambda',x)$, it follows that $\theta_i(\lambda * x) = \theta_j(\lambda' * x)$. For $i \neq j$, the sets $X_i$ and $X_j$ are disjoint. So, $i=j$ and $\lambda * x = \lambda' * x$. Being a Bernoulli action of an infinite group, $\Lambda \overset{*}{{\curvearrowright}} X_1$ is essentially free and we conclude that $\lambda = \lambda'$. This proves the claim.
Since for a.e. $x \in X_1$ the elements ${\varphi}_1(x),\ldots,{\varphi}_\kappa(x)$ are distinct, it follows from Lemma \[lem.indep\] that the random variables $(x,y) \mapsto \pi(x)$ and $(x,y) \mapsto y_{{\varphi}_i(x)}$, $i=1,\ldots,\kappa$, are all independent. Since they are all measure preserving as well, we conclude that $\rho$ is measure preserving. Note that $$\rho(\lambda * (x,y)) = \bigl(\pi(\lambda * x),y_{{\varphi}_1(\lambda * x) {\omega}(\lambda,x)},\ldots,y_{{\varphi}_\kappa(\lambda * x) {\omega}(\lambda,x)}\bigr) \; .$$ It therefore remains to prove that $$\{(x,y) \mapsto \pi(\lambda * x) \mid \lambda \in \Lambda\} \cup \{(x,y) \mapsto y_{{\varphi}_i(\lambda * x) {\omega}(\lambda,x)} \mid \lambda \in \Lambda, i = 1,\ldots,\kappa\}$$ is an independent family of random variables that generate, up to null sets, the Borel $\sigma$-algebra of $X_1 \times Y$. The factor map $\pi$ was chosen in such a way that the random variables $\{ x \mapsto \pi(\lambda * x) \mid \lambda \in \Lambda\}$ are independent and generate, up to null sets, the Borel $\sigma$-algebra of $X_1$. So, we must prove that $$\label{eq.onzefamilie}
\{(x,y) \mapsto y_{{\varphi}_i(\lambda * x) {\omega}(\lambda,x)} \mid \lambda \in \Lambda, i = 1,\ldots,\kappa\}$$ forms a family of independent random variables that are independent of $(x,y) \mapsto x$ and that, together with $(x,y) \mapsto x$, generate up to null sets the Borel $\sigma$-algebra of $X_1 \times Y$. We apply Lemma \[lem.indep\] to the countable set $\Gamma$, the direct product $X_1 \times Y_0^\Gamma$ and the family of maps $X_1 {\rightarrow}\Gamma : x \mapsto {\varphi}_i(\lambda * x) {\omega}(\lambda,x)$ indexed by $\lambda \in \Lambda, i = 1,\ldots,\kappa$. Since for a.e. $x \in X_1$, the set ${\mathcal{F}}$ in is an enumeration of $\Gamma$, it follows from Lemma \[lem.indep\] that is indeed a family of independent random variables that are moreover independent of $(x,y) \mapsto x$.
Denote by ${\mathcal{B}}_1$ the smallest $\sigma$-algebra on $X_1 \times Y$ such that the map $(x,y) \mapsto x$ and the random variables in are measurable. It remains to prove that, up to null sets, ${\mathcal{B}}_1$ is the Borel $\sigma$-algebra of $X_1 \times Y$. So, it remains to prove that for all $g \in \Gamma$, the random variables $(x,y) \mapsto y_g$ are ${\mathcal{B}}_1$-measurable. Put ${\mathcal{J}}= \{1,\ldots,\kappa\} \times \Lambda$ and define the Borel map $\eta : {\mathcal{J}}\times X_1 {\rightarrow}\Gamma$ given by $\eta(i,\lambda,x) := {\varphi}_i(\lambda * x) {\omega}(\lambda,x)$. Since for a.e. $x \in X_1$, the family ${\mathcal{F}}$ in is an enumeration of $\Gamma$, we can take a Borel map $\gamma : \Gamma \times X_1 {\rightarrow}{\mathcal{J}}$ such that $\eta(\gamma(g,x),x) = g$ for all $g \in \Gamma$ and a.e. $x \in X_1$. By the definition of ${\mathcal{B}}_1$ and $\eta$, we know that the map $$\label{eq.eenmap}
{\mathcal{J}}\times X_1 \times Y {\rightarrow}Y_0 : (j,x,y) \mapsto y_{\eta(j,x)}$$ is ${\mathcal{B}}_1$-measurable. Fix $g \in \Gamma$. Since $(x,y) \mapsto x$ is ${\mathcal{B}}_1$-measurable, also $(x,y) \mapsto (\gamma(g,x),x,y)$ is ${\mathcal{B}}_1$-measurable. The composition with the map in yields $(x,y) \mapsto y_g$ a.e. So $(x,y) \mapsto y_g$ is ${\mathcal{B}}_1$-measurable. This concludes the proof of the lemma.
Isomorphisms of factors of Bernoulli actions of free products
=============================================================
Before proving Theorem \[thm.B\], we need the following elementary lemma.
\[lem.factor\] Let $\Gamma,\Lambda$ be countable groups and $K$ a nontrivial second countable compact group equipped with its normalized Haar measure. Consider the action $(\Gamma * \Lambda) \times K {\curvearrowright}X := K^{\Gamma \backslash \Gamma * \Lambda}$ where $\Gamma * \Lambda$ shifts the indices and $K$ acts by diagonal left translation. The resulting factor action $\Gamma * \Lambda {\curvearrowright}X/K$ is isomorphic with the co-induced action of $\Lambda {\curvearrowright}K^\Lambda / K$ to $\Gamma * \Lambda$.
Define the factor map $\rho : K^{\Gamma \backslash \Gamma * \Lambda} {\rightarrow}K^\Lambda$ given by $\rho(x)_\lambda = x_{\Gamma \lambda}$. Note that $\rho$ is $(\Lambda \times K)$-equivariant. Denote $X := K^{\Gamma \backslash \Gamma * \Lambda}$ and denote by $x \mapsto {\overline{x}}$ the factor map of $X$ onto $X/K$. So we get the $\Lambda$-equivariant factor map ${\overline{\rho}}: X/K {\rightarrow}K^\Lambda/K : {\overline{\rho}}({\overline{x}}) = \overline{\rho(x)}$. We prove that $\Gamma * \Lambda {\curvearrowright}X/K$ together with ${\overline{\rho}}$ satisfies the abstract characterization of the co-induced action of $\Lambda {\curvearrowright}K^\Lambda/K$ to $\Gamma * \Lambda$.
For $g \in \Gamma * \Lambda$, denote by $|g|$ the number of letters from $\Gamma - \{e\}$ appearing in a reduced expression for $g$. Define the subsets $I_n \subset \Gamma * \Lambda$ given by $I_0 := \{e\}$ and $$I_n := \bigl\{g \in \Gamma * \Lambda \;\big| \; |g| = n \;\;\text{and the reduced expression of $g$ starts with a letter from $\Gamma - \{e\}$} \; \bigr\} \; .$$ Note that $\bigcup_{n=0}^\infty I_n$ is a right transversal for $\Lambda < \Gamma * \Lambda$. So we have to prove that $$\label{eq.mynicefam}
\{{\overline{x}}\mapsto {\overline{\rho}}(g \cdot {\overline{x}}) \mid n \in {\mathbb{N}}, g \in I_n \}$$ is a family of independent random variables that generate, up to null sets, the whole $\sigma$-algebra of $X/K$.
For $i \in \Gamma \backslash \Gamma * \Lambda$, we write $|i|=n$ if $i = \Gamma g$, where $|g|=n$ and the reduced expression for $g$ starts with a letter from $\Lambda - \{e\}$. For every $\lambda \in \Lambda - \{e\}$, define $\theta_\lambda : K^\Lambda/K {\rightarrow}K : \theta_\lambda({\overline{x}}) = x_e^{-1} x_\lambda$. Observe that for all $g \in I_n$ and $\lambda \in \Lambda-\{e\}$, we have $$\label{eq.consequence}
\theta_\lambda({\overline{\rho}}(g \cdot {\overline{x}})) = x_{\Gamma g}^{-1} \, x_{\Gamma \lambda g} \; .$$ Since $g \in I_n$ starts with a letter from $\Gamma-\{e\}$, we have $|\Gamma \lambda g| = |g| = n$, while $|\Gamma g| = n-1$. Write ${\mathcal{I}}_n := \{i \in \Gamma \backslash \Gamma * \Lambda \mid |i| \leq n\}$. We apply Lemma \[lem.indep\] to the countable set ${\mathcal{I}}_n - {\mathcal{I}}_{n-1}$, the direct product $$Z := K^{{\mathcal{I}}_{n-1}} \times K^{{\mathcal{I}}_n - {\mathcal{I}}_{n-1}}$$ and the family of maps ${\omega}_{g,\lambda} : Z {\rightarrow}K \times ({\mathcal{I}}_n - {\mathcal{I}}_{n-1})$, indexed by $g \in I_n, \lambda \in \Lambda - \{e\}$, only depending on the coordinates $x_i$, $i \in {\mathcal{I}}_{n-1}$, and given by $${\omega}_{g,\lambda} : x \mapsto (x_{\Gamma g}^{-1}, \Gamma \lambda g) \; .$$ Since the elements $\Gamma \lambda g$, for $g \in I_n,\lambda \in \Lambda - \{e\}$, enumerate ${\mathcal{I}}_n - {\mathcal{I}}_{n-1}$, it follows from Lemma \[lem.indep\] that the random variables $$\{X {\rightarrow}K : x \mapsto x_{\Gamma g}^{-1} x_{\Gamma \lambda g} \mid g \in I_n , \lambda \in \Lambda - \{e\} \; \}$$ are independent, only depend on the coordinates $x_i$, $|i| \leq n$, and are independent of the coordinates $x_i$, $|i| \leq n-1$. In combination with , it follows that is indeed a family of independent random variables.
Denote by ${\mathcal{B}}_0$ the smallest $\sigma$-algebra on $X/K$ for which all the functions ${\overline{x}}\mapsto {\overline{\rho}}(g \cdot {\overline{x}})$, $g \in \Gamma * \Lambda$, are ${\mathcal{B}}_0$-measurable. Formula and an induction on $n$ show that ${\overline{x}}\mapsto x_{\Gamma e}^{-1} \, x_i$ is ${\mathcal{B}}_0$-measurable for every $i \in \Gamma \backslash \Gamma * \Lambda$ with $|i|\leq n$. Hence, ${\mathcal{B}}_0$ is the entire $\sigma$-algebra on $X/K$.
Theorem \[thm.B\] will be an immediate corollary of the following general result.
\[thm.stability\] Let $\Gamma_i$, $i=0,1$, be countable groups and $K$ a nontrivial second countable compact group equipped with its normalized Haar measure. Assume that $\Gamma_i {\curvearrowright}K^{\Gamma_i} / K$ is isomorphic with the Bernoulli action $\Gamma_i {\curvearrowright}Y_i^{\Gamma_i}$ with base space $(Y_i,\mu_i)$. Write $G := \Gamma_0 * \Gamma_1$. Then $G {\curvearrowright}K^{G}/K$ is isomorphic with the Bernoulli action $G {\curvearrowright}(Y_0 \times Y_1)^{G}$ with base space $Y_0 \times Y_1$.
Put $A := K^{\Gamma_0}$ and denote by ${\alpha}$ the action $\Gamma_0 \times K \overset{{\alpha}}{{\curvearrowright}} A$ where $\Gamma_0$ shifts the indices and $K$ acts by diagonal left translation. Put $B := Y_0^{\Gamma_0} \times K$ and denote by $\beta$ the action $\Gamma_0 \times K \overset{\beta}{{\curvearrowright}} B$ where $\Gamma_0$ only acts on the factor $Y_0^{\Gamma_0}$ in a Bernoulli way and $K$ only acts on the factor $K$ by translation. Our assumptions say that $\Gamma_0 {\curvearrowright}A/K$ and $\Gamma_0 {\curvearrowright}B/K$ are isomorphic actions. We apply Theorem \[thm.coinduced\] to these two actions of $\Gamma_0$.
So denote $G = \Gamma_0 * \Gamma_1$ and denote by $G {\curvearrowright}{\tilde{A}}$, resp. $G {\curvearrowright}{\tilde{B}}$, the co-induced actions of $\Gamma_0 {\curvearrowright}A$, resp. $\Gamma_0 {\curvearrowright}B$, to $G$. Note that these actions come together with natural actions $K {\curvearrowright}{\tilde{A}}$ and $K {\curvearrowright}{\tilde{B}}$ that commute with $G$-actions. By Theorem \[thm.coinduced\], the actions $G {\curvearrowright}{\tilde{A}}/ K$ and $G {\curvearrowright}{\tilde{B}}/ K$ are isomorphic.
We now identify the actions $G \times K {\curvearrowright}{\tilde{A}}$ and $G \times K {\curvearrowright}{\tilde{B}}$ with the following known actions. First, the action $G \times K {\curvearrowright}{\tilde{A}}$ is canonically isomorphic with $G \times K {\curvearrowright}K^G$ where $G$ acts in a Bernoulli way and $K$ acts by diagonal left translation. Secondly, using Remark \[rem.bete\].3, the action $G \times K {\curvearrowright}{\tilde{B}}$ is isomorphic with the action $G \times K {\curvearrowright}Y_0^{G} \times K^{\Gamma_0 \backslash G}$ where $G$ acts diagonally in a Bernoulli way and $K$ only acts on the second factor by diagonal left translation. In combination with the previous paragraph, it follows that the action $G {\curvearrowright}K^{G}/K$ is isomorphic with the diagonal action $G {\curvearrowright}Y_0^G \times (K^{\Gamma_0 \backslash G})/K$.
From Lemma \[lem.factor\], we know that $G {\curvearrowright}(K^{\Gamma_0 \backslash G})/K$ is isomorphic with the co-induced action of $\Gamma_1 {\curvearrowright}K^{\Gamma_1}/K$ to $G$. Since we assumed that $\Gamma_1 {\curvearrowright}K^{\Gamma_1}/K$ is isomorphic with the Bernoulli action $\Gamma_1 {\curvearrowright}Y_1^{\Gamma_1}$, it follows that $G {\curvearrowright}(K^{\Gamma_0 \backslash G})/K$ is isomorphic with the Bernoulli action $G {\curvearrowright}Y_1^G$. In combination with the previous paragraph, it follows that $G {\curvearrowright}K^G/K$ is isomorphic with the Bernoulli action $G {\curvearrowright}(Y_0 \times Y_1)^G$.
Proof of Theorem \[thm.B\] {#proof-of-theorem-thm.b .unnumbered}
--------------------------
Since the action $\Lambda_i {\curvearrowright}K^{\Lambda_i} / K$ arises as the factor of a Bernoulli action and $\Lambda_i$ is amenable, it follows from [@OW86] that $\Lambda_i {\curvearrowright}K^{\Lambda_i} / K$ is isomorphic with a Bernoulli action $\Lambda_i {\curvearrowright}Y_i^{\Lambda_i}$. Repeatedly applying Theorem \[thm.stability\], it follows that $\Gamma {\curvearrowright}K^{\Gamma}/K$ is isomorphic with the Bernoulli action $\Gamma {\curvearrowright}(Y_1 \times \cdots \times Y_n)^\Gamma$.
The special case $\Gamma = {\mathbb{F}}_n$ is a very easy generalization of [@OW86 Appendix C.(b)]. Denote by $x \mapsto {\overline{x}}$ the quotient map from $K^{{\mathbb{F}}_n}$ to $K^{{\mathbb{F}}_n}/K$. Denote by $a_1,\ldots,a_n$ the free generators of ${\mathbb{F}}_n$. Define the measurable map $$\theta : K^{{\mathbb{F}}_n}/K {\rightarrow}(K \times \cdots \times K)^{{\mathbb{F}}_n} : \theta({\overline{x}})_g = (x_g^{-1} \, x_{a_1 g},\ldots, x_g^{-1} \, x_{a_n g}) \; .$$ We shall prove that $\theta$ is an isomorphism between ${\mathbb{F}}_n {\curvearrowright}K^{{\mathbb{F}}_n}/K$ and ${\mathbb{F}}_n {\curvearrowright}(K \times \cdots \times K)^{{\mathbb{F}}_n}$. First note that $\theta$ is indeed ${\mathbb{F}}_n$-equivariant. It remains to prove that $$\label{eq.myfamily}
\{{\overline{x}}\mapsto x_g^{-1} \, x_{a_i g} \mid i=1,\ldots,n \; , \; g \in {\mathbb{F}}_n \}$$ is a family of independent random variables on $K^{{\mathbb{F}}_n}/K$ that generate up to null sets the whole $\sigma$-algebra of $K^{{\mathbb{F}}_n}/K$. Denote by $|g|$ the word length of an element $g \in {\mathbb{F}}_n$. Define for $i \in \{1,\ldots,n\}$, ${\varepsilon}= \pm 1$ and $k \in {\mathbb{N}}$, the subsets $I^{i,{\varepsilon}}_k \subset {\mathbb{F}}_n$ given by $$I^{i,{\varepsilon}}_k := \bigl\{g \in {\mathbb{F}}_n \;\big|\; |g| = k \; , \; |a_i^{{\varepsilon}} g|=k+1 \bigr\} \; .$$ If $|g| = k$ and $|a_i g| = k-1$, we compose the random variable ${\overline{x}}\mapsto x_g^{-1} \, x_{a_i g}$ by the map $K {\rightarrow}K : y \mapsto y^{-1}$ and observe that $a_i g \in I^{i,-1}_{k-1}$. So we need to prove that $$\label{eq.secondfamily}
\{{\overline{x}}\mapsto x_g^{-1} \, x_{a_i^{\varepsilon}g} \mid i = 1,\ldots,n \; , \; {\varepsilon}= \pm 1 \; , \; k \in {\mathbb{N}}\; , \; g \in I^{i,{\varepsilon}}_k \}$$ is a family of independent random variables that generate up to null sets the whole $\sigma$-algebra of $K^{{\mathbb{F}}_n}/K$.
Write ${\mathcal{I}}_k = \{g \in {\mathbb{F}}_n \mid |g| \leq k\}$ and fix $k \in {\mathbb{N}}$. We apply Lemma \[lem.indep\] to the countable set ${\mathcal{I}}_{k+1}- {\mathcal{I}}_k$, the direct product $$Z := K^{{\mathcal{I}}_k} \times K^{{\mathcal{I}}_{k+1} - {\mathcal{I}}_k}$$ and the family of maps ${\omega}_{i,{\varepsilon},g} : Z {\rightarrow}K \times ({\mathcal{I}}_{k+1} - {\mathcal{I}}_k)$ indexed by the set $${\mathcal{F}}:= \{(i,{\varepsilon},g) \mid i = 1,\ldots,n \; , \; {\varepsilon}= \pm 1 \; , \; g \in I^{i,{\varepsilon}}_k \} \; ,$$ only depending on the coordinates $x_i$, $i \in {\mathcal{I}}_k$, and given by $${\omega}_{i,{\varepsilon},g} : x \mapsto (x_g^{-1}, a_i^{\varepsilon}g) \; .$$ Since the elements $a_i^{\varepsilon}g$ with $(i,{\varepsilon},g) \in {\mathcal{F}}$ enumerate ${\mathcal{I}}_{k+1} - {\mathcal{I}}_k$, it follows from Lemma \[lem.indep\] that $\{x \mapsto x_g^{-1} \, x_{a_i^{\varepsilon}g} \mid i=1,\ldots,n \; , \; {\varepsilon}= \pm 1 \; , \; g \in I^{i,{\varepsilon}}_k\}$ is a family of independent random variables that are independent of the coordinates $x_h$, $|h| \leq k$. By construction, these random variables only depend on the coordinates $x_h$, $|h|\leq k+1$. This being proven for all $k \in {\mathbb{N}}$, it follows that is a family of independent random variables. Hence the same is true for . These random variables can be easily seen to generate up to null sets the whole $\sigma$-algebra of $K^{{\mathbb{F}}_n}/K$.
Appendix: essentially free actions of locally compact groups {#appendix-essentially-free-actions-of-locally-compact-groups .unnumbered}
============================================================
A p.m.p. action of a second countable locally compact group $G$ on a standard probability space $(X,\mu)$ is an action of the group $G$ on the set $X$ such that $G \times X {\rightarrow}X : (g,x) \mapsto g \cdot x$ is a Borel map and such that for all $g \in G$ and all Borel sets $A \subset X$, we have $\mu(g \cdot A) = \mu(A)$.
For every $x \in X$, we define the subgroup ${\operatorname{Stab}}x$ of $G$ given by ${\operatorname{Stab}}x = \{g \in G \mid g \cdot x = x\}$. For the sake of completeness, we give a proof for the following folklore lemma.
\[lem.compact-free\] Let $G {\curvearrowright}(X,\mu)$ be a p.m.p. action of a second countable locally compact group $G$ on a standard probability space $(X,\mu)$, as above.
1. The set $X_0 := \{x \in X \mid {\operatorname{Stab}}x = \{e\} \; \}$ is a $G$-invariant Borel subset of $X$.
2. Assume that $\mu(X_0) = 1$ and that $G$ is compact. Denote by $m$ the normalized Haar measure on $G$. There exists a standard probability space $(Y_0,\eta)$ and a bijective Borel isomorphism $\theta : G \times Y_0 {\rightarrow}X_0$ such that $\theta(gh,y) = g \cdot \theta(h,y)$ for all $g,h \in G$, $y \in Y_0$, and such that $\theta_*(m \times \eta) = \mu$.
A p.m.p. action $G {\curvearrowright}(X,\mu)$ is called essentially free if the Borel set $\{x \in X \mid {\operatorname{Stab}}x = \{e\}\}$ has measure $1$.
By [@Va62 Theorem 3.2], there exists a continuous action of $G$ on a Polish space $Y$ and an injective Borel map $\psi : X {\rightarrow}Y$ satisfying $\psi(g \cdot x) = g \cdot \psi(x)$ for all $g \in G$ and $x \in X$. Since $\psi$ is injective, $\psi(X)$ is a Borel subset of $Y$ and $\psi$ is a Borel isomorphism of $X$ onto $\psi(X)$ (see e.g. [@Ke95 Theorem 15.1]). So, we actually view $X$ as a $G$-invariant Borel subset of $Y$.
To prove 1, fix a sequence of compact subsets $K_n \subset G - \{e\}$ such that $G-\{e\} = \bigcup_{n=1}^\infty K_n$. Also fix a metric $d$ on $Y$ that induces the topology on $Y$. Define $$f_n : X {\rightarrow}{\mathbb{R}}: f_n(x) = \min_{g \in K_n} d(g \cdot x,x) \; .$$ Whenever ${\mathcal{F}}_n \subset K_n$ is a countable dense subset, we have $f_n(x) = \inf_{g \in {\mathcal{F}}_n} d(g \cdot x,x)$, so that $f_n$ is Borel. Since ${\operatorname{Stab}}x = \{e\}$ if and only if $f_n(x) > 0$ for all $n$, statement 1 follows.
To prove 2, assume that $\mu(X_0) = 1$ and that $G$ is compact. Since $G$ acts continuously on $Y$ and $G$ is compact, all orbits $G \cdot y$ are closed. By [@Ke95 Theorem 12.16], we can choose a Borel subset $Y_1 \subset Y$ such that $Y_1 \cap G \cdot y$ is a singleton for every $y \in Y$. Define $Y_0 := Y_1 \cap X_0$. By construction, the map $$\theta : G \times Y_0 {\rightarrow}X_0 : \theta(g,y) = g \cdot y$$ is Borel, bijective and satisfies $\theta(gh,y) = g \cdot \theta(h,y)$ for all $g,h \in G$ and $y \in Y_0$. Then also $\theta^{-1}$ is Borel (see e.g. [@Ke95 Theorem 15.1]). The formula $\eta_0 := (\theta^{-1})_*(\mu)$ yields a $G$-invariant probability measure on $G \times Y_0$. Defining the probability measure $\eta$ on $Y_0$ as the push forward of $\eta_0$ under the quotient map $(g,y) \mapsto y$, the $G$-invariance of $\eta_0$ together with the Fubini theorem, imply that $\eta_0 = m \times \eta$.
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abstract: 'We describe digital tracking, a method for asteroid searches that greatly increases the sensitivity of a telescope to faint unknown asteroids. It has been previously used to detect faint Kuiper Belt objects using the Hubble Space Telescope and large ground-based instruments, and to find a small, fast-moving asteroid during a close approach to Earth. We complement this earlier work by developing digital tracking methodology for detecting asteroids using large-format CCD imagers. We demonstrate that the technique enables the ground-based detection of large numbers of new faint asteroids. Our methodology resolves or circumvents all major obstacles to the large-scale application of digital tracking for finding main belt and near-Earth asteroids. We find that for both asteroid populations, digital tracking can deliver a factor of ten improvement over conventional searches. Digital tracking has long been standard practice for deep Kuiper Belt surveys, but even there our methodology enables deeper integrations than have yet been attempted. Our search for main belt asteroids using a one-degree imager on the 0.9m WIYN telescope on Kitt Peak validates our methodology, delivers sensitivity to asteroids in a regime previously probed only with 4-meter and larger instruments, and leads to the detection of 156 previously unknown asteroids and 59 known objects in a single field. Digital tracking has the potential to revolutionize searches for faint moving objects ranging from the Kuiper Belt through main belt and near-Earth asteroids, and perhaps even anthropogenic space debris in low Earth orbit.'
author:
- 'Aren N. Heinze, Stanimir Metchev, and Joseph Trollo'
title: Digital Tracking Observations Can Discover Asteroids Ten Times Fainter than Conventional Searches
---
Introduction
============
Thousands of new asteroids are discovered every year by dedicated searches such as LINEAR [@Stokes2000], Spacewatch [@Gehrels1996], the Catalina Sky Survey [@Larson2007], NEAT [@Helin1997; @Pravdo1999], and others; by more general sky surveys such as Pan-STARRS [@Denneau2013] and WISE [@mainzer2012]; and by advanced amateur astronomers. The fundamental methodology of the detection is the same in all cases: an asteroid is identified as an object that changes its position in a systematic fashion relative to the starfield, and is detected individually in each of three to five different images [@Larson2007]. A good example of a highly successful automated implementation of this methodology is the tracklet creation module of the Pan-STARRS Moving Object Processing System (MOPS; Denneau et al. 2013).
The fundamental methodology described above implies that an asteroid can only be discovered if it is detected above some noise threshold on each one of a series of individual images. For a given telescope and detector, sensitivity to faint stars increases with exposure time $t$, being proportional to $\sqrt{t}$ in the typical, background-limited case. By contrast, sensitivity to faint asteroids generally ceases to increase for exposures longer than a maximum useful exposure time $\tau_M$, which is equal to the time an asteroid takes to move an angular distance corresponding to the resolution of the system being used to detect it. On images with exposures longer than $\tau_M$, asteroids blur out into streaks that fade into the background noise (see Figure \[fig:trackexamp\]). For typical asteroid-search telescopes delivering 1–1.5 arcsecond resolution, $\tau_M$ is about 2 minutes for main-belt asteroids (MBAs) at opposition. Kuiper Belt objects (KBOs) move more slowly and allow longer exposures of up to 15 minutes; near-Earth objects (NEOs) can move so fast that $\tau_M$ gets as short as one second [@Shao2014]. By contrast, 4–6 hour combined integration times are routinely used for imaging faint stars and galaxies using ground-based telescopes. The maximum useful exposure time thus imposes a severe handicap on our sensitivity to NEOs, MBAs, and even KBOs. Digital tracking overcomes the handicap imposed by the maximum useful exposure time. It enables the detection of previously unknown moving objects using integrations lasting up to eight hours (i.e., one entire night) or even spanning several nights in the case of KBOs. With digital tracking, sensitivity to faint moving objects increases as the square root of the integration time just as it does for stars and galaxies. Digital tracking represents a fundamentally different detection paradigm from that employed in MOPS [@Denneau2013] and all other major asteroid surveys. We first became aware of the method’s potential from reading the work of @Bernstein2004, who used it to detect extremely faint KBOs using long integrations with the Hubble Space Telescope. More recently, @Shao2014 and @Zhai2014 have developed the technique in a different context: detection of NEOs during close approaches using a specialized CCD imager with very fast readtime.
Such results have demonstrated the power of digital tracking, but they have only begun to exploit its potential for discovering new Solar System objects. We demonstrate this by developing and applying digital tracking methods in a new context: the use of large format, mosaic CCD imagers to detect hundreds of previously unknown asteroids in a single field. In Section \[sec:basic\] we describe the basics of digital tracking, and outline its capabilities and limitations. We briefly review past applications of the method in Section \[sec:overview\], in Sections \[sec:Data\]–\[sec:results\] we report our new methodology and results, and in Section \[sec:maincomp\] we discuss the advantages and disadvantages of digital tracking as compared to conventional surveys. We conclude in Section \[sec:conc\] with a summary of our results and their implications for future surveys.
![Usefulness of digital tracking. **Panel 1:** Single two-minute exposure showing a previously unknown asteroid (circled) and similar-brightness Star A. **Panel 2:** Median stack of 20 two-minute exposures registered on stars. The asteroid has blurred into a faint streak, while Star A and several fainter stars are clearly seen. Rectangular vs. circular apertures around the asteroid illustrate two different (both unsatisfactory) ways of trying to detect the streak. **Panel 3:** Stack of same 20 images digitally tracked to register on the asteroid. The asteroid is very bright, while stars fainter than Star A leave undetectable streaks. **Panel 4:** Like Panel 3 but with stars subtracted to allow rapid automatic detection of asteroids. \[fig:trackexamp\]](f01.eps)
How Digital Tracking Works {#sec:basic}
==========================
Here we will briefly outline the basics of digital tracking. Although we will use the word ‘asteroid’, the reader should take this as a generic term for a moving object in the sky, remembering that the technique is applicable not only to asteroids (both NEOs and MBAs) but also to KBOs, to comets, and to anthropogenic satellites or space debris.
Digital tracking begins with the acquisition of a large number of images with individual exposure times $ \le \tau_M$. These images should all target approximately the same position in the sky, and should be obtained consecutively (or, at least, all within a limited time span; see Section \[sec:skymotion\]). We then parameterize the space of possible asteroid motions (which we will refer to as ‘angular motion phase space’) in such a way that we can define a region that contains the motions of the asteroids of interest. We search this region of angular motion phase space using a finely spaced grid of sampling points. Each point corresponds to a possible asteroid trajectory that is traced out in the time spanned by the acquisition of the images, and that is fully specified except for an arbitrary translation that allows it to begin or end anywhere in the field of view covered by the images. For each of these sampling points in angular motion phase space, we create a separate, sigma-clipped median stack of shifted input images (a ‘trial stack’), where the shifts applied to each image are calculated to correctly register any asteroids moving on the trajectory corresponding to that sampling point. Such an asteroid will appear as a point of light on the stacked image, while stars, galaxies, and asteroids with different motions will blur into streaks and fade into the background[^1]. The final step in digital tracking is therefore the identification of point sources on the trial stacks, each of which corresponds to an asteroid. Accurate measurements of the position, motion, and brightness of such asteroids are natural products of the detection process.
Two important facts render our specific implementation of digital tracking practical. First, the read noise of modern CCDs is negligible in the context of broadband optical imaging to detect asteroids, allowing us to stack large numbers of short-exposure images and obtain the same sensitivity as a single equivalent long exposure. It is this property that makes digital tracking far superior to other ways of detecting faint asteroids (Section \[sec:noise\]). Second, for observations obtained from Earth’s surface, the motions of asteroids and KBOs can be usefully approximated as linear motion in the plane of the sky, at constant angular velocity (Section \[sec:skymotion\]). This allows us to use a simple two-dimensional parameterization of the angular motion phase space, as illustrated in Figure \[fig:drift\].
![Angular motion phase space for our observations on April 19, 2013. The measured angular velocities of asteroids detected in our data are shown as black hexagons. The gray squares plotted to suggest the locus of main belt asteroids are known objects over a much larger region of the sky (a disk 6 degrees in diameter) centered on our field at the time of our observations. These motions were obtained from the Minor Planet Center. The rectangular outline is the boundary of the region of angular motion phase space that we have searched. Note that we have no detections in the corner regions far from the locus of known asteroids. This provides evidence that our false positive rate is extremely low, as we further demonstrate through the tests described in Sections \[sec:mancheck\] and \[sec:falsepos\]. \[fig:drift\]](f02.eps)
Sensitivity of Various Detection Methods {#sec:noise}
----------------------------------------
We will now consider how the sensitivity of asteroid-search observations depends on integration time for different detection methods, and demonstrate that alternative methods to digital tracking do not overcome the sensitivity limit imposed by the maximum useful exposure. We note that @Shao2014 have presented a similar analysis, illustrated especially by their Figures 5–6.
We will refer to the cumulative exposure time of a stack of images as the *integration time*, while we will use *exposure time* to apply strictly to a single image. Thanks to the low read noise of modern CCDs, there is no significant noise penalty for dividing a long integration up into many shorter exposures. A digital stack of sixty images each taken with a one-minute exposure has the same sensitivity as a single image taken with a one-hour exposure. In fact, stacks of short exposures usually have superior sensitivity because they are not plagued by artifacts from severely saturated stars, and because cosmic rays can be rejected by sigma-clipping. The dominant source of noise in broad-band images with astronomical CCDs is the Poisson noise of the sky background. Let the sky background have a surface brightness of $F_{sky}$ in photons/s/arcsec$^2$, while a point source (star or asteroid) has a brightness of $F_{source}$ in photons/s. The source is to be detected based on its flux within an aperture of radius $r$ in arcseconds. The optimal value of $r$ for the detection and accurate measurement of faint sources is usually comparable to half the full width at half-maximum (FWHM) of the telescope’s point spread function (PSF). Note that $r$ remains a meaningful concept even when sources are being detected using using PSF profile-fitting rather than aperture photometry: the detection is still effectively performed within a roughly circular region corresponding to the bright core of the PSF. In either case, where readnoise and dark current are negligible, the signal-to-noise (SNR) level at which effectively stationary objects are detected in integration lasting $t$ seconds is given by:
$$SNR(t) = \frac{t F_{source}}{\sqrt{\pi r^2 t F_{sky}}} \propto \sqrt{t}
\label{eq:snr01}$$
This equation describes the detection of stars on Panel 1 of Figure \[fig:trackexamp\], and of the asteroid since two minutes does not exceed the maximum useful exposure time $\tau_M$. Note that the circles drawn around the asteroid in the figure do not correspond to the optimal photometric radius $r$: they are exaggerated for clarity. Equation \[eq:snr01\] accurately predicts that on Panel 2, where 20 images are stacked for an effective 40-minute integration, the SNR of stellar images (or, equivalently, the minimum stellar flux that can be detected) is improved by a factor of $\sqrt{20}$. The same is not true of the moving asteroid, which is elongated into a streak on this panel.
The circular aperture in Panel 2 illustrates how one could measure the SNR of the asteroid’s streaked image within an aperture with the same optimal radius $r$ used to detect un-blurred point sources. The SNR for detecting the asteroid in this way is:
$$SNR(t) = \frac{\tau_M F_{source}}{\sqrt{\pi r^2 t F_{sky}}} \propto t^{-1/2}
\label{eq:snr02}$$
for $t > \tau_M$. Equation \[eq:snr02\] accurately predicts that for $t > \tau_M$, the SNR of any given PSF-sized portion of the asteroid trail actually *decreases* with increasing integration time. This is seen in Figure \[fig:trackexamp\]: the streak in Panel 2 is a less obvious signal than the point source in Panel 1. However, Panel 2 also illustrates an alternative way of measuring the asteroid: one could use an elongated aperture with width 2$r$ and length $(t/\tau_M) 2r$ to encompass the whole streak. The SNR of this type of measurement is:
$$SNR(t) = \frac{t F_{source}}{\sqrt{4 r^2 (t/\tau_M) t F_{sky}}} \propto t^{0}.
\label{eq:snr03}$$
Since the SNR does not decrease with increasing integration time, this method is an improvement on that described by Equation \[eq:snr02\]. It is the best way of detecting asteroids on images with exposures longer than $\tau_M$. However, it still indicates that asteroids, unlike stars, cannot be detected with increasing sensitivity using exposures of increasing length. By contrast, digital tracking restores the $\sqrt{t}$ dependence of Equation \[eq:snr01\] for moving objects on integrations with lengths of up to hundreds of times $\tau_M$.
The Approximation of Linear Motion at Constant Velocity {#sec:skymotion}
-------------------------------------------------------
Digital tracking is possible even for objects that exhibit nonlinear sky motions [@Bernstein2004], but for such objects it becomes more computationally challenging, especially for the high data volumes produced by large format CCD imagers. To keep the analysis computationally tractable, our current implementation uses the approximation of linear motion at a constant velocity. This means that the angular motion phase space we search has only two dimensions, which we parameterize in terms of on-sky angular velocities in right ascension (RA) and declination (DEC), as illustrated by Figure \[fig:drift\]. The price of simplifying the angular motion phase space in this way comes in the form of a maximum duration $\tau_{lin}$ for digital tracking integrations targeting a given population of objects. The limit is reached when the on-sky tracks of the target objects deviate from the best-fit linear, constant velocity approximation by one resolution element (e.g., one arcsecond). Thus, while the maximum useful individual exposure $\tau_M$ is set by the angular velocity itself, $\tau_{lin}$ is set by the first time derivative of the angular velocity, corresponding to curvature and/or acceleration in the sky motion of the target object. As we determine below, $\tau_{lin}$ is large relative to $\tau_M$ for all target populations. The linear, constant velocity approximation is sufficient for very long, sensitive digital tracking integrations.
For NEOs and MBAs, the dominant cause of acceleration and curvature in on-sky tracks is the Earth’s rotation. This imposes a 24-hr sinusoidal variation in the asteroid’s on-sky velocity, and causes a waviness its trajectory with the same period (Figure \[fig:trochoid\]). The amplitudes of the acceleration and curvature due to Earth’s rotation are inversely dependent on the asteroid’s distance from the Earth, since they are angular effects produced by the observer’s motion. For MBAs, the acceleration and curvature are weak enough that $\tau_{lin}$ is always longer than the $\sim 8$ hr limit imposed on digital tracking integrations by the ordinary observing considerations of twilight and high airmass[^2]. Eight-hour integrations are therefore possible on MBAs with residuals considerably less than one arcsecond (Figure \[fig:skymotions\]), and the maximum integration time for these objects will be set by twilight or airmass rather than $\tau_{lin}$. This is true for MBAs regardless of viewing geometry: it applies equally to objects at opposition and far from opposition.
All-night integrations are not possible for NEOs making close approaches to the Earth, but 1–2 hr integrations work well up to very close distances. The acceleration and curvature induced by Earth’s rotation both reach a stationary point when the asteroid transits the observer’s meridian: that is, at zero hour angle. Thus, digital tracking integrations on NEOs can be longer if the observations are centered on zero hour angle. For a closely-approaching NEO at a distance of 0.1 AU from Earth, a two hour integration is possible if centered on zero hour angle, as shown in Figure \[fig:skymotions\]. For observations far from the meridian, $\tau_{lin}$ drops to about one hour.
KBOs are sufficiently distant that the diurnal oscillation imposed on their tracks by Earth’s rotation has an amplitude considerably less than 1 arcsecond, as shown in Figure \[fig:skymotions\]. The dominant cause of acceleration and curvature in the on-sky tracks of KBOs is therefore the curvature of Earth’s orbit. It follows that $\tau_{lin}$ is greater than 24 hours: digital tracking integrations spanning multiple nights are possible. The effects from Earth’s orbit reach a stationary point for objects exactly at opposition, so KBO integrations centered on opposition can last up to twelve nights (Figure \[fig:skymotions\]). One month after opposition, $\tau_{lin}$ shortens to three nights. To sum up, the $\tau_{lin}$ values for NEOs, MBAs, and KBOs are all hundreds of times longer than $\tau_M$ for the corresponding objects. Since the sensitivity gain from digital tracking observations is a factor of $\sqrt{t/\tau_M}$, digital-tracking observations obeying the requirement that $t < \tau_{lin}$ can detect objects at least ten times fainter than conventional searches with the same telescope and imager at the same SNR level.
Numbers of Trial Vectors to Search for MBAs, NEOs, and KBOs {#sec:points}
-----------------------------------------------------------
The computational challenge of digital tracking depends on the number of trial points in angular motion phase space that must be probed in the analysis. In the context of our approximation of linear, constant-velocity motion, each point in angular motion phase space corresponds to a two dimensional vector; hence we will also refer to them as trial vectors. We have already introduced the term ‘trial stack’ for the image stack corresponding to a given trial vector. The required number of trial vectors is determined by the size of the region in angular motion phase space that is to be explored and the maximum permissible spacing between sampling points (vectors) in this region. We consider the spacing of the sampling grid and the size of the region of angular motion phase space in turn. We restrict the current discussion to the case of linear motion at a constant velocity as described in Section \[sec:skymotion\].
Trial vectors in the resulting two-dimensional angular motion phase space should be spaced finely enough that every object will be imaged in at least one trial stack with a blur length less than some maximum acceptable value $b_{max}$, which will normally be about one arcsecond. Let the grid spacing in angular motion phase space be $\Delta_m$ and the total time spanned by the integration be $t_{int}$ (note that observational overheads always make this somewhat longer than the cumulative integration time $t$). The worst possible blur will be for an object centered between four grid points. Its blur length on each of the four corresponding trial stacks is identical and is equal to $t_{int} \Delta_m/\sqrt{2}$. The optimal grid spacing is therefore:
$$\Delta_m = \frac{\sqrt{2} \; b_{max}}{t_{int}}
\label{eq:blur}$$
While the grid spacing depends only on the temporal span $t_{int}$ of the digital tracking integration, the size of the grid that should be explored depends on the type of object being sought. We will now consider the requirements for targeting MBAs, NEOs, and KBOs.
### Trial Vectors for Main Belt Asteroids {#sec:mbatrial}
MBAs near opposition are in their retrograde loops and thus have negative (westward) motions in RA, with speeds between 20 and 50 arcsec/hr. Their motions in DEC depend on their orbital inclinations and the trendline of the ecliptic (e.g., the DEC motions skew northward near 12:00 RA, southward at 00:00, and have mean zero at 06:00 or 18:00). In any case, a range of $\pm16$ arcsec/hr centered on the mean value encompasses most of them (however, as illustrated by Figure \[fig:drift\], we choose a slightly larger north-south range when analyzing our April 2013 observations, to increase the chance of including interesting outliers). For a more typical north-south range, the rectangular region of angular motion phase space to be explored has a size of about 30$\times$32 arcsec/hr. If we consider an eight-hour digital tracking integration, with $b_{max}=1$ arcsec, the grid spacing $\Delta_m$ is 0.18 arcsec/hr and the number of points required to span the grid is 31,000. Where each point requires the stacking of a few hundred large-format CCD images, the computational demand is considerable — but nonetheless feasible even with modern desktop workstations.
As an example of the computational requirements, the April 2013 test survey we describe in Sections \[sec:Data\]–\[sec:results\] consists of two data sets of 126 and 130 images. After processing, each image is 12,000 pixels square and thus totals 144 megapixels, where each pixel is 32 bits. Our digital tracking analysis probed 28,086 distinct trial vectors for each data set (equivalently, 112,344 trial vectors per data set with the images split into quadrants as described in Section \[sec:digipars\]). We can quantify the processing task in terms of ‘vector-pixels’, where the number of vector-pixels is defined as the number of input images times the number of pixels per image times the number of trial vectors. Our test survey probed $1.035\times10^{15}$ vector-pixels (i.e. $1.44\times10^8$ pixels per image times 256 total images times 28,086 trial vectors). Using a 32-core desktop workstation, our processing rate was $8.1\times 10^{11}$ vector-pixels per hour, allowing us to process the entire two-night survey in about 50 days. For comparison, the Tightly Coupled System (TCS) supercomputer at the SciNet HPC Consortium[^3] (funded by the Canada Foundation for Innovation), to which we have access for future work, has well over 100 times the processing power of our desktop and could re-analyze the entire two-night survey in only a few hours.
### Trial Vectors for Near-Earth Objects: A Statistical Survey {#sec:neostat}
We will present two examples of hypothetical digital tracking surveys targeting NEOs within 0.25 AU of the Earth. Both surveys involve digital tracking integrations lasting one hour or less, as is appropriate for NEOs (Section \[sec:skymotion\]), and hence more than one digital tracking integration can be obtained each night.
The first hypothetical NEO survey is presented in this Section, and is aimed at detecting and calculating approximate orbits for extremely small NEOs to probe their population statistics. Most of these objects will be too faint for extensive followup observations, which is acceptable because the 2–4 day span of the survey itself will yield orbits sufficiently accurate for statistical analysis. Also, the faint objects will be too small to pose a significant impact risk to humans on Earth, so long-term followup observations to predict impacts or retire impact risk are unnecessary. A second hypothetical survey, this time focused on risk retirement, is described in Section \[sec:risk\].
NEOs can have a wide range of possible motions depending on their distances and Earth-approach geometries. Our hypothetical statistical survey will target asteroids close to Earth, at distances between 0.1 and 0.25 AU. This is a challenging case due to the fast sky velocities of such closely-approaching objects, and therefore it provides a good example both of the limitations and the power of digital tracking.
We have obtained ephemerides for known NEOs from the Minor Planet Center (MPC) to determine the appropriate range of angular motion phase space for our hypothetical survey. Using the full list of 1,915 numbered NEOs (Amors, Atens, and Apollos) given by the MPC as of June 2015, we search for Earth encounters at distances between 0.1 and 0.25 AU where the celestial coordinates of the asteroids involved are within 20$^{\circ}$ of the antisolar point[^4]. We confine our search to the years 2000 through 2020, and use an ephemeris sampling interval of one week. We find 699 encounters, involving 246 distinct asteroids, which satisfy our criteria. In this sample of 699 encounters, the median on-sky velocity of the asteroids is 140 arcsec/hr and the 90th percentile velocity is 340 arcsec/hr. We note that if we do not confine the sample to encounters where the asteroid is near the antisolar point, the median and 90th percentile angular velocities go up to 200 arcsec/hr and 460 arcsec/hr, respectively. We suspect this is due to the inclusion of more high-inclination objects that are rarely found near the ecliptic or the antisolar point, and hence are less likely to appear in the initial sample. A target field near the antisolar point (hence detecting asteroids that are near opposition) is strongly to be preferred for digital tracking observations, because asteroid phase functions and observability constraints result in considerably lower sensitivity with any other targeting. The angular velocities from the sample of encounters that is restricted to large solar elongation are therefore more representative of those in a real digital-tracking survey of NEOs, and we will use them to calculate the number of trial vectors.
The maximum useful exposure ($\tau_M$) is typically about 30 seconds for objects moving at 140 arcsec/hr (given 1.2 arcsecond seeing), and 13 seconds at 340 arcsec/hr. The read times for the current generation of large-format imagers (e.g. 20 seconds for DECam at Cerro Tololo; Flaugher et al. 2012) render observations with exposures much shorter than 30 seconds quite inefficient due to readout overhead. Thus, rather than adopting an exposure time that is less than or equal to $\tau_M$ for all objects in our target sample, our hypothetical NEO survey must use an exposure time that is a compromise between inefficiency due to readout overhead on the one hand, and reduced sensitivity due to trailing losses on the other. Even for fast-moving objects whose trailing losses are considerable, digital tracking will still yield a large increase in sensitivity over conventional methods. Given an exposure time of 30 seconds (less than $\tau_M$ for half the target population), trailing losses will amount to only about 0.6 magnitudes for objects moving at 340 arcsec/hr. This is calculated by comparing Equations \[eq:snr01\] and \[eq:snr03\]: the loss in SNR due to trailing is a factor of $\sqrt{(4/\pi)(t/\tau_M)}$. In the present case we have $\sqrt{(4/\pi)(30/13)}$ = 1.71, which is 0.59 magnitudes.
The digital tracking analysis for our hypothetical, statistical NEO survey should then search a region of motion space described by a circular disk of radius 340 arcsec/hr centered on the origin. Such a search will motion-match 90% of NEOs at geocentric distances between 0.1 AU and 0.25 AU, and will also deliver sensitivity to slower-moving objects at even smaller distances. Given a one-hour digital tracking integration with $b_{max}=1$ arcsec and thus $\Delta_m$ equal to 1.4 arcsec/hr, 185,000 trial points are required. Given 120 images of 100 megapixels each, the computational requirement is $2.22 \times 10^{15}$ vector-pixels, and would take four months with our multi-core desktop but only about 24 hours with SciNet’s TCS supercomputer. This is for a digital tracking integration lasting only one hour. Since several such integrations could be obtained each night, and orbital statistics might require two or more nights, access to a supercomputer is a practical necessity for a statistical survey of this type.
Future generations of large-format CCD imagers will have shorter readout times and may thus enable digital tracking searches to use very short (1–5 sec) exposures and probe faster-moving NEOs for which $\tau_M$ is only a few seconds [@Shao2014]. However, targeting faster moving objects with shorter exposure times greatly increases computational requirements, if the length of the digital tracking integration is held fixed. Using a small-format but fast-readout CCD, Zhai et al. (2014) have demonstrated that interesting sensitivity regimes may be probed for very fast-moving objects even with quite short digital tracking integrations. The potential sensitivity of long (e.g. one-hour) digital tracking integrations such as we have proposed in this section will remain even greater, and we may hope that supercomputer capabilities will continue to increase in parallel with the development of next-generation CCD imagers with large formats and fast readtimes. If computer capabilities increase sufficiently, long digital tracking integrations targeting very fast-moving NEOs may be computationally feasible by the time large-format detectors capable of observing with the required very short exposures are available.
Digital tracking analyses requiring days or even weeks of computer time to process each night’s data are acceptable for surveys aimed at probing the statistics of an asteroid population. The science goals of such a survey do not require followup imaging by other observatories, nor the calculation of sufficiently precise orbits to allow long-term recovery observations or impact prediction. Such surveys will, however, generally make at least a serendipitous contribution to long-term orbit calculation for some objects, by producing recoveries (or precoveries) at relatively large distances of objects that were (or will be) discovered by other, less sensitive surveys during closer approaches to Earth. To reap the benefits of such detections, it is very desirable that all asteroid detections should be reported to the MPC, even in the case of a statistical survey.
### Trial Vectors for Near-Earth Objects: A Risk-Retirement Survey {#sec:risk}
The processing times discussed in Section \[sec:neostat\] above are too long for an NEO survey aimed at calculating precise orbits, making long-term impact predictions, and retiring terrestrial impact risk. Such surveys should run continuously and process all data promptly in order to give other observers the opportunity to followup potentially hazardous asteroids and refine their orbits. The data from each night should therefore be processed before the next night’s observations. Several surveys of this type are currently underway (for example, Gehrels & Jedicke 1996; Helin et al. 1997; Pravdo et al. 1999; Stokes et al. 2000; Larson 2007), but none use digital tracking at present. We will now explore what role digital tracking could play, if any, in such a survey.
We will consider a system like that of the Catalina Sky Survey, which uses a 16 megapixel CCD to cover an 8.2 square-degree field of view with 2.6-arcsecond pixels, with a standard exposure time of 30 seconds [@Larson2007]. Although the acquisition of a dedicated supercomputer would be a reasonable step for a survey of this type transitioning to a digital tracking mode, we will assume that only a more modest processing capacity equal to twelve times that of our own desktop workstation is available. Twelve such computers would cost about about \$60,000 in 2015, and would enable the processing of $10^{13}$ vector-pixels per hour.
Like that described in Section \[sec:neostat\], our hypothetical risk-retirement survey will use digital tracking integrations spanning one hour or less, and will acquire several integrations per night. However, while the integration length of one hour used in Section \[sec:neostat\] was set by $\tau_{lin}$ (see Section \[sec:skymotion\]), here the integration length will be set by the limits on processing power. We will assume that 8 hours of useful data are acquired each night and that computer processing of this data runs continuously 24 hours each day. Thus, 24 hours are available to process each 8 hours of data, and processing of each digital tracking integration must take no longer than three times as long as data acquisition.
We will now attempt to determine an optimum value for the length of an individual integration. Specifically, we will seek to optimize $n$, the number of images per digital tracking integration. The larger the number of images, the longer the integration and the greater the sensitivity (which scales as $\sqrt n$). On the other hand, a longer integration time $t_{int}$ also requires a finer sampling interval $\Delta_m$ for trial vectors in angular motion phase space (Equation \[eq:blur\]). A finer sampling interval means the available compute power is only sufficient to cover a smaller region of angular motion phase space, which means we will detect a smaller fraction of the target asteroids. Additionally, a longer integration time $t_{int}$ on each field means fewer fields can be covered each night.
We find that if the readout time and other overheads are negligible, $n=40$ appears as a good compromise. This increases the sensitivity by a factor of 6.3 (2.0 magnitudes) relative to single exposures, and the digital tracking analysis could motion-match over half the asteroids within 0.05 AU and 90% of those between 0.05 and 0.1 AU. These statistics are calculated using ephemerides from the MPC for 1,098 close encounters of known asteroids between 2000 and 2020, and unlike those used in Section \[sec:neostat\] they are not restricted to encounters where the asteroid is near the anti-solar point.
The 40 images, each with a 30-second exposure, would be taken over a time interval of 20 minutes. As each image has 16 megapixels, the total number of pixels is $6.4 \times 10^8$. Three times the integration time, or 1 hour, is available to process the data, meaning that $10^{13}$ vector-pixels may be processed and thus $10^{13}/6.4 \times 10^8 = 15,625$ trial vectors may be probed. Given the 2.6-arcsecond pixels of the detector, we set the allowable blur $b_{max}$ to 3.0 arcsec. Equation \[eq:blur\] then gives a sampling interval $\Delta_m$ of 14.2 arcsec/hr. Given this sampling interval, with 15,625 trial vectors we can probe a circular region of radius 1,000 arcsec/hr in angular motion phase space. Centering this region on the origin, it follows that all asteroids with angular velocities slower than 1,000 arcsec/hr can be detected. Based on the statistics of known NEOs, half of all asteroids within 0.05 AU of Earth; 90% of those between 0.05 and 0.1 AU, and 99% of those between 0.1 and 0.2 AU are moving slower than this.
To facilitate the calculation of orbits and confirm lower-significance detections, our hypothetical NEO survey should obtain at least two digital tracking integrations of each field per night: that is, 80 images per field per night with the parameters described thus far. The Catalina Sky Survey currently acquires only four images per field per night. Transitioning to a digital tracking mode would therefore reduce the survey’s nightly sky coverage by a factor of twenty. On the other hand, it would create sensitivity to a large number of asteroids that are currently completely undetectable. As more and more of the large NEOs are discovered and their impact risk is retired through accurate orbit calculation, digital tracking may present an avenue for current surveys to go on detecting new objects which, though smaller than the surveys’ original targets, remain large enough to pose a regional impact hazard to humans on Earth.
The survey parameters described above could be adjusted to accommodate non-negligible readout overheads and/or a need to detect faster-moving objects. For example, if the readout overhead were 10 seconds, 34 images could be acquired over a slightly longer interval of 22.7 minutes, and all asteroids with angular speeds below 1,000 arcsec/hr could still be detected in the 68.1 minute processing time available for the digital tracking analysis. The sensitivity increase relative to single exposures would be $\sqrt 34 = 5.8$, or 1.9 magnitudes. Alternatively, to detect all asteroids with motions up to 2,000 arcsec/hr (a threshold which includes 88% of those within 0.05 AU), we could stack just 17 images, and the sensitivity increase would be a factor of 4.1 (1.5 magnitudes).
Asteroids moving at 2,000 arcsec/hr will be 17-arcsecond streaks on individual 30-second exposures. With 2.6 arcsecond pixels, this corresponds to a sensitivity reduction of about 1.1 magnitudes due to trailing losses[^5], regardless of whether digital tracking is used. New-generation imagers with very short readout times will allow efficient observations with exposures much shorter than 30 seconds. These will be able to detect faster-moving asteroids with reduced trailing losses, at the expense of increased computational requirements for digital tracking integrations. As we indicated in Section \[sec:neostat\], it is possible that the development of faster computers will keep pace with that of fast-readout astronomical imagers, and thus cutting-edge digital tracking integrations will become more rather than less tractable over time. A dedicated supercomputer may still be required for a risk-retirement survey seeking to push to the limits of detection for faint, fast-moving NEOs.
### Trial Vectors for Kuiper Belt Objects
The sky motions of KBOs are slow and are dominated not by their own space velocities but by the much faster orbital motion of the Earth. The region to be searched is thus quite small. If we seek objects ranging from 30 to 100 AU in geocentric distance, in orbits of arbitrary eccentricity with (non-retrograde) inclinations up to $90 \degree$, the motions of all target objects in this very generous dynamical range can still be enclosed in a rectangular region of motion space with dimensions 3.7 arcsec/hr by 1.3 arcsec/hr (see the derivation in Parker and Kavelaars 2010). For a three-night integration with $b_{max}=1$ arcsec, only 8,000 points are required to search this region. Even the twelve-night integration we found to be possible in Section \[sec:skymotion\] would require just 180,000 points. While this is comparable to the number of trial vectors required for one of the NEO integrations described in Section \[sec:neostat\], the NEO integrations spanned one hour, as compared to 12 days for the Kuiper Belt. Thus, the computational requirements are relatively modest even for the most ambitious digital tracking integrations targeting KBOs.
Comparing Digital Tracking Parameters for NEOs, MBAs, and KBOs {#sec:comp3}
--------------------------------------------------------------
We have considered digital tracking as applied to three very different target populations: NEOs, MBAs, and KBOs, listed here in order of decreasing angular velocity. In Section \[sec:skymotion\] we calculated $\tau_{lin}$, the maximum duration of a digital tracking integration using the approximation of linear motion at constant angular velocity. We found that for all three populations, $\tau_{lin}$ is hundreds of times longer than the maximum useful exposure $\tau_M$ for single images. Thus, digital tracking offers a potential sensitivity increase of at least a factor of 10 for all three populations. In Section \[sec:points\], we found that the limits on digital tracking integration lengths imposed by $\tau_{lin}$ for each class of objects also translate into digital tracking analyses with numbers of trial vectors that are not greatly different: roughly 30,000 for MBAs, and a range of about 10,000 – 200,000 for both NEOs and KBOs depending on the survey parameters. The computational requirements are greatest for NEOs because a single survey should include many, relatively short digital tracking integrations — but even for NEOs, the required processing appears to be within the capabilities of modern clusters or supercomputers.
To give concrete examples, digital tracking can be used to target fast moving NEOs with a 1-hour integration composed of 120 individual 30-second exposures[^6]; MBAs using a 6-hour integration composed of 180 two-minute exposures; or KBOs near opposition using a 36-hour integration composed of 216 ten-minute exposures acquired over six nights. The computational requirement is the same to within a factor of a few for each of these integrations, and would require less than a day of supercomputer time. Each integration will detect objects more than ten times fainter than conventional observations using the same telescope and SNR threshold. Although conventional surveys can typically adopt a lower SNR threshold (e.g. 5$\sigma$ for @Denneau2013 versus 7$\sigma$ for our own observations; see Section \[sec:comp\]), this reduces the advantage of a digital tracking survey only slightly. For example, an object detected at 7$\sigma$ on a stack of 180 images is $\sqrt{180} = 13.4$ times fainter than an equal-significance detection on a single image, and remains 9.6 times fainter even than a 5$\sigma$ detection on a single image.
Digital tracking can also increase the sensitivity of surveys aimed at retiring terrestrial impact risk by discovery and prompt followup of previously unknown NEOs. For such surveys, the need for rapid data processing restricts the length of digital tracking integrations: e.g., only 15-40 images would be stacked per integration (Section \[sec:risk\]). This reduces the likely sensitivity increase from digital tracking to factors of four to six, rather than ten to fifteen for statistical surveys stacking over a hundred images. Even with these limitations, digital tracking will still bring numerous previously undetectable asteroids into range.
Review of Past Work {#sec:overview}
===================
Kuiper Belt Objects and Outer Planet Moons
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@Tyson1992 are to be credited with the first application of digital tracking to Kuiper Belt searches, even though they did not detect any objects. There followed several other results in which a few KBOs were detected in deep, small-field digital tracking searches using large telescopes. @Gladman1998 used the CFHT (Mauna Kea) and the 5m Hale Telescope (Palomar) to search 0.025 to 0.13 deg$^2$ per night for ten nights, finding 5 KBOs in all; @Luu1998 used the 10m Keck Telescope (Mauna Kea) to search 0.01 deg$^2$ per night for three nights and found one object; and @Chiang1999 used Keck to search 0.01 deg$^2$ over a single night and found two objects. All of these surveys used the linear, constant-velocity approximation for the motions of their objects, and searched between 20 and 100 distinct trial stacks. All of them also parameterized the motion space to be searched by the total angular velocity and its position angle, rather than by the eastward and northward components of the velocity vector. The former is appropriate in the context of the Kuiper Belt surveys, but the latter is preferable for the larger areas of motion space that must be searched in order to target asteroids.
As larger-format CCD imagers became available on large telescopes in the early 2000’s, digital tracking surveys targeting KBOs put them to use. @Allen2001 used the 4-meter Blanco telescope to carry out the first (and, thus far, the only) survey to attempt digital tracking integrations spanning more than one night. Because they also targeted three fields per night, their two-night integrations did not attain greater sensitivity than would a single night’s observations targeting only one field; nevertheless their success proves the feasibility of multi-night integrations. Their digital tracking analysis used thousands of trial stacks, in order to search their angular motion phase space with sufficiently fine sampling to register the images of moving objects across a two-night span. Covering about 0.5 deg$^2$ per pair of nights over three different pairs, they detected 24 new objects. Other observers confined their digital tracking integrations to one night, but nevertheless obtained excellent results using the new large-format imagers. @Gladman2001 used the CFHT and the 8-meter VLT for two nights each, but analyzed the data from each night independently, detecting several new KBOs including some beyond 48 AU. @Fraser2008 used the CFHT and the 4-meter Blanco Telescope to cover 0.32–0.85 deg$^2$ per night over a total of nine nights. Using only 25 digital tracking trial stacks for each night’s data, they covered a total of 3 deg$^2$ and discovered 70 new KBOs. @Fraser2009 used the 8-meter Subaru Telescope (Mauna Kea) with its new Suprime-cam imager for a digital tracking search of 0.33 deg$^2$ of sky over two nights, finding 36 new KBOs. @Fuentes2009 used archival data from the same instrument, combined with a digital-tracking analysis probing 736 trial stacks, to find 20 KBOs as faint as $R=26.8$. @Kavelaars2004 and @Holman2004 used digital tracking observations with the CFHT and the Blanco Telescope to find new irregular satellites of Uranus and Neptune, respectively, with each search covering of order 1 deg$^2$.
All of the above digital tracking surveys used the approximation of linear motion at a constant velocity. Except for @Allen2001, none required more than a few hundred digital tracking trial stacks to search the appropriate region in motion space. By contrast, @Bernstein2004, building on an earlier attempt by @Cochran1995, used the Hubble Space Telescope to search a multidimensional angular motion phase space requiring about nine million digital tracking trial stacks per field. Covering a total area of 0.02 deg$^2$ with an effective integration time of 10.6 hr, they detected three new KBOs, including one at magnitude $m_{\mathrm{606W}}=28.38$ which remains the faintest Solar System object ever accurately measured.
Asteroids
---------
None of the above digital tracking observations targeted objects closer than the orbit of Uranus. Except for @Kavelaars2004 and @Holman2004, all of them targeted KBOs. Indeed, until 2014 there was almost no attempt to apply digital tracking to asteroids. The only exception we are aware of is the work @Gural2005, who developed a mathematically sophisticated matched-filter technique for asteroid detection. They employed it in a manner equivalent to digital tracking, and demonstrated that they could detect previously-missed faint objects in existing data from the Spacewatch survey. As these data were optimized for conventional asteroid search methods, only 3–5 images of each field were available. The modest sensitivity increase possible in this context, and the differing optimal methodology for realizing it, renders the @Gural2005 technique distinct from and complementary to digital tracking as discussed herein, even though it is closely analogous conceptually. Consistent with their own terminology, we shall not henceforth refer to the @Gural2005 method as digital tracking, reserving the term for analyses that aim for large increases in sensitivity by stacking tens to hundreds of images of each field.
We are aware of only one published detection of a previously unknown asteroid using digital tracking as defined above. @Zhai2014 achieved this detection using the 5m Hale telescope with the CHIMERA instrument described by @Shao2014. CHIMERA uses new EMCCDs that deliver extremely fast read times while maintaining low read noise. The field of view at present is limited to 0.002 deg$^2$, but the instrument enables efficient observations with only a one-second exposure time. This allows digital tracking observations targeting NEOs passing very close to Earth, with motions of several degrees per day. In fact, while the single-frame exposure times for other digital tracking surveys are two minutes or longer, @Zhai2014 stacked 30 one-second exposures for digital tracking stacks with an effective *integration* of only 30 seconds. Despite this extremely short integration time, they detected a previously unknown 23rd magnitude asteroid moving at $6.32 \degree$/day (948 arcsec/hr). Due to the instrument’s small field, which can only be partly compensated by the large number of short digital tracking integrations that can be carried out each night, this object is the only previously unknown asteroid to be detected with CHIMERA so far. As the maximum useful exposure ($\tau_M$) for such an object is less than four seconds, it would have been extremely difficult to detect with any other instrument.
The methodology of @Shao2014 and @Zhai2014 is highly complementary to our own, which uses large format CCD imagers to detect asteroids with larger geocentric distances and slower angular motions. While we have not targeted NEOs thus far, our methodology has the potential to detect them with good sensitivity at distances as small as 0.1 AU in fields of view hundreds of times larger than that of CHIMERA (Section \[sec:points\]). For very small asteroids passing extremely close to Earth, CHIMERA and proposed next-generation instruments with larger fields [@Shao2014] will remain uniquely capable. If pushed to even shorter individual exposures, digital tracking using similar instruments would also detect small pieces of anthropogenic space debris in low Earth orbit.
Past Work and the Potential of Digital Tracking
-----------------------------------------------
We are now in a position to compare the actual digital tracking surveys that have been carried out with our careful consideration of the limits of digital tracking in Sections \[sec:skymotion\]–\[sec:points\]. We find that although highly successful surveys have obtained scientifically valuable results on KBOs, the full potential of digital tracking has not been exploited even for them: considerably longer and more sensitive integrations are possible than have ever been attempted from the ground. Furthermore, while @Zhai2014 have obtained a remarkable result in their detection of an NEO closer and faster-moving even than we have considered targeting, large-format CCD imagers can probe larger volumes of space and larger statistical samples of NEOs. No previous digital tracking survey has targeted either NEOs or MBAs with large-format imagers, except the MBA-optimized observations we report herein. For all three classes of objects, digital tracking holds yet-to-be exploited potential for extremely sensitive surveys.
Our Survey of MBAs: Observations and Data Reduction {#sec:Data}
===================================================
Observations
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Our observations were obtained using the WIYN 0.9m telescope[^7] on Kitt Peak, and the MOSAIC CCD imager. MOSAIC is a 65 megapixel, 8-detector imager originally developed for (and mostly used on) the 4m Mayall telescope. It has excellent cosmetic, QE, and noise characteristics, and a good readout time of 22 seconds. The unusual opportunity to use such an instrument on a relatively small telescope was the result of an agreement between the WIYN consortium and NOAO. On the WIYN 0.9m, the MOSAIC imager delivers a 59x59 arcminute field of view with 0.43 arcsecond pixels.
Since this was to be the first digital tracking survey for asteroids, we targeted MBAs rather than the less abundant and more challenging NEO population. We optimized our observations for MBAs both in terms of the integration lengths (all night) and the individual exposure times (2 minutes: i.e., $\sim \tau_M$ for MBAs). Although in the current work we analyze only MBA detections based on WIYN 0.9m data, we note that NEOs can be detected by digital tracking analysis even of MBA-optimized data sets, and we present such detections from a DECam data set in a forthcoming work (Heinze et al, in prep).
To detect the maximum possible number of faint asteroids, we centered our WIYN 0.9m observations on the ecliptic, at a point directly opposite the Sun. Such a field would be highest in the sky and allow the longest possible integration times in December (for the Northern Hemisphere), when the coordinates would be near RA 06:00 and DEC +23:15. However, near RA 06:00 the ecliptic is crossing the Milky Way, producing extremely rich starfields that create challenges for asteroid detection. Although we believe that the star subtraction methods we are developing (Section \[sec:starsub\]) will allow digital tracking to perform well even in rich starfields, we desired sparse starfields for our first survey. Thus, we chose to observe in the spring even though antisolar fields are then at more southerly declinations.
On the nights of April 19 and April 20, we observed a single ecliptic field all night, accumulating roughly five hours’ worth of integration. Moonlight was bright enough to reduce our sensitivity appreciably (full Moon was on April 24), but the weather was good and the seeing averaged 1.6–1.7 arcseconds through the majority of each night. On April 19, we acquired 150 two-minute R-band exposures of a field centered near RA 13:56:13, DEC -11:53:22; and on the following night we acquired 158 exposures of a field centered near 13:55:23, -11:49:14. Both fields are very near the antisolar point on their respective dates. The offset between them tracks the mean sky motion of main belt asteroids, determined based on an average of known asteroids near these fields on the dates of our observations. As the motion of such asteroids over 24 hours was much less than the field of the MOSAIC imager, the two fields overlap heavily; nevertheless the offsetting is desirable to avoid unnecessary loss of asteroids near the field edges from one night to the next.
We paid careful attention to dithering the images, both to fill the gaps between the detectors in the MOSAIC imager and to allow the construction of star-free night sky flats from a median stack of the science images. Dithering is also desirable because it tends to convert systematic effects into random effects, and greatly reduces the chance that detector artifacts will produce spurious detections in stacked images. We performed a dither offset every two images, using a quasi-random pattern centered on the coordinates given above and spatially constrained so that all pointings lay within 5 arcminutes of the nominal position. The dithering was quasi-random in that it consisted of many sets of regular dither patterns (e.g. hollow squares and linear dithers along various vectors), but the spacing and angles describing these patterns were changed to avoid redundant pointings and thoroughly sample the spatially constrained dither region.
Image Processing
----------------
### Basic Processing
We begin our image processing by subtracting the mean in an overscan region for each detector, interpolating across the few cosmetic defects in the MOSAIC detectors, and correcting for electronic crosstalk. Crosstalk in the MOSAIC detector is relatively simple: bright sources in one half of each detector produce spurious mirror-images of themselves at greatly reduced brightness in the other half. It is easy to remove these spurious images by subtracting from each detector half-image a reflection of the other half multiplied by the appropriate crosstalk coefficient. We determine the crosstalk coefficients for each detector empirically from our data, finding that they range from 0.0016 to 0.0021 and exhibit no significant variations over time.
Following the crosstalk correction, we use the Laplacian edge detection algorithm of @lacos to remove cosmic rays[^8]. Following these operations, the individual images from the eight MOSAIC detectors are tiled into single-extension FITS images with pixel dimensions 8,192$\, \times \,$8,192. This allows us to perform dark subtraction and flatfielding using simple routines designed for single-detector cameras — in particular, it means that a simple normalized flat automatically corrects for variations in sensitivity between the detectors. The flatfield we use is a clipped median-stack of dark-subtracted science images. Due to the thorough dithering described above, the stars vanish from this stack. After correction using this flatfield, our images have fairly uniform sky backgrounds. We find that removing the sky background by simply subtracting a constant value equal to the clipped median over all pixels in the image is sufficient for our purposes.
### Astrometric Registration
At this point, our tiled images are fully corrected and ready for astrometric registration. We begin by separating them again into the tiles corresponding to each of the eight detectors of the MOSAIC imager. Based on a manually selected reference star, for each tile we identify stars in the UCAC4 astrometric catalog and use them to construct an astrometric solution mapping pixel coordinates to celestial coordinates. We use a third-order polynomial mapping with cross terms, yielding ten degrees of freedom. To reduce the danger of roundoff errors in fitting, we measure both pixel and celestial coordinates relative to the center of the image, and we apply scaling factors so that the absolute values of all coordinates are near 1.0 at the image edges. We iteratively reject stars from the fit until the worst remaining outlier deviates by less than 0.3 arcsec. Since typically 50-80 stars per detector survive this clipping, the fits remain well constrained.
Having obtained an astrometric fit for each detector, we resample each image onto a consistent astrometric grid using bilinear interpolation. We define this grid to have a constant pixel size $s_{pix}$ of exactly 0.4 arcsec, such that the $x$ and $y$ pixel coordinates are given using the simple mapping
$$\label{eq:mapping}
\begin{array}{lcl}
x &=& x_0 - (\alpha - \alpha_0) \cos(\delta)/s_{pix} \\
y &=& y_0 + (\delta - \delta_0)/s_{pix} \\
\end{array}$$
where $\alpha$ is the RA; $\delta$ is the DEC; and $x_0,y_0$ and $\alpha_0,\delta_0$ both refer to the center of the new, resampled image (which may be freely chosen by the user); and the difference in sign between the equations is needed to avoid mirror imaging and to produce a final image with east left and north up. The output image, which contains the data from all eight individual detectors resampled onto the single self-consistent grid defined by Equation \[eq:mapping\], measures 12,000$\, \times \,$12,000 pixels. This includes the gaps between the detectors, as well as generous zero-padding that we include around the outside of the mosaic to prevent any data from dithered images from being lost off the edges of the array. Figure \[fig:ditherdat\] illustrates our resampled images, with excess zero-padding trimmed away.
### Preservation of Geometric Linearity under Resampling
The astrometric grid on which we have resampled our images constitutes a very simple map projection of the spherical sky onto a two-dimensional plane. Our digital tracking algorithm makes the approximation that the objects being sought will move with constant angular velocity in a straight line. In Section \[sec:skymotion\], we explored the range of validity of this assumption in terms of the sky motion of the objects, and found it sufficient to allow extremely long digital tracking integrations. Now, however, we must consider another question: whether Equation \[eq:mapping\] projects linear, constant-velocity sky motion onto linear, constant-velocity motion across the pixel grid with sufficient accuracy.
In evaluating a map projection for digital tracking images, only two types of distortion are relevant: spatial variations in the pixel scale, and failure to map linear (i.e. Great Circle) trajectories onto straight lines on the pixel grid. No projection is simultaneously free from both types of distortion. We have chosen the projection described by Equation \[eq:mapping\] because it is mathematically simple and is free from distortion of the first type: all of the pixels have exactly the same angular size. However, it does suffer from distortion of the second type. While lines of constant RA are Great Circles and lines of constant DEC (other than the equator) are not, our projection renders lines of constant DEC as straight and lines of constant RA as curved except for the central one that vertically bisects the image. In the following, we quantify the importance of this distortion for digital tracking analyses.
The relevant value is the maximum extent by which the projection of a track that is a segment of a Great Circle deviates from a straight line. We have calculated this deviation for a wide range of cases in order to probe the limits of our map projection, fixing the track length to be 20 arcminutes because the maximum digital tracking integration lengths described in Section \[sec:skymotion\] correspond to tracks of this length or shorter for all classes of objects. The deviation is strongly dependent on the celestial position angle of the track, being in general much worse for east-west motion than for north-south motion, due to the characteristic of Equation \[eq:mapping\] that lines of constant DEC are rendered as straight lines even though they are not Great Circles. Great Circle tracks with a locally east-west orientation therefore show deviations at all nonzero declinations even if they pass through the center of the field. By contrast, north-south trajectories follow RA lines, which Equation \[eq:mapping\] renders correctly at the center of the field, so such tracks have nonzero deviations only when they are offset in RA from the field center.
Objects can exhibit east-west trajectories in any digital tracking observation, so this worst-case scenario is relevant. The maximum deviation for a 20 arcminute Great Circle track is zero on the celestial equator, but reaches 0.5 arcsec at $40 \degree$ DEC and 1.0 arcsec at $60 \degree$. The deviation of north-south trajectories is independent of DEC, is always zero at the center of the field, and rises only to 0.05 arcsec for an RA offset of $5 \degree$ from field center. Thus, the map projection given by Equation \[eq:mapping\] is adequate for digital tracking applications up to $40$–$60 \degree$ DEC with fields spanning more than $10 \degree$ in RA — far larger than the field of view of any major telescope. For our own observations using a one square-degree imager at $-12 \degree$ DEC, targeting asteroids whose track lengths are less than 6 arcminutes, track deviations from the Equation \[eq:mapping\] projection are entirely negligible, of order 0.01 arcsec.
The projection given in Equation \[eq:mapping\] is not appropriate for digital tracking observations at declinations greater than $40$–$60 \degree$. Alternatives may easily be envisioned: for example, one can perform the astrometric resampling in the local tangent plane and then transform the positions and motions of detected asteroids back to the celestial coordinate system once the digital tracking analysis is complete. This is equivalent to using Equation \[eq:mapping\] on the celestial equator: even with a $10 \degree$ field, the maximum distortion is only 0.05 arcsec. Mapping distortion therefore does not necessitate any reduction in the maximum durations for digital tracking observations that were calculated in Section \[sec:skymotion\].
### Subtraction of Stationary Objects {#sec:starsub}
While not all previous digital tracking searches subtracted stationary objects prior to the digital tracking analysis, our large-scale implementation renders such subtraction essential. This is because we need to be able to avoid unmanageable numbers of false positives while at the same time using a fast and simple automated algorithm to search each of our thousands of trial stacks — and simple algorithms are easily confused by the noisy streaks that unsubtracted stars and galaxies leave behind, even in clipped-median stacks.
@Alard1998 and many others have developed sophisticated techniques for subtracting constant stars and galaxies from a series of images in order to find variable objects even in highly confused fields. To measure variable stars in a consistent way, these methods require the use of a consistent reference image for subtraction throughout the data set. Normalized convolution kernels are then identified that will blur the reference image to match each successive image from which it is to be subtracted. Our case is different in two ways. First, we are not trying to measure stationary objects, only to remove them as completely as possible: we do not have to use a consistent reference image. Second, we place a very high priority on achieving the subtraction with the least possible increase of sky background noise.
Rather than the convolution strategy of @Alard1998, we choose to model each image in our data set as a linear combination of other images taken at a large enough temporal separation that moving objects in our target populations (the MBAs) cannot self-subtract. We find that conventional least-squares methods for determining the optimal linear combination severely amplify the noise in some cases by producing large positive and negative coefficients with similar absolute values. Hence, we use a downhill simplex method from @nrc to find the best least-squares match with coefficients restricted to positive values. To further reduce noise and speed processing, we divide the set of images from each night into several (typically seven) contiguous subsets and create a clipped median stack of each subset, producing a smaller number of cleaner and lower-noise star images. The downhill simplex analysis then matches each individual image using a linear combination of these seven low-noise star images. Since the PSF can vary across an image, we perform the downhill simplex matching individually within sixteen regions defined by an approximately rectangular Voronoi tessellation across our image. In order to avoid fitting irrelevant sky noise and accurately match the PSFs of stars, pixels with values below a threshold on the master stack (shown in the right panel of Figure \[fig:ditherdat\]) are excluded from the fit. As a final step, a constant value is subtracted to set the median sky value to 0.0 in each Voronoi region.
This subtraction eliminates faint stars and galaxies completely, but noisy residuals remain near the cores of bright objects, as well as inconstant artifacts that are probably electronic in origin emanating from saturated stellar images. We mask these residuals aggressively based on pixel brightnesses in the master star image, adding rectangular regions by hand for the worst saturated objects. Figure \[fig:subtraction\] illustrates our results.
![Subtraction and masking of stationary sources illustrated by a small region trimmed from a single frame from our April 20, 2013 data set. The unsubtracted object boxed in the rightmost panel is an asteroid bright enough to be detected without digital tracking. (A) Original resampled image. (B) The same image after subtraction. Artifacts, probably electronic in origin, can be seen emanating from the brightest stars. (C) The same image after the final masking step, ready for use in our digital tracking analysis. \[fig:subtraction\]](f06.eps)
While the image subtraction methodology described above was entirely satisfactory for our April 2013 data, we have continued developing image subtraction for future, more challenging applications. Full discussion of this development is beyond the scope of the current work, but we will make two important observations. First, our Voronoi tessellation turns out to be an irrelevant frill: a simple rectangular grid works fine. Second, a hybrid method including the kernel convolution developed by @Alard1998 produces much better results in crowded fields at low galactic latitude. Although this method greatly increases the computer time for image subtraction, it remains small compared to that required for the digital tracking analysis itself.
Digital Tracking Analysis
=========================
Basic Parameters {#sec:digipars}
----------------
As discussed in Section \[sec:skymotion\], digital tracking integrations targeting main belt asteroids can last 8–9 hr within a single night, but cannot span multiple nights under the approximation of linear, constant-velocity motion. Thus we analyze our data from April 19 and April 20 separately. The two data sets consist of 150 and 158 images, respectively, each a 2-minute exposure in the $R$ band.
We excluded some images on each night that exhibited elevated sky backgrounds (due to twilight) or had unusually bad seeing, in order to avoid wasting compute time stacking poor-quality images that would add little or nothing to our final sensitivity. On April 19, we excluded the final 24 images of our data sequence on account of very bad seeing (averaging 2.8 arcsec vs. 1.6 arcsec for the 126 remaining images). On April 20, we excluded 13 images from the beginning of the data sequence due to elevated background levels (averaging 1,600 ADU vs. 1,000 ADU for the remaining images) and 15 images from near the end of the sequence due to bad seeing (averaging 2.5 arcsec vs. 1.7 arcsec for the 130 remaining images). Thus we arrived at final data sets consisting of 126 images for April 19 and 130 for April 20.
The time spanned by these data sets is 5.76 hr in each case. Equation \[eq:blur\] indicates an optimal grid spacing of $\Delta_m = 0.25$ arcsec/hr for the digital tracking search, and we conservatively adopt $\Delta_m = 0.20$ arcsec/hr. We search a rectangular region in angular motion phase space extending from -50 to -20 arcsec/hr eastward and -7 to +30 arcsec/hr northward, which spans the range of plausible motion vectors for MBAs (Figure \[fig:drift\]). In all, we probe 28,086 trial vectors.
We perform our digital tracking analysis using a desktop workstation having 64 gigabytes of memory. This memory is insufficient to simultaneously load and manipulate 130 full-size resampled images, so we divide each image into four overlapping quadrants of 6,000 $\times$ 6,000 pixels. We are able to make the quadrants overlap by discarding some of the excess zero-padding in our original 12,000 $\times$ 12,000 pixel resampled images. The overlapping is necessary in order to maintain full sensitivity for asteroids that cross a quadrant boundary.
Thus, for each data set we carry out four digital tracking runs, each processing a single quadrant. Since each quadrant is processed through 28,086 trial vectors, our search over four quadrants probes 112,344 trial stacks with a size of 6,000 $\times$ 6,000 pixels for each of the two data sets. The total number of vector-pixels (number of images $\times$ pixels per image $\times$ number of trial stacks; see Section \[sec:mbatrial\]) probed in our two data sets of 126 and 130 images is $1.035\times10^{15}$. We use nearest-neighbor interpolation for our image shifts and a median with 5$\sigma$ clipping for our stacks. Nearest-neighbor interpolation speeds processing relative to bilinear interpolation, and is not expected to reduce sensitivity since our images are well sampled. Our full digital tracking analysis requires a total of about 50 days of runtime for our multi-core desktop, which processes the data at a rate of $8.1 \times 10^{11}$ vector-pixels per hour.
Automated Detection of Asteroids
--------------------------------
### Choosing a Detection Threshold {#sec:autothresh}
Taking into account the zero-padding and overlapping of the quadrants, our search of 224,688 trial stacks (that is, 112,344 stacks per night for two nights of data) spans about $5.6\times10^{12}$ non-overlapping non-zero pixels in the final stacks. These correspond to $6.2\times10^{11}$ square, 3 $\times$ 3 pixel boxes that each sample the noise over a region equal to the size of an asteroid’s PSF. If all of these noise realizations were independent and the noise were Gaussian, detection thresholds of 8$\sigma$, 7$\sigma$, or 6$\sigma$ would result the expectation of $4 \times 10^{-4}$, 0.8, or 600 false positives, respectively, for our entire survey. Our sampling of angular motion phase space is fine enough that adjacent trial stacks are not completely independent, which produces a modest reduction (no more than a factor of two) in the expected false positive numbers. While false positives due to non-Gaussian effects such as edge noise, star subtraction residuals, or cosmic rays would be rejected by the manual checking described below (Section \[sec:mancheck\]), pure Gaussian outliers might not exhibit any signatures of problematic data and hence could pass all the tests. We wish to detect as many genuine asteroids as possible while not reporting any spurious detections as real objects, and for simplicity we prefer to choose an integer threshold. Thus, we adopt a 7$\sigma$ threshold for our automated detection: it is the integer value that yields the greatest sensitivity while still producing an expected number of false positives less than 1.0. We will conclude below (Section \[sec:falsepos\]) that none of the asteroids we finally confirmed is a false positive.
### The Automated Detection Algorithm
Saving trial stacks for later analysis would require prohibitive amounts of hard drive space, so our digital tracking code searches and then discards each stack, ultimately outputting not a set of images but a detection log. To streamline processing, we use a simple and fast algorithm for source detection. First, the trial stack is smoothed with a square boxcar approximating the size of the PSF: in this case, 3$\times$3 pixels or 1.2$\times$1.2 arcseconds. A new image that maps the standard deviation of this smoothed image is then constructed with a resolution of 3 pixels. We obtain the value for each pixel in this map by calculating the standard deviation in a square annulus around the corresponding point on the smoothed image: the annulus has inner dimension 15 pixels and outer dimension 27 pixels. Within this annulus, only every ninth pixel (every third pixel in each dimension) is used in calculating the standard deviation, to preserve independence. The smoothed image is then scanned for pixels whose brightness indicates a detection above 7$\sigma$ based on the standard deviation map. If such a pixel is also the brightest one within a ‘redundancy radius’ of 5 pixels, it is written to the log as a possible asteroid detection.
### Duplicate Detections of Real Asteroids
Our simple and fast detection algorithm results in numerous duplicate detections of real asteroids, and these are the dominant type of ‘false positive’ encountered in the automated detection logs. The brightest objects produce many hundreds of duplicate detections because even on trial stacks far from the correct motion vector, the asteroids’ streaked images remain above the detection threshold. Rather than adopting a more sophisticated (and therefore slower) automated detection algorithm, we use several techniques to winnow down the thousands of automatically logged detections to a much smaller set consisting exclusively of real asteroids.
The first step is to gather duplicate detections into clusters and retain only the brightest detection at the center of each cluster. Our clustering code begins with the very highest-significance detection in the whole log, which corresponds to a bright real asteroid on an accurately motion-matched trial stack. A radius is defined surrounding this asteroid, and all objects with pixel coordinates lying inside this radius are provisionally classified as duplicate detections. Finally, the standard deviation of motion rates among these provisional duplicates is calculated, and 3$\sigma$ outliers are re-classified as possibly real, fainter asteroids that passed near the much brighter object. The program then proceeds to the most significant remaining un-clustered detection, and makes it the center of a new cluster.
The radii used for clustering are determined as follows. First, the cluster program calculates the mean density of detections in pixel space based on the entire log. Next, the user specifies a threshold factor above this background density[^9]. When defining a cluster, the program expands or contracts the bounding radius until the density of detections within it reaches the specified threshold above the background density. When the detection log is fully clustered, the program outputs the fraction of the entire 4-dimensional[^10] volume of the digital tracking space that was ultimately included in a cluster. This represents an estimate of the false-negative rate (FNR) due to distinct real asteroids being incorrectly clustered with brighter objects. If the threshold factor above the background density is too high, clusters are too small and the manual effort required to weed out the resulting large number of duplicate detections for bright asteroids becomes excessive. If the factor is too low, clusters are too large and the FNR becomes excessive. We optimize the factor manually to obtain a manageably small number of duplicate detections without raising the FNR above 1%. In some cases, a satisfactory clustering may be obtained while the estimated, cluster-induced FNR is still as low as 0.1%. In others, in order to reduce the numerous duplicate detections we lower the threshold factor until the FNR rises to 1%, but in no case do we allow the FNR to exceed 1%. The final output of our clustering code is a greatly refined list of detections which are likely to be real, motion-matched asteroids.
Manual Checking of Detections {#sec:mancheck}
-----------------------------
Each detection output by our cluster code is specified by its motion rates and its pixel coordinates on a reference image from the middle of our data sequence. We have produced visualization software that uses this information to quickly reconstruct a small region of the appropriate trial stack, centered on the asteroid. Using the output of the cluster code, this allows us to reconstruct — in a few minutes — images that show all the plausible asteroids detected in a digital tracking search that ran for weeks. We manually examine such images to confirm or reject each candidate asteroid. Since our visualization software uses bilinear interpolation for the image shifts, this test also allows us to check reported detections using a different interpolation scheme from the nearest-neighbor method adopted in the initial digital tracking search.
### Motion Rate Check Images {#sec:checkim}
For our first test, we produce a small (e.g. 41$\times$41 pixel) stacked image centered on the asteroid under investigation using the motion vector corresponding to its logged detection, and several other identically-centered images using motion vectors corresponding to adjacent gridpoints in angular motion phase space. We tile these small images together, placing the nominally motion-matched tile in the center and arranging the others around it such that that if the detection is real, the streaked images in the mismatched tiles will point back toward the sharp image at the center[^11]. For convenience, we will refer to such tiled images as check images. Looking at a check image allows us immediately to determine if the asteroid is (1) a real, motion-matched detection; (2) a duplicate, velocity-mismatched detection of a bright asteroid that slipped through the clustering code; (3) an apparently real detection that is too faint to confidently classify; or (4) a completely spurious detection due to a noisy edge or other artifact in the trial stack. The top row of Figure \[fig:check\] shows check images of three real asteroids, while the bottom row shows dubious or spurious detections: one example each of categories (2), (3), and (4).
Our program for making check images also performs a quadratic fit to the flux of the central source in each tile to determine the true flux-maximizing motion rates with greater accuracy than can be achieved by eye. The flux is measured within a small aperture of radius only 2.0 pixels (0.8 arcsec), so even small motion errors blur the image enough to reduce the measured flux. We iterate with the check image program as necessary, refining the motion rates to ensure that the optimal motion identified by the quadratic fit lies near the center of the fitting region. Finally, we pass all detections in categories (1) and (3) on to the second stage of manual verification.
![Examples of check images used to manually verify automatically detected asteroids. In each panel, the different tiles are offset from one another successively by 0.4 arcsec/hr in angular velocity space, and the central one has the velocity of a particular automated detection. All objects in the top row are real, and only 2013 EW$_{149}$ was known prior to our analysis. Asteroid pt2124 was detected only on April 19 and thus lacks size and distance measurements, which require two nights. Objects in the bottom row are dubious or spurious, exemplifying categories (2), (3), and (4) from Section \[sec:checkim\]. Candidate t1 0060 is a badly motion-matched duplicate detection of a known $R=20.4$ mag asteroid. Candidate t1 0070a may be real, but is very faint. Candidate t1 0073a does not fade uniformly toward every corner: it may be a noise artifact. \[fig:check\]](f07.eps)
### Verification in Independent Subsets of the Data: Pyramid Images {#sec:pyrim}
In the second round of verification, we again create multiple small images from digital tracking stacks, but instead of using different motion rates, we keep the rates fixed at the optimal values and use different subsets of the input images. We divide the full data set into first two, then three, then four, etc. consecutive independent subsets. For easy manual investigation, we arrange these images in a pyramid with the stack of all images by itself at the top; the two half-stacks below it; the three one-third stacks below that, etc. We use five-layer pyramids for our analysis, but the bottom layer is needed only in special cases (see below). Figure \[fig:pyr\] gives examples of four-layer pyramid images from our April 19 data. As this data set contains 126 images, the numbers of images per tile in the respective layers is 126, 63, 42, and 31 or 32. The corresponding integration times are 4.2, 2.1, 1.4, and 1.05 hours.
![Examples of pyramid verification images from our April 19 data. As described in the text, each layer of a pyramid represents a different division of our 126 images into equal independent subsets. Appearance in multiple tiles of the same layer confirms an object’s reality. Objects in the top row are confirmed real, pt2124 and 2013 HZ$_{154}$ being new discoveries. Those in the middle row are not confirmed: t1 0064 is probably real but is too faint for secure detection, 0070a is less plausible, and 0073a appears to be an artifact of noise from the masking of a bright star. We do not include unconfirmed objects in Table \[tab:asteroids\], nor have we reported them to the MPC. The objects in the bottom row are all spurious: they are the three most significant detections from our ‘scrambled’ data set (Section \[sec:falsepos\]). Significance values quoted in units of $\sigma$ are not from the raw detection logs but are re-calculated more accurately using master stacks produced by our visualization software. \[fig:pyr\]](f08.eps)
The purpose of the pyramid images is to test if a detection can be confirmed in multiple independent subsets of the data. It is difficult to conceive of an artifact that could imitate a real asteroid well enough to pass this test. A spurious detection caused by a coincidence of cosmic rays, CCD defects, or star-subtraction residuals would reveal its spurious nature by vanishing from some tiles but getting brighter (and likely changing morphology) on others. Our thorough dithering, star subtraction, and cosmic ray removal renders such events vanishingly rare in our data. More common are spurious detections caused by locally noisy regions close to smoother data that cause our automated algorithm to underestimate the noise. Candidate t1 0073a in Figure \[fig:pyr\] is an example: it appears similar to confirmed asteroid pt2124 in the full stack, but in the lower rows of the pyramid we see that it coincides with the masked region surrounding a bright star, and may be an artifact of the locally higher noise due to reduced data coverage at this location.
All other things being equal, a real object that is detected with with significance $s$ in the top tile of a pyramid image should appear at significance $s/\sqrt{2}$ in each of the half stacks, $s/\sqrt{3}$ in the one-third stacks, etc. This idealized situation frequently fails to be realized in practice. In extreme cases (e.g. objects that entered or left the field during our observations), only half the exposures may have supplied image data, but the asteroids can still be confirmed as real if they are bright enough. This is our reason for using five-layer pyramids in our analysis: the bottom layers can be necessary for verifying such objects even though four-layer images like those shown in Figure \[fig:pyr\] are sufficient for most others. More commonly, rotation can cause asteroids to change brightness considerably on a timescale of hours, and changes in the seeing and sky background can change the sensitivity in different subsets of the data. Our confirmation criterion for pyramid images is therefore simply that the object must be consistently detected and readily apparent to the human investigator on at least two different tiles in a single layer of the pyramid. Provided this condition is satisfied, we do not penalize objects for not being readily visible in additional tiles in the same layer. It would be reasonable to apply such a penalty only if the implied change in brightness was implausibly large, which is a scenario we have not encountered.
Results for Detected Asteroids {#sec:results}
==============================
Initial Matching to Known Objects {#sec:known1}
---------------------------------
The criteria described in Section \[sec:pyrim\] resulted in 199 confirmed asteroids in our April 19 data (Figure \[fig:tracks\]), and 181 in our data from April 20. The slight reduction in sensitivity from one night to the next is likely due to the waxing of the moon. Even on April 19, the 10-day-old moon was bright enough to reduce the sensitivity of our observations significantly from their full dark-sky potential, and on April 20 the moon was brighter and closer to our field.
We compare our final detection lists against ephemerides from the Minor Planet Center (MPC), matching objects within a 20 arcsecond radius and also checking for consistent sky motions. We find that on April 19, we detected 47 known asteroids and 152 new objects; on April 20 the respective numbers are 49 and 132. We performed this cross matching in October 2014, and hence a ‘known’ objects in this context means one for which, on that date, the MPC possessed an orbit sufficient to determine April 2013 positions with 20 arcsecond accuracy. Some of these asteroids had in fact been discovered since our observations, but as our analysis software was still under development, we were not able to report them to the Minor Planet Center in time to obtain discovery credit.
Night-to-Night Linkages {#sec:link}
-----------------------
How many of the previously unknown asteroids we detected on April 19 were also detected on April 20? To answer this question, we must match the positions and motion rates of asteroids between the two nights. This is an important problem for all asteroid surveys, but the highly accurate sky motions that come ‘for free’ along with digital tracking detections make it easier. Simply extrapolating forward (or backward) for 24 hours assuming linear motion at constant velocity is often sufficient for matching, but such positions are systematically incorrect by about 30 arcseconds to the west (east) if extrapolating forward (backward) by one night. The source of this offset is the Earth’s rotation, which causes the eastward space velocity of an observer on Earth’s surface at night to be faster than the orbital velocity that would apply to idealized observations made from the geocenter.
Much more accurate extrapolations can be obtained using the angular velocity the asteroid would have had if viewed from the geocenter. To obtain an approximate value for this geocentric angular velocity, we simply estimate the component of the angular velocity that is due to the Earth’s rotation and subtract it from the measured angular velocity. The angular velocity contributed by Earth’s rotation is given by the physical velocity of the observer relative to the geocenter, projected onto the plane perpendicular to the line-of-sight to the asteroid and divided by the asteroid’s estimated distance (see Heinze & Metchev 2015 for a more detailed discussion). The fact that geocentric distance is inversely correlated with total angular velocity allows us to obtain a crude approximation for the distance that is sufficient for our current purpose. We perform a linear fit to distance as a function of inverse angular velocity for known asteroids in our field. The RMS error of these distances is 0.3 AU. This crude distance estimation should not be confused with the far more accurate method of @curves, but the latter cannot be used at this stage because it requires that measurements of the asteroid have already been linked across two subsequent nights.
As an example of a night-to-night linkage, analyzing our April 19 data reveals a previously unknown $R$ = 21.7 mag asteroid[^12] with a motion of -37.36 arcsec/hr east and 10.32 arcsec/hr north, for a total angular velocity of 38.76 arcsec/hr. Our linear fit based on known asteroids maps this angular velocity to an approximate geocentric distance of 1.55 AU. At 1.55 AU, the projected rotational velocity of Kitt Peak during our observations (1340.0 km/hr eastward and -4.8 km/hr northward) produces a reflected angular velocity of -1.19 arcsec/hr eastward and 0.00 arcsec/hr northward. Thus, if measured from the geocenter, the angular velocity of this object would have been $-37.36 + 1.19 = -36.17$ arcsec/hr eastward and 10.32 arcsec/hr northward. To predict the asteroid’s location in our April 20 data, we simply use these velocities to extrapolate linearly forward by the elapsed time between the reference time for the April 19 digital tracking integration and that for the April 20 integration. There is an $R$ = 21.8 magnitude April 20 detection only 2.4 arcseconds away from the resulting position, and its motion vector matches that of the April 19 asteroid to within 0.09 arcsec/hr. Using the linkage method illustrated above, we find that 165 of the asteroids detected in our April 19 observations were independently recovered by our detection software in the April 20 data. Thus 34 asteroids (including 1 known object) are unique to the April 19 data and 16 (including 3 known objects) are unique to the April 20 data. The full count of confirmed asteroids is therefore 215 objects, of which 50 had accurate orbits as of October 2014, and the remaining 165 appeared to be new discoveries of our survey. Of these 165 new asteroids, 33 were automatically detected only in the April 19 data, 13 only in the April 20 data, and 119 were automatically detected in the data sets from both nights. The reality of the 46 single-night objects is not in doubt, since they were confirmed on multiple independent subsets of the data within the night of their discovery. Nevertheless, we attempted to recover them manually by creating check images and pyramid images centered on their extrapolated locations for the nights on which they were not automatically detected. This was highly successful: all but 12 of the single-night asteroids were recovered on a second night. Of the 12 objects not recovered on a second night, in many cases the cause was obvious: the object had moved out of the field or become superimposed on a masked star. In the remaining cases the non-detection may be due to rotational variability or increasing moonlight.
Verification of a Negligible False Positive Rate {#sec:falsepos}
------------------------------------------------
We have performed a test to probe whether we have achieved our goal of zero false positives among confirmed objects, and also whether a different set of criteria for confirmation would result in improved sensitivity while maintaining a very low false positive rate. This test consists of a full re-analysis of the April 19 data with the temporal order of the images scrambled by randomly reassigning the image acquisition times. Real asteroids cannot be registered with the timestamps scrambled, so all detections in this analysis must be false positives. At the same time, the images used are identical to those in the real analysis, so the rate and statistics of false positives in the ‘scrambled’ data set should match those in the real data.
Similar to the real data, the scrambled data set exhibits about 40 false positives that correspond to locally noisy regions near cleaner data that caused the automated algorithm to use an inappropriately low sky noise value in estimating the significance of the detection. The bottom left panel in Figure \[fig:pyr\] shows an example of this. Other false positives appear against a smooth background and accurately mimic a real but low-significance point source. The two strongest examples are shown in the bottom center and bottom right panels of Figure \[fig:pyr\]. Both have automatically logged significance values of 7.2$\sigma$. We have re-calculated these significance values by creating 301$\times$301 pixel images with our visualization software, allowing us to use a wider background annulus to more accurately measure the noise. The new, more accurate values are 6.9$\sigma$ and 6.5$\sigma$, respectively. One 6.9$\sigma$ false positive in a single night’s data is not surprising, and suggests that once obvious false positives have been discounted, the remaining noise in our images is almost perfectly Gaussian.
For comparison to the 6.9$\sigma$ significance of the strongest false positive in our scrambled data set, the identically calculated value for the least-significant one-night asteroid confirmed in our real data (pt2124) is 7.2$\sigma$ — a level at which a false positive is nine times less likely under Gaussian statistics, and only 0.18 false positives would be expected in our entire survey even if the trial stacks were strictly independent. The pyramid image for pt2124 is much more visually convincing than those for any of the detections in the scrambled data set (see Figure \[fig:pyr\]). Additionally, the sky motions of pt2124 (-33.80 arcsec/hr east and 14.56 arcsec/hr north) put it near the densest concentration of real asteroids in Figure \[fig:drift\], rather than off in one of the empty corners where a truly spurious detection would have an equal probability of falling — a consideration which reduces its false positive probability by a further factor of at least 2. This asteroid therefore appears to have no more that about a 5% chance of being a false positive. All of our other confirmed asteroids are detected with far smaller false positive probability than pt2124, due to greater significance and/or consistent detection on both nights.
The scrambled-data analysis supports the reality of all our confirmed asteroids. However, it also validates the expectation from Section \[sec:autothresh\] that our analysis probes a sufficiently large number of noise realizations that multiple spurious point sources will appear with significance above 6.5$\sigma$, even with pure Gaussian noise. Indeed, the spurious sources from the scrambled data are indistinguishable from many unconfirmed candidates in the real data set, including those shown in the middle row of Figure \[fig:pyr\]. While the greater abundance of such detections in the real data relative to the scrambled analysis makes it certain that some of them are genuine asteroids, they cannot be confidently identified. Thus it does not seem that our detection criteria can be significantly relaxed without incurring numerous false positives.
Reporting Objects to the MPC {#sec:mpc}
----------------------------
We have reported all 215 of our detected asteroids to the MPC: 50 as previously known objects, 153 as two-night discoveries, and 12 as single-night objects. For all but the lowest SNR detections, we reported two positions per night. All of our observations have been accepted and processed. The MPC holds single-night asteroids in a file for later matching, but does not assign discovery designations for them. Thus we still refer to single-night objects by our own temporary designations: e.g., pt2124 in Figures \[fig:check\] and \[fig:pyr\].
Of our 153 previously unknown two-night objects, the MPC has issued discovery designations for 144. The remaining nine have been matched by the MPC to previously-known objects with orbits too inaccurate for our own matching to have identified them. Our data have resulted in significant improvements to the orbital accuracy of these nine objects. One example is 2013 EW$_{149}$, shown in Figures \[fig:check\] and \[fig:pyr\]. It was discovered March 13, 2013 and measured three times over a period of two nights. Apart from these discovery observations, 2013 EW$_{149}$ is known only through our measurements on April 19 and 20. Without our data, no meaningful orbit would be known for this object.
Our 215 detected asteroids can thus be divided into four categories: 50 asteroids with well-known orbits independent of our observations; 9 objects to which our observations contributed significant orbital information; 144 two-night discoveries with designations from the MPC, and 12 single-night objects. Table \[tab:asteroids\] gives the MPC designation for each of our asteroids, where applicable, and indicates the category to which each object belongs.
Measurements of Detected Asteroids {#sec:motion}
----------------------------------
For each asteroid, we measure the angular velocity, brightness, and celestial coordinates at reference times selected to be near the midpoints of the digital tracking integration on each respective night. For the positions and fluxes, we simply measure the best motion-matched image stacks. Except for the faintest objects, the precision of the positions is better than 0.1 arcsec and is probably limited by astrometric calibration error. We convert our measured asteroid fluxes into magnitudes using a calibration derived from known asteroids in our data, obtaining the magnitudes of these known objects from ephemerides generated by the MPC. The results are likely good to about 0.1 mag except for the faintest objects. Obtaining a more accurate calibration from photometric standard stars would require careful corrections for PSF variations, and the precise magnitudes it would yield are not required at present. The coordinates and magnitudes of all 215 detected asteroids are given in Table \[tab:asteroids\].
To measure the angular velocities of our asteroids, we use quadratic fits to a grid of digital tracking stacks with motions near the optimal value, as described in Section \[sec:checkim\]. We calculate the uncertainties based on the residuals from these quadratic fits. The measured tracks of our asteroids relative to the background starfield are shown in Figure \[fig:tracks\], and the measured drift rates are plotted and compared to known objects in Figure \[fig:drift\].
These precisely measured angular velocities allow us to calculate the geocentric distance of each two-night object using Earth rotational velocity reflex as we describe in @curves. The mean error of our distance determinations is only 1.5% for known objects. For newly discovered objects, our distances and flux measurements allow us to calculate absolute magnitudes and approximate sizes for the first time. These values are given in Table \[tab:asteroids\], and the histogram of absolute magnitudes is shown in Figure \[fig:distmag\]. The smallest asteroids we have detected have absolute magnitudes $H_R \sim 21.5$, and hence diameters of 130–300 meters depending on their albedo. While asteroids of this size range among the NEOs are routinely detected during close approaches to Earth, our survey is the first to detect them in the main belt with a telescope smaller than 4 meters.
![The real on-sky tracks of asteroids detected in our April 19 data. The background image against which the tracks are plotted is a master star image made by stacking the same data, and the field of view shown (including the narrow rim of zero padding) is 1.13$\times$1.10 degrees. Of 199 asteroids, 154 are new discoveries, and all are verified by consistent detection in multiple independent subsets of the images. Their magnitudes extend past $R$ = 23: digital tracking enables the 0.9-meter telescope to probe a regime previously accessible only to 4-meter and larger instruments. \[fig:tracks\]](f09.eps)
[lcccccccc]{} Hrabe & 13:57:07.56 & -11:31:06.8 & 13:56:11.76 & -11:27:19.2 & 18.1 & $1.101 \pm 0.010$ & 16.3 & 1.5–3.3\
(12231) & 13:54:38.48 & -11:58:03.5 & 13:53:43.78 & -11:53:50.4 & 18.2 & $2.019 \pm 0.023$ & 14.3 & 3.7–8.2\
(123940) & 13:54:57.46 & -12:19:10.5 & 13:54:08.12 & -12:17:12.3 & 18.3 & $1.901 \pm 0.033$ & 14.6 & 3.2–7.1\
(20458) & 13:55:42.60 & -12:12:08.0 & 13:54:45.43 & -12:04:35.5 & 18.5 & $1.554 \pm 0.013$ & 15.5 & 2.1–4.7\
(154215) & 13:56:30.30 & -12:02:10.2 & 13:55:48.45 & -11:52:57.1 & 18.6 & $1.951 \pm 0.020$ & 14.8 & 2.9–6.5\
(82140) & 13:55:53.89 & -11:55:07.6 & 13:54:59.55 & -11:48:52.6 & 18.7 & $1.771 \pm 0.021$ & 15.2 & 2.4–5.4\
(121685) & 13:56:04.95 & -11:53:36.2 & 13:55:20.69 & -11:48:28.3 & 18.7 & $2.054 \pm 0.022$ & 14.7 & 3.1–6.9\
(304006) & 13:54:17.66 & -11:59:45.4 & 13:53:16.84 & -11:58:08.1 & 18.7 & $0.907 \pm 0.005$ & 17.5 & 0.8–1.9\
(93774) & 13:55:17.52 & -12:16:18.1 & 13:54:28.34 & -12:12:31.1 & 18.9 & $1.869 \pm 0.022$ & 15.2 & 2.4–5.3\
(203904) & 13:56:47.67 & -11:33:29.6 & 13:55:51.46 & -11:26:41.1 & 18.9 & $1.048 \pm 0.008$ & 17.3 & 0.9–2.1\
(211984) & 13:55:50.42 & -11:22:54.4 & 13:54:53.85 & -11:19:12.2 & 18.9 & $1.149 \pm 0.015$ & 17.0 & 1.1–2.4\
(162579) & & & 13:56:58.35 & -11:38:29.4 & 19.0 & & &\
(178125) & 13:57:45.22 & -11:21:01.7 & 13:56:48.01 & -11:18:10.6 & 19.2 & $1.546 \pm 0.062$ & 16.2 & 1.5–3.4\
(251570) & 13:56:56.95 & -12:20:19.2 & 13:55:58.88 & -12:16:08.6 & 19.3 & $1.205 \pm 0.009$ & 17.2 & 1.0–2.2\
(357974) & 13:55:35.08 & -11:29:06.8 & 13:54:29.56 & -11:26:58.3 & 19.3 & $1.097 \pm 0.009$ & 17.5 & 0.8–1.8\
(79454) & 13:54:51.03 & -11:59:13.6 & 13:53:55.27 & -11:57:06.3 & 19.5 & $2.069 \pm 0.019$ & 15.5 & 2.1–4.7\
(187832) & 13:56:04.07 & -12:02:17.2 & 13:55:20.34 & -11:56:07.0 & 19.5 & $2.231 \pm 0.028$ & 15.2 & 2.5–5.5\
2009 FD$_{72}$ & 13:57:16.55 & -12:02:19.2 & 13:56:07.56 & -12:04:58.2 & 19.6 & $1.230 \pm 0.009$ & 17.4 & 0.9–2.0\
(257628) & 13:54:15.51 & -12:18:30.2 & 13:53:30.88 & -12:11:35.8 & 19.6 & $1.978 \pm 0.027$ & 15.8 & 1.9–4.2\
(212989) & 13:54:34.23 & -11:30:49.8 & 13:53:35.48 & -11:26:27.6 & 19.6 & $1.424 \pm 0.012$ & 16.9 & 1.1–2.4\
2013 JA$_{53}$ & 13:54:23.25 & -11:52:56.0 & 13:53:26.27 & -11:49:05.7 & 19.6 & $1.039 \pm 0.008$ & 18.0 & 0.7–1.5\
pt2132 & 13:54:07.36 & -12:12:29.4 & & & 20.0 & & &\
(361542) & 13:54:34.12 & -11:36:24.9 & 13:53:34.70 & -11:28:57.1 & 20.2 & $1.398 \pm 0.012$ & 17.6 & 0.8–1.8\
(398106) & 13:54:10.32 & -12:08:55.8 & 13:53:15.19 & -12:08:56.6 & 20.2 & $2.474 \pm 0.044$ & 15.5 & 2.1–4.6\
(209656) & 13:55:58.91 & -11:36:28.9 & 13:55:00.61 & -11:31:55.5 & 20.2 & $1.605 \pm 0.032$ & 17.1 & 1.0–2.2\
(152105) & 13:57:11.68 & -11:59:11.0 & 13:56:25.40 & -11:55:31.3 & 20.3 & $2.417 \pm 0.029$ & 15.7 & 1.9–4.3\
(403699) & 13:55:00.01 & -11:47:54.1 & 13:54:09.99 & -11:45:45.1 & 20.3 & $2.231 \pm 0.024$ & 16.0 & 1.7–3.7\
(257171) & 13:57:09.89 & -12:05:37.9 & 13:56:22.59 & -11:59:41.7 & 20.3 & $1.950 \pm 0.021$ & 16.5 & 1.3–2.9\
2005 SC$_{42}$ & 13:55:57.02 & -11:49:26.4 & 13:55:08.49 & -11:42:24.4 & 20.4 & $2.016 \pm 0.024$ & 16.5 & 1.3–3.0\
(364373) & 13:56:43.78 & -11:56:52.8 & 13:55:49.26 & -11:54:10.2 & 20.4 & $1.805 \pm 0.020$ & 16.9 & 1.1–2.5\
(172698) & 13:55:50.71 & -12:08:05.6 & 13:54:57.13 & -12:02:53.9 & 20.4 & $1.946 \pm 0.025$ & 16.6 & 1.3–2.8\
(311762) & 13:54:22.91 & -11:47:58.1 & 13:53:28.80 & -11:44:58.3 & 20.4 & $1.751 \pm 0.017$ & 17.0 & 1.1–2.4\
2013 HJ$_{153}$ & 13:55:43.71 & -11:31:27.1 & 13:55:03.36 & -11:24:49.2 & 20.4 & $2.197 \pm 0.022$ & 16.2 & 1.5–3.4\
2013 HQ$_{152}$ & 13:55:22.20 & -11:49:43.6 & 13:54:28.26 & -11:49:07.7 & 20.4 & $1.604 \pm 0.014$ & 17.2 & 0.9–2.1\
2013 HC$_{155}$ & 13:57:01.16 & -11:50:09.4 & 13:56:07.83 & -11:43:03.5 & 20.4 & $0.965 \pm 0.008$ & 19.0 & 0.4–1.0\
2013 HC$_{154}$ & 13:56:12.78 & -11:59:30.0 & 13:55:06.71 & -11:58:57.5 & 20.5 & $1.524 \pm 0.011$ & 17.5 & 0.8–1.9\
2013 HK$_{153}$ & 13:55:43.10 & -11:22:42.3 & 13:54:45.15 & -11:19:34.3 & 20.5 & $1.441 \pm 0.019$ & 17.8 & 0.7–1.6\
(412517) & 13:54:36.93 & -11:52:58.2 & 13:53:41.57 & -11:47:50.6 & 20.6 & $1.855 \pm 0.023$ & 17.0 & 1.1–2.3\
2013 HL$_{153}$ & 13:55:43.86 & -12:09:13.6 & 13:54:48.74 & -12:08:39.2 & 20.7 & $2.248 \pm 0.031$ & 16.4 & 1.4–3.1\
2013 HW$_{154}$ & 13:56:53.62 & -12:00:12.3 & 13:56:10.75 & -11:53:46.7 & 20.7 & $2.161 \pm 0.024$ & 16.5 & 1.3–3.0\
2014 RB$_{7}$ & 13:58:04.43 & -11:32:31.5 & 13:57:05.82 & -11:27:53.0 & 20.7 & $1.612 \pm 0.034$ & 17.6 & 0.8–1.8\
2005 JW$_{17}$ & 13:56:17.44 & -12:21:14.4 & 13:55:21.70 & -12:15:14.8 & 20.8 & $1.421 \pm 0.014$ & 18.1 & 0.6–1.4\
(363018) & 13:56:07.10 & -11:40:50.3 & 13:55:05.22 & -11:37:26.0 & 20.8 & $1.815 \pm 0.029$ & 17.2 & 0.9–2.1\
2013 HD$_{152}$ & 13:55:05.71 & -11:50:47.7 & 13:54:08.88 & -11:46:49.8 & 20.8 & $1.041 \pm 0.010$ & 19.1 & 0.4–0.9\
(252335) & 13:57:41.02 & -12:16:19.7 & 13:56:50.53 & -12:11:10.8 & 20.9 & $2.039 \pm 0.025$ & 17.0 & 1.1–2.4\
2013 EJ$_{149}$ & 13:55:15.19 & -12:05:18.5 & 13:54:21.95 & -12:03:58.5 & 20.9 & $1.994 \pm 0.023$ & 17.0 & 1.1–2.4\
2013 HT$_{152}$ & 13:55:25.39 & -12:18:20.0 & 13:54:42.73 & -12:11:54.8 & 20.9 & $2.091 \pm 0.024$ & 16.9 & 1.1–2.5\
(290287) & 13:54:03.09 & -12:14:31.2 & 13:53:14.64 & -12:09:45.1 & 20.9 & $1.866 \pm 0.047$ & 17.3 & 0.9–2.1\
(266193) & 13:55:51.80 & -11:34:01.8 & 13:55:00.94 & -11:28:34.3 & 20.9 & $1.825 \pm 0.034$ & 17.3 & 0.9–2.0\
2007 TO$_{45}$ & 13:57:28.75 & -11:33:26.0 & 13:56:28.68 & -11:26:38.1 & 20.9 & $1.176 \pm 0.009$ & 18.8 & 0.5–1.0\
2013 HS$_{155}$ & 13:57:29.86 & -11:56:05.6 & 13:56:39.91 & -11:53:21.0 & 20.9 & $1.782 \pm 0.020$ & 17.4 & 0.9–1.9\
2013 HG$_{154}$ & 13:56:20.80 & -12:18:22.2 & 13:55:25.95 & -12:18:09.3 & 20.9 & $2.091 \pm 0.046$ & 16.8 & 1.2–2.6\
2013 HQ$_{156}$ & 13:58:08.94 & -12:05:07.7 & 13:57:12.76 & -11:59:24.1 & 20.9 & & &\
2013 HX$_{150}$ & 13:54:03.63 & -11:36:45.1 & 13:53:21.99 & -11:27:33.2 & 21.0 & $1.881 \pm 0.045$ & 17.4 & 0.9–2.0\
2006 QG$_{123}$ & 13:56:07.54 & -11:23:39.2 & 13:55:20.15 & -11:16:16.6 & 21.1 & $2.384 \pm 0.117$ & 16.5 & 1.3–3.0\
pt2166 & & & 13:56:29.89 & -11:17:54.3 & 21.1 & & &\
2013 HO$_{151}$ & 13:54:33.31 & -11:55:59.7 & 13:53:38.87 & -11:53:11.9 & 21.2 & $1.838 \pm 0.025$ & 17.6 & 0.8–1.8\
2013 EL$_{135}$ & 13:56:45.77 & -11:27:08.8 & 13:55:40.59 & -11:19:46.5 & 21.2 & $1.184 \pm 0.013$ & 19.1 & 0.4–0.9\
2013 HO$_{152}$ & 13:55:18.94 & -12:13:50.1 & 13:54:23.16 & -12:09:13.6 & 21.2 & $1.466 \pm 0.017$ & 18.4 & 0.5–1.2\
2013 HH$_{156}$ & 13:58:00.78 & -12:20:14.8 & 13:57:11.96 & -12:14:06.5 & 21.3 & $1.987 \pm 0.048$ & 17.4 & 0.9–1.9\
2013 HE$_{154}$ & 13:56:16.70 & -11:49:21.6 & 13:55:21.92 & -11:46:20.5 & 21.3 & $2.090 \pm 0.039$ & 17.3 & 0.9–2.1\
2014 QX$_{347}$ & 13:57:41.05 & -11:59:03.3 & 13:56:42.29 & -11:54:47.9 & 21.3 & $1.791 \pm 0.024$ & 17.8 & 0.7–1.6\
2005 SC$_{212}$ & 13:57:57.96 & -12:16:01.2 & 13:57:07.44 & -12:10:49.7 & 21.3 & $1.897 \pm 0.028$ & 17.6 & 0.8–1.8\
2013 HZ$_{153}$ & 13:56:04.72 & -11:59:18.2 & 13:55:06.00 & -11:53:40.2 & 21.3 & $1.375 \pm 0.016$ & 18.7 & 0.5–1.1\
2013 HN$_{155}$ & 13:57:20.33 & -11:59:56.9 & 13:56:31.24 & -11:55:04.5 & 21.3 & $1.929 \pm 0.033$ & 17.5 & 0.8–1.9\
2013 HX$_{155}$ & 13:57:36.20 & -11:25:36.7 & 13:56:45.26 & -11:22:11.2 & 21.3 & $1.882 \pm 0.027$ & 17.6 & 0.8–1.8\
2013 HB$_{154}$ & 13:56:12.13 & -11:40:37.8 & 13:55:18.18 & -11:37:30.7 & 21.3 & $1.472 \pm 0.016$ & 18.4 & 0.5–1.2\
2013 HY$_{152}$ & 13:55:29.52 & -12:03:23.6 & 13:54:28.42 & -12:01:12.8 & 21.3 & $0.987 \pm 0.009$ & 19.8 & 0.3–0.7\
2013 HO$_{153}$ & 13:55:46.85 & -11:48:48.2 & 13:54:50.63 & -11:47:12.7 & 21.3 & $1.167 \pm 0.012$ & 19.2 & 0.4–0.8\
2013 HR$_{153}$ & 13:55:50.88 & -11:59:58.9 & 13:54:53.75 & -11:56:10.9 & 21.3 & $1.050 \pm 0.011$ & 19.6 & 0.3–0.7\
2013 HN$_{151}$ & 13:54:33.39 & -11:54:20.1 & 13:53:42.55 & -11:48:20.8 & 21.4 & $2.141 \pm 0.039$ & 17.2 & 1.0–2.1\
(313002) & 13:56:59.35 & -12:02:55.6 & 13:56:13.74 & -11:59:05.5 & 21.4 & $2.548 \pm 0.047$ & 16.6 & 1.3–2.8\
2005 NG$_{1}$ & 13:55:12.84 & -11:47:57.8 & 13:54:23.57 & -11:40:26.0 & 21.4 & $2.182 \pm 0.050$ & 17.2 & 1.0–2.1\
2013 HA$_{155}$ & 13:56:57.33 & -11:32:29.3 & 13:55:57.97 & -11:28:57.0 & 21.5 & $1.474 \pm 0.022$ & 18.7 & 0.5–1.1\
2013 HW$_{151}$ & 13:54:49.35 & -11:27:27.0 & 13:54:00.57 & -11:22:30.6 & 21.5 & $1.798 \pm 0.028$ & 18.0 & 0.7–1.5\
2013 HH$_{152}$ & 13:55:07.25 & -12:17:40.8 & 13:54:09.87 & -12:10:23.4 & 21.5 & $1.005 \pm 0.010$ & 19.9 & 0.3–0.6\
2013 EL$_{150}$ & 13:57:46.50 & -12:06:40.1 & 13:56:58.71 & -12:03:42.5 & 21.6 & $2.491 \pm 0.063$ & 16.9 & 1.1–2.4\
2013 HF$_{155}$ & 13:57:03.68 & -11:25:08.1 & 13:56:13.84 & -11:20:48.6 & 21.6 & $2.025 \pm 0.039$ & 17.7 & 0.8–1.7\
2013 HG$_{153}$ & 13:55:37.73 & -11:35:17.0 & 13:54:49.50 & -11:31:22.4 & 21.6 & $2.013 \pm 0.035$ & 17.7 & 0.8–1.7\
2013 HB$_{156}$ & 13:57:41.68 & -12:18:53.5 & 13:56:54.80 & -12:14:04.2 & 21.6 & $2.010 \pm 0.040$ & 17.7 & 0.8–1.7\
2013 HK$_{156}$ & 13:58:02.77 & -11:45:57.0 & 13:57:11.64 & -11:38:29.3 & 21.6 & $1.055 \pm 0.021$ & 20.0 & 0.3–0.6\
2013 HH$_{155}$ & 13:57:08.66 & -12:05:25.5 & 13:56:08.04 & -12:06:24.0 & 21.6 & $1.047 \pm 0.013$ & 19.9 & 0.3–0.6\
2007 TA$_{195}$ & 13:57:00.64 & -11:49:27.2 & 13:56:05.20 & -11:43:50.6 & 21.6 & $1.737 \pm 0.030$ & 18.2 & 0.6–1.3\
pt2159 & & & 13:56:07.89 & -11:20:00.7 & 21.6 & & &\
2013 HU$_{151}$ & 13:54:44.40 & -11:34:40.1 & 13:53:54.44 & -11:33:05.3 & 21.7 & $2.449 \pm 0.050$ & 17.1 & 1.0–2.3\
2013 HY$_{153}$ & 13:56:03.69 & -12:19:37.7 & 13:55:04.63 & -12:15:29.8 & 21.7 & $1.479 \pm 0.025$ & 18.9 & 0.4–1.0\
2013 HP$_{152}$ & 13:55:19.16 & -11:45:47.3 & 13:54:25.72 & -11:45:08.8 & 21.7 & $1.825 \pm 0.032$ & 18.1 & 0.6–1.4\
2013 HH$_{154}$ & 13:56:21.46 & -12:14:44.3 & 13:55:34.85 & -12:10:39.2 & 21.7 & $1.797 \pm 0.030$ & 18.1 & 0.6–1.4\
2013 HJ$_{151}$ & 13:54:24.81 & -11:41:45.1 & 13:53:21.65 & -11:40:27.3 & 21.7 & $1.044 \pm 0.017$ & 20.0 & 0.3–0.6\
2013 HM$_{155}$ & 13:57:16.50 & -11:32:35.0 & 13:56:26.00 & -11:24:38.1 & 21.8 & $1.916 \pm 0.042$ & 18.1 & 0.6–1.4\
2013 EW$_{149}$ & 13:56:01.59 & -12:14:41.7 & 13:55:08.01 & -12:13:56.3 & 21.8 & $2.197 \pm 0.046$ & 17.5 & 0.8–1.8\
2013 HB$_{153}$ & 13:55:33.52 & -12:00:36.8 & 13:54:38.64 & -11:57:42.5 & 21.8 & $1.758 \pm 0.031$ & 18.4 & 0.6–1.3\
2007 TM$_{173}$ & 13:55:15.62 & -12:13:06.7 & 13:54:15.69 & -12:08:02.3 & 21.8 & $1.638 \pm 0.030$ & 18.7 & 0.5–1.1\
2013 HT$_{154}$ & 13:56:44.08 & -11:33:59.5 & 13:55:44.17 & -11:29:59.4 & 21.8 & $1.397 \pm 0.026$ & 19.1 & 0.4–0.9\
2013 HC$_{156}$ & 13:57:45.47 & -12:12:27.6 & 13:56:56.62 & -12:07:59.5 & 21.8 & $1.922 \pm 0.039$ & 18.0 & 0.7–1.5\
2013 HJ$_{156}$ & 13:58:01.75 & -11:26:58.7 & 13:57:12.88 & -11:23:45.4 & 21.8 & $1.890 \pm 0.071$ & 18.1 & 0.6–1.4\
2013 HJ$_{152}$ & 13:55:07.32 & -12:12:40.1 & 13:54:10.93 & -12:09:50.6 & 21.8 & $0.910 \pm 0.009$ & 20.6 & 0.2–0.4\
2013 HF$_{151}$ & 13:54:23.57 & -11:48:42.2 & 13:53:25.93 & -11:44:39.3 & 21.8 & $1.856 \pm 0.037$ & 18.2 & 0.6–1.4\
(329364) & 13:55:25.22 & -11:30:00.6 & 13:54:36.11 & -11:25:08.5 & 21.9 & $2.329 \pm 0.060$ & 17.4 & 0.9–2\
2011 WH$_{135}$ & 13:56:21.56 & -12:08:58.9 & 13:55:24.31 & -12:03:50.9 & 21.9 & $1.801 \pm 0.038$ & 18.4 & 0.6–1.2\
2013 HC$_{153}$ & 13:55:33.91 & -12:02:51.3 & 13:54:33.38 & -11:57:35.3 & 21.9 & $1.371 \pm 0.021$ & 19.3 & 0.4–0.8\
2013 HV$_{153}$ & 13:56:00.26 & -11:24:23.9 & 13:55:08.09 & -11:21:06.0 & 21.9 & $1.939 \pm 0.044$ & 18.2 & 0.6–1.4\
2013 HP$_{153}$ & 13:55:48.78 & -11:39:53.2 & 13:54:55.22 & -11:33:29.5 & 21.9 & $1.631 \pm 0.042$ & 18.7 & 0.5–1.1\
2013 HF$_{153}$ & 13:55:37.25 & -12:15:22.7 & 13:54:43.82 & -12:14:18.2 & 21.9 & $1.774 \pm 0.031$ & 18.5 & 0.5–1.2\
2013 HU$_{152}$ & 13:55:25.65 & -11:34:11.5 & 13:54:37.24 & -11:29:19.5 & 21.9 & $2.000 \pm 0.046$ & 18.1 & 0.7–1.5\
2013 HZ$_{155}$ & 13:57:38.29 & -11:35:19.6 & 13:56:47.16 & -11:30:05.6 & 21.9 & $1.759 \pm 0.034$ & 18.4 & 0.5–1.2\
2013 HE$_{152}$ & 13:55:06.48 & -12:06:11.1 & 13:54:19.45 & -12:02:22.9 & 21.9 & $2.029 \pm 0.047$ & 18.0 & 0.7–1.5\
2013 HH$_{153}$ & 13:55:41.87 & -11:59:14.3 & 13:54:54.97 & -11:54:55.4 & 21.9 & $1.881 \pm 0.038$ & 18.2 & 0.6–1.3\
2013 HT$_{153}$ & 13:55:57.40 & -11:41:45.6 & 13:54:56.89 & -11:38:36.9 & 21.9 & $1.190 \pm 0.018$ & 19.8 & 0.3–0.7\
2013 HA$_{151}$ & 13:54:10.97 & -11:38:53.1 & 13:53:12.43 & -11:35:44.6 & 21.9 & $0.968 \pm 0.015$ & 20.5 & 0.2–0.5\
2013 HQ$_{153}$ & 13:55:49.08 & -11:36:22.1 & 13:54:59.41 & -11:34:40.6 & 22.0 & $2.487 \pm 0.086$ & 17.3 & 0.9–2.0\
2013 HV$_{155}$ & 13:57:35.91 & -11:31:55.9 & 13:56:55.17 & -11:21:18.7 & 22.0 & $1.955 \pm 0.059$ & 18.2 & 0.6–1.4\
2013 HY$_{154}$ & 13:56:56.16 & -11:25:45.1 & 13:56:03.42 & -11:24:15.9 & 22.0 & $1.990 \pm 0.052$ & 18.1 & 0.6–1.4\
2013 HM$_{153}$ & 13:55:44.24 & -11:42:03.3 & 13:54:55.46 & -11:38:49.2 & 22.0 & $1.924 \pm 0.040$ & 18.2 & 0.6–1.3\
2013 HM$_{154}$ & 13:56:33.84 & -11:37:34.3 & 13:55:46.69 & -11:33:38.7 & 22.0 & $1.732 \pm 0.039$ & 18.6 & 0.5–1.1\
2013 HU$_{153}$ & 13:55:59.94 & -12:12:39.1 & 13:55:03.91 & -12:07:47.1 & 22.0 & $1.131 \pm 0.022$ & 20.1 & 0.3–0.6\
2013 HY$_{150}$ & 13:54:04.82 & -11:45:19.9 & 13:53:11.24 & -11:41:44.6 & 22.0 & $1.531 \pm 0.077$ & 19.1 & 0.4–0.9\
2013 HD$_{154}$ & 13:56:15.68 & -11:49:38.3 & 13:55:29.28 & -11:45:32.4 & 22.0 & $1.830 \pm 0.041$ & 18.4 & 0.5–1.2\
2013 HN$_{156}$ & 13:54:03.28 & -12:13:59.3 & 13:53:16.27 & -12:04:08.6 & 22.0 & $1.494 \pm 0.050$ & 19.2 & 0.4–0.9\
2013 HG$_{151}$ & 13:54:24.44 & -12:08:17.1 & 13:53:35.37 & -11:58:55.6 & 22.1 & $1.766 \pm 0.042$ & 18.7 & 0.5–1.1\
2013 HT$_{151}$ & 13:54:43.42 & -11:32:19.6 & 13:54:00.95 & -11:25:37.7 & 22.1 & $1.990 \pm 0.048$ & 18.2 & 0.6–1.3\
2013 HL$_{151}$ & 13:54:29.31 & -11:34:18.6 & 13:53:30.53 & -11:30:50.9 & 22.1 & $1.114 \pm 0.020$ & 20.3 & 0.2–0.5\
2013 HZ$_{150}$ & 13:54:09.88 & -11:47:12.0 & 13:53:16.54 & -11:40:51.0 & 22.1 & $1.046 \pm 0.033$ & 20.4 & 0.2–0.5\
2013 HO$_{154}$ & 13:56:34.35 & -11:29:43.4 & 13:55:47.68 & -11:25:57.1 & 22.2 & $2.255 \pm 0.072$ & 17.9 & 0.7–1.6\
2004 VA$_{38}$ & 13:55:49.82 & -11:26:32.1 & 13:55:05.00 & -11:22:46.5 & 22.2 & $2.567 \pm 0.151$ & 17.4 & 0.9–2.0\
2013 HV$_{151}$ & 13:54:49.27 & -11:32:35.0 & 13:54:02.84 & -11:28:41.0 & 22.2 & $2.241 \pm 0.102$ & 17.9 & 0.7–1.6\
2013 HM$_{156}$ & 13:58:04.09 & -11:37:09.8 & 13:57:04.94 & -11:32:26.3 & 22.2 & $1.748 \pm 0.101$ & 18.8 & 0.5–1.0\
2013 HN$_{154}$ & 13:56:33.27 & -12:08:33.4 & 13:55:36.75 & -12:01:27.7 & 22.3 & $1.582 \pm 0.043$ & 19.3 & 0.4–0.8\
2010 WU$_{71}$ & 13:57:14.36 & -11:37:30.5 & 13:56:29.51 & -11:33:37.4 & 22.3 & $2.649 \pm 0.099$ & 17.3 & 0.9–2.0\
2013 HM$_{152}$ & 13:55:11.52 & -11:30:36.6 & 13:54:15.56 & -11:26:38.4 & 22.3 & $1.895 \pm 0.068$ & 18.6 & 0.5–1.1\
2013 HS$_{152}$ & 13:55:25.03 & -11:58:57.5 & 13:54:40.73 & -11:52:23.2 & 22.3 & $2.027 \pm 0.054$ & 18.3 & 0.6–1.3\
2013 HU$_{155}$ & 13:57:34.02 & -11:46:28.5 & 13:56:40.76 & -11:42:46.7 & 22.3 & $1.863 \pm 0.051$ & 18.6 & 0.5–1.1\
2013 HP$_{155}$ & 13:57:23.23 & -11:47:28.4 & 13:56:35.55 & -11:42:50.5 & 22.3 & $1.832 \pm 0.058$ & 18.7 & 0.5–1.1\
2013 EH$_{150}$ & 13:55:45.70 & -12:04:16.0 & 13:54:43.82 & -12:00:25.0 & 22.3 & $1.205 \pm 0.033$ & 20.2 & 0.2–0.5\
2013 HP$_{151}$ & 13:54:36.22 & -12:18:46.5 & 13:53:44.50 & -12:11:59.2 & 22.3 & $1.449 \pm 0.044$ & 19.5 & 0.3–0.7\
2013 HT$_{155}$ & 13:57:33.09 & -11:43:17.2 & 13:56:36.24 & -11:41:13.1 & 22.3 & $1.334 \pm 0.031$ & 19.8 & 0.3–0.6\
pt2129 & 13:54:11.62 & -12:12:03.4 & & & 22.3 & & &\
2013 HR$_{151}$ & 13:54:38.05 & -11:25:36.4 & 13:53:52.59 & -11:22:25.6 & 22.4 & $2.555 \pm 0.097$ & 17.6 & 0.8–1.8\
2013 HL$_{156}$ & 13:58:02.87 & -11:57:45.9 & 13:57:10.24 & -11:55:42.9 & 22.4 & $2.128 \pm 0.383$ & 18.3 & 0.6–1.3\
2013 HO$_{155}$ & 13:57:20.78 & -12:07:19.2 & 13:56:21.71 & -12:01:27.1 & 22.4 & $1.675 \pm 0.069$ & 19.1 & 0.4–0.9\
2013 EG$_{150}$ & 13:56:36.71 & -11:40:02.0 & 13:55:47.16 & -11:36:19.5 & 22.4 & $2.026 \pm 0.068$ & 18.4 & 0.6–1.2\
2013 HE$_{155}$ & 13:57:03.44 & -12:19:07.1 & 13:56:17.10 & -12:15:06.6 & 22.4 & $1.990 \pm 0.069$ & 18.5 & 0.5–1.2\
2013 HA$_{156}$ & 13:57:40.07 & -12:22:20.9 & 13:56:52.98 & -12:18:19.3 & 22.4 & $2.066 \pm 0.200$ & 18.3 & 0.6–1.3\
2013 HK$_{155}$ & 13:57:10.91 & -12:05:06.9 & 13:56:27.89 & -11:59:07.3 & 22.4 & $2.317 \pm 0.101$ & 18.0 & 0.7–1.5\
2014 QA$_{423}$ & 13:54:47.40 & -12:15:01.2 & 13:53:50.85 & -12:10:06.5 & 22.5 & $1.860 \pm 0.054$ & 18.9 & 0.5–1.0\
2013 HW$_{153}$ & 13:56:03.16 & -11:56:43.4 & 13:55:14.57 & -11:52:36.5 & 22.5 & $1.756 \pm 0.065$ & 19.0 & 0.4–0.9\
pt2122 & 13:54:56.09 & -11:23:26.6 & & & 22.5 & & &\
2013 HE$_{153}$ & 13:55:36.40 & -12:10:36.9 & 13:54:46.17 & -12:09:00.4 & 22.6 & $2.170 \pm 0.062$ & 18.4 & 0.6–1.3\
2013 HP$_{154}$ & 13:56:34.39 & -11:44:38.9 & 13:55:31.11 & -11:39:44.8 & 22.6 & $1.274 \pm 0.047$ & 20.3 & 0.2–0.5\
2013 HL$_{155}$ & 13:57:10.94 & -11:48:30.6 & 13:56:11.14 & -11:43:57.4 & 22.6 & $1.476 \pm 0.041$ & 19.8 & 0.3–0.7\
2013 HL$_{154}$ & 13:56:30.44 & -12:18:36.2 & 13:55:38.82 & -12:13:14.6 & 22.6 & $1.623 \pm 0.053$ & 19.4 & 0.4–0.8\
2013 HG$_{155}$ & 13:57:04.92 & -12:11:30.0 & 13:56:17.61 & -12:07:24.6 & 22.6 & $1.823 \pm 0.059$ & 19.0 & 0.4–0.9\
2013 HK$_{151}$ & 13:54:29.78 & -11:23:19.6 & 13:53:43.85 & -11:18:55.2 & 22.6 & $1.756 \pm 0.088$ & 19.1 & 0.4–0.9\
2013 HC$_{151}$ & 13:54:17.68 & -12:12:15.3 & 13:53:21.01 & -12:05:56.8 & 22.6 & $1.132 \pm 0.051$ & 20.7 & 0.2–0.4\
2013 HN$_{152}$ & 13:55:12.12 & -11:57:29.1 & 13:54:23.94 & -11:47:35.9 & 22.6 & $0.898 \pm 0.020$ & 21.4 & 0.1–0.3\
2013 HF$_{152}$ & 13:55:07.05 & -11:46:25.3 & 13:54:19.07 & -11:41:35.3 & 22.6 & & &\
2013 HW$_{152}$ & 13:55:27.14 & -12:01:02.5 & 13:54:29.10 & -11:53:31.2 & 22.7 & $1.317 \pm 0.070$ & 20.3 & 0.2–0.5\
2013 HD$_{155}$ & 13:57:03.40 & -12:11:21.4 & 13:56:18.82 & -12:03:36.3 & 22.7 & $2.129 \pm 0.174$ & 18.6 & 0.5–1.1\
2013 HD$_{156}$ & 13:57:47.80 & -11:24:26.4 & 13:56:54.87 & -11:21:32.0 & 22.7 & $2.157 \pm 0.135$ & 18.6 & 0.5–1.2\
2010 RM$_{180}$ & 13:55:20.65 & -11:45:09.1 & 13:54:32.96 & -11:43:18.2 & 22.7 & $3.096 \pm 0.178$ & 17.1 & 1.0–2.2\
2013 HW$_{155}$ & 13:57:35.76 & -11:58:56.2 & 13:56:45.70 & -11:53:13.0 & 22.7 & $1.821 \pm 0.118$ & 19.2 & 0.4–0.9\
2013 HX$_{154}$ & 13:56:53.55 & -11:37:13.9 & 13:56:01.34 & -11:28:46.9 & 22.7 & $1.061 \pm 0.032$ & 21.0 & 0.2–0.4\
2013 HS$_{153}$ & 13:55:56.67 & -12:11:00.5 & 13:55:01.15 & -12:06:43.1 & 22.7 & $1.119 \pm 0.038$ & 20.8 & 0.2–0.4\
pt2156 & & & 13:56:06.69 & -12:04:07.8 & 22.7 & & &\
2013 HL$_{152}$ & 13:55:10.49 & -11:52:17.2 & 13:54:13.06 & -11:51:32.5 & 22.8 & $1.920 \pm 0.103$ & 19.0 & 0.4–0.9\
2013 HV$_{152}$ & 13:55:27.61 & -12:13:15.7 & 13:54:44.99 & -12:04:23.8 & 22.8 & $1.798 \pm 0.173$ & 19.2 & 0.4–0.8\
2013 HJ$_{154}$ & 13:56:24.06 & -11:37:12.6 & 13:55:34.78 & -11:33:29.7 & 22.8 & $1.969 \pm 0.095$ & 19.0 & 0.4–1\
2013 HH$_{151}$ & 13:54:25.63 & -12:08:34.0 & 13:53:42.43 & -12:02:02.7 & 22.8 & $1.955 \pm 0.156$ & 19.0 & 0.4–0.9\
2013 HE$_{151}$ & 13:54:22.04 & -11:45:24.0 & 13:53:21.47 & -11:41:15.4 & 22.8 & $1.326 \pm 0.040$ & 20.3 & 0.2–0.5\
2013 HZ$_{152}$ & 13:55:30.35 & -12:14:49.5 & 13:54:33.42 & -12:09:05.3 & 22.8 & $1.224 \pm 0.061$ & 20.6 & 0.2–0.4\
2013 HS$_{151}$ & 13:54:41.40 & -11:50:29.6 & 13:53:51.12 & -11:42:12.8 & 22.8 & $1.237 \pm 0.038$ & 20.6 & 0.2–0.5\
2013 HX$_{153}$ & 13:56:02.85 & -12:19:21.2 & 13:54:57.47 & -12:18:12.4 & 22.8 & $1.365 \pm 0.051$ & 20.2 & 0.2–0.5\
2013 HS$_{154}$ & 13:56:37.27 & -12:09:21.3 & 13:55:33.88 & -12:08:08.9 & 22.8 & $0.958 \pm 0.024$ & 21.4 & 0.1–0.3\
pt2138 & 13:57:54.43 & -11:25:52.5 & & & 22.8 & & &\
pt2147 & 13:54:13.36 & -11:51:02.5 & & & 22.8 & & &\
2013 HE$_{156}$ & 13:57:51.18 & -12:03:42.0 & 13:57:02.97 & -11:58:57.4 & 22.9 & $2.240 \pm 0.112$ & 18.5 & 0.5–1.2\
2013 HF$_{156}$ & 13:57:50.80 & -11:54:57.0 & 13:56:50.85 & -11:52:57.8 & 22.9 & $1.817 \pm 0.149$ & 19.4 & 0.4–0.8\
2013 HG$_{156}$ & 13:57:51.20 & -12:18:40.0 & 13:56:54.08 & -12:14:09.9 & 22.9 & $1.892 \pm 0.105$ & 19.1 & 0.4–0.9\
2013 HD$_{153}$ & 13:55:34.45 & -11:34:24.9 & 13:54:35.30 & -11:30:18.9 & 22.9 & $1.890 \pm 0.075$ & 19.1 & 0.4–0.9\
2013 HX$_{151}$ & 13:54:55.85 & -11:58:48.6 & 13:54:13.51 & -11:50:58.4 & 22.9 & $1.765 \pm 0.103$ & 19.5 & 0.3–0.8\
2013 HJ$_{155}$ & 13:57:09.87 & -12:19:18.2 & 13:56:22.23 & -12:15:18.8 & 22.9 & & &\
2013 HX$_{152}$ & 13:55:29.35 & -11:25:13.9 & 13:54:45.05 & -11:20:00.2 & 22.9 & $2.111 \pm 0.110$ & 18.8 & 0.5–1.0\
2013 HQ$_{154}$ & 13:56:35.52 & -12:04:42.1 & 13:55:48.92 & -11:59:55.0 & 23.0 & $2.425 \pm 0.230$ & 18.4 & 0.6–1.2\
2013 HR$_{152}$ & 13:55:24.32 & -11:50:19.7 & 13:54:35.81 & -11:41:35.5 & 23.0 & $1.659 \pm 0.108$ & 19.8 & 0.3–0.7\
2013 HR$_{154}$ & 13:56:36.48 & -11:25:56.3 & 13:55:41.57 & -11:20:48.5 & 23.0 & $1.730 \pm 0.167$ & 19.7 & 0.3–0.7\
2013 HM$_{151}$ & 13:54:33.27 & -11:50:22.8 & 13:53:51.21 & -11:43:12.3 & 23.0 & $1.997 \pm 0.433$ & 19.1 & 0.4–0.9\
2013 HY$_{151}$ & 13:54:55.15 & -11:23:54.7 & 13:54:00.28 & -11:20:30.6 & 23.0 & $1.770 \pm 0.105$ & 19.5 & 0.3–0.7\
2013 HK$_{152}$ & 13:55:08.72 & -12:15:41.9 & 13:54:18.06 & -12:11:53.1 & 23.0 & $1.674 \pm 0.052$ & 19.7 & 0.3–0.7\
2013 HR$_{155}$ & 13:57:29.21 & -12:11:44.5 & 13:56:39.31 & -12:08:02.4 & 23.0 & $1.695 \pm 0.264$ & 19.7 & 0.3–0.7\
2013 HF$_{154}$ & 13:56:18.62 & -12:03:46.5 & 13:55:21.57 & -11:58:59.1 & 23.0 & $1.254 \pm 0.084$ & 20.7 & 0.2–0.4\
2013 HY$_{155}$ & 13:57:36.92 & -12:15:30.7 & 13:56:51.25 & -12:11:21.9 & 23.0 & $3.247 \pm 0.210$ & 17.2 & 1.0–2.1\
2013 HZ$_{151}$ & 13:55:00.07 & -11:31:32.3 & 13:54:11.15 & -11:29:29.2 & 23.0 & & &\
pt2139 & 13:54:26.57 & -12:13:12.5 & & & 23.0 & & &\
pt2140 & 13:57:28.41 & -11:49:26.6 & & & 23.0 & & &\
2013 HC$_{152 }$ & 13:55:05.29 & -11:23:31.1 & 13:54:16.42 & -11:20:59.5 & 23.0 & $2.613 \pm 0.202$ & 18.1 & 0.6–1.4\
2013 HB$_{151}$ & 13:54:17.93 & -11:57:05.5 & 13:53:32.50 & -11:52:32.4 & 23.0 & $1.740 \pm 0.365$ & 19.6 & 0.3–0.7\
2014 SN$_{155}$ & 13:56:29.85 & -11:59:20.3 & 13:55:34.53 & -11:53:18.4 & 23.0 & $2.153 \pm 0.148$ & 18.7 & 0.5–1.1\
2013 HU$_{154}$ & 13:56:46.71 & -12:03:59.6 & 13:55:54.22 & -12:01:53.2 & 23.0 & $1.795 \pm 0.215$ & 19.5 & 0.3–0.8\
2013 HN$_{153}$ & 13:55:47.18 & -11:40:58.5 & 13:55:01.88 & -11:36:47.4 & 23.1 & & &\
2013 HD$_{151}$ & 13:54:19.64 & -12:17:35.3 & 13:53:35.22 & -12:09:42.7 & 23.1 & $2.112 \pm 0.373$ & 18.9 & 0.4–1.0\
2013 HB$_{152}$ & 13:55:04.86 & -11:29:55.8 & 13:54:18.64 & -11:25:47.8 & 23.1 & $2.022 \pm 0.111$ & 19.1 & 0.4–0.9\
2013 HA$_{152}$ & 13:55:00.25 & -11:47:23.8 & 13:54:09.56 & -11:38:21.2 & 23.1 & $1.258 \pm 0.038$ & 20.8 & 0.2–0.4\
2013 HA$_{153}$ & 13:55:32.82 & -11:30:33.2 & 13:54:33.72 & -11:29:28.3 & 23.1 & $1.153 \pm 0.045$ & 21.1 & 0.2–0.4\
2013 HQ$_{151}$ & 13:54:36.51 & -11:59:46.8 & 13:53:31.39 & -11:57:26.9 & 23.1 & $1.113 \pm 0.037$ & 21.2 & 0.2–0.3\
2013 HA$_{154}$ & 13:56:10.65 & -11:29:33.1 & 13:55:13.52 & -11:23:42.3 & 23.1 & $1.512 \pm 0.236$ & 20.2 & 0.2–0.5\
2013 HV$_{154}$ & 13:56:47.77 & -11:53:52.5 & 13:55:44.63 & -11:49:13.1 & 23.2 & $1.343 \pm 0.093$ & 20.7 & 0.2–0.4\
2013 HK$_{154}$ & 13:56:28.18 & -11:56:36.0 & 13:55:32.34 & -11:54:28.7 & 23.2 & $1.483 \pm 0.088$ & 20.3 & 0.2–0.5\
2013 HB$_{155}$ & 13:56:58.87 & -12:11:39.2 & 13:56:03.76 & -12:05:34.0 & 23.2 & $1.197 \pm 0.062$ & 21.1 & 0.2–0.4\
2013 HQ$_{155}$ & 13:57:24.31 & -11:54:55.2 & 13:56:30.76 & -11:51:50.6 & 23.2 & $1.768 \pm 0.136$ & 19.7 & 0.3–0.7\
2013 HG$_{152}$ & 13:55:06.92 & -11:28:03.9 & 13:54:13.46 & -11:25:01.8 & 23.2 & $1.690 \pm 0.089$ & 19.9 & 0.3–0.6\
pt2124 & 13:57:00.21 & -12:12:17.9 & & & 23.3 & & &\
2013 HZ$_{154}$ & 13:56:55.40 & -12:06:09.2 & 13:55:49.12 & -12:04:04.1 & 23.3 & $1.313 \pm 0.052$ & 20.9 & 0.2–0.4\
pt2150 & 13:55:23.10 & -11:25:56.6 & & & 23.3 & & &\
2013 HO$_{156}$ & 13:55:23.01 & -11:39:47.1 & 13:54:26.10 & -11:39:11.5 & 23.4 & $2.073 \pm 0.120$ & 19.2 & 0.4–0.9\
2013 HP$_{156}$ & 13:57:20.99 & -11:56:28.1 & 13:56:37.64 & -11:50:10.9 & 23.4 & $2.110 \pm 0.462$ & 19.3 & 0.4–0.8\
![Histograms of the absolute magnitudes (and corresponding physical diameters) for all asteroids (solid line) and for previously known objects (dashed line) measured in our data. The current census of the main belt becomes substantially incomplete at a diameter of about 2 km. By contrast, we have detected dozens of new asteroids in the 200-500 meter size range. \[fig:distmag\]](f10.eps)
As illustrated by Figure \[fig:distmag\], we have detected a large number of asteroids smaller than 500 meters, a size regime where the current census of the main belt has very low completeness. It would be possible to probe the statistics of asteroid sizes and absolute magnitudes in this regime using our data, but this would require an extensive completeness analysis that is beyond the scope of the current work. It would also be of limited scientific value, since the statistics of MBAs in this regime have been probed by @Gladman2009 and @Yoshida2007 using 4–8 meter telescopes. The fact that we have reached a comparable regime using only an 0.9-meter telescope shows the power of digital tracking. A digital tracking survey on a 4–8 meter telescope would easily probe fainter asteroids than have ever previously been detected, and would open up a new regime for statistical studies of the main belt — especially when combined with the distance determination method we describe in @curves.
Comparison of Digital Tracking and Conventional Methods {#sec:maincomp}
=======================================================
A Specific Case: 0.9-meter Digital Tracking and a 4-meter Conventional Survey
-----------------------------------------------------------------------------
To compare the sensitivity of our digital tracking analysis to that of conventional asteroid surveys, we use only the set of asteroids we have confirmed as genuine. Each was initially detected automatically as a $\ge 7\sigma$ source in one of our digital tracking trial stacks. They have been further subjected to manual evaluations, using conservative criteria as detailed in Section \[sec:mancheck\]. The final result is that some probably-real objects are rejected, but no (or no more than one) false positive could plausibly exist in the final list. Thanks to this conservatism, our asteroids could be used, e.g., to probe main belt size statistics without followup observations to confirm their reality. This is an important consideration because most of our new asteroids are faint enough that followup from other observatories cannot reasonably be expected, and this will be even more true of asteroids detected in future digital tracking surveys with larger telescopes.
Within the $\sim$1 deg$^2$ field of our observations, a total of 211 asteroids are confirmed in our April 19 data, including twelve that were recovered based on an initial detection in the April 20 images. The faintest of these 211 asteroids have $R$-band magnitudes of 23.4. The SKADS asteroid survey of @Gladman2009 used the 4-meter Mayall Telescope at Kitt Peak without digital tracking to reach a limiting magnitude of $R=23.5$ under much more favorable lunar conditions (lunar ages 2–6 days versus 10–11 days for us), and found an on-sky density of 210 asteroids/deg$^2$ at magnitudes brighter than $R=23.0$. The $R=23.5$ limiting magnitude of @Gladman2009 is not directly comparable to our value of $R=23.4$, since the former corresponds to roughly 50% completeness while the latter represents the faintest objects detected at any completeness. A robust derivation of our own 50% completeness limit, while feasible, requires a statistical analysis beyond the scope of the present work. Nevertheless, the similarity in on-sky number densities between our observations and those of @Gladman2009 suggests our true limiting magnitude at 50% completeness is close to $R=23.0$. Thus, while our observations are not quite as sensitive as those of the SKADS survey, we have probed a similar regime using a telescope with less than one-sixteenth the collecting area.
More General Advantages and Disadvantages of Digital Tracking {#sec:comp}
-------------------------------------------------------------
Some disadvantages of digital tracking should be considered when deciding whether to apply it or conventional methods to a particular science case. Digital tracking requires much longer times spent on each field, so a conventional survey can cover a larger angular area each night. Long digital tracking integrations average the seeing throughout the night; by contrast, conventional surveys sometimes serendipitously obtain very sharp images on which unusually faint objects can be detected.
Initial detection thresholds for digital tracking stacks can never be much below 7$\sigma$, or the false positive rate based on pure Gaussian statistics will be non-negligible (Section \[sec:autothresh\]). By contrast, conventional surveys that take at least three images per field can use lower thresholds for detecting objects on individual images because real objects will distinguish themselves from false positives by appearing in subsequent images along a consistent motion vector. In principle, the Gaussian false positive probability for a sequence of three independent 4$\sigma$ detections is lower than that for a single detection at 7$\sigma$. This advantage of the conventional method is partially balanced by the fact that cosmic rays, ghosts, and other artifacts typically enforce a higher detection threshold by making the statistics of individual images much more non-Gaussian than those of our trial stacks, which are made through a clipped median combine of over one hundred individual images. As an example, Pan-STARRS currently uses a 5$\sigma$ threshold [@Denneau2013].
The $\sqrt{t/\tau_M}$ sensitivity advantage for digital tracking versus conventional searches applies to detections at the same significance, and thus the true advantage of digital tracking is less than this factor when compared to conventional searches that use at least three images per field and apply a well-optimized moving object detection methodology like that of @Denneau2013. However, this does not contradict our claim of a factor of ten sensitivity increase for digital tracking. For example, under good conditions it is easy to obtain a digital tracking integration consisting of 180 individual exposures of length $\tau_M$ (Section \[sec:comp3\]), so that $\sqrt{t/\tau_M} = \sqrt{180} = 13.4$. An object detected at 7$\sigma$ in such an integration would therefore be 13.4 times fainter than an object detected at the same significance on a single image, and it would be 9.6 times fainter even than an object detected at 5$\sigma$ on a single frame. Thus, while the true advantage of the digital tracking survey in terms of discovery sensitivity is less than the nominal value of $\sqrt{180}$, it remains approximately a factor of ten.
The biggest advantage of digital tracking is that it enables the detection of asteroids and Kuiper Belt objects that are simply out of reach by any other method: too faint and/or moving too quickly. It thus allows statistical analyses of populations of extremely small objects, and could also extend the reach of surveys aimed at retiring terrestrial impact risk from potentially hazardous NEOs. Another significant advantage is that with digital tracking, every detection comes with a precise measurement of the asteroid’s motion, which can aid in determining distances [@curves] and orbital information even for extremely faint objects unlikely to be recovered over longer timescales of weeks to years.
Conclusion {#sec:conc}
==========
We have described the technique of digital tracking, focusing on its application to searches for faint asteroids and Kuiper Belt objects by stacking tens to hundreds of images from the current generation of large-format CCD mosaic imagers. Digital tracking is suitable for observations of near-Earth objects (NEOs), main belt asteroids (MBAs), and Kuiper Belt objects (KBOs). For all three classes of objects, it typically yields a factor of $\ge 10$ increase in the sensitivity of a telescope to faint moving objects, as compared to conventional techniques.
The linear, constant velocity approximation for objects’ motions greatly simplifies digital tracking computations, and is sufficient for all-night integrations on MBAs; two-hour integrations targeting NEOs transiting the observer’s meridian at 0.1 AU geocentric distance; and 12-night integrations centered on opposition and targeting KBOs. Integrations must be reduced to one hour for NEOs at a distance of 0.1 AU when they are observed far from the meridian (i.e. hour angle far from zero), and to three nights for KBOs one month from opposition. For MBAs, multi-night integrations are never possible using the linear constant velocity approximation, but all-night (8–9 hour) integrations are possible even when the objects are far from opposition. While previous digital tracking observations of outer Solar System objects have been very valuable scientifically, and @Zhai2014 have made a remarkable detection of a very small asteroid during a close approach to Earth, no previous work has exploited the full potential of digital tracking with large format CCD imagers.
The computational challenge of digital tracking depends on the size of the region of angular motion phase space that must be searched, and the fineness of the required sampling over this region. We have determined the appropriate parameters for digital tracking surveys targeting various classes of objects, and demonstrated that all of them are computationally tractable. Rapid analysis (hours to days) can be achieved with a supercomputer. In many cases, a high-end desktop workstation is sufficient if a few weeks are available to analyze each night’s data, and for a risk-retirement survey requiring faster processing a small cluster may be sufficient.
We have carried out the first digital tracking survey to target asteroids using a large-format CCD imager. Using an 0.9-meter telescope, we have detected MBAs fainter than 23rd magnitude in the $R$ band, thus probing a regime previously explored only with 4-meter and larger telescopes. Within a field of view of approximately 1 square degree, observed with all-night digital tracking integrations on the nights of April 19 and 20, 2013, we have detected a total of 215 asteroids (see Table \[tab:asteroids\]), of which only 59 were previously known. All 156 of the previously unknown asteroids were manually checked and confirmed to be real based on significant detection in multiple independent subsets of the data.
Among our 215 detected asteroids, 197 were measured on both April 19 and April 20 with sufficient precision that we could derive meaningful distances for them using Earth rotational reflex velocities as described in @curves. The resulting precise distances allow us to calculate absolute magnitudes and hence approximate diameters for our newly discovered asteroids. Our faintest objects have $H_R \sim 21.5$ mag and hence diameters of 130–300 meters depending on their albedo. While the current census of the main belt becomes substantially incomplete at a diameter of about 2 km, we have detected dozens of new asteroids in the 200–500 meter size range with an 0.9-meter telescope.
To conclude, we have successfully employed digital tracking to detect large numbers of previously unknown asteroids with a very small telescope. We have described the enormous potential of digital tracking, and demonstrated solutions to all significant problems with its large-scale implementation. Our methodology is completely scalable up to the largest telescopes in existence, and will allow them to detect fainter asteroids than have ever previously been imaged. Furthermore, the precise motion measurements that are intrinsically included in digital tracking detections allow accurate geocentric distances to be obtained for any asteroids observed on two nights [@curves]. While asteroidal science cases remain for which digital tracking is not the optimal solution, the time is ripe for its widespread deployment in the next generation of asteroid surveys.
Acknowledgments
===============
Based on observations at Kitt Peak National Observatory, National Optical Astronomy Observatory (NOAO Prop. ID: 2013A-0501; PI: Aren Heinze), which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
This publication makes use of the SIMBAD online database, operated at CDS, Strasbourg, France, and the VizieR online database (see @vizier).
This publication makes use of data products 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 have also made extensive use of information and code from @nrc. We have used digitized images from the Palomar Sky Survey (available from <http://stdatu.stsci.edu/cgi-bin/dss_form>), which were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope.
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[^1]: That is, in principle. In practice, stars, galaxies, and other stationary celestial objects should be removed using image subtraction before creating the digital tracking stacks.
[^2]: Note, however, that $\tau_{lin}$ for MBAs is always less than 24 hours: digital tracking integrations spanning multiple nights are not possible for these objects.
[^3]: http://www.scinethpc.ca/
[^4]: As the antisolar point is 180$^{\circ}$ from the sun by definition, we apply this constraint to the MPC ephemerides by requiring solar elongation $\ge 160^{\circ}$.
[^5]: We assume here that the effective resolution is approximately equal to the pixel scale of 2.6 arcsec, and thus $\tau_M$ for an object moving at 2,000 arcsec/hr is 2.6/2,000 = 0.0013 hr = 4.7 seconds.
[^6]: Strictly speaking this would only work with a next-generation imager having very short readtime. With a 20-second readtime (e.g. DECam), one could obtain 72 individual 30-second exposures in an hour, and the digital tracking advantage would still be $\sqrt{72} = 8.5$, close to a factor of 10.
[^7]: The WIYN Observatory is a joint facility of the University of Wisconsin-Madison, Indiana University, the National Optical Astronomy Observatory and the University of Missouri.
[^8]: The order of operations is important here: if the @lacos cosmic ray removal is applied first, it will trigger on the edges of the crosstalk artifacts from saturated stars and partially remove them, after which the crosstalk correction will introduce artifacts by attempting to subtract artifacts that have already been partially removed.
[^9]: We find the optimal threshold factor is near 15 for the current data set, but it could be widely different in other contexts.
[^10]: The four dimensions are the ordinary x and y dimensions of the images plus the two dimensions of the angular velocity space.
[^11]: A similar way of checking digital tracking detections was independently developed by @Fraser2008; see their Figure 2.
[^12]: One of the 144 two-night discoveries of our survey, it has been designated 2013 HY$_{153}$ by the MPC.
|
---
abstract: 'Measurements of electron drift properties in liquid and gaseous xenon are reported. The electrons are generated by the photoelectric effect in a semi-transparent gold photocathode driven in transmission mode with a pulsed ultraviolet laser. The charges drift and diffuse in a small chamber at various electric fields and a fixed drift distance of 2.0 cm. At an electric field of 0.5 kV/cm, the measured drift velocities and corresponding temperature coefficients respectively are $1.97 \pm 0.04$ mm/$\mu$s and $(-0.69\pm0.05)$%/K for liquid xenon, and $1.42 \pm 0.03$ mm/$\mu$s and $(+0.11\pm0.01)$%/K for gaseous xenon at 1.5 bar. In addition, we measure longitudinal diffusion coefficients of $25.7 \pm 4.6$ cm$^2$/s and $149 \pm 23$ cm$^2$/s, for liquid and gas, respectively. The quantum efficiency of the gold photocathode is studied at the photon energy of 4.73 eV in liquid and gaseous xenon, and vacuum. These charge transport properties and the behavior of photocathodes in a xenon environment are important in designing and calibrating future large scale noble liquid detectors.'
address:
- 'Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, USA'
- 'Brookhaven National Laboratory, Upton, NY 11973, USA'
- 'Physics Department, Colorado State University, Fort Collins, CO 80523, USA'
- 'SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA'
- 'Physics Department, McGill University, Montréal, Québec H3A 2T8, Canada'
- 'Erlangen Centre for Astroparticle Physics (ECAP), Friedrich-Alexander University Erlangen-Nürnberg, Erlangen 91058, Germany'
- 'Pacific Northwest National Laboratory, Richland, WA 99352, USA'
- 'Department of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada'
- 'Department of Physics, Duke University, and Triangle Universities Nuclear Laboratory (TUNL), Durham, NC 27708, USA'
- 'Physics Department, University of Illinois, Urbana-Champaign, IL 61801, USA'
- 'Institute for Theoretical and Experimental Physics named by A. I. Alikhanov of National Research Center “Kurchatov Institute”, Moscow 117218, Russia'
- 'Department of Physics, University of South Dakota, Vermillion, SD 57069, USA'
- 'Lawrence Livermore National Laboratory, Livermore, CA 94550, USA'
- 'Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, NY 12180, USA'
- 'TRIUMF, Vancouver, British Columbia V6T 2A3, Canada'
- 'Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China'
- 'Department of Physics, Laurentian University, Sudbury, Ontario P3E 2C6 Canada'
- 'Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada'
- 'Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China'
- 'Physics Department, Stanford University, Stanford, CA 94305, USA'
- 'Department of Physics and Physical Oceanography, University of North Carolina at Wilmington, Wilmington, NC 28403, USA'
- 'Department of Physics and CEEM, Indiana University, Bloomington, IN 47405, USA'
- 'Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada'
- 'Department of Physics, Drexel University, Philadelphia, PA 19104, USA'
- 'Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA'
- 'Amherst Center for Fundamental Interactions and Physics Department, University of Massachusetts, Amherst, MA 01003, USA'
- 'Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA'
- 'Wright Laboratory, Department of Physics, Yale University, New Haven, CT 06511, USA'
- 'Department of Physics, Colorado School of Mines, Golden, CO 80401, USA'
- 'IBS Center for Underground Physics, Daejeon 34126, Korea'
- 'LHEP, Albert Einstein Center, University of Bern, Bern CH-3012, Switzerland'
author:
- 'O. Njoya,'
- 'T. Tsang,'
- 'M. Tarka,'
- 'W. Fairbank,'
- 'K.S. Kumar,'
- 'T. Rao'
- 'T. Wager'
- 'S. Al Kharusi,'
- 'G. Anton,'
- 'I.J. Arnquist,'
- 'I. Badhrees,'
- 'P.S. Barbeau,'
- 'D. Beck,'
- 'V. Belov,'
- 'T. Bhatta,'
- 'J.P. Brodsky,'
- 'E. Brown,'
- 'T. Brunner,'
- 'E. Caden,'
- 'G.F. Cao,'
- 'L. Cao,'
- 'W.R. Cen,'
- 'C. Chambers,'
- 'B. Chana,'
- 'S.A. Charlebois,'
- 'M. Chiu,'
- 'B. Cleveland,'
- 'M. Coon,'
- 'A. Craycraft,'
- 'J. Dalmasson,'
- 'T. Daniels,'
- 'L. Darroch,'
- 'S.J. Daugherty,'
- 'A. De St. Croix,'
- 'A. Der Mesrobian-Kabakian,'
- 'R. DeVoe,'
- 'M.L. Di Vacri,'
- 'J. Dilling,'
- 'Y.Y. Ding,'
- 'M.J. Dolinski,'
- 'A. Dragone,'
- 'J. Echevers,'
- 'M. Elbeltagi,'
- 'L. Fabris,'
- 'D. Fairbank,'
- 'J. Farine,'
- 'S. Ferrara,'
- 'S. Feyzbakhsh,'
- 'R. Fontaine,'
- 'A. Fucarino,'
- 'G. Gallina,'
- 'P. Gautam,'
- 'G. Giacomini,'
- 'D. Goeldi,'
- 'R. Gornea,'
- 'G. Gratta,'
- 'E.V. Hansen,'
- 'M. Heffner,'
- 'E.W. Hoppe,'
- 'J. Hö[ß]{}l,'
- 'A. House,'
- 'M. Hughes,'
- 'A. Iverson,'
- 'A. Jamil,'
- 'M.J. Jewell,'
- 'X.S. Jiang,'
- 'A. Karelin,'
- 'L.J. Kaufman,'
- 'D. Kodroff,'
- 'T. Koffas,'
- 'R. Krücken,'
- 'A. Kuchenkov,'
- 'Y. Lan,'
- 'A. Larson,'
- 'K.G. Leach,'
- 'B.G. Lenardo,'
- 'D.S. Leonard,'
- 'G. Li,'
- 'S. Li,'
- 'Z. Li,'
- 'C. Licciardi,'
- 'Y.H. Lin,'
- 'P. Lv,'
- 'R. MacLellan,'
- 'T. McElroy,'
- 'M. Medina-Peregrina,'
- 'T. Michel,'
- 'B. Mong,'
- 'D.C. Moore,'
- 'K. Murray,'
- 'P. Nakarmi,'
- 'C.R. Natzke,'
- 'R.J. Newby,'
- 'Z. Ning,'
- 'F. Nolet,'
- 'O. Nusair,'
- 'K. Odgers,'
- 'A. Odian,'
- 'M. Oriunno,'
- 'J.L. Orrell,'
- 'G.S. Ortega,'
- 'I. Ostrovskiy,'
- 'C.T. Overman,'
- 'S. Parent,'
- 'A. Piepke,'
- 'A. Pocar,'
- 'J.-F. Pratte,'
- 'V. Radeka,'
- 'E. Raguzin,'
- 'S. Rescia,'
- 'F. Retière,'
- 'M. Richman,'
- 'A. Robinson,'
- 'T. Rossignol,'
- 'P.C. Rowson,'
- 'N. Roy,'
- 'J. Runge,'
- 'R. Saldanha,'
- 'S. Sangiorgio,'
- 'K. Skarpaas VIII,'
- 'A.K. Soma,'
- 'G. St-Hilaire,'
- 'V. Stekhanov,'
- 'T. Stiegler,'
- 'X.L. Sun,'
- 'J. Todd,'
- 'T. Tolba,'
- 'T.I. Totev,'
- 'R. Tsang,'
- 'F. Vachon,'
- 'V. Veeraraghavan,'
- 'S. Viel,'
- 'G. Visser,'
- 'C. Vivo-Vilches,'
- 'J.-L. Vuilleumier,'
- 'M. Wagenpfeil,'
- 'M. Walent,'
- 'Q. Wang,'
- 'M. Ward,'
- 'J. Watkins,'
- 'M. Weber,'
- 'W. Wei,'
- 'L.J. Wen,'
- 'U. Wichoski,'
- 'S.X. Wu,'
- 'W.H. Wu,'
- 'X. Wu,'
- 'Q. Xia,'
- 'H. Yang,'
- 'L. Yang,'
- 'Y.-R. Yen,'
- 'O. Zeldovich,'
- 'J. Zhao,'
- 'Y. Zhou'
- 'and T. Ziegler'
bibliography:
- 'LongitudinalDiffusion\_nEXOCollaboration.bib'
title: 'Measurements of electron transport in liquid and gas Xenon using a laser-driven photocathode'
---
\[sec:level1\]Introduction
==========================
In recent years liquid xenon (LXe) time projection chambers (TPCs) have proven to be excellent detectors in the searches for neutrinoless double beta decay [@albert2014search; @albert2017searches] and dark matter [@aprile2016xenon100; @aprile2016low; @akerib2017results; @aprile2017xenon1t] as well as for other low-background physics searches [@albert2017searches; @albert2018searchnucleon]. Xenon is attractive because it can be chemically and radiologically purified to very high levels and its high density and atomic number provide substantial shielding against background radiation [@albert2014improved]. Detectors ranging from a few to hundreds of kilograms have produced high-quality results, paving the way for future tonne-scale detectors; XENONnT, LZ, and PandaX are multi-tonne LXe detectors for the direct detection of dark matter [@aprile2017xenon1t; @collaboration2019observation; @akerib2018projected; @mount2017lux; @cui2017dark]. nEXO, the proposed successor to EXO-200, is a ton-scale experiment that aims to perform a search for neutrinoless double beta decay of $^{136}$Xe, with a design half-life sensitivity of $\sim 10^{28}$ years [@albert2017sensitivity; @kharusi2018nexo]. Proposals for large GXe TPCs for neutrinoless double beta decay of $^{136}$Xe have also been put forth.
A distinctive attribute of liquid noble elements, and xenon in particular, is the simultaneous production of ionization electrons and scintillation photons when exposed to ionizing radiation [@conti2003correlated; @neilson2009characterization]. The longitudinal position of events in a LXe TPC is reconstructed using the delay between primary scintillation and the detection of ionization charge. However this can be complicated not only by electron losses but also by effects of electron diffusion which smear the spatial resolution of event localization; for this reason understanding electron diffusion is important. This is especially relevant when the drift distances are large (>1 m) as in ton-scale experiments [@albert2017sensitivity; @kharusi2018nexo; @aprile2017xenon1t; @mount2017lux]. Because the diffusion of electrons in high electric fields is generally anisotropic [@robson1972thermodynamic], longitudinal (in the direction of the drift field) and transverse diffusion need to be measured separately. For most TPCs the longitudinal diffusion is smaller than the transverse diffusion due to the longitudinal confinement along the electric field [@aprile2010liquid]. Literature on measurement of the longitudinal diffusion of electric charges in LXe is sparse.
The inherent self-shielding of next generation LXe detectors presents new challenges for calibration and the monitoring of small time variations in LXe properties such as the electron lifetime. With a drift distance of 1.3 m in nEXO, an electron lifetime better than 10 ms is desired and the uncertainty in the lifetime correction must be at most 3% [@albert2014improved; @kharusi2018nexo]. *In-situ* continuous monitoring of LXe properties will be one of the important factors to obtain optimal performance. The investigation of the feasibility of producing calibrated amounts of charge with laser pulses transported to gold photocathodes embedded in the main TPC cathode is an important facet of the nEXO R&D effort.
A gold cathode was used in a laboratory-scale setup as a laser-driven electron source to perform measurements of the longitudinal diffusion coefficient and drift speed, and their temperature dependence, of electrons as they drift in electric fields ranging from 70 V/cm to 1000 V/cm. These measurements are reported in this paper. Section \[sec:level2\] contains a description of the experimental apparatus. In section \[sec:level3\], the data acquisition and analysis are described. The results are shown and discussed in section \[sec:level4\] before concluding remarks in section \[sec:level5\].
\[sec:level2\]Experimental Apparatus
====================================
Three methods exist in the literature for the measurement of longitudinal electron diffusion: un-gridded [@hunter1986electron; @kusano2012electron], gridded [@davies1989measurements], and shuttered [@takatou2011drift] drift cells. In this work, the gridded cell arrangement was employed because of its simplicity and similarity to LXe TPCs.
In this experiment, pulses of electrons are generated by back-illuminating a semitransparent gold photocathode with a pulsed UV laser. A schematic diagram of the drift chamber and accompanying electronics is shown in Figure \[fig:grid\]. The drift chamber is housed in a 0.5 L cylinder with an inner diameter of 5 cm and a height of 27 cm. Elements of the cell include an optical fiber for photon delivery, a drift stack composed of a gold (Au) photocathode, three copper field shaping rings, and an anode grid followed by a copper anode. The photocathode is a 22-nm thick gold film ($\sim$50% transmission at 266 nm) thermally evaporated on a 1-mm thick, 10-mm diameter sapphire disk. The sapphire disk sits in a macor holder at the top of the drift stack. The photon source is a pulsed frequency-quadrupled [(Spectra Physics: Evolution X,)]{} Nd:YLF laser with 4.73-eV photon energy and 71-ns pulse-width FWHM at 100-Hz repetition rate. The pulsed UV photons are coupled into a 4.35-m long 600-$\mu$m core diameter solarization-resistant UV fused silica optical fiber (ThorLabs: UM22-600) with a numerical aperture (NA) of 0.22 [@thorlabs]. One end of the portion of fiber that is in vacuum directly contacts the back surface of the sapphire disk. The laser light back-illuminates the Au photocathode (which has a work function of 4.2 eV in vacuum [@li2016measurement]), releasing photoelectrons into the LXe, where they drift along the lines of the uniform electric field. The UV laser energy per pulse is measured using a J3-02 Molectron pyroelectric energy detector fitted with a fiber-coupled adapter. The maximum laser energy deposited on the surface of the photocathode per pulse is $\sim$0.64 $\mu$J; this corresponds to an energy density of approximately 0.2 mJ/cm$^2$, far below the measured 6 mJ/cm$^2$ damage threshold of the thin gold film. The overall fiber transmission in the UV is $\sim$30%. Therefore, after taking into account the $\sim$50% attenuation of the photocathode, the optical throughput of the fiber to the surface of the photocathode is $\sim$15% at UV wavelengths.
![\[fig:grid\]A schematic diagram of the experimental setup. A UV laser back-illuminates a gold photocathode via a 600-$\mu$m fused-silica fiber. The space between the photocathode and the grid defines the drift region, where a uniform drift field is maintained with the help of copper field shaping rings. Electrons are collected on a Cu anode and their signal is amplified with the charge-sensitive preamplifier.](drift_stack.png)
A 280-$\mu$m thick phosphor-bronze disk with a clear inner diameter (ID) of 8 mm is negatively biased and makes electrical contact with the photocathode; it is securely fastened to a macor holder with ceramic screws. To ensure the uniformity of the drift field, three copper field shaping rings (2 mm thick, 21.3-mm outer diameter, 15.7-mm ID) are spaced at a 5-mm pitch between the cathode and the anode. These rings are precisely locked in place by four slotted alumina rods. The drift distance can be modified by adding or removing field shaping rings as needed and by shifting the cathode and anode assembly. A series of 1-G$\Omega$ cryogenic- and vacuum-compatible resistors electrically connect the photocathode, the field shaping rings, and the grounded anode grid. A 300-$\mu$m thick Kapton spacer is sandwiched between the anode grid and the anode. The bias voltage of +300 V is applied on the anode disk for the collection of drifted electrons. The grid (35-$\mu$m Ni-Cu wire width with 350-$\mu$m pitch) is mounted onto a 250-$\mu$m thick phosphor-bronze disk with an ID of 8 mm, and all three components are secured by ceramic screws to a macor holder at the bottom of the drift stack. Due to the low thermal expansion coefficients of macor and alumina, the change in drift distance (and hence the drift field) with temperature is negligible. The anode grid is kept at ground. The collection field—the field between the anode and the anode grid—is kept constant at 10000 V/cm to maximize transmission of electrons. The drift field between the cathode and the anode grid is varied from 70 V/cm to 1000 V/cm.
To perform the quantum efficiency measurements described in section \[sec:level4\]-D, a second grid (not shown in Figure \[fig:grid\]) identical to the anode grid is added in front the cathode. This grid is mounted on an additional phosphor-bronze plate and is separated from the cathode by a 1-mm thick macor insulator; the net distance from the surface of the cathode to the grid is calculated to be $1.28 \pm 0.02$ mm, accounting for the combined thickness of the plate between the cathode and the macor insulator. The extraction field between the photocathode surface and the cathode/upper grid is tuned to always be half the drift field. This is done to ensure optimal transmission through the grid [@bunemann1949design].
Charges arriving at the anode are converted to voltage by a BNL IO535 or a Cremat-CR-Z-110-HV charge-sensitive preamplifier followed by an Ortec 474 timing filter amplifier with 100-ns shaping time. The preamplifier is calibrated by studying its response when a known amount of charge from a function generator is injected into it. Preamplifier and amplified shaped signals are both recorded on a 5 GS/s Agilent digital oscilloscope. The sweep trigger is the output of a <1-ns risetime fast photodiode that intercepts a small portion of the frequency doubled Nd:YLF green laser beam. At the laser energy of $\sim$0.64 $\mu$J the typical number of drifting electrons ranges from $8\times10^4$ to $4\times10^5$ per pulse at bias fields of 70 to 1000 V/cm.
We employ all stainless steel pipes and valves in the xenon gas purification and recovery system. Furthermore, macor and alumina were chosen for drift stack construction to minimize out-gassing. All HV feedthroughs and thermocouples are Kapton-insulated. With these considerations and proper UHV handling techniques, vacuum levels in the low $10^{-6}$ torr range are achieved before the cell is filled with xenon.
The xenon is liquefied by immersing the cell in a cold ethanol bath (155 K). Two thermocouples are mounted in the cell, one near the anode and the other near the cathode, to monitor the temperature of the liquid; they also serve as level sensors during the initial fill-up (when the drift region is completely filled with LXe both thermocouples will read the same temperature). Liquefaction is also confirmed visually through a glass viewport located at the top of the cell. Note that during the gas measurements the cell can only be pressurized to a maximum of 2 bar because of the glass viewport. Prior to liquefaction, 99.999% pure GXe is fed through a dry ice cold trap to remove water vapor, followed by a SAES purifier (a heated Zr/Al alloy getter capable of achieving ppb impurity levels)[@saes]. The purity level as determined from an estimate of the electron lifetime from a double-gridded measurement ranges from a few $\mu$s to about 35 $\mu$s. At the end of each run the xenon is recovered via cryopumping to a clean stainless steel tank. There is no active gas xenon recirculation.
\[sec:level3\]Raw Waveform Analysis and Systematic Uncertainties
================================================================
Waveforms of collected electron bunches are recorded for various drift fields, laser energies, and temperatures using the digital oscilloscope. These waveforms are analyzed and experimental parameters such as electron signal amplitude, delay, and width are extracted.
![\[fig:traces1\]A typical raw preamplifier electron-charge signal trace (black), a background trace (red), and a background-subtracted trace (blue) in LXe are shown.](raw_vs_bkgd_traces.png)
A typical raw preamplifier trace (black) is shown in Figure \[fig:traces1\] along with a background (red) trace and a background-subtracted (blue) trace. The laser pulse trigger defines ‘time-zero’ $t = 0$. The small rise of the signal near $t = 0$ within region 1 in Figure \[fig:traces1\] is due to electrons generated at the anode grid by the $\sim$50% laser light penetrating through the photocathode. This signal, which does not depend on the drift field and is proportional to the laser pulse energy, has no impact on the electrons generated from the photocathode.
The dominant step in the preamplifier signal occurs when the photoelectron bunch originating from the photocathode is collected by the anode. The time delay of the step and its risetime can be used to determine the drift time and the longitudinal width of the electron bunch, respectively. In the absence of longitudinal diffusion and with no Coulomb repulsion of the electron bunch, this step would have a fast risetime limited only by the response of the preamplifier convoluted with the initial electron bunch temporal width which is dictated by the laser pulse-width. The slow exponential decay (>100 $\mu$s) is due to the RC time constant of the preamplifier.
The growth of the raw signal in region 2 of Figure \[fig:traces1\] between 100 ns and the arrival time of the electrons at the anode indicates the presence of an additional background signal. There is qualitative evidence that this is the induction signal due to the motion of the electron bunch between the cathode and anode grid that results from imperfect shielding of the anode by the anode grid. To obtain a best approximation to the drift signal of the electron bunch region 3, the induction background is subtracted from the raw signal. The induction background signal is collected independently for each drift field by setting the anode bias voltage to 0V while maintaining the cathode at the same negative bias and the anode grid at ground; this voltage configuration ensures that a minimal amount of the electron bunch originating from the photocathode is collected at the anode.
![\[fig:traces2\]A representative set of background-subtracted preamplifier (top panel) and shaping amplifier (bottom panel) electron-charge signal traces at different drift fields in LXe is shown.](sample_preampandshaper_traces.png)
A set of background-subtracted preamplifier signal traces taken at various drift fields is shown in Figure \[fig:traces2\]. It can be seen that decreasing drift fields are accompanied by increased drift times and longer risetimes. The reduction in signal height with lower drift field is the result of decreased quantum efficiency of the photocathode in LXe at lower electric fields. The longer risetime is a manifestation of increased electron bunch longitudinal spread at the anode that results primarily from longitudinal diffusion, as discussed in section \[sec:level4\]-D.
Separate fits to the preamplifier and the shaping amplifier signals are performed for consistency. The preampflier trace is fitted with:
$$f(t,\sigma,\tau,A) = A \times (1+\operatorname{erf}(\frac{t-t_d}{\sigma\sqrt{2}})) \times \exp(-\frac{t-t_d}{\tau}),$$
where $A$ is the amplitude, $t_d$ is the arrival time at the anode, the width $\sigma$ is the standard deviation of the Gaussian, and $\tau$ is the preamplifier RC time. The parameters $A$, $t_d$, and $\sigma$ for each waveform are determined by a least-squares fit. Similarly, the shaped signal is fitted with
$$g(t,\sigma,\tau,A) = A \times \exp(-\frac{(t-t_d)^{2}}{2\sigma^{2}}) \times \exp(-\frac{t-t_d}{\tau}).$$
The preamplifier and the shaping-amplifier fit results agree to within 0.8% for the time delays and 1.1% for the widths.
The drift time is calculated by taking the difference between the arrival time at the anode $t_d$ and an offset of $t_0 = 47 \pm 7$ ns. This offset time accounts for the travel delay in the optical fiber as well as the combined instrument delay of the preamplifier, oscilloscope, and cable mismatch, and is derived from the photoemission signal in vacuum. There is an additional delay time from the anode grid to the anode of 0.1 $\mu$s which is treated as a systematic error. The value of 0.1 $\mu$s is obtained by assuming a drift speed of 3 mm/$\mu$s at 10000 V/cm [@yoshino1976effect]. The resulting error ranges from 0.4% to 1.2% depending on the drift field. This is the largest systematic error on the drift time.
Other sources of systematic errors were considered and are summarized in Table \[tab:table1\]. In the case of the electron signal width $\sigma$ the largest average systematic error contribution (6.6%) comes from the background subtraction. This value is the difference between $\sigma$ extracted from the raw traces and $\sigma$ from the background-subtracted traces; it thus accounts for any potential error in signal width measurements due to the background subtraction.
Temperature variations between cathode and anode are maintained to less than 0.3 K and contribute less than 1% uncertainty to the delay and the width of the signal. The impact of temperature is discussed in detail in section \[sec:level4\]-C.
The laser energy was found to have a significant impact on the signal width in LXe due to electron Coulomb repulsion and will be discussed in section \[sec:level4\]-B. The width uncertainty associated with a correction applied for Coulomb repulsion is listed in Table \[tab:table1\].
--------------------------------- ---------- ----------
**error source** **** ****
laser shot to shot fluctuations 0.11% 2.8%
background subtraction 0.1% 6.6%
anode grid to anode distance 0.4-1.2% NA
temperature 0.36%/K NA
waveform model error 0% 0.5%
Coulomb repulsion $<$ 0.1% 1.0-4.1%
--------------------------------- ---------- ----------
: \[tab:table1\] Summary of systematic uncertainties.
\[sec:level4\]Results and Discussion
====================================
Electron Drift Velocity
-----------------------
The drift velocity is given by
$$v = \frac{d}{t},$$
where $d$ is the drift distance and $t=t_d-t_0$ is the drift time. Here the drift distance between the cathode and the anode grid mesh is $d = 20.0 \pm 0.1$ mm. Figure \[fig:vvsE\] shows the measured electron drift velocity as a function of drift field in LXe and GXe. The error bars include statistical (average of multiple runs) and the systematic uncertainties described in the previous section and summarized in Table \[tab:table1\]. At a drift field of 500 V/cm the measured drift velocity is 1.97 $\pm$ 0.04 mm/$\mu$s in LXe and 1.42 $\pm$ 0.03 mm/$\mu$s in GXe. It is evident that electrons drift faster in LXe than in GXe, which is in agreement with the literature [@aprile2010liquid; @miller1968charge; @yoshino1976effect; @pack1992longitudinal].
![\[fig:vvsE\]The field dependence of electron drift velocity in LXe is shown. The present work is shown in blue squares with error bars. Other published values are also displayed with measurement temperature: EXO-200 (Albert [@albert2017measurement]), XENON100 (Aprile [@aprile2014analysis]), LUX (Akerib [@akerib2014first]), Gushchin et al. [@gushchin1982electron], and Miller et al [@miller1968charge]. Measurements of drift velocity in GXe are also shown for comparison (red squares).](speed_vs_field.png)
Previous reported measurements [@miller1968charge; @gushchin1982electron; @aprile2014analysis; @akerib2014first; @albert2017measurement] are also shown in Figure \[fig:vvsE\]. The variance in reported electron drift velocity values for LXe at a given electric field is not well understood. The weak temperature dependence of the LXe drift velocity, discussed in section \[sec:level4\]-C, is insufficient to account for the spread in reported measurements.
![\[fig:GvvsE\]The electron drift velocity versus reduced field in GXe is shown. The reduced field, given by $E/N$ is in Townsend (Td) and $N$ is the GXe number density in the cell. Our measurements (blue circles) are in good agreement with those reported in [@kusano2012electron] (green crosses), [@pack1992longitudinal] (black diamonds), and [@hunter1988low] (red squares).](GXe_speed_vs_rfield.png)
For GXe, the measured drift velocity plotted against the reduced field $E/N$ is shown in Figure \[fig:GvvsE\] where $E$ is the drift field and $N$ is the GXe number density. The reduced field (in units of Townsend or 10$^{-17}$ Vcm$^{2}$) ranges from 0.1 Td to 2.6 Td. [@pack1992longitudinal]. The results reported here are in agreement with previously published measurements [@kusano2012electron; @pack1992longitudinal; @hunter1988low].
Longitudinal Diffusion
----------------------
A key feature of TPCs is the ability to accurately reconstruct events in 3D. The spread of the intrinsic electron bunch as it propagates due to diffusion can reduce the reconstruction accuracy thereby limiting the position resolution. The phenomenon is discussed in [@li2016measurement; @agnes2018electroluminescence] and references therein. Transverse diffusion of electrons in LXe has been measured at a range of electric fields [@albert2017measurement; @tadayoshi1982recent]. Four measurements of longitudinal diffusion in LXe have been reported in the literature [@sorensen2011anisotropic; @mei2011thesis; @shibumura2009; @hogenbirk2018field].
In the absence of Coulomb repulsion, the longitudinal diffusion coefficient $D_L$ in terms of the drift distance $d$, the drift time $t$, and the temporal width (standard deviation) in the longitudinal direction $\sigma_L$ is given by:
$$D_{L} = \frac{d^{2}\sigma_{L}^{2}}{2t^{3}}.
\label{eq:DL}$$
In Equation \[eq:DL\], $\sigma_L$ is given by:
$$\sigma_L^2 = \sigma^2 - \sigma_0^2,
\label{eq:DL2}$$
where $\sigma$ is the measured electron pulse width obtained by fitting the anode preamplifier signal and $\sigma_0$ is the initial broadening due to the laser pulse width (30 ns) and intrinsic preamplifier risetime of 35.3 $\pm$ 0.3 ns from fits to the calibrated preamplifier waveforms. This gives $\sigma_0 = 46.3 \pm 5.3$ ns.
Electrons are generated in bunches of about 10$^5$ or more before they begin to drift. Because of the short laser pulses used, the charge density is high enough in LXe that Coulomb repulsion becomes a significant factor in the growth of the bunch from its initial width to the width at the time of the measurement. To gauge this effect, the growth of the electron bunch is modeled as the electrons drift. This model is used to determine the Coulomb contribution to the measured $\sigma$.
Our simplified model of Coulomb interactions assumes an initial uniform charge distribution of ellipsoidal shape where the initial radius $w_{\parallel}$(0) along the longitudinal dimension is given by:
$$w_{\parallel}(t=0) = v(E) \times \Delta_{l},
\label{eq:w_para}$$
where $v(E)$ is the electron drift speed at the applied drift field $E$ and $\Delta_l$ is the laser pulse temporal $1/e$ half-width. The initial transverse radius $w_{\perp}$(0) is the calculated laser spatial $1/e$ half-width after propagation through the 1-mm thick sapphire plate (on which the gold is evaporated). The propagation of such an ellipsoid in an electric field in vacuum is described in detail in [@grech2011coulomb]. The electrons at the front of the ellipsoid experience a greater longitudinal field $E+E_{\parallel}$, and move faster than the electrons at the back of the ellipsoid, which experience field $E-E_{\parallel}$ where $E_{\parallel}$ is the longitudinal Coulomb field at the surface of the ellipsoid. Thus the ellipsoid spreads under the combined effects of diffusion and Coulomb repulsion according to:
$$\frac{dw_{\parallel}}{dt} = {\beta}E_{\parallel}+\frac{2D_L}{w_{\parallel}},
\label{eq:CMw_para}$$
$$\frac{dw_{\perp}}{dt} = {\mu}E_{\perp}+\frac{2D_T}{w_{\perp}},
\label{eq:CMw_perp}$$
where $\beta = dv/dE$ is the slope of the drift velocity change at a particular E field, $E_{\perp}$ is the transverse Coulomb field, $\mu$ is the low-field electron mobility, and $D_T$ is the transverse diffusion coefficient. The mobility $\mu$ is calculated at low fields to approximate the fact that there is no drift field in the transverse direction. $\beta$ is calculated from the measured dependence of drift velocity on the E field (Figure \[fig:vvsE\]). The diffusion terms in Equations \[eq:CMw\_para\] and \[eq:CMw\_perp\] account phenomenologically for radius change with time due to diffusion. $E_{\parallel}$ and $E_{\perp}$ are extracted from fits to the numerical simulations given in [@grech2011coulomb]:
$$E_{\perp} = \frac{3{\lambda}Q}{4{\pi\epsilon}w_{\perp}w_{\parallel}}\frac{0.5}{1+0.76\alpha}
\label{eq:CME_perp}$$
and
$$E_{\parallel} = \frac{3{\lambda}Q}{4{\pi\epsilon}w_{\parallel}^2}(1-\frac{1}{1+1.54\alpha^{0.36}}),
\label{eq:CME_para}$$
with $\alpha = w_{\perp}/w_{\parallel}$. $Q$ is the electron bunch total charge, $\lambda$ is a charge scale factor to account for differences between the uniform charge distribution with sharp edge of the model and the approximately Gaussian charge distribution of reality, and $\epsilon = \kappa\epsilon_0$ is the permittivity of LXe with dielectric constant $\kappa$ = 1.9 [@lide1995crc]. The temporal width $\sigma_{th}$ (standard deviation) of the ellipsoid at the anode is:
$$\sigma_{th} = \frac{w_{\parallel}(t)}{v\sqrt{2}}.$$
The model outputs a $D_L$ value that minimizes the residuals between $\sigma_{th}$ (model) and $\sigma$ (data) at each drift field. The input parameters are $\beta$, $D_T$, $\mu$, $Q$, $\lambda$, and the drift time t. The charge $Q$ is set by the laser energy and applied drift field; $\beta$ and the drift time are set by the drift field. A range of $\mu$ values corresponding to previous drift speed measurements (see Figure \[fig:vvsE\]) were tried and found to have no significant effect on $\sigma_{th}$. $D_T$ is taken from [@albert2017measurement] and has only a small effect on $\sigma_{th}$.
To determine the appropriate $\lambda$ value to use in the model, a SIMION [@simion] simulation of electron transport in GXe at 1.5 bar and 295K was compared to the Coulomb model for similar conditions. The simulation tracks the path of each electron with an average of 5 steps between collisions. The global Coulomb field of a specified number of electrons is included in the calculations. Gaussian-distributed electrons are released near the cathode and begin to drift. A low applied electric field of 50 V/cm was chosen so that a significant Coulomb repulsion effect could be seen in GXe. A collision cross section value of $1.2 \times 10^{-15}$ cm$^2$ was selected so that the drift time in the simulation agreed with the measured value at 500 V/cm. The transverse broadening in model and simulation were matched by adjusting the $D_T$ value in the Coulomb model.
For each of the four total charges simulated in GXe, the temporal width of the bunch and an error limit were determined by a Gaussian fit to a histogram of the electron arrival times at the anode. The results are shown in red circles in Figure \[fig:wvsLE2\]. For Q $\sim$ 2$\times$10$^{5}$ electrons, the broadening due to Coulomb repulsion is not very large; however the broadening becomes more significant as Q increases to 8$\times$10$^{5}$ electrons. The model is fitted to the simulated GXe data by varying $D_L$ and $\lambda$ to minimize $\chi^2$. The best fit is shown by the solid blue triangles in Figure \[fig:wvsLE2\]. The error limits on $\lambda$ are determined from the $\pm1\sigma$ limit on $\chi^2(\lambda)$. The model curves representing these error limits are shown by the $\pm1\sigma$ dashed blue lines in Figure \[fig:wvsLE2\]. The Coulomb effect with the full charge $\lambda=1$ is clearly much larger than in the simulation. The result is $\lambda = 0.30\pm0.10$. For comparison, one might expect that the appropriate charge fraction to use in the simplified model, should correspond to that within the $1/e$ radius ($w$) of the Gaussian distribution. For an ellipsoidal Gaussian distribution, this charge fraction is 0.23.
![\[fig:wvsLE2\]Electron signal temporal widths versus total electron charge in GXe, for various charge fractions $\lambda$. The temporal widths calculated from the model (blue) include Coulomb repulsion and are in good agreement with those obtained from simulations (red).](width_vs_charge.png)
The value $\lambda = 0.30 \pm 0.10$ is used to determine $D_L$ at each field in LXe by matching the model $\sigma_{th}$ to the experimental $\sigma$. The Coulomb correction (relative to Coulomb-free calculation of $D_L$) ranges from 6.1% to 17.4%. The Coulomb repulsion model error for $D_L$ at each drift field is determined from the error limits on $\lambda$ and ranges from 1.0% to 4.1% (last entry in Table \[tab:table1\]).
The longitudinal diffusion coefficient $D_L$ in LXe is plotted as a function of drift field in Figure \[fig:DLvsE\]. At 500 V/cm, $D_L = 25.7\pm4.6$ cm$^2$/s. The error bars include both statistical and systematic errors. Each data point on Figure \[fig:DLvsE\] is an average of two distinct measurements. The statistical error is calculated as the standard deviation of the mean for the pair of measurements at each field.
![\[fig:DLvsE\] Electron longitudinal diffusion coefficient $D_L$ versus drift field in LXe. Values from this work (blue circles) and measurements from [@hogenbirk2018field] (magenta squares), [@sorensen2011anisotropic] (black triangle), [@mei2011thesis] (gray triangle), and Shibamura (green triangle) [@shibumura2009]. Also shown are the transverse diffusion coefficient $D_T$ from EXO-200 [@albert2017measurement] (hollow red diamonds), and [@tadayoshi1982recent] (hollow black squares).](LXe_DL_vs_E.png)
Previous measurements of $D_L$ include a set of $D_L$ values for fields ranging from 15 V/cm to 493 V/cm [@hogenbirk2018field], a single $D_L$ value at 700 V/cm [@mei2011thesis], a single $D_L$ value at 730 V/cm [@sorensen2011anisotropic], and a set of $D_L$ values for fields ranging from 3.6 kV/cm to 6.8 kV/cm [@shibumura2009]. These are shown in Figure \[fig:DLvsE\]. Our measurements agree with those of [@hogenbirk2018field] at higher fields but are somewhat higher at low fields. For comparison, the transverse diffusion coefficient $D_T$ values at various fields are shown [@albert2017measurement; @tadayoshi1982recent]. It is notable that $D_L$ rises as the field decreases and becomes comparable to $D_T$ at $\sim$ 100 V/cm.
In the case of GXe, the measured ratio $D_L/\mu$ as a function of reduced field $E/N$, where $\mu = v/E$ is the electron mobility, is shown in Figure \[fig:GDLvsE\]. Here $D_L$ = 149 $\pm$ 23 cm$^2$/s at 0.5 kV/cm. The Coulomb repulsion model gave no significant Coulomb correction for the GXe data. The values from [@pack1992longitudinal] are overlaid in Figure \[fig:GDLvsE\] for comparison. The overall drift field dependence of $D_L/\mu$ for electrons in GXe are in good agreement with [@pack1992longitudinal]. Measurements at lower $E/N$ values have also been reported [@kusano2012electron; @mcdonald2019electron].
![\[fig:GDLvsE\] GXe $D_L/\mu$ versus reduced field $E/N$ at room temperature is shown. The reduced field is given by $E/N$. Results from Pack [@pack1992longitudinal] are shown in black squares.](GXe_DL_vs_rE.png)
Temperature Dependence
----------------------
The temperature dependence of the drift speed and longitudinal diffusion coefficient were studied to help quantify the importance of the temperature uniformity inside a LXe TPC and assess whether it could explain the spread of drift velocity values found in the literature. These results are summarized in Figure \[fig:vvsT\]. At the drift field of 500 V/cm, the electron drift speed in LXe decreases linearly with temperature at a rate of $-0.69 \pm 0.05$ %/K, in qualitative agreement with previous measurements at various drift fields [@benetti1993simple; @tadayoshi1982recent]. On the other hand, no significant temperature dependence of $D_L$ in LXe (also shown in Figure \[fig:vvsT\]) is found within the uncertainties of the measurements.
 Temperature dependence of the electron drift velocity in LXe (blue circles). Also included are measurements of Doke (green pyramids) [@tadayoshi1982recent] and Benetti (red diamonds) [@benetti1993simple]. (b) Temperature dependence of the electron longitudinal diffusion coefficient in LXe (blue squares).](LXe_vs_temperature.png)
In GXe, both $D_L$ and the drift velocity increase linearly with temperature; this is shown in Figure \[fig:GvvsT\]. The drift velocity increases with temperature in GXe, while it decreases in LXe. A similar behavior has been reported in LAr and GAr [@li2016measurement].
![\[fig:GvvsT\]Temperature dependence of electron drift velocity (top panel) and longitudinal diffusion coefficient $D_L$ (bottom panel) in GXe at 500 V/cm drift field and pressure $P = 1.53 \pm 0.01$ bar.](GXe_vs_temperature.png)
Quantum Efficiency of Au Photocathode
-------------------------------------
The quantum efficiency ($QE$) is defined as the number of electrons leaving the photocathode per UV photon incident on the back surface of the photocathode. To perform this measurement, charges leaving the cathode were measured. This was enabled by the addition a mesh 1.28 $\pm$ 0.02 mm away from the cathode, as described in Section-\[sec:level2\]. The $QE$ was studied as a function of extraction field in vacuum, room temperature GXe (1.5 bar), and in LXe. Due to changes in absolute photocathode response, possibly due to environmental changes, only qualitative behavior and ranges of values are reported. Measurements were made at extraction fields ranging from 25 V/cm to 1000 V/cm. In vacuum the measured $QE$ depends only very weakly on extraction field $E$; however it grows monotonically in both LXe and GXe.
QEs of $\sim 5 \times 10^{-6}$ are obtained in vacuum. This is consistent with values reported by [@li2016measurement] for gold photocathodes similar to the ones used here. The $QE$ in LXe and the $QE$ in GXe range between $(1-5) \times 10^{-7}$. Remarkably, the $QE$ in LXe and GXe were nearly identical at each extraction field. This is in contrast to the case of argon [@li2016measurement] where it was found that the $QE$ in GAr is an order of magnitude higher than the $QE$ in LAr.
Future measurements will be performed to assess the stability and more precisely measure the $QE$ and work function of these gold photocathodes in LXe.
\[sec:level5\]Conclusions
=========================
A small drift cell for the study of electron drift properties in LXe was built and operated. A gold photocathode was back-illuminated and the released photoelectrons were investigated in a LXe environment and GXe. The longitudinal diffusion coefficient $D_L$ was measured in LXe as a function of drift field for st[the first time at fields below 1 kV/cm]{} a range of fields. The increase in $D_L$ with decreasing drift fields was qualitatively consistent with theoretical predictions [@robson1972thermodynamic; @lowke1969theory; @skullerud1969longitudinal]. Within the experimental uncertainty, no significant variation of $D_L$ with respect to temperature was observed. The field dependence of electron drift velocity in LXe and GXe agrees well with previously published values.
The use of calibrated charge bunches using a gold photocathode as a laser-driven electron source for *in-situ* monitoring of electron lifetime is being further investigated for the nEXO design. It is important to assess the long term stability of the photocathode quantum efficiency as well as the precision and accuracy of the technique. A new cell will feature improved laser power monitoring, simultaneous measurements from two photocathodes, and *in-situ* source calibration crosschecks.
Acknowledgment
==============
This work has been supported by the Offices of Nuclear and High Energy Physics within the DOE Office of Science, and NSF in the United States, by NSERC, CFI, FRQNT, and NRC in Canada, by IBS in Korea, by RFBR (18-02-00550) in Russia, and by CAS and NSFC in China. This work was supported in part by Laboratory Directed Research and Development (LDRD) programs at Brookhaven National Laboratory (BNL).
References
==========
|
---
abstract: 'We present an experimental study of the movement of individual particles in a layer of vertically shaken granular material. High-speed imaging allows us to investigate the motion of beads within one vibration period. This motion consists mainly of vertical jumps, and a global ordered drift. The analysis of the system movement as a whole reveals that the observed bifurcation in the flight time is not adequately described by the Inelastic Bouncing Ball Model. Near the bifurcation point, friction plays and important role, and the branches of the bifurcation do not diverge as the control parameter is increased. We quantify the friction of the beads against the walls, showing that this interaction is the underlying mechanism responsible for the dynamics of the flow observed near the lateral wall.'
address:
- |
Depto. de Física y Mat. Apl., Facultad de Ciencias ,\
Universidad de Navarra, 31080 Pamplona, Spain
- 'CPMOH, Université de Bordeaux I, 33405 Talence Cedex, France'
author:
- 'J.M. Pastor'
- 'D. Maza'
- 'I. Zuriguel'
- 'A. Garcimartín'
- 'J.-F. Boudet'
title: |
Time resolved particle dynamics\
in granular convection
---
,
,
,
Granular flow ,Convection
45.70.-n ,45.70.Qj
Introduction
============
Granular convection is a patent example of how collective movement of grains can give rise to an ordered yet complex behavior. As soon as 1831, M. Faraday [@Faraday] reported a long range flow developed in a vertically shaken granular layer. This flow is called *granular convection* because of the likeness between it and the movement of a liquid layer heated from below. Although many works have dwelt on this topic, the origin of this convective movement, and in particular the role of the lateral walls or the boundaries, is not fully understood. In 1989, P. Evesque and J. Rajchenbach [@evesque] published an article where they showed experimentally that the threshold for collective motions to appear corresponds to the acceleration of gravity $g$. This is why the acceleration of the external driving is often given in the form of an adimensional number $\Gamma=\frac{A \omega
^2}{g}$, where $A$ is the amplitude and $\omega$ is the frequency of the forcing. They also described that a heap grows changing the shape of the *free surface* of the medium, as a consequence of the grains circulating in a “convective” fashion.
Almost at the same time, C. Laroche [*[et al.]{}*]{} [@laroche1] reported both the importance of interstitial air for the deformation of the granular layer and the development of a compactation front that splits the layer into two zones, a *solid* one and a *liquid* one. The origin of convection, according to these authors, would be directly related to the air circulating among the grains. This effect determines the rising of material at the center of the layer and a flow of grains going down near the walls, which influence the material by increasing its porosity with respect to the central zone. Nevertheless, subsequent studies [@clément][@durand] have revealed that the walls can by themselves provide the driving force for convection, at least for a two-dimensional geometry. This point was finally demonstrated by the works of the Chicago group [@nagel], who used NMR techniques to show that wall friction does affect the velocity profile of the particles. It should be noted that the shaking was conveyed in this case in the form of short pulses, or “taps”, separated by rest periods much longer than the pulse itself. At the same time, enlightening ideas were put forward, and tested numerically, setting the framework in which to understand the collective behavior of granular matter. Following an analogy with hydrodynamics, models were developed that qualitatively predicted the long range ordering of a shaken granular media, even though simplifications sometimes made them unrealistic [@kadanoff] [@Bourzosky] [@Hayakawa] [@rosa].
Above the convective threshold, a granular layer can also undergo a rich array of instabilities. In 1989 S. Douady *et al.* [@douady] showed that beyond a certain value of $\Gamma$ the flight of the grains experiences a period doubling bifurcation, in a way essentially equivalent to a solid body that is placed on a vibrating plate [@holmes]. As a layer of granular material is strongly dissipative, it can be considered perfectly inelastic, and its behavior as a whole can be modeled by an inelastic ball on a vibrating plate. This simple model, known as the Inelastic Bouncing Ball Model (IBBM) has been discussed by several authors [@pieranski][@tufillaro][@mehta] and successfully used to describe the temporal dynamics of a shallow granular layer (without convection) [@pancho1] as well as the dilation of a thick granular layer [@vandoorn]. Moreover, as the system is spatially extended, it can also undergo spatial instabilities associated to the breaking of translational symmetries between different zones of the layer, that can oscillate with different phases [@pancho1].
More recently, it has been shown how the convective velocity field depends on the adimensional acceleration and other parameters, such as frequency and the air pressure [@nos] [@squires], and new theoretical models have been developed [@behringer3], [@merson], [@rodrigo].
In this article we present an experimental study of the motion that a dense granular system develops when it is submitted to a vibration in the same direction than gravity. By detecting the time at which the layer collides with the shaking plate, flight times of the granular layer as a whole are measured, and the temporal bifurcations are described. At the same time, the movement of the particles near the container walls has been tracked with a high-speed recording system. By tracking the grains within an oscillation period, the friction effects caused by walls can be quantified and its influence on the global circulation is assessed.
Experimental set-up
===================
![Experimental set-up. A granular layer is vibrated by means of a shaker (S), which is in turn controlled by a function generator (FG). The acceleration is measured with an accelerometer (A). An oscilloscope (O) is used to monitor the instantaneous acceleration. The movement of the particles adjacent to the wall is recorded with a high-speed camera (C). The devices are controlled from a PC.[]{data-label="fig.setup"}](montaje2.eps){width="9cm"}
Convection can be observed with almost any granular material, irrespective of shape, size or surface features. We have used glass beads with a diameter of $0.5 \pm 0.1$ mm, but we checked that sand gives the same qualitative results. The relative effect of cohesive forces (such as humidity or static charges) is important if the beads are much smaller than this. On the other hand, it is desirable to have as many grains as possible, and this particular size offers a good compromise. We put a big number of beads (typically of the order of $10^4$) inside a cylindrical box made of glass. The fact that both the beads and the box are of the same material reduces the amount of electrical charges created by friction. We also sprinkled the box with antistatic spray. The diameter of the box is 52 mm and it is high enough to avoid grains falling over when vibrated. This box is attached to a TiraVib 52100 magnetic shaker capable of delivering a sinusoidal acceleration of up to 15 $g$ with a distortion smaller than 0.05 $g$. The shaker is commanded by a Stanford Research DS345 function generator. The vibration is characterized with an accelerometer attached to the box that has a sensitivity of $100\;mV/g$, whose signal is picked by a Hewlett-Packard HP54510 digitizing oscilloscope. Both the oscilloscope and the function generator are connected to a PC. A sketch is provided in Fig. \[fig.setup\].
The frequency of the external vibration $f$ was kept constant at $f=110\;Hz$. We had previously found that the features of convection do not change qualitatively with frequency [@nos] provided that it is higher than 60 Hz. The acceleration was therefore changed by regulating the vibration amplitude $A$. The size of the granular layer is given in terms of the dimensionless height $N=h/\phi$, where $h$ is the thickness of the layer and $\phi$ the particle diameter.
We used a high-speed camera (Motionscope Redlake, model 1105-0003) with a macro lens and a VCR to record the movement of the grains at 1000, 2000 or 4000 frames per second. Under proper illumination, each glass sphere will reflect a bright spot that has been tracked with the following procedure. Once transferred to the computer, the movie was split into individual frames. A morphological image processing was performed on each frame to obtain the centroid of one bright spot in the first recorded frame. As the spheres budge less than one diameter from one frame to the next, the position of the bright spot is easily identifiable in the subsequent frame. Repeating the procedure for all the recorded frames and by tracking several beads, a set of grain positions versus time was obtained from each movie. Note that only spheres adjacent to the walls are accessible with this method, and we can only measure the velocity in the plane of the wall. An alternative method that has also been used, yielding the same results, is to calculate the correlation function between consecutive frames. In this case, the averaged velocity of all the beads in the frame is obtained.
Motion of the center of mass
============================
Let us begin by describing the motion of the layer as a whole without considering the movement of the individual grains. Under this assumption, and considering the layer as a perfectly inelastic solid, its center of mass will begin to fly when its acceleration overcomes the gravity. From then on, the material will perform a free flight, and will lose all its energy when it collides with the plate. If the acceleration of the container is at that moment smaller than $g$, as in Fig. \[fig.ibbm\].a, the layer gets stuck to the container base until it departs from the base again when the acceleration exceeds the gravity. The layer spends therefore a time $\tau$ in the flight and a time $T - \tau$ (where $T$ is the period of the container oscillation) stuck to the container base in each cycle.
![ Parabolic flight predicted by the Inelastic Bouncing Ball Model ([*top*]{}) and the acceleration measured by the oscilloscope for $\Gamma=2.37$ and $N=33$ ([*bottom*]{}). The collisions of the granular layer against the base of the container are evident in the signal from the accelerometer. The value of the acceleration equal to $-g$ is marked with a horizontal dotted line; the granular layer gets loose at this coordinate. Remarkably, the flight time measured in the experiment, $\tau$ suffers a phase delay with regard to the predicted by the IBBM. Take off and collision are marked with vertical dotted lines.[]{data-label="fig.ibbm"}](figibbm.eps){width="14"}
The flight time $\tau$ grows with $\Gamma$ until it reaches the value of oscillation period of the forcing $T$. If the granular layer is considered as a point mass, this happens for $\Gamma=\sqrt{1+\pi^{2}}$. At that point, the flight time undergoes a saddle-node bifurcation with the stable branch corresponding to a flight time $\tau=T$ [@holmes]. This lasts until $\Gamma=\sqrt{4+\pi^{2}}$, where a period doubling bifurcation takes place. Beyond that point, the particle can either perform a *long* flight (longer than $T$) or a *short* flight (shorter than $T$), depending on the container acceleration at the time of the collision. As $\Gamma$ increases, the long flight grows longer and the short flight shorter. Above $\Gamma= \sqrt{1+4\pi^{2}}$ only the long flight survives, and when it reaches the value $2T$ it bifurcates again.
The validity of this model to reproduce the interaction of the granular layer as a whole with the vibrating plate can be assessed by comparing its predictions to the experimental measurement of $\tau$ (see Fig. \[fig.ibbm\].b). In order to do this, we have taken the value predicted by the IBBM for the phase at the beginning of the flight: $\phi_{i}=arcsin(1/\Gamma)$. There is no way to obtain this value from the acceleration signal, as the take off does not leave any trace on it. In principle this value is not prone to be affected by the friction between the grains and the container or the presence of interstitial gas, because at take off the relative velocity between the granular layer and the vibrating plate is zero. The collision time can be obtained experimentally from the measured acceleration, as in Fig. \[fig.ibbm\].b. The flight times obtained in this way, subtracting the take-off times from the collision times, are displayed in Fig. \[fig.bifurcacion\].a along with the bifurcation diagram predicted by the IBBM. Clearly the model reproduces quite well the flight times of the center of mass for $\Gamma \lesssim 2.7$, as has already been demonstrated [@vandoorn]. Above this value, the model is not valid anymore. The measured flight times are shorter than those predicted, and the bifurcation point is at $\Gamma=4.8\pm0.1$. Beyond that point, the branches grow but eventually they seem to saturate. Another remark is that the system bifurcates directly from a monotonically growing solution to a period two solution, without showing the saddle-node bifurcation predicted by the model.
![(a) Dimensionless flight time $\tau$ of the granular layer measured in the acceleration signal (*symbols*). When $\tau$ reaches $T$ the flight time undergoes a period doubling: a long and a short flight are performed every two cycles. The line indicates the values predicted by the IBBM. (b) The same data but now the solid line is a numerical simulation including the air effects (as in the Kroll analysis). This analysis improves the fit somewhat. The dotted line is the same than the solid one in (a). []{data-label="fig.bifurcacion"}](bifurcacion2.eps){width="14cm"}
It is conceivable that one cause for this behaviour could be associated to the effects of the interstitial gas on the granular layer. The layer should then be considered as a porous medium whose porosity changes as it detaches from the base. A pioneering study of those dynamics has been done by Kroll [@kroll] and refined by Gutman [@gutman], who introduced air compressibility and a coupling with the porosity of the medium.
Let us introduce the hypothesis of Kroll (which is easier than Gutman’s to perform) in the numerical simulation of our problem. Considering the inelastic ball as a porous piston interacting with the air in the cell (a similar analysis has been recently reported for the case of granular segregation [@colombia]), the numerical analysis of the flight time suggests that air effects on the systems should be measurable but ought not to change the dynamics (see Fig. \[fig.bifurcacion\].b), *i. e.* the saddle-node bifurcation is still present (its range of stability is even increased) and the branches diverge as the flight time approaches $2T$.
In order to test the air influence on the flight time, we evacuated the container to $10^{-2}Torr$ and we collected data for the same range of $\Gamma$. Results are shown on Fig. \[fig.vacio\].a. The agreement of measured data in vacuum with the IBBM is excellent up to $\Gamma \sim 2.3$, better than in the presence of air. Up to that value, flight times are a bit shorter if there is interstitial air. Nevertheless, above $\Gamma \sim 2.3$ the measured flight times do not fit to the IBBM even in vacuum. The bifurcation point is noticeably changed, even though it is still beyond the place predicted by the IBBM; the saddle-node bifurcation is neither observed. Beyond the bifurcation, the branches behave similarly in both cases (in vacuum and in air); a remarkable feature in vacuum is a region where flight times are bivaluated ($4.7<\Gamma<5$). It is interesting to note the similarity of the branches between them and the likeness to the branch before the bifurcation point; this will be analyzed elsewhere.
![Dimensionless flight time $\tau$ of the granular layer in an evacuated container (*filled symbols*). The agreement with the IBBM is better than in air, but only for $\Gamma<2.3$. Above this value and up to the bifurcation, the dynamics of the system does not conform to the IBBM and coincides with the behaviour in the presence of air. (b) The collision retrieved from the measured acceleration, normalized to the maximum height (for $\Gamma=3$). The collision between the layer and the vibrating plate takes place during a finite time that can be measured from these data.[]{data-label="fig.vacio"}](vacio.eps){width="14"}
From the comparison of both curves (see Fig. \[fig.vacio\].a), the one in air and the one in an evacuated container, it is evident that although air does affect granular convection, this is manifested basically in the location of the bifurcation point, that gets closer to the predicted by the IBBM. It is reasonable to think that a higher vacuum would lead to an even better agreement. But it still does not explain why the branches do not diverge and why the saddle-node bifurcation goes unobserved.
Another feature that is lacking in the model is the finite duration of the collision between the inelastic ball and the plate. The extent of this time is a considerable portion of the oscillation period $T$, as can be appreciated in Fig. \[fig.vacio\].b. This implies that the velocity of the center of mass at take off is not necessarily well defined. If the collision lasts for some time, it is reasonable to think that there is a delay that leads to a decrease in the initial velocity of the center of mass, and therefore to shorter flight times. This phenomenon is associated to the propagation of a shock wave front [@bougie] that will be described elsewhere, and it has significant consequences for $\Gamma>3$, when the duration of the collision becomes similar to the time that the granular layer spends stuck to the vibrating plate. For flight times shorter than $80\%$ of the period, however, a finite collision time should not affect the flight time and we should search for another cause.
The key could be the interaction of the particles with the lateral wall of the container. Thus, the wall would exert an effective force on the inelastic ball larger than gravity that would affect not only the initial phase of the flight [@durand] (its effects on the initial velocity being negligible) but the acceleration during the entire flight as well, resulting in an effective force applied to the grains during the flight bigger than gravity. Assuming that this force is independent of the relative movement between the particles and the container wall, we can estimate its value by comparing the measured flight times with those predicted by the IBBM. We therefore introduce an effective control parameter such as $\Gamma_{eff}=\frac{A\omega^{2}}{g_{eff}}$ which depends on an effective acceleration $g_{eff}$ whose value is $10.6\;m/s^2$. This is the value that must be introduced in the IBBM in order to recover the measured flight time. This will be described in detail in the next section, where the movement of individual grains is dealt with.
The motion of individual grains
===============================
Till here we have described the motion of the granular layer as a whole. But there is motion in the frame of reference of the layer: the convective flow. Let us now study the movement of individual particles. The convective motion –it has been described previously [@nos],[@nagel] – is much slower than the vibration, so it could somehow be expected that the motion of individual beads is a combination of flights similar to those of the IBBM coupled with a slow drift.
In our experiment we have tracked the position of individual beads near the lateral wall. The measurement have been performed near the surface, where the downward velocity peaks. The paths of the beads in the plane of the wall do not divert much from the vertical: azimuthal velocities are typically less than $10 \%$ of the vertical velocities. Note, however, that there can be motion in the radial direction; this component is not accessible in our experiment. Therefore we will restrict in the following to the vertical direction except when explicitly indicated.
![The vertical position of a bead, tracked during about 10 cycles at 2000 samples per second. The container was being vibrated at $\Gamma = 2.59 $ ([*a*]{}) and at $\Gamma = 5.81 $ ([*b*]{}). The height of the layer was $N=100$ in both cases. The origin of distances is arbitrary. Note the increasing in the drift velocity when $\Gamma$ grows.[]{data-label="fig.saltos"}](trazas.eps){width="14"}
The vertical position of a single bead at $\Gamma$ below and above the period doubling point is plot on Fig. \[fig.saltos\]. The beads roughly follow the same sort of movement described by the IBBM, with a conspicuous difference: there is a distinct drift downwards. We can consider the motion as consisting of two components: a fast one (the jumps at frequency $f$) and a slow one, which is the drift. The velocity of the latter (the convective motion) is about an order of magnitude smaller than the peak value of the former. Below $\Gamma\simeq6$, we have observed that this configuration always forms a toroidal convective roll: the beads go down near the wall and they rise near the center of the container (see Fig. \[fig.campo\]).
![ The convective velocity field at the top of the layer, near the lateral wall and at the bottom of the layer ([*from top to bottom; only part of the layer is represented.*]{}) Note the different scales. This figure corresponds to $N=33$ and $\Gamma=1.90$. The velocity field has been obtained by particle tracking at a small sampling rate (25 frames per second) effectively filtering out the rapid movement at the excitation frequency. The small crosses in the top and bottom plots mark the center of the container. []{data-label="fig.campo"}](campo.eps){width="8cm"}
Friction against the walls
==========================
![(a) The trajectories of many grain flights are plotted together. The origins of the positions are the places where velocity changes sign. The data corresponds to $\Gamma=2.5$, a value of the forcing where flight times for an evacuated container and a container with air are the same. It is difficult to fit these data because of their dispersion. (b) The same data, but now the maximum height has been chosen as the common reference. In order to relate them to the movement of the vibrating plate, the phase at the moment of collision has been adjusted to the experimental data (this point is marked with a star in the plot). From this phase, the experimental points have been displaced so their trajectories are tangent to the base at the beginning of the flight. The dotted line corresponds to the flight as predicted by the IBBM. Particles do reach a lower height, and they finally go down further than expected from a “free” flight. The difference $\Delta h$ divided by the oscillation period gives the velocity of the convective flow. The dashed line is a quadratic fit. Although the fit could be deemed god, the initial phase for the flight as obtained from the fit (arrow) does not coincide with $\Gamma=g$. []{data-label="fig.vuelos"}](vuelos2.eps){width="14cm"}
We are now able to discuss the origin of the downward movements near the lateral wall. The motion of a single particle has been shown to consist of a “fast” component (the jumps at the driving frequency $f$ or $f/2$) and a “slow” component (the drift giving rise to convection). Clearly, there must be some mechanism imposing a net shear on the grains in order to induce the flow depicted in Fig. \[fig.campo\]. With the aim of investigating this subject, we took a closer look at the trajectories of the particles during each cycle.
A large number of trajectories of *single grain flights* recorded during one cycle, such as those displayed in Fig. \[fig.saltos\], is shown in Fig. \[fig.vuelos\].a The common origin has been chosen as the moment when the particles collide with the base. Obviously, the measured positions of individual beads are too noisy to obtain clean paths. This is mainly due to the rearrangements of grains during the flight and to the fact that beads rotate. In order to regroup all the trajectories, the origin for all the paths has been arbitrarily chosen at the maximum of the flight. Near this point, almost all of the trajectories should have the same dynamics. We implemented an algorithm to find the maximum that chooses seven points around this zone an fits the trajectory to a parabolic flight, and then regroups all the trajectories using the calculated maximum as the reference. Using this method, the temporal coordinate where the particle touches the base has a lower dispersion. We use the mean value of these coordinates and the corresponding time determined from the oscilloscope to adjust the phase between the vibrating plate and the flights of the grains.
In figure \[fig.vuelos\].b we show the regrouped trajectories compared with the trajectory predicted by the IBBM. (We have displaced vertically the measured trajectories so that they are tangent to the plate oscillation). It seems clear that although the particles begin the flight for a phase almost equal to the one predicted by the IBBM, they finish the flight well below the position were they would land after a free flight. This difference, averaged for an oscillation period, is just the velocity of the slow drift giving rise to the convective flow [@nos].
From the comparison between the experimental data and the trajectory predicted by the IBBM it is clear that an acceleration larger than gravity is acting on the particles (as can be seen in Fig. \[fig.vuelos\]. b, this can be easily deduced because the heights reached in the flights, with the same initial condition, are rather different in the measured data and the IBBM). If this acceleration is constant during the flight time, the path of the particles in a space-time diagram should fit to a parabolic trajectory. We have fit a parabola, leaving all the parameters free except the maximum, which we have taken as a common origin for all the grains. This fit is shown in Fig. \[fig.vuelos\]. b. The fit seems acceptable, and the effective acceleration during the flight would be $g_{f}=10.63\pm0.04\;m/s^2$. This value matches the $g_{eff}$ estimated from the flight times. Although the data are fitted reasonably well, the match is not completely satisfying. As stated above, the position of the maximum is fixed and then the acceleration and the initial velocity are free parameters of the fit. The value obtained for the initial velocity does not agree with the one predicted by the IBBM, and differs from the one that can be extracted from the data.
In order to improve the quality of the fitting, we have tried another approach by introducing a viscous dissipation which depends on the velocity of the grains. Then the ballistic trajectory of the grain will be modeled by the expression $\ddot{z}+\gamma \dot{z}=-g_{\nu}$ where $z$ is the position of the particle and $\gamma$ represents the dissipative coefficient. The fit is shown in Fig \[fig.vueloviscoso\].a. The value obtained for the initial velocity is in close agreement with the experimental data, and the value for the effective acceleration $g_{\nu}=12.77\pm0.07\;m/s^2$ is larger than the obtained from the flight times. The dissipation can also be taken into account by introducing a term which depends on the relative velocity between the grains and the wall, so that $\ddot{z}+b\dot{z} -g_{\nu} \left(\Gamma \; \mbox{sin}(\omega t + \phi)\; - 1 \right) =0$, with $\dot z$ being now the relative velocity between the grains and the lateral wall. The fit is shown in Fig \[fig.vueloviscoso\].b. The value obtained for the effective acceleration is now $g_{\nu}=11.9\pm0.1\;m/s^2$. In both cases, the initial velocity is correctly reproduced.
![(a) The same data than in Fig. \[fig.vuelos\] but with a viscous dissipation that depends on the velocity of the grains included in the simulation. Now the fit (dot-dashed line) is better, and the initial phase for the flight is correctly predicted. The value obtained for the effective acceleration is now $g_{\nu}=12.77\pm0.07\;m/s^2$. The dotted line corresponds to the IBBM. (b) Same data than in (a) with a viscous dissipation proportional to the relative velocity between the grains and the lateral wall. The effective acceleration obtained from the fit is $g_{\nu}=11.9\pm0.1\;m/s^2$.[]{data-label="fig.vueloviscoso"}](vueloviscoso2.eps){width="14cm"}
It is not easy to choose which one of the fits is better. In any case, the introduction of a dissipation which depends on the velocity clearly improves the quality of the analysis, and the effective acceleration $g_{\nu}$ that is found for the particles near the wall, where the trajectories of the grains have been tracked, is larger than the one found from the flight times for all the granular layer as a whole, $g_{eff}$. Therefore, the difference between $g_{eff}$ and $g_{\nu}$ must be due to the wall-particle interaction. This difference between both values would amount to the net stress that the walls cause on the granular layer, thus imposing the downward flow near the wall.
A similar fit could be performed for the region after the period doubling (Fig. \[fig.saltos\].b), but a better temporal resolution is needed.
Conclusions and discussion
==========================
From the analysis carried out we can conclude the following.
For $\Gamma \lesssim 2.7$ the Inelastic Bouncing Ball Model reproduces quite well the jumps of the granular medium as a whole, just as had been previously reported for the dilation of the upper layer [@vandoorn]. Nevertheless, flight times are a little bit shorter than predicted by the model. We have verified that the cause for this is the interstitial air. When the container is evacuated, the agreement between the modified model and the measured data is excellent. We have therefore introduced in the model the interaction between air and the grains so that the agreement when there is air in the container is improved.
Above $\Gamma \sim 2.7$ the model is unable to faithfully reproduce the dynamics observed in the experiment. According to the model, the flight time should increase monotonically until a saddle-node bifurcation appears; then it should remain constant for a finite range of the control parameter. Afterwards it should suffer a period-doubling bifurcation. The saddle-node bifurcation has not been observed in our experiments. Besides, flight times are much lower than predicted both in the presence of air and in vacuum. The finite duration of the collision between the granular layer and the vibrating plate can be the cause for this discrepancy. A deeper study of this subject will be presented elsewhere.
The critical value for the control parameter where the period doubling bifurcation takes place is noticeably influenced by the presence of interstitial gas. For a moderate vacuum, such as the one reached in our experiment, the critical value of $\Gamma$ approaches the predicted one for an inelastic model but is still larger than it. Remarkably, flight times in vacuum below the bifurcation are almost indistinguishable from those measured in air. This suggests that the presence of air affects mainly the stability of the solutions for the flight times after the bifurcation.
Just above the period doubling bifurcation, the model fits quite well the branches if the container is evacuated. In air, the flight times are smaller than those predicted by the IBBM. In neither case, however, the divergence predicted by the model as $\Gamma$ grows larger is observed: flight times change slowly. This behaviour is probably caused by the finite duration of the collision as well.
Therefore, even though near the points where the solutions change (*i. e.* $\Gamma=1$ and the period doubling bifurcation) the system can be modeled as a perfectly inelastic body, when the control parameter is increased there is a certain point where the model loses validity. Finite duration of the collision is suspected to be involved.
In this work we have focused in the region below the bifurcation point, where flight times in air and in vacuum are almost the same. From the study of the trajectories described by the particles near the lateral wall we have checked that they are subjected during their flight to a net force larger than gravity. A quadratic fit of the flight allows us to estimate the value of effective gravity, that agrees quite well with the one inferred from the measurement of the flight time. The quadratic fit implies a constant external force.
The quadratic fit, however, does not yield a correct value for the initial velocities (*i. e.* the velocities with which the grains take off when they begin the flight). In order to improve this, we have introduced a viscous dissipation. The trajectory is correctly reproduced, including the initial velocity. Remarkably, the value obtained for the effective acceleration is bigger than that obtained from the flight time measurements. The difference between both effective accelerations suggests that there is an extra force acting on the grains near the walls larger than the average over all the granular layer. This would be the cause of the downward movement near the walls that sets the sense of the convective flow.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been funded by Spanish Government project BFM2002-00414 and FIS2005-03881 as well as Acción Integrada HF2002-0015, by the local Government of Navarre, and by the Universidad de Navarra (PIUNA). I.Z. and J.M.P. thank the Asociación de Amigos de la Universidad de Navarra for a grant. The authors wish to thank H. Kellay for his hospitality and his comments and R. Arévalo for his useful comments about the numerical simulation.
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abstract: 'Let $F$ be a non-archimedean local field of characteristic $0$, and ${{\mathfrak o}}$ the ring of integers in $F$. We give an explicit formula for the Siegel series of a half-integral matrix over ${{\mathfrak o}}$. This formula expresses the Siegel series of a half-integral matrix $B$ explicitly in terms of the Gross-Keating invariant of $B$ and its related invariants.'
address:
- 'Graduate school of mathematics, Kyoto University, Kitashirakawa, Kyoto, 606-8502, Japan'
- 'Muroran Institute of Technology 27-1 Mizumoto, Muroran, 050-8585, Japan'
author:
- Tamotsu IKEDA
- Hidenori KATSURADA
date: '27 September, 2017'
title: 'An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field '
---
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Introduction
============
The Siegel series is one of the simplest but most important subjects in number theory, and is related with various types of arithmetic theories of modular forms. It appears in the Fourier coefficients of the Hilbert-Siegel Eisenstein series, and it is also related with the Fourier coefficients of the lift, called the Duke-Imamoglu-Ikeda lift, constructed by the first named author in [@Ike1] (see also [@Ike-Ya]). It also plays very important roles in the study of various types of L-functions associated with cusp forms through the pullback formula(cf. [@B], [@B-S], [@Ko], [@Or], [@Sh3]). Moreover, it is closely related to arithmetic algebraic geometry (cf. [@Ku1], [@Ku2], [@K-R-Y]). In all cases, precise information on the Siegel series is necessary. Thus it is very important to give an explicit form of the Siegel series. In [@Kat1], the second named author gave an explicit formula for the Siegel series of a half-integral matrix over ${{\mathbb Z}}_p$ with any prime number $p$ of any degree. The formula is useful for a practical computation of the Siegel series, and has several interesting applications. Indeed, the formula was used to give special values of the standard L-functions of Siegel modular forms and the triple product L-functions of elliptic modular forms exactly (cf. [@D-I-K],[@I-K-P-Y],[@Ib-Kat], [@Kat2],[@Kat3],[@Kat-Mi]). These computations played important roles not only in confirming several conjectures on such values numerically but also in proposing new conjecture on them. The formula was also one of key ingredients in proving the conjecture on the period of the Duke-Imamoglu-Ikeda lift proposed in [@Ike2] (cf. [@Kat-Kaw2]). Moreover, it was used to relate the local intersection multiplicities on certain Shimura varieties to the derivatives of certain local Whittaker functions in [@Ra-Wed].
However, it is not satisfactory in the following reasons. Firstly, the formula is complicated in the case $p=2$, and it seems difficult to unify it with the formula in the case that $p$ is odd as it is. Secondly, it does not seem clear what invariants determine the Siegel series. Though there are other explicit formulas for local densities (cf. [@Hi-Sa], [@Ya]), it seems difficult to resolve the above problems using them. In [@Wed], Wedhorn reformulated the formula in [@Kat1] for the Siegel series of a half-integral matrix of degree three in terms of the Gross-Keating invariant in [@G-K]. In [@Ot], Otsuka gave an explicit formula for the Siegel series of a half-integral matrix of degree two over the ring of integers of any non-archimedean local field of characteristic $0$. In this paper, we give an explicit formula of the Siegel series of a half-integral matrix of any degree over any non-archimedean local field of characteristic $0$.
We explain our main result more precisely. Let $F$ be a non-archimedean local field of characteristic $0$ with the residue field ${{\mathfrak k}}$ and let ${{\mathfrak o}}$ be the ring of integers in $F$. Put $q=\#({{\mathfrak k}})$. For a non-degenerate half-integral matrix $B$ of degree $n$ over ${{\mathfrak o}}$, let $b(B,s)$ be the Siegel series of $B$. Then, as will be explained in Section 2, we obtain a polynomial $\widetilde F(B,X)$ in $X^{1/2}$ and $X^{-1/2}$ attached to $b(B,s)$. Let ${\mathrm{GK}}(B)$ be the Gross-Keating invariant of $B$. We then define a set ${\mathrm{EGK}}(B)$ of invariants of $B$ (cf. Definition \[def.3.5\]), which will be called the extended GK datum of $B$. In the non-dyadic case, $B$ has a diagonal Jordan decomposition: $$B \sim {\varpi}^{m_1} U_1 \bot \cdots \bot {\varpi}^{m_r}U_r$$ with $m_1,\ldots,m_r$ non-negative integers such that $m_1<\cdots<m_r$ and $U_i$ a diagonal unimodular matrix of degree $n_i$ for $i=1,\ldots,r$. Then $${\mathrm{GK}}(B)=(\underbrace{m_1,\ldots,m_1}_{n_1},\ldots,\underbrace{m_r,\ldots,m_r}_{n_r}).$$ For each $i=1,\ldots,r$, we define $\zeta_i$ as $$\zeta_i=\begin{cases}
\xi_{B^{(n_1+\cdots+n_i)}} & \text{ if } \deg B^{(n_1+\cdots+n_i)} \text{ is even}\\
\eta_{B^{(n_1+\cdots+n_i)}} & \text{ if } \deg B^{(n_1+\cdots+n_i)} \text{ is odd},
\end{cases}$$ where $B^{(k)}$ is the upper left $k \times k$ block of $B$, and $\xi_A$ and $\eta_A$ are the invariants of a half-integral matrix $A$, which will be defined in Section 2. Then ${\mathrm{EGK}}(B)$ is defined as $(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$. In the dyadic case, the invariants ${\mathrm{GK}}(B)$ and ${\mathrm{EGK}}(B)$ of $B$ are more elaborately defined. Then we express $\widetilde F(B,X)$ explicitly in terms of ${\mathrm{EGK}}(B)$. This polynomial is universal in the following sense. We define an ${\mathrm{EGK}}$ datum $G$ of length $n$ as an element $(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$ of ${{\mathbb Z}}^r_{>0} \times {{\mathbb Z}}^r_{\ge 0} \times \{0,1,-1 \}^r$ with $n_1+\cdots +n_r=n$ satisfying certain conditions (cf. Definition \[def.4.5\]). The ${\mathrm{EGK}}$ datum is defined by axiomatizing some properties of the extended GK datum of a half-integral matrix, and naturally ${\mathrm{EGK}}(B)$ is an EGK datum (cf. Theorem \[th.4.2\]). We define a Laurent polynomial $\widetilde {{\mathcal F}}(G;Y,X)$ in $X^{1/2},Y$ attached to $G$. An explicit formula for $\widetilde {{\mathcal F}}(G;Y,X)$ will be given in Proposition \[prop.4.1\]. Then, our main result in this paper is as follows:
\[th.1.1\] Let $B$ be a non-degenerate half-integral matrix of degree $n$ over ${{\mathfrak o}}$. Then we have $$\widetilde F(B,X)=\widetilde {{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X).$$
This unifies the formula for $p=2$ with that for an odd prime $p$ in [@Kat1]. Therefore, our result not only gives a generalization of the main result in [@Kat1] but also reformulates it in a satisfactory way, and is new even in the case $F={{\mathbb Q}}_p$. By the above theorem we immediately have
Let $B$ be as above. Then $\widetilde F(B,X)$ is determined by ${\mathrm{EGK}}(B)$.
By using the above theorem, Cho, Yamana, and Yamauchi [@C-Yamana-Yamauchi] give a formula for the Fourier coefficients of the derivative of Eisenstein series of weight $2$ and genus $4$, and show that they are related with the intersection number of $4$ modular correspondences. Moreover, by using the above corollary, Cho and Yamauchi [@C-Y] give an induction formula for the Siegel series, which is different from that in the present paper. It gives a description of the local intersection multiplicities of the special cycles on the special fiber of Shimura varieties for $SO(2,n)$ with $n \le 3$ over a finite field in terms of Siegel series directly. These results shed a new light on Kudla’s program [@Ku2]. Therefore, our result plays a crucial role also in arithmetic algebraic geometry.
A proof of Theorem \[th.1.1\] will be given in Sections 6 and 7. To explain the method of the proof of our main result, first we review the proof of the main result in [@Kat1] with some modification. For simplicity let $q$ be odd, and let $B$ be a non-degenerate half-integral matrix of degree $n$ over the ring ${{\mathfrak o}}$. Then $B$ has the following diagonal Jordan decomposition $$B \sim {\varpi}^{a_1}u_1 \bot \cdots \bot {\varpi}^{a_{n}}u_n$$ with $a_1 \le \cdots \le a_n$ and $u_1,\ldots, u_n \in {{\mathfrak o}}^{\times}$. Then, in the case that ${{\mathfrak o}}$ is the ring ${{\mathbb Z}}_p$ of $p$-adic integers, by the induction formulas for local densities (cf. Theorem \[th.5.1\]), and the functional equation of the Siegel series (cf. Proposition \[prop.2.1\]), we can express $\widetilde F(B,X)$ in terms of $\widetilde F(B^{(n-1)},X)$ (cf. \[[@Kat1], Theorem 4.1)\]). This argument works for any Siegel series over a non-dyadic field. To be more precise, for an integer $1 \le i \le n$, we define ${{\mathfrak e}}_i$ as $${{\mathfrak e}}_i=
\begin{cases} a_1+\cdots +a_i & \text{ if $i$ is odd} \\
2[(a_1+\cdots+a_i)/2] & \text{ if $i$ is even.}
\end{cases}$$ Then we have the following [(cf. Theorem \[th.6.1\])]{} :\
[ *Under the above notation and the assumption, we have $$\begin{aligned}
\widetilde F(B,X)&=D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1};\xi_{B^{(n-1)}};q^{1/2},X)\widetilde F(B^{(n-1)},q^{1/2}X) \\
&+\eta_B D({{\mathfrak e}}_n, {{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};q^{1/2},X^{-1})\widetilde F(B^{(n-1)},q^{1/2}X^{-1})\end{aligned}$$ if $n$ is odd, and $$\begin{aligned}
\widetilde F(B,X)&=C({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_B;q^{1/2},X)\widetilde F(B^{(n-1)},q^{1/2}X) \\
&+ C({{\mathfrak e}}_n, {{\mathfrak e}}_{n-1},\xi_B;q^{1/2},X^{-1})\widetilde F(B^{(n-1)},q^{1/2}X^{-1})\end{aligned}$$ if $n$ is even. In particular if $n=1$ we have $$\widetilde F(B,X)=\sum_{i=0}^{r_1} X^{i-(r_1/2)},$$ where $\xi_*$ and $\eta_*$ are the invariants stated above, and $C(*,*,*;Y,X)$ and $D(*,*,*;Y,X)$ are rational functions in $X^{1/2}$ and $Y^{1/2}$, which will be defined in Definition \[def.4.3\].*]{}
Using the induction formulas stated above repeatedly, we get an explicit formula for $\widetilde F(B,X)$. However, in the case $p=2$, we do not necessarily have a diagonal Jordan decomposition for $B$, and therefore, the formula for $p=2$ becomes complicated, and it is no hope to generalize it to any dyadic field as it is. To overcome this obstacle, we adopt a reduced decomposition of $B$ (cf. Definition \[def.3.5\]) instead of a diagonal Jordan decomposition. Let $B$ be a reduced form of degree $n$. Then, in the case that $B^{(n-1)}$ is a reduced form, and we can express $\widetilde F(B,X)$ in terms of $\widetilde F(B^{(n-1)},X)$ as (\[eq.1.1\]) and (\[eq.1.2\]) in the proof of Theorem 1.1 in the dyadic case. In the other cases, we can also express $\widetilde F(B,X)$ in terms of $\widetilde F(B^{(n-2)},X)$ as (\[eq.2.3\]) and (\[eq.3.1\]) therein. From the induction formulas we prove the explicit formula stated above. A key ingredient for proving such induction formulas is a stability of the extended GK datum of a reduced form (cf. Theorem \[th.3.3\]), which was essentially proved in [@Ike-Kat]. Therefore the extended GK datum plays a very important roll not only in formulating main results but also in proving them.
It seems interesting to consider a Hermitian version of the main result in this paper.
We would like to thank Sungmun Cho, Shunsuke Yamana and Takuya Yamauchi for many fruitful discussions and suggestions. The research was partially supported by the JSPS KAKENHI Grant Number 26610005, 24540005, and 16H03919.
[**Notation**]{} Let $R$ be a commutative ring. We denote by $R^{\times}$ the group of units in $R$. We denote by $M_{mn}(R)$ the set of $(m,n)$ matrices with entries in $R$, and especially write $M_n(R)=M_{nn}(R)$. We often identify an element $a$ of $R$ and the matrix $(a)$ of degree 1 whose component is $a$. If $m$ or $n$ is 0, we understand an element of $M_{mn}(R)$ is the [*empty matrix*]{} and denote it by $\emptyset$. Let $GL_n(R)$ be the group consisting of all invertible elements of $M_n(R)$, and ${\rm Sym}_n(R)$ the set of symmetric matrices of degree $n$ with entries in $R$. For a semigroup $S$ we put $S^{\Box}=\{s^2 \ | \ s \in S \}$. Let $R$ be an integral domain of characteristic different from $2$, and $K$ its quotient field. We say that an element $A $ of $\mathrm{Sym}_n(K)$ is non-degenerate if the determinant $\det A$ of $A$ is non-zero. For a subset $S$ of $\mathrm{Sym}_n(K)$, we denote by $S^{{\rm{nd}}}$ the subset of $S$ consisting of non-degenerate matrices. We say that a symmetric matrix $A=(a_{ij})$ of degree $n$ with entries in $K$ is half-integral if $a_{ii} \ (i=1,...,n)$ and $2a_{ij} \ (1 \le i \not= j \le n)$ belong to $R$. We denote by ${{\mathcal H}}_n(R)$ the set of half-integral matrices of degree $n$ over $R$. We note that ${{\mathcal H}}_n(R)={\rm Sym}_n(R)$ if $R$ contains the inverse of $2$. We denote by ${{\mathbb Z}}_{> 0}$ and ${{\mathbb Z}}_{\ge 0}$ the set of positive integers and the set of non-negative integers, respectively. For an $(m,n)$ matrix $X$ and an $(m,m)$ matrix $A$, we write $A[X] ={}^tXAX$, where $^t X$ denotes the transpose of $X$. Let $G$ be a subgroup of $GL_n(K)$. Then we say that two elements $B$ and $B'$ in $\mathrm{Sym}_n(K)$ are $G$-equivalent if there is an element $g$ of $G$ such that $B'=B[g]$. For two square matrices $X$ and $Y$ we write $X \bot Y =\mattwo(X;O;O;Y)$. We often write $x \bot Y$ instead of $(x) \bot Y$ if $(x)$ is a matrix of degree 1. For a square matrix $B$ of degree $n$ and integers $1 \le i_1, \ldots, i_r \le n, 1 \le j_1 , \ldots, j_r \le n$ such that $i_k \not=i_l \ (k \not=l)$ and $j_{k'} \not= j_{l'} \ (k' \not= l')$ we denote by $B(i_1,\ldots,i_r;j_1,\ldots,j_r)$ the matrix obtained from $B$ by deleting its $i_1,\ldots,i_r$-th rows and $j_1,\ldots,j_r$-th columns. In particular, put $T^{(k)}=T(k+1,\ldots,n;k+1,\ldots, n)$. We make the convention that $T^{(k)}$ is the empty matrix if $k=0$. We denote by $1_m$ the unit matrix of degree $m$ and by $O_{m,n}$ the zero matrix of type $(m,n)$. We sometimes abbreviate $O_{m,n}$ as $O$ if there is no fear of confusion.
Siegel series
=============
Let $F$ be a non-archimedean local field of characteristic $0$, and ${{\mathfrak o}}={{\mathfrak o}}_F$ its ring of integers. The maximal ideal and the residue field of ${{\mathfrak o}}$ is denoted by ${{\mathfrak p}}$ and ${{\mathfrak k}}$, respectively. We fix a prime element ${\varpi}$ of ${{\mathfrak o}}$ once and for all. The cardinality of ${{\mathfrak k}}$ is denoted by $q$. Let ${\mathrm{ord}}={\mathrm{ord}}_{{{\mathfrak p}}}$ denote additive valuation on $F$ normalized so that ${\mathrm{ord}}({\varpi})=1$. If $a=0$, We write ${\mathrm{ord}}(0)=\infty$ and we make the convention that ${\mathrm{ord}}(0) > {\mathrm{ord}}(b)$ for any $b \in F^{\times}$. We also denote by $|*|_{{{\mathfrak p}}}$ denote the valuation on $F$ normalized so that $|{\varpi}|_{{{\mathfrak p}}}=q^{-1}$. We put $e_0={\mathrm{ord}}_{{{\mathfrak p}}}(2)$.
For a non-degenerate element $B\in{{\mathcal H}}_n({{\mathfrak o}})$, we put $D_B=(-4)^{[n/2]}\det B$. If $n$ is even, we denote the discriminant ideal of $F(\sqrt{D_B})/F$ by ${{\mathfrak D}}_B$. We also put $$\xi_B=
\begin{cases}
1 & \text{ if $D_B\in F^{\times 2}$,} \\
-1 & \text{ if $F(\sqrt{D_B})/F$ is unramified quadratic,} \\
0 & \text{ if $F(\sqrt{D_B})/F$ is ramified quadratic.}
\end{cases}$$ Put $${{\mathfrak e}}_B=
\begin{cases}
{\mathrm{ord}}(D_B)-{\mathrm{ord}}({{\mathfrak D}}_B) & \text{ if $n$ is even} \\
{\mathrm{ord}}(D_B) & \text{ if $n$ is odd.}
\end{cases}$$
Let $\langle \ {} \ , \ {} \ \rangle=\langle \ {} \ , \ {} \ \rangle_F$ be the Hilbert symbol on $F$. Let $B$ be a non-degenerate symmetric matrix with entries in $F$ of degree $n$. Then $B$ is $GL_n(F)$-equivalent to $b_1 \bot \cdots \bot b_n$ with $b_1,\ldots,b_n \in F^{\times}$. Then we define $\varepsilon_B$ as $$\varepsilon_B=\prod_{1 \le i < j \le n} \langle b_i,b_j \rangle.$$ This does not depend on the choice of $b_1,\ldots,b_n$. We also denote by $\eta_B$ the Clifford invariant of $B$ (cf. [@Ike3]). Then we have $$\eta_B=
\begin{cases}
\langle -1,-1 \rangle^{m(m+1)/2} \langle (-1)^m,\det B \rangle \varepsilon_B & \ \text{if $n=2m+1$}\\
\langle -1,-1 \rangle^{m(m-1)/2} \langle (-1)^{m+1},\det B \rangle \varepsilon_B & \ \text{if $n=2m$.}
\end{cases}$$ (cf. \[[@Ike3], Lemma 2.1\]). We make the convention that $\xi_B=1, {{\mathfrak e}}_B=0$ and $\eta_B=1$ if $B$ is the empty matrix. Once for all, we fix an additive character $\psi$ of $F$ of order zero, that is, a character such that $${{\mathfrak o}}=\{ a \in F \ | \ \psi(ax)=1 \ \text{ for any} \ x \in {{\mathfrak o}}\}.$$ For a half-integral matrix $B$ of degree $n$ over ${{\mathfrak o}}$ define the local Siegel series $b_{{{\mathfrak p}}}(B,s)$ by $$b_{{{{\mathfrak p}}}}(B,s)= \sum_{R} \psi({\rm tr}(BR))\mu(R)^{-s},$$ where $R$ runs over a complete set of representatives of ${\rm Sym}_n(F)/{\rm Sym}_n({{\mathfrak o}})$ and $\mu(R)=[R{{\mathfrak o}}^n+{{\mathfrak o}}^n:{{\mathfrak o}}^n]$.
Now for a non-degenerate half-integral matrix $B$ of degree $n$ over ${{\mathfrak o}}$ define a polynomial $\gamma_q(B,X)$ in $X$ by $$\gamma_q(B,X)=
\begin{cases}
(1-X)\prod_{i=1}^{n/2}(1-q^{2i}X^2)(1-q^{n/2}\xi_B X)^{-1} & \text{ if $n$ is even} \\
(1-X)\prod_{i=1}^{(n-1)/2}(1-q^{2i}X^2) & \text{ if $n$ is odd.} \end{cases}$$ Then it is shown by [@Sh1] that there exists a polynomial $F_{{{\mathfrak p}}}(B,X)$ in $X$ with coefficients in ${{\mathbb Z}}$ such that $$F_{{{\mathfrak p}}}(B,q^{-s})={b_{{{\mathfrak p}}}(B,s) \over \gamma_q(B,q^{-s})}.$$ We define a symbol $X^{1/2}$ so that $(X^{1/2})^2=X$. We define $\widetilde F_{{{\mathfrak p}}}(B,X)$ as $$\widetilde F_{{{\mathfrak p}}}(B,X)=X^{-{{\mathfrak e}}_B/2}F(B,q^{-(n+1)/2}X).$$ We note that $\widetilde F_{{{\mathfrak p}}}(B,X) \in {{\mathbb Q}}[q^{1/2}][X,X^{-1}]$ if $n$ is even, and $\widetilde F_{{{\mathfrak p}}}(B,X) \in {{\mathbb Q}}[X^{1/2},X^{-1/2}]$ if $n$ is odd. We sometimes write $F_{{{\mathfrak p}}}(B,X)$ and $\widetilde F_{{{\mathfrak p}}}(B,X)$ as $F(B,X)$ and $\widetilde F(B,X)$, respectively.
The following proposition is due to \[[@Ike3], Theorem 4.1\].
\[prop.2.1\] We have $$\widetilde F(B,X^{-1})=\zeta_B\widetilde F(B,X),$$ where $\zeta_{B}=\eta_B$ or 1 according as $n$ is odd or even.
The Gross-Keating invariant and related invariants {#sec:1}
==================================================
We first recall the definition of the Gross-Keating invariant [@G-K] of a quadratic form over ${{\mathfrak o}}$.
For two matrices $B, B'\in{{\mathcal H}}_n({{\mathfrak o}})$, we sometimes write $B\sim B'$ if $B$ and $B'$ are $GL_n({{\mathfrak o}})$-equivalent. The $GL_n({{\mathfrak o}})$-equivalence class of $B$ is denoted by $\{B\}$. Let $B=(b_{ij}) \in {{\mathcal H}}_n({{\mathfrak o}})^{\rm nd}$. Let $S(B)$ be the set of all non-decreasing sequences $(a_1, \ldots, a_n)\in{{{\mathbb Z}}_{\geq 0}^n}$ such that $$\begin{aligned}
{\mathrm{ord}}(b_i)&\geq a_i, \\
{\mathrm{ord}}(2 b_{ij})&\geq (a_i+a_j)/2\qquad (1\leq i,j\leq n).\end{aligned}$$ Set $$S(\{B\})=\bigcup_{B'\in\{B\}} S(B')=\bigcup_{U\in{{\mathrm{GL}}}_n({{\mathfrak o}})} S(B[U]).$$ The Gross-Keating invariant (or the GK-invariant for short) ${{\underline{a}}}=(a_1, a_2, \ldots, a_n)$ of $B$ is the greatest element of $S(\{B\})$ with respect to the lexicographic order $\succ$ on ${{{\mathbb Z}}_{\geq 0}^n}$. Here, the lexicographic order $\succ$ is, as usual, defined as follows. For $(y_1, y_2, \ldots, y_n), (z_1, z_2, \ldots, z_n)\in {{\mathbb Z}}_{\geq 0}^n$, let $j$ be the largest integer such that $y_i=z_i$ for $i<j$. Then $(y_1, y_2, \ldots, y_n)\succ (z_1, z_2, \ldots, z_n)$ if $y_j>z_j$. The Gross-Keating invariant is denoted by ${\mathrm{GK}}(B)$. A sequence of length $0$ is denoted by $\emptyset$. When $B$ is a matrix of degree $0$, we understand ${\mathrm{GK}}(B)=\emptyset$.
By definition, the Gross-Keating invariant ${\mathrm{GK}}(B)$ is determined only by the $GL_n({{\mathfrak o}})$-equivalence class of $B$. We say that $B\in{{\mathcal H}}_n({{\mathfrak o}})$ is an optimal form if ${\mathrm{GK}}(B)\in S(B)$. Let $B \in {{\mathcal H}}_n({{\mathfrak o}})$. Then $B$ is $GL_n({{\mathfrak o}})$-equivalent to an optimal form $B'$. Then we say that $B$ has an optimal decomposition $B'$. We say that $B \in {{\mathcal H}}_n({{\mathfrak o}})$ is a diagonal Jordan form if $B$ is expressed as $$B={\varpi}^{a_1} u_1 \bot \cdots \bot {\varpi}^{a_n}u_n$$ with $a_1 \le \cdots \le a_n$ and $u_1,\cdots,u_n \in {{\mathfrak o}}^{\times}$. Then, in the non-dyadic case, the diagonal Jordan form $B$ above is optimal, and ${\mathrm{GK}}(B)=(a_1,\ldots,a_n)$. Therefore, the diagonal Jordan decomposition is an optimal decomposition. However, in the dyadic case, not all half-integral symmetric matrices have a diagonal Jordan decomposition, and the Jordan decomposition is not necessarily an optimal decomposition.
\[def.3.1\] Let ${{\underline{a}}}=(a_1,\ldots,a_n)$ be a non-decreasing sequence of non-negative integers. Write ${{\underline{a}}}$ as $${{\underline{a}}}=(\underbrace{m_1,\ldots,m_1}_{n_1},\ldots,\underbrace{m_r,\ldots,m_r}_{n_r})$$ with $m_1<\cdots<m_r$ and $n=n_1+\cdots+n_{r-1}+n_r$. For $s=1,2,\ldots,r$ put $$n_s^\ast=\sum_{u=1}^sn_u,$$ and $$I_s=\{n_{s-1}^\ast+1,n_{s-1}^\ast+2,\ldots,n_s^\ast\}.$$
\[def.3.2\] Let $B \in {{\mathcal H}}_n({{\mathfrak o}})^{\rm nd}$ with ${\mathrm{GK}}(B)=(a_1,\ldots,a_n)$, and $n_1,\ldots,n_r,n_1^\ast,\ldots,n_r^\ast$ and $m_1,\ldots,m_r$ be those in Definition \[def.3.1\]. Take an optimal decomposition $C$ of $B$, and for $s=1,\ldots,r$ we put $$\zeta_s(C)=\zeta(C^{(n_s^\ast)}),$$ where $\zeta(C^{(n_s^\ast)})=\xi_{C^{(n_s^\ast)}}$ or $\zeta(C^{(n_s^\ast)})=\eta_{C^{(n_s^\ast)}}$ according as $n_s^\ast$ is even or odd. Then $\zeta_s(C)$ does not depend on the choice of $C$ (cf. \[[@Ike-Kat], Theorem 0.4\]), which will be denoted by $\zeta_s=\zeta_s(B)$. Then we define ${\mathrm{EGK}}(B)$ as ${\mathrm{EGK}}(B)=(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$, and we call it the extended GK datum of $B$.
From now on, until the end of this section, we assume that $q$ is even. We denote by ${{\mathfrak S}}_n$ the symmetric group of degree $n$. Recall that a permutation ${\sigma}\in {{\mathfrak S}}_n$ is an involution if ${\sigma}^2=\mathrm{id}$.
\[def.3.3\] For an involution ${\sigma}\in {{\mathfrak S}}_n$ and a non-decreasing sequence ${{\underline{a}}}=(a_1,\ldots,a_n)$ of non-negative integers , we set $$\begin{aligned}
{{\mathcal P}}^0&={{\mathcal P}}^0(\sigma)=\{i\,| 1\leq i\leq n,\; i={\sigma}(i)\}, \\
{{\mathcal P}}^+&={{\mathcal P}}^+(\sigma)=\{i\,| 1\leq i\leq n,\; a_i>a_{{\sigma}(i)}\}, \\
{{\mathcal P}}^-&={{\mathcal P}}^-(\sigma)=\{i\,| 1\leq i\leq n,\; a_i<a_{{\sigma}(i)}\}. \end{aligned}$$ We say that an involution ${\sigma}\in{{\mathfrak S}}_n$ is an $\underline{a}$-admissible involution if the following two conditions are satisfied.
- ${{\mathcal P}}^0$ has at most two elements. If ${{\mathcal P}}^0$ has two distinct elements $i$ and $j$, then $a_i\not\equiv a_j \text{ mod $2$}$. Moreover, if $i \in I_s\cap {{\mathcal P}}^0$, then $i$ is the maximal element of $I_s$, and $$i=\max\{j \ | \ j \in {{\mathcal P}}^0 \cup {{\mathcal P}}^+ , a_j \equiv a_i \text{ mod } 2 \}.$$
- For $s=1, \ldots, r$, there is at most one element in $I_s\cap{{\mathcal P}}^-$. If $i \in I_s\cap{{\mathcal P}}^-$, then $i$ is the maximal element of $I_s$ and $${\sigma}(i)=\min\{j\in {{\mathcal P}}^+ \,| \, j>i,\, a_j\equiv a_i \text{ mod } 2\}.$$
- For $s=1, \ldots, r$, there is at most one element in $I_s\cap{{\mathcal P}}^+$. If $i \in I_s\cap{{\mathcal P}}^+$, then $i$ is the minimal element of $I_s$ and $${\sigma}(i)=\max\{j\in {{\mathcal P}}^- \,| \, j<i,\, a_j\equiv a_i \text{ mod } 2\}.$$
- If $a_i=a_{{\sigma}(i)}$, then $|i -{\sigma}(i)| \le 1$.
This is called a standard $\underline{a}$-admissible involution in [@Ike-Kat], but in this paper we omit the word “standard”, since we do not consider an $\underline{a}$-admissible involution which is not standard.
\[def.3.4\] For ${{\underline{a}}}=(a_1, \ldots, a_n)\in{{\mathbb Z}}_{\geq 0}^n$, put $$\begin{aligned}
{{\mathcal M}}({{\underline{a}}})&=\left\{B=(b_{ij})\in{{\mathcal H}}_n({{\mathfrak o}})\,\vrule\, \begin{array}{ll}{\mathrm{ord}}(b_{ii})\geq a_i, \\ {\mathrm{ord}}(2 b_{ij})\geq (a_i+a_j)/2 \ (1\leq i < j\leq n)\end{array} \right\}, \\
{{\mathcal M}}^0({{\underline{a}}})&=\left\{B=(b_{ij})\in{{\mathcal H}}_n({{\mathfrak o}})\,\vrule\, \begin{array}{ll}{\mathrm{ord}}(b_{ii}) > a_i, \\ {\mathrm{ord}}(2 b_{ij}) > (a_i+a_j)/2 \ (1\leq i < j\leq n)\end{array} \right\}.\end{aligned}$$
\[def.3.5\] Let ${\sigma}\in{{\mathfrak S}}_n$ be an ${{\underline{a}}}$-admissible involution. We say that $B=(b_{ij})\in {{\mathcal M}}({{\underline{a}}})$ is a reduced form with GK-type $({{\underline{a}}}, {\sigma})$ if the following conditions are satisfied.
- If $i\notin{{\mathcal P}}^0$ and $j={\sigma}(i)$, then $${\mathrm{ord}}(2 b_{i \,j})=\frac{a_i+a_j}{2}$$
- If $i\in{{\mathcal P}}^0 \cup {{\mathcal P}}^- $, then $${\mathrm{ord}}(b_{ii})=a_i.$$
- If $j\neq i, {\sigma}(i)$, then $${\mathrm{ord}}(2 b_{ij})>\frac{a_i+a_j}2.$$
We often say that $B$ is a reduced form with GK-type ${{\underline{a}}}$ without mentioning ${\sigma}$. We formally think of a matrix of degree $0$ as a reduced form with GK-type $\emptyset$. The following theorems are fundamental in our theory.
\[th.3.1\] [(\[[@Ike-Kat], Corollary 5.1\])]{} Let $B$ be a reduced form of ${\mathrm{GK}}$ type $({{\underline{a}}},\sigma)$. Then we have ${\mathrm{GK}}(B)={{\underline{a}}}$.
\[th.3.2\] [(cf. \[[@Ike-Kat], Theorem 4.3\])]{} Assume that $\mathrm{GK}(B)=\underline{a}$ for $B\in {{\mathcal H}}_n({{\mathfrak o}})^\mathrm{nd}$. Then $B$ is $GL_n({{\mathfrak o}})$-equivalent to a reduced form of ${\mathrm{GK}}$ type $(\underline{a}, \sigma)$ for some $\underline{a}$-admissible involution $\sigma$.
By Theorem \[th.3.2\], any non-degenerate half-integral symmetric matrix $B$ over ${{\mathfrak o}}$ is $GL_n({{\mathfrak o}})$-equivalent to a reduced form $B'$. Then we say that $B$ has a reduced decomposition $B'$.
The following theorem plays an important role in proving our main result.
\[th.3.3\] Let $B$ and $B'$ be elements of ${{\mathcal H}}_n({\mathfrak o})$. Assume that $B$ is a reduced form of ${\mathrm{GK}}$-type $({{\underline{a}}},{\sigma})$ and that $B'-B \in {{\mathcal M}}^0({{\underline{a}}})$. Then $B'$ is also a reduced form of the same type as $B$. Moreover we have ${\mathrm{EGK}}(B')={\mathrm{EGK}}(B)$.
The first assertion can easily be proved. The second assertion follows from \[[@Ike-Kat], Proposition 3.3\].
Laurent polynomial attached to EGK datum
========================================
We recall the definition of naive EGK datum (cf. [@Ike-Kat]). Let ${\mathcal Z}_3=\{0,1,-1 \}$.
\[def.4.1\] An element $(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ of ${{\mathbb Z}}_{\ge 0}^n \times {\mathcal Z}_3^n$ is said to be a naive EGK datum of length $n$ if the following conditions hold:
- $a_1 \le \cdots \le a_n$.
- Assume that $i$ is even. Then ${\varepsilon}_i \not=0$ if and only if $a_1+\cdots+a_i$ is even.
- Assume that $i$ is odd. Then ${\varepsilon}_i \not=0$.
- ${\varepsilon}_1=1$.
- Let $i \ge 3$ be an odd integer and assume that $a_1+\cdots + a_{i-1}$ is even. Then ${\varepsilon}_i={\varepsilon}_{i-1}^{a_i+a_{i-1}}{\varepsilon}_{i-2}$.
We denote by $\mathcal{NEGK}_n$ the set of all naive ${\mathrm{EGK}}$ data of length $n$.
\[def.4.2\] For integers $e,\widetilde e$, a real number $\xi$, and $i=0,1$ define rational functions $C(e,\widetilde e,\xi;Y,X)$ and $D(e,\widetilde e,\xi;Y,X)$ in $Y^{1/2}$ and $X^{1/2}$ by $$C(e,\widetilde e,\xi;Y,X)={Y^{\widetilde e/2}X^{-(e- \widetilde e)/2-1}(1-\xi Y^{-1} X) \over X^{-1}-X}$$ and $$D(e,\widetilde e,\xi;Y,X)= {Y^{\widetilde e/2}X^{-(e-\widetilde e)/2} \over 1- \xi X} .$$
For a positive integer $i$ put $$C_i(e,\widetilde e,\xi;Y,X)= \begin{cases}
C(e,\widetilde e,\xi;Y,X) & \text { if $i$ is even } \\
D(e,\widetilde e,\xi;Y,X) & \text{ if $i$ is odd.}
\end{cases}.$$
\[def.4.3\] For a sequence $\underline a=(a_1,\ldots,a_n)$ of integers and an integer $1 \le i \le n$, we define ${{\mathfrak e}}_i={{\mathfrak e}}_i(\underline a)$ as $${{\mathfrak e}}_i=
\begin{cases} a_1+\cdots +a_i & \text{ if $i$ is odd} \\
2[(a_1+\cdots+a_i)/2] & \text{ if $i$ is even.}
\end{cases}$$ We also put ${{\mathfrak e}}_0=0$.
\[def.4.4\] For a naive EGK datum $H=(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ we define a rational function ${{\mathcal F}}(H;Y,X)$ in $X^{1/2}$ and $Y^{1/2}$ as follows: First we define $${{\mathcal F}}(H;Y,X)=X^{-a_1/2}+X^{-a_1/2+1}+\cdots+X^{a_1/2-1}+X^{a_1/2}$$ if $n=1$. Let $n>1$. Then $H'= (a_1,\ldots,a_{n-1};{\varepsilon}_1,\ldots,{\varepsilon}_{n-1})$ is a naive EGK datum of length $n-1$. Assume that ${{\mathcal F}}(H';Y,X)$ is defined for $H'$. Then, we define ${{\mathcal F}}(H;Y,X)$ as $$\begin{aligned}
&{{\mathcal F}}(H;Y,X)=C_n({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi;Y,X){{\mathcal F}}(H';Y,YX)\\
&+\zeta C_n({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi;Y,X^{-1}){{\mathcal F}}(H';Y,YX^{-1}),\end{aligned}$$ where $\xi={\varepsilon}_n$ or ${\varepsilon}_{n-1}$ according as $n$ is even or odd, and $\zeta=1$ or ${\varepsilon}_n$ according as $n$ is even or odd.
By the definition of ${{\mathcal F}}(H;Y,X)$ we easily see the following.
\[prop.4.1\] Let $H=(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ be a naive ${\mathrm{EGK}}$ datum of length $n$. Then we have $$\begin{aligned}
& {{\mathcal F}}(H;Y,X) \\
&=\sum_{(i_1,\ldots,i_n) \in \{\pm 1 \}^n} \eta_n^{(1-i_n)/2} C_{n}({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_n;Y,X^{i_n})\\
& \times \prod_{j=1}^{n-1}\eta_j^{(1-i_j)/2} C_{j}({{\mathfrak e}}_j,{{\mathfrak e}}_{j-1},\xi_j;Y,Y^{i_j+i_ji_{j+1}+\cdots+i_ji_{j+1}\cdots i_{n-1}}X^{i_j\cdots i_n}),\end{aligned}$$ where $$\xi_j=\begin{cases} {\varepsilon}_j & \text{ if $j$ is even} \\
{\varepsilon}_{j-1} & \text{ if $j$ is odd},
\end{cases}$$ and $$\eta_j=\begin{cases} 1 & \text{ if $j$ is even} \\
{\varepsilon}_j & \text{ if $j$ is odd}
\end{cases}$$ for $1 \le j \le n$. In particular, $${{\mathcal F}}(H;Y,X^{-1})=\eta_n {{\mathcal F}}(H;Y,X).$$
\[prop.4.2\] Let $H=(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ be a naive ${\mathrm{EGK}}$ datum of length $n$. Then ${{\mathcal F}}(H;Y,X)$ is a Laurent polynomial in $X^{1/2}$ with coefficients in ${{\mathbb Z}}[Y,Y^{-1}]$.
Put ${{\mathcal F}}(H;Y,X)'=X^{{{\mathfrak e}}_n/2}{{\mathcal F}}(H;Y,X)$. It suffices to show that ${{\mathcal F}}(H;Y,X)'$ is a polynomial in $X$ with coefficients in ${{\mathbb Z}}[Y,Y^{-1}]$. By definition, we easily see that ${{\mathcal F}}(H;Y,X)'$ belongs to ${{\mathbb Q}}(X,Y)$. Moreover, by Proposition \[prop.4.1\], ${{\mathcal F}}(H;Y,X)'$ can be expressed as $${{\mathcal F}}(H;Y,X)'=\frac{P(X,Y)}{Q(X,Y)}$$ with $P(X,Y),Q(X,Y) \in {{\mathbb Z}}[X,Y,Y^{-1}]$ such that $Q(X,Y)$ is a monic polynomial in $X$ with coefficients in ${{\mathbb Z}}[Y,Y^{-1}]$. By \[[@Ike-Kat], Remark 6.1, Proposition 6.3 \], and \[[@Kat1], Theorem 4.3\], $Q[X,p^{1/2}]$ divides $P[X,p^{1/2}]$ for any odd prime $p$ in ${{\mathbb Q}}[X,p^{1/2}]$. It follows that $Q(X,Y) $ divides $P(X,Y)$ in ${{\mathbb Z}}[X,Y,Y^{-1}]$. This proves the assertion.
\[prop.4.3\] Let $H=(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ be a naive ${\mathrm{EGK}}$ datum of length $n$ and $H''=(a_1,\ldots,a_{n-2};{\varepsilon}_1,\ldots,{\varepsilon}_{n-2})$. Then $H''$ is a naive ${\mathrm{EGK}}$ datum of length $n-2$. Assume that $a_{n-1}=a_n$. Then the following assertions hold.
- Assume that $n$ is odd and $a_1+\cdots +a_{n-1}$ is even. Then we have $$\begin{aligned}
{{\mathcal F}}(H;Y,X)&=Y^{{{\mathfrak e}}_{n-2}-1} \Biggl \{ {X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2 -1} \over (YX)^{-1}-YX}{{\mathcal F}}(H'';Y,Y^2X)\\
&+{{\varepsilon}_n X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2 +1} \over (YX^{-1})^{-1}-YX^{-1}}{{\mathcal F}}(H'';Y,Y^2X^{-1})\Biggr \}\\
&+ {Y^{{{\mathfrak e}}_{n-1}} (Y^2-Y^{-2}) {\varepsilon}_n \over ((YX)^{-1}-YX)((YX^{-1})^{-1}-YX^{-1})}{{\mathcal F}}(H'';Y,X) .\end{aligned}$$ In particular, ${{\mathcal F}}(H;Y,X)$ does not depend on ${\varepsilon}_{n-1}$.\
- Assume that $n$ is even and $a_1+\cdots +a_n$ is odd. Then we have $$\begin{aligned}
{{\mathcal F}}(H;Y,X)&=Y^{{{\mathfrak e}}_{n-2}} \Biggl \{{X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2 -1} \over X^{-1}-X}{{\mathcal F}}(H'';Y,Y^2X)\\
&+{X^{({{\mathfrak e}}_n- { e}_{n-2})/2 +1} \over X-X^{-1}}{{\mathcal F}}(H'';Y,Y^2X^{-1})\Biggr \}.\end{aligned}$$ In particular, ${{\mathcal F}}(H;Y,X)$ does not depend on ${\varepsilon}_{n-1}$.
We note that $(H')'=H''$, and hence $H''$ is a naive EGK datum of length $n-2$. Assume that $n$ is odd and $a_1+\cdots +a_{n-1}$ is even. Hence, we have $$\begin{aligned}
{{\mathcal F}}(H;Y,X) &=C_n({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},{\varepsilon}_{n-1};Y,X)\\
& \times \Biggl \{C_{n-1}({{\mathfrak e}}_{n-1},{{\mathfrak e}}_{n-2},{\varepsilon}_{n-1};Y,YX){{\mathcal F}}(H'';Y,Y^2X) \\
& +C_{n-1}({{\mathfrak e}}_{n-1},{{\mathfrak e}}_{n-2},{\varepsilon}_{n-1};Y,(YX)^{-1}){{\mathcal F}}(H'';Y,X^{-1})\Biggr \}\\
&+{\varepsilon}_n C_n({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},{\varepsilon}_{n-1};Y,X^{-1})\\
&\times \Biggl \{C_{n-1}({{\mathfrak e}}_{n-1},{{\mathfrak e}}_{n-2},{\varepsilon}_{n-1};Y,YX^{-1}){{\mathcal F}}(H'';Y,Y^2X^{-1})\\
& +C_{n-1}({{\mathfrak e}}_{n-1},{{\mathfrak e}}_{n-2},{\varepsilon}_{n-1};Y,(YX^{-1})^{-1}){{\mathcal F}}(H'';Y,X)\Biggr \},\end{aligned}$$ By definition, we have ${\varepsilon}_{n-2}={\varepsilon}_n$, and by Proposition \[prop.4.1\], we have $${{\mathcal F}}(H'',Y,X^{-1})={\varepsilon}_n{{\mathcal F}}(H'';Y,X).$$ Then the assertion (1) can be proved by a direct calculation. Similarly the assertion (2) can be proved.
Now we recall the definition of EGK datum (cf. [@Ike-Kat]).
\[def.4.5\] Let $G=(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$ be an element of ${{\mathbb Z}}_{>0}^r \times {{\mathbb Z}}_{\ge 0}^r \times
{\mathcal Z}_3^r$. Put $n_s^\ast=\sum_{i=1}^s n_i$ for $s \le r$. We say that $G$ is an EGK datum of length $n$ if the following conditions hold:
- $n_r^\ast=n$ and $m_1 <\cdots <m_r$.
- Assume that $n_s^\ast$ is even. Then $\zeta_s \not=0$ if and only if $m_1n_1+\cdots+m_sn_s$ is even.
- Assume that $n_s^\ast$ is odd. Then $\zeta_s \not=0$. Moreover we have
- Assume that $n_i^\ast$ is even for any $i<s$. Then $$\zeta_s=\zeta_{s-1}^{m_s+m_{s-1}}\cdots\zeta_2^{m_2+m_1} \zeta_1^{m_2+m_1}.$$ In particular, $\zeta_1=1$ if $n_1$ is odd.
- Assume that $n_1m_1+\cdots+(n_{s-1}-1)m_{s-1} $ is even and that $n_i^\ast$ is odd for some $i<s$. Let $t<s$ be the largest number such that $n_t^\ast$ is odd. Then $$\zeta_s=\zeta_{s-1}^{m_s+m_{s-1}}\cdots \zeta_{t+2}^{m_{t+3}+m_{t+2}} \zeta_{t+1}^{m_{t+2}+m_{t+1}} \zeta_t.$$ In particular, $\zeta_s=\zeta_t$ if $t=s-1$.
We denote by $\mathcal{EGK}_n$ the set of all ${\mathrm{EGK}}$ data of length $n$.
\[def.4.6.\] Let $H=(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ be an naive ${\mathrm{EGK}}$ datum of length $n$, and $n_1,\ldots,n_r,n_1^\ast,\ldots,n_r^\ast$ and $m_1,\ldots,m_r$ be those defined in Definition \[def.3.1\]. Then put $\zeta_s={\varepsilon}_{{n_s}^\ast}$ for $s=1,\ldots,r$. Then $G_H:=(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$ is an ${\mathrm{EGK}}$ datum (cf. \[[@Ike-Kat], Proposition 6.2\]). We then define a mapping ${\Upsilon}_n$ from $\mathcal{NEGK}_n$ to $\mathcal{EGK}_n$ by ${\Upsilon}_n(H)=G_H$.
The mapping ${\Upsilon}_n$ is surjective (cf. \[[@Ike-Kat], Proposition 6.3\]), but it is not injective in general.
\[prop.4.4\] Let $G\in\mathcal{EGK}_n$ be an ${\mathrm{EGK}}$ datum. Choose $H=(a_1,\ldots, a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)\in\mathcal{NEGK}_n$ such that ${\Upsilon}_n(H)=G$ and put $H'=(a_1,\ldots,a_{n-1};{\varepsilon}_1,\ldots,{\varepsilon}_{n-1})$ and $G'={\Upsilon}_{n-1}(H')$. Then $G'$ is uniquely determined by $G$ except in the following two cases:
- $n\geq 3$ is odd, $a_{n-1}=a_n$, and $a_1+\cdots+a_{n-1}$ is even.
- $n$ is even, $a_{n-1}=a_n$, and $a_1+\cdots+a_n$ is odd.
Moreover, in these exceptional cases, put $H''=(a_1,\ldots,a_{n-2};{\varepsilon}_1,\ldots,{\varepsilon}_{n-2})$ and $G''={\Upsilon}_{n-2}(H'')$. Then $G''$ is uniquely determined by $G$.
For the first part, it is enough to prove ${\varepsilon}_{n-1}$ is uniquely determined by $G$ except in (Case 1) and (Case 2). If $a_{n-1}<a_n$, then the assertion is obvious. Assume that $a_{n-1}=a_n$. If both $n\geq 3$ and $a_1+\cdots+a_{n-1}$ are odd, then we have ${\varepsilon}_{n-1}=0$ by (N2). Assume that both $n$ and $a_1+\cdots+a_n$ are even. Write $G=(n_1^\ast,\ldots,n_r^\ast;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$. Then we have $${\varepsilon}_{n-1}=
\begin{cases}
\zeta_{r-1} \ & \text{ if $n_r$ is odd,} \\
\zeta_{r-1}^{m_i+m_{r-1}}\cdots \zeta_1^{m_2+m_1} \ & \text{ if $n_1,\ldots,n_r$ are even,}\\
\zeta_{r-1}^{m_r+m_{r-1}}\cdots \zeta_{k+1}^{m_{k+2}+m_{k+1}}\zeta_k \
& \text{ if $n_r$ is even and}
\\
{ } & \ n_{k+1} \text{ is odd with some $k \le r -2$.}
\end{cases}$$ by (E3) and (N5). The latter part can be proved in a similar way.
\[th.4.1\] Let $G$ be an ${\mathrm{EGK}}$ datum of length $n$, take $H\in {\Upsilon}_n^{-1}(G)$. Then the Laurent polynomial ${{\mathcal F}}(H;Y,X)$ in $X^{1/2}, Y$ is uniquely determined by $G$, and does not depend on the choice of $H$.
We prove the assertion by the induction on $n$. The assertion holds for $n=1$. Let $n >1$ and assume that the assertion holds for any $l<n$. Write $$G=(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r), \quad
H=(a_1,\ldots, a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n).$$ We note that $(a_1,\ldots, a_n)$ is uniquely determined by $G$, and ${\varepsilon}_n=\zeta_r$. For a positive integer $i \le n$, let ${{\mathfrak e}}_i={{\mathfrak e}}_i(\widetilde m)$ be that defined in Definition \[def.4.3\]. Then except in (Case 1) and (Case 2) in Proposition \[prop.4.4\], we have $$\begin{aligned}
{{\mathcal F}}(H;Y,X)=&C_n({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi;Y,X){{\mathcal F}}(H';Y,YX)\\
&\quad +\zeta C_n({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi;Y,X^{-1}){{\mathcal F}}(H';Y,YX^{-1}) \end{aligned}$$ where $H'$ is as in Proposition \[prop.4.4\] and $$\begin{aligned}
\zeta=&
\begin{cases}
1 & \text{ if $n$ is even} \\
{\varepsilon}_n & \text{ if $n$ is odd,}
\end{cases}
\\\
\xi=&
\begin{cases}
{\varepsilon}_{n-1} & \text{ if $n$ is odd} \\
{\varepsilon}_n & \text{ if $n$ is even.}
\end{cases} \end{aligned}$$ By the induction assumption, ${{\mathcal F}}(H',Y,X)$ is uniquely determined by $G'={\Upsilon}_{n-1}(H')$, and hence ${{\mathcal F}}(H;Y,X)$ is uniquely determined by $G$ by Proposition \[prop.4.4\].
Next, we consider (Case 1), i.e., $n\geq 3$ is odd, $a_{n-1}=a_n$, and $a_1+\cdots a_{n-1}$ is even. Then, by (2) of Proposition \[prop.4.3\], we have $$\begin{aligned}
{{\mathcal F}}(H;Y,X)&=Y^{{{\mathfrak e}}_{n-2}-1} \Biggl \{ {X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2 -1} \over (YX)^{-1}-YX}{{\mathcal F}}(H'',Y^2X)\\
&+{X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2 +1} \over (YX^{-1})^{-1}-YX^{-1}}{{\mathcal F}}(H'';Y,Y^2X^{-1})\Biggr \}\\
&+ {Y^{{{\mathfrak e}}_{n-1}}(Y^2-Y^{-2}) {\varepsilon}_r \over ((YX)^{-1}-YX)((YX^{-1})^{-1}-YX^{-1})}{{\mathcal F}}(H'';Y,X).\end{aligned}$$ By the induction assumption, ${{\mathcal F}}(H'',Y,X)$ is uniquely determined by $G''={\Upsilon}_{n-2}(H'')$, and hence ${{\mathcal F}}(H;Y,X)$ is uniquely determined by $G$ by Proposition \[prop.4.4\].
Now we consider (Case 2), i.e., $n$ is even, $a_{n-1}=a_n$, and $a_1+\cdots +a_n$ is odd. Then, by (1) of Proposition \[prop.4.3\], we have $$\begin{aligned}
{{\mathcal F}}(H;Y,X)&=Y^{{{\mathfrak e}}_{n-2}} \Biggl \{{X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2-1} \over X^{-1} -X}{{\mathcal F}}(H'';Y,Y^2X)\\
&+{X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2+1} \over X -X^{-1}}{{\mathcal F}}(H'';Y,Y^2X^{-1}) \Biggr \}.\end{aligned}$$ By the induction assumption, ${{\mathcal F}}(H'',Y,X)$ is uniquely determined by $G''={\Upsilon}_{n-2}(H'')$, and hence ${{\mathcal F}}(H;Y,X)$ is uniquely determined by $G$ by Proposition \[prop.4.4\].
For an EGK datum $G$ we define $\widetilde {{\mathcal F}}(G;Y,X)$ as ${{\mathcal F}}(H;Y,X)$, where $H$ is a naive ${\mathrm{EGK}}$ datum of length $n$ such that ${\Upsilon}_n(H)=G$. For later purpose we recall the following theorem.
\[th.4.2\] [(cf. \[[@Ike-Kat], Theorem 6.1\])]{} Let $B \in {{\mathcal H}}_n({{\mathfrak o}})^{\rm{nd}}$. Then ${\mathrm{EGK}}(B)$ is an ${\mathrm{EGK}}$ datum of length $n$.
The following proposition will be used later.
\[prop.4.5\] Assume that $q$ is even. Let $B\in{{\mathcal H}}_n({{\mathfrak o}})$ be a reduced form of ${\mathrm{GK}}$ type $({{\underline{a}}},\sigma)$, where ${{\underline{a}}}=(a_1, \ldots, a_n)$.
- Assume that $B^{(n-1)}$ is a reduced form with $GK(B^{(n-1)})={{\underline{a}}}^{(n-1)}=(a_1, \ldots, a_{n-1})$. Then there exists $H=(\underline{a};{\varepsilon}_1, \ldots, {\varepsilon}_n)\in\mathcal{NEGK}_n$ such that ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$ and ${\Upsilon}_{n-1}(H')={\mathrm{EGK}}(B^{(n-1)})$.
- Assume that ${\mathrm{GK}}(B)$ satisfies the condition either in ${\rm (Case 1)}$ or ${\rm (Case2)}$ of Proposition \[prop.4.4\] and $\sigma(a_{n-1})=a_n$. Then $B^{(n-2)}$ is a reduced form with ${\mathrm{GK}}(B^{(n-2)})={{\underline{a}}}^{(n-2)}=(a_1, \ldots, a_{n-2})$, and there exists $H=(\underline{a};{\varepsilon}_1, \ldots, {\varepsilon}_n)\in\mathcal{NEGK}_n$ such that ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$ and ${\Upsilon}_{n-2}(H'')={\mathrm{EGK}}(B^{(n-2)})$.
Put ${\mathrm{EGK}}(B)=(n_1^\ast,\ldots,n_r^\ast;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$.\
(1) The assertion follows from [(cf. \[[@Ike-Kat], Theorem 0.4\])]{} if $a_{n-1}=a_n$. In the case that $n\geq 3$ is odd, $a_{n-1}=a_n$, and $a_1+\cdots+a_{n-1}$ is odd, we have ${\varepsilon}_{n-1}=\xi_{B^{(n-1)}}=0$. Similarly, in the case that $n$ is even, $a_{n-1}=a_n$, and $a_1+\cdots+a_n$ is even, we have $${\varepsilon}_{n-1}=
\begin{cases}
\zeta_{r-1} \ & \text{ if $n_r$ is odd,} \\
\zeta_{r-1}^{m_i+m_{r-1}}\cdots \zeta_1^{m_2+m_1} \ & \text{ if $n_1,\ldots,n_r$ are even,}\\
\zeta_{r-1}^{m_r+m_{r-1}}\cdots \zeta_{k+1}^{m_{k+2}+m_{k+1}}\zeta_k \
& \text{ if $n_r$ is even and}
\\
{ } & \ n_{k+1} \text{ is odd with some $k \le r -2$.}
\end{cases}$$ by Proposition \[prop.4.4\]. By (E3), we have $\eta_{B^{(n-1)}}={\varepsilon}_{n-1}$.
In (Case 1) of Proposition \[prop.4.4\], choose any $H=(\underline{a};{\varepsilon}_1, \ldots, {\varepsilon}_n)\in\mathcal{NEGK}_n$ such that ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$. Then we have $$(\underline{a};{\varepsilon}_1, \ldots, {\varepsilon}_{n-2}, 1, {\varepsilon}_n), \, (\underline{a};{\varepsilon}_1, \ldots, {\varepsilon}_{n-2}, -1, {\varepsilon}_n)\in\mathcal{NEGK}_n.$$ Thus one can find $H=(\underline{a};{\varepsilon}_1, \ldots, {\varepsilon}_n)\in\mathcal{NEGK}_n$ such that ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$ and ${\varepsilon}_{n-1}=\xi_{B^{(n-1)}}$. Thus we have ${\Upsilon}_{n-1}(H')={\mathrm{EGK}}(B^{(n-1)})$. Similarly, the assertion holds in (Case 2) of Proposition \[prop.4.4\].\
(2) Let $H''=(a_1,\ldots,a_{n-2};{\varepsilon}_1,\ldots,{\varepsilon}_{n-2})$ be a naive EGK datum of length $n-2$ such that ${\Upsilon}_{n-2}(H'')={\mathrm{EGK}}(B^{(n-2)})$. Assume that ${\mathrm{GK}}(B)$ satisfies the condition in (Case 1). Then put ${\varepsilon}_n={\varepsilon}_{n-2}$, and take ${\varepsilon}_{n-1}= \pm1$ arbitrary. Then $H:=({{\underline{a}}};{\varepsilon}_1,\ldots,{\varepsilon}_n)$ is a naive EGK data. Moreover, by \[[@Ike-Kat], Lemma 6.2 (2)\], we have ${\varepsilon}_n=\eta_{B}$, and hence we have ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$. This proves the assertion. Similarly the assertion holds in (Case 2).
Induction formulas of the local densities and Siegel series
===========================================================
Let $dY$ be the Haar measure of ${\rm Sym}_n(F)$ normalized so that $$\int_{{\rm Sym}_n({{\mathfrak o}})}dY =1.$$ For a measurable subset $C$ of ${\rm Sym}_n(F)$, we define the volume ${\rm vol}(C)$ of $C$ as $${\rm vol}(C)=\int_C dY.$$ We also normalize the Haar measure $dX$ of $M_{mn}(F)$ so that $$\int_{M_{mn}({{\mathfrak o}})}dX =1.$$ Let $m,n$ and $r$ be non-negative integers such that $m \ge n \ge r \ge 1$. For an element $S \in {{\mathcal H}}_m({{\mathfrak o}})^{\rm{nd}}, A \in {{\mathcal H}}_r({{\mathfrak o}}), R \in M_{r,n-r}({{\mathfrak o}})$, and $ T \in {{\mathcal H}}_{n-r}({{\mathfrak o}})$, we define the modified local density with respect to $S,T,R$, and $A$ as $${{\mathcal A}}_e(S,T,R,A)=\{(X_1,X_2) \in M_{m,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) \ |$$ $$S[X_1]-\left(\begin{matrix} T & 2^{-1}{}^tR \\ 2^{-1}R & A \end{matrix}\right) \left[\begin{matrix} 1_{n-r} \\ X_2 \end{matrix}\right] \in {\varpi}^e{{\mathcal H}}_{n-r}({{\mathfrak o}}) \},$$ and $$\alpha_{{{\mathfrak p}}}(S,T,R,A)=\lim_{e \rightarrow \infty}q^{e(n-r)(n-r+1)/2}{\rm vol}({{\mathcal A}}_e(S,T,R,A)).$$ As for the existence of the above limit, see Lemma \[lem.5.1\] and Theorem \[th.5.1\]. The modified local density $\alpha_{{{\mathfrak p}}}(S,T,R,A)$ can also be expressed as the following improper integral: $$\alpha_{{{\mathfrak p}}}(S,T,R,A)$$ $$=\lim_{e \rightarrow \infty} \int_{{{\mathfrak p}}^{-e}{\rm Sym}_{n-r}({{\mathfrak o}})}\int_{M_{m,n-r}({{\mathfrak o}}) \times
M_{r,n-r}({{\mathfrak o}})} \psi({\rm tr}(Y(F(X_1,X_2)))) dX_1dX_2dY,$$ where $$F(X_1,X_2)=S[X_1]-\left(\begin{matrix} T & 2^{-1}{}^tR \\ 2^{-1}R & A \end{matrix}\right) \left[\begin{matrix} 1_{n-r} \\ X_2 \end{matrix}\right].$$ We sometimes write the above improper integral as $$\int_{{\rm Sym}_{n-r}(F)}\int_{M_{m,n-r}({{\mathfrak o}}) \times
M_{r,n-r}({{\mathfrak o}})} \psi({\rm tr}(Y(F(X_1,X_2)))) dX_1dX_2dY.$$ In the case $r=0$, we write $\alpha_{{{\mathfrak p}}}(S,T,R,A)$ as $\alpha_{{{\mathfrak p}}}(S,T)$, and call it the local density representing $T$ by $S$. We have $$\alpha_{{{\mathfrak p}}}(S,T)=\lim_{e \rightarrow \infty} q^{en(n+1)/2}{\rm vol}({{\mathcal A}}_e(S,T)),$$ where $${{\mathcal A}}_e(S,T) = \{ X =(x_{ij}) \in M_{mn}({{\mathfrak o}}) \ | \ S[X] - T \in {\varpi}^e{{\mathcal H}}_n({{\mathfrak o}}) \}.$$ The local density can also be expressed as $$\alpha_{{{\mathfrak p}}}(S,T)=\int_{{\rm Sym}_n(F)}\int_{M_{mn}({{\mathfrak o}})}\psi({\rm tr}(Y(S[X]-T)))dXdY.$$ An element $X$ of $M_{mn}({{\mathfrak o}})$ with $m \ge n$ is said to be primitive if there is a matrix $Y \in M_{m,m-n}({{\mathfrak o}})$ such that $\left(\begin{matrix} X & Y \end{matrix}\right) \in GL_m({{\mathfrak o}})$. We denote by $M_{mn}({{\mathfrak o}})^\mathrm{{prm}}$ the subset of $M_{mn}({{\mathfrak o}})$ consisting of all primitive matrices. Let $m,n$ and $l$ be non-negative integers such that $m \ge n \ge l \ge 1$. For $S \in {{\mathcal H}}_m({{\mathfrak o}})^{\rm{nd}}$ and $ T \in {{\mathcal H}}_n({{\mathfrak o}})$, put $${{\mathcal B}}_e(S,T)^{(l)} = \{ X =(x_{ij}) \in {{\mathcal A}}_e(S,T) \ | \ (x_{ij})_{1 \le i \le m,n-l+1 \le j \le n} \text{ is primitive} \},$$ and we define the modified primitive local density $$\beta_{{{\mathfrak p}}}(S,T)^{(l)} = \lim_{e \rightarrow \infty} q^{en(n+1)/2}{\rm vol}({{\mathcal B}}_e(S,T)^{(l)}).$$ In particular put $$\beta_{{{\mathfrak p}}}(S,T)=\beta_{{{\mathfrak p}}}(S,T)^{(n)},$$ and call it the primitive local density. We make the convention that ${{\mathcal B}}_e(S,T)^{(0)}={{\mathcal A}}_e(S,T)$ and $\beta_{{{\mathfrak p}}}(S,T)^{(0)}=\alpha_{{{\mathfrak p}}}(S,T)$. We can also express $\beta_{{{\mathfrak p}}}(S,T)^{(l)}$ as $$\beta_{{{\mathfrak p}}}(S,T)^{(l)} =
\int_{\mathrm{Sym}_n(F)}
\int_{M_{m,n-l}({{\mathfrak o}})}
\int_{M^\mathrm{prm}_{m,l}({{\mathfrak o}})}
\psi(\mathrm{tr} (Y(S\left[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right) \right]-T)))\, dX_2\,dX_1\, dY.$$
Now let $H_k = \overbrace{H \bot \cdots \bot H}^k$ with $H =\left(\begin{matrix}
0 & 1/2 \\
1/2 & 0
\end{matrix}\right)$. We note that $$b_{{{\mathfrak p}}}(B,k)=\alpha_{{{\mathfrak p}}}(H_k,B)$$ for any positive integer $k \ge n$.
\[def.5.1\] Let $T \in {{\mathcal H}}_{n-r}({{\mathfrak o}})^{\rm{nd}}$. For $X=(x_{ij}) \in M_{r,n-r}({{\mathfrak o}})$ and $A \in {{\mathcal H}}_r({{\mathfrak o}}), R \in M_{r,n-r}({{\mathfrak o}})$ we define a matrix $T(R,A,X)$ by $$T(R,A,X)=
\begin{pmatrix} T & {}^t\!R/2 \\ R/2 & A\end{pmatrix}\left[
\begin{pmatrix} 1_{n-r} \\ X \end{pmatrix} \right]=
T+ A[X]+ \frac{1}2 \left({}^t\! RX+ {}^t\!X R\right).$$
For $B \in {{\mathcal H}}_n({{\mathfrak o}})^{\rm nd}$, we denote by ${{\mathfrak l}}_B$ the least integer such that $p^{{{\mathfrak l}}_B}B^{-1} \in {{\mathcal H}}_n({{\mathfrak o}})$.
\[lem.5.1\] Let $S \in {{\mathcal H}}_m({{\mathfrak o}})^{{\rm nd}}$.
- Let $T \in {{\mathcal H}}_{n-r}({{\mathfrak o}})$, and $A \in {{\mathcal H}}_r({{\mathfrak o}}), R \in M_{r,n-r}({{\mathfrak o}})$. Assume that there is a positive integer $l_0=l_0(T,R,A)$ depending only on $T,R,A$ such that $${\mathrm{ord}}(\det (2T(R,A,X))) \le l_0(T,R,A)$$ for any $X \in M_{r,n-r}({{\mathfrak o}})$. Put $m_0=2l_0+1$. Then the limit $\alpha_{{\mathfrak p}}(S,T,R,A)$ exists, and $$\alpha_{{\mathfrak p}}(S,T,R,A)=q^{-er(n-r)}\sum_{X \in M_{r,n-r}({{\mathfrak o}})/{{\mathfrak p}}^{e}M_{r,n-r}({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(S,T(R,A,X))$$ for any $e \ge m_0$. Here we use the same symbol $X$ to denote the coset of $X \in M_{r,n-r}({{\mathfrak o}})$ modulo ${{{\mathfrak p}}}^e$.
- Let $B \in {{\mathcal H}}_n({{\mathfrak o}})$.
- Assume that $B$ is non-degenerate. Then for any $0 \le r \le n$ the limit $\beta_{{\mathfrak p}}(S,B)^{(r)}$ exists and $$\beta_{{\mathfrak p}}(S,B)^{(r)}=q^{en(n+1)/2}{\rm vol}({{\mathcal B}}_{e}(S,B)^{(r)})$$ for any integer $e \ge 2{\mathrm{ord}}(\det (2B))+1$.
- Assume that $2S \in GL_m({{\mathfrak o}})$. Then $\beta_{{\mathfrak p}}(S,B)$ exists and $$\beta_{{\mathfrak p}}(S,B)=q^{en(n+1)/2}{\rm vol}({{\mathcal B}}_{e}(S,B))$$ for any $e \ge 1$.
- Assume that $B$ is non-degenerate. Then, there is a positive integer $\lambda_{{{\mathfrak l}}_B}$ depending on ${{\mathfrak l}}_B$ satisfying the following condition:\
If $A \in {{\mathcal H}}_n({{\mathfrak o}})$ satisfies $A \equiv B \ {\rm mod} \ {\varpi}^{\lambda_{{{\mathfrak l}}_B}} S_m({{\mathfrak o}})$, $A$ is $GL_m({{\mathfrak o}})$-equivalent to $B$.
In particular if $q$ is odd, we can take ${{\mathfrak l}}_B+1$ as $\lambda_{{{\mathfrak l}}_B}$.
\(1) For each $e$ put ${\bf A}_e= q^{e(n-r)(n-r+1)/2}{\rm vol}({{\mathcal A}}_e(S,T,R,A))$. Then we have $${\bf A}_e=\int_{M_{r,n-r}({{\mathfrak o}})} q^{e(n-r)(n-r+1)/2}{\rm vol}({{\mathcal A}}_e(S,T(R,A,X_2)))dX_2.$$ If $X_2 \equiv X_2' \in {{{\mathfrak p}}}^eM_{nr}({{\mathfrak o}})$, then ${\rm vol}({{\mathcal A}}_e(S,T(R,A,X_2)))={\rm vol}({{\mathcal A}}_e(S,T(R,A,X_2')))$. Hence we have $${\bf A}_e=\sum_{X \in M_{r,n-r}({{\mathfrak o}})/{{{\mathfrak p}}}^e M_{r,n-r}({{\mathfrak o}})}\int_{X+{{{\mathfrak p}}}^eM_{r,n-r}({{\mathfrak o}})} q^{e(n-r)(n-r+1)/2}{\rm vol}({{\mathcal A}}_e(S,T(R,A,X)))dX$$ $$=q^{-er(n-r)}\sum_{X \in M_{r,n-r}({{\mathfrak o}})/{{{\mathfrak p}}}^e M_{r,n-r}({{\mathfrak o}})}q^{e(n-r)(n-r+1)/2}{\rm vol}({{\mathcal A}}_e(S,T(R,A,X))).$$ Let $e \ge m_0$. Then, by using the same argument as in the proof of \[[@Sh1], Proposition 14.3\], we can prove that $$\alpha_{{{\mathfrak p}}}(S,T(R,A,X))=q^{e(n-r)(n-r+1)/2}{\rm vol}({{\mathcal A}}_e(S,T(R,A,X))).$$ Hence we have $${\bf A}_e=q^{-er(n-r)}\sum_{X \in M_{r,n-r}({{\mathfrak o}})/{{\mathfrak p}}^{e}M_{r,n-r}({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(S,T(R,A,X)).$$ We also have $${\bf A}_{e+1}=q^{-(e+1)r(n-r)}\sum_{X \in M_{r,n-r}({{\mathfrak o}})/{{\mathfrak p}}^{e+1}M_{r,n-r}({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(S,T(R,A,X)).$$ We have $\alpha_{{{\mathfrak p}}}(S,T(R,A,X))= \alpha_{{{\mathfrak p}}}(S,T(R,A,X'))$ if $X \equiv X' \text{ mod } {{\mathfrak p}}^eM_{r,n-r}({{\mathfrak o}})$. Hence $${\bf A}_{e+1}=q^{-(e+1)r(n-r)}q^{r(n-r)} \sum_{X \in M_{r,n-r}({{\mathfrak o}})/{{\mathfrak p}}^{e}M_{r,n-r}({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(S,T(R,A,X))={\bf A}_e.$$ This proves the assertion.\
(2) The assertion (2.1) can be proved by using the same argument as in the proof of \[[@Sh1], Proposition 14.3\]. The assertions (2.2) and (2.3) can easily be proved.
\[lem.5.2\] Let $S \in {{\mathcal H}}_m({{\mathfrak o}})^{{\rm nd}}, T \in {{\mathcal H}}_n({{\mathfrak o}})$, and let $l$ be a non-negative integer such that $l \le n$. Assume that the $\beta_{{{\mathfrak p}}}(S,T)^{(l)}$ exists. Then, for any $W \in M_n({{\mathfrak o}})^{{\rm nd}}$, we have $$\begin{aligned}
&\int_{{\rm Sym}_n(F)} \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M_{ml}^{\mathrm{prm}}({{\mathfrak o}})} \psi({\rm tr}(YW(S[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right)]-T)))dX_2dX_1dY
\\
&=|\det W|_{{{\mathfrak p}}}^{-n-1}\beta_{{{\mathfrak p}}}(S,T)^{(l)}.\end{aligned}$$
For a subset ${{\mathcal S}}$ of ${\rm Sym}(F)$ and an element $V \in GL_n(F)$ put $${{\mathcal S}}[V]=\{ X[V] \ | \ X \in {{\mathcal S}}\}.$$ Then we have $$\begin{aligned}
& \int_{{\rm Sym}_n(F)} \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M_{ml}^{\mathrm{prm}}({{\mathfrak o}})} \psi({\rm tr}(YW(S[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right)]-T)))dX_2dX_1dY \\
& = \lim_{e \rightarrow \infty} \int_{{{{\mathfrak p}}}^{-e} {\rm Sym}_n({{\mathfrak o}})} \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M_{ml}^{\mathrm{prm}}({{\mathfrak o}})} \psi({\rm tr}(YW(S[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right)]-T)))dX_2dX_1dY \\
&=|\det W|_{{{\mathfrak p}}}^{-n-1} \\
& \times \lim_{e \rightarrow \infty} \int_{{{{\mathfrak p}}}^{-e} {\rm Sym}_n({{\mathfrak o}})[W] } \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M_{ml}^{\mathrm{prm}}({{\mathfrak o}})} \psi({\rm tr}(Y(S[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right)]-T)))dX_2dX_1dY \\
&=|\det W|_{{{\mathfrak p}}}^{-n-1} \\
& \times \int_{{\rm Sym}_n(F) } \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M_{ml}^{\mathrm{prm}}({{\mathfrak o}})} \psi({\rm tr}(Y(S[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right)]-T)))dX_2dX_1dY.\end{aligned}$$ This proves the assertion.
\[prop.5.1\] Let $A$ and $B$ be non-degenerate half-integral matrices of degree $m$ and $n$, respectively, over ${{\mathfrak o}}$ such that $m \ge n$. Then we have $$\alpha_{{{\mathfrak p}}}(A,B)=\sum_{W \in GL_l({{\mathfrak o}}) \backslash M_l({{\mathfrak o}})^{{\rm nd}}} q^{{\mathrm{ord}}(\det W) (-m+n+1)}\beta_{{{\mathfrak p}}}(A,B[1_{n-l} \bot W^{-1}])^{(l)}.$$
We have $$\begin{aligned}
&\alpha_{{{\mathfrak p}}}(A,B) \\
=&\int_{\mathrm{Sym}_n(F)} \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M_{m,l}({{\mathfrak o}})} \psi(\mathrm{tr} (Y(A\left[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right) \right]-B)))dX_2dX_1dY \\
=&\int_{\mathrm{Sym}_n(F)} \sum_{W\in GL_l({{\mathfrak o}})\backslash M_l({{\mathfrak o}})^{{\rm nd}}} |\det W|_{{{\mathfrak p}}}^m \\
&\times \int_{M_{m,n-l}({{\mathfrak o}})} \int_{M^\mathrm{prm}_{m,l}({{\mathfrak o}})} \psi(\mathrm{tr} (Y\!A\left[\left(\begin{matrix} X_1 & X_2\!W \end{matrix}\right) \right]- Y\!B)) dX_2dX_1dY \\
=& \sum_{W\in GL_l({{\mathfrak o}})\backslash M_l({{\mathfrak o}})^{{\rm nd}}} |\det W|_{{{\mathfrak p}}}^{m} \\
&\times \int_{\mathrm{Sym}_n(F)} \int_{M_{m,n-l}({{\mathfrak o}})}\int_{M^\mathrm{prm}_{m,l}({{\mathfrak o}})}\psi\left(\mathrm{tr}(Y[1_{n-l} \bot \,{}^t W]A\left[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right) \right]-Y\!B)\right)
dX_2dX_1dY \\
=& \sum_{W\in GL_l({{\mathfrak o}})\backslash M_l({{\mathfrak o}})^{{\rm nd}}}|\det W|_{{{\mathfrak p}}}^{m} \\
&\times \int_{\mathrm{Sym}_n(F)} \int_{M_{m,n-l}({{\mathfrak o}})}\int_{M^\mathrm{prm}_{m,l}({{\mathfrak o}})}\psi\left(\mathrm{tr}( Y[1_{n-l} \bot \,{}^t W]g(X_1,X_2))\right)dX_2dX_1dY ,\end{aligned}$$ with $g(X_1,X_2)=A\left[\left(\begin{matrix} X_1 & X_2 \end{matrix}\right) \right]-B[1_{n-l} \bot W^{-1}]$. Then the assertion follows from Lemma \[lem.5.2\].
\[def.5.2\] Put ${\bf D}_{l,i}= GL_l({{\mathfrak o}}) ({\varpi}1_i \bot 1_{l-i}) GL_l({{\mathfrak o}})$ for $0 \le i \le l$. We define the function $\pi_l$ on $M_l({{\mathfrak o}})^{{\rm nd}}$ as $$\pi_l(W) =q^{i(i-1)/2}(-1)^i \text{ for } W \in {\bf D}_{l,i}$$ and $$\pi_l(W)=0 \text{ if } W\not\in \bigcup_{i=0}^l {\bf D}_{l,i}.$$
Then, using the same argument as in the proof of \[[@Ki1], Theorem 3.1\], we obtain the following.
\[cor.5.1\] Let $A$ and $B$ be non-degenerate half-integral matrices of degree $m$ and $n$, respectively, over ${{\mathfrak o}}$ such that $m \ge n$. Then we have $$\sum_{W \in GL_l({{\mathfrak o}}) \backslash M_l({{\mathfrak o}})^{{\rm nd}}} \pi_l(W)q^{(-m+n+1){\mathrm{ord}}(\det W)}\alpha_{{{\mathfrak p}}}(A,B[1_{n-l} \bot W^{-1}])=\beta_{{{\mathfrak p}}}(A,B)^{(l)}.$$
Let $M$ be a free module of rank $n$ over ${{\mathfrak o}}$, and $Q$ a non-degenerate quadratic form on $M$ with values in ${{\mathfrak o}}$. The pair $(M, Q)$ is called a quadratic module over ${{\mathfrak o}}$. The symmetric bilinear form $(x, y)=(x, y)_Q$ associated to $Q$ is defined by $$(x, y)_Q=Q(x+y)-Q(x)-Q(y), \qquad x, y\in M.$$ When there is no fear of confusion, we simply denote $(x, y)$. We denote by $\mathrm{s}(M)$ the fractional ideal of $F$ generated by $\{(x,y)\,|\, x,y\in M\}$, and call it the scale of $M$. For a basis $\{z_1, \ldots, z_n\}$, we define a matrix $B=(b_{ij})\in{{\mathcal H}}_n({{\mathfrak o}})$ by $$b_{ij}=\frac12 (z_i,z_j).$$ Two matrices are $GL_n({{\mathfrak o}})$-equivalent if and only if the associated quadratic modules are isomorphic.
An ${{\mathfrak o}}$-submodule $M'$ of a free ${{\mathfrak o}}$-free module $M$ is said to be primitive if $M'$ is a direct summand of $M$. Let $\{u_1,\ldots,u_m\}$ be a basis of $M$, and let $\{v_1,\ldots,v_l\}$ be a basis of a submodule $M'$. Write $$v_j=\sum_{i=1}^m a_{ij} u_j \ (j=1,\ldots, l).$$ Then $M'$ is primitive if and only if $(a_{ij})_{1 \le i \le m, 1 \le j \le l}$ is primitive. For an element $B \in {{\mathcal H}}_n({{\mathfrak o}})$ we denote by $\langle B \rangle$ the quadratic module $(M,Q)$ with a basis $\{z_1,\ldots,z_n\}$ such that $$\left({1 \over 2}(z_i,z_j)\right)_{1 \le i,j \le n}=B.$$ We note that the isomorphism class of $\langle B \rangle$ is uniquely determined by the $GL_n({{\mathfrak o}})$-equivalence class of $B$. To prove an induction formula for the local densities, we first prove the following lemma:
\[lem.5.3\] Let $k$ be a positive integer. Let $(M,Q)$ be a quadratic module over ${{\mathfrak o}}$ with a basis $\{z_1,\ldots, z_{2k}\}$ such that $$\left({1 \over 2} (z_i,z_j)\right)_{1 \le i,j \le 2k}=H_k .$$ Let $z_{2k-r+1}',\ldots,z_{2k}'$ be elements of $M$, and put $$A=\left({1 \over 2}(z_{2k-r+i}',z_{2k-r+j}')\right)_{1 \le i,j \le r}.$$ Assume that $\sum_{i=1}^r{{\mathfrak o}}z_{2k-r+i}'$ is primitive, and $A \in {{\mathfrak p}}{{\mathcal H}}_r({{\mathfrak o}})$. Then there exist elements $z_{2k-2r+1}',\ldots,z_{2k-r}'$ of $M$ and submodules $M_1'$ and $M_2'$ of $M$ such that
- $M_2'=\sum_{i=1}^{2r}{{\mathfrak o}}z_{2k-2r+i}'$ and $M_2'= \left\langle \mattwo(O;2^{-1}1_r;2^{-1}1_r;A) \right\rangle$.
- $M_1'$ is isometric to $\langle H_{k-r} \rangle$
- $M= M_1' \bot M_2'$.
[*Proof.*]{} Since $\sum_{i=1}^r{{\mathfrak o}}z_{2k-2r+2i}'$ is primitive, $A \in {{\mathfrak p}}{{\mathcal H}}_r({{\mathfrak o}})$, and $M$ is an orthogonal sum of hyperbolic spaces, there exist elements $z_{2k-2r+1}',\ldots,z_{2k-r}'$ of $M$ such that $$\left({1 \over 2}(z_{2k-2r+i}',z_{2k-2r+j}')\right)_{1 \le i,j \le 2r}=\mattwo(O;2^{-1}1_r;2^{-1}1_r;A).$$ Hence the submodule $$M_2'=\sum_{i=1}^{2r}{{\mathfrak o}}z_{2k-2r+i}'$$ satisfies the condition (1). Moreover since $s(M)={1 \over 2}{{\mathfrak o}}$ and $${1 \over 2}(u,v) \in {1 \over 2}{{\mathfrak o}}$$ for any $u \in M_2'$ and $v \in M$, Hence we have $$M=M_2'^{\bot} \bot M_2'.$$ The module $M_2'$ is isometric to $\langle H_r \rangle$ and $M$ is isometric to $\langle H_k \rangle$. Hence, by the cancellation theorem, there exists a submodule $M_1'$ of $M$ such that $$M_1' \cong \langle H_{k-r}\rangle,$$ and $$M=M_1' \bot M_2'.$$ This proves the assertion.
A similar assertion has been proved in \[[@Kat1], Lemma 2.3\].
\[cor.5.2\] Let $\Xi \in M_{2k,r}^{\mathrm{prm}}({{\mathfrak o}})$ such that $H_k[\Xi] \in {{\mathfrak p}}{{\mathcal H}}_r({{\mathfrak o}})$. Then there is an element $U \in GL_{2k}({{\mathfrak o}})$ of such that
- $H_k[U]=H_{k-r} \bot \mattwo(O;2^{-1}1_r;2^{-1}1_r;H_k[\Xi])$,
- $U^{-1}\Xi=\left(\begin{matrix} O \\ 1_r \end{matrix}\right)$.
\[th.5.1\] Let $B=\left(\begin{matrix} T & 2^{-1}{}^t R \\ 2^{-1}R & A \end{matrix}\right) \in {{\mathcal H}}_n({{\mathfrak o}})^{\rm{nd}}$ with $T \in {{\mathcal H}}_{n-r}({{\mathfrak o}}), R \in M_{r,n-r}({{\mathfrak o}})$ and $A \in {{\mathcal H}}_{r}({{\mathfrak o}})$. Then the limit $\alpha_{{{\mathfrak p}}}(H_{k-r},T,{\varpi}R,{\varpi}^2 A)$ exists and $$\beta_{{{\mathfrak p}}}(H_k,B[1_{n-r} \bot {\varpi}1_r])^{(r)}=\beta(H_k,{\varpi}^2 A) \alpha_{{{\mathfrak p}}}(H_{k-r},T,{\varpi}R,{\varpi}^2 A).$$
The above theorem can be proved in the same manner as \[[@Kat1], Proposition 2.4\]. (See also the proof of \[[@Kat-Kaw1], Lemma 3.2\].) But for the sake of completeness we give another proof. Put $B'=B[1_{n-r} \bot {\varpi}1_r]$ and write $X \in M_{2k,n}({{\mathfrak o}})$ and $Y \in {\rm Sym}_n(F)$ as $X=\left(\begin{matrix} X_1 & X_2 \end{matrix}\right)$ with $X_1 \in M_{2k,n-r}({{\mathfrak o}}), X_2 \in M_{2k,r}({{\mathfrak o}})$, and $Y=\mattwo(Y_{11};Y_{12};{}^tY_{12};Y_{22})$ with $Y_{11} \in {\rm Sym}_{n-r}(F), Y_{22} \in {\rm Sym}_r(F)$, and $Y_{12} \in M_{n-r,r}(F)$. For each non-negative integer $e$ and $X_2 \in M_{2k,r}^{\mathrm{prm}}({{\mathfrak o}})$ put $$\begin{aligned}
I_{X_2,e}&=&\int_{{{{\mathfrak p}}}^{-e}{\rm Sym}_{n-r}({{\mathfrak o}}) \times {{{\mathfrak p}}}^{-e}M_{n-r,r}({{\mathfrak o}})} \int_{M_{2k,n-r}({{\mathfrak o}})}
\psi({\rm tr}(Y_{11}(H_k[X_1]-T)))\\
&\times& \psi({\rm tr}(Y_{12}(2 {}^t\! X_2H_kX_1-{\varpi}R)))dX_1dY_{11}dY_{12}.
\end{aligned}$$ Then $$\begin{aligned}
& q^{en(n+1)/2}{\rm vol}({{\mathcal B}}_e(H_k,B')^{(r)})\\
= &\int_{{{{\mathfrak p}}}^{-e}{\rm Sym}_r({{\mathfrak o}})}\int_{M_{2k,r}^{\mathrm{prm}}({{\mathfrak o}})}\psi({\rm tr}(Y_{22}( H_k[X_2]-{\varpi}^2 A))) I_{X_2,e} dX_2dY_{22}\\
=&\int_{{{\mathcal B}}_e(H_k,{\varpi}^2 A)} I_{X_2,e}dX_2.\end{aligned}$$ We shall show that $$I_{\Xi,e}=q^{e(n-r)(n-r+1)/2}\mathrm{vol}({{\mathcal A}}_e(H_{k-r},T,{\varpi}R,{\varpi}^2A))$$ for any $\Xi \in {{\mathcal B}}_e(H_k,{\varpi}^2A)$. Let $U$ be the matrix in Corollary \[cor.5.2\], and write $$U^{-1}\left(\begin{matrix} X_1 & \Xi \end{matrix}\right)=\left(\begin{matrix} Y_1 & O\\
Y_2 & O \\
Y_3 & 1_r
\end{matrix}\right),$$ with $Y_1 \in M_{2k-2r,n-r}({{\mathfrak o}}), Y_2,Y_3 \in M_{r,n-r}({{\mathfrak o}})$. Then, by the change of variable given by $$M_{2k,r}({{\mathfrak o}}) \ni X_1 \longrightarrow (Y_1,Y_2,Y_3) \in M_{2k-2r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}),$$ we have $$\begin{aligned}
I_{\Xi,e}&=\int_{{{{\mathfrak p}}}^{-e}{\rm Sym}_{n-r}({{\mathfrak o}}) \times {{{\mathfrak p}}}^{-e}M_{n-r,r}({{\mathfrak o}})} \Bigl(\int_{M_{2k-2r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) } \\
& \psi(\mathrm{tr}(Y_{11}(H_{k-r}[Y_1]+2^{-1}({}^tY_2Y_3+{}^tY_3Y_2)+H_k[\Xi Y_3]-T)))\\
&\times \psi(Y_{12}(Y_2+2H_k[\Xi]Y_3-{\varpi}R))dY_1dY_2dY_3\Bigr) dY_{11}dY_{12}.\end{aligned}$$ Since $H_k[\Xi] \equiv {\varpi}^2 A \text{ mod } {\varpi}^e{{\mathcal H}}_r({{\mathfrak o}})$, we have $$\begin{aligned}
I_{\Xi,e}&=\int_{{{{\mathfrak p}}}^{-e}{\rm Sym}_{n-r}({{\mathfrak o}}) \times {{{\mathfrak p}}}^{-e}M_{n-r,r}({{\mathfrak o}})} \Bigl(\int_{M_{2k-2r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) } \\
& \psi({\rm tr}(Y_{12} W)) \psi({\rm tr}(Y_{11}g(Y_1,Y_2,Y_3))) dY_1dY_2dY_3\Bigr)dY_{11}dY_{12},\end{aligned}$$ where $$g(Y_1,Y_2,Y_3)=H_{k-r}[Y_1]+2^{-1}{}^tY_2Y_3+2^{-1}{}^tY_3Y_2+{\varpi}^2 A[Y_3] -T,$$ and $$W=Y_2+2{\varpi}^2 AY_3 -{\varpi}R.$$ By the change of variables given by $(Y_1,Y_2,Y_3) \mapsto (Y_1,W,Y_3),$ we have $$\begin{aligned}
I_{\Xi,e} &=\int_{{{{\mathfrak p}}}^{-e}{\rm Sym}_{n-r}({{\mathfrak o}}) \times {{{\mathfrak p}}}^{-e}M_{r,n-r}({{\mathfrak o}})} \Bigl(\int_{M_{2k-2r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}})}\\
&\psi({\rm tr}(Y_{11}f(Y_1,Y_3) )) \psi({\rm tr}((Y_{11}{}^tY_3+Y_{12})W))dY_1dW dY_3\Bigr) dY_{11}dY_{12},\end{aligned}$$ where $$\begin{aligned}
f(Y_1,Y_3)&=H_{k-r}[Y_1]- {\varpi}^2 A[Y_3]-2^{-1}{\varpi}{}^tR Y_3-2^{-1}{\varpi}{}^tY_3R-T\\
&=H_{k-r}[Y_1]-B'\left[\begin{pmatrix} 1_{n-r} \\ Y_3 \end{pmatrix}\right].\end{aligned}$$ Put $Z_{12}=Y_{11}\,{}^t Y_3+Y_{12}$. Then we have $$\begin{aligned}
I_{\Xi,e}&=\int_{{{{\mathfrak p}}}^{-e}{\rm Sym}_{n-r}({{\mathfrak o}}) } \int_{M_{2k-2r,n-r}({{\mathfrak o}}) \times M_{r,n-r}({{\mathfrak o}})}
\psi({\rm tr}(Y_{11}f(Y_1,Y_3)))dY_{11} dY_1dY_3\\
& \times \int_{{{{\mathfrak p}}}^{-e} M_{n-r,r}({{\mathfrak o}})} \int_{M_{r,n-r}({{\mathfrak o}})} \psi({\rm tr}(Z_{12}W))dZ_{12}dW .\end{aligned}$$ We have $$\int_{{{{\mathfrak p}}}^{-e} M_{n-r,r}({{\mathfrak o}})} \int_{M_{r,n-r}({{\mathfrak o}})} \psi({\rm tr}(Z_{12}W))dZ_{12}dW =1,$$ and hence $$I_{\Xi,e}= q^{e(n-r)(n-r+1)/2} {\rm vol}({{\mathcal A}}_e(H_{k-r},T,{\varpi}R,{\varpi}^2 A))$$ for any $\Xi \in M_{2k,r}^{\mathrm{prm}}({{\mathfrak o}})$. Hence we have $$\begin{aligned}
&q^{en(n+1)/2}{\rm vol}({{\mathcal B}}_e(H_k,B')^{(r)} \\
=&q^{er(r+1)/2} {\rm vol}({{\mathcal B}}_e(H_k,{\varpi}^2 A)) q^{e(n-r)(n-r+1)/2} {\rm vol}({{\mathcal A}}_e(H_{k-r},T,{\varpi}R,{\varpi}^2 A)).\end{aligned}$$ By Lemma \[lem.5.1\], the limits $\alpha_{{{\mathfrak p}}}(H_k,B')$ and $\beta_{{{\mathfrak p}}}(H_k,{\varpi}^2A)^{(r)}$ exist, and this proves the assertion.
Let $F={{\mathbb Q}}_p$ and $R=O$. Then the above theorem is nothing but \[[@Kat1], Proposition 2.4\].
\[def.5.3\] Let $T=(t_{ij}) \in {{\mathcal H}}_n({{\mathfrak o}})^{\rm{nd}}$ and $n_1$ be a positive integer such that $n_1 \le n$. Then put $R_T^{(n_1)}=(t_{ij})_{n-n_1+1 \le i \le n,1 \le j \le n-n_1}$ and $A_T^{(n_1)}=(t_{ij})_{n-n_1+1 \le i,j \le n}$, and for $x \in M_{n_1,n-n_1}({{\mathfrak o}})$ put $$T_x=T^{(n-n_1)}({\varpi}R_T^{(n_1)},{\varpi}^2 A_T^{(n_1)},x),$$ where $T^{(n-n_1)}({\varpi}R_T^{(n_1)},{\varpi}^2 A_T^{(n_1)},x)$ is the matrix in Definition \[def.5.1\]. Here we make the convention that $T_x$ is the empty matrix if $n_1=n$.
Clearly $T_{O}=T^{(n-n_1)}$ for the zero matrix $O$ of type $n_1 \times (n-n_1)$. We note that $$T_x=T\left[\left(\begin{matrix} 1_{n-n_1} \\ {\varpi}x \end{matrix}\right)\right].$$
\[lem.5.4\] Let $k$ be a positive integer. Then for any $T \in {{\mathfrak p}}{{\mathcal H}}_r({{\mathfrak o}})$, we have $$\beta_{{{\mathfrak p}}}(H_k,T)=(1-q^{-k})(1+q^{-k+r})\prod_{i=1}^{r-1}(1-q^{-2k+2i}).$$
By (2.2) of Lemma \[lem.5.1\], we have $$\begin{aligned}
& \beta_{{{\mathfrak p}}}(H_k,T) \\
&=q^{r(r+1)/2}{\rm vol}({{\mathcal B}}_1(H_k,T))\\
&=q^{-2k+r(r+1)/2}\#\{X \in M_{2k,r}({{\mathfrak o}})/{{\mathfrak p}}M_{2k,r}({{\mathfrak o}}) \ | \ X \in {{\mathcal B}}_1(H_k,T)\}\end{aligned}$$ Thus the assertion follows from \[[@Ki2], Lemma 5.6.9\].
\[th.5.2\] Let $B=(b_{ij})$ be an element of ${{\mathcal H}}_n({{\mathfrak o}})^{{\rm nd}}$.
- Assume that there is a positive integer $l_0$ such that $${\mathrm{ord}}(\det (2B_x)) \le l_0 \text{ for any } x \in M_{1,n-1}({{\mathfrak o}}).$$ Then there is a positive integer $m_0$ such that $$\begin{aligned}
& \alpha_{{{\mathfrak p}}}(H_k, B[1_{n-1} \bot {\varpi}])- q^{n-2k+1}\alpha_{{{\mathfrak p}}}(H_k,B) \\
&=\beta_{{{\mathfrak p}}}(H_k, {\varpi}^2 b_{nn})\alpha_{{{\mathfrak p}}}(H_{k-1},B^{(n-1)},({\varpi}b_{n,1},\ldots, {\varpi}b_{n,n-1}),{\varpi}^2 b_{nn}) \\
& =q^{-m_0(n-1)}(1-q^{-k})(1+q^{-k+1})\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(H_{k-1},B_x).\end{aligned}$$ Here we understand that $\alpha_{{{\mathfrak p}}}(H_{k-1},B_x)=1$ if $n=1$.
- Assume that there is a positive integer $l_0$ such that $${\mathrm{ord}}(\det (2B_x)) \le l_0 \text{ for any } x \in M_{2,n-2}({{\mathfrak o}}).$$ Then there is a positive integer $m_0$ such that $$\begin{aligned}
& \alpha_{{{\mathfrak p}}}(H_k,B[1_{n-2} \bot {\varpi}1_2]) +q^{2n-4k+3}\alpha_{{{\mathfrak p}}}(H_k,B) \\
& -q^{n-2k+1}\sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(H_k,B[1_{n-2} \bot W]) \\
&=\beta_{{{\mathfrak p}}}(H_k,{\varpi}^2 A)\alpha_{{{\mathfrak p}}}(H_{k-2},B^{(n-2)},{\varpi}R,{\varpi}^2 A) \\
&=q^{-2m_0(n-2)}(1-q^{-k})(1+q^{-2k+2})(1+q^{-k+2})\\
& \times \sum_{x \in M_{2,n-2}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{2,n-2}({{\mathfrak o}})} \alpha_{{{\mathfrak p}}}(H_{k-2},B_x),\end{aligned}$$ where $R=(b_{ij})_{n-1 \le i \le n, 1 \le j \le n-2}, A=\mattwo(b_{n-1,n-1};b_{n-1,n};b_{n-1,n};b_{nn})$. Here we understand that $\alpha_{{{\mathfrak p}}}(H_{k-2},B_x)=1$ if $n=2$.
[*Proof.*]{} By Lemma \[lem.5.4\], for $a \in {{\mathfrak o}}$, we have $$\beta_{{\mathfrak p}}(H_k,{\varpi}^2 a)= (1-q^{-k})(1+q^{-k+1})$$ and for $A \in {{\mathcal H}}_2({{\mathfrak o}})$. We note that $${{\mathfrak l}}_{B_x} \le {\mathrm{ord}}(\det (2B_x)) \le l_0 \text{ for any } x \in M_{1,n-1}({{\mathfrak o}}).$$ Hence, by (2.3) of Lemma \[lem.5.1\], there is a positive integer $c_{l_0}$ depending on $l_0$ satisfying the following condition:
If $B_{x'} \equiv B_x \text{ mod } {{\mathfrak p}}^{c_{l_0}} S_{n-1}({{\mathfrak o}})$ for $x, x' \in M_{1,n-1}({{\mathfrak o}})$, then $B_{x'} \sim B_x$.
Put $m_0=\max({\mathrm{ord}}(\det 2B)+1, c_{l_0})$. Then, by Lemma \[lem.5.1\], Corollary \[cor.5.1\], and Theorem \[th.5.1\], we see that $m_0$ satisfies the required condition in (1). We also have $$\beta_{{\mathfrak p}}(H_k,{\varpi}^2 A)=(1-q^{-k})(1-q^{-2k+2})(1+q^{-k+2}).$$ Hence the assertion (2) can be proved similarly.
\[cor.5.3\] Let $B$ be as above.
- Let the notation and the assumption be as in (1) of Theorem \[th.5.2\]. Then $$\begin{aligned}
F(B[1_{n-1} \bot {\varpi}],X) &=q^{n+1}X^2F(B,X)\\
&+q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})} K(X,x)F(B_x,qX),\end{aligned}$$ where $$K(X,x)={(1-X)(1+qX)\gamma_q(B_{x},qX) \over \gamma_q(B,X)}.$$ Here we understand that $F(B_x,qX)=1$ and $\gamma_q(B_x,qX)=1$ if $n=1$.
- Let the notation and the assumption be as in (2) of Theorem \[th.5.2\]. Then $$\begin{aligned}
& F(B[1_{n-2} \bot {\varpi}1_2],X) \\
& = -q^{2n+3}X^4 F(B,X)+q^{n+1}X^2\sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})} F(B[1_{n-2} \bot W],X) \\
& +q^{-2m_0(n-2)}\sum_{x \in M_{2,n-2}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{2,n-2}({{\mathfrak o}})} K(X,x)F(B_x,q^2X),\end{aligned}$$ where $$K(X,x)={(1-X)(1-q^2X^2)(1+q^2X)\gamma_q(B_{x},q^2X) \over \gamma_q(B,X)}.$$ Here we understand that $F(B_x,q^2X)=1$ and $\gamma_q(B_x,q^2X)=1$ if $n=2$.
Let $C(e,\widetilde e,\xi;Y,X)$ and $D(e,\widetilde e,\xi;Y,X)$ be rational functions in $X^{1/2}$ and $Y^{1/2}$ defined in Definition \[def.4.2\]. We often write $$\begin{aligned}
C(e,\widetilde e,\xi;X)=&C(e,\widetilde e,\xi;q^{1/2},X),\\
D(e,\widetilde e,\xi;X)=&D(e,\widetilde e,\xi;q^{1/2},X)\end{aligned}$$ if there is no fear of confusion. From now on we make the convention that we have $\widetilde F(A,X)=1$ if $\deg A=0$.
\[th.5.3\] Let the notation and the assumption be as in (1) of Theorem \[th.5.2\]. Put ${{{\mathfrak e}}}={{\mathfrak e}}_B$ and $\widetilde{{{\mathfrak e}}}_x={{\mathfrak e}}_{B_x}$ for $x \in M_{1,n-1}({{\mathfrak o}})$.
- Let $n$ be odd, and put $\xi_x=\xi_{B_x}$. Then we have $$\begin{aligned}
& \widetilde F(B,X) \\
&=q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\Bigl\{D({{\mathfrak e}},\widetilde {{{\mathfrak e}}}_x,\xi_x;X)\widetilde F(B_x,q^{1/2}X)\\
&+\eta_B D({{{\mathfrak e}}},\widetilde {{{\mathfrak e}}}_x,\xi_x;X^{-1})\widetilde F(B_x,q^{1/2}X^{-1})\Bigr\}.\end{aligned}$$ Here we make the convention that $ \xi_x=1, \widetilde {{\mathfrak e}}_x=0$ and $\widetilde F(B_x,q^{1/2}X)=\widetilde F(B_x,q^{1/2}X^{-1})=1$ if $n=1$.
- Let $n$ be even. Then we have $$\begin{aligned}
& \widetilde F(B,X) \\
&=q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\Bigl\{C({{{\mathfrak e}}},\widetilde {{{\mathfrak e}}}_x,\xi_B;X)\widetilde F(B_x,q^{1/2}X)\\
&+C({{{\mathfrak e}}},\widetilde {{{\mathfrak e}}}_x,\xi_B;X^{-1})\widetilde F(B_x,q^{1/2}X^{-1})\Bigr\}.\end{aligned}$$
[*Proof.*]{} Put $B'=B[1_{n-1} \bot {\varpi}]$. Let $n$ be odd. Then, by Corollary \[cor.5.3\], we have $$\begin{aligned}
& \widetilde F(B',X)=X\widetilde F(B,X) \\
&+q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})} {1-X^2 \over 1- \xi_x X}q^{\widetilde {{\mathfrak e}}_x/4}X^{(-{{\mathfrak e}}+\widetilde {{\mathfrak e}}_x)/2-1}\widetilde F(B_x,q^{1/2}X).\end{aligned}$$ We also have $$\begin{aligned}
& \widetilde F(B',X^{-1})=X^{-1}\widetilde F(B,X^{-1}) \\
&+ q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}{1-X^{-2} \over 1-q^{-1} \xi_x X^{-1}}q^{\widetilde {{\mathfrak e}}_x/4}X^{({{\mathfrak e}}-\widetilde {{\mathfrak e}}_x)/2+1}\widetilde F(B_x,q^{1/2}X^{-1}).\end{aligned}$$ By Proposition \[prop.2.1\] we have $\widetilde F(B',X^{-1})=\eta_B\widetilde F(B',X)$ and $\widetilde F(B,X^{-1})=\eta_B\widetilde F(B,X)$. This proves the assertion.
Let $n$ be even. Then, by (2) of Corollary \[cor.5.3\], we have $$\begin{aligned}
& \widetilde F(B',X)=X\widetilde F(B,X) \\
&+ q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}(1-q^{-1/2} \xi X)q^{\widetilde {{\mathfrak e}}_x/4}X^{(-{{\mathfrak e}}+\widetilde {{\mathfrak e}}_x)/2-1}\widetilde F(B_x,q^{1/2}X).\end{aligned}$$ We also have $$\begin{aligned}
\widetilde F(B',X^{-1})&=X^{-1}\widetilde F(B,X^{-1})\\
& + q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}(1-q^{-1/2} \xi X^{-1})\\
&\times q^{\widetilde {{\mathfrak e}}_x/4}X^{({{\mathfrak e}}-\widetilde {{\mathfrak e}}_x)/2+1}\widetilde F(B_x,q^{1/2}X^{-1}).\end{aligned}$$ By Proposition \[prop.2.1\] we have $\widetilde F(B',X^{-1})=\widetilde F(B',X)$ and $\widetilde F(B,X^{-1})=\widetilde F(B,X)$. This proves the assertion.
\[th.5.4\] Let $B$ be an element of ${{\mathcal H}}_n({{\mathfrak o}})$. Assume that $B^{(n-2)}$ is non-degenerate and that $\widetilde F(B_y,X)=\widetilde F(B^{(n-2)},X)$ for any $y \in M_{2,n-2}({{\mathfrak o}})$. Put ${{\mathfrak e}}={{\mathfrak e}}_B$ and $\hat {{\mathfrak e}}={{\mathfrak e}}_{B^{(n-2)}}$.
- Let $n$ be an even integer such that $n \ge 4$, and assume that $\xi_B=\xi_{B^{(n-2)}}=0$.
- We have $$\begin{aligned}
&\widetilde F(B[1_{n-2} \bot {\varpi}1_2],X)=q \widetilde F(B,X)\\
&+{q^{\hat {{\mathfrak e}}/2} \over X^{-1}-X} \Bigl\{X^{(\hat {{\mathfrak e}}-{{\mathfrak e}})/2-3}\widetilde F(B^{(n-2)},qX)(1-qX^2)\\
&-X^{(-\hat {{\mathfrak e}}+{{\mathfrak e}})/2+3}\widetilde F(B^{(n-2)},qX^{-1})(1-qX^{-2})\Bigr\}.\end{aligned}$$
- Assume that $B^{(n-1)}$ is non-degenerate and that ${\mathrm{ord}}(\det B_x)={\mathrm{ord}}(\det B^{(n-1)})$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. Put $\widetilde {{\mathfrak e}}={{\mathfrak e}}_{B^{(n-1)}}$. Then $$\begin{aligned}
& \widetilde F(B,X)(X^{(-{{\mathfrak e}}+2\widetilde{{\mathfrak e}}- \hat{{\mathfrak e}})/2-2} -X^{({{\mathfrak e}}-2\widetilde{{\mathfrak e}}+ \hat{{\mathfrak e}})/2 +2})\\
& =\widetilde F(B[1_{n-1} \bot {\varpi}],X)(X^{(-{{\mathfrak e}}+2\widetilde{{\mathfrak e}}- \hat{{\mathfrak e}})/2-1}
-X^{({{\mathfrak e}}-2\widetilde{{\mathfrak e}}+ \hat{{\mathfrak e}})/2+1}) \\
&+q^{\hat {{\mathfrak e}}/2}X^{\hat {{\mathfrak e}}-\widetilde {{\mathfrak e}}}\widetilde F(B^{(n-2)},qX)
-q^{\hat {{\mathfrak e}}/2}X^{-\hat {{\mathfrak e}}+\widetilde {{\mathfrak e}}}\widetilde F(B^{(n-2)},qX^{-1}).\end{aligned}$$
- Let $n$ be an odd integer such that $n \ge 3$. Then $$\begin{aligned}
& \widetilde F(B,X)=(q(X^{-1}+X))^{-1}\Bigl\{ \sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})}\widetilde F(B[1_{n-2} \bot W],X) \\
& -q^{\hat {{\mathfrak e}}/2}X^{-({{\mathfrak e}}-\hat {{\mathfrak e}})/2-1}\widetilde F(B^{(n-2)},qX)-\eta_Bq^{\hat {{\mathfrak e}}/2}X^{({{\mathfrak e}}-\hat {{\mathfrak e}})/2+1} \widetilde F(B^{(n-2)},qX^{-1}) \Bigr\}.\end{aligned}$$
\(1) We note that ${{\mathfrak e}}_{B_y}={{\mathfrak e}}_{B^{(n-2)}}$ for any $y \in M_{2,n-2}({{\mathfrak o}})$, and hence there is a positive integer $l_0$ such that $${\mathrm{ord}}(\det (2B_y)) \le l_0 \text{ for any } y \in M_{2,n-2}({{\mathfrak o}}).$$ Hence by Corollary \[cor.5.3\], we have, $$\begin{aligned}
& \widetilde F(B[1_{n-2} \bot {\varpi}1_2],X)X^{-1} \\
& = -qX \widetilde F(B,X)+ \sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})} \widetilde F(B[1_{n-2} \bot W],X), \\
& +q^{\hat e/2}X^{\hat e/2 -e/2-3} (1-qX^2) \widetilde F(B^{(n-2)},qX),\end{aligned}$$ and $$\begin{aligned}
& \widetilde F(B[1_{n-2} \bot {\varpi}1_2],X)X \\
& = -qX^{-1} \widetilde F(B,X)+ \sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})} \widetilde F(B[1_{n-2} \bot W],X) \\
& +q^{\hat e/2}X^{-\hat e/2 +e/2+3} (1-qX^{-2})\widetilde F(B^{(n-2)},qX^{-1}). \end{aligned}$$ Then the assertion (1.1) follows from these equality.
We prove (1.2). By the assumption, we can take a positive integer $m_0$ in (1) of Theorem \[th.5.3\]. Then, as in the proof of Theorem \[th.5.3\], we have $$\begin{aligned}
&\widetilde F(B[1_{n-1} \bot {\varpi}],X)=X\widetilde F(B,X) \\
&+ q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}q^{\widetilde {{\mathfrak e}}/4}X^{(-{{\mathfrak e}}+\widetilde {{\mathfrak e}})/2-1}\widetilde F(B_x,q^{1/2}X)\end{aligned}$$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. By the remark after Definition \[def.5.3\], for any $y \in M_{1,n-2}({{\mathfrak o}})$ we have $$\begin{aligned}
&(B_x)_y=(B\left[\left(\begin{matrix} 1_{n-1} \\ {\varpi}x \end{matrix}
\right)\right])\left[\left(\begin{matrix} 1_{n-2} \\ {\varpi}y \end{matrix}
\right)\right]=B\left[\left(\begin{matrix} 1_{n-2} \\ {\varpi}z \end{matrix}
\right)\right],\end{aligned}$$ where for $x=(x_1,\ldots,x_{n-1})$ and $y=(y_1,\ldots,y_{n-2})$ we define $z$ by $$z=\left(\begin{matrix} y_1 & \ldots & y_{n-2} \\
x_1+{\varpi}x_{n-1}y_1 &\ldots & x_{n-2}+{\varpi}x_{n-1}y_{n-2}
\end{matrix}\right).$$ Hence, by the assumption we have $F((B_x)_y,X)=F(B^{(n-2)},X)$. Hence, by (1) of Theorem \[th.5.3\], we have $$\begin{aligned}
& \widetilde F(B_x,q^{1/2}X) \\
&=q^{{\widetilde e}/4}(q^{1/2}X)^{(-\widetilde {{\mathfrak e}}+\hat {{\mathfrak e}})/2} \widetilde F(B^{(n-2)},qX)+\eta_x
q^{{\widetilde e}/4}(q^{1/2}X)^{(\widetilde {{\mathfrak e}}-\hat {{\mathfrak e}})/2} \widetilde F(B^{(n-2)},X^{-1}),\end{aligned}$$ where $\eta_x=\eta_{B_x}$. Hence $$\begin{aligned}
&\widetilde F(B[1_{n-1} \bot {\varpi}],X)=X\widetilde F(B,X) \\
&+q^{\hat {{\mathfrak e}}/2}X^{(-{{\mathfrak e}}+\hat {{\mathfrak e}})/2-1}\widetilde F(B^{(n-2)},qX) \\
& +q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\eta_x
q^{\widetilde {{\mathfrak e}}/2}X^{(-{{\mathfrak e}}+2\widetilde {{\mathfrak e}}-\hat {{\mathfrak e}})/2-1}\widetilde F(B^{(n-2)},X^{-1}).\end{aligned}$$ We also have $$\begin{aligned}
&\widetilde F(B[1_{n-1} \bot {\varpi}],X^{-1})=X^{-1}\widetilde F(B,X^{-1})\\
&+q^{\hat {{\mathfrak e}}/2}X^{({{\mathfrak e}}-\hat {{\mathfrak e}})/2+1}\widetilde F(B^{(n-2)},qX^{-1})\\
&+q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\eta_x
q^{\widetilde {{\mathfrak e}}/2}X^{({{\mathfrak e}}-2\widetilde {{\mathfrak e}}+\hat {{\mathfrak e}})/2+1}\widetilde F(B^{(n-2)},X).\end{aligned}$$ By Proposition \[prop.2.1\] we have $$\widetilde F(B[1_{n-1} \bot {\varpi}],X^{-1})=\widetilde F(B[1_{n-1} \bot {\varpi}],X),$$ $$\widetilde F(B,X^{-1})=\widetilde F(B,X),$$ and $$\widetilde F(B^{(n-2)},X^{-1})=\widetilde F(B^{(n-2)},X).$$ Thus the assertion (1.2) holds.\
(2) The assertion (2) can be proved by using the same argument as in the proof of (1.2).
Proof of Theorem \[th.1.1\]-non dyadic case-
=============================================
In this section and the next, we give a proof of Theorem 1.1. Main tools for the proof are refined versions of Theorem \[th.5.3\], from which we can obtain an explicit formula of $\widetilde F(B,X)$ for any $B \in {{\mathcal H}}_n({{\mathfrak o}})$. In this section, we assume that $q$ is odd.
[**Proof of Theorem \[th.1.1\].**]{} We prove the induction on $n$. If $n=1$, by Theorem \[th.5.3\], we easily see that $$\widetilde F(B,X)=\sum_{i=0}^{a_1} X^{i-(a_1/2)}$$ This proves the assertion for $n=1$. Let $n \ge 2$ and assume that the assertion holds for $n'=n-1$. We may assume that $B \in {{\mathcal H}}_n({{\mathfrak o}})$ is a diagonal Jordan form with ${\mathrm{GK}}(B)=(a_1,\ldots,a_n)$. Then $B^{(n-1)}$ is also a diagonal Jordan form with ${\mathrm{GK}}(B^{(n-1)})=(a_1,\ldots,a_{n-1})$. Then, we prove the following theorem, by which we prove Theorem 1.1.
\[th.6.1\] Under the above notation and the assumption, we have $$\begin{aligned}
&\widetilde F(B,X)=D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X)\widetilde F(B^{(n-1)},q^{1/2}X) \\
&+\eta_B D({{\mathfrak e}}_n, {{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X^{-1})\widetilde F(B^{(n-1)},q^{1/2}X^{-1})\end{aligned}$$ if $n \ge 3$ is odd, and $$\begin{aligned}
&\widetilde F(B,X)=C({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_B;X)\widetilde F(B^{(n-1)},q^{1/2}X) \\
&+ C({{\mathfrak e}}_n, {{\mathfrak e}}_{n-1},\xi_B;X^{-1})\widetilde F(B^{(n-1)},q^{1/2}X^{-1})\end{aligned}$$ if $n$ is even.
Let $n$ be odd. Then, by \[[@Ike-Kat], Theorem 0.1\], we have ${{\mathfrak e}}_B={{\mathfrak e}}_n$. For $x \in M_{1,n-1}({{\mathfrak o}})$ let $B_x$ be the matrix in Definition \[def.5.3\], and put ${{\mathfrak e}}_x={{\mathfrak e}}_{B_x}$ and $\xi_x=\xi_{B_x}$. We note that $$B_x \equiv B^{(n-1)} \text{ mod } {{\mathfrak p}}^{{{\mathfrak l}}_{B^{(n-1)}}+1}S_{n-1}({{\mathfrak o}}) \text{ for any } x \in M_{1,n-1}({{\mathfrak o}}).$$ Hence, by (2.3) of Lemma \[lem.5.1\], $B_x \sim B^{(n-1)}$ and in particular $${{\mathfrak e}}_x={{\mathfrak e}}_{n-1}, \ \xi_x=\xi_{B^{(n-1)}} \text{ and } \widetilde F(B_x,X)=\widetilde F(B^{(n-1)},X)$$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. Hence, by Theorem \[th.5.3\], we have $$\begin{aligned}
&\widetilde F(B,X)=D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X)\widetilde F(B^{(n-1)},q^{1/2}X)\\
&+\eta_B D({{\mathfrak e}}_n, {{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X^{-1})\widetilde F(B^{(n-1)},q^{1/2}X^{-1}).\end{aligned}$$ This proves the formula in the case that $n \ge 3$ is odd. Similarly the induction formula can be proved in the case that $n \ge 2$ is even.
Proof of Theorem \[th.1.1\]-dyadic case-
=========================================
Next we must consider a more complicated case where $q$ is even. Throughout this section we assume that $q$ is even. Let $B$ be a reduced form in ${{\mathcal H}}_n({\mathfrak o})$ with GK-type $((a_1,\ldots,a_n),{\sigma})$. Put ${{\underline{a}}}=(a_1,\ldots,a_n)$. We say that $({{\underline{a}}},{\sigma})$ belongs to category (I) if $n={\sigma}(n-1)$ and $a_{n-1}=a_n$. We say that ${\sigma}$ belongs to category (II) if $B$ does not belong to category (I). We note that $({{\underline{a}}},{\sigma})$ belongs to category (II) if and only if $a_{n-1}<a_n$ or ${\sigma}(n)=n$. In particular, $({{\underline{a}}},{\sigma})$ belongs to category (II) if $n=1$. We also say that $B$ belongs to category (I) or (II) according as $({{\underline{a}}},{\sigma})$ belongs to category (I) or (II). We note that if two reduced forms are of the same GK-type, then they belong to the same category.
\[def.7.1\] Let $B=(b_{ij})$ be a reduced form in ${{\mathcal H}}_n({{\mathfrak o}})$ with GK-type $({{\underline{a}}},{\sigma})$, and put ${{\underline{a}}}=(a_1,\ldots,a_n)$, and ${{\mathcal P}}^0$ be the set in Definition \[def.3.3\]. Let $n$ be odd. Then ${{{\mathcal P}}}^0$ consists of exactly one element, which will be denoted by $i_0=i_0(B)$. Let $n$ be even. Then $\#({{{\mathcal P}}}^0)=0$ or $2$. If $a_1+\cdots+a_n$ is odd, then ${{{\mathcal P}}}^0$ consists of two elements, which will be denoted by $i_1=i_1(B)$ and $i_2=i_2(B)$. In this case we note that for $k=1,2$, we have $b_{i_ki_k}={\varpi}^{a_{i_k}}u_{i_ki_k}$ with $u_{i_ki_k} \in {{\mathfrak o}}^{\times}$, and for $j \not=i_k$, we have $b_{i_kj}=2^{-1}{\varpi}^{[(a_{i_k}+a_j+2)/2]}u_{i_kj}$ with $u_{i_kj} \in {{\mathfrak o}}$. We say that $({{\underline{a}}},{\sigma})$ is good if ${{\mathcal P}}^0$ is empty. We also say that $B$ is a good form in this case. If $B$ is a good form, then we remark that $$\langle (-1)^{n/2} \det B,x \rangle =\xi_B^{{\mathrm{ord}}(x)}$$ for any $x \in F^{\times}$. We also note that $B$ is a good form if and only if both $n$ and $a_1+\cdots +a_n$ are even.
We say that an element $K$ of ${{\mathcal H}}_2({{\mathfrak o}})$ is a primitive unramified binary form if ${\mathrm{GK}}(B)=(0,0)$. We note that $K=(c_{ij}) \in {{\mathcal H}}_2({{\mathfrak o}})$ is a primitive unramified binary form if and only if $2c_{12} \in {{\mathfrak o}}^{\times}$. The following assertion can easily be proved.
\[lem.7.1\] Let $K=(c_{ij})$ be a primitive unramified binary form such that $c_{11}c_{22} \in {{\mathfrak p}}$. Then, for any $a,b \in {{\mathfrak o}}$ such that $ab \in {{\mathfrak p}}$, there is an element $U \in GL_2({{\mathfrak o}})$ such that $B[U]=\left(\begin{smallmatrix} a & 1/2 \\ 1/2 &b \end{smallmatrix}\right)$.
For a non-decreasing sequence ${{\underline{a}}}=(a_1,\ldots,a_n)$ of integers, let $G_{{{\underline{a}}}}$ be the group defined by $$G_{{{\underline{a}}}}=
\{g=(g_{ij})\in {{\mathrm{GL}}}_n({{\mathfrak o}}) \,|\, \text{ ${\mathrm{ord}}(g_{ij})\geq (a_j-a_i)/2$, if $a_i<a_j$}\}.$$ We say a reduced form $B=(b_{ij})$ of GK type $({{\underline{a}}}, {\sigma})$ is a strongly reduced form if the following condition hold:
- If $i\notin {{\mathcal P}}^0$, then $b_{ij}=0$ for $j>\max\{ i, {\sigma}(i)\}$.
We note that a reduced form of GK type $({{\underline{a}}}, {\sigma})$ is $G_{{\underline{a}}}$-equivalent to a strongly reduced form of GK type $({{\underline{a}}}, {\sigma})$ (see \[[@Ike-Kat], Remark 4.1\]). Let ${\bf D}_{2,1}$ be the subset of $M_2({{\mathfrak o}})$ in Definition \[def.5.2\].
\[lem.7.2\] Let $n$ be an odd integer, and let $B =(b_{ij}) \in {{\mathcal H}}_n({{\mathfrak o}})$ be a strongly reduced form of ${\mathrm{GK}}$-type $({{\underline{a}}},{\sigma})$ with ${{\underline{a}}}=(a_1,\ldots,a_n)$. Assume that $B$ belongs to category [(I)]{} and that $a_1+\cdots+a_{n-1}$ is even. Then for any $W \in {\bf D}_{2,1}$ the matrix $B[1_{n-2} \bot W]$ is $GL_n({{\mathfrak o}})$-equivalent to a reduced form $C$ belonging to category [(I)]{} of ${\mathrm{GK}}$ type $(a_1,\ldots,a_{n-2},a_{n-1}+1,a_{n-1}+1)$ such that $C^{(n-2)}=B^{(n-2)}$.
Take $U_1, U_2 \in GL_2({{\mathfrak o}})$ such that $W=U_1\smallmattwo(1;0;0;{\varpi})U_2$. Then, $B[1_{n-2} \bot U_1]$ is also a strongly reduced form, and $B[1_{n-2} \bot W]$ is $GL_n({{\mathfrak o}})$-equivalent to $B[1_{n-2} \bot U_1]\left[1_{n-2} \bot \smallmattwo(1;0;0;{\varpi})\right]$. Therefore, it suffices to prove the assertion for $W= \smallmattwo(1;0;0;{\varpi})$. Let $i_0=i_0(B)$ be the integer defined in Definition \[def.7.1\]. Put $a_i'= a_i$ for $1 \le i \le n-2$, and $a_{n-1}'=a_n'=a_{n-1}+1$. Put $B'=(b_{ij}')=B\left[1_{n-2} \bot \smallmattwo(1;0;0;{\varpi}) \right]$. Then $$b_{ij}'=b_{ij}$$ for any $(i,j)$ such that $1 \le i,j \le n-1$, $$b_{i,n}'={\varpi}b_{i,n}$$ for any $1 \le i \le n-1$, and $$b_{n,n}'={\varpi}^2 b_{n,n}.$$ Hence, $${\mathrm{ord}}(2b_{n-1,n}') =a_{n-1}', \ {\mathrm{ord}}(b_{n,n}') \ge a_n',$$ and $${\mathrm{ord}}(2b_{i,n}') > (a_i'+a_{n}')/2$$ for any $1 \le i \le n-2$. Since $B$ is strongly reduced and $a_{i_0}+a_{n-1}$ is even, we have $$b_{i,n-1}' =b_{i,n-1}=0$$ for any $1 \le i \le n-2$ such that $i \not=i_0$, and $${\mathrm{ord}}(2b_{i_0,n-1}')={\mathrm{ord}}(2b_{i_0,n-1}) \ge (a_{i_0} +a_{n-1})/2+1 >(a_{i_0}+a_{n-1}')/2.$$ We note that $B^{(n-2)}$ is a reduced form by \[[@Ike-Kat], Lemma 4.1\].
Now first assume that ${\mathrm{ord}}(b_{n-1,n-1}) >a_{n-1}$. Then, ${\mathrm{ord}}(b_{n-1,n-1}') \ge a_{n-1}'$, and this implies that $B'$ is a reduced form with ${\mathrm{GK}}(B')=(a_1',\ldots,a_n')$.
Next assume that ${\mathrm{ord}}(b_{n-1,n-1}) =a_{n-1}$. Since we have ${\sigma}(i_0)=i_0$, we have ${\mathrm{ord}}(b_{i_0,i_0})=a_{i_0}$ and ${\mathrm{ord}}(2b_{i_0,n-1}) > {a_{i_0}+a_{n-1} \over 2}$. Moreover since $a_{n-1}-a_{i_0}$ is even, by \[[@Ike-Kat], Lemma 4.4\], we can take $x \in {{\mathfrak o}}$ such that $${\mathrm{ord}}(x) \ge {a_{n-1} -a_{i_0} \over 2}, \ {\mathrm{ord}}(b_{n-1,n-1} +2b_{i_0,n-1}x+b_{i_0,i_0}x^2) > a_{n-1}.$$ Take $$V_1=
\left(\begin{smallmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\
\noalign{\vskip -2pt}
0 & \smallddots & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & \!\! u_{i_0,n-1} & 0 \\
\noalign{\vskip -6pt}
0 & 0 & 0 & \;\smallddots\; &0 &0 \\
0 & 0 & 0 & 0 &1 &0 \\
\noalign{\vskip 2pt}
0 & 0 & 0 & 0 &0 &1
\end{smallmatrix} \right)$$ with $u_{i_0,n-1}=x$. Put $B''=(b_{ij}'')=B'[V_1]$. Then we have $$b_{ij}'' =b_{ij}' \ \text{ if } 1 \le i,j \le n-2, \text { or } i=n, \text { or } j=n,$$ and $${\mathrm{ord}}(2^{1-\delta_{i,n-1}}b_{i,n-1}'')={\mathrm{ord}}(2^{1-\delta_{i,n-1}}b_{n-1,i}'') > (a_i+a_{n-1})/2$$ for any $1 \le i \le n-1$. Then, by \[[@Ike-Kat], Lemma 4.3\], $B''$ is $GL_n({{\mathfrak o}})$-equivalent to a reduced form $C=(c_{ij})$ such that $$\begin{aligned}
&c_{ij} =b_{ij}'' \ \text{ if } 1 \le i,j \le n-2, \text { or } i=n, \text { or } j=n,\\
&c_{i,n-1}=c_{n-1,i}=0 \text{ if } 1 \le i \le n-2 \text{ and } i\not=i_0,\end{aligned}$$ and $${\mathrm{ord}}(2^{1-\delta_{i,n-1}}c_{i,n-1})={\mathrm{ord}}(2^{1-\delta_{i,n-1}}c_{n-1,i}) > (a_i+a_{n-1})/2$$ if $i=i_0$ or $n-1$. Hence we can prove that $C$ satisfies the required conditions in the same way as in the first case.
\[def.7.2\] Let $B=(b_{ij}) \in {{\mathcal H}}_n({{\mathfrak o}})$ be a strongly reduced form with ${\rm GK}(B) =(a_1,\ldots,a_n)$. Assume that $B$ belongs to category (I) and that $a_1+\cdots+a_{n-2}$ is odd, and let $i_1=i_1(B)$ and $i_2=i_2(B)$ be those defined in Definition \[def.7.1\]. We then define $\kappa(B)$ as $$\kappa(B)=\min\{2{\mathrm{ord}}(2b_{i_k,m})-a_{i_k}-a_m \ | \ k=1,2, m=n-1,n \} .$$ Here we make the convention that $\kappa(B)=\infty$ if $b_{i_k,m}=0$ for any $k=1,2$ and $m=n-1,n$. We remark that $$\kappa(B[1_{n-2} \bot {\varpi}^rU])=\kappa(B)$$ for any non-negative integer $r$ and $U \in GL_2({{\mathfrak o}})$.
\[lem.7.3\] Let $m$ be an even integer . Let $C =(c_{ij}) \in {{\mathcal H}}_m({{\mathfrak o}})$ be a reduced form of ${\mathrm{GK}}$ type $({{\underline{a}}},\sigma)$ with ${{\underline{a}}}=(a_1,\ldots,a_m)$. Assume that $a_1+\cdots+a_m$ is odd, and let $i_1=i_1(C),i_2=i_2(C)$ be the integers $i_1, i_2 $ defined in Definition \[def.7.1\]. Let $A$ be an integer such that $A \ge a_m$ and $A+a_{i_1}+{\mathrm{ord}}({{\mathfrak D}}_C)$ is odd. For each $i$ such that $1 \le i \le m$ put $$x_i=\begin{cases} 2^{-1}c{\varpi}^{(A+a_{i_1}+{\mathrm{ord}}({{\mathfrak D}}_C)-1)/2} & \text{ if } i=i_1 \\
0 & \text{ otherwise},
\end{cases}$$ with $c \in {{\mathfrak o}}^{\times}$ and ${\bf y}=\left(\begin{matrix}y_1 \\ \vdots \\ y_m \end{matrix}\right)=C^{-1}\left(\begin{matrix} x_1 \\ \vdots \\ x_m \end{matrix}\right)$. Then $${\mathrm{ord}}(y_i) \ge (A-a_i)/2$$ for any $1 \le i \le m$, and $${\mathrm{ord}}(C[{\bf y}]) =A.$$
Write $C^{-1}=(c_{ij}^*)_{1 \le i,j \le m}$ and put ${{\mathfrak d}}={\mathrm{ord}}({{\mathfrak D}}_{C})$. Then by \[[@Ike-Kat], Lemma 3.12\], we have the following. $$\begin{aligned}
\label{eq.7.3.1}
&{\mathrm{ord}}(c_{ii}^*)= 2e_0+1-{{\mathfrak d}}-a_i & (i =i_k \text{ with } k=1,2), \tag{7.3.1} \\
\label{eq.7.3.2}
&{\mathrm{ord}}(c_{ii}^*) > 2e_0+1-{{\mathfrak d}}-a_i & (i\not=i_1,i_2), \tag{7.3.2}\\
\label{eq.7.3.3}
&
{\mathrm{ord}}(c_{ij}^*) \ge (2e_0+1-{{\mathfrak d}}-a_i-a_j)/2 & (i=i_1, j=i_2), \tag{7.3.3}\\
\label{eq.7.3.4}
&
{\mathrm{ord}}(c_{ij}^*) > (2e_0+1-{{\mathfrak d}}-a_i-a_j)/2 & (i \not=j, \{ i,j \} \not=\{i_1,i_2\}). \tag{7.3.4}\end{aligned}$$ Hence, by a simple computation, we have ${\mathrm{ord}}(y_i) \ge (A-a_i)/2$ for any $1 \le i \le m$, and $${\mathrm{ord}}(C[{\bf y}])={\mathrm{ord}}\left(C^{-1}\left[\left(\begin{matrix} x_1 \\ \vdots \\ x_m \end{matrix}\right)\right]\right)={\mathrm{ord}}(c_{i_1,i_1}^*x_{i_1}^2)=A.$$
\[lem.7.4\] Let $n$ be an even integer . Let $B$ be a reduced form of ${\mathrm{GK}}$ type $({{\underline{a}}},\sigma)$ with ${{\underline{a}}}=(a_1,\ldots,a_n)$. Assume that $B$ belongs to category [(I)]{} and that $a_1+\cdots+a_n$ is odd, and let $i_1=i_1(B),i_2=i_2(B)$ be the integers $i_1, i_2 $ defined in Definition \[def.7.1\]. Then, we have $n \ge 4$ and $i_1, i_2 \le n-2$, and there is a strongly reduced form $C=(c_{ij})_{n \times n}$ which is $G_{{{\underline{a}}}}$-equivalent to $B$ and satisfies the following three conditions:
- $C^{(n-2)}=B^{(n-2)}$.\
- $ \ \ (c_{ij})_{n-1 \le i,j \le n}={\varpi}^{a_{n-1}} \left(\begin{smallmatrix} 0 & 1/2 \\ 1/2 & 0 \end{smallmatrix}\right).$
- $\kappa(C)=2{\mathrm{ord}}(2c_{i_k,n-1})-a_{i_k}-a_{n-1}<{\mathrm{ord}}({{\mathfrak D}}_{C^{(n-2)}})$ with some $k=1,2$.
Since $B$ belongs to category (I), $B^{(n-2)}$ is a reduced form with GK-type $(a_1,\ldots,a_{n-2})$ and $i_1,i_2 \le n-2$. Then either $a_{i_1} \equiv a_{n-1} \text{ mod $2$}$ or $a_{i_2} \equiv a_{n-1} \text{ mod $2$}$. We note that the matrix $B^{(n-2)}(i_1,i_2;i_1,i_2)$ is a good form. Hence, in view of \[[@Ike-Kat], Lemma 4.3\] and Lemma \[lem.7.1\], in the same way as in the proof of Lemma \[lem.7.2\], we may assume that $B$ is a strongly reduced form such that $ (b_{ij})_{n-1 \le i,j \le n}={\varpi}^{a_{n-1}} \left(\begin{smallmatrix} a & 1/2 \\ 1/2 & 0 \end{smallmatrix}\right)$ with some $a \in {{\mathfrak o}}$. First assume that $\kappa(B)<{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})$. Then, again by Lemma \[lem.7.1\], we may assume that $a=0$. Moreover, if ${\mathrm{ord}}(2b_{i_l,n-1}) -a_{i_l}/2 > {\mathrm{ord}}(2b_{i_k,n})-a_{i_k}/2$ for any $l=1,2$, replacing $B$ by $B\left[1_{n-2} \bot \smallmattwo(0;1;1;0)\right]$, we obtain the matrix $C$ satisfying the condition (CEI-2). Next assume that $\kappa(B) \ge {\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})$. Without loss of generality, we may assume that $a_{i_1}+a_{n-1}+{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})$ is odd. For each $i$ such that $1 \le i \le n-2$ put $$x_i=\begin{cases} 2^{-1}{\varpi}^{(a_{n-1}+a_{i_1}+{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1)/2} & \text{ if } i=i_1 \\
0 & \text{ otherwise},
\end{cases}$$ and ${\bf y}=\left(\begin{matrix}y_1 \\ \vdots \\ y_{n-2} \end{matrix}\right)=(B^{(n-2)})^{-1}\left(\begin{matrix} x_1 \\ \vdots \\ x_{n-2} \end{matrix}\right)$. Then, by Lemma\[lem.7.3\], we have $${\mathrm{ord}}(y_i) \ge (a_{n-1}-a_i)/2 \text{ for any } 1 \le i \le n-2,$$ $$2{\mathrm{ord}}(2(x_{i_1}+b_{i_1,n-1}))-a_{i_1}-a_{n-1}<{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})$$ and $${\mathrm{ord}}(B^{(n-2)}[{\bf y}]+2b_{i_1,n-1}y_{i_1}+2b_{i_2,n-1}y_{i_2}) \ge a_{n-1}.$$ This implies that the matrix $B'=B\left[\left(\begin{matrix} 1_{n-2} & {\bf y} & O \\ O & 1 & 0 \\ O & 0 & 1 \end{matrix}\right)\right]$ is a strongly reduced form, satisfies the conditions (CEI-1) and (CEI-2), and its lower right $2 \times 2$ block is ${\varpi}^{a_n}\left(\begin{matrix} a' & 1/2 \\ 1/2 & 0 \end{matrix}\right)$ with $a' \in {{\mathfrak o}}$. Thus, by Lemma \[lem.7.1\], we obtain the matrix $C$ satisfying the condition (CEI-1).
\[lem.7.5\] Let $n$ be an even integer. Let $B \in {{\mathcal H}}_n({{\mathfrak o}})$ be a strongly reduced form belonging to category [(I)]{} with ${\mathrm{GK}}(B)=(a_1,\ldots,a_n)$. Assume that $a_1+\cdots +a_n$ is odd and let $i_1=i_1(B)$ and $i_2=i_2(B)$ be the integers in Definition \[lem.7.1\], and put $r_k={\mathrm{ord}}(2b_{i_k,n-1})$ for $k=1,2$. Moreover assume that $B$ satisfies the conditions [(CEI-1)]{} and [(CEI-2)]{}. Then we have the following:
- $B^{(n-1)}$ is non-degenerate and $$e_{B^{(n-1)}}=a_{n-1}+e_{B^{(n-2)}}+\kappa(B)+1,$$ and $$\begin{aligned}
&\eta_{B^{(n-1)}}=\eta_{B^{(n-2)}} \langle D_{B^{(n-2)}(i_1;i_1)},D_{B^{(n-2)}} \rangle. \end{aligned}$$
- Assume that $$a_{n-1} \ge \max(2\lambda_{{{\mathfrak l}}_{B^{(n-2)}}}+2e_0, 2{\mathrm{ord}}(\det B^{(n-2)}) +2e_0(n+1)+2),$$ where $\lambda_{{{\mathfrak l}}_{B^{(n-2)}}}$ is the integer defined in (2.3) of Lemma \[lem.5.1\]. Then for any $x =(x_1,\ldots,x_{n-1}) \in M_{1,n-1}({{\mathfrak o}})$, $B_x $ is non-degenerate, and $$e_{B_x}=a_{n-1}+e_{B^{(n-2)}}+\kappa(B)+1.$$ Moreover, if $\kappa(B)=1$, then $$\eta_{B_x}=\eta_{B^{(n-1)}}
\langle 1+x_{n-1} v_1 {\varpi}^{{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1}, D_{B^{(n-2)}} \rangle,$$ where $v_1$ is an element of ${{\mathfrak o}}^{\times}$ independent of $x$. In particular, $\eta_{B_x}$ is uniquely determined by $x_{n-1} \ {\rm mod} \ {{\mathfrak p}}$.
Without loss of generality we may assume that $\kappa(B)=2r_1-a_{n-1}-a_{i_1}$.\
(1) Put $A_{kl}=-\det B^{(n-2)}(i_k;i_l)$, and $D_{kl}=A_{kl}b_{i_k,n-1}b_{i_l,n-1}$ for $k,l=1,2$. Then we have $$\begin{aligned}
&\det B^{(n-1)}=A_{11}b_{i_1,n-1}^2+2A_{12}(-1)^{i_1+i_2}b_{i_1,n-1}b_{i_2,n-1}+A_{22}b_{i_2,n-1}^2. \end{aligned}$$ First we prove $$\begin{aligned}
\label{eq.7.4.1}
&{\mathrm{ord}}(\det B^{(n-1)})={\mathrm{ord}}(A_{11})+2(r_1-e_0).\tag{7.4.1}\end{aligned}$$ To prove this, put $(b_{ij}^*)_{1 \le i,j \le n-2}=(B^{(n-2)})^{-1}$. Then we note that $A_{kl}=(-1)^{i_k+i_l+1}b_{i_l,i_k}^* \det B^{(n-2)}$ and by \[[@Ike-Kat], Theorem 0.1\] we have $$\begin{aligned}
\label{eq.7.4.2}
{\mathrm{ord}}(\det B^{(n-2)})=\sum_{i=1}^{n-2} a_i +{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1-(n-2)e_0.\tag{7.4.2}\end{aligned}$$ Hence, by (\[eq.7.3.1\]), we have $$\begin{aligned}
\label{eq.7.4.3}
{\mathrm{ord}}(A_{kk})=\sum_{1 \le i \le n-2 \atop i \not=i_k} a_i -(n-4)e_0 \tag{7.4.3}\end{aligned}$$ and $$\begin{aligned}
\label{eq.7.4.4}
&{\mathrm{ord}}(D_{kk}) \tag{7.4.4} \\
&={\mathrm{ord}}(A_{kk})+2(r_k-e_0) \notag\\
&=\sum_{1 \le i \le n-2 \atop i \not=i_k} a_i -(n-2)e_0+ 2r_k \notag\end{aligned}$$ for $k=1,2$, and by (\[eq.7.3.3\]) $$\begin{aligned}
{\mathrm{ord}}(2D_{12})&= e_0+{\mathrm{ord}}(A_{12})+(r_1-e_0)+(r_2-e_0) \\
& \ge \sum_{i=1}^{n-2}a_i -(a_{i_1}+a_{i_2})/2+r_1+r_2\\
&+({\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1)/2-(n-2)e_0.\end{aligned}$$ We note that we have $r_1-a_{i_1}/2 <r_2 -a_{i_2}/2$. Hence we have ${\mathrm{ord}}(D_{11}) < {\mathrm{ord}}(D_{22})$ and ${\mathrm{ord}}(D_{11}) < {\mathrm{ord}}(2D_{12})$, and this proves (\[eq.7.4.1\]). Hence we prove the former half of (1) in view of (\[eq.7.4.4\]). We prove the latter half of (1). To prove this, we recall that $$\eta_{B^{(n-1)}}=\eta_{B^{(n-2)}}\langle (-1)^{(n-2)/2}\det B^{(n-1)},(-1)^{(n-2)/2}\det B^{(n-2)}\rangle$$ in view of \[[@Ike-Kat], Lemma 3.4\]. By Example 3 in Chapter II, Section 5 of [@Sat], we have $$\det \left(\begin{matrix} A_{11} & A_{12} \\
A_{12} & A_{22}
\end{matrix}\right)
=\det B^{(n-2)} \det B^{(n-2)}(i_1,i_2;i_1,i_2).$$ Hence we have $$\det B^{(n-1)}=A_{11}u^2+A_{11}\det B^{(n-2)}\det B^{(n-2)}(i_1,i_2;i_1,i_2)v^2$$ with some $u,v \in F$. Hence we have $$\langle A_{11}^{-1} \det B^{(n-1)},-\det B^{(n-2)}\det B^{(n-2)}(i_1,i_2;i_1.i_2)\rangle=1.$$ Moreover we note that $(b_{i_1,n-1}^2A_{11})^{-1}\det B^{(n-1)} \in {{\mathfrak o}}^{\times}$ and $B^{(n-2)}(i_1,i_2;i_1,i_2)$ is a good form (for the definition of ‘good form’, see Definition \[def.7.1\]). Then, by the remark in Definition \[def.7.1\], we have $$\langle A_{11}^{-1} \det B^{(n-1)}, (-1)^{(n-4)/2}\det B^{(n-2)}(i_1,i_2;i_1,i_2) \rangle=1.$$ This proves the latter half of (1).\
(2) For $x =(x_1,\ldots,x_{n-1}) \in M_{1,n-1}({{\mathfrak o}})$, write $B_x=(b_{x;ij})_{(n-1) \times (n-1)}$. We have $$B_x=B^{(n-1)}+ {\varpi}{}^tx {\bf b} + {\varpi}{}^t{\bf b}x,$$ where ${\bf b}=(b_{i,n})_{1 \le i \le n-1} \in M_{1,n-1}({{\mathfrak o}})$. Hence, for any $1 \le i,j \le n-1$, we have $$b_{x;ij}=b_{ij}+x_i{\varpi}b_{jn}+x_j{\varpi}b_{in}.$$ We have $$b_{in}=
\begin{cases}
b_{i_k,n} & \text{ if } i=i_k \text{ with } k=1,2 \\
0 & \text{ if } i\not=i_1,i_2, i \le n-2 \\
2^{-1} {\varpi}^{a_n} & \text{ if } i=n-1,
\end{cases}$$ and ${\mathrm{ord}}(b_{i_k,n}) > (a_n+a_{i_k})/2-e_0$. We note that $a_n=a_{n-1}$. Hence, by the assumption on $a_{n-1}$, we have $${\mathrm{ord}}(b_{i_k,n}) \ge {\mathrm{ord}}(\det B^{(n-2)}) +(n-2)e_0$$ for any $1 \le k \le 2$, and $${\mathrm{ord}}(2^{-1}{\varpi}^{a_n}) \ge {\mathrm{ord}}(\det B^{(n-2)}) +(n-2)e_0.$$ Hence we have $$\begin{aligned}
\label{eq.7.4.5}
b_{x;i_1,n-1} b_{i_1,n-1}^{-1} \equiv 1 \text{ mod } {{\mathfrak p}}, \tag{7.4.5}\end{aligned}$$ and $$b_{x;ij} \equiv b_{ij} \ {\rm mod} \ \det B^{(n-2)}{{\mathfrak p}}^{(n-2)e_0+1}$$ for any $1 \le i,j \le n-2$. We note that $b_{ij}, b_{x;ij} \in 2^{-1}{{\mathfrak o}}$, and hence $$\begin{aligned}
\label{eq.7.4.6}
\det B_x^{(n-2)}(i;j) \equiv \det B^{(n-2)}(i;j) \ {\rm mod} \ \det B^{(n-2)}{{\mathfrak p}}^{2e_0+1}
\tag{7.4.6}\end{aligned}$$ for any $1 \le i,j \le n-2$, and $$\begin{aligned}
\label{eq.7.4.7}
\det B_x^{(n-2)} \equiv \det B^{(n-2)} \ {\rm mod} \ \det B^{(n-2)}{{\mathfrak p}}^{e_0+1}.
\tag{7.4.7}\end{aligned}$$ Hence, by (\[eq.7.4.2\]) and (\[eq.7.4.3\]), we have $${\mathrm{ord}}(\det B_x^{(n-2)}(i_k;i_k))={\mathrm{ord}}(\det B^{(n-2)}(i_k;i_k))$$ for $k=1,2$. Moreover we have $$B_x^{(n-2)} \equiv B^{(n-2)} \ {\rm mod} \ {\varpi}^{\lambda_{{{\mathfrak l}}_{B^{(n-2)}}}}S_{n-2}({{\mathfrak o}}),$$ and hence, by (2.3) of Lemma \[lem.5.1\], $B_x^{(n-2)}$ is $GL_{n-2}({{\mathfrak o}})$-equivalent to $B^{(n-2)}$. We have $$b_{x;i,n-1}=
\begin{cases} 2^{-1}{\varpi}^{a_n+1}x_i & \text{ if } i\not=i_1,i_2, i \le n-2\\
b_{i_k,n-1}+ {\varpi}b_{i_k,n}x_{n-1}+ 2^{-1}{\varpi}^{a_n+1}x_{i_k} & \text{ if } i=i_k \text{ with } k=1,2\\
{\varpi}^{a_{n-1}+1} x_{n-1}& \text { if } i=n-1,
\end{cases}$$ $${\mathrm{ord}}(b_{i_k,n-1}) > (a_{n-1}+a_{i_k})/2-e_0$$ and $$a_{n-1}/2 \ge {\mathrm{ord}}(\det B^{(n-2)})+e_0(n-2)+2e_0+1 \ge 2e_0+1,$$ and hence ${\mathrm{ord}}(b_{i_k,n-1}) \ge e_0+1$. Similarly ${\mathrm{ord}}(b_{i_k,n}) \ge e_0+1$ and ${\mathrm{ord}}(2^{-1}\varpi^{a_n}) \ge e_0+1$. Hence ${\mathrm{ord}}(b_{x;i,n-1}) \ge e_0+1$ for any $1 \le i \le n-2$. Hence $$\det B_x^{(n-2)}(i;j) b_{x;i,n-1}b_{x;j,n-1} \equiv 0 \text { mod } \det B^{(n-2)}{{\mathfrak p}}^{a_{n-1}+2}$$ for any integers $1 \le i,j \le n-2$ such that $i \not= i_1,i_2$ or $j \not= i_1,i_2$. Similarly we have $$\begin{aligned}
&\det B_x^{(n-2)}(i_k;i_l)b_{x;i_k,n-1}b_{x;i_l,n-1} \\
&\equiv (b_{i_k,n-1}+{\varpi}x_{n-1}b_{i_k,n})(b_{i_l,n-1}+{\varpi}x_{n-1}b_{i_l,n})\\
&\times \det B^{(n-2)}(i_k;i_l) \text { mod } \det B^{(n-2)}{{\mathfrak p}}^{a_{n-1}+2} \end{aligned}$$ for any $k,l=1,2$, and $$\det B_x^{(n-2)}b_{x;n-1,n-1} \equiv \det B^{(n-2)} {\varpi}^{a_{n-1}+1} x_{n-1} \text { mod } \det B^{(n-2)}{{\mathfrak p}}^{a_{n-1}+2} .$$ Put $$\widetilde {B_x}= \mattwo( {B_x^{(n-2)}} ; {\begin{matrix}O \\ b_{x;i_1,n-1} \\ O \\ b_{x;i_2,n-1}\\ O \end{matrix}};{\begin{matrix}O & b_{x;i_1,n-1} & O & b_{x;i_2,n-1}& O \end{matrix}}; 0).$$ Then we have $$\det B_x=\det \widetilde B_x +\det B_x^{(n-2)}b_{x;n-1,n-1}.$$ By (\[eq.7.3.2\]),(\[eq.7.3.3\]),(\[eq.7.3.4\]), (\[eq.7.4.6\]) and (\[eq.7.4.7\]), we have $$\begin{aligned}
\label{eq.7.4.8}
&\det \widetilde B_x \tag{7.4.8} \\
&=\sum_{1 \le i,j \le n-2} (-1)^{i+j+1} \det B_x^{(n-2)}(i;j)b_{x;i,n-1}b_{x;j,n-1} \notag \\
& \equiv \sum_{1 \le k,l \le 2} (b_{i_k,n-1}+{\varpi}x_{n-1}b_{i_k,n})(b_{i_l,n-1}+{\varpi}x_{n-1}b_{i_l,n}) \notag \\
& \times (-1)^{i_k+i_l}A_{kl} \text{ mod }
\det B^{(n-2)}{{\mathfrak p}}^{a_{n-1}+2}. \notag\end{aligned}$$ and $$\begin{aligned}
&\det B_x^{(n-2)}b_{x;n-1,n-1} \equiv \det B^{(n-2)} x_{n-1}{\varpi}^{a_{n-1}+1} \text{ mod } \det B^{(n-2)}{{\mathfrak p}}^{a_{n-1}+2},\end{aligned}$$ where $A_{kl}$ is that in the proof of (1). Hence, by (1) and (\[eq.7.4.2\]), we have $${\mathrm{ord}}(\det B_x^{(n-2)} b_{x;n-1,n-1})- {\mathrm{ord}}(\det \widetilde {B_x})={\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-\kappa(B),$$ and in particular $${\mathrm{ord}}(\det \widetilde {B_x})={\mathrm{ord}}(\det B^{(n-1)})<{\mathrm{ord}}({\varpi}^{a_{n-1}+1}\det B_x^{(n-2)}).$$ Hence ${\mathrm{ord}}(\det B_x)={\mathrm{ord}}(\det B^{(n-1)})$. This proves the former half of (2).
Now assume that $\kappa(B)=1$. Then $$\begin{aligned}
\label{eq.7.4.9}
&{\mathrm{ord}}(\det B^{(n-2)} {\varpi}^{a_{n-1}+1})-{\mathrm{ord}}(b_{i_1,n-1}^2A_{11})={\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1 \tag{7.4.9} \end{aligned}$$ By (\[eq.7.4.5\]) and (\[eq.7.4.8\]) we have $$\det \widetilde {B_x}(b_{i_1,n-1}^2A_{11})^{-1} \equiv 1 \ {\rm mod} \ {{\mathfrak p}}.$$ Put $v_1=(b_{i_1,n-1}^2A_{11}{\varpi}^{{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1} )^{-1}\det B^{(n-2)} {\varpi}^{a_{n-1}+1}$. Then, by (\[eq.7.4.9\]), we have $v_1 \in {{\mathfrak o}}^{\times}$ and $$\det B_x (\det \widetilde {B_x})^{-1} \equiv 1+x_{n-1}v_1 {\varpi}^{{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1} \text{ mod } {{\mathfrak D}}_{B^{(n-2)}}.$$ We have $$\begin{aligned}
\eta_{B_x} &=\eta_{B_x^{(n-2)}}\langle B_{B_x},D_{B_x^{(n-2)}} \rangle \\
&=\eta_{B^{(n-2)}}\langle (-1)^{(n-2)/2}( \det \widetilde B_x +\det B_x^{(n-2)}b_{x;n-1,n-1}),D_{B^{(n-2)}} \rangle.\end{aligned}$$ By using the same argument as in the proof of the latter half of (1), we have $$\eta_{B^{(n-2)}}\langle D_{\widetilde {B_x}},D_{B^{(n-2)}}\rangle =\eta_{B^{(n-1)}},$$ and this proves the equality of in the latter half of (2). We note that the conductor of the character $${{\mathfrak o}}^{\times} \ni z \mapsto \langle z, D_{B^{(n-2)}} \rangle$$ is ${{\mathfrak D}}_{B^{(n-2)}}$. Hence we prove that $\eta_{B_x}$ is uniquely determined by $x_{n-1} \text{ mod } {{\mathfrak p}}$.
For integers $e,\widetilde e$, a real number $\xi$, let $C(e,\widetilde e,\xi;Y,X)$ and $D(e,\widetilde e,\xi;Y,X)$ be the rational functions in $Y^{1/2},X^{1/2}$ defined in Definition \[def.4.3\], and for an ${\mathrm{EGK}}$ datum $G$, let $\widetilde {{\mathcal F}}(G;Y,X)$ be the Laurent polynomial defined in Section 4.
[**Proof of Theorem \[th.1.1\].**]{}
We prove $$\begin{aligned}
\tag{$\mathrm{EF}_n$}
&\widetilde F(B,X)=\widetilde {{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X) \\
&\text{ for any reduced form } B \in {{\mathcal H}}_n({{\mathfrak o}})\end{aligned}$$ by induction on $n$. This proves Theorem \[th.1.1\] in view of Theorem \[th.3.2\]. Let $B\in {{\mathcal H}}_n({{\mathfrak o}})$ be a reduced form of type $({{\underline{a}}}, {\sigma}). $ Put ${{\underline{a}}}=(a_1,\ldots,a_n)$. For a non-negative integer $i \le n$ let ${{\mathfrak e}}_i={{\mathfrak e}}({{\underline{a}}})_i$ be the integer in Definition \[def.4.3\], and let ${{\mathcal M}}^{0}({{\underline{a}}})$ be the set in Definition \[def.3.4\]. By \[[@Ike-Kat], Theorem 0.1\], we have ${{\mathfrak e}}_B={{\mathfrak e}}_n$.
Assume that $n=1$. Then, by Theorem \[th.5.3\], we have $$\widetilde F(B,X)=\sum_{i=0}^{a_1} X^{i-(a_1/2)}.$$ Thus ($\mathrm{EF}_1$) holds. Now assume that $n \ge 2$, and assume that $(\mathrm{EF}_{n'}$) holds for any positive integer $n'<n$.
[**Step 1:**]{} First we assume that $B$ belongs to category (II) or that $B$ belongs to category (I) and $n+a_1+\cdots+a_{2[n/2]}$ is even. Then, $B$ is $GL_n({{\mathfrak o}})$-equivalent to a reduced form $B_1$ such that $B_1^{(n-1)}$ is a reduced form with ${\mathrm{GK}}(B_1^{(n-1)})={{\underline{a}}}^{(n-1)}$. In fact, $B^{(n-1)}$ is automatically a reduced form and ${\mathrm{GK}}(B^{(n-1)})={{\underline{a}}}^{(n-1)}$ if $B$ belongs to category (II). Assume that $B$ belongs to category (I). Assume that both $n$ and $a_1+\cdots+a_{n-1}$ are odd. By Lemma \[lem.7.1\], we may assume that ${\mathrm{ord}}(b_{n-1, n-1})=a_{n-1}$. Then $B^{(n-1)}$ is a reduced form with ${\mathrm{GK}}(B^{(n-1)})={{\underline{a}}}^{(n-1)}$. The case that both $n$ and $a_1+\cdots+a_{n-1}$ are even is similar. Therefore we may assume that $B^{(n-1)}$ is a reduced form with ${\mathrm{GK}}(B^{(n-1)})={{\underline{a}}}^{(n-1)}$. By Proposition \[prop.4.5\], there exists $H\in\mathcal{NEGK}_n$ such that ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$ and ${\Upsilon}_{n-1}(H')={\mathrm{EGK}}(B^{(n-1)})$.
For each $x \in M_{1,n-1}({{\mathfrak o}})$, let $B_x$ be the matrix in Definition \[def.5.3\]. By definition, $B_{x}-B^{(n-1)} \in {{\mathcal M}}^0({{\underline{a}}}^{(n-1)})$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. Hence, by Theorem \[th.3.3\], $B_x$ is also a reduced form with ${\mathrm{EGK}}(B_x)={\mathrm{EGK}}(B^{(n-1)})$. In particular, we have ${{\mathfrak e}}_{B_x}={{\mathfrak e}}_{B^{(n-1)}}={{\mathfrak e}}_{n-1}$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. Hence, ${\mathrm{ord}}(\det (2B_x)) \le {{\mathfrak e}}_{n-1}+2e+1$ for any $x \in M_{1,n-1}({{\mathfrak o}})$, and we can take a positive integer $m_0$ satisfying the condition in (1) of Theorem \[th.5.2\].
Assume that $n$ is odd. Then, by Theorem \[th.5.3\], we have $$\begin{aligned}
&\widetilde F(B,X) \\
=&q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0}M_{1,n-1}({{\mathfrak o}})}\Bigl\{ D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X)\widetilde F(B_{x},q^{1/2}X) \\
&\quad +\eta_B D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X^{-1})\widetilde F(B_{x},q^{1/2}X^{-1})\Bigr\}. \end{aligned}$$ By ($\mathrm{EF}_{n-1}$), we have $$\begin{aligned}
\widetilde F(B_{x},X)&=\widetilde {{\mathcal F}}({\mathrm{EGK}}(B^{(n-1)});q^{1/2},X) \\
&=\widetilde F(B^{(n-1)},X)={{\mathcal F}}(H', q^{1/2}, X) \end{aligned}$$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. Hence we have $$\begin{aligned}
\label{eq.1.1}
\widetilde F(B,X)
=&D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X)\widetilde F(B^{(n-1)}, q^{1/2} X) \tag{1.1}\\
&+\eta_B D({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B^{(n-1)}};X^{-1})\widetilde F(B^{(n-1)},q^{1/2} X). \notag\end{aligned}$$ Thus by Definition \[def.4.4\] $$\begin{aligned}
\widetilde F(B,X)={{\mathcal F}}(H, q^{1/2}, X) ={{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X).\end{aligned}$$ In the case that $n$ is even, similarly we have $$\begin{aligned}
\label{eq.1.2}
\widetilde F(B,X)
=&C({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B};X)\widetilde F(B^{(n-1)}, q^{1/2} X) \tag{1.2}\\
&+ C({{\mathfrak e}}_n,{{\mathfrak e}}_{n-1},\xi_{B};X^{-1})\widetilde F(B^{(n-1)},q^{1/2} X), \notag\end{aligned}$$ and $$\begin{aligned}
\widetilde F(B,X)={{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X).\end{aligned}$$
[**Step 2:**]{} Let $n$ be odd, and assume that $B$ belongs to category (I) and that $a_1+\cdots+a_{n-1}$ is even. Then, ${\mathrm{GK}}(B)$ satisfies the condition of (Case 1) of Proposition \[prop.4.4\]. In this case, $B^{(n-2)}$ is a reduced form with ${\mathrm{GK}}(B^{(n-2)})={{\underline{a}}}^{(n-2)}$. For any $y \in M_{2,n-2}({{\mathfrak o}})$, we have $B_y - B^{(n-2)} \in {{\mathcal M}}^0({\mathrm{GK}}(B^{(n-2)}))$, and by Theorem \[th.3.1\] and the induction assumption ($\mathrm{EF}_{n-2}$), we have $$\widetilde F(B_y,X)=\widetilde F(B^{(n-2)},X).$$ By (2) of \[[@Ike-Kat], Lemma 6.2 (2)\], we have $\eta_{B^{(n-2)}}=\eta_B$. In view of Lemma \[lem.7.2\], we may assume that $B$ is a strongly reduced. By (2) of Theorem \[th.5.4\], we have $$\begin{aligned}
\label{eq.2.1}
& \widetilde F(B,X) \tag{2.1}
\\
&=q^{-1}(X^{-1}+X)^{-1}\Bigl\{ \sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})} \widetilde F(B[1_{n-2} \bot W],X) \notag \\
&\quad -q^{{{\mathfrak e}}_{n-2}/2}X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2-1}\widetilde F(B^{(n-2)},qX) \notag \\
&\quad -\eta_B q^{{{\mathfrak e}}_{n-2}/2}X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2+1}\widetilde F(B^{(n-2)},qX^{-1})
\vphantom{\sum_{W \in {\bf D}_{2,1}/GL_2({{\mathfrak o}})}}
\Bigr\}.\notag\end{aligned}$$ By (2) of Lemma \[lem.7.2\], $B[1_{n-2} \bot W]$ is $GL_n({{\mathfrak o}})$-equivalent to a reduced form whose GK invariant is $(a_1,\ldots,a_{n-2},a_{n-1}+1,a_{n-1}+1)$ for any $W \in {\bf D}_{2,1}$. Moreover, we have $\eta_{B[1_{n-2} \bot W]}=\eta_B$ for any $W \in {\bf D}_{2,1}$. Hence, by a direct calculation using (\[eq.1.1\]) and (\[eq.1.2\]), we have $$\begin{aligned}
\label{eq.2.2}
&\widetilde F(B[{1}_{n-2}\bot W],X) \tag{2.2} \\
&=q^{({{\mathfrak e}}_{n-2}-1)/2}
\Biggl\{
\frac{X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2-2}} {(q^{1/2}X)^{-1}-q^{1/2}X }
\widetilde F(B^{(n-2)},qX) \notag\\
&\quad +\eta_B
\frac{X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2+2}} {(q^{1/2}X^{-1})^{-1}-q^{1/2}X^{-1} }\widetilde F(B^{(n-2)},qX^{-1})\Biggr\} \notag \\
&+\eta_B \frac{q^{{{\mathfrak e}}_{n-1}/2}(q-1)(X+X^{-1}) } { ((q^{1/2}X)^{-1}-q^{1/2}X)((q^{1/2}X^{-1})^{-1}-q^{1/2}X^{-1})} \notag \\
&\times \widetilde F(B^{(n-2)},X) . \notag
\notag\end{aligned}$$ for any $W \in {\bf D}_{2,1}$. Hence, by (\[eq.2.1\]) and (\[eq.2.2\]) we have $$\begin{aligned}
\label{eq.2.3}
\widetilde F(B,X)=&q^{{{\mathfrak e}}_{n-2}/2-1/2}
\Biggl\{{X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2-1} \over (q^{1/2}X)^{-1}-q^{1/2}X } \widetilde F(B^{(n-2)},qX) \tag{2.3}\\
&+ \eta_B{X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2+1} \over (q^{1/2}X^{-1})^{-1}-q^{1/2}X^{-1} }\widetilde F(B^{(n-2)},qX^{-1})\Biggr\} \notag\\
&+\eta_B {q^{{{\mathfrak e}}_{n-1}/2}(q-q^{-1}) \over ((q^{1/2}X)^{-1}-q^{1/2}X)((q^{1/2}X^{-1})^{-1}-q^{1/2}X^{-1})} \notag\\
&\times \widetilde F(B^{(n-2)},X). \notag\end{aligned}$$ Hence $$\widetilde F(B, X)=\widetilde {{\mathcal F}}({\mathrm{EGK}}(B), q^{1/2}, X)$$ by ($\mathrm{EF}_{n-2}$), (1) of Proposition \[prop.4.3\] and (2) of Proposition \[prop.4.5\].
[**Step 3:**]{} Let $n$ be even, and assume that $B$ belongs to category (I) and that $a_1+\cdots +a_n$ is odd. Then, ${\mathrm{GK}}(B)$ satisfies the condition of (Case 2) of Proposition \[prop.4.4\]. Then we prove the following equality: $$\begin{aligned}
\label{eq.3.1}
\widetilde F(B,X) &=q^{{{\mathfrak e}}_{n-2}/2} \Biggl \{{X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2 -1} \over X^{-1}-X}\widetilde F(B^{(n-2)},qX) \tag{3.1} \\
&+{X^{({{\mathfrak e}}_n- {{\mathfrak e}}_{n-2})/2 +1} \over X-X^{-1}}\widetilde F(B^{(n-2)},qX^{-1})\Biggr \}.\notag\end{aligned}$$ This combined with (2) of Proposition \[prop.4.3\] and (2) of Proposition \[prop.4.5\] proves that we have $$\begin{aligned}
\widetilde F(B,X)={{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X).\end{aligned}$$ Let $i_1=i_1(B)$ and $i_2=i_2(B)$ be those defined in Definition \[def.7.1\]. To prove (\[eq.3.1\]), we may assume that $B$ is a strongly reduced form and that it satisfies the conditions (CEI-1) and (CEI-2) in Lemma \[lem.7.4\]. Similarly to Step 2, by Theorem \[th.3.1\] and the induction assumption ($\mathrm{EF}_{n-2}$), we have $$\widetilde F(B_y,X)=\widetilde F(B^{(n-2)},X).$$ Let $r$ be a positive integer. Then, by (1.1) of Theorem \[th.5.4\] and by a simple computation, we see that (3.1) holds for $\widetilde F(B[1_{n-2} \bot {\varpi}^{r-1} 1_2],X)$ if it holds for $\widetilde F(B[1_{n-2} \bot {\varpi}^r 1_2],X)$. Therefore, to show (\[eq.3.1\]) we may assume that
(CEI-3) $a_{n-1} \ge \max(2\lambda_{{{\mathfrak l}}_{B^{(n-2)}}}+2e_0, 2{\mathrm{ord}}(\det B^{(n-2)}) +2e_0(n+2))$.
Let $k=\kappa(B)$. Then $k$ is a positive integer such that $k<{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})$. We prove (\[eq.3.1\]) by induction on $k$. First we prove (\[eq.3.1\]) for any strongly reduced form $B \in {{\mathcal H}}_n({{\mathfrak o}})$ with $\kappa(B)=1$ satisfying the conditions (CEI-1),(CEI-2) and (CEI-3). In this case, by (2) of Lemma \[lem.7.5\], ${{\mathfrak e}}_{B_x}=2+({{\mathfrak e}}_{n-2}+{{\mathfrak e}}_n)/2$ for any $x \in M_{1,n-1}({{\mathfrak o}})$ and it does not depend on the choice of $x$, which will be denoted by $\widetilde {{\mathfrak e}}$. Hence we can take a positive integer satisfying the condition in Theorem \[th.5.2\]. Then, as in the proof of Theorem \[th.5.3\], we have $$\begin{aligned}
\widetilde F(B[1_{n-1} \bot {\varpi}],X)&=X\widetilde F(B,X) \\
&+q^{\widetilde {{\mathfrak e}}/4}X^{(-{{\mathfrak e}}_n+\widetilde {{\mathfrak e}})/2-1}\\
&\times q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\widetilde F(B_x,q^{1/2}X).\end{aligned}$$ By (2) of Theorem \[th.5.3\], we have $$\begin{aligned}
\widetilde F(B_x,q^{1/2}X) &=q^{{\widetilde {{\mathfrak e}}}/4}(q^{1/2}X)^{(-\widetilde {{\mathfrak e}}+{{\mathfrak e}}_{n-2})/2} \widetilde F(B^{(n-2)},qX)\\
&+\eta_x
q^{{\widetilde {{\mathfrak e}}}/4}(q^{1/2}X)^{(\widetilde {{\mathfrak e}}-{{\mathfrak e}}_{n-2})/2} \widetilde F(B^{(n-2)},X^{-1}),\end{aligned}$$ where $\eta_x=\eta_{B_x}$. Hence $$\begin{aligned}
\widetilde F(B[1_{n-1} \bot {\varpi}],X)&=X\widetilde F(B,X) \\
&+q^{{{\mathfrak e}}_{n-2}/2}X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2-1}\widetilde F(B^{(n-2)},qX) \\
& +\widetilde F(B^{(n-2)},X^{-1})q^{\widetilde {{\mathfrak e}}/2}X^{(-{{\mathfrak e}}_n+2\widetilde {{\mathfrak e}}-{{\mathfrak e}}_{n-2})/2-1}\\
&\times q^{-m_0(n-1)}\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\eta_x.\end{aligned}$$ Put $$I=\sum_{x \in M_{1,n-1}({{\mathfrak o}})/{{\mathfrak p}}^{m_0} M_{1,n-1}({{\mathfrak o}})}\eta_x.$$ Then, by (2) of Lemma \[lem.7.5\], we have $$\begin{aligned}
I =cq^{m_0(n-1)-1}\sum_{y \in {{\mathfrak o}}/{{\mathfrak p}}} \langle 1+y{\varpi}^{{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1},D_{B^{(n-2)}}\rangle,\end{aligned}$$ where $c$ is a constant independent of $y$. By (2) of Lemma \[lem.7.5\], the homomorphism $${{\mathfrak o}}/{{\mathfrak p}}\ni y \mapsto \langle 1+y{\varpi}^{{\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})-1}, (-1)^{(n-2)/2} \det B^{(n-2)} \rangle \in \{\pm 1 \}$$ is non-trivial, and we have $I=0$. Hence $$\begin{aligned}
\widetilde F(B[1_{n-1} \bot {\varpi}],X)&=X\widetilde F(B,X) \\
&+q^{{{\mathfrak e}}_{n-2}/2}X^{(-{{\mathfrak e}}_n+{{\mathfrak e}}_{n-2})/2-1}\widetilde F(B^{(n-2)},qX) .\end{aligned}$$ We also have $$\begin{aligned}
\widetilde F(B[1_{n-1} \bot {\varpi}],X^{-1})&=X^{-1}\widetilde F(B,X^{-1})\\
&+q^{ {{\mathfrak e}}_{n-2}/2}X^{({{\mathfrak e}}_n-{{\mathfrak e}}_{n-2})/2+1}\widetilde F(B^{(n-2)},qX^{-1}).\end{aligned}$$ By Proposition \[prop.2.1\] we have $$\widetilde F(B[1_{n-1} \bot {\varpi}],X^{-1})=\widetilde F(B[1_{n-1} \bot {\varpi}],X),$$ $$\widetilde F(B,X^{-1})=\widetilde F(B,X),$$ and $$\widetilde F(B^{(n-2)},X^{-1})=\widetilde F(B^{(n-2)},X).$$ Hence we prove the equality (\[eq.3.1\]) in the case $\kappa(B)=1$.\
Let $2 \le k < {\mathrm{ord}}({{\mathfrak D}}_{B^{(n-2)}})$, and assume that (\[eq.3.1\]) holds for any $B' \in {{\mathcal H}}_n({{\mathfrak o}})$ with $\kappa(B')<k$ satisfying the conditions (CEI-1),(CEI-2) and (CEI-3). Let $B$ be a strongly reduced form satisfying the conditions (CEI-1),(CEI-2) and (CEI-3) such that $\kappa(B)=k$. Then, by (2) of Lemma \[lem.7.5\], ${{\mathfrak e}}_{B_x}=k+({{\mathfrak e}}_{n-2}+{{\mathfrak e}}_n)/2+1$ for any $x \in M_{1,n-1}({{\mathfrak o}})$. Hence, by (1.2) of Theorem \[th.5.4\], $$\begin{aligned}
\widetilde F(B,X)(X^{k-1}-X^{1-k}) &=\widetilde F(B[1_{n-1} \bot {\varpi}],X)(X^k-X^{-k}) \\
&+q^{{{\mathfrak e}}_{n-2}/2}X^{-k-1+({{\mathfrak e}}_{n-2}-{{\mathfrak e}}_n)/2}\widetilde F(B^{(n-2)},qX)\\
&-q^{{{\mathfrak e}}_{n-2}/2}X^{k+1+(-{{\mathfrak e}}_{n-2}+{{\mathfrak e}}_n)/2}\widetilde F(B^{(n-2)},qX^{-1}).\end{aligned}$$ We note that $B[1_{n-1} \bot {\varpi}]$ is a strongly reduced form satisfying the conditions (CEI-1), (CEI-2) and (CEI-3) with $\kappa(B[1_{n-1} \bot {\varpi}])=k-1$. Hence, by the induction assumption, we prove (\[eq.3.1\]) for $B$. This completes the induction. $\hfill\Box$
Examples
========
\(1) Let $G=(n_1,\ldots,n_r;m_1,\ldots,m_r;\zeta_1,\ldots,\zeta_r)$ be an EGK datum of length $n$. For $1 \le i \le n$ we define $\widetilde m_i$ as $$\widetilde m_i =m_j \text{ if } n_1+\cdots +n_{j-1} +1 \le i \le n_1+\cdots +n_j,$$ and for such $\widetilde m_1,\ldots,\widetilde m_n$ we define the integers ${{\mathfrak e}}_1,\ldots,{{\mathfrak e}}_n$ as in Definition \[def.4.3\].
(1.1) An EGK datum of length $2$ is one of the following forms
- $G=(1,1;m_1,m_2;1,\zeta_2)$ with $m_1 <m_2$ and $\zeta_2 \in {{\mathcal Z}}_3$
- $G=(2;m_1;\zeta_1)$ with $\zeta_2 \in \{\pm 1\}$.
Put $\xi=\zeta_2$ or $\xi=\zeta_1$ according as case (a) or case (b). Then $$H=(\widetilde m_1,\widetilde m_2;1,\xi)$$ is a naive EGK datum such that ${\Upsilon}_2(H)=G$, and by a simple calculation combined with Proposition \[prop.4.1\], $\widetilde {{\mathcal F}}(G;Y,X)$ can be expressed as $$\begin{aligned}
\widetilde {{\mathcal F}}(G;Y,X) &=\sum_{i=0}^{{{\mathfrak e}}_1} Y^i X^{-{{\mathfrak e}}_2/2+i}\sum_{j=0}^{{{\mathfrak e}}_2/2-i} X^{2j}\\
&- \xi \sum_{i=0}^{{{\mathfrak e}}_1} Y^{i-1}X^{-{{\mathfrak e}}_2/2+i+1}\sum_{j=0}^{{{\mathfrak e}}_2/2-i-1}X^{2j}. \end{aligned}$$ Let $B \in {{\mathcal H}}_2({{\mathfrak o}})^{\rm{nd}}$. Then by Theorem \[th.1.1\], we have $$\widetilde F(B,X)=\widetilde {{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X).$$ This coincides with \[[@Ot], Corollary 5.1\].
(1.2) An EGK datum of length $3$ is one of the following forms:
- $G=(1,1,1;m_1,m_2,m_3;1,\zeta_2,\zeta_3)$ with $\zeta_2 \in {{\mathcal Z}}_3$, and $\zeta_3 \in \{\pm 1 \}$
- $G=(1,2;m_1,m_2;1,\zeta_2)$ with $\zeta_2 \in \{\pm 1 \}$
- $G=(2,1;m_1,m_2;\zeta_1,\zeta_2)$ with $\zeta_1 \in {{\mathcal Z}}_3$ and $\zeta_2 \in \{\pm 1 \}$
- $G=(3;m_1;1)$.
We put $$\xi=
\begin{cases} \zeta_2 & \text{ in case (a) }\\
\zeta_1 & \text{ in case (c)} \\
1 & \text { in case (b) or case (d), and $\widetilde m_1+\widetilde m_2$ is even } \\
0 & \text { in case (b) or case (d), and $\widetilde m_1+\widetilde m_2$ is odd, }
\end{cases}$$ and $$\eta=
\begin{cases}
\zeta_3 & \text{ in case (a) }\\
\zeta_2 & \text{ in case (b) or (c)} \\
1 & \text { in case (d).}
\end{cases}$$ Moreover put ${{\mathfrak e}}_2'=2[(a_1+a_2+1)/2]$. Then, $$H=(\widetilde m_1,\widetilde m_2,\widetilde m_3;1,\xi,\eta)$$ is a naive EGK datum such that ${\Upsilon}_3(H)=G$, and by a simple calculation combined with Proposition \[prop.4.1\], $\widetilde {{\mathcal F}}(G;Y,X)$ can be expressed as $$\begin{aligned}
\widetilde {{\mathcal F}}(G;Y,X)&=X^{-{{\mathfrak e}}_3/2}\Biggl \{\sum_{i=0}^{{{\mathfrak e}}_1} (Y^2X)^i \sum_{j=0}^{{{\mathfrak e}}_2'/2-i-1} (YX)^{2j} \\
&+ \eta X^{{{\mathfrak e}}_3}\sum_{i=0}^{{{\mathfrak e}}_1} (Y^2X^{-1})^i \sum_{j=0}^{{{\mathfrak e}}_2'/2-i-1} (YX^{-1})^{2j} \\
&+\xi^2 Y^{{{\mathfrak e}}_2'}X^{{{\mathfrak e}}_2'-{{\mathfrak e}}_1} \sum_{j=0}^{{{\mathfrak e}}_3-2{{\mathfrak e}}_2'+{{\mathfrak e}}_1} (\xi X)^j \sum_{i=0}^{{{\mathfrak e}}_1}X^i \Biggr \}.\end{aligned}$$ Let $B \in {{\mathcal H}}_3({{\mathfrak o}})^{\rm{nd}}$. Then by Theorem \[th.1.1\], we have $$\widetilde F(B,X)=\widetilde {{\mathcal F}}({\mathrm{EGK}}(B);q^{1/2},X).$$ This essentially coincides with \[[@Kat1], Example (3)\] and \[[@Wed], (2.8)\] in the case $F={{\mathbb Q}}_p$.
\(2) Let $q$ be odd, and let $$B \sim {\varpi}^{a_1}u_1 \bot \cdots \bot {\varpi}^{a_n}u_n \quad (a_1 \le \cdots \le a_n, \ u_1,\ldots, u_n \in {{\mathfrak o}}^{\times})$$ be a diagonal Jordan decomposition of $B \in {{\mathcal H}}_n({{\mathfrak o}})^{{\rm nd}}$. Put $${\varepsilon}_i=\begin{cases} \xi_{B^{(i)}} & \text{ if $i$ is even} \\
\eta_{B^{(i)}} & \text{ if $i$ is odd.}
\end{cases}$$ Then $H=(a_1,\ldots,a_n;{\varepsilon}_1,\ldots,{\varepsilon}_n)$ is a naive EGK datum such that ${\Upsilon}_n(H)={\mathrm{EGK}}(B)$, and by Proposition \[prop.4.1\] and Theorem \[th.1.1\], we can get an explicit formula for $\widetilde F(B,X)$ in terms of $H$, which essentially coincides with \[[@Kat1], Theorem 4.3\] in the case $F={{\mathbb Q}}_p$. If $F$ is an unramified extension of ${{\mathbb Q}}_2$, we have an algorithm for giving a naive EGK datum associated with ${\mathrm{EGK}}(B)$ for $B \in {{\mathcal H}}_n({{\mathfrak o}})^{{\rm nd}}$ from its Jordan decomposition, and we can also give an explicit formula for $\widetilde F(B,X)$ in terms of it . This essentially coincides with \[[@Kat1], Theorem 4.3\] in the case $F={{\mathbb Q}}_2$ (cf. [@C-I-K-Y]).
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---
abstract: 'Based on the recently introduced model of Ref. [@Blaschke:2009e] for [[non-commutative]{}]{} $U_\star(1)$ gauge fields, a generalized version of that action for $U_\star(N)$ gauge fields is put forward. In this approach to [[non-commutative]{}]{} gauge field theories, [[UV/IR mixing]{}]{} effects are circumvented by introducing additional “soft breaking” terms in the action which implement an IR damping mechanism. The techniques used are similar to those of the well-known Gribov-Zwanziger approach to QCD.'
author:
- 'Daniel N. Blaschke'
date: '[July 1, 2010]{}'
title: 'A New Approach to Non-Commutative $U_\star(N)$ Gauge Fields'
---
Faculty of Physics, University of Vienna\
Boltzmanngasse 5, A-1090 Vienna (Austria)\
Introduction
============
For a long time quantum field theories formulated in a Groenewold-Moyal deformed (or ${\theta}$-deformed) space [@Groenewold:1946; @Moyal:1949] suffered from new types of divergences arising due to a phenomenon referred to as [[UV/IR mixing]{}]{} [@Minwalla:1999; @Susskind:2000]. For a review on the topic see Refs. [@Szabo:2001; @Rivasseau:2007a; @Blaschke:2010kw]. Only some years ago, Grosse and Wulkenhaar were able to resolve the [[UV/IR mixing]{}]{} problem in the case of a scalar field theory by adding an oscillator-like term to the (Euclidean) action [@Grosse:2003; @Grosse:2004b], thereby rendering it renormalizable to all orders of perturbation theory [@Grosse:2004a; @Rivasseau:2006a; @Rivasseau:2006b]. Eventually, an alternative approach was put forward by Gurau et al. [@Rivasseau:2008a] by replacing the oscillator term with one of type $\phi(-p){\frac{1}{p^2}}\phi(p)$. The authors were able to prove renormalizability of this “${\frac{1}{p^2}}$-model” to all orders by means of Multiscale Analysis.
Inspired by these successes, similar approaches were tried for $U_\star(1)$ gauge theories[^1] in Euclidean space [@Grosse:2007; @Wulkenhaar:2007; @Blaschke:2007b; @Wallet:2008a; @Blaschke:2008a; @Vilar:2009; @Blaschke:2009b; @Blaschke:2009e]. The latest approach, the model presented in Ref. [@Blaschke:2009e], seems to be a very promising candidate for a renormalizable $U_\star(1)$ gauge theory on ${\theta}$-deformed space.
In the present work I generalize this model to the $U_\star(N)$ gauge group. It is formulated on Euclidean ${\mathds{R}}_{\theta}^4$ with the Moyal-deformed product $$\begin{aligned}
{\left[ x_{\mu}\stackrel{\star}{,}x_{\nu}\right] } \equiv x_{\mu} \star
x_{\nu} -x_{\nu} \star x_{\mu} = {\mathrm{i}}{\varepsilon}{\theta}_{\mu \nu}\,,\end{aligned}$$ of regular commuting coordinates $x_\mu$. The real parameter ${\varepsilon}$ has mass dimension $-2$, rendering the constant antisymmetric matrix ${\theta}_{{\mu}{\nu}}$ dimensionless.
In the following I will use the abbreviations $\tilde{v}_{\mu}\equiv {\theta}_{{\mu}{\nu}}v_{\nu}$ for vectors $v$ and $\tilde{M} \equiv {\theta}_{{\mu}{\nu}}M_{{\mu}{\nu}}$ for matrices $M$. For the deformation, I furthermore consider the simplest block-diagonal form $$\begin{aligned}
\label{eq:def-eth}
{\theta}_{\mu\nu}=\left(\begin{array}{cccc}
0&1&0&0\\
-1&0&0&0\\
0&0&0&1\\
0&0&-1&0
\end{array}\right)\, ,\end{aligned}$$ for the dimensionless matrix describing non-commutativity.
The U(N) gauge field action
===========================
The main ideas that in a series of papers [@Blaschke:2008a; @Blaschke:2009a; @Vilar:2009; @Blaschke:2009b; @Blaschke:2009c; @Blaschke:2009d] led to the construction of a model for $U_\star(1)$ gauge fields [@Blaschke:2009e], which has a good chance of being fully renormalizable, are:
- to implement a damping mechanism similar to the one present in the scalar ${\frac{1}{p^2}}$-model of Gurau et al. [@Rivasseau:2008a],
- to endow the tree level action with counter terms for the quadratic and linear one-loop infrared divergent terms of type [@Hayakawa:1999; @Armoni:2000xr; @Ruiz:2000; @Blaschke:2005b]: $$\begin{aligned}
\label{eq:generic-IR-div}
\Pi^{\text{IR}}_{{\mu}{\nu}}(k)&\propto\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{({\varepsilon}{\tilde{k}}^2)^2}\,,\end{aligned}$$ and $$\begin{aligned}
\Gamma^{3A,\text{IR}}_{{\mu}{\nu}{\rho}}(p_1,p_2,p_3)
&\propto\cos\left({\varepsilon}\frac{p_1{\tilde{p}}_2}{2}\right)\sum\limits_{i=1,2,3}\frac{{\tilde{p}}_{i,{\mu}}{\tilde{p}}_{i,{\nu}}{\tilde{p}}_{i,{\rho}}}{{\varepsilon}({\tilde{p}}_i^2)^2}\,,
\label{eq:counterterm-3A}\end{aligned}$$
- and to keep the model as simple as possible.
The greatest difficulty in the early approaches turned out to be the implementation of the IR damping: In Ref. [@Blaschke:2008a] an additional gauge invariant term was added to the action which ultimately led to an infinite number of vertices. Localization of that term through the introduction of auxiliary fields[^2] could remedy the situation with respect to the tree level vertices. However, other problems concerning renormalizability due to additional new Feynman rules for the auxiliary fields, remained — cf. [@Vilar:2009; @Blaschke:2009b; @Blaschke:2009d]. Therefore, an alternative approach was proposed in Ref. [@Blaschke:2009e], where the required extension to the $U_\star(1)$ gauge field action was implemented by means of a “soft breaking” technique similar to the Gribov-Zwanziger action in QCD — see Refs. [@Gribov:1978; @Zwanziger:1989; @Zwanziger:1993; @Baulieu:2009] for details.
Here I will follow the same ideas. Furthermore, it must also be taken into account, that in [[non-commutative]{}]{} $U_\star(N)$ gauge field models only the $U_\star(1)$ subsector is responsible for [[UV/IR mixing]{}]{} (cf. Refs. [@Armoni:2000xr; @Armoni:2001; @Armoni:2002fh]). This means, that infrared divergent terms only appear in Feynman graphs which have at least one external leg in the $U_\star(1)$ subsector. The key point here is that, by employing the soft breaking mechanism, one only modifies the infrared regime of the model while keeping the UV intact. Both UV divergences, on the one hand, as well as IR terms originating from [[UV/IR mixing]{}]{} in e.g. one-loop corrections, on the other hand, are caused by the UV regime of the integrand in a Feynman loop graph [@Minwalla:1999; @Susskind:2000; @Szabo:2001]. Therefore, the same one-loop results to leading order are expected for the current model as in the literature [@Armoni:2000xr] (see the discussion in [Section \[sec:one-loop\]]{} below and in Ref. [@Blaschke:2009e]). In other words, we need to implement counter terms of type and in the soft breaking part of our action for $U_\star(1)$ gauge fields, but not for the pure $SU_\star(N)$ sector.
#### Notation.
Throughout the remainder of this paper, the following notation will be used: Following Ref. [@Armoni:2000xr] I denote $U_\star(N)$ indices with capital letters $A,B,C,\ldots$ and $SU_\star(N)$ indices with $a,b,c,\ldots$. Finally, the index $0$ is used for fields which are $U_\star(1)$, and whenever an index is omitted, the according field including the $U(N)$ gauge group generator $T^A$ is meant. Furthermore, all products are implicitly assumed to be deformed (i.e. star products).
#### $\mathbf{U_\star(N)}$ gauge fields.
The covariant derivative $D_{\mu}$ and the field strength $F_{{\mu}{\nu}}$ are defined as $$\begin{aligned}
D_{\mu}\bullet&={\partial}_{\mu}\bullet-{\mathrm{i}g}{\left[A_{\mu},\bullet\right]}\,, & A_{\mu}&=A_{\mu}^AT^A\,, \nonumber\\
F_{{\mu}{\nu}}&={\partial}_{\mu}A_{\nu}-{\partial}_{\nu}A_{\mu}-{\mathrm{i}g}{\left[A_{\mu},A_{\nu}\right]}\,,\end{aligned}$$ where $T^A$ are the generators of the $U(N)$ gauge group. They are normalized as ${\text{Tr}}(T^AT^B)={\frac{1}{2}}{\delta}^{AB}$, and $T^0={\frac{1}{\sqrt{2N}}}{\mathds{1}}_N$ (cf. [@Armoni:2000xr]). Due to the star product, the field strength tensor $F_{{\mu}{\nu}}$ exhibits additional couplings between the $U_\star(1)$ and the $SU_\star(N)$ sector, i.e. we have $$\begin{aligned}
F_{{\mu}{\nu}}&=\left({\partial}_{\mu}A^0_{\nu}-{\partial}_{\nu}A^0_{\mu}-\frac{{\mathrm{i}g}}{2}d^{AB0}{\left[A^A_{\mu},A^B_{\nu}\right]}\right)T^0\nonumber\\
&\quad +\left({\partial}_{\mu}A^c_{\nu}-{\partial}_{\nu}A^c_{\mu}+\frac{g}{2}f^{abc}{\left\{A^a_{\mu},A^b_{\nu}\right\}}-\frac{{\mathrm{i}g}}{2}d^{ABc}{\left[A^A_{\mu},A^B_{\nu}\right]}\right)T^c\nonumber\\
&\equiv F^0_{{\mu}{\nu}}T^0+F^c_{{\mu}{\nu}}T^c \,,\end{aligned}$$ where $f^{abc}$ and $d^{ABC}$ are (anti)symmetric structure constants of the gauge group. The terms proportional to $d^{AB0}=\sqrt{\frac{2}{N}}{\delta}^{AB}$ contain both types of fields, i.e. $U_\star(1)$ and $SU_\star(N)$, and hence giving rise to the additional couplings. In the commutative limit, the star commutators would vanish and the two sectors would decouple once more.
Similarly, one has for the covariant derivative of e.g. a ghost field $c$: $$\begin{aligned}
D_{\mu}c&=\left({\partial}_{\mu}c^0-\frac{{\mathrm{i}g}}{2}d^{AB0}{\left[A^A_{\mu},c^B\right]}\right)T^0 +\left({\partial}_{\mu}c^c+\frac{g}{2}f^{abc}{\left\{A^a_{\mu},c^b\right\}}-\frac{{\mathrm{i}g}}{2}d^{ABc}{\left[A^A_{\mu},c^B\right]}\right)T^c
\,.\end{aligned}$$
#### Proposed action.
In light of the considerations above, the following generalized BRST invariant $U_\star(N)$ gauge field action formulated in Euclidean ${\mathds{R}}_{\theta}^4$ including additional $U_\star(1)$ auxiliary fields is suggested: [$$\begin{aligned}
\label{eq:renormalizable_action}
{S}&={S}_{\text{inv}}+{S}_{\text{gf}}+{S}_{\text{aux}}+{S}_{\text{soft}}+{S}_{\text{ext}}\,,\nonumber\\*
{S}_{\text{inv}}&={\int\!\! {\rm d}^4x}{\tfrac{1}{4}}F^A_{{\mu}{\nu}}F^A_{{\mu}{\nu}}\,,\nonumber\\*
{S}_{\text{gf}}&={\int\!\! {\rm d}^4x}\,s\left({\bar{c}}^A\,{\partial}_{\mu}A^A_{\mu}\right)={\int\!\! {\rm d}^4x}\left(b^A\,{\partial}_{\mu}A^A_{\mu}-{\bar{c}}^A\,{\partial}_{\mu}(D_{\mu}c)^A\right)\,,\nonumber\\
{S}_{\text{aux}}&=-{\int\!\! {\rm d}^4x}\,s\left({\bar{\psi}}^0_{{\mu}{\nu}}B^0_{{\mu}{\nu}}\right)={\int\!\! {\rm d}^4x}\left(-{\bar{B}}^0_{{\mu}{\nu}}B^0_{{\mu}{\nu}}+{\bar{\psi}}^0_{{\mu}{\nu}}\psi^0_{{\mu}{\nu}}\right)\,,\nonumber\\
{S}_{\text{soft}}&={\int\!\! {\rm d}^4x}\,s\Bigg[\!\left(\bar{Q}^0_{{\mu}{\nu}{\alpha}{\beta}}B^0_{{\mu}{\nu}}+Q^0_{{\mu}{\nu}{\alpha}{\beta}}{\bar{B}}^0_{{\mu}{\nu}}\right){\frac{1}{{\widetilde{\square}}}}\left(\!f^0_{{\alpha}{\beta}}+{\sigma}\frac{{\theta}_{{\alpha}{\beta}}}{2}\tilde{f}^0\!\right)\nonumber\\*
&\quad\qquad\qquad
+e^{ABC}Q'^0{\left\{A^A_{\mu},A^B_{\nu}\right\}} \frac{\tilde{{\partial}}_{\mu}\tilde{{\partial}}_{\nu}\tilde{{\partial}}_{\rho}}{{\widetilde{\square}}^2}A^C_{\rho}\Bigg]\nonumber\\*
&={\int\!\! {\rm d}^4x}\bigg[\!\!\left(\bar{J}^0_{{\mu}{\nu}{\alpha}{\beta}}B^0_{{\mu}{\nu}}+J^0_{{\mu}{\nu}{\alpha}{\beta}}{\bar{B}}^0_{{\mu}{\nu}}\right)\!{\frac{1}{{\widetilde{\square}}}}\!\left(\!f^0_{{\alpha}{\beta}}+{\sigma}\frac{{\theta}_{{\alpha}{\beta}}}{2}\tilde{f}^0\!\right)\! - \bar{Q}^0_{{\mu}{\nu}{\alpha}{\beta}}\psi^0_{{\mu}{\nu}}{\frac{1}{{\widetilde{\square}}}}\!\left(\!f^0_{{\alpha}{\beta}}+{\sigma}\frac{{\theta}_{{\alpha}{\beta}}}{2}\tilde{f}^0\!\right)\nonumber\\*
& \qquad \qquad -\left(\bar{Q}^0_{{\mu}{\nu}{\alpha}{\beta}}B^0_{{\mu}{\nu}}+Q^0_{{\mu}{\nu}{\alpha}{\beta}}\bar{B}^0_{{\mu}{\nu}}\right){\frac{1}{{\widetilde{\square}}}}\mathop{s}\left(\!f^0_{{\alpha}{\beta}}+{\sigma}\frac{{\theta}_{{\alpha}{\beta}}}{2}\tilde{f}^0\!\right)\nonumber\\*
&\qquad\qquad
+e^{ABC}J'^0{\left\{A^A_{\mu},A^B_{\nu}\right\}} \frac{\tilde{{\partial}}_{\mu}\tilde{{\partial}}_{\nu}\tilde{{\partial}}_{\rho}}{{\widetilde{\square}}^2}A^C_{\rho}-
e^{ABC}Q'^0s\left({\left\{A^A_{\mu},A^B_{\nu}\right\}} \frac{\tilde{{\partial}}_{\mu}\tilde{{\partial}}_{\nu}\tilde{{\partial}}_{\rho}}{{\widetilde{\square}}^2}A^C_{\rho}\right)\!\bigg]\,, \nonumber\\
{S}_{\text{ext}}&={\int\!\! {\rm d}^4x}\left({\Omega}^A_{\mu}(sA_{\mu})^A+{\omega}^A (sc)^A\right)\,,\end{aligned}$$ where ]{} $$\begin{aligned}
\label{eq:def-eABC}
e^{ABC}&\equiv d^{ABC}-d^{abc}{\delta}^{aA}{\delta}^{bB}{\delta}^{cC}\,,
&& {\widetilde{\square}}= \tilde\partial_\mu \tilde\partial_\mu \,.\end{aligned}$$ The abbreviation $e^{ABC}$ denotes all symmetric structure constants $d^{ABC}$ where at least one index is $0$, i.e. in the $U_\star(1)$ subsector of the gauge group. The reason for this restriction has already been mentioned above: In loop calculations terms of type only appear when at least one external leg is in the $U_\star(1)$ subsector, as was first shown by A. Armoni [@Armoni:2000xr; @Armoni:2001].
Finally, $f^0_{{\mu}{\nu}}$ denotes the free part of $F^0_{{\mu}{\nu}}$, i.e. $$\begin{aligned}
f^0_{{\mu}{\nu}}&={\partial}_{\mu}A^0_{\nu}-{\partial}_{\nu}A^0_{\mu}\,, &
\tilde{f}^0&={\theta}_{{\mu}{\nu}}f^0_{{\mu}{\nu}}\,,\end{aligned}$$ the multiplier field $b$ implements the Landau gauge fixing ${\partial}_{\mu}A_{\mu}=0$, ${\bar{c}}$/$c$ denote the (anti)ghost, and ${\sigma}$ is a dimensionless parameter. The complex $U_\star(1)$ field $B^0_{{\mu}{\nu}}$, its complex conjugate ${\bar{B}}^0_{{\mu}{\nu}}$ and the associated additional ghosts ${\bar{\psi}}^0$, $\psi^0$ are introduced in order to implement the IR damping mechanism explained in Ref. [@Blaschke:2009e] on the according $U_\star(1)$ gauge model. The additional $U_\star(1)$ sources ${\bar{Q}}^0,Q^0,Q'^0,{\bar{J}}^0,J^0,J'^0$ are needed in order to ensure BRST invariance of the action in the ultraviolet. In the infrared they take the “physical” values $$\begin{aligned}
&{\bar{Q}}^0_{{\mu}{\nu}{\alpha}{\beta}}\Big|_{\text{phys}}=Q^0_{{\mu}{\nu}{\alpha}{\beta}}\Big|_{\text{phys}}=Q'^0\Big|_{\text{phys}}=0\,,
&& J'^0\Big|_{\text{phys}}={\mathrm{i}}g{\gamma}'^2\,,\nonumber\\
&{\bar{J}}^0_{{\mu}{\nu}{\alpha}{\beta}}\Big|_{\text{phys}}=J^0_{{\mu}{\nu}{\alpha}{\beta}}\Big|_{\text{phys}}=\frac{{\gamma}^2}{4}\left({\delta}_{{\mu}{\alpha}}{\delta}_{{\nu}{\beta}}-{\delta}_{{\mu}{\beta}}{\delta}_{{\nu}{\alpha}}\right)\,,
\label{eq:physical-values}\end{aligned}$$ where ${\gamma}$ and ${\gamma}'$ are Gribov-like parameters of mass dimension 1 (cf. [@Gribov:1978; @Zwanziger:1989; @Zwanziger:1993; @Baulieu:2009]). The action is hence invariant under the BRST transformations $$\begin{aligned}
\label{eq:BRST_of_renorm_action}
&sA_\mu=D_\mu c\,, && sc={\mathrm{i}g}{c}{c}\, ,\nonumber\\
&s{\bar{c}}=b\,, && sb=0\, , \nonumber\\
&s{\bar{\psi}}_{\mu\nu}={\bar{B}}_{{\mu}{\nu}}\,, && s{\bar{B}}_{{\mu}{\nu}}=0\,,\nonumber\\
&sB_{{\mu}{\nu}}=\psi_{{\mu}{\nu}}\,, && s\psi_{{\mu}{\nu}}=0\,,\nonumber\\
&s\bar{Q}_{{\mu}{\nu}{\alpha}{\beta}}=\bar{J}_{{\mu}{\nu}{\alpha}{\beta}}\,, && s\bar{J}_{{\mu}{\nu}{\alpha}{\beta}}=0\,, \nonumber\\
&sQ_{{\mu}{\nu}{\alpha}{\beta}}=J_{{\mu}{\nu}{\alpha}{\beta}}\,, && sJ_{{\mu}{\nu}{\alpha}{\beta}}=0\,,\nonumber\\
&sQ'=J'\,, && sJ'=0\,,\end{aligned}$$ and for the non-linear transformations $sA_{\mu}$ and $sc$, external sources ${\Omega}_{\mu}$ and ${\omega}$ have been introduced, respectively. Notice, that the auxiliary fields form BRST doublets reflecting their unphysical nature. Dimensions and ghost numbers of the fields involved are given in Table \[tab:field\_prop\].
[lcccccccccccccccc]{}
------------------------------------------------------------------------
Field & $A$ & $c$ & ${\bar{c}}$ & $B$ & ${\bar{B}}$ & $\psi$ & ${\bar{\psi}}$ & $J$ & $\bar{J}$ & $J'$ & $Q$ & $\bar{Q}$ & $Q'$ & ${\Omega}$ & ${\omega}$ & b\
$g_\sharp$ & 0 & 1 & -1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -2 & 0\
Mass dim. & 1 & 0 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 3 & 4 & 2\
Finally, the Slavnov-Taylor identity describing the BRST symmetry content of the model is given by $$\begin{aligned}
\mathcal{B}({S})&={\int\!\! {\rm d}^4x}\Bigg({\frac{{\delta}{S}}{{\delta}{\Omega}_{\mu}}}{\frac{{\delta}{S}}{{\delta}A_{\mu}}}+{\frac{{\delta}{S}}{{\delta}{\omega}}}{\frac{{\delta}{S}}{{\delta}c}}+b{\frac{{\delta}{S}}{{\delta}{\bar{c}}}}+{\bar{B}}_{{\mu}{\nu}}{\frac{{\delta}{S}}{{\delta}{\bar{\psi}}_{{\mu}{\nu}}}}+\psi_{{\mu}{\nu}}{\frac{{\delta}{S}}{{\delta}B_{{\mu}{\nu}}}}\nonumber\\*
&\phantom{={\int\!\! {\rm d}^4x}\Bigg(} +\bar{J}_{{\mu}{\nu}{\alpha}{\beta}}{\frac{{\delta}{S}}{{\delta}\bar{Q}_{{\mu}{\nu}{\alpha}{\beta}}}}+J_{{\mu}{\nu}{\alpha}{\beta}}{\frac{{\delta}{S}}{{\delta}Q_{{\mu}{\nu}{\alpha}{\beta}}}}+J'{\frac{{\delta}{S}}{{\delta}Q'}}\Bigg)=0\,.\end{aligned}$$
Discussion of one-loop properties {#sec:one-loop}
=================================
The gauge field propagator (using [Eqn. (\[eq:physical-values\])]{}, cp. Ref. [@Blaschke:2009e]) takes the form $$\begin{aligned}
\label{eq:prop_aa}
G^{A^0A^0}_{{\mu}{\nu}}(k)&=\left[k^2+\frac{{\gamma}^4}{{\tilde{k}}^2}\right]^{-1}\left[{\delta}_{{\mu}{\nu}}-\frac{k_{\mu}k_{\nu}}{k^2}-\frac{{\bar{\sigma}}^4}{\left(k^2+\left({\bar{\sigma}}^4+{\gamma}^4\right){\frac{1}{{\tilde{k}}^2}}\right)}\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{({\tilde{k}}^2)^2}\right]
\,, \nonumber\\
G^{A^aA^b}_{{\mu}{\nu}}(k)&=\frac{{\delta}^{ab}}{k^2}\left({\delta}_{{\mu}{\nu}}-\frac{k_{\mu}k_{\nu}}{k^2}\right)\,,\end{aligned}$$ where we have introduced the abbreviation $$\begin{aligned}
{\bar{\sigma}}^4\equiv 2(1+{\sigma}){\sigma}{\gamma}^4\,,\end{aligned}$$ and considered the case where ${\theta}_{{\mu}{\nu}}$ has the simple block diagonal form given in so that ${\tilde{k}}^2=k^2$ and ${\theta}_{{\mu}{\nu}}{\theta}_{{\mu}{\nu}}=4$. Notice that the soft-breaking terms in the action lead to an IR modified propagator in the $U_\star(1)$ sector. Two limits are of special interest: the IR limit $k^2\to0$ and the UV limit $k^2\to\infty$. A simple analysis reveals that $$G^{A^0A^0}_{{\mu}{\nu}}(k)\approx\begin{cases}\frac{{\tilde{k}}^2}{{\gamma}^4}\left[{\delta}_{{\mu}{\nu}}-\frac{k_{\mu}k_{\nu}}{k^2}-\frac{{\bar{\sigma}}^4}{\left({\bar{\sigma}}^4+{\gamma}^4\right)}\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{{\tilde{k}}^2}\right], & \text{for } {\tilde{k}}^2\to0\,,\\[0.8em]
{\frac{1}{k^2}}\left({\delta}_{{\mu}{\nu}}-\frac{k_{\mu}k_{\nu}}{k^2}\right), & \text{for } k^2\to\infty\,.\\
\end{cases}
\label{eq:prop_aa_limits}$$ From [Eqn. (\[eq:prop\_aa\_limits\])]{} one can nicely see the appearance of a term of the same type as in the IR limit. This, by construction, admits the absorption of the problematic divergent terms appearing in the one loop results [@Blaschke:2009d]. Another advantageous property of the gauge propagator is that the UV limit (from which divergences originate), admits to neglect the term proportional to ${\gamma}$ which reduces the number of terms in Feynman integrals considerably, especially since both $U_\star(1)$ and $SU_\star(N)$ propagators are of the same form in this limit.
The ghost propagator takes the usual simple form $G^{{\bar{c}}c}(k)=-\frac{{\delta}^{AB}}{k^2}$. Since $A_{\mu}$ does not couple to the auxiliary fields ($B,{\bar{B}},\psi,{\bar{\psi}}$) no other propagator will contribute to physical results, and they are hence omitted at this point.
Additionally, the model features several vertices. Similar to , one may consider their UV approximations, which for the purpose of one-loop calculations would be sufficient. Within this approximation the vertices are given by the same expressions as in e.g. [@Armoni:2000xr]. In other words, terms proportional to ${\gamma}'$ in the $3A$ vertex may be omitted as they are negligible for large momenta and hence play no role in the one-loop divergences. In the infrared, they scale as , but such linear IR divergent terms are always compensated by quadratically IR damping gauge field propagators — cp. [Eqn. (\[eq:prop\_aa\_limits\])]{}.
Considering the scaling behaviour of all these Feynman rules for large momenta, one derives an estimate for the superficial degree of ultraviolet divergences, which is the well-known result $$\begin{aligned}
\label{eq:powercounting}
d_\gamma=4-E_A-E_{c{\bar{c}}}\,, \end{aligned}$$ where $E$ denotes the number of external legs of the various field types in a Feynman graph.
When extracting the resulting UV divergences of the various one-loop corrections, only those terms in the Feynman rules contribute which survive the UV approximations discussed above. Hence, these computations reduce to exactly the same ones already done in the literature, i.e. see [@Armoni:2000xr; @Ruiz:2000; @Hayakawa:1999; @Blaschke:2009e]. The vacuum polarization hence exhibits a logarithmic UV divergence of the form $$\begin{aligned}
\Pi^{\text{UV}}_{{\mu}{\nu}}&\propto {\delta}^{AB}g^2\left(p^2{\delta}_{{\mu}{\nu}}-p_{\mu}p_{\nu}\right)\ln|{\Lambda}^2{\tilde{p}}^2|+\text{finite},\end{aligned}$$ where ${\Lambda}$ denotes an ultraviolet cutoff. In addition, as discussed in the literature, there is also a UV finite contribution to the vacuum polarization which diverges quadratically for vanishing external momentum. It is of type and appears *only* in graphs where the external legs are in the $U_\star(1)$ sector. In fact, this can be easily seen by considering the according phase factors when the free colour indices $a,b\in SU_\star(N)$. In that case, one has phase factors of the form $$\begin{aligned}
d^{aCD}d^{bCD}\sin^2(k{\tilde{p}}/2)+f^{acd}f^{bcd}\cos^2(k{\tilde{p}}/2)=N{\delta}^{ab}\,,\end{aligned}$$ since $d^{aCD}d^{bCD}=f^{acd}f^{bcd}=N{\delta}^{ab}$. Clearly, they are phase-independent and hence lead to purely planar contributions.
The $3A$-vertex corrections exhibit UV divergences of the form [@Armoni:2000xr; @Blaschke:2009e] $$\begin{aligned}
\label{3A_correction_UV}
\Gamma^{3A,\text{UV}}_{{\mu}{\nu}{\rho}}(p_1,p_2,p_3)&\propto-g^2N\ln (\Lambda) \widetilde{V}^{3A,\text{tree}}_{{\mu}{\nu}{\rho}}(p_1,p_2,p_3)\,,\end{aligned}$$ as well as finite contributions. The latter exhibit IR divergences in the external momenta of type only if at least one external leg is in the $U_\star(1)$ subsector, as emphasized above [@Armoni:2000xr; @Armoni:2002fh]. In fact, one can argue that infrared divergent terms appear also in $n$-point graphs *only* if at least one of the external legs is in the $U_\star(1)$ subsector by considering, for example, graphical representations of non-planar Feynman diagrams using the ’t Hooft double index notation [@Armoni:2001; @Armoni:2002fh; @Levell:2003ta]. What one finds, is that it is impossible to construct a non-planar graph without having at least one external $U_\star(1)$ leg, and since [[UV/IR mixing]{}]{} occurs only in non-planar graphs [@Minwalla:1999; @Susskind:2000; @Szabo:2001], as is well-known, the same is true for the appearance of IR terms.
Renormalization
===============
A renormalizable action must be form-invariant under quantum corrections, and its parameters are fixed by renormalization conditions on the vertex functions. So far, we have worked in Landau gauge with ${\alpha}=0$ (cf. [Eqn. (\[eq:prop\_aa\])]{}). However, for the following considerations, where we follow the steps of Ref. [@Blaschke:2009e], an arbitrary gauge parameter ${\alpha}\neq0$ will be more advantageous[^3], as the inverse of the gauge field propagator diverges in the limit ${\alpha}\to0$ due to elimitaion of the multiplier field $b$.
The renormalized propagator
---------------------------
Recall that the tree-level gauge field propagator (using a compact and intuitive notation) has the form $$\begin{aligned}
\label{eq:propAA-general}
G^{A^AA^B}_{{\mu}{\nu}}(k)&=\frac{{\delta}^{AB}}{k^2\mathcal{D}(k)}\left({\delta}_{{\mu}{\nu}}-\left(1-{\alpha}\mathcal{D}(k)\right)\frac{k_{\mu}k_{\nu}}{k^2}-\mathcal{F}(k)\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{{\tilde{k}}^2}\right)\,,\end{aligned}$$ where we have introduced the abbreviations $$\begin{aligned}
\mathcal{D}(k)&\equiv\left(1+{\delta}^{A0}{\delta}^{B0}\frac{{\gamma}^4}{({\tilde{k}}^2)^2}\right)
\,, &
\mathcal{F}(k)&\equiv\frac{{\delta}^{A0}{\delta}^{B0}}{{\tilde{k}}^2}\frac{{\bar{\sigma}}^4}{\left(k^2+\left({\bar{\sigma}}^4+{\gamma}^4\right){\frac{1}{{\tilde{k}}^2}}\right)}\,,\end{aligned}$$ i.e. terms including parameters ${\bar{\sigma}}$ or ${\gamma}$ only appear in the $A^0A^0$-propagator. Its inverse, the tree-level two-point vertex function, is given by $$\begin{aligned}
{\Gamma}^{AA,\text{tree}}_{{\mu}{\nu}}(k)&=\left(G_{AA}^{-1}\right)_{{\mu}{\nu}}(k)={\delta}^{AB}k^2\mathcal{D}(k)\left({\delta}_{{\mu}{\nu}}+\left({\frac{1}{{\alpha}\mathcal{D}(k)}}-1\right)\frac{k_{\mu}k_{\nu}}{k^2}+\frac{{\delta}^{A0}{\delta}^{B0}{\bar{\sigma}}^4}{k^2{\tilde{k}}^2\mathcal{D}(k)}\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{{\tilde{k}}^2}\right)\,.\end{aligned}$$ As discussed in Section \[sec:one-loop\], its (divergent) one-loop corrections are qualitatively given by $$\begin{aligned}
{\Gamma}^{AA,\text{corr.}}_{{\mu}{\nu}}(k)&=\varPi_1\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{({\tilde{k}}^2)^2}+\varPi_2\left(k^2{\delta}_{{\mu}{\nu}}-k_{\mu}k_{\nu}\right)\,,\nonumber\\
\textrm{with }\quad\varPi_1&\propto{\delta}^{A0}{\delta}^{B0}\frac{Ng^2}{{\varepsilon}^2}
\,,\qquad
\varPi_2\propto{\delta}^{AB}{Ng^2}\ln{\Lambda}\,,\end{aligned}$$ where ${\Lambda}$ is an ultraviolet cutoff. Hence, we find that in introducing the wave-function renormalization $Z_A$ and the renormalized parameters ${\gamma}_r$ and ${\bar{\sigma}}_r$ according to $$\begin{aligned}
\label{eq:renormalized_parameters}
Z_A&={\frac{1}{\sqrt{1-\varPi_2}}}
\,, &
{\gamma}_r^4&={{\gamma}^4}{Z_A^2}
\,, &
{\bar{\sigma}}_r^4&=\left({{\bar{\sigma}}^4-\varPi_1}\right){Z_A^2}\,,\end{aligned}$$ the one-loop two-point vertex function can be cast into the same form as its tree-level counter part, i.e. $$\begin{aligned}
{\Gamma}^{AA,\text{ren}}_{{\mu}{\nu}}(k)&={\Gamma}^{AA,\text{tree}}_{{\mu}{\nu}}(k)-{\Gamma}^{AA,\text{corr.}}_{{\mu}{\nu}}(k)\nonumber\\
&=\frac{k^2\mathcal{D}_r}{Z_A^2}\left({\delta}_{{\mu}{\nu}}+\left(\frac{Z_A^2}{{\alpha}\mathcal{D}_r}-1\right)\frac{k_{\mu}k_{\nu}}{k^2}+\frac{{\delta}^{A0}{\delta}^{B0}{\bar{\sigma}}_r^4}{k^2{\tilde{k}}^2\mathcal{D}_r}\frac{{\tilde{k}}_{\mu}{\tilde{k}}_{\nu}}{{\tilde{k}}^2}\right)\,,\nonumber\\
\mathcal{D}_r(k)&\equiv\left(1+{\delta}^{A0}{\delta}^{B0}\frac{{\gamma}_r^4}{({\tilde{k}}^2)^2}\right)\,.\end{aligned}$$ Perhaps the most important result of this calculation is that the wave-function renormalization $Z_A$ is exactly the same for the $U_\star(1)$ and the $SU_\star(N)$ gauge field because it is independent of $\varPi_1$. In fact, the quadratic IR divergence $\varPi_1$ only enters the renormalization of the newly introduced parameter ${\bar{\sigma}}$. For the sake of completeness, we note that the renormalized propagator takes the same form as apart from an additional prefactor $Z_A^2$, but with all parameters replaced by their renormalized counter parts.
We also need to provide renormalization conditions for the two-point vertex function for the gauge boson $$\begin{aligned}
\Gamma^{AA}_{\mu\rho}
& = \Gamma^{AA,T} ({\delta}_{{\mu}{\rho}} - \frac{k_{\mu}k_{\rho}}{k^2}) +(\Gamma^{AA,NC})\, \frac{\tilde k_{\mu}\tilde k_{\rho}}{\tilde k^2} + (\Gamma^{AA,L})\, \frac{k_{\mu}k_{\rho}}{k^2}\,,\end{aligned}$$ where the vertex function has been split into a transversal and longitudinal part following Ref. [@Blaschke:2009e]. We have used the identifications $$\begin{aligned}
\Gamma^{AA,T} & = k^2 \mathcal D\,,\qquad
\Gamma^{AA,NC} = {\delta}^{A0}{\delta}^{B0}\frac{{\bar{\sigma}}^4}{{\tilde{k}}^2}\,,\qquad (\Gamma^{AA,L}) = \frac{k^2}{\alpha} \,,\end{aligned}$$ which finally allow to formulate the following renormalization conditions: [$$\begin{aligned}
\frac{({\tilde{k}}^2)^2}{k^2} \Gamma^{AA,T} \Big|_{k^2=0} & = {\delta}^{A0}{\delta}^{B0}\gamma^4\,,
&
{\frac{1}{2 k^2}} \frac{{\partial}(k^2 \Gamma^{AA,T})}{{\partial}k^2} \Big|_{k^2=0} & = 1
\,, \nonumber\\
\Gamma^{AA,L}\Big|_{k^2=0} & = 0\,,
&
\frac{{\partial}\Gamma^{AA,L}}{{\partial}k^2} \Big|_{k^2=0} & = {\frac{1}{{\alpha}}}
\,, \nonumber\\
{\tilde{k}}^2 \Gamma^{AA,NC}\Big|_{k^2=0} & = {\delta}^{A0}{\delta}^{B0}{\bar{\sigma}}^4
\,.\end{aligned}$$ ]{}
The beta-function and renormalization of gamma-prime
----------------------------------------------------
As usual, the $\beta$-function is given by the logarithmic derivative of the bare coupling $g$ with respect to the cut-off for fixed $g_r$, i.e. $$\begin{aligned}
\label{eq:betafunction_def}
{\beta}(g,{\Lambda}) &= {\Lambda}\frac{ {\partial}g }{ {\partial}{\Lambda}}\Big|_{g_r\, \text{fixed}}
\,, &
{\beta}(g) &= \lim_{{\Lambda}\to\infty} {\beta}(g,{\Lambda})\,.\end{aligned}$$ The renormalized coupling is obtained from the relation $g_r = g Z_g Z_A^3$, where $Z_g$ denotes the multiplicative UV correction to the three-gluon vertex , and $Z_A$ is the wave function renormalization introduced in [Eqn. (\[eq:renormalized\_parameters\])]{}. Note that the wave function renormalization $Z_A$ enters $g_r$ since the vertex correction was computed with the unrenormalized fields $A_{\mu}$, which hence need to be replaced by their renormalized counter parts $A^r_\mu = Z^{-1}_A A_\mu$. One eventually obtains ${\beta}\sim-Ng^3<0$, i.e. a ${\beta}$-function with negative sign [@Blaschke:2009e; @Martin:1999aq; @Armoni:2000xr; @Minwalla:1999; @Ruiz:2000] which indicates asymptotic freedom and the absence of a Landau ghost.
Finally, the linear IR divergence in the 3-gluon vertex correction, which qualitatively results to $N$ times the expression , leads to a renormalized parameter $$\begin{aligned}
{\gamma}'^2_r&={\gamma}'^2Z_{{\gamma}'}Z_A^3
\,, & \text{with } \qquad
1-Z_{{\gamma}'}&\propto Ng^2\,.\end{aligned}$$ Hence, it is absorbed by the according counter term.
Conclusion and Outlook
======================
In this paper, the [[non-commutative]{}]{} $U_\star(N)$ gauge field action has been proposed as a generalization of the previously put forward $U_\star(1)$ counter part of Ref. [@Blaschke:2009e]. At least at one-loop level, all dangerous IR divergent terms can be absorbed into according counter terms, and one may hope that this is true also for higher loop orders. Furthermore, the wave-function renormalization $Z_A$ is exactly the same for the $U_\star(1)$ and the $SU_\star(N)$ gauge fields, which is crucial if the present model is to be taken seriously as a candidate for a renormalizable [[non-commutative]{}]{} $U_\star(N)$ gauge field model. In a next step towards a general proof of renormalizability, however, further explicit (higher-)loop calculations are required.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Discussions with H. Grosse, H. Steinacker and M. Wohlgenannt, as well as correspondence with A. Armoni are gratefully acknowledged. This work was supported by the “Fonds zur Förderung der Wissenschaftlichen Forschung” (FWF) under contract P21610-N16.
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[^1]: Note, that the non-commutativity of the space coordinates alters the gauge group, which is why I have denoted the deformed $U(1)$ group by $U_\star(1)$.
[^2]: By “localization” it is only meant that the inverse of covariant derive operators leading to infinitely many vertices no longer enters the action explicitly. Of course the star products remain non-local nonetheless.
[^3]: Note, that the quadratic IR divergence is independent of the gauge fixing [@Blaschke:2005b; @Ruiz:2000; @Hayakawa:1999b].
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=23.7cm=16.5cm=-2.0cm=-1.4cm
[*One-loop divergences of quantum gravity* ]{}
[**Guilherme de Berredo-Peixoto**]{}[^1]\
Dep. Campos e Partculas (DCP)-Centro Brasileiro de Pesquisas Físicas-CBPF\
[^2]\
Dep. Campos e Partculas (DCP)-Centro Brasileiro de Pesquisas Físicas-CBPF\
Faculdade de Educação da Universidade Federal do Rio de Janeiro, (UFRJ)\
[^3]\
Departamento de Física – ICE, Universidade Federal de Juiz de Fora\
Juiz de Fora, 36036-330, MG, Brazil
[*Abstract*]{}.$\,\,\,\,$ [*We calculate the one-loop divergences for quantum gravity with cosmological constant, using new parametrization of quantum metric. The conformal factor of the metric is treated as an independent variable. As a result the theory possesses an additional degeneracy and one needs an extra conformal gauge fixing. We verify the on shell independence of the divergences from the parameter of the conformal gauge fixing, and find a special conformal gauge in which the divergences coincide with the ones obtained by t’Hooft and Veltman (1974). Using conformal invariance of the counterterms one can restore the divergences for the conformal metric-scalar gravity.*]{}
Introduction
============
The renormalization of quantum gravity and in particular the calculation of the one-loop divergences for quantum General Relativity is considered as a problem of special interest. The non-renormalizability of quantum gravity has been established after the pioneer one-loop calculation by t’Hooft and Veltman [@hove] and Deser and van Nieuwenhuizen [@dene], who derived the divergences for pure gravity and also for the gravity coupled to scalar, vector and spinor fields. In both [@hove] and [@dene] the background field method has been used such that the splitting of the metric was performed according to $\,\,\,g_{\mu\nu}\to g_{\mu\nu} + h_{\mu\nu}$.
Later, the derivations of the one-loop divergences have been carried out many times, using different parametrizations of quantum metric and non-minimal gauge fixing conditions. The calculations were also done for gravity coupled to various kinds of matter fields. One can mention: the first calculation for the pure gravity in a non-minimal gauge [@ktt]; $\,\,\,$ the calculations using plane Feynman diagrams with various parametrizations of the quantum metric [@capper]; $\,\,\,$ in [@local_mom] the result identical to the one of [@hove] has been achieved using local momentum representation technique; $\,\,\,$ the calculation for gravity coupled to Majorana spinor using the (slightly modified) Schwinger-DeWitt technique [@bavi1]; $\,\,\,$ the calculations in the first order formalism (with affine connection independent on the metric) using plane Feynman diagrams [@diplom] and background field method and Schwinger-DeWitt technique [@acta]. Ref. [@acta] contains also the one-loop result for the $\,g^{\mu\nu}\to g^{\mu\nu} + h^{\mu\nu}\,$ parametrization, different from the one of [@hove]. The generalized Schwinger-DeWitt technique has been applied in [@bavi] to confirm the gauge fixing dependence found in [@ktt]. The calculation for gravity with cosmological constant has been done in [@chrisduff] and for the Einstein-Cartan theory with external spinor current in [@EC]. Recently, the one-loop calculations for the pure gravity has been performed in [@kalmyk] where the parametrizations like $\,g_{\mu\nu}\to g_{\mu\nu} + (-g)^r\,h_{\mu\nu}\,$ have been applied. The parametrizations of [@kalmyk] are more general than the ones used in both [@hove] and [@acta], so that [@kalmyk] reproduces both results in the limiting cases.
The interest to the gauge fixing dependence of the in quantum gravity has been revealed in the last years, when some more complicated linear gauges have been studied [@lavrov] (one can consult this paper for the list of references concerning the problem of gauge dependence in quantum field theory and quantum gravity). The purpose of the present letter is to report about the calculation of the one-loop divergences in quantum gravity, in some new parametrization which is different from those which have been used before. This parametrization is based on the separation of the conformal factor from the metric and is related to the well known conformal structure of gravity (see, for example, [@deser; @conf]). In part, our parametrization resembles the one which has been applied in [@kani] for the derivation of the divergences in $\,2+\ep\,$ space-time dimensions. As usual, there is the possibility to conduct an efficient auto-verification of the result, using the on shell gauge fixing independence. One has to notice that the study of conformal gauge in four dimensions has some special importance, since its use permits partial verification of the gauge fixing procedure $\,h^\mu_\mu=0$, which is usually applied in conformal quantum gravity [@frts; @anmamo]. It is worth to notice that the divergences for the Weyl gravity calculated in [@frts] and [@anmamo] differ unlike one uses the so-called conformal regularization introduced in [@frts]. The result of our calculation, which is intended to check the applicability of the conformal gauge $\,\,h_\mu^\mu=0$, can be relevant in the general context of conformal quantum gravity theories in four dimensions.
The present letter is organized as follows. In the next section we present the details of the one-loop calculations. The analysis of the results, including the on-shell gauge fixing independence is performed in section 3, and in the last section we draw our conclusions.
One-loop calculation in a conformal gauge
=========================================
Our starting point is the gravity action with the cosmological constant S = d\^4x (R+2), \[iniact\] In order to illustrate how the degeneracy related to the conformal symmetry appears, let us briefly repeat the consideration of [@deser; @conf].
Performing conformal transformation $\,g_{\mu\nu} \to {\hat g}_{\mu\nu}=g_{\mu\nu}\cdot e^{2\si(x)}$, one meets relations between geometric quantities of the original and transformed metrics: = e\^[4]{}, = e\^[-2]{}. \[n1\] Substituting (\[n1\]) into (\[iniact\]), after integration by parts, we arrive at: S = d\^4x { e\^[2]{}()\^2 + e\^[2]{}R + e\^[4]{}}, \[n2\] where $(\na \si)^2
= g^{\mu\nu}\partial_\mu\si \partial_\nu\si $. If one denotes = e\^ , \[n3\] the action (\[iniact\]) becomes S=d\^4x { ()\^2+R\^2+ \^4 }, \[n5\] that is the action of conformal metric-scalar theory. This theory is conformally equivalent to General Relativity with cosmological constant. Contrary to General Relativity, the theory (\[n5\]) possesses extra local conformal symmetry, for it is invariant under the transformation g\^\_=g\_e\^[2(x)]{} , \^ = e\^[-(x)]{}. \[nnn\] This symmetry compensates an extra (with respect to (\[iniact\])) scalar degree of freedom.
Let us now discuss the relation between two theories on quantum level. In case of renormalizable field theory the difference between two conformally equivalent theories appears on quantum level because of conformal anomaly. For quantum gravity one can not go so far because both theories are non-renormalizable and therefore anomaly is ambiguous [^4]. At the same time, we can investigate the difference in quantization of two theories and the resulting difference in divergences. One has to notice that, despite the derivation of divergences in the theory (\[n5\]) is possible using the techniques developed in [@bavi] and [@odsh], such a calculation would be quite difficult. Technically it is much more cumbersome than similar derivation for the non-minimal, non-conformal metric-scalar theory [@bakaka; @spec]. In this paper we do not try to perform this calculation directly, but instead consider the derivation of the one-loop divergences in the theory (\[iniact\]) using special conformal parametrization.
Since the theory (\[iniact\]) is diffeomorphism invariant, it should be quantized as a gauge theory. On the other hand, the theory (\[n5\]) has an extra conformal symmetry, and thus its quantization requires an extra gauge fixing which is called to remove corresponding degeneracy. As we shall see later, this is also true for the quantization of (\[iniact\]) in conformal variables.
In the framework of the background field method, let us consider the following shift of the metric g\_ g\^\_ = e\^[2]{} \[g\_+h\_\], \[param\] where $h_{\mu\nu}$ and $\si$ are quantum fields and $g_{\mu\nu}$ is the background metric. All raising and lowering of indices is done through $g_{\mu\nu}$. The parametrization (\[param\]) resembles the conformal transformation which led to the conformal form of the action (\[n2\]). Then one can expect to meet an additional degeneracy for the quantum field, related to the conformal symmetry.
For the one-loop divergences, one needs only the bilinear, in the quantum fields $h^{\mu\nu}$ and $\si$, part of the action. This part can be presented in the symbolic form: S\^[(2)]{}= d\^4x (
[cc]{} h\^ &
) (
[c]{} h\^\
). Now, one has to introduce the gauge fixing for the diffeomorphism. We choose the gauge fixing term in the form S\_[GF]{}=- d\^4x \_\^ \[gafite\] with \_=\_h\^\_+ \_h-\_, \[gafipa\] where $h=h^{\mu}\mbox{}_{\mu}$ and $\al$, $\be, \ga$ are gauge fixing parameters. It is useful to choose them in such a way that the bilinear form becomes minimal second order operator.
One can find that this can be achieved by taking $\al =2$, $\be =-1/2$ and $\ga =2$. Then the bilinear form of the action with the gauge fixing term becomes $$S^{(2)}+S_{GF}^{(2)} = \int\; d^4x\sqrt{-g}\, \left\{
h^{\mu\nu}\,\left[\,
K_{\mu\nu\, ,\,\al\be}(\Box-2\La)+
M_{\mu\nu\, ,\,\al\be}\,\right] \, h^{\al\be}+
\right.$$ . + ( -4+2R+16) + h\^( -g\_- 2R\_ + g\_R + 4g\_ ) }, \[minimal\_form\] where K\_[ ,]{}=( \_[ , ]{}- g\_g\_) and $$M_{\mu\nu\, ,\,\al\be}= - \frac{1}{4}\,\de _{\mu\nu\, , \,\al\be}R+
\frac{1}{8}\,\left(\,g_{\nu\al}R_{\mu\be}
+g_{\mu\al}R_{\nu\be}+g_{\mu\be}R_{\nu\al}+g_{\nu\be}R_{\mu\al}
\,\right)-$$ -(g\_R\_+g\_R\_) +( R\_+R\_+R\_+R\_ ) +g\_g\_R, where we have used standard notation $\,\,\de _{\mu\nu\, , \,\al\be}=\frac{1}{2}(g_{\mu\al}g_{\nu\be}+
g_{\mu\be}g_{\nu\al})$.
It proves useful to separate the field $h^{\mu\nu}$ into the trace and the traceless part, $h^{\mu\nu}=\bar{h}^{\mu\nu}+\frac{1}{4}g^{\mu\nu}h$. Then the bilinear form (\[minimal\_form\]) becomes S\^[(2)]{}+S\_[GF]{}\^[(2)]{} & = & d\^4x { |[h]{}\^ |[h]{}\^+ .\
& + & . |[h]{}\^ + h h + .\
& + & . h + ( -4+ 2R + 16\] }. \[degen\] Here $${\bar \de}_{\mu\nu , \al\be} = \de_{\mu\nu , \al\be}
- \frac14\,g_{\mu\nu} g_{\al\be}$$ is the projector to the traceless states. The expression (\[degen\]) exhibits the degeneracy in the mixed $\,\,h-\si\,\,$ sector, and hence further calculation requires some additional restriction on the quantum fields. This degeneracy is a direct consequence of the conformal symmetry (\[nnn\]) and thus we have to fix this symmetry. Let us choose the conformal gauge fixing in the form $\si =\la h$ with $\,\la\,$ being the gauge fixing parameter. Then (\[degen\]) becomes: S\^[(2)]{}+S\_[GF]{}\^[(2)]{} & = & d\^4x { |[h]{}\^|[h]{}\^+ .\
& + & . |[h]{}\^ h + h\[ b\_[1]{}+2b\_[2]{}+ b\_[3]{}R ) h} \[hgrav\] where we introduced the notations b\_1 = --- 4\^2; b\_2=+2+8\^2; b\_3=+2\^2.
The total one-loop divergences will be given by \^[(1)]{}\_[[div]{}]{}= Tr[ln]{} \_[[grav]{}]{}|\_[[div]{}]{} -iTr[ln]{} |\_[[div]{}]{} where the last term is the contribution from the ghost fields, and $\hat{H}_{{\rm grav}}$ is the operator corresponding to eq. (\[hgrav\]). The standard Schwinger-DeWitt algorithm enables one to derive $$\frac{i}{2}
Tr\,{\rm ln}\, \hat{H}_{{\rm grav}}|_{{\rm div}}
= -\frac{1}{\vp}\int\; d^4x\sqrt{-g}\, \left\{\,
\frac{19}{18}R^2_{\rho\la\si\ta}+
\left( \frac{4}{b_1}\la ^2-\frac{55}{18}\right)\,
R^2_{\rho\la}+
\right.$$ . ( -++ ) R\^2+ (++9) R+ (+18) \^2 } \[newforma\] where $\,\vp=(4\pi )^2(n-4)$. Also, the operator of the ghost action $\hat{M}$ is \_\^= - \_\^-R\_\^. \[ghost\_oper\] We remark that the ghost operator does not depend on the gauge transformation of the field $\si$, because at the one-loop level, in the background field method, the generator of the gauge transformations which enters into the expression for $\,\hat{M}_\mu^\nu \,$ is the one for the background (not quantum!) fields [@hove] (see also [@book]) and in case of $\si$ this operator is zero.
Calculation of the ghost contribution yields standard result [@hove] -iTr[ln]{} |\_[[div]{}]{} & = & d\^4x { - E + R\^2\_+ R\^2} , where $E=R^2_{\mu\nu\al\be}-4R_{\mu\nu}^2+R^2$. Finally, one arrives at the following one-loop divergences: \^[(1)]{}\_[[div]{}]{} =- d\^4x { p\_1()E+p\_2()C\^2+p\_3()R\^2+p\_4()R+p\_5()\^2 } \[naschitali\] where $C^2$ is the square of the Weyl tensor $\,C^2=E+2(R_{\mu\nu}^2-\frac13\,R^2)\,$ and p\_1() & = & ,\
p\_2() & = & ,\
p\_3() & = & ,\
p\_4() & = &\
p\_5() & = & 4 . \[1-loop\_divs\] The above expression (\[naschitali\]), (\[1-loop\_divs\]) contains complicated dependence on the gauge fixing parameter $\,\la\,$. Besides, the one-loop divergences may depend on others gauge fixing parameters $\,\al,\,\be,\,\ga\,$ from (\[gafipa\]). Here we are interested only in the dependence on $\,\la$, and keep $\,\al,\,\be,\,\ga\,$ fixed as before.
Analysis of the results
=======================
The expression (\[naschitali\]), (\[1-loop\_divs\]) looks quite cumbersome and somehow chaotic because of the complicated dependence on the gauge fixing parameter $\la$. But, in fact, there are a few possibilities to check and analyze it. First of all, for the value $\la=0$, all the $\si$-field contributions drop and we arrive at the well-known result [@hove; @chrisduff] \^[(1)]{}\_[[div]{}]{} =- d\^4x { R\^2+R\_\^2+E + R+ 10\^2 }. \[hv\] For other values of $\la$ the divergences are different and one can check that the $\la$-dependence can not be compensated by the change of other gauge fixing parameters $\,\al,\be\,$ or by the change of parameter $\,r\,$ introduced in [@kalmyk].
If we take a limit $\la\to\infty$, the result is not conformal invariant, as one could naively expect. Let us give some additional comment on this point. The above calculation can be regarded as a particular case of the much more complicated derivation of the one-loop divergences in the theory (\[n5\]), which was mentioned in the Introduction. In general calculation one is supposed to shift both fields $\ph$ (or $\si$, this is equivalent) and $g_{\mu\nu}$, while in this paper we took the background scalar to be constant. Let us imagine, for a moment, that we shifted both fields + , g\_ g\_ + h\_. \[total\_shift\] As far as we believe into conformal invariance of the one-loop divergences[^5], the result for conformal metric-scalar theory is [@alter] $$\Ga_{div}=\frac{1}{\ep}\,\int d^{4}x\sqrt{-g}\left\{ p_1\,E
+ p_2\,C^{2} +
\right.$$ . + p\_3\^2 + p\_4+ p\_5\^2 }, \[conf2\] where $\zeta =\zeta (\chi )$ is some function of $\,\chi$. The procedure accepted in this paper is equivalent to taking $\,\chi=const$, and therefore (\[1-loop\_divs\]) should be regarded as (\[conf2\]) with constant $\zeta$. Obviously, constant $\zeta$ does not transform and the conformal invariance is lost.
In order to verify the result of the calculation, one can use classical equations of motion $R_{\mu\nu} = -\La\,g_{\mu\nu}$. On shell the divergences become \^[(1)[on-shell]{}]{}\_[[div]{}]{}=- d\^4x { E-\^2 }, \[onshell\] independent on the gauge fixing parameter $\la$. As a consequence, the on shell renormalization group equation for the dimensionless cosmological constant $\,\ka^2\Lambda\,\,$ [@frts] is gauge fixing independent in our conformal parametrization. One has to notice that the coefficients of (\[onshell\]) are linear combinations of all five functions (\[1-loop\_divs\]), and thus the complete cancellation of the $\la$-dependence, together with (\[hv\]), provide a very confident verification of the result (\[naschitali\]).
Conclusions
===========
We have studied the equivalence between General Relativity and conformal metric-scalar theory on quantum level. The one-loop divergences were calculated for quantum gravity, for the first time this was done in the conformal parametrization for quantum metric. We have found that the dependence on the new gauge fixing parameter disappears on shell. This gives an efficient check to the whole procedure based on fixing the conformal symmetry by using the trace of quantum metric $h=h_{\mu}^{\mu}$. The results of our work show that the source of the discrepancy in the results for the quantum Weyl gravity is not caused by this conformal gauge fixing. Finally, the supposition of conformal invariance of the counterterms enables one to restore the divergences for the gravity coupled to conformal scalar field (\[conf2\]).
[**Acknowledgements**]{}. I.Sh. and G.B.P. are grateful to CNPq for for permanent support and for the scholarship correspondingly. A.P.F. is grateful to DCP-CBPF for hospitality and support.
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[^1]: Electronic address: peixoto@cbpf.br
[^2]: Electronic address: andrepf@cbpf.br
[^3]: Electronic address: shapiro@fisica.ufjf.br. On leave from Tomsk Pedagogical University, Russia.
[^4]: For instance, the divergences of (\[iniact\]) vanish for the special gauge fixing, and then anomaly vanishes.
[^5]: Some remarkable example of the opposite one can meet in the Weyl gravity, where the results of two one-loop calculations [@frts] and [@anmamo] coincide only after the use of the so-called conformal regularization [@frts]. The lack of equivalence between the results of [@bakaka] and [@spec] may indicate to the similar problem.
|
---
abstract: 'We consider an M server system in which each server can service at most one update packet at a time. The system designer controls (1) *scheduling* - the order in which the packets get serviced, (2) *routing* - the server that an arriving update packet joins for service, and (3) the *service time distribution* with fixed service rate. Given a fixed update generation process, we prove a strong age-delay and age-delay variance tradeoff, wherein, as the average AoI approaches its minimum, the packet delay and its variance approach infinity. In order to prove this result, we consider two special cases of the M server system, namely, a single server system with last come first server with preemptive service and an infinite server system. In both these cases, we derive sufficient conditions to show that three heavy tailed service time distributions, namely Pareto, log-normal, and Weibull, asymptotically minimize the average AoI as their tail gets heavier, and establish the age-delay tradeoff results. We provide an intuitive explanation as to why such a seemingly counter intuitive age-delay tradeoff is natural, and that it should exist in many systems.'
author:
- 'Rajat Talak and Eytan Modiano [^1] [^2] [^3]'
title: 'Age-Delay Tradeoffs in Queueing Systems'
---
Introduction {#sec:intro}
============
Information freshness and low latency communication is gaining increasing relevance in many futuristic communication systems, such as industrial automation, autonomous driving, tele-surgery, financial markets, and virtual reality [@2018_LowLatencySurvey_Fischione; @2016_5G_Enabled_Tactile_Internet; @2018_Haptic_and_5G; @2016_Tactile_Internet_Vision_Progress_Challenges]. The latency requirements vary depending on the application. While applications such as autonomous driving, tele-surgery, virtual reality, financial markets are envisioned to require a latency of a few milliseconds, other systems such as industry automation, control signalling in power grids aim at a latency of 10-100 milliseconds [@2018_LowLatencySurvey_Fischione; @2016_5G_Enabled_Tactile_Internet]. In many of these applications, seeking the most recent status update is crucial to the overall system performance. In a network of unmanned aerial vehicles, for example, exchanging the most recent position, speed, and other control information [@talakCDC16; @FANETs2013]; in operations monitoring systems, accessing the most recent sensor measurement; and in cellular systems, obtaining the timely channel state information from the mobile users [@LTE_book], is important and can lead to significant performance improvements.
![Age evolution in time. Update packets generated at times $t_i$ and received, by the destination, at times $t^{'}_{i}$. Packet $3$ is received out of order, and thus, doesn’t contribute to age.[]{data-label="fig:age"}](age){width="0.95\linewidth"}
Age of information (AoI) is a metric for information freshness that measures the time that elapses since the last received fresh update was generated at the source [@2011SeCON_Kaul; @2012Infocom_KaulYates]. It is, therefore, a destination-centric measure, and is suitable for applications that necessitate timely updates. A typical evolution of AoI for a single source-destination system is shown in Figure \[fig:age\]. The AoI increases linearly in time, until the destination receives a fresh update packet. Upon reception of a fresh update packet $i$, at time $t^{'}_{i}$, the AoI drops to the time since the packet $i$ was generated, which is $t^{'}_i - t_i$; here $t_i$ is the time of generation of the update packet $i$. Unlike the traditional latency metrics such as packet delay, AoI only accounts for the update packets that deliver fresh updates to the destination – such packets are called *informative packets* [@2014ISIT_KamKomEp]. For example, in Figure \[fig:age\], packet $3$ is an informative packet but packet $2$ is not. This is because packet $3$ reaches the destination before packet $2$, which is therefore rendered stale by time $t^{'}_{2}$.
AoI was first studied for the first come first serve (FCFS) M/M/1, M/D/1, and D/M/1 queues in [@2012Infocom_KaulYates]. Since then, AoI has been analyzed for several queueing systems with the goal to minimize AoI [@2011SeCON_Kaul; @2012CISS_KaulYates; @2012Infocom_KaulYates; @2012Infocom_TIT_KaulYates; @2015ISIT_LongBoEM; @talak17_allerton; @talak18_Mobihoc; @talak19_ToN_Mobihoc; @vishrant19_AoI_discreteQ; @2016X_Najm; @2016_ISIT_YinSun_AoI_Thput_Delay_LCFS; @2016_ISIT_TIT_YinSun_Thput_Delay_LCFS; @2017_ISIT_YinSun_LCFSopt_MultiHop; @2017_ISIT_TIT_YinSun_LCFSopt_MultiHop; @2014ISIT_KamKomEp; @2014ISIT_CostaEp; @Inoue17_FCFS_AoIDist; @2018_Ulukus_GG11; @2018ISIT_Yates_AoI_ParallelLCFS; @2018_Yates_LCFS_Multihop; @2016_ISIT_Ep_AoI_Deadlines; @2018_ISIT_Inoue_AoI_Deadline; @2016_MILCOM_Ep_AoI_Buffer_Deadline_Replace; @YinSun_2019_AoI_book]. Two time average metrics of AoI, namely, peak and average age are generally considered. The analysis mostly relies on the specificities of the queueing model under consideration. Typically, a peak or average age expression is derived and then optimized over the update generation rate. However, progress has been made recently towards a more general analysis of AoI. A general formula for the stationary distribution of AoI for a single-server systems was recently developed in [@Inoue17_FCFS_AoIDist; @Inoue18_FCFS_LCFS_AoIDist], while [@2018_Yates_SHS] used the theory of stochastic hybrid systems to systematically derive expressions for the average age and its higher moments.
Despite the difficulty in analyzing age for general queueing systems several approaches that reduce or minimize AoI have been brought to light. The advantage of having parallel servers in reducing AoI was demonstrated in [@2014ISIT_KamKomEp; @2014ISIT_CostaEp; @2018ISIT_Yates_AoI_ParallelLCFS]. Methods such as limiting the buffer sizes [@2011SeCON_Kaul; @2016_MILCOM_Ep_AoI_Buffer_Deadline_Replace] or introducing packet deadlines [@2016_MILCOM_Ep_AoI_Buffer_Deadline_Replace; @2016_ISIT_Ep_AoI_Deadlines; @2018_ISIT_Inoue_AoI_Deadline] have also been shown to reduce AoI. In a general queueing system, with exponentially distributed service times, the last come first serve (LCFS) queue scheduling discipline with preemptive service was proved to be age optimal in [@2016_ISIT_YinSun_AoI_Thput_Delay_LCFS; @2016_ISIT_TIT_YinSun_Thput_Delay_LCFS; @2017_ISIT_YinSun_LCFSopt_MultiHop; @2017_ISIT_TIT_YinSun_LCFSopt_MultiHop]. In [@yin17_tit_update_or_wait], an optimal update generation policy was investigated, and it was discovered that an intuitively apt zero-wait policy, which sends the next update right after the previous one is received, is not always age optimal. More recently, minimizing age metrics over update generation and service time distribution has been of interest [@talak18_determinacy; @Inoue18_FCFS_LCFS_AoIDist]. In a related work [@talak18_determinacy], we considered the problem of minimizing peak and average age over packet generation and service time distributions, given a particular update generation and service rate. We showed that determinacy in packet generation and/or service does not necessarily minimize age. Similar results were independently obtained in [@Inoue18_FCFS_LCFS_AoIDist].
Packet delay and delay variance (jitter), on the other hand, have traditionally been considered as measures of communication latency. Minimizing packet delay in a queueing system is known to be a hard problem. Several works have focused on the problem of reducing or minimizing the packet delay and its variance [@1962_Kingman_Q_Discipline_Wait_Variance; @1975_Rolski_Stoyan_Comparison_Waiting_Times_GG1; @1977_Wolff; @1980_Whitt; @1984_Whitt; @1983_Hajek; @1983_Whitt; @2019_Ness_DelayOptimal_EnergyEffi_Commun_MarkovArrivals; @2017_PingChun_IHong_DelayOptimalityQueues_Switching_Overhead; @2017_Ness_FastConvergent_LowDelay_LowComplexity_NetwOpt; @2016_YinSun_Ness_DelayOptimal_Scheduling_Queues_Pkt_Replications; @2016_Infocom_HeavyBall_to_Tame_Delay; @2013_Neely_Delay_Based_NUM; @2013_Ness_Delay_Based_MaxWeight; @2013_Neely_LIFO_Backpressure; @2011_Ness_DelayAnalysis_MultiHop]. It is widely believed that AoI and delay are closely related, and hence, can be minimized simultaneously. For example, in a simple FCFS queue under Poisson arrivals, less variability in service time distribution minimizes both packet delay and peak age [@talak18_determinacy; @Inoue18_FCFS_LCFS_AoIDist]. For a system of $M$ parallel servers with exponential service times, minimum age and delay can be simultaneously attained by resorting to the LCFS with preemptive service [@2016_ISIT_YinSun_AoI_Thput_Delay_LCFS; @2016_ISIT_TIT_YinSun_Thput_Delay_LCFS].
*Is it then always possible to minimize age and packet delay simultaneously, or are there systems in which minimizing one does not imply minimizing the other?*
In this work, we answer this question by considering an $M$ server queueing system. We show that as we tailor the queue scheduling discipline, routing, and the service time distributions to minimize average age, the packet delay and its variance approach infinity.
![Plot of achieved age-delay points for various single server systems, Poisson packet generation at rate $\lambda = 0.5$, and service at rate $\mu = 0.8$. Scheduling disciplines: FCFS, LCFS with preemptive service. Service time distributions: Deterministic, Exponential, and Heavy Tailed distributions in Table \[tbl:heavy\_tail\].[]{data-label="fig:AoI_Delay_tradeoff1_plot1"}](AoI_Delay_tradeoff1_plot1){width="0.95\linewidth"}
As an example, consider a single server queue with a fixed update generation and service rate. Updates are generated according to a Poisson process. In Figure \[fig:AoI\_Delay\_tradeoff1\_plot1\], we plot the achieved packet delay and average age attained under various queue scheduling policies (FCFS and LCFS with preemptive service) and service time distributions (deterministic, exponential, log-normal, Pareto, and Weibull). For the three heavy tailed service time distributions, namely, log-normal, Pareto, and Weibull we plot the age-delay points for various values of the free parameter; see Table \[tbl:heavy\_tail\]. It appears that there is a strong age-delay tradeoff, i.e. a lower average age can be achieved at the cost of higher delay.
Intuitively, this tradeoff can be explained as follows: *In order to achieve minimum age the system has to prioritize informative packets. In doing so, the non-informative updates lag behind in the system thereby incurring a large waiting time cost. The delay and its variance get dominated by the large delays incurred by the non-informative update packets, thus leading to the tradeoff curve in Figure \[fig:AoI\_Delay\_tradeoff1\_plot1\].*
Given this intuition, we suspect that an age-delay tradeoff should exist in many systems. In this paper, we prove it for an $M$ server queueing system.
Name Distribution Free Parameter
------------ -------------------------------------------------------------------------------------------- ----------------
Log-normal $S = \exp\left(- \log \mu - \frac{\sigma^2}{2} + \sigma N\right)$ $\sigma > 0$
$N \sim \mathcal{N}(0, 1)$
Pareto $F_S(x) = 1 - \left(\theta(\alpha)/x\right)^{\alpha}\mathbb{I}_{\{ x > \theta(\alpha) \}}$ $\alpha > 1$
$\theta(\alpha) = (\alpha - 1)/(\mu \alpha)$
Weibull ${\mathbf{P}\left[ S > x \right]} = \exp\left\{- (x/\beta(k))^{k} \right\}$ $k > 0$
$\beta(k) = \left[ \mu \Gamma(1 + 1/k)\right]^{-1}$
: Heavy tailed service time distributions with mean ${\mathbb{E}\left[{S}\right]} = 1/\mu$.[]{data-label="tbl:heavy_tail"}
Contributions
-------------
We consider an $M$ server system in which each server can service at most one update packet at any given time. Update packets are generated according to a renewal process at a fixed rate. The system designer decides the queue *scheduling discipline*, i.e. the order in which the packets get serviced, the *routing*, which determines the server for each arriving update packet, and the *service time distribution*. In order to observe the age-delay tradeoff, we consider the problem of minimizing packet delay (and packet delay variance), subject to an average age constraint, over the space of all queue scheduling disciplines, routing, and service time distributions, with a fixed mean service time budget of $1/\mu$ for each queue. When the updates are generated according to a Poisson process we show that there is a strong age-delay tradeoff, namely, as the average age approaches its minimum, the delay approaches infinity. When the updates are generated according to a general renewal process, we show that there is a strong age-delay variance tradeoff, i.e. as the average age approaches its minimum, the variance in packet delay approaches infinity.
The proof of this result involves first proving the same result in two special cases of the $M$ server system: (1) A single server system, i.e. $M = 1$, in which the queue scheduling discipline is fixed to LCFSp, and (2) An infinite server system, i.e. $M = \infty$. In both these cases, we derive sufficient conditions on the average age minimizing service time distribution. We then show that these sufficient conditions are satisfied by the three heavy tailed service time distributions, namely Pareto, log-normal, and Weibull, asymptotically in its tail parameter. This helps us establish the age-delay tradeoff results in the two special cases of the $M$ server system. We also observe a certain age-delay disparity in these two cases in which the delay (or delay variance) minimizing service time distributions result in the worst case average age.
The results derived for the two special cases are then used to prove the strong age-delay and age-delay variance tradeoffs for the $M$ server system. To the best of our knowledge, this is the first work to establish an age-delay tradeoff result. A preliminary version of the this work was available on arXiv [@talak18_determinacy] and appeared in ISIT 2019 [@talak19_AoI_age_delay; @talak19_AoI_heavytail]. This work builds upon the results in [@talak19_AoI_age_delay; @talak19_AoI_heavytail; @talak18_determinacy].
Organization {#sec:org}
------------
In Section \[sec:model\], we describe the system model and provide a general definition of AoI. In Section \[sec:age\_delay\_tradeoff\], we formulate the age-delay and age-delay variance problems for the $M$ server system. The age-delay tradeoff result for the $M$ server system is also stated and discussed here. In Sections \[sec:lcfs\] and \[sec:inf\_serv\], we prove the age-delay tradeoff result in the two special cases of the single server LCFSp and infinite server systems. The paper culminates in Section \[sec:M\_ServSyst\] with a proof of the age-delay tradeoff for the $M$ server system. We conclude in Section \[sec:conclusion\].
System Model {#sec:model}
============
![Illustration of the $M$ server queueing system.[]{data-label="fig:queueing_system"}](queueing_system){width="0.95\linewidth"}
A source generates update packets according to a renewal process, at a given rate $\lambda$. The update packets enter a queueing system, which consists of $M$ servers shown in Figure \[fig:queueing\_system\]. Each server has a rate $\mu$, and can service at most one update packet at any given time. The service times are independent and identically distributed across update packets. A scheduler determines routing and scheduling of update packets, which upon service reach the destination. Our goal is to ensure minimum age and/or minimum delay at the destination.
The system designer has control over three things:
1. *service*: it can decide the service time distribution, given the mean service time of $1/\mu$;
2. *routing*: it can determine the server that an update packet connects for service; and
3. *scheduling*: it can decide the order in which the update packets get serviced at each server.
A *scheduler* implements the routing and scheduling policy. The scheduler is also not allowed to drop any packets. Further, in determining the order of service of generated packets, we assume that the scheduler is not privy to the service times of the individual packets. We also assume that only the service time distribution can be set before hand by the system designer, and not the service times of individual packets.
We use $X$ to denote the inter-generation time of update packets with distribution $F_X$, and $S$ to denote the service time random variable, with distribution $F_S$. Note that ${\mathbb{E}\left[{X}\right]} = 1/\lambda$ and ${\mathbb{E}\left[{S}\right]} = 1/\mu$ is fixed. We assume that $\lambda < \mu$, i.e. there is enough serving capacity in the network to service the generated updates.We use Minimize or $\min$, instead of the technically correct $\inf$, for ease of presentation. We now define the latency measures of packet delay, delay variance, and average age.
Delay and Age of Information {#sec:delay_aoi_def}
----------------------------
Let the update packets be generated at times $t_1, t_2, \ldots$, and let the update packet $i$ reach the destination at time $t^{'}_{i}$. The update packets may not reach the destination in the same order as they were generated. In Figure \[fig:age\], packet $3$ reaches the destination before packet $2$, i.e. $t^{'}_{3} < t^{'}_{2}$. Delay for the $i$th packet is $D_i = t^{'}_{i} - t_{i}$, and the packet delay for the system is given by $$\label{eq:delay}
D = \limsup_{N \rightarrow \infty} {\mathbb{E}\left[{\frac{1}{N}\sum_{i=1}^{N}D_i}\right]},$$ where the expectation is over the update generation, service times, and scheduling discipline. We skip a formal definition, but will use the notation ${\text{VarD}}$ to denote variance in packet delay. For the $M$ server queueing system considered, we note that ${\text{VarD}}$ is lower-bounded by the variance in service time distribution $F_S$.
Age of a packet $i$ is defined as the time since it was generated: $A^{i}(t) = (t-t_{i})\mathbb{I}_{\{ t > t_i\}}$, which is $0$ by definition for time prior to its generation $t < t_i$. Age of information at the destination node, at time $t$, is defined as the minimum age across all received packets up to time $t$: $$\label{eq:age_t}
A(t) = \min_{i \in \mathcal{P}(t)} A^{i}(t),$$ where $\mathcal{P}(t) \subset \{1, 2, 3, \ldots \}$ denotes the set of packets received by the destination, up to time $t$. Notice that AoI increases linearly, and drops only at the times of certain packet receptions: $t^{'}_{1}, t^{'}_{3}, t^{'}_{4}, \ldots$, but not $t^{'}_{2}$ in Figure \[fig:age\]. Such an age drop occurs only when an update packet with a lower age, than all packets received thus far, is received by the destination. We refer to such packets, that result in age drops, as the *informative packets* [@2014ISIT_KamKomEp].
We consider a time averaged metrics of age of information, namely, the average age. The average age is defined to be the time averaged area under the age curve: $$\label{eq:Aave}
A^{\text{ave}} = \limsup_{T \rightarrow \infty} {\mathbb{E}\left[{\frac{1}{T}\int_{0}^{T}A(t) dt}\right]},$$ where the expectation is over the packet generation and packet service processes.
It is important to note that the age $A(t)$, and therefore the average age, is defined from the view of the destination, and not a packet. $A(t)$ is the time since the last received informative packet was generated at the source. It, therefore, does not matter how long the non-informative packets take to reach the destination. This is unlike packet delay, which accounts for every packet in the system equally.
We use the notation $D(F_S, \pi_Q)$, ${\text{VarD}}(F_S, \pi_Q)$, and $A^{\text{ave}}(F_S, \pi_Q)$ to make explicit the dependency of delay, its variance, and average age on the service time distribution $F_S$ and the queue scheduling policy $\pi_Q$.
In the next section, we pose the age-delay tradeoff problems. Age-delay tradeoff results are then proved for two special cases in Section \[sec:lcfs\] and Section \[sec:inf\_serv\], before arriving at the result for the $M$ server system in Section \[sec:M\_ServSyst\].
Age-Delay Tradeoff {#sec:age_delay_tradeoff}
==================
Motivated by the example in Figure \[fig:AoI\_Delay\_tradeoff1\_plot1\], we define two age-delay tradeoff problems. One, minimizes delay while the other minimizes delay variance, both over an average age constraint. The age-delay tradeoff is defined as: $$\begin{aligned}
\label{eq:Aave_Delay_Tradeoff}
\begin{aligned}
T({\text{AoI}}) &= \underset{F_S, \pi_{Q}}{\text{Minimize}}
& & D(F_S, \pi_Q) \\
& \text{subject to} & & A^{\text{ave}}(F_S, \pi_Q) \leq {\text{AoI}}, \\
& & & {\mathbb{E}\left[{S}\right]} = 1/\mu.
\end{aligned}\end{aligned}$$ Here, the function $T({\text{AoI}})$ denotes the minimum delay that can be achieved for the $M$ server queueing system, with an average age constraint of $A^{\text{ave}}(F_S, \pi_Q) \leq {\text{AoI}}$. It might seem that both minimum age and delay could be attained simultaneously. We will show that, $T({\text{AoI}}) \rightarrow \infty$ as ${\text{AoI}}$ approaches the minimum average age $$\begin{aligned}
\label{eq:Amin}
\begin{aligned}
A_{\min} &= \underset{F_S, \pi_{Q}}{\text{Minimize}}
& & A^{\text{ave}}(F_S, \pi_Q), \\
& \text{subject to} & & {\mathbb{E}\left[{S}\right]} = 1/\mu.
\end{aligned}\end{aligned}$$
Variability in packet delay is also an important metric in system performance. We define the age-delay variance tradeoff problem to be: $$\begin{aligned}
\label{eq:Aave_DelayVar_Tradeoff}
\begin{aligned}
V({\text{AoI}}) &= \underset{F_S, \pi_{Q}}{\text{Minimize}}
& & {\text{VarD}}(F_S, \pi_Q) \\
& \text{subject to} & & A^{\text{ave}}(F_S, \pi_Q) \leq {\text{AoI}}, \\
& & & {\mathbb{E}\left[{S}\right]} = 1/\mu.
\end{aligned}\end{aligned}$$ Here, the function $V({\text{AoI}})$ denotes the minimum delay variance that can be achieved for the $M$ server queueing system, with an average age constraint of $A^{\text{ave}}(F_S, \pi_Q) \leq {\text{AoI}}$. Counter to our intuition, we show that $V({\text{AoI}}) \rightarrow +\infty$ as approaches its minimum value $A_{\min}$.
Ideally, we would like to obtain every point on the tradeoff curves, i.e., completely characterize the functions $T({\text{AoI}})$ and $V({\text{AoI}})$. The following result shows that the tradeoff curves can be done by optimizing a linear combination of average age and packet delay.
\[thm:aoi\_delay\_min\] The points on the age-delay tradeoff curve $({\text{AoI}}, T({\text{AoI}}))$ can be obtained by solving $$\begin{aligned}
\label{eq:aoi_delay_min}
\begin{aligned}
& \underset{F_S, \pi_{Q}}{\text{Minimize}}
& & D(F_S, \pi_Q) + \nu A^{\text{ave}}(F_S, \pi_Q)\\
& \text{subject to} & & {\mathbb{E}\left[{S}\right]} = 1/\mu,
\end{aligned}\end{aligned}$$ for all $\nu \geq 0$. Similarly, the points on the age-delay variance tradeoff curve $({\text{AoI}}, V({\text{AoI}}))$ can be obtained by solving , by replacing $D(F_S, \pi_Q)$ with ${\text{VarD}}(F_S, \pi_Q)$.
This follows from Theorem II.2 in [@Neely_NetworkOpt].
Theorem \[thm:aoi\_delay\_min\] motivates optimization of a latency metric that is a linear combination of average age and packet delay (or packet delay variance). This problem, however, is not easy to solve. For instance, in the case of a singe server, i.e. $M = 1$, with Poisson arrivals, the delay is minimized with deterministic service times and the variance in delay is minimized under the FCFS service discipline [@1962_Kingman_Q_Discipline_Wait_Variance]. Exactly the opposite is true for the metric of average age. We will show in Section \[sec:lcfs\] that the LCFS queue scheduling policy with heavy tailed service minimizes average age. It, therefore, appears that the delay term and the average age term in are pulling the decision variables in opposite directions. We say that a *strong age-delay tradeoff* exists for $T({\text{AoI}})$ if $T({\text{AoI}}) \rightarrow +\infty$ and ${\text{AoI}}$ approaches $A_{\min}$. Conversely, *no age-delay tradeoff* exists for $T({\text{AoI}})$ if the minimum average age and the minimum packet delay can be achieved simultaneously. Similar definition apply for age-delay variance tradeoff $V({\text{AoI}})$. Figure \[fig:traeoff\_illustration\] illustrates a strong age-delay tradeoff. Note that this matches with our numerical results in Figure \[fig:AoI\_Delay\_tradeoff1\_plot1\].
![Illustration of strong age-delay tradeoff.[]{data-label="fig:traeoff_illustration"}](strong_tradeoff){width="0.75\linewidth"}
We show that for the $M$ server system defined above there is a strong age-delay tradeoff when the update generation is Poisson, and a strong age-delay variance tradeoff for the general update generation process.
\[thm:M\_tradeoff\] For the $M$ server system, the following statements are true:
1. The minimum achievable average age is $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$.
2. When update generation is Poisson, there is a strong age-delay tradeoff.
3. When the update generation is a general renewal process, there is a strong age-delay variance tradeoff.
In Theorem \[thm:M\_tradeoff\], the update generation process is kept fixed. Thus, the strong age-delay tradeoffs hold even if we could control the inter-generation time distribution $F_X$, with a mean budget constraint of ${\mathbb{E}\left[{X}\right]} = 1/\lambda$. The optimal update generation, with the mean constraint ${\mathbb{E}\left[{X}\right]} = 1/\lambda$, that minimizes the $A_{\min}$ is the periodic update generation.
It seems counterintuitive at first that a strong tradeoff should exist between delay, or delay variance, and average age. However, a close examination reveals the following insight:
*For age minimization it becomes necessary that the informative packets, the packets that reduce age, get serviced as soon as they arrive, while the non-informative packets, may incur as long a service time and queueing delay, as they do not matter in the age calculations. As we do this, the packet delay gets dominated by the delay of the non-informative packets, resulting in the two age-delay tradeoffs.*
In what follows, we prove this strong tradeoff between age-delay and age-delay variance. The proof of Theorem \[thm:M\_tradeoff\] is given in Section \[sec:M\_ServSyst\]. It relies on age-delay tradeoff results in two special cases, which are first studied in Sections \[sec:lcfs\] and \[sec:inf\_serv\].
In Section \[sec:lcfs\], we consider the special case of a single server system, i.e. $M = 1$, in which the scheduling policy $\pi_Q$ is fixed to the last come first server with preemptive service (LCFSp). We prove the statements of Theorem \[thm:M\_tradeoff\] for this special case. Namely, we we show that a age-delay tradeoff exists when the update generation is Poisson, and a age-delay variance tradeoff exists when the updates are generated according to a general renewal process.
In Section \[sec:inf\_serv\], we consider the special case of an infinite server queue, i.e. $M = \infty$, in which the scheduling policy $\pi_Q$ assigns every newly generate update to a new server. We again prove the statements of Theorem \[thm:M\_tradeoff\] for this case.
Our approach in both these sections is as follows: We first derive an expression for the average age, and use it to obtain the minimum average age $A_{\min}$. We then use this to prove the two strong age-delay tradeoffs. Using these two special cases, in Section \[sec:M\_ServSyst\], we finally prove Theorem \[thm:M\_tradeoff\].
LCFSp Queue {#sec:lcfs}
===========
In this section, we consider a special case of the $M$ server system. We consider a single server system, i.e. $M = 1$, in which the queue scheduling discipline $\pi_Q$ is fixed to the LCFSp. The age and delay metrics, therefore, are just a function of the service time distribution $F_S$. The age-delay and the age-delay variance problem, for this case, reduces to $$\begin{aligned}
\label{eq:LCFS_Aave_Delay_Tradeoff}
\begin{aligned}
T({\text{AoI}}) &= \underset{F_S}{\text{Minimize}}
& & D(F_S) \\
& \text{subject to} & & A^{\text{ave}}_{\text{G/G/1}}(F_S) \leq {\text{AoI}}, \\
& & & {\mathbb{E}\left[{S}\right]} = 1/\mu,
\end{aligned}\end{aligned}$$ and $$\begin{aligned}
\label{eq:LCFS_Aave_DelayVar_Tradeoff}
\begin{aligned}
V({\text{AoI}}) &= \underset{F_S}{\text{Minimize}}
& & {\text{VarD}}(F_S) \\
& \text{subject to} & & A^{\text{ave}}_{\text{G/G/1}}(F_S) \leq {\text{AoI}}, \\
& & & {\mathbb{E}\left[{S}\right]} = 1/\mu,
\end{aligned}\end{aligned}$$ where we use the notation $A^{\text{ave}}_{\text{G/G/1}}(F_S)$ to denote the average age for the LCFSp queue. The optimization is only over the service time distribution. We omit the dependence on $F_S$, whenever convenient.
The rest of this section is organized as follows. In Section \[sec:lcfs\_min\_age\], we derive an expression for average age, and characterize the minimum average age. We also show that heavy tailed service time distributions achieve the minimum average age. In Section \[sec:lcfs\_age\_delay\_tradeoff\], we then use these results to prove that there is a strong age-delay and age-delay variance tradeoff. In Section \[sec:lcfs\_age\_delay\_disparity\] point out a distinct age-delay disparity when the update generation is Poisson.
Minimizing Age {#sec:lcfs_min_age}
--------------
We first derive explicit expression average age for general inter-generation and service time distributions. We assume at least one of the distributions $F_{X}$ and $F_{S}$ to be continuous.
\[lem:LCFS\_gg1\] The average age $A^{\text{ave}}_{\text{G/G/1}}(F_S)$ is given by $$\nonumber
A^{\text{ave}}_{\text{G/G/1}}(F_S) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + \frac{{\mathbb{E}\left[{\min\left(X, S\right) }\right]}}{{\mathbf{P}\left[ S < X \right]}},$$ where $X$ and $S$ denotes the independent inter-generation and service time distributed random variables, respectively.
![Age $A(t)$ evolution in time $t$ for the LCFS queue with preemptive service.[]{data-label="fig:lcfs"}](lcfs){width="\linewidth"}
Let $X_i$ denote the inter-generation time between the $i$th and $(i+1)$th update packet. Due to preemption, not all packets get serviced on time to contribute to age reduction. We illustrate this in Figure \[fig:lcfs\]. Observe that packets $2$ and $3$ arrive before packet $4$. However, packet $2$ is preempted by packet $3$, which is subsequently preempted by packet $4$. Thus, packet $4$ is serviced before $2$ and $3$. Service of packet $2$ and $3$ (not shown in figure) does not contribute to age curve $A(t)$ because they contain stale information.
In order to analyze this, define $S_{i}$ to be the virtual service time for packet $i$, such that $\{ S_i \}_{i \geq 1}$ are i.i.d., and distributed according to the service time distribution $F_{S}$. If $S_{i} < X_{i}$, then the packet $i$ is serviced, and the age $A(t)$ drops to $S_{i}$, which is the time since generation of the packet $i$. In Figure \[fig:lcfs\], we observe this for packets $1$, $4$ and $5$. However, if $S_{i} > X_{i}$, the service of packet $i$ is preempted, and the server starts serving the newly arrived packet $(i+1)$. In Figure \[fig:lcfs\], observe that $S_2 > X_2$ and $S_3 > X_3$, while $S_4 < X_4$, and thus, packet $4$ gets serviced before $2$ and $3$.
For computing the average age, which is nothing but the time averaged area under the age curve $A(t)$, we compute the sum $\sum_{i=1}^{M} R_i$, where $R_i$ is the area under $A(t)$ between the $i$th and $(i+1)$th generation of update packets; see Figure \[fig:lcfs\]. To do so, we obtain a recursion for $B_i$, the age $A(t)$ at the time of generation of the $i$th update packet: define $Z_i \triangleq \sum_{k=0}^{i-1} X_k$ and $B_i = A(Z_i)$, and show that $$R_i = \left\{ \begin{array}{cc}
B_i X_i + \frac{1}{2}X_{i}^{2} & \text{if}~X_i < S_i \\
B_i S_i + \frac{1}{2}X_{i}^{2} & \text{if}~X_i \geq S_i
\end{array}\right..$$ The detailed proof is given in Appendix \[pf:lem:LCFS\_gg1\].
![Plotted is the average age under deterministic, exponential, and Pareto ($\alpha = 1.5, 1.1, 1.01,$ and $1.001$) distributed service times distributions for the LCFS queue with preemptive service. Service rate $\mu = 1$, while the packet generation rate $\lambda$ varies from $0.5$ to $0.99$.[]{data-label="fig:LCFS_MG1_ave_Par"}](LCFS_MG1_ave_Par){width="\linewidth"}
We now prove that a heavy tailed continuous service time distribution minimizes the average age. In Figure \[fig:LCFS\_MG1\_ave\_Par\], we plot average age as a function of packet generation rates $\lambda$, for three different service time distributions: deterministic service, exponential service, and Pareto service. The cumulative distribution function for a Pareto service distribution, with mean $1/\mu$, is given by $$\label{eq:Par_Dist}
F_{S}(s) = \left\{ \begin{array}{cc}
1 - \left( \frac{\theta(\alpha)}{s}\right)^{\alpha} &~\text{if}~s \geq \theta(\alpha) \\
0 &~\text{otherwise}
\end{array}\right.,$$ where $\theta(\alpha) = \frac{1}{\mu}\left( 1 - \frac{1}{\alpha}\right)$ and $\alpha > 1$ is the shape parameter. The shape parameter $\alpha$ determines the tail of the distribution. The closer the shape parameter is to $1$, the heavier is the tail.
We observe in Figure \[fig:LCFS\_MG1\_ave\_Par\] that the Pareto service yields better age than the exponential service. Furthermore, observe that the heavier the tail of the Pareto distribution, i.e. the closer $\alpha$ is to $1$, the lower is the age. Also plotted is the age lower-bound $1/\lambda$, as no matter what the service, the age cannot decrease below the inverse rate at which packets are generated.
We observe similar behavior not just for Pareto distributed service, but also for other heavy tailed distributions. In Figure \[fig:LCFS\_MG1\_ave\_LN\], we plot average age for log-normal service distribution, another heavy-tail distribution, with mean $1/\mu$ given by: $$\label{eq:log_normal}
S = \exp\left\{ -\log\mu - \frac{\sigma^2}{2} + \sigma N\right\},$$ where $N \sim \mathcal{N}(0,1)$ is the standard normal distribution and $\sigma$ is a parameter that determines the tail of the distribution $F_S$. Higher $\sigma$ implies heavier tail, and in Figure \[fig:LCFS\_MG1\_ave\_LN\] we observe that it results in smaller age, that approaches the age lower-bound of $1/\lambda$ as $\sigma \rightarrow +\infty$. We observe similar behavior for Weibull distributed service, with mean $1/\mu$: $$\label{eq:Weibull}
F_{S}(s) = 1 - e^{-\left( s /\beta \right)^{\kappa}},$$ for all $s \geq 0$, where $\beta = \left[ \mu \Gamma(1 + 1/\kappa) \right]^{-1}$, as $\kappa \downarrow 0$; here $\Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt$ is the gamma function.
![Plotted is the average age under deterministic, exponential, and log-normal ($\sigma = 1, 2, 4,$ and $50$) distributed service times distributions for the LCFS queue with preemptive service. Service rate $\mu = 1$, while the packet generation rate $\lambda$ varies from $0.5$ to $0.99$.[]{data-label="fig:LCFS_MG1_ave_LN"}](LCFS_MG1_ave_LN){width="\linewidth"}
We now show that the minimum average age is given by $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$. We first prove that the average age is lower-bounded by $\frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ for any service time distribution, and then show that this lower-bound is in fact achievable for the three heavy tailed service time distributions.
\[thm:LCFS\_GG1\_heavy\_tail\_opt\] The average age is lower bounded by $$\nonumber
A^{\text{ave}}_{\text{G/G/1}}(F_S) \geq \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}.$$ Further, this lower-bound is achieved for
1. Pareto distributed service as $\alpha \rightarrow 1$,
2. Log-normal distributed service as $\sigma \rightarrow +\infty$, and
3. Weibull distributed service as $\kappa \rightarrow 0$.
The lower-bound on the average age follows directly from the age expressions obtained in Lemma \[lem:LCFS\_gg1\], and noticing that $\frac{{\mathbb{E}\left[{\min\{ X, S \}}\right]}}{{\mathbf{P}\left[ S < X \right]}} \geq 0$. The distributions, namely the Pareto, log-normal, and Weibull, are all parametric distributions parameterized here by $\alpha$, $\sigma$, and $\kappa$, respectively. We, therefore, prove the following generic result, which gives us a sufficient conditions for the optimality of the average age for a general, parametric continuous service time distribution $F_{S}$, parameterized by $\eta$. We hide the dependence of the parameter $\eta$ on $S$ and $F_S$ for notational convenience.
\[lem:suff\_cond\_lcfs\] Let a parametric, continuous service time $S$, with parameter $\eta$, satisfy
1. ${\mathbb{E}\left[{S}\right]} = 1/\mu$ for all $\eta$,
2. ${\mathbb{E}\left[{\mathbb{I}_{\{ S > x\}}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$, and
3. ${\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x\}}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$,
for all $x > 0$ and for some $\eta^{\ast}$. Then, $$\lim_{\eta \rightarrow \eta^{\ast}} A^{\text{ave}}_{\text{G/G/1}}(F_{S}) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}.$$
Let for a parametric, continuous service time distribution $F_{S}$ the stated properties hold. Note that $$\nonumber
{\mathbb{E}\left[{\min\{X, S\}}\right]} = {\mathbb{E}\left[{X\mathbb{I}_{\{ S \geq X\}}}\right]} + {\mathbb{E}\left[{S\mathbb{I}_{\{S < X \}}}\right]},$$ Using conditions 2 and 3 in the Lemma, and the bounded convergence theorem [@Durrett], we have ${\mathbb{E}\left[{X\mathbb{I}_{\{ S \geq X\}}}\right]} \rightarrow 0$, ${\mathbb{E}\left[{S \mathbb{I}_{\{S < X \}}}\right]} \rightarrow 0$, and ${\mathbf{P}\left[ S < X \right]} \rightarrow 1$ as $\eta \rightarrow \eta^{\ast}$. Substituting all this in the average age expression in Lemma \[lem:LCFS\_gg1\], we obtain $A^{\text{ave}}_{\text{G/G/1}}(F_{S}) \rightarrow \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ as $\eta \rightarrow \eta^{\ast}$.
It, therefore, suffices to prove that the sufficient conditions in Lemma \[lem:suff\_cond\_lcfs\] are satisfied by the Pareto, log-normal, and Weibull distributions. We know, by definition, that all these distributions are continuous and have mean ${\mathbb{E}\left[{S}\right]} = 1/\mu$ for all parameter values. The other conditions are verified in Appendix \[pf:heavy\_tail\].
We showed that the minimum average age $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ is achieved under the three heavy tailed service time distributions. Such heavy tailed services’ induce maximum variation in the service times, and thus, will yield a worse delay and delay variance. However, it is not clear whether these are the only distributions that can achieve minimum age. Perhaps, we may be able to find a distribution $F_S$, that minimizes age and well as delay and delay variance. In the next section, we prove that this is not so, and that there is a strong age-delay and age-delay variance tradeoff.
Age-Delay Tradeoff {#sec:lcfs_age_delay_tradeoff}
------------------
We now prove that there exists a strong age-delay and age-delay variance tradeoff for the single server system, when the queue scheduling is fixed to LCFSp.
For a single server system under LCFSp scheduling policy, the following statements are true:
1. When the update generation is Poisson, there is a strong age-delay tradeoff.
2. When the update generation is a general renewal process, there is a strong age-delay variance tradeoff.
Let $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ denote the minimum average age. We have to show that as ${\text{AoI}}\rightarrow A_{\min}$ in , $T({\text{AoI}}) \rightarrow +\infty$, when the updates are generated according to a Poisson process. We also have to show that as ${\text{AoI}}\rightarrow A_{\min}$ in , $V({\text{AoI}}) \rightarrow +\infty$ for general update generation process.
When the update generation is Poisson, the queue is a M/G/1 LCFSp queue. The packet delay for this queue is given by [@data_nets]: $$D(F_S) = \frac{\lambda}{2}\frac{{\mathbb{E}\left[{S^2}\right]}}{1-\rho} + {\mathbb{E}\left[{S}\right]}.$$ Furthermore, the delay variance is lower-bounded by the variance in service time, namely, $\text{VarD}(F_S) \geq {\mathbb{E}\left[{S^2}\right]} - {\mathbb{E}\left[{S}\right]}^2$. Therefore, it suffices to show that as ${\text{AoI}}\rightarrow A_{\min}$ in and we have ${\mathbb{E}\left[{S^2}\right]} \rightarrow +\infty$.
In the following, we prove the strong age-delay tradeoff. The arguments are exactly the same for establishing the strong age-delay variance tradeoff, as we only have to show that ${\mathbb{E}\left[{S^2}\right]} \rightarrow +\infty$.
To establish the strong age-delay tradeoff, we use the expressions for average age derived in Lemma \[lem:LCFS\_gg1\]. Let $S_{{\text{AoI}}}$ denote the service time, and $F_{S_{{\text{AoI}}}}$ the corresponding service time distribution, that solves . Now, as ${\text{AoI}}\rightarrow A_{\min}$ in we must have $$\begin{gathered}
A^{\text{ave}}_{\text{G/G/1}}(F_{S_{{\text{AoI}}}}) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + \frac{{\mathbb{E}\left[{\min\left(X, S_{{\text{AoI}}}\right) }\right]}}{{\mathbf{P}\left[ S_{{\text{AoI}}} < X \right]}} \\
\rightarrow \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} = A_{\min},\end{gathered}$$ which implies $$\label{eq:nuance1}
\lim_{{\text{AoI}}\rightarrow A_{\min}} \frac{{\mathbb{E}\left[{\min\left(X, S_{{\text{AoI}}}\right) }\right]}}{{\mathbf{P}\left[ S_{{\text{AoI}}} < X \right]}} = 0.$$ Now, notice that ${\mathbf{P}\left[ S_{{\text{AoI}}} < X \right]}$, being probability, is bounded by $1$. Therefore, for to hold, it must be the case that $$\label{eq:nuance2}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{\min\left(X, S_{{\text{AoI}}}\right)}\right]} = 0.$$
Substituting the fact ${\mathbb{E}\left[{\min\left(X, S_{{\text{AoI}}}\right)}\right]} = {\mathbb{E}\left[{X\mathbb{I}_{\{X < S_{{\text{AoI}}}\}}}\right]} + {\mathbb{E}\left[{S_{{\text{AoI}}} \mathbb{I}_{\{ X \geq S_{{\text{AoI}}}\}}}\right]}$ in we get $$\label{eq:nuance3}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{X\mathbb{I}_{\{X < S_{{\text{AoI}}}\}}}\right]} = 0,$$ and $$\label{eq:nuance3b}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{S_{{\text{AoI}}} \mathbb{I}_{\{ X \geq S_{{\text{AoI}}}\}}}\right]} = 0.$$ Now, and implies that there exists a $x_0 > 0$ such that $$\label{eq:nuance4}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{\mathbb{I}_{\{x_0 < S_{{\text{AoI}}}\}}}\right]} = 0,$$ and $$\label{eq:nuance4b}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{S_{{\text{AoI}}} \mathbb{I}_{\{ x_0 \geq S_{{\text{AoI}}}\}}}\right]} = 0.$$ This can be established by a short proof by contradiction. Using Lemma \[lem:exists\_to\_all\] in Appendix \[app:serv\_dist\_prop\], along with and , we obtain $$\label{eq:nuance5}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{\mathbb{I}_{\{x < S_{{\text{AoI}}}\}}}\right]} = 0,$$ and $$\label{eq:nuance5b}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{S_{{\text{AoI}}} \mathbb{I}_{\{ x \geq S_{{\text{AoI}}}\}}}\right]} = 0,$$ for all $x \geq x_0$. Lemma \[lem:C\_implies\_S2\] in Appendix \[app:serv\_dist\_prop\] shows that these two conditions in and imply $$\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{S_{{\text{AoI}}}^2}\right]} = +\infty,$$ which proves our result.
In the next subsection, we bring out an even stronger contrast between age and delay, than the strong age-delay tradeoff. We show that the the delay minimizing service time distribution results in the worst case average age, and that the average age minimizing service time distribution results in the worst case delay.
Age-Delay Disparity under Poisson Update Generation {#sec:lcfs_age_delay_disparity}
---------------------------------------------------
To bring out the contrast between packet delay and AoI metrics, we consider the special case in which the update packets are generated according to a Poisson process. In this case, the single server system is nothing but a M/G/1 LCFSp queue. We now show that deterministic service yields the worst average age, across all service time distributions. We use the notation $A^{\text{ave}}_{G/G/1}$ for average age for the G/G/1 LCFSp queue, and omit the dependence on service time distribution $F_S$ for convenience.
\[thm:opt\_LCFS\_mg1\] For a single server system under LCFSp scheduling policy and Poisson update generation, the deterministic service yields the worst case average age: $$\begin{aligned}
A^{\text{ave}}_{\text{M/G/1}} &\leq A^{\text{ave}}_{\text{M/D/1}}. \nonumber\end{aligned}$$
See Appendix \[pf:thm:opt\_LCFS\_mg1\].
It should be intuitive that if the packets in service are often preempted, then very few packets complete service on time, and this results in a very high AoI. It turns out that deterministic service maximizes the probability of preemption. For the LCFS M/G/1 queue, the probability of preemption is given by ${\mathbf{P}\left[ S > X \right]} = 1 - {\mathbb{E}\left[{e^{-\lambda S}}\right]}$, as $X$ is exponentially distributed with rate $\lambda$. This can be upper-bounded by $1 - e^{-\lambda{\mathbb{E}\left[{S}\right]}} = {\mathbf{P}\left[ {\mathbb{E}\left[{S}\right]} > X \right]}$, using Jensen’s inequality, which is nothing but the probability of preemption under deterministic service: $S = {\mathbb{E}\left[{S}\right]}$ almost surely.
Comparing age with packet delay for the LCFSp queue results in a peculiar conclusion. The packet delay for a M/G/1 LCFSp queue is given by [@data_nets]: $$\nonumber
D = \frac{\lambda}{2}\frac{{\mathbb{E}\left[{S^2}\right]}}{1-\rho} + {\mathbb{E}\left[{S}\right]}.$$ Note that this expression of packet delay $D$ is minimized when the service time $S$ is deterministic, namely $S = {\mathbb{E}\left[{S}\right]}$ almost surely; follows from Jensen’s inequality ${\mathbb{E}\left[{S^2}\right]} \geq {\mathbb{E}\left[{S}\right]}^2$. However, from Theorem \[thm:opt\_LCFS\_mg1\] we know that deterministic service time maximizes age. This leads to the conclusion that, for the M/G/1 LCFSp queue, *the service time distribution that minimizes delay, maximizes age of information*.
Infinite Servers {#sec:inf_serv}
================
In this section, we consider the case when $M = \infty$, i.e. there are infinite servers in the system. The queue scheduling policy $\pi_Q$ is also fixed, and it assigns a new server to every arriving update packet. We call this the *work conserving* scheduling policy. The infinite server system, under the work conserving policy, is nothing but the G/G/$\infty$ queue. With the scheduling policy $\pi_Q$ fixed, the age and delay metrics are just a function of the service time distribution $F_S$.
Note that under the above scheduling policy, the packet delay incurred equals the service time, and thus, $D = {\mathbb{E}\left[{S}\right]} = 1/\mu$. This implies that the minimum age and minimum delay, which is $1/\mu$, can be achieved simultaneously, and the service time distribution that achieves this can be obtained by solving .
The age-delay variance tradeoff problem, on the other hand, is not so trivial. This can be written as $$\begin{aligned}
\label{eq:Inf_Aave_DelayVar_Tradeoff}
\begin{aligned}
V({\text{AoI}}) &= \underset{F_S}{\text{Minimize}}
& & {\text{VarD}}(F_S) \\
& \text{subject to} & & A^{\text{ave}}_{\text{G/G/}\infty}(F_S) \leq {\text{AoI}}, \\
& & & {\mathbb{E}\left[{S}\right]} = 1/\mu,
\end{aligned}\end{aligned}$$ where the notation $A^{\text{ave}}_{\text{G/G/}\infty}(F_S)$ denotes the average age for the G/G/$\infty$ queue. The optimization is only over the service time distribution. We omit the dependence on $F_S$, whenever convenient.
The rest of this section is organized as follows. In Section \[sec:inf\_min\_age\], we derive an expressions for the average age, and characterize its minimum. We also show that heavy tailed service time distributions achieve the minimum average age. In Section \[sec:inf\_age\_delay\_tradeoff\], we use these results to prove that there is a strong age-delay variance tradeoff. In Section \[sec:inf\_age\_delay\_disparity\], we obtain an average age maximizing service time distribution, and point to the disparity between the average age metric and delay variance.
Minimizing Age {#sec:inf_min_age}
--------------
We first derive an expression for average age for the system.
\[lem:gginf\] For the G/G/$\infty$ queue, the average is given by $$A^{\text{ave}}_{\text{G/G/}\infty}(F_S) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + {\mathbb{E}\left[{\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} }\right]}, \nonumber$$ where $X$ and $\{ X_{k} \}_{k \geq 1}$ are i.i.d. distributed according to $F_X$, while $\{ S_{k} \}_{k \geq 1}$ are i.i.d. distributed according to $F_{S}$.
For the G/G/$\infty$ queue, each arriving packet is serviced by a different server. As a result, the packets may get serviced in an out of order fashion. Figure \[fig:gginf\], which plots age evolution for the G/G/$\infty$ queue, illustrates this. In Figure \[fig:gginf\], observe that packet $3$ completes service before packet $2$. As a result, the age doesn’t drop at the service of packet $3$, as it now contains stale information. To analyze average age, it is important to characterize these events of out of order service.
![Age $A(t)$ evolution over time $t$ for G/G/$\infty$ queue.[]{data-label="fig:gginf"}](gginf){width="\linewidth"}
Let $X_i$ denote the inter-generation time between the $i$th and $(i+1)$th packet, and $S_i$ denote the service time for the $i$th packet. In Figure \[fig:gginf\], $X_2 + S_3 < S_2$, and therefore, packet $3$ completes service before packet $2$. To completely characterize this, define $Z_i \triangleq \sum_{k=0}^{i-1} X_k$ to be the time of generation of the $i$th packet. Note that the $i$th packet gets serviced at time $Z_i + S_i$, the $(i+1)$th packet gets services at time $Z_i + X_i + S_{i+1}$, and similarly, the $(i+l)$th packet gets serviced at time $Z_i + \sum_{k=1}^{l}X_{i+k-1} + S_{i+l}$, for all $l \geq 1$. Let $D_i$ denote the time from the $i$th packet generation to the time there is a service of the $i$th packet, or a packet that arrived after the $i$th packet, whichever comes first. Thus, $$\begin{aligned}
D_i &= \min\{S_i, X_{i} + S_{i+1}, X_{i} + X_{i+1} + S_{i+2}, \ldots \} \nonumber \\
&= \min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{i+k-1} + S_{i+l}\right\}.\end{aligned}$$ In Figure \[fig:gginf\], note that $D_1 = S_1$, $D_2 = X_2 + S_3$, $D_3 = S_3$, and $D_4 = S_4$.
The area under the age curve $A(t)$ is nothing but the sum of the areas of the trapezoids $Q_i$ (see Figure \[fig:gginf\]). Applying the renewal reward theorem [@wolff], by letting the reward for the $i$th renewal, namely $[Z_i, Z_i + X_i)$, be the area $Q_i$, we get the average age to be: $$\label{eq:z0}
A^{\text{ave}}_{\text{G/G/}\infty}(F_S) = \frac{{\mathbb{E}\left[{Q_i}\right]}}{{\mathbb{E}\left[{X_i}\right]}}.$$ It is easy to see that $$Q_i = \frac{1}{2}(X_i + D_{i+1})^2 - \frac{1}{2}D^{2}_{i+1}, \label{eq:z1}$$ as the trapezoid $Q_i$ extends from the time of the $i$th packet generation to the time at which the $(i+1)$th, or a packet that arrives after the $(i+1)$th packet, is served; which is nothing but $X_{i} + D_{i+1}$. For illustration, note that $Q_1 = \frac{1}{2}(X_1 + X_2 + S_3)^2 - \frac{1}{2}(X_2 + S_3)^2$, which is same as , for $i = 1$, since $D_2 = X_2 + S_3$. Substituting in , we obtain $$A^{\text{ave}}_{\text{G/G/}\infty}(F_S) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + \frac{{\mathbb{E}\left[{X_i D_{i+1}}\right]}}{{\mathbb{E}\left[{X_i}\right]}}.$$ We obtain the result by noting that $X_i$ and $D_{i+1}$ are independent.
In Figure \[fig:InfServ\_ave\_Par\], we plot the average age for the M/G/$\infty$ queue under three service distributions: deterministic, exponential, and Pareto distribution (given in ), with mean $1/\mu$. We observe that the heavy tail Pareto distributed service performs better than the exponential service. Also, heavier tail or decreasing $\alpha$ results in improvement in age. It appears, like in the LCFSp queue, that as $\alpha \downarrow 1$ the average age approaches the lower bound $1/\lambda$. Similar observations are made for the log-normal distributed service and Weibull distributed service , we $\sigma \rightarrow +\infty$ and $\kappa \rightarrow 0$, respectively.
![Plotted is the average age under deterministic, exponential, and Pareto ($\alpha = 1.5, 1.1, 1.01,$ and $1.001$) distributed service times distributions for the infinite server M/G/$\infty$ queue. Service rate $\mu = 1$, while the packet generation rate $\lambda$ varies from $0.5$ to $0.99$.[]{data-label="fig:InfServ_ave_Par"}](InfServ_ave_Par){width="\linewidth"}
We now prove a simple lower bound on the average age, and show that the average age converges to this lower bound for the three heavy tailed service time distribution.
\[thm:opt2\_gginf\] For the infinite server G/G/$\infty$ queue, the average age is lower-bounded by $$\nonumber
A^{\text{ave}}_{\text{G/G/}\infty}(F_S) \geq \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}.$$ Further, the lower-bound is achieved for
1. Pareto distributed service as $\alpha \rightarrow 1$,
2. Log-normal distributed service as $\sigma \rightarrow +\infty$, and
3. Weibull distributed service as $\kappa \rightarrow 0$.
The lower-bound immediately follows from the average age expression in Lemma \[lem:gginf\]. We use a similar approach to that followed in the proof of Theorem \[thm:LCFS\_GG1\_heavy\_tail\_opt\]. We show that the same sufficient conditions as in Lemma \[lem:suff\_cond\_lcfs\] suffices for the average age optimality for the G/G/$\infty$ queue.
\[lem:suff\_cond\_inf\_serv\] Let a parametric, continuous, service time distribution, with parameter $\eta$, satisfy
1. ${\mathbb{E}\left[{S}\right]} = 1/\mu$,
2. ${\mathbb{E}\left[{\mathbb{I}_{\{ S > x\}}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$, and
3. ${\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x\}}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$,
for some $\eta^{\ast}$, and all $x > 0$. Then $$\lim_{\eta \rightarrow \eta^{\ast}} A^{\text{ave}}_{\text{G/G/}\infty}(F_S) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}.$$
It suffices to argue that when the above conditions hold for a parametric random variable $S$, we have $$\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} }\right]} = 0.$$ This is proved in Lemma \[lem:gginf\_iff\_C\] in Appendix \[app:serv\_dist\_prop\].
It, now, suffices to argue that the three heavy tailed service time distributions satisfy the conditions in Lemma \[lem:suff\_cond\_inf\_serv\]. All the three heavy tailed distributions are continuous, and have mean ${\mathbb{E}\left[{S}\right]} = 1/\mu$, by definition. The other two properties are verified in Appendix \[pf:heavy\_tail\].
Thus, the minimum age can be achieved by the three heavy tailed service time distribution. For these three distributions, the second moment approaches infinity, as their tails get heavier; namely as $\alpha \rightarrow 1$, $\sigma \rightarrow +\infty$, and $\kappa \rightarrow 0$. This implies that the delay variance, which is lower-bounded by the variance in service time, also approaches infinity. However, it is not known that there is no other distribution that can simultaneously minimize average age and delay variance. In the next sub-section, we prove just that, and show that there is a strong age-delay variance tradeoff.
Age-Delay Tradeoff {#sec:inf_age_delay_tradeoff}
------------------
We now prove that there is a strong age-delay variance tradeoff.
For the infinite server system, under the work conserving scheduling policy, there is strong age-delay variance tradeoff.
Let $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ denote the minimum average age. We have to show that as ${\text{AoI}}\rightarrow A_{\min}$ in , $V({\text{AoI}}) \rightarrow +\infty$. Note that the delay variance is lower-bounded by the variance in service time, namely, $\text{VarD}(F_S) \geq {\mathbb{E}\left[{S^2}\right]} - {\mathbb{E}\left[{S}\right]}^2$. Therefore, it suffices to show that as ${\text{AoI}}\rightarrow A_{\min}$ in we have ${\mathbb{E}\left[{S^2}\right]} \rightarrow +\infty$.
To establish this, we use the average age expression derived in Lemma \[lem:gginf\]. Let $S_{{\text{AoI}}}$ denote the service time, and $F_{S_{{\text{AoI}}}}$ the corresponding service time distribution, that solves . Now, as ${\text{AoI}}\rightarrow A_{\min}$ in we must have $$\begin{gathered}
A^{\text{ave}}_{\text{G/G/}\infty}(F_{S_{{\text{AoI}}}}) = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + {\mathbb{E}\left[{\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} }\right]} \\
\rightarrow \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} = A_{\min},\end{gathered}$$ where $S_l$ and $X_k$ are independent and distributed according to $F_{S_{{\text{AoI}}}}$ and $F_X$, respectively. This implies $$\label{eq:xnuance1}
\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} }\right]} = 0.$$ From Lemma \[lem:gginf\_iff\_C\] and Lemma \[lem:C\_implies\_S2\], in Appendix \[app:serv\_dist\_prop\], implies that $$\lim_{{\text{AoI}}\rightarrow A_{\min}} {\mathbb{E}\left[{S^{2}_{{\text{AoI}}}}\right]} = +\infty,$$ which proves the result.
In the next sub-section, we prove an even stronger disparity between average age and delay. We show that the service time distribution that minimizes delay variance, i.e. deterministic service, yields the worst case age.
Age-Delay Disparity {#sec:inf_age_delay_disparity}
-------------------
We first prove that deterministic service yields the worst average age, across all service time distributions.
\[thm:opt\_gginf\] For the infinite server G/G/$\infty$ queue, $$\nonumber
A^{\text{ave}}_{\text{G/G/}\infty}(\lambda, \mu) \leq A^{\text{ave}}_{\text{G/D/}\infty}(\lambda, \mu),$$ for all packet generation and service rates, $\lambda$ and $\mu$, respectively.
See Appendix \[pf:thm:opt\_gginf\].
The intuition is as follows: In the G/G/$\infty$ queue, packets do not get serviced in the same order as they are generated. However, a swap in order helps improve age, because it means that a packet that arrived later was served earlier. Therefore, the service that swaps the packet order the least maximizes age. Under deterministic service, the packet order is retained exactly, with probability $1$, and therefore, yields the maximizes age.
Notice that, for the G/G/$\infty$ queue, packet delay equals the service time, and therefore, deterministic service minimizes delay variance. This observation, along with Theorem \[thm:opt\_gginf\], imply that for the G/G/$\infty$ queue, the *service time distribution that reduces packet delay variance, maximizes average age of information*.
The next section considers the general $M$ server system, and proves the age-delay tradeoff result of Theorem \[thm:M\_tradeoff\].
$M$ Server System {#sec:M_ServSyst}
=================
In this section, we consider the $M$ server system and prove Theorem \[thm:M\_tradeoff\], which asserts a strong tradeoff between age-delay and age-delay variance. Recall that $A^{\text{ave}}(F_S, \pi_Q)$, $D(F_S, \pi_Q)$, and $\text{VarD}(F_S, \pi_Q)$ denote the average age, delay, and delay variance, respectively, under the scheduling policy $\pi_Q$ and the service time distribution $F_S$.
We first derive the minimum average age $A_{\min}$, over the space of all scheduling policies $\pi_Q$ and service time distributions $F_S$.
\[lem:AoI\_min\] The minimum average age $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$.
The fact that $\frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ is a lower-bound on the average age, can be proved by pretending that each update packet spends zero time in the system, i.e. $t_i = t^{'}_i$. This provides a sample path lower bound for the age process. In this sample path, the age drops to $0$ at every $t_i$, and increases to $t_{i+1} - t_{i}$, just before dropping to $0$ again at $t_{i+1}$. The average age of this artificially constructed age process is $\frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$, and since it is a sample path wise lower-bound, we have $A^{\text{ave}}(F_S, \pi_Q) \geq \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$. This lower-bound $\frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ is independent of the scheduling policy $\pi_Q$ and the service time distribution $F_S$. Therefore, we have $A_{\min} \geq \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$.
In Section \[sec:lcfs\], we showed that this lower-bound can be achieved by a single server system, i.e. $M = 1$, under the LCFSp scheduling policy with heavy tailed service. Therefore, choosing to route update packets only through a single sever, scheduling packets in that server with LCFSp scheduling policy with heavy tailed service, we can achieve this lower bound average age. Thus, $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$.
We now prove the strong age-delay tradeoff and age-delay variance tradeoff.
\[thm:strong\_age\_delay\] For the $M$ server system, the following statements are true:
1. When the update generation is Poisson, there is a strong age-delay tradeoff.
2. When the update generation is a general renewal process, there is a strong age-delay variance tradeoff.
Let $A_{\min} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}}$ denote the minimum average age. Note that the variance in packet delay is lower-bounded by the variance in service time, under any scheduling policy $\pi_Q$. Therefore, $$\label{eq:new_lb1}
\text{VarD}(F_S, \pi_Q) \geq {\mathbb{E}\left[{S^2}\right]} - {\mathbb{E}\left[{S}\right]}^2.$$
It is known that the minimum delay can be attained by any work conserving scheduling policy [@2016_ISIT_YinSun_AoI_Thput_Delay_LCFS; @2016_ISIT_TIT_YinSun_Thput_Delay_LCFS]. We define the following work conserving scheduling policy $\pi_{Q}^{\ast}$:
1. Generated updates are queued in a single FCFS queue.
2. Whenever a server is free, an update packet at the head of the FCFS queue, is assigned to that server.
This scheduling policy, begin work conserving, attains minimum average delay for a given service time distribution, i.e. $$\label{eq:new_lb2_pre}
D(F_S, \pi_Q) \geq D(F_S, \pi_Q^{\ast}),$$ for all scheduling policies $\pi_Q$. The $M$ server system under Poisson update generation and scheduling policy $\pi_Q^{\ast}$ is nothing but the M/G/k queue. For the M/G/k queue, the average delay, namely $D(F_S, \pi_Q^{\ast})$, is lower-bounded by a constant times the variance in service time [@2010QS_gupta_MGk_Queue]: $$\label{eq:new_lb2}
D(F_S, \pi_Q^{\ast}) \geq c \left( {\mathbb{E}\left[{S^2}\right]} - {\mathbb{E}\left[{S}\right]}^2\right),$$ where in [@2010QS_gupta_MGk_Queue] the constant relates to the delay in M/M/k queue. From and , we have $$\label{eq:new_lb3}
D(F_S, \pi_Q) \geq c \left( {\mathbb{E}\left[{S^2}\right]} - {\mathbb{E}\left[{S}\right]}^2\right),$$ for any scheduling policy $\pi_Q$.
From and , it is clear, that in order to prove a strong age-delay tradeoff, for Poisson update generation, or to prove the strong age-delay variance tradeoff, for general update generation, it suffices to argue that ${\mathbb{E}\left[{S^2}\right]} \rightarrow +\infty$. In the rest of the proof, we prove just this.
**1. Age-delay tradeoff:** We consider Poisson update generation. Let $S_{{\text{AoI}}}$ and $F_{S_{{\text{AoI}}}}$ denote the service time random variable and its distribution, respectively, that solves . We argue that as ${\text{AoI}}\rightarrow A_{\min}$ in we must have ${\mathbb{E}\left[{S_{{\text{AoI}}}^2}\right]} \rightarrow +\infty$.
We first note that the average age $A^{\text{ave}}(F_S, \pi_Q)$, under any queue scheduling policy $\pi_Q$, is lower-bounded by the average age for the G/G/$\infty$ queue: $$\label{eq:lb}
A^{\text{ave}}(F_S, \pi_Q) \geq A^{\text{ave}}_{\text{G/G/}\infty}(F_S).$$ This is because, in G/G/$\infty$ queue, an arriving packet is immediately put to service, and therefore, incurs no queueing delay. Due to this the average age for the G/G/$\infty$ queue serves as a lower-bound for any $M$ server queue, in a stochastic ordering sense. Taking expected value yields .
We know the average age for the G/G/$\infty$ queue to be: $$\label{eq:Aave_gginf}
A^{\text{ave}}_{\text{G/G/}\infty} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + {\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]},$$ where $S_l$ and $X_k$ are independent random variables with distributions $F_{S_{{\text{AoI}}}}$ and $F_X$, respectively. Notice that the first term in is nothing but $A_{\min}$. Therefore, as ${\text{AoI}}\rightarrow A_{\min}$ in , it must be the case that ${\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]} \rightarrow 0$. Lemmas \[lem:gginf\_iff\_C\] and \[lem:C\_implies\_S2\], in Appendix \[app:serv\_dist\_prop\], prove that ${\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]} \rightarrow 0$ implies ${\mathbb{E}\left[{S_{{\text{AoI}}}^2}\right]} \rightarrow +\infty$.
**2. Age-delay variance tradeoff:** Let $S_{{\text{AoI}}}$ and $F_{S_{{\text{AoI}}}}$ denote the service time random variable and its distribution that solves . We have to argue that as ${\text{AoI}}\rightarrow A_{\min}$ in we have ${\mathbb{E}\left[{S_{{\text{AoI}}}^2}\right]} \rightarrow +\infty$. We just proved this in establishing the age-delay tradeoff.
Conclusion {#sec:conclusion}
==========
We considered an $M$ server system in which each server serves at most one update packet at any given time. Updates are generated according to a renewal process and the system designer controls the scheduling discipline, routing, and the service time distribution. When the updates are generated according to a Poisson process, we show that there is a strong age-delay tradeoff, i.e. as the average age approaches its minimum the packet delay tends to infinity. However, for a general update generation process, we prove a strong age-delay variance tradeoff. The proof involves first establishing similar age-delay tradeoff results for two special cases of the $M$ server system, namely, the single server system with LCFSp service and the infinite server system. For the two cases, we also show that heavy tailed service time distributions asymptotically minimize average age, as their tail gets heavier.
Though seemingly counterintuitive, the age-delay tradeoff is natural and occurs due to the delays incurred by the non-informative packets. When the system attempts to minimize age, it minimizes waiting and service times for the informative packets. This results in very high waiting and service times for the non-informative packets, which dominate the packet delay, and causes it to increase unboundedly. We therefore expect similar age-delay tradeoffs to exist in other communication systems as well, and investigating them is an open question for future research.
Proof of Lemma \[lem:LCFS\_gg1\] {#pf:lem:LCFS_gg1}
--------------------------------
Let $A(t)$ denote the age at time $t$. Let $B_i$ denote the age at the generation of the $i$th update packet, i.e. $Z_i = \sum_{k=0}^{i-1}X_k$: $$B_{i} = A( Z_i ).$$ Then, we have the following recursion for $B_i$: $$B_{i+1} = \left\{ \begin{array}{cc}
X_i & \text{if}~S_i < X_i \\
B_i + X_i & \text{if}~S_i \geq X_i
\end{array}\right.,$$ for all $i \geq 0$. This can be written as $$B_{i+1} = X_i + B_{i}\left( 1 - \mathbb{I}_{S_i < X_i}\right).$$ Note that $B_i$ is independent of $S_i$ and $X_i$. Further, $\{ B_i \}_{i \geq 1}$ is a Markov process, and can be shown to be positive recurrent using the drift criteria [@meyn_markov_chains_stability]; using the fact that $X_i$ and $S_i$ are continuous random variables and ${\mathbf{P}\left[ S_i < X_i \right]} < 1$. Taking expected value, and noting that at stationarity ${\mathbb{E}\left[{B_i}\right]} = {\mathbb{E}\left[{B_{i+1}}\right]}$, we get $$\label{eq:B_exp}
{\mathbb{E}\left[{B}\right]} = \frac{{\mathbb{E}\left[{X}\right]}}{{\mathbf{P}\left[ S < X \right]}}.$$
We now compute the average age. Let $R_i$ denote the area under the age curve $A(t)$ between the generation of packet $i$ and packet $i+1$: $$R_i \triangleq \int_{Z_i}^{Z_i + X_i} A(t) dt,$$ where $Z_i = \sum_{k=0}^{i-1}X_k$ is the time of generation of the $i$th update packet. This $R_i$ can be computed explicitly to be $$R_i = \left\{ \begin{array}{cc}
B_i X_i + \frac{1}{2}X_{i}^{2} & \text{if}~X_i < S_i \\
B_i S_i + \frac{1}{2}X_{i}^{2} & \text{if}~X_i \geq S_i
\end{array}\right.,$$ which can be written compactly as $$\label{eq:Ri}
R_i = \frac{1}{2}X^{2}_{i} + B_{i}\min\left( X_i, S_i\right).$$ Since, $B_i$ is independent of $X_i$ and $S_i$, taking expected value at stationarity we obtain $$\label{eq:R_exp}
{\mathbb{E}\left[{R}\right]} = \frac{1}{2}{\mathbb{E}\left[{X^2}\right]} + {\mathbb{E}\left[{B}\right]}{\mathbb{E}\left[{\min\left( X, S\right)}\right]}.$$
Using renewal theory, the average age can be obtained to be $$\begin{aligned}
A^{\text{ave}}_{\text{G/G/1}} &= \frac{{\mathbb{E}\left[{R}\right]}}{{\mathbb{E}\left[{X}\right]}} , \\
&= \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + \frac{{\mathbb{E}\left[{B}\right]}}{{\mathbb{E}\left[{X}\right]}}{\mathbb{E}\left[{\min\left( X, S\right)}\right]}.\end{aligned}$$ Substituting we get the result.
Properties of the Heavy Tailed Distributions {#pf:heavy_tail}
--------------------------------------------
For any $x > 0$, we have ${\mathbf{P}\left[ S > x \right]} \rightarrow 0$ and ${\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x \}}}\right]} \rightarrow 0$ for:
1. Pareto distributed service $S$, as $\alpha \rightarrow 1$; see .
2. Log-normal distributed service $S$, as $\sigma \rightarrow +\infty$; see .
3. Weibull distributed service $S$, as $\kappa \rightarrow 0$; see .
**1. Pareto Service:** Choose a $x > 0$. Then there exists a $\overline{\alpha}_x > 1$ such that $\theta(\alpha) = \frac{1}{\mu}\frac{\alpha - 1}{\alpha} < x$ for all $\alpha < \overline{\alpha}_{x}$. For such any $\alpha < \overline{\alpha}_{x}$, we have ${\mathbf{P}\left[ S > x \right]} = \left( \frac{\theta(\alpha)}{x} \right)^{\alpha} \rightarrow 0$ as $\alpha \downarrow 1$, since $\theta(\alpha) \rightarrow 0$ as $\alpha \downarrow 1$.
For the second part, we first compute ${\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x\}}}\right]}$ for $\alpha < \overline{\alpha}_{x}$: $$\begin{aligned}
{\mathbb{E}\left[{S \mathbb{I}_{S \leq x}}\right]} &= \int_{\frac{1}{\mu}\left(1 - \frac{1}{\alpha}\right)}^{x} s f_{S}(s) ds = \frac{\alpha}{\mu^{\alpha}}\int_{\frac{1}{\mu}\left(1 - \frac{1}{\alpha}\right)}^{x} \frac{\left( 1 - \frac{1}{\alpha}\right)^{\alpha}}{s^{\alpha}} ds. \nonumber\end{aligned}$$ Substituting $y = \alpha s/(\alpha - 1)$, and solving the definite integral, we get $${\mathbb{E}\left[{S \mathbb{I}_{S \leq x}}\right]} = \frac{1}{\mu} - \frac{1}{\mu} \frac{(\alpha/\mu)^{\alpha - 1}}{(\alpha - 1)^{\alpha - 1}} x^{\alpha - 1}.$$ From the above expression, it can be deduced that ${\mathbb{E}\left[{S \mathbb{I}_{S \leq x}}\right]} \rightarrow 0$ as $\alpha \downarrow 1$.
**2. Log-normal Service:** Choose a $x > 0$. From notice that $$\nonumber
{\mathbf{P}\left[ S > x \right]} = {\mathbf{P}\left[ N > \frac{\log(x \mu)}{\sigma} + \frac{\sigma}{2} \right]} \rightarrow 0,$$ as $\sigma \rightarrow +\infty$.
For the second part, using the relation between the log-normal service time and normal random variable $N$, we can compute the expectation ${\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x\}}}\right]}$ to be $$\nonumber
{\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x\}} }\right]} = \frac{1}{\mu} - \frac{1}{\mu}\Phi\left( - \frac{\log(x \mu)}{\sigma} + \frac{\sigma}{2}\right),$$ where $\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}dt$. Taking the limit $\sigma \rightarrow +\infty$ we get $\Phi\left( - \frac{\log(x \mu)}{\sigma} + \frac{\sigma}{2}\right) \rightarrow 1$, and therefore, ${\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x\}} }\right]} \rightarrow 0$.
**3. Weibull Service:** Choose a $x > 0$. Using the distribution function , we can conclude ${\mathbf{P}\left[ S > x \right]} = e^{-(x\mu)^{\kappa}}e^{-\left[ \Gamma(1 + 1/\kappa)\right]^{\kappa}}$. Using Sterling’s formula, $\left[ \Gamma(1 + 1/\kappa)\right]^{\kappa} \geq 1/\kappa$, and therefore $\left[ \Gamma(1 + 1/\kappa)\right]^{\kappa} \rightarrow +\infty$ as $\kappa \rightarrow 0$. Therefore, we have ${\mathbf{P}\left[ S > x \right]} \rightarrow 0$ as $\kappa \rightarrow 0$.
For the second part, we can explicitly derive the conditional expectation ${\mathbb{E}\left[{S\mathbb{I}_{\{S \leq x\}}}\right]}$ using the distribution : $$\begin{aligned}
{\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x \}}}\right]} &= \int_{0}^{x} \frac{\kappa}{\beta} \left( \frac{t}{\beta}\right)^{\kappa - 1} e^{- (t/\beta)^{\kappa}} t dt, \nonumber \\
&= \frac{1}{\mu \Gamma(1 + 1/\kappa)} \int_{0}^{\left(x \mu \Gamma(1 + 1/\kappa) \right)^{\kappa}} y^{1/\kappa} e^{-y} dy, \label{eq:no_to}\end{aligned}$$ which is obtained by substituting $\beta = \left[ \mu \Gamma(1 + 1/\kappa)\right]^{-1}$ and changing variables $y = (t/\beta)^{\kappa}$. Using lower-bounds given by Sterling approximation on Gamma function, we can deduce that , and therefore ${\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x \}}}\right]}$, approaches $0$ as $\kappa \rightarrow 0$.
Proof of Theorem \[thm:opt\_LCFS\_mg1\] {#pf:thm:opt_LCFS_mg1}
---------------------------------------
We first show that the average age for Poisson update generation is given by $$\label{eq:a0}
A^{\text{ave}}_{\text{M/G/1}} = \frac{{\mathbb{E}\left[{S}\right]}}{{\mathbf{P}\left[ S < X \right]}}.$$ Let $A(t)$ be the age at time $t$, and $B_i$ be the age at the time of generation of the $i$th update packet $Z_i = \sum_{k=0}^{i-1}X_k$: $$B_{i} = A(Z_i).$$ Let $B$ denote the distribution of $B_i$ at stationarity. By PASTA property and ergodicity of the age process $A(t)$ we have $A^{\text{ave}}_{\text{M/G/1}} = {\mathbb{E}\left[{B}\right]}$, as update generation process is a Poisson process. Substituting the expression for ${\mathbb{E}\left[{B}\right]}$ in , from Appendix \[pf:lem:LCFS\_gg1\], we obtain .
Now, substituting $S = {\mathbb{E}\left[{S}\right]}$ almost surely we get the average age expression for the $M/D/1$ LCFSp queue to be $$\label{eq:a1}
A^{\text{ave}}_{\text{M/D/1}} = \frac{{\mathbb{E}\left[{S}\right]}}{{\mathbf{P}\left[ {\mathbb{E}\left[{S}\right]} < X \right]}} = \frac{{\mathbb{E}\left[{S}\right]}}{e^{-\lambda {\mathbb{E}\left[{S}\right]}}},$$ where we have used the fact that the packet inter-generation time $X$ is exponentially distributed. We obtain $A^{\text{ave}}_{\text{M/G/1}} \leq A^{\text{ave}}_{\text{M/D/1}}$ by noting that $${\mathbf{P}\left[ S < X \right]} = {\mathbb{E}\left[{e^{-\lambda S}}\right]} \geq e^{-\lambda {\mathbb{E}\left[{S}\right]}},$$ by Jensen’s inequality.
Proof of Theorem \[thm:opt\_gginf\] {#pf:thm:opt_gginf}
-----------------------------------
From Lemma \[lem:gginf\], it is clear that the average age depend on service time through the term: $${\mathbb{E}\left[{\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} }\right]}.$$ We show that this quantity is maximized when service times are deterministic, i.e. $S = {\mathbb{E}\left[{S}\right]}$ almost surely.
First, notice that $$\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} = S_{1},$$ if $S_{k}$ are all equal and deterministic. This is because $X_k \geq 0$ almost surely. Thus, the peak and average age for the G/D/$\infty$ queue is given by $$\label{eq:m0}
A^{\text{p}}_{\text{G/D/}\infty} = {\mathbb{E}\left[{X}\right]} + {\mathbb{E}\left[{S}\right]},$$ and $$\label{eq:m1}
A^{\text{ave}}_{\text{G/D/}\infty} = \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + {\mathbb{E}\left[{S}\right]}.$$ Furthermore, we must have $$\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} \leq S_{1},$$ since $S_{1}$ is the first term in the minimization. Therefore, $${\mathbb{E}\left[{\min_{l \geq 0}\left\{ \sum_{k=1}^{l}X_{k} + S_{l+1}\right\} }\right]} \leq {\mathbb{E}\left[{S_{1}}\right]} = {\mathbb{E}\left[{S}\right]}.$$ Applying this to the peak and average age expression from Lemma \[lem:gginf\], we get $$\label{eq:n0}
A^{\text{p}}_{\text{G/G/}\infty} \leq {\mathbb{E}\left[{X}\right]} + {\mathbb{E}\left[{S}\right]},$$ and $$\label{eq:n1}
A^{\text{ave}}_{\text{G/G/}\infty} \leq \frac{1}{2}\frac{{\mathbb{E}\left[{X^2}\right]}}{{\mathbb{E}\left[{X}\right]}} + {\mathbb{E}\left[{S}\right]}.$$ The result follows from , , , and .
Properties of Service Time Random Variable $S$ {#app:serv_dist_prop}
----------------------------------------------
Here, we derive several asymptotic properties of the service time distributions and their implications. These properties are used throughout the paper.
Let $S$ be a continuous random variable with distribution $F_S$, with parameter $\eta$, such that ${\mathbb{E}\left[{S}\right]} = 1/\mu$ for all $\eta$. For notational convenience, we hide the dependence of $S$ and $F_S$ on $\eta$. We are interested in the $S$ and $F_S$ as $\eta \rightarrow \eta^{\ast}$, for some specific $\eta^{\ast}$.
\[lem:exists\_to\_all\] If $\exists$ a $x_0 > 0$ such that ${\mathbf{P}\left[ S > x_0 \right]} \rightarrow 0$ and ${\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x_0 \}}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$ then $$\label{eq:S_asymp_prop}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbf{P}\left[ S > x \right]} = 0~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x \}}}\right]} = 0,$$ for all $x \geq x_0$.
Let there be a $x_0 > 0$ such that $$\label{eq:S_asymp_prop1}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbf{P}\left[ S > x_0 \right]} = 0~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S\mathbb{I}_{\{ S < x_0 \}}}\right]} = 0.$$ Take a $x > x_0$. Then, $\mathbb{I}_{\{ S > x \}} \leq \mathbb{I}_{\{ S > x_0 \}}$, and therefore, ${\mathbf{P}\left[ S > x \right]} \leq {\mathbf{P}\left[ S > x_0 \right]}$. This and implies $$\label{eq:S_asymp_prop2}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbf{P}\left[ S > x \right]} = 0.$$ For a $x > x_0$, we can re-write ${\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x \}}}\right]}$ as $$\begin{aligned}
{\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x \}}}\right]} &= {\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x_0 \}}}\right]} + {\mathbb{E}\left[{S\mathbb{I}_{\{ x_0 < S \leq x \}}}\right]}, \\
&\leq {\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x_0 \}}}\right]} + x {\mathbb{E}\left[{\mathbb{I}_{\{ x_0 < S \leq x \}}}\right]}, \\
&\leq {\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x_0 \}}}\right]} + x {\mathbf{P}\left[ S > x_0 \right]}. \label{eq:S_asymp_prop3}\end{aligned}$$ Using , which states that both the terms in tend to $0$ as $\eta \rightarrow \eta^{\ast}$, we get $$\label{eq:S_asymp_prop4}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S\mathbb{I}_{\{ S \leq x \}}}\right]} = 0.$$
Since and hold for any $x > x_0$, we have the result.
In the infinite server case, the average age expression in Lemma \[lem:gginf\] has a term $${\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]},$$ where $S_l$ and $X_k$ are independent, distributed according to $F_S$ and $F_X$, respectively. We would like to derive conditions on $S$ such that $$\nonumber
{\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]} \rightarrow 0,$$ as $\eta$ approaches certain $\eta^{\ast}$, for a given distribution $F_X$. The following result, derives an equivalent condition that only requires verifying certain properties of $F_S$.
\[lem:gginf\_iff\_C\] For $S_l$ and $X_k$ that are i.i.d. distributed according to $F_S$ and $F_X$, respectively, we have $$\label{eq:min_term}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]} = 0,$$ if and only if, for all $x > 0$, we have $$\label{eq:suff_cond}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbf{P}\left[ S > x \right]} = 0,~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x\}}}\right]} = 0.$$
**(a)** We first prove that implies . Let $Z = \min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)$. We first lower-bound $Z$ as follow: $$\nonumber
Z = \min\{ S_1, X_1 + S_2, X_1 + X_2 + S_3, \ldots \} = \min\{ S_1, X_1 + Z' \},$$ where $Z' = \min\{ S_2, X_2 + S_3, X_2 + X_3 + S_4, \ldots \}$. Since $Z' \geq 0$, we must have $Z \geq \min\{S_1, X_1\}$. Without loss of generality, we can loose the subscripts and write $Z \geq \min\{S, X\}$, where $S \sim F_S$ and $X \sim F_X$.
If ${\mathbb{E}\left[{Z}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$ then clearly ${\mathbb{E}\left[{\min\{S, X\}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$. Pick a $x_0 > 0$ such that ${\mathbf{P}\left[ X \geq x_0 \right]} > 0$. Note that such an $x_0 > 0$ always exists since ${\mathbb{E}\left[{X}\right]} = 1/\lambda > 0$. Now construct $\hat{X}$ such that: $$\nonumber
\hat{X} = \left\{ \begin{array}{cc}
0 &~\text{if}~X < x_0 \\
x_0 &~\text{if}~X \geq x_0
\end{array}\right..$$ Clearly, $\hat{X} \leq X$, and thus, $\min\{S, \hat{X}\} \leq \min\{S, X\}$, which implies ${\mathbb{E}\left[{\min\{S, \hat{X}\}}\right]} \rightarrow 0$. Since $\hat{X}$ takes only two values, namely $0$ and $x_0$, we have ${\mathbb{E}\left[{\min\{S, \hat{X}\}}\right]} = {\mathbb{E}\left[{\min\{S, x_0 \}}\right]}{\mathbf{P}\left[ X \geq x_0 \right]}$. Further, ${\mathbf{P}\left[ X \geq x_0 \right]}$ does not depend on $S$, and therefore, is independent of the parameter $\eta$. Therefore, ${\mathbb{E}\left[{\min\{S, \hat{X}\}}\right]} \rightarrow 0$ implies $$\label{eq:nuclear2}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{\min\{S, x_0\}}\right]} = 0.$$ Now, notice that $$\label{eq:nuclear3}
{\mathbb{E}\left[{\min\{S, x_0\}}\right]} = {\mathbb{E}\left[{S\mathbb{I}_{ \{ S \leq x_0 \} }}\right]} + x_0 {\mathbb{E}\left[{\mathbb{I}_{ \{ S > x_0\} }}\right]},$$ Substituting in we get $$\label{eq:nuclear4}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S\mathbb{I}_{ \{ S \leq x_0 \} }}\right]} = 0~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{\mathbb{I}_{ \{ S > x_0\} }}\right]} = 0.$$ Using Lemma \[lem:exists\_to\_all\], implies $$\label{eq:nuclear5}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S\mathbb{I}_{ \{ S \leq x \} }}\right]} = 0~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{\mathbb{I}_{ \{ S > x\} }}\right]} = 0,$$ for all $x \geq x_0$.
Now, we had chosen $x_0$ to be such that ${\mathbf{P}\left[ X \geq x_0 \right]} > 0$. Since ${\mathbb{E}\left[{X}\right]} = 1/\lambda > 0$, the choice of $x_0$ could be as small, and close to $0$, as possible. This and yield the result in .
**(b)** We now prove that implies . First, note that along with the bounded convergence theorem [@Durrett] imply $$\label{eq:suo1}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbf{P}\left[ S > X \right]} = 0~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq X \}}}\right]} = 0.$$ Using the same arguments we also have $$\label{eq:suo2}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{X \mathbb{I}_{\{ S > X\}}}\right]} = 0.$$ Secondly, note that $$\label{eq:suo3}
{\mathbb{E}\left[{\min_{l \geq 0} \left( \sum_{k=1}^{l} X_k + S_{l+1} \right)}\right]} \leq {\mathbb{E}\left[{\min\{ S_1, X_1 + S_2 \}}\right]}.$$ It suffices to show that ${\mathbb{E}\left[{\min\{ S_1, X_1 + S_2 \}}\right]} \rightarrow 0$ as $\eta \rightarrow \eta^{\ast}$. To see this, we write ${\mathbb{E}\left[{\min\{ S_1, X_1 + S_2 \}}\right]}$ as: $$\begin{aligned}
\nonumber
&{\mathbb{E}\left[{ \min\{S_1, X_1 + S_2 \} }\right]} \nonumber \\
&= {\mathbb{E}\left[{ S_1 \mathbb{I}_{\{ S_1 \leq X_1 \}} }\right]} + {\mathbb{E}\left[{\left[ X_1 + \min\{ S_1 - X_1, S_2\}\right]\mathbb{I}_{\{ S_1 > X_1\}} }\right]},\nonumber \\
&\leq {\mathbb{E}\left[{ S_1 \mathbb{I}_{\{ S_1 \leq X_1 \}} }\right]} + {\mathbb{E}\left[{\left[ X_1 + S_2\right]\mathbb{I}_{\{ S_1 > X_1\}} }\right]}, \nonumber \\
&= {\mathbb{E}\left[{ S_1 \mathbb{I}_{\{ S_1 \leq X_1 \}} }\right]} + {\mathbb{E}\left[{X_1\mathbb{I}_{\{ S_1 > X_1\}}}\right]} + {\mathbb{E}\left[{S_2 \mathbb{I}_{\{ S_1 > X_1\}}}\right]}, \nonumber \\
&\rightarrow 0,~~\text{as}~~\eta \rightarrow \eta^{\ast}, \nonumber\end{aligned}$$ where the last equation follows from and .
We now give a sufficient condition on the service time distributions $F_S$, parameterized by $\eta$, to have its second moment approach infinity. This result is used in proving the strong age-delay and age-delay variance tradeoffs.
\[lem:C\_implies\_S2\] For the parameterized, service time random variable $S$, we have $\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S^2}\right]} = +\infty$ if $$\label{eq:suff_cond2}
\lim_{\eta \rightarrow \eta^{\ast}} {\mathbf{P}\left[ S > x \right]} = 0,~\text{and}~\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x\}}}\right]} = 0,$$ for all $x \geq x_0$, and some $x_0 > 0$.
Let the two conditions hold for $S$. First, note that ${\mathbb{E}\left[{S^2}\right]} \geq {\mathbb{E}\left[{S^2 \mathbb{I}_{\{ S > x\}}}\right]} \geq x {\mathbb{E}\left[{S \mathbb{I}_{\{ S > x\}}}\right]}$, for all $x > 0$. We can write ${\mathbb{E}\left[{S \mathbb{I}_{\{ S > x\}}}\right]}$ as ${\mathbb{E}\left[{S}\right]} - {\mathbb{E}\left[{S \mathbb{I}_{\{ S \leq x\}}}\right]} \rightarrow 1/\mu$ as $\eta \rightarrow \eta^{\ast}$ by and the fact that ${\mathbb{E}\left[{S}\right]} = 1/\mu$. Therefore, we have $\liminf_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S^2}\right]} \geq x/\mu$ for all $x \geq x_0$. This can only be true if $\lim_{\eta \rightarrow \eta^{\ast}} {\mathbb{E}\left[{S^2}\right]} = +\infty$.
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[^1]: This work was supported by NSF Grants AST-1547331, CNS-1713725, and CNS-1701964, and by Army Research Office (ARO) grant number W911NF- 17-1-0508.
[^2]: A preliminary version of the this work was available on arXiv [@talak18_determinacy] and appeared in ISIT 2019 [@talak19_AoI_age_delay; @talak19_AoI_heavytail].
[^3]: The authors are with the Laboratory for Information and Decision Systems (LIDS) at the Massachusetts Institute of Technology (MIT), Cambridge, MA. [{talak, modiano}@mit.edu]{}
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---
abstract: 'Abstract dialectical frameworks (ADFs) are a powerful generalisation of Dung’s abstract argumentation frameworks. In this paper we present an answer set programming based software system, called DIAMOND (DIAlectical MOdels eNcoDing). It translates ADFs into answer set programs whose stable models correspond to models of the ADF with respect to several semantics (i.e. admissible, complete, stable, grounded).'
author:
- Stefan Ellmauthaler and Hannes Strass
bibliography:
- 'paper.bib'
title: 'The DIAMOND System for Argumentation: Preliminary Report[^1]'
---
Introduction {#sec:introduction}
============
Formal argumentation has established itself as a vibrant subfield of artificial intelligence, contributing to such diverse topics as legal decision making and multi-agent interactions. A particular, well-known formalism to model argumentation scenarios are Dung’s abstract argumentation frameworks [@dung95acceptability]. In that model, arguments are treated as abstract atomic entities. The only information given about them is a binary relation expressing that one argument attacks another.
A criticism often advanced against Dung frameworks is their restricted expressive capability of allowing only attacks between arguments. This leads to quite a number of extensions of Dung AFs, for example with attacks from sets of arguments [@NielsenP06], attacks on attacks [@Modgil09] and meta-argumentation [@DBLP:journals/sLogica/BoellaGTV09]. Unifying these and other extensions to AFs, Brewka and Woltran [@brewka-woltran10adfs] proposed a general model, abstract dialectical frameworks (ADFs). In ADFs, attack, support, joint support, combined attacks and many more relations between arguments (called statements there) can be expressed, while the whole formalism stays on the same abstraction level as Dung’s.
In this paper we present the DIAMOND software system that computes models of ADFs with respect to several different semantics. The name DIAMOND abbreviates “DIAlectical MOdels eNcoDing” and hints at the fact that DIAMOND is built on top of the state of the art in answer set programming: abstract dialectical frameworks are encoded into logic programs, and an answer set solver is used to compute the models of the ADF. By providing an expressive argumentation formalism with an implementation, we pave the way for practical applications of ADFs in scenarios where dialectical aspects are of interest, for example in group decision making.
The paper proceeds as follows. We first introduce the necessary background in abstract dialectical frameworks and answer set programming. We then present the DIAMOND system – how ADFs are represented there, and how the ADF semantics are encoded into answer set programs. We conclude with a contrasting discussion of the most significant related work.
Background {#sec:background}
==========
An abstract dialectical framework (ADF) [@brewka-woltran10adfs] is a directed graph whose nodes represent statements or positions that can be accepted or not. The links represent dependencies: the status of a node $s$ only depends on the status of its parents (denoted ${\mathit{par}(s)}$), that is, the nodes with a direct link to $s$. In addition, each node $s$ has an associated acceptance condition $C_s$ specifying the exact conditions under which $s$ is accepted. $C_s$ is a function assigning to each subset of ${\mathit{par}(s)}$ one of the truth values ${\mathbf{t}}$, ${\mathbf{f}}$. Intuitively, if for some $R \subseteq {\mathit{par}(s)}$ we have $C_s(R) = {\mathbf{t}}$, then $s$ will be accepted provided the nodes in $R$ are accepted and those in ${\mathit{par}(s)} \setminus R$ are not accepted.
\[def:ADF\] An *abstract dialectical framework* is a tuple $D = (S, L, C)$ where
- $S$ is a set of statements (positions, nodes),
- $L \subseteq S \times S$ is a set of links,
- $C = \{C_s\}_{s \in S}$ is a set of total functions $C_s : 2^{{\mathit{par}(s)}}\rightarrow \{{\mathbf{t}},
{\mathbf{f}}\}$.
In many cases it is convenient to represent the acceptance condition of a statement $s$ by a propositional formula $\varphi_s$, as is done in our running example.
\[exa:ADF\] Consider the ADF with a support cycle and one attack relation: . This ADF can also be represented as a graph, where the nodes are statements and the relations between them are directed edges. The boxes below each node are the acceptance conditions for the particular statement.
In recent work [@brewka13adfs], we redefined several standard ADF semantics and defined additional ones. In this paper, we use these revised definitions, which are based on three-valued logic.[^2] The three truth values true (${\mathbf{t}}$), false (${\mathbf{f}}$) and unknown (${\mathbf{u}}$) are partially ordered by ${\leq_i}$ according to their information content: we have and and no other pair in ${<_i}$, which intuitively means that the classical truth values contain more information than the truth value unknown. On the set of truth values, we define a meet operation, *consensus*, which assigns , , and returns ${\mathbf{u}}$ otherwise. The information ordering ${\leq_i}$ extends in a straightforward way to valuations $v_1,v_2$ over $S$ in that iff for all . Obviously, a three-valued interpretation $v$ is two-valued if all statements are mapped to either true or false. For a three-valued interpretation $v$, we say that a two-valued interpretation $w$ [*extends*]{} $v$ iff . We denote by $[v]_2$ the set of all two-valued interpretations that extend $v$. A three-valued interpretation $E_v$ has an [*associated extension*]{} .
Brewka and Woltran [@brewka-woltran10adfs] defined an operator $\Gamma_D$ over three-valued interpretations. For each statement $s$, the operator returns the consensus truth value for its acceptance formula $\varphi_s$, where the consensus takes into account all possible two-valued interpretations $w$ that extend the input valuation $v$.
\[def:semantics\] Let $D$ be an ADF and $v$ be a three-valued interpretation. Then the interpretation $\Gamma_D(v)$ is given by . Furthermore $v$ is [*admissible*]{} iff $v{\leq_i}\Gamma_D(v)$; [*complete*]{} iff $\Gamma_D(v) = v$, that is, $v$ is a fixpoint of $\Gamma_D$; [*grounded*]{} iff $v$ is the ${\leq_i}$-least fixpoint of $\Gamma_D$.
A two-valued interpretation $v$ is a [*model of $D$*]{} iff $\Gamma_D(v)=v$; it is a [*stable model*]{} of iff $v$ is a model of $D$ and $E_v$ equals the grounded extension of the reduced ADF , where and for we set .
We will now show the models with respect to the different semantics for the ADF introduced in Example \[exa:ADF\]. For readability, we write interpretations $v$ as sets of literals . There are
- five admissible interpretations: $\emptyset$, $\{a,b\}$, $\{a,b,\lnot c\}$ $\{\lnot a,\lnot b,c\}$, $\{\lnot a,\lnot b\}$,
- three complete models: $\{\lnot a,\lnot b,c\}$, $\emptyset$, $\{a,b,\lnot c\}$; of which $\emptyset$ is grounded;
- two models: $\{a,b,\neg c\}$, $\{\neg a, \neg b, c\}$, of which one is stable: $\{\neg a, \neg b, c\}$
Brewka et al. [@brewka13adfs] also defined an approach to handle preferences in ADFs. The approach generalises the one for AFs from Amgoud and Cayrol [@AmgoudC98]. Since DIAMOND also implements this treatment of preferences, we recall it here. For this approach, the links are restricted to links that are attacking or supporting.
A prioritised ADF (PADF) is a tuple $P = (S, L^+, L^-, >)$ where $S$ is the set of nodes, $L^+$ and $L^-$ are subsets of $S \times
S$, the supporting and attacking links, and $>$ is a strict partial order (irreflexive, transitive, antisymmetric) on $S$ representing preferences among the nodes.
Here (alternatively: ) expresses that $a$ is preferred to $b$. The semantics of prioritised ADFs is given by a translation to standard ADFs: $P$ translates to , where for each statement $s \in S$ the acceptance condition $C_s$ is defined as: $C_s(M) = {\mathbf{t}}$ iff for each $a \in M$ such that $(a,s) \in L^-$ and not $s > a$ we have: for some $b \in M$, $(b,s) \in
L^+$ and $b > a$. Intuitively, an attacker does not succeed if the attacked node is more preferred or if there is a more preferred supporting node.
Answer Set Programming {#sec:asp}
----------------------
A [*propositional normal logic program*]{} $\Pi$ is a set of finite rules $r$ over a set of ground atoms $\mathcal{A}$. A rule $r$ is of the form , where $\alpha\in\mathcal{A}$, $\beta_i\in\mathcal{A}$ are ground atoms and $m\leq n\leq 0$. Each rule consists of a [*body*]{} $B(r)=\{\beta_1,\ldots,\beta_m, \text{not }\beta_{m+1}\ldots, \text{not }\beta_n\}$ and a [*head*]{} $H(r)=\{\alpha\}$, divided by the $\leftarrow$-symbol. We will split up the body into two parts, $B(r)=B^+(r)\cup B^-(r)$, where $B^+(r)=\{\beta_1,\ldots,\beta_m\}$ and $B^-(r)=\{\text{not }\beta_{m+1}\ldots, \text{not }\beta_n\}$. A rule $r$ is said to be positive if $B^-(r)=\emptyset$ and a program $\Pi$ is positive if every rule $r\in\Pi$ is positive. For a positive program $\Pi$, its immediate consequence operator $T_\Pi$ is defined for $S\subseteq\mathcal{A}$ by $T_\Pi(S) = {\left\{ H(r)\in\mathcal{A} {\ \middle\vert\ }r\in \Pi, B^+(r)\subseteq S \right\}}$. A set $A\subseteq\mathcal{A}$ of ground atoms is a [*minimal model*]{} of a positive propositional logic program $\Pi$ iff $A$ is the least fixpoint of $T_\Pi$. To allow rules with negative body atoms, Gelfond and Lifschitz [@Gelfond1988] proposed the stable model semantics (also called answer set semantics).
Let $A\subseteq\mathcal{A}$ be a set of ground atoms. $A$ is a [*stable model*]{} for the propositional normal logic program $\Pi$ iff $A$ is the minimal model of the reduced program $\Pi^A$, where .
We use [*clasp*]{} from the Potsdam Answer Set Solving Collection [*Potassco*]{}[^3] [@Gebser2011] as the back-end answer set solver for our software system. Potassco allows us to use an enriched input language where in addition to the above pictured propositional logic programs we can use first-order variables and predicates. Ground atoms are generally written in lower case while variables are represented with upper case characters. Additionally [*Potassco*]{} offers features like aggregates, cardinality constraints, choice rules and conditional literals. For further details we refer to the recent book by Gebser et al. [@Gebser2012].
DIAMOND {#sec:diamond}
=======
Our software system DIAMOND is a collection of answer set programming encodings and tools to compute the various models with respect to the semantics for a given ADF. The different encodings are designed around the *Potsdam Answer Set Solving Collection (Potassco)* [@Gebser2011] and the additional provided tools utilise clasp as solver, too. Note that the encodings for DIAMOND are built in a modular way. To compute the models of an ADF with respect to a semantics, different modules need to be grounded together to get the desired behaviour. DIAMOND is available for download and experimentation at the web page <http://www.informatik.uni-leipzig.de/~ellmau/diamond>. There we also provide further documentation on its usage. In short, DIAMOND is a Python-script,[^4] which can be invoked via the command line. Different switches are used to designate the desired semantics, and the input file is given as a file name or via the standard input. The options for the command line are as follows:
usage: diamond.py \[-h\] \[-cf\] \[-m\] \[-sm\] \[-g\] \[-c\] \[-a\]\
\[–transform\_pform | –transform\_prio\] \[-all\] \[–version\] instance\
positional arguments:\
instance File name of the ADF instance\
optional arguments:\
-h, –help show this help message and exit\
-cf, –conflict-free compute the conflict free sets\
-m, –model compute the two-valued models\
-sm, –stablemodel compute the stable models\
-g, –grounded compute the grounded model\
-c, –complete compute the complete models\
-a, –admissible compute the admissible models\
–transform\_pform transform a propositional formula ADF before the computation\
–transform\_prio transform a prioritized ADF before the computation\
-all, –all compute all sets and models\
–version prints the current version
We next describe how specific ADF instances are represented in DIAMOND.
Instance Representation {#sec:instancerepresentation}
-----------------------
In order to represent an ADF for DIAMOND its acceptance conditions need to be in the functional representation as given in Definition \[def:ADF\]. The statements of an ADF are declared by the predicate [s]{}, and the links are represented by the binary predicate [l]{}, such that [l(b,a)]{} reflects that there is a link from $b$ to $a$. The acceptance condition is modelled via the unary and tertiary predicates [ci]{} and [co]{}. Intuitively [ci]{} (resp. [co]{}) identifies the parents which need to be accepted, such that the acceptance condition maps to true (i.e. *in*) (resp. false (i.e. *out*)). To achieve a flat representation of each set of parent statements, we use an arbitrary third term in the predicate to identify them. To express what happens to a statement when none of the parents is accepted we use the unary versions of [ci]{} and [co]{}. Here is the DIAMOND representation of Example \[exa:ADF\]:
s(a). s(b). s(c). l(b,a). l(a,b). l(b,c).\
co(a). ci(a,1,b). co(b). ci(b,1,a). ci(c). co(c,1,b).
The first line declares the statements and links. The second line expresses the acceptance conditions: statement $a$ is ${\mathit{out}}$ if $b$ is ${\mathit{out}}$ and ${\mathit{in}}$ if $b$ is; likewise $b$ gets the same status as $a$; statement $c$ is ${\mathit{in}}$ if $b$ is ${\mathit{out}}$, and $c$ is ${\mathit{out}}$ if $b$ is ${\mathit{in}}$.
As a part of the DIAMOND software bundle, we also provide an ECL$^i$PS$^e$ Prolog[^5] [@SchimpfS2010] program that transforms acceptance functions given as formulas into the functional representation used by DIAMOND.
We have chosen this functional representation of acceptance conditions for pragmatic reasons. An alternative would have been to represent acceptance conditions by propositional formulas. In this case, computing a single step of the operator would entail solving several [$\mathbf{NP}$]{}-hard problems. The standard way to solve these is the use of saturation [@EiterG95], which however causes undesired side-effects when employed together with meta-ASP [@GebserKS2011]. Furthermore, other ADF semantics (e.g. preferred) utilise concepts like $\subseteq$-minimality, which also require the use of meta-argumentation. We plan to extend DIAMOND to further semantics and therefore chose the functional representation of acceptance conditions to forestall potential implementation issues.
Due to compatibility considerations, it is possible for DIAMOND to understand the propositional formula representation as well as a PADF. The propositional formula representation uses the unary predicate [statement]{} to identify statements. The binary predicate [ac(s,$\phi$)]{} associates to each statement [s]{} one formula $\phi$. Each formula [$\phi$]{} is constructed in the usual inductive way, where atomic formulae are other statements and the truth constants (i.e. [c(v)]{} and [c(f)]{}) and the operators are written as functions. The allowed operators are [neg, and, or, imp,]{} and [iff]{} for their respective logical operators. To describe a PADF, we use the unary predicate [s]{} to describe the set of statements. In addition the support (i.e. $L^+$) and attack (i.e. $L^-$) links are represented by the binary predicates [lp]{} and [lm]{} (i.e. positive resp. negative links). To express a preference, such as $a > b$, we use the predicate [pref(a,b)]{}. Note that DIAMOND provides a method to translate propositional formula ADFs and PADFs into ADFs with total functions and only computes the models using the functional representation.
For illustration, let us look at another, slightly more complicated example.
\[exa:adf2\] Consider the ADF $D_2=(S_2,L_2,C_2)$ with $S_2=\{a,b,c,d\}$, $L_2=\{(a,c),(b,b),(b,c),(b,d)\}$, and $C_2=\{\varphi_a={\mathbf{t}}, \varphi_b=b, \varphi_c=a \land b, \varphi_d=\lnot b\}$.
For this ADF there are
- 16 admissible interpretations: $\emptyset$, $\{a\}$, $\{b\}$, $\{\lnot b\}$, $\{b,\lnot d\}$, $\{a,b\}$, $\{a,\lnot b\}$, $\{\lnot b,d\}$, $\{\lnot b,\lnot d\}$, $\{a,b,c\},$ $\{a,b,\lnot d\}$, $\{a,\lnot b,d\}$, $\{a,\lnot b,\lnot c\}$, $\{\lnot b,\lnot c,d\}$, $\{a,b,c,\lnot d\}$, $\{a,\lnot b,\lnot c,d\}$
- three complete models: $\{a\}$, $\{a,b,c,\lnot d\}$, $\{a,\lnot b,\lnot c,d\}$; of these, $\{a\}$ is the grounded model;
- two models: $\{a,b,c,\lnot d\}$, $\{a,\lnot b,\lnot c, d\}$, of which one is stable: $\{a,\lnot b,\lnot c,d\}$. Its propositional formula representation for DIAMOND (inherited from [ADFsys]{}) is given by the following ASP code:
statement(a). statement(b). statement(c). statement(d).\
ac(a,c(v)).\
ac(b, b).\
ac(c, and(a,b)).\
ac(d, neg(b)).
The functional ASP representation of the same ADF looks thus:
s(a). s(b). s(c). s(d).\
l(a,c). l(b,b). l(b,c). l(b,d).\
ci(a).\
co(b). ci(b,1,b).\
co(c). co(c,1,a). co(c,2,b). ci(c,3,a). ci(c,3,b).\
ci(d). co(d,1,b).
Arguably, the formula representation is easier to read for humans.
Implementation of $\Gamma_D$
----------------------------
Since all of the semantics are defined via the operator $\Gamma_D$, we will now present how the implementation of the operator is done in DIAMOND. The unary predicate [step]{} with an arbitrary term is used to apply the operator several times. The input for the operator is given by the predicates [in]{} and [out]{} to represent mappings to ${\mathbf{t}}$ and ${\mathbf{f}}$. The resulting interpretation can be read off the predicates [valid]{} and [unsat]{}. Predicates [fp]{} and [nofp]{} denote whether a fixpoint is reached or not. First, DIAMOND decides which of the mappings to ${\mathbf{t}}$ are still of interest ([cii]{}) (i.e. which of those can still be satisfied under the given interpretation):
ciui(S,J,I) :- lin(X,S,I), not ci(S,J,X), ci(S,J).\
ciui(S,J,I) :- lout(X,S,I), ci(S,J,X).\
cii(S,J,I) :- not ciui(S,J,I), ci(S,J), step(I).
The predicates [lin]{} and [lout]{} are those links between arguments which are already decided by the given three-valued interpretation. The binary predicate [ci]{} (resp. [co]{}) is just the projection of its tertiary version to express that at least one predicate with a specific statement occurs in a specific acceptance condition. The treatment of the interesting mappings to ${\mathbf{f}}$ ([coi]{}) is dual:
coui(S,J,I) :- lin(X,S,I), not co(S,J,X), co(S,J).\
coui(S,J,I) :- lout(X,S,I), co(S,J,X).\
coi(S,J,I) :- not coui(S,J,I), co(S,J), step(I).
Afterwards it is checked whether there exist two-valued extensions of the given interpretation that are a model or not, which is denoted by the predicates [pmodel]{} (resp. [imodel]{}). Then a statement can be seen to be [valid]{} (resp. [unsat]{}) if there does not exist an interpretation which is not a model (is a model). The predicate [verum]{} (resp. [falsum]{}) represents that the acceptance condition is always true (resp. false).
pmodel(S,I) :- cii(S,J,I). pmodel(S,I) :- verum(S), step(I).\
pmodel(S,I) :- not lin(S,I), ci(S), step(I).\
pmodel(S,I) :- not lin(S,I), ci(S), step(I).\
valid(S,I) :- pmodel(S,I), not imodel(S,I).\
\
imodel(S,I) :- coi(S,J,I).\
imodel(S,I) :- falsum(S), step(I).\
imodel(S,I) :- not lin(S,I), co(S), step(I).\
unsat(S,I) :- imodel(S,I), not pmodel(S,I).
At last, either [nofp]{} or [fp]{} is deduced. To achieve this, DIAMOND checks whether the application of the operator resulted in an interpretation that is different from the given one.
nofp(I) :- in(X,I), not valid(X,I), step(I).\
nofp(I) :- valid(X,I), not in(X,I), step(I).\
nofp(I) :- out(X,I), not unsat(X,I), step(I).\
nofp(I) :- unsat(X,I), not out(X,I), step(I).\
fp(I) :- not nofp(I), step(I).
Semantics {#sec:semantics}
---------
The admissible model is computed by the use of a guess and check approach. At first a three-valued interpretation is guessed, by an assignment of the statements to be [in]{}, [out]{}, or neither. The last two lines remove all guesses which violate the definition of the admissible model (i.e. check which guesses are right):
step(0).\
{in(S,0):s(S)}.\
{out(S,0):s(S)}.\
:- in(S,0), out(S,0).\
:- in(S), not valid(S,0).\
:- out(S), not unsat(S,0).
The complete model encoding uses the same concept as used for the admissible model. The only difference is that the guessed model needs to be a fixpoint. To this effect the last two rules of the above encoding are replaced by the constraint “[:- nofp(0).]{}”.
To compute the grounded model, we need to apply $\Gamma_D$ until a fixpoint is reached. This is done via a sequence of steps, where the result of one step is taken as the used given interpretation for the next step:
maxit(I) :- I:={s(S)}. step(0).\
in(S,I+1) :- valid(S,I). out(S,I+1) :- unsat(S,I).\
step(I+1) :- step(I), not maxit(I).\
in(S) :- fp(I), in(S,I).\
out(S) :- fp(I), out(S,I).\
udec(S) :- fp(I), s(S), not in(S), not out(S).
Note that we use the number of statements as the upper bound on the number of operator applications as this is the maximal number of steps needed to reach a fixpoint.
To implement the model semantics, the operator is not essential: as the model is only two-valued, there do not remain undecided parts. So each variable is mapped to a truth-value and therefore every acceptance condition may only map to one value (i.e. ${\mathbf{t}}$ or ${\mathbf{f}}$). The encoding just guesses a two-valued interpretation and checks whether the guessed interpretation agrees with the acceptance conditions of each statement or not. The stable model combines the encoding for models with the operator encoding to check for each model whether it is also the grounded extension of its reduced ADF or not.
Discussion and Future Work {#sec:discussion}
==========================
We presented the DIAMOND software system that uses answer set programming to compute models of abstract dialectical frameworks under various semantics. DIAMOND can be seen as a continuation of the trend to utilise ASP for implementing abstract argumentation. The most important existing tool in this line of work is the ASPARTIX system[^6] [@EglyGW10] for computing extensions of Dung argumentation frameworks.
Quite recently, Ellmauthaler and Wallner presented their system [ADFsys]{}[^7] for determining the semantics of ADFs [@Ellmauthaler2012]. Since their system likewise uses answer set programming, it is natural to ask where the differences lie. For one, after the discovery of several examples where some original ADF semantics do not behave as intended, Brewka et al. [@brewka13adfs] proposed revised and generalised versions of these semantics. The DIAMOND system implements the new semantics while [ADFsys]{} still computes the old versions. For another, [ADFsys]{} relies solely on the representation of acceptance conditions via propositional formulas, while DIAMOND can additionally deal with functional representations. Due to the new semantics it is not trivial to compare those two systems. In fact only the model and the grounded semantics have not changed. During preliminary tests, we used different methods to generate randomised ADF instances. Depending on the used generation method, DIAMOND could compete with [ADFsys]{} and even outperform it. Alas, there were also instances for which [ADFsys]{} outperformed DIAMOND. We consider it an important future task to determine specific classes of ADFs that distinguish the two systems, and to connect these ADF classes to possible real-world applications.
To adapt [ADFsys]{} to the new semantics, it would be needed to decide at each operation of $\Gamma_D$ which acceptance formulae are (under the given three-valued interpretation) irrefutable (resp. unsatisfiable). To solve such an embedded co-[$\mathbf{NP}$]{} problem it would be necessary to use the saturation technique or similar concepts, which will make the use of disjunctive logic programs obligatory. Therefore there would also be issues with more complex semantics (like the preferred semantics). There the use of meta-ASP would conflict with the use of saturation in the disjunctive program.
Apart from the semantics implemented in this paper, there are also ADF semantics that DIAMOND cannot yet deal with – these remain for future work. For example, the preferred semantics is based on maximisation, and so we will need meta-ASP to implement that. In general, ADFs are a quite new formalism, and we expect that further ADF semantics will be defined in the future. Naturally, we plan to implement these new semantics using the infrastructure already available through DIAMOND.
Another future research interest concerns a possible practical application for ADFs: We intend to analyse discussions in social media, where opinions and viewpoints can be modelled by statements that are in some relation to each other. ADF semantics can guide the respective online community, for example as to what positions everybody can agree on, or how a group decision can be justified. Such an approach was proposed by Toni and Torroni [@Toni2011] as a possible application of assumption-based argumentation frameworks [@BondarenkoDKT97]. However, assumption-based argumentation inherits the expressiveness limitations of abstract argumentation, that is, it can also express only attack relationships between statements. We expect that ADFs with their greater expressiveness are better suited to model online interactions in social media.
A similar application of argumentation in online social communities is the approach by Snaith et al. [@Snaith2012]. They utilise their database for arguments in the Argument Interchange Format [@Rahwan2009] to capture discussions via different blogging-sites and use their tool TOAST [@Snaith2012a] to compute an acceptable consensus about the issues under discussion. Again we think that ADFs are more suitable for this application due to their expressiveness.
[^1]: This research has been supported by DFG projects BR 1817/7-1 and FOR 1513.
[^2]: For further details on those newly introduced semantics we refer the interested reader to Brewka et al. [@brewka13adfs].
[^3]: Available at <http://potassco.sourceforge.net>
[^4]: Python is available at <http://www.python.org>.
[^5]: ECL$^i$PS$^e$ is available at <http://eclipseclp.org/>.
[^6]: ASPARTIX is available at <http://www.dbai.tuwien.ac.at/research/project/argumentation/systempage/>
[^7]: [ADFsys]{} is available at <http://www.dbai.tuwien.ac.at/research/project/argumentation/adfsys/>
|
---
abstract: |
We use several quasar samples (LBQS, HBQS, Durham/AAT and EQS) to determine the density and luminosity evolution of quasars. Combining these different samples and accounting for varying selection criteria require tests of correlation and the determination of density functions for multiply truncated data. We describe new non-parametric techniques for accomplishing these tasks, which have been developed recently by Efron and Petrosian (1998). With these methods, the luminosity evolution can be found through an investigation of the correlation of the bivariate distribution of luminosities and redshifts. We use matter dominated cosmological models with either zero cosmological constant or zero spatial curvature to determine luminosities from fluxes. Of the two most commonly used models for luminosity evolution, $L \propto e^{k t(z)} $ and $L \propto (1+z)^{k'}$, we find that the second form is a better description of the data at all luminosities; we find $k' = 2.58 $ ($[2.14,2.91]$ one $\sigma$ region) for the Einstein - de Sitter cosmological model.
Using this form of luminosity evolution we determine a global luminosity function and the evolution of the co-moving density for the two types of cosmological models. For the Einstein - de Sitter cosmological model we find a relatively strong increase in co-moving density up to $z \lesssim 2$, at which point the density peaks and begins to decrease rapidly. This is in agreement with results from high redshift surveys. We find some co-moving density evolution for all cosmological models, with the rate of evolution lower for models with lower matter density. We find that the local cumulative luminosity function $\Phi(L_o)$ exhibits the usual double power law behavior. The luminosity density ${\cal L} (z) = \int_0^\infty L \Psi(L,z) dL$, where $\Psi(L,z)$ is the luminosity function, is found to increase rapidly at low redshift and to reach a peak at around $z \approx 2$. Our results for ${\cal L} (z)$ are compared to results from high redshift surveys and to the variation of the star formation rate with redshift.
author:
- Alexander Maloney and Vahé Petrosian
title: THE EVOLUTION AND LUMINOSITY FUNCTION OF QUASARS FROM COMPLETE OPTICAL SURVEYS
---
Center for Space Science and Astrophysics, Stanford University, Stanford, CA 94305
INTRODUCTION
============
Investigations of the evolution of the quasar population have played a major role in the development of our ideas about the nature of these sources and their connection to other extragalactic objects. Ever since the first complete survey of 3C radio quasars by Schmidt (1968) and the subsequent survey of 4C quasars by Lynds and Wills (1972) it has been evident that the population of quasars as a whole has undergone rapid evolution. Using the so-called $\langle V / V_{max} \rangle$ method these authors interpreted the evolution with redshift $z$ as caused by an increase in the co-moving density of quasars with redshift. However, both the source counts (Giaconni et al. 1979, Tananbaum et al. 1979) and the redshift distribution of optically selected samples of quasars (see e.g. Marshall 1985) clearly showed that such pure density evolution (PDE) models, for which the luminosity function is separable as \[eq:pde\] (L,z) = (L) (z) , cannot be correct. As more data was accumulated the pure luminosity evolution (PLE) model, with (L,z) = (L/g(z)) / g(z) , gained more popularity. The function $g(z)$ describes the luminosity evolution of the population and $L_o = L/g(z)$ is the luminosity adjusted to its present epoch ($z=0$) value. This model, while providing a better fit to the data than that of pure density evolution, also appears to be inadequate in many cases (see e.g. Petrosian 1973, Schmidt and Green 1978, Koo and Kron 1988, Caditz and Petrosian 1990). Without loss of generality, we can write the luminosity function as \[eq:gle\] (L,z) = (z) (L/g(z),\_i) / g(z) . With $\psi$ normalized such that $\int_0^\infty \psi(L,\alpha_i) dL=1$, $\rho (z)$ gives the density and its evolution and $\psi (L_o, \alpha_i)$ describes the local luminosity function (with $g(0) = 1$). Here we explicitly include the $\alpha_i$, parameters such as spectral index and break luminosity that describe the shape of the luminosity function. In general, these parameters may vary with redshift. A surprising, and a priori unexpected, result has been the absence of evidence for strong shape variation.
Such results imply that the density and luminosity functions $\rho (z)$ and $g(z)$ describe the [*physical evolution*]{} of sources; e.g. the rate of birth and death of sources and the changes in source luminosity with time. Cavaliere and colleagues (see Cavaliere and Padovani 1988 and references cited therein) were the first to emphasize this fact. A more complete description of the relation between the physical evolution and the functions describing the generalized luminosity function (or statistical evolution) can be found in Caditz and Petrosian (1990). Unfortunately, this relation is not unique, thus any test of quasar models via their expected evolution with cosmic epoch must usually involve additional assumptions. For example, quasars could be long lived (compared to the Hubble time) sources created during a relatively short period at high redshift undergoing continuous luminosity evolution. This is the model used in Caditz, Petrosian and Wandel (1991). Alternatively, quasars could be short lived phenomena with birth rate, death rate and luminosity that vary systematically with redshift (see Siemiginowska and Elvis 1997). Most of the work along these lines assumes what is now the standard model for the source of energy of quasars and other active galactic nuclei (or AGNs); namely, accretion onto a massive black hole.
Another interesting aspect of quasar evolution is the relation between the evolution of the luminosity density of quasars, ${\cal L} = \int L \Psi(L,z) dL$, and similar functions describing the evolution of galaxies, such as the star formation rate (SFR). As shown by the high redshift surveys of Schmidt et al. (1995) and others (e.g. Warren et al. 1994), the rapid rise with redshift of the luminosity of quasars stops around redshift $2$ or $3$ and is expected to drop, perhaps mimicking the SFR (see Shaver et al. 1998, Cavaliere and Vitorini 1998, and Hawkins and Véron 1996).
In this paper we determine the luminosity function and its evolution for combined samples of quasars from various surveys, described in $\S 2$. For a complete review of the various ways of accomplishing this task, see Petrosian (1992). We use here non-parametric methods based on Lynden-Bell’s (1971) idea which was generalized by Efron and Petrosian (1992) to allow not only a determination of the functional forms but also a test of correlation between redshifts and luminosities. This is essential for any determination of the functional form of the luminosity evolution $g(z)$. The above papers deal with magnitude limited samples with only an upper magnitude (lower flux) limit. However, because of observational constraints, some of the samples are truncated in redshift space and some have an upper magnitude (lower flux) limit. The methods described in the above papers cannot be used for such multiply truncated data. New methods for treating this type of data were developed by Efron and Petrosian (1998). In $\S 3$ we describe these new techniques and their relation to the older methods. The choice of cosmological model also plays a crucial role in such determinations. It is well known that one can not determine both the evolution of the luminosity function and the parameters of the cosmological model from magnitude limited samples alone. Only by assuming values for the cosmological parameters such as matter density, curvature and cosmological constant can one calculate the form of $\Psi(L,z)$. Alternatively, with some assumptions about $\Psi(L,z)$ one can test various cosmological models. In $\S 4$ we describe the cosmological model parameters and in $\S 5$ we present the results of applications of the new statistical methods to the data described in $\S 2$. Finally, in $\S 6$ we summarize these results and present our conclusions.
THE DATA {#sec:data}
========
There have been many surveys of quasars, ranging from the original radio selected samples of the 3C (Schmidt 1963) and 4C (Lynds and Wills 1972) surveys to a variety of other optically and X-ray selected samples. In this paper, we will focus only on optically selected data. We use data from four samples with relatively similar selection criteria that provide a somewhat homogeneous data set spanning a large area of the $L-z$ plane. In order to combine the samples, all magnitudes were transformed to the $B$ band. Some of the samples were artificially truncated in order to insure completeness within the truncation limits. The combined sample consisted of 1552 objects in the range $ 15.5 \lesssim B \lesssim 21.2 $ and $0.3 < z < 2.2$, with well defined upper and lower magnitude limits for each object.
Figure $1$ shows the distribution of the complete surveys in the $B-z$ plane, which at first sight shows little evidence for a Hubble relation of $B \propto 5 \log (z)$. However, as we shall discuss below, there is evidence for cosmological dimming with redshift of the sources.
The Large Bright QSO Survey
---------------------------
The Large Bright QSO survey (LBQS) contains 1055 QSOs in the magnitude range $16.0 < B_J < 18.85$ and redshift range $0.2 < z < 3.4$ (Hewett et al. 1995). The faint magnitude limits for the 18 fields range from $18.41$ to $18.85$ and the bright magnitude limit is $16.0$ for the entire sample. In order to transform from $B_J$ to $B$ magnitudes the color equation of Blair and Gilmore (1982) \[bj\] B = B\_J + 0.28 (B - V) was used assuming an average $B-V$ of $0.3$ for the entire sample. When we combine this sample with the AAT sample and others we introduce artificial cutoffs at $z = 0.3$ and $z = 2.2$, which are the completeness limits for the AAT data. In addition, we removed all objects brighter than $B = 16.5$ in order to insure completeness near the bright magnitude limit, leaving a total of 871 objects.
The Homogeneous Bright QSO Survey
---------------------------------
Data from the six deepest fields of the Homogeneous Bright QSO Survey (HBQS) were published by Cristiani et al. (1995) The six deepest fields of the HBQS contain either $B_J$ or $B'$ magnitudes for 285 QSOs in the range $ 15.5 < B_J < 18.85$, with faint magnitude limits ranging from $18.25$ to $18.85$. The bright magnitude limits are generally lower than those of the LBQS, varying from $14.0$ to $16.0$. For observations in the $B_J$ band equation (\[bj\]) was used and for observations in the $B'$ band the equation B = B’ + 0.11 (B-V) of Blair and Gilmore (1982) was used assuming $B - V = 0.3$. Again, when combining this sample only objects in the redshift range $0.3 < z < 2.2$ were used, leaving a total of $254$ objects.
The Durham/AAT Survey
---------------------
The Durham/AAT survey contains $419$ QSOs in the magnitude range $17 < b < 21.27$ (Boyle et al. 1990) and redshift range $0.3 < z < 2.2$, giving information about the QSO luminosity function in a different regime than the previous two samples. The faint magnitude limit ranges from $20.25$ to $21.27$ and the bright magnitude limit ranges from $16.4$ to $18.0$. The data are given in the “$b$” magnitude system, which according to Boyle et al. (1990) may be converted into the $B$ system by the relation B = b + 0.23 (B - V - 0.9) . As above, the average value $B-V = 0.3$ was used to determine $B$.
The Edinburgh QSO Survey
------------------------
A subsample of the Edinburgh QSO Survey (EQS) consisting of $12$ QSOs brighter than $B = 16.5$ was published by Goldschmidt et al. in 1992. Of these, the $8$ that fall in the redshift range $0.3 < z < 2.2$ were added to the combined samples to give information about the luminosity function at the bright end.
There have been several previous analyses of the above quasar surveys. Boyle et al. (1990) used binning techniques to fit the the Durham/AAT data to the PLE model. More recently, La Franca and Cristiani (1996) fit the LBQS and HBQS data to a more complex luminosity function (involving 3 or more parameters) with no density evolution. Hatziminaoglou et al. (1998) examined the PLE and PDE cases separately using both the Durham/AAT and LBQS data. As described below, we combine all of these data and determine both the luminosity evolution $g(z)$ and the density evolution $\rho(z)$.
STATISTICAL METHODS {#sec:statistics}
===================
The statistical problem at hand is the determination of the true distribution of luminosities and redshifts of the sources from a biased or truncated data set, such as a flux limited sample. Several different techniques exist that give such a determination. The most common method is to bin the data and fit to some parametric forms of $\psi(L)$ or $\rho(z)$. However, it is preferable to avoid binning and to use non-parametric methods whenever possible. For a review of the various methods see Petrosian (1992). Assuming the general form of equation (\[eq:gle\]) for the luminosity function we must determine the functional forms of the local luminosity function $\psi(L_o,\alpha_i)$, density evolution $\rho(z)$ and the luminosity evolution $g(z)$ as well as the changes in the parameters $\alpha_i$ with redshift, if any. This last aspect is a higher order effect and will not be within the scope of this paper. We will discuss the certainty with which this can be ignored in the final analysis.
All non-parametric techniques for determining the distribution in a bivariate setting require that the data be expressed in terms of two uncorrelated variables, i.e. that we use variables $x$ and $y$ for which the density function is separable: $\Psi (x,y) = \rho(x) \psi(y)$. Thus before applying non-parametric methods one must first determine the degree of correlation of the data in the $x - y$ plane. This determination and the process of removing the correlation is equivalent to a determination of the functional form of the luminosity evolution $g(z)$ and the subsequent transformation $L \to L_o = L / g(z)$. This cannot be accomplished easily by non-parametric methods. Therefore, we chose two parametric forms for the luminosity evolution, $g_k(z)$ and $g_{k'}(z)$, and find the values of the parameters $k$ and $k'$ for which $L_o$ and $z$ are uncorrelated. Once this is done, the non-parametric methods described in Petrosian (1992) may be used to determine $\rho(z)$ and $\psi(L_o)$.
The methods normally used to test correlation and determine the distributions are suited for simple truncations, such as $y < f(x)$ (which by defining $x' = f(x)$ can be reduced to the generic case of $y < x'$). This is sufficient for simple flux limited data. However, the majority of astronomical data, and quasar data in particular, suffer from more than one truncation. The data may have an upper as well as a lower truncation, $f^-(x) < y < f^+(x)$, or there may be similar truncations in the value of the other variable. In addition, the functions $f^-(x)$ and $f^+(x)$ may not be continuous or even single valued. In general, multiply truncated redshift - magnitude data may be written as $\{z_i,m_i, [z^-_i,z^+_i], [m^-_i,m^+_i]\}_{i=1}^N$ where $[z^-_i,z^+_i]$ and $[m^-_i,m^+_i]$ are the observational limits on $z$ and $m$ for the $i^{th}$ object, respectively. Given a cosmological model $\Omega$ this gives data of the form $\{z_i,L_i, [z^-_i,z^+_i], [L^-_i,L^+_i]\}_{i=1}^N$. If one assumes that each object has the same redshift limits $[z^-_i,z^+_i]$, which is the case in the majority of our analyses, then the problem is to test the correlation and distribution of $x$ and $y$ from a data set $\{x_i,y_i\}_{i=1}^N$ given truncation limits $[y^-_i,y^+_i]$ for each point. The previous methods developed for this test (see Petrosian 1992 and Efron and Petrosian 1992) are suited for one sided truncations. In a more recent work (Efron and Petrosian 1998) we have developed methods, which are a generalization of the earlier methods, for dealing with doubly truncated data. We will briefly review these new methods of testing correlation and non-parametrically determining the density evolution and luminosity function.
Tests of Correlation and Determination of Luminosity Evolution
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### Untruncated data
If $x$ and $y$ are independent then the rank $R_i$ of $x_i$ in an untruncated sample (i.e. a sample truncated parallel to the $x$ and $y$ axes, so that $y_i^\pm$ are independent of $x_i$) will be distributed uniformly between $1$ and $N$ with an expected mean $E={1 \over 2} (N+1) $ and variance $V = {1 \over 12} (N^2 - 1)$. One may then normalize $R_i$ to have a mean of $0$ and a variance of $1$ by defining the statistic $T_i = (R_i - E) / V $. One then rejects or accepts the hypothesis of independence based on the distribution of the $T_i$.
One simple way of doing so is by defining a single statistic $t_{data}$ based on the $T_i$ with a mean of $0$ and a variance of $1$. One then rejects the hypothesis of independence if $|t_{data}|$ is too large (e.g. $|t_{data}| \geq 1$ for rejection of independence at the $ 1 \ \sigma$ level). The quantity \[taudef\_1a\] = [ [\_i ( R\_i - E )]{} ]{} is one choice of such a test statistic. This $\tau$ is equivalent to Kendell’s $\tau$ statistic, which is defined in the following manner. Consider all possible pairings ${\cal P} = \{(i,j)\}$ between data points and call a pairing $(i,j)$ positive if $(x_i - x_j) (y_i - y_j) > 0 $ and negative if $(x_i - x_j) (y_i - y_j) < 0 $. If there are no ties then the $\tau$ of equation (\[taudef\_1a\]) is equivalent to Kendell’s $\tau$ statistic \[eq:taudef\_2\] = [ \# [ positive ]{} (i,j) - \# [ negative ]{} (i,j) \# (i,j) ]{}.
### Data with One-Sided Truncation
A straightforward application of this method to truncated data will clearly give a false correlation signal. Efron and Petrosian (1992), and independently Tsai (1990), describe how this method can be applied to data with one-sided truncation, i.e. either $y_i^-= -\infty,\ i=1,\ldots, N$ or $y_i^+ = \infty, \ i=1,\ldots,N$. For example, if $y_i^+ = \infty$ but $y_i^-$ varies with $x_i$ then the above procedure is modified as follows. For each object define the [*comparable*]{} or [*associated*]{} set \[eq:jdef\] J\_i = { j : y\_j > y\_i, y\_j\^- < y\_i } to consist of all objects of greater $y$ for which the value $y=y_i$ could possibly be observed. This is the same set as defined in Lynden-Bell’s $C^-$ method and, unlike the definition of Efron and Petrosian (1992, e.q. (2.9)), does not include the object in question. In the case of luminosity-redshift data this is the largest subset of luminosity (not magnitude or flux) and volume limited data that can be constructed for each given $(L_i, z_i)$. If $x$ and $y$ are independent then we expect the rank $R_i$ of $x_i$ in the eligible set, \[Ri\] R\_i = \# {j J\_i : x\_j x\_i}, to be uniformly distributed between $1$ and $N_i$, where $N_i$ is the number of points in $J_i$. The rest of the procedure follows as for untruncated data. The normalized statistic $T_i$ is defined here as $T_i = (R_i - E_i) / V_i$ where $E_i = {1 \over 2} (N_i + 1)$ and $V_i = {1 \over 12} (N_i^2 - 1)$. The test statistic $\tau$ is then defined by \[taudef\_1\] = [ [\_i ( R\_i - E\_i )]{} ]{} and is equivalent to a version of Kendell’s $\tau$ statistic defined by equation (\[eq:taudef\_2\]). In this case, however, we consider only the set of possible pairings allowed by truncation ${\cal P} = \{(i,j): y_i > y_j^-, y_j > y_i^-\}$.
### Multiply Truncated Data
A generalization of the above method to doubly (or multiply) truncated data was developed recently by Efron and Petrosian (1998). The method is equivalent to the previous method, with the eligible set defined as J\_i = { j : y\_j > y\_i, y\_i (y\_j\^-, y\_j\^+)} and the set of allowed pairings = { (i,j) : y\_i (y\^-\_j,y\^+\_j),y\_j (y\^-\_i,y\^+\_i) } defined such that each object lies within the truncation region of the other.
In this case, however, the distribution of the rankings (or of $\tau$) is unknown. If the data are uncorrelated then $\tau$ must still have a mean of zero and a bootstrap method may be used to determine the variance $V_\tau$ as follows. By assuming the data $\{x_i,y_i\}$ are uncorrelated one can use the methods of the next section to determine $\psi(y)$, the underlying (i.e. non-truncated) probability distribution of $y$. Once $\psi(y)$ is found one can simulate $N_{sim}$ sets of data with underlying probability density $\psi(y)$ and truncation limits $[y^-_i,y^+_i]$ for the $i^{th}$ object in each set. For each simulated set of data ${\cal D}_k$ one may find $\tau_k$ as in equation (\[eq:taudef\_2\]) and estimate $V_\tau$ from the distribution of $\{ \tau_k \}_{k=1}^{N_{sim}}$. For large numbers of simulations $N_{sim}$ the error in this determination of $V_\tau$ is approximately $V_\tau / \sqrt{N_{sim}}$. Given $V_\tau$ one can define a normalized test statistic $\tau / V_\tau$ with a mean of $0$ and a variance of $1$.
### The Luminosity Evolution
If $x$ and $y$, in this case the luminosity and redshift, prove to be independent, which would be the case if $|\tau| < 1$, one may assume that there is no luminosity evolution and proceed with the determination of the univariate distributions $\psi(L)$ and $\rho(z)$ of equation (\[eq:pde\]) using the methods described below. However, if $|\tau| \geq 1$ then $L$ and $z$ cannot be considered independent and one may assume that the most likely explanation is the presence of luminosity evolution ($g(z) \neq $ constant). Another possibility is the variation of the shape parameters $\alpha_i$ with $z$. We will return to this possibility below. One can determine the function $g(z)$ parametrically as follows.
Given a parametric form for luminosity evolution $g_k(z)$ one can transform the luminosities into $L_o (k) = L / g_k(z)$ and proceed with the determination of the correlation $\tau(k)$ between $L_o$ and $z$ as a function of the parameter $k$. If $\tau$ is normalized to have a standard deviation of $1$, then the values of $k$ allowed by the data at the $1 \ \sigma$ confidence level are $\{ k : |\tau(k)| < 1 \}$, and the most likely values of $k$ are those with $\tau(k) = 0$. Although in principle it is possible for the function $\tau (k)$ to have several zeroes, this did not occur for the specific cases described in $\S 5$.
Non-Parametric Determination of Distribution Functions
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Once a function $g$ is found that removes the correlation between $x$ and $y$ (or $z$ and $L$), the task is to find the underlying distributions $\rho(x)$ (the density evolution) and $\psi(y')$ (the local luminosity function) given uncorrelated data $\{ x_i,y'_i = y_i / g(x_i) \}_{i=1}^N$ and truncation limits $[y'^-_i,y'^+_i]$. If the truncation is one-sided (e.g. $y'^+_i = \infty$ for all $i$) then a variety of non-parametric methods can be used to determine the univariate distribution functions. As shown by Petrosian (1992) all non-parametric methods reduce to Lynden-Bell’s (1971) method. As with the tests of correlation described above, the gist of this method is to find for each point the comparable set defined in equation (\[eq:jdef\]) and the number $N_i$ of points in this set. For example, the cumulative distribution in $y'$, $\Phi(y') = \int^\infty _{y'} \psi(t) dt $, is given by (y’\_i) = \_[j < i]{} (1 + [1 N\_j]{}) . For doubly truncated data the comparable set is not completely observed, thus a simple analytic method such as the one described above is not possible. However, it turns out that a simple iterative procedure can lead to a maximum likelihood estimate of the distributions. Here we give a brief description of this method; for details the reader is referred to Efron and Petrosian (1998).
Assume that the underlying density function $\psi(y)$ is discretely distributed over the $N$ observed values of $y$. If we let $\psi_i = \psi(y_i)$ be the probability density at $y_i$ then $\Phi_i = \sum_j \psi_j$, where the summation includes all data points for which $y_j \in [y_i^-, y_i^+]$, is the total probability density for the truncation region $[y^-_i,y^+_i]$. If we define the matrix $\bf J$ by \[Jdef\] J\_[ij]{} = then the definition of $\Phi_i$ is equivalent to ${\bf \Phi} = {\bf J} \cdot {\bf \Psi}$ where ${\bf \Psi} = (\psi_1,\dots,\psi_N)$ and ${\bf \Phi} = (\Phi_1,\dots,\Phi_N)$. This matrix ${\bf J}$ contains all of the information about the data and the truncation limits needed to find the vector of probability densities ${\bf \Psi}$. We also have the normalization condition on ${\bf \Psi}$, \[norm\] \_[i=1]{}\^N \_i = 1 . From the definition of $\bf \Phi$ it follows that the conditional probability $\psi(y_i|[y^-_j,y^+_j])$ of observing a value $y_i$ within the truncation region $[y^-_j,y^+_j]$ is \[condf\] (y\_i|\[y\^-\_j,y\^+\_j\]) = . The final condition on ${\bf \Psi}$ is determined by maximizing the likelihood of observing the actual data, \[prob\] P\_[data]{} = \_[i=1]{}\^N (y\_i|\[y\^-\_i,y\^+\_i\]) = \_[i=1]{}\^N [\_i \_j J\_[ij]{} \_j]{}. By setting ${\partial P_{data} \over \partial {\bf \Psi}} = 0$ it follows that $\psi_k^{-1} = \sum_j J_{jk} \Phi_j^{-1}, k=1, \dots, N$. This may be written compactly as \[likely\] [1 ]{} = [**J**]{}\^ with the notation ${1 \over {\bf a}} = (a_1^{-1},\dots,a_N^{-1})$. Thus we have reduced the problem of finding the density function for data with arbitrary truncation to the “moral equivalent” of an eigenvalue problem for the matrix $\bf J$.
In practice, this condition may be used as a recursive formula to determine ${\bf \Psi}$. One starts with an initial guess for the density vector ${\bf \Psi}^o$. Equation (\[likely\]) then gives the recursion relation \[recursive\] [1 \^[(j+1)]{}]{} = [**J\^**]{} [1 \^[(j)]{}]{} + c\^[(j)]{} where the constant $c^{(j)}$ is determined by the normalization condition $\sum_i \psi^{(j+1)}_i = 1$. One may use as an initial guess the untruncated solution $\psi_i^o = {1 \over N}$. In most problems, however, one of the two truncations will have a more pronounced effect than the other. In this case, one may ignore the weaker truncation and use the result based on the one-sided method as an an initial guess. We found that with this initial guess and data confined to one region of the $L - z$ plane the sequence of ${\bf \Psi}^{(j)}$ defined by equation (\[recursive\]) usually converged quickly. However, for combined samples spanning different regions of the $L - z$ plane it was helpful to use an algorithm to accelerate the convergence of the series of ${\bf \Psi}^{(j)}$. For this purpose we used Aitken’s $\delta^2$ method, which gives an improved estimate for the terms of series by assuming approximately geometric convergence (see, e.g. Press et al. 1992, chapter 5.1).
COSMOLOGY AND MODELS OF LUMINOSITY EVOLUTION {#sec:cosmology}
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In order to determine the intrinsic parameters of an object from the observed data one must assume a certain cosmological model. Many cosmological models may be described in terms of a few fundamental parameters, which (see e.g. Peebles 1993) for a matter dominated (non-relativistic) universe are the matter density $\rho_o$, the cosmological constant $\Lambda$ and the curvature of space $k$ (which is $+1$, $0$ or $-1$ for closed, flat or open universes, respectively). Using Hubble’s constant $H_o$ these parameters may be written in dimensionless form as \_M = [8 G 3 H\_o \^2]{}, \_k = - [k c\^2 (H\_o R\_o)\^2 ]{} [and]{} \_= [3 H\_o\^2]{} where $R_o$ is the value of the expansion parameter of the universe at the present epoch. These three parameters are related via the Friedman-Lemaitre equation $\Omega_M + \Omega_k + \Omega_\Lambda = 1 $, allowing us to eliminate one of them in favor of Hubble’s constant. For example, the curvature term may be written \_k = 1 - (\_M + \_). We will consider the two classes of cosmological models given by $\Omega_\Lambda=0$ (no cosmological constant) and $\Omega_k = 0$ (flat universe with cosmological constant). For calculations of the luminosity function we will pay particular attention to the two cases $\Omega_k = \Omega_\Lambda = 0$ and $\Omega_\Lambda = 0.85$, $\Omega_M=0.15$. The first of these is the standard Einstein - de Sitter model, and the second is an inflationary model with parameters in accordance with current observations. For definiteness, Hubble’s constant was assumed to be $H_o = 70 \ {\rm km / (s \ Mpc)}$, although most results are independent of this assumption.
Once the values of the cosmological parameters are fixed, calculations of intrinsic parameters are relatively straightforward (Peebles 1993). The absolute luminosity, for example, takes the form \[eq:ldef\] L = f 4 d\_L\^2 K (z) where $f$ is the observed flux, the luminosity distance $d_L$ is d\_L = [c H\_o]{} (1 + z) [ ]{} and $K(z)$ is the K-correction term. The co-moving coordinate distance is u(z) = \_0\^z [dz ]{} and the co-moving volume contained within a sphere of radius corresponding to redshift $z$ is \[eq:vdef\] V(z) = 4 ([c H\_o]{}) \_0\^z [du dz]{} [d\_L\^2 (1+z)\^2]{} dz . In general, these integrals must be evaluated numerically.
In order to determine the K-correction term $K(z)$ one must make an assumption about the quasar spectrum in the optical region. The general practice here is to assume a power law spectrum $L_{optical} \propto \nu^\alpha$ with spectral index $\alpha \simeq -0.5$, which gives $K(z) = (1+z)^{1+\alpha} \simeq \sqrt {(1+z)}$.
Two models for luminosity evolution were used: $g_k(z) \propto e^{kt(z)}$ and $g_k(z) \propto (1+z)^k$. The first of these assumes an exponential dependence on the fractional lookback time $t(z)$, which is defined as $t(z) = 1 - {T(z) / T(0)}$, where $T(z)$ is the age of the universe at redshift $z$, T(z) = H\_o\^[-1]{} \_z\^ . The second model assumes a power law dependence on the scale factor of the universe (or the expansion parameter $R$), which is independent of the cosmological parameters. Analyses of earlier data (see e.g. Caditz and Petrosian 1990) have traditionally given estimates of quasar evolution for these two parametric forms of $g(z) \approx e^{7.5 t(z)}$ and $g(z) \approx (1+z)^{3}$.
THE EVOLUTION OF THE LUMINOSITY FUNCTION {#sec:results1}
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In what follows we apply the procedures of $\S 3$ to determine the correlation between the luminosities and redshifts and test parametric forms for the evolution of the luminosity, the function $g(z)$. Then we transform all luminosities to their present epoch values $L_o = L/g(z)$ and determine the co-moving density evolution $\rho(z)$ and the present epoch luminosity function $\psi(L_o)$ for the cosmological models described in $\S 4$. We apply these tests to the surveys described in $\S 2$ individually and in various combinations. Before presenting these results we discuss briefly the redshift - magnitude data, i.e. the Hubble diagram shown in Figure $1$.
The Quasar Hubble Diagram
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As is evident from Figure $1$, at first glance there seems to be very little correlation between the redshifts and magnitudes (or fluxes) of quasars; i.e. there is no obvious evidence for a Hubble type relation. This result is well known. For example, a preliminary test of correlation between $m$ and $z$ in a small subsample by Efron and Petrosian (1992, Figure 6) showed no correlation. Earlier, the absence of a clear Hubble relation was used as an argument against the cosmological origin of quasar redshift (see, e.g. Burbidge and O’Dell 1973). This is not the only possible interpretation, however. One would not expect a simple Hubble diagram for sources with a broad luminosity function (non-standard candle sources, see e.g. Petrosian 1974). The absence of an obvious Hubble relation can also arise from approximate cancelation between cosmological dimming and luminosity evolution. Exact cancelation of these two effects is highly implausible and could bring into question the basic assumptions about the distribution of sources. To clarify this situation we have applied the correlation tests described in $\S 3$ to the surveys of $\S 2$. First ignoring the high flux (low magnitude) limit, i.e. with $m_{min} = - \infty$, we use the one-sided tests and find the results labeled $\tau_1$ shown in Table 1. When using the double-sided tests we find the results $\tau_2$, which indicate slightly less correlation, as expected because the one-sided methods ignore the slight truncation induced by the high flux limits. These tests, when applied to the combined data sets, give a correlation of $\tau = 3.63$. This result rejects the hypothesis of independence between $B$ and $z$ at the $ 99.97 \%$ confidence level and is independent of any cosmological parameters. In addition, we may test the parametric fit $B(z) = B - \beta \log ( d^2_L(z, \Omega) K(z)) + $ constant for the data using the methods for multiple truncations to determine the best value of $\beta$, i.e. the value for which $B(z)$ and $z$ are uncorrelated ($\tau = 0$). A value of $\beta = 2.5$ is what one would expect for standard candle sources with a very narrow luminosity function, while a value of $\beta = 0$ would mean the complete absence of a Hubble relation and the exact cancelation described above. The results shown in Table $1$ indicate that $\beta$, while clearly less than $2.5$, differs significantly from $0$ for the cosmological models discussed in $\S 4$. The best parametric fit for the Einstein - de Sitter cosmological model ($\beta = 0.84$) is shown in Figure 1 along with the expected relation for standard candle sources ($\beta = 2.5$).
--------------- ------ ---------- ------- ---------- ------- ----------- -----------
Sample $N$ $\tau_1$ $P_1$ $\tau_2$ $P_2$ $\beta_1$ $\beta_2$
Durham/AAT 419 -0.65 48.71 -0.45 35.02 -0.22 -0.18
LBQS 871 4.65 99.99 3.75 99.98 1.11 0.92
HBQS 254 1.37 82.92 1.27 79.42 0.71 0.59
LBQS and HBQS 1125 4.80 99.99 3.98 99.99 1.01 0.83
Combined Data 1552 3.98 99.99 3.63 99.97 0.84 0.70
--------------- ------ ---------- ------- ---------- ------- ----------- -----------
We now turn to a determination of the evolution of the luminosity.
The Luminosity Evolution $g(z)$
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We examine the two commonly used parametric forms for the luminosity evolution, the exponential and power law forms. These two different forms emphasize the evolution in different regions of the luminosity $-$ redshift ($L-z$) plane. A correct parameterization will have the same value for its parameters when applied to samples with different limits (i.e. different coverage of the $L - z$ plane). This fact can be used to test a given parametric form. Table 2 summarizes the results described in the subsequent sections. Figure 2 gives an example of the variation of the test statistic $\tau$ as a function of the evolution parameter $k$. The optimal value of $k$, i.e. the value which indicates that $L_o$ and $z$ are not correlated, is given by the condition $\tau = 0$ and the $1 \ \sigma$ range of this parameter is given by the condition $|\tau| < 1$. These values are shown in Figure 2.
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Sample $N$ $k$ $k_{min}$ $k_{max}$ $k'$ $k'_{min}$ $k'_{max}$
Durham/AAT 419 8.72 6.66 10.07 3.53 2.57 5.05
HBQS 254 5.39 $- \infty$ 6.49 3.20 $-\infty$ 3.94
LBQS 871 4.28 2.66 5.17 2.02 1.24 2.53
Combined Data 1552 5.15 4.36 5.70 2.58 2.14 2.91
--------------- ------ ------ ------------ ----------- ------ ------------ ------------
### Evolution of the form $g_k(z) \propto e^{k t(z)}$
Figures 3a and 3b display the best values and the $1 \ \sigma$ ranges for the evolution parameter $k$ assuming both $\Omega_\Lambda = 0$ and $\Omega_k = 0$ cosmological models for the different samples. The Durham/AAT data exhibit much stronger evolution than the other data, indicating that this form of evolution does not adequately describe the data in the entire $L - z$ plane. The values of $k$ found for most of the data sets are considerably less than the previously determined value $k \approx 7.5$ by Caditz and Petrosian (1990), which was dominated by the Durham/AAT data.
### Evolution of the form $g_{k'}(z) \propto (1 + z) ^ {k'}$
Figures 4a and 4b give results for the evolution parameter $k'$ for the same cosmological models as above. The allowed ranges obtained from the different samples are much closer, thus this form of evolution is shown to be a closer approximation to the actual evolution than the exponential form. For the Einstein - de Sitter model, the best value of evolution parameter is $k' = 2.58$ with a one $\sigma$ range of $k' \in [2.14,2.91]$. As with the previous case, the values of $k'$ are somewhat less than the previously found value of $k' \approx 3$ of Caditz and Petrosian (1990).
It is clear that a better fit can be achieved with a different functional form for $g(z)$ with two or more parameters. However, in what follows we assume the simpler form of evolution $g(z) \propto (1+z)^{k'}$ with $k' (\Omega)$ the optimal value of $k'$ for a given cosmological model $\Omega$. We therefore transform the data to $\{[L_{o \\ i},z_i]\}_{i=1}^N$ with $L_o(z,m,\Omega) = L(z,m,\Omega) / (1+z)^{k'}$ and apply the method of $\S 3$ to find non-parametric estimates for the density functions $\rho(z)$ and $\psi(L_o)$. This method now gives directly the cumulative functions $\sigma (z) = \int_0^z \rho (z) {dV \over dz} dz$ and $\Phi (L_o) = \int _{L'}^\infty \psi(L') dL'$.
The Density Evolution $\rho(z)$
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The cumulative density function $\sigma(z)$ is the total number of objects within the angular area of the survey up to redshift $z$. If there is no density evolution, i.e. $\rho(z) = \rho_o$ is a constant, then $\sigma \propto V$ where $V(z)$ is the co-moving volume up to redshift $z$. We determine $\sigma (z)$ and $\rho(z)$ using the new method for doubly truncated data. In order to determine if density evolution exists we fit $\sigma$ to $V$ by a simple power law $\sigma(z) \propto V^\lambda$, where $\lambda \not = 1$ indicates the presence of density evolution. If the density increases with redshift we expect $\lambda > 1$ and if the density decreases with redshift we expect $\lambda < 1$. Even if $\lambda = 1$, however, density evolution may be present: the density may increase and decrease in such a way as to cancel and give a fit of $\lambda = 1$. Figures 5a and 5b show the variation of $\sigma$ and $\rho$ with $V$ for the combined sample for three different cosmological models. The dotted lines show the best fits to the form $\sigma \propto [V(z,\Omega)]^\lambda$. For the Einstein - de Sitter model (top curves in Figure 5) we have $\lambda = 1.19$, indicating that the co-moving density increases with redshift roughly as $\sigma \propto V^{1.19}$. This would indicate a simple power law density evolution $\rho \propto V^{0.19}$. The density evolution shown in Figure 5b exhibits this average behavior, but in detail is more complex: the density increases more rapidly at low $z$, reaches a plateau at $z\approx 2$ and possibly decreases at higher $z$. This behavior will be discussed below in more detail and for higher redshift data.
As mentioned in §1, one cannot determine the evolution of sources (the function $\rho (z)$) and the evolution of the universe (the parameters $\Omega_i$) simultaneously. Given one of these (e.g. the cosmological model), the other (the density evolution) may be determined from the data. The variation of $\lambda = d \ln \sigma / d \ln V$ with $\Omega_M$ for the two different classes of cosmological models is shown in Figures 6a and 6b. It is evident that there is a monotonic variation of $\lambda$ with $\Omega_M$. In particular, $\lambda = 1$ for the two cosmological models: $\Omega_M \approx 0$, $\Omega_\Lambda \approx 1$ and $\Omega_M \approx 0.15$, $\Omega_\Lambda \approx 0.85$. This second set of parameters is quite close to those currently favored by many observations. As above, this does not imply the complete absence of density evolution. The lower two curves in Figure 5b show the variation of $\rho$ for these two models. Clearly, there is less variation than for the Einstein - de Sitter model, but the general behavior is similar.
To further analyze the variation of $\rho(z)$, in Figure 7 we show our non-parametric determination of $\rho(z)$ for the Einstein - de Sitter cosmological model. Two sets of results are given. The first of these (depicted by squares) shows $\rho(z)$ in the region $0.3 < z < 2.2$ from the combined data. The second (triangles) shows $\rho(z)$ in the region $0.3 < z < 3.3$ from the LBQS data (which do not have a high redshift cutoff at $z = 2.2$) alone. As evident in Figure 7, the density increases relatively slowly at low redshift ($\rho \sim (1 + z)^{2.5}$) before reaching a peak at $z \approx 2$ and decreasing rapidly ($\rho \sim (1 + z)^{-5}$) at higher redshift. As discussed further in §5.5, the decrease in density present in this data at redshift of about 2 is in agreement with high redshift survey results (Schmidt et al. 1995, Warren et al. 1994).
The Luminosity Function $\psi(L_o)$
-----------------------------------
In a similar fashion we may obtain the cumulative luminosity function $\Phi(L_o)$ from the uncorrelated data set $\{L_o,z\}$. Figures 8a and 8b show $\Phi(L_o)$ for the combined data set with $k'=2.58$, along with the best fits to a double power law form \[eq:doublepow\] (L\_o) = [\_o (L\_o / L\_)\^[k\_1]{} + (L\_o / L\_)\^[k\_2]{}]{} . We used the cosmological models $\Omega_M=1$, $\Omega_\Lambda =0$ (Einstein - de Sitter model) and $\Omega_M = 0.15$, $\Omega_\Lambda = 0.85$ (pure luminosity evolution with cosmological constant). In both cases, the results for the combined samples exhibit roughly double power law dependence on $L_o$ with similar values of the fitting parameters. The primary differences between the two models are that the Einstein - de Sitter model gives a slightly gentler slope above the break luminosity, $k_2 = 3.17$ as opposed to $k_2 = 3.59$, and a lower break luminosity, $L_\ast = 6.19 \times 10^{29}$ erg / (sec Hz) as opposed to $L_\ast = 9.48 \times 10^{29}$ erg / (sec Hz).
We check for possible variation in the shape of $\psi(L_o)$ by dividing the data into three redshift bins: $ 0.3 < z < 0.86 $, $ 0.86 < z < 1.48 $, and $ 1.48 < z < 2.2$. We then find the differential luminosity function $\psi(L_o)$ for these three redshift bins, first assuming no luminosity evolution ($g(z) = $ constant) and then assuming luminosity evolution $g(z) \propto (1+z)^{k'}$. These luminosity functions (with arbitrary vertical normalization) are shown in Figures 9a and 9b, respectively. The presence of a strong shift to higher luminosities is clearly evident for $g(z) = $ constant. However, when the evolution $g(z) \propto (1+z)^{k'}$ is taken out the luminosity function seems to exhibit little variation; the slopes appear roughly the same at low $L_o$ and high $L_o$ and the break luminosity does not vary as much with redshift. Although imprecise, these results indicate that our choice of $g(z)$ removes most of the variation of the parameters $\alpha_i$ with redshift, i.e. the shape of the luminosity function is almost invariant.
The Luminosity Density ${\cal L} (z)$
-------------------------------------
Finally, we determine the luminosity density function ${\cal L} (z)$. This quantity is defined as the total rate of energy production by quasars in the optical range as a function of redshift; ${\cal L} (z) = \int_0^\infty L \psi(L) dL$. If the shape of the luminosity function is invariant then ${\cal L} (z) \propto \rho(z) g(z)$. This rate depends in a complicated way on the distribution of masses of the central black holes and the variation of the accretion rate, both of which are related to the formation of galaxies and their evolution through mergers or collisions. Using the above results we can evaluate ${\cal L}(z)$ up to a redshift of $2.2$. We extend this further to a redshift of $3.3$ using the LBQS data, which is claimed to be complete up to this redshift. These results, with arbitrary vertical normalization, are shown in Figure 10. We first note the good agreement in the $z < 2$ region, indicating that perhaps the LBQS result at higher redshift is a representative behavior. We may also use high redshift surveys of quasars (Schmidt et al. 1995, Warren et al. 1994) to study ${\cal L}(z)$ in this range. Unfortunately, the selection of high $z$ quasars in these samples is more complicated and the subsequent analyses involve more assumptions. For example, Schmidt et al. use the $V / V_{max}$ method to determine $\rho(z)$, tacitly assuming that $g(z) = $ constant (as well as $\alpha_i = $ constant) so that ${\cal L} (z) \propto \rho (z)$. We show these results (again with arbitrary vertical normalization) in Figure 10. These results agree with the general trend of decline in ${\cal L} (z)$ at high redshifts.
It has been claimed (Cavaliere and Vittorini 1998, Shaver et al. 1998) that this rise and fall of ${\cal L} (z)$ with redshift is similar to the behavior of the star formation rate (SFR), which has recently been extended to high redshift (see, e.g. Madau 1997). We have shown this rate in Figure 10 as well. Although the general trend of rise and fall of the SFR and ${\cal L} (z)$ is the same, there is considerable difference in the detailed variation. The similarity may indicate some relation between the SFR and the feeding of the central engine of the quasars (e.g. both are affected by mergers). However, considering the many differences between star formation and the generation of energy by quasars, the observed difference between the SFR and ${\cal L} (z)$ in Figure 10 is not surprising.
SUMMARY AND CONCLUSIONS {#sec:discussion}
=======================
Although there have been several analyses of quasar evolution in the past, our results differ from these previous results in two important respects:
$\bullet$ We have used non-parametric statistical methods for multiply truncated data that allow us to combine samples with different selection criteria.
$\bullet$ We have used the data to study models of the luminosity function that take into account both luminosity evolution $g(z)$ and density evolution $\rho (z)$.
The new non-parametric statistical methods differ from those used in the past in the following ways:
$\bullet$ No binning is required and most of the characteristics of the distribution functions are determined non-parametrically.
$\bullet$ The methods are not limited to simply truncated data such as that found in flux limited surveys and can account for selection biases in generally truncated data where each data point has different truncation limits. In particular, these methods can treat samples with both upper and lower flux limits and redshift limits.
$\bullet$ This versatility allows one to combine data from different surveys with different selection criteria.
$\bullet$ The first of our techniques, a generalized non-parametric test of independence, allows one to determine the degree of correlation between luminosity and redshift, giving an indication of the luminosity evolution in the luminosity function. The evolution may then be determined parametrically.
$\bullet$ The second of our techniques provides a non-parametric estimate for the univariate distributions in redshift and luminosity, i.e. the co-moving density evolution and the local luminosity function, respectively.
We have applied these methods to the combined data from several large surveys and determined the luminosity evolution $g(z)$, the density evolution $\rho(z)$ and the luminosity function $\psi(L_o = L/g(z))$ of the generalized luminosity function of equation (\[eq:gle\]) for flat ($\Omega_k = 0$ and $\Omega_M = 1 - \Omega_\Lambda$) and zero cosmological constant ($\Omega_\Lambda = 0$ and $\Omega_M = 1 - \Omega_k$) cosmological models. We assume a shape invariant luminosity function, $\alpha_i = $ constant. More complex luminosity functions, $\alpha_i \not = $ constant, or those with luminosity dependent density evolution, etc., can be tested if the simpler prescription used here is not consistent with all of the data. We found that the scenario of equation (\[eq:gle\]) provides an adequate description of the existing data.
Our results may be summarized as follows:
$\bullet$ We found a strong correlation between luminosity and redshift, indicating the presence of rapid luminosity evolution.
$\bullet$ The parametric model of luminosity evolution $(1+z)^{k'}$ provides a better description of the data than the model $e^{k t(z)}$, although neither parameterization perfectly removes the correlation in all areas of the $L - z$ plane. In order to better model this evolution future analyses of quasar evolution could consider other parametric forms, including those with more than one free parameter.
$\bullet$ The cumulative co-moving density of quasars may be approximately modeled as $\sigma (z) \propto V^\lambda$, where the value of $\lambda$ depends on the cosmological model. For example, $\lambda = 1.19$ for the Einstein - de Sitter model and $\lambda = 1$ for the cosmological models with $\Omega_M \approx 0$, $\Omega_k \approx 1$ and $\Omega_M \approx 0.15$, $\Omega_\Lambda \approx 0.85$. This simple parameterization $\sigma \propto V^\lambda$ does not describe precisely the variation of $\rho$ with redshift. When examined in greater detail, the co-moving density shows a relatively slow increase ($\rho \sim (1 + z)^{2.5}$) for low redshifts and a rapid decline ($\rho \sim (1 + z)^{-5}$) for $z > 2$ (for the Einstein - de Sitter model). This is in qualitative agreement with the observed density evolution of high $z$ quasars (Schmidt et al. 1995, Warren et al. 1994). Qualitatively similar behavior is found even for models that show no overall density evolution ($\lambda = 1$). A more rigorous comparison cannot be made at this stage because the analysis of the high $z$ data ignores possible $L - z$ correlation and assumes pure density evolution.
$\bullet$ The cumulative local luminosity function $\Phi(L_o) = \int _{L_o} ^ \infty \psi(x) dx$ has a double power law form. In the Einstein - de Sitter model the break luminosity is $L_\ast = 6 \times 10^{29}$ erg / (sec Hz) and the low and high luminosity power law indices are $k_1 = 1.05$ and $k_2 = 3.17$. There appears to be little variation with redshift of the shape of the cumulative and differential luminosity functions, thus the $\alpha_i$ = constant prescription seems adequate. With more data one could determine precisely the variation with redshift of the shape of $\phi(L_o)$.
$\bullet$ The above description of the luminosity function allows us to determine the rate of energy generation per unit co-moving volume of quasars as a function of redshift. We show that this function ${\cal L}(z) \propto \rho (z) g(z)$ increases rapidly with $z$ at low redshift, peaks around $z \approx 2$ and then decreases. This is also in rough agreement with the high $z$ survey results mentioned above. This variation of ${\cal L}(z)$ is similar to but significantly different from recent determinations of the star formation rate.
We would like to thank Bradley Efron for his invaluable help with the statistical methods used in this paper, Paul Hewett for help with the analysis of the LBQS data and Nicole Lloyd for helpful discussions. A. M. would like to acknowledge support from the Stanford University Undergraduate Research Opportunity program.
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---
abstract: 'We study the deformations of a smooth curve $C$ on a smooth projective $3$-fold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of $C$ in $V$ to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, we give a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of $(-2)$-curves and elliptic curves on $S$. Applying this result, we prove that the Hilbert scheme ${\operatorname{Hilb}}^{sc} V_4$ of smooth connected curves on a smooth quartic $3$-fold $V_4 \subset \mathbb P^4$ contains infinitely many generically non-reduced irreducible components, which are variations of Mumford’s example for ${\operatorname{Hilb}}^{sc} \mathbb P^3$.'
address: ' Department of Mathematical Sciences, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, JAPAN'
author:
- Hirokazu Nasu
title: |
Obstructions to deforming curves on a 3-fold, III:\
Deformations of curves lying on a $K3$ surface
---
Introduction
============
Let $V$ be a smooth projective $3$-fold over an algebraically closed field $k$. This paper is a sequel to the preceding papers [@Mukai-Nasu; @Nasu4], in which the embedded deformations of a smooth curve $C$ in $V$ have been studied under the presence of an intermediate smooth surface $S$ satisfying $C \subset S \subset V$. As is well known, first order (infinitesimal) deformations $\tilde C \subset V \times {\operatorname{Spec}}k[t]/(t^2)$ of $C$ in $V$ are in one-to-one correspondence with global sections $\alpha$ of the normal bundle $N_{C/V}$ of $C$ in $V$. Then the obstruction ${\operatorname{ob}}(\alpha)$ to lifting $\tilde C$ to a second order deformation $\tilde {\tilde C} \subset V \times {\operatorname{Spec}}k[t]/(t^3)$ of $C$ in $V$ is contained in $H^1(C,N_{C/V})$, and ${\operatorname{ob}}(\alpha)$ is computed as a cup product $\alpha \cup \alpha$ by the map $$H^0(C,N_{C/V})\times H^0(C,N_{C/V}) \overset{\cup}\longrightarrow H^1(C,N_{C/V})$$ (cf. Theorem \[thm:original cup product\]). It is generally difficult to compute ${\operatorname{ob}}(\alpha)$ directly. Mukai and Nasu [@Mukai-Nasu] introduced [*the exterior components*]{} of $\alpha$ and ${\operatorname{ob}}(\alpha)$, which are defined as the images of $\alpha$ and ${\operatorname{ob}}(\alpha)$ in $H^i(C,N_{S/V}\big{\vert}_C)$ ($i=0,1$) by the natural projection $\pi_{C/S}: N_{C/V} \rightarrow N_{S/V}\big{\vert}_C$, and denoted by $\pi_{C/S}(\alpha)$ and ${\operatorname{ob}}_{S}(\alpha)$, respectively (cf. §\[subsec:exterior\]). They gave a sufficient condition for ${\operatorname{ob}}_S(\alpha)$ to be nonzero, which implies the non-liftability of $\tilde C$ to any $\tilde {\tilde C}$.
In this paper, we generalize their result and give a weaker condition for ${\operatorname{ob}}_S(\alpha)\ne 0$. Let $\tilde C$ or $\alpha \in H^0(C,N_{C/V})$ be a first order deformation of $C$ in $V$, and $\pi_{C/S}(\alpha) \in H^0(C,N_{S/V}\big{\vert}_C)$ the exterior component of $\alpha$. Suppose that the image of $\pi_{C/S}(\alpha)$ in $H^0(C,N_{S/V}(mE)\big{\vert}_C)$ lifts to a section $$\beta \in H^0(S,N_{S/V}(mE))
\qquad$$ for an integer $m \ge 1$ and an effective Cartier divisor $E$ on $S$. In other words, we have $\pi_{C/S}(\alpha)= \beta\big{\vert}_C$ in $H^0(C,N_{S/V}(mE)\big{\vert}_C)$. Here $\beta$ is called an [*infinitesimal deformations with a pole*]{} (along $E$) of $S$ in $V$ (cf. §\[subsec:with pole\]). The following is a generalization of [@Mukai-Nasu Theorem 2.2], in which, it was assumed that $m=1$, $E$ is smooth and irreducible with negative self-intersection number $E^2<0$ on $S$, and furthermore, $E$ was assumed to be a $(-1)$-curve on $S$ (i.e. $E \simeq \mathbb P^1$ and $E^2=-1$) in its application.
\[thm:main1\] Let $\tilde C$, $\alpha$, $\beta$ be as above. Suppose that the natural map $H^1(S,\mathcal O_S(kE))\rightarrow H^1(S,\mathcal O_S((k+1)E))$ is injective for every integer $k \ge 1$. If the following conditions are satisfied, then the exterior component ${\operatorname{ob}}_S(\alpha)$ of ${\operatorname{ob}}(\alpha)$ is nonzero:
1. the restriction map $
H^0(S,\Delta) \overset{\vert_E}\longrightarrow
H^0(E,\Delta\big{\vert}_E)
$ is surjective for $\Delta:=C+K_V\big{\vert}_S-2mE$, a divisor on $S$, and
2. we have $$m\partial_E(\beta\big{\vert}_E) \cup
\beta\big{\vert}_E \ne 0
\qquad
\mbox{in}
\qquad
H^1(E,N_{S/V}((2m+1)E-C)\big{\vert}_E),$$ where $\beta\big{\vert}_E \in H^0(E,N_{S/V}(mE)\big{\vert}_E)$ is the principal part of $\beta$ along the pole $E$, and $\partial_E$ is the coboundary map of the exact sequence $$\label{ses:normal bundle of E}
[0 \longrightarrow \underbrace{N_{E/S}}_{\simeq \mathcal O_E(E)}
\longrightarrow N_{E/V}
\overset{\pi_{E/S}}\longrightarrow N_{S/V}\big{\vert}_E
\longrightarrow 0]
\otimes_{\mathcal O_E} \mathcal O_E(mE).$$
The relation between $\alpha$ and $\beta\big{\vert}_E$ is explained with Figure \[fig:relation\] in §\[sect:obstruction\].
Given a projective scheme $V$, let ${\operatorname{Hilb}}^{sc} V$ denote the Hilbert scheme of smooth connected curves in $V$. Mumford [@Mumford] first proved that ${\operatorname{Hilb}}^{sc} \mathbb P^3$ contains a generically non-reduced (irreducible) component. Later, many examples of such non-reduced components of ${\operatorname{Hilb}}^{sc} \mathbb P^3$ were found in [@Kleppe87; @Ellia87; @Gruson-Peskine; @Floystad; @Nasu1], etc. More recently, Mumford’s example was generalized in [@Mukai-Nasu] and it was proved that for many uniruled $3$-folds $V$, ${\operatorname{Hilb}}^{sc} V$ contains infinitely many generically non-reduced components. (See [@Vakil] for a different generalization.) In the construction of the components, $(-1)$-curves $E \subset V$ on a surface $S \subset V$ play a very important role. In this paper, as an application we study the deformations of curves $C$ on a smooth Fano $3$-fold $V$ when $C$ is contained in a smooth $K3$ surface $S\subset V$. On a $K3$ surface, $(-2)$-curves $E$ (i.e. $E \simeq \mathbb P^1$ and $E^2=-2$) and elliptic curves $F$ (then $F^2=0$) play a role very similar to that of a $(-1)$-curve. If we have $m=1$ in the exact sequence , then the sheaf homomorphism $\pi_{E/S}\otimes_{\mathcal O_E} \mathcal O_E(E)$ tensored with $\mathcal O_E(E)$ induces a map (called the “$\pi$-map” for $(E,S)$) $$\label{map:pi-map}
\pi_{E/S}(E):
H^0(E,N_{E/V}(E))
\longrightarrow
H^0(E,N_{S/V}(E)\big{\vert}_E)$$ on the cohomology groups. From now on, we assume that ${\operatorname{char}}(k)=0$.
\[thm:main2\] Let $V$ be a smooth Fano $3$-fold, $S \subset V$ a smooth $K3$ surface, and $C \subset S$ a smooth connected curve, and put $D:=C+K_V\big{\vert}_S$ a divisor on $S$. Suppose that there exists a first order deformation $\tilde S$ of $S$ which does not contain any first order deformations $\tilde C$ of $C$.
1. If $D \ge 0$ and there exist no $(-2)$-curves and no elliptic curves on $S$, or more generally, if $H^1(S,D)=0$, then ${\operatorname{Hilb}}^{sc} V$ is nonsingular at $[C]$.
2. If $D \ge 0$, $D^2 \ge 0$ and there exists a $(-2)$-curve $E$ on $S$ such that $E.D=-2$ and $H^1(S,D-3E)=0$, then we have $h^1(S,D)=1$. If moreover, $\pi_{E/S}(E)$ is not surjective, then ${\operatorname{Hilb}}^{sc} V$ is singular at $[C]$.
3. If there exists an elliptic curve $F$ on $S$ such that $D \sim mF$ for an integer $m \ge 2$, then we have $h^1(S,D)=m-1$. If moreover, $\pi_{F/S}(F)$ is not surjective, then ${\operatorname{Hilb}}^{sc} V$ is singular at $[C]$.
Note that $H^1(S,D)\simeq H^1(S,N_{S/V}(-C))^{\vee}$ (cf. ). If $H^1(S,D)=0$, then $C$ is unobstructed in $V$. By using Theorem \[thm:main1\], we partially prove that $C$ is obstructed in $V$ if $H^1(S,D)\ne 0$. Under the assumption of Theorem \[thm:main2\], the Hilbert-flag scheme ${\operatorname{HF}}V$ of $V$ (cf. §\[subsec:flag schemes\]) is nonsingular at $(C,S)$ (cf. Lemma \[lem:flag of k3fano\]). Therefore $(C,S)$ belongs to a unique irreducible component $\mathcal W_{C,S}$ of ${\operatorname{HF}}^{sc} V$. The image $W_{C,S}$ of $\mathcal W_{C,S}$ in ${\operatorname{Hilb}}^{sc} V$ is called the [*$S$-maximal family*]{} of curves containing $C$ (cf. Definition \[dfn:S-maximal\]). Then $W_{C,S}$ is of codimension $h^1(S,D)$ in the tangent space $H^0(C,N_{C/V})$ of ${\operatorname{Hilb}}^{sc} V$ at $[C]$ (cf. ).
\[cor:main3\] In (1),(2),(3) of Theorem \[thm:main2\], we have furthermore that
1. If $h^1(S,D)\le 1$, then $W_{C,S}$ is an irreducible component of $({\operatorname{Hilb}}^{sc} V)_{{\operatorname{red}}}$.
2. ${\operatorname{Hilb}}^{sc} V$ is generically smooth along $W_{C,S}$ if $h^1(S,D)=0$, and generically non-reduced along $W_{C,S}$ if $h^1(S,D)=1$.
3. If $H^0(S,-D)=0$, then $\dim_{[C]} {\operatorname{Hilb}}^{sc} V
=(-K_V\big{\vert}_S)^2/2+g(C)+1$, where $g(C)$ is the genus of $C$.
The following is a simplification or a variation of Mumford’s example. (See Examples \[ex:non-reduced V\_4\] and \[ex:non-reduced P\^3\] for more examples.)
\[ex:non-reduced\] In the following examples, the closure $\overline W$ of $W$ is an irreducible component of $({\operatorname{Hilb}}^{sc} V)_{{\operatorname{red}}}$ and ${\operatorname{Hilb}}^{sc} V$ is generically non-reduced along $W$. We have $h^0(C,N_{C/V})=\dim W+1$ at the generic point $[C]$ of $W$.
1. Let $V$ be a smooth quartic $3$-fold $V_4 \subset \mathbb P^4$, $E$ a smooth conic on $V$ with trivial normal bundle $N_{E/V} \simeq \mathcal O_E^{2}$, $S$ a smooth hyperplane section of $V$ containing $E$ and such that ${\operatorname{Pic}}S=\mathbb Z\mathbf h\oplus \mathbb Z E$, where $\mathbf h\sim \mathcal O_S(1)$. Then a general member $C$ of the complete linear system $|2\mathbf h+2E|$ on $S$ is a smooth connected curve of degree $12$ and genus $13$. Such curves $C$ are parametrised by a locally closed irreducible subset $W$ of ${\operatorname{Hilb}}^{sc} V$ of dimension $16$.
2. Let $V=\mathbb P^3$ and let $F$ be a smooth plane cubic (elliptic) curve, $S$ a smooth quartic surface containing $F$. Then a general member $C$ of $|4\mathbf h+2F|$ ($\mathbf h \sim \mathcal O_S(1)$) is a smooth connected curve of degree $22$ and genus $57$ on $S$. Such curves $C$ are parametrised by a locally closed irreducible subset $W$ of ${\operatorname{Hilb}}^{sc} \mathbb P^3$ of dimension $90$.
The organization of this paper is as follows. The proof of Theorem \[thm:main1\] heavily depends on the analysis of the singularity of [*polar $d$-maps*]{}. Given a projective scheme $V$ and its hypersurface $S \subset V$, there exists a so-called “Hilbert-Picard” morphism $\psi_{S}: {\operatorname{Hilb}}^{cd} V \rightarrow {\operatorname{Pic}}S$ from the Hilbert scheme of effective Cartier divisors on $V$ to the Picard scheme of $S$, sending a hypersurface $S' \subset V$ to the invertible sheaf $\mathcal O_V(S')\big{\vert}_S$ on $S$ (cf. §\[subsec:hypersurface\]). The tangent map $d_S: H^0(S,N_{S/V}) \rightarrow H^1(S,\mathcal O_S)$ of $\psi_S$ at $[S]$ is called the [*$d$-map*]{} for $S\subset V$. In §\[subsec:with pole\], we show that this map is extended into a version $d_{S^\circ}: H^0(S,N_{S/V}(mE)) \rightarrow H^1(S,\mathcal O_S((m+1)E))$ with a pole along a divisor $E\ge 0$ on $S$. We also prove that for any $\beta \in H^0(S,N_{S/V}(mE))$ the restriction $d_{S^\circ}(\beta)\big{\vert}_E$ to $E$ of the image $d_{S^\circ}(\beta)$ coincides with the coboundary image $\partial_E(\beta\big{\vert}_E)$ of up to constant (cf. Proposition \[prop:key\]). In §\[sect:obstruction\], applying this result to a $3$-fold $V$, we prove Theorem \[thm:main1\]. In § \[sect:k3\], we prove Theorem \[thm:main2\] and Corollary \[cor:main3\] as an application of Theorem \[thm:main1\]. In §\[subsec:mori cone of quartics\], we study the Mori cone ${\overline{\operatorname{NE}}}(S_4)$ of a smooth quartic surface $S_4$ of Picard number $2$ (cf. Lemmas \[lem:mori cone1\], \[lem:mori cone2\]). Applying the results on Mori cones, in §\[subsec:hilb\], for curves $C$ lying on $S_4$, we study the deformations of $C$ in $\mathbb P^3$ and $C$ in a smooth quartic $3$-fold $V_4 \subset \mathbb P^4$. In particular, we give a sufficient condition for $W_{C,{S_4}}$ to be a generically non-reduced (or generically smooth) component of ${\operatorname{Hilb}}^{sc} \mathbb P^3$ and ${\operatorname{Hilb}}^{sc} V_4$ (cf. Theorems \[thm:Hilb with rational curve\], \[thm:Hilb with elliptic curve\] and \[thm:Hilb without rational nor elliptic curve\]).
Preliminaries {#sect:preliminarlies}
=============
We start by recalling some basic facts on the deformation theory of closed subschemes, as well as setting up Notations. We work over an algebraically closed field $k$ of characteristic $p \ge 0$. Let $V\subset \mathbb P^n$ be a closed subscheme of $\mathbb P^n$ with the embedding invertible sheaf $\mathcal O_V(1)$ on $V$, and $X \subset V$ a closed subscheme of $V$ with the Hilbert polynomial $P_X=\chi(\mathcal O_X(n))$. Then as is well known, the [*Hilbert scheme*]{} ${\operatorname{Hilb}}_{P} V$ of $V$ parametrises all closed subscheme $X'$ of $V$ with $P_{X'}=P_X$ (cf. [@Grothendieck]). We denote by ${\operatorname{Hilb}}V$ the (full) Hilbert scheme $\bigsqcup_{P} {\operatorname{Hilb}}_P V$ of $V$. Let $\mathcal I_X$ and $N_{X/V}=(\mathcal I_X/\mathcal I_X^2)^{\vee}$ denote the ideal sheaf and the normal sheaf of $X$ in $V$, respectively. The symbol $[X]$ represents the point of ${\operatorname{Hilb}}V$ corresponding to $X$. Then the tangent space of ${\operatorname{Hilb}}V$ at $[X]$ is known to be isomorphic to the group ${\operatorname{Hom}}(\mathcal I_X,\mathcal O_X)$ of sheaf homomorphisms from $\mathcal I_X$ to $\mathcal O_X$, which is isomorphic to $H^0(X,N_{X/V})$. Every obstruction to deforming $X$ in $V$ is contained in the group ${\operatorname{Ext}}^1(\mathcal I_X,\mathcal O_X)$, and if $X$ is a locally complete intersection in $V$, then it is contained in a smaller subgroup $H^1(X,N_{X/V}) \subset {\operatorname{Ext}}^1(\mathcal I_X,\mathcal O_X)$. If $H^1(X,N_{X/V})=0$, then ${\operatorname{Hilb}}V$ is nonsingular at $[X]$ of dimension $h^0(X,N_{X/V})$. A [*first order (infinitesimal) deformation*]{} of $X$ in $V$ is a closed subscheme $X' \subset X \times {\operatorname{Spec}}D$, flat over the ring $D=k[t]/(t^2)$ of dual numbers, with a central fiber $X'_0=X$. By the universal property of the Hilbert scheme, there exists a one-to-one correspondence between the set of $D$-valued points $\gamma: {\operatorname{Spec}}D \rightarrow {\operatorname{Hilb}}V$ sending $0$ to $[X]$, and the set of the first order deformations of $X$ in $V$. By the infinitesimal lifting property of smoothness (cf. [@Hartshorne10 Proposition 4.4, Chap. 1]), if there exists a first order deformation of $X$ in $V$ not liftable to a deformation over ${\operatorname{Spec}}k[t]/(t^n)$ for some $n \ge 3$, then ${\operatorname{Hilb}}V$ is singular at $[X]$. We say $X$ is [*unobstructed*]{} (resp. [*obstructed*]{}) in $V$ if ${\operatorname{Hilb}}V$ is nonsingular (resp. singular) at $[X]$, and for an irreducible closed subset $W$ of ${\operatorname{Hilb}}V$, we say ${\operatorname{Hilb}}V$ is [*generically smooth*]{} (resp. [*generically non-reduced*]{}) along $W$ if ${\operatorname{Hilb}}V$ is nonsingular (resp. singular) at the generic point $X_{\eta}$ of $W$.
Primary obstructions {#subsec:primary}
--------------------
Let $V$ be a (projective) scheme over $k$, $X$ a closed subscheme of $V$, $\alpha$ a global section of $N_{X/V}$. We define a cup product ${\operatorname{ob}}(\alpha) \in {\operatorname{Ext}}^1(\mathcal I_X,\mathcal O_X)$ by $${\operatorname{ob}}(\alpha):= \alpha \cup \mathbf e \cup \alpha,$$ where $\mathbf e \in {\operatorname{Ext}}^1(\mathcal O_X,\mathcal I_X)$ is the extension class of the standard short exact sequence $$\label{ses:standard}
0 \longrightarrow \mathcal I_X \longrightarrow \mathcal O_V
\longrightarrow \mathcal O_X \longrightarrow 0$$ on $V$. Though the following fact is well-known to the experts, we give a proof for the reader’s convenience.
\[thm:original cup product\] Let $\tilde X$ be a first order deformation of $X$ in $V$ corresponding to $\alpha$. If $X$ is a locally complete intersection in $V$, then $\tilde X$ lifts to a deformation ${\tilde {\tilde X}}$ over $k[t]/(t^3)$, if and only if ${\operatorname{ob}}(\alpha)$ is zero.
First we fix some notations for the proof. Let $\mathfrak U:=\left\{U_i \bigm| i \in I\right\}$ be an open affine covering of $V$, $R_i$ the coordinate ring of $U_i$, $I_i$ the defining ideal of $X\cap U_i$ in $U_i$. We take a covering $\mathfrak U$ such that for all $i,j$, (i) the intersections $U_{ij}:=U_i \cap U_j$ are affine, and (ii) $I_i$ are generated by $m$ elements $f_{i1},\dots,f_{im}$ in $R_i$, where $m$ denotes the codimension of $X$ in $V$. (Such covering exists by assumption.) Let $R_{ij}$ be the coordinate ring of $U_{ij}$. Then since $I_i$ and $I_j$ agree on the overlap $U_{ij}$, there exists a $m\times m$ matrix $A_{ij}$ with entries in $R_{ij}$ (i.e., $A_{ij} \in M(m,R_{ij})$) such that $$\label{eqn:transition1}
\mathbf f_j
=A_{ij} \mathbf f_i, \quad
\mbox{where $\mathbf f_i:=
\begin{pmatrix}
f_{i1} \\ \vdots \\ f_{im}
\end{pmatrix}
$}.$$ Here and later, for a ring $R$, we denote by $M(m,R)$ the set of $m\times m$ matrices with entries in $R$. For an object $o$ in $V$, we denote by $\overline o$ the restriction of $o$ to $X$. For example, if $u$ is a section of a sheaf $\mathcal F$ on $V$, $\overline u$ denotes the image of $u$ in $\mathcal F\big{\vert}_X=\mathcal F\otimes_{\mathcal O_V} \mathcal O_X$. We note that the restriction $\overline{A_{ij}}$ to $X$ of $A_{ij}$ represents the transition matrix of $N_{X/V}$ over $U_{ij}$.
Secondly we recall the correspondence between $\tilde X$ and $\alpha$. Since $X$ is a locally complete intersection in $V$, so is $\tilde X$ in $V \times {\operatorname{Spec}}k[t]/(t^2)$ (cf. [@Hartshorne10 §9]). Then for each $i$, the defining ideal $J_i$ of $\tilde X$ over $U_i\times {\operatorname{Spec}}k[t]/(t^2)$ is generated by $$f_{i1}+tu_{i1},\quad \dots, \quad f_{im}+tu_{im}$$ in $R_i[t]/(t^2)$ for some $u_{ik} \in R_i$ ($k=1,\dots,m$). Since $J_i$ and $J_j$ agree on $U_{ij}\times {\operatorname{Spec}}k[t]/(t^2)$, there exists a matrix $B_{ij} \in M(m,R_{ij})$ such that $$\mathbf f_j + t\mathbf u_j
=(A_{ij}+tB_{ij})(\mathbf f_i + t\mathbf u_i),
\qquad
\mbox{where $\mathbf u_i:=
\begin{pmatrix}
u_{i1} \\ \vdots \\ u_{im}
\end{pmatrix}
$}.$$ Comparing the coefficient of $t$, we have $$\label{eqn:transition2}
\mathbf u_j=A_{ij}\mathbf u_i+B_{ij}\mathbf f_i,$$ which implies that $\overline{\mathbf u_j}=\overline{A_{ij}\mathbf u_i}$ in $\mathcal O_{X_{ij}}^{\oplus m}$. Let $\alpha_i$ be the section of $N_{X/V}\simeq \mathcal Hom(\mathcal I_X,\mathcal O_X)$ over $U_i$ sending each $f_{ik} \in I_i$ to $\overline{u_{ik}}\in R_i/I_i$ ($k=1,\dots,m$), respectively. Then by and , the local sections $\alpha_i$ ($i \in I$) over $U_i$ agree on $U_{ij}$ for every $i,j$ and define a global section of $N_{X/V}$, which is nothing but $\alpha$. In the rest of the proof, for convenience, we write as $\alpha_i(\mathbf f_i)=\overline{\mathbf u_i}$ instead of writing $\alpha_i(f_{ik})=\overline{u_{ik}}$ ($k=1,\dots,m$).
Now we consider liftings of $\tilde X$ to a second order deformation $\tilde{\tilde X}$ of $X$ in $V$ (over $k[t]/(t^3)$). If there exists such a ${\tilde{\tilde X}}$, then its defining ideal $K_i$ over $U_i\times {\operatorname{Spec}}k[t]/(t^3)$ is generated by $$f_{i1}+tu_{i1}+t^2v_{i1},\quad \dots, \quad f_{im}+tu_{im}+t^2v_{im}$$ in $R_i[t]/(t^3)$ for some $v_{ik} \in R_i$ ($k=1,\dots,m$). Then there exists a matrix $C_{ij} \in M(m,R_{ij})$ such that $$\mathbf f_j + t\mathbf u_j +t^2\mathbf v_j
=(A_{ij}+tB_{ij}+t^2C_{ij})(\mathbf f_i + t\mathbf u_i +t^2\mathbf v_i),
\qquad
\mbox{where $\mathbf v_i:=
\begin{pmatrix}
v_{i1} \\ \vdots \\ v_{im}
\end{pmatrix}
$},$$ which is equivalent to that $$\label{eqn:cohomologous to 0}
\overline{\mathbf v_j-A_{ij}\mathbf v_i}
=\overline{B_{ij}\mathbf u_i}$$ in $\mathcal O_{X_{ij}}^{\oplus m}$ by comparison of the coefficient of $t^2$. We see that $\tilde {\tilde X}$ is defined as a subscheme of $V \times k[t]/(t^3)$, flat over $k[t]/(t^3)$ if and only if we can solve the equation for $\mathbf v_i$. On the other hand, let us define a $1$-cochain $\beta:=\left\{\beta_{ij}\right\} \in C^1(\mathfrak U,N_{X/V})$, where $\beta_{ij}$ is the section of $N_{X/V}\simeq \mathcal Hom(\mathcal I_X,\mathcal O_X)$ over $U_{ij}$ with $$\label{eqn:gamma}
\beta_{ij}(\mathbf f_i)
=\overline{B_{ij}\mathbf u_i}.$$ Then implies that $\beta$ is cohomologous to zero, since $\overline{A_{ij}}$ is the transition matrix of $N_{X/V}$ over $U_{ij}$. In fact, if we have , then $\beta$ is equal to the coboundary of the $0$-cochain $\alpha'=\left\{\alpha'_i\right\}
\in C^0(\mathfrak U,N_{X/V})$ defined by $\alpha'_i(\mathbf f_i)=\overline{\mathbf v_i}$. Thus for the proof, it suffices to prove the next claim.
The cohomology class in $H^1(X,N_{X/V})$ represented by $\beta$ equals ${\operatorname{ob}}(\alpha)$.
#### **Proof of Claim.**
The functor ${\operatorname{Hom}}(\mathcal I_X,*)$ induces a coboundary map $\delta: {\operatorname{Hom}}(\mathcal I_X,\mathcal O_X) \rightarrow
{\operatorname{Ext}}^1(\mathcal I_X,\mathcal I_X)$. We also deduce from an exact sequence of [Č]{}ech complexes $$0 \longrightarrow
C^\bullet(\mathfrak U,\mathcal Hom(\mathcal I_X,\mathcal I_X)) \longrightarrow
C^\bullet(\mathfrak U,\mathcal Hom(\mathcal I_X,\mathcal O_V)) \longrightarrow
C^\bullet(\mathfrak U,\mathcal Hom(\mathcal I_X,\mathcal O_X)) \longrightarrow
0.$$ We compute the image $\delta(\alpha) (=\alpha \cup \mathbf e)$ of $\alpha$ by a diagram chase. Let $\alpha_i:=\alpha\big{\vert}_{U_i}$ for $i\in I$. Then as we see before, we have $\alpha_i(\mathbf f_i)=\overline{\mathbf u_i}$. If we define a section $\hat \alpha_i$ of $\mathcal Hom(\mathcal I_X,\mathcal O_V)$ over $U_i$ by $\hat \alpha_i(\mathbf f_i)=\mathbf u_i$, then $\hat \alpha_i$ is a local lift of $\alpha$ over $U_i$. Since $\alpha$ is globally defined, $\delta(\alpha)_{ij}=\tilde \alpha_j - \tilde \alpha_i$ becomes a section of $\mathcal Hom(\mathcal I_X,\mathcal I_X)$ over $U_{ij}$ for every $i,j$. Then by , we have $\delta(\alpha)_{ij}(\mathbf f_i)=B_{ij}\mathbf f_i$. Thus we have computed $\delta(\alpha)$ as an element of $H^1(V,\mathcal Hom(\mathcal I_X,\mathcal I_X))
\subset {\operatorname{Ext}}^1(\mathcal I_X,\mathcal I_X)$. Since ${\operatorname{ob}}(\alpha)=\alpha \cup \mathbf e \cup \alpha=\delta(\alpha)\cup \alpha$, ${\operatorname{ob}}(\alpha)$ is represented by the $1$-cocycle $\left\{\alpha_i \circ\delta(\alpha)_{ij}\right\}$ of $N_{X/V}$. Therefore ${\operatorname{ob}}(\alpha)$ is contained in $H^1(X,N_{X/V}) \subset {\operatorname{Ext}}^1(\mathcal I_X,\mathcal O_X)$. Since we have $$\alpha_i \circ\delta(\alpha)_{ij}(\mathbf f_i)
=\alpha_i (B_{ij} \mathbf f_i)
=B_{ij} \alpha_i (\mathbf f_i)
=\overline{B_{ij} \mathbf u_i},$$ we conclude that $\alpha_i \circ\delta(\alpha)_{ij}=\beta_{ij}$ by . Thus we have proved the claim and have finished the proof of Theorem \[thm:original cup product\].
Here ${\operatorname{ob}}(\alpha)$ is called the [*(primary) obstruction*]{} for $\alpha$ (or $\tilde X$).
Hypersurface case and $d$-map {#subsec:hypersurface}
-----------------------------
Let $X$ be an effective Cartier divisor on $V$, i.e., a closed subscheme of $V$ whose ideal sheaf is locally generated by a single equation. We denote by ${\operatorname{Hilb}}^{cd} V$ the Hilbert scheme of effective Cartier divisors on $V$. There exists a natural morphism $\varphi: {\operatorname{Hilb}}^{cd} V \rightarrow {\operatorname{Pic}}V$ to the Picard scheme ${\operatorname{Pic}}V$ of $V$, sending a divisor $D$ on $V$ to the invertible sheaf $\mathcal O_V(D)$ associated to $D$. We define a morphism $\psi_X: {\operatorname{Hilb}}^{cd} V \rightarrow {\operatorname{Pic}}X$ by the composition of $\varphi$ with the morphism ${\operatorname{Pic}}V \overset{{\vert}_X}\longrightarrow {\operatorname{Pic}}X$ defined by the restriction to $X$. By definition, the tangent map $d_X$ of $\psi_X$ at $[X]$ is the composite $$\label{map:divisorial d-map}
d_X: H^0(X,N_{X/V}) \overset{\delta}\longrightarrow H^1(V,\mathcal O_V)
\overset{|_X}\longrightarrow H^1(X,\mathcal O_X),$$ where $\delta$ is the coboundary map of the exact sequence $0 \rightarrow \mathcal O_V \rightarrow
\mathcal O_V(X) \rightarrow N_{X/V}\rightarrow 0$. We call $d_X$ the [*$d$-map*]{} for $X$. Let $\tilde X$ be a first order deformation of $X$ in $V$, corresponding to a global section $\beta$ of $N_{X/V}$. Then by [@Nasu4 Lemma 2.9], the primary obstruction ${\operatorname{ob}}(\beta)$ of $\tilde X$ equals the cup product $d_X(\beta)\cup \beta$ by the map $
H^1(X,\mathcal O_X)\times H^0(X,N_{X/V}) \overset{\cup}{\rightarrow}
H^1(X,N_{X/V}).
$
Exterior components {#subsec:exterior}
-------------------
We recall the definition of the exterior components (cf. [@Mukai-Nasu; @Nasu4]), which is useful for computing the obstructions to deforming subschemes of codimension greater than $1$. Let $X$ and $Y$ be two closed subschemes of $V$ such that $X \subset Y$, $\pi_{X/Y}: N_{X/V} \rightarrow N_{Y/V}\big{\vert}_X$ the natural projection. Then we have the induced maps $H^i(\pi_{X/Y}): H^i(X,N_{X/V}) \rightarrow H^i(X,N_{Y/V}\big{\vert}_X)$ on the cohomology groups for $i=0,1$. The two images $$\pi_{X/Y}(\alpha):=H^0(\pi_{X/Y})(\alpha)
\qquad
\mbox{and}
\qquad
{\operatorname{ob}}_Y(\alpha):=H^1(\pi_{X/Y})({\operatorname{ob}}(\alpha))$$ are called the [*exterior component*]{} of $\alpha$ and ${\operatorname{ob}}(\alpha)$, respectively. These objects respectively correspond to the deformation of $X$ in $V$ into the normal direction to $Y$ and its obstruction. Now we assume that $Y$ is an effective Cartier divisor on $V$, and $X$ is a locally complete intersection in $Y$. The $d$-map $d_X$ in is generalized and defined for a pair $(X,Y)$. (In fact, we have $d_Y=d_{Y,Y}$.)
\[dfn:d-map\] Let $\delta$ be the coboundary map of $[0 \rightarrow \mathcal I_X \rightarrow
\mathcal O_V \rightarrow \mathcal O_X \rightarrow 0]\otimes
\mathcal O_V(Y)$. The composition $$d_{X,Y}:
H^0(X,N_{Y/V}\big{\vert}_X) \overset{\delta} \longrightarrow
H^1(V,\mathcal I_X \otimes \mathcal O_V(Y)) \overset{|_X}\longrightarrow
H^1(X,{N_{X/V}}^{\vee} \otimes N_{Y/V}\big{\vert}_X)$$ of $\delta$ with the restriction map $|_X$ to $X$ is called the [*$d$-map*]{} for $(X,Y)$.
Then the two $d$-maps $d_{X,Y}$ and $d_Y$ are related by the following commutative diagram: $$\label{diag:d-map}
\begin{array}{ccccc}
H^0(Y,N_{Y/V}) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{d_Y}}} &
H^1(Y,\mathcal O_Y) \\
& & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_X\,$}}$ }}\\
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_X\,$}}$ }} && H^1(X,\mathcal O_X) \\
&& {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyleH^1(\iota)\,$}}$ }} \\
H^0(X,N_{Y/V}\big{\vert}_X) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{d_{X,Y}}}} &
H^1(X,{N_{X/V}}^{\vee}\otimes N_{Y/V}\big{\vert}_X),
\end{array}$$ where $\iota: \mathcal O_X \rightarrow {N_{X/V}}^{\vee}\otimes N_{Y/V}\big{\vert}_X$ is the sheaf homomorphism induced by ${\pi_{X/Y}}$.
\[lem:exterior\] Let $\tilde X$ and $\tilde Y$ be first order deformations of $X$ and $Y$ in $V$, with the corresponding global sections $\alpha$ and $\beta$ of $N_{X/V}$ and $N_{Y/V}$, respectively. If we have $\pi_{X/Y}(\alpha) = \beta\big{\vert}_X$ in $H^0(X,N_{Y/V}\big{\vert}_X)$, then we have $${\operatorname{ob}}_Y(\alpha) = d_{X,Y}(\pi_{X/Y}(\alpha))\cup_1 \alpha\\
= d_Y(\beta)\big{\vert}_X \cup_2 \pi_{X/Y}(\alpha)$$ in $H^1(X,N_{Y/V}\big{\vert}_X)$, where $\cup_1$ and $\cup_2$ are the cup product maps $$\begin{aligned}
H^1(X,{N_{X/V}}^{\vee}\otimes N_{Y/V}\big{\vert}_X) \times H^0(X,N_{X/V})
\overset{\cup_1}{\longrightarrow} H^1(X,N_{Y/V}\big{\vert}_X), \\
H^1(X,\mathcal O_X)\times H^0(X,N_{Y/V}\big{\vert}_X)
\overset{\cup_2}{\longrightarrow}
H^1(X,N_{Y/V}\big{\vert}_X)
\end{aligned}$$ respectively.
Infinitesimal deformations with poles and polar $d$-maps {#subsec:with pole}
--------------------------------------------------------
In this section, we recall the theory of [*infinitesimal deformations with poles*]{}, which was introduced in [@Mukai-Nasu]. Here we develop the study [@Mukai-Nasu §2.4] on the polar $d$-maps further. The infinitesimal deformations with poles are defined as rational sections of some sheaves admitting a pole along some divisor, and they are usually regarded as the deformations of the open objects complementary to the poles (cf. [@Nasu4]).
Let $V$ be a projective scheme, $Y$ and $E$ effective Cartier divisors on $V$ and $Y$, respectively. Put $Y^{\circ}:=Y \setminus E$ and $V^{\circ}:=V \setminus E$ and let $\iota: Y^{\circ} \hookrightarrow Y$ be the open immersion. Since $\iota_*\mathcal O_{Y^{\circ}}$ contains $\mathcal O_Y(mE)$ as a subsheaf for any $m \ge 0$, there exist natural inclusions $
\mathcal O_Y \subset \mathcal O_Y(E) \subset \cdots
\subset \mathcal O_Y(mE) \subset \cdots \subset \iota_*\mathcal O_{Y^{\circ}}
$ of sheaves on $Y$. Similarly, since $N_{Y/V}(mE) \subset \iota_* N_{Y^{\circ}/V^{\circ}}$ for any $m \ge 0$, we regard $H^0(Y,N_{Y/V}(mE))$ as a subgroup of $H^0(Y^{\circ},N_{Y^{\circ}/V^{\circ}})$ by the natural injective map $$H^0(Y,N_{Y/V}(mE)) \hookrightarrow
H^0(Y^{\circ}, N_{Y^{\circ}/V^{\circ}}).$$ A rational section $\beta$ of $N_{Y/V}$ admitting a pole along $E$, i.e. $$\beta \in H^0(Y,N_{Y/V}(mE))$$ for some integer $m\ge 1$ is called an [*infinitesimal deformation of $Y$ with a pole*]{} along $E$. Every infinitesimal deformation of $Y$ in $V$ with a pole induces a first order deformation of $Y^{\circ}$ in $V^{\circ}$ by the above injection.
Now we assume that the natural map $$\label{map:inclusion of H^1}
H^1(Y,\mathcal O_Y(mE)) \longrightarrow H^1(Y,\mathcal O_Y((m+1)E))$$ is injective for any integer $m \ge 1$. Then since $V$ is projective, by the same argument as in [@Mukai-Nasu Lemma 2.5], the natural map $$H^1(Y, \mathcal O_Y(mE))
\longrightarrow H^1(Y^{\circ}, \mathcal O_{Y^{\circ}})$$ is injective. By this map, we regard $H^1(Y,\mathcal O_Y(mE))$ as a subgroup of $H^1(Y^{\circ},\mathcal O_{Y^{\circ}})$. Given an invertible sheaf $L$ on $Y$, we identify an element of $H^1(Y,\mathcal O_Y(mE))$ as a first order deformation of the invertible sheaf $L^{\circ}:=\iota_* L$ on $Y^{\circ}$, and call it an [*infinitesimal deformation of $L$ with a pole*]{} along $E$.
Let $m \ge 1$ be an integer and $\beta \in H^0(Y,N_{Y/V}(mE))$ an infinitesimal deformation of $Y$ with a pole along $E$. Let $d_{Y^{\circ}}: H^0(Y^{\circ},N_{Y^{\circ}/V^{\circ}})
\rightarrow H^1(Y^{\circ},\mathcal O_{Y^{\circ}})$ be the $d$-map for $Y^{\circ} \subset V^{\circ}$. The following is a generalization of [@Mukai-Nasu Proposition 2.6], which enables us to compute the singularity of $d_{Y^{\circ}}(\beta) \in H^1(Y^{\circ},\mathcal O_{Y^{\circ}})$ along the boundary $E$.
\[prop:key\] Let $m \ge 1$ be an integer. Then
1. $d_{Y^{\circ}}(H^0(Y,N_{Y/V}(mE)))
\subset H^1(Y,\mathcal O_Y((m+1)E)).$
2. Let $d_Y$ be the restriction of $d_{Y^{\circ}}$ to $H^0(Y,N_{Y/V}(mE))$, and let $\partial_E$ be the coboundary map of . Then the diagram $$\begin{CD}
H^0(Y,N_{Y/V}(mE)) @>d_{Y}>> H^1(Y,\mathcal O_Y((m+1)E)) \\
@V{|_E}VV @V{|_E}VV \\
H^0(E,N_{Y/V}(mE)\big{\vert}_E) @>{m\partial_E}>>
H^1(E,\mathcal O_E((m+1)E))
\end{CD}$$ is commutative.
In other words, if $Y$ is a hypersurface in $V$, then every infinitesimal deformation of $Y \subset V$ with a pole induces that of the invertible sheaf $N_{Y/V}$. The principal part of $d_{Y^\circ}(\beta)$ along $E$ coincides with the coboundary $\partial_E(\beta\big{\vert}_E)$ of the principal part $\beta\big{\vert}_E$, up to constant.
The proof is similar to the one in [@Mukai-Nasu], where $Y$ is a surface by assumption. Let $\mathfrak U:=\{ U_i \}_{i \in I}$ be an open affine covering of $V$ and let $x_i=y_i=0$ be the local equation of $E$ over $U_i$ such that $y_i$ defines $Y$ in $U_i$. Through the proof, for a local section $t$ of a sheaf $\mathcal F$ on $V$, $\bar t$ denotes the restriction $t\big{\vert}_Y \in \mathcal F\big{\vert}_Y$ for conventions. Let $D_{x_i}$ and $D_{\bar x_i}$ denote the open affine subsets of $U_i$ and $U_i \cap Y$ defined by $x_i \ne 0$ and $\bar x_i \ne 0$, respectively. Then $\left\{ D_{\bar x_i} \right\}_{i \in I}$ is an open affine covering of $Y^{\circ}$, since $D_{\bar x_i}=D_{x_i} \cap Y=U_i \cap Y^{\circ}$.
Let $\beta$ be a global section of $N_{Y/V}(mE) \simeq \mathcal O_Y(Y)(mE)$. Then the product $\bar x_i^m \beta$ is contained in $H^0(U_i,\mathcal O_Y(Y))$ and lifts to a section $s_i' \in \Gamma(U_i, \mathcal O_V(Y))$ since $U_i$ is affine. In particular, $\beta$ lifts to the section $s_i:=s_i'/x_i^m$ of $\mathcal O_{V^{\circ}}(Y^{\circ})$ over $D_{x_i}$. Let $\delta: H^0(Y^{\circ},N_{Y^{\circ}/V^{\circ}})
\rightarrow H^1(Y^{\circ},\mathcal O_{V^{\circ}})$ be the coboundary map in the definition of $d_{Y^{\circ}}$. Then we have $$\delta(\beta)_{ij}= s_j-s_i
=\dfrac{s_j'}{x_j^m} -\dfrac{s_i'}{x_i^m}
\qquad
\mbox{in}
\qquad
\Gamma(D_{x_i} \cap D_{x_j}, \mathcal O_{V^{\circ}}(Y^{\circ}))$$ for every $i,j$. Since $\beta$ is a global section of $N_{Y^{\circ}/V^{\circ}}$, $\delta(\beta)_{ij}$ is contained in $\Gamma(D_{x_i} \cap D_{x_j}, \mathcal O_{V^{\circ}})$. Now we put $$f_{ij}:=x_i^mx_j^m\delta(\beta)_{ij}=x_i^ms_j'-x_j^ms_i'.$$ Since $x_i^ms_i=s_i' \in \Gamma(U_i,\mathcal O_V(Y))$ for every $i$, $f_{ij}$ is a section of $\mathcal O_{U_{ij}}$. Now we recall the relations between the local equations $x_i,y_i$ of $E$ over $U_i$. Since the two ideals $(x_i,y_i)$ and $(x_j,y_j)$ agree on the overlap $U_{ij}$, there exist elements $b_{ij}$ and $c_{ij}$ of $\mathcal O_{U_{ij}}$ satisfying $x_i = b_{ij} y_j + c_{ij} x_j$. Then we have $$\begin{aligned}
f_{ij}&=(x_i^m-c_{ij}^mx_j^m)s_j'+x_j^m(c_{ij}^ms_j'-s_i')\notag \\
&=\sum_{k=0}^{m-1}x_i^{m-1-k}(c_{ij}x_j)^k(x_i-c_{ij}x_j)s_j'
+x_j^m(c_{ij}^ms_j'-s_i')\notag \\
&=\sum_{k=0}^{m-1}x_i^{m-1-k}(c_{ij}x_j)^kb_{ij}y_js_j'
+x_j^m(c_{ij}^ms_j'-s_i')\label{eqn:f_ij}.\end{aligned}$$ Since $y_j \in \Gamma(U_j,\mathcal O_V(-Y))$ and $s_j' \in \Gamma(U_j,\mathcal O_V(Y))$, $b_{ij}y_js_j'$ is a section of $\mathcal O_{U_{ij}}$, while $c_{ij}^ms_j'-s_i' \in \Gamma(U_{ij},\mathcal O_V(Y))$ is also a section of $\mathcal O_{U_{ij}}$, because we have $${\overline{c_{ij}^ms_j'-s_i'}}
=\bar c_{ij}^m\bar x_j^m\beta-\bar x_i^m\beta
=(\bar c_{ij}^m\bar x_j^m-\bar x_i^m)\beta=0$$ in $\Gamma(Y\cap U_{ij},N_{Y/V})$. Therefore, $f_{ij}$ is contained in $\Gamma(U_{ij},{\mathcal I_E}^{m-1})$ by , and hence $\bar f_{ij}$ is contained in $\Gamma(Y\cap U_{ij},\mathcal O_Y((-m+1)E))$. This implies that $$\label{eqn:d_Y}
d_{Y^{\circ}}(\beta)_{ij} =(\delta(\beta)_{ij})\big{\vert}_{Y^\circ}=
\dfrac{\bar f_{ij}}{\bar x_i^m \bar x_j^m}
\qquad
\mbox{in}
\qquad
\Gamma(D_{\bar x_i} \cap D_{\bar x_j}, \mathcal O_{Y^{\circ}}),$$ is contained in $\Gamma(Y\cap U_{ij}, \mathcal O_Y((m+1)E))$. Thus we have proved (1).
Now we compute the image of $d_Y(\beta)=d_{Y^{\circ}}(\beta)$ by the restriction map $H^1(Y,\mathcal O_Y((m+1)E)) \rightarrow H^1(E,\mathcal O_E((m+1)E))$, regarding $\mathcal O_E((m+1)E)$ as the quotient sheaf $\mathcal O_Y((m+1)E)/\mathcal O_Y(mE)$. Since ${\overline{x}}_i={\overline{c_{ij}x_j}}$ and $\beta={\overline{s}}_j'/{\overline{x}}_j^m$, it follows from and that $$d_Y(\beta)_{ij}
= \sum_{k=0}^{m-1}\left(
\dfrac{{\overline{c_{ij}x_j}}}{{\overline{x}}_i}
\right)^k
\dfrac{\overline{b_{ij}y_j}}{{\overline{x}}_i}
\dfrac{{\overline{s}}_j'}{{\overline{x}}_j^m}
+\dfrac{{\overline{c_{ij}^ms_j'-s_i'}}}{{\overline{x}}_i^m}
= m\dfrac{\overline{b_{ij}y_j}}{{\overline{x}}_i}\beta
+\dfrac{{\overline{c_{ij}^ms_j'-s_i'}}}{{\overline{x}}_i^m}$$ in $\Gamma(Y\cap U_{ij}, \mathcal O_Y((m+1)E))$. Since $c_{ij}^ms_j' - s_i'$ a section of $\mathcal O_{U_{ij}}$, ${\overline{c_{ij}^ms_j'-s_i'}}/\bar x_i^m$ is contained in $\Gamma(Y\cap U_{ij},\mathcal O_Y(mE))$. On the other hand, the restriction of the $1$-cochain $\left\{ {\overline{b_{ij}y_j}}/\bar x_i \right\}_{i,j \in I}$ to $E$ is a cocycle and represents the extension class $\mathbf e' \in H^1(E,\mathcal O_E(-Y+E))$ of the exact sequence (cf. [@Mukai-Nasu]). Therefore $d_Y(\beta)\big{\vert}_E$ is equal to $m\mathbf e' \cup (\beta\big{\vert}_E)=m\partial_E(\beta\big{\vert}_E)$, which implies (2).
We finish this section by giving a refinement of Proposition \[prop:key\], which will be used in the proof of Theorem \[thm:refinement\]. Let $E_i$ ($1\le i\le k$) be irreducible Cartier divisors on $Y$. Suppose that $E_i$ are mutually disjoint, i.e., $E_i \cap E_j=\emptyset$ for all $i,j$. We suppose furthermore that for any two effective divisors $D,D'$ on $S$ with supports on $\bigcup_{i=1}^k E_i$, if $D \le D'$, then the natural map $H^1(Y,\mathcal O_Y(D)) \rightarrow H^1(Y,\mathcal O_Y(D'))$ is injective. Then as we have seen before, for any such divisor $D$, $H^1(Y,\mathcal O_Y(D))$ is regarded as a subgroup of $H^1(Y^{\circ},\mathcal O_{Y^{\circ}})$, where $Y^{\circ}:=Y\setminus \bigcup_{i=1}^k E_i$. Let $E=\sum_{i=1}^k m_i E_i$ be an effective divisor on $Y$ with coefficients $m_i \ge 1$, and let $\beta \in H^0(Y,N_{Y/V}(E))$. We put $V^{\circ}:=V\setminus \bigcup_{i=1}^k E_i$ and denote by $d_{Y^{\circ}}$ the $d$-map for $Y^{\circ} \subset V^{\circ}$. If $H^1(Y,N_{Y/V})=0$, then by the following lemma, $\beta\in H^0(Y,N_{Y/V}(E))$ is written as a $k$-linear combination $\sum_{i=1}^k c_i \beta_i$ of $\beta_i \in H^0(Y,N_{Y/V}(m_iE_i))$.
\[lem:k-linear combination\] Let $L$ be an invertible sheaf on $Y$ with $H^1(Y,L)=0$, and let $E,E'$ be two effective divisors on $Y$ whose supports are mutually disjoint. Then the natural map $
H^0(Y,L(E))\oplus H^0(Y,L(E')) \rightarrow H^0(Y,L(E+E'))
$ is surjective.
It follows from the exact sequence $[0 \rightarrow \mathcal O_Y
\rightarrow \mathcal O_Y(E) \oplus \mathcal O_Y(E')
\rightarrow \mathcal O_Y(E+E')
\rightarrow 0]\otimes L$ on $Y$ of Koszul type.
Since the $d$-map $d_{Y^{\circ}}$ is $k$-linear, we find $d_{Y^{\circ}}(\beta)=\sum_{i=1}^k c_i d_{Y^{\circ}}(\beta_i)$. Because for each $i$, $d_{Y^{\circ}}(\beta_i)$ is contained in $H^1(Y,\mathcal O_Y((m_i+1)E_i))$ by Proposition \[prop:key2\], $d_{Y^{\circ}}(\beta)$ is contained in $H^1(Y,\mathcal O_Y(\sum_{i=1}^k (m_i+1)E_i))$. Furthermore, since $E_i$’s are mutually disjoint, we have $d_{Y^{\circ}}(\beta_i)\big{\vert}_{E_j}=0$ if $i \ne j$ and $d_{Y^{\circ}}(\beta_i)\big{\vert}_{E_i}
=m_i\partial_{E_i}(\beta_i\big{\vert}_{E_i})$ by the same proposition. Thus we conclude that
\[prop:key2\] Let $m_i \ge 1$ be integers. If $H^1(Y,N_{Y/V})=0$, then
1. $d_{Y^{\circ}}(H^0(Y,N_{Y/V}(\sum_{i=1}^k m_i E_i)))
\subset H^1(Y,\mathcal O_Y(\sum_{i=1}^k (m_i+1)E_i))$.
2. Let $d_Y$ be the restriction of $d_{Y^{\circ}}$ to $H^0(Y,N_{Y/V}(\sum_{i=1}^k m_i E_i))$. Then the diagram $$\begin{CD}
H^0(Y,N_{Y/V}(\sum_{i=1}^k m_i E_i)) @>d_{Y}>>
H^1(Y,\mathcal O_Y(\sum_{i=1}^k (m_i+1)E_i)) \\
@V{|_{E_i}}VV @V{|_{E_i}}VV \\
H^0(E_i,N_{Y/V}(m_iE_i)\big{\vert}_{E_i})
@>{m_i\partial_{E_i}}>>
H^1(E_i,\mathcal O_{E_i}((m_i+1)E_i))
\end{CD}$$ is commutative for any $i=1,\dots,k$.
Hilbert-flag schemes {#subsec:flag schemes}
--------------------
In this section, we recall some basic results on Hilbert-flag schemes. For the construction (the existence), the local properties, etc., of the Hilbert-flag schemes, we refer to [@Kleppe81; @Kleppe87; @Hartshorne10; @Sernesi].
Let $V$ be a projective scheme, and let $X, Y$ be two closed subschemes of $V$ such that $X \subset Y$, with the Hilbert polynomials $P, Q$, respectively. Then there exists a projective scheme ${\operatorname{HF}}_{P,Q} V$, called the Hilbert-flag scheme of $V$, parametrising all pairs $(X',Y')$ of closed subschemes $X' \subset Y' \subset V$ with the Hilbert polynomials $P$ and $Q$, respectively. There exists a natural diagram of the Hilbert(-flag) schemes $$\raisebox{20pt}{
\xymatrix{
{\operatorname{HF}}_{P,Q} V \ar[r]^{pr_2} \ar[d]_{pr_1} & {\operatorname{Hilb}}_Q V \\
{\operatorname{Hilb}}_P V &
}}
\quad
\left(
\raisebox{20pt}{
\xymatrix{
(X,Y) {\ar@{|->}[r]} \ar@{|->}[d] & Y \\
X &
}}
\right),$$ where $pr_i$ ($i=1,2$) are the forgetful morphisms, i.e., the projections. We denote the tangent space of ${\operatorname{HF}}V$ at $(X,Y)$ by $A^1(X,Y)$. Then there exists a Cartesian diagram $$\label{diag:cartesian}
\raisebox{25pt}{
\xymatrix{
A^1(X,Y) \ar[d]_{p_1} \ar[r]^{p_2} \ar@{}[dr]|\square &
H^0(Y,N_{Y/V}) \ar[d]_{\rho} \\
H^0(X,N_{X/V}) \ar[r]^{\pi_{X/Y}} & H^0(X,N_{Y/V}\big{\vert}_X) \\
}}$$ where $p_i$ is the tangent map of $pr_i$ ($i=1,2$), $\rho$ is the restriction map, and $\pi_{X/Y}$ is the projection. In what follows, we assume that $X$ and $Y$ are smooth and ${\operatorname{Hilb}}V$ is nonsingular at $[Y]$. Let $$\partial_X: H^0(X,N_{Y/V}\big{\vert}_X)
\rightarrow H^1(X,N_{X/Y})$$ be the coboundary map of the exact sequence $0\rightarrow N_{X/Y} \rightarrow N_{X/V}
\overset{\pi_{X/Y}}\rightarrow N_{Y/V}\big{\vert}_X
\rightarrow 0$ on $V$ and let $\alpha_{X/Y}$ be the composition $\partial_X \circ \rho$ of $\rho$ with $\partial_X$. Then since ${\operatorname{Hilb}}V$ is nonsingular at $[Y]$, every obstruction to deforming a pair $(X,Y)$ of subschemes $X,Y$ with $X \subset Y \subset V$ is contained in the group $$\label{eqn:obst.sp.flag}
A^2(X,Y):={\operatorname{coker}}\alpha_{X/Y},$$ and we have $$\label{ineq:dimension of flag}
\dim A^1(X,Y) - \dim A^2(X,Y) \le \dim_{(X,Y)} {\operatorname{HF}}V \le \dim A^1(X,Y)$$ (cf. [@Kleppe81 Theorem 1.3.2], [@Kleppe87 §2]). Thus $A^2(X,Y)$ represents the obstruction space of ${\operatorname{HF}}V$ at $(X,Y)$. There exists an exact sequence $$\begin{aligned}
\label{seq:flag to hilb}
0 &\longrightarrow H^0(Y,\mathcal I_{X/Y}\otimes N_{Y/V})
\longrightarrow A^1(X,Y)
\longrightarrow H^0(X,N_{X/V}) \notag \\
&\longrightarrow {\operatorname{coker}}\rho
\longrightarrow A^2(X,Y)
\longrightarrow H^1(X,N_{X/V})
\longrightarrow H^1(X,N_{Y/V}\big{\vert}_X)
$$ of cohomology groups, which connects the tangent spaces and the obstruction spaces of Hilbert(-flag) schemes (see [@Kleppe81; @Kleppe87] for the proof). If we have $H^i(Y,N_{Y/V})=0$ for $i=1,2$, then we deduce from the exact sequence $
[0 \rightarrow \mathcal I_{X/Y}
\rightarrow \mathcal O_Y \rightarrow \mathcal O_X
\rightarrow 0]\otimes N_{Y/V}
$ the two isomorphisms ${\operatorname{coker}}\rho \simeq H^1(Y,\mathcal I_{X/Y}\otimes N_{Y/V})$ and $H^1(X,N_{Y/V}\big{\vert}_X) \simeq H^2(Y,\mathcal I_{X/Y}\otimes N_{Y/V})$. If $\dim X=1$ then the last map of is surjective. Thus we obtain (3) of the next lemma.
1. If $\rho$ is surjective (cf. ), then $pr_1: {\operatorname{HF}}V \rightarrow {\operatorname{Hilb}}V$ is smooth at $(X,Y)$ (cf. [@Kleppe87 Lemma A10]).
2. If $H^0(Y,\mathcal I_{X/Y}\otimes N_{Y/V})=0$, then $pr_1$ is a local embedding at $(X,Y)$.
3. If $\dim X=1$, $\dim Y=2$ and $H^i(Y,N_{Y/V})=0$ ($i=1,2$), then we have $$\begin{aligned}
\label{eqn:expected dimension}
\dim A^1(X,Y)- \dim A^2(X,Y)
&=\chi(X,N_{X/V})+\chi(Y,\mathcal I_{X/Y}\otimes N_{Y/V}) \notag \\
&=\chi(X,N_{X/Y})+\chi(Y,N_{Y/V}).
\end{aligned}$$
\[lem:flag to hilb\]
The number represents the expected dimension of the Hilbert-flag scheme ${\operatorname{HF}}V$ at $(X,Y)$. If $A^2(X,Y)=0$, then ${\operatorname{HF}}V$ is nonsingular at $(X,Y)$ by . If moreover $H^1(Y,\mathcal I_{X/Y}\otimes N_{Y/V})=0$, then so is ${\operatorname{Hilb}}V$ at $[X]$ by Lemma \[lem:flag to hilb\](1). The following lemma will be essentially used in the proof of Theorem \[thm:main2\] (cf. §\[sect:k3\]).
\[lem:flag of k3fano\] Let $V$ be a smooth Fano $3$-fold, $S$ a smooth $K3$ surface contained in $V$, $C$ a smooth curve on $S$. Then
1. $H^i(S,N_{S/V})=0$ for all $i\ge 1$. In particular, ${\operatorname{Hilb}}V$ is nonsingular at $[S]$.
2. We have an isomorphism $$\label{isom:cohomology of abnormality}
H^i(S,\mathcal I_{C/S}\otimes N_{S/V}) \simeq H^i(S,-D)$$ for every integer $i$, where $D:=C+K_V\big{\vert}_S$ is a divisor on $S$.
3. Suppose that there exists a first order deformation $\tilde S$ of $S$ which does not contain any first order deformations $\tilde C$ of $C$. Then we have $A^2(C,S)=0$. In particular, the Hilbert-flag scheme ${\operatorname{HF}}V$ is nonsingular at $(C,S)$ of dimension $(-K_V\big{\vert}_S)^2/2+g(C)+1$, where $g(C)$ is the genus of $C$.
Since $K_S$ is trivial, by adjunction, we have $N_{S/V}\simeq -K_V\big{\vert}_S$ and $N_{C/S}\simeq K_C$. Then (1) follows from the ampleness of $-K_V$, and (2) from $\mathcal I_{C/S}\otimes N_{S/V} \simeq
N_{S/V}(-C)\simeq -K_V\big{\vert}_S-C = -D$. On the other hand, we have $H^1(C,N_{C/S})\simeq k$. Thus the obstruction group $A^2(C,S)$ (cf. ) of ${\operatorname{HF}}V$ at $(C,S)$ is of dimension at most $1$. For proving (3), let $\beta$ be the global section of $N_{S/V}$ corresponding to $\tilde S$. Then $\rho(\beta)$ is not contained in the image of $\pi_{C/S}$, because the diagram is Cartesian. Hence the map $\alpha_{C/S}$ is nonzero and we conclude that $A^2(C,S)=0$. By Lemma \[lem:flag to hilb\](3), $\dim_{(C,S)} {\operatorname{HF}}V=\dim A^1(C,S)
=\chi(-K_V\big{\vert}_S)+\chi(K_C)=(-K_V\big{\vert}_S)^2/2+g(C)+1$.
The first projection $pr_1$ induces a morphism $pr_1': {\operatorname{HF}}^{sc} V \rightarrow {\operatorname{Hilb}}^{sc} V$, where ${\operatorname{HF}}^{sc} V:=pr_1^{-1} ({\operatorname{Hilb}}^{sc} V)$. If $X$ is a smooth connected curve and ${\operatorname{HF}}V$ is nonsingular at $(X,Y)$, then there exists a unique irreducible component $\mathcal W_{X,Y}$ of ${\operatorname{HF}}^{sc} V$ passing through $(X,Y)$.
\[dfn:S-maximal\] The image $W_{X,Y}$ of $\mathcal W_{X,Y}$ by $pr_1'$ is called the [**$Y$-maximal family**]{} of curves containing $X$.
$K3$ surfaces and quartic surfaces {#subsec:K3 and quartic}
----------------------------------
We recall some basic results on $K3$ surfaces and quartic surfaces.
\[lem:K3\] Let $S$ be a smooth projective $K3$ surface, $D\ne 0$ an effective divisor on $S$.
1. If $D$ is nef, then the complete linear system $|D|$ has a base point if and only if there exist curves $E$ and $F$ on $S$ and an integer $k \ge 2$ such that $D\sim E+kF$, $E^2=-2$, $F^2=0$ and $E.F=1$.
2. Let $D^2\ge 0$. Then $H^1(S,D)\ne 0$ if and only if (i) $D.\Delta \le -2$ for some divisor $\Delta \ge 0$ with $\Delta ^2=-2$, or (ii) $D \sim kF$ for some nef and primitive divisor $F\ge 0$ with $F^2=0$ and an integer $k \ge 2$. (We have $h^1(S,D)=k-1$ in (ii).)
\(1) follows from [@Saint-Donat 2.7] and (2) from [@Knutsen-Lopez].
The following lemma will be used in §\[sect:non-reduced\] to show the existence of quartic surfaces of Picard number two containing a rational curve or an elliptic curve.
\[lem:mori\] We assume that ${\operatorname{char}}k=0$.
1. There exists a smooth curve $C$ of degree $d>0$ and genus $g\ge 0$ on a smooth quartic surface $S \subset \mathbb P^3$ if and only if (i) $g=d^2/8+1$, or (ii) $g<d^2/8$ and $(d,g)\ne (5,3)$.
2. If there exists a smooth quartic surface $S_0$ containing smooth curve $C_0$ of degree $d$ and genus $g$, then there exists a smooth quartic surface $S_1$ containing a smooth curve $C_1$ of the same degree and genus, with the property that ${\operatorname{Pic}}S_1$ is generated by $C_1$ and the class $\mathbf h$ of hyperplane sections.
The following lemma will be used in §\[sect:non-reduced\] to apply Theorem \[thm:main2\].
\[lem:non-surjective\] Let $V$ be a smooth projective variety, $S$ a smooth $K3$ surface contained in $V$, and $E$ a smooth curve on $S$. If
1. $V\simeq \mathbb P^n$ and $E$ is rational (i.e. $E \simeq \mathbb P^1$) or elliptic, \[item:projective space\]
2. $E$ is rational and $N_{E/V}$ is globally generated, or \[item:rational and globally generated\]
3. $E$ is elliptic and there exists a first order deformation $\tilde S$ of $S$ not containing any first order deformation $\tilde E$ of $E$, \[item:elliptic\]
then the $\pi$-map $\pi_{E/S}(E)$ in is not surjective.
There exists an exact sequence $$[0 \longrightarrow N_{E/S}
\overset{\iota}\longrightarrow N_{E/V}
\overset{\pi_{E/S}}\longrightarrow N_{S/V}\big{\vert}_E
\longrightarrow 0]\otimes \mathcal O_E(E),$$ on $E$, where $\iota$ is a natural inclusion. We note that $N_{E/S}\simeq \mathcal O_E(E)\simeq K_E$, since $K_S$ is trivial. Thus we have $H^1(E,N_{E/S}(E)) \simeq H^1(E,2K_E) \ne 0$ by assumption. We prove $H^1(E,N_{E/V}(E))=0$ in the case . Suppose that $V\simeq \mathbb P^n$. Then by the Euler sequence $0 \rightarrow \mathcal O_{\mathbb P^n}
\rightarrow \mathcal O_{\mathbb P^n}(1)^{n+1}
\rightarrow \mathcal T_{\mathbb P^n}
\rightarrow 0$ on $\mathbb P^n$, there exists a surjective homomorphism $[\mathcal O_E(1)^{n+1}
\twoheadrightarrow
\mathcal T_{\mathbb P^n}\big{\vert}_E]
\otimes_{\mathcal O_E} \mathcal O_E(E)$. We have $H^1(E,\mathcal O_E(E)(1))\simeq H^1(E,K_E(1))=0$ and $E$ is a curve, we have $H^1(E,\mathcal T_{\mathbb P^n}\big{\vert}_E(E))=0$ and hence $H^1(E,N_{E/\mathbb P^n}(E))=0$. Suppose that is satisfied. Then we have an inequality $h^1(E,N_{E/V}(E))\le 2$, while $h^1(E,N_{E/S}(E))=h^1(\mathbb P^1,\mathcal O_{\mathbb P^1}(-4))=3$. Therefore $H^1(E,\iota \otimes_{\mathcal O_E} \mathcal O_E(E))$ is not injective. Suppose that is satisfied. As we see in the proof of Lemma \[lem:flag of k3fano\], by assumption, $H^0(E,N_{E/V}) \overset{\pi_{E/S}}\rightarrow H^0(E,N_{S/V}\big{\vert}_E)$ is not surjective, and so is $\pi_{E/S}(E)$, because $\mathcal O_E(E)$ is trivial.
Obstructedness criterion {#sect:obstruction}
========================
In this section, we compute obstructions to deforming curves on a $3$-fold, and prove Theorem \[thm:main1\] and its refinement Theorem \[thm:refinement\].
Let $C \subset S \subset V$ be a sequence of a curve $C$, a surface $S$, a $3$-fold $V$, $E$ an effective Cartier divisor on $S$. We assume that $C$ and $S$ are Cartier divisors on $S$ and $V$, respectively. Let $Z:=C\cap E$ be the scheme-theoretic intersection of $C$ and $E$. In this section, given a coherent sheaf $\mathcal F$ on $S$ (resp. $C$), integers $i,m \ge 0$, and a cohomology class $*$ in $H^i(S,\mathcal F)$ (resp. $H^i(C,\mathcal F)$), we denote by $\overline *_{(m)}$ the image of $*$ in $H^i(S,\mathcal F(mE))$ (resp. $H^i(C,\mathcal F(mZ))$). We define $\mathbf k_C \in {\operatorname{Ext}}^1(\mathcal O_C,\mathcal O_S(-C))$ as the extension class of the exact sequence $$\label{ses:CE}
0 \longrightarrow \mathcal O_S(-C)
\longrightarrow \mathcal O_S
\longrightarrow \mathcal O_C
\longrightarrow 0,$$ and the class $\mathbf k_E$ similarly.
\[lem:restriction to C and E\] Let $L$ be an invertible sheaf on $S$ and $\gamma$ a global section of $L\big{\vert}_C=L\otimes_{\mathcal O_S} \mathcal O_C$.
1. Let $m \ge 0$ be an integer. Then $\overline \gamma_{(m)}\in H^0(C,L(mE)\big{\vert}_C)$ lifts to a section $\beta \in H^0(S,L(mE))$ on $S$ if and only if $\overline \gamma_{(m)}\cup \mathbf k_C=0$ in $H^1(S,L(mE-C))$.
2. If $\overline \gamma_{(m)}$ lifts to a section $\beta \in H^0(S,L(mE))$ for $m \ge 1$, then the principle part $\beta\big{\vert}_E$ of $\beta$ is contained in $H^0(E,L(mE-C)\big\vert_E)$, and hence $\beta$ is contained in $H^0(S,\mathcal I_{Z/S}\otimes L(mE))$. Here $\beta\big{\vert}_E$ is nonzero if and only if $\beta \not\in H^0(S,L((m-1)E))$, equivalently, $\overline \gamma_{(m-1)}\cup \mathbf k_C \ne 0$ in $H^1(S,L((m-1)E-C))$.
\(1) follows from the short exact sequence $\eqref{ses:CE}\otimes L(mE)$, whose coboundary map coincides with the cup product map $\cup \mathbf k_C: H^0(C,L(mE)\big{\vert}_C) \rightarrow H^1(S,L(mE-C))$ with $\mathbf k_C$. (2) follows from a diagram chase on the commutative diagram $$\begin{array}{ccccc}
& & 0 & & 0\\
& & \downarrow & & \downarrow \\
& & H^0(S,L(mE-C)) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{|_E}}} & H^0(E,L(mE-C)\big{\vert}_E) \\
& & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }} & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }} \\
H^0(S,L((m-1)E))& \longrightarrow & H^0(S,L(mE)) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{|_E}}} & H^0(E,L(mE)\big{\vert}_E)\\
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_C\,$}}$ }} & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_C\,$}}$ }} & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_C\,$}}$ }} \\
H^0(C,L((m-1)E)\big{\vert}_C) & \longrightarrow & H^0(C,L(mE)\big{\vert}_C) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{|_E}}} & k(Z)\\
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \mathbf k_C\,$}}$ }} & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \mathbf k_C\,$}}$ }} & & \\
H^1(S,L((m-1)E-C))& \longrightarrow & H^1(S,L(mE-C)) & & \\
\end{array}$$ of cohomology groups, which is exact both vertically and horizontally.
#### **Proof of Theorem \[thm:main1\].**
We first recall the relation between $\alpha$ and $\partial_E(\beta\big{\vert}_E)$ by Figure \[fig:relation\].
$$\begin{array}{ccccccccc}
H^0(N_{C/V}) & \ni & \alpha & & & & & & H^0(N_{E/V}(mE)) \\
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\pi_{C/S}\,$}}$ }} && \big\downarrow && && &&{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\pi_{E/S}(mE)\,$}}$ }} \\
H^0(N_{S/V}\big{\vert}_C) & \ni & \beta\big{\vert}_C
& \overset{res}\longmapsfrom & \beta & \overset{res}\longmapsto &
\beta\big{\vert}_E & \in & H^0(N_{S/V}(mE)\big{\vert}_E) \\
\bigcap &&&& \rotatebox{90}{$\ni$} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\partial_E\,$}}$ }}\\
H^0(N_{S/V}(mE)\big{\vert}_C) && \overset{res}\longleftarrow &&
H^0(N_{S/V}(mE)) && \partial_E(\beta\big{\vert}_E) & \in & H^1(\mathcal O_E((m+1)E))
\end{array}$$
We use the same strategy as [@Mukai-Nasu], in which the proof is separated into 3 steps. We follow the same steps and prove $\overline{{\operatorname{ob}}_S(\alpha)}_{(m+1)}\ne 0$ in $H^1(C,N_{S/V}((m+1)E)\big{\vert}_C)$ instead of proving ${\operatorname{ob}}_S(\alpha)\ne 0$ in $H^1(C,N_{S/V}\big{\vert}_C)$.
Let $\alpha$ be a global section of $N_{C/V}$, and $\pi_{C/S}(\alpha)\in H^0(C,N_{S/V}\big{\vert}_C)$ the exterior component of $\alpha$. Suppose that the image $\overline{\pi_{C/S}(\alpha)}_{(m)}$ of $\pi_{C/S}(\alpha)$ in $H^0(C,N_{S/V}(mE)\big{\vert}_C)$ lifts to a section $\beta \in H^0(S,N_{S/V}(mE))\setminus H^0(S,N_{S/V}((m-1)E))$ for some integer $m \ge 1$, i.e., an infinitesimal deformation with a pole along $E$. We need the relation between the two $d$-maps $d_{C,S}$ and $d_S$ (cf. Definition \[dfn:d-map\]), allowing a pole along $E$. The polar version of the diagram is the following [*partially commutative*]{} diagram:
$$\label{diag:d-map with pole}
\begin{array}{ccccc}
\beta & \in & H^0(S,N_{S/V}(mE)) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{d_S}}} &
H^1(S,\mathcal O_S((m+1)E)) \\
&& & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_C\,$}}$ }}\\
&& {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_C\,$}}$ }} && H^1(C,\mathcal O_C((m+1)Z)) \\
&& && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyleH^1(\iota)\,$}}$ }} \\
&& H^0(C,N_{S/V}(mE)\big{\vert}_C)& &
H^1(C,{N_{C/V}}^{\vee}\otimes N_{S/V}((m+1)E)\big{\vert}_C)\\
&& \cup &&\uparrow\\
\gamma & \in & H^0(C,N_{S/V}\big{\vert}_C) & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{d_{C,S}}}} &
H^1(C,{N_{C/V}}^{\vee}\otimes N_{S/V}\big{\vert}_C),
\end{array}$$
in which, the commutativity holds only for $\gamma \in H^0(C,N_{S/V}\big{\vert}_C)$ which has a lift $\beta \in H^0(S,N_{S/V}(mE))$. More precisely, for such a pair $\gamma$ and $\beta$, we have $$\label{eq:d-map with pole}
\overline{d_{C,S} (\gamma)}_{(m+1)} = H^1(\iota)(d_S(\beta)\big{\vert}_C).$$
[**Step 1**]{} $\overline{{\operatorname{ob}}_S(\alpha)}_{(m+1)}
=d_{S}(\beta)\big{\vert}_C \cup \pi_{C/S}(\alpha)
\quad \text{in} \quad H^1(C,N_{S/V}((m+1)E)\big{\vert}_C)$.
By Lemma \[lem:exterior\], we have $\overline{{\operatorname{ob}}_S(\alpha)}_{(m+1)}
=\overline{d_{C,S}(\pi_{C/S}(\alpha))}_{(m+1)} \cup \alpha$. Then it follows from that $\overline{d_{C,S}(\pi_{C/S}(\alpha))}_{(m+1)}=
H^1(\iota)(d_S(\beta)\big{\vert}_C)$. By the commutative diagram $$\begin{array}{ccccc}
H^1({N_{C/V}}^{\vee}\otimes N_{S/V}((m+1)E)) & \times
& H^0(N_{C/V}) & \overset{\cup}{\longrightarrow}
& H^1(N_{S/V}((m+1)E)\big{\vert}_C)\\
{\Big\uparrow
\rlap{$\vcenter{\hbox{$\scriptstyleH^1(\iota)$}}$ }} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\pi_{C/S}\,$}}$ }} && \Vert \\
H^1(\mathcal O_C((m+1)Z)) & \times & H^0(N_{S/V}\big{\vert}_C)
& \overset{\cup}{\longrightarrow} & H^1(N_{S/V}((m+1)E)\big{\vert}_C),
\end{array}$$ we have the required equation.
Next we relate ${\operatorname{ob}}_S(\alpha)$ with a cohomology class on $E$. Let $\mathbf k_C$ and $\mathbf k_E$ be the extension classes defined by .
[**Step 2**]{} $\overline{{\operatorname{ob}}_S(\alpha)}_{(2m)} \cup \mathbf k_C
=(d_S(\beta)\big{\vert}_E \cup \beta\big{\vert}_E) \cup \mathbf k_E
\quad \text{in} \quad H^2(S,N_{S/V}(2mE-C))$.
We note that for every integers $i$, $n \ge 0$ and for any coherent sheaf $\mathcal F$ on $S$, the map $H^i(S,\mathcal F)\rightarrow H^i(S,\mathcal F(nE))$, $* \mapsto \overline *_{(n)}$, and the cup product maps are compatible. For example, the diagram $$\begin{CD}
H^0(C,N_{S/V}\big{\vert}_C) @>{d_S(\beta)\cup}>> H^1(C,N_{S/V}((m+1)E)\big{\vert}_C)\\
@VVV @VVV \\
H^0(C,N_{S/V}((m-1)E)\big{\vert}_C) @>{d_S(\beta)\cup}>> H^1(C,N_{S/V}(2mE)\big{\vert}_C)
\end{CD}$$ is commutative. Therefore, by Step 1 we have $$\overline{{\operatorname{ob}}_S(\alpha)}_{(2m)}
=\overline{d_{S}(\beta)\big{\vert}_C \cup \pi_{C/S}(\alpha)}_{(m-1)}
=d_{S}(\beta)\big{\vert}_C \cup \overline{\pi_{C/S}(\alpha)}_{(m-1)}$$ in $H^1(C,N_{S/V}(2mE)\big{\vert}_C)$. Since there exists a commutative diagram $$\label{diag:res-coboundary1}
\begin{array}{ccccc}
&& \overline{\pi_{C/S}(\alpha)}_{(m-1)} && \overline{{\operatorname{ob}}_S(\alpha)}_{(2m)}\\ [-10pt]
&& \rotatebox{-90}{$\in$} && \rotatebox{-90}{$\in$} \\ [4pt]
H^1(\mathcal O_C((m+1)Z)) & \times & H^0(N_{S/V}((m-1)E)\big{\vert}_C)
& \overset{\cup}{\longrightarrow} & H^1(N_{S/V}(2mE)\big{\vert}_C)\\
{\Big\uparrow
\rlap{$\vcenter{\hbox{$\scriptstyle|_C$}}$ }} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_C\,$}}$ }} &&
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_C\,$}}$ }} \\
H^1(\mathcal O_S((m+1)E)) & \times & H^1(N_{S/V}((m-1)E-C))
& \overset{\cup}{\longrightarrow} & H^2(N_{S/V}(2mE-C)), \\
\rotatebox{90}{$\in$} && && \\ [-4pt]
d_S(\beta) && && \\
\end{array}$$ we have $$\overline{{\operatorname{ob}}_S(\alpha)}_{(2m)} \cup \mathbf k_C
=(d_{S}(\beta)\big{\vert}_C \cup
\overline{\pi_{C/S}(\alpha)}_{(m-1)}) \cup \mathbf k_C
= d_{S}(\beta) \cup (\overline{\pi_{C/S}(\alpha)}_{(m-1)} \cup \mathbf k_C)$$ in $H^2(S,N_{S/V}(2mE-C))$. Then by Lemma \[lem:restriction to C and E\], $\beta$ is contained in the subgroup $H^0(S,\mathcal I_{Z/S} \otimes N_{S/V}(mE)) \subset H^0(S,N_{S/V}(mE))$, and its restriction $\beta\big{\vert}_C$ to $C$ is a global section of the invertible sheaf $$\mathcal I_{Z/S} \otimes N_{S/V}(mE)\big{\vert}_C
\simeq \mathcal O_C(-Z)\otimes N_{S/V}(mE)\big{\vert}_C
\simeq N_{S/V}((m-1)E)\big{\vert}_C$$ on $C$ and we have $\beta\big{\vert}_C = \overline{\pi_{C/S}(\alpha)}_{(m-1)}$ by assumption. Therefore we obtain $$d_{S}(\beta) \cup (\overline{\pi_{C/S}(\alpha)}_{(m-1)} \cup \mathbf k_C)
=d_{S}(\beta) \cup (\beta\big{\vert}_C \cup \mathbf k_C).$$ Then [@Mukai-Nasu Lemma 2.8] shows that we have $\beta\big{\vert}_C \cup \mathbf k_C
=\beta\big{\vert}_E \cup \mathbf k_E$ in $H^1(S,N_{S/V}((m-1)E-C))$. Hence we have $$d_{S}(\beta) \cup (\beta\big{\vert}_C \cup \mathbf k_C)
=d_{S}(\beta) \cup (\beta\big{\vert}_E \cup \mathbf k_E)
=(d_{S}(\beta)\big{\vert}_E \cup \beta\big{\vert}_E) \cup \mathbf k_E,$$ where the last equality follows from the commutative diagram $$\label{diag:res-coboundary2}
\begin{array}{ccccc}
&& \beta\big{\vert}_E && \\ [-10pt]
&& \rotatebox{-90}{$\in$} && \\ [4pt]
H^1(\mathcal O_E((m+1)E)) & \times & H^0(N_{S/V}(mE-C)\big{\vert}_E)
& \overset{\cup}{\longrightarrow} & H^1(N_{S/V}((2m+1)E-C)\big{\vert}_E) \\
{\Big\uparrow
\rlap{$\vcenter{\hbox{$\scriptstyle|_E$}}$ }} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_E\,$}}$ }} &&
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_E\,$}}$ }} \\
H^1(\mathcal O_S((m+1)E)) & \times & H^1(N_{S/V}((m-1)E-C))
& \overset{\cup}{\longrightarrow} & H^2(N_{S/V}(2mE-C)), \\
\rotatebox{90}{$\in$} && && \\ [-4pt]
d_S(\beta) && && \\
\end{array}$$ similar to . Thus we obtain the equation required.
[**Step 3**]{} Let $\partial_E$ be the coboundary map of . Then by Proposition \[prop:key\] (2), we have $d_{S}(\beta)\big{\vert}_E\cup \beta\big{\vert}_E
=m\partial_E (\beta\big{\vert}_E)\cup \beta\big{\vert}_E$, which is nonzero by the assumption (b). Consider the coboundary map $$\cup\, \mathbf k_E: H^1(E,N_{S/V}((2m+1)E-C)\big{\vert}_E) \longrightarrow
H^2(S,N_{S/V}(2mE-C)),$$ which appears in . By the Serre duality, it is dual to the restriction map $$H^0(S,C+K_V\big\vert_S -2mE) \overset{|_E}{\longrightarrow} H^0(E,(C+K_V-2mE)\big\vert_E),$$ which is surjective by the assumption (a). Hence the coboundary map $\cup\, \mathbf k_E$ is injective. Therefore we obtain $d_{S}(\beta)\big{\vert}_E \cup \beta\big{\vert}_E \cup \mathbf k_E\ne 0$ and hence by Step 2 we conclude that $\overline{{\operatorname{ob}}_S(\alpha)}_{(2m)}\ne 0$ in $H^1(C,N_{S/V}(2mE)\big{\vert}_C)$, and hence we have finished the proof of Theorem \[thm:main1\].
Let $\pi_{E/S}(mE): H^0(E,N_{E/V}(mE)) \rightarrow H^0(E,N_{S/V}(mE)\big{\vert}_E)$ be the map induced by . If this map is not surjective, then the sections $\gamma$ in its image span a proper linear subsystem $$\Lambda':=\left\{
{\operatorname{div}}(\gamma) \bigm|
\gamma \in
{\operatorname{im}}\pi_{E/S}(mE)
\right\}$$ of the complete linear system $\Lambda:=|N_{S/V}(mE)\big{\vert}_E|$ on $E$, where ${\operatorname{div}}(\gamma)$ denotes the divisor of zeros for $\gamma$. The condition (b) in Theorem \[thm:main1\] can be replaced with the following conditions (b1), (b2) and (b3) in the next corollary, which is more accessible in many situations.
\[cor:obstruction\] Let $C \subset S\subset V$, $E$, $\alpha$, $\beta$, $\Delta$ be as in Theorem \[thm:main1\]. Suppose that $\beta\big{\vert}_E \ne 0$. If the following conditions are satisfied, then the exterior component ${\operatorname{ob}}_S(\alpha)$ of ${\operatorname{ob}}(\alpha)$ is nonzero.
1. The restriction map $
H^0(S,\Delta) \overset{\vert_E}\longrightarrow
H^0(E,\Delta\big{\vert}_E)
$ is surjective,
2. $m$ is not divisible by the characteristic $p$ of the ground field $k$,
3. $E$ is irreducible curve of arithmetic genus $g(E)$ and $(\Delta.E)=2g(E)-2-(m+1)E^2$.
4. $Z:=C\cap E$ is not a member of $\Lambda'$.
It suffices to prove that the condition (b) of Theorem \[thm:main1\] follows from the conditions (b1), (b2) and (b3) of this corollary. Since we have $N_{S/V}(mE-C) \simeq -K_V\big{\vert}_S+K_S+mE-C = K_S-\Delta-mE$, there exists an isomorphism $$\label{isom:triviality}
N_{S/V}(mE-C)\big{\vert}_E
\simeq \mathcal O_E(K_E-\Delta-(m+1)E)$$ of invertible sheaves on $E$, whose degree is zero by (b2). By Lemma \[lem:restriction to C and E\], $\beta\big{\vert}_E$ is a nonzero global section of $N_{S/V}(mE-C)\big{\vert}_E$. Since $E$ is irreducible, the invertible sheaves in are trivial. Hence as a section of $N_{S/V}(mE)\big{\vert}_E$, the divisor ${\operatorname{div}}(\beta\big{\vert}_E)$ of zeros associated to $\beta\big{\vert}_E$ coincides with $Z$. It follows from (b3) that $\partial_E(\beta\big{\vert}_E)\ne 0$ in $H^1(E,\mathcal O_E((m+1)E))$. Since $\beta\big{\vert}_E$ is a nonzero section of the trivial sheaf $N_{S/V}(mE-C)\big{\vert}_E$, the cup product $m\partial_E(\beta\big{\vert}_E)\cup \beta\big{\vert}_E$ is nonzero.
We finish this section by giving a refinement of Theorem \[thm:main1\]. Let $E_i$ ($1 \le i \le k$) be irreducible curves on $S$, which are mutually disjoint. We assume that for any two effective divisors $D,D'$ on $S$ with supports on $\bigcup_{i=1}^k E_i$, if $D \le D'$ then the map $H^1(S,\mathcal O_S(D)) \rightarrow H^1(S,\mathcal O_S(D'))$ is injective.
\[thm:refinement\] Let $E=\sum_{i=1}^k m_i E_i$ be a divisor on $S$ with $m_i \ge 1$. Let $\tilde C$ or $\alpha \in H^0(C,N_{C/V})$ be a first order deformation of $C$. Suppose that $H^1(S,N_{S/V})=0$ and the image of the exterior component $\pi_{C/S}(\alpha)$ in $H^0(C,N_{S/V}(E)\big{\vert}_C)$ lifts to an infinitesimal deformation $\beta \in H^0(S,N_{S/V}(E))$ with poles along $E$. If there exists an integer $1 \le i \le k$ satisfying the following conditions, then the exterior component ${\operatorname{ob}}_S(\alpha)$ of ${\operatorname{ob}}(\alpha)$ is nonzero:
1. Let $\Delta:=C+K_V\big{\vert}_S-\sum_{j \ne i}(m_j+1)E_j-2m_iE_i$. Then the restriction map $$H^0(S,\Delta) \overset{\vert_{E_i}}\longrightarrow
H^0(E_i,\Delta\big{\vert}_{E_i})$$ to $E_i$ is surjective, and
2. Let $\beta\big{\vert}_{E_i} \in H^0(E_i,N_{S/V}(m_iE_i)\big{\vert}_{E_i})$ be the principal part of $\beta$ along $E_i$, and let $\partial_{E_i}$ be the coboundary map of the exact sequence for $E=E_i$. Then we have $$m_i\partial_{E_i}(\beta\big{\vert}_{E_i}) \cup
\beta\big{\vert}_{E_i} \ne 0
\qquad
\mbox{in}
\qquad
H^1(E_i,N_{S/V}((2m_i+1){E_i}-C)\big{\vert}_{E_i}).$$
The proof is similar to that of Theorem \[thm:main1\]. We follow the same steps in the proof. Let $d_{S^\circ}: H^0(S^{\circ},N_{S^{\circ}/V^{\circ}})
\rightarrow H^1(S^{\circ},\mathcal O_{S^\circ})$ be the $d$-map for $S^{\circ} \subset V^{\circ}$. Then the image $d_S(\beta)$ of $\beta \in H^0(S,N_{S/V}(E))$ is contained in $H^1(S,\mathcal O_S(E'))
\subset H^1(S^{\circ},\mathcal O_{S^{\circ}})$ by Proposition \[prop:key2\], where $E':=E+\sum_{i=1}^kE_i=\sum_{i=1}^k(m_i+1)E_i$. There exists a partially commutative diagram similar to , which connects the two polar $d$-maps $d_{C,S}$ and $d_S$ with poles along $E$. By using this diagram, we find $$\overline{d_{C,S}(\pi_{C/S}(\alpha))}=
H^1(\iota)(d_S(\beta)\big{\vert}_C)
\qquad
\mbox{in}
\qquad
H^1(C,{N_{C/V}}^{\vee}\otimes N_{S/V}(E')\big{\vert}_C).$$ We will show that the image $\overline{{\operatorname{ob}}_S(\alpha)}$ of ${\operatorname{ob}}_S(\alpha)$ in $H^1(C,N_{S/V}(E')\big{\vert}_C)$ is nonzero. Using the argument in Step 1 before, we get $\overline{{\operatorname{ob}}_S(\alpha)}=d_S(\beta)\big{\vert}_C \cup
\pi_{C/S}(\alpha)$, where the cup product is taken by the map $$H^1(C,\mathcal O_C(E'))\times
H^0(C,N_{S/V}\big{\vert}_C)
\overset{\cup}\longrightarrow
H^1(C,N_{S/V}(E')\big{\vert}_C).$$ We consider a cup product $$\overline{{\operatorname{ob}}_S(\alpha)}\cup \mathbf k_C\in
H^2(S,N_{S/V}(E'-C))$$ of $\overline{{\operatorname{ob}}_S(\alpha)}$ with $\mathbf k_C \in {\operatorname{Ext}}^1(\mathcal O_C,\mathcal O_S(-C))$. There exists a commutative diagram $$\begin{array}{ccccc}
H^1(\mathcal O_C(E')) & \times & H^0(N_{S/V}\big{\vert}_C)
& \overset{\cup}{\longrightarrow} & H^1(N_{S/V}(E')\big{\vert}_C)\\
\Vert && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }} \\
H^1(\mathcal O_C(E')) & \times & H^0(N_{S/V}((m_i-1)E_i)\big{\vert}_C)
& \overset{\cup}{\longrightarrow} & H^1(N_{S/V}(E'+(m_i-1)E_i)\big{\vert}_C)\\
{\Big\uparrow
\rlap{$\vcenter{\hbox{$\scriptstyle|_C$}}$ }} && {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_C\,$}}$ }} &&
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_C\,$}}$ }} \\
H^1(\mathcal O_S(E')) & \times & H^1(N_{S/V}((m_i-1)E_i-C))
& \overset{\cup}{\longrightarrow} & H^2(N_{S/V}(E'+(m_i-1)E_i-C)).
\end{array}$$ By using this diagram, we compute the image of $\overline{{\operatorname{ob}}_S(\alpha)}\cup \mathbf k_C$ in $H^2(S,N_{S/V}(E'+(m_i-1)E_i-C))$ as $$\begin{aligned}
\overline{\overline{{\operatorname{ob}}_S(\alpha)}\cup \mathbf k_C}
&= (\overline{d_S(\beta)\big{\vert}_C \cup \pi_{C/S}(\alpha)) \cup \mathbf k_C} \\
&= (d_S(\beta)\big{\vert}_C \cup \overline{\pi_{C/S}(\alpha)}) \cup \mathbf k_C \\
&= d_S(\beta)\cup (\overline{\pi_{C/S}(\alpha)} \cup \mathbf k_C) \\
&= d_S(\beta)\cup (\beta\big{\vert}_C \cup \mathbf k_C),\end{aligned}$$ where $\overline{\pi_{C/S}(\alpha)}$ is the image of $\pi_{C/S}(\alpha)$ in $H^0(C,N_{S/V}((m_i-1)E_i)\big{\vert}_C)$. There exists a commutative diagram $$\begin{array}{ccccc}
H^1(\mathcal O_S(E')) & \times & H^1(N_{S/V}((m_i-1)E_i-C))
& \overset{\cup}{\longrightarrow} & H^2(N_{S/V}(E'+(m_i-1)E_i-C)) \\
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle|_{E_i}\,$}}$ }} && {\Big\uparrow
\rlap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_{E_i}$}}$ }} &&
{\Big\uparrow
\rlap{$\vcenter{\hbox{$\scriptstyle\cup \, \mathbf k_{E_i}$}}$ }} \\
H^1(\mathcal O_{E_i}((m_i+1)E_i)) & \times &
H^0(N_{S/V}(m_iE_i-C)\big{\vert}_{E_i})
& \overset{\cup}{\longrightarrow} & H^1(N_{S/V}((2m_i+1)E_i-C)\big{\vert}_{E_i}).
\end{array}$$ Then by [@Mukai-Nasu Lemma 2.8] and Proposition \[prop:key2\](2), we compute as $$\begin{aligned}
d_S(\beta)\cup (\beta\big{\vert}_C \cup \mathbf k_C)
&= d_S(\beta)\cup (\beta\big{\vert}_{E_i} \cup \mathbf k_{E_i}) \\
&= (d_S(\beta)\big{\vert}_{E_i}\cup \beta\big{\vert}_{E_i}) \cup \mathbf k_{E_i} \\
&= (m_i\partial_{E_i}(\beta\big{\vert}_{E_i})\cup \beta\big{\vert}_{E_i}) \cup \mathbf k_{E_i},\end{aligned}$$ where $m_i\partial_{E_i}(\beta\big{\vert}_{E_i})\cup
\beta\big{\vert}_{E_i}\ne 0$ by assumption. It follows from the assumption (a) that the cup product map $$H^1(E_i,N_{S/V}((2m_i+1)E_i-C)\big{\vert}_{E_i}) \overset{\cup \mathbf k_{E_i}}{\longrightarrow}
H^2(S,N_{S/V}(E'+(m_i-1)E_i-C))$$ with $\mathbf k_{E_i} \in {\operatorname{Ext}}^1(\mathcal O_{E_i},\mathcal O_S(-E_i))$ is injective. Therefore, we have $(m_i\partial_{E_i}(\beta\big{\vert}_{E_i})\cup \beta\big{\vert}_{E_i}) \cup \mathbf k_{E_i}\ne 0$ and thus we have completed the proof.
Obstructions to deforming curves lying on a $K3$ surface {#sect:k3}
========================================================
In this section, we prove Theorem \[thm:main2\] and Corollary \[cor:main3\]. In this and later sections, we assume that ${\operatorname{char}}k=0$. Let $C \subset S \subset V$ be as in the theorem. Then by Lemma \[lem:flag of k3fano\], the Hilbert-flag scheme ${\operatorname{HF}}V$ is nonsingular at $(C,S)$, and moreover, $A^2(C,S)=0$. Put $D:=C+K_V\big{\vert}_S$, a divisor on $S$. Then by the same lemma together with the Serre duality, we have $H^i(S,N_{S/V}(-C)) \simeq H^i(S,-D)\simeq H^{2-i}(S,D)^{\vee}$ for any integer $i$.
#### [**Proof of Theorem \[thm:main2\]**]{}.
\(1) It is known that if there exist no $(-2)$-curves and no elliptic curves on a smooth $K3$ surface $X$, then every nonzero effective divisor on $X$ is ample. (Then the effective cone ${\operatorname{NE}}(X)$ and the ample cone ${\operatorname{Amp}}(X)$ coincides.) Therefore we have $H^1(S,D)=0$ by assumption. Then the restriction map $\rho: H^0(S,N_{S/V})\rightarrow H^0(C,N_{S/V}\big{\vert}_C)$ is surjective. Then by Lemma \[lem:flag to hilb\](1), ${\operatorname{Hilb}}V$ is nonsingular at $[C]$.
(2)We show that the tangent map $p_1$ of $pr_1: {\operatorname{HF}}V \rightarrow {\operatorname{Hilb}}V$ at $(C,S)$ is not surjective and its cokernel is of dimension $1$. Since $D \ge 0$ and $D\ne 0$, we have $H^0(S,-D)=0$. Therefore by , there exists an exact sequence $$\label{ses:flag to hilb}
0 \longrightarrow A^1(C,S) \overset{p_1}
\longrightarrow H^0(C,N_{C/V})
\longrightarrow H^1(S,-D)
\longrightarrow 0.$$
\[claim:non-S-linear\] $H^1(S,-D)\simeq k$ and $H^1(S,-D+E)=0$
#### **Proof of Claim \[claim:non-S-linear\].**
Since $D.E=-2$ and $E^2=-2$, there exists an exact sequence $$\label{ses:(-2)-P^1}
0 \longrightarrow \mathcal O_S(D-(l+1)E)
\longrightarrow \mathcal O_S(D-lE)
\longrightarrow \mathcal O_{\mathbb P^1}(2l-2)
\longrightarrow 0$$ for every integer $l$. Since $H^1(\mathbb P^1,\mathcal O_{\mathbb P^1}(2l-2))=0$ for $l \ge 1$ and $H^1(S,D-3E)=0$, it follows from this exact sequence that $H^1(S,D-lE)=0$ for $l=1,2$. We prove $H^2(S,D-E)=0$. In fact, if $H^2(S,D-E)\ne 0$, then by the Serre duality, $-D+E$ is effective and we have $(-D+E)^2=D^2+2 \ge 2 >0$. Then it follows from the signature theorem (cf. [@BHPV IV.2, Thm. 2.14 and VIII.1]) that $0 \le (D.-D+E) =-D^2-2<0$ and hence we get a contradiction. Therefore by , we have $H^1(S,D) \simeq H^1(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2)) \simeq k$. By the Serre duality, we have proved the claim.
Let $\alpha$ be a global section of $N_{C/V}$. It suffices to prove the next claim.
\[claim:obstruction\] ${\operatorname{ob}}_S(\alpha)\ne 0$ if $\alpha \not \in {\operatorname{im}}p_1$.
#### **Proof of Claim \[claim:obstruction\].**
Let $\pi_{C/S}(\alpha)\in H^0(C,N_{S/V}\big{\vert}_C)$ be the exterior component of $\alpha$. There exists a commutative diagram $$\begin{CD}
0 @>>> A^1(C,S) @>{p_1}>> H^0(C,N_{C/V}) @>>> H^1(S,-D) @>>> 0 \\
@. @V{p_2}VV @V{\pi_{C/S}}VV \Vert \\
0 @>>> H^0(S,N_{S/V}) @>{\rho}>> H^0(C,N_{S/V}\big{\vert}_C) @>{\cup \mathbf k_C}>> H^1(S,-D) @>>> 0,
\end{CD}$$ where $\mathbf k_C$ is the extension class of . By this diagram, we see that $\pi_{C/S}(\alpha)$ is not contained in ${\operatorname{im}}\rho$, and hence $\pi_{C/S}(\alpha)\cup \mathbf k_C\ne 0$ in $H^1(S,-D)$. On the other hand, since $H^1(S,-D+E)=0$, we have $\overline{\pi_{C/S}(\alpha)}_{(1)}\cup \mathbf k_C=0$ and $\pi_{C/S}(\alpha)$ lifts to an infinitesimal deformation $\beta \in H^0(S,N_{S/V}(E))$ with a pole along $E$ by Lemma \[lem:restriction to C and E\](1). Then by (2) of the same lemma, the principal part $\beta\big{\vert}_E$ of $\beta$ is a nonzero global section of $N_{S/V}(E)\big{\vert}_E$, and its divisor of zeros contains $Z:=C\cap E$ (i.e. $Z \subset {\operatorname{div}}(\beta\big{\vert}_E)$).
Now we verify that the four conditions (a), (b1), (b2) and (b3) of Corollary \[cor:obstruction\] are satisfied. Put $\Delta=C+K_V\big{\vert}_S-2E$, a divisor on $S$. Since $H^1(S,\Delta-E)=H^1(S,D-3E)=0$, (a) follows from the exact sequence $0 \rightarrow \mathcal O_S(\Delta-E)
\rightarrow \mathcal O_S(\Delta)
\rightarrow \mathcal O_E(\Delta)
\rightarrow 0.
$ (b1) is clear. Since $E$ is a $(-2)$-curve, we compute that $\Delta.E=(D-2E.E)=2=2g(E)-2-2E^2$, and hence (b2) follows. Then by , this implies that $N_{S/V}(E-C)\big{\vert}_E$ is trivial and hence we have $Z ={\operatorname{div}}(\beta\big{\vert}_E)$. Finally, for (b3), we show that $H^1(S,C-E)=0$. In fact, we have $C-E=D-E-K_V\big{\vert}_S$. Since $H^1(S,D-E)=0$, we have $(D-E.E')\ge -1$ for any $(-2)$-curve $E'$ on $S$ by Lemma \[lem:K3\](2), which implies that $C-E$ is nef because $-K_V\big{\vert}_S$ is ample. Then $C-E$ is big by $$(C-E)^2=(D-E)^2+2(D-E.-K_V\big{\vert}_S)+(-K_V\big{\vert}_S)^2
>(D-E)^2>0.$$ Therefore $H^1(S,C-E)=0$ by the Kodaira-Ramanujam vanishing theorem. Then the rational map $$|C| \dashrightarrow |\mathcal O_E(C)|, \qquad C' \longmapsto Z=C'\cap E$$ is dominant. By assumption, $\Lambda'=\left\{{\operatorname{div}}(\gamma) \bigm|
\gamma \in {\operatorname{im}}\pi_{E/S}(E)
\right\}$ is a proper linear subsystem of $\Lambda=|N_{S/V}(E)\big{\vert}_E|$. Therefore, if necessary, by replacing $C$ with a general member $C'$ of $|C|$, we may assume that $Z=C\cap E$ is not contained in $\Lambda'$. In fact, by the upper semicontinuity, if a general member $C'\in |C|$ is obstructed, then so is $C$. Hence (b3) follows. By Corollary \[cor:obstruction\], we have proved the claim.
(3)The proof is very similar to that of (2). Suppose that $D\sim mF$ for $m \ge 2$ and an elliptic curve $F$. Then by Lemma \[lem:K3\](2), we have $H^1(S,-D)\simeq k^{m-1}$. Thus the cokernel of the tangent map $p_1$ of $pr_1$ is nonzero. Since $H^1(S,-D+F)\simeq k^{m-2}$, the kernel of the natural map $H^1(S,-D)\rightarrow H^1(S,-D+F)$ is of dimension at least one. Hence by , there exists a global section $\alpha$ of $N_{C/V}$ whose exterior component $\pi_{C/S}(\alpha)$ satisfies $\pi_{C/S}(\alpha)\cup \mathbf k_C \ne 0$ in $H^1(S,-D)$, while $\overline{\pi_{C/S}(\alpha)}_{(1)}\cup \mathbf k_C=0$ in $H^1(S,-D+F)$. We fix such a global section $\alpha$ and prove that ${\operatorname{ob}}_{S}(\alpha)\ne 0$. Again by Lemma \[lem:restriction to C and E\], there exists an infinitesimal deformation $\beta \in H^0(S,N_{S/V}(F))$ with a pole along $F$ such that $\beta\big{\vert}_C = \overline{\pi_{C/S}(\alpha)}_{(1)}$ and $\beta\big{\vert}_F$ is a nonzero global section of $N_{S/V}(F)\big{\vert}_F$. Thus it suffices to verify the conditions (a), (b1), (b2) and (b3) of Corollary \[cor:obstruction\]. (a) follows from $\Delta=(m-2)F$ and $\mathcal O_F(F)\simeq \mathcal O_F$. (b1) is clear. (b2) follows from $\Delta.F=2g(F)-2-2F^2=0$. Since $C-F=-K_V+(m-1)F$ is ample, we have $H^1(S,C-F)=0$ by the Kodaira vanishing theorem. Hence (b3) follows. The rest of the proof is same as that of (2).
#### [**Proof of Corollary \[cor:main3\]**]{}.
Let $W_{C,S}$ be the $S$-maximal family of curves containing $C$. If $H^1(S,D)=0$, then by Lemma \[lem:flag to hilb\](1), $pr_1$ is smooth. Since a smooth morphism is flat and a flat morphism maps a generic point onto a generic point, we have the conclusion of (a) (cf. [@Kleppe81 Corollary 1.3.5]). Suppose that $h^1(S,D)=1$. Then it follows from that $$\dim W_{C,S}
\le \dim_{[C]} {\operatorname{Hilb}}^{sc} V
\le h^0(C,N_{C/V})
=\dim W_{C,S}+1.$$ Since $C$ is obstructed by the theorem, we have $\dim_{[C]} {\operatorname{Hilb}}^{sc} V=\dim W_{C,S}$. Therefore (a) follows. Since $C$ is a generic member of $W_{C,S}$, we obtain (b). If $H^0(S,-D)=0$, then by Lemma \[lem:flag to hilb\](2) and Lemma \[lem:flag of k3fano\], we obtain $\dim_{[C]} {\operatorname{Hilb}}^{sc} V=
\dim W_{C,S} = (-K_V\big{\vert}_S)^2/2+g+1$. Thus we have completed the proof.
Non-reduced components of the Hilbert scheme {#sect:non-reduced}
============================================
In this section, as an application, we study the deformations of curves lying on a smooth quartic surface $S$ in $\mathbb P^3$, or a smooth hyperplane section $S$ of a smooth quartic $3$-fold $V_4 \subset \mathbb P^3$ (assuming ${\operatorname{char}}k=0$). We give some examples of generically non-reduced components of the Hilbert schemes ${\operatorname{Hilb}}^{sc} \mathbb P^3$ and ${\operatorname{Hilb}}^{sc} V_4$ (cf. Examples \[ex:non-reduced V\_4\] and \[ex:non-reduced P\^3\]). As is well known, $S$ is a $K3$ surface. Here we consider $S$ (i) of Picard number two ($\rho(S)=2$), (ii) containing a curve $E$ not a complete intersection in $S$, and such that (iii) ${\operatorname{Pic}}S$ is generated by $E$ and the class $\mathbf h$ of hyperplane sections of $S$ (i.e., ${\operatorname{Pic}}S \simeq \mathbb Z \mathbf h \oplus \mathbb Z E$). Kleppe and Ottem [@Kleppe-Ottem] have studied the deformations of space curves $C \subset \mathbb P^3$ lying on such a quartic surface $S$ by assuming that $E$ is a line, or a conic. They have also produced examples of generically non-reduced components of ${\operatorname{Hilb}}^{sc} \mathbb P^3$ by a different method (cf. Remark \[rmk:Kleppe-Ottem\]).
Mori cone of quartic surfaces {#subsec:mori cone of quartics}
-----------------------------
Let $S \subset \mathbb P^3$ be a smooth quartic surface. If $S$ is general, then we have $\rho(S)=1$ and ${\operatorname{Pic}}S$ is generated by $\mathbf h=-\frac 14 K_{\mathbb P^3}\big{\vert}_S$. Then every curve $C$ on $S$ is a complete intersection of $S$ with some other surface in $\mathbb P^3$, and hence $C$ is arithmetically Cohen-Macaulay. Thus we see that $C$ is unobstructed, thanks to a result of Ellingsrud [@Ellingsrud].
First we consider a quartic surface $S$ containing a (smooth) rational curve $E$. Then by its genus, $E$ is not a complete intersection in $S$. It follows from Lemma \[lem:mori\] that for every integer $e\ge 1$, there exists a smooth quartic surface $S \subset \mathbb P^3$ containing a rational curve $E$ of degree $e$, and such that ${\operatorname{Pic}}S \simeq \mathbb Z \mathbf h \oplus \mathbb Z E$. Let $(S,E)$ be such a pair of a surface $S$ and a curve $E\simeq \mathbb P^1$. Then every divisor $D$ on $S$ is linearly equivalent to $x\mathbf h-yE$ for some $x,y \in \mathbb Z$. Since we have $\mathbf h^2=4, \mathbf h.E=e$ and $E^2=-2$, we compute the self intersection number of $D$ on $S$ as $D^2=4x^2-2exy-2y^2$. Recall that for a projective surface $X$, the effective cone ${\operatorname{NE}}(X)$ of $X$ is defined as ${\operatorname{NE}}(X)=\left\{
\sum_{i=1}^n a_i[C_i] \bigm|
\mbox{$C_i$ is an irreducible curve on $X$ and $a_i \in \mathbb R_{\ge 0}$}
\right\}$ and the Mori cone ${\overline{\operatorname{NE}}}(X)$ is defined as the closure of ${\operatorname{NE}}(X)$ in ${\operatorname{NS}}(X)_{\mathbb R}$. Kovács [@Kovacs] proved that for every $K3$ surface $X$ with $\rho(X)=2$, ${\overline{\operatorname{NE}}}(X)$ has two extremal rays, which can be generated by the classes of two $(-2)$-curves, one $(-2)$-curve and an elliptic curve, two elliptic curves, or two non-effective classes $x_1,x_2$ with $x_i^2=0$. Applying Kovács’s result, we have the following lemma.
\[lem:mori cone1\] Let $S$ be a smooth quartic surface, $E$ a smooth rational curve of degree $e \ge 2$ on $S$ such that ${\operatorname{Pic}}S=\mathbb Z\mathbf h\oplus \mathbb ZE$, $D$ a divisor on $S$. Then
1. If $D$ is not linearly equivalent to $0$, then $D^2\ne 0$.
2. There exists a (unique) $(-2)$-curve $E'$ on $S$ such that ${\overline{\operatorname{NE}}}(S)=\mathbb R_{\ge 0}[E] +\mathbb R_{\ge 0}[E']$.
3. $D$ is nef if and only if $D.E\ge 0$ and $D.E'\ge 0$.
4. Suppose that $D\ge 0$ and $D \ne 0$. Then a general member $C$ of $|D|$ is a smooth connected curve if and only if $D$ is nef, $D=E$, or $D=E'$.
5. If (i) $D\ge 0$ and $D$ is nef or (ii) $D=E,E'$, then $H^1(S,D)=0$.
Since $E$ spans one of the two extremal rays, for proving (1) and (2), it suffices to prove that there exist no elliptic curves on $S$. Suppose that there exists a nonzero divisor $D\sim x\mathbf h-yE$ on $S$ with $D^2=-2(y^2+exy-2x^2)=0$. Then the discriminant $d=(ex)^2-4(-2x^2)=(8+e^2)x^2$ of this quadratic equation (with a variable $y$) equals a power of an integer. Since $D\not\sim 0$ in ${\operatorname{Pic}}S$, we have $8+e^2=k^2$ for some integer $k \ge 1$. By solving this equation, we have $(k,e)=(3,1)$, or $(9/2,7/2)$, which are both impossible by assumption. Since the nef cone and the Mori cone are dual to each other, we have (3). If $C \in |D|$ is irreducible and $C \ne E,E'$, then $C$ is nef. Conversely, if $D$ is nef, then $|D|$ is base point free by Lemma \[lem:K3\](1), and a general member $C \in |D|$ is a smooth connected curve by Bertini’s theorem. Hence (4) follows. (5) is clear if $D\sim 0$, $E$, or $E'$. Otherwise it follows from (1) and the Kodaira-Ramanujam vanishing theorem.
1. If $e=1$, then ${\overline{\operatorname{NE}}}(S)$ is generated by the classes of a line $E$ and a smooth elliptic curve $F\sim \mathbf h -E$ of degree $3$, contained in a plane in $\mathbb P^3$ (cf. [@Kleppe-Ottem Proposition 5.1]).
2. Once $e\ge 2$ is given, the class $x\mathbf h-yE$ of the curve $E'\simeq \mathbb P^1$ spanning the other ray of ${\overline{\operatorname{NE}}}(S)$ can be explicitly computed as the minimal nonzero solution $(x,y)$ of the equation $$y^2+exy-2x^2=1,$$ which is equivalent to $(E')^2=-2$. If $e$ is even, i.e., $e=2m$ for $m \ge 1$, then we have $E'=mh-E$. If $e$ is odd, then we see that $y$ is odd and $x$ is even ($x=2x'$), and the above equation is reduced to the Pell equation $X^2-(e^2+8)Y^2=1$, by putting $X:=y+ex'$ and $Y:=x'$. Thus the class $x\mathbf h-yE$ of $E'$ in ${\operatorname{Pic}}S$ is obtained as the minimal solution of this Pell equation and computed as in Table \[table:class of E’\].
$e$ 2 3 4 5 6 7 8 9 $\cdots$
--------- ---------- ---------- --------- --------- --------- ----------- --------- ------------------ ----------
$(x,y)$ $(1,1) $ $(16,9)$ $(2,1)$ $(8,3)$ $(3,1)$ $(40,11)$ $(4,1)$ $(106000,23001)$ $\cdots$
$d(E')$ $2$ $37$ $4$ $17$ $6$ $83$ $8$ $216991$ $\cdots$
: Classes of $E'$[]{data-label="table:class of E’"}
\[rmk:class of E’\]
Secondly, we consider elliptic quartic surfaces. It follows from Lemma \[lem:mori\] that for every integer $e\ge 3$, there exists a smooth quartic surface $S \subset \mathbb P^3$ containing a smooth elliptic curve $F$ of degree $e$ such that ${\operatorname{Pic}}S \simeq \mathbb Z \mathbf h \oplus \mathbb Z F$.
\[lem:mori cone2\] Let $S$ be a smooth quartic surface, $F$ a smooth elliptic curve of degree $e \ge 4$ on $S$ such that ${\operatorname{Pic}}S=\mathbb Z\mathbf h\oplus \mathbb ZF$, $D$ a divisor on $S$. Then
1. $D^2\ne -2$. In particular, there exists no $(-2)$-curve on $S$.
2. There exists a smooth elliptic curve $F'$ on $S$ such that ${\overline{\operatorname{NE}}}(S)=\mathbb R_{\ge 0}[F] +\mathbb R_{\ge 0}[F']$. Moreover, the class of $F'$ in ${\operatorname{Pic}}S$ equals $e\mathbf h-2F$ if $e$ is odd, and $(e/2)\mathbf h-F$ otherwise.
3. $D$ is nef if and only if $D \ge 0$.
4. Suppose that $D \ge 0$ and $D\ne 0$. Then the following are equivalent: (i) every general member $C$ of $|D|$ is a smooth connected curve, (ii) $D \not\sim kF$ and $D \not\sim kF'$ for $k \ge 2$, (iii) $H^1(S,D)= 0$
If $D\sim x\mathbf h-yF$ with $x,y \in \mathbb Z$, then we have $D^2=2x(2x-ye)$ by $h^2=4, h.F=e$ and $F^2=0$. The Diophantine equation $x(2x-ye)=-1$ on $(x,y)$ has no solutions if $e\ne 1,3$. Thus we obtain (1). Since $F$ spans one of the extremal ray, by Kovács’s result, the other ray is spanned by the class of a smooth elliptic curve $F'$ on $S$. In fact, by solving the equation $x(2x-ye)=0$, we see that if $D^2=0$ and $D$ is not spanned by $F$, then $D$ is spanned by the (primitive) classes given in the lemma. Thus we obtain (2). (3) follows from (2), because ${\overline{\operatorname{NE}}}(S)$ is dual to itself. For (4), suppose that $D \ge 0$ and $D\ne 0$. Then $|D|$ has no fixed component. Hence by [@Saint-Donat Prop. 2.6], if $D^2>0$, then every general member $C$ of $|D|$ is a smooth connected curve and $h^1(S,D)=0$, and otherwise $D$ is a multiple $kF$ ($k \ge 1$) of a smooth elliptic curve $F$ on $S$ and $h^1(S,D)=k-1$.
Finally, we consider quartic surfaces without $(-2)$-curves nor elliptic curves. Let $S$ be a smooth quartic surface not containing any $(-2)$-curves nor smooth elliptic curves. Then ${\operatorname{NE}}(S)$ coincides with the ample cone ${\operatorname{Amp}}(S)$ of $S$. In particular, $H^1(S,D)$ vanishes for every effective divisor $D$ on $S$. If $\rho(S)=2$, then ${\overline{\operatorname{NE}}}(S)=\mathbb R_{\ge 0}[x_1] +\mathbb R_{\ge 0}[x_2]$ for two non-effective classes $x_i$ ($i=1,2$) with $x_i^2=0$. We give an example of such a quartic surface.
\[ex:quartic without P\^1’s nor elliptics\] It follows from Lemma \[lem:mori\] that there exists a smooth quartic surface $S \subset \mathbb P^3$ and a smooth connected curve $\Gamma$ on $S$ of degree $6$ and genus $2$ such that ${\operatorname{Pic}}(S)=\mathbb Z \mathbf h \oplus \mathbb Z \Gamma$. Then on $S$ there exist no divisors $D$ with $D^2=-2$ and no divisors $D$ with $D^2=0$ and $D \not\sim 0$. In fact, let $D\sim x\mathbf h-y\Gamma$ for $x,y \in \mathbb Z$. Then we have $D^2=4x^2-12xy+2y^2=(2x-3y)^2-7y^2$. We have $D^2\ne -2$, because $-2\equiv 5$ is not a quadratic residue modulo $7$. If $D\not\sim 0$, then we have $D^2\ne 0$ as well. Thus we conclude that there exist no $(-2)$-curves and no smooth elliptic curves on $S$.
Hilbert schemes of $\mathbb P^3$ and $V_4$ {#subsec:hilb}
------------------------------------------
Let $V$ be $\mathbb P^3$ or a smooth quartic $3$-fold $V_4 \subset \mathbb P^4$, $S$ a smooth quartic surface in $\mathbb P^3$ or a smooth hyperplane section of $V_4$. It is known that if $S$ is general (in $|\mathcal O_{\mathbb P^3}(4)|$ or in $|\mathcal O_{V_4}(1)|$), the Picard group of $S$ is generated by the class $\mathbf h$ of hyperplane sections of $S$ (see e.g. [@Moisezon] for $V=V_4$). Let $C$ be a smooth connected curve of degree $d$ ($=C.\mathbf h$) and genus $g$ in $S$, not a complete intersection in $S$. Then there exists a first order deformation $\tilde S$ of $S$ not containing any first order deformation $\tilde C$ of $C$. Then by Lemma \[lem:flag of k3fano\], ${\operatorname{HF}}V$ is nonsingular at $(C,S)$ of dimension $a^1(C,S)=(-K_V\big{\vert}_S)^2/2+g+1$. Since $K_{\mathbb P^3}\big{\vert}_S=-4\mathbf h$ and $K_{V_4}\big{\vert}_S=-\mathbf h$, the number $a^1(C,S)$ equals $g+33$ if $V=\mathbb P^3$, and $g+3$ if $V=V_4$. Let $W_{C,S} \subset {\operatorname{Hilb}}^{sc} V$ be the $S$-maximal family of curves containing $C$ (cf. Definition \[dfn:S-maximal\]). Then if $d>16$ (resp. $d>4$), then we have $H^0(S,N_{S/V}(-C))=0$ and hence $\dim W_{C,S}=a^1(C,S)$. In what follows, we assume that $d >16$ if $V=\mathbb P^3$ and $d>4$ if $V=V_4$.
\[thm:Hilb with rational curve\] Let $V=\mathbb P^3$ or $V=V_4$, and let $S,C$ be as above. Suppose that there exists a smooth rational curve $E$ of degree $e \ge 2$ on $S$ such that ${\operatorname{Pic}}S=\mathbb Z\mathbf h\oplus \mathbb ZE$. Let $E'$ be as in Lemma \[lem:mori cone1\] and suppose that $D:=C+K_V\big{\vert}_S$ is effective.
1. If $D$ is nef, or $D=E,E'$, then $W_{C,S}$ is a generically smooth component of ${\operatorname{Hilb}}^{sc} V$.
2. Suppose that $N_{E/V}$ is globally generated if $V=V_4$. If $D.E=-2$ and $D\ne E$, then $W_{C,S}$ is a generically non-reduced component of ${\operatorname{Hilb}}^{sc} V$.
\(1) follows from Corollary \[cor:main3\] and Lemma \[lem:mori cone1\]. We prove (2). For an invertible sheaf $L\sim x\mathbf h+yE$ on $S$, we have $L.E=0$ if and only if $ex-2y=0$. This implies that if $L.E=0$, then the class of $L$ in ${\operatorname{Pic}}S$ is spanned by the class $2\mathbf h+eE$ if $e$ is odd, and the class $\mathbf h+(e/2)E$ otherwise. By assumption, there exists an integer $k \ge 1$ such that $D$ is linearly equivalent to the class $$\begin{cases}
k(2\mathbf h+eE)+E & (\mbox{if $e$ is odd}), \\
k(\mathbf h+(e/2)E)+E & (\mbox{otherwise}).
\end{cases}$$ Since $(D-E.E)=0$, we have $D^2=(D-E)^2+E^2=(D-E)^2-2>0$. We show that $H^1(S,D-3E)=0$. Suppose that $e$ is even and put $e=2e'$ ($e'\ge 1$). If $k \ge 2$ or $e'\ge 2$, then $D-3E$ is nef and effective, because $(D-3E.E)=4$ and $(D-3E.E')=k\mathbf h.E'+(ke'-2)E.E'>0$. If $k=1$ and $e'=1$, then $D-3E=\mathbf h-E=E'$. Thus we conclude that $H^1(S,D-3E)=0$ by Lemma \[lem:mori cone1\]. Similarly, we can show that $H^1(S,D-3E)=0$ if $e$ is odd. It follows from Lemma \[lem:non-surjective\] that $\pi_{E/S}(E)$ is not surjective. Therefore, by Corollary \[cor:main3\], $W_{C,S}$ is a generically non-reduced component of ${\operatorname{Hilb}}^{sc} V$.
In Theorem \[thm:Hilb with rational curve\], if $C\sim a\mathbf h+bE$ ($b \ne 0$ by assumption), then we have $d=4a+be$ and $g=2a^2+abe-b^2+1$. Given a class $(a,b)$ of $C$ in ${\operatorname{Pic}}S$, one can check whether $D$ is nef or not by using Lemma \[lem:mori cone1\](3) together with the method to compute the class of $E'$ shown in Remark \[rmk:class of E’\](2).
\[rmk:Kleppe-Ottem\] Kleppe and Ottem [@Kleppe-Ottem] have also studied the deformations of space curves lying on a smooth quartic surface. They have considered a smooth quartic surface $S\subset \mathbb P^3$ containing a line $E$, and such that ${\operatorname{Pic}}S \simeq \mathbb Z\mathbf h\oplus \mathbb ZE$, and a curve $C \subset S$ of degree $d>16$ and genus $g$, not a complete intersection in $S$, satisfying $D:=C-4\mathbf h\ge 0$. Then they have proved that if $D.E \ge -1$ or $D\sim E$, then $W_{C,S}$ becomes a generically smooth component of ${\operatorname{Hilb}}^{sc} \mathbb P^3$, and if $D.E \le -2$ (i.e., $D.E=-2,-3$, or $-4$), $D\ne E$, $d \ge 21$ and $g> \min\left\{G(d,5)-1,d^2/10+21\right\}$, then $W_{C,S}$ becomes a generically non-reduced component of ${\operatorname{Hilb}}^{sc} \mathbb P^3$. Here $G(d,5)$ denotes the maximum genus of curves of degree $d$ not contained in a surface of degree $4$. More recently, we have been informed by Kleppe that if $D.E \le -2$, then the assumption on the genus $g$ is almost always satisfied (with several exceptions of the classes of $C$ in ${\operatorname{Pic}}S \simeq \mathbb Z^2$). See [@Kleppe-Ottem] for the details.
For a space curve $C \subset \mathbb P^3$, $s(C)$ denotes the minimal degree of surfaces containing $C$. An irreducible closed subset $W$ of ${\operatorname{Hilb}}^{sc} \mathbb P^3$ is called [*$s$-maximal*]{}, if a general member $C$ of $W$ is contained in a surface of degree $s(C)=s$ (i.e. $s(W)=s$), and if we have $s(W')>s(W)$ for any closed irreducible subset $W' \subset {\operatorname{Hilb}}^{sc} \mathbb P^3$ strictly containing $W$ (cf. [@Kleppe87]). If $C \subset \mathbb P^3$ is contained in a surface $S\subset \mathbb P^3$ of degree $s=s(C)$, then every $s$-maximal subset is a $S$-maximal family, i.e. the image of some irreducible component of ${\operatorname{HF}}^{sc} \mathbb P^3$ passing through $(C,S)$. It is known that if a very general curve $C$ of a $4$-maximal family $W$ sits on a smooth quartic surface $S$ and $d(C)>16$, then the Picard number is at most $2$ (cf. [@Kleppe-Ottem Remark 2.3]).
We give infinitely many examples of generically non-reduced components of ${\operatorname{Hilb}}^{sc} V_4$, which contains Example \[ex:non-reduced\](1) as a special case ($n=2$).
\[ex:non-reduced V\_4\] Let $V_4 \subset \mathbb P^4$ be a smooth quartic $3$-fold, $E \simeq \mathbb P^1$ a conic on $V_4$ with trivial normal bundle $N_{E/V_4} \simeq \mathcal O_{\mathbb P^1}^2$, $S$ a smooth hyperplane section of $V_4$ containing $E$ and such that ${\operatorname{Pic}}S=\mathbb Z\mathbf h\oplus \mathbb ZE$. We consider a complete linear system $\Lambda_n:=|n(\mathbf h+E)|$ on $S$ for an integer $n \ge 2$. Then $\Lambda_n$ is base point free and a general member $C$ of $\Lambda_n$ is a smooth connected curve of degree $6n$ and genus $3n^2+1$. Let $W_n$ be the $S$-maximal family $W_{C,S}$ of curves containing $C$. Since $D.E=(C-\mathbf h.E)=-2$ and $N_{E/{V_4}}$ is globally generated, $W_n$ becomes generically non-reduced components of ${\operatorname{Hilb}}^{sc} V_4$ of dimension $3n^2+4$ by Theorem \[thm:Hilb with rational curve\].
It was proved in [@Mukai-Nasu Theorem 1.3] that for every smooth cubic $3$-fold $V_3 \subset \mathbb P^4$, the Hilbert scheme ${\operatorname{Hilb}}^{sc} V_3$ contains a generically non-reduced component $W$ of dimension $16$. Then every general member $C$ of $W$ is a smooth connected curve of degree $8$ and genus $5$, and for each $C$, there exist a smooth hyperplane section $S_3$ of $V_3$ and a line $E$ on $S_3$ satisfying the linear equivalence $C \sim 2\mathbf h+2E \sim -K_{V_3}\big{\vert}_{S_3}+2E$. It was also proved that if $N_{E/V_3}$ is trivial (in other words, $E$ is a [*good line*]{} on $V_3$), then $C$ is primarily obstructed. We refer to [@Nasu4] for a generalization to a smooth del Pezzo $3$-fold $V_n$ of degree $n$.
We have a similar result for curves lying on an elliptic quartic surface. We have the next theorem as a consequence of Lemma \[lem:mori cone2\], Lemma \[lem:non-surjective\], and Corollary \[cor:main3\].
\[thm:Hilb with elliptic curve\] Let $V=\mathbb P^3$ or $V=V_4$, and let $S,C$ be as above. Suppose that there exists a smooth elliptic curve $F$ of degree $e \ge 4$ on $S$ such that ${\operatorname{Pic}}S=\mathbb Z\mathbf h\oplus \mathbb ZF$. Let $F'$ be as in Lemma \[lem:mori cone2\] and we assume that $D:=C+K_V\big{\vert}_S$ is effective.
1. If $D \not\sim kF$ and $D \not\sim kF'$ for $k \ge 2$, then $W_{C,S}$ is a generically smooth component of ${\operatorname{Hilb}}^{sc} V$.
2. If $D \sim 2F$ or $D\sim 2F'$, then $W_{C,S}$ is a generically non-reduced component of ${\operatorname{Hilb}}^{sc} V$.
In Theorem \[thm:Hilb with elliptic curve\], we have
1. If $C\sim a\mathbf h+bF$ ($b \ne 0$ by assumption), then $d=4a+be$ and $g=2a^2+abe+1$.
2. Theorem \[thm:main2\] shows that if $D\sim mE$ for some integer $m\ge 2$ and an elliptic curve $E$ on $S$, then $C$ is obstructed. In this case, we have $h^1(S,D)=m-1$. However, Theorems \[thm:main1\] and \[thm:main2\] are not sufficient for proving that $W_{C,S}$ is an irreducible component of $({\operatorname{Hilb}}^{sc} \mathbb P^3)_{{\operatorname{red}}}$ (or $({\operatorname{Hilb}}^{sc} V_4)_{{\operatorname{red}}}$) for $m > 2$.
\[ex:non-reduced P\^3\] Let $V=\mathbb P^3$, $S$ a smooth quartic surface containing a smooth elliptic curve $F$. Then the complete linear system $\Lambda:=|4\mathbf h+2F|$ on $S$ is base point free and every general member $C$ of $\Lambda$ is a smooth connected curve on $S$. Then $C$ is obstructed by Theorem \[thm:main2\], and moreover, $W_{C,S}$ is a generically non-reduced component of ${\operatorname{Hilb}}^{sc} \mathbb P^3$ by Corollary \[cor:main3\].
One can compare this example with Mumford’s example [@Mumford] of a generically non-reduced component $W$ of ${\operatorname{Hilb}}^{sc} \mathbb P^3$, whose general member $C$ is contained in a smooth cubic surface $S$ and $C\sim 4\mathbf h+2E$ on $S$ for a line $E \subset S$.
\[thm:Hilb without rational nor elliptic curve\] Let $V=\mathbb P^3$ or $V=V_4$, and let $S,C$ be as above. Suppose that there exist no $(-2)$-curves nor elliptic curves on $S$. If $D:=C+K_V\big{\vert}_S$ is effective, then $W_{C,S}$ is a generically smooth component of ${\operatorname{Hilb}}^{sc} V$.
See Example \[ex:quartic without P\^1’s nor elliptics\] for an example of a smooth quartic surface not containing any $(-2)$-curves nor elliptic curves.
Acknowledgments {#acknowledgments .unnumbered}
===============
I should like to thank Prof. Eiichi Sato for asking me a useful question concerning the existence of a non-reduced component of the Hilbert scheme of a smooth Fano $3$-fold of index one. I am grateful to Prof. Jan Oddvar Kleppe for very helpful comments and a discussion on non-reduced components of the Hilbert scheme of space curves. Also thanks to the referee for his/her helpful comments. This work was partially supported by JSPS Grant-in-Aid (C), No 25400048, and JSPS Grant-in-Aid (S), No 25220701.
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abstract: 'Conformal classical Yang-Baxter equation and $S$-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely, $\mathcal O$-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map $T$ where $T_0=T_\lambda\mid_{\lambda=0}$ is an $\mathcal O$-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang-Baxter equation, whereas the symmetric part is a symmetric solution of conformal $S$-equation. One byproduct is that a finite left-symmetric conformal algebra which is a free $\mathbb{C}[\partial]$-module gives a natural $\mathcal O$-operator and hence there is a construction of solutions of conformal classical Yang-Baxter equation and conformal $S$-equation from the former. Another byproduct is that the non-degenerate solutions of these two equations correspond to 2-cocycles of Lie conformal algebras and left-symmetric conformal algebras respectively. We also give a further study on a special class of $\mathcal{O}$-operators called Rota-Baxter operators on Lie conformal algebras and some explicit examples are presented.'
address:
- 'College of Science, Zhejiang Agriculture and Forestry University, Hangzhou, 311300, P.R.China'
- 'Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China'
author:
- Yanyong Hong
- Chengming Bai
title: 'Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators'
---
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Introduction
============
The notion of Lie conformal algebra, formulated by Kac in [@K1; @K2], gives an axiomatic description of the operator product expansion (or rather its Fourier transform) of chiral fields in conformal field theory. It appears as an useful tool to study vertex algebras ([@K1]) and has many applications in the theory of infinite-dimensional Lie algebras satisfying the locality property in [@K] and Hamiltonian formalism in the theory of nonlinear evolution equations (see [@Do] and the references therein, and also [@BDK; @GD; @Z; @X4]). The structure theory ([@DK1]), representation theory ([@CK1; @CK2]) and cohomology theory ([@BKV]) of finite Lie conformal algebras have been well developed. On the other hand, a vertex algebra is a “combination" of a Lie conformal algebra and another algebraic structure, namely a left-symmetric algebra, satisfying certain compatible conditions ([@BK]). Moreover, for studying whether there exist compatible left-symmetric algebra structures on formal distribution Lie algebras, the definition of left-symmetric conformal algebra was introduced in [@HL], which can be used to construct vertex algebras.
Motivated by the study of Lie bialgebras ([@Dr]), a theory of Lie conformal bialgebra was established in [@L]. The notion of a finite Lie conformal bialgebra which is free as a $\mathbb{C}[\partial]$-module was introduced to be equivalent to a conformal Manin triple associated to a non-degenerate symmetric invariant conformal bilinear form. The notion of conformal classical Yang-Baxter equation was also introduced to construct (coboundary) Lie conformal bialgebras and hence as a byproduct, the conformal Drinfeld’s double was constructed. Explicitly, let $R$ be a Lie conformal algebra and $r=\sum_i a_i\otimes b_i\in R\otimes R$. Set $\partial^{\otimes^3}=\partial\otimes 1\otimes 1
+1\otimes \partial\otimes 1+1\otimes 1\otimes \partial$. The equation $$\begin{aligned}
[[r,r]]:&=&\sum\limits_{i,j}([{a_i}_\mu a_j]\otimes b_i\otimes b_j|_{\mu=1\otimes \partial \otimes 1}-a_i\otimes [{a_j}_\mu b_i]\otimes b_j|_{\mu=1\otimes 1\otimes \partial}
-a_i\otimes a_j\otimes [{b_j}_\mu b_i]|_{\mu=1\otimes \partial \otimes 1})\nonumber\\
&=&0\;\;\text{mod}\; (\partial^{\otimes^3}) ~~~~~~~\text{in}\; R\otimes R\otimes R,\end{aligned}$$ is called [**conformal classical Yang-Baxter equation (conformal CYBE)**]{} in $R$. Then the (skew-symmetric) solutions of conformal CYBE can be used to construct Lie conformal bialgebras and many interesting examples beyond the classical Lie bialgebras due to this construction were given in [@L]. Therefore how to find solutions of conformal CYBE becomes an important problem.
As both an analogue of the above Lie conformal bialgebra in the context of left-symmetric conformal algebra and a conformal analogue of left-symmetric bialgebra given in [@Bai1], the notion of a finite left-symmetric conformal bialgebra which is free as a $\mathbb{C}[\partial]$-module was introduced in [@HL1]. It is equivalent to a parak$\ddot{\text{a}}$hler Lie conformal algebra which is an analogue of the above Manin triple for Lie conformal algebras, but associated to a non-degenerate skew-symmetric conformal bilinear form satisfying the so-called “2-cocycle" condition. There is also the corresponding analogue of the classical Yang-Baxter equation, namely conformal $S$-equation. Explicitly, let $A$ be a left-symmetric conformal algebra and $r=\sum_{i}r_i\otimes l_i\in A\otimes A$. Then $$\begin{gathered}
\{\{r,r\}\}:=\sum_{i,j}({l_j}_\mu r_i\otimes r_j\otimes l_i)|_{\mu=1\otimes \partial\otimes 1}-\sum_{i,j}(r_j\otimes {l_j}_\mu r_i\otimes l_i)|_{\mu=\partial\otimes 1\otimes1}\nonumber\\
-\sum_{i,j}(r_i\otimes r_j\otimes [{l_i}_\mu l_j])|_{\mu=\partial\otimes 1\otimes 1}=0 ~~~\text{mod}(\partial^{\otimes^3})\;\;{\rm in}\;\;A\otimes A\otimes A,\end{gathered}$$ is called [**conformal $S$-equation**]{} in $A$. The (symmetric) solutions of conformal $S$-equation can be used to construct left-symmetric conformal bialgebras. Thus it is also natural and important to find solutions of conformal $S$-equation.
On the other hand, in the classical case, for the CYBE and $S$-equation in a Lie algebra and a left-symmetric algebra respectively, an important idea to find solutions is to replace the tensor form by an operator form ([@S; @Ku; @Bai; @Bai1]). It is Semonov-Tian-Shansky who gave the first operator form of CYBE in a Lie algebra $\mathfrak g$ as a linear transformation on $\mathfrak g$ satisfying the condition that is known as a Rota-Baxter operator in the context of Lie algebra now ([@S]). The notion of $\mathcal O$-operator was introduced by Kupershmidt in [@Ku] replacing the linear transformation by a linear map, which is a natural generalization of CYBE. Both of them are equivalent to the tensor form of CYBE under certain conditions. Moreover, a systematic study involving $\mathcal O$-operators and CYBE was given in [@Bai], where an equivalence between the operator forms and tensor form of CYBE was obtained in a more general extent. There are two direct consequences.
1. There is a relationship between the CYBE and $S$-equation in terms of $\mathcal O$-operator, that is, the skew-symmetric part of an $\mathcal{O}$-operator gives a skew-symmetric solution of CYBE ([@Bai]), whereas a symmetric part of an $\mathcal{O}$-operator gives a symmetric solution of $S$-equation ([@Bai1]). Thus the solutions of both CYBE and $S$-equation can be obtained from the construction of $\mathcal O$-operators.
2. A left-symmetric algebra gives a natural $\mathcal O$-operator and hence there are solutions of CYBE and $S$-equation obtained from left-symmetric algebras.
In a summary, the operator forms ($\mathcal O$-operators) and the algebraic structures (left-symmetric algebras) behind indeed provide a practical construction for solutions of CYBE and $S$-equation.
Therefore it is natural to consider the “conformal analogues" of the above construction, which is the main aim of this paper, that is, we study what are the operator forms of conformal CYBE and $S$-equation. We would like to point out that the conformal generalization is not trivial. For example, we define the conformal analogue of an $\mathcal O$-operator to be a conformal linear map $T$ where $T_0=T_\lambda\mid_{\lambda=0}$ is an $\mathcal{O}$-operator in the conformal sense and in particular, assume $T_0\ne 0$, whereas the case $T_0=0$ is meaningless for the construction of Lie conformal bialgebras and left-symmetric conformal bialgebras. Furthermore, many results in the conformal sense are obtained and hence these results will be useful to provide solutions of conformal CYBE and $S$-equation, to construct conformal Manin triples and parak$\ddot{\text{a}}$hler Lie conformal algebras and then to study some related geometry in the conformal sense.
This paper is organized as follows. In Section 2, we recall some necessary definitions, notations and some results about Lie conformal algebras and left-symmetric conformal algebras. In Section 3, the operator forms of conformal CYBE are investigated. We introduce the definitions of Rota-Baxter operator and $\mathcal{O}$-operator of a Lie conformal algebra and obtain some relations between Rota-Baxter operator, $\mathcal{O}$-operator and conformal CYBE. Moreover, a relation between the non-degenerate skew-symmetric solutions of conformal CYBE and 2-cocycles of Lie conformal algebras is presented in terms of $\mathcal{O}$-operators. In Section 4, we study the relations between conformal CYBE and left-symmetric conformal algebras. Section 5 is devoted to investigating the operator forms of conformal $S$-equation. Moreover, similar results as those in the case of conformal CYBE are obtained. In Section 6, Rota-Baxter operators on a class of Lie conformal algebras named quadratic Lie conformal algebras are studied.
Throughout this paper, denote by $\mathbb{C}$ the field of complex numbers; $\mathbb{N}$ the set of natural numbers, i.e. $\mathbb{N}=\{0, 1, 2,\cdots\}$; $\mathbb{Z}$ the set of integer numbers. All tensors over $\mathbb{C}$ are denoted by $\otimes$. Moreover, if $A$ is a vector space, the space of polynomials of $\lambda$ with coefficients in $A$ is denoted by $A[\lambda]$.
Preliminaries on conformal algebras
===================================
In this section, we recall some definitions, notations and results about conformal algebras. Most of the results in this section can be found in [@K1; @HL].
\[def1\]
A [**conformal algebra**]{} $R$ is a $\mathbb{C}[\partial]$-module endowed with a $\mathbb{C}$-bilinear map $R\times R\rightarrow
R[\lambda]$ denoted by $a\times b\rightarrow
a_{\lambda} b$ satisfying $$\begin{aligned}
\partial a_{\lambda}b=-\lambda a_{\lambda}b, \quad
a_{\lambda}\partial b=(\partial+\lambda)a_{\lambda}b.\end{aligned}$$
A [**Lie conformal algebra**]{} $R$ is a conformal algebra with the $\mathbb{C}$-bilinear map $[\cdot_\lambda \cdot]: R\times R\rightarrow R[\lambda]$ satisfying $$\begin{aligned}
&&[a_\lambda b]=-[b_{-\lambda-\partial}a],~~~~\text{(skew-symmetry)}\\
&&[a_\lambda[b_\mu c]]=[[a_\lambda b]_{\lambda+\mu} c]+[b_\mu[a_\lambda c]],~~~~~~\text{(Jacobi identity)}\end{aligned}$$ for $a$, $b$, $c\in R$.
A [**left-symmetric conformal algebra**]{} $R$ is a conformal algebra with the $\mathbb{C}$-bilinear map $\cdot_\lambda \cdot: R\times R\rightarrow R[\lambda]$ satisfying $$\begin{aligned}
(a_{\lambda}b)_{\lambda+\mu}c-a_{\lambda}(b_\mu
c)=(b_{\mu}a)_{\lambda+\mu}c-b_\mu(a_\lambda c),\end{aligned}$$ for $a$, $b$, $c\in R$.
A conformal algebra is called [**finite**]{} if it is finitely generated as a $\mathbb{C}[\partial]$-module. The [**rank**]{} of a conformal algebra $R$ is its rank as a $\mathbb{C}[\partial]$-module.
If $A$ is a left-symmetric conformal algebra, then the $\lambda$-bracket $$\label{3}[a_\lambda b]=a_\lambda
b-b_{-\lambda-\partial}a,~~~\text{for any }~~~a, ~~b \in A,$$ defines a Lie conformal algebra $\mathfrak{g}(A)$, which is called [**the sub-adjacent Lie conformal algebra of $A$**]{}. In this case, $A$ is also called [**a compatible left-symmetric conformal algebra structure on the Lie conformal algebra $\mathfrak{g}(A)$**]{}.
\[def:module\] [A [**module $M$ over a Lie conformal algebra $R$**]{} is a $\mathbb{C}[\partial]$-module endowed with a $\mathbb{C}$-bilinear map $R\times M\longrightarrow M[\lambda]$, $(a, v)\mapsto a_\lambda v$, satisfying the following axioms $(a, b\in R, v\in M)$:\
(LM1)$\qquad\qquad (\partial a)_\lambda v=-\lambda a_\lambda v,~~~a_\lambda(\partial v)=(\partial+\lambda)a_\lambda v,$\
(LM2)$\qquad\qquad [a_\lambda b]_{\lambda+\mu}v=a_\lambda(b_\mu v)-b_\mu(a_\lambda v).$]{}
An $R$-module $M$ is called [**finite**]{} if it is finitely generated as a $\mathbb{C}[\partial]$-module.
[Let $U$ and $V$ be two $\mathbb{C}[\partial]$-modules. A [**conformal linear map**]{} from $U$ to $V$ is a $\mathbb{C}$-linear map $a: U\rightarrow V[\lambda]$, denoted by $a_\lambda: U\rightarrow V$, such that $[\partial, a_\lambda]=-\lambda a_\lambda$. Denote the $\mathbb C$-vector space of all such maps by $\text{Chom}(U,V)$. It has a canonical structure of a $\mathbb{C}[\partial]$-module: $$(\partial a)_\lambda =-\lambda a_\lambda.$$ Define the [**conformal dual**]{} of a $\mathbb{C}[\partial]$-module $U$ as $U^{\ast c}=\text{Chom}(U,\mathbb{C})$, where $\mathbb{C}$ is viewed as the trivial $\mathbb{C}[\partial]$-module, that is $$U^{\ast c}=\{a:U\rightarrow \mathbb{C}[\lambda]~~|~~\text{$a$ is $\mathbb{C}$-linear and}~~a_\lambda(\partial b)=\lambda a_\lambda b\}.$$]{}
Let $U$ and $V$ be finite modules over a Lie conformal algebra $R$. Then the $\mathbb{C}[\partial]$-module $\text{Chom}(U,V)$ has an $R$-module structure defined by: $$\begin{aligned}
\label{oo2}
(a_\lambda \varphi)_\mu u=a_\lambda(\varphi_{\mu-\lambda}u)-\varphi_{\mu-\lambda}(a_\lambda u).\end{aligned}$$ for $a\in R, \varphi\in \text{Chom}(U,V), u\in U$. Hence, one special case is the contragradient conformal $R$-module $U^{\ast c}$, where $\mathbb{C}$ is viewed as the trivial $R$-module and $\mathbb{C}[\partial]$-module. In particular, when $U=V$, set $\text{Cend}(V)=\text{Chom}(V,V)$. The $\mathbb{C}[\partial]$-module $\text{Cend}(V)$ has a canonical structure of an associative conformal algebra defined by $$\begin{aligned}
(a_\lambda b)_\mu v=a_\lambda (b_{\mu-\lambda} v),~~~~\text{ $a$, $b\in \text{Cend}(V)$, $v\in V$.}\end{aligned}$$ Therefore $\text{gc}(V):=\text{Chom}(V,V)$ has a Lie conformal algebra structure defined by $$\begin{aligned}
[a_\lambda b]_\mu v=a_\lambda (b_{\mu-\lambda} v)-b_{\mu-\lambda}(a_{\lambda} v), ~~~ a,~b\in \text{gc}(V), v\in V.\end{aligned}$$ $\text{gc}(V)$ is called the [**general Lie conformal algebra**]{} of $V$.
\[rrm\][By Definition \[def:module\], it is easy to see that a module over a Lie conformal algebra $R$ in a finite $\mathbb{C}[\partial]$-module $V$ is the same as a homomorphism of Lie conformal algebra $\rho: R\rightarrow \text{gc}(V)$ which is called a [**representation**]{} of $R$. Denote the [**adjoint representation**]{} of $R$ by $ad$, i.e. $ad(a)_\lambda b=[a_\lambda b]$, where $a$, $b\in R$. Moreover, the contragradient conformal $R$-module $V^{\ast c}$ is the same as the representation $\rho^\ast: R\rightarrow
\text{gc}(V^{\ast c})$ which is dual to $\rho$. By Eq. (\[oo2\]), the relation between $\rho^\ast$ and $\rho$ is given as follows. $$\begin{aligned}
(\rho^\ast(a)_\lambda \varphi)_\mu u=-\varphi_{\mu-\lambda}(\rho(a)_\lambda u),\;\;\forall a\in R, \varphi\in V^{\ast c}, u\in V.\end{aligned}$$]{}
[([@HL1])]{} Let $R$ be a Lie conformal algebra. Let $V$ be a $\mathbb{C}[\partial]$-module of finite rank and $\rho: R\rightarrow gc(V)$ be a representation of $R$. Then $R\oplus V$ is endowed with a $\mathbb{C}[\partial]$-module structure given by $$\partial (a+v)=\partial a+\partial v,\;\;\forall a\in R, v\in V.$$ Hence the $\mathbb{C}[\partial]$-module $R\oplus V$ is endowed with a Lie conformal algebra structure as follows. $$\begin{aligned}
[{(a+u)}_\lambda (b+v)]=[a_\lambda b]+\rho(a)_\lambda v-\rho(b)_{-\lambda-\partial} u,~~~~~~~\text{for any $a$, $b\in R$ and $u$, $v\in V$.}\end{aligned}$$ This Lie conformal algebra is called [**the semi-direct sum of $R$ and $V$**]{}, denoted by $R\ltimes_\rho V$.
The tensor product $U\otimes V$ can be naturally endowed with an $R$-module structure as follows. $$\partial(u\otimes v)=\partial u\otimes v+u\otimes \partial v,$$ and $$r_\lambda(u\otimes v)=r_\lambda u\otimes v+u\otimes r_\lambda v,$$ where $u\in U$, $v\in V$ and $r\in R$.
\[prop1\] [([@BKL])]{} Let $U$ and $V$ be two $R$-modules. Suppose that $U$ is a $\mathbb{C}[\partial]$-module of finite rank. Then $U^{\ast c}\otimes V\cong \text{Chom}(U,V)$ as $R$-modules with the identification $(f\otimes v)_\lambda (u)=f_{\lambda+\partial}(u)v$ where $f\in U^{\ast c}$, $u\in U$ and $v\in V$.
\[555\][A [**module $M$ over a left-symmetric conformal algebra $A$**]{} is a $\mathbb{C}[\partial]$-module with two $\mathbb{C}$-bilinear maps $A\times M\rightarrow M[\lambda]$, $a\times v\rightarrow a_\lambda v$ and $M\times A\rightarrow M[\lambda]$, $v\times a\rightarrow v_\lambda a$ such that $$(\partial a)_\lambda v=[\partial, a_\lambda]v=-\lambda a_\lambda v,\quad (\partial v)_\lambda a=[\partial, v_\lambda]a=-\lambda v_\lambda a,$$ $$(a_\lambda b)_{\lambda+\mu}v-a_\lambda(b_\mu v)=(b_\mu a)_{\lambda+\mu}v-b_\mu (a_\lambda v),$$ $$(a_\lambda v)_{\lambda+\mu}b-a_\lambda(v_\mu b)=(v_\mu a)_{\lambda+\mu}b-v_\mu(a_\lambda b)$$ hold for $a$, $b\in A$ and $v\in M$.]{}
Similarly, an $A$-module is called [**finite**]{} if it is finitely generated as a $\mathbb{C}[\partial]$-module.
[Suppose $M$ is finite. Let $a_\lambda v=l_A(a)_\lambda v$ and $v_\lambda a=r_A(a)_{-\lambda-\partial} v$. It is easy to show that the structure of a module $M$ over a left-symmetric conformal algebra $A$ is the same as a pair $\{l_A, r_A\}$, where $l_A, r_A: A\rightarrow \text{Cend}(M)$ are two $\mathbb{C}[\partial]$-module homomorphisms such that the following conditions hold $$\begin{aligned}
\label{101}
l_A(a_\lambda b)_{\lambda+\mu}v-l_A(a)_\lambda(l_A(b)_\mu v)=l_A(b_\mu a)_{\lambda+\mu}v-l_A(b)_\mu(l_A(a)_\lambda v),\end{aligned}$$ $$\begin{aligned}
\label{102}
r_A(b)_{-\lambda-\mu-\partial}(l_A(a)_\lambda v)-l_A(a)_\lambda(r_A(b)_{-\mu-\partial}v)
=r_A(b)_{-\lambda-\mu-\partial}(r_A(a)_\lambda v)-r_A(a_\lambda b)_{-\mu-\partial} v,\end{aligned}$$ for any $a$, $b\in A$ and $v\in M$. Denote this module by $(M,l_A, r_A)$.]{}
Throughout this paper, we mainly deal with $\mathbb{C}[\partial]$-modules which are finitely generated. So, for convenience, we use the notions of representations of conformal algebras instead of those of modules of conformal algebras.
[([@HL1])]{} Let $(M,l_A, r_A)$ be a finite module over a left-symmetric conformal algebra $A$. Then\
(i) $l_A:A\rightarrow \text{gc}(M)$ is a representation of the sub-adjacent Lie conformal algebra $\mathfrak{g}(A)$.\
(ii) $\rho=l_A-r_A$ is a representation of the Lie conformal algebra $\mathfrak{g}(A)$.\
(iii) For any representation $\sigma:\mathfrak{g}(A)\rightarrow \text{gc}(M)$ of the Lie conformal algebra $\mathfrak{g}(A)$, $(M,\sigma,0)$ is an $A$-module.
Let $A$ be a finite left-symmetric conformal algebra. Define two $\mathbb{C}[\partial]$-module homomorphisms $L_A$ and $R_A$ from $A$ to $\text{Cend}(A)$ by $L_A(a)_\lambda b=a_\lambda b$ and $R_A(a)_\lambda b=b_{-\lambda-\partial}a$ for any $a$, $b\in A$. Then $L_A:A\rightarrow \text{gc}(A)$ and $\rho=L_A-R_A$ are two representations of the Lie conformal algebra $\mathfrak{g}(A)$.
[$L_A$ is called the [**regular representation**]{} of $\mathfrak{g}(A)$.]{}
[([@HL1])]{} Let $A$ be a left-symmetric conformal algebra and $(M,l_A, r_A)$ be a module of $A$. Then the $\mathbb{C}[\partial]$-module $A\oplus M$ is a left-symmetric conformal algebra with the following $\lambda$-product $$\begin{aligned}
(a+u)_\lambda (b+v)=a_\lambda b+l_A(a)_\lambda v+r_A(b)_{-\lambda-\partial}u,~~a,~b\in A,~~u,~v\in M.\end{aligned}$$ Denote it by $A\ltimes_{l_A,r_A} M$, which is called [**the semi-direct sum of $A$ and $M$**]{}.
Finally, let us recall the definition of coefficient algebra of a conformal algebra. Let $\chi$ be a variety of algebras (Lie, left-symmetric, etc) and $R$ be a $\chi$-conformal algebra. There is an associated $\chi$-algebra constructed as follows. Set $$a_\lambda b=\sum_{n\in \mathbb{N}}\frac{\lambda^n}{n!}a_{(n)}b,$$ where $a_{(n)}b$ is called [**the $n$-th product**]{} of $a$ and $b$. Let Coeff$R$ be the quotient of the vector space with basis $a_n$ $(a\in R, n\in\mathbb{Z})$ by the subspace spanned over $\mathbb{C}$ by elements: $$(\alpha a)_n-\alpha a_n,~~(a+b)_n-a_n-b_n,~~(\partial
a)_n+na_{n-1},~~~\text{where}~~a,~~b\in R,~~\alpha\in \mathbb{C},~~n\in
\mathbb{Z}.$$ The operation on $\text{Coeff}R$ is defined as follows. $$\begin{aligned}
\label{eq1}
a_m\cdot b_n=\sum_{j\in \mathbb{N}}\left(\begin{array}{ccc}
m\\j\end{array}\right)(a_{(j)}b)_{m+n-j}.\end{aligned}$$ Then $\text{Coeff}R$ is a $\chi$-algebra ([@K1]), which is called the [**coefficient algebra**]{} of $R$.
The operator forms of conformal CYBE
====================================
In this section, we study the operator forms of conformal CYBE and give a relation between the non-degenerate skew-symmetric solutions of conformal CYBE and 2-cocycles of Lie conformal algebras.
Let $R$ be a Lie conformal algebra and $r=\sum_{i} a_i\otimes b_i\in R\otimes R$. Set $r^{21}=\sum_{i} b_i\otimes a_i$. We say $r$ is [**skew-symmetric**]{} if $r=-r^{21}$, whereas $r$ is called [**symmetric**]{} if $r=r^{21}$.
Let $V$ be free and of finite rank as a $\mathbb{C}[\partial]$-module. Set $\{v_i\}\mid_{i=1,\cdots,m}$ be a $\mathbb{C}[\partial]$-basis of $V$. Obviously, as an $R$-module, $V\cong {V^{\ast c}}^{\ast c}$ though the $\mathbb{C}[\partial]$-module homomorphism $v_i\rightarrow {v_i}^{\ast\ast}$ where ${y^\ast}_\lambda(v_i)={v_i^{\ast\ast}}_{-\lambda-\partial}(y^\ast)$ for any $y^\ast \in V^{\ast c}$.
Now suppose that $R$ is a finite Lie conformal algebra which is free as a $\mathbb{C}[\partial]$-module. Then by the discussion above, as $R$-modules, $$R\otimes R\cong {R^{\ast c}}^{\ast c}\otimes R \cong
\text{Chom}(R^{\ast c},R).$$ Define $$\{u,a\}_\lambda=u_\lambda(a),\;\; {\rm and}\;\;
\{u\otimes v, a\otimes b\}_{(\lambda,\mu)}=\{u,a\}_\lambda \{v,b\}_\mu.$$ where $a$, $b\in R$ and $u$, $v\in R^{\ast c}$. Then by Proposition \[prop1\], for any $r\in R\otimes R$, we associate a conformal linear map $T^r\in \text{Chom}(R^{\ast c},R)$ as follows. $$\begin{aligned}
\label{eqn5}
\{ f, T^r_{-\mu-\partial}(g)\}_\lambda =\{ g\otimes f, r\}_{(\mu,\lambda)},~~~~~~~~~~~~\text{$f$, $g\in R^{\ast c}$.}\end{aligned}$$ Set $r=\sum_{i} a_i\otimes b_i\in R\otimes R$. Then by Eq. (\[eqn5\]), we get $$T_{\lambda}^r(u)=\sum_i \{ u, a_i\}_{-\lambda-\partial} b_i, ~~~~u\in R^{\ast c}.$$
\[tthe3\] Let $R$ be a finite Lie conformal algebra which is free as a $\mathbb{C}[\partial]$-module and $r\in R\otimes R$ be skew-symmetric. Then $r$ is a solution of conformal CYBE if and only if the $T^r\in \text{Chom}(R^{\ast c},R)$ corresponding to $r$ satisfies $$\begin{aligned}
[T_0^r(u)_\lambda T_0^r(v)]= T_0^r(ad^{\ast}(T_0^r(u))_\lambda v
-ad^{\ast}(T_0^r(v))_{-\lambda-\partial} u),\;\;\forall u, v\in R^{\ast c},\end{aligned}$$ where $T_0^r=T_\lambda^r\mid_{\lambda=0}$.
Define $\langle a, u\rangle_\lambda=\{u,a\}_{-\lambda}=u_{-\lambda}(a)$ where $a\in R$ and $u\in R^{\ast c}$. Obviously, it is easy to see that $$\begin{aligned}
\label{kh1}
\langle \partial a, u\rangle_\lambda=-\lambda\langle a, u\rangle_\lambda.\end{aligned}$$ Moreover, we define $$\langle a\otimes b\otimes c, u\otimes v\otimes w\rangle_{(\lambda, \nu,\theta)}=\langle a,u\rangle_\lambda \langle b,v\rangle_\nu\langle c,w\rangle_\theta,$$ where $a$, $b$, $c\in R$, and $u$, $v$, $w\in R^{\ast c}$. Moreover, by the definition of dual representation in Remark \[rrm\], we can easily get $$\begin{aligned}
\label{x2}
\langle [a_\mu b], u\rangle_\lambda=\langle a, ad^\ast(b)_{\lambda-\partial} u\rangle_\mu,\end{aligned}$$ for any $a$, $b\in R$, and $u\in R^{\ast c}$.
Set $r=\sum_{i} a_i\otimes b_i\in R\otimes R$. By the discussion above, $T^r \in \text{Chom}(R^{\ast c},R)$ corresponding to $r$ is given by $$T_{\lambda}^r(u)=\sum_i \langle a_i, u\rangle_{\lambda+\partial} b_i, ~~~~u\in R^{\ast c}.$$ Similarly, since $r$ is skew-symmetric, one can obtain $T_{\lambda}^r(v)=-\sum_i \langle b_i, v\rangle_{\lambda+\partial} a_i, ~~~~v\in R^{\ast c}$.
Then we consider $$\begin{aligned}
\label{q1}\langle [[r,r]]~~\text{mod}~(\partial^{\otimes^3}), u\otimes v\otimes w\rangle_{(\lambda,\nu,\theta)}=0 ~~~\text{ for any $u$, $v$, $w\in R^{\ast c}$.}\end{aligned}$$ By Eq. (\[kh1\]), Eq. (\[q1\]) is equivalent to the following equality $$\begin{aligned}
\label{q2}\langle [[r,r]], u\otimes v\otimes w\rangle_{(\lambda,\nu,\theta)}=0 ~~~~ \text{mod}~(\lambda+\nu+\theta) ~~~\text{ for any $u$, $v$, $w\in R^{\ast c}$.}\end{aligned}$$ By a direct computation, we have $$\begin{aligned}
&&\langle \sum_{i,j}[{a_i}_\mu a_j]\otimes b_i\otimes b_j\mid_{\mu=1\otimes \partial \otimes 1}, u\otimes v\otimes w\rangle_{(\lambda,\nu,\theta)}\\
&&=\sum_{i,j}\langle [{a_i}_{-\nu} a_j], u\rangle_\lambda \langle b_i, v\rangle_\nu \langle b_j, w\rangle_\theta\\
&&=\sum_{i,j} \langle [(\langle b_i, v\rangle_\nu {a_i})_{-\nu} a_j], u\rangle_\lambda \langle b_j, w\rangle_\theta=-\sum_{i,j}\langle [T_0^r(v)_{-\nu} a_j], u\rangle_{\lambda}\langle b_j, w\rangle_\theta\\
&&=\sum_{i,j}\langle [{a_j}_{\lambda+\nu}T_0^r(v)],u\rangle_\lambda \langle b_j, w\rangle_\theta=\sum_{i,j}\langle a_j, ad^\ast (T_0^r(v))_{-\nu} u\rangle_{\lambda+\nu}
\langle b_j, w\rangle_\theta\\
&&= \sum_{i,j}\langle \langle a_j, ad^\ast (T_0^r(v))_{-\nu} u\rangle_{\lambda+\nu} b_j, w\rangle_\theta=\langle T_{\lambda+\nu-\partial}^r(ad^\ast (T_0^r(v))_{-\nu} u), w\rangle_\theta\\
&&=\langle T_{\lambda+\nu+\theta}^r(ad^\ast (T_0^r(v))_{-\nu} u), w\rangle_\theta.\end{aligned}$$ Similarly, one can get $$\begin{aligned}
\langle \sum_{i,j}
a_i\otimes[{a_j}_\mu b_i]\otimes b_j\mid_{\mu=1\otimes 1\otimes \partial}, u\otimes v\otimes w\rangle_{(\lambda,\nu,\theta)}
=\langle T_0^r(ad^\ast(T_{\lambda+\nu+\theta}^r(u))_{\nu+\theta} v), w \rangle_\theta,\end{aligned}$$ and $$\begin{aligned}
\langle \sum_{i,j}a_i\otimes a_j \otimes [{b_j}_\mu b_i]\mid_{\mu=1\otimes \partial\otimes 1}, u\otimes v\otimes w\rangle_{(\lambda,\nu,\theta)}
=-\langle [T_{\lambda+\nu+\theta}^r(u)_{\nu+\theta}T_0^r(v)], w\rangle_\theta.\end{aligned}$$ By Eq. (\[q2\]) and the above discussion, the conformal CYBE is equivalent to $$\begin{gathered}
\langle [T_{\lambda+\nu+\theta}^r(u)_{\nu+\theta}T_0^r(v)]-T_0^r(ad^\ast(T_{\lambda+\nu+\theta}^r(u))_{\nu+\theta} v)\nonumber\\
\label{q3}-T_{\lambda+\nu+\theta}^r(ad^\ast (T_0^r(v))_{-\nu} u), w\rangle_\theta=0 ~~~\text{mod}~~(\lambda+\nu+\theta).\end{gathered}$$ Therefore $(\ref{q3})$ is equivalent to the following equality $$\begin{aligned}
\label{q4}[T_0^r(u)_{-\lambda} T_0^r(v)]=T_0^r(ad^\ast(T_{0}^r(u))_{-\lambda} v)+T_{0}^r(ad^\ast (T_0^r(v))_{\lambda-\partial} u).\end{aligned}$$ Then we get the conclusion replacing $-\lambda$ by $\lambda$.
\[def:bl\] [([@L])A [**conformal bilinear form**]{} on $R$ is a $\mathbb{C}$-bilinear map $\langle,\rangle_\lambda: R\times R\rightarrow \mathbb{C}[\lambda]$ satisfying $$\begin{aligned}
\langle \partial a, b\rangle_\lambda =-\lambda\langle a,b \rangle_\lambda
=-\langle a,\partial b\rangle_\lambda~~~~~\text{ for all $a$, $b\in R$.}\end{aligned}$$ If $\langle a, b\rangle_\lambda=\langle b,a\rangle_{-\lambda}$ for all $a$, $b\in R$, we say this conformal bilinear form is [**symmetric**]{}. $\langle,\rangle_\lambda$ is called [**invariant**]{} if for any $a$, $b$, $c\in R$, $$\begin{aligned}
\langle [a_\mu b], c\rangle_\lambda=\langle a, [b_{\lambda-\partial}c]\rangle_\mu=-\langle a, [c_{-\lambda}b]\rangle_\mu.\end{aligned}$$ Suppose $R$ is a free and of finite rank $\mathbb{C}[\partial]$-module. Given a conformal bilinear form on $R$. If the $\mathbb{C}[\partial]$-module homomorphism $L: R\rightarrow R^{\ast c},
a\rightarrow L_a$ given by $(L_a)_\lambda b=\langle a, b\rangle_\lambda$, $b\in R$, is an isomorphism, then we call the bilinear form [**non-degenerate**]{}.]{}
Let $R$ have a non-degenerate conformal bilinear form. For any $a\otimes b$, $c\otimes d\in R\otimes R$, set $$\begin{aligned}
\langle a\otimes b, c\otimes d\rangle_{(\lambda,\mu)}
=\langle a, c\rangle_\lambda \langle b, d\rangle_\mu.\end{aligned}$$ Let $r=\sum_{i} a_i\otimes b_i\in R\otimes R$. Define a linear map $P^r: R\rightarrow R[\lambda]$ as $$\begin{aligned}
\langle r, u\otimes v\rangle_{(\lambda,\mu)}=\langle P_{\lambda-\partial}^r(u), v\rangle_\mu.\end{aligned}$$ It is easy to see that $P^r\in \text{Cend}(R)$.
\[t1\] Let $R$ be a finite Lie conformal algebra which is free as a $\mathbb{C}[\partial]$-module. Suppose that there exists a non-degenerate symmetric invariant conformal bilinear form on $R$ and $r\in R\otimes R$ is skew-symmetric. Then $r$ is a solution of conformal CYBE if and only if the element $P^r\in
\text{Cend}(R)$ corresponding to $r$ satisfies the following equality $$\begin{aligned}
\label{et1}[P_0^r(a)_\lambda P_0^r(b)]=P_0^r([a_\lambda P_0^r(b)])+P_0^r([P_0^r(a)_\lambda b]),~~~~\forall a, b\in R,\end{aligned}$$ where $P_0^r=P_\lambda^r\mid_{\lambda=0}$.
Since $R$ has a non-degenerate symmetric invariant conformal bilinear form, $R^{\ast c}$ is isomorphic to $R$ as a $\mathbb{C}[\partial]$-module through this conformal bilinear form. Then it can be directly obtained from Theorem \[tthe3\].
Note that $P_0^r$ is a $\mathbb{C}[\partial]$-module homomorphism, which motivates us to give the following definition.
\[RB0\][Let $R$ be a Lie conformal algebra. If $T:R\rightarrow R$ is a $\mathbb{C}[\partial]$-module homomorphism satisfying $$\begin{aligned}
\label{oo3}
[T(a)_\lambda T(b)]=T([a_\lambda T(b)])+T([T(a)_\lambda b]),~~\forall~~\text{$a$, $b\in R$,}\end{aligned}$$ then $T$ is called a [**Rota-Baxter operator (of weight 0)** ]{} on $R$.]{}
[Let $R$ be a Lie conformal algebra and $\rho: R\rightarrow gc(V)$ be a representation. If a $\mathbb{C}[\partial]$-module homomorphism $T: V\rightarrow R$ satisfies $$\begin{aligned}
\label{eqn3}
[T(u)_\lambda T(v)]=T(\rho(T(u))_\lambda v-\rho(T(v))_{-\lambda-\partial}u),~~\forall~~u,~v\in V,\end{aligned}$$ then $T$ is called an [**$\mathcal{O}$-operator**]{} associated with $\rho$.]{}
[By Theorem \[tthe3\], a skew-symmetric solution of conformal CYBE in $R$ is equivalent to $T\in \text{Chom}(R^{\ast c},R)$ where $T_0$ is an $\mathcal{O}$-operator associated to $ad^\ast$ and $R$ is a finite Lie conformal algebra and free as a $\mathbb{C}[\partial]$-module.]{}
Next, we study the $\mathcal{O}$-operators of Lie conformal algebras in a more general extent.
In the following, let $R$ and $V$ be free $\mathbb{C}[\partial]$-modules of finite ranks. Let $\{e_i\}_{i=1}^n$ be a $\mathbb{C}[\partial]$-basis of $R$, $\{v_j\}_{j=1}^m$ be a $\mathbb{C}[\partial]$-basis of $V$ and $\{v_j^\ast\}_{j=1}^m$ be the dual $\mathbb{C}[\partial]$-basis in $V^{\ast c}$. Then there is a natural representation $\rho^\ast: R\rightarrow gc(V^{\ast c})$ which is dual to $\rho$ given by $$\begin{aligned}
\label{eqn2}
\rho^\ast(e_i)_\lambda v_j^{\ast}=-\sum_{k=1}^m{v_j^\ast}_{-\lambda-\partial}
(\rho(e_i)_\lambda (v_k))v_k^\ast.\end{aligned}$$ By Proposition \[prop1\], as $R$-modules, $\text{Chom}(V, R)\cong V^{\ast c}\otimes R\cong R\otimes V^{\ast c}$. Therefore through this isomorphism, any conformal linear map $T\in \text{Chom}(V, R)$ corresponds to an element $r_{T}\in
R\otimes V^{\ast c}\in R\ltimes_{\rho^\ast}V^{\ast c}\otimes R\ltimes_{\rho^\ast}V^{\ast c}$. Set $T_\lambda(v_i)=\sum_{j=1}^na_{ij}(\lambda,\partial)e_j$ for $i=1$, $\cdots$, $m$. Then by the definition of the isomorphism, we get $$\begin{aligned}
\label{eqn1}r_T=\sum_{i=1}^m\sum_{j=1}^n a_{ij}(-1\otimes \partial-\partial\otimes 1,\partial\otimes 1)e_j\otimes v_i^\ast.\end{aligned}$$
\[t2\] With the conditions above, $r=r_T-r_T^{21}$ is a solution of conformal CYBE in $R\ltimes_{\rho^\ast}V^{\ast c}$ if and only if for $T\in \text{Chom}(V, R)$, $T_0=T_\lambda\mid_{\lambda=0}$ is an $\mathcal{O}$-operator.
By Eq. (\[eqn1\]), we get $$\begin{aligned}
r=\sum_{i,j}a_{ij}(-\partial\otimes 1-1\otimes \partial,\partial\otimes 1)e_j\otimes v_i^\ast-\sum_{i,j}a_{ij}(-\partial\otimes 1-1\otimes \partial,1\otimes \partial)v_i^\ast \otimes e_j.\end{aligned}$$ Then by a direct computation, we get $$\begin{aligned}
[[r,r]]
&=&(\sum_{i,j,k,l} a_{ij}(0,-1\otimes \partial\otimes 1)a_{kl}(0,-1\otimes 1\otimes \partial)[{e_j}_\mu e_l]\otimes v_i^\ast \otimes v_k^\ast\\
&&-\sum_{i,j,k,l}a_{ij}(0,1\otimes \partial \otimes 1) a_{kl}(0,-1\otimes 1\otimes \partial)[{v_i^\ast}_\mu e_l]\otimes e_j\otimes v_k^\ast\\
&&-\sum_{i,j,k,l}a_{ij}(0,-1\otimes \partial \otimes 1)a_{kl}(0,1\otimes 1\otimes \partial)[{e_j}_\mu v_k^\ast]\otimes v_i^\ast \otimes e_l)\mid_{\mu=1\otimes \partial \otimes 1}\\
&&-(\sum_{i,j,k,l}a_{ij}(0,\partial\otimes 1\otimes 1)a_{kl}(0,-1\otimes 1\otimes \partial)e_j\otimes [{e_l}_\mu v_i^\ast] \otimes v_k^\ast\\
&&-\sum_{i,j,k,l}a_{ij}(0,-\partial\otimes 1\otimes 1)a_{kl}(0,-1\otimes 1\otimes \partial)v_i^\ast\otimes [{e_l}_\mu e_j]\otimes v_k^\ast\\
&&+\sum_{i,j,k,l}a_{ij}(0,-\partial\otimes 1\otimes 1)a_{kl}(0,1\otimes 1\otimes \partial)v_i^\ast\otimes [{v_k^\ast}_\mu e_j]\otimes v_l)\mid_{\mu=1\otimes 1\otimes \partial}\\
&&-(-\sum_{i,j,k,l}a_{ij}(0,\partial\otimes 1\otimes 1)a_{kl}(0,-1\otimes \partial\otimes 1)e_j\otimes v_k^\ast\otimes [{e_l}_\mu v_i^\ast]\\
&&-\sum_{i,j,k,l}a_{ij}(0,-\partial\otimes 1\otimes 1)a_{kl}(0,1\otimes \partial\otimes 1)v_i^\ast\otimes e_l\otimes [{v_k^\ast}_\mu e_j]\\
&&+\sum_{i,j,k,l}a_{ij}(0,-\partial\otimes 1\otimes 1)a_{kl}(0,-1\otimes \partial\otimes 1)v_i^\ast\otimes v_k^\ast\otimes [{e_l}_\mu e_j])\mid_{\mu=1\otimes \partial \otimes 1}~~~\text{mod}~~(\partial^{\otimes^3}).\end{aligned}$$ Note that $T_0(v_i)=\sum_{j=1}^na_{ij}(0,\partial)e_j$. Thus we obtain $$\begin{aligned}
[[r,r]]
&=&\sum_{i,k}([T_0(v_i)_\mu T_0(v_k)]\otimes v_i^\ast \otimes v_k^\ast
+\rho^\ast(T_0(v_k))_{-\mu-\partial}v_i^\ast\otimes T_0(v_i)\otimes v_k^\ast\\
&&-\rho^\ast(T_0(v_i))_{\mu}v_k^\ast\otimes v_i^\ast\otimes T_0(v_k))\mid_{\mu=1\otimes \partial \otimes 1}\\
&&-\sum_{i,k}(T_0(v_i)\otimes \rho^\ast(T_0(v_k))_{\mu}v_i^\ast\otimes v_k^\ast
-v_i^\ast\otimes [T_0(v_k)_\mu T(v_i)]\otimes v_k^\ast\\
&&-v_i^\ast\otimes \rho^\ast(T_0(v_i))_{-\mu-\partial}v_k^\ast\otimes T(v_k))\mid_{\mu=1\otimes 1\otimes \partial}\\
&&-\sum_{i,k}(-T_0(v_i)\otimes v_k^\ast\otimes \rho^\ast(T_0(v_k))_{\mu}v_i^\ast
+v_i^\ast\otimes T_0(v_k)\otimes \rho^\ast(T_0(v_i))_{-\mu-\partial}v_k^\ast\\
&&+v_i^\ast \otimes v_k^\ast \otimes [T_0(v_k)_\mu T_0(v_i)])\mid_{\mu=1\otimes \partial \otimes 1}~~~~~\text{mod} ~~(\partial^{\otimes^3}).\end{aligned}$$ Moreover, by Eq. (\[eqn2\]) and the fact that $T_0$ commutes with $\partial$, we get $$\begin{aligned}
&&\sum_{i,k}T_0(v_i)\otimes \rho^\ast(T_0(v_k))_{\mu}v_i^\ast\otimes v_k^\ast\mid_{\mu=1\otimes 1\otimes \partial}\\
&=&-\sum_{i,k}T_0(v_i)\otimes \sum_{j}{v_i^\ast}_{-\mu-\partial}(\rho(T_0(v_k))_\mu v_j)v_j^\ast\otimes v_k^\ast\mid_{\mu=1\otimes 1\otimes \partial}\\\end{aligned}$$ $$\begin{aligned}
&=&-\sum_{i,j,k}{v_j^\ast}_\partial(\rho(T_0(v_k))_\mu(v_i))T_0(v_j)\otimes v_i^\ast\otimes v_k^\ast\mid_{\mu=1\otimes 1\otimes \partial}\\
&=&-\sum_{i,k} T_0(\sum_{j}({v_j^\ast}_\partial(\rho(T_0(v_k))_\mu(v_i)))v_j)\otimes v_i^\ast\otimes v_k^\ast\mid_{\mu=1\otimes 1\otimes \partial}\\
&=&-\sum_{i,k}T_0(\rho(T_0(v_k))_\mu(v_i))\otimes v_i^\ast\otimes v_k^\ast\mid_{\mu=1\otimes 1\otimes \partial}.\end{aligned}$$ Therefore by a similar study, we get $$\begin{aligned}
&&[[r,r]]~~~~~~\text{mod}~~(\partial^{\otimes^3})\\
&=&\sum_{i,k}((-[T_0(v_k)_\mu T_0(v_i)]+T_0(\rho(T_0(v_k))_\mu(v_i))+T_0(\rho(T_0(v_i))_{-\mu-\partial}(v_k)))\otimes v_i^\ast\otimes v_k^\ast\mid_{\mu=1\otimes 1\otimes \partial}\\
&&+(v_i^\ast\otimes (-T_0(\rho(T_0(v_k))_\mu(v_i))+[T_0(v_k)_\mu T_0(v_i)]
+T_0(\rho(T_0(v_i))_{-\mu-\partial}(v_k)))\otimes v_k^\ast)\mid_{\mu=1\otimes 1\otimes \partial}\\
&&+\sum_{i,k}(v_i^\ast\otimes v_k^\ast\otimes (-T_0(\rho(T_0(v_k))_\mu(v_i))+[T_0(v_k)_\mu T_0(v_i)]
+T_0(\rho(T_0(v_i))_{-\mu-\partial}(v_k)))\mid_{\mu=1\otimes \partial\otimes 1}\\
&=&0.\end{aligned}$$ Therefore $r$ is a solution of conformal CYBE if and only if $T_0$ satisfies Eq. (\[eqn3\]).
[In fact, from the proof, when $r=r_T-r_T^{21}$ is replaced by $$r=\sum_{i,j}b_{i,j}(-\partial^{\otimes^2},\partial\otimes 1)e_j\otimes v_i^\ast-\sum_{i,j}c_{i,j}(-\partial^{\otimes^2},1\otimes \partial) v_i^\ast \otimes e_j,$$ where $b_{i,j}(0,\partial)=c_{i,j}(0,\partial)=a_{i,j}(0,\partial)$, the conclusion still holds.]{}
\[rem1\]
For any $T\in \text{Chom}(V,R)$, set $T_\lambda =
T_0+\lambda T_1+\cdots +\lambda^n T_n$ where $T_i(V)\subset R$. Suppose $T_0$ satisfies Eq. (\[eqn3\]). Obviously, $T_0=0$ is an $\mathcal{O}$-operator. Then by Theorem \[t2\], no matter what $T_1$, $\cdots$, $T_n$ are, the element $r=r_T-r^{21}_T \in R\ltimes_{\rho^\ast}V^{\ast c}\otimes R\ltimes_{\rho^\ast}V^{\ast c}$ where $r_T\in
R\otimes V^{\ast c}$ corresponds to $T_\lambda$ are all solutions of conformal CYBE in $R\ltimes_{\rho^\ast}V^{\ast c}$.
Assume that $r_T$ is given by (\[eqn1\]) and $r=r_T-r_T^{21}$. The Lie conformal bialgebra structures are obtained through these solutions of conformal CYBE as follows ([@L]). $$\begin{aligned}
\delta(a)&=&a_\lambda r\mid_{\lambda=-\partial^{\otimes^2}}\\
&=&a_\lambda (\sum_{i,j}a_{ij}(-\partial\otimes 1-1\otimes \partial,\partial\otimes 1)e_j\otimes v_i^\ast\\
&&-\sum_{i,j}a_{ij}(-\partial\otimes 1-1\otimes \partial,1\otimes \partial)v_i^\ast \otimes e_j)\mid_{\lambda=-\partial^{\otimes^2}}\\
&=&a_\lambda (\sum_{i,j}a_{ij}(0,\partial\otimes 1)e_j\otimes v_i^\ast-\sum_{i,j}a_{ij}(0,1\otimes \partial)v_i^\ast \otimes e_j)\mid_{\lambda=-\partial^{\otimes^2}}\\
&=& a_\lambda(r_{T_0}-r_{T_0}^{21})\mid_{\lambda=-\partial^{\otimes^2}}.\end{aligned}$$ Therefore the solutions of conformal CYBE corresponding to $T_\lambda=\lambda T_1+\cdots +\lambda^n T_n$ do not take effect here. Hence in the sense of Lie conformal bialgebras, the unique useful solution corresponding to $T_\lambda$ is determined by the $\mathcal{O}$-operator $T_0$.
Finally, let us study the relation between the non-degenerate skew-symmetric solutions of conformal CYBE and 2-cocycles of Lie conformal algebras.
\[def11\][Let $R$ be a finite Lie conformal algebra which is free as a $\mathbb{C}[\partial]$-module. If $T_\lambda^r=T_0^r$ defined by Eq. (\[eqn5\]) is a $\mathbb{C}[\partial]$-module isomorphism from $R^{\ast c}$ to $R$, then $r$ is called [**non-degenerate**]{}. Note that in this case, Eq. (\[eqn5\]) becomes $$\begin{aligned}
\{ f, T_0^r(g)\}_\lambda =\{ g\otimes f, r\}_{(-\lambda,\lambda)}~~~~~~~~~~~~\text{$f$, $g\in R^{\ast c}$.}\end{aligned}$$]{}
[Let $R$ be a Lie conformal algebra. The $\mathbb{C}$-linear map $\alpha_\lambda: R\otimes R\rightarrow \mathbb{C}[\lambda]$ is called a [**2-cocycle**]{} of $R$ if $\alpha_\lambda$ satisfies the following conditions $$\begin{aligned}
\label{eqn6}\alpha_\lambda(\partial a,b)=-\lambda \alpha_\lambda(a,b),~~\alpha_\lambda(a,\partial b)=\lambda \alpha_\lambda(a,b),\\
\label{eqn7}\alpha_\lambda(a,b)=-\alpha_{-\lambda}(b,a),\\
\label{eqn8}\alpha_\lambda(a,[b_\mu c])-\alpha_\mu(b,[a_\lambda c])
=\alpha_{\lambda+\mu}([a_\lambda b],c),\end{aligned}$$ where $a$, $b$, $c\in R$.]{}
\[th2\] Let $R$ be a Lie conformal algebra. Then $r\in R\otimes R$ is a skew-symmetric and non-degenerate solution of conformal CYBE in $R$ if and only if the bilinear form defined by $$\begin{aligned}
\alpha_\lambda(a,b)=\{ (T_0^r)^{-1}(a), b\}_\lambda,~~~~~~~a,~~b\in R,\end{aligned}$$ is a 2-cocycle on $R$, where $T_0^r\in \text{Chom}(R^{\ast c}, R)$ is the element corresponding to $r$ through the isomorphism $R\otimes R\cong \text{Chom}(R^{\ast c}, R)$.
Obviously, $\alpha_\lambda$ satisfies Eq. (\[eqn6\]). Since $T_0^r$ is a $\mathbb{C}[\partial]$-module isomorphism from $R^{\ast c}$ to $R$, there exist $f$ and $g\in R^{\ast c}$ such that $T_0^r(f)=a$ and $T_0^r(g)=b$. Therefore by the correspondence of $r$ and $T_0^r$, $$\begin{aligned}
\alpha_\lambda(T_0^r(f),T_0^r(g))=\{ f, T_0^r(g)\}_\lambda=\{ g\otimes f ,r \}_{(-\lambda,\lambda)}.\end{aligned}$$ Since $r=-r^{21}$, we get $$\begin{aligned}
\alpha_\lambda(T_0^r(f),T_0^r(g))&=&\{ g\otimes f ,r \}_{(-\lambda,\lambda)}
=\{f \otimes g ,r^{21} \}_{(\lambda,-\lambda)}
=-\{ f \otimes g ,r \}_{(\lambda,-\lambda)}=-\{ g, T_0^r(f)\}_{-\lambda}\\
&=&-\alpha_{-\lambda}(T_0^r(g),T_0^r(f)).\end{aligned}$$ Therefore $\alpha_\lambda(a,b)=-\alpha_{-\lambda}(b,a)$ for any $a$, $b\in R$. Conversely, it is also easy to see that if Eq. (\[eqn7\]) holds, $r=-r^{21}$. Moreover, note that $$\begin{aligned}
\label{x1}
\{ f, T_0^r(g)\}_\lambda=-\{ g, T_0^r(f)\}_{-\lambda},~~~f,~g\in R^{\ast c}.\end{aligned}$$
By Theorem \[tthe3\], $r$ is a solution of conformal CYBE in $R$ if and only if $T_0^r$ is an $\mathcal{O}$-operator. Therefore we only need to show that Eq. (\[eqn8\]) is equivalent to that $T_0^r$ is an $\mathcal{O}$-operator. Replacing $a$, $b$, $c$ by $T_0^r(f)$, $T_0^r(g)$ and $T_0^r(h)$ respectively in Eq. (\[eqn8\]) and according to Eq. (\[x1\]) and Eq. (\[x2\]), we get $$\begin{aligned}
&&\alpha_\lambda(T_0^r(f),[T_0^r(g)_\mu T_0^r(h)])-\alpha_\mu(T_0^r(g),[T_0^r(f)_\lambda T_0^r(h)])
-\alpha_{\lambda+\mu}([T_0^r(f)_\lambda T_0^r(g)],T_0^r(h))\\
&=&\{ f, [T_0^r(g)_\mu T_0^r(h)]\}_\lambda -\{ g, [T_0^r(f)_\lambda T_0^r(h)]\}_\mu+\{ h, [T_0^r(f)_\lambda T_0^r(g)]\}_{-\lambda-\mu}\\
&=&-\{ ad^\ast (T_0^r(g))_\mu f, T_0^r(h)\}_{\lambda+\mu}
+\{ad^\ast (T_0^r(f))_\lambda g, T_0^r(h)\}_{\lambda+\mu}
+\{ h, [T_0^r(f)_\lambda T_0^r(g)]\}_{-\lambda-\mu}\\
&=&\{ h, T_0^r(ad^\ast (T_0^r(g))_\mu f)\}_{-\lambda-\mu}
-\{ h, T_0^r( ad^\ast (T_0^r(f)_\lambda) g)\}_{-\lambda-\mu}
+\{h, [T_0^r(f)_\lambda T_0^r(g)]\}_{-\lambda-\mu}\\
&=& \{ h, [T_0^r(f)_\lambda T_0^r(g)]-T_0^r(ad^\ast ( T_0^r(f))_\lambda g)+T_0^r(ad^\ast (T_0^r(g))_{-\lambda-\partial} f) \}_{-\lambda-\mu}.\end{aligned}$$ Since $h\in R^{\ast c}$ is arbitrary, Eq. (\[eqn8\]) is equivalent to that $T_0^r$ is an $\mathcal{O}$-operator. This completes the proof.
Conformal CYBE and left-symmetric conformal algebras
====================================================
In this section, we investigate the relation between conformal CYBE and left-symmetric conformal algebras.
\[ll1\] There is a compatible left-symmetric conformal algebra structure on a Lie conformal algebra $R$ if and only if there exists a bijective $\mathcal{O}$-operator $T: V\rightarrow R$ associated with a certain representation $\rho$.
If there is a compatible left-symmetric conformal algebra $(R,\cdot_\lambda \cdot)$, then $R$ is an $R$-module through the left multiplication operators of the left-symmetric conformal algebra. Then $id: R\rightarrow R$ is a bijective $\mathcal{O}$-operator of $R$ associated to this representation.
Conversely, suppose there exists a bijective $\mathcal{O}$-operator $T: V\rightarrow R$ of $R$ associated with a representation $\rho$. Then $$\begin{aligned}
a_\lambda b=T(\rho(a)_\lambda T^{-1}(b)),\;\;\forall a,b\in R,\end{aligned}$$ defines a compatible left-symmetric conformal algebra structure on $R$.
\[the1\] Let $A$ be a left-symmetric conformal algebra which is free and of finite rank as a $\mathbb{C}[\partial]$-module. Then $$\begin{aligned}
\label{eqn4}
r=\sum_{i=1}^n(e_i\otimes e_i^\ast-e_i^\ast\otimes e_i)\end{aligned}$$ is a solution of conformal CYBE in $\mathfrak{g}(A)\ltimes_{L_A^\ast} \mathfrak{g}(A)^{\ast c}$, where $\{e_1,\cdots,e_n\}$ is a basis of $A$ and $\{e_1^\ast,\cdots, e_n^\ast\}$ is the dual basis of $A^{\ast c}$.
By Proposition \[ll1\], $T=id:\mathfrak{g}(A)\rightarrow \mathfrak{g}(A) $ is an $\mathcal{O}$-operator associated to $L_A$. Then by Theorem \[t2\], the conclusion holds.
[By Theorem \[t2\] and Remark \[rem1\], there are infinitely many solutions of conformal CYBE obtained from the $\mathcal{O}$-operator $id: A\rightarrow A$. But the $r$ given by Eq. (\[eqn4\]) is the unique non-degenerate solution corresponding to $id$.]{}
Let $A$ be a left-symmetric conformal algebra which is free and of finite rank as a $\mathbb{C}[\partial]$-module. Then there is a natural 2-cocycle $\alpha_\lambda$ on $\mathfrak{g}(A)\ltimes_{L_A^\ast} \mathfrak{g}(A)^{\ast c}$ given by $$\begin{aligned}
\alpha_\lambda(a+f, b+g)=\{ f,b\}_\lambda -\{ g,a\}_{-\lambda},~~~~~~~a,~~b\in \mathfrak{g}(A),~~f,~~g\in \mathfrak{g}(A)^{\ast c}.\end{aligned}$$
It follows from Theorem \[th2\] and Theorem \[the1\].
\[proo1\] Let $R$ be a Lie conformal algebra and $\rho: R\rightarrow gc(V)$ be a representation of $R$. Let $T: V\rightarrow R$ be a $\mathbb{C}[\partial]$-module homomorphism. Then $$\begin{aligned}
u\ast_\lambda v=\rho(T(u))_\lambda v,~~~~~u,~v\in V,\end{aligned}$$ defines a left-symmetric conformal algebra structure on $V$ if and only if $$\begin{aligned}
[T(u)_\lambda T(v)]-T(\rho(T(u))_\lambda v-\rho(T(v))_{-\lambda-\partial}u)\in \text{ker}(\rho)[\lambda]\end{aligned}$$ for any $u$, $v\in V$.
For any $u$, $v$, $w\in V$, by a direct computation, we can get $$\begin{aligned}
&&(u\ast_\lambda v)_{\lambda+\mu}w-u\ast_\lambda(v\ast_\mu w)
-(v\ast_\mu u)\ast_{\lambda+\mu}w+v\ast_\mu(u\ast_\lambda w)\\
&=&-\rho([T(u)_\lambda T(v)]-T(\rho(T(u))_\lambda v-\rho(T(v)_{-\lambda-\partial} u)))_{\lambda+\mu}w.\end{aligned}$$ Then the conclusion holds.
\[cor1\] Let $R$ be a Lie conformal algebra and $\rho: R\rightarrow gc(V)$ be a representation of $R$. Suppose $T: V\rightarrow R$ is an $\mathcal{O}$-operator associated to $\rho$. Then the following $\lambda$-product $$\begin{aligned}
u\ast_\lambda v=\rho(T(u))_\lambda v,~~~~~~\text{$u$, $v\in V$,}\end{aligned}$$ endows a left-symmetric conformal algebra structure on $V$. Therefore $V$ is a Lie conformal algebra which is the sub-adjacent Lie conformal algebra of this left-symmetric conformal algebra, and $T:V\rightarrow R$ is a homomorphism of Lie conformal algebra. Moreover, $T(V)\subset R$ is a Lie conformal subalgebra of $R$ and there is also a natural left-symmetric conformal algebra structure on $T(V)$ defined as follows $$\begin{aligned}
\label{ww1}
T(u)_\lambda T(v)=T(u\ast_\lambda v)=T(\rho(T(u))_\lambda v),~~~\text{$u$, $v\in V$.}\end{aligned}$$ In addition, the sub-adjacent Lie conformal algebra of this left-symmetric conformal algebra is a subalgebra of $R$ and $T: V\rightarrow R$ is a homomorphism of left-symmetric conformal algebra.
It can directly obtained from Proposition \[proo1\].
\[x3\] Let $R$ be a Lie conformal algebra and $T: R\rightarrow R$ is a Rota-Baxter operator of weight zero. Then there is a left-symmetric conformal algebra structure on $R$ with the following $\lambda$-product $$\begin{aligned}
a_\lambda b=[T(a)_\lambda b],~~~~~a,~b\in R.\end{aligned}$$
It follows directly from Corollary \[cor1\].
$\mathcal{O}$-operator and conformal $S$-equation
=================================================
\[the3\] Let $A$ be a finite left-symmetric conformal algebra which is free as a $\mathbb{C}[\partial]$-module and $r\in A\otimes A$ be symmetric. Then $r$ is a solution of conformal $S$-equation if and only if the $T\in \text{Chom}(A^{\ast c},A)$ corresponding to $r$ satisfies $$\begin{aligned}
[T_0(a^\ast)_\lambda T_0(b^\ast)]= T_0(L_A^{\ast}(T_0(a^\ast))_\lambda b^\ast
-L_A^{\ast}(T_0(b^\ast))_{-\lambda-\partial} a^\ast),\;\;\forall a^\ast, b^\ast\in A^{\ast c},\end{aligned}$$ where $T_0=T_\lambda\mid_{\lambda=0}$.
It follows from a similar proof as the one in Theorem \[tthe3\].
Let $A$ be a finite left-symmetric conformal algebra which is free as a $\mathbb{C}[\partial]$-module. Suppose $r$ is a symmetric solution of conformal $S$-equation. Then $T\in \text{Chom}(A^{\ast c},A)$ corresponding to $r$ is an $\mathcal{O}$-operator associated to $L_A^\ast$. Therefore there is a left-symmetric conformal algebra structure on $A^{\ast c}$ given as follows $$\begin{aligned}
a^\ast \circ_\lambda b^\ast=L_A^\ast(T(a^\ast))_\lambda b^\ast, ~~~~a^\ast,~~b^\ast\in A^{\ast c}.\end{aligned}$$
It follows from Corollary \[cor1\] and Theorem \[the3\].
\[the5\] Let $R$ be a finite Lie conformal algebra which is free as a $\mathbb{C}[\partial]$-module and $V$ a free and finitely generated $\mathbb{C}[\partial]$-module. Set $\{e_i\}\mid_{i=1,\cdots,n}$ and $\{v_i\}\mid_{i=1,\cdots,m}$ be the $\mathbb{C}[\partial]$-basis of $R$ and $V$ respectively. Suppose $\rho:R\rightarrow gc(V)$ is a representation of $R$ and $\rho^\ast: R\rightarrow gc(V^{\ast c})$ is its dual representation. Let $T\in \text{Chom}(V,R)$ with $T_0=T_\lambda\mid_{\lambda=0}: V\rightarrow R$ be an $\mathcal{O}$-operator associated to $\rho$ where $T_\lambda(v_i)=\sum_{j=1}^na_{ij}(\lambda,\partial)e_j$ for any $i\in \{1,\cdots,m\}$. Then $$\begin{aligned}
r=\sum_{i,j}(a_{ij}(-1\otimes \partial-\partial\otimes 1,\partial\otimes 1)e_j\otimes v_i^\ast+a_{ij}(-1\otimes \partial-\partial\otimes 1,1\otimes \partial)v_i^\ast \otimes e_j),\end{aligned}$$ is a symmetric solution of conformal $S$-equation in $T_0(V)\ltimes_{\rho^\ast,0} V^{\ast c}$ where the left-symmetric conformal algebra structure on $T_0(V)$ is given by Eq. (\[ww1\]).
Obviously, $r$ is symmetric. Then we only need to show that $r$ satisfies the conformal $S$-equation in $T_0(V)\ltimes_{\rho^\ast,0} V^{\ast c}$. Note that $T_0(u)_\lambda T_0(v)=T_0(\rho(T_0(u))_\lambda v)$. Then with a similar proof as of Theorem \[t2\], we have $$\begin{aligned}
&&\{\{r,r\}\}~~~\text{mod} ~~(\partial^{\otimes^3})\\
&=&\sum_{i,k}(T_0(v_k)_\mu T_0(v_i)\otimes v_k^\ast \otimes v_i^\ast
+\rho^\ast(T_0(v_k))_\mu v_i^\ast \otimes v_k^\ast \otimes T_0(v_i))\mid_{\mu=1\otimes \partial\otimes 1}\\
&&-\sum_{i,k}(v_k^\ast\otimes T_0(v_k)_\mu T_0(v_i)\otimes v_i^\ast
+v_k^\ast\otimes \rho^\ast(T_0(v_k))_\mu(v_i^\ast)\otimes T_0(v_i))\mid_{\mu=\partial \otimes 1\otimes 1}\\
&&+\sum_{i,k}(T_0(v_i)\otimes v_k^\ast\otimes \rho^\ast(T_0(v_k))_{-\mu-\partial} v_i^\ast
-v_i^\ast\otimes T_0(v_k)\otimes \rho^\ast(T_0(v_i))_\mu v_k^\ast\\
&&-v_i^\ast\otimes v_k^\ast\otimes [T_0(v_i)_\mu T_0(v_k)])\mid_{\mu=\partial \otimes 1\otimes 1} ~~~\text{mod} ~~(\partial^{\otimes^3})\\
&=&\sum_{i,k}(T_0(\rho(T_0(v_k))_\mu v_i)\otimes v_k^\ast\otimes v_i
-v_i^\ast\otimes v_k^\ast\otimes T_0(\rho(T_0(v_k))_\mu v_i))\mid_{\mu=1\otimes \partial\otimes 1}\\
&&-\sum_{i,k}(v_k^\ast\otimes T_0(\rho(T(v_k))_\mu v_i)\otimes v_i^\ast
-v_k^\ast\otimes v_i^\ast\otimes T_0(\rho(T_0(v_k))_\mu(v_i)))\mid_{\mu=\partial\otimes 1\otimes 1}\\
&&-\sum_{i,k}T_0(\rho(T_0(v_k))_\mu v_i)\otimes v_k^\ast\otimes v_i\mid_{\mu=1\otimes \partial\otimes 1}\\
&&+\sum_{i,k}(v_k^\ast\otimes T_0(\rho(T_0(v_k))_\mu v_i)\otimes v_i^\ast-v_k^\ast\otimes v_i^\ast\otimes T_0(\rho(T_0(v_k))_\mu v_i))\mid_{\mu=\partial\otimes 1\otimes 1}\\
&&+\sum_{i,k}v_i^\ast\otimes v_k^\ast \otimes T_0(\rho(T_0(v_k))_\mu v_i\mid_{\mu=1\otimes \partial\otimes 1}=0.\end{aligned}$$ This completes the proof.
\[cv\] Let $A$ be a finite left-symmetric conformal algebra which is free as a $\mathbb{C}[\partial]$-module. Then $$\begin{aligned}
r=\sum_{i=1}^n(e_i\otimes e_i^\ast+e^\ast\otimes e_i)\end{aligned}$$ is a non-degenerate symmetric solution of conformal $S$-equation in $A \ltimes_{L_A^\ast,0} A^{\ast c}$ where $\{e_i\}\mid_{i,\cdots,n}$ is a $\mathbb{C}[\partial]$-basis of $A$ and $\{e_i^\ast\}\mid_{i,\cdots,n}$ is its dual basis of $A^{\ast c}$.
It follows from Theorem \[the5\] by letting $V=A$, $\rho=L_A$ and $T=id_A$.
[Let $A$ be a left-symmetric conformal algebra. A bilinear form $\beta_\lambda:A\otimes A\rightarrow \mathbb{C}[\lambda]$ is called a [**2-cocycle**]{} of $A$ if $\beta_\lambda$ satisfies the following properties: $$\begin{aligned}
\label{ww9}\beta_\lambda(\partial a, b)=-\lambda\beta_\lambda(a, b),~~~\beta_\lambda( a, \partial b)=\lambda\beta_\lambda(a, b),\\
\label{ww2}\beta_{\lambda+\mu}(a_\lambda b,c)-\beta_\lambda(a,b_\mu c)
=\beta_{\lambda+\mu}(b_\mu a,c)-\beta_\mu(b,a_\lambda c),\end{aligned}$$ for any $a$, $b$, $c\in A$. A 2-cocycle $\beta_\lambda: A\otimes A\rightarrow \mathbb{C}[\lambda]$ of $A$ is called [**symmetric**]{} if $$\begin{aligned}
\beta_\lambda(a,b)=\beta_{-\lambda}(b,a),~~~~~~~~~~~~\text{for any $a$, $b\in A$.}\end{aligned}$$]{}
[It is shown in [@HL] that a 2-cocycle of $A$ is equivalent to a central extension of $A$ by the trivial $\mathbb{C}[\partial]$-module $\mathbb{C}$.]{}
\[th9\] Let $A$ be a finite left-symmetric conformal algebra which is free as a $\mathbb{C}[\partial]$-module. Then $r$ is a symmetric non-degenerate solution of conformal $S$-equation if and only if the bilinear form defined by $$\begin{aligned}
\label{ww3}\beta_\lambda(a,b)=\{ T^{-1}(a), b\}_\lambda,~~~~~~~~a,~~b\in A,\end{aligned}$$ is a symmetric 2-cocycle of $A$ where $T:A^{\ast c}\rightarrow A$ is the $\mathbb{C}[\partial]$-module homomorphism corresponding to $r$ by the $\mathfrak{g}(A)$-module isomorphism $A\otimes A\cong \text{Chom}(A^{\ast c},A)$. Here, $A$ is seen as a module of $\mathfrak{g}(A)$ via the left multiplication operators of $A$.
By Eq. (\[ww3\]), Eq. (\[ww9\]) is satisfied. Suppose that $r$ is symmetric. Then $$\begin{aligned}
\label{ww4}\{f, T(g)\}_\lambda=\{ g\otimes f, r\}_{(-\lambda,\lambda)}
=\{f\otimes g, r\}_{(\lambda,-\lambda)}
=\{ g, T(f)\}_{-\lambda},\;\;\forall f, g \in A^{\ast c}.\end{aligned}$$ Since $T$ is a $\mathbb{C}[\partial]$-module isomorphism from $A^{\ast c}\rightarrow A$, there exist $f$ and $g$ such that $T(f)=a$ and $T(g)=b$. By Eq. (\[ww4\]), we get $$\beta_\lambda(a,b)=\{ f, T(g)\}_\lambda=\{ g, T(f)\}_{-\lambda}
=\beta_{-\lambda}(b,a).$$ Therefore $\beta_\lambda$ is symmetric. With a similar process, it is easy to see that if $\beta_\lambda$ is symmetric, then $r$ is symmetric. Therefore $r$ is symmetric if and only if $\beta_\lambda$ is symmetric.
Set $r=\sum_ir_i\otimes l_i\in A\otimes A$. Since $r$ is symmetric, $r=\sum_i r_i\otimes l_i=\sum_i l_i\otimes r_i$. Since $\{ f, T(g)\}_\lambda=\{ g\otimes f, r\}_{(-\lambda,\lambda)}$, we get $T(g)=\sum_ig_{-\partial}(r_i)l_i=\sum_ig_{-\partial}(l_i)r_i$ for any $g\in A^{\ast c}$. Since $T$ is invertible, there exist $f$, $g$, $h\in A^{\ast c}$ such that $T(f)=a$, $T(g)=b$ and $T(h)=c$. Then $$\begin{aligned}
&&\beta_{\lambda+\mu}(a_\lambda b,c)-\beta_\lambda(a,b_\mu c)
-\beta_{\lambda+\mu}(b_\mu a,c)+\beta_\mu(b,a_\lambda c)\\
&=&\{ h, T(f)_\lambda T(g)\}_{-\lambda-\mu}
-\{ f,T(g)_\mu T(h)\}_\lambda
-\{ h, T(g)_\mu T(f)\}_{-\lambda-\mu}
+\{g, T(f)_\lambda T(h)\}_\mu\\
&=&\{ f\otimes g\otimes h,\sum_{ij} r_i\otimes r_j
\otimes {l_i}_\lambda l_j\}_{(\lambda, \mu,-\lambda-\mu)}
-\{f\otimes g\otimes h,\sum_{ij} {r_i}_\mu r_j
\otimes {l_i}\otimes l_j\}_{(\lambda, \mu,-\lambda-\mu)}\\
&&-\{ f\otimes g\otimes h,\sum_{ij} l_j\otimes l_i
\otimes {r_i}_\mu r_j\}_{(\lambda, \mu,-\lambda-\mu)}
+\{ f\otimes g\otimes h,\sum_{ij} l_i\otimes {r_i}_\lambda r_j
\otimes l_j\}_{(\lambda, \mu,-\lambda-\mu)}\\
&=& \{f\otimes g\otimes h, \sum_{ij} r_i\otimes r_j
\otimes [{l_i}_{\mu^{'}} l_j]\mid_{\mu^{'}=\partial\otimes 1\otimes 1}\\
&&-({l_j}_{\mu^{'}} r_i\otimes r_i\otimes l_i)\mid_{\mu^{'}=1\otimes \partial\otimes 1}
+(r_j\otimes {l_i}_{\mu^{'}} r_i\otimes l_i)\mid_{\mu^{'}=\partial\otimes 1\otimes 1}\}_{(\lambda, \mu,-\lambda-\mu)}.\end{aligned}$$ Therefore $\beta_\lambda$ is a 2-cocycle of $A$ if and only if $r=0~~\text{mod} (\partial^{\otimes^3})$. This competes the proof.
Let $A$ be a finite left-symmetric conformal algebra which is free as a $\mathbb{C}[\partial]$-module. Then there is a natural 2-cocycle $\beta_\lambda$ of $A\ltimes_{L^{\ast},0} A^{\ast c}$ given by $$\begin{aligned}
\beta_\lambda(a+f,b+g)=\{ f,b\}_\lambda+\{ g, b\}_{-\lambda},~~~a,~b\in A,~~f,~g\in A^{\ast c}.\end{aligned}$$
It can be directly obtained from Corollary \[cv\] and Theorem \[th9\].
Rota-Baxter operators on Lie conformal algebras
===============================================
It is known that Rota-Baxter operators (of weight zero) on Lie conformal algebras are a class of $\mathcal{O}$-operators and hence they can be used to provide the solutions of conformal CYBE and $S$-equation in the previous sections. In this section, we give a further study of Rota-Baxter operators with any weight on Lie conformal algebras.
Recall the definition of Rota-Baxter operator of weight $\alpha$ on an associative (or a non-associative) algebra.
[Let $A$ be an associative (or a nonassociative) algebra with the operation $\cdot: A\otimes A\rightarrow A$ over $\mathbb{C}$. A linear operator $P:A\rightarrow A$ is called a [**Rota-Baxter operator of weight $\alpha$**]{} $(\alpha \in \mathbb{C})$ on $A$ if $P$ satisfies the following condition: $$\begin{aligned}
P(x)\cdot P(y)=P(P(x)\cdot y+x\cdot P(y))+\alpha P(x\cdot y),~~\forall~~\text{$x$,~~$y\in A.$}\end{aligned}$$ If $\alpha=0$, we call $P$ a [**Rota-Baxter operator**]{} simply.]{}
[In particular, let $R$ be a Lie conformal algebra. If $T:R\rightarrow R$ is a $\mathbb{C}[\partial]$-module homomorphism satisfying $$\begin{aligned}
\label{ooo3}
[T(a)_\lambda T(b)]=T([a_\lambda T(b)])+T([T(a)_\lambda b])+\alpha T[a_\lambda b],~~~\forall~~\text{$a$, $b\in R$, }\end{aligned}$$ then $T$ is called a [**Rota-Baxter operator of weight $\alpha$**]{} on $R$, where $\alpha \in \mathbb{C}$. In fact, the case of weight zero has already been studied in the previous sections (for example, see Definition \[RB0\]). ]{}
\[p1\] Let $R$ be a Lie conformal algebra and $T$ be a Rota-Baxter operator of weight $\alpha$ on $R$. Define the linear operator $\mathcal{T}: \text{Coeff}(R)\rightarrow \text{Coeff}(R)$ as follows. $$\begin{aligned}
\mathcal{T}(a_n)=T(a)_n.\end{aligned}$$ Then $\mathcal{T}$ is a Rota-Baxter operator of weight $\alpha$ on the Lie algebra $\text{Coeff}(R)$.
It is directly obtained from Eqs. (\[oo3\]) and (\[eq1\]).
[Note that a similar study about averaging operators on Lie conformal algebras has been given in [@Ko].]{}
[Let $R=\mathbb{C}[\partial]V$ be a free $\mathbb{C}[\partial]$-module. Then by the definition of $\mathcal{T}$ in Proposition \[p1\], $\mathcal{T}: \mathbb{C} a_n\rightarrow \mathbb{C} a_n $ if and only if $T(a)=f(a) a$ for any $a\in V$ where $f(a)\in \mathbb{C}$. If $T: R\rightarrow R$ satisfies this condition, $T$ is called [**homogeneous**]{}.]{}
Next we study Rota-Baxter operators on a class of Lie conformal algebras named quadratic Lie conformal algebras ([@X1]).
[ If $R=\mathbb{C}[\partial]V$ is a Lie conformal algebra as a free $\mathbb{C}[\partial]$-module and the $\lambda$-bracket is of the following form: $$\begin{aligned}
[a_{\lambda} b]=\partial u+\lambda v+ w,~~~~~~\text{$a$, $b\in V$,}\end{aligned}$$ where $u$, $v$, $w\in V$, then $R$ is called a [**quadratic Lie conformal algebra**]{}.]{}
[A [**Novikov algebra**]{} $(A, \circ)$ is a vector space $A$ with an operation “$\circ$" satisfying the following axioms: for $a$, $b$, $c\in A$, $$\begin{aligned}
&(a\circ b)\circ c=(a\circ c)\circ b,\\
&(a\circ b)\circ c-a\circ (b\circ c)=(b\circ a)\circ c-b\circ(a\circ c).\end{aligned}$$]{}
[([@X1]) A [**Gel’fand-Dorfman bialgebra**]{} $(A, [\cdot, \cdot], \circ)$ is a vector space $A$ with two algebraic operations $[\cdot,\cdot]$ and $\circ$ such that $(A,[\cdot,\cdot])$ forms a Lie algebra, $(A,\circ)$ forms a Novikov algebra and the following compatibility condition holds: $$\begin{aligned}
\label{xx1}
[a\circ b, c]+[a,b]\circ c-a\circ [b,c]-[a\circ c, b]-[a,c]\circ b=0,\end{aligned}$$ for $a$, $b$, and $c\in A$. If a linear operator $P:A\rightarrow A$ is a Rota-Baxter operator of weight $\alpha$ on $(A,[\cdot,\cdot])$ as well as $(A,\circ)$, then $P$ is called a [**Rota-Baxter operator of weight $\alpha$** ]{} on the Gel’fand-Dorfman bialgebra $(A, [\cdot, \cdot], \circ)$.]{}
[([@X1] or [@GD])]{} A Lie conformal algebra $R=\mathbb{C}[\partial]V$ is quadratic if and only if $V$ is a Gel’fand-Dorfman bialgebra. The correspondence is given as follows. $$\begin{aligned}
[a_\lambda b]=\partial(b\circ a)+\lambda(a\ast b)+[b,a], ~~~\text{for any $a$. $b\in V$.}\end{aligned}$$ Here $a\ast b=a\circ b+b\circ a$ and $(V,[\cdot,\cdot],\circ)$ is a Gel’fand-Dorfman bialgebra.
Then we study the Rota-Baxter operators of weight $\alpha$ on quadratic Lie conformal algebras.
\[tt1\] Let $R=\mathbb{C}[\partial]V$ be the finite quadratic Lie conformal algebra corresponding to a Gel’fand-Dorfman bialgebra $(V,[\cdot,\cdot],\circ)$. If the algebra $(V,\ast)$ has no zero divisors, then any Rota-Baxter operator $T$ of weight $\alpha$ on $R$ is homogenous and $T$ is just the Rota-Baxter operator of weight $\alpha$ on the Gel’fand-Dorfman bialgebra $(V,[\cdot,\cdot],\circ)$.
Since $R$ is free and finite as a $\mathbb{C}[\partial]$-module, $V$ is a finite-dimensional vector space. Set $$T=\sum_{i=0}^n\partial^iT_i,$$ where $T_i: V\rightarrow V$ $(i=0, \cdots, n)$ are linear maps and there exists some $a\in V$ such that $T_n(a)\neq 0$. Therefore $$\begin{aligned}
[T(a)_\lambda T(b)]
&=&[\sum_{i=0}^n\partial^iT_i(a)_\lambda \sum_{j=0}^n\partial^jT_j(b)]\\
&=&\sum_{i,j=0}^n(-\lambda)^i(\lambda+\partial)^j(\partial(T_j(b)\circ T_i(a))
+\lambda(T_j(b)\ast T_i(a))+[T_j(b),T_i(a)]),\end{aligned}$$ and $$\begin{aligned}
&&T([T(a)_\lambda b]+[a_\lambda T(b)])+\alpha T[a_\lambda b]\\
&&=\sum_{i=0}^n(-\lambda)^i(\partial T(b\circ T_i(a))+\lambda T(T_i(a)\ast b)
+T([b, T_i(a)]))\\
&&+\sum_{j=0}^n(\lambda+\partial)^j(\partial T(T_j(b)\circ a)
+\lambda T(T_j(b)\ast a)+T([T_j(b),a])\\
&&+\alpha(\partial T(b\circ a)+\lambda T(b\ast a)+T([b,a])).\end{aligned}$$
If $n\geq 1$, by comparing the coefficients of $\lambda^{2n+1}$, we get $T_n(b)\ast T_n(a)=0$ for any $a$, $b\in V$. By the assumption, there exists $a\in V$ such that $T_n(a)\neq 0$. Hence $T_n(a)\ast T_n(a)=0$, which contradicts with the assumption that $(V,\ast)$ has no zero divisors. Therefore in this case, $T(a)=T_0(a)$ for any $a\in V$. Hence $T_0$ is exactly the Rota-Baxter operator $T$ of weight $\alpha$ on the Gel’fand-Dorfman bialgebra $(V,[\cdot,\cdot],\circ )$. This completes the proof.
\[y1\] Let $T$ be a Rota-Baxter operator of weight $\alpha$ on the a Gel’fand-Dorfman bialgebra $(V,[\cdot,\cdot],\circ )$. Then $\widetilde{T}: R=\mathbb{C}[\partial]V\rightarrow \mathbb{C}[\partial]V$ given by $\widetilde{T}(a)=T(a)$ for any $a\in V$ is a Rota-Baxter operator of weight $\alpha$ on $R$.
It can be directly obtained from Theorem \[tt1\].
\[y2\] Let $T$ be a Rota-Baxter operator $T$ of weight $\alpha$ on a Gel’fand-Dorfman bialgebra $(V,[\cdot,\cdot],\circ )$. Let $R=\mathbb{C}[\partial]V$ be the corresponding quadratic Lie conformal algebra. Then $\mathcal{T}: \text{Coeff}(R)\rightarrow \text{Coeff}(R)$ given by $ \mathcal{T}(a_n)=T(a)_n$ where $a\in V$ and $n\in \mathbb{Z}$ is a Rota-Baxter operator $T$ of weight $\alpha$.
It can be obtained by Corollary \[y1\] and Proposition \[p1\].
Finally two examples about Rota-Baxter operators of weight $0$ are given.
The Virasoro Lie conformal algebra $\text{Vir}$ is the simplest nontrivial example of Lie conformal algebras. It is defined by $$\text{Vir}=\mathbb{C}[\partial]L, ~~[L_\lambda L]=(\partial+2\lambda)L.$$ Coeff$\text{(Vir)}$ is just the Witt algebra.
Now we consider the Rota-Baxter operators on $\text{Vir}$. In fact, $\text{Vir}$ is a quadratic Lie conformal algebra corresponding to the Novikov algebra $(V=\mathbb{C}L, \circ)$ where $L\circ L=L$. Obviously, $(V,\ast)$ has no zero divisors. Therefore by Theorem \[tt1\], we only need to consider Rota-Baxter operators on $V$. It is easy to see that any Rota-Baxter operator on $V$ is of the form $T(L)=0$. Therefore all Rota-Baxter operators on $\text{Vir}$ are trivial.
Let $R=\mathbb{C}[\partial]L\oplus \mathbb{C}[\partial]W$ be a Lie conformal algebra of rank 2 with the $\lambda$-bracket given by $$\begin{aligned}
[L_\lambda L]=(\partial+2\lambda)L,~~[L_\lambda W]=(\partial+\lambda)W,~~
[W_\lambda W]=0.\end{aligned}$$ $\text{Coeff}(R)$ is isomorphic to the Heisenberg-Virasoro Lie algebra which is spanned by $\{L_n, W_n\mid n\in \mathbb{Z}\}$ satisfying $$\begin{aligned}
[L_m, L_n]=(m-n)L_{m+n},~~~[L_m,W_n]=-nW_{m+n},~~[W_m,W_n]=0.\end{aligned}$$ Obviously, the Gel’fand-Dorfman bialgebra corresponding to the Heisenberg-Virasoro Lie conformal algebra does not satisfy the condition in Theorem \[tt1\]. By a direct computation, we show that any Rota-Baxter operator on $R$ is one of the following two forms: $$\begin{aligned}
T(L)=-b(L+W),~~T(W)=b(L+W),~~~\text{where $b\in \mathbb{C}\backslash\{0\}$;}\end{aligned}$$ and $$\begin{aligned}
T(L)=g(\partial)W,~~T(W)=0,~~~\text{where $g(\partial)\in \mathbb{C}[\partial]$.}\end{aligned}$$ By Proposition \[p1\], we obtain two classes of Rota-Baxter operators on the Heisenberg-Virasoro Lie algebra as follows. One is $$\begin{aligned}
T(L_m)=-b(L_m+W_m),~~T(W_m)=b(L_m+W_m)~~~~\text{where $b\in \mathbb{C}\backslash\{0\}$};\end{aligned}$$ another is $$\begin{aligned}
T(L_m)=a,~~T(W_m)=0,~~~\text{where $a\in \mathbb{C}\{W_m\mid m\in \mathbb{Z}\}$.}\end{aligned}$$
In addition, by Corollary \[x3\], from the Rota-Baxter operators on the Heisenberg-Virasoro Lie conformal algebra, we can naturally get the following two left-symmetric conformal algebras structures on $A=\mathbb{C}[\partial]L\oplus \mathbb{C}[\partial]W$: one is $$\begin{aligned}
L_\lambda L=-b((\partial+2\lambda)L+\lambda W), ~~~~L_\lambda W=-b(\partial+\lambda) W,\\
W_\lambda L=b((\partial+2\lambda)L+\lambda W),~~~~W_\lambda W=b(\partial+\lambda) W;\end{aligned}$$ another is $$\begin{aligned}
L_\lambda L=g(-\lambda)\lambda W, ~~~~L_\lambda W=W_\lambda L=W_\lambda W=0.\end{aligned}$$
[**Acknowledgments**]{} [ This work was supported by the National Natural Science Foundation of China (11425104, 11501515), the Zhejiang Provincial Natural Science Foundation of China (LQ16A010011) and the Scientific Research Foundation of Zhejiang Agriculture and Forestry University (2013FR081). This work was carried out during the first author’s stay at Chern Institute of Mathematics, Tianjin, China, from April 10st to April 24th, 2016, and he would like to thank the CIM for its support and hospitality.]{}
[99]{}
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[^1]: \#1
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---
author:
- |
Veniamin BEREZINSKY\
[*Laboratori Nazionali del Gran Sasso, Istituto Nazionale di Fisica Nucleare\
I-67010 Assergi (AQ), Italy*]{}\
Vyacheslav DOKUCHAEV\
[*Institute for Nuclear Research of the Russian Academy of Sciences,\
60th Anniversary of October Prospect 7a, Moscow 117312, Russia*]{}\
Yury EROSHENKO\
[*Institute for Nuclear Research of the Russian Academy of Sciences,\
60th Anniversary of October Prospect 7a, Moscow 117312, Russia*]{}
---
Cosmological Origin of Small-Scale Clumps\
and DM Annihilation Signal
==========================================
Abstract {#abstract .unnumbered}
--------
We study the cosmological origin of small-scale DM clumps in the hierarchical scenario with the most conservative assumption of adiabatic Gaussian fluctuations. The mass spectrum of small-scale clumps with $M
\leq 10^3 M_{\odot}$ is calculated with tidal destruction of the clumps taken into account within the hierarchical model of clump structure. Only $0.1-0.5$% of small clumps survive the stage of tidal destruction in each logarithmic mass interval $\Delta\ln M\sim1$. The mass distribution of clumps has a cutoff at $M_{\rm min}$ due to diffusion of DM particles out of a fluctuation and free streaming at later stage. $M_{\rm min}$ is a model dependent quantity. In the case the neutralino DM, considered as a pure bino, $M_{\rm min}
\sim 10^{-8} M_{\odot}$. The evolution of density profile in a DM clump does not result in the singularity because of formation of the core under influence of tidal interaction. The radius of the core is $R_c
\sim 0.1 R$, where $R$ is radius of the clump. The applications for annihilation of DM particles in the Galactic halo are studied. The number density of clumps as a function of their mass, radius and distance to the Galactic center is presented. The enhancement of annihilation signal due to clumpiness, valid for arbitrary DM particles, is calculated. In spite of small survival probability, the global annihilation signal in most cases is dominated by clumps, with major contribution given by small clumps. The enhancement due to large clumps with $M\geq 10^6 M_{\odot}$ is very small.
Introduction
------------
The gravitationally bound structures in the universe are developed from primordial density fluctuations $\delta(\vec x,t)=\delta\rho/\rho$. They are produced at inflation from quantum fluctuations. The predicted power spectrum of these fluctuations has a nearly universal form $P(k)\propto
k^{n_p}$, with $n_p\simeq1$. At radiation-dominated epoch the fluctuations grow logarithmically slowly. After transition at $t=t_{\rm
eq}$ to the matter-dominated epoch, the fluctuations grow as $\delta\propto t^{2/3}$. The gravitationally bound objects are formed and detached from cosmological expansion when fluctuations enter the non-linear stage $\delta\geq1$. The non-linear stage of fluctuation growth has been studied both by analytic calculations [@Zeldovich] and [@ufn1] and in numerical simulations [@NFW; @moore99; @JingSuto] for Large Scale Structure (LSS). The density profile in the inner part of these objects is given by $\rho(r) \propto r^{-\beta}$, with with $\beta
\approx 1.7 - 1.9$ in analytic calculations [@ufn1], $\beta =1$ in simulations of NFW [@NFW] and $\beta =1.5$ in simulations of Moore et al. [@moore99] and Jing and Suto [@JingSuto]. In this work we apply this approach to the smallest DM objects in the universe, which we shall call [*clumps*]{}. The clumps, being the smallest structures, are produced first in the universe, and it makes difference of our consideration from LSS formation. The theoretical observation of this work is the importance of tidal interaction in the process of DM clump formation: the central nonsingular core is formed in the clumps and large fraction of clumps are tidally disrupted.
We use in the calculations the hierarchical model, in which due to merging of objects a small clump is hosted by the bigger one, the latter is submerged to more bigger etc. We use the standard cosmology with WMAP parameters. The primordial spectrum index is $n_p=0.99 \pm 0.04$ (WMAP) or $n_p=0.93 \pm 0.03$ (WMAP+2dF+Ly$\alpha$).
Tidal Destruction of Clumps in Hierarchical Model {#destruction}
-------------------------------------------------
The destruction of clumps by the tidal interaction occurs at the epoch of their hierarchical formation, long before the formation of galaxies. This interaction arises when two clumps pass near each other and when a clump moves in the external gravitational field of the bigger host to which this clump belongs. In both cases a clump is exited by the external gravitational field, i. e. its constituent particles obtain additional velocities in the c. m. system. The clump is destroyed if its internal energy increase $\Delta E$ exceeds the corresponding total energy $|E|
\sim GM^2/2R$. In [@bde03] we have calculated the rate of excitation energy production by both aforementioned processes. The dominating process is given by tidal interaction in the gravitational field of the host clumps, with the main contribution from the smallest host clump. We use the Press-Schechter formalism [@ps74; @cole] for hierarchical clustering. A small-scale clump during its life can be a constituent part of many host clumps of successively larger masses. After tidal disruption of the lightest host, a small clump becomes a constituent part of the larger host etc. Transition of a small clump from one host to another occurs nearly continuously in time up to the formation of a big enough host, where tidal destruction becomes inefficient.
The fraction of mass in the form of clumps which escape the tidal destruction in each logarithmic mass interval $\Delta\ln M\sim1$ is found as $$\xi_{\rm int}\simeq0.01(n+3).
\label{xitot}$$ In other words the mass function of clumps is $\xi_{\rm int} dM/M$. Since $n$ is close to $- 3$, only a small fraction of clumps about $0.1-0.5$% survive the stage of tidal destruction. However, this fraction is enough to dominate the total annihilation rate in the Galactic halo.
Core of Dark Matter Clump {#core}
-------------------------
We use the following parameterization of the density profile in a clump: $$\rho_{\rm int}(r)=\left\{ \begin{array}{lr}
\rho_c, & r<R_c; \\ \displaystyle{
\rho_c\left(\frac{r}{R_c}\right)^{-\beta} },
& R_c<r<R;\\ 0, & r>R.
\label{rho}
\end{array} \right.$$ In [@ufn1] the relative core radius of the clump is estimated as $x_c=R_c/R\simeq\delta^3_{\rm{eq}}\ll1$ from consideration of the perturbation of the velocity field due to damped mode of the cosmological density perturbations. Here $\delta_{\rm{eq}}$ is an initial density fluctuation value at the end of radiation dominated epoch. In [@BBM97] the core is considered to be produced for spherically symmetric clump by inverse flow caused by annihilation of DM particles. We show [@bde03] that these phenomena are not the main effects and that much stronger disturbance of the velocity field in the central part of clumps is produced by tidal forces. The tidal forces influence the nearly radial motion of DM particles at the time of clump formation. As a result these particles obtain some angular momentum which prevents the formation of singularity. Once the core is produced it is not destroyed in the evolution followed. The core formation proceeds mainly near the time of the clump maximal expansion $t_s$. At this moment the clump decouples from an expansion of the universe and contracts in the non-linear regime. Soon after this period a clump enters the hierarchical stage of evolution, when the tidal forces can destroy it, but surviving clumps retain their cores.
The calculations proceed in the following way (see [@bde03] for details). The background gravitational field (including that of the host clumps) is expanded in series in respect to the distance from the point with maximum density in a fluctuation. The motion of a DM particle in this field is studied. The spherically symmetric term of the expansion causes the radial motion of a particle in the oscillation regime. Spherically non-symmetric term describes the tidal interaction. It results in deflection of a particle trajectory from a center (point with maximum density). The average (over statistical ensemble) deflection gives the radius of the core $R_c$. After statistical averaging, $R_c$ is expressed through the amplitude of the fluctuation $\delta_{\rm eq}$ and the variance $\sigma_{\rm eq}$ (or $\nu=\delta_{\rm eq}/\sigma_{\rm eq}$) as $$x_c=R_c/R\approx 0.3 \nu^{-2}f^2(\delta_{\rm eq}),
\label{x_c}$$ where the function $f(\delta_{\rm{eq}})\sim1$ is given in Ref [@bde03]. The fluctuations with $\nu \sim 0.5 - 0.6$ have $x_c \sim 1$, i.e. they are practically destroyed by tidal interactions. Most of galactic clumps are formed from $\nu \sim 1$ peaks, but the main contribution to the annihilation signal is given by the clumps with $\nu\simeq2.5$ for which $x_c\simeq0.05$.
Clumps of Minimal Mass {#smmin}
----------------------
The mass spectrum of clumps has a low-mass cutoff at $M=M_{\rm min}$, which value is determined by a leakage of DM particles from the overdense fluctuations in the early universe. CDM particles at high temperature $T>T_f \sim 0.05 m_{\chi}$ are in the thermodynamical (chemical) equilibrium with cosmic plasma. After freezing at $t>t_f$ and $T<T_f$, the DM particles remain for some time in [*kinetic*]{} equilibrium with plasma, when the temperature of CDM particles $T_{\chi}$ is equal to temperature of plasma $T$. At this stage the CDM particles are not perfectly coupled to the cosmic plasma. Collisions between a CDM particle and fast particles of ambient plasma result in exchange of momenta and a CDM particle diffuses in the space. Due to diffusion the DM particles leak from the small-scale fluctuations and thus their distribution obtain a cutoff at the minimal mass $M_D$.
The DM particles get out of the kinetic equilibrium when the energy relaxation time for DM particles $\tau_{\rm rel}$ becomes larger than the Hubble time $H^{-1}(t)$. This conditions determines the time of kinetic decoupling $t_d$. At $t \geq t_d$ the CDM matter particles are moving in the free streaming regime and all fluctuations on the scale of free-streaming length $\lambda_{fs}$ and smaller are washed away. The corresponding minimal mass $M_{\rm fs} =
(4\pi/3)\rho_{\chi}(t_0)\lambda_{\rm fs}^3$, is much larger than $M_D$ and therefore $M_{\rm min}=M_{\rm fs}$. In [@bde03] we have performed the calculations using two methods: the transparent physical method, based on the description of diffusion and free streaming, and more formal method based on solution of kinetic equation for DM particles starting from the period of chemical equilibrium. Both methods agree perfectly. The minimal mass in the DM mass distribution is determined by the process of free-streaming. For the case of neutralino (bino) as DM particle this minimal mass equals to $M_{\rm min} = 1.5\times10^{-8}M_{\odot}$ for neutralino mass $m_{\chi}=100$ GeV and the mass of selectron and sneutrino $\tilde{M}=1$ TeV. Our calculations agree reasonably well with that of [@bino01], while $M_{min}$ from [@gzv12] coincides with our value for $M_D$.
Annihilation Signal Due to Small Clumps {#enhancement}
---------------------------------------
There is distribution of clumps in the Galactic halo over three parameters, mass $M$, radius $R$, and distance from the Galactic Center $l$: $n_{\rm cl}(M,R,l)$. This distribution also depends on the parameters which describe the internal structure of the clumps, $\beta$ and $x_c=x_c(M,R)$, from Eq. (\[rho\]). With the number density of clumps in the halo written as $dN_{\rm{cl}} =
n_{\rm{cl}}(l,M,R)d^3ldMdR$, the observed signal at the position of the Earth from DM particle annihilation in the clumps is given by quantity $$\begin{aligned}
I_{\rm cl}&=&\frac{1}{4\pi}
\int\limits_{0}^{\pi}d\zeta\sin\zeta
\int\limits_{0}^{r_{\rm{max}}(\zeta)}\frac{2\pi r^2dr}{r^2}
\int\limits_{M_{\rm{min}}}^{M_{\rm{max}}}dM
\int\limits_{R_{\rm{min}}}^{R_{\rm{max}}} dR \nonumber \\
&& \times\,\, n_{\rm{cl}}(l(\zeta,r),M,R) \dot N_{\rm cl}(M,R),
\label{ihal}\end{aligned}$$ where $r$ is distance from the Sun (Earth) to a clump and $\zeta$ is angle between the line of observation and the direction to the Galactic center, $r_{\rm{max}}$ is the distance from the Sun to the halo’s outer border and $\dot N_{\rm cl}(M,R)$ is annihilation rate in the single clump of mass $M$ and radius $R$.
Additional annihilation signal is given by unclumpy DM in the halo with homogeneous ([*i.e.*]{} smoothly spread) density $\rho_{\rm
DM}(l)$, where $l$ is a distance to the Galactic Center. $$I_{\rm{hom}}=\frac{\langle\sigma_{\rm ann}v\rangle}{2}
\int\limits_{0}^{\pi}\!
d\zeta\sin\zeta\!\!\!\int\limits_{0}^{r_{\rm{max}}(\zeta)}
\!\!\!\!dr\rho_{\rm{DM}}^2(l(\zeta,r))/m_{\chi}^2.
\label{hom}$$ The [*enhancement*]{} $\eta$ of the signal due to a presence of clumps is given by $$\eta=\frac{I_{\rm cl}+I_{\rm{hom}}}{I_{\rm{hom}}} \label{eta}$$ This quantity describes the global enhancement of the annihilation signal observed at the Earth ([*e. g.*]{} the flux of radio, gamma, and neutrino radiations) as compared with usual calculations from annihilation of unclumpy DM.
![The global enhancement $\eta$ of the annihilation signal from the Galactic halo as a function of the minimal clump mass $M_{\rm{min}}$, for clump density profile with index $\beta=1.5$ and for different indices $n_p$ of primeval perturbation spectrum. The curves are marked by the values of $n_p$.[]{data-label="fig3r"}](fig4beta.eps){height="21.5pc"}
We assume that space density of clumps in the halo, $n_{\rm cl}(l)$ is proportional to the unclumpy DM density, $\rho_{\rm DM}(l)$: $n_{\rm{cl}}(l)=\xi\rho_{\rm DM}(l)/M$ with $\xi \ll 1$. This assumption holds with a good accuracy for the small-scale clumps. The signal from small clumps is determined mainly by clumps of the minimal mass. In calculations [@bde03] we used different density profiles in the clumps, the distribution of DM clumps over their masses $M$ and radii $R$, and the distribution of clumps in the galactic halo. The enhancement depends on the nature of DM particle only through $M_{\rm min}$. For $\beta=1.5$ enhancement is given in Fig. \[fig3r\] Details of calculations and the plots for other values of $\beta$ can be found in [@bde03]. The enhancements $\eta$ for $n_p=1$ or less is not large: typically it is 2-5 for $M_{\rm min}
\sim 10^{-8}M_{\odot}$. For example, $\eta=5$ for $n_p=1.0$ and $M_{\rm{min}}= 2\cdot 10^{-8}M_{\odot}$. It strongly increases for smaller $M_{\rm{min}}$ and larger $n_p$. For example, for $n_p= 1.1$ and $M_{\rm{min}}=2\cdot 10^{-8}M_{\odot}$, enhancement is $\eta=130$. Our approach is based on the hierarchical clustering model in which smaller mass objects are formed earlier than the larger ones, i. e. $\sigma_{\rm{eq}}(M)$ diminishes with the growing of $M$. This condition is satisfied for objects with mass $M>M_{min} \simeq 2\cdot10^{-8}M_{\odot}$ only if the primordial power spectrum has the power index $n_p>0.84$. The enhancement of the annihilation signal is absent, e.g $\eta\simeq1$, for $n_p<0.9$.
Annihilation Signal Due to Big Clumps {#enhancement1}
-------------------------------------
Numerical simulations reveal in the galactic halos the big clumps with masses $10^8-10^{10}M_{\odot}$. At $ t \sim t_{\rm eq}$ these clumps are characterized by an effective power spectrum $P(k) \propto k^{n}$ with $n\approx -2$ (in contrast to $n\approx -3$ for small clumps), and thus the survival probability given by Eq. (\[xitot\]) is larger for the big clumps. Indeed, the effective power spectrum index $$n= -3\left[1+2 \,\frac{\partial\ln \sigma_{\rm eq}(M)}{\partial \ln
M}\right]
\label{n}$$ tends to $n= n_p-4=-3$ for small $M$ and to $n \approx -2$ for $M$ in the interval $10^6 - 10^9 M_{\odot}$. On the other hand the mean internal number density of DM particles in the big clumps is much smaller in comparison with that in the small clumps, and it compensates the first effect.
The number density $n(M)$ of the big clumps with $M\geq 10^8 M_{\odot}$ in the numerical simulations found [@moore99] as $n(M)dM \propto dM/M^{\gamma}$ with $\gamma \approx 1.9 - 2.0$ and with a mass fraction of clumps $\varepsilon \sim 0.1 - 0.2$. Observations of halo lensing [@clobs] give smaller values $\varepsilon \sim 0.06-0.07$. It is interesting to note that the mass function of clumps, obtained from Eq. (\[xitot\]), is close (including the normalization coefficient) to that obtained in the numerical simulations for big clumps with mass $M\geq10^8M_{\odot}$. Strictly speaking our calculations are not valid for big clumps, because of their destruction in the halo up to the present epoch $t_0$ and accretion of new clumps into the halo. Nevertheless, for the small interval of masses, where the power-law spectrum can be used as a rather good approximation, our approach appears to be roughly valid.
Calculations of the enhancement factor $\eta$ from simplified Eqs. (\[ihal\]) and (\[hom\]) are performed by using $\varepsilon= 0.1$, the number density distribution of clumps in the Galaxy $n_{\rm cl}(l) \propto \rho_{\rm DM}(l)$, and internal density distribution of the DM particles in clumps $$\rho(r)=\rho_c (r/a)^{-\beta}(1+r/a)^{-\kappa}
\label{profile}$$ valid down to the core radius $R_c$. The core is defined as $\rho(r)
=\rho_c=const$ at $r \leq R_c$. The NFW profile has $\beta =1$ and $\kappa = 2$, while the Moore et al. profile has $\beta=\kappa=1.5$.
The other important parameter is clump radius $R$, which determines the average density $\bar{\rho}$ at the epoch of DM virialization in the clump. This quantity is calculated from the value of overdensity $\delta$ at the linear stage of clump formation. The values of $\delta$ have Gaussian distribution and the normalization of fluctuation spectrum was performed as usual to the value of r.m.s. fluctuation at the 8 Mpc scale $\sigma_8\simeq1$. This approach corresponds to the picture that the big clumps in the Galactic halo are similar to the small protogalaxies (galactic building blocks), which escape the tidal destruction when capturing by the Galaxy. The tidal stripping of the outer parts of the clumps change their structure. Nevertheless this process is not important for clumps with mass $M\ll 10^{10} M_{\odot}$. For minimal and maximal masses of the big clump we use $M_{\rm min}=10^8 M_{\odot}$ and $M_{\rm max}=10^{10} M_{\odot}$.
As a result of calculations, identical to those in Section 5, we have found the enhancement factor $\eta=1.02$ for the NFW profile [@NFW] with $\gamma=1.9$, $\varepsilon=0.1$, $R_c=a$, $R/a=5$ (the case of existence of the core), and $\eta=1.14$ for the case of absence of the core $R_c=0$.
For the Moore et al. profile [@moore99], $\eta=1.06$ for $\varepsilon=0.1$, $R_c/a=0.5$, $R/a=5$ and $\eta=1.16$ for a smaller core $R_c/a=0.2$.
Diminishing of $M_{\rm min}$ increases the enhancement weakly, approximately as $\eta(M_{\rm min}) \propto M_{\rm min}^{-0.35}$.
We conclude that enhancement of the annihilation signal due to the big clumps is small.
We are grateful to Ben Moore for providing us with the data which allow us to normalize the density distribution of DM in the clumps.
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---
abstract: 'Due to its low atomic mass hydrogen is the most promising element to search for high-temperature phononic superconductors. However, metallic phases of hydrogen are only expected at extreme pressures (400GPa or higher). The measurement of a record superconducting critical temperature of 190K in a hydrogen-sulfur compound at 200GPa of pressure [@Eremets_arxiv2014], shows that metallization of hydrogen can be reached at significantly lower pressure by inserting it in the matrix of other elements. In this work we re-investigate the phase diagram and the superconducting properties of the H-S system by means of minima hopping method for structure prediction and Density Functional theory for superconductors. We also show that Se-H has a similar phase diagram as its sulfur counterpart as well as high superconducting critical temperature. We predict [H$_3$Se]{} to exceed 120K superconductivity at 100GPa. We show that both [H$_3$Se]{} and [H$_3$S]{}, due to the critical temperature and peculiar electronic structure, present rather unusual superconducting properties.'
author:
- 'José A. Flores-Livas'
- Antonio Sanna
- 'E. K. U. Gross'
bibliography:
- 'paper.bib'
title: High temperature superconductivity in sulfur and selenium hydrides at high pressure
---
Under high pressure conditions, insulating and semiconducting materials tend to become metallic, because, with increasing electronic density, the kinetic energy grows faster than the potential energy. As metallicity is a necessary condition for superconductivity, superconductivity becomes more likely under pressure [@Shimizu_EHP2005; @superconductingelements_2005]. Wigner and Huntington [@Wigner_JCP1935], already in 1935 suggested the possibility of a metallic modification of hydrogen under very high pressures. Ashcroft and Richardson predicted [@Ashcroft_PRL1968; @RichardsonAshcroft_PRL97] hydrogen to become metallic under pressure and also the possibility to be a high temperature superconductor. The high critical temperature ([T$_{\textmd C}$]{}) of hydrogen [@Cudazzo_PRL2008; @Zhang_H_SSC2007; @Cudazzo_1_PRB2010] is a consequence of its low atomic mass leading to high energy vibrational modes and in turn to a large phase space available for electron-phonon scattering to induce superconductivity [@BCS_1957]. However, the estimated pressure of metallization [@pickard_structure_2007; @Cudazzo_2_PRB2010] is beyond the current experimental capabilities and it has been a challenge to confirm this hypothesis [@LeToullec2002; @Eremets_NatMat2011; @Hemley_PRL2012; @HRussell_hydrogenJACS2014].
It was only recently that hydrogen-rich compounds started to be explored as a way to decrease the tremendous pressure of metallization in pure hydrogen [@Ashcroft_PRL2004], essentially performing a chemical pre-compression. The first system explored experimentally was silane (SiH$_4$) [@Eremets_Science2008]. Soon after, many other pre-compressed hydrogen rich materials have been explored experimentally [@chen_pressure-induced_2008; @degtyareva_formation_2009; @Hanfland_PRL2011; @Strobel_PRL2011] and theoretically [@tse_novel_2007; @Chen_PNAS2008; @Kim_PNAS2008; @Wang_PNAS2009; @Yao_PNAS2010; @gao_high-pressure_2010; @Kim_PNAS2010; @Li_PNAS2010; @Zhou_PRB2012; @LiYanmingMa_JCP2014; @Yanming_JCP2014; @Hooper_JPC-2014; @Duan_SciRep2014]. The importance of a systematic search for a crystalline ground state has been put in evidence for disilane (SiH$_6$), where structures enthalpically higher lead to transition temperatures of the order of 130K. Interesting structures have been proved not to be the global minimum and for the correct ground state was found a rather moderate electron-phonon coupling and [T$_{\textmd C}$]{} of 25K [@Disilane_JAFL]. In agreement with experimental evidence.
Recently it was reported that sulfur hydride (SH$_2$), when pressurized, becomes metallic and superconducting. For pressures above 180GPa an extremely high transition temperature of about 190K was measured [@Eremets_arxiv2014]. This [T$_{\textmd C}$]{} is higher than in other superconductors known so far, including cuprates and pnictides. The experimental evidence is supported theoretically [@Duan_SciRep2014; @LiYanmingMa_JCP2014; @Bernstein_arxiv_2014], and crystal prediction methods suggest that the system becomes superconducting with a SH$_3$ stoichiometry. In this work we re-investigate extensively the S-H phases with state of the art *ab-initio* material search minima hopping methods [@Goedecker_2004; @Goedecker_2005; @Amsler_2010] (MHM) and compute the superconducting properties with the completely parameter-free Density Functional Theory for Superconductors (SCDFT). We also extend the analysis to the Se-H system, predicting a fairly similar phase diagram and comparable superconducting properties.
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Methods
=======
Electronic and phononic structure calculations are performed within density-functional theory as implemented in the two plane-wave based codes [abinit]{} [@gonze_abinit_2009], and [espresso]{} [@QE-2009] within the local density approximation LDA exchange correlation functional. The core states were accounted for by norm-conserving Troullier-Martins pseudopotentials [@FHI_Fuchs]. The pseudopotential accuracy has been checked against all-electron (LAPW+lo) method as implemented in the [elk]{} code (http://elk.sourceforge.net/). In order to predict the ground state structure of sulfur/selenium hydride compounds we use the minima hopping method [@Goedecker_2004; @Goedecker_2005; @Amsler_2010] for the prediction of low-enthalpy structures. This method has been successfully used for global geometry optimization in a large variety of applications [@MA_JAFL; @LiAlH_Maxmotif; @BJAFL_PRB2012]. Given only the chemical composition of a system, the MHM aims at finding the global minimum on the enthalpy surface while gradually exploring low-lying structures. Moves on the enthalpy surface are performed by using variable cell shape molecular dynamics with initial velocities approximately chosen along soft mode directions. We have used 1,2,3 formula units of [H$_3$S]{} and [H$_3$Se]{} at selected pressures of 100, 150, 200, 250 and 300GPa. The relaxations to local minima are performed by the fast inertia relaxation engine [@FIRE_2006] and both atomic and cell degrees of freedom are taking into account. Final structural relaxations and enthalpy calculations were performed with the [vasp]{} code [@VASP_Kresse]. The plane-wave cutoff energy was set to 800eV, and Monkhorst-Pack $k$-point meshes with grid spacing denser than $2\pi\times 0.01$Å$^{-1}$, resulting in total energy convergence to better than 1meV/atom. Superconducting properties have been computed within density-functional theory for superconductors (SCDFT) [@OGK_SCDFT_PRL1988; @Lueders_SCDFT_PRB2005; @Marques_SCDFT_PRB2005]. This theory of superconductivity is completely *ab-initio*, fully parameter-free and proved to be rather accurate and successful in describing phononic superconductors[@Floris_Pb_PRB2007; @Gonnelli_CaC6_PRL2008; @Sanna_CaC6_RapCom2007; @Profeta_LiKAl_PRL2006]. It allows to compute all superconducting properties including the critical temperature and the excitation spectrum of the system [^1].
Crystal structure prediction and enthalpies
===========================================
Experimentally little is known on the high pressure stability and composition of the S-H system and, to the best of our knowledge, nothing is known about the Se-H. Therefore we investigate their low temperature phase diagram by means of the MHM for the prediction of low-enthalpy structures. Computed enthalpies as a function of pressure are reported in Fig. \[fig:enthalpy\]. We consider the H$_3$S stoichiometry as well as its elemental decomposition (sulfur + hydrogen), its decomposition into H$_2$S + hydrogen and H$_4$S - hydrogen [^2]. At low pressure we find the $Cccm$ structure up to 95GPa and the $R3m$ ($\beta$-Po-type) rhombohedral structure between 95 and 150GPa. Above 150GPa, we confirm[@LiYanmingMa_JCP2014; @Bernstein_arxiv_2014] the cubic [$Im$3$m$ ]{}(bcc) as the most stable lattice.
In a similar way we have studied the Se-H phase diagram. Chemically, selenium is known to have very similar physical properties to sulfur and this system is not an exception. The enthalpies of the phases found in our MHM runs are shown in Fig. \[fig:enthalpy\]b. Once again we use the $Cccm$ structure as reference since, as in the case of sulfur, it is the most stable at low pressures and up to 70GPa. Between 70GPa to 100GPa, we find that the H$_2$Se + hydrogen decomposition is more stable than the [H$_3$Se]{} stoichiometry. [H$_3$Se]{} returns to be the most stable composition above 100GPa and at least up to 250GPa.
Therefore from our analysis both systems in the range 50GPa to 250GPa show, with increasing pressure, two phase transitions. The S-H system, always stable in the [H$_3$S]{} stoichiometry, has a first order phase transition from $Cccm$ to $R3m$ at $\sim$100GPa, then the $R3m$ rhombohedral distortion decreases continuously up to 150GPa where it transforms into the [$Im$3$m$ ]{}cubic structure. The Se-H system at low pressure is also stable in the [H$_3$Se]{} stoichiometry but becomes unstable to a phase separation into H$_2$S + hydrogen in the range from 70GPa to 100GPa. Above 100GPa another discontinuous phase transition occurs, directly into the [$Im$3$m$ ]{}cubic structure. Note that 100GPa is also the pressure below which the [$Im$3$m$ ]{}structure would distort into the $\beta$-Po $R3m$, therefore depending on experimental conditions this rhombohedral phase may occur as a metastable one. The sequence of transformation is highlighted in Fig. \[fig:enthalpy\] by means of shaded areas.
electronic and phononic properties of [H$_3$Se]{} and [H$_3$S]{} at high pressure {#sec:bandsandcoupling}
=================================================================================
We focus now on the properties of [H$_3$S]{} and [H$_3$Se]{} in the pressure range of stability of the [$Im$3$m$ ]{}structure. The two materials present very similar properties. At 200GPa electronic band dispersions and Fermi surfaces are barely distinguishable, as seen in Fig. \[fig:bands\_and\_fs\]. And in the range of pressure between 100 to 200GPa there are no significant changes in the electronic properties apart from the overall bandwidth that increases with pressure. An important aspect of the electronic structure is the presence of several Fermi surface sheets, with no marked nesting features and with Fermi states both at low and high momentum vector. At small momentum (close to the $\Gamma$-point, center of the Brillouin zone in Fig. \[fig:bands\_and\_fs\]) there are three small Fermi surfaces (only the green larger one can be seen in the figure, smaller ones being inside it). However, these provide a rather small contribution to the total density of states (DOS) at the Fermi level which mostly comes from the two larger Fermi surface sheets. These are of *hybrid* character, meaning that their Kohn-Sham (KS) states overlap both with H and S/Se states (the overlap is expressed in the figure by the color-scale of the band lines), suggesting that they will be coupled with both hydrogen and S/Se lattice vibrations (more details on this point will be given in the next section). Overall the DOS shows a square root behavior of the 3D electron gas, the main deviation from this occurs close to the Fermi energy where a peak with an energy width of about 2eV is present. This structure will play a relevant role in the superconducting properties.
![(Color online) Fermi Surfaces (top) and electronic band structures (bottom) of [H$_3$S]{} and [H$_3$Se]{} at high pressure in the [$Im$3$m$ ]{}phase. The color-scale in the band lines indicates the projection of the KS states on the atomic orbitals of the sulfur/selenium atom normalized by the maximum total atomic projection of these valence states that is of about 70%.[]{data-label="fig:bands_and_fs"}](fig2.png){width="0.97\columnwidth"}
Unlike the electronic structure, phonons are strongly pressure and material dependent. Clearly a key role is played by the occurrence of the II order $R3m$ to [$Im$3$m$ ]{}phase transition. Far away from it (i.e. at very high pressure) we have three sets of well separated phonon modes: acoustic (below 60meV), optical modes that are transverse with respect to the S/Se-H bond (between 100 to 200meV) and, above 200meV, stretching modes of the Se/S-H bond. These are clearly seen in Fig. \[fig:phonons\]b for [H$_3$Se]{} at 200GPa. As pressure reduces, the bond structure of the system tends to destabilize because, from a four-fold coordination in the [$Im$3$m$ ]{}structure, it goes to a three-fold coordination in the $R3m$ one. This means that one of the high-energy stretching mode slowly softens at $\Gamma$. This softening can be clearly seen in [H$_3$S]{} at 200GPa (Fig. \[fig:phonons\]a) where a H-S stretching mode went down to about 60meV. Eventually, as pressure lowers this softens to zero energy, marking the occurrence of the phase transition, at about 150GPa in [H$_3$S]{} and slightly below 100GPa in [H$_3$Se]{}. In fact, at 100GPa this mode has, in [H$_3$Se]{}, almost zero energy (see Fig. \[fig:phonons\]c).
In spite of these important changes in the phononic energy dispersion, the overall coupling strength[@Carbotte_RMP1990; @AllenMitrovic1983] $\lambda$ does not show large variations over the pressure range, as we can see from Tab. \[tab:tc\]. Naturally the coupling increases near the phase transition due the optical mode softening, however, as this is restricted to a relatively small region near the $\Gamma$ point, the effect is not dramatic. On the other hand there is definitively a difference in the coupling strength of the Se ($\lambda \sim 1.5$) with respect to the S system ($\lambda \sim 2.5$), indicating that selenium, due to its larger ionic size, provides a better electronic screening of the hydrogen vibrations.
Superconducting properties {#sec:sc}
==========================
We have computed, by means of SCDFT, critical temperatures of [H$_3$S]{} and [H$_3$Se]{} in the pressure range of stability of the [$Im$3$m$ ]{}structure, these are collected in Tab. \[tab:tc\]. The predicted [T$_{\textmd C}$]{} for the [H$_3$S]{} system is 180K at 200GPa and 195K at 180GPa, in agreement with the measured value of 185K at 177GPa. On the other hand our prediction for the deuterium substituted system [D$_3$S]{} is 141K, at 200GPa. That is much larger than the measured[@Eremets_arxiv2014] [T$_{\textmd C}$]{} of 90K. This huge experimental isotope effect is therefore not consistent with our calculations. However the good agreement obtained with the [T$_{\textmd C}$]{} of the [H$_3$S]{} system seem to exclude an explanation in terms of anharmonic effects in the hydrogen vibrations, as suggested in Ref. . Nevertheless, the theoretical isotope coefficients $\alpha^{S}=0.05$ and $\alpha^H=0.4$ (defined as $\alpha^A=-\frac{M^A}{Tc}\frac{\partial T_C}{\partial M^A}$, and computed at 200GPa with a three point numerical differentiation) clearly indicate and confirm [@Pickett_Arxiv2015] the dominant contribution of hydrogen phonon modes to the superconducting pairing.
Our prediction for [H$_3$Se]{} at 200GPa is of 131K, this reduction of [T$_{\textmd C}$]{} is clearly not an isotopic effect of the substitution S to Se. As mentioned in the previous section, it is caused, instead, by a different coupling strength of the hydrogen modes in the Se environment. In spite of the much lower coupling strength $\lambda$ the reduction of [T$_{\textmd C}$]{} is not very large with respect to the sulfur system, as expected from the fact that the critical temperature at high coupling increases with the square root of $\lambda$ (while is exponential at low coupling) [@Carbotte_RMP1990; @AllenMitrovic1983].
To compute the critical temperatures we have used SCDFT, this choice allow us to deal with the unusual superconducting properties of these systems from first principles without relying on conventional assumptions coming from low pressure experience. There are two aspects of these systems that are uncommon, that make the use of conventional [^3] Eliashberg methods difficult to apply to these systems. First, the strong variation of the electronic density of states at the Fermi level, that is pinned to a rather sharp peak in the DOS, second the extremely large el-ph coupling and phonon frequencies that lead to a very broad region around [E$_{\rm F}$]{} where the interaction is dominated by phonons over Coulomb repulsion.
$\lambda$ [$\omega_{\textmd log}$]{} [T$_{\rm c}^{\rm SCDFT}$]{} [$\Delta$(T=0)]{} [T$_{\rm c}^{\rm SCDFT,ph}$]{} [T$_{\rm c}^{\rm AD,\mu^*=0.1}$]{} [T$_{\rm c}^{\rm AD,\mu^*=0}$]{}
-------------- ------------ ----------- ---------------------------- ----------------------------- ------------------- -------------------------------- ------------------------------------ ----------------------------------
[H$_3$S]{} [200GPa]{} 2.41 109meV [**180K**]{} 43.8meV 284K 255K 338K
[D$_3$S]{} [200GPa]{} 2.41 82meV [**141K**]{} 32.9meV 216K 188K 247K
[H$_3$S]{} [180GPa]{} 2.57 101meV [**195K**]{} 44.8meV 297K 250K 331K
[H$_3$Se]{} [200GPa]{} 1.45 120meV [**131K**]{} 28.4meV 234K 174K 246K
[H$_3$Se]{} [150GPa]{} 1.38 107meV [**110K**]{} 23.4meV 195K 145K 209K
[H$_3$Se]{} [100GPa]{} 1.76 87meV [**123K**]{} 27.0meV 198K 156K 214K
![(Color online) Phonon dispersion and $\alpha^2F$ functions [@Carbotte_RMP1990; @AllenMitrovic1983] of [H$_3$S]{} and [H$_3$Se]{} at high pressure in the cubic [$Im$3$m$ ]{}structure. The color coding gives the projection of the mode displacement on the S/Se atom. Displacements are visible dominated by H due to its lighter mass.[]{data-label="fig:phonons"}](fig3.png){width="1.0\columnwidth"}
The effect of the energy dependence of the DOS can be appreciated by comparing Eliashberg results with SCDFT when neglecting the Coulomb interaction (see Tab. \[tab:tc\]). At 200GPa the two theories [^4] disagree by 54K, SCDFT giving 284K while Eliashberg gives 338K. This difference comes from the energy dependence of the DOS, while Eliashberg assumes a flat DOS, in the SCDFT we can easily check this assumption by assuming a flat DOS, and for this case the SCDFT calculation would lead to 334K, in agreement with the Eliashberg result.
Physically the reduction of [T$_{\textmd C}$]{}, occurring when the real DOS is considered, arises from the fact that the phononic pairing extends in a rather large region around the Fermi level, over the DOS peak structure of these systems (see Fig. \[fig:bands\_and\_fs\]). Beyond the range of the phononic pairing the coupling is dominated by the Coulomb interaction. As, in the static limit, this is repulsive, a superconducting system compensates it by a phase shift in the gap (i.e. in the quasiparticle orbitals), therefore making this repulsion contribute to the condensation (in unconventional superconductors exactly the same happens but directly at the Fermi level). This mechanism is called Coulomb renormalization [@AllenMitrovic1983] since it *renormalizes* the repulsive Coulomb scattering that occurs at low energy. The phase shift occurs at $|\epsilon_{\bf k}|\gtrsim$[$\omega_{\textmd log}$]{} but the scattering processes become less and less important as $|\epsilon_{\bf k}|$ increases (going down as 1/$\epsilon$). Therefore the most important energy region is where the DOS of the [H$_3$S]{} and [H$_3$Se]{} systems shows a dip, implying that the phase space available for this process is small and its effect weak. Note that in order to reproduce the [T$_{\textmd C}$]{} coming from SCDFT within the Allen-Dynes (AD) formula one should assume a [$\mu^*$]{} of 0.16, that is actually much larger than the value of $\mu$ itself ($\simeq$0.1). Making clear that the the Morel-Anderson theory[@MorelAnderson_1962] can not even be applied.
The superconducting pairing is distributed over many phonon modes and over the Brillouin zone in ${\bf q}$-space, despite the presence of several Fermi surface sheets and with different orbital character across the Fermi level, we obtained a isotropic (weakly ${\bf k}$-dependent) gap at the Fermi level and the effect of anisotropy [@SMW_multibandBCS_PRL1959] on [T$_{\textmd C}$]{} is negligible ($<$ 1K).
Conclusions
===========
We have presented a theoretical investigation on the crystal structure and superconductivity of [H$_3$S]{}. An extensive structural search confirms the [H$_3$S]{} stoichiometry as the most stable configuration at high pressure. By means of parameter-free SCDFT we have predicted a [T$_{\textmd C}$]{} of 180K at 200GPa, in excellent agreement with experimental results. This confirms [H$_3$S]{} to be the material with the highest known superconducting critical temperature. The mechanism of superconductivity is clearly the same that was predicted for metallic hydrogen[@Ashcroft_PRL1968; @Cudazzo_PRL2008; @Cudazzo_1_PRB2010]: the combinate effect of high characteristic frequency due to hydrogen light mass and strong coupling due to the lack of electronic core in hydrogen. Still the working pressures of this superconductor is too high for any technological application [@PaulMc_NatHPMat]. Nevertheless the discovery of metallic superconducting hydrogenic bands already at 150GPa gives hope that further theoretical and experimental research in this direction may lead to even lower hydrogen metallization pressures and higher temperature superconductivity. Here we predict that [H$_3$S]{} is stable in the cubic [$Im$3$m$ ]{}already at 100GPa with a very high [T$_{\textmd C}$]{} of 123K, a value which is comparable to the cuprate superconductors.
J.A.F.L. acknowledge fruitful discussion with Maximilian Amsler on crystal prediction and the financial support from EU’s 7th framework Marie-Skłodowska-Curie scholarship program within the “ExMaMa” Project (329386)
[^1]: The phononic functional we use is an improved version with respect to Ref. [@Lueders_SCDFT_PRB2005; @Marques_SCDFT_PRB2005] and is discussed in Ref. [@Sanna_Migdal]. In this work Coulomb interactions are included within static RPA [@Sanna_CaC6_RapCom2007], therefore excluding magnetic source of coupling[@Frank_SF_PRB2014]
[^2]: The decomposition enthalpies have been computed from the predicted structures of hydrogen $P6_3m$ and $C2/c$ [@pickard_structure_2007] and for sulfur and selenium on the $R3m$ and [$Im$3$m$ ]{}reported to occur at high pressure. [@Akahama_S_PRB1993; @Akahama_Se_PRB1993; @Akahama_Se_met_PRB1997; @Shimizu_EHP2005]
[^3]: Conventional implementation of the Eliashberg equations due to their computational cost, usually assume a $k$-independent pairing and a flat density of states. Anisotropic implementations [@Margine_anisoEliashberg_PRB2013] are not intrinsically affected by this limit.
[^4]: We are actually not reporting the Eliashberg result but that coming from the Allen-Dynes (AD) formula. The reason for this choice is that the two approaches agree perfectly (the difference being less than 1K) for the phonon case. But in addition when including [$\mu^*$]{} the AD formula depends only on it, while the Eliashberg equations also depend on the Coulomb frequency cut-off (that changes the meaning of the $*$ in [$\mu^*$]{}. If we want to use a conventional value of [$\mu^*$]{} between 0.1 and 0.15 [@AllenMitrovic1983] it is then better to use the parametrized AD version of the Eliashberg method
|
---
abstract: 'We introduce a new class of mixed finite element methods for $2$D and $3$D compressible nonlinear elasticity. The independent unknowns of these conformal methods are displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor. The so-called edge finite elements of the $\mathrm{curl}$ operator is employed to discretize the trial space of displacement gradients. Motivated by the differential complex of nonlinear elasticity, this choice guarantees that discrete displacement gradients satisfy the Hadamard jump condition for the strain compatibility. We study the stability of the proposed mixed finite element methods by deriving some inf-sup conditions. By considering $32$ choices of simplicial conformal finite elements of degrees $1$ and $2$, we show that $10$ choices are not stable as they do not satisfy the inf-sup conditions. We numerically study the stable choices and conclude that they can achieve optimal convergence rates. By solving several $2$D and $3$D numerical examples, we show that the proposed methods are capable of providing accurate approximations of strain and stress.'
author:
- 'Arzhang Angoshtari[^1]'
- 'Ali Gerami Matin[^2]'
bibliography:
- '/home/arzhang/Dropbox/LaTexBibliography/biblio.bib'
---
Keywords.
: Nonlinear elasticity; mixed finite element methods; inf-sup conditions; differential complex; finite element exterior calculus.
Introduction
============
Although modeling deformations of nonlinearly elastic solids is an old problem [@TrNo65], designing stable computational methods for predicting nonlinear deformations in some modern engineering applications such as electroactive polymers and biological tissues is still a challenging task. A simple strategy is to extend well-performing computational methods of linearized elasticity to nonlinear elasticity, however, it is well-known that due to the occurrence of various unphysical instabilities, such extensions may have a very poor performance [@AuDaLoReTaWr2013; @AuDaLoRe2005].
It was shown that mixed finite element methods provide a useful framework for studying compressible and incompressible nonlinear elasticity [@Wr2009]. Mixed finite element methods involve several independent unknowns and are usually defined as finite element methods which are based on a primal-dual problem or a saddle-point variational problem [@BoBrFo2013]. A different definition of mixed methods is methods that simultaneously approximate an unknown and some of its derivatives [@Od1972], see also [@Ciarlet1978 Chapter 7] for a general classification of finite element methods.
Different mixed methods exist for compressible nonlinear elasticity; For example, two-field methods based on the Hellinger-Reissner principle in terms of displacement and stress [@Wr2009] and three-field methods based on the Hu-Washizu principle in terms of displacement, strain, and stress [@SiAr1992; @AnFSYa2017]. Some potential advantages of mixed finite element methods in nonlinear elasticity include locking free behavior for thin solids and for the (near) incompressible regime, good performance for problems involving large bending, accurate approximations of strains and stresses, insensitivity to mesh distortions, and simple implementation of constitutive equations [@Wr2009; @Re2015]. On the other hand, a disadvantage of mixed methods is that they are computationally more expensive comparing to standard single-field methods as there are more degrees of freedom per element. Another disadvantage of mixed methods is that the well-posedness of the underlying mixed formulation is not necessarily inherited by its discretizations. This aspect of mixed methods is usually studied in the context of inf-sup conditions [@BoBrFo2013].
In this paper, we introduce a new class of conformal mixed finite element methods for $2$D and $3$D compressible nonlinear elasticity based on a three-field formulation in terms of displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor. The main idea is to discretize the trial space of displacement gradients by employing finite elements suitable for the $\mathrm{curl}$ operator. This choice can be readily justified by using a mathematical structure called the differential complex of nonlinear elasticity [@AngoshtariYavari2014I; @AngoshtariYavari2014II] and guarantees that the Hadamard jump condition for the strain compatibility is satisfied on the discrete level as well. A relation between the nonlinear elasticity complex and a well-known complex from differential geometry called the de Rham complex allows one to discretize the former by using finite element spaces suitable for the discretization of the latter. We employ these finite element spaces to derive finite element methods for nonlinear elasticity. We show that even for hyperelastic materials, the underlying weak form does not correspond to a saddle-point problem. However, the resulting finite element methods are still called mixed methods in the sense that displacement and its derivative are approximated simultaneously.
Similar ideas were employed in [@AnFSYa2017] to obtain a class of mixed finite element methods for $2$D compressible nonlinear elasticity. In contrary to the present work, one can show that the underlying weak form of [@AnFSYa2017] is associated to a saddle-point of a Hu-Washizu-type functional for hyperelastic matrerials. Numerical examples suggested that the resulting finite element methods have good features such as optimal convergence rates, good bending performance, accurate approximations of strains and stresses, and the lack of the hourglass instability that may occur in non-conformal enhanced strain mixed methods [@WrRe1996]. However, those mixed methods suffer from at least two drawbacks: On the one hand, only a limited number of finite element choices lead to a stable method and on the other hand, and more importantly, their extension to $3$D problems is hard.
In comparison to [@AnFSYa2017], the present mixed methods work well for both $2$D and $3$D problems and also are stable for broader choices of elements. For example, in [@AnFSYa2017] it was observed that only $7$ out of the $32$ possible choices of first-order and second-order triangular elements lead to stable methods while $22$ choices are stable in the present work. The main difference is among the test spaces of the constitutive relation: While curl-based spaces are used in the previous work, divergence-based spaces are employed in this work. It is not hard to see that in the formulation of [@AnFSYa2017], instead of seeking stresses in divergence-based spaces, they are implicitly sought in the intersection of curl-based and divergence-based spaces. The present formulation does not impose this unphysical restriction on stress.
We employ the general framework of [@PoRa1994; @CaRa1997] for the Galerkin approximation of regular solutions of nonlinear problems to study the stability of the proposed methods. In particular, we write a sufficient inf-sup condition and two other weaker inf-sup conditions. By considering $32$ choices of simplicial finite elements of degrees $1$ and $2$ in $2$D and $3$D, the performance of mixed methods are studied. We show that $10$ choices are not stable as they violate the inf-sup conditions. Our numerical examples suggest that the proposed mixed methods are capable of attaining optimal convergence rates and approximate strains and stresses accurately.
This paper is organized as follows: In Section \[Sec\_MixFEM\], we first briefly review the differential complex of nonlinear elasticity and then we introduce a mixed formulation for nonlinear elasticity. This formulation is then discretized by employing suitable conformal finite element spaces. In Section \[Sec\_Conv\], a convergence analysis for regular solutions is presented and suitable inf-sup conditions are written. Also we rigorously show that some choices of simplicial finite elements do not satisfy these inf-sup conditions. By considering several $2$D and $3$D numerical examples, the performance of the proposed finite element methods is studied in Section \[Sec\_Examples\]. We present a numerical study of the inf-sup conditions as well. Some final remarks will be made in Section \[Sec\_Conc\].
A Class of Mixed Finite Element Methods for Nonlinear Elasticity {#Sec_MixFEM}
================================================================
The mixed finite element methods introduced in this work are closely related to the nonlinear elasticity complex. We begin with a brief description of this complex for $3$D nonlinear elasticity and then, we employ this complex to introduce a mixed formulation for nonlinear elasticity. Mixed finite element methods are then defined by discretizing this mixed formulation by using suitable conformal finite element spaces. We assume $\{\mathrm{X}^{I}\}_{I=1}^{n}$ and $\{\mathbf{E}^{I}\}_{I=1}^{n}$ are respectively the Cartesian coordinates and the standard orthonormal basis of $\mathbb{R}^{n}$, $n=2,3$. Since covariant and contravariant components of tensors are the same in $\{\mathrm{X}^{I}\}_{I=1}^{n}$, we will only use contravariant components of tensors. Also unless stated otherwise, we use the summation convention on repeated indices.
The Nonlinear Elasticity Complex
--------------------------------
Let $\mathcal{B}$ represent the reference configuration of a $3$D elastic body with the boundary $\partial \mathcal{B}$. The unit outward normal vector field of $\partial\mathcal{B}$ is denoted by $\boldsymbol{N}$ and we assume $\partial\mathcal{B}=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_{1}$ and $\Gamma_{2}$ have disjoint interiors. A second-order tensor field $\boldsymbol{T}$ on $\mathcal{B}$ is said to be normal to $\Gamma_{i}$, $i=1,2$, if $\boldsymbol{T}(\mathbf{Y}):=T^{IJ}Y^{J}\mathbf{E}_{I}=\boldsymbol{0}$, for any vector $\mathbf{Y}$ parallel to $\Gamma_{i}$. Similarly, $\boldsymbol{T}$ is said to be parallel to $\Gamma_{i}$ if $\boldsymbol{T}(\mathbf{Y})=\boldsymbol{0}$, for any vector $\mathbf{Y}$ normal to $\Gamma_{i}$.
Given a vector field $\boldsymbol{U}$ and a tensor field $\boldsymbol{T}$, one can define the operators $\mathbf{grad}$, $\mathbf{curl}$, and $\mathbf{div}$ as $$(\mathbf{grad}\,\boldsymbol{U})^{IJ}=\partial_{J}U^{I},~~ (\mathbf{curl}\,\boldsymbol{T})^{IJ}=\varepsilon_{JKL}\partial_{K}T^{IL}, ~~(\mathbf{div}\,\boldsymbol{T})^{I}=\partial_{J}T^{IJ},$$ where “$\partial_{J}$” denotes $\partial/\partial \mathrm{X}^{J}$ and $\varepsilon_{JKL}$ is the standard permutation symbol. Suppose $[H^{1}(\mathcal{B})]^{3}$ is the standard space of $H^{1}$ vector fields on $\mathcal{B}$ (i.e. the space of vector fields such that their components and first derivatives of their components are square integrable) and let $[H^{1}_{i}(\mathcal{B})]^{3}$ be the space of $H^{1}$ vector fields that vanish on $\Gamma_{i}$. By $H^{\mathbf{c}}(\mathcal{B})$, we denote the space of second-order tensor fields $\boldsymbol{T}$ such that both $\boldsymbol{T}$ and $\mathbf{curl}\, \boldsymbol{T}$ are of $L^{2}$-class (i.e. have square integrable components). The space of $H^{\mathbf{c}}$ tensor fields that are normal to $\Gamma_{i}$ is denoted by $H^{\mathbf{c}}_{i}(\mathcal{B})$. Similarly, the space of $L^{2}$ second-order tensor fields with $L^{2}$ divergence is denoted by $H^{\mathbf{d}}(\mathcal{B})$ and $H^{\mathbf{d}}_{i}(\mathcal{B})$ indicates $H^{\mathbf{d}}$ tensor fields that are parallel to $\Gamma_{i}$.
It is possible to define continuous operators $\mathbf{grad}: [H^{1}_{i}(\mathcal{B})]^{3}\rightarrow H^{\mathbf{c}}_{i}(\mathcal{B})$, $\mathbf{curl}: H^{\mathbf{c}}_{i}(\mathcal{B})\rightarrow H^{\mathbf{d}}_{i}(\mathcal{B})$, and $\mathbf{div}: H^{\mathbf{d}}_{i}(\mathcal{B}) \rightarrow [L^{2}(\mathcal{B})]^{3}$ that satisfy the relations $\mathbf{curl}(\mathbf{grad}\,\boldsymbol{Y})=\boldsymbol{0}$, and $\mathbf{div}(\mathbf{curl}\,\boldsymbol{T})=\boldsymbol{0}$. These facts are usually expressed by writing the differential complex $$\label{gcdTenHilb3D}
\scalebox{1}{\xymatrix@C=3.5ex{0 \ar[r] & [H^{1}_{i}(\mathcal{B})]^{3} \ar[r]^-{\mathbf{grad}} &H^{\mathbf{c}}_{i}(\mathcal{B}) \ar[r]^-{\mathbf{curl}} & H^{\mathbf{d}}_{i}(\mathcal{B}) \ar[r]^-{\mathbf{div}} &[L^{2}(\mathcal{B})]^{3} \ar[r] &0. } }$$ The above complex is called the nonlinear elasticity complex as it describes the kinematics and the kinetics of nonlinearly elastic bodies in the following sense [@AngoshtariYavari2014I; @AngoshtariYavari2014II]: Let $\varphi:\mathcal{B}\rightarrow\mathbb{R}^{3}$ be a deformation of $\mathcal{B}$ and let $\boldsymbol{U}$ be the associated displacement field. Then, the displacement gradient is $\boldsymbol{K}:=\mathbf{grad}\,\boldsymbol{U}$, and $\mathbf{curl}\,\boldsymbol{K}=\boldsymbol{0}$, is the necessary condition for the compatibility of $\boldsymbol{K}$. On the other hand, $\mathbf{div}\,\boldsymbol{P}=\boldsymbol{0}$, is the equilibrium equation in terms of the first Piola-Kirchhoff stress tensor $\boldsymbol{P}$. This equation is also the necessary condition for the existence of a stress function $\boldsymbol{\Psi}$ such that $\boldsymbol{P}=\mathbf{curl}\,\boldsymbol{\Psi}$. By considering the $2$D curl operator $(\mathbf{curl}\,\boldsymbol{T})^{I}=\partial_{1}T^{I2}-\partial_{2}T^{I1}$, one can also write similar results for $2$D nonlinear elasticity. One can show that provides a connection between solutions of certain partial differential equations and the topologies of $\mathcal{B}$ and $\Gamma_{i}$.
A Three-Field Mixed Formulation
-------------------------------
Motivated by the complex , we write a mixed formulation for nonlinear elasticity in terms of the displacement $\boldsymbol{U}$, the displacement gradient $\boldsymbol{K}$, and the first Piola-Kirchhoff stress tensor $\boldsymbol{P}$. Let $\boldsymbol{P}=\mathbb{P}(\boldsymbol{K})$ express the constitutive equation of the elastic body $\mathcal{B}\subset\mathbb{R}^{n}$, $n=2,3$. The boundary value problem of nonlinear elastostatics can be written as: Given a body force $\boldsymbol{B}$, a displacement $\overline{\boldsymbol{U}}$ of $\mathcal{B}$, and a traction vector field $\overline{\boldsymbol{T}}$ on $\Gamma_{2}$, find $(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})$ such that
\[NonLinStrong\]
[rlrll]{} & = - , & && \[NonLinStrong1\]\
&-=, & & & , \[NonLinStrong2\]\
& -()=, & & & \[NonLinStrong3\]\
& =, & & & \_[1]{}, \[NonLinStrong4\]\
& ()=, & & & \_[2]{}. \[NonLinStrong5\]
To write a weak formulation for the above problem, we proceed as follows: Let “$\boldsymbol{\cdot}$” denote the standard inner product of $\mathbb{R}^{n}$ and let $\llangle,\rrangle$ denote both the $L^{2}$-inner product of vector fields $\llangle\boldsymbol{Y},\boldsymbol{Z}\rrangle:=\int_{\mathcal{B}}Y^{I}Z^{I}dV$, and the $L^{2}$-inner product of tensor fields $\llangle\boldsymbol{S},\boldsymbol{T}\rrangle:=\int_{\mathcal{B}}S^{IJ}T^{IJ}dV$. By taking the $L^{2}$-inner product of with an arbitrary $\boldsymbol{\Upsilon}\in [H^{1}_{1}(\mathcal{B})]^{n}$ and using Green’s formula $$\llangle \boldsymbol{P},\mathbf{grad}\,\boldsymbol{\Upsilon} \rrangle = - \llangle \mathbf{div}\,\boldsymbol{P},\boldsymbol{\Upsilon} \rrangle + \int_{\partial\mathcal{B}} \boldsymbol{P}(\boldsymbol{N})\boldsymbol{\cdot} \boldsymbol{\Upsilon} dA,$$ one concludes that $$\llangle \boldsymbol{P},\mathbf{grad}\,\boldsymbol{\Upsilon} \rrangle = \llangle \boldsymbol{B},\boldsymbol{\Upsilon} \rrangle + \int_{\Gamma_{2}} \boldsymbol{P}(\boldsymbol{N})\boldsymbol{\cdot} \boldsymbol{\Upsilon} dA.$$ We also take the $L^{2}$-inner product of and with arbitrary $\boldsymbol{\lambda}$ of $H^{\mathbf{c}}$-class and arbitrary $\boldsymbol{\pi}$ of $H^{\mathbf{d}}$ class and obtain the following mixed formulation for nonlinear elastostatics:
[*Given a body force $\boldsymbol{B}$, a displacement $\overline{\boldsymbol{U}}$ of $\mathcal{B}$, and a boundary traction vector field $\overline{\boldsymbol{T}}$ on $\Gamma_{2}$, find $(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})\in [H^{1}(\mathcal{B})]^{n}\times H^{\mathbf{c}}(\mathcal{B}) \times H^{\mathbf{d}}(\mathcal{B})$ such that $\boldsymbol{U}= \overline{\boldsymbol{U}}$, on $\Gamma_1$ and* ]{} $$\label{3DElasMixF}
\begin{alignedat}{3}
\llangle\boldsymbol{P},\mathbf{grad}\,\boldsymbol{\Upsilon}\rrangle&=\llangle \boldsymbol{B},\boldsymbol{\Upsilon} \rrangle + \int_{\Gamma_{2}}\overline{\boldsymbol{T}}\boldsymbol{\cdot}\boldsymbol{\Upsilon} dA, &\quad &\forall \boldsymbol{\Upsilon}\in [H^{1}_{1}(\mathcal{B})]^{n},\\
\llangle\mathbf{grad}\,\boldsymbol{U},\boldsymbol{\lambda}\rrangle - \llangle \boldsymbol{K},\boldsymbol{\lambda}\rrangle &= 0,& &\forall \boldsymbol{\lambda}\in H^{\mathbf{c}}(\mathcal{B}),\\
\llangle \mathbb{P}(\boldsymbol{K}),\boldsymbol{\pi}\rrangle - \llangle \boldsymbol{P}, \boldsymbol{\pi} \rrangle &= 0, & &\forall \boldsymbol{\pi}\in H^{\mathbf{d}}(\mathcal{B}).
\end{alignedat}$$
The mixed formulation is different from that of [@AnFSYa2017 Equation (2.8)]: Here, test functions associated to the definition of the displacement gradient and the constitutive relation are respectively of classes $H^{\mathbf{c}}$ and $H^{\mathbf{d}}$. In [@AnFSYa2017], $H^\mathbf{c}$ test functions are employed for the constitutive relation and $H^{\mathbf{d}}$ test functions for the definition of the displacement gradient. Later we will show that the mixed formulation of [@AnFSYa2017] imposes an unphysical constraint on stresses.
For hyperelastic materials, the mixed formulation of [@AnFSYa2017] is a saddle-point problem associated to a Hu-Washizu-type functional. However, the mixed formulation does *not* correspond to a stationary point of any functional $J:Z\rightarrow \mathbb{R}$ with $Z=[H^{1}(\mathcal{B})]^{n}\times H^{\mathbf{c}}(\mathcal{B}) \times H^{\mathbf{d}}(\mathcal{B})$. To show this, let $u,v,w\in Z$, where $u=(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})$, $v=(\boldsymbol{\Upsilon},\boldsymbol{\lambda},\boldsymbol{\pi})$, $w=(\boldsymbol{V},\boldsymbol{M},\boldsymbol{Q})$, and notice that the problem can be written as: Find $u\in Z$ such that $G(u,v)=0$, $\forall v\in Z$, where $$\begin{alignedat}{3}
G(u,v) &= \llangle\boldsymbol{P},\mathbf{grad}\,\boldsymbol{\Upsilon}\rrangle + \llangle\mathbf{grad}\,\boldsymbol{U},\boldsymbol{\lambda}\rrangle - \llangle \boldsymbol{K},\boldsymbol{\lambda}\rrangle\\
&+ \llangle \mathbb{P}(\boldsymbol{K}),\boldsymbol{\pi}\rrangle - \llangle \boldsymbol{P}, \boldsymbol{\pi} \rrangle - \llangle \boldsymbol{B},\boldsymbol{\Upsilon} \rrangle - \int_{\Gamma_{2}}\overline{\boldsymbol{T}}\boldsymbol{\cdot}\boldsymbol{\Upsilon} dA.
\end{alignedat}$$ If corresponds to a stationary point of $J:Z\rightarrow \mathbb{R}$, then $G(u,v)=\mathrm{D}J(u)v$, where $\mathrm{D}J(u)v$ is the (Fréchet) derivative of $J$ at $u$ in the direction of $v$. Since the second derivative of $J$ has the symmetry $\mathrm{D}^{2}J(u)(v,w)=\mathrm{D}^{2}J(u)(w,v)$ [@Av1986], the first derivative of $G$ should satisfy $\mathrm{D}_{1}G(u,v)w=\mathrm{D}_{1}G(u,w)v$, with $$\mathrm{D}_{1}G(u,v)w = \llangle\boldsymbol{Q},\mathbf{grad}\,\boldsymbol{\Upsilon}\rrangle + \llangle\mathbf{grad}\,\boldsymbol{V},\boldsymbol{\lambda}\rrangle - \llangle \boldsymbol{M},\boldsymbol{\lambda}\rrangle + \llangle \mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M},\boldsymbol{\pi}\rrangle - \llangle \boldsymbol{Q}, \boldsymbol{\pi} \rrangle,$$ where $\mathsf{A}(\boldsymbol{K})$ is the elasticity tensor in terms of the displacement gradient and $(\mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M})^{IJ}:=A^{IJRS}M^{RS}$. However, it is easy to check that $\mathrm{D}_{1}G(u,v)w\neq\mathrm{D}_{1}G(u,w)v$, and therefore, the formulation does not correspond to a saddle-point of any functional. Despite this fact, we still call a finite element method based on a mixed method in the general sense that displacement and its derivative are independent unknowns of this formulation; See the discussion of [@Ciarlet1978 Page 417] regarding definitions of mixed methods.
\[NotWellDefined\] In general, the response function $\mathbb{P}$ of a nonlinearly elastic body $\mathcal{B}$ is not a well-defined mapping $\mathbb{P}:H^{\mathbf{c}}(\mathcal{B})\rightarrow H^{\mathbf{d}}(\mathcal{B})$, i.e. $\boldsymbol{K}\in H^{\mathbf{c}}(\mathcal{B})$ does not necessarily imply that $\mathbb{P}(\boldsymbol{K})\in H^{\mathbf{d}}(\mathcal{B})$ [@An2018]. Roughly speaking, this means that it is impossible to induce arbitrary continuous deformations in nonlinearly elastic bodies by using external loads. To study the well-posedness of the problem by using approaches based on the implicit function theorem, it is sufficient to assume that the restriction $\mathbb{P}:O\rightarrow [L^{2}(\mathcal{B})]^{3}$ is a well-defined mapping, where $O$ is an open subset of $H^{\mathbf{c}}(\mathcal{B})$.
Mixed Finite Element Methods {#Sec_CSFEM}
----------------------------
The complex has a close relation with a well-known complex from differential geometry called the de Rham complex [@AngoshtariYavari2014II]. One can employ this relation to obtain conformal mixed finite element methods for approximating solutions of as follows. It was shown that the finite element exterior calculus (FEEC) provides a systematic method for discretizing the de Rham complex using finite element spaces [@Arnold2010; @Arnold2006]. The relation between and the de Rham complex allows one to obtain $H^{\mathbf{c}}$- and $H^{\mathbf{d}}$-conformal finite element spaces by using FEEC. For example, to obtain $H^{\mathbf{c}}$-conformal finite element spaces over a $3$D body $\mathcal{B}$, we proceed as follows: Let the (row) vector field $\boldsymbol{K}_{I}=(K^{I1},K^{I2},K^{I3})$ denote the $I$-th row of the displacement gradient $\boldsymbol{K}$. One can write $$\mathbf{curl}\,\boldsymbol{K}=\mathbf{curl}\left[ \begin{array}{c} \boldsymbol{K}_{1} \\ \boldsymbol{K}_{2} \\ \boldsymbol{K}_{3} \end{array}\right] = \left[ \begin{array}{c} \mathrm{curl}\,\boldsymbol{K}_{1} \\ \mathrm{curl}\,\boldsymbol{K}_{2} \\ \mathrm{curl}\,\boldsymbol{K}_{3} \end{array}\right],$$ where $\mathrm{curl}$ is the standard curl operator of vector fields. Consequently, $H^{\mathbf{c}}(\mathcal{B})$ can be identified with three copies of the standard curl space $H^{c}(\mathcal{B})$ for vector fields, i.e. $H^{\mathbf{c}}(\mathcal{B})=[H^{c}(\mathcal{B})]^{3}$. On the other hand, the space $H^{c}(\mathcal{B})$ of vector fields can be identified with a space of differential $1$-forms, which can be discretized using FEEC. Thus, conformal finite element spaces for $H^{\mathbf{c}}(\mathcal{B})$ can be obtained by using three copies of conformal finite element spaces of differential $1$-forms. Similarly, since $H^{\mathbf{d}}(\mathcal{B})=[H^{d}(\mathcal{B})]^{3}$, where $H^{d}(\mathcal{B})$ is the divergence space of vector fields, one can obtain conformal finite element spaces for $H^{\mathbf{d}}(\mathcal{B})$ by using three copies of conformal finite element spaces for $2$-forms. In $2$D, by noting that $H^{\mathbf{c}}(\mathcal{B})=[H^{c}(\mathcal{B})]^{2}$ and $H^{\mathbf{d}}(\mathcal{B})=[H^{d}(\mathcal{B})]^{2}$, one can obtain conformal finite element spaces for $H^{\mathbf{c}}(\mathcal{B})$ and $H^{\mathbf{d}}(\mathcal{B})$ by using two copies of conformal finite element spaces for $1$-forms.
![Conventional finite element diagrams of the first and the second degree $H^{1}$, $H^{c}$, and $H^{d}$ elements on triangles and tetrahedra. In this figure, $\mathrm{LE}_i$ stands for the Lagrange element of degree $i$, $\mathrm{NED}^{j}_i$ stands for the $i$-th degree Nédélec element of the $j$-th kind, $\mathrm{RT}_i$ stands for the Raviart-Thomas element of degree $i$, and $\mathrm{BDM}_i$ stands for the Brezzi-Douglas-Marini element of degree $i$. Arrows parallel (normal) to an edge or a face denote degrees of freedom associated to tangent (normal) components of vector fields along that edge or face. Only degrees of freedom associated to visible edges and faces are shown.[]{data-label="FEO12"}](FEO12.jpg)
Let $\mathcal{B}$ be a polyhedral domain with a simplicial mesh $\mathcal{B}_{h}$, i.e. $\mathcal{B}_{h}$ is a triangular mesh in $2$D and a tetrahedral mesh in $3$D. The above discussion implies that one can associate a tensorial finite element space $V^{\mathbf{c}}_{h}:=[V^{c}_{h}]^{n}\subset H^{\mathbf{c}}(\mathcal{B})$ to any vectorial finite element space $V^{c}_{h}\subset H^{c}(\mathcal{B})$ with $\dim V^{\mathbf{c}}_{h}=n\dim V^{c}_{h}$. Similarly, for any $H^{d}$-conformal finite element space $V^{d}_{h}\subset H^{d}(\mathcal{B})$, one obtains $H^{\mathbf{d}}$-conformal finite element space $V^{\mathbf{d}}_{h}:=[V^{d}_{h}]^{n}\subset H^{\mathbf{d}}(\mathcal{B})$ with $\dim V^{\mathbf{d}}_{h}=n\dim V^{d}_{h}$. FEEC provides a systematic approach for obtaining finite element spaces $V^{c}_{h}$ and $V^{d}_{h}$ of arbitrary order. For example, Figure \[FEO12\] shows conventional finite element diagrams of some $H^{1}$-, $H^{c}$-, and $H^{d}$-conformal elements of degrees 1 and 2. Notice that for $H^{c}$ elements, some degrees of freedom are associated to tangent components of vectors fields along faces and edges, whereas degrees of freedom of $H^{d}$ elements are associated to normal components of vector fields along faces; See [@loMaWe2012 Chapter 3] for more details about these elements.
Let $[V^{1}_{h}]^{n}$, $V^{\mathbf{c}}_{h}$, $V^{\mathbf{d}}_{h}$ be finite element spaces as described above and let $V^{1}_{h,i}=V^{1}_{h}\cap H^{1}_{i}(\mathcal{B})$. Also suppose $\mathcal{I}^{1}_{h}$ is the canonical interpolation operators associated to the $H^{1}$ elements. Then, we consider the following mixed finite element methods for :
[*Given a body force $\boldsymbol{B}$, a displacement $\overline{\boldsymbol{U}}$, and a boundary traction vector field $\overline{\boldsymbol{T}}$ on $\Gamma_{2}$, find $(\boldsymbol{U}_{h},\boldsymbol{K}_{h},\boldsymbol{P}_{h})\in [V^{1}_{h}]^{n}\times V^{\mathbf{c}}_{h} \times V^{\mathbf{d}}_{h}$ such that $\boldsymbol{U}_{h}= \mathcal{I}^{1}_{h}(\overline{\boldsymbol{U}})$, on $\Gamma_1$ and* ]{}
\[CSFEM23\] $$\begin{aligned}
{3}
\llangle\boldsymbol{P}_{h},\mathbf{grad}\,\boldsymbol{\Upsilon}_{h}\rrangle&=\llangle \boldsymbol{B},\boldsymbol{\Upsilon}_{h} \rrangle + \int_{\Gamma_{2}}\overline{\boldsymbol{T}}\boldsymbol{\cdot}\boldsymbol{\Upsilon}_{h} dA, &\quad &\forall \boldsymbol{\Upsilon}_{h}\in [V^{1}_{h,1}]^{n}, \label{CSFEM23_1}\\
\llangle\mathbf{grad}\,\boldsymbol{U}_{h},\boldsymbol{\lambda}_{h}\rrangle - \llangle \boldsymbol{K}_{h},\boldsymbol{\lambda}_{h}\rrangle &= 0,& &\forall \boldsymbol{\lambda}_{h}\in V^{\mathbf{c}}_{h}, \label{CSFEM23_2}\\
\llangle \mathbb{P}(\boldsymbol{K}_{h}),\boldsymbol{\pi}_{h}\rrangle - \llangle \boldsymbol{P}_{h}, \boldsymbol{\pi}_{h} \rrangle &= 0, & &\forall \boldsymbol{\pi}_{h}\in V^{\mathbf{d}}_{h}. \label{CSFEM23_3}\end{aligned}$$
As mentioned earlier in Remark \[NotWellDefined\], generally speaking, the response function $\mathbb{P}$ is not well-defined as a mapping $H^{\mathbf{c}}(\mathcal{B})\rightarrow H^{\mathbf{d}}(\mathcal{B})$. By considering simple piecewise polynomial deformation gradients, it is easy to see that $\mathbb{P}$ is not necessarily well-defined as a mapping $V^{\mathbf{c}}_{h}\rightarrow V^{\mathbf{d}}_{h}$ as well; For example, see [@An2018 Section 4]. Thus, in general, we have $\mathbb{P}(\boldsymbol{K}_{h})\notin V^{\mathbf{d}}_{h}$ in . This equation simply defines the approximate stress $\boldsymbol{P}_{h}$ as the unique $L^{2}$-orthogonal projection of $\mathbb{P}(\boldsymbol{K}_{h})$ on $V^{\mathbf{d}}_{h}$. In the mixed methods introduced in [@AnFSYa2017 Section 3.2], $\boldsymbol{P}_{h}\in V^{\mathbf{d}}_{h}$ is the $L^{2}$-orthogonal projection of $\mathbb{P}(\boldsymbol{K}_{h})$ on $V^{\mathbf{c}}_{h}$, which means that $\boldsymbol{P}_{h}\in V^{\mathbf{c}}_{h}\cap V^{\mathbf{d}}_{h}$, and therefore, unlike members of $V^{\mathbf{d}}_{h}$ which can be discontinuous along internal faces of $\mathcal{B}_{h}$, $\boldsymbol{P}_{h}$ is forced to be continuous on $\mathcal{B}_{h}$. The implicit assumption $\boldsymbol{P}_{h}\in V^{\mathbf{c}}_{h}$ is unphysical and severely restricts the solution space of $\boldsymbol{P}_{h}$. As a consequence, it was observed that the extension of the finite element method of [@AnFSYa2017] to the $3$D case is very challenging.
In , the approximate displacement gradient $\boldsymbol{K}_{h}\in V^{\mathbf{c}}_{h}$ is defined as the unique $L^{2}$-orthogonal projection of $\mathbf{grad}\,\boldsymbol{U}_{h}$ on $V^{\mathbf{c}}_{h}$, with $\boldsymbol{K}_{h}\neq\mathbf{grad}\,\boldsymbol{U}_{h}$, in general. The relation between the complex and the de Rham complex allows to discretize by using FEEC. In particular, the discrete de Rham complexes introduced in [@Arnold2010 Section 5.1] implies that can be discretized as $$\label{gcdDis3D}
\scalebox{1}{\xymatrix@C=3.5ex{0 \ar[r] & [V^{1}_{h}]^{3} \ar[r]^-{\mathbf{grad}} &V^{\mathbf{c}}_{h} \ar[r]^-{\mathbf{curl}} & V^{\mathbf{d}}_{h} \ar[r]^-{\mathbf{div}} &[V_{h}]^{3} \ar[r] &0, } }$$ where the finite element spaces $(V^{1}_{h},V^{\mathbf{c}}_{h},V^{\mathbf{d}}_{h},V_{h})$ are associated to one of the following choices of finite elements: $$\begin{aligned}
{5}
\big(\mathrm{LE}_{i},&\,\mathrm{NED}^{2}_{i-1}&&,\mathrm{BDM}_{i-2}&,\,\mathrm{DE}_{i-3}\big), ~ i\geq3, \\
\big(\mathrm{LE}_{i},&\,\mathrm{NED}^{2}_{i-1}&&,\mathrm{RT}_{i-1}&,\,\mathrm{DE}_{i-2}\big), ~ i\geq2,\\
\big(\mathrm{LE}_{i},&\,\mathrm{NED}^{1}_{i}&&,\mathrm{BDM}_{i-1}&,\,\mathrm{DE}_{i-2}\big),~ i\geq2,\\
\big(\mathrm{LE}_{i},&\,\mathrm{NED}^{1}_{i}&&,\mathrm{RT}_{i}&,\,\mathrm{DE}_{i-1}\big), ~ i\geq1,\end{aligned}$$ where $\mathrm{LE}_i$ is the Lagrange element of degree $i$, $\mathrm{NED}^{j}_{i}$ is the $i$-th degree Nédélec element of the $j$-th kind [@Ned1986], $\mathrm{BDM}_{i}$ is the Brezzi-Douglas-Marini element of degree $i$ [@BDM1985], $\mathrm{RT}_{i}$ is the Raviart-Thomas element of degree $i$ [@RT1977], and $\mathrm{DE}_i$ is the discontinuous element of degree $i$, see Figure \[FEO12\]. Thus, if the finite element spaces $(V^{1}_{h},V^{\mathbf{c}}_{h})$ are induced by $(\mathrm{LE}_{i},\mathrm{NED}^{1}_{i})$ or $(\mathrm{LE}_{i+1},\mathrm{NED}^{2}_{i})$, $i\geq1$, the mapping $\mathbf{grad}: [V^{1}_{h}]^{n}\rightarrow V^{\mathbf{c}}_{h}$ will be well-defined. For these choices of finite element spaces, we have $\boldsymbol{K}_{h}=\mathbf{grad}\,\boldsymbol{U}_{h}$ as $\mathbf{grad}\,\boldsymbol{K}_{h}\in V^{\mathbf{c}}_{h}$ for $\boldsymbol{K}_{h}\in[V^{1}_{h}]^{n}$, i.e. the projection of $\mathbf{grad}\,\boldsymbol{U}_{h}\in V^{\mathbf{c}}_{h}$ on $V^{\mathbf{c}}_{h}$ is equal to itself.
Stability Analysis {#Sec_Conv}
==================
We employ the general theory introduced in [@PoRa1994; @CaRa1997] for the Galerkin approximation of nonlinear problems to study the convergence of solutions of to regular solutions of the problem . This theory is summarized in the Appendix. In particular, we write a sufficient inf-sup condition and two other weaker inf-sup conditions. The former condition is a necessary and sufficient condition for the uniqueness of solutions of the linearization of . We mention a computational framework for studying these inf-sup conditions as well and rigorously show that certain choices of finite elements violate these inf-sup conditions. The following analysis is not valid for singular solutions, which may be studied based on the general approximation framework of [@BrRaRa1981].
A Sufficient Stability Condition
--------------------------------
For simplicity and without loss of generality, we assume that $\overline{\boldsymbol{U}}=0$ in . To apply the theory of [@PoRa1994; @CaRa1997], we write the problem in the abstract form as follows: Let $Z=[H^{1}_{1}(\mathcal{B})]^{n}\times H^{\mathbf{c}}(\mathcal{B}) \times H^{\mathbf{d}}(\mathcal{B})$, and let $u,y,z\in Z$, where $u=(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})$, $y=(\boldsymbol{\Upsilon},\boldsymbol{\lambda},\boldsymbol{\pi})$, $z=(\boldsymbol{V},\boldsymbol{M},\boldsymbol{Q})$. Then, can be stated as: Find $u\in Z$ such that $$\label{Abs_MixProblem}
\begin{alignedat}{3}
\langle H(u), y \rangle &= \llangle\boldsymbol{P},\mathbf{grad}\,\boldsymbol{\Upsilon}\rrangle + \llangle\mathbf{grad}\,\boldsymbol{U},\boldsymbol{\lambda}\rrangle - \llangle \boldsymbol{K},\boldsymbol{\lambda}\rrangle\\
&+ \llangle \mathbb{P}(\boldsymbol{K}),\boldsymbol{\pi}\rrangle - \llangle \boldsymbol{P}, \boldsymbol{\pi} \rrangle - \llangle \boldsymbol{B},\boldsymbol{\Upsilon} \rrangle - \int_{\Gamma_{2}}\overline{\boldsymbol{T}}\boldsymbol{\cdot}\boldsymbol{\Upsilon} dA = 0, \quad \forall y \in Z.
\end{alignedat}$$ To write an inf-sup condition for the stability of approximations of the above problem, we consider the bilinear form $$\label{BiLinCSFEM}
b(z, y) = \llangle\boldsymbol{Q},\mathbf{grad}\,\boldsymbol{\Upsilon}\rrangle + \llangle\mathbf{grad}\,\boldsymbol{V},\boldsymbol{\lambda}\rrangle - \llangle \boldsymbol{M},\boldsymbol{\lambda}\rrangle + \llangle \mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M},\boldsymbol{\pi}\rrangle - \llangle \boldsymbol{Q}, \boldsymbol{\pi} \rrangle, \quad \forall z,y \in Z.$$ where $\mathsf{A}(\boldsymbol{K})$ is the elasticity tensor in terms of the displacement gradient and $(\mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M})^{IJ}:=A^{IJRS}M^{RS}$. This bilinear form is the derivative of the mapping $H$ in , see .
Suppose $Z_{h}:=[V^{1}_{h,1}]^{n}\times V^{\mathbf{c}}_{h} \times V^{\mathbf{d}}_{h}$, where the finite element spaces $V^{1}_{h,1}$, $V^{\mathbf{c}}_{h}$, and $V^{\mathbf{d}}_{h}$ were introduced in Section \[Sec\_CSFEM\]. Since $Z_{h}$ is a $Z$-conformal finite element space with the approximability property, the conditions (i) and (ii) of the abstract theory of the Appendix are satisfied and therefore, close to a regular solution $u=(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})$ of , the discrete problem has a unique solution $u_{h}=(\boldsymbol{U}_{h},\boldsymbol{K}_{h},\boldsymbol{P}_{h})$ that converges to $u$ as $h\rightarrow 0$ if the inf-sup condition holds, that is, if there exists a mesh-independent number $\beta>0$ such that $$\label{CSFEM-infsup}
\underset{y_{h}\in Z_{h}}{\inf}\, \underset{z_{h}\in Z_{h}}{\sup} \frac{b(z_{h},y_{h})}{\|z_{h}\|_{Z} \|y_{h}\|_{Z} } \geq \beta>0,$$ where $y_{h}=(\boldsymbol{\Upsilon}_{h},\boldsymbol{\lambda}_{h},\boldsymbol{\pi}_{h})\in Z_{h}$, $z_{h}=(\boldsymbol{V}_{h},\boldsymbol{M}_{h},\boldsymbol{Q}_{h})\in Z_{h}$, the bilinear form $b(z_{h},y_{h})$ is given in , and $\|z_{h}\|^{2}_{Z}=\|\boldsymbol{V}_{h}\|^{2}_{1}+\|\boldsymbol{M}_{h}\|^{2}_{\mathbf{c}}+\|\boldsymbol{Q}_{h}\|^{2}_{\mathbf{d}}$, with $$\begin{aligned}
\|\boldsymbol{V}_{h}\|^{2}_{1} &= \llangle \boldsymbol{V}_{h}, \boldsymbol{V}_{h} \rrangle + \llangle \mathbf{grad}\,\boldsymbol{V}_{h}, \mathbf{grad}\,\boldsymbol{V}_{h} \rrangle,\\
\|\boldsymbol{M}_{h}\|^{2}_{\mathbf{c}} &= \llangle \boldsymbol{M}_{h}, \boldsymbol{M}_{h} \rrangle + \llangle \mathbf{curl}\,\boldsymbol{M}_{h}, \mathbf{curl}\,\boldsymbol{M}_{h} \rrangle,\\
\|\boldsymbol{Q}_{h}\|^{2}_{\mathbf{d}} &= \llangle \boldsymbol{Q}_{h}, \boldsymbol{Q}_{h} \rrangle + \llangle \mathbf{div}\,\boldsymbol{Q}_{h}, \mathbf{div}\,\boldsymbol{Q}_{h} \rrangle.\end{aligned}$$ Notice that depends on the material properties. If the abstract inf-sup condition of the Appendix holds, then the discrete linear system has a unique solution for any given data. Using the bilinear form , this linear system reads: Given $\boldsymbol{f}^{1}$, $\boldsymbol{f}^{\mathbf{c}}$, and $\boldsymbol{f}^{\mathbf{d}}$ of $L^{2}$-class, find $(\boldsymbol{Y}_{\!\!h},\boldsymbol{M}_{h},\boldsymbol{Q}_{h})\in Z_{h}$ such that
\[LinCSFEM23\] $$\begin{aligned}
{3}
\llangle\boldsymbol{Q}_{h},\mathbf{grad}\,\boldsymbol{\Upsilon}_{h}\rrangle&= \llangle \boldsymbol{f}^{1},\boldsymbol{\Upsilon}_{h}\rrangle, &\quad &\forall \boldsymbol{\Upsilon}_{h}\in [V^{1}_{h,1}]^{n}, \label{LinCSFEM23_I}\\
\llangle\mathbf{grad}\,\boldsymbol{Y}_{\!\!h},\boldsymbol{\lambda}_{h}\rrangle - \llangle \boldsymbol{M}_{h},\boldsymbol{\lambda}_{h}\rrangle &= \llangle \boldsymbol{f}^{\mathbf{c}},\boldsymbol{\lambda}_{h}\rrangle, & &\forall \boldsymbol{\lambda}_{h}\in V^{\mathbf{c}}_{h}, \label{LinCSFEM23_II}\\
\llangle \mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M}_{h},\boldsymbol{\pi}_{h}\rrangle - \llangle \boldsymbol{Q}_{h}, \boldsymbol{\pi}_{h} \rrangle &= \llangle \boldsymbol{f}^{\mathbf{d}},\boldsymbol{\pi}_{h}\rrangle, & &\forall \boldsymbol{\pi}_{h}\in V^{\mathbf{d}}_{h}. \label{LinCSFEM23_III}\end{aligned}$$
Thus, if the material-dependent inf-sup condition holds, the linear system will have a unique solution for any input data $\boldsymbol{f}^{1}$, $\boldsymbol{f}^{\mathbf{c}}$, and $\boldsymbol{f}^{\mathbf{d}}$.
Following the computational framework discussed in the Appendix, to computationally investigate the inf-sup condition , we write its matrix form. Given a mesh $\mathcal{B}_{h}$ of the body $\mathcal{B}$, let $\{\boldsymbol{\Psi}_{i}\}_{i=1}^{n_{1}}$, $\{\boldsymbol{\Lambda}_{i}\}_{i=1}^{n_{\mathbf{c}}}$, and $\{\boldsymbol{\Phi}_{i}\}_{i=1}^{n_{\mathbf{d}}}$ be respectively the global shape functions of $[V^{1}_{h,1}]^{n}$, $V^{\mathbf{c}}_{h}$, and $V^{\mathbf{d}}_{h}$ and let $n_{t}=n_{1}+n_{\mathbf{c}}+n_{\mathbf{d}}$ denote the total number of degrees of freedom. Using the relations $$\boldsymbol{Y}_{\!\!h}=\sum_{j=1}^{n_{1}}y_{j}\boldsymbol{\Psi}_{j},\quad \boldsymbol{M}_{h}=\sum_{j=1}^{n_{\mathbf{c}}}m_{j}\boldsymbol{\Lambda}_{j}, \text{ and } \boldsymbol{Q}_{h}=\sum_{j=1}^{n_{\mathbf{d}}}q_{j}\boldsymbol{\Phi}_{j},$$ one can write in the matrix form $$\label{LinSysCSFEM}
\mathbb{S}_{n_{t}\times n_{t}}\cdot\mathbf{z}_{n_{t}\times 1}=\mathbf{f}_{n_{t}\times 1},$$ with $$\label{CoefDef}
\mathbb{S}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c;{2pt/2pt}c;{2pt/2pt}c} \mathbf{0} & \mathbf{0} & \mathbb{S}^{1\mathbf{d}}_{n_{1}\times n_{\mathbf{d}}} \\ \hdashline[2pt/2pt]
\mathbb{S}^{\mathbf{c}1}_{n_{\mathbf{c}}\times n_{1}} & \mathbb{S}^{\mathbf{c}\mathbf{c}}_{n_{\mathbf{c}}\times n_{\mathbf{c}}} &\mathbf{0} \\ \hdashline[2pt/2pt]
\mathbf{0} & \mathbb{S}^{\mathbf{d}\mathbf{c}}_{n_{\mathbf{d}}\times n_{\mathbf{c}}} & \mathbb{S}^{\mathbf{d}\mathbf{d}}_{n_{\mathbf{d}}\times n_{\mathbf{d}}}\end{array} \right], \quad
\mathbf{z}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c} \mathbf{y}_{n_{1}\times1} \\ \mathbf{m}_{n_{\mathbf{c}}\times1} \\ \mathbf{q}_{n_\mathbf{d}\times1} \end{array} \right], \text{ and }
\mathbf{f}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c} \mathbf{f}^{1}_{n_{1}\times1} \\ \mathbf{f}^{\mathbf{c}}_{n_{\mathbf{c}}\times1} \\ \mathbf{f}^{\mathbf{d}}_{n_\mathbf{d}\times1} \end{array} \right],$$ where the components of the above matrices and vectors are given by $$\begin{aligned}
{5}
& \mathbb{S}^{1\mathbf{d}}_{ij}=\llangle \boldsymbol{\Phi}_{j},\mathbf{grad}\,\boldsymbol{\Psi}_{i}\rrangle, &~~&\mathbb{S}^{\mathbf{c}1}_{ij}=\llangle \mathbf{grad}\,\boldsymbol{\Psi}_{j},\boldsymbol{\Lambda}_{i}\rrangle, &~~&\mathbb{S}^{\mathbf{c}\mathbf{c}}_{ij}= -\llangle \boldsymbol{\Lambda}_{i},\boldsymbol{\Lambda}_{j}\rrangle,\\
& \mathbb{S}^{\mathbf{d}\mathbf{c}}_{ij}= \llangle \mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{\Lambda}_{j},\boldsymbol{\Phi}_{i}\rrangle, && \mathbb{S}^{\mathbf{d}\mathbf{d}}_{ij}= -\llangle \boldsymbol{\Phi}_{i},\boldsymbol{\Phi}_{j}\rrangle, && \\
& \mathbf{y}_{i}= y_{i}, && \mathbf{m}_{i}=m_{i}, &&\mathbf{q}_{i}=q_{i}, \\
&\mathbf{f}^{1}_{i}=\llangle \mathbf{f}^{1}, \boldsymbol{\Psi}_{i} \rrangle, && \mathbf{f}^{\mathbf{c}}_{i}=\llangle \mathbf{f}^{\mathbf{c}}, \boldsymbol{\Lambda}_{i} \rrangle, &&\mathbf{f}^{\mathbf{d}}_{i}= \llangle \mathbf{f}^{\mathbf{d}}, \boldsymbol{\Phi}_{i} \rrangle. \end{aligned}$$ By replacing the matrix $\mathbb{B}$ of the inf-sup condition with $\mathbb{S}$, one obtains the matrix form of as $$\label{infsupCondMat_CSFEM}
\underset{\mathbf{w}\in\mathbb{R}^{n_{t}}}{\inf}\, \underset{\mathbf{z}\in\mathbb{R}^{n_{t}}}{\sup} \frac{\mathbf{w}^{T}\mathbb{S}\, \mathbf{z}}{\|\mathbf{w}\|_{Z} \|\mathbf{z}\|_{Z}} \geq \beta>0,$$ with $\|\mathbf{z}\|^{2}_{Z}=\mathbf{z}^{T}\mathbb{D}\,\mathbf{z}$, where the symmetric, positive definite matrix $\mathbb{D}$ is given by $$\mathbb{D}_{n_{t}\times n_{t}}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c;{2pt/2pt}c;{2pt/2pt}c} \mathbb{D}^{1}_{n_{1}\times n_{1}} &\mathbf{0} & \mathbf{0} \\ \hdashline[2pt/2pt]
\mathbf{0} & \mathbb{D}^{\mathbf{c}}_{n_{\mathbf{c}}\times n_{\mathbf{c}}} &\mathbf{0} \\ \hdashline[2pt/2pt]
\mathbf{0} & \mathbf{0} & \mathbb{D}^{\mathbf{d}}_{n_{\mathbf{d}}\times n_{\mathbf{d}}}\end{array} \right],$$ and the components of the symmetric, positive definite matrices $\mathbb{D}^{1}$, $\mathbb{D}^{\mathbf{c}}$, and $\mathbb{D}^{\mathbf{d}}$ are $$\label{MetMat}
\begin{aligned}
\mathbb{D}^{1}_{ij} &= \llangle \boldsymbol{\Psi}_{i}, \boldsymbol{\Psi}_{j} \rrangle + \llangle \mathbf{grad}\,\boldsymbol{\Psi}_{i}, \mathbf{grad}\,\boldsymbol{\Psi}_{j} \rrangle,\\
\mathbb{D}^{\mathbf{c}}_{ij} &= \llangle \boldsymbol{\Lambda}_{i}, \boldsymbol{\Lambda}_{j} \rrangle + \llangle \mathbf{curl}\,\boldsymbol{\Lambda}_{i}, \mathbf{curl}\,\boldsymbol{\Lambda}_{j} \rrangle,\\
\mathbb{D}^{\mathbf{d}}_{ij} &= \llangle \boldsymbol{\Phi}_{i}, \boldsymbol{\Phi}_{j} \rrangle + \llangle \mathbf{div}\,\boldsymbol{\Phi}_{i}, \mathbf{div}\,\boldsymbol{\Phi}_{j} \rrangle.
\end{aligned}$$ The discussion of the Appendix then implies that the inf-sup condition holds if and only if the smallest singular value of $\mathbb{M}^{Z}\mathbb{S}\,\mathbb{M}^{Z}$ is positive and bounded from below by a positive number $\beta$ as $h\rightarrow 0$, where $\mathbb{M}^{Z}$ is the unique symmetric, positive definite matrix that satisfies $(\mathbb{M}^{Z})^{2}=\mathbb{D}$.
Weaker Stability Conditions
---------------------------
If the inf-sup condition holds, then will have a unique solution for any input data. In particular, must have a solution for any $\boldsymbol{f}^{1}$, or equivalently, the left-hand side of must define an onto mapping. This condition is equivalent to the following material-independent inf-sup condition: There exists $\alpha_{h}>0$ such that $$\label{CSFEM-infsup_Sur}
\underset{\boldsymbol{\Upsilon}_{h}\in [V^{1}_{h,1}]^{n}}{\inf}\, \underset{\boldsymbol{Q}_{h}\in V^{\mathbf{d}}_{h}}{\sup} \frac{\llangle\boldsymbol{Q}_{h},\mathbf{grad}\,\boldsymbol{\Upsilon}_{h}\rrangle}{\|\boldsymbol{Q}_{h}\|_{\mathbf{d}}\, \|\boldsymbol{\Upsilon}_{h}\|_{1} } \geq \alpha_{h}.$$ On the other hand, and must have a solution for any $\boldsymbol{f}^{1}$ and $\boldsymbol{f}^{\mathbf{d}}$, which means that the left-hand side of and should define an onto mapping. This latter condition can be stated by another inf-sup condition: There should be $\gamma_{h}>0$ such that $$\label{CSFEM-infsup_Sur2}
\underset{(\boldsymbol{\Upsilon}_{h},\boldsymbol{\pi}_{h})\in Z_{1\mathbf{d}}}{\inf}\, \underset{(\boldsymbol{M}_{h},\boldsymbol{Q}_{h})\in Z_{\mathbf{cd}}}{\sup} \frac{\llangle\boldsymbol{Q}_{h},\mathbf{grad}\,\boldsymbol{\Upsilon}_{h}\rrangle + \llangle \mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M}_{h},\boldsymbol{\pi}_{h}\rrangle - \llangle \boldsymbol{Q}_{h}, \boldsymbol{\pi}_{h} \rrangle }{\|(\boldsymbol{M}_{h},\boldsymbol{Q}_{h})\|_{\mathbf{cd}} \, \|(\boldsymbol{\Upsilon}_{h},\boldsymbol{\pi}_{h})\|_{1\mathbf{d}} } \geq \gamma_{h},$$ with $\|(\boldsymbol{M}_{h},\boldsymbol{Q}_{h})\|^{2}_{\mathbf{cd}} = \|\boldsymbol{M}_{h}\|^{2}_{\mathbf{c}} + \|\boldsymbol{Q}_{h}\|^{2}_{\mathbf{d}}$, and $\|(\boldsymbol{\Upsilon}_{h},\boldsymbol{\pi}_{h})\|^{2}_{\mathbf{1d}} = \|\boldsymbol{\Upsilon}_{h}\|^{2}_{1} + \|\boldsymbol{\pi}_{h}\|^{2}_{\mathbf{d}}$. The inf-sup conditions and are weaker than in the sense that they are only necessary for the validity of .
The material-independent inf-sup condition admits the matrix form $$\label{CSFEM-infsup_SurMat}
\underset{\mathbf{y}\in\mathbb{R}^{n_{1}}}{\inf}\, \underset{\mathbf{q}\in\mathbb{R}^{n_{\mathbf{d}}}}{\sup} \frac{\mathbf{y}^{T}\mathbb{S}^{1\mathbf{d}} \mathbf{q}}{\|\mathbf{y}\|_{1} \|\mathbf{q}\|_{\mathbf{d}}} \geq \alpha_{h}>0,$$ where $\mathbb{S}^{1\mathbf{d}}_{n_{1}\times n_{\mathbf{d}}}$ is defined in . Let $\mathbb{M}^{1}$ and $\mathbb{M}^{\mathbf{d}}$ be the unique symmetric and positive definite matrices such that $(\mathbb{M}^{1})^{2}=\mathbb{D}^{1}$, and $(\mathbb{M}^{\mathbf{d}})^{2}=\mathbb{D}^{\mathbf{d}}$, where $\mathbb{D}^1$ and $\mathbb{D}^{\mathbf{d}}$ are given in . Then, holds if and only if the smallest singular value of $\mathbb{M}^{1}\mathbb{S}^{1\mathbf{d}}\mathbb{M}^{\mathbf{d}}$ is positive. Similarly, the matrix form of reads $$\label{CSFEM-infsup_SurMat2}
\underset{\mathbf{u}\in\mathbb{R}^{n_{1}+n_{\mathbf{d}}}}{\inf}\, \underset{\mathbf{x}\in\mathbb{R}^{n_{\mathbf{c}}+n_{\mathbf{d}}}}{\sup} \frac{\mathbf{u}^{T}\mathbb{G} \mathbf{x}}{\|\mathbf{u}\|_{1\mathbf{d}} \|\mathbf{x}\|_{\mathbf{cd}}} \geq \gamma_{h}>0,$$ with $$\mathbb{G}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c;{2pt/2pt}c} \mathbf{0} & \mathbb{S}^{1\mathbf{d}}_{n_{1}\times n_{\mathbf{d}}} \\ \hdashline[2pt/2pt]
\mathbb{S}^{\mathbf{d}\mathbf{c}}_{n_{\mathbf{d}}\times n_{\mathbf{c}}} & \mathbb{S}^{\mathbf{d}\mathbf{d}}_{n_{\mathbf{d}}\times n_{\mathbf{d}}}\end{array} \right],$$ where $\mathbb{S}^{1\mathbf{d}}$, $\mathbb{S}^{\mathbf{dc}}$, and $\mathbb{S}^{\mathbf{dd}}$ are defined in . Suppose $\mathbb{M}^{1\mathbf{d}}$ and $\mathbb{M}^{\mathbf{cd}}$ are positive definite matrices that satisfy $$(\mathbb{M}^{1\mathbf{d}})^{2}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c;{2pt/2pt}c} \mathbb{D}^{1}_{n_{1}\times n_{1}} & \mathbf{0} \\ \hdashline[2pt/2pt]
\mathbf{0} & \mathbb{D}^{\mathbf{d}}_{n_{\mathbf{d}}\times n_{\mathbf{d}}} \end{array} \right], \text{ and } (\mathbb{M}^{\mathbf{cd}})^{2}=\left[ \arraycolsep=1.1pt\def\arraystretch{1.2} \begin{array}{c;{2pt/2pt}c} \mathbb{D}^{\mathbf{c}}_{n_{\mathbf{c}}\times n_{\mathbf{c}}} & \mathbf{0} \\ \hdashline[2pt/2pt]
\mathbf{0} & \mathbb{D}^{\mathbf{d}}_{n_{\mathbf{d}}\times n_{\mathbf{d}}} \end{array} \right],$$ where $\mathbb{D}^{1}$, $\mathbb{D}^{\mathbf{c}}$, and $\mathbb{D}^{\mathbf{d}}$ were introduced in . The condition holds if and only if the smallest singular value of $\mathbb{M}^{1\mathbf{d}}\mathbb{G}\,\mathbb{M}^{\mathbf{cd}}$ is positive.
The inf-sup condition is equivalent to the surjectivity of the linear mapping $\mathbb{S}^{1\mathbf{d}}:\mathbb{R}^{n_{\mathbf{d}}}\rightarrow\mathbb{R}^{n_{1}}$, that is, the matrix $\mathbb{S}^{1\mathbf{d}}$ being full ranked. This result can be directly deduced from the structure of the matrix $\mathbb{S}$ in as well. As a consequence of the rank-nullity theorem, it is easy to see that $\mathbb{S}^{1\mathbf{d}}$ is not full rank if $n_{\mathbf{d}}<n_{1}$. The upshot can be stated as follows.
\[SimplInfSup\] Suppose $n_{1} = \dim([V^{1}_{h,1}]^{n})$, and $n_{\mathbf{d}} = \dim V^{\mathbf{d}}_{h}$. The inf-sup conditions and do not hold if $n_{\mathbf{d}}<n_{1}$.
The condition is equivalent to the surjectivity of $\mathbb{G}: \mathbb{R}^{n_{\mathbf{c}}+ n_{\mathbf{d}}} \rightarrow \mathbb{R}^{n_{\mathbf{1}}+ n_{\mathbf{d}}}$, that is, $\mathbb{G}$ being full rank. Due to the rank-nullity theorem, this result does not hold if $n_{\mathbf{c}}+ n_{\mathbf{d}} < n_{1}+ n_{\mathbf{d}}$. Thus, one concludes that:
\[SimplInfSup2\] Suppose $n_{1} = \dim([V^{1}_{h,1}]^{n})$, and $n_{\mathbf{c}} = \dim V^{\mathbf{c}}_{h}$. The inf-sup conditions and do not hold if $n_{\mathbf{c}}<n_{1}$.
Notice that if the inf-sup condition fails, then the discrete nonlinear problem is not stable as it may not have any solution for some body forces and boundary tractions. Assume that $\mathcal{B}$ is a polyhedral domain with a triangular (2D) or a tetrahedral (3D) mesh $\mathcal{B}_{h}$, which is geometrically conformal. Let $N_{v}$, $N_{ed}$, and $N_{f}$ be respectively the number of vertices, edges, and faces of $\mathcal{B}_{h}$ (in $2$D, we have $N_{f}=N_{ed}$). For the $n$-dimensional elements $\text{LE}_{2}$, $\text{NED}^{1}_{1}$, and $\text{RT}_{1}$, $n=2,3$, of Figure \[FEO12\], it is straightforward to show that $n_{1}= n(N_{v}+N_{ed})$, $n_{\mathbf{c}} = n\,N_{ed}$, and $n_{\mathbf{d}} = n\,N_{f}$. These relations imply the following corollary of Theorems \[SimplInfSup\] and \[SimplInfSup2\].
\[Cor\_infsup\] Let $\text{FE}_{\mathbf{c}}$ and $\text{FE}_{\mathbf{d}}$ respectively be arbitrary $H^{\mathbf{c}}$- and $H^{\mathbf{d}}$-conformal finite elements. We have:
1. In $2$D, the finite element choice $(\text{LE}_{2},\text{FE}_{\mathbf{c}},\text{RT}_{1})$ for mixed finite element methods does not satisfy the inf-sup conditions and .
2. In $2$D and $3$D, the finite element choice $(\text{LE}_{2},\text{NED}^{1}_{1},\text{FE}_{\mathbf{d}})$ for mixed finite element methods does not satisfy the inf-sup conditions and .
In [@Re2015], a three-field formulation for linearized elasticity in terms of displacement, strain, and stress was introduced, which is similar to the system but by using discontinuous $L^{2}$-elements instead of $H^{\mathbf{c}}$- and $H^{\mathbf{d}}$-conformal elements. For that linear system, it was shown that the ellipticity of the elasticity tensor and the analogue of the inf-sup condition in terms of $L^{2}$ finite element spaces are sufficient for the well-posedness [@Re2015 Theorem 5.2]. The inf-sup condition is a stronger condition in the sense that it is both necessary and sufficient for the well-posedness of .
The condition is similar to the Babuška-Brezzi condition for the Stokes problem. For choices of finite elements that fails, one can use strategies similar to those for the Babuška-Brezzi condition to enrich $V^{\mathbf{d}}_{h}$, e.g. employing bubble functions or using a finer mesh for $V^{\mathbf{d}}_{h}$, see [@ErnGuermond2004 Chapter 4].
Numerical Results {#Sec_Examples}
=================
To study the performance of the mixed finite element method , we employ the finite elements of Figure \[FEO12\] and solve several $2$D and $3$D numerical examples in this section. Numerical simulations are performed by using FEniCS [@loMaWe2012], which is an open-source platform with the high-level Python and C++ interfaces. For our simulations, we consider compressible Neo-Hookean materials with the stored energy function $$W(\boldsymbol{F})=\frac{\mu}{2}(I_{1}-3) - \frac{\mu}{2}\ln I_{3} + \frac{\lambda}{2}(\ln I_{3})^{2}, \quad \mu,\lambda>0,$$ where $\boldsymbol{F}$ is the deformation gradient, $I_{1}=\mathrm{tr}\,\boldsymbol{C}$, and $I_{3}=\det \boldsymbol{C}$, with $\boldsymbol{C}=\boldsymbol{F}^{T}\boldsymbol{F}$. The constitutive equation in terms of $\boldsymbol{F}$ then reads $$\label{NeoHook}
\mathbb{P}(\boldsymbol{F})= \mu\boldsymbol{F} -\mu \boldsymbol{F}^{-T} + 2\lambda (\ln I_{3}) \boldsymbol{F}^{-T}.$$ Substituting $\boldsymbol{F}=\boldsymbol{I}+\boldsymbol{K}$ in the above equation yields the constitutive equation $\mathbb{P}(\boldsymbol{K})$ in terms of the displacement gradient $\boldsymbol{K}$, where $\boldsymbol{I}$ is the identity matrix. Moreover, the tensor $\mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M}$ in the bilinear form becomes $$\mathsf{A}(\boldsymbol{K})\boldsymbol{:}\boldsymbol{M}= \mu \boldsymbol{M} + (\mu-2\lambda\ln I_{3}) \boldsymbol{F}^{-T}\boldsymbol{M}^{T}\boldsymbol{F}^{-T} + 4\lambda(\mathrm{tr}\, \boldsymbol{F}^{-1}\boldsymbol{M}) \boldsymbol{F}^{-T}.$$
#### Convention
To concisely refer to a choice of the elements of Figure \[FEO12\] for the mixed finite element methods , we use the following convention: $\mathrm{L}i$, $\mathrm{N}ji$, $\mathrm{R}i$, and $\mathrm{B}i$ respectively denote $\text{LE}_i$, $\text{NED}^{j}_{i}$, $\text{RT}_{i}$, and $\mathrm{BDM}_{i}$. For example, $\mathrm{L1N12B2}$ denotes the choice $(\text{LE}_{1},\text{NED}^{1}_{2},\text{BDM}_{2})$.
![Numerical analysis of the inf-sup conditions (the left panel) and (the right panel) using unstructured meshes of the unit square (2D) and the unit cube (3D) similar to the second row of Figures \[PlateStUnst\] and \[CubeStUnstM\]. Left Panel: The ratio $\mathrm{rank}(\mathbb{S}^{1\mathbf{d}})/n_{1}$ versus the maximum diameter of elements $h$ for the finite element choices $\mathrm{L2R1}$ and $\mathrm{L2B1}$ for $(\boldsymbol{U},\boldsymbol{P})$. As this ratio is smaller than $1$, $\mathbb{S}^{1\mathbf{d}}$ is rank deficient and the condition does not hold. Right Panel: Values of the lower bound $\beta_h$ of the inf-sup condition versus $h$ associated to $2$D and $3$D meshes and the choice of elements $\mathrm{L1N11R1}$ for $(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})$. The results suggest that $\beta_h$ does not decrease as $h$ decreases, and hence there is a positive lower bound $\beta$ that satisfies .[]{data-label="InfSupR"}](InfSupR.png)
![Meshes of a unit square where the number of elements $N_{e}$ and the maximum diameter of elements $h$ are given by $(N_{e},h)= (8,0.707)$, $(32,0.354)$, $(72,0.236)$, $(128,0.177)$, for the structured meshes of the first row and $(N_{e},h) = (16, 0.559)$, $(32,0.358)$, $(88,0.249)$, $(146,0.181)$, for the unstructured meshes of the second row.[]{data-label="PlateStUnst"}](PlateStUnst.png)
Stability Analysis {#subsec_StAn}
------------------
We begin by numerically investigating the inf-sup conditions introduced earlier by using their matrix forms. Of course, the following numerical results are not mathematical proofs; Rather, they provide strong evidences for obtaining mathematical proofs.
As discussed earlier, the inf-sup conditions and are respectively equivalent to $\mathbb{S}^{1\mathbf{d}}$ and $\mathbb{G}$ being full rank. Therefore, to study the validity of these conditions, one can study the rank of the associated matrices. Due to Corollary \[Cor\_infsup\], we already know that the choice $\mathrm{L2R1}$ for $(\boldsymbol{U},\boldsymbol{P})$ in $2$D does not satisfy the inf-sup condition . Our numerical studies show that this choice does not satisfy in $3$D as well. Moreover, the choice $\mathrm{L2B1}$ does not satisfy neither in $2$D nor in $3$D. For example, the left panel of Figure \[InfSupR\] depicts the ratio $\mathrm{rank}(\mathbb{S}^{1\mathbf{d}})/n_{1}$ versus the maximum diameter of elements $h$ for the choices $\mathrm{L2R1}$ and $\mathrm{L2B1}$ calculated using unstructured meshes of the unit square and the unit cube. The matrix $\mathbb{S}^{1\mathbf{d}}$ is not full rank in these cases since $\frac{\mathrm{rank}(\mathbb{S}^{1\mathbf{d}})}{n_{1}} <1$. It is interesting to note that unlike $\mathrm{L2R1}$, we have $n_{\mathbf{d}}> n_{1}$ for $\mathrm{L2B1}$.
Corollary \[Cor\_infsup\] implies that the choice $\mathrm{L2N11}$ for $(\boldsymbol{U},\boldsymbol{K})$ does not satisfy the inf-sup condition . Notice that unlike , the inf-sup condition is material-dependent. Our numerical experiments suggest that for Neo-Hookean materials with regular deformations, all other choices of the elements of Figure \[FEO12\] for $(\boldsymbol{U},\boldsymbol{K})$ satisfy .
The inf-sup condition is sufficient for the convergence of solutions of to regular solutions of . Our numerical results for Neo-Hookean materials discussed in the remainder of this section suggest that all choices of elements of Figure \[FEO12\] for $(\boldsymbol{U},\boldsymbol{K},\boldsymbol{P})$ that satisfy the inf-sup conditions and also satisfy the inf-sup condition . As an example, the right panel of Figure \[InfSupR\] shows the values of the lower bound $\beta_h$ of the matrix form for the choice of elements $\mathrm{L1N11R1}$ in $2$D and $3$D. Results are calculated using unstructured meshes of the unit square and the unit cube with the material parameters $\mu=\lambda =1$ near the reference configuration, that is, $\boldsymbol{K}=0$. To approximate $\beta_h$ for each mesh, one can employ the smallest singular value of $\mathbb{M}^{Z}\mathbb{S}\,\mathbb{M}^{Z}$ or equivalently, the square root of the smallest eigenvalue of $\mathbb{D}\,\mathbb{S}^{T}\mathbb{D}\,\mathbb{S}$, with $\mathbb{D} = (\mathbb{M}^{Z})^{2}$. The results suggests that the values of $\beta_h$ are bounded from below as $h$ decreases and therefore, there is a lower bound $\beta>0$ that satisfies .
The validity of the material-dependent inf-sup conditions and is dependent to properties of the elasticity tensor $\mathsf{A}(\boldsymbol{K})$. To rigorously study these inf-sup conditions, one needs to impose some additional restrictions on $\mathsf{A}$ which are physically reasonable. The classical inequalities for $\mathsf{A}$ [@TrNo65 Section 51] and suitable assumptions on the stored energy functional $W$ such as polyconvexity [@Ci1988] are relevant here.
=3.0pt
Deformation of a $2$D Plate {#Ex_2DPlate}
---------------------------
To studying the convergence rate of solutions, we consider a unit-square plate with the material parameters $\mu=\lambda=1$ and solve by employing the body force and the boundary conditions that induce the displacement field $$\label{2Dplate_Exact}
\boldsymbol{U}_{e} = \left[\begin{array}{c} \frac{1}{2}Y^{3} + \frac{1}{2}\sin(\frac{\pi}{2}Y) \\ 0 \end{array} \right],$$ where $(X,Y)$ denotes the Cartesian coordinates in $\mathbb{R}^{2}$. We use Newton’s method to solve the resulting nonlinear systems. The linear system solved in each Newton iteration is similar to the linear system where $\boldsymbol{K}$ is replaced with the solution of the previous iteration. Therefore, the coefficient matrix of each Newton iteration is similar to the matrix $\mathbb{S}$ of the inf-sup condition and consequently, Newton iterations become singular if any of the inf-sup conditions introduced earlier (with the solution of the previous iteration instead of $\boldsymbol{K}$) is not satisfied.
Table \[Pl1ConvError\] shows $L^{2}$-errors and the associated convergence rates of the mixed method which are calculated by using the structured meshes in the first row of Figure \[PlateStUnst\] with different combinations of the $2$D elements of degrees 1 and 2 of Figure \[FEO12\]. The convergence rate $r$ means the error is $O(h^{r})$ as $h\rightarrow 0$, where $h$ is the maximum diameter of elements of a mesh. We observe that $22$ combinations out of $32$ possible combinations of the $2$D elements of Figure \[FEO12\] are stable. More specifically, the $10$ unstable cases include $\mathrm{L2N11R}i$, $\mathrm{L2N11B}i$, $\mathrm{L2N}ij\mathrm{R1}$, and $\mathrm{L2N}ij\mathrm{B1}$, $i,j=1,2$. Following Corollary \[Cor\_infsup\] and the results of Section \[subsec\_StAn\], we already know that the cases $\mathrm{L2N}ij\mathrm{R1}$ and $\mathrm{L2N}ij\mathrm{B1}$ are unstable as they do not satisfy the inf-sup condition and that $\mathrm{L2N11R}i$ and $\mathrm{L2N11B}i$ are unstable as they do not satisfy the inf-sup condition . Thus, the inf-sup conditions and are sufficient for studying the stability of this example.
Table \[Pl1ConvError\] suggests that methods with the element $\mathrm{L1}$ for displacement have very close errors and convergence rates regardless of the degrees of their elements for $\boldsymbol{K}$ and $\boldsymbol{P}$. A similar conclusion also holds for methods with the element $\mathrm{L2}$. This suggests that the degree of the element for displacement has a significant effect on the overall performance of these mixed finite element methods. The optimal convergence rate (that is, the convergence rate of finite element interpolations of sufficiently smooth functions) of $\mathrm{L}i$, $\mathrm{N2i}$ and $\mathrm{B}i$ is $i+1$ while that of $\mathrm{N1i}$ and $\mathrm{R}i$ is $i$ [@BoBrFo2013]. Table \[Pl1ConvError\] shows that the convergence rates for displacement gradient and stress may not be optimal but the convergence rate of displacement is always optimal.
![$L^{2}$-errors of displacement $\|\boldsymbol{U}_{h} - \boldsymbol{U}_{e}\|$, displacement gradient $\|\boldsymbol{K}_{h} - \boldsymbol{K}_{e}\|$, and stress $\|\boldsymbol{P}_{h} - \boldsymbol{P}_{e}\|$ associated to the structured meshes (the solid lines) and the unstructured meshes (the dashed lines) of Figure \[PlateStUnst\]. The data marked by $\times$ and $\bullet$ are respectively calculated by the first-order elements $\mathrm{L1N21B1}$ and the second-order elements $\mathrm{L2N22B2}$.[]{data-label="2DConvStUnst"}](2DConvStUnst.png)
![The geometry and deformed configurations of $2$D Cook’s membrane. The deformed configurations are computed using the elements $\mathrm{L2N22B2}$ and the shear force $f = 24\,\mathrm{N}/\mathrm{mm}$. Colors in the deformed configuration depict the distribution of the Frobenius norm of stress $\|\boldsymbol{P}\|_{f}$.[]{data-label="CookConfig"}](CookConfig.png)
To compare the formulation of this paper with that of [@AnFSYa2017], we notice that the latter mixed formulation is stable only for $7$ out of $32$ possible combinations of the elements of Figure \[FEO12\]. A comparison between Table \[Pl1ConvError\] and Table 3 of [@AnFSYa2017] suggests that the performance of $\mathrm{L1N11R1}$, $\mathrm{L1N12B1}$, and $\mathrm{L1N22B1}$ is nearly similar in these two formulations while the performance of $\mathrm{L1N12R1}$, $\mathrm{L1N22R1}$, $\mathrm{L1N22R2}$, and $\mathrm{L2N22R2}$ is better using the formulation of this paper. As will be shown in the sequel, the main advantage of the present formulation is that unlike the formulation of [@AnFSYa2017] which is only stable in $2$D, it is stable in both 2D and 3D.
For the brevity, we only consider the choices $\mathrm{L1N21B1}$ and $\mathrm{L2N22B2}$ to solve the other 2D examples of this section. To study the effect of mesh irregularities on the performance of these finite element methods, the $L^{2}$-norm of errors corresponding to structured and unstructured meshes of Figure \[PlateStUnst\] are shown in Figure \[2DConvStUnst\]. These results suggest that comparing to the accuracy of approximate displacement and displacement gradient, mesh irregularities may have more impact on the accuracy of approximate stress. Notice that the slope of the curves in Figure \[2DConvStUnst\] which are associated to the structured meshes are the convergence rates of Table \[Pl1ConvError\].
![The $L^{2}$-norm of approximate solutions versus the number of elements of meshes $N_{e}$ for $2$D Cook’s membrane. The dashed and the solid lines correspond to $\mathrm{L1N21B1}$ and $\mathrm{L2N22B2}$, respectively. Results are computed using unstructured meshes and the shear forces $F1 = 24\,\mathrm{N/mm}$ and $F2 = 32\,\mathrm{N/mm}$.[]{data-label="L2Norm"}](L2Norm.png)
![The geometry and deformed configurations of the inhomogeneous compression example. The deformed configurations are computed using the elements $\mathrm{L2N22B2}$ and the force $f = 600\,\mathrm{N}/\mathrm{mm}$. Colors in the deformed configuration depict the distribution of the Frobenius norm of stress $\|\boldsymbol{P}\|_{f}$. []{data-label="2D_Inhomo"}](2D_inhomo.png)
![The percentage of the compression of the point $A$ of Figure \[2D\_Inhomo\] versus the number of elements $N_{e}$. Results are computed using the elements $\mathrm{L2N22B2}$. The associated results of @Re2002 are also shown for the comparison.[]{data-label="Inhomo_comp"}](2DInho_comp.png)
$2$D Cook’s Membrane
--------------------
Consider the $2$D Cook’s membrane problem with the geometry shown in Figure \[CookConfig\]. This example is usually used to study the performance in bending and in the near-incompressible regime [@Re2002]. The material properties are $\mu = 80.194\,\mathrm{N}/\mathrm{mm}^{2}$, and $\lambda = 400889.8\,\mathrm{N}/\mathrm{mm}^{2}$.
Figure \[CookConfig\] shows deformed configurations calculated using the element $\mathrm{L1N21B1}$ and the load $f = 24\,\mathrm{N}/\mathrm{mm}$. Colors in the deformed configurations depict the distribution of the Frobenius norm of stress $\|\boldsymbol{P}\|_{f} = \sqrt{\mathrm{tr}\,\boldsymbol{P}^{T}\boldsymbol{P}} = \sqrt{\sum_{I,J}|P^{IJ}|^{2}}$. Figure \[L2Norm\] shows the convergence of the $L^{2}$-norms of approximate solutions. Results are calculated using the elements $\mathrm{L1N21B1}$ and $\mathrm{L2N22B2}$ with two different loads of magnitudes $24$ and $32\,\mathrm{N/mm}$. These results suggest that the mixed formulation can provide accurate approximations of stress in bending and in the near-compressible regime.
![The $L^{2}$-norms of solutions of the $2$D inhomogeneoud compression problem versus the number of elements $N_{e}$. The elements $\mathrm{L2N22B2}$ were used for computing these results.[]{data-label="InHo_norm"}](2DIn_norm.png)
Inhomogeneous Compression
-------------------------
Enhanced strain methods are nonconformal three-field methods for small and finite deformations [@SiAr1992]. It is well-known that in some cases, these methods may become unstable due to the so-called hourglass instability [@Re2002]. One example for this type of instability is the inhomogeneous compression problem shown in Figure \[2D\_Inhomo\]. The horizontal displacement at the top of the domain and the vertical displacement at the bottom are assumed to be zero and the material properties are the same as the previous example.
Deformed configurations of this problem associated to two different meshes which are calculated using the elements $\mathrm{L2N22B2}$ and the load $f = 600\,\mathrm{N}/\mathrm{mm}$ are shown in Figure \[2D\_Inhomo\]. Colors in the deformed configurations show the distribution of the Frobenius norm of stress. Figure \[Inhomo\_comp\] depicts the percentage of compression versus the number of elements for different loads $f$. The compression level is calculated using the vertical displacement of the point $A$ of Figure \[2D\_Inhomo\], which is located at the midpoint of the top boundary. The results are consistent with those of [@Re2002]. Figure \[InHo\_norm\] shows the convergence of the $L^{2}$-norm of solutions by refining meshes. We do not observe any numerical instability in our computations.
![Meshes of the unit cube where the number of elements $N_{e}$ and the maximum diameter of elements $h$ are given by $(N_{e},h)= (48,0.866)$, $(384,0.433)$, $(750,0.346)$, for the structured meshes of the first row and $(N_{e},h) = (242, 0.636)$, $(502,0.443)$, $(867,0.363)$, for the unstructured meshes of the second row.[]{data-label="CubeStUnstM"}](CubeStUnstM.png)
=3.0pt
![$L^{2}$-errors of displacement $\|\boldsymbol{U}_{h} - \boldsymbol{U}_{e}\|$, displacement gradient $\|\boldsymbol{K}_{h} - \boldsymbol{K}_{e}\|$, and stress $\|\boldsymbol{P}_{h} - \boldsymbol{P}_{e}\|$ associated to the structured meshes (the solid lines) and the unstructured meshes (the dashed lines) of Figure \[CubeStUnstM\]. The data marked by $\times$ and $\bullet$ are respectively calculated by the first-order elements $\mathrm{L1N21B1}$ and the second-order elements $\mathrm{L2N22B2}$.[]{data-label="3DConvStUnst"}](3DConvStUnst.png)
![The $3$D Cook-type beam example: The first row shows the geometry (the left panel) and the deformed configuration induced by the uniform in-plane load $F1 = 300\, \mathrm{N/mm^2}$ in the $Y$-direction imposed at the right end (the right panel). The second row shows two different angles of view of the deformed configuration induced by the out-of-plane load $F2 = 600\,\mathrm{N}/\mathrm{mm}^{2}$ in the $Z$-direction applied at the right end of the beam. These results are calculated using the elements $\mathrm{L2N22B2}$ and the underlying mesh has $767$ elements. Colors in these figures depict the distribution of the Frobenius norm of stress.[]{data-label="3Dcook_deformed"}](3Dcook_deformed.png)
![The $L^{2}$-norms of solutions of the $3$D Cook-type beam example associated to the in-plane load $F1 = 300\,\mathrm{N}/\mathrm{mm}^{2}$ and the out-of-plane load $F2 = 600\,\mathrm{N}/\mathrm{mm}^{2}$ versus the number of elements $N_{e}$. The loads $F1$ and $F2$ are imposed at the right end of the beam in the $Y$- and the $Z$-directions, respectively. The elements $\mathrm{L1N21B1}$ and $\mathrm{L2N22B2}$ were used for computing these results.[]{data-label="3DCOOK_norm"}](3DCOOK_norm.png)
Deformation of a Cube
---------------------
To study the convergence rates in the $3$D case, we study the $3$D analogue of the plate problem of Section \[Ex\_2DPlate\]. More specifically, we consider the unit cube with the material parameters $\mu=\lambda=1$ and solve the mixed method by using the body force and the boundary conditions that induce the displacement field $$\label{cube_Exact}
\boldsymbol{U}_{e} = \left[\begin{array}{c} \frac{1}{2}Y^{3} + \frac{1}{2}\sin(\frac{\pi}{2}Y) \\ 0 \\ 0 \end{array} \right].$$ Table \[CubeConvError\] shows $L^{2}$-errors and convergence rates of the solutions of , which are calculated by using different combinations of the $3$D elements of degrees 1 and 2 of Figure \[FEO12\] and the structured meshes shown in the first row of Figure \[CubeStUnstM\]. Similar to the $2$D plate example, one observes that the degree of the element for displacement has a significant effect on the overall performance of these mixed finite element methods. Moreover, Table \[CubeConvError\] suggests that the convergence rates of displacement gradient and stress may not be optimal.
Our numerical results suggest that similar to $2$D cases, $22$ combinations out of $32$ possible combinations of the $3$D elements of Figure \[FEO12\] are stable. The $10$ unstable cases are the same as those of $2$D cases and are those that do not satisfy the inf-sup conditions and . The extension of the mixed formulation to the $3$D case is straightforward. This is the main advantage of this formulation comparing to the mixed formulation of [@AnFSYa2017].
For the brevity, we consider the choices $\mathrm{L1N21B1}$ and $\mathrm{L2N22B2}$ in the remainder of this work. Figure \[3DConvStUnst\] depicts the $L^{2}$-errors of approximate solutions corresponding to the structured and unstructured meshes of Figure \[CubeStUnstM\]. The slopes of curves of the structured meshes are the convergence rates of Table \[CubeConvError\]. As the $2$D case, these results suggest that mesh irregularities have more impact on the accuracy of approximate stresses.
A Near-Incompressible Cook-Type Beam
------------------------------------
Next, we study the $3$D analogue of Cook’s membrane under in-plane and out-of-plane loads. The geometry of this problem in the $XY$-plane is similar to that of the $2$D case shown in Figure \[CookConfig\] with the thickness $10\,\mathrm{mm}$ in the $Z$-direction, see Figure \[3Dcook\_deformed\]. We use the near-incompressible material properties of the $2$D Cook’s membrane.
The configuration in the right panel of the first row of Figure \[3Dcook\_deformed\] is the deformed configuration under a uniform load $F1 = 300\,\mathrm{N}/\mathrm{mm}^{2}$ imposed at the right end of the beam in the $Y$-direction. The second row of Figure \[3Dcook\_deformed\] shows two different angles of view of a deformed configuration due to the out-of-plane load $F2 = 600\,\mathrm{N}/\mathrm{mm}^{2}$ in the $Z$-direction applied at the right end of the beam. These results are computed using $\mathrm{L2N22B2}$ and colors in the deformed configurations depict the distribution of the Frobenius norm of stress. Figure \[3DCOOK\_norm\] shows the convergence of the $L^{2}$-norms of finite element solutions associated to the above in-plane and out-of-plane loads. The elements $\mathrm{L1N21B1}$ and $\mathrm{L2N22B2}$ were used for these computations. Our results suggest that similar to the $2$D case, the $3$D mixed formulation can provide accurate approximations of stress in bending and in the near-compressible regime.
Conclusion {#Sec_Conc}
==========
We introduced a new mixed formulation for $2$D and $3$D nonlinear elasticity in terms of displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor. We showed that even for hyperelastic solids, this formulation does not correspond to a stationary point of any functional, in general. For obtaining conformal mixed finite element methods based on this formulation, finite element spaces suitable for the $\mathrm{curl}$ and the $\mathrm{div}$ operators are respectively employed for displacement gradient and stress. Discrete displacement gradients and stresses satisfy suitable jump conditions due to these choices.
We studied stability of these mixed finite element methods by writing suitable inf-sup conditions. We examined the performance of these methods for $32$ combinations of $2$D and $3$D simplicial elements of degree $1$ and $2$ and showed that $10$ combinations are not stable as they violate the inf-sup conditions. Several $2$D and $3$D numerical examples were solved to study convergence rates, the effect of mesh distortions, and the performance for bending problems and the near-incompressible regime. These examples suggest that it is possible to achieve the optimal convergence rates and obtain accurate approximations of strains and stresses. Moreover, we did not observe the hourglass instability that may occur in enhanced strain methods.
An Abstract Theory for the Galerkin Approximation {#Sec_infsup}
=================================================
In the following, we summarize the general framework for the Galerkin approximation of nonlinear problems introduced in [@PoRa1994; @CaRa1997]. Let $H:Z\rightarrow Y'$ be a mapping, where $Z$ and $Y$ are Banach spaces with the norms $\|\cdot\|_{Z}$ and $\|\cdot\|_{Y}$, respectively, and $Y'$ is the dual space of $Y$. Also let the linear operator $\mathrm{D}H(u):Z\rightarrow Y'$ be the (Fréchet) derivative of $H$ at $u\in Z$, i.e. $\mathrm{D}H(u)z=\frac{d}{ds}|_{s=0} H(u+sz)$, $\forall z\in Z$. The goal is to approximate a regular solution $u\in Z$ of the problem $H(u)=0$, where regular means the derivative of $H$ at $u$ is “nonzero” in the sense that the linear mapping $\mathrm{D}H(u)$ is one-to-one and onto. The relation $H(u)=0$ is equivalent to $$\label{AbsProb}
\langle H(u),y\rangle = 0, \quad \forall y\in Y,$$ where $\langle f,y\rangle:=f(y)$, $\forall f\in Y'$. Given finite element spaces $Z_{h}\subset Z$ and $Y_{h}\subset Y$, a Galerkin approximation of the problem reads: Find $u_{h}\in Z_{h}$ such that $$\label{GalAbsProb}
\langle H(u_{h}),y_{h}\rangle = 0, \quad \forall y_{h}\in Y_{h}.$$ To express sufficient conditions for the existence and the convergence of solutions of as $h\rightarrow 0$, we consider the bilinear form $b:Z\times Y\rightarrow \mathbb{R}$ defined as $$\label{BiLinF}
b(z,y):=\langle \mathrm{D}H(u)z,y\rangle, \quad \forall z\in Z,~y\in Y.$$ Then, one can show that the following result holds [@CaRa1997 Theorem 7.1]: Roughly speaking, for sufficiently small $h>0$, the problem has a unique solution $u_{h}$ in a neighborhood of a regular solution $u$ of and $u_{h}\rightarrow u$ as $h\rightarrow 0$ if: (i) Any element of $Z$ can be approximated by $Z_{h}$ as $h\rightarrow 0$ (approximibility); (ii) $\dim Z_{h}=\dim Y_{h}$; and (iii) There exists a mesh-independent number $\beta>0$ such that the following inf-sup condition holds: $$\label{infsupCond}
\underset{y_{h}\in Y_{h}}{\inf}\, \underset{z_{h}\in Z_{h}}{\sup} \frac{b(z_{h},y_{h})}{\|z_{h}\|_{Z} \|y_{h}\|_{Y} } \geq \beta>0.$$
It is also possible to write a priori and a posteriori estimates for the error $\|u-u_{h}\|_{Z}$ [@CaRa1997 Theorem 7.1]. In particular, the a priori estimate provides an upper bound for $\|u-u_{h}\|_{Z}$ which is proportional to $\beta^{-1}$. If the constant of the inf-sup condition is a mesh-dependent number $\beta_{h}$ such that $\beta_{h}\rightarrow 0$ as $h\rightarrow0$, then $u_{h}$ may converge poorly or diverge as $h\rightarrow0$ even if the inf-sup condition holds for all meshes. Since $u$ is a regular solution of , the linearized problem $$\label{LinProb}
\langle \mathrm{D} H(u) z,y\rangle = \langle f,y\rangle, \quad \forall y\in Y,$$ has a unique solution $z\in Z$ for any $f\in Y'$. The inf-sup condition together with the condition (ii) imply that the discrete linear problem $$\label{DisLinProb}
\langle \mathrm{D} H(u) z_{h},y_{h}\rangle = \langle f,y_{h}\rangle, \quad \forall y_{h}\in Z_{h},$$ also has a unique solution $z_{h}\in Z_{h}$ for any $f\in Y'$.
A simple approach to numerically investigate the inf-sup condition is as follows: Let $\{\zeta_{i}\}_{i=1}^{n_{Z}}$ and $\{\theta_{i}\}_{i=1}^{n_{Y}}$ respectively be global shape functions for $Z_{h}$ and $Y_{h}$. Then, we have $z_{h}=\sum_{i=1}^{n_{Z}}z_{i}\zeta_{i}$, $\forall z_{h}\in Z_{h}$, and $y_{h}=\sum_{i=1}^{n_{Y}}y_{i}\theta_{i}$, $\forall y_{h}\in Y_{h}$. We associate the vector $\mathbf{z}=(z_{1},\dots,z_{n_{Z}})^{T}\in \mathbb{R}^{n_{Z}}$ ($\mathbf{y}=(y_{1},\dots,y_{n_{Y}})^{T}\in \mathbb{R}^{n_{Y}}$) to $z_{h}$ ($y_{h}$) and define $\|\mathbf{z}\|_{Z}:= \|z_{h}\|_{Z}$ ($\|\mathbf{y}\|_{Y}:= \|y_{h}\|_{Y}$). Assume that there exist symmetric and positive definite matrices $\mathbb{M}^{Z}_{n_{Z}\times n_{Z}}$ and $\mathbb{M}^{Y}_{n_{Y}\times n_{Y}}$ such that $$\begin{aligned}
\|\mathbf{z}\|^{2}_{Z}&= (\mathbb{M}^{Z}\mathbf{z})^{T}(\mathbb{M}^{Z}\mathbf{z})= \mathbf{z}^{T}(\mathbb{M}^{Z})^{2}\mathbf{z}, \\
\|\mathbf{y}\|^{2}_{Y}&= (\mathbb{M}^{Y}\mathbf{y})^{T}(\mathbb{M}^{Y}\mathbf{y})= \mathbf{y}^{T}(\mathbb{M}^{Y})^{2}\mathbf{y}.\end{aligned}$$ By using the vectors $\mathbf{y}$ and $\mathbf{z}$, the inf-sup condition can be expressed in the matrix form $$\label{infsupCondMat}
\underset{\mathbf{y}\in\mathbb{R}^{n_{Y}}}{\inf}\, \underset{\mathbf{z}\in\mathbb{R}^{n_{Z}}}{\sup} \frac{\mathbf{y}^{T}\mathbb{B}\, \mathbf{z}}{\|\mathbf{y}\|_{Y} \|\mathbf{z}\|_{Z}} \geq \beta>0,$$ where the matrix $\mathbb{B}_{n_{Y}\times n_{Z}}$ is given by $\mathbb{B}_{ij}=b(\zeta_{j},\theta_{i})$. Recall that the singular values of an arbitrary matrix $\mathbb{M}$ are the square root of the eigenvalues of $\mathbb{M}^{T}\mathbb{M}$. Then, one can show that the inf-sup condition holds if and only if the smallest singular value of $\mathbb{M}^{Y}\mathbb{B}\,\mathbb{M}^{Z}$ is positive and bounded from below by a positive constant $\beta$ as $h\rightarrow 0$ [@BoBrFo2013 Proposition 3.4.5].
[^1]: Department of Civil and Environmental Engineering, The George Washington University, Washington, DC 20052. E-mail: aangoshtari@gwu.edu.
[^2]: Department of Civil and Environmental Engineering, The George Washington University, Washington, DC 20052. E-mail: agerami@gwu.edu.
|
---
abstract: 'The Internet of Things (IoT) field has gained much attention from industry and academia, being the main subject for numerous research and development projects. Frequently, the dense amount of generated data from IoT applications is sent to a cloud service, that is responsible for processing and storage. Many of these applications demand security and privacy for their data because of their sensitive nature. This is specially true when such data must be processed in entities hosted in public clouds, where the environment in which applications run may not be trusted. Some concerns are then raised since it is not trivial to provide the needed protection for these sensitive data. We present a solution that considers the security components of FIWARE and the Intel SGX capabilities. FIWARE is a platform created to support the development of Smart Applications, including IoT systems, and SGX is the Intel solution for Trusted Execution Environment (TEE). We propose a new component for key management that, together with other FIWARE components, can be used to provide privacy, confidentiality, and integrity guarantees for IoT data. A case study illustrates how this proposed solution can be employed in a realistic scenario, which allows the dissemination of sensitive data through public clouds without risking privacy issues. The results of the experiments provide evidence that our approach does not harm scalability or availability of the system. In addition, it presents acceptable memory costs when considering the benefit of the privacy guarantees achieved.'
author:
- |
Dalton Cézane Gomes Valadares$^{1,2}$, Matteus Sthefano Leite da Silva$^{2}$,\
Andrey Elísio Monteiro Brito$^{2}$ and Ewerton Monteiro Salvador$^{3}$ [^1][^2] [^3], [^4] [^5]
bibliography:
- 'iscc.bib'
title: '**Achieving Data Dissemination with Security using FIWARE and Intel Software Guard Extensions (SGX)**'
---
[^1]: \*This research was partially funded by EU-BRA SecureCloud project (EC, MCTIC/RNP, and SERI, 3rd Coordinated Call, H2020-ICT-2015 Grant agreement no. 690111)
[^2]: $^{1}$Federal University of Campina Grande, Informatics and Electrical Engineering Center, Computer Science, Campina Grande, Paraíba, Brazil [cezane@lsd.ufcg.edu.br, silvamatteus@lsd.ufcg.edu.br, andrey@computacao.ufcg.edu.br]{}
[^3]: $^{2}$Federal Institute of Pernambuco, Mechanical Engineering Department, Caruaru, Pernambuco, Brazil [dalton.valadares@caruaru.ifpe.edu.br]{}
[^4]: $^{3}$Federal University of Paraíba, João Pessoa, Paraíba, Brazil [esalvador@lsd.ufcg.edu.br]{}
[^5]: ©©2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
|
ITP-UU-11/18\
SPIN-11/13\
CCTP-2011-15\
UFIFT-QG-11-03
[**The Graviton Propagator in de Donder Gauge on de Sitter Background**]{}
S. P. Miao$^*$
*[Institute for Theoretical Physics & Spinoza Institute, Utrecht University\
Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht, NETHERLANDS]{}\
*
N. C. Tsamis$^{\dagger}$
*[Institute of Theoretical Physics & Computational Physics, Dept. of Physics\
University of Crete, GR-710 03 Heraklion, HELLAS]{}*
R. P. Woodard$^{\ddagger}$
*[Department of Physics, University of Florida\
Gainesville, FL 32611, UNITED STATES]{}*
ABSTRACT
We construct the graviton propagator on de Sitter background in exact de Donder gauge. We prove that it must break de Sitter invariance, just like the propagator of the massless, minimally coupled scalar. Our explicit solutions for its two scalar structure functions preserve spatial homogeneity and isotropy so that the propagator can be used within the larger context of inflationary cosmology, however, it is simple to alter the residual symmetry. Because our gauge condition is de Sitter invariant (although no solution for the propagator can be) renormalization should be simpler using this propagator than one based on a noncovariant gauge. It remains to be seen how other computational steps compare.
PACS numbers: 04.62.+v, 04.60-m, 98.80.Cq
$^*$ e-mail: S.Miao@uu.nl\
$^{\dagger}$ e-mail: tsamis@physics.uoc.gr\
$^{\ddagger}$ e-mail: woodard@phys.ufl.edu
Introduction
============
There are bound to be interesting lessons when two plausible arguments lead to opposite conclusions. The conflict we have in mind concerns the graviton propagator on de Sitter background and the arguments about it derive, respectively, from cosmology and mathematical physics:
- [From the perspective of cosmology, the graviton propagator cannot be de Sitter invariant because the unique Fourier mode sum for an invariant propagator is infrared divergent as a consequence of the scale invariance of the tensor power spectrum [@MTW1].]{}
- [From the perspective of mathematical physics, the graviton propagator must be de Sitter invariant because explicit, de Sitter invariant solutions for the graviton propagator equation have been obtained when covariant gauge fixing terms are added to the action [@INVPROP].]{}
The cosmology argument turns out to be correct. Seeing this teaches:
- [There is a topological obstacle to adding covariant gauge fixing terms on any manifold, and for any gauge theory, which possesses a linearization instability [@MTW1].]{}
- [One can impose covariant gauges in which the field obeys exact conditions, but previous solutions employed analytic continuation techniques that incorrectly subtract off power law infrared divergences [@MTW2].]{}
- [Solutions exist to the propagator equation which do not correspond to propagators in the sense of being the expectation value of the time-ordered product of two fields in the presence of some state [@TW1].[^1]]{}
The first point occurs even for flat space electromagnetism on the manifold $T^3 \times R$: the invariant equations’ linearization instability requires the total charge to vanish, whereas the Feynman gauge equations can be solved for any charge. The second point is familiar to everyone who has encountered the automatic subtraction of dimensional regularization. And a trivial example of the third point comes from multiplying the entirely real, retarded propagator by a factor of $i$.
These insights resolve a number of puzzles in the literature. For example, employing the Feynman gauge fixing term for scalar quantum electrodynamics on de Sitter [@AJ] produces a one loop self-mass-squared which possesses on-shell singularities [@KW1]. These singularities seem to be the quantum field theory representation of what one would expect classically from an $A_0
J^0$ interaction energy in view of the erroneous temporal growth of $A_0$ in Feynman gauge. The simplest noncovariant gauge [@RPW] fails to show on-shell singularities [@KW1]. Nor is there any problem using the de Sitter invariant, Lorentz gauge propagator [@TW2; @PTW1]. The conclusion for de Sitter electromagnetism is therefore that one must avoid adding covariant gauge fixing terms, but no physical breaking of de Sitter invariance occurs.
The situation for gravitons is different owing to infrared divergences. It has long been noted that certain discrete choices of the two covariant gauge fixing parameters result in infrared divergences if one insists on a de Sitter invariant solution [@IAEM; @Folacci]. These choices had been dismissed as unphysical, “singular gauges” which must simply be avoided [@Higuchi]. However, we can now see that they are precisely the cases for which the order of the omnipresent infrared divergence in the formal, de Sitter invariant mode sum changes from power law to logarithmic [@MTW2]. The power law infrared divergences of other choices were automatically — but incorrectly — subtracted by analytic regularization techniques to produce solutions of the propagator equations that are not true propagators. The correct procedure in all cases is to allow free gravitons to resolve their infrared problem by breaking de Sitter invariance.
The purpose of this note is to construct the graviton propagator in an allowed covariant gauge, without employing analytic continuation techniques to subtract off infrared divergences. Our procedure is to express the propagator in terms of covariant projectors acting on scalar structure functions, without making any assumption about the eventual de Sitter invariance of the result. These structure functions obey completely de Sitter invariant equations, but they fail to possess de Sitter invariant solutions on account of infrared divergences. The procedure is so general that we implement it as well for a vector particle of general mass $M_V$, and check that it agrees with the known de Sitter invariant solutions [@MTW2] for $M_V^2 > -2(D-1) H^2$ in the transverse sector and $M_V^2 > 0$ in the longitudinal sector. When de Sitter breaking must occur we have chosen to give explicit solutions which preserve the symmetries of homogeneity and isotropy that are relevant to cosmology. However, our equations for the structure functions are invariant, so one can easily derive solutions which respect any of the allowed subgroups.
Our notation is laid out in section 2. Section 3 presents a general treatment for minimally coupled scalars of any mass $M_S$. In section 4 we solve for the propagator of a vector with general mass $M_V$, including longitudinal and transverse parts. Section 5 applies the same technique to solve for the graviton propagator in de Donder gauge. Our results are summarized and discussed in section 6.
Because this work represents a long and mostly technical exercise we have thought it right to briefly discuss the physical motivation. The point is to facilitate the study of quantum effects from the epoch of primordial inflation for which the de Sitter geometry provides an excellent paradigm.[^2] The source of these effects is particle production. The small amount of particle production which has long been known to occur in an expanding universe [@old] becomes explosive during inflation for any particle which is both massless and not conformally invariant on the classical level [@Parker]. This includes massless, minimally coupled scalars and gravitons [@Grishchuk]. Of course this phenomenon is the origin of the tensor [@Starobinsky] and scalar [@Mukhanov] perturbations which are such an exciting tree order prediction of inflation. Our motivation is getting at the fascinating loop effects which should also be present.
There have been extensive studies of the quantum loop effects from inflation producing massless, minimally coupled scalars. In three models there are complete, dimensionally regulated and fully renormalized results:
- [For a real scalar with a quartic self-interaction there are one and two loop results for the expectation value of the stress tensor [@OW] and the self-mass-squared [@BOW]. These show that inflationary particle production pushes the scalar up its potential, which increases the vacuum energy and leads to a violation of the weak energy condition on cosmological scales without any instability.]{}
- [For a massless fermion which is Yukawa-coupled to a real scalar there are one loop results for the fermion self-energy [@PWGP], for the scalar self-mass-squared [@DW], and for the effective potential [@MW1]. There is also a two loop computation of the coincident vertex function [@MW1]. These results show that the inflationary production of scalars endows super-horizon fermions with a mass, which decreases the vacuum energy without bound in such a way that the universe eventually undergoes a Big Rip singularity [@MW1].]{}
- [For scalar quantum electrodynamics there are one loop results for the vacuum polarization [@PTW] and the scalar self-mass-squared [@KW1; @PTW1], and there are two loop results for coincident scalars [@PTW1], for coincident field strength tensors [@PTW2], and for the expectation value of the stress tensor [@PTW2]. These results show that the inflationary production of charged scalars causes super-horizon photons develop a mass, while the scalar remains light and the vacuum energy decreases slightly [@PTW3].]{}
Scalar effects are simpler than those from gravitons because there is no issue about general coordinate invariance. They are also generally stronger because they can avoid derivative interactions. However, scalar effects are correspondingly less universal and less reliable because they depend upon the existence of light, minimally coupled scalars at inflationary scales. In four models with gravitons there are complete, dimensionally regulated and fully renormalized results:
- [For pure quantum gravity the graviton 1-point function has been computed at one loop order [@TW3]. This result shows that the effect of inflationary gravitons at one loop order is a slight increase in the cosmological constant.]{}
- [For quantum gravity plus a massless fermion the fermion self-energy has been computed at one loop order [@MW2]. This result shows that spin-spin interactions with inflationary gravitons drive the fermion field strength up by an amount that increases without bound [@MW3].]{}
- [For quantum gravity plus a massless, minimally coupled scalar there are one loop computations of the scalar self-mass-squared [@KW2] and the graviton self-energy [@PW]. The scalar effective field equations reveal that the scalar kinetic energy redshifts too rapidly for there to be a significant interaction with inflationary gravitons [@KW2]. The effects of inflationary scalars on dynamical gravitons, and on the force of gravity, are still under study [@PW].]{}
- [The nonlinear sigma model has been exploited to better understand the derivative interactions of quantum gravity [@TW4], and explicit two loop results have been obtained for the expectation value of the stress tensor [@KK].]{}
There are also a variety of other, sometimes less complete results, including:
- [In pure quantum gravity, a very early, approximate computation of the one loop graviton 1-point function was made [@Ford], as well as a later evaluation using adiabatic regularization [@Finelli]. Momentum cutoff computations of the one loop graviton self-energy [@TW5] and the two loop graviton 1-point function [@TW6] have also been done.]{}
- [For gravity plus a scalar the one loop scalar contribution to the noncoincident (and hence unregulated) graviton self-energy has been computed [@Albert].]{}
- [In scalar-driven inflation there have been computations of the one loop back-reaction [@back], culminating in the realization that a physical measure of the expansion rate shows no significant effect at one loop order [@measure]. (This issue is still open at two loop order, and as well in pure gravity [@puremeasure].) There has also been a vast amount of work on loop corrections to the scalar power spectrum [@many], including corresponding work on how to correctly quantify effects [@gauge], and a powerful theorem by Weinberg which limits the maximum rate at which corrections can display secular growth [@Weinberg]. Recently there has been renewed attention to the problem of untangling infrared effects from ultraviolet divergences [@UVIR].]{}
- [In gravity plus generic matter much interest has been devoted to the recent proposal by Polyakov [@Polyakov] (following numerous antecedents [@oldclaims; @Ford; @IAEM; @TW6; @back]) that runaway particle production may destabilize de Sitter space [@destab].]{}
Notation
========
In the coming sections we shall study the de Sitter background propagators of three kinds of fields: minimally coupled scalars with arbitrary mass $M_S$, vectors with arbitrary mass $M_V$, and gravitons. The respective Lagrangians are, $$\begin{aligned}
\mathcal{L}_{S} & = & -\frac12 \partial_{\mu} \varphi \partial_{\nu}
\varphi g^{\mu\nu} \sqrt{-g} - \frac12 M_S^2 \varphi^2 \sqrt{-g} \;
, \label{scalarL} \\
\mathcal{L}_{V} & = & -\frac12 \partial_{\mu} A_{\rho}
\partial_{\nu} A_{\sigma} g^{\mu\nu} g^{\rho\sigma} \sqrt{-g}
-\frac12 \Bigl[(D \!-\! 1) H^2 \!+\! M_V^2 \Bigr] A_{\rho}
A_{\sigma} g^{\rho\sigma} \sqrt{-g} \; , \label{vectorL} \\
\mathcal{L}_G & = & \frac1{16 \pi G} \Bigl[R - (D\!-\!2) (D\!-\!1)
H^2\Bigr] \sqrt{-g} \; . \label{gravitonL}\end{aligned}$$ Here $D$ is the dimension of spacetime, $H$ is the Hubble constant (which gives the cosmological constant $(D \!-\! 1) H^2 = \Lambda$) and $G$ is Newton’s constant. We make no assumption that the vector is transverse, although the form of its mass term in (\[vectorL\]) obviously derives from partially integrating and commuting covariant derivatives in the Maxwell Lagrangian, and then adding a spurious longitudinal kinetic term. The propagator of such a field appears in projection operators, even though the associated field cannot be dynamical.
We define the graviton field as the perturbation of the full metric $g_{\mu\nu}(x)$ about its background value $\overline{g}_{\mu\nu}$, $$h_{\mu\nu}(x) \equiv g_{\mu\nu}(x) - \overline{g}_{\mu\nu}(x) \; .
\label{graviton}$$ Once this definition has been made, there is no more point to distinguishing the background from the full metric so we drop the overbar and refer to the de Sitter background as simply $g_{\mu\nu}(x)$. Graviton indices are raised and lowered using this background field, for example, $h^{\mu}_{~\nu} \equiv g^{\mu\rho}
h_{\rho\nu}$. Covariant derivative operators $D_{\mu}$ and other geometrical quantities are similarly constructed with respect to the background. Of special importance is the Lichnerowicz operator which, when simplified using the de Sitter result for the curvature $R_{\rho\sigma\mu\nu} = H^2 (g_{\mu \rho} g_{\nu\sigma} -
g_{\mu\sigma} g_{\nu\rho})$, takes the form, $$\begin{aligned}
\lefteqn{\mathbf{D}^{\mu\nu\rho\sigma} \equiv D^{(\rho} g^{\sigma)
(\mu} D^{\nu)} -\frac12 \Bigl[ g^{\mu\nu} D^{\rho} D^{\sigma} \!+\!
g^{\rho\sigma} D^{\mu} D^{\nu} \Bigr] } \nonumber \\
& & \hspace{2cm} + \frac12 \Bigl[ g^{\mu\nu} g^{\rho\sigma} \!-\!
g^{\mu (\rho} g^{\sigma) \nu}\Bigr] \square + (D\!-\!1) \Bigl[
\frac12 g^{\mu\nu} g^{\rho\sigma} \!-\! g^{\mu (\rho} g^{\sigma)
\nu} \Bigr] H^2 \; . \qquad \label{Lichnerowicz}\end{aligned}$$ Here and henceforth parenthesized indices are symmetrized, and $\square \equiv g^{\mu\nu} D_{\mu} D_{\nu}$ is the covariant d’Alembertian operator. With the help of (\[Lichnerowicz\]) we can express the free part of the gravitational Lagrangian (\[gravitonL\]) in a convenient form, $$\mathcal{L}_{G} = \frac{(D \!-\! 1) H^2}{8 \pi G} \sqrt{-g} + \Bigl(
{\rm Surface\ Term}\Bigr) - \frac12 h_{\mu\nu}
\mathbf{D}^{\mu\nu\rho\sigma} h_{\rho\sigma} \sqrt{-g} + O(h^3) \; .$$
Much of our work will be valid in any coordinate realization of de Sitter space. However, when breaking de Sitter is necessary we shall always do so on the $D$-dimensional open conformal submanifold in which de Sitter can be imagined as a special case of the larger class of homogeneous, isotropic and spatially flat geometries relevant to cosmology. A spacetime point $x^{\mu} = (x^0, x^i)$ takes values in the ranges, $$-\infty < x^0 < 0 \quad {\rm and} \quad -\infty < x^i < +\infty
\quad {\rm for} \quad i = 1,\ldots,(D\!-\!1) \; .$$ In these coordinates the invariant element is, $$ds^2 \equiv g_{\mu\nu} dx^{\mu} dx^{\nu} = a_x^2 \Bigl[-(dx^0)^2 +
d\vec{x} \!\cdot\! d\vec{x} \Bigr] = a_x^2 \eta_{\mu\nu} dx^{\mu}
dx^{\nu}\; ,$$ where $\eta_{\mu\nu}$ is the Lorentz metric and $a_x \equiv -1/Hx^0$ is the scale factor.
Although important de Sitter breaking occurs, it turns out that the vast majority of our propagator is de Sitter invariant. This suggests to express it in terms of the de Sitter invariant length function $y(x;z)$, $$y(x;z) \equiv a_x a_z H^2 \Biggl[ \Bigl\Vert \vec{x} \!-\! \vec{z}
\Bigr\Vert^2 - \Bigl(\vert x^0 \!-\! z^0\vert \!-\! i
\epsilon\Bigr)^2 \Biggr]\; . \label{ydef}$$ Except for the factor of $i\epsilon$ (whose purpose is to enforce Feynman boundary conditions) the function $y(x;z)$ is closely related to the invariant length $\ell(x;z)$ from $x^{\mu}$ to $z^{\mu}$, $$y(x;z) = 4 \sin^2\Bigl( \frac12 H \ell(x;z)\Bigr) \; .$$
With this de Sitter invariant quantity $y(x;z)$, we can form a convenient basis of de Sitter invariant bi-tensors. Note that because $y(x;z)$ is de Sitter invariant, so too are covariant derivatives of it. With the metrics $g_{\mu\nu}(x)$ and $g_{\mu\nu}(z)$, the first three derivatives of $y(x;z)$ furnish a convenient basis of de Sitter invariant bi-tensors [@KW1], $$\begin{aligned}
\frac{\partial y(x;z)}{\partial x^{\mu}} & = & H a_x \Bigl(y
\delta^0_{\mu}
\!+\! 2 a_z H \Delta x_{\mu} \Bigr) \; , \label{dydx} \\
\frac{\partial y(x;z)}{\partial z^{\nu}} & = & H a_z \Bigl(y
\delta^0_{\nu}
\!-\! 2 a_x H \Delta x_{\nu} \Bigr) \; , \label{dydz} \\
\frac{\partial^2 y(x;z)}{\partial x^{\mu} \partial z^{\nu}} & = &
H^2 a_x a_z \Bigl(y \delta^0_{\mu} \delta^0_{\nu} \!+\! 2 a_z H
\Delta x_{\mu} \delta^0_{\nu} \!-\! 2 a_x \delta^0_{\mu} H \Delta
x_{\nu} \!-\! 2 \eta_{\mu\nu}\Bigr) \; . \qquad \label{dydxdx'}\end{aligned}$$ Here and subsequently we define $\Delta x_{\mu} \equiv \eta_{\mu\nu}
(x \!-\!z)^{\nu}$. Acting covariant derivatives generates more basis tensors, for example [@KW1], $$\begin{aligned}
\frac{D^2 y(x;z)}{Dx^{\mu} Dx^{\nu}}
& = & H^2 (2 \!-\!y) g_{\mu\nu}(x) \; , \label{covdiv} \\
\frac{D^2 y(x;z)}{Dz^{\mu} Dz^{\nu}} & = & H^2 (2 \!-\!y)
g_{\mu\nu}(z) \; .\end{aligned}$$ The contraction of any pair of the basis tensors also produces more basis tensors [@KW1], $$\begin{aligned}
g^{\mu\nu}(x) \frac{\partial y}{\partial x^{\mu}} \frac{\partial
y}{\partial x^ {\nu}} & = & H^2 \Bigl(4 y - y^2\Bigr) =
g^{\mu\nu}(z) \frac{\partial y}{
\partial z^{\mu}} \frac{\partial y}{\partial z^{\nu}} \; ,
\label{contraction1}\\
g^{\mu\nu}(x) \frac{\partial y}{\partial x^{\nu}} \frac{\partial^2
y}{
\partial x^{\mu} \partial z^{\sigma}} & = & H^2 (2-y) \frac{\partial y}{
\partial z^{\sigma}} \; ,
\label{contraction2}\\
g^{\rho\sigma}(z) \frac{\partial y}{\partial z^{\sigma}}
\frac{\partial^2 y}{\partial x^{\mu} \partial z^{\rho}} & = & H^2
(2-y) \frac{\partial y}{\partial x^{\mu}} \; ,
\label{contraction3}\\
g^{\mu\nu}(x) \frac{\partial^2 y}{\partial x^{\mu} \partial
z^{\rho}} \frac{\partial^2 y}{\partial x^{\nu} \partial z^{\sigma}}
& = & 4 H^4 g_{\rho\sigma}(z) - H^2 \frac{\partial y}{\partial
z^{\rho}} \frac{\partial y}{\partial z^{\sigma}} \; ,
\label{contraction4}\\
g^{\rho\sigma}(z) \frac{\partial^2 y}{\partial x^{\mu}\partial
z^{\rho}} \frac{\partial^2 y}{\partial x^{\nu} \partial z^{\sigma}}
& = & 4 H^4 g_{\mu\nu}(x) - H^2 \frac{\partial y}{\partial x^{\mu}}
\frac{\partial y}{\partial x^{\nu}} \; . \label{contraction5}\end{aligned}$$
Scalar Propagators
==================
Scalar propagator equations play an important role in our analysis because our strategy is to enforce the de Donder gauge condition, without making assumptions about de Sitter invariance, using covariant derivative projectors acting on scalar structure functions. The graviton propagator equation will then be used to infer invariant equations for these scalar structure functions. The point of this section is to review and systematize previous work [@MTW1; @MTW2] about how to solve such equations. We begin giving a general scalar propagator equation and explaining why infrared divergences for $M_S^2 \leq 0$ preclude a de Sitter invariant solution. We review the two fixes in the literature, and then give a simple approximate implementation for our favorite one. The section closes with some powerful results for integrating propagators.
One can see from (\[scalarL\]) that the propagator of a minimally coupled scalar with mass $M_S$ obeys the equation, $$\Bigl[ \square - M_S^2\Bigr] i\Delta(x;z) = \frac{i \delta^D(x \!-\!
z)}{\sqrt{-g}} \; . \label{nuprop}$$ The plane wave mode function corresponding to Bunch-Davies vacuum is [@BD], $$u_{\nu}(x^0,k) \equiv \sqrt{\frac{\pi}{4 H}} \; a_x^{-\frac{D-1}2}
\, H^{(1)}_{\nu}(-k x^0) \quad {\rm where} \quad \nu =
\sqrt{\Bigl(\frac{D \!-\!1}2\Bigr)^2 \!-\! \frac{M_S^2}{H^2}} \; .
\label{unu}$$ The Fourier mode sum for the propagator on infinite space is, $$\begin{aligned}
\lefteqn{i\Delta^{\rm dS}_{\nu}(x;z) = \int \!\!
\frac{d^{D-1}k}{(2\pi)^{D-1}} \, e^{i \vec{k} \cdot (\vec{x} -
\vec{z})} \Biggl\{ \theta(x^0 \!-\! z^0)
u_{\nu}(x^0,k) u^*_{\nu}(z^0,k) } \nonumber \\
& & \hspace{6cm} + \theta(z^0 \!-\! x^0) u_{\nu}(x^0,k)
u_{\nu}(z^0,k) \Biggr\} . \qquad \label{modesum}\end{aligned}$$ The result is de Sitter invariant when the integral converges [@CT; @JMPW1], $$\begin{aligned}
\lefteqn{i\Delta^{\rm dS}_{\nu}(x;z) } \nonumber \\
& & \hspace{-.5cm} = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}}
\frac{\Gamma(\frac{D-1}2 \!+\! \nu) \Gamma(\frac{D-1}2 \!-\! \nu)}{
\Gamma(\frac{D}2)} \, \mbox{}_2 F_1\Bigl( \frac{D-1}2 \!+\! \nu,
\frac{D-1}2 \!-\! \nu; \frac{D}2;1 \!-\! \frac{y}4\Bigr) \; , \qquad \\
& & \hspace{-.5cm} = \frac{H^{D-2} \Gamma(\frac{D}2 \!-\! 1)}{(4
\pi)^{\frac{D}2}} \Biggl\{ \Bigl( \frac{4}{y}\Bigr)^{\frac{D}2 -1}
\mbox{}_2 F_1\Bigl(\frac12 \!+\!\nu,\frac12 \!-\! \nu ; 2 \!-\!
\frac{D}2 ; \frac{y}4\Bigr) \qquad \nonumber \\
& & + \frac{\Gamma(\frac{D-1}2 \!+\!\nu) \Gamma(\frac{D-1}2 \!-\!
\nu) \Gamma(1 \!-\! \frac{D}2)}{\Gamma(\frac12 \!+\! \nu)
\Gamma(\frac12 \!-\! \nu) \Gamma(\frac{D}2 \!-\! 1)} \, \mbox{}_2
F_1\Bigl(\frac{D-1}2 \!+\! \nu , \frac{D-1}2 \!-\! \nu ;
\frac{D}2 ; \frac{y}4\Bigr) \Biggr\} , \qquad \\
& & \hspace{-.5cm} = \frac{H^{D-2}}{(4 \pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\! 1\Bigr)
\Bigl(\frac{4}{y}\Bigr)^{\frac{D}2-1}
\nonumber \\
& & \hspace{2cm} - \frac{\Gamma(\frac{D}2) \Gamma(1 \!-\!
\frac{D}2)}{ \Gamma(\frac12 \!+\! \nu) \Gamma(\frac12 \!-\! \nu)}
\sum_{n=0}^{\infty} \Biggl[ \frac{\Gamma(\frac32 \!+\! \nu \!+\! n)
\Gamma(\frac32 \!-\! \nu \!+\! n)}{ \Gamma(3 \!-\! \frac{D}2 \!+\!
n) \, (n \!+\! 1)!} \Bigl(\frac{y}4
\Bigr)^{n - \frac{D}2 +2} \nonumber \\
& & \hspace{5cm} - \frac{\Gamma(\frac{D-1}2 \!+\! \nu \!+\! n)
\Gamma(\frac{D-1}2 \!-\! \nu \!+\! n)}{\Gamma(\frac{D}2 \!+\! n) \,
n!} \Bigl(\frac{y}4\Bigr)^n \Biggr] \Biggr\} . \qquad
\label{expansion}\end{aligned}$$
The gamma function $\Gamma(\frac{D-1}2 -\nu + n)$ on the final line of (\[expansion\]) diverges for, $$\nu = \Bigl(\frac{D\!-\!1}{2}\Bigr) + N \qquad \Longleftrightarrow
\qquad M_S^2 = - N(D \!-\!1 \!+\! N) H^2 \; . \label{logprob}$$ Its origin can be understood by performing the angular integration in the naive mode sum (\[modesum\]) and then changing to the dimensionless variable $\tau \equiv k/H\sqrt{a_x a_z}$, $$\begin{aligned}
\lefteqn{ i\Delta^{\rm dS}_{\nu}(x;z) = \frac{ (a_x
a_z)^{-(\frac{D-1}2)}}{ 2^D \pi^{\frac{D-3}2} H} \int_0^{\infty}
\!\! dk \, k^{D-2} \Bigl( \frac12 k \Delta x\Bigr)^{-(\frac{D-3}2)}
J_{\frac{D-3}2}(k \Delta x) } \nonumber \\
& & \times \Biggl\{ \theta(x^0 \!-\! z^0) H_{\nu}^{(1)}(-k x^0)
H_{\nu}^{(1)}(-k z^0)^* + \theta(z^0 \!-\! x^0) \Bigl({\rm
conjugate}
\Bigr) \Biggr\} , \qquad \\
& & \hspace{-.5cm} = \frac{H^{D-2}}{2^D \pi^{\frac{D-3}2}}
\int_0^{\infty} \!\! d\tau \, \tau^{D-2} \Bigl( \frac12 \sqrt{a_x
a_z} \, H \Delta x \tau \Bigr)^{-(\frac{D-3}2)}
J_{\frac{D-3}2}\Bigl( \sqrt{a_x a_z} \,
H \Delta x \tau\Bigr) \nonumber \\
& & \times \Biggl\{ \theta(x^0 \!-\! z^0) H_{\nu}^{(1)}\Bigl(
\sqrt{\frac{a_z}{a_x}} \, \tau\Bigr)
H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr)^* +
\theta(z^0 \!-\! x^0) \Bigl({\rm conjugate}\Bigr) \Biggr\} . \qquad
\label{integral}\end{aligned}$$ In these and subsequent expressions we define $\Delta x \equiv \Vert
\vec{x} \!-\! \vec{z}\Vert$. That the divergence at (\[logprob\]) is infrared can be seen from the small argument expansion of the Bessel function and from its relation to the Hankel function, $$\begin{aligned}
J_{\nu}(x) & = & \sum_{n=0}^{\infty} \frac{ (-1)^n (\frac12 x)^{\nu
+2n}}{
n! \Gamma(\nu \!+\! n \!+\! 1)} \; , \\
H_{\nu}^{(1)}(x) & = & \frac{i \Gamma(\nu) \Gamma(1 \!-\! \nu)}{\pi}
\Bigl\{ e^{-i\nu \pi} J_{\nu}(x) \!-\! J_{-\nu}(x)\Bigr\} \; .\end{aligned}$$ The small $\tau$ behavior of the integrand (\[integral\]) derives from three factors, the first being $\tau^{D-2}$. The second factor takes the form, $$\Bigl( \frac12 \sqrt{a_x a_z} \, H \Delta x
\tau\Bigr)^{-(\frac{D-3}2)} J_{\frac{D-3}2}\Bigl( \sqrt{a_x a_z} \,
H \Delta x \tau\Bigr) = \frac1{\Gamma(\frac{D-1}2)}
\sum_{n=0}^{\infty} C_1(n) \tau^{2n} \; .$$ And the final factor from the Hankel functions is, $$H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_z}{a_x}} \, \tau\Bigr)
H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr)^* = \frac{2
\Gamma(\nu) \Gamma(2 \nu)}{\pi^{\frac32} \Gamma(\nu \!+\! \frac12)
\tau^{2\nu} } \sum_{n=0}^{\infty} C_2(n) \tau^{2n} \; .$$ Hence the small $\tau$ expansion of the integrand has the form, $$\begin{aligned}
\lefteqn{\tau^{D-2} \times \frac1{\Gamma(\frac{D-1}2)}
\sum_{k=0}^{\infty} C_1(k) \tau^{2k} \times \frac{\Gamma^2(\nu)
2^{2\nu}}{\pi^2 \tau^{2\nu} }
\sum_{\ell=0}^{\infty} C_2(\ell) \tau^{2\ell} } \nonumber \\
& & \hspace{4cm} = \frac{2 \Gamma(\nu) \Gamma(2\nu)}{\pi^{\frac32}
\Gamma(\frac{D-1}2) \Gamma(\nu \!+\! \frac12)} \, \tau^{D-2-2\nu}
\sum_{n=0}^{\infty} C_3(n) \tau^{2n} \; . \qquad \label{IR}\end{aligned}$$ The naive mode sum (\[modesum\]) is infrared divergent for $$D - 2 - 2\nu \leq -1 \qquad \Longleftrightarrow \qquad M_S^2 \leq 0
\; . \label{truediv}$$ However, there will only be a [*logarithmic*]{} infrared divergence, either from the leading term in (\[IR\]) or from one of the series corrections at $n = N$, if one has, $$D - 2 - 2\nu + 2N = -1 \qquad \Longleftrightarrow \qquad M_S^2 = -N
(D \!-\! 1 \!+\! N) H^2 \; .$$ This is precisely the condition (\[logprob\]) for the formal, de Sitter invariant mode sum (\[expansion\]) to diverge.
The infrared divergence we have just seen was first noted in 1977 for the special case of $M_S = 0$ by Ford and Parker [@FP]. The appearance of an infrared divergence signals that something is unphysical about the quantity being computed. The correct response to an infrared divergence is not to subtract it off, either explicitly or implicitly with the automatic subtraction of some analytic regularization technique. One must instead understand the physical problem which caused the divergence and then fix that problem. As we will see, the fix involves breaking de Sitter invariance, which was realized in 1982 for the special case of $M_S
= 0$ [@classic]. Allen and Folacci later gave a rigorous proof that de Sitter invariance must be broken [@AF].
The divergence (\[truediv\]) occurs because of the way the Bunch-Davies mode functions (\[unu\]) depend upon $k$ for small $k$. The unphysical thing about having Bunch-Davies vacuum for arbitrarily small $k$ is that no experimentalist can causally enforce it (or any other condition) for super-horizon modes. This has led to two fixes:
1. [One can continue to work on the spatial manifold $R^{D-1}$ but assume the initial state is released with its super-horizon modes in some less singular condition [@AV]; or]{}
2. [One can work on the compact spatial manifold $T^{D-1}$ with its coordinate radius chosen so the initial state has no super-horizon modes [@TW7].]{}
We will adopt the latter fix. This makes the mode sum discrete, but the integral approximation should be excellent, and gives a simple expression for the propagator which differs from (\[modesum\]) only by an infrared cutoff at $k = H$.
From the preceding discussion we see that the infrared corrected propagator $i\Delta(x;z)$ is just (\[integral\]) with the lower limit cutoff at $\tau = 1/\sqrt{a_x a_z}$, $$\begin{aligned}
\lefteqn{i\Delta(x;z) = \frac{H^{D-2}}{2^D \pi^{\frac{D-3}2}}
\int_{\frac1{\sqrt{a_x a_z}}}^{\infty} \!\!\!\!\! d\tau \,
\tau^{D-2} \frac{J_{\frac{D-3}2}( \sqrt{a_x a_z} \, H \Delta x
\tau)}{(\frac12
\sqrt{a_x a_z} \, H \Delta x \tau)^{\frac{D-3}2} } } \nonumber \\
& & \times \Biggl\{ \theta(x^0 \!-\! z^0) H_{\nu}^{(1)}\Bigl(
\sqrt{\frac{a_z}{a_x}} \, \tau\Bigr)
H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr)^* +
\theta(z^0 \!-\! x^0) \Bigl({\rm conjugate}\Bigr) \Biggr\} . \qquad\end{aligned}$$ Of course we can express the truncated integral as the full one minus an integral over just the infrared, $$\int_{\frac1{\sqrt{a_x a_z}}}^{\infty} \!\!\!\!\! d\tau =
\int_0^{\infty} \!\! d\tau - \int_0^{\frac1{\sqrt{a_x a_z}}} \!\!\!
d\tau \quad \Longleftrightarrow \quad i\Delta(x;z) \equiv
i\Delta^{\rm dS}_{\nu}(x;z) + \Delta^{\rm IR}_{\nu}(x;z) \; .$$ In this case it does not matter if dimensional regularization is used to evaluate both $i\Delta^{\rm dS}_{\nu}(x;z)$ and $\Delta^{\rm
IR}_{\nu}(x;z)$ because the errors we make at the lower limits will cancel.
A further simplification is that $\Delta^{\rm IR}_{\nu}(x;z)$ only needs to include the infrared singular terms which grow as $a_x a_z$ increases. These terms come entirely from the $J_{-\nu}$ parts of the Hankel function and they are entirely real, $$\begin{aligned}
\lefteqn{\Delta^{\rm IR}_{\nu}(x;z) =
-\frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{2 \Gamma(\nu)
\Gamma(2\nu)}{\Gamma(\nu \!+\! \frac12)} \int_0^{\frac1{\sqrt{a_x
a_z}}} \!\!\! d\tau \, \tau^{D-2} \frac{J_{\frac{D-3}2}( \sqrt{a_x
a_z} \, H \Delta x \tau)}{(\frac12
\sqrt{a_x a_z} \, H \Delta x \tau)^{\frac{D-3}2} } } \nonumber \\
& & \hspace{5.5cm} \times \frac{\Gamma^2(1\!-\!\nu)}{2^{2\nu}} \,
J_{-\nu}\Bigl( \sqrt{\frac{a_z}{a_x}} \, \tau\Bigr) J_{-\nu}\Bigl(
\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr) \; . \qquad \label{homo}\end{aligned}$$ The final result is [@JMPW2; @MTW2], $$\begin{aligned}
\lefteqn{\Delta^{\rm IR}_{\nu}(x;z) =
\frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{\Gamma(\nu)
\Gamma(2\nu)}{\Gamma(\frac{D-1}2)
\Gamma(\nu \!+\! \frac12)} } \nonumber \\
& & \hspace{.7cm} \times \sum_{N=0}^{\infty} \frac{(a_x a_z)^{\nu -
(\frac{D-1}2) - N}}{\nu \!-\! (\frac{D-1}2) \!-\! N} \sum_{n=0}^N
\Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)^n \sum_{m=0}^{
[\frac{N-n}2]} C_{Nnm} (y \!-\!2)^{N-n-2m} \; , \qquad
\label{series}\end{aligned}$$ where the coefficients $C_{Nnm}$ are, $$\begin{aligned}
\lefteqn{C_{Nnm} = \frac{(-\frac14)^N}{m! n! (N \!-\!n \!-\! 2m)!}
\times \frac{\Gamma(\frac{D-1}2 \!+\! N \!+\! n \!-\!
\nu)}{\Gamma(\frac{D-1}2
\!+\! N \!-\! \nu)} } \nonumber \\
& & \hspace{2cm} \times
\frac{\Gamma(\frac{D-1}2)}{\Gamma(\frac{D-1}2 \!+\! N\!-\! 2m)}
\times \frac{\Gamma(1 \!-\!\nu)}{\Gamma(1 \!-\! \nu \!+\! n \!+\!
2m)} \times \frac{\Gamma(1 \!-\! \nu)}{\Gamma(1 \!-\! \nu \!+\! m)}
\; . \qquad \label{cdef}\end{aligned}$$ Of course there is no point in extending the sum over $N$ to values $N > \nu -(\frac{D-1}2)$ for which the exponent of $a_x a_z$ becomes negative. Those terms rapidly approach zero, and they can be dropped without affecting the propagator equation because they are separately annihilated by $\square - M_s^2$.
It might be worried that the approximations made in deriving the infrared correction do violence to delicate consistency relations in quantum field theory, but this is not the case. For the $M_S = 0$ scalar renormalization has been successfully implemented at one and two loop orders [@KW1; @PTW1; @OW; @BOW; @PWGP; @MW1; @PTW; @PTW2; @PTW3]. Because the physical graviton polarizations obey the same mode functions as massless, minimally coupled scalars [@Grishchuk], one can also test the integral approximation with the graviton propagator. There is no disruption of powerful consistency checks such as the Ward identity at tree order [@TW8] and one loop [@TW6]. Nor is there any problem with the allowed one loop counterterms [@TW3; @MW2; @KW2; @PW].
It is worthwhile to summarize these results in the context of a consistent notation. Consider a general scalar whose mass obeys $M_s^2/H^2 = (\frac{D-1}2)^2 - b^2$. Its propagator $i\Delta_b(x;z)$ obeys the equation, $$\Bigl[ \square_x + b^2 H^2 - \Bigl(\frac{D \!-\! 1}2\Bigr)^2 H^2
\Bigr] i \Delta_b(x;z) = \frac{i\delta^D( x\!-\! z)}{\sqrt{-g}} \; .
\label{bprop}$$ We define the final result for $i\Delta_b(x;z)$ as the limit as $\nu$ approaches $b$ of two functions which we wish to consider for general index $\nu$. The first term in the sum is $i\Delta^{\rm
dS}_{\nu}(x;z)$ as defined by expression (\[expansion\]). The second term is $\Delta^{\rm IR}_{\nu}(x;z)$, as defined by expression (\[series\]), except that the sum over $N$ is cut off at the largest nonnegative integer for which $N \leq b -
(\frac{D-1}2)$, with $\Delta^{\rm IR}_{\nu}(x;z)$ defined as zero for $b < (\frac{D-1}2)$. Hence our final result is, $$i\Delta_b(x;z) = \lim_{\nu \rightarrow b} \Bigl[ i\Delta^{\rm
dS}_{\nu}(x;z) + \Delta^{\rm IR}_{\nu}(x;z) \Bigr] \; .$$
We shall make significant use of four special cases for which a separate notation has been introduced: $$\begin{aligned}
b_B = \Bigl(\frac{D \!-\! 3}2\Bigr) & \Longleftrightarrow & i\Delta_B(x;z) = B(y) \; , \\
b_A = \Bigl(\frac{D \!-\! 1}2\Bigr) & \Longleftrightarrow &
i\Delta_A(x;z) = A(y) + \delta A(a_x,a_z,y) \; , \label{Adef} \\
b_W = \Bigl(\frac{D \!+\! 1}2\Bigr) & \Longleftrightarrow &
i\Delta_W(x;z) = W(y) + \delta W(a_x,a_z,y) \; , \\
b_M = \frac12 \sqrt{(D \!-\! 1) (D \!+\! 7)} & \Longleftrightarrow &
i\Delta_M(x;z) = M(y) + \delta M(a_x,a_z,y) \; . \label{Mdef}\end{aligned}$$ Although the $B$-type propagator is de Sitter invariant, its $A$-type, $W$-type and $M$-type cousins have de Sitter breaking parts, $$\begin{aligned}
\delta A & = & k \ln(a_x a_z) \; , \label{dsbA} \\
\delta W & = & k \Biggl\{ (D\!-\!1)^2 a_x a_z -
\Bigl(\frac{D\!-\!1}2\Bigr) \ln(a_x a_z) (y \!-\! 2) -
\Bigl(\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) \Biggr\} \; ,
\qquad \\
\delta M & = & k_M \Biggl\{ \frac{ (a_x a_z)^{b_M - b_A}}{b_M \!-\!
b_A} - \frac{ (a_x a_z)^{b_M - b_A - 1}}{b_M \!-\! b_A \!-\! 1}
\times \frac{(y \!-\! 2)}{4 b_A} \nonumber \\
& & \hspace{5cm} + \frac{ (a_x a_z)^{b_M - b_A - 1}}{4 b_A (b_M
\!-\! 1)} \times \Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)
\Biggr\} \; . \qquad\end{aligned}$$ The constants $k$ and $k_M$ are, $$k \equiv \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \, \frac{\Gamma(D \!-\!
1)}{ \Gamma(\frac{D}2)} \qquad , \qquad k_M \equiv
\frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{ \Gamma(b_M)
\Gamma(2b_M)}{\Gamma(b_A) \Gamma(b_M \!+\! \frac12)} \; .$$ The main, de Sitter invariant parts of each propagator consist of a few, potentially ultraviolet divergent terms (at $y=0$), plus an infinite series. For the $M$-type propagator there are no cancelations with the de Sitter breaking terms: just replace $\nu$ everywhere by $b_M$ in expression (\[expansion\]) to find $M(y) =
i\Delta^{\rm dS}_{b_M}(x;z)$. However, there are cancelations when this replacement is done for the $A$-type and $W$ propagators, $$\begin{aligned}
\lefteqn{B(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\!1\Bigr)
\Bigl(\frac{4}{y}\Bigr)^{\frac{D}2 -1} }
\nonumber \\
& & \hspace{1.5cm} + \sum_{n=0}^{\infty} \Biggl[
\frac{\Gamma(n\!+\!\frac{D}2)}{(n \!+\! 1)!} \Bigl(\frac{y}4
\Bigr)^{n - \frac{D}2 +2} \!\!\!\!\! - \frac{\Gamma(n \!+\! D \!-\!
2)}{\Gamma(n \!+\! \frac{D}2)}
\Bigl(\frac{y}4 \Bigr)^n \Biggr] \Biggr\} , \qquad \label{DeltaB} \\
\lefteqn{A(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\!1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{
\frac{D}2 -1} \!+\! \frac{\Gamma(\frac{D}2 \!+\! 1)}{\frac{D}2 \!-\!
2}
\Bigl(\frac{4}{y} \Bigr)^{\frac{D}2-2} \!+\! A_1 } \nonumber \\
& & \hspace{1.5cm} - \sum_{n=1}^{\infty} \Biggl[
\frac{\Gamma(n\!+\!\frac{D}2\!+\!1)}{(n\!-\!\frac{D}2\!+\!2) (n
\!+\! 1)!} \Bigl(\frac{y}4 \Bigr)^{n - \frac{D}2 +2} \!\!\!\!\! -
\frac{\Gamma(n \!+\! D \!-\! 1)}{n \Gamma(n \!+\! \frac{D}2)}
\Bigl(\frac{y}4 \Bigr)^n \Biggr] \Biggr\} , \qquad \label{DeltaA} \\
\lefteqn{W(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\!1\Bigr)
\Bigl(\frac{4}{y}\Bigr)^{\frac{D}2 -1} \!+\! \frac{\Gamma(\frac{D}2
\!+\! 2)}{(\frac{D}2 \!-\! 2) (\frac{D}2 \!-\!1)}
\Bigl(\frac{4}{y} \Bigr)^{\frac{D}2-2} } \nonumber \\
& & \hspace{3cm} \!+\! \frac{\Gamma(\frac{D}2 \!+\! 3)}{2 (\frac{D}2
\!-\! 3) (\frac{D}2\!-\!2)} \Bigl(\frac{4}{y} \Bigr)^{\frac{D}2-3}
\!+\! W_1 \!+\!
W_2 \Bigl(\frac{y \!-\!2}4\Bigr) \nonumber \\
& & \hspace{1.5cm} + \sum_{n=2}^{\infty} \Biggl[
\frac{\Gamma(n\!+\!\frac{D}2\!+\!2)
(\frac{y}4)^{n-\frac{D}2+2}}{(n\!-\! \frac{D}2\!+\!2) (n \!-\!
\frac{D}2 \!+\!1) (n \!+\! 1)!} - \frac{\Gamma(n \!+\! D)
(\frac{y}4)^n }{n (n \!-\!1) \Gamma(n \!+\! \frac{D}2)} \Biggr]
\Biggr\} , \qquad \label{DeltaW}\end{aligned}$$ And the $D$-depdendent constants $A_1$, $W_1$ and $W_2$ are, $$\begin{aligned}
A_1 & = & \frac{\Gamma(D\!-\!1)}{\Gamma(\frac{D}2)} \Biggl\{
-\psi\Bigl(1 \!-\! \frac{D}2\Bigr) + \psi\Bigl(\frac{D\!-\!1}2\Bigr)
+
\psi(D \!-\!1) + \psi(1) \Biggr\} , \\
W_1 & = & \frac{\Gamma(D\!+\!1)}{\Gamma(\frac{D}2 \!+\!1)} \Biggl\{
\frac{D \!+\!1}{2 D} \Biggr\} , \\
W_2 & = & \frac{\Gamma(D\!+\!1)}{\Gamma(\frac{D}2 \!+\!1)} \Biggl\{
\psi\Bigl(-\frac{D}2\Bigr) - \psi\Bigl(\frac{D\!+\!1}2\Bigr) -
\psi(D \!+\!1) - \psi(1) \Biggr\} .\end{aligned}$$
A problem we shall often encounter consists of integrating one propagator against another. The result can be represented as the solution of a modified propagator equation, $$\Bigl[ \square + b^2 H^2 - \Bigl(\frac{D \!-\! 1}2\Bigr)^2
H^2 \Bigr] i \Delta_{bc}(x;z) = i\Delta_c(x;z) \; . \label{Int1}$$ The solution is easily seen to be [@MTW1; @MTW2], $$i\Delta_{bc}(x;z) = \frac1{(b^2 \!-\! c^2) H^2} \Bigl[
i\Delta_c(x;z) \!-\! i\Delta_b(x;z)\Bigr] = i\Delta_{cb}(x;z) \; .
\label{Int2}$$ For the special case that the indices $b$ and $c$ agree one gets a derivative, $$i\Delta_{bb}(x;z) = -\frac1{2 b H^2} \frac{\partial}{\partial b}
i\Delta_b(x;z) \; . \label{Int3}$$
We can obviously continue the process [*ad infinitum*]{}. For example, consider the case where the source is not a propagator but rather a singly integrated propagator, $$\Bigl[ \square + b^2 H^2 - \Bigl(\frac{D \!-\! 1}2\Bigr)^2 H^2
\Bigr] i \Delta_{bcd}(x;z) = i\Delta_{cd}(x;z) \; . \label{Int4}$$ The solution can be written in a form which is manifestly symmetric under any interchange of the three indices $a$, $b$ and $c$, $$\begin{aligned}
i\Delta_{bcd}(x;z) & = & \frac{ i\Delta_{bd}(x;z) \!-\!
i\Delta_{bc}(x;z)}{(c^2 \!-\! d^2) H^2} \; , \qquad \label{Int5a} \\
& = & \frac{ (d^2 \!-\! c^2) i\Delta_{b}(x;z) \!+\! (b^2 \!-\! d^2)
i\Delta_{c}(x;z) \!+\! (c^2 \!-\! b^2) i\Delta_{d}(x;z) }{(b^2 \!-\!
c^2) (c^2 \!-\! d^2) (d^2 \!-\! b^2) H^4} \; . \qquad \label{Int5b}\end{aligned}$$ The case in which two of the indices are the same gives, $$i\Delta_{bcc}(x;z) = -\frac1{2 c H^2} \frac{\partial}{\partial c}
i\Delta_{bc}(x;z) = \frac{ i\Delta_{cc}(x;z) \!-\!
i\Delta_{bc}(x;z)}{ (b^2 \!-\! c^2) H^2} \; . \label{Int6}$$ And equating all three indices produces, $$\begin{aligned}
i\Delta_{bbb}(x;z) & = & -\frac1{2 b H^2} \frac{\partial}{\partial
b} i\Delta_{bc}(x;z) \Bigl\vert_{c = b} \; , \qquad \label{Int7a} \\
& = & -\frac1{8 b^3 H^4} \Biggl[ \frac{\partial}{\partial b}
i\Delta_b(x;z) \!-\! b \Bigl(\frac{\partial}{\partial b}\Bigr)^2
i\Delta_b(x;z) \Biggr] \; . \label{Int7b}\end{aligned}$$
Vector Propagators
==================
One can see from (\[vectorL\]) that the vector propagator obeys the equation, $$\Bigl[ \square \!-\! (D \!-\! 1) H^2 \!-\! M_V^2 \Bigr] i
\Bigl[\mbox{}_{\mu} \Delta_{\rho}\Bigr](x;z) = \frac{i g_{\mu\rho}
\delta^D( x\!-\! z)}{\sqrt{-g}} \; . \label{veceqn}$$ Note that we do not assume transversality; indeed, the full vector propagator cannot be transverse because the right hand side of equation (\[veceqn\]) is not transverse. The first part of this section describes how to decompose the full propagator into its transverse and longitudinal parts, [*without*]{} making any assumptions about its eventual de Sitter invariance. Our technique is to express these parts using projectors formed from covariant derivative operators, acting on scalar structure functions. In the second part we derive a scalar equation for the longitudinal structure function and solve it using the techniques of section 3. In the final part we carry out the same analysis for the transverse structure function. The techniques employed here are a paradigm for the work of the subsequent section on the graviton propagator.
Transverse and Logitudinal Parts
--------------------------------
The full vector propagator can be written as the sum of a transverse part and a longitudinal part, $$i\Bigl[\mbox{}_{\mu} \Delta_{\rho}\Bigr](x;z) = i\Bigl[\mbox{}_{\mu}
\Delta^T_{\rho}\Bigr](x;z) + i\Bigl[\mbox{}_{\mu}
\Delta^L_{\rho}\Bigr](x;z) \; . \label{decomp}$$ In previous studies [@AJ; @TW2] the vector propagator was expressed as a linear combination of de Sitter invariant basis tensors like those introduced at the end of section 2. Then the coefficient functions were chosen to enforce transversality (or longitudinality). This method is not open to us because we cannot assume de Sitter invariance for general mass $M_V$. What we require instead is a covariant decomposition which entails no assumption about de Sitter invariance.
The longitudinal part is easy, $$i\Bigl[\mbox{}_{\mu} \Delta^L_{\rho}\Bigr](x;z) \equiv
\frac{\partial}{\partial x^{\mu}} \frac{\partial}{\partial z^{\rho}}
\Bigl[\mathcal{S}_{L}(x;z) \Bigr] \; . \label{longform}$$ This expression is longitudinal for any choice of the longitudinal structure function $\mathcal{F}_{L}(x;z)$. After much consideration we decided to express the transverse part as, $$i\Bigl[\mbox{}_{\mu} \Delta^T_{\rho}\Bigr](x;z) =
\mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \times \Bigl[ \mathcal{R}_{\beta
\lambda}(x;y) \; \mathcal{S}_{T}(x;z)\Bigr] \; . \label{transform}$$ These symbols require explanation. The differential operator $\mathcal{P}^{\alpha\beta}_{\mu}$ is defined by writing the Maxwell field strength tensor as $F^{\alpha\beta} =
\mathcal{P}^{\alpha\beta}_{\mu} A^{\mu}$, $$\mathcal{P}^{\alpha\beta}_{\mu} \equiv \delta^{\beta}_{\mu}
D^{\alpha} \!-\! \delta^{\alpha}_{\mu} D^{\beta} \; .$$ Note that acting $\mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z)$ on any 4-index, symmetric function of $x$ and $z$ produces something with the right properties to be a transverse propagator. Of course some choices for the 4-index function give simpler final results than others! The best selection seems to be taking two of the four indices to be more covariant derivatives and the other two to belong to the section 2 basis tensor (\[dydxdx’\]) which gives a Lorentz metric in the flat space limit. This corresponds to form (\[transform\]) with the definitions, $$\begin{aligned}
\mathcal{Q}_{\alpha\kappa}(x;z) & \equiv & -\frac1{2 H^2}
\frac{D}{D x^{\alpha}} \frac{D}{D z^{\kappa}} \; , \label{Qdef} \\
\mathcal{R}_{\beta\lambda}(x;z) & \equiv & -\frac1{2 H^2}
\frac{\partial^2 y(x;z)}{\partial x^{\beta} \partial z^{\lambda}} \;
. \label{Rdef}\end{aligned}$$
Solution for the Longitudinal Part
----------------------------------
To derive an equation for the longitudinal structure function we take the divergence of the full propagator equation (\[veceqn\]), substitute relations (\[decomp\]), (\[longform\]) and (\[transform\]), and then commute the derivative to the left, $$\begin{aligned}
\lefteqn{D_z^{\rho} \Bigl[\square_x \!-\! (D\!-\!1) H^2 \!-\!
M_V^2\Bigr] i\Bigl[\mbox{}_{\mu}\Delta_{\rho}\Bigr](x;z) } \nonumber
\\
& & \hspace{4cm} = \Bigl[\square_x \!-\! (D\!-\!1) H^2 \!-\!
M_V^2\Bigr] \frac{\partial}{\partial
x^{\mu}} \square_z \mathcal{S}_{L}(x;z) \; , \qquad \\
& & \hspace{4cm} = D_{\mu}^x \Bigl[\square_x \!-\! M_V^2\Bigr]
\square_z \mathcal{S}_{L}(x;z) \; , \qquad \\
& & \hspace{4cm} = -D_{\mu}^x \Biggl( \frac{i \delta^D(x \!-\!
z)}{\sqrt{-g}} \Biggr) \; .\end{aligned}$$ Hence we conclude, $$\Bigl[\square_x \!-\! M_V^2\Bigr] \square_z \mathcal{S}_{L}(x;z) =
-\frac{i\delta^D(x \!-\! z)}{\sqrt{-g}} \; .$$ From relation (\[bprop\]) of section 3 this implies, $$\square_z \mathcal{S}_{L}(x;z) = -i\Delta_b(x;z) \qquad {\rm for}
\qquad b^2 = \Bigl(\frac{D\!-\!1}2\Bigr)^2 - \frac{M_V^2}{H^2} \; .$$ The final solution for $\mathcal{S}_{L}$ follows from relations (\[Int1\]-\[Int2\]), $$\mathcal{S}_{L}(x;z) = \frac1{M_V^2} \Bigl[ i\Delta_A(x;z) \!-\!
i\Delta_b(x;z)\Bigr] = -i\Delta_{Ab}(x;z) \; . \label{longsol}$$ We remind the reader of special case $A$ with index $b_A =
(\frac{D-1}2)$ and the explicit expansion for $i\Delta_{A}(x;z)$ given by expressions (\[Adef\]), (\[dsbA\]) and (\[DeltaA\]).
Solution for the Transverse Part
--------------------------------
Substituting our explicit solution (\[longsol\]) for the longitudinal structure function into the full propagator equation (\[veceqn\]) allows us to derive an equation for the transverse part that was previously guessed [@TW2], $$\Bigl[ \square \!-\! (D \!-\! 1) H^2 \!-\! M_V^2 \Bigr] i
\Bigl[\mbox{}_{\mu} \Delta^T_{\rho}\Bigr](x;z) = \frac{i g_{\mu\rho}
\delta^D( x\!-\! z)}{\sqrt{-g}} + \frac{\partial}{\partial x^{\mu}}
\frac{\partial}{\partial z^{\rho}} i\Delta_A(x;z) \; .
\label{transeqn}$$ The most effective technique for solving this equation is to reduce each side of the equation to the standard transverse form (\[transform\]). We then read off a scalar equation for the transverse structure function $\mathcal{S}_{T}(x;z)$, which can be solved by the methods of section 3.
It is best to begin by establishing some important properties of the quadratic differential operator, $$\mathbf{P}_{\mu}^{~\beta} \equiv \mathcal{P}^{\alpha\beta}_{\mu}
\times D_{\alpha} = \delta_{\mu}^{~\beta} \square - D^{\beta}
D_{\mu} \; . \label{bfP1}$$ We shall always contract $\mathbf{P}_{\mu}^{~\beta}$ into some vector $T_{\beta}$, so it is possible to commute the final covariant derivatives to reach the form, $$\mathbf{P}_{\mu}^{~\beta} T_{\beta} = \Biggl( \delta_{\mu}^{~\beta}
\Bigl[\square \!-\! (D\!-\! 1)H^2\Bigr] - D_{\mu} D^{\beta}\Biggr)
T_{\beta} \; . \label{commuted}$$ It is tedious but straightforward to show that the covariant d’Alembertian commutes with $\mathbf{P}_{\mu}^{~\beta}$, $$\square \mathbf{P}_{\mu}^{~\beta} T_{\beta} =
\mathbf{P}_{\mu}^{~\beta} \square T_{\beta} \; . \label{dalcom}$$ Note also that $\mathbf{P}_{\mu}^{~\beta}$ is transverse on both left and right, $$D^{\mu} \mathbf{P}_{\mu}^{~\beta} T_{\beta} = 0 =
\mathbf{P}_{\mu}^{~\beta} D_{\beta} T \; . \label{doubleT}$$ $\mathbf{P}_{\mu}^{~\beta}$ must therefore be proportional to transverse projection operator. The proportionality factor can be found by squaring. Comparing relations (\[doubleT\]) and (\[commuted\]) implies, $$\mathbf{P}_{\mu}^{~\alpha} \times \mathbf{P}_{\alpha}^{~\beta}
T_{\beta} = \Bigl[ \square \!-\! (D\!-\!1) H^2\Bigr]
\mathbf{P}_{\mu}^{~\beta} T_{\beta} = \mathbf{P}_{\mu}^{~\beta}
\Bigl[\square \!-\! (D\!-\!1) H^2 \Bigr] T_{\beta} \; .
\label{Psquare}$$
The relevance of $\mathbf{P}_{\mu}^{~\beta}$ is that it gives the differential operators in front of the general transverse form (\[transform\]), $$\mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) = -\frac1{2 H^2}
\mathbf{P}_{\mu}^{~\beta}(x) \times \mathbf{P}_{\rho}^{~\lambda}(z)
\; . \label{Prel}$$ Substituting (\[transform\]) into equation (\[transeqn\]), and making use or relations (\[Prel\]) and (\[dalcom\]), implies, $$\begin{aligned}
\lefteqn{\Bigl[ \square \!-\! (D \!-\! 1) H^2 \!-\! M_V^2 \Bigr] i
\Bigl[\mbox{}_{\mu} \Delta^T_{\rho}\Bigr](x;z) } \nonumber \\
& & \hspace{.5cm} = \mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Bigl[ \square \!-\! (D \!-\! 1)H^2
\!-\! M_V^2\Bigr] \Bigl[ \mathcal{R}_{\beta\lambda} \mathcal{S}_T
\Bigr] \; . \qquad\end{aligned}$$ We need next to consider what the d’Alembertian gives when acting on the factors to the far right, $$\square_x \Bigl[ \mathcal{R}_{\beta \lambda} \mathcal{S}_T \Bigr] =
\Bigl[ \square_x \mathcal{R}_{\beta\lambda} \Bigr] \mathcal{S}_T + 2
g^{\alpha\gamma} \frac{D \mathcal{R}_{\beta\lambda}}{D x^{\alpha}}
\frac{\partial \mathcal{S}_T}{\partial x^{\gamma}} +
\mathcal{R}_{\beta\lambda} \square_x \mathcal{S}_T \; . \label{actD}$$ Recalling the definition (\[Rdef\]) of $\mathcal{R}_{\beta\lambda}(x;z)$, and making use of relation (\[covdiv\]) from section 2, gives two identities we shall use in this section and the next, $$\begin{aligned}
\frac{D}{D x^{\alpha}} \mathcal{R}_{\beta\lambda}(x;z) & = & \frac12
\, g_{\alpha\beta}(x) \frac{\partial y}{\partial z^{\lambda}} \; ,
\qquad \label{useful1} \\
\square \mathcal{R}_{\beta\lambda}(x;z) & = & -H^2
\mathcal{R}_{\beta\lambda}(x;z) \; . \label{useful2}\end{aligned}$$ Substitute these in (\[actD\]) and pass the single derivative back outside to obtain, $$\begin{aligned}
\square_x \Bigl[ \mathcal{R}_{\beta\lambda} \mathcal{S}_T \Bigr] & =
& \frac{\partial y}{\partial z^{\lambda}} \frac{\partial
\mathcal{S}_T}{\partial x^{\beta}} + \mathcal{R}_{\beta\lambda}
\Bigl[ \square_x \!-\! H^2\Bigr] \mathcal{S}_T \; , \qquad \\
& = & \frac{\partial}{\partial x^{\beta}} \Bigl[ \frac{\partial
y}{\partial z^{\lambda}} \mathcal{S}_T \Bigr] - \frac{\partial^2
y}{\partial x^{\beta} \partial z^{\lambda}} \mathcal{S}_T +
\mathcal{R}_{\beta\lambda} \Bigl[\square_x \!-\! H^2\Bigr]
\mathcal{S}_T \; , \qquad \\
& = & \frac{\partial}{\partial x^{\beta}} \Bigl[ \frac{\partial
y}{\partial z^{\lambda}} \mathcal{S}_T \Bigr] + \mathcal{R}_{\beta
\lambda} \Bigl[ \square_x \!+\! H^2\Bigr] \mathcal{S}_T \; . \qquad
\label{fullpass}\end{aligned}$$ The first term on the right of equation (\[fullpass\]) is longitudinal. In view of relation (\[doubleT\]) we therefore conclude, $$\begin{aligned}
\lefteqn{\Bigl[ \square \!-\! (D \!-\! 1) H^2 \!-\! M_V^2 \Bigr] i
\Bigl[\mbox{}_{\mu} \Delta^T_{\rho}\Bigr](x;z) } \nonumber \\
& & \hspace{.5cm} = \mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Biggl[ \mathcal{R}_{\beta\lambda}
\Bigl[ \square \!-\! (D \!-\! 2)H^2 \!-\! M_V^2\Bigr] \mathcal{S}_T
\Biggr] . \qquad \label{finalleft}\end{aligned}$$
It remains to reduce the right hand side of (\[transeqn\]) to the standard form (\[transform\]) we have adopted for transverse bi-tensors, $$\begin{aligned}
i\Bigl[\mbox{}_{\mu} P_{\rho}\Bigr](x;z) & \equiv & \frac{i
g_{\mu\rho} \delta^D( x\!-\! z)}{\sqrt{-g}} +
\frac{\partial}{\partial x^{\mu}} \frac{\partial}{\partial z^{\rho}}
i\Delta_A(x;z) \; , \label{form1} \qquad \\
& = & \mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times \mathcal{Q}_{\alpha
\kappa}(x;z) \Bigl[ \mathcal{R}_{\beta\lambda}(x;z)
\mathcal{P}_1(x;z) \Bigr] \; . \qquad \label{form2}\end{aligned}$$ This is easily accomplished by acting $\mathbf{P}_{\nu}^{~\mu}(x)
\times \mathbf{P}_{\sigma}^{~\rho}(z)$ on both forms. Acting this operator on (\[form1\]) and making use of relation (\[doubleT\]) gives, $$\begin{aligned}
\lefteqn{\mathbf{P}_{\nu}^{~\mu}(x) \times
\mathbf{P}_{\sigma}^{~\rho}(z) \, i\Bigl[ \mbox{}_{\mu}
P_{\rho}\Bigr](x;z) } \nonumber \\
& & \hspace{1cm} = -2 H^2 \mathcal{P}^{\alpha\beta}_{\nu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\sigma}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Bigl[ g_{\beta\lambda} \frac{i
\delta^D(x \!-\! z)}{\sqrt{-g}} \Bigr] \; , \qquad \\
& & \hspace{1cm} = -2 H^2 \mathcal{P}^{\alpha\beta}_{\nu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\sigma}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Bigl[
\mathcal{R}_{\beta\lambda}(x;z) \frac{i \delta^D(x \!-\!
z)}{\sqrt{-g}} \Bigr] \; . \qquad \label{exp1}\end{aligned}$$ Acting instead on (\[form2\]) and making use of relations (\[Psquare\]) and (\[fullpass\]) gives, $$\begin{aligned}
\lefteqn{ \mathbf{P}_{\nu}^{~\mu}(x) \times
\mathbf{P}_{\sigma}^{~\rho}(z) \, i\Bigl[ \mbox{}_{\mu}
P_{\rho}\Bigr](x;z) } \nonumber \\
& & \hspace{1cm} = \mathcal{P}^{\alpha\beta}_{\nu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\sigma}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Bigl[ \square \!-\! (D \!-\! 1)
H^2\Bigr]^2 \Bigl[ \mathcal{R}_{\beta\lambda} \mathcal{P}_1 \Bigr]
\; , \qquad \\
& & \hspace{1cm} = \mathcal{P}^{\alpha\beta}_{\nu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\sigma}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Biggl[ \mathcal{R}_{\beta\lambda}
\Bigl[ \square \!-\! (D \!-\! 2) H^2\Bigr]^2 \mathcal{P}_1 \Biggr]
\; . \qquad \label{exp2}\end{aligned}$$ Comparing expressions (\[exp1\]) and (\[exp2\]) implies, $$\Bigl[ \square \!-\! (D\!-\!2) H^2\Bigr]^2 \mathcal{P}_1(x;z) = -2
H^2 \frac{i \delta^D(x \!-\! z)}{\sqrt{-g}} \; .$$ Relation (\[bprop\]) from section 3 — with the special case of $b = (\frac{D-3}2)$ — to infer, $$\Bigl[\square \!-\! (D\!-\!2)H^2\Bigr] \mathcal{P}_1(x;z) = -2 H^2
i\Delta_B(x;z) \; .$$ Now apply relations (\[Int1\]-\[Int3\]) to finally obtain the structure function for the transverse projection functional, $$\mathcal{P}_1(x;z) = -2 H^2 i\Delta_{BB}(x;z) \; .$$
We have now reduced the transverse propagator equation to the form, $$\begin{aligned}
\lefteqn{\mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Biggl[ \mathcal{R}_{\beta\lambda}
\Bigl[ \square \!-\! (D \!-\! 2)H^2 \!-\! M_V^2\Bigr] \mathcal{S}_T
\Biggr] } \nonumber \\
& & \hspace{2.5cm} = \mathcal{P}^{\alpha\beta}_{\mu}(x) \times
\mathcal{P}^{\kappa\lambda}_{\rho}(z) \times
\mathcal{Q}_{\alpha\kappa}(x;z) \Biggl[ \mathcal{R}_{\beta\lambda}
\Bigl[-2 H^2 i\Delta_{BB}\Bigr] \Biggr] \; . \qquad\end{aligned}$$ The transverse structure function therefore obeys, $$\Bigl[\square \!-\! (D \!-\! 2) H^2 \!-\! M_V^2 \Bigr] \mathcal{S}_T
= -2H^2 i\Delta_{BB}(x;z) \; .$$ Again making use of relations (\[Int1\]-\[Int3\]), our solution for it is, $$\mathcal{S}_T = + \frac{2 H^2}{M_V^2} i\Delta_{BB} + \frac{2
H^2}{M_V^4} \Bigl[ i\Delta_B - i\Delta_c\Bigr] \quad {\rm where}
\quad c = \sqrt{ \Bigl( \frac{D \!-\! 3}2\Bigr)^2 -
\frac{M_V^2}{H^2} } \; . \label{transsol}$$
The Graviton Propagator
=======================
The previous section provides a model for the analysis of this section, except that we immediately specialize to gravitons which obey de Donder gauge, $$D^{\mu} h_{\mu\nu} - \frac12 D_{\nu} h^{\mu}_{~\mu} = 0 \; .
\label{deDonder}$$ The first task is to express the propagator of such a graviton in terms of covariant projectors acting on scalar structure functions. With just a small extension of our previous results we can then commute the differential operator to act directly on the structure functions. The final step is identifying the de Donder gauge projection functionals.
Enforcing de Donder Gauge
-------------------------
In de Donder gauge (\[deDonder\]) the propagator must obey the gauge condition on either coordinate and its associated index group, $$\begin{aligned}
\Bigl[ \delta^{\alpha}_{\mu} D_x^{\beta} - \frac12 D^x_{\mu}
g^{\alpha\beta}(x)\Bigr] \times i \Bigl[\mbox{}_{\alpha\beta}
\Delta_{\rho\sigma} \Bigr](x;z) & = & 0 \; , \label{dD1} \\
\Bigl[ \delta^{\alpha}_{\rho} D_z^{\beta} - \frac12 D^z_{\rho}
g^{\alpha\beta}(z)\Bigr] \times i \Bigl[\mbox{}_{\mu\nu}
\Delta_{\alpha\beta} \Bigr](x;z) & = & 0 \; . \label{dD2}\end{aligned}$$ Just as was the case for the vector propagator with the analogous conditions of transversality and longitudinality, we seek here to enforce (\[dD1\]-\[dD2\]) by acting covariant projectors on scalar structure functions. It turns out there are two ways to do it, corresponding to the spin zero and spin two parts of the graviton propagator, $$i \Bigl[\mbox{}_{\alpha\beta} \Delta_{\rho\sigma} \Bigr](x;z) = i
\Bigl[\mbox{}_{\alpha\beta} \Delta^0_{\rho\sigma} \Bigr](x;z) + i
\Bigl[\mbox{}_{\alpha\beta} \Delta^2_{\rho\sigma} \Bigr](x;z) \; .
\label{gravdecomp}$$
The spin zero part of the graviton propagator is almost as simple as the longitudinal part of the vector propagator. It is a linear combination of longitudinal and trace terms from each index group, $$i\Bigl[\mbox{}_{\mu\nu} \Delta^0_{\rho\sigma}\Bigr](x;z) =
\mathcal{P}_{\mu\nu}(x) \times \mathcal{P}_{\rho\sigma}(z)
\Bigl[\mathcal{S}_0(x;z) \Bigr] \; . \label{Spin0}$$ The projector $\mathcal{P}_{\mu\nu}$ is, $$\mathcal{P}_{\mu\nu} \equiv D_{\mu} D_{\nu} + \frac{g_{\mu\nu}}{D \!-\!2}
\Bigl[ \square \!+\! 2 (D \!-\! 1) H^2 \Bigr] \; . \label{P0op}$$
Unlike the spin zero part of the graviton propagator, the spin two part is both transverse and also traceless within each index group. Recall that we obtained the key projector for the transverse part of the photon propagator by writing the Maxwell field strength tensor as $F^{\alpha\beta} = \mathcal{P}^{\alpha\beta}_{\mu} A^{\mu}$. We similarly define the projector $\mathcal{P}^{\alpha\beta\gamma\delta}_{\mu\nu}$ by expanding the Weyl tensor in powers of the graviton field $C^{\alpha\beta\gamma\delta} =
\mathcal{P}^{\alpha\beta\gamma\delta}_{\mu\nu} h^{\mu\nu} + O(h^2)$. The final result takes the form [@PW], $$\begin{aligned}
\lefteqn{\mathcal{P}_{\mu\nu}^{\alpha\beta\gamma\delta} \equiv
\mathcal{D}_{\mu\nu}^{\alpha\beta\gamma\delta} + \frac1{D \!-\!2}
\Bigl[ g^{\alpha\delta} \mathcal{D}^{\beta\gamma}_{\mu\nu} \!-\!
g^{\beta\delta} \mathcal{D}^{\alpha\gamma}_{\mu\nu} \!-\!
g^{\alpha\gamma} \mathcal{D}^{\beta\delta}_{\mu\nu} \!+\!
g^{\beta\gamma} \mathcal{D}^{\alpha\delta}_{\mu\nu} \Bigr] } \nonumber \\
& & \hspace{5cm} + \frac1{(D \!-\! 1) (D \!-\! 2)} \Bigl[ g^{\alpha
\gamma} g^{\beta \delta} \!-\! g^{\alpha\delta} g^{\beta \gamma}
\Bigr] \mathcal{D}_{\mu\nu} \; . \qquad \label{CPdef}\end{aligned}$$ The various pieces of this are, $$\begin{aligned}
\mathcal{D}^{\alpha\beta\gamma\delta}_{\mu\nu} &\equiv & \frac12 \Bigl[
\delta^{\alpha}_{(\mu} \delta^{\delta}_{\nu)} D^{\gamma} D^{\beta} \!-\!
\delta^{\beta}_{(\mu} \delta^{\delta}_{\nu)} D^{\gamma} D^{\alpha} \!-\!
\delta^{\alpha}_{(\mu} \delta^{\gamma}_{\nu)} D^{\delta} D^{\beta} \!+\!
\delta^{\beta}_{(\mu} \delta^{\gamma}_{\nu)} D^{\delta} D^{\alpha} \Bigr]
\; , \qquad \label{CDdef1} \\
\mathcal{D}^{\beta\delta}_{\mu\nu} & \equiv & g_{\alpha\gamma}
\mathcal{D}^{\alpha\beta\gamma\delta}_{\mu\nu} = \frac12 \Bigl[
\delta^{\delta}_{(\mu} D_{\nu)} D^{\beta} \!-\! \delta^{\beta}_{(\mu}
\delta^{\delta}_{\nu)} \square \!-\! g_{\mu\nu} D^{\delta} D^{\beta}
\!+\! \delta^{\beta}_{(\mu} D^{\delta} D_{\nu)} \Bigr] \; , \label{CDdef2} \\
\mathcal{D}_{\mu\nu} & \equiv & g_{\alpha\gamma} g_{\beta\delta}
\mathcal{D}^{\alpha\beta\gamma\delta}_{\mu\nu} = D_{(\mu} D_{\nu)} -
g_{\mu\nu} \square \; . \label{CDdef3}\end{aligned}$$ Acting $\mathcal{P}^{\alpha\beta\gamma\delta}_{\mu\nu}(x) \times
\mathcal{P}^{\kappa\lambda\theta\phi}_{\rho\sigma}(z)$ on any eight index, symmetric function of $x$ and $z$ would produce a transverse and traceless tensor but, as with the vector, it pays to select a simple form. The best choice seems to be taking half the indices in the form of more covariant derivative operators, and the other half from two factors of the mixed second derivative (\[dydxdx’\]) of the length function, $$i\Bigl[\mbox{}_{\mu\nu} \Delta^2_{\rho\sigma}\Bigr](x;z) =
\mathcal{P}_{\mu\nu}^{\alpha\beta\gamma\delta}(x) \times
\mathcal{P}_{\rho\sigma}^{\kappa\lambda\theta\phi}(z) \times
\mathcal{Q}_{\alpha\kappa} \times \mathcal{Q}_{\gamma\theta} \Bigl[
\mathcal{R}_{\beta\lambda} \mathcal{R}_{\delta\phi} \,
\mathcal{S}_2(x;z) \Bigr] \; . \label{Spin2}$$ We remind the reader of the definitions (\[Qdef\]-\[Rdef\]) of $\mathcal{Q}_{\alpha\kappa}(x;z)$ and $\mathcal{R}_{\beta\lambda}(x;z)$.
We close this subsection by giving the propagator equation. Acting the Lichnerowicz operator (\[Lichnerowicz\]) on the graviton field and making use of the de Donder gauge condition (\[deDonder\]) gives, $$-\mathbf{D}^{\mu\nu\rho\sigma} h_{\rho\sigma} = \frac12 \Bigl[
\square \!-\! 2 H^2\Bigr] h^{\mu\nu} - \frac14 g^{\mu\nu} \Bigl[
\square \!+\! 2 (D \!-\! 3) H^2 \Bigr] h^{\rho}_{~\rho} \; .$$ This means the propagator obeys a relation of the form, $$\begin{aligned}
\lefteqn{\frac12 \Bigl[ \square_x \!-\! 2 H^2\Bigr]
i\Bigl[\mbox{}_{\mu\nu} \Delta_{\rho\sigma}\Bigr](x;z) - \frac14
g_{\mu\nu}(x) \Bigl[ \square_x \!+\! 2(D \!-\!3) H^2\Bigr] i\Bigl[
\mbox{}^{\alpha}_{~\alpha} \Delta_{\rho\sigma}\Bigr](x;z) }
\nonumber \\
& & \hspace{4.5cm} =\! g_{\mu (\rho} g_{\sigma) \nu} \times
\frac{i\delta^D(x \!-\!z)}{\sqrt{-g}} + \Bigl( {\rm Other\ Terms}
\Bigr) , \qquad \label{propeqn1}\end{aligned}$$ where the “Other Terms” make the right hand side consistent with the gauge condition. However, the right hand side of (\[propeqn1\]) cannot be symmetric under the interchange of $x^{\mu}$ and $z^{\mu}$ (and their associated indices) because the left hand side of the equation obeys de Donder gauge on $z^{\mu}$ but not on $x^{\mu}$. It is better to subtract off a term proportional to the trace, $$\begin{aligned}
\lefteqn{\frac12 \Bigl[ \square_x \!-\! 2 H^2\Bigr]
i\Bigl[\mbox{}_{\mu\nu} \Delta_{\rho\sigma}\Bigr](x;z) - \frac14
g_{\mu\nu}(x) \Bigl[ \square_x \!+\! 2(D \!-\!3) H^2\Bigr] i\Bigl[
\mbox{}^{\alpha}_{~\alpha} \Delta_{\rho\sigma}\Bigr](x;z) }
\nonumber \\
& & \hspace{2.5cm} -\frac{g_{\mu\nu}}{D\!-\!2} \times -\Bigl(\frac{D
\!-\! 2}4\Bigr) \Bigl[ \square \!+\! 2 (D \!-\! 1) H^2 \Bigr] \,
i\Bigl[\mbox{}^{\alpha}_{\alpha} \Delta_{\rho\sigma}\Bigr](x;z) \; ,
\qquad \\
& & \hspace{1cm} = \frac12 \Bigl[ \square_x \!-\! 2 H^2\Bigr]
i\Bigl[\mbox{}_{\mu\nu} \Delta_{\rho\sigma}\Bigr](x;z) + H^2
g_{\mu\nu}(x) i\Bigl[\mbox{}^{\alpha}_{~\alpha} \Delta_{\rho\sigma}
\Bigr](x;z) \; . \qquad \label{goodform}\end{aligned}$$ It is easily checked that (\[goodform\]) obeys the de Donder gauge condition on both $x^{\mu}$ and $z^{\mu}$. Hence the right hand side of the equation is symmetric under interchange of $x^{\mu}$ and $z^{\mu}$ and can in fact be guessed [@MTW2], $$\begin{aligned}
\lefteqn{\frac12 \Bigl[ \square_x \!-\! 2 H^2\Bigr]
i\Bigl[\mbox{}_{\mu\nu} \Delta_{\rho\sigma}\Bigr](x;z) + H^2
g_{\mu\nu}(x) i\Bigl[ \mbox{}^{\alpha}_{~\alpha}
\Delta_{\rho\sigma}\Bigr](x;z) } \nonumber \\
& & \hspace{-.7cm} =\! \Bigl[g_{\mu (\rho} g_{\sigma) \nu} \!-\!
\frac{g_{\mu\nu} g_{\rho\sigma}}{D \!-\!2} \Bigr] \frac{i\delta^D(x
\!-\!z)}{\sqrt{-g}} + \frac12 \! \left\{ \!\matrix{ D^x_{\mu}
D^z_{\rho} i [\mbox{}_{\nu} \Delta^W_{\sigma}] + D^x_{\mu}
D^z_{\sigma} i [\mbox{}_{\nu} \Delta^W_{\rho}] \cr + D^x_{\nu}
D^z_{\rho} i [\mbox{}_{\mu} \Delta^W_{\sigma}] + D^x_{\nu}
D^z_{\sigma} i [\mbox{}_{\mu} \Delta^W_{\rho}] \cr} \!\right\} .
\qquad \label{propeqn2}\end{aligned}$$ Here $i[\mbox{}_{\mu} \Delta^W_{\rho}]$ is the full vector propagator for the tachyonic mass $M_V^2 = -2(D-1) H^2$, which obeys the equation, $$\Bigl[ \square \!+\! (D \!-\! 1) H^2\Bigr] i \Bigl[ \mbox{}_{\mu}
\Delta^W_{\rho}\Bigr](x;z) = \frac{i g_{\mu\rho} \delta^D(x \!-\!
z)}{\sqrt{-g}} \; .$$ Recall from section 4 that it has the form given by equations (\[decomp\]), (\[longform\]) and (\[transform\]). From equations (\[longsol\]) and (\[transsol\]) we see that the longitudinal and transverse structure functions are, $$\begin{aligned}
\mathcal{S}_{L}(x;z) & = & -i\Delta_{AM}(x;z) \; , \qquad \label{SL} \\
\mathcal{S}_{T}(x;z) & = & \frac1{D \!-\! 1} \Bigl[
-i\Delta_{BB}(x;z) \!+\! i\Delta_{BW}(x;z)\Bigr] \; . \qquad
\label{ST}\end{aligned}$$
The Spin Zero Part
------------------
To derive an equation for the spin zero structure function we simply take the trace of the propagator equation (\[propeqn2\]). Tracing on the left hand side and making use of relations (\[gravdecomp\]-\[P0op\]) gives, $$\begin{aligned}
\lefteqn{\frac12 \Bigl[ \square_x \!+\! 2 (D \!-\! 1) H^2 \Bigr] \,
i\Bigl[\mbox{}^{\alpha}_{~\alpha} \Delta_{\rho\sigma}\Bigr](x;z) }
\nonumber \\
& & \hspace{1cm} = \Bigl( \frac{D \!-\! 1}{D \!-\! 2}\Bigr)
\Bigl[\square_x \!+\! 2 (D \!-\! 1) H^2\Bigr] \Bigl[ \square_x \!+\!
D H^2 \Bigr] \mathcal{P}_{\rho\sigma}(z) \Bigl[ \mathcal{S}_{0}(x;z)
\Bigr] \; . \label{Trace1} \qquad\end{aligned}$$ Tracing the right hand side of (\[propeqn2\]) and making use of relations (\[SL\]) and (\[Int1\]-\[Int3\]) implies, $$\begin{aligned}
\lefteqn{\frac12 \Bigl[ \square_x \!+\! 2 (D\!-\!1) H^2 \Bigr] i
\Bigl[ \mbox{}^{\alpha}_{~\alpha} \Delta_{\rho\sigma}\Bigr](x;z) } \nonumber \\
& & \hspace{3cm} = -\frac{2}{D \!-\!2} \frac{g_{\rho\sigma} i
\delta^D(x \!-\! z)}{\sqrt{-g}} - 2 D^z_{\rho} D^z_{\sigma} i\Delta_M(x;z) \; , \qquad \\
& & \hspace{3cm} = -2 \mathcal{P}_{\rho\sigma}(z) i\Delta_{M}(x;z)
\; . \label{Trace2}\end{aligned}$$ The equation for $\mathcal{S}_0(x;z)$ derives from comparing expressions (\[Trace1\]) and (\[Trace2\]), $$\Bigl[ \square \!+\! 2 (D \!-\! 1) H^2 \Bigr] \Bigl[ \square \!+\! D
H^2\Bigr] \mathcal{S}_0(x;z) = -2 \Bigl( \frac{D \!-\! 2}{D \!-\!
1}\Bigr) i\Delta_M(x;z) \; . \label{S0eqn}$$ Its solution follows easily from relations (\[Int1\]-\[Int7b\]), $$\mathcal{S}_0(x;z) = \frac{2 i\Delta_{MM}(x;z) \!-\! 2 i
\Delta_{MW}(x;z)}{(D \!-\! 1) H^2} = -2 \Bigl( \frac{D \!-\! 2}{D
\!-\! 1}\Bigr) i\Delta_{WMM}(x;z) \; . \label{spin0sol}$$
The Spin Two Part
-----------------
This is the most complicated analysis we shall have to make and it is greatly facilitated by the analogy with what was done for the transverse part of the vector propagator in section 4.3. Here, as for that case, the first step is to derive an equation for the remaining (spin two) part of the propagator by subtracting off the part we already have. We then establish some identities for a differential projector which comprises the exterior operators of the spin two part (\[Spin2\]) of the propagator. These properties allow us to pass the d’Alembertian in the propagator equation through to act on the spin two structure function $\mathcal{S}_2(x;z)$. Squaring this operator also allows us to express the right hand side of the propagator equation in the same form (\[Spin2\]) with a known structure function. Comparing the two sides of the equation leads to a scalar differential equation which can be solved by the techniques of section 3.
We derive an equation for the pure spin two part of the propagator from the full equation (\[propeqn2\]) by substituting the spin zero structure function (\[spin0sol\]), with definitions (\[Spin0\]-\[P0op\]). Now move everything known to right hand side to reach the form, $$\begin{aligned}
\lefteqn{\frac12 \Bigl[ \square_x \!-\! 2 H^2\Bigr] \, i\Bigl[
\mbox{}_{\mu\nu} \Delta^2_{\rho\sigma}\Bigr](x;z) \equiv i \Bigl[
\mbox{}_{\mu\nu} P^2_{\rho\sigma} \Bigr](x;z) \; , }
\label{spin2eqn} \\
& & \hspace{.2cm} = \Bigl[ g_{\mu (\rho} g_{\sigma) \nu} -
\frac{g_{\mu\nu} g_{\rho\sigma}}{D \!-\! 2} \Bigr] \frac{i
\delta^D(x \!-\! z)}{\sqrt{-g}} + \Bigl( \frac{D \!-\!2}{D \!-\!
1}\Bigr) \mathcal{P}_{\mu\nu}(x) \times \mathcal{P}_{\rho\sigma}(z)
i\Delta_{WM}(x;z) \nonumber \\
& & \hspace{3cm} + \frac12 \left( \matrix{ D^x_{\mu} D^z_{\rho} i
[\mbox{}_{\nu} \Delta^{W}_{\sigma}](x;z) + D^x_{\mu} D^z_{\sigma} i
[\mbox{}_{\nu} \Delta^{W}_{\rho}](x;z) \cr + D^x_{\nu} D^z_{\rho} i
[\mbox{}_{\mu} \Delta^W_{\sigma}](x;z) + D^x_{\nu} D^z_{\sigma} i
[\mbox{}_{\mu} \Delta^W_{\rho}](x;z) \cr} \right) \; . \qquad
\label{P2}\end{aligned}$$ It can easily be checked that the right hand side of (\[P2\]) is transverse and traceless on each index group. We will eventually reduce this transverse-traceless projector to standard form, $$i\Bigl[\mbox{}_{\mu\nu} P^2_{\rho\sigma}\Bigr](x;z) =
\mathcal{P}^{\alpha\beta\gamma\delta}_{\mu\nu}(x) \times
\mathcal{P}^{\kappa\lambda\theta\phi}_{\rho\sigma}(z) \times
\mathcal{Q}_{\alpha\kappa} \times \mathcal{Q}_{\gamma\theta}
\Bigl[\mathcal{R}_{\beta\lambda} \mathcal{R}_{\delta\phi}
\mathcal{P}_2\Bigr] \; . \label{standard}$$ However, it is best to first concentrate on the left hand side of the propagator equation (\[P2\]).
In analogy with the transverse projector $\mathbf{P}_{\mu}^{~\beta}$ defined in equation (\[bfP1\]), we define the transverse-traceless projector, $$\mathbf{P}_{\mu\nu}^{~~ \beta\delta} \equiv
\mathcal{P}^{\alpha\beta\gamma\delta}_{\mu\nu} \times D_{\alpha}
D_{\gamma} \; . \label{bfP2}$$ We shall always consider this acted on a second rank tensor $F_{\beta\delta}$. From the expressions (\[CPdef\]-\[CDdef3\]) which define $\mathcal{P}^{\alpha\beta\gamma\delta}_{\mu\nu}$ it is straightforward but tedious to reach the form, $$\begin{aligned}
\lefteqn{ \mathbf{P}_{\mu\nu}^{~~ \beta\delta} \times
F_{\beta\delta} = \frac12 \Bigl(\frac{D \!-\! 3}{D \!-\! 2}\Bigr)
\Biggl\{ D_{\mu} \square D^{\alpha} F_{\alpha \nu} + D_{\mu} \square
D^{\beta} F_{\nu\beta} - \square^2
F_{\mu \nu} } \nonumber \\
& & \hspace{0cm} - D_{\mu} D_{\nu} D^{\alpha} D^{\beta} F_{\alpha
\beta} + \frac1{D \!-\! 1} \Bigl[ D_{\mu} D_{\nu} \!-\! g_{\mu\nu}
\square\Bigr] \Bigl[ D^{\alpha} D^{\beta} F_{\alpha\beta} \!-\!
\square F_{\alpha}^{~\alpha} \Bigr]
\nonumber \\
& & \hspace{.5cm} + H^2 \Bigl[ -2 g_{\mu\nu} D^{\alpha} D^{\beta}
F_{\alpha\beta} - g_{\mu\nu} \square F_{\alpha}^{~\alpha} -2
D_{\mu} D_{\nu} F_{\alpha}^{~\alpha} + D_{\mu} D^{\alpha} F_{\alpha
\nu} \nonumber \\
& & \hspace{2cm} + D_{\mu} D^{\beta} F_{\nu\beta} + (D \!+\! 2)
\square F_{\mu\nu} \Bigr] + H^4 \Bigl[ 2 g_{\mu\nu}
F_{\alpha}^{~\alpha} - 2 D F_{\mu\nu} \Bigr] \Biggr\} . \qquad
\label{bigresult}\end{aligned}$$ (Note the multiplicative factor of $D-3$ which derives from the fact that the Weyl tensor vanishes for $D=3$.) It is easy to see from (\[bigresult\]) that $\mathbf{P}_{\mu\nu}^{~~ \beta\delta}$ is traceless on both the left and the right, $$g^{\mu\nu} \mathbf{P}_{\mu\nu}^{~~\beta\delta} F_{\beta\delta} = 0 =
\mathbf{P}_{\mu\nu}^{~~\beta\delta} ( g_{\beta\delta} F) \; .
\label{traceless}$$ It is also transverse on any index, both on the left and the right, $$\begin{aligned}
D^{\mu} \Bigl( \mathbf{P}_{\mu\nu}^{~~\beta\delta} F_{\beta\delta}
\Bigr) = & 0 & = D^{\nu} \Bigl( \mathbf{P}_{\mu\nu}^{~~\beta\delta}
F_{\beta\delta} \Bigr) \; , \label{transverse1} \\
\mathbf{P}_{\mu\nu}^{~~\beta\delta} (D_{\beta} F_{\delta}) = & 0 & =
\mathbf{P}_{\mu\nu}^{~~\beta\delta} (D_{\delta} F_{\beta}) \; .\end{aligned}$$ These two properties are very important because the only terms in expression (\[bigresult\]) which don’t involve either divergences or traces are, $$\begin{aligned}
\lefteqn{\frac12 \Bigl(\frac{ D \!-\! 3}{D \!-\! 2} \Bigr) \Biggl\{
-\square^2 F_{\mu\nu} + (D \!+\!2) H^2 \square F_{\mu\nu} - 2 D H^4
F_{\mu\nu} \Biggr\} } \nonumber \\
& & \hspace{4.8cm} = -\frac12 \Bigl( \frac{D \!-\! 3}{D \!-\!
2}\Bigr) \Bigl[ \square \!-\! 2 H^2\Bigr] \Bigl[ \square \!-\! D
H^2\Bigr] F_{\mu\nu} \; . \qquad\end{aligned}$$ Hence squaring $\mathbf{P}_{\mu\nu}^{~~\beta\delta}$ gives, $$\begin{aligned}
\mathbf{P}_{\mu\nu}^{~~\alpha\gamma} \times
\mathbf{P}_{\alpha\gamma}^{~~\beta\delta} F_{\beta\delta} & = &
-\frac12 \Bigl( \frac{D \!-\! 3}{D \!-\! 2}\Bigr) \Bigl[ \square
\!-\! 2 H^2\Bigr] \Bigl[ \square \!-\! D H^2\Bigr]
\mathbf{P}_{\mu\nu}^{~~\beta\delta} F_{\beta\delta} \; , \qquad \\
& = & -\frac12 \Bigl( \frac{D \!-\! 3}{D \!-\! 2}\Bigr)
\mathbf{P}_{\mu\nu}^{~~\beta\delta} \Bigl[ \square \!-\! 2 H^2\Bigr]
\Bigl[ \square \!-\! D H^2\Bigr] F_{\beta\delta} \; . \qquad
\label{2ndsquare}\end{aligned}$$ We note in passing that the covariant d‘Alembertian commutes with $\mathbf{P}_{\mu\nu}^{~~\beta\delta}$, just as it did for the transverse projector $\mathbf{P}_{\mu}^{~\beta}$.
Of course the relevance of the transverse-traceless projector $\mathbf{P}_{\mu\nu}^{~~~\beta\lambda}$ is that two factors of it give the exterior operators of the spin two part of the propagator, $$i\Bigl[ \mbox{}_{\mu\nu} \Delta^2_{\rho\sigma}\Bigr](x;z) = \frac1{4
H^4} \mathbf{P}_{\mu\nu}^{~~\beta\delta}(x) \times
\mathbf{P}_{\rho\sigma}^{~~\lambda\phi}(z) \Bigl[
\mathcal{R}_{\beta\lambda}(x;z) \mathcal{R}_{\delta\phi}(x;z)
\mathcal{S}_2(x;z) \Bigr] \; .$$ From the fact that the d’Alembertian commutes with $\mathbf{P}_{\mu\nu}^{~~\beta\delta}$ we see, $$\begin{aligned}
\lefteqn{ \frac12 \Bigl[\square \!-\! 2 H^2\Bigr] i\Bigl[
\mbox{}_{\mu\nu} \Delta^2_{\rho\sigma}\Bigr](x;z) } \nonumber \\
& & \hspace{2cm} = \frac1{4 H^4} \mathbf{P}_{\mu \nu}^{~~\beta
\delta}(x) \times \mathbf{P}_{\rho\sigma}^{~~\lambda\phi}(z) \times
\frac12 \Bigl[\square \!-\! 2 H^2\Bigr] \Bigl[ \mathcal{R}_{\beta
\lambda} \mathcal{R}_{\delta \phi} \mathcal{S}_2 \Bigr] \; . \qquad\end{aligned}$$ The next step is to pass the differential operator through to the structure function, making use of identities (\[useful1\]-\[useful2\]) from section 4, $$\begin{aligned}
\lefteqn{\square \Bigl[\mathcal{R}_{\beta\lambda}
\mathcal{R}_{\delta\phi} \mathcal{S}_2\Bigr] =
\mathcal{R}_{\beta\lambda} \mathcal{R}_{\delta\phi} \square
\mathcal{S}_2 + 2 g^{\alpha\gamma}(x) \Bigl[\frac{D
\mathcal{R}_{\beta\lambda}}{D x^{\alpha}} \mathcal{R}_{\delta\phi} +
\mathcal{R}_{\beta\lambda} \frac{D \mathcal{R}_{\delta\phi}}{D
x^{\alpha}} \Bigr] \frac{\partial \mathcal{S}_2}{\partial
x^{\gamma}}} \nonumber \\
& & \hspace{1.5cm} + 2 g^{\alpha\gamma}(x) \frac{D
\mathcal{R}_{\beta\lambda}}{D x^{\alpha}} \frac{D
\mathcal{R}_{\delta \phi}}{D x^{\gamma}} \mathcal{S}_2 + \Bigl[
(\square \mathcal{R}_{\beta\lambda}) \mathcal{R}_{\delta\phi} +
\mathcal{R}_{\beta\lambda} (\square \mathcal{R}_{\delta\phi})\Bigr]
\mathcal{S}_2 \; , \qquad \\
& & = \mathcal{R}_{\beta\lambda} \mathcal{R}_{\delta\phi}
\Bigl[\square \!+\! 2 H^2\Bigr] \mathcal{S}_2 \nonumber \\
& & \hspace{1cm} + \frac{D}{D x^{\beta}} \Bigl[\frac{\partial
y}{\partial z^{\lambda}} \mathcal{R}_{\delta\phi} \mathcal{S}_2
\Bigr] + \frac{D}{D x^{\delta}} \Bigl[ \mathcal{R}_{\beta\lambda}
\frac{\partial y}{\partial z^{\phi}} \mathcal{S}_2\Bigr] - \frac12
g_{\beta\delta}(x) \frac{\partial y}{\partial z^{\lambda}}
\frac{\partial y}{\partial z^{\phi}} \mathcal{S}_2 \; . \qquad
\label{finalline}\end{aligned}$$ When the external operators are contracted into this the terms on the final line of (\[finalline\]) all drop by virtue of either transversality or tracelessness. Hence we have, $$\frac12 \Bigl[\square \!-\! 2 H^2\Bigr] i\Bigl[ \mbox{}_{\mu\nu}
\Delta^2_{\rho\sigma}\Bigr](x;z) = \frac1{4 H^4} \mathbf{P}_{\mu
\nu}^{~~\beta \delta}(x) \times \mathbf{P}_{\rho \sigma}^{~~\lambda
\phi}(z) \Bigl[ \mathcal{R}_{\beta \lambda} \mathcal{R}_{\delta
\phi} \times \frac{\square}2 \mathcal{S}_2 \Bigr] \; .
\label{leftpass}$$
It is now time to reduce transverse-traceless projection functional (\[P2\]) to standard form (\[standard\]). Just as we did with the transverse projection functional of section 4, this is accomplished by acting $\mathbf{P}_{\alpha\gamma}^{~~\mu\nu}(x)
\times \mathbf{P}_{\kappa\theta}^{~~\rho\sigma}(z)$ on both forms. When acting on expression (\[P2\]) tracelessness or transversality make all but the first term drop out, $$\begin{aligned}
\lefteqn{\mathbf{P}_{\alpha\gamma}^{~~\mu\nu}(x) \times
\mathbf{P}_{\kappa\theta}^{~~\rho\sigma}(z) \, i\Bigl[
\mbox{}_{\mu\nu} P^2_{\rho\sigma}\Bigr](x;z) } \nonumber \\
& & \hspace{4cm} = \mathbf{P}_{\alpha\gamma}^{~~\mu\nu}(x) \times
\mathbf{P}_{\kappa\theta}^{~~\rho\sigma}(z) \Biggl[ g_{\mu\rho}
g_{\nu\sigma} \frac{i \delta^D(x \!-\! z)}{\sqrt{-g}} \Biggr] \; ,
\qquad \\
& & \hspace{4cm} = \mathbf{P}_{\alpha\gamma}^{~~\mu\nu}(x) \times
\mathbf{P}_{\kappa\theta}^{~~\rho\sigma}(z) \Biggl[
\mathcal{R}_{\mu\rho} \mathcal{R}_{\nu\sigma} \frac{i \delta^D(x
\!-\! z)}{\sqrt{-g}} \Biggr] \; . \qquad \label{rel1}\end{aligned}$$ On the other hand, acting the same operator on (\[standard\]), and making use of relations (\[2ndsquare\]) and (\[leftpass\]), tells us, $$\begin{aligned}
\lefteqn{\mathbf{P}_{\alpha\gamma}^{~~\mu\nu}(x) \times
\mathbf{P}_{\kappa\theta}^{~~\rho\sigma}(z) \, i\Bigl[
\mbox{}_{\mu\nu} P^2_{\rho\sigma}\Bigr](x;z) = \frac1{4 H^4}
\mathbf{P}_{\alpha\gamma}^{~~ \mu \nu}(x) \times
\mathbf{P}_{\kappa\theta}^{~~\rho\sigma}(z) } \nonumber \\
& & \hspace{3.5cm} \times \Biggl[ \mathcal{R}_{\mu\rho}
\mathcal{R}_{\nu\sigma} \frac14 \Bigl(\frac{D \!-\! 3}{D \!-\!
2}\Bigr)^2 \square^2 \Bigl[ \square \!-\! (D\!-\! 2) H^2\Bigr]^2
\mathcal{P}_2 \Biggr] \; . \qquad \label{rel2}\end{aligned}$$ Comparing (\[rel1\]) with (\[rel2\]) we infer an equation for the structure function of the transverse-traceless projection functional, $$\square^2 \Bigl[ \square \!-\! (D \!-\!2) H^2\Bigr]^2
\mathcal{P}_2(x;z) = 16 H^4 \Bigl( \frac{D \!-\! 2}{D \!-\!
3}\Bigr)^2 \times \frac{i \delta^D(x \!-\! z)}{\sqrt{-g}} \; .$$ The solution is easily constructed using relation (\[bprop\]) and successive applications of (\[Int1\]-\[Int3\]), $$\mathcal{P}_2(x;z) = \Bigl( \frac{4}{D \!-\! 3} \Bigr)^2 \Biggl[
i\Delta_{AA}(x;z) - 2 i\Delta_{AB}(x;z) + i\Delta_{BB}(x;z) \Biggr]
\; . \label{P2answer}$$
The long-sought equation for the spin two structure function derives from the substitution in equation (\[spin2eqn\]) of relations (\[leftpass\]) and (\[P2answer\]), $$\frac12 \square \mathcal{S}_2(x;z) = \Bigl( \frac{4}{D \!-\! 3}
\Bigr)^2 \Biggl[ i\Delta_{AA}(x;z) - 2 i\Delta_{AB}(x;z) +
i\Delta_{BB}(x;z) \Biggr] \; . \label{S2eqn}$$ The solution can be found using relations (\[Int4\]-\[Int7b\]) from the end of section 3, $$\mathcal{S}_2(x;z) = \frac{32}{(D \!-\! 3)^2} \Bigl[
i\Delta_{AAA}(x;z) \!-\! 2 i\Delta_{AAB} + i\Delta_{ABB}(x;z) \Bigr]
\; . \label{spin2sol}$$
Discussion
==========
We have constructed the graviton propagator on de Sitter background in exact de Donder gauge (\[deDonder\]). Our result takes the form (\[gravdecomp\]) of a spin zero part and a spin two part. Both parts are represented in terms of covariant differential projectors which automatically enforce the gauge condition, acting on scalar structure functions. Our form for the spin zero part is given by relations (\[Spin0\]) and (\[P0op\]). The spin two part (\[Spin2\]) has a complicated definition involving relations (\[CPdef\]-\[CDdef3\]) and (\[Qdef\]-\[Rdef\]). By taking appropriate traces and commuting differential operators we eventually derive scalar equations (\[S0eqn\]) and (\[S2eqn\]) for the structure functions of the respective parts. These equations are then solved using the general scalar techniques explained and summarized in section 3.
We emphasize that our forms for the spin zero and spin two parts of the propagator involve no assumption about de Sitter invariance, nor specialization to any particular portion of the de Sitter manifold. The equations (\[S0eqn\]) and (\[S2eqn\]) we derive for the two structure functions are scalar equations, valid in any coordinate system and with no inherent assumption about de Sitter invariance. To emphasize this, we act extra derivatives so as to make the source on the right hand side proportional to a delta function in each case, $$\begin{aligned}
\Bigl[ \square \!+\! D H^2\Bigr] \Bigl[ \square \!+\! 2 (D \!-\! 1)
H^2 \Bigr]^2 \mathcal{S}_0(x;z) & = & -2 \Bigl( \frac{D \!-\! 2}{D
\!-\! 1}\Bigr) \frac{i \delta^D(x \!-\! z)}{\sqrt{-g}} \; , \qquad
\label{EQN0} \\
\square^3 \Bigl[ \square \!-\! (D \!-\! 2) H^2 \Bigr]^2
\mathcal{S}_2(x;z) & = & 32 \Bigl(\frac{D \!-\! 2}{D \!-\! 3}
\Bigr)^2 H^4 \frac{i \delta^D(x \!-\! z)}{\sqrt{-g}} \; . \qquad
\label{EQN2}\end{aligned}$$ It happens that neither the spin zero structure function (\[spin0sol\]) nor its spin two counterpart (\[spin2sol\]) is de Sitter invariant. For the spin zero case this is obvious from the presence of tachyonic mass terms in both of the differential operators on the left hand side of equation (\[EQN0\]). The mass $M_S^2 = -D H^2$ includes a logarithmic singularity which shows up even in analytic regularization techniques. For the spin two equation (\[EQN2\]) the squared operator has positive mass-squared $M_S^2 = (D-2) H^2$ and would not lead to breaking of de Sitter invariance were it alone. However, the cubed operator is the same as that for a massless, minimally coupled scalar — as might have been expected from Grishchuk’s old result [@Grishchuk]. Allen and Folacci long ago proved that this has no de Sitter invariant solution [@AF].
Exact de Donder gauge is interesting because de Sitter invariant constructions based on analytic continuation methods had previously dismissed it as an infrared divergent special case [@Higuchi]. In fact, all valid gauges show infrared divergences. The special thing about de Donder gauge is that some of its infrared divergences are logarithmic so that they are not automatically (and incorrectly) subtracted by analytic continuation. In all cases the right way to resolve the infrared divergence is by breaking de Sitter invariance.
We have gone to considerable lengths — in previous work [@MTW1; @MTW2] and again in section 3 — to elucidate precisely what goes wrong with previous constructions [@INVPROP] which seemed to produce de Sitter invariant results. However, it worth pointing out that the fact of de Sitter breaking was already obvious to cosmologists from the scale invariance of the tensor power spectrum, which becomes exact in the de Sitter limit [@RPW]. It was also obvious from the explicit form of a propagator constructed by mode sums on the open submanifold (for which there is no linearization instability) [@TW9; @RPW]. On the open submanifold the $\frac12 D (D+1)$ elements of the de Sitter group break down into four parts:
1. [$(D-1)$ spatial translations;]{}
2. [$\frac12 (D-2)(D-1)$ spatial rotations;]{}
3. [A single dilatation; and]{}
4. [$(D-1)$ spatial special conformal transformations.]{}
The gauge condition only breaks the last of these, but the solution for the propagator additionally breaks dilatation invariance [@TW9; @RPW]. The physical de Sitter breaking of this propagator was demonstrated by Kleppe, who augmented a naive de Sitter transformation by the compensating gauge transformation needed to restore the gauge condition [@Kleppe]. Had the propagator been physically invariant this technique would have revealed it.
We should also comment on the apparent conflict of our result with the pro-invariance argument given by Marolf and Morrison [@DMIAM], based on work by Higuchi [@Higuchi2]. They dealt with free dynamical gravitons in a noncovariant gauge on the full de Sitter manifold and they were able to construct the complete panoply of mode solutions and inner products. This should imply a vacuum which is physically de Sitter invariant — that is, invariant once the compensating gauge transformation is included. We know of no problem with this work but it should be noted that the propagator one gets using only dynamical gravitons (that is, the spatial, transverse-traceless polarizations) is not complete. It is like the purely spatial and transverse photon propagator of flat space electrodynamics in Coulomb gauge. To fully describe electromagnetic interactions also requires the instantaneous Coulomb interaction. Both of these are part of the same propagator in a covariant gauge such as the one we employ here.
The constrained, spin zero part of our propagator — which is missing from the transverse-traceless part — provides the largest source of the de Sitter breaking we found. It is relatively simple to show that the de Sitter breaking terms in $\mathcal{S}_0(x;z)$ do not drop out when acted upon by the spin zero projector $\mathcal{P}_{\mu\nu}(x) \times \mathcal{P}_{\rho \sigma}(z)$. The spin two structure function contains less severe de Sitter breaking terms of the form, $$\Bigl[\mathcal{S}_2(x;z) \Bigr]_{\rm de\ Sitter\ \atop breaking} =
\sum_{k=1}^3 s_k \Bigl[ \ln(a_x a_z) \Bigr]^k \; .$$ It is possible that these drop out from the spin two part of the propagator (\[Spin2\]) after all eight of the derivatives have been taken. In that case our work would be fully consistent with that of Marolf and Morrison. However, what we expect is that one of the infrared logarithms survives, which seems to be indicated by the scale invariance of the tensor power spectrum.
The fact of de Sitter breaking in this system cannot be disputed, but there is wide freedom as to how one chooses to manifest that breaking. This freedom amounts to picking the initial state. We have chosen the explicit solutions of section 3 so as to preserve the symmetries of homogeneity and isotropy, which allow one to view de Sitter as a special case of a spatially flat, FRW geometry. This choice is known in the literature as the “$E(3)$ vacuum.” Readers who prefer to preserve another subgroup can do so by starting from our scalar equations (\[EQN0\]) and (\[EQN2\]).
We wrote this paper to help resolve the long-standing controversy about de Sitter breaking for free gravitons, however, it has other applications. One of these is to test for gauge dependence in quantum gravitational loop corrections from primordial inflation. Of course gauge-fixed Green’s functions will show such dependence, mingled with valid physical information. In flat space we would sift out the gauge dependence by forming the S-matrix. That observable is not available in cosmology [@Witten], and there is not yet any consensus for what replaces it. One technique is simply to carry out computations in different gauges. It may be that the leading infrared logarithm contributions (e.g., the one loop contribution to the fermion field strength from inflationary gravitons [@MW2]) are independent of the choice of gauge. Now we can test this conjecture using a completely different gauge from the one [@TW9; @RPW] employed in all previous computations.
Our propagator should also make renormalization simpler because it precludes the appearance of noninvariant counterterms. These complicated the analysis for previous computations [@MW2; @KW2]. It may also be that the gauge condition (\[deDonder\]) and the special properties of the differential projectors in our propagator make actual computations simpler. That turned out to be the case with the vector propagator in Lorentz gauge [@TW2] for a variety of one and two loop computations [@PTW1; @PTW2; @PTW3].
**Acknowledgements**
This work was partially supported by FQXi Mini Grant \#MGB-08-008, by NWO Veni Project \# 680-47-406, by European Union Grant FP-7-REGPOT-2008-1-CreteHEPCosmo-228644, by NSF grant PHY-0855021, and by the Institute for Fundamental Theory at the University of Florida.
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[^1]: It has been conjectured that this shows up in mathematical physics as a violation of reflection positivity [@Marolf].
[^2]: Just how good can be quantified using the deceleration parameter $q(t) =
-a\ddot{a}/\dot{a}^2$, which measures minus the fractional cosmic acceleration. Its value for de Sitter is $q = -1$, and the threshold between inflation and deceleration occurs at $q=0$. If one assumes single scalar inflation then the measured result for the scalar amplitude [@WMAP], and the bound on the tensor-to-scalar ratio [@WMAP], imply $95\%$ confidence that $q(t) < -0.986$ when the largest observable perturbations experienced first horizon crossing [@KOW]. Because this would have been near the end of inflation, when $q(t)$ was growing, most of the inflationary epoch was likely even closer to de Sitter.
|
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abstract: 'We investigate the scalar field perturbations of the $4+1$-dimensional Schwarzschild black hole immersed in a Gödel Universe, described by the Gimon-Hashimoto solution. This may model the influence of the possible rotation of the Universe upon the radiative processes near a black hole. In the regime when the scale parameter $j$ of the Gödel background is small, the oscillation frequency is linearly decreasing with $j$, while the damping time is increasing. The quasinormal modes are damping, implying stability of the Schwarzschild-Gödel space-time against scalar field perturbations. The approximate analytical formula for large multipole numbers is found.'
---
łŁ § ø ¶
\#1[(\#1)]{} \#1[(\#1)]{}
**Scalar field perturbations of the Schwarzschild**
**black hole in the Gödel Universe**
**R.A. Konoplya and Elcio Abdalla**
Instituto de Fisica, Universidade de São Paulo,
C.P.66318, CEP 05315, São Paulo SP, Brazil
konoplya@fma.if.usp.br
Introduction
============
Black hole’s behavior is often crucially dependent upon the cosmological background in which the black hole is immersed. The simple case of a black hole immersed in an asymptotically flat space-time is described by the Schwarzschild solution. A natural extension is to consider a cosmological term which is described in terms of the Schwarzschild-de Sitter or Schwarzschild - anti-de Sitter solutions. Recent investigations of radiative processes around such black holes show that the radiative features are dependent upon the asymptotic conditions on infinity. Thus, for example, the quasinormal spectrum and late time tails are totally different for Schwarzschild [@S], Schwarzschild-de Sitter [@SdS] and Schwarzschild - anti-de Sitter black holes [@SAdS].
The cosmological background we are interested in here, is the rotating Universe. The rotation seems to be a universal phenomenon: all compact objects in the Universe rotate. Yet the standard Friedman-Robertson-Walker metric represents rather idealized model of isotropic homogeneous world filled with perfect fluid. It looks improbable that such a finely tuned universe can exist starting from the Big Bang to the present stage. In the beginning of the investigations of rotating cosmological models it was suggested that one should observe the anisotropy of the Microwave Background Radiation (MBR), yet, as shown later, that the rotating models with no anisotropy of MBR or broken causality can exist [@obukhov1]. In addition, apparent anisotropy in distribution of the observed angles between the polarization vectors and position of the major axis of radio sources can be related to a possible rotation of the Universe [@Brich]. For further advance in the possibility of observation of global rotation see the review [@obukhov2].
An exact solution for the rotating Universe was found by Gödel [@Godel]. His solution was originally proposed for a four dimensional space-time. It possesses, among others, the following properties: it is homogeneous, has rotational symmetry, and allows the definition of the direction of positive time consistently in the whole solution, and, what was in the focus of further research, it allows closed time-like curves, i.e. the time machine.
Recently, the Gödel Universes have been of considerable interest [@kucha1], because in five-dimensional minimal supergravity the maximally supersymmetric Gödel-type universes are U-dual to pp-waves. Thereby, the Gödel-type universes are important as a an opportunity of quantizing strings in this background a well as due to its relation to the corresponding limit of super-Yang Mills theories. On the gravitational side, the pp-waves dual to the Gödel Universe corresponds to the Penrose limit of near-horizon geometries.
As far as we are aware, until recently, an exact solution for a stationary black hole immersed in a rotating Universe was not known. Nevertheless, such a solution has been obtained by Gimon and Hashimoto within the above mentioned five-dimensional minimal supergravity [@gimon]. This solution represents the $4+1$-dimensional Schwarzschild space-time when the scale parameter of the Gödel background $j$ goes to zero, and to the five dimensional Gödel Universe when the black hole mass vanishes, thereby giving us the model for the Schwarzschild black hole immersed in the rotating Universe. The different features of this solution have been investigated recently in a series of papers [@kucha2]. The generalizations of the Gimon-Hashimoto solution were obtained in [@kucha3].
Yet, here, we are interested in this solution from a rather different point of view, namely, we would like to find out what will happen with black hole (classical) radiation in the rotating Universe. The straightforward way to know it, is to investigate the quasinormal modes which govern the black hole response to external perturbations at late times. The quasinormal spectrum is sensitive to boundary conditions both at the event horizon and at spatial infinity, so the spectrum must be considerably affected by the rotating cosmological background.
In the present paper we had to be limited by the case of “slow rotation”, i.e. the case when the influence of the cosmological background is weak. This happens when the above mentioned parameter $j$ is small. We found that at least in the regime of small $j$, the black hole is stable against scalar field perturbations and as a result all found modes are damping. Due to cosmological rotation the real oscillation frequencies are decreasing and are roughly proportional to $j$, while the damping rates are decreasing non-linearly with $j$. In addition we derive an approximate analytical formula for QN modes with large multipole number $L$. Fortunately the regime of small $j$ seems to be the most reasonable phenomenologically.
The paper is organized as follows: in Sec.2 we give some introductory material on the Schwarzschild-Gödel metric. In Sec 3., the scalar field equations is obtained in the limit of small scale parameter $j$ of the cosmological background. For any $j$, the Klein-Gordon equation is not separable at least in the coordinates we have considered. In Sec.IV we find the quasinormal frequencies for the scalar field perturbations. In the end we discuss the obtained results and future perspectives.
Preliminaries of the Schwarzschild-Gödel space-time.
====================================================
The bosonic fields of the minimal (4+1)- supergravity theory consist of the metric and the one-form gauge field, which are governed by the equations of motion $$R_{\mu \nu} =2 \left(F_{\mu \alpha} F_{\nu}^{\alpha} -\frac{1} {6} g_{\mu \nu}
F^{2}\right)$$ $$D_{\mu} F^{\mu \nu} = \frac{1}{2 \sqrt{3}} \varepsilon^{\alpha \lambda
\gamma \mu \nu} F_{\alpha \lambda} F_{\gamma \mu}$$ Here, $ \varepsilon_{\alpha \lambda \gamma \mu \nu} = \sqrt{-g} \epsilon_{\alpha \lambda \gamma \mu \nu}$.
In the Euler coordinates $(t, \theta', \psi', \phi')$, the solution of the equations of motion (1), (2), describing the Gödel universe, has the form [@gimon]: $$ds^2 = - (dt + j (r^2) \sigma_{L}^{3})^2 + dr^{2} + \frac{r^2}{4}
(d \theta'^{2} + d \psi'^{2} + d \phi'^{2} + 2 cos \theta' d \psi' d \phi'),$$ where $\sigma_{L}^{3}= d \phi' + cos \theta d \psi'$. The parameter $j$ defines the scale of the Gödel background. At $j=0$ we have the Minkowski space-time. The solution for the Schwarzschild black hole in the Gödel universe is given by [@gimon] $$ds^2 = - f(r) dt^2 -g(r) r \sigma_{L}^{3} d t + h(r) r^2
(\sigma_{L}^{3})^2 + k(r) d r^2 +$$ $$\frac{r^2}{4}
(d \theta'^{2} + d \psi'^{2} + d \phi'^{2} + 2 cos \theta' d \psi' d \phi'),$$ where $$f(r)=1- \frac{2 M}{r^2}, \quad g(r) = 2 j r,$$ $$h(r) =j^2 (r^2 + 2 M), \quad k(r) = \left(1 - \frac{2 M}{r^2} + \frac{16 j^2 M^2}{r^2}\right)^{-1}.$$ The radius of the event horizon is also corrected by parameter $j$, $$r_{BH} = \sqrt{2 M (1- 8 j^2 M)}.$$ Note that the maximal value of the black hole mass $M$ is $1/8
j^2$. For a larger mass the horizon area vanishes and one has a naked singularity.
The above black hole metric keeps five of the nine isometries of the Gödel universe, generated by $\partial_{t}$, and by four generators of the $SU(2) \times U(1)$ subgroup of the $SO(4)$ isometry group acting on $S^3$ [@gimon].
In the limit $j =0$ we have the (4+1)-dimensional Schwarzschild solution, while in the limit of $m=0$ the pure Gödel space-time is recovered. To treat the scalar field perturbations around such a Schwarzschild-Gödel black hole let us rewrite the metric in the bi-spherical coordinates ($\phi$, $\psi$, $\theta$), which are connected with the Euler angles ($\phi'$, $\psi'$, $\theta'$) by $$\phi' = \psi + \phi, \quad \psi'= \psi - \phi, \quad, \theta'= 2
\theta.$$ Then, in the regime of small $j$, i.e.discarding terms of order $O(j^2)$, the metric takes the form $$ds^2 \approx - f(r) dt^2 -2 g(r) r ((sin \theta)^2 d \phi + (cos
\theta)^2 d \psi) d t + k(r) d r^2 +$$ $$r^2 (d \theta^{2} + (cos \theta)^2 d \psi^{2} + (sin \theta)^2 d \phi^{2}).$$ Note that in the above equation $k(r) = \left(1 - \frac{2
M}{r^2}\right)^{-1}$.
Up to $O(j^2)$, the inverse metric $g^{\mu \nu}$ has components $$g^{11}=-\left(1 - \frac{2 M}{r^2} \right)^{-1} \quad g^{22}=1 - \frac{2 M}{r^2}
\quad g^{33} = \frac{1}{r^2}$$ $$g^{44}=r^{-2} (cos \theta)^{-2} , \quad g^{55}=r^{-2} (sin \theta)^{-2},
\quad g^{14}=g^{15}=- 2 j \left(1- \frac{2 M}{r^2} \right)^{-1}.$$
Scalar field perturbations of the Schwarzschild - Gödel space-time.
===================================================================
The scalar field perturbations in a curved background are governed by the Klein-Gordon equation $$\Box \Phi \equiv \frac{1}{\sqrt{-g}} \left(g^{ \mu \nu} \sqrt{-g}
\Phi,_{\mu}\right),_{ \nu} = 0.$$ Since the background metric has the Killing vectors $\partial_{t}$, $\partial_{\psi}$, $\partial_{\phi}$, the wave function $\Psi$ can be represented in the form $$\Phi \sim e^{i \omega t + i k \psi + i m \phi} Y(\theta) R(r).$$
Unfortunately, variables in the Klein-Gordon equation are not separable, at least in the considered coordinates for the full Gimon-Hashimoto metric. The separability is connected with the existence of the Killing tensor [@bagrov], and, it is possible that one can separate variables in some other privileged coordinate systems. Here we were limited to small values of $j$, for which the variables in the Klein-Gordon equation can be separated even in ordinary bi-spherical or Euler coordinates.
Using the expressions for metric coefficients (8-10), the scalar field equation (11) takes the form: $$r^{-3} \frac{\partial }{\partial r} \left(\left(1-\frac{2 M}{r^2}\right)
r^{3} \frac{\partial R(r)}{\partial r} \right) + (\omega^2 + 4 j
\omega (k + m)) f^{-1}(r) R(r) + \frac{\lambda}{r^2} R(r) =0,$$ where the separation constant comes from the equation for angular variables, $$\frac{1}{cos \theta sin \theta} \frac{\partial}{\partial \theta}
\left(sin \theta cos \theta \frac{\partial Y}{\partial \theta}
\right)
- \left(\frac{k^2}{(cos \theta)^2}+ \frac{m^2}{(sin \theta)^2} \right)
Y = \lambda Y.$$
Going over to the tortoise coordinate $d r^{*} =d r/f(r)$ and to the new wave function $\Psi = R(r) r ^{3/2}$, the equation (13) can be reduced, after some algebra, to the wave-like form
\[Wave-like-equation\]( + \^2 + 4 j (k + m) - V(r\^\*))= 0.
The effective potential has the form $$V(r)=f(r)\left(\frac{3}{4 r^{2}}f(r)+ \frac{3}{2r} f'(r)+ \frac{(2 l + k + m) (2 l + k + m + 2)}{r^{2}}\right).$$ Here $l$, $k$, and $m$ run over the values $0, 1, 2, ...$ The tortoise coordinate $r^{*}$ is defined on the interval $(- \infty, +
\infty)$ in such a way, that the spatial infinity $r=+\infty $ corresponds to $r^{*} =\infty$, while the event horizon corresponds to $r^{*} =-\infty$. The above effective potential is positively defined and has the form of the potential barrier which approaches constant values at both spatial infinity and event horizon. In fact, the potential $V(r)$ coincides with that for (4+1)-dimensional Schwarzschild black hole when taking the multipole number $L$ to be $2 l + k + m$, yet, the spectrum is different, due to the term $4 j \omega (k + m)$ in (5), which depend not only on the final value $L$, but also on terms $l$, $k$, $m$.
Quasinormal modes of the Schwarzschild-Gödel black hole.
========================================================
If choosing a positive sign for the real part of $\omega$ ($\omega = Re \omega - i Im \omega$), QNMs satisfy the following boundary conditions \[bounds\] ¶(r\^\*) \~C\_(r\^\*), r, corresponding to purely in-going waves at the event horizon and purely out-going waves at infinity.
In order to find quasinormal frequencies of the black hole with an effective potential in the form of the potential barrier (16), we use the WKB approach. The WKB approach was used for calculations of quasinormal modes in the first order beyond eikonal approximation by Schutz and Will [@schutz-will], extended by Iyer and Will to the third order [@IyerWill], and recently extended to the sixth WKB order [@KonoplyaWKB6]. The WKB approach up to the 6th WKB order has been used used recently in a series of papers [@cardoso-acustic], [@berti-kok], [@konoplya03-3], where QN frequencies of different black holes were considered, and, the comparison with accurate numerical values showed very good agreement. The accuracy of the WKB results is the better, the larger the multipole number $L$ and the less the overtone number $n$. In fact for $n$ larger then $L$ the WKB formula cannot be applied.
From here we shall use the units such that $2 M =1$.
The WKB formula has the form [@KonoplyaWKB6]:
\[WKB\] - Ł\_2 - Ł\_3 - Ł\_4 - Ł\_5 - Ł\_6 = n + , where $V_0$ is the height and $V_0^{\prime\prime}$ is the second derivative with respect to the tortoise coordinate of the potential at the maximum. $\L_2$ and $\L_3$ can be found in [@IyerWill], $\L_4$, $\L_5$ and $\L_6$ are presented in [@KonoplyaWKB6]; the corrections depend on the value of the potential and higher derivatives of it at the maximum.
The 6th order WKB values of the quasinormal frequencies are shown on Figures 1-4 and Table I. From Fig. 1 one can see that the real part of $\omega$ is decreasing with growing of $j$, being roughly proportional to $j$, for a fixed black hole mass and fixed values of $l$, $m$, $k$. This can be easily explained in the following way: from the wave equation (15) one can learn that if one discards small values of order $O(j^2)$, then $\omega^{2} + 4 j (m+k) \omega = \omega_{0}^2$, where $\omega_{0}$ is the Schwarzschild value of $\omega$ under some fixed $M$, $l$, $m$, $k$, $n$. Furthermore this can be represented as $(\omega + 2 j (k+m))^{2} - 8 j^2 (k+m)^2=\omega_{0}^2$, i.e. the real part of $\omega$ is roughly increased by $2 j
(k+m)$, while the change in imaginary part comes from the term $8 j^2 (k+m)^2$. More accurately, Fig. 2 shows that the imaginary part is decreasing when $j$ is increasing. Therefore, the influence of rotating cosmological background, represented by parameter $j$, gives rise to decreasing of the oscillation frequency and of the damping rate. Thus, in the rotating Universe the QN modes damp more slowly, but, because of the considerable falling down of $Re \omega$ and slight falling down of the $Im \omega$, the resulting quality factor $Re \omega/2 Im \omega$ is decreasing and, thereby, the black hole in a rotating Universe is a worse oscillator than in a non-rotating one.
In Table 1. we put the low overtones of the QN spectrum for different values of $k$, $m$, $l$, and $j$. From that table we can learn that the higher any of the values $k$, $l$, or $m$ and the lower the overtone $n$, the better the accuracy of the WKB formula, and, as a result, the less is the difference between the 6th and 3th order WKB data. An intrinsic fact for any black hole quasinormal spectrum, when the overtone number grows $Re \omega$ falls down, while the damping rate grows. We cannot judge what will happen with asymptotically high overtones ($n \rightarrow \infty$), since we analyze here only an approximate solution.
In the eikonal (high frequency) approximation, we can use the first order WKB formula for finding the lower overtones. Thus, for large $l$, and thereby for large $L= 2 l + k + m$, in units $2 (2 M)^{-1}$ we obtain: $$\omega = \frac{L+1}{2} + 2 j (k + m) - i \frac{2 n +1}{2 \sqrt{2}}.$$ Note that there are two limitations on this formula. First, when $k$ or $m$ is also large, the general relation $$\omega^2 - 4 j (k+m) \omega = \left(\frac{L+1}{2} - i \frac{2 n +1}{2 \sqrt{2}}\right)^2$$ holds. Moreover, we should be careful when interpreting this formula at asymptotically large $L$, since it uses an approximate metric and $j^{2}$ corrections in metric may produce different asymptotic values. Yet we expect it should be correct for moderately large values of $L$. Note that the above formulas are accurate enough even for not very large values of $L$, for instance, for $L=4$ the relative error is about several percents.
The massless scalar QNMs analysis can easily be extended to the massive case, in which one has the same wave-like equation but with the effective potential $$V(r)=f(r)\left(\frac{3}{4 r^{2}}f(r)+ \frac{3}{2r} f'(r)+ \frac{(2 l +
k + m) (2 l + k + m + 2)}{r^{2}} +\mu^2 \right).$$ Yet we can use the above WKB formula only for small values of the field mass $\mu$, since for large $\mu$ the effective potential has three turning points. The 6th order WKB frequencies are presented in Fig.3 and 4 as functions of $\mu$. Thus, we see that the larger the field mass, the larger is the real part of $\omega$, and the smaller the imaginary part. In other words, the “massive” QN modes decay more slowly and have greater real frequency of oscillation. It is known that at asymptotically high overtones the mass of the field does not affect the QN modes [@SMF]. Note that all the above features were observed for massive scalar field of the Schwartzshild [@SMF] and Reissner-Nordstrom black holes [@RMF], [@Xue].Since these features were found also for massive Dirac field (see [@Karlucio] and references therein), it is possible that they are generic for massive fields of arbitrary spin.
TABLE I: WKB values for QNMs at fixed $j=1/8$. $2 M=1$.
$k$ $m$ $l$ $n$ 3th WKB order 6th WKB order
----- ----- ----- ----- ------------------------- -------------------------
$0$ $0$ $0$ $0$ $ 0.49123 - 0.41100 i $ $ 0.54633 - 0.36087 i $
$1$ $0$ $0$ $0$ $ 0.78080 - 0.35384 i $ $ 0.79151 - 0.35575 i $
$1$ $0$ $0$ $1$ $ 0.59794 - 1.15113 i $ $ 0.61846 - 1.14902 i $
$0$ $0$ $1$ $0$ $ 1.50707 - 0.35787 i $ $ 1.51050 - 0.35770 i $
$0$ $0$ $1$ $1$ $ 1.38524 - 1.10754 i $ $ 1.39249 - 1.10537 i $
$0$ $0$ $1$ $2$ $ 1.19442 - 1.90690 i $ $ 1.18639 - 1.94829 i $
$1$ $0$ $1$ $0$ $ 1.77155 - 0.35330 i $ $ 1.77293 - 0.35317 i $
$1$ $0$ $1$ $1$ $ 1.67618 - 1.07942 i $ $ 1.67927 - 1.07838 i $
$1$ $0$ $1$ $2$ $ 1.51426 - 1.84424 i $ $ 1.50511 - 1.86038 i $
$1$ $0$ $1$ $3$ $ 1.30766 - 2.63920 i $ $ 1.27639 - 2.73256 i $
$1$ $1$ $1$ $0$ $ 2.05408 - 0.34832 i $ $ 2.05475 - 0.34826 i $
$1$ $1$ $1$ $1$ $ 1.97396 - 1.05821 i $ $ 1.97552 - 1.05774 i $
$1$ $1$ $1$ $2$ $ 1.83110 - 1.79890 i $ $ 1.82449 - 1.80621 i $
$1$ $1$ $1$ $3$ $ 1.64511 - 2.57051 i $ $ 1.61726 - 2.61762 i $
$1$ $1$ $1$ $4$ $ 1.42547 - 3.36434 i $ $ 1.37379 - 3.51096 i $
Discussions
===========
We have investigated the decay of (generally speaking, massive) scalar field around a Schwartzshild black hole immersed in a rotating cosmological background. In the limit of the small cosmological parameter $j$, the QNMs, which govern the decay of the scalar field at late times, have been found. It was found that the cosmological rotation gives rise the decreasing of the real frequencies of oscillations (proportional to the cosmological parameter) and of damping rates. The quality factor of the black hole as an oscillator is smaller in the presence of cosmological rotation. The massive scalar field damps more slowly and have greater oscillation frequency. All found modes are damping what supports the stability of the Schwartzshild-Gödel space-time against scalar field perturbations. Yet, within the approximate solution we analyzed, one cannot judge about stability eventually. Note also that the stability of the metric as such is determined by the gravitational perturbations, although the scalar field perturbations may coincide with tensor type gravitational perturbations [@ishibashi] which are decisive in gravitational stability [@gibbons]. The present analysis can also be extended to the case of scalar field interacting electromagnetically with the charge of the black hole [@charged_scalar], i.e. to the case of the decay of charged scalar field around a Reissner-Nordstrem-Gödel black hole.
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![Real part of $\omega$ as a function of the Gödel background scale parameter $j$ (diamond $k=2$, $m=1$, $l=1$), (box $k=1$, $m=1$, $l=1$), (star $k=0$, $m=1$, $l=1$).[]{data-label="1"}](godel1.eps)
![Imaginary part of $\omega$ as a function of the Gödel background scale parameter $j$ (star $k=2$, $m=1$, $l=1$), (box $k=1$, $m=1$, $l=1$), (diamond $k=0$, $m=1$, $l=1$).[]{data-label="2"}](godel2.eps)
![Imaginary part of $\omega$ as a function of the mass $\mu$ for $l=1$, $k=1$, $m=1$ (box), $l=1$, $k=0$, $m=1$ (star), $l=1$, $k=0$, $m=0$ (diamond); $j=1/8$.[]{data-label="3"}](godel3.eps)
![Real part of $\omega$ as a function of the mass $\mu$ for $l=1$, $k=1$, $m=1$ (box), $l=1$, $k=0$, $m=1$ (star), $l=1$, $k=0$, $m=0$ (diamond); $j=1/8$.[]{data-label="4"}](godel4.eps)
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abstract: 'Peer prediction mechanisms are often adopted to elicit truthful contributions from crowd workers when no ground-truth verification is available. Recently, mechanisms of this type have been developed to incentivize effort exertion, in addition to truthful elicitation. In this paper, we study a sequential peer prediction problem where a data requester wants to dynamically determine the reward level to optimize the trade-off between the quality of information elicited from workers and the total expected payment. In this problem, workers have homogeneous expertise and heterogeneous cost for exerting effort, both unknown to the requester. We propose a sequential posted-price mechanism to dynamically learn the optimal reward level from workers’ contributions and to incentivize effort exertion and truthful reporting. We show that (1) in our mechanism, workers exerting effort according to a non-degenerate threshold policy and then reporting truthfully is an equilibrium that returns highest utility for every worker, and (2) The regret of our learning mechanism w.r.t. offering the optimal reward (price) is upper bounded by $\tilde{O}(T^{3/4})$ where $T$ is the learning horizon. We further show the power of our learning approach when the reports of workers do not necessarily follow the game-theoretic equilibrium.'
author:
- |
Yang Liu and Yiling Chen\
Harvard University, Cambridge MA, USA\
{yangl,yiling}@seas.harvard.edu
bibliography:
- 'myref.bib'
- 'library.bib'
title: 'Sequential Peer Prediction: Learning to Elicit Effort using Posted Prices'
---
Introduction {#sec:intro}
============
Crowdsourcing has arisen as a promising option to facilitate machine learning via eliciting useful information from human workers. For example, such a notion has been widely used for labeling training samples, e.g., Amazon Mechanical Turk. Despite its simplicity and popularity, one salient feature or challenge of crowdsourcing is the lack of evaluation for the collected answers, because ground-truth labels often are either unavailable or too costly to obtain. This problem is called [*information elicitation without verification*]{} (IEWV) [@Waggoner:14]. A class of mechanisms, collectively called [*peer prediction*]{}, has been developed for the IEWV problem [@Prelec:2004; @MRZ:2005; @jurca2007collusion; @jurca2009mechanisms; @witkowski2012robust; @witkowski2012peer; @radanovic2013]. In peer prediction, an agent is rewarded according to how his answer compares with those of his peers and the reward rules are designed so that everyone truthfully reporting their information is a game-theoretic equilibrium.
More recent work [@Witkowski_hcomp13; @dasgupta2013crowdsourced; @2016arXiv160303151S] on peer prediction concerns *effort elicitation*, where the goal is not only to induce truthful report, but also to induce high quality answers by incentivizing agents to exert effort. In such work, the mechanism designer is assumed to know workers’ expertise level and their cost for effort exertion and designs reward rules to induce optimal effort levels and truthful reporting at an equilibrium.
This paper also focuses on the effort elicitation of peer prediction. But different from prior work, our mechanism designer knows neither workers’ expertise level nor their cost for effort exertion. We introduce a *sequential peer prediction* problem, where the mechanism proceeds in rounds and the mechanism designer wants to learn to set the optimal reward level (that balances the amount of effort elicited and the total payment) while observing the elicited answers in previous rounds. There are several challenges to this problem. First, effort exertion is not observable and no ground-truth answers are available for evaluating contributions. Hence, it is not immediately clear what information the mechanism designer can learn from the observed answers in a sequential mechanism. Second, forward-looking workers may have incentives to mislead the learning process, hoping for better future returns.
The main contributions of this paper are the following: (1) We propose a *sequential peer prediction* mechanism by combining ideas from peer prediction with multi-armed bandit learning [@LR85; @Auer:2002:FAM:599614.599677]. (2) In this mechanism, workers exerting effort according to a non-degenerate threshold policy and then reporting truthfully in each round is an equilibrium that returns highest utility for every worker. (3) We show that the regret of this mechanism w.r.t. offering the optimal reward is upper bounded by $\tilde{O}(T^{3/4})$ where $T$ is the learning horizon. We also show that under a “mean-field” assumption, the sequential learning mechanism can be extended to a setting where workers may not be fully rational. (4) Our sequential peer prediction mechanism is [*minimal*]{} in that reported labels are the only information we need from workers. In the rest of the paper, we first survey the most related work in Section \[sec:related\]. Section \[sec:formulate\] introduces our problem formulation. We then present a game-theoretic analysis of worker behavior in a one-stage static setting in Section \[sec:static\]. Based on the equilibrium analysis of the one-stage setting, we propose and analyze a learning mechanism to learn the optimal bonus level using posted price in Section \[sec:learn\]. We also discuss an extension of our learning mechanism to a setting where workers may not be fully rational. Section \[sec:conclude\] concludes this paper. All omitted details can be found in the full version of the paper [@fullversion_aaai17].
Related work {#sec:related}
------------
Eliciting high-quality data from effort-sensitive workers hasn’t been addressed within the literature of peer prediction until recently. @Witkowski_hcomp13 \[[-@Witkowski_hcomp13]\] and @dasgupta2013crowdsourced \[[-@dasgupta2013crowdsourced]\] formally introduced costly effort into models of IEWV. The costs for effort exertion were assumed to be homogeneous and known and static, one-shot mechanisms were developed for effort elicitation and truthful reporting. Our setting allows participants to have heterogeneous cost of effort exertion drawn from a common unknown distribution and hence we consider a sequential setting that enables learning over time. @fullversion \[[-@fullversion]\] is the closest to this work. It considered the same general setting and partially resolved the problem of learning the optimal reward level sequentially. There are two major differences however. First, the method developed in @fullversion \[[-@fullversion]\] required workers to report their private cost in addition to their answer, which is arguably undesirable for practical applications. Our learning mechanism in contrast is “minimal” [@segal2007communication; @Witkowski_ec13] and only asks for answers (for tasks) from workers. Second, the mechanism of @fullversion \[[-@fullversion]\] was built upon the output agreement mechanism as the single-round mechanism. Output agreement and hence the mechanism of @fullversion \[[-@fullversion]\] suffer from potential, simple collusions of workers: colluding by reporting an uninformative signal will lead to a better equilibrium (higher utility) for workers. By building upon the mechanism of @dasgupta2013crowdsourced \[[-@dasgupta2013crowdsourced]\], which as a one-shot mechanism is resistant to such simple collusion, we develop a collusion-resistant sequential learning mechanism.
Generally speaking, when there is a lack of knowledge of agents, the design problem needs to incorporate learning from prior outputs from running the mechanism – see @chawla2014mechanism \[[-@chawla2014mechanism]\] for specific examples on learning with auction data. And this particular topic has also been studied within the domain of crowdsourcing. For example, [@roth2012conducting \[[-@roth2012conducting]\] and @abernethy2015actively \[[-@abernethy2015actively]\] consider strategic data acquisition for estimating the mean and for online learning respectively. ]{} [ Our problem differs from above in that both agents’ action (effort exertion) and ground-truth outcomes are unavailable.]{}
Problem Formulation {#sec:formulate}
===================
Formulation and settings
------------------------
Suppose in our system we have one data requester (or a mechanism designer), and there are $N$ candidate workers denoted by $\mathcal C = \{1,2,...,N\}$, where $N \geq 4$. In all we have $N+1$ interactive agents. The data requester has binary-answer tasks, with answer space $\{-1,+1\}$, that she’d like to get labels for. The requester assigns tasks to workers.
Label generated by worker $i \in \mathcal C$ comes from a distribution that depends both on the ground-truth label and an effort variable $e_i$. Suppose there are two effort levels, High and Low, that a worker can potentially choose from: $e_i \in \{H,L\}$. We model the cost $c$ for exerting High effort for each (worker, task) pair as drawn from a distribution with c.d.f. $F(c)$ on a bounded support $[0, c_{\max}]$; while exerting Low effort incurs no cost. We assume such costs are drawn in an i.i.d. fashion. Denote worker $i$’s probability of observing $s' \in \{-1,+1\}$ when the ground-truth label is $s \in \{-1,+1\}$ as $p_{i,e_i} = \Pr(s'=s|s,e_i)$, under effort level $e_i$. Note with above we have assumed that the labeling accuracy is symmetric, and is independent of the ground-truth label $s$. Further for simplicity of analysis, we will assume all workers have the same set of $p_{i,H}, p_{i,L}$, denoting as $p_H, p_L$. With higher effort, the expertise level is higher: $1 \geq p_H >p_L \geq 0.5$ – we also assume the labeling accuracy is no less than 0.5. [The above are common knowledge among workers, while the mechanism designer doesn’t know the form of $F(\cdot)$; neither does she know $p_H,p_L$. But we assume the mechanism designer knows the structural information, such as costs are i.i.d., workers are effort sensitive, and there are two effort levels etc.]{}
[The goal of the learner is to design a sequential peer prediction mechanism for effort elicitation via observing contributed data from workers, such that the mechanism will help the learner converge to making the optimal action (will be defined later). ]{}
Reward mechanism
----------------
Once assigned a task, worker $i$ has a guaranteed base payment $b>0$ for each task he completes. In addition to the base payment, the worker receives a reward $B_i (k)$ for task $k$ that he has provided an answer for. The reward is determined using the mechanism proposed by @dasgupta2013crowdsourced \[[-@dasgupta2013crowdsourced]\], where in this paper, we denote this specific peer prediction mechanism for effort elicitation as (`DG13`). In this mechanism, for each (worker, task) pair $(i,k)$, first a reference worker $j \neq i$ is selected randomly from the set of workers who are also assigned task $k$. [Suppose any pair of such workers have been assigned $d$ other distinct tasks $\{i_1,...,i_d\}$ and $\{j_1,...,j_d\}$ respectively. Then the mechanism pays $B'_i(k)$ to worker $i$ on task $k$ in the following way: the payment consists of two terms; one term that rewards agreement on task $k$, and another that penalizes on uninformative agreement on other tasks: $$\begin{aligned}
B'_i(k) = 1&\biggl (L_i(k)=L_j(k)\biggr) - L^d_i \cdot L^d_j -\overline{L}^d_i \cdot \overline{L}^d_i,~\label{bonus:index}\end{aligned}$$ where we denote reports from worker $i$ on task $n$ as $L_i(n)$ and $L^d_i := \sum_{n=1}^{d}(1+ L_i(i_n))/(2d), \overline{L}^d_i=1-L^d_i$. Our bonus rule follows exactly the same idea except that we will multiply $B'_i(k)$ by a constant $B \in [0,\bar{B}]$ (which we can choose): $B_i(k) := B\cdot B'_i(k)$. ]{}
#### Task assignment:
We’d like all workers to work on the same number of tasks, all tasks are assigned to the same number of workers and any pair of workers have distinct tasks. In particular, each worker is assigned $M>1$ tasks – denote the set of tasks assigned to worker $i$ as $\mathcal T_i: |\mathcal T_i|=M$. This is to simplify the computation of workers’ utility and payment functions. Each task is assigned to at least two workers. For any pair of workers who has been assigned a common task, they have at least $1 \leq d < M$ distinct tasks. These are to ensure that the (`DG13`) mechanism is applicable. We also set the number of assignments for each task to be the same – denote this number as $1 \leq K < N$, so that when we evaluate the accuracy of aggregated labels later, all tasks receive the same level of accuracy. But note we do not assign all tasks to all workers, i.e., $K \neq N$. Suppose each assigned task $k$ appears in $D \leq d$ tasks’ distinct set for each worker. The described assignment can be achieved by assigning $N$ different tasks in each round. For more details on assignments please refer to our full version.
Worker model
------------
After receiving each task $k$, worker $i$ first realizes the cost $c_i(k)$ for exerting High effort. Then worker $i$ decides his effort level $e_i(k) \in \{H,L\}$ and observes a signal $L_i(k)$ (label of the task). Worker $i$ can decide either to truthfully report his observation $r_i(k) = 1$ (denote by $r_i(k)$ the decision variable on reporting) or to revert the answer $r_i(k) = 0$: $$L^r_i(k) =\left\{
\begin{array}{ll}
L_i(k), ~\text{if}~r_i(k)=1\\
-L_i(k), ~\text{if}~r_i(k)=0
\end{array}
\right.$$ Workers are utility maximizers. Denote the utility function at each time (or step) for each worker as $u_i$, which is assumed to have the following form (payment $-$ cost):$$\begin{aligned}
u_i = Mb + \sum_{k \in \mathcal T_i} B_i(k) - \sum_{k \in \mathcal T_i} c_i(k),~\forall i.\end{aligned}$$
Data requester model
--------------------
After collecting labels for each task, the data requester will aggregate labels via majority voting. Denote workers who labeled task $k$ as $w_k(1),...,w_k(K)$. Then the aggregate label for $k$ is given by $$L^A(k) = 1\biggl(\sum_{n=1}^K L^r_{w_k(n)}(k)/K > 0\biggr) \cdot 2-1.~$$
The data requester’s objective is to find a bonus level $B$ (as in $B_i(k)$) that balances the accuracy of labels collected from workers, and the total payment. Denote requester’s objective function at each step as $\mathcal U(B)$ (assigning $N$ tasks):$$\begin{aligned}
\mathcal U(B) := \sum_{k=1}^N \biggl [ \Pr[L^A(k) = L(k)]-\eta \sum_{n=1}^K \mathbb E[B_{w_k(n)}(k)]\biggr],\end{aligned}$$ where $L(k)$ denotes the true label of task $k$, and $\eta>0$ is a weighting constant balancing the two terms in the objective.
Since we have assumed that all tasks have been assigned the same number of workers, and workers are homogeneous in their labeling accuracy and cost (i.i.d.), we know all tasks enjoy the same probability of having a correct label (a-priori). We denote this probability as $\mathcal P(B) := \Pr[L^A(k) = L(k)], \forall k$. Further as workers do not receive payment when a task is not assigned to him, $\mathcal U(B)$ can be simplified (and normalized [^1]) to the following form: $$\begin{aligned}
\mathcal U(B) = \mathcal P(B) - \frac{\eta}{N} \sum_{i \in \mathcal C} \sum_{k=1}^N \mathbb E[B_i(k)],~\label{learner:objective}\end{aligned}$$ Suppose there exists a maximizer $
B^* = \text{argmax}_B \mathcal U(B)~.
$
Sequential learning setting
---------------------------
[Suppose our sequential learning algorithm goes for $T$ stages. At each stage $t=1,...,T$, learner assigns a certain number of tasks $M_i(t)$ to a set of selected workers $i \in S(t)$[^2]. The learner offers a bonus bundle $B_{i,t}$ to each worker $i \in S(t)$ (the bonus constant in reward mechanism). The regret of offering $\{B_{i,t}\}_{i,t}$ w.r.t. $B^*$ is defined as follows: $$\begin{aligned}
R(T) = T\cdot\mathcal U(B^*)- \sum_{t=1}^T \sum_{i \in S(t)}\frac{M_i(t) \cdot \mathbb E[\mathcal U(B_{i,t})] }{\sum_{j \in S(t)} M_j(t)}.\label{regret:defn}\end{aligned}$$ ]{} Note we normalize $\mathbb E[\mathcal U(B_{i,t})]$ using the number of assignments – intuitively the more the requester assigned with a wrong price, the more regret will be incurred. The goal of the data requester is to design an algorithm such that $R(T) = o(T)$. We can also define $R(T)$ as being un-normalized, which will add a constant (bounded number of assignments at each step) in front of our results later.
One stage game-theoretic analysis {#sec:static}
=================================
From the data requester’s perspective, we need to first understand workers’ actions towards effort exertion and reporting under different bonus levels, in order to figure out the optimal $B^*$. We start with the case that the data requester knows the cost distribution, and we characterize the equilibria for effort exertion and reporting, i.e. $(\mathbf{e_i,r_i})$, on workers’ side. Note $\mathbf{e_i,r_i}$ are both vectors defined over all tasks – this is a simplification of notation as workers do not receive all tasks. We are safe as if task $k$ is not assigned to $i$, worker $i$ does not make decisions on $(e_i(k),r_i(k))$. We define Bayesian Nash Equilibrium (BNE) in our context as follows:
We say $\{(\mathbf{e^*_i,r^*_i})\}_{i \in \mathcal C}$ is a BNE if $\forall j, (\mathbf{e_j,r_j})$: $$\begin{aligned}
\mathbb E[u_j| \{(\mathbf{e^*_i,r^*_i})\}_{i \in \mathcal C}] \geq
\mathbb E[u_j| \{(\mathbf{e^*_i,r^*_i})\}_{i \neq j},(\mathbf{e_j, r_j})]~. \end{aligned}$$
[In this paper, we restrict our attention to symmetric BNE.]{} For each assigned task, we have a Bayesian game among workers in $\mathcal C$: a worker’s decision on effort exertion is a function of $\mathbf{c_i}$, $\mathbf{e_i(c_i)}: [0, c_{\text{max}}]^M \rightarrow \{H, L\}^M$, which specifies the effort levels for worker $i$ when his realized cost is $\mathbf{c_i}$ and $\mathbf{r_i(e_i)}: \{0, 1\}^M \rightarrow \{0, 1\}^M$ gives the reporting strategy for the chosen effort level. We focus on threshold policies: that is, there is a threshold $c^*$ such that $e_i(k) =H$ for all $c_i(k) \leq c^*$ and $e_i(k) =L$ otherwise. [In fact, players must play a threshold strategy for effort exertion at any symmetric BNE: workers’ outputs do not depend on $c_i(k)$ and worker $i$’s chance of getting a bonus will not change when he has a cost $c'_i(k) < c_i(k)$; so a worker will choose to exert effort, if it is a better move for an even higher cost.]{} We will use $(c^*,r_i(k))$ to denote this threshold ($c^*$) strategy for workers. Denote $r_i(\cdot) \equiv 1$ the reporting strategy that $r_i(H) = r_i(L) =1$, i.e. reporting truthfully regardless of the choice of effort.
When $p_L>0.5$ and $F(c)$ is concave, there exists a unique threshold $c^*(B)>0$ for $B>0$ such that $(c^*(B), 1)$ is a symmetric BNE for all workers on all tasks.\[bne:static\]
#### Other equilibrias:
The above threshold policy is unique only in non-degenerate effort exertion ($c^*>0$). There exist other equilibria. We summarize them here:
- *Un-informative equilibrium:* Colluding by always reporting the same answer to all tasks is an equilibrium. Similarly as mentioned in [@dasgupta2013crowdsourced], when colluding (pure or mixed strategies) the bonus index defined in Eqn. (\[bonus:index\]) reduces to 0 for each worker, which leads to a worse equilibrium.
- *Low effort:* When $p_L = 1/2$, $c^*=0$, i.e., no effor exertion (followed by either truthful or untruthful reporting) is also an equilibrium: when no one else is exerting effort, each worker’s answer will be compared to a random guess. So there would be no incentive for effort exertion.
- *Permutation:* Exerting the same amount of effort and then reverting the reports ($r_i \equiv 0$) is also an equilibrium.
But we would like to note that though there may exist multiple equilibria, all others lead to strictly less utility for each worker at equilibrium compared to the threshold equilibrium with $c^*>0$ followed by truthful reporting, except for the permutation equilibria, which gives the same expected utility to workers.
*Solve for optimal $B^*$:* After characterizing the equilibria $c^*$ on effort exertion as a function of $B$, we can compute $\mathcal P(B)$ and $ \mathbb E[B_i(k)]$ for each reward level $B$. Then solving for the optimal reward level becomes a programming problem in $B$, which can be solved efficiently when certain properties, e.g. convexity, can be established for $\mathcal U(\cdot)$.
Sequential Peer Prediction {#sec:learn}
==========================
In this section we propose an adaptive learning mechanism to learn to converge to the optimal or nearly optimal reward level. As mentioned earlier, a recent work [@fullversion] attempted to resolve this challenge. But besides the output labels, workers are also required to report the private costs, in which sense the proposed learning mechanism is not “minimal”. We try to remove this requirement by learning only through the label information reported by the workers. In this section, we assume the requirements for Theorem \[bne:static\] hold, and workers will follow an equilibrium that returns the highest utility.
Challenges
----------
In designing the mechanism, we face two main challenges. The first challenge is on the learning part. In order to select the best $B^*$, we need to compute $\mathcal U(B), \forall B$, which can be computed as a function of $B$ and $\bar{p}(B)$, the probability of labeling accurately when $B$ is offered and the threshold policy $c^*(B)$ is adopted by workers: $$\begin{aligned}
\bar{p}(B):=F(c^*(B))p_H+[1-F(c^*(B))]p_L. \label{pb}\end{aligned}$$ The dependency on $B$ is straight-forward. For $\bar{p}(B)$, e.g. when using Chernoff bound for approximating $\mathcal P(B)$: $$\begin{aligned}
\mathcal P(B)&= \Pr\biggl[\frac{\sum_i 1(\text{worker i is correct})}{M} \geq 0.5\biggr]\\
&\geq 1-\text{exp}(-2(\bar{p}(B)-0.5)^2 M),\end{aligned}$$ it is clear $\mathcal P(B)$ is a function of $\bar{p}(B)$. [In fact both $\mathbb E[B_i(k)]$ and $\mathcal P(B)$ are functions of $\bar{p}(B)$, so is $\mathcal U(\cdot)$. For details, please see Appendix of [@fullversion_aaai17]. ]{} The question pings down to learn $\bar{p}(B)$. Since we do not have the ground-truth labels, we have no way to directly evaluate $\bar{p}(B)$ via checking workers’ answers. Also since we do not elicit reports on private costs, we are un-able to estimate the amount of induced effort for each reward level.
The second challenge we have is that when workers are returning and participating in a/an sequential/adaptive learning mechanism, they have incentives to mislead the learning process by deviating from the one-shot BNE strategy for a task, so to create untruthful samples (and then collected by learner), which will lead the learner into believing that inducing certain amount of effort requires a much higher reward level. The cost-reporting mechanism described in [@fullversion] proposes a method to deter such a deviation by eliminating workers who over-reported from receiving potentially higher bonus. We will describe a two-fold cross validation approach to decouple such incentives, which aims to remove the requirement of reporting additional information.
### Learning w/o ground-truth
The following observation inspires our method for learning without ground-truth. For each bonus level $B$, we can estimate $\bar{p}(B)$ (at equilibrium) through the following experiments: the probability of observing a pair of matching answers for any pair of workers $i,j$ (denoted by $p_{m}(B)$ for each bonus level $B$) on equilibrium can be written as follows: $$\begin{aligned}
p_m(B) &= \underbrace{\bar{p}^2(B)}_{\text{match on correct label}}+\underbrace{(1-\bar{p}(B))^2}_{\text{match on wrong label}}. \label{infer:equilibria}\end{aligned}$$ The above matching formula forms a quadratic equation of $\bar{p}(B)$. From Eqn. (\[pb\]) we know $
\bar{p}(B) \geq 0.5,~\forall B,~\text{when~} p_H, p_L \geq 0.5
$. Then the only solution to the matching Eqn. (\[infer:equilibria\]) that is larger than $0.5$ is $$\begin{aligned}
\bar{p}(B) =1/2+\sqrt{2p_m(B)-1}/2.\end{aligned}$$ Above solution is well defined, as from Eqn. (\[infer:equilibria\]) we can also deduce that $p_m(B) \geq 1/2$. Therefore though we cannot evaluate each worker’s labeling accuracy directly, we can make such an inference using the matching probability, which is indeed observable.
### Decoupling incentives via cross validation
To solve the incentive issue, we propose the following cross validation approach (illustrated in Fig. \[mechanism:ill\]). First the entire crowd $\mathcal C$ is separated into two groups $G_1, G_{-1}$ uniformly random, but with equal size (when $N$ is even) or their sizes differ by at most 1 (when $N$ is odd). Suppose we have at least $N \geq 4$. Denote worker $i$’s group ID as $g(i) \in \{-1,1\}$. Then we have $|G_1|, |G_{-1}| \geq 2$. For our learning algorithm, only the data/samples collected from group $-g(i)$ will be used to reward any worker $i$ in group $g(i)$. Secondly when selecting reference worker for comparing answers for mechanism (`DG13`), we select from the same group $g(i)$.
![Illustration of our mechanism.[]{data-label="mechanism:ill"}](hcomp1.eps){width="40.00000%"}
Mechanism
---------
We would like to treat each bonus level as an “arm” (as in standard MAB context) to explore with. Since we have a continuous space of bonus level $B$, we separate the support of bonus level $B$ ($[0,\bar{B}]$) into finite intervals. Then we treat each bonus interval as an arm. Our goal is to select the best one of them, and bound the performance in such a selection.
We set up $\lceil T^z \rceil$ arms as follows: chooses a $0<z<1$, separate $[0,\bar{B}]$ into $N_a = \lceil T^z \rceil$ uniform intervals: $$[0, \bar{B}/N_a], ..., [(k-1)\bar{B}/N_a, k\bar{B}/N_a],....,[(N_a-1)\bar{B}/N_a, \bar{B}]$$ For each interval we take its right end point as the bonus level to offer: $B_k = k\bar{B}/N_a.$ Denote by $\tilde{p}^{g}_{m,t}(B_k)$ the estimated matching probability for agents in group $g$ under bonus level $B_k$, and $\tilde{p}^g_t(B_k)$s the estimated $\bar{p}(B_k)$ for group $g$, at stage $t$; and we use $\tilde{\mathcal U}_{\tilde{p}}(B)$ to denote the estimated utility function when using a noisy $\bar{p}(B)$ ($\tilde{p}$), instead of the true ones. We present Mechanism \[m:simple\].[^3]
**Initialization:** $t=1$. $D(t):=t^{\theta}\log t, 0\leq \theta\leq 1$. Explore each bonus level $B_k$ once and update $\tilde{p}^{g}_{m,t}(B_k)$ (for details please refer to the exploration phases), and set the number of explorations as $n_i(t) = 1$. Set $\mathcal E(t):=\{i: n_i(t) < D(t)\} $. *Exploration*: randomly pick $B_k, k \in \mathcal E(t)$ to offer. Follow subroutine (`Explore_Crowd`). *Exploitation*: $S(t) = \mathcal C$. Offer $B^{*}_{g,t}$ to $i \in G_g$: $$\begin{aligned}
B^{*}_{g,t} = \text{argmax}_{B_k, k=1,2,...,\lceil T^z \rceil} \tilde{\mathcal U}_{\tilde{p}^g_t(B_k)}(B_k),\end{aligned}$$ Follow (`DG13`) for workers in each group $G_g$.
At exploration phases,
- Randomly select two workers $S_g(t) = \{i^g(t),j^g(t)\}$ from each group $G_g$; $S(t) := \cup_{g} S_g(t)$.
- Assign 1 common and $d$ distinct tasks to each $S_g(t)$.
- Denote by $\mathcal E_k(t)$ the set of time steps the algorithm enters exploration and offered price $B_k$ by time $t$. Estimate the probability of matching $\tilde{p}^{g}_{m,t}(B_k)$ for each crowd $G_g, g \in \{-1,1\}$ by averaging:$$\begin{aligned}
\tilde{p}^{g}_{m,t}(B_k) = \overline{\sum_{t' \in \mathcal E_k(t)} 1(L_{i^{-g}(t')}(t') = L_{j^{-g}(t')}(t'))},$$ and reset $\tilde{p}^{g}_{m,t}(B_k)$ to $\max\{\tilde{p}^{g}_{m,t}(B_k),1/2\}$. $L_{i}(t')$ denotes the label for the common task at time $t'$.
- Compute $\tilde{p}^g_t(B_k)$ (estimate for $\bar{p}(B_k)$ at time $t$): $$\begin{aligned}
\tilde{p}^g_t(B_k) = 1/2+\sqrt{2\tilde{p}^{g}_{m,t}(B_k)-1}/2.\end{aligned}$$ Reset $\tilde{p}^g_t(B_k):=\max\{\tilde{p}^g_t(B_k),p_L\}$.
Note since we assign the same number of tasks to each labeler at all stages, we have the regret defined in Eqn. (\[regret:defn\]) become equivalent with the following form: $$\begin{aligned}
R(T) = \sum_{t=1}^T | \sum_{g \in \{1,-1\}} \omega_g(t) \cdot \mathbb E[\mathcal U(B^{*}_{g,t})] - \mathcal U(B^*))|,\end{aligned}$$ where when $t$ is in exploration stages $
\omega_g(t) \equiv 1/2,
$ otherwise $
\omega_g(t) := |G_g|/N.
$
Equilibrium analysis: workers’ side
-----------------------------------
Denote a worker’s action profile at step $t$ as $\mathbf{a_i}(t):=(\mathbf{e_i}(t),\mathbf{r_i}(t))$. We adopt BNE as our solution concept:
A set of reporting strategy $\{\mathbf{\tilde{a}_i}:= \{\mathbf{\tilde{a}}_i(t)\}_{t=1,...,T}\}_{i \in \mathcal C}$ is a BNE if for any $i$, $\forall \mathbf{\tilde{a}'_i} \neq \mathbf{\tilde{a}_i}$ we have $$\begin{aligned}
&\sum_{t} \mathbb E[\max_{\mathbf{e_i}(t),\mathbf{r_i}(t)}u_i(\mathbf{\tilde{a}_i}, \mathbf{\tilde{a}_{-i}})|\mathbf{\tilde{a}_i}(1:t-1), \mathbf{\tilde{a}_{-i}}(1:t-1)] \\
&\geq \sum_{t} \mathbb E[\max_{\mathbf{e_i}(t),\mathbf{r_i}(t)} u_i(\mathbf{\tilde{a}'_i}, \mathbf{\tilde{a}_{-i}})|\mathbf{\tilde{a}'_i}(1:t-1), \mathbf{\tilde{a}_{-i}}(1:t-1)] ~.\end{aligned}$$
We first characterize the equilibrium strategy for workers’ effort exertion and reporting with (`SPP_PostPrice`).
At a symmetric BNE, the strategy for workers can be decoupled into a composition of myopic equilibrium strategies, that is $
e_i(t) = c^*(B^{*,g(i)}(t)),
$ combined with $r_i(t) \equiv 1, \forall i,t$.
W.l.o.g., consider worker $i$ in $G_1$. We are going to reason that deviating from one step BNE strategy for effort exertion is non-profitable for worker $i$, when other players are following the equilibria. Since the one stage equilibrium strategy maximizes the utility at current stage, and it does not affect the past utilities that have been already collected, the potential gain by deviating comes from the future gains in utilities in: (1) the offered bonus level (2) matching probability from other peers. For the bonus level offered to worker $i$, it will only be computed using observed data from workers in $G_{-1}$ at exploration phases. Note for our online learning process, the exploration phases only depend on the pre-defined parameter $D(t)$, which does not depend on worker $i$’s data (deterministic exploration). Similarly for all other workers $j \in G_1$ (reference workers), their future utility gain is not affected by worker $i$’s data. Therefore an unilateral deviation from worker $i$ will not affect the matching probability from other peers. So no deviation is profitable.
Again consider colluding workers. Potentially when offered a certain bonus level, workers can collude by not exerting effort regardless of their cost realization, so to mislead the learner into believing that in order to incentivize certain amount of effort, a much higher bonus level is needed.[^4] There potentially exist infinitely many colluding strategies for workers to game against a sequential learning algorithm. We focus on the following easy-to-coordinate (for workers), yet powerful strategy (`Collude_Learn`):
Workers collude by agreeing to exert effort when the offered bonus level $B^{*}_{g(i),t}$ satisfies $B^{*}_{g(i),t} \geq B(c_{i,t}(k)) + \nabla B,~
$where $c_{i,t}(k)$ is the cost for workers $i$ to exert effort for task $k$ at time $t$. $\nabla B > 0$ is a collusion constant.
In doing so workers mislead the learner into believing that a higher bonus (differs by $\nabla B$) is needed to induce certain effort. The next lemma establishes the collusion-proofness:
Colluding in *(`SPP_PostPrice`)* via *`(Collude_Learn)`* is not an equilibrium. \[collusion:price\]
In fact the reasoning for removing *all* symmetric colluding equilibria is similar – regardless of how others collude on effort exertion, when a worker’s realized cost is small enough, he will deviate.
Performance of (`SPP_PostPrice`)
--------------------------------
We impose the following assumptions [^5]:
$\mathcal U(\cdot)$ is Lipschitz in both $\bar{p}(B)$ and $B$: $$\begin{aligned}
|\tilde{\mathcal U}_{\tilde{p}}(\tilde{B})-\mathcal U(B)| \leq L_1 |\tilde{p}(\tilde{B})-\bar{p}(\tilde{B})|+L_2|\tilde{B}-B|,\end{aligned}$$ $L_1, L_2>0$ are the Lipschitz constants.
We have the following theorem:
The regret of *(`SPP_PostPrice`)* is: $$\begin{aligned}
R(T) \leq & O(\lceil T^{\theta+z}\log T \rceil) + \frac{L_1C(\theta)}{\sqrt{2p_L-1}}T^{1-\theta/2} \\
&+ L_2 C(z)\bar{B} T^{1-z} + \text{const.}
$$ $0<\theta,z<1$ are tunable parameters. $C(\theta), C(z) >0$ are constants. The optimal regret is $R(T) \leq \tilde{O}(T^{3/4})$ when setting $z=1/4,\theta=1/2$. \[thm:learn\]
(Sketch) First notice by triangular inequality we know $
R(T) \leq \sum_{g \in \{1,-1\}}\omega_g R_g(T),
$ i.e., the total regret is upper bounded by the weighted sum of each group’s regret. Since the two learning processes for the two groups $G_1,G_{-1}$ parallel each other, we omit the $g$ super- or sub-script. We analyze for the regret incurred at the exploration and exploitation stages respectively. For exploitation regret, we first characterize the estimation error for estimating each $U(B_k)$. First by mean value theorem we can show: $$\begin{aligned}
|\bar{p}(B_k)-\tilde{p}_t(B_k)| \leq \frac{1}{2\sqrt{2p_L-1}}|p_m(B_k)-\tilde{p}_{m,t}(B_k)|.\end{aligned}$$ At time $t$, an exploitation phase, there are $D(t)$ number of samples guaranteed; so the estimation $\tilde{p}_{m,t}(B_k)$ satisfies: $
\Pr[|p_m(B_k)-\tilde{p}_{m,t}(B_k)| \geq \frac{1}{t^{\theta/2}}] \leq \frac{2}{t^2}~.
$ Then w.h.p., we have established that $
|\bar{p}(B_k)-\tilde{p}_t(B_k)| \leq \frac{ t^{-\theta/2}}{2\sqrt{2p_L-1}}.
$ Further by Lipschitz condition we know $|\tilde{\mathcal U}(B_k)-\mathcal U(B_k)| \leq \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}.$ For any $B$ that falls in the same interval as $B_k$ we know: $
|\mathcal U(B)-\mathcal U(B_k)| \leq L_2 |B-B_k|\leq L_2 \bar{B}T^{-z}.
$
Denote by $B^*_t = \text{argmax}_{B_k, k=1,2,...,\lceil T^z \rceil} \tilde{\mathcal U}_t(B_k)~,$ i.e., $B^*_t$ is the estimated optimal bonus level at time $t$ – at any time $t$, by searching through all arm and we can find the one maximizes the utility function from the empirically estimated function the learner currently has. Combine above arguments, we can prove $
\mathcal U(B^*)-\mathcal U(B^*_t) \leq \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}+L_2 \bar{B}T^{-z}.
$ Then the exploration regret can be bounded as $$\begin{aligned}
&\sum_{t=1}^T ( \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}+L_2 \bar{B}T^{-z}) + O(\sum_{t=1}^T \frac{2}{t^2})\\
&\leq \frac{L_1 \cdot C(\theta)}{2\sqrt{2p_L-1}}T^{1-\theta/2} + L_2 \cdot C(z) \bar{B} T^{1-z} + \text{const.} \end{aligned}$$ where we have used the fact that for any $0 < \alpha < 1$, there exists a constant $C(\alpha)$ such that $\sum_{t=1}^T \frac{1}{t^{\alpha}} \leq C(\alpha) T^{1-\alpha}~.
$ The total number of explorations are ($T^{\theta}\log T$ number of explorations needed for each of the $\lceil T^z \rceil$ arms): $
\lceil T^z \rceil \cdot T^{\theta}\log T = \lceil T^{\theta+z}\log T \rceil~.
$ Sum over we finish the proof.
Beyond game theoretical model {#sec:nongame}
-----------------------------
So far we have modeled the workers as being fully rational, and the reports as coming from game theoretical responses. Consider the case workers’ responses do not necessarily follow a game (or arguably no one is fully rational). Instead we assume each worker has a labeling accuracy $p_i(B)$ for different $B$, where $p_i(B)$ can come from different models, being game theory driven, behavioral model driven or decision theory driven, and can be different for different workers.
*Challenges and a mean filed approach*: With this model, we can again write $\mathcal U(B)$ as a function of $\{p_i(B)\}_i$ and $B$. In order to select $B^*$, again we need to learn $p_i(B)$. We re-adopt the bandit model we described earlier, and estimate $p_i(B)$s via observing the matching probability between worker $i$ and a randomly selected reference worker $j$: For each $i,B$ we define $\bar{p}_{-i}(B) := \sum_{j \in G_{g(i)}\backslash i} p_j(B)/|G_{g(i)}\backslash i |, ~\text{and we have } $ $$\begin{aligned}
&p^{i}_{m}(B) = p_i(B)\bar{p}_{-i}(B) + (1-p_i(B))(1-\bar{p}_{-i}(B)),\label{eqn:match}\end{aligned}$$ where $p^{i}_{m}(B) $ is the probability of observing a matching for worker $i$, when a random reference worker is drawn (uniformly) from his group. The above forms a system of quadratic equations in $\{p_i(B)\}_i$ when $\{p^{i}_{m}(B)\}_i$s are known. We then need to solve a perturbed quadratic equations for $\{p_i(B)\}_i$, with $\{p^{i}_{m}(B)\}_i$s being estimated via observations (Step 3 of (`Explore_Crowd`)). The following challenges arise for analysis: (1) it is hard to tell whether the solution for above quadratic equations is unique or not. (2) Solving a set of perturbed (error in estimating $\{p^{i}_{m}(B)\}_i$s) quadratic equations for each $B$ incurs heavy computations.[^6]
Instead, by observing the availability of relatively large and diverse population of crowd workers, we make the following mean filed assumption:
For any worker $i$, $\bar{p}_{-i}(B) \equiv \bar{p}_{g(i)}(B)~.$
That is one particular worker’s expertise level does not affect the crowd’s mean. This is not a entirely unreasonable assumption to make, as the candidate pool of crowd workers is generally large. With above $p^{i}_{m}(B) $ then becomes $$p^{i}_{m}(B) = p_i(B)\bar{p}_{g(i)}(B)+ (1-p_i(B))(1-\bar{p}_{g(i)}(B)).$$ Averaging over $i \in G_{g(i)}$ we have:\
$
\sum_{i \in G_{g(i)}} p^{i}_{m}(B)/|G_{g(i)}| =\bar{p}^2_{g(i)}(B) + (1-\bar{p}_{g(i)}(B))^2,
$ which is very similar to the matching equation we derived earlier on. Again we can solve for $\bar{p}_{g(i)}(B)$ as a function of $\sum_{i \in G_{g(i)}} p^{i}_{m}(B)/|G_{g(i)}|$. Plugging $\bar{p}_{g(i)}(B)$ back into Eqn. (\[eqn:match\]), we obtain an estimate of $p^i(B)$ as follows: $$\begin{aligned}
p_i(B) = (p^{i}_{m}(B)+\bar{p}_{g(i)}(B)-1)/(2\bar{p}_{g(i)}(B)-1).\end{aligned}$$ Similar regret can be obtained – the difference only lies in estimating $p_i(B_k)$s. Details can be found in [@fullversion_aaai17].
Conclusion {#sec:conclude}
==========
We studied the sequential peer prediction mechanism for eliciting effort using posted price. We improve over status quo towards making the peer prediction mechanism for effort elicitation more practical: (1) we propose a posted-price and “minimal” sequential peer prediction mechanism with bounded regret. The mechanism does not require workers to report additional information, except their answers for assigned tasks. Further we show our learning results can generalize to the case when workers may not necessarily be fully rational, under a mean-filed assumption. (2) Workers exerting effort according to an informative threshold strategy and reporting truthfully is an equilibria that returns highest utility.
#### Acknowledgement:
We acknowledge the support of NSF grant CCF-1301976.
{#section .unnumbered}
Randomized task assignment
==========================
We explain on why we need a well structured random task assignment. First we make sure for each task, it has been assigned at least to two workers, so each of the assignment can serve as a peer evaluation for the other. Secondly for any pair of workers that share the same task, they also need to have distinct tasks, which is motivated by the mechanism (`DG13`). A reader may notice that by simply assigning each task to *all* worker both of above conditions will be satisfied satisfied. (For example, assign 4 tasks $\{1,2,3,4\}$ to all 4 workers, and when distinct tasks are needed for worker 1 and 2, we can compute using only task 3 and 4 for each worker respectively.) But instead we will make our assignment process random, which will help exclude the possibility of more complicated collusion strategies (e.g. such as colluding on subset of tasks but not on the rest, see the example below), especially when also considering we can randomly shuffle labels (or IDs) of both workers and tasks. Therefore in this paper we only consider one type of collusion, that is if workers decide to contribute uninformative signals, they will report the same labels for all tasks.
#### Example on more sophisticated collusion:
Suppose workers are assigned the same set of tasks with the same ID. And for simplicity we assume there are even number of tasks. Workers agree on the IDs of tasks, and they will agree on reporting -1 for odd number ID tasks, and +1 for even IDs. Then $$\begin{aligned}
B_i(k) &= 1(L_i(k)=L_j(k)) - L^d_i \cdot L^d_j -\overline{L}^d_i \cdot \overline{L}^d_i\\
&=1-\frac{1}{2}\cdot\frac{1}{2}-\frac{1}{2}\cdot\frac{1}{2} =\frac{1}{2},\end{aligned}$$ which is the maximum score that can be achieved as $$\begin{aligned}
&1(L_i(k)=L_j(k)) \leq 1,\\
&L^d_i \cdot L^d_j + \overline{L}^d_i \cdot \overline{L}^d_i \geq 1/2.\end{aligned}$$ To see this, denote by $x:=L^d_i, y:= L^d_j$ – we know $0 \leq x,y \leq 1$. The following holds: $$\begin{aligned}
xy+(1-x)(1-y) \leq (x^2+y^2)/2 + ((1-x)^2+(1-y)^2)/2 = \frac{2x^2-2x+1}{2} + \frac{2y^2-2y+1}{2} \geq 1/2.\end{aligned}$$
#### A feasible assignment:
Now we demonstrate that such an assignment that meets all requirements can be achieved. For example, suppose we have $4$ workers, set $M=2, d=1, K=2$ and we assign 4 tasks $\{1,2,3,4\}$ each time as follows: $$\begin{aligned}
\text{Worker 1:}~\{1,2\},~\text{Worker 2:}~\{1,3\},~\text{Worker 3:}~\{2,4\},~\text{Worker 4:}~\{3,4\}~\end{aligned}$$ Not hard to verify the above assignment satisfies all constraints we enforced. More generally when we have $N$ workers, we can prepare $N$ tasks to assign for each time. Again set $M=2, d=1, K=2$. Denote the tasks received by worker $i$ as $t_i(1),t_i(2)$. Then the assignment can be adaptively decided as follows: $$\begin{aligned}
t_1(1) = &1,~t_1(2) = 2,~t_2(1) = 1, t_2(2) = 3,\\
t_i(1) =& t_{i-2}(2), t_i(2) = \min\{t_{i-1}(2)+1,N\}, \forall i > 2~.\end{aligned}$$ It is easy to verify the above assignment rule satisfies our requirements: each tasks is assigned at least twice; workers receiving the same task also receive different tasks; all tasks are assigned the same number of times; not all tasks are assigned to all workers.
Proof of Theorem \[bne:static\]
===============================
Denote by $\mathcal P_{+1}, \mathcal P_{-1}$ the priors for labels, and the probability of observing label +1 and -1 of each worker $i$ under effort level $e_i$ as $$p_{+,e_i} := \Pr[L_i = +1|e_i] = \mathcal P_{+} p_{e_i} + \mathcal P_{-} (1-p_{e_i}),$$ and $p_{-,e_i} := 1-p_{+,e_i}$. W.l.o.g. consider task $k$ of worker $i$ (when task $k$ is indeed assigned to worker $i$). Exerting effort or not on task $k$ will affect $u_i$ in two ways:
First on $\mathbb E[B_i(k)]$: notice the decision on $k$ does not affect $\mathbb E[ L^d_i \cdot L^d_j]$. For $\mathbb E[ 1(L_i(k)=L_j(k))]$, consider the fact that every other player is following the threshold policy that $e_j(k) = H$ if $c_j(k) \leq c^*$, and truthfully reporting. Then $$\begin{aligned}
\mathbb E[p_{e_j(k)}] &= F(c^*)p_H+(1-F(c^*))(1-p_L),~\\
\mathbb E[p_{+,e_j(k)}] &=F(c^*)p_{+,H}+(1-F(c^*))p_{+,L}.\end{aligned}$$ From which we have the utility difference between exerting and not exerting effort becomes: $$\begin{aligned}
&\mathbb E[B_i(k)|e_i=H] -\mathbb E[B_i(k)|e_i=L]\\
=& \mathbb E[p_{H} p_{e_j(k)} + (1-p_{H})(1-p_{e_j(k)})] \\
&~~~~- \mathbb E[p_{L} p_{e_j(k)} + (1-p_{L})(1-p_{e_j(k)})] \\
=& (p_H-p_L) (2\mathbb E[p_{e_j(k)}]-1), ~\end{aligned}$$ where $$\mathbb E[p_{e_j(k)}] = F(c^*)p_{H}+(1-F(c^*))p_{L}.$$ Now consider the effect of $e_i(k)$ on $E[B_i(k')], k' \neq k$. Suppose after the randomized assignment, $k$ appears in $D \leq M-1$ other tasks’ distinct set. Denote the set as $\mathcal D$. For $k' \in \mathcal D$, $e_i(k)$ affects the “penalty term”: first we know by independence that $$\begin{aligned}
\mathbb E[ L^d_i \cdot L^d_j] &= \mathbb E[ L^d_i]\cdot \mathbb E[ L^d_j]~. $$ Then $$\begin{aligned}
&\mathbb E[ L^d_i \cdot L^d_j|e_i(k)=H]- \mathbb E[ L^d_i \cdot L^d_j|e_i(k)=L] \\
=& \mathbb E[ L^d_j] \biggl(\mathbb E[ L^d_i |e_i(k)=H]- \mathbb E[ L^d_i|e_i(k)=L]\biggr)\\
=&\mathbb E[p_{+,e_j}]\frac{p_{+,H}-p_{+,L}}{d}~.\end{aligned}$$ And similarly $$\begin{aligned}
&\mathbb E[ \overline{L}^d_i \cdot \overline{L}^d_j|e_i(k)=H]- \mathbb E[ \overline{L}^d_i \cdot \overline{L}^d_j|e_i(k)=L] = (1-\mathbb E[p_{+,e_j}])\frac{p_{+,L}-p_{+,H}}{d}\end{aligned}$$ Summarize above difference we know:$$\begin{aligned}
&\mathbb E[u_i|e_i(k) = H]-\mathbb E[u_i|e_i(k)=L]=V_1 \cdot F(c^*)+V_2,\end{aligned}$$ where $$\begin{aligned}
&V_1:= 2(p_H-p_L)^2[1-\frac{D}{d}(\mathcal P_{+}-\mathcal P_{-})^2]\\
&V_2 := (2p_L-1) [1-\frac{D}{d}(\mathcal P_{+}-\mathcal P_{-})^2] (1-2\mathcal P_{-})^2~.\end{aligned}$$ The equilibrium equation establishes itself when the above equals to the cost $c^*$: (after re-arrange) $$\begin{aligned}
B[V_1 \cdot F(c^*)+V_2] =c^*.\label{eqn:ne}\end{aligned}$$ When $D$ is chosen such that $D \leq d$ we know $V_1, V_2>0$ (as $\mathcal P_{+}, \mathcal P_{-} > 0$). We claim when $F(\cdot)$ is concave, there exists a unique solution if $p_L > 0.5$: first $\text{LHS}(c^*=0) > \text{RHS}(c^*=0)$, and when $$\begin{aligned}
&B[V_1 \cdot F(c^*=\bar{c})+V_2] \leq c_{\max},\end{aligned}$$ we have LHS of Eqn. (\[eqn:ne\]) and the RHS intersects exactly once. So this unique intersecting point $c^* \leq c_{\max}$ is the unique solution to Eqn. (\[eqn:ne\]), and o.w., we have $c^* \equiv c_{\max}$, that is $B$ is large enough so that exerting effort is always the best action to take.
Also reporting by reverting the answer, i.e., $r_i = 0$, the probability of matching the true label becomes $1-p_H < p_L$, which leads to even less utility. So deviating from truthfully reporting is not profitable.
Proof of Lemma \[collusion:price\]
==================================
This lemma is due to the fact that if everyone else is colluding to mis-lead the learner into believing a wrong price, a particular worker has no incentive to also do so: first his reported data will not affect his own price in the future. And as a rational worker, he should not care about the prices received by others. Due to the index rule we adopted, workers can do better than colluding: deviating from colluding to not exert effort possibly increases his current stage payment, when $p_L > 0.5$, and when cost $c_i$ is small enough: $$\begin{aligned}
&~~~~\mathbb E[B_i(k)|e_i=H] -\mathbb E[B_i(k)|e_i=L] \\
&= (p_H-p_L) (2\mathbb E[p_{e_j(k)}]-1), \\
&=(p_H-p_L) (2p_{L}-1) > 0~.\end{aligned}$$
Lipschitz assumption on $\mathcal U(B)$
=======================================
Detailed argument for establishing the Lipschitz conditions can be similarly found in [@fullversion]. We briefly mention it here: when we use lower bound approximation for $\mathcal P(B)$ we have $$\begin{aligned}
\mathcal U(B) = 1-\text{exp}(-2(\bar{p}(B)-0.5)^2 M) - \eta \cdot B \cdot M \biggl[\mathcal P_{-}(2\bar{p}(B)-1)(\mathcal P_{+}(2\bar{p}(B)-1)+1) \biggr]. \end{aligned}$$ The second part $\mathcal P_{-}(2\bar{p}(B)-1)(\mathcal P_{+}(2\bar{p}(B)-1)+1)$ is obtained by computing $ \mathbb E[B_i(k)]$. It is easy to see both $\text{exp}(-2(\bar{p}(B)-0.5)^2 N)$ and the second quadratic terms are Lipschitz in $\bar{p}(B)$. The rest remains to prove is that $\bar{p}(B)$ is also Lipschitz in $B$, as the composition of bounded Lipschitz functions are also Lipschitz. Since $$\begin{aligned}
\bar{p}(B):=F(c^*(B))p_H+[1-F(c^*(B))]p_L= (p_H-p_L)F(c^*(B)) + p_L,\end{aligned}$$ and as $F(\cdot)$ is concave and Lipschitz in $c$, we only need to prove that $c^*(B)$ is Lipschitz in $B$. The proof can be similarly established as Lemma 13.2 in [@fullversion] with similar assumptions, based on the equilibrium equation we characterized in the proof of Theorem \[bne:static\].
Proof of Theorem \[thm:learn\]
==============================
First notice by triangle inequality we know $$\begin{aligned}
R(T) &= \sum_{t=1}^T | \sum_{g \in \{1,-1\}} \omega_g \cdot \mathbb E[\mathcal U(B^{*}_{g,t})] - \mathcal U(B^*)|\\
&\leq \sum_{g \in \{1,-1\}} \omega_g \sum_{t=1}^T |\mathbb E[\mathcal U(B^{*}_{g,t})]-\mathcal U(B^*)|\\
&:=\sum_{g \in \{1,-1\}}\omega_g R_g(T),\end{aligned}$$ i.e., the total regret is upper bounded by the weighted sum of each crowd’s regret. Since the two learning processes for the two groups $G_1,G_{-1}$ parallel each other, we omit the $g$ super- or sub-script: we analyze the learning performance for each sub-group and the ones for the other one follows exactly the same.
We separate the regret analysis into exploration regret and exploitation regret, that is the regret incurred at the exploration and exploitation stages respectively. We start with analyzing the exploitation regret. In order to characterize the exploitation regret, we need to characterize the estimation error for estimating each $U(B_k)$. First by mean value theorem we know $\exists p \in [\min\{p_m(B_k),\tilde{p}_{m,t}(B_k)\}, \max\{p_m(B_k),\tilde{p}_{m,t}(B_k)\}]$ $$\begin{aligned}
|\bar{p}(B_k)-\tilde{p}_t(B_k)| &=\frac{|\sqrt{2p_m(B_k)-1}-\sqrt{2\tilde{p}_{m,t}(B_k)}|}{2} \\
&=\frac{1}{2\sqrt{1-2(1-p)}}|p_m(B_k)-\tilde{p}_{m,t}(B_k)|, \\
&\leq \frac{1}{2\sqrt{2p_L-1}}|p_m(B_k)-\tilde{p}_{m,t}(B_k)|.\end{aligned}$$ The inequality comes from the fact $p \geq \min\{p_m(B_k),\tilde{p}_{m,t}(B_k)\} \geq p_L > 0.5,
$ so the bound is well defined. [^7] Now consider the estimation error in $p_m(B_k)$. At time $t$, an exploitation phase, there are $D(t)$ number of exploration/samples guaranteed, so the estimation for $p_m(B_k)$ at time $t$ satisfies: $$\begin{aligned}
\Pr&\biggl [|p_m(B_k)-\tilde{p}_{m,t}(B_k)| \geq \frac{1}{t^{\theta/2}} \biggr ] \leq 2\text{exp}(-2(\frac{1}{t^{\theta/2}})^2 t^{\theta}\log t) = \frac{2}{t^2}~.\end{aligned}$$ Then w.h.p., we have established that $$|\bar{p}(B_k)-\tilde{p}_t(B_k)| \leq \frac{ t^{-\theta/2}}{2\sqrt{2p_L-1}}.$$ By Lipschitz condition we know $$\begin{aligned}
|\tilde{\mathcal U}(B_k)-\mathcal U(B_k)| \leq \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}} ~.\end{aligned}$$ For any $B$ that falls in the same interval as $B_k$ we know: $$\begin{aligned}
|\mathcal U(B)&-\mathcal U(B_k)| \leq L_2 |B-B_k|\leq L_2 \bar{B}T^{-z}.\end{aligned}$$ Combine above arguments, we know for any $B$, we have the estimated utility function $\tilde{\mathcal U}(B_k)$ (the same interval as $B$) satisfy the follows $$\begin{aligned}
|\tilde{\mathcal U}(B_k)-\mathcal U(B)| \leq \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}+L_2 \bar{B}T^{-z}.\end{aligned}$$ Denote by $B^*_t = \text{argmax}_{B_k, k=1,2,...,\lceil T^z \rceil} \tilde{\mathcal U}_t(B_k)~,$ i.e., $B^*_t$ is the estimated optimal bonus level at time $t$ – at any time $t$, by searching through all arm and we can find the one maximizes the utility function from the empirically estimated function the learner currently has. Then $$\begin{aligned}
&~~~~~\mathcal U(B^{*}_t) - \mathcal U(B^*) \\
&\geq \tilde{\mathcal U}(B^{*}_t)- \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}} - \mathcal U(B^*)\\
&\geq \tilde{\mathcal U}(B^*_k) - \mathcal U(B^*)- \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}\\
&\geq - \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}-L_2 \bar{B}T^{-z}~,\end{aligned}$$ where $B^*_k$ is the bonus level that is in the same interval as $B^*$. Combine with the fact $\mathcal U(\tilde{B}^*) \leq \mathcal U(B)$ we have $$\begin{aligned}
\mathcal U(B^*)-\mathcal U(B^*_t) \leq \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}+L_2 \bar{B}T^{-z}.\end{aligned}$$
Then the regret for exploration phases can be bounded as $$\begin{aligned}
&\sum_{t=1}^T ( \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}+L_2 \bar{B}T^{-z}) + O(\sum_{t=1}^T \frac{2}{t^2})\leq \frac{L_1 \cdot C(\theta)}{2\sqrt{2p_L-1}}T^{1-\theta/2} + L_2 \cdot C(z) \bar{B} T^{1-z} + \text{const.} \end{aligned}$$ where we have used the fact that for any $0 < \alpha < 1$, there exists a constant $C(\alpha)$ such that $\sum_{t=1}^T \frac{1}{t^{\alpha}} \leq C(\alpha) T^{1-\alpha}~.
$ The total number of explorations are ($T^{\theta}\log T$ number of explorations needed for each of the $\lceil T^z \rceil$ arms) $$\begin{aligned}
\lceil T^z \rceil \cdot T^{\theta}\log T = \lceil T^{\theta+z}\log T \rceil~.\end{aligned}$$ Then the total accumulated regret incurred (exploration regret + exploitation regret) is bounded by $$\begin{aligned}
R(T)& \leq | \max \mathcal U - \min \mathcal U| \cdot\lceil T^{\theta+z}\log T \rceil + \frac{L_1C(\theta)}{\sqrt{2p_L-1}}T^{1-\theta/2} + L_2 C(z)\bar{B} T^{1-z} + \text{const.}
$$ The above regret achieves the optimal order when the three exponent term matches each other: $$\begin{aligned}
\theta + z = 1 - \theta/2, ~1-\theta/2 = 1 - z,\end{aligned}$$ which leads to the solution of $z=1/4, \theta=1/2$, which further results to the optimal order of regret $T^{3/4}$.
The assumption that $p_L > 0.5$
-------------------------------
The assumption that $p_L>0.5$ is somewhat bothering. There are two places this assumption is needed. The first place is in proving the collusion proof for our learning mechanism (Lemma \[collusion:price\]). The reason for this particular assumption is that when $p_L = 0.5$, the matching probability becomes independent of worker $i$’s labeling accuracy (as intuitively one’s answer is compared to a random guess. So there is no reason to exert effort in such a case, regardless of workers’ realized costs.) This requirement can be relaxed by considering the fact a $0<\beta<1$ fraction of workers is independent of the collusion.
The second place that such an assumption is needed is in bounding the estimation errors (when use the mean value theorem, we need a bounded graident). We hope to find other bounds to get around of the requirement $p_L > 0.5$.
Performance for (`SPP_PostPrice`) under non-game theoretical model
==================================================================
When run *(`SPP_PostPrice`)* for the non-game theoretical model, $R(T)$ is bounded as: $$\begin{aligned}
R(T)& \leq O(\lceil T^{\theta+z}\log T \rceil) +2C(\theta)L_1 (\frac{2}{(2p_L-1)^2}+\frac{1}{2p_L-1}) T^{1-\theta/2} + L_2 T^{1-z}+\text{const.}\end{aligned}$$ \[thm:nongame\]
Similar with the proof for Theorem \[thm:learn\], we are going to bound a number of estimation errors in evaluating $U(\cdot)$. First of all notice at time $t$, $$\begin{aligned}
&|\frac{\sum_{i \in G_{g(i)}} p^{i}_{m}(B_k)}{|G_{g(i)}|}- \frac{\sum_{i \in G_{g(i)}} \tilde{p}^{i}_{m}(B_k)}{|G_{g(i)}|}|\leq \sum_{i \in G_{g(i)}}| p^{i}_{m}(B_k)- \tilde{p}^{i}_{m}(B_k)|/|G_{g(i)}|.\end{aligned}$$ Again when the algorithm enters exploitation phases (so $D(t)$ number of samples guaranteed for all arms) we have $\forall k$ by Chernoff bound that $$\begin{aligned}
\Pr\biggl [| p^{i}_{m}(B_k)- \tilde{p}^{i}_{m}(B_k)| \geq t^{\theta/2}\biggr] \leq 2/t^2,\end{aligned}$$ Then via union bound we know with probability being at least $1-\frac{N+1}{t^2}$($\forall i$): $$\begin{aligned}
&|\frac{\sum_{i \in G_{g(i)}} p^{i}_{m}(B_k)}{|G_{g(i)}|}- \frac{\sum_{i \in G_{g(i)}} \tilde{p}^{i}_{m}(B_k)}{|G_{g(i)}|}|\leq t^{-\theta/2}.\end{aligned}$$ Then we know the following holds: $$\begin{aligned}
&~~~~~|\tilde{p}_{i,t}(B_k)-p_i(B_k)|\\
& \leq |\frac{p^{i}_{m}(B_k)+\bar{p}(B_k)-1}{2\tilde{p}_t(B_k)-1}-p_i(B_k)|+|\frac{p^{i}_{m,t}(B_k)-p^{i}_m(B_k)+\tilde{p}_t(B_k)-\bar{p}(B_k)}{2\tilde{p}_t(B)-1}|\\
&\leq |\frac{p^{i}_{m}(B_k)+\bar{p}(B_k)-1}{2\tilde{p}(B_k)-1}-p^i(B_k)| + \frac{2t^{-\theta/2}}{2p_L-1}\\
&\leq 2 \frac{p^{i}_{m}(B_k)+\bar{p}(B_k)-1}{(2p_L-1)^2}|\tilde{p}_{t}(B_k)-\bar{p}(B_k)|+ \frac{2t^{-\theta/2}}{2p_L-1} \\
&\leq \frac{2}{(2p_L-1)^2} \frac{L_1 \cdot t^{-\theta/2}}{2\sqrt{2p_L-1}}+\frac{2}{2p_L-1} t^{-\theta/2}\\
&=\frac{ t^{-\theta/2}}{2p_L-1}(\frac{L_1}{(2p_L-1)^{1.5}}+2)~.\end{aligned}$$ The second inequality uses the concentration bound, as well as the fact that $
\bar{p}(B) \geq p_L > \frac{1}{2}, \forall B~
$ so that the estimation should never go smaller than this quantity. The third inequality applies mean value theorem to function $1/(2x-1)$; while the last one uses fact $|p^{i}_{m}(B_k)+\bar{p}(B_k)-1| < 1$ and concentration bound. Then for any $B$, we have the estimated utility function $\tilde{\mathcal U}(B)$ satisfy the follows $$\begin{aligned}
|\tilde{\mathcal U}(B)-\mathcal U(B)| \leq &L_1 \frac{ t^{-\theta/2}}{2p_L-1}(\frac{L_1}{(2p_L-1)^{1.5}}+2)~.\end{aligned}$$ The rest of analysis is very similar to the one we presented for Theorem \[thm:learn\]. We will not re-state the derivation details, but we are led to: $$\begin{aligned}
&\mathcal U(B^*_t) - \mathcal U(B^*)\geq -L_1 \frac{ t^{-\theta/2}}{2p_L-1}(\frac{L_1}{(2p_L-1)^{1.5}}+2)- L_2 \bar{B} \cdot T^{-z}~.
$$ And the exploitation regret accumulated up to time $t$ is bounded by $$\begin{aligned}
&\sum_t \biggl( L_1 \frac{ t^{-\theta/2}}{2p_L-1}(\frac{L_1}{(2p_L-1)^{1.5}}+2)+ L_2 \bar{B} \cdot T^{-z}\biggr)+\text{const.}\leq \frac{C(\theta)L_1}{2p_L-1}(\frac{L_1}{(2p_L-1)^{1.5}}+2)T^{1-\theta/2}+ L_2C(z) T^{1-z}+\text{const.}\end{aligned}$$ And the exploration regret is upper bounded as $O(T^z D(t)) = O(T^{z+\theta}\log T)$.
[^1]: which does not affect optimizing the utility function.
[^2]: For details please refer to our algorithm.
[^3]: We assume we know $p_L$ or a non-trivial lower bound on $p_L>0.5$.
[^4]: This is similar to the colluding strategy that contributes uninformative signals we studied in Section \[sec:static\].
[^5]: Please refer to our full version for justification.
[^6]: We do not claim this is impossible to do. Rather, analyzing the output from such a system of perturbed quadratic equations merits a further study.
[^7]: In the learning section we simply consider the case $p_L>0.5$.
|
---
abstract: 'A new defense mechanism for different jamming attack on [*Wireless Sensor Network (WSN)*]{} based on ant system it is introduced. The artificial sensitive ants react on network attacks in particular based on their sensitivity level. The information is re-directed from the attacked node to its appropriate destination node. It is analyzed how are detected and isolated the jamming attacks with mobile agents in general and in particular with the newly ant-based sensitive approach.'
address: 'Technical University Cluj Napoca, North University Center Baia Mare, Romania'
author:
- 'Camelia-M. Pintea'
- 'Petrica C. Pop'
title: |
Sensitive Ants for Denial Jamming Attack\
on Wireless Sensor Network
---
Introduction
============
Jamming attacks on wireless networks, special cases of [*Denial of Service (DoS)*]{} are the attacks that disturb the transceivers’ operations on wireless networks [@adamy04]. A [*Radio Frequency (RF)*]{} signal emitted by a jammer corresponds to the ‘useless’ information received by the sensor nodes of a network.
Nowadays are several techniques used to reduce the effect of jamming attacks in wireless networks. In [@yujin12] is proposed a traffic rerouting scheme for [*Wireless Mesh Network (WMN)*]{}. There are determined multiple candidates of a detour path which are physically disjoint. In a stochastic way, is selected just one candidate path as a detour path to distribute traffic flows on different detour paths. The mechanism on packet delivery ratio and end-to-end delay is improved when compared with a conventional scheme.
Securing [*WSN*]{}s against jamming attacks is a very important issue. Several security schemes proposed in the [*WSN*]{} literature are categorized in: [*detection techniques, proactive countermeasures, reactive countermeasures*]{} and [*mobile agent-based countermeasures*]{}. The advantages and disadvantages of each method are described in [@Mpitziopoulos]. Our current interest is in [*mobile agent-based solutions*]{}.
Artificial intelligence has today a great impact in all computational models. Multi-agent systems are used for solving difficult Artificial Intelligence problems [@wooldridge; @Iantovics]. Multi-agents characteristics includes organization, communication, negotiation, coordination, learning, dependability, learning and cooperation [@his; @wooldridge; @stoean]. Other features are their knowledge [@Popescu] and their actual/future relation between self awareness and intelligence. There are also specific multi-agents with their particular properties, as robots-agents [@pintea2010; @pinteacisis13]. One of the commonly paradigm used with [*MAS*]{} systems is the pheromone, in particular cases artificial ants pheromone. Ant-based models are some of the most successfully nowadays techniques used to solve complex problems [@crisan07; @crisan; @dorigo97; @pintea2010; @Reihaneh; @stoean2010].
Our goal is to improve the already existing ant systems on solving jamming attacks on [*WSN*]{} using the ants sensitivity feature. The second section describes the already known mechanisms of Jamming Attack on Wireless Sensor Network and the particular unjamming techniques based on Artificial Intelligence. The next section is about ants sensitivity. The section includes also the newly introduced sensitive ant-model for detecting and isolated jamming in a [*WSN*]{}. Several discussions about the new methods and the future works concludes the paper.
About Jamming Attack on Wireless Sensor Network using Artificial Intelligence
=============================================================================
The current section describes the main concepts and the software already implemented on Jamming Attack on Wireless Sensor Network including the [*Artificial Intelligence*]{} models.
Jamming Attack on Wireless Sensor Network
-----------------------------------------
Jamming attacks are particular cases of Denial of Service (DoS) attacks[@wood02]. The main concepts related to this domain follows. At first several considerations on Wireless Sensor Network (WSN).
A Wireless Sensor Network (WSN) consists of hundreds/thousands of sensor nodes randomly deployed in the field forming an infrastructure-less network.
A sensor node of a [*WSN*]{} collects data and routes it back to the Processing Element (PE) via ad-hoc connections with neighbor sensor nodes.
A Denial of Service attack is any event that diminishes or eliminates a network’s capacity to perform its expected function.
Jamming is defined as the emission of radio signals aiming at disturbing the transceivers’ operation.
There are differences between jamming and radio frequency interference.
- the jamming attack is [*intentional*]{} and [*against a specific target*]{};
- the radio frequency interference is unintentional, as a result of nearby transmitters, transmitting in the same or very close frequencies levels.
An example of radio frequency interference is the coexistence of multiple wireless networks on the same area with the same frequency channel [@Mpitziopoulos].
Noise is considered the undesirable accidental fluctuation of electromagnetic spectrum, collected by an antenna.
The Signal-to-Noise Ratio is $$SNR= \frac{P_{signal}}{P_{noise}}$$ where $P$ is the average power.
A jamming attack can be considered [*effective*]{} if the [*Signal-to-Noise Ratio(SNR)*]{} is less than one ($SNR<1$).
There are several jamming techniques [@Mpitziopoulos]: Spot Jamming, Sweep Jamming, Barrage Jamming and Deceptive Jamming.
‘Jammer’ refers to the equipment and its capabilities that are exploited by the adversaries to achieve their goal.
There are several types of jammers, from simple transmitter or jamming stations with special equipment, used against wireless networks.[@xu05]
- the constant jammer - emitting totally random continuous radio signals; [*target*]{}: keeping the [*WSN*]{}s channel busy; disrupting nodes’ communication; causing interference to nodes that have already commenced data transfers and corrupt their packets.
- the deceptive jammer;
- the random jammer - sleeps for a random time and jams for a random time;
- the reactive jammer - in case of activity in a [*WSN*]{} immediately sends out a random signal to collide with the existing signal on the channel.
Sensitive Ant-based Technique for Jamming Attack on Wireless Sensor Network
===========================================================================
The current section describes the concept of sensitive ants proposed in [@chira07; @pintea2010]. Effective metaheuristics for complex problems, as large scale routing problems (e.g. the [*Generalized Traveling Salesman Problem*]{}) based on sensitive ants are illustrated in [@chira07; @his; @pintea2010]. Several concepts are defined and described further.
Sensitive ants refers to artificial ants with a Pheromone Sensitivity Level (PSL) expressed by a real number in the unit interval $0\leq PSL \leq 1$.
A pheromone blind ant is an ant completely ignoring stigmergic information, with the smallest PSL value, zero.
A maximum pheromone sensitivity ant has the highest possible PSL value, one.
The sensitive-explorer ants have small\
Pheromone Sensitivity Level values indicating that they normally choose very high pheromone level moves.
The sensitive-exploiter ants have high\
Pheromone Sensitivity Level values indicating that they normally choose any pheromone marked move.
The sensitive-explorer ants are also called small PSL-ants, hPSL and sensitive-exploiter ants are called high PSL-ants, hPSL. They intensively exploit the promising search regions already identified by the sensitive-explorer ants. For some particular problems the sensitivity level of hPSL ants have been considered to be distributed in the interval (0.5, 1) and for the sPSL ants the sensitivity level in the interval (0, 0.5).
Based on the already described notions it is introduced a new ant-based concept with sensitivity feature. The [*Sensitive Ant Algorithm for Denial Jamming Attack on Wireless Sensor Network*]{} is further called [*Sensitive Ant Denial Jamming*]{} on [*WSN*]{} algorithm.
As we know, not all ants react in the same way to pheromone trails and their sensitivity levels are different there are used several groups of ants with different levels of sensitivity. For an easy implementation are used just two groups-colonies.
In [@Muraleedharan] is firstly introduced an ant system for jamming attack detection on [*WSN*]{}. The performance of the ant system is given by the node spacing and several parameters: $Q$ an arbitrary parameter, $\rho$ trail memory, $\alpha$ power applied to the pheromones in probability function and $\beta$ power of the distance in probability function.
It is considered a [*WSN*]{} in a two dimensional Euclidean space. There are several key elements of AS for keeping the network robust and de-centralized. One is the information on the resource availability on every node used to predict the link for the ant’s next visit.
Other key elements are the pheromones intensity and dissipate energy of ants as they traverse the nodes based on path probabilities. The key factor for making decisions in [@Muraleedharan] is the transition probability (\[eq:1\]). In the newly [*Sensitive Ant Denial Jamming*]{} on [*WSN*]{} only the ants with small pheromone level are using this probability.
$$\label{eq:1}
P_{ij}=\frac{(\varphi_{ij}\cdot\eta_{ij})^\alpha\cdot(\frac{1}{D_{ij}})^\beta}{\sum_k (\varphi_{ik}\cdot\eta_{ik})^\alpha\cdot(\frac{1}{D_{ik}})^\beta}$$
where $\eta_{ij}$ is the normalized value (\[eq:2\]) of Hop, $H_{ij}$, Energy, $E_{ij}$, Bit Error Rate, $B_{ij}$, Signal to Noise ratio, $
SNR_{ij}$, Packet Delivery, $Pd_{ij}$ and Packet Loss, $Pl_{ij}$ [@Liang]. $$\label{eq:2}
\eta_{ij}=H_{ij}\cdot E_{ij}\cdot B_{ij}\cdot SNR_{ij}\cdot Pd_{ij}\cdot Pl_{ij}$$
In the sensitive ant model the ants with high pheromone level are choosing the next node based on (\[eq:3\]) from the neighborhood $J$ of node $j$. $$\label{eq:3}
j=argmax_{u\in J_{ik}}\{(\varphi_{iu}\cdot\eta_{iu})^\alpha\cdot(\frac{1}{D_{iu}})^\beta\}.$$
$\varphi_{ij}$is the pheromone intensity between the source node $i$ and destination node $j$; The normalized value is the difference between total and actual value of the performance parameters. The performance value is used to compute the transition probabilities in a route. The link being active or dead in a tour taken by an ant is incorporated in the pheromone. The pheromone is globally updated [@dorigo] following each complete tour by ant system with the update rule (\[eq:4\]) is following [@Muraleedharan]. $$\label{eq:4}
\varphi_{ij}(t)=\rho(\varphi_{ij}(t-1))+\frac{Q}{D_t\cdot \eta_t}$$ where $D_t$ is the total distance traveled by ants during the current tour. The trails formed by the ant is dependent on the link factor. The tabu list includes now updated values of the energy available in the nodes for a particular sub-optimal route with high reach-ability. A run of the algorithm returns the valid path of the wireless network. That is how the information in [*WSN*]{} is re-routed. Termination criteria is given by a given number of iterations.
[**Discussions.**]{} Sensitive ants with lower pheromone level are able to explore the wireless network and the ants with high pheromone level intensively exploit the promising search regions already identified in the network. The ant’s behavior emphasizes search intensification. The ants “learn” during their lifetime and are capable to improve their performances. That is how they modifies their level of sensitivity: the PSL value increase or decrease based on the search space topology encoded in the ant’s experience.
The introduced model [*Sensitive Ant Algorithm for Denial Jamming Attack on WSN*]{} seems to improve the jamming attack detection and re-routing in wireless sensor network.
Conclusions
===========
The paper shows the main jamming attacks and several countermeasure. It is introduced a new [*Sensitive Ant Denial Jamming*]{} on [*Wireless Sensor Network*]{} based on ant system as mobile-agents class. In general mobile agent techniques including ant models proved to have a medium defense effectiveness, a medium cost but a good compatibility with existing software. The introduced sensitive model brings a new feature that improves the reactions of agents in the network in case of jamming attacks and redirect the information also to the processing element in order to re-routing information.
Acknowledgement. {#acknowledgement. .unnumbered}
================
This research is supported by Grant PN II TE 113/2011, New hybrid metaheuristics for solving complex network design problems, funded by CNCS Romania.
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bibliography:
- 'Biblio.bib'
title: Enhancement of electron mobility at oxide interfaces induced by overlayers
---
00\_Authors 00\_Abstract
The formation of a two-dimensional electron system (2DES) at the interface between band insulators (STO) and (LAO) is among the most intriguing effects studied in oxide electronics [@ohtomo2004high]. Gate tunable superconductivity [@reyren2007superconducting; @caviglia2008electric], strong spin-orbit coupling [@caviglia2010tunable; @diez2015giant] and magnetism [@bert2011direct; @li2011coexistence] are some of the many phenomena observed. The origin of this 2DES is a long standing question in the solid state community and recent results indicate that a consistent picture should take into account both the built-in polar field and the presence of point defects [@nakagawa2006some; @gunkel2012influence; @yu2014polarity]. Among these, oxygen vacancies and cation off-stoichiometry in STO are capable of inducing a 2DES [@kalabukhov2007effect; @warusawithana2013laalo3]. However, defects residing in the conductive channel are usually responsible for a decreased electronic mobility [@bristowe2011surface]. In order to promote high electron mobility, it is crucial to confine donor sites away from the conducting plane, without preventing the 2DES formation in the STO top layers. Previous attempts to control the defect concentration profile and thus enhance the mobility involved the use of crystalline insulating overlayers [@huijben2013defect; @chen2015extreme], adsorbates [@xie2013enhancing], amorphous materials [@chen2013high] and even thin metallic layers [@wu2013nonvolatile; @lesne2014suppression]. A promising material to control defect formation is tungsten oxide . The several possible oxidations states of tungsten make particularly active in undergoing redox reactions. For this reason this material is often utilized in electrochemical applications and electrochromic devices [@deb2008opportunities; @meng2015electrolyte; @cong2016tungsten]. Also, both crystalline and amorphous can host vacancies and interstitial atoms, thus allowing cation accommodation and diffusion, with a tendency to form compounds such as tungsten bronzes [@arab2013strontium; @he2016atomistic]. Recent progress demonstrated the high-quality growth of thin films on perovskite materials [@du2014strain; @leng2015epitaxial; @altendorf2016facet].
In this work we combine the use of a crystalline LAO/STO interface with the high reactivity of amorphous to realise a high-mobility metallic 2DES in heterostructures. Our approach is based on the tendency of to undergo redox reactions, whose contribution is primarily manifested by the reduction of the critical LAO thickness required for the formation of a 2DES. We characterise the transport properties of this system as a function of and thickness and find multi-band conduction and an increased electron mobility up to . The multi-band conduction leads to a remarkably strong classical magnetoresistance, which reaches 900% at and . Furthermore, the analysis of Shubnikov-de Haas oscillations unveils an unusually large effective mass of the highly mobile electrons.
figs/01\_DFT figs/02\_MIT Ultra-thin heterostructures of amorphous and crystalline LAO are grown on -terminated STO (001) substrates by pulsed laser deposition (details on the growth are provided in the supplementary information). We denote by $(m,n)$ the crystalline equivalent number of unit cells (u.c.) of and LAO, respectively, that form the heterostructure. To investigate structurally the heterostructure, we perform high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM). The HAADF-STEM images in \[fig:TEM\_large,fig:TEM\_zoom\] acquired from a $(4,2)$ heterostructure show uniform layers of amorphous corresponding to 4 u.c. in thickness, followed by 2 u.c. of crystalline LAO. Due to the atomic number difference between La and Sr, the HAADF signal from the LAO is more intense than from the underlying STO substrate, as expected. To further confirm the growth of LAO, electron energy loss spectroscopy (EELS) is subsequently performed. With an energy dispersion of , the Ti-L$_\mathrm{2,3}$ and La-M$_\mathrm{4,5}$ edges are recorded simultaneously, providing atomic-resolution Ti and La elemental maps as presented in \[fig:EELS\_Ti,fig:EELS\_La,fig:EELS\_Ti\_La\]. By averaging the La map parallel to the interface in \[fig:EELS\_profile\], two clear peaks are shown for La, consistent with the growth of 2 LAO layers in our heterostructure. However, significant diffusion of La into the is also observed. The surface of all heterostructures is additionally measured by atomic force microscopy (\[fig:AFM\]), revealing the same regular steps and terraces of the underlying STO substrate, indicating uniform film growth.
The series of resistance versus temperature curves of $(m,n)$ heterostructures in \[fig:MIT\_thicknesses\], shows a sharp tickness-dependent insulator-to-metal transition. The transport measurements are performed in a Van der Pauw configuration (see methods for details). For different $(m,n)$ combinations the samples show either insulating (orange curves) or metallic (blue curves) character, with a sharp transition between the two regimes as function of layer thickness. This can be noted comparing the $(4,1)$ and $(4,2)$ curves, where a variation of a single u.c. of the LAO interlayer determines a three orders of magnitude resistivity difference at room temperature, which diverges upon cooling. It is noteworthy that the onset of the metallic state corresponds to the sheet resistance value $h/e^2$ (dotted line in \[fig:MIT\_thicknesses\]) which is the quantum limit for metallicity in 2D [@mott1985minimum], suggesting this electronic system has a two-dimensional nature.
The interplay between and LAO thicknesses is summarised in the phase diagram of \[fig:PhaseDiagram\], where we indicate with a shaded orange background the $(m,n)$ combinations resulting in insulating samples. For LAO-only films $(0,n)$ we reproduce the well-known critical thickness for metallicity of 4 unit cells in crystalline interfaces, while samples with only $(m,0)$ are always insulating. Heterostructures with 1 u.c. of LAO $(m,1)$ show an insulating state, independently of the layer thickness. When $n=2$ the insulating state persists for thickness $m\leq 2$ only, above which a metallic state is induced. With 3 cells of LAO a single layer of is enough to trigger the metallic state. We can compare the metallicity of the conducting heterostructures by evaluating their residual resistivity ratio, defined as $\mathrm{RRR}=\rho_{xx}(\SI{300}{\kelvin})/\rho_{xx}(\SI{1.5}{\kelvin})$. Higher RRR values indicate more pronounced metallic behaviour and are represented by the colour map in \[fig:PhaseDiagram\]. Our reference heterostructure $(0,4)$ has $\mathrm{RRR}=110$, similarly to previous reports [@gariglio2009superconductivity; @warusawithana2013laalo3]. In the system we find higher values for decreasing thickness of the LAO interlayer. As an example, the $(4,2)$ combination shows $\mathrm{RRR}= 700$. A simple interpretation for this trend can be provided by considering two competing effects. On the one hand the spatially closer the amorphous overlayer is to the STO, the more effective it is in controlling defect formation and maintaining a clean conductive channel. On the other hand a sufficiently thick LAO interlayer is required to provide the polar electric field necessary for driving charge carriers at the interface. The optimal balance of these two effects seems to be achieved for 2 u.c. of LAO, where we measure the highest RRR value. In this picture we are thus able to combine the mobility enhancement provided by the amorphous overlayer with the advantage of a crystalline conductive interface.
figs/03\_MT\_thickness figs/01\_Table\_HE The characteristics of this metallic state are investigated by performing magnetotransport measurements on a $(4,2)$ heterostructure, which shows a high RRR value. In \[fig:MR\] we present its magnetoresistance (MR) which is defined as $\mathrm{MR}=\frac{\rho_{xx}(B)-\rho_0}{\rho_0}$, where $\rho_0$ is the sheet resistance at $B=0$ and the magnetic field is applied perpendicular to the interface plane. At the MR is positive and reaches 900% at , corresponding to one order of magnitude increase in sheet resistance. This response is very different from what is usually observed in LAO/STO heterostructures $(0,n)$ as can be seen from the comparison with a $(0,4)$ sample in \[fig:MR\]. The LAO/STO, in fact, shows a positive MR of only at .
The Hall resistance of the $(4,2)$ heterostructure(\[fig:HE\]) is negative, indicating electronic transport, with a kink at about . A non-linear component in the Hall effect is typically related to multiple conduction channels contributing to the transport. In the simplest approximation of two independent channels in parallel, the classical magnetoresistance and the Hall resistance are given by
$$\label{eq:MR_2bands}
\rho_{xx}=
\frac{({n_\mathtt{I}}{\mu_\mathtt{I}}+{n_\mathtt{II}}{\mu_\mathtt{II}}) + ({n_\mathtt{I}}{\mu_\mathtt{II}}+ {n_\mathtt{II}}{\mu_\mathtt{I}}){\mu_\mathtt{I}}{\mu_\mathtt{II}}B^2}
{({n_\mathtt{I}}{\mu_\mathtt{I}}+{n_\mathtt{II}}{\mu_\mathtt{II}})^2+(n{\mu_\mathtt{I}}{\mu_\mathtt{II}}B)^2}
\cdot\frac{1}{e},$$
$$\label{eq:HE_2bands}
\rho_{xy}=
\frac{(\pm {n_\mathtt{I}}{\mu_\mathtt{I}}^2 \pm {n_\mathtt{II}}{\mu_\mathtt{II}}^2) + n({\mu_\mathtt{I}}{\mu_\mathtt{II}}B)^2}
{({n_\mathtt{I}}{\mu_\mathtt{I}}+{n_\mathtt{II}}{\mu_\mathtt{II}})^2+(n{\mu_\mathtt{I}}{\mu_\mathtt{II}}B)^2}
\cdot\frac{B}{e},$$
where $n_i$, $\mu_i$ are the carrier density and mobility of the $i$-th channel, $n=(\pm {n_\mathtt{I}}\pm {n_\mathtt{II}})$ and the $\pm$ sign indicates hole or electron carriers, respectively. We use \[eq:HE\_2bands\] to fit with good agreement the $(4,2)$ Hall data (dashed line in \[fig:HE\]) and extract in \[tab:HE\_params\] the corresponding transport parameters (see methods for details). The $(4,2)$ heterostructure presents two channels of electrons: one with lower-mobility ${\mu_\mathtt{I}}=\SI{3600}{\centi\metre\squared\per\volt\per\second}$, ${n_\mathtt{I}}=\SI{1.7e13}{\per\centi\metre\squared}$ and one with higher mobility ${\mu_\mathtt{II}}=\SI{80000}{\centi\metre\squared\per\volt\per\second}$, ${n_\mathtt{II}}= \SI{9.3e12}{\per\centi\metre\squared}$. Higher mobility values are observed for lower carrier densities, consistent with previous studies of STO-based 2DES [@gunkel2016defect]. We note that the sheet resistance of the higher-mobility channel ${\rho_\mathtt{II}}$ is one order of magnitude smaller than ${\rho_\mathtt{I}}$, suggesting that it dominates the low-temperature transport. The $(4,2)$ mobility is about two orders of magnitude higher than what observed in the reference $(0,4)$ sample (${\mu_\mathtt{I}}=\SI{840}{\centi\metre\squared\per\volt\per\second}$). The absence of a higher-mobility channel is coherent with the higher resistivity at and the lower RRR value usually found in heterostructures.
Using $n_i$, $\mu_i$ extracted from the Hall effect we calculate with \[eq:MR\_2bands\] the classical two-channels MR (dashed line in \[fig:MR\]). The resulting curve accounts for a good extent of the measured signal, which is thus the dominant MR contribution, in particular for small magnetic fields. The residual MR can arise from the presence of further conduction channels or disorder (Supplementary Figure 5). Quantum corrections might also be present, but considering their typical magnitude, they are negligible compared to the other contributions. figs/04\_MT\_temperature A better insight into the effects of these parallel conduction channels is given by tracking the resistivity and Hall coefficient as a function of temperature. The measurements are performed in a Hall bar geometry (inset of \[fig:FieldCool\]), where the conductive regions are defined using an insulating hard mask, as described in the methods. In \[fig:FieldCool\] we compare the resistivity versus temperature curve measured with a magnetic field of and applied perpendicular to the interface plane. At the curves are well separated, underscoring a strong positive MR of . On warming, the MR decreases and disappears below our measurement limit around room temperature.
By tracking the Hall effect as a function of temperature (Supplementary Figure 4) we can investigate the temperature dependence of $n_i$, $\mu_i$ for the different channels. A non-linear Hall effect is observed between and , while a linear trend is seen at higher temperatures. The extracted mobilities and carrier densities are presented in \[fig:mu\_vs\_T,fig:n\_vs\_T\]. In this patterned sample we measure ${\mu_\mathtt{I}}=\SI{2500}{\centi\metre\squared\per\volt\per\second}$ and ${\mu_\mathtt{II}}=\SI{27000}{\centi\metre\squared\per\volt\per\second}$ at . With increasing temperature, at first ${\mu_\mathtt{I}}$ retains an almost constant value while ${\mu_\mathtt{II}}$ decreases. Above the high-mobility channel disappears and the Hall effect becomes linear, signalling the cross-over to single channel transport. At higher temperatures ${\mu_\mathtt{I}}$ decreases several orders of magnitude and reaches ${\mu_\mathtt{I}}=\SI{7}{\centi\metre\squared/\volt\second}$ at room temperature. This trend is similar to what has previously been reported for LAO/STO heterostructures [@fete2015growth].
The strong MR in our system can be explained by considering the peculiar characteristics of the two conduction channels. In general, the classical theory of MR gives a strong resistivity increase with applied magnetic field whenever the charge carriers possess high mobility. To observe high MR in systems with multiple channels it is also required that the high mobility channel is dominant in the electronic conduction (i.e. ${\rho_\mathtt{II}}/{\rho_\mathtt{I}}\ll 1$). Both conditions are met in our system, where we find a direct correlation between the ratio ${\rho_\mathtt{II}}/{\rho_\mathtt{I}}$ and the MR magnitude at : with ${\rho_\mathtt{II}}/{\rho_\mathtt{I}}\sim \SI{e-1}{}$ in \[fig:MagnetoTransport\] we measure $\mathrm{MR} \sim 900\%$, and with ${\rho_\mathtt{II}}/{\rho_\mathtt{I}}\sim 1$ in \[fig:MT\_temperature\] we have a lower $\mathrm{MR} \sim 200\%$. A further confirmation of this behaviour is given by considering that ${\rho_\mathtt{I}}$ values in \[fig:FieldCool\] well represent the resistivity versus temperature curve at $B=\SI{12}{\tesla}$. This indicates that the high-mobility channel is suppressed in the transport at high magnetic field.
figs/05\_SdH
The carrier density of the two conduction channels present opposite trends as a function of temperature. At we find that the lower-mobility channel has a higher density ${n_\mathtt{I}}=\SI{2.2e13}{\per\centi\metre\squared}$, and the higher-mobility has a lower-density ${n_\mathtt{II}}=\SI{2.4e12}{\per\centi\metre\squared}$. Upon warming, ${n_\mathtt{I}}$ maintains an almost constant value, while ${n_\mathtt{II}}$ undergoes a sharp drop above and subsequently disappears. This disappearance might be due to the activation of interband scattering processes at higher temperatures, which cause a mixing of ${\rho_\mathtt{I}}$, ${\rho_\mathtt{II}}$, so that their populations cannot be independently resolved in Hall effect measurements [@gunkel2016defect]. Another possible interpretation for this trend is that the two conduction channels are situated in STO at two different distances from the interface. The first channel might be spatially closer to the LAO layer, where electrons experience more defects and a stronger polar electric field, resulting in lower mobility and higher carrier density. The second channel, instead, could be further away from the interface, where a less-defected STO determines a higher electron mobility. In this picture, the depopulation of ${\rho_\mathtt{II}}$ might be linked to the drop of the STO dielectric constant upon warming [@sakudo1971dielectric] (Supplementary Fig. 6).
The electronic state confined in our heterostructures shows Shubnikov-de Haas (SdH) oscillations superimposed on the background of strong positive MR. The SdH as a function of temperature are shown in \[fig:SdH\_dRxx\], where their signal was extracted by fitting the background with a 3^rd^ order polynomial (dashed line in \[fig:SdH\_Rxx\]). The oscillations disappear when the magnetic field is applied parallel to the interface plane, as expected for a two-dimensional system. SdH oscillations in 2DES can be modelled by $$\label{eq:SdH}
\Delta \rho_{xx}=
4\rho_\mathrm{c}\mathrm{e}^{-\alpha T_\mathrm{D}}\frac{\alpha T}{\sinh(\alpha T)}\sin\left(2\pi \frac{\omega_\mathrm{SdH}}{B}\right),$$ where $\rho_\mathrm{c}$ is the classical sheet resistance in zero magnetic field, $\alpha=2\pi^2 k_\mathrm{B}/\hbar \omega_\mathrm{c}$ with cyclotron frequency $\omega_\mathrm{c}=eB/m^*$, Boltzmann’s constant $k_B$, reduced Planck’s constant $\hbar$, carrier effective mass $m^*$ and Dingle temperature $T_D$. Fourier analysis in \[fig:SdH\_FFT\] reveals that the oscillations are periodic in $B^{-1}$, with a single frequency peak at $\omega_\mathrm{SdH}=\SI{49}{\tesla}$. Assuming a 2DES with circular sections of the Fermi surface, we can estimate the carrier density as $n_\mathrm{SdH}=\omega_\mathrm{SdH}\nu_\mathrm{s}e/h$, where $\nu_\mathrm{s}$ indicates the spin degeneracy. By considering $\nu_\mathrm{s}=2$ we find $n_\mathrm{SdH}=\SI{2.4e12}{\per\centi\metre\squared}$. We note that even if Hall effect measurements indicate the presence of two conduction channels (values in \[fig:SdH\_dRxx\]), only one channel contributes to the quantum oscillations. Furthermore, the obtained $n_\mathrm{SdH}$ is lower than both ${n_\mathtt{II}}$, ${n_\mathtt{I}}$ for this sample, so that it is not possible to associate the SdH oscillation to one specific channel. A discrepancy between $n_\mathrm{SdH}$ and $n_\mathrm{Hall}$ in interfaces has already been reported and its origin remains unknown [@caviglia2010two; @shalom2010shubnikov].
To extract the mass of the electrons showing the SdH effect, in \[fig:SdH\_ampl\] we track the oscillation amplitude at $B=\SI{11.85}{\tesla}$ as a function of temperature (similar results are obtained using different values of $B$). Fitting the trend with \[eq:SdH\], we find a surprisingly high value $m^*=5.6\, m_\mathrm{e}$. Considering the enhanced mobility of carriers in the system, in fact, one would expect a decreased effective mass, while in this case $m^*$ is three times larger than typical observations in heterostructures [@chen2013high; @mccollam2014quantum].
A possible explanation of this electron mass renormalization can be lead back to strong electron-phonon coupling, which is enhanced by the tight spatial confinement of the 2DEG. Such coupling was previously found to produce large phonon-drag [@pallecchi2015giant; @pallecchi2016large] and polaronic effects in both interfaces and amorphous thin films [@cancellieri2016polaronic; @berggren2001polaron]. Another possibility is that the modified defect profile with respect to conventional interfaces determines a mass enhancement of the 2DES bands [@wunderlich2009enhanced].
Finally, from the Dingle plot in \[fig:SdH\_dingle\] we extract $T_D=\SI{.45}{\kelvin}$. This value points to an ordered electronic system with sharp Landau levels, considering that their energy smearing $k_\mathrm{B}T_\mathrm{D}\sim\SI{40}{\micro\electronvolt}$ is much smaller than their spacing $\hbar\omega_\mathrm{C}\sim\SI{250}{\micro\electronvolt}$. The extracted value $\rho_\mathrm{C}=\SI{14}{\ohm/square}$ is in good agreement with $\rho_0=\SI{35}{\ohm/square}$, corroborating the performed analysis. Using $T_D = \hbar / 2\pi k_\mathrm{B}\tau$ and $\tau = m^*\mu_\mathrm{SdH}/e$ we calculate the elastic scattering time $\tau=\SI{2.7}{\pico\second}$ and the quantum mobility $\mu_\mathrm{SdH}=\SI{851}{\centi\metre\squared\per\volt\per\second}$. Even though $\mu_\mathrm{SdH}$ is lower than both the Hall effect values ${\mu_\mathtt{II}}$, ${\mu_\mathtt{I}}$, it further confirms the formation of a high mobility 2DES in the heterostructure.
To conclude, we have demonstrated that amorphous is an effective overlayer to form 2DES with enhanced mobility and effective mass at interfaces. Reducing the crystalline critical thickness from 4 to 2 unit cells, the overlayer determined a metallic system with high RRR and increased electron mobility. We ascribed the insurgence of a strong classical magnetoresistance to the peculiar characteristics of the multiple conduction channels observed in the system. Quantum oscillations of conductance confirmed the realisation of high-quality heterostructures, where a strong two-dimensional confinement of carriers is achieved. All these results are achieved using an amorphous overlayer, which does not require crystal matching. Our work thus demonstrates a new approach for defect control at oxide interfaces, which can be exploited to induce high-mobility 2DES in a broad variety of oxide materials.
Experimental Section
====================
*Samples growth:* heterostructures were grown by pulsed laser deposition on commercially available (001) substrates, with surface termination. The laser ablation was performed using a KrF excimer laser (Coherent COMPexPro 205, $\lambda=\SI{248}{\nano\metre}$) with a repetition rate and fluence. The target-substrate distance was fixed at . For the thin films a crystalline target was employed and the deposition performed at substrate temperature and oxygen pressure. film thickness was monitored *in-situ* during growth by intensity oscillations of reflection high-energy electron diffraction (RHEED). The samples were annealed for at in of atmosphere to compensate for the possible formation of oxygen vacancies. The amorphous thin films were deposited from a sintered target at substrate temperature and oxygen pressure. film thickness was calibrated by depositing crystalline on and monitoring the growth by RHEED. The thickness value was then confirmed by X-ray diffraction and transmission electron microscopy measurements (results to be published elsewhere). At the end of the growth the heterostructures were cooled down to ambient temperature in oxygen pressure (further details in Supplementary Figure 1).
*Hall bar geometry fabrication:* substrates were patterned prior to thin films deposition with standard e-beam lithography followed by the evaporation of an insulating mask. The mask was deposited at room temperature by RF sputtering in a atmosphere, resulting in amorphous alumina.
*Electrical measurements:* The measurements in \[fig:PhaseDiagram\_MIT,fig:MagnetoTransport\] were carried out in van der Pauw configuration, while for the ones in \[fig:MT\_temperature,fig:SdH\] a Hall bar geometry was used. In both measurement configurations the metallic interface was directly contacted by ultrasonically wire-bonded .
*Non-linear Hall effect fits:* The fits are performed with the least squared method using data in the magnetic field range . The constraint $1/\rho_0=1/{\rho_\mathtt{I}}+1/{\rho_\mathtt{II}}$ is applied to the fitting parameters, and $\rho_0$ is extracted from the $\rho_{xx}(B)$ measurement. With the assumption $1/\rho_i = n_i e \mu_i$, only three free parameters among [$\rho_i$, $n_i$, $\mu_i$]{}, with $i={\mathtt{I},\mathtt{II}}$, are varied in the fitting procedure.
Acknowledgments
===============
We thank P. Zubko for valuable feedback and for performing XRD measurements; Y. M. Blanter and D. J. Groenendijk for fruitful discussions. This work was supported by The Netherlands Organisation for Scientific Research (NWO/OCW) as part of the Frontiers of Nanoscience program (NanoFront), the Dutch Foundation for Fundamental Research on Matter (FOM), the European Research Council under the European Union’s H2020 programme/ ERC GrantAgreement n. \[677458\] and the Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR-1120296). The FEI Titan Themis 300 TEM was acquired through NSF-MRI-1429155, with additional support from Cornell University, the Weill Institute and the Kavli Institute at Cornell. A. F. thanks TU Delft and Kavli Institute for the access to Computing Center resources, and computational support from the CRS4 Computing Center (Piscina Manna, Pula, Italy).
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abstract: 'Full counting statistics is a fundamentally new concept in quantum transport. After a review of basic statistics theory, we introduce the powerful Green’s function approach to full counting statistics. To illustrate the concept we consider a number of examples. For generic two-terminal contacts we show how counting statistics elucidates the common (and different) features of transport between normal and superconducting contacts. Finally, we demonstrate how correlations in multi-terminal structures are naturally included in the formalism.'
author:
- Wolfgang Belzig
title: Full Counting Statistics in Quantum Contacts
---
Introduction {#sec:intro}
============
The probabilistic interpretation is a fundamental ingredient of quantum mechanics. While the wave function determines the full quantum state a system and its evolution in time, observable quantities are related to hermitian operators. Expectation values of these operators determine the average value of a large number of identical measurements. However, an individual measurement yields in general a different result. Applying this idea to a current measurement in a quantum conductor, leads directly to the concept of *full counting statistics* (FCS): during a given time interval a certain number of charges will pass the conductor. To predict the statistical properties of the number of transfered charges we need a probability distribution. The theoretical goal is to find this distribution.
### Overview
In this article we give an introduction to the field of *full counting statistics in mesoscopic electron transport*. We will concentrate on the powerful technique – using Keldysh-Green’s functions – which at the same time also is based on microscopic theory. To accomplish this goal we will first review concepts of basic statistics, which are relevant for counting statistics. In the next section we address the microscopic derivation of FCS using Keldysh-Green’s functions. In the rest of the article we demonstrate the use of counting statistics in a number of examples, like two-terminal contacts with normal and superconducting leads, diffusive metals and, finally, multi-terminal structures. But first we review briefly the development of the field.
### History
Full counting statistics has its roots in quantum optics [@mandelwolf], where the number statistics of photons is used, e. g., to characterize coherence properties of photon sources. The major step to adopt the concept to mesoscopic electron transport has been undertaken by Levitov and Lesovik [@levitov:93-fcs]. Since then the theory of FCS of charge transport in mesoscopic conductors has advanced substantially, see Refs. [@blanter; @nazarov:03-book]. In Ref. [@levitov:93-fcs] it was shown that scattering between uncorrelated Fermi leads with probability $T$ is described by a binomial statistics $P(N)={ M \over N} T^N (1-T)^{M-N}$. Here, $P(N)$ is the probability, that out of $M=2et_0V/h$ independent attempts $N$ charges are transfered. Furthermore, Levitov and coworkers studied the counting statistics of diffusive conductors [@levitov:96-diffusive], time-dependent problems [@levitov:96-coherent] and of a tunnel junction [@levitov:95-tunnel]. A theory of full counting statistics based on the powerful Keldysh-Green’s function method was initiated by Nazarov [@yuli:99-annals]. This formulation allows a straightforward generalization to systems containing superconductors [@belzig:01-super; @belzig:01-diff] and multi-terminal structures [@nazarov:02-multi; @boerlin:02]. Classical approaches to FCS were recently put forward for Coulomb blockade systems [@dejong:96; @bagrets:03-coulomb], and, for chaotic cavities based on a stochastic path-integral approach [@pilgram:03-stochastic]. The field of counting statistics in the quantum regime is closely related to the fundamental measuring problem of quantum mechanics, which has been addressed in a number of works [@levitov:96-coherent; @makhlin:00-readout; @nazarov:01-measuring; @shelankov:03; @klich]. Expressing the FCS of charge transport by the counting statistics of photons emitted from the conductor provides an interesting alternative to classical counting of electrons [@beenakker:01-photon]. Counting statistics has been addressed by now for many different phenomena
- Andreev contacts [@muzykantskii:94]
- generic quantum conductors [@dejong:96; @blanter:01; @levitov:01-cumulant; @gutman:02-thirdcumulant]
- adiabatic quantum pumping [@andreev:00-pump; @levitov:01-pump; @makhlin:01-pump; @muzykanstkii:02-pump]
- qubit-readout [@makhlin:00-readout; @choi:02; @engel:02; @clerk:03-resonantcooperpair]
- superconducting contacts in equilibrium [@belzig:01-super]
- proximity effect structures [@belzig:01-diff; @samuelsson:03-counting; @reulet:03-diff; @belzig:03-incoherent; @bezuglyi:03-interferometer]
- cross-correlations with normal [@yuli:01-multi] or superconducting contacts [@boerlin:02; @samuelsson:02-cavity]
- entangled electron pairs [@taddei:02-entanglerfcs; @taddei:03-clauserhorne].
- phonon counting [@kindermann:02-phonon]
- relation between photon counting and electron counting [@kindermann:02-photoncounting]
- current biased conductors [@kindermann:03-voltage]
- interaction effects: weak and strong Coulomb blockade [@bagrets:03-coulomb; @bagrets:03-weakcoulomb; @kindermann:03-coulomb].
- multiple Andreev reflections in superconducting contacts [@cuevas:03-marfcs; @johansson:03-marfcs]
Very recently, an important experimental step forward was achieved. Reulet, Senzier, and Prober measured for the first time the third cumulant of current fluctuations produced by a tunnel junction [@reulet:03-thirdcumulant]. Surprisingly the measured voltage dependence deviated from the expected voltage-independent third cumulant of a simple tunnel contact [@levitov:93-fcs; @levitov:01-cumulant]. A subsequent theoretical explanation is that the third cumulant is in fact susceptible to environmental effects [@beenakker:03-thirdcumulant]. This experiment has already triggered some theoretical activity [@gutman:02-thirdcumulant; @pilgram:03-thirdcumulant; @galaktionov:03-thirdcumulant].
Full Counting Statistics {#sec:fcs}
========================
The fundamental quantity of interest in quantum transport is the probability distribution $$P_{t_0}(N_1,N_2,\ldots,N_M)\equiv P(\vec N)\,,$$ which denotes for a $M$-terminal conductor the probability that during a certain period of time $t_0$ $N_1$ charges enter through terminal 1, $N_2$ charges enter through terminal 2, $\ldots$, and $N_M$ charges enter through terminal $M$ (negative $N_i$ correspond to charges leaving the respective terminal). The same information is contained in the cumulant generating function (CGF), defined by $$\label{eq:cgf}
S(\chi)=\ln\left[\sum\nolimits_{\vec N} e^{i\vec N\vec \chi} P(\vec N)\right]\,,$$ where we introduced the vector of counting fields $\vec
\chi=(\chi_1,\chi_2,\ldots,\chi_N)$. The normalization condition requires $\sum_{\vec N} P(\vec N)=1\,\leftrightarrow\,
S(\vec \chi=\vec 0)=0$.
### Charge conservation
We are interested in the long-time limit of the charge counting statistics, which means that no extra charges remain inside the conductor after the counting interval. If we count only the total number of transfered charges, we simply have to consider $P(N)=\sum_{\vec N} \delta_{\sum N_\alpha,N} P(\vec \chi)$, or, equivalently, to put all counting fields equal $S(\chi_1=\chi ,
\chi_2=\chi,\ldots,\chi_N=\chi)$. Charge conservation now means that $S(\chi_1=\chi , \chi_2=\chi,\ldots,\chi_N=\chi)=0$. As a consequence the CGF depends only on differences between counting fields. This has the direct interpretation, that a difference $\chi_\alpha-\chi_\beta$ is related to a charge transfer between terminal $\alpha$ and $\beta$. In general, this means that we need only $M-1$ counting fields to describe a $M$-terminal structure. If one of the counting fields, e. g. $\chi_M$, has been eliminated, the charge transfer into terminal $M$ can be restored from the CGF, in which all other $\chi_\alpha$ are equal $\chi_\alpha-\chi_M$. In the special case of a two-terminal device, the CGF depends only on $\chi\equiv\chi_1-\chi_2$. We denote this below with $S(\chi)$. Later we will see that the CGF’s are in general *periodic* functions of $\chi$, i. e. $S(\chi+2\pi)=S(\chi)$. This ensures that the total charge transfered is an integer multiple of the electron charge $e$, which makes sense, since we are talking about electron transport and want to neglect transient effects.
However, the interesting question, what the charge of an elementary event is, can be answered by FCS. Suppose the a CGF has the property $S(\chi+2\pi/n)=S(\chi)$. Direct calculation shows that $$\begin{aligned}
P(Q) = \int \frac{d\chi}{2\pi} e^{-iN\chi+S(\chi)} =
\left\{
\begin{array}[c]{lll}
P_n(Q/n) &,& (Q \textrm{ mod } n) = 0 \\
0 &,& (Q \textrm{ mod } n) \neq 0
\end{array}\right.,\end{aligned}$$ where $P_n(N)$ is the distribution $S_n(\chi)=S(\chi/n)$. The probability distribution vanishes for all $N$ which are not multiples of $n$, thus the elementary charge transfer is in units of $n e$, where $e$ is the electron charge. This has interesting consequences in the context of superconductivity, in which multiple charge transfers can occur [@muzykantskii:94; @cuevas:03-marfcs; @johansson:03-marfcs], or for fractional charge transfer [@levitov:01-cumulant].
### Correlations
One commonly addressed question is, if two different events (say the charges transfered into terminals $\alpha$ and $\beta$) are independent or not. For independent events the probability distributions are separable and we find that $\langle N_\alpha^k
N_\beta^l \rangle = \langle N_\alpha^k\rangle\langle N_\beta^l
\rangle$. In terms of the CGF this means that the CGF is the sum of two terms: one which depends only on $\chi_\alpha$ and a second one, which depends only on $\chi_\beta$. On the contrary, if the CGF can not be written as such a sum, the charge transfers in terminals $\alpha$ and $\beta$ are correlated.
### Special distributions (two terminals)
If the elementary events are uncorrelated, the probability distribution is *Poissonian*. With the average number of events is $\bar N$ we have $$\label{eq:poisson}
P_{Poisson}=\frac{\bar N}{N!}e^{-\bar N}
\leftrightarrow
S(\chi)=\bar N\left(e^{i\chi}-1\right)\,.$$ In the context of electron transport we encounter this distribution mostly for *tunnel junctions* with an almost negligible transmission probability at low temperatures. Here $\bar N=G_TVt_0/e$ is simply related to the voltage bias and the tunnel conductance.
As second example we consider the binomial (or Bernoulli) distribution. This is obtained if an event occurs with a probability T and the number of tries is fixed to $N_0$: $$\label{eq:binomial}
P_{binomal} = \binom{N_0}{N} T^N(1-T)^{N_0-N}
\leftrightarrow
S(\chi)=N_0 \ln\left[1+T\left(e^{i\chi}-1\right)\right]\,.$$ In some sense this is the most fundamental distribution in quantum transport: it gives the statistics of a voltage biased single channel quantum conductor if we identify $N_0=eVt_0/h$.
### Special distributions (many terminals)
For uncorrelated processes the CGF takes the simple form $$S(\vec \chi)=\sum_{\alpha,\beta} \bar N_{\alpha,\beta}
\left(e^{i(\chi_\alpha-\chi_\beta)}-1\right) \,.$$ The resulting distribution is just the product of Poisson distributions, taking into account total charge conservation. An important example is a multinomial distribution for $N_0$ independent attempts, which can have different outcomes with probabilities $T_\alpha $. It has the form $$S(\vec \chi)=N_0\ln\left[1+\sum_{\alpha}
T_{\alpha}
\left(e^{i\chi_\alpha}-1\right)\right] \,.$$
Theoretical approach to full counting statistics {#sec:theory}
================================================
### General theory
We will follow here the approach to FCS using the Green’s function technique [@yuli:99-annals]. Quantum-mechanically we define the cumulant generating function by [@yuli:99-annals; @belzig:01-super; @belzig:01-diff; @levitov:01-cumulant] $$\label{eq:cgf-general}
e^{S(\chi)} =
\langle {\cal T}_K e^{-i\frac{1}{2e}\int_{C_K} dt \chi(t) I(t)}
\rangle\,,$$
![Keldysh time-ordering contour.[]{data-label="fig:timepath"}](timepath){width="8cm"}
Here ${\cal T}_K$ denotes time ordering along the Keldysh-contour $C_K$, depicted in Fig. \[fig:timepath\]. The time-dependent field $\chi(t)$ is defined as $\pm \chi$ for $t\in C_{1(2)}$, i.e. $\chi(t)$ changes sign between the upper and the lower branch of $C_K$. $\hat
I(t)$ is the usual operator of the current through a certain cross section. Expansion in the *counting field* yields the cumulants. In the second order we find the $2^{nd}$ cumulant as $$\label{eq:second-cumulant}
C_2(t_0)=\int_0^{t_0} dt \int_0^{t_0} dt^\prime \left\langle \delta\hat
I(t)\delta \hat I(t^\prime)\right\rangle\,.$$ Higher cumulants yield more complicated expressions.
### Current Correlation Functions
The cumulants $C_n(t_0)$ are directly related to experimentally accessible quantities like current noise or the third cumulant of the current fluctuations. Let us demonstrate the relation for the low-frequency current noise, defined by $$\label{eq:noise}
S_I=2 \Delta f \int_{-\infty}^{\infty} d\tau
\left\langle \delta \hat I(\tau) \delta \hat I(0) \right\rangle\,,$$ where $\delta \hat I(\tau)=\hat I(\tau)-\langle \hat I \rangle$ and $\Delta f=f_{max}-f_{min}$ is the frequency band width, in which the noise is measured. The factor of 2 enters here to conform to the review article [@blanter]. We now transform in (\[eq:second-cumulant\]) the integration variables from $t,t^\prime$ to $T=(t+t^\prime)/2,\tau=t-t^\prime$. In the limit $t_0\equiv
(\Delta f)^{-1}$ much larger than the correlation time of current-fluctuations, the integral over $T$ can be evaluated and we obtain from (\[eq:second-cumulant\]) the desired result $S_I/2$. Similar arguments hold for higher cumulants, for which the expression corresponding to \[eq:second-cumulant\] are less trivial, however. In Ref. [@reulet:03-thirdcumulant] it was noted that $C_3$ depends in an quite unusual way on the frequency band measured, i.e. it is proportional to $2f_{max}-f_{min}$, which made it possible to prove experimentally that the third cumulant is actually measured.
### Keldysh-Green’s Functions
So far we have formally defined the CGF quantum mechanically. The relation to standard quantum-field theory methods is made in the following way. We introduce the standard Green’s function [@rammer:86] in the presence of a time-dependent Hamiltonian $$\label{eq:hpert}
H_{c}(t) = H_0 + \frac{1}{2e} \chi(t) \hat I \,,$$ where the time-dependence is only in the ’counting’ field $\chi(t)$. The counting field couples to the operator $\hat I$ of the current through a cross section, which intersects the conductor entirely. The single-particle operators corresponding to $H_0$ and $I$ are denoted by $h_0$ and $j$.
Using the matrix notation for the Keldysh-Green’s functions, we arrive at the equation of motion $$\label{eq:eom}
\left[i\frac{\partial}{\partial t} - \hat h_0 -
\frac{\chi}{2e} \bar\tau_3 \hat j_c\right]
\check G(t,t';\chi)
=\delta(t-t^\prime)\,.$$ Here $\bar\tau_3$ denotes the third Pauli matrix in the Keldysh space and is a result of the unusual time-dependence of the counting field. The relation of the Green’s function (\[eq:eom\]) to the CGF (\[eq:cgf-general\]) is obtained from a diagrammatic expansion in $\chi$ (the calculation is formally equivalent to the calculation of the thermodynamic potential in an external field, see e. g. [@agd]). One obtains the relation [@yuli:99-annals] $$\label{eq:chi-current}
\frac{\partial S(\chi)}{\partial \chi} = \frac{it_0}{e}
\textrm{Tr}\left[\bar\tau_3\hat j
\check G(t,t;\chi)\right] \equiv \frac{it_0}{e} I(\chi)\,,$$ where we have restricted us to a static situation, for which $\check
G(t,t)$ is independent of time. Note, that the *counting current* $I(\chi)$ should not be confused with the standard electrical current, which is actually given by $I_{el}=I(0)$. Rather, $I(\chi)$ contains (via an expansion in $\chi$) *all current-correlators* at once. It nevertheless resembles a current in the usual sense. E. g., it follows from Eq. (\[eq:eom\]) that the counting current is conserved.
### A simplification
In a typical mesoscopic transport problem we can access the full counting statistics based on the separation into terminals (or reservoirs) and a scattering region. Terminals provide boundary conditions to Green’s function far away from the scattering region. These are usually determined by external current or voltage sources and include material properties like superconductivity. Let us now take the following parameterization of the current operator in Eq. (\[eq:eom\]) $$\label{eq:countfield-choice}
\hat j({\mathbf x})=(\nabla F({\mathbf x}))
\lim_{{\mathbf x}\to{\mathbf x'}}
\frac{ie}{2m}\left(\nabla_{\mathbf x}-
\nabla_{{\mathbf x}'}\right)\hat\sigma_3\,.$$ $F({\mathbf x})$ is chosen such that it changes from 0 to 1 across a cross section C, which intersects the terminal, but is of arbitrary shape. Here we have introduced a matrix $\hat\sigma_3$ in the current operator, occurring e. g. in the context of superconductivity. We assume that the change from 0 to 1 should occur on a length scale $\Lambda$, for which we assume $\lambda_F\ll\Lambda\ll l_{imp},\xi_0$ (Fermi wave length $\lambda_F$, impurity mean free path $l_{imp}$, and coherence length $\xi_0=v_F/2\Delta$). With this assumption we can reduce Eq. (\[eq:eom\]) *inside the terminal* to its quasiclassical version (see Ref. [@rammer:86]) $$\label{eq:eilenberger-counting}
{\mathbf v_F }\nabla \check g({\mathbf{x}},{\mathbf v_F},t,t',\chi)
= \left[-i\frac{\chi}{2} (\nabla F({\mathbf x})) {\mathbf v_F} \check \tau_K
\,,\,\check g({\mathbf{x}},{\mathbf v_F},t,t',\chi) \right] \,.$$ Here $\check\tau_K=\bar\tau_3\hat\sigma_3$ is the matrix of the current operator and $\check g$ obeys the normalization condition $\check g^2=1$. Other terms can be neglected due to the assumptions we have made for $\Lambda$. The counting field can then be eliminated by the gauge-like transformation $$\check g({\mathbf x},{\mathbf v_F},t,t',\chi)
= e^{-i\chi F({\mathbf x})\check \tau_K/2}
\check g({\mathbf x},{\mathbf v_F},t,t',0)
e^{i\chi F({\mathbf x})\check \tau_K/2}\,.$$ We assume now that the terminal is a diffusive metal of negligible resistance. Then the Green’s functions are constant in space (except in the vicinity of the cross section C) and isotropic in momentum space. Applying the diffusive approximation [@rammer:86] in the terminal leads to a transformed terminal Green’s function $$\label{eq:countrot}
\check G(\chi) = e^{-i\chi\check \tau_K/2}
\check G(0) e^{i\chi\check \tau_K/2}\,,$$ on the right of the cross section $C$ (where $F({\mathbf x})=1$) with respect to the case without counting field. Consequently, the counting field is entirely incorporated into a *modified boundary condition* imposed by the terminal onto the mesoscopic system.
### Summary of Theoretical Approach
This concludes the theoretical approach to counting statistics of mesoscopic transport. Let us briefly summarize the scheme to follow. The FCS can be obtained by a slight extension of the usual Keldysh-Green’s function approach, which is widely employed to treat quantum transport problems. Making use of the separation of the mesoscopic structure into *terminals* and a *scattering* region, the formalism boils down to a very powerful, but nevertheless simple rule: we have to apply the *counting rotation* (\[eq:countrot\]) to a terminal, thus providing new boundary conditions (now depending on the *counting field* $\chi$) to the scattering problem. We then proceed ’as usual’ and calculated the current in the terminal, which again depends on $\chi$. Finally the counting statistics is obtained from Eq. (\[eq:chi-current\]).
Two-Terminal contacts {#sec:qpc}
=====================
### Tunnel contact
To illustrate the theoretical method we first calculate the counting statistics of a tunnel junction. As usual the system is described by a tunnel Hamiltonian $H=H_1+H_2+H_T$, where $H_{1(2)}$ describe the left(right) terminal and $H_T$ describes the tunneling. The current is calculated in second order in the tunneling amplitudes and we obtain $I(\chi)=\frac{G_T}{8e}\int dE \textrm{Tr}\left(
\check\tau_K\left[\check G_1(\chi),\check G_2\right]\right)$, where $G_T$ is the conductance of the tunnel junction and we have included the counting field in $\check G_1$. The CGF is (using $ (\partial/\partial\chi) G_1(\chi)=(i/2)\left[\check\tau_K,\check G_1(\chi)\right]$) $$S(\chi)=i\frac{t_0}{e}\int_0^\chi d\chi^\prime I(\chi^\prime)
= \frac{G_Tt_0}{4e^2} \int dE \textrm{Tr}\left\{
\check G_1(\chi),\check G_2\right\}\,,
\label{eq:cgf-tunnel-general}$$ which is the general expression for the FCS of a tunnel junction. We use the pseudo-unitarity $\check\tau_K^2=\check 1$ to write $$\label{eq:cgf-tunnel}
S(\chi) = N_{12}(e^{i\chi}-1)+N_{21}(e^{-i\chi}-1)\,,$$ where $ N_{ij} = (t_0G_T/16e^2)\int dE \textrm{Tr}
\left[(1+\check\tau_K) \check G_i (1-\check\tau_K)\check G_j\right]$ denotes the average number of charges tunnel from $i$ to $j$. The statistics is therefore a bidirectional Poisson distribution [@levitov:01-cumulant]. It is easy to see that the cumulants are $C_{n}=N_{12}+(-1)^n N_{21}$. If either $N_{21}=0$ or $N_{12}=0$ we obtain the Schottky limit. Furthermore, in equilibrium $N_{12}=N_{21}$ and the FCS is $(2G_Tk_BT t_0/e^2)(\cos(\chi)-1)$, which is non-Gaussian, remarkably.
### General CGF for quantum contacts
Using the method presented in the previous section, we can find the counting statistics for all conductors, which are characterized by a set of transmission coefficients $\{T_n\}$. Nazarov has shown that the transport properties of such a contact are described by a [*matrix current*]{} [@nazarov:99-circuit] $$\label{eq:matrix-current}
\check{I}_{12}=-\frac{e^2}{\pi}\sum_n
\frac{2T_n\left[\check G_1,\check G_2\right]}{
4+T_n\left(\{\check G_1,\check G_2\}-2\right)}\,.$$ Here $\check G_{1(2)}$ denote the matrix Green’s functions on the left and the right of the contact. We should emphasize that the matrix form of (\[eq:matrix-current\]) is crucial to obtain the FCS, since it is valid for any matrix structure of the Green’s functions. The *scalar current* is obtained from the matrix current by $$\label{eq:el-current}
I_{12}=\frac{1}{4e}\int dE {\rm Tr} \check\tau_K\check I_{12}\,.$$ To find the FCS, we apply the (\[eq:countrot\]) to terminal 1, i. e. $\check G_1$ becomes $\chi$-dependent. It turns out that the CGF can then be found generally from the relations (\[eq:chi-current\]), (\[eq:matrix-current\]), and (\[eq:el-current\]). To integrate (\[eq:chi-current\]) with respect to $\chi$, we need the relations $i(\partial/\partial\chi)\check G_1(\chi)=[\check\tau_K,\check
G_1(\chi)]$ and $\textrm{tr}[\check G_1(\chi),\{\check
G_1(\chi),\check G_2\}^n]=0$. We find [@belzig:01-super] $$\label{eq:cgf-two-terminal}
S(\chi)=\frac{t_0}{2\pi}\sum_n\int dE {\textrm{Tr}}
\ln\left[1+\frac{T_n}{4}
\left(\{\check G_1(\chi),\check G_2\}-2\right)\right]\;.$$ This is a very important result. It shows that the counting statistics of a large class of constrictions can be cast in a common form, independent of the contact types. Note, that Eq. (\[eq:cgf-two-terminal\]) is just the sum over CGF’s of all eigenchannels. Thus, we can obtain the CGF’s of all constrictions from a known transmission eigenvalue density. These are known for a number of generic contacts (see e.g. [@beenakker:97-rmp] and Table \[tab:cgf\]), can be determined numerically, or can be taken from experiment. Below we will discuss several illustrative examples for a single channel contacts.
$\displaystyle\rho(T) [G/G_Q] $ $\displaystyle \check s(\Lambda )$
--------------------- ------------------------------------------------------------ ---------------------------------------------------------
Single channel $\displaystyle\delta(T-T_1)$ $\displaystyle\ln(1-T_1(\Lambda-1)/2)$
Diffusive connector $ \displaystyle\frac{1}{2}\frac{1}{T\sqrt{1-T}}$ $\displaystyle\frac{1}{4}\textrm{acosh}^2(\Lambda)$
Dirty interface $\displaystyle\frac{1}{\pi}\frac{1}{T^{3/2}\sqrt{1-T}}$ $\displaystyle \sqrt{2(1+\Lambda)}$
Chaotic cavity $\displaystyle \frac{2}{\pi}\frac{1}{\sqrt{T}\sqrt{1-T}} $ $\displaystyle 4 \ln\left(2+\sqrt{2(1+\Lambda)}\right)$
: Characteristic functions of some generic conductors. The transmission eigenvalue densities are normalized to $G/G_Q$, where $G_Q=2e^2/h$ is the quantum conductance. The third column displays the CGF-density, which determines the CGF via $S(\chi)=(t_0G/4eh) \int
dE \textrm{tr}\check s(\{\check G_1(\chi),\check G_2\}/2)$.[]{data-label="tab:cgf"}
### Normal contacts
Consider first a single channel with transmission T between two normal reservoirs. They are characterized by occupation factors $f_{1(2)} =
[\exp((E-\mu_{1(2)})/k_B T_e)+1]^{-1}$ ($T_e$ is the temperature). We obtain the result [@levitov:93-fcs; @levitov:96-coherent] (see Appendix) $$\begin{aligned}
\label{eq:cgf-normalcontact}
S(\chi)=\frac{2t_0}{h}\int dE
\ln\left[1+T_{12}(E)\left(e^{i\chi}-1\right)
+T_{21}(E)\left(e^{-i\chi}-1\right)\right]\,.\end{aligned}$$ Here we introduced the probabilities $T_{12}=T
f_1(E)\left(1-f_2(E)\right)$ for a tunneling event from 1 to 2 and $T_{21}(E)$ for the reverse process. We see that the FCS (for each energy) is a trinomial of an electron going from left to right, from right to left, or no scattering at all. The [*counting factors*]{} $e^{\pm i\chi}-1$ thus correspond to single charge transfers from 1 to 2 (2 to 1).
At zero temperature and $\mu_1-\mu_2=eV\ge 0$ the argument of the energy integral is constant in the interval $\mu_1<E<\mu_2$ and we obtain the *binomial form* $S(\chi)=\frac{2et_0|V|}{h}
\ln\left[1+T \left(e^{i\chi}-1\right)\right]$. Note that for reverse bias $\mu_2>\mu_1$ the CGF has the same form, but with a counting factor $e^{-i\chi}-1$. The prefactor denotes the *number of attempts* $M=e t_0V/h $ to transfer an electron [^1]. If the transmission probability is unity the FCS is non-zero only for $N=M$, which therefore constitutes the maximal number of electrons occupying an energy strip $eV$ that can be sent through one (spin-degenerate) channel in a time interval $t_0$. In equilibrium it follows from Eq. (\[eq:cgf-normalcontact\]) that the counting statistics is [@levitov:03-noisebook] $$\label{eq:cgf-equilibrium}
S(\chi) = -\frac{2 t_0 k_B T_{el}}{h}
\textrm{asin}^2\left(\sqrt T \sin\frac{\chi}{2}\right)\,.$$ The fluctuations are non-Gaussian, except for $T=1$, when $S(\chi)=-\frac{t_0 k_B T_{el}}{h}\chi^2$.
### SN-contact
The FCS of a contact between a superconductor and a normal metal also follows from the general expression Eq. (\[eq:cgf-two-terminal\]). Using the Green’s functions given in the Appendix we find the result [@muzykantskii:94] $$\label{eq:cgf-sncontact}
S(\chi)=\frac{t_0}{2\pi}\sum_n\int dE
\ln\left[1+\sum\limits_{q=-2}^{2}
A_{nq}(E)\left(e^{iq\chi}-1\right)\right]\,.$$ The coefficients $A_{nq}(E)$ are related to a charge transfer of $q\times e$. For example, a term $\exp(2i\chi)-1$ corresponds just to an Andreev reflection process, in which two charges are transfered simultaneously. Explicit expressions for the various coefficients are given in Refs. [@muzykantskii:94; @belzig:03-book]. The most interesting regime is that of pure Andreev reflection: $eV,k_BT\ll\Delta$. Here we obtain $$\label{eq:cgf-andreev-general}
S(\chi)=\frac{t_0}{h}\int dE
\ln\left[1+R_A f_+f_-\left(e^{i2\chi}-1\right)+
R_A(1-f_+)(1-f_-))\left(e^{-i2\chi}-1\right)\right]\,,$$ where $R_A=T^2/(2-T)^2$ is just the Andreev reflection probability and $f_\pm=f(\pm E)$ denotes the occupation with electrons above(below) the chemical potential of the superconductor. For low temperatures $k_BT_{e}\ll eV \ll\Delta$, the CGF becomes $$S(\chi)=\frac{2et_0 |V|}{h}
\ln\left[1+R_A\left(e^{i2\chi}-1\right)\right]\,.
\label{eq:cgf-andreev}$$ The CGF is now $\pi$-periodic, which according to Sec. \[sec:fcs\] reflects that the charge transfer of an elementary event is now $2e$, a consequence of Andreev reflection. Quite remarkably, the statistics is again a simple binomial distribution. In equilibrium, we can adapt the result from Eq. \[eq:cgf-equilibrium\] to find $$S(\chi) = -\frac{2 t_0 k_B
T_{el}}{h} \textrm{asin}^2\left(\sqrt{R_A}\sin\chi\right)\quad
(\textrm{for} \chi\in[-\pi/2,\pi/2])\,.$$ The counting statistics is also non-Gaussian, except for $R_A=1$.
### Superconducting Contact {#sec:mar}
Now we turn to a slightly more involved problem: a contact between two superconductors biased at a finite voltage $V$. For $eV < 2 \Delta$ the transport is dominated by multiple Andreev reflections (MAR). The microscopic analysis of the average current and the shot noise calculations suggest that the current at subgap energies proceeds in “giant" shots, with an effective charge $q \sim e(1 + 2\Delta/|eV|)$. However, the question of size of the charge transfered in an elementary event can only be rigorously resolved by the FCS. The answer was given by Cuevas and the author [@cuevas:03-marfcs] based on a microscopic Green’s function approach. Independently, Johansson, Samuelsson and Ingerman [@johansson:03-marfcs] arrived at the same conclusion using a different method.
Now, what would we like to have? In Sec. \[sec:fcs\] we have discussed that one can speak of multiple charge transfers if the CGF allows an interpretation in terms of elementary events, which are described by counting factors $e^{in\chi}-1$, where $n$ denotes the charge transfered in the process. How can we ever hope to obtain this from the general formula (\[eq:cgf-two-terminal\])? We have to calculate the determinant of a 4$\times$4-matrix, which can give only factors of the type $e^{i2\chi}$ or even smaller charges. The answer to this puzzle is that we have to reinterpret the matrix structure in (\[eq:cgf-two-terminal\]), since the Green’s functions of superconductors at a finite bias voltage are essentially nonlocal in energy. The general result for the CGF can be written as $S(\chi)=\textrm{Tr}\ln \check Q$, where Tr$=\int_0^{t_0} dt$tr and $\check Q(t)=1+(T/4)(\{\check G_1\stackrel{\otimes}{,}\check
G_2\}-2)(t,t)$. Here $\check G_1\otimes\check
G_2(t,t^\prime)=\int dt^{\prime\prime} \check G_1(t,t^{\prime\prime}\check
G_2(t^{\prime\prime},t^\prime)$. Let us set the chemical potential of the right electrode to zero and represent the Green’s functions by $\check G_1(t,t^{\prime}) = e^{i eVt \bar \tau_3} \check
G_S(t-t^{\prime}) e^{-i eVt^\prime \bar \tau_3}$ and $\check
G_2(t,t^{\prime}) = \check G_S(t-t^{\prime})$. Here, we have not included the dc part of the phase, since it can be shown that it drops from the expression of the dc FCS at finite bias. The Fourier transform leads to a representation of the form $\check
G(E,E^{\prime}) = \sum_{n} \check G_{0,n}(E) \delta(E - E^{\prime}
+neV)$, where $n=0,\pm 2$. Restricting the fundamental energy interval to $E-E^\prime \in [0,eV]$ we can represent the convolution as *matrix product*, i.e. $(G_1 \otimes G_2) (E,E^{\prime})
\to (\check G_1 \check G_2)_{n,m} (E,E^{\prime}) = \sum_k
(G_1)_{n,k}(E,E^\prime) (G_2)_{k,m}(E,E^{\prime})$. The trace in this new representation is written as $\int_0^{eV} dE
\sum_n \textrm{Tr} \ln \left(\check Q\right)_{nn}$. In this way, the functional convolution is reduced to matrix algebra for the infinite-dimensional matrix $\check Q$. From these arguments it is clear that the statistics is a *multinomial* distribution of *multiple* charge transfers: $$\label{eq:marfcs}
S(\chi) = \frac{t_0}{h}\int_0^{eV} dE \ln \left[
1+\sum_{n=-\infty}^{\infty}P_n(E,V)\left(e^{in\chi}-1\right)\right]\,.$$ General expressions for the probabilities $P(E,V)$ have been derived in Ref. [@cuevas:03-marfcs].
Here we will pursue a different path and study a toy model. Let us neglect all set $f^{R,A}(|E|<\Delta)=1$, $g^{R(A)}(|E|>\Delta)=\pm 1$, and equal to zero otherwise. Physically, this means that we neglect Andreev reflections above the gap and replace the quasiparticle density of states by a constant $|E|>\Delta$. This simplifies the calculation a lot, since the matrix trace now becomes finite. Let us for example consider a voltage $eV=2\Delta/4$. In that case, we consider the determinant of the matrix $$\textrm{det}\left[1-\frac{\sqrt{T}}{2}
\left(
\begin{array}[c]{ccccc}
\hat Q_-(\chi) & 1 \\
1 & 0 & e^{- i\chi\hat\tau_3} \\
& e^{i\chi\hat\tau_3} & 0 & 1\\
& & 1 & 0 & e^{- i\chi\hat\tau_3} \\
& & & e^{i\chi\hat\tau_3} & \hat Q_+\\
\end{array}\right)\right]\,,$$ where $ Q_\pm(\chi)$ describe quasiparticle emission (injection) and off-diagonal pairs $e^{\pm\chi}$ are associated with Andreev reflection. Evaluating the determinant we find $S(\chi)=\frac{\Delta
t_0}{2h} \ln\left[1+P_5\left(e^{in\chi}-1\right)\right]$, where $P_5=T^5/(16-20T+5T^2)^2$. This expression describes binomial transfers of $5$ charges with probability $P_5$. For general subharmonic voltages $2\Delta/(n-1)$ we find $$S(\chi)=\frac{2\Delta t_0}{(n-1)h}
\ln\left[1+P_n\left(e^{in\chi}-1\right)\right]\,,$$ where the probabilities are given by $$\begin{array}{l}
P_2 = \frac{T^2}{(2-T)^2} \,,\,
P_3 = \frac{T^3}{(4-3T)^2} \,,\,
P_4 = \frac{T^4}{(8-8T+T^2)^2}\,,\,
P_5 = \frac{T^5}{(16-20T+5T^2)^2}\\
P_6 = \frac{T^6}{(2-T)^2(16-16T+T^2)^2}\,,
P_7 = \frac{T^7}{(64-112T+56T^2-7T^3)^2}\,.
\end{array}$$ Note the limiting cases of these probabilities $P_n\sim T^n/4^{n-1}$ for $T\ll 1$ and $P_n=1$ for $T\to 1$. We conclude this section by saying that the general results for the CGF [@cuevas:03-marfcs] allow for a fast and efficient calculation of all dc-transport properties of contacts between superconductors (which may contain magnetic impurities, phonon broadening or other imperfections).
Quantum Noise in Diffusive SN-Structures {#sec:diff}
========================================
In this section, we illustrate a further advantage of the Keldysh-Green’s functions approach to counting statistics. We consider a normal metallic diffusive wire connected on one end to a normal metal reservoir and on the other side to a superconductor. The wire is supposed to have a mean free path $l\gg\lambda_F$, a corresponding diffusion coefficient $D=v_F l/3$, and a length $L$. For $eV,k_BT\ll\Delta$ the transport occurs through Andreev reflection at the interface to the superconductor. This system shows a quite remarkable property, which is the so-called reentrance effect of the conductance. The energy difference $2E$ of electron-hole pairs leads to a dephasing on a length scale $\xi_E=\sqrt{D/2E}$. This has the consequence that the (otherwise) normal wire becomes partially superconducting and the conductance increases with decreasing energy. However, once the coherence length $\xi_E$ reaches $L$ the conductance *decreases* again. Finally for $E=0$ the conductance is *exactly* equal to the conductance in the normal state. This is the reentrance effect occurring at an energy of the order of the Thouless energy $E_c=\hbar D/L^2$. In Fig. \[fig:diff\] (left panel, dotted curve) the resulting differential conductance at zero temperature is plotted.
The transport in this system is described by a matrix diffusion equation for the Keldysh Green’s functions, the so-called Usadel equation $$\label{eq:usadel}
-\frac{D}{\sigma}\nabla \check I =
\left[ -i E\hat\tau_3 , \check G \right]\;,\;
\check I = -\sigma \check G \nabla\check G\,.$$ In these equations $\sigma=2e^2N_0D$ is the conductivity. The boundary conditions for this equation are that the Green’s functions in the terminal approach the bulk solution $\check G_N$ or $\check G_S$, respectively. This equation is in general difficult to solve, even if one is interested in the average current only. However, we can calculate the noise and the counting statistics using the recipe outlined in Sec. \[sec:theory\] and obtain the noise in the full parameter range of Eq. (\[eq:usadel\]).
Before considering Eq. (\[eq:usadel\]) in its full generality, we consider the limiting cases of low and high energies (compared to $E_c$). For $E=0$ the r.h.s. is absent and the system is completely analogous to a diffusive connector as discussed in \[sec:qpc\]. From Table \[tab:cgf\] and using the eigenvalues (\[eq:sneigenvalues\]) we find $$\label{eq:cgf-diffusive-andreev}
S(\chi) = \frac{t_0 G}{16e^2} \int dE\textrm{acosh}^2\left[2\left(f_+f_-(e^{2i\chi}-1) +
(1-f_+)(1-f_-)(e^{-2i\chi}-1)\right)-1\right].$$ This result shows, once again, that the charges are transfered in pairs. It is interesting to compare with the CGF for a diffusive wire between two normal metals, for which we obtain [@levitov:96-diffusive; @yuli:99-annals] $$\label{eq:cgf-diffusive-normal}
S(\chi) = \frac{t_0 G}{4e^2} \int dE
\textrm{acosh}^2\left[2\left(f_1(1-f_2)(e^{i\chi}-1) +
f_2(1-f_1)(e^{-i\chi}-1)\right)-1\right].$$ We see that the only difference in the CGF between the SN- and the NN-case is in the counting factors, and a prefactor $1/4$. Note, that this coincidence only occurs for the diffusive connector, but is by no means a general rule. At zero temperature the results simplify and we find $$\label{eq:cgf-diff-zerotemp}
S^{\textrm{SN}}(\chi)=\frac12 S^{\textrm{NN}}(2\chi) \; ,
\;
S^{\textrm{NN}}(\chi)=\frac{t_0 G V}{4e} \textrm{acosh}^2\left(2e^{i\chi}-1\right)\,,$$ a surprising simple relation between the CGF for the Andreev wire and the normal diffusive wire. It is easy to see that the cumulants obey the general relation $ C_n^{\rm SN}=2^{n-1} C_n^{\rm NN}$. We observe that we can read off the effective charge from the ratio $C_n^{\rm
SN}/C_n^{\rm NN}$ $=$ $(q_{eff}/e)^{n-1}$ and, indeed, find $q_{eff}=2e$. This result for the effective charge is a special property of the *diffusive connector*.
At energies large compared to $E_c$ it is also possible to find the CGF for the Andreev wire in general. Then the proximity effect in the wire is absent and it turns out [@belzig:03-incoherent] that the wire can be effectively mapped on a normal circuit, consisting of two identical wires in series to which twice the voltage is applied and twice the counting field. Thus, for $E \gg E_c$ we obtain $S^{SN}(\chi)$ from $S^{NN}(\chi)$ by the replacement $\chi\to2\chi$ and $G\to G/2$, which exactly brings us to Eq. \[eq:cgf-diffusive-andreev\] and shows that the counting statistics is again the same in the incoherent limit.
![Noise in diffusive SN-systems. Left panel a): the differential conductance and the noise show a reentrant behavior. The effective charge, defined as $q_{eff}(E)=(3/2)dS/dI$ reveals that the correlated Andreev pair transport suppresses the noise below the uncorrelated Boltzmann-Langevin result $2e$. Right panels b) and c): Effective charge of the Andreev interferometer shown in the inset (realized experimentally in Ref. [@reulet:03-diff]). The upper panel b) shows the theoretical predictions and the lower panel c) the experimental results. The theoretical results contain no fitting parameter (the Thouless energy $E_c=30\mu$eV was extracted from the sample geometry and the experimental temperature of $43$mK was included in the calculation). Therefore, it is reasonable that the deviations between experimental and theoretical results comes from possible heating effects in the experiment, which are not accounted for in the theoretical calculation.[]{data-label="fig:diff"}](wire_comb){width="90.00000%"}
The full quantum-mechanical calculation of the energy-dependent shot noise can be performed on the basis of the approach of Sec. \[sec:theory\] [@belzig:01-diff]. We expand up to linear order in $\chi$, i.e. $\check G(\chi)=\check G_0-i(\chi/2) \check
G_1$ and $\check I(\chi)=\check I_0-i(\chi/2) \check I_1$. Substituting in (\[eq:usadel\]) we find $$\frac{D}{\sigma}\frac{\partial}{\partial x}\check I_1 =
\left[-iE\bar\tau_3\,,\,\check G_1\right]\,,\,
\check I_1 =
- \sigma \left( \check G_0\frac{\partial}{\partial x}\check G_1
+ \check G_1\frac{\partial}{\partial x}\check G_0\right).
\label{eq:usadelnoise}$$ The boundary conditions at the reservoirs read $\check
G_1(0)=\left[\check\tau_{\text{K}},\check G_{\text{L}}\right]$ at the left end and $\check G_1(L)=0$ at the right end. Finally the noise is $S_{\text{I}}=-e\int dE \text{Tr}\check\tau_{\text{K}}\check I_1(x)$. By taking the trace of Eq. (\[eq:usadelnoise\]) multiplied with $\check\tau_{\text{K}}$ it follows that it does not matter, where the noise is evaluated, as it should be. From these equations the generalization of the Boltzmann-Langevin equation to superconductors can be derived [@pistolesi:03], which allows for a faster numerical solution. The results for the energy dependent noise is shown in the left panel Fig. \[fig:diff\]. A direct comparison of the differential shot noise and the differential conductance (for zero temperature) shows the difference in the energy dependence. The effective charge defined as $q_{eff}=(3/2) dS/dI$ displays the clear deviation of the quantum noise from the Boltzmann-Langevin result of $2e$. At energies below the Thouless energy $E_c$ the effective charge is suppressed below $2e$. This shows that the correlated Andreev pair transport suppresses the noise below the uncorrelated Boltzmann-Langevin result.
To experimentally probe the pair correlations in diffusive superconductor-normal metal-heterostructures it is most convenient to use an Andreev interferometer. An example is shown in the left part of Fig. \[fig:diff\]. A diffusive wire connected to a normal terminal is split into two parts, which are connected to two different points of a superconducting terminal. By passing a magnetic flux through the loop one can effectively vary the phase difference between the two connections to the superconductor. Such a structure has been experimentally realized by the Yale group [@reulet:03-diff]. In Fig. \[fig:diff\] we present a direct comparison between our theoretical predictions and the experimentally obtained effective charge. Note, that we have included the experimental temperature in the theoretical modeling. The finite temperature explains the strong decrease of the effective charge in the regime $|eV|\leq k_BT$, where the noise is fixed by the fluctuation-dissipation theorem. The disagreement between theory and experiment in this regime stems solely from differences in the measured temperature-dependent conductance from the theoretical prediction. We attribute this to heating effects. The qualitative agreement in the shot-noise regime $|eV|\geq k_BT$ is satisfactory, if one takes into account, that we have no free parameters for the theoretical calculation. Both, experiment and theory show a suppression of the effective charge for some finite energy, which is of the order of the Thouless energy and depends on flux in a qualitative similar manner. Remarkably for half-integer flux the effective charge is completely flat, in contrast to what one would expect from circuit arguments based on the conductance distribution in the fork geometry. Currently we have no explanation for this behavior, and therefore more work is needed in this direction.
Multi Terminal Circuits {#sec:split}
=======================
In circuits with more than two terminals it is of particular interest to study non-local correlations of currents in different terminals. For that purpose we need a slight extension of the theoretical approach of Sec. \[sec:theory\], suitable for multi-terminal circuits. We will now introduce this method.
### Circuit Theory {#sec:circuit}
To study transport in general mesoscopic multi-terminal structures the so-called circuit theory for quantum transport was developed by Nazarov [@nazarov:99-circuit; @yuli:94-circuit]. Its main idea, borrowed from Kirchhoff’s classical circuit theory, is to represent a mesoscopic device by discrete elements, which resemble the known elements of electrical transport. We briefly repeat the essentials of the circuit theory. Topologically, one distinguishes three elements: terminals, nodes and connectors. Terminals are the connections to the external voltage or current sources and provide boundary conditions, specifying externally applied voltages, currents or phase differences in the case of superconductors. The actual circuit is represented by a network of nodes and connectors, the first determining the approximate layout and the second describing the connections between different nodes, respectively.
To describe quantum effects it is necessary to represent the variables describing a node by *matrix Green’s function* $\check G$, which can be either Nambu or Keldysh matrices, or a combination thereof. Consequently, we describe the current through a connector by a *matrix current* $\check I$, which relates the fluxes of all elements of $\check G$ on neighboring nodes. The current has been derived by Nazarov [@nazarov:99-circuit] and is given by Eq. (\[eq:matrix-current\]) for a connector, characterized by a set of transmission coefficients $\{T_n\}$. Note that the *electrical current* is obtained from $ I_{12}=\frac{1}{4e}\int dE {\rm Tr}
\check\tau_K\check I_{12}$. The boundary condition are given in terms of fixed matrix Green’s functions $\check G_i$, which are determined by the applied potential, the temperature, the type of lead, and a counting field $\chi_i$.
Once the network is determined and all connectors are specified, the transport properties can be found by means of the following circuit rules. We associate an (unknown) Green’s function $\check G_j$ to each node $j$. The two rules are
1. $\check G^2_j=\check 1$ for the Green’s functions of all internal nodes $j$.
2. the total matrix current in a node is conserved: $\sum_i \check
I_{ij}=0$, where the sum goes over all nodes or terminals connected to node $j$ and each matrix current is given by (\[eq:matrix-current\]).
Finally, the observable currents into the terminals are given by $I_i=\sum_j I_{ij}$, where the sum runs over all nodes connected to the terminal $i$. To obtain the counting statistics, we finally integrate all currents $I_i(\vec\chi)=(\partial/\partial\chi_i)S(\vec\chi)$ to find the CGF $S(\vec\chi)$.
![Multi tunnel junction structure: a) general setup with K terminals connected to a common node. b) beam splitter setup in which terminal 3 is either a normal metal or a superconductor.[]{data-label="fig:multitunnel"}](multi-tunnel){width="8cm"}
### Multi tunnel junction structure
A general expression of $S(\vec\chi)$ can be obtained for a system of an arbitrary number of terminals connected to one common node by tunnel contacts, see Fig. \[fig:multitunnel\] [@yuli:01-multi; @boerlin:02]. At the same time it nicely demonstrates the application of the circuit theory rules, presented above. Let us denote the unknown Green’s function of the central node by $\check G_c(\vec\chi)$. The matrix current from a terminal $\alpha$ ($\alpha=1,\ldots,K$) into the central node is given by the relation $$\check I_\alpha(\vec\chi) = \frac{g_\alpha}{2} \left[ \check
G_c(\vec\chi) , \check G_\alpha(\chi_\alpha) \right]\,,$$ where $g_\alpha=G_Q\sum_n T_n$ is the conductance of the respective tunnel junction junction, for which we have assumed that all $T_n\ll
1$ and $g_\alpha\gg G_Q$ to avoid Coulomb blockade. The Green’s function of the central node is determined by matrix current conservation, reading $ \sum_{\alpha=1}^K \check I_\alpha = [
\sum_{\alpha=1}^K g_\alpha \check G_\alpha , \check G_c ]/2=0$. Employing the normalization condition $\check G_c^2=1$, the solution is $$\check G_c(\vec\chi) = \frac{\sum_{\alpha=1}^K g_\alpha \check G_\alpha(\chi_\alpha)}{
\sqrt{\sum\nolimits_{\alpha,\beta=1}^K g_\alpha g_\beta
\left\{\check G_\alpha(\chi_\alpha),\check G_\beta(\chi_\beta)\right\}}}\,.$$ To find the cumulant-generating function (CGF) $S(\vec\chi)$ we integrate the equations $\partial S(\vec\chi)/\partial
\chi_\alpha = (-it_0/4e^2) \int dE \mbox{Tr}\check\tau_K\check
I_{\alpha}(\vec\chi)$ [@nazarov:02-multi]. We obtain $$\label{eq:cgf-general-multi}
S(\vec\chi)
= \frac{t_0}{2e^2}\int dE \mbox{Tr}
\sqrt{\sum\nolimits_{\alpha,\beta=1}^M g_\alpha g_\beta
\left\{\check G_\alpha(\chi_\alpha),\check G_\beta(\chi_\beta)\right\}}\,.$$ This is the general result for an M-terminal geometry in which all terminals are tunnel-coupled to a common node. It is valid for arbitrary combinations of normal metal and superconductor, fully accounting for the proximity effect. Note, that we have dropped the normalization of $S(\vec\chi)$ to write the expression more compact.
### Normal metals
If all terminals are normal metals, the matrices in Eq. (\[eq:cgf-general-multi\]) are all diagonal and trace is trivial. We obtain $$\label{eq:cgf-multi-normal}
S(\vec\chi)= \frac{t_0}{ 2 e^2 } \int dE
\sqrt{g_\Sigma^2 + \sum_{\alpha \neq \beta} g_\alpha g_\beta
f_\alpha(E)(1-f_\beta(E))\left(e^{i(\chi_\alpha-\chi_\beta)}-1\right)}$$ where $f_\alpha$ is the occupation function of terminal $\alpha$. Here, we introduced the abbreviation $g_{\Sigma}=\sum_{\alpha=1}^Ng_\alpha$ for the sum of all conductances. We note, that the statistics is essentially non-Poissonian, despite the fact the we are considering tunnel junctions.
We now restrict us to two terminals (in which case we have to consider only one counting field $\chi=\chi_1-\chi_2$). For zero temperature and voltage bias $V$ the CGF reads then $$\label{eq:cgf-double-tunnel-general}
S(\chi)=\frac{t_0 V}{2e}\sqrt{g_\Sigma^2+ 4 g_1g_2(e^{i\chi}-1)},$$ the result for a double tunnel junction first obtained by de Jong [@dejong:96] using a master equation approach. We obtain as limiting cases for an asymmetric junction (either $g_1\ll
g_2$ or $g_1\gg g_2$) Poisson statistics $S(\chi)=(t_0V
g_1g_2/(g_1+g_2))(\exp(i\chi)-1)$.
Next we consider a three terminal structure, which is voltage biased such that the mean current $\bar{I}_3$ in lead $3$ vanishes (voltage probe) and a transport current $\bar I=g_1g_2/(g_1+g_2)V$ flows between terminals 1 and 2. The CGF is [@boerlindiplom] $$\begin{aligned}
\label{eq:ntgtn_s}
S(\vec\chi) & = & \frac{t_0 |V|}{2e}\left(
g_2 \sqrt{g_\Sigma^2+ 4g_3g_1(e^{-i\chi_1}-1) +
4g_1g_2(e^{i\chi_2-i\chi_1}-1)}\right.\nonumber\\
&&\left. +g_1\sqrt{g_\Sigma^2+ 4g_3g_2
(e^{i\chi_2}-1) + 4g_1g_2(e^{i\chi_2-i\chi_1}-1)}\right)\,.\end{aligned}$$ It is interesting to note that the presence of the voltage probe makes the CGF asymmetric under the transformation $g_1\leftrightarrow g_2$, whereas the current is symmetric. In certain limits in which the square roots in Eq. \[eq:ntgtn\_s\] can be expanded one is able to find the counting statistics. E. g in the strong-coupling limit $g_3\gg
(g_1+g_2)$ we find $$S(\vec\chi) = \bar{N}\left[ e^{-i\chi_1}
+ e^{i\chi_2}-2\right] \label{eq:ntgtn_s_lim1}\,.$$ The CGF is simply the sum of two Poisson distribution, demonstrating drastically the effect of the voltage probe. It completely suppresses the correlation between electrons entering and leaving the central node.
Another interesting geometry is a beam splitter configuration, in which a voltage bias is applied between one terminal and the other two. We find $$S^N(\chi_1,\chi_2)=\frac{t_0|V|}{2e}
\sqrt{g_\Sigma^2+g_1g_3\left(e^{i\chi_1}-1\right)
+ g_3g_2\left(e^{i\chi_2}-1\right)}\,.
\label{eq:cgf-splitter-normal}$$ In the limit that $g_1+g_2$ and $g_3$ are very different, we can expand the CGF and find for the CGF $S(\chi)=N_1e^{i\chi_1}+N_2e^{i\chi}$, i. e., the tunneling processes into the two terminals are uncorrelated. The corresponding probability distribution is simply the product of two Poisson distributions.
### SN-contact
We now consider the case of a double tunnel junction, in which one of the terminals is superconducting. From the general result (\[eq:cgf-general-multi\]) and (\[eq:sneigenvalues\]) we find after some algebra $$\label{eq:cgf-double-tunnel-andreev}
S(\chi)=\frac{t_0|V|}{e\sqrt 2}
\sqrt{g_1^2+g_2^2+\sqrt{\left(g_1^2+g_2^2\right)^2
+4g_1^2g_2^2(e^{i2\chi}-1)}}\,.$$ Remarkably, the statistics is fundamentally different from the corresponding normal case (\[eq:cgf-double-tunnel-general\]). Still, the elementary events are transfers of pairs of electrons, which, however are correlated in a more complicated way than normal electrons. If the junction is very asymmetric, the FCS reduces to Poissonian transfer of electron pairs. This is similar to the effect of decoherence between electrons and holes for energies of the order of the Thouless energy [@samuelsson:03-counting].
For the beam splitter configuration we are also able to find the FCS analytically. The CGF is [@boerlin:02] $$\begin{aligned}
\label{eq:cgf-splitter-andreev}
\lefteqn{S(\chi_1,\chi_2)=
\frac{Vt_0}{\sqrt{2}e}\times}\\\nonumber
&&
\sqrt{g_S^2+\sqrt{g_S^4+
4g_3^2g_1^2(e^{i2\chi_1}-1)+
4g_3^2g_2^2(e^{i2\chi_2}-1)+
8g_3^2g_1g_2(e^{i(\chi_1+\chi_2)}-1)}},\end{aligned}$$ where we abbreviated $g_S^2=g_3^2+(g_1+g_2)^2$. From this result we see that the elementary processes are now double charge transfers to either terminal of a splitting of a Cooper pair among the two terminal. It is interesting to note, that, if we assume that $g_1+g_2$ and $g_3$ are very different (but $g_1\approx g_2$), we obtain non-separable statistics $$\label{eq:cgf-splitter-andreev-limit}
S(\chi)=N_{11} e^{i2\chi_1}+N_{22} e^{i2\chi_2}+N_{12} e^{i(\chi_1+\chi_2)}\,.$$ This expression can not be written as a sum of two independent terms. Furthermore, the last term is positive, which implies that current crosscorrelation $S_{12}=-(2e^2/t_0)
(\partial^2/\partial\chi_1\partial\chi_2)
S(\chi_1,\chi_2)|_{\chi_1,\chi_2\to 0}$ are *positive*. Eq. (\[eq:cgf-splitter-andreev-limit\]) provides a simple explanation for this surprising effect: it is a consequence of independence of the different events, contributing to the current. This result, in fact, holds for a large class of superconducting beam splitters [@belzig:03-incoherent; @samuelsson:02-cavity; @samuelsson:02-crosscorrelation; @taddei:02-ferro].
Conclusion {#sec:sum}
==========
We have tried to give a pedagogical introduction to the field of counting statistics. Many technical details were left out, but we have tried to cover the essence of the derivation and concentrated on looking at concrete examples. For a more thorough study we recommend the recent book *Quantum Noise in Mesoscopic Physics* [@nazarov:03-book] or the original literature. While a number of aspects have already been explored, many open questions remain, e. g., experimental strategies to measure FCS, strongly interacting systems, or spin-dependent problems. For the future, we expect even more activity in the field and, consequently, even more interesting results will emerge.
### Acknowledgement
The ideas presented here are the results of numerous discussions with many people. In particular I would like to mention D. Bagrets, C. Bruder, J. C. Cuevas, Yu. V. Nazarov, and P. Samuelsson. This work was supported by the Swiss NSF and the NCCR Nanoscience.
Appendix
========
We summarize here the matrix-Green’s function for superconducting and normal contact, as they were used in the text. The time-dependent Green’s functions are expressed by their Fourier transforms $\check
G_0(t-t^\prime) = \int (dE/2\pi)$ $e^{-iE(t-t^\prime)}$$\check G_0(E)$. The energy-dependent Green’s functions in the Keldysh$\times$Nambu-space have the form $$\label{eq:reservoir}
\check G(E)= \left( \begin{array}[c]{cc}
(\bar A - \bar R) \bar f + \bar R & (\bar A - \bar R) \bar f \\
(\bar A - \bar R) (1 - \bar f) & (\bar R - \bar A) \bar f + \bar A
\end{array}\right)\,,$$ where the advanced, retarded and occupation Nambu matrices are $$\label{eq:nambu-matrices}
\bar A(\bar R)=
\left( \begin{array}[c]{cc}
g_{A(R)} & f_{A(R)} \\
f_{A(R)} & -g_{A(R)} \\
\end{array}\right)
\quad,\quad
\bar f(E)=
\left(\begin{array}[c]{cc}
f(E) & 0 \\
0 & f(-E)
\end{array}\right)\,.$$ The phase $\varphi$ of the superconducting order parameter as well as the electrical potential $eV$ enter via the gauge transformation $\check
G(t,t^\prime)=\check U(t) \check G_0(t-t^\prime) \check
U^\dagger(t^\prime)$. Here $\check U(t)=\exp\left[i\phi(t)\bar\tau_3/2\right]$, where $\phi(t)=\varphi+eVt$.
In the calculation of the FCS of contacts between normal metals and superconductors we frequently need the eigenvalues of anticommutators of two Green’s functions. For two normal metals $\{\check
G_{N1}(\chi),\check G_{N2}\}/2$ is diagonal and the eigenvalue is $$\label{eq:normaleigenvalues}
\left[1+2f_{1}(E)\left(1-f_{2}(E)\right)\left(e^{i\chi}-1\right)
+2f_{2}(E)\left(1-f_{1}(E)\right)\left(e^{-i\chi}-1\right)\right],$$ for the electron block and the same expression with $E\to -E$ for the ’hole’-block in Nambu space.
In the case of Andreev reflection, i. e. for $eV,k_BT_{el}\ll\Delta$, we find for $\{\check G_{N}(\chi),\check G_{S}\}/2$ the two eigenvalues $$\label{eq:sneigenvalues}
\pm \sqrt{f_N(E)f_N(-E)\left(1-e^{i2\chi}\right)+
(1-f_N(E))(1-f_N(-E))\left(1-e^{-i2\chi}\right)}.$$
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|
---
author:
- 'E. Villaver'
- 'A. Niedzielski'
- 'A. Wolszczan'
- 'G. Nowak'
- 'K. Kowalik'
- 'M. Adamów'
- 'G. Maciejewski'
- 'B. Deka-Szymankiewicz'
- 'J. Maldonado'
bibliography:
- 'villaver.bib'
date: 'Received;accepted'
subtitle: ' [**A**]{} Massive Jupiter orbiting the very low metallicity giant star BD+03 2562 [**and a possible planet around**]{} HD 103485.'
title: 'Tracking Advanced Planetary Systems (TAPAS) with HARPS-N. V. [^1] [^2] '
---
[We present two evolved stars (and ) from the TAPAS (Tracking Advanced PlAnetary Systems) with HARPS-N project devoted to RV precision measurements of identified candidates within the PennState - Toruń Centre for Astronomy Planet Search.]{} [Evolved stars with planets are crucial to understand the dependency of the planet formation mechanism on the mass and metallicity of the parent star and to study star-planet interactions.]{} [The paper is based on precise radial velocity (RV) measurements. For we collected 57 epochs over 3317 days with the Hobby-Eberly Telescope and its High Resolution Spectrograph and 18 ultra-precise HARPS-N data over 919 days. For we collected 46 epochs of HET data over 3380 days and 19 epochs of HARPS-N data over 919 days.]{} [We present the analysis of the data and the search for correlations between the RV signal and stellar activity, stellar rotation and photometric variability. Based on the available data, we interpret the RV variations measured in both stars as Keplerian motion. Both stars have masses close to Solar (1.11and 1.14), very low metallicities (\[Fe/H$]= -0.50$ and $-0.71$ for and ), and, both have Jupiter planetary mass companions ($m_2\sin i\thinspace=\thinspace7$ and 6.4 for and resp.), in close to terrestrial orbits (1.4 au and 1.3 au ), with moderate eccentricities ($e=0.34$ and 0.2 for and ). However, we cannot totally exclude that the signal in the case of HD 103485 is due to rotational modulation of active regions.]{} [Based on the current data, we conclude that BD+03 2562 has a bona fide planetary companion while for HD 103485 we cannot totally exclude that the best explanation for the RV signal modulations is not the existence of a planet but stellar activity. If, the interpretation remains that both stars have planetary companions they represent systems orbiting very evolved stars with very low metallicities, a challenge to the conditions required for the formation of massive giant gas planets.]{}
Introduction
============
With the discovery of 51 Peg [@Mayor1995] we arrived to the realisation that planet formation as we understood it had to be revised to account for the existence of “Hot Jupiters”. Since then, every planetary system discovered has added to our understanding of the physics of planet formation (see e.g. ). In this regard, planets orbiting evolved stars hold the key to several processes related not only to how planet formation operates around stars more massive than the Sun but also to understand star-planet interactions [@Villaver2007; @Villaver2014; @Privitera2016c; @Privitera2016a]. In this context, planets around evolved stars have revealed a lack of hot Jupiters that most likely reflect effects induced by stellar evolution [@VillaverLivio2009; @Mustill2012; @Villaver2014; @Privitera2016b].
In the main sequence, the presence of giant planets has shown to be very sensitive to the metallicity \[Fe/H\] of the host star [@Gonzalez1997; @Santos2004; @FischerValenti2005]. The precise functional form of the correlation still remain elusive, despite the fact it is one of the fundamental parameters to help constrain the planet formation models (see e.g. @Mordasini2012). But most important, the evolved hosts of planets have shown to present some chemical peculiarities with respect to their main sequence counterparts (see e.g. @Pasquini1997 [@daSilva2006; @Ghezzi2010; @Maldonado2013; @Mortier2013; @Jofre2015; @Jones2014; @Reffert2015; @Maldonado2016]). In particular, the planet occurrence rate does seem to depend on both stellar mass and stellar metallicity [@Maldonado2013] and cannot be explained by sample contamination [@Maldonado2016] as it has been argued by [@Reffert2015].
The established picture of giant planet formation, the basis of the core accretion model [@Perri1974; @Cameron1978; @Mizuno1980; @Pollack1996], begins with the building of km-sized or larger planetesimals from the growth of 1$-$10 mm pebbles (see e.g. @Youdin2011 [@Simon2016]). The planetesimals growth continues until a solid core, big enough for gravity to accrete gas from the protoplanetary disk, is formed. The metallicity dependency in the core accretion model thus comes from the need of a fast core growth before disk dissipation occurs [@Ida2004]. Furthermore, it has been shown that the alternative scenario, giant planet formation via gravitational instability in the protoplanetary disk [@Boss1997; @Mayer2002; @Boss2004] does not carry a metallicity dependency that can explain the observed relation.
Over the last $\approx 10$ years we have embarked in a quest for substellar/planetary companions to giant stars that started with the [PennState - Toruń Centre for Astronomy Planet Search]{} (PTPS, @Niedzielski2007 [@NiedzielskiWolszczan2008; @Niedzielski2015b; @Niedzielski2016a]) program and has continued with the high precision RV follow up of previous selected PTPS candidates program [Tracking Advanced Planetary Systems (TAPAS) program with HARPS-N]{} [@Niedzielski2015a; @Adamow2015; @Niedzielski2016b; @Niedzielski2016c]. In this paper, we present the latest finding of our TAPAS program: two very evolved giant stars with very low metallicities that host [**a**]{} massive “warm” Jupiters and a possible one and thus represent rather extreme outliers to the general planet-metallicity relation.
The paper is organised as follows: a summary of the observations, radial velocity and activity measurements is given in Section \[observations\] together with a description of the general procedure and the basic properties of the two stars; in Section \[rot\] and \[phot\] we show the analysis of the stellar rotation and photometry together with a discussion of the activity indicators; in Section \[results-g\] we present the Keplerian analysis of the radial velocity measurements, and our results are summarised and further discussed in Section \[conclusions\].
Observations, radial velocities, line bisectors and activity indicators\[observations\]
=======================================================================================
HD 103485 (BD+02 2493) and BD+03 2562 (TYC 0276-00507-1) belong to a sample of about 300 planetary or brown dwarf (BD) candidates identified from a sample of about 1000 stars searched for radial velocity (RV) variations with the 9.2m Hobby-Eberly Telescope (HET, @Ramsey1998). The full sample have been monitored since 2004 using the High-Resolution Spectrograph (HRS, @Tull1998) at HET within the PTPS program. Targets were selected for a more intense precise RV follow-up within the TAPAS program with the High Accuracy Radial velocity Planet Searcher in the North hemisphere (HARPS-N, @Cosentino2012).
The spectroscopic observations presented in this paper are thus a combination of data taken with the HRS at HET in the queue scheduled mode [@Shetrone2007], and with HARPS-N at the 3.58 meter Telescopio Nazionale Galileo (TNG).
For HET HRS spectra we use a combined gas-cell [@MarcyButler1992; @Butler1996], and cross-correlation [@Queloz1995; @Pepe2002] method for precise RV and spectral bisector inverse slope (BIS) measurements, respectively. The implementation of this technique to our data is described in [@Nowak2012] and [@Nowak2013].
HARPS-N radial velocity measurements and their uncertainties as well as BIS measurements were obtained with the standard user pipeline, which is based on the weighted CCF method [@1955AcOpt...2....9F; @1967ApJ...148..465G; @1979VA.....23..279B; @Queloz1995; @Baranne1996; @Pepe2002], using the simultaneous Th-Ar calibration mode of the spectrograph and the K5 cross-correlation mask.
A summary of the available data for HD 103485 and BD+03 2562 is given in Tables \[Parameters1\] and \[Parameters2\] respectively.
Parameter value reference
-------------------------------------------- ------------------------ -------------------------
$V$ \[mag\] 8.28$\pm$0.01 [@Hog2000]
$B-V$ \[mag\] 1.56$\pm$ 0.03 [@Hog2000]
$(B-V)_0$ \[mag\] 1.395 [@Zielinski2012]
$M_\mathrm{V}$ \[mag\] -2.51 [@Zielinski2012]
$T_{\mathrm{eff}}$ \[K\] 4097$\pm$20 [@Zielinski2012]
$\log g$ 1.93$\pm$0.08 [@Zielinski2012]
$[Fe/H]$ -0.50$\pm$0.09 [@Zielinski2012]
RV $[\kms]$ 27.56$\pm$0.08 [@Zielinski2012]
$v_{\mathrm{rot}} \sin i_{\star}$ $[\kms]$ 2.9$\pm$0.4 [@Adamow2014]
$\ALi $ $ -0.84$ [@Adamow2014]
$[$O/H$]$ -0.40$\pm$0.31 [@Adamow2014]
$[$Mg/H$]$ -0.27$\pm$0.15 [@Adamow2014]
$[$Al/H$]$ -0.10$\pm$0.09 [@Adamow2014]
$[$Ca/H$]$ -0.54$\pm$0.18 [@Adamow2014]
$[$Ti/H$]$ -0.07$\pm$0.25 [@Adamow2014]
$M/M_{\odot}$ 1.11$\pm$0.21 [@Adamczyk2015]
$\log L/L_{\odot}$ 2.51$\pm$0.13 [@Adamczyk2015]
$R/R_{\odot}$ 27.37$\pm$6.69 [@Adamczyk2015]
$\log \mathrm{age}$ \[yr\] 9.79$\pm$0.25 [@Adamczyk2015]
$d$ \[pc\] 1134$\pm$ 145 calculated from M$_{V}$
$V_{\mathrm{osc}}$ \[$\ms$\] 68.2$^{+52.1}_{-28.2}$ this work
$P_{\mathrm{osc}}$ \[d\] 2.2$^{+2.0}_{-1.1}$ this work
$P_{\mathrm{rot}}/ \sin i_{\star}$ \[d\] 477$\pm$134 this work
: Summary of the available data on HD 103485
\[Parameters1\]
Parameter value reference
-------------------------------------------- ------------------------- -------------------------
$V$ \[mag\] 9.58$\pm$0.01 [@Hog2000]
$B-V$ \[mag\] 1.27 $\pm$ 0.09 [@Hog2000]
$(B-V)_0$ \[mag\] 1.38 [@Zielinski2012]
$M_\mathrm{V}$ \[mag\] -2.51 [@Zielinski2012]
$T_{\mathrm{eff}}$ \[K\] 4095$\pm$20 [@Zielinski2012]
$\log g$ 1.89$\pm$0.10 [@Zielinski2012]
$[Fe/H]$ -0.71$\pm$0.09 [@Zielinski2012]
RV $[\kms]$ 50.88$\pm$0.06 [@Zielinski2012]
$v_{\mathrm{rot}} \sin i_{\star}$ $[\kms]$ 2.7$\pm$0.3 [@Adamow2014]
$\ALi $ $-0.56$ [@Adamow2014]
$[$O/H$]$ -0.23$\pm$0.22 [@Adamow2014]
$[$Mg/H$]$ -0.01$\pm$0.13 [@Adamow2014]
$[$Al/H$]$ -0.21$\pm$0.10 [@Adamow2014]
$[$Ca/H$]$ -0.68$\pm$0.18 [@Adamow2014]
$[$Ti/H$]$ -0.34$\pm$0.24 [@Adamow2014]
$M/M_{\odot}$ 1.14$\pm$0.25 [@Adamczyk2015]
$\log L/L_{\odot}$ 2.70$\pm$0.14 [@Adamczyk2015]
$R/R_{\odot}$ 32.35$\pm$8.82 [@Adamczyk2015]
$\log \mathrm{age}$ \[yr\] 9.72$\pm$0.28 [@Adamczyk2015]
$d$ \[pc\] 2618 $\pm$ 564 calculated from M$_{V}$
$V_{\mathrm{osc}}$ \[$\ms$\] 102.9$^{+89.9}_{-45.4}$ this work
$P_{\mathrm{osc}}$ \[d\] 2.9$^{+2.0}_{-1.1}$ this work
$P_{\mathrm{rot}} / \sin i_{\star}$ \[d\] 606$\pm$179 this work
: Summary of the available data on BD+03 2562
\[Parameters2\]
RV and BIS
----------
The 57 epochs of HET/HRS data for HD 103485 show RV variations of $553 \ms$ with average uncertainty of $5.6 \ms$ and BIS variations of $104\ms$ with an average uncertainty of $16 \ms$. No correlation between RV and BIS exists (Pearson’s r=0.09). HARPS-N RV 18 epochs of data show an amplitude of $553 \ms$ (average uncertainty of $1.7 \ms$). The BIS shows a peak-to-peak amplitude of $85 \ms$ and no correlation with RV (r=-0.02). Lomb-Scargle (LS) periodogram analysis [@1976ApJSS..39..447L; @1982ApJ...263..835S; @1992nrfa.book.....P] of combined HET/HRS and HARPS-N RV data reveals a strong periodic signal in RV at 557 days.
For BD+03 2562 the 46 epochs of HET/HRS data show RV variations of $575\ms$ with average uncertainty of $7\ms$ and BIS variations of $136\ms$ with average uncertainty of $22\ms$. The 19 epochs of HARPS-N data show RV amplitude of $444\ms$ and average uncertainty of $2.1\ms$. The BIS shows a peak-to-peak amplitude of $58\ms$. There is no correlation between RV and BIS in either HET/HRS data (r=0.23) or HARPS-N (r=0.13) data. A strong periodic signal in RV at 482 days appears in the LS periodogram analysis of combined HET/HRS and HARPS-N RV data.
HET/HRS and HARPS-N BIS have to be considered separately due to their different definition (see @Niedzielski2016b for more details). The RV and BIS data for both stars are presented in Tables \[HETdata1\], \[HETdata2\], \[HARPSdata1\] and \[HARPSdata2\].
Activity indicators: the Ca H$\&$K lines
----------------------------------------
The Ca II H, and K line profiles (see @Noyes1984 [@Duncan1991]) and the reversal profile, typical for active stars [@EberhardSchwarzschild1913] are widely accepted as stellar activity indicators. The Ca II H and K lines are only available to us in the TNG HARPS-N spectra. The signal-to-noise of our red giants in that spectral range is low, 3-5 in this particular case, but we found no trace of reversal. To quantify the observations, we calculated an instrumental $\shk$ index according to the prescription of [@Duncan1991] for HARPS-N data. The $\shk$ index for TNG HARPS-N spectra was calibrated to the Mt Wilson scale with the formula given by [@2011arXiv1107.5325L]. For HD 103485, we obtained a value of $0.20\pm0.05$ and for BD+03 2562 of $0.14\pm0.07$, rather typical values for non-active stars.
The $\shk$ for both stars show no statistically significant correlation with the RV ($r=-0.41$ and $r=-0.25$, respectively). In order to dig dipper into the posible correlation we have performed further statistical Bayesian tests following the prescription given in [@Figueira2016]. For the Pearson’s coefficient of the data is $0.408$ with a $0.093$ 2-sided p-value and the Spearman’s rank coefficient is 0.514 (0.029 2-sided p-value). The distribution of the parameter of interest, $\rho$, characterizing the strength of the correlation is $0.327$ with a standard deviation of $0.187$ and 95% credible interval \[-0.049 0.663\]. For the Pearson’s coefficient is $0.254$, $0.310$ 2-sided p-value and the Spearman’s rank coefficient $0.051$ with a $0.841$ 2-sided p-value. $\rho$ = 0.2 with a $0.197$ and 95% credible interval \[-0.201 0.559\]. For both stars a correlation is not conclusively seen and seems unlikely, with the 95% credible interval lower limit being above $\rho$= 0. However, a note of warning is in place here given that in the case of rotational modulation, it is expected to have non-linear relations between RV and activity. This is caused by a phase shift between the activity maximum (that occurs when the active regions are at the centre of the disk) and the maximum RV effect, that happens at a phase of $\approx$ 60$^{\circ}$.
Thus, we conclude that the Ca II H and K line profile analysis reveals that, over the period covered by TNG observations, both giants are quite inactive and there is no trace of activity influence upon the observed RV variations.
Activity indicators: $\mathbf{H\alpha}$ analysis
------------------------------------------------
showed that the calcium and hydrogen lines indices do not always correlate and cannot be used interchangeably as activity indicators. We thus measured the H$\alpha$ activity index ($I_{H\alpha}$) in both HET/HRS and TNG/HARPS-N spectra, following the procedure described in detail by [@2013AJ....146..147M], which based on the approach presented by and @2013ApJ...764....3R [and references therein]. We also measured the index in the Fe I 6593.883 [Å]{} control line ($I_{Fe}$) which is insensitive to stellar activity to take possible instrumental effects into account. Moreover, in the case of HET/HRS spectra, that may still contain weak $\mathrm{I_{2}}$ lines in the wavelength regime relevant to the H$\alpha$ and Fe I 6593.883 [Å]{} lines, we also measured the H$\alpha$ and Fe I indices for the iodine flat-field spectra ($I_{\mathrm{I_{2}}, H\alpha}$ and $I_{\mathrm{I_{2}}, {Fe}}$ respectively).
### BD+03 2562
The marginal rms variations of the $I_{\mathrm{I_{2}}, H\alpha}$ = 0.11% and $I_{\mathrm{I_{2}}, {Fe}}$ = 0.33% in comparison to the relative scatter of $I_{H\alpha,HRS}$ = 3.03% and $I_{Fe,HRS}$ = 1.08% assure us of the negligible contribution of the weak iodine lines to H$\alpha$ and Fe I 6593.883 [Å]{} line indices measured from HET/HRS spectra of BD+03 2562. The rms variation of $H\alpha$ activity index measured from 18 TNG/HARPS-N spectra is slightly larger than the one measured from the HET/HRS spectra ($I_{H\alpha,HARPS-N}$ = 3.68%). The rms variation of TNG/HARPS-N Fe I 6593.883 [Å]{} line index is more than two times larger than that measured from HET/HRS spectra ($I_{Fe,HARPS-N}$ = 2.39%). The larger rms variations of the line indices measured from the TNG/HARPS-N spectra compared to the HET/HRS spectra measurements might be a consequence of the lower SNR of the TNG/HARPS-N spectra (40–70), compared to that of the HET/HRS spectra (120–220). There is no correlation between neither HET/HRS $I_{H\alpha,HRS}$ and RVs (the Pearson coefficient, $r$ = 0.27), nor between TNG/HARPS-N $I_{H\alpha,HARPS-N}$ and RVs, (the same value of $r$ = 0.27). There is no any significant signal in the Lomb-Scargle periodograms of BD+03 2562 $H\alpha$ indices.
### HD 103485
The iodine flat-field HET/HRS spectra of HD 103485 show marginal rms variations ($I_{\mathrm{I_{2}}, H\alpha}$ = 0.1% and $I_{\mathrm{I_{2}}, {Fe}}$ = 0.13%) in the H$\alpha$ and Fe I 6593.883 [Å]{} indices. Comparing to the relative scatter of H$\alpha$ and Fe I indices ($I_{H\alpha,HRS}$ = 3.16% and $I_{Fe,HRS}$ = 0.79%) we are assured of the negligible contribution of the weak iodine lines to H$\alpha$ and Fe I 6593.883 [Å]{} line indices. The Pearson coefficient between HET/HRS H$\alpha$ index and RVs is $r$ = 0.11. The rms variations of H$\alpha$ and Fe I 6593.883 [Å]{} line indices measured from 18 TNG/HARPS-N spectra are similar to those measured for BD+03 2562: $I_{H\alpha,HARPS-N}$ = 3.86% and $I_{Fe,HARPS-N}$ = 2.73%. However, the Pearson coefficient between TNG/HARPS-N H$\alpha$ index and RVs, $r$ = 0.66, while the critical value of the Pearson correlation coefficient at the confidence level of 0.01, $r_{16,0.01}$ = 0.59. On the other hand, this value ($r$ = 0.66) is lower than the critical value of the Pearson correlation coefficient at the confidence level of 0.001 ($r_{16,0.001}$ = 0.71). Given the small number of epochs the correlation may very well be spurious. The H$\alpha$ indices versus HET/HRS and TNG/HARPS-N radial velocities are presented on Figure \[hd103485-rv\_vs\_hai\]. Figure \[hd103485-rv-hai-tng\_harpsn\] presents the TNG/HARPS-N radial velocity and H$\alpha$ activity index curves of HD 103485. Figure \[hd103485-lsp\] presents LS periodograms of TNG/HARPS-N and HET/HRS radial velocities and H$\alpha$ indices.
![H$\alpha$ activity index of HD 103485 measured from TNG/HARPS-N spectra (top panel) and from HET/HRS ones (lower panel). Solid lines presents linear fits to the data. Although H$\alpha$ activity index measured from HET/HRS does not present correlation with HET/HRS radial velocities, the one measured form TNG/HARPS-N spectra shows clear correlation with TNG/HARPS-N radial velocities. \[hd103485-rv\_vs\_hai\]](hd103485-rv_vs_hai){width="\columnwidth"}
![The TNG/HARPS-N radial velocity and H$\alpha$ activity index curves of HD 103485. \[hd103485-rv-hai-tng\_harpsn\]](hd103485-rv-hai-tng_harpsn){width="\columnwidth"}
![LS periodograms of (a) TNG/HARPS-N RVs, (b) TNG/HARPS-N H$\alpha$ activity index, (c) HET/HRS RVs, and (d) HET/HRS H$\alpha$ activity index of HD 103485. The levels of FAP = 1.0% and 0.1% are shown. \[hd103485-lsp\]](hd103485-lsp){width="\columnwidth"}
Wavelength dependence of the radial velocity signal
---------------------------------------------------
As both HD 103485 and BD+03 2562 exhibit significant scatter both in radial velocities and H$\alpha$ activity indices, the unambiguous interpretation of their origin is very difficult. Therefore, we analysed the wavelength dependence of the radial velocity peak-to-peak amplitude ($A$). The value of $A$ should be constant in the case of spectral shifts induced by the gravitational pull of the companion. In the case of a RV signal generated by the rotation of a spotted stellar photosphere, the value of $A$ should decrease with increasing wavelength as the temperature difference between the stellar photosphere and a stellar spot decreases at longer wavelengths .
Figure \[rrv\_vs\_rno\] shows, $A$, the peak-to-peak RV amplitude as a function of the TNG/HARS-N order number for HD 103485, BD+03 2562, and the multiple planetary host PTPS target TYC 1422-00614-1 presented in (TAPAS-I paper). TYC 1422-00614-1 does not show any significant stellar activity related to any of the two signals reported in its radial velocity curve. Therefore, it is a good benchmark to test the wavelength dependence of the radial velocity signal peak-to-peak amplitudes of the stars in this paper HD 103485 and BD+03 2562. As shown in Figure \[rrv\_vs\_rno\], both HD 103485 and BD+03 2562 show chromatic dependence of $A$, although its unambiguous interpretation is not straightforward, especially if we note that the peak-to-peak RV amplitude is systematically higher in TNG/HARPS-N orders 10–16.
Equation (5) of gives the relation between the peak-to-peak amplitude of the RV variation ($A$), the projected rotation velocity of the star ($v_{\mathrm{rot}} \sin i_{\star}$) and the fraction of the visible hemisphere of the star that might be covered by the spot (parameter $f_{r}$, see for its definition and relation to the fraction of the projected area covered by the spot, $f_{p}$, on the 2D stellar disk used by other authors). Using the above mentioned equation we computed the parameter $f_{r}$ for HD 103485 and BD+03 2562. As inputs in equation (5) we used the values of the projected rotation velocities of both stars from Tables 1 and 2 and the values of the $K$ semi-amplitudes from Tables 3 and 4 ($A$ is $\approx$ $2K$). For HD 103485 we obtained $f_{r} = 5.93$ % and for BD+03 2562 $f_{r} = 5.71$%. Then, using equation (6) of , that gives the relation between the peak-to-peak amplitude of the bisector inverse slope peak-to-peak variation ($S$), the parameter $f_{r}$, $v_{\mathrm{rot}} \sin i_{\star}$ and the instrumental width of the spectrograph ($v_0 = 3$ for both HARPS and HAPRS-N) we computed the values of $S$ for both of our targets. We obtain $S = 96$ for HD 103485 and $S = 75.5$ for BD+03 2562. The computed value of the peak-to-peak amplitude of bisector inverse slope for HD 103458 is consistent with its peak-to-peak amplitude of TNG/HARPS-N BIS ($85$ , see section 2.1.), while in the case of BD+03 2562, it is significantly higher (see section 2.1. TNG/HARPS-N BIS for BD+03 2562 is 58 ). We have to remember though, that equations (5) and (6) were derived for main sequence K5 type stars and HD 103458 and BD+03 2562 are giant stars.
![Radial velocity peak-to-peak amplitude ($A$) as a function of TNG/HARPS-N order number. \[rrv\_vs\_rno\]](rrv_vs_rno){width="\columnwidth"}
Stellar rotation and solar-like oscillations\[rot\]
===================================================
The one sigma limit of the rotation period (${P_\mathrm{rot} (\sin\,i_{\star})^{-1}}$) is equal to 477 $\pm$ 134 days for HD 103485 and 606 $\pm$ 179 days for BD+03 2562. Thus, the true rotation period (${P_\mathrm{rot}}$) of HD 103485 at one sigma is then lower than 611 days and ${P_\mathrm{rot}}$ is lower than 785 days for BD+03 2562. Based on the upper limits of the rotation periods of HD 103485 and BD+03 2562 we then cannot exclude that the signals in their radial velocity curves are generated by rotational modulation of active photospheres.
The amplitudes ($V_{osc}$) and periods ($P_{osc}$) of solar-like oscillations computed using equations (7) and (10) of are equal to $68.2^{+52.1}_{-28.2}$ and $2.2^{+2.0}_{-1.1}$ days for HD 103485 and to $102.9^{+89.9}_{-45.4}$ and $2.9^{+3.2}_{-1.7}$ days for BD+03 2562. Computed values of solar-like oscillations amplitudes are consistent with the values of stellar jitter ($\sigma_{jitter}$) and post-fit rms ($\mathrm{RMS}$) presented in Tables 3 and 4. Both stars exhibit extremely high stellar jitter.
Photometry and discussion of activity indicators\[phot\]
========================================================
For HD 103485 two extensive sets of photometric observations are available from Hipparcos and ASAS [@1997AcA....47..467P]. 133 epochs of Hipparcos data were gathered over 1140 days between JD 2447878.4 and 2449019.0, long before our monitoring of this star. The average brightness is v$_{Hip}=8.422 \pm 0.013$ mag and they show no trace of variability. 392 epochs of ASAS photometry were collected between JD 2451871.9 and 2455040.5 (3169 days), partly during our HET observations. These show average brightness of v$_{ASAS}=8.281\pm 0.01$ and trace of a 27d period, possibly due to the Moon. No significant photometric variability similar to that shown in RV is present in the available data as illustrated by the LS periodogram in Figure \[LSP\_1\].
BD+03 2562 was observed within ASAS over 3169 days between JD 2451871.9 and 2455040.5, partly covering the timespan of our HET observations. The average brightness is v$_{ASAS}=9.497\pm0.015 $ mag and we find no trace of activity in these data (see Figure \[LSP\_2\]).
The spectral line bisectors and the calcium H$\&$K line shape show that we are dealing with a Keplerian motion that alters the position of the observed absorption lines in the spectra of both stars. Both weak and uncorrelated variations of $\mathbf{H\alpha}$ and the lack of photometric variability in the case of support that conclusion. In the case of the $\mathbf{H\alpha}$ variations, weakly correlated with the observed RV in the TNG/HARPS-N data suggest that the observed RV variations may be due to a spot, but no such spot is visible in the photometric data, partly contemporaneous with our spectroscopic observations.
We can therefore conclude that, although in the case of the activity should be studied in more detail in the future, there exists no inexorable evidence that contradicts the interpretation of the observed RV variations as Doppler displacements due to the presence of a companion.
Keplerian analysis \[results-g\]
================================
![Keplerian best fit to combined HET HRS (green points) and TNG HARPS-N (red points) data for . The estimated jitter due to p-mode oscillations has been added to the uncertainties.[]{data-label="Fit_1"}](fitp_PTPS_0081.pdf){width="50.00000%"}
![From top to bottom Lomb-Scarge periodograms for (a) the original HET HRS and HARPS-N RV data of , (b) ASAS photometry, (c) Bisector analysis, and (d) RV residua (HET and TNG) after the best Keplerian planet fit.[]{data-label="LSP_1"}](LSP_PTPS_0081.pdf){width="50.00000%"}
![Keplerian best fit to combined HET HRS and TNG HARPS-N data for . The jitter is added to the uncertainties.[]{data-label="Fit_2"}](fitp_PTPS_0086.pdf){width="50.00000%"}
![Same as Fig.\[LSP\_1\] for []{data-label="LSP_2"}](LSP_PTPS_0086.pdf){width="50.00000%"}
The Keplerian orbital parameters have been derived using a hybrid approach (e.g. ), in which the PIKAIA-based, global genetic algorithm (GA; @Charbonneau1995) was combined with the MPFit algorithm [@Markwardt2009], to find the best-fit Keplerian orbit delivered by RVLIN [@WrightHoward2009] modified to allow the stellar jitter to be fitted as a free parameter [@2007ASPC..371..189F; @2011ApJS..197...26J]. The RV bootstrapping method is employed to assess the uncertainties of the best-fit orbital parameters (see TAPAS I for more details). The results of the Keplerian analysis for are presented in Table \[KeplerianFit1\] and in Figure \[Fit\_1\] and for in Table \[KeplerianFit2\] and in Figure \[Fit\_2\].
\[KeplerianFit1\]
\[KeplerianFit2\]
Discussion and conclusions\[conclusions\]
=========================================
In this paper, the fifth of our TAPAS series, we present a planetary mass companion and a possible one to two very metal poor giant stars in the constellation of Virgo. We have interpreted the RV variations measured in both stars as Keplerian motion, and while there is a lack of compelling evidence that the signal is originated by stellar activity for it is not so clear for for which based on the available data both interpretations for the RV signal (planet and activity) are possible. In the meanwhile more data is gathered for it is hard to justify or disprove both possible interpretations so we keep as a working hypothesis for the following discussion that the RV signal is originated by a planet. In this case, both giant stars have masses close to Solar (1.11 and 1.14 ), very low metallicities ($[Fe/H]= -0.50$ and $-0.71$ for and respectively) and Jupiter planetary mass companions ($m_2\sin i\thinspace=\thinspace7$ and 6.4 for and resp.) in close to terrestrial orbits (1.4 au and 1.3 au ) with moderate eccentricities ($e=0.34$ and 0.2 for and ).
In Fig.\[Evo\] we show the location of all the stars included in the PTPS sample on the Hertzsprung-Russel (HR) diagram where we have marked the location of (blue) and (red) (and their corresponding uncertainties). The [@Bertelli2008] evolutionary tracks of a 1stars at two different very low metallicities are also shown for comparison. From the figure is clear that the two stars presented in this paper are among the most evolved stars of the whole PTPS sample. With ages of 6.17 and 5.25 Gyr for and (see Tables\[Parameters1\] and \[Parameters2\]), these stars are certainly above the mean age value of 3.37 Gyr obtained for Giant stars with planets in [@Maldonado2016].
Again under the interpretation that the RV signal measured in both stars is due to Keplerian motion and based on the derived orbital parameters we compute the orbital solution under stellar evolution. None of the planets are expected to have experienced orbital decay caused by stellar tides at their current location (with $a/R_{\star}=10.99$ and 8.64 for and respectively) (see e.g. @VillaverLivio2009 [@Villaver2014]). Given the tidal dissipation mode that operates in these giant stars, the planets should experience eccentricity decay together with orbital decay [@Villaver2014], thus their moderate eccentricities and their $a/R_{\star}$ ratios are consistent with both planets being yet too far from the star to have experience tidal forces. Both planets reported in this paper are located in similar regions in $M_{\star}$ versus orbital distance or the $a-e$ plane as most of the other planets orbiting giant stars reported in the literature [@Villaver2014]. Neither b nor b, are expected to survive engulfment when the star evolves up the tip of the RGB [@VillaverLivio2009]. Assuming an average value of $\sin i$ both these companions stay within the planetary-mass range.
Thus the orbital characteristics of the substellar object and the posible one we report in this paper do not appear to be different from the ones shown by the bulk population of planets found orbiting giant or subgiant stars. These two objects, however, clearly stand out in two important aspects: i) they are among the few very massive planets found around metal poor stars, and ii) in particular populates a region in the $M_{\star} -[\mathrm{Fe/H}]_{\star}$ plane where only another star has been found to host planets BD +20 2457 (see Figs.\[ZMp\] and \[ZMs\]).
From Figs.\[ZMp\],\[ZMs\] it is clear that the two planet/star combinations reported in this paper are very special. First, they are two of the very few massive planets orbiting around stars close to the mass of the Sun to be found at very low metallicity. At lower metallicities than , only two other planets have been reported in the literature orbiting the stars BD +20 2457 [@Niedzielski2009b] and HD 11755 . The similarities among these systems are striking: giant, close to Solar mass, evolved stars with radii close to 30 , with $\approx$ 7 minimum mass planets and in $\approx 1 au$ orbits. Note also that the planet around is the only massive planet in the region around stars with metallicity in the range $-0.6< [\mathrm{Fe/H}]_{\star} < -0.4$. and stand out even more in Fig.\[ZMs\] where very few planets are known with $[Fe/H]_{\star} < -0.48$ and $M_{\star}>1$.
Our current understanding of massive planet formation offers two channels. First, core accretion $-$ and the growing of planets from the accretion of a gas envelope into a massive core $-$ has problems to easily explain systems formed at low metallicity. Models of planet formation by core accretion require a protoplanetary disk with a high density of solids to form planetary cores which accrete gas before the primordial gas disk dissipates (see @Ida2004). The probability of a star hosting a planet that is detectable in radial velocity surveys increases $P_{pl} (Z) = 0.03 \times 10^{2\times Z}$, where $Z = [\mathrm{Fe/H}]$ is the stellar metallicity between $-0.5$ and 0.5 dex [@Gonzalez1997; @FischerValenti2005]. Thus although, core accretion does not exclude the formation at low metallicity, the probability of finding such planets is low $P_{pl}$ = 0.2% for and 0.1% for .
The alternative mechanism, in-situ fragmentation via gravitational instability (GI) (see e.g., @Cameron1978 [@Boss1997]) posses such strong requirements in the characteristics of a protoplanetary disk at $1\thinspace \mathrm{au}$ that has been shown cannot exit on dynamical grounds (see e.g. @Rafikov2005 [@Stamatellos2008]). The planet could form by GI at larger distances [@Rafikov2005; @Matzner2005] and experience subsequent migration to $\approx 400 d$ orbital periods in relatively short timescales. In fact, the fragments formed by instability at $100\thinspace\mathrm{au}$ are expected to have minimum masses of $10 \Mjup$ [@Rafikov2005] which is approximately the typical mass of the planets found around the low metallicity stars reported in this paper. Note that although it has been shown that GI cannot be the main channel for planet formation as it cannot reproduce the overall characteristics of the bulk of the planet detections, nothing prevents it to be the preferred mechanism under certain circumstances. In fact, it has been reported that protoplanetary disks with low metallicities generally cool faster and show stronger overall GI activity [@Mejia2005; @Cai2006] although the lowest metallicity consider in these models is a quarter Solar (still much larger than the ones shown by the stars presented in this paper). So the question still remains on how gas cooling in the disk operates at the low metallicity of these stars given the disk needs to be atypically cold for GI and whether these planets represent indeed the low-mass tail of the distribution of disk-born companions [@Kratter2010].
Planet host giants, in fact, have been reported to show peculiar characteristics regarding the planet-metallicity relation. In particular, [@Maldonado2013] show that, whilst the metallicity distribution of planet-hosting giant stars with stellar masses $M > 1.5\Msun$ follows the general trend that has been established for main sequence stars hosting planets, giant planet hosts in the mass domain $M \leqslant 1.5\Msun$ do not show metal enrichment. Similar results were found by [@Mortier2013]. Note that [@Reffert2015] challenged these results based on a discussion of planet contamination but it has been shown by[@Maldonado2016] based on their planet list that the result is sustained using the [@Reffert2015] list of candidates.
The two objects presented in this paper add two more points to a already puzzling relation between giant planets and giant stars that might help understand planet formation mechanisms for low metallicity stars.
![Hertzsprung-Russell diagram for the complete PTPS sample with (in blue) and (in red) and the evolutionary tracks from [@Bertelli2008] for a star with 1 and metallicities $Z=0.008$ and $Z=0.004$ (see the legend in the bottom left corner of the plot).[]{data-label="Evo"}](hr_tapas6.pdf){width="50.00000%"}
![Planet minimum mass ($M \sin i$ in Jupiter mass) versus stellar metallicity (\[Fe/H\]) for all the confirmed planets (blue points) as taken from the Exoplanet encyclopedia (exoplanet.eu, exoplanets.org). The two red points represent the location of the planets reported in this paper. The horizontal lines are to guide the eye in the logarithmic scale in the vertical axis.[]{data-label="ZMp"}](Z_Mp.pdf){width="50.00000%"}
![Stellar mass (in ) versus stellar metallicity (\[Fe/H\]) for all the stars with confirmed planets (green points) according to the Exoplanet encyclopedia (exoplanet.eu, exoplanets.org). The two red points represent the location of the stars reported in this paper. []{data-label="ZMs"}](Z_Mstar.pdf){width="50.00000%"}
We thank the HET and IAC resident astronomers and telescope operators for their support. EV acknowledges support from the Spanish Ministerio de Economía y Competitividad under grant AYA2014-55840P.
MA acknowledges the Mobility+III fellowship from the Polish Ministry of Science and Higher Education.
AN, BD-S and MiA were supported by the Polish National Science Centre grant no. UMO-2012/07/B/ST9/04415 and UMO-2015/19/B/ST9/02937.
KK was funded in part by the Gordon and Betty Moore Foundation’s Data-Driven Discovery Initiative through Grant GBMF4561.
This research was supported in part by PL-Grid Infrastructure.
The HET is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig- Maximilians-Universität München, and Georg-August-Universität Göttingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium.
This work made use of NumPy [@numpy], Matplotlib [@mpl], Pandas [@pandas] and `yt` [@yt] and of the Exoplanet Orbit Database and the ExoplanetData Explorer at exoplanets.org and exoplanet.eu.
\[HETdata1\]
\[HARPSdata1\]
\[HETdata2\]
\[HARPSdata2\]
[^1]: Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen.
[^2]: Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.
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abstract: 'After the submission of the paper, three strong earthquakes with magnitude around 6.0-units occurred on October 17 and October 20, 2005, with epicenters in the Aegean Sea, at a distance [*only*]{} 100km from MYT station at which the intense signals $M_1$ to $M_4$ -analyzed in the main text- have been recorded. This confirms experimentally the proposed criterion we used for the classification of these signals as Seismic Electric Signals (SES). Moreover, we show that, if we follow the procedure described in \[P.A. Varotsos, N. V. Sarlis, H. K. Tanaka and E. S. Skordas [*Phys. Rev. E*]{} [**72**]{}, 041103 (2005)\], the analysis in the natural time of the seismicity after the SES initiation allows the estimation of the time window of the impending earthquakes with very good accuracy.'
author:
- 'P. A. Varotsos'
- 'N. V. Sarlis'
- 'E. S. Skordas'
- 'H. K. Tanaka'
- 'M. S. Lazaridou'
title: 'Additional information[^1] for the paper ‘Entropy of seismic electric signals: Analysis in natural time under time-reversal’'
---
On October 17, 2005 two strong earthquakes (EQs) with magnitude around 6.0-units occurred at 05:45:20 and 12:46:57 UT with epicenters (according to USGS) at 38.15$^oN$,26.68$^oN$ and 38.13$^oN$,26.65$^oN$, respectively (see Fig.\[f1\]). At the same epicenter, a third almost equally strong earthquake occurred at 21:40 UT on October 20,2005. All the three epicenters lie at a distance of around 100km from Lesvos island, at which the MYT station -on the dipoles of which the intense signals $M_1$ to $M_4$ (Fig.1(a),(b) of the main text) have been recorded- is located. This verifes that these signals are actually Seismic Electric Signals (SES) as classified in advance (i.e., upon the initial submission of the paper on April 16,2005) in the main text on the basis of the entropy criterion proposed.
![Map of the area surrounding the measuring station of MYT and the epicenters of the three strong EQs that occurred on October 17 and October 20,2005. The earthquake mechanisms of all three EQs are also shown. The seismicity subsequent to the SES initiation has been studied in the gray shaded regions.[]{data-label="f1"}](Fig1)
We now follow the procedure described in Refs.[@NAT01; @prlop], in order to investigate whether the time window of the impending strong EQs could have been estimated. We consider either the region A:$N_{37.0}^{39.5}E_{25.5}^{28.0}$ or the region B:$N_{37.5}^{39.5}E_{26.0}^{28.0}$, which surround the EQ epicenters and the MYT station (see Fig.\[f1\]), and study how the seismicity evolved after the SES initiation. If we set the natural time for seismicity zero at the initiation of the concerned SES activities, we form time series of seismic events in natural time for various time windows as the number $N$ of consecutive (small) EQs increases. We now compute the normalized power spectrum [@NAT01; @prlop] in natural time $\Pi (\phi )$ for each of the time windows and the results are depicted in Fig.\[f2\]. As examples we consider in this figure two magnitude thresholds (herafter referring to the local magnitude $M_L$ or the ‘duration’ magnitude $M_D$) 3.4 (upper) and 3.6 (lower). In the same figure, we plot in blue the power spectrum obeying the relation $$\Pi ( \omega ) = \frac{18}{5 \omega^2}
-\frac{6 \cos \omega}{5 \omega^2}
-\frac{12 \sin \omega}{5 \omega^3}
\label{fasma}$$ which holds[@NAT01; @NAT02; @NAT02A] when the system enters into the [*critical*]{} stage ($\omega = 2\pi \phi$, where $\phi$ stands for the natural frequency[@NAT01; @NAT02; @newbook]). An inspection of Fig.\[f2\] reveals that the red line approaches the blue line as $N$ increases and a [*coincidence*]{} occurs at the last small event which had a magnitude 3.6 and occurred at 04:31 UT on October 17, 2005, i.e., roughly one hour before the first strong EQ. To ensure that this coincidence is a [*true*]{} one[@NAT01; @prlop; @newbook] we also calculate the evolution of $\kappa_1$,$S$ and $S_{-}$ (cf. $\kappa_1$ stands for the variance $\kappa_1\equiv \langle \chi^2 \rangle -\langle \chi \rangle^2$ as explained in Refs.[@NAT01; @NAT02]) and the results are depicted in Fig.\[f4\] for three magnitude thresholds 3.4, 3.5 and 3.6.
![The normalized power spectrum(red) $\Pi (\phi )$ of the seismicity in area A as it evolves event by event after the initiation of the SES activities $M_1$ to $M_4$. The two examples presented correspond to the two different magnitude thresholds 3.4 and 3.6 in (a),(b) respectively. In each case only the region $\phi \in [0,0.5]$ is depicted (for the reasons discussed in Refs.[@NAT01; @prlop]), whereas the $\Pi (\phi )$ of Eq.(\[fasma\]) is depicted by blue color. []{data-label="f2"}](Fig2)
![Evolution of $\kappa_1$, $S$ and $S_{-}$ upon using the USGS catalogue for various magnitude ($M_L$ or $M_D$) thresholds for the two regions A and B.[]{data-label="f4"}](Fig3)
We now further comment on the aforementioned results. Since the strong EQs occurred in the border between Greece and Turkey, the seismicity catalogues of neither Greek nor the Turkish Institutes can be considered as complete for small magnitudes. Hence, we preferred here to make the calculations by using the United States Geological Survey (USGS) catalogue (see Table \[tab1\]). Irrespective if we use the seismicity in the region $N_{37.0}^{39.5}E_{25.5}^{28.0}$ or in the smaller region $N_{37.5}^{39.5}E_{26.0}^{28.0}$, the coincidence occurs upon the occurence of the aforementioned 3.6 EQ (almost 1 hour before the first strong EQ). The magnitude of this EQ comes from the European-Mediterranean Seismological Centre (EMSC) (see the corresponding announcement in Fig.\[f3\]) since it has not yet been reported by USGS. Note that if we take the magnitude of this EQ to be somewhat larger, then the first box in Fig.\[f4\] (which has been plotted for magnitude threshold 3.4) shows that the coincidence occurs on the last but one event, i.e., on October 13,2005 (almost three days before the first strong EQ).
The following comment might be useful: In the analysis of signals depicted in Figs1(a),(b) we proceeded as follows: We first analyzed $M_1$ (recorded on March 21, see Fig1(a)) and found that it obeys the criterion (i.e., $S$ and $S_{-}$ smaller than $S_u$). We then turned to the recordings on March 23: we first considered $M_4$ -which is well distinguishable from the others in view of its opposite polarity- and found again that the criterion is fullfilled. As for the remaining recordings of March 23(comprising $M_2$ and $M_3$) we checked both possibilities that is: (1) we considered $M_2$ and $M_3$ together and found that not only the criterion was violated but also the $M_2+M_3$ signal (altogether) behaved like signals obtained from a “uniform” distribution(i.e., $\kappa_1$ was around $1/12$). (2)on the other hand, if we analyse separately $M_2$ and $M_3$ then both results obeyed the criterion. (thus being consistent with the conclusions drawn from $M_1$ and $M_4$). Hence, we preferred to draw in Fig.1b the latter possibility. It has been earlier suggested (see pages 18 and 114 of Ref.\[10\] of the main text) that the change of the SES polarity might be associated with a different EQ source mechanism. The first three SES activities $M_1$, $M_2$ and $M_3$ -which have been plotted in Figs. 1(a,b) do have the same polarity and hence- might be associated with the 3 EQs that have already occurred having the [*same*]{} mechanism. If this suggestion is correct, the last signal, i.e., $M_4$, with opposite polarity, should correspond to another EQ with different mechanism which has not occurred yet.
![The detailed European-Mediterranean Seismological Centre (EMSC) report for the 3.6 EQ that occured almost one hour before the first strong EQ.[]{data-label="f3"}](Fig4)
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[^1]: Submitted to Physical Review E on October, 21, 2005
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abstract: |
We present measurements of branching fractions and charge asymmetries in -meson decays to $\rho^+ \pi^{0}$, $\rho^{0}\pi^+$ and $\rho^0\pi^0$. The data sample comprises $89 \times 10^6$ $\FourS \to B\Bbar$ decays collected with the detector at the 2 asymmetric-energy $B$ Factory at SLAC. We find the charge-averaged branching fractions ${\cal B}( B^{+}\rightarrow \rho^{+}\pi^0) = (10.9 \pm 1.9{\rm (stat)}
\pm 1.9{\rm (syst)})\times
10^{-6}$ and ${\cal B}( B^{+} \rightarrow \rho^0 \pi^{+}) = (9.5 \pm
1.1 \pm 0.8) \times 10^{-6}$, and we set a $90\%$ confidence-level upper limit ${\cal B}( \B^0 \rightarrow \rho^0\pi^0) < 2.9 \times 10^{-6}$. We measure the charge asymmetries $A_{CP}^{\rho^{+}\pi^0} = 0.24 \pm 0.16 \pm 0.06$ and $A_{CP}^{\rho^0\pi^{+}} = -0.19 \pm 0.11 \pm 0.02$.
title: |
[**Measurement of Branching Fractions and Charge Asymmetries\
in $B^\pm\to\rho^\pm \pi^0$ and $B^\pm\to\rho^0 \pi^\pm$ Decays, and Search for $B^0\to\rho^0 \pi^0$**]{}
---
-PUB-[03]{}/[037]{}\
SLAC-PUB-[10236]{}\
authors\_sep2003.tex
The study of -meson decays into charmless hadronic final states plays an important role in the understanding of violation in the system. Recently, the experiment performed a search for -violating asymmetries in neutral $B$ decays to $\rho^\pm\pi^\mp$ final states [@bib:350PRL], where the mixing-induced asymmetry is related to the angle $\alpha \equiv \arg\left[-V_{td}^{}V_{tb}^{*}/V_{ud}^{}V_{ub}^{*}\right]$ of the Unitarity Triangle [@unitarity]. The extraction of $\alpha$ from $\rho^\pm\pi^\mp$ is complicated by the interference of decay amplitudes with differing weak and strong phases. One strategy to overcome this problem is to perform an SU(2) analysis that uses all $\rho\pi$ final states [@bib:Nir]. Assuming isospin symmetry, the angle $\alpha$ can be determined free of hadronic uncertainties from a pentagon relation formed in the complex plane by the five decay amplitudes $B^0\rar\rho^+\pi^-$, $B^0\rar\rho^-\pi^+$, $B^0\rar\rho^0\pi^0$, $B^+\rar\rho^+\pi^0$ and $B^+\rar\rho^0\pi^+$ [@footn]. These amplitudes can be determined from measurements of the corresponding decay rates and -asymmetries. The branching fractions have been measured for $B^0\rar\rho^+\pi^-$ and $B^+\rar\rho^0\pi^+$, and an upper limit has been set for $B^0\rar\rho^0\pi^0$ [@bib:350PRL; @bib:cleo_rhopi].
In this letter we present measurements of the branching fractions of the decay modes $B^+\rar\rho^+\pi^0$ and $B^+\rar\rho^0\pi^+$, and a search for the decay $B^0\rar\rho^0\pi^0$. All three analyses follow a quasi-two-body approach [@roy; @bib:350PRL]. For the charged modes we also measure the charge asymmetry, defined as $$\label{equ:Acp}
A_{CP} \equiv
\frac{\Gamma(B^- \rightarrow f) \,-\, \Gamma(B^+ \rightarrow \overline f)}
{\Gamma(B^- \rightarrow f) \,+\, \Gamma(B^+ \rightarrow \overline f)}\;,$$ where $f$ and $\overline f$ are the final state and its charge-conjugate, respectively.
The data used in this analysis were collected with the detector [@bib:babarNim] at the 2 asymmetric-energy $e^+e^-$ storage ring at SLAC. The sample consists of $(88.9\pm1.0)\times10^{6}$ $B\Bbar$ pairs collected at the resonance (“on-resonance”), and an integrated luminosity of 9.6 collected about 40 [$\mathrm{\,Me\kern -0.1em V}$]{}below the (“off-resonance”). Each signal candidate is reconstructed from three-pion final states that must be $\pi^{+}\pi^0\pi^0$, $\pi^{+}\pi^-\pi^+$, or $\pi^{+}\pi^{-}\pi^0$. Charged tracks must have ionization-energy loss and Cherenkov-angle signatures inconsistent with those expected for electrons, kaons, protons, or muons [@bib:babarNim]. The $\pi^0$ candidate must have a mass that satisfies $0.11<m(\gamma\gamma)<0.16{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$, where each photon is required to have an energy greater than $50{\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$ in the laboratory frame and to exhibit a lateral profile of energy deposition in the electromagnetic calorimeter consistent with an electromagnetic shower [@bib:babarNim]. The mass of the reconstructed $\rho$ candidate must satisfy $0.4<m(\pi^{+}\pi^0)<1.3{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ for $\rho^+$ and $0.53<m(\pi^+\pi^-) < 0.9{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ for $\rho^0$. The tight upper $m(\pi^+\pi^-)$ cut at 0.9${\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ is to remove contributions from the scalar $f_0(980)$ resonance, and the tight lower cut is to reduce the contamination from $\KS$ decays. To reduce contributions from $B^0\to\rho^+\pi^-$ decays, a $\btorp$ candidate is rejected if $0.4<m(\pi^{\pm}\pi^0)<1.3{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$. For the $\bchtorchp$ and $\btorp$ modes, the invariant mass of any charged track in the event and the $\piz$ must be less than $5.14{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ to reject $\B^+\rar\pi^+\pi^0$ background. For the $\bchtorpch$ mode, we remove background from charmed decays $B\rightarrow \Dzb X$, $\Dzb\rightarrow K^{+}\pi^{-}$ or $\pi^{+}\pi^{-}$, by requiring the masses $m(\pi^+\pi^-)$ and $m(K^+\pi^-)$ to be less than $1.844{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ or greater than $1.884{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$. We take advantage of the helicity structure of $B \rar \rho \pi$ decays by requiring that $|\cos\theta_{\rho}|>0.25$, where $\theta_{\rho}$ is the angle between the $\pi^0$ ($\pi^+$) momentum from the $\rho^{+}$ $(\rho^{0})$ decay and the momentum in the $\rho$ rest frame.
Two kinematic variables, $\dE$ and $\mes$, allow the discrimination of signal decays from random combinations of tracks and $\pi^0$ candidates. The energy difference, $\dE$, is the difference between the $\ee$ center-of-mass (CM) energy of the candidate and $\sqrt{s}/2$, where $\sqrt{s}$ is the total CM energy. The beam-energy-substituted mass, $\mes$, is defined by $\sqrt{(s/2+{\mathbf {p}}_i\cdot{\mathbf{p}}_B)^2/E_i^2-{\mathbf {p}}_B^2},$ where the $B$ momentum, ${\mathbf {p}}_B$, and the four-momentum of the initial state ($E_i$, ${\mathbf {p}}_i$) are measured in the laboratory frame. For $\bchtorpch$ we require that $-0.05< \dE < 0.05{\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$ while for both modes containing a $\pi^0$ we relax this requirement to $-0.15<\dE< 0.10{\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$. For both $\bchtorpch$ and $\btorp$ we require that $5.23<\mes<5.29{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ while for $\bchtorchp$ it is relaxed to $5.20 < \mes < 5.29{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$
Continuum $e^+e^-\to q\bar{q}$ ($q = u,d,s,c$) events are the dominant background. To enhance discrimination between signal and continuum, we use neural networks (NN) to combine six discriminating variables: the reconstructed $\rho$ mass, $|\cos\theta_{\rho}|$, the cosine of the angle between the momentum and the beam direction in the CM frame, the cosine of the angle between the thrust axis and the beam direction in the CM frame, and the two event-shape variables that are used in the Fisher discriminant of Ref. [@bib:babarsin2b]. The event shape variables are sums over all particles $i$ of $p_i\times|cos\theta_i|^n$, where $n=0$ or $2$ and $\theta_i$ is the angle between momentum $i$ and the $B$ thrust axis. The NN for each analysis weighs the discriminating variables differently, according to training on off-resonance data and the relevant Monte Carlo (MC) simulated signal events. The final $\rho\pi$ candidate samples are selected with cuts on the corresponding NN outputs.
To further discriminate further between signal and continuum background, for the $\btorp$ mode, we use the separation between the vertex of the reconstructed $B$ and the vertex reconstructed for the remaining tracks. This separation is related to $\deltat$, the difference between the two decay times, by $\Delta z = c\beta\gamma \deltat$, where for 2 the boost is $\beta\gamma=0.56$.
Approximately $33\%$, $7\%$, and $8\%$ of the events have more than one candidate satisfying the selection in the $\bchtorchp$, $\bchtorpch$, and $\btorp$ decay mode, respectively. In such cases we choose the candidate with the reconstructed $\rho$ mass closest to the nominal value of $0.77{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$. Table \[tab:sumtab\] summarizes the numbers of events selected from the data sample and the signal efficiencies estimated from MC simulation. Some of the actual signal events are misreconstructed; this is primarily due to the presence of random combinations involving low momentum pions. For the charged modes we distinguish misreconstructed signal events with correct charge assignment from those with incorrect charge assignment. These numbers, estimated from MC, are also listed in Table \[tab:sumtab\].
------------------- ---------------- ---------------- ----------------
\[-0.3cm\] $\bchtorchp$ $\bchtorpch$ $\btorp$
Selected events 13177 8551 7048
Signal efficiency $17.5\pm0.1\%$ $28.3\pm0.1\%$ $20.0\pm0.1\%$
Misreconstructed $38.6\pm0.2\%$ $7.1\pm0.1\%$ $9.1\pm0.2\%$
Wrong charge $8.1\pm0.1\%$ $1.6\pm0.1\%$ -
------------------- ---------------- ---------------- ----------------
: Numbers of selected events from on-resonance data, signal efficiencies, relative fraction of misreconstructed and wrong charge events from MC.
\[tab:sumtab\]
We use MC-simulated events to study the background from other $B$ decays, (-background), which include both charmed ($b \to c$) and charmless decays. In the selected $\rho^+\pi^0$ ($\rho^0\pi^+$, $\rho^0\pi^0$) sample we expect $205\pm46$ ($73\pm19$, $59\pm18$) $b \to c$ and $228\pm77$ ($92\pm11$, $74\pm22$) charmless background events. All the three analyses share the major -background modes: $B^0\to\rho^+\pi^-$, longitudinally polarized $B^0\to\rho^+\rho^-$, and $B^+\to\rho^+\rho^0$. Other important modes include $\bchtorchp$ (for $\btorp$), $B^+\to(\a_1\pi)^+$ (for $\bchtorchp$), $B^+\to K^{*}(892)^{0}\pi^+$ (for $\bchtorpch$), and background modes containing higher kaon resonances.
An unbinned maximum likelihood fit is used for each analysis to determine event yields and charge asymmetries. To enhance discrimination between signal and background events, we use the $B$-flavor-tagging algorithm developed for the measurement of the $CP$-violating amplitude $\sin2\beta$ [@bib:babarsin2b], where events are separated into categories based on the topology of the event and the probability of misassigning the $B$-meson flavor. The likelihood for the $N_\cat$ candidates tagged in category $k$ is $$\label{eq:pdfsum}
{\cal L}_k = e^{-N^{\prime}_\cat}\!\prod_{i=1}^{N_\cat}
\bigg\{ N^{\rho \pi}
\epsilon_\cat {\cal P}_{i,\cat}^{\rho \pi}
+ N_\cat^{\cont} {\cal P}_{i,\cat}^{\cont}
+ \sum_{j=1}^{N_B} {\cal L}^{\B}_{ij, \cat}\bigg\}\;,$$ where $N^{\rho \pi}$ is the number of signal events in the entire sample, $\epsilon_\cat$ is the fraction of signal events tagged in category $\cat$, $N^{\cont}_\cat$ is the number of continuum background events that are tagged in category $\cat$, and $N_B$ is the number of -background modes. $N^{\prime}_\cat$ is the sum of the expected event yields for signal ($\epsilon_\cat N^{\rho \pi}$), continuum ($ N_\cat^{\cont}$) and fixed background. For the charged modes the asymmetries are introduced by multiplying the signal yields by $\frac{1}{2}(1-Q_{i}A_{CP})$, where $Q_i$ is the charge of $B$-candidate $i$. The likelihood term ${\cal L}^{\B}_{ij,\,\cat}$ corresponds to the $j_{th}$ -background contribution of the $N_B$ -background classes. The total likelihood is the product of likelihoods for each tagging category.
The probability density functions (PDF) for signal and continuum, ${\cal P}_{\cat}^{\rho\pi}$ and ${\cal P}_{\cat}^{q\bar q}$, are the products of the PDFs of the discriminating variables. The signal PDFs are given by ${\cal P}^{(\rho\pi)^+}_\cat \equiv {\cal P}^{(\rho\pi)^+}(\mes)
\cdot {\cal P}^{(\rho\pi)^+} (\dE)\cdot {\cal P}^{(\rho\pi)^+}_\cat (\NN)$ for the charged decay modes, and by ${\cal P}^{\rho^0\pi^0}_\cat \equiv {\cal P}^{\rho^0\pi^0}(\mes)
\cdot {\cal P}^{\rho^0\pi^0} (\dE)\cdot {\cal P}^{\rho^0\pi^0}_\cat (\NN)
\cdot {\cal P}^{\rho^0\pi^0}_\cat (\deltat)$ for $\Bz \to \rho^0\piz$. Each signal PDF is decomposed into two parts with distinct distributions: signal events that are correctly reconstructed and signal events that are misreconstructed. For the charged modes, each PDF for the misreconstructed events is further divided into a right-charge and wrong-charge part. The $\mes$, $\dE$, and NN PDFs for signal and for background are taken from MC simulation. For continuum, the yields and PDF parameters are determined simultaneously in the fit to on-resonance data.
In the $\btorp$ decay the $\dt$ distributions for signal and background are modeled from fully reconstructed $B^0$ decays from data control samples [@bib:babarsin2b]. The continuum $\deltat$ parameters are free in the fit to on-resonance data.
--------------------------- -------------------------- ----------------------------- ---------------- ---------------------------------------- ------------------------
$\rho^+ \pi^0$ $\rho^0 \pi^+$ $\rho^0 \pi^0$ [1]{}[ c ]{}[$A_{CP}^{\rho^+\pi^0}$]{} $A_{CP}^{\rho^0\pi^+}$
[3]{}[ c ]{}[(events)]{} [2]{}[ c ]{}[($10^{-2}$)]{}
Signal model 10.7 3.8 3.3 3.4 0.3
Fit procedure bias 14.4 8.2 2.0 - -
background 11.2 2.3 3.3 5.0 2.2
Detector charge bias - - - 1.0 0.9
Total fit error 21.1 9.3 5.1 6.1 2.4
Relative efficiency error 11.6$\%$ 7.2$\%$ 7.0$\%$ - -
--------------------------- -------------------------- ----------------------------- ---------------- ---------------------------------------- ------------------------
: Summary of the systematic uncertainties.
\[tab:sys\_table\]
To validate the fit procedure, we perform fits on large MC samples that contain the measured number of signal and continuum events and the expected -background. Biases observed in these tests are largely due to correlations between the discriminating variables, which are not accounted for in the PDFs. For $\rho^+ \pi^0$ and $\rho^0 \pi^+$ they are not negligible and are used to correct the fitted signal yields. In addition, the full fit biases are assigned as systematic uncertainties on all three signal yields.
Contributions to the systematic errors are summarized in Table \[tab:sys\_table\]. Uncertainties in the signal MC simulation are obtained from a topologically similar control sample of fully reconstructed $B^{0} \rightarrow D^{-} \rho^{+}$ decays. For the $\bchtorchp$ channel we also use $B^{+} \rightarrow K^{+} \pi^{0}$ decays to estimate the uncertainty in the $\dE$ model. We vary the signal parameters, that are fixed in the fit, within their estimated errors and assign the effects on the signal yields and charge asymmetries as systematic errors. The expected yields from the -background modes are varied according to the uncertainties in the measured or estimated branching fractions. Since -background modes may exhibit direct violation, the corresponding charge asymmetries are varied within their physical ranges. For $B^0\to\rho^0\pi^0$, the systematic uncertainty due to interference with $B^0\to\rho^+\pi^-$ is found to be 1.5 events. This is obtained by repeating the fit to data, after removing the cut on $m(\pi^{\pm}\pi^0)$. Systematic errors due to possible nonresonant $B^0\to\pi^+ \pi^-\pi^0$ decays are derived from experimental limits [@bib:cleo_rhopi]. Contributions from nonresonant $B^+\rightarrow \pi^+\pi^0\pi^0$ for the $\rho^+\pi^0$ mode and $\B^+ \rightarrow \pi^+\pi^-\pi^+$ for the $\rho^0\pi^+$ mode are estimated to be negligible. For the $\bchtorpch$ and $\btorp$ decay modes, systematic uncertainties due to interference between $\rho^0$ and $f_0(980)$ or a possible broad scalar $\sigma(400-1200)$ were also studied and found to be negligible. Repeating the selection and fit for all three modes, without using the $\rho$-candidate mass and helicity angle, gives results that are compatible with those reported here. In the $\bchtorpch$ case, the analysis was repeated in the region $|\cos\theta_{\rho}|<0.25$, and the resulting signal yield was consistent with zero.
4.5cm 4.5cm
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After correcting for the fit biases we find from the maximum likelihood fits the event yields, $
N({\rho^+\pi^0}) = 169.0\pm28.7$, $N({\rho^0 \pi^+}) = 237.9\pm26.5$, and $N({\rho^0 \pi^0}) = 24.9\pm11.5$, where the errors are statistical only. Figure \[fig:ProjMesDE\] shows distributions of $\mes$ and $\dE$, enhanced in signal content by cuts on the signal-to-continuum likelihood ratios of the other discriminating variables. The statistical significance of the previously unobserved $\bchtorchp$ signal amounts to $7.3 \sigma$, computed as $\sqrt{2\Delta \mathrm{log}{\cal L}}$, where $\Delta$log${\cal L}$ is the log-likelihood difference between a signal hypothesis corresponding to the bias-corrected yield and a signal hypothesis corresponding to a yield that equals one standard deviation of the systematic error. We find the branching fractions to be $$\begin{aligned}
{\cal B}(\B^+\to\rho^{+}\pi^{0}) &=&
(10.9 \pm 1.9 \pm1.9)\times 10^{-6}\,, \\
{\cal B}(\B^+\to\rho^{0}\pi^{+}) &=&
\phantom{1}(9.5 \pm 1.1 \pm 0.8)\times 10^{-6}\,, \\
{\cal B}( \B^0\rightarrow\rho^0\pi^0) &=&
\phantom{1}(1.4 \pm 0.6 \pm 0.3)\times 10^{-6}\,, \end{aligned}$$ where the first errors are statistical and the second systematic. The systematic errors include the uncertainties in the efficiencies, which are dominated by the uncertainty in the $\pi^0$ reconstruction efficiency and in the case of $\rho^0\pi^+$, by the uncertainty due to particle identification.
Here we define the $B^0\to\rho^0\pi^0$ branching ratio by including those events that pass our selection and are fitted as signal but excluding those events that can be interpreted as $B^0\to\rho^+\pi^-$ with a $\rho^+$, whose mass is closer to $0.77{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ than the mass of the reconstructed $\rho^0$. The signal significance for $\rho^0 \pi^0$, including statistical and systematic errors, is $2.1 \sigma$, and we use a limit setting procedure similar to Ref. [@Frequentist] to obtain a $90\%$ Confidence-Level upper limit on its branching fraction. Fits on MC samples are used to find the signal hypothesis for which the ratio of the probablity that the fitted signal yield is less than that observed in data, and the probablity that the fitted yield is less than that in data under the null signal hypothesis, is 0.1. This signal hypothesis is shifted up by one sigma of the systematic error and the efficiency is shifted down also by one sigma. This method gives an upper limit of ${\cal B}( \B^0\rightarrow\rho^0\pi^0) < 2.9\times 10^{-6}$.
Theoretical predictions of the ratio of branching fractions $R\equiv {\cal B}(\Bz\to\rho^{\pm}\pi^{\mp})/{\cal B}(\B^{+}\to\rho^{0}\pi^{+})$, vary over a wide range. Tree level estimates suggest $R\simeq 6$ [@bib:Bauer], while the inclusion of penguin contributions, off-shell $B^*$ excited states and scalar $\pi^+\pi^-$ resonances leads to lower values, $R\simeq2-3$ [@bib:Gardner]. Using the measured $\bchtorpch$ branching fraction and the $B^0\to\rho^{\pm}\pi^{\mp}$ branching fraction from Ref. [@bib:350PRL] we find $R=2.38^{+0.37}_{-0.31}({\rm stat})^{+0.24}_{-0.20}({\rm syst})$, which is in agreement with previous experimental results [@bib:cleo_rhopi].
For the charged decays we find the charge asymmetries, $ A^{\rho^{+}\pi^0}_{CP} = 0.24 \pm 0.16 \pm 0.06,
A^{\rho^{0}\pi^{+}}_{CP} = -0.19 \pm 0.11 \pm0.02$, with contributions to the systematic errors listed in Table \[tab:sys\_table\].
In summary, we have presented measurements of branching fractions and -violating charge asymmetries in $\bchtorchp$ and $\bchtorpch$ decays, and a search for the decay $\btorp$. We observe the decay $\B^{+}\rightarrow\rho^{+}\pi^0$ with a statistical significance of $ 7.3\sigma$. We also find a branching fraction for $\B^{+}\to\rho^0\pi^{+}$ that is consistent with previous measurements [@bib:cleo_rhopi], and set an upper limit for $\btorp$. We do not observe evidence for direct violation.
acknow\_PRL.tex
[99]{}
Collaboration, B. Aubert , hep-ex/0306030, submitted to Phys. Rev. Lett. (2003).
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---
abstract: 'This is an expanded version of my talk given at the International Conference “Algebra and Number Theory” dedicated to the 80th anniversary of V. E. Voskresenskii, which was held at the Samara State University in May 2007. The goal is to give an overview of results of V. E. Voskresenskii on arithmetic and birational properties of algebraic tori which culminated in his monograph “Algebraic Tori” published in Russian 30 years ago. I shall try to put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the English version of the monograph “Algebraic Groups and Their Birational Invariants” appeared.'
address: 'Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, ISRAEL'
author:
- Boris Kunyavskiĭ
title: '[**Algebraic tori — thirty years after**]{}'
---
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\[theorem\][Proposition]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Sublemma]{}
\[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Examples]{} \[theorem\][Remarks]{} \[theorem\][Remark]{} \[theorem\][Algorithm]{} \[theorem\][Question]{} \[theorem\][Problem]{} \[theorem\] \[theorem\][Claim]{}
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{#intro .unnumbered}
Rationality and nonrationality problems {#sec:rat}
=======================================
A classical problem, going back to Pythagorean triples, of describing the set of solutions of a given system of polynomial equations by rational functions in a certain number of parameters ([*rationality problem*]{}) has been an attraction for many generations. Although a lot of various techniques have been used, one can notice that after all, to establish rationality, one usually has to exhibit some explicit parameterization such as that obtained by stereographic projection in the Pythagoras problem. The situation is drastically different if one wants to establish non-existence of such a parameterization ([*nonrationality problem*]{}): here one usually has to use some known (or even invent some new) [*birational invariant*]{} allowing one to detect nonrationality by comparing its value for the object under consideration with some “standard” one known to be zero; if the computation gives a nonzero value, we are done. Evidently, to be useful, such an invariant must be (relatively) easily computable. Note that the above mentioned subdivision to rationality and nonrationality problems is far from being absolute: given a class of objects, an ultimate goal could be to introduce some computable birational invariant giving [*necessary and sufficient*]{} conditions for rationality. Such a task may be not so hopeless, and some examples will be given below.
In this section, we discuss several rationality and nonrationality problems related to [*[algebraic tori]{}*]{}. Since this is the main object of our consideration, for the reader’s convenience we shall recall the definition.
\[def:tor\] Let $k$ be a field. An algebraic $k$-torus $T$ is an algebraic $k$-group such that over a (fixed) separable closure $\bar k$ of $k$ it becomes isomorphic to a direct product of $d$ copies of the multiplicative group: $$T\times _k\bar k\cong {{\mathbb G}}_{\text{\rm{m}},\bar k}^d$$ (here $d$ is the dimension of $T$).
We shall repeatedly use the duality of categories of algebraic $k$-tori and finite-dimensional torsion-free ${{\mathbb Z}}$-modules viewed together with the action of the Galois group $\gg :=\Gal (\kb /k)$ which is given by associating to a torus $T$ its $\gg$-module of characters $\Th :=\Hom (T\times \kb, {{\mathbb G}}_{\text{\rm{m}},\kb })$. Together with the fact that $T$ splits over a finite extension of $k$ (we shall denote by $L$ the smallest of such extensions and call it the [*minimal splitting field*]{} of $T$), this allows us to reduce many problems to considering (conjugacy classes of) finite subgroups of $GL(d,{{\mathbb Z}})$ corresponding to $\Th$.
Tori of small dimension
-----------------------
There are only two subgroups in $GL(1,{{\mathbb Z}})$: $\{(1)\}$ and $\{(1),(-1)\}$, both corresponding to $k$-rational tori: ${{\mathbb G}}_{{\text{\rm{m}}},k}$ and $R_{L/k}{{\mathbb G}}_{{\text{\rm{m}}},L}/{{\mathbb G}}_{{\text{\rm{m}}},k}$, respectively (here $L$ is a separable quadratic extension of $k$ and $R_{L/k}$ stands for the Weil functor of restriction of scalars).
For $d=2$ the situation was unclear until the breakthrough obtained by Voskresenskiĭ [@Vo67] who proved
\[th:two\] All two-dimensional tori are $k$-rational.
For $d=3$ there exist nonrational tori, see [@Ku87] for birational classification.
For $d=4$ there is no classification, some birational invariants were computed in [@Po].
\[rem-class-free\] The original proof of rationality of the two-dimensional tori in [@Vo67] is based on the classification of finite subgroups in $GL(2,{{\mathbb Z}})$ and case-by-case analysis. In the monographs [@Vo-rus] and [@Vo-eng] a simplified proof is given, though using the classification of [*maximal*]{} finite subgroups in $GL(2,{{\mathbb Z}})$. Merkurjev posed a question about the existence of a classification-free proof. (Note that one of his recent results on 3-dimensional tori [@Me] (see Section \[sec:R\] below) was obtained without referring to the birational classification given in [@Ku87].) During discussions in Samara, Iskovskikh and Prokhorov showed me a proof only relying on an “easy” part of the classification theorem for rational surfaces.
\[rem:cremona\] One should also note a recent application of Theorem \[th:two\] to the problem of classification of elements of prime order in the Cremona group $Cr(2,{{\mathbb Q}})$ of birational transformations of plane (see [@Se 6.9]). The problem was recently solved by Dolgachev and Iskovskikh [@DI]. See the above cited papers for more details.
Invariants arising from resolutions {#sec:res}
-----------------------------------
Starting from the pioneering works of Swan and Voskresenskiĭ on Noether’s problem (see Section \[sec:N\]), it became clear that certain resolutions of the Galois module $\Th$ play an important role in understanding birational properties of $T$. These ideas were further developed by Lenstra, Endo and Miyata, Colliot-Thélène and Sansuc (see [@Vo-rus], [@Vo-eng] for an account of that period); they were pursued through several decades, and some far-reaching generalizations were obtained in more recent works (see the remarks at the end of this and the next sections).
For further explanations we need to recall some definitions. Further on “module” means a finitely generated ${{\mathbb Z}}$-free $\gg$-module.
\[def:fl\] We say that $M$ is a [*permutation*]{} module if it has a ${{\mathbb Z}}$-basis permuted by $\gg$. We say that modules $M_1$ and $M_2$ are similar if there exist permutation modules $S_1$ and $S_2$ such that $M_1\oplus S_1\cong M_2\oplus S_2$. We denote the similarity class of $M$ by $[M]$. We say that $M$ is a [*coflasque*]{} module if $H^1(\hh , M)=0$ for all closed subgroups $\hh$ of $\gg$. We say that $M$ is a [*flasque*]{} module if its dual module $M^{\circ}:=\Hom (M,{{\mathbb Z}})$ is coflasque.
The following fact was first established in the case when $k$ is a field of characteristic zero by Voskresenskiĭ [@Vo69] by a geometric construction (see below) and then in [@EM], [@CTS77] in a purely algebraic way (the uniqueness of $[F]$ was established independently by all above authors).
\[th:fl\] Any module $M$ admits a resolution of the form $$0\to M\to S \to F \to 0,
\label{eq:fl}$$ where $S$ is a permutation module and $F$ is a flasque module. The similarity class $[F]$ is determined uniquely.
We denote $[F]$ by $p(M)$. We call (\[eq:fl\]) the flasque resolution of $M$. If $T$ is a $k$-torus with character module $M$, the sequence of tori dual to $(\ref{eq:fl})$ is called the flasque resolution of $T$.
Seeming strange at the first glance, the notion of flasque module (and the corresponding flasque torus) turned out to be very useful. Its meaning is clear from the following theorem due to Voskresenskiĭ [@Vo-rus 4.60]:
\[th:stable\] If tori $T_1$ and $T_2$ are birationally equivalent, then $p(\hat T_1)=p(\hat T_2)$. Conversely, if $p(\hat T_1)=p(\hat T_2)$, then $T_1$ and $T_2$ are [*s*tably equivalent]{}, i.e. $T_1\times {{\mathbb G}}_{\text{\rm{m}},k}^{d_1}$ is birationally equivalent to $T_2\times {{\mathbb G}}_{\text{\rm{m}},k}^{d_2}$ for some integers $d_1$, $d_2$.
Theorem \[th:stable\] provides a birational invariant of a torus which can be computed in a purely algebraic way. Moreover, it yields other invariants of cohomological nature which are even easier to compute. The most important among them is $H^1(\gg ,F)$. One should note that it is well-defined in light of Shapiro’s lemma (because $H^1(\gg ,S)=0$ for any permutation module $S$) and, in view of the inflation-restriction sequence, can be computed at a finite level: if $L/k$ is the minimal splitting field of $T$ and $\Gamma=\Gal (L/k)$, we have $H^1(\gg,F)=H^1(\Gamma ,F)$. The finite abelian group $H^1(\Gamma ,F)$ is extremely important for birational geometry and arithmetic, see below.
\[rem:fl-res\] Recently Colliot-Thélène [@CT07] discovered a beautiful generalization of the flasque resolution in a much more general context, namely, for an arbitrary connected linear algebraic group $G$.
Geometric interpretation of the flasque resolution
--------------------------------------------------
As mentioned above, the flasque resolution (\[eq:fl\]) was originally constructed in a geometric way. Namely, assuming $k$ is of characteristic zero, one can use Hironaka’s resolution of singularities to embed a given $k$-torus $T$ into a smooth complete $k$-variety $V$ as an open subset and consider the exact sequence of $\gg$-modules $$0\to \hat T \to S \to \operatorname{Pic\,}\Vb \to 0,
\label{eq:Pic}$$ where $\Vb =V\times_k\kb$, $\operatorname{Pic\,}\Vb$ is the Picard module, and $S$ is a permutation module (it is generated by the components of the divisors of $\Vb$ whose support is outside $\overline T$). With another choice of an embedding $T\to V'$ we are led to an isomorphism $\operatorname{Pic\,}\Vb \oplus S_1\cong \operatorname{Pic\,}\Vb '\oplus S_2$, so the similarity class $[\operatorname{Pic\,}\Vb ]$ is well defined. Voskresenskiĭ established the following property of the Picard module (see [@Vo-rus 4.48]) which is of utmost importance for the whole theory:
\[th:fl-mod\] The module $\operatorname{Pic\,}\Vb$ is flasque.
As mentioned above, this result gave rise to a purely algebraic way of constructing the flasque resolutions, as well as other types of resolutions; flasque tori and torsors under such tori were objects of further thorough investigation [@CTS87a], [@CTS87b].
\[rem:bryl\] The geometric method of constructing flasque resolutions described above can be extended to arbitrary characteristic. This can be done in the most natural way after the gap in Brylinski’s proof [@Br] of the existence of a smooth complete model of any torus have been filled in [@CTHS].
\[rem:fl-mod\] Recently Theorem \[th:fl-mod\] was extended to the Picard module of a smooth compactification of an arbitrary connected linear algebraic group [@BK04] and, even more generally, of a homogeneous space of such a group with connected geometric stablizer [@CTK06]. In each case, there is a reduction to the case of tori (although that in [@CTK06] is involved enough).
Noether’s problem {#sec:N}
-----------------
One of the most striking applications of the birational invariant described above is a construction of counter-examples to a problem of E. Noether on rationality of the field of rational functions invariant under a finite group $G$ of permutations. Such an example first appeared in a paper by Swan [@Sw] where tori were not mentioned but a resolution of type (\[eq:fl\]) played a crucial role; at the same time Voskresenskiĭ [@Vo69] considered the same resolution to prove that a certain torus is nonrational. In a later paper [@Vo70] he formulated in an explicit way that the field of invariants under consideration is isomorphic to the function field of an algebraic torus. This discovery yielded a series of subsequent papers (Endo–Miyata, Lenstra, and Voskresenskiĭ himself) which led to almost complete understanding of the case where the finite group $G$ acting on the function field is abelian; see [@Vo-rus Ch. VII] for a detailed account. Moreover, the idea of realizing some field of invariants as the function field of a certain torus proved useful in many other problems of the theory of invariants, see works of Beneish, Hajja, Kang, Lemire, Lorenz, Saltman, and others; an extensive bibliography can be found in the monograph [@Lo], see [@Ka] and references therein for some more recent development; note also an alternative approach to Noether’s problem based on cohomological invariants (Serre, in [@GMS]).
Note a significant difference in the proofs of nonrationality for the cases $G={{\mathbb Z}}_{47}$ (the smallest counter-example for a cyclic group of prime order) and $G={{\mathbb Z}}_8$ (the smallest counter-example for an arbitrary cyclic group). If $G$ is a cyclic group of order $q=p^n$, and $k$ is a field of characteristic different from $p$, the corresponding $k$-torus splits over the cyclotomic extension $k(\zeta_q)$. If $p>2$, the extension $k(\zeta_q)/k$ is cyclic. According to [@EM], if a $k$-torus $T$ splits over an extension $L$ such that all Sylow subgroups of $\Gal (L/k)$ are cyclic, then the $\Gamma$-module $F$ in the flasque resolution for $T$ is a direct summand of a permutation module. Thus to prove that it is not a permutation module, one has to use subtle arguments. If $p=2$ and $n\ge 3$, the Galois group $\Gamma$ of $k(\zeta_q)/k$ may be noncyclic and contain a subgroup $\Gamma'$ such that $H^1(\Gamma ',F)\ne 0$ which guarantees that $F$ is not a permutation module and hence $T$ is not rational. This important observation was made in [@Vo73]. Another remark should be done here: for the tori appearing in Noether’s problem over a field $k$ with cyclic $G$ such that $\char (k)$ is prime to the exponent of $G$, triviality of the similarity class $[F]$ is necessary and [*sufficient*]{} condition for rationality of $T$ (and hence of the corresponding field of invariants). This is an important instance of the following phenomenon: in a certain class of tori any stably rational torus is rational. The question whether this principle holds in general is known as Zariski’s problem for tori and is left out of the scope of the present survey.
The group $H^1(\Gamma ,\operatorname{Pic\,}\Xb )$, where $X$ is a smooth compactification of a $k$-torus $T$, admits another interpretation: it is isomorphic to $\operatorname{Br\,}X/\operatorname{Br\,}k$, where $\operatorname{Br\,}X$ stands for the Brauer–Grothendieck group of $X$. This birational invariant, later named the unramified Brauer group, played an important role in various problems, as well as its generalization for higher unramified cohomology (see [@CT96a], [@Sa96] for details). Let us only note that the main idea here is to avoid explicit construction of a smooth compactification of an affine variety $V$ under consideration, trying to express $\operatorname{Br\,}X/\operatorname{Br\,}k$ in terms of $V$ itself. In the toric case this corresponds to the formula [@CTS87b] $$\operatorname{Br\,}X/\operatorname{Br\,}k = \ker [H^2(\Gamma ,\Th) \to \prod_C H^2(C,\Th )],
\label{eq:sha}$$ where the product is taken over all cyclic subgroups of $\Gamma$. Formulas of similar flavour were obtained for the cases where $V=G$, an arbitrary connected linear algebraic group [@CTK98], [@BK00], and $V=G/H$, a homogeneous space of a simply connected group $G$ with connected stabilizer $H$ [@CTK06]; the latter formula was used in [@CTKPR] for proving nonrationality of the field extensions of the form $k(V)/k(V)^G$, where $k$ is an algebraically closed field of characteristic zero, $G$ is a simple $k$-group of any type except for $A_n$, $C_n$, $G_2$, and $V$ is either the representation of $G$ on itself by conjugation or the adjoint representation on its Lie algebra.
It is also interesting to note that the same invariant, the unramified Brauer group of the quotient space $V/G$, where $G$ is a finite group and $V$ its faithful complex representation, was used by Saltman [@Sa84] to produce the first counter-example to Noether’s problem over ${{\mathbb C}}$. In the same spirit as in formula (\[eq:sha\]) above, this invariant can be expressed solely in terms of $G$: it equals $$B_0(G):= \ker [H^2(G ,{{\mathbb Q}}/{{\mathbb Z}}) \to \prod_A H^2(A,{{\mathbb Q}}/{{\mathbb Z}})],
\label{eq:Bog}$$ where the product is taken over all abelian subgroups of $G$ (and, in fact, may be taken over all bicyclic subgroups of $G$) [@Bo87]. This explicit formula yielded many new counter-examples (all arising for nilpotent groups $G$, particularly from $p$-groups of nilpotency class 2). The reader interested in historical perspective and geometric context is referred to [@Sh], [@CTS07], [@GS 6.6, 6.7], [@Bo07]. We only mention here some recent work [@BMP], [@Ku07] showing that such counter-examples cannot occur if $G$ is a simple group; this confirms a conjecture stated in [@Bo92].
Generic tori {#sec:gen}
------------
Having at our disposal examples of tori with “good” and “bad” birational properties, it is natural to ask what type of behaviour is typical. Questions of such “nonbinary” type, which do not admit an answer of the form “yes-no”, have been considered by many mathematicians, from Poincaré to Arnold, as the most interesting ones. In the toric context, the starting point was the famous Chevalley–Grothendieck theorem stating that the variety of maximal tori in a connected linear algebraic group $G$ is rational. If $G$ is defined over an algebraically closed field, its underlying variety is rational. However, if $k$ is not algebraically closed, $k$-rationality (or nonrationality) of $G$ is a hard problem. The Chevalley–Grothendieck theorem gives a motivation for studying generic tori in $G$: if this torus is rational, it gives the $k$-rationality of $G$. The notion of generic torus can be expressed in Galois-theoretic terms: these are tori whose minimal splitting field has “maximal possible” Galois group $\Gamma$ (i.e. $\Gamma$ lies between $W(R)$ and $\operatorname{Aut}(R)$ where $R$ stands for the root system of $G$). This result, going back to E. Cartan, was proved in [@Vo88]. In this way, Voskresenskiĭand Klyachko [@VoKl] proved the rationality of all adjoint groups of type $A_{2n}$. The rationality was earlier known for the adjoint groups of type $B_n$, the simply connected groups of type $C_n$, the inner forms of the adjoint groups of type $A_n$, and (after Theorem \[th:two\]) for all groups of rank at most two. It turned out that for the adjoint and simply connected groups of all remaining types the generic torus is not even stably rational [@CK]; in most cases this was proved by computing the birational invariant $H^1(\Gamma ',F)$ for certain subgroups $\Gamma'\subset\Gamma$. In the case of inner forms of simply connected groups of type $A_n$, corresponding to generic norm one tori, this confirms a conjecture by Le Bruyn [@LB] (independently proved later in [@LL]). The above theorem was extended to groups which are neither simply connected nor adjoint and heavily used in the classification of linear algebraic groups admitting a rational parameterization of Cayley type [@LPR].
The above mentioned results may give an impression that except for certain types of groups the behaviour of generic tori is “bad” from birational point of view. However, there is also a positive result: if $T$ is a generic torus in $G$, then $H^1(\Gamma ,F)=0$, This was first proved in [@VoKu] for generic tori in the classical simply connected (type $A_n$) and adjoint groups, and in [@Kl89] in the general case. (An independent proof for the simply connected case was communicated to the author by M. Borovoi.) This result has several number-theoretic applications, see Section \[sec:arith\].
Yet another approach to the notion of generic torus was developed in [@Gr05b] where the author, with an eye towards arithmetic applications, considered maximal tori in semisimple simply connected groups arising as the centralizers of regular semisimple stable conjugacy classes.
Relationship with arithmetic {#sec:arith}
============================
Global fields: Hasse principle and weak approximation
-----------------------------------------------------
According to a general principle formulated in [@Ma], the influence of (birational) geometry of a variety on its arithmetic (diophantine) properties may often be revealed via some algebraic (Galois-cohomological) invariants. In the toric case such a relationship was discovered by Voskresenskiĭ and stated as the exact sequence (see [@Vo-rus 6.38]) $$0\to A(T) \to H^1(k, \operatorname{Pic\,}\Vb )\,\tilde{} \to {\mbox{\rus{\fontsize{11}{11pt}\selectfont{SH}}}}(T) \to 0,
\label{eq:a-sh}$$ where $k$ is a number field, $T$ is a $k$-torus, $V$ is a smooth compactification of $T$, $A(T)$ is the defect of weak approximation, ${\mbox{\rus{\fontsize{11}{11pt}\selectfont{SH}}}}(T)$ is the Shafarevich–Tate group, and $\tilde{}$ stands for the Pontrjagin duality of abelian groups. The cohomological invariant in the middle, being a purely algebraic object, governs arithmetic properties of $T$.
On specializing $T$ to be the norm one torus corresponding to a finite field extension $K/k$, we get a convenient algebraic condition sufficient for the Hasse norm principle to hold for $K/k$. In particular, together with results described in Section \[sec:gen\], this shows that the Hasse norm principle holds “generically”, i.e. for any field extension $K/k$ of degree $n$ such that the Galois group of the normal closure is the symmetric group $S_n$ [@VoKu]. (Another proof was independently found for $n>7$ by Yu. A. Drakokhrust.) Another application was found in [@KS]: combining the above mentioned theorem of Klyachko with the Chevalley–Grothendieck theorem and Hilbert’s irreducibility theorem, one can produce a uniform proof of weak approximation property for all simply connected, adjoint and absolutely almost simple groups. (Another uniform proof was found by Harari [@Ha] as a consequence of a stronger result on the uniqueness of the Brauer–Manin obstruction, see the next paragraph.) Yet another interesting application of sequence (\[eq:a-sh\]) refers to counting points of bounded height on smooth compactifications of tori [@BT95], [@BT98]: the constant appearing in the asymptotic formula of Peyre [@Pe] must be corrected by a factor equal to the order of $H^1(k, \operatorname{Pic\,}\Vb )$ and arising on the proof as the product of the orders of $A(T)$ and ${\mbox{\rus{\fontsize{11}{11pt}\selectfont{SH}}}}(T)$ (I thank J.-L. Colliot-Thélène for this remark).
The sequence (\[eq:a-sh\]) was extended by Sansuc [@San] to the case of arbitrary linear algebraic groups. On identifying the invariant in the middle with $\operatorname{Br\,}V/\operatorname{Br\,}k$, as in Section \[sec:N\], one can put this result into more general context of the so-called Brauer–Manin obstruction to the Hasse principle and weak approximation (which is thus the only one for principal homogeneous spaces of linear algebraic groups). This research, started in [@Ma], gave many impressive results. It is beyond the scope of the present survey.
\[rem:strong\] Other types of approximation properties for tori have been considered in [@CTSu] (weaker than weak approximation), [@Ra], [@PR] (strong approximation with respect to certain infinite sets of primes with infinite complements — so-called generalized arithmetic progressions). In the latter paper generic tori described above also played an important role. They were also used in [@CU] in a quite different arithmetic context.
Arithmetic of tori over more general fields {#sec:fields}
-------------------------------------------
Approximation properties and local-global principles were studied for some function fields. In [@An] the exact sequence (\[eq:a-sh\]) has been extended to the case where the ground field $k$ is pseudoglobal, i.e. $k$ is a function field in one variable whose field of constants $\kappa$ is pseudofinite (this means that $\kappa$ has exactly one extension of degree $n$ for every $n$ and every absolutely irreducible affine $\kappa$-variety has a $\kappa$-rational point). In [@CT96b] weak approximation and the Hasse principle were established for any torus defined over $\R (X)$ where $X$ is an irreducible real curve. This allows one to establish these properties for arbitrary groups over such fields, and, more generally, over the fields of virtual cohomological dimension 1 [@Sch]. The same properties for tori defined over some geometric fields of dimension 2 (such as a function field in two variables over an algebraically closed field of characteristic zero, or the fraction field of a two-dimensional. exce;;ent, Henselian, local domain with algebraically closed residue field, or the field of Laurent series in one variable over a field of characteristic zero and cohomological dimension one) were considered in [@CTGP]. Here one can note an interesting phenomenon: there are counter-examples to weak approximation but no counter-example to the Hasse principle is known. One can ask whether there exists some Galois-cohomological invariant of tori defined over more general fields whose vanishing would guarantee weak approximation property for the torus under consideration. Apart from the geometric fields considered in [@CTGP], another interesting case could be $k={{\mathbb Q}}_p(X)$, where $X$ is an irreducible ${{\mathbb Q}}_p$-curve; here some useful cohomological machinery has been developed in [@SvH].
Integral models and class numbers of tori
-----------------------------------------
The theory of integral models of tori, started by Raynaud who constructed an analogue of the Néron smooth model [@BLR], has been extensively studied during the past years, and some interesting applications were found using both Néron–Raynaud models and Voskresenskiĭ’s “standard” models. The interested reader is referred to the bibliography in [@VKM]. Some more recent works include standard integral models of toric varieties [@KM] and formal models for some classes of tori [@DGX].
Main results on class numbers of algebraic tori are summarized in [@Vo-eng]. One can only add that the toric analogue of Dirichlet’s class number formula established in [@Shyr] suggests that a toric analogue of the Brauer–Siegel theorem may also exist. A conjectural formula of the Brauer–Siegel type for constant tori defined over a global function field can be found in a recent paper [@KT].
$R$-equivalence and zero-cycles {#sec:R}
===============================
$R$-equivalence on the set of rational points of an algebraic variety introduced in [@Ma] turned out to be an extremely powerful birational invariant. Its study in the context of algebraic groups, initiated in [@CTS77], yielded many striking achievements. We shall only recall here that the first example of a simply connected group whose underlying variety is not $k$-rational is a consequence of an isomorphism, established by Voskresenskiĭ, between $G(k)/R$, where $G=SL(1,D)$, the group of norm 1 elements in a division algebra over $k$, with the reduced Whitehead group $SK_1(D)$; as the latter group may be nonzero because of a theorem by Platonov [@Pl], this gives the needed nonrationality of $G$. This breakthrough gave rise to dozens of papers on the topic certainly deserving a separate survey. For the lack of such, the interested reader is referred to [@Vo-eng Ch. 6], [@Gi07a Sections 24–33], [@Gi07b].
As to $R$-equivalence on tori, the most intriguing question concerns relationship between $T(k)/R$ and the group $A_0(X)$ of classes of 0-cycles of degree 0 on a smooth compactification $X$ of $T$. In a recent paper [@Me] Merkurjev proved that these two abelian groups are isomorphic if $T$ is of dimension 3 (for tori of dimension at most 2 both groups are zero because of their birational invariance and Theorem \[th:two\]). For such tori he also obtained a beautiful formula expressing $T(k)/R$ in “intrinsic” terms, which does not require constructing $X$ as above nor a flasque resolution of $\Th$ as in [@CTS77]: $T(k)/R\cong H^1(k,T^{\circ})/R$, where $T^{\circ}$ denotes the dual torus (i.e. $\hat{T^{\circ }}=\Hom (\Th ,{{\mathbb Z}})$). (In the general case, it is not even known whether the map $X(k)/R\to A_0(X)$ is surjective, see [@CT05 §4] for more details.) As a consequence, Merkurjev obtained an explicit formula for the Chow group $CH_0(T)$ of classes of 0-cycles on a torus $T$ of dimension at most 3: $CH_0(T)\cong T(k)/R\oplus {{\mathbb Z}}_{i_T}$ where $i_T$ denotes the greatest common divisor of the degrees of all field extensions $L/k$ such that the torus $T_L$ is isotropic. The proofs, among other things, use earlier results [@Kl82], [@MP] on the $K$-theory of toric varieties. As mentioned above, they do not rely on the classification of 3-dimensional tori.
To conclude this section, one can also add the same references [@An], [@CTGP] as in Section \[sec:fields\] for $R$-equivalence on tori over more general fields.
Applications in information theory
==================================
Primality testing
-----------------
One should mention here several recent papers [@Gr05a], [@Ki] trying to interpret in toric terms some known methods for checking whether a given integer $n$ is a prime. In fact, this approach goes back to a much older paper [@CC] where the authors noticed symmetries in the sequences of Lucas type used in such tests (though algebraic tori do not explicitly show up in [@CC]).
Public-key cryptography
-----------------------
A new cryptosystem based on the discrete logarithm problem in the group of rational points of an algebraic torus $T$ defined over a finite field was recently invented by Rubin and Silverberg [@RS03], [@RS04a], [@RS04b]. Since this cryptosystem possesses a compression property, i.e. allows one to use less memory at the same security level, it drew serious attention of applied cryptographers and yielded a series of papers devoted to implementation issues [@DGPRSSW], [@DW], [@GV], [@Ko] (in the latter paper another interesting approach is suggested based on representing a given torus as a quotient of the generalized jacobian of a singular hyperelliptic curve). In [@GPS] the authors propose to use a similar idea of compression for using tori in an even more recent cryptographic protocol (so-called pairing-based cryptography). It is interesting to note that the efficiency (compression factor) of the above mentioned cryptosystems heavily depends on [*rationality*]{} of tori under consideration (more precisely, on an explicit rational parameterization of the underlying variety). As the tori used by Rubin and Silverberg are known to be stably rational, the seemingly abstract question on rationality of a given stably rational torus is moving to the area of applied mathematics. The first challenging problem here is to obtain an explicit rational parameterization of the 8-dimensional torus $T_{30}$, defined over a finite field $k$ and splitting over its cyclic extension $L$ of degree 30, whose character module $\Th _{30}$ is isomorphic to ${{\mathbb Z}}[\zeta_{30}]$, where ${{\mathbb Z}}[\zeta_{30}]$ stands for a primitive 30th root of unity. (Here is an alternative description of $T_{30}$: it is a maximal torus in $E_8$ such that $\Gal (L/k)$ acts on $\Th _{30}$ as the Coxeter element of $W(E_8)$; this can be checked by a direct computation or using [@BF].)
This is a particular case of a problem posed by Voskresenskiĭ [@Vo-rus Problem 5.12] 30 years ago. Let us hope that we will not have to wait another 30 years for answering this question on a degree 30 extension.
[*Acknowledgements*]{}. The author’s research was supported in part by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics and by a grant from the Ministry of Science, Culture and Sport, Israel, and the Russian Foundation for Basic Research, the Russian Federation. This paper was written during the visit to the MPIM (Bonn) in August–September 2007. The support of these institutions is highly appreciated. I thank J.-L. Colliot-Thélène for many helpful remarks.
[DGPRSSW]{}
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|
---
abstract: 'We present the first statistical analysis of 27 Ultra-violet Optical Telescope (UVOT) optical/ultra-violet lightcurves of Gamma-Ray Burst (GRB) afterglows. We have found, through analysis of the lightcurves in the observer’s frame, that a significant fraction rise in the first 500s after the GRB trigger, that all lightcurves decay after 500s, typically as a power-law with a relatively narrow distribution of decay indices, and that the brightest optical afterglows tend to decay the quickest. We find that the rise could either be produced physically by the start of the forward shock, when the jet begins to plough into the external medium, or geometrically where an off-axis observer sees a rising lightcurve as an increasing amount of emission enters the observers line of sight, which occurs as the jet slows. We find that at 99.8% confidence, there is a correlation, in the observed frame, between the apparent magnitude of the lightcurves at 400s and the rate of decay after 500s. However, in the rest frame a Spearman Rank test shows only a weak correlation of low statistical significance between luminosity and decay rate. A correlation should be expected if the afterglows were produced by off-axis jets, suggesting that the jet is viewed from within the half-opening angle $\theta$ or within a core of uniform energy density $\theta_c$. We also produced logarithmic luminosity distributions for three rest frame epochs. We find no evidence for bimodality in any of the distributions. Finally, we compare our sample of UVOT lightcurves with the X-ray Telescope (XRT) lightcurve canonical model. The range in decay indices seen in UVOT lightcurves at any epoch is most similar to the range in decay of the shallow decay segment of the XRT canonical model. However, in the XRT canonical model there is no indication of the rising behaviour observed in the UVOT lightcurves.'
author:
- |
S. R. Oates$^{1}$, M. J. Page$^{1}$, P. Schady$^{1}$, M. de Pasquale$^{1}$, T. S. Koch$^{2}$, A. A. Breeveld$^{1}$, P. J. Brown$^{2}$, M. M. Chester$^{2}$, S. T. Holland$^{3,4,5}$, E. A. Hoversten$^{2}$, N. P. M. Kuin$^{1}$, F. E. Marshall$^{3}$, P. W. A. Roming$^{2}$, M. Still$^{1}$, D. E. Vanden Berk$^{2}$, S. Zane$^{1}$ and J. A. Nousek$^{2}$\
$^{1}$ Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking Surrey, RH5 6NT, UK; sro@mssl.ucl.ac.uk\
$^{2}$ Department of Astronomy and Astrophysics, Pennsylvania State University, 104 Davey Laboratory, University Park, PA 16802\
$^{3}$ Astrophysics Science Division, Code 660.1, NASA Goddard Space Flight Centre, 8800 Greenbelt Road, Greenbelt, Maryland 20771, USA\
$^{4}$ Universities Space Research Association, 10211 Wincopin Circle, Suite 500, Columbia, Maryland 21044, USA\
$^{5}$ Centre for Research and Exploration in Space Science and Technology, Code 668.8, NASA Goddard Space Flight Centre,\
8800 Greenbelt Road, Greenbelt, Maryland 20771, USA
bibliography:
- 'OatesSR.bib'
title: 'A statistical study of gamma-ray burst afterglows measured by the [*Swift*]{} Ultra-violet Optical Telescope'
---
\[firstpage\]
gamma-rays: bursts
Introduction {#intro}
============
Gamma-ray bursts (GRBs) release between $10^{52}$ and $10^{54}$ ergs during the prompt emission, which lasts from a few milliseconds to a few thousand seconds, and is followed by an afterglow, which is observed in the X-ray to radio range from as little as a few tens of seconds up to several months after the GRB trigger.
The energy is transported in a relativistic outflow [@mes97] that is likely anisotropic [@sar99] and the energy is expected to be released by internal and external shocks. Internal shocks [@rees94] are thought to produce the prompt gamma-ray emission, while external shocks [@rees92] are thought to produce the afterglow. Internal shocks occur when shells of material, which are thrown violently from the progenitor at different Lorentz factors, overtake each other. The external shocks are produced when the shells of material are decelerated by the external medium.
The short duration of the gamma-ray emission and the rapid decay of the afterglow motivated the construction and launch of [*Swift*]{}, a rapid response satellite. [*Swift*]{} houses 3 instruments: the Burst Alert Telescope (BAT; [@bar05]), the X-ray Telescope (XRT; [@bur05]) and the Ultra-violet Optical Telescope (UVOT; [@roming]). The energy ranges of the BAT and the XRT instruments are 15 keV - 350 keV and 0.2 - 10 keV, respectively, and the wavelength range of the UVOT is 1600Å-8000Å. The large field of view of the BAT (2 str), enables $\rm 1/6^{th}$ of the sky to be searched for GRBs at any one time. Once a GRB has been detected by the BAT, [*Swift*]{} rapidly slews allowing the XRT and UVOT to observe the afterglow within a few tens of seconds after the BAT trigger.
Since launch, [*Swift*]{} has produced a large sample of UV/optical and X-ray lightcurves which begin soon after the trigger. The high detection rate with the XRT [96%; @bur08] allowed a large number of XRT lightcurves to be obtained within the first year. The systematic reduction of this sample resulted in the discovery of a 4 segment canonical XRT lightcurve [@zhang05; @nousek]. After 2 years of operation the UVOT, with a much lower detection rate than the XRT [26%; @roming08], has detected more than 50 optical afterglows. This allows for the first time a systematic reduction and analysis of a significant sample of GRBs with optical afterglows observed with the UVOT and allows an investigation of their generic characteristics.
In this paper we present and analyze a sample of 27 UVOT lightcurves of GRB afterglows. In Section 2 we explain how we selected the sample of UVOT lightcurves and in Section 3 we describe how we systematically reduced and analyzed them. In Section 4 we present the results and in Section 5 we discuss our findings. Throughout the paper we will use the following flux convention, $ F\,\propto\,t^{\alpha}\,\nu^{\beta+1}$ with $\alpha$ and $\beta$ being the temporal and photon indices respectively. We assume the Hubble parameter $ H_0\,=\,70$$\rm km\,s^{-1}\,Mpc^{-1}$ and density parameters $\Omega_\Lambda$=0.7 and $\Omega_m$=0.3. Unless stated otherwise, all uncertainties are quoted at 1$\sigma$.
The Sample
==========
To investigate the nature of GRB optical/UV lightcurves a large number of well sampled, good quality UVOT lightcurves were required. The sample was selected according to the following specific criteria: the optical/UV lightcurves must be observed in the v filter of the UVOT with a magnitude of $\leq$17.8, UVOT observations must have commenced within the first 400s after the BAT trigger and the afterglow must have been observed until at least $10^5$s after the trigger. These selection criteria ensure the lightcurves have adequate signal to noise and cover both early and late time evolution. In addition to the above criteria, the colour of the afterglows must not evolve significantly with time, meaning that at no stage should the lightcurve from a single filter significantly deviate from any of the other filter lightcurves when normalized to the v filter. This ensures that a single lightcurve can be constructed from the UVOT multi-filter observations. Three GRBs, GRB 060218, GRB 060614 and GRB 060729 were excluded as they showed significant colour evolution.
In total 27 GRBs, which occurred between 1st January 2005 and 1st August 2007, fit the selection criteria. As there were no short GRBs that met the selection criteria, all the GRBs in this sample are long. Observations, for the majority of GRBs in this sample, began within the first 100s and the optical afterglow was detected until at least $10^5$s. Formally, GRB 050820a meets our selection criteria, but we have excluded this burst from the sample because the BAT triggered on a precursor of this GRB, and the main GRB took place as [*Swift*]{} entered the South Atlantic Anomaly (SAA) with the consequence that UVOT completely missed the early phase of the afterglow.
Data Reduction
==============
After the BAT has triggered on a GRB and [*Swift*]{} has slewed, the UVOT performs a sequence of pre-programmed exposures of varying length in multiple observing modes and filters designed to balance good time resolution and spectral coverage. Observations are performed in either event mode, where the arrival time and position are recorded for every photon detected, or in image mode, in which the data are recorded as an image accumulated over a fixed period of time. The pre-programmed observations begin with the settling and finding chart exposures. The settling exposure is not included in this analysis because the cathode voltage may still increase during the first few seconds. Two finding charts follow immediately after the settling exposure and these are observed in event mode with the v and white filters (as of the 7th November 2008 the finding charts are observed in u and white). The rest of the pre-programmed observations are a combination of event and image mode observations until $\sim$2700s after the trigger, after which, only image mode is used. These observations are taken as a series of short, followed by medium and then long exposures, which are usually observed with all seven filters of the UVOT (white, v, b, u, uvw1, uvm2, uvw2). However, for some targets, the pre-programmed observation sequence may change or not be executed fully because of an observing constraint.
For GRB observations, the earliest part of the lightcurve is expected to show variability over the shortest timescales. As the finding charts, which are exposures of 100-400s, contain the earliest observations of the GRB, it is essential to obtain lightcurves from these event lists as well as obtaining lightcurves from the images.
Event and Image Data Reduction
------------------------------
To obtain the best possible lightcurves, we refined the astrometry of the event files before the count rates were obtained. The astrometry of the event list was refined by extracting an image every 10s and cross-correlating the stars in the image with those found in the USNO-B1 catalogue. The differences in RA and DEC between the stars in the image and the catalogue were converted into pixels and then applied to the position of every event in the event list during that particular 10s interval. This process was repeated until the end of each event list.
The images used in this paper were reprocessed by the the Swift Science Data Center (SDC) for the UVOT GRB catalogue [@roming08]. These images were used because not all the image files in the Swift archive have been corrected for modulo-8 fixed pattern noise; only those processed with [*Swift*]{} processing script version 3.9.9 and later have had this correction applied. For a small number of images, the aspect correction failed during the SDC processing and in these cases, the images were corrected using in-house aspect correction software.
An aperture of 5$\arcsec$, selected in order to be compatible with the UVOT calibration [@poole], was used to obtain the source count rates. However, for sources with a low count rate, it is more precise to use a smaller aperture [@poole]. Therefore, below a threshold of 0.5 counts per second the source count rates were determined from 3$\arcsec$ radius apertures and the count rates were corrected to 5$\arcsec$ using a table of aperture correction factors contained within the calibration. The background counts were obtained from circular regions with radii, typically, of 20$\arcsec$. These regions were positioned, in each GRB field, over a blank area of sky. For each GRB, the same source and background regions were used to determine the count rates from the event lists and the images. The software used can be found in the software release, [headas]{} 6.3.2 and the version 20071106 of the UVOT calibration files.
For each GRB, to maximise the signal to noise of the observed optical afterglow, the lightcurves from each filter in which the burst was detected were combined into one overall lightcurve. The lightcurves corresponding to the different filters were normalized to that in the v filter. This was possible because there was no significant colour evolution between the filters, which was one of the selection criteria described in Section 2. The normalization was determined by fitting a power-law to each of the lightcurves in a given time range simultaneously. The power-law indices were constrained to be the same for all the filters and the normalizations were allowed to vary between the filters. The ratios of the power-law normalizations were then used to normalize the count rates in each filter. The time range used to normalize the lightcurves was selected on the basis that the time range included data from all filters in which the burst was detected and (as far as possible) the lightcurves were in a power-law decay phase. Once the lightcurves were normalized, they were binned by taking the weighted average of the normalized count rates in time bins of $\delta$T/T=0.2. The overall lightcurves were converted from v count rate to v magnitude using the zero point 17.89 [@poole]. For many GRBs, the trigger time does not represent the true start of the gamma-ray emission. Therefore, the start of the gamma-ray emission was chosen as the start time of the T90 parameter. This parameter corresponds to the time in which 90% of the counts in the 15 keV - 350 keV band arrive at the detector [@sak07] and is determined from the gamma-ray event data for each GRB, by the BAT processing script. The results of the processing are publicly available and are provided for each trigger at http://gcn.gsfc.nasa.gov/swift\_gnd\_ana.html. The difference between the trigger time and the start time of the T90 parameter is typically less than a few seconds. However, in a minority of cases the difference is much larger, with the largest difference being 133.1s for GRB 050730.
Photometric Redshifts
---------------------
Spectroscopic redshifts were obtained from the literature for 19 of the GRBs in the sample (see Table \[norm\]). For a further 4 GRBs, it was possible to determine the redshift using an instantaneous UVOT-XRT Spectral Energy Distribution (SED), which was created using the method of [@schady]. The SEDs were fit with the best fitting model of either a power-law or broken power-law, with Galactic and host galaxy absorption and extinction. The Galactic $N_{H}$ was taken from the Leiden/Argentine/Bonn (LAB) Survey of Galactic HI [@kalberla] and the Galactic extinction was taken from a composite 100$\mu$m map of COBE/DIRBE and IRAS/ISSA observations [@schlegel]. For the host extinction the Small Magellanic Cloud (SMC) extinction curve [@pei92] was assumed. The host reddening and absorption, E(B-V), $N_{H}$, and the redshift were left to vary. The resulting redshifts can be found in Table \[phot\].
Luminosity Lightcurve
---------------------
Luminosity lightcurves were produced for all the GRBs whose host E(B-V) value could be determined, except for GRBs with photometric redshifts which have a 1$\sigma$ error on the redshift that corresponds to an uncertainty in log luminosity of $>0.1$. In total, luminosity lightcurves were produced for 21 of the GRBs in the sample.
For the 21 GRBs, the observed count rate was converted into luminosity at a common rest frame wavelength. In order to select the common wavelength and determine the resulting k-correction factor for each lightcurve, an SED was computed for each GRB. The SED was produced by multiplying the relative count rates in each filter, given in Table \[norm\], by the count rate-to-flux conversion factors given by [@poole]. These relative flux densities were corrected for Galactic extinction and positioned at the central wavelength of the filter in the rest frame of the GRB; the SEDs are shown in Fig. \[flux\_SED\]. The common rest frame wavelength at which to determine the luminosities was selected to maximise the number of GRBs with SEDs that include this wavelength and to be relatively unaffected by host extinction. The wavelength which best satisfies these conditions is 1600[Å]{}. The k-correction factor was taken as the flux density in the rest frame at 1600[Å]{}, $F_{1600}$, divided by the flux density at the observed central wavelength of the v filter (5402[Å]{}), $F_{\rm v}$. In the case where a GRB’s SED did not cover 1600[Å]{}, an average k-correction was determined from the other GRBs in the sample, whose SEDs covered both 1600[Å]{} and the rest frame wavelength corresponding to the v filter.
To produce the luminosities, the lightcurves in count rate were corrected for Galactic extinction, converted into flux density and then into luminosity using the following equation: $$L(1600)=4\pi D_L^2 F_{\rm v}k\label{luminsoity}$$ where $L(1600)$ is the luminosity at a 1600[Å]{}, $D_L$ is the luminosity distance, $k=(1+z)(F_{1600}/F_{\rm v})$ is the k-correction factor, and $z$ is the redshift of the GRB. Finally, the luminosity lightcurves were corrected for host extinction using the $A_{1600}$ values given in Table \[norm\]. These values were determined for the GRBs, using the $A_{\rm v}$ values reported in [@schady].
Bolometric Energy and GRB Classification
----------------------------------------
The k-corrected isotropic energy of the prompt gamma-ray emission $E_{k,iso}$, was calculated for each GRB with known redshift using Eq. 4 from [@bloo01]. The energies were corrected to a rest frame bandpass of 0.1 keV to 10000 keV using one of three spectral templates: a power-law, cut-off power-law or Band function [@band93]. As Konus-Wind has a larger energy range than BAT, the spectral analysis of the prompt emission, observed by Konus-Wind, better represents the spectral behaviour. These results were taken from the literature. The spectra observed with Konus-Wind tend to be best fit by a cut-off power-law, therefore the spectral template chosen for the GRBs that were not observed with Konus-wind, with power-law spectra of photon index $\Gamma>-2$ in the 15 keV - 150 keV energy range was a cut-off power-law with $E_{peak}$=162.2 keV [@D'alessio06]. For the GRBs with a power-law spectrum with a photon index of $\Gamma<-2$ in the 15 keV - 150 keV energy range, a Band function was used with $\Gamma_1=-0.99$ [@D'alessio06] and $E_{peak}$=15 keV. The resulting k-corrected energies can be found in Table \[redshift\].
The GRBs in this sample were classified into three categories depending on the ratio $R$ of the fluence in the 25 keV - 50 keV and 50 - 100 keV BAT energy bands, which are given in [@sak07]. The categories and their respective ratios are: an X-ray flash (XRF) for $R>1.32$; an X-ray rich GRB (XRR) for $0.72<R\leq1.32$; or a classical GRB (C-GRB) for $R\leq0.72$. Table \[fits\] lists the GRBs with their classifications. In total, there are 12 C-GRBs, 13 XRR and 1 XRF, which is GRB 060512.
--------- ------------------- ------------ ------ ------ ------ ------ ------ -------
GRB Redshift $A_{1600}$ b u w1 m2 w2 white
050319 $ 3.24^{a} $ 0.27 1.36 - - - - N/A
050525 $ 0.606^{b} $ 0.25 2.29 2.72 1.05 0.46 0.56 4.59
050712 - - - 0.68 - - - N/A
050726 - - 0.63 - - - - N/A
050730 $ 3.97^{c} $ 0.64 0.78 - - - - N/A
050801 $ 1.38^{*}$ 0.00 2.25 2.46 0.81 0.29 0.13 N/A
050802 $ 1.71^{d} $ 0.47 2.75 2.34 0.52 0.05 0.07 4.79
050922c $ 2.198^{e} $ 0.35 1.99 1.83 0.11 - - N/A
051109a $ 2.346^{f} $ 0.04 1.98 0.88 0.21 - - 7.96
060206 $ 4.04795^{g}$ 0.00 0.42 - - - - N/A
060223a $ 4.41^{h} $ - 0.26 - - - - 4.18
060418 $ 1.4901^{i} $ 0.34 1.82 1.40 0.36 0.07 0.06 6.21
060512 $ 0.4428^{j} $ 1.93 3.03 2.31 0.20 - - 8.46
060526 $ 3.221^{k} $ 0.00 1.39 - - - - 2.99
060605 $ 3.8^{l} $ 1.08 0.79 - - - - 2.74
060607a $ 3.082^{m} $ 0.21 1.56 0.19 - - - 4.36
060708 $ 1.92^{*}$ 0.56 1.85 1.76 0.25 - - 7.49
060804 - - 2.10 0.00 - - - N/A
060908 $ 2.43^{n} $ 0.08 2.07 2.23 0.40 - - 9.27
060912 $ 0.937^{o} $ 2.09 1.58 2.16 0.41 0.17 0.27 8.83
061007 $ 1.262^{p} $ 2.77 1.84 1.35 0.25 0.07 0.06 6.03
061021 $ 0.77^{*}$ 0.00 2.38 3.14 1.74 0.81 1.23 11.97
061121 $ 1.314^{q}$ 1.35 2.70 2.45 0.58 0.17 0.14 9.03
070318 $ 0.836^{r} $ 1.72 1.75 1.50 0.29 0.08 0.07 6.69
070420 $ 3.01^{*} $ 1.53 1.52 0.47 0.19 - - 6.35
070529 $ 2.4996^{s} $ 0.57 3.42 0.92 - - - N/A
--------- ------------------- ------------ ------ ------ ------ ------ ------ -------
Results
=======
In this Section, the flux lightcurves in the observer frame and their properties shall be investigated first, then the luminosity lightcurves in the rest frame and their properties shall be examined.
Observer Frame Flux Lightcurves
-------------------------------
The lightcurves, shown in Fig. \[lightcurves\], are ordered by peak magnitude from the brightest, GRB 061007, to the faintest, GRB 050712. Data points with signal to noise below 2 are shown as 3$\sigma$ upper limits. The peak magnitude was taken as the maximum magnitude in each binned lightcurve and is given in Table \[fits\]. A trend is observed in the figure, with the brightest GRBs decaying more quickly than the faintest GRBs. The lightcurves generally follow one of two types of behaviour. Either they rise to a peak within the first 1000s and then decay, or they decay from the beginning of the observations. There are 6 lightcurves that appear to rise to a peak between 200s to 1000s after the burst trigger. The peak times for the 6 GRBs were determined from a Gaussian fit to each lightcurve in log time. The fit was performed between the brightest data point and the data points on either side to the point at which the count rate is 60% of the peak value. The mean of these peak times is 397s. For the remaining GRBs, the beginning of the lightcurve was taken as the upper limit to the peak time and the mean of these upper limits is 132s.
To classify the behaviour of the lightcurves, two power-laws were fit to each lightcurve, covering the time ranges from the start of observations until 500s and from 500s until the end of the observations; the results are given in Table \[fits\]. A time of 500s was chosen as it ensures that the early power law fits are performed on at least 100s of each lightcurve and because the rising phase tends to occur during the first 500s.
A comparison of the temporal indices before and after 500s is shown in Fig. \[versus\]. The figure is divided into three groups of behaviour, which are: lightcurves which rise before 500s, lightcurves which decay more steeply after 500s and lightcurves which decay less steeply after 500s. Each of these groups contains a similar number of GRBs. From Fig. \[versus\], there are 4 lightcurves that are clearly rising with an $\alpha_{<500s}$ ranging from $0.26 \pm 0.13$ (GRB 060605) to $0.73 \pm 0.14$ (GRB 070420). A further 7 lightcurves are consistent with $\alpha_{<500s} \sim 0$ or have large errors and thus it is not clear if these lightcurves are rising, constant or decaying before 500s. The remaining 15 lightcurves decay with temporal indices of between $\alpha_{<500s}=-0.12\pm 0.05$ (GRB 061121) and $\alpha_{<500s}=-2.67 \pm 0.80$ (GRB 050726). After 500s, all the lightcurves decay with values of $\alpha_{>500s}$ ranging from $-0.50\pm0.05$ (GRB 050712) to $-1.67\pm0.15$ (GRB 070420), except for GRB 050726 where, due to the poor signal to noise, it is not possible to tell if the lightcurve after 500s is rising or decaying.
The mean and intrinsic dispersion of the temporal indices was determined using the maximum likelihood method [@mac88], which assumes a Gaussian distribution. The mean for $\alpha_{<500s}$ is $-0.48^{+0.15}_{-0.19}$ with a dispersion of $0.69^{+0.19}_{-0.06}$ and the mean for $\alpha_{>500s}$ is $-0.88^{+0.08}_{-0.07}$ with a dispersion of $0.31^{+0.07}_{-0.03}$. To see if $\alpha_{<500s}$ and $\alpha_{>500s}$ are independent parameters, a Spearman rank test was performed. This test gives a coefficient of -0.22 with a probability of 73$\%$, indicating no evidence for a correlation and suggesting that the behaviour after 500s is independent of the behaviour before 500s.
Since the lightcurves in Fig. \[lightcurves\] suggest a connection between the brightness and the decay rate, a Spearman rank correlation was performed between the temporal indices and the interpolated magnitudes of the lightcurves at 400s. The test performed between $\alpha_{<500s}$ and the magnitude at 400s, indicates that these parameters are not related, as the coefficient is -0.28 at 84$\%$ confidence. However, the Spearman rank test performed on $\alpha_{>500s}$ and the magnitude at 400s gives a coefficient of 0.59 at 99.8$\%$ confidence (see Fig. \[after\]). The correlation is statistically significant at $>3\sigma$ and therefore implies that brighter GRBs tend to have faster decays.
The mean redshifts of the two columns in Fig. \[lightcurves\] are $\langle{z}\rangle=1.63$ and $\langle{z}\rangle=2.98$, suggesting that the decay rate and magnitude are also correlated with redshift because the optical afterglows that are brighter and decay more steeply tend to have lower redshifts. A Spearman Rank test performed between $\alpha_{>500s}$ and redshift gives no evidence for a correlation with a correlation coefficient of 0.07 and a insignificant probability of 23%. However, a Spearman Rank test performed between the redshift and magnitude at 400s, provides a weak correlation with a coefficient of 0.32 and a statistical significance of 87%. Moreover, if the Spearman Rank test is performed between the peak magnitude and redshift, the link between the redshift and magnitude is stronger and more significant as the coefficient is 0.55 and the confidence is 99.3%. These correlations imply that the correlation between magnitude at 400s and $\alpha_{>500s}$ is only weakly dependent on redshift.
The lightcurves of the GRBs after 500s, in a few cases, appear to show a change in their temporal behaviour. To quantify this behaviour, a broken power-law was fitted to each GRB from 500s until the end of the observations. The broken power-law is considered an improvement if the $\chi^2/D.O.F$ has decreased and the probability of chance improvement is small ($<1\%$), as determined using an F-test. In five cases a broken power-law was an improvement compared with a single power-law. The results of the broken power-law fits for the 5 GRBs are given in Table \[brkn\]. In four of these five cases, the broken power-law shows a transition from a shallow to a steeper decay. In the fifth case, GRB 070318, the decay became shallower at late times. To test if a single break is sufficient for the decay after 500s, a doubly broken power-law was fit to these 5 GRBs. As with the broken power-law, the doubly broken power-law was considered an improvement if the $\chi^2/D.O.F$ decreased and the probability of chance improvement is small ($<1\%$). The doubly broken power-law was an improvement for only GRB 070318. The best fit values for this model are: $\alpha_1=-1.08\pm0.01$, $t_{break,1}=53800^{+6800}_{-6100}$, $\alpha_2=-0.11^{+0.12}_{-0.14}$, $t_{break,2}=197000^{+22000}_{-15000}$, $\alpha_3=-1.72\pm0.18$ with $\chi^2/D.O.F=62/23$.
For the 4 GRBs where the broken power-law was the best fit, the mean decay index before the break is $-0.60\pm0.14$ with a dispersion of $0.19^{+0.18}_{-0.07}$ and the mean decay index after the break is -1.53$\pm$0.19 with a dispersion of $0.22^{+0.28}_{-0.09}$. The break times range from $6000^{+1000}_{-1100}$s to $(4.9\pm0.25)\times 10^{4}$s. If the mean decay index after 500s is recalculated including only those lightcurves that decay as a single power-law after 500s, the mean is $-0.87^{+0.10}_{-0.09}$ with a dispersion of $0.35^{+0.10}_{-0.04}$. This mean is similar to the mean decay index determined using all the GRBs in the sample. For the lightcurves that show a break, the mean decay before the break is consistent within 2$\sigma$ with these mean values.
GRB Rest Frame Luminosity Lightcurves
-------------------------------------
GRB Photometric Redshift $\chi^2/D.O.F$
----------- ------------------------ ----------------
GRB050801 1.38 $\pm$ 0.07 (14/12)
GRB060708 1.92 $\pm$ 0.12 (21/20)
GRB061021 $0.77^{+0.06}_{-0.01}$ (229/174)
GRB070420 $3.01^{+0.96}_{-0.68}$ (58/60)
: Photometric redshifts for 4 of the GRBs without spectroscopic redshifts.[]{data-label="phot"}
GRB $D_L $ (cm) $\beta_1$ $\beta_2$ $E_{peak}$ $E_{iso,k}$
--------- ------- ---------------------- ------------- ------------- ------------- -------------
050319 BAND $8.60\times 10^{28}$ $-0.99$ $-2.02^{a}$ $162.0 $ 1.44e+53
050525 CPL $1.10\times 10^{28}$ $-1.10^{b}$ $-2.31$ $ 84.1^{b}$ 1.59e+53
050730 CPL $1.04\times 10^{29}$ $-1.53^{a}$ $-2.31$ $162.2 $ 3.85e+54
050801 CPL $3.04\times 10^{28}$ $-1.99^{a}$ $-2.31$ $162.2 $ 7.67e+53
050802 CPL $3.96\times 10^{28}$ $-1.54^{a}$ $-2.31$ $162.2 $ 5.86e+53
050922c CPL $5.38\times 10^{28}$ $-1.55^{c}$ $-2.31$ $162.2 $ 4.30e+54
051109a CPL $5.82\times 10^{28}$ $-1.25^{d}$ $-2.31$ $161.0^{d}$ 6.43e+53
060206 CPL $1.12\times 10^{29}$ $-1.20^{a}$ $-2.31$ $ 78.0^{a}$ 3.42e+53
060223a CPL $1.24\times 10^{29}$ $-1.74^{a}$ $-2.31$ $162.2 $ 4.47e+54
060418 CPL $3.34\times 10^{28}$ $-1.50^{e}$ $-2.31$ $230.0^{e}$ 2.94e+54
060512 BAND $7.56\times 10^{27}$ $-0.99$ $-2.48^{a}$ $162.2 $ 5.04e+50
060526 BAND $8.54\times 10^{28}$ $-0.99$ $-2.01^{a}$ $162.2 $ 1.37e+53
060605 CPL $1.04\times 10^{29}$ $-1.55^{a}$ $-2.31$ $162.2 $ 1.15e+54
060607a CPL $8.10\times 10^{28}$ $-1.47^{a}$ $-2.31$ $162.2 $ 1.81e+54
060708 CPL $4.56\times 10^{28}$ $-1.68^{a}$ $-2.31$ $162.2 $ 3.79e+53
060908 CPL $6.08\times 10^{28}$ $-1.00^{a}$ $-2.31$ $151.0^{a}$ 1.88e+53
060912 POWER $1.88\times 10^{28}$ $-1.74^{f}$ $-2.31$ $162.2 $ 2.52e+54
061007 BAND $2.72\times 10^{28}$ $-0.70^{g}$ $-2.61^{g}$ $399.0^{g}$ 2.00e+53
061021 CPL $1.48\times 10^{28}$ $-1.22^{h}$ $-2.31$ $777.0^{h}$ 9.65e+52
061121 CPL $2.86\times 10^{28}$ $-1.32^{i}$ $-2.31$ $606.0^{i}$ 2.75e+54
070318 CPL $1.63\times 10^{28}$ $-1.42^{a}$ $-2.31$ $162.2 $ 8.78e+52
070420 CPL $7.87\times 10^{28}$ $-1.23^{j}$ $-2.31$ $147.0^{j}$ 6.39e+54
070529 CPL $6.14\times 10^{28}$ $-1.34^{a}$ $-2.31$ $162.2 $ 6.46e+53
: Properties of the GRBs with spectroscopic or photometric redshifts. This table contains the luminosity distance, the gamma-ray photon indices and peak energies used to determine the k-corrected isotopic energy for each GRB in the energy range 10 keV - 10 MeV. References: a)[@sak07], b)[@3474], c)[@4030] d)[@4238], e)[@4989], f)[@5570], g)[@5722], h)[@5748], i)[@5837], j)[@6344] []{data-label="redshift"}
The luminosity lightcurves at 1600[Å]{}, in units of erg s$^{-1}$ cm$^{-1}$ $\rm \AA^{-1}$, before and after correction for host extinction, are shown in Fig. \[luminosity\]. Panel (a) shows the luminosity lightcurves before any correction for the host extinction has been applied. In both panels of Fig. \[luminosity\], GRB 060512 lies significantly below all the other lightcurves. We suspect that this is caused by either an incorrect determination of the host extinction or of the redshift. The redshift of this GRB could be wrong because it was not determined from the afterglow spectra, but was based on the alignment of the GRB with a galaxy [@5217]. In [@schady], the best fitting model to the SED of GRB 060512 gives a poor fit with a $\chi^2/D.O.F=84/23$ and a host extinction of $E(B-V)_{host}=0.16^{+0.01}_{-0.00}$. A photometric redshift was determined for this GRB using the method described in Section 2, of $z=2.279^{+0.09}_{-0.18}$ and an extinction of $A_{1600}=0.00^{+0.02}_{-0.00}$. Using these values a luminosity lightcurve for GRB 060512 at 1600[Å]{} was produced. This photometric redshift changes the rest frame relative flux SED and consequently the k-correction factor. The result is that the luminosity lightcurve increases by $\sim3$ orders of magnitude, which means that this GRB is no longer separated from the rest of the GRBs in the sample. Nonetheless, this GRB may be intrinsically different to all the other GRBs in the sample as this GRB is the only XRF in the sample and it may be that XRFs are a class of sub-luminous GRBs. However, as it is uncertain whether the redshift of GRB 060512 is correct, the luminosity lightcurve for this GRB will be excluded from any further analysis.
To produce luminosity distributions, the luminosities were interpolated from the lightcurves before and after correction for host extinction at the 3 rest frame epochs: 100s, 1000s and 10 ks. The logarithmic distribution of the luminosities at the three epochs are shown in Panels (a) to (f) of Fig. \[lum\_hist\]. The distributions consisting of the luminosities at 100s contain 18 GRBs whereas the distributions for the luminosities at 1000s and 10 ks contain 20 GRBs. Panels (a) to (c), show the logarithmic distribution of luminosities before correction for host extinction. The means of these distributions at 100s, 1000s and 10 ks in the rest frame are 11.08, 10.29 and 9.39, respectively. The standard deviations at the three rest frame epochs are 0.65 at 100s, 0.71 at 1000s, and 0.68 at 10 ks. Panels (d) to (f) show the logarithmic distributions of the luminosity lightcurves that have been corrected for host extinction. The mean of the host extinction corrected distributions at 100s, 1000s and 10 ks in the rest frame are 11.29, 10.55 and 9.64, respectively and the standard deviations are 0.57 at 100s, 0.67 at 1000s, and 0.62 at 10 ks.
Name Classification Peak Magnitude Peak Time (s) $\alpha_{<500s}$ $(\chi^2/D.O.F)$ $\alpha_{>500s}$ ($\chi^2/D.O.F$)
--------- ---------------- ---------------- --------------- ------------------ ------------------ ------------------ ------------------
050319 XRR 17.09 < 234 0.09 $\pm$ 0.35 0/ 2 -0.63 $\pm$ 0.03 34/17
050525 XRR 13.57 < 78 -1.25 $\pm$ 0.03 21/ 9 -1.00 $\pm$ 0.01 398/26
050712 C-GRB 17.77 < 178 0.11 $\pm$ 0.64 4/ 3 -0.50 $\pm$ 0.05 52/26
050726 C-GRB 17.21 < 159 -2.67 $\pm$ 0.80 1/ 3 0.13 $\pm$ 0.29 21/13
050730 C-GRB 17.22 744 0.15 $\pm$ 0.50 6/ 3 -0.89 $\pm$ 0.05 72/16
050801 XRR 15.26 < 66 -0.46 $\pm$ 0.04 43/10 -1.17 $\pm$ 0.03 41/15
050802 XRR 17.07 < 289 0.07 $\pm$ 0.48 0/ 1 -0.75 $\pm$ 0.01 53/29
050922c C-GRB 14.34 < 109 -1.01 $\pm$ 0.05 21/ 6 -0.94 $\pm$ 0.01 88/14
051109a C-GRB 16.33 < 122 -0.47 $\pm$ 0.45 0/ 2 -0.68 $\pm$ 0.02 80/16
060206 XRR 16.64 < 57 -0.02 $\pm$ 0.50 9/ 4 -0.70 $\pm$ 0.03 205/22
060223a XRR 17.33 < 88 -1.06 $\pm$ 0.33 4/ 6 -1.53 $\pm$ 0.60 31/14
060418 XRR 14.69 260 0.01 $\pm$ 0.03 138/ 6 -1.22 $\pm$ 0.01 58/21
060512 XRF 16.50 < 114 -0.74 $\pm$ 0.08 3/ 6 -0.99 $\pm$ 0.05 10/14
060526 XRR 16.59 < 82 -0.29 $\pm$ 0.08 36/ 8 -0.69 $\pm$ 0.02 141/22
060605 XRR 16.50 459 0.26 $\pm$ 0.13 6/ 4 -0.94 $\pm$ 0.03 27/11
060607a C-GRB 14.50 254 0.50 $\pm$ 0.03 702/ 8 -1.17 $\pm$ 0.03 152/20
060708 XRR 17.19 < 72 -0.10 $\pm$ 0.10 8/ 9 -0.89 $\pm$ 0.03 39/23
060804 XRR 17.43 < 231 -1.76 $\pm$ 0.49 0/2 -0.57 $\pm$ 0.08 21/9
060908 C-GRB 15.21 < 88 -1.19 $\pm$ 0.05 6/ 8 -1.03 $\pm$ 0.04 20/18
060912 XRR 16.44 < 114 -0.97 $\pm$ 0.09 2/ 6 -0.53 $\pm$ 0.01 149/27
061007 C-GRB 12.68 < 298 -1.63 $\pm$ 0.11 1/ 1 -1.59 $\pm$ 0.01 129/28
061021 C-GRB 15.64 < 79 -0.92 $\pm$ 0.06 3/ 9 -0.88 $\pm$ 0.01 556/29
061121 C-GRB 15.67 < 53 -0.12 $\pm$ 0.05 181/11 -0.67 $\pm$ 0.01 179/31
070318 C-GRB 15.37 316 0.42 $\pm$ 0.03 19/ 9 -1.02 $\pm$ 0.01 261/27
070420 C-GRB 17.16 347 0.73 $\pm$ 0.14 13/ 5 -1.67 $\pm$ 0.15 56/23
070529 C-GRB 15.95 < 131 -1.64 $\pm$ 0.14 5/ 5 -0.53 $\pm$ 0.04 42/19
The distribution of rest frame peak times and upper limits is shown in Fig. \[time\_hist\]. The peak times of the GRBs with observed peaks overlap with the upper limits of the GRBs without observed peaks. Therefore, it is not possible to tell if the GRBs with observed peaks are a separate class, or if they belong to the tail end of a distribution where the majority of GRB peaks occur before the UVOT can observe them.
To determine if the relationship between the brightness of the afterglow and the late time decay rate is intrinsic, the luminosity lightcurves were fitted with a power-law from 150s onwards $\alpha_{>150s,rest}$, where $150{\rm s}\simeq 500{\rm s}/(1+\langle z\rangle)$ and $\langle z\rangle=2.21$ is the mean redshift of the GRBs in the sample, and a Spearman rank test was performed between this decay and the extinction corrected luminosity at 100s in the rest frame. These parameters are shown in Fig. \[after\_150s\]. The Spearman rank test does not support or refute a correlation between these parameters because the coefficient is -0.29 and the probability of correlation is not significant at 76%.
GRB $\alpha_{1}$ Break Time $\alpha_{2}$ ($\chi^2/D.O.F$) $\Delta\chi^2$
------------- ------------------ ---------------------------- ------------------------- ------------------ ----------------
GRB 050525 $-0.80 \pm 0.01$ $16400 ^{+1200 }_{-1400}$ $-1.70 \pm 0.08$ ( 120/24 ) 278
GRB 050922c $-0.76 \pm 0.03$ $6000 ^{+1100}_{-1000}$ $-1.20 \pm 0.05 $ ( 16/12 ) 59
GRB 060526 $-0.33 \pm 0.04$ $30800 ^{+4700}_{-5800}$ $-2.33^{+0.65}_{-0.47}$ ( 32/20 ) 93
GRB 061021 $-0.51 \pm 0.02$ $49300 ^{+2500}_{-2500}$ $-1.60^{+0.07}_{-0.06}$ ( 49/27 ) 507
GRB 070318 $-1.09 \pm 0.01$ $16000 ^{+3600}_{-3000}$ $-0.78 \pm 0.03 $ ( 131/25 ) 129
GRB Lorentz Factor at Peak $M_{fb}$ $R_{dec}$
--------- ------------------------ -------------- --------------
050319 >168 <2.40e-04 <9.33e+16
050525 >179 <2.49e-04 <9.25e+16
050730 174 6.18e-03 2.72e+17
050801 >275 <7.81e-04 <1.17e+17
050802 >156 <1.05e-03 <1.56e+17
050922c >308 <3.91e-03 <1.93e+17
051109a >236 <7.61e-04 <1.22e+17
060206 >338 <2.83e-04 <7.80e+16
060223a >409 <3.06e-03 <1.62e+17
060418 193 4.27e-03 2.32e+17
060512 > 72 <1.95e-06 <2.48e+16
060526 >247 <1.55e-04 <7.09e+16
060605 177 1.82e-03 1.80e+17
060607a 220 2.30e-03 1.81e+17
060708 >256 <4.14e-04 <9.72e+16
060908 >231 <2.27e-04 <8.23e+16
060912 >235 <3.00e-03 <1.94e+17
061007 >126 <4.42e-04 <1.26e+17
061021 >173 <1.56e-04 <7.99e+16
061121 >337 <2.28e-03 <1.57e+17
070318 103 2.38e-04 1.09e+17
070420 228 7.84e-03 2.69e+17
070529 >234 <7.71e-04 <1.23e+17
: Properties of the GRBs, derived assuming the rise of the forward shock is the cause of the rise observed in the UVOT lightcurves. The initial Lorentz factors were determined using the peak times and Eq. 1 of [@mol06]. Where only an upper limit to the peak time is known, only a lower limit to the Lorentz factor is given. The fraction of mass as baryons and the deceleration radius are determined using the Lorentz factor.[]{data-label="energies"}
Discussion
==========
In this section, we shall discuss the possible mechanisms that could produce the rising behaviour of the early afterglow viewed in the observer frame before 500s, and we shall discuss the late afterglow from 500s onwards. We will also discuss the implications of the luminosity distribution and compare the UVOT lightcurves with the XRT canonical lightcurve model.
The early UVOT afterglow
------------------------
There are several physical mechanisms and geometric scenarios that may produce a rise in the early optical afterglow. In this section the following mechanisms and scenarios shall be discussed: (i) passage of the peak synchrotron frequency of the forward shock $\nu_{m,f}$, through the observing band, (ii) a reverse shock, (iii) decreasing extinction with time, (iv) the onset of the forward shock in the cases of an isotropic outflow or a jet viewed in a region of uniform energy density, (v) the rise produced by an off-axis jet, which may be structured, and finally, (vi) a two component outflow.
### Passage of the synchrotron frequency, $\nu_{m,f}$
The first mechanism, the passage of the peak frequency of the forward shock which moves with time as $\nu_{m,f}\propto t^{-3/2}$, through the observing band is expected to produce a chromatic peak in the optical lightcurve evolving from the shortest wavelengths through to the longest wavelengths. For 5 of the lightcurves with a peak, the UVOT observed the majority of the rise and the peak during the two finding chart exposures observed in white and v (see Fig. \[early\_afterglows\]). If the peak were due to the passage of the synchrotron frequency, a stepped decrease in flux would be observed in the normalized lightcurves at the transition between the white and v observations, which has not been observed in any of these GRBs. For the 6th GRB with an observed peak, GRB 050730, the rise was observed in the v and b filters. If the passage of $\nu_{m,f}$ was the cause of this rise, the afterglow would appear to be brighter in the b filter than in the v filter during the rise, and begin to decay earlier. However, the afterglow is not brighter in the b filter than in the v filter during the rise. Therefore, the passage of $\nu_{m,f}$ through the optical band is not responsible for any of the peaks observed in these optical afterglows.
During the passage of $\nu_m$ from the shortest wavelengths through to the longest wavelengths, the spectrum of the optical afterglow changes (assuming slow cooling) from $\nu^{1/3}$, for $\nu<\nu_m$ to $\nu^{-(p-1)/2}$, for $\nu_m<\nu<\nu_c$ [@sari98]. As $\nu_m$ passes from high frequencies to lower frequencies, there will be a change in colour. The colour change between the white and v filters can be calculated using the central wavelengths of the white and v filters, given in [@poole], converted in to frequency: $\nu_{\rm white}=8.64\times10^{14}$ Hz, $\nu_{\rm v}=5.49\times10^{14}$Hz and assuming $p=2.3$ where $p$ is the electron energy index. The colour change between white and v as $\nu_m$ moves from above the white frequency to below the v frequency is 0.48 magnitudes. In the lightcurves of Fig. \[early\_afterglows\], which have been normalized using the late time data, the colour difference between the white and the v filter after the peak is zero. None of the 5 lightcurves, in Fig. \[early\_afterglows\], with early white and v observations, show such a large offset between the lightcurves in two filters during the rise. Furthermore, the equation for $\nu_{m,f}$ in a constant density medium, as given by [@zhang05] as: $$\nu_{m,f}\,=(6\times10^{15}Hz)(1+z)^{1/2}E_{52}^{1/2}\epsilon_e^2\epsilon_B^{1/2}(t/1 day)^{-3/2}$$ where $E_{52}=10^{52}E$ is the isotropic energy in units of $10^{52}$ ergs, $\epsilon_e$ is the fraction of energy in the electrons, $\epsilon_B$ is the fraction of energy in the magnetic field, $z$ is the redshift and $t$ is the time. Assuming $\epsilon_e$ and $\epsilon_B$ are not evolving with time and given the time $t_1$ at which $\nu_{m,f}$ is at a given frequency $\nu_1$, the time $t_2$ at which $\nu_{m,f}$ is at the frequency $\nu_2$ is given by: $t_2=t_1(\nu_2/\nu_1)^{(-2/3)}$. Using the central wavelengths of the white and v filters, the v filter should peak 1.35$\times$ later than the white filter. There does not appear to be any time difference between the peak in the white and v filters in the 5 lightcurves shown in \[early\_afterglows\], therefore, the passage of $\nu_m$ through the optical band is not the cause the rise in the optical band.
For $\nu_{m,f}$, to have passed below the v filter (5402[Å]{}) by the time the UVOT has begun observations ($t\sim100$s), using $\langle E_{k,iso}\rangle=1.5\times10^{54}$ erg and $\langle z\rangle=2.21$, then $\epsilon_e^2\epsilon_B^{1/2}<1.7\times10^{-5}$. The values $\epsilon_e$ and $\epsilon_B$ provided in [@pan02] for 10 GRBs give values for $\epsilon_e^2\epsilon_B^{1/2}$ ranging from $3\times10^{-3}$ to $2\times10^{-7}$, suggesting that $1.7\times10^{-5}$ is consistent with values found for other GRBs.
\
\
\
### The reverse shock
Considering a constant density medium, there are two main cases of the reverse shock that depend on the position of the peak synchrotron frequency of the reverse shock $\nu_{m,r}$ relative to the optical band $\nu_{opt}$. If $\nu_{m,r}<\nu_{opt}$ then the lightcurve produced by the reverse shock is expected to decay immediately after the peak with $\alpha=(3p+1)/4$ [@zhang03], where $p$ is the electron energy index. The value of $p$ is typically taken to lie between 2 and 3, therefore $\alpha$ is expected to range between $\alpha=-1.75$ for $p=2$ to $\alpha=-2.5$ for $p=3$. Within the sample, GRB 050726, GRB 061007 and GRB 070529 are the only GRBs with decays before 500s that are consistent at 2$\sigma$ with the slowest expected reverse shock decay index $\alpha=-1.75$; GRB 050726 has a decay of $\alpha=-2.67 \pm 0.80$, GRB 061007 decays with $\alpha=-1.63 \pm 0.11$ and GRB 070529 decays with $\alpha=-1.64\pm 0.14$. The other lightcurves in the sample are shallower than the reverse shock prediction with $>2\sigma$ confidence. The second case arises if $\nu_{m,r}>\nu_{opt}$, then immediately after the peak there is an intermediate stage where $\alpha\sim~-0.5$, which is followed by $\alpha=(3p+1)/4$. In the sample, there are 7 GRBs that before 500s have temporal indices consistent with $\alpha=- 0.5$ at $2\sigma$ confidence. However, of these GRBs only GRB 060223a has a decay after 500s ($\alpha=-1.53 \pm 0.60$) that is consistent at the $2\sigma$ level with the slowest expected reverse shock decay index $\alpha=-1.75$.
The reverse shock in a wind medium is expected to produce a much steeper decay immediately after the peak with $\alpha\sim-3.5$ [@kobay04]. Only GRB 050726, has a value of $\alpha_{<500s}$, which is consistent to within 2$\sigma$ confidence. All other GRBs are inconsistent at $>5\sigma$.
The inconsistency of the temporal indices of the GRBs in this sample to the temporal decay expected during a reverse shock for both types of medium implies that the reverse shock is not the main contributor to the optical emission at early times, and therefore is not responsible for the rise. Still the reverse shock is expected to occur for all relativistic outflows that interact with the external medium. For a number of GRBs, the reverse shock may not be observed as it can be suppressed by high levels of magnetisation in the outflow [@zhang05; @giannios08] or if the forward and reverse shock have comparable energy, the sharp decay in the reverse shock may be masked by the flux produced by the forward shock [@mcmahon06; @mundell07].
### Dust Destruction
If there are high levels of extinction at the beginning of the afterglow [@klotz08 and references within], the lightcurve produced would be dim and reddened at the beginning. As the dust is destroyed by the radiation, a chromatic peak would be observed as the afterglow brightens and becomes less red. The bluer filters would be expected to rise more steeply when compared to the red filters as would the white filter, because the sensitivity of this filter is skewed to the blue. However, the amount of dust destroyed is highly dependent on the environment of the GRB, in particular to the density and the size of the region surrounding the progenitor, and simulations suggest that most of the dust destruction is expected to occur within the first few tens of seconds after the trigger [@perna02]. Therefore, it is unlikely that the UVOT is observing the afterglow while dust destruction is occurring.
However, as the duration of dust destruction and the quantity of dust destroyed are only theoretical predictions, we must rule out dust destruction using observations. Therefore, the 6 UVOT lightcurves with a rise were examined to see if the bluer filters, including white, rise more steeply when compared to the v filter. The UVOT observed five of the lightcurves in white and v (the reddest UVOT filter) during the rise, see Fig. \[early\_afterglows\]. GRB 060607a has the only lightcurve where there appears to be a significant excess in v compared with the white filter. However, the H band lightcurve given in [@mol06] rises at the same rate as the UVOT lightcurve, which was observing in white during the rise. If dust destruction was the cause of the rise, the H band would be expected to rise less steeply than the UVOT lightcurve. The 6th rising GRB, GRB 050730, was observed with the v and b filters during the rise and peak. If the peak in this case were due to dust destruction, an excess in v compared to b would be observed. However, the lightcurves of the v and b filters are consistent within 1$\sigma$ errors. Therefore, there is no evidence to suggest that dust destruction is the cause of the rise for this GRB or for any of the GRBs in this sample.
![Distribution of the rest frame peak times. The red filled area represents the GRBs with known rest frame peak times, whereas the unfilled area contains the GRBs with known rest frame peak times and those GRBs with only upper limits to their peak time. Only 21 GRBs, for which luminosity lightcurves were produced, are included in this figure.[]{data-label="time_hist"}](./Images/Restframe_Peak_time_hist.ps)
### Start of the forward shock
At the start of the forward shock, a rise will be observed in the lightcurves as the jet ploughs into the external medium [@sari99]. The lightcurves for an observer viewing within a uniform jet, or within a cone of uniform energy density will be the same as those observed within an isotropic outflow [@granot02 and references within]. The temporal index of the rise will vary according to the thickness of the shell and the density of the external medium. Assuming the synchrotron self absorption frequency $\nu_{a}<\nu_{opt}$, then the thickness of the shell and the density profile of the external medium affect the rate at which the lightcurve rises. For the thick shell case, the temporal index is $\alpha = 1$ for a constant density medium, or $\alpha=1/9$ for a wind medium. For the thin shell case, in a constant density medium the temporal index is either $\alpha=2$ for $\nu_{c}<\nu_{opt}$, or $\alpha=3$ for $\nu_{c}>\nu_{opt}$. Lastly, the temporal index is $\alpha=0.5$ if the shell is thin and in a wind medium [@pan08].
Given the peak time, the Lorentz factor $\Gamma$ of the shell at the moment of the peak for a constant density medium, can be derived using the following equation [@mol06; @sari97]: $$\Gamma(t_{peak})=\left(\frac{3E(1+z)^3}{32\pi nm_pc^5\eta (t_{peak}/100{\rm s})^3}\right)^{1/8}$$ where $t_{peak}$ is the peak time, $\eta$ is the radiative efficiency and $n$ is the density of the medium. Here we will assume $\eta=0.2$ and $n=1$ cm$^{-3}$. However, changing the values of $\eta$ and $n$ has a minor effect on the final values of $\Gamma(t_{peak})$ as the dependence of $\Gamma(t_{peak})$ on these parameters is small: $\Gamma(t_{peak})\propto (\eta n)^{-1/8}$. For each GRB, the k-corrected energy, given in Table \[redshift\], was used in the equation above to determine $\Gamma(t_{peak})$ and the resulting values of $\Gamma(t_{peak})$ for the individual GRBs can be found in Table \[redshift\]. The mean of the $\Gamma(t_{peak})$ for the GRBs with a peak in their lightcurve is $\langle{\Gamma(t_{peak})}\rangle\sim 180$, which is consistent with the expectation that the initial Lorentz factor of the jet $\Gamma_0$, where $\Gamma_0\sim2\Gamma(t_{peak})$ [@pan00; @mes06], of GRBs is $>100$ [@fen93]. For the GRBs where only an upper limit to their peak time is known, the mean value for $\Gamma(t_{peak})$ is a lower limit, $\langle{\Gamma(t_{peak})}\rangle >230$. This suggests that the GRBs with observed peaks typically have lower Lorentz factors than the GRBs with upper limits to their peak times.
Using the derived values of $\Gamma(t_{peak})$, it is possible to deduce two more quantities: the isotropic-equivalent mass of the baryonic load $M_{fb}=E/(\Gamma_0 c^2)$, and the deceleration radius $R_{dec}\simeq2ct_{peak}\Gamma(t_{peak})^2/(1+z)$ [@mol06]. These quantities were determined for each GRB and the results are given in Table \[energies\]. The mean mass of the baryonic load and the mean deceleration radius for the GRBs with an observed rise are $\langle{M_{fb}}\rangle=3.8\times10^{-3}M_{\odot}$ and $\langle{R_{dec}}\rangle=2.1\times10^{17}$cm and for the GRBs without an observed rise the quantities are $\langle{M_{fb}}\rangle<1.1\times 10^{-3}M_{\odot}$ and $\langle{R_{dec}}\rangle<1.2\times 10^{17}$cm. The deceleration radii are in agreement with $R_{dec}\sim10^{16}\rm cm$ as predicted by theory [@rees92]. Therefore the forward shock is consistent with our observations.
### Off-axis and structured outflows
A rise may be produced in the lightcurve if the observer’s viewing angle is $\theta_{obs}>\theta$, where $\theta_{obs}$ is the observers viewing angle and $\theta$ is the half opening angle of the jet. In the case of a uniform jet, the ejecta are released into a cone of angle $\theta$ and due to relativistic effects, the emission in the jet is beamed as $\Gamma^{-1}$. If $\theta_{obs}>\theta$, then the emission is strongly beamed away from the observer, but as $\Gamma$ decreases, the emission entering the line of sight increases and the observed lightcurve will rise. The lightcurve will peak when $\Gamma\sim(\theta_{obs}-\theta)^{-1}$ [@granot02]. Larger observing angles will view a later peak and the peak magnitude will be lower [@granot05]. A structured jet, in which the energy per solid angle decreases around a uniform core of angular size $\theta_c$, viewed off-axis ($\theta_{obs}>\theta_c$) can produce a rise in the optical afterglow, where the behaviour of the rise may vary depending on the distribution of energy around the core. In this case, the more diffuse the energy per solid angle the slower the rise and the later the peak time [@pan08; @granot02]. The peak time of the lightcurve is also dependent on the viewing angle of the jet and the peak will occur when $\Gamma\sim(\theta_{obs}-\theta_c)^{-1}$.
### Two Component Outflows
A two component outflow consists of a narrow jet surrounded by a wide jet. The narrow component will be denoted by a subscript $n$ and the wide component will be denoted by a subscript $w$. Both components move at relativistic speeds, but the narrow jet will have a larger $\Gamma$ (i.e $\Gamma_w<\Gamma_n$) and the wide component will have a larger half opening angle than the inner narrow jet (i.e $\theta_w<\theta_n$). The optical emission is expected to be produced either within the wide component [@depas08; @oates07] or more traditionally from both the narrow and wide components [@peng05; @huang04].
In the case where the optical emission is only produced by the wide component, the rise observed will occur as in Section 5.1.4, provided that the jet is viewed within $1.5\theta_w$, as has been demonstrated by [@granot02]. At angles larger than $1.5\theta_w$, the peak of the optical lightcurve will occur when $\Gamma\sim(\theta_{obs}-\theta_{w})^{-1}$. This model of the two component outflow could produce the rises observed in this sample.
In the case where the optical emission is produced in both the narrow and the wide components, the dominance of emission from one component over the other, depends mostly on the energy within each component and on the viewing angle of the observer (for a more detailed description see [@peng05]). However, in this case as the emission is produced in both components, it is likely that on-axis and off-axis viewers will observe two peaks. This effect is not seen in the UVOT lightcurves in this paper and therefore, the jet is unlikely to have two components where both produce optical/UV emission.
The Late UVOT afterglow
-----------------------
A correlation has been observed between the observed magnitude at 400s and the decay after 500s. However, there is no significant evidence from a Spearman Rank test performed between $\alpha_{>150s,rest}$ and the luminosity in the rest frame at 100s, for a similar correlation in the rest frame. For an off-axis jet, a correlation is expected between the luminosity and the decay of the lightcurve when the viewing angle is changed, with fainter, shallower afterglows having a larger viewing angle [@pan08]. As this data do not show a strong correlation of this type, this suggests that the outflows are viewed within the half-opening angle $\theta$ or within a core of uniform energy density $\theta_c$. This supports the idea that the start of the forward shock produces the rises observed in the UVOT lightcurves. However, this does not give an indication of the geometry of the jet which may be uniform, comprise of two components, or be structured.
UVOT afterglow luminosity
-------------------------
The luminosity distribution found in this work shows no evidence for bimodality, which is in contradiction with the results of [@nard06; @nard08], [@liang] and [@kann07], who all claim a bimodal distribution within their samples. However, the lack of evidence for bimodality within the distributions in this sample is consistent with the work of [@cenko08] and [@mel08]. [@cenko08] present a luminosity distribution at 1000s in the rest frame from a sample of 17 GRBs, with known redshift, observed with the Palomar 60 inch telescope. Their distribution shows no evidence for bimodality. [@mel08] produce 3 luminosity distributions for 16 optical afterglows observed with the Liverpool and Faulkes telescopes. They find a single peaked luminosity distribution for three rest frame epochs: 10mins, 0.5 days and 1 day, which are well fit by a log-normal function. [@mel08] do not correct their lightcurves for host extinction, but as discussed in this paper the correction for host extinction appears to have minimal effect on the luminosity distribution. Therefore, it is possible to compare the distributions of [@mel08] with the distributions produced with this sample.
Comparison of the XRT and UVOT canonical lightcurves
----------------------------------------------------
There are three segments which are usually found in X-ray lightcurves within the first $\sim10^{5}$s [@zhang05; @nousek]. The first segment is a fast, early decay with $-5<\alpha_{X1}<-3$, typically ending within 100s-1000s after the trigger. The fast decay is thought to be caused by the tail of the prompt emission [@zhang05; @nousek]. The second segment is shallow with $-1.0<\alpha_{X2}<-0.5$ [although this range should now be considered as $-1.0<\alpha_{X2}<0.0$, @liang07], ceasing between 1000s and 10000s and is attributed to energy injection [@zhang05; @nousek]. The third segment decays as $-1.5<\alpha_{X3}<-1$ and is expected to occur at the end of energy injection [@zhang05; @nousek].
The range in temporal index of the optical lightcurve before 500s, taken as the mean plus or minus the dispersion, is $-1.17<\alpha<0.21$. This range is clearly inconsistent with the first segment of the XRT canonical lightcurve. The range of temporal index before 500s is most similar to the range of the second segment of the XRT canonical model. However, none of the XRT canonical lightcurve segments indicate rising behaviour for the X-ray lightcurves. Applying the theoretical interpretations for the individual segments of the XRT canonical lightcurve, provided by [@zhang05] and [@nousek], to the UVOT canonical lightcurve suggests that before 500s, the emission producing the optical lightcurves is from the forward shock and that a number of the UVOT afterglows during this period are energy injected. The lack of corresponding rising behaviour in the X-ray lightcurves, presuming an achromatic rise ($\nu_m<\nu_{opt}$), suggests that the rise of the forward shock is masked in the X-rays, possibly by the contribution of the prompt emission. This is consistent with the model of [@willingale07] who suggest that the steep and shallow decays of the X-ray lightcurves are dominated by the prompt emission and afterglow emission respectfully, and that they do not observe the rise of the afterglow emission as this is masked by the prompt emission.
In Section 4, the lightcurves after 500s were fitted with power-laws and broken power-laws. The GRBs which were best fitted by broken power-laws are discussed separately. For the optical lightcurves that decay as power-laws, the range in decay is $-1.22<\alpha<-0.52$. Like the range in decay before 500s, the range in decay after 500s is most similar to the decay range of the second segment of the XRT canonical model. Assuming the same reasoning as for the XRT segments, this suggests that after 500s the optical lightcurves are consistent with emission from the forward shock and most of these are energy injected.
For the four GRBs that are best fit with a broken power-law after 500s, the range in temporal decay before the break is $-0.74<\alpha<-0.46$, which is consistent with the range given for the second segment of the canonical XRT model. This suggests that before the break the optical lightcurves are energy injected. The range in the temporal decay after the break is $-1.72<\alpha<-1.34$, which is consistent with the third decay of the XRT canonical lightcurve, which has been suggested to be the decay following the end of energy ejection.
Conclusions
===========
In this paper we systematically reduced and analyzed a sample of 27 GRBs, which met a strict set of selection criteria. We note that the temporal behaviour of the optical afterglows in the sample is varied, with the greatest variation occurring in the early phase of the lightcurves: before 500s the lightcurves may rise or decay. The mean for $\alpha_{<500s}$ is $-0.48^{+0.15}_{-0.19}$ with a dispersion of $0.69^{+0.19}_{-0.06}$. However, after 500s, all the lightcurves decay. The lightcurves were fitted with power-laws and broken power-laws. A broken power-law was deemed to be an improvement, if the $\chi^2/D.O.F$ decreased and the probability of chance improvement was small ($<1\%$) and in 5 cases a broken power-law was considered a better fit. The mean decay index after 500s, when including only those that decay with a single power-law, is $-0.87^{0.10}_{-0.09}$ with a dispersion of $0.35^{+010}_{-0.04}$. There is a correlation at $99.8\%$ probability, between the magnitude at 400s and the temporal decay after 500s, with the brightest optical afterglows decaying the fastest.
We investigated the cause of the rising behaviour in the early afterglow and discussed the following physical mechanisms and geometric scenarios: the passage of the synchrotron frequency $\nu_m$, reverse shock, dust destruction, the start of the forward shock, the viewing angle of a (possibly structured) jet and a two component outflow. The rise in the optical lightcurves may be attributed to either the start of the forward shock, or to an off-axis viewing angle where the observer sees an increasing amount of emission as the Lorentz factor of the jet decreases. We also investigated the correlation between magnitude and decay after 500s. We determined that a correlation observed between the magnitude at 400s and the decay after 500s is only weakly dependent on redshift. A Spearman rank test performed between the luminosities at 100s and the decay after 150s in the rest frame did not reveal a significant correlation. However, a luminosity-decay correlation would be expected for jets viewed off-axis, where the more off-axis the jet the fainter and shallower the lightcurve. We do not observe a strong correlation of this type, suggesting that the optical lightcurves are produced by jets viewed on-axis and that the rise observed in the optical lightcurves is caused by the start of the forward shock.
We produced luminosity lightcurves for the 21 GRBs in the sample with known redshift. The luminosity lightcurves were produced at a common wavelength of 1600[Å]{} in the rest frame. We find that the logarithmic distribution of the luminosities at three rest frame epochs: 100s, 1000s and 10 ks do not show evidence for bimodality. Correcting the lightcurves for the host extinction increases the mean luminosities of the distributions, but does not considerably alter their appearance and the change in standard deviations of the logarithmic luminosity distributions is no greater than 0.08 for any of the three epochs. The lack of evidence for bimodality is consistent with the findings of [@mel08] and [@cenko08].
Finally, we compared the temporal behaviour of the optical afterglows in this sample with the XRT canonical model. We have found that the temporal indices before 500s and the temporal indices of the lightcurves after 500s are most consistent with the the shallow decaying segment of the XRT canonical model. [@nousek] and [@zhang05] suggest that the shallow decay segment of the XRT canonical model is energy injected. This would suggest that the optical lightcurves are energy injected as well. The lack of rises observed in X-ray afterglows could be due to the prompt emission masking them.
Acknowledgments
===============
This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC) and the Leicester Database and Archive Service (LEDAS), provided by NASA’s Goddard Space Flight Center and the Department of Physics and Astronomy, Leicester University, UK, respectively. SRO acknowledges the support of an STFC Studentship. SZ thanks STFC for its support through an STFC Advanced Fellowship.
|
---
abstract: 'Gamma-ray burst sources with a high luminosity can produce electron-positron pair cascades in their environment as a result of back-scattering of a seed fraction of their original hard spectrum. The resulting spectral modifications offer the possibility of diagnosing not only the compactness parameter of the $\gamma$-ray emitting region but also the baryonic density of the environment external to the burst.'
author:
- 'P. $^{1,2,3}$ , E. Ramirez-Ruiz$^1$, M.J. Rees$^1$'
title: ' $e^\pm$ Pair Cascades and Precursors in Gamma-Ray Bursts'
---
2si Ł50[L\_[w50]{}]{} 51[E\_[w51]{}]{} 1[t\_[w1]{}]{} 3[t\_[v-3]{}]{}
$^1$[Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.]{}\
$^2$[Dpt. of Astronomy & Astrophysics, Pennsylvania State University, University Park, PA 16803]{}\
$^3$[Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030]{}
Introduction
============
The non-thermal $\gamma$-ray spectrum of Gamma-Ray Burst sources (GRBs) is thought to arise in shocks which develop beyond the radius at which the relativistic fireball from the initial event has become optically thin to scattering. However, the observed spectra are hard, with a significant fraction of the energy above the $\gamma\gamma \to \epm$ formation energy threshold, and a high compactness parameter can result in new pairs being formed outside the originally optically thin shocks responsible for the primary radiation. New pairs can be made as some of the initial energetic photons are backscattered and interact with other incoming photons. Such effects have been considered by Madau & Thompson (2000) and Thompson & Madau (2000), who investigated the acceleration of new pairs for a particular fireball model. Dermer & Böttcher (2000) considered the effect of pair formation for an external shock model of GRB, in which the compactness parameter is relatively low, while Madau, Blandford & Rees (2000) investigated the effects of Compton echos produced by pairs.
Here we present a simplified discussion of the generic effects of pair formation in a specific class of models. We suppose that the central engine gives rise to an unsteady baryonic wind, which is relativistic, carries a magnetic field, and lasts for a time $t_w$. A fraction of the wind energy is converted into $\gamma$-rays via internal shocks; the remaining wind energy drives a blast wave which decelerates as it sweeps up the external medium, and gives rise to the afterglow emission. The $\gamma$-rays would propagate ahead of the blast wave, leading to pair production (and an associated deposition of momentum) into the external medium. The pair cascades saturate after the external (pair-enriched) medium reaches a critical bulk Lorentz factor, which is generally below that of the original relativistic wind. For external baryonic densities similar to those in molecular clouds the pairs can achieve scattering optical depths $\tau_\pm \siml 1$. Even for less extreme external densities the effect of the additional pairs can be substantial, increasing the radiative efficiency of the blast wave and leading to distortions of the original spectrum. This provides a potential tool for diagnosing the compactness parameter of the bursts and thus the radial distance at which shocks producing an observed luminosity can occur. It also provides a tool for diagnosing the baryonic density of the external environment of the bursts, and testing the association with star-forming regions.
Scattering and Two-photon Pair Formation
========================================
Consider an initial input radiation spectrum produced by the fireball of the form ()= [ F m\_e c\^2 \_o\^2 q ]{} (/\_o)\^[-]{} 2si ; \_o , \[eq:spin\] where $F=L/4\pi d_L^2$ is the observed energy flux, $\eps=h\nu/m_e c^2$ is photon energy in electron rest-mass units, $\eps_o\sim 0.2-1.0$ is the break energy above which the photon number spectral index $\beta\sim 2-3$, and below which the spectral slope is flatter, e.g. $\alpha\sim -2/3$ for a simple low-energy cutoff synchrotron spectrum (although for the present purposes the exact low energy slope is unimportant), with $q\sim 1$ a normalization constant. For most values of the low and high energy slopes, the majority of the photons in the spectrum are near the break frequency.
For an $\epm$ moving away radially from the source of radiation with a velocity characterized by a Lorentz factor $\Gamma$, taking into account the Klein-Nishina drop-off in the scattering cross section above $\eps\sim
1$, the effective fraction of photons contributing to accelerating the electron is $\int_{\eps_o}^\Gamma \phi(\eps) d\eps =q^{-1}
\eps_o^{\beta-2}(\beta-1)^{-1}
[\Gamma^{-1}-\eps_o^{-1}] \simeq q^{-1} \eps_o^{-1} (\beta-1)^{-1}$ for $\beta>1$. At a distance $r$ in front of the radiation source, an $\epm$ can be accelerated to a maximum value of $\Gamma$ satisfying (/2) t m\_e c\^2 , \[eq:scatt\] where the cross section near $\eps\sim 1$ is approximately $\st/2$, $\mu m_e$ is the effective mass per scatterer (equation \[\[eq:mu\]\]), and $\Delta t$ is the effective duration of the light pulse as seen by the electron. The latter is either $\Delta t\sim r/2c\Gamma^2$ (impulsive regime at small radii), or $\Delta t \simeq t_w$ (wind regime), depending on which one is smallest at a given radius, where $t_w$ is the duration of the wind, or essentially the burst duration as seen by a distant observer. Equation (\[eq:scatt\]) says that the total time-integrated momentum of the radiation intercepted by the electrons in the Thompson limit is converted into their kinetic energy of motion, taking into account the effective mean mass per scatterer. (This follows directly from the mean rate of momentum transfer per particle in a steady flow, e.g., as discussed in Madau & Thompson, 2000, their equation (2), (10) and (23), noting that our definition of $\mu$ differs from theirs.) Here we define $\mu$ as the effective mass per scattering electron or positron in units of electron mass, m\_e =[[mass]{} ]{}= m\_p [ \[1+2(/n\_p)(m\_e/m\_p)\] ]{} \[eq:mu\] Defining a reference radius r\_= L/16m\_e c\^3=510\^[19]{}Ł50 , \[eq:rast\] from equation (\[eq:scatt\]) the maximum $\epm$ Lorentz factor is (r) \[eq:Gammasc\] The transition from the $\Gamma\propto r^{-1/3}$ to the steeper $\Gamma\propto r^{-2}$ occurs at a critical radius $r_{c}\ll \rast$, for which the Lorentz factor is $\Gamma_{c}$, r\_[c]{}= & 2.210\^[14]{}\^[-2/5]{}(q \_o (-1))\^[-2/5]{} Ł50\^[2/5]{}t\_w\^[3/5]{}\
\_[c]{}= & 6.110\^1 \^[-1/5]{}(q \_o (-1))\^[2/15]{} Ł50\^[1/5]{}t\_w\^[-1/5]{}. \[eq:rcsc\]
A second criterion for a maximum $\epm$ Lorentz factor comes from the pair formation threshold, since the incident photon $\eps$ and the back-scattered photon $\eps_r$ must satisfy $\eps\eps_r\geq 2$. The back-scattered photon, has, in the reference frame of the scattering $\epm$ moving with $\Gpm$, an energy of at most $\eps_r'\siml 1/2$; this photon cannot give rise to a further pair unless it collides with another photon with, in the lab frame, an energy exceeding $\sim 4 \Gpm $. Thus the fraction of incident photons able to make pairs through the two-photon mechanism against target photons backscattered from $\epm$ moving with $\Gpm$ is $q^{-1}\eps_o^{-2}\int_{4\Gamma}^\infty \phi(\eps)d\eps=
q^{-1}\eps_o^{\beta-2} (\beta-1)^{-1} (4\Gamma)^{1-\beta}$. The maximum Lorentz factor achievable by pairs before the two-photon cascade cuts off is that for which the compactness parameter has dropped to unity, (r) = [L4r\^2]{} [\_o\^[-2]{} q (-1)]{} [/3 m\_e c\^2]{} [1(4)\^[-1]{}]{} t 1 , \[eq:compact\] where the effective duration is as before. The maximum Lorentz factor for pair formation is then (r) . \[eq:Gammapp\] This yields a critical radius and Lorentz factor for the transition between the impulsive and wind dominated regimes of r\_[c]{}= & ( [2\^[7-3]{} 9(-1)\^2]{})\^[1+3]{} ([\_oq]{})\^[2-4+3]{} ([ct\_w]{})\^[+1+3]{} 510\^[14]{}Ł50\^[2/5]{}1\^[3/5]{} ,\
\_[c]{}= & ([2\^[-2(-1)]{} 3(-1)]{})\^[1+3]{} ([\_oq]{})\^[-2+3]{} ([ct\_w]{})\^[1+3]{} 310\^1 Ł50\^[1/5]{}1\^[-1/5]{}, \[eq:rcpp\] where the numerical values in the second equation of both lines are calculated for $\beta=2$, $\eps_o \simeq q \simeq 1$, $L=10^{50}\L50$ erg s$^{-1}$ and $t_w=10\tw1$ s. (At early times $t<t_w$ the radius $r_c$ is smaller, $\propto t^{3/5}$, since it takes $t_w$ for the entire photon energy to build up). The maximum pair Lorentz factor is shown in Figure \[fig:fig1\] for nominal parameter values. The pair formation limit (\[eq:rcpp\]) is somewhat more restrictive than the scattering limit (\[eq:rcsc\]). Also, unlike the scattering limit, the pair formation limit depends exponentially on the photon number slope $\beta$. However, for the canonical value $\beta=2$ the two power law dependences are the same and the numerical values are close to each other. In what follows we shall use the pair formation limit of equations (\[eq:Gammapp\]) and (\[eq:rcpp\]).
Pair-precursor and Kinematics
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Given a certain external baryon density $n_p$ at a radius $r$ outside the shocks producing the GRB primordial spectrum, the initial Thomson scattering optical depth is $\tau_i \sim n_p\st r$ and a fraction $\tau_i$ of the primordial photons will be scattered back, initiating a pair cascade. Since the photon flux drops as $r^{-2}$, for a uniform (or decreasing) external ion density most of the scattering occurs between $r$ and $r/2$, and the scattering and pair formation may be approximated as a local phenomenon. This pair-dominated plasma, as long as its density $\npm \siml
n_p(m_p/2m_e)$, is initially held back by the inertia of its constituent ions component, provided that the pairs remain coupled to the baryons. (The latter is likely to be the case in the presence of even weak magnetic fields, e.g. Madau & Thompson 2000, or plasma wave scattering, e.g. Lightman, 1982. Moreover, the initial magnetic field strength can also increase as consequence of instabilities caused by an initial pair streaming relative to ions. In what follows, we assume that this coupling is effective). The pairs (together with the ions) start acquiring a significant velocity $\Gpm\simg 1$ only after their density $\npm\simg n_p(m_p/2m_e)$, when the effective mass per scatterer $\mu m_e \sim m_e$. After the pairs start being accelerated, the pair cascade can continue multiplying as long as the compactness parameter of equation (\[eq:Gammapp\]) is $\ell\simg 1$. Both the compactness $\ell$ and the maximum $\Gpm$ decrease for increasing radius according to equations (\[eq:compact\]) and (\[eq:Gammapp\]), and the cascade process shuts off for $\ell\siml 1$ at a radius where the maximum $\Gpm \sim 1$, r\_\~(4 c t\_w /3 )\^[1/2]{}\~4 10\^[15]{}Ł50\^[1/2]{}1\^[1/2]{} . \[eq:rell\] On the other hand, the blast wave producing the shock(s) responsible for the afterglow spectrum starts decelerating (i.e. its initial Lorentz factor $\Gamma_f\sim\eta =10^2\eta_2$ starts to decrease as a negative power law in radius) at the deceleration radius r\_d(3 L t\_w/4n\_p m\_p c\^2 \^2)\^[1/3]{} 310\^[15]{}Ł50\^[1/3]{}1\^[1/3]{}\_2\^[-2/3]{}n\_3\^[-1/3]{} , \[eq:rdec\] where we normalized to an external baryon density $n_p=10^3n_3\cmcui$. (This expression is valid provided that the deceleration radius exceeds $ct_w \eta^2$. Otherwise the deceleration starts before the outflow is over, and $t_w$ should be replaced by $t$ in the expression). A larger value $n_p \geq 10^{3}n_3\cmcui$ would give $r_d\siml r_{\ell}$. (Such a higher density would, e.g., be required for pair effects to be important in an external shock model, where the $\gamma$-rays arise at $r_d$). In an internal shock model the primary $\gamma$-rays themselves arise at a smaller radius r\_i\~c t\_v \^2 \~310\^[13]{} t\_[v -1]{} \_2\^2 \[eq:rint\] where $t_v =10^{-1}t_{v-1} \simg 10^{-3}$ s is the variability timescale in the outflow. The criterion for an $\epm$ cascade to form ahead of the fireball producing a primary spectrum (\[eq:spin\]) is that $r_i$ (or $r_d$ for an external shock model) is smaller than $r_\ell$. Pair formation occurs then for $r \leq r_\ell$.
The minimum number of pairs is formed just inside $r_\ell$, where they are created essentially at rest, and their density can reach $\npm\sim
(m_p/2 m_e) n_p \sim 10^3 n_p$, at which point the mass per scatterer has become comparable to the electron mass. This represents an increase in the electron (or positron) scattering opacity by a multiplication factor of $k_p\sim
2\times 10^3$. As the density exceeds this value at $r_\ell$ the pairs are pushed by scattering beyond $r_\ell$, where the condition $\ell <1$ prevents further multiplication.
For radii $r<r_\ell$, and as long as the blast wave has not reached that radius, pair formation exceeding the above value $\npm\sim 10^3 n_p$ can continue to occur, with the pairs being accelerated by scattering until they reach the maximum Lorentz factor (\[eq:Gammapp\]). Pair production will be most copious at smaller radii $r_i <r < r_{c}$, where the pairs are accelerated to a maximum Lorentz factor $\Gpm(r)\siml
\Gamma_{c}(r/r_{c})^{-1/3}$. At each step of the cascade the Lorentz factor of the new generation of pairs approximately doubles, and the maximum pair multiplication factor of the cascade can be estimated from the number $s$ of pair generations required to go from $\Gamma\sim 1$ to $\Gpm(r)$ given by equation (\[eq:Gammapp\]). At $r_{c}$ this is $s(r_c) \sim \log \Gamma_c /\log 2 \sim 5$, and for $r_i \siml r $ it is limited to $s(r_i) \siml 7.5 $. This represents an extra multiplication factor $k_a \sim 2^s \sim \Gamma_c (r/r_c)^{-1/3} \sim 30-170$ in the electron opacity for $r_i \leq r \leq r_c$, in addition to the previous factor $k_p \sim 2\times 10^3$ achieved before acceleration starts. The critical value is k\_c = k\_p k\_a(r\_c) \~(m\_p/m\_e) \_c \~ 510\^4 Ł50\^[1/5]{}1\^[-1/5]{} , \[eq:kc\] with $k(r)=k_c (r/r_c)^{-1/3}$ for $r_i\leq r_c$, while $k(r) \sim k_c
(r/r_c)^{-2}$ for $r_c\leq r \leq r_\ell$ drops to $m_p/m_e$ at $r_\ell$, and to zero beyond that. In a more detailed calculation, the number of cascades depends somewhat on the radiation spectral index $\beta$, here set equal to 2. For a steeper spectrum, the number of cascades is reduced ( e.g. Ramirez-Ruiz [*et al.*]{} 2000). The maximum pair optical depth is achieved at $r_c$, \_[c]{} \~k\_c n\_p r\_c = . \[eq:tauc\] The pair opacity scales as $\tau_\pm\propto r^{2/3}$ for $r_i\siml r \siml
r_c$ and $\tau_\pm\propto r^{-1}$ for $r_c\siml r \siml r_\ell$. For an external density greater than n\_[p,c]{}10\^5 Ł50\^[-3/5]{} 1\^[-2/5]{} , \[eq:npc\] the pair density inside $r_c$ is prevented from multiplying beyond a value corresponding to $\tau_\pm\sim 1$, due to self-shielding. The number of pairs created at $r_c$ is N\_[c]{}\~n\_[p]{} k\_c (4/3)r\_c\^3 \~310\^[49]{} n\_p Ł50\^[7/5]{} 1\^[8/5]{} , \[eq:Npmc\] which scales as $N_\pm\propto r^{8/3},~r$ for $r$ below or above $r_c$. The maximum value for $n_p=n_{p,c}$ is $N_{\pm,c,mx}\sim 3\times 10^{54}\L50^{4/5} \tw1^{6/5}$ at $r=r_c$, and $N_{\pm,\ell ,mx}\sim 3\times 10^{55}\L50^{9/10} \tw1^{11/10}$ at $r_\ell$. The pair cloud extends over a range of radii comparable to $r$, much larger than the photon pulse radial width $c t_w$.
Figure \[fig:fig2\] shows the schematic world-lines of three blast waves expanding into an external medium. At each radius $r_i < r < r_\ell$ pairs form ahead of $r_i$. While for $r\sim r_\ell$ the pairs are sub-relativistic and do not propagate significantly, for smaller radii they move relativistically with an initial speed $\Gpm$, with $\Gpm<\eta$ in general, although at very small radii, and for times $t<t_w$, $\Gpm> \eta$ may be possible. If the blast wave did not exist, then the pair-enriched material (moving out with a Lorentz factor that is larger at smaller r) would pile up at a radius $r_l$ after a time $r_l/c$. However, the pair-rich external medium would generally carry less energy and inertia than the relativistic wind itself; it therefore starts being decelerated by the external medium at a smaller radius than a blast wave with the same Lorentz factor would be, so that the latter always overtakes it and sweeps up the pair-enriched external medium. The resultant outer shock is more likely to be radiative, but may be weaker, because the ambient material might already be moving outward relativistically before the blast wave hits it.
For a relatively low external density such as $n_p=10^3\cmcui$, as in curve (a) of Figure \[fig:fig2\], the radius $r_c <r_d$, and the pairs are swept up by a fireball expanding at its original speed $\eta\sim$ constant, and deceleration of the fireball begins sometime after the pairs have been swept up. In this case the main afterglow, produced by the deceleration of the pair-enriched fireball will be characterized by a larger emission measure than in the usual case, due to the extra pairs in the swept-up matter, but will not otherwise exhibit any irregularities in its time development. For a larger external density such as $10^6\cmcui$, corresponding to curve (b) of Figure \[fig:fig2\], the radius $r_c\sim r_d$ and the main part of the afterglow, caused by the deceleration of the fireball baryons and the pairs formed at early stages, continues to sweep up further pairs which were formed at or beyond the deceleration radius (in curve (c), with low $\eta$, deceleration occurs at radii $r>\;r_\ell$). These new pairs change the emission measure of the deceleration shock, since their optical depth $\tau_\pm\sim 1$ is larger than that of the swept up baryonic electrons. If $\tau_\pm \sim 1$ is reached, the original spectrum may be modified by Comptonization, and may approach a quasi-thermal shape. We note that the optical depth cannot substantially exceed unity. This is because high-energy non-thermal photons would be comptonised by pairs , and after a few scatterings would be degraded below the pair-production threshold. So, however intense the radiation incident on its inner edge, a shell of pair-dominated plasma cannot build up an opacity much larger than unity. It will come to a quasi-steady state in which the luminosity from its outer surface (comptonised synchrotron radiation and annihilation radiation) equals the non-thermal radiation shining on its inner surface.
Observable Pair-precursor Effects {#sec:prec}
=================================
The energy in the accelerated pairs is $E_\pm \propto N_\pm \Gamma_\pm r^3
\propto
r^{7/3}$ for $r\siml r_c$ and $E_\pm \propto r^{-1}$ for $r_c\siml r \siml
r_\ell$. The maximum number of pairs created is E\_[c]{} N\_m\_e c\^2\_c = [erg]{}, \[eq:Epm\] where the second number is the limit $N_{\pm ,mx}$ obtained for $n_p=n_{p c}$, at which the saturation value of $\tau_\pm\sim 1$ is obtained. The pair energy never exceeds more than a fraction of the initial wind energy, so the deceleration radius (equation \[\[eq:rdec\]\]) is unaffected, while the pairs which are produced and accelerated ahead of the blast wave, are decelerated at a (smaller) radius r\_[d]{}\~2.510\^[15]{} E\_[50]{}\^[1/3]{} n\_3\^[-1/3]{}(/30)\^[-2/3]{} [cm]{} , \[eq:rdecpm\] before the ejecta itself starts to decelerate.
The total number of protons in the ejecta wind is $N_p =E/\eta m_p c^2=
(2/3)\times 10^{52}\L50\tw1\eta_2^{-1}$. The number of protons in the (pair enriched) external medium which are swept up by the time the blast wave reaches its deceleration radius $r_d$ is $N_{ps}=N_p\eta^{-1}=
(2/3)\times 10^{50}\L50\tw1 \eta_2^{-2}$. The maximum number of swept-up pairs can exceed the number of baryons by a large factor. However the inertia of the pairs is less than that of the baryonic ejecta, since $(N_{\pm,c,mx}/N_{p})\sim 5\times 10^2 \L50^{-1/5}\tw1^{1/5}\eta_2 \siml
(m_p/m_e)(\eta/\Gamma_c) =6\times 10^3 \L50^{-1/5}\tw1^{1/5}\eta_2$ at $r_c$, and $(N_{\pm,\ell, mx}/N_p)\sim 5\times 10^3\L50^{-1/10}
\tw1^{1/10}\eta_2 \leq (m_p/m_e)\eta=2\times 10^5\eta_2$ at $r_\ell$.
The pairs will modify the usual properties of the deceleration shocks and the afterglow emission. The total energy available in the afterglow is not changed: it is still, essentially, the kinetic energy of the relativistic wind, minus the fraction dissipated (and converted into prompt gamma rays) in internal shocks. But the lepton/proton ratio in the ejecta can be much larger than usual. This increases the radiative efficiency significantly, since most of the particles are $\epm$. The resultant radiation will thus be softer than in the usual picture, because the same energy density has to be shared among a larger number of particles ($N_\pm/N_p \gg 1$). In the reverse shock that occurs after the baryonic+pair ejecta starts being decelerated by the external medium, the comoving frame peak random electron Lorentz factor is $\gamma_{\pm, m,r} \sim (E \eta^{-1}/ N_\pm m_e c^2)$. Even for $\tau_\pm$ substantially less than unity, e.g. for an external density $n=10^3 n_3\cmcui$ (below the critical value $n_{p,c}$) for which the deceleration radius $r_d\sim r_c$, the reverse shock random lepton Lorentz factor is $\gamma_{\pm, m, r} \sim
30\E51^{-1/5}\tw1^{-3/5}n_3^{-1} \eta_2^{-1}$. The pairs should not affect, however, the random turbulent magnetic field strength, which is in pressure equilibrium with the forward shock at some fraction $\eps_B$ of the equipartition value, $B'\sim (\eps_B 8\pi n m_p c^2)^{1/2}\eta
\sim 6\times \eps_B^{1/2} n_3^{1/2}\eta_2$ G. Thus the observed reverse synchrotron peak frequency \_[,sy,r]{}\~10\^6 B’\^2 (1+z)\^[-1]{} \~610\^[13]{} 51\^[-2/5]{}1\^[-6/5]{}\_B\^[1/2]{} n\_3\^[-3/2]{}\_2\^[-1]{} (1+z)\^[-1]{} [Hz]{} \[eq:nusyr\] would be in the far IR, as opposed to the optical/UV of the usual baryon-dominated prompt reverse flash.
The forward shock afterglow radiation, normally in the $\gamma$/X-ray range, would be unaffected outside $r_\ell$. However, up to about the time when deceleration starts, there will be a “pair pick-up" photon pulse, when the protons and electrons of the relativistic wind moving with $\eta >\Gamma_c$ sweep up the pair-enriched external medium. The protons are not decelerated by the pairs, so there is only a subsonic reverse compression wave in the proton ejecta, but there will be a mildly relativistic forward shock in the picked-up pairs. The $\epm$ random Lorentz factor will be equal to the bulk kinetic energy it has in the proton frame, e.g. $\gamma_{\pm, m, f} \sim
\eta/\Gpm \sim 3 \L50^{-1/5}\tw1^{1/5}\eta_2$ near $r_c$. The comoving random magnetic field in the shocked pair fluid is $B'\sim
(8\pi \eps_B N_{\pm,c} m_e c^2 /4 r_c^3)^{1/2} (\eta/\Gamma_\pm)$ $\sim 20 \eps_B^{1/2}\L50^{1/5}\tw1^{1/10} n_3^{1/2}\eta_2$ G for $n_p=10^3 n_3\cmcui$, corresponding to a pair-pickup pulse synchrotron peak frequency in the observer frame of \_[,sy,f]{} \~210\^[10]{} \_B\^[1/2]{}Ł50\^[-1/5]{}1\^[5/10]{}n\_3\^[1/2]{}\_2\^3 (1+z)\^[-1]{} [Hz]{}, \[eq:nusyf\] and a synchrotron power law above this frequency.
There could also be additional pair-precursor signatures which are not associated with the ejecta blast wave, but with the dynamics of the pre-accelerated pair-enriched plasma. The r-dependent $\Gpm$ of the pair-enriched external plasma will lead to internal shocks. Pair regions at $r<r_c$ whose $\Gpm$ differ by order unity will collide at radii $\propto r \Gpm^2 \propto r^{1/3}$, so for sufficiently short variability times these shocks would occur between $r_i$ and $r_c$, i.e. up to an observer time $t_c\sim r_c/c\Gamma_c^2\sim 15 \tw1$ s, independent of the wind luminosity $L$. (For $r_c <r <r_\ell$ the collision radii are $\propto r^{-3}$, so the shocks would tend to pile up at $r_\ell$ at an observer timescale $t_\ell\sim r_\ell/c \sim$ 1 day, provided the main part of the fireball wind has not caught up with it before. This could be the case for a jet-like wind for pairs in a rim of angular width $\Gamma_c^{-1}$ around the jet). The pair internal shocks at $r\sim r_c$ can produce a Fermi accelerated power law $\epm$ spectrum above $\gamma_{\pm m} \sim 1$ and a turbulent comoving field $B'_{\pm } \siml 2\times 10^2$ G, leading to a synchrotron spectrum whose observer-frame peak frequency $\nu_{\pm m}
\sim 6\times 10^9$ Hz would be self-absorbed, with a power law extension above it which would be optically thin at higher frequencies.
The above observational signatures would be present even if $\tau_\pm$ is low, as expected for external ion densities $n_p \siml n_{p,c} \sim
10^5 \L50^{-3/5} \tw1^{-2/5} \cmcui$ around the burst. An additional effect of interest, for external densities in excess of this which lead to a pair screen of optical depth $\tau_\pm \sim 1$, is that a quasi-thermal pulse of X-rays could accompany the burst, caused by upscattering of diffuse progenitor stellar photons. A collapsing massive progenitor leading to a GRB is likely to be highly super-Eddington for sometime after the GRB event, e.g. $L_\ast \sim 10^4 L_{Ed}\sim 10^{43} L_{\ast 43}$ erg/s. The density of photons of energy $\eps_\ast\sim 10$ eV near $r_c$, which would be quasi-isotropized due to the condition $\tau_\pm \sim 1$ at $r_c$ (and beyond, where $\Gamma_\pm \to 1$ at $r_\ell$), would be $n_\ast \sim 10^{13} L_{\ast 43} \eps_{\ast 10}^{-1}\L50^{-4/5}
\tw1^{-6/5}\cmcui$ near $r_c$. The screen with $\tau_\pm\sim 1$ moving with $\Gamma_c\sim 30$ sweeps up a total number of photons $N_{\ast}\sim
3\times 10^{57}L_{\ast 43}\eps_{\ast 10}^{-1} \L50^{2/5}\tw1^{3/5}$. The mean energy per photon and the time-integrated total energy of the upscattered precursor is \_X\~& 10 \_[10]{} Ł50\^[2/5]{}1\^[-2/5]{}(1+z)\^[-1]{} [keV]{}\
E\_X\~& 510\^[49]{} L\_[43]{}\_[10]{}\^[-1]{}Ł50\^[4/5]{} 1\^[1/5]{} (1+z)\^[-1]{} [erg]{}. \[eq:xrprec\] This would last until the pairs are swept up by the ejecta, $t_X \siml t_c\sim r_c/c\Gamma_c^2\sim 15 \tw1$ s.
Discussion
==========
We have discussed, in the context of a standard internal/external shock model of gamma-ray bursts (which are normally assumed to occur after the original fireball has become optically thin) the $\gamma\gamma\to \epm$ cascades triggered by the back-scattering of seed gamma-ray photons on the external medium. This effect can modify the initial scattering optical depth of the outflow at radii $r\siml r_\ell \simeq 4\times
10^{15}\L50^{1/2}\tw1^{1/2}~\hbox{cm}$ (equation \[\[eq:rell\]\]), which is comparable to the radii of external shocks of equation (\[eq:rdec\]) at which the afterglow begins, and is generally larger than the typical internal shock radii given by equation (\[eq:rint\]). The spectral effects of the pairs on the burst and the afterglow can be substantial, and within radii $\sim r_l$ they can affect the dynamics.
Pair production can increase the optical depth outside of the shocks by up to $\siml 10^5$; self-shielding ensures that the maximum scattering optical depths achievable by the pairs is $\tau_\pm\sim 1$. For typical interstellar densities the pair opacity $\tau_\pm \ll 1$, which does not significantly affect the gamma-ray spectrum. The number of pairs may nonetheless be large enough to increase the radiative efficiency and soften significantly the radiation spectrum of the afterglow reverse shock, where the same energy is shared among a number of $\epm$ which can exceed that of the original $e^-$ and $p^+$ of the ejecta.
The pair production processes themselves (determining $r_c,~ r_l$, equations \[\[eq:rcpp\]\], \[\[eq:rell\]\]) just depend on the “seed" $\gamma$-ray photon flux (which are here postulated to come from internal shocks). The manifestations depend on the external density and on the initial dimensionless entropy or bulk Lorentz factor $\eta$. The external baryon density $n_{ext}$ determines the optical depth that can be built up through back-scattering and pair multiplication. This affects whether the pair optical depth gets up to unity, with smearing and reprocessing of the primordial $\gamma$-ray spectrum, or whether it merely makes the blast wave more radiative. Madau and Thompson (2000) have made this point, in the context of a specific fireball model.
The external density (along with the initial Lorentz factor $\eta$) determines when the outer shock and the reverse shock become important and whether this happens within the radius already polluted with pairs (and pre-accelerated by radiation pressure before the shock hits). There are two rather different cases depending on whether or not $\eta^2$ is less than $r_l/ct_w$). In the former case the external shock responsible for the afterglow occurs beyond the region “polluted" by new pairs, while in the second case the afterglow shock may experience, after starting out in the canonical manner, a “resurgence" or second kick as its radiative efficiency is boosted by running into an $\epm$-enriched gas. Internal shocks in the pair-dominated external plasma can lead to self-absorbed radiation at $\sim 10^9-10^{10}$ Hz, while the swept-up pairs can also contribute a $10^{11}-10^{12}$ Hz ‘prompt’ signal, which precedes the onset of the standard deceleration afterglow phase.
Additional effects are expected when $\tau_\pm \to 1$. This requires external baryon densities at radii $r<r_\ell$ of $n_p \simg n_{c,p} \sim
10^5 \L50^{-2/5} \tw1^{-3/5} \cmcui$. Such high densities would only be expected if the burst is associated with a massive star in which prior mass loss led to a dense circumstellar envelope The pair optical depth saturates to $\tau_\pm\sim 1$ and in addition to an increased efficiency and softer spectrum of the afterglow reverse shock, the original gamma-ray spectrum of the GRB will be modified as well. The specific nature of this spectral modification depends on the value of the luminosity, which influences (equation \[\[eq:rcpp\]\]) the bulk Lorentz factor of the reprocessing pair cloud before it has been swept up by the ejecta, and also on the extent to which the outflow is beamed. One of the consequences of such a critical external density leading to $\tau_\pm\sim 1$ would be the presence of an X-ray quasi-thermal pulse, whose total energy may be a few percent of the total burst energy. In the case of even more extreme densities, there are other interesting possibilities. For instance, dense blobs of Fe-enriched thermal plasma would emit strong recombination features, as well as annihilation radiation features, if the normal electron density were augmented by extra pairs.
Even in isotropic situations, the spectrum would be modified by transmission through a pair plasma of optical depth unity. The effect is maximal for photons of energy $m_e c^2$ in the frame of the pair plasma: for higher energies, the Klein-Nishina cross-section is smaller; for lower energies, the scatterings are almost elastic. If, for instance, $\Gamma_c\sim 40$, the gamma-ray spectrum around photon energies $\eps\sim \Gamma_c m_e c^2 \sim 20$ MeV would be depressed by a factor $\sim 1$ relative to its original value, smoothly rejoining its original value above and below that energy. For a beamed primary output, however, there would be a suppression at lower energies (where the scattering is in the Thompson regime) because the scattered photons would be spread over a wider angle. Pair-induced processes would therefore yield evidence on the beaming properties of the bursts. Irrespective of the external density, the processes discussed here suggest that bursts and afterglows may have a more complex spectrum and time-structure than ’standard’ models suggest. But the effects are especially interesting when the external density is high: they probe the environment of GRBs, and thus can offer clues to the nature of the progenitor stars, and their location within the host galaxy. For instance, a quasi-thermal X-ray pulse accompanying the gamma-ray emission could be indicative of an external circum-burst density of at least $10^4-10^5~\cmcui$. While quasi-thermal X-ray pulses might also arise due to other reasons, e.g. from an underlying optically thick central engine, if the X-ray luminosity scales as equation (\[eq:xrprec\]) and is accompanied by radio or far-IR signals such as in equations (\[eq:nusyr\])(\[eq:nusyf\]), this could be indicative of birth in a dense environment from a massive progenitor. Detailed Monte Carlo simulations (Ramirez-Ruiz , 2001) should provide a more detailed assessment of the self-consistent spectrum of a GRB in the presence of self-induced pair formation.
Dermer, C & Böttcher, M., 2000 ApJ 534, L155 Lightman, A.P, 1982, ApJ 253, 842. Madau, P & Thompson, C, 2000 ApJ, 534, 239 Madau, P, Blandford, R & Rees, M.J., 2000, ApJ, 541, 712. Ramirez-Ruiz, E, , 2001, in preparation Thompson, C & Madau, P, 2000 ApJ, 538, 105
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---
abstract: 'We present a magnetotransport study of the low–carrier density ferromagnet EuB$_{6}$. This semimetallic compound, which undergoes two ferromagnetic transitions at $T_{l}$$=$15.3K and $T_{c}$$=$12.5K, exhibits close to $T_{l}$ a colossal magnetoresistivity (CMR). We quantitatively compare our data to recent theoretical work [@maj], which however fails to explain our observations. We attribute this disagreement with theory to the unique type of magnetic polaronic formation in EuB$_{6}$.'
address:
- '$^{1}$Department of Physics, 2071 Randall Laboratory, University of Michigan, Ann Arbor, Michigan 48109–1120'
- '$^{2}$Inst. für Metallphysik, TU Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig, Germany'
- '$^{3}$National High Magnetic Field Laboratory, 1800 E. Paul Dirac Drive, Florida State University, Tallahassee, Florida 32310'
author:
- 'S. Süllow$^{1,2}$, I. Prasad$^{1,}$[@pra], S. Bogdanovich$^{1,}$[@bog], M.C. Aronson$^{1}$, J.L. Sarrao$^{3,}$[@sar], and Z. Fisk$^{3}$'
title: 'Magnetotransport in the low carrier density ferromagnet EuB$_{6}$'
---
Recently, critical fluctuations have been proposed by Majumdar and Littlewood [@maj] as mechanism causing the colossal magnetoresistivity (CMR) in a number of non–manganite materials, like in the pyrochlores or chalcogenide spinels [@ramirez; @shimikawa]. The authors of Ref. [@maj] argued that, because in a ferromagnetic metal close to its critical point the dominant magnetic fluctuations are those with a wave vector $q$$\rightarrow$0, the contributions from these fluctuations to the resistivity $\rho$ should grow as Fermi number $k_{f}$ and the carrier density $n$ decrease. For sufficiently small $n$, like in ferromagnetic semimetals, a major part of the zero–field resistivity close to $T_{c}$ would be caused by magnetic fluctuations. Suppressing these in magnetic fields should generate the CMR in such materials.
The resistivity of a low–carrier density system might also be affected by magnetic polarons. But magnetic polarons disappear ([*i.e.*]{} delocalize) if the magnetically correlated regions overlap, implying that critical magnetic scattering should dominate the resistivity for $k_{f}$$\xi(T)$$\gg$1 ($\xi(T)$: magnetic correlation length). In this regime of dominant critical scattering and in the clean limit, [*i.e.*]{} $k_{f}$$\lambda$$\gg$1, the low–field magnetoresistivity is quantitatively predicted to $(\rho (T, B) -\rho_{0} )/\rho_{0} = \Delta \rho /\rho = C (M/M_{sat})^{2}$, with $C$$\approx$($k_{f}$$\xi_{0}$)$^{-2}$ ($\lambda$: mean free path; $M$: magnetization; $\xi_{0}$: magnetic lattice spacing). Then, for a free electron gas a relationship between magnetoresistive coefficient and carrier density, $C$$\propto$($n^{2/3}$$\xi_{0}^{2}$)$^{-1}$, emerges as central result of Ref. [@maj], with a proportionality constant $\approx$1/38.
To test this prediction we performed a detailed study of the magnetoresistive properties of the divalent cubic hexaboride EuB$_{6}$ [@fisk]. This semimetal, with a carrier density determined from quantum oscillation experiments of 8.8$\times$10$^{-3}$ electrons per unit cell, undergoes two ferromagnetic transitions at $T_{c}$=12.5K and $T_{l}$=15.3K [@sullow], derived here from the maxima in the temperature derivative of the resistivity, $d \rho/ d T$. The effective carrier masses are slightly smaller than the free electron mass, and the Fermi surface is almost spherical. Hence, a free electron model appropriately describes this compound. The zero–field resistivity is metallic, and in free electron approximation we find $k_{f}$$\lambda$$\gg$1 up to room temperature [@lambda]. Further, with $\xi_{0}$$=$$\xi_{300 \rm K}$$=$4.185Å, in Oernstein–Zernicke approximation, $\chi (0)$$\propto$$\xi^{2}(T)$, and with the experimentally determined dc–susceptibility $\chi_{0}$ from Ref. [@sullow] the condition $k_{f}$$\xi(T)$$\gg$1 is fulfilled below 17K. Hence, EuB$_{6}$ fulfills all requirements of the model of Ref. [@maj].
Here, we present resistivity and magnetoresistivity measurements employing a standard 4–probe ac–technique on the crystal studied in Ref. [@sullow], with the current applied along the \[100\] and the field along the \[010\] of the crystalline unit cell. For a quantitative analysis of the magnetoresistivity we use the magnetization from Ref. [@sullow].
In Fig. \[fig:fig1\](a) and (b) we plot our raw data: the temperature ($T$) dependent resistivity $\rho$ of EuB$_{6}$ in fields $B$ up to 5T and the normalized magnetoresistivity $\Delta \rho (B) /\rho$ between 5.5 and 20K, corrected for demagnetization effects. The field dependence of $\rho$ reveals two different magnetoresistive regimes: For small $B$ a rapid decrease of $\rho (B)$ close to $T_{l}$ occurs, while hardly any effect on $\rho$ is observable below $\simeq$10K. The suppression of $\rho$ close to $T_{l}$, $\Delta \rho / \rho$$\approx$$-0.9$ in 2T, is comparable in size to other CMR compounds [@ramirez; @shimikawa]. In contrast, for large fields $\rho (B)$ increases with $B$, this in particular at low $T$. The positive magnetoresistivity represents the normal metallic contribution $\rho_{met}$ to $\rho (B)$.
We extract the magnetic scattering contribution from the total magnetoresistivity by subtracting the metallic magnetoresistivity $\rho_{met}$. To do that we parametrize the high–field magnetoresistivity with $\rho_{met}$=$\rho_{0} + a B^{x}$, $x$$\approx$2 and derive the magnetic part $\rho_{mag}$$=$$\rho (B) - \rho_{met}$. The field dependence of $\rho_{met}$ thus established for the data at 13K is included in Fig. \[fig:fig1\](b) as dashed line.
At any given temperature the minimum value of $\rho$ as function of $B$, $\rho_{min}$, constitutes an upper limit for the phonon contribution to $\rho$. We have included the values $\rho_{min}$ as function of $T$ in Fig. \[fig:fig1\](a) as shaded area, illustrating that at and above $T_{c}$ phonons contribute less than 15% to the zero–field resistivity.
To examine the dependence of the magnetic magnetoresistivity $\Delta \rho_{mag} /\rho$ on the normalized magnetization $M/M_{sat}$ we plot the two quantities at different temperatures in Fig. \[fig:fig2\] in a log–log plot, with the magnetic field as implicit variable. In order to compare with the model of Ref. [@maj], which is valid only in the paramagnetic phase, we restrict our analysis to temperatures $T$$\geq$$T_{l}$. As is illustrated in Fig. \[fig:fig2\], at these temperatures all data sets collapse on a universal curve. In particular, for $M/M_{sat}$$\leq$0.07 we find $\Delta \rho_{mag} /\rho$$=$$C$$(M/M_{sat})^{2}$, with $C$=75 (solid line).
The value of $C$ is in striking contrast to the prediction of Ref. [@maj]. With the carrier density $n$$=$1.2$\times$10$^{-4}$/Å$^{3}$ for our crystal EuB$_{6}$ we compute $C$$\approx$(38$n^{2/3} \xi_{0}^{2}$)$^{-1}$=0.62 rather than the observed 75. More generally, following Ref. [@maj] we plot $C$ vs. $n$$\xi_{0}^{3}$ for metallic ferromagnets and manganites in Fig. \[fig:fig3\], together with the data for EuB$_{6}$ of our crystal and from previous works [@guy; @cooley]. In the plot we include the predicted values $C$$=$(38$n^{2/3} \xi_{0}^{2}$)$^{-1}$. The data for EuB$_{6}$ deviate by an order of magnitude from those of the other materials, emphasizing the vastly different magnetoresistive behavior of this compound.
We believe that the unique type of magnetic polaron formation feature in EuB$_{6}$ causes the failure of the model of Ref. [@maj] to account for the observed behavior. As we have proposed elsewhere [@sul2], at $T_{l}$ polaron metallization via magnetic polaron overlap leads to a drop of $\rho (T)$. The polaron metallization is accompanied by a filamentory type of ferromagnetic ordering, which arises from internal structure of the polarons. The bulk magnetic transition occurs at $T_{c}$. The field dependence of the resistivity close to $T_{l}$ then is mainly governed by the increase of the polaron size with magnetic field (rather than by the suppression of critical scattering, as suggested in Ref. [@maj]), causing the metallization to occur at a higher temperature, and leading to the reduction of the resistivity at $T_{l}$ in magnetic fields.
Work at the University of Michigan was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under grant 94–ER–45526 and 97–ER–2753. Work at the TU Braunschweig was supported by the Deutsche Forschungsgemeinschaft DFG
Present address: Harvard Univ., Cambridge, MA 02138. Present address: NHMFL, Tallahassee, FL 32310. Present address: LANL, Los Alamos, NM 87545. P. Majumdar and P. Littlewood, Phys. Rev. Lett. [**81**]{}, 1314 (1998); Nature (London) [**395**]{}, 479 (1998). A.P. Ramirez, J. Phys. Condens. Matter [**9**]{}, 8171 (1997); A.J. Millis, B.I. Shraiman, and R. Mueller, Phys. Rev. Lett. [**77**]{}, 175 (1996). Y. Shimikawa, Y. Kubo and T. Manako, Nature (London) [**379**]{}, 53 (1996); M.A. Subramanian [*et al.*]{}, Science [**273**]{}, 81 (1996); A.P. Ramirez, R.J. Cava, and J. Krajewski, Nature [**386**]{}, 156 (1997). L. Degiorgi [*et al.*]{}, Phys. Rev. Lett. [**79**]{}, 5134 (1997); D.P. Young [*et al.*]{}, Nature [**397**]{}, 412 (1999). S. Süllow [*et al.*]{}, Phys. Rev. B [**57**]{}, 5860 (1998); M.C. Aronson [*et al.*]{}, Phys. Rev. B [**59**]{}, 4720 (1999). We calculate at room temperature, with $\rho \approx 700 \mu \Omega$cm, in free electron approximation $k_{f} \lambda$=$\frac{(3 \pi^{2})^{2/3} \hbar}{e^{2} \rho n^{1/3}}$$\approx$10. C.N. Guy [*et al.*]{}, Solid State Commun. [**33**]{}, 1055 (1980). J. Cooley [*et al.*]{}, Phys. Rev. B [**56**]{}, 14541 (1997); J. Cooley, Ph.D. Thesis, Ann Arbor (1997), unpublished. S. Süllow [*et al.*]{}, submitted.
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abstract: 'In the usual optomechanical cooling, even if the system has no thermal component, it still has a quantum limitknown as the quantum backaction limit (QBL)on the minimum phonon number related to shot noise. By studying the side-band cooling regime in optomechanical system (OMS), we find that the cooling can be improved significantly when the frequency modulation (FM) that can suppress the Stokes heating processes is introduced into the system. We analyze and demonstrate the reasons of the phonon number below the QBL redefined in the whole stable region of the standard OMSs. The above analyses are further checked by numerically solving the differential equations of second order moments derived from the quantum master equation with broad system parameters, ranging from weak coupling (WC) to ultra-strong coupling (USC) and resolved side-band (RSB) to unresolved side-band (USB) regimes. Comparing with the cases of those without FM, the stable ground-state cooling can also be achieved even in the conventional unstable region.'
author:
- 'Dong-Yang Wang'
- 'Cheng-Hua Bai'
- 'Shutian Liu[^1]'
- 'Shou Zhang[^2]'
- 'Hong-Fu Wang[^3]'
title: Optomechanical cooling beyond the quantum backaction limit with frequency modulation
---
Introduction {#sec.1}
============
Over the past decades, special attention has been paid to studying the micromechanical resonators which are novel quantum devices and are used to explore various of quantum mechanics questions of the macroscopic scale, such as quantum-classical mechanics boundary [@PhysRevLett.88.120401; @PhysRevLett.97.237201; @POOT2012273], ultrahigh precision metrology [@RevModPhys.52.341; @RevModPhys.68.755], and gravitational-wave detection [@PhysRevLett.116.061102], etc. However, for overcoming thermal noise to achieve those applications, the mechanical resonator needs to be cooled to its ground state at first. So far, various of proposals of cooling resonator have been proposed and discussed, including feedback control [@PhysRevLett.80.688; @Wilson2015], modulating coupling strength [@PhysRevLett.95.097204; @PhysRevLett.107.177204], and cavity side-band cooling [@PhysRevLett.99.093901; @PhysRevLett.99.093902]. In these proposals mentioned above, cavity side-band cooling method is one of the most promising proposals, which has been studied widely and realized experimentally [@nature.444.67; @Nat.Phys.4.415; @Nature.463.72; @1367-2630-14-9-095015; @PhysRevLett.116.013602]. However, there has a QBL on the minimum phonon number related to shot noise even if the systems have no any coupling with thermal environment [@PhysRevLett.99.093902; @CPB.22.114213]. Recently, the QBL of cooling mechanical resonator has also been reached in cavity OMS via the side-band cooling approach [@PhysRevLett.116.063601]. It is natural to think that the cooling limit can be broken through some ways.
To this end, some theoretical proposals have been proposed to cool the mechanical resonator below the QBL, including dissipative coupling [@PhysRevLett.102.207209], pulsed cooling [@PhysRevLett.110.153606], electromagnetically-induced-transparency-like [@PhysRevA.90.013824; @PhysRevA.90.053841; @Liu:18; @PhysRevA.93.033845], trapping optical parametric amplifier [@PhysRevA.79.013821], and injecting squeezed light [@PhysRevA.94.051801], etc. And the proposal utilizing squeezed light to improve cooling has been proved experimentally [@nature.541.191]. Moreover, another interesting experiment also cools the mechanical resonator below the QBL via feedback-controllably engineering the pump field [@PhysRevLett.119.123603]. On the other hand, the modulated OMSs exhibit rich behaviors and have attracted amount of attention [@PhysRevLett.103.213603; @PhysRevA.83.043804; @PhysRevA.86.013820; @PhysRevA.93.033853; @PhysRevA.89.023843; @PhysRevA.92.013822; @PhysRevA.94.053807; @PhysRevA.95.053861]. However, in those modulation proposals, the studying about mechanical resonator cooling is relatively rare. In Ref. [@PhysRevA.91.023818], the authors study the optomechanical cooling theoretically using an adiabatic approximation when the frequency and damping of mechanical resonator are modulated periodically. As far as we know, the approach of directly using FM to improve mechanical cooling below the QBL has not yet been reported.
In this paper, we propose a proposal to improve micromechanical resonator cooling in OMS via modulating frequencies of both the optical and mechanical components. The FM of optical component is easy to implement and the modulation of micromechanical resonators has also been reported [@PhysRevA.91.023818; @JAP.1.2785018; @nnano.2013.232; @s151026478; @nnano.2014.168; @nl500879k; @1367-2630-9-2-035]. Here we provide a complete and simple understanding of the physical processes about improving mechanical cooling, which allows us to illustrate the deep reasons of the lower mechanical cooling. We consider a conventional OMS in which the mechanical resonator is coupled to optical cavity field via radiation pressure, and the cavity and mechanical resonator are modulated periodically. In such an OMS, we show that the Stokes heating processes can be fully suppressed via FM, and the physical mechanism is also explained and demonstrated through the Raman-scattering and frequency domain pictures. In addition, since the conventional QBL defined as in Ref. [@RevModPhys.86.1391] is satisfied only in the WC region, we recalculate the QBL and steady-state final mean phonon number (MPN) of the standard OMSs in the absence of FM. Moreover, the dynamical evolution of MPN with FM is also obtained by numerically solving the differential equations of second order moments, which are derived strictly from the quantum master equation. And the results are shown respectively with very broad system parameters, e.g., the optomechanical coupling strength ranging from WC to USC and cavity decay rate ranging from RSB to USB regimes. As we all know, the ground-state cooling cannot be achieved when either the coupling strength or cavity decay rate is too large in the standard OMSs without FM. However, the stable lower MPN can be achieved in the presence of FM even in the conventional unstable region of the standard OMSs. Different from the previous methods in Refs. [@PhysRevA.90.013824; @PhysRevA.90.053841; @Liu:18; @PhysRevA.93.033845; @PhysRevA.79.013821; @PhysRevA.94.051801], the mechanical resonator cooling is achieved below QBL only through the FM, which does not need extra technologies, such as trapping optical parametric amplifier, injecting squeezed light, or transparent window induced by auxiliary qubit, etc.
The paper is organized as follows: In Sec. \[sec.2\], we derive the linearized Hamiltonian of the OMS with synchronous FM and explain the physical mechanism of suppressing the Stokes processes through the Raman-scattering and frequency domain pictures. In Sec. \[sec.3\], firstly, we give the analytical expression of the steady-state final MPN in the absence of FM and discuss the QBL in the whole stable region. Then we study the dynamical evolution of MPN with broad system parameters through solving numerically the differential equations of second order moments derived from the quantum master equation. Lastly, a conclusion is given in Sec. \[sec.4\].
System and Hamiltonian {#sec.2}
======================
We consider a generic OMS in which the mechanical resonator is coupled to a driven optical cavity via radiation pressure. The simplest possible cavity OMS model has been used to describe successfully most of the experiments to date. In the rotating frame at the driven laser frequency $\omega_{l}$, the Hamiltonian of the system is written as $$\begin{aligned}
\label{e01}
H=\Delta_{c}a^{\dagger}a+\omega_{m}b^{\dagger}b-ga^{\dagger}a(b^{\dagger}+b)+(Ea^{\dagger}+E^{\ast}a),\end{aligned}$$ where $a~(b)$ and $a^{\dagger}~(b^{\dagger})$ represent the annihilation and creation operators for optical (mechanical) mode with frequency $\omega_{c}~(\omega_{m})$, respectively. $\Delta_{c}=\omega_{c}-\omega_{l}$ is the cavity laser detuning, $g$ is the single photon optomechanical coupling strength [@PhysRevApplied.9.064006], and $E$ is the driving amplitude of laser. Following the usual linearization approach, $a=\langle a\rangle+\delta a=\alpha+\delta a$ and $b=\langle b\rangle+\delta b=\beta+\delta b$, the linearized Hamiltonian reads $$\begin{aligned}
\label{e02}
H_{L}=\Delta_{c}^{'}\delta a^{\dagger}\delta a+\omega_{m}\delta b^{\dagger}\delta b
-(G\delta a^{\dagger}+G^{\ast}\delta a)(\delta b^{\dagger}+\delta b),\end{aligned}$$ where $\Delta_{c}^{'}=\Delta_{c}-g(\beta+\beta^{\ast})$ and $G=g\alpha$ is linearized optomechanical coupling strength. The Hamiltonian in Eq. (\[e02\]) corresponds to the standard quantum Rabi model [@PhysRevA.97.033807] including the beam-splitter interaction $(G\delta a^{\dagger}\delta b+G^{\ast}\delta a\delta b^{\dagger})$ and two-mode squeezing interaction $(G\delta a^{\dagger}\delta b^{\dagger}+G^{\ast}\delta a\delta b)$. In the usual side-band cooling mechanism, the resonant conditions of the beam-splitter interaction are an essential prerequisite to enhance (suppress) the anti-Stokes (Stokes) process, where the strict restrictions of small cavity decay and weak optomechanical coupling strength need to be satisfied. To break those restrictions, we introduce a cosine modulation of the free terms into the initial Hamiltonian. For simplicity, the modulation Hamiltonian is given by $$\begin{aligned}
\label{e03}
H_{M}=\frac{1}{2}\xi\nu\cos(\nu t)(a^{\dagger}a+b^{\dagger}b),\end{aligned}$$ where $\xi$ is the normalized modulation amplitude and $\nu$ is the modulation frequency. In the superconducting OMSs, the modulation of optical mode can be realized easily by tuning the magnetic flux in superconducting systems. Meanwhile, the modulation of mechanical mode can be realized by tuning the voltage of the gate electrode for graphene resonator [@nnano.2013.232; @nnano.2014.168; @nl500879k], which can also be coupled superconducting qubit. In the presence of FM, we can derive the linearized Hamiltonian using the usual linearization approach (see Appendix \[App1\]) $$\begin{aligned}
\label{e04}
H_{\mathrm{LM}}=\Delta_{c}^{'}\delta a^{\dagger}\delta a+\omega_{m}\delta b^{\dagger}\delta b-(G\delta a^{\dagger}+G^{\ast}\delta a)
(\delta b^{\dagger}+\delta b)+\frac{1}{2}\xi\nu\cos(\nu t)(\delta a^{\dagger}\delta a+\delta b^{\dagger}\delta b).\end{aligned}$$
To study the effect of FM on the system dynamics clearly, it is useful to perform the rotating transformation defined by $$\begin{aligned}
\label{e05}
V_{2}&=&\mathcal{T}\exp\left\{-i\int_{0}^{t}d\tau\left[\Delta_{c}^{'}\delta a^{\dagger}\delta a+\omega_{m}\delta b^{\dagger}\delta b+\frac{1}{2}\xi\nu\cos(\nu\tau)(\delta a^{\dagger}\delta a+\delta b^{\dagger}\delta b)\right]\right\}\cr\cr
&=&\exp\left[-i\Delta_{c}^{'}t\delta a^{\dagger}\delta a-i\omega_{m}t\delta b^{\dagger}\delta b-\frac{i}{2}\xi\sin(\nu t)(\delta a^{\dagger}\delta a+\delta b^{\dagger}\delta b)\right],\end{aligned}$$ where $\mathcal{T}$ denotes the time ordering operator. In the rotating frame defined by the transformation operator $V_{2}$, the transformed Hamiltonian becomes $$\begin{aligned}
\label{e06}
\widetilde{H}_{\mathrm{LM}}&=&V_{2}^{\dag}H_{\mathrm{LM}}V_{2}-iV_{2}^{\dag}\dot{V}_{2}\cr\cr
&=&-G\left(\delta a^{\dagger}\delta b^{\dagger}e^{i[(\Delta_{c}^{'}+\omega_{m})t+\xi\sin(\nu t)]}
+\delta a^{\dagger}\delta be^{i(\Delta_{c}^{'}-\omega_{m})t}\right)+\mathrm{H.c.}.\end{aligned}$$ Using the Jacobi-Anger expansions: $e^{i\xi\sin(\nu t)}=\sum_{k=-\infty}^{\infty}J_{k}(\xi)e^{ik\nu t}$, the transformed Hamiltonian in Eq. (\[e06\]) can be rewritten as $$\begin{aligned}
\label{e07}
\widetilde{H}_{\mathrm{LM}}=-G\delta a^{\dagger}\delta be^{i(\Delta_{c}^{'}-\omega_{m})t}
-\sum_{k=-\infty}^{\infty}GJ_{k}(\xi)\delta a^{\dagger}\delta b^{\dagger}e^{i(\Delta_{c}^{'}+\omega_{m}+k\nu)t}+\mathrm{H.c.},\end{aligned}$$ where $J_{k}(\xi)$ is the Bessel function of the first kind with $k$ being an integer. It is worth noting that, except for the usual linearized approach, our calculations have not taken any other approximations. The further understanding of the physics process can be explained in a Raman-scattering picture, as shown in Fig. \[fig:raman-scattering\], where $|n, m\rangle$ denotes the state of $n$ photons and $m$ phonons in the displaced frame. Under the red detuning side-band resonant condition $(\Delta_{c}^{'}=\omega_{m})$, the anti-Stokes process (the green arrows) is on resonance leading to enhance for the mechanical cooling. Different from the anti-Stokes process, the Stokes processes (the red arrows) exist the detuning $(2\omega_{m}+k\nu)$ with coupling strength $GJ_{k}(\xi)$, which are able to be modulated independently by choosing appropriate parameters $\xi$ and $\nu$. Naturally, if the ratio $GJ_{k}(\xi)/(2\omega_{m}+k\nu)$ can be reduced significantly by modulating those two parameters, the heating (Stokes) process will be suppressed efficiently. However, in the absence of FM, the ratio is a constant $G/(2\omega_{m})$, which implies that the efficient mechanical motion cooling needs to satisfy the restriction condition $G\ll2\omega_{m}$ to suppress the Stokes heating process.
![(Color online) Level diagram of the linearized Hamiltonian with FM (Eq. \[e07\]). $|n, m\rangle$ denotes the state of $n$ photons and $m$ phonons in the displaced frame. The green (red) arrows represent the cooling (heating) corresponding to the anti-Stokes (Stokes) process.[]{data-label="fig:raman-scattering"}](fig1.eps){width="0.6\linewidth"}
![(Color online) Frequency domain interpretation of the optomechanical interactions in the presence of FM.[]{data-label="fig:Red-side-band"}](fig2.eps){width="0.6\linewidth"}
Furthermore, the physical mechanism of improving cooling can also be understood by using the frequency domain interpretation, as shown in Fig. \[fig:Red-side-band\], where the Stokes heating processes (the red peaks) are discrete and have been separated with modulation frequency $\nu$ space. For a given value of $\nu$, there exists a corresponding $k=k_{0}$ to make sure the detuning $(2\omega_{m}+k_{0}\nu)$ minimum in all side-bands of Stokes. If the modulation frequency $\nu$ is large enough, the other Stokes processes will be far away the resonant condition and the corresponding photons also cannot exist in the cavity, which lead to those contributions to heating mechanical resonator negligible. The primary purpose is to adopt an appropriate $\xi$ to reduce the ratio $GJ_{k_{0}}(\xi)/(2\omega_{m}+k_{0}\nu)$ for suppressing the nearest resonant heating process, which is the major obstacle for cooling the mechanical resonator. That is easy to achieve because there are always a series of parameters $\xi$ to satisfy $J_{k_{0}}(\xi)=0$, as shown in Fig. \[fig:Besselfun\]. Therefore, the nearest resonant heating process will also be negligible. Different from the usual side-band cooling proposals, which rely on the RSB regime $\kappa<\omega_{m}$, the method we proposed can suppress the nearest resonant heating process efficiently through manipulating its coupling strength $GJ_{k_{0}}(\xi)$ even in USB regime, namely, even if the photons of resonating to Stokes processes can exist abundantly in the cavity, the photons cannot interact with the mechanical resonator.
![(Color online) The absolute value of the Bessel function of the first kind $|J_{k}(\xi)|$ versus the variables $\xi$.[]{data-label="fig:Besselfun"}](fig3.eps){width="0.6\linewidth"}
Based on the above analyses, all the Stokes heating processes can be suppressed via FM approach, which strongly indicates that the QBL of mechanical cooling can be broken in the presence of FM. Next, we will verify that how it can be broken via calculating the MPN.
Improving mechanical cooling via frequency modulation {#sec.3}
=====================================================
In this section, we study the dynamical evolution behavior of the phonon number under the linearized Hamiltonian in Eq. (\[e04\]). It is not necessary to calculate the whole density matrix $\rho$ to obtain the dynamical evolution of MPN. Here, we just need to solve a set of differential equations about the mean values of all the second order moments $\langle\delta a^{\dagger}\delta a\rangle$, $\langle\delta b^{\dagger}\delta b\rangle$, $\langle\delta a^{\dagger}\delta b\rangle$, $\langle\delta a\delta b\rangle$, $\langle\delta a^{2}\rangle$, and $\langle\delta b^{2}\rangle$, which can be derived via the quantum master equation $$\begin{aligned}
\label{e08}
\dot{\rho}=-i\left[H_{\mathrm{LM}},\rho\right]+\kappa\mathcal{L}[a]\rho+\gamma\left(n_{\mathrm{th}}+1\right)\mathcal{L}[b]\rho
+\gamma n_{\mathrm{th}}\mathcal{L}[b^{\dag}]\rho,\end{aligned}$$ where $\mathcal{L}[o]\rho=o\rho o^{\dag}-(o^{\dag}o\rho+\rho o^{\dag}o)/2$ is the standard Lindblad operators. $\kappa$ and $\gamma$ are the decay rate of optical cavity and the damping rate of mechanical resonator, respectively. Through the quantum master equation in Eq. (\[e08\]) and $\mathrm{d}\langle o_{i}o_{j}\rangle/\mathrm{d}t=\mathrm{Tr}(\dot{\rho}o_{i}o_{j})$, where $o_{i}$ and $o_{j}$ are the arbitrary operators of the system, the derivatives of those second order moments are given as [@PhysRevLett.110.153606; @NJP.10.095007] $$\begin{aligned}
\label{e09}
\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta a^{\dagger}\delta a\rangle&=&i\left(G\langle\delta a^{\dagger}\delta b\rangle
-G^{\ast}\langle\delta a^{\dagger}\delta b\rangle^{\ast}+G\langle\delta a\delta b\rangle^{\ast}
-G^{\ast}\langle\delta a\delta b\rangle\right)-\kappa\langle\delta a^{\dagger}\delta a\rangle,\cr\cr
\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta b^{\dagger}\delta b\rangle&=&i\left(-G\langle\delta a^{\dagger}\delta b\rangle
+G^{\ast}\langle\delta a^{\dagger}\delta b\rangle^{\ast}+G\langle\delta a\delta b\rangle^{\ast}
-G^{\ast}\langle\delta a\delta b\rangle\right)-\gamma\langle\delta b^{\dagger}\delta b\rangle+\gamma n_{\mathrm{th}},\cr\cr
\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta a^{\dagger}\delta b\rangle&=&
\left[i\left(\Delta-\Omega_{m}\right)-\frac{\kappa+\gamma}{2}\right]\langle\delta a^{\dagger}\delta b\rangle
+i\left(G^{\ast}\langle\delta a^{\dagger}\delta a\rangle-G^{\ast}\langle\delta b^{\dagger}\delta b\rangle
+G\langle\delta a^{2}\rangle^{\ast}-G^{\ast}\langle\delta b^{2}\rangle\right),\cr\cr
\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta a\delta b\rangle&=&
-\left[i\left(\Delta+\Omega_{m}\right)+\frac{\kappa+\gamma}{2}\right]\langle\delta a\delta b\rangle
+i\left(G\langle\delta a^{\dagger}\delta a\rangle+G\langle\delta b^{\dagger}\delta b\rangle+G
+G^{\ast}\langle\delta a^{2}\rangle+G\langle\delta b^{2}\rangle\right),\cr\cr
\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta a^{2}\rangle&=&
-\left(2i\Delta+\kappa\right)\langle\delta a^{2}\rangle+2i\left(G\langle\delta a^{\dagger}\delta b\rangle^{\ast}
+G\langle\delta a\delta b\rangle\right),\cr\cr
\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta b^{2}\rangle&=&
-\left(2i\Omega_{m}+\gamma\right)\langle\delta b^{2}\rangle+2i\left(G\langle\delta a^{\dagger}\delta b\rangle
+G^{\ast}\langle\delta a\delta b\rangle\right),\end{aligned}$$ where $\Delta=\Delta_{c}^{'}+\frac{1}{2}\xi\nu\cos(\nu t)$ and $\Omega_{m}=\omega_{m}+\frac{1}{2}\xi\nu\cos(\nu t)$ are the renormalized parameters. It is worth noting that the dynamical evolution of MPN is exact via solving the differential equations and the dynamics dimension is also not limited as the original quantum master equation. The system stability can also be determined by observing the dynamical behaviors of MPN in the presence of FM.
Cooling Limits in the absence of frequency modulation
-----------------------------------------------------
Firstly, as a comparison, we give the quantum cooling limit for the whole stable region in the absence of FM $(\xi=0)$, which can be derived from Eq. (\[e09\]) when the system finally reaches the steady state, namely, $\mathrm{d}\langle o_{i}o_{j}\rangle/\mathrm{d}t=0$. Under the red side-band resonant condition $(\Delta_{c}^{'}=\omega_{m})$ and the cooperativity parameter $C\equiv4|G|^{2}/(\gamma\kappa)\gg1$, the steady-state final MPN is $$\begin{aligned}
\label{e010}
\langle\delta b^{\dagger}\delta b\rangle_{\mathrm{lim}}\simeq\frac{4|G|^{2}+\kappa^{2}}{4|G|^{2}(\kappa+\gamma)}\gamma n_{\mathrm{th}}
+\frac{(4\omega_{m}^{2}-\kappa^{2})(8|G|^{2}+\kappa^{2})+2\kappa^{4}}{16\omega_{m}^{2}(4\omega_{m}^{2}
+\kappa^{2}-16|G|^{2})},\end{aligned}$$ where the first term represents the classical cooling limit corresponding to the thermal environment, while the second term represents the quantum cooling limit, as the redefined QBL, corresponding to the cavity decay and quantum backaction. Under the red side-band resonant condition, the conventional steady-state final MPN is defined as (see Appendix \[App2\])
![(Color online) The cooling limit varies with the optomechanical coupling strength $G/\omega_{m}$ (red solid line, black dashed line, and blue dot line). The black solid lines represent the dynamical evolution of MPN for different optomechanical coupling strengths in the absence of FM. The other parameters are set as $\Delta_{c}^{'}=\omega_{m}$, $\gamma=10^{-5}\omega_{m}$, $n_{\mathrm{th}}=10^{3}$, and $\kappa=0.2\omega_{m}$.[]{data-label="fig:phonon_num_lim_G"}](fig4.eps){width="0.6\linewidth"}
$$\begin{aligned}
\label{e011}
n_{f}=\frac{A_{+}+n_{\mathrm{th}}\gamma}{\Gamma_\mathrm{opt}+\gamma},\end{aligned}$$
with $$\begin{aligned}
\label{e012}
\Gamma_\mathrm{opt}=A_{-}-A_{+},~~~~A_{-}=\frac{4|G|^2}{\kappa},~~~~A_{+}=\frac{|G|^{2}\kappa}{4\omega_{m}^{2}+\frac{\kappa^{2}}{4}},\end{aligned}$$ where $A_{-}~(A_{+})$ is the emitting (absorbing) rate of phonon and $\Gamma_\mathrm{opt}$ is the cooling rate. Different from $n_{f}$, which requires very weak optomechanical coupling strength (see Fig. \[fig:phonon\_num\_lim\_G\]), the expression of Eq. (\[e010\]) is satisfied in the whole stable region $(|G|^{2}<\omega_{m}^{2}/4+\kappa^{2}/16)$, which can be derived via Routh-Hurwitz criterion [@PhysRevA.35.5288]. In order to clearly compare the two steady-state final MPNs with the dynamical results, we numerically solve the differential equations (Eq. (\[e09\])) in the absence of FM and show those results in Fig. \[fig:phonon\_num\_lim\_G\]. Here, for the dynamical results, we have assumed that the MPN of initial system equals to the average phonon number in thermal equilibrium bath $\langle\delta b^{\dagger}\delta b\rangle(t=0)=n_{\mathrm{th}}=10^{3}$, and the other second order moments are zero. The red solid line is the analytical steady-state final MPN which is given by Eq. (\[e010\]), the black dashed line is the exactly numerical result which is obtained by solving Eq. (\[e09\]) for the steady-state system, and the blue dot line is the conventional steady-state final MPN which is only satisfied with the very weak optomechanical coupling strength, as shown in Fig. \[fig:phonon\_num\_lim\_G\]. In addition, we also give the relationship between the steady-state final MPNs and cavity decay rate, as shown in Fig. \[fig:phonon\_num\_lim\_kappa\]. The results indicate that the ground-state cooling of mechanical resonator cannot be achieved when the cavity decay rate is too large or too small in the standard OMSs, which is consistent with previous studies.
![(Color online) The cooling limit varies with cavity decay rate $\kappa/\omega_{m}$ (red solid line, black dashed line, and blue dot line). The black solid lines represent the dynamical evolution of MPN for different cavity decay rates in the absence of FM. Here, $G=0.2\omega_{m}$ and the other parameters are the same as in Fig. \[fig:phonon\_num\_lim\_G\].[]{data-label="fig:phonon_num_lim_kappa"}](fig5.eps){width="0.6\linewidth"}
In the whole stable region, we give a more general result of the cooling limit in the standard OMSs without FM, which can describe the mechanical cooling better in the broad parameter region. Next, we study the phonon number via solving the differential equations in the presence of FM, which will indicate that the cooling limit can be broken due to the existence of FM.
Weak coupling
-------------
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ by solving Eq. (\[e09\]) for different system parameters with or without FM. The other parameters are chosen as $\Delta_{c}^{'}=\omega_{m}$, $\gamma=10^{-5}\omega_{m}$, $n_{\mathrm{th}}=10^{3}$, and $\nu=5\omega_{m}$.[]{data-label="fig:phonon_num_weak"}](fig6.eps){width="0.6\linewidth"}
For the WC regime, the ground-state cooling of mechanical resonator has been studied extensively in the RSB regime [@PhysRevB.69.125339; @PhysRevB.79.193407; @PhysRevB.80.144508; @PhysRevA.83.013816]. When the decay rate of the cavity is small enough $(\kappa\ll\omega_{m})$, the Stokes heating processes can be suppressed well in the usual red side-band cooling regime without FM. Therefore, the further suppression effect is unobvious via FM, namely, the improving rate of introducing the FM is very small. The result can be verified by showing the dynamical evolution of MPN in Fig. \[fig:phonon\_num\_weak\]. One can see that there is not obvious difference between modulation and no modulation for WC and small cavity decay rate (see Table \[t01\]). However, for larger coupling strength $G=0.1\omega_{m}$ and cavity decay $\kappa=0.1\omega_{m}$, the MPN with FM is lower obviously than that without FM (see the cyan diamond line and magenta hexagram line in Fig. \[fig:phonon\_num\_weak\]). This is due to the fact that the Stokes heating processes cannot be suppressed fully for larger coupling strength and cavity decay in the absence of FM.
----------- ---------------------------- --------------------------------- ---------------- ---------------------------- ---------
\*[$G/\omega_{m}$]{} \*[$\kappa/\omega_{m}$]{} \*[improving rate]{}
without FM with FM
\*[WC]{} 0.02 0.05 0.5128 0.5123 0.98
0.1 0.1 0.1319 0.125 5.23%
\*[SC]{} 0.3 0.1 0.1876 0.1032 44.99%
0.5 0.2 13.9419 0.0527 99.62%
0.6 0.3 fail (diverge) 0.0363 success
0.9 0.5 fail (diverge) 0.0233 success
\*[USC]{} 1.2 0.5 fail (diverge) 0.0239 success
1.5 0.5 fail (diverge) 0.0253 success
\*[USB]{} 0.2 4 1.5225 0.2565 83.15%
0.5 10 7.1013 0.1212 98.29%
----------- ---------------------------- --------------------------------- ---------------- ---------------------------- ---------
: The steady-state final MPN and the improving rate for different system parameters with or without FM. The other parameters are chosen as $\Delta_{c}^{'}=\omega_{m}$, $\gamma=10^{-5}\omega_{m}$, $n_{\mathrm{th}}=10^{3}$, and $\nu=30\omega_{m}$ when the system is in WC, strong coupling (SC), USC, and USB regimes.[]{data-label="t01"}
In a word, the further cooling from FM is unobvious compared with the usual side-band cooling for the weaker optomechanical coupling in RSB regime. Based on the above analysis, we can easily infer that the cooling approach of utilizing FM to suppress Stokes scattering will stand out with the increment of optomechanical coupling strength and decay rate.
Strong coupling
---------------
In the SC regime $(|G|>\kappa)$, the improving rate becomes obvious no matter the decay rate of cavity is large or small. This is very easy to understand based on the above analysis. It is worth noting that the standard OMSs without FM exist the dynamical stability condition $(|G|^{2}<\omega_{m}^{2}/4+\kappa^{2}/16)$. Therefore, the SC region can be roughly divided to the stable $(|G|<0.5\omega_{m})$ and unstable $(|G|>0.5\omega_{m})$ regions, where the MPN of standard OMSs will converge or diverge corresponding to different regions in the absence of FM. Here, we should point out that the OMS with FM can be stable even in the unstable region mentioned above, which can be determined by observing the dynamical behaviors of MPN. Next, we study the mechanical resonator cooling in two different coupling regions, respectively.
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ corresponding to different optomechanical coupling strengths with or without FM. Here, $\kappa=0.1\omega_{m}$, $\nu=10\omega_{m}$, and the other parameters are the same as in Fig. \[fig:phonon\_num\_weak\].[]{data-label="fig:phonon_num3D_G_t"}](fig7.eps){width="0.6\linewidth"}
In the stable region of the standard OMSs, we show the dynamical evolution of MPN corresponding to different optomechanical coupling strengths in Fig. \[fig:phonon\_num3D\_G\_t\], when the FM exists or not. One can see from Fig. \[fig:phonon\_num3D\_G\_t\] that the different values of the final MPNs with and without FM increase with increasing the coupling strength. The results indicate that the advantages of cooling with FM are becoming more and more obvious as the system enters the SC regime (see Table \[t01\]). We can also see that the final MPN without FM increases with the increase of coupling strength. One reason is the saturation effect of the cooling rate [@PhysRevA.80.063819] and the another is the enhanced Stokes processes. However, the final MPN with FM almost does not change with the increase of coupling strength. This is because the small decay rate of the optical cavity limits the final mechanical resonator cooling in the SC regime.
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ corresponding to different optomechanical coupling strengths with different modulation frequencies. Here, $\kappa=0.5\omega_{m}$ and the other parameters are the same as in Fig. \[fig:phonon\_num\_weak\].[]{data-label="fig:phonon_num3D_G_large"}](fig8.eps){width="0.6\linewidth"}
In the unstable region of the standard OMSs, where the ground-state cooling of mechanical resonator cannot be achieved due to the divergent behavior of phonon number (see $G>0.5\omega_{m}$ in Table \[t01\]), we just show the dynamical evolution of MPN in the presence of FM corresponding to different optomechanical coupling strengths and modulation frequencies, as shown in Fig. \[fig:phonon\_num3D\_G\_large\]. The results show that the system is stable due to the existence of FM even in the conventional unstable region. With the increase of modulation frequency, the system becomes more stable and the final MPN is smaller. The fundamental reason is that the large modulation frequency can suppress the Stokes heating processes efficiently. Furthermore, we also note that, for the small modulation frequency in Fig. \[fig:phonon\_num3D\_G\_large\], the final MPN increases with the increase of optomehcanical coupling strength. This is because the small modulation frequency cannot suppress the Stokes heating processes fully when the coupling strength is very large. Moreover, we also show the dynamical evolution of MPN in the USC regime $(|G|>\omega_{m})$, as shown in Fig. \[fig:phonon\_num3D\_G\_large\]. We find that the ground-state cooling can also be achieved in USC regime with FM. So we can infer that, for more larger coupling strength, more larger modulation frequency will be needed to achieve the better ground-state cooling of the mechanical resonator.
Unresolved side-band regime
---------------------------
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ for different cavity decay rates and modulation frequencies. Here, $G=0.2\omega_{m}$, $\nu=10\omega_{m}$, and the other parameters are the same as in Fig. \[fig:phonon\_num\_weak\].[]{data-label="fig:phonon_num3D_kappa_t"}](fig9.eps){width="0.6\linewidth"}
For the USB regime $(\kappa>\omega_{m})$, the Stokes processes cannot be suppressed well in the standard OMSs, which results in an unsatisfactory ground-state cooling of mechanical resonator (see Table \[t01\]). However, in the presence of FM, the nearest resonant Stokes process can be fully suppressed and the other Stokes processes can also be neglected when the modulation frequency is large enough. Similarly, we show the dynamical evolution of MPN corresponding to different cavity decay rates in Fig. \[fig:phonon\_num3D\_kappa\_t\]. We find that, in the absence of FM, the ground-state cooling cannot be achieved when the cavity decay rate is too large. However, in the presence of FM, the ground-state cooling can be achieved very well even in the USB regime.
![(Color online) The final stable MPN $\langle\delta b^{\dagger}\delta b\rangle$ in relation to the optomechanical coupling strength and cavity decay rate. Here, $\nu=30\omega_{m}$ and the other parameters are the same as in Fig. \[fig:phonon\_num\_weak\]. The black smooth surface represents the final MPN without FM, the red mesh surface represents the final MPN with FM, and the yellow vertical plane is the roughly stable boundary of the standard OMSs without FM.[]{data-label="fig:phonon_num_G_kappa"}](fig10.eps){width="0.6\linewidth"}
In the above discussions, the study of the MPN dynamical evolution is only related to one of the system parameters, i.e., the optomechanical coupling strength or cavity decay rate. To clearly demonstrate the results of their synergistic interaction, we plot the average value of the final MPN after the system is stable, as shown in Fig. \[fig:phonon\_num\_G\_kappa\], where the final stable MPN changes with the optomechanical coupling strength and cavity decay rate, with or without the FM. In the presence of FM, the final MPN of the mechanical resonator is lower than that without FM in the stable region. In addition, we can also achieve the ground-state cooling of mechanical resonator even in the conventional unstable region, as shown the left side of the yellow plane in Fig. \[fig:phonon\_num\_G\_kappa\]. The results indicate that the system with FM is still stable and the ground-state cooling can also be achieved even in the conventional unstable region for the standard OMSs.
Small modulation frequency
--------------------------
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ corresponding to different system parameters with or without the modulation. Here $\nu=2\omega_{m}$, $k_{0}=-1$, and the other parameters are the same as in Fig. \[fig:phonon\_num\_weak\].[]{data-label="fig:phonon_number_small_modu"}](fig11.eps){width="0.6\linewidth"}
In the above subsections, in order to suppress the Stokes heating processes better, the large modulation frequency is necessary. In practice, however, too large modulation frequency may not be conducive to experimental implementation, although various of schemes about the FM of mechanical resonator have been reported theoretically and experimentally [@PhysRevA.91.023818; @JAP.1.2785018; @nnano.2013.232; @s151026478; @nnano.2014.168; @nl500879k; @1367-2630-9-2-035]. Therefore, we consider the effect of a small modulation frequency on the system in this subsection. We set the modulation frequency as $\nu=2\omega_{m}$, where one of the Stokes processes $(k_{0}=-1)$ can be resonant ideally like the anti-Stokes process in the red detuning side-band resonant regime. Here, the resonant Stokes process can also be suppressed by making coupling strength zero, for example, with the choice of $\xi=3.8317$, the effective optomechanical coupling strength $GJ_{-1}(3.8317)=0$. However, the other Stokes processes $(k\neq-1)$ cannot be suppressed well due to $J_{k\neq-1}(3.8317)\neq0$ at this time. The dynamical evolution of MPN with the small modulation frequency is shown in Fig. \[fig:phonon\_number\_small\_modu\] for different system parameters. The results indicate that the cooling effect resorting to FM is still better than the standard OMSs without FM under these system parameters. The influences of MPN by changing system parameters are also similar to the situation of large modulation frequency.
Asynchronous frequency modulation
---------------------------------
In previous discussions, we have assumed that the modulations of optical and mechanical components are identical and synchronous to demonstrate clearly the physical mechanism of the further cooling via FM. Here, we will study the effects of asynchronous FM, e.g., different amplitudes, phases, and frequencies of the modulation. The Hamiltonian of the modulation parts changes to $H_{M}=[\xi_{1}\nu_{1}\cos(\nu_{1}t)a^{\dagger}a+\xi_{2}\nu_{2}\cos(\nu_{2}t+\theta)b^{\dagger}b]/2$, where $\xi_{1}(\xi_{2})$ and $\nu_{1}(\nu_{2})$ are normalized amplitude and frequency of the optical (mechanical) modulation, respectively, and $\theta$ is the relative phase of two modulations.
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ corresponding to different amplitudes of the FM. Here $G=0.4\omega_{m}$, $\kappa=0.1\omega_{m}$, $\nu_{1}=\nu_{2}=10\omega_{m}$, $\xi_{2}=2\xi_{0}-\xi_{1}(\xi_{2}=\xi_{1}-2\xi_{0})$ for $\theta=0(\theta=\pi)$, and the other parameters are the same as in Fig. \[fig:phonon\_num\_weak\].[]{data-label="fig:different_amp"}](fig12.eps){width="0.6\linewidth"}
Firstly, we study the effects of different optical and mechanical modulation amplitudes, where the other modulation parameters are identical, i.e., $\xi_{1}\neq\xi_{2}$, $\theta=0$, and $\nu_{1}=\nu_{2}$. The interaction Hamiltonian can be written as $$\begin{aligned}
\label{e013}
\widetilde{H}_{\xi}&=&\sum_{k=-\infty}^{\infty}\Bigg[-GJ_{k}\left(\frac{\xi_{1}-\xi_{2}}{2}\right)\delta a^{\dagger}\delta be^{i(\Delta_{c}^{'}-\omega_{m}+k\nu)t}\cr\cr
&&-GJ_{k}\left(\frac{\xi_{1}+\xi_{2}}{2}\right)\delta a^{\dagger}\delta b^{\dagger}e^{i(\Delta_{c}^{'}+\omega_{m}+k\nu)t}+\mathrm{H.c.}\Bigg].\end{aligned}$$ Based on the previous analyses, the mechanical cooling can be improved if the nearest resonant Stokes heating process is suppressed with $(\xi_{1}+\xi_{2})/2=\xi_{0}$, where $J_{k_{0}}(\xi_{0})=0$. It is worth noting that the beam-splitter interactions are also related to the different modulation amplitudes. Therefore, the improving mechanical cooling cannot be achieved if the difference value $(\xi_{1}-\xi_{2})/2$ is close to $\xi_{0}$, which greatly reduces the optomechanical coupling. In addition, the similar results can also be obtained with $\theta=\pi$, where the beam-splitter and two-mode squeezing interactions exchange their coupling strengths. The dynamical results of MPN are shown in Fig. \[fig:different\_amp\] with different amplitudes.
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ corresponding to different phases of the FM. Here $\xi_{1}=\xi_{2}=\xi_{0}$ and the other parameters are the same as in Fig. \[fig:different\_amp\].[]{data-label="fig:different_pha"}](fig13.eps){width="0.6\linewidth"}
For the different phases of FM, we also study the effects by changing $\theta$ and show the results in Fig. \[fig:different\_pha\]. We find that the improving mechanical cooling can also be achieved when $\theta$ is far from $\pi$ with $\xi_{1}=\xi_{2}$ and $\nu_{1}=\nu_{2}$. That is easy to understand due to the exchange of coupling strengths for Eq. (\[e013\]) when the relative phase $\theta$ equals to $\pi$, where the coupling strength of the nearest resonant two-mode squeezing interaction is $GJ_{0}(0)=G$. And the Stokes processes cannot be suppressed so that the improving mechanical cooling is not achieved.
![(Color online) The time evolution of MPN $\langle\delta b^{\dagger}\delta b\rangle$ corresponding to different frequencies of the FM. Here $\theta=0$, $\nu_{1}=10\omega_{m}$, and the other parameters are the same as in Fig. \[fig:different\_amp\].[]{data-label="fig:different_fre"}](fig14.eps){width="0.6\linewidth"}
Lastly, we study the effects of different modulation frequencies, i.e., $\nu_{1}\neq\nu_{2}$. We find that the improving mechanical cooling cannot be achieved when the two modulation frequencies satisfy $|\nu_{1}-\nu_{2}|\simeq2\omega_{m}$, as shown in Fig. \[fig:different\_fre\]. However, the steady-state final MPN with FM is still lower than that in the conventional OMS without FM when the difference value of the two modulation frequencies is far from $2\omega_{m}$. The foremost reason is that the detuning of the Stokes processes is decreased and results in the suppressed effect reducing. Furthermore, even if all the modulation parameters are different, e.g., $\xi_{1}=1$, $\xi_{2}=3.8096$, $\theta=\pi/4$, $\nu_{1}=10\omega_{m}$, and $\nu_{2}=15\omega_{m}$, the average steady-state final MPN is $\langle\delta b^{\dagger}\delta b\rangle=0.1327$, which is still lower than the result of conventional OMS without FM $\langle\delta b^{\dagger}\delta b\rangle=0.3677$.
Conclusions {#sec.4}
===========
In conclusion, we have proposed a proposal for improving the cooling of micromechanical resonator in OMS through introducing FM method. We proposed a deeper insight of the optomechanical cooling when the frequencies of cavity and mechanical resonator are modulated periodically, which would provide a guide for optomechanical cooling experiments to cool the mechanical resonator below the QBL. Here, we have redefined a more general QBL in the standard OMSs without FM, which is satisfied in the whole stable region. In the presence of FM, we analyze and demonstrate the reasons of cooling mechanical resonator to the lower MPN, which are due to the fact that the Stokes heating processes can be suppressed fully such that the QBL of mechanical cooling can be broken via the FM. In addition, we also study the dynamical evolution of MPN by solving the differential equations of second order moments numerically, which are derived strictly from the quantum master equation. We find that the lower MPN can be reached when the modulation frequency is large enough, even if the coupling strength ranges from WC to USC regimes and the cavity decay rate ranges from RSB to USB regimes. And the improving rate of mechanical cooling with FM becomes more and more larger with the increase of optomechanical coupling strength or cavity decay rate. However, in the standard OMSs without FM, the ground-state cooling of mechanical resonator cannot be achieved when either the coupling strength or cavity decay rate is too large. The results show that, in the presence of FM, the mechanical ground-state cooling is not limited to the conventional stability boundary and the RSB regime, and in the meanwhile the proposed proposal is still feasible even with more board system parameters. Moreover, we also give the mechanical cooling results when the modulation frequency is not large enough, where the Stokes processes cannot be suppressed well. At last, we discuss respectively the effects of asynchronous FM and find that the improving mechanical can be achieved even with different modulation. Our proposal would open up the possibility for cooling the mechanical resonator beyond the QBL of the standard optomechanical cooling.
[**[ACKNOWLEDGMENTS]{}**]{}
This work was supported by the National Natural Science Foundation of China under Grant Nos. 11465020, 11264042, 61465013, 61575055, and the Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project under Grant No. 20160519022JH.
Linearizing the system Hamiltonian in the presence of frequency modulation {#App1}
==========================================================================
In the presence of FM, the system Hamiltonian of the standard cavity OMS reads $$\begin{aligned}
\label{Ae01}
H=\omega_{c}a^{\dagger}a+\omega_{m}b^{\dagger}b-ga^{\dagger}a(b^{\dagger}+b)+(Ea^{\dagger}e^{-i\omega_{l}t}+E^{\ast}ae^{i\omega_{l}t})
+\frac{1}{2}\xi\nu\cos(\nu t)(a^{\dagger}a+b^{\dagger}b),\end{aligned}$$ where $a~(b)$ and $a^{\dagger}~(b^{\dagger})$ represent the annihilation and creation operators for optical (mechanical) mode with the corresponding frequency $\omega_{c}~(\omega_{m})$, respectively. The parameter $g$ is the single photon optomechanical coupling rate. The parameters $E$ and $\omega_{l}$ are the driving amplitude and frequency, respectively. By performing a rotating transformation defined by $V_{1}=\exp[-i\omega_{l}ta^{\dagger}a]$, the transformed Hamiltonian $\widetilde{H}=V_{1}^{\dag}HV_{1}-iV_{1}^{\dag}\dot{V}_{1}$ becomes $$\begin{aligned}
\label{Ae02}
\widetilde{H}=\Delta_{c}a^{\dagger}a+\omega_{m}b^{\dagger}b-ga^{\dagger}a(b^{\dagger}+b)+(Ea^{\dagger}+E^{\ast}a)
+\frac{1}{2}\xi\nu\cos(\nu t)(a^{\dagger}a+b^{\dagger}b),\end{aligned}$$ where $\Delta_{c}=\omega_{c}-\omega_{l}$ is the cavity laser detuning.
The quantum Langevin equations are given by $$\begin{aligned}
\label{Ae03}
\dot{a}&=&-i\Delta_{c}a-\frac{\kappa}{2}a+iga(b^{\dagger}+b)-iE-i\frac{1}{2}\xi\nu\cos(\nu t)a-\sqrt{\kappa}a_{\mathrm{in}},\cr\cr
\dot{b}&=&-i\omega_{m}b-\frac{\gamma}{2}b+iga^{\dagger}a-i\frac{1}{2}\xi\nu\cos(\nu t)b-\sqrt{\gamma}b_{\mathrm{in}},\end{aligned}$$ where $\kappa$ and $\gamma$ are the decay rate of optical cavity and the damping rate of mechanical resonator, respectively. $a_{\mathrm{in}}$ and $b_{\mathrm{in}}$ are the corresponding noise operators. Under the strongly coherent laser driving, we can apply a displacement transformation to linearize Eq. (\[Ae03\]), i.e., $a=\alpha+\delta a$ and $b=\beta+\delta b$, where $\alpha$ and $\beta$ are $c$-numbers representing the displacement mean values of the cavity and mechanical resonator modes. $\delta a$ and $\delta b$ are the operators relating to the quantum fluctuations of the cavity and mechanical resonator modes. Equation (\[Ae03\]) can be separated to two different sets of equations, one for the mean values, and the other for the fluctuations, which are given by $$\begin{aligned}
\label{Ae04}
\dot{\alpha}&=&-i\Delta_{c}^{'}\alpha-\frac{\kappa}{2}\alpha-iE-i\frac{1}{2}\xi\nu\cos(\nu t)\alpha,\cr\cr
\dot{\beta}&=&-i\omega_{m}\beta-\frac{\gamma}{2}\beta+ig|\alpha|^{2}-i\frac{1}{2}\xi\nu\cos(\nu t)\beta,\cr\cr
\dot{\delta a}&=&-i\Delta_{c}^{'}\delta a-\frac{\kappa}{2}\delta a+iG(\delta b^{\dagger}+\delta b)
-i\frac{1}{2}\xi\nu\cos(\nu t)\delta a-\sqrt{\kappa}a_{\mathrm{in}},\cr\cr
\dot{\delta b}&=&-i\omega_{m}\delta b-\frac{\gamma}{2}\delta b+iG\delta a^{\dagger}+iG^{\ast}\delta a
-i\frac{1}{2}\xi\nu\cos(\nu t)\delta b-\sqrt{\gamma}b_{\mathrm{in}},\end{aligned}$$ where $\Delta_{c}^{'}=\Delta_{c}-g(\beta^{\ast}+\beta)$ is the effective detuning modified by optomechanical coupling, $G=g\alpha$ is linearized optomechanical coupling strength, and we have neglected the nonlinear terms $ig\delta a(\delta b^{\dagger}+\delta b)$ and $ig\delta a^{\dagger}\delta a$ due the strong coherent driving conditions. Then we obtain the linearized Hamiltonian (see Eq. (\[e04\]) in the main text) in the presence of FM.
The conventional steady-state final mean phonon number {#App2}
======================================================
For the standard OMSs, the MPN can be derived and calculated by using the rate equations of the mechanical resonator, where we need calculate the spectral density of the optical force using the quantum noise approach. Firstly, we give the effective Hamiltonian of the standard optomechanical (see Eq. (\[e02\]) in the main text) in the absence of FM, $$\begin{aligned}
\label{Be01}
H_{L}=\Delta_{c}^{'}\delta a^{\dagger}\delta a+\omega_{m}\delta b^{\dagger}\delta b
-(G\delta a^{\dagger}+G^{\ast}\delta a)(\delta b^{\dagger}+\delta b),\end{aligned}$$ where we can obtain the optical force $F=(G\delta a^{\dagger}+G^{\ast}\delta a)/x_{\mathrm{ZPF}}$. The quantum noise spectrum of the optical force is given by the Fourier transform of the autocorrelation function $S_{FF}(\omega)=\int dte^{i\omega t}\langle F(t)F(0)\rangle$, which is calculated easily in the frequency domain. In the WC regime, the optomechanical coupling can be regarded as a perturbation to the optical field, where the spectral density of the optical force can be calculated properly without considering optomechanical coupling. In the frequency domain, the cavity mode fluctuation operator $\delta a(\omega)$ can be obtained in the absence of the optomechanical coupling, $$\begin{aligned}
\label{Be02}
\delta a(\omega)=\frac{\sqrt{\kappa}a_{\mathrm{in}}(\omega)}{i(\omega-\Delta_{c}^{'})-\frac{\kappa}{2}}.\end{aligned}$$
According to the Hamiltonian in Eq. (\[Be01\]), we can write the rate equations of the mechanical resonator as $$\begin{aligned}
\label{Be03}
\dot{P}_{n}&=&\Gamma_{n+1\rightarrow n}P_{n+1}+\Gamma_{n-1\rightarrow n}P_{n-1}-\Gamma_{n\rightarrow n-1}P_{n}
-\Gamma_{n\rightarrow n+1}P_{n}\cr\cr
&&+\gamma(n_{\mathrm{th}}+1)(n+1)P_{n+1}+\gamma n_{\mathrm{th}}nP_{n-1}-\gamma(n_{\mathrm{th}}+1)nP_{n}-\gamma n_{\mathrm{th}}(n+1)P_{n},\end{aligned}$$ where $P_{n}$ is the occupation probability of the mechanical Fock staet $|n\rangle$ with $n$ phonons. $\Gamma_{n\rightarrow m}$ is the transition rate of phononic state from $|n\rangle$ to $|m\rangle$ induced by the optomechanical coupling. Using the Fermi golden rule and the above calculations, we can obtain $$\begin{aligned}
\label{Be04}
\Gamma_{n\rightarrow n-1}&=&nA_{-}=nS_{FF}(\omega_{m})x_{\mathrm{ZPF}}^{2}
=\frac{n|G|^{2}\kappa}{(\omega_{m}-\Delta_{c}^{'})^{2}+\frac{\kappa^{2}}{4}},\cr\cr
\Gamma_{n-1\rightarrow n}&=&nA_{+}=nS_{FF}(-\omega_{m})x_{\mathrm{ZPF}}^{2}
=\frac{n|G|^{2}\kappa}{(-\omega_{m}-\Delta_{c}^{'})^{2}+\frac{\kappa^{2}}{4}},\end{aligned}$$ where $A_{-}~(A_{+})$ is the emitting (absorbing) rate of phonon. By solving Eq. (\[Be03\]) with $\dot{P}_{n}=0$, the steady-state final MPN can be obtained as $$\begin{aligned}
\label{Be05}
n_{f}=\frac{A_{+}+n_{\mathrm{th}}\gamma}{\Gamma_{\mathrm{opt}}+\gamma}.\end{aligned}$$ In the absence of intrinsic mechanical damping rate $(\gamma=0)$, we also obtain the fundamental quantum limit of cooling $$\begin{aligned}
\label{Be06}
n_{c}=\frac{A_{+}}{A_{-}-A_{+}}.\end{aligned}$$
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[^1]: E-mail: stliu@hit.edu.cn
[^2]: E-mail: szhang@ybu.edu.cn
[^3]: E-mail: hfwang@ybu.edu.cn
|
---
abstract: 'Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.'
address: 'Department of Mathematics, Sungkyunkwan University (SKKU), Suwon, Gyeonggi-do 16419, South Korea'
author:
- Jihyeug Jang and Jang Soo Kim
title: Volumes of flow polytopes related to caracol graphs
---
Introduction
============
The main objects in this paper are flow polytopes, which are certain polytopes associated to acyclic directed graphs with net flow vectors. Flow polytopes have interesting connections with representation theory, geometry, analysis, and combinatorics. A well known flow polytope is the Chan–Robbins–Yuen polytope, which is the flow polytope of the complete graph $K_{n+1}$ with net flow vector $(1,0,\dots,0,-1)$. Chan, Robbins, and Yuen [@CRY2000] conjectured that the volume of this polytope is a product of Catalan numbers. Their conjecture was proved by Zeilberger [@Zeilberger1999] using the Morris constant term identity [@Morris1982], which is equivalent to the famous Selberg integral [@Selberg1944].
Since the discovery of the Chan–Robbins–Yuen polytope, researchers have found many flow polytopes whose volumes have nice product formulas, see [@Benedetti2019; @CKM2017; @Meszaros2015; @MMR2017; @MMS2019; @MSW2019; @Yip2019] and references therein. In this paper we add another flow polytope to this list by proving a product formula for the volume of flow polytope coming from a caracol graph, which was recently conjectured by Benedetti et al. [@Benedetti2019]. In order to state our results we introduce necessary definitions.
We denote $[n]:=\{1,2,\dots,n\}$. Throughout this paper, we only consider connected directed graphs in which every vertex is an integer and every directed edge is of the form $(i,j)$ with $i<j$.
Let $G$ be a directed graph on vertex set $[n+1]$ with $m$ directed edges. We allow $G$ to have multiple edges but no loops. Let ${\mathbf{a}}=(a_1,a_2,\dots
,a_n)\in {\mathbb{Z}}^{n}$. An $m$-tuple $(b_{ij})_{(i,j)\in E}\in \mathbb{R}_{\geq 0}^m$ is called an *${\mathbf{a}}$-flow of $G$* if $$\sum_{(i,j)\in E}b_{ij}({\mathbf{e}}_i-{\mathbf{e}}_j)=\left(a_1,\dots,a_n, -\sum_{i=1}^n a_i\right),$$ where ${\mathbf{e}}_i$ is the standard basis vector in $\mathbb{R}^{n+1}$ with a one in the $i$th entry and zeroes elsewhere. The *flow polytope ${\mathcal{F}}_{G}({\mathbf{a}})$ of $G$ with net flow ${\mathbf{a}}$* is defined as the set of all ${\mathbf{a}}$-flows of $G$.
In this paper we consider the following two graphs, see Figures \[fig:PS\] and \[fig:Car\]:
- The *Pitman-Stanley graph* ${\operatorname{PS}}_{n+1}$ is the graph with vertex set $[n+1]$ and edge set $$\{(i,i+1): i=1,2,\dots, n\}\cup\{(i,n+1):i=1,2,\dots ,n-1\}.$$
- The *Caracol graph* ${\operatorname{Car}}_{n+1}$ is the graph with vertex set $[n+1]$ and edge set $$\{(i,i+1):i=1,2,\dots,n\}\cup\{(1,i):i=3,4,\dots,n\}\cup\{(i,n+1):i=2,3,\dots,n-1\}.$$
(0,0) – (0.96,0); (1,0) – (1.96,0); (2,0) – (2.96,0); (4,0) – (4.96,0); (5,0) – (5.96,0); (6,0) – (6.96,0); (0,0) to \[out=20,in=160\] (6.96,0); (1,0) to \[out=18,in=162\] (6.96,0); (2,0) to \[out=16,in=164\] (6.96,0); (5,0) to \[out=14,in=166\] (6.96,0); (0,0) circle \[radius=0.04\]; at(0,0) [1]{}; (1,0) circle \[radius=0.04\]; at(1,0) [2]{}; (2,0) circle \[radius=0.04\]; at(2,0) [3]{}; (5,0) circle \[radius=0.04\]; at(5,0) [$n-1$]{}; (6,0) circle \[radius=0.04\]; at(6,-0.06) [$n$]{}; (7,0) circle \[radius=0.04\]; at(7,0) [$n+1$]{}; at (3.5,0) [$\cdots$]{};
(0,0) – (0.96,0); (1,0) – (1.96,0); (2,0) – (2.96,0); (4,0) – (4.96,0); (5,0) – (5.96,0); (6,0) – (6.96,0); (1,0) to \[out=18,in=162\] (6.96,0); (2,0) to \[out=16,in=164\] (6.96,0); (5,0) to \[out=14,in=166\] (6.96,0); (0,0) to \[out=-14,in=-166\] (1.96,0); (0,0) to \[out=-16,in=-164\] (4.96,0); (0,0) to \[out=-18,in=-162\] (5.96,0); (0,0) circle \[radius=0.04\]; at(0,0) [1]{}; (1,0) circle \[radius=0.04\]; at(1,0) [2]{}; (2,0) circle \[radius=0.04\]; at(2,0) [3]{}; (5,0) circle \[radius=0.04\]; at(5,0) [$n-1$]{}; (6,0) circle \[radius=0.04\]; at(6,-0.06) [$n$]{}; (7,0) circle \[radius=0.04\]; at(7,0) [$n+1$]{}; at (3.5,0) [$\cdots$]{};
We note that the flow polytope ${\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a_1,\dots,a_n)$ are affinely equivalent to the polytope $$\Pi_{n-1}(a_1,\dots,a_{n-1}):= \{(x_1,\dots,x_{n-1}): x_i\ge0,
x_1+\dots+x_{i}\le a_1+\dots+a_{i}, 1\le i\le n-1\},$$ considered in [@PitmanStanley]. Pitman and Stanley [@PitmanStanley] found volume formulas for certain polytopes, which can be restated as normalized volumns of flow polytopes as follows: $$\begin{aligned}
\label{eq:PS11}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a,b^{n-2},d) &= a(a+(n-1)b)^{n-2}, \\
\label{eq:PS12}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a,b^{n-3},c,d) &= a(a+(n-1)b)^{n-2} + (n-1)a(c-b)(a+(n-2)b)^{n-3}, \\
\label{eq:PS13}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a,b^{n-m-2},c,0^{m-1},d) &= a\sum_{j=0}^m \binom nj (c-(m+1-j)b)^j(a+(n-1-j)b)^{n-j-2},\end{aligned}$$ where $b^k$ means the sequence $b,b,\dots,b$ of $k$ $b$’s. We note that ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a_1,\dots,a_{n})$ is independent of $a_n$.
In [@Benedetti2019], Benedetti et al. introduced combinatorial models called gravity diagrams and unified diagrams to compute volumes of flow polytopes. Using these models they showed $$\begin{aligned}
\label{eq:5}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+1}}(a^n) &=C_{n-2}a^n n^{n-2},\\
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+1}}(a,b^{n-1})&=C_{n-2} a^{n-2}(a+(n-1)b)^{n-2},\end{aligned}$$ where $C_k:=\frac{1}{k+1}\binom{2k}{k}$ is the $k$th Catalan number.
For a positive integer $k$ and a directed graph $G$ on $[n+1]$, let ${\widehat{G}}(k)$ be the directed graph obtained from $G$ by adding a new vertex $0$ and $k$ multiple edges $(0,i)$ for each $1\le i\le
n+1$. Then we define $$\label{eq:EG}
E_G(k) = {\operatorname{vol}}\mathcal{F}_{{\widehat{G}}(k)}(1,0^{n}).$$
In [@Benedetti2019], Benedetti et al. showed that $E_G(k)$ is a polynomial function in $k$. Therefore we can consider the polynomial $E_G(x)$. They also showed that these polynomials $E_G(x)$ have similar properties as Ehrhart polynomials. For example, the leading coefficient of $E_G(x)$ is the normalized volume of ${\mathcal{F}}_G(1^n)$. For this reason, they called $E_G(x)$ an Ehrhart-like polynomial. In the same paper they proved the following theorem.
\[thm:PS\] We have $$E_{{\operatorname{PS}}_{n+1}}(k)=\frac{1}{kn-1}\binom{(k+1)n-2}{n}.$$
Our main result is the following theorem, which was conjectured in [@Benedetti2019].
\[thm:main\] We have $$E_{{\operatorname{Car}}_{n+1}}(k)=\frac{1}{kn+n-3}\binom{kn+2n-5}{n-1}\binom{n+k-3}{k-1}.$$
In this paper we prove both Theorems \[thm:PS\] and \[thm:main\].
The remainder of this paper is organized as follows. In Section \[sec:const-term-ident\] we use the Lidskii volume formula to interpret $E_{{\operatorname{Car}}_{n+1}}(k)$ as a Kostant partition function, which is equal to the constant term of a Laurent series. In Section \[sec:labeled-dyck-paths\] we introduce labeled Dyck paths and show that the constant term is equal to the number of certain labeled Dyck paths. In Section \[sec:cyclic-lemma\] we enumerate these labeled Dyck paths using a cyclic lemma. In Section \[sec:more-prop-label\] using our combinatorial models we show the following volume formulas: $$\begin{aligned}
\label{eq:PS3}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a,b,c^{n-2}) & =(a+b-c)(a+b+(n-2)c)^{n-2}+(-b+c)(b+(n-2)c)^{n-2},\\
\label{eq:PS4}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a,b,c,d^{n-3})&=(a+b+c-2d)(a+b+c+(n-3)d)^{n-2} \\
\notag & \qquad \qquad -(b+c-2d)(b+c+(n-3)d)^{n-2}\\
\notag&\qquad\qquad -(n-1)a(c-d)(c+(n-3)d)^{n-3} ,\\
\label{eq:conj}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+1}}(a,b,c^{n-2}) &=C_{n-2}a^{n-1}(a+b(n-1))(a+b+c(n-2))^{n-3},\end{aligned}$$ where was conjectured by Benedetti et al. in [@Benedetti2019].
Constant term identities {#sec:const-term-ident}
========================
In this section we review the Lidskii volume formula and restate Theorems \[thm:PS\] and \[thm:main\] as constant term identities.
Let $G$ be a directed graph on $[n+1]$ and ${\mathbf{a}}\in{\mathbb{Z}}^n$. The *Kostant partition function* $K_G({\mathbf{a}})$ of $G$ at ${\mathbf{a}}$ is the number of integer points of ${\mathcal{F}}_{G}({\mathbf{a}})$, *i.e.*, if $G$ has $m$ edges, $$K_G({\mathbf{a}})=|{\mathcal{F}}_G({\mathbf{a}}) \cap {\mathbb{Z}}^m|.$$ We denote by $G|_{n}$ the restriction of $G$ to the vertices in $[n]$. Let ${\operatorname{outdeg}}(i)$ denote the out-degree of vertex $i$ in $G$. The following formula, known as the Lidskii volume formula, allows us to express the (normalized) volume of the flow polytope ${\mathcal{F}}_G({\mathbf{a}})$ in terms of Kostant partition functions, see [@Baldoni2008 Theorem 38].
\[thm:lidskii\] Let $G$ be a connected directed graph on $[n+1]$ with $m$ directed edges, where every directed edge is of the form $(i, j)$ with $i<j$ and let ${\mathbf{a}}=(a_1,a_2,\dots ,a_n)\in {\mathbb{Z}}^{n}$. Denoting ${\mathbf{t}}=(t_1,\dots ,t_n):=({\operatorname{outdeg}}(1)-1,\dots,{\operatorname{outdeg}}(n)-1)$, we have $${\operatorname{vol}}{\mathcal{F}}_{G}({\mathbf{a}}) = \sum_{\substack{|{\mathbf{s}}|=m-n\\ {\mathbf{s}}\ge {\mathbf{t}}}}\binom{m-n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{n}^{s_n}K_{G|_{n}}({\mathbf{s}}-{\mathbf{t}}),$$ where the sum is over all sequences ${\mathbf{s}}=(s_1,\dots ,s_n)$ of nonnegative integers such that $|{\mathbf{s}}|=s_1+\dots+s_n=m-n$ and ${\mathbf{s}}\geq {\mathbf{t}}$ in dominance order, *i.e.*, $\sum_{i=1}^{k}s_{i} \geq \sum_{i=1}^{k}t_{i}$ for $k=1,2,\dots,n$.
Note that if ${\mathbf{a}} = (1,0^n)$ in Theorem \[thm:lidskii\] there is only one term in the sum giving the following corollary, see [@Baldoni2008], [@PitmanStanley], or [@VolumesandEhrhart Corollary 1.4].
For a directed graph $G$ on $[n+1]$, we have $$\label{eq:outdeg}
{\operatorname{vol}}{\mathcal{F}}_G(1,0^n) = K_G(p,1-{\operatorname{outdeg}}(2),1-{\operatorname{outdeg}}(3),\dots,1-{\operatorname{outdeg}}(n),0),$$ where $p={\operatorname{outdeg}}(2)+{\operatorname{outdeg}}(3)+\cdots+{\operatorname{outdeg}}(n)-n+1$.
For a multivariate rational function $f(x_1,x_2,\dots,x_n)$ we denote by ${\operatorname{CT}}_{x_i}f$ the constant term of the Laurant series expansion of $f$ with respect to $x_i$ by considering other variables as constants. Since ${\operatorname{CT}}_{x_1}
f$ is a rational function in $x_2,\dots,x_n$, we can apply ${\operatorname{CT}}_{x_2}$ to it. Repeating in this way the constant term ${\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}} f$ is defined. We also define $[x_n^{a_n}\dots x_1^{a_1}]f$ to be the coefficient of the monomial $x_n^{a_n}\dots x_1^{a_1}$ in the Laurent expansion of $f$ when expanded in the variables $x_1, x_2,\dots,x_n$ in this order. Note that we have $$\label{eq:1}
[x_n^{a_n}\dots x_1^{a_1}]f = {\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}} \left(x_n^{-a_n}\dots x_1^{-a_1} f\right).$$
Let $G$ be a directed graph on $[n+1]$. Then for ${\mathbf{a}}=(a_1,\dots,a_{n}) \in {\mathbb{Z}}^{n}$ and $a_{n+1}=-(a_1+\cdots+a_n)$, the Kostant partition function $K_G({\mathbf{a}})$ can be computed by $$\label{eq:KG}
K_G({\mathbf{a}}) = [x_{n+1}^{a_{n+1}} \cdots x_1^{a_1}] \prod_{(i,j)\in E(G)}\left( 1-\frac{x_i}{x_j}\right)^{-1}.$$
Now we are ready to express $E_{{\operatorname{PS}}_{n+1}}(k)$ and $E_{{\operatorname{Car}}_{n+1}}(k)$ as constant terms of Laurent series. Throughout this paper the factor $(x_j-x_i)^{-1}$, where $i<j$, means the Laurent expansion $$(x_j-x_i)^{-1} = \frac{1}{x_j} \left( 1-\frac{x_i}{x_j} \right)^{-1}
= \frac{1}{x_j}\sum_{l\ge0} \left( \frac{x_i}{x_j}\right)^l.$$
\[prop:car\] We have $$\begin{aligned}
\label{eq:CT1}
E_{{\operatorname{PS}}_{n+1}}(k) &={\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1},\\
\label{eq:CT2}
E_{{\operatorname{Car}}_{n+1}}(k) &= {\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}}\frac{1}{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{n}-x_{i})^{-1}\prod _{i=1}^{n-2}(x_{i+1}-x_{i})^{-1}. \end{aligned}$$
We will only prove since can be proved similarly. Let $G={\operatorname{Car}}_{n+1}$ and $H={\widehat{G}}(k)$. Then $H$ is a graph with vertices $0,1,2,\dots,n+1$, and by and , $$E_{{\operatorname{Car}}_{n+1}}(k)=K_H(p,1-{\operatorname{outdeg}}_H(1),1-{\operatorname{outdeg}}_H(2),\dots,1-{\operatorname{outdeg}}_H(n),0),$$ where $p={\operatorname{outdeg}}_H(1)+{\operatorname{outdeg}}_H(2)+\cdots+{\operatorname{outdeg}}_H(n)-n$. Since ${\operatorname{outdeg}}_{H}(1)=n-1$, ${\operatorname{outdeg}}_{H}(n)=1$, and ${\operatorname{outdeg}}_{H}(i)=2$ for $2\le i\le n-1$, we can rewrite the above equation as $$E_{{\operatorname{Car}}_{n+1}}(k)=K_H(2n-4,2-n,(-1)^{n-2},0,0).$$ Then, by , we obtain $$\label{eq:3}
E_{{\operatorname{Car}}_{n+1}}(k)= [x_{n-1}^{-1} \cdots x_2^{-1} x_1^{2-n} x_0^{2n-4}] \prod_{(i,j)\in E(H)}\left( 1-\frac{x_i}{x_j}\right)^{-1}.$$
Since every term in the expansion of $$\prod_{(i,j)\in E(H)}\left( 1-\frac{x_i}{x_j}\right)^{-1}=
\prod_{i=1}^{n}\left( 1-\frac{x_0}{x_{i}} \right)^{-k}
\prod_{i=2}^n\left( 1-\frac{x_1}{x_{i}} \right)^{-1} \left( 1-\frac{x_i}{x_{n+1}} \right)^{-1}
\prod_{i=2}^{n-1}\left( 1-\frac{x_i}{x_{i+1}} \right)^{-1}$$ is homogeneous of degree 0 in the variables $x_0,x_1,\dots,x_{n+1}$, we can set $x_{0}=1$ in . Moreover, since every term in the expansion of $(1-x_i/x_{n+1})^{-1}$ has a negative power of $x_{n+1}$ except for the constant term $1$, we can omit the factors involving $x_{n+1}$ in . Then, by the same argument, we can also omit the factors involving $x_n$ in to obtain $$E_{{\operatorname{Car}}_{n+1}}(k) =[x_1^{2-n}x_2^{-1}\cdots x_{n-1}^{-1}]
\prod_{i=1}^{n-1}\left( 1-\frac{1}{x_{i}} \right)^{-k}
\prod_{i=2}^{n-1}\left( 1-\frac{x_1}{x_{i}} \right)^{-1}
\prod_{i=2}^{n-2}\left( 1-\frac{x_i}{x_{i+1}} \right)^{-1}.$$ By replacing $x_i$ by $x_{n-i}^{-1}$ for each $1\le i\le n-1$ we have $$E_{{\operatorname{Car}}_{n+1}}(k) = [x_{n-1}^{n-2} x_{n-2} \cdots x_1]
\prod_{i=1}^{n-1}\left( 1-x_{i} \right)^{-k}
\prod_{i=1}^{n-2}\left( 1-\frac{x_i}{x_{n-1}} \right)^{-1}
\prod_{i=1}^{n-3}\left( 1-\frac{x_{i}}{x_{i+1}} \right)^{-1},$$ which is equivalent to by .
By Proposition \[prop:car\], we can restate Theorems \[thm:PS\] and \[thm:main\] as follows.
\[Thm:PS\] We have $${\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1} = \frac{1}{kn-1}\binom{(k+1)n-2}{n}.$$
\[Thm:Car\] We have $$\begin{gathered}
{\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}}\frac{1}{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{n}-x_{i})^{-1}\prod _{i=1}^{n-2}(x_{i+1}-x_{i})^{-1}\\
=\frac{1}{k(n+1)+n-2}\binom{kn+k+2n-3}{n}\binom{n+k-2}{k-1}.\end{gathered}$$
Labeled Dyck Paths {#sec:labeled-dyck-paths}
==================
In this section we give combinatorial meanings to the constant terms in Theorems \[Thm:PS\] and \[Thm:Car\] using labeled Dyck paths.
A *Dyck path of length $2n$* is a lattice path from $(0,0)$ to $(2n,0)$ consisting of *up-steps* $(1,1)$ and *down-steps* $(1,-1)$ lying on or above the line $y=0$. The set of Dyck paths of length $2n$ is denoted by ${\operatorname{Dyck}}_n$.
Let $k$ be a positive integer. A *$k$-labeled Dyck path* is a Dyck path with a labeling on the down-steps such that the label of each down-step is an integer $0\le i\le k$ and the labels of any consecutive down-steps are in weakly decreasing order, see Figure \[fig:LD\]. A *doubly $k$-labeled Dyck path* is a $k$-labeled Dyck path together with an additional labeling on the down-steps labeled $0$ and the up-steps with integers from $\{1,2,\dots,k\}$ such that the additional labels on these steps are weakly increasing, see Figure \[fig:DLD\].
We denote by ${{\operatorname{Dyck}}}_n(k)$ (resp. ${{\operatorname{Dyck}}^{(2)}}_n(k)$) the set of $k$-labeled Dyck paths (resp. doubly $k$-labeled Dyck paths) of length $2n$. We also denote by ${{\operatorname{Dyck}}}_n(k,d)$ the set of $k$-labeled Dyck paths of length $2n$ with exactly $d$ down-steps labeled $0$.
(0,0) grid (20,6); (0,0)– ++(1,1)–++(1,-1)–++(1,1)–++(1,1)– ++(1,1)– ++(1,1)– ++(1,-1)– ++(1,1)– ++(1,-1)– ++(1,-1)– ++(1,1)– ++(1,1)– ++(1,1)– ++(1,-1)– ++(1,-1)– ++(1,-1)– ++(1,-1)– ++(1,1)– ++(1,-1)– ++(1,-1); (0,0)–(20,0); at (1,0) [0]{}; at (6,3) [5]{}; at (8,3) [3]{}; at (9,2) [0]{}; at (13,4) [4]{}; at (14,3) [2]{}; at (15,2) [0]{}; at (16,1) [0]{}; at (18,1) [1]{}; at (19,0) [1]{}; at (0,0) [$(0,0)$]{}; at (20,0) [$(20,0)$]{};
(0,0) grid (20,6); (0,0)– ++(1,1)–++(1,-1)–++(1,1)–++(1,1)– ++(1,1)– ++(1,1)– ++(1,-1)– ++(1,1)– ++(1,-1)– ++(1,-1)– ++(1,1)– ++(1,1)– ++(1,1)– ++(1,-1)– ++(1,-1)– ++(1,-1)– ++(1,-1)– ++(1,1)– ++(1,-1)– ++(1,-1); (0,0)–(1,1)–(2,0)–(6,4) (7,3)–(8,4) (9,3)–(10,2)–(13,5) (15,3)–(17,1)–(18,2); (0,0)–(20,0); at (1,0) [0]{}; at (6,3) [5]{}; at (8,3) [3]{}; at (9,2) [0]{}; at (13,4) [4]{}; at (14,3) [2]{}; at (15,2) [0]{}; at (16,1) [0]{}; at (18,1) [1]{}; at (19,0) [1]{}; at (0,1) [1]{}; at (2,1) [1]{}; at (2,1) [1]{}; at (3,2) [1]{}; at (4,3) [2]{}; at (5,4) [3]{}; at (7,4) [3]{}; at (10,3) [3]{}; at (10,3) [4]{}; at (11,4) [4]{}; at (12,5) [5]{}; at (16,3) [5]{}; at (17,2) [5]{}; at (17,2) [5]{}; at (0,0) [$(0,0)$]{}; at (20,0) [$(20,0)$]{};
A *multiset* is a set with repetitions allowed. Let $\multiset{n}{m}:=\binom{n+m-1}{m}$. Then $\multiset{n}{m}$ is the number of multisets with $m$ elements taken from $[n]$. Equivalently, $\multiset{n}{m}$ is the number of nonnegative integer solutions $(a_{1},a_{2},\dots ,a_{n})$ to $a_{1}+a_{2}+\dots +a_{n}=m$ and also the number of $m$-tuples $(i_{1},i_{2},\dots ,i_{m})$ of nonnegative integers satisfying $1\leq i_{1} \leq i_{2} \leq \dots \leq i_{m} \leq n$.
The following proposition is immediate from the definitions of ${{\operatorname{Dyck}}^{(2)}}_n(k)$ and ${{\operatorname{Dyck}}}_n(k,d)$.
We have \[prop:DLD&LD\] $$\begin{aligned}
|{{\operatorname{Dyck}}^{(2)}}_n(k)|=\sum_{d=0}^{n}|{{\operatorname{Dyck}}}_n(k,d)| \Multiset{k}{n+d}.\end{aligned}$$
We now show that the constant terms in Theorems \[Thm:PS\] and \[Thm:Car\] have the following combinatorial interpretations.
We have \[Thm:PS=LD\] $$\begin{aligned}
{\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1} = |{{\operatorname{Dyck}}}_{n-1}(k,0)|.\end{aligned}$$
Consider that we choose $x_i^{a_{i1}}x_i^{a_{i2}}\cdots x_i^{a_{ik}}$ in $(1-x_{i})^{-k}=(1+x_{i}+x_{i}^{2}+\cdots)\cdots (1+x_{i}+x_{i}^{2}+\cdots)$ for $i=1,2,\dots ,n$ and we choose $x_{i}^{b_{i}}/x_{i+1}^{b_{i}+1}$ in $(x_{i+1}-x_{i})^{-1}=1/x_{i+1}+x_{i}/x_{i+1}^{2}+x_{i}^{2}/x_{i+1}^{3}+\cdots$ for $i=1,2,\dots,n-1$. Then $\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{i+1}-x_{i})^{-1}=\sum\prod_{i=1}^{n}(x_{i}^{a_{i1}+\dots+ a_{ik}+b_{i}-b_{i-1}-1})$, where the sum is over all nonnegative integers $a_{ij},b_{i}$ for $1\leq i\leq n$ and $1\leq j \leq k$ with $b_{0}=-1$ and $b_{n}=0$. Hence the left-hand side is the number of the nonnegative integer solutions to the equations $a_{i1}+\dots +a_{ik}=b_{i-1}-b_{i}+1$ for $i=1,2,\dots,n$ with $b_{0}=-1,b_{n}=0$. If we set $r_{i}=b_{i-1}-b_{i}+1$, so that $r_1+r_2+\dots +r_j=j+b_0-b_j\leq j-1$, then the number of solutions is $\sum\prod_{i=1}^{n}\multiset{k}{r_{i}}$, where the sum is over all nonnegative integers $r_{1},\dots,r_{n}$ with $r_{1}+\dots +r_{j}\leq j-1$ for $j=1,2,\dots, n-1$ and $r_{1}+\dots +r_{n}=n-1$. For such an $n$-tuple $(r_1,\dots,r_n)$, let $D$ be the Dyck path of length $2(n-1)$ such that the number of consecutive down-steps after the $i$th up-step is $r_{i+1}$ for $i=1,\dots,n-1$. The map $(r_1,\dots,r_n) \mapsto D$ is a bijection from the set of $n$-tuples satisfying the above conditions to ${\operatorname{Dyck}}_{n-1}$. Under this correspondence, $\prod_{i=1}^{n}\multiset{k}{r_i}$ is the number of $k$-labeled Dyck paths in ${{\operatorname{Dyck}}}_{n-1}(k,0)$ whose underlying Dyck path is $D$. Therefore we obtain the result.
We have \[Thm:Car=DLD\] $$\begin{aligned}
{\operatorname{CT}}_{x_{n}}\dots {\operatorname{CT}}_{x_{1}}\frac{1}{x_{1}}\prod_{i=1}^{n}(1-x_{i})^{-k}\prod _{i=1}^{n-1}(x_{n}-x_{i})^{-1}\prod _{i=1}^{n-2}(x_{i+1}-x_{i})^{-1} = |{{\operatorname{Dyck}}^{(2)}}_{n-1}(k)|.\end{aligned}$$
Similarly to the previous theorem, considering $x_i^{a_{i1}}x_i^{a_{i2}}\cdots x_i^{a_{ik}}$ in $(1-x_{i})^{-k}$ and $x_{i}^{b_{i}}/x_{i+1}^{b_{i}+1}$ in $(x_{i+1}-x_i)^{-1}$ and $x_{i}^{c_{i}}/x_{n}^{c_{i}+1}$ in $(x_n-x_i)^{-1}$, we get that the left-hand side is the number of the nonnegative integer solutions to the equations $a_{i1}+\dots +a_{ik}+b_{i}+c_{i}=1+b_{i-1}$ for $i=1,2,\dots,n-1$ and $a_{n1}+\dots+ a_{nk}+b_{n}+c_{n}=n-1+c_{1}+\dots +c_{n-1}$ with $b_{0}=b_{n-1}=b_{n}=c_{n}=0$. If we set $r_{i}=b_{i-1}-b_{i}+1$ for $i=1,2,\dots,n-1$, then the number of solutions is $\sum_{r_i,c_i}\prod_{i=1}^{n-1}\multiset{k}{r_{i}-c_i}\multiset{k}{n-1+c_{1}+\dots +c_{n-1}}$ where the sum is over all nonnegative integers $r_i,c_i$ for $i=1,2,\dots, n-1$ with $r_{1}+\dots +r_{j}\leq j$ for $j=1,2,\dots, n-2$ and $r_{1}+\dots +r_{n-1}=n-1$. For such an $n$-tuple $(r_1,\dots,r_n)$, let $D$ be the Dyck path of length $2(n-1)$ such that the number of consecutive down-steps after the $i$th up-step is $r_{i}$ for $i=1,\dots,n-1$. The map $(r_1,\dots,r_n) \mapsto D$ is a bijection from the set of $n$-tuples satisfying the above conditions to ${\operatorname{Dyck}}_{n-1}$. Regard $\multiset{k}{r-c}$ as the number of $r$-tuples $(i_1,\dots,i_r)$ of integers with $k\geq i_1 \geq \dots \geq i_{r-c} \geq 1$ and $ i_{r-c+1} =\dots=i_r= 0$. Then $\sum_{c_i}\prod_{i=1}^{n-1}\multiset{k}{r_{i}-c_i}\multiset{k}{n-1+c_{1}+\dots +c_{n-1}}$, where the sum is over all nonnegative integers $c_i$ for $i=1,2,\dots,n-1$ with $c_i \leq r_i$, is the number of doubly $k$-labeled Dyck paths whose underlying Dyck path is $D$. Therefore we obtain the result.
Note that by Proposition \[prop:DLD&LD\], we can compute the constant terms in Theorems \[Thm:PS=LD\] and \[Thm:Car=DLD\] if we have a formula for the cardinality $|{{\operatorname{Dyck}}}_n(k,d)|$. Therefore our next step is to find this number.
A cyclic lemma {#sec:cyclic-lemma}
==============
Let ${{\operatorname{Dyck}}}_{n}(k;a_0,a_1,\dots,a_k)$ denote the set of $k$-labeled Dyck paths of length $2n$ such that the number of down-steps with label $i$ is $a_i$ for $0\le i\le k$. In this section we prove the following theorem using a cyclic lemma.
\[thm:dyck\] We have $$|{{\operatorname{Dyck}}}_n(k;a_0,a_1,\dots,a_k)|=\frac{1}{n+1}\prod_{i=0}^{k}\Multiset{n+1}{a_i}.$$
A *parking function* of length $n$ is a tuple $(p_1,p_2,\dots,p_n)\in {\mathbb{Z}}^n_{>0}$ with a condition that $q_i\leq i$ for $i=1,2,\dots,n$ where $(q_1,q_2,\dots,q_n)$ is the rearrangement of $(p_1,p_2,\dots,p_n)$ in weakly increasing order. Let $PF_n$ be the set of parking function of length $n$. There is a well-known bijection between $PF_n$ and $n$-labeled Dyck paths of length $2n$ which the number of each label from $1$ to $n$ equals $1$. Thus, using Theorem \[thm:dyck\], we have $$|PF_n|=|{{\operatorname{Dyck}}}_n(n;0,1,1,\dots,1)|=(n+1)^{n-1}.$$
Recently, Yip [@Yip2019 Theorem 3.18] considered a set $\mathcal{T}_k(n,i)$ of certain labeled Dyck paths and found a simple formula for its cardinality using a cyclic lemma. Using our notation, this set can be written $$\mathcal{T}_k(n,i) = \bigcup_{a_0+\dots+a_{k-1}=n-i}{\operatorname{Dyck}}_n(k+i-1;a_0,\dots,a_{k-1},1^i).$$ The proof of Theorem \[thm:dyck\] in this section is essentially the same as that in [@Yip2019 Theorem 3.18].
A *$k$-labeled Dyck word* of length $2n$ is a sequence $w = w_1\dots w_{2n}$ of letters in $\{U,D_0,D_1,\dots,D_k\}$ satisfying the following conditions:
- The number of $U$’s is equal to $n$.
- For any prefix $w_1\dots w_j$, the number of $U$’s is greater than or equal to the total number of $D_i$’s for $0\le i\le k$.
- The labels of any consecutive $D_i$’s are in weakly decreasing order, i.e., if $w_i=D_a$ and $w_{i+1}=D_b$, then $a\ge b$.
Replacing each up step by $U$ and each down step labeled $i$ by $D_i$ is an obvious bijection from $k$-labeled Dyck paths to $k$-labeled Dyck words. For example, the $k$-labeled Dyck word corresponding to the $k$-labeled Dyck path in Figure \[fig:LD\] is $$\label{eq:w}
UD_0UUUUD_5UD_3D_0UUUD_4D_2D_0D_0UD_1D_1.$$
From now on, we will identify $k$-labeled Dyck paths with $k$-labeled Dyck words using this bijection. Note that ${{\operatorname{Dyck}}}_n(k;a_0,a_1,\dots,a_k)$ is then the set of $k$-labeled Dyck words of length $2n$ in which the number of $D_i$’s is equal to $a_i$ for $0\le i\le k$. We can count such words by using a well-known cyclic argument. We first need another definition.
An *extended $k$-labeled word* of length $2n+1$ is a sequence $w = w_1\dots w_{2n+1}$ of letters in $\{U,D_0,D_1,\dots,D_k\}$ with $w_1=U$ and exactly $n+1$ $U$’s that satisfies the third condition of a $k$-labeled Dyck word: the labels of any consecutive $D_i$’s are in weakly decreasing order, i.e., if $w_i=D_a$ and $w_{i+1}=D_b$, then $a\ge b$. The set of extended $k$-labeled words of length $2n+1$ is denoted by ${\operatorname{EW}}_n(k)$.
Let $w = w_1\dots w_{2n+1}\in {\operatorname{EW}}_n(k)$. We define the integer ${\operatorname{index}}(w)$ using the following algorithm. Here, $w = w_1\dots w_{2n+1}$ is cyclically ordered, which means that $w_1$ is followed by $w_2$, $w_2$ is followed by $w_3$, and so on, and $w_{2n+1}$ is followed by $w_1$.
- Find a letter $U$ followed by a $D_i$ for some $0\le i\le k$ in cyclic order and delete this pair $U$ and $D_i$ from $w$. Repeat this until there is only one letter left, which must be $U$.
- If the remaining $U$ is the $j$th $U$ in the original word $w$ then define ${\operatorname{index}}(w)=j$.
We also define the *shifting operator* $s:{\operatorname{EW}}_n(k)\to {\operatorname{EW}}_n(k)$ by $$s(w): = w_{i}w_{i+1}\cdots w_{2n+1} w_1 \dots w_{i-1},$$ where $i$ is the largest integer with $w_i=U$.
Let $w=UD_1D_0UUD_1U \in {\operatorname{EW}}_3(1)$. Then by the algorithm, $$\underline{UD_1}D_0UUD_1U \rightarrow D_0U\underline{UD_1}U \rightarrow \underline{D_0}U\underline{U} \rightarrow U,$$ we get ${\operatorname{index}}(w)=2$ since the remaining $U$ is the second $U$ in $w$.
Since the above algorithm treats the word $w$ cyclically one can easily see that the following lemma holds.
\[lem:cyclic\] For any element $w\in {\operatorname{EW}}_n(k)$, we have $${\operatorname{index}}(s(w)) \equiv {\operatorname{index}}(w) + 1 \mod n+1.$$
Observe that for $w=w_1w_2\dots w_{2n+1}\in {\operatorname{EW}}_n(k)$ we have $w_1\dots w_{2n}\in{\operatorname{Dyck}}_n(k)$ if and only if ${\operatorname{index}}(w)=n+1$. Therefore, by Lemma \[lem:cyclic\], for each $w\in {\operatorname{EW}}_n(k)$ there is a unique integer $0\le j\le n$ such that $s^j(w) = w'U$ for some $k$-labeled Dyck word $w'$ of length $2n$. This defines a map $p: {\operatorname{EW}}_n(k)\to{\operatorname{Dyck}}_n(k)$ sending $w$ to $p(w)=w'$. Again, by Lemma \[lem:cyclic\], this is a $(n+1)$-to-$1$ map. Note that $w$ and $w'$ have the same the number of steps $D_i$ for each $0\le i\le k$. We have proved the following proposition.
\[prop:n-to-1\] There is an $(n+1)$-to-$1$ map $p: {\operatorname{EW}}_n(k)\to{\operatorname{Dyck}}_n(k)$ preserving the number of $D_i$’s for $0\le i\le k$.
We now can prove Theorem \[thm:dyck\] easily.
By Proposition \[prop:n-to-1\], $(n+1)|{{\operatorname{Dyck}}}_n(k;a_0,a_1,\dots,a_k)|$ is the number of elements $w\in {\operatorname{EW}}_n(k)$ in which $D_i$ appears $a_i$ times for $0\le i\le k$. Since consecutive $D_i$’s are always ordered according to their subscripts, such elements $w$ are obtained from the sequence $U\dots U$ of $n+1$ $U$’s by inserting $a_i$ $D_i$’s after $U$’s in $\multiset{n+1}{a_i}$ ways for $0\le i\le k$ independently. Thus we have $$(n+1)|{{\operatorname{Dyck}}}_n(k;a_0,a_1,\dots,a_k)|=\prod_{i=0}^{k}\Multiset{n+1}{a_i},$$ which completes the proof.
As corollaries we obtain formulas for $|{{\operatorname{Dyck}}}_{n}(k,d)|$ and $|{{\operatorname{Dyck}}^{(2)}}_{n-1}(k)|$.
\[cor:LD\] We have $$|{{\operatorname{Dyck}}}_{n}(k,d)|=\frac{1}{n+1}\Multiset{n+1}{d} \Multiset{k(n+1)}{n-d}.$$
By Theorem \[thm:dyck\], $$|{{\operatorname{Dyck}}}_{n}(k,d)|=\frac{1}{n+1} \Multiset{n+1}{d} \sum_{a_1+\dots+a_k=n-d}\prod_{i=1}^{k}\Multiset{n+1}{a_i}.$$ The above sum is equal to the number of $k$-tuples $(A_1,\dots,A_k)$ of multisets such that each element $x\in A_i$ satisfies $(n+1)(i-1)+1\le x\le (n+1)i$ and $\sum_{i=1}^k |A_i| = n-d$. Since such a $k$-tuple is completely determined by $A:=A_1\cup \cdots\cup A_k$, the sum is equal to $\multiset{k(n+1)}{n-d}$, the number of multisets of size $n-d$ whose elements are in $[k(n+1)]$. Thus we obtain the formula.
\[cor:DLD\] We have $$\begin{aligned}
|{{\operatorname{Dyck}}^{(2)}}_{n-1}(k)|=\frac{1}{k(n+1)+n-2}\binom{kn+k+2n-3}{n}\binom{n+k-2}{k-1}.\end{aligned}$$
We will use the standard notation in hypergeometric series, see for example [@AAR_SP Chapter 2]. By Proposition \[prop:DLD&LD\] and Corollary \[cor:LD\], $$\begin{aligned}
\notag |{{\operatorname{Dyck}}^{(2)}}_{n-1}(k)|&=\sum_{d=0}^{n-1}|{{\operatorname{Dyck}}}_{n-1}(k,d)|\Multiset{k}{d+n-1}
\\ \notag &=\sum_{d=0}^{n-1}\frac{1}{n}\Multiset{kn}{n-d-1}\Multiset{n}{d}\Multiset{k}{d+n-1}
\\ \label{eq:2} &=\frac{(kn+n-2)!(k+n-2)!}{n!(kn-1)!(n-1)!(k-1)!}{
{}_{2}F_{1} \left(
\begin{matrix}
-n+1, k+n-1\\
-kn-n+2\\
\end{matrix}
; 1
\right)
} .\end{aligned}$$ By the Vandermonde summation formula [@AAR_SP Corollary 2.2.3] $${
{}_{2}F_{1} \left(
\begin{matrix}
-n,b\\
c\\
\end{matrix}
; 1
\right)
} = \frac{(c-b)_n}{(c)_n},$$ we have $$\label{eq:4}
{
{}_{2}F_{1} \left(
\begin{matrix}
-n+1, k+n-1\\
-kn-n+2\\
\end{matrix}
; 1
\right)
} = \frac{(-kn-2n-k+3)_{n-1}}{(-kn-n+2)_{n-1}}
=\frac{(kn+2n+k-3)!}{(kn+n+k-2)!}\frac{(kn-1)!}{(kn+n-2)!}.$$ By and we obtain the result.
The constant term identities in Theorems \[Thm:PS\] and \[Thm:Car\] follow immediately from Theorems \[Thm:PS=LD\], \[Thm:Car=DLD\] and Corollary \[cor:DLD\]. This completes the proof of Theorems \[thm:PS\] and \[thm:main\] in the introduction.
More Properties of Labeled Dyck Paths {#sec:more-prop-label}
=====================================
In this section we find volumes of flow polytopes of Pitman–Stanley graph ${\operatorname{PS}}_{n+1}$ and Caracol graph ${\operatorname{Car}}_{n+1}$ for certain flow vectors using Lidskii’s formula and $k$-labeled Dyck prefixes.
A *$k$-labeled Dyck prefix* is the part of a $k$-labeled Dyck path from $(0,0)$ to $(a,b)$ for some point $(a,b)$ in the path. The set of $k$-labeled Dyck prefixes from $(0,0)$ to $(2n-i,i)$ is denoted by ${\operatorname{Dyck}}_{n,i}$. We also denote by ${\operatorname{Dyck}}_{n,i}(k;a_0,a_1,\dots,a_k)$ the set of $k$-labeled Dyck prefixes in ${\operatorname{Dyck}}_{n,i}$ such that the number of down-steps labeled $j$ is $a_j$ for $0\le j\le k$.
Recall that ${\operatorname{Dyck}}_n(k)$ is in bijection with the set of $k$-Dyck words of length $2n$. Therefore one can consider an element in ${\operatorname{Dyck}}_{n,i}$ as a $k$-Dyck word of length $2n$ whose last $i$ letters are $D_0$’s. See Figure \[fig:LD2\].
(\#1,\#2)\[\#3\][at (\#1+0.4,\#2) [\#3]{};]{} (0,0) grid (20,6); (0,0)– ++(1,1)–++(1,-1)–++(1,1)–++(1,1)– ++(1,1)– ++(1,1)– ++(1,-1)– ++(1,1)– ++(1,-1)– ++(1,-1)– ++(1,1)– ++(1,1)– ++(1,1)– ++(1,-1)– ++(1,-1)– ++(1,-1)– ++(1,1)– ++(1,-1); (18,2)– ++(1,-1)– ++(1,-1); (1,1)\[0\]; (6,4)\[3\]; (8,4)\[1\]; (9,3)\[1\]; (13,5)\[2\]; (14,4)\[2\]; (15,3)\[0\]; (17,3)\[1\]; at (0,0) [$(0,0)$]{}; at (18,2) [$(18,2)$]{};
Now we find the cardinality of ${\operatorname{Dyck}}_{n,i}(k;a_0,a_1,\dots,a_k)$.
\[lem:dyck\_n,i\] We have $$|{\operatorname{Dyck}}_{n,i}(k;a_0,a_1,\dots,a_k)|=\dfrac{i+1}{n+1}\prod_{j=0}^{k}\Multiset{n+1}{a_j}.$$
Let ${\operatorname{EW}}_{n,i}(k)$ be the set of words $w=w_1\dots w_{2n-i+1}$ of letters in $\{U,D_0,D_1,\dots,D_k\}$ with $w_1=U$ and exactly $n+1$ $U$’s that satisfies the third condition of a $k$-labeled Dyck word: the labels of any consecutive $D_i$’s are in weakly decreasing order, i.e., if $w_i=D_a$ and $w_{i+1}=D_b$, then $a\ge b$. For $w \in {\operatorname{EW}}_{n,i}(k)$, an *index candidate* of $w$ is an integer $j$ satisfying the following condition:
- Find a letter $U$ followed by a $D_i$ for some $0\le i\le k$ in cyclic order and delete this pair $U$ and $D_i$ from $w$. Repeat this until there are $i+1$ letters left, which must be all $U$’s. Then the $j$th $U$ in the original word $w$ is one of the remaining $U$’s.
Note that there are $i+1$ index candidates for any $w\in {\operatorname{EW}}_{n,i}(k)$.
Let ${\operatorname{EW}}_{n,i}'(k)$ be the set of words obtained from a word $w\in{\operatorname{EW}}_{n,i}(k)$ by adding $i$ $D_0$’s to the left of the $j$th $U$ in $w$ for an index candidate $j$ of $w$. Note that ${\operatorname{EW}}_{n,i}'(k)$ is a subset of ${\operatorname{EW}}_{n}(k)$ which is defined in Section \[sec:cyclic-lemma\]. Thus every $w'\in {\operatorname{EW}}_{n,i}'(k)$ has length $2n+1$ and the unique index ${\operatorname{index}}(w')$ exists. Then by the map $p$ defined in Proposition \[prop:n-to-1\], there is an $(n+1)$-to-$1$ map from ${\operatorname{EW}}_{n,i}'(k)$ to ${\operatorname{Dyck}}_{n,i}(k)$. Since there are $(i+1)$ ways to choose an index candidate for $w\in {\operatorname{EW}}_{n,i}(k)$, we have $$(i+1)\prod_{j=0}^k\Multiset{n+1}{a_j}=(n+1)|{\operatorname{Dyck}}_{n,i}(k;a_0,a_1,\dots,a_k)|,$$ which completes the proof.
Volumes of flow polytopes for the Pitman–Stanley graph. {#subsec:PS}
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Recall that ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a^n)$ and ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a,b^{n-1})$ were computed in [@Benedetti2019] and [@PitmanStanley]. In this subsection using Lemma \[lem:dyck\_n,i\] we compute ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+1}}(a_1,\dots ,a_k,b^{n-k})$ for $0\le k\le 3$. For simplicity, we will consider ${\operatorname{PS}}_{n+2}$ instead of ${\operatorname{PS}}_{n+1}$.
Note that ${\operatorname{PS}}_{n+2}$ has $2n+1$ edges and ${\mathbf{t}}:=({\operatorname{outdeg}}(1)-1,\dots,{\operatorname{outdeg}}(n+1)-1)=(1,1,\dots,1,0)$. Since ${\operatorname{PS}}_{n+2}|_{n+1}$ is the path graph on $[n+1]$ with edges $(i, i+1)$ for $1\le i\le n$, one can easily see that $K_{{\operatorname{PS}}_{n+2}|_{n+1}}({\mathbf{s}}-{\mathbf{t}})=1$ for any sequence ${\mathbf{s}}\ge {\mathbf{t}}$. Moreover, if ${\mathbf{s}}=(s_1,\dots,s_{n+1})\ge {\mathbf{t}}$, then $s_{n+1}=0$. Thus Lidskii’s formula (Theorem \[thm:lidskii\]) implies $${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a_1,\dots ,a_{n+1})=\sum_{\substack{s_1+\dots+s_{n}=n\\ (s_1,\dots,s_{n})\ge
(1^{n})}}\binom{n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{n}^{s_{n}} .$$ Thus, we have $$\begin{aligned}
\notag & {\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a_1,\dots ,a_k,b^{n-k+1}) \\
\notag &=\sum_{\substack{s_1+\dots+s_{n}=n\\ (s_1,\dots,s_{n})\ge (1^{n})}}
\binom{n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{k}^{s_{k}} b^{s_{k+1}+\dots+s_{n}}\\
\notag &=\sum_{m=0}^{n} \sum_{\substack{s_1+\dots+s_{k}=m\\ (s_1,\dots,s_{k})\ge (1^k)}}
\sum_{\substack{s_{k+1}+\dots+s_{n}=n-m\\ (m,s_{k+1},\dots,s_{n})\ge (k,1^{n-k})}}
\binom{n}{s_{1},s_{2},\dots ,s_{n}}a_{1}^{s_1}\dots a_{k}^{s_{k}} b^{n-m}\\
\label{eq:PSA} &=\sum_{m=0}^{n} \binom nm b^{n-m} A_{k,m}(a_1,\dots,a_k)B_{n,k,m},\end{aligned}$$ where $$\begin{aligned}
A_{k,m}(a_1,\dots,a_k) &= \sum_{\substack{s_1+\dots+s_{k}=m\\ (s_1,\dots,s_{k})\ge (1^k)}}
\binom{m}{s_{1},s_{2},\dots ,s_{k}}a_{1}^{s_1}\dots a_{k}^{s_{k}}, \\
B_{n,k,m}&= \sum_{\substack{s_{k+1}+\dots+s_{n}=n-m\\ (m,s_{k+1},\dots,s_{n})\ge (k,1^{n-k})}}
\binom{n-m}{s_{k+1},\dots ,s_{n}}.\end{aligned}$$ The following lemma shows that $B_{n,k,m}$ has a simple formula.
We have \[lem:B\] $$B_{n,k,m} = (m-k+1)(n-k+1)^{n-m-1}.$$
For a sequence $(s_{k+1},\dots,s_n)$ of nonnegative integers, we have $s_{k+1}+\dots+s_n=n-m$ and $(m,s_{k+1},\dots,s_n)\ge (k,1^{n-k})$ if and only if $UD^{s_n}UD^{s_{n-1}}\dots UD^{s_{k+1}}U^kD^m$ is a Dyck path from $(0,0)$ to $(2n,0)$, or equivalently, $UD^{s_n}UD^{s_{n-1}}\dots UD^{s_{k+1}}$ is a Dyck prefix from $(0,0)$ to $(2n-m-k,m-k)$. Moreover, if such a sequence $(s_{k+1},\dots, s_n)$ is given, $\binom{n-m}{s_{k+1},\dots ,s_{n}}$ is the number of ways to label the down steps of this Dyck prefix with labels from $\{0,1,\dots,n-m-1\}$ such that there is exactly one down step labeled $j$ for each $0\le j\le n-m-1$ and the labels of consecutive down steps are in decreasing order. Thus $$B_{n,k,m} =|{\operatorname{Dyck}}_{n-k,m-k}(n-m-1;1^{n-m})|.$$ By Lemma \[lem:dyck\_n,i\] we obtain the formula.
By and Lemma \[lem:B\], we obtain the following proposition.
\[prop:vPS\] We have $$\begin{gathered}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a_1,\dots ,a_k,b^{n-k+1}) \\
=\sum_{m=0}^{n} \binom nm b^{n-m} (m-k+1)(n-k+1)^{n-m-1} A_{k,m}(a_1,\dots,a_k),\end{gathered}$$ where $$A_{k,m}(a_1,\dots,a_k) = \sum_{\substack{s_1+\dots+s_{k}=m\\ (s_1,\dots,s_{k})\ge (1^k)}}
\binom{m}{s_{1},s_{2},\dots ,s_{k}}a_{1}^{s_1}\dots a_{k}^{s_{k}}.$$
By Proposition \[prop:vPS\], in order to compute ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a_1,\dots ,a_k,b^{n-k+1})$, it is enough to find $A_{k,m}(a_1,\dots,a_k)$. For $k=0,1$, using this method we can easily recover the following formulas in [@Benedetti2019; @PitmanStanley]: $$\begin{aligned}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a^{n+1}) & =a^{n}(n+1)^{n-1},\\
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a,b^{n}) & =a(a+nb)^{n-1}.\end{aligned}$$
We now find a formula for this volume for ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a_1,\dots ,a_k,b^{n-k+1})$ for $k=2,3$.
For positive integers $a$, $b$, and $c$, we have $${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a,b,c^{n-1})=(a+b-c)(a+b+(n-1)c)^{n-1}-(b-c)(b+(n-1)c)^{n-1}.$$
By Proposition \[prop:vPS\], $${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a,b,c^{n-1}) =
\sum_{m=0}^n \binom{n}{m}c^{n-m}(m-1)(n-1)^{n-m-1} A_{2,m}(a,b),$$ where $A_{2,0}(a,b) =A_{2,1}(a,b)=0$ and for $m>2$, $$A_{2,m}(a,b)=\sum_{\substack{i+j=m\\ (i,j)\ge (1,1)}}\binom{m}{i,j}a^{i}b^j
=(a+b)^m-b^m.$$ Thus $$\begin{aligned}
\notag{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a,b,c^{n-1})
&= \sum_{m=2}^n \binom{n}{m}c^{n-m}\left((a+b)^m-b^m\right)(m-1)(n-1)^{n-m-1}\\
\label{eq:vol(a,c,b,)_1} &=\dfrac{1}{n-1}\left(g_n(a+b,c^{n-1})-g_n(b,c^{n-1})-f_n(a+b,c^{n-1})+f_n(b,c^{n-1})\right),\end{aligned}$$ where $$\begin{aligned}
f_n(x,y)&=\sum_{m=0}^n \binom{n}{m}x^my^{n-m} = (x+y)^n,\\
g_n(x,y)&=\sum_{m=0}^nm \binom{n}{m}x^my^{n-m} = nx(x+y)^{n-1}.\end{aligned}$$ Simplifying we obtain the result.
In a similar way one can check $A_{3,m}(a,b,c)=(a+b+c)^m-(b+c)^m-ac^{m-1}$ and obtain the following proposition. We omit the details.
For positive integers $a$, $b$, $c$, and $d$, we have $$\begin{gathered}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{PS}}_{n+2}}(a,b,c,d^{n-2})=(a+b+c-2d)(a+b+c+(n-2)d)^{n-1}\\
-(b+c-2d)(b+c+(n-2)d)^{n-1}-na(c-d)(c+(n-2)d)^{n-2}.\end{gathered}$$
Volumes of flow polytopes for the Caracol graph.
------------------------------------------------
In [@Benedetti2019], Benedetti et al. computed ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+1}}(a^n)$ and ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+1}}(a,b^{n-1})$ using unified diagrams and conjectured a formula for ${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+1}}(a,b,c^{n-2})$, see Proposition \[prop:VolCar(a,b,c)\] below. In this subsection we prove their conjecture. As before, for simplicity, we consider ${\operatorname{Car}}_{n+2}$ instead of ${\operatorname{Car}}_{n+1}$.
The Caracol graph ${\operatorname{Car}}_{n+2}$ has $3n-1$ edges and ${\mathbf{t}}':=({\operatorname{outdeg}}(1)-1,\dots,{\operatorname{outdeg}}(n+1)-1)=(n-1,1,1,\dots,1,0)$. Note that ${\mathbf{s}}=(s_1,\dots,s_{n+1}) \geq {\mathbf{t}}'$ implies $s_{n+1}=0$. Thus, by Lidskii’s formula, $$\begin{gathered}
{\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+2}}(a_1,\dots ,a_{n+1}) \\
= \sum_{\substack{s_1+\dots+s_{n}=2n-2\\(s_1,\dots,s_{n})\geq(n-1,1^{n-1})}}
\binom{2n-2}{s_{1},\dots ,s_{n}}a_{1}^{s_1}\dots a_{n}^{s_{n}}K_{{\operatorname{Car}}_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(n-1,1^{n-1})) .\end{gathered}$$
Our goal is to find a formula for $X:={\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+2}}(a,b,c^{n-1})$. By the above equation, $$\begin{gathered}
X= \sum_{\substack{s_1+\dots+s_{n}=2n-2\\ (s_1,\dots,s_{n})\ge (n-1,1^{n-1})}}
\binom{2n-2}{s_{1},s_{2},\dots ,s_{n}}a^{s_1}b^{s_{2}} c^{s_{3}+\dots+s_{n}}\\
\times K_{{\operatorname{Car}}_{n+2}|_{n+1}}(s_1-n+1,s_2-1,\dots,s_n-1).\end{gathered}$$ By replacing $s_1$ by $s_1+n-2$, we obtain $$\begin{gathered}
X=\sum_{\substack{s_1+\dots+s_{n}=n\\ (s_1,\dots,s_{n})\ge (1^{n})}}
\binom{2n-2}{s_{1}+n-2,s_{2},\dots ,s_{n}}a^{s_1+n-2}b^{s_{2}} c^{s_{3}+\dots+s_{n}}\\
\times K_{{\operatorname{Car}}_{n+2}|_{n+1}}(s_1-1,\dots,s_n-1).\end{gathered}$$ Considering $p=s_1$, $q=s_2$, and $r=s_3+\dots+s_n$ separately, we can rewrite the above equation as $$\label{eq:8}
X = \sum_{\substack{p+q+r=n\\ (p,q)\ge (1,1)}}
\binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r A(p,q,r),$$ where $$A(p,q,r)=\sum_{\substack{s_3+\dots+s_{n}=r\\(p,q,s_3,\dots,s_{n})\geq (1^{n})}}
\binom{r}{s_3,\dots,s_{n}} K_{{\operatorname{Car}}_{n+2}|_{n+1}}(p-1,q-1,s_3-1,\dots,s_n-1).$$
In the next two lemmas we find a formula for $A(p,q,r)$ using labeled Dyck paths.
Note that every Dyck path of length $2n$ can be expressed uniquely as a sequence $UD^{d_n}UD^{d_{n-1}}\dots UD^{d_1}$ of up steps $U$ and down steps $D$ for some $n$-tuple $(d_1,\dots,d_n)\in{\mathbb{Z}}_{\ge0}^n$ such that $d_1+\dots+d_n=n$ and $(d_1,\dots,d_n)\ge (1^n)$. For nonnegative integers $a_1,\dots,a_n$ whose sum is at most $n$, let $$D_n(a_1,\dots,a_n) := \{UD^{d_n}UD^{d_{n-1}}\dots UD^{d_1} \in {\operatorname{Dyck}}_n: d_i\ge a_i\}.$$
\[lem:Car1\] Let $(s_1,\dots,s_n)\in {\mathbb{Z}}_{\geq 0}^n$ with $\sum_{i=1}^n s_i=n$ and $(s_1,\dots,s_n)\geq (1^n)$. Then $$K_{{\operatorname{Car}}_{n+2}|_{n+1}}(s_1-1,\dots,s_{n}-1) = |D_{n-1}(s_2,\dots,s_n)|.$$
Note that ${\operatorname{Car}}_{n+2}|_{n+1}$ is a directed graph on $[n+1]$ with edges $(1,i)$ for $2\leq i \leq n+1$ and $(j,j+1)$ for $2\leq j \leq n$. By definition of Kostant partition function, $K_{{\operatorname{Car}}_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(1^n))$ is the number of nonnegative integer solutions $\{b_{1,i}, b_{j,j+1}: 2\leq i \leq n+1, 2\leq j \leq n \}$ satisfying $$\begin{aligned}
b_{1,2}+b_{1,3}+\dots+b_{1,n+1}&=s_1-1,
\\b_{2,3}-b_{1,2}&=s_2-1,
\\b_{j,j+1}-b_{j-1,j}-b_{1,j}&=s_{j}-1, \qquad (3\le j\le n)
\\-b_{n,n+1}-b_{1,n+1}&=-(s_1+\dots+s_n)+n.\end{aligned}$$ Since $\sum_{i=1}s_i = n$, we must have $b_{1,n+1}=b_{n,n+1}=0$ and the above equations are equivalent to $$\begin{aligned}
b_{1,2}+b_{1,3}+\dots+b_{1,n}&=s_1-1,\\
b_{j,j+1}&=(s_2+\dots+s_{j})+(b_{1,2}+\dots+b_{1,j})-(j-1), \qquad (2\le j\le n).\end{aligned}$$ Thus the integers $b_{j,j+1}$ for $2\le j\le n$ are completely determined by the integers $b_{1,i}$ for $2\le i\le n+1$. Moreover, the condition $b_{j,j+1}\ge0$ for $2\le j\le n$ is equivalent to $(s_2,\dots,s_{n})+(b_{12},\dots,b_{1n})\geq (1^{n-1})$ in dominance order. Hence $K_{{\operatorname{Car}}_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(1^n))$ is the number of $(n-1)$-tuples $(b_{12},b_{13},\dots,b_{1n})\in {\mathbb{Z}}_{\geq
0}^{n-1}$ such that $b_{12}+b_{13}+\dots+b_{1n}=s_1-1$ and $(s_2,\dots,s_{n})+(b_{12},\dots,b_{1n})\geq (1^{n-1})$.
Now let $d_i=s_{i+1}+b_{1,i+1}$ for $1\le i\le n-1$. Then we can reinterprete $K_{{\operatorname{Car}}_{n+2}|_{n+1}}((s_1,\dots,s_{n})-(1^n))$ as the number of $(n-1)$-tuples $(d_1,\dots,d_{n-1})\in {\mathbb{Z}}_{\geq 0}^{n-1}$ such that $d_1+\dots+d_{n-1}=n-1$, $(d_1,\dots,d_{n-1})\geq (1^{n-1})$ and $d_i\ge s_{i+1}$ for $1\le i\le
n-1$. Since the condition $(d_1,\dots,d_{n-1})\geq (1^{n-1})$ is equivalent to the condition $UD^{d_{n-1}} UD^{d_{n-2}}\cdots UD^{d_{1}}\in{\operatorname{Dyck}}_{n-1}$, we obtain the desired result.
Let $p,q$ and $r$ be fixed nonnegative integers with $p+q+r=n$ and $(p,q)\geq (1,1)$. Then \[lem:Car2\] $$A(p,q,r) =(p+q-1)\binom{n+p-2}{n-1}(n-1)^{r-1}-\binom{n+p-2}{n}(n-1)^r.$$
By Lemma \[lem:Car1\], $$A(p,q,r)= \sum_{\substack{s_3+\dots+s_{n}=r\\(p,q,s_3,\dots,s_{n})\geq (1^{n})}}
\binom{r}{s_3,\dots,s_{n}} |D_{n-1}(q,s_3,\dots,s_n)|.$$
We will give a combinatorial interpretation of each summand in the above formula using labeled Dyck paths. Let $s_3,\dots,s_n$ be nonnegative integers satisfying $s_3+\dots+s_{n}=r$ and $(p,q,s_3,\dots,s_{n})\geq (1^{n})$. Consider a Dyck path $\pi=UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{1}}\in D_{n-1}(q,s_3,\dots,s_n)$. Then $d_1\ge q$ and $d_i\ge s_{i+1}$ for $2\le i\le n-1$. Now we label the down steps of $\pi$ except the last consecutive down steps $D^{d_1}$ as follows:
- Distribute the $r$ labels $1,2,\dots,r$, each label occurring exactly once, to the sequences $D^{d_{n-1}}, D^{d_{n-2}},\dots, D^{d_2}$ consecutive down steps of $\pi$ so that the sequence $D^{d_i}$ gets $s_{i+1}$ labels. There are $\binom{r}{s_3,\dots,s_{n}}$ ways to do this.
- Add $d_i-s_{i+1}$ zero labels to the sequence $D^{d_i}$ and arrange the labels in weakly decreasing order.
Considering the resulting objects of this process we obtain that $\binom{r}{s_3,\dots,s_{n}} |D_{n-1}(q,s_3,\dots,s_n)|$ is the number of Dyck paths $\pi=UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{1}}$ together with a labeling on the down steps except the last consecutive down steps $D^{d_{1}}$ satisfying the following conditions:
1. $d_i\ge s_{i+1}$ for $2\le i\le n-1$.
2. $q\le d_1\le n-1-r$.
3. The number of down steps labeled $i$ is $1$ for $1\le i\le r$.
4. The number of down steps labeled $0$ is $n-1-r-d_1$.
5. The labels of any consecutive down steps are weakly decreasing.
Summing over all possible $s_3,\dots,s_n$ we obtain that $A(p,q,r)$ is the number of Dyck paths $\pi=UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{1}}$ together with a labeling on the down steps of its prefix $UD^{d_{n-1}}UD^{d_{n-2}}\dots UD^{d_{2}}$ from $(0,0)$ to $(2n-3-d_1,d_1-1)$ satisfying the above conditions except (1). This implies $$A(p,q,r)= \sum_{d_1=q}^{n-1-r} |{\operatorname{Dyck}}_{n-2,d_1-1}(r;n-1-r-d_1,1^{r})|.$$ By Lemma \[lem:dyck\_n,i\], $$\begin{aligned}
A(p,q,r) & = \sum_{d_1=q}^{n-1-r} \frac{d_1}{n-1} \Multiset{n-1}{n-1-r-d_1} (n-1)^r\\
&= (n-1)^{r-1}\sum_{d_1=q}^{n-1-r} d_1 \binom{2n-3-r-d_1}{n-2}.\end{aligned}$$ Replacing $d_1$ by $n-1-r-i$, we have $$A(p,q,r) = (n-1)^{r-1} \sum_{i=0}^{p-1} (n-1-r-i) \binom{n-2+i}{n-2}.$$ Since $$\begin{aligned}
(n-1-r-i) \binom{n-2+i}{n-2} &= \left(
(2n-2-r) - (n-1+i) \right) \binom{n-2+i}{n-2}\\
&= (2n-2-r) \binom{n-2+i}{n-2} - (n-1) \binom{n-1+i}{n-1}\\
&= (n-1-r) \binom{n-2+i}{n-2} - (n-1) \binom{n-2+i}{n-1},\end{aligned}$$ we have $$A(p,q,r) = (n-1)^{r-1} \left((p+q-1) \sum_{i=0}^{p-1} \binom{n-2+i}{n-2}
-(n-1)\sum_{i=0}^{p-2}\binom{n-1+i}{n-1} \right).$$ Finally the identity $\sum_{i=0}^{k}\binom{m+i}{m}=\binom{m+k+1}{m+1}$ finishes the proof.
Now we are ready to compute $X={\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+2}}(a,b,c^{n-1})$.
[@Benedetti2019 Conjecture 6.16] \[prop:VolCar(a,b,c)\] For positive integers $a$, $b$, and $c$, we have $${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+2}}(a,b,c^{n-1})=C_{n-1}a^{n-1}(a+nb)(a+b+(n-1)c)^{n-2}.$$
By and Lemma \[lem:Car2\], we have $${\operatorname{vol}}{\mathcal{F}}_{{\operatorname{Car}}_{n+2}}(a,b,c^{n-1}) = X = Y-Z,$$ where $$Y = \sum_{\substack{p+q+r=n\\ (p,q)\ge (1,1)}}
\binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r (p+q-1)\binom{n+p-2}{n-1}(n-1)^{r-1},$$ $$Z = \sum_{\substack{p+q+r=n\\ (p,q)\ge (1,1)}}
\binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r \binom{n+p-2}{n}(n-1)^r.$$ Note that in the above two sums, the condition $(p,q)\ge(1,1)$ can be omitted since the summand is zero if $p=0$ or $(p,q)=(1,0)$. Thus $$\begin{aligned}
Y &= \frac{a^{n-1}}{n-1}\binom{2n-2}{n-1} \sum_{p+q+r=n}
(p+q-1) \binom{n-1}{p-1,q,r}a^{p-1}b^q (c(n-1))^r,\\
Z &= \sum_{p+q+r=n}
\binom{2n-2}{p+n-2,q,r}a^{p+n-2}b^q c^r \binom{n+p-2}{n}(n-1)^r\\
&=a^{n}\binom{2n-2}{n} \sum_{p+q+r=n}
\binom{n-2}{p-2,q,r}a^{p-2}b^q (c(n-1))^r .\end{aligned}$$ Using the multinomial theorem $$\sum_{i+j+k=m} \binom{m}{i,j,k}x^iy^j z^k t^{i+j} = (xt+yt+z)^m,$$ and its derivative with respect to $t$, i.e, $$\sum_{i+j+k=m} (i+j) \binom{m}{i,j,k} x^i y^j z^k t^{i+j-1} = m(x+y)(xt+yt+z)^{m-1},$$ we obtain $$\begin{aligned}
Y &= C_{n-1}a^{n-1}n(a+b)(a+b+(n-1)c)^{n-2},\\
Z &=C_{n-1}a^n(n-1)(a+b+(n-1)c)^{n-2},\end{aligned}$$ and the proof follows.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Alejandro Morales for informing them that Theorems \[thm:PS\] and \[thm:main\] are equivalent to Theorems \[Thm:PS\] and \[Thm:Car\]. They also thank Nathan Williams for helpful discussion.
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|
Introduction. {#section-un}
=============
Motivations of this work. {#sub1-1}
-------------------------
One of the most striking property of some strongly correlated systems is fractionalization, that is the existence of elementary excitations carrying only part of the quantum numbers of the constituent particles of the system. The most famous example is probably the charge one-third Laughlin quasiparticle, which is the elementary excitation of the fractional quantum Hall fluid at filling $\nu =1/3$.[@laughlin] Its existence was recently confirmed in a beautiful set of shot noise experiments[@glattli]. The earliest example of fractionalization in condensed matter physics is however found in one dimension: the exact solution of the Hubbard model[@hubbard] by Bethe Ansatz[@bethe] revealed that the charge and spin of the electron split into two excitations with independent dynamics, known as the holon and the spinon. Faddeev and Takhtajan later showed that the same spinon is also the elementary excitation of the 1D Heisenberg model: the magnon (the usual Goldstone boson) is replaced by two spinons generating a continuum for $\Delta S=1$ excitations[@fadeev; @johnson]. This property of the Hubbard model is known as spin-charge separation and is generic of so-called Luttinger liquids (LL): LL constitute a universality class for gapless one dimensional models such as the Heisenberg chain, the Hubbard and $t-J$ models[@haldane79]. Luttinger liquids are non-Fermi liquids: Landau quasiparticles[@landau] are not elementary excitations of the LL and as a consequence the electron Green’s function shows no quasiparticle pole (this property is true both for the LL with spin and for the spinless LL). Haldane, who coined the name of LL, conjectured that 1D gapless models would have the same low-energy physics as that of the Tomonaga-Luttinger model. For energies smaller than the bandwidth[@luttinger; @tomonaga; @mattis-lieb], the latter model is a fixed point of the renormalization group (RG)[@shankar]. In 1D, bosonization allows to transform the Tomonaga-Luttinger model into a gaussian acoustic hamiltonian describing free phonons [@f1]; the considerable success and popularity of bosonization stems from the fact that all the computations are straightforward because the effective hamiltonian is that of a free bosonic field. Another perspective on the LL is provided by conformal field theory (CFT) which describes two dimensional (or $1+1$) critical theories with conformal invariance; this has allowed to identify the Luttinger liquid universality class as the set of $c=1$ CFTs[@anomaly], i.e. the set of all models which flow under RG towards the gaussian free boson hamiltonian[@cftref]. CFT has allowed to formalize the finite-size analysis of Luttinger liquids first introduced by Haldane[@cardy; @kawakami; @haldane79]. In terms of the gaussian hamiltonian, the LL theory can be described as a phenomenological theory characterized by the following parameters: $u$ which is a velocity for collective modes and $K$ which is proportional to the compressibility of the system.
Yet, although the LL description is supposedly quite well established through the formalisms of bosonization or CFT, and despite the fact that exact solutions (Bethe Ansatz) show the existence of fractional states in the spectrum of several Luttinger liquids, there exists no systematic study of fractional excitations in the LL to the best of the authors’ knowledge. What’s more, in the framework of the bosonization formalism, it is sometimes stated that the only physically relevant excitations of a LL are phonons, since the effective hamiltonian is just that of acoustic phonons. As we show below this statement is incorrect. Conformal field theory is an alternative to bosonization which does stress the spectroscopic aspects: yet, application to the study of fractional excitations in a LL has been limited to the spinon in the case of $SU(2)$ symmetry, which is the situation relevant for the Heisenberg chain[@schoutens]. Fractional excitations must exist in a Luttinger liquid if Bethe Ansatz is correct but as far as the authors are aware, the characterization of these very unconventional fractional states, through either bosonization or CFT, is mostly [*terra incognita*]{} as the following list of issues may show:
1\) $\Delta S=1$ excitations for the Heisenberg chain form a continuum of pairs of spinons. When an Ising anisotropy is introduced, in the massless regime (with obvious notations: $\left| J_{z}\right| \leq J_{x}$ $(=J_{y})$ ) the continuum still exists and evolves smoothly as a function of the anisotropy $\Delta =J_{z}/J_{x}$[@johnson]. The continuum is again ascribed to pairs of spinons[@spinon]. It is intuitively clear that these spinons should be in some sense deformations of the $SU(2)$ spinon. We will derive in this paper creation operators for these non-$SU(2)$ spinons using the bosonization method.
In the case of $SU(2)$ symmetry, trial wavefunctions for these spinons can be found by making use of the exact solvability of the Haldane-Shastry (HS) chain[@hs]. The HS model shares the properties of the Heisenberg chain: it is a gapless $SU(2)$ symmetric spin chain with a continuum of spinon excitations. Its ground state and spinon wavefunctions[@hal94] are remarkably similar to those one would write for bosonic Laughlin states at filling $\nu =1/2$:
$$\begin{aligned}
\Psi _{gs}(x_{1},..,x_{N}) &=&\prod_{i<j}(z_{i}-z_{j})^{2}, \\
\Psi _{spinon}(z_{0}) &=&\prod_{i}(z_{i}-z_{0})\prod_{i<j}(z_{i}-z_{j})^{2},
\\
z_{i} &=&\exp i\frac{2\pi }{L}x_{i},\end{aligned}$$
where $x_{i}$ is the coordinate a spin down, and $x_{0}$ that of the spinon[@phase]. We will exhibit a similar wavefunctions for the spinon in the absence of $SU(2)$ symmetry. $SU(2)$ spinons are semions[@hal94] (anyons with a statistics intermediate between that of fermions and bosons): we will show that the statistics is affected when an anisotropy is introduced.
A continuum is also found by Bethe Ansatz for $\Delta S_{z}=0$ transitions [@woy82]. Low-lying excitations are described in that approach as two-strings states in the string formalism customary to Bethe Ansatz. This description is similar to that given for the $\Delta S=1$ continuum of the isotropic chain. In the latter case it is quite clear that a spin one-half should be ascribed to each of the (pseudo) ”hole” states in the string, which leads to the spinon interpretation since each state should contribute symmetrically to the spin-flip[@fadeev]. For $\Delta S_{z}=0$ transitions, the total $z$ spin components of the excitations add up to $0$ and the continuum results from the excitation of particle-hole pairs. For the isotropic chain, this continuum is generated by spin $1/2$ spinon-antispinon pairs. By contrast, for the $XY$ chain the $\Delta S_{z}=0$ continuum is due to particle-hole pairs of magnon-like spin $S_{z}=1$ excitations. The case of the isotropic Heisenberg chain for which spin $1/2$ spinons are involved both in the $\Delta S_{z}=0$ and the $\Delta S_{z}=1$ continuum, is therefore incidental. The important lesson to be learned is that in the presence of an Ising anisotropy, the $%
\Delta S_{z}=1$ and $\Delta S_{z}=0$ continua may involve different fractional spin states: in the first case we have spinons[@spinon], but in the second case the spinon identification is not always correct. What happens in the case of an arbitrary anisotropy will be dealt with in this paper.
2\) The holon appearing in the exact solution of the Hubbard model is a spinless charge one excitation[@lieb]. The issues raised for the spinon (operator, wavefunction, statistics) extend naturally to the holon.
3\) Spin-charge separation is an asymptotic property of the Hubbard model valid in the low-energy limit. When a magnetic field is applied Frahm and Korepin found that spin-charge decoupling was not realized even in the low-energy limit[@frahm]. In this paper we derive the new excitations replacing the holon and the spinon.
4\) In the context of the Calogero-Sutherland (CS) model[@cs] the existence of fractional excitations similar to Laughlin quasiparticles was suggested[@zirn]. In a variant of the standard LL known as the chiral LL used to describe edges of a FQHE sample, Laughlin quasiparticles do appear but the existence of such states follows from that of the same excitations in the bulk[@wen]. In the CS model the proposal was triggered by the similarity of the ground state with that of the 2D Laughlin wavefunctions and by special selection rules. The ground state is[@cs]:
$$\begin{aligned}
\Psi (x_{1},..,x_{N}) &=&\prod_{i<j}\left| z_{i}-z_{j}\right| ^{\lambda }, \\
z_{i} &=&\exp i\frac{2\pi }{L}x_{i}\end{aligned}$$
where $\lambda $ is a coupling constant for the $1/r^{2}$ interaction potential of the CS model. The CS model is a LL[@kawakami] and the LL parameter is just $K=1/\lambda $. A pseudo-particle formalism similar to that of Bethe Ansatz can be introduced and for the restricted case of rational couplings $\lambda =p/q$ special selection rules are found for the dynamical structure factor: $p$ pseudo-holes (a pseudo-hole is a hole in the Fermi sea of pseudo-momenta) must be accompanied in any excitation by $q$ pseudo-particles[@ha]. For a charge $(-1)$ pseudo-particle, this means one has a charge $1/\lambda =K$ for the pseudo-hole. In the interpretation of those selection rules it is proposed to view the CS model as a gas of non-interacting pseudo-particles with anyonic statistics $\pi \lambda $ and one rewrites the ground state as an anyonic wavefunction $\prod_{i<j}\left(
z_{i}-z_{j}\right) ^{\lambda }$. The pseudo-holes are particle-hole conjugate of these anyons: the main modification with the non-interacting case being the new selection rule; a wavefunction for these pseudo-holes consistent with those interpretations is then $\prod_{i}\left(
z_{i}-z\right) \prod_{i<j}\left( z_{i}-z_{j}\right) ^{\lambda }$ which has the correct charge and statistics. The pseudo-hole is therefore identified as a Laughlin quasiparticle. It exists for rational couplings and carries the rational charge $1/\lambda $.
There are however several limitations to those views. Firstly, these considerations are only valid for rational couplings (the pseudo-particle selection rules can not be extended to irrational $\lambda $): the physics of the CS model is by contrast completely continuous with the coupling and does not discriminate between rational and irrational couplings. The impossibility to describe rational and irrational couplings on the same footing means the representation is not adequate. Secondly a disymmetry between particle and hole excitations is introduced.
In a parallel strand of ideas, Laughlin quasiparticles were also proposed in studies of transport in a LL. The basis of the argument is that for a LL with an impurity potential a charge $K$ and not a charge unity is backscattered at the impurity location (where $K$ is the LL parameter, i.e. the conductance of the LL)[@fisher]. The impurity potential can be rewritten as a hopping potential for a charge $K$ state whose exchange statistics is $\pi K$ as seen from the commutation relations (i.e. anyonic). But for $K=1/(2n+1)$ these states are just those given by Wen for the Laughlin quasiparticles of his chiral LL: this suggests to identify these states as Laughlin quasiparticles. The main difficulty in that argument is that it relies on the introduction of an impurity potential in the LL: this obscures the question of the existence or not of a Laughlin quasiparticle in the pure non-chiral Luttinger liquid. In summary what is missing is a proof that states similar to Laughlin quasiparticles might be exact eigenstates of the LL boson hamiltonian (i.e. of the RG fixed point in the low-energy limit). The existence of Laughlin quasiparticles for the non-chiral LL for arbitrary couplings must then be considered at this point as an unproved conjecture.
The long list of issues we have brought up in points $(1-4)$ above should convince the reader that a thorough discussion of fractional excitations for Luttinger liquids within the formalisms of bosonization or CFT remains to be done. This is what motivated us to re-examine in that paper the spectrum of Luttinger liquids. We want to stress that although the previous examples concern integrable models, the detailed physics of such integrable models is not really our main interest: what matters for us is the universal low-energy content of these theories and of course we will be unable to tell anything through bosonization on the high-energy physics. Although it seems to be taken for granted that the excitations of a LL are the holon and the spinon on account of Bethe Ansatz studies of the Hubbard model, we are not aware of any existence proof of such fractional excitations whenever the model is non-integrable: this is so because any proof must resort to the universality hypothesis, that is to the LL and bosonization frameworks. The theoretical formalism we wish to introduce aims at bridging that gap by focusing on the universal structure of fractional excitations of Luttinger liquids through the bosonization method.
The structure of the paper will be as follows: section \[section-un\] is an introduction to the topics considered in the paper. In subsection \[sub1-2\] we give a short review of the LL physics in order to set the notations used throughout the paper; the issues discussed in the present subsection \[sub1-1\] will be amplified in subsection \[sub1-23\] in which we present the standard view on excitations of the LL. In the next section \[ch-two\] we will show that an alternative eigenstate basis can be built: that quasiparticle basis allows a natural discussion of fractional states. In section \[ch-3\], we will generalize our analysis to the LL with spin. When a magnetic field is added on to the Hubbard model, spin-charge separation no longer occurs[@frahm]. The standard spin-charge separated Luttinger liquid theory is not applicable any more. We will introduce in subsection \[ch-32\] a general framework related to the $K$ topological matrix of Wen’s chiral Luttinger liquids[@wen], which yields simple criteria of spin-charge separation in terms of a $Z_{2}$ symmetry: we will be able then in subsection \[ch-33\] to derive the fractional excitations which replace the holon and the spinon. The general LL theory we have introduced will then be applied to the Hubbard model in a magnetic field in sub-section \[ch-34\] in which we explain the relation between our approach and the formalism of the dressed charge matrix due to Frahm and Korepin. Let us mention that section \[ch-two\] of this paper expands on a short version which contained results in the case of a spinless LL[@short], whereas part \[ch-3\] presents totally new material.
The Luttinger liquid. {#sub1-2}
---------------------
### Notations.
This section will define the notations employed throughout the paper. We exclusively deal with Luttinger liquids and therefore when considering some specific models such as the Heisenberg spin chain or the Hubbard model we implicitly assume that we are working in the LL part of their phase diagrams. The whole physics of the LL is embodied in the following hamiltonian: $$H_{B}=\frac{u}{2}\int_{0}^{L}dx\ K^{-1}(\nabla \Phi (x))^{2}+K(\nabla \Theta
(x))^{2} \label{boson}$$ supplemented by the so-called bosonization formulas. We work on a ring of length $L$. $u$ and $K$ are the LL parameters. $\Phi $ can be interpreted as a displacement field for phonons, while $\Theta $ is a superfluid phase; indeed the particle and current densities are defined as:
$$\begin{aligned}
\rho (x)-\rho _{0} &=&-\frac{1}{\sqrt{\pi }}\nabla \Phi (x), \\
j(x) &=&\frac{1}{\sqrt{\pi }}\nabla \Theta (x).\end{aligned}$$
[*Renormalized current:*]{} Actually $j(x)$ is a [*bare current density*]{} which corresponds to the correct one only in the non-interacting case $K=1$: the continuity equation shows that the correct current density is renormalized and is $$j_{R}(x)=uKj(x)$$ (the Fermi velocity has been set to unity). We will discuss in section \[ch-two\] the meaning of such a renormalization.
The particle operators for bosons and right and left moving fermions respectively are given as:
$$\begin{aligned}
\Psi _{B}(x) &=&:\exp i\sqrt{\pi }\Theta (x) :, \\
\Psi _{F,R}(x) &=&:\exp i\sqrt{\pi }\left( \Theta (x) -\Phi (x)\right) :\exp
ik_{F}x, \\
\Psi _{F,L}(x) &=&:\exp i\sqrt{\pi }\left( \Theta (x)+\Phi (x) \right) :\exp
-ik_{F}x\end{aligned}$$
(in the following we will assume that these operators are normal ordered). $%
k_{F}=\pi N_{0}/L$ is the Fermi momentum where $N_{0}$ is the number of particles which is fixed by the chemical potential. $\Theta $ and $\Pi
=\nabla \Phi $ are canonical conjugate boson fields: $$\left[ \Theta (x),\Pi (y)\right] =i\delta (x-y).$$
The zero modes of the charge and current density are respectively:
$$\begin{aligned}
\widehat{N} &=&N_{0}+\widehat{Q}=\int_{0}^{L}\rho (x)dx=N_{0}-\int_{0}^{L}%
\frac{1}{\sqrt{\pi }}\nabla \Phi dx, \\
\widehat{J} &=&\int_{0}^{L}j(x)dx=\int_{0}^{L}\frac{1}{\sqrt{\pi }}\nabla
\Theta (x)dx.\end{aligned}$$
$\widehat{Q}$ has integral eigenvalues as befits a charge operator; in the bosonization mapping, the charge quantization is taken into account by the topological quantization of the phase field $\Phi $. Similarly, since $%
\int_{0}^{L}j(x)dx$ is a closed line integral (around the LL), it is a quantized number: this is just the topological quantization of the superfluid phase; the normalization of the fields have been chosen so that $%
\widehat{J}$ is an integer. For fermions, $\widehat{Q}=N_{+}+N_{-}$ and $%
\widehat{J}=N_{+}-N_{-}$ where $N_{+}$ and $N_{-}$ are respectively the (integral) number of (bare) electrons added to the ground state at the right and left Fermi points. The construction we have reviewed above is due to Haldane[@hal81].
Integrating the Fourier expansions of the charge and current density gives: $$\begin{aligned}
\Theta (x) &=&\Theta _{0}+\frac{\sqrt{\pi }}{L}\widehat{J}x+\frac{1}{\sqrt{L}%
}\sum_{n\neq 0}\Theta _{n}\exp i\frac{2\pi n}{L}x, \label{theta} \\
\Phi (x) &=&\Phi _{0}-\frac{\sqrt{\pi }}{L}\widehat{Q}x+\frac{1}{\sqrt{L}}%
\sum_{n\neq 0}\Phi _{n}\exp i\frac{2\pi n}{L}x. \label{phi}\end{aligned}$$ Note that these fields are not periodic: this allows for the above mentioned topological excitations. We demand that the boson or fermion operators are physical objects and be periodic on the ring: $\Psi
_{B/F}(x)=\Psi _{B/F}(x+L)$; this then implies the following selection rules on the eigenvalues $Q$ and $J$ of the zero modes $\widehat{Q}$ and $\widehat{%
J}$:
$$\begin{aligned}
Bosons &:&\;J\;even\;%
%TCIMACRO{\func{integer} }
%BeginExpansion
\mathop{\rm integer}%
%EndExpansion
, \\
Fermions &:&\;Q-J\;\;even\;%
%TCIMACRO{\func{integer} }
%BeginExpansion
\mathop{\rm integer}%
%EndExpansion
.\end{aligned}$$
Both $Q$ and $J$ are integers. The zero modes are sometimes extracted from the definition of the fermion operator which defines the $U_{\pm }$ operators, first built by Heidenreich and Haldane for the Tomonaga-Luttinger model[@heid; @haldane79]: $$U_{\pm }=\exp i\sqrt{\pi }\left( \Theta _{0}\pm \Phi _{0}\right) .$$ It will be useful to consider the commutation properties of the following operators: $$V_{\alpha ,\beta }(x)=:\exp -i\sqrt{\pi }\left( \alpha \Theta (x)-\beta \Phi
(x)\right) :.$$ Using Campbell-Haussdorf formula, one finds:
$$\begin{aligned}
&&V_{\alpha ,\beta }(x)V_{\alpha ,\beta }(y) \nonumber \\
&=&V_{\alpha ,\beta }(y)V_{\alpha ,\beta }(x)e^{-i\pi \alpha \beta
\;sgn(y-x)} \label{anyon}\end{aligned}$$
where $sgn(x)$ is the sign function, which shows in particular that $\Psi _{F}(x)$ is a fermionic operator. We define the exchange statistics of an operator per: $$\begin{aligned}
&&O(x)O(y) \nonumber \\
&=&O(y)O(x)\exp -i\theta \;sgn(y-x).\end{aligned}$$ For instance $\theta =\pi $ for fermions.
### Excitations. {#sub1-23}
Until the work of Heidenreich[@heid] and subsequently [ of]{} Haldane[@haldane79], the only excitations considered in the gaussian model were the bosonic phonon (or plasmon[@f1]) modes. But the hamiltonian contains a second part corresponding to the energies of states with non-zero charge or current with respect to the ground state. In reciprocal space, the gaussian hamiltonian becomes: $$\begin{aligned}
H_{B} &=&\frac{u}{2}\sum_{q\neq 0}K^{-1}\Pi _{q}\Pi _{-q}+Kq^{2}\Theta
_{q}\Theta _{-q} \nonumber \\
&&+\frac{\pi u}{2L}\left( \frac{\widehat{Q}^{2}}{K}+K\widehat{J}^{2}\right) .
\label{qj}\end{aligned}$$ We have split the hamiltonian into the phonon part and the non-bosonic zero mode part. The first term can indeed be rewritten as: $$H_{phonon}=\sum_{q\neq 0}u\left| q\right| \left( b_{q}^{+}b_{q}+\frac{1}{2}%
\right)$$ with the phonon operators: $$\begin{aligned}
b_{q} &=&\sqrt{\frac{K|q|}{2}}\left( \Theta _{q}-\frac{q}{K\left| q\right| }%
\Phi _{q}\right) , \label{p1} \\
b_{q}^{+} &=&\sqrt{\frac{K|q|}{2}}\left( \Theta _{-q}-\frac{q}{K\left|
q\right| }\Phi _{-q}\right) . \label{p2}\end{aligned}$$ The second term in the hamiltonian is standard in conformal field theory; it corresponds to finite size corrections to the energy when one adds particles or creates persistent currents in the Luttinger liquid. The corresponding states are built by means of Haldane’s $U_{\pm }$ operators which act as ladder operators in Fock space[@haldane79]. This $(Q,J)$ part of the hamiltonian is often called in CFT a zero mode part. The corresponding excitations [*may*]{} however carry momentum. A non zero $J$ excitation creates indeed a persistent current with momentum $Jk_{F}$. These states are therefore non-dispersive since their momentum may only assume the discrete values $Jk_{F}$.
The spectrum of the hamiltonian results from a convolution of plasmon excitations and of these $(Q,J)$ excitations as is apparent in figures (\[fig1\],\[fig2\]): two linear plasmon branches rise from each local minimum of the energy obtained for the zero-mode states $(Q,J)$. It is important to note that there are selection rules on the allowed values of $(Q,J)$, which refer back to the quantum statistics of the particles: as reviewed in the previous section, the gaussian model can be considered either for bosons or fermions, which results in different bosonization formulas. For bosons, $J$ is constrained to be an even integer while for fermions, $Q$ and $J$ must have the same parity. This then leads to two different spectra as can be seen from figures (\[fig1\]) and (\[fig2\]): for instance, for bosons the state $(Q=1,J=0)$ is available while it is forbidden for fermions; conversely $(Q=1,J=1)$ is available to fermions but not to bosons. Thus we have two different theories: the same hamiltonian leads to different properties depending on whether we consider a Fock space of bosons or a Fock space of fermions[@hal81]. We will call the LL with bosonic (resp. fermionic) selection rules: the bosonic (resp. fermionic) LL. For the bosonic LL, as depicted in figure (\[fig1\]) the spectrum in arbitrary charge sectors has the same form but for a shift in energies: in the charge sector $Q$ one must add the constant $\pi
uQ^{2}/(2L)$ to the energy. The same energies are found for the fermionic LL in charge sectors for which $Q$ is an even integer, but if $Q$ is an odd integer there is a new spectrum with local minima at momenta $\pm k_{F}$ and not $k=0$ (figure \[fig2\]).
In the rest of the paper we refer to this parametrization of the spectrum in terms of phonons and zero modes as the zero mode basis; this is to be distinguished from the quasiparticle basis which we will build later. A property which will prove crucial for the rest of the discussion is the fact that in the free-fermion case a quasiparticle basis exists as an alternative to the zero mode basis: instead of the zero modes basis, it is indeed possible to parametrize the spectrum in terms of the usual Landau quasiparticles. Below we show that a similar quasiparticle basis can be built in the interacting case. While fractional quasiparticles do occur in exactly solvable models (the holon, the spinon), scant contact had been made with the bosonization approach as mentioned earlier. In the low-energy limit, using the bosonization formalism, we will directly recover the fractional excitations predicted in Bethe Ansatz, with the advantage that the simplifications brought by the low-energy limit will allow a complete characterization, giving for instance easy access to statistical phases.
Fractional excitations of the spinless Luttinger Liquid. {#ch-two}
========================================================
This section is divided as follows: first, we discuss the property of chiral separation which is central to the physics of fractionalization; then, we exhibit fractional quasiparticles for the bosonic LL before turning to the fermionic LL for which we will find a different set of elementary excitations.
Chiral separation. {#sec2-1}
------------------
### Chiral vertex operators and fractionalization. {#cvo}
The gaussian model is endowed with a very basic property which is that of chiral separation, i.e. we can split it into two commuting parts corresponding to right or left propagation of the fields. This is a property which is systematically used by CFT in the analysis of conformally invariant systems. Indeed: $$H_{B}=\frac{u}{2}\int_{0}^{L}dx\ K^{-1}\Pi (x)^{2}+K(\nabla \Theta (x))^{2},$$ $$\Rightarrow \left[ \partial _{x}^{2}-\frac{1}{u^{2}}\partial _{t}^{2}\right]
\Theta (x,t)=0$$ More precisely we introduce the following chiral fields: $$\Theta _{\pm }(x)=\Theta (x)\mp \frac{\Phi (x)}{K}$$ which are related to the phonon operators by: $$\begin{aligned}
q &>&0:\;b_{q}=\sqrt{\frac{K|q|}{2}}\Theta _{+,q}, \\
q &<&0:\;b_{q}=\sqrt{\frac{K|q|}{2}}\Theta _{-,q}.\end{aligned}$$ In terms of these fields the hamiltonian becomes:
$$\begin{aligned}
H_{B} &=&H_{+}+H_{-}, \\
H_{\pm } &=&\frac{uK}{4}\int_{0}^{L}dx\;:\left( \partial _{x}\Theta _{\pm
}(x)\right) ^{2}: \\
&=&\sum_{\pm q>0}u\left| q\right| :b_{q}^{+}b_{q}:+\frac{\pi u}{LK}\left(
\frac{\widehat{Q}\pm K\widehat{J}}{2}\right) ^{2}.\end{aligned}$$
$H_{+}$ only contains right-moving phonons and similarly for $H_{-}$ with left-moving phonons. It is clear also that $\left[ H_{+},H_{-}\right] =0$. Let us show now that these fields $\Theta _{\pm }$ are chiral; they obey the equal-time commutation relations: $$\left[ \Theta _{\pm }(x),\mp \frac{K}{2}\partial _{y}\Theta _{\pm
}(y)\right] =i\delta (x-y),$$ which implies that the momentum canonically conjugate to $\Theta _{\pm }$ is $\Pi _{\Theta _{\pm }}=\mp \frac{K}{2}\partial _{x}\Theta _{\pm }$. The equations of motions for these fields are: $$u\partial _{x}\Theta _{\pm }=\mp \partial _{t}\Theta _{\pm }.$$ Thus: $\Theta _{\pm }(x,t)=\Theta _{\pm }(x\mp ut)$ which means that we have chiral fields indeed. The superfluid phase has therefore been parametrized as: $\Theta (x,t)=\frac{1}{2}\left[ \Theta _{+}(x-ut)+\Theta
_{-}(x+ut)\right] $.
One may define chiral density operators as well as the corresponding chiral charges as: $$\begin{aligned}
\rho _{\pm }(x) &=&\frac{1}{2\sqrt{\pi }}\partial _{x}\Phi _{\pm }(x)=\frac{%
\delta \rho (x)\pm Kj(x)}{2}, \label{chiral} \\
\widehat{Q}_{\pm } &=&\frac{\widehat{Q}\pm K\widehat{J}}{2}.\end{aligned}$$ Those chiral densities obey the anomalous (Kac-Moody) commutation relations: $$\left[ \rho _{\pm }(x),\rho _{\pm }(y)\right] =\mp \frac{iK}{2\pi }\partial
_{x}\delta (x-y).$$
Let us now consider the injection of $Q$ particles with a momentum $q$ and current $J$. In that case, the plasmon total momentum is equal to $q-J(k_{F}+\frac{\pi Q}{L})$. In the bosonization formalism, the operator creating this state is: $$V_{Q,J}(q)=\frac{1}{\sqrt{L}}\int_{0}^{L}dxe^{i(q-Jk_{F})x}:\exp -i\sqrt{\pi
}(Q\Theta -J\Phi ):.$$ This can also be rewritten as: $$\begin{aligned}
V_{Q,J}(q) &=&\frac{1}{\sqrt{L}}\int_{0}^{L}dxe^{i(q-Jk_{F})x}\exp -i\sqrt{%
\pi }Q_{+}\Theta _{+}(x) \nonumber \\
&&\times \exp -i\sqrt{\pi }Q_{-}\Theta _{-}(x).\end{aligned}$$ As a function of time: $$\begin{aligned}
V_{Q,J}(q,t) &=&\frac{1}{\sqrt{L}}\int_{0}^{L}dxe^{i(q-Jk_{F})x}\exp -i\sqrt{%
\pi }Q_{+}\Theta _{+}(x-ut) \nonumber \\
&&\times \exp -i\sqrt{\pi }Q_{-}\Theta _{-}(x+ut).\end{aligned}$$ There is therefore a splitting into two counter-propagating states. For non-interacting electrons the chiral charges $Q_{\pm }$ are integers since $%
K=1$ and the operators $\exp -i\sqrt{\pi }Q_{\pm }\Theta _{\pm }(x\mp ut)$ are just those of $Q_{\pm }$ Landau quasiparticless. But in the general case this is not true anymore: we will therefore have states carrying fractional charges.
We now define the chiral vertex operators which appeared in the previous expression as: $$V_{Q_{\pm }}^{\pm }(x)=\exp -i\sqrt{\pi }Q_{\pm }\Theta _{\pm }(x),$$ where the upperscript $\pm $ refers to the direction of propagation. They obey the following commutation rules:
$$\begin{aligned}
\left[ \rho (x),V_{Q_{\pm }}^{\pm }(y)\right] &=&Q_{\pm }\delta
(x-y)\;V_{Q_{\pm }}^{\pm }(x), \\
\left[ \widehat{Q},V_{Q_{\pm }}^{\pm }(x)\right] &=&Q_{\pm }\;V_{Q_{\pm
}}^{\pm }(x), \\
\left[ \widehat{J},V_{Q_{\pm }}^{\pm }(x)\right] &=&\frac{Q_{\pm }}{K}%
\;V_{Q_{\pm }}^{\pm }(x),\end{aligned}$$
which shows they carry charges $Q_{\pm }=\frac{Q\pm KJ}{2}$ which are non-integral in general. The above operator identity means that the charge is ’sharp’: by ’sharp’ we mean that the charge found is not a quantum average ( $<Q>$ is not necessarily quantized of course). This is a point we want to stress because this means that these quantum states are genuinely fractional. This shows then that if one injects $Q$ particles with current $%
J $ in a LL, one should observe a charge $Q_{+}=\frac{Q+KJ}{2}$ state propagating to the right at velocity $u$ and a charge $Q_{-}=\frac{Q-KJ}{2}$ going to the left with velocity $-u$. For instance, let us inject an electron exactly at the right Fermi point: this is a $(Q=1,J=1)$ excitation (with no plasmon excited); there would then be fractionalization into a charge $\frac{1+K}{2}$ state going to the right and a charge $\frac{1-K}{2}$ going to the left.
The most important property of these fractional states is that they are exact eigenstates of the gaussian hamiltonian. The proof requires a proper definition of their Fourier transform because they are anyons, as will be shown shortly: from equation (\[anyon\]) it is clear indeed that the commutation relations are anyonic with an anyonic phase $$\theta =\pm \pi \frac{Q_{\pm }^{2}}{K}.$$ Due to its anyonic character $V_{Q_{\pm }}^{\pm }(x)$ does not obey periodic boundary conditions; if we use the expressions of the fields $\Phi $ and $%
\Theta $ (equations \[phi\], \[theta\]), we immediately find that:
$$V_{Q_{\pm }}^{\pm }(x+L)=\exp \pm i2\pi \frac{Q_{\pm }^{2}}{K}\;V_{Q_{\pm
}}^{\pm }(x).$$
The Fourier transform is then defined as: $$V_{Q_{\pm }}^{\pm }(q_{n})=\frac{1}{\sqrt{L}}\int_{0}^{L}dx\exp -i\left(
\frac{2\pi }{L}n\pm \frac{2\pi }{L}\frac{Q_{\pm }^{2}}{K}\right)
x\;V_{Q_{\pm }}^{\pm }(x),$$ with a pseudo-momentum $q_{n}$ quantized as: $$\begin{aligned}
q_{n} &=&\frac{2\pi }{L}n\pm \frac{2\pi }{L}\frac{Q_{\pm }^{2}}{K} \\
&=&\overline{q_{n}}\pm \frac{2\pi }{L}\frac{Q_{\pm }^{2}}{K},\end{aligned}$$ (where we have defined a phonon part $\overline{q_{n}}$ of the momentum).
The operators $V_{Q_{\pm }}^{\pm }(q_{n})$ are such that:
1\) $V_{Q_{\pm }}^{\pm }(q)|\Psi _{0}>$ is an exact eigenstate of the chiral hamiltonian $H_{\pm }$ with energy: $$E(Q_{\pm },\overline{q_{n}})=\left[ u\left| \overline{q_{n}}\right| +\frac{%
\pi u}{2L}\frac{Q_{\pm }^{2}}{K}\right] .$$ where $|\Psi _{0}>$ is the interacting ground state (see appendix).
It has a linear dispersion.
2\) The states [created by the $V_{Q_{\pm }}^{\pm }(q_{n})$ to which one adds the phonon excitations form a complete set]{}. This is obvious because the states $%
V_{Q,J}(x)$ span the full Fock space[@basis].
### The LL spectrum in terms of fractional quasiparticles.
Let us consider figures (\[fig1\],\[fig2\]) which show the spectrum of the LL hamiltonian in various charge sectors [and ask the following question]{}: what happens when one adds $Q$ particles to the system (i.e. in the charge sector $Q$)? In the standard view of the LL spectrum based on the phonon and zero modes basis, the dynamics of the charge added to the LL is unclear because it is concealed in the zero modes. The parametrization of the spectrum in terms of the zero modes and the phonons does not allow to find [what happens once the charge $Q$ is ]{} added to the system because that choice of basis involves the use of non-dynamical states (Haldane’s $U_{pm}$ operators, which describe the zero modes). [By]{} contrast the quasiparticle basis only involves states which have a dynamics (the phonons and the fractional states) and we are therefore able to tell what happens to the charge, how much of it will move to the right, and so forth: if we consider the two branches starting from $k=Jk_{F}$ in the charge sector $Q$, the right branch corresponds to a right-moving fractional excitation with linear dispersion and with charge $Q_{+}=\frac{%
Q+KJ}{2}$, while the left branch is due to a left-moving fractional state with charge $Q_{-}=\frac{Q-KJ}{2}$. The continuum in between the branches [simply]{} results from the creation of the two fractional excitations with both non-zero momentum $\overline{q_{+,n}}$ and $\overline{q_{-,n}}$ (on the right branch, a charge $Q_{-}=\frac{Q-KJ}{2}$ is also created but it has zero momentum $\overline{q_{-,n}}=0$, and conversely on the left branch).
The direct way to find out how the charge $Q$ will behave, is to exhibit the quantum states which will describe the propagation of the charge. This is what the quasiparticle basis does because it directly considers the states involved in the dynamics of the charge. Of course, the two bases (the quasiparticle basis and the zero mode basis) are mathematically equivalent and therefore lead to identical physics: therefore the charge dynamics can also in principle be determined in the zero mode basis, but in the quasiparticle basis, we have the benefit that the spectroscopy immediately [tells us]{} the fate of the charge added to the system. In sharp contrast, in the zero mode basis, the spectroscopy is not useful because the states used in that basis are the phonons (which have no charge) and the $U_{\pm}$ operators (which have charge but no dynamics). We will give such an argument in the next section: this will prove in an independent manner the fractionalization of the LL spectrum (in a way which does not depend on the explicit construction of the fractional states operators).
### Selection rules and fractionalization. {#frac-current}
The fractional charges carried by the fractional excitations considered above are not arbitrary: they must take [ on]{} the values $$Q_{\pm }=\frac{Q\pm KJ}{2}, \label{spec-charge}$$ where both $Q$ and $J$ are integers. We may view these constraints on the allowed spectrum of fractional charges as selection rules. These selection rules have however a clear physical meaning which we discuss now.
Although [these excitations do not carry the electron quantum numbers because of the fractionalization of the spectrum, nevertheless the elementary constituents of our system are electrons (they are the high energy elementary particles of our systems)]{}: this means that they alone define the structure of Fock space, with the implication that all physical states must consist [of]{} an integrer number of electrons. [Despite the fact that there are]{} fractional states, the previous remark implies that these fractional states will be created in appropriate combinations so that the total charge is always an integer. This is the explanation of the previous selection rules we found, which [*in fine*]{} [enforce]{} the basic constraint that [we started out with]{} electrons. We may view these selection rules as being topological since they are directly related to the structure of the Fock space.
It is easy to show that eq. (\[spec-charge\]) [immediately follows from the requirement]{} that all states are electronic. We consider two counterpropagating states with arbitrary charge $Q_{+}$ and $Q_{-}$; we make no hypothesis on the values of the charges, nor on the nature of the chiral states (we do [*not*]{} assume they correspond to $V_{Q_{+}}^{+}$ and $V_{Q_{-}}^{-}$). The only assumptions we make are the following: (a) the one-dimensionality which means that the eigenstates have momenta in one of either two directions and (b) that the current density operator is renormalized. We then have two constraints on the values that the charges $Q_{+}$ and $Q_{-}$ may assume: since our Fock space is that of electrons, [all the states contain an integer number]{} of electrons i.e. $Q_{+}+Q_{-}=Q$ is an integer. The second constraint stems from the renormalization of the current density operator: $$j_{R}(x)=uKj(x)=uK\left( \frac{1}{\sqrt{\pi }}\partial _{x}\Theta (x)\right)$$ where $j(x)$ is the current density in the non-interacting case. This expression can be derived from the continuity equation[@haldane79]. Going around a ring of length $L$ in the LL we get a (persistent) current which must be quantized: $$J_{R}=\int_{0}^{L}dxj_{R}(x)=uKJ$$ where $J$ is an integer[@f4]; but the current carried by the states with charges $Q_{+}$ and $Q_{-}$ is $J_{R}=u(Q_{+}-Q_{-}).$ Therefore: $$(Q_{+}-Q_{-})=KJ,\;J\;%
%TCIMACRO{\func{integer}}
%BeginExpansion
\mathop{\rm integer}%
%EndExpansion$$ while: $$(Q_{+}+Q_{-})=Q,\;Q\;%
%TCIMACRO{\func{integer}}
%BeginExpansion
\mathop{\rm integer}%
%EndExpansion$$ Solving for these constraints, one recovers the selection rules [(\[spec-charge\]), i.e the spectrum of fractional charges]{}. We observe in passing that this argument [*does not*]{} depend on our formal algebraic derivation of subsection (\[cvo\]) and [provides an alternative proof of]{} the existence of fractional states as well as it yields the allowed charge spectrum. In that argument, [*fractionalization follows from the renormalization of the current in the presence of interactions*]{}.
Elementary excitations of the bosonic LL. {#sec2-2}
-----------------------------------------
### Elementary excitations.
We now establish a series of new results concerning the elementary chiral excitations of a non-chiral LL. We would like to find a basis of elementary excitations, i.e. identify objects from which all the other excitations can be built. It will be useful to use a spinor notation to represent the fractional states: $$\label{toto}
\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$%
\end{tabular}
\right) =\left(
\begin{tabular}{l}
$\frac{Q+KJ}{2}$ \\
$\frac{Q-KJ}{2}$%
\end{tabular}
\right) .$$ (\[toto\]) should be understood as follows: the fractional state $%
V_{Q_{+}}^{+}$ which is an anyon propagating with velocity $u$ is created along with the fractional state $V_{Q_{-}}^{-}$ which propagates in the [opposite]{} direction with the velocity $-u$. The selection rules are encoded in the second spinor: the equation is then read as meaning that addition of $Q$ particles with (persistent) current $J$ will result in a splitting into the two counterpropagating fractional states $V_{Q_{+}}^{+}$ and $V_{Q_{-}}^{-}$.
We must carefully distinguish between Bose and Fermi statistics because of the constraints on $Q$ and $J$. Let us consider bosons first: since $J$ is even we can rewrite it as $J=2n$ where $n$ is now an arbitrary integer. But then for [*bosons this implies that the spinor can be written in terms of two other independent spinors*]{}:
$$\begin{aligned}
\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$%
\end{tabular}
\right) &=&\left(
\begin{tabular}{l}
$\frac{Q+KJ}{2}$ \\
$\frac{Q-KJ}{2}$%
\end{tabular}
\right) \nonumber \\
&=&Q\left(
\begin{tabular}{l}
$\frac{1}{2}$ \\
$\frac{1}{2}$%
\end{tabular}
\right) +n\left(
\begin{tabular}{l}
$K$ \\
$-K$%
\end{tabular}
\right) . \label{bose-q}\end{aligned}$$
This implies that in real space the fractional charge excitation is: $$V_{Q_{\pm }}^{\pm }(x)=\left[ V_{1/2}^{\pm }(x)\right] ^{Q}\left[ V_{\pm
K}^{\pm }(x)\right] ^{n},$$ where $Q$ and $n$ are now [*independent integers*]{} of arbitrary sign: $%
(Q,n)\in Z^{2}$. In reciprocal space, one has a convolution for the exact fractional eigenstate: $$\begin{aligned}
V_{Q_{\pm }}^{\pm }(\overline{q}) &=&\int ..\int \prod_{i=1}^{Q}d\overline{%
q_{i}}\left[ V_{1/2}^{\pm }(\overline{q_{i}})\right] \prod_{j=1}^{n}d%
\overline{p_{j}}\left[ V_{\pm K}^{\pm }(\overline{p_{j}})\right] \nonumber
\\
&&\times \delta (\sum_{i=1}^{Q}\overline{q_{i}}+\sum_{j=1}^{n}\overline{p_{j}%
}-\overline{q})\end{aligned}$$ (the momenta in that expression are the phonon parts $\overline{q_{n}}$ of the momentum of the operator: for $V_{Q_{\pm }}^{\pm }(q_{n}),$ $q_{n}=%
\overline{q_{n}}\pm \frac{2\pi }{L}\frac{Q_{\pm }^{2}}{K}$ and $\overline{%
q_{n}}=\frac{2\pi n}{L}$)[@f6].
The above equation demonstrates clearly that the excitation $V_{Q_{\pm
}}^{\pm }(\overline{q})$ can be built from $Q$ charge $1/2$ states $%
V_{1/2}^{\pm }$ and $n$ charge $\pm K$ states $V_{\pm K}^{\pm }$. The whole spectrum of fractional excitations is thus built by repeated creation of $%
V_{1/2}^{\pm }$ and $V_{\pm K}^{\pm }$ which means that they are the [*elementary excitations* ]{}we were seeking. These two elementary excitations will be identified in the following as [respectively]{} the spinon (for spin systems) and a (1D) Laughlin quasiparticle.
### Wavefunctions of the fractional excitations. {#sec-222}
[To be complete, we]{} compute the wavefunctions of all the chiral excitations. We will first need the ground state wavefunction which is simply a Jastrow wavefunction: this is of course expected since the gaussian hamiltonian is the 1D version of the acoustic hamiltonian of Chester and Reatto’s Jastrow theory of He4[@chester] and is also identical to Bohm-Pines RPA plasmon hamiltonian adapted to 1D[@bohm-pines]. Since the gaussian hamiltonian is a sum of oscillators, the ground state is a gaussian function of the densities:
$$\begin{aligned}
&&\Psi _{0,B}(\{\rho _{q}\}) \nonumber \\
&=&\exp (-\sum_{q\neq 0}\frac{\pi }{2K\left| q\right| }\rho _{q}\rho _{-q})
\\
&=&\exp \frac{1}{2K}\int \int dxdx^{\prime }\;\widehat{\rho }(x)\ln \left|
\sin \frac{\pi }{L}(x-x^{\prime })\right| \widehat{\rho }(x^{\prime }).\end{aligned}$$
This expression is valid for the bosonic LL; for the fermionic LL, antisymmetry is recovered by observing that the fermionic LL [simply]{} derives from the bosonic LL through a singular gauge transformation on the bosonic LL (the Jordan-Wigner transformation)[@f8]: this is exactly as in the composite boson Chern-Simons theory for which the hamiltonian is plasmon-like at the one-loop level (RPA) and whose ground state is of course symmetric (the modulus of Laughlin wavefunction); in that theory, the Laughlin state is then found after undoing the Chern-Simons gauge transformation[@kiv]. Similarly undoing the Jordan-Wigner transformation amounts to multiplying the bosonic ground state by the phase factor $%
\prod_{i<j}sgn(x_{i}-x_{j})=\prod_{i<j}\left( \frac{(x_{i}-x_{j})}{\left|
x_{i}-x_{j}\right| }\right) $ (that phase factor is found by applying the Jordan-Wigner operator on the ground state). The wavefunction is [the 1D analog of]{} the 2D Laughlin state of FQHE if we rewrite the previous expression in terms of the particles’ positions: $\rho (x)=\sum_{i}\delta (x-x_{i})$, and by introducing the circular coordinates $z=\exp i\frac{2\pi }{L}x$: $${\psi }_{0,B}(\{x_{i}\})=\prod_{i<j}\mid z_{i}-z_{j}{\mid }^{1/K}.
\label{bosonic}$$
The [wavefunctions of the excited states]{} can now be computed[@f9]. Let us consider first the operator $\exp -i\sqrt{\pi }\alpha \Theta (x_{0})$; since $\Theta $ is the canonical conjugate of the field $\Pi =\partial _{x}\Phi =-\sqrt{\pi }%
\delta \widehat{\rho }$, $$\Theta (x)=-\frac{1}{i\sqrt{\pi }}\frac{\delta }{\delta \widehat{\rho }(x)},$$ and therefore: $$\exp -i\sqrt{\pi }\alpha \Theta (x_{0})|\Psi _{0,B}>$$ $$\begin{aligned}
&=&\exp \alpha \frac{\delta }{\delta \widehat{\rho }(x_{0})} \nonumber \\
&&\times \exp \frac{1}{2K}\int \int dxdx^{\prime }\;\widehat{\rho }(x)\ln
\left| \sin \frac{\pi }{L}(x-x^{\prime })\right| \widehat{\rho }(x^{\prime })
\nonumber \\
&=&\exp \frac{\alpha }{K}\int dx\;\widehat{\rho }(x)\ln \left| \sin \frac{%
\pi }{L}(x-x_{0})\right| \;\Psi _{0,B} \nonumber \\
&=&C\prod_{i}\left| z_{i}-z_{0}\right| ^{\alpha /K}\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{1/K}\end{aligned}$$ where $C$ is an unessential constant.
Similarly:
$$\begin{aligned}
&&\exp \pm i\sqrt{\pi }\frac{Q_{\pm }}{K}\Phi (x)\;\exp \mp i\frac{Q_{\pm }}{%
K}k_{F}x \nonumber \\
&=&\exp\mp i\pi \frac{Q_{\pm }}{K}\int_{0}^{x}\widehat{\rho }(y)dy \nonumber\\
&=&\exp \mp i\pi \frac{Q_{\pm }}{K}\int_{0}^{L}\widehat{\rho }(y)\theta
(x-y)dy \nonumber \\
&=&\prod_{i}\left[ \frac{(x_{i}-x)}{\left| x_{i}-x\right| }\right] ^{\mp
Q_{\pm }/K} \nonumber \\\end{aligned}$$
$$=\prod_{i}\left[ \frac{(z_{i}-z)}{\left| z_{i}-z\right| }\right] ^{\mp
Q_{\pm }/K}\exp \pm ik_{F}\frac{Q_{\pm }}{K}\left( \frac{\sum_{i}x_{i}}{N_{0}%
}+x\right) ,$$
where we [use the definitions of $\rho$ and $z$ introduced above and where]{} $$\theta (x)=\frac{1}{i\pi }\ln \left[ \frac{-x}{\left| x\right| }\right] ,$$ is the Heaviside step function.
The above operator can thus be seen as a generalized Jordan-Wigner operator, since it multiplies wavefunctions by a singular phase factor; in this manner we recover the phase $\prod_{i<j}\left( \frac{(x_{i}-x_{j})}{\left|
x_{i}-x_{j}\right| }\right) $ of the ground state of the fermionic LL. Finally we have that: $$\begin{aligned}
&&V_{Q_{+}}^{+}(x)\Psi _{0,B}(x_{1},..,x_{N}) \nonumber \\
&=&C\prod_{i}(\overline{z_{i}}-\overline{z})^{Q_{+}/K}\;\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{1/K} \nonumber \\
&&\times \exp ik_{F}\frac{Q_{+}}{2K}\left( \frac{\sum_{i}x_{i}}{N_{0}}%
+x\right) ,\end{aligned}$$ with a similar expression for $V_{Q_{-}}^{-}$ (the bar over $z$ denotes complex conjugation). It is noteworthy that these wavefunctions are obtained by multiplying a Jastrow ground state with a Laughlin-like prefactor $\prod_{i}\left| z_{i}-z\right| ^{Q_{\pm }/K}$ which generalizes the Laughlin quasihole factor $\prod_{i}(z_{i}-z)$. We can now write down the wavefunctions of the two elementary excitations: $$\begin{aligned}
&&V_{1/2}^{+}(x)\Psi _{0,B}(x_{1},..,x_{N}) \nonumber \\
&=&C\prod_{i}(z_{i}-z)^{1/2K}\;\prod_{i<j}\left| z_{i}-z_{j}\right| ^{1/K}
\nonumber \\
&&\times \exp -i\frac{k_{F}}{2K}\left( \frac{\sum_{i}x_{i}}{N_{0}}+x\right) ,\end{aligned}$$ and, $$\begin{aligned}
&&V_{K}^{+}(x)\Psi _{0,B}(x_{1},..,x_{N}) \nonumber \\
&=&C\prod_{i}(z_{i}-z)\;\prod_{i<j}\left| z_{i}-z_{j}\right| ^{1/K}
\nonumber \\
&&\times \exp -ik_{F}\left( \frac{\sum_{i}x_{i}}{N_{0}}+x\right) .\end{aligned}$$ We see that $V_{K}^{+}(x)$ is nothing but the 1D counterpart of the 2D Laughlin quasi-hole wavefunction, provided we make the following correspondence between 1D and 2D wavefunctions: $%
K\Longleftrightarrow \nu $, $z=\exp i2\pi x/L\Longleftrightarrow z=x+iy$ (up to a galilean boost absorbing the factor $\exp -ik_{F}\left( \frac{%
\sum_{i}x_{i}}{N_{0}}+x\right) $): [*in view of the formal analogy we will call that state a 1D Laughlin state*]{}.
### The spinon. {#sec-223}
We found an elementary excitation $V_{1/2}^{\pm }(x)$ for the bosonic LL carrying a charge $1/2$. When we consider spins, which are hard-core bosons, this result translates into having a state carrying a spin $\Delta S_{z}=1/2$ with respect to the ground state. In spin language, adding a particle into the system $\left( Q=1\right) $ corresponds to flipping a spin $\left( \Delta S_{z}=1\right) $. But it follows from eq.(\[bose-q\]) that this excitation is a composite of two elementary excitations, each carrying a charge $1/2$. Therefore a pair of states with spin $S_{z}=1/2$ is created when one flips a spin $\left( \Delta
S_{z}=1\right) $. We naturally identify this fractional spin excitation as a spinon. The spinon can be generated without any Laughlin quasiparticless if the spin current is zero $\left( J=0\right) $: this is a process which we [term]{} a [*pure spin process*]{}, to be distinguished from a [*pure spin current process*]{} $(S_{z}=0)$ which generates Laughlin quasiparticle-quasihole pairs (see below).
The properties of the spinon specifically depend on the LL parameter $K$. Although the spin is always $S_{z}=1/2$ the exchange statistics varies continuously with $K$ (i.e. when one varies the interaction): $$\theta _{spinon}=\frac{\pi }{4K}.$$ For instance for $K=1/2$ (which corresponds to $SU(2)$ symmetric spin interactions) the spinon is a semion. In that special case, the spinon wavefunction we obtain coincides exactly with that proposed by Haldane for the Haldane-Shastry spin chain[@hal94]: $$\begin{aligned}
\Psi _{spinon}(z) &=&\prod_{i}(z_{i}-z)\;\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{2} \nonumber \\
&&\times \exp -ik_{F}\left( \frac{\sum_{i}x_{i}}{N_{0}}+x\right)\end{aligned}$$ In this expression the coordinates are those of the down spins. For $K=1/2$ the spinon and the spin $K$ Laughlin quasiparticles are identical. [Although we have discussed fractional excitations for spin systems]{}, the previous considerations apply of course to bosons: the ”spinon” is then a [*charge*]{} $1/2$ excitation. For convenience we will call the excitation a spinon even when we consider bosons.
### The LL Laughlin quasiparticle.
The second elementary excitation we found has the following wavefunction: $$\begin{aligned}
\Psi _{Laughlin-qp}(z_{0}) &=&\prod_{i}(z_{i}-z_{0})\;\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{1/K} \nonumber \\
&&\times \exp -ik_{F}\left( \frac{\sum_{i}x_{i}}{N_{0}}+x_{0}\right) ,\end{aligned}$$ which leads us to identify it with a Laughlin quasiparticles. The parallels which can be drawn between the 2D Laughlin quasi-hole and the Luttinger liquid Laughlin quasiparticles are indeed very strong. For instance as in 2D one can use the plasma analogy to find the fractional charge: $$\begin{aligned}
&&\left| \Psi _{Laughlin-qp}(z_{0})\right| ^{2} =\left|
\prod_{i}(z_{i}-z_{0})\prod_{i<j}\left| z_{i}-z_{j}\right| ^{1/K}\right| ^{2}
\nonumber \\
&=&\exp \frac{1}{K}\int \int dxdx^{\prime }\left[ \widehat{\rho }(x)+K\delta
(x-x_{0})\right] \nonumber \\
&&\ln \left| \sin \frac{\pi }{L}(x-x^{\prime })\right| \left[ \widehat{\rho }%
(x^{\prime })+K\delta (x^{\prime }-x_{0})\right] .\end{aligned}$$ The above expression clearly shows that the charge carried by the excitation is $K$ in agreement with the direct algebraic determination (using the operator $%
V_{K}$). There are however several differences between the LL Laughlin quasiparticles and its 2D famous counter-part; first, there is no analyticity [requirement in the 1D problem, since we do not have to project into ]{}the lowest Landau level: we have two chiralities and the LL Laughlin quasi-electron is simply $$\begin{aligned}
\Psi_{Laughlin-qe}(z_{0}) &=&\prod_{i}(z_{i}-z_{0})^{-1}\;\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{1/K} \nonumber \\
&&\times \exp -ik_{F}\left( \frac{\sum_{i}x_{i}}{N_{0}}+x_{0}\right) .\end{aligned}$$
Second, while topological quantization forces the 2D FQHE Laughlin quasiparticles to have a [*rational*]{} charge, the [charge of the 1D LL Laughlin quasiparticles can take on]{} [*arbitrary*]{} real positive values, in particular [*irrational*]{}. This is a very startling property: irrational spin had already been proposed for solitons in coexisting CDW-SDW systems by B. Horowitz[@horowitz], but in a sense this is perhaps less surprising since in one dimension there is no quantization axis for spin which can therefore take a continuum of values. We show below that the Laughlin quasiparticles also exist for the fermionic LL; furthermore we will find that for the fermionic LL there is another elementary excitation which may have an irrational charge.
How are Laughlin quasiparticles created in a LL? They are generated whenever $J\neq 0$; they are always created as quasiparticle-quasihole pairs. In particular in pure current processes $(Q=0)$ no ”spinon” is created and we have only Laughlin quasiparticle-quasihole (qp-qh) pairs. For a persistent current $J$ excitation with $Q=0$ it follows from the expression $Q_{\pm }=\frac{Q\pm KJ}{2}$ that $J/2$ quasiparticle-quasihole pairs are generated.
From the above analysis we now can give a physical interpretation to the renormalization of the current density operator in the presence of interactions:
$$\begin{aligned}
j(x) &=&\frac{1}{\sqrt{\pi }}\partial _{x}\Theta (x), \\
&\longrightarrow &j_{R}(x)=\frac{uK}{\sqrt{\pi }}\partial _{x}\Theta (x).\end{aligned}$$
The velocity $u$ has been normalized to the Fermi velocity so that $u=1$ in the absence of interactions for fermions ($K=1$). We have found that current excitations were due to Laughlin quasiparticles. The natural explanation of the renormalization is therefore that the current is no longer carried by Landau quasiparticles but by Laughlin quasiparticles with velocity $u$ and charge $K$.
### The bosonic LL spectrum in terms of fractional elementary excitations.
For the bosonic LL we can now add the following precisions to the description of the spectrum.
For $Q=0$ excitations (see figure (\[fig1\])), [*the continuum is due to multiple Laughlin quasiparticle-quasihole pairs*]{}: the right branch starting at $k=2k_{F}$ corresponds to the propagation of a 1D Laughlin quasielectron while the left branch is due to a Laughlin quasihole; in between the two lines, we have a continuum generated by these two excitations. More generally at $k=2nk_{F}$ where $n$ is an arbitrary integer, the two branches create a continuum of $n$ Laughlin quasiparticle-quasihole pairs. (Note that for $n=0$, which is an exceptional case, we have multiphonon processes.) Therefore the spectrum in the zero charge sector is [*not*]{} a Landau quasiparticle-quasihole pair continuum except at the special value $K=1$ which describes indeed in the low energy limit a gas of hard-core bosons.
In the charge sector $Q=1$, pairs of charge one-half excitations are created: they correspond to the ”spinons” of spin chains; the pairs are superimposed on the previous Laughlin quasiparticle continuum: for instance a $2k_{F}$ excitation generates a Laughlin quasiparticle-quasihole pair in addition to the ”spinon” pair.
[The Laughlin quasiparticle and the spinon are dual states for the bosonic LL; by duality we mean electromagnetic duality which exchanges charge and current processes. Indeed the ”spinon” is associated with charge processes while the Laughlin quasiparticles is due to current excitations. The duality operation which maps a bosonic LL onto another bosonic LL is:$%
K=1/(4K^{\prime })$ with $\Theta =2\Phi ^{\prime }$ and $\Phi =\Theta
^{\prime }/2$; zero modes then transform as $J=2Q^{\prime }$ et $Q=J^{\prime
}/2$. With these relations, the selection rule remains bosonic ($J^{\prime }$ even) while the hamiltonian $H_{B}\left[ K,\Theta ,\Phi \right] =H_{B}\left[
K^{\prime },\Theta ^{\prime },\Phi ^{\prime }\right] $ retains a gaussian form. It is clear then that $K=1/2$ is a self-dual point while $V_{K}$ and $%
V_{1/2}$ create dual quasiparticles. This is not true for the fermionic LL.]{}
### The $XXZ$ spin chain.
Let us illustrate these results on the specific example of the anisotropic Heisenberg $XXZ$ spin chain. The hamiltonian of the $XXZ$ spin chain with anisotropy $\Delta $, after a bipartite rotation is: $$H\left[ J,\Delta \right] =J\sum_{i}\left\{ -\frac{1}{2}\left(
S_{i}^{+}S_{i+1}^{-}+S_{i}^{-}S_{i+1}^{+}\right) +\Delta
S_{i}^{z}S_{i+1}^{z}\right\} .$$ [As $\Delta$ is varied]{}, one finds three phases: i) for $\Delta >1$ one gets an Ising antiferromagnet [the twofold degenerate groundstate of which leads to solitonic excitations with spin one-half $1/2$ domain walls]{}; ii) for $\Delta <1$ one has an Ising ferromagnet; iii) for -$1\leq \Delta \leq 1$ we have the so-called $XY$ phase: this is the Luttinger liquid phase we are interested in. The isotropic Heisenberg chain with $SU(2)$ invariance corresponds to $\Delta =1$ . The Luttinger liquid parameter was determined exactly by Luther and Peschel on the basis of a comparison with the Baxter model [@luther]: $$K(\Delta )=\frac{\pi }{2\arccos \left( -\Delta \right) }.$$ The spectrum in the sector $\Delta S_{z}=1$ is shown in figure (\[fig3\]) for the Heisenberg model; its linearization through bosonization is also shown in figure.
[Given that a spin one-half can be mapped onto a hard-core boson:[@f12] through the Holstein-Primakov transformation, we can transpose the results we found for the bosonic LL to the $XXZ$ spin chain.]{}
[If we want to compare the bosonization linearized spectrum to the exact one there are however two provisos:]{} (a) we have to shift the bosonization spectrum by a momentum $\pi $ : this is due to the bipartite transformation one makes in the bosonization of the $XXZ$ spin chain (in order to change the sign of the $XY$ term) and (b), there is a Brillouin zone: therefore we have to identify momenta modulo $2\pi $ and since the Fermi vector is $k_{F}=\pi /2,$ excitations with $J/2$ odd (resp. even) correspond to the same harmonics $k=\pi +Jk_{F}\equiv 0$ (resp. $\pi $). [Taking (a) and (b) into account]{} , we can use the results of the previous section pertaining to the bosonic LL, to recover the linearized spectrum of the $XXZ$ chain.
We first consider the spin sector $\Delta S_{z}=0$. In figure (\[fig3\]) starting from momentum $\pi $ we have two straight lines corresponding to left and right moving phonons, [bounding]{} a continuum; due to the folding of the continuum spectrum of the bosonic LL, one superimposes on these lines the lines due to the creation of any even number of Laughlin qp-qh pairs (the qp dispersion being given by one line, and that of the qh by the other; if the qp is right-handed, its dispersion is that of the right line, etc...). Similarly the lines starting from momentum zero or $2\pi $ correspond to the creation of an odd number of Laughlin qp-qh pairs. The continuum is therefore seen to be parametrized entirely in terms of the phonons and Laughlin quasiparticle-quasihole pairs while the zero mode basis relies on phonons and zero modes. The $\Delta S_{z}=1$ continuum is described in a similar manner but for the substitution of the phonons by a pair of counterpropagating spinons. In the special case of $SU(2)$ symmetry, the Laughlin quasiparticle and the spinon become identical operators. The previous parametrization reduces then to one involving only pairs of spinons because a pair of counterpropagating spinon plus a pair of counterpropagating spinon-antispinon is equivalent to a pair of spinons propagating in arbitrary directions. One then recovers the Bethe Ansatz result.
[*In the low-energy limit, we can now answer the various questions raised in the introduction about the spectrum of the* ]{}$XXZ$[* chain:*]{}
[*-what is the nature of the* ]{}$\Delta S_{z}=1$[* continuum? It is indeed a spinon pair continuum; but superimposed on them Laughlin quasiparticle-quasihole pairs can exist. The spinon changes when the anisotropy is varied: it acquires a statistical phase* ]{}$\pi /4K=\arccos
(-\Delta )/2$[*. Therefore the spinons at different anisotropy are not adiabatically connected: they are orthogonal states;*]{}[@f13][* this is consistent with numerical computations of the spectral density of the* ]{}$%
SU(2)$[* spinon, where it is found that the* ]{}$SU(2)$[* spinon has a zero quasiparticles weight for the* ]{}$XY$[* chain*]{}[@talstra][*.*]{}
[*-what is the nature of the* ]{}$\Delta S_{z}=0$[* continuum? It is a Laughlin qp-qh pair continuum with an unquantized spin* ]{}$S_{z}=\pm K=\pm \pi
/(2\arccos \left( -\Delta \right) )$[*; in the* ]{}$SU(2)$[* symmetric case, they are identical to the spinons. In the* ]{}$XY$[* limit, one recovers the standard spin one continuum predicted through a Jordan-Wigner transformation (*]{}$K=1,$[* *]{}$S_{z}=\pm 1$[*). But in between these two points, the elementary excitation is neither a spinon nor a Jordan-Wigner fermion.*]{}
The fermionic Luttinger Liquid. {#sec2-3}
-------------------------------
### Elementary excitations : the Laughlin quasiparticles and the ”hybrid state”. {#sec-231}
We now turn to fermions; the analysis of the elementary excitations will differ from that found for the bosonic LL because the allowed ($Q,J$) states obey different selection rules, namely $J$ is not constrained any more to be an even integer, but must have the same parity as $Q$. We may therefore write $Q-J=2n$. Then for [*fermions*]{} using eq.\[spec-charge\]: $$\left( Q_{+},Q_{-}\right) =Q\;\left( \frac{1+K}{2},\frac{1-K}{2}\right)
-n\;\left( K,-K\right) . \label{first}$$
[The most general excitation once again, is built by applying $Q$ times $V_{\frac{1\pm K}{2}}^{\pm }$ and/or $n$ times $V_{\pm K}^{\pm
}$ to the ground state ]{}; this means that we have identified a set of elementary excitations for the fermionic LL. [Here too]{} we find Laughlin quasiparticles $V_{\pm K}^{\pm }$, but [instead of]{} the spinon [we get a]{} ”hybrid state”: this is a consequence of statistics; as we will show below, that hybrid state is self-dual and is intermediate between the Laughlin quasiparticle and its dual state.
The Laughlin quasiparticle is created by current excitations: for a pure current process ($Q=0,J\neq 0$) one indeed generates Laughlin qp-qh pairs as the above equation shows. The continuum for zero charge excitations ($Q=0$) is often depicted as a (Landau) particle-hole continuum as in the non-interacting system ($K=1$): this is incorrect; we have instead a Laughlin quasiparticle-quasihole continuum. The latter does reduce to the standard Landau quasiparticle continuum when $K=1$. For $k=Jk_{F}$ there is a local minimum of the energy from which two linear branches rise corresponding to $J/2=-n$ pairs of Laughlin quasiparticles and quasiholes. For the fermionic LL, the Laughlin quasiparticle is not the only state which may have an irrational charge: this is also possible for the hybrid state.
[The hybrid state]{} is created in mixed charge and current processes: this is the main difference with the bosonic LL for which the decoupling between charge and current processes is complete. As reviewed in the introduction, there are even-odd effects in the fermionic spectrum: the spectra [obtained by adding]{} an even or an odd number of particles are qualitatively different for the fermionic LL. The $Q=1$ continuum is understood as follows: the two branches at $k_{F}$ correspond to a pair of hybrid excitations carrying a charge $%
\frac{1-K}{2}$ and $\frac{1+K}{2},$ [and propagating with velocities respectively $-u$ and $u$]{} . At $-k_{F}$ the correspondence is reversed. More generally at $%
Jk_{F}$ ($J$ is an odd integer if $Q=1$), in addition to the hybrid quasiparticles, one also creates $J/2-Q$ pairs of Laughlin quasiparticle and quasihole. When $K=1$ the hybrid states reduce to Landau quasiparticles. It is interesting to note an evidence for these states in the work of Safi and Schulz who considered the evolution of a charge $1$ wavepacket injected at $k_{F}$ in a LL: they found that there was a splitting with an average charge $<Q>=(1+K)/2$ propagating to the right and an average charge $%
<Q>=(1-K)/2$ going [to the left]{}[@safi]. This is exactly [what we predict]{}. Note however a crucial difference: the charge they find is a quantum average while we deal with elementary excitations (exact eigenstates); this has an important consequence: while it is clear that on average a charge may assume irrational values, our result goes beyond that observation since it proves that there may exist in condensed matter systems a genuine good quantum state with sharp irrational charge.
In this section, we have found that for the fermionic LL there are two elementary excitations. One is the Laughlin quasiparticle already found for the bosonic LL. The second one is a hybrid state intermediate between the spinon and the Laughlin quasiparticle. The excitations corresponding to $Q=0$ transitions (they are particle-hole excitations in the non-interacting case), form a Laughlin quasiparticle-quasihole pair continuum when $K\neq 1$.
### Dual basis and the dual quasiparticles.
The elementary excitations we have derived form a basis from which all the LL spectrum is recovered; by no means [is this choice of basis unique]{}: other bases of elementary excitations are [generated with matrices associated with a basis change having]{} integer entries whose inverses are also integer-valued: this ensures that all excitations are integral linear combinations of the elementary excitations. (The matrices belong to $SL(2,Z)$.) For instance for fermions, another basis of elementary excitations consists of states $V_{\frac{1\pm K}{2}}^{\pm }$and $V_{1}^{\pm }$: $$\left( Q_{+},Q_{-}\right) =J\;\left( \frac{1+K}{2},\frac{1-K}{2}\right)
+n\;\left( 1,1\right) . \label{dual}$$ It is actually a dual basis to the previous one: for fermions the electro-magnetic duality which exchanges charge and current excitations [ is expressed by]{} $K\longleftrightarrow 1/K$ and $\Phi
\longleftrightarrow \Theta $. This is a canonical transformation; it results in: $H_{B}\left[ K,\Theta ,\Phi \right] =H_{B}\left[ 1/K,\Phi ,\Theta
\right] $. The fermionic selection rule $Q-J$ even is obviously preserved: the transformation is therefore a duality operation for the fermionic LL. Observe that the transformations differ from those for the bosonic LL.
What is the nature of the elementary excitation $V_{1}^{\pm }$? Under the duality transformation $V_{K}^{\pm }\longrightarrow V_{-1}^{\pm }$ and $%
V_{-K}^{\pm }\longrightarrow V_{1}^{\pm }$. Therefore $V_{1}^{\pm }$ is an excitation dual to the Laughlin quasiparticle. It carries a charge unity and its wavefunction is: $$V_{1}^{+}(z_{0})\Psi _{F}=\prod_{i}(z_{i}-z_{0})^{1/K}\;\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{1/K}\;\prod_{i<j}\frac{\left( z_{i}-z_{j}\right) }{%
\left| z_{i}-z_{j}\right| }$$ We stress that although $V_{1}^{\pm }$ carries a unit charge, it is [*not*]{} an electron: the statistical exchange phase is $\pi /K$ which means $%
V_{1}^{\pm }$ is an anyon ([it can be a fermion, in the special case]{} $K=1/(2n+1)$). The difference with the electron is quite clear since the electron [creation operator]{} is:
$$\Psi (x)=\sum_{n}\exp i(2n+1)k_{F}x$$ $$\exp i\left( \sqrt{\pi }\Theta (x)-\sqrt{\pi }(2n+1)\Phi (x)\right) ,$$ while: $$V_{1}^{+}(x)=:\exp i\sqrt{\pi }(\Theta (x)-\frac{\Phi (x)}{K}):.$$ [For $K\neq 1/(2n+1)$ the dual excitation appears to be a]{} non-linear soliton of the electron. This excitation is interesting in many respects. If $K=1/(2n+1)$ (the Laughlin fractions) the excitation is fermionic and the exchange statistics of the operator is $\pi (2n+1)$. The dual quasiparticle corresponds then to a sub-dominant harmonic of the electron Fourier expansion around $k\simeq
(2n+1)k_{F}$. If one attaches $2n$ flux tubes to the electron (i.e. multiplies the electron operator by the Jordan-Wigner phase $\exp i\sqrt{\pi
}2n\Phi $) the dual state becomes the dominant $k=k_{F}$ harmonics: this is exactly the composite fermion construction and it may then be more fitting to speak of a composite fermion (indeed, the statistics of the operator is $%
(2n+1)\pi $ and not $\pi $). [Because of the similar long distance behavior of their Green functions, Stone proposed to identify such a sub-dominant operator -which he calls a hyperfermion- with Wen’s electron operator introduced in the chiral LL[@stone]. This hyperfermion is identical to the dual state for $K=1/(2n+1)$]{}. In general the dual state and the electron are however orthogonal: this is quite clear when one considers the LL with spin. The dual state is then generalized to a state with the same quantum numbers as the electron (carrying a unit charge and a spin one-half), but with again anyonic statistics. But due to spin-charge separation, that state is not stable and decays into a spinless charge one quasiparticle which is none other than the holon, and a spin one-half excitation, which is just the spinon. The dual excitation we have found is therefore the analog of the spinon and the holon for the spinless LL and has nothing to do (in general) with an electron. In the following in accordance with the previous remarks we will call these states holons (for the spinless LL) or dual states.
These dual holon states also occur in Haldane’s interpretation of the Calogero-Sutherland model: he proposed that a natural interpretation of such a model was not in terms of electrons or bosons but as a gas of non-interacting anyons[@hal94; @zirn]. The basis for that interpretation is the finding that for rational values of the coupling $\lambda =p/q$ ($%
\lambda $ is related to the LL parameter by the simple relation $\lambda
=1/K $), the dynamical structure factor obeys simple selection rules: $q$ ”charge $-1$ bare particles” (the anyon -it has anyonic statistics $\pi
\lambda =\pi /K$) are created with $p$ ”holes” which therefore carry a charge $1/\lambda $. The particles and holes appear as pseudo-particles in a pseudo-momenta parametrization of the spectrum. There are however difficulties with that interpretation: it is not clear how the selection rules are generalized when the coupling $\lambda $ is irrational; the physics is indeed completely continuous with the coupling while the selection rules are only valid for rational couplings; besides, an asymmetry is introduced between particles and holes. This means that this parametrization which relies on pseudoparticles is probably inadequate. In the low energy limit, the Calogero-Sutherland model has the properties of a Luttinger liquid. It is therefore possible to describe its quasiparticle spectrum in terms of the fractional excitations we have found in this paper: we do find the charge $1$ anyon proposed (this is our dual state); however our selection rules are quite different. First they depend on the statistics (electrons or bosons): in contrast the pseudo-particle based selection rules do not involve spinons; this runs contrary to results on the bosonic LL for which the dual state must actually be seen as a composite state made out of two spinons. Second, our selection rules are valid even for irrational couplings [that is imply the existence of]{} quantum states with sharp irrational charges. Third our selection rules respect particle-hole duality: there exist both a charge one anyon with statistics $\pi /K$ (the dual holon) and a similar charge $-1$ anyon; the same applies to Laughlin quasiparticle for which we have both quasiholes and quasielectrons. Our selection rules involve the hybrid state and the Laughlin quasiparticles, and the holon only appears in the dual basis where it is accompanied again by the hybrid quasiparticle.
Finally, a [symmetric basis]{} can be associated with the hybrid excitation: $$\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$%
\end{tabular}
\right) =m\left(
\begin{tabular}{l}
$\frac{1+K}{2}$ \\
$\frac{1-K}{2}$%
\end{tabular}
\right) +n\left(
\begin{tabular}{l}
$\frac{1-K}{2}$ \\
$\frac{1+K}{2}$%
\end{tabular}
\right) ,$$ where $m=\frac{Q+J}{2}$ and $n=\frac{Q-J}{2}$; $m$ and $n$ are again independent integers. They physically correspond to the number of electrons added to the system at the right and left Fermi points respectively. That self-dual basis reduce to Landau quasiparticles when $K=1$. The physical processes [ generated]{} in that symmetric basis are not charge or current excitations but addition of electrons at the Fermi surface. Note that the arbitrariness in the choice of a basis simply reflects the possibility to stress various specific physical processes as elementary. But experiments probe $Q_{+}$ and $Q_{-}$; for a given set of $Q$ and $J$, $Q_{\pm}$ assume the same value irrespective of the basis choice.
The Luttinger Liquid with spin. {#ch-3}
===============================
In this section we generalize the construction of fractional excitations developed in section (\[ch-two\]) to the full Luttinger liquid with spin. One of the main properties exhibited by the effective theory is spin-charge separation, the complete decoupling of spin and charge dynamics. In the exact solution of the Hubbard model by Bethe Ansatz, excitations display such a spin-charge separation: [one state, the holon is a spinless particle carrying the charge of the electron, while the other, the spinon, is a neutral spin one-half state[@lieb]]{}. This is an asymptotic property only valid in the low-energy limit (the Hubbard model in the large $U$ limit is an exception because spin-charge separation is realized at all energy scales). However this property is not [obtained for all the gapless itinerant 1D models, in the low energy limit]{}. According to the universality hypothesis they should be described by the Luttinger liquid framework if the interaction is not too long-ranged. One such example is the Hubbard model in a magnetic field which does not display spin-charge separation even in the low-energy limit, although it is a short range gapless model. This model was analyzed by Frahm and Korepin in the framework of Bethe Ansatz plus conformal field theory[@frahm]: they were able to compute the anomalous exponents for the correlation functions. Several issues remain unclear for such models in a magnetic field: [in particular what the excitations are]{}. Since spin-charge separation [does not occur]{}, the holon and the spinon cannot be the elementary excitations of the system anymore. [To answer the question we have to turn to the low-energy effective theory]{}. Frahm and Korepin’s results imply that an effective description in terms of the gaussian model should be possible since conformal invariance is realized. We will find a generalization of the spin-charge separated gaussian hamiltonian [suitable for a description of]{} the Hubbard model in a magnetic field. [Our formalism is]{} very similar to Wen’s $K$ matrix approach to edge states of the FQHE. This will enable us to characterize very precisely, in the low-energy limit, the properties of 1D gapless models with or without spin-charge separation such as the Hubbard model in a magnetic field: we will find that in the latter case, although there is no spin-charge separation, there is still a generalized decoupling. The excitations are again fractional; as expected the holon and the spinon [are no longer present in]{} the spectrum and we will give the general framework allowing the description of the fractional states which replace them.
Spin-charge separated Luttinger liquid. {#ch-31}
---------------------------------------
We start with the standard case when spin-charge separation exists. Although fractional excitations are clearly present in Bethe Ansatz, no description of these special states was attempted in the low-energy limit through bosonization. [In the following we answer several questions:]{} how does the holon evolve with interaction? What would be an effective wavefunction for it? Is it a semion? First, we consider the ground state of the two-component gaussian model, because it will suggest to us a possible generalization of the gaussian model which will prove to be the correct one for the description of gapless models without spin-charge separation such as the Hubbard model in a magnetic field.
### Ground state of the gaussian hamiltonian.
We consider a two component model by introducing an internal quantum number such as the $SU(2)$ spin. We consider the charge and spin densities as well as their associated phase fields:
$$\begin{aligned}
\rho _{c} &=&\rho _{\uparrow }+\rho _{\downarrow };\;\rho _{s}=\rho
_{\uparrow }-\rho _{\downarrow }, \\
\rho _{\sigma } &=&-\frac{1}{\sqrt{\pi }}\partial _{x}\Phi _{\sigma
},\;\sigma =\uparrow ,\downarrow \\
\left[ \Theta _{\sigma }(x),\partial _{x}\Phi _{\sigma ^{\prime }}(y)\right]
&=&i\delta _{\sigma \sigma ^{\prime }}\delta (x-y), \\
\Phi _{c/s} &=&\frac{\Phi _{\uparrow }\pm \Phi _{\downarrow }}{\sqrt{2}}, \\
\left[ \Theta _{\tau }(x),\partial _{x}\Phi _{\tau ^{\prime }}(y)\right]
&=&i\delta _{\tau \tau ^{\prime }}\delta (x-y);\;\tau =c,s.\end{aligned}$$
The effective hamiltonian derived for instance from the Hubbard model in the absence of a magnetic field is:
$$\begin{aligned}
H &=&H_{c}+H_{s}, \\
H_{\tau } &=&\frac{u_{\tau }}{2}\int_{0}^{L}dx\ K_{\tau }^{-1}\left( \nabla
\Phi _{\tau }\right) ^{2}+K_{\tau }(\nabla \Theta _{\tau })^{2};\;\tau =c,s.\end{aligned}$$
$\nabla \Phi _{\tau }=\Pi _{\tau }$ is canonically conjugate to the field $%
\Theta _{\tau }$. One easily extracts the ground state which is simply a product of gaussians: $$\Psi _{0}(\{\Pi _{\tau ,q_{n}}\})=\prod_{\tau =c,s}\exp (-\frac{1}{2K_{\tau }%
}\sum_{n\neq 0}\frac{1}{\left| q_{n}\right| }\Pi _{\tau ,q_{n}}\Pi _{\tau
,-q_{n}}).$$ In terms of charge and spin densities: $$\Psi _{0}(\{\rho _{\tau ,q_{n}}\})=\prod_{\tau =c,s}\exp (-\frac{1}{4K_{\tau
}}\sum_{n\neq 0}\frac{\pi }{\left| q_{n}\right| }\rho _{\tau ,q_{n}}\rho
_{\tau ,-q_{n}}).$$ The ground state displays of course a complete decoupling of spin and charge as is apparent from the previous expression. This is also a Jastrow wavefunction. In real space: $$\Psi _{0}(\rho _{\tau })=\prod_{\tau =c,s}\exp \frac{1}{4K_{\tau }}[$$ $$\int \int dxdx^{\prime }\rho _{\tau }(x)\ln \left| \sin \frac{\pi
(x-x^{\prime })}{L}\right| \rho _{\tau }(x^{\prime })].$$ We define the charge and spin parts of the ground state per: $$\begin{aligned}
\Psi _{c/s} &=&\exp \frac{1}{4K_{c/s}}[ \nonumber \\
&&\int \int dxdx^{\prime }\rho _{c/s}(x)\ln \left| \sin \frac{\pi
(x-x^{\prime })}{L}\right| \rho _{c/s}(x^{\prime })]. \label{chargespin}\end{aligned}$$ The previous ground state may be rewritten in terms of the densities of each species: $$\Psi _{0}(\{\rho _{\sigma }\})=\exp \frac{1}{2}[$$ $$\int \int dxdx^{\prime }\rho _{\sigma }(x)g_{\sigma \sigma ^{\prime }}\ln
\left| \sin \frac{\pi (x-x^{\prime })}{L}\right| \rho _{\sigma ^{\prime
}}(x^{\prime })], \label{plasma}$$ where we have introduced the following $\widehat{g}$ matrix: $$g_{\sigma \sigma ^{\prime }}=\left(
\begin{tabular}{ll}
$\frac{K_{c}^{-1}+K_{s}^{-1}}{2}$ & $\frac{K_{c}^{-1}-K_{s}^{-1}}{2}$ \\
$\frac{K_{c}^{-1}-K_{s}^{-1}}{2}$ & $\frac{K_{c}^{-1}+K_{s}^{-1}}{2}$%
\end{tabular}
\right)$$ The eigenvalues of that matrix are simply the inverses of the Luttinger liquid parameters $K_{c}^{-1}$ and $K_{s}^{-1}$. If one rewrites the wavefunction in terms of individual electron coordinates $\rho _{\sigma
}(x)=\sum_{i,\sigma }\delta (x-x_{i})$ (the sum is restricted to particles with spin $\sigma $) and if one sets $z=\exp i\frac{2\pi }{L}x$, one easily finds: $$\label{multiboson}
\Psi _{0}(\{x_{i},\sigma _{i}\})=\prod_{i<j}\left| z_{i}-z_{j}\right|
^{g_{\sigma _{i}\sigma _{j}}}.$$ The wavefunction is bosonic; for the fermionic LL, one undoes the Jordan-Wigner transformation which leads to: $$\Psi _{F,0}(\{x_{i},\sigma _{i}\})=\prod_{i<j}\left| z_{i}-z_{j}\right|
^{g_{\sigma _{i}\sigma _{j}}}$$ $$\label{multifermion}
\times \prod_{i<j}\left\{ \left( \frac{(z_{i}-z_{j})}{\left|
z_{i}-z_{j}\right| }\right) ^{\delta _{\sigma _{i}\sigma _{j}}}\exp i\frac{%
\pi }{2}sgn(\sigma _{i}-\sigma _{j})\right\} .$$ (The antisymmetrizing factor [consists]{} of two parts, one which ensures [that particles of the same species anticommute]{}, and a second part known as a Klein factor which allows antisymmetry for particles of different spin.) Let us redefine the matrix elements of $\widehat{g}$ per:
$$\begin{aligned}
g_{\sigma \sigma ^{\prime }} &=&\left(
\begin{tabular}{ll}
$\lambda $ & $\mu $ \\
$\mu $ & $\lambda $%
\end{tabular}
\right) \\
K_{c}^{-1} &=&\lambda +\mu \\
K_{s}^{-1} &=&\lambda -\mu\end{aligned}$$
If we denote the coordinates of particles with spin $\uparrow $ and $%
\downarrow $ respectively by $u$ and $v$ then the ground state can be rewritten as (for convenience the antisymmetrizing factor is omitted):
$$\Psi _{0}(\{u_{i},v_{i}\})=\prod_{i<j}\left| u_{i}-u_{j}\right| ^{\lambda
}\prod_{i<j}\left| v_{i}-v_{j}\right| ^{\lambda }\prod_{i,j}\left|
u_{i}-v_{j}\right| ^{\mu },$$
$$\begin{aligned}
\Psi _{c} &=&\left[ \prod_{i<j}\left| u_{i}-u_{j}\right| \left|
v_{i}-v_{j}\right| \left| u_{i}-v_{j}\right| \right] ^{1/2K_{c}} \nonumber
\\
&=&\prod_{i<j}\left| z_{i}-z_{j}\right| ^{1/2K_{c}}, \label{psi-c}\end{aligned}$$
$$\begin{aligned}
\Psi _{s} &=&\left[ \prod_{i<j}\left| u_{i}-u_{j}\right| \left|
v_{i}-v_{j}\right| /\left| u_{i}-v_{j}\right| \right] ^{1/2K_{s}} \nonumber
\\
&=&\prod_{i<j}\left| z_{i}-z_{j}\right| ^{\sigma _{i}\sigma _{j}/2K_{s}}.
\label{psi-s}\end{aligned}$$
For the fermionic LL the charge part gets an additional factor $%
\left( \frac{(x_{i}-x_{j})}{\left| x_{i}-x_{j}\right| }\right) ^{1/2}$ and the spin part, a factor $%
\left( \frac{(x_{i}-x_{j})}{\left| x_{i}-x_{j}\right| }\right) ^{\sigma
_{i}\sigma _{j}/2}\exp i\frac{\pi }{2}sgn(\sigma _{i}-\sigma _{j})$.
These are 1D Laughlin multi-component wavefunctions. In 2D they are known as Halperin wavefunctions which describe multi-component systems of the FQHE[@halperin]. In that context the $\widehat{g}$ matrix is known as Wen’s topological $K$ matrix[@wen]. The main difference between the two matrices is that the the entries of the $K$ matrix are integers while $%
\widehat{g}$ matrix elements are arbitrary real numbers ([only constrained to yield positive and real eigenvalues]{}). The $\widehat{g}$ matrix does not allow for a topological interpretation either since there is no topological quantization as in the FQHE. We will call the $\widehat{g}$ matrix, the [*charge matrix*]{} because it corresponds to the couplings between particles in the plasma analogy (see eq.(\[plasma\]))[@f14].
### Elementary excitations : the holon, the spinon, Laughlin quasiparticles.
We generalize the approach followed for the spinless LL. There is a decoupling of the dynamics at two levels: chiral separation as well as spin-charge separation. In particular both the charge and the spin hamiltonians - $H_{c}$ and $H_{s}$ - display chiral separation: $%
H_{c}=H_{c+}+H_{c-}$ and $H_{s}=H_{s+}+H_{s-}$ where the four hamiltonians all commute ($\left[ H_{c/s\pm },H_{c/s\pm }\right] =0$). The following operators create the exact eigenstates of the relevant chiral hamiltonians: $$V_{\tau }^{\pm }(Q_{\tau ,\pm },q)=\int dx\exp iqx\exp -i\sqrt{\pi /2}%
Q_{\tau ,\pm }\Theta _{\tau ,\pm }$$ $$\begin{aligned}
\Theta _{\tau ,\pm } &=&\Theta _{\tau }\mp \Phi _{\tau }/K_{\tau };\;\tau
=c,s, \\
q &=&\frac{2\pi n}{L}\mp \frac{2\pi }{L}\frac{Q_{\tau ,\pm }^{2}}{K_{\tau }},
\\
Q_{\tau ,\pm } &=&\frac{Q_{\uparrow }+\tau Q_{\downarrow }}{2}\pm K_{\tau }%
\frac{J_{\uparrow }+\tau J_{\downarrow }}{2}.\end{aligned}$$ (The square root $\sqrt{2}$ in the exponential comes from the normalization of the charge and spin fields; $c$ and $s$ index charge and spin respectively; $\tau =\pm 1$ for charge and spin respectively.) One easily checks that: $$\left[ \widehat{Q}_{\uparrow }+\tau \widehat{Q}_{\downarrow },V^{\pm
}(Q_{\tau ^{\prime },\pm })\right] =\delta _{\tau \tau ^{\prime }}Q_{\tau
,\pm }V^{\pm }(Q_{\tau ^{\prime },\pm }).$$ This implies that these excitations either carry a charge $Q=Q_{c,\pm }$ but then have no spin (the operators $V_{c}^{\pm }$), or that they have a spin $%
S_{z}=S_{\pm }=Q_{s,\pm }/2$ but no charge (operators $V_{s}^{\pm }$).As expected the fractional states come in two brands: the first corresponds to charge excitations and the second to spin excitations. Hereafter we will note the charge and the spins of these excitations as
$$\begin{aligned}
Q_{\pm } &=&Q_{c\pm }, \\
S_{\pm } &=&\frac{Q_{s,\pm }}{2}.\end{aligned}$$
Because of the obvious relevance to physical systems we focus first on elementary excitations for a fermionic LL; we will consider the case of a bosonic LL later in section (\[ch-33\]). Solving the constraints on the charge and current which again obey the selection rule $Q_{\uparrow
}-J_{\uparrow }=2n_{\uparrow }$ and $Q_{\downarrow }-J_{\downarrow
}=2n_{\downarrow }$ ( the $n$ are integers), we find: $$\begin{aligned}
\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$ \\
$S_{+}$ \\
$S_{-}$%
\end{tabular}
\right) &=&n_{\uparrow }\left(
\begin{tabular}{l}
$1$ \\
$1$ \\
$1/2$ \\
$1/2$%
\end{tabular}
\right) +n_{\downarrow }\left(
\begin{tabular}{l}
$1$ \\
$1$ \\
$-1/2$ \\
$-1/2$%
\end{tabular}
\right) \nonumber \\
&&+J_{\uparrow }\left(
\begin{tabular}{l}
$\frac{1+K_{c}}{2}$ \\
$\frac{1-K_{c}}{2}$ \\
$\frac{1+K_{s}}{4}$ \\
$\frac{1-K_{s}}{4}$%
\end{tabular}
\right) +J_{\downarrow }\left(
\begin{tabular}{l}
$\frac{1+K_{c}}{2}$ \\
$\frac{1-K_{c}}{2}$ \\
$-\frac{1+K_{s}}{4}$ \\
$-\frac{1-K_{s}}{4}$%
\end{tabular}
\right) \label{states}\end{aligned}$$ This compact equation must be read as follows. Each entry represents a fractional excitation; the first two lines are charge spinless excitations, while the last two lines represent spin excitations. For instance the entry $%
Q_{+}$ is associated with a fractional excitation with charge $Q=Q_{+}$ which carries no spin, and propagates in the right direction: therefore $%
\frac{1+K_{c}}{2}$ in the first line means a spinless state with charge $Q=%
\frac{1+K_{c}}{2}$ going to the right. Likewise the second line characterizes charge excitations propagating to the left. The line $S_{+}$ means that the states have no charge, a spin component $S_{z}=S_{+}$ and propagate to the right: for instance $1/2$ is a spin one-half fractional state. Each line gives the decomposition of a given fractional excitation into elementary excitations: for instance the $Q_{+}$ excitation is made up of $n_{\uparrow }+n_{\downarrow }$ excitations $V_{c}^{+}(Q=1)$, and $%
J_{\uparrow }+J_{\downarrow }$ excitations $V_{c}^{+}(Q=\frac{1+K_{c}}{2})$. The previous equation summarizes the selection rules which are obeyed by the elementary excitations.
Let us give an example. Suppose one adds a spin up electron at the Fermi level in the Luttinger liquid. This is a $Q_{\uparrow }=1=J_{\uparrow }$ and $Q_{\downarrow }=0=J_{\downarrow }$ excitation or in terms of $n_{\uparrow }$ and $n_{\downarrow }$, this is a ($n_{\uparrow }=0$, $n_{\downarrow }=0,$ $%
J_{\uparrow }=1,$ $J_{\downarrow }=0$) state. Equation (\[states\]) shows that the spinor $$\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$ \\
$S_{+}$ \\
$S_{-}$%
\end{tabular}
\right) =\left(
\begin{tabular}{l}
$\frac{1+K_{c}}{2}$ \\
$\frac{1-K_{c}}{2}$ \\
$\frac{1+K_{s}}{4}$ \\
$\frac{1-K_{s}}{4}$%
\end{tabular}
\right)$$ is created; this means that the spin up electron added at the Fermi level $%
k_{F}$ splits into four fractional states: a charge $\frac{1+K_{c}}{2%
}$ anyon propagating at velocity $u_{c}$; this state has no spin; a second charge anyon with charge $\frac{1-K_{c}}{2}$ and velocity $-u_{c}$ and then two spin anyons with velocities $u_{s}$ and $-u_{s}$ and respective spin $S_{z}=\frac{1\pm K_{s}}{4}$. In the special case of spin rotational invariance ($K_{s}=1$) there is only one spin anyon: the spinon with spin $S_{z}=1/2$ which propagates to the right with velocity $u_{s}$ ( or to the left with velocity $-u_{s}$ if the electron had been added at the left Fermi point $-k_{F}$). Likewise, if $K_{c}=1$, there is a single charge state, which has charge $Q=1$. In the non-interacting case, the charge velocity $u_{c}$ and the spin velocity $u_{s}$ are equal and therefore the spinon and the charge $1$ state subsume into a single state since they move in the same direction with the same velocity: we have just recovered the spin up electron.
#### Holon and spinon.
Let us identify the content of the elementary excitations, starting with $%
V_{c}^{\pm }(Q=1)$ and $V_{s}^{\pm }(S_{z}=1/2)$. The charge and the spin carried by these fractional excitations make it reasonable to interpret them as the holon and the spinon respectively.
The physical processe involved in the creation of each of these states confirms this identification. Indeed the minimal operation which involves $%
V_{c}^{\pm }(Q=1)$ is obtained when $J_{\uparrow }=0=J_{\downarrow }$ and set $n_{\uparrow }=1=n_{\downarrow }$ in equation (\[states\]). This is an excitation for which $Q_{\uparrow }=1=Q_{\downarrow }$ and $J_{\uparrow
}=0=J_{\downarrow }$ which means that this is a pure charge process (no spin variation $S_{z}=(Q_{\uparrow }-Q_{\downarrow })/2=0$, no spin current nor charge current). $V_{c}^{\pm }(Q=1)$ is an excitation associated with the addition of charge in the LL. All [the transitions in Fock space which occur after adding a charge to the ground state therefore involve $V_{c}^{\pm }(Q=1).$ It is then consistent to identify $V_{c}^{\pm }(Q=1)$ as the holon[@holon-charge]]{}.
Likewise the minimal excitation generating $V_{s}^{\pm }(S_{z}=1/2)$ is a spin one transition which is a pure spin process. All excitations for which there is a spin flip will therefore create $V_{s}^{\pm }(S_{z}=1/2)$ (in pairs). This is what we expect from a spinon.
Notice that both for the holon and spinon there are even-odd effects arising in the low-energy gaussian theory. Indeed, equation (\[states\]) shows that an excitation with includes a spin one-half transition will not create a spinon (we have instead a ”hybrid” spin excitation): one needs at least a spin one transition to crate a spinon. This means that adding a single electron does not create a spinon: [even number of electrons are required]{}. [This makes sense ]{} since for a spin chain the minimal spin excitation is also a spin-flip which involves two electrons and not just one. The same behaviour is observed with the holon: the minimal process which creates it [adds two electrons]{} ($Q=2$ since $%
Q_{\uparrow }=1=Q_{\downarrow }$). These even-odd effects are a direct consequence of statistics and would not be observed with a bosonic two component LL.
With eq.(\[chargespin\]), we find that the wavefunctions for the holon and the spinon are simply (with $z=\exp i\frac{2\pi }{L}x$): $$\begin{aligned}
\Psi _{holon}(z_{0}) &=&V_{c}^{+}(Q_{c}=1,z_{0})\Psi _{c} \nonumber \\
&=&\prod_{i}\left( \frac{(x_{i}-x_{0})}{\left| x_{i}-x_{0}\right| }\right) ^{%
\frac{1}{2K_{c}}}\prod_{i}\left| z_{i}-z_{0}\right| ^{1/2K_{c}} \nonumber\end{aligned}$$
$$\begin{aligned}
\label{hoho}
&&\times \prod_{i<j}\left| z_{i}-z_{j}\right|
^{1/2K_{c}}\prod_{x_{0},i<j}\left\{ \frac{(x_{i}-x_{j})}{\left|
x_{i}-x_{j}\right| }\right\} ^{1/2} \\
&=&\prod_{i}\left( z_{i}-z_{0}\right) ^{1/2K_{c}}\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{1/2K_{c}} \nonumber \\
&&\times \exp -i\frac{2k_{F}}{K_{c}}\left( \frac{\sum_{i}x_{i}}{N}%
+x_{0}\right) \prod_{x_{0},i<j}\left\{ \frac{(x_{i}-x_{j})}{\left|
x_{i}-x_{j}\right| }\right\} ^{1/2},\end{aligned}$$
and: $$\begin{aligned}
\label{spispi}
&&\Psi _{spinon}(\sigma _{0},z_{0}) \nonumber \\
&=&V_{s}^{+}(S_{z}=\sigma _{0}/2,z_{0})\Psi _{s} \nonumber \\
&=&\prod_{i}\left( \frac{(x_{i}-x_{0})}{\left| x_{i}-x_{0}\right| }\right)
^{\sigma _{0}\sigma _{i}/2K_{s}}\prod_{i}\left| z_{i}-z_{0}\right| ^{\sigma
_{0}\sigma _{i}/2K_{s}} \nonumber \\
&&\times \prod_{i<j}\left| z_{i}-z_{j}\right| ^{\sigma _{i}\sigma
_{j}/2K_{s}}\prod_{x_{0},i<j}\left\{ \frac{(x_{i}-x_{j})}{\left|
x_{i}-x_{j}\right| }\right\} ^{\sigma _{i}\sigma _{j}/2}.\end{aligned}$$
The holon and the spinon are both anyons with exchange statistics:
$$\begin{aligned}
\theta _{c} &=&\frac{\pi }{2K_{c}}, \\
\theta _{s} &=&\frac{\pi }{2K_{s}}\end{aligned}$$
(the statistics were computed in subsection \[sub1-2\]). [Except for the special case $K_{\tau}=1$, these objects are not semions]{}; in addition, for holons we must also require $v_{c}\neq
v_{s}$ to ensure spin-charge separation. [Contrast our results (eqns (\[hoho\]) and (\[spispi\])) with the commonly used but]{} incorrect charge-spin decoupling of the electron operator[@f15]. The holon and the spinon generalize the dual excitation found for the spinless LL; in the same way, duality transforms the holon and the spinon into the two-component generalizations of the Laughlin quasiparticles. This is a most remarkable yet simple result because it shows that two seemingly unrelated fractional excitations -the holon (or spinon) and the Laughlin quasiparticle- occurring in two very different contexts are actually deeply connected.
Similarly to the spinless LL, in addition to the holon and to the spinon, we hybrid excitations complete the basis of fractional excitations. Their charge and spin are intermediate between those of the holon and spinon and those of their dual excitations, the Laughlin quasiparticles which we discuss now.
#### Laughlin quasiparticles.
We can choose another basis of elementary excitations dual to the previous one which will parametrize the excitations in terms of current processes and electron addition at the Fermi surface. This basis, emphasizing Laughlin quasiparticles as elementary excitations reads: $$\begin{aligned}
\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$ \\
$S_{+}$ \\
$S_{-}$%
\end{tabular}
\right) &=&n_{\uparrow }\left(
\begin{tabular}{l}
$K_{c}$ \\
$-K_{c}$ \\
$K_{s}/2$ \\
$-K_{s}/2$%
\end{tabular}
\right) +n_{\downarrow }\left(
\begin{tabular}{l}
$K_{c}$ \\
$-K_{c}$ \\
$-K_{s}/2$ \\
$K_{s}/2$%
\end{tabular}
\right) \nonumber \\
&&+Q_{\uparrow }\left(
\begin{tabular}{l}
$\frac{1+K_{c}}{2}$ \\
$\frac{1-K_{c}}{2}$ \\
$\frac{1+K_{s}}{4}$ \\
$\frac{1-K_{s}}{4}$%
\end{tabular}
\right) +Q_{\downarrow }\left(
\begin{tabular}{l}
$\frac{1+K_{c}}{2}$ \\
$\frac{1-K_{c}}{2}$ \\
$-\frac{1+K_{s}}{4}$ \\
$-\frac{1-K_{s}}{4}$%
\end{tabular}
\right) , \label{excite}\end{aligned}$$ where again $Q_{\uparrow }-J_{\uparrow }=2n_{\uparrow }$ and $Q_{\downarrow
}-J_{\downarrow }=2n_{\downarrow }.$ In addition to the hybrid quasiparticles $V_{c}^{\pm }(Q=\frac{1\pm K_{c}}{2})$ and $V_{s}^{\pm
}(S_{z}=\frac{1\pm K_{s}}{4})$ which already existed in the previous basis, we have two excitations associated with pure charge current or spin current processes: $V_{c}(Q_{c}=K_{c})$[* *]{}and[* *]{}$V_{s}(S_{z}=K_{s}/2)$[*.* ]{}Under electromagnetic duality the latter are conjugate to the holon and spinon respectively. Actually they are obtained by spin-charge separation of the two-component Laughlin quasiparticle[*,*]{} and we may call them a Laughlin holon and a Laughlin spinon. Let us compute the wavefunctions of these two excitations; one finds: $$\begin{aligned}
\label{creho}
&&V_{c}(Q_{c} =K_{c},z_{0})\Psi _{c} \nonumber \\
&=&\prod_{i}(z_{i}-z_{0})^{1/2} \nonumber\\
&&\times \exp -i2k_{F}\left( \frac{\sum x_{i}}{N}+x_{0}\right) \Psi _{c},\end{aligned}$$ $$\begin{aligned}
&&V_{s}(S^{z}=\sigma _{0}K_{s}/2,z_{0})\Psi _{s} \nonumber \\
&=&\prod_{i}(z_{i}-z_{0})^{\sigma _{i}\sigma _{0}/2} \nonumber \\
&\times & \exp -i2\sigma _{0}(k_{\uparrow }-k_{\downarrow })\left( \frac{\sum
x_{i\uparrow }-x_{j\downarrow }}{M}+x_{0}\right) \Psi _{s} \\
\label{crespi}\end{aligned}$$
In the previous expression $\sigma _{0}$ takes on the values $\pm 1$ and $%
k_{\uparrow },k_{\downarrow }$ are the Fermi vectors associated with particles of spin up and down: $k_{\sigma }=\frac{\pi }{L}N_{\sigma }$ and $%
N=N_{\uparrow }-N_{\downarrow }$, $M=N_{\uparrow }-N_{\downarrow }$. $\Psi
_{c}$ and $\Psi _{s}$ are given for the bosonic LL in eq.(\[psi-c\]) and (\[psi-s\]); for the fermionic LL, there are additional phase factors given in the text following eq.(\[psi-c\]) and (\[psi-s\]). Plasma analogy allows to get the charge and spin of the Laughlin holon eq.(\[creho\]) and spinon eq.(\[crespi\])). [The Laughlin holon and spinon are the stable excitations into which the Laughlin quasiparticle decays as a result of spin-charge separation]{}; [in the localized Wannier basis we have considered throughout, the product of the wavefunctions of the two excitations yields indeed]{}: $$\Psi _{qp}(z_{0},\sigma _{0})=\prod_{i}(z_{i}-z_{0})^{\delta _{\sigma
_{0}\sigma _{i}}}\prod_{i<j}\left| z_{i}-z_{j}\right| ^{g_{\sigma _{i}\sigma
_{j}}},$$ which is just the generalization of the Laughlin quasiparticle to two component systems: when we add spin, the Laughlin quasiparticle comes in two flavours (up or down) and the Laughlin correlation hole acts only on particles of the same flavour. It therefore carries both fractional charge and fractional spin. (It would be spinless if the Laughlin prefactor were $%
\prod_{i}(z_{i}-z_{0})$ instead of $\prod_{i}(z_{i}-z_{0})^{\delta _{\sigma
_{0}\sigma _{i}}}$).
The Laughlin holon and spinon have statistical phases $\theta _{c}=\pi K_{c}$ and $\theta _{s}=\pi K_{s}/2$. For a spin-rotational invariant system, the Laughlin spinon and its dual conjugate - the spinon - are identical states ($%
K_{s}=1$ is the self-dual point for spin excitations).
Luttinger liquid without spin-charge separation. {#ch-32}
------------------------------------------------
### The general LL and the charge matrix.
We now generalize the standard LL theory to include situations with no spin-charge separation. We start from the ground state of the gaussian hamiltonian: $$\Psi _{0}\left[ \widehat{g}\right] =\exp \frac{1}{2}\int \int dxdx^{\prime
}\rho _{\sigma }(x)g_{\sigma \sigma ^{\prime }}\ln \left| \sin \frac{\pi
(x-x^{\prime })}{L}\right| \rho _{\sigma ^{\prime }}(x^{\prime }),$$
$$\begin{aligned}
g_{\sigma \sigma ^{\prime }} &=&\left(
\begin{tabular}{ll}
$\lambda $ & $\mu $ \\
$\mu $ & $\lambda $%
\end{tabular}
\right) , \\
K_{c}^{-1} &=&\lambda +\mu , \\
K_{s}^{-1} &=&\lambda -\mu .\end{aligned}$$
and relax the constraint $g_{\uparrow \uparrow }=g_{\downarrow \downarrow
}$ (while $\widehat{g}$ is kept symmetric). We consider the charge matrix: $%
g_{\sigma \sigma ^{\prime }}=\left(
\begin{tabular}{ll}
$$ & $$ \\
$$ & $\^$%
\end{tabular}
\right) $ and the associated wavefunction $\Psi _{0}\left[ \widehat{g}%
\right] $. We introduce for convenience the eigenvalues of the charge matrix and the unitary matrix $P$ :
$$\begin{aligned}
P^{-1}\widehat{g}P &=&\widehat{D}=\left(
\begin{tabular}{ll}
$1/K_{1}$ & $0$ \\
$0$ & $1/K_{2}$%
\end{tabular}
\right) \\\label{changebaz}
P_{\sigma \tau }P_{\sigma ^{\prime }\tau ^{\prime }}g_{\sigma \sigma
^{\prime }} &=&\frac{\delta _{\tau \tau ^{\prime }}}{K_{\tau }}\;;\quad \tau
=1,2\end{aligned}$$
The normal modes of the charge matrix are simply: $$\label{dansite}
\rho _{\tau }=P_{\sigma \tau }\rho _{\sigma }\Leftrightarrow \rho _{\sigma
}=P_{\sigma \tau }\rho _{\tau }\;\tau =1,2$$ $$\sum_{\sigma \sigma ^{\prime }}\rho _{\sigma }(x)g_{\sigma \sigma ^{\prime
}}\rho _{\sigma ^{\prime }}(x^{\prime })=\sum_{\tau }\rho _{\tau }(x)\frac{1%
}{K_{\tau }}\rho _{\tau }(x^{\prime }).$$ If the charge matrix obeys a $Z_{2}$ symmetry then the normal modes are just the charge and spin density (up to a normalization factor) $\rho _{1}=\rho
_{c}/\sqrt{2}$ and $\rho _{2}=\rho _{s}/\sqrt{2}$). The wavefunction [now reads]{}: $$\begin{aligned}
&&\Psi _{0}[\widehat{g}] \nonumber \\
&=&\exp \frac{1}{2K_{1}}\int \int dxdx^{\prime }\rho _{1}(x)\ln \left| \sin
\frac{\pi (x-x^{\prime })}{L}\right| \rho _{1}(x^{\prime }) \nonumber\end{aligned}$$ $$\times \exp \frac{1}{2K_{2}}\int \int dxdx^{\prime }\rho _{2}(x)\ln \left|
\sin \frac{\pi (x-x^{\prime })}{L}\right| \rho _{2}(x^{\prime }).$$
We have expressed the ground state in this decoupled form because this allows us to directly write down a gaussian hamiltonian with ground state $\Psi _{0}[%
\widehat{g}]$. This generalizes the spin-charge decoupled gaussian theory. We introduce the phase fields associated with the normal densities $%
\rho _{1}$ and $\rho _{2}$$$\label{dansitebis}
\rho _{\tau }=-\frac{1}{\sqrt{\pi }}\partial _{x}\Phi _{\tau },\;j_{\tau }=%
\frac{1}{\sqrt{\pi }}\partial _{x}\Theta _{\tau },$$ $$\left[ \Phi _{\tau }(x),\partial _{x}\Theta _{\tau ^{\prime }}(y)\right]
=i\delta _{\tau \tau ^{\prime }}\delta (x-y).$$ It is then clear that $\Psi _{0}[\widehat{g}]$ is the exact ground state of the following family of two component gaussian hamiltonians for arbitrary velocities $u_{1},u_{2}$: $$\begin{aligned}
\label{hache}
&&H[\widehat{g},u_{1},u_{2}] \nonumber \\
&=&H_{B}[u_{1},K_{1}]+H_{B}[u_{2},K_{2}] \nonumber \\
&=&\sum_{\tau =1,2}\frac{u_{\tau }}{2}\int_{0}^{L}dx\left[ K_{\tau
}^{-1}\left( \partial _{x}\Phi _{\tau }\right) ^{2}+K_{\tau }\left( \partial
_{x}\Theta _{\tau }\right) ^{2}\right] .\end{aligned}$$ The two hamiltonians $H_{B}[u_{1},K_{1}]$ and $H_{B}[u_{2},K_{2}]$ commute by construction. The next section will be devoted to the properties of that generalized LL theory. [Let us stress here]{} the main property of this general LL now: by construction that theory corresponds to [*a generalized separation*]{}: the normal modes will not be charge and spin modes but mix charge and spin in a proportion fixed in time. This will translate for the fractional excitations to states with both fractional charge and fractional spin.
### Main properties.
The compressibility and spin susceptibility are easily computed and one finds: $$\kappa ^{-1}=\frac{1}{L}\frac{\partial ^{2}E_{0}}{\partial \rho _{0}^{2}}=%
\frac{\pi }{4}\sum_{\tau }\frac{u_{\tau }}{K_{\tau }}\left( \sum_{\sigma
}P_{\sigma \tau }\right) ^{2},$$
$$\chi _{s}^{-1}=\frac{1}{L}\frac{\partial ^{2}E_{0}}{\partial \rho _{s}^{2}}=%
\frac{\pi }{4}\sum_{\tau }\frac{u_{\tau }}{K_{\tau }}\left( \sum_{\sigma
}\sigma P_{\sigma \tau }\right) ^{2}.$$
$\rho _{0}$ and $\rho _{s}$ are the charge and spin mean densities, while $%
E_{0}$ is the ground state energy. The Drude peak is: $$D=\frac{1}{L}\frac{\partial ^{2}E_{0}}{\partial \phi ^{2}}=\sum_{\tau
}u_{\tau }K_{\tau }\left( \sum_{\sigma }P_{\sigma \tau }\right) ^{2},$$ where $\phi $ is a flux threading the LL ring of length $L$. This expression can also be recovered using the Kubo formula; one then needs the expression of the current density which one finds with the continuity equation. The current is renormalized as for the spinless LL and the LL with spin-charge separation. [In contrast to the case of the LL with spin-charge separation, here]{} the expression involves both $K_{1}$ and $K_{2}$ because both modes one and two involve charge: $$j_{R}(x)=\sum_{\sigma }\left( \sum_{\sigma ^{\prime },\tau }u_{\tau }K_{\tau
}P_{\sigma \tau }P_{\sigma ^{\prime }\tau }\right) \frac{\partial _{x}\Theta
_{\sigma }}{\sqrt{\pi }}.$$
Anomalous exponents are easily computed as functions of the charge matrix $\widehat{g}$ which leads to compact expressions valid both for the LL with spin-charge separation or for the more general LL; one introduces the Fermi vectors: $k_{F\sigma }=\frac{\pi N_{\sigma }}{L}.$ [For instance density-density correlators are obtained with $H[\widehat{g}]$ and with the bosonization formulas, yielding the static structure factor]{}; the dominant Fourier components are $k=0$, $k=2k_{F\uparrow }$, $%
k=2k_{F\downarrow }$ and $k=2k_{F\uparrow }+2k_{F\downarrow }$. Near $k=0$:
$$\begin{aligned}
&<&\delta \rho _{\sigma }(0)\delta \rho _{\sigma ^{\prime }}(x)>_{k=0}=\frac{%
\widehat{g^{-1}}_{\sigma \sigma ^{\prime }}}{2(\pi x)^{2}}, \\
&<&\delta \rho (0)\delta \rho (x)>=\frac{A_{k=0}}{(\pi x)^{2}}%
;\;A_{k=0}=\sum_{\sigma \sigma ^{\prime }}\frac{\widehat{g^{-1}}_{\sigma
\sigma ^{\prime }}}{2}.\end{aligned}$$
For the higher harmonics one includes a mode at $2k_{F\uparrow
}+2k_{F\downarrow }=2\pi \rho $ (which appears in the Hubbard model in a magnetic field):
$$\begin{aligned}
&<&\delta \rho (0)\delta \rho (x)>=\frac{A_{k=0}}{(\pi x)^{2}}+a_{\uparrow }%
\frac{\cos (2k_{F\uparrow }x)}{x^{2+\alpha (2k_{F\uparrow })}}+a_{\downarrow
}\frac{\cos 2k_{F\downarrow }x}{x^{2+\alpha (2k_{F\downarrow })}} \nonumber
\\
&&+b\frac{\cos (2k_{F\uparrow }+2k_{F\downarrow })x}{x^{2+\alpha
(2k_{F\uparrow }+2k_{F\downarrow })}},\end{aligned}$$
$$\begin{aligned}
2+\alpha (2k_{F\sigma }) &=&2\widehat{g}_{\sigma \sigma }^{-1}, \\
2+\alpha (2k_{F\uparrow }+2k_{F\downarrow }) &=&2\sum_{\sigma \sigma
^{\prime }}\widehat{g^{-1}}_{\sigma \sigma ^{\prime }}.\end{aligned}$$
$A_{k=0}$ is fixed in the low-energy limit but the other constants $%
a_{\uparrow }$,$a_{\downarrow },b$ are non-universal and depend on high-energy processes. When there is spin-charge separation, the exponent for $4k_{F}$ oscillations and the constant $A_{k=0}$ are related by the equation $2+\alpha (4k_{F})=4A_{k=0}$; [ in the general case,]{} we find: $$2+\alpha (2k_{F\uparrow }+2k_{F\downarrow })=4A_{k=0}$$
The derivation of the exponents is done in exactly the same manner as in the spin-charge separated LL.
Spin-spin correlation functions are: $$\begin{aligned}
&<&S_{z}(0)S_{z}(x)>=\frac{\sum_{\sigma \sigma ^{\prime }}\left( \sigma
\sigma ^{\prime }\widehat{g^{-1}}_{\sigma \sigma ^{\prime }}\right) }{2(\pi
x)^{2}} \nonumber \\
&&+\sum_{\sigma }\frac{\cos 2k_{\sigma }x}{\left| x\right| ^{2\widehat{g}%
_{\sigma \sigma }^{-1}}},\end{aligned}$$
$$\begin{aligned}
&<&S^{+}(0)S^{-}(x)>=\frac{\cos (k_{F\uparrow }+k_{F\downarrow })x}{\left|
x\right| ^{\gamma }}, \\
\gamma &=&\left[ g_{\uparrow \downarrow }^{-1}-g_{\uparrow \downarrow }+%
\frac{1}{2}\sum_{\sigma }(\widehat{g}^{-1}+\widehat{g})_{\sigma \sigma
}\right] .\end{aligned}$$
Electronic Green functions decay as: $$<\Psi _{\sigma }(0)\Psi _{\sigma }^{+}(x)>=\frac{\exp ik_{\sigma }x}{\left|
x\right| ^{1+\alpha (\sigma )}};\;1+\alpha _{F}(\sigma )=\frac{1}{2}(%
\widehat{g}+\widehat{g}^{-1})_{\sigma \sigma }.$$ For bosons the exponent is modified as: $1+\alpha _{B}(\sigma )=\frac{1}{2}%
\widehat{g}_{\sigma \sigma }.$ These exponents are derived with the bosonization formulas but can also be found by plasma analogy.
### The charge matrix: a summary.
For the two-component LL there are three interesting situations which we summarize below:
i\) in the general case, the (symmetric) $\widehat{g}$ matrix has arbitrary entries; there is no spin-charge separation but a more general two-mode separation: $$\widehat{g}=\left(
\begin{array}{ll}
\lambda & \mu \\
\mu & \lambda ^{\prime }
\end{array}
\right) .$$ As will be shown below, the Hubbard model in a magnetic field can be described by such a theory.
ii\) the $\widehat{g}$ matrix has a $Z_{2}$ symmetry; [this case pertains to]{} spin-charge separation: $$\widehat{g}=\left(
\begin{array}{ll}
\lambda & \mu \\
\mu & \lambda
\end{array}
\right) ,\;K_{\rho }=\frac{1}{\lambda +\mu },\;K_{\sigma }=\frac{1}{\lambda
-\mu }$$ Indeed the symmetry under the exchange of up and down spins implies that the normal modes of the charge matrix are just the charge and spin modes. The LL parameters are then the eigenvalues of the inverse of the charge matrix. This situation describes models with spin-charge separation but with a spin anisotropy, for instance a Hubbard model to which one would add some Ising term $\sum_{n}S_{z}(n)S_{z}(n+1)$.
iii\) the $\widehat{g}$ matrix corresponds to a $SU(2)$ symmetric case ($K_{\sigma }=1$): $$\widehat{g}=\left(
\begin{array}{ll}
\mu +1 & \mu \\
\mu & \mu +1
\end{array}
\right) ,\qquad K_{\rho }=\frac{1}{2\mu +1},K_{\sigma }=1$$ This situation describes the low-energy limit of the Hubbard model. It is noteworthy that the wavefunctions $\Psi \left[ \widehat{g}\right] $ for that sub-case were used in a variational approach of the 1D $t-J$ model giving very good results although it was not realized they were the exact ground states of the gaussian model[@hel]. [The reason why it is so is now transparent.]{}
Elementary excitations for the generalized LL. {#ch-33}
----------------------------------------------
We now consider the excitations of the general bosonic and fermionic LL with or without spin-charge separation. Let us inject particles in the Luttinger liquid. In real space this is described by the operator: $$V(x)=\exp -i\sqrt{\pi }\sum_{\sigma }\left( Q_{\sigma }\Theta _{\sigma
}(x)-J_{\sigma }\Phi _{\sigma }(x)\right) .$$ Fractionalization stems from two decouplings: chiral separation and a separation for the internal quantum number generalizing spin-charge separation. In terms of the normal modes fields $\Theta _{\tau }$ and $\Phi
_{\tau }$ $(\tau =1,2$): $$V=\exp -i\sqrt{\pi }\sum_{\tau }\left( (\sum_{\sigma }P_{\sigma \tau
}Q_{\sigma })\Theta _{\tau }-(\sum_{\sigma }P_{\sigma \tau }J_{\sigma })\Phi
_{\tau }\right) .$$ The chiral fields are: $$\Theta _{\tau ,\pm }(x)=\Theta _{\tau }(x)\mp \Phi _{\tau }(x)/K_{\tau },$$ and therefore: $$V(x)=\prod_{\tau ,\pm }\exp -i\sqrt{\pi }Q_{\tau ,\pm }\Theta _{\tau ,\pm
}(x).$$ This expression explicitly shows a decoupling into four components. We have defined in the above the chiral charges: $$Q_{\tau ,\pm }=\frac{1}{2}\left[ (\sum_{\sigma }P_{\sigma \tau }Q_{\sigma
})\pm K_{\tau }(\sum_{\sigma }P_{\sigma \tau }J_{\sigma })\right] .$$ The following operators are exact eigenstates of each chiral hamiltonian $H_{\pm ,\tau }$ $\tau =1,2$:
$$\begin{aligned}
V_{\tau }^{\pm }(Q_{\tau ,\pm },q) &=&\int dx\exp iqx\exp -i\sqrt{\pi }%
Q_{\tau ,\pm }\Theta _{\tau ,\pm }(x), \\
q &=&\frac{2\pi n}{L}\mp \frac{2\pi }{L}\frac{Q_{\tau ,\pm }^{2}}{K_{\tau }}.\end{aligned}$$
The chiral charges correspond to the charge and spin carried by each of these excitations up to a normalization factor: $$\left[ \widehat{Q}_{\sigma },V_{\tau }^{\pm }(Q_{\tau ,\pm },q)\right]
=Q_{\tau ,\pm }P_{\sigma \tau }V_{\tau }^{\pm }(Q_{\tau ,\pm },q),$$ which implies that the charge and spin of $V_{\tau }^{\pm }(Q_{\tau ,\pm
},q) $ are:
$$\begin{aligned}
Q &=&Q_{\tau ,\pm }\left( \sum_{\sigma }P_{\sigma \tau }\right) , \label{c1}
\\
S_{z} &=&Q_{\tau ,\pm }\left( \frac{1}{2}\sum_{\sigma }\sigma P_{\sigma \tau
}\right) . \label{c2}\end{aligned}$$
Thus for an arbitrary charge matrix, fractional excitations carry both charge and spin. However the ratio of charge to spin is constant for each given mode $\tau =1,2$: $\left( \frac{1}{2}\sum_{\sigma }\sigma
P_{\sigma \tau }\right) Q=S_{z}\left( \sum_{\sigma }P_{\sigma \tau }\right) $; of course the phonons associated to each mode mix charge and spin in exactly the same proportions since:
$$\begin{aligned}
\rho _{\tau }(x) &=&\left( \frac{1}{2}\sum_{\sigma }P_{\sigma \tau }\right)
\rho _{c}(x)+\left( \frac{1}{2}\sum_{\sigma }\sigma P_{\sigma \tau }\right)
\rho _{s}(x) \nonumber \\
&=&\left( \frac{1}{2}\sum_{\sigma }P_{\sigma \tau }\right) \rho
_{c}(x)+\left( \sum_{\sigma }P_{\sigma \tau }\right) s_{z}(x)\end{aligned}$$
where $s_{z}(x)$ is a spin density.
It is convenient to define the charge to spin ratio: $$r=\frac{Q}{2S_{z}},$$ for each mode. Unitary implies that if for the first mode: $$r=\frac{Q}{2S_{z}}=p$$ then for the second mode: $$r=\frac{Q}{2S_{z}}=-\frac{1}{p}.$$ where $p$ is arbitrary. Note that for a Fermi liquid these ratios are $r=\pm 1$ (we are characterizing Landau quasiparticles (or holes) of either spin) and when spin-charge separation is realized the ratio is either $r=0$ or $r=\pm \infty $.
Let us give the elementary excitations. The simplest case is that of bosons: $$Q_{\tau ,\pm }=Q_{\uparrow }\left( \frac{P_{\uparrow \tau }}{2}\right)
+Q_{\downarrow }\left( \frac{P_{\downarrow \tau }}{2}\right)$$ $$+\frac{J_{\uparrow }}{2}\left( \pm P_{\uparrow \tau }K_{\tau }\right) +\frac{%
J_{\downarrow }}{2}\left( \pm P_{\downarrow \tau }K_{\tau }\right) .$$
To simplify the notation, we have only written a single line, but $Q_{\tau ,\pm
} $ and the other entries should be read as four-vectors. $Q_{\sigma }$ and $%
\frac{J_{\sigma }}{2}$ are arbitrary independent integers, which shows that the states $V_{\tau }^{\pm }(\widetilde{Q}_{\tau ,\pm },q)$ where $%
\widetilde{Q}_{\tau ,\pm }=\frac{P_{\uparrow \tau }}{2},\frac{P_{\downarrow
\tau }}{2},\pm P_{\uparrow \tau }K_{\tau }$ or $\pm P_{\downarrow \tau
}K_{\tau }$ are elementary excitations. As an illustration let us consider the simple case of a $Z_{2}$ symmetric charge matrix for bosons which have a pseudo-spin index. The unitary matrix $P$ is: $$P_{\sigma \tau }=\left(
\begin{tabular}{ll}
$1/\sqrt{2}$ & $1/\sqrt{2}$ \\
$1/\sqrt{2}$ & $-1/\sqrt{2}$%
\end{tabular}
\right)$$ Then it follows from eq.(\[c1\]) and (\[c2\]) that for mode $\tau =1$ (the charge mode) $Q=\sqrt{2}Q_{\tau ,\pm }$ and $S_{z}=0$; for mode $\tau
=2 $ $Q=0$ but $S_{z}=Q_{\tau ,\pm }/\sqrt{2}$. If we take these normalizations into account: $$\begin{aligned}
\left(
\begin{tabular}{l}
$Q_{+}$ \\
$Q_{-}$ \\
$S_{+}$ \\
$S_{-}$%
\end{tabular}
\right) &=&Q_{\uparrow }\left(
\begin{tabular}{l}
$1/2$ \\
$1/2$ \\
$1/4$ \\
$1/4$%
\end{tabular}
\right) +Q_{\downarrow }\left(
\begin{tabular}{l}
$1/2$ \\
$1/2$ \\
$-1/4$ \\
$-1/4$%
\end{tabular}
\right) \nonumber \\
&&+\frac{J_{\uparrow }}{2}\left(
\begin{tabular}{l}
$K_{c}$ \\
$-K_{c}$ \\
$K_{s}/2$ \\
$-K_{s}/2$%
\end{tabular}
\right) +\frac{J_{\downarrow }}{2}\left(
\begin{tabular}{l}
$K_{c}$ \\
$-K_{c}$ \\
$-K_{s}/2$ \\
$K_{s}/2$%
\end{tabular}
\right)\end{aligned}$$
Once again we find a charge $1/2$ particle and a charge $K_{c}$ Laughlin quasiparticle as for the bosonic spinless LL. But in addition we find new states resulting from a fractionalization of ”pseudo-spin” for the bosons (half-spinons for instance).
For the fermionic LL the elementary excitations are obtained by the equation: $$\begin{aligned}
Q_{\tau ,\pm } &=&Q_{\uparrow }\left( P_{\uparrow \tau }\frac{1\pm K_{\tau }%
}{2}\right) +Q_{\downarrow }\left( P_{\downarrow \tau }\frac{1\pm K_{\tau }}{%
2}\right) \nonumber \\
&&+n_{\uparrow }\left( \mp P_{\uparrow \tau }K_{\tau }\right) +n_{\downarrow
}\left( \mp P_{\downarrow \tau }K_{\tau }\right) ,\end{aligned}$$ where we have resolved the constraint: $Q_{\sigma }-J_{\sigma }=2n_{\sigma }$. [This fully characterizes the low-energy elementary excitations of a LL in a magnetic field (see below)]{}.
Application to the Hubbard model. {#ch-34}
---------------------------------
To illustrate the previous results we discuss the Hubbard model in one dimension. The model was solved exactly by Bethe Ansatz by Lieb and Wu. [In zero magnetic field, for repulsive ($U>0$) interactions, a LL metallic phase exists both for weak and strong coupling, except at half-filling.]{} For very large $U$ the spin-charge decoupling is valid at all energy scales. This was shown by Ogata and Shiba who also found that the Bethe Ansatz ground state then took a remarkable factorized form[@ogata]: it is the product of a charge part (a Slater determinant for free fermions involving all electrons ) and a Bethe wavefunction similar to that of the Heisenberg model on a reduced lattice from which one has removed the holes $$\begin{aligned}
&&\Psi _{Hubbard}(x_{i},\sigma _{i}) \nonumber \\
&=&\det (\exp ik_{j}r_{i},\left| k_{j}\right| \leq k_{F})\;\Psi
_{Heisenberg}(y_{i},\sigma _{i})\end{aligned}$$ ($y_{i}$ is the coordinate in the reduced lattice of particle $i$ whose real position is $x_{i}$).
It is instructive to compare it to the two-component Jastrow wavefunctions which are also explicitly spin-charge decoupled. The Slater determinant is rewritten as (in terms of the circular coordinates $z$): $$\Psi _{Hubbard}(x_{i},\sigma _{i})=\prod_{i<j}\left( z_{i}-z_{j}\right)
\;\Psi _{Heisenberg}(y_{i},\sigma _{i})$$ This is to be compared with: $$\begin{aligned}
\Psi &=&\prod_{i<j}\left| z_{i}-z_{j}\right| ^{1/2K_{c}}\;\prod_{i<j}\left|
z_{i}-z_{j}\right| ^{\sigma _{i}\sigma _{j}/2K_{s}}\; \nonumber \\
&&\prod_{i<j}\left\{ \left( \frac{(z_{i}-z_{j})}{\left| z_{i}-z_{j}\right| }%
\right) ^{\delta _{\sigma _{i}\sigma _{j}}}\exp i\frac{\pi }{2}sgn(\sigma
_{i}-\sigma _{j})\right\}\end{aligned}$$ Or if we separate spin and charge:
$$\begin{aligned}
\Psi _{c} &=&\prod_{i<j}\left[ \left| z_{i}-z_{j}\right| ^{1/2K_{c}}\left(
\frac{(z_{i}-z_{j})}{\left| z_{i}-z_{j}\right| }\right) ^{1/2}\right] , \\
\Psi _{s} &=&\prod_{i<j}[\left| z_{i}-z_{j}\right| ^{\sigma _{i}\sigma
_{j}/2K_{s}} \nonumber \\
&&\times \left( \frac{(z_{i}-z_{j})}{\left| z_{i}-z_{j}\right| }\right)
^{\sigma _{i}\sigma _{j}/2}\exp i\frac{\pi }{2}sgn(\sigma _{i}-\sigma _{j})].\end{aligned}$$
The spin part of the Laughlin ground state is just the Haldane-Shastry wavefunction if $K_{s}=1$ (rotational invariance) which has the same large-distance physics as the Heisenberg ground state. We can also determine $K_{c}$ without any computation by just reading off its value from the wavefunctions: the charge parts of the two wavefunctions coincide if $%
K_{c}=1/2$ which indeed is the known value of the LL parameter for large $U$.
Bethe Ansatz gives the spectrum and the eigenstates; however it is very difficult to compute correlation functions. An important advance came however with the works of Frahm and Korepin who used CFT in conjunction with Bethe Ansatz to compute critical exponents[@frahm]. If a theory is conformally invariant, one can show that the finite-size energies of excitations are directly related to their operator dimension (which is one-half of the anomalous dimension of their correlation function). By using Woynarovich’s Bethe Ansatz calculations for the finite-size spectrum to order $1/L$ which he computed within a so-called ”dressed charge matrix formalism”[@woy], Frahm and Korepin were able to extract critical exponents for the correlation functions of the Hubbard model. In particular they found that in the presence of a magnetic field, spin-charge separation was not realized. Penc and Solyom later showed that in $1/L$ the same spectrum derived by Woynarovich could be expressed in terms of a generalized Tomonaga-Luttinger model with interactions described in the g-ology framework; using equation of motion methods they also derived the anomalous exponents[@penc]. These two approaches [give little insight into the nature]{} of the elementary excitations: how are the holon and spinon modified as a function of microscopic parameters? The description of spin-charge separation (or its absence) is not transparent either: the dressed charge matrix tell us little about spin-charge separation; its changes are not easy to relate to that property. This is to be contrasted with our charge matrix formalism in which spin-charge separation is directly connected to a symmetry of the charge matrix $\widehat{g}$ ($Z_{2}$ symmetry). We will show that the ”dressed charge matrix” of Bethe Ansatz and the charge matrix $\widehat{g}$ are in fact related: the inverse of the symmetric charge matrix is roughly the square of the $Z$ matrix. We will proceed in the following manner: we will show that Woynarovich’s finite-size spectrum is identical to that of our generalized LL. [This yields the charge matrix $\widehat{g}$ in terms of the dressed charge matrix $Z$ and gives us both the anomalous exponents and the fractional excitations]{} since we already derived them for the generalized LL. The relation of our charge matrix formalism to Penc and Solyom g-ology approach is the following: it can be understood as a bosonization of their generalized Tomonaga-Luttinger model; it is much simpler however to work directly within the gaussian hamiltonian framework. Our approach has several advantages in addition to making an explicit contact with the seemingly unrelated physics of Laughlin states: (a) we avoid an ambiguity in the determination of anomalous exponents in Frahm’s and Korepin approach;[@f16] (b), we can give the nature of elementary excitations ( and show that they are fractional states in the first place ) and (c) we are able to give a clear criterion of spin-charge separation.
Woynarovich’s finite-size spectrum in Frahm and Korepin’s notations is the following: $$\begin{aligned}
&&E({ \Delta N},{ D,}N_{c}^{\pm },N_{s}^{\pm })-E_{0} \nonumber \\
&=&\frac{2\pi }{L}\left[
v_{c}(N_{c}^{+}+N_{c}^{-})+v_{s}(N_{s}^{+}+N_{s}^{-})\right] \nonumber \\
&&+\frac{2\pi }{L}\left[ \frac{1}{4}{ \Delta N}^{T}(Z^{-1})^{T}{ V}%
Z^{-1}{ \Delta N+D}^{T}Z{ V}Z^{T}{ D}\right] \nonumber \\
&&+O(\frac{1}{L}),\end{aligned}$$
$$\begin{aligned}
&&P({ \Delta N},{ D,}N_{c}^{\pm },N_{s}^{\pm })-P_{0} \nonumber \\
&=&\frac{2\pi }{L}\left[
v_{c}(N_{c}^{+}-N_{c}^{-})+v_{s}(N_{s}^{+}-N_{s}^{-})\right] \nonumber \\
&&+\frac{2\pi }{L}\left[ { \Delta N}^{T}{ D}\right]
+2D_{c}k_{F\uparrow }+2(D_{c}+D_{s})k_{F\downarrow }.\end{aligned}$$
$k_{F\uparrow }=\frac{2\pi }{L}N_{\uparrow }$ and $k_{F\downarrow }=\frac{%
2\pi }{L}N_{\downarrow }$ are the Fermi momentum for particles of spin up and spin down. The energy and the momentum are those of a state with the (integer) quantum numbers (${ \Delta N},{ D,}N_{c}^{\pm },N_{s}^{\pm
} $); there are two modes indexed by $c$ and $s$: these two modes [*do not*]{} in general correspond to charge and spin. $Z$ is a $2$ by $2$ matrix: $$Z=\left(
\begin{tabular}{ll}
$Z_{cc}$ & $Z_{cs}$ \\
$Z_{sc}$ & $Z_{ss}$%
\end{tabular}
\right)$$ and ${ \Delta N}$ and ${ D}$ are two-vectors: ${ \Delta N=(}%
N_{c}=N_{\uparrow }+N_{\downarrow },N_{s}=N_{\downarrow })$ and ${ D=(}%
D_{c},D_{s})$.[@f17] In these expressions $N_{c/s}^{+-}$ are integers: they are simply the modulus of phonon momenta in units of $2\pi /L$ for the two modes $c$ and $s$; the index $\pm $ refers to the sign of the momentum. The phonon velocities for the two modes are $(v_{c},v_{s})$.
The spectrum of the general gaussian model $H\left[ u_{\tau },\widehat{g}%
\right] $ is: $$\begin{aligned}
&&E(Q_{\sigma },J_{\sigma },N_{\tau }^{\pm }) \nonumber \\
&=&\frac{2\pi }{L}\left[ v_{\tau =1}(N_{\tau =1}^{+}+N_{\tau
=1}^{-})+v_{\tau =2}(N_{\tau =2}^{+}+N_{\tau =2}^{-})\right] \nonumber \\
&&+\frac{\pi }{2L}\sum_{\tau =1,2}v_{\tau }\left( \frac{Q_{\tau }^{2}}{%
K_{\tau }}+K_{\tau }J_{\tau }^{2}\right) , \\
&&P(Q_{\sigma },J_{\sigma },N_{\tau }^{\pm }) \nonumber \\
&=&\frac{2\pi }{L}\left[ (N_{\tau =1}^{+}-N_{\tau =1}^{-})+(N_{\tau
=2}^{+}-N_{\tau =2}^{-})\right] \nonumber \\
&&+\sum_{\sigma =\uparrow \downarrow }\frac{\pi Q_{\sigma }}{L}J_{\sigma
}+k_{F\sigma }J_{\sigma }.\end{aligned}$$ $N_{\tau =1}^{\pm },N_{\tau =2}^{\pm }$ are again the moduli of phonon momenta. The charges and currents $(Q_{\tau },J_{\tau })$ are related to $%
(Q_{\sigma },J_{\sigma })$ by $Q_{\tau }=P_{\sigma \tau }Q_{\sigma }$ and $%
J_{\tau }=P_{\sigma \tau }J_{\sigma }$. We can now identify the parameters of both theories:
$$\begin{aligned}
J_{\uparrow } &=&2D_{c},\;J_{\downarrow }=2(D_{c}+D_{s}),\; \\
Q_{\uparrow } &=&N_{\uparrow },\;Q_{\downarrow }=N_{\downarrow },\;N_{\tau
}^{\pm }=N_{c/s}^{\pm }, \\
v_{\tau } &=&v_{c/s}.\end{aligned}$$
The zero modes can be identified term by term; it is sufficient to consider the current terms to uniquely determine the charge matrix. The charge zero modes yield extra relations which lead to the very same expression for $\widehat{g}$. Indeed expanding the squares gives:
$$\begin{aligned}
K_{1}P_{\uparrow 1}^{2} &=&(Z_{cc}-Z_{sc})^{2}, \nonumber \\
K_{1}P_{\downarrow 1}^{2} &=&(Z_{sc})^{2}, \nonumber \\
K_{1}P_{\uparrow 1}P_{\downarrow 1} &=&(Z_{cc}-Z_{sc})Z_{sc}, \nonumber \\
K_{2}P_{\uparrow 2}^{2} &=&(Z_{cs}-Z_{ss})^{2}, \nonumber \\
K_{2}P_{\downarrow 2}^{2} &=&(Z_{ss})^{2}, \nonumber \\
K_{2}P_{\uparrow 2}P_{\downarrow 2} &=&(Z_{cs}-Z_{ss})Z_{ss}.\end{aligned}$$
Since: $$\widehat{g^{-1}}_{\sigma \sigma ^{\prime }}=\sum_{\tau }K_{\tau }P_{\sigma
\tau }P_{\sigma ^{\prime }\tau }$$ it follows that the inverse of the charge matrix is: $$\begin{aligned}
\widehat{g^{-1}}_{\uparrow \uparrow }
&=&(Z_{cc}-Z_{sc})^{2}+(Z_{cs}-Z_{ss})^{2} \nonumber \\
\widehat{g^{-1}}_{\downarrow \downarrow } &=&(Z_{sc})^{2}+(Z_{ss})^{2}
\nonumber \\
\widehat{g^{-1}}_{\uparrow \downarrow } &=&\widehat{g^{-1}}_{\downarrow
\uparrow }=(Z_{cc}-Z_{sc})Z_{sc}+(Z_{cs}-Z_{ss})Z_{ss}\end{aligned}$$ We define the matrix $\widetilde{Z}$ obtained from the dressed charge matrix $Z$ by subtracting the second line from the first: $$\widetilde{Z}=\left(
\begin{tabular}{ll}
$Z_{cc}-Z_{sc}$ & $Z_{cs}-Z_{ss}$ \\
$Z_{sc}$ & $Z_{ss}$%
\end{tabular}
\right)$$ Then: $$\widehat{g}^{-1}=\widetilde{Z}\widetilde{Z}^{T}$$
[This is the most important result of the present section: the low-energy properties of the Hubbard model are expressed in terms of quantities which can be computed from the microscopic parameters and spin-charge separation simply follows from the $Z_{2}$ symmetry of our charge matrix. In the framework of the dressed charge matrix $Z$ approach, the second feature is not easily decoded from the structure of $Z$ which is then triangular]{} with some relations between its matrix elements whose physical interpretation is quite unclear.[@f18] We can use the results of the previous sections on the elementary excitations and those on the various properties of the charge matrix hamiltonian such as the Drude peak, the susceptibility, the anomalous exponents. In particular the new modes replacing the spin and charge modes are simply the eigenvectors of the charge matrix.
The charge matrix is obtained by inversion: $$\widehat{g}=\frac{1}{\left( \det Z\right) ^{2}}$$ $$\times \left(
\begin{tabular}{ll}
$\widehat{g^{-1}}_{\downarrow \downarrow }$ & $-\widehat{g^{-1}}_{\uparrow
\downarrow }$ \\
$-\widehat{g^{-1}}_{\uparrow \downarrow }$ & $\widehat{g^{-1}}_{\uparrow
\uparrow }$%
\end{tabular}
\right) . \label{g}$$
Term by term identification of the charge zero modes $Q_{\sigma }$ would lead to exactly the same expression for $\widehat{g}$; indeed:
$$\begin{aligned}
\frac{1}{K_{1}}P_{\uparrow 1}^{2} &=&\frac{(Z_{ss})^{2}}{\left( \det
Z\right) ^{2}}, \nonumber \\
\frac{1}{K_{1}}P_{\downarrow 1}^{2} &=&\frac{(Z_{cs}-Z_{ss})^{2}}{\left(
\det Z\right) ^{2}}, \nonumber \\
\frac{1}{K_{1}}P_{\uparrow 1}P_{\downarrow 1} &=&\frac{-(Z_{cs}-Z_{ss})Z_{ss}%
}{\left( \det Z\right) ^{2}}, \nonumber \\
\frac{1}{K_{2}}P_{\uparrow 2}^{2} &=&\frac{(Z_{sc})^{2}}{\left( \det
Z\right) ^{2}}^{2}, \nonumber \\
\frac{1}{K_{2}}P_{\downarrow 2}^{2} &=&\frac{(Z_{cc}-Z_{sc})^{2}}{\left(
\det Z\right) ^{2}}, \nonumber \\
\frac{1}{K_{2}}P_{\uparrow 2}P_{\downarrow 2} &=&\frac{-(Z_{cc}-Z_{sc})Z_{sc}%
}{\left( \det Z\right) ^{2}},\end{aligned}$$
and since $\widehat{g}_{\sigma \sigma ^{\prime }}=\sum_{\tau }K_{\tau
}^{-1}P_{\sigma \tau }P_{\sigma ^{\prime }\tau }$ one recovers equ.(\[g\]). As it should be, one can check that the anomalous exponents predicted for $H\left[ u_{\tau },\widehat{g}\right] $ agree then completely with Frahm and Korepin’s results.
[Let us illustrate these results in two situations, one with spin-charge separation, the other without. From these, we can exhibit the criterion for spin-charge separation within the matrix formalism.]{}
In the presence of spin-charge separation, Frahm and Korepin find that the dressed charge matrix $Z$ is $Z=\left(
\begin{tabular}{ll}
$Z\_[cc]{}=$ & $Z\_[cs]{}=0$ \\
$Z\_[sc]{}=/2$ & $Z\_[ss]{}=1/$%
\end{tabular}
\right) $. This implies that $\widehat{g}$ and its inverse $\widehat{g}%
^{-1}$ are:
$$\begin{aligned}
\widehat{g}^{-1} &=&\left(
\begin{tabular}{ll}
$\frac{\xi ^{2}}{4}+\frac{1}{2}$ & $\frac{\xi ^{2}}{4}-\frac{1}{2}$ \\
$\frac{\xi ^{2}}{4}-\frac{1}{2}$ & $\frac{\xi ^{2}}{4}+\frac{1}{2}$%
\end{tabular}
\right) \\
\widehat{g} &=&\left(
\begin{tabular}{ll}
$\frac{1}{\xi ^{2}}+\frac{1}{2}$ & $\frac{1}{\xi ^{2}}-\frac{1}{2}$ \\
$\frac{1}{\xi ^{2}}-\frac{1}{2}$ & $\frac{1}{\xi ^{2}}+\frac{1}{2}$%
\end{tabular}
\right)\end{aligned}$$
The charge matrix explicitly exhibits spin-charge separation and takes the form characteristic of $SU(2)$ symmetry. The eigenvalues of $\widehat{g}%
^{-1} $ are $K_{c}=\frac{\xi ^{2}}{2}$ and $K_{s}=1$.
The $Z$ matrix can also be explicitly computed in the limit of infinite repulsion with a magnetic field close to the critical field $h_{c}$ for which all the spins are polarized (i.e. close to the ferromagnetic phase). In terms of the parameter $$\delta =\sqrt{\frac{h_{c}-h}{h_{c}}}$$ the dressed charge matrix is: $$Z=\left(
\begin{tabular}{ll}
$1$ & $0$ \\
$\frac{2}{\pi }\delta $ & $1-\frac{1}{\pi }\delta $%
\end{tabular}
\right)$$ which implies that the (inverse of the) charge matrix $\widehat{g}$ is: $$\widehat{g}^{-1} =$$ $$\left(
\begin{tabular}{ll}
$\left( 1-\frac{1}{\pi }\delta \right) ^{2}+\left( 1-\frac{2}{\pi }\delta
\right) ^{2}$ & $\left( 1-\frac{2}{\pi }\delta \right) \left( \frac{2}{\pi }%
\delta \right) -\left( 1-\frac{1}{\pi }\delta \right) ^{2}$ \\
$\left( 1-\frac{2}{\pi }\delta \right) \left( \frac{2}{\pi }\delta \right)
-\left( 1-\frac{1}{\pi }\delta \right) ^{2}$ & $\left( 1-\frac{1}{\pi }%
\delta \right) ^{2}+\left( \frac{2}{\pi }\delta \right) ^{2}$%
\end{tabular}
\right) .$$ This expression shows explicitly the breakdown of spin-charge separation (except for $\delta =\pi /4$).
Conclusions and Perspectives. {#ch-4}
=============================
The main goal of our paper was to establish and describe fractional excitations for the Luttinger liquid within the bosonization scheme. [ bf Bethe Ansatz gives exact eigenstates and shows the existence of some fractional excitations: however their description is quite complex in that framework and it is unclear how to generate systematically a complete set of excitations.]{} [In the low-energy limit, the Luttinger liquid approach allows a very precise characterization of the fractional states already known from exact solutions but what’s more allows us to discover novel fractional]{} excitations which may carry irrational charges (the 1D Laughlin quasiparticle, the hybrid state). In section \[ch-two\] the low-energy spectrum of Luttinger liquids can be reinterpreted in terms of fractional states: for instance, the particle-hole continuum consists of a Laughlin quasiparticles-quasihole continuum. The quasiparticle perspective clarifies many properties of the LL: the renormalization of the current operator is a direct consequence of fractionalization; for spin chains, we present the correct description of the spinon excitation in the generic case of a violation of $SU(2)$ invariance. We also show that the $S_{z}=0$ continuum of spin chains involves the analogs of Laughlin quasiparticles. In section \[ch-3\] we describe fractional excitations such as the holon or the spinon for the Luttinger liquid with spin; we also present a generalization of the gaussian theory valid for Luttinger liquids without spin-charge separation and display in that situation the new fractional states replacing the holon and spinon (see \[ch-33\]).
An important test, of course, would be to observe experimentally (or numerically) all these fractional states. Although the existence of the holon and the spinon was ascertained theoretically quite a long time ago[@lieb] no experiment has yet allowed their detection: in fact, the property of spin-charge separation itself is not yet established experimentally. The observation of two of the fractional states discussed in this paper would be particularly important: the LL Laughlin quasiparticle and the hybrid state. Indeed they may assume irrational charges. The precise spectroscopy of fractional excitations we have done in this paper allows us to determine which processes are involved in their creation: for the Laughlin states, current probes are needed, while the hybrid particle is created by addition of an odd number of electrons. For Laughlin quasiparticles shot noise is likely an adequate probe: the shot noise coefficient for Luttinger liquids can be computed exactly and is predicted to be equal to $K$;[@ludwig] in the two-dimensional electron gas at filling $\nu =1/3$ this yields a charge $1/3$[@glattli]. The latter situation involves Wen’s chiral Luttinger liquid. The identification of the shot noise coefficient with the charge of a carrier has been debated because the coefficient one measures might actually be the conductance rather than a quasiparticle charge (at $\nu =1/3$ the conductance also assumes the value $1/3$). For the non-chiral Luttinger liquid our spectroscopy of fractional states allows to resolve that ambiguity: the shot noise coefficient is indeed identical to the conductance $K$ of the LL but the (backscattering) current-current correlation function measured in shot noise involves charge $K$ excitations, because charge $K$ LL Laughlin quasiparticles are precisely generated by current excitations. These might thus be detected in any physical realization of the Luttinger liquid: quantum wires, or possibly nanotubes. An intriguing possibility is also suggested by recent experiments of tunneling at the edge of a two-dimensional gas in a magnetic field[@grayson]. $I-V$ characteristics measured at the edge showed very surprising non-Fermi liquid behaviour compatible with a chiral LL with unquantized LL parameter $K$; the $I-V$ curves evolve smoothly when one varies the filling fraction and do not show a plateau structure. There seems to be a continuum of Luttinger liquids living at the edge: this caused quite a stir because the chiral LL theory can presumably be derived only for incompressible filling fractions. These puzzling results are sofar unexplained, but if an interpretation in terms of a single-boson mode chiral LL with unquantized parameter $K$ can be in some manner justified, according to the results given in the present paper this would imply that there exists charge $K$ Laughlin quasiparticles in that experimental setting: such a chiral LL is identical to the chiral half of the non-chiral gaussian hamiltonian considered throughout our paper.
The authors wish to acknowledge the late Heinz Schulz for insightful remarks on the results of this paper. We also thank Vincent Pasquier for a discussion on the Calogero-Sutherland model. Bernard Jancovici gave us information on one dimensional plasmas for which we are grateful.
Dispersion of the fractional states.
====================================
We show here that $V_{Q_{\pm }}^{\pm }(q)|\Psi _{0}>$ (where $|\Psi
_{0}>$ is the interacting ground state) is an exact eigenstate of the chiral hamiltonian $H_{\pm }$ with energy: $$E(Q_{\pm },\overline{q_{n}})=\left[ u\left| \overline{q_{n}}\right| +\frac{%
\pi u}{2L}\frac{Q_{\pm }^{2}}{K}\right] .$$
Let us rewrite the state considered $V_{Q_{\pm }}^{\pm }(q_{n})|\Psi _{0}>$ in terms of zero modes and phonon operators (we use equ. (\[theta\]), (\[phi\]), (\[p1\]) and (\[p2\])): $$V_{Q_{\pm }}^{\pm }(x)|\Psi _{0}>=\exp iQ_{\pm }\left[ \mp \frac{2\pi }{K}%
\widehat{Q}_{\pm }x-\sqrt{\pi }\left( \Theta _{0}\mp \frac{\Phi _{0}}{K}%
\right) \right]$$ $$\times \exp -i\sqrt{\pi }Q_{\pm }\sum_{n\neq 0}\Theta _{\pm ,n}\exp i\frac{%
2\pi n}{L}x|\Psi _{0}>$$
Taking into account the fact that the operators $b_{q}$ annihilate the ground state, it follows that: $$V_{Q_{+}}^{+}(x)|\Psi _{0}>=\exp iQ_{+}\left[ -\frac{2\pi }{K}\widehat{Q}%
_{+}x-\sqrt{\pi }\left( \Theta _{0}-\frac{\Phi _{0}}{K}\right) \right]$$ $$\times \exp -i\sqrt{\pi }Q_{+}\sum_{n>0}\sqrt{\frac{L}{K\pi \left| n\right| }%
}b_{n}^{+}\exp i\frac{2\pi n}{L}x|\Psi _{0}>,$$ with a similar expression for $V_{Q_{-}}^{-}(x)|\Psi _{0}>$ (the sum is then over negative momentum phonons). Going back to reciprocal space: $$\begin{aligned}
V_{Q_{\pm }}^{\pm }(q_{n})|\Psi _{0} &>&=\frac{1}{\sqrt{L}}%
\int_{0}^{L}dx\exp -i\frac{2\pi }{L}nx \\
&&\times \exp iQ_{+}\left[ -\sqrt{\pi }\left( \Theta _{0}\mp \frac{\Phi _{0}%
}{K}\right) \right]\end{aligned}$$ $$\times \exp -i\sqrt{\pi }Q_{+}\sum_{\pm p>0}\sqrt{\frac{L}{K\pi \left|
p\right| }}b_{p}^{+}\exp i\frac{2\pi p}{L}x|\Psi _{0}>,$$ which shows that $V_{Q_{\pm }}^{\pm }(q_{n})|\Psi _{0}>$ only spans chiral phonons with momentum $\pm n>0$ ; in other words this state is obtained by the action of the zero mode of the chiral field plus the creation of phonons with [*momenta of the same sign.* ]{}When one expands the phonon exponential, the integral over position will select configuration of phonons with identical total momentum $\overline{q_{n}}=\frac{2\pi n}{L}$. All these configurations consist of phonons of identical chirality and total momentum which means that they are eigenstates with the same eigenvalue of the appropriate chiral hamiltonian ($%
H_{+}$ or $H_{-}$). This is enough to prove that $V_{Q_{\pm }}^{\pm
}(q_{n})|\Psi _{0}>$ is an exact eigenstate of $H_{\pm }$:
$$\begin{aligned}
H_{\pm }V_{Q_{\pm }}^{\pm }(q_{n})|\Psi _{0} &>&=\left[
\sum_{q>0}u|q|a_{q}^{+}a_{q}+\frac{\pi u}{2L}\frac{\widehat{Q}_{\pm }^{2}}{K}%
\right] \\
\times V_{Q_{\pm }}^{\pm }(q_{n})|\Psi _{0} &>& \\
&=&\left[ u\left| \frac{2\pi n}{L}\right| +\frac{\pi u}{2L}\frac{Q_{\pm }^{2}%
}{K}\right] V_{Q_{\pm }}^{\pm }(q_{n})|\Psi _{0}>,\end{aligned}$$
where $\overline{q_{n}}=\frac{2\pi n}{L}$ is the momentum due to phonons.
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The zero mode basis (phonons and $U_{\pm}$ operators) forms a complete family of states: this was proved by Haldane in his computation of the grand canonical partition function of the Tomonaga-Luttinger model[@haldane79]. Therefore $V_{Q,J}$ operators span the Fock space since $%
V_{Q,J} $ operators are products of Haldane’s $U_{\pm}$ operators and of an exponential of phonons. Since the fractional states $V_{Q_{\pm }}^{\pm }$ come from the decay of the $V_{Q,J}$ operators, they, along with the phonons span the Fock space.
This is just the quantization of the superfluid phase: the boson operator is $\Psi _{B}\propto \exp i\sqrt{\pi }%
\Theta $ and therefore $\sqrt{\pi }\left( \Theta (L)-\Theta (0)\right) $ must be an integer multiple of $\pi $. The total current around the ring is defined as the integral $\int_{0}^{L}dx\;j_{R}(x)=\widehat{J}_{R}$; therefore $\widehat{J}_{R}=\frac{uK}{\sqrt{\pi }}\left( \Theta (L)-\Theta
(0)\right) $ and we may then write $\widehat{J}_{R}=uKJ$ where $J$ is an integer.
The previous equation assumes that both $Q$ and $n$ are positive; if that is not so, it suffices to take their absolute value and replace the operators by their conjugate: for instance $Q\longrightarrow
\left| Q\right| $ and $\left[ W_{1/2}^{\pm }(x)\right] \longrightarrow
\left[ W_{-1/2}^{\pm }(x)\right] $.
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Q$ tubes to $Q$ particles, which increases $J$ by $Q$ units). Since $J$ is even for the bosonic LL, $Q+J^{\prime }$ is even for the fermionic LL.
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The $\widehat{g}$ matrix is distinguished from the $K$ matrix to avoid a possible confusion because its eigenvalues are $1/K_{c}$ and $%
1/K_{s}$.
It would be more correct to call this operator an anti-holon since its electrical charge is $Q_{e}=(-e)Q=-e$; but apart from the sign of the charge all fractional particles considered in this paper and their particle-hole conjugates have the same properties. So we call this state a holon.
A naive decoupling of the electron operator would lead to: $$\begin{aligned}
\Psi _{\sigma } &=&\exp -i\sqrt{\pi }\left( \Theta _{\sigma }-\Phi _{\sigma
}\right) \\
&=&\exp -i\sqrt{\pi /2}\left( \Theta _{c}-\Phi _{c}\right) \\
&&\exp -i\sqrt{\pi /2}\left( \Theta _{s}-\Phi _{s}\right)\end{aligned}$$ The first term is sometimes identified as the holon and it has a semionic statistics. The second term is likewise supposed to be the spinon. These identifications are in general incorrect because these states are unstable because of chiral separation (see the anisotropic spin chain). In the special case of $SU(2)$ symmetry, the above expression for the spinon is accidentally correct.
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The ambiguity which is rather technical is the following: there are four kinds of anomalous exponents (two chiralities and two modes). For a given excitation in a given mode, the sum of anomalous exponents for the two chiralities is fixed but their difference is not. [The dimensions are therefore only known up to a constant; to fix their values, Frahm and Korepin make the unproven assumption that they are perfect squares if the excitation corresponds to a primary field.]{}
Note that in Frahm and Korepin’ notations $%
N_{s}=N_{\downarrow }$ is not the spin density.
The charge matrix and its inverse have the same properties, $%
Z_{2}$ symmetry or $SU(2)$ symmetry in the relevant cases.
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|
---
abstract: 'The Lie superalgebra $sl(r+1|s+1)$ admits several inequivalent choices of simple root systems. We have carried out analytic Bethe ansatz for any simple root systems of $sl(r+1|s+1)$. We present transfer matrix eigenvalue formulae in dressed vacuum form, which are expressed as the Young supertableaux with some semistandard-like conditions. These formulae have determinant expressions, which can be viewed as quantum analogue of Jacobi-Trudi and Giambelli formulae for $sl(r+1|s+1)$. We also propose a class of transfer matrix functional relations, which is specialization of Hirota bilinear difference equation. Using the particle-hole transformation, relations among the Bethe ansatz equations for various kinds of simple root systems are discussed.'
author:
- |
Zengo Tsuboi [^1]\
Institute of Physics, University of Tokyo\
Komaba 3-8-1, Meguro-ku, Tokyo 153 Japan
title: 'Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra $sl(r+1|s+1)$'
---
\[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{}
PACS numbers: 02.20.Qs, 02.20.Sv, 03.20.+i, 05.50.+q\
Keywords: analytic Bethe ansatz, Lie superalgebra, solvable lattice model, transfer matrix, T-system, Young superdiagram\
\
Journal-ref: Physica A 252 (1998) 565-585\
DOI: 10.1016/S0378-4371(97)00625-0
Introduction
============
In reference [@KS1], analytic Bethe ansatz [@R1; @R2] was carried out systematically for fundamental representations of the Yangians $Y({\cal G})$ [@D] associated with classical simple Lie algebras ${\cal G}=B_{r}$, $C_{r}$ and $D_{r}$. That is, eigenvalue formulas in dressed vacuum form were presented for the commuting transfer matrices of solvable vertex models. These formulae are Yangian analogues of the Young tableaux for ${\cal G}$ and obey some semi-standard like conditions. It had been proven that they do not have poles under the Bethe ansatz equation. Furthermore, for ${\cal G}=B_{r}$ case, these formulae were generalized [@KOS] to the case of finite dimensional modules labeled by skew-Young diagrams $\lambda \subset \mu$. The eigenvalue formulae of the transfer matrices in dressed vacuum form labeled by rectangular Young diagrams $\lambda =\phi, \mu=(m^a)$ obey a class of functional relations, the T-system [@KNS1] (see also, references [@BR; @KLWZ; @K; @KP; @KS2; @LWZ; @S2; @KNS2]). Making use of the T-system, we are able to calculate [@KNS2] various kinds of physical quantities such as the correlation lengths of the vertex models and central charges of RSOS models. The T-system is not only a class of transfer matrix functional relations but also a two-dimensional Toda field equation on discrete space time. Solving it recursively, we can express its solutions in terms of pfaffians or determinants [@KOS; @KNH; @TK; @T1].
In contrast to above mentioned successful story in the T-system and the analytic Bethe ansatz for simple Lie algebras, systematic treatment of them for Lie superalgebras [@Ka] had not been studied yet until quite recently. Studying supersymmetric integrable models is significant not only in mathematical physics but also in condensed matter physics (see for example, reference [@KE]). For instance, the supersymmetric $t-J$ model received much attention in connection with high $T_{c}$ superconductivity. As is well known, there are several choices of simple root systems for a superalgebra. We can construct all the simple root systems, from any one of them by applying repeatedly the reflections with respect to the elements of the Weyl supergroup $ {\cal SW}({\cal G})$ [@LSS]. The simplest system of simple roots is so called distinguished one [@Ka]. Recently we had executed [@T2] analytic Bethe ansatz associated mainly with the distinguished simple root system of the Lie superalgebra $sl(r+1|s+1)$ and then established functional relations for commuting family of transfer matrices.
The purpose of this paper is to extend our previous results [@T2] to any simple root systems of ${\cal G}=sl(r+1|s+1)$. One can reproduce many of the earlier results [@T2] if one set the grading parameters (\[grading\]) to $p_{a}=1 : 1\le a \le r+1; p_{a}=-1:r+2 \le a \le r+s+2$. Throughout this paper, we often use similar notation presented in references [@KS1; @KOS; @TK; @T2]. We execute analytic Bethe ansatz based upon the Bethe ansatz equation (\[BAE\]) associated with any simple root systems of $sl(r+1|s+1)$. The observation that the Bathe ansatz equation can be expressed by the root system of a Lie algebra is traced back to reference [@RW] (see also, reference [@Kul] for $sl(r+1|s+1)$ case). Moreover, Kuniba et.al. [@KOS] conjectured that the left hand side of the Bethe ansatz equation (\[BAE\]) can be written as a ratio of some Drinfeld polynomials’ [@D]. In addition, extra signs appear in the Bethe ansatz equation. This is because in the supersymmetric models, the R-matrix satisfies the graded Yang-Baxter equation [@KulSk] and then the transfer matrix is defined as a supertrace of the monodromy matrix. There are several sets of Bethe ansatz equations corresponding to the fact that there are several choices of simple root systems for a Lie superalgebra. However these sets of the Bethe ansatz equations are connected with each other under the particle-hole transformation. In fact, the eqivalence of these sets of the Bethe ansatz equations was established for $sl(1|2)$ case in references [@BCFH; @EK] and for $sl(2|2)$ case in reference [@EKS2]. Then we discuss relations among these sets of the Bethe ansatz equations for $sl(r+1|s+1)$ and we point out that the particle-hole transformation is related with the reflection with respect to the element of the Weyl supergroup for odd simple root $\alpha $ with $(\alpha | \alpha)=0$.
We introduce the Young superdiagram [@BB1]. To put it more precisely, this Young superdiagram is different from the classical one in that it carries spectral parameter $u$. In contrast to ordinary Young diagram, there is no limitation on the number of rows. We define semi-standard like tableau on it. Making use of this tableau, we introduce the function ${\cal T}_{\lambda \subset \mu}(u)$ (\[Tge1\]), which should be the fusion transfer matrix whose auxiliary space is finite dimensional module of super Yangian $Y(sl(r+1|s+1))$ [@N] or quantum affine superalgebra $U_{q}(sl(r+1|s+1)^{(1)})$ [@Y1; @Y2], labeled by skew-Young superdiagram $\lambda \subset \mu$. We can trace the origin of the function ${\cal T}^{1}(u)$ back to the eigenvalue formula of transfer matrix of the Perk-Schultz model [@PS1; @PS2; @Sc], which is a multi-component generalization of the six-vertex model (see also reference [@Kul]). Furthermore, the function ${\cal T}^{1}(u)$ reduces to the eigenvalue formula of transfer matrix derived by algebraic Bethe ansatz (For instance, reference [@FK]: $r=1,s=0$ case; reference [@EK]: $r=0,s=1$ case; references [@EKS1; @EKS2]: $r=s=1$ case). We prove pole-freeness of ${\cal T}^{a}(u)={\cal T}_{(1^{a})}(u)$, essential property in the analytic Bethe ansatz. Owing to the same mechanism presented in reference [@KOS], the function ${\cal T}_{\lambda \subset \mu}(u)$ has determinant expressions whose matrix elements are only the functions associated with Young superdiagrams with shape $\lambda = \phi $; $\mu =(m)$ or $(1^{a})$. They can be viewed as quantum analogue of Jacobi-Trudi and Giambelli formulae for $sl(r+1|s+1)$. Then we can easily show that the function ${\cal T}_{\lambda \subset \mu}(u)$ is free of poles under the Bethe ansatz equation (\[BAE\]). We present a class of transfer matrix functional relations among the above-mentioned eigenvalue formulae of transfer matrix in dressed vacuum form associated with rectangular Young superdiagrams. It is specialization of Hirota bilinear difference equation [@H], which can be proved by the Jacobi identity.
The outline of this paper is given as follows. In section 2, we brefly review the Lie superalgebra ${\cal G}=sl(r+1|s+1)$. In section 3, we execute the analytic Bethe ansatz based upon the Bethe ansatz equation (\[BAE\]) associated with any simple root systems. We prove pole-freeness of the function ${\cal T}^{a}(u)={\cal T}_{(1^{a})}(u)$. In section 4, we propose functional relations, the T-system, associated with the transfer matrces in dressed vacuum form defined in the previous section. In section 5, using the particle-hole transformation, relations among the sets of the Bethe ansatz equations for various kinds of simple root systems are discussed. Section 6 is devoted to summary and discussion.
The Lie superalgebra $sl(r+1|s+1)$
==================================
In this section, we brefly review the Lie superalgebra ${\cal G}=sl(r+1|s+1)$. A Lie superalgebra [@Ka] is a ${\bf Z}_2$ graded algebra ${\cal G} ={\cal G}_{\bar{0}} \oplus {\cal G}_{\bar{1}}$ with a product $[\; , \; ]$, whose homogeneous elements $a\in {\cal G_{\alpha}},b\in {\cal G_{\beta}}$ $(\alpha, \beta \in {\bf Z}_2=\{\bar{0},\bar{1} \})$ and $c\in {\cal G}$ obey the following relations. $$\begin{aligned}
\left[a,b\right] & \in & {\cal G}_{\alpha+\beta}, \nonumber \\
\left[a,b\right]&=&-(-1)^{\alpha \beta}[b,a], \\
\left[a,[b,c]\right]&=&[[a,b],c]+(-1)^{\alpha \beta} [b,[a,c]].
\nonumber \end{aligned}$$ We can divide the set of non-zero roots into the set of non-zero even roots (bosonic roots) $\Delta_0$ and the set of odd roots (fermionic roots) $\Delta_1$. For $sl(r+1|s+1)$ case, they have the following form $$\Delta_0=
\{ \epsilon_{i}-\epsilon_{j} \} \cup \{\delta_{i}-\delta_{j}\},
i \ne j ;\quad \Delta_1=\{\pm (\epsilon_{i}-\delta_{j})\}$$ where $\epsilon_{1},\dots,\epsilon_{r+1};\delta_{1},\dots,\delta_{s+1}$ are basis of dual space of the Cartan subalgebra with the bilinear form $(\ |\ )$ such that $$(\epsilon_{i}|\epsilon_{j})=\delta_{i\, j}, \quad
(\epsilon_{i}|\delta_{j})=(\delta_{i}|\epsilon_{j})=0 , \quad
(\delta_{i}|\delta_{j})=-\delta_{i\, j}.$$ The Weyl group ${\cal W}({\cal G})$ of a Lie superalgebra ${\cal G}$ is generated by the Weyl reflections with respect to the even roots : $$\omega_{\alpha}(\beta)=\beta
-\frac{2(\alpha | \beta)}{(\alpha | \alpha)} \alpha$$ where $\alpha \in \Delta_0, \beta \in
\Delta_0 \cup \Delta_1 $. Moreover the Weyl group ${\cal W}({\cal G})$ can be extended to the Weyl supergroup ${\cal SW}({\cal G})$ [@LSS] by adding the reflections with respect to the odd roots: $$\omega_{\alpha}(\beta)=
\left\{
\begin{array}{@{\,}ll}
\beta -\frac{2(\alpha | \beta)}{(\alpha | \alpha)} \alpha &
\mbox{for} \quad (\alpha | \alpha) \ne 0 \\
\beta +\alpha & \mbox{for} \quad (\alpha | \alpha) = 0 \quad
\mbox{and} \quad (\alpha | \beta) \ne 0 \\
\beta & \mbox{for} \quad (\alpha | \alpha)=(\alpha | \beta) =0 \\
-\alpha & \mbox{for} \quad \beta=\alpha
\end{array}
\right.$$ where $\alpha \in \Delta_1, \beta \in
\Delta_0 \cup \Delta_1 $. Note that $\Delta_0$ and $\Delta_1 $ are invariant under $\omega_{\alpha} \in {\cal W}({\cal G})$; are not invariant under $\omega_{\alpha} \in {\cal SW}({\cal G})$ with $(\alpha | \alpha)=0$. There are several choices of simple root systems depending on choices of Borel subalgebras. The simplest system of simple roots is so called distinguished one [@Ka]. For example, the distinguished simple root system $\{\alpha_1,\dots,\alpha_{r+s+1} \}$ of $sl(r+1|s+1)$ has the form $$\begin{aligned}
&&\alpha_i = \epsilon_{i}-\epsilon_{i+1}
\quad i=1,2,\dots,r, \nonumber \\
&&\alpha_{r+1} = \epsilon_{r+1}-\delta_{1} \\
&& \alpha_{j+r+1} = \delta_{j}-\delta_{j+1} ,
\quad j=1,2,\dots,s \nonumber
\end{aligned}$$ where $\{\alpha_i \}_{i \ne r+1}$ are even roots and $\alpha_{r+1}$ is an odd root with $(\alpha_{r+1} | \alpha_{r+1})=0$. One can construct all the simple root systems, unequivalent with respect to ${\cal W}({\cal G})$, from any one of them by applying repeatedly the reflections with respect to $\omega_{\alpha} \in {\cal SW}({\cal G})$ with $(\alpha | \alpha )=0$ (see, figure \[dynkin\]). We define the sets $$\begin{aligned}
J=\{ 1,2,\dots,r+s+2\}
\label{set}\end{aligned}$$ with the total order $$\begin{aligned}
1\prec 2 \prec \cdots \prec r+s+2 . \label{order}\end{aligned}$$ Divide the set $J$ into two disjoint sets $$\begin{aligned}
J=J_{+} \bigcup J_{-}, & \qquad &
J_{+} \bigcap J_{-} = \phi, \nonumber \\
J_{+}=\{ i_{1},i_{2},\dots,i_{r+1}\} , &\quad &
J_{-}=\{ j_{1},j_{2},\dots,j_{s+1}\}
\label{disj} \end{aligned}$$ with the ordering $$\begin{aligned}
i_{1} \prec i_{2} \prec \cdots \prec i_{r+1},\quad
j_{1} \prec j_{2} \prec \cdots \prec j_{s+1} . \end{aligned}$$ For any element of $J$, we introduce the grading $$p_{a}=\left\{
\begin{array}{@{\,}ll}
1 & \mbox{for $a \in J_{+}$} \\
-1 & \mbox{for $a \in J_{-}$ }
\quad .
\end{array}
\right. \label{grading}$$ Using this grading parameters $\{p_{j} \}$, one can express Cartan matrix as follows $$(\alpha_{k}|\alpha_{l})=(p_{k}+p_{k+1})\delta_{k\>l}
-p_{k+1}\delta_{k+1\> l}
-p_{k}\delta_{k\> l+1}. \label{cartangr}$$
(380,340) (130,20) (140,20)[(1,0)[40]{}]{} (190,20) (200,20)[(1,0)[40]{}]{} (250,20) (182.929,12.9289)[(1,1)[14.14214]{}]{} (182.929,27.07107)[(1,-1)[14.14214]{}]{} (112,0)[$\delta_{1}-\delta_{2}$]{} (172,0)[$\delta_{2}-\epsilon_{1}$]{} (232,0)[$\epsilon_{1}-\epsilon_{2}$]{} (190,75)[(0,-1)[40]{}]{} (130,100) (140,100)[(1,0)[40]{}]{} (190,100) (200,100)[(1,0)[40]{}]{} (250,100) (122.929,92.9289)[(1,1)[14.14214]{}]{} (122.929,107.07107)[(1,-1)[14.14214]{}]{} (182.929,92.9289)[(1,1)[14.14214]{}]{} (182.929,107.07107)[(1,-1)[14.14214]{}]{} (242.929,92.9289)[(1,1)[14.14214]{}]{} (242.929,107.07107)[(1,-1)[14.14214]{}]{} (112,80)[$\delta_{1}-\epsilon_{1}$]{} (172,80)[$\epsilon_{1}-\delta_{2}$]{} (232,80)[$\delta_{2}-\epsilon_{2}$]{} (160,160)[(3,-2)[75]{}]{} (220,160)[(-3,-2)[75]{}]{} (10,180) (20,180)[(1,0)[40]{}]{} (70,180) (80,180)[(1,0)[40]{}]{} (130,180) (122.929,172.9289)[(1,1)[14.14214]{}]{} (122.929,187.07107)[(1,-1)[14.14214]{}]{} (2.929,172.9289)[(1,1)[14.14214]{}]{} (2.929,187.07107)[(1,-1)[14.14214]{}]{} (-8,160)[$\delta_{1}-\epsilon_{1}$]{} (52,160)[$\epsilon_{1}-\epsilon_{2}$]{} (112,160)[$\epsilon_{2}-\delta_{2}$]{} (250,180) (260,180)[(1,0)[40]{}]{} (310,180) (320,180)[(1,0)[40]{}]{} (370,180) (242.929,172.9289)[(1,1)[14.14214]{}]{} (242.929,187.07107)[(1,-1)[14.14214]{}]{} (362.929,172.9289)[(1,1)[14.14214]{}]{} (362.929,187.07107)[(1,-1)[14.14214]{}]{} (232,160)[$\epsilon_{1}-\delta_{1}$]{} (292,160)[$\delta_{1}-\delta_{2}$]{} (352,160)[$\delta_{2}-\epsilon_{2}$]{} (280,240)[(3,-2)[75]{}]{} (100,240)[(-3,-2)[75]{}]{} (130,260) (140,260)[(1,0)[40]{}]{} (190,260) (200,260)[(1,0)[40]{}]{} (250,260) (122.929,252.9289)[(1,1)[14.14214]{}]{} (122.929,267.07107)[(1,-1)[14.14214]{}]{} (182.929,252.9289)[(1,1)[14.14214]{}]{} (182.929,267.07107)[(1,-1)[14.14214]{}]{} (242.929,252.9289)[(1,1)[14.14214]{}]{} (242.929,267.07107)[(1,-1)[14.14214]{}]{} (112,240)[$\epsilon_{1}-\delta_{1}$]{} (172,240)[$\delta_{1}-\epsilon_{2}$]{} (232,240)[$\epsilon_{2}-\delta_{2}$]{} (190,315)[(0,-1)[40]{}]{} (130,340) (140,340)[(1,0)[40]{}]{} (190,340) (200,340)[(1,0)[40]{}]{} (250,340) (182.929,332.9289)[(1,1)[14.14214]{}]{} (182.929,347.07107)[(1,-1)[14.14214]{}]{} (112,320)[$\epsilon_{1}-\epsilon_{2}$]{} (172,320)[$\epsilon_{2}-\delta_{1}$]{} (232,320)[$\delta_{1}-\delta_{2}$]{}
Analytic Bethe ansatz
=====================
Consider the following type of the Bethe ansatz equation (cf. references [@Kul; @RW; @KOS; @Sc]). $$\begin{aligned}
-\frac{P_{a}(u_k^{(a)}+\zeta_{a})}
{P_{a}(u_k^{(a)}-\zeta_{a})}
&=&(-1)^{{\rm deg}(\alpha_a)}
\prod_{b=1}^{r+s+1}\frac{Q_{b}(u_k^{(a)}+(\alpha_a|\alpha_b))}
{Q_{b}(u_k^{(a)}-(\alpha_a|\alpha_b))}, \label{BAE} \\
Q_{a}(u)&=& \prod_{j=1}^{N_{a}}[u-u_j^{(a)}],
\label{Q_a} \\
P_{a}(u)&=& \prod_{j=1}^{N}P_{a}^{(j)}(u), \\
P_{a}^{(j)}(u)&=&[u-w_{j}]^{\delta_{a,1}} \label{drinfeld}\end{aligned}$$ where $[u]=(q^u-q^{-u})/(q-q^{-1})$; $N_{a} \in {\bf Z }_{\ge 0}$; $u, w_{j}\in {\bf C}$; $a,k \in {\bf Z}$ ($1\le a \le r+s+1$,$\ 1\le k\le N_{a}$), $\zeta_{1}=p_{1}$ and $$\begin{aligned}
{\rm deg}(\alpha_a)&=&\left\{
\begin{array}{@{\,}ll}
0 & \mbox{for even root} \\
1 & \mbox{for odd root}
\end{array}
\right. \\
&=& \frac{1-p_{a}p_{a+1}}{2} . \nonumber \end{aligned}$$ Particularly for distinguished simple root of $sl(r+1|s+1)$, we have ${\rm deg}(\alpha_{a})=\delta_{a,r+1}$. In the present paper, we suppose that $q$ is generic. The left hand side of the Bethe ansatz equation (\[BAE\]) is connected with the quantum space. We suppose that it is the ratio of some Drinfeld polynomials’ labeled by skew-Young diagrams $\tilde{\lambda} \subset \tilde{\mu}$ (cf. reference [@KOS]). For simplicity, we deal only with the case $\tilde{\lambda}=\phi, \tilde{\mu}=(1) $. The generalization to the case for any skew-Young diagram will be accomplished by the empirical procedures given in reference [@KOS]. The factor $(-1)^{{\rm deg}(\alpha_a)}$ of the Bethe ansatz equation (\[BAE\]) exists so as to make the transfer matrix to be the supertrace of the monodromy matrix. Note that the Bethe ansatz equation (\[BAE\]) is invariant under the Weyl group ${\cal W}({\cal G})$ since $(\omega_{\beta}(\alpha)|\omega_{\beta}(\gamma))=
(\alpha|\gamma)$ for $\alpha,\gamma \in \Delta_{0} \cup \Delta_{1}$ and $\beta \in \Delta_{0}$.
For any $a \in J $, set $$\begin{aligned}
z(a;u)=\psi_{a}(u)
\frac{Q_{a-1}(u+\sum_{j=1}^{a-1}p_{j}+2p_{a})
Q_{a}(u+\sum_{j=1}^{a}p_{j}-2p_{a})}
{Q_{a-1}(u+\sum_{j=1}^{a-1}p_{j})Q_{a}(u+\sum_{j=1}^{a}p_{j})} \end{aligned}$$ where $Q_{0}(u)=1, Q_{r+s+2}(u)=1$ and $$\psi_{a}(u)=
\left\{
\begin{array}{@{\,}ll}
P_{1}(u+2p_{1}) & \mbox{for } \quad a=1 \\
P_{1}(u) & \mbox{for } \quad a \in J-\{1\}
\end{array} \label{psi}
\right. .$$ In the present paper, we frequently express the function $z(a;u)$ as the box $\framebox{a}_{u}$, whose spectral parameter $u$ will often be abbreviated. Under the Bethe ansatz equation (\[BAE\]), the following relations are valid $$\begin{aligned}
Res_{u=-\sum_{j=1}^{b}p_{j}+u_{k}^{(b)}}
(p_{b}z(b;u)+p_{b+1}z(b+1;u))=0 , \quad
b \in J-\{r+s+2 \} . \label{res1} \end{aligned}$$ It was pointed out [@KS1] that the dressed vacuum form in the analytic Bethe ansatz for fundamental representations of Yangians $Y({\cal G})$ associated with simple Lie algebras ${\cal G}=B_{r},C_{r},D_{r}$ have similar structure of the crystal graph [@KN; @Na]. This is also the case with ${\cal G}=sl(r+1|s+1)$. Actually, we can express the relation ($\ref{res1}$) schematically as follows: $$p_{1} \framebox{1} \stackrel{1}{\longrightarrow}
p_{2} \framebox{2} \stackrel{2}{\longrightarrow}
\cdots \stackrel{r+s+1}{\longrightarrow} p_{r+s+2} \framebox{r+s+2}
\label{cg}$$ where the number $b$ on the arrow represents the color (superscript of $u_{k}^{(b)}$) of the common pole $-\sum_{j=1}^{b}p_{j}+u_{k}^{(b)}$ of the functions $z(b;u)$ and $z(b+1;u)$.
We will use the functions ${\cal T}^{a}(u)$ and ${\cal T}_{m}(u)$ ($a,m \in {\bf Z }$; $u \in {\bf C }$) determined by the non-commutative generating series of the form $$\begin{aligned}
& & (1+z(r+s+2;u)X)^{p_{r+s+2}}\cdots (1+z(r+2;u)X)^{p_{r+2}}
\nonumber \\
&&\times (1+z(r+1;u)X)^{p_{r+1}}\cdots (1+z(1;u)X)^{p_{1}}
\nonumber \\
&&=\sum_{a=-\infty}^{\infty}
{\cal T}^{a}(u+a-1)X^{a},
\label{generating}\end{aligned}$$ $$\begin{aligned}
&& (1-z(1;u)X)^{-p_{1}}\cdots (1-z(r+1;u)X)^{-p_{r+1}} \nonumber \\
&&\times (1-z(r+2;u)X)^{-p_{r+2}}\cdots (1-z(r+s+2;u)X)^{-p_{r+s+2}}
\nonumber \\
&&= \sum_{m=-\infty}^{\infty} {\cal T}_{m}(u+m-1)X^{m} \label{generating2}\end{aligned}$$ where $X$ is a shift operator $X=e^{2\partial_{u}}$. In particular, we have ${\cal T}^{0}(u)=1$; ${\cal T}_{0}(u)=1$; ${\cal T}^{a}(u)=0$ for $a<0$; ${\cal T}_{m}(u)=0$ for $m<0$. We note that the origin of the functions ${\cal T}^{1}(u),{\cal T}_{1}(u)$ and the Bethe ansatz equation (\[BAE\]) with (\[cartangr\]) trace back to the eigenvalue formula of transfer matrix and the Bethe ansatz equation for the Perk-Schultz model [@Sc] but the vacuum part, some gauge factors and extra signs after some redefinition. (See also, reference [@Kul]). As for the relation between fundamental $L$ operator and the function ${\cal T}^{1}(u)$, see, for example, Appendix A in reference [@T2].
Let $\lambda \subset \mu$ be a skew-Young superdiagram labeled by the sequences of non-negative integers $\lambda =(\lambda_{1},\lambda_{2},\dots)$ and $\mu =(\mu_{1},\mu_{2},\dots)$ such that $\mu_{i} \ge \lambda_{i}: i=1,2,\dots;$ $\lambda_{1} \ge \lambda_{2} \ge \dots \ge 0$; $\mu_{1} \ge \mu_{2} \ge \dots \ge 0$ and $\lambda^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots)$ be the conjugate of $\lambda $. We assign a coordinates $(i,j)\in {\bf Z}^{2}$ on this skew-Young superdiagram $\lambda \subset \mu$ such that the row index $i$ increases as we go downwords and the column index $j$ increases as we go from left to right and that $(1,1)$ is on the top left corner of $\mu$. We define an admissiable tableau $b$ on the skew-Young superdiagram $\lambda \subset \mu$ as a set of element $b(i,j)\in J$ labeled by the coordinates $(i,j)$ mentioned above, with the following rule (admissibility conditions).
1. For any elements of $J$, $$b(i,j) \preceq b(i,j+1),\quad b(i,j) \preceq b(i+1,j).$$
2. For any elements of $J_{+}$, $$b(i,j) \prec b(i+1,j).$$
3. For any elements of $J_{-}$, $$b(i,j) \prec b(i,j+1).$$
Let $B(\lambda \subset \mu)$ be the set of admissible tableaux on $\lambda \subset \mu$. For any skew-Young superdiagram $\lambda \subset \mu$, define the function ${\cal T}_{\lambda \subset \mu}(u)$ as follows $${\cal T}_{\lambda \subset \mu}(u)=
\sum_{b \in B(\lambda \subset \mu)}
\prod_{(i,j) \in (\lambda \subset \mu)}
p_{b(i,j)}
z(b(i,j);u-\mu_{1}+\mu_{1}^{\prime}-2i+2j)
\label{Tge1}$$ where the product is taken over the coordinates $(i,j)$ on $\lambda \subset \mu$. The following relations should be valid by the same reason mentioned in [@KOS], that is, they will be verified by induction on $\mu_{1}$ or $\mu_{1}^{\prime}$. $$\begin{aligned}
{\cal T}_{\lambda \subset \mu}(u)&=&{\rm det}_{1 \le i,j \le \mu_{1}}
({\cal T}^{\mu_{i}^{\prime}-\lambda_{j}^{\prime}-i+j}
(u-\mu_{1}+\mu_{1}^{\prime}-\mu_{i}^{\prime}-\lambda_{j}^{\prime}+i+j-1))
\label{Jacobi-Trudi1} \\
&=&{\rm det}_{1 \le i,j \le \mu_{1}^{\prime}}
({\cal T}_{\mu_{j}-\lambda_{i}+i-j}
(u-\mu_{1}+\mu_{1}^{\prime}+\mu_{j}+\lambda_{i}-i-j+1)) .
\label{Jacobi-Trudi2} \end{aligned}$$ For instance, for $sl(2|1)$: $\lambda=\phi; \mu=(2^{1},1^{1});
J_{+}=\{1,3\}; J_{-}=\{2\}$ case, we obtain $$\begin{aligned}
{\cal T}_{(2^{1},1^{1})}(u) &=&
-\> \begin{array}{|c|c|}\hline
1 & 1 \\ \hline
2 \\ \cline{1-1}
\end{array}
+\begin{array}{|c|c|}\hline
1 & 1 \\ \hline
3 \\ \cline{1-1}
\end{array}
+
\begin{array}{|c|c|}\hline
1 & 2 \\ \hline
2 \\ \cline{1-1}
\end{array}
-\begin{array}{|c|c|}\hline
1 & 2 \\ \hline
3 \\ \cline{1-1}
\end{array} \nonumber \\
&-&\begin{array}{|c|c|}\hline
1 & 3 \\ \hline
2 \\ \cline{1-1}
\end{array}
+\begin{array}{|c|c|}\hline
1 & 3 \\ \hline
3 \\ \cline{1-1}
\end{array}
+
\begin{array}{|c|c|}\hline
2 & 3 \\ \hline
2 \\ \cline{1-1}
\end{array}
-\begin{array}{|c|c|}\hline
2 & 3 \\ \hline
3 \\ \cline{1-1}
\end{array} \nonumber \\
& =&
P_1(u-2)P_1(u+2)
\Bigl\{
-P_1(u+4)\frac{Q_1(u-3)Q_2(u)}{Q_1(u+3)Q_2(u-2)} \nonumber \\
&+&P_1(u+4) \frac{Q_1(u-1)Q_2(u)}{Q_1(u+3)Q_2(u-2)} \nonumber \\
&+& P_1(u+2)\frac{Q_1(u-3)Q_2(u)Q_2(u+4)}
{Q_1(u+3)Q_2(u-2)Q_2(u+2)} \nonumber \\
&-& P_1(u+2)\frac{Q_1(u-1)Q_2(u)Q_2(u+4)}{Q_1(u+3)Q_2(u-2)Q_2(u+2)}
\nonumber \\
&-&P_1(u+2)\frac{Q_1(u-3)Q_2(u)Q_2(u+4)}
{Q_1(u+1)Q_2(u-2)Q_2(u+2)} \nonumber \\
&+&P_1(u+2) \frac{Q_1(u-1)Q_2(u)Q_2(u+4)}{Q_1(u+1)Q_2(u-2)Q_2(u+2)}
\nonumber \\
&+& P_1(u)\frac{Q_1(u-3)Q_2(u+4)}
{Q_1(u+1)Q_2(u-2)} \nonumber \\
&-& P_1(u)\frac{Q_1(u-1)Q_2(u+4)}{Q_1(u+1)Q_2(u-2)}
\Bigr\} \label{example} \\
&=&\begin{array}{|cc|}
{\cal T}^{2}(u-1) & {\cal T}^{3}(u) \\
1 & {\cal T}^{1}(u+2) \\
\end{array}
\nonumber
\end{aligned}$$ where $$\begin{aligned}
{\cal T}^{1}(u)
&=&\begin{array}{|c|}\hline
1 \\ \hline
\end{array}
-\begin{array}{|c|}\hline
2 \\ \hline
\end{array}
+\begin{array}{|c|}\hline
3 \\ \hline
\end{array} \nonumber \\
&=&
P_1(u+2)\frac{Q_1(u-1)}{Q_1(u+1)}
-P_1(u)\frac{Q_1(u-1)Q_2(u+2)}{Q_1(u+1)Q_2(u)} \nonumber \\
&+&P_1(u)\frac{Q_2(u+2)}{Q_2(u)}, \\
{\cal T}^{2}(u)
&=&
-\> \begin{array}{|c|}\hline
1 \\ \hline
2 \\ \hline
\end{array}
+ \begin{array}{|c|}\hline
1 \\ \hline
3 \\ \hline
\end{array}
+ \begin{array}{|c|}\hline
2 \\ \hline
2 \\ \hline
\end{array}
- \begin{array}{|c|}\hline
2 \\ \hline
3 \\ \hline
\end{array} \nonumber \\
&=& P_1(u-1)
\Bigl\{
-P_1(u+3)\frac{Q_1(u-2)Q_2(u+1)}{Q_1(u+2)Q_2(u-1)} \nonumber \\
&+&P_1(u+3)\frac{Q_1(u)Q_2(u+1)}{Q_1(u+2)Q_2(u-1)} \\
&+&P_1(u+1)\frac{Q_1(u-2)Q_2(u+3)}{Q_1(u+2)Q_2(u-1)}
-P_1(u+1)\frac{Q_1(u)Q_2(u+3)}{Q_1(u+2)Q_2(u-1)}
\Bigr\}, \nonumber \\
{\cal T}^{3}(u)
&=&
\begin{array}{|c|}\hline
1 \\ \hline
2 \\ \hline
2 \\ \hline
\end{array}
- \begin{array}{|c|}\hline
1 \\ \hline
2 \\ \hline
3 \\ \hline
\end{array}
- \begin{array}{|c|}\hline
2 \\ \hline
2 \\ \hline
2 \\ \hline
\end{array}
+ \begin{array}{|c|}\hline
2 \\ \hline
2 \\ \hline
3 \\ \hline
\end{array}
\nonumber \\
&=&P_1(u-2)P_1(u)
\Bigl\{
P_1(u+4)\frac{Q_1(u-3)Q_2(u+2)}{Q_1(u+3)Q_2(u-2)} \nonumber \\
&-&P_1(u+4)\frac{Q_1(u-1)Q_2(u+2)}{Q_1(u+3)Q_2(u-2)} \\
&-&P_1(u+2)\frac{Q_1(u-3)Q_2(u+4)}{Q_1(u+3)Q_2(u-2)}
+P_1(u+2)\frac{Q_1(u-1)Q_2(u+4)}{Q_1(u+3)Q_2(u-2)}
\Bigr\} \nonumber \\
&=& -\> \frac{{\cal T}_{(2^2)}(u)}{P_{1}(u+2)}. \nonumber
\end{aligned}$$ Remark1: If we drop the $u$ dependence of (\[Jacobi-Trudi1\]) and (\[Jacobi-Trudi2\]) for $p_{1}=p_{2}=\cdots =p_{r+1}=1,
p_{r+2}=p_{r+3}=\cdots =p_{r+s+2}=-1$, they reduce to classical Jacobi-Trudi and Giambelli formulae for $sl(r+1|s+1)$ [@BB1; @PT], which give us classical (super) characters. This fact confirms character limit [@KS1] of the eigenvalue formula for the transfer matrix.\
Remark2: In the case $\lambda =\phi$ and $s=-1$, $(\ref{Jacobi-Trudi1})$ and $(\ref{Jacobi-Trudi2})$ reduce to the quantum analogue of Jacobi-Trudi and Giambelli formulae for $sl_{r+1}$ presented in reference [@BR].\
Remark3: $(\ref{Jacobi-Trudi1})$ and $(\ref{Jacobi-Trudi2})$ have the same form as the quantum Jacobi-Trudi and Giambelli formulae for $U_{q}(B_{n}^{(1)})$ in reference [@KOS], but the function ${\cal T}^{a}(u)$ is quite different as we can be easily seen from (\[generating\]) and (\[generating2\]).
The following Theorem is a generalization of the Theorem in reference [@T2]. We will present a detailed proof here partly because it is essential in the analytic Bethe ansatz and partly because for reader’s convenience.
\[polefree\] For any integer $a$, the function ${\cal T}^a(u)$ is free of poles under the condition that the Bethe ansatz equation [(\[BAE\])]{} is valid.
Proof. For simplicity, we assume that the vacuum parts are formally trivial, that is, the left hand side of the Bethe ansatz equation (\[BAE\]) is constantly $-1$. We prove that ${\cal T}^a(u)$ is free of color $b$ pole, namely, $Res_{u=u_{j}^{(b)}+\cdots}{\cal T}^a(u)=0$ for any $b \in J-\{r+s+2 \}$ under the condition that the Bethe ansatz equation (\[BAE\]) is valid. The function $z(c;u)=\framebox{$c$}_{u}$ with $c\in J $ has the color $b$ pole only for $c=b$ or $b+1$, so we shall trace only or . Denote $S_{k}$ the partial sum of ${\cal T}^a(u)$, which contains k boxes among or . Apparently, $S_{0}$ does not have color $b$ pole. Now we examine $S_{1}$, which is the summation of the tableaux of the following form $$\begin{array}{|c|}\hline
\xi \\ \hline
b \\ \hline
\zeta \\ \hline
\end{array}
\qquad \qquad
\begin{array}{|c|}\hline
\xi \\ \hline
b+1 \\ \hline
\zeta \\ \hline
\end{array} \label{tableaux1}$$ where and are columns whose total length are $a-1$ and they do not involve and . Thanks to the relation (\[res1\]), color $b$ residues in these tableaux (\[tableaux1\]) cancel each other under the Bethe ansatz equation (\[BAE\]). Then we deal with $S_{k}$ only for $2 \le k \le a $ from now on.\
The case $ b, b+1 \in J_{+}$ : In this case, only the case for $k=2$ should be considered because or appear at most twice in one column. $S_{2}$ is the summation of the tableaux of the following form $$\begin{array}{|c|l}\cline{1-1}
\xi & \\ \cline{1-1}
b & _v \\ \cline{1-1}
b+1 & _{v-2} \\ \cline{1-1}
\zeta & \\ \cline{1-1}
\end{array}
= \frac{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j}+2)Q_{b+1}(v+\sum_{j=1}^{b+1}p_{j}-4)}
{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j})Q_{b+1}(v+\sum_{j=1}^{b+1}p_{j}-2)}X_{1}$$ where and are columns whose total length are $a-2$, which do not involve and ; $v=u+h_1$: $h_1$ is some shift parameter; the function $X_{1}$ does not contain the function $Q_{b}$. Obviously, $S_{2}$ is free of color $b$ pole.\
The case $ b \in J_{+}, b+1 \in J_{-} $ : $S_{k} (k\ge 2)$ is the summation of the tableaux of the following form $$\begin{aligned}
\begin{array}{|c|l}\cline{1-1}
\xi & \\ \cline{1-1}
b & _v \\ \cline{1-1}
b+1 & _{v-2} \\ \cline{1-1}
\vdots & \\ \cline{1-1}
b+1 & _{v-2k+2} \\ \cline{1-1}
\zeta & \\ \cline{1-1}
\end{array}
&=& \frac{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j}+2)
Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2k+2)}
{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j})
Q_{b}(v+\sum_{j=1}^{b-1}p_{j}+2)} \label{tabk1} \\
&\times & \frac{Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j})}
{Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}-2k+2)}X_{2} \nonumber \end{aligned}$$ and $$\begin{array}{|c|l}\cline{1-1}
\xi & \\ \cline{1-1}
b+1 & _v \\ \cline{1-1}
b+1 & _{v-2} \\ \cline{1-1}
\vdots & \\ \cline{1-1}
b+1 & _{v-2k+2}\\ \cline{1-1}
\zeta & \\ \cline{1-1}
\end{array}
=\frac{Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2k+1)
Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}+2)}
{Q_{b}(v+\sum_{j=1}^{b-1}p_{j}+1)
Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}-2k+2)}X_{2}
\label{tabk2}$$ where and are columns with total length $a-k$, which do not involve and ; $v=u+h_2$: $h_2$ is some shift parameter; the function $X_{2}$ does not contain the function $Q_{b}$. Obviously, color $b$ residues in (\[tabk1\]) and (\[tabk2\]) cancel each other under the Bethe ansatz equation (\[BAE\]).\
The case $ b \in J_{-}, b+1 \in J_{+} $ : $S_{k} (k\ge 2)$ is the summation of the tableaux of the following form $$\begin{aligned}
\begin{array}{|c|l}\cline{1-1}
\xi & \\ \cline{1-1}
b & _v \\ \cline{1-1}
\vdots & \\ \cline{1-1}
b & _{v-2k+4} \\ \cline{1-1}
b+1 & _{v-2k+2} \\ \cline{1-1}
\zeta & \\ \cline{1-1}
\end{array}
&=& \frac{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j}-2k+2)
Q_{b}(v+\sum_{j=1}^{b-1}p_{j}+1)}
{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j})
Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2k+1)} \label{tabk31} \\
&\times & \frac{Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}-2k)}
{Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}-2k+2)}X_{3} \nonumber \end{aligned}$$ and $$\begin{array}{|c|l}\cline{1-1}
\xi & \\ \cline{1-1}
b & _v \\ \cline{1-1}
\vdots & \\ \cline{1-1}
b & _{v-2k+4} \\ \cline{1-1}
b & _{v-2k+2}\\ \cline{1-1}
\zeta & \\ \cline{1-1}
\end{array}
=\frac{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j}-2k)
Q_{b}(v+\sum_{j=1}^{b-1}p_{j}+1)}
{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j})
Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2k+1)}X_{3}
\label{tabk32}$$ where and are columns with total length $a-k$, which do not involve and ; $v=u+h_3$: $h_3$ is some shift parameter; the function $X_{3}$ does not contain the function $Q_{b}$. Obviously, color $b$ residues in (\[tabk31\]) and (\[tabk32\]) cancel each other under the Bethe ansatz equation (\[BAE\]).\
The case $b,b+1 \in J_{-}$: $S_{k} (k \ge 2)$ is the summation of the tableaux of the following form $$\begin{aligned}
&& f(k,n,\xi,\zeta,u):=
\begin{array}{|c|l}\cline{1-1}
\xi & \\ \cline{1-1}
b & _v \\ \cline{1-1}
\vdots & \\ \cline{1-1}
b & _{v-2n+2}\\ \cline{1-1}
b+1 & _{v-2n} \\ \cline{1-1}
\vdots & \\ \cline{1-1}
b+1 & _{v-2k+2}\\ \cline{1-1}
\zeta & \\ \cline{1-1}
\end{array} \nonumber \\
&=&\frac{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j}-2n)Q_{b}(v+\sum_{j=1}^{b-1}p_{j}+1)}
{Q_{b-1}(v+\sum_{j=1}^{b-1}p_{j})Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2n+1)} \\
&\times & \frac{Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2k-1)
Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}-2n)}
{Q_{b}(v+\sum_{j=1}^{b-1}p_{j}-2n-1)
Q_{b+1}(v+\sum_{j=1}^{b-1}p_{j}-2k)} X_{4}
,\quad 0 \le n \le k \nonumber
\label{tableauxk3}\end{aligned}$$ where and are columns with total length $a-k$, which do not involve and ; $v=u+h_4$: $h_4$ is some shift parameter and is independent of $n$; the function $X_{4}$ does not have color $b$ pole and is independent of $n$. $f(k,n,\xi,\zeta,u)$ has color $b$ poles at $u=-h_4-\sum_{j=1}^{b-1}p_{j}+2n-1+u_{l}^{(b)}$ and $u=-h_4-\sum_{j=1}^{b-1}p_{j}+2n+1+u_{l}^{(b)}$ for $1 \le n \le k-1$; at $u=-h_4-\sum_{j=1}^{b-1}p_{j}+1+u_{l}^{(b)}$ for $n=0$ ; at $u=-h_4-\sum_{j=1}^{b-1}p_{j}+2k-1+u_{l}^{(b)}$ for $n=k$. Evidently, color $b$ residue at $u=-h_4-\sum_{j=1}^{b-1}p_{j}+2n+1+u_{l}^{(b)}$ in $f(k,n,\xi,\zeta,u)$ and $f(k,n+1,\xi,\zeta,u)$ for $0\le n \le k-1$ cancel each other under the Bethe ansatz equation (\[BAE\]). Thus, under the Bethe ansatz equation (\[BAE\]), $\sum_{n=0}^{k}f(k,n,\xi,\zeta,u)$ is free of color $b$ poles, so is $S_{k}$.
------------------------------------------------------------------------
\
Applying Theorem \[polefree\] to (\[Jacobi-Trudi1\]), one can show that ${\cal T}_{\lambda \subset \mu}(u)$ is free of poles under the Bethe ansatz equation (\[BAE\]). Thus each term in ${\cal T}_{\lambda \subset \mu}(u)$ has a counterterm which cancel the common pole under the Bethe ansatz equation (\[BAE\]). Futhermore the set of all the terms in ${\cal T}_{\lambda \subset \mu}(u)$ forms Bethe-strap structure, which bears a resemblance to a weight space diagram. See Figure \[best\] and the relation (\[example\]) for $\lambda=\phi$ and $\mu=(2^{1},1^{1})$ case; the diagram (\[cg\]) for $\lambda=\phi$ and $\mu=(1^{1})$ case.
(100,180) (40,0)[(1,0)[10]{}]{} (40,10)[(1,0)[20]{}]{} (40,20)[(1,0)[20]{}]{} (40,0)[(0,1)[20]{}]{} (50,0)[(0,1)[20]{}]{} (60,10)[(0,1)[10]{}]{} (22,38)[(1,-1)[15]{}]{} (30,33) (78,38)[(-1,-1)[15]{}]{} (60,33) (32,9)[$-$]{} (43,3)[$3$]{} (43,13)[$2$]{} (53,13)[$3$]{} (0,40)[(1,0)[10]{}]{} (0,50)[(1,0)[20]{}]{} (0,60)[(1,0)[20]{}]{} (0,40)[(0,1)[20]{}]{} (10,40)[(0,1)[20]{}]{} (20,50)[(0,1)[10]{}]{} (10,77)[(0,-1)[14]{}]{} (12,69) (3,43)[$3$]{} (3,53)[$1$]{} (13,53)[$3$]{} (80,40)[(1,0)[10]{}]{} (80,50)[(1,0)[20]{}]{} (80,60)[(1,0)[20]{}]{} (80,40)[(0,1)[20]{}]{} (90,40)[(0,1)[20]{}]{} (100,50)[(0,1)[10]{}]{} (90,77)[(0,-1)[14]{}]{} (92,69) (83,43)[$2$]{} (83,53)[$2$]{} (93,53)[$3$]{} (0,80)[(1,0)[10]{}]{} (0,90)[(1,0)[20]{}]{} (0,100)[(1,0)[20]{}]{} (0,80)[(0,1)[20]{}]{} (10,80)[(0,1)[20]{}]{} (20,90)[(0,1)[10]{}]{} (10,117)[(0,-1)[14]{}]{} (12,109) (-8,89)[$-$]{} (3,83)[$3$]{} (3,93)[$1$]{} (13,93)[$2$]{} (80,80)[(1,0)[10]{}]{} (80,90)[(1,0)[20]{}]{} (80,100)[(1,0)[20]{}]{} (80,80)[(0,1)[20]{}]{} (90,80)[(0,1)[20]{}]{} (100,90)[(0,1)[10]{}]{} (90,117)[(0,-1)[14]{}]{} (92,109) (72,89)[$-$]{} (83,83)[$2$]{} (83,93)[$1$]{} (93,93)[$3$]{} (0,120)[(1,0)[10]{}]{} (0,130)[(1,0)[20]{}]{} (0,140)[(1,0)[20]{}]{} (0,120)[(0,1)[20]{}]{} (10,120)[(0,1)[20]{}]{} (20,130)[(0,1)[10]{}]{} (38,158)[(-1,-1)[15]{}]{} (19,152) (3,123)[$3$]{} (3,133)[$1$]{} (13,133)[$1$]{} (80,120)[(1,0)[10]{}]{} (80,130)[(1,0)[20]{}]{} (80,140)[(1,0)[20]{}]{} (80,120)[(0,1)[20]{}]{} (90,120)[(0,1)[20]{}]{} (100,130)[(0,1)[10]{}]{} (62,158)[(1,-1)[15]{}]{} (71,152) (83,123)[$2$]{} (83,133)[$1$]{} (93,133)[$2$]{} (40,160)[(1,0)[10]{}]{} (40,170)[(1,0)[20]{}]{} (40,180)[(1,0)[20]{}]{} (40,160)[(0,1)[20]{}]{} (50,160)[(0,1)[20]{}]{} (60,170)[(0,1)[10]{}]{} (32,169)[$-$]{} (43,163)[$2$]{} (43,173)[$1$]{} (53,173)[$1$]{} (72,81)[(-2,-1)[49]{}]{} (44,74) (72,121)[(-2,-1)[49]{}]{} (44,114)
Consult the references [@KS1; @S2] for detailed accounts on the Bethe-strap.
Functional equations
====================
Consider the following Jacobi identity: $${D}\left[
\begin{array}{c}
b \\
b
\end{array}
\right]
{D}
\left[
\begin{array}{c}
c \\
c
\end{array}
\right]-
{D}\left[
\begin{array}{c}
b \\
c
\end{array}
\right]
{D}\left[
\begin{array}{c}
c \\
b
\end{array}
\right]=
{D}\left[
\begin{array}{cc}
b & c\\
b & c
\end{array}
\right]
{D},
\quad b \ne c
\label{jacobi}$$ where $D$ is the deterement of a matrix and ${D}\left[
\begin{array}{ccc}
a_{1} & a_{2} & \dots \\
b_{1} & b_{2} & \dots
\end{array}
\right]$ is its minor removing $a_{\alpha}$’s rows and $b_{\beta}$’s columns. Set $\lambda = \phi$, $ \mu =(m^a)$ in (\[Jacobi-Trudi1\]). From the relation (\[jacobi\]), we have $${\cal T}_{m}^{a}(u-1) {\cal T}_{m}^{a}(u+1) =
{\cal T}_{m+1}^{a}(u) {\cal T}_{m-1}^{a}(u)+
{\cal T}_{m}^{a-1}(u) {\cal T}_{m}^{a+1}(u)
\label{t-sys1}$$ where $a,m \ge 1$; ${\cal T}_{m}^{a}(u)={\cal T}_{(m^a)}(u)$: $a,m \ge 1$; ${\cal T}_{m}^{0}(u)=1$: $m \ge 0$; ${\cal T}_{0}^{a}(u)=1$: $a \ge 0$. The functional equation (\[t-sys1\]) is a special case of Hirota bilinear difference equation [@H]. For $s=-1$, this functional equation (\[t-sys1\]) reduces to a discretized Toda field equation of $A_{r}$ type. Furthermore, there is a restriction on it, which we consider below.
\[vanish\] ${\cal T}_{\lambda \subset \mu}(u)=0$ if $\lambda \subset \mu$ contains a rectangular subdiagram with $r+2$ rows and $s+2$ columns.
Proof. Consider a tablau $b$ on this Young superdiagram $\lambda \subset \mu$. Decompose the set $J_{+}$ and $J_{-}$ (\[disj\]) as a union of the disjoint sets: $$\begin{aligned}
J_{+}=\bigcup_{k=1}^{\alpha} J_{+}^{(k)}: \quad
J_{+}^{(k)}=\{i_{1}^{(k)},i_{2}^{(k)},\cdots,i_{a_{k}}^{(k)} \}, \\
J_{-}=\bigcup_{k=1}^{\alpha} J_{-}^{(k)}: \quad
J_{-}^{(k)}=\{j_{1}^{(k)},j_{2}^{(k)},\cdots,j_{b_{k}}^{(k)} \}\end{aligned}$$ where we assumed, for any $k \in \{ 1,2,\dots ,\alpha \}$, $$\begin{aligned}
i_{\gamma}^{(k)} & = & \sum_{\delta=1}^{k-1}
(a_{\delta}+b_{\delta})+\gamma :
\quad \gamma \in \{ 1,2,\dots ,a_k \}, \\
j_{\gamma}^{(k)} & = & \sum_{\delta=1}^{k-1}
(a_{\delta}+b_{\delta})+a_{k}+\gamma :
\quad \gamma \in \{ 1,2,\dots ,b_k \} .\end{aligned}$$ Note that $J_{+}^{(1)}=\phi \quad (a_{1}=0)$, if the minimal element in the set $J$ is a member of the set $J_{-}$; $J_{-}^{(\alpha)}=\phi \quad (b_{\alpha}=0)$, if the maximal element in the set $J$ is a member of the set $J_{+}$. On this rectangular subdiagram, consider a strip, which is a union of $a_{k}\times 1$ rectangular subdiagrams, $1\times b_{k}$ rectangular subdiagrams and $1\times 1$ square subdiagram. Fill this strip by the elements $\{h_{t}^{(k)}, l_{t}^{(k)} \}$ of $J$ so as to meet the admissibility conditions (i), (ii) and (iii) (see, Figure \[strip\]).
(120,120) (0,0)[(0,1)[40]{}]{} (-0.5,45)[$\vdots$]{} (0,55)[(0,1)[55]{}]{} (10,80)[(0,1)[30]{}]{} (30,55)[(0,1)[15]{}]{} (40,55)[(0,1)[25]{}]{} (70,0)[(0,1)[30]{}]{} (80,10)[(0,1)[30]{}]{} (110,0)[(0,1)[40]{}]{} (109.3,45)[$\vdots$]{} (110,55)[(0,1)[55]{}]{} (0,0)[(1,0)[40]{}]{} (44,-1.5)[$\cdots$]{} (55,0)[(1,0)[55]{}]{} (80,10)[(1,0)[30]{}]{} (55,30)[(1,0)[15]{}]{} (55,40)[(1,0)[25]{}]{} (0,70)[(1,0)[30]{}]{} (10,80)[(1,0)[30]{}]{} (0,110)[(1,0)[40]{}]{} (44,108.5)[$\cdots$]{} (55,110)[(1,0)[55]{}]{} (1,103)[$h_{1}^{(1)}$]{} (4,93)[$\vdots$]{} (1,83)[$h_{a_{1}}^{(1)}$]{} (3,73)[$l_{1}^{(1)}$]{} (12.5,73.5)[$\cdots$]{} (23,73)[$l_{b_{1}}^{(1)}$]{} (31,73)[$h_{1}^{(2)}$]{} (31,63)[$h_{2}^{(2)}$]{} (34,56)[$\cdot$]{} (43,43)[$\ddots$]{} (54,34)[$\cdot$]{} (59,33)[$l_{b_{\alpha-1}} ^{(\alpha-1)}$]{} (71,32)[$h_{1}^{(\alpha)}$]{} (74,23)[$\vdots$]{} (71,13)[$h_{a_{\alpha}}^{(\alpha)}$]{} (73,3)[$l_{1}^{(\alpha)}$]{} (82.5,3.5)[$\cdots$]{} (93,3)[$l_{b_{\alpha}}^{(\alpha)}$]{} (103,3)[$y$]{}
For any $k \in \{1,2,\dots \alpha \}$, we find $$\begin{aligned}
h_{t}^{(k)} \succeq i_{t}^{(k)} &:& \quad t \in \{1,2,\dots a_{k}\}, \\
l_{t}^{(k)} \succeq j_{t}^{(k)} &:& \quad t \in \{1,2,\dots b_{k} \}. \end{aligned}$$ If $J_{-}^{(\alpha)} \ne \phi \quad (b_{\alpha} \ne 0)$, there is no admissible element $y \in J$ since $l_{b_{\alpha}}^{(\alpha)}
\succeq j_{b_{\alpha}}^{(\alpha)}=r+s+2 \in J_{-}$. If $J_{-}^{(\alpha)} = \phi \quad (b_{\alpha} = 0)$, there is no admissible element $y \in J$ since $h_{a_{\alpha}}^{(\alpha)}
\succeq i_{a_{\alpha}}^{(\alpha)}=r+s+2 \in J_{+}$. Then there is no admissible tableau on this Young superdiagram.
------------------------------------------------------------------------
\
Remark: The spectrum of fusion model was discussed in references [@DM; @MR] from the point of view of representation theory and the corresponding theorem was discussed.\
As a corollary, we have $${\cal T}_{m}^{a}(u)=0 \quad {\rm for} \quad
a \ge r+2 \quad {\rm and} \quad m \ge s+2. \label{vanish2}$$ Applying the relation (\[vanish2\]) to (\[t-sys1\]), we obtain $${\cal T}_{m}^{r+1}(u-1) {\cal T}_{m}^{r+1}(u+1) =
{\cal T}_{m+1}^{r+1}(u) {\cal T}_{m-1}^{r+1}(u)
\quad m \ge s+2,
\label{laplace1}$$ $${\cal T}_{s+1}^{a}(u-1) {\cal T}_{s+1}^{a}(u+1) =
{\cal T}_{s+1}^{a-1}(u) {\cal T}_{s+1}^{a+1}(u)
\quad a \ge r+2.
\label{laplace2}$$
On the equivalence of the Bethe ansatz equations
================================================
Bares et. al. [@BCFH] showed that Lai’s [@L] representation of the Bethe ansatz equation on the supersymmetric $t-J$ model is equivalent to Sutherland ’s one [@Su] under the particle-hole transformation. Moreover, following reference [@BCFH], Essler and Korepin [@EK] showed that Sutherland ’s [@Su] representation of the Bethe ansatz equation on the supersymmetric $t-J$ model is equivalent to the one originate from the grading $(p_{1},p_{2},p_{3})=(-1,1,-1)$ for Lie superalgebra $sl(1|2)$. Then the eqivalence of three different sets of Bethe ansatz equations on the supersymmetric $t-J$ model, which originate from $\frac{3!}{1! 2!}=3$ different gradings for $sl(1|2)$ was established. Futhermore Essler et. al. [@EKS2] established the equivalence of six different sets of Bethe ansatz equations on the supersymmetric extended Hubbard model, which originate from $\frac{4!}{2! 2!}=6$ different gradings for Lie superalgebra $sl(2|2)$ (see, Figure \[dynkin\]). Now, following reference [@BCFH], we discuss relations among the sets of Bethe ansatz equations (\[BAE\]) for different $\frac{(r+s+2)!}{(r+1)!(s+1)!}$ gradings $\{p_{j}\}$ (\[grading\]) or different sets of simple root systems inequivalent under the Weyl group ${\cal W}({\cal G})$ of Lie superalgebra $sl(r+1|s+1)$.
In this section, we assume that $q=1$. For some $b$ ($2 \le b \le r+s $), we assume $p_{b}p_{b+1}=-1$. Namely, $b$ th simple root $\alpha_{b}$ is an odd root with $(\alpha_{b}|\alpha_{b})=0$. In this case, $b$ th Bethe ansatz equation in (\[BAE\]) has the following form $$1=\frac{Q_{b-1}(u_{k}^{(b)}-p_{b}) Q_{b+1}(u_{k}^{(b)}-p_{b+1})}
{Q_{b-1}(u_{k}^{(b)}+p_{b}) Q_{b+1}(u_{k}^{(b)}+p_{b+1})},\quad
k=1,2,\dots,N_{b}.\label{bae2}$$ Define the polynomial $$f(z)=Q_{b-1}(z+p_{b}) Q_{b+1}(z+p_{b+1})
-Q_{b-1}(z-p_{b}) Q_{b+1}(z-p_{b+1}).\label{ffun}$$ Among the roots of the equation $f(z)=0$, $N_{b}$ of which are $\{u_{k}^{(b)}\}_{1 \le k \le N_{b}}$. So $\{f(u_{k}^{(b)})=0\}_{1 \le k \le N_{b}}$ reproduces the Bethe ansatz equation (\[bae2\]). Let the rest of the roots be $\{\tilde{u}_{k}^{(b)}\}_{
1 \le k \le \tilde{N}_{b}}$. Then $\{f(\tilde{u}_{k}^{(b)})=0\}_{
1 \le k \le \tilde{N}_{b}}$ reduces to the Bethe ansatz equation of the form $$1=\frac{Q_{b-1}(\tilde{u}_{k}^{(b)}+p_{b})
Q_{b+1}(\tilde{u}_{k}^{(b)}+p_{b+1})}
{Q_{b-1}(\tilde{u}_{k}^{(b)}-p_{b})
Q_{b+1}(\tilde{u}_{k}^{(b)}-p_{b+1})},\quad
k=1,2,\dots,\tilde{N}_{b}. \label{baet2}$$ Thanks to the residue theorem, the following relation holds $$\begin{aligned}
&& \sum_{j=1}^{{N}_{b}}\frac{1}{2\pi i} \int_{C_{j}}dz \frac{1}{i}
{\bf Log}
\frac{z-u_{l}^{(b-1)}-p_{b}}{z-u_{l}^{(b-1)}+p_{b}} \
\frac{d}{dz} {\bf Log} f(z) \nonumber \\
& =& \sum_{j=1}^{{N}_{b}} \frac{1}{i} {\bf Log}
\frac{u_{j}^{(b)}-u_{l}^{(b-1)}-p_{b}}
{u_{j}^{(b)}-u_{l}^{(b-1)}+p_{b}} \label{log1} \end{aligned}$$ where $C_{j}$ denotes contor around $u_{j}^{(b)}$. We assume the branch cut of the logarithm in the left hand side of (\[log1\]) extends from $u_{l}^{(b-1)}-p_{b}$ to $u_{l}^{(b-1)}+p_{b}$. The left hand side of the relation (\[log1\]) can be rewritten as follows $$\begin{aligned}
-\sum_{j=1}^{{\tilde{N}}_{b}} \frac{1}{i} {\bf Log}
\frac{\tilde{u}_{j}^{(b)}-u_{l}^{(b-1)}-p_{b}}
{\tilde{u}_{j}^{(b)}-u_{l}^{(b-1)}+p_{b}}
+\frac{1}{i} {\bf Log}
\frac{f(u_{l}^{(b-1)}+p_{b})}{f(u_{l}^{(b-1)}-p_{b})}.\end{aligned}$$ Then the following relation holds $$-1=\frac{Q_{b-1}(u_{l}^{(b-1)}+2p_{b})
\tilde{Q}_{b}(u_{l}^{(b-1)}-p_{b})
Q_{b}(u_{l}^{(b-1)}-p_{b})}
{Q_{b-1}(u_{l}^{(b-1)}-2p_{b})
\tilde{Q}_{b}(u_{l}^{(b-1)}+p_{b})
Q_{b}(u_{l}^{(b-1)}+p_{b})},\quad
l=1,2,\dots,N_{b-1}. \label{baem1}$$ where $\tilde{Q}_{b}(u)=
\prod_{j=1}^{\tilde{N}_{b}}(u-\tilde{u}_{j}^{(b)})$. Noting that the relation $$\frac{Q_{b-1}(u_{l}^{(b-1)}+p_{b-1}+p_{b})}
{Q_{b-1}(u_{l}^{(b-1)}-p_{b-1}-p_{b})}
=\frac{Q_{b-1}(u_{l}^{(b-1)}+2p_{b})}{Q_{b-1}(u_{l}^{(b-1)}-2p_{b-1})},$$ we find that the $b-1$th Bethe ansatz equation in (\[BAE\]) has the form: $$\begin{aligned}
-1=(-1)^{{\rm deg}(\alpha_{b-1})}
\frac{Q_{b-2}(u_{l}^{(b-1)}-p_{b-1})
Q_{b-1}(u_{l}^{(b-1)}+2p_{b})
Q_{b}(u_{l}^{(b-1)}-p_{b})}
{Q_{b-2}(u_{l}^{(b-1)}+p_{b-1})
Q_{b-1}(u_{l}^{(b-1)}-2p_{b-1})
Q_{b}(u_{l}^{(b-1)}+p_{b})}, \label{bae1} \\
l=1,2,\dots,N_{b-1}.\nonumber\end{aligned}$$ Combining these two equations (\[bae1\]) and (\[baem1\]), we obtain $$\begin{aligned}
-1=(-1)^{{\rm deg}(\tilde{\alpha}_{b-1})}
\frac{Q_{b-2}(u_{l}^{(b-1)}-p_{b-1})
Q_{b-1}(u_{l}^{(b-1)}-2p_{b})
\tilde{Q}_{b}(u_{l}^{(b-1)}+p_{b})}
{Q_{b-2}(u_{l}^{(b-1)}+p_{b-1})
Q_{b-1}(u_{l}^{(b-1)}-2p_{b-1})
\tilde{Q}_{b}(u_{l}^{(b-1)}-p_{b})}, \label{baet1} \\
l=1,2,\dots,N_{b-1} \nonumber\end{aligned}$$ where ${\rm deg}(\tilde{\alpha}_{b-1})={\rm deg}(\alpha_{b-1})+1
\quad mod \> 2$. The following relation is valid $$\begin{aligned}
&& \sum_{j=1}^{{N}_{b}}\frac{1}{2\pi i} \int_{C_{j}}dz \frac{1}{i}
{\bf Log}
\frac{z-u_{l}^{(b+1)}-p_{b+1}}{z-u_{l}^{(b+1)}+p_{b+1}} \
\frac{d}{dz} {\bf Log} f(z) \nonumber \\
&=& \sum_{j=1}^{{N}_{b}} \frac{1}{i} {\bf Log}
\frac{u_{j}^{(b)}-u_{l}^{(b+1)}-p_{b+1}}
{u_{j}^{(b)}-u_{l}^{(b+1)}+p_{b+1}} \label{log2} \\
&=& -\sum_{j=1}^{{\tilde{N}}_{b}} \frac{1}{i} {\bf Log}
\frac{\tilde{u}_{j}^{(b)}-u_{l}^{(b+1)}-p_{b+1}}
{\tilde{u}_{j}^{(b)}-u_{l}^{(b+1)}+p_{b+1}}
+\frac{1}{i}
{\bf Log} \frac{f(u_{l}^{(b+1)}+p_{b+1})}{f(u_{l}^{(b+1)}-p_{b+1})}
\nonumber \end{aligned}$$ where $C_{j}$ denotes contor around $u_{j}^{(b)}$. We assume the branch cut of the logarithm in left hand side of (\[log2\]) extends from $u_{l}^{(b+1)}-p_{b+1}$ to $u_{l}^{(b+1)}+p_{b+1}$. This equation reduces to the following equation: $$\begin{aligned}
-1=\frac{Q_{b+1}(u_{l}^{(b+1)}+2p_{b+1})
\tilde{Q}_{b}(u_{l}^{(b+1)}-p_{b+1})
Q_{b}(u_{l}^{(b+1)}-p_{b+1})}
{Q_{b+1}(u_{l}^{(b+1)}-2p_{b+1})
\tilde{Q}_{b}(u_{l}^{(b+1)}+p_{b+1})
Q_{b}(u_{l}^{(b+1)}+p_{b+1})}, \label{baem2} \\
l=1,2,\dots,N_{b+1}. \nonumber \end{aligned}$$ $b+1$th Bethe ansatz equation in (\[BAE\]) has the form: $$\begin{aligned}
-1=(-1)^{{\rm deg}(\alpha_{b+1})}
\frac{Q_{b}(u_{l}^{(b+1)}-p_{b+1})
Q_{b+1}(u_{l}^{(b+1)}+2p_{b+2})
Q_{b+2}(u_{l}^{(b+1)}-p_{b+2})}
{Q_{b}(u_{l}^{(b+1)}+p_{b+1})
Q_{b+1}(u_{l}^{(b+1)}-2p_{b+1})
Q_{b+2}(u_{l}^{(b+1)}+p_{b+2})}, \label{bae3} \\
l=1,2,\dots,N_{b+1}.\nonumber\end{aligned}$$ Combining these two equations (\[baem2\]) and (\[bae3\]), we obtain $$\begin{aligned}
-1=(-1)^{{\rm deg}(\tilde{\alpha}_{b+1})}
\frac{\tilde{Q}_{b}(u_{l}^{(b+1)}+p_{b+1})
Q_{b+1}(u_{l}^{(b+1)}+2p_{b+2})
Q_{b+2}(u_{l}^{(b+1)}-p_{b+2})}
{\tilde{Q}_{b}(u_{l}^{(b+1)}-p_{b+1})
Q_{b+1}(u_{l}^{(b+1)}+2p_{b+1})
Q_{b+2}(u_{l}^{(b+1)}+p_{b+2})}, \label{baet3} \\
l=1,2,\dots,N_{b+1}.\nonumber\end{aligned}$$ where ${\rm deg}(\tilde{\alpha}_{b+1})={\rm deg}(\alpha_{b+1})+1
\quad mod \> 2$. Note that the set of the equations (\[bae1\]), (\[bae2\]) and (\[bae3\]) is transfered to the equivalent set of the equations (\[baet1\]), (\[baet2\]) and (\[baet3\]) under the following transformation: $$(p_{b},p_{b+1},N_{b},\{u_{k}^{(b)}\}) \longrightarrow
(-p_{b},-p_{b+1},\tilde{N}_{b},\{\tilde{u}_{k}^{(b)}\}). \label{trans}$$ One can also develop similar argument for $p_{1}p_{2}=-1$ case using the function $$f(z)=P_{1}(z+p_{1}) Q_{2}(z+p_{2})
-P_{1}(z-p_{1}) Q_{2}(z-p_{2})$$ instead of the function (\[ffun\]) and for $p_{r+s+1}p_{r+s+2}=-1$ case using the function $$f(z)=Q_{r+s}(z+p_{r+s+1})-Q_{r+s}(z-p_{r+s+1})$$ instead of the function (\[ffun\]). Then the set of the Bethe ansatz equations (\[BAE\]) is transfered to the equivalent set of the Bethe ansatz equations under the transformation (\[trans\]) for $p_{b}p_{b+1}=-1$. Therefore, taking notice of a change of the grading $\{p_{j}\}$ or odd simple root $\alpha_{b}$ with $(\alpha_{b}|\alpha_{b})=0$ and applying the transformation (\[trans\]) repeatedly to the set of the Bethe ansatz equations (\[BAE\]) with any one of the grading $\{p_{j}\}$ for $sl(r+1|s+1)$, one can get the set of the Bethe ansatz equations with any other grading $\{p_{j}\}$ for $sl(r+1|s+1)$. Furthermore, we note that the transformation (\[trans\]) corresponds to the reflection $\omega_{b} \in {\cal SW}({\cal G})$ for odd simple root $\alpha_{b}$ with $(\alpha_{b}|\alpha_{b})=0$. This fact follows from the relations: $(\omega_{\alpha_{b}}(\alpha_{b})|\omega_{\alpha_{b}}(\alpha_{b+1}))=
-(-p_{b+1})$, $(\omega_{\alpha_{b}}(\alpha_{b-1})|\omega_{\alpha_{b}}(\alpha_{b-1}))=
p_{b-1}+(-p_{b})$, etc.
Summary and discussion
======================
In the present paper, we have executed analytic Bethe ansatz based upon the Bethe ansatz equations (\[BAE\]) with any simple root systems of the Lie superalgebra $sl(r+1|s+1)$. Pole-freeness of eigenvalue formula of transfer matrix in dressed vacuum form was shown for a wide class of finite dimensional representations labeled by skew-Young superdiagrams. Functional relation has been given especially for the eigenvalue formulae of transfer matrices in dressed vacuum form labeled by rectangular Young superdiagrams, which is a special case of Hirota bilinear difference equation with a restrictive relation. There are earlier results [@T2] for the distinguished simple root system of $sl(r+1|s+1)$, many of which are special case of the results in the present paper. We discussed how the set of the Bethe ansatz equations for any simple root system of $sl(r+1|s+1)$ is related to the one for any other simple root system of $sl(r+1|s+1)$ under the particle-hole transformation. And then, we pointed out that the particle-hole transformation is connected with the reflection with respect to the element of the Weyl supergroup for odd simple root $\alpha $ with $(\alpha | \alpha)=0$.
It should be emphasized that our method explained in the present paper is still valid even if such factors like gauge factor, extra sign (different from $(-1)^{{\rm deg}(\alpha_{a})}$ in (\[BAE\])), etc. appear in the Bethe ansatz equation (\[BAE\]) as long as such factors do not influence the analytical property of the right hand side of the Bethe ansatz equation (\[BAE\]).
In reference [@LWZ], functional relations for any fusion type transfer matrices associated with any (not always rectangular) Young diagrams of simple Lie algebra $A_{r}$ was given. Similar functional relations for suitable boundary conditions will be also valid for $sl(r+1|s+1)$ case.
In reference [@FR], coincidence between the free field realization of the generators of $U_{q}({\cal G}^{(1)})$ and eigenvalue formulae [@KS1] of transfer matrices in dressed vacuum form in the analytic Bethe ansatz was discussed associated with classial simple Lie algebras ${\cal G}$. As for a Lie superalgebra ${\cal G}$ case, nobody has studied such a relation so far. A deeper inspection will be desirable.
It will be interesting problems to extend a similar analysis discussed in this paper for other Lie superalgebras, such as $B(m|n),C(n)$ and $D(m|n)$.
Finally we note that functional relations among fusion transfer matrices at [*finite*]{} temperatures have been given in the preprint [@JKS] quite recently using quantum transfer matrix approach. In addition, these functional relations are transformed into TBA equations without using string hypothesis.\
[**Acknowledgments**]{}\
The author would like to thank Professor A. Kuniba for encouragement. He also thanks Dr J. Suzuki for discussions.
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|
---
abstract: 'We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schrödinger equation is the only possible choice, up to a shift of the global phase.'
author:
- Jonathan Steinberg
- 'H. Chau Nguyen'
- Matthias Kleinmann
bibliography:
- 'the.bib'
title: 'Quaternionic quantum theory admits universal dynamics only for two-level systems'
---
Introduction
============
Our understanding of quantum theory has significantly improved by investigating alternatives to quantum theory and analyzing how these alternatives would or would not be at variance with observations or expectations on the structure of a physical theory. Recently, these investigations are mostly based on the formalism of generalized probabilistic theories, where the fundamental objects are the convex sets of states and measurements. Different sets of assumptions have been found which are sufficient to single out quantum theory as the only possible theory . A special role in these set of assumptions plays the analysis of the dynamics of such generalizations of quantum theory, see, for example, Ref. . Specifically, in quantum mechanics (and also in classical mechanics) there is an intimate relation between the Hamiltonian $H$ as the energy observable and the generator of time shifts $-\frac i\hbar H$ as it occurs in the Schrödinger equation.
Maybe the most notable early alternatives to quantum theory that have been studied in great detail are real and quaternionic quantum mechanics. Those are based on the question, why the wave function in quantum theory is complex valued and whether it would also be possible to formulate quantum theory over different fields, in those cases using real valued or quaternionic valued wave functions. The two main concerns for the real and quaternionic case are the composition of systems via a tensor product and a suitable modification of the Schrödinger equitation. For real quantum theory, both topics lead to basically the same conclusion, namely that there must be a superselection rule . In quaternionic quantum theory, the tensor product is complicated, at best [@Horwitz1984; @Baez2012; @Joyce2001; @Ng2007]. However, the need for composing systems has also been questioned recently [@Chiribella2018]. A consistent dynamics in quaternionic quantum theory has been formulated, however, also at the price of a superselection rule, where only a subspace of all self-adjoint operators can be used as the Hamiltonian of a system [@Finkelstein1962]. Nonetheless, Peres [@Peres1979] suggested a possible experiment on the basis of noncommuting phases which could reveal the characteristics of quaternionic quantum theory. Corresponding experiments were realized using neutron interferometry [@Kaiser1984] and using single photon interferometry .
In this paper we use the standard quaternionic formulation [@Adler1995] of states and observables, that is, states are normalized vectors and observables are self-adjoint matrices over the quaternions. We ask which dynamical evolution is admissible in this case. For canonical quantum theory, the Schrödinger equation implies that the state evolves according to a unitary group parametrized by time. Each such group is determined by the Hamiltonian of the system. We seek for a similar construction with the aim to derive a Schrödinger-type equation for quaternionic quantum theory. In contrast to previous work [@Finkelstein1962; @Adler1995], we are interested in the case where the set of Hamiltonians is unrestricted, that is, every self-adjoint operator must induce some dynamics. We find that this is only possible for one-level or two-level systems and that the corresponding Schrödinger-type equation is necessarily of the form $$\label{eq:qseq}
\hbar\dot\psi(t) = [AH+HA-\operatorname{tr}(H)A]\psi(t),$$ where $H$ is the Hamiltonian and $A$ is a skew-adjoint operator which is independent of $H$. The term in the square brackets replaces here $-iH$ from the canonical Schrödinger equation. We arrive at this result assuming that the term in the square brackets is an ${\mathds{R}}$-linear expression in $H$ and that it commutes with $H$.
The paper is organized as follows. In Section \[sec:canonicalqt\] we consider the case of canonical quantum theory. We review the connection between the Schrödinger equation, generators of time shifts, and Stone’s theorem and develop then the axioms for the correspondence between the Hamiltonian and the generator of time shifts. In Theorem \[thm:cphi\] we establish that these axioms are sufficient to reproduce the usual Schrödinger equation. In Section \[sec:quaternionicqt\] we then turn to the quaternionic case. We first summarize quantum theory over the quaternions. In Theorem \[thm:qrep\] we characterize the possible dynamics in quaternionic quantum theory and subsequently discuss alternatives to the axioms that lead to Eq. before we conclude in Section \[sec:conclusions\].
Time evolution in canonical quantum theory {#sec:canonicalqt}
==========================================
In quantum mechanics, the time evolution of a system is described by the Schrödinger equation, $$i\hbar \dot\psi(t) = H_S \psi(t).$$ For a time-independent Hamiltonian this gives rise to the unitary time evolution operator $$U^S_t = {\mathrm{e}}^{-\frac i\hbar H_St},$$ which provides a solution of the Schrödinger equation via $\psi(t_0+t)= U^S_t
\psi(t_0)$.
Unitary groups and Stone’s theorem
----------------------------------
There is a different way to obtain a time evolution operator $U_t$ of the same structure, without building on the Schrödinger equation. This is based on the assumptions that the transformation $U_t\colon
\psi(t_0)\mapsto \psi(t_0+t)$ is linear in $\psi(t_0)$, preserves the norm of $\psi(t_0)$, is independent of $t_0$, and is continuous in $t$. More precisely, $(U_t)_{t\in {\mathds{R}}}$ must be a strongly continuous unitary group, that is,
$U_t$ is unitary for all $t$, with $U_0={\mathds{1}}$,
$U_{s+t}= U_s U_t$ for all $s$, $t$, and
$t\mapsto U_t$ is strongly continuous.
Condition (i) expresses linearity and isometry and $U_0={\mathds{1}}$ is used to implement the identity $\psi(t_0)= U_0\psi(t_0+0)$. (We do not consider antiunitary transformations as allowed by Wigner’s theorem.) Condition (ii) follows from the independence of $t_0$ and condition (iii) is a specification of the assumption of continuity. Strong continuity refers to the strong operator topology and reduces because of (i) and (ii) to $\lim\limits_{t\to 0} \norm{U_t\psi- \psi}= 0$ for all $\psi$.
The fundamental representation theorem of strongly continuous one-parameter unitary groups is due to Stone (see, for example, Ref. ). For every such group $(U_t)_{t\in{\mathds{R}}}$ there exists a unique self-adjoint operator $G$, such that $$\label{eq:cstone}
U_t = {\mathrm{e}}^{-iGt}$$ holds. This result is valid for general Hilbert spaces, with subtleties occurring if $t\mapsto U_t\psi$ is not differentiable at $t=0$ for some $\psi$. Clearly, in the finite-dimensional case this function is always differentiable.
Hamiltonians as generators of time shifts {#sec:cgens}
-----------------------------------------
From a physical perspective, the skew-adjoint operator $-iG$ in Eq. is responsible for time shifts and in this sense it is the generator of time shifts. The Schrödinger equation implies that the correspondence between the Hamiltonian and the generator of time shifts is obtained as $-iG=-\frac i\hbar
H$. But is this the only way to establish a correspondence between the generator of time shifts and the Hamiltonian and if not, how can we classify the different possibilities?
To answer this question, we write the Schrödinger equation in the form $$\dot\psi(t) = \Phi_S(H) \psi(t)$$ yielding the time evolution operator $U_t^S= {\mathrm{e}}^{\Phi_S(H)t}$. Here $\Phi_S(H)$ is the generator of time shifts associated to the Hamiltonian $H$, that is $\Phi_S$ is a linear map from the self-adjoint matrices to the skew-adjoint matrices, $$\Phi_S\colon H \mapsto -\tfrac i\hbar H.$$ We denote by $\operatorname{{\mathbf H}}({\mathds{C}}^n)$ \[$\operatorname{{\mathbf A}}(n,{\mathds{C}})$\] the ${\mathds{R}}$-vector space of self-adjoint (skew-adjoint) complex $n\times n$ matrices. The skew-adjoint matrices obey $A= -A^\dag$ and together with the self-adjoint matrices, which satisfy $H= H^\dag$, they span the set of all matrices, that is, $$\operatorname{Mat}(n,{\mathds{C}})= \operatorname{{\mathbf H}}(n,{\mathds{C}}) \oplus_{\mathds{R}}\operatorname{{\mathbf A}}(n,{\mathds{C}}).$$ The dimension of $\operatorname{{\mathbf H}}(n,{\mathds{C}})$ equals the dimension of $\operatorname{{\mathbf A}}(n,{\mathds{C}})$ and $\Phi_S$ is a one-to-one mapping between these two vector spaces. This coincidence of dimensions is a peculiarity of the complex matrices. Over the reals as well as over the quaternions ${\mathds{H}}$, the dimensions differ, specifically, $\dim\operatorname{{\mathbf H}}(n,{\mathds{R}})= \dim\operatorname{{\mathbf A}}(n,{\mathds{R}})+n$ and $\dim\operatorname{{\mathbf H}}(n,{\mathds{H}})= \dim\operatorname{{\mathbf A}}(n,{\mathds{H}})-2n$.
In order to generalize $\Phi_S$, we consider an arbitrary relation $\varphi\subset \operatorname{{\mathbf H}}(n,{\mathds{C}})\times \operatorname{{\mathbf A}}(n,{\mathds{C}})$ between the Hamiltonians and the generators of time shifts. In literature, $\varphi$ is sometimes called a dynamical correspondence [@Alfsen1998; @Barnum2014; @Branford2018]. We add several restrictions to this general case by requiring that $\varphi$ is
${\mathds{R}}$-homogeneous, $(H,A)\in \varphi$ implies $(\lambda H,\lambda
A)\in\varphi$ for all $\lambda\in {\mathds{R}}$,
additive, $(H,A)\in \varphi$ and $(H',A')\in \varphi$ implies $(H+H',A+A')\in
\varphi$, and
commuting, $(H,A)\in \varphi$ implies $HA=AH$.
These assumptions have physical motivations and appeared earlier in literature, see, for example, Refs. . The relation $\varphi$ should be ${\mathds{R}}$-homogeneous (DC1) to match the intuition that higher energies correspond in direct proportion to faster time evolutions and vice versa. The assumption of additivity (DC2) can be easily justified for the case where $H$ and $H'$ commute since in this case we have the addition of two Hamiltonians that differ only in spectrum. Then for any eigenstate of $H$ and $H'$ an argument similar to the motivation for assumption (DC1) can be applied. Noncommuting Hamiltonians most prominently appear in the form of interaction Hamiltonians, where $H=H_A+H_B+\mu H_I$. In many situations the interaction strength $\mu$ is an external experimental parameter, for example, the strength of a magnetic field, and hence additivity is a reasonable assumption. This reasoning might be more difficult to apply, if the Hamiltonian does not emerge as an effective description, but is an unalterable property of the system. Finally, the relation $\varphi$ should also be commuting (DC3), so that the Hamiltonian is invariant under time shifts. That is, the observable $H$ should be a conserved quantity under its own time evolution, $$U_t^\dag H U_t = {\mathrm{e}}^{At} H {\mathrm{e}}^{-At} = H,$$ for all $(H, A)\in \varphi$. The properties (DC1) and (DC2) make $\varphi$ an ${\mathds{R}}$-linear space and any relation $\varphi$ obeys (DC1)–(DC3) if and only if the set $\set{H+A|(H,A)\in \varphi}$ is an ${\mathds{R}}$-linear subspace of the normal matrices. Note that we do not include another familiar condition on $\varphi$, namely, that it should be
covariant under unitary transformations,
that is, $(H, A)\in \varphi$ implies $(V^\dag HV,V^\dag A V)\in \varphi$, for any unitary $V$.
It is conceivable that the Hamiltonian alone does not determine the time evolution completely and that the map $\varphi_1\colon H\mapsto \set{A|
(H,A)\in \varphi}$ is multivalued. The set $\varphi_1(H)$ could also be empty, in which case $H$ would not correspond to any dynamics, that is, $H$ would be unphysical. Conversely, the same dynamics might arise from different Hamiltonians so that $\varphi_2: A\mapsto \set{H|(H,A)\in \varphi}$ is multivalued. Also the set $\varphi_2(A)$ could be empty and hence the corresponding time evolution $t\mapsto U_t$ would be unphysical. In this paper, we focus on the first case and we only consider maps $\Phi\colon
\operatorname{{\mathbf H}}(n,{\mathds{C}})\to \operatorname{{\mathbf A}}(n,{\mathds{C}})$ such that $\varphi_1(H)= \set{\Phi(H)}$. The latter condition is not a restriction, since we can always obtain $\varphi$ from a family of maps $(\Phi_y)_y$ via $\varphi= \set{(H,\Phi_y(H))}_y$.
The conditions (DC1)–(DC3) are equivalent to the condition that $\Phi$ is a commuting ${\mathds{R}}$-linear map, that is, $\Phi$ is ${\mathds{R}}$-linear with $H\Phi(H)=\Phi(H)H$. We now demonstrate that such maps must have a very specific form. For this we use the following result by Brešar.
\[l:bresar\] Let $\mathcal A$ be a simple unital ring. Then every commuting additive map $f$ on $ \mathcal A$ is of the form $f(x) =
zx + g(x)$, where $z\in \mathcal Z (\mathcal A)$ and $g\colon \mathcal A \to
\mathcal Z(\mathcal A)$ is an additive map.
Here, $\mathcal Z(\mathcal A)$ denotes the center of $\mathcal A$, that is, the elements of $\mathcal A$ that commute with all of $\mathcal A$. Since $\operatorname{Mat}(n,{\mathds{C}})$ is a simple unital ring with center ${\mathds{C}}{\mathds{1}}$, we can in principle apply Lemma \[l:bresar\] in order to characterize all maps $\Phi$. However, we first need to extend the domain of $\Phi$ to be all of $\operatorname{Mat}(n,{\mathds{C}})$. This is readily achieved by the canonical extension $\Phi_c$ of $\Phi$ via $$\Phi_c(M_1 + iM_2)= \Phi(M_1) + i\Phi(M_2),$$ where $X= M_1+iM_2$ is the unique decomposition of $X\in \operatorname{Mat}(n,{\mathds{C}})$ into its self-adjoint and skew-adjoint part and $M_1$, $M_2$ are self-adjoint matrices. Clearly this extension is additive (${\mathds{R}}$-linear) if $\Phi$ is additive (${\mathds{R}}$-linear). The extension is also commuting if $\Phi$ is additive and commuting. Indeed we have $[\Phi(M_1),M_2]+ [\Phi(M_2),M_1]= 0$ due to $[\Phi(M_1+M_2),M_1+M_2]= 0$ and hence $[\Phi_c(X), X]= i([\Phi (M_1), M_2]+
[\Phi(M_2),M_1])= 0$ holds. This embedding together with Lemma \[l:bresar\] yields the following characterization of all maps obeying (DC1)–(DC3).
\[thm:cphi\] Every commuting ${\mathds{R}}$-linear map $\Phi\colon \operatorname{{\mathbf H}}(n, {\mathds{C}}) \to \operatorname{{\mathbf A}}(n,
{\mathds{C}})$ is of the form $$\Phi(H)= i \lambda H + i\operatorname{tr}(B H){\mathds{1}},$$ where $\lambda\in {\mathds{R}}$ and $B\in\operatorname{{\mathbf H}}(n,{\mathds{C}})$.
Applying Lemma \[l:bresar\] to the extended map $\Phi_c$ yields $\Phi_c(H)=
\eta H+\operatorname{tr}(QH)$ with $\eta\in {\mathds{C}}$ and $Q\in \operatorname{Mat}(n,{\mathds{C}})$, if $H\in
\operatorname{{\mathbf H}}(n,{\mathds{C}})$. Note that we used the ${\mathds{R}}$-homogeneity of $\Phi_c$ to write $g(H)=
\operatorname{tr}(QH)$. Since $\Phi_c(H)$ must be skew-adjoint, $\eta+\eta^*=0$ and $Q+Q^\dag=0$ follows, that is, $\eta\in i{\mathds{R}}$ and $Q\in i\operatorname{{\mathbf H}}(n,{\mathds{C}})$.
Note that $\Phi(H)$ is covariant if and only if $B$ is real multiple of ${\mathds{1}}$. In quantum mechanics, we have $\lambda= -\frac1\hbar$ and $B=0$. The value of $\lambda$ constitutes a constant of nature, including a sign convention. The term involving $B$ cannot be measured on a single system since it would only cause a global phase shift on the quantum state. In an interferometer-type setup even this phase shift would be accessible but is in contradiction to observation.
Quaternionic quantum theory {#sec:quaternionicqt}
===========================
We have outlined the formalism for obtaining the Schrödinger equation for the familiar case of complex quantum mechanics. Our choices and assumptions have been made such that we can extend our considerations to construct a dynamical quantum theory over the quaternions. We start by summarizing a quaternionic version of quantum theory (see, for example, Ref. ) and we then proceed by characterizing possible expressions for the Schrödinger equation.
The quaternions
---------------
The quaternions ${\mathds{H}}$ are an extension of the real and complex numbers. They form an associative division ring where multiplication is noncommutative. Any quaternion $q \in {\mathds{H}}$ can be written in the form $$q = a_1 + a_2 i + a_3 j + a_4 k,$$ where the coefficients $a_\ell$ are real numbers and $i,j,k$ are the quaternion units, which play a role similar to the complex unit $i$. The real part of $q$ is $\operatorname{Re}(q) = a_1$ and the imaginary part is the triple $\operatorname{Im}(q) = (a_2,a_3,a_4)$. The multiplication on ${\mathds{H}}$ is commutative for the real numbers and otherwise determined by $$i^2 = j^2 = k^2 =-1 \quad \text{and} \quad ijk=-1,$$ yielding $ij=k=-ji$, $jk=i=-kj$, $ki=j=-ik$. Similar to the complex numbers, conjugation is defined by $$q^* = a_1- a_2i- a_3j- a_4k,$$ yielding the rules $(uv)^* = v^*u^*$, $qq^*=q^*q$, and $(q^*)^* = q$. The modulus $\abs{q}=\sqrt{q q^*}$ induces the euclidean norm $\norm{(a_1,a_2,a_3,a_4)}= \abs{a_1+a_2i+a_3j+a_4k}$. This way, the quaternions are a complete normed ${\mathds{R}}$-algebra.
We identify the complex numbers as a subset of the quaternions by identifying the complex unit $i$ with the quaternion unit $i$. This allows us to write uniquely $q=a+bj$ for $a,b\in{\mathds{C}}$. Similar to the representation of complex numbers as real $2 \times 2$ matrices, the quaternions can be represented as complex matrices, $$a+bj \mapsto \begin{pmatrix} a & b \\ -b^* & a^* \end{pmatrix}.$$ This map is a $\dag$-monomorphism of the corresponding real algebras, were the involution $\dag$ reduces on the quaternions to the conjugation $*$.
Modules and matrices
--------------------
Since quantum mechanics is formulated on the basis of complex Hilbert spaces, we use a similar structure over the quaternions, but taking into account the the noncommutativity of the quaternions. We consider here the $n$-fold direct product of quaternions, denoted by ${\mathds{H}}^n$. It forms a free bimodule and possesses, apart from commutativity, most properties of a vector space. In particular, since it arises from a direct product, it can be equipped with the canonical basis $(e^{(1)},e^{(2)},\dotsc,e^{(n)})$.
For $x,y\in {\mathds{H}}^n$ with $x=\sum_i x_i e^{(i)}$ and $y$ similar we define the inner product to be of the canonical form, $\exv{x,y}= \sum_i x^*_i y_i =
\exv{y,x}^*$, giving also rise to the norm $\norm{x}=\sqrt{\exv{x,x}}$ and turning ${\mathds{H}}^n$ into a Hilbert module. For scalar multiplication with $\alpha\in {\mathds{H}}$ we obtain the rules $\exv{x\alpha,y}=\alpha^* \exv{x,y}$ and $\exv{x,y\alpha}= \exv{x,y}\alpha$. This suggests that scalar multiplication in ${\mathds{H}}^n$ is preferably taken from the right, although, technically, ${\mathds{H}}^n$ is a bimodule.
We take linear maps $M\colon {\mathds{H}}^n\to {\mathds{H}}^m$ to be right-homogeneous, $M(x\alpha)=M(x)\alpha$ which allows for a representation of $M$ as an $m\times n$ matrix $(M_{i,j})_{i,j}$ via $M_{i,j}= \exv{e^{(i)},Me^{(j)}}$. Then $\exv{x,M(y\alpha)}= \sum_{i,j} x_i^* M_{i,j} y_j \alpha$. We consider $\operatorname{Mat}(n, {\mathds{H}})$ as an ${\mathds{R}}$-algebra, ignoring that it can be treated consistently as a left ${\mathds{H}}$-module. Where unambiguous, we use $\alpha\in {\mathds{H}}$ also as the linear map $x\mapsto
\sum_i \alpha x_i e^{(i)}$. The adjoint $\dag$ of a linear map is defined as usual, $\exv{x,M(y)}=
\exv{M^\dag(x),y}$, and therefore $(M^\dag)_{i,j}=(M_{j,i}^*)_{i,j}$. Since linear maps are well-represented by matrices, we mostly use the latter notion. Self-adjoint and skew-adjoint matrices are defined in the obvious way. For unitary matrices $U\in \operatorname{Mat}(n,{\mathds{H}})$, we note that $U^\dag U={\mathds{1}}$ is equivalent to [@Zhang1997] $U U^\dag = {\mathds{1}}$. If $M$ is normal, $M^\dag M=M M^\dag$, then there exits a diagonal matrix $D$ with entries in $\set{a+bi | a,b\in {\mathds{R}},\; b\ge 0}$ and a unitary $U$ such that [@Zhang1997] $M=U^\dag D U$. The trace $\operatorname{tr}(M)=\sum_i M_{i,i}$ for self-adjoint matrices is invariant under unitary transformations, $\operatorname{tr}(UMU^\dag)=\operatorname{tr}(M)$, and thus equivalent to the sum of diagonal elements of $D$. This follows from the fact that one can readily construct matrices of the form $(b_ib_j^*)_{i,j}$ the real span of which are the self-adjoint matrices.
It is sometimes convenient to use the embedding $\Lambda\colon \operatorname{Mat}(n, {\mathds{H}})
\to \operatorname{Mat}(2n, {\mathds{C}})$, $$\Lambda\colon A+Bj \mapsto
\begin{pmatrix} A & B \\ -B^* & A^* \end{pmatrix}.$$ This map is a $\dag$-monomorphism of the corresponding ${\mathds{R}}$-algebras. In particular, we have $\Lambda[(r A+BC^\dag)]= r\Lambda(A)+
\Lambda(B)\Lambda(C)^\dag$ for $r\in {\mathds{R}}$ and $A,B,C\in \operatorname{Mat}(n,{\mathds{H}})$. The map $\Lambda^{-1}\colon \operatorname{Mat}(2n,{\mathds{C}})\to \operatorname{Mat}(n,{\mathds{H}})$, $$\Lambda^{-1}\colon \begin{pmatrix} A&B \\ C&D\end{pmatrix}
\mapsto A+Bj,$$ is an ${\mathds{R}}$-linear left inverse of $\Lambda$, which, however, it is not preserving the algebraic properties of $\operatorname{Mat}(2n,{\mathds{C}})$.
Stone’s theorem
---------------
In order to study the dynamics in quaternionic quantum theory we proceed similar to the complex case by studying continuous unitary groups $(U_t)_t$. Then, an analogous result to Stone’s theorem holds.
\[thm:qstone\] For every continuous unitary group $(U_t)_{t \in {\mathds{R}}} \subset \operatorname{Mat}(n,
{\mathds{H}})$ there exist a unique skew-adjoint matrix $A\in \operatorname{Mat}(n, {\mathds{H}})$ such that $U_t = {\mathrm{e}}^{At}$.
This theorem has been proved in Ref. . For completeness, we provide here a proof for finite dimensions.
Since the embedding $\Lambda$ is a mapping between finite-dimensional vector spaces, the family $[\Lambda(U_t)]_t\subset \operatorname{Mat}(2n,{\mathds{C}})$ is also a continuous unitary group. By virtue of Stone’s theorem there exists a skew-adjoint matrix $B\in
\operatorname{Mat}(2n,{\mathds{C}})$ such that $\Lambda(U_t)= {\mathrm{e}}^{Bt}$. The map $\mathcal W\colon t\mapsto \Lambda(U_t)$ is also differentiable, so that we can write for $\mathcal U\colon t\mapsto U_t$, $$\Lambda^{-1}(\dot{\mathcal W})
= \frac{{\mathrm{d}}}{{\mathrm{d}}t} \Lambda^{-1} (\mathcal W)
= \dot{\mathcal U}.$$ The left hand side exists, proving that also $\mathcal U$ is differentiable. By letting $A= \dot{\mathcal U}(0)$ and using $U_{t+\delta}= U_\delta U_t$, we have $$\dot{\mathcal U}(t) = \lim_{\delta \to 0}
\frac{U_{t + \delta} - U_t}{\delta}= AU_t.$$ It remains to show that $A$ is skew-adjoint and satisfies $U_t={\mathrm{e}}^{At}$. Since $\Lambda(U_t)$ is unitary, the identity $\Lambda(A)\Lambda(\mathcal U)=
\Lambda(A\mathcal U) = \Lambda(\dot{\mathcal U})=\frac{\mathrm{d}}{{\mathrm{d}}t}
\Lambda(\mathcal U) = \dot{\mathcal W}= B\mathcal W = B\Lambda(\mathcal U)$, allows us to conclude that $\Lambda(A)=B$. This implies that $A$ is skew-adjoint, since $B$ is. Finally, applying $\Lambda^{-1}$ to $$\Lambda(U_t)= {\mathrm{e}}^{Bt}= {\mathrm{e}}^{\Lambda(A)t}=\Lambda({\mathrm{e}}^{At})$$ from the left gives us $U_t={\mathrm{e}}^{At}$. Uniqueness then follows immediately from $\dot{\mathcal U}(0)=A$.
We mention that the smoothness of the map $t\mapsto U_t$, which we show in the first part of the above proof, is a simple consequence of the fact that the unitary matrices form a Lie group. Indeed, for any Lie group $G$ a continuous homomorphism ${\mathds{R}}\to G$ is necessarily smooth [@Lee2012]. Also note that in contrast to the complex case, we consider here only the finite-dimensional case and therefore it is not necessary to use strong continuity.
Observables and generators of time shifts
-----------------------------------------
We now head for the characterization of the dynamics in a quaternionic version of quantum theory. So far we have obtained a result about the structure of all possible dynamical evolutions. But for a dynamical evolution to be useful we need to specify a notion of states and observables. Here, we proceed in complete analogy to quantum mechanics, that is, states are represented by normalized vectors and observables by self-adjoint matrices [@Adler1995]. The expectation value of an observable $H$ for a system in state $\psi$ is then given by $\exv{H}= \exv{\psi,H\psi}$. Clearly, the expectation value is always real and all states $\psi\alpha$ are equivalent for all $\alpha\in {\mathds{H}}$ with $\abs\alpha= 1$. The spectral theorem for self-adjoint matrices can also be written as $H=
\sum_k h_k \Pi_k$, with distinct eigenvalues $h_k\in {\mathds{R}}$ and self-adjoint projections $\Pi_k$ such that $\sum_k\Pi_k={\mathds{1}}$. The expectation values of the projections correspond then to the probability $p_k$ for observing the eigenvalue $h_k$, that is, $p_k=\exv{\Pi_k}$. This way we recover a large bit of the structure and physical interpretation of quantum theory.
With the same arguments as in the complex case, we assume that the time evolution of a state is generated by a continuous unitary group $(U_t)_t$ and by virtue of Theorem \[thm:qstone\] we have $U_t={\mathrm{e}}^{At}$. It is worth mentioning here a significant difference [@Adler1995] to the complex case, which occurs if we add a global, time-dependent phase to a state, $\psi(t)\to \psi_\alpha(t)= \psi(t)\alpha(t)$. We obtain $\dot\psi_\alpha(t)= A\psi_\alpha(t)+ \psi_\alpha(t)\varphi(t)$ where $\varphi(t)=\alpha^*(t)\dot\alpha(t)$. Since $\varphi(t)$ in general does not commute with $\psi_\alpha$, we cannot simply write $\dot\psi_\alpha(t)= (A+\varphi(t))\psi_\alpha(t)$, as it would be the case in the complex case.
In analogy to the complex case we are interested in the correspondence between observables and the generators of time shifts, in particular for the case where the observable is the Hamiltonian of the system. In the complex case, the multiplication with a purely imaginary number $i\lambda$ is the right choice to establish this correspondence. The matrix $A$ in Theorem \[thm:qstone\] can written in the polar decomposition as [@Finkelstein1962] $A=-XH$, where $X$ is unitary and skew-adjoint, $H$ is self-adjoint and positive semidefinite, and $[X,H]= 0$ holds. It is conceivable to identify $H$ in this decomposition with the Hamiltonian of the system while $X$ is kept constant. This limits the possible set of Hamiltonians to those with $[X,H]= 0$ which basically reduces quaternionic quantum theory to complex quantum theory [@Finkelstein1962].
Here, we are interested in the case where the Hamiltonian of the system can be any self-adjoint operator. The discussion in Section \[sec:cgens\] remains valid and leaves us with the task to characterize the commuting ${\mathds{R}}$-linear maps $\Phi\colon
\operatorname{{\mathbf H}}(n,{\mathds{H}}) \to \operatorname{{\mathbf A}}(n,{\mathds{H}})$. However, we cannot proceed similar to above to obtain a result akin to Theorem \[thm:cphi\]. The main difficulty here is that it is not possible to use an extension of $\Phi$ as in Section \[sec:cgens\], since such an extended map would be no longer commuting. We hence resort to a case-by-case study for different dimensions $n$.
For this it is useful to note that determining all admissible maps $\Phi$ can be reduced to finding the kernel of an ${\mathds{R}}$-linear map. Indeed, $\Phi$ is commuting if and only if the ${\mathds{R}}$-bilinear map $\mathcal
Q_\Phi\colon (X,Y)\mapsto [\Phi(X),Y]+[\Phi(Y),X]$ is trivial for all $X,Y\in
\operatorname{{\mathbf H}}(n,{\mathds{H}})$, that is, $\mathcal Q_\Phi=0$. Here, sufficiency follows immediately for $Y=X$ and necessity from $[\Phi(X+Y),X+Y]=0$. Thus, the set of commuting maps $\Phi$ is determined by the kernel of the ${\mathds{R}}$-linear map $\Phi\mapsto \mathcal Q_\Phi$. We perform this calculation with the help of a computer algebra system for $n=1,2,3$ with the following results. For $n=1$, all admissible maps are (obviously) given by $\Phi_1(H)= \alpha H$ for any $\alpha\in {\mathds{H}}$ with $\operatorname{Re}(\alpha)=0$. For $n=2$, all admissible maps are of the form $\Phi_2(H)= AH+HA-\operatorname{tr}(H)A$, where $A\in \operatorname{{\mathbf A}}(2,{\mathds{H}})$ is arbitrary. For $d=3$, only the trivial map $\Phi=0$ is commuting. This implies also that for $d>3$ all commuting maps must be trivial, as it follows from the following contradiction.
If there would exist a nontrivial commuting ${\mathds{R}}$-linear map $\Phi\colon
\operatorname{{\mathbf H}}(n,{\mathds{H}})\to \operatorname{{\mathbf A}}(n,{\mathds{H}})$ for $n\ge 3$, then there would also exist a nontrivial commuting ${\mathds{R}}$-linear map $\Phi_3\colon \operatorname{{\mathbf H}}(3,{\mathds{H}})\to \operatorname{{\mathbf A}}(3,{\mathds{H}})$.
The real span of the matrices of the form $(u_iu_j^*)_{i,j}$ with $u\in
{\mathds{H}}^n$ is $\operatorname{{\mathbf H}}(n,{\mathds{H}})$. Hence we can choose linearly independent vectors $x, y, z\in {\mathds{H}}^n$ such that $\exv{y, \Phi(X) z}\ne 0$ with $X_{i,j}=x_i x_j^*$. Then there is an isometry $\tau\colon {\mathds{H}}^3\to {\mathds{H}}^n$ such that $x,y,z\in
\tau{\mathds{H}}^3$ and $\pi=\tau\tau^\dag$ acts as identity on $x$, $y$, and $z$. Such an isometry can be constructed by means of the Gram–Schmidt procedure, yielding orthonormal vectors which are then used as columns of $\tau$. We define the map $\Phi_3$ as $\Phi_3\colon H\mapsto \tau^\dag \Phi(\tau
H\tau^\dag)\tau$. By construction, this map is ${\mathds{R}}$-linear and maps self-adjoint matrices to skew-adjoint matrices. Since $\Phi$ is commuting, we have $[ \Phi(\tau H\tau^\dag), \tau
H\tau^\dag]=0$ for any $H\in \operatorname{{\mathbf H}}(3,{\mathds{H}})$ and by multiplication with $\tau^\dag$ from the left and $\tau$ from the right, it follows that $\Phi_3$ is also commuting. Finally, $\Phi$ is nontrivial for $X'=\tau^\dag X\tau$, $y'=\tau^\dag y$, and $z'=\tau^\dag z$ in that we have $\exv{y',\Phi_3(X')z'}= \exv{\pi y, \Phi(\pi
X\pi)\pi z} =\exv{y,\Phi(X)z}\ne 0$, where $\pi X\pi=X$ follows from $\pi X
\pi v= \pi x\exv{x,\pi v}= x\exv{\pi x, v}= Xv$ for all $v\in {\mathds{H}}^n$.
In summary we have the following characterization.
\[thm:qrep\] For $n=1,2$, every commuting ${\mathds{R}}$-linear map $\Phi\colon \operatorname{{\mathbf H}}(n,{\mathds{H}})\to
\operatorname{{\mathbf A}}(n,{\mathds{H}})$ is of the form $$\Phi(H)= AH+HA-\operatorname{tr}(H)A,$$ where $A\in \operatorname{{\mathbf A}}(n,{\mathds{H}})$. Conversely, for $n=1,2$ every map of this form is commuting. For $n>2$ every commuting ${\mathds{R}}$-linear map is trivial, $\Phi=0$.
For a two-level system, $n=2$, the rank of $\Phi$ can be at most $\dim[\operatorname{{\mathbf H}}(n,{\mathds{H}})]-1=5$ due to $\Phi({\mathds{1}})=0$, for any choice of $A$. The maximal rank is achieved, for example, for $A$ being the diagonal matrix with diagonal $(i,0)$. A more intuitive choice for $A$ might be $-\frac i{2\hbar}$, which yields the quaternionic Schrödinger equation $$i\hbar\dot\psi(t)= [H_c- \tfrac12\operatorname{tr}(H_c){\mathds{1}}]\psi(t),$$ where $H_c=\frac12(H+iHi^*)$ is the complex part of the matrix $H$. Hence the corresponding map $\Phi$ has an ${\mathds{R}}$-rank of only 3.
Theorem \[thm:qrep\] has been obtained using the conditions (DC1)–(DC3). In contrast to the complex case, the resulting map $\Phi$ only obeys the covariance condition (DC4) if it is trivial. This can be seen by defining $\Phi_V\colon H\mapsto V\Phi(V^\dag HV)V^\dag=
A_VH+HA_V-\operatorname{tr}(H)A_V$ with $A_V=V A V^\dag$. Covariance requires then $\Phi_V=\Phi$ for any unitary $V$. Without loss of generality, we can choose $A=iD$ with $D$ a real diagonal matrix and then the case $V=j$ leads to $\Phi_V=-\Phi$.
However, the no-go statement in Theorem \[thm:qrep\] can be avoided by loosening our assumptions. For example, one can drop the assumption of additivity (DC2). Then a rather natural such candidate can be achieved as follows. We fix a spectral decomposition $H=U_H^\dag D_H U_H$ for every $H$ such that $U_{rH}$ is independent of $r\in {\mathds{R}}$. This allows us to define the ${\mathds{R}}$-homogeneous map $\Psi\colon
\operatorname{{\mathbf H}}(n,{\mathds{H}})\to \operatorname{{\mathbf A}}(n, {\mathds{H}})$ as $$\Psi\colon H\mapsto U_H^\dag i D_H U_H,$$ which can be easily seen to be commuting, but, in general, fails to be additive, due to Theorem \[thm:qrep\]. We can even satisfy the covariance condition (DC4) by requiring that $U_{VHV^\dag}=VU_H$ for all unitaries $V$. As mentioned before, another way to evade Theorem \[thm:qrep\] is to limit the set of Hamiltonians to be purely complex [@Finkelstein1962], $[H,i]=
0$. Then $\Phi_S\colon H\mapsto -\frac i\hbar H$ is admissible under (DC1)–(DC3) and we obtain the usual Schrödinger equation also in the quaternionic case.
Conclusions {#sec:conclusions}
===========
We studied the structure of universal dynamics in quantum theory using three main axioms (DC1)–(DC3). These axioms proved to be sufficient in order to recover the Schrödinger equation for the case of canonical quantum theory but when applied to quaternionic quantum theory they yield nontrivial dynamics only for dimension two (and one). For two-level systems, the resulting Schrödinger equation is not unique but can be modified by the choice of a skew-adjoint operator, see Eq. .
This makes quaternionic quantum theory for two-level systems exceptionally interesting. For higher dimensions, a possible conclusion from our analysis is to discard quaternionic quantum theory. However, it should be noted that the main reason for our no-go result is axiom (DC2), which requires that the sum of two Hamiltonian should correspond to the sum of two generators of time shifts. While this axiom is natural at least in canonical quantum theory it is an open question whether it is expendable nonetheless, and what this would imply for canonical quantum theory as well as quaternionic quantum theory.
This work was supported by the DFG, the FQXi Large Grant “The Observer Observed: A Bayesian Route to the Reconstruction of Quantum Theory”, and the ERC (Consolidator Grant 683107/TempoQ).
|
---
abstract: 'Reasoning is a crucial part of natural language argumentation. To comprehend an argument, one must analyze its [*warrant*]{}, which explains why its claim follows from its premises. As arguments are highly contextualized, warrants are usually presupposed and left implicit. Thus, the comprehension does not only require language understanding and logic skills, but also depends on common sense. In this paper we develop a methodology for reconstructing warrants systematically. We operationalize it in a scalable crowdsourcing process, resulting in a freely licensed dataset with warrants for 2k authentic arguments from news comments.[^1] On this basis, we present a new challenging task, the *argument reasoning comprehension task*. Given an argument with a claim and a premise, the goal is to choose the correct implicit warrant from two options. Both warrants are plausible and lexically close, but lead to contradicting claims. A solution to this task will define a substantial step towards automatic warrant reconstruction. However, experiments with several neural attention and language models reveal that current approaches do not suffice.'
author:
- |
Ivan Habernal$^{\dagger}$ Henning Wachsmuth$^{\ddagger}$ Iryna Gurevych$^{\dagger}$ Benno Stein$^{\ddagger}$\
$^\dagger$ Ubiquitous Knowledge Processing Lab (UKP) and Research Training Group AIPHES\
Department of Computer Science, Technische Universität Darmstadt, Germany\
[www.ukp.tu-darmstadt.de]{}\
$^{\ddagger}$ Faculty of Media, Bauhaus-Universität Weimar, Germany\
[.<lastname>@uni-weimar.de]{}
bibliography:
- 'bibliography.bib'
title: |
The Argument Reasoning Comprehension Task:\
Identification and Reconstruction of Implicit Warrants
---
**The Argument Reasoning Comprehension Task: Identification and Reconstruction of Implicit Warrants**
Ivan Habernal, Henning Wachsmuth, Iryna Gurevych, Benno Stein
This is a **pre-print non-final version** of the article accepted for publication at the *2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (NAACL 2018)*. The final official version along with the supplementary materials will be published on the ACL Anthology website in June 2018: <http://aclweb.org/anthology/>
Please cite this pre-print version as follows.
@InProceedings{habernal.et.al.2018.NAACL.arct,
title = {The Argument Reasoning Comprehension Task: Identification
and Reconstruction of Implicit Warrants},
author = {Habernal, Ivan and Wachsmuth, Henning and
Gurevych, Iryna and Stein, Benno},
publisher = {Association for Computational Linguistics},
booktitle = {Proceedings of the 2018 Conference of the North American
Chapter of the Association for Computational Linguistics:
Human Language Technologies},
pages = {(to appear)},
month = jun,
year = {2018},
address = {New Orleans, LA, USA}
url = {https://arxiv.org/abs/1708.01425}
}
Introduction
============
*Most house cats face enemies*. *Russia has the opposite objectives of the US*. *There is much innovation in 3-d printing and it is sustainable*.
What do the three propositions have in common? They were never uttered but solely [presupposed]{} in arguments made by the participants of online discussions. Presuppositions are a fundamental pragmatic instrument of natural language argumentation in which parts of arguments are left unstated. This phenomenon is also referred to as common knowledge [@Macagno.Walton.2014 p. 218], enthymemes [@Walton.2007a p. 12], tacit major premises [@Amossy.2009 p. 319], or implicit [*warrants*]{} [@Newman.1991 p. 8]. suggest that, when we comprehend arguments, we reconstruct their warrants driven by the cognitive principle of relevance. In other words, we go straight for the interpretation that seems most relevant and logical within the given context [@Hobbs.et.al.1993]. Although any incomplete argument can be completed in different ways [@Plumer.2016], it is assumed that certain knowledge is shared between the arguing parties [@Macagno.Walton.2014 p. 180].
Filling the gap between the claim and premises (aka reasons) of a natural language argument empirically remains an open issue, due to the inherent difficulty of reconstructing the world knowledge and reasoning patterns in arguments. In a direct fashion, let annotators write down implicit warrants, but they concluded only with a preliminary analysis due to large variance in the responses. In an indirect fashion, implicit warrants correspond to major premises in argumentation schemes; a concept heavily referenced in argumentation theory [@Walton.2012]. However, mapping schemes to real-world arguments has turned out difficult even for the author himself.
Our main hypothesis is that, even if there is no limit to the tacit length of the reasoning chain between claims and premises, it is possible to systematically reconstruct a meaningful warrant, depending only on what we take as granted and what needs to be explicit. As warrants encode our current presupposed world knowledge and connect the reason with the claim in a given argument, we expect that other warrants can be found which connect the reason with a different claim. In the extreme case, there may exist an [*alternative warrant*]{} in which the same reason is connected to the opposite claim.
The intuition of alternative warrants is key to the systematic methodology that we develop in this paper for reconstructing a warrant for the original claim of an argument. In particular, we first ‘twist’ the stance of a given argument, trying to plausibly explain its reasoning towards the opposite claim. Then, we twist the stance back and use a similar reasoning chain to come up with a warrant for the original argument. As we discuss further below, this works for real-world arguments with a missing piece of information that is taken for granted and considered as common knowledge, yet, would lead to the opposite stance if twisted.
We demonstrate the applicability of our methodology in a large crowdsourcing study. The study results in 1,970 high-quality instances for a new task that we call *argument reasoning comprehension*: Given a reason and a claim, identify the correct warrant from two opposing options. An example is given in Figure \[fig:example1\]. A solution to this task will represent a substantial step towards automatic warrant reconstruction. However, we present experiments with several neural attention and language models which reveal that current approaches based on the words and phrases in arguments and warrants do not suffice to solve the task.
**Title:** Is Marijuana a Gateway Drug? **Description:** Does using marijuana lead to the use of more dangerous drugs, making it too dangerous to legalize?
**Reason:** Milk isn’t a gateway drug even though most people drink it as children. And since *{Warrant 1 $|$ Warrant 2}*, **Claim:** Marijuana is not a gateway drug.
**Warrant 1:** milk is similar to marijuana
**Warrant 2:** milk is not marijuana\
The main contributions of this paper are (1) a *methodology* for obtaining implicit warrants realized by means of scalable crowdsourcing and (2) a new *task* along with a high-quality dataset. In addition, we provide (a) 2,884 user-generated arguments annotated for their stance, covering 50+ controversial topics, (b) 2,026 arguments with annotated reasons supporting the stance, (c) 4,235 rephrased reason gists, useful for argument summarization and sentence compression, and (d) a method for checking the reliability of crowdworkers in document and span labeling using traditional inter-annotator agreement measures.
Related Work
============
It is widely accepted that an argument consists of a *claim* and one or more *premises* (reasons) [@Damer.2013]. elaborated on a model of argument in which the reason supports the claim on behalf of a *warrant*. The abstract structure of an argument then is *Reason* $\rightarrow$ (since) *Warrant* $\rightarrow$ (therefore) *Claim*. The warrant takes the role of an inference rule, similar to the *major premise* in Walton’s terminology [@Walton.2007].
In principle, the chain *Reason* $\rightarrow$ [*Warrant*]{} $\rightarrow$ [*Claim*]{} is applicable to deductive arguments and syllogisms, which allows us to validate arguments properly formalized in propositional logic. However, most natural language arguments are in fact inductive [@Govier.2010 p. 255] or defeasible [@Walton.2007a p. 29].[^2] Accordingly, the unsuitability of formal logic for natural language arguments has been discussed by argumentation scholars since the 1950’s [@Toulmin.1958]. To be clear, we do not claim that arguments cannot be represented logically (e.g., in predicate logic), however the drift to *informal logic* in the 20th century makes a strong case that natural language argumentation is more than modus ponens [@vanEemeren.et.al.2014].
In argumentation theory, the notion of a *warrant* has also been contentious. Some argue that the distinction of warrants from premises is clear only in Toulmin’s examples but fails in practice, i.e., it is hard to tell whether the reason of a given argument is a premise or a warrant [@vanEemeren.et.al.1987 p. 205]. However, provides alternative views on modeling an argument. Given a claim and two or more premises, the argument structure is *linked* if the reasoning step involves the logical conjunction of the premises. If we treat a warrant as a simple premise, then the linked structure fits the intuition behind Toulmin’s model, such that premise and warrant combined give support to the claim. For details, see [@Freeman.2011 Chap. 4].
What makes comprehending and analyzing arguments hard is that claims and warrants are usually implicit [@Freeman.2011 p. 82]. As they are ‘taken for granted’ by the arguer, the reader has to infer the contextually most relevant content that she believes the arguer intended to use. To this end, the reader relies on common sense knowledge [@Oswald.2016; @Wilson.Sperber.2004].
The reconstruction of implicit premises has already been faced in computational approaches. In light of the design of their argument diagramming tool, pointed out that the automatic reconstruction is a task that skilled analysts find both taxing and hard to explain. More recently, as well as outlined the reconstruction of missing enthymemes or warrants as future work, but they never approached it since. To date, the most advanced attempt in this regard is from . The authors let annotators ‘reconstruct’ several propositions between premises and claims and investigated whether the number of propositions correlates with the semantic distance between the claim and the premises. However, they conclude that the written warrants heavily vary both in depth and in content. By contrast, we explore cases with a missing single piece of information that is considered as common knowledge, yet leading to the opposite conclusion if twisted. Recently, also experimented with reconstructing implicit knowledge in short German argumentative essays. In contrast to our work, they used expert annotators who iteratively converged to a single proposition.
As the task we propose involves natural language comprehension, we also review relevant work outside argumentation here. In particular, the goal of the semantic inference task *textual entailment* is to classify whether a proposition entails or contradicts a hypothesis [@Dagan.et.al.2009]. A similar task, *natural language inference*, was boosted by releasing the large SNLI dataset [@Bowman.et.al.2015] containing 0.5M entailment pairs crowdsourced by describing pictures. While the understanding of semantic inference is crucial in language comprehension, argumentation also requires coping with phenomena beyond semantics. presented a large dataset for reading comprehension by answering questions over Wikipedia articles (SQuAD). In an analysis of this dataset found, though, that only 6.2% of the questions require causal reasoning, 1.2% logical reasoning, and 0% analogy. In contrast, these reasoning types often make up the core of argumentation [@Walton.2007]. introduced the *cloze story test*, in which the appropriate ending of a narrative has to be selected automatically. The overall context of this task is completely different to ours. Moreover, the narratives were written from scratch by explicitly instructing crowd workers, whereas our data come from genuine argumentative comments. Common-sense reasoning was also approached by who targeted the inference of common-sense facts from a large knowledge base. Since their logical formalism builds upon an enhanced version of Aristotle’s syllogisms, its applicability to natural language argumentation remains limited (see our discussion above). In contrast to our data source, a few synthetic datasets for general natural language reasoning have been recently introduced, such as answers to questions over a described physical world [@Weston.et.al.2016.ICLR] or an evaluation set of 100 questions in the Winograd Schema Challenge [@Levesque.et.al.2012].
Finally, we note that, although being related, research on argument mining, argumentation quality, and stance classification is not in the immediate scope of this paper. For details on these, we therefore refer to recent papers from or .
Argument Reasoning Comprehension {#sec:argument-reasoning-task}
================================
Let $R$ be a reason for a claim $C$, both of which being propositions extracted from a natural language argument. Then there is a warrant $W$ that justifies the use of $R$ as support for $C$, but $W$ is left implicit.
For example, in a discussion about whether declawing a cat should be illegal, an author takes the following position (which is her claim $C$): ‘It should be illegal to declaw your cat’. She gives the following reason ($R$): ‘They need to use their claws for defense and instinct’.[^3] The warrant $W$ could then be ‘If cat needs claws for instincts, declawing would be against nature’ or similar. $W$ remains implicit, because $R$ already implies $C$ quite obviously and so, according to common sense, any further explanation seems superfluous.
Now, the question is how to find the warrant $W$ for a given reason $R$ and claim $C$. Our key hypothesis in the definition of the argument reasoning comprehension task is the existence of an *alternative warrant* $AW$ that justifies the use of $R$ as support for the opposite $\neg C$ of the claim $C$ (regardless of the question of how strong this justification is).
For the example above, assume that we ‘twist’ $C$ to ‘It should be *legal* to declaw your cat’ ($\neg C$) but use the same reason $R$. Is it possible to come up with an alternative warrant $AW$ that justifies $R$? In the given case, ‘most house cats don’t face enemies’ would bridge $R$ to $\neg C$ quite plausibly. If we now use a reasoning based on $AW$ but twist $AW$ again such that it leads to the claim $C$, we get ‘most house cats face enemies’, which is a plausible warrant $W$ for the original argument containing $R$ and $C$. [^4]
Constructing an alternative warrant is not possible for all reason/claim pairs; in some reasons the arguer’s position is deeply embedded. As a result, trying to give a plausible reasoning for the opposite claim $\neg C$ either leads to nonsense or to a proposition that resembles a rebuttal rather than a warrant [@Toulmin.1958]. However, if both $W$ and $AW$ are available, they usually capture the core of a reason’s relevance and reveal the implicit presuppositions (examples follow further below).
Based on our key hypothesis, we define the argument reasoning comprehension task as:
*Given a reason $R$ and a claim $C$ along with the title and a short description of the debate they occur in, identify the correct warrant $W$ from two candidates: the correct warrant $W$ and an incorrect alternative warrant $AW$.*
An instance of the task is thus basically given by a tuple $(R, C, W, AW)$. The debate title and description serve as the context of $R$ and $C$. As it is binary, we propose to evaluate the task using accuracy.
Reconstruction of Implicit Warrants {#sec:dataset-construction}
===================================
We now describe our methodology to systematically reconstruct implicit warrants, along with the scalable crowdsourcing process that operationalizes the methodology. The result of the process is a dataset with authentic instances $(R, C, W, AW)$ of the argument reasoning comprehension task.
Source Data
-----------
Instead of extending an existing dataset, we decided to create a new one from scratch, because we aimed to study a variety of controversial issues in user-generated web comments and because we sought for a dataset with a permissive license.
As a source, we opted for the *Room for Debate* section of the New York Times.[^5] It provides authentic argumentation on contemporary issues with good editorial work and moderation — as opposed to debate portals such as *createdebate.com*, where classroom assignments, silly topics, and bad writing prevail. We manually selected 188 debates with polar questions in the title. These questions are controversial and provoking, giving a stimulus for stance-taking and argumentation.[^6] For each debate we created two explicit opposing claims, e.g., ‘It should be illegal to declaw your cat’ and ‘It should be legal to declaw your cat’. We crawled all comments from each debate and sampled about 11k high-ranked, root-level comments.[^7]
Methodology and Crowdsourcing Process
-------------------------------------
{width="\textwidth"}
The methodology we propose consists of eight consecutive steps that are illustrated in Figure \[fig:annotation-process\] and detailed below. Each step can be operationalized with crowdsourcing. For our dataset, we performed crowdsourcing on 5,000 randomly sampled comments using Amazon Mechanical Turk (AMT) from December 2016 to April 2017. Before, each comment was split into elementary discourse units (EDUs) using SistaNLP [@Surdeanu.2015].
#### 1. Stance Annotation
For each comment, we first classify what stance it is taking (recall that we always have two explicit claims with opposing stance). Alternatively, it may be neutral (considering both sides) or may not take any stance.[^8]
All 2,884 comments in our dataset classified as stance-taking by the crowdworkers were then also annotated as to whether being sarcastic or ironic; both pose challenges in analyzing argumentation not solved so far [@Habernal.Gurevych.2017.COLI].
#### 2. Reason Span Annotation
For all comments taking a stance, the next step is to select those spans that give a reason for the claim (with a single EDU as the minimal unit).
In our dataset, the workers found 5,119 reason spans, of which 2,026 lay within arguments. About 40 comments lacked any explicit reason.
#### 3. Reason Gist Summarization
This new task is, in our view, crucial for downstream annotations. Each reason from the previous step is rewritten, such that the reason’s gist in the argument remains the same but the clutter is removed (examples are given in the supplementary material which is available both in the ACL Anthology and the project GitHub site). Besides, wrongly annotated reasons are removed in this step. The result is pairs of reason $R$ and claim $C$.
All 4,294 gists in our dataset were summarized under Creative Commons Zero license (CC-0).
#### 4. Reason Disambiguation
Within our methodology, we need to be able to identify to what extent a reason itself implies a stance: While ‘$C$ because $R$’ allows for many plausible interpretations (as discussed above), whether $R \rightarrow C$ or $R \rightarrow \neg C$ depends on how much presupposition is encoded in $R$. In this step, we decide which claim ($C$ or $\neg C$) is most plausible for $R$, or whether both are similarly plausible (in the given data, respective reasons turned out to be rather irrelevant though).
We used only those 1,955 instances where $R$ indeed implied $C$ according to the workers, as this suggests at least some implicit presupposition in $R$.
#### 5. Alternative Warrant
This step is the trickiest, since it requires both creativity and ‘brain twisting’. As exemplified in Section \[sec:argument-reasoning-task\], a plausible explanation needs to be given why $R$ supports $\neg C$ (i.e., the alternative warrant $AW$). Alternatively, this may be classified as being impossible.
Exact instructions for our workers can be found in the provided sources. All 5,342 alternative warrants in our dataset are written under CC-0 license.
#### 6. Alternative Warrant Validation
As the previous step produces largely uncontrolled writings, we validate each fabricated alternative warrant $AW$ as to whether it actually relates to the reason $R$. To this end, we show $AW$ and $\neg C$ together with two alternatives: $R$ itself and a distracting reason. Only instances with correctly validated $R$ are kept.
For our dataset, we sampled the distracting reason from the same debate topic, using the most dissimilar to $R$ in terms of skip-thought vectors [@kiros2015skip] and cosine similarity. We kept 3,791 instances, for which the workers also rated how ‘logical’ the explanation of $AW$ was (0–2 scale).
#### 7. Warrant For Original Claim
This step refers to the second task in the example from Section \[sec:argument-reasoning-task\]: Given $R$ and $C$, make minimal modifications to the alternative warrant $AW$, such that it becomes an actual warrant $W$ (i.e., such that $R \rightarrow W \rightarrow C$).
For our dataset, we restricted this step to those 2,613 instances that had a ‘logic score’ of at least 0.68 (obtained from the annotations mentioned above), in order to filter out nonsense alternative warrants. All resulting 2,447 warrants were written by the workers again under CC0 license.
#### 8. Warrant Validation
To ensure that each tuple $(R, C, W, AW)$ allows only one logical explanation (i.e., either $R \rightarrow W \rightarrow C$ or $R \rightarrow AW \rightarrow C$ is correct, not both), all instances are validated again.
Disputed cases in the dataset (according to our workers) were fixed by an expert to ensure quality. We ended up with 1,970 instances to be used for the argument reasoning comprehension task.
[@p[0.1em]{}YYp[2em]{}>p[9em]{}p[1.9em]{}Y>p[10em]{}@]{} **\#** &**Methodology Step** &**Input Sata** &**Size** &**Output Data** &**Size** &**Quality Assurance** &**Use of Data**\
1 &Stance annotation &Comment, topic &5,000 &Stance-taking arguments &2,884 &Cohen’s $\kappa$ 0.58 &Argument stance detection; sarcastic argument detection\
2 &Reason span annotation &Stance-taking argument &2,884 &Reason spans (in arguments) &5,119 (2,026) &Krippendorff’s $\alpha_{\textrm{u}}$ 0.51 &Argument component detection; argumentative text segmentation\
3 &Reason gist summarization &Claim, reason span &5,119 &Summarized reason gists (in arguments) & 4,294 (1,927) &Qualified workers, manual inspection & Abstractive argument summarization; reason clustering; empirical analysis of controversies\
4 &Reason disambiguation &Reason gist, both claims &4,235 &Reasons implying original stance &1,955 &Cohen’s $\kappa$ 0.42 (task-important categories) &Argument component stance detection\
5 &Writing alternative warrant &Reason gist, opposing claim &1,955 &Fabricated warrant for reason and opposing claim &5,342 &Qualified workers, manual inspection &–\
6 &Alternative warrant validation &Opposing claim, alternative warrant, reason, distracting reason &5,342 &Plausible triple of reason, alternative warrant, and opposing claim &3,791 & – &Reason/Warrant relevance detection\
7 &Writing warrant for original claim &Claim, reason, alternative warrant &2,613\* &Warrant similar to alternative warrant for reason and claim &2,447 &Qualified workers, manual inspection &–\
8 &Warrant validation &Claim, reason, warrant, alternative warrant &2,447 &Validated triple of reason, warrant, and claim &1,970 &Qualified workers, experts for hard cases &Argument reasoning comprehension (our main task)\
Agreement and Dataset Statistics
--------------------------------
To strictly assess quality in the entire crowdsourcing process, we propose an evaluation method that enables ‘classic’ inter-annotator agreement measures for crowdsourcing, such as Fleiss’ $\kappa$ or Krippendorff’s $\alpha$. Applying $\kappa$ and $\alpha$ directly to crowdsourced data has been disputed [@Passonneau.Carpenter.2014]. For estimating gold labels from the crowd, several models have been proposed; we rely on MACE [@Hovy.et.al.2013]. Given a number of noisy workers, MACE outputs best estimates, outperforming simple majority votes. At least five workers are recommended for a crowdsourcing task, but how reliable is the output really?
We hence collected 18 assignments per item and split them into two groups (9+9) based on their submission time. We then considered each group as an independent crowdsourcing experiment and estimated gold labels using MACE for each group, thus yielding two ‘experts from the crowd.’ Having two independent ‘experts’ from the crowd allowed us to compute standard agreement scores. We also varied the size of the sub-sample from each group from 1 to 9 by repeated random sampling of assignments. This revealed how the score varies with respect to the crowd size per ‘expert’.
Figure \[fig:agreement-graph-stance\] shows the Cohen’s $\kappa$ agreement for stance annotation with respect to the crowd size computed by our method. As MACE also includes a threshold for keeping only the most confident predictions in order to benefit precision, we tuned this parameter, too. Deciding on the number of workers per task is a trade-off between the desired quality and the budget. For example, reason span annotation is a harder task; however, the results for six workers are comparable to those for the expert annotations of .[^9]
Table \[tab:pipeline-overview\] lists statistics of the entire crowdsourcing process carried out for our dataset, including tasks for which we created data as a by-product.
![\[fig:agreement-graph-stance\] Cohen’s $\kappa$ agreement for stance annotation on 98 comments. As a trade-off between reducing costs (i.e., discarding fewer instances) and increasing reliability, we chose 5 annotators and a threshold of 0.95 for this task, which resulted in $\kappa$ = 0.58 (moderate to substantial agreement). ](figures/agreement-graph-stance.pdf){width="\columnwidth"}
Examples {#sec:examples}
--------
Below, we show three examples in which implicit common-sense presuppositions were revealed during the construction of the alternative warrant $AW$ and the original warrant $W$. For brevity, we omit the debate title and description here. A full walk-through example is found in the supplementary material.
- Cooperating with Russia on terrorism ignores Russia’s overall objectives.
- Russia cannot be a partner.
- Russia has the same objectives of the US.
- Russia has the opposite objectives of the US.
<!-- -->
- Economic growth needs innovation.
- 3-D printing will change the world.
- There is no innovation in 3-d printing since it’s unsustainable.
- There is much innovation in 3-d printing and it is sustainable.
<!-- -->
- College students have the best chance of knowing history.
- College students’ votes do matter in an election.
- Knowing history doesn’t mean that we will repeat it.
- Knowing history means that we won’t repeat it.
Experiments {#sec:experiments}
===========
Given the dataset, we performed first experiments to assess the complexity of argument reasoning comprehension. To this end, we split the 1,970 instances into three sets based on the year of the debate they were taken from: 2011–2015 became the training set (1,210 instances), 2016 the development set (316 instances), and 2017 the test set (444 instances). This follows the paradigm of learning on past data and predicting on new ones. In addition, it removes much lexical and topical overlap.
Human Upper Bounds
------------------
To evaluate human upper bounds for the task, we sampled 100 random questions (such as those presented in Section \[sec:examples\]) from the test set and distributed them among 173 participants of an AMT survey. Every participant had to answer 10 questions. We also asked the participants about their highest completed education (six categories) and the amount of formal training they have in reasoning, logic, or argumentation (no training, some, or extensive). In addition, they specified for each question how familiar they were with the topic (3-point scale).
![\[fig:human-upper-bound\] Human upper bounds on the argument reasoning comprehension task with respect to education and formal training in reasoning, logic, or argumentation. For each configuration, the mean values are displayed together with the number of participants (above the bar) and with their standard deviations (error bars).](figures/human-upper-bound){width="\columnwidth"}
#### How Hard is the Task for Humans?
It depends, as shown in Figure \[fig:human-upper-bound\]. Whereas education had almost negligible influence on the performance, the more extensive formal training in reasoning the participants had, the higher their score was. Overall, 30 of the 173 participants scored 100%. The mean score for those with extensive formal training was 90.9%. For all participants, the mean was 79.8%. However, we have to note that some of the questions are more difficult than others, for which we could not control explicitly.
#### Does Topic Familiarity Affect Human Performance?
Not really, i.e., we found no significant (Spearman) correlation between the mean score and familiarity of a participant in almost all education/training configurations. This suggests that argument reasoning comprehension skills are likely to be independent of topic-specific knowledge.
Computational Models
--------------------
To assess the complexity of computationally approaching argument reasoning comprehension, we carried out first experiments with systems based on the following models.
The simplest considered model was the [*random baseline*]{}, which chooses either of the candidate warrants of an instance by chance. As another baseline, we used a 4-gram Modified Kneser-Ney *language model* trained on 500M tokens (100k vocabulary) from the C4Corpus [@Habernal.et.al.2016.LREC]. The effectiveness of language models was demonstrated by for the narrative cloze test where they achieved state-of-the-art results. We computed log-likelihood of the candidate warrants and picked the one with lower score.[^10]
To specifically appoach the given task, we implemented two neural models based on a bidirectional LSTM. In the standard [*attention*]{} version, we encoded the reason and claim using a BiLSTM and provided it as an attention vector after max-pooling to LSTM layers from the two available warrants $W_0$ and $W_1$ (corresponding to $W$ and $AW$, see below). Our more elaborated version used [*intra-warrant attention*]{}, as shown in Figure \[fig:network\]. Both versions were also extended with the debate title and description added as context to the attention layer (*w/ context*). We trained the resulting four models using the ADAM optimizer, with heavy dropout (0.9) and early stopping (5 epochs), tuned on the development set. Input embeddings were pre-trained word2vec’s [@Mikolov.2013]. We ran each model three times with random initializations.
![\[fig:network\] Intra-warrant attention. Only the attention vector for the warrant $W_1$ is shown; the attention vector for $W_0$ is constructed analogously. Grey areas represent a modification with additional context.](figures/network.pdf){width="0.9\columnwidth"}
To evaluate all systems, each instance in our dataset is represented as a tuple $(R, C, W_0, W_1)$ with a label (0 or 1). If the label is 0, $W_0$ is the correct warrant, otherwise $W_1$. Recall that we have two warrants $W$ and *AW* whose correctness depends on the claim: $W$ is correct for $R$ and the original claim $C$, whereas $AW$ would be correct for $R$ and the opposite claim $\neg C$. We thus doubled the training data by adding a permuted instance $(R, C, W_1, W_0)$ with the respective correct label; this led to increased performance. The overall results of all approaches (humans and systems) are shown in Table \[tab:results\]. Intra-warrant attention with rich context outperforms standard neural models with a simple attention, but it only slightly beats the language model on the dev set. The language model is basically random on the test set.
A manual error analysis of 50 random wrong predictions (a single run of the best-performing system on the dev set) revealed no explicit pattern of encountered errors. Drawing any conclusions is hard given the diversity of included topics and the variety of reasoning patterns. A possible approach would be to categorize warrants using, e.g., argumentation schemes [@Walton.2008] and break down errors accordingly. However, this is beyond the scope here and thus left for future work.
#### Can We Benefit from Alternative Warrants and Opposite Claims?
Since the reasoning chain $R \rightarrow AW \rightarrow \neg C$ is correct, too, we also tried adding respective instances to the training set (thus doubling the size). In this configuration, however, the neural models failed to learn anything better than a random guess. The reason behind is probably that the opposing claims are lexically very close, usually negated, and the models cannot pick this up. This underlines that argument reasoning comprehension cannot be solved by simply looking at the occurring words or phrases.
[@Yl@ll@l@]{} **Approach** & **Dev** & $(\pm)$ & **Test** & $(\pm)$\
Human average & & &.798 &.162\
Human w/ training in reasoning & & &.909 &.114\
Random baseline &.473 & .039 & .491 & .031\
Language model &.617 & & .500 &\
Attention & .488 & .006 & .513 & .012\
Attention w/ context &.502 & .031 & .512 & .014\
Intra-warrant attention & **.638** & .024 & .556 & .016\
Intra-warrant attent. w/ context & .637 & .040 & **.560** & .055\
Conclusion and Outlook
======================
We presented a new task called *argument reasoning comprehension* that tackles the core of reasoning in natural language argumentation — implicit warrants. Moreover, we proposed a methodology to systematically reconstruct implicit warrants in eight consecutive steps. So far, we implemented the methodology in a manual crowdsourcing process, along with a strategy that enables standard inter-annotator agreement measures in crowdsourcing.
Following the process, we constructed a new dataset with 1,970 instances for the task. This number might not seem large (e.g., compared to 0.5M from SNLI), but tasks with hand-crafted data are of a similar size (e.g., 3,744 Story Cloze Test instances). Also, the crowdsourcing process is scalable and is limited only by the budget.[^11] Moreover, we created several data ‘by-products’ that are valuable for argumentation research: 5,000 comments annotated with stance, which outnumbers the 4,163 tweets for stance detection of ; 2,026 arguments with 4,235 annotated reasons, which is six times larger than the 340 documents of ; and 4,235 summarized reason gists — we are not aware of any other hand-crafted dataset for abstractive argument summarization built upon authentic arguments.
Based on the dataset, we evaluated human performance in argument reasoning comprehension. Our findings suggest that the task is harder for people without formal argumentation training, while being solvable without knowing the topic. We also found that neural attention models outperform language models on the task.
In the short run, we plan to draw more attention to this topic by running a SemEval 2018 shared task.[^12] A deep qualitative analysis of the warrants from the theoretical perspective of reasoning patterns or argumentation schemes is also necessary. In the long run, an automatic generation and validation warrants can be understood as the ultimate goal in argument evaluation. It has been claimed that for reconstructing and evaluating natural language arguments, one has to fully ‘roll out’ their implicit premises [@vanEemeren.et.al.2014 Chap. 3.2] and leverage knowledge bases [@Wyner.et.al.2016.ArgCompJournal]. We believe that a system that can distinguish between the wrong and the right warrant given its context will be helpful in filtering out good candidates in argument reconstruction.
For the moment, we just made a first empirical step towards exploring how much common-sense reasoning is necessary in argumentation and how much common sense there might be at all.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the ArguAna Project GU 798/20-1 (DFG), and by the DFG-funded research training group “Adaptive Preparation of Information form Heterogeneous Sources” (AIPHES, GRK 1994/1).
[^1]: Available at <https://github.com/UKPLab/argumentreasoning-comprehension-task/>, including source codes and supplementary materials.
[^2]: A recent empirical example is provided by who propose possible approaches to identify patterns of inference from premises to claims in vaccine court cases. The authors conclude that it is extremely rare that a reasoning is explicitly laid out in a deductively valid format.
[^3]: The example is taken from our dataset introduced below.
[^4]: This way, we also reveal the weakness of the original argument that was hidden in the implicit premise. It can be challenged by asking the arguer whether house cats really face enemies.
[^5]: <https://www.nytimes.com/roomfordebate>
[^6]: Detailed theoretical research on polar and alternative questions can be found in [@vanRooy.Safarova.2003]; analyze bias and presupposition in polar questions.
[^7]: To remove ‘noisy’ candidates, we applied several criteria, such as the absence of quotations or URLs and certain lengths. For details, see the source code we provide. We did not check any quality criteria of arguments, as this was not our focus; see, e.g., [@Wachsmuth.et.al.2017.ACL] for argumentation quality.
[^8]: We also experimented with approaching the annotations top-down starting by annotating explicit claims, but the results were unsatisfying. This is in line with empirical observations made by who showed that the majority of claims in user-generated arguments are implicit.
[^9]: The supplementary material contains a detailed figure; not to be confused with Figure \[fig:agreement-graph-stance\] which refers to stance annotation.
[^10]: This might seem counterintuitive, but since $W$ is created by rewriting $AW$, it may suffer from some dis-coherency, which is then caught by the language model.
[^11]: In our case, the total costs were about \$6,000 including bonuses and experiments with the workflow set-up.
[^12]: <https://competitions.codalab.org/competitions/17327>
|
---
bibliography:
- 'biblio\_exemple.bib'
---
Introduction
============
Les requêtes approximatives (RA) représentent une solution pertinente qui permet d’améliorer le temps de réponse aux dépens de l’exactitude. Celles-ci sont adaptées aux requêtes avec agrégats pour lesquelles la précision au dernier décimal n’est pas exigée. La contribution de ce travail est le calcul en ligne de l’agrégation pour les requêtes flexibles avec groupement d’attributs et jointure en se basant sur l’Analyse Formelle de Concepts (AFC) de [@will:1982ID5] et le formalisme des sous-ensembles flous.
Agrégation en ligne des requêtes flexibles avec groupement d’attributs et jointure
==================================================================================
La première de notre approche consiste à générer la base de connaissances (BC) à partir de la base de données (BD). Elle est assurée par une procédure de classification non supervisée [@sassi12] sur les attributs relaxables[^1]. La deuxième phase comporte deux étapes. La première est une réécriture de la requête qui aura la forme suivante d’une RA:
[***SELECT** Fonction(Attribut), DegréConfiance As Confidence,*]{}
[*FonctionInterval(DegréDeConfiance) **FROM** Tables*]{}
[***WHERE** Attribut1 IS ConditionFlexible1 \[And Attribut2 IS ConditionFlexible2...\]*]{}
[***And** Table1.attribut=Table2.attribut... **GROUP BY** Table.Attribut;*]{}
L’étape suivante consiste à construire un échantillon par jointure à partir de la BC. En effet, au lieu d’interroger toute la BC qui contient des milliers d’enregistrements, nous interrogeons un échantillon de la BC qui est constitué par un sous ensemble de tuples de la BD, ce qui permet d’améliorer le temps de réponse. Cet échantillon sera transformé en un treillis de concepts, noyau de l’AFC, qui sera la clé du parcours et du calcul de la fonction d’agrégation ainsi que du taux d’erreur de cet échantillon. Pour le calcul de l’agrégation, nous adoptons les fonctions d’agrégation définies initialement par @hass99 et étendus par @sassi12.
Évaluation
==========
Nous avons utilisé un exemple de jeu de données qui gèrent un ensemble de patients afin d’étudier les facteurs de risque d’athérosclérose. Considérons la requête suivante “Lister le nombre de décès régulièrement alcooliques des patients scolaires par année”. Les expérimentations faites prouvent que l’approche proposée favorise le temps de réponse au détriment de l’exactitude du résultat obtenu suite à une requête flexible. La figure \[fig1\] présente la variation du nombre de patients par rapport au temps de réponse.
![Comparaison entre l’interrogation flexible avec et sans traitement approximatif.[]{data-label="fig1"}](Figure1)
Comme le montre la figure \[fig1\], le temps de réponse sans TRA est de l’ordre de 6 secondes (6000ms), tandis que, avec TRA, est de l’ordre de 2 secondes (2000ms). Pour tester l’exactitude de la réponse, nous extractions les valeurs exactes de la BD et nous les comparons avec celles obtenues par notre approche. Les résultats obtenus prouve l’efficience de l’approche proposée dans l’exactitude des réponses retournées.
[^1]: Attributs que les utilisateurs peuvent utiliser dans un prédicat de comparaison contenant un terme linguistique.
|
---
abstract: 'We examine interferometric measurements of the topological charge of (non-Abelian) anyons. The target’s topological charge is measured from its effect on the interference of probe particles sent through the interferometer. We find that superpositions of distinct anyonic charges $a$ and $a^{\prime}$ in the target decohere (exponentially in the number of probes particles used) when the probes have nontrivial monodromy with the charges that may be fused with $a$ to give $a^{\prime}$.'
author:
- Parsa Bonderson
- Kirill Shtengel
- 'J. K. Slingerland'
bibliography:
- 'corr.bib'
title: Decoherence of Anyonic Charge in Interferometry Measurements
---
Quantum physics in two spatial dimensions allows for the existence of particles which are neither bosons nor fermions. Instead, the exchange interactions of such “anyons” are described by representations of the braid group [@Leinaas77; @Wilczek82a; @Wilczek82b], which may even be non-Abelian [@Goldin85; @Froehlich90]. Recently, there has been a resurgence of interest in anyons, due to increased experimental capabilities in systems believed to harbor them, and also their potential application to topologically protected quantum computation [@Kitaev03; @Preskill98; @Freedman02a]. In this quantum computing scheme, qubits are encoded in non-localized, topological charges carried by clusters of non-Abelian anyons. Topological charges decouple from local probes, affording them protection from decoherence. However, this also makes their measurement, which is vital for qubit readout, more difficult, typically requiring interferometry. The most promising candidate system for discovering non-Abelian statistics is the fractional quantum Hall (FQH) state observed at filling fraction $\nu =5/2$ [@Willett87; @Pan99], which is widely expected to be described by the Moore-Read state [@Moore91; @Nayak96c]. Interference experiments, similar to that proposed [@Chamon97] and only recently implemented [@Camino05a; @Camino05b] for *Abelian* FQH states, may soon verify the braiding statistics of the $\nu =5/2$ state [@Fradkin98; @DasSarma05; @Stern06a; @Bonderson06a]. The analyses in these treatments assume the target particle to be in an eigenstate of topological charge. We show that, when this is not the case, the density matrix of the target particle is diagonalized in the charge basis during the experiment if a simple criterion on the braiding of source and target particles is satisfied: superpositions of distinct anyonic charges $a$ and $a^{\prime }$ decohere as long as the probe particles have nontrivial monodromy with the charge differences between $a$ and $a^{\prime }$, that is, with the charges that fuse with $a$ to give $a^{\prime }$.
![A Mach-Zehnder interferometer containing the target anyon(s) $A$, to be probed by the anyons $B_k$. (Detectors not shown.)[]{data-label="fig:interferometer"}](Mach-Zehnder.eps){width="2in"}
We consider a Mach-Zehnder type interferometer (see Fig. \[fig:interferometer\]), though the same methods may be applied to other types, with similar conclusions. A target “particle” $A$ carrying a superposition of anyonic charges [^1] is located in the region between the two paths of the interferometer. A beam of probe particles $B_{k}$, $k=1,\ldots ,N$ may be sent into two possible input channels, is passed through a beam splitter $%
T_{1}$, reflected by mirrors around the central region, passed through a second beam splitter $T_{2}$, and finally detected at two possible output channels. The state acquires a phase $e^{i\theta _{\text{I}}}$ or $e^{i\theta
_{\text{II}}}$ when a probe particle passes through the bottom or top path around the central region (this may come from background flux, path length differences, phase shifters, etc.) and a separate, independent contribution strictly from the braiding of the probe and target particles, which, for non-Abelian anyons, will be more complicated than a mere phase. If the phases $e^{i\theta _{\text{I}}}$ and $e^{i\theta _{\text{II}}}$ are fixed, or closely monitored, this provides a non-demolitional measurement of the anyonic charge of $A$ [^2]. This admittedly idealized setup is similar to one experimentally realized for quantum Hall systems [@Ji03], the primary difference being that the number of quasiparticle excitations in the central interferometry region is not fixed in that experiment. While unsuitable for measuring a target charge, this situation may still be used to detect the presence of non-Abelian statistics [@Feldman06].
The experiment we describe was also considered in the paper [@Overbosch01], where it was referred to as the “many-to-one” experiment. In that paper, the authors use a quantum group inspired approach, where individual particles are assumed to have internal Hilbert spaces, and they study what happens to the internal state of the target particle. In our descriptions of the systems examined, we use the theory of general anyon models (unitary braided tensor categories), which does not ascribe individual particles internal degrees of freedom. Instead, the relevant observables are the overall anyonic charges of groups of particles (our main result will be stated in terms of the density matrix of an anyon pair $A$–$\overline{A}$). This is the situation relevant to the topological systems (e.g. FQH states) that we have in mind. We also remove some constraints imposed in [@Overbosch01], specifically, that the probe particles are all identical and have trivial self-braiding.
Let us recall some information about anyon models (see e.g. [@Preskill-lectures; @Kitaev06a] for additional details). States in these models may be represented by superpositions of oriented worldline diagrams that give a history of splitting and fusion of particles carrying an anyonic charge. Each allowed fusion/splitting vertex is associated with a (possibly multi-dimensional) vector space containing normalized bra/ket vectors$$\begin{aligned}
\left( d_{c} / d_{a}d_{b} \right) ^{1/4}
\pspicture[0.4](0,0)(1.4,-1)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline{-<}(0.7,0)(0.7,-0.35)
\psline(0.7,0)(0.7,-0.55)
\psline(0.7,-0.55) (0.25,-1)
\psline{-<}(0.7,-0.55)(0.35,-0.9)
\psline(0.7,-0.55) (1.15,-1)
\psline{-<}(0.7,-0.55)(1.05,-0.9)
\rput[tl]{0}(0.4,0){$c$}
\rput[br]{0}(1.4,-0.95){$b$}
\rput[bl]{0}(0,-0.95){$a$}
\rput[bl]{0}(0.85,-0.5){$\mu$}
\endpspicture
&=&\left\langle a,b;c,\mu \right| \in
V_{ab}^{c}
\label{eq:bra}\\
\left( d_{c} / d_{a}d_{b}\right) ^{1/4}
\pspicture[0.4](0,0)(1.4,1)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline{->}(0.7,0)(0.7,0.45)
\psline(0.7,0)(0.7,0.55)
\psline(0.7,0.55) (0.25,1)
\psline{->}(0.7,0.55)(0.3,0.95)
\psline(0.7,0.55) (1.15,1)
\psline{->}(0.7,0.55)(1.1,0.95)
\rput[bl]{0}(0.4,0){$c$}
\rput[br]{0}(1.4,0.8){$b$}
\rput[bl]{0}(0,0.8){$a$}
\rput[bl]{0}(0.85,0.35){$\mu$}
\endpspicture
&=&\left| a,b;c,\mu \right\rangle \in
V_{c}^{ab},
\label{eq:ket}\end{aligned}$$where $\mu $ labels the basis states of the splitting space $V_{c}^{ab}$ of the charges $a$ and $b$ from charge $c$ and the number $d_{a}\ge 1$ is the quantum dimension of $a$. The factors of $\left( d_{c} / d_{a}d_{b}\right)
^{1/4}$ are necessary for isotopy invariance, i.e. so the meaning of the diagrams is not changed by continuous deformation. The vacuum is labeled $1$, and has $d_{1}=1$. Since $\dim V_{c}^{ab}=1$ when any of $a,b,c$ equals $1$, the basis label in this case is redundant, and will be dropped. In fact, the meaning of diagrams is invariant under addition/removal of vacuum lines, so we may drop them and smooth out their vertices. The charge conjugate, or antiparticle, of $a$ is denoted $\overline{a}$, and may also be denoted by reversing the arrow on a line labeled by $a$. Diagrams with multiple vertices correspond to tensor products of vertex spaces. Density matrices may be represented by diagrams with the same numbers of lines emerging at the top and bottom (being combinations of kets and bras). Conjugation of states and operators corresponds to reflecting their diagrams through the horizontal plane while reversing orientations \[e.g. Eqs. (\[eq:bra\]),(\[eq:ket\])\]. One may diagrammatically trace out a charge that enters and exits a diagram at the same spatial position by connecting the lines at these positions with an arc that does not interfere with the rest of the diagram (giving zero if the charges do not match). This is actually the *quantum* trace, which equals the ordinary trace with each sector of overall charge $f$ multiplied by $d_f$. Here are some important diagrammatic relations:$$\pspicture[0.4](0,-0.4)(1.1,1)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline(0.3,-0.45)(0.3,1)
\psline{->}(0.3,-0.45)(0.3,-0.05)
\psline{->}(0.3,0.5)(0.3,0.85)
\psline(0.8,-0.45)(0.8,1)
\psline{->}(0.8,-0.45)(0.8,-0.05)
\psline{->}(0.8,0)(0.8,0.85)
\psline(0.8,0.05)(0.3,0.45)
\psline{->}(0.8,0.05)(0.45,0.33)
\rput[bl]{0}(0.48,0.4){$e$}
\rput[br]{0}(1.05,0.8){$b$}
\rput[bl]{0}(0,0.8){$a$}
\rput[bl]{0}(0.02,0.35){$\alpha$}
\rput[br]{0}(1.1,-0.4){$c$}
\rput[bl]{0}(-0.05,-0.4){$d$}
\rput[bl]{0}(0.85,-0.1){$\beta$}
\endpspicture
=\sum\limits_{f,\mu
,\nu }\left[ F_{d,c}^{a,b}\right] _{\left( e,\alpha ,\beta \right) ,\left(
f,\mu ,\nu \right)}
\pspicture[0.4](0,-0.4)(1.4,1)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline{->}(0.7,0)(0.7,0.45)
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\rput[br]{0}(1.4,-0.4){$c$}
\rput[bl]{0}(0,-0.4){$d$}
\rput[bl]{0}(0.85,-0.05){$\nu$}
\endpspicture,$$ $$R_{ab}=
\pspicture[0.4](0,0)(1.2,1)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline(0.96,0.05)(0.2,1)
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\psline(0.24,0.05)(1,1)
\psline[border=2pt]{->}(0.24,0.05)(0.92,0.9)
\rput[bl]{0}(0,0){$a$}
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\endpspicture
,\qquad
R_{ab}^{\dag}= R_{ab}^{-1}=
\pspicture[0.4](0,0)(1.2,1)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline{->}(0.24,0.05)(0.92,0.9)
\psline(0.24,0.05)(1,1)
\psline(0.96,0.05)(0.2,1)
\psline[border=2pt]{->}(0.96,0.05)(0.28,0.9)
\rput[bl]{0}(0,0){$b$}
\rput[br]{0}(1.2,0){$a$}
\endpspicture
,$$ $$S_{ab}=\frac{1}{D}
\pspicture[0.5](2.4,1.3)
\psarc[linewidth=0.9pt,linecolor=black,arrows=<-,arrowscale=1.5,
arrowinset=0.15] (1.6,0.7){0.5}{165}{363}
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arrowinset=0.15] (0.9,0.7){0.5}{180}{375}
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\rput[bl]{0}(0.15,0.3){$a$}
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, \qquad
\pspicture[0.4](1,1.3)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline(0.4,0)(0.4,1.2)
\psarc[linewidth=0.9pt,linecolor=black] (0.4,0.5){0.3}{3}{180}
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arrowinset=0.15] (0.4,0.5){0.3}{180}{375}
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arrowinset=0.15](0.4,0.5)(0.4,1.1)
\rput[bl]{0}(0.8,0.4){$a$}
\rput[tl]{0}(0.5,1.2){$b$}
\endpspicture
=\frac{S_{ab}}{S_{1b}}
\pspicture[0.4](0.3,0)(0.6,1.3)
\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
\psline(0.4,0)(0.4,1.2)
\psline[linewidth=0.9pt,linecolor=black,arrows=->,arrowscale=1.5,
arrowinset=0.15](0.4,0.5)(0.4,0.9)
\rput[tl]{0}(0.5,1.0){$b$}
\endpspicture
\label{eq:loopaway}$$where $d_a=DS_{1a}$ is the value of an unknotted loop carrying charge $a$, and $D=\sqrt{\sum_{a}d_{a}^{2}}$ is the total quantum dimension. Another useful quantity, especially for interference experiments [@Bonderson06b], is the monodromy matrix element $M_{ab}=\frac{S_{ab}S_{11}}{S_{1a}S_{1b}}$. It has the property $\left| M_{ab}\right| \leq 1$, with $M_{ab}=1$ corresponding to trivial monodromy, i.e. the state is unchanged by taking the charges $a$ and $b$ all the way around each other.
Using this formalism, it is important to keep track of all particles involved in a process. We invoke the physical assumption that the particles $A$ and all $B_{k}$ are initially unentangled. This means there are no non-trivial charge lines connecting them, and to achieve this, they must each be created separately from vacuum, with their own antiparticles [^3]. We write the initial state of the $A$–$\overline{A}$ system as$$\left| \Psi _{0}\right\rangle =\sum\limits_{a}A_{a}\left| a,\overline{a}%
;1\right\rangle$$and that of each $B_{k}$–$\overline{B}_{k}$ system as$$\label{eq:stateBk}
\left| \varphi _{k}\right\rangle =\sum\limits_{b,s}B_{b,s}^{\left( k\right)
}\left| \overline{b},b;1;s\right\rangle$$where $s=\rightarrow ,\uparrow $ indicates in which direction the probe particle is traveling. The probes’ antiparticles, $\overline{B}_{k}$, will be taken off to the left and do not participate in the interferometry. The location of the target’s antiparticle $\overline{A}$ with respect to the interferometer is important and we will let it be located below the central region, as in Fig. \[fig:interferometer\].
Utilizing the two-component vector notation $
\left(
\begin{smallmatrix}
1 \\
0
\end{smallmatrix}
\right) =\left| \rightarrow \right\rangle$, $\left(
\begin{smallmatrix}
0 \\
1
\end{smallmatrix}
\right) =\left| \uparrow \right\rangle$, the two beam splitters, which (along with the mirrors) are assumed to be lossless, are represented by the unitary operators $T_{j}= \left[ \begin{smallmatrix}
t_{j} & r_{j}^{\ast } \\
r_{j} & -t_{j}^{\ast }
\end{smallmatrix} \right]$, with $\left| t_{j}\right| ^{2}+\left|
r_{j}\right| ^{2}=1$. The evolution operator for passing the probe particle $B_{k}$ through the interferometer is$$\begin{aligned}
U_{k} &=&T_{2}\Sigma _{k}T_{1} \\
\Sigma _{k} &=&\left[
\begin{array}{cc}
0 & e^{i\theta _{\text{II}}}R_{A,B_{k}}^{-1} \\
e^{i\theta _{\text{I}}}R_{B_{k},A} & 0
\end{array}%
\right].\end{aligned}$$Diagrammatically, this takes the form$$\begin{gathered}
\label{eq:Udiag}
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\rput[bl](0.25,0.24){\psframebox{$U_k$}}
\endpspicture
=e^{i\theta _{\text{I}}}\left[
\begin{array}{cc}
t_{1}r_{2}^{\ast } & r_{1}^{\ast }r_{2}^{\ast } \\
-t_{1}t_{2}^{\ast } & -r_{1}^{\ast }t_{2}^{\ast }
\end{array}
\right]
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\endpspicture
\\
+e^{i\theta _{\text{II}}}\left[
\begin{array}{cc}
r_{1}t_{2} & -t_{1}^{\ast }t_{2} \\
r_{1}r_{2} & -t_{1}^{\ast }r_{2}
\end{array}
\right]
\pspicture[0.4](-0.2,0)(1.25,1)
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\rput[tl]{0}(-0.2,0.2){$B_k$}
\rput[tr]{0}(1.25,0.2){$A$}
\endpspicture
.\end{gathered}$$Keeping track of antiparticles, we need $V_{k}=R_{\overline{A},B_{k}}^{-1}$ for braiding the probe particles with $\overline{A}$ [^4], and, adding in each successive $\left| \varphi _{k}\right\rangle$ from the left, we also use the operators$$W_{k}=R_{\overline{B}_{k},\overline{B}_{k-1}}R_{B_{k},\overline{B}
_{k-1}}\ldots R_{\overline{B}_{k},\overline{B}_{1}}R_{B_{k},\overline{B}_{1}}$$(and $W_1=1$), which move the $\overline{B}_{k}B_{k}$ pair from the left to the center of the configuration $\overline{B}_{1}\ldots \overline{B}_{k-1}B_{k-1}\ldots B_{1}$. This may be viewed either as spatial sorting after creation, or, as shown suggestively in Eq. (\[eq:state\]), as the temporal condition that each $\overline{B}B$ pair is utilized before creating the next.
The state of the combined system after $N$ probe particles have passed through the interferometer (but have not yet been detected) may now be defined iteratively as$$\left| \Psi _{N}\right\rangle =V_{N}U_{N}W_{N}\left| \varphi
_{N}\right\rangle \otimes \left| \Psi _{N-1}\right\rangle
.$$
Focusing on the $A$–$\overline{A}$ system, the reduced density matrix, $\rho _{N}^{A}=Tr_{B^{\otimes N}}\left[ \left| \Psi_{N}\right\rangle \left\langle \Psi _{N}\right| \right] $, is obtained by tracing over the $B_{k}$ and $\overline{B}_{k}$ particles. This may be interpreted as ignoring the detection results. Given the placement of $\overline{A}$, one sees that this averaging over detector measurements makes the second beam splitter irrelevant. If we kept track of the measurement outcomes $s_k$, we would project with $ \left| s_{k}\right\rangle \left\langle s_{k}\right| $ after the $k^{th}$ probe particle. In $\left| \Psi_{N}\right\rangle$, we did not include braidings between the $B_{k}$, but they may be added without changing the results, as they drop out of $\rho _{N}^{A} $ [^5].
We will first assume that the probe particles all have the same, definite anyonic charge $b$ and enter through the horizontal leg, so that $\left| \varphi _{k}\right\rangle =\left| \overline{b},b;1;\rightarrow
\right\rangle $ for all $k$, and then later return to the general case. This results in the state$$\left| \Psi_N \right. \rangle =
\sum_{a} A_a \frac{1}{\sqrt{d_a d_b^N}}
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\scriptsize
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\rput[bl](0,2){\rnode{B1}{$\overline{b}$}}
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\nccurve[nodesepA=2pt,angleA=-90,angleB=135]{B1}{O1}
\nccurve[nodesepA=2pt,angleA=-90,angleB=135]{Bn}{On}
\nccurve[angleA=-115,angleB=45]{U1}{O1}
\nccurve[angleA=-105,angleB=45]{Un}{On}
\nccurve[angleA=108,angleB=-72,linestyle=dotted,linewidth=1pt,dotsep=1.4pt]
{U1} {Un}
\nccurve[nodesepA=2pt,angleA=-105,angleB=75]{B1n}{U1}
\nccurve[nodesepA=2pt,angleA=-105,angleB=55]{B11}{Un}
\nccurve[nodesepA=2pt,angleA=-95,angleB=65,border=2pt]{A2}{Oa}
\nccurve[angleA=-75,angleB=125]{U1}{Oa}
\nccurve[nodesepA=2pt,angleA=-85,angleB=105]{A1}{Un}
%%%
\psline{->}(0.0705,1.775)(0.065,1.825)
\psline{->}(0.556,1.775)(0.552,1.825)
\psline{->}(1.026,1.775)(1.008,1.825)
\psline{->}(2.138,1.675)(2.1425,1.725)
\psline{->}(1.75,1.706)(1.8,1.7175)
\psline{->}(2.825,1.675)(2.89,1.725)
\endpspicture
\label{eq:state}$$ (with directional indices left implicit).
We first consider the case $N=1$. Tracing out the $b$ and $\overline{b}$ lines of $\left| \Psi_{1}\right\rangle \left\langle \Psi _{1}\right|$, and using Eq. (\[eq:Udiag\]), one finds that terms cancel to give$$\begin{gathered}
\rho_1^A =
\sum_{a,a'} \frac{A_a A_{a'}^{\ast}}{\sqrt{d_a d_{a'}} d_b }
\\
\times
\left[
\left|r_1 \right|^2\,
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%
\rput[bl](-0.18,0.57){$\overline{b}$}
\rput[bl](0.24,0.84){$b$}
\rput[bl](-0.18,0.09){$\overline{b}$}
\rput[bl](0.24,-0.09){$b$}
%
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\ncarc[nodesepB=2pt]{->}{OA1}{A1}
\ncarc[nodesepA=2pt]{<-}{A2}{OA1}
%
\rput[bl](0.45,-0.42){\rnode{A3}{$a'$}}
\rput[bl](0.93,-0.42){\rnode{A4}{$\overline{a'}$}}
\ncarc[nodesepA=2pt,border=2pt]{A3}{OA2}
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+\left|t_1 \right|^2\,
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\nccurve[angleA=155,angleB=-55]{>-}{ML}{OB2}
\nccurve[angleA=120,angleB=-25]{RB}{ML}
%
\rput[bl](-0.18,0.57){$\overline{b}$}
\rput[bl](0.24,0.84){$b$}
\rput[bl](-0.18,0.09){$\overline{b}$}
\rput[bl](0.24,-0.09){$b$}
%
\rput[bl](0.45,1.14){\rnode{A1}{$a$}}
\rput[bl](0.93,1.14){\rnode{A2}{$\overline{a}$}}
\ncarc[nodesepB=2pt]{->}{OA1}{A1}
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\ncarc[nodesepA=2pt,linewidth=0,border=2.5pt]{A2}{OA1}
\ncarc[nodesepA=2pt]{<-}{A2}{OA1}
\ncarc[nodesepA=1.5pt]{A2}{OA1}
\pnode(0.684,0.72){WW}
\ncarc{OA1}{WW}
%
\rput[bl](0.45,-0.42){\rnode{A3}{$a'$}}
\rput[bl](0.93,-0.42){\rnode{A4}{$\overline{a'}$}}
\ncarc[nodesepA=2pt,border=2pt]{A3}{OA2}
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\right]
\label{eq:density}\end{gathered}$$This result simply reflects the fact that all that matters after averaging over measurement outcomes is that the probe particle passes between $A$ and $\overline{A}$ with probability $\left| t_{1}\right| ^{2}$, and passes around them with probability $\left| r_{1}\right| ^{2}$. Since they are initially unentangled, each additional probe particle has the same analysis as the first, and just results in another loop that passes between $A$ and $\overline{A}$ with probability $\left| t_{1}\right| ^{2}$. Noting that an unlinked $b$ loop may be replaced by a factor $d_{b}$, we see that the reduced density matrix for $A$ after passing $N$ probe particles through the interferometer is
$$\begin{aligned}
\rho _{N}^{A}&=&\sum\limits_{a,a^{\prime }}\frac{A_{a}A_{a^{\prime }}^{\ast }%
}{\sqrt{d_{a}d_{a^{\prime }}}}\sum\limits_{n=0}^{N}\dbinom{N}{n}\left|
r_{1}\right| ^{2\left( N-n\right) }\left| t_{1}\right| ^{2n}\frac{1}{%
d_{b}^{n}}
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\\
&=&\sum\limits_{a,a^{\prime }}\frac{A_{a}A_{a^{\prime }}^{\ast }}{\sqrt{d_{a}d_{a^{\prime }}}}\sum\limits_{\left( e,\alpha ,\beta \right) }\left[
\left( F_{a^{\prime },\overline{a^{\prime }}}^{a,\overline{a}}\right) ^{-1}%
\right] _{1,\left( e,\alpha ,\beta \right) }\sum\limits_{n=0}^{N}\dbinom{N}{n%
}\left| r_{1}\right| ^{2\left( N-n\right) }\left| t_{1}\right| ^{2n}\frac{1}{%
d_{b}^{n}}
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&=&\sum\limits_{a,a^{\prime }}\frac{A_{a}A_{a^{\prime }}^{\ast }}{\sqrt{d_{a}d_{a^{\prime }}}}\sum\limits_{\left( e,\alpha ,\beta \right) ,\left(
f,\mu ,\nu \right) }\left[ \left( F_{a^{\prime },\overline{a^{\prime }}}^{a,%
\overline{a}}\right) ^{-1}\right] _{1,\left( e,\alpha ,\beta \right) }\left(
\left| r_{1}\right| ^{2}+\left| t_{1}\right| ^{2}M_{be}\right) ^{N}\left[
F_{a^{\prime },\overline{a^{\prime }}}^{a,\overline{a}}\right] _{\left(
e,\alpha ,\beta \right) ,\left( f,\mu ,\nu \right) }
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\label{eq:rhoAN}\end{aligned}$$
where the relations in Eq. (\[eq:loopaway\]) were used to remove all the $b$ loops, allowing us to perform the sum over $n$, before applying $F$ in the last step. The intermediate charge label $e$ represents the difference between the charges $a$ and $a^{\prime }$, taking values that may be fused with $a^{\prime} $ to give $a$ (the $F$-symbols vanish otherwise). Notice the potential for this process to transfer an overall anyonic charge $f$ to the $A$–$\overline{A}$ system.
From this result we see, noting $\left| t_{1}\right| ^{2}+\left|
r_{1}\right| ^{2}=1$, that taking the limit $N\rightarrow \infty $ will exponentially kill off the $e$-channels with $M_{be}\neq 1$, and preserve only those which have trivial monodromy with $b$, $M_{be}=1$. The interpretation of $M_{be}=1$ is that $a$ and $a^{\prime }$ have a difference charge $e$ that is invisible (in the sense of monodromy) to the charge $b$, and so the corresponding fusion channel remains unaffected by the probe. In general, the only $e$-channels guaranteed to always survive this process (even for the most general $B_{k}$ states) have trivial monodromy with all charges. This always includes $e=1$ (and for modular theories/TQFTs only includes $e=1$), which requires that $a=a^{\prime }$. Tracing over the $A$ and $\overline{A}$ particles gives $Tr\left[ \rho _{N}^{A}\right] =1$ as expected, but by considering the intermediate channels, one also finds that the entire contribution to this trace is from $e=1$. We should also note that some terms may alternatively be killed off due to their corresponding $F$-symbols having zero values.
Defining $\rho ^{A}\equiv \lim_{N\rightarrow \infty }\rho _{N}^{A}$, and denoting by $e_{b}$ the intermediate charges that have trivial monodromy with $b$, we get the final result (converted back into bra/ket notation, with an extra factor of $d_f$ inserted for compatibility with the ordinary trace) $$\begin{aligned}
\rho ^{A} =\sum\limits_{a,a^{\prime }}A_{a}A_{a^{\prime }}^{\ast
}\sum\limits_{\left( e_{b},\alpha ,\beta \right) ,\left( f,\mu ,\nu \right) }
\left[ \left( F_{a^{\prime },\overline{a^{\prime }}}^{a,\overline{a}}\right)
^{-1}\right] _{1,\left( e_{b},\alpha ,\beta \right) }
\nonumber
\\
\times \left[ F_{a^{\prime },\overline{a^{\prime }}}^{a,\overline{a}}\right]
_{\left( e_{b},\alpha ,\beta \right) ,\left( f,\mu ,\nu \right) } \sqrt{d_{f}}
\left| a,\overline{a};f,\mu \vphantom{\overline{a^{\prime }}}\right\rangle
\left\langle a^{\prime },\overline{a^{\prime }};f,\nu \right|\!.\end{aligned}$$
We now return to the case of general probe particle states as given in Eq. (\[eq:stateBk\]). Since tracing requires the charge on a line to match up, a similar analysis as before applies. For the result, we simply replace $\left(
\left| r_{1}\right| ^{2}+\left| t_{1}\right| ^{2}M_{be}\right) ^{N}$ in Eq. (\[eq:rhoAN\]), with$$\prod\limits_{k=1}^{N}\left[ 1-\sum\limits_{b}\left| B_{b,\rightarrow
}^{\left( k\right) }t_{1}+B_{b,\uparrow }^{\left( k\right) }r_{1}^{\ast
}\right| ^{2}\left( 1-M_{be}\right) \right] .$$This term determines the rate at which the $A$ system decoheres, and will generically vanish exponentially as $N\rightarrow \infty $ unless $e$ has trivial monodromy (in which case this term simply equals $1$). In some cases, complete decoherence may even be achieved with a single probe step. By setting $\left| r_{1}\right| = 0$ and $\left| t_{1}\right| = 1$ in Eq. (\[eq:rhoAN\]), we may do away with the interferometer and interpret the result as decoherence from stray anyons passing between $A$ and $\overline{A}$, which is important to consider as a source of errors in a quantum computation.
As a practical example, we apply the results to the Ising anyon model (see e.g. Table 1 of [@Kitaev06a] for details), which captures the essence of the Moore-Read state’s non-Abelian statistics. For the initial state $\left|
\Psi _{0}\right\rangle =A_{1}\left| 1,1;1\right\rangle +A_{\psi }\left| \psi
,\psi ;1\right\rangle +A_{\sigma }\left| \sigma ,\sigma ;1\right\rangle $, using $b=\sigma $ probes (which have trivial monodromy only with $e=1$) gives$$\begin{aligned}
\rho ^{A} = \left| A_{1}\right| ^{2}\left| 1,1;1\right\rangle \left\langle
1,1;1\right| +\left| A_{\psi }\right| ^{2}\left| \psi ,\psi ;1\right\rangle
\left\langle \psi ,\psi ;1\right|
\nonumber
\\
+\left| A_{\sigma }\right| ^{2}\frac{1}{2}\left( \left| \sigma ,\sigma
;1\right\rangle \left\langle \sigma ,\sigma ;1\right| +\left| \sigma ,\sigma
;\psi \right\rangle \left\langle \sigma ,\sigma ;\psi \right| \right)\end{aligned}$$which exhibits loss of all coherence. For $b=\psi $ probes (which have trivial monodromy with both $e=1$ and $\psi$) the result$$\begin{aligned}
\rho ^{A} &=&\left| A_{1}\right| ^{2}\left| 1,1;1\right\rangle \left\langle
1,1;1\right| +A_{\psi }A_{1}^{\ast }\left| \psi ,\psi ;1\right\rangle
\left\langle 1,1;1\right|
\nonumber
\\
&&+A_{1}A_{\psi }^{\ast }\left| 1,1;1\right\rangle \left\langle \psi ,\psi
;1\right| +\left| A_{\psi }\right| ^{2}\left| \psi ,\psi ;1\right\rangle
\left\langle \psi ,\psi ;1\right|
\nonumber
\\
&&+\left| A_{\sigma }\right| ^{2}\left| \sigma ,\sigma ;1\right\rangle
\left\langle \sigma ,\sigma ;1\right|\end{aligned}$$shows decoherence only between $\sigma$ and the other charges.
For another example, we consider the Fibonacci anyon model (see e.g. [@Preskill-lectures] for details). The initial state $\left| \Psi _{0}\right\rangle =A_{1}\left| 1,1;1\right\rangle
+A_{\varepsilon }\left| \varepsilon ,\varepsilon ;1\right\rangle $ probed by $b=\varepsilon $ particles gives$$\begin{aligned}
\rho ^{A} &=&\left| A_{1}\right| ^{2}\left| 1,1;1\right\rangle \left\langle
1,1;1\right| \\
&&+\left| A_{\varepsilon }\right| ^{2}\left( \phi ^{-2}\left| \varepsilon
,\varepsilon ;1\right\rangle \left\langle \varepsilon ,\varepsilon ;1\right|
+\phi ^{-1}\left| \varepsilon ,\varepsilon ;\varepsilon \right\rangle
\left\langle \varepsilon ,\varepsilon ;\varepsilon \right| \right)
\nonumber\end{aligned}$$(where $\phi = \frac{1+\sqrt{5}}{2}$), which exhibits loss of all coherence.
The decoherence effect described in this letter is due to measurements being made by probe particles. Keeping track of these measurement outcomes, one generically finds collapse of the target system state into subspaces where the difference charge has trivial monodromy with the probes [@Bonderson07b]. If this includes only the $e=1$ subspaces, the target collapses onto a state of definite charge. One may also consider completely general initial $A$ and $B_{k}$ systems described by density matrices, but as long as they are all still unentangled, the resulting behavior is qualitatively similar. It may also be physically relavant in some cases to allow initial entanglement between the probes, though this greatly complicates the analysis and results. These generalizations will be addressed in [@Bonderson07b].
We thank A. Kitaev, I. Klich, and especially J. Preskill for illuminating discussions, and the organizers and participants of the KITP Workshop on Topological Phases and Quantum Computation where this work was initiated. We would also like to acknowledge the hospitality of the IQI, the KITP, and Microsoft Project Q. This work was supported in part by the NSF under Grant No. PHY-0456720 and PHY99-07949, and the NSA under ARO Contract No. W911NF-05-1-0294.
[^1]: Since localized charges cannot be superimposed, when we refer to “a particle having a superposition of charges,” we really mean several (quasi-)particles, treated collectively.
[^2]: *Decoherence* in the charge basis is actually independent of these phases, as they drop out of the density matrix for $A$.
[^3]: These “particle-antiparticle pairs” may really be multiple pair-created particles that are made to interact amongst each other as needed and then split into two groups that are henceforth treated collectively.
[^4]: If $\overline{A}$ is located above, rather than below, the central region of the interferometer, we would instead use $V_{k}=R_{B_{k},\overline{A}}$. This essentially interchanges $r_{1}$ with $t_{1}$ and conjugates $M_{be}$ in the result, Eq. (\[eq:rhoAN\]). If however, $\overline{A}$ is placed between the two output legs, the situation is complicated by the resulting $V_{k}= \left[ \protect\begin{smallmatrix}
R_{B_{k},\overline{A}} & 0 \\
0 & R_{\overline{A},B_{k}}^{-1}
\protect\end{smallmatrix} \right]$, which makes evaluation more difficult, and gives a different limiting behavior. If $\overline{A}$ is situated in the central region (with $A$), there will, of course, be no effect.
[^5]: *Superpositions* of braiding may however change these results.
|
---
abstract: |
In this study, all rings are commutative with non-zero identity and all modules are considered to be unital. Let $M$ be a left $R$-module. A proper submodule $N$ of $M$ is called an $S$-$weakly$ $prime$ submodule if $0_{M}\neq
f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in
S=End(M)$ and $m\in M.$ Some results concerning $S$-prime and $S$-weakly prime submodules are obtained. Then we study $S$-prime and $S$-weakly prime submodules of multiplication modules. Also for $R$-modules $M_{1}$ and $M_{2},$ we examine $S$-prime and $S$-weakly prime submodules of $M=M_{1}\times M_{2},$ where $S=S_{1}\times S_{2},$ $S_{1}=End(M_{1})$ and $S_{2}=End(M_{2}).$
address: 'Department of Mathematics, Marmara University, Kadikoy, Istanbul, 34722, Turkey'
author:
- Emel ASLANKARAYIGIT UGURLU
date: '2 September, 2019'
title: 'S-PRIME AND S-WEAKLY PRIME SUBMODULES'
---
Introduction
============
Throughout this paper $R$ will denote a commutative ring with a non-zero identity and $M$ is considered to be unital left $R$-module. A proper ideal $P$ of $R$ is said to be *prime* if $ab\in P$ implies $a\in P$ or $b\in
P$, [@atiyah]. Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied by D. D. Anderson and E. Smith in [@DE2003]. A proper ideal $P$ of $R$ is said to be *weakly prime* if $0_{R}\neq ab\in P$ implies $a\in P$ or $b\in P$. Several authors have extended the notion of prime ideals to modules, see, for example, [@JD1978; @L; @MM]. In [@RA2003], a proper submodule $N$ of a module $M$ over a commutative ring $R$ is said to be *prime* submodule if whenever $rm\in N$ for some $r\in R,m\in M$, then $m\in N$ or $rM\subseteq N.$ Then in [@SF2007], S. Ebrahimi and F. Farzalipour introduced weakly prime submodules over a commutative ring $R$ as following: A proper submodule $N$ of $M$ is called *weakly prime* if for $r\in R$ and $m\in M$ with $0_{M}\neq rm\in N$, then $m\in N$ or $rM\subseteq N.$ Clearly, every prime submodule of a module is a weakly prime submodule. However, since $0_{M}$ is always weakly prime, a weakly prime submodule need not be prime. Various properties of weakly prime submodules are considered in [@SF2007].
Now we define the concepts the residue of $N$ by $M$. If $N$ is a submodule of an $R$-module $M$, the ideal $\{r\in R:rM\subseteq N\}$ is called the residue of $N$ by $M$ and it is denoted by $(N:_{R}M).$ In particular, $(0_{M}:_{R}M)$ is called the *annihilator* of $M$ and denoted by $Ann(M),$ see [@ZP1988]. If the annihilator of $M$ equals to $0_{R},$ then $M$ is called a *faithful module*. Also, for a proper submodule $N$ of $M$, the *radical* of $N,$ denoted by $\sqrt{N},$ is defined to be the intersection of all prime submodules of $M$ containing $N.$ If there is no prime submodule containing $N$, then $\sqrt{N}=M,$ see [@ZP1988].
An $R$-module $M$ is called a *multiplication module* if every submodule $N$ of $M$ has the form $IM$ for some ideal $I$ of $R$, see [@ZP1988]. Note that, since $I\subseteq(N:_{R}M)$ then $N=IM\subseteq
(N:_{R}M)M\subseteq N$. So, if $M$ is multiplication, $N=(N:_{R}M)M$, for every submodule $N$ of $M.$ Let $N$ and $K$ be submodules of a multiplication $R$-module $M$ with $N=I_{1}M$ and $K=I_{2}M$ for some ideals $I_{1}$ and $I_{2}$ of $R$. The product of $N$ and $K$ denoted by $NK$ is defined by $NK=I_{1}I_{2}M$. Then by [@RA2003 Theorem 3.4], the product of $N$ and $K$ is independent of presentations of $N$ and $K$. Note that, for $m,m^{\prime}\in M$, by $mm^{\prime}$, we mean the product of $Rm$ and $Rm^{\prime}$. Clearly, $NK$ is a submodule of $M$ and $NK\subseteq N\cap K,$ see [@RA2003]. Also, if $M$ is multiplication module, in Theorem 3.13 of [@RA2003], R. Ameri showed that $\sqrt{N}=\{m\in M:m^{k}\subseteq N$ for some positive integer $k\}.$
This paper is inspired by the notion of $S$-prime submodule which appears in [@GG2000; @SOD2010]. The authors defined the concept as following: A proper submodule $N$ of an $R$-module $M$ is said to be $S$-$prime$ submodule of $M$ if $f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in
S=End(M)$ and $m\in M.$ Every $S$-$prime$ submodule is prime, see [@GG2000]. For more information one can examine [@SOD2010].
In this study we introduce the concept of *S-weakly prime* submodule as following: A proper submodule $N$ of an $R$-module $M$ is said to be $S$-$weakly$ $prime$ submodule of $M$ if $0_{M}\neq f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in S=End(M)$ and $m\in M.$ Clearly, every $S$-$prime$ submodule is an $S$-$weakly$ $prime$ submodule. In Proposition \[pro1\], it is obtained that every $S$-$weakly$ $prime$ submodule of an $R$-module $M$ is a $weakly$ $prime$ submodule. However, we show that the opposite of Proposition \[pro1\] is not correct, see Example \[exa1\]. Then we prove in Proposition \[pro4\] (Proposition \[pro5\]) that $N$ is an $S$-prime ($S$-weakly prime) submodule if and only if $f(K)\subseteq N$ $(0_{M}\neq f(K)\subseteq N)$ implies that either $K\subseteq N$ or $f(M)\subseteq N,$ where $f\in S=End(M)$ and $K$ is a submodule of $M.$ Also, we give characterizations of $S$-$prime$ submodule and $S$-$weakly$ $prime$ submodule (Theorem \[theS\], Theorem \[the1\], recpectively). In Corollary \[COR1\], by the help of Proposition \[pro3\], it is proved that when $N$ is an $S$-weakly prime submodule, then $(N:_{R}M)$ is an $S$-weakly prime ideal of $R.$ In multiplication module, we obtain another characterizations for $S$-$prime$ submodule and $S$-$weakly$ $prime$ submodule (Theorem \[the mult1\], Theorem \[the mult2\], recpectively). Among the other results, some properties of $S$-$prime$ and $S$-$weakly$ $prime$ submodules in multiplication modules are obtained. Moreover, we characterize $S$-$prime$ and $S$-$weakly$ $prime$ submodules of $M=M_{1}\times
M_{2}$ over $R$-module, where $M_{1},M_{2}$ be $R$-modules, see Theorem \[prokart\], Theorem \[thekart\], Proposition \[pro1in2\], Proposition \[pro2in2\]. Finally, we obtain a relation between $S$-$prime$ and $S$-$weakly$ $prime$ submodules of $M=M_{1}\times M_{2}$ over $R=R_{1}\times
R_{2}$-module, where $M_{1},M_{2}$ are $R_{1}$-module and $R_{2}$-module, recpectively, see, Theorem \[theD1\] and Theorem \[theD2\].
S-PRIME AND S-WEAKLY PRIME SUBMODULES
=====================================
Throughout this study $End(M)$ is denoted by $S.$
A proper submodule $N$ of an $R$-module $M$ is said to be $S$-$weakly$ $prime
$ submodule of $M$ if $0_{M}\neq f(m)\in N$ implies that either $m\in N $ or $f(M)\subseteq N,$ where $f\in S=End(M)$ and $m\in M.$
It is clear that every $S$-$prime$ submodule is an $S$-$weakly$ $prime$ submodule. However, since $\{0_{M}\}$ is an $S$-$weakly$ $prime$ submodule of $M,$ then an $S$-$weakly$ $prime$ submodule may not be an $S$-$prime$ submodule.
Note that if we consider any $R$ as an $R$-module, then a proper ideal $I$ of $R$ is said to be $S$-$prime$ $(S$-$weakly$ $prime)$ ideal if $f(a)\in I$ $(0_{R}\neq f(a)\in I$) implies that either $a\in I$ or $f(R)\subseteq I,$ where $f\in S=End(R)$ and $a\in R.$
\[pro1\]Every $S$-$weakly$ $prime$ submodule of an $R$-module $M$ is a $weakly$ $prime$ submodule.
Let $N$ be an $S$-$weakly$ $prime$ submodule of an $R$-module $M.$ Suppose that for some $r\in R$ and $m\in M$ such that $0_{M}\neq rm\in N$ and $m\notin
N.$ We show that $r\in(N:_{R}M).$
Define $h:M\rightarrow M$ such that $h(x)=rx$ for all $x\in M.$ Clearly, $h\in
End(M).$ Since definition of $h$ and our assumption, then $0_{M}\neq
h(m)=rm\in N.$ Then as $m\notin N$ and $N$ is an $S$-$weakly$ $prime$ submodule, we get $h(M)\subseteq N.$ Thus $h(M)=rM\subseteq N,$ i.e., $r\in(N:_{R}M).$
Note that generally a $weakly$ $prime$ submodule is not an $S$-$weakly$ $prime$ submodule. For this one can see the following example:
\[exa1\]Let consider the submodule $N=2\mathbb{Z}
\oplus\mathbb{Z}
$ of $\mathbb{Z}
$-module $M=\mathbb{Z}
\oplus\mathbb{Z}
.$ Since $N$ is a maximal submodule, $N$ is a prime, so weakly prime submodule. But $N$ is not an $S$-$weakly$ $prime$ submodule. Indeed, let define $f:M\longrightarrow M$ such that $f((x,y))=(y,x)$ for all $(x,y)\in M.$ Then we get $f\in S=End(M).$ Thus we can easily see $0_{M}\neq
f((1,2))=(2,1)\in N,$ but $(1,2)\notin N$ and $f(M)=M\nsubseteq N.$ Consequently, $N$ is not $S$-$weakly$ $prime.$
\[pro4\]Let $M$ be an $R$-module and $N$ be a proper submodule of $M.$ Then the followings are equivalent:
1. $N$ is an $S$-prime submodule.
2. $f(K)\subseteq N$ implies that either $K\subseteq N$ or $f(M)\subseteq
N,$ where $f\in S$ and $K$ is a proper submodule of $M.$
$(1)\Longrightarrow(2):$ Let $N$ be an $S$-prime submodule. Assume that $f(K)\subseteq N$ and $K\nsubseteq N.$ Then there exists $k\in K-N.$ Thus $f(k)\in f(K)\subseteq N.$ Since $N$ is $S$-prime, $f(M)\subseteq N.$
$(2)\Longrightarrow(1):$ Let $f(m)\in N.$ We show that either $m\in N$ or $f(M)\subseteq N.$ Since $f(Rm)\subseteq N,$ by our hypothesis we obtain either $Rm\subseteq N$ or $f(M)\subseteq N.$ Consequently, $m\in N$ or $f(M)\subseteq N.$
Let $M$ be an $R$-module and $N$ be a proper submodule of $M.$ Then the followings are equivalent:
1. $N$ is an $S$-prime submodule.
2. $f(Rm)\subseteq N$ implies that either $Rm\subseteq N$ or $f(M)\subseteq
N,$ where $f\in S$ and $m\in M.$
By Proposition \[pro4\].
\[pro babei\]Let $M$ be an $R$-module and $N$ be a proper submodule of $M.$ Then $N$ is an $S$-prime submodule if and only if $f^{-1}(N)=M$ or $f^{-1}(N)\subseteq N,$ for all $f\in S.$
$\Longrightarrow:$ Let $N$ be an $S$-prime submodule. Assume that $f(M)\subseteq N.$ Then it is clear that $f^{-1}(N)=M.$ So suppose that $f(M)\nsubseteq N.$ Take $m\in f^{-1}(N).$ Then $f(m)\in N.$ Since $N$ is an $S$-prime submodule and $f(M)\nsubseteq N,$ we have $m\in N.$ Consequently, $f^{-1}(N)\subseteq N.$
$\Longleftarrow:$ Assume that $f^{-1}(N)\subseteq N$ or $f^{-1}(N)=M$ for all $f\in End(M).$ Let $f(m)\in N.$ If $f^{-1}(N)\subseteq N,$ then $m\in
f^{-1}(N)\subseteq N,$ so we are done. On the other hand, if $f^{-1}(N)=M$, then we get $f(M)\subseteq N.$ Thus $N$ is an $S$-prime submodule.
The zero submodule $\{0_{M}\}$ of $M$ is an $S$-prime submodule if and only if $f$ is one-to-one, for all $0\neq f\in S.$
By Proposition \[pro babei\].
\[the1\]Let $M$ be an $R$-module and $N$ be a proper submodule of $M.$ For all $f\in S,$ the followings are equivalent:
1. $N$ is an $S$-$weakly$ $prime$ submodule of $M.$
2. $(N:_{R}f(x))=(N:_{R}f(M))\cup(0_{M}:_{R}f(x))$ for all $x\notin N.$
3. $(N:_{R}f(x))=(N:_{R}f(M))$ or $(N:_{R}f(x))=(0_{M}:_{R}f(x))$ for all $x\notin N.$
$(1)\Longrightarrow(2):$ Assume that $N$ is $S$-$weakly$ $prime$. Let $r\in(N:_{R}f(x))$ and $x\notin N.$ Then we get $rf(x)\in N.$ If $rf(x)=0_{M},$ then $r\in(0_{M}:_{R}f(x)).$ Suppose that $rf(x)\neq0_{M}.$ Define $h:M\rightarrow M$ such that $h(m)=rf(m),$ for all $m\in M.$ Clearly $h\in End(M)$, also $0_{M}\neq h(x)=rf(x)\in N.$ Since $N$ is an $S$-$weakly$ $prime$ submodule and $x\notin N,$ then we obtain $h(M)\subseteq N.$ Thus $h(M)=rf(M)\subseteq N$ and so $r\in(N:_{R}f(M)).$
$(2)\Longrightarrow(3):$ Clear.
$(3)\Longrightarrow(1):$ Suppose that $h\in End(M)$ and $m\notin N$ such that $0_{M}\neq h(m)\in N.$ We prove that $h(M)\subseteq N.$ Since $0_{M}\neq
h(m),$ we get $(N:_{R}h(m))\neq(0_{M}:_{R}h(m)).$ Indeed, if $(N:_{R}h(m))=(0_{M}:_{R}h(m)),$ then we obtain $(N:_{R}h(m))=R=(0_{M}:_{R}h(m)),$ i.e., $1_{R}h(m)=0_{M},$ a contradiction. Thus we have $(N:_{R}h(m))=(N:_{R}h(M)),$ by our hypothesis $(3)$. Since $(N:_{R}h(m))=R,$ we get $h(M)\subseteq
N.$
\[pro5\]Let $M$ be an $R$-module and $N$ be a proper submodule of $M$. Then the followings are equivalent:
1. $N$ is an $S$-weakly prime submodule.
2. $0_{M}\neq f(K)\subseteq N$ implies that either $K\subseteq N$ or $f(M)\subseteq N,$ where $f\in S$ and $K$ is a submodule of $M.$
$(1)\Longrightarrow(2):$ Let $N$ be an $S$-weakly prime submodule. Suppose that $f(K)\subseteq N$, $K\nsubseteq N$ and $f(M)\nsubseteq N.$ Then we show $f(K)=0_{M}.$ For every $k\in K,$ we have 2 cases:
Case 1: Let $k\in K-N.$ By Theorem \[the1\], we can see that $(N:_{R}f(k))=(N:_{R}f(M))$ or $(N:_{R}f(k))=(0_{M}:_{R}f(k)).$ Since $f(k)\in
f(K)\subseteq N,$ one get $1_{R}\in(N:_{R}f(k)).$ Thus either $1_{R}\in
(N:_{R}f(M))$ or $1_{R}\in(0_{M}:_{R}f(k)).$ The first one contradicts our assumption. Thus we obtain $f(k)=0_{M}.$
Case 2: Let $k\in K\cap N.$ If $f(k)=0_{M},$ we are done. Let suppose $f(k)\neq0_{M}.$ Since $K\nsubseteq N$, there exists $0_{M}\neq y\in K-N$. Then $f(y)\in f(K)\subseteq N,$ one get $1_{R}\in(N:_{R}f(y)).$ Thus either $1_{R}\in(N:_{R}f(M))$ or $1_{R}\in(0_{M}:_{R}f(y)).$ The first one contradicts our assumption. Thus we obtain $f(y)=0_{M}.$ Then one can see $0_{M}\neq f(y+k)=f(y)+f(k)\in f(K)\subseteq N.$ Since $N$ is $S$-weakly prime, we get $y+k\in N$ or $f(M)\subseteq N.$ So, $y\in N$ or $f(M)\subseteq
N,$ i.e., contradiction.
Consequently, for every $k\in K,$ we obtain $f(k)=0_{M}.$
$(2)\Longrightarrow(1):$ Let $0_{M}\neq f(m)\in N.$ We show that either $m\in
N$ or $f(M)\subseteq N.$ Since $0_{M}\neq f(Rm)\subseteq N,$ by our hypothesis we obtain either $Rm\subseteq N$ or $f(M)\subseteq N.$ Consequently, $m\in N$ or $f(M)\subseteq N.$
Let $M$ be an $R$-module and $N$ be a proper submodule of $M$. Then the followings are equivalent:
1. $N$ is an $S$-weakly prime submodule.
2. $0_{M}\neq f(Rm)\subseteq N$ implies that either $Rm\subseteq N$ or $f(M)\subseteq N,$ where $f\in S$ and $m\in M.$
By Proposition \[pro5\].
\[theS\]Let $M$ be an $R$-module and $N$ be a proper submodule of $M.$ For all $f\in S,$ the followings are equivalent:
1. $N$ is an $S$-$prime$ submodule of $M.$
2. $(N:_{R}f(x))=(N:_{R}f(M))$ for all $x\notin N.$
$(1)\Longrightarrow(2):$ Assume that $N$ is $S$-$prime$ and $x\notin N$. Let $r\in(N:_{R}f(x)).$ Then we get $rf(x)\in N.$ Define $h:M\rightarrow M$ such that $h(m)=rf(m)$ for all $m\in M.$ Clearly $h\in End(M)$, also $h(x)=rf(x)\in
N.$ Since $N$ is an $S$-$prime$ submodule and $x\notin N,$ then we obtain $h(M)\subseteq N.$ Thus $h(M)=rf(M)\subseteq N$ and so $r\in(N:_{R}f(M)).$ The other containment is always hold.
$(2)\Longrightarrow(1):$ Suppose that $h\in End(M)$ and $m\notin N$ such that $h(m)\in N.$ We prove that $h(M)\subseteq N.$ Since $1_{R}h(m)\in N,$ we get $1_{R}\in(N:_{R}h(m))=$ $(N:_{R}h(M)).$ Thus $h(M)\subseteq N$ by the assumption.
To avoid losing the integrity, we give the following proposition.
\[pro2\]([@SF2007], Proposition 2.1) Let $M$ be a faithful cyclic $R$-module. If $N$ is a weakly prime submodule, then $(N:_{R}M)$ is a weakly prime ideal of $R.$
However, Proposition \[pro2\] is not true for “$S$-weakly prime situation”. So we mean if $N$ is an $S$-weakly prime submodule, then $(N:_{R}M)$ may not be an $S$-weakly prime ideal of $R.$
Note that for a subset $A$ of $M$, we denote the submodule generated by $A$ in $M$ as $<A>$. In particular, if $X=\{a\},$ then it is denoted by $<a>$. If $M$ is an $R$-module such that $M=<a>,$ then $M$ is called cyclic module. It is clear that every cyclic module is a multiplication module, see [@PF1988].
\[pro3\]Let $M$ be a cyclic $R$-module such that $M=<a>$ and $N$ be a proper submodule of $M.$ Then the followings are equivalent:
1. $N$ is a weakly prime submodule.
2. $N$ is an $S$-weakly prime submodule.
$(1)\Longrightarrow(2):$ Assume that $N$ is a weakly prime submodule. Let choose $m\in M$ and $f\in End(M)$ such that $0_{M}\neq f(m)\in N$ and $m\notin
N.$ We prove that $f(M)\subseteq N.$ Let $f(x)\in f(M).$ Since $M=<a>,$ there exist $r_{1},r_{2}\in R\ $such that $x=r_{1}a$ and $m=r_{2}a.$ Then we get $0_{M}\neq f(m)=f(r_{2}a)=r_{2}f(a)\in N.$ Since $N$ is weakly prime, then $r_{2}\in(N:_{R}M)$ or $f(a)\in N.$ If $r_{2}\in(N:_{R}M),$ then we get $m=r_{2}a\in N,$ i.e., a contradiction. Thus $f(a)\in N,$ so $f(x)=r_{1}f(a)\in N.$ As $x$ is an arbitrary element of $M$, we obtain $f(M)\subseteq
N.$
$(2)\Longrightarrow(1):$ By Proposition \[pro1\].
\[COR1\]Let $M$ be a faithful cyclic $R$-module. If $N$ is an $S$-weakly prime submodule, then $(N:_{R}M)$ is an $S$-weakly prime ideal of $R.$
Assume that $N$ is an $S$-weakly prime submodule. Then $N$ is an weakly prime submodule. By Proposition \[pro2\], $(N:_{R}M)$ is a weakly prime ideal of $R.$ Since $R$ is a cyclic $R$-module, $(N:_{R}M)$ is an $S$-weakly prime ideal of $R$ by Proposition \[pro3\].
For the integrity of our study, we give the following Lemma:
\[lemma fg\]([@PF1988],Corollary in page 231) Let $I$, $J$ be two ideals of $R$ and $M$ be a finitely generated multiplication $R$-module. Then $IM\subseteq JM$ if and only if $I\subseteq J+Ann(M).$
\[the mult1\]Let $M$ be a finitely generated multiplication $R$-module and $N$ be a proper submodule of $M.$ Then the followings are equivalent:
1. $N$ is an $S$-prime submodule.
2. $(N:_{R}M)$ is an $S$-prime ideal of $R.$
3. $N=IM,$ for some $S$-prime ideal $I$ of $R$ with $Ann(M)\subseteq I.$
$(1)\Longrightarrow(2):$ By Corollary 2.1.5 in [@SOD2010].
$(2)\Longrightarrow(3):$ Since $M$ is a multiplication module, $N=(N:_{R}M)M,$ so we are done.
$(3)\Longrightarrow(1):$ Suppose that $N=IM,$ for some $S$-prime ideal $I$ of $R$ with $Ann(M)\subseteq I.$ To use Proposition \[pro4\], assume that $K$ is a submodule of $M$ such that $f(K)\subseteq N,$ for any $f\in S.$ Since $M$ is a multiplication module, there exist two ideals $J_{1},J_{2}$ of $R$ such that $K=J_{1}M$ and $f(M)=J_{2}M.$ Then $f(J_{1}M)=J_{1}f(M)=J_{1}J_{2}M\subseteq N=IM.$ By Lemma \[lemma fg\], $J_{1}J_{2}\subseteq
I+Ann(M)=I.$ As $I$ is an $S$-prime ideal, so prime, we get $J_{1}\subseteq I$ or $J_{2}\subseteq I.$ It implies $J_{1}M\subseteq N$ or $J_{2}M\subseteq N.$ Consequently, $K\subseteq N$ or $f(M)\subseteq N.$
\[the mult2\]Let $M$ be a cyclic faithful $R$-module and $N$ be a proper submodule of $M$. Then the followings are equivalent:
1. $N$ is an $S$-weakly prime submodule.
2. $(N:_{R}M)$ is an $S$-weakly prime ideal of $R.$
3. $N=IM,$ for some $S$-weakly prime ideal $I$ of $R.$
$(1)\Longrightarrow(2):$ By Corollary \[COR1\].
$(2)\Longrightarrow(3):$ Since $M$ is a multiplication module, $N=(N:_{R}M)M,$ so we are done.
$(3)\Longrightarrow(1):$ By the help of Proposition \[pro5\] and Lemma \[lemma fg\], as the previous proof one can prove easily.
([@SOD2010], Definition 2.1.1)A proper submodule $N$ of an $R$-module $M$ is said to be fully invariant submodule of $M$ if $f(N)\subseteq N,$ for every $f\in S$.
\[the3\]Let $M$ be an $R$-module and $N$ be a fully invariant and $S$-weakly prime submodule of $M$ that is not $S$-prime. If $I$ is an ideal of $R$ such that $I\subseteq(N:_{R}M),$ then $If(N)=0_{M},$ for any $f\in S.$ In particular, $(N:_{R}M)f(N)=0_{M}.$
Suppose that $If(N)\neq0_{M}.$ We show that $N$ is an $S$-prime submodule. Let $f(m)\in N,$ where $f\in End(M)$ and $m\in M.$ If $f(m)\neq0_{M},$ since $N$ is $S$-weakly prime, we are done. So, assume that $f(m)=0_{M}.$ Then we have 2 cases for $f(N).$
Case 1: $f(N)\neq0_{M}.$ Then there exists $n\in N$ such that $f(n)\neq0_{M}.$ Thus $0_{M}\neq f(n+m).$ As $N$ is fully invariant, one see $0_{M}\neq
f(n+m)\in N.$ Since $N$ is $S$-weakly prime, $m+n\in N,$ i.e., $m\in N$ or $f(M)\subseteq N.$
Case 2: $f(N)=0_{M}.$ As $If(N)\neq0_{M},$ contradiction.
\[cor n2\]Let $M$ be an $R$-module and $N$ be a fully invariant and $S$-weakly prime submodule of $M$ that is not $S$-prime. If $M$ is multiplication, then $f(N)^{2}=0_{M},$ for any $f\in S.$
Let $M$ be multiplication. Then $f(N)^{2}=(f(N):_{R}M)M(f(N):_{R}M)M=(f(N):_{R}M)(f(N):_{R}M)M=(f(N):_{R}M)f(N)\subseteq(N:_{R}M)f(N)$, since $N$ is fully invariant. By Theorem \[the3\], $(N:_{R}M)f(N)=0_{M},$ so $f(N)^{2}=0_{M}.$
Let $M$ be a multiplication $R$-module and $N,K$ be fully invariant and $S$-weakly prime submodules of $M$ that are not $S$-prime. Then $f(N)f(K)\subseteq\sqrt{0_{M}}.$
Assume that $f(b)\in f(K).$ Then $f(b)^{2}=Rf(b)Rf(b)\subseteq f(K)^{2}=0_{M},$ by Corollary \[cor n2\]. Then we get $f(K)\subseteq\sqrt{0_{M}}.$ Similarly, $f(N)\subseteq\sqrt{0_{M}}.$ Then we obtain $f(N)f(K)\subseteq
\sqrt{0_{M}}.$
For the next proof, we will need the following Lemma:
\[lemma atani\]([@SF2007], Lemma 2.5) Let $M$ be a multiplication module over $R$. Let $N$ and $K$ be submodules of $M$. Then the followings are hold:
1. If for every $a\in N,aK=0_{M},$ then $NK=0_{M}.$
2. If for every $b\in K,Nb=0_{M},$ then $NK=0_{M}.$
3. If for every $a\in N,b\in K,ab=0_{M},$ then $NK=0_{M}.$
\[the4\]Let $M$ be a finitely generated faithful multiplication $R$-module and $N$ be a fully invariant and $S$-weakly prime submodule of $M$ that is not $S$-prime. If $f\in S$ is onto, then $f(N)f(\sqrt{0_{M}})=0_{M}.$
Let $y=f(x)\in f(\sqrt{0_{M}})$ such that $x\in\sqrt{0_{M}}.$ Then there exists two ideals $I,J$ in $R$ such that $f(N)=IM$ and $Rx=JM$. Then as $f$ is onto, one see that $Rf(x)=f(Rx)=f(JM)=Jf(M)=JM.$ For $f(x),$ we have 2 cases :
Case 1: Let $f(x)\in f(N).$ Then $Rf(x)\subseteq f(N).$ By Lemma \[lemma fg\], we get $J\subseteq I.$ Thus with Corollary \[cor n2\], we have $f(N)Rf(x)=IJM\subseteq f(N)^{2}=0_{M}.$ By Lemma \[lemma atani\], $f(N)f(\sqrt{0_{M}})=0_{M}.$
Case 2: Let $f(x)\notin f(N).$ Then we get $x\notin N.$ By Theorem \[the1\], $(N:_{R}f(x))=(0_{M}:_{R}f(x))$ or $(N:_{R}f(x))=(N:_{R}f(M)).$
Assume that $(N:_{R}f(x))=(0_{M}:_{R}f(x)).$ Thus $(N:_{R}M)M\subseteq
(N:_{R}f(x))M=(0_{M}:_{R}f(x))M,$ so, as $N$ is fully invariant, $f(N)\subseteq(0_{M}:_{R}f(x))M$. Then $f(N)Rf(x)\subseteq(0_{M}:_{R}f(x))Rf(x)=0_{M},$ i.e., $f(N)f(x)=0_{M}.$ By Lemma \[lemma atani\], $f(N)f(\sqrt{0_{M}})=0_{M}.$
Now, suppose that $(N:_{R}f(x))=(N:_{R}f(M)).$ As $x\in\sqrt{0_{M}},$ there exists a smallest positive integer $n$ such that $x^{n}=0_{M}$ and $x^{n-1}\neq0_{M}.$ Then we see $Rx^{n}=J^{n}M=0_{M}$ and $Rx^{n-1}=J^{n-1}M\neq0_{M}.$ Hence, since $Rf(x)=JM,$ one get $Rf(x)^{n}=J^{n}M=0_{M}$ and $Rf(x)^{n-1}=J^{n-1}M\neq0_{M}.$ Moreover, we have $J^{n}M=0_{M}$ implies $J^{n}=0_{R},$ by Lemma \[lemma fg\]. Then it is clear that $J^{n-1}\subseteq(I:_{R}J).$ Hence, as $N$ is fully invariant, $0_{M}\neq
Rf(x)^{n-1}=J^{n-1}M\subseteq(IM:_{R}JM)M=(f(N):_{R}Rf(x))M\subseteq
(N:_{R}Rf(x))M.$ Then by our hypothesis and as $f$ is onto, $0_{M}\neq
Rf(x)^{n-1}\subseteq(N:_{R}Rf(x))M=(N:_{R}f(M))M=(N:_{R}M)M=N,$ this implies $0_{M}\neq f(x^{n-1})\subseteq N.$ Since $N$ is $S$-weakly prime, we get $0_{M}\neq x^{n-1}\subseteq N$ or $f(M)\subseteq N.$ The second one contradicts with $f(x)\notin N.$ As every $S$-weakly prime is a weakly prime submodule and by Theorem 2.6 in [@SF2007], $0_{M}\neq Rx^{n-1}\subseteq N$ implies $Rx\subseteq N,$ so $f(x)\in f(N),$ a contradiction.
Let $M$ be a finitely generated faithful multiplication $R$-module and $N,K$ be fully invariant and $S$-weakly prime submodules of $M$ that are not prime. If $f\in S$ is onto, then $f(N)f(K)=0_{M}.$
Assume that $f(b)\in f(K).$ Then $f(b)^{2}\subseteq f(K)^{2}=0_{M},$ by Corollary \[cor n2\]. Then we get $f(K)\subseteq\sqrt{0_{M}}.$ As $N$ is fully invariant, one see $f(N)f(K)\subseteq f(N)\sqrt{0_{M}}\subseteq
N\sqrt{0_{M}}.$ Since $N$ is $S$-weakly prime (so weakly prime) and not prime, we know $N\sqrt{0_{M}}=0_{M}$, by the help of Theorem 2.7 in [@SF2007]. Consequently, $f(N)f(K)=0_{M}.$
\[cor0\]Let $M$ be a finitely generated faithful multiplication module over $R$ with unique maximal submodule $K$ and every prime of $M$ is maximal. Let $N$ be a fully invariant and $S$-weakly prime submodule of $M$. If $f\in
S$ is onto, then $N=K$ or $f(N)f(K)=0_{M}.$
If $N$ is $S$-prime, so prime, by our hypothesis $N=K.$ If $N$ is not $S$-prime, one see $\sqrt{0_{M}}={\displaystyle\bigcap\limits_{N_{i}\in Spec(M)}}
N_{i}=K$, where $Spec(M)$ is the set of all prime submodules of $M.$ Then we obtain $f(\sqrt{0_{M}})=f(K).$ Thus $f(N)f(K)=$ $f(N)f(\sqrt{0_{M}})=0_{M},$ by Theorem \[the4\].
Let $M$ be a finitely generated faithful module over a local ring $R$ with unique maximal submodule $K$ and every prime of $M$ is maximal. Let $N$ be a fully invariant and $S$-weakly prime submodule of $M$. If $f\in S$ is onto, then $N=K$ or $f(N)f(K)=0_{M}.$
By Corollary 1 in [@CPL1995], $M$ is cyclic. Thus $M$ is multiplication $R$-module. By Corollary \[cor0\], it is done.
Let $M_{1},M_{2}$ be $R$-modules and we know that $M=M_{1}\times M_{2}$ is an $R$-module. For every $f_{i}\in End(M_{i}),$ let define $f:M\rightarrow M$ with $f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))$, for every $(m_{1},m_{2})\in M,$ $i=1,2.$ Then one can easily see, $f\in End(M)$ and $f(M)=f_{1}(M_{1})\times f_{2}(M_{2}).$ Also, we use the following notations: $S_{1}=End(M_{1})$, $S_{2}=End(M_{2})$ and $S=S_{1}\times S_{2}.$
\[prokart\]Let $M_{1},M_{2}$ be $R$-modules and $N_{1}$ be a proper submodule of $M_{1}$. Then the followings are equivalent:
1. $N=N_{1}\times M_{2}$ is an $S$-prime submodule of $M=M_{1}\times
M_{2}.$
2. $N_{1}$ is an $S_{1}$-prime submodule of $M_{1}.$
$(1)\Longrightarrow(2):$ Suppose that $N=N_{1}\times M_{2}$ is an $S$-prime submodule of $M=M_{1}\times M_{2}.$ Let $f_{1}(m_{1})\in N_{1},$ for some $m_{1}\in M_{1}$ and $f_{1}\in End(M_{1}).$ Then for every $m_{2}\in M_{2}$ and $f_{2}\in End(M_{2}),$ we get $f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N_{1}\times M_{2}=N.$ So, as $N=N_{1}\times M_{2}$ is an $S$-prime submodule of $M=M_{1}\times M_{2},$ we get $(m_{1},m_{2})\in N$ or $f(M_{1}\times M_{2})\subseteq N.$ Thus, by $f(M)=f_{1}(M_{1})\times
f_{2}(M_{2}),$ one see $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1},$ i.e., $N_{1}$ is $S_{1}$-prime.
$(2)\Longrightarrow(1):$ Let $N_{1}$ be an $S_{1}$-prime submodule of $M_{1}.$ Assume that $f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N=N_{1}\times
M_{2},$ for some $(m_{1},m_{2})\in M,f_{1}\in End(M_{1})$ and $f_{2}\in
End(M_{2}).$ Then $f_{1}(m_{1})\in N_{1}.$ As $N_{1}$ is $S_{1}$-prime, we have $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1}.$ Thus $(m_{1},m_{2})\in
N$ or $f(M_{1}\times M_{2})=f_{1}(M_{1})\times f_{2}(M_{2})\subseteq
N=N_{1}\times M_{2}.$
\[thekart\]Let $M_{1},M_{2}$ be $R$-modules and $N_{1}$ be a proper submodule of $M_{1}$. Then the followings are equivalent:
1. $N=N_{1}\times M_{2}$ is an $S$-weakly prime submodule of $M=M_{1}\times
M_{2}.$
2. $N_{1}$ is $S_{1}$-weakly prime and for every $(m_{1},m_{2})\in M$, $f_{1}\in S_{1}$ and $f_{2}\in S_{2},$ if $f_{1}(m_{1})=0_{M_{1}},m_{1}\notin
N_{1}$ and $f_{1}(M_{1})\nsubseteq N_{1},$ then $f_{2}(m_{2})=0_{M_{2}}.$
$(1)\Longrightarrow(2):$ Assume that $N=N_{1}\times M_{1}$ is an $S$-weakly prime submodule of $M=M_{1}\times M_{2}.$ Firstly, we show that $N_{1}$ is $S_{1}$-weakly prime. Let $0_{M_{1}}\neq f_{1}(m_{1})\in N_{1},$ for some $m_{1}\in M_{1}.$ Then for every $m_{2}\in M_{2},$ we get $0_{M}\neq
f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N.$ So, as $N=N_{1}\times
M_{2}$ is an $S$-weakly prime submodule of $M=M_{1}\times M_{2},$ we get $(m_{1},m_{2})\in N$ or $f(M_{1}\times M_{2})\subseteq N.$ Thus $m_{1}\in
N_{1}$ or $f_{1}(M_{1})\subseteq N_{1},$ i.e., $N_{1}$ is $S_{1}$-weakly prime. Let $0_{M_{1}}=f_{1}(m_{1})\in N_{1}$ such that $m_{1}\notin N_{1}$ and $f_{1}(M_{1})\nsubseteq N_{1}.$ Then for every $m_{2}\in M_{2},$ we say $(m_{1},m_{2})\notin N$ and $f(M_{1}\times M_{2})\nsubseteq N.$ Moreover, if $f_{2}(m_{2})\neq0_{M_{2}},$ we have $0_{M}\neq f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N,$ so $(m_{1},m_{2})\in N$ or $f(M_{1}\times
M_{2})\subseteq N,$ a contradiction. Consequently, $f_{2}(m_{2})=0_{M_{2}}.$
$(2)\Longrightarrow(1):$ Suppose that the condition (2) is true.
Let $0_{M}\neq f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N,$ for some $(m_{1},m_{2})\in M.$ Then for $f_{1}(m_{1}),$ we have 2 cases:
Case 1: Let $0_{M_{1}}\neq f_{1}(m_{1}).$ Since $f(m_{1})\in N_{1}$ and $N_{1}$ is $S_{1}$-weakly prime, we get $m_{1}\in N_{1}$ or $f(M_{1})\subseteq
N_{1},$ i.e., $(m_{1},m_{2})\in N$ or $f(M_{1}\times M_{2})\subseteq N.$ Thus, it is done.
Case 2: Let $0_{M_{1}}=f_{1}(m_{1}).$ Then $0_{M_{2}}\neq f_{2}(m_{2}).$ Assume that $m_{1}\notin N_{1}$ and $f_{1}(M_{1})\nsubseteq N_{1}.$ Then by (2), $f_{2}(m_{2})=0_{M_{2}},$ a contradiction. So $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1}.$ Thus $(m_{1},m_{2})\in N$ or $f(M)\subseteq N.$
\[pro1in2\]Let $M_{1},M_{2}$ be $R$-modules and $N_{1},N_{2}$ be a proper submodule of $M_{1},M_{2},$ resceptively. If $N=N_{1}\times N_{2}$ is an $S$-prime submodule of $M=M_{1}\times M_{2}$, then $N_{1}$ is an $S_{1}$-prime submodule of $M_{1}$ and $N_{2}$ is an $S_{2}$-prime submodule of $M_{2}.$
Suppose that $N=N_{1}\times N_{2}$ is an $S$-prime submodule of $M=M_{1}\times
M_{2}.$ Now, we will show that $N_{1}$ is an $S_{1}$-prime submodule. Take $f_{1}\in S_{1}$ such that $f_{1}(m_{1})\in N_{1}$ for some $m_{1}\in M_{1}.$ Also take $f_{2}\in S_{2}.$ Then $f((m_{1},0_{M_{2}}))=(f_{1}(m_{1}),f_{2}(0_{M_{2}}))=(f_{1}(m_{1}),0_{M_{2}})\in N_{1}\times N_{2}=N.$ Since $N$ is $S$-prime, $(m_{1},0_{M_{2}})\in N$ or $f(M_{1}\times M_{2})\subseteq
N.$ This implies that $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1}.$ Similar argument shows that $N_{2}$ is an $S_{2}$-prime submodule.
\[pro2in2\]Let $M_{1},M_{2}$ be $R$-modules and $N_{1},N_{2}$ be a proper submodule of $M_{1},M_{2},$ resceptively. If $N=N_{1}\times N_{2}$ is an $S$-weakly prime submodule of $M=M_{1}\times M_{2}$, then $N_{1}$ is an $S_{1}$-weakly prime submodule of $M_{1}$ and $N_{2}$ is an $S_{2}$-weakly prime submodule of $M_{2}.$
Assume that $N=N_{1}\times N_{2}$ is an $S$-weakly prime submodule of $M=M_{1}\times M_{2}.$ Let $f_{1}\in S_{1}$ such that $0_{M_{1}}\neq
f_{1}(m_{1})\in N_{1}$ for some $m_{1}\in M_{1}.$ Take $f_{2}\in S_{2}.$ Then $0_{M}\neq f((m_{1},0_{M_{2}}))\in N.$ Since $N$ is $S$-weakly prime, we get either $(m_{1},0_{M_{2}})\in N$ or $f(M_{1}\times M_{2})\subseteq N.$ This yields that $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1}.$ Similarly, one can see that $N_{2}$ is $S_{2}$-weakly prime.
Let $R_{i}$ be a commutative ring with identity and $M_{i}$ be an $R_{i}$-module for $i=1,2.$ Let $R=R_{1}\times R_{2}.$ Then $M=M_{1}\times M_{2}$ is an $R$-module and each submodule of $M$ is in the form of $N=N_{1}\times
N_{2},$ for some submodules $N_{1}$ of $M_{1}$ and $N_{2}$ of $M_{2},$ see [@hojjat].
\[theD1\]Let $R=R_{1}\times R_{2}$ be a decomposable ring and $M=M_{1}\times M_{2}$ be an $R$-module, where $M_{1}$ is an $R_{1}$-module and $M_{2}$ is an $R_{2}$-module. Suppose that $N=N_{1}\times M_{2}$ is a proper submodule of $M.$ Then the followings are equivalent:
1. $N_{1}$ is an $S_{1}$-prime submodule of $M_{1}.$
2. $N$ is an $S$-prime submodule of $M.$
3. $N$ is an $S$-weakly prime submodule of $M.$
$(1)\Longrightarrow(2):$ Let $N_{1}$ be an $S_{1}$-prime submodule of $M_{1}.$
Assume that $f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N=N_{1}\times
M_{2}$ for some $(m_{1},m_{2})\in M.$ Then $f_{1}(m_{1})\in N_{1}.$ As $N_{1}$ is $S_{1}$-prime, we have $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1}.$ Thus $(m_{1},m_{2})\in N$ or $f(M_{1}\times M_{2})\subseteq N.$
$(2)\Longrightarrow(3):$ It is clear.
$(3)\Longrightarrow(1):$ Suppose that $N$ is an $S$-weakly prime submodule of $M.$ Let $0_{M_{1}}\neq f_{1}(m_{1})\in N_{1}$ for some $m_{1}\in M_{1}.$ Put $f_{2}=id_{M_{2}},$ where $id_{M_{2}}$ denotes the identity homomorphism of $M_{2}.$ Then for every $m_{2}\in M_{2},$ we get $f((m_{1},m_{2}))=(f_{1}(m_{1}),f_{2}(m_{2}))\in N_{1}\times M_{2}=N.$ As $0_{M_{1}}\neq
f_{1}(m_{1}),$ we get $0_{M}\neq(f_{1}(m_{1}),f_{2}(m_{2})).$ By our hypothesis, $(m_{1},m_{2})\in N$ or $f(M_{1}\times M_{2})\subseteq N.$ Consequently, $m_{1}\in N_{1}$ or $f_{1}(M_{1})\subseteq N_{1}.$
\[theD2\]Let $R=R_{1}\times R_{2}$ be a decomposable ring and $M=M_{1}\times M_{2}$ be an $R$-module, where $M_{1}$ is an $R_{1}$-module and $M_{2}$ is an $R_{2}$-module. Suppose that $N_{1},N_{2}$ be a proper submodule of $M_{1},M_{2},$ resceptively. Then the followings are hold:
1. If $N=N_{1}\times N_{2}$ is an $S$-prime submodule of $M=M_{1}\times
M_{2}$, then $N_{1}$ is an $S_{1}$-prime submodule of $M_{1}$ and $N_{2}$ is an $S_{2}$-prime submodule of $M_{2}.$
2. If $N=N_{1}\times N_{2}$ is an $S$-weakly prime submodule of $M=M_{1}\times M_{2}$, then $N_{1}$ is an $S_{1}$-weakly prime submodule of $M_{1}$ and $N_{2}$ is an $S_{2}$-weakly prime submodule of $M_{2}.$
$(1)$ : It can be easily proved similar to Proposition \[pro1in2\].
$(2)$ : Similar to Proposition \[pro2in2\].
[99]{}
S. Ebrahimi and F. Farzalipour, 2007, On weakly prime submodules, Tamkang journal of Mathematics, 38(3), 247-252.
Shireen Ouda Dakheel, 2010, S-prime submodules and some related concepts(Thesis).
Z. A. El-Bast and P. F. Smith, 1988, Multiplication modules, Comm. in Algebra, 16, 755-779.
C. P. Lu, 1995, Spectra of modules, Comm. in Algebra, 23, 3741-3752.
J. Dauns, 1978, Prime modules, J. reine Angew. Math., 2, 156–181.
G. Gungoroglu, 2000, Strongly Prime Ideals in CS-Rings, Turk. J. Math., 24, 233-238.
D. D. Anderson and E. Smith, 2003, Weakly prime ideals, Houston J. of Math., 29, 831–840.
R. Ameri, 2003, On the prime submodules of multiplication modules, Inter. J. of Math. and Mathematical Sciences, 27, 1715–1724.
P. F. Smith, 1988, Some remarks on Multiplication modules, Arch Math. 50, 223-235.
Atiyah M. F., MacDonald I. G., 1969, Introduction to commutative algebra, CRC Press.
Lu C. P., 1984, Prime submodules of modules, Comm. Math. Univ. Sancti Pauli, 33: 61–69.
McCasland R. L. and Moore M. E., 1992, Prime submodules, Comm. Algebra, 20: 1803–1817.
Mostafanasab H., Tekir U. and Oral K. H., 2016, Weakly classical prime submodules, Kyunhpook Math. J. 56, 10085-1101.
|
---
abstract: 'We study the evolution of the configuration entropy of a biased tracer in the flat $\Lambda$CDM model assuming different time evolution of linear bias. We describe the time evolution of linear bias using a simple form $b(a)=b_{0} a^{n}$ with different index $n$. The derivative of the configuration entropy rate is known to exhibit a peak at the scale factor corresponding to the $\Lambda$-matter equality in the unbiased $\Lambda$CDM model. We show that in the $\Lambda$CDM model with time-dependent linear bias, the peak shifts to smaller scale factors for negative values of $n$. This is related to the fact that the growth of structures in the tracer density field can significantly slow down even before the onset of $\Lambda$ domination in presence of a strong time evolution of the linear bias. We find that the shift is linearly related to the index $n$. We obtain the best fit relation between these two parameters and propose that identifying the location of this peak from observations would allow us to constrain the time evolution of linear bias within the framework of the $\Lambda$CDM model.'
author:
- |
Biswajit Das[^1] and Biswajit Pandey[^2]\
Department of Physics, Visva-Bharati University, Santiniketan, Birbhum, 731235, India\
title:
-
- 'Can we constrain the evolution of linear bias using configuration entropy?'
---
\[firstpage\]
methods: analytical - cosmology: theory - large scale structure of the Universe
Introduction
============
Our knowledge of the present day galaxy distribution in the nearby Universe has been revolutionized by the modern galaxy survey (SDSS, @york; 2dFGRS, @colles; 2MRS, @huchra) carried out over the last few decades. However cosmological observations suggest that most of the mass in the Universe is in the form of an unseen dark matter which is yet to be directly detected by observations. The galaxies are known to be a biased tracer of the underlying dark matter distribution. On large scales, it is believed that the fluctuations in the galaxy distribution and the dark matter distribution are linearly related by a bias parameter [@kaiser84; @dekel]. The linear bias parameter is known to be scale-independent on large scales [@mann] but is expected to evolve with time [@fry; @tegmark]. The time evolution of the linear bias parameter determine the evolution of the large scale distribution of the tracer relative to the underlying mass distribution. However the galaxies have not always been in place. They are the product of the non-linear evolution of the cosmic density field. Thanks to the improvement of computing power and algorithms, modern day N-body simulations [@springel; @vogelsberger] can give us a clear idea about the emergence of structures through non-linear evolution. In fact, the understanding of the process of structure formation has become so good that it has become a standard tool for testing cosmological models.
Early measurements of the two point correlation function for galaxies and galaxy clusters did not match, indicating that both cannot be unbiased tracers of the underlying matter distribution. @kaiser84 pointed out that the density field of clusters are the peaks in the galaxy density field and therefore they are more strongly clustered than the galaxies. The relative clustering strength of any tracer with respect to the dark matter distribution is quantified through the linear bias parameter measured from observations. One can employ the two-point correlation function and power spectrum to determine the linear bias parameter [@nor; @teg; @zehavi10]. The redshift space distortions of the two-point correlation function and power spectrum [@kaiser87; @hamilton] can be also employed to measure the linear bias parameter [@haw; @teg]. The other alternatives which have been successfully used to compute the linear bias parameter are the three-point correlation function and bispectrum [@feldman; @verde; @gaztanaga], filamentarity [@pandey07] and information entropy [@pandey17a].
Galaxies do not exist at high redshift whereas the neutral Hydrogen (HI) is present throughout the history of the Universe since its formation after the recombination at $z \sim 1100$. The redshifted 21 cm line from neutral Hydrogen would reveal a wealth of information about the formation and evolution of structures in the Universe. A number of surveys (HIPASS, @zwaan; ALFALFA, @martin) have been designed to map the HI content of galaxies in the nearby Universe. A significant effort has been also directed to detect the redshifted 21 cm signal using different ongoing and upcoming radio interferometric facilities (GMRT, @paciga; LOFAR, @vanharlem ; MWA @bowman ; SKA, @mallema). The redshifted 21 cm line can be used as a promising probe of the large scale structures over a wide redshift range [@bharad01; @bharadsethi]. The knowledge about the HI bias and its time evolution is also important in understanding the uncertainties associated with the measured intensity fluctuation power spectrum. Several studies have been carried out to measure the HI bias [@martin; @masui; @switzer] at low redshifts ($z<1$) but presently the evolution of HI bias with redshift is not known. Some theoretical and observational constraints on the evolution of HI bias over the redshift range $0-3.5$ is summarized in @hamsa and references therein.
Recently, it has been suggested that the measurement of the configuration entropy [@pandey1; @pandey3] of the mass distribution in the Universe can be used to test the different cosmological models [@das1], determine the mass density parameter and cosmological constant [@pandey2] and constrain the dark energy equation of state parameters [@das2]. In the present work, we propose a theoretical framework based on the study of configuration entropy which would allow us to probe the evolution of linear bias from future redshifted 21 cm observations.
Theory
======
Evolution of configuration entropy
----------------------------------
We consider a large comoving volume $V$ of the Universe and divide it into sub-volumes $dV$. Let the density of the tracer in each of these sub-volumes at time $t$ be $\rho_T (\vec{x}, t)$ where $\vec {x}$ is the comoving coordinate of the sub-volume defined with respect to an arbitrary origin. The configuration entropy of the matter density field can be defined as [@pandey1], $$\begin{aligned}
S_c(t) = - \int \rho_T(\vec{x}, t) \log \rho_T (\vec{x}, t)\, dV.
\label{eq:one}\end{aligned}$$ The definition of configuration entropy is motivated from the definition of information entropy [@shannon48].
The mass distribution of the Universe is often treated as an ideal fluid to a good approximation. The continuity equation of this fluid in an expanding Universe can be written as, $$\begin{aligned}
\frac{\partial \rho_T}{\partial t} + 3 \frac{\dot a}{a} \rho_T +
\frac{1}{a} \nabla \cdot (\rho_T \vec {v_T}) = 0.
\label{eq:two}\end{aligned}$$ In , $a$ is the cosmological scale factor and $\vec{v_T}$ is the peculiar velocity of the tracer fluid element. We can combine and to get, $$\begin{aligned}
\frac{dS_c(t)}{dt} + 3 \frac{\dot a}{a} S_c(t) - \frac{1}{a} \int
\rho_T (3 \dot a + \nabla \cdot \vec {v_T})\, dV = 0.
\label{eq:three}\end{aligned}$$ We rewrite as, $$\begin{aligned}
\frac{dS_c(a)}{da} \dot a + 3 \frac{\dot a}{a} S_c(a) - 3 \frac{\dot
a}{a} \int \rho_T(\vec {x}, a)\, dV \nonumber \\ - \frac{1}{a} \int
\rho_T(\vec {x}, a) \nabla \cdot \vec {v_T}\, dV = 0,
\label{eq:four}\end{aligned}$$ where the variable of differentiation has been changed from $t$ to $a$. Here $\int \rho_T(\vec {x}, a)\, dV = M_T$ is the total mass of the tracer contained inside the comoving volume $V$. The density of tracer at comoving location $\vec{x}$ can be expressed as $\rho_T(\vec
{x}, a) = \bar \rho_T (1 + \delta_T (\vec {x}, a))$, where $\delta_T(\vec {x}, a)$ is the density contrast at location $\vec{x}$ and $\bar \rho_T = \frac{M_T}{V}$ is the average density of tracer. In linear perturbation theory, one can write $\delta_m(\vec
{x}, a) = D(a) \delta_m (\vec {x})$ and $\nabla \cdot \vec {v_T} = - a
\frac{\partial \delta_T(\vec{x}, a)}{\partial t}$. Here, $D(a)$ is the growing mode and $\delta_m(\vec{x})$ is the initial mass density perturbation at location $\vec{x}$. We simplify using these relations to get, $$\begin{aligned}
\frac{dS_c(a)}{da} \dot a + 3 \frac{\dot a}{a} (S_c(a) - M_T)
- \frac{\bar \rho_T}{a} \int \nabla \cdot \vec{v_T}\, dV \nonumber \\
- \frac{\bar \rho_T}{a} \int \delta_T(\vec{x}, a) \nabla \cdot \vec{v_T}\, dV = 0.
\label{eq:five}\end{aligned}$$
In the linear bias assumption, $$\begin{aligned}
\delta_{T} (\vec{x}, a) = b(a) \delta_m (\vec{x}, a),
\label{eq:six}\end{aligned}$$ where $b(a)$ is the scale-independent linear bias parameter and $\delta_{T} (\vec{x}, a)$ and $\delta_m (\vec{x}, a)$ are the density contrast corresponding to the tracer and the underlying mass density field respectively. So, $$\begin{aligned}
\nabla \cdot \vec{v} = - a \dot a \left[D(a)
\frac{db(a)}{da} + b(a) \frac{dD(a)}{da}\right] \delta_m (\vec{x}).
\label{eq:seven}\end{aligned}$$ We combine and and simplify to get, $$\begin{aligned}
\frac{dS_c(a)}{da} + \frac{3}{a} (S_c(a) - M_T) + \bar \rho_T B(a) \int
\delta^2_m(\vec{x})\, dV = 0.
\label{eq:eight}\end{aligned}$$ Here, $B(a) = b(a)D(a)\left[D(a)\frac{db(a)}{da} +
b(a)\frac{f(a)D(a)}{a}\right]$ where $f(a) = \frac{a}{D(a)}
\frac{dD(a)}{da}$ is the dimensionless linear growth rate.
This equation governs the evolution of configuration entropy of the tracer in presence of time evolution of linear bias. One can integrate to get $$\begin{aligned}
\frac{S_c(a)}{S_c(a_i)} = \frac{M_T}{S_c(a_i)} + \left[1 -
\frac{M_T}{S_c(a_i)}\right]\Big(\frac{a_i}{a}\Big)^3 \nonumber \\-
\Bigg(\frac{\bar \rho_T \int
\delta^2_m(\vec{x})\,dV}{S_c(a_i)a^3}\Bigg) \int_{a_i}^a da^{\prime}
a^{\prime 3} B(a^{\prime}).
\label{eq:nine}\end{aligned}$$ Here $a_i$ is some initial scale factor and $S_c(a_i)$ is the initial configuration entropy. In our analysis we have chosen $a_i =
10^{-3}$.
We find the evolution of the ratio of configuration entropy to its initial value by numerically calculating the integral in the third term for different time evolution of bias and substituting back at . We set the product $\bar \rho_T \int
\delta^2_m(\vec{x})\,dV = 1$ for simplicity. The choices of $S_c(a_i)$ and $M_T$ are arbitrary and in no way depend on the cosmological model concerned. Choosing $S_c(a_i)>M$ or $S_c(a_i)<M$ causes a sudden growth or decay in $\frac{S_c(a)}{S_c(a_i)}$ near the initial scale factor $a_i$, respectively. We have chosen $S_c(a_i) = M_T$ in our analysis to ignore these transients caused by the initial conditions. The integral in the third term of involves evolution of growing mode, time dependent bias and their derivatives which are discussed in the next two subsections.
\
Growth rate of density perturbations
------------------------------------
The CMBR observations suggest that the Universe was highly isotropic at $z \sim 1100$. But the present day Universe is not homogeneous and isotropic on small scales. We find galaxies and clusters of galaxies where huge mass is accumulated over a small region whereas there are large empty regions or voids with very little amount of mass. The linear perturbation theory provides a theoretical framework to understand the growth of structures from tiny fluctuations seeded in a homogeneous and isotropic distribution in the early Universe. In the currently accepted paradigm, gravitational instability is the primary mechanism behind the formation of structures in the Universe. CMBR observations indicate that inhomogenities of very small magnitude were present in the matter distribution at the time of recombination. These tiny inhomogeneities get amplified by the gravitational instability over time. When the density contrast is much smaller than $1$, its evolution can be described by the following differential equation, $$\begin{aligned}
\frac{\partial^2 \delta_m(\vec{x}, t)}{\partial t^2} + 2 H(a) \frac{\partial \delta_m(\vec{x}, t)}{\partial t} - \frac{3}{2} \Omega_{m0} H^2_0 \frac{1}{a^3} \delta_m(\vec{x}, t) = 0.
\label{eq:ten}\end{aligned}$$ Here we have considered perturbation to only matter component. $\Omega_{m0}$ and $H_0$ are the present value of density parameter for matter and Hubble parameter, respectively. This equation governs the growth of density perturbation in the underlying matter distribution. The equation has a growing mode solution of the form $\delta_m(\vec{x}, t) = D(t)\delta_m(\vec{x})$. The growing mode solution can be expressed as [@peebles] $$\begin{aligned}
D(a) = \frac{5}{2} \Omega_{m0} X^{\frac{1}{2}}(a) \int_0^a \frac{da^{\prime}}{a^{\prime 3} X^{\frac{3}{2}}(a^{\prime})},
\label{eq:eleven}\end{aligned}$$ where $X(a) = \frac{H^2(a)}{H_0^2} = [\Omega_{m0} a^{-3} +
\Omega_{\Lambda0}]$. Here $\Omega_{\Lambda0}$ is the present value of the density parameter corresponding to cosmological constant.
The dimensionless linear growth rate $f(a) = \frac{d\log D(a)}{d \log
a}$ in a universe with no curvature can be approximated as [@lahav] $$\begin{aligned}
f(a) = \Omega_m(a)^{0.6} + \frac{1}{70}\left[1 - \frac{1}{2}
\Omega_m(a) (1 + \Omega_m(a))\right].
\label{eq:twelve}\end{aligned}$$ Here $\Omega_m(a) = \frac{\Omega_{m0} a^{-3}}{X(a)}$. We have used $\Omega_{m0} = 0.3$ and $\Omega_{\Lambda0} = 0.7$ throughout the present work.
Evolution of linear bias
------------------------
The time evolution of the linear bias parameter is expected to affect the time evolution of the configuration entropy of the tracer density field. We consider a simple power law of the form $b(a)=b_{0} a^{n}$ with different possible values of $n$. The functional form is motivated by @bagla where $b(z) \propto (1 + z)^{0.5}$ was reported to give a reasonably good description of the evolution of HI bias in the simulated HI distributions from the N-body simulations. We consider the following values of $n$ in our analysis: $n = -1, -0.75,
-0.5, -0.25, 0.5, 1$. We also incorporate the unbiased $\Lambda$CDM model in this framework by putting $b(a) = b_0$. We set $b_0 = 1$ in all the models considered here.
Results and Conclusions
=======================
We show the evolution of the linear bias with scale factor for different values of $n$ in the top left panel of . The amplitude of the bias at any given scale factor depends on the index $n$. The linear bias monotonically decreases with increasing scale factor for negative $n$. A negative value of $n$ indicates that the tracer density field was strongly biased in the past which decreases with time and eventually reaches unity at present. The decrease in bias corresponds to an overall dilution in the clustering of the tracer mass distribution. The evolution of $\frac{S_c(a)}{S_c(a_i)}$ with scale factor for all these models is shown in the top right panel of . The evolution of the configuration entropy is primarily governed by the growth of density perturbations which in turn is affected by the dynamics of the expansion of the Universe. Expansion of the Universe slows down the growth of perturbations. Besides the expansion, the time evolution of bias would also play an important role in controlling the dissipation of the configuration entropy of the Universe. For example, all the models with negative $n$ show a decrease in the configuration entropy at earlier times. However the dissipation slows down with time and in some cases it may even reverse its behaviour and starts to grow with time. The time of reversal from dissipation to growth depends on the index $n$. More negative index leads to an early reversal in the behaviour of the configuration entropy.
The lower left panel of shows the entropy rate as a function of scale factor in models with different $n$. The entropy rate is decided by the function $B(a) =
b(a)D(a)\left[D(a)\frac{db(a)}{da} + b(a)\frac{f(a)D(a)}{a}\right]$ which consists of two terms and the combined contribution from these two terms decides the behaviour of the entropy rate at any given time for any specific model. The two terms are separately plotted as function of the scale factor for different models in the left and right panels of . Clearly a growth in entropy is expected when $B(a)$ is negative and a positive $B(a)$ is associated with entropy dissipation. For example $B(a)$ is negative at all scale factor for $n=-1$ and this implies that there will be no dissipation of entropy in this model. On the other hand the model with $n=1$ and $n=0.5$ have positive $B(a)$ at all scale factors and there is a continuous dissipation of entropy in these models. All the other models considered here show dissipation of entropy at some scale factors and growth of entropy at some other scale factors. A zero up crossing in the entropy rate corresponds to a local minimum in the configuration entropy. Clearly this zero up crossing appears at a smaller scale factor for more negative values of $n$.
We show the derivative of the entropy rate in these models in the lower right panel of . The derivative of the entropy rate exhibits a peak in all the models with negative $n$. We find that the location of the peak is sensitive to the index $n$ and it appears at a smaller scale factor for models with smaller index. In an earlier work, @pandey2 noted that in the unbiased $\Lambda$CDM model, this peak is exactly located at the scale factor corresponding to the $\Lambda$-matter equality. We have used $\Omega_{m0}=0.3$ and $\Omega_{\Lambda}=0.7$ in the $\Lambda$CDM model. So in the unbiased $\Lambda$CDM model the peak is expected to appear at $a=0.754$. This can be clearly seen in the result shown for the unbiased $\Lambda$CDM model in the same panel. Now the location of this peak is shifted towards a smaller scale factor when time evolution of bias is considered within the $\Lambda$CDM model. The shift is measured with reference to the location of the peak in the unbiased $\Lambda$CDM model. The magnitude of the entropy rate slows down after the occurrence of this peak. In the unbiased $\Lambda$CDM model, the structure formation starts to slow down after the onset of $\Lambda$ domination. The bias models with negative value of $n$ dilute the clustering and slows down the structure formation even before the $\Lambda$-matter equality. This effect would manifest in a more prominent way in the models with more negative $n$. So the peak in the derivative of the entropy rate is expected to exhibit a larger shift in these models. We measure the location of the peak in the models with different negative index and find them to be linearly related. We show the index $n$ as a function of the location and the shift of the peak in the left and right panels of respectively. The best fit relations between these parameters are also provided in the same figure.
We also consider two positive values of $n$ in the time evolution of bias. A positive value of $n$ in this case ($b_0=1$) indicates that the tracer density field is anti biased with respect to the underlying mass density field and the bias slowly increases from a very small positive value to unity at present. A decrease in anti biasing with time would enhance the clustering of the tracer leading to a continuous dissipation of the configuration entropy. In these models, initially entropy show a slower decrease than that of $\Lambda$CDM model but then decrease quite quickly in the later part. We do not observe the peak in the derivative of the entropy rate in these models and they can be easily distinguished from the models with negative values of $n$. These models are not realistic and we consider them only for the sake of completeness.
In this work, we calculate the evolution of the configuration entropy for a biased tracer of the density field (e.g. neutral Hydrogen) assuming different time evolution of bias. We consider the flat $\Lambda$CDM model as the benchmark model of the Universe and within it consider the time evolution of linear bias as, $b(a)=b_{0} a^{n}$ with different values of the index $n$. We show that the time evolution of bias alters the position of the peak in the derivative of the entropy rate. The peak shifts towards a smaller scale factor for negative index and is absent when the index is positive. We find that the shift is linearly related with the index $n$ and a larger shift is observed for a smaller index. We find the best fit relation between these two parameters and propose that identifying the location of this peak from observations would allow us to constrain the time evolution of linear bias within the framework of the $\Lambda$CDM model. Finally we conclude that the analysis presented in this work provides an alternative method to constrain the evolution of linear bias using configuration entropy.
Acknowledgement
===============
BP acknowledges financial support from the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India through the project EMR/2015/001037. BP would also like to acknowledge IUCAA, Pune for providing support through the associateship programme.
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\[lastpage\]
[^1]: E-mail: bishoophy@gmail.com
[^2]: E-mail: biswap@visva-bharati.ac.in
|
---
abstract: 'The behaviour of limits of weak morphisms in 2-dimensional universal algebra is not 2-categorical in that, to fully express the behaviour that occurs, one needs to be able to quantify over strict morphisms amongst the weaker kinds. F-categories were introduced to express this interplay between strict and weak morphisms. We express doctrinal adjunction as an ${\ensuremath{\mathcal{F}}\xspace}$-categorical lifting property and use this to give monadicity theorems, expressed using the language of ${\ensuremath{\mathcal{F}}\xspace}$-categories, that cover each weaker kind of morphism.'
address: 'Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, Brno 60000, Czech Republic'
author:
- John Bourke
bibliography:
- 'bibdata.bib'
title: Two Dimensional Monadicity
---
[^1]
=2em
Introduction
============
The category of monoids sits over the category of sets via a forgetful functor $U:{\ensuremath{\textnormal{Mon}}\xspace}\to {\ensuremath{\textnormal{Set}}\xspace}$. This functor is *monadic* in the sense that it has a left adjoint $F$ and the canonical comparison $E:{\ensuremath{\textnormal{Mon}}\xspace}\to {\ensuremath{\textnormal{Set}}\xspace}^{T}$ to the category of algebras for the induced monad $T=UF$ is an equivalence of categories. So if you are interested in monoids you can set about proving some theorem about algebras for an abstract monad $T$ and be sure it holds for monoids, or any variety of universal algebra for that matter: this is the categorical approach to universal algebra via monads.\
Before going down this path one thing must be established – namely, the monadicity of $U$. To this end the fundamental theorem is *Beck’s monadicity theorem* [@Beck1967Triples] which asserts that a functor $U:{\ensuremath{\mathcal{A}}\xspace} \to {\ensuremath{\mathcal{B}}\xspace}$ is monadic just when it admits a left adjoint, is conservative and creates $U$-absolute coequalisers. What makes the theorem so useful in practice is that the conditions, up to the *existence* of a left adjoint, are cast entirely in terms of the typically simple $U$ – these conditions are clearly met for monoids or indeed any variety (see Section 6.8 of [@Mac-Lane1971Categories]).\
Now our interest is not in universal algebra, but in two dimensional universal algebra and 2-monads, and monadicity as appropriate to this setting. What do we mean by the varieties of 2-dimensional universal algebra? Monoidal structure borne by categories provides a basic example: one observes that associated to this notion are several kinds of structure. On the objects front we have at least strict monoidal categories and monoidal categories of interest. Between these are strict, strong, lax and colax monoidal functors all commonly arising, between which we have just one kind of monoidal transformation. Restricting ourselves to just one kind of object, let us take the monoidal categories, we still find that we are presented with four related 2-categories ${\ensuremath{\textnormal{MonCat}}\xspace}_{w}$, where $w \in \{s,p,l,c\}$, living over the 2-category of categories [$\textnormal{Cat}$]{}as on the left below. $$\xy
(-30,0)*+{\txt{(1)}}="a";
(-15,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{s}}="00";
(10,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{p}}="10"; (30,10)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{l}}="11";(30,-10)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{c}}="1-1";(10,-25)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="1-2";
{\ar^{} "00"; "10"};
{\ar^{} "10"; "11"};
{\ar^{} "10"; "1-1"};
{\ar_{V_{s}} "00"; "1-2"};
{\ar_{V_{p}} "10"; "1-2"};
{\ar@{.>}^<<<<<<<{V_{l}} "11"; "1-2"};
{\ar^{V_{c}} "1-1"; "1-2"};
\endxy
\xy
(-15,0)*+{{\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}}="00";
(10,0)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{p}}="10"; (30,10)*+{{\ensuremath{\textnormal{T-Alg}_{\textnormal{l}}}\xspace}}="11";(30,-10)*+{{\ensuremath{\textnormal{T-Alg}_{\textnormal{c}}}\xspace}}="1-1";(10,-25)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="1-2";
{\ar^{} "00"; "10"};
{\ar^{} "10"; "11"};
{\ar^{} "10"; "1-1"};
{\ar_{U_{s}} "00"; "1-2"};
{\ar_{U_{p}} "10"; "1-2"};
{\ar@{.>}^<<<<<<<{U_{l}} "11"; "1-2"};
{\ar^{U_{c}} "1-1"; "1-2"};
\endxy$$ The objects in each of these are monoidal categories; the morphisms are respectively strict, strong (or pseudo), lax and colax monoidal functors with monoidal transformations between them in each case. The inclusions witness that strict morphisms can be viewed as pseudomorphisms $(s \leq p)$, which can in turn be viewed as lax or colax $(p \leq l)$ and ($p \leq c)$.\
The situation corresponds with that of a 2-monad $T$ based on [$\textnormal{Cat}$]{}, associated with which are several kinds of algebra, including strict and pseudo-algebras. We will only ever consider the *strict algebras*: note that even *non-strict* monoidal categories are the *strict* algebras for a 2-monad, as further discussed below. Between strict algebras are strict, pseudo, lax and colax morphisms, with again a single notion of algebra transformation. As before we obtain a diagram of 2-categories
[The objects in each of these 2-categories are the strict algebras.]{}
and 2-functors over [$\textnormal{Cat}$]{}, as on the right above.\
Comparing the diagrams left and right above we see that a monadicity theorem in this setting ought to match each 2-category on the left with the corresponding one on the right in a compatible way. More precisely, there should exist a 2-monad $T$ on [$\textnormal{Cat}$]{}and, for each $w \in \{s,p,l,c\}$, an equivalence of 2-categories $E_{w}:{\ensuremath{\textnormal{MonCat}}\xspace}_{w} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ over [$\textnormal{Cat}$]{}, as left below $$\xy
(-30,0)*+{\txt{(2)}}="a";
(0,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{w}}="00";
(30,0)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{w}}="10";(15,-20)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="1-2";
{\ar^{E_{w}} "00"; "10"};
{\ar_{V_{w}} "00"; "1-2"};
{\ar^{U_{w}} "10"; "1-2"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{w_{1}}}="00"; (30,0)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{w_{1}}}="10";
(0,-20)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{w_{2}}}="01"; (30,-20)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{w_{2}}}="11";
{\ar^{E_{w_{1}}} "00";"10"};
{\ar ^{}"10";"11"};
{\ar _{}"00";"01"};
{\ar _{E_{w_{2}}} "01";"11"};
\endxy$$ and these equivalences should be natural in the inclusions $w_{1} \leq w_{2}$ for $w_{1},w_{2} \in \{s,p,l,c\}$, as on the right.\
Now it is well known that such a 2-monad does indeed exist, with, moreover, each comparison $E_{w}:{\ensuremath{\textnormal{MonCat}}\xspace}_{w} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ an isomorphism of 2-categories. Likewise many of the other varieties of 2-dimensional universal algebra are monadic in this sense. With such varieties as the primary object of study the subject of 2-dimensional monad theory was developed, notably in [@Blackwell1989Two-dimensional]. General results were obtained such as those enabling one to deduce that each inclusion ${\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{MonCat}}\xspace}_{w}$ has a left 2-adjoint, or establish the bicategorical completeness and cocompleteness of ${\ensuremath{\textnormal{MonCat}}\xspace}_{p}$.\
Of course to apply such abstract results one must first establish monadicity. How is this known? Here the subject diverges substantially from the 1-dimensional approach of Beck’s theorem. The standard approach is that of colimit presentations [@Lack2010A-2-categories][@Kelly1993Adjunctions]. Here one explicitly constructs the intended 2-monad $T$ as an iterated colimit of free ones, and then performs lengthy calculations with the universal property of the colimit $T$ to establish monadicity in the sense described for monoidal categories above.\
Now although the natural analogue of Beck’s theorem has been obtained for *pseudomonads* [@LeCreurer2002Beck] and pseudoalgebra pseudomorphisms this does not specialise to capture monadicity in the precise sense described above, even when $w=p$.\
Our objective in the present paper is to establish 2-dimensional monadicity theorems, in which monadicity is recognised not by using an explicit description of a 2-monad, but by analysing the manner in which the varieties of 2-dimensional universal algebra sit over the base 2-category – as in Diagram 1. Apart from characterising such monadic situations our main results, Theorem \[thm:EMExtension\] through Theorem \[thm:monadicity\], enable one to establish monadicity when a workable description of a 2-monad is not easily forthcoming.\
In seeking to understand monadicity in the above sense the first important observation is that the world of *strict morphisms* is easily understood: one can check that $V_{s}:{\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{Cat}}\xspace}$ has a left 2-adjoint – say, by an adjoint functor theorem – and then apply the enriched version of Beck’s monadicity theorem [@Dubuc1970Kan-extensions] to establish strict monadicity. Having observed this to be the case the natural question to ask is: *Which properties of the commutative triangle $$\xy
(0,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{s}}="00";
(30,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{w}}="10";(15,-20)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="1-2";
{\ar^{} "00"; "10"};
{\ar_{V_{s}} "00"; "1-2"};
{\ar^{V_{w}} "10"; "1-2"};
\endxy$$ ensure that the canonical isomorphism $E:{\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{s}$ extends to an isomorphism $E_{w}:{\ensuremath{\textnormal{MonCat}}\xspace}_{w} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ over the base?* We answer this question in Theorem \[thm:monadicity\] but not using the language of 2-categories. For it turns out that these determining properties are not 2-categorical in nature: they cannot be expressed as properties of the 2-categories or 2-functors in the above diagram considered individually. Rather, to express these properties we must be able to *single out strict morphisms amongst each weaker kind*. Consequently we treat the inclusion ${\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{MonCat}}\xspace}_{w}$ as a single entity, an *F-category* ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}$, and the above triangle as a single *F-functor* $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w} \to {\ensuremath{\textnormal{Cat}}\xspace}$.\
Let us now give an overview of the paper and of our line of argument. F-categories were introduced in [@Lack2011Enhanced] in order to explain certain relationships between strict and weak morphisms in 2-dimensional universal algebra. We recall some basic facts about F-categories in Section 2, in particular discussing F-categories ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ of algebras for a 2-monad and F-categories ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}$ of monoidal categories – we will use monoidal categories as our running example throughout the paper.\
Given a strict monoidal functor $F$ and an adjunction $(\epsilon, F \dashv G, \eta)$ the right adjoint $G$ obtains a unique lax monoidal structure such that the adjunction becomes a monoidal adjunction. This is an instance of *doctrinal adjunction*, the main topic of Section 3, and the first key F-categorical property that we meet. We express three variants of doctrinal adjunction – $w$-doctrinal adjunction for $w \in \{l,p,c\}$ – as lifting properties of an F-functor, so that the case just described asserts that the forgetful F-functor $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ satisfies $l$-doctrinal adjunction. We define the closely related class ${\ensuremath{w\textnormal{-Doct}}\xspace}$ of *$w$-doctrinal* F-functors and analyse the relationships between the different notions for $w \in \{l,p,c\}$; each such class of F-functor is shown to be an orthogonality class in the category of F-categories.\
In the fourth section we turn to the reason F-categories were introduced in [@Lack2011Enhanced] – namely, because of the interplay between strict and weak morphisms, *tight* and *loose*, that occurs when considering limits of weak morphisms in 2-dimensional universal algebra. The crucial limits are $\bar{w}$-limits of loose morphisms for $w \in \{l,p,c\}$ – after defining these we describe the illuminating case of the colax limit of a lax monoidal functor. We then examine how such limits allow one to represent loose morphisms by *tight spans* – the nature of this representation is analysed in detail.\
This analysis allows us to prove the key result of the paper, Theorem \[thm:orthogonality\] of Section 5, an orthogonality result which has nothing to do with 2-monads at all. Its immediate consequence, Corollary \[thm:decomposition\], ensures that the decomposition in [$\mathcal{F}\textnormal{-CAT}$]{}$$\xy
(0,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{s}}="00";(25,0)*+{{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}}="10";(50,0)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="20";
{\ar^{j} "00"; "10"};
{\ar^<<<<<{V} "10"; "20"};
\endxy
\xy
(0,0)*+{=};
\endxy
\xy
(0,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{s}}="00";(25,8)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{s}}="11";(25,-7)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{w}}="1-1";(50,0)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="20";
{\ar^{1} "00"; "11"};
{\ar_{j} "00"; "1-1"};
{\ar^{j} "11"; "1-1"};
{\ar^{V_{s}} "11"; "20"};
{\ar_{V_{w}} "1-1"; "20"};
\endxy$$ of the forgetful 2-functor $V_{s}:{\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{Cat}}\xspace}$ is an orthogonal $(^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace},{\ensuremath{w\textnormal{-Doct}}\xspace})$-decomposition. Likewise for a 2-monad $T$ on [$\textnormal{Cat}$]{}the F-category ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ is obtained as a $(^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace},{\ensuremath{w\textnormal{-Doct}}\xspace})$-factorisation of $U_{s}:{\ensuremath{\textnormal{T-Alg}}\xspace}_{s} \to {\ensuremath{\textnormal{Cat}}\xspace}$.\
Our monadicity results, given in Section 6, use the uniqueness of $(^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace},{\ensuremath{w\textnormal{-Doct}}\xspace})$-decompositions to extend our understanding of monadicity in the strict setting to cover each weaker kind of morphism. For instance, the isomorphism $E:{\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{s}$ over [$\textnormal{Cat}$]{}induces a commuting diagram as on the outside of $$\xy
(0,0)*+{{\ensuremath{\textnormal{MonCat}}\xspace}_{s}}="00";(30,0)*+{{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}}="10";(60,0)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="20";
{\ar^{j} "00"; "10"};
{\ar^{V} "10"; "20"};
(0,-15)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{s}}="01";(30,-15)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="11";(60,-15)*+{{\ensuremath{\textnormal{Cat}}\xspace}}="21";
{\ar_{E} "00"; "01"};
{\ar@{.>}|{E_{w}} "10"; "11"};
{\ar_{j} "01"; "11"};
{\ar_{U} "11"; "21"};
{\ar^{1} "20"; "21"};
\endxy$$ with each horizontal path an orthogonal $(^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace},{\ensuremath{w\textnormal{-Doct}}\xspace})$-decomposition. The two outer vertical isomorphisms then induce a unique invertible filler $E_{w}:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$, so establishing monadicity in each weaker context. This is the idea behind the main monadicity result, Theorem \[thm:monadicity\]. Naturality in different weaknesses (as in Diagram 2 above) is treated in Theorem \[thm:naturality\].\
In the seventh and final section we describe examples and applications of our results. We begin by completing the example of monoidal categories before moving on to more complex cases. In Theorem \[thm:monoids\] we give an example of the kind of monadicity result that cannot be established using techniques, like presentations, that require explicit knowledge of a 2-monad.
The author thanks Richard Garner, Stephen Lack, Ignacio López Franco, Michael Shulman and Lukáš Vokřínek for useful feedback during the preparation of this work, and Michael Shulman for carefully reading a preliminary draft. Thanks are due to the referee whose insightful observations enabled us to remove unnecessary hypotheses from the main results. Thanks also to the organisers of the *PSSL93* and the *Workshop on category theory, in honour of George Janelidze* for providing the opportunity to present it, and the members of the *Brno Category Theory Seminar* for listening to a number of talks about it.
F-categories in 2-dimensional universal algebra
===============================================
In this section we recall the notion of F-category, introduced in [@Lack2011Enhanced], and a few basic facts about them.
F-categories
------------
An *F-category* A is a 2-category with two kinds of 1-cell: those of the 2-category itself which are called *loose* and a subcategory of *tight* morphisms containing all of the identities. A second perspective is that an F-category A is specified by a pair of 2-categories ${\ensuremath{\mathcal{A}}\xspace}_{\tau}$ and ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ connected by a 2-functor $$j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda}$$ which is the *identity on objects, faithful and locally fully faithful*. Here ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ contains all of the morphisms, which is to say the loose ones, and all 2-cells between them, whilst ${\ensuremath{\mathcal{A}}\xspace}_{\tau}$ contains the tight morphisms together with all 2-cells between them in ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$. Loose morphisms in A are drawn with wavy arrows $A \rightsquigarrow B$ and tight morphisms with straight arrows $A \to B$, so that a typical diagram in A would be $$\xy
(0,0)*+{A}="00"; (20,0)*+{B}="10";
(20,-20)*+{C}="11";
{\ar@{~>}^{f} "00";"10"};
{\ar ^{g}"10";"11"};
{\ar @{~>}_{h}"00";"11"};
{\ar@{=>}^{\alpha}(17,-4)*+{};(12,-9)*+{}};
\endxy$$ For each pair of objects $A,B \in {\ensuremath{\mathbb{A}}\xspace}$ the inclusion of hom categories $$j_{A,B}:{\ensuremath{\mathcal{A}}\xspace}_{\tau}(A,B) \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda}(A,B)$$ constitutes an *injective on objects fully faithful functor*. In fact F-categories are precisely categories enriched in F, the full subcategory of the arrow category ${\ensuremath{\textnormal{Cat}}\xspace}^{{\textbf{2}\xspace}}$ whose objects are those functors which are both injective on objects and fully faithful. ${\ensuremath{\mathcal{F}}\xspace}$ is a complete and cocomplete cartesian closed category so that the full theory of enriched categories [@Kelly1982Basic] can be applied to the study of F-categories.\
To begin with we have [$\mathcal{F}\textnormal{-CAT}$]{}, the 2-category of F-categories, F-functors and F-natural transformations. An F-functor $F:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ consists of a pair of 2-functors $F_{\tau}:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{B}}\xspace}_{\tau}$ and $F_{\lambda}:{\ensuremath{\mathcal{A}}\xspace}_{\lambda} \to {\ensuremath{\mathcal{B}}\xspace}_{\lambda}$ rendering commutative the square $$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00"; (15,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\lambda}}="10";
(0,-15)*+{{\ensuremath{\mathcal{B}}\xspace}_{\tau}}="01"; (15,-15)*+{{\ensuremath{\mathcal{B}}\xspace}_{\lambda}}="11";
{\ar^{j_{A}} "00";"10"};
{\ar ^{F_{\lambda}}"10";"11"};
{\ar _{F_{\tau}}"00";"01"};
{\ar _{j_{B}} "01";"11"};
\endxy$$ This equally amounts to a 2-functor $F_{\lambda}:{\ensuremath{\mathcal{A}}\xspace}_{\lambda} \to {\ensuremath{\mathcal{B}}\xspace}_{\lambda}$ which *preserves tightness*. An F-natural transformation $\eta:F \to G$ is a 2-natural transformation $\eta:F_{\lambda} \to G_{\lambda}$ with *tight components*.
2-categories as F-categories
----------------------------
Each 2-category A may be viewed as an F-category in two extremal ways: as an F-category in which only the identities are tight, or as an F-category in which *all morphisms are tight*, whereupon the induced F-category has the form $$1:{\ensuremath{\mathcal{A}}\xspace} \to {\ensuremath{\mathcal{A}}\xspace}$$ When we view a 2-category A as an F-category it will *always be in this second sense* and we again denote it by A.\
With this convention established we can treat 2-categories as special kinds of F-categories and unambiguously speak of F-functors $F:{\ensuremath{\mathcal{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ from 2-categories to F-categories, or from F-categories to 2-categories as in $G:{\ensuremath{\mathbb{B}}\xspace} \to {\ensuremath{\mathcal{C}}\xspace}$. These F-functors appear as triangles $$\xy
(10,0)*+{{\ensuremath{\mathcal{A}}\xspace}}="00";
(0,-15)*+{{\ensuremath{\mathcal{B}}\xspace}_{\tau}}="01"; (20,-15)*+{{\ensuremath{\mathcal{B}}\xspace}_{\lambda}}="11";
{\ar ^{F_{\lambda}}"00";"11"};
{\ar _{F_{\tau}}"00";"01"};
{\ar _{j_{B}} "01";"11"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{{\ensuremath{\mathcal{B}}\xspace}_{\tau}}="00"; (20,0)*+{{\ensuremath{\mathcal{B}}\xspace}_{\lambda}}="10";
(10,-15)*+{{\ensuremath{\mathcal{C}}\xspace}}="01";
{\ar^{j_{A}} "00";"10"};
{\ar ^{G_{\lambda}}"10";"01"};
{\ar _{G_{\tau}}"00";"01"};
\endxy$$ with an F-functor between 2-categories just a 2-functor. Observe that each ${\ensuremath{\mathcal{F}}\xspace}$-category ${\ensuremath{\mathbb{A}}\xspace}$ induces an F-functor from its 2-category of tight morphisms $$j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}$$ which is the identity on tight morphisms and $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ on loose ones – we abuse notation by using $j$ in either situation.
F-categories of monoidal categories
-----------------------------------
In two-dimensional universal algebra one encounters morphisms of four different flavours and so F-categories naturally arise. Here we recall the various F-categories associated to the notion of monoidal structure – we recall the defining equations for monoidal functors as we will use these later on.\
The data for a monoidal category consists of a tuple $\overline{A}=(A,\otimes,i^{A},\lambda^{A},\rho^{A}_{l},\rho^{A}_{r})$ where we use juxtaposition for the tensor product. A lax monoidal functor $(F,f,f_{0}):\overline{A} \rightsquigarrow \overline{B}$ consists of a functor $F:A \to B$, coherence constraints $f_{a,b}:Fa\otimes Fb \to F(a\otimes b)$ natural in $a$ and $b$ and a comparison $f_{0}:i^{B} \to Fi^{A}$, all satisfying the three conditions below $$\xy
(0,0)*+{(Fa\otimes Fb)\otimes Fc}="10"; (40,0)*+{F(a\otimes b)\otimes Fc}="20"; (80,0)*+{F((a\otimes b)\otimes c)}="30";
(0,-12)*+{Fa\otimes (Fb\otimes Fc)}="11"; (40,-12)*+{Fa\otimes F(b\otimes c)}="21"; (80,-12)*+{F(a\otimes (b\otimes c))}="31";
{\ar ^{f_{a,b}\otimes 1}"10";"20"};
{\ar^{f_{a\otimes b,c}}"20";"30"};
{\ar _{1\otimes f_{b,c}}"11";"21"};
{\ar_{f_{a,b\otimes c}}"21";"31"};
{\ar_{\lambda_{Fa,Fb,Fc}^{B}}"10";"11"};
{\ar^{F\lambda_{a,b,c}^{A}}"30";"31"};
\endxy$$ $$\xy
(00,0)*+{i^{B}\otimes Fa}="10"; (25,0)*+{Fi^{A}\otimes Fa}="20"; (50,0)*+{F(i^{A}\otimes a)}="30";
(0,-12)*+{Fa}="11"; (50,-12)*+{Fa}="31";
{\ar^<<<<{f_{0}\otimes 1}"10";"20"};
{\ar^{f_{i,a}}"20";"30"};
{\ar@{=}_{}"11";"31"};
{\ar_{\rho_{l}^{B}}"10";"11"};
{\ar^{F\rho_{l}^{A}}"30";"31"};
\endxy
\hspace{0.3cm}
\xy
(00,0)*+{(Fa)\otimes i^{B}}="10"; (25,0)*+{Fa\otimes Fi^{A}}="20"; (50,0)*+{F(a\otimes i^{A})}="30";
(0,-12)*+{Fa}="11"; (50,-12)*+{Fa}="31";
{\ar ^{1\otimes f_{0}}"10";"20"};
{\ar^{f_{a,i}}"20";"30"};
{\ar@{=}_{}"11";"31"};
{\ar_{\rho_{r}^{B}}"10";"11"};
{\ar^{F\rho_{r}^{A}}"30";"31"};
\endxy$$ which we call the *associativity, left unit* and *right unit* conditions. We call $(F,f,f_{0})$ strong or strict monoidal just when the constraints $f_{a,b}$ and $f_{0}$ are invertible or identities respectively; reversing these constraints we obtain the notion of a colax monoidal functor. $$\xy
(00,0)*+{Fa\otimes Fb}="10"; (30,0)*+{Ga\otimes Gb}="20";
(0,-12)*+{F(a\otimes b)}="11"; (30,-12)*+{G(a\otimes b)}="21";
{\ar ^{\eta_{a}\otimes \eta_{b}}"10";"20"};
{\ar_{\eta_{a\otimes b}}"11";"21"};
{\ar_{f_{a,b}}"10";"11"};
{\ar^{g_{a,b}}"20";"21"};
\endxy
\hspace{1cm}
\xy
(00,0)*+{i^{B}}="10"; (20,0)*+{Fi^{A}}="20"; (20,-12)*+{Gi^{A}}="21";
{\ar ^{f_{0}}"10";"20"};
{\ar_{g_{0}}"10";"21"};
{\ar^{\eta_{i^{A}}}"20";"21"};
\endxy
$$ Between lax monoidal functors are monoidal transformations $\eta:(F,f,f_{0}) \to (G,g,g_{0})$: these are natural transformations $\eta:F \to G$ satisfying the two conditions above, which we call the *tensor* and *unit* conditions for a monoidal transformation.\
For $w \in \{s,l,p,c\}$ we thus have $w$-monoidal functors (with $p$-monoidal meaning strong monoidal). Together with monoidal categories and monoidal transformations these form a 2-category ${\ensuremath{\textnormal{MonCat}}\xspace}_{w}$ which sits over [$\textnormal{Cat}$]{}via a forgetful 2-functor $V_{w}:{\ensuremath{\textnormal{MonCat}}\xspace}_{w} \to {\ensuremath{\textnormal{Cat}}\xspace}$. Now strict monoidal functors are strong $(s \leq p)$ and strong monoidal functors can be viewed as lax ($p \leq l$) or colax $(p \leq c)$. Whenever $w_{1} \leq w_{2}$ it follows that we have an F-category ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w_{1},w_{2}}$ of monoidal categories with tight and loose morphisms the $w_{1}$ and $w_{2}$-monoidal functors respectively, as specified by the inclusion $j:{\ensuremath{\textnormal{MonCat}}\xspace}_{w_{1}} \to {\ensuremath{\textnormal{MonCat}}\xspace}_{w_{2}}$. Each such F-category comes equipped with a forgetful F-functor $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w_{1},w_{2}} \to {\ensuremath{\textnormal{Cat}}\xspace}$ where $V_{\tau}=V_{w_{1}}$ and $V_{\lambda}=V_{w_{2}}$ – see the commuting triangles in Diagram 1 of the Introduction. Of particular importance will be those F-categories ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{s,w}$ for $s \leq w$, which we denote by ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}$.
F-categories of algebras
------------------------
Of prime importance are those F-categories associated to a 2-monad $T$ on a 2-category C. Each 2-monad has various associated flavours of algebra and morphism. We will only be interested in *strict algebras* and will call them algebras. Between algebras we have strict, pseudo, lax and colax morphisms – as with monoidal functors we specify these using $s$, $p$, $l$ and $c$.\
Drawn in turn from left to right below is the data $(f,\overline{f})$ for a strict, pseudo, lax or colax morphism of algebras $(f,\overline{f}):(A,a) \rightsquigarrow (B,b)$ $$\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar _{f} "01";"11"};
\endxy
\hspace{0.75cm}
\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar _{f} "01";"11"};
(10,-10)*+{\cong};
(10,-7)*+{^{\overline{f}}};
\endxy
\hspace{0.75cm}
\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar _{f} "01";"11"};
{\ar@{=>}^{\overline{f}}(10,-7)*+{};(10,-13)*+{}};
\endxy
\hspace{0.75cm}
\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar _{f} "01";"11"};
{\ar@{=>}_{\overline{f}};(10,-13)*+{};(10,-7)*+{}}
\endxy$$ Thus $\overline{f}$ is an identity 2-cell in the first case, invertible in the second, and points into or out of $f$ in the lax or colax cases; in all cases these 2-cells are required to satisfy two coherence conditions [@Kelly1974Review].\
There is a single notion of 2-cell between any kind of algebra morphisms; for instance given a pair of lax morphisms $(f,\overline{f}),(g,\overline{g}):(A,a) \rightsquigarrow (B,b)$ an algebra 2-cell $\alpha:(f,\overline{f}) \Rightarrow (g,\overline{g})$ is a 2-cell $\alpha:f \Rightarrow g$ satisfying $$\xy
(0,0)*+{TA}="11";
(20,0)*+{TB}="31"; (0,-20)*+{A}="12";(20,-20)*+{B}="32";
{\ar^{b} "31"; "32"};
{\ar@/^1pc/^{f} "12"; "32"};
{\ar@/_1pc/_{g} "12"; "32"};
{\ar_{a} "11"; "12"};
{\ar^{Tf} "11"; "31"};
{\ar@{=>}_{\alpha}(10,-18)*+{};(10,-23)*+{}};
{\ar@{=>}_{\overline{f}}(10,-4)*+{};(10,-10)*+{}};
\endxy
\hspace{1cm}
\xy
(0,-10)*+{=};
\endxy
\hspace{1cm}
\xy
(0,0)*+{TA}="11";
(20,0)*+{TB}="31"; (0,-20)*+{A}="12";(20,-20)*+{B}="32";
{\ar^{b} "31"; "32"};
{\ar_{g} "12"; "32"};
{\ar_{a} "11"; "12"};
{\ar@/^1pc/^{Tf} "11"; "31"};
{\ar@/_1pc/_{Tg} "11"; "31"};
{\ar@{=>}^{\overline{g}}(10,-10)*+{};(10,-16)*+{}};
{\ar@{=>}_{T\alpha}(10,3)*+{};(10,-2)*+{}};
\endxy$$ whilst the equation in the colax case looks like the lax case with the directions reversed.
Algebras, $w$-algebra morphisms and transformations live in 2-categories ${\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ for $w \in \{s,p,l,c\}$, each of which comes equipped with an evident forgetful 2-functor to the base, which we always denote by $U_{w}:{\ensuremath{\textnormal{T-Alg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$. Each strict morphism is a pseudo morphism ($s \leq p$), and each pseudomorphism can be viewed either as lax $(p \leq l)$ or colax $(p \leq c)$. It follows that for each pair $w_{1},w_{2} \in \{s,p,l,c\}$ satisfying $w_{1} \leq w_{2}$ we have an F-category ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w_{1},w_{2}}$ with tight morphisms the $w_{1}$-morphisms, and loose morphisms the $w_{2}$-morphisms. Each comes equipped with a forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w_{1},w_{2}} \to {\ensuremath{\mathcal{C}}\xspace}$ (so $U_{\tau}=U_{w_{1}}$ and $U_{\lambda}=U_{w_{2}}$) – see the commuting triangles of Diagram 1 of the Introduction.\
Of particular importance are those F-categories ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{s,w}$ whose tight morphisms are the strict ones, and following [@Lack2011Enhanced] we abbreviate these by ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$. As well as using $j:{\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}\to {\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ for the defining inclusion and $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ for the forgetful F-functor, we occasionally use $j_{w}$ or $U_{w}$ if we are in the presence of multiple $j$’s or $U$’s.
Duality between lax and colax morphisms
---------------------------------------
Colax algebra morphisms are lax algebra morphisms with 2-cells reversed. This statement can be made precise using the covariant duality 2-functor $(-)^{co}:{\ensuremath{\textnormal{2-CAT}}\xspace}\to {\ensuremath{\textnormal{2-CAT}}\xspace}$ which takes a 2-category ${\ensuremath{\mathcal{C}}\xspace}$ to the 2-category ${\ensuremath{\mathcal{C}}\xspace}^{co}$ with the same underlying category but with 2-cells reversed. Likewise it takes a 2-monad $T:{\ensuremath{\mathcal{C}}\xspace} \to {\ensuremath{\mathcal{C}}\xspace}$ to a 2-monad $T^{co}:{\ensuremath{\mathcal{C}}\xspace}^{co} \to {\ensuremath{\mathcal{C}}\xspace}^{co}$ and one then has, as noted in [@Kelly1974Doctrinal], an equality ${\ensuremath{\textnormal{T}^{co}\textnormal{-Alg}}\xspace}_{l}={\ensuremath{\textnormal{T-Alg}}\xspace}_{c}^{co}$ which restricts to ${\ensuremath{\textnormal{T}^{co}\textnormal{-Alg}}\xspace}_{s}={\ensuremath{\textnormal{T-Alg}}\xspace}_{s}^{co}$. The $(-)^{co}$ duality naturally extends to a 2-functor $$(-)^{co}:{\ensuremath{\mathcal{F}\textnormal{-CAT}}\xspace}\to {\ensuremath{\mathcal{F}\textnormal{-CAT}}\xspace}$$ under whose action we have that ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{c}^{co}= {\ensuremath{\textnormal{T}^{co}\textnormal{-}\mathbb{A}\textnormal{lg}}\xspace}_{l}$ and moreover that $$U^{co}:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{c}^{co} \to {\ensuremath{\mathcal{C}}\xspace}^{co}\hspace{0.5cm}\textnormal{equals}\hspace{0.5cm}U:{\ensuremath{\textnormal{T}^{co}\textnormal{-}\mathbb{A}\textnormal{lg}}\xspace}_{l} \to {\ensuremath{\mathcal{C}}\xspace}^{co}\hspace{0.2cm}.$$ A consequence of this duality is that each theorem about lax morphisms has a dual version concerning colax morphisms. Indeed all of our definitions and results in the colax case will be dual to those in the lax setting – though we will *state* these results for colax morphisms it will always suffice to *prove* results only in the lax setting.
Equivalence of F-categories
---------------------------
Our monadicity theorems in Section 6 will assert that certain F-categories are *equivalent* to F-categories of algebras for a 2-monad. By an equivalence of F-categories we mean an equivalence in the 2-category of F-categories [$\mathcal{F}\textnormal{-CAT}$]{}, which is to say an equivalence of V-categories for ${\ensuremath{\mathcal{V}}\xspace} = {\ensuremath{\mathcal{F}}\xspace}$.\
Recall from [@Kelly1982Basic] that a V-functor $F:{\ensuremath{\mathcal{A}}\xspace} \to {\ensuremath{\mathcal{B}}\xspace}$ is an equivalence just when it is *essentially surjective on objects* and *fully faithful* in the enriched sense. When ${\ensuremath{\mathcal{V}}\xspace} = {\ensuremath{\mathcal{F}}\xspace}$ the first condition amounts to $F_{\tau}:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{B}}\xspace}_{\tau}$ being essentially surjective on objects, in the usual sense. This also implies the weaker statement that $F_{\lambda}:{\ensuremath{\mathcal{A}}\xspace}_{\lambda} \to {\ensuremath{\mathcal{B}}\xspace}_{\lambda}$ is essentially surjective on objects. Enriched fully faithfulness of $F$ amounts to ${F_{\tau}}_{A,B}:{\ensuremath{\mathcal{A}}\xspace}_{\tau}(A,B) \to {\ensuremath{\mathcal{B}}\xspace}_{\tau}(FA,FB)$ and ${F_{\lambda}}_{A,B}:{\ensuremath{\mathcal{A}}\xspace}_{\lambda}(A,B) \to {\ensuremath{\mathcal{B}}\xspace}_{\lambda}(FA,FB)$ being isomorphisms of categories for each pair $A,B \in {\ensuremath{\mathbb{A}}\xspace}$.\
We conclude that $F$ is an equivalence of F-categories just when both 2-functors $F_{\tau}$ and $F_{\lambda}$ are essentially surjective on objects and 2-fully faithful, which is to say that both $F_{\tau}$ and $F_{\lambda}$ are 2-equivalences, or equivalences of 2-categories.
Doctrinal adjunction and F-categorical lifting properties
=========================================================
If a strict monoidal functor has a right adjoint that right adjoint admits a unique lax monoidal structure such that the adjunction lifts to a monoidal adjunction. This is an instance of *doctrinal adjunction* – the topic of the present section. We begin by recalling Kelly’s treatment of doctrinal adjunction in the setting of 2-monads, recasting the notion in F-categorical terms, so that the above special case becomes the assertion that the forgetful F-functor $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ satisfies $l$-doctrinal adjunction – we treat the cases $w \in \{l,p,c\}$. In Section 3.2 we define the closely related notion of a $w$-doctrinal F-functor before showing that the $w$-doctrinal F-functors form an orthogonality class in [$\mathcal{F}\textnormal{-CAT}$]{}.
Doctrinal adjunction F-categorically
------------------------------------
Doctrinal adjunction was first studied in Kelly’s paper [@Kelly1974Doctrinal] of the same name. Motivated by the example of adjunctions between monoidal categories amongst others, all known to be describable using 2-monads via clubs [@Kelly1974On-clubs] or other techniques, he gave his treatment in the setting of 2-dimensional monad theory. Let us now recall the relevant aspects of this. Given $T$-algebras $(A,a)$ and $(B,b)$ and a morphism $f:A \to B$ together with an adjunction $(\epsilon,f \dashv g,\eta)$ in the base, his Theorem 1.2 asserts that there is a bijection between colax algebra morphisms of the form $(f,\overline{f}):(A,a) \rightsquigarrow (B,b)$ and lax morphisms of the form $(g,\overline{g}):(B,b) \rightsquigarrow (A,a)$. The structure 2-cells $\overline{f}:f.a \Rightarrow b.Tf$ and $\overline{g}:a.Tg \Rightarrow g.b$ are expressed in terms of one another as mates as below $$\xy
(00,0)*+{TA}="11";(30,0)*+{TB}="31"; (00,-35)*+{A}="12";(30,-35)*+{B}="32";
(00,-15)*+{TA}="a";(30,-20)*+{B}="b";
{\ar^{b} "31"; "b"};
{\ar_{f} "12"; "32"};
{\ar_{a} "a"; "12"};
{\ar^{Tf} "11"; "31"};
{\ar_{1} "11"; "a"};
{\ar^{Tg} "31"; "a"};
{\ar_{g} "b"; "12"};
{\ar^{1} "b"; "32"};
{\ar@{=>}_{\overline{g}}(13,-14)*+{};(18,-19)*+{}};
{\ar@{=>}^{T\eta}(8,-6)*+{};(13,-6)*+{}};
{\ar@{=>}_{\epsilon}(18,-30)*+{};(23,-30)*+{}};
\endxy
\hspace{3cm}
\xy
(00,0)*+{TB}="11";(30,0)*+{TA}="31"; (00,-35)*+{B}="12";(30,-35)*+{A}="32";
(00,-15)*+{TB}="a";(30,-20)*+{A}="b";
{\ar^{a} "31"; "b"};
{\ar_{g} "12"; "32"};
{\ar_{b} "a"; "12"};
{\ar^{Tg} "11"; "31"};
{\ar_{1} "11"; "a"};
{\ar^{Tf} "31"; "a"};
{\ar_{f} "b"; "12"};
{\ar^{1} "b"; "32"};
{\ar@{=>}_{\overline{f}}(18,-20)*+{};(13,-15)*+{}};
{\ar@{=>}_{T\epsilon}(13,-6)*+{};(8,-6)*+{}};
{\ar@{=>}_{\eta}(23,-30)*+{};(18,-30)*+{}};
\endxy$$ Since lax and colax morphisms cannot be composed this relationship cannot be expressed 2-categorically or indeed F-categorically – it can be captured using *double categories* as in Example 5.4 of [@Shulman2011Comparing]. However if we start with $(f,\overline{f})$ a pseudomorphism then it does live in the same 2-category as the resultant lax morphism $(g,\overline{g}):(B,b) \rightsquigarrow (A,a)$. Moreover it is shown in Proposition 1.3 of [@Kelly1974Doctrinal] that the unit and counit $\eta$ and $\epsilon$ then become algebra 2-cells $\eta:1 \Rightarrow (g,\overline{g}) \circ (f,\overline{f})$ and $\epsilon:(f,\overline{f}) \circ (g,\overline{g}) \Rightarrow 1$ in [$\textnormal{T-Alg}_{\textnormal{l}}$]{}and so yield an adjunction $(\epsilon,(f,\overline{f}) \dashv (g,\overline{g}),\eta)$ in [$\textnormal{T-Alg}_{\textnormal{l}}$]{}.\
Dually, if $(g,\overline{g})$ were a pseudomorphism, then upon equipping $f$ with the corresponding colax structure $(f,\overline{f})$ the adjunction $(\epsilon,f \dashv g,\eta)$ lifts to an adjunction $(\epsilon,(f,\overline{f}) \dashv (g,\overline{g}),\eta)$ in the 2-category ${\ensuremath{\textnormal{T-Alg}}\xspace}_{c}$.\
The invertibility of $\overline{f}$ does not imply the invertibility of its mate $\overline{g}$, or vice-versa. However, if both $\eta$ and $\epsilon$ are invertible, then $\overline{f}$ is invertible just when $\overline{g}$ is, which is to say that if $(\epsilon,f \dashv g,\eta)$ is an adjoint equivalence and either $f$ or $g$ admits the structure of a pseudomorphism the adjoint equivalence lifts to an adjoint equivalence in ${\ensuremath{\textnormal{T-Alg}}\xspace}_{p}$.\
Let us now abstract these lifting properties of adjunctions and adjoint equivalences into properties of the forgetful F-functors $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w_{1},w_{2}} \to {\ensuremath{\mathcal{C}}\xspace}$. By an *adjunction or adjoint equivalence in an F-category* A we will mean an adjunction or adjoint equivalence in its 2-category ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ of loose morphisms. Given an F-functor $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ and an adjunction $(\epsilon,f \dashv g, \eta)$ in B a *lifting of this adjunction along $H$* is an adjunction $(\epsilon^{\prime},f^{\prime} \dashv g^{\prime},\eta^{\prime})$ in A such that $Hf^{\prime}=f, Hg^{\prime}=g, H\epsilon^{\prime}=\epsilon$ and $H\eta^{\prime}=\eta$. We will likewise speak of liftings of adjoint equivalences along $H$. An F-functor $H: {\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ is said to satisfy
- *weak $l$-doctrinal adjunction* if for each tight arrow $f:A \to B \in {\ensuremath{\mathbb{A}}\xspace}$ each adjunction $(\epsilon,Hf \dashv g, \eta)$ in B lifts along $H$ to an adjunction in ${\ensuremath{\mathbb{A}}\xspace}$ with left adjoint $f$.
- *weak $p$-doctrinal adjunction* if for each tight arrow $f:A \to B \in {\ensuremath{\mathbb{A}}\xspace}$ each adjoint equivalence $(\epsilon,Hf \dashv g, \eta)$ in B lifts along $H$ to an adjoint equivalence in ${\ensuremath{\mathbb{A}}\xspace}$ with left adjoint $f$.
[This lifting property appears biased but of course is not, since the left adjoint of an adjoint equivalence is equally its right adjoint.]{}
- *weak $c$-doctrinal adjunction* if for each tight arrow $f:A \to B \in {\ensuremath{\mathbb{A}}\xspace}$ each adjunction $(\epsilon,g \dashv Hf, \eta)$ in B lifts along $H$ to an adjunction in A with right adjoint $f$.
The lifting properties for algebras described above can be rephrased as asserting exactly that *the forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p,w} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfies weak $w$-doctrinal adjunction for $w \in \{l,p,c\}$*. Each of these statements asserts that if we are given a pseudomorphism of algebras whose underlying arrow has some kind of adjoint, then that adjunction lifts in a certain way; of course if the starting pseudomorphism were in fact strict then the same lifting will exist so that, in particular, *the forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfies weak $w$-doctrinal adjunction for $w \in \{l,p,c\}$*.\
In fact such forgetful F-functors lift these adjunctions *uniquely* – as will follow upon considering the following simple lifting properties. Recall that a 2-functor $H:{\ensuremath{\mathcal{A}}\xspace} \to {\ensuremath{\mathcal{B}}\xspace}$ *reflects identity 2-cells* or is *locally conservative* when it reflects the property of a 2-cell being an identity or an isomorphism, and is *locally faithful* if it reflects the equality of parallel 2-cells. Let us say that an F-functor $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ has any of these three local properties when its loose part $H_{\lambda}:{\ensuremath{\mathcal{A}}\xspace}_{\lambda} \to {\ensuremath{\mathcal{B}}\xspace}_{\lambda}$ has them: this means that $H$ has these properties with respect to all 2-cells and not just those between tight morphisms. The following is evident from the definition of an algebra 2-cell.
Given $w_{1},w_{2} \in \{s,l,p,c\}$ satisfying $w_{1} \leq w_{2}$ the forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w_{1},w_{2}} \to {\ensuremath{\mathcal{C}}\xspace}$ reflects identity 2-cells, is locally conservative and locally faithful. In particular these properties are true of each $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$.
Let $w \in \{l,p,c\}$. An F-functor $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ is said to satisfy *$w$-doctrinal adjunction* if it satisfies the unique form of weak $w$-doctrinal adjunction.
For instance $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ satisfies $l$-doctrinal adjunction if for each tight arrow $f:A \to B \in {\ensuremath{\mathbb{A}}\xspace}$ each adjunction $(\epsilon,Hf \dashv g, \eta)$ in B lifts *uniquely* along $H$ to an adjunction in ${\ensuremath{\mathbb{A}}\xspace}$ with left adjoint $f$. Let us note that, since the $(-)^{co}$ duality interchanges left and right adjoints in an F-category, $H$ satisfies $c$-doctrinal adjunction just when $H^{co}$ satisfies $l$-doctrinal adjunction. Since adjoint equivalences are fixed $H$ satisfies $p$-doctrinal adjunction just when $H^{co}$ does.
\[prop:ids\] Let $w \in \{l,p,c\}$ and consider $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$.
1. If $H$ satisfies weak $w$-doctrinal adjunction and reflects identity 2-cells it satisfies $w$-doctrinal adjunction.
2. If $H$ is locally conservative and satisfies $l$-doctrinal adjunction or $c$-doctrinal adjunction then $H$ satisfies $p$-doctrinal adjunction.
<!-- -->
1. To prove the cases $w=l$ and $w=p$ it will suffice to show that any two adjunctions $(\epsilon,f \dashv g,\eta)$ and $(\epsilon^{\prime},f \dashv g^{\prime},\eta^{\prime})$ in A with common left adjoint and common image under $H$ necessarily coincide in A. Since adjoints are unique up to isomorphism we have $m:g \cong g^{\prime}$ given by the composite $$\xy
(-20,-15)*+{B}="e0";
(0,0)*+{A}="a0"; (20,0)*+{A}="b0";(0,-15)*+{B}="c0";
{\ar^{1} "a0"; "b0"};
{\ar@{~>}_{f} "a0"; "c0"};
{\ar@{~>}^{g} "e0"; "a0"};
{\ar_{1} "e0"; "c0"};
{\ar@{~>}_{g^{\prime}} "c0"; "b0"};
{\ar@{=>}_{\epsilon}(-8,-8)*+{};(-8,-13)*+{}};
{\ar@{=>}_{\eta^{\prime}}(7,-2)*+{};(7,-7)*+{}};
\endxy$$ Using the triangle equations for $f \dashv g$ and $f \dashv g^{\prime}$ we see that $(m{.}f)\circ \eta=\eta^{\prime}$ and that $\epsilon^{\prime}\circ(f{.}m)=\epsilon$. Now the image of the 2-cell $m$ under $H$ is an identity by one of the triangle equations for $Hf \dashv Hg = Hg^{\prime}$. Therefore $m$ is an identity and $g=g^{\prime}$. Since $m{.}f$ and $f{.}m$ are identities it follows that $\eta=\eta^{\prime}$ and $\epsilon=\epsilon^{\prime}$ too. The case $w=c$ is dual.
2. Suppose that $H$ satisfies $l$-doctrinal adjunction and is locally conservative. Then given a tight arrow $f \in {\ensuremath{\mathbb{A}}\xspace}$ each adjoint equivalence $(\epsilon,Hf \dashv g,\eta)$ in B lifts uniquely to an adjunction $(\epsilon^{\prime},f \dashv g^{\prime},\eta^{\prime})$ in A. Since $H$ is locally conservative both $\epsilon^{\prime}$ and $\eta^{\prime}$ are invertible because their images are. Therefore the lifted adjunction is an adjoint equivalence so that $H$ satisfies $p$-doctrinal adjunction. The $c$-case is dual.
Let $w \in \{l,p,c\}$. Then $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfies $w$-doctrinal adjunction. Furthermore both $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{l} \to {\ensuremath{\mathcal{C}}\xspace}$ and $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{c} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfy $p$-doctrinal adjunction.
Since $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfies weak $w$-doctrinal adjunction and reflects identity 2-cells it follows from Proposition \[prop:ids\].1 that $U$ satisfies $w$-doctrinal adjunction. Since it is locally conservative the second part of the claim follows from Proposition \[prop:ids\].2.
\[thm:DoctrinalMonoidal\] In the concrete setting of monoidal categories doctrinal adjunction is well known. Here we describe only those aspects relevant to our needs: namely, that the forgetful F-functors $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w} \to {\ensuremath{\textnormal{Cat}}\xspace}$ satisfy $w$-doctrinal adjunction for $w \in \{l,p,c\}$. Consider then a strict monoidal functor $F:\overline{A} \to \overline{B}$ and an adjunction of categories $(\epsilon, F \dashv G,\eta)$. The right adjoint $G$ obtains the structure of a lax monoidal functor $\overline{G}=(G,g,g_{0}):\overline{B} \rightsquigarrow \overline{A}$ with constraints $g_{x,y}$ and $g_{0}$ given by the composites $$\xy
(0,0)*+{Gx\otimes Gy}="00"; (35,0)*+{GF(Gx\otimes Gy)}="10"; (70,0)*+{G(FGx\otimes FGy)}="20";(105,0)*+{G(x\otimes y)}="30";
{\ar^<<<<<<{\eta_{Gx\otimes Gy}} "00";"10"};
{\ar@{=}^{}"10";"20"};
{\ar^<<<<<<{G(\epsilon_{x}\otimes \epsilon_{y})} "20";"30"};
\endxy$$ and $\eta_{i^{A}}:i^{A} \to GFi^{A} = Gi^{B}$. It is straightforward to show that, with respect to these constraints, the natural transformations $\epsilon$ and $\eta$ become monoidal transformations. Therefore we obtain the lifted adjunction $(\epsilon, F \dashv (G,g,g_{0}),\eta)$ in ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l}$ whose uniqueness follows, using Proposition \[prop:ids\].1, from the fact that $V$ reflects identity 2-cells; thus $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ satisfies $l$-doctrinal adjunction. The cases $w=p$ and $w=c$ are entirely analogous.
Note that unless $w = l$ the forgetful F-functor $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w} \to {\ensuremath{\textnormal{Cat}}\xspace}$ does not satisfy $l$-doctrinal adjunction. That $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ itself does so is due to the fact that the constraints $f_{a,b}:Fa\otimes Fb \to F(a\otimes b)$ and $f_{0}:i^{B} \to Fi^{A}$ point in the correct direction – into $F$ – and are not invertible. Whilst $l$-doctrinal adjunction captures, to some extent, this *laxness* it does not determine in any way the *coherence axioms* for a lax monoidal functor. This is illustrated by the fact that there is an F-category of monoidal categories, strict and *incoherent* lax monoidal functors (these have components $f$ and $f_{0}$ oriented as above but satisfying no equations) and this too sits over [$\textnormal{Cat}$]{}via a forgetful F-functor satisfying $l$-doctrinal adjunction.
Reflections and doctrinal F-functors
------------------------------------
Although each forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfies $w$-doctrinal adjunction it turns out that, insofar as this property characterises such F-functors, the only relevant adjunctions take a more specialised form. We will begin by treating the cases $w=l$ and $w=p$, before treating the case $w=c$ by duality. Let us call an adjunction $(1,f \dashv g,\eta)$ with tight left adjoint and identity counit an *l-reflection*. If, in addition, the unit $\eta$ is invertible then we call the adjunction a *p-reflection*. A $p$-reflection is of course an adjoint equivalence. We remark that any adjunction $(\epsilon,f \dashv g,\eta)$ is determined by three of its four parts: in particular the unit $\eta$ is the unique 2-cell $1 \Rightarrow g{.}f$ satisfying the triangle equation $(\epsilon {.}f)\circ (f{.}\eta) =1$. In a $w$-reflection $(1,f \dashv g,\eta)$ the unit $\eta$ is therefore uniquely determined by the adjoints $f \dashv g$ and the knowledge that the counit is the identity – we can thus faithfully abbreviate a $w$-reflection $(1,f \dashv g,\eta)$ by $f \dashv g$ if convenient. Whilst we only need to consider liftings of $w$-reflections we also need to consider liftings of *morphisms of $w$-reflections*. Consider $w$-reflections $(1,f_{1} \dashv g_{1},\eta_{1})$ and $(1,f_{2} \dashv g_{2},\eta_{2})$ and a *tight* commutative square $(r,s):f_{1} \to f_{2}$ as left below. $$\xy
(0,0)*+{A}="00"; (15,0)*+{C}="10";
(0,-10)*+{B}="01"; (15,-10)*+{D}="11";
{\ar^{r} "00";"10"};
{\ar ^{f_{2}}"10";"11"};
{\ar _{f_{1}}"00";"01"};
{\ar _{s} "01";"11"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{A}="00"; (15,0)*+{C}="10";
(0,-10)*+{B}="01"; (15,-10)*+{D}="11";
{\ar^{r} "00";"10"};
{\ar@{~>} _{g_{2}}"11";"10"};
{\ar@{~>} ^{g_{1}}"01";"00"};
{\ar _{s} "01";"11"};
\endxy$$ We call $(r,s):f_{1} \to f_{2}$ a *morphism of $w$-reflections* if the above square of right adjoints also commutes and furthermore the compatibility with units $r{.}\eta_{1}=\eta_{2}{.}r$ is met. Compatibility with the identity counits is automatic. The following lemma is sometimes useful for recognising morphisms of $w$-reflections.
\[prop:morphAdj\] Let $w \in \{l,p\}$. Consider $w$-reflections $(1,f_{1} \dashv g_{1},\eta_{1})$ and $(1,f_{2} \dashv g_{2},\eta_{2})$ and a tight commuting square $(r,s):f_{1} \to f_{2}$. Then $(r,s)$ is a morphism of $w$-reflections just when its mate $m(r,s):g_{2}{.}s \Rightarrow r{.}g_{1}$ is an identity 2-cell.
It suffices to consider the case $w=l$. Given a morphism of $l$-reflections $(r,s):f_{1} \to f_{2}$ $$\xy
(0,0)*+{A}="00"; (15,0)*+{B}="10";
(0,-15)*+{C}="01"; (15,-15)*+{D}="11";
{\ar^{r} "00";"10"};
{\ar ^{f_{2}}"10";"11"};
{\ar _{f_{1}}"00";"01"};
{\ar _{s} "01";"11"};
\endxy
\hspace{0.3cm}
\xy
(-20,-15)*+{C}="e0";(35,0)*+{B}="f0";
(0,0)*+{A}="a0"; (15,0)*+{B}="b0";(0,-15)*+{C}="c0";(15,-15)*+{D}="d0";
{\ar^{r} "a0"; "b0"};
{\ar_{s} "c0"; "d0"};
{\ar^{f_{1}} "a0"; "c0"};
{\ar_{f_{2}} "b0"; "d0"};
{\ar@{~>}^{g_{1}} "e0"; "a0"};
{\ar_{1} "e0"; "c0"};
{\ar^{1} "b0"; "f0"};
{\ar@{~>}_{g_{2}} "d0"; "f0"};
{\ar@{=>}_{\eta_{2}}(22,-2)*+{};(22,-7)*+{}};
(30,-15)*+{C}="e0";
(50,0)*+{A}="a0"; (70,0)*+{A}="b0";(50,-15)*+{C}="c0";(85,0)*+{B}="f0";(65,-15)*+{D}="d0";
{\ar^{1} "a0"; "b0"};
{\ar_{f_{1}} "a0"; "c0"};
{\ar@{~>}^{g_{1}} "e0"; "a0"};
{\ar^{r} "b0"; "f0"};
{\ar_{1} "e0"; "c0"};
{\ar@{~>}_{g_{1}} "c0"; "b0"};
{\ar@{=>}_{\eta_{1}}(57,-2)*+{};(57,-7)*+{}};
{\ar_{s} "c0"; "d0"};
{\ar@{~>}_{g_{2}} "d0"; "f0"};
\endxy$$ consider the mate of the left square: the central composite above. As $(r,s)$ is a morphism of $l$-reflections we have $\eta_{2}{.}r=r{.}\eta_{1}$ so that the mate reduces to the rightmost composite: this is an identity by the triangle equation for $f_{1} \dashv g_{1}$. Conversely suppose that the mate of $(r,s):f_{1} \to f_{2}$ is an identity. Then $(r,s)$ certainly commutes with the right adjoints so that it remains to verify compatibility with the units. This is straightforward.
Note that each F-functor preserves morphisms of $w$-reflections. The following lifting properties are crucial. For $w \in \{l,p\}$ we say that an F-functor $H: {\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ satisfies:
- *$w$-Refl* if given a tight arrow $f:A \to B \in {\ensuremath{\mathbb{A}}\xspace}$ each $w$-reflection $(1,Hf \dashv g,\eta)$ lifts uniquely along $H$ to a $w$-reflection $(1,f \dashv g^{\prime},\eta^{\prime})$ in ${\ensuremath{\mathbb{A}}\xspace}$.
- *$w$-Morph* if given $w$-reflections $(1,f_{1} \dashv g_{1},\eta_{1})$ and $(1,f_{2} \dashv g_{2},\eta_{2})$ a tight commuting square $(r,s):f_{1} \to f_{2}$ is a morphism of $w$-reflections just when its image $(Hr,Hs):Hf_{1} \to Hf_{2}$ is one.
We say that $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ satisfies *$c$-Refl* or *$c$-Morph* if $H^{co}:{\ensuremath{\mathbb{A}}\xspace}^{co} \to {\ensuremath{\mathbb{B}}\xspace}^{co}$ satisfies $l$-Refl or $l$-Morph respectively. We remark that these $c$-variants concern *$c$-reflections* which are adjunctions with *tight right adjoint* and *identity unit*.
Let $w \in \{l,p,c\}$. An F-functor $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ is said to be *$w$-doctrinal* if it satisfies $w$-Refl, $w$-Morph and is locally faithful. We denote the class of $w$-doctrinal F-functors by $w$-Doct.
The condition that $W$ be locally faithful may seem somewhat unnatural – see the discussion after Theorem \[thm:orthogonality\] for our reasons for including it.\
Evidently we have that $H$ is $c$-doctrinal just when $H^{co}$ is $l$-doctrinal. Furthermore $H$ is $p$-doctrinal just when $H^{co}$ is $p$-doctrinal. Let us now compare these lifting properties with those of Section 3.1.
\[thm:doctrinalLemma\] Let $w \in \{l,p,c\}$ and consider $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$.
1. If $H$ satisfies $w$-doctrinal adjunction and reflects identity 2-cells then it satisfies $w$-Refl and $w$-Morph.
2. If $H$ satisfies $w$-doctrinal adjunction, reflects identity 2-cells and is locally faithful then it is $w$-doctrinal.
3. If $H$ is locally conservative, reflects identity 2-cells, is locally faithful and satisfies either $l$ or $c$-doctrinal adjunction then it is $p$-doctrinal.
We will only consider the $l$-case of (1) and (2), all being essentially identical, with the $l$ and $c$ cases dual.
1. Consider a tight morphism $f:A \to B \in {\ensuremath{\mathbb{A}}\xspace}$ and $l$-reflection $(1,Hf \dashv g,\eta)$. Because $H$ satisfies $l$-doctrinal adjunction this lifts uniquely to an adjunction $(\epsilon^{\prime},f \dashv g^{\prime},\eta^{\prime})$ in A. Since $H\epsilon^{\prime}=1$ and $H$ reflects identity 2-cells $\epsilon^{\prime}$ is an identity. This verifies $l$-Refl. For $l$-Morph consider $l$-reflections $(1,f_{1} \dashv g_{1},\eta_{1})$ and $(1,f_{2} \dashv g_{2},\eta_{2})$ in A and a tight morphism $(r,s):f_{1} \to f_{2}$ such that $(Hr,Hs):Hf_{1} \to Hf_{2}$ is a morphism of $l$-reflections; we must show $(r,s):f_{1} \to f_{2}$ is one too. By Lemma \[prop:morphAdj\] this is equally to say that the mate $m_{r,s}:r{.}g_{1} \Rightarrow g_{2}{.}s$ of this square is an identity 2-cell, so giving a commutative square $(s,r):g_{1} \to g_{2}$. So it will suffice to check $Hm_{r,s}$ is an identity. But $Hm_{r,s}=m_{Hr,Hs}$ which is an identity since $(Hr,Hs)$ is a morphism of $l$-reflections.
2. Since $w$-doctrinal just means $w$-Refl and $w$-Morph together with local faithfulness this follows immediately from Part 1.
3. Since by Proposition \[prop:ids\].2 such an F-functor satisfies $p$-doctrinal adjunction this follows from Part 2.
\[thm:algdoctrinal\] Let $T$ be a 2-monad on C. For each $w \in \{l,p,c\}$ the forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ is $w$-doctrinal. Furthermore both $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{l} \to {\ensuremath{\mathcal{C}}\xspace}$ and $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{c} \to {\ensuremath{\mathcal{C}}\xspace}$ are $p$-doctrinal.
Since $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ satisfies $w$-doctrinal adjunction, reflects identity 2-cells and is locally conservative the first claim follows from Lemma \[thm:doctrinalLemma\].2; since each such $U$ is also locally conservative the second claim follows from Lemma \[thm:doctrinalLemma\].3.
Let us conclude by mentioning a further evident example of $w$-doctrinal F-functors.
\[prop:equivalences\] Let $w \in \{l,p,c\}$. If $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{B}}\xspace}$ is such that $H_{\lambda}:{\ensuremath{\mathcal{A}}\xspace}_{\lambda} \to {\ensuremath{\mathcal{B}}\xspace}_{\lambda}$ is 2-fully faithful then $H$ is $w$-doctrinal for each $w$. In particular each equivalence of F-categories is $w$-doctrinal for each $w$.
A small orthogonality class
---------------------------
Though not strictly necessary in what follows let us remark that, with the exception of weak $w$-doctrinal adjunction, all of the lifting/reflection properties so far considered are expressible as orthogonal lifting properties in [$\mathcal{F}\textnormal{-CAT}$]{}. Certainly it is not hard to see that this is true of the property of reflecting identity 2-cells, or of being locally faithful or locally conservative. Less obvious is that this is true of the notion of $w$-doctrinal adjunction or of the conditions $w$-Refl and $w$-Morph, in particular of the condition $w$-Morph concerning liftings of morphisms of adjunctions. We describe the conditions $l$-Refl and $l$-Morph here – the $p$ and $c$ cases being similar. To this end consider the following ${\ensuremath{\mathcal{F}}\xspace}$-category ${\ensuremath{\mathbb{A}}\xspace}dj_{l}$ depicted in its entirety on the left below $$\xy
(0,7)*+{0}="00"; (0,-7)*+{0}="01";
(20,7)*+{1}="10";(20,-7)*+{1}="11";
{\ar_{1} "00"; "01"};
{\ar^{f} "00"; "10"};
{\ar@{~>}^{g} "10"; "01"};
{\ar_{f} "01"; "11"};
{\ar^{1} "10"; "11"};
{\ar@{=>}^{\eta}(3,2)*+{};(8,2)*+{}};
\endxy
\hspace{0.5cm}
\xy
(0,7)*+{{\textbf{2}\xspace}}="a0"; (20,7)*+{{\ensuremath{\mathbb{A}}\xspace}dj_{l}}="b0";(0,-7)*+{{\ensuremath{\mathbb{A}}\xspace}}="c0";(20,-7)*+{{\ensuremath{\mathbb{B}}\xspace}}="d0";
{\ar^{j} "a0"; "b0"};
{\ar_{} "a0"; "c0"};
{\ar^{} "b0"; "d0"};
{\ar_{H} "c0"; "d0"};
{\ar@{.>}|{} "b0"; "c0"};
\endxy
\hspace{0.5cm}
\xy
(0,7)*+{{\ensuremath{\mathcal{F}\textnormal{-CAT}}\xspace}({\ensuremath{\mathbb{A}}\xspace}dj_{l},{\ensuremath{\mathbb{A}}\xspace})}="a0"; (35,7)*+{{\ensuremath{\mathcal{F}\textnormal{-CAT}}\xspace}({\ensuremath{\mathbb{A}}\xspace}dj_{l},{\ensuremath{\mathbb{B}}\xspace})}="b0";(0,-7)*+{{\ensuremath{\mathcal{F}\textnormal{-CAT}}\xspace}({\textbf{2}\xspace},{\ensuremath{\mathbb{A}}\xspace})}="c0";(35,-7)*+{{\ensuremath{\mathcal{F}\textnormal{-CAT}}\xspace}({\textbf{2}\xspace},{\ensuremath{\mathbb{B}}\xspace})}="d0";
{\ar^{H_{*}} "a0"; "b0"};
{\ar_{j^{*}} "a0"; "c0"};
{\ar^{j^{*}} "b0"; "d0"};
{\ar_{H_{*}} "c0"; "d0"};
\endxy$$ where $g{.}f$ is loose, $f{.}g=1$, $f{.}\eta=1$ and $\eta {.}g=1$. This is the free adjunction with identity counit and tight left adjoint $f$. It has a single non-identity tight arrow $f$ so that ${{\ensuremath{\mathbb{A}}\xspace}dj_{l}}_{\tau}$ equals the free tight arrow ${\textbf{2}\xspace}$; therefore the inclusion ${{\ensuremath{\mathcal{A}}\xspace}dj_{l}}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}dj_{l}$ is the F-functor $j:{\textbf{2}\xspace}\to {\ensuremath{\mathbb{A}}\xspace}dj_{l}$ selecting $f$. Now to give a commutative square as in the middle above is to give a tight arrow $f \in {\ensuremath{\mathbb{A}}\xspace}$ and adjunction $(1,Hf \dashv g,\eta)$ in B. To give a filler is to lift the adjunction along $H$ to an adjunction $(1,f \dashv g^{\prime},\eta^{\prime})$ in A; thus $H$ is orthogonal to $j$ when condition $l$-Refl is met. The conditions $l$-Refl and $l$-Morph together assert exactly that $j$ and $H$ are orthogonal in [$\mathcal{F}\textnormal{-CAT}$]{}as a 2-category – this means that the right square above is a pullback in [$\textnormal{CAT}$]{}. In fact by a standard argument the conditions $l$-Refl and $l$-Morph jointly amount to ordinary (1-categorical) orthogonality against the single F-functor $j \times 1:{\textbf{2}\xspace}\times {\textbf{2}\xspace}\to {\ensuremath{\mathbb{A}}\xspace}dj_{l} \times {\textbf{2}\xspace}$, of which $j$ is a retract. Therefore the class of $w$-doctrinal F-functors, ${\ensuremath{w\textnormal{-Doct}}\xspace}$, forms an orthogonality class in the category of F-categories and F-functors. The following section is geared towards establishing sufficient conditions on an F-category A under which the inclusion $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}$ belongs to $^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace}$ – is orthogonal to each $w$-doctrinal F-functor.
Loose morphisms as spans
========================
We now consider completeness properties of F-categories appropriate to those of the form ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$. In an F-category A with such completeness properties we can represent loose morphisms in A by certain tight spans. In the lax setting, for instance, each loose morphism $f:A \rightsquigarrow B$ is represented as a tight span $C_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}B$ $$\xy
(0,0)*+{C_{f}}="a0"; (-20,0)*+{A}="b0";(20,0)*+{B}="c0";
{\ar_{p_{f}\dashv r_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
\endxy$$ by taking its *colax* limit. The tight left leg $p_{f}:C_{f} \to A$ is part of an $l$-reflection $(1,p_{f} \dashv r_{f},\eta_{f})$ from which $f$ can be recovered as the composite $q_{f}{.}r_{f}:A \rightsquigarrow C_{f} \to B$. In the present section we analyse this representation of loose morphisms by tight spans in detail, beginning with a discussion of the relevant limits.
Limits of loose morphisms
-------------------------
The main reason that F-categories were introduced in [@Lack2011Enhanced] was to capture the behaviour of limits in 2-categories of the form ${\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ for $w \in \{l,p,c\}$. In such 2-categories many *2-categorical* limits have an F-categorical aspect – namely, that the projections from the limit are *strict and jointly detect strictness*. Interpreted in the F-category ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ this asserts that the *limit projections are tight and jointly detect tightness*. This latter property is exactly that which distinguishes F-categorical limits (in the sense of ${\ensuremath{\mathcal{F}}\xspace}$-enriched category theory) from 2-categorical ones (see Proposition 3.6 of [@Lack2011Enhanced]). We will only need a few basic F-categorical limits and our treatment, in what follows, is elementary. There are three limits to consider here – *colax, pseudo and lax limits of loose morphisms* – these correspond, in turn, to lax, pseudo and colax morphisms. As we always lead with the case of lax morphisms we focus here primarily on colax limits. Given a loose morphism $f:A \rightsquigarrow B \in {\ensuremath{\mathbb{A}}\xspace}$ its (colax/pseudo/lax)-limit consists of an object $C_{f}/P_{f}/L_{f}$ and a (colax/pseudo/lax)-cone $(p_{f},\lambda_{f},q_{f})$ as below $$\xy
(0,0)*+{C_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\lambda_{f}}(0,-5)*+{};(0,-11)*+{}};
\endxy
\hspace{1cm}
\xy
(0,0)*+{P_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
(0,-8)*{{\cong}_{\lambda_{f}}};
\endxy
\hspace{1cm}
\xy
(0,0)*+{L_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\lambda_{f}}(0,-11)*+{};(0,-5)*+{}};
\endxy$$ with both projections $p_{f}$ and $q_{f}$ *tight*. The colax limit $C_{f}$
[Colax limits of arrows are usually called oplax limits of arrows. We prefer colax limits here since they are lax limits in ${\ensuremath{\mathbb{A}}\xspace}^{co}$ rather than ${\ensuremath{\mathbb{A}}\xspace}^{op}$ and, similarly, sit better with our usage of lax and colax morphisms.]{}
is required to be the usual 2-categorical *colax limit of an arrow* [@Lack2002Limits] in ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$: this means that it has the following two universal properties.
1. Given any *colax cone* $(r,\alpha,s)$ as below $$\xy
(0,0)*+{X}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar@{~>}_{r} "a0"; "b0"};
{\ar@{~>}^{s} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\alpha}(0,-5)*+{};(0,-11)*+{}};
\endxy$$ there exists a unique $t:X \rightsquigarrow C_{f}$ satisfying $$p_{f}{.}t=r\textnormal{, }q_{f}{.}t=s\textnormal{ and }\lambda_{f}{.}t=\alpha$$
2. Given a pair of colax cones $(r,\alpha,s)$ and $(r^{\prime},\alpha^{\prime},s^{\prime})$ with common base $X$ together with 2-cells $\theta_{r}:r \Rightarrow r^{\prime} \in {\ensuremath{\mathcal{A}}\xspace}_{\lambda}(X,A)$ and $\theta_{s}:s \Rightarrow s^{\prime} \in {\ensuremath{\mathcal{A}}\xspace}_{\lambda}(X,B)$ satisfying $$\xy
(0,0)*+{s}="00";
(15,0)*+{s^{\prime}}="10"; (0,-10)*+{fr}="01";
(15,-10)*+{fr^{\prime}}="11";
{\ar@{=>}^{\theta_{s}} "00"; "10"};
{\ar@{=>}^{\alpha^{\prime}} "10"; "11"};
{\ar@{=>}_{\alpha} "00"; "01"};
{\ar@{=>}_{f\theta_{r}} "01"; "11"};
\endxy$$ there exists a unique 2-cell $\phi:t \Rightarrow t^{\prime} \in {\ensuremath{\mathcal{A}}\xspace}_{\lambda}(X,C_{f})$ between the induced factorisations such that $$p_{f}{.}\phi=\theta_{r}\textnormal{ and }q_{f}{.}\phi=\theta_{s} \textnormal{ .}$$
For $C_{f}$ to be the colax limit of $f$ in the F-categorical sense we must also have
1. A morphism $t:X \rightsquigarrow C_{f}$ is tight just when $p_{f}{.}t$ and $q_{f}{.}t$ are tight – *the projections jointly detect tightness.*
In the case of the pseudolimit $P_{f}$ the 2-cell $\lambda_{f}$ is required to be invertible. If we call those colax cones with an invertible 2-cell *pseudo-cones* then the universal properties of (1) and (2) above are only changed by replacing colax cones by pseudo-cones – thus $(p_{f},\lambda_{f},q_{f})$ is the universal pseudo-cone. The F-categorical aspect of (3) remains the same. The lax limit of $f$ is simply the colax limit in ${\ensuremath{\mathbb{A}}\xspace}^{co}$. For $w \in \{l,p,c\}$ let us write *$w$-limit of a loose morphism* as an abbreviation, so that $c$-limit stands for colax limit, for instance. Now we mentioned above that lax morphisms correspond to colax limits and so on. To capture this let us set $\overline{l}=c$, $\bar{p}=p$ and $\bar{c}=l$ as in [@Lack2011Enhanced]: the correspondence is captured by the following result.
\[prop:limits\] Let $w \in \{l,p,c\}$. The forgetful F-functor $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ creates $\overline{w}$-limits of loose morphisms.
In its F-categorical formulation above this is a specialisation of Theorem 5.13 of [@Lack2011Enhanced] which characterises those limits created by the forgetful F-functors $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$. However for $\overline{w}$-limits of loose morphisms, as concern us, the result goes back to [@Blackwell1989Two-dimensional] and [@Lack2002Limits]. Because lax limits are colax limits in ${\ensuremath{\mathbb{A}}\xspace}^{co}$ we can and will avoid them entirely. In order to work with colax and pseudolimits simultaneously let us introduce a final piece of notation. For $w \in \{l,p\}$ we will use $$\xy
(0,0)*+{\bar{W}_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\lambda_{f}}(0,-5)*+{};(0,-11)*+{}};
\endxy$$ to denote the $\overline{w}$-limit of $f$ and universal $\overline{w}$-cone: so $C_{f}$ and its colax cone when $w=l$ and $P_{f}$ when $w=p$. When $w=p$ the 2-cell $\lambda_{f}$ should be interpreted as invertible.
In [$\textnormal{Cat}$]{}the colax limit of a functor $F:A \to B$ is given by the comma category $B/F$: this has objects $(x,\alpha:x \to Fa,a)$ and morphisms $(r,s):(x,\alpha,a) \to (y,\beta,b)$ given by pairs of arrows $r:x \to y \in {\ensuremath{\mathcal{B}}\xspace}$ and $s:a \to b \in {\ensuremath{\mathcal{A}}\xspace}$ rendering commutative the square on the left. $$\xy
(0,0)*+{x}="00"; (20,0)*+{Fa}="10"; (0,-12)*+{y}="01"; (20,-12)*+{Fb}="11"; {\ar^{\alpha} "00";"10"};
{\ar_{\beta} "01";"11"};
{\ar_{r}"00";"01"};
{\ar^{Fs}"10";"11"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{B/F}="a0"; (-15,-12)*+{A}="b0";(15,-12)*+{B}="c0";
{\ar_{p} "a0"; "b0"};
{\ar^{q} "a0"; "c0"};
{\ar_{F} "b0"; "c0"};
{\ar@{=>}_{\lambda}(0,-4)*+{};(0,-9)*+{}};
\endxy$$ The projections $p:B/F \to A$ and $q:B/F \to B$ of the colax cone $(p,\lambda,q)$ act on a morphism $(r,s)$ of $B/F$ as $p(r,s)=s:a \to b$ and $q(r,s)=r:x \to y$; the value of $\lambda:q \Rightarrow pF$ at $(x,\alpha,a)$ is simply the morphism $\alpha:x \to Fa$ itself.\
The pseudolimit of $F$ is the full subcategory of $B/F$ whose objects are those pairs $(\alpha:x \to Fa,a)$ with $\alpha$ invertible, whilst the lax limit of $F$ is the comma category $F/B$.
\[thm:LimitsMonoidal\] It is illuminating to consider the colax limit of a lax monoidal functor $F=(F,f,f_{0}):\overline{A} \rightsquigarrow \overline{B}$. The forgetful F-functor $U:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ creates these limits: to see how this goes first consider the colax limit of the functor $F$, the comma category $B/F$ equipped with its colax cone $(p,\lambda,q)$ described above. The crux of the argument is to show that this lifts uniquely to a colax cone in ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l}$: that *$B/F$ admits a unique monoidal structure such that $p$ and $q$ become strict monoidal and $\lambda$ a monoidal transformation*. So consider two objects $(x,\alpha,a)$ and $(y,\beta,b)$ of $B/F$: if $p$ and $q$ are to be strict monoidal the tensor product $(x,\alpha,a)\otimes (y,\beta,b)$ must certainly be of the form $(x\otimes y,\theta,a\otimes b)$; furthermore the tensor condition for $\lambda$ to be a monoidal transformation interpreted at this pair asserts precisely that $(x,\alpha,a)\otimes (y,\beta,b)$ equals $$\xy
(0,0)*+{x\otimes y}="00"; (30,0)*+{Fa\otimes Fb}="10"; (60,0)*+{F(a\otimes b)}="20";
{\ar^{\alpha \otimes \beta} "00";"10"};
{\ar^{f_{a,b}}"10";"20"};
\endxy$$ Likewise the unit condition for a monoidal transformation forces us to define the unit of $B/F$ to be $(f_{0}:i^{B} \to Fi^{A},i^{A})$. For $p$ and $q$ to preserve tensor products of morphisms we must define the tensor product as $(r,s)\otimes (r^{\prime},s^{\prime})=(r\otimes r^{\prime},s\otimes s^{\prime})$ at morphisms of $B/F$ – to say the resulting pair is a morphism of $B/F$ is then to say that the following square is commutative. $$\xy
(0,0)*+{x\otimes y}="00"; (30,0)*+{Fa\otimes Fb}="10"; (60,0)*+{F(a\otimes b)}="20";(0,-12)*+{x^{\prime}\otimes y^{\prime}}="01"; (30,-12)*+{Fa^{\prime}\otimes Fb^{\prime}}="11"; (60,-12)*+{F(a^{\prime}\otimes b^{\prime})}="21";
{\ar^{\alpha \otimes \beta} "00";"10"};
{\ar ^{f_{a,b}}"10";"20"};
{\ar_{\alpha^{\prime} \otimes \beta^{\prime}} "01";"11"};
{\ar _{f_{a^{\prime},b^{\prime}}}"11";"21"};
{\ar_{r\otimes r^{\prime}}"00";"01"};
{\ar^{F(s\otimes s^{\prime})}"20";"21"};
{\ar^{Fs\otimes Fs^{\prime}}"10";"11"};
\endxy$$ The left square trivially commutes and the right square commutes by naturality of the $f_{a,b}$. It remains to give the associator and the left and right unit constraints for the monoidal structure on $B/F$ – this is where the coherence axioms for a lax monoidal functor finally come into play. Certainly if $p$ and $q$ are to be strict monoidal they must preserve the associators strictly: this means that the associator at a triple of objects $((x,\alpha,a),(y,\beta,b),(z,\gamma,c))$ of $B/F$ must be given by $(\lambda^{B}_{x,y,z},\lambda^{A}_{a,b,c})$. To say that this is a morphism of $B/F$ is equally to say that the composite square $$\xy
(0,0)*+{(x\otimes y)\otimes z}="00"; (40,0)*+{(Fa\otimes Fb)\otimes Fc}="10"; (78,0)*+{F(a\otimes b)\otimes Fc}="20"; (113,0)*+{F((a\otimes b)\otimes c)}="30";
(0,-12)*+{x\otimes (y\otimes z)}="01"; (40,-12)*+{Fa\otimes (Fb\otimes Fc)}="11"; (78,-12)*+{Fa\otimes F(b\otimes c)}="21"; (113,-12)*+{F(a\otimes (b\otimes c))}="31";
{\ar^<<<<<<<{(\alpha \otimes \beta)\otimes \gamma} "00";"10"};
{\ar ^{f_{a,b}\otimes 1}"10";"20"};
{\ar^{f_{a\otimes b,c}}"20";"30"};
{\ar_<<<<<<<{\alpha \otimes (\beta \otimes \gamma)} "01";"11"};
{\ar _{1\otimes f_{b,c}}"11";"21"};
{\ar_{f_{a,b\otimes c}}"21";"31"};
{\ar_{\lambda^{B}_{x,y,z}}"00";"01"};
{\ar^{\lambda^{B}_{Fa,Fb,Fc}}"10";"11"};
{\ar^{F\lambda^{A}_{a,b,c}}"30";"31"};
\endxy$$ is commutative. The left square commutes by naturality of the associators in $B$ with the right square asserting exactly the associativity condition for a lax monoidal functor. Similarly the left and right unit constraints at $(x,\alpha,a)$ must be given by $(\rho^{B}_{l}x,\rho^{A}_{l}a)$ and $(\rho^{B}_{r}x,\rho^{A}_{r}a)$ – that these lift to isomorphisms of $B/F$ likewise correspond to the left and right unit conditions for a lax monoidal functor. Having given the monoidal structure for $B/F$ it remains to check it verifies the axioms for a monoidal category, but all of these clearly follow from the corresponding axioms for $\overline{A}$ and $\overline{B}$ because $p$ and $q$ preserve the structure strictly and are jointly faithful.\
Finally one needs to verify that this uniquely lifted colax cone in ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l}$ satisfies the universal property of the colax limit of $(F,f,f_{0})$ therein. That $p$ and $q$ jointly detect tightness follows from the fact that they jointly reflect identity arrows – from here it is straightforward to verify that $B/F$ has the universal property of the colax limit in ${\ensuremath{\textnormal{MonCat}}\xspace}_{l}$. For $w \in \{p,c\}$ one constructs the $\bar{w}$-limit of a loose morphism in ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}$ in an entirely similar way.
Observe that in lifting the colax cone $(p,\lambda,q)$ to ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l}$ we used all of the coherence axioms for a lax monoidal functor, and indeed these generating coherence axioms are required for the colax cone to lift. Thus while $l$-doctrinal adjunction is related to the *laxness* – orientation and non-invertibility – of our lax monoidal functors, colax limits of loose morphisms concern the *coherence* axioms these lax morphisms must satisfy.
Loose morphisms as tight spans
------------------------------
For $w \in \{l,p\}$ we suppose that A admits $\bar{w}$-limits of loose morphisms. Then given $f:A \rightsquigarrow B$ we have the commutative triangle on the left below. $$\xy
(0,5)*+{A}="a0"; (-15,-10)*+{A}="b0";(15,-10)*+{B}="c0";
{\ar_{1} "a0"; "b0"};
{\ar@{~>}^{f} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
\endxy
\hspace{1.5cm}
\xy
\textnormal{=}
\endxy
\hspace{1.5cm}
\xy
(0,15)*+{A}="d0";
(0,0)*+{\bar{W}_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar@{~>}_{r_{f}} "d0"; "a0"};
{\ar@/_1.5pc/_{1} "d0"; "b0"};
{\ar@{~>}@/^1.5pc/^{f} "d0"; "c0"};
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\lambda_{f}}(0,-5)*+{};(0,-11)*+{}};
\endxy$$ By the universal property of $\bar{W}_{f}$ we obtain a unique 1-cell $r_{f}:A \rightsquigarrow \bar{W}_{f}$ satisfying $$\label{eq:conR}
p_{f}{.}r_{f}=1, \textnormal{ } q_{f}{.}r_{f}=f \textnormal{ and }\lambda_{f}{.}r_{f}=1$$ as expressed in the equality of pasting diagrams above. Since $p_{f}$ and $q_{f}$ jointly detect tightness we also have that $$\label{eq:rTight}
r_{f}\textnormal{ is tight just when }f\textnormal{ is.}$$
\[prop:factor\] Given $f:A \rightsquigarrow B$ as above we have a $w$-reflection $(1,p_{f} \dashv r_{f},\eta_{f})$ where $\eta_{f}:1 \Rightarrow r_{f}p_{f}$ is the unique 2-cell satisfying $$\label{eq:adjR}
p_{f}{.}\eta_{f}=1\textnormal{ and }q_{f}{.}\eta_{f}=\lambda_{f}.$$
Let us consider firstly the case $w=l$. We need to give a unit $\eta_{f}:1 \Rightarrow r_{f}.p_{f}$. To give such a 2-cell is, by the 2-dimensional universal property of $C_{f}$, equally to give 2-cells $\theta_{1}:p_{f}{.}(1) \Rightarrow p_{f}{.}(r_{f}{.}p_{f})$ and $\theta_{2}:q_{f}{.}(1) \Rightarrow q_{f}{.}(r_{f}{.}p_{f})$ satisfying $\theta_{1} \circ \lambda_{f}=(\lambda_{f}{.}r_{f}{.}p_{f}) \circ \theta_{2}$. We take $\theta_{1}$ to be the identity and $\theta_{2}$ to be $\lambda_{f}:q_{f} \Rightarrow f{.}p_{f}$; the required equality involving $\theta_{1}$ and $\theta_{2}$ is then the assertion that $\lambda_{f}$ equals itself. We thus obtain a unique $\eta_{f}:1 \Rightarrow r_{f}{.}p_{f}$ such $p_{f}{.}\eta_{f}=1$ and $q_{f}{.}\eta_{f}=\lambda_{f}$. If the identity 2-cell $p_{f}{.}q_{f}=1$ is to be the counit of the adjunction then the triangle equations become $p_{f}{.}\eta_{f}=1$ and $\eta_{f}{.}r_{f}=1$. So it remains to check that $\eta_{f}{.}r_{f}=1$ for which it suffices, again by the 2-dimensional universal property of $C_{f}$, to show that $p_{f}{.}\eta_{f}{.}r_{f}=1$ and $q_{f}{.}\eta_{f}{.}r_{f}=1$. The first of these holds since $p_{f}{.}\eta_{f}=1$; the second since $q_{f}{.}\eta_{f}=\lambda_{f}$ and $\lambda_{f}{.}r_{f}=1$. The case $w=p$ is essentially identical – the key point is that the 2-cells $\theta_{1}=1$ and $\theta_{2}=\lambda_{f}$ used above to construct $\eta_{f}$ are now both invertible. That $\eta_{f}$ is itself invertible follows from the fact that $p_{f}$ and $q_{f}$ are jointly conservative – this conservativity follows from the 2-dimensional universal property of $P_{f}$.
The above constructions have their genesis in the proof of Theorem 4.2 of [@Blackwell1989Two-dimensional], in which F-categorical aspects of pseudolimits of arrows in ${\ensuremath{\textnormal{T-Alg}}\xspace}_{p}$ were used to study establish properties of pseudomorphism classifiers. If we ignore F-categorical aspects then the above constructions and resulting factorisations $f=q_{f}{.}r_{f}$ have appeared in other contexts too. In the pseudolimit case the factorisation is the (trivial cofibration, fibration)-factorisation of the natural model structure on a 2-category [@Lack2007Homotopy-theoretic]. In [$\textnormal{Cat}$]{}the factorisation $q_{f}{.}r_{f}:A \to C_{f} \to B$ of a functor $f$ through its colax limit coincides with its factorisation $A \to B/f \to B$ through the comma category $B/f$ – this is the factorisation $(Lf,Rf)$ of a natural weak factorisation system on [$\textnormal{Cat}$]{}described in [@Grandis2006Natural].
Let us return to $f:A \rightsquigarrow B$ as in Proposition \[prop:factor\]. By that result we have a span of tight morphisms (*a tight span*) as below: $$\xy
(0,0)*+{A}="0";(20,0)*+{\bar{W}_{f}}="1";(40,0)*+{B}="2";
{\ar_{p_{f} \dashv r_{f}} "1"; "0"}; {\ar^{q_{f}} "1"; "2"};
\endxy$$ whose left leg $p_{f}$ is equipped with the structure of a $w$-reflection $(1,p_{f} \dashv r_{f},\eta_{f})$. More generally let us use the term *$w$-span* to refer to a tight span equipped with the structure of a $w$-reflection on its left leg. Now let $\bar{W}_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}B$ denote the $w$-span just described. Given $f:A \rightsquigarrow B$ and $g:B \rightsquigarrow C$ we are going to show that $\bar{W}_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}B$ and $\bar{W}_{g}:B {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}C$ can be composed to give a $w$-span $\bar{W}_{g}\bar{W}_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}C$ and furthermore we will study the relationship between $\bar{W}_{g}\bar{W}_{f}$ and $\bar{W}_{gf}$. In order to consider composition of such spans we will need to consider *tight pullbacks*. Given tight morphisms $f:A \to C$ and $g:B \to C$ in A the tight pullback $D$ of $f$ and $g$ $$\xy
(0,0)*+{D}="a0"; (15,0)*+{A}="b0";(0,-10)*+{B}="c0";(15,-10)*+{C}="d0";
{\ar^{p} "a0"; "b0"};
{\ar_{g} "c0"; "d0"};
{\ar_{q} "a0"; "c0"};
{\ar^{f} "b0"; "d0"};
\endxy$$ is the pullback in the 2-category ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ with, moreover, both projections $p$ and $q$ tight and jointly detecting tightness. This is equally to say that $D$ is a pullback in ${\ensuremath{\mathcal{A}}\xspace}_{\tau}$ which is preserved by the inclusion $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda}$.
\[prop:pullbacks1\] Let $w \in \{l,p\}$ and A admit $\bar{w}$-limits of loose morphisms. At $f:A \rightsquigarrow B$ consider the induced tight projection $p_{f}:\bar{W}_{f} \to A$ from the limit. The tight pullback of $p_{f}$ along any tight morphism $g:C \to A$ exists.
In fact the tight pullback is given by $\bar{W}_{fg}$, the $w$-limit of the composite $fg:C \rightsquigarrow A$. For observe that by the universal property of $\bar{W}_{f}$ the $\overline{w}$-cone $(g{.}p_{fg},\lambda_{fg},q_{fg})$ induces a unique tight map $t:\bar{W}_{fg} \to \bar{W}_{g}$ such that the left square below commutes $$\xy
(0,0)*+{\bar{W}_{fg}}="00"; (20,0)*+{\bar{W}_{f}}="10"; (40,0)*+{B}="20";(0,-12)*+{C}="01"; (20,-12)*+{A}="11"; (40,-12)*+{B}="21";
{\ar^{t} "00";"10"};
{\ar ^{q_{f}}"10";"20"};
{\ar_{g} "01";"11"};
{\ar @{~>}_{f}"11";"21"};
{\ar_{p_{fg}}"00";"01"};
{\ar^{1}"20";"21"};
{\ar_{p_{f}}"10";"11"};
{\ar@{=>}_{\lambda_{f}}(32,-3)*+{};(28,-9)*+{}};
\endxy$$ and such that $\lambda_{f}{.}t=\lambda_{gf}$. Now the universal property of $\bar{W}_{fg}$ implies that the left square is a tight pullback.
\[prop:pullbacks2\] For $w \in \{l,p\}$ consider a tight pullback square $$\xy
(0,0)*+{A}="00"; (15,0)*+{B}="10";(0,-10)*+{C}="01";(15,-10)*+{D}="11";
{\ar^{r} "00"; "10"};
{\ar_{s} "01"; "11"};
{\ar_{f_{1}} "00"; "01"};
{\ar^{f_{2}} "10"; "11"};
\endxy$$ and $w$-reflection $(1,f_{2} \dashv g_{2},\eta_{2})$. There exists a unique $w$-reflection $(1,f_{1} \dashv g_{1},\eta_{1})$ such that $(r,s):f_{1} \to f_{2}$ is a morphism of $w$-reflections.
First suppose that $w=l$. Then the left square below $$\xy
(0,0)*+{C}="00"; (15,0)*+{B}="10";(0,-10)*+{C}="01";(15,-10)*+{D}="11";
{\ar@{~>}^{g_{2}{.}s} "00"; "10"};
{\ar_{s} "01"; "11"};
{\ar_{1} "00"; "01"};
{\ar^{f_{2}} "10"; "11"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{C}="00"; (15,0)*+{D}="10";(0,-10)*+{A}="01";(15,-10)*+{B}="11";
{\ar^{s} "00"; "10"};
{\ar_{r} "01"; "11"};
{\ar@{~>}_{g_{1}} "00"; "01"};
{\ar@{~>}^{g_{2}} "10"; "11"};
\endxy$$ commutes and so induces a unique loose morphism $g_{1}:C \rightsquigarrow A$ such that $f_{1}{.}g_{1}=1$ and such that the right square commutes. These necessary commutativities will ensure the claimed uniqueness. By the 2-dimensional universal property of the pullback there exists a unique 2-cell $\eta_{1}:1 \Rightarrow g_{1}{.}f_{1}$ such that $r{.}\eta_{1}=\eta_{2}{.}r$ and $f_{1}{.}\eta_{1}=1$. The other triangle equation $\eta_{1}{.}f_{1}=1$ also follows from the universal property of the pullback. Since $(r,s)$ commutes with both adjoints and the units it is a morphism of $l$-reflections. Note that if $\eta_{2}$ is invertible then, since the pullback projections are jointly conservative, so too is $\eta_{1}$. This gives the case $w=p$.
Now given $f:A \rightsquigarrow B$ and $g:B \rightsquigarrow C$ we can form the composite span $$\xy
(0,0)*+{A}="00";(15,15)*+{\bar{W}_{f}}="11";(30,0)*+{B}="20";(45,15)*+{\bar{W}_{g}}="31";(60,0)*+{C}="40";(30,30)*+{\bar{W}_{g}\bar{W}_{f}}="22";
{\ar_{p_{f}\dashv r_{f}} "11"; "00"};
{\ar_{q_{f}} "11"; "20"};
{\ar_{p_{g}\dashv r_{g}} "31"; "20"};
{\ar^{q_{g}} "31"; "40"};
{\ar_{p_{g,f}\dashv r_{g,f}} "22"; "11"};
{\ar^{q_{g,f}} "22"; "31"};
\endxy$$ in which the central square is the tight pullback of $p_{g}$ along $q_{f}$. (This pullback exists by Lemma \[prop:pullbacks1\]). By Lemma \[prop:pullbacks2\] there exists a unique $w$-reflection $(1,p_{g,f}\dashv r_{g,f},\eta_{g,f})$ such that $$\label{eq:spancomp}
(q_{g,f},q_{f}):p_{g,f} \to p_{g}\textnormal{ is a morphism of }w\textnormal{-reflections.}$$ We can then compose the $w$-reflections $p_{f} \dashv r_{f}$ and $p_{g,f} \dashv r_{g,f}$ to obtain another $w$-reflection $p_{f}{.}p_{g,f} \dashv r_{g,f}{.}r_{f}$, so that the outer span becomes a $w$-span $\bar{W}_{g}\bar{W}_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}C$. Let us consider the relationship between $\bar{W}_{g}\bar{W}_{f}$ and $\bar{W}_{gf}$. By the universal property of $\bar{W}_{gf}$ the $\overline{w}$-cone left below $$\xy
(15,15)*+{\bar{W}_{g}\bar{W}_{f}}="f0";
(0,0)*+{\bar{W}_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\lambda_{f}}(0,-5)*+{};(0,-11)*+{}};
(30,0)*+{\bar{W}_{g}}="d0";(45,-15)*+{C}="e0";
{\ar_{p_{g}} "d0"; "c0"};
{\ar^{q_{g}} "d0"; "e0"};
{\ar@{~>}_{g} "c0"; "e0"};
{\ar@{=>}_{\lambda_{g}}(30,-5)*+{};(30,-11)*+{}};
{\ar_{p_{g,f}} "f0"; "a0"};
{\ar^{q_{g,f}} "f0"; "d0"};
\endxy
\hspace{0.3cm}
\textnormal{=}
\hspace{0.3cm}
\xy
(15,15)*+{\bar{W}_{g}\bar{W}_{f}}="f0";
(0,0)*+{\bar{W}_{f}}="a0"; (-15,-15)*+{A}="b0";(15,-15)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar@{~>}_{f} "b0"; "c0"};
{\ar@{=>}_{\lambda_{gf}}(15,-5)*+{};(15,-11)*+{}};
(30,0)*+{\bar{W}_{g}}="d0";(45,-15)*+{C}="e0";
{\ar^{q_{g}} "d0"; "e0"};
{\ar@{~>}_{g} "c0"; "e0"};
{\ar_{p_{g,f}} "f0"; "a0"};
{\ar^{q_{g,f}} "f0"; "d0"};
(15,0)*+{\bar{W}_{gf}}="g0";
{\ar|{k_{g,f}} "f0"; "g0"};
{\ar^{p_{gf}} "g0"; "b0"};
{\ar_{q_{gf}} "g0"; "e0"};
\endxy$$ induces a unique tight arrow $k_{g,f}:\bar{W}_{g}\bar{W}_{f} \to \bar{W}_{gf}$ satisfying the equations $$\label{eq:comp}
p_{gf}{.}k_{g,f}=p_{f}{.}p_{g,f}\textnormal{, }q_{gf}{.}k_{g,f}=q_{g}{.}q_{g,f}
\textnormal{ and }
\lambda_{gf}{.}k_{g,f}=(g{.}\lambda_{f}{.}p_{g,f})\circ (\lambda_{g}{.}q_{g,f})$$ equally expressed in the equality of pasting diagrams above. Furthermore
\[prop:spanmap\] Let $w \in \{l,p\}$. In the span map $$\xy
(0,0)*+{A}="00";(0,12)*+{A}="01";(25,12)*+{\bar{W}_{g}\bar{W}_{f}}="11";(25,0)*+{\bar{W}_{gf}}="10";(50,0)*+{B}="20";(50,12)*+{B}="21";
{\ar_{p_{f}{.}p_{g,f}} "11"; "01"};
{\ar_{p_{gf}} "10"; "00"};
{\ar^{q_{g}{.}q_{g,f}} "11"; "21"};
{\ar^{q_{gf}} "10"; "20"};
{\ar^{k_{g,f}} "11"; "10"};
{\ar^{1} "01"; "00"};
{\ar^{1} "21"; "20"};
\endxy$$ the commuting square $(k_{g,f},1):p_{f}{.}p_{g,f} \to p_{gf}$ is a morphism of $w$-reflections.
To show that $(k_{g,f},1):p_{f}{.}p_{g,f} \to p_{gf}$ is a morphism of $w$-reflections it suffices, by Lemma \[prop:morphAdj\], to show that the mate of this square is an identity. Because the $w$-reflection $p_{f}{.}p_{g,f} \dashv r_{g,f}{.}r_{f}$ has identity counit the mate of $(k_{g,f},1)$ is simply $$\xy
(-20,-15)*+{A}="e0";(40,0)*+{\bar{W}_{gf}}="f0";
(0,0)*+{\bar{W}_{g}\bar{W}_{f}}="a0"; (20,0)*+{\bar{W}_{gf}}="b0";(0,-15)*+{A}="c0";(20,-15)*+{A}="d0";
{\ar^{k_{g,f}} "a0"; "b0"};
{\ar_{1} "c0"; "d0"};
{\ar|{p_{f}{.}p_{g,f}} "a0"; "c0"};
{\ar_{p_{gf}} "b0"; "d0"};
{\ar@{~>}^{r_{g,f}{.}r_{f}} "e0"; "a0"};
{\ar_{1} "e0"; "c0"};
{\ar^{1} "b0"; "f0"};
{\ar@{~>}_{r_{gf}} "d0"; "f0"};
{\ar@{=>}_{\eta_{gf}}(24,-1)*+{};(28,-7)*+{}};
\endxy$$ Now the 2-dimensional universal property of $\bar{W}_{gf}$ implies that the projections $p_{gf}$ and $q_{gf}$ jointly reflect identity 2-cells: see Lemma 3.1 of [@Lack2002Limits] in the case of the colax limit. Therefore it suffices to show that the composite of the above 2-cell with both $p_{gf}$ and $q_{gf}$ yields an identity. One of the triangle equations for $p_{gf}\dashv r_{gf}$ gives $p_{gf}{.}\eta_{gf}$=1; thus it remains to show $q_{gf}{.}\eta_{gf}{.}k_{g,f}{.}r_{g,f}{.}r_{f}=1$. By this equals $\lambda_{gf}{.}k_{g,f} {.}r_{g,f}{.}r_{f}$ and by definition of $k_{g,f}$ (as in ) we have $\lambda_{gf}{.}k_{g,f}=(g{.}\lambda_{f}{.}p_{g,f})\circ (\lambda_{g}{.}q_{g,f})$. Therefore it suffices to show that the 2-cells $(g{.}\lambda_{f}{.}p_{g,f}){.}r_{g,f}{.}r_{f}$ and $(\lambda_{g}{.}q_{g,f}){.}r_{g,f}{.}r_{f}$ are identities separately. With regards the former we have that $\lambda_{f}{.}p_{g,f}{.}r_{g,f}{.}r_{f}=\lambda_{f}{.}r_{f}=1$ where we first use that $p_{g,f}{.}r_{g,f}=1$ and then ; for the other composite we have $(\lambda_{g}{.}q_{g,f}){.}r_{g,f}{.}r_{f}=\lambda_{g}{.}r_{g}{.}q_{f}{.}r_{f}=1$. The first equation holds by and the second equation by .
Given any F-category A one can define a bicategory $Span_{w}({\ensuremath{\mathbb{A}}\xspace})$ of $w$-spans in A. Their composition extends that described for $w$-spans of the form $\bar{W}_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}B$ above. To ensure composites exist one allows only those $w$-spans whose left legs admit tight pullbacks along arbitrary tight maps. Now when A admits $w$-limits of loose morphisms the assigment of $\bar{W}_{f}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}B$ to $f$ can be extended to an identity on objects lax functor from the underlying category of ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ to $Span_{w}({\ensuremath{\mathbb{A}}\xspace})$ with the span map $k_{g,f}:\bar{W}_{g}\bar{W}_{f} \to \bar{W}_{gf}$ describing one of the coherence constraints.
Representing 2-cells via span transformations
---------------------------------------------
Lastly we consider how to represent a 2-cell $\alpha:f \Rightarrow g$ by a span map $\bar{W}_{f} \to \bar{W}_{g}$. Here the cases $w=l$ and $w=p$ diverge.
### The case w=l
Suppose that A admits colax limits of loose morphisms. Given $\alpha:f \Rightarrow g$ the colax cone left below $$\xy
(0,10)*+{C_{f}}="a0";
(-20,-10)*+{A}="b0";(20,-10)*+{B}="c0";
{\ar_{p_{f}} "a0"; "b0"};
{\ar^{q_{f}} "a0"; "c0"};
{\ar@/^1pc/@{~>}^{f} "b0"; "c0"};
{\ar@{=>}_{\alpha}(0,-7)*+{};(0,-13)*+{}};
{\ar@/_1pc/@{~>}_{g} "b0"; "c0"};
{\ar@{=>}_{\lambda_{f}}(0,3)*+{};(0,-2)*+{}};
\endxy
\hspace{1cm}
\xy
\textnormal{=}
\endxy
\hspace{1cm}
\xy
(0,15)*+{C_{f}}="d0";
(0,0)*+{C_{g}}="a0"; (-20,-15)*+{A}="b0";(20,-15)*+{B}="c0";
{\ar_{c_{\alpha}} "d0"; "a0"};
{\ar@/_1.5pc/_{p_{f}} "d0"; "b0"};
{\ar@/^1.5pc/^{q_{f}} "d0"; "c0"};
{\ar_{p_{g}} "a0"; "b0"};
{\ar^{q_{g}} "a0"; "c0"};
{\ar@{~>}_{g} "b0"; "c0"};
{\ar@{=>}_{\lambda_{g}}(0,-5)*+{};(0,-11)*+{}};
\endxy$$ induces a unique *tight* arrow $c_{\alpha}:C_{f} \to C_{g}$ satisfying $$\label{eq:l2}
p_{f}=c_{\alpha}{.}p_{g}, \textnormal{ } q_{f}=c_{\alpha}{.}q_{g} \textnormal{ and }(\alpha {.}p_{f})\circ \lambda_{f}=\lambda_{g}{.}c_{\alpha}$$ In particular the 2-cell $\alpha$ is represented by a span map $c_{\alpha}:C_{f} \to C_{g}$ as below. $$\xy
(0,0)*+{A}="00";(0,12)*+{A}="01";(20,12)*+{C_{f}}="11";(20,0)*+{C_{g}}="10";(40,0)*+{B}="20";(40,12)*+{B}="21";
{\ar_{p_{f}} "11"; "01"};
{\ar_{p_{g}} "10"; "00"};
{\ar^{q_{f}} "11"; "21"};
{\ar^{q_{g}} "10"; "20"};
{\ar^{c_{\alpha}} "11"; "10"};
{\ar^{1} "01"; "00"};
{\ar^{1} "21"; "20"};
\endxy$$
\[prop:l2\] Let $c_{\alpha}:C_{f} \to C_{g}$ be as above and let $m_{\alpha}$ denote the mate of the square $(c_{\alpha},1):p_{f} \to p_{g}$ through the adjunctions $p_{f} \dashv r_{f}$ and $p_{g} \dashv r_{g}$. $$\xy
(0,0)*+{A}="00";(0,12)*+{A}="01";(20,12)*+{C_{f}}="11";(40,0)*+{B}="20";(20,0)*+{C_{g}}="10";(40,12)*+{B}="21";
{\ar@{~>}^{r_{f}} "01"; "11"};
{\ar@{~>}_{r_{g}} "00"; "10"};
{\ar^{q_{f}} "11"; "21"};
{\ar_{q_{g}} "10"; "20"};
{\ar^{c_{\alpha}} "11"; "10"};
{\ar@{=>}^{m_{\alpha}}(8,9)*+{};(8,4)*+{}};
{\ar_{1} "01"; "00"};
{\ar^{1} "21"; "20"};
\endxy$$ The composite 2-cell $q_{g}{.}m_{\alpha}$ equals $\alpha$.
Because the counit of $p_{f} \dashv r_{f}$ is an identity the mate $m_{\alpha}$ is simply given by $$\xy
(00,0)*+{A}="11";(30,0)*+{C_{f}}="31"; (00,-25)*+{A}="12";(30,-25)*+{C_{g}}="32";
(00,-15)*+{A}="a";(30,-10)*+{C_{g}}="b";
{\ar^{c_{\alpha}} "31"; "b"};
{\ar@{~>}_{r_{g}} "12"; "32"};
{\ar_{1} "a"; "12"};
{\ar@{~>}^{r_{f}} "11"; "31"};
{\ar_{1} "11"; "a"};
{\ar^{p_{f}} "31"; "a"};
{\ar_{p_{g}} "b"; "12"};
{\ar^{1} "b"; "32"};
{\ar@{=>}_{\eta_{g}}(24,-19)*+{};(18,-22)*+{}};
\endxy$$ Therefore $q_{g}{.}m_{\alpha}=q_{g}{.}\eta_{g}{.}c_{\alpha}{.}r_{f}=\lambda_{g}{.}c_{\alpha}{.}r_{f}=(\alpha {.}p_{f}{.}r_{f})\circ (\lambda_{f}{.}r_{f})=\alpha \circ 1=\alpha$ where the second, third and fourth equalities use , and respectively.
### The case w=p
Suppose that A admits pseudolimits of loose morphisms. If $\alpha:f \Rightarrow g$ is *invertible* then we may construct a map $P_{f} \to P_{g}$ in essentially the same way as for colax limits of loose morphisms. However this approach does not work in general. The following lemma describes a representation that works for non-invertible 2-cells. It is based upon the notion of a *transformation* of *anafunctors* [@Makkai1996Avoiding] – see also [@Roberts2012Internal] and [@Bartels2006Higher]. Anafunctors can be viewed as spans of categories and functors in which the left leg is a surjective on objects equivalence. If we take ${\ensuremath{\mathbb{A}}\xspace}= {\ensuremath{\textnormal{Cat}}\xspace}$ and functors $F,G:A \to B$ then the associated spans $P_{F},P_{G}:A {\ensuremath{\, \mathaccent\shortmid\rightarrow\,}}B$ are anafunctors. In this setting the 2-cell $\rho_{\alpha}$ described below specifies precisely a transformation between the anafunctors $P_{F}$ and $P_{G}$.
\[prop:p2\] Given $\alpha:f \Rightarrow g$ consider the tight pullback left below (existing by Lemma \[prop:pullbacks1\]) $$\xy
(0,-5)*+{K_{f,g}}="a0"; (20,-5)*+{P_{f}}="b0";(0,-25)*+{P_{g}}="c0";(20,-25)*+{A}="d0";
{\ar^{s_{f,g}} "a0"; "b0"};
{\ar_{p_{g}} "c0"; "d0"};
{\ar_{t_{f,g}} "a0"; "c0"};
{\ar^{p_{f}} "b0"; "d0"};
{\ar^{u_{f,g}} "a0"; "d0"};
\endxy
\hspace{1cm}
\xy
(-5,0)*+{A}="00"; (15,-15)*+{K_{f,g}}="1-1";(30,-15)*+{P_{f}}="2-1";(15,-30)*+{P_{g}}="1-2";(30,-30)*+{A}="2-2";
{\ar@{~>}^{v_{f,g}} "00"; "1-1"};
{\ar@/^1pc/@{~>}^{r_{f}} "00"; "2-1"};
{\ar@/_1pc/@{~>}_{r_{g}} "00"; "1-2"};
{\ar^{s_{f,g}} "1-1"; "2-1"};
{\ar^{q_{f}} "2-1"; "2-2"};
{\ar_{t_{f,g}} "1-1"; "1-2"};
{\ar_{q_{g}} "1-2"; "2-2"};
{\ar@{=>}^{\rho_{\alpha}}(25,-20)*+{};(20,-25)*+{}}
\endxy$$ with diagonal denoted $u_{f,g}$.
1. There is a unique $p$-reflection $u_{f,g} \dashv v_{f,g}$ such that $(s_{f,g},1):u_{f,g} \to p_{f}$ and $(t_{f,g},1):u_{f,g} \to p_{g}$ are morphisms of $p$-reflections. If $f$ and $g$ are tight so too is $v_{f,g}$.
2. There exists a unique 2-cell $\rho_{\alpha}:q_{f}{.}s_{f,g} \Rightarrow q_{g}{.}t_{f,g}$ such that $\rho_{\alpha}.v_{f,g}=\alpha$.
<!-- -->
1. Consider the diagram $$\xy
(-5,0)*+{A}="00"; (15,-15)*+{K_{f,g}}="1-1";(30,-15)*+{P_{f}}="2-1";(15,-30)*+{P_{g}}="1-2";(30,-30)*+{A}="2-2";
{\ar@{~>}^{v_{f,g}} "00"; "1-1"};
{\ar@/^0.7pc/@{~>}^{r_{f}} "00"; "2-1"};
{\ar@/_0.5pc/@{~>}^{r_{g}} "00"; "1-2"};
{\ar_{s_{f,g}} "1-1"; "2-1"};
{\ar^{p_{f}} "2-1"; "2-2"};
{\ar^{t_{f,g}} "1-1"; "1-2"};
{\ar_{p_{g}} "1-2"; "2-2"};
\endxy$$ in which $v_{f,g}$ is the unique loose map satisfying $s_{f,g}{.}v_{f,g}=r_{f}$ and $t_{f,g}{.}v_{f,g}=r_{g}$. It follows that $u_{f,g}{.}v_{f,g}=1$. To give an invertible 2-cell $\theta_{f,g}:1 \cong v_{f,g}{.}u_{f,g}$ is, by the universal property of $K_{f,g}$, equally to give invertible 2-cells $\theta_{1}:s_{f,g} \cong s_{f,g}{.}v_{f,g}{.}u_{f,g}$ and $\theta_{2}:t_{f,g} \cong t_{f,g}{.}v_{f,g}{.}u_{f,g}$ satisfying $p_{f}{.}\theta_{1}=p_{g}{.}\theta_{2}$. We set $\theta_{1}=\eta_{f}{.}s_{f,g}$ and $\theta_{2}=\eta_{g}{.}t_{f,g}$ noting that $p_{f}{.}\eta_{f}{.}s_{f,g}=1=p_{g}{.}\eta_{g}{.}s_{f,g}$. The triangle equations for the $p$-reflection follow using the universal property of the pullback $K_{f,g}$ and that $s_{f,g}:u_{f,g} \to p_{f}$ and $t_{f,g}:u_{f,g} \to p_{g}$ are morphisms of $p$-reflections follows from the construction of $u_{f,g}$ and $\theta_{f,g}$. If $f$ and $g$ are tight so too, by , are $r_{f}$ and $r_{g}$. By the universal property of the tight pullback $K_{f,g}$ it then follows that $v_{f,g}$ is tight.
2. From the first part we have $s_{f,g}{.}v_{f,g}=r_{f}$ and $t_{f,g}{.}v_{f,g}=r_{g}$ as in the two triangles above. Now by we have $q_{f}{.}r_{f}=f$ and $q_{g}{.}r_{g}=g$. Therefore we can write $\alpha:(q_{f}{.}s_{f,g}){.}v_{f,g} \Rightarrow (q_{g}{.}t_{f,g}){.}v_{f,g}$. Since $v_{f,g}:A \rightsquigarrow K_{f,g}$ is an equivalence in ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ the functor ${\ensuremath{\mathcal{A}}\xspace}_{\lambda}(v_{f,g},A):{\ensuremath{\mathcal{A}}\xspace}_{\lambda}(K_{f,g},A) \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda}(A,A)$ is an equivalence of categories – using its fully faithfulness we obtain $\rho_{\alpha}:q_{f}{.}s_{f,g} \Rightarrow q_{g}{.}t_{f,g}$.
Orthogonality
=============
The following theorem is the crucial result of the paper. The monadicity theorems of Section 6 follow easily from it. We note that both this theorem and the corollary that follows it are independent of the formalism of 2-monads.
\[thm:orthogonality\] Let $w \in \{l,p,c\}$. Consider an F-category ${\ensuremath{\mathbb{A}}\xspace}$ with $\bar{w}$-limits of loose morphisms. Then the inclusion of tight morphisms $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}$ is orthogonal to each $w$-doctrinal F-functor.
Consider a commuting square in [$\mathcal{F}\textnormal{-CAT}$]{}$$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="a0"; (20,0)*+{{\ensuremath{\mathbb{A}}\xspace}}="b0";(0,-10)*+{{\ensuremath{\mathbb{B}}\xspace}}="c0";(20,-10)*+{{\ensuremath{\mathbb{C}}\xspace}}="d0";
{\ar^{j} "a0"; "b0"};
{\ar_{R} "a0"; "c0"};
{\ar^{S} "b0"; "d0"};
{\ar_{H} "c0"; "d0"};
{\ar@{.>}|{K} "b0"; "c0"};
\endxy$$ in which $H$ is $w$-doctrinal. We must show there exists a unique diagonal filler $K$. We begin by noting that the cases $w=c$ and $w=l$ are dual since A satisfies the $c$-criteria of the theorem just when ${\ensuremath{\mathbb{A}}\xspace}^{co}$ satisfies the $l$-criteria with, equally, $H$ $c$-doctrinal just when $H^{co}$ is $l$-doctrinal. Therefore it will suffice to suppose $w \in \{l,p\}$.
1. Before constructing the diagonal we fix some notation and make some observations about lifted adjunctions that will be repeatedly used in what follows. Given a $w$-reflection $(1, f \dashv g,\eta) \in {\ensuremath{\mathbb{A}}\xspace}$ we obtain a $w$-reflection $(1, Sf \dashv Sg,S\eta)$ in C with $Sf=Hf)$ since $f$ is tight. As $H$ is $w$-doctrinal this lifts uniquely along $H$ to a $w$-reflection in B which we denote by $(1,Rf \dashv \overline{g},\overline{\eta})$. Next consider $w$-reflections $(1, f_{1} \dashv g_{1},\eta_{1})$ and $(1, f_{2} \dashv g_{2},\eta_{2})$ in A and a tight commuting square $(r,s):f_{1} \to f_{2}$ in A as left below $$\hspace{0.2cm}
\xy
(0,0)*+{A}="a0"; (20,0)*+{C}="b0";(0,-10)*+{B}="c0";(20,-10)*+{D}="d0";
{\ar^{r} "a0"; "b0"};
{\ar_{s} "c0"; "d0"};
{\ar_{f_{1}} "a0"; "c0"};
{\ar^{f_{2}} "b0"; "d0"};
\endxy
\hspace{1cm}
\xy
(0,0)*+{RA}="a0"; (20,0)*+{RC}="b0";(0,-10)*+{RB}="c0";(20,-10)*+{RD}="d0";
{\ar^{Rr} "a0"; "b0"};
{\ar_{Rs} "c0"; "d0"};
{\ar_{Rf_{1}} "a0"; "c0"};
{\ar^{Rf_{2}} "b0"; "d0"};
\endxy
\hspace{1cm}
\xy
(0,0)*+{RB}="a0"; (20,0)*+{RD}="b0";(0,-10)*+{RA}="c0";(20,-10)*+{RC}="d0";
{\ar@{~>}_{\overline{g_{1}}} "a0"; "c0"};
{\ar@{~>}^{\overline{g_{2}}} "b0"; "d0"};
{\ar_{Rr} "c0"; "d0"};
{\ar^{Rs} "a0"; "b0"};
\endxy$$ Since F-functors preserves morphisms of $w$-reflections the square $(Sr,Ss):Sf_{1} \to Sf_{2}$ is one in C. Now the tight commutative square $(Rr,Rs):Rf_{1} \to Rf_{2}$ has image under $H$ the morphism of $w$-reflections $(Sr,Ss):Sf_{1} \to Sf_{2}$. Because $H$ is $w$-doctrinal it follows that $(Rr,Rs):Rf_{1} \to Rf_{2}$ is a morphism between the lifted $w$-reflections in B. In particular we obtain a commuting square of right adjoints $(Rs,Rr):\overline{g_{1}} \to \overline{g_{2}}$.\
Finally consider a composable pair of tight left adjoints $f_{1}:A \to B$ and $f_{2}:B \to C$ with associated $w$-reflections $(1, f_{1} \dashv g_{1},\eta_{1})$ and $(1, f_{2} \dashv g_{2},\eta_{2})$ in A. We can form the $w$-reflections $(1,Rf_{1} \dashv \overline{g_{1}},\overline{\eta_{1}})$ and $(1,Rf_{2} \dashv \overline{g_{2}},\overline{\eta_{2}})$ in B and compose these to obtain a further $w$-reflection $f_{2}{.}f_{1} \dashv \overline{g_{1}}{.}\overline{g_{2}}$ in B. It is clear that this is a lifting of the $w$-reflection $S(f_{2}{.}f_{1}) \dashv S(g_{1}{.}g_{2})$ so that, by uniqueness of liftings, the $w$-reflections $Rf_{2}{.}Rf_{1} \dashv \overline{g_{1}}{.}\overline{g_{2}}$ and $R(f_{2}{.}f_{1}) \dashv \overline{g_{1}{.}g_{2}}$ coincide.
2. Now to begin constructing $K$ observe that for the left triangle to commute we must define $KA=RA$ for each $A \in {\ensuremath{\mathbb{A}}\xspace}$.
3. Given $f:A \rightsquigarrow B \in {\ensuremath{\mathbb{A}}\xspace}$ we recall from its factorisation as $q_{f}{.}r_{f}:A \rightsquigarrow C_{f} \to B$ where $(1,p_{f} \dashv r_{f},\eta_{f})$. Since $p_{f}$ is tight we have the lifted $w$-reflection $(1,Rp_{f} \dashv \overline{r_{f}},\overline{\eta}_{f})$ in B living over $(1,Sp_{f} \dashv Sr_{f},S\eta_{f})$. We define $Kf:RA \rightsquigarrow RB$ as the composite $Rq_{f} {.}\overline{r}_{f}:RA \rightsquigarrow R\bar{W}_{f} \to RB$.
4. Observe that $HKf=HRq_{f} {.}H\overline{r}_{f}= Sq_{f} {.}Sr_{f}=S(q_{f} {.}r_{f})=Sf$ as required. To see that $K$ extends $R$ observe that if $f:A \to B$ is tight then, by , $r_{f}$ is tight too, so that we have a $w$-reflection $(1,Rp_{f} \dashv Rr_{f},R\eta_{f})$ living over $(1,HRp_{f} \dashv Sr_{f},\eta_{f})$; thus $(1,Rp_{f} \dashv Rr_{f},R\eta_{f})=(1,Rp_{f} \dashv \overline{r_{f}},\overline{\eta_{f}})$ so that $Kf=Rq_{f} {.}Rr_{f} = Rf$. Thus $K$ coincides with $R$ on tight morphisms.
5. As $K$ agrees with $R$ on tight morphisms we already know that it preserves identity 1-cells. To see that it preserves composition of 1-cells it will suffice to show that all of the regions of the diagram below commute. $$\xy
(15,30)*+{R\bar{W}_{gf}}="f0";
(-25,-15)*+{RA}="b0";(15,-15)*+{RB}="c0";
(0,-2.5)*+{R\bar{W}_{f}}="a0";
{\ar_{Rq_{f}} "a0"; "c0"};
(30,-2.5)*+{R\bar{W}_{g}}="d0";(55,-15)*+{RC}="e0";
{\ar_{Rq_{g}} "d0"; "e0"};
{\ar@{~>}_{\overline{r}_{g}} "c0"; "d0"};
{\ar^{Rq_{gf}} "f0"; "e0"};
(15,10)*+{R\bar{W}_{g,f}}="g0";
{\ar|{Rk_{g,f}} "g0"; "f0"};
{\ar@{~>}_{\overline{r}_{f}} "b0"; "a0"};
{\ar@{~>}^{\overline{r}_{gf}} "b0"; "f0"};
{\ar@{~>}^{\overline{r}_{g,f}} "a0"; "g0"};
{\ar^{Rq_{g,f}} "g0"; "d0"};
\endxy$$ The rightmost quadrilateral certainly commutes as it is the image of a commutative diagram in ${\ensuremath{\mathcal{A}}\xspace}_{\tau}$, from , under $R$. To see that the central square commutes recall from the morphism of $w$-reflections $(q_{g,f},q_{f}):p_{g,f} \to p_{g}$ in A. By Part 1 $(Rq_{g,f},Rq_{f}):Rp_{g,f} \to p_{g,f}$ is a morphism of $w$-reflections in B: now commutativity of the central square simply asserts commutativity with the right adjoints. With regards the leftmost quadrilateral recall from Lemma \[prop:spanmap\] the morphism of $w$-reflections in A given by $(k_{g,f},1):p_{f}{.}p_{g,f} \to p_{gf}$. By Part 1 we know that $(Rk_{g,f},1):R(p_{f}{.}p_{g,f}) \to Rp_{gf}$ is a morphism of the lifted $w$-reflections so that, in particular, we have a commuting square of right adjoints $(1,Rk_{g,f}):\overline{r_{g,f}{.}r_{f}} \to \overline{r}_{gf}$ in B. Again by Part 1 we have $\overline{r_{g,f}{.}r_{f}} = \overline{r}_{g,f} \overline{r}_{f}$ and therefore the desired commutativity.
6. We define $K$ differently on 2-cells depending upon whether $w=l$ or $w=p$. Consider $\alpha:f \Rightarrow g$ and the case $w=l$. The commuting square $(c_{\alpha},1):p_{f} \to p_{g}$ of Lemma \[prop:l2\] has image $(Rc_{\alpha},1):Rp_{f} \to Rp_{g}$. We denote the mate of this square through the adjunctions $Rp_{f} \dashv \overline{r}_{f}$ and $Rp_{g} \dashv \overline{r}_{g}$ by $\overline{m}_{\alpha}:Rc_{\alpha}{.}\overline{r}_{f} \Rightarrow \overline{r}_{g}$. $$\xy
(-10,0)*+{}="-10";
(0,0)*+{RA}="00";(0,12)*+{RA}="01";(20,12)*+{RC_{f}}="11";(40,0)*+{RB}="20";(20,0)*+{RC_{g}}="10";(40,12)*+{B}="21";
{\ar@{~>}^{\overline{r}_{f}} "01"; "11"};
{\ar@{~>}_{\overline{r}_{g}} "00"; "10"};
{\ar^{Rq_{f}} "11"; "21"};
{\ar_{Rq_{g}} "10"; "20"};
{\ar^{Rc_{\alpha}} "11"; "10"};
{\ar@{=>}^{\overline{m}_{\alpha}}(10,9)*+{};(10,4)*+{}};
{\ar_{1} "01"; "00"};
{\ar^{1} "21"; "20"};
\endxy$$ Now we set $K\alpha=Rq_{g}{.}\overline{m}_{\alpha}$ as depicted above. Note that we have $\overline{m}_{\alpha}=\overline{\eta}_{g}{.}Rc_{\alpha}{.}\overline{r}_{f}$ since the counit of the $l$-reflection $Rp_{f} \dashv \overline{r}_{f}$ is an identity. Let us show that $HK\alpha=S\alpha$. We have from Lemma \[prop:l2\] the decomposition in A of $\alpha$ as $q_{g}{.}m_{\alpha}= q_{g}{.}\eta_{g}{.}c_{\alpha}{.}r_{f}$ and also have $K\alpha=Rq_{g}{.}\overline{\eta}_{g}{.}Rc_{\alpha}{.}\overline{r}_{f}$ from above. Thus $HK\alpha=Sq_{g}{.}S\eta_{g}{.}Sc_{\alpha}{.}Sr_{f}=S(q_{g}{.}\eta_{g}{.}c_{\alpha}{.}r_{f})=S\alpha$. To see that $K$ extends $R$ observe that if both $f$ and $g$ are tight then by Part 4 we have $(1,Rp_{f} \dashv \overline{r}_{f},\overline{\eta}_{f})=(1,Rp_{f} \dashv Rr_{f},R\eta_{f})$ and likewise for $g$. Using the same decomposition of $\alpha$ we have $R\alpha=Rq_{g}{.}R\eta_{g}{.}Rc_{\alpha}{.}Rr_{f}=K\alpha$. For $w=p$ consider the $p$-reflection $u_{f,g} \dashv v_{f,g}$ and the morphisms of $p$-reflections $(s_{f,g},1):u_{f,g} \to p_{f}$ and $(t_{f,g},1):u_{f,g} \to p_{g}$ of Lemma \[prop:p2\]. By Part 1 their images $(Rs_{f,g},1):Ru_{f,g} \to Rp_{f}$ and $(Rt_{f,g},1):Ru_{f,g} \to Rp_{g}$ are morphisms of $p$-reflections in B so that the two triangles in the diagram below $$\xy
(-5,0)*+{RA}="00"; (15,-15)*+{RK_{f,g}}="1-1";(35,-15)*+{RP_{f}}="2-1";(15,-30)*+{RP_{g}}="1-2";(35,-30)*+{A}="2-2";
{\ar@{~>}^{\overline{v}_{f,g}} "00"; "1-1"};
{\ar@/^1pc/@{~>}^{\overline{r}_{f}} "00"; "2-1"};
{\ar@/_1pc/@{~>}_{\overline{r}_{g}} "00"; "1-2"};
{\ar^{Rs_{f,g}} "1-1"; "2-1"};
{\ar^{Rq_{f}} "2-1"; "2-2"};
{\ar_<<<{Rt_{f,g}} "1-1"; "1-2"};
{\ar_{Rq_{g}} "1-2"; "2-2"};
{\ar@{=>}^{R\rho_{\alpha}}(27,-20)*+{};(22,-25)*+{}}
\endxy$$ commute. We set $K\alpha =R\rho_{\alpha}{.}\overline{v}_{f,g}$. By Lemma \[prop:p2\] we have that $\alpha=\rho_{\alpha}{.}v_{f,g}$. Therefore $S\alpha=S\rho_{\alpha}{.}Sv_{f,g}=HR\rho_{\alpha} {.}H\overline{v}_{f,g}=HK\alpha$. If now both $f$ and $g$ are tight then so is $v_{f,g}$, again by Lemma \[prop:p2\]. It follows that the $p$-reflections $Ru_{f,g} \dashv \overline{v}_{f,g}$ and $Ru_{f,g} \dashv Rv_{f,g}$ coincide, and therefore that $R\alpha=R\rho_{\alpha}{.}Rv_{f,g}=K\alpha$.
7. Treating the cases $w=l$ and $w=p$ together again observe that because $H$ is locally faithful the functoriality of $K$ on 2-cells trivally follows from the functoriality of $HK=S$ on 2-cells. Thus $K$ is an F-functor.
8. For uniqueness let us observe that the definition of the filler is forced upon us by the constraints. Consider any diagonal filler $L$. We must have $Lf=L(q_{f} {.}r_{f})=Lq_{f} {.}Lr_{f}= Rq_{f} {.}Lr_{f}$. Since $HL=S$ we would need that the image of the adjunction $(1,p_{f} \dashv r_{f},\eta_{f})$ under $L$ were a lifting of its image $(1,Sp_{f} \dashv Sr_{f},S\eta_{f})$ under $HL$. But by uniqueness of liftings we would then have $(1,Lp_{f} \dashv Lr_{f},L\eta_{f})=(1,Rp_{f} \dashv \overline{r_{f}},\overline{\eta_{f}})$ so that $Lf=Rq_{f}{.}\overline{r}_{f}$. Now for $w=l$ we have, by Lemma \[prop:l2\], the decomposition of $\alpha:f \Rightarrow g$ as $q_{g}{.}m_{\alpha}=q_{g}{.}\eta_{g} {.}c_{\alpha} {.}r_{f}$ and so must have $L\alpha=Rq_{g}{.}\overline{\eta}_{g}{.}Rc_{\alpha}{.}\overline{r}_{f}$. The argument when $w=p$ is similar but uses the decomposition $\alpha=\rho_{\alpha}{.}v_{f,g}$ of Lemma \[prop:p2\] instead.
Let us note that Theorem \[thm:orthogonality\] remains true even if we remove the assumption that $w$-doctrinal F-functors are locally faithful. The only place that we used this assumption was in establishing the functoriality of the diagonal $K$ on 2-cells. This can alternatively be established by carefully analysing the functoriality of the assignment in Section 4.3 which represents a 2-cell by a span transformation. However the proof becomes significantly longer and more technical, very significantly when $w=p$. Since local faithfulness is true of those F-functors $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ that we seek to characterise (see Theorem \[thm:monadicity\]) and because the property is easily verified in practice, we have chosen to include it in our definition of $w$-doctrinal F-functor.
As an immediate consequence of Theorem \[thm:orthogonality\] we have:
\[thm:decomposition\] Let $w \in \{l,p,c\}$. Consider an F-functor $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathcal{B}}\xspace}$ to a 2-category and suppose that A has $\bar{w}$-limits of loose morphisms and that $H$ is $w$-doctrinal. Then the decomposition in [$\mathcal{F}\textnormal{-CAT}$]{}$$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00";(15,0)*+{{\ensuremath{\mathbb{A}}\xspace}}="10";(30,0)*+{{\ensuremath{\mathcal{B}}\xspace}}="20";
{\ar^{j} "00"; "10"};
{\ar^<<<<<{H} "10"; "20"};
\endxy$$ of the 2-functor $H_{\tau}:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{B}}\xspace}$ is an orthogonal $(^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace},{\ensuremath{w\textnormal{-Doct}}\xspace})$-decomposition.
Since orthogonal decompositions are essentially unique, this asserts that an F-category satisfying some completeness properties and sitting over a base 2-category in a certain way – such as ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w}$ or ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over ${\ensuremath{\textnormal{Cat}}\xspace}$ – is *uniquely determined* by how its tight part sits over the base. This is the core idea behind our monadicity results of Section 6.
A note on alternative hypotheses
--------------------------------
There are other hypotheses upon A which ensure that $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}$ belongs to $^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace}$. Let us focus on the lax case. Say that A admits *loose morphism classifiers* if the inclusion $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda}$ has a left 2-adjoint $Q$ and write $p_{A}:A \rightsquigarrow QA$ and $q_{A}:QA \to A$ for the unit and counit. Each loose $f:A \rightsquigarrow B$ corresponds to a tight morphism $f^{\prime}:QA \to B$ such that $f^{\prime}{.}p_{A}=f$. One triangle equation gives $q_{A}{.}p_{A}=1$; if for each $A$ this happens to be the counit of an $l$-reflection $q_{A} \dashv p_{A}$ then each $f$ is represented by an $l$-span $$\xy
(0,0)*+{A}="0";(20,0)*+{QA}="1";(40,0)*+{B}="2";
{\ar_{q_{A} \dashv p_{A}} "1"; "0"}; {\ar^{f^{\prime}} "1"; "2"};
\endxy$$ and it can be easily shown that $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace} \in ^{\bot}{\ensuremath{l\textnormal{-Doct}}\xspace}$. Of course these hypotheses are strong: it is not easy to check whether a given F-category admits loose morphism classifiers.
Monadicity
==========
In this section we give our monadicity theorems. We begin by extending Eilenberg-Moore comparison 2-functors to F-functors. In Theorem \[thm:naturality\] we show these comparisons to be natural in $w$, in the sense of Diagram 2 of the Introduction. Our main result on monadicity is Theorem \[thm:monadicity\].
Extending the Eilenberg-Moore comparison
----------------------------------------
\[thm:EMExtension\] Let $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathcal{B}}\xspace}$ be an F-functor to a 2-category whose tight part $H_{\tau}:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{B}}\xspace}$ has a left adjoint and consider the induced Eilenberg-Moore comparison 2-functor $E$ left below. $$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="a0"; (15,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="b0";(30,0)*+{{\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}}="c0";
{\ar_{H_{\tau}} "a0"; "b0"};
{\ar^{E} "a0"; "c0"};
{\ar^{U_{s}} "c0"; "b0"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00";
(25,0)*+{{\ensuremath{\mathbb{A}}\xspace}}="10";
(0,-20)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{s}}="0-2";
(25,-20)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="1-2";
(40,-10)*+{{\ensuremath{\mathcal{B}}\xspace}}="2-1";
{\ar^{j} "00"; "10"};
{\ar^{j_{w}} "0-2"; "1-2"};
{\ar^{H} "10"; "2-1"};
{\ar_{E} "00"; "0-2"};
{\ar@{.>}_{E_{w}} "10"; "1-2"};
{\ar_{U} "1-2"; "2-1"};
\endxy$$ Let $w \in \{l,p,c\}$ and suppose that A admits $\overline{w}$-limits of loose morphisms. Then $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ admits a unique extension to an F-functor $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over B, as depicted on the right above.
The commutativity of the outside of the right diagram just says that $U_{s} E = H_{\tau}$. By Corollary \[thm:algdoctrinal\] and Theorem \[thm:orthogonality\] we know that $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{B}}\xspace}$ is $w$-doctrinal and that $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}$ is orthogonal to each $w$-doctrinal F-functor, in particular $U$. Therefore there exists a unique F-functor $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ satisfying the depicted equations $E_{w}j=j_{w}E$ and $UE_{w}=H$. These respectively assert that $E_{w}$ extends $E$ and lives over the base ${\ensuremath{\mathcal{B}}\xspace}$.
In order to understand the naturality in $w$ of the above Eilenberg-Moore extensions $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ it will be convenient to briefly consider ${\ensuremath{\mathcal{F}}\xspace}_{2}$-categories. An ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category consists of a 2-category equipped with three kinds of morphism: tight, loose and very loose, all satisfying the expected axioms. For instance we have the ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category of monoidal categories, strict, strong and lax monoidal functors; likewise of algebras together with strict, pseudo and lax morphisms for a 2-monad. One presentation of an ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category is as a triple on the left below $$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00";
(20,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\lambda}}="10";
(40,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\phi}}="20";
{\ar^{j} "00"; "10"};
{\ar^{j} "10"; "20"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00";
(20,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="10";
{\ar^{j} "00"; "10"};
\endxy$$ in which each inclusion is the identity on objects, faithful and locally fully faithful. Thus an ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category has three associated ${\ensuremath{\mathcal{F}}\xspace}$-categories ${\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}$, ${\ensuremath{\mathbb{A}}\xspace}_{\lambda,\phi}$ and ${\ensuremath{\mathbb{A}}\xspace}_{\tau, \phi}$, and is determined by the first and third of these together with the inclusion F-functor, right above, which views tight and loose morphisms as tight and very loose respectively.\
We commonly encounter ${\ensuremath{\mathcal{F}}\xspace}_{2}$-categories sitting over a base 2-category as on the left below. See how monoidal categories, strict, strong and lax monoidal functors sit over [$\textnormal{Cat}$]{}for instance. $$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00";
(20,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\lambda}}="10";
(40,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\phi}}="20";
(20,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="1-2";
{\ar^{j} "00"; "10"};
{\ar^{j} "10"; "20"};
{\ar_{H_{\tau}} "00"; "1-2"};
{\ar|{H_{\lambda}} "10"; "1-2"};
{\ar^{H_{\phi}} "20"; "1-2"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00";
(25,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="20";
(12.5,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="1-2";
{\ar^{j} "00"; "20"};
{\ar_{H_{\tau,\lambda}} "00"; "1-2"};
{\ar^{H_{\tau,\phi}} "20"; "1-2"};
\endxy$$ To give such a diagram is equally to give a commutative triangle in [$\mathcal{F}\textnormal{-CAT}$]{}as on the right above – here the F-functors $H_{\tau,\lambda}$ and $H_{\tau,\phi}$ agree as $H_{\tau}$ on tight parts, and have loose parts $H_{\lambda}$ and $H_{\phi}$ respectively.
\[thm:naturality\] Let $w \in \{l,c\}$. Consider an ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category over a 2-category as below $$\xy
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00";
(25,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="20";
(12.5,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="1-2";
{\ar^{j} "00"; "20"};
{\ar_{H_{\tau,\lambda}} "00"; "1-2"};
{\ar^{H_{\tau,\phi}} "20"; "1-2"};
\endxy$$ Suppose that $H_{\tau}$ admits a left adjoint and that ${\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}$ and ${\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}$ satisfy the $(p/w)$ variants of the completeness criteria of Theorem \[thm:EMExtension\] so that the Eilenberg-Moore comparison 2-functor $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ extends uniquely to F-functors $E_{p}:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}$ and $E_{w}:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over the base B.\
When all of this holds the square $$\xy
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00"; (20,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="10";
(0,-15)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}}="01"; (20,-15)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="11";
{\ar^{j} "00";"10"};
{\ar ^{E_{w}}"10";"11"};
{\ar _{E_{p}}"00";"01"};
{\ar _{j} "01";"11"};
\endxy$$ commutes.
Consider the diagram left below $$\xy
(-20,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="-10";
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00"; (20,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="10";
(0,-15)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}}="01"; (20,-15)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="11";(40,-15)*+{{\ensuremath{\mathcal{B}}\xspace}}="21";
{\ar^{j} "-10";"00"};
{\ar^{j} "00";"10"};
{\ar ^{E_{w}}"10";"11"};
{\ar _{E_{p}}"00";"01"};
{\ar _{j} "01";"11"};
{\ar _{U_{w}} "11";"21"};
\endxy
\hspace{2cm}
\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00"; (20,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="10";
(0,-15)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="01"; (20,-15)*+{{\ensuremath{\mathcal{B}}\xspace}}="11";
{\ar^{j} "00";"10"};
{\ar ^{H_{\tau,\lambda}}"10";"11"};
{\ar _{jE}"00";"01"};
{\ar@{.>}_{}"10";"01"};
{\ar _{U_{w}} "01";"11"};
\endxy$$ Now $E_{p}$ and $E_{w}$ agree as $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ on tight morphisms. Consequently both paths of the square coincide upon precomposition with $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}$ as the composite $jE:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}\to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$. Because both Eilenberg-Moore extensions $E_{p}$ and $E_{w}$ lie over the base B we find that postcomposing both paths of the square with $U_{w}$ yields the common composite $H_{\tau,\lambda}:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda} \to {\ensuremath{\mathcal{B}}\xspace}$. Therefore both paths of the square are diagonal fillers for the square on the right above. Now by Corollary \[thm:algdoctrinal\] we know that $U_{w}:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{B}}\xspace}$ is $p$-doctrinal. But by Theorem \[thm:orthogonality\] $j:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}$ is orthogonal to each $p$-doctrinal F-functor so that both paths coincide as the unique filler.
2-categorical monadicity
------------------------
We now turn to monadicity. Let us say that a 2-functor with a left adjoint is *monadic* if the induced Eilenberg-Moore comparison is a 2-equivalence and *strictly monadic* if the comparison is an isomorphism.
\[thm:monadicity\] Let $H:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathcal{B}}\xspace}$ be an F-functor to a 2-category B. Let $w \in \{l,p,c\}$ and suppose that
1. $H_{\tau}:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{B}}\xspace}$ is monadic.
2. A admits $\overline{w}$-limits of loose morphisms.
3. B admits $\overline{w}$-limits of arrows.
4. $H$ is $w$-doctrinal (it suffices that $H$ satisfies $w$-doctrinal adjunction, is locally faithful and reflects identity 2-cells).
Then the 2-equivalence $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ extends uniquely to an equivalence of F-categories $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over B. Moreover if $H_{\tau}$ is strictly monadic then $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ is an isomorphism of F-categories.
As in Theorem \[thm:EMExtension\] we have our extension $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$, unique in filling the square $$\xy
(0,10)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00"; (30,10)*+{{\ensuremath{\mathbb{A}}\xspace}}="10";(0,-20)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="02";(30,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="11";(0,-5)*+{{\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}}="01";
{\ar_{E} "00"; "01"};
{\ar_{j_{w}} "01"; "02"};
{\ar_{U} "02"; "11"};
{\ar^{j} "00"; "10"};
{\ar^{H} "10"; "11"};
{\ar@{.>}^{E_{w}} "10"; "02"};
\endxy$$ Recall that this filler exists because $U$ is $w$-doctrinal and $j$ orthogonal to such F-functors.\
We begin by proving the theorem in the case of strict monadicity and deduce the general case from that – so suppose that $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ is an isomorphism of 2-categories. By Proposition \[prop:limits\] $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{B}}\xspace}$ creates $\overline{w}$-limits of loose morphisms so that ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ admits them. Now Theorem \[thm:orthogonality\] implies that the inclusion $j_{w}:{\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}\to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ is orthogonal to each $w$-doctrinal F-functor; since $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ is an isomorphism $j_{w}E:{\ensuremath{\mathcal{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ is also orthogonal to $w$-doctrinal F-functors. Now $H$ is $w$-doctrinal by assumption so that the two outer paths of the square are orthogonal decompositions of a common F-functor. Therefore the unique filler $E_{w}:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ is an isomorphism.\
Suppose now that $E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ is only an equivalence of 2-categories. The problem now is that $E$ will no longer be orthogonal to $w$-doctrinal F-functors. We will rectify this by factoring the composite 2-functor $j_{w} E:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}\to {\ensuremath{\textnormal{T-Alg}}\xspace}_{w}$ as the identity on objects followed by 2-fully faithful through a 2-category ${\ensuremath{\mathcal{\overline{A}}}\xspace}_{\lambda}$ as in the commutative square below. $$\xy
(0,10)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="a0"; (30,10)*+{{\ensuremath{\mathcal{\overline{A}}}\xspace}_{\lambda}}="b0";(0,-10)*+{{\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}}="c0";(30,-10)*+{{\ensuremath{\textnormal{T-Alg}}\xspace}_{w}}="d0";
{\ar^{\overline{j}} "a0"; "b0"};
{\ar_{E} "a0"; "c0"};
{\ar^{K} "b0"; "d0"};
{\ar^{j_{w}} "c0"; "d0"};
\endxy$$ Then $K$ is 2-fully faithful by construction but also essentially surjective on objects since each of $E$, $\overline{j}$ and $j_{w}$ are; as such $K$ is a 2-equivalence. Moreover the composite $K \overline{j} = j_{w} E$ is both faithful and locally fully faithful since both $j_{w}$ and $E$ are, whilst $K$ is 2-fully faithful. Therefore $\overline{j}$ is faithful and locally fully faithful too and since it is identity on objects by construction it is the inclusion of an F-category $\overline{{\ensuremath{\mathbb{A}}\xspace}}:{\ensuremath{\mathcal{A}}\xspace}_{\tau} \to \overline {{\ensuremath{\mathcal{A}}\xspace}}_{\lambda}$. It now follows that the above commutative square, whose vertical legs are 2-equivalences, exhibits $L=(E,K):\overline{{\ensuremath{\mathbb{A}}\xspace}} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ as an equivalence of F-categories.\
Consider the diagram defining the extension $E_{w}$ again, now drawn on the left below $$\xy
(0,10)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00"; (30,10)*+{{\ensuremath{\mathbb{A}}\xspace}}="10";(0,-20)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="02";(30,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="11";(0,-5)*+{{\ensuremath{\mathbb{\overline{A}}}\xspace}}="01";
{\ar_{\overline{j}} "00"; "01"};
{\ar_{L} "01"; "02"};
{\ar_{U} "02"; "11"};
{\ar^{j} "00"; "10"};
{\ar^{H} "10"; "11"};
{\ar@{.>}^{E_{w}} "10"; "02"};
{\ar@{.>}_{\overline{E}} "10"; "01"};
\endxy
\hspace{2cm}
\xy
(0,10)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00"; (40,10)*+{{\ensuremath{\mathbb{A}}\xspace}}="10";(20,-20)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="02";(40,-20)*+{{\ensuremath{\mathcal{B}}\xspace}}="11";(0,-20)*+{{\ensuremath{\mathbb{\overline{A}}}\xspace}}="01";
{\ar_{\overline{j}} "00"; "01"};
{\ar_{L} "01"; "02"};
{\ar_{U} "02"; "11"};
{\ar^{j} "00"; "10"};
{\ar^{H} "10"; "11"};
{\ar@{.>}^{\overline{E}} "10"; "01"};
\endxy$$ with the left leg rewritten as the composite $L \circ \overline{j}$. Since $L$ is F-fully faithful (*2-fully faithful on tight and loose parts*) it is orthogonal to each bijective on objects F-functor, in particular $j$, so that we obtain a unique diagonal filler $\overline{E}:{\ensuremath{\mathbb{A}}\xspace} \to \overline{{\ensuremath{\mathbb{A}}\xspace}}$ making the two leftmost triangles commute.\
Our goal is to show that $E_{w}$ is an equivalence of F-categories - but since $L$ is an equivalence this is, by 2 out of 3, equivalently to show that $\overline{E}$ is an equivalence of F-categories. Now consider the square on the right. The bottom leg is $w$-doctrinal as both of its components are, $L$ by Proposition \[prop:equivalences\]. Since the F-category $\overline{{\ensuremath{\mathbb{A}}\xspace}}$ is equivalent to ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ it has the same completeness properties, so that, using Theorem \[thm:orthogonality\] again, the left leg is orthogonal to each $w$-doctrinal F-functor. Therefore the right commutative square consists of two orthogonal decompositions of a common F-functor and we conclude that $\overline{E}$ is an isomorphism.
Note that although Theorem \[thm:monadicity\], in each of its variants, only asks that ${\ensuremath{\mathbb{A}}\xspace}$ admits certain limits it follows that $H$ creates those limits: for $U:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathcal{B}}\xspace}$ does so by Proposition \[prop:limits\] and $E:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ is an equivalence of F-categories. In applying the theorem this is often useful in that it tells us how these limits must be constructed in A.
F-categorical monadicity
------------------------
Our focus has been upon 2-monads but indeed the above results extend in a routine way to cover F-categorical monadicity too. Let us briefly explain how this goes. An F-monad [@Lack2011Enhanced] is a monad in the 2-category [$\mathcal{F}\textnormal{-CAT}$]{}and so consists of an F-functor $T:{\ensuremath{\mathbb{A}}\xspace} \to {\ensuremath{\mathbb{A}}\xspace}$ and two F-natural transformations satisfying the usual equations. 2-monads are just F-monads on 2-categories viewed as F-categories. An F-monad $T$ induces 2-monads $T_{\tau}$ and $T_{\lambda}$ and so we have strict $T_{\tau}$ and $T_{\lambda}$-algebra morphisms as in the two leftmost diagrams below. $$\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar _{f} "01";"11"};
\endxy
\hspace{1cm}
\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar@{~>}^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar@{~>}_{f} "01";"11"};
\endxy
\hspace{1cm}
\xy
(0,0)*+{TA}="00"; (20,0)*+{TB}="10";
(0,-20)*+{A}="01"; (20,-20)*+{B}="11";
{\ar@{~>}^{Tf} "00";"10"};
{\ar ^{b}"10";"11"};
{\ar _{a}"00";"01"};
{\ar@{~>}_{f} "01";"11"};
{\ar@{=>}^{\overline{f}}(10,-7)*+{};(10,-13)*+{}};
\endxy$$ These are the tight and loose morphisms of the Eilenberg-Moore F-category which is denoted by ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{s}$. We also have pseudo, lax and colax $T_{\lambda}$-morphisms – a lax $T_{\lambda}$-morphism is drawn above. These are the loose morphisms of the F-categories ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ whose tight morphisms are the strict $T_{\tau}$-algebra maps. Now for each $w \in \{l,p,c\}$ we have the ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category whose tight and loose morphisms are the strict $T_{\tau}$ and $T_{\lambda}$-morphisms and whose very loose morphisms are the $w\textnormal{-}T_{\lambda}$-morphisms: this is captured by the inclusion F-functor $j_{w}:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{s} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$. We have the evident forgetful F-functors $U_{s}:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{s} \to {\ensuremath{\mathbb{A}}\xspace}$ and $U_{w}:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathbb{A}}\xspace}$ commuting with $j_{w}$ over the base. The key point for our applications is that these have the same properties as in the 2-monads case: namely $U_{s}$ creates all limits, $U_{w}$ creates $\overline{w}$-limits of loose morphisms (this follows from Theorem 5.13 of [@Lack2011Enhanced]) and $U_{w}$ is $w$-doctrinal. With these facts in place we can give our monadicity theorem for F-monads – we leave it to the reader to formulate the naturality of the Eilenberg-Moore extensions, which can be done using *${\ensuremath{\mathcal{F}}\xspace}_{3}$-categories*.
\[thm:fmonadicity\] Consider an ${\ensuremath{\mathcal{F}}\xspace}_{2}$-category ${\ensuremath{\mathcal{A}}\xspace}_{\tau} \to {\ensuremath{\mathcal{A}}\xspace}_{\lambda} \to {\ensuremath{\mathcal{A}}\xspace}_{\phi}$ over an F-category B as below $$\xy
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00";
(25,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="20";
(12.5,-20)*+{{\ensuremath{\mathbb{B}}\xspace}}="1-2";
{\ar^{j} "00"; "20"};
{\ar_{G} "00"; "1-2"};
{\ar^{H} "20"; "1-2"};
\endxy$$ (this means that ${G}_{\tau}=H_{\tau}$). Suppose that $G$ has a left F-adjoint so that we have the Eilenberg-Moore comparison F-functor $E:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{s}$ over the base B.\
Now let $w \in \{l,p,c\}$ and suppose that both ${\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}$ and ${\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}$ admit $\overline{w}$-limits of loose morphisms. Then there exists a unique F-functor $E_{w}:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ extending $E$ and living over the base, as depicted by the following everywhere commutative diagram. $$\xy
(0,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="00";
(25,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="10";
(0,-20)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{s}}="0-2";
(25,-20)*+{{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}}="1-2";
(40,-10)*+{{\ensuremath{\mathbb{B}}\xspace}}="2-1";
{\ar^{j} "00"; "10"};
{\ar^{j_{w}} "0-2"; "1-2"};
{\ar^{H} "10"; "2-1"};
{\ar_{E} "00"; "0-2"};
{\ar@{.>}_{E_{w}} "10"; "1-2"};
{\ar_{U_{w}} "1-2"; "2-1"};
\endxy$$ If $G$ is F-monadic, B admits $\overline{w}$-limits of loose morphisms and $H$ is $w$-doctrinal then $E_{w}$ is an equivalence of F-categories, and an isomorphism of F-categories whenever $G$ is strictly monadic.
The outside of the diagram clearly commutes – since $Hj=G$ and $U_{w}j_{w}=U_{s}$ this just amounts to the fact that the Eilenberg-Moore comparison $E$ satisfies $U_{s}E=G$. Now $U_{w}:{\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w} \to {\ensuremath{\mathbb{B}}\xspace}$ is $w$-doctrinal so that if we can show that the inclusion $j:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda} \to {\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}$ belongs to $^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace}$ then we will obtain $E_{w}$ as the unique filler. Now have a commutative triangle of inclusions $$\xy
(0,0)*+{{\ensuremath{\mathcal{A}}\xspace}_{\tau}}="00";
(25,0)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda}}="20";
(25,-15)*+{{\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi}}="1-2";
{\ar^{j} "00"; "20"};
{\ar_{j} "00"; "1-2"};
{\ar^{j} "20"; "1-2"};
\endxy$$ in which the two $j$’s moving from left to right belong to $^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace}$ by Theorem \[thm:orthogonality\]; thus by 2 out of 3 $j:{\ensuremath{\mathbb{A}}\xspace}_{\tau,\lambda} \to {\ensuremath{\mathbb{A}}\xspace}_{\tau,\phi} \in ^{\bot}{\ensuremath{w\textnormal{-Doct}}\xspace}$. As such we obtain the Eilenberg-Moore extension $E_{w}$ as the unique filler. The remainder of the proof is a straightforward modification of the proof of Theorem \[thm:monadicity\].
Examples and applications
=========================
We now turn to examples. We begin by completing our running example of monoidal categories. Of course it is well known that monoidal categories, and each flavour of morphism between them, can be described using 2-monads – see Section 5.5 of [@Lack2010A-2-categories] for an argument via colimit presentations – although this has not previously been established by application of a monadicity theorem. We then turn to more complex examples. Our final example, in 7.3, is new and typical of the kind of result which cannot be established using techniques, such as colimit presentations, that require explicit knowledge of a 2-monad.
Monoidal categories
-------------------
Let us focus, as usual, on the lax monoidal functors of ${\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l}$. From Example \[thm:DoctrinalMonoidal\] we know that $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ satisfies $l$-doctrinal adjunction. It is clearly locally faithful and reflects identity 2-cells. From Example \[thm:LimitsMonoidal\] we know that $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ creates colax limits of loose morphisms. Therefore to apply Theorem \[thm:monadicity\] and establish monadicity it remains to verify that the 2-functor $V_{s}:{\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{Cat}}\xspace}$ is monadic. That $V_{s}$ strictly creates $V_{s}$-absolute coequalisers (in the enriched sense) is true by essentially the same argument given for groups in 6.8 of [@Mac-Lane1971Categories]; by Beck’s theorem in the enriched setting [@Dubuc1970Kan-extensions] it follows that $V_{s}$ is strictly monadic so long as it has a left 2-adjoint. Proposition 3.1 of [@Blackwell1989Two-dimensional] asserts that a 2-functor preserving *cotensors with [**2**]{}* admits a left 2-adjoint just when its underlying functor admits a left adjoint. Now cotensors with [**2**]{}are, in fact, just colax limits of identity arrows. That these are created by $V_{s}$ follows from the fact that $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{Cat}}\xspace}$ creates colax limits of loose morphisms. It therefore remains to show that the underlying functor $(V_{s})_{0}$ admits a left adjoint. Since this functor creates all limits it suffices to show that $(V_{s})_{0}$ satisfies the solution set condition. For this it suffices to show that given a small category $A$ each functor $F:A \to C=U\overline{C}$ to a monoidal category factors as $ME:A \to B \to C$ with $B$ monoidal, $M$ strict monoidal and the cardinality of the set of morphisms $Mor(B)$ bounded by that of $Mor(A)$. Here $B$ will be the monoidal subcategory of $C$ generated by the image of $F$ and $M$ the inclusion of this monoidal subcategory. For the induced 2-monad $T$ on [$\textnormal{Cat}$]{}we now conclude, by Theorem \[thm:monadicity\], that the isomorphism of 2-categories $E:{\ensuremath{\textnormal{MonCat}}\xspace}_{s} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ over [$\textnormal{Cat}$]{}extends uniquely to an isomorphism of F-categories $E_{l}:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{l} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{l}$ over [$\textnormal{Cat}$]{}. Likewise one can verify, in an entirely similar way, that $V:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w} \to {\ensuremath{\textnormal{Cat}}\xspace}$ satisfies the conditions of Theorem \[thm:monadicity\] in the cases $w \in \{p,c\}$. It follows that we have isomorphisms $E_{w}:{\ensuremath{\mathbb{M}\textnormal{onCat}}\xspace}_{w} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over [$\textnormal{Cat}$]{}for each $w \in \{l,p,c\}$. By Theorem \[thm:naturality\] these isomorphisms are natural in $p \leq l$ and $p \leq c$ in the sense of Diagram 2 of the Introduction.
Categories with structure and variants
--------------------------------------
Of course there was nothing special about our taking monoidal categories in the preceding section. The same arguments can be used to establish monadicity of categories with any kind of algebraically specified structure and their various flavours of morphisms: categories with chosen limits of some kind for instance, distributive categories and so forth. All of these cases are well known to be monadic using colimit presentations of 2-monads, although it requires substantial and laborious calculation to use that theory to establish monadicity in the detailed manner above. Such a detailed treatment using colimit presentations, one expressed in terms of isomorphisms of 2-categories or F-categories, will not be found in the literature. Colimit presentations are one of the two standard techniques for understanding the monadicity of weaker kinds of morphisms; the other is direct calculation with a 2-monad known to exist. As a representative example of this technique consider a small 2-category ${\ensuremath{\mathcal{J}}\xspace}$ and the forgetful 2-functor $U:[{\ensuremath{\mathcal{J}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}] \to [ob {\ensuremath{\mathcal{J}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]$ which restricts presheaves to families along the inclusion $ob {\ensuremath{\mathcal{J}}\xspace} \to {\ensuremath{\mathcal{J}}\xspace}$. $U$ has a left 2-adjoint $F$ given by left Kan extension and is strictly monadic by Beck’s theorem; moreover the induced 2-monad $T=UF$ admits a simple pointwise description: at $X \in [ob {\ensuremath{\mathcal{J}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]$ we have $TX(j) = \Sigma_{i}{\ensuremath{\mathcal{J}}\xspace}(i,j) \times Xi$. Using this formula (as in [@Blackwell1989Two-dimensional]) one directly calculates that $T$-pseudomorphisms bijectively correspond to pseudonatural transformations and so on, eventually deducing an isomorphism $Ps({\ensuremath{\mathcal{J}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}) \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{p}$. It is in such cases, more specifically when $T$ is not so simple, that our results have most value.\
An example of this kind was given in [@Kelly2000On-the]. Given a complete and cocomplete symmetric monoidal closed category V the authors take as base the 2-category [$\textnormal{V-Cat}$]{}of small V-categories. For a small class of weights $\Phi$ they consider the 2-category ${\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}$ (we will write ${\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}_{s}$) of V-categories with *chosen* $\Phi$-weighted colimits, V-functors preserving those colimits strictly and V-natural transformations. This 2-category lives over ${\ensuremath{\textnormal{V-Cat}}\xspace}$ via a forgetful 2-functor $U_{s}:{\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}_{s} \to {\ensuremath{\textnormal{V-Cat}}\xspace}$. One also has V-functors preserving colimits in the usual, up to isomorphism, sense: these are the loose morphisms of the F-category ${\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p}:{\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}_{s} \to {\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}_{p}$ which sits over [$\textnormal{V-Cat}$]{}via a forgetful F-functor $U:{\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p} \to {\ensuremath{\textnormal{V-Cat}}\xspace}$. The authors show that $U_{s}:{\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}_{s} \to {\ensuremath{\textnormal{V-Cat}}\xspace}$ is strictly monadic and then, by calculating directly with the induced 2-monad $T$, show that one obtains an isomorphism of 2-categories $U:{\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}$. Let us show how this can be deduced from Theorem \[thm:monadicity\]. Firstly observe that $U$ satisfies $p$-doctrinal adjunction: this follows from the fact that any equivalence of V-categories preserves colimits. Again since $U$ is locally fully faithful it is certainly locally faithful and reflects identity 2-cells. It is not hard to see that ${\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p}$ admits pseudo-limits of loose morphisms – indeed this is shown in Section 5 of [@Kelly2000On-the]. It then follows immediately from Theorem \[thm:monadicity\] that the isomorphism of 2-categories $E:{\ensuremath{\Phi\textnormal{-}\textnormal{Colim}}\xspace}_{s} \to {\ensuremath{\textnormal{T-Alg}}\xspace}_{s}$ extends uniquely to an isomorphism of F-categories $E:{\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}$ over [$\textnormal{V-Cat}$]{}.\
Furthermore, because left adjoints preserve colimits, $U:{\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p} \to {\ensuremath{\textnormal{V-Cat}}\xspace}$ has the additional property of satisfying $c$-doctrinal adjunction; since it is isomorphic to ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}$ it also admits, by Theorem 2.6 of [@Blackwell1989Two-dimensional], lax limits of loose morphisms. Therefore Theorem \[thm:monadicity\] ensures that the composite ${\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{p} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{c}$ is also an isomorphism, thus explaining why the colax and pseudo $T$-morphisms coincide as those V-functors which preserve $\Phi$-colimits. Again one easily applies Theorem \[thm:monadicity\] to show that ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{l}$ is isomorphic to the F-category ${\ensuremath{\Phi\textnormal{-}\mathbb{C}\textnormal{olim}}\xspace}_{l}$ whose loose morphisms are arbitary V-functors. This final isomorphism highlights the fact that $U_{l}:{\ensuremath{\textnormal{T-Alg}}\xspace}_{l} \to {\ensuremath{\textnormal{V-Cat}}\xspace}$ is 2-fully faithful: 2-monads with this property are called *lax idempotent/Kock-Zöberlein* and have been carefully studied in [@Kelly1997On-property-like].
In a monoidal 2-category
------------------------
In a monoidal category C one can consider the category of monoids ${\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})$ or of commutative monoids. If the forgetful functor $U:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace}) \to {\ensuremath{\mathcal{C}}\xspace}$ has a left adjoint then Beck’s theorem can be applied, with no further information, to show that $U$ is monadic. For our final example we study the analogous situation in the context of a monoidal 2-category C, in which one can consider monoids, pseudomonoids (generalising monoidal categories), braided pseudomonoids and so on. We consider only the simplest case of monoids because, in the absence of a suitable graphical calculus, it is difficult to encode diagrams compactly. We have the 2-category of monoids, strict monoid morphisms and monoid transformations ${\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{s}$ and a forgetful 2-functor $U_{s}:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{s} \to {\ensuremath{\mathcal{C}}\xspace}$. Just as before, the enriched version of Beck’s theorem [@Dubuc1970Kan-extensions] can be applied to show that if $U_{s}$ has a left 2-adjoint then it is monadic. However we now also have (lax/pseudo/colax)-morphisms of monoids and we would like to understand that these too are monadic in the appropriate sense: this is the content, under completeness conditions on the base, of the present example.\
By a monoidal 2-category C we will mean a monoidal V-category where ${\ensuremath{\mathcal{V}}\xspace}={\ensuremath{\textnormal{Cat}}\xspace}$: this satisfies the same axioms as a monoidal category with the exception that the tensor product, the associator and the other data involved are now 2-functorial and 2-natural. In working with C we will write as though it were strict monoidal – this is justified in the theorem that follows – and will use juxtaposition for the tensor product. A monoid in C is just a monoid in the usual sense. Given monoids $(X,m_{X},i_{X})$ and $(Y,m_{Y},i_{Y})$ a lax monoid map $(f,\overline{f},f_{0}):(X,m_{X},i_{X}) \to (Y,m_{Y},i_{Y})$ consists of an arrow $f:X \to Y$ and 2-cells as below $$\xy
(0,0)*+{X^{2}}="00"; (20,0)*+{Y^{2}}="10"; (0,-12)*+{X}="01"; (20,-12)*+{Y}="11";
{\ar^{f^{2}} "00";"10"};
{\ar_{f} "01";"11"};
{\ar_{m_{X}}"00";"01"};
{\ar^{m_{Y}}"10";"11"};
{\ar@{=>}_{\overline{f}}(10,-4)*+{};(10,-9)*+{}};
\endxy
\hspace{1cm}
\xy
(0,0)*+{I}="00"; (-10,-12)*+{X}="10"; (10,-12)*+{Y}="11";
{\ar_{i_{X}} "00";"10"};
{\ar^{i_{Y}} "00";"11"};
{\ar_{f}"10";"11"};
{\ar@{=>}_{f_{0}}(0,-4)*+{};(0,-9)*+{}};
\endxy$$ such that the equation $$\xy
(0,0)*+{X^{3}}="00"; (20,0)*+{XY^{2}}="10"; (40,0)*+{Y^{3}}="20";
(0,-12)*+{X^{2}}="01"; (20,-12)*+{XY}="11"; (40,-12)*+{Y^{2}}="21"; (0,-24)*+{X}="02"; (40,-24)*+{Y}="22";
{\ar^{1ff} "00";"10"};
{\ar^{f11} "10";"20"};
{\ar_{1m_{X}} "00";"01"};
{\ar^{1m_{Y}}"10";"11"};
{\ar^{1m_{Y}}"20";"21"};
{\ar@{=>}_{1\overline{f}}(10,-4)*+{};(10,-9)*+{}};
{\ar_{1f} "01";"11"};
{\ar_{f1} "11";"21"};
{\ar_{f} "02";"22"};
{\ar_{m_{X}}"01";"02"};
{\ar^{m_{Y}}"21";"22"};
{\ar@{=>}_{\overline{f}}(20,-16)*+{};(20,-21)*+{}};
\endxy
\hspace{0.25cm}
\xy
(0,-12)*+{=};
\endxy
\hspace{0.25cm}
\xy
(0,0)*+{X^{3}}="00"; (20,0)*+{X^{2}Y}="10"; (40,0)*+{Y^{3}}="20";
(0,-12)*+{X^{2}}="01"; (20,-12)*+{XY}="11"; (40,-12)*+{Y^{2}}="21"; (0,-24)*+{X}="02"; (40,-24)*+{Y}="22";
{\ar^{11f} "00";"10"};
{\ar^{ff1} "10";"20"};
{\ar_{m_{X}1} "00";"01"};
{\ar_{m_{X}1}"10";"11"};
{\ar^{m_{Y}1}"20";"21"};
{\ar@{=>}_{\overline{f}1}(30,-4)*+{};(30,-9)*+{}};
{\ar_{1f} "01";"11"};
{\ar_{f1} "11";"21"};
{\ar_{f} "02";"22"};
{\ar_{m_{X}}"01";"02"};
{\ar^{m_{Y}}"21";"22"};
{\ar@{=>}_{\overline{f}}(20,-16)*+{};(20,-21)*+{}};
\endxy$$ holds and such that both composite 2-cells $$\xy
(0,0)*+{X}="00"; (20,0)*+{X}="10"; (40,0)*+{Y}="20";
(0,-12)*+{X^{2}}="01"; (20,-12)*+{XY}="11"; (40,-12)*+{Y^{2}}="21"; (0,-24)*+{X}="02"; (40,-24)*+{Y}="22";
{\ar^{1} "00";"10"};
{\ar^{f} "10";"20"};
{\ar_{1i_{X}} "00";"01"};
{\ar_{1i_{Y}}"10";"11"};
{\ar^{1i_{Y}}"20";"21"};
{\ar@{=>}_{1f_{0}}(10,-4)*+{};(10,-9)*+{}};
{\ar_{1f} "01";"11"};
{\ar_{f1} "11";"21"};
{\ar_{f} "02";"22"};
{\ar_{m_{X}}"01";"02"};
{\ar^{m_{Y}}"21";"22"};
{\ar@{=>}_{1\overline{f}}(20,-16)*+{};(20,-21)*+{}};
\endxy
\hspace{1cm}
\xy
(0,0)*+{X}="00"; (20,0)*+{Y}="10"; (40,0)*+{Y}="20";
(0,-12)*+{X^{2}}="01"; (20,-12)*+{XY}="11"; (40,-12)*+{Y^{2}}="21"; (0,-24)*+{X}="02"; (40,-24)*+{Y}="22";
{\ar^{f} "00";"10"};
{\ar^{1} "10";"20"};
{\ar_{i_{X}1} "00";"01"};
{\ar_{i_{X}1}"10";"11"};
{\ar^{i_{Y}1}"20";"21"};
{\ar@{=>}_{f_{0}1}(30,-4)*+{};(30,-9)*+{}};
{\ar_{1f} "01";"11"};
{\ar_{f1} "11";"21"};
{\ar_{f} "02";"22"};
{\ar_{m_{X}}"01";"02"};
{\ar^{m_{Y}}"21";"22"};
{\ar@{=>}_{1\overline{f}}(20,-16)*+{};(20,-21)*+{}};
\endxy$$ are identities. A monoid transformation $\alpha:(f,\overline{f},f_{0}) \Rightarrow (g,\overline{g},g_{0})$ is a 2-cell $\alpha:f \Rightarrow g$ satisfying the equations $$\xy
(0,0)*+{X^{2}}="11";
(20,0)*+{Y^{2}}="31"; (0,-15)*+{X}="12";(20,-15)*+{Y}="32";
{\ar^{m_{Y}} "31"; "32"};
{\ar@/_1pc/_{g} "12"; "32"};
{\ar_{m_{X}} "11"; "12"};
{\ar@/^1pc/^{f^{2}} "11"; "31"};
{\ar@/_1pc/_{g^{2}} "11"; "31"};
{\ar@{=>}^{\overline{g}}(10,-9)*+{};(10,-15)*+{}};
{\ar@{=>}_{\alpha^{2}}(10,2)*+{};(10,-3)*+{}};
\endxy
\xy
(0,-7.5)*+{=};
\endxy
\xy
(0,0)*+{X^{2}}="11";
(20,0)*+{Y^{2}}="31"; (0,-15)*+{X}="12";(20,-15)*+{Y}="32";
{\ar^{m_{Y}} "31"; "32"};
{\ar@/^1pc/^{f} "12"; "32"};
{\ar@/_1pc/_{g} "12"; "32"};
{\ar_{m_{X}} "11"; "12"};
{\ar@/^1pc/^{f^{2}} "11"; "31"};
{\ar@{=>}_{\alpha}(10,-13)*+{};(10,-18)*+{}};
{\ar@{=>}_{\overline{f}}(10,0)*+{};(10,-6)*+{}};
\endxy
\hspace{0.5cm}
\xy
(0,2)*+{I}="00"; (-10,-15)*+{X}="10"; (10,-15)*+{Y}="11";
{\ar@/_0.5pc/_{i_{X}} "00";"10"};
{\ar@/^0.5pc/^{i_{Y}} "00";"11"};
{\ar@/^1pc/^{f}"10";"11"};
{\ar@/_1pc/_{g}"10";"11"};
{\ar@{=>}_{f_{0}}(0,-2)*+{};(0,-7)*+{}};
{\ar@{=>}_{\alpha}(0,-13)*+{};(0,-18)*+{}};
\endxy
\xy
(0,-7.5)*+{=};
\endxy
\xy
(0,2)*+{I}="00"; (-10,-15)*+{X}="10"; (10,-15)*+{Y}="11";
{\ar@/_0.5pc/_{i_{X}} "00";"10"};
{\ar@/^0.5pc/^{i_{Y}} "00";"11"};
{\ar@/_1pc/_{g}"10";"11"};
{\ar@{=>}_{g_{0}}(0,-5)*+{};(0,-11)*+{}};
\endxy$$ These are the 2-cells of the F-category ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l}:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{s} \to {\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l}$ of monoids, strict and lax monoid morphisms which sits over C via a forgetful F-functor $U:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l} \to {\ensuremath{\mathcal{C}}\xspace}$. Likewise we have pseudo and colax monoid morphisms and forgetful F-functors $U:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ for each $w \in \{l,p,c\}$. For a simple statement we assume in the following result that C admits *pie limits* [@Power1991A-characterization]: these are a good class of limits containing $w$-limits of arrows for each $w$.
\[thm:monoids\] Let C be a monoidal 2-category admitting pie limits and suppose that $U_{s}:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{s} \to {\ensuremath{\mathcal{C}}\xspace}$ has a left 2-adjoint. Let $T$ be the induced 2-monad on ${\ensuremath{\mathcal{C}}\xspace}$. Then for each $w \in \{l,p,c\}$ we have isomorphisms of F-categories ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over C and these are natural in $w$.
Let us begin by showing that it suffices to suppose C to be strict monoidal. A straightforward extension of the usual argument for monoidal categories shows that C is equivalent to a strict monoidal 2-category D via a strong monoidal 2-equivalence $E:{\ensuremath{\mathcal{C}}\xspace} \to {\ensuremath{\mathcal{D}}\xspace}$. $$\xy
(0,0)*+{{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w}}="00"; (30,0)*+{{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{D}}\xspace})_{w}}="10";
(0,-10)*+{{\ensuremath{\mathcal{C}}\xspace}}="01"; (30,-10)*+{{\ensuremath{\mathcal{D}}\xspace}}="11";
{\ar^{E_{*}} "00";"10"};
{\ar ^{U_{D}}"10";"11"};
{\ar _{U_{C}}"00";"01"};
{\ar_{E} "01";"11"};
\endxy$$ Such an equivalence naturally lifts to a 2-equivalence $E_{*}:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w} \to {\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{D}}\xspace})_{w}$ for each $w$ and so induces a commuting square of F-categories and F-functors as above with both horizontal legs equivalences of F-categories. Now to apply Theorem \[thm:monadicity\] we must show that $U_{C}:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w} \to {\ensuremath{\mathcal{C}}\xspace}$ is $w$-doctrinal and that ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w}$ has $\overline{w}$-limits of loose morphisms. So suppose that $U_{D}$ and ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{D}}\xspace})_{w}$ have these properties and let us deduce from these the corresponding properties for C. Certainly if $U_{D}$ were $w$-doctrinal then $U_{C}$ would be too; for both horizontal legs, being equivalences, are $w$-doctrinal and such F-functors, being defined by lifting properties (as in Section 3.3), are closed under 2 out of 3. Likewise any limits existing in ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{D}}\xspace})_{w}$ exist in the F-equivalent ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w}$. Therefore it suffices to suppose that C is strict monoidal.\
Now certainly $U:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l} \to {\ensuremath{\mathcal{C}}\xspace}$ is locally faithful and reflects identity 2-cells. Moreover given a strict monoid map $(f,\overline{f},f_{0}):X \to Y$ and adjunction $(\epsilon, f \dashv g,\eta) \in {\ensuremath{\mathcal{C}}\xspace}$ taking mates gives 2-cells $$\xy
(00,0)*+{Y^{2}}="11";(30,0)*+{X^{2}}="31"; (00,-25)*+{Y}="12";(30,-25)*+{X}="32";
(00,-15)*+{Y^{2}}="a";(30,-10)*+{X}="b";
{\ar^{m_{X}} "31"; "b"};
{\ar_{g} "12"; "32"};
{\ar_{m_{Y}} "a"; "12"};
{\ar^{g^{2}} "11"; "31"};
{\ar_{1} "11"; "a"};
{\ar^{f^{2}} "31"; "a"};
{\ar_{f} "b"; "12"};
{\ar^{1} "b"; "32"};
{\ar@{=>}_{\epsilon^{2}}(14,-3)*+{};(8,-6)*+{}};
{\ar@{=>}_{\eta}(24,-19)*+{};(18,-22)*+{}};
\endxy
\hspace{1cm}
\xy
(00,-7.5)*+{I}="a";(30,-7.5)*+{X}="b";
(00,-22.5)*+{Y}="12";(30,-22.5)*+{X}="32";
{\ar^{i_{X}} "a"; "b"};
{\ar_{i_{Y}} "a"; "12"};
{\ar_{g} "12"; "32"};
{\ar_{f} "b"; "12"};
{\ar^{1} "b"; "32"};
{\ar@{=>}_{\eta}(24,-16)*+{};(18,-20)*+{}};
\endxy$$ It is straightforward to see, by cancelling mates, that these give $g$ the structure of a lax monoid map $(g,\overline{g},g_{0})$, with respect to which $(\epsilon, f \dashv (g,\overline{g},g_{0}),\eta)$ is an adjunction in ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l}$. Uniqueness of the lifted adjunction follows from Proposition \[prop:ids\].1. Therefore $U$ satisfies $l$-doctrinal adjunction.\
We will show that $U:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l} \to {\ensuremath{\mathcal{C}}\xspace}$ creates colax limits of loose morphisms. Consider a lax monoid map $(f,\overline{f},f_{0}):X \to Y$ and the colax limit $C$ of $f$ in C with colax cone as below $$\xy
(0,0)*+{C}="00"; (-10,-12)*+{X}="10"; (10,-12)*+{Y}="11";
{\ar_{p} "00";"10"};
{\ar^{q} "00";"11"};
{\ar_{f}"10";"11"};
{\ar@{=>}_{\lambda}(0,-4)*+{};(0,-9)*+{}};
\endxy$$ By the universal property of $C$ the composite colax cone left below induces a unique map $m_{C}:CC \to C$ such that $p{.}m_{C}=m_{X}{.}p^{2}$, $q{.}m_{C}=m_{Y}{.}q^{2}$ and such that the left equation below holds. Likewise we obtain a unique $i_{C}:I \to C$ such that $p{.}i_{C}=i_{X}$, $q{.}i_{C}=i_{Y}$ and satisfying $\lambda {.}i_{C}=f_{0}$ as on the right below. $$\xy
(10,10)*+{C^{2}}="-10";
(0,0)*+{X^{2}}="00"; (20,0)*+{Y^{2}}="10"; (0,-12)*+{X}="01"; (20,-12)*+{Y}="11";
{\ar_{p^{2}} "-10";"00"};
{\ar^{q^{2}} "-10";"10"};
{\ar_{f^{2}} "00";"10"};
{\ar@{=>}_{\lambda^{2}}(10,7)*+{};(10,2)*+{}};
{\ar_{f} "01";"11"};
{\ar_{m_{X}}"00";"01"};
{\ar^{m_{Y}}"10";"11"};
{\ar@{=>}_{\overline{f}}(10,-5)*+{};(10,-10)*+{}};
\endxy
\xy
(0,0)*+{=};
\endxy
\xy
(10,10)*+{C^{2}}="-10";
(0,0)*+{X^{2}}="00"; (20,0)*+{Y^{2}}="10";
(10,-2)*+{C}="-1A";
(0,-12)*+{X}="01"; (20,-12)*+{Y}="11";
{\ar_{p^{2}} "-10";"00"};
{\ar^{q^{2}} "-10";"10"};
{\ar_{p} "-1A";"01"};
{\ar^{q} "-1A";"11"};
{\ar|{m_{C}} "-10";"-1A"};
{\ar@{=>}_{\lambda}(10,-5)*+{};(10,-10)*+{}};
{\ar_{f} "01";"11"};
{\ar_{m_{X}}"00";"01"};
{\ar^{m_{Y}}"10";"11"};
\endxy
\hspace{0.5cm}
\xy
(10,10)*+{I}="-10";
(0,-12)*+{X}="01"; (20,-12)*+{Y}="11";
{\ar@/_0.5pc/_{i_{X}} "-10";"01"};
{\ar@/^0.5pc/^{i_{Y}} "-10";"11"};
{\ar@{=>}_{f_{0}}(10,-1)*+{};(10,-7)*+{}};
{\ar_{f} "01";"11"};
\endxy
\xy
(0,0)*+{=};
\endxy
\xy
(10,10)*+{I}="-10";
(10,-2)*+{C}="-1A";
(0,-12)*+{X}="01"; (20,-12)*+{Y}="11";
{\ar@/_1pc/_{i_{X}} "-10";"01"};
{\ar@/^1pc/^{i_{Y}} "-10";"11"};
{\ar_{p} "-1A";"01"};
{\ar^{q} "-1A";"11"};
{\ar|{i_{C}} "-10";"-1A"};
{\ar@{=>}_{\lambda}(10,-5)*+{};(10,-10)*+{}};
{\ar_{f} "01";"11"};
\endxy$$ If we can show $(C,m_{C},i_{C})$ to be a monoid then these combined equations will assert exactly that $p$ and $q$ are strict monoid maps and $\lambda:q \Rightarrow (f,\overline{f},f_{0}).p$ a monoid transformation. To show that $m_{C}{.}(m_{C}1) = m_{C}{.}(1m_{C}):C^{3} \to C$ amounts to showing that both paths coincide upon postcomposition with the components $p$, $q$ and $\lambda$ of the universal colax cone. We have that $p{.}m_{C}{.}(m_{C}1)=m_{X}{.}p^{2}{.}(m_{C}1)=m_{X}{.}(m_{X}1){.}p^{3}=m_{X}{.}(1m_{X}){.}p^{3}=m_{X}{.}p^{2}{.}(1m_{C})=p{.}m_{C}{.}(1m_{C})$ and similarly for $q$ so that associativity of $m_{C}$ will follow if we can show that both paths coincide upon postcomposition with $\lambda$. Consider the following series of equalities $$\xy
(10,30)*+{C^{3}}="00";
(10,15)*+{C^{2}}="11";
(10,0)*+{C}="12";
(0,-12)*+{X}="03"; (20,-12)*+{Y}="13";
{\ar|{1m_{C}} "00";"11"};
{\ar|{m_{C}} "11";"12"};
{\ar_{p} "12";"03"};
{\ar^{q} "12";"13"};
{\ar_{f} "03";"13"};
{\ar@{=>}_{\lambda}(10,-5)*+{};(10,-10)*+{}};
(25,22)*+{=};
\endxy
\xy
(12,30)*+{C^{3}}="00";
(0,15)*+{X^{3}}="01";(12,15)*+{C^{2}}="11";(24,15)*+{Y^{3}}="21";
(0,0)*+{X^{2}}="02"; (24,0)*+{Y^{2}}="12";
(0,-12)*+{X}="03"; (24,-12)*+{Y}="13";
{\ar_{ppp} "00";"01"};
{\ar^{qqq} "00";"21"};
{\ar|{1m_{C}} "00";"11"};
{\ar_{1m_{X}} "01";"02"};
{\ar^{1m_{Y}} "21";"12"};
{\ar_{pp} "11";"02"};
{\ar^{qq} "11";"12"};
{\ar_{ff} "02";"12"};
{\ar@{=>}_{\lambda\lambda}(12,9)*+{};(12,4)*+{}};
{\ar_{f} "03";"13"};
{\ar_{m_{X}}"02";"03"};
{\ar^{m_{Y}}"12";"13"};
{\ar@{=>}_{\overline{f}}(12,-5)*+{};(12,-10)*+{}};
(34,22)*+{=};
\endxy
\xy
(26,30)*+{C^{3}}="00";
(13,22)*+{XC^{2}}="01";(39,22)*+{CY^{2}}="11";
(0,12)*+{X^{3}}="02"; (26,18)*+{C^{2}}="12"; (52,12)*+{Y^{3}}="22";
(13,10)*+{XC}="03"; (39,10)*+{CY}="13";
(0,0)*+{X^{2}}="04"; (26,0)*+{XY}="14"; (52,0)*+{Y^{2}}="24";
(0,-12)*+{X}="05"; (52,-12)*+{Y}="15";
{\ar_{p11} "00";"01"};
{\ar_{1pp} "01";"02"};
{\ar|{p1} "12";"03"};
{\ar^{1qq} "00";"11"};
{\ar|{1q}"12";"13"};
{\ar^{q11}"11";"22"};
{\ar|{1m_{C}} "01";"03"};
{\ar_{1p} "03";"04"};
{\ar^{1q} "03";"14"};
{\ar|{1m_{Y}}"11";"13"};
{\ar_{p1}"13";"14"};
{\ar^{q1}"13";"24"};
{\ar@{=>}_{1\lambda}(13,7)*+{};(13,2)*+{}};
{\ar@{=>}_{\lambda1}(39,7)*+{};(39,2)*+{}};
{\ar_{1m_{X}} "02";"04"};
{\ar|{1m_{C}}"00";"12"};
{\ar^{1m_{Y}}"22";"24"};
{\ar_{1f} "04";"14"};
{\ar_{f1} "14";"24"};
{\ar_{f} "05";"15"};
{\ar_{m_{X}}"04";"05"};
{\ar^{m_{Y}}"24";"15"};
{\ar@{=>}_{\overline{f}}(26,-4)*+{};(26,-9)*+{}};
\endxy$$ $$\xy
(0,14)*+{=};
(26,18)*+{C^{3}}="00";
(13,10)*+{XC^{2}}="01";(39,10)*+{CY^{2}}="11";
(0,0)*+{X^{3}}="02"; (26,0)*+{XY^{2}}="12"; (52,0)*+{Y^{3}}="22";
(13,-2)*+{XC}="03"; (39,-2)*+{CY}="13";
(0,-12)*+{X^{2}}="04"; (26,-12)*+{XY}="14"; (52,-12)*+{Y^{2}}="24";
(0,-24)*+{X}="05"; (52,-24)*+{Y}="15";
{\ar_{p11} "00";"01"};
{\ar_{1pp} "01";"02"};
{\ar^{1qq} "01";"12"};
{\ar^{1qq} "00";"11"};
{\ar_{p11}"11";"12"};
{\ar^{q11}"11";"22"};
{\ar|{1m_{C}} "01";"03"};
{\ar_{1p} "03";"04"};
{\ar|{1q} "03";"14"};
{\ar|{1m_{Y}}"11";"13"};
{\ar|{p1}"13";"14"};
{\ar^{q1}"13";"24"};
{\ar@{=>}_{1\lambda}(13,-5)*+{};(13,-10)*+{}};
{\ar@{=>}_{\lambda1}(39,-5)*+{};(39,-10)*+{}};
{\ar_{1m_{X}} "02";"04"};
{\ar|{1m_{Y}}"12";"14"};
{\ar^{1m_{Y}}"22";"24"};
{\ar_{1f} "04";"14"};
{\ar_{f1} "14";"24"};
{\ar_{f} "05";"15"};
{\ar_{m_{X}}"04";"05"};
{\ar^{m_{Y}}"24";"15"};
{\ar@{=>}_{\overline{f}}(26,-16)*+{};(26,-21)*+{}};
(60,14)*+{=};
\endxy
\xy
(32,18)*+{C^{3}}="00";
(20,10)*+{XC^{2}}="01";(44,10)*+{CY^{2}}="11";
(0,0)*+{X^{3}}="02";(16,0)*+{X^{2}Y}="12"; (32,0)*+{XY^{2}}="22"; (55,0)*+{Y^{3}}="32";
(0,-12)*+{X^{2}}="03"; (32,-12)*+{XY}="13"; (55,-12)*+{Y^{2}}="23";
(0,-24)*+{X}="04"; (55,-24)*+{Y}="14";
{\ar_{p11} "00";"01"};
{\ar_{1pp} "01";"02"};
{\ar^{1qq} "01";"22"};
{\ar^{1qq} "00";"11"};
{\ar_{p11}"11";"22"};
{\ar^{q11}"11";"32"};
{\ar@{=>}_{1\lambda\lambda}(20,7)*+{};(20,2)*+{}};
{\ar@{=>}_{\lambda11}(45,7)*+{};(45,2)*+{}};
{\ar_{11f} "02";"12"};
{\ar_{1f1} "12";"22"};
{\ar_{f11} "22";"32"};
{\ar_{1m_{X}} "02";"03"};
{\ar^{1m_{Y}}"22";"13"};
{\ar^{1m_{Y}}"32";"23"};
{\ar@{=>}_{1\overline{f}}(20,-4)*+{};(20,-9)*+{}};
{\ar_{1f} "03";"13"};
{\ar_{f1} "13";"23"};
{\ar_{f} "04";"14"};
{\ar_{m_{X}}"03";"04"};
{\ar^{m_{Y}}"23";"14"};
{\ar@{=>}_{\overline{f}}(32,-16)*+{};(32,-21)*+{}};
\endxy$$ The first holds by definition of $m_{C}$, the second merely rewrites the tensor product $\lambda\lambda$ and the third rewrites the commuting central diamond. The final equality of pasting composites does two things at once: on its left side it again applies the definition of $m_{C}$ postcomposed with $\lambda$; we rewrite the right hand side using $(\lambda1).(1m_{Y}) = (1m_{Y}).(\lambda11)$. By an entirely similar diagram chase we obtain the equality $$\xy
(10,30)*+{C^{3}}="00";
(10,15)*+{C^{2}}="11";
(10,0)*+{C}="12";
(0,-12)*+{X}="03"; (20,-12)*+{Y}="13";
{\ar|{m_{C}1} "00";"11"};
{\ar|{m_{C}} "11";"12"};
{\ar_{p} "12";"03"};
{\ar^{q} "12";"13"};
{\ar_{f} "03";"13"};
{\ar@{=>}_{\lambda}(10,-5)*+{};(10,-10)*+{}};
(40,20)*+{=};
\endxy
\hspace{1cm}
\xy
(30,30)*+{C^{3}}="00";
(17,22)*+{X^{2}C}="01";(43,22)*+{C^{2}Y}="11";
(0,12)*+{X^{3}}="02";(26,12)*+{X^{2}Y}="12"; (43,12)*+{XY^{2}}="22"; (60,12)*+{Y^{3}}="32";
(0,0)*+{X^{2}}="03"; (26,0)*+{XY}="13"; (60,0)*+{Y^{2}}="23";
(0,-12)*+{X}="04"; (60,-12)*+{Y}="14";
{\ar_{pp1} "00";"01"};
{\ar_{11p} "01";"02"};
{\ar^{11q} "01";"12"};
{\ar^{11q} "00";"11"};
{\ar_{pp1}"11";"12"};
{\ar^{qq1}"11";"32"};
{\ar@{=>}_{11\lambda}(17,19)*+{};(17,14)*+{}};
{\ar@{=>}_{\lambda\lambda1}(45,19)*+{};(45,14)*+{}};
{\ar_{11f} "02";"12"};
{\ar_{1f1} "12";"22"};
{\ar_{f11} "22";"32"};
{\ar_{m_{X}1} "02";"03"};
{\ar_{m_{X}1}"12";"13"};
{\ar^{m_{Y}1}"32";"23"};
{\ar@{=>}_{\overline{f}1}(42,7)*+{};(42,2)*+{}};
{\ar_{1f} "03";"13"};
{\ar_{f1} "13";"23"};
{\ar_{f} "04";"14"};
{\ar_{m_{X}}"03";"04"};
{\ar^{m_{Y}}"23";"14"};
{\ar@{=>}_{\overline{f}}(30,-4)*+{};(30,-9)*+{}};
\endxy$$ This final composite and the one above it each constitute $\lambda^{3}$ sat atop either side of the first equation for a lax monoid morphism: as such they agree and $m_{C}$ is associative. Much smaller, though similar, diagram chases show that $m_{C}{.}(i_{C}1)=1$ and that $m_{C}{.}(1i_{C})=1$; thus $C$ is a monoid and the colax cone $(p,\lambda:q \Rightarrow pf,q)$ lifts (uniquely) to a colax cone in ${\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l}$.\
That the lifted colax cone satisfies the universal property of the colax limit is relatively straightforward and left to the reader. That $p$ and $q$ detect strict monoid morphisms is a consequence of the fact that they jointly detect identity 2-cells in C.\
Now if $U_{s}:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{s} \to {\ensuremath{\mathcal{C}}\xspace}$ has a left adjoint it is automatically strictly monadic by the enriched version of Beck’s monadicity theorem [@Dubuc1970Kan-extensions]. Therefore, using the above, Theorem \[thm:monadicity\] asserts that the isomorphism of 2-categories $E:{\ensuremath{\textnormal{Mon}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{s} \to {\ensuremath{\textnormal{T-Alg}_{\textnormal{s}}}\xspace}$ over C extends uniquely to an isomorphism of F-categories $E_{l}:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{l} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}_{\textnormal{l}}}\xspace}$ over C. In a similar way one verifies the conditions of Theorem \[thm:monadicity\] when $w \in \{p,c\}$ to obtain isomorphisms of F-categories $E_{w}:{\ensuremath{\mathbb{M}\textnormal{on}}\xspace}({\ensuremath{\mathcal{C}}\xspace})_{w} \to {\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{w}$ over C for each $w$; by Theorem \[thm:naturality\] these isomorphisms are natural in $w$.
A non-example
-------------
All of the examples we have seen are of the strictly monadic variety and indeed this is the case whenever one studies structured objects over some same base 2-category. Now Theorem \[thm:monadicity\] is general enough to cover ordinary monadicity – up to equivalence of F-categories – but in fact there exist situations of a weaker kind. Here is one such case. Let ${\ensuremath{\textnormal{Cat}}\xspace}_{f} \subset {\ensuremath{\textnormal{Cat}}\xspace}$ be a full sub 2-category of [$\textnormal{Cat}$]{}whose objects form a skeleton of the finitely presentable categories (the finitely presentable objects in [$\textnormal{Cat}$]{}) and let $[{\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]_{f} \subset [{\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]$ be the full sub 2-category consisting of those endo 2-functors preserving filtered colimits: this is the tight part of the F-category ${\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace})_{f}:[{\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]_{f} \to {\ensuremath{\textnormal{Ps}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace})_{f}$ whose loose morphisms are pseudonatural transformations. Likewise we have an F-category ${\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace}):[{\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace}] \to {\ensuremath{\textnormal{Ps}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace})$ and now restriction along the inclusion ${\ensuremath{\textnormal{Cat}}\xspace}_{f} \to {\ensuremath{\textnormal{Cat}}\xspace}$ induces a forgetful F-functor $R:{\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace})_{f} \to {\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace})$. Further restricting along the inclusion $ob{\ensuremath{\textnormal{Cat}}\xspace}_{f} \to {\ensuremath{\textnormal{Cat}}\xspace}_{f}$ gives a commuting triangle $$\xy
(0,0)*+{{\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace})_{f}}="00";
(35,0)*+{{\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace})}="20";
(17.5,-20)*+{[ob {\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace}]}="1-2";
{\ar^{R} "00"; "20"};
{\ar_{SR} "00"; "1-2"};
{\ar^{S} "20"; "1-2"};
\endxy$$ The composite $S_{\tau}R_{\tau}:[{\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]_{f} \to [ob{\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace}]$ is monadic though not strictly so: for the induced 2-monad $T$ we have ${\ensuremath{\textnormal{T-}\mathbb{A}\textnormal{lg}}\xspace}_{p}$ isomorphic to ${\ensuremath{\mathbb{P}\textnormal{s}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace})$ with $R_{\tau}:[{\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace}]_{f} \to [{\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace}]$ the Eilenberg-Moore comparison 2-functor – that this is a 2-equivalence follows from [$\textnormal{Cat}$]{}’s being locally finitely presentable as a 2-category. Whilst $R_{\tau}$ is a 2-equivalence the 2-functor $R_{\lambda}:{\ensuremath{\textnormal{Ps}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace})_{f} \to {\ensuremath{\textnormal{Ps}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace})$ is not: indeed ${\ensuremath{\textnormal{Ps}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace}_{f},{\ensuremath{\textnormal{Cat}}\xspace})$ is locally small whereas ${\ensuremath{\textnormal{Ps}}\xspace}({\ensuremath{\textnormal{Cat}}\xspace},{\ensuremath{\textnormal{Cat}}\xspace})_{f}$ is not. Yet $R_{\lambda}$ turns out to be a biequivalence and $R$ the uniquely induced F-functor to the F-category of algebras.
[^1]: Supported by the Grant agency of the Czech Republic under the grant P201/12/G028.
|
gr-qc/0603135\
UFIFT-QG-06-01
[**Gravitons Enhance Fermions during Inflation**]{}
S. P. Miao$^{\dagger}$ and R. P. Woodard$^{\ddagger}$
*[Department of Physics\
University of Florida\
Gainesville, FL 32611]{}*
ABSTRACT
We solve the effective Dirac equation for massless fermions during inflation in the simplest gauge, including all one loop corrections from quantum gravity. At late times the result for a spatial plane wave behaves as if the classical solution were subjected to a time dependent field strength renormalization of $Z_2(t) = 1 - \frac{17}{4 \pi} G H^2 \ln(a) + O(G^2)$. We show that this also follows from making the Hartree approximation, although the numerical coefficients differ.
PACS numbers: 04.30.Nk, 04.62.+v, 98.80.Cq, 98.80.Hw
$^{\dagger}$ e-mail: miao@phys.ufl.edu\
$^{\ddagger}$ e-mail: woodard@phys.ufl.edu
Introduction
============
Gravitons and massless, minimally coupled scalars can mediate vastly enhanced quantum effects during inflation because they are simultaneously massless and not conformally invariant [@RPW1]. One naturally wonders how interactions with these quanta affect themselves and other particles. The first step in answering this question on the linearized level is to compute the one particle irreducible (1PI) 2-point function for the field whose behavior is in question. This has been done at one loop order for gravitons in pure quantum gravity [@TW1], for photons [@PTW1; @PTW2] and charged scalars [@KW] in scalar quantum electrodynamics (SQED), for fermions [@PW1; @GP] and Yukawa scalars [@DW] in Yukawa theory, for fermions in Dirac + Einstein [@MW1] and, at two loop order, for scalars in $\phi^4$ theory [@BOW]. The next step is using the 1PI 2-point function to correct the linearized equation of motion for the field in question. That is what we shall do here for the fermions of massless Dirac + Einstein.
It is worth reviewing the conventions used in computing the fermion self-energy [@MW1]. We worked on de Sitter background in conformal coordinates, $$ds^2 = a^2(\eta) \Bigl(-d\eta^2 + d\vec{x} \!\cdot\! d\vec{x}\Bigr) \qquad
{\rm where} \qquad a(\eta) = -\frac1{H \eta} = e^{H t} \; .$$ We used dimensional regularization and obtained the self-energy for the conformally re-scaled fermion field, $$\Psi(x) \equiv a^{(\frac{D-1}2)} \psi(x) \; .$$ The local Lorentz gauge was fixed to allow an algebraic expression for the vierbein in terms of the metric [@RPW2]. The general coordinate gauge was fixed to make the tensor structure of the graviton propagator decouple from its spacetime dependence [@TW2; @RPW3]. The result we obtained is, $$\begin{aligned}
\lefteqn{\Bigl[\Sigma^{\rm ren}\Bigr](x;x') \!=\!\frac{i \kappa^2
H^2}{2^6 \pi^2} \Biggl\{\frac{\ln(a a')}{H^2 a a'} \hspace{-.1cm}
\not{\hspace{-.1cm}
\partial} \partial^2 \!+\! \frac{15}2 \ln(a a') \hspace{-.1cm} \not{\hspace{
-.1cm} \partial} \!-\! 7 \ln(a a') \; \hspace{-.1cm}
\overline{\not{\hspace{
-.1cm} \partial}} \Biggr\} \delta^4(x \!-\! x') } \nonumber \\
& & \hspace{.2cm} + \frac{\kappa^2}{2^8 \pi^4} (a a')^{-1}
\hspace{-.1cm} \not{\hspace{-.1cm} \partial} \partial^4 \Bigl[
\frac{\ln(\mu^2 \Delta x^2)}{ \Delta x^2} \Bigr] + \frac{\kappa^2
H^2}{2^8 \pi^4} \Biggl\{\Bigl(\frac{15}2 \hspace{-.1cm}
\not{\hspace{-.1cm} \partial} \, \partial^2 - \hspace{-.1cm}
\overline{\not{\hspace{-.1cm} \partial}} \, \partial^2 \Bigr)
\Bigl[ \frac{\ln(\mu^2 \Delta x^2)}{\Delta x^2} \Bigr] \nonumber \\
& & \hspace{1.5cm} + \Bigl(-8 \; \hspace{-.1cm}
\overline{\not{\hspace{-.1cm}
\partial}} \partial^2 \!+\! 4 \hspace{-.1cm} \not{\hspace{-.1cm} \partial}
\nabla^2 \Bigr) \Bigl[ \frac{\ln(\frac14 H^2\Delta x^2)}{\Delta x^2}
\Bigr] \!+\! 7 \hspace{-.1cm} \not{\hspace{-.1cm} \partial} \,
\nabla^2 \Bigl[ \frac1{\Delta x^2} \Bigr]\!\Biggr\} + O(\kappa^4) \; ,
\qquad \label{ren}\end{aligned}$$ where $\kappa^2 \equiv 16 \pi G$ is the loop counting parameter of quantum gravity. The various differential and spinor-differential operators are, $$\partial^2 \equiv \eta^{\mu\nu} \partial_{\mu} \partial_{\nu} \;\; , \;\;
\nabla^2 \equiv \partial_i \partial_i \;\; , \;\; \hspace{-.1cm}
\not{\hspace{-.1cm} \partial} \equiv \gamma^{\mu} \partial_{\mu} \;\; {\rm and}
\;\; \hspace{-.1cm} \overline{\not{\hspace{-.1cm} \partial}} \, \equiv \gamma^i
\partial_i \; ,$$ where $\eta^{\mu\nu}$ is the Lorentz metric and $\gamma^{\mu}$ are the gamma matrices. The conformal coordinate interval is basically $\Delta x^2 \equiv
(x\!-\!x')^{\mu} (x\!-\!x')^{\nu} \eta_{\mu\nu}$, up to a subtlety about the imaginary part which will be explained shortly.
The linearized, effective Dirac equation we will solve is, $$i\hspace{-.1cm}\not{\hspace{-.08cm} \partial}_{ij} \Psi_{j}(x) -
\int d^4x' \, \Bigl[\mbox{}_i \Sigma_j \Bigr](x;x') \, \Psi_{j}(x')
= 0 \; . \label{Diraceqn}$$ In judging the validity of this exercise it is important to answer five questions:
1. [What is the relation between the $\comp$-number, effective field equation (\[Diraceqn\]) and the Heisenberg operator equations of Dirac + Einstein?]{}
2. [How do solutions to (\[Diraceqn\]) change when different gauges are used?]{}
3. [How do solutions to (\[Diraceqn\]) depend upon the finite parts of counterterms?]{}
4. [What is the imaginary part of $\Delta x^2$? and]{}
5. [What can we do without the higher loop contributions to the fermion self-energy?]{}
Issues 1 and 2 are closely related, and require a lengthy digression that we have consigned to section 2 of this paper. In this Introduction we will comment on issues 3-5.
Dirac + Einstein is not perturbatively renormalizable [@DVN], so we could only obtain a finite result by absorbing divergences in the BPHZ sense [@BP; @H; @Z1; @Z2] using three higher derivative counterterms, $$-\kappa^2 H^2 \Bigl\{\frac{\alpha_1}{H^2 a a'} \hspace{-.1cm}
\not{\hspace{-.1cm} \partial} \partial^2 + \alpha_2 D (D\!-\!1) \hspace{-.1cm}
\not{\hspace{-.1cm} \partial} + \alpha_3 \; \hspace{-.1cm} \overline{\not{
\hspace{-.1cm} \partial}} \Bigr\} \delta^D(x\!-\!x') \; . \label{genctm}$$ No physical principle seems to fix the finite parts of these counterterms so any result which derives from their values is arbitrary. We chose to null local terms at the beginning of inflation ($a = 1$), but any other choice could have been made and would have affected the solution to (\[Diraceqn\]). Hence there is no point in solving the equation exactly. However, each of the three counterterms is related to a term in (\[ren\]) which carries a factor of $\ln(a a')$, $$\begin{aligned}
\frac{\alpha_1}{H^2 a a'} \hspace{-.1cm} \not{\hspace{-.1cm} \partial}
\partial^2 & \Longleftrightarrow & \frac{\ln(a a')}{H^2 a a'} \hspace{-.1cm}
\not{\hspace{-.1cm} \partial} \partial^2 \; , \label{1stlog} \\
\alpha_2 D (D\!-\!1) \hspace{-.1cm} \not{\hspace{-.1cm} \partial}
& \Longleftrightarrow & \frac{15}2 \ln(a a') \hspace{-.1cm} \not{\hspace{-.1cm}
\partial} \; , \label{2ndlog} \\
\alpha_3 \; \hspace{-.1cm} \overline{\not{\hspace{-.1cm} \partial}}
& \Longleftrightarrow & -7 \ln(a a') \; \hspace{-.1cm} \overline{\not{\hspace{-
.1cm} \partial}} \; . \label{3rdlog}\end{aligned}$$ Unlike the $\alpha_i$’s, the numerical coefficients of the right hand terms are uniquely fixed and completely independent of renormalization. The factors of $\ln(a a')$ on these right hand terms mean that they dominate over any finite change in the $\alpha_i$’s at late times. It is in this late time regime that we can make reliable predictions about the effect of quantum gravitational corrections.
The analysis we have just made is a standard feature of low energy effective field theory, and has many distinguished antecedents [@BN; @SW; @FS; @HS; @CDH; @CD; @DMC1; @DL; @JFD1; @JFD2; @MV; @HL; @ABS; @KK1; @KK2]. Loops of massless particles make finite, nonanalytic contributions which cannot be changed by counterterms and which dominate the far infrared. Further, these effects must occur as well, with precisely the same numerical values, in whatever fundamental theory ultimately resolves the ultraviolet problems of quantum gravity.
We must also clarify what is meant by the conformal coordinate interval $\Delta x^2(x;x')$ which appears in (\[ren\]). The in-out effective field equations correspond to the replacement, $$\Delta x^2(x;x') \longrightarrow \Delta x^2_{\scriptscriptstyle ++}(x;x')
\equiv \Vert \vec{x} - \vec{x}' \Vert^2 - (\mid\e-\e'\mid-i\d)^2 \; .
\label{D++}$$ These equations govern the evolution of quantum fields under the assumption that the universe begins in free vacuum at asymptotically early times and ends up the same way at asymptotically late times. This is valid for scattering in flat space but not for cosmological settings in which particle production prevents the in vacuum from evolving to the out vacuum. Persisting with the in-out effective field equations would result in quantum correction terms which are dominated by events from the infinite future! This is the correct answer to the question being asked, which is, “what must the field be in order to make the universe to evolve from in vacuum to out vacuum?” However, that question is not very relevant to any observation we can make.
A more realistic question is, “what happens when the universe is released from a prepared state at some finite time and allowed to evolve as it will?” This sort of question can be answered using the Schwinger-Keldysh formalism [@JS; @KTM; @BM; @LVK; @CSHY; @RDJ; @CH]. For a recent derivation in the position-space formalism we are using, see [@FW]. We confine ourselves here to noting four simple rules:
- [The endpoints of lines in the Schwinger-Keldysh formalism carry a $\pm$ polarity, so every n-point 1PI function of the in-out formalism gives rise to $2^n$ 1PI functions in the Schwinger-Keldysh formalism;]{}
- [The linearized effective Dirac equation of the Schwinger-Keldysh formalism takes the form (\[Diraceqn\]) with the replacement, $$\Bigl[\mbox{}_i \Sigma_j\Bigr](x;x') \longrightarrow
\Bigl[\mbox{}_i \Sigma_j\Bigr]_{\scriptscriptstyle ++}\!\!\!\!(x;x') +
\Bigl[\mbox{}_i \Sigma_j\Bigr]_{\scriptscriptstyle +-}\!\!\!\!(x;x') \; ;$$]{}
- [The ${\scriptscriptstyle ++}$ fermion self-energy is (\[ren\]) with the replacement (\[D++\]); and]{}
- [The ${\scriptscriptstyle +-}$ fermion self-energy is, $$\begin{aligned}
\lefteqn{
- \frac{\kappa^2}{2^8 \pi^4 a a'} \hspace{-.1cm} \not{\hspace{-.1cm} \partial}
\partial^4 \Bigl[\frac{\ln(\mu^2 \Delta x^2)}{ \Delta x^2} \Bigr] -
\frac{\kappa^2 H^2}{2^8 \pi^4} \Biggl\{\Bigl(\frac{15}2 \hspace{-.1cm}
\not{\hspace{-.1cm} \partial} \, \partial^2 - \hspace{-.1cm}
\overline{\not{\hspace{-.1cm} \partial}} \, \partial^2 \Bigr)
\Bigl[ \frac{\ln(\mu^2 \Delta x^2)}{\Delta x^2} \Bigr]} \nonumber \\
& & \hspace{.5cm} + \Bigl(-8 \; \hspace{-.1cm} \overline{\not{\hspace{-.1cm}
\partial}} \partial^2 \!+\! 4 \hspace{-.1cm} \not{\hspace{-.1cm} \partial}
\nabla^2 \Bigr) \Bigl[ \frac{\ln(\frac14 H^2\Delta x^2)}{\Delta x^2}
\Bigr] \!+\! 7 \hspace{-.1cm} \not{\hspace{-.1cm} \partial} \, \nabla^2
\Bigl[ \frac1{\Delta x^2} \Bigr]\!\Biggr\} + O(\kappa^4) \; , \qquad\end{aligned}$$ with the replacement, $$\Delta x^2(x;x') \longrightarrow \Delta x^2_{\scriptscriptstyle +-}(x;x')
\equiv \Vert \vec{x} - \vec{x}' \Vert^2 - (\e-\e' + i\d)^2 \; . \label{D+-}$$]{}
The difference of the ${\scriptscriptstyle ++}$ and ${\scriptscriptstyle +-}$ terms leads to zero contribution in (\[Diraceqn\]) unless the point $x^{\prime \mu}$ lies on or within the past light-cone of $x^{\mu}$.
We can only solve for the one loop corrections to the field because we lack the higher loop contributions to the self-energy. The general perturbative expansion takes the form, $$\Psi(x) = \sum_{\ell = 0}^{\infty} \kappa^{2\ell} \Psi^{\ell}(x)
\qquad {\rm and}\,\,\, \Bigl[\Sigma\Bigr](x;x') =
\sum_{\ell=1}^{\infty} \kappa^{2\ell}
\Bigl[\Sigma^{\ell}\Bigr](x;x') \; .$$ One substitutes these expansions into the effective Dirac equation (\[Diraceqn\]) and then segregates powers of $\kappa^2$, $$i\hspace{-.1cm}\not{\hspace{-.08cm} \partial} \Psi^0(x) = 0 \qquad , \qquad
i\hspace{-.1cm}\not{\hspace{-.08cm} \partial} \Psi^1(x) = \int d^4x'
\Bigl[\Sigma^1\Bigr](x;x') \Psi^0(x') \qquad {\rm et\ cetera.}$$ We shall work out the late time limit of the one loop correction $\Psi^1_i(\eta,\vec{x};\vec{k},s)$ for a spatial plane wave of helicity $s$, $$\Psi^0_i(\eta,\vec{x};\vec{k},s) = \frac{e^{-i k \eta}}{\sqrt{2k}}
u_i(\vec{k},s) e^{i \vec{k} \cdot \vec{x}} \qquad {\rm where} \qquad k^{\ell}
\gamma^{\ell}_{ij} u_j(\vec{k},s) = k \gamma^0_{ij} u_j(\vec{k},s)
\; .\label{freefun}$$
In the next section we derive the effective field equation. In section 3 we derive some key simplifications. In section 4 we solve for the late time limit of $\Psi^1_i(\eta,\vec{x};\vec{k},s)$. The result takes the surprising form of a time dependent field strength renormalization of the tree order solution. In section 5 we show that this can be understood qualitatively using mean field theory. Our results are summarized and discussed in section 6.
The Effective Field Equations
=============================
The purpose of this section is to elucidate the relation between the Heisenberg operators of Dirac + Einstein — $\overline{\psi}_i(x)$, $\psi_i(x)$ and $h_{\mu\nu}(x)$ — and the $\comp$-number plane wave mode solutions $\Psi_i(x;\vec{k},s)$ of the linearized, effective Dirac equation (\[Diraceqn\]). After explaining the relation we work out an example, at one loop order, in a simple scalar analogue model. Finally, we return to Dirac + Einstein to explain how $\Psi_i(x;\vec{k},s)$ changes with variations of the gauge.
Heisenberg operators and effective field equations
--------------------------------------------------
The invariant Lagrangian of Dirac + Einstein in $D$ spacetime dimensions is, $$\mathcal{L} = \frac1{16 \pi G} \Bigl(R \!-\! (D\!-\!1) (D\!-\!2) H^2\Bigr)
\sqrt{-g} \!+\! \overline{\psi} e^{\mu}_{~b} \gamma^b \Bigl(i\partial_{\mu}
\!-\! \frac12 A_{\mu cd} J^{cd}\Bigr) \psi \sqrt{-g} \; .$$ Here $e_{\mu b}$ is the vierbein field and $g_{\mu\nu} \equiv e_{\mu b}
e_{\nu c} \eta^{bc}$ is the metric. The metric and vierbein-compatible connections are, $$\Gamma^{\rho}_{~\mu\nu} \equiv \frac12 g^{\rho\sigma} \Bigl(g_{\sigma \mu ,
\nu} + g_{\nu \sigma , \mu} - g_{\mu \nu , \sigma}\Bigr) \qquad {\rm and}
\qquad A_{\mu cd} \equiv e^{\nu}_{~c} \Bigl( e_{\nu d , \mu} -
\Gamma^{\rho}_{~\mu\nu} e_{\rho d}\Bigr) \; .$$ The Ricci scalar is, $$R \equiv g^{\mu\nu} \Bigl( \Gamma^{\rho}_{~\nu\mu , \rho} - \Gamma^{\rho}_{
~\rho \mu , \nu} + \Gamma^{\rho}_{~\rho \sigma} \Gamma^{\sigma}_{~\nu\mu}
- \Gamma^{\rho}_{~\nu \sigma} \Gamma^{\sigma}_{~\rho \mu} \Bigr) \; .$$ The gamma matrices $\gamma^b_{ij}$ have spinor indices $i, j \in \{1,2,3,4\}$ and obey the usual anti-commutation relations, $$\{\gamma^b , \gamma^c\} = -2 \eta^{bc} I \; .$$ The Lorentz generators of the bispinor representation are, $$J^{bc} \equiv \frac{i}4 [\gamma^b ,\gamma^c] \; .$$
We employ the Lorentz symmetric gauge, $e_{\mu b} = e_{b \mu}$, which permits one to perturbatively determine the vierbein in terms of the metric and their respective backgrounds (denoted with overlines) [@RPW2], $$e_{\mu b}[g] = \Bigl(\sqrt{g {\overline{g}}_0^{-1}} \Bigr)_{\mu}^{~\nu}
\; \overline{e}_{\nu b} \; .$$ We define the graviton field $h_{\mu\nu}$ in de Sitter conformal coordinates as follows, $$g_{\mu\nu}(x) \equiv a^2 \Bigl( \eta_{\mu\nu} + \kappa h_{\mu\nu}(x)\Bigr)
\qquad {\rm where} \qquad a = -\frac1{H \eta} \; .$$ By convention the indices of $h_{\mu\nu}$ are raised and lowered with the Lorentz metric. We fix the general coordinate freedom by adding the gauge fixing term, $$\mathcal{L}_{\scriptscriptstyle {\rm GF}} = -\frac12 a^{D-2} \eta^{\mu\nu}
F_{\mu} F_{\nu} \quad {\rm where} \quad F_{\mu} = \eta^{\rho\sigma} \Bigl(
h_{\mu\rho , \sigma} \!-\! \frac12 h_{\rho \sigma , \mu} \!+\! (D\!-\!2) H a
h_{\mu \rho} \delta^0_{\sigma}\Bigr) .$$
One solves the gauge-fixed Heisenberg operator equations perturbatively, $$\begin{aligned}
h_{\mu\nu}(x) & = & h^0_{\mu\nu}(x) + \kappa h^1_{\mu\nu}(x) + \kappa^2
h^2_{\mu\nu}(x) + \ldots \; , \qquad \\
\psi_i(x) & = & \psi^0_i(x) + \kappa \psi^1_i(x) + \kappa^2 \psi^2_i(x) +
\ldots \; .\end{aligned}$$ Because our state is released in free vacuum at $t=0$ ($\eta = -1/H$), it makes sense to express the operator as a functional of the creation and annihilation operators of this free state. So our initial conditions are that $h_{\mu\nu}$ and its first time derivative coincide with those of $h^0_{\mu\nu}(x)$ at $t=0$, and also that $\psi_i(x)$ coincides with $\psi^0_i(x)$. The zeroth order solutions to the Heisenberg field equations take the form, $$\begin{aligned}
h^0_{\mu\nu}(x) & = & \int \frac{d^{D-1}k}{(2\pi)^{D-1}} \sum_{\lambda}
\Bigl\{ \epsilon_{\mu\nu}(\eta;\vec{k},\lambda) e^{i \vec{k} \cdot \vec{x}}
\alpha(\vec{k},\lambda) \nonumber \\
& & \hspace{5cm} + \epsilon^*_{\mu\nu}(\eta;\vec{k},\lambda)
e^{-i \vec{k} \cdot \vec{x}} \alpha^{\dagger}(\vec{k},\lambda) \Bigr\} \; ,
\qquad \\
\psi^0_i(x) & = & a^{-(\frac{D-1}2)} \int \frac{d^{D-1}k}{(2\pi)^{D-1}}
\sum_s \Bigl\{ \frac{e^{-ik \eta}}{\sqrt{2 k}} u_i(\vec{k},s) e^{i \vec{k}
\cdot \vec{x}} b(\vec{k},s) \nonumber \\
& & \hspace{5cm} + \frac{e^{ik \eta}}{\sqrt{2 k}} v_i(\vec{k},\lambda)
e^{-i \vec{k} \cdot \vec{x}} c^{\dagger}(\vec{k},s) \Bigr\} \; . \qquad\end{aligned}$$ The graviton mode functions are proportional to Hankel functions whose precise specification we do not require. The Dirac mode functions $u_i(\vec{k},
s)$ and $v_i(\vec{k},s)$ are precisely those of flat space by virtue of the conformal invariance of massless fermions. The canonically normalized creation and annihilation operators obey, $$\begin{aligned}
\Bigl[\alpha(\vec{k},\lambda), \alpha^{\dagger}(\vec{k}',\lambda')\Bigr]
& = & \delta_{\lambda \lambda'} (2\pi)^{D-1} \delta^{D-1}(\vec{k} \!-\!
\vec{k}') \label{alpop} \; , \\
\Bigl\{b(\vec{k},s), b^{\dagger}(\vec{k}',s')\Bigr\} & = & \delta_{s s'}
(2\pi)^{D-1} \delta^{D-1}(\vec{k} \!-\! \vec{k}') =
\Bigl\{c(\vec{k},s), c^{\dagger}(\vec{k}',s')\Bigr\} \; . \qquad \label{bcop}\end{aligned}$$
The zeroth order Fermi field $\psi^0_i(x)$ is an anti-commuting operator whereas the mode function $\Psi^0(x;\vec{k},s)$ is a $\comp$-number. The latter can be obtained from the former by anti-commuting with the fermion creation operator, $$\Psi^0_i(x;\vec{k},s) = a^{\frac{D-1}2} \Bigl\{\psi^0_i(x),
b^{\dagger}(\vec{k},s)\Bigr\} = \frac{e^{-i k \eta}}{\sqrt{2k}} u_i(\vec{k},s)
e^{i \vec{k} \cdot \vec{x}} \; .$$ The higher order contributions to $\psi_i(x)$ are no longer linear in the creation and annihilation operators, so anti-commuting the full solution $\psi_i(x)$ with $b^{\dagger}(\vec{k},s)$ produces an operator. The quantum-corrected fermion mode function we obtain by solving (\[Diraceqn\]) is the expectation value of this operator in the presence of the state which is free vacuum at $t=0$, $$\Psi_i(x;\vec{k},s) = a^{\frac{D-1}2} \Bigl\langle \Omega \Bigl\vert
\Bigl\{ \psi_i(x), b^{\dagger}(\vec{k},s) \Bigr\} \Bigr\vert \Omega
\Bigr\rangle \; . \label{SKop}$$ This is what the Schwinger-Keldysh field equations give. The more familiar, in-out effective field equations obey a similar relation except that one defines the free fields to agree with the full ones in the asymptotic past, and one takes the in-out matrix element after anti-commuting.
A worked-out example
--------------------
It is perhaps worth seeing a worked-out example, at one loop order, of the relation (\[SKop\]) between the Heisenberg operators and the Schwinger-Keldysh field equations. To simplify the analysis we will work with a model of two scalars in flat space, $$\mathcal{L} = - \partial_{\mu} \varphi^* \partial^{\mu} \varphi -
m^2 \varphi^* \varphi - \lambda \chi \!:\! \varphi^* \varphi \!:\! -
\frac12 \partial_{\mu} \chi \partial^{\mu} \chi \; . \label{2sL}$$ In this model $\varphi$ plays the role of our fermion $\psi_i$, and $\chi$ plays the role of the graviton $h_{\mu\nu}$. Note that we have normal-ordered the interaction term to avoid the harmless but time-consuming digression that would be required to deal with $\chi$ developing a nonzero expectation value. We shall also omit discussion of counterterms.
The Heisenberg field equations for (\[2sL\]) are, $$\begin{aligned}
\partial^2 \chi - \lambda \!:\! \varphi^* \varphi \!:\! & = & 0 \; , \\
(\partial^2 - m^2) \varphi - \lambda \chi \varphi & = & 0 \; .\end{aligned}$$ As with Dirac + Einstein, we solve these equations perturbatively, $$\begin{aligned}
\chi(x) & = & \chi^0(x) + \lambda \chi^1(x) + \lambda^2 \chi^2(x) +
\ldots \; , \\
\varphi(x) & = & \varphi^0(x) + \lambda \varphi^1(x) + \lambda^2 \varphi^2(x)
+ \ldots \; .\end{aligned}$$ The zeroth order solutions are, $$\begin{aligned}
\chi^0(x) & = & \int \frac{d^{D-1}k}{(2\pi)^{D-1}} \Bigl\{ \frac{e^{-ikt}}{
\sqrt{2k}} e^{i \vec{k} \cdot \vec{x}} \alpha(\vec{k}) + \frac{e^{ikt}}{
\sqrt{2k}} e^{-i \vec{k} \cdot \vec{x}} \alpha^{\dagger}(\vec{k}) \Bigr\}
\; , \\
\varphi^0(x) & = & \int \frac{d^{D-1}k}{(2\pi)^{D-1}} \Bigl\{ \frac{e^{-i
\omega t}}{\sqrt{2 \omega}} e^{i \vec{k} \cdot \vec{x}} b(\vec{k}) +
\frac{e^{i \omega t}}{\sqrt{2 \omega}} e^{-i \vec{k} \cdot \vec{x}}
c^{\dagger}(\vec{k}) \Bigr\} \; .\end{aligned}$$ Here $k \equiv \Vert \vec{k} \Vert$ and $\omega \equiv \sqrt{k^2 + m^2}$. The creation and annihilation operators are canonically normalized, $$\Bigl[\alpha(\vec{k}),\alpha^{\dagger}(\vec{k}')\Bigr] =
\Bigl[b(\vec{k}),b^{\dagger}(\vec{k}')\Bigr] =
\Bigl[c(\vec{k}),c^{\dagger}(\vec{k}')\Bigr] = (2\pi)^{D-1} \delta^{D-1}(
\vec{k} - \vec{k}') \; .$$ We choose to develop perturbation theory so that all the operators and their first time derivatives agree with the zeroth order solutions at $t=0$. The first few higher order terms are, $$\begin{aligned}
\chi^1(x) & \!\!\!\!\! = \!\!\!\!\! & \int_0^t \!\! dt' \!\! \int d^{D-1}x' \,
\Bigl\langle x \Bigl\vert \frac1{\partial^2} \Bigr\vert x'
\Bigr\rangle_{\rm ret} \!:\! \varphi^{0*}(x') \varphi^0(x') \!:\! \; , \\
\varphi^1(x) & \!\!\!\!\! = \!\!\!\!\! & \int_0^t \!\! dt' \!\! \int d^{D-1}x'
\, \Bigl\langle x \Bigl\vert \frac1{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\rm ret} \chi^0(x') \varphi^0(x') \; , \\
\varphi^2(x) & \!\!\!\!\! = \!\!\!\!\! & \int_0^t \!\! dt' \!\! \int d^{D-1}x'
\Bigl\langle x \Bigl\vert \frac1{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\rm ret} \Bigl\{\chi^1(x') \varphi^0(x') \!+\! \chi^0(x')
\varphi^1(x') \Bigr\} . \qquad\end{aligned}$$
The commutator of $\varphi^0(x)$ with $b^{\dagger}(\vec{k})$ is a $\comp$-number, $$\Bigl[ \varphi^0(x) , b^{\dagger}(\vec{k}) \Bigr] = \frac{e^{-i \omega t}}{
\sqrt{2 \omega}} \, e^{i \vec{k} \cdot \vec{x}} \equiv \Phi^0(x;\vec{k}) \; .
\label{Phi^0}$$ However, commuting the full solution with $b^{\dagger}(\vec{k})$ leaves operators, $$\begin{aligned}
\lefteqn{\Bigl[ \varphi(x) , b^{\dagger}(\vec{k}) \Bigr] = \Phi^0(x;\vec{k})
+ \lambda \int_0^t \!\! dt' \!\! \int d^{D-1}x' \,
\Bigl\langle x \Bigl\vert \frac1{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\rm ret} \chi^0(x') \Phi^0(x';\vec{k}) } \nonumber \\
& & \hspace{-.5cm} + \lambda^2 \int_0^t \!\! dt' \!\! \int d^{D-1}x' \,
\Bigl\langle x \Bigl\vert \frac1{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\rm ret} \Biggl\{ \Bigl[\chi^1(x') , b^{\dagger}(\vec{k})\Bigr]
\varphi^0(x') + \chi^1(x') \Phi^0(x';\vec{k}) \nonumber \\
& & \hspace{6cm} + \chi^0(x') \Bigl[ \varphi^1(x') , b^{\dagger}(\vec{k})
\Bigr] \Biggr\} + O(\lambda^3) \; . \qquad \label{com} \end{aligned}$$ The commutators in (\[com\]) are easily evaluated, $$\begin{aligned}
\lefteqn{\Bigl[\chi^1(x') , b^{\dagger}(\vec{k})\Bigr] \varphi^0(x') }
\nonumber \\
& & \hspace{1.5cm} = \int_0^{t'} \!\! dt'' \!\! \int d^{D-1}x'' \, \Bigl\langle
x' \Bigl\vert \frac1{\partial^2} \Bigr\vert x'' \Bigr\rangle_{\rm ret}
\varphi^{0*}(x'') \varphi^0(x') \Phi^0(x'';\vec{k}) \; , \qquad \\
\lefteqn{\chi^0(x') \Bigl[\varphi^1(x') , b^{\dagger}(\vec{k}) \Bigr] }
\nonumber \\
& & \hspace{1.5cm} = \int_0^{t'} \!\! dt''\!\!\int d^{D-1}x'' \, \Bigl\langle x'
\Bigl\vert \frac1{\partial^2 \!-\! m^2} \Bigr\vert x'' \Bigr\rangle_{\rm ret}
\chi^0(x') \chi^0(x'') \Phi^0(x'';\vec{k}) \; . \qquad\end{aligned}$$ Hence the expectation value of (\[com\]) gives, $$\begin{aligned}
\lefteqn{\Bigl\langle \Omega \Bigl\vert \Bigl[ \varphi(x), b^{\dagger}(\vec{k})
\Bigr] \Bigr\vert \Omega \Bigr\rangle = \Phi^0(x;\vec{k}) + \lambda^2 \int_0^t
\!\! dt' \!\! \int d^{D-1}x' \, \Bigl\langle x \Bigl\vert \frac1{\partial^2
\!-\! m^2} \Bigr\vert x' \Bigr\rangle_{\rm ret} } \nonumber \\
& & \times \int_0^{t'} \!\! dt'' \!\! \int d^{D-1}x'' \, \Biggl\{ \Bigl\langle
x' \Bigl\vert \frac1{\partial^2} \Bigr\vert x'' \Bigr\rangle_{\rm ret}
\Bigl\langle \Omega \Bigl\vert \varphi^{0*}(x'') \varphi^0(x') \Bigr\vert
\Omega \Bigr\rangle \nonumber \\
& & \hspace{1.3 cm} + \Bigl\langle x' \Bigl\vert \frac1{\partial^2 \!-\! m^2}
\Bigr\vert x'' \Bigr\rangle_{\rm ret} \Bigl\langle \Omega \Bigl\vert
\chi^0(x') \chi^0(x'') \Bigr\vert \Omega \Bigr\rangle \Biggr\} \Phi^0(x'';
\vec{k}) + O(\lambda^4) \; . \qquad \label{expcom}\end{aligned}$$
To make contact with the effective field equations we must first recognize that the retarded Green’s functions can be written in terms of expectation values of the free fields, $$\begin{aligned}
\lefteqn{\Bigl\langle x' \Bigl\vert \frac1{\partial^2} \Bigr\vert x''
\Bigr\rangle_{\rm ret} = -i \theta(t' \!-\! t'') \Bigl[ \chi^0(x') ,
\chi^0(x'')\Bigr] } \\
& & \hspace{1cm} = -i \theta(t' \!-\! t'') \Biggl\{ \Bigl\langle \Omega
\Bigl\vert \chi^0(x') \chi^0(x'') \Bigr\vert \Omega \Bigr\rangle - \Bigl\langle
\Omega \Bigl\vert \chi^0(x'') \chi^0(x') \Bigr\vert \Omega \Bigr\rangle
\Biggr\} \; , \qquad \\
\lefteqn{\Bigl\langle x' \Bigl\vert \frac1{\partial^2 \!-\! m^2} \Bigr\vert x''
\Bigr\rangle_{\rm ret} = -i \theta(t' \!-\! t'') \Bigl[ \varphi^0(x') ,
\varphi^{0*}(x'')\Bigr] } \\
& & \hspace{1cm} = -i \theta(t' \!-\! t'') \Biggl\{ \Bigl\langle \Omega
\Bigl\vert \varphi^0(x') \varphi^{0*}(x'') \Bigr\vert \Omega \Bigr\rangle -
\Bigl\langle \Omega \Bigl\vert \varphi^{*0}(x'') \varphi^0(x') \Bigr\vert
\Omega \Bigr\rangle \Biggr\} \; . \qquad\end{aligned}$$ Substituting these relations into (\[expcom\]) and canceling some terms gives the expression we have been seeking, $$\begin{aligned}
\lefteqn{\Bigl\langle \Omega \Bigl\vert \Bigl[ \varphi(x), b^{\dagger}(\vec{k})
\Bigr] \Bigr\vert \Omega \Bigr\rangle = \Phi^0(x;\vec{k}) -i \lambda^2 \int_0^t
\!\! dt' \!\! \int d^{D-1}x' \, \Bigl\langle x \Bigl\vert \frac1{\partial^2
\!-\! m^2} \Bigr\vert x' \Bigr\rangle_{\rm ret} } \nonumber \\
& & \times \int_0^{t'} \!\! dt'' \!\! \int d^{D-1}x'' \, \Biggl\{ \Bigl\langle
\Omega \Bigl\vert \chi^0(x') \chi^0(x'') \Bigr\vert \Omega \Bigr\rangle
\Bigl\langle \Omega \Bigl\vert \varphi^0(x') \varphi^{0*}(x'') \Bigr\vert
\Omega \Bigr\rangle \nonumber \\
& & \hspace{.9cm} - \Bigl\langle \Omega \Bigl\vert \chi^0(x'') \chi^0(x')
\Bigr\vert \Omega \Bigr\rangle \Bigl\langle \Omega \Bigl\vert \varphi^{0*}(x'')
\varphi^0(x') \Bigr\vert \Omega \Bigr\rangle \Biggr\} \Phi^0(x'';\vec{k})
+ O(\lambda^4) \; . \qquad \label{fexp}\end{aligned}$$
We turn now to the effective field equations of the Schwinger-Keldysh formalism. The $\comp$-number field corresponding to $\varphi(x)$ at linearized order is $\Phi(x)$. If the state is released at $t=0$ then the equation $\Phi(x)$ obeys is, $$(\partial^2 - m^2) \Phi(x) - \int_0^t \!\! dt' \!\! \int d^{D-1}x'
\Bigl\{ M^2_{\scriptscriptstyle ++}(x;x') + M^2_{\scriptscriptstyle +-}(x;x')
\Bigr\} \Phi(x') = 0 \; . \label{Phieqn}$$ The one loop diagram for the self-mass-squared of $\varphi$ is depicted in Fig. 1.
(300,70)(0,0) (150,20)(40,0,180)[5]{} (190,20)(110,20) (110,20)(50,20) (110,20)[3]{} (110,10)\[b\][$x$]{} (250,20)(190,20) (190,20)[3]{} (191,10)\[b\][$x'$]{}
\
[Fig. 1: Self-mass-squared for $\varphi$ at one loop order. Solid lines stands for $\varphi$ propagators while dashed lines represent $\chi$ propagators.]{}
Because the self-mass-squared has two external lines, there are $2^2 = 4$ polarities in the Schwinger-Keldysh formalism. The two we require are [@DW; @FW], $$\begin{aligned}
-i M^2_{\scriptscriptstyle ++}(x;x') & = & (-i \lambda)^2 \Bigl\langle x
\Bigl\vert \frac{i}{\partial^2} \Bigr\vert x' \Bigr\rangle_{\scriptscriptstyle
++} \Bigl\langle x \Bigl\vert \frac{i}{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\scriptscriptstyle ++} + O(\lambda^4) \; , \label{M++} \\
-i M^2_{\scriptscriptstyle +-}(x;x') & = & (-i \lambda) (+i \lambda)
\Bigl\langle x \Bigl\vert \frac{i}{\partial^2} \Bigr\vert x' \Bigr\rangle_{
\scriptscriptstyle +-} \Bigl\langle x \Bigl\vert \frac{i}{\partial^2 \!-\! m^2}
\Bigr\vert x' \Bigr\rangle_{\scriptscriptstyle +-} + O(\lambda^4) \; . \qquad
\label{M+-}\end{aligned}$$
To recover (\[fexp\]) we must express the various Schwinger-Keldysh propagators in terms of expectation values of the free fields. The ${\scriptscriptstyle ++}$ polarity gives the usual Feynman propagator [@FW], $$\begin{aligned}
\lefteqn{\Bigl\langle x \Bigl\vert \frac{i}{\partial^2} \Bigr\vert x'
\Bigr\rangle_{\scriptscriptstyle ++}
= \theta(t \!-\! t') \Bigl\langle \Omega \Bigl\vert
\chi^0(x) \chi^0(x') \Bigr\vert \Omega \Bigr\rangle \!+\! \theta(t' \!-\! t)
\Bigl\langle \Omega \Bigl\vert \chi^0(x') \chi^0(x) \Bigr\vert \Omega
\Bigr\rangle \; , \qquad } \\
\lefteqn{\Bigl\langle x \Bigl\vert \frac{i}{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\scriptscriptstyle ++} } \nonumber \\
& & \hspace{1.3cm} = \theta(t \!-\! t') \Bigl\langle \Omega
\Bigl\vert \varphi^0(x) \varphi^{0*}(x') \Bigr\vert \Omega \Bigr\rangle \!+\!
\theta(t' \!-\! t) \Bigl\langle \Omega \Bigl\vert \varphi^{0*}(x') \varphi^0(x)
\Bigr\vert \Omega \Bigr\rangle \; . \qquad\end{aligned}$$ The ${\scriptscriptstyle +-}$ polarity propagators are [@FW], $$\begin{aligned}
\Bigl\langle x \Bigl\vert \frac{i}{\partial^2} \Bigr\vert x'
\Bigr\rangle_{\scriptscriptstyle +-}
& = & \Bigl\langle \Omega \Bigl\vert \chi^0(x') \chi^0(x) \Bigr\vert \Omega
\Bigr\rangle \; , \\
\Bigl\langle x \Bigl\vert \frac{i}{\partial^2 \!-\! m^2} \Bigr\vert x'
\Bigr\rangle_{\scriptscriptstyle +-}
& = & \Bigl\langle \Omega \Bigl\vert \varphi^{0*}(x') \varphi^0(x) \Bigr\vert
\Omega \Bigr\rangle \; . \qquad\end{aligned}$$ Substituting these relations into (\[M++\]-\[M+-\]) and making use of the identity $1 = \theta(t \!-\! t') \!+\! \theta(t' \!-\! t)$ gives, $$\begin{aligned}
\lefteqn{ M^2_{\scriptscriptstyle ++}(x;x') + M^2_{\scriptscriptstyle +-}(x;x')
= -i \lambda^2 \theta(t \!-\! t') \Biggl\{
\Bigl\langle \Omega \Bigl\vert \chi^0(x) \chi^0(x') \Bigr\vert \Omega
\Bigr\rangle } \nonumber \\
& & \hspace{-.7cm} \times \Bigl\langle \Omega \Bigl\vert \varphi^0(x)
\varphi^{0*}(x') \Bigr\vert \Omega \Bigr\rangle \!-\! \Bigl\langle \Omega
\Bigl\vert \chi^0(x') \chi^0(x) \Bigr\vert \Omega \Bigr\rangle \Bigl\langle
\Omega \Bigl\vert \varphi^{0*}(x') \varphi^0(x) \Bigr\vert \Omega \Bigr\rangle
\!\Biggr\} \!+\! O(\lambda^4) \, . \qquad \label{oneloop}\end{aligned}$$
We now solve (\[Phieqn\]) perturbatively. The free plane wave mode function (\[Phi\^0\]) is of course a solution at order $\lambda^0$. With (\[oneloop\]) we easily recognize its perturbative development as, $$\begin{aligned}
\lefteqn{\Phi(x;\vec{k}) = \Phi^0(x;\vec{k}) -i \lambda^2 \int_0^t
\!\! dt' \!\! \int d^{D-1}x' \, \Bigl\langle x \Bigl\vert \frac1{\partial^2
\!-\! m^2} \Bigr\vert x' \Bigr\rangle_{\rm ret} } \nonumber \\
& & \times \int_0^{t'} \!\! dt'' \!\! \int d^{D-1}x'' \, \Biggl\{ \Bigl\langle
\Omega \Bigl\vert \chi^0(x') \chi^0(x'') \Bigr\vert \Omega \Bigr\rangle
\Bigl\langle \Omega \Bigl\vert \varphi^0(x') \varphi^{0*}(x'') \Bigr\vert
\Omega \Bigr\rangle \nonumber \\
& & \hspace{.9cm} - \Bigl\langle \Omega \Bigl\vert \chi^0(x'') \chi^0(x')
\Bigr\vert \Omega \Bigr\rangle \Bigl\langle \Omega \Bigl\vert \varphi^{0*}(x'')
\varphi^0(x') \Bigr\vert \Omega \Bigr\rangle \Biggr\} \Phi^0(x'';\vec{k})
+ O(\lambda^4) \; . \qquad \end{aligned}$$ That agrees with (\[fexp\]), so we have established the desired connection, $$\Phi(x;\vec{k}) = \Bigl\langle \Omega \Bigl\vert \Bigl[ \varphi(x),
b^{\dagger}(\vec{k}) \Bigr] \Bigr\vert \Omega \Bigr\rangle \; ,$$ at one loop order.
The gauge issue
---------------
The preceding discussion has made clear that we are working in a particular local Lorentz and general coordinate gauge. We are also doing perturbation theory. The function $\Psi^0_i(x;\vec{k},s)$ describes how a free fermion of wave number $\vec{k}$ and helicity $s$ propagates through classical de Sitter background in our gauge. What $\Psi^1_i(x;\vec{k},s)$ gives is the first quantum correction to this mode function. It is natural to wonder how the effective field $\Psi_i(x;\vec{k},s)$ changes if a different gauge is used.
The operators of the original, invariant Lagrangian transform as follows under diffeomorphisms ($x^{\mu} \rightarrow x^{\prime \mu}$) and local Lorentz rotations ($\Lambda_{ij}$),[^1] $$\begin{aligned}
\psi'_i(x) & = & \Lambda_{ij}\Bigl(x^{\prime -1}(x)\Bigr) \psi_j\Bigl(
x^{\prime -1}(x)\Bigr) \; , \\
e'_{\mu b}(x) & = & \frac{\partial x^{\nu}}{\partial x^{\prime \mu}}
\Lambda_b^{~c}\Bigl(x^{\prime -1}(x)\Bigr) e_{\nu c}\Bigl( x^{\prime -1}(x)
\Bigr) \; .\end{aligned}$$ The invariance of the theory guarantees that the transformation of any solution is also a solution. Hence the possibility of performing local transformations precludes the existence of a unique initial value solution. This is why no Hamiltonian formalism is possible until the gauge has been fixed sufficiently to eliminate transformations which leave the initial value surface unaffected.
Different gauges can be reached using field-dependent gauge transformations [@TW7]. This has a relatively simple effect upon the Heisenberg operator $\psi_i(x)$, but a complicated one on the linearized effective field $\Psi_i(x;\vec{k},s)$. Because local Lorentz and diffeomorphism gauge conditions are typically specified in terms of the gravitational fields, we assume $x^{\prime \mu}$ and $\Lambda_{ij}$ depend upon the graviton field $h_{\mu\nu}$. Hence so too does the transformed field, $$\psi'_i[h](x) = \Lambda_{ij}[h]\Bigl(x^{\prime -1}[h](x)\Bigr)
\psi_j\Bigl(x^{\prime -1}[h](x)\Bigr) \; .$$ In the general case that the gauge changes even on the initial value surface, the creation and annihilation operators also transform, $$b'[h](\vec{k},s) = \frac1{\sqrt{2k}} u^*_i(\vec{k},s) \int d^{D-1}x \,
e^{-i \vec{k} \cdot \vec{x}} \psi'_i[h](\eta_i,\vec{x}) \; ,$$ where $\eta_i \equiv -1/H$ is the initial conformal time. Hence the linearized effective field transforms to, $$\Psi'_i(x;\vec{k},s) = a^{\frac{D-1}2} \Bigl\langle \Omega \Bigl\vert
\Bigl\{ \psi'_i[h](x), b^{\prime\dagger}[h](\vec{k},s) \Bigr\} \Bigr\vert
\Omega \Bigr\rangle \; . \label{SKprime}$$ This is quite a complicated relation. Note in particular that the $h_{\mu\nu}$ dependence of $x^{\prime \mu}[h]$ and $\Lambda_{ij}[h]$ means that $\Psi'_i(x;\vec{k},s)$ is not simply a Lorentz transformation of the original function $\Psi_i(x;\vec{k},s)$ evaluated at some transformed point.
Some Key Reductions
===================
The purpose of this section is to derive three results that are used repeatedly in reducing the nonlocal contributions to the effective field equations. We observe that the nonlocal terms of (\[ren\]) contain $1/\Delta x^2$. We can avoid denominators by extracting another derivative, $$\f{1}{\D x^2}=\frac{\del^2}4 \ln(\D x^2) \qquad {\rm and} \qquad
\f{\ln(\D x^2)}{\D x^2} = \frac{\del^2}8 \Bigl[\ln^2(\D x^{2}) - 2
\ln(\D x^2) \Bigr] \; . \label{id2}$$ The Schwinger-Keldysh field equations involve the difference of ${\scriptscriptstyle ++}$ and ${\scriptscriptstyle +-}$ terms, for example, && -\
& & = { \^2(\^2x\^2\_[++]{}) - 2 (\^2x\^2\_[++]{}) - \^2(\^2x\^2\_[ +-]{})+2(\^2x\^2\_[+-]{}) }. \[r1\] We now define the coordinate intervals $\D\eta \equiv \e \!-\! \e'$ and $\D x\equiv \Vert \vec{x} \!-\! \vec{x}' \Vert$ in terms of which the ${\scriptscriptstyle ++}$ and ${\scriptscriptstyle +-}$ intervals are, x\^2\_[++]{} = x\^2 - (- i)\^2 [and]{} x\^2\_[+-]{} = x\^2 - (+ i)\^2 . When $\e'>\e$ we have $\D x^2_{\scriptscriptstyle ++} = \D x^2_{
\scriptscriptstyle +-}$ , so the ${\scriptscriptstyle ++}$ and ${\scriptscriptstyle +-}$ terms in (\[r1\]) cancel. This means there is no contribution from the future. When $\e'<\e$ and $\D x>\D\e$ (past spacelike separation) we can take $\delta = 0$, $$\ln(\mu^2\D x^2_{\scriptscriptstyle ++}) = \ln[\mu^2(\D x^2 \!-\! \D \e^2 )]
= \ln(\mu^2\D x^2_{\scriptscriptstyle +-}) \qquad (\Delta x > \Delta \eta > 0)
\,\, . \label{ln1}$$ So the ${\scriptscriptstyle ++}$ and ${\scriptscriptstyle +-}$ terms again cancel. Only for $\eta'<\eta$ and $\D x<\D\eta$ (past timelike separation) are the two logarithms different, $$\ln(\mu^2\D x^2_{\scriptscriptstyle +\pm}) = \ln[\mu^2(\D\eta^2 \!-\! \D x^2)]
\pm i \pi \qquad (\Delta \eta > \Delta x > 0) \,\, .$$ Hence equation (\[r1\]) can be written as, - = 2 \^2 {(- x) } . \[r2\] This step shows that the Schwinger-Kledysh formalism is causal.
To integrate (\[r2\]) up against the plane wave mode function (\[freefun\]) we first pull the $x^{\mu}$ derivatives outside the integration, then make the change of variables $\vec{x}' \!=\! \vec{x} \!+\! \vec{r}$ and perform the angular integrals, $$\begin{aligned}
\lefteqn{\int d^4x' \Biggl\{ \f{\ln(\mu^2\D x^2_{++})}{\D x^2_{++}} -
\f{\ln(\mu^2\D x^2_{+-})}{\D x^2_{+-}} \Biggr\} \Psi^0_i(\e',\vec{x},\vec{k},s)}
\nn \\
&& = \f{i 2\pi^2}{k} u_i(\vec{k},s) \del^2 e^{i\vec{k}\cdot\vec{x}}
\int^{\e}_{\e_{i}} d\e' \frac{e^{-ik \eta'}}{\sqrt{2k}} \!\! \int^{\D\e}_{0}
\!\!\! dr r \sin(kr) \Bigl\{ \ln[\mu^2(\D\e^2 \!-\! r^2)] \!-\! 1\Bigr\} \nn \\
&& = \f{i2\pi^2}{k \sqrt{2k}} e^{i\vec{k} \cdot \vec{x}} u_i(\vec{k},s)
[-\del^2_{0} \!-\! k^2] \int^{\e}_{\e_{i}} \!\! d\e' e^{-ik \eta'} {\Delta
\eta}^2 \nn \\
&& \hspace{4.5cm} \times \int_{0}^{1} \!\!\! dz z \sin(\a z) \Bigl\{
\ln(1 \!-\! z^2) \!+\! 2 \ln(\f{\mu\a}{k}) \!-\! 1 \Bigr\} . \qquad \label{r3}\end{aligned}$$ Here $\alpha \equiv k \Delta \eta$ and $\e_{i} \equiv -1/H$ is the initial conformal time, corresponding to physical time $t=0$. The integral over $z$ is facilitated by the special function, $$\begin{aligned}
\lefteqn{\xi(\a) \equiv \int_{0}^{1} \!\! dz z \sin(\a z) \ln(1 \!-\! z^2) =
\f{2}{\a^2} \sin(\a) - \f{1}{\a^2} \Bigl[\cos(\a) \!+\! \a \sin(\a) \Bigr] }
\nn \\
& & \hspace{2cm} \times \Bigl[{\rm si}(2\a) \!+\! \f{\pi}{2} \Bigr] +
\Bigl[\sin(\a) \!-\! \a \cos(\a)\Bigr] \Bigl[{\rm ci}(2\a) \!-\! \gamma \!-\!
\ln(\f{\a}{2})\Bigr] \,\, . \qquad\end{aligned}$$ Here $\gamma$ is the Euler-Mascheroni constant and the sine and cosine integrals are, $$\begin{aligned}
{\rm si}(x) & \equiv & -\int_{x}^{\infty} \!\! dt \, \f{\sin(t)}{t} =
- \f{\pi}{2} + \int_{0}^{x} \!\! dt \, \f{\sin t}{t} \,\, , \\
{\rm ci}(x) & \equiv & -\int_{x}^{\infty} \!\! dt \, \f{\cos t}{t} =
\g + \ln(x) + \int_{0}^{x} \!\! dt \, \Bigl[\f{\cos(t) - 1}{t}\Bigr] \,\, .\end{aligned}$$ After substituting the $\xi$ function and performing the elementary integrals, (\[r3\]) becomes, $$\begin{aligned}
\lefteqn{\int d^4x' \Biggl\{ \f{\ln(\mu^2\D x^2_{++})}{\D x^2_{++}} -
\f{\ln(\mu^2\D x^2_{+-})}{\D x^2_{+-}} \Biggr\} \Psi^0_i(\e',\vec{x},\vec{k},s)
= \f{i 2 \pi^2}{k \sqrt{2k}} e^{i \vec{k} \cdot \vec{x}} u_i(\vec{k},s) } \nn \\
& & \hspace{.5cm} \times (\del^2_{k \eta} \!+\! 1) \hspace{-.1cm} \int_{\e_i}^{
\e} \hspace{-.3cm} d\e' e^{-ik\e'} \Biggl\{\a^2\xi(\a) \!+\! \Bigl[2\ln(\f{
\mu\a}{k}) \!-\! 1\Bigr]\Bigl[\sin(\a) \!-\! \a \cos(\a) \Bigr] \Biggr\} .
\qquad \label{r4}\end{aligned}$$
One can see that the integrand is of order $\a^3 \ln(\alpha)$ for small $\alpha$, which means we can pass the derivatives through the integral. After some rearrangements, the first key identity emerges, $$\begin{aligned}
\lefteqn{ \int d^4x' \Biggl\{ \frac{\ln(\mu^2 \Delta
x^2_{\scriptscriptstyle ++})}{\Delta x^2_{\scriptscriptstyle ++}} -
\frac{\ln(\mu^2 \Delta x^2_{ \scriptscriptstyle +-})}{\Delta
x^2_{\scriptscriptstyle +-}} \Biggr\}
\Psi^0(\eta',\vec{x}';\vec{k},s) } \nonumber \\
& & = -i 4 \pi^2 k^{-1} \Psi^0(\eta,\vec{x};\vec{k},s)
\int_{\eta_i}^{\eta}\!\!\! d\eta' e^{i k \Delta \eta}
\Biggl\{-\cos(k \Delta \eta) \int_{\scriptscriptstyle 0}^{2 k \Delta
\eta} \!\!\!\!\!\! dt \, \frac{\sin(t)}{t} \nonumber \\
& & \hspace{3cm} + \sin(k \Delta \eta) \Biggl[
\int_{\scriptscriptstyle 0}^{2 k \Delta \eta} \!\!\!\!\!\! dt \,
\Bigl(\frac{\cos(t) \!-\! 1}{t}\Bigr) \!+\! 2 \ln(2 \mu \Delta
\eta)\Biggr] \Biggr\} . \label{key1}\end{aligned}$$ Note that we have written $e^{-i k \eta'} = e^{-i k \eta} \times e^{-i k
\Delta \eta}$ and extracted the first phase to reconstruct the full tree order solution $\Psi^0(\eta,\vec{x};\vec{k},s) = \frac{e^{-i k \eta}}{
\sqrt{2k}} u_i(\vec{k},s) e^{i \vec{k} \cdot \vec{x}}$.
The second identity derives from acting a d’Alembertian on (\[key1\]). The d’Alembertian passes through the tree order solution to give, $$\partial^2 \Psi^0(\eta,\vec{x};\vec{k},s) = \Psi^0(\eta,\vec{x};\vec{k},s)
\partial_{\eta} (\partial_{\eta} \!-\! 2 i k) \; .$$ Because the integrand goes like $\alpha \ln(\alpha)$ for small $\alpha$, we can pass the first derivative through the integral to give, $$\begin{aligned}
\lefteqn{ \partial^2 \int d^4x' \Biggl\{ \frac{\ln(\mu^2 \Delta
x^2_{\scriptscriptstyle ++})}{\Delta x^2_{\scriptscriptstyle ++}} -
\frac{\ln(\mu^2 \Delta x^2_{ \scriptscriptstyle +-})}{\Delta
x^2_{\scriptscriptstyle +-}} \Biggr\}
\Psi^0(\eta',\vec{x}';\vec{k},s) } \nonumber \\
& & \hspace{1.5cm} = i 4 \pi^2 \Psi^0(\eta,\vec{x};\vec{k},s)
\partial_{\eta} \int_{\e_i}^{\e} \!\! d\e' \Biggl\{
\int_{\scriptscriptstyle 0}^{2\a} \!\! dt \Bigl(\f{e^{it} \!-\!
1}{t} \Bigr) + 2 \ln(\f{2\mu\a}{k}) \Biggr\} \,\, . \qquad\end{aligned}$$ We can pass the final derivative through the first integral but, for the second, we must carry out the integration. The result is our second key identity, $$\begin{aligned}
\lefteqn{ \partial^2 \int d^4x' \Biggl\{ \frac{\ln(\mu^2 \Delta
x^2_{ \scriptscriptstyle ++})}{\Delta x^2_{\scriptscriptstyle ++}} -
\frac{\ln(\mu^2 \Delta x^2_{ \scriptscriptstyle +-})}{\Delta
x^2_{\scriptscriptstyle +-}}
\Biggr\} \Psi^0(\eta',\vec{x}';\vec{k},s) } \nonumber \\
& & \hspace{1.5cm} = i 4 \pi^2 \Psi^0(\eta,\vec{x};\vec{k},s) \Biggl\{ 2
\ln\Bigl[\frac{2\mu}{H} (1 \!+\! H \eta)\Bigr] \!+\!
\int_{\eta_i}^{\eta} \!\!\! d\eta' \, \Bigl(\frac{e^{i 2 k \Delta
\eta} \!-\! 1}{\Delta \eta}\Bigr) \Biggr\} . \qquad \label{key2}\end{aligned}$$
The final key identity is derived through the same procedures. Because they should be familiar by now we simply give the result, $$\begin{aligned}
\lefteqn{ \int d^4x' \Biggl\{ \frac1{\Delta x^2_{\scriptscriptstyle
++}} - \frac1{\Delta x^2_{\scriptscriptstyle +-}} \Biggr\}
\Psi^0(\eta',\vec{x}'; \vec{k},s) } \nonumber \\
& & \hspace{3cm} = - i 4 \pi^2 k^{-1} \Psi^0(\eta,\vec{x};\vec{k},s)
\int_{\e_i}^{\e} \!\! d\e' \, e^{ik\D\e}\sin(k\D\e) \,\, . \qquad \label{key3}\end{aligned}$$
Solving the Effective Dirac Equation
====================================
In this section we first evaluate the various nonlocal contributions using the three identities of the previous section. Then we evaluate the vastly simpler and, as it turns out, more important, local contributions. Finally, we solve for $\Psi^1(\eta,\vec{x};\vec{k},s)$ at late times.
The various nonlocal contributions to (\[Diraceqn\]) take the form, $$\begin{aligned}
\lefteqn{\int \!\! d^4x' \sum_{I=1}^5 U^I_{ij} \Biggr\{\f{\ln(\a_{I}^2\D x^2_{
\scriptscriptstyle ++})}{\D x^2_{\scriptscriptstyle ++}} - \f{\ln(\a_{I}^2\D
x^2_{\scriptscriptstyle +-})}{\D x^2_{\scriptscriptstyle +-}} \Biggl\}
\Psi^0_j(\eta',\vec{x}';\vec{k},s) } \nonumber \\
& & \hspace{4cm} + \int \!\! d^4x' U^6_{ij} \Biggr\{\f1{\D x^2_{
\scriptscriptstyle ++}} - \f1{\D x^2_{\scriptscriptstyle +-}} \Biggl\}
\Psi^0_j(\eta',\vec{x}';\vec{k},s) \,\, . \qquad \label{analyticform}\end{aligned}$$ The spinor differential operators $U^I_{ij}$ are listed in Table \[nond\]. The constants $\a_{I}$ are $\mu$ for $I = 1,2,3$, and $\f12 H$ for $I=4,5$.
As an example, consider the contribution from $U^2_{ij}$: $$\begin{aligned}
\lefteqn{\f{15}{2} \f{\kappa^2H^2}{2^8\pi^4} \not{\hspace{-.08cm}\del} \del^2
\!\! \int \!\! d^4x' \Biggr\{\f{\ln(\mu^2\D x^2_{\scriptscriptstyle ++})}{\D
x^2_{\scriptscriptstyle ++}} - \f{\ln(\mu^2\D x^2_{\scriptscriptstyle +-})}{
\D x^2_{\scriptscriptstyle +-}}\Biggl\} \Psi^0(\eta',\vec{x}';\vec{k},s) }\nn\\
&& \hspace{-.5cm} = \f{15}{2} \f{\kappa^2H^2}{2^8\pi^4} \!\not{\hspace{-.1cm}
\del} \!\times\! i 4 \pi^2 \Psi^{0}(\e,\vec{x};\vec{k},s)\Biggl\{\!2\ln\Bigl[
\f{2\mu}{H}(1 \!+\! H\e) \Bigr] \!+\!\! \int_{\e_i}^{\e} \!\! d\e' \Bigl(\f
{e^{2ik\D\e} \!-\! 1}{\D\e} \Bigr)\! \Biggr\} , \qquad \\
&& \hspace{-.5cm} = \f{\kappa^2H^2}{2^6\pi^2} i H \g^0 \Psi^0(\eta,\vec{x};
\vec{k},s) \times \f{15}{2} \f{1}{
1 \!+\! H\e} \Biggl\{e^{2i\f{k}{H}(1+H\e)} \!+\! 1\Biggr\} . \label{non2}\end{aligned}$$ In these reductions we have used $i \hspace{-.1cm} \not{\hspace{-.1cm}\del} \Psi^0(\eta,\vec{x};\vec{k},s) \!=\!
i \gamma^0 \Psi^0(\eta,\vec{x};\vec{k},s) \, \partial_{\eta}$ and (\[key2\]). Recall from the Introduction that reliable predictions are only possible for late times, which corresponds to $\eta \rightarrow 0^-$. We therefore take this limit, $$\begin{aligned}
\lefteqn{\f{15}{2} \f{\kappa^2H^2}{2^8\pi^4} \not{\hspace{-.08cm}\del} \del^2
\!\! \int \!\! d^4x' \Biggr\{\f{\ln(\mu^2\D x^2_{\scriptscriptstyle ++})}{\D
x^2_{\scriptscriptstyle ++}} - \f{\ln(\mu^2\D x^2_{\scriptscriptstyle +-})}{
\D x^2_{\scriptscriptstyle +-}}\Biggl\} \Psi^0(\eta',\vec{x}';\vec{k},s)}\nn \\
& & \hspace{3cm} \longrightarrow \f{\kappa^2H^2}{2^6\pi^2} i H \g^0
\Psi^0(\eta,\vec{x};\vec{k},s) \times \f{15}{2} \Bigl\{\exp(2i\f{k}{H}) + 1
\Bigr\} . \qquad\end{aligned}$$
The other five nonlocal terms have very similar reductions. Each of them also goes to $\frac{\kappa^2 H^2}{2^6 \pi^2} \times i H \gamma^0 \Psi^0(\eta,
\vec{x};\vec{k},s)$ times a finite constant at late times. We summarize the results in Table \[noncon\] and relegate the details to an appendix.
The next step is to evaluate the local contributions. This is a straightforward exercise in calculus, using only the properties of the tree order solution (\[freefun\]) and the fact that $\partial_{\mu} a = H a^2 \delta^0_{\mu}$. The result is, $$\begin{aligned}
\lefteqn{\f{i\kappa^2 H^2}{2^6\pi^2} \!\!\int \!\! d^4x' \Biggl\{ \f{\ln(aa')}{
H^2 aa'} \! \not{\hspace{-.1cm}\del} \del^2 \!+\! \f{15}{2} \ln(aa') \! \not{
\hspace{-.1cm}\del} \!-\! 7 \ln(aa') \!\! \not{\hspace{-.08cm} \bar{\del}}
\Biggr\} \d^4(x \!-\! x') \Psi^0(\eta',\vec{x}';\vec{k},s) } \nonumber \\
& & = \frac{i \kappa^2 H^2}{2^6 \pi^2} \Biggl\{ \frac{\ln(a)}{H^2 a}
\!\not{\hspace{-.1cm}\del} \del^2 \Bigl(\frac1{a} \Psi^0(\eta,\vec{x};\vec{k},s)
\Bigr) + \frac1{H^2 a} \! \not{\hspace{-.1cm}\del} \del^2 \Bigl(\frac{\ln(a)}{
a} \Psi^0(\eta,\vec{x};\vec{k},s)\Bigr) \nonumber \\
& & \hspace{1.1cm} + \frac{15}2 \Bigl(\ln(a) \! \not{\hspace{-.1cm}\del} \!+\!
\not{\hspace{-.1cm}\del} \ln(a)\Bigr) \Psi^0(\eta,\vec{x};\vec{k},s)
- 14 \ln(a) \!\not{\hspace{-.1cm} \bar{\del}}
\Psi^0(\eta,\vec{x};\vec{k},s) \Biggr\} , \qquad \label{localgen} \\
& & = \frac{\kappa^2 H^2}{2^6 \pi^2} iH \gamma^0 \Psi^0(\eta,\vec{x};\vec{k},s)
\times \Biggl\{ \frac{17}2 a - 14 i \frac{k}{H} \ln(a) - 2 i \frac{k}{H}
\Biggr\} . \label{localtotal}\end{aligned}$$
The local quantum corrections (\[localtotal\]) are evidently much stronger than their nonlocal counterparts in Table \[noncon\]! Whereas the nonlocal terms approach a constant, the leading local contribution grows like the inflationary scale factor, $a = e^{H t}$. Even factors of $\ln(a)$ are negligible by comparison. We can therefore write the late time limit of the one loop field equation as, $$\begin{aligned}
i\hspace{-.1cm}\not{\hspace{-.08cm}\del} \kappa^2 \Psi^{1}(\eta,\vec{x};
\vec{k},s) \longrightarrow \f{\kappa^2 H^2}{2^6\pi^2} \f{17}{2} i H a \g^0
\Psi^0(\eta,\vec{x};\vec{k},s) \,\, .\end{aligned}$$ The only way for the left hand side to reproduce such rapid growth is for the time derivative in $i\hspace{-.1cm}\not{\hspace{-.08cm}\del}$ to act on a factor of $\ln(a)$, $$i\gamma^{\mu}\partial_{\mu}\ln(a)
=i\gamma^{\mu}\frac{Ha^2}{a}\delta^0_{\mu}=iHa\gamma^0\;.$$ We can therefore write the late time limit of the tree plus one loop mode functions as, $$\Psi^{0}(\eta,\vec{x};\vec{k},s) + \kappa^2 \Psi^{1}(\eta,\vec{x};\vec{k},s)
\longrightarrow \Biggl\{1 \!+\! \f{\kappa^2 H^2}{2^6 \pi^2} \f{17}{2} \ln(a)
\Biggr\} \Psi^0(\eta,\vec{x};\vec{k},s) \,\, . \label{modefun}$$ All other corrections actually fall off at late times. For example, those from the $\ln(a)$ terms in (\[localtotal\]) go like $\ln(a)/a$.
There is a clear physical interpretation for the sort of solution we see in (\[modefun\]). When the corrected field goes to the free field times a constant, that constant represents a field strength renormalization. When the quantum corrected field goes to the free field times a function of time that is independent of the form of the free field solution, it is natural to think in terms of a [*time dependent field strength renormalization*]{}, $$\Psi(\eta,\vec{x};\vec{k},s) \longrightarrow \frac{\Psi^0(\eta,\vec{x};
\vec{k},s)}{\sqrt{Z_2(t)}} \quad {\rm where} \quad Z_2(t) = 1 \!-\! \frac{17
\kappa^2 H^2}{2^6 \pi^2} \ln(a) \!+\! O(\kappa^4) \; .\label{fieldrenor}$$ Of course we only have the order $\kappa^2$ correction, so one does not know if this behavior persists at higher orders. If no higher loop correction supervenes, the field would switch from positive norm to negative norm at $\ln(a) = 2^6 \pi^2/17 \kappa^2 H^2$. In any case, it is safe to conclude that perturbation theory must break down near this time.
Hartree Approximation
=====================
The appearance of a time-dependent field strength renormalization is such a surprising result that it is worth noting we can understand it on a simple, qualitative level using the Hartree, or mean-field, approximation. This technique has proved useful in a wide variety of problems from atomic physics [@DRH] and statistical mechanics [@RKP], to nuclear physics [@HoS] and quantum field theory [@HJS]. Of particular relevance to our work is the insight the Hartree approximation provides into the generation of photon mass by inflationary particle production in SQED [@DDPT; @DPTD; @PW2].
The idea is that we can approximate the dynamics of Fermi fields interacting with the graviton field operator, $h_{\mu\nu}$, by taking the expectation value of the Dirac Lagrangian in the graviton vacuum. To the order we shall need it, the Dirac Lagrangian is [@MW1], $$\begin{aligned}
\lefteqn{\mathcal{L}_{\rm Dirac} = \overline{\Psi} i\hspace{-.1cm}\not{
\hspace{-.08cm} \partial} \Psi + \frac{\kappa}2 \Bigl\{h \overline{\Psi}
i\hspace{-.1cm} \not{\hspace{-.08cm} \partial} \Psi \!-\! h^{\mu\nu}
\overline{\Psi} \gamma_{\mu} i \partial_{\nu} \Psi \!-\! h_{\mu\rho , \sigma}
\overline{\Psi} \gamma^{\mu} J^{\rho\sigma} \Psi\Bigr\} } \nonumber \\
& & + \kappa^2 \Bigl[\frac18 h^2 \!-\! \frac14 h^{\rho\sigma} h_{\rho\sigma}
\Bigr] \overline{\Psi} i\hspace{-.1cm} \not{\hspace{-.08cm} \partial} \Psi
+ \kappa^2 \Bigl[-\frac14 h h ^{\mu\nu} \!+\! \frac38 h^{\mu\rho} h_{\rho}^{~\nu}
\Bigr] \overline{\Psi} \gamma_{\mu} i \partial_{\nu} \Psi \nonumber \\
& & \hspace{-.5cm} + \kappa^2 \Bigl[-\frac14 h h_{\mu\rho , \sigma}
\!+\! \frac18 h^{\nu}_{~\rho} h_{\nu \sigma , \mu} \!+\! \frac14
(h^{\nu}_{~\mu} h_{\nu \rho})_{,\sigma} \!+\! \frac14
h^{\nu}_{~\sigma} h_{\mu\rho , \nu}\Bigr] \overline{\Psi}
\gamma^{\mu} J^{\rho\sigma} \Psi + O(\kappa^3) .
\qquad\label{DiracL}\end{aligned}$$ Of course the expectation value of a single graviton field is zero, but the expectation value of the product of two fields is the graviton propagator [@TW2; @RPW3], $$\begin{aligned}
\lefteqn{\langle\Om\mid
T\Bigl[ h_{\mu\nu}(x) h_{\rho\sigma}(x') \Bigr] \mid\Om\rangle }
\nonumber \\
& & \hspace{1cm} =
i\D_{A}(x;x')\Bigl[\mbox{}_{\mu\nu}T^{A}_{\rho\sigma}\Bigr]+
i\D_{B}(x;x')\Bigl[\mbox{}_{\mu\nu}T^{B}_{\rho\sigma}\Bigr]+
i\D_{C}(x;x')\Bigl[\mbox{}_{\mu\nu}T^{C}_{\rho\sigma}\Bigr]
\,\, . \qquad \label{gprop}\end{aligned}$$ The various tensor structures are, $$\begin{aligned}
\lefteqn{\Bigl[\mbox{}_{\mu\nu}T^{A}_{\rho\sigma}\Bigr] =
2\bar{\e}_{\mu(\rho}\bar{\e}_{\sigma)\nu} - \frac{2}{D \!-\! 3}
\bar{\e}_{\mu\nu}\bar{\e}_{\rho\sigma} \quad , \quad
\Bigl[\mbox{}_{\mu\nu}T^{B}_{\rho\sigma}\Bigr] =
-4\d^0\mbox{}_{(\mu}\bar{\e}_{\nu)}\mbox{}_{(\rho}\d^0_{\sigma)}
\,\, , } \\
& & \Bigl[\mbox{}_{\mu\nu}T^{C}_{\rho\sigma}\Bigr] =
\frac2{(D \!-\! 2) (D \!-\! 3)} \Bigl[(D \!-\! 3) \d^{0}_{\mu}\d^{0}_{\nu}
+ \bar{\e}_{\mu\nu}\Bigr] \Bigl[(D \!-\! 3) \d^{0}_{\rho}\d^{0}_{\sigma}
+ \bar{\e}_{\rho\sigma}\Bigr] \,\, . \qquad \label{gtensor}\end{aligned}$$ Parenthesized indices are symmetrized and a bar over a common tensor such as the Kronecker delta function denotes that its temporal components have been nulled, $$\overline{\delta}^{\mu}_{\nu} \equiv \delta^{\mu}_{\nu} -
\delta^{\mu}_{\scriptscriptstyle 0} \delta^0_{\nu} \qquad , \qquad
\overline{\eta}_{\mu\nu} \equiv \eta_{\mu\nu} + \delta^0_{\mu}
\delta^0_{\nu} \; .$$
The three scalar propagators that appear in (\[gprop\]) have complicated expressions which we omit in favor of simply giving their coincidence limits and the coincidence limits of their first derivatives [@TW3], $$\begin{aligned}
\lim_{x' \rightarrow x} \, {i\Delta}_A(x;x') & = & \frac{H^{D-2}}{(4\pi)^{
\frac{D}2}} \frac{\Gamma(D-1)}{\Gamma(\frac{D}2)} \left\{-\pi \cot\Bigl(
\frac{\pi}2 D \Bigr) + 2 \ln(a) \right\} , \\
\lim_{x' \rightarrow x} \, \partial_{\mu} {i\Delta}_A(x;x') & = &
\frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{\Gamma(D-1)}{\Gamma(\frac{D}2)}
\times H a \delta^0_{\mu} = \lim_{x' \rightarrow x} \, \partial_{\mu}'
{i\Delta}_A(x;x') \; , \qquad \\
\lim_{x' \rightarrow x} \, {i\Delta}_B(x;x') & = & \frac{H^{D-2}}{(4\pi)^{
\frac{D}2}} \frac{\Gamma(D-1)}{\Gamma(\frac{D}2)}\times -\frac1{D\!-\!2} \; ,\\
\lim_{x' \rightarrow x} \, \partial_{\mu} {i\Delta}_B(x;x') & = & 0 =
\lim_{x' \rightarrow x} \, \partial_{\mu}' {i\Delta}_B(x;x') \; , \\
\lim_{x' \rightarrow x} \, {i\Delta}_C(x;x') & = & \frac{H^{D-2}}{(4\pi)^{
\frac{D}2}} \frac{\Gamma(D-1)}{\Gamma(\frac{D}2)}\times \frac1{(D\!-\!2)
(D\!-\!3)} \; ,\\
\lim_{x' \rightarrow x} \, \partial_{\mu} {i\Delta}_C(x;x') & = & 0 =
\lim_{x' \rightarrow x} \, \partial_{\mu}' {i\Delta}_C(x;x') \; .\end{aligned}$$ We are interested in terms which grow at late times. Because the $B$-type and $C$-type propagators go to constants, and their derivatives vanish, they can be neglected. The same is true for the divergent constant in the coincidence limit of the $A$-type propagator. In the full theory it would be absorbed into a constant counterterm. Because the remaining, time dependent terms are finite, we may as well take $D \!=\! 4$. Our Hartree approximation therefore amounts to making the following replacements in (\[DiracL\]), $$\begin{aligned}
h_{\mu\nu} h_{\rho\sigma} & \longrightarrow & \frac{H^2}{4 \pi^2} \ln(a)
\Bigl[ \overline{\eta}_{\mu\rho} \overline{\eta}_{ \nu\sigma} \!+\!
\overline{\eta}_{\mu\sigma} \overline{\eta}_{\nu\rho} \!-\! 2
\overline{\eta}_{\mu\nu} \overline{\eta}_{\rho\sigma} \Bigr] \; ,
\label{Atensor1} \\
h_{\mu\nu} h_{\rho\sigma, \alpha} & \longrightarrow &
\frac{H^2}{8 \pi^2} H a \delta^0_{\alpha} \, \Bigl[\overline{\eta}_{\mu\rho}
\overline{\eta}_{\nu\sigma} \!+\! \overline{\eta}_{\mu\sigma}
\overline{\eta}_{\nu\rho} \!-\! 2 \overline{\eta}_{\mu\nu}
\overline{\eta}_{\rho\sigma} \Bigr] \; . \label{Atensor2}\end{aligned}$$
It is now just a matter of contracting (\[Atensor1\]-\[Atensor2\]) appropriately to produce each of the quadratic terms in (\[DiracL\]). For example, the first term gives, $$\begin{aligned}
\f{\kappa^2}{8} h^2 \overline{\Psi} i \hspace{-.1cm} \not{
\hspace{-.08cm}\del} \Psi &\!\!\!\! \longrightarrow \!\!\!\!&
\f{\kappa^2 H^2}{2^5\pi^2}
\ln(a) \Bigl[\e^{\mu\nu}\e^{\rho\sigma}\Bigr]\Bigl[\bar{\e}_{\mu\rho}
\bar{\e}_{\nu\sigma} +\bar{\e}_{\mu\sigma}\bar{\e}_{\nu\rho}
-2\bar{\e}_{\mu\nu}\bar{\e}_{\rho\sigma}\Bigr]
\overline{\Psi}i\hspace{-.1cm}\not{\hspace{-.08cm}\del}\Psi ,
\qquad \\
& \!\!\!\! = \!\!\!\! &
\f{\kappa^2 H^2}{2^5\pi^2}\ln(a)\Bigl[3+3-18\Bigr]
\overline{\Psi}i\hspace{-.1cm}\not{\hspace{-.08cm}\del}\Psi \,\,.\end{aligned}$$ The second quadratic term gives a proportional result, $$\begin{aligned}
\f{-\kappa^2}{4}h^{\rho\sigma}h_{\rho\sigma}
\overline{\Psi}i\hspace{-.1cm}\not{\hspace{-.08cm}\del}\Psi
\longrightarrow\f{-\kappa^2 H^2}{2^4\pi^2}\ln(a)\bigl[9+3-6\Bigr]
\overline{\Psi}i\hspace{-.1cm}\not{\hspace{-.08cm}\del}\Psi \,\, .\end{aligned}$$ The total for these first two terms is $\f{-3\kappa^2 H^2}{4\pi^2} \ln(a)
\overline{\Psi}i\hspace{-.1cm} \not{\hspace{-.08cm}\del}\Psi$.
The third and fourth of the quadratic terms in (\[DiracL\]) result in only spatial derivatives, $$\begin{aligned}
\f{-\kappa^2H^2}{4}hh^{\mu\nu}\overline{\Psi}\g_{\mu}i\del_{\nu}\Psi
& \longrightarrow & \f{-\kappa^2 H^2}{2^4\pi^2}\ln(a)\Bigl[1+1-6\Bigr]
\overline{\Psi}i\hspace{-.1cm}\not{\hspace{-.08cm}\bar{\del}}\Psi
\,\, , \\
\f{3}{8}\kappa^2h^{\mu\rho}h_{\,\rho}^{\nu}
\overline{\Psi}\g_{\mu}i\del_{\nu}\Psi & \longrightarrow &
\f{3\kappa^2H^2}{2^5\pi^2}\ln(a)\Bigl[3+1-2\Bigr]
\overline{\Psi}i\hspace{-.1cm}\not{\hspace{-.08cm}\bar{\del}}\Psi \,\, .\end{aligned}$$ The total for this type of contribution is $\f{7\kappa^2 H^2}{2^4\pi^2}
\ln(a) \overline{\Psi} i \hspace{-.1cm} \not{\hspace{-.08cm} \bar{\del}}
\Psi$.
The final four quadratic terms in (\[DiracL\]) involve derivatives acting on at least one of the two graviton fields, $$\begin{aligned}
-\f{\kappa^2}{4}hh_{\mu\rho,\si}\overline{\Psi}\g^{\mu}J^{\rho\si}\Psi
& \longrightarrow & \f{-\kappa^2 H^2}{2^5\pi^2}Ha\Bigl[1+1-6\Bigr]
\bar{\e}_{\mu\rho}\overline{\Psi}\g^{\mu}J^{\rho0}\Psi \; , \\
\f{\kappa^2}{8}h^{\nu}_{\,\rho}h_{\nu\si,\mu}
\overline{\Psi}\g^{\mu}J^{\rho\si}\Psi & \longrightarrow &
\f{\kappa^2H^2}{2^6\pi^2}Ha\Bigl[3+1-2\Bigr] \bar{\e}_{\rho\si}
\overline{\Psi}\g^0J^{\rho\si}\Psi \; , \\
\f{\kappa^2}{4}\Bigl(h^{\nu}_{\,\mu}h_{\nu\rho}\Bigr)_{,\sigma}
\overline{\Psi}\g^{\mu}J^{\rho\si}\Psi & \longrightarrow &
\f{\kappa^2H^2}{2^4\pi^2}Ha\Bigl[3+1-2\Bigr]\e_{\mu\rho}
\overline{\Psi}\g^{\mu}J^{\rho0}\Psi \; , \\
\f{\kappa^2}{4}h^{\nu}_{\,\si}h_{\mu\rho,\nu}
\overline{\Psi}\g^{\mu}J^{\rho\si}\Psi & \longrightarrow & 0 \; .\end{aligned}$$ The second of these contributions vanishes owing to the antisymmetry of the Lorentz representation matrices, $J^{\mu\nu} \equiv \frac{i}4
[\gamma^{\mu},\gamma^{\nu}]$, whereas $\overline{\eta}_{\mu \rho}
\gamma^{\mu} J^{\rho 0} = -\frac{3i}2 \gamma^0$. Hence the sum of all four terms is $\f{-3\kappa^2H^2}{8\pi^2} H a \overline{\Psi} i \g^0 \Psi$.
Combining these results gives, $$\begin{aligned}
\lefteqn{\Bigl\langle \mathcal{L}_{\rm Dirac} \Bigr\rangle =
\overline{\Psi} i\!\hspace{-.1cm}\not{\hspace{-.08cm} \partial} \Psi
-\frac{3 \kappa^2 H^2}{4 \pi^2} \ln(a) \overline{\Psi}
i\hspace{-.1cm} \not{\hspace{-.08cm} \partial}
\Psi } \nonumber \\
& & \hspace{3cm} - \frac{3 \kappa^2 H^2}{8 \pi^2} H a \overline{\Psi} i\gamma^0
\Psi \!+\! \frac{7 \kappa^2 H^2}{16 \pi^2} \ln(a) \overline{\Psi} i \,
\hspace{-.1cm} \overline{\not{\hspace{-.08cm} \partial}} \Psi \!+\! O(\kappa^4)
, \\
& & \hspace{-.7cm} = \!\overline{\Psi} \Bigl[1 \!-\! \frac{3
\kappa^2 H^2}{8 \pi^2} \ln(a)\Bigr]
i\!\hspace{-.1cm}\not{\hspace{-.08cm} \partial} \Bigl[1 \!-\!
\frac{3 \kappa^2 H^2}{8 \pi^2} \ln(a)\Bigr] \Psi \!+\! \frac{7
\kappa^2 H^2}{16 \pi^2} \ln(a) \overline{\Psi} i \, \hspace{-.1cm}
\overline{\not{ \hspace{-.08cm} \partial}} \Psi \!+\! O(\kappa^4) .
\qquad\label{Hartreesum}\end{aligned}$$ If we express the equations associated with (\[Hartreesum\]) according to the perturbative scheme of Section 1, the first order equation is, $$\begin{aligned}
i\hspace{-.1cm}\not{\hspace{-.08cm}\del} \kappa^2 \Psi^{1}(\eta,\vec{x};
\vec{k},s) = \f{\kappa^2 H^2}{2^6\pi^2} i H \g^0 \Psi^0(\eta,\vec{x};\vec{k},s)
\Bigl\{ 24 a - 28 i \frac{k}{H}\ln(a) \Bigr\} \,\, . \label{Hareqn}\end{aligned}$$ This is similar, but not identical to, what we got in expression (\[localtotal\]) from the delta function terms of the actual one loop self-energy (\[ren\]). In particular, the exact calculation gives $\frac{17}{2} a
\!-\! 14 i \frac{k}H \ln(a)$, rather than the Hartree approximation of $24 a \!-\! 28 i \frac{k}H \ln(a)$. Of course the $\ln(a)$ terms make corrections to $\Psi^1$ which fall like $\ln(a)/a$, so the real disagreement between the two methods is limited to the differing factors of $\frac{17}{2}$ versus $24$.
We are pleased that such a simple technique comes so close to recovering the result of a long and tedious calculation. The slight discrepancy is no doubt due to terms in the Dirac Lagrangian (\[DiracL\]) which are linear in the graviton field operator. As described in relation (\[SKop\]) of section 2, the linearized effective field $\Psi_i(x;\vec{k},s)$ represents $a^{\frac{D-1}2}$ times the expectation value of the anti-commutator of the Heisenberg field operator $\psi_i(x)$ with the free fermion creation operator $b(\vec{k},s)$. At the order we are working, quantum corrections to $\Psi_i(x;\vec{k},s)$ derive from perturbative corrections to $\psi_i(x)$ which are quadratic in the free graviton creation and annihilation operators. Some of these corrections come from a single $h h \overline{\psi} \psi$ vertex, while others derive from two $h \overline{\psi} \psi$ vertices. The Hartree approximation recovers corrections of the first kind, but not the second, which is why we believe it fails to agree with the exact result. Yukawa theory presents a fully worked-out example [@PW1; @GP; @MW2] in which the [*entire*]{} lowest-order correction to the fermion mode functions derives from the product of two such linear terms, so the Hartree approximation fails completely in that case.
Discussion
==========
We have used the Schwinger-Keldysh formalism to include one loop, quantum gravitational corrections to the Dirac equation, in the simplest local Lorentz and general coordinate gauge, in the locally de Sitter background which is a paradigm for inflation. Because Dirac + Einstein is not perturbatively renormalizable, it makes no sense to solve this equation generally. However, the equation should give reliable predictions at late times when the arbitrary finite parts of the BPHZ counterterms (\[genctm\]) are insignificant compared to the completely determined factors of $\ln(a a')$ on terms (\[1stlog\]-\[3rdlog\]) which otherwise have the same structure. In this late time limit we find that the one loop corrected, spatial plane wave mode functions behave as if the tree order mode functions were simply subject to a time-dependent field strength renormalization, $$\begin{aligned}
Z_2(t) = 1 - \f{17}{4\pi} G H^2 \ln(a) + O(G^2) \,\,\,\,{\rm where}\,\,\,\,
G=16\pi\kappa^2\,\,.\end{aligned}$$ If unchecked by higher loop effects, this would vanish at $\ln(a) \simeq
1/G H^2$. What actually happens depends upon higher order corrections, but there is no way to avoid perturbation theory breaking down at this time, at least in this gauge.
Might this result be a gauge artifact? One reaches different gauges by making field dependent transformations of the Heisenberg operators. We have worked out the change (\[SKprime\]) this induces in the linearized effective field, but the result is not simple. Although the linearized effective field obviously changes when different gauge conditions are employed to compute it, we believe (but have not proven) that the late time factors of $\ln(a)$ do not change.
It is important to realize that the 1PI functions of a gauge theory in a fixed gauge are not devoid of physical content by virtue of depending upon the gauge. In fact, they encapsulate the physics of a quantum gauge field every bit as completely as they do when no gauge symmetry is present. One extracts this physics by forming the 1PI functions into gauge independent and physically meaningful combinations. The S-matrix accomplishes this in flat space quantum field theory. Unfortunately, the S-matrix fails to exist for Dirac + Einstein in de Sitter background, nor would it correspond to an experiment that could be performed if it did exist [@TW8; @EW; @AS].
If it is conceded that we know what it means to release the universe in a free state then it would be simple enough — albeit tedious — to construct an analogue of $\psi_i(x)$ which is invariant under gauge transformations that do not affect the initial value surface. For example, one might extend to fermions the treatment given for pure gravity by [@TW9]:
- [Propagate an operator-valued geodesic a fixed invariant time from the initial value surface;]{}
- [Use the spin connection $A_{\mu cd} J^{cd}$ to parallel transport along the geodesic; and]{}
- [Evaluate $\psi$ at the operator-valued geodesic, in the Lorentz frame which is transported from the initial value surface.]{}
This would make an invariant, as would any number of other constructions [@GMH]. For that matter, the gauge-fixed 1PI functions also correspond to the expectation values of invariant operators [@RPW5]. Mere invariance does not guarantee physical significance, nor does gauge dependence preclude it.
What is needed is for the community to agree upon a relatively simple set of operators which stand for experiments that could be performed in de Sitter space. There is every reason to expect a successful outcome because the last few years have witnessed a resolution of the similar issue of how to measure quantum gravitational back-reaction during inflation, driven either by a scalar inflaton [@WU; @AW1; @AW2; @GB] or by a bare cosmological constant [@TW10]. That process has begun for quantum field theory in de Sitter space [@EW; @AS; @GMH] and one must wait for it to run its course. In the meantime, it is safest to stick with what we have actually shown: perturbation theory must break down for Dirac + Einstein in the simplest gauge.
This is a surprising result but we were able to understand it qualitatively using the Hartree approximation in which one takes the expectation value of the Dirac Lagrangian in the graviton vacuum. The physical interpretation seems to be that fermions propagate through an effective geometry whose ever-increasing deviation from de Sitter is controlled by inflationary graviton production. At one loop order the fermions are passive spectators to this effective geometry.
It is significant that inflationary graviton production enhances fermion mode functions by a factor of $\ln(a)$ at one loop. Similar factors of $\ln(a)$ have been found in the graviton vacuum energy [@TW4; @TW5]. These infrared logarithms also occur in the vacuum energy of a massless, minimally coupled scalar with a quartic self-interaction [@OW1; @OW2], and in the VEV’s of almost all operators in Yukawa theory [@MW2] and SQED [@PTW]. A recent all orders analysis was not even able to exclude the possibility that they might contaminate the power spectrum of primordial density fluctuations [@SW2]!
The fact that infrared logarithms grow without bound raises the exciting possibility that quantum gravitational corrections may be significant during inflation, in spite of the minuscule coupling constant of $G H^2 \ltwid 10^{-12}$. However, the only thing one can legitimately conclude from the perturbative analysis is that infrared logarithms cause perturbation theory to break down, in our gauge, if inflation lasts long enough. Inferring what happens after this breakdown requires a nonperturbative technique.
Starobinskiĭ has long advocated that a simple stochastic formulation of scalar potential models serves to reproduce the leading infrared logarithms of these models at each order in perturbation theory [@AAS]. This fact has recently been proved to all orders [@RPW4; @TW6]. When the scalar potential is bounded below it is even possible to sum the series of leading infrared logarithms and infer their net effect at asymptotically late times [@SY]! Applying Starobinskiĭ’s technique to more complicated theories which also show infrared logarithms is a formidable problem, but solutions have recently been obtained for Yukawa theory [@MW2] and for SQED [@PTW]. It would be very interesting to see what this technique gives for the infrared logarithms we have exhibited, to lowest order, in Dirac + Einstein. And it should be noted that even the potentially complicated, invariant operators which might be required to settle the gauge issue would be straightforward to compute in such a stochastic formulation.
Appendix: nonlocal terms from section 4
=======================================
It is important to establish that the nonlocal terms make no significant contribution at late times, so we will derive the results summarized in Table \[noncon\]. For simplicity we denote as $[U^I]$ the contribution from each operator $U^I_{ij}$ in Table \[nond\]. We also abbreviate $\Psi^0(\e,\vec{x};\vec{k},s)$ as $\Psi^0(x)$.
Owing to the factor of $1/a'$ in $U^1_{ij}$, and to the larger number of derivatives, the reduction of $[U^1]$ is atypical, $$\begin{aligned}
\lefteqn{[{\rm U^1}] \equiv \f{\kappa^2}{2^8\pi^4} \f{1}{a}
\hspace{-.1cm} \not{\hspace{-.08cm}\del}\del^4\int d^4x'\f{1}{a'}
\Biggr\{\f{\ln(\mu^2\D x^2_{\scriptscriptstyle ++})}{\D x^2_{
\scriptscriptstyle ++}}-\f{\ln(\mu^2\D x^2_{\scriptscriptstyle +-})}{
\D x^2_{\scriptscriptstyle +-}}\Biggl\}\Psi^0(x') \; , } \\
&&=\f{-i\kappa^2}{2^6\pi^2a}\g^0\Psi^0(x)\Bigl[-2ik\del_{\e}
+\del_{\e}^{2}\Bigr]\Biggl\{\del_{\e}\int_{\e_i}^{\e} \!\! d\eta'
(-H\e') \Bigl(\f{e^{2ik\D\e}-1}{\Delta \eta}\Bigr) \nonumber \\
&& \hspace{6cm} + \del_{\e}^2 \int_{\e_i}^{\e} \!\! d\eta'
(-2H\e')\ln(2\mu\D\e)\Biggr\} \; , \\
&& = \f{-i\kappa^2}{2^6\pi^2a}\g^0\Psi^0\Bigl(-2ik+\del_{\e}\Bigr)
\Biggl\{-\f{e^{2ik(\e+\f{1}{H})}-1}{(\e+\f{1}{H})^2}\nn\\
&& \hspace{2cm} +\f{(2ik-H)e^{2ik(\e+\f{1}{H})}}{\e+\f{1}{H}}
-\f{3H^2}{(1+H\e)}+\f{2H^3\e}{(1+H\e)^2}\Biggr\}\; , \\
&& = \f{\kappa^2H^2}{2^6\pi^2}(H\e)iH\g^0\Psi\Biggl\{\f{2\Bigl[
e^{\f{2ik}{H}(1+H\e)}-1-2H\e\Bigr]}{(1+H\e)^3}+\f{(1-\f{2ik}{H})
e^{\f{2ik}{H}(1+H\e)}}{(1+H\e)^2}\nn\\
&& \hspace{6cm} + \f{5-4ik\e-\f{2ik}{H}}{(1+H\e)^2}+
\f{\f{6ik}{H}}{1+H\e}\Biggr\} \,\, .\end{aligned}$$ This expression actually vanishes in the late time limit of $\eta \!
\rightarrow \! 0^-$.
$[U^2]$ was reduced in Section 4 so we continue with $[U^3]$, $$\begin{aligned}
\lefteqn{[U^3] \equiv -\f{\kappa^2H^2}{2^8\pi^4} \hspace{-.1cm}
\not{\hspace{-.08cm}\bar{\del}} \del^2 \!\! \int \! d^4x'\Biggr\{
\f{\ln(\mu^2\D x^2_{\scriptscriptstyle ++})}{\D x^2_{\scriptscriptstyle
++}} - \f{\ln(\mu^2\D x^2_{\scriptscriptstyle +-})}{\D x^2_{
\scriptscriptstyle +-}} \Biggl\} \Psi^0(x') \; , } \\
&& = -\f{\kappa^2H^2}{2^8\pi^4}\hspace{-.1cm}
\not{\hspace{-.08cm}\bar{\del}}i4\pi^2\Psi^0(x)\Biggl\{
2\ln\Bigl[\f{2\mu}{H}(1+H\e)\Bigr] + \int_{\e_i}^{\e} d\e'
\Bigl(\f{e^{2ik\D\e}-1}{\D\e}\Bigr)\Biggr\} \; , \\
&& = \f{\kappa^2H^2}{2^6\pi^2}k\g^0\Psi^0(x)\Biggl\{2\ln\Bigl[
\f{2\mu}{H}(1+H\e)\Bigr]+\int_{\e_i}^{\e}d\e'
\Bigl(\f{e^{2ik\D\e}-1}{\D\e}\Bigr)\Biggr\} \; , \\
&& \longrightarrow \f{\kappa^2H^2}{2^6\pi^2}iH\g^0\Psi^0(x) \times
-\f{i k}{H} \Biggl\{2\ln(\f{2\mu}{H})-\int_{\e_i}^{0}d\e'
\Bigl(\f{e^{-2ik\e'}-1}{\e'}\Bigr)\Biggr\} . \qquad \label{U3}\end{aligned}$$ $U^4_{ij}$ has the same derivative structure as $U^3_{ij}$, so $[U^4]$ follows from (\[U3\]), $$\begin{aligned}
\lefteqn{[U^4] \equiv -\f{\kappa^2H^2}{2^8\pi^4} \times 8 \hspace{-.1cm}
\not{\hspace{-.08cm}\bar{\del}} \del^2 \!\! \int \! d^4x'\Biggr\{
\f{\ln(\frac14 H^2 \D x^2_{\scriptscriptstyle ++})}{\D x^2_{
\scriptscriptstyle ++}} - \f{\ln(\frac14 H^2 \D x^2_{\scriptscriptstyle
+-})}{\D x^2_{\scriptscriptstyle +-}} \Biggl\} \Psi^0(x') \; , } \\
&& \hspace{1.3cm} = \f{\kappa^2H^2}{2^6\pi^2}8k\g^0\Psi^0(x)\Biggl\{2
\ln\Bigl[ (1+H\e)\Bigr]+\int_{\e_i}^{\e}d\e'
\Bigl(\f{e^{2ik\D\e}-1}{\D\e}\Bigr)\Biggr\} \; , \qquad \\
&& \hspace{1.3cm} \longrightarrow \f{\kappa^2H^2}{2^6\pi^2} i H \g^0
\Psi^0(x) \times 8 i \f{k}{H} \int_{\e_i}^{0} \!\! d\e'
\Bigl(\f{e^{-2ik\e'}-1}{\e'}\Bigr) \; .\end{aligned}$$
$U^5_{ij}$ has a Laplacian rather than a d’Alembertian so we use identity (\[key1\]) for $[U^5]$. We also employ the abbreviation $k\D\e \!=\! \a$, $$\begin{aligned}
\lefteqn{[U^5] \equiv 4 \f{\kappa^2H^2}{2^8\pi^4} \hspace{-.1cm}
\not{\hspace{-.08cm} \del} \nabla^2 \!\! \int \! d^4x'\Biggr\{\f{
\ln(\mu^2\D x^2_{\scriptscriptstyle ++})}{\D x^2_{\scriptscriptstyle
++}} - \f{\ln(\mu^2\D x^2_{\scriptscriptstyle +-})}{\D x^2_{
\scriptscriptstyle +-}} \Biggl\} \Psi^0(x') \; , } \\
&& \hspace{-.5cm} = 4\f{\kappa^2H^2}{2^8\pi^4}\hspace{-.1cm}
\not{\hspace{-.08cm}\del} \nabla^2 \Bigl( \f{-4i\pi^2}{k} \Bigr)
\Psi^0(x) \int_{\e_i}^{\e}d\e' e^{i\a} \nn \\
&& \hspace{0cm} \times \Biggl\{\!-\cos(\a) \! \int_{0}^{2\a} \!\!\!
dt \, \f{\sin(t)}{t} + \sin(\a) \Bigl[\int_{0}^{2\a} \!\!\! dt \Bigl(
\f{\cos(t) \!-\! 1}{t}\Bigr) + 2\ln\Bigl(\f{H\a}{k}\Bigr)\Bigr] \!
\Biggr\} , \qquad \\
&& \hspace{-.5cm} = \f{\kappa^2H^2}{2^6\pi^2} i H \g^0 \Psi^0(x)
\times 4\f{k^2}{H} \int_{\e_i}^{\e} \!\! d\e'e^{2i\a}\Bigl[\int_{0}^{2\a}
\!\!\! dt\Bigl( \f{e^{-it}-1}{t}\Bigr)+\ln(H\D\e)^2\Bigr] \; , \qquad \\
&& \hspace{-.5cm} \longrightarrow \f{\kappa^2H^2}{2^6\pi^2}iH\g^0
\Psi^0(x) \times 4 \f{k^2}{H} \int_{\e_i}^{0} \!\! d\e'e^{2i\a} \Bigl[
\int_{0}^{2\a} \!\!\! dt \Bigl(\f{e^{-it}-1}{t}\Bigr)+\ln(H\e')^2\Bigr] .\end{aligned}$$ $U^6_{ij}$ has the same derivative structure as $U^5_{ij}$ but it acts on a different integrand. We therefore apply identity (\[key3\]) for $[U^6]$, $$\begin{aligned}
\lefteqn{[U^6] \equiv 7 \f{\kappa^2H^2}{2^8\pi^4} \hspace{-.1cm}
\not{\hspace{-.08cm}\del} \nabla^2 \!\! \int \! d^4x' \Biggr\{
\f{1}{\D x^2_{++}}-\f{1}{\D x^2_{+-}}\Biggl\}\Psi^0(x') \; , } \\
&& = 7\f{\kappa^2H^2}{2^8\pi^4}\hspace{-.1cm}
\not{\hspace{-.08cm}\del}\nabla^2\times(-i4\pi^2)k^{-1}
\Psi^0(x)\int_{\e_i}^{\e}d\e'e^{ik\D\e}\sin(k\D\e) \; , \\
&&= \f{\kappa^2H^2}{2^6\pi^2}iH\g^0\Psi^0(x)\times-\f{7}{2}\f{ik}{H}
\Bigl[e^{\f{2ik}{H}(1+H\e)}-1\Bigr] \; , \\
&&\longrightarrow\f{\kappa^2H^2}{2^6\pi^2}iH\g^0\Psi^0(x)
\times-\f{7}{2}\f{ik}{H} \Bigl[e^{\f{2ik}{H}}-1\Bigr] \; .\end{aligned}$$
.3cm
**Acknowledgements**
This work was partially supported by NSF grant PHY-0244714 and by the Institute for Fundamental Theory at the University of Florida.
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[^1]: Of course the spinor and vector representations of the local Lorentz transformation are related as usual, with same parameters $\omega_{cd}(x)$ contracted into the appropriate representation matrices, $$\begin{aligned}
\Lambda_{ij} \equiv \delta_{ij} - \frac{i}2 \omega_{cd} J^{cd}_{~~ij} + \ldots
\qquad {\rm and} \qquad
\Lambda_b^{~c} = \delta_b^{~c} - \omega_b^{~c} + \ldots \; . \nonumber\end{aligned}$$
|
---
author:
- 'B. Sánchez-Andrade Nuño'
- 'R. Centeno'
- 'K. G. Puschmann'
- 'J. Trujillo Bueno'
- |
\
J. Blanco Rodríguez
- 'F. Kneer'
date: 'Received date; accepted date'
title: Spicule emission profiles observed in 10830 Å
---
[Off-the-limb observations with high spatial and spectral resolution will help us understand the physical properties of spicules in the solar chromosphere.]{} [Spectropolarimetric observations of spicules in the 10830Å multiplet were obtained with the Tenerife Infrared Polarimeter on the German Vacuum Tower Telescope at the Observatorio del Teide (Tenerife, Spain). The analysis shows the variation of the off-limb emission profiles as a function of the distance to the visible solar limb. The ratio between the intensities of the blue and the red components of this triplet $({\cal R}=I_{\rm blue}/I_{\rm red})$ is an observational signature of the optical thickness along the light path, which is related to the intensity of the coronal irradiation. ]{} [We present observations of the intensity profiles of spicules above a quiet Sun region. The observable ${\cal R}$ as a function of the distance to the visible limb is also given. We have compared our observational results to the intensity ratio obtained from detailed radiative transfer calculations in semi-empirical models of the solar atmosphere assuming spherical geometry. The agreement is purely qualitative. We argue that future models of the solar chromosphere and transition region should account for the observational constraints presented here. ]{}
Introduction
============
The [ solar]{} chromosphere is in a highly dynamic state. Its small-scale structures evolve in timescales of minutes or even less. Due to its low density, the chromosphere is transparent in most of the optical spectrum. Nevertheless, in lines such as H$\alpha$ and He [i]{} 10830Å, the significant absorption provides a mean for direct studies of the chromosphere’s peculiar characteristics, such as bright plages in active regions, dark and bright fibrils on the disc in both the quiet Sun’s network and around sunspots, as well as spicules above the limb. Recent studies [e.g. @tziotziou03] claimed that it is possible that many of these chromospheric features have the same physical nature within different scenarios.
In the chromosphere, between the photosphere and the much hotter corona, the average temperature starts to rise outwards, along with a decrease of the density. The dynamics of a magnetised gas depends on the ratio of the gas pressure and magnetic pressure, the plasma $\beta$ parameter, $\beta$=$P_\mathrm{gas}/P_\mathrm{mag}$, with $P_\mathrm{mag}$=$B^2/(8\pi)$. From the low chromosphere into the extended corona, this plasma parameter decreases from values $\beta>1$ to a low-beta regime, $\beta\ll1$, where the plasma motions are magnetically driven.
Spicules, known for more than 130 years [see hand drawings in @secchi1877], represent a prominent example of the dynamic chromosphere. We refer the reader to reviews by @beckers68 [@beckers72] and to the paper by @wilhelm00 on UV properties. Spicules are seen at and outside the limb of the Sun as thin, elongated features that develop speeds of 10–30 kms$^{-1}$ and reach heights of 5–9 Mm on average, during their lifetimes of 3–15 minutes. As pointed out by @sterling00, a key impediment to develop a satisfactory understanding has been the lack of reliable observational data. Many theoretical models have been developed to understand the nature of spicules, using a wide variety of motion triggers and driving mechanisms. In this study we focus on the 10830Å triplet emission line using recent technical improvements in observational facilities. We are able to provide observational evidence of the link between the corona and the infrared emission of this line, in the frame of the current theoretical models of the solar atmosphere.
The 10830 Å multiplet consists of the three transitions from the upper term [ $^3$P$_{2,1,0}$]{}, which has three levels, to the lower [ metastable]{} term [ $^3$S$_{1}$]{}, which has one single level. The two transitions from the J=2 and J=1 upper levels appear blended at typical chromospheric temperatures, and form the so-called red component, at 10830.3 Å. [ (Note that the two red transitions are only 0.09 Å apart).]{} The blue component, at 10829.1 Å, corresponds to the transition from the upper level with J=0 [ to the lower level with J=1]{}. The energy levels that take part in these transitions are basically populated via an ionization-recombination process [@avrett94]. The EUV coronal irradiation (CI) at wavelengths lambda $\lambda<504$ Å ionizes the neutral helium, and subsequent recombinations of singly ionized helium with free electrons lead to an overpopulation of these levels.
[ Alternative theories suggest other mechanisms that might also contribute to the formation of the helium lines relying on the collisional excitation of the electrons in regions with higher temperature [e.g., @a97]]{}
@Centeno06 modelled the ionisation and recombination processes using various amounts of CI, non-LTE radiative transfer, and different atmospheric models [ [see also @cente07 in preparation]]{}. They have simulated limb observations for different heights, obtaining synthetic emission profiles in spherically symmetric models of the solar atmosphere. One conclusion of their study is that the ratio of intensities $({\cal R}=I_{\rm blue}/I_{\rm red})$ of the ‘blue’ to the ‘red’ components of the 10830 Å emission is a very good candidate for diagnosing the CI. The population of the metaestable level depends on optical thickness, whose variation with height governs the change in the ratio $\cal R$ as a function of the distance to the limb.
@truj05 measured the four Stokes parameters of quiet-Sun chromospheric spicules and could show evidence of the Hanle effect by the action of inclined magnetic fields with an average strength of the order of 10 G. They modelled the 10830 Åprofiles assuming the medium along the integrated line of sight as a slab of constant properties and with its optical thickness as a free parameter. @truj05 showed that the observed intensity profiles and their ensuing $\cal R$ values can be reproduced by choosing an optical thickness significantly larger than unity. @Centeno06 demonstrated that this optical thickness is related to the coronal irradiance (through the ratio $\cal R$), thus providing a physical meaning to the free parameter in the slab model [ (see also Centeno et al. 2007)]{}.
We present novel observations showing the spectral emission of 10830 Å and its dependence on the height of the spicules above a quiet region. We compare the deduced observational $\cal R$ with that obtained from detailed non-LTE numerical calculations using available atmospheric profiles.
Observations
============
The observations were carried out on December 4$^{th}$, 2005, at the Vacuum Tower Telescope (VTT) at Observatorio del Teide. They were supported by the Kiepenheuer Adaptive Optics System [KAOS, @luehe03]. [ We used the Echelle spectrograph of the VTT and the new version (TIP-2) of the Tenerife Infrared Polarimeter [@pilllet99], which has a larger CCD camera [@collados07] ]{}. The seeing conditions were good [ (average seeing after KAOS correction around 7 cm, maximum 12 cm)]{}.
[ The strong darkening close to the solar limb and the actual presence of the limb make it difficult to use KAOS for off-limb observations, since the correlation algorithm was not developed for this kind of observation.]{} Fortunately, we could use a nearby facula inside the limb for the tracking lock point of the adaptive optics system.
Although we carried out full Stokes vector observations, in this study we have used only the intensity profiles. The spectral region covered by TIP-2 spans from 10826 to 10837 Å and contains the 10830 Å multiplet. [The spectral sampling was $11$ mÅ/px]{}, and the slit was 40 long and 05 wide. It was oriented parallel to the NE limb, in a region far from major activity at that time, at 598 from North. We scanned the full height of the spicules, up to 7 off the visible solar limb, with a step size of 035.
At each position, 5 consecutive spectra were measured with 10 accumulations of 250 ms each, with a total integration time of 2.5 s per off-limb slit position. A nearby disc profile was also taken to help remove the scattered light from the off-limb spectra [ via the data analysis]{}.
Data reduction and processing \[ajuste\]
========================================
We applied usual flatfielding and dark current corrections to the data. A high frequency, quasi-periodic, spurious electronic pattern in the profiles was removed using a low-pass Fourier filter, which left the frequencies containing the spectral line information untouched.
We define the position of the solar limb as the height of the first scanning position where the helium line appears in emission. For increasing distances to the solar limb a decreasing amount of sunlight is added by scattering in the Earth’s atmosphere and by the telescope optical surfaces to the signal. Since the true off-limb continuum must be close to zero, i.e. below our detection limit, the observed continuum signal measures the spurious light. Therefore, we removed the spurious continuum intensity level by using the information given by a nearby average disc spectrogram. This first subtraction estimates the continuum level on a region 6 Å away from the 10830 Å emission lines. After this correction with a coarse estimate of the spurious light, a second correction was applied to remove the residual continuum level seen around the emission lines. This was needed since the transmission curve of the used prefilter is not flat but variable with wavelength.
[ Figure \[fig:spe\] shows the emission profiles of the 10830 Å (after the reduction process) for different heights above the limb. Figure \[fig:3d\] illustrates this in three dimensions, as a function of wavelength and the distance to the solar limb, clearly showing a change in the intensity ratio of the blue and red components of the multiplet $({\cal R}=I_{\rm blue}/I_{\rm red})$ with height. This will be discussed in Sect. \[results\].]{}
![Measured 10830 Å emission profiles for increasing distances to the solar limb, scanning a broad range of the height extension of the spicules. Each profile is the average of the 312 pixels along the slit (which was always kept parallel to the limb)[]{data-label="fig:spe"}](7936fig1.ps){width="50.00000%"}
![3D representation of the measured 10830 Å emission profiles for increasing distances to the solar visible limb. Note that the x-axis is wavelength, the y-axis the height above solar limb and the z-axis the intensity normalised to the maximum emission in the line centre of the red component.[]{data-label="fig:3d"}](7936fig2.ps){width="47.00000%"}
For the calculation of $\cal{R}$ we need to determine the amplitudes of the blue and red components of the emission profile (as shown in Fig. \[fig:ajuste\]).
To determine the core wavelength of the red component of the triplet we fitted a Gaussian profile to its core, in a 1.3 Å range around the maximum. After symmetrising the observed profile around this maximum, using the values on the red side of the red component, we fitted another Gaussian function to the resulting symmetric profile. Subtraction of the fitted symmetric profile from the data leaves the emission profile of the blue component, which was also approximated by a Gaussian to determine its central wavelength. Our tests trying to fit directly both profiles using two Gaussians failed in a number of cases, probably due to the following reasons: (a) the red component is in fact the result of two blended lines, (b) the much weaker blue component was almost hidden in the broadened red component, and (c) the presence of noise. Our technique determines first the red component and then, after subtraction of the fitted profile, the blue one.
We have thus separated the helium emissions into their red and blue components assuming only 2 that both are present and that they are both symmetric. We can now measure their widths and intensities and also check that the line core positions coincide with the theoretical ones. After the fitting process the residuals between measured and observed profiles were small, the largest errors occurring in the determination of the core intensities of the red line. This happens because the red component consists of two blended lines (with a separation of 0.09 Å), a fact that flattens the emission profile near the core as opposed to a more peaked Gaussian function. Nevertheless, the differences between fitted profiles and data are only significant in the red core and are always lower than +0.08 of the maximum normalised intensity, with a mean difference of $\sim$0.02. To avoid systematic errors, we used the observational values for the centre of the red component when calculating $\cal R$.
![Determination of the blue and red components of the 10830 Å triplet from the observed emission profiles. In this example the slit was placed at 14 off the solar visible limb. See text for details. The solid line [ represents]{} the average emision profile. The dotted line is the Gaussian fit to the symmetrised red component. Subtraction of this from the observed profile leaves the blue component, which is also fitted by a Gaussian profile (thin solid line). The sum of both Gaussians (dashed line) gives the fit to the observed profile.[]{data-label="fig:ajuste"}](7936fig3.ps){width="50.00000%"}
Results\[results\]
==================
The chromospheric temperature and density are too low to populate the ortho-helium levels via collisions [@avrett94]. The EUV irradiation from the corona (CI) ionises the para-helium, and the subsequent recombinations lead to an overpopulation of all the ortho-helium levels, in particular those involved in the 10830 Åtransitions. @Centeno06 and Centeno et al. (2007) have modelled the off-the-limb emission profiles and concluded that the ratio $\cal{R}$=$I_{blue}/I_{red}$ is a function of the height and a direct tracer of the amount of CI. Here we compare the results from the theoretical modelling with observations.
@truj05 [modelled their spectropolarimetric observations assuming a slab with constant physical properties with a given optical thickness]{}. In the optically thin regime $\cal R$$\sim$0.12, which is the ratio of the relative oscillator strengths of the triplet. As the optical thickness (at the line-centre of the red blended component) grows, this ratio also increases until it reaches a saturation value slightly larger than 1 for $\tau\sim10$. (This type of calculation can be done and improved as explained in Trujillo Bueno & Asensio Ramos 2007). To reproduce the observed emission profile @truj05 had to choose $\tau \sim 3$. Interestingly, [ the values of $\tau$ yielded by this modelling strategy are consistent]{} with the more realistic approach of Centeno (2006), where non-LTE radiative transfer calculations in semi-empirical models of the solar atmosphere are presented, using spherical geometry and taking into account the ionising coronal irradiation. With our data we are able to test such theoretical calculations by comparing the measured values of $\cal R$ with those resulting from various chromospheric models. This way we may eventually trace the amount of CI inciding on the spicules. The analysis described in Sect. \[ajuste\] yielded the values of $\cal R$ for the observed profiles. The resulting dependence on the distance to the solar limb, for each pixel along the slit and each position of the slit above the limb, are presented in Fig. \[fig:ratios\]. The solid black line gives the average value of $\cal R$.
![Measured ratio $\cal R$=$I_{blue}/I_{red}$ as a function of distance to the solar limb. Thin lines come from each pixel along the slit. The thick solid line represents the average and the dashed line the value of the optically thin regime.[]{data-label="fig:ratios"}](7936fig4.ps){width="48.00000%"}
The dependence of $\cal R$ with height can be understood in a qualitative way as follows: in the outer layers of the chromosphere the density is so low that the transitions occur in the optically thin regime. With decreasing altitude the ratio $\cal R$ increases (proportionally with density) until a maximum optical thickness is reached. At even lower layers, although the density still continues to rise, the extinction of the coronal irradiance leads to a reduction in the number of ionizations, which results in a decrease of the optical thickness in the core wavelength of the red component, [ and thus in a decrease of $\cal R$.]{}
For a quantitative comparison with theoretical modelling we have used the results from @Centeno06 and [@cente07] where they calculated the ratios $\cal R$ for different standard model atmospheres: FAL-C and FAL-P [@fontenla91] and FAL-X [@avrett95]. The FAL-C and FAL-X models may be considered as illustrative of the thermal conditions in the quiet Sun, while the FAL-P model of a plage region. The FAL-X model has a relatively cool atmosphere in order to explain the molecular CO absorption at 4.6 $\mu$.
The comparison is shown in Fig. \[fig:comp\]. We notice that the modelled height variations of $\cal{R}$ agree only in a qualitative manner with what is found in our observations. However, the calculations from different models of the solar atmosphere are unable to reproduce the measured ratio. Higher values of the coronal irradiance lead to an increase of the optical thickness (at the line centres of the multiplet) and an upward shift in the run of $\cal{R}$ vs. height. Yet the shape of the height dependence is mainly given by the atmospheric density profile and the attenuation of the ionising radiation as it reaches the lower layers of the chromosphere. It is also clear from Fig. \[fig:comp\] that the models do not extend high enough.
![Observed (average) vs. theoretical variation of the ratio $\cal R$$ =
I_{blue}/I_{red}$ with height.[]{data-label="fig:comp"}](7936fig5.ps){width="50.00000%"}
Conclusions\[conclusion\]
=========================
The theoretical behaviour of the ratio $\cal R$ agrees qualitatively with observations. Yet, a quantitative comparison shows poor agreement. Also, the simulated ratios are highly model dependent. As already explained, [ the failure to reproduce the observed profiles is very likely due to the density stratification not being adequate for spicule modelling and to the limited vertical extension of the atmospheric models.]{} Modelling of the intensity ratio $\cal R$ in the infrared triplet should account for the fact that the solar chromosphere is inhomogeneous on small scales and that the spicules are small-scale intrusions of chromospheric matter into the hot corona.
New data of spicule regions near the poles and the equator, below coronal holes or coronal active regions should help us to understand the detailed behaviour of the 10830 Å lines. In further work, we will extend this study to the full Stokes vector, in order to see the variation of the linear polarization - or even the variation of the Hanle effect - with height. [ It would also be interesting to use the most recent models of active region fibrils and spicules [e.g., @hegg07] in order to see whether or not they agree with our observations.]{} Future models of the solar chromosphere should be constrained by the observational evidences presented here.
We thank M. Collados for the extensive help and discussions, as well as for the reduction software. The help from A. Lagg was very valuable during the reduction phase. BSAN acknowledges a PhD fellowship at the International Max Planck Research School [*On Physical Processes in the Solar System and Beyond*]{}. KGP thanks the Deutsche Forchungsgemeinschaft for support through grant KN 152/29. RC and JTB acknowledge the support of the Spanish Ministerio de Educación y Ciencia through project AYA2004-05792. JTB thanks the Akademie der Wissenschaften zu Göttingen for the Gauss-Professur during a sabatical stay at the Institut für Astrophysik of the University of Göttingen. The Vacuum Tower Telescope is operated by the Kiepenheuer-Institut für Sonnenphysik, Freiburg, at the Spanish Observatorio del Teide of the Instituto de Astrofísica de Canarias. The National Center for Atmospheric Research is sponsored by the National Science Foundation
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|
---
author:
- 'Jose Barrionuevo and Michael T. Lacey[^1]\'
title: ' A Weak–type Orthogonality Principle '
---
=2
Introduction and Principal Inequalities
=======================================
We are interested in the relationships between three different concepts, first and foremost is that of the phase space, by which we generally mean the Euclidean space formed from the cross product of the spatial variable with the dual frequency variable. Next, we want to associate subsets of that space with functions, the subset describing the location of the function in natural ways. And finally, we want to understand the extent to which orthogonality of the functions can be quantified by geometric conditions on the corresponding sets in the phase plane.
These concerns are not currently very much in the forefront of harmonic analysis, but rather the means towards an end. We treat them as a subject in their own right because the inequalities that we obtain are of an optimal nature and they refine the basic orthogonality issues in the proof of the bilinear Hilbert transform inequalities [@lt.pnas], and complement investigations into “best basis" signal or image processing [@w], including the directional issues that arise in the context of brushlets [@brush].
We state the principal results, and then turn to complementary issues and discussion. For a set $W\subset\ZR^d$ of finite volume we define ${\lambda}W:=\{c(W)+{\lambda}(x-c(W))\mid x\in W\}$ where $c(W)$ is the center of mass of $W$. We say that $W$ is [*symmetric*]{} if $-(W-c(W))=W-c(W)$. We call the product $s=W_s\times{\Omega}_s$ of a symmetric convex set $W_s$ and a second set a [*tile*]{}. Here, we need not assume that the second set lies in $\ZR^d$, it could lie in some other set altogether.
We use tiles to study the connection between geometry of the phase plane and orthogonality which is the intention of the following definition.
d.adapted Let $\S$ be a set of tiles. The functions ${\bf {\Phi}}=\{{\varphi}_s\mid s\in\S\}$ are said to be [*adapted to*]{} $\S$ if the functions ${\varphi}_s$ are Schwartz functions, and for all $s\in\S$, 5 e.=1 \_s.2.=1,\
\[e.perp\] \_s,\_[s’]{}.=0\
\[e.space\] (1+(x,s ))\^[-2d-5]{},x\^d,\
(x,W):={a>0xaW}.
We call a collection of sets $\z0G$ a [*grid*]{} if for all $G,G'\in\z0G$, we have $G\cap
G'=\emptyset, G$ or $G'$.
t.count1 Let $\S$ be a set of tiles such that 5 e.grid {s s§}\
e.disjoint {s§}\
e.geo s,s’§, s’ s s’ -c(s’ )s -c(s ). Let $\{\svf s \mid s\in\S\}$ be adapted to $\S$, and for ${\lambda}>0$ and $f\in L^2(\ZR^d)$ let 1 e.> §\_:={s§}. Then we have the inequality 0 \_[s§\_]{}(2+KC\_0\^2)\^[-2]{} f\_2\^2. Here and throughout $K$ denotes a constant depending only on dimension $d$. This inequality can be rephrased as 0 { s§} .L\^[2,]{}(\^d§). f.2. . In this inequality, $\ind A$ denotes the indicator function for the set $A$, $L^{2,{\infty}}$ denotes the weak $L^2$ space, and we assign $\ZR^d\times \S$ the product measure of Lebesque measure times counting measure on $\S$.
Note that in the last inequality if the weak $L^2$ norm could be replaced by the $L^2$ norm, we would have a Bessel inequality. However, the weak $L^2$ space cannot be replaced by any smaller Lorentz space. We demonstrate this with an example at the conclusion of the paper. However, in the context of computation, one cannot distinguish between $L^2$ and weak–$L^2$.
It is worth noting that the concluding Lemmas of [@lt.pnas] and [@lt] study exactly the question of orthogonality for more restrictive class of functions $\svf s $; therein the tiles are rectangles in the phase plane. And the analysis shows that orthogonality is linked to the boundedness of the the related maximal function.
There is a second form of this Theorem in which a multiscale object plays the role of a single tile. We call that object a [*cluster*]{}.
d.cluster Let $\{\svf s \mid s\in\C\}$ be adapted to a set of tiles $\C$. We call $\{\svf s \mid
s\in\C\}$ a [*cluster with shadow* ]{} $I$ if the following four conditions are met. $(a)$ $\{s\in\C\}$ are pairwise disjoint, $(b)$ for all $s\in\C$, $\sw s \subset I$, 4 (c) s=s’, s s’ = s ,s’ .=0,\
(d) \_[s]{}\^[-1]{}\^2 C\_1.
t.cluster Let $\S$ be a collection of tiles which is a disjoint union of subcollections $\{\C_I\mid I\in\I\}$. Let $\{\svf s \mid s\in\S\}$ be adapted to $\S$, so that for each $I\in\I$, $\{ \svf s \mid s\in\C_I\}$ is a cluster. Suppose that .grid/ and .geo/ hold. Finally, suppose that the clusters $\C_I$ satisfy this condition. 1 e.cl.disjoint s=s’§, s\_I, s s’ s’\_Is’ I=. Define 0 SQ(I,f)\^2:=1 \_[s\_I]{}\^2,I. Then, under the assumptions $SQ(I,f)\ge{\lambda}$ for all $I\in\I$ and $\abs{\ip f,\svf s .}\le2{\lambda}\sqw s $ for all $s\in\S$, we have 0 \_[I]{}\^[-2]{}K(1+ C\_1\^[1/2]{}C\_0\^2)f\_2\^2.
The non–linear form of the hypotheses of this last Lemma preclude a natural formulation of a weak–type inequality.
These Theorems can also be used to study two complicated operators of harmonic analysis, namely Carleson’s operator controlling the maximum partial Fourier sums [@carleson] and the bilinear Hilbert transform [@lt.pnas]. A result clearly related to these Theorems can be found in a neglected paper of Prestini [@prestini]. But the first forms of these Theorems, again for special tiles, is in [@lacey].
Proofs of Theorems
==================
We will need a precise estimate of $\ip \svf s ,\svf s' .$, which is the purpose of
l.ip Let $s$ and $s'$ be two tiles with $\sw s' -c(\sw s' )\subset\sw s -c(\sw s )$ but $\sw
s' \cap\sw s =\emptyset$. Let $\{\svf s ,\svf s' \}$ be adapted to $\{s,s'\}$. Then there is a constant $K$ so that 1e.ss’ K[C\_0\^2]{} \_[xs’ ]{}(1+(x,s ))\^[-d-5]{}.
Heuristically, the Lemma follows from the estimate 0 \_[s’ ]{}dx s .L\^(s’ ).s’ .1. and .space/. Of course the first step must be made precise.
Observe that 0 K\_0\^(x), which is a consequence of .space/. We can estimate 2 & K\_0\^\_0\^,.\
& KC\_0\^2\_0\^\_0\^ . The point to exploit is that $a\sw s \cap{\alpha}\sw s' =\emptyset$ for $a,{\alpha}\le{}\z1{\sigma}:=\sup\{ a\mid
aW_s\cap aW_{s'}\not=\emptyset\}$.
We bound the double integral above by breaking the region of integration into four pieces. Take $R^0=[0,\z1{\sigma})$, $R^1=[\z1{\sigma},{\infty})$ and define regions in the $(a,{\alpha})$ plane by 0 R\_[\_1\_2]{}=R\^[\_1]{}R\^[\_2]{},\_i{0,1}. The integral over $R_{00}$ is zero by the choice of $\z1{\sigma}$. The integral over $R_{01}$ is 0 \_0\^[1]{}\_[1]{}\^ K (1+1)\^[-d-5]{} . We have a similar estimate for the integral over $R_{10}$. Finally, the estimate over $R_{11}$ is 0 \_[1]{}\^\_[1]{}\^ K (1+1)\^[-d-5]{}. These estimates supply us with 0 K C\_0\^2 (1+1)\^[-d-5]{}, which is quite close to our claim.
To finish, observe that $\z1{\sigma}\ge1$ as $\sw s $ and $\sw s' $ are disjoint. We shall also see that ${\sigma}(x,\sw s )\le1+2\z1{\sigma}$ for all $x\in\sw
s' $. These two points finish the proof of the Lemma.
Indeed, we can assume that $\sw s $ and $\sw s' $ are closed sets. Let $\z1x$ be a point in $\z1{\sigma}\sw s \cap\z1{\sigma}\sw s' $. Note that for any point $x\in\sw s' $, 0 x-c(s )=(x-c(s’ ))+(c(s’ )-1x)+(1x -c(s )). Of the three terms on the right, the first is in $\sw s' -c(\sw s' )\subset \sw s -c(\sw s )$, by assumption, the second is in $\z1{\sigma}(\sw s' -c(\sw s' ))\subset\z1{\sigma}(\sw s -c(\sw s ))$ by symmetry of $\sw s' $ and assumption, and the third is in $\z1{\sigma}(\sw s -c(\sw s ))$. The convexity of $\sw s $ then implies that $x\in(2\z1{\sigma}+1)\sw s $. Hence, ${\sigma}(x,\sw s )\le1+2\z1{\sigma}$ for all $x\in\sw
s' $ as was to be shown.
It is sufficient to prove a different assertion. For $f\in L^2(\ZR^d)$, we assume that 1 e.b 12,s§, and prove that 1e.-2 \_[s§]{}(1+KC\_0\^2)f\_2\^[2]{}. For $k\ge0$ define $\S_k:=\{s\in\S\mid 2^{k}\le{}\abs{\sw s }^{-1/2}\abs{\ip f,\svf s
.}\le{}2^{k+1}\}$. One sees that the sum over this set of tiles is at most $2^{-2k}$ times the upper bound in .-2/. The Theorem is then established in the case of ${\lambda}=1$, but this is sufficient as $f\in L^2(\ZR^d)$ is arbitrary.
For the proof of .-2/ we can assume that $\S$ is a finite collection, so that [*a priori*]{} the quantity 0 B:=\_[s§]{}f,s s .2. is finite. It suffices to estimate $B$, for by using .b/ we see that 1e.W \_[s§]{}\_[s§]{}\^2 = f,\_[s§]{}f,s s .Bf.2..
We expand $B^2$ into diagonal and off–diagonal terms. 1e.B2 B\^2=\_[s§]{}f,s s \_2\^2\_[s§]{}\^2+ 2\_[s§]{}|f,s |0O(s), where we define $\S(s)=\{s'\in\S-\{s\}\mid \szw s \subset\szw s' ,\langle \svf s
,\svf s' \rangle\not=0 \},$ and 1e.O 0O(s):=\_[s’§(s)]{}. Recall that $\ip \svf s
,\svf s' .\not=0$ only if $\szw s \cap \szw s' \not=\emptyset$. But then from the grid structure, we may assume that $\szw s \subset\szw s' $.
We have already seen that the diagonal term is dominated by $B\norm f.2.$, so that the term $\z0O(s)$ is our concern. Using .ss’/ and the upper bound on $\ip \svf s' ,f.$ we have 2 0O(s)&KC\_0\^2\^[-1/2]{}\_[s’§(s)]{} \_[xs’ ]{}(1+(x,s ))\^[-d-1]{}\
&KC\_0\^2\^[-1/2]{}\_[(s )\^c]{}(1+(x,s ))\^[-d-1]{}dx\
&KC\_0\^2\^[1/2]{}. For the middle line above, the sets $\szw s' $ for $s'\in\S(s)$ contain $\szw s $. But the tiles are disjoint, thus the sets $\sw s' $ are pairwise disjoint and contained in $(\sw s )^c$.
Therefore, the off diagonal term is, by .b/ and .W/, 0 \_[s§]{}|f,s |0O(s) KC\_0\^2 \_[s§]{} KC\_0\^2Bf.2.. Combining this with .B2/ we see that 0 B\^2Bf.2.+ KC\_0\^2Bf.2., which gives the desired upper bound $B$.
It suffices to consider the case of ${\lambda}=1$. The initial steps are just as before. We assume that the collection of tiles is finite and set 0 B:=\_[s§]{}f,s s .2. and estimate $B$. This is sufficient since 0 \_[I]{}\_[s§]{}\^2Bf.2.. Then by Cauchy–Schwartz, 2 B\^2&\_[s§]{}\^2+ 2\_[s§]{}|f,s |0O(s),\
&Bf.2.+\[Bf.2.\]\^[1/2]{}\^[1/2]{} Here as before, $\S(s)$ and $\z0O(s)$ are as in .O/. Then by .cl.disjoint/ and .cluster/ (c) the sets $\{\sw s' \mid s'\in\S(s)\}$ are contained in $I^c$ if $s\in\C_I$. But they are also pairwise disjoint. Indeed, consider $s',s''\in\S(s)$. If they fall into the same cluster, they are disjoint by assumption on clusters. Suppose they are in different clusters. Assuming as we may that $\szw s' \subset \szw s'' $, we see that .cl.disjoint/ implies $\sw s' \cap \sw s'' =\emptyset$.
Recalling ł.ip/ and that we have the upper bound $\abs{\ip f,\svf s'
.}\le2\sqw s' $ by assumption, we see that 2 0O(s)& KC\_0\^2\^[-1/2]{}\_[s’§(s)]{}\_[xs’ ]{}(1+(x,s ))\^[-d-1]{}\
& KC\_0\^2\^[-1/2]{}\_[I\^c]{}(1+(x,s ))\^[-d-1]{}dx if $s\in\C_I$. Hence by the definition of a cluster 2 \_[s\_I]{}0O(s)\^2& KC\_0\^4\_[s\_I]{}\^[-1]{}\^2\
& KC\_1C\_0\^4. And so, $B^2\le{}B\norm f.2.+KC_1^{1/2}C_0^2B\norm f.2.$, which proves the Theorem.
Counterexample
==============
We demonstrate the optimality of the $L^{2,{\infty}}$ norm in our Theorem. It suffice to consider the first Theorem on $\ZR$. And this we will do with the collection of disjoint rectangles 0 §:={\[2\^j,2\^[j+1]{})\[(n-1/2)2\^[-j]{},(n+1/2)2\^[-j]{})j,n}. Let ${\varphi}$ denote a Schwartz function with ${\varphi}(x)>0$ for all $x$, $L^2$ norm one, and $\hat{\varphi}$ supported on $[-1/2,1/2]$. For $s=W_s\times{\Omega}_s\in\S$ define 0 \_s(x):=e\^[2i c(\_s)x]{}\^[-1/2]{}(). Clearly, $\{{\varphi}_s\mid s\in\S\} $ is adapted to $\S$.
Take $f=\ind{[-1,0)}$. For integers $j\ge0$ and $\abs{n}<2^{j-1}$, let $s=[2^j,2^{j+1})\times[(n-1/2)2^{-j},(n+1/2)2^{-j})$. We have 2 2\^[-j/2]{}f,\_s.=&2\^[-j]{}\_[-1]{}\^0e\^[2i n2\^[-j]{}x]{}()dx\
=&e\^[3in]{}\_[--2\^[-j]{}]{}\^[-]{}e\^[2i nx]{}(x)dx Since ${\varphi}(-3/2)>0$, we see that $\abs{ 2^{-j/2}\ip f,{\varphi}_s.}\ge{}c2^{-j}$. This estimate is uniform in $n$ and $j$ as we have specified them. Thus, for all $0<{\lambda}<1$, 0 \^2\_[s§]{}{ }c’. Thus, for any finite $t$, 0 { s§} .L\^[2,t]{}(\^d§).=.
[99]{} L. Carleson. [ “On convergence and growth of partial sums of Fourier series."]{} [*Acta Math.*]{} [**116**]{} (1966) [pp. 135-157]{}.
M.T. Lacey. [ “The bilinear Hilbert transform is pointwise finite."]{} [*Rev. Mat.*]{} [**13**]{} (1997) 403—469.
M.T. Lacey, C.M. Thiele. [ “Bounds for the bilinear Hilbert transform on $L^p$." ]{} Proc. Nat. Acad. Sci. [**94**]{} (1997) 33—35.
M.T. Lacey, C.M. Thiele. [“$L^p$ Bounds for the bilinear Hilbert transform, $p>2$."]{} [*Ann. Math.*]{} [**146**]{} (1997) 693—724.
F.G. Meyer, R.R. Coifman. “Brushlets: A tool for directional image analysis and image compression." [*Appl. Comp. Harmonic Anal.*]{} [**4**]{} (1997) 188—221.
E. Prestini. [ “On the two proof of pointwise convergence of Fourier series."]{} [*Amer. J. Math.*]{} [**104**]{} (1982) 127—139.
M. V. Wickerhauser. [*Adapted Wavelet Analysis from Theory to Software*]{} A K Peters Press, 1994.
[ll]{} Jose Barrionuevo & .45in Michael Lacey\
Department of Mathematics and Statistics & .45in School of Mathematics\
University of South Alabama &.45in Georgia Institute of Technology\
Mobile AL 36688 &.45in Atlanta GA 30332\
\
jose@mathstat.usouthal.edu &.45inlacey@math.gatech.edu\
&.45in http://www.math.gatech.edu/\~lacey\
[^1]: This author has been supported by an NSF grant DMS–9706884
|
---
abstract: 'We introduce the class of operator $p$-compact mappings and completely right $p$-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator $\infty$-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster’s class.'
address:
- 'Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA'
- 'Departamento de Matemática y Ciencias, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires, Argentina and CONICET'
- 'Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina and CONICET'
author:
- 'Javier Alejandro Chávez-Domínguez'
- Verónica Dimant
- Daniel Galicer
title: 'Operator $p$-compact mappings'
---
[^1]
Introduction {#introduction .unnumbered}
============
Alexander Grothendieck, one of the most influential mathematicians of the 20th century, took his first steps in research in the area of functional analysis. In his early work, he developed a systematic theory of tensor products and norms [@grothendieck1956resume], where he set the basis of what was later known as local theory — that is, the study of Banach spaces in terms of their finite-dimensional subspaces — and exhibited the importance of the use of tensor products in the theory of Banach (and locally convex) spaces. Grothendieck’s approach, although at first it received little notice given its complexity and the language in which it was written, was rediscovered at the end of the sixties by Lindenstrauss and Pe[ł]{}czy[ń]{}ski [@lindenstrauss1968absolutely] and inspired huge developments in Banach space theory. In particular, it inspired the creation of a whole new theory: the study of operator ideals (systematized by Pietsch and the German school [@pietsch1968ideale; @pietsch1978operator]). In order to develop the metric theory of tensor products, Grothendieck gave a characterization of compactness in Banach spaces of independent interest [@grothendieck1955produits Chap. I, p. 112]: relatively compact sets are precisely those that lie within the absolutely convex hull of a null sequence. That is, a set $K$ in a Banach space $\mathbf X$ is relatively compact if and only if there is a sequence $(x_n)_{n \in \mathbb{N}}$ in $\mathbf X$ such that
$$\label{groth definicion de compacto}
K\subseteq \left\{\sum_{n=1}^{\infty} \alpha_n x_n \colon \sum_{n =1}^\infty \vert \alpha_n \vert \leq 1\right\} \mbox{ and } \lim_{n\to \infty} \Vert x_n \Vert = 0.$$
Inspired by Grothendieck’s result, Sinha and Karn [@sinha2002compact] introduced the notion of relatively $p$-compact sets. This definition describes relatively $p$-compact sets as those which are determined in a way similar to , but by $p$-summable sequences instead of null ones. That is, if $1 \leq p < \infty$ and $\frac{1}{p} + \frac{1}{p'} = 1$, a subset $K \subset \textbf X$ is called *relatively $p$-compact* if there exists a sequence $(x_n)_n$ in $\mathbf X$ such that $$\label{p-compacto}
K\subseteq \left\{\sum_{n=1}^{\infty} \alpha_n x_n \colon \sum_{n =1}^\infty \vert \alpha_n \vert^{p'} \leq 1\right\} \mbox{ and } \sum_{n=1}^{\infty} \Vert x_n \Vert^p < \infty.$$ Of course, in the limiting case $p = 1$ we replace the definition above by $K\subseteq \left\{\sum_{n=1}^{\infty} \alpha_n x_n \colon \vert \alpha_n \vert \leq 1\right\}$ and $\sum_{n=1}^{\infty} \Vert x_n \Vert < \infty.$ Thus, classical compact sets can then be seen as “$\infty$-compact”. Moreover, the following monotonicity relation holds: if $1 \leq q \leq p \leq \infty$, any relatively $q$-compact set is in fact relatively $p$-compact. Therefore, $p$-compactness reveals “finer and subtle” structures on compact sets.
Having this definition at hand, it is natural to think about extending the notion of compactness to linear mappings. By simple analogy, $p$-compact mappings arise as those which map the unit ball into a relatively $p$-compact set. This definition, which comes from Sinha and Karn’s original paper [@sinha2002compact], turns out to yield a normed operator ideal, denoted by $\mathcal K_p$ (see the precise definition in Section \[section completely compact mappings\]). In recent years there has been great interest in this class — and in some notions that naturally relate to it — from a variety of perspectives including: the theory of operator ideals [@prasad2008compact; @delgado2010operators; @delgado2010density; @ain2012compact; @pietsch2014ideal; @turco2016mathcal; @fourie2018injective], the theory of tensor norms [@galicer2011ideal], the interplay with various approximation properties [@delgado2009p; @choi2010dual; @lassalle2012p; @lassalle2013banach; @lassalle2016weaker; @lassalle2017null; @oja2012remark; @oja2012grothendieck], structural properties of sets and sequences [@pineiro2011p; @ain2015p; @kim2014unconditionally; @fourie2017weak], infinite dimensional complex analysis [@aron2010p; @aron2011p; @aron2012ideals; @munoz2015alpha; @aron2016behavior].
The main goal of the present paper is to initiate the study of a noncommutative version of $p$-compact mappings, specifically in the context of operator spaces. Recall that an operator space is a Banach space $\textbf X$ with an extra “matricial norm structure”: in addition to the norm on $\textbf X$, we have norms on all the spaces $M_n(\textbf X)$ of $n \times n$ matrices with entries from $\textbf X$ (and these norms must satisfy certain consistency requirements). The morphisms are no longer just bounded maps, but the *completely bounded ones*: they are required to be uniformly bounded on all the matricial levels. Operator spaces are thus a quantized or noncommutative version of Banach spaces, giving rise to a theory that not only is mathematically attractive but it is also naturally well-positioned to have applications to quantum physics. The systematic study of operator spaces started with Ruan’s thesis, and was developed mainly by Effros and Ruan, Blecher and Paulsen, Pisier and Junge (see the monographs [@junge1999factorization; @Pisier-Asterisque-98; @Blecher-LeMerdy-book; @Effros-Ruan-book; @Pisier-Operator-Space-Theory] and the references therein).
Due to their very definition, it is natural to investigate to what extent the classical theory of Banach spaces can be translated to the noncommutative context of operator spaces. Though some properties do carry over, many do not and these differences are one of the reasons making the new theory so interesting. In particular, both ideals of operators and tensor norms have inspired important developments in the operator space setting. Noncommutative versions of nuclear, integral, summing, and other ideals of operators have played significant roles in, e.g., [@Effros-Ruan-book; @Pisier-Asterisque-98; @Junge-Habilitationschrift; @Effros-Junge-Ruan; @Junge-Parcet-Maurey-factorization], whereas in addition to the usefulness of various specific tensor norms for operator spaces (most notably the Haagerup one), Blecher and Paulsen [@Blecher-Paulsen-Tensors] have showed that the elementary theory of tensor norms of Banach spaces carries over to operator spaces, initiating a “tensor norm program” for operator spaces further developed in [@Blecher-1992]. However, the abstract tool of associating ideals of operators and tensor norms in the sense of [@Defant-Floret Chap. 17] does not appear to have been developed yet. In the present paper we take some small steps in this direction, which we need throughout our definition and study of the new class of *operator $p$-compact mappings* (which we denote by $\mathcal K_p^{o}$). A much more thorough study of the relationships between tensor norms and ideals of mappings in the operator space category will appear in [@CDDG].
One way to define this class is to rely on certain factorizations that characterize $p$-compact mappings via right $p$-nuclear maps. To this end, we study and define the ideal of *completely right $p$-nuclear mappings*, $\mathcal{N}^{p}_{o}$ (which are, in some way, a transposed version of the class given in [@Junge-Habilitationschrift Definition 3.1.3.1.] called $p$-nuclear operators). In particular, $\mathcal K_p^{o}$ is the surjective hull of $\mathcal{N}^p_{o}$, as shown by Delgado, Piñeiro and Serrano [@delgado2010operators Prop. 3.11] and by Pietsch [@pietsch2014ideal Thm. 1] in the Banach space context. This relation among these two classes enables us to give an appropriate operator space structure to $\mathcal K_p^{o}$.
At the end of the nineties, Webster (a student of Effros) proposed in his doctoral thesis [@webster1997local] several ways to extend the notion of compactness to the operator space framework (see also [@Webster1998]). One of these notions, which he called *operator compactness*, is in effect a noncommutative version of Grothendieck’s characterization (being in the absolute convex hull of a null sequence as in Equation ). This also gave raise to the notion of *operator compact mappings*. Therefore, another path to translate the concept of $p$-compactness to the category of operator spaces is based on Webster’s ideas. We show that these two possible ways to define $p$-compactness in the operator space setting (the one which involves the factorization through completely right $p$-nuclear mappings and the one based on Webster’s work) in fact coincide. This allows us to endow Webster’s class (of operator compact mappings) with an operator space structure.
The article is organized as follows. In Section \[Preliminaries\] we present the notation and some basic concepts in the theory of operator spaces. In Section \[section right $p$-nuclear\] we introduce the notion of completely right $p$-nuclear mappings, first in tensor terms and then proving a characterization in terms of factorizations. It should be mentioned that for $p=1$, this concept coincides with the completely nuclear mappings already studied in [@Effros-Ruan-book Section 12.2]. In Section \[section completely compact mappings\] we introduce a notion of operator $p$-compact mappings based on factorization schemes, and give an operator space structure to this class by relating it to the aforementioned right $p$-nuclear mappings. Additionally, we characterize operator $p$-compact mappings in terms of a noncommutative notion of $p$-compact sets. We end the article with some open questions.
Preliminaries {#Preliminaries}
=============
Our Banach space notation is quite standard, and follows closely that of [@Defant-Floret; @Ryan]. The injective tensor product of Banach spaces will be denoted by $\otimes_\varepsilon$. For a Banach space $\mathbf{X}$ and $1\le p < \infty$, we denote by $\ell_p(\mathbf{X})$ and $\ell_{p}^w(\textbf X)$ the spaces of $p$-summable and weakly $p$-summable sequences in $\mathbf{X}$, respectively, with their usual norms $$\Vert (x_n) \Vert_{\ell_{p}(\textbf X)} = \bigg( \sum_{n=1}^\infty { \left\|x_n\right\| }^p \bigg)^{1/p} , \qquad
\Vert (x_n) \Vert_{\ell_{p}^w(\textbf X)} = \sup_{{ \left\|x'\right\| } \le 1} \bigg( \sum_{n=1}^\infty \big|x'(x_n)\big|^p \bigg)^{1/p}$$ where $x'$ is taken in $\mathbf{X}'$, the dual of $\mathbf{X}$. The obvious modifications are adapted to the case $p=\infty$. The unit ball of $\mathbf{X}$ is denoted by $B_{\mathbf{X}}$.
We only assume familiarity with the basic theory of operator spaces; the books [@Pisier-Operator-Space-Theory] and [@Effros-Ruan-book] are excellent references. Our notation follows closely that from [@Pisier-Asterisque-98; @Pisier-Operator-Space-Theory], with one main exception: we denote the dual of a space $E$ by $E'$.
Throughout the article $E$ and $F$ will be operator spaces. For each $n$, $M_n(E)$ will denote the space of $n\times n$ matrices of elements of $E$. We will just write $M_n$ when $E$ is the scalar field. The spaces $M_{n \times m}(E)$ and $M_{n\times m}$ are defined analogously. For a linear mapping $T:E\to F$, we will denote its $n$-amplification by $T_n:M_n(E)\to M_n(F)$. The space of completely bounded linear mappings from $E$ into $F$ will be noted by $\operatorname{\mathcal{CB}}(E,F)$. The corresponding bilinear notion is the following: a bilinear map $T : E_1 \times E_2 \to F$ is jointly completely bounded if there exists a constant $C \ge 0$ such that for any matrices $(x_{i,j})_{i,j =1}^n \in M_n(E_1)$ and $(y_{k,\ell})_{k,\ell =1}^m \in M_m(E_2)$ we have $${ \left\| \big(T(x_{i,j},y_{k,\ell})) \right\| }_{M_{nm}(F)} \le { \left\| (x_{i,j})_{i,j =1}^n \right\| }_{M_n(E_1)} { \left\|(y_{k,\ell})_{k,\ell =1}^m\right\| }_{M_m(E_2)}.$$ The least such $C$ is denoted ${ \left\|T\right\| }_{jcb}$. As in [@Effros-Ruan-book Sec. 10.1] we will denote by $M_{\infty}(E)$ the space of all infinite matrices $(x_{i,j})$ with coefficients in $E$ such that the truncated matrices are uniformly bounded, i.e., $\sup_{n \in {{\mathbb{N}}}} \Vert (x_{i,j})_{i,j =1}^n \Vert_{M_n(E)} < \infty$. This supremum corresponds to the norm of the element $(x_{i,j})$ in $M_{\infty}(E)$. The space $M_\infty$ of bounded scalar infinite matrices is naturally identified with $\mathcal B(\ell_2)$.
Our notation for the minimal and the projective operator space tensor products will be, respectively, $\otimes_{\min}$ and $\otimes_{\operatorname{proj}}$. By an operator space cross norm $\alpha$ we mean an assignment of an operator space $E \otimes_\alpha F$ to each pair $(E,F)$ of operator spaces, in such a way that $E \otimes_\alpha F$ is the algebraic tensor product $E \otimes F$ together with a matricial norm structure on $E \otimes F$ that we write as $\alpha_n$ or ${ \left\|\cdot\right\| }_{\alpha_n}$, and such that $\alpha_{nm}(x \otimes y) = { \left\|x\right\| }_{M_n(E)} \cdot { \left\|y\right\| }_{M_m(F)}$ for every $x \in M_n(E), y \in M_m(F)$. This implies that the identity map $E \otimes_{\operatorname{proj}} F \to E \otimes_{\alpha} F$ is completely contractive [@Blecher-Paulsen-Tensors Thm. 5.5]. If in addition the identity map $E \otimes_\alpha F \to E \otimes_{\min} F$ is also completely contractive, we say that $\alpha$ is an *operator space tensor norm*. Moreover, an operator space tensor norm $\alpha$ is called *uniform* if additionally for any operator spaces $E_1$, $E_2$, $F_1$, $F_2$, the map
$\otimes_\alpha : \operatorname{\mathcal{CB}}(E_1,E_2) \times \operatorname{\mathcal{CB}}(F_1,F_2) \to \operatorname{\mathcal{CB}}(E_1 \otimes_\alpha E_2, F_1 \otimes_\alpha F_2)$ given by $(S,T) \mapsto S \otimes_\alpha T$ is jointly completely contractive.
If $\alpha$ is an operator space tensor norm, the completion of the tensor product $E \otimes_\alpha F$ will be denoted by $E \widehat{\otimes}_\alpha F$. A degree of caution is required when consulting different works dealing with operator space tensor products, since the term “tensor norm” is not always taken to have the exact same meaning. We point out that our definitions are slightly different than those of [@Blecher-Paulsen-Tensors; @Effros-Ruan-book; @dimant2015biduals], though not in any significant way. An operator space tensor norm $\alpha$ is said to be *finitely generated* if for any operator spaces $E$ and $F$ and $u \in M_n(E \otimes F)$, ${\alpha}_n(u ; E, F) = \inf\left\{\alpha_n(u; E_0,F_0) \colon u \in M_n(E_0 \otimes F_0)\right\} $, where the infimum is taken over finite-dimensional subspaces $E_0 \subset E$ and $F_0 \subset F$. For operator space tensor norms $\alpha$ and $\beta$ and a constant $c$, we write “$\alpha \le c \beta$ on $E\otimes F$” to indicate that the identity map $E \otimes_\beta F \to E \otimes_\alpha F$ has $\operatorname{cb}$-norm at most $c$. If no reference to spaces is made, we mean that the inequality holds for any pair of operator spaces. A linear map $Q : E \to F$ between operator spaces is called a complete 1-quotient if it is onto and the associated map from $E/\ker(Q)$ to $F$ is a completely isometric isomorphism. These maps are called complete metric surjections in [@Pisier-Operator-Space-Theory Sec. 2.4], where it is proved that a linear map $u:E \to F$ is a complete 1-quotient if and only if its adjoint $u' : F' \to E'$ is a completely isometric embedding.
For an operator space $E$ and $1\le p \le \infty$, let us define the spaces $S_p$ and $S_p[E]$ following [@Pisier-Asterisque-98]. For $1 \le p <\infty$, $S_p$ denotes the space of Schatten $p$-class operators on $\ell_2$. In the case $p=\infty$, we denote by $S_\infty$ the space of all compact operators on $\ell_2$. We endow $S_\infty$ with the operator space structure inherited from $B(\ell_2)$, and $S_1$ with the one induced by the duality $S_1'= B(\ell_2)$; this then determines an operator space structure on $S_p$ for each $1<p < \infty$ via complex interpolation. More generally, we define $S_\infty[E]$ as the minimal operator space tensor product of $S_\infty$ and $E$, and $S_1[E]$ as the operator space projective tensor product of $S_1$ and $E$. In the case $1 < p < \infty$, $S_p[E]$ is defined via complex interpolation between $S_\infty[E]$ and $S_1[E]$. For $1 < p \le \infty$, the dual of $S_p[E]$ can be canonically identified with $S_{p'}[E']$, where $p'$ satisfies $1/p + 1/p' = 1$.
Recall that, given Banach spaces $\mathbf X$ and $\mathbf Y$ and $1 \leq p \leq \infty$, the $p$-right Chevet-Saphar tensor norm ([@Ryan Chapter 6], [@Defant-Floret Chapter 12]) $d_p$ of a tensor $u \in \mathbf X \otimes \mathbf Y$ is defined by $$\label{dp-Banach}
d_p(u) = \inf \left\{\Vert (x_j) \Vert_{\ell_{p'}\otimes_{\varepsilon} X } \Vert(y_j)\Vert_{\ell_p(Y)} : u = \sum_{j} x_j \otimes y_j \right\},$$ where the infimum runs over all the possible ways in which the tensor $u$ can be written as a finite sum as above.
Moving to the operator space realm, for $1 \leq p \leq \infty$, in [@CD-Chevet-Saphar-OS] the $p$-right Chevet-Saphar operator space tensor norm $d_p^o$ is defined in the following way: given operator spaces $E$ and $F$, for $u \in E \otimes F$, $$\label{dp-os}
d_p^o(u) = \inf \left\{\Vert (x_{i,j}) \Vert_{S_{p'}\otimes_{min } E} \; \Vert(y_{i,j})\Vert_{S_p[F]} : u = \sum_{i,j} x_{i,j} \otimes y_{i,j} \right\},$$ where the infimum runs over all the possible ways in which the tensor $u$ can be written as a finite sum as above.
The operator space structure of the tensor $E \widehat \otimes_{d_p^o} F$ is given by the following 1-quotient (see [@CD-Chevet-Saphar-OS Section 3]) $$Q^p : (S_{p'} \widehat{\otimes}_{\min} E) \widehat{\otimes}_{\operatorname{proj}} S_p[F] \to E \widehat{\otimes}_{d^o_p} F,$$ where $\widehat \otimes $ means the completion of the corresponding tensor product.
An operator space $E$ is called projective if, for any $\varepsilon>0$, any completely bounded map $T:E \to F/S$ into a quotient space (here $F$ is any operator space and $S \subset F$ any closed space) admits a lifting $\widetilde T: E \to F$ with $\Vert \widetilde{T} \Vert_{\operatorname{cb}} \leq (1+ \varepsilon) \Vert T \Vert_{\operatorname{cb}}$.
Following [@Effros-Ruan-book Sec. 12.2], a *mapping ideal* $(\mathfrak{A},\mathbf{A})$ is an assignment, for each pair of operator spaces $E, F$, of a linear space $\mathfrak{A}(E,F) \subseteq \operatorname{\mathcal{CB}}(E,F)$ together with an operator space matrix norm $\mathbf{A}$ on $\mathfrak{A}(E,F)$ such that
1. The identity map $\mathfrak{A}(E,F) \to \operatorname{\mathcal{CB}}(E,F)$ is a complete contraction.
2. The ideal property: whenever $T \in M_n(\mathfrak{A}(E,F))$, $r \in \operatorname{\mathcal{CB}}(E_0,E)$ and $s \in \operatorname{\mathcal{CB}}(F,F_0)$, it follows that $
\mathbf{A}_n( s_n \circ T \circ r ) \le { \left\|s\right\| }_{\operatorname{cb}} \mathbf{A}_n(T) { \left\|r\right\| }_{\operatorname{cb}}.
$
Finally, a remark about our use of the word *operator*. In the Banach space literature it is usual to speak of $p$-compact *operators* rather than $p$-compact *mappings*, and similarly for other various classes of bounded linear transformations between normed spaces. In order to avoid confusions, throughout this paper we reserve the word *operator* for notions that are noncommutative in nature (as in *operator space*), and use the word *mapping* to refer to bounded linear transformations.
Completely right $p$-nuclear mappings {#section right $p$-nuclear}
=====================================
In this section we introduce the notion of completely right $p$-nuclear mappings, inspired by the definitions in the Banach space setting. For $p=1$, this concept coincides with the completely nuclear mappings studied in [@Effros-Ruan-book Section 12.2]. Recall that in the Banach space setting a linear mapping $T: \mathbf X \to \mathbf Y$ is *right $p$-nuclear* ([@reinov2000linear],[@Ryan Section 6.2]) if $T$ can be written as $$T = \sum_{n=1}^\infty x_n' {\otimes} y_n,$$ where $(x_n') \in \ell_{p'}^w(\textbf X')$, $(y_n) \in \ell_{p}(\textbf Y)$. Moreover, the right $p$-nuclear norm of $T$ is defined as
$$\nu^p(T):=\inf \{ \Vert (x_n) \Vert_{\ell_{p'}^w(\textbf X')} \cdot \Vert (y_n) \Vert_{\ell_{p}(\textbf Y)}\},$$ where the infimum is taken all over possible representations of $T$ as above. The class of all right $p$-nuclear mappings is denoted by $\mathcal N^p$. This definition is known to be equivalent to having a factorization $$\label{right p-nuclear}
\xymatrix{
\mathbf X \ar[r]^T \ar[d]_{U} &\mathbf Y \\
\ell_{p'} \ar[r]_{D_\lambda} & \ell_1 \ar[u]_{V},
}$$ where $U$ and $V$ are bounded mappings, $\lambda \in \ell_p$, and $D_\lambda$ stands for the diagonal multiplication mapping $(x_n) \mapsto (\lambda_nx_n)$. Moreover, $
\nu^p(T) = \inf \{ \Vert U \Vert \Vert D_{\lambda} \Vert \Vert V \Vert \}
$ where the infimum runs over all factorizations as above. It is well-known that right $p$-nuclear mappings between the Banach spaces $\mathbf X$ and $\mathbf Y$ are exactly those mappings which are in the range of the canonical inclusion $
\mathbf X' \widehat \otimes_{d_p}\mathbf Y \to \mathbf X' \widehat \otimes_{\varepsilon} \mathbf Y,
$ and the right $p$-nuclear norm coincides with the quotient norm inherited from this inclusion (see also the analogous results given in [@Defant-Floret 22.3], [@Ryan Section 6.2] and [@Diestel-Jarchow-Tonge Proposition 5.23]).
Motivated by this, we introduce a corresponding notion in the category of operator spaces.
\[p-nuclear def\] Let $1 \leq p \leq \infty$. We say that a linear mapping $T : E \to F$ between operator spaces $E$ and $F$ is *completely right $p$-nuclear* if it corresponds to an element in the range of the canonical inclusion $$J^p : E' \widehat{\otimes}_{d_p^o} F \to E' \widehat{\otimes}_{\min} F.$$ We denote the space of all such mappings by $\mathcal{N}^p_o(E,F)$, and we endow it with the quotient structure $(E' \widehat{\otimes}_{d_p^o} F)/ \ker J^p$. In particular, its norm – that we denote by $\nu^p_o$ – is the quotient norm.
For future reference, it is important to remark that since the operator space structure on $E' \widehat{\otimes}_{d_p^o} F$ is itself coming from a quotient, the above definition is equivalent to being in the range of $$J^p \circ Q^p : (S_{p'} \widehat{\otimes}_{\min} E') \widehat{\otimes}_{\operatorname{proj}} S_p[F] \to E' \widehat{\otimes}_{\min} F,$$ and this still induces the same quotient structure.
Let us start by proving that $\mathcal{N}^p_o$ is a mapping ideal. We observe, also, that each space $\mathcal N^p_o (E,F)$ is complete.
\[proposition:ideal-property\] $\mathcal N^p_o$ is a mapping ideal, for $1 \leq p \leq \infty$.
Given $T \in M_n (\mathcal N_o^p (E;F))$, the definition of right $p$-nuclearity yields $\Vert T \Vert_{M_n(\mathcal CB(E;F))} \leq \Vert T \Vert_{M_n(\mathcal N_o^p(E;F))}$. Also for $S \in \operatorname{\mathcal{CB}}(E_0,E)$ and $R \in \operatorname{\mathcal{CB}}(F,F_0)$ we have $R_nTS \in M_n\left(\mathcal{N}^p_o(E_0,F_0)\right)$ and $$\Vert R_nTS \Vert_{M_n(\mathcal N^p_o(E_0,F_0))} \le { \left\|R\right\| }_{\operatorname{cb}} \Vert T \Vert_{M_n(\mathcal N^p_o(E;F))} { \left\|S\right\| }_{\operatorname{cb}}.$$ Indeed, by definition there exists $t \in M_n(E' \widehat{\otimes}_{d_p^o} F)$ such that $(J^p)_n(t) = T$. Since $d_p^o$ is a uniform operator space tensor norm, $$d_p^o\big( (S' \otimes R)_n(t) \big) \le { \left\|S'\right\| }_{\operatorname{cb}} d_p^o(t) { \left\|R\right\| }_{\operatorname{cb}} = { \left\|S\right\| }_{\operatorname{cb}} d_p^o(t) { \left\|R\right\| }_{\operatorname{cb}}.$$ Note that $(J^p)_n \big( (S' \otimes R)_n(t) \big) = R_nTS$, so $R_nTS \in M_n(\mathcal{N}^p_o(E_0,F_0))$ and $$\Vert R_nTS \Vert_{M_n(\mathcal N^p_o(E_0,F_0))} \le { \left\|S\right\| }_{\operatorname{cb}} d_p^o(t) { \left\|R\right\| }_{\operatorname{cb}}.$$ Taking the infimum over all $t$ we get the desired conclusion.
The following result shows that the formula given in Equation can be extended for tensors that lie in the completion $ E \widehat{\otimes}_{d_p^o} F$.
\[thm:hard-Chevet-Saphar\] Let $1 \leq p \leq \infty$, let $E$ and $F$ be operator spaces, and $u \in E \widehat{\otimes}_{d_p^o} F$. Then $${ \left\|u\right\| }_{d_p^o} = \inf \big\{ { \left\|(x_{ij})\right\| }_{S_{p'}\widehat{\otimes}_{\min} E} { \left\|(y_{ij})\right\| }_{S_{p}[F]} \; : \; u = \sum_{i,j} x_{ij} \otimes y_{ij} \big\}.$$
Let $u \in E \widehat{\otimes}_{d_p^o} F$. It is clear that ${ \left\|u\right\| }_{d_p^o} \le { \left\|(x_{ij})\right\| }_{S_{p'}\widehat{\otimes}_{\min} E} { \left\|(y_{ij})\right\| }_{S_{p}[F]}$, for any representation of $u= \sum_{i,j} x_{ij} \otimes y_{ij}= Q^p\left(( x_{ij})\otimes( y_{ij})\right)$, since $Q^p$ is a 1-quotient mapping.
Also, for every $\eta >0$ there is a sequence $(u_m) \in E \otimes F$ such that $u = \sum_{m} u_m \in E \widehat{\otimes}_{d_p^o} F$ with $d_p^o(u_1) < d_p^o(u) + \eta$ and $ d_p^o(u_m) \leq \frac{\eta^2}{4^m}$ for every $ m \geq 2$. Now, we can write $u_1 := \sum_{i,j = 1}^{k_1} x_{i,j} \otimes y_{i,j}$ where ${ \left\|(x_{ij})_{i,j=1}^{k_1}\right\| }_{S_{p'}^{k_1} \otimes_{\min} E} \leq d_p^o(u) + \eta$ and ${ \left\|(y_{ij})_{i,j=1}^{k_1}\right\| }_{S_p[F]} \leq 1$. Also, for $m \geq 2$, we may represent $u_m : = \sum_{i,j = k_{m-1} + 1}^{k_m} x_{i,j} \otimes y_{i,j}$ with ${ \left\|(x_{ij})_{i,j={k_{m-1} + 1}}^{k_2}\right\| }_{S_{p'}^{k_2} \otimes_{\min} E} \leq \frac{\eta}{2^m}$ and ${ \left\|(y_{ij})_{i,j={k_{m-1} + 1}}^{k_2}\right\| }_{S_p[F]} \leq \frac{\eta}{2^m}.$ By the triangle inequality we derive ${ \left\|(y_{ij})\right\| }_{S_{p}[F]} \leq 1+ \eta \sum_{m=2}^\infty 2^{-m}$ and also ${ \left\|(x_{ij})\right\| }_{S_{p'}\widehat{\otimes}_{\min} E} \leq d_p^o(u) + \eta + \eta \sum_{m=2}^\infty 2^{-m}$, which concludes the proof.
We now introduce a non-commutative version of the sequence space $\ell_p^w(E)$, namely
$$S_{p}^w[E]:=\{ x \in M_{\infty}(E) : \sup_{N} \Vert (x_{ij})_{i,j=1}^N \Vert_{S_p^N \widehat{\otimes}_{\min} E } < \infty \}.$$
It can be easily seen that this is an operator space endowed with the matricial norm structure given by
$$\Vert \big (x_{ij}^{k,l})_{i,j} \big)_{k,l=1}^n \Vert_{M_n(S_{p}^w[E])} := \sup_{N} \Vert \big((x_{ij}^{k,l})_{i,j=1}^N \big)_{k,l=1}^{n} \Vert_{M_n(S_p^N \widehat{\otimes}_{\min} E) }.$$
Recall that $\ell_p^w(E)$ can be identified with the space of bounded linear mappings from $\ell_{p'}$ to $E$ [@Defant-Floret Prop. 8.2.(1)]. The following is the analogous statement for $S_{p}^w[E]$.
\[lemma-Spweak-as-mappings\] For $1 \leq p \leq \infty$, we have the following completely isometric identification $$S_{p}^w[E] = \operatorname{\mathcal{CB}}(S_{p'},E).$$
Let us see first that the spaces $S_{p}^w[E]$ and $\operatorname{\mathcal{CB}}(S_{p'},E)$ are isometrically isomorphic. For each $N\in\mathbb N$, we denote $\rho_N:M_{\infty}(E)\to M_N(E)$ the $N$-truncation given by $\rho_N\left((x_{ij})_{ij}\right)=(x_{i,j})_{i,j=1}^N$. Each $X \in S_p^w[E]$ satisfies that $$\|X\|_{S_p^w[E]}=\sup_N \|\rho_N(X)\|_{S_p^N \widehat{\otimes}_{\min} E}=\sup_N \|\rho_N(X)\|_{\operatorname{\mathcal{CB}}(S_{p'}^N,E)}$$ since the spaces $S_p^N \widehat{\otimes}_{\min} E$ and $\operatorname{\mathcal{CB}}(S_{p'}^N,E)$ are isometric. Thus, the mapping $T_X: \bigcup_{N=1}^\infty S_{p'}^N \to E$ is well defined (given $A \in S_{p'}^N$ for some $N \in \mathbb{N}$, $T_X(A):=\rho_N(X) (A)$) and uniquely extends to $S_{p'}$ by density. The same holds with the $n$-amplification $(T_X)_n:M_n(\bigcup_{N=1}^\infty S_{p'}^N) \to M_n(E)$ (whose norm is obviously bounded by $\Vert X \Vert_{S_p^w[E]}$), and extends to $M_n(S_{p'})$. Hence, $T_X\in \operatorname{\mathcal{CB}}(S_{p'},E)$ with $\|T_X\|_{\operatorname{\mathcal{CB}}(S_{p'},E)}\le \Vert X \Vert_{S_p^w[E]}$.
Conversely, for $T\in \operatorname{\mathcal{CB}}(S_{p'},E)$, denoting by $i_N:S_{p'}^N\to S_{p'}$ the canonical completely isometric inclusion, we derive that $T\circ i_N\in \operatorname{\mathcal{CB}}(S_{p'}^N,E)$ with $\|T\circ i_N\|_{\operatorname{\mathcal{CB}}(S_{p'}^N,E)}\le \|T\|_{\operatorname{\mathcal{CB}}(S_{p'},E)}$, for each $N\in\mathbb N$. Now, defining $X_T\in M_\infty (E)$ by $X_T=\left(Te_{ij}\right)_{i,j}$ it is plain, through the identity $S_p^N \widehat{\otimes}_{\min} E=\operatorname{\mathcal{CB}}(S_{p'}^N,E)$, that $$\|X_T\|_{S_p^w[E]}=\sup_N\|\rho_N(X_T)\|_{S_p^N \widehat{\otimes}_{\min} E}=\sup_N\|T\circ i_N\|_{\operatorname{\mathcal{CB}}(S_{p'}^N,E)}\le \|T\|_{\operatorname{\mathcal{CB}}(S_{p'},E)}.$$
Hence, we have proved the isometry $S_{p}^w[E] = \operatorname{\mathcal{CB}}(S_{p'},E)$. To see that this isometry is complete, we need to show that, for each $n$, the spaces $M_n\left(S_{p}^w[E]\right)$ and $M_n\left(\operatorname{\mathcal{CB}}(S_{p'},E)\right)$ are isometric. Recall that $$M_n\left(\operatorname{\mathcal{CB}}(S_{p'},E)\right) \overset{1}{=} \operatorname{\mathcal{CB}}(S_{p'},M_n(E)) \overset{1}{=} S_{p}^w[M_n(E)],$$ where the last equality follows from the first part of the proof. To finish it only remains to see that $ S_{p}^w[M_n(E)]\overset{1}{=} M_n\left(S_{p}^w[E]\right)$. Indeed, by [@Effros-Ruan-book Cor. 8.1.3 and Cor. 8.1.7], for all $N$ the spaces $S_p^N\widehat{\otimes}_{\min} M_n(E)$ and $M_n\left(S_p^N \widehat{\otimes}_{\min} E\right)$ are completely isometric. Therefore, for each $X\in M_n\left(S_{p}^w[E]\right)$, it holds $$\|X\|_{M_n\left(S_{p}^w[E]\right)}=\sup_N \|(\rho_N)_n(X)\|_{M_n\left(S_p^N \widehat{\otimes}_{\min} E\right)} =\sup_N \|(\rho_N)_n(X)\|_{S_p^N\widehat{\otimes}_{\min} M_n(E)} = \|X\|_{S_{p}^w[M_n(E)]}.$$
As in the Banach space setting, we can replace $S_{p'}\widehat{\otimes}_{\min} E$ by $S_{p'}^w[E]$ in the quotient description of the norm $d_p^o$.
\[Thm:Q-tilde\] For $1 \leq p \leq \infty$, the mapping $$\widetilde{Q}^p: S_{p'}^w[E] \widehat{\otimes}_{\operatorname{proj}} S_p[F] \to E \widehat{\otimes}_{d^o_p} F,$$ is a complete 1-quotient.
We have just to prove that $\widetilde{Q}^p$ defines a complete contraction. The fact that $Q^p$ is a complete 1-quotient and the complete isometry $S_{p'}\widehat{\otimes}_{\min} E\hookrightarrow S_{p'}^w[E]$ do the rest.
Let us denote by $\tau^k: S_{p'}^w[E]\to S_{p'}^k\otimes_{\min} E$ the truncation mapping given by $\tau^k\left((x_{ij})_{i,j}\right)=(x_{ij})_{i,j=1}^k$. It is plain, due to the operator space structure of $S_{p'}^w[E] $, that $\tau^k$ is a complete contraction. We note by $\widetilde{\tau}^k$ the “same” truncation mapping but on a different space, $\widetilde{\tau}^k:S_p[F]\to S_p[F]$.
Let $(x_{ij}) \in S_{p'}^w[E]$ and $(y_{ij}) \in S_{p}[F]$. To see that $\sum_{i,j} x_{ij} \otimes y_{ij}$ converges in $E \widehat{\otimes}_{d_p^o} F$ we prove that $(u_k)$ is a Cauchy sequence, where $u_k = \sum_{i,j=1}^n x_{ij} \otimes y_{ij}$. By definition, $$d_p^o(u_k - u_l ) \leq { \left\|\tau^k(x_{ij})-\tau^l(x_{ij})\right\| }_{S_{p'}^m \otimes_{\min} E} { \left\|\widetilde{\tau}^k(y_{ij})-\widetilde{\tau}^l(y_{ij})\right\| }_{S_p[F]}.$$ Note that ${ \left\|\tau^k(x_{ij})-\tau^l(x_{ij})\right\| }_{S_{p'}^m\otimes_{\min} E} \leq 2{ \left\|(x_{ij})\right\| }_{S_{p'}^w[E]} $ and by [@Pisier-Asterisque-98 Lemma 1.12], ${ \left\|\widetilde{\tau}^k(y_{ij})-\widetilde{\tau}^l(y_{ij})\right\| }_{S_p[F]} \leq \varepsilon$ for $k,\ l$ sufficiently large. So, $(u_k)_k$ converges to $u = \sum_{i,j} x_{ij} \otimes y_{ij} $ and $d_p^o(u) \leq { \left\|(x_{ij})\right\| }_{S_{p'}^w[E]} { \left\|(y_{ij})\right\| }_{S_{p}[F]}$. Thus, $\widetilde{Q}^p$ is well defined.
Now, let $n\in\mathbb N$ fixed. We need to prove that $$\left\|(\widetilde{Q}^p)_n: M_n\left(S_{p'}^w[E] \widehat{\otimes}_{\operatorname{proj}} S_p[F]\right)\to M_n\left( E \widehat{\otimes}_{d^o_p} F\right)\right\|\le 1.$$
Recall that, given $r\in\mathbb N$ and $Y\in M_r(S_p[F])$, by [@Pisier-Asterisque-98 Lemma 1.12], it holds $$\left\|(\widetilde\tau^k)_r(Y)-Y\right\|_{M_r(S_p[F])}\underset{k\to\infty}{\longrightarrow} 0.$$
For $A\in M_n\left(S_{p'}^w[E] {\otimes}_{\operatorname{proj}} S_p[F]\right)$ take any representation of the form $A=\alpha (X\otimes Y)\beta$, with $X\in M_q\left(S_{p'}^w[E]\right)$, $Y\in M_r\left(S_p[F]\right)$, $\alpha\in M_{n,q\cdot r}$, $\beta\in M_{q\cdot r, n}$. By arguing as in the beginning of the proof we obtain $$(Q^p)_n (\tau^k\otimes\widetilde\tau^k)_n(A)=(\widetilde Q^p)_n (\tau^k\otimes\widetilde\tau^k)_n(A) \underset{k\to\infty}{\longrightarrow} (\widetilde Q^p)_n (A), \textrm{ in } M_n (E \widehat{\otimes}_{d^o_p} F).$$ Also, the fact that $Q^p$ is a complete 1-quotient implies, for each $k$, $$\begin{aligned}
\left\|(Q^p)_n (\tau^k\otimes\widetilde\tau^k)_n(A)\right\|_{M_n (E \widehat{\otimes}_{d^o_p} F)} & = & \left\|(Q^p)_n \left(\alpha (\tau_q^k(X) \otimes \widetilde\tau^k_r(Y)\right)\beta\right\|_{M_n (E \widehat{\otimes}_{d^o_p} F)} \\
&\le & \|\alpha\|_{M_{n,q\cdot r}} \cdot \|\tau_q^k(X) \|_{M_q(S_{p'}^k\otimes_{\min} E)} \cdot \|\widetilde\tau^k_r(Y)\|_{M_r(S_p[F])} \cdot \|\beta\|_{M_{q\cdot r, n}}\\
&\le & \|\alpha\|_{M_{n,q\cdot r}} \cdot \|X \|_{M_q\left(S_{p'}^w[E]\right)} \cdot \|\widetilde\tau^k_r(Y)\|_{M_r(S_p[F])} \cdot \|\beta\|_{M_{q\cdot r, n}}.\end{aligned}$$ Taking limit as $k\to\infty$ we obtain $$\left\|(\widetilde Q^p)_n (A)\right\|_{M_n (E \widehat{\otimes}_{d^o_p} F)} \le \|\alpha\|_{M_{n,q\cdot r}} \cdot \|X \|_{M_q\left(S_{p'}^w[E]\right)} \cdot \|Y\|_{M_r(S_p[F])} \cdot \|\beta\|_{M_{q\cdot r, n}}.$$ Since this holds for every representation of $A$ it is clear that $$\left\|(\widetilde Q^p)_n (A)\right\|_{M_n (E \widehat{\otimes}_{d^o_p} F)} \le \|A\|_{M_n\left(S_{p'}^w[E] {\otimes}_{\operatorname{proj}} S_p[F]\right)}.$$
Two direct consequences of the previous theorem are stated in the next corollary:
\[defB\_p\] Let $1 \leq p \leq \infty$ then:
1. The bilinear mapping associated to $\widetilde{Q}^p$, $B^p: S_{p'}^w[E]\times S_p[F]\to E\widehat\otimes_{d_p^o} F$ given by $$B^p\left((x_{ij}), (y_{ij})\right)= \sum_{i,j} x_{ij} \otimes y_{ij}$$ is jointly completely bounded with $\|B^p\|_{jcb}=1$.
2. For each $u \in E \widehat{\otimes}_{d_p^o} F$, $${ \left\|u\right\| }_{d_p^o} = \inf \big\{ { \left\|(x_{ij})\right\| }_{S_{p'}^w[E]} { \left\|(y_{ij})\right\| }_{S_{p}[F]} \; : \; u = \widetilde{Q}^p\left( (x_{ij})\otimes (y_{ij})\right) \big\}.$$
Now we show that certain two-sided multiplication operators are completely right $p$-nuclear. This is closely related to the results in [@Oikhberg-10], and the argument here follows closely [@Effros-Ruan-book Prop. 12.2.2]. As in [@Pisier-Asterisque-98 p. 21], if $E$ is an operator space, $x \in M_\infty(E)$ and $a \in M_\infty$, we denote by $a \cdot x$ (resp. $x \cdot a$) the matrix product, that is, $$(a \cdot x)_{ij} = \sum_k a_{ik}x_{kj}, \qquad \bigg( \text{resp. } \quad (x\cdot a)_{ij} = \sum_k x_{ik}a_{kj} \bigg).$$ If $b \in M_\infty$, we denote $a \cdot x \cdot b = a \cdot (x \cdot b) = (a \cdot x) \cdot b$.
\[proposition:multiplication-operators-are-nuclear\] Let $1 \leq p \leq \infty$ and $a,b \in S_{2p}$. Then the multiplication operator $
M(a,b) : x \mapsto a \cdot x \cdot b
$ belongs to $\mathcal{N}^p_o( S_{p'} , S_1 )$ and satisfies $\nu^p_o(M(a,b)) \le { \left\|a\right\| }_{S_{2p}} { \left\|b\right\| }_{S_{2p}}$.
Write $a = (a_{ij})$ and $b = (b_{ij})$. Let us first observe that we may assume that only finitely many entries in each of these two infinite matrices are nonzero. To that effect, suppose that we know the result for finitely supported matrices, and let $a, b \in S_{2p}$. We can find sequences $(a^k), (b^k)$ in $S_{2p}$ of finitely supported matrices converging to $a$ and $b$, respectively, in the $S_{2p}$ norm. Since for $x \in S_{p'}$ we have $$a^kxb^k - a^lxb^l = (a^k-a^l)xb^k + a^lx(b^k-b^l),$$ it follows that $\big( M(a^k,b^k) \big)_k$ is a Cauchy sequence in $\mathcal{N}^p_o( S_{p'} , S_1)$ and therefore converges in the same space. Since the norm in $\operatorname{\mathcal{CB}}( S_{p'} , S_1 )$ is dominated by that of $\mathcal{N}^p_o( S_{p'} , S_1)$, it follows that the limit of $\big( M(a^k,b^k) \big)_k$ has to be $M(a,b)$, and therefore $M(a,b) \in \mathcal{N}^p_o( S_{p'} , S_1)$ with $\nu^p_o(M(a,b)) \le { \left\|a\right\| }_{S_{2p}} { \left\|b\right\| }_{S_{2p}}$.
Therefore, from now on we assume $a=(a_{ij})_{i,j=1}^n$ and $b=(b_{ij})_{i,j=1}^n$. Let $\varepsilon = [\varepsilon_{ij}]_{i,j=1}^n$ be the matrix of matrix units. For $x = (x_{ij}) \in S_{p'}$, $$\begin{aligned}
(a x b)_{ij} &= \sum_{k,l} \varepsilon_{ij} \otimes a_{ik} x_{kl} b_{lj} \\
&= \sum_{k,l} \varepsilon_{ij} \otimes a_{ik} \varepsilon_{kl}(x) b_{lj}.\end{aligned}$$ From the above calculation $$M(a,b)(x) = \widetilde{Q}^p\big( \varepsilon \otimes (a \cdot \varepsilon \cdot b) \big) (x),$$ hence $M(a,b) = \widetilde{Q}^p\big( \varepsilon \otimes (a \cdot \varepsilon \cdot b) \big)$ and $$\nu^p_o\big( M(a,b) \big) \le { \left\|\varepsilon\right\| }_{S_{p'}^w[S_p]} { \left\| a \cdot \varepsilon \cdot b \right\| }_{S_p[S_1]}.$$ Note that $${\varepsilon = [\varepsilon_{ij}]_{i,j=1}^n \in S_{p'}^w[S_p] } = \operatorname{\mathcal{CB}}(S_{p},S_{p})$$ is just the projection onto an initial block $S_p^n$, and thus ${ \left\|\varepsilon\right\| }_{S_{p'}^w[S_p]} = 1$.
On the other hand, by [@Pisier-Asterisque-98 Theorem 1.5 and Lemma 1.6], $${ \left\|a \cdot \varepsilon \cdot b\right\| }_{S^n_p[S_1]} \le { \left\|a\right\| }_{S^n_{2p}} { \left\|\varepsilon\right\| }_{M_n(S_1)} { \left\|b\right\| }_{S^n_{2p}}.$$ Now, $$\varepsilon = [\varepsilon_{ij}]_{i,j=1}^n \in M_n(S_1) = M_n(S_\infty') \subseteq \operatorname{\mathcal{CB}}(S_\infty, M_n)$$ is once again just a projection onto an initial block, so ${ \left\|\varepsilon\right\| }_{M_n(S_1)} = 1$. Therefore, we conclude that $\nu^p_o\big( M(a,b) \big) \le { \left\|a\right\| }_{S_{2p}} { \left\|b\right\| }_{S_{2p}}$.
The multiplication operators defined in the previous proposition are the canonical prototypes of completely right $p$-nuclear mappings: we show below that a mapping is completely right $p$-nuclear if and only if it admits a factorization through one such multiplication operator (similar to the Banach space framework as in ).
\[right p nuclear factorization\] For a linear map $T : E \to F$ and $1\le p\le\infty$, the following are equivalent:
1. $T$ is completely right $p$-nuclear.
2. There exist $a,b \in S_{2p}$ such that $T$ admits a factorization $$\xymatrix{
E \ar[r]^T \ar[d]_{U} &F \\
S_{p'} \ar[r]_{M(a,b)} &S_1 \ar[u]_{V}
}$$
Moreover, in this case $$\nu^p_o(T) = \inf \big\{ { \left\|U\right\| }_{\operatorname{cb}} { \left\|V\right\| }_{\operatorname{cb}} { \left\|a\right\| }_{S_{2p}} { \left\|b\right\| }_{S_{2p}} \big\}$$ where the infimum is taken over all factorizations as in (b).
\(b) $\Rightarrow$ (a): This follows from Propositions \[proposition:ideal-property\] and \[proposition:multiplication-operators-are-nuclear\].\
\
(a) $\Rightarrow$ (b): Let $1\le p<\infty$, assume that $T \in \mathcal{N}^p_o(E;F)$ with $\nu^p_o(T) < 1$. Then there exists $u \in E' \widehat{\otimes}_{d_p^o} F$ such that $J^p(u) = T$ and ${ \left\|u\right\| }_{d_p^o} < 1$. By Theorem \[thm:hard-Chevet-Saphar\], we can in turn find $X \in S_{p'}\widehat{\otimes}_{\min} E'$, $Y \in S_p[F]$ such that $u = Q^p( X\otimes Y )$, ${ \left\|X\right\| }_{S_{p'}\widehat{\otimes}_{\min} E'}<1$ and ${ \left\|Y\right\| }_{S_{p}[F]}<1$. On the one hand, the representation of the minimal tensor product and its symmetry allows us to think of $X$ as a mapping $U$ in $\mathcal{CB}(E, S_{p'})$ with ${ \left\|U\right\| }_{\operatorname{cb}}<1 $. On the other hand, by [@Pisier-Asterisque-98 Thm. 1.5], we can write $Y = a \cdot \bar{Y} \cdot b$ with ${ \left\|a\right\| }_{S_{2p}}<1$, ${ \left\|b\right\| }_{S_{2p}}<1$ and ${ \left\|\bar{Y}\right\| }_{S_\infty[F]} < 1$. Also, since $S_\infty[F] = S_\infty \widehat{\otimes}_{\min} F$ completely isometrically embeds into $\operatorname{\mathcal{CB}}(S_1,F)$, $\bar{Y}$ canonically induces a linear map $V : S_1 \to F$ with ${ \left\|V\right\| }_{\operatorname{cb}} = { \left\|\bar{Y}\right\| }_{S_\infty[F]}$. Chasing down the formulas it is easy to see that $T = V \circ M(a,b) \circ U$ (in the spirit of [@Effros-Ruan-book Prop. 12.2.3]), giving us the desired factorization. Finally, for $p=\infty$ this is derived directly from the definition of right $\infty$-nuclear with $M(a,b)=Id$.
operator $p$-compact mappings {#section completely compact mappings}
=============================
Following [@sinha2002compact; @delgado2010operators] we say that a linear mapping between Banach spaces $T: \textbf X \to \textbf Y$ is $p$-compact if $T(B_{\textbf X})$ is a relatively $p$-compact set. That is, there exists a sequence $(y_n)_n \in \textbf Y$ such that $$\label{T bola p-compacto}
T(B_{\textbf X})\subset \left\{\sum_{n=1}^{\infty} \alpha_n y_n \colon \sum_{n =1}^\infty \vert \alpha_n \vert^{p'} \leq 1\right\} \mbox{ and } \sum_{n=1}^{\infty} \Vert y_n \Vert^p < \infty.$$
The class of $p$-compact mappings is denoted by $\mathcal K_p$, and endowed with the norm $$\label{norma kp Banach}
\kappa_p(T):=\inf\{\Vert (y_n)_n \Vert_{\ell_p(\textbf Y)}\},$$ where the infimum runs all over the sequences $(y_n)_n \in \textbf Y$ as in Equation .
This notion should not be confused with the homonymous concept studied in the late seventies and early eighties by [@Pietsch-Operator-Ideals] and [@fourie1979banach] (see also [@Defant-Floret]). Nowadays, this older class is sometimes referred to as “classical $p$-compact” mappings (as proposed in [@oja2012remark]). We will not deal with such mappings in this paper.
The following characterization of $p$-compactness in terms of commutative diagrams is well-known to experts, but we were unable to find an explicit reference for it. Thus, we include a sketch of its proof for completeness.
Let $\textbf X$ and $\textbf Y$ be Banach spaces, and $T: \textbf X \to \textbf Y$ a linear mapping. The following are equivalent:
1. $T: \textbf X \to \textbf Y$ is $p$-compact
2. There exist a Banach space $\textbf Z$, a mapping $\Theta \in \mathcal N^p(\textbf Z; \textbf Y)$ and a bounded mapping $R \in \mathcal{L}(\textbf X,\textbf Z/\ker\Theta)$ with $\Vert R \Vert \leq 1$ such that the following diagram commutes
$$\label{p-compact factor}
\xymatrix{
\textbf X \ar[r]^T \ar[rd]_R & \textbf Y & \ar[l]_{\Theta } \ar@{->>}[ld]^{\pi} \textbf Z \\
& \textbf Z/\ker\Theta \ar[u]^{\widetilde{\Theta}}, &
}$$
Moreover, $$\kappa_p(T) = \inf\{ \nu^p(\Theta): \Theta \in \mathcal N^p(\textbf Z; \textbf Y) \text{ as in } \eqref{p-compact factor}\}.$$
$(1) \Rightarrow (2)$: This is essentially contained in the proof of [@galicer2011ideal Proposition 2.9.] (which is based on [@sinha2002compact Theorem 3.2.]). A careful look shows that it also holds $$\inf\{ \nu^p(\Theta): \Theta \in \mathcal N^p(\textbf Z; \textbf Y) \text{ as in } \eqref{p-compact factor}\} \leq \kappa_p(T).$$
$(2) \Rightarrow (1)$: Given $\varepsilon >0$ we have the following representation for $\Theta$:
$$\Theta = \sum_{n\in \mathbb N} x_n' \otimes y_n,$$ where $(x_n)_{n \in \mathbb N} \in \ell_{p'}^w(\textbf Z')$, $(y_n)_{n \in \mathbb N} \in \ell_{p}(\textbf Y)$ such that $\Vert (x_n) \Vert_{\ell_{p'}^w(\textbf Z')} \cdot \Vert (y_n) \Vert_{\ell_{p}(\textbf Y)} \leq (1+\varepsilon) \nu^p(\Theta)$.
Let us show that $T= \tilde \Theta R$ is $p$-compact. Indeed, if $x \in B_{\textbf X}$ then $\Vert Rx \Vert_{\textbf Z/\ker\Theta} \leq 1$. Thus, there is $z \in \textbf Z$ such that $\pi z = Rx$, $\Vert z \Vert_{\textbf Z} \leq 1 + \varepsilon$. Then, $Tx=\tilde \Theta Rx= \tilde \Theta \pi z = \Theta z = \sum_{n \in \mathbb N} x_n'(z)y_n$. Since $$\Vert(x_n'(z))\Vert_{\ell_{p'}} \leq \Vert (x_n) \Vert_{\ell_{p'}^w(\textbf Z')} \Vert z \Vert_{\textbf Z} \leq (1 + \varepsilon) \Vert (x_n) \Vert_{\ell_{p'}^w(\textbf Z')},$$ this shows that $Tx$ lies in the $p$-convex hull of the sequence $(y_n \cdot (1+\varepsilon) \Vert (x_n) \Vert_{\ell_{p'}^w(\textbf Z')})_{n \in\mathbb N}$. This implies that $T$ is $p$-compact and moreover, $\kappa(T) \leq (1+ \varepsilon)^2 \nu^p(\Theta)$.
We now look for a definition of $p$-compact mappings in the noncommutative framework. We choose to define it as in Equation since this will allow us to provide the ideal with an operator space structure. Later on, we will see an equivalent description which is more obviously similar to the original Banach space definition (see Theorem \[thm:characterization-p-compact-mapping-via-p-compact-sets\]).
Let $E$ and $F$ be operator spaces. A mapping $T \in \mathcal{CB}(E,F)$ is called *operator $p$-compact* if there exist an operator space $G$, a completely right $p$-nuclear mapping $\Theta \in \mathcal{N}^p_o(G,F)$ and a completely bounded mapping $R \in \mathcal{CB}(E,G/\ker\Theta)$ with $\Vert R \Vert_{cb} \leq 1$ such that the following diagram commutes
$$\label{p-compact o.s.}
\xymatrix{
E \ar[r]^T \ar[rd]_R & F & \ar[l]_{\Theta } \ar@{->>}[ld]^{\pi} G \\
& G/\ker\Theta \ar[u]^{\widetilde{\Theta}}, &
}$$
where $\pi$ stands for the natural 1-quotient mapping and $\widetilde{\Theta}$ is given by $\widetilde{\Theta}(\pi(g)) = \Theta(g)$.
The set of all operator $p$-compact mappings from $E$ to $F$ is denoted by $\mathcal{K}_p^o(E,F)$. For $T \in \mathcal{K}_p^o(E,F)$, we also define $$\kappa_p^o(T):= \inf \{ \nu^p_o(\Theta) \},$$ where the infimum runs over all possible completely right $p$-nuclear mappings $\Theta \in \mathcal{N}^p_o(G,F)$ as in .
Let $E$ and $F$ be operator spaces. The set $\mathcal{K}_p^o(E,F)$ is a linear subspace of $\mathcal{CB}(E,F)$ and $\kappa_p^o$ defines a norm for this space.
It is clear that $\kappa_p^o(T)\ge 0$, for all $T\in \mathcal{K}_p^o(E,F)$. Also, $\kappa_p^o(T)= 0$ implies, for each $\varepsilon>0$, the existence of a commutative diagram as with $\nu^p_o(\Theta) <\varepsilon$. Then, $\|\Theta\|_{cb}<\varepsilon$ and so $\|\widetilde\Theta\|_{cb}<\varepsilon$ from which it is derived that $T=0$.
If $T\in \mathcal{K}_p^o(E,F)$ and $\lambda$ is a scalar, it is easy to see that $\lambda T\in \mathcal{K}_p^o(E,F)$ with $\kappa_p^o(\lambda T)=|\lambda| \kappa_p^o(T)$.
Now, let us consider $T_1, T_2 \in \mathcal{K}^o_p(E,F)$. For each $i=1,2$ we have a commutative diagram:
$$\xymatrix{
E \ar[r]^{T_i} \ar[rd]_{R_i} & F & \ar[l]_{\Theta_i } \ar@{->>}[ld]^{\pi_i} G_i \\
& G_i/\ker\Theta_i \ar[u]^{\widetilde{\Theta_i}} &
}$$
where $G_i$ is an operator space, $R_i\in \mathcal{CB}(E, G_i)$ with $\|R_i\|_{cb}\le 1$ and $\Theta_i \in \mathcal{N}^p_o(G_i,F)$. Define $G=G_1\oplus_\infty G_2$ and consider $\Theta:G\to F$ given by $\Theta (g_1,g_2)=\Theta_1 g_1+ \Theta_2 g_2$. If we denote, for each $i=1,2$, $\rho_i:G\to G_i$ the canonical restriction mapping we can write $\Theta=\Theta_1 \rho_1+ \Theta_2 \rho_2$. Since $\mathcal N^p_o$ is a mapping ideal, we derive that $\Theta\in \mathcal{N}^p_o(G,F)$ with $\nu^p_o(\Theta)\le \nu^p_o(\Theta_1)+\nu^p_o(\Theta_2)$.
We define $\Lambda:G_1/\ker\Theta_1\oplus_\infty G_2/\ker\Theta_2\to G/\ker\Theta$ by $\Lambda(\pi_1 g_1, \pi_2 g_2)=\pi (g_1, g_2)$, where $\pi:G\to G/\ker\Theta$ is the natural 1-quotient mapping. It is clear that $\Lambda$ is well defined and that it is completely bounded with $\|\Lambda\|_{cb}\le 1$. And the same is true for the mapping $R:E\to G/\ker\Theta$, given by $Rx= \Lambda(R_1x,R_2x)$, for any $x\in E$.
Now, we can assemble the following diagram:
$$\xymatrix{
E \ar[r]^{T_1+T_2} \ar[rd]_R & F & \ar[l]_{\Theta } \ar@{->>}[ld]^{\pi} G \\
& G/\ker\Theta \ar[u]^{\widetilde{\Theta}} &
}$$
Straightforward computations show that the diagram is commutative and hence $T_1+T_2$ belongs to $\mathcal{K}_p^o(E,F)$. Also, $\kappa_p^o(T_1+T_2)\le \nu^p_o(\Theta)\le \nu^p_o(\Theta_1)+\nu^p_o(\Theta_2)$, for every $\Theta_1$ and $\Theta_2$ “admissible” for the factorization of $T_1$ and $T_2$. Therefore, $\kappa_p^o(T_1+T_2)\le \kappa_p^o(T_1) + \kappa_p^o(T_2)$.
From the definition, it is straightforward to show that any $T \in \mathcal{N}^p_o(E,F)$ belongs to $\mathcal{K}_p^o(E,F)$ with $\kappa_p^o(T) \leq \nu^p_o(T)$.
We are now interested in giving an operator space structure to $\mathcal{K}_p^o$ based on the relation, in the Banach space setting, of the right $p$-nuclear and $p$-compact ideals.
A mapping ideal $\mathcal{A}$ is *surjective* if for each completely 1-quotient mapping $Q: G \twoheadrightarrow E$ we have $T \in \mathcal{A}(E,F)$ whenever $T Q \in \mathcal{A}(G,F)$. In this case, we have $\Vert T \Vert_{\mathcal{A}} = \Vert T Q \Vert_{\mathcal{A}}$ for every $T \in M_n(\mathcal{A}(E,F))$.
Given a mapping ideal $\mathcal{A}$, its *surjective hull* $\mathcal{A}^{sur}$ is the smallest surjective mapping ideal that contains $\mathcal{A}$.
We recall [@Pisier-Operator-Space-Theory Proposition 2.12.2] that for every operator space $E$ there is a set $I$ and a family $(n_i)_{i \in I} \subset \mathbb{N}$ such that $E$ is the quotient of $\ell_1(\{S_1^{n_i} : i \in I \})$. We denote the latter space by $Z_E$ and $Q_E : Z_E \twoheadrightarrow E$ the corresponding completely 1-quotient mapping. The space $Z_E$ is projective (see for example [@Pisier-Operator-Space-Theory Chapter 24]).
For every $E$ and $F$ operator spaces we have the following characterization of the *surjective hull* $$\mathcal{A}^{sur}(E,F)= \{ T \in \mathcal{CB}(E,F) : T Q_E \in \mathcal{A}(E,F)\},$$ with $\Vert T \Vert_{\mathcal{A}^{sur}} = \Vert T Q_E \Vert_{\mathcal{A}}$.
We define $$\mathcal{U}(E,F) = \{ T \in \mathcal{CB}(E,F) : T Q_E \in \mathcal{A}(E,F)\},$$ endowed with the operator space norm given by $\Vert T \Vert_{\mathcal{U}} = \Vert T Q_E \Vert_{\mathcal{A}}$, for every $T \in M_n(\mathcal{U}(E,F))$.
We now show that $\mathcal{U}$ is a mapping ideal. Indeed, if $T \in M_n(\mathcal{U}(E,F))$ we obviously have $$\Vert T \Vert_{\mathcal{U}(E,F)} = \Vert T Q_E \Vert_{\mathcal{A}} \geq \Vert T Q_E \Vert_{cb} = \Vert T \Vert_{cb}.$$ Let $T \in M_n(\mathcal{U}(E,F))$, $R \in \mathcal{CB}(E_0,F)$, $S \in \mathcal{CB}(E,F_0)$. Since $Z_{E_0}$ is projective, given $\varepsilon >0$, there is a lifting $L_{\varepsilon} \in \mathcal{CB}(Z_{E_0},Z_E)$ of $R Q_{E_0}$ with $\Vert L_{\varepsilon} \Vert_{cb} \leq (1+\varepsilon) \Vert R \Vert_{cb}$ such that the following diagram commutes
$$\xymatrix{
E_0 \ar[r]^R & E \ar[r]^{T_{i,j} } & F \ar[r]^S & F_0 \\
Z_0 \ar@{->>}[u]^{Q_{E_0}} \ar@{.>}[r]_{L_{\varepsilon}} & Z_E \ar@{->>}[u]_{Q_{E}} & &
}.$$
Then, for every $1 \leq i, j \leq n$ we have $ST_{i,j}RQ_{E_0} = ST_{i,j}Q_{E}L_{\varepsilon} \in \mathcal{A}(Z_{E_0},F_0)$ and hence $ ST_{i,j}R \in \mathcal{U}(E_0,F_0)$. Moreover, by the ideal property of $\mathcal{A}$,
$$\begin{aligned}
\Vert S_n T R \Vert_{M_n(\mathcal{U}(E_0,F_0))} & = \Vert S_n T R Q_{E_0} \Vert_{M_n(\mathcal{A}(Z_{E_0},F_0))} = \Vert S_n T Q_{E} L_{\varepsilon} \Vert_{M_n(\mathcal{A}(Z_{E_0},F_0))} \\
& \leq \Vert S_n T Q_{E} \Vert_{M_n(\mathcal{A}(Z_{E},F_0))} (1+\varepsilon) \Vert R \Vert_{cb} \\
& \leq (1+\varepsilon) \Vert S \Vert_{cb} \Vert T \Vert_{M_n(\mathcal{U}(E,F))} \Vert R \Vert_{cb}.\end{aligned}$$
We now prove that the mapping ideal $\mathcal{U}$ is surjective. Let $Q : G \twoheadrightarrow E$ be a complete 1-quotient mapping and $T \in \mathcal{CB}(E,F)$ such that $TQ \in \mathcal{U}(G,F)$. Since $Z_E$ is projective and $QQ_G : Z_G \twoheadrightarrow E$ is a complete 1-quotient mapping, given $\varepsilon >0$, there is a lifting of $Q_E$, $L_{\varepsilon}: Z_E \to Z_G$ with cb-norm less than or equal to $1+ \varepsilon$.
We have $TQ_E = T Q Q_G L_{\varepsilon} \in \mathcal{A}(Z_E,F)$ since $TQQ_G \in \mathcal{A}(Z_G,F)$. Then, $T \in \mathcal{U}(E,F)$. Also, if $T \in M_n(\mathcal{U}(E,F))$ we obtain
$$\begin{aligned}
\Vert T \Vert_{\mathcal{U}} & = \Vert TQ_E \Vert_{\mathcal{A}} = \Vert TQ Q_G L_{\varepsilon} \Vert_{\mathcal{A}} \leq (1 + \varepsilon) \Vert TQ Q_G \Vert_{\mathcal{A}} \\
& = (1 + \varepsilon) \Vert TQ \Vert_{\mathcal{U}} \leq (1+ \varepsilon) \Vert T \Vert_{\mathcal{U}}.\end{aligned}$$
Note that, by the definition, $\mathcal{U}(E,F) \subset \mathcal{A}^{sur}(E,F)$: if $T \in \mathcal{U}(E,F)$, we have $TQ_E \in \mathcal{A}(Z_E,F) \subset \mathcal{A}^{sur}(Z_E,F)$. Since $\mathcal{A}^{sur}$ is surjective we obtain that $T \in \mathcal{A}^{sur}(E,F)$.
We have shown that $\mathcal{U}$ is a surjective mapping ideal (which obviously contains $\mathcal A$). Then, by minimality we have the reverse inclusion.
\[projectivo coinciden\] Let $E$ be a projective operator space. Then, $T \in \mathcal{K}_p^o(E,F)$ if and only if ${T \in \mathcal{N}^p_o(E,F)}$ and $\kappa_p^o(T) = \nu^p_o(T)$.
Let $T \in \mathcal{K}_p^o(E,F)$, and consider a factorization as in . Given $\varepsilon>0$, since $E$ is projective, there is a lifting of $R$, $\widetilde R_{\varepsilon} \in \mathcal{CB}(E,G)$ with $\Vert \widetilde R_{\varepsilon} \Vert \leq 1 + \varepsilon$, such that the following diagram commutes
$$\xymatrix{
E \ar[r]^T \ar@{.>}@(ur,ul)[rr]^{\widetilde R_{\varepsilon}} \ar[rd]_R & F & \ar[l]_{\Theta } \ar@{->>}[ld]^{\pi} G \\
& G/\ker\Theta \ar[u]^{\widetilde{\Theta}}. &
}$$
Then, $T = \Theta \widetilde R_{\varepsilon} \in \mathcal{N}^p_o(E,F)$ and $\nu^p_o(T) \leq (1 + \varepsilon) \nu^p_o(\Theta)$. Observe that this holds for every $\Theta$ verifying Equation . Thus, by definition of the norm $\kappa_p^o(T)$, we obtain $\nu^p_o(T) \leq (1 + \varepsilon) \kappa_p^o(T)$ and the result follows.
Let $E$ and $F$ be operator spaces. Then $T \in \mathcal{K}_p^o(E,F)$ if and only if $T Q_E \in \mathcal{N}^p_o(Z_E,F)$ and $\kappa_p^o(T) = \nu^p_o(T Q_E)$.
If $T \in \mathcal{K}_p^o(E,F)$, then $TQ_E \in \mathcal{K}_p^o(Z_E,F)$. Now by Proposition \[projectivo coinciden\], $TQ_E \in \mathcal{N}^p_o(Z_E,F)$ and also $\nu^p_o(TQ_E) = \kappa_p^o(TQ_E) \leq \kappa_p^o(T)$, since $Z_E$ is projective.
Reciprocally, if $TQ_E \in \mathcal{N}^p_o(Z_E,F)$, we define $R \in \mathcal{CB}(E,Z_E/\ker\Theta)$ in the following way: $Rx := \pi y$ where $Q_Ey=x$. It is not difficult to check that $R$ is well defined and also $\Vert R \Vert_{cb} \leq 1$. If we denote $\Theta :=TQ_E$ we have $\widetilde \Theta R x = \widetilde \pi y = \Theta y = TQ_Ey = Tx$ for every $x \in E$. Thus, the following diagram commutes
$$\xymatrix{
E \ar[r]^T \ar@{.>}[rd]^{R} & F & \ar[l]_{\Theta } \ar@{->>}[ld]^{\pi} Z_E \\
& Z_E/\ker\Theta \ar[u]^{\widetilde{\Theta}}. &
}$$
Therefore, $T$ is an operator $p$-compact mapping and $\kappa_p^o(T) \leq \nu^p_o(TQ_E)$. This concludes the proof.
We have shown that for every $E$ and $F$ operator spaces, we have the equality (as Banach spaces) $$\mathcal{K}_p^o(E,F) = (\mathcal{N}^p_o)^{sur}(E,F).$$
This induces a natural operator space structure for $\mathcal{K}_p^o(E,F)$. Indeed, if $T \in M_n(\mathcal{K}_p^o(E,F))$, we define $$\Vert T \Vert_{M_n(\mathcal{K}_p^o(E,F))} := \Vert T \Vert_{M_n((\mathcal{N}^p_o)^{sur}(E,F))} = \Vert T Q_E \Vert_{M_n(\mathcal{N}^p_o(Z_E,F))}.$$
As a consequence we can say that $\mathcal{K}_p^o$ is a mapping ideal through the following identification:
$$\label{Npsur}
\mathcal{K}_p^o=(\mathcal{N}^p_o)^{sur}.$$
In the Banach space setting, a continuous linear mapping is compact if it sends the unit ball into a relatively compact set. Equivalently, it sends the unit ball into the closure of the convex hull of a null sequence. In the operator space framework, if we look at how a mapping acts on the “matrix unit ball” (see definition below) the previous conditions are not equivalent. Webster [@webster1997local; @Webster1998] noticed this and studied several notions of compactness: operator compactness, strong operator compactness, and matrix compactness. Let us recall the definition of “operator compactness” that later we will naturally extend to the case of $p$-compactness.
If $E$ is an operator space, a *matrix set* is a sequence of sets $\mathbf{K}=(K_n)$, where $K_n\subset M_n(E)$, for all $n$. The closure of a matrix set means the closure of each set of the sequence in the corresponding matrix space. The *matrix unit ball* is the matrix set $\left(B_{M_n(E)}\right)$.
Given $X \in \mathcal S_\infty[E]$, Webster defined the *absolutely matrix convex hull* of $X$ to be the matrix set $\mathbf{co}(X)$ where $$\big(\mathbf{co}(X)\big)_k = \{v \in M_k(E) : \exists\sigma \in M_k(M_{\infty}^{fin}), \; \Vert \sigma \Vert_{M_k(S_{1})} \leq 1 \textrm{ such that } (\sigma \otimes id)X=v \}.$$
A mapping $T\in\mathcal{CB}(E,F)$ is *operator compact* if the image by $T$ of the matrix unit ball of $E$ is contained in $\overline{\mathbf{co}(Y)}$, for some $Y\in \mathcal S_\infty[F]$.
Based on the work of Webster we now define the notion of $p$-absolutely matrix convex hull.
Let $E$ be an operator space and $X\in S_p [E]$, for $1 \leq p \leq \infty$, we define the $p$-absolutely matrix convex hull of $X$ to be the matrix set $\mathbf{co}_p(X)$ where $$\big(\mathbf{co}_p(X)\big)_k = \{v \in M_k(E) : \exists\sigma \in M_k(M_{\infty}^{fin}), \; \Vert \sigma \Vert_{M_k(S_{p'})} \leq 1 \textrm{ such that } (\sigma \otimes id)X=v \}.$$
We say that a matrix set $\mathbf{K}=(K_n)$ in $E$ is *operator $p$-compact* if $\mathbf{K}$ is is contained in $\overline{\mathbf{co}_p(X)}$, for some $X\in S_p [E]$.
We are using the term “operator $p$-compact” for two seemingly different notions, but they will turn out to be the same. If we denote by $\mathbf{B}_E$ the matrix unit ball of $E$, we will prove that $T: E \to F$ is *operator $p$-compact* if and only if the image under $T$ of the matrix unit ball of $E$ is an operator $p$-compact matrix set in $F$.
Note that the $\infty$-absolutely matrix convex hull is Webster’s absolutely matrix convex hull, and thus operator $\infty$-compact mappings are the same as operator compact mappings.
\[remark capsula Webster\] Given $1\le p\le\infty$, for any $X \in S_p[E]$ and $k \in {{\mathbb{N}}}$ we have $$\left(\overline{\mathbf{co}_p(X)}\right)_k = \{v \in M_k(E) : \exists\sigma \in M_k(S_{p'}), \; \Vert \sigma \Vert_{M_k(S_{p'})} \leq 1 \textrm{ such that } (\sigma \otimes id)X=v \},$$ where, in the case $p=1$ the space $S_{p'}$ should be replaced by $M_\infty$.
Indeed, let us denote by $C_k$ the set on the right hand side. For $1<p\le\infty$, it is enough to show that $C_k$ is closed. Consider $\Psi: M_k(S_{p'}) \to M_k(E)$ defined by $\sigma \mapsto (\sigma \otimes id)(X)$. Note that the image by $\Psi$ of $B_{M_k(S_{p'})}$, the closed unit ball of $M_k(S_{p'})$, is exactly $C_k$. Take a sequence $(\sigma_l)_l \in B_{M_k(S_{p'})}$ such that $\Psi(\sigma_l) \to v \in M_k(E)$. We have to see that $v= \Psi(\sigma)$ for certain $\sigma \in M_k(S_p')$ with $\Vert \sigma \Vert_{M_k(S_{p'})} \leq 1$. Since $B_{M_k(S_{p'})}$ is weak$^{*}$ sequentially compact (since the predual of $M_k(S_{p'})$ is separable) there is a subsequence $(\sigma_{l_j})_j$ weak$^{*}$ convergent to an element $\sigma \in B_{M_k(S_{p'})}$. By uniqueness of the limit our conclusion follows once proving that for all matrices $v' \in M_k(E')$ we have the convergence of the scalar pairing (in the sense of [@Effros-Ruan-book 1.1.24]) $$\langle \Psi(\sigma_{l_j}),v' \rangle \to \langle \Psi(\sigma),v' \rangle.$$ Now, $$\langle \Psi(\sigma_{l_j}),v' \rangle = \langle ( \sigma_{l_j} \otimes id)(X),v' \rangle = \langle \sigma_{l_j}, (id \otimes v')(X) \rangle \to \langle \sigma, (id \otimes v')(X) \rangle = \langle \Psi(\sigma),v' \rangle.$$ For $p=1$, the previous argument (replacing $S_{p'}$ by $M_\infty$) shows that $\left(\overline{\mathbf{co}_1(X)}\right)_k\subset C_k$. The other inclusion runs as follows. Recall, using the notation as in Theorem \[Thm:Q-tilde\], that any $X \in S_1[E]$ can be approximated by its “truncations” $\{\widetilde{\tau}^m(X)\}_m$ [@Pisier-Asterisque-98 Lemma 1.12] and denote by $\rho^m:M_\infty\to M_m$ the truncated mapping between these matrix spaces. Now, for $v=(\sigma \otimes id)X \in C_k$ with $\sigma\in M_k(M_\infty)$, we have to see that $\{((\rho^m)_k\sigma \otimes id)X\}_m\subset \mathbf{co}_1(X)$ approximates $v$. Indeed, $$\begin{aligned}
\|v-((\rho^m)_k\sigma \otimes id)X\|_{M_k(E)} & = \|((\sigma-(\rho^m)_k\sigma)\otimes id)X\|_{M_k(E)} = \|(\sigma\otimes id) (X-(\widetilde{\tau}^m)_k(X))\|_{M_k(E)} \\
& \le \|X-(\widetilde{\tau}^m)_k(X)\|_{M_k(S_1[E])} \underset{m\to\infty}{\longrightarrow} 0.\end{aligned}$$
The following result shows that the first definition of $p$-compactness (the one which involves the factorization through completely right $p$-nuclear mappings) and the definition based on Webster’s work are the same. We highlight the analogy presented between Equations and , below.
\[thm:characterization-p-compact-mapping-via-p-compact-sets\] Let $T: E \to F$ be a completely bounded mapping and $1\le p\le \infty$. Then $T$ is operator $p$-compact if and only if $T(\mathbf{B}_E)$ is an operator $p$-compact matrix set in $F$. Moreover,
$$\label{kpo}
\kappa_p^o(T) = \inf\{ \Vert Y \Vert_{S_p[F]} : T(\mathbf{B}_E) \subset \overline{\mathbf{co}_p(Y)}\}.$$
Suppose $T$ is operator $p$-compact. Without loss of generality we suppose that $\kappa_p^o(T) < 1$. By the commutative diagram given in there is a completely right $p$-nuclear mapping $\Theta:G\to F$ with $\nu^p_o$-norm less than $1$. It suffices to see that $\Theta$ is operator $p$-compact, since $T(\mathbf{B}_E)$ is contained in $\Theta(\mathbf{B}_G)$. By Theorem \[thm:hard-Chevet-Saphar\] we can write $\Theta = J^p\circ Q^p (X\otimes Y)$ where $\Vert X \Vert_{S_{p'}\widehat\otimes_{\min} G'} < 1$ and $\Vert Y \Vert_{S_p[F]} < 1$. For $g \in M_n(G)$ of norm less than one, we will show that $\Theta_n g \in (\overline{\mathbf{co}_p(Y)})_n$. Indeed, for $1<p\le \infty$, since $X \in S_{p'}\widehat\otimes_{\min} G'$ we derive $X'\circ \iota_G\in \mathcal{CB}(G, S_{p'})$. For $p=1$, being $X \in S_{\infty}\widehat\otimes_{\min} G'$ we obtain $X'\circ \iota_G\in \mathcal{CB}(G, M_{\infty})$. Consider $\sigma := \left(X'\circ \iota_G\right)_n g \in M_n(S_{p'})$ (replacing $S_{p'}$ by $M_\infty$ in the case $p=1$), then, $\Vert \sigma \Vert_{M_n(S_{p'})} < 1$ and it is not hard to check that $$\Theta_n g = (\sigma \otimes id) (Y).$$ Then, $T(\mathbf{B}_E) \subset \Theta(\mathbf{B}_G) \subset \overline{\mathbf{co}_p(Y)}$ and hence $ \inf\{ \Vert Y \Vert_{S_p[F]} : T(\mathbf{B}_E) \subset \overline{\mathbf{co}_p(Y)}\} < 1$.
For the converse, let $T: E \to F$ be an operator $p$-compact mapping with $ \inf\{ \Vert Y \Vert_{S_p[F]} : T(\mathbf{B}_E) \subset \overline{\mathbf{co}_p(Y)}\}< 1$. We begin by considering the case $1<p<\infty$. Denote by $\Theta : S_{p'} \to F$, the operator given by $J^p \circ B^p (id, Y)$, which by definition is in $\mathcal N^p_o$ since $id \in S_{p'}^w[S_p]=\mathcal{CB}(S_p, S_p)$. Observe that by Corollary \[defB\_p\] (1), $\nu_o^p(\Theta) < 1$. Remark \[remark capsula Webster\] asserts that for each $x \in E$ there is $\sigma \in S_{p'}$ such that $Tx = (\sigma \otimes id) (Y)$. Defining $Rx:=[\sigma]$ we have the following commutative diagram
$$\xymatrix{
E \ar[r]^T\ar[rd]_R & F & \ar[l]_{\Theta } \ar@{->>}[ld]^{\pi} S_{p'} \\
& S_{p'}/\ker\Theta \ar[u]^{\widetilde{\Theta}}, &
}.$$
Note that $R$ is well defined. Moreover, if $x \in B_{M_k(E)}$ by hypothesis $T_k x =( \widetilde{ \sigma} \otimes id)(Y)$, for certain $\widetilde \sigma $ with $\Vert \widetilde \sigma \Vert_{M_k(S_{p'})} \leq 1$. Checking coordinates we have $T(x_{i_j})= ( \widetilde{ \sigma}_{i,j} \otimes id)(Y)$ and therefore $R(x_{i,j}) = [ \widetilde{ \sigma}_{i,j} ]$. Hence, $R_k x = ([\widetilde{ \sigma}_{i,j}])_{i,j})$ and then $\Vert R \Vert_{cb} \leq 1$. We conclude that $T \in \mathcal K^o_p(E;F)$ and $\kappa_p^o(T) \leq \nu_o^p(\Theta) < 1$.
For $p=1$, we have $Y \in S_1[F]=S_1\widehat\otimes_{proj} F$. Since the projective tensor norm satisfies the Embedding Lemma (see [@dimant2015biduals] or [@CDDG]), there is a completely isometric embedding $\kappa: S_1\widehat\otimes_{proj} F\to S_1^{''}\widehat\otimes_{proj} F$. Now, $\Theta= J^1\circ \kappa(Y)$ belongs to $\mathcal N^1_o(M_\infty,F)$, where $J^1: S_1^{''}\widehat\otimes_{proj} F\to S_1^{''}\widehat\otimes_{\min} F$ is the canonical mapping. The rest of the argument follows as in the previous case.
For $p=\infty$, we have $Y \in S_\infty [F]$ and so $B^\infty (id, Y)$ belongs to $S_\infty \widehat\otimes_{d_\infty^o} F$. Since $d_\infty^o$ is finitely generated and $S_\infty$ is locally reflexive, we can appeal again to the Embedding Lemma [@CDDG] to get a completely isometric embedding $\kappa: S_\infty\widehat\otimes_{d_\infty^o} F\to M_\infty\widehat\otimes_{d_\infty^o} F$. Hence, $\Theta=J^\infty \circ\kappa\circ B^\infty (id, Y)$ belongs to $\mathcal N_o^\infty(S_1,F)$ and the proof finishes as in the two previous cases.
The previous theorem, together with the relation presented in Equation , allows us to endow Webster’s class (of operator compact mappings) with a natural operator space structure, showing that this class is indeed a mapping ideal. Namely, it is exactly the ideal $\mathcal{K}_{\infty}^o$.
Some final questions {#section questions .unnumbered}
====================
We present some open questions regarding the mapping ideal $\mathcal K_p^o$.
In parallel to the Banach space case, we say that a mapping ideal $(\mathfrak{A},\mathbf{A})$ and an operator space tensor norm $\alpha$ are *associated*, denoted $(\mathfrak{A},\mathbf{A}) \sim \alpha$ if for every pair of finite-dimensional operator spaces $(E,F)$ we have a complete isometry $
\mathfrak{A}(E,F) = E' \otimes_{\alpha} F,
$ given by the canonical map. Notice that since this definition is based only on finite-dimensional spaces, two different mapping ideals can be associated to the same tensor norm (and vice versa). For example, from Definition \[p-nuclear def\] it is clear that $d_p^o$ is associated to the mapping ideal $\mathcal N_o^p$.
Recall that an operator space tensor norm $\alpha$ is called left-injective if for any operator spaces $E_1, E_2, F$ and a complete isometry $i:E_1 \to E_2$, the mapping $$i\otimes id_F : E_1 \otimes_{\alpha} F \to E_2 \otimes_{\alpha} F$$ is completely isometric. Given an operator space tensor norm we denote by $/ \alpha$ the greatest left-injective tensor norm that is dominated by $\alpha$. This essentially mimics the notion given in the context of normed spaces deeply described in [@Defant-Floret Chapter 20] and [@Ryan Section 7.2].
In the Banach space setting, it is shown in [@galicer2011ideal Theorem 3.3] that $/ d_p$ is the tensor norm associated to $\mathcal K_p$. Therefore, is natural to ask if an analogous result holds in the operator space framework.
\[preg1\] Is the operator space tensor norm $/ d_p^o$ associated to the mapping ideal $\mathcal{K}_p^o$?
This question is a particular case of the following more general one (whose statement is valid in the Banach space context):
\[preg2\] Let $\mathcal A$ be a mapping ideal and $\alpha$ its associated operator space tensor norm. Is $/ \alpha$ associated to $\mathcal A^{sur}$?
Note that $d_p^o \sim \mathcal N_o^p$ and, by Equation , we have $\mathcal K_p^o = (\mathcal N_o^p)^{sur}$. Thus an affirmative answer to this question would provide also an answer to the first one. Result of this kind for the Banach space setting, although sometimes hard to see it at first glance, are based on *local techniques* (e.g., [@Defant-Floret Proposition 20.9]) which are no longer valid in the operator space framework.
In [@CDDG] it is shown that if $\beta \sim \mathcal A^{sur}$ then $\beta$ is left-injective. Therefore, since $\mathcal A \subset \mathcal A^{sur}$ we have that $\beta \leq / \alpha$. An affirmative answer to Question \[preg2\] is obtained in [@CDDG] for the particular case where $\alpha$ is left accessible (definition similar to the Banach space case). It is known in the Banach space setting that $d_p$ satisfies this property ([@Defant-Floret Theorem 21.5], [@Ryan Proposition 7.21.]), but unfortunately we do not know yet whether this also holds for the operator space counterpart $d_p^o$.
Another interesting question, which involves the mapping ideal $\mathcal K_{\infty}^o$, was posed by Webster in his thesis [@webster1997local Section 4.1]:
[Webster:]{}\[problema webster\] “It is an open question as to whether the space of operator compact maps must be closed in the cb-norm topology”.
In other words, if a sequence of operator compact mappings $(T_n)_{n \in {{\mathbb{N}}}}$ converges to $T$ in the completely bounded norm ($\Vert T - T_n \Vert_{cb} \to 0$), does it imply that $T$ is also an operator compact map?
By Theorem \[thm:characterization-p-compact-mapping-via-p-compact-sets\], as a direct consequence of the Open Mapping Theorem we see that Problem \[problema webster\] can be reformulated in terms of the following question.
Are the norms $\kappa_{\infty}^o$ and $\Vert \cdot \Vert_{cb}$ equivalent on $\mathcal{K}_{\infty}^o$?
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[^1]: The first author was partially supported by NSF grant DMS-1400588. The second author was partially supported by CONICET PIP 11220130100483 and ANPCyT PICT 2015-2299. The third author was partially supported by CONICET PIP 11220130100329 and ANPCyT PICT 2015-3085.
|
---
abstract: 'We have performed a FCI-quality benchmark calculation for the tetramethyleneethane molecule in cc-pVTZ basis set employing a subset of CASPT2(6,6) natural orbitals for the FCIQMC calculation. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calculations by Pozun *et al.* and available experimental results. Our computations verified that there is a maximum on PES of the ground singlet state ($^1\text{A}$) $45^{\circ}$ torsional angle and the corresponding vertical singlet-triplet energy gap is $0.01$ eV. We have employed this benchmark for assessment of the accuracy of MkCCSDT and DMRG-tailored CCSD (TCCSD) methods. MR MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with an excellent accuracy, but for the correct ordering requires computation of the zero-spin-projection component of the triplet state ($^3\text{B}_1$).'
author:
- Libor Veis
- Andrej Antalík
- Örs Legeza
- Ali Alavi
- Jiří Pittner
bibliography:
- 'references.bib'
title: Full configuration interaction quantum Monte Carlo benchmark and multireference coupled cluster studies of tetramethyleneethane diradical
---
Introduction {#section_introduction}
============
Tetramethyleneethane (TME), the simplest disjoint non-Kekulé diradical firstly synthesized by Dowd [@dowd_1970], has due to its complex electronic structure been often used as a benchmark system for the state-of-the-art multireference computational methods [@pittner_2001; @bhaskaran-nair_2011; @chattopadhyay_2011; @pozun_2013; @demel_2015]. Its complexity comes out of the fact that it contains a nearly degenerate pair of the frontier orbitals, which tend to be localized on separate allyl subunits [@pozun_2013] and are occupied by two electrons. Moreover, TME possesses a degree of freedom corresponding to the rotation about the central C-C bond (maintaining $D_2$ symmetry, see Figure \[tme\_rotation\]) and the energetic ordering of these two frontier orbitals and consequently their occupation in the lowest singlet state changes along the rotation [^1]. As a result, determining the relative stability of the lowest singlet and triplet states turned out to be a big challenge for both experimental and theoretical methods.
-0.5cm ![The studied process of a rotation of the TME allyl subunits about the central C-C bond. Carbon atoms are colored brown, hydrogens are white.[]{data-label="tme_rotation"}](./tme.pdf "fig:"){width="45.00000%"}
The first experimental electron paramagnetic resonance (EPR) results predicted TME to have a triplet ground state [@dowd_1970], when stabilized in a matrix with a torsional angle being approximately 45$^{\circ}$ [@dowd_1986; @dowd_1987]. The predicted triplet ground state attracted a lot of interest in TME for its potential use as an organic magnet [@berson_1999]. However, photo-electron spectroscopy of the TME$^{-}$ ion strongly suggested TME to have the singlet ground state at the torsional angle corresponding to 90$^{\circ}$ [@clifford_1998], similarly as the EPR experiments on TME derivatives [@bush_1997a; @bush_1997b].
Several theoretical studies using different level of approximations [@du_1987; @nachtigall_1992; @nachtigall_1993; @filatov_1999; @rodriguez_2000; @pittner_2001; @bhaskaran-nair_2011; @chattopadhyay_2011] have step by step contributed to understanding of the electronic structure of the TME diradical. Nevertheless, only the work of Pozun *et al.* employing the large scale diffusion Monte Carlo (DMC) calculations [@pozun_2013] finally reliably established the magnitude of the singlet triplet gap and also the shape of the singlet potential energy surface (PES).
The main conclusions tell us that the correct theoretical description of the multireference singlet state ($^{1}\text{A}$) requires all of the following conditions being fulfilled, namely the flexible-enough atomic basis set, the proper description of the static correlation with the minimum active space comprising of six $\pi$-orbitals, and a proper treatment of the dynamic correlation (at least at the level of the second-order perturbation theory). All this make TME a very delicate molecule and indeed the perfect benchmark system for state-of-the-art multireference methods. Moreover, TME serves as a model system for more complicated disjoint diradicals.
In the present work, we follow [@pozun_2013] and compute the singlet, as well as triplet twisting PESs of TME. Firstly, we provide the full configuration interaction (FCI) quality data by the FCI quantum Monte Carlo (FCIQMC) method [@booth_2009; @cleland_2010; @petruzielo_2012; @overy_2015], whose accuracy is justified by an excellent agreement with DMC results of Pozun *et al.* [@pozun_2013] and available experimental data. Secondly, we compare the results of the Hilbert space Mukherjee’s multireference coupled clusters (MR MkCC) [@mahapatra_1999; @evangelista_2006; @evangelista_2007; @evangelista_2008; @evangelista_2010; @das_2010; @bhaskaran-nair_2008; @bhaskaran-nair_2010; @demel_2010] and the recently developed coupled clusters with single and double excitations tailored by the density matrix renormalization group method (DMRG-TCCSD) [@veis_2016] against the FCIQMC benchmark.
The paper is organized as follows: in Sec. II, we give a very brief overview of the used computational approaches and the actual computational details, the next Section summarizes the results with discussion, and the final Section closes with conclusions and outlook.
Overview of computational approaches {#section_computational_details}
====================================
In this Section we, for completeness, sketch the main concepts and ideas of the employed computational approaches.
FCI quantum Monte Carlo
-----------------------
The FCIQMC method [@booth_2009; @cleland_2010; @booth_2014; @overy_2015], originally developed by one of us, is a stochastic approach performing a long time integration of the imaginary-time Schrödinger equation which is capable of converging onto the FCI solution for much larger orbital spaces than the exact diagonalization allows. In contrast to DMC, FCIQMC sample the *Slater determinant space* by an ensemble of walkers that move around randomly.
Master equations governing walkers’ population dynamics are given by
$$-\frac{\text{d} N_i}{\text{d} \tau} = (H_{ii} - S)N_i + \sum_{j \neq i} H_{ij} N_j,
\label{qmc_dynamics}$$
where $\tau$ is imaginary time, $N_i$ the walker population on determinant $i$, $S$ the energy shift parameter controlling the total walker population, and $H_{ij}$ Hamiltonian matrix elements in the basis of Slater determinants. When employing the stochastic approach, individual walkers evolve according to a simple set of rules which include spawning, death/cloning and most importantly also annihilation processes [@booth_2009].
We have used the semi-stochastic method with real walker weights [@petruzielo_2012], in which part of the imaginary-time propagation (Eq. \[qmc\_dynamics\]) is performed exactly (deterministic space) and the rest stochastically. Such an approach in fact greatly reduces stochastic errors.
Mukherjee’s coupled clusters
----------------------------
The MR MkCC approach formulated by Mukherjee *et al.* [@mahapatra_1999] and later on developed by others, including one of us [@evangelista_2006; @evangelista_2007; @evangelista_2008; @evangelista_2010; @das_2010; @bhaskaran-nair_2008; @bhaskaran-nair_2010; @demel_2010; @brabec_2012a; @brabec_2012; @demel_2015], is a state specific Hilbert-space multireference coupled cluster method. Consequently, the MkCC wave function $|\Psi_{\text{MkCC}}\rangle$ is expressed by means of the Jeziorski-Monkhorst ansatz
$$|\Psi_{\text{MkCC}}\rangle = \sum_{\mu = 1}^{M} c_{\mu} e^{T(\mu)} |\Phi_{\mu}\rangle.
\label{jeziorski-monkhorst}$$
In Eq. \[jeziorski-monkhorst\], $|\Phi_{\mu}\rangle$ are the reference functions spanning the model space (in our case complete) and $T(\mu)$ the reference-dependent cluster operators. The $c_{\mu}$ coefficients as well as the desired energy are obtained by diagonalization of the effective Hamiltonian matrix, whose elements read
$$H_{\mu \nu}^{\text{eff}} = \langle \Phi_{\mu} | e^{-T(\nu)} H e^{T(\nu)} | \Phi_{\nu} \rangle,$$
with $H$ being the Hamiltonian operator.
The MkCC method is superior to the related Hilbert-space multireference method based on the Brillouin-Wigner CC theory due to its exact size extensivity. Though not completely free of problems, the MkCC approach is reliable for small model spaces and indeed the method of choice for electronic structure studies of diradicals. In the present work, we have employed the MR MkCC methods including single and double (MkCCSD) and single, double, and triple excitations (MkCCSDT).
DMRG-based tailored coupled clusters
------------------------------------
The tailored CC (TCC) approach was formulated by Kinoshita *et al.* [@kinoshita_2005] and belongs to the class of so called externally corrected CC methods. The TCC wave function expansion employes the split-amplitude ansatz used previously by Piecuch *et al.* [@piecuch_1993; @piecuch_1994]
$$\label{eq:TCC}
| \Psi_\text{TCC} \rangle = e^{T} | \Phi_\text{0} \rangle = e^{T_\text{ext}+T_\text{CAS}} | \Phi_\text{0} \rangle = e^{T_\text{ext}} e^{T_\text{CAS}} | \Phi_\text{0} \rangle,$$
i.e. the cluster operator is split up into its active space part ($T_\text{CAS}$) and the remaining external part ($T_\text{ext}$). Since $| \Phi_\text{0} \rangle$ is a single-determinant reference wave function, both of the aforementioned cluster operators mutually commute, which keeps the methodology very simple.
The $T_\text{CAS}$ amplitudes are supposed to be responsible for a proper description of the static correlation. They are computed from the complete active space configuration interaction (CASCI) wave function coefficients and are kept frozen during the CC iterations. Only the $T_\text{ext}$ part, which is responsible for a proper description of the dynamic correlation, is being optimized.
Recently, some of us have developed the DMRG-TCCSD method, i.e. coupled clusters with single and double excitations tailored by matrix product state (MPS) wave functions (wave functions produced by the DMRG algorithm [@white_1992; @schollwock_2011]) [@veis_2016]. This approach replaces CASCI of the original TCC method by DMRG and thus allows employing much larger active spaces. It has indeed proven itself a reliable method suitable for difficult multireference problems requiring larger active spaces [@veis_2016].
Computational details
---------------------
We have performed constrained geometry optimizations for seven values of the torsional angle along the twisting process. Geometries were optimized for both states ($^1\text{A}$, $^3\text{B}_1$) with the complete active space second order perturbation theory (CASPT2) as implemented in the MOLPRO package [@molpro]. The CASPT2 calculations were carried out using the active space comprising of six $\pi$ orbitals, CAS(6,6), and cc-pVTZ basis [@dunning_basis]. Only the first 60 CASPT2(6,6) natural orbitals sorted according to their occupation numbers were kept for the correlation treatment by the FCIQMC, MR MkCC, DMRG, and DMRG-TCCSD methods, the rest was dropped. We have chosen this strategy rather than employing a smaller basis, e.g. 6-31G/6-31G\*, which would still be manageable by the massively parallel FCIQMC implementation [@booth_2014], as it was clearly demonstrated in [@pozun_2013], that a triple-$\zeta$ basis with f functions on the C atoms is essential for the proper description of the singlet ($^{1}\text{A}$) PES (see Section with results and discussion for further comments).
For the FCIQMC calculations, we have employed the following computational protocol: (1) equilibration computations with 10 million walkers; (2) generation of FCIQMC natural orbitals [@overy_2015] (for faster convergence with the number of walkers) with 50-million-walker computations; (3) subsequent 100, 500, and 1000-million-walker computations with the FCIQMC natural orbitals. We have used the initiator version of the FCIQMC method as implemented in the NECI program package [@neci]. Moreover, to greatly reduce stochastic errors, we have employed the semi-stochastic method with real walker weights [@petruzielo_2012] and, in case of the largest 1000-million-walker computations, 50 thousand most populated determinants in the deterministic space.
MR MkCC calculations were performed with the complete model space comprising of the frontier orbitals, CAS(2,2).
In all production DMRG calculations (those used for generation of the active space CC amplitudes), we have employed the dynamical block state selection (DBSS) procedure [@legeza_2003a; @legeza_2004] with the truncation error criterion set to $5 \cdot 10^{-6}$, which resulted in bond dimensions varying in the range of $1000-8000$. The orbitals for DMRG active spaces were chosen according to their single-orbital entropies ($S_i$), in particular for CAS(6,6) $S_i > 0.3$, in case of CAS(12,12) $S_i > 0.1$, and for CAS(24,25) $S_i > 0.075$. As usually, the DMRG active space orbitals were split-localized [@olivares-amaya_2015]. The Fiedler method [@barcza_2011; @fertitta_2014] was used for optimization of the orbital ordering and DMRG runs were initialized using the CI-DEAS procedure [@legeza_2003b; @legeza_review].
In all DMRG-TCCSD calculations, we have employed the frozen core approximation. Apart from the high spin triplet ($m_s = 1$), for the reasons discussed below, we have also calculated the low spin triplet components ($m_s = 0$). Such calculations were indeed realized by swapping (rotation) of the open-shell $\beta$ spin-orbitals and finally closed-shell computations employing the unrestricted versions of the DMRG [^2] and TCCSD codes (the molecular orbital integrals become spin-dependent).
Results and discussion {#section_results}
======================
The FCIQMC PESs of the singlet ($^1\text{A}$) as well as the triplet state ($^3\text{B}_1$) corresponding to the twisting process are shown in Figure \[fciqmc\_plot\]. We do not present the absolute energies as they may not be fully converged with the number of walkers [^3], however 1000 million walkers was the maximum we could afford with 2000 CPU cores and the relative energies are definitely not affected giving excellent agreement with the DMC energies by Pozun *et al.* [@pozun_2013] and available experimental data (see Table \[tab\_results\]).
One can observe a very similar shape of the singlet PES as demonstrated by Pozun *et al.* [@pozun_2013], i.e. with its maximum corresponding to the torsional angle of 45$^{\circ}$. The height of this “hump” \[$E(45^{\circ}) - E(0^{\circ})$\] calculated by the FCIQMC method equals 0.05 eV. Pozun *et al.* [@pozun_2013] demonstrated that a triple-$\zeta$ basis with f functions on the C atoms is essential to obtain a correct shape of the singlet PES with the maxima at 45$^{\circ}$. We have performed additional FCIQMC calculations for the torsional angles of $0^{\circ}$, $45^{\circ}$, and $90^{\circ}$ with the hybrid 6-31G/6-31G\* including polarization functions only on the two central C atoms (74 molecular orbitals in total) to verify this conclusion. In fact, the energy difference between the points at $45^{\circ}$ and $0^{\circ}$ that we obtained was zero within the statistical errors, which is in agreement with [@pozun_2013].
The magnitude of the TME singlet-triplet energy gap is indeed very small, corresponding to $0.01$ eV as obtained by the FCIQMC method. This is also the reason for the originally wrong ground state triplet assignment by EPR spectroscopy [@dowd_1970]. Weak EPR signal was apparently caused by small population of the triplet state allowed by the rotation about the central C-C bond.
![The FCIQMC singlet ($^1\text{A}$) and triplet state ($^3\text{B}_1$) twisting PESs of TME. Vertical lines correspond to errors calculated by the blocking analysis [@flyvbjerg_1989].[]{data-label="fciqmc_plot"}](./plot1.pdf){width="45.00000%"}
In Figure \[mrcc\_plot\], we present the MR MkCCSD, MkCCSDT and DMRG(24,25) singlet ($^1\text{A}$) and triplet state ($^3\text{B}_1$) TME PESs. TCCSD PESs are shown in Figure \[tcc\_plots\].
As can be observed in Figure \[mrcc\_plot\], the MR MkCCSD method gives wrong state ordering for all points along the twisting process except for one ($0^{\circ}$). Inclusion of triple excitations (MR MkCCSDT) in fact corrects this behavior. Nevertheless, nor the MR MkCCSDT method with CAS(2,2) model space is able to properly describe the PES of the singlet state with the apparent maximum at the torsional angle of $45^{\circ}$. There is actually an indication of an arising maximum on the MR MkCCSDT singlet PES close to $45^{\circ}$, however it is still too flat. Either enlargement of the model space as suggested by Pozun *et al.* [@pozun_2013] or inclusion of higher (quadruple) excitations is probably necessary for the correct singlet state description. We have not pursued any of these possibilities, mainly due to considerably higher computational demands. Moreover, larger model spaces \[in our case ideally CAS(6,6)\] are not recommended for the Hilbert space multireference coupled cluster methods due to the so called proper residual problem [@lyakh_2012].
Figure \[mrcc\_plot\] also depicts the DMRG(24,25) PESs. One can see the correct ordering of both spin states, however the singlet ($^1\text{A}$) PES also does not possess the right shape. The singlet state energy is correctly increasing when going from the torsional angle of $0^{\circ}$ to $45^{\circ}$, but does not sufficiently decrease for $45^{\circ}$ to $90^{\circ}$. Apparently, the missing dynamic correlation has an important effect on this part of the singlet PES, changing its shape qualitatively.
![The MR MkCCSD, MkCCSDT, and DMRG(24,25) singlet ($^1\text{A}$) and triplet state ($^3\text{B}_1$) twisting PESs of TME.[]{data-label="mrcc_plot"}](./plot2.pdf){width="45.00000%"}
The TCC results from Figure \[tcc\_plots\] indicate that the TCCSD method is successful in recovering a major part of the missing dynamic correlation and thus properly describes the singlet PES. The effect of enlarging CAS is graphically depicted in Figure \[tcc\_plot1\] and numerically by comparison with the FCIQMC benchmark in Table \[tcc\_tab\]. One can observe that the value of the twisting energy barrier \[$E(45^{\circ}) - E(0^{\circ})$\] is decreasing with enlarging CAS and also improving towards the FCIQMC benchmark, eventually giving an excellent agreement for TCCSD(24,25) with the error of $-0.005$ kcal/mol.
Method $E$ \[kcal/mol\] $\Delta E$ \[kcal/mol\]
-------------- ------------------ -------------------------
TCCSD(6,6) $2.527$ $\phantom{-}1.395$
TCCSD(12,12) $1.577$ $\phantom{-}0.445$
TCCSD(24,25) $1.127$ $-0.005$
: The TME singlet state ($^1\text{A}$) twisting energy barrier calculated by the TCCSD method with various CASs and the energy differences from the FCIQMC benchmark.[]{data-label="tcc_tab"}
Nevertheless, the shape of the singlet PES is only a part of the story. In case of the spin state ordering, even the TCCSD method is not completely free of problems. When calculating the high spin triplet component ($m_s = 1$), TCCSD(24,25) gives a wrong ordering of both spin states near the torsional angle of $45^{\circ}$ (see Figure \[tcc\_plot2\]). The reason for this behavior is obviously a fact that at $45^{\circ}$, both spin states very much differ in their character, triplet dominated by a single determinant, whereas singlet being strongly multireference with the two determinants (HOMO$^2$LUMO$^0$ and HOMO$^0$LUMO$^2$ [^4]) of practically equal weight. This is actually the worst case scenario for the TCC method, which even though we call a multireference CC, strictly speaking uses a single reference determinant and may be slightly biased in such “degenerate” situations. Taking into account that the TME singlet-triplet energy gap is really small ($0.2$ kcal/mol by FCIQMC), the aforementioned fact results in wrong state ordering.
To verify our assumptions, we have also calculated the low spin triplet component ($m_s = 0$). It is strongly multireference as well, since it can be qualitatively described by a combination of two determinants (HOMO$^{\alpha}$LUMO$^{\beta}$ and HOMO$^{\beta}$LUMO$^{\alpha}$) with equal weights. Our aim was to eliminate to some extent the bias towards one of the two equally important determinants by calculating the states of a similar character. Figure \[tcc\_plot2\] proves that such an approach gives correct spin state ordering.
Last but not least, Table \[tab\_results\] compares the singlet-triplet energy gaps for the torsional angles of $45^{\circ}$ and $90^{\circ}$ calculated by different methods with the DMC result of Pozun *et al.* [@pozun_2013] and available experimental data. One can notice results of a similar quality for the MR MkCCSDT and TCCSD(24,25)$_{m_s = 0}$ methods, where the fact that the former method includes single, double, and triple excitations whereas the later only single and double excitations should be emphasized.
-------------------------------------- ----------------------- --------------------
$45^\circ$ $90^\circ$
MkCCSD $-0.12$ $-0.03$
MkCCSDT $\phantom{-}0.07$ $\phantom{-}0.15$
DMRG(24,25) $\phantom{-}0.05$ $\phantom{-}0.13$
TCCSD(24,25)$_{\text{m}_\text{s}=1}$ $-0.09$ $\phantom{-}0.04$
TCCSD(24,25)$_{\text{m}_\text{s}=0}$ $\phantom{-}0.07$ $\phantom{-}0.20$
FCIQMC $\phantom{-}0.01$ $\phantom{-}0.13$
best available $\phantom{-^a}0.02^a$ $0.13 \pm 0.013^b$
-------------------------------------- ----------------------- --------------------
: The TME singlet-triplet energy gaps corresponding to the torsional angles of $45^{\circ}$ and $90^{\circ}$ calculated by different methods. $^a$DMC result [@pozun_2013], $^b$photo-electron spectroscopy result [@clifford_1998].[]{data-label="tab_results"}
Conclusions {#section_conclusions}
===========
We have presented the FCIQMC benchmark data for the twisting process of the TME diradical which give an excellent agreement with the previous DMC and available experimental results. Our computations verified that there is a maximum on PES of the ground singlet state ($^1\text{A}$) corresponding to the torsional angle of $45^{\circ}$. At this point, there is also the smallest vertical singlet-triplet energy gap of $0.01$ eV as provided by FCIQMC.
Against the FCIQMC benchmark data, we have critically assessed the accuracy of the MR MkCC and TCC methods. We have found out that the MR MkCCSD method is not able to correctly predict the ordering of both lowest lying spin states and that the MR MkCCSDT, though giving good values for the singlet-triplet energy gap, is, due to the small CAS(2,2) model space, not able to properly describe the shape of the multireference singlet PES. On the other hand, the TCCSD method describes the ground singlet state with an excellent accuracy, but for the correct ordering requires computation of the low-spin component of the triplet state ($^3\text{B}_1$).
Taking into account strengths and weaknesses of the employed MR CC approaches, we propose a combination of both of them, namely a multireference generalization of the tailored CC method. Such an approach, based on the Jeziorski-Monkhorst ansatz (Eq. \[jeziorski-monkhorst\]), i.e. employing different sets of CC amplitudes for each reference determinant, and MR cluster analysis [@paldus-ccanalysis] of the MPS wave function, is currently being developed by some of us and will be a subject of the follow-up paper.
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to thank J. Brabec for helpful discussions. This work has been supported by the Czech Science Foundation (grant no. 16-12052S), Czech Ministry of Education, Youth and Sports (project no. LTAUSA17033), the Hungarian-Czech Joint Research Project MTA/16/05, the Hungarian National Research, Development and Innovation Office (grant no. K120569), and the Hungarian Quantum Technology National Excellence Program (project no. 2017-1.2.1-NKP-2017-00001). FCIQMC computations were carried out on the Salomon supercomputer in Ostrava, we would therefore like to acknowledge the support by the Czech Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center - LM2015070".
[^1]: At the torsional angle matching 45$^{\circ}$, the occupation of both frontier orbitals is approaching one.
[^2]: Huge flexibility of the Budapest QC-DMRG program [@budapest_qcdmrg] allows among others use of unrestricted molecular orbital integrals, as well as more general relativistic ones [@knecht_2014] or those appearing in nuclear structure calculations [@legeza_2015].
[^3]: The MR MkCCSDT energies are in fact lower by approximately 10 mHartree, nevertheless the CC method is generally not variational and the error coming out from this fact is questionable.
[^4]: We use the standard notation of HOMO being the highest occupied molecular orbital and LUMO the lowest unoccupied molecular orbital.
|
---
abstract: 'Starting from realistic nuclear forces, the chiral N$^3$LO and JISP16, we have applied many-body perturbation theory (MBPT) to the structure of closed-shell nuclei, $^4$He and $^{16}$O. The two-body N$^3$LO interaction is softened by a similarity renormalization group transformation while JISP16 is adopted without renormalization. The MBPT calculations are performed within the Hartree-Fock (HF) bases. The angular momentum coupled scheme is used, which can reduce the computational task. Corrections up to the third order in energy and up to the second order in radius are evaluated. Higher-order corrections in the HF basis are small relative to the leading-order perturbative result. Using the anti-symmetrized Goldstone diagram expansions of the wave function, we directly correct the one-body density for the calculation of the radius, rather than calculate corrections to the occupation propabilities of single-particle orbits as found in other treatments. We compare our results with other methods where available and find good agreement. This supports the conclusion that our methods produce reasonably converged results with these interactions. We also compare our results with experimental data.'
author:
- 'B. S. Hu (胡柏山)'
- 'F. R. Xu (许甫荣)'
- 'Z. H. Sun (孙中浩)'
- 'J. P. Vary'
- 'T. Li (李通)'
bibliography:
- 'references.bib'
title: '[*Ab initio*]{} nuclear many-body perturbation calculations in the Hartree-Fock basis'
---
[^1]
\[sec:level1\] Introduction
===========================
A fundamental and challenging problem in nuclear structure theory is the calculation of finite nuclei starting from realistic nucleon-nucleon ($NN$) interactions. The realistic nuclear forces, such as CD-Bonn [@PhysRevC.63.024001], Nijmegen [@PhysRevC.49.2950], Argonne V18 (AV18) [@PhysRevC.51.38], INOY [@PhysRevC.69.054001] and chiral potential [@PhysRevC.68.041001; @Machleidt20111], contain strong short-range correlations which cause convergence problems in the calculations of nuclear structures. To deal with the strong short-range correlations and speed up the convergence, realistic forces are usually processed by certain renormalizations. A traditional approach is the G-matrix renormalization in the Brueckner-Bethe-Goldstone theory [@PhysRev.97.1353; @goldstone1957; @PhysRev.129.225] in which all particle ladder diagrams are summed. Recently, a new class of renormalization methods has been developed, including $V_{\text{low-}k}$ [@PhysRevC.65.051301; @Bogner20031], Similarity Renormalization Group (SRG) [@PhysRevC.75.061001], Okubo-Lee-Suzuki [@okubo01111954; @Suzuki01121980; @Suzuki01071982; @Suzuki01081983; @Suzuki01121982; @Suzuki01121994] and Unitary Correlation Operator Method (UCOM) [@PhysRevC.72.034002; @Roth201050]. The renormalizations soften realistic $NN$ interactions and generate effective Hamiltonians, while all symmetries and observables are preserved in the low-energy domain. The renormalization process also generates effective multi-nucleon interactions (sometimes called “induced” interactions) that are typically dropped for four or more nucleons interacting simultaneously. We will neglect three-nucleon and higher multi-nucleon interactions both “bare” and “induced”. There is another class of “bare” $NN$ forces which are sufficiently soft that they can be used without renormalization, e.g., the JISP interaction which is obtained by the $J$-matrix inverse scattering technique [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733]. These interactions can often be used directly for nuclear structure calculations.
A renormalized $NN$ interaction should retain its description of the experimental phase shifts up to a cutoff. At the same time, the renormalized interaction provides better convergence in nuclear structure calculations without involving parameter refitting or additional parameters. The calculations based on realistic forces are called [*ab initio*]{} methods when they retain predictive power and accurate treatment of the first principles of quantum mechanics. There have been several [*ab initio*]{} many-body methods, such as No-Core Shell Model (NCSM) [@PhysRevC.62.054311; @PhysRevLett.84.5728; @0954-3899-36-8-083101; @Caprio2013179; @Barrett2013131], Green’s Function Monte Carlo (GFMC) [@PhysRevC64014001; @PhysRevC70054325; @PhysRevC76064319; @PhysRevC78065501] and Coupled Cluster (CC) [@PhysRevLett101092502; @PhysRevLett103062503; @PhysRevC82034330]. However, due to the limit of computer capability, the NCSM and GFMC calculations are currently limited to light nuclei (e.g., $\leq ^{16}$O), while the CC calculations are limited to nuclei near double closed shells.
While renormalization methods typically address short-range correlations, the Hartree-Fock (HF) approach is used to treat long-range correlations. However, the conventional HF method that takes only one Slater determinant describes the motion of nucleons in the average field of other nucleons and neglects higher-order correlations. For a phenomenological potential, one can adjust parameters to improve the agreement of the HF results with data. For realistic $NN$ interactions, one needs to go beyond the HF approach to include the intermediate-range correlations which are missing in the lowest order HF approach. The many-body perturbation theory (MBPT) is a powerful tool to include the missing correlations [@bartlett2009; @PhysRevC.68.034320; @PhysRevC.69.034332; @PhysRevC.73.044312]. The perturbation method starts from a solvable mean-field problem and derives a correlated perturbed solution. The most well-known perturbation expansions are the Brillouin-Wigner (BW) [@Brillouin1932; @Wigner1935] and Rayleigh-Schr[ö]{}dinger (RS) [@Rayleigh1894; @Schrodinger1926] methods. MBPT calculations are usually performed with an order-by-order expansion represented in the form of groups of diagrams [@bartlett2009]. The diagrams of MBPT proliferate as one goes to higher orders but some techniques, such as those introduced by Bruekner [@PhysRev.100.36], lead to useful cancellations of entire classes of diagrams. This leads to the linked-diagram theorem which simplifies greatly perturbation calculations up to high orders. Goldstone first proved the theorem valid to all orders in the non-degenerate case [@goldstone1957]. Later, the theorem was extended to the degenerate case [@RevModPhys39771; @Johnson1971172; @Kuo197165; @sandar1969]. The linked-diagram theorem in the degenerate case is often referred to as the folded-diagram method.
Some recent works [@PhysRevC.68.034320; @PhysRevC.69.034332; @PhysRevC.73.044312] show that the MBPT corrections to HF can significantly improve calculations which were based on realistic forces. The authors used different renormalization schemes, $V_{\text{low-}k}$, OLS and UCOM, and obtained the convergence of low-order MBPT calculations [@PhysRevC.68.034320; @PhysRevC.69.034332; @PhysRevC.73.044312]. In the present work, we perform similar MBPT calculations with the SRG-renormalized chiral N$^3$LO potential [@PhysRevC.68.041001; @Machleidt20111] and the “bare” JISP16 interaction [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733]. We also calculate the MBPT corrections to the nuclear radius with the anti-symmetrized Goldstone (ASG) diagrams of the one-body density (up to the second order). We note that, in Ref. [@PhysRevC.68.034320], the same ASG diagrams for the corrections to energy were used for the corrections to the radius. In Refs. [@PhysRevC.69.034332; @PhysRevC.73.044312], corrections to the radius were approximated through corrections to occupation probabilities. In order to reduce computational task, we calculate the diagrams in the angular momentum coupling representation. Our MBPT corrections to energy are up to the third order, while our MBPT corrections to the radius are up to the second order.
\[sec:1\]Theoretical framework
==============================
The effective Hamiltonian
---------------------------
The intrinsic Hamiltonian of the $A$-nucleon system used in this work reads $$\begin{array}{ll}
\text{\^{H}}= \displaystyle\sum_{i<j}^{A} \frac{(\vec{p}_{i}-\vec{p}_{j})^{2}}{2mA} +
\displaystyle\sum_{i<j}^{A} V_{NN,ij},
\end{array}
\label{eq1}$$ where the notation is standard. The first term on the right is the intrinsic kinetic energy, and $V_{NN,ij}$ is the $NN$ interaction including the Coulomb interaction between the protons. We do not include a three-body interacton. In the present work, two different $NN$ interactions have been adopted for comparison. One is the chiral potential N$^3$LO developed by Entem and Machleidt [@PhysRevC.68.041001]. Another one is the “bare” interaction JISP16 [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733].
The N$^3$LO potential is renormalized by using the SRG technique to soften the short-range repulsion and short-range tensor components. The SRG method is based on a continuous unitary transformation that suppresses off-diagonal matrix elements and drives the Hamiltonian towards a band-diagonal form [@PhysRevC.75.061001]. The process leads to high- and low-momentum parts of the Hamiltonian being decoupled. This implies that the renormalized potential becomes softer and more perturbative than the original one. In principle, the SRG method generates three-body, four-body, etc., effective interactions. We neglect these induced terms for the purposes of examining the similarities and differences of results with NN interactions alone. After the renormalization, the Coulomb interaction between protons is added.
The “bare” JISP16 interaction is obtained by the phase-equivalent transformations of the $J$-matrix inverse scattering potential. The parameters are determined by fitting to not only the $NN$ scattering data but also the binding energies and spectra of nuclei with $A\leq16$ [@Shirokov200733]. In the JISP16 potential, the off-shell freedom is exploited to improve the description of light nuclei by phase-equivalent transformations. Polyzou and Glockle [@latePolyzou] have shown that changing the off-shell properties of the two-body potential is equivalent to adding many-body interactions. Therefore, the phase-equivalent transformation can minimize the need of three-body interactions. The “bare” JISP16 interaction has been used extensively and successfully in configuration interaction calculations of light nuclei [@Maris:2013poa; @Shirokov:2014] and in nuclear matter [@Shirokov:2014kqa].
Spherical Hartree-Fock formulation
----------------------------------
With the effective Hamiltonian established, we first perform the HF calculation and then calculate the MBPT corrections to the HF result. For simplicity of computational effort, we limit our investigations here to the spherical, closed-shell, nuclei $^4$He and $^{16}$O. These systems are sufficient to gain initial insights into the convergence rates of the ground-state energy and radius with these realistic interactions.
The spherical symmetry preserves the quantum numbers of the orbital angular momentum ($l$), the total angular momentum ($j$) and its projection ($m_j$) for the HF single-particle states. In the spherical harmonic oscillator (HO) basis $|n l j m_j m_t \rangle$, the HF single-particle state $|\alpha \rangle$ can be written as $$| \alpha \rangle=| \nu l j m_{j} m_{t} \rangle
=\displaystyle\sum_{n} D_{n}^{(\nu ljm_{j}m_{t})} | n l j m_{j} m_{t} \rangle,$$ where the labels are standard with $n$ and $m_t$ for the radial quantum number of the HO basis and isospin projection, respectively. The HF wave function for the $A$-body nucleus is then represented by an anti-symmetrized Slater determinant constructed with the HF single-particle states. By varying the HF energy expectation value (with respect to the coefficients $D_n^{(\nu l j m_j m_t)}$), we obtain the HF single-particle eigen equations, $$\begin{aligned}
\displaystyle\sum_{n_{2}}
h_{n_{1}n_{2}}^{(ljm_{j}m_{t})}D_{n_{2}}^{(\nu ljm_{j}m_{t})}
=\varepsilon_{\nu ljm_{j}m_{t}}D^{(\nu ljm_{j}m_{t})}_{n_{1}},\end{aligned}$$ where $\varepsilon_{\nu l j m_j m_t}$ represents the HF single-particle eigen energies, and $h_{n_{1} n_{2}}^{(l j m_{j} m_{t})}$ designates the matrix elements of the HF single-particle Hamiltonian given by $$\begin{aligned}
h_{n_{1}n_{2}}^{(ljm_{j}m_{t})}=
\displaystyle\sum_{l'j'm'_{j}m'_{t}}
\displaystyle\sum_{n'_{1}n'_{2}}
H_{n_{1}n'_{1}n_{2}n'_{2}}^{(ljm_{j}m_{t};l'j'm'_{j}m'_{t})}
\rho_{n'_{1}n'_{2}}^{(l'j'm'_{j}m'_{t})},
\label{hfsph}\end{aligned}$$ where $H^{(l j m_j m_t, l^\prime j^\prime m_j^\prime m_t^\prime)}_{n_1 n_1^\prime n_2 n_2^\prime}$ and $\rho^{( l^\prime j^\prime m_j^\prime m_t^\prime)}_{ n_1^\prime n_2^\prime}$ are the matrix elements of the two-body effective Hamiltonian $\hat{H}$ and one-body density, respectively. They can be written $$\begin{aligned}
H_{n_{1}n'_{1}n_{2}n'_{2}}^{(ljm_{j}m_{t};l'j'm'_{j}m'_{t})}=
\langle n_{1}ljm_{j}m_{t},n'_{1}l'j'm'_{j}m'_{t}|\hat{H} |
n_{2}ljm_{j}m_{t},n'_{2}l'j'm'_{j}m'_{t}\rangle\end{aligned}$$ and $$\begin{aligned}
\rho_{n'_{1}n'_{2}}^{(l'j'm'_{j}m'_{t})}=
\displaystyle\sum_{u}\mathscr{N}^{(ul'j'm'_{j}m'_{t})}
D_{n'_{1}}^{\ast (ul'j'm'_{j}m'_{t})}D_{n'_{2}}^{(ul'j'm'_{j}m'_{t})},\end{aligned}$$ where $\mathscr{N}^{(\mu l^\prime j^\prime m_j ^\prime m_t^\prime)}$ is the occupation number of the HF single-particle orbit, i.e., $\mathscr{N}^{(\mu l^\prime j^\prime m_j ^\prime m_t^\prime)}
=1$ (occupied) or $0$ (unoccupied).
In practice, we diagonalize the following equation to solve the HF single-particle eigenvalue problem $$\begin{aligned}
\displaystyle\sum_{n_{2}} \left[ \displaystyle\sum_{n'_{1}n'_{2}}
\displaystyle\sum_{l'j'm'_{j}m'_{t}}
H^{ (ljm_{j}m_{t},l'j'm'_{j}m'_{t})}_{n_{1}n'_{1},n_{2}n'_{2}}
\rho^{ ( l'j'm'_{j}m'_{t})}_{n'_{1}n'_{2}} \right]
D^{(\nu ljm_{j}m_{t})}_{n_{2} }
=\varepsilon_{\nu ljm_{j}m_{t}}D^{(\nu ljm_{j}m_{t})}_{n_{1}}.
\label{hfe}\end{aligned}$$ This is a nonlinear equation with respect to variational coefficients $D_{n}^{(\nu l j m_{j} m_{t})}$. In the spherical closed shell, the HF single-particle eigenvalues are independent of the magnetic quantum number $m_j$, which leads to a $2j+1$ degeneracy. In this case, we can rewrite the eigenvalues by omitting $m_{j}$, i.e., $ D_{n}^{(\nu l j m_{t})}=D_n^{(\nu l j m_{j} m_{t})}$ and $\varepsilon_{\nu l j m_{t}}=\varepsilon_{\nu l j m_{j} m_{t}}$. Then we can simplify Eq. (\[hfe\]) in the angular momentum coupled representation as follows[@PhysRevC.73.044312], $$\begin{array}{ll}
\displaystyle\sum_{n_{2}} \Bigg[ \displaystyle\sum_{n'_{1}n'_{2}}
\displaystyle\sum_{l'j'm'_{t}} \displaystyle\sum_{J}
\frac{2J+1}{(2j+1)(2j'+1)}
\sqrt{1+\delta_{k_{1}k'_{1}}}
\sqrt{1+\delta_{k_{2}k'_{2}}}
\\ \times
\left\langle n_{1}ljm_{t}, n'_{1}l'j'm'_{t};J|\text{\^{H}}|n_{2}ljm_{t}, n'_{2}l'j'm'_{t};J\right\rangle
\rho^{ (l'j'm'_{t})}_{n'_{1}n'_{2}} \Bigg]
\\ \times
D^{(\nu ljm_{t})}_{n_{2}}
=\varepsilon_{\nu ljm_{t}}D^{(\nu ljm_{t})}_{n_{1}}
\end{array}$$ with $\delta_{kk'}=\delta_{nn'}\delta_{ll'}\delta_{jj'}\delta_{m_{t}m'_{t}}$ and one-body density matrix $$\begin{array}{ll}
\rho^{ (l'j'm'_{t})}_{n'_{1}n'_{2}}=
\displaystyle\sum_{\mu}O^{(\mu l'j'm'_{t})}
D^{ \text{\textasteriskcentered} (\mu l'j'm'_{t}) }_{n'_{1} } D^{ (\mu l'j'm'_{t})}_{n'_{2} },
\end{array}$$ where $\displaystyle O^{(\mu l'j'm'_{t})}$ is the number of the occupied magnetic subshell, i.e., $O^{(\mu l'j'm'_{t})}=2j'+1$ (occupied) or 0 (unoccupied).
Rayleigh-Schrödinger perturbation theory
-----------------------------------------
We can separate the $A$-nucleon Hamiltonian Eq. (\[eq1\]) into a zero-order part $\hat{H_{0}}$ and a perturbation $\hat{V}$, $$\text{\^{H}}=\text{\^{H}}_{0}+(\text{\^{H}}-\text{\^{H}}_{0})=\text{\^{H}}_{0}+\text{\^{V}}.$$ The exact solutions of the $A$-nucleon system are $$\text{\^{H}}\Psi_{n}=E_{n} \Psi_{n}, \qquad n=0,1,2,...$$ For the zero-order part, we write $$\text{\^{H}}_{0} \Phi_{n}=E^{(0)}_{n} \Phi_{n}, \qquad n=0,1,2,...$$
If we choose the HF single-particle Hamiltonian Eq. (\[hfsph\]) as $H_{0}$, the zero-order energy $E_{0}^{(0)}$ is simply the summation of the single-particle energies up to the Fermi level. In the present work, we only investigate the ground states of closed-shell nuclei. For simplicity, we denote the ground-state energy $E_{0}$ and wave function $\Psi_{0}$ by $E$ and $\Psi$, respectively, omitting the subscript. For the ground state $(n=0)$, we formulate the Rayleigh-Schr[ö]{}dinger perturbation theory (RSPT), as follows, $$\chi= \Psi - \Phi_{0},$$ $$\Delta E= E- E^{(0)},
\label{r0}$$ $$\Psi= \displaystyle\sum_{m=0}^{\infty} \big{[} \text{\^{R}}_{0}(E^{(0)}) (\text{\^{V}}-\Delta E)\big{]}^{m} \Phi_{0},
\label{r1}$$ $$\Delta E= \displaystyle\sum_{m=0}^{\infty} \langle \Phi_{0} |\text{\^{V}} \big{[}\text{\^{R}}_{0}(E^{(0)})
(\text{\^{V}}-\Delta E)\big{]}^{m} |\Phi_{0}\rangle,
\label{r2}$$ where $\text{\^{R}}_{0}=\displaystyle\sum_{i \neq 0} \frac{|\Phi_{i}\rangle\langle\Phi_{i}|}{E_{0}^{(0)}-E_{i}^{(0)} }$ is called the resolvent of $\text{\^{H}}_{0}$. Here we use intermediate normalization $$\begin{array}{ll}
\langle\Phi_{n}|\Phi_{n}\rangle=1, \qquad \langle\chi_{n}|\Phi_{n}\rangle=0,
\\
\langle\Psi_{n}|\Phi_{n}\rangle=1, \qquad
\langle\Psi_{n}|\Psi_{n}\rangle=1+\langle\chi_{n}|\chi_{n}\rangle.
\end{array}
\label{norm}$$ Arranging the above expressions according to the perturbation orders of $\text{\^{V}}$, we have $$E=E^{(0)}+E^{(1)}+E^{(2)}+E^{(3)}+\ldots$$ The first-, second-, third-order corrections are $$E^{(1)}=\langle \Phi_{0} |\text{\^{V}} |\Phi_{0}\rangle,$$ $$E^{(2)}=\langle \Phi_{0} |\text{\^{V}} \text{\^{R}}_{0} \text{\^{V}} |\Phi_{0}\rangle,$$ $$E^{(3)}=\langle \Phi_{0} |\text{\^{V}} \text{\^{R}}_{0}
(\text{\^{V}}-\langle \Phi_{0} |\text{\^{V}} |\Phi_{0}\rangle) \text{\^{R}}_{0} \text{\^{V}} |\Phi_{0}\rangle.$$ Similarly, the wave function can be written in the perturbation scheme $$\Psi=\Phi_{0}+\Psi^{(1)}+\Psi^{(2)}+\ldots$$ with $$\Psi^{(1)}=\text{\^{R}}_{0} \text{\^{V}} |\Phi_{0}\rangle$$ and $$\Psi^{(2)}=\text{\^{R}}_{0} (\text{\^{V}}-E^{(1)})\text{\^{R}}_{0} \text{\^{V}} |\Phi_{0}\rangle$$ for the first- and second-order corrections to the wave function, respectively. We can use the diagrammatic approach to describe various terms in RSPT. The ASG diagrams are the most commonly-used method of the diagrammatic representation.
Diagrammatic expansion for Rayleigh-Schrödinger perturbation theory in the Hartree-Fock basis
----------------------------------------------------------------------------------------------
If we choose the HF Hamiltonian as an auxiliary zero-order one-body Hamiltonian $\text{\^{H}}_{0}$, many of the ASG diagrams are cancelled [@bartlett2009]. Only a small number of low-order ASG diagrams for RSPT remain. In this subsection, we give the remaining AGS diagrams for the energy and wave function written in the standard perturbation theory [@kuo1990]. We consider corrections up to third order for the energy and second order for the wave function. To evaluate other observables that can be expressed by one-body operators, we calculate the corrections up to second order for the one-body density. It has been shown that the corrections up to third order for the energy in the HF basis give well-converged results for soft interactions [@Tichai:2016joa]. Spherical HF (SHF) produces degenerate single-particle states, so we can evaluate the vacuum-to-vacuum linked diagrams in angular momentum coupled representation [@Kuo1981237] which is computationally efficient.
Fig. \[fig:1\] displays the ASG diagrams corresponding to the first-, second- and third-order corrections to the energy in RSPT. The vertices, i.e., the dashed lines, represent $\hat H$ in Eq. (\[eq1\]). The diagrams (a) and (b) are for $E^{(1)}$ and $E^{(2)}$, respectively, while the diagrams (c), (d) and (e) sum up for $E^{(3)}$. The zero-order energy $E^{(0)}$ is the simple summation of the HF single-particle energies up to the Fermi level, i.e., $E^{(0)}=\displaystyle\sum_{i=1}^{A} \varepsilon_{i}$, where $\varepsilon_{i}$ represents the HF single-particle energy. The summation of the $E^{(0)}$ and $E^{(1)}$ gives the HF energy, i.e., $E_{\text{HF}}=E^{(0)}+E^{(1)}=\dfrac{1}{2}\displaystyle\sum_{i=1}^{A} \varepsilon_{i}$, since the initial Hamiltonian is entirely expressed in relative coordinates [@PhysRevC.37.1240; @PhysRevC.69.034332].
### Corrections to the one-body density
MBPT corrections to the wave function bring configuration mixing. The convergence can be discussed in order-by-order perturbation calculations. Any observable that is expressed by one-body operators can be calculated by using the One-Body Density Matrix (OBDM). By definition, the local one-body density operator in an $A$-body Hilbert space is written as [@PhysRevC.86.034325] $$\hat{\rho} (\vec{r})=\displaystyle\sum_{k=1}^{A}\delta^{3} \left(\vec{r}- \vec{r}_{k} \right)
=\displaystyle\sum_{k=1}^{A} \frac{\delta \left(r- r_{k} \right) }{r^{2}}
\displaystyle\sum_{lm} Y_{lm}^{\ast}(\text{\^{r}}_{k}) Y_{lm}(\text{\^{r}}),$$ where $\text{\^{r}}$ is the unit vector in the direction $\vec{r}$, and $Y_{lm}(\text{\^{r}})$ is the spherical harmonic function.
We can write the density operator in the second quantization representation in the HO basis as $$\begin{aligned}
\hat{\rho}(\vec{r})
=&&\displaystyle\sum_{K}
\displaystyle\sum_{n_{1}l_{1}j_{1}}
\displaystyle\sum_{n_{2}l_{2}j_{2}}\displaystyle\sum_{m_{j}}
R_{n_{1}l_{1}}(r)R_{n_{2}l_{2}}(r)
\frac{-Y_{K0}^{\ast}(\text{\^{r}})}{\sqrt{2K+1}}
\nonumber \\ &&\times
\left\langle l_{1} \frac{1}{2} j_{1} \left|| Y_{K} | \right| l_{2} \frac{1}{2} j_{2} \right\rangle
\left\langle j_{1} m_{j} j_{2} -m_{j} | K 0 \right\rangle
\nonumber \\ &&\times
(-1)^{j_{2}+m_{j}}
a_{n_{1}l_{1}j_{1}m_{j}}^{\dag}a_{n_{2}l_{2}j_{2}m_{j}}\end{aligned}$$ with $$\begin{aligned}
\left\langle l_{1} \frac{1}{2} j_{1} \left|| Y_{K} | \right| l_{2} \frac{1}{2} j_{2} \right\rangle
=&& \frac{1}{\sqrt{4 \pi}} \text{\^{j}}_{1}\text{\^{j}}_{2}\text{\^{l}}_{1}\text{\^{l}}_{2}(-1)^{j_{1}+\frac{1}{2}}
\left\langle l_{1} 0 l_{2} 0 | K 0 \right\rangle
\nonumber \\ &&\times
\left\{
\begin{array}{ccc}
j_{1} & j_{2} & K\\
l_{2} & l_{1} & \frac{1}{2}\\
\end{array}
\right\}.\end{aligned}$$ The $R_{nl}$’s are the radial components of the HO wave function. We use the Condon-Shortley convention for the Clebsch-Gordan coefficients. Since we are dealing with a spherically symmetric system (K=0), we can obtain a simple form, $$\hat{\rho} (\vec{r})=\displaystyle\sum_{n_{1}n_{2}}\displaystyle\sum_{ljm_{j}}
\left[ \frac{R_{n_{1}l}(r)R_{n_{2}l}(r)}{4 \pi} \right]
a_{n_{1}ljm_{j}}^{\dag}a_{n_{2}ljm_{j}}.$$
By introducing the normally-ordered product relative to the SHF ground state $|\Phi_{0} \rangle$, the local one-body density operator can be written as $$\hat{\rho} (\vec{r})= \rho_{0}(\vec{r}) +\hat{\rho}_{N}
=\rho_{0}(\vec{r}) +
\displaystyle\sum_{i,j} \rho_{ij}:c_{i}^{\dag}c_{j}:,
\label{normal-order}$$ where $\rho_{0}(\vec{r})=\langle \Phi_{0}| \hat{\rho} (\vec{r}) | \Phi_{0}\rangle$ gives the HF density, while $\hat{\rho}_{N}=\displaystyle\sum_{i,j} \rho_{ij}:c_{i}^{\dag}c_{j}:$ brings corrections to the density. $\rho_{ij}$ is the density matrix elements $\langle i|\rho (\vec{r})|j \rangle$, and $:c_{i}^{\dag}c_{j}:$ indicates the normally-ordered product of the creation and annihilation operators. It is required that all annihilation and creation operators which take $|\Phi_{0}\rangle$ to zero when acting on it are to the right of all other operators which do not take $|\Phi_{0}\rangle$ to zero. The expectation value of the density is obtained with the corrected wave function through Eq. (\[normal-order\]). In the present work, we consider the first- and second-order wave function corrections.
The ASG diagrams for the first- and second-order corrections to the wave function [@bartlett2009] are displayed in Fig. \[fig:wave\]. The first-order wave function diagram, i.e., panel (a) in Fig. \[fig:wave\], produces the second-order correction to the density. While diagrams (b) and (c) of the second-order wave function correction produce second-order corrections to the density, other diagrams of the second-order wave function correction contribute to higher-order corrections to the density. The first- and second-order wave function corrections which correct the density up to the second order can be written as $$\begin{aligned}
\Psi^{(1)}= &&-\displaystyle\frac{1}{4}
\displaystyle\sum_{h_{1}h_{2}} \displaystyle\sum_{p_{1}p_{2}}
\frac{\langle p_{1} p_{2} |\text{\^{H}}| h_{1} h_{2}\rangle}
{(\varepsilon_{h_{1}}+\varepsilon_{h_{2}}-\varepsilon_{p_{1}}-\varepsilon_{p_{2}})}
\nonumber \\ &&\times
\left( c_{p_{1}}^{\dagger} c_{p_{2}}^{\dagger} c_{h_{2}} c_{h_{1}}| \Phi_{0}\rangle \right),\end{aligned}$$ $$\begin{aligned}
\Psi^{(2)}_{b}= && \displaystyle\frac{1}{2}
\displaystyle\sum_{h_{1}h_{2}} \displaystyle\sum_{p_{1}p_{2}p_{3}}
\frac{\langle p_{1} h_{2} |\text{\^{H}}| p_{2} p_{3}\rangle
\langle p_{2} p_{3} |\text{\^{H}}| h_{1} h_{2}\rangle}
{(\varepsilon_{h_{1}}-\varepsilon_{p_{1}})
(\varepsilon_{h_{1}}+\varepsilon_{h_{2}}-\varepsilon_{p_{2}}-\varepsilon_{p_{3}})}
\nonumber \\ &&\times
\left(c_{p_{1}}^{\dagger} c_{h_{1}}| \Phi_{0}\rangle \right),\end{aligned}$$ $$\begin{aligned}
\Psi^{(2)}_{c}= &&-\displaystyle\frac{1}{2}
\displaystyle\sum_{h_{1}h_{2}h_{3}} \displaystyle\sum_{p_{1}p_{2}}
\frac{\langle h_{2} h_{3} |\text{\^{H}}| h_{1} p_{2}\rangle
\langle p_{1} p_{2} |\text{\^{H}}| h_{2} h_{3}\rangle}
{(\varepsilon_{h_{1}}-\varepsilon_{p_{1}})
(\varepsilon_{h_{2}}+\varepsilon_{h_{3}}-\varepsilon_{p_{1}}-\varepsilon_{p_{2}})}
\nonumber \\ &&\times
\left(c_{p_{1}}^{\dagger} c_{h_{1}}| \Phi_{0}\rangle \right).\end{aligned}$$ The total wave function that corrects the density up to the second order is $$\begin{aligned}
\begin{array}{ll}
\Psi=\Phi_{0}+\Psi^{(1)}+\Psi^{(2)}_{b}+\Psi^{(2)}_{c}.
\end{array}\end{aligned}$$ Then, the corrected density is written as $$\begin{aligned}
\rho (\vec{r})&&=\langle \Psi| \hat{\rho} (\vec{r}) | \Psi \rangle
\nonumber \\ &&
=\langle \Phi_{0}| \hat{\rho} (\vec{r}) | \Phi_{0}\rangle+
\langle \Phi_{0}| \hat{\rho} (\vec{r}) | \Phi_{0}\rangle
\langle \Psi^{(1)}| \Psi^{(1)}\rangle
\nonumber \\ &&
\quad +2 \langle \Phi_{0}| \hat{\rho}_{N}| \Psi_{b}^{(2)}\rangle+
2 \langle \Phi_{0}|\hat{\rho}_{N}| \Psi_{c}^{(2)}\rangle+
\langle \Psi^{(1)}|\hat{\rho}_{N}| \Psi^{(1)}\rangle
\nonumber \\ &&
=\langle \Phi_{0}| \hat{\rho} (\vec{r}) | \Phi_{0}\rangle+
\langle \Phi_{0}| \hat{\rho} (\vec{r}) | \Phi_{0}\rangle
\langle \Psi^{(1)}| \Psi^{(1)}\rangle
\nonumber \\ &&
\quad +2 \rho_{a}+2 \rho_{b}+ \rho_{c_{1}}+ \rho_{c_{2}},\end{aligned}$$ where $\rho_{a}=\langle \Phi_{0}| \hat{\rho}_{N}| \Psi_{b}^{(2)}\rangle$, $\rho_{b}=\langle \Phi_{0}|\hat{\rho}_{N}| \Psi_{c}^{(2)}\rangle$ and $\rho_{c_{1}}+\rho_{c_{2}}=\langle \Psi^{(1)}|\hat{\rho}_{N}| \Psi^{(1)}\rangle$. They are displayed using the language of the diagram in Fig. \[fig:density\]. Dashed lines with cross contribute to the reduced matrix elements $\langle \nu_{1} l j\| \rho \| \nu_{2} l j\rangle=
\sqrt{2j+1}\langle \nu_{1} l j m_{j} | \rho | \nu_{2} l j m_{j}\rangle$.
The detailed formulae of the density correction terms in the angular momentum coupled scheme are written as $$\begin{aligned}
\rho_{a}=&& \frac{1}{2}\displaystyle\sum_{h_{1},h_{2}}
\displaystyle\sum_{p_{1},p_{2},p_{3}}
\frac{(-1)^{j_{h_{1}}+j_{h_{2}}}\sqrt{2j_{h_{2}}+1}}
{(\varepsilon_{h_{1}}-\varepsilon_{p_{1}})
(\varepsilon_{h_{1}}+\varepsilon_{h_{2}}-\varepsilon_{p_{2}}-\varepsilon_{p_{3}})}
\nonumber \\ \times &&
\displaystyle\sum_{J}(-1)^{J}(2J+1)
\left\{
\begin{array}{ccc}
j_{h_{1}} & j_{p_{1}} & 0\\
j_{h_{2}} & j_{h_{2}} & J
\end{array}
\right\}
\langle (h_{1} h_{2})J |\text{\^{H}}|(p_{2} p_{3})J\rangle
\nonumber \\ \times &&
\langle (p_{2} p_{3})J |\text{\^{H}}|(p_{1} h_{2})J\rangle
\langle h_{1} \| \rho \| p_{1}\rangle,\end{aligned}$$ $$\begin{aligned}
\rho_{b}=&& -\frac{1}{2}\displaystyle\sum_{h_{1},h_{2},h_{3}}
\displaystyle\sum_{p_{1},p_{2}}
\frac{(-1)^{j_{h_{1}}+j_{p_{2}}}\sqrt{2j_{p_{2}}+1}}
{(\varepsilon_{h_{1}}-\varepsilon_{p_{1}})
(\varepsilon_{h_{2}}+\varepsilon_{h_{3}}-\varepsilon_{p_{1}}-\varepsilon_{p_{2}})}
\nonumber \\ \times &&
\displaystyle\sum_{J}(-1)^{J}(2J+1)
\left\{
\begin{array}{ccc}
j_{h_{1}} & j_{p_{1}} & 0\\
j_{p_{2}} & j_{p_{2}} & J
\end{array}
\right\}
\langle (p_{1} p_{2})J |\text{\^{H}}|(h_{2} h_{3})J\rangle
\nonumber \\ \times &&
\langle (h_{2} h_{3})J |\text{\^{H}}|(h_{1} p_{2})J\rangle
\langle h_{1} \| \rho \| p_{1}\rangle,\end{aligned}$$ $$\begin{aligned}
\rho_{c_{1}}=&& -\frac{1}{2}\displaystyle\sum_{h_{1},h_{2},h_{3}}
\displaystyle\sum_{p_{1},p_{2}}
\frac{(-1)^{j_{h_{1}}+j_{h_{2}}}\sqrt{2j_{h_{1}}+1}}
{(\varepsilon_{h_{1}}+\varepsilon_{h_{2}}-\varepsilon_{p_{1}}-\varepsilon_{p_{2}})
(\varepsilon_{h_{1}}+\varepsilon_{h_{3}}-\varepsilon_{p_{1}}-\varepsilon_{p_{2}})}
\nonumber \\ \times &&
\displaystyle\sum_{J}(-1)^{J}(2J+1)
\left\{
\begin{array}{ccc}
j_{h_{1}} & j_{h_{1}} & 0\\
j_{h_{2}} & j_{h_{3}} & J
\end{array}
\right\}
\langle (h_{1} h_{2})J |\text{\^{H}}|(p_{1} p_{2})J\rangle
\nonumber \\ \times &&
\langle (p_{1} p_{2})J |\text{\^{H}}|(h_{1} h_{3})J\rangle
\langle h_{3} \| \rho \| h_{2}\rangle,\end{aligned}$$ $$\begin{aligned}
\rho_{c_{2}}=&& \frac{1}{2}\displaystyle\sum_{h_{1},h_{2}}
\displaystyle\sum_{p_{1},p_{2},p_{3} }
\frac{(-1)^{j_{p_{1}}+j_{p_{3}}}\sqrt{2j_{p_{1}}+1}}
{(\varepsilon_{h_{1}}+\varepsilon_{h_{2}}-\varepsilon_{p_{1}}-\varepsilon_{p_{3}})
(\varepsilon_{h_{1}}+\varepsilon_{h_{2}}-\varepsilon_{p_{1}}-\varepsilon_{p_{2}})}
\nonumber \\ \times &&
\displaystyle\sum_{J}(-1)^{J}(2J+1)
\left\{
\begin{array}{ccc}
j_{p_{1}} & j_{p_{1}} & 0\\
j_{p_{3}} & j_{p_{2}} & J
\end{array}
\right\}
\langle (p_{1} p_{3})J |\text{\^{H}}|(h_{1} h_{2})J \rangle
\nonumber \\ \times &&
\langle (h_{1} h_{2})J |\text{\^{H}}|(p_{1} p_{2})J\rangle
\langle p_{2} \| \rho \| p_{3}\rangle,\end{aligned}$$ where $\left\{
\begin{array}{ccc}
j_{1} & j_{2} & j_{3}\\
j_{4} & j_{5} & j_{6} \\
\end{array}
\right\}$ is Wigner 6-j symbol. The letters $h_{1},h_{2},...$ indicate occupied single-particle levels in $|\text{HF}\rangle$ (i.e., hole states), the letters $p_{1},p_{2},...$ for unoccupied levels (i.e., particle states). $\varepsilon_{h}$ or $\varepsilon_{p}$ is the energy of particle or hole state, respectively. States $h$ or $p$ includes the quantum numbers of the orbital angular momentum $l$, total angular momentum $j$, isospin projection quantum number $m_{t}$, and additional quantum number $\nu$, i.e., $|h\rangle$ or $|p\rangle=|\nu ljt_{z}\rangle$. We define an anti-symmetrized two-particle state (unnormalized) coupled to a good angular momentum $J$ with a projection $M$, $$\begin{aligned}
| (j_{1} j_{2})J M\rangle=\displaystyle\sum_{m_{1},m_{2}}
\langle j_{1} m_{1}j_{2} m_{2}| J M \rangle
| (j_{1} m_{1}) (j_{2} m_{2})\rangle.\end{aligned}$$
### Root-mean-square radii
The root-mean-square (rms) radius is an important global indicator for the change of the density distribution arising from correlations beyond HF. The squares of the rms radii for point-like proton, neutron and nucleon (matter) distributions are the averaged values of the operators [@PhysRevC.56.191], respectively, $$\begin{aligned}
\begin{array}{ll}
{\hat{r}_{\text{pp}}}^{2}=\displaystyle\frac{1}{Z} \displaystyle\sum_{i=1}^{Z} (\vec{r_{i}}-\vec{r_{0}})^{2},
\end{array}
\label{r2-1}\end{aligned}$$ $$\begin{aligned}
\begin{array}{ll}
{\hat{r}_{\text{nn}}}^{2}=\displaystyle\frac{1}{N} \displaystyle\sum_{i=1}^{N} (\vec{r_{i}}-\vec{r_{0}})^{2},
\end{array}
\label{r2-2}\end{aligned}$$ $$\begin{aligned}
\begin{array}{ll}
{\hat{r}_{\text{m}}}^{2}=\displaystyle\frac{1}{A} \displaystyle\sum_{i=1}^{A} (\vec{r_{i}}-\vec{r_{0}})^{2}
= \frac{1}{A^{2}} \displaystyle\sum_{i<j}^{A} (\vec{r_{i}}-\vec{r_{j}})^{2},
\end{array}
\label{r2-3}\end{aligned}$$ with the c.m. position $\vec{r_{0}}=\frac{1}{A} \displaystyle\sum_{i=1}^{A}\vec{r_{i}}$. The charge radius $r_{\text{ch}}$ obtained from the point-proton radius $r_{\text{pp}}$ using the standard expression [@PhysRevC.91.051301] $$\begin{aligned}
\label{charge}
\langle r_{\text{ch}}^{2}\rangle =\langle r_{\text{pp}}^{2}\rangle +
R_{\text{p}}^{2} + \dfrac{N}{Z} R_{\text{n}}^{2}
+\dfrac{3 \hbar^{2}}{4m_{\text{p}}^{2}c^{2}},\end{aligned}$$ where $\dfrac{3 \hbar^{2}}{4m_{\text{p}}^{2}c^{2}} \approx 0.033 \; \text{fm}^{2}$, $R_{\text{n}}^{2}=-0.1149(27)\; \text{fm}^{2}$, $R_{\text{p}}=0.8775(51)$ fm. The point-proton or point-neutron rms radius operator is a two-body operator. The squares of the rms radii can be calculated either from the translational invariant local density or directly using the two-body operators \[ i.e., Eqs. (\[r2-1\]), (\[r2-2\]) and (\[r2-3\]) \]. Since we adopt MBPT with intermediate normalization \[ i.e., Eqs. (\[norm\]) \], the perturbed wave function is unnormalized. In the present work, we use the one-body local density to calculate the radius, as $$\begin{aligned}
\label{Rpp}
\langle{R^{2}_{\text{pp}}}\rangle=\displaystyle\frac{\int r^{2} \rho_{\text{p}}(\vec{r}) d^{3}r }{\int \rho_{\text{p}}(\vec{r}) d^{3}r}.\end{aligned}$$
The wave function is written in the laboratory HO coordinate, starting from an anti-symmetrized Slater determinant which contains the component of the center-of-mass (c.m.) motion. Consequently, the local one-body density calculated with the wave function includes contribution from the c.m. motion. The c.m. correction to the radius can be approximated as follows. Eq. (\[r2-3\]) gives $$\begin{aligned}
\begin{array}{ll}
{\hat{r}_{\text{m}}}^{2}= \displaystyle\frac{1}{A^{2}} \displaystyle\sum_{i<j}^{A} (\vec{r_{i}}-\vec{r_{j}})^{2}
=\left( 1-\frac{1}{A} \right)\cdot \left({\displaystyle\sum_{i=1}^{A}\vec{r_{i}}^{2}}/{A} \right)
-\frac{2}{A^{2}}\cdot \left( \displaystyle\sum_{i<j}^{A} \vec{r_{i}} \cdot \vec{r_{j}} \right).
\end{array}\end{aligned}$$ If the cross term $ \displaystyle\sum_{i<j}^{A} \vec{r_{i}} \cdot \vec{r_{j}} $ is neglected, we have $$\begin{aligned}
\begin{array}{ll}
{\hat{r}_{\text{m}}}^{2}\approx \left( 1-\displaystyle\frac{1}{A} \right)\cdot \left({\displaystyle\sum_{i=1}^{A}\vec{r_{i}}^{2}}/{A} \right)
\end{array}.\end{aligned}$$ Similarly for the proton radius, $$\begin{aligned}
\begin{array}{ll}
{\hat{r}_{\text{pp}}}^{2}\approx \left( 1-\displaystyle\frac{1}{A} \right)\cdot \left({\displaystyle\sum_{i=1}^{Z}\vec{r_{i}}^{2}}/{Z} \right)
\end{array}.\end{aligned}$$ This gives an approximate c.m. correction to the point-proton rms radius, $$\begin{aligned}
\label{r-com}
\begin{array}{ll}
\Delta r_{\text{c.m.}}=\left[\left(1- \displaystyle\frac{1}{A}\right)\cdot \langle R_{\text{pp}}^{2} \rangle \right]^{1/2}
- \langle R_{\text{pp}}^{2} \rangle ^{1/2},
\end{array}\end{aligned}$$ where $\langle R_{\text{pp}}^{2} \rangle ^{1/2}$ is the point-proton rms radius calculated by Eq (\[Rpp\]). Then the rms radius of the point-proton distribution is obtained by $$\begin{aligned}
r_{\text{pp}}=\langle R_{\text{pp}}^{2} \rangle ^{1/2}+\Delta r_{\text{c.m.}}.\end{aligned}$$
Calculations and discussions
============================
In this section, we apply the method outlined in Section \[sec:1\] to two light closed-shell nuclei, $^{4}$He and $^{16}$O. The SRG-softened chiral N$^{3}$LO and the “bare” JISP16 interactions are adopted for the effective Hamiltonians.
Calculations with chiral N$^3$LO interaction
--------------------------------------------
The SHF is carried out within the HO basis. The HO basis is truncated by a cutoff according to the number $N_{\rm shell} = \text{max}(2n+l +1)$, where $N_{\rm shell}$ indicates how many major HO shells are included in the truncation. After the SHF calculation, the MBPT corrections are calculated in the SHF basis. In the present calculations, the basis spaces employed take $N_{\rm shell}$=7, 9, 11 and 13. We verify that such a truncation is sufficient for the converged calculations of the ground state energies for these magic nuclei $^4$He and $^{16}$O.
![\[fig:he4\] HF-MBPT calculations of $^{4}\text{He}$ ground-state energy through third order as a function of oscillator parameter $\hbar \Omega$ with the chiral N$^{3}$LO potential [@PhysRevC.68.041001; @Machleidt20111] renormalized by SRG at different softening parameters $\lambda=1.5, 2.0, 2.5, 3.0$ $\text{fm}^{-1}$. The dashed line represents the experimental ground-state energy.](he4_mbpt_srg.eps)
![\[fig:he4\_n3lo\_rms\] Point-proton rms radius of $^{4}$He as a function of oscillator parameter $\hbar \Omega$ with different $N_{\text{shell}}$. The chiral N$^{3}$LO potential [@PhysRevC.68.041001; @Machleidt20111] is softened by the SRG method.](he4_mbpt_n3lo_srg_20_r_c.eps)
------------------------------ --------- --------- --------- --------- --
Expt. [@1674-1137-36-12-001] -28.296 -28.296 -28.296 -28.296
NCSM [@PhysRevC.87.054312] -28.20 -28.41 -27.43 -26.80
SHF -25.754 -21.864 -15.854 -10.278
PT2 -1.788 -5.088 -9.652 -13.783
PT3 -0.391 -0.899 -1.523 -1.953
SHF+PT2+PT3 -27.933 -27.850 -27.029 -26.013
------------------------------ --------- --------- --------- --------- --
: \[tab:n3lo-he4-e\] Ground-state energy (in MeV) of $^4$He, analyzed in order-by-order HF-MBPT calculations with N$^3$LO softened at different SRG-softening parameter values ($\lambda$). PT2 and PT3 represent the second- and third-order corrections to energy, respectively. We take $N_{\text{shell}}=13$ and $\hbar\Omega=35$ MeV.
---------------------------------- -------- -------- -------- -------- --
Expt. 1.477 1.477 1.477 1.477
SHF 1.677 1.652 1.714 1.816
PT2 0.007 0.001 -0.021 -0.065
$\Delta r_{\text{c.m.}}$ -0.226 -0.222 -0.227 -0.235
SHF+PT2+$\Delta r_{\text{c.m.}}$ 1.458 1.431 1.466 1.516
---------------------------------- -------- -------- -------- -------- --
: \[tab:n3lo-he4-r\] Point-proton rms radius (in fm) of $^4$He in the HF-MBPT calculations with N$^3$LO softened at different SRG-softening parameter values. PT2 designates the second-order correction to the radius. $N_{\text{shell}}=13$ and $\hbar\Omega=35$ MeV are taken. The experimental point-proton rms radius is obtained using Eq. (\[charge\]) with the experimental charge radius taken from [@Angeli201369].
Fig. \[fig:he4\] shows the MBPT calculated ground-state energy of $^4$He. The calculations were done with the chiral N$^3$LO interaction which was renormalized by SRG. We see that good convergence of the calculated energy by virtue of independence from the oscillator parameter $\hbar\Omega$ and $N_{\text{shell}}$ is obtained at least for the truncations $N_{\text{shell}}=11$ and 13. We note that the dependence on the parameter $\hbar\Omega$ displays behavior similar to NCSM calculations [@Bogner200821; @PhysRevC.87.054312]. The softening parameter $\lambda=3.0$ fm$^{-1}$ seems to be insufficient to produce an interaction soft enough for good convergence in MBPT. Jurgenson [*et al.*]{}, have investigated the SRG evolution with the softening parameter $\lambda$ in $^4$He at $\hbar\Omega=36$ MeV [@PhysRevLett.103.082501; @PhysRevC.87.054312]. They found that $\lambda \approx 2.0$ fm$^{-1}$ can reasonably reproduce the experimental $^4$He ground-state energy with the $NN$-only interaction (without requiring a three-body force).
Fig. \[fig:he4\_n3lo\_rms\] shows the radius calculations at different $\hbar\Omega$ with $\lambda=2.0$ fm$^{-1}$. Tables \[tab:n3lo-he4-e\] and \[tab:n3lo-he4-r\] give the details of the HF-MBPT calculations with different $\lambda$ values. We see that both second- and third-order corrections to energy decrease with decreasing $\lambda$. This is easily understood because MBPT mainly treats intermediate-range correlations and these correlations are weakened with decreasing $\lambda$. With sufficiently small $\lambda$, higher-order corrections to the energy can be neglected. The second-order correction to the radius is already small, which decreases with decreasing $\lambda$ in $^4$He. The c.m. correction to the radius is larger than the MBPT correction. It may be concluded that, at least for $^4$He, MBPT corrections up to third order in energy and up to second order in radius within the HF basis should give converged results for $\lambda$ below about 3.0 fm$^{-1}$. It has been pointed out that the MBPT calculation within the HO basis could be divergent even for softened interactions [@Tichai:2016joa]. The Hamiltonian (1) is written already in the relative coordinate, and SHF can preserve the translational invariance for the ground state energy [@epja14413] so that no c.m. correction is needed for the ground state energy.
![\[fig:o16\] HF-MBPT calculations of $^{16}\text{O}$ as a function of oscillator parameter $\hbar \Omega$ with the chiral N$^{3}$LO potential [@PhysRevC.68.041001; @Machleidt20111] renormalized by SRG at different softening parameters $\lambda=1.5, 2.0, 2.5, 3.0$ $\text{fm}^{-1}$.](o16_mbpt_srg.eps)
![\[fig:o16\_n3lo\_rms\] Point-proton rms radius of $^{16}$O as a function of oscillator parameter $\hbar \Omega$ with different $N_{\text{shell}}$. The chiral N$^{3}$LO potential [@PhysRevC.68.041001; @Machleidt20111] is softened by the SRG method.](o16_mbpt_n3lo_srg_25_r_c.eps)
------------------------------ ---------- ---------- ---------- ---------- --
Expt. [@1674-1137-36-12-001] -127.619 -127.619 -127.619 -127.619
SHF -169.968 -133.169 -85.173 -44.102
PT2 -10.132 -29.497 -59.617 -88.326
PT3 -0.794 -1.931 -4.630 -7.339
SHF+PT2+PT3 -180.893 -164.597 -149.419 -139.767
------------------------------ ---------- ---------- ---------- ---------- --
: \[tab:n3lo-o16-e\] Ground-state energy (in MeV) of $^{16}$O, analyzed in order-by-order HF-MBPT calculations with N$^3$LO softened at different SRG-softening parameter values ($\lambda$). We take $N_{\text{shell}}=13$ and $\hbar\Omega=35$ MeV.
---------------------------------- -------- -------- -------- -------- --
Expt. 2.581 2.581 2.581 2.581
SHF 2.098 2.096 2.201 2.345
PT2 0.011 0.011 -0.006 -0.042
$\Delta r_{\text{c.m.}}$ -0.067 -0.067 -0.070 -0.073
SHF+PT2+$\Delta r_{\text{c.m.}}$ 2.042 2.040 2.125 2.230
---------------------------------- -------- -------- -------- -------- --
: \[tab:n3lo-o16-r\] Point-proton rms radius (in fm) of $^{16}$O in the HF-MBPT calculations with N$^3$LO softened at different SRG-softening parameter values. $N_{\text{shell}}=13$ and $\hbar\Omega=35$ MeV are taken. The experimental point-proton rms radius is obtained using Eq. (\[charge\]) with the experimental charge radius taken from [@Angeli201369].
Fig. \[fig:o16\] shows the energy calculations for $^{16}$O. The convergence behavior is similar to that in $^4$He. The $N_{\rm shell}=11$ and 13 calculations appear nearly convergent. However, calculations with small $\lambda$ values (e.g., $\leq 2.0$ fm$^{-1}$) give over-binding, compared with data. This phenomenon should be more obvious for heavier nuclei. The main reason is that the three-body and higher-order forces are omitted in these calculations. The emergence of induced three-body forces and beyond is related to the SRG softening parameter $\lambda$. A larger $\lambda$ value evolves a harder effective $NN$ potential. In large $\lambda$ cases (e.g., $\lambda>3.0$ fm$^{-1}$), effects from induced three-body and higher-order forces are small. But a large $\lambda$ value may not sufficiently soften the short-range correlations of the realistic force, leading to demands for an excessively large model space and increased dependence on higher-order corrections. While a small $\lambda$ value may sufficiently soften the potential, the contribution from induced three-body force may be not ignorable. Within SRG, $\lambda\sim 2.0-2.5$ fm$^{-1}$ seems to be an optimal range in which the $NN$ interaction can be softened reasonably and the combined three-body (initial plus induced) effects are greatly reduced [@PhysRevC.87.054312; @PhysRevC.75.061001; @Tichai:2016joa].
The calculation of the radius for $^{16}$O is displayed in Fig. \[fig:o16\_n3lo\_rms\]. Reasonable convergence is obtained for $N_{\text{shell}}=11$ and 13. But the calculated radius is smaller than the experimental value. It seems that other [*ab initio*]{} results yield radii that are systematically smaller than experiment [@PhysRevC.73.044312; @PhysRevC.91.051301]. In Tables \[tab:n3lo-o16-e\] and \[tab:n3lo-o16-r\], we give the order-by-order results of the HF-MBPT $^{16}$O calculations with the same parameters as those in $^4$He (i.e., $N_{\text{shell}}=13$ and $\hbar\Omega=35$ MeV) at different $\lambda$ values. The situation is similar to that in $^4$He. We can see that smaller contributions from the neglected higher-order corrections decrease with decreasing $\lambda$, and good convergence is obtained for the MBPT calculations within the HF basis at small $\lambda$ values. It has pointed out that in the HF basis the fourth- and higher-order MBPT corrections are known to be negligible in some cases [@Tichai:2016joa].
Calculations with the “bare” JISP16 potential
---------------------------------------------
As mentioned in the Introduction, the JISP16 interaction is established by the $J$-matrix technique, and its parameters were determined by fitting both $NN$ scattering data and nuclear structure data up to $A=16$ [@Shirokov200733]. It is called “bare” because we, along with others, do not apply renormalization procedures in order to use it in nuclear structure calculations. To fit selected nuclear properties, the interaction has been tuned with phase-equivalent transformations to minimize the role of neglected many-body interactions. This tuning exploits the residual freedoms in the off-shell properties of the NN interaction [@latePolyzou].
![\[fig:jispo16\] Ground-state binding energies of $^{4}$He and $^{16}$O as a function of the oscillator parameter $\hbar \Omega$ for different $N_{\text{shell}}$. The “bare” JISP16 potential [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733] is used. The dashed lines represent the experimental ground state energies.](mbpt_jisp16.eps)
![\[fig:he4\_jisp\_rms\] Point-proton rms radius of $^{4}$He as a function of the oscillator parameter $\hbar \Omega$ for different $N_{\text{shell}}$. The JISP16 potential [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733] is used.](he4_mbpt_jisp16_r_c.eps)
![\[fig:o16\_jisp\_rms\] Point-proton rms radius of $^{16}$O as a function of the oscillator parameter $\hbar \Omega$ for different $N_{\text{shell}}$. The JISP16 potential [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733] is used.](o16_mbpt_jisp16_r_c.eps)
[lcdr]{} & &\
Expt.& 1.477 & -28.296\
NCSM& $1.418$ & -28.222\
SHF & 1.562 & -22.462\
PT2& 0.015 & -4.373\
PT3& $-$ & -0.803\
$\Delta r_{\text{c.m.}}$ & -0.211 & $-$\
HF-MBPT totally& 1.366 & -27.638\
-------------------------- -------- ---------- --
Expt. 2.581 -127.619
NCSM 1.836 -131.091
SHF 1.852 -71.638
PT2 0.052 -58.873
PT3 $-$ -4.260
$\Delta r_{\text{c.m.}}$ -0.061 $-$
HF-MBPT totally 1.843 -134.771
-------------------------- -------- ---------- --
: \[tab:4\]Ground-state binding energy and point-proton radius of $^{16}$O with the “bare” JISP16 interaction [@PhysRevC.70.044005; @Shirokov200596; @Shirokov200733] at $\hbar \Omega=35$ MeV. The results of HF-MBPT are obtained with $N_{\text{shell}}=10$. The NCSM results with $N_{\text{max}}=8$ are taken from Ref. [@PhysRevC.79.014308; @Gianina]. The experimental energy is from Ref. [@1674-1137-36-12-001], and the experimental radius is obtained as in Table \[tab:n3lo-o16-r\].
Similar to the investigations with the chiral N$^3$LO potential, we have applied the “bare” two-body JISP16 interaction to $^4$He and $^{16}$O. Figs. \[fig:jispo16\] show calculated binding energies for these two closed-shell nuclei. Figs. \[fig:he4\_jisp\_rms\] and \[fig:o16\_jisp\_rms\] are the radii calculations. Good convergence is obtained as indicated by the improved independence of $\hbar \Omega$ and $N_{\text{shell}}$ with increasing $N_{\text{shell}}$. The JISP16 potential without three-body force gives reasonable ground state energies compared with data. Tables \[tab:3\] and \[tab:4\] give the details of the HF-MBPT calculations with JISP16. To see how well the HF-MBPT approach does, we have made a comparison with the benchmark given by the NCSM calculation with the same JISP16 [@PhysRevC.79.014308; @Gianina]. For the NCSM calculation, we introduce the model space truncation parameter $N_{max}$ that measures the maximal allowed HO excitation energy above the unperturbed lowest zero-order reference state. We choose to compare out results with $N_{max}$=10 for $^{4}$He calculations, impling that a total of 11 major HO shells are involved. Such a model space is sufficient for $^{4}$He. For the HF-MBPT calculation, fast convergence with increasing the size of the model space $N_{\text{shell}}$ has been shown in Fig. \[fig:jispo16\]. We use the results of HF-MBPT with $N_{\text{shell}}$=10 to compare with the results of NCSM with $N_{\text{max}}=10$ as in Table \[tab:3\]. We see that HF-MBPT and NCSM calculations give similar results for the energy and radius of $^4$He, in good agreement with data. For $^{16}$O, we use $N_{\text{max}}$=8, which corresponds to a total of 10 major HO shells involved. The results of HF-MBPT with $N_{\text{shell}}$=10 truncation is used to compare with the NCSM results as in Table \[tab:4\]. Both HF-MBPT and the NCSM give larger binding energies but smaller radii than experimental data. The MBPT convergence with perturbative order in the “bare” JISP16 calculation is similar to that in the chiral N$^3$LO calculation. With the calculations based on N$^3$LO and JISP16, we may conclude that the MBPT method can give fairly converged results in the HF single-particle basis for these realistic $NN$ interactions.
Summary
=======
We have performed the HF-MBPT calculations with the realistic $NN$ interactions chiral N$^3$LO and “bare” JISP16. The detailed formulation and anti-symmetrized Goldstone diagram expansions are given. While the bare N$^3$LO potential is softened using the SRG method, the “bare” JISP16 is employed without softening.. The MBPT corrections are performed based on the spherical Hartree-Fock approach. The spherical symmetry preserves the quantum numbers of angular momenta. The angular momentum coupled scheme can significantly reduce the model dimension and save the computational resources. As an improvement, we correct the one-body density for the calculation of the radius using anti-symmetrized Goldstone diagram expansions through second order.
The closed-shell nuclei, $^4$He and $^{16}$O, have been chosen as examples for the present HF-MBPT calculations. Convergence with respect to the SRG-softening parameter, harmonic oscillator frequency and model space truncation have been discussed in detail. Our results are consistent with other works published with MBPT or with other [*ab initio*]{} methods. We discussed the MBPT convergence order by order, showing that corrections up to the third order in energy and up to the second order in radius appear to be reasonable when one performs the HF-MBPT calculations within the Hartree-Fock single-particle basis. It is demonstrated that smaller contributions from the neglected higher orders decrease with decreasing SRG-softening parameter $\lambda$. In the present calculations, three-body and higher-order forces are not considered. To check the convergence of the MBPT calculation, we have made comparisons with benchmarks given by NCSM calculations with the same $NN$ potential. Consistent results have been obtained. In general, the calculated radii are smaller than experimental values, which is a common problem in current [*ab initio*]{} calculations with these interactions..
Valuable discussions with R. Machleidt and L. Coraggio are gratefully acknowledged. This work has been supported by the National Key Basic Research Program of China under Grant No. 2013CB834402; the National Natural Science Foundation of China under Grants No. 11235001, No. 11320101004 and NO. 11575007; the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) funded by the U.S. Department of Energy, Office of Science under grant number DE-SC0009971; the Department of Energy under Grant No. DE-FG02-87ER40371; and National Training Program of Innovation for Undergraduates.
[^1]: frxu@pku.edu.cn
|
---
title: Diffusion coefficient and radial gradient of galactic cosmic rays
---
Introduction
============
The galactic cosmic rays (GCRs) transport in the heliosphere is governed by the four important processes: outward convection by the solar wind, inward diffusion, particle drifts (gradient, curvature and on the neutral sheet) in the turbulent interplanetary magnetic field (IMF) and adiabatic cooling. Estimation of the local electromagnetic conditions near the Earth orbit is possible by establishing modulation parameters as diffusion coefficient, density gradients etc. It is especially essential when in situ measurements are absent. It is the basic knowledge needed to study space weather prediction. A major issue in GCR transport research and space weather studies is how GCR particles propagate through the heliosphere, and how interact with the interplanetary space especially in the inner heliosphere near the Earth’s orbit. The essential significance of characterizing GCR propagation is evident, because this will lead to a practical capability in space weather forecasting which has important consequences for life and technology on the Earth and also in the interplanetary space.
The first theoretical description of cosmic ray transport coefficients was done by Jokipii [@bib:jokipii66] by formulation the quasilinear theory (QLT) for GCR diffusion. One limiting assumption of the QLT is that in the guiding center approximation transport of the GCR particles is not perturbed by the IMF turbulence. This assumption is inaccurate especially for the highly anisotropic strong turbulent heliosphere. This classical approach has been improved by higher-order theories of GCR particles turbulent flow. Thus the theories considering nonlinear effects have been introduced [@bib:matheus95]-[@bib:shalchi09]. A validity of the QLT for the GCR particles of the energy $>1 GeV$ is confirmed by the weakly nonlinear theory (WNLT) [@bib:schalchi04a], nonlinear parallel diffusion theory (NLPA) [@bib:Qin07] and in papers [@bib:droge03; @bib:shalchi04b; @bib:shalchi09] (see e.g. [@bib:WA10]).
However, for selecting correct set of modulation parameters used in theoretical modelling (especially diffusion coefficients, etc.), one criterion remains the most important, if it is possible, to estimate them from the experimental data, supported by appropriate theory.
Observations of GCR intensity and anisotropy by neutron monitors (NMs) and IMF fluctuations can be successfully used for establishment of various parameters characterizing modulation of GCR by the solar wind. In this paper we present the temporal evaluation of the parallel diffusion coefficient of GCR particles for rigidities to which NMs respond. Parallel diffusion coefficient $K_{\parallel}$, equivalent to parallel mean free path (MFP) $\lambda_{\parallel}=\frac{3K_{\parallel}}{v}$, determined by physical properties of interplanetary medium, is a very important parameter to study the transport of energetic particles in the heliosphere, especially for a solar event (SEP) connected with the space weather prediction [@bib:Qin09; @bib:Qin11].
Using data of the product $\lambda_{\parallel} \nabla_{r} n$ of the parallel MFP $\lambda_{\parallel}$ and radial gradient $\nabla_{r} n$ of GCR calculated based on the GCR anisotropy data (Ahluwalia et al., this conference ICRC 2013, poster ID: 487 [@bib:Ahl13]), we estimate the temporal changes of the radial gradient $\nabla_{r} n$ of GCR at the Earth’s orbit. As a final point, determination of the parallel diffusion coefficient $K_{\parallel}$ (equivalent to parallel MFP $\lambda_{\parallel}$) of GCR particles according to Quenby [@bib:Quenby] and Hedgecock [@bib:hedg75] formulas will be performed for minimum conditions of solar activity.
Convection-diffusion approximation
==================================
It has been shown that $\sim 75-80\%$ of the 11-year variation of the GCR intensity can be interpreted based on the diffusion$-$convection model of GCR propagation [@bib:Dorman01; @bib:Alania08]. So, the long-term variations of GCR intensity can be described by the Parker transport equation, invoking the isotropic convection-diffusion approximation [@bib:parker63]. In scope of this approximation one can calculate changes of the parallel diffusion coefficient $K_{\parallel}$ as follows: $$\begin{aligned}
I=I_{0}exp(-\int_{r_{0}} ^{r_{E}} \frac{Vdr}{K_{\parallel}})\nonumber \\
dI=\frac{I_{0}-I}{I_{0}}\nonumber \\
dI\approx \int _{r_{0}} ^{r_{E}}\frac{Vdr}{K_{\parallel}}\nonumber \\
K_{\parallel}\propto\frac{Vr}{dI}\end{aligned}$$ where $dI$ is variation of the GCR intensity, $V$ solar wind velocity and $r$ radial distance.
![Temporal changes of the annual SSN, GCR intensity for Moscow NM, solar wind velocity $V$, magnitude $B$ of the IMF, product $V B$ and estimated parallel diffusion coefficient $K_{\parallel}$ for 1965-2011.[]{data-label="simp_fig1"}](Fig1.eps){width="45.00000%"}
Figure \[simp\_fig1\] shows a plot of the temporal changes of the annual sunspot numbers (SSN), GCR intensity for Moscow NM, the solar wind velocity $V$, the magnitude $B$ of IMF, product $V B$ and the estimated parallel diffusion coefficient $K_{\parallel}$ according to expression (1) for the period of 1965-2011. One notes that the parallel diffusion coefficient $K_{\parallel}$ exhibits $\sim11$-year variation, but a stronger solar polarity dependence ($\sim22$-year variation); a significant increase is observed in the minimum epochs of solar activity, especially in the $A<0$ magnetic polarity period. An anomalous increase of $K_{\parallel}$ for the recent solar minimum $23/24$ is clearly seen, as well.
Parallel mean free path
=======================
It has been suggested that parallel mean free path (MFP) $\lambda_{\parallel}$ can be expressed in terms of interplanetary magnetic field $B$ as follows [@bib:zank98]-[@bib:bazilevskaya13]:
![Temporal changes of the 13-month smoothed magnitude $B$ and $B_{x}$, $B_{y}$ and $B_{z}$ components of the IMF for 1965-2011.[]{data-label="simp_fig2"}](Fig2.eps){width="45.00000%"}
$$\begin{aligned}
\lambda_{\parallel}\propto \frac{B^{\frac{5}{3}}}{\delta B^{2}}\end{aligned}$$
where $\delta B$ is the standard deviation. In our calculations the formula (2) is replaced by an equivalent expression: $$\begin{aligned}
\lambda_{\parallel}\propto \frac{B}{\delta B}\end{aligned}$$ We calculate parallel MFP $\lambda_{\parallel}$ from the $27-$day running averages of the observed radial $B_{x}$, azimuthal $B_{y}$, and latitudinal $B_{z}$ components of IMF according to the formulas: $$\begin{aligned}
\lambda_{x}\propto \frac{B_{x}}{\delta B_{x}}, \lambda_{y}\propto \frac{B_{y}}{\delta B_{y}}, \lambda_{z}\propto \frac{B_{z}}{\delta B_{z}},\end{aligned}$$
![Temporal changes of the 13-month smoothed SSN and parallel MFP $\lambda_{\parallel}$ calculated according to expressions (3) and (4) for 1965-2011.[]{data-label="simp_fig3"}](Fig3.eps){width="45.00000%"}
Data sets of the IMF magnitude $B$ and $B_{x}$, $B_{y}$, $B_{z}$ componets used in this calculation are presented in figure \[simp\_fig2\]. The results of MFP calculations for 13-month smoothed data for the corresponding components and magnitude of the IMF according to expressions (3) and (4) are presented in figure \[simp\_fig3\]. One notes that the MFP oscillates with a period of $\sim 11$ year solar activity cycle with a significant increase in the minimum periods of solar activity. Also, MFP is strongly polarity dependent in accord with the drift theory with a considerable enhancement especially in the minimum epoch of solar activity in the $A<0$ magnetic polarity period.
Radial gradient of GCR
======================
On the basis of the long term changes of the product $\lambda_{\parallel} \nabla_{r} n$ calculated based on the GCR anisotropy data (Ahluwalia et al., this conference ICRC 2013, poster ID: 487 [@bib:Ahl13]) and parallel diffusion coefficient $K_{\parallel}$ of GCR found above (figure \[simp\_fig1\]), we estimate also the radial gradient $\nabla_{r} n$ of GCR at the Earth’s orbit. The results of our calculations are presented in figure \[simp\_fig4\]. One notes that the radial gradient $\nabla_{r} n$ of GCR oscillates with a period $\sim 11$year solar activity cycle with a weaker solar polarity dependence, being in agreement with the previous calculation reported by Chen and Bieber [@bib:Chen93] and with PIONEER/VOYAGER observations [@bib:fuji97].
![Time variation of the annual GCR radial gradient $\nabla_{r} n$ for 1965-2011 with errors bars.[]{data-label="simp_fig4"}](Fig4.eps){width="45.00000%"}
Transport coefficient for minimum conditions of solar activity
==============================================================
Transport coefficients (e. g. diffusion coefficient of GCR) may be derived from a precise knowledge of the regular interplanetary magnetic field values and its fluctuations (turbulence) [@bib:Quenby]. The derivation of the parallel diffusion coefficient given by Jokipii [@bib:jokipii66; @bib:jokipii67] and Hasselmann and Wibberenz [@bib:hasselman68] is best illustrated in a simple way by following the Kennel and Petschek [@bib:kennel66] formulation given in the context of magnetospheric particle scattering.
In order to accurately obtain the parallel mean free path (diffusion coefficient) of GCR particles in the heliosphere, a method of power spectrum density of the interplanetary magnetic field turbulence has been used [@bib:hedg75]. In this paper we compare the values of parallel MFP obtained by means of formulation of Hedgecock [@bib:hedg75] and Quenby [@bib:Quenby]. We consider frequency range $10^{-6}-10^{-5} Hz$, responding for modulation of the GCR particles detected by NMs.
The appropriate parallel MFP can be expressed in terms of the power spectrum density of the interplanetary magnetic field fluctuations according to Hedgecock [@bib:hedg75] and Quenby [@bib:Quenby], respectively: $$\begin{aligned}
\lambda_{\parallel}\propto \frac{2\nu(\nu+2)cR^{2}}{9V \cdot P(f)}\end{aligned}$$ $$\begin{aligned}
\lambda_{\parallel}\propto \frac{V B^{2}}{4 \pi \cdot P(f)} \frac{1}{f^{2}}\end{aligned}$$ Where $V$ is the solar wind velocity, $B$ magnitude of IMF, $R$ - magnetic rigidity of GCR particles to which NM respond (in this case $R=15GV$), $f$ is the resonant frequency of GCR scattering, $P(f)$ is the power spectrum density at the resonant frequency $f$ with the spectral index $\nu$ for the frequency range $10^{-6}-10^{-5} Hz$.
![The time variation of the resonant frequency $f$ for the 1965-2011 period.[]{data-label="simp_fig5"}](Fig5.eps){width="45.00000%"}
The time variation of the resonant frequency $f=\frac{V}{2 \pi}\frac{300 B}{R}$ for the period 1975-2011 is shown in figure \[simp\_fig5\]. Figure \[simp\_fig5\] shows clear $11-$year variation in accordance with the solar activity cycle.
![Values of the parallel mean free path $\lambda_{\parallel}$ calculated based on expression (5) according to Hedgecock [@bib:hedg75] for the consecutive minimum epochs of solar activity with different signs of global magnetic polarity for the period of 1975-2011. Each point corresponds to the average value for three years around each solar minimum period.[]{data-label="simp_fig6"}](Fig6.eps){width="45.00000%"}
![Values of the parallel mean free path $\lambda_{\parallel}$ calculated based on expression (6) according to Quenby [@bib:Quenby] for the consecutive minimum epochs of solar activity with different signs of global magnetic polarity for 1975-2011. Each point corresponds to the average value for three years around each solar minimum period.[]{data-label="simp_fig7"}](Fig7.eps){width="45.00000%"}
Figures \[simp\_fig6\] and \[simp\_fig7\] present the values of the parallel mean free path $\lambda_{\parallel}$ calculated based on expression (5) and (6) according to Hedgecock [@bib:hedg75] and Quenby [@bib:Quenby], respectively, for the consecutive minimum epochs of solar activity with different signs of global magnetic polarity for the period of 1975-2011. The parallel mean free path $\lambda_{\parallel}$ is calculated based on the transverse components $B_{y}$ and $B_{z}$ of the IMF. Each point corresponds to the average value for three years around each solar minimum period (e.g., 1986 corresponds to the time interval 1985-1987). Calculations for parallel MFP according to both formulas (5) and (6) are in good agreement with each other. One can see that parallel MFP calculated based on the $B_{y}$ transverse component of the IMF is strong polarity dependent with large increase in $A<0$ for 1985-1987. Unfortunately this statement for $B_{y}$ component is not satisfied in the last minimum 2007-2009 ($A<0$). On the other hand in the last minimum period with record level of the GCR intensity ever measured by NMs, calculations for $B_{z}$ transverse component show an increase in the changes of the parallel MFP.
Conclusions
===========
1. The parallel diffusion coefficient $K_{\parallel}$, obtained based on the isotropic convection-diffusion GCR modulation model, generally displays $\sim 11$-year variation, but with strong polarity dependence ($\sim 22$ years). A significant increase is observed in the minimum epochs of solar activity, especially in the $A<0$ magnetic polarity period. An anomaly increase of $K_{\parallel}$ in recent solar minimum $23/24$ is clearly seen, as well.
2. We calculate parallel mean free path $\lambda_{\parallel}$ of GCR based on the experimental data of the IMF. Its value is polarity dependent in accord with drift theory and oscillate with a period of $\sim 11$-years solar activity cycle.
3. On the basis of the long term changes of the GCR anisotropy we show the $\sim 11$-year variation of the radial gradient $\nabla_{r} n$ of GCR being in good agreement with the PIONEER/VOYAGER observations.
4. Parallel mean free path $\lambda_{\parallel}$ calculated based on the $B_{y}$ transverse component of the IMF is strong polarity dependent with large increase in $A<0$ for 1985-1987. In the last minimum epoch of solar activity with record level of the GCR intensity ever measured by NMs, calculations for $B_{z}$ transverse component of the IMF show an increase in the changes of the parallel mean free path.
E.N. Parker, Planet. Space Sci., 13, (1965) 9-49. G. P. Zank, W. H. Matthaeus, J. W. Bieber and H. Moraal, J. Geophys. Res., 103, A2, (1998) 2085-2097. R.A. Mewaldt, A.J. Davis, K.A. Lave, R.A. Leske, E.C. Stone, et al., Astrophys. J. Lett., 723, (2010) L1-L6. G. A. Bazilevskaya, M. B. Krainev, A. K. Svirzhevskaya and N. S. Svirzhevsky, J. Phys.: Conf. Ser. 409, (2013) 012191, doi:10.1088/1742-6596/409/1/012191 H.S. Ahluwalia, M.V. Alania and R. Modzelewska, ICRC 2013 conference, poster ID: 487. J. R. Jokipii, Astrophys. J. 146, (1966) 480-487. W. H. Matthaeus, P. C. Gray, D. H. Pontius, Jr., and J.W. Bieber, Physical Review Letters, 75, (1995), 2136-2139. J.W. Bieber and W. H. Matthaeus, 485, (1997), 655-659, doi: 10.1086/304464. W. H. Matthaeus, G. Qin, J. W. Bieber, and G. P. Zank, 590, (2003) L53-L56, doi: 10.1086/376613. A. Shalchi, J. W. Bieber, W. H. Matthaeus, and G. Qin, 616, (2004) 617-629, doi: 10.1086/424839. A. Shalchi, Nonlinear Cosmic Ray Diffusion Theories, Astrophysics and Space Science Library, vol. 362. (2009) Springer-Verlag, Berlin. G. Qin, Astrophys. J. 656, (2007) 217-221. W. Droge, Astrophys. J. 589, (2003) 1027-1039. A. Shalchi and R. Schlickeiser, Astrophys. J. 604, (2004) 861-873. A. Wawrzynczak and M.V. Alania, Adv. Space Res., 45, (2010) 622-631. G. Qin, M. Zhang and H. K. Rassoul, Astrophys. J., 114, (2009) A09104 H.Q. He and G. Qin, Astrophys. J., 730, 46, (2011) 1-6, doi: 10.1088/0004-637X/730/1/46 J.J. Quenby, Space Science Rev., 37, (1984) 201-267. P.C. Hedgecock, Solar Phys., 42, (1975) 497-527. L.I. Dorman, Adv. Space Res., 27, (2001) 601-606. M. V. Alania, K. Iskra and M. Siłuszyk, Adv. Space Res., 41, (2008) 267-274. E.N. Parker, Interplanetary Dynamical Processes, Interscience Publishers, New York, (1963). J. Chen and J. W. Bieber, Astrophys. J., 405, (1993) 375-389. Fuji, Z., and F. B. McDonald, J. Geophys. Res., 102, A11, (1997) 24201-24208 J. R. Jokipii, Astrophys. J. 149, (1967) 405-416. K. Hasselman and G. Wibberentz, Z. Geophys. 34, (1968) 353. C.F. Kennel, and H. E. Petscheck, J. Geophys. Res. 71, (1966) 1-61.
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\#1[\#1]{} \#1[\#1]{} 2[ GeV$^2$]{} 2[$\chi^2_{d.o.f}$]{} \#1
epsf =-2cm =-1.5cm
February 1998 LYCEN 9808\
0.8cm
**INTERPOLATING BETWEEN SOFT AND HARD DYNAMICS**
**IN DEEP INELASTIC SCATTERING**
**P. Desgrolard ([[^1]]{}), L. Jenkovszky ([ [^2]]{}), F. Paccanoni ([ [^3] ]{})**
($^1$) [*Institut de Physique Nucléaire de Lyon, IN2P3-CNRS et Université Claude Bernard, F69622 Villeurbanne Cedex, France.*]{}
($^2$) [*Bogolyubov Institute for Theoretical Physics, Kiev-143, Ukraine.*]{}
($^3$) [*Dipartimento di Fisica, Universitá di Padova, Istituto Nationale di Fisica Nucleare, Sezione di Padova, via F.Marzolo, I-35131 Padova, Italy.*]{}
**Abstract**
An explicit model for the proton structure function is suggested, interpolating between low-$Q^2$ vector meson dominance and Regge behavior, on the one hand, and the high-$Q^2$ solution of the Gribov-Lipatov-Altarelli-Parisi evolution equation, on the other hand. The model is fitted to the experimental data in a wide range of the kinematical variables with emphasis on the low-$x$ HERA data. The boundaries, transition region and interface between various regimes are quantified.
[**1 INTRODUCTION** ]{}
In deep inelastic scattering the dynamics of low - and high virtualities, $Q^2$ is usually treated in a disconnected way, by using different methods. The structure function (SF) $F_2(x,Q^2)$ at small $Q^2$ (and small $x$, where $x$ is the fraction of the momentum carried by a parton) is known to be Regge-behaved and satisfying vector meson dominance (VMD) with the limit $F_2(x,Q^2)\underarrow{Q^2\rightarrow 0}0$, imposed by gauge invariance. At large $Q^2$, on the other hand, $F_2(x,Q^2)$ obeys the solutions of the Gribov-Lipatov-Altarelli-Parisi (GLAP) evolution equation \[1\].
One important problem remains open: where do these two regimes meet and how do they interpolate? In the present paper we seek answers to these questions.
For definiteness, we deal with the proton SF to be denoted $F_2$. Our emphasis is on the small $x$ region, dominated by gluodynamics. The valence quark contribution will be added at large-$x$ in a phenomenological way to make the fits complete.
The forthcoming presentation has also an important aspect relevant to quantum chromodynamics (QCD), namely in clarifying the range of applicability and the interface between the GLAP and the Balitsky-Fadin-Kuraev-Lipatov (BFKL) \[2\] evolutions. While the GLAP equation describes the evolution of the SF in $Q^2$ starting from a given $x-$ dependence, the BFKL evolution means variation of the SF in $x$ for fixed $Q^2$, both implying large enough $Q^2$ for the perturbative expansion to be valid. QCD leaves flexible the relevant limits and boundaries. Moreover, the onset of their asymptotic solutions depends on details of the calculations. In this paper we try to make some of these limits explicit and quantitative.
HERA is an ideal tool to verify the above theories. The relevant data extend over a wide range of $Q^2$ - a fruitful test field for the GLAP evolution, on one hand, and to low enough $x$, where the SF is dominated by a Pomeron contribution, expected to be described by the BFKL evolution (see below).
For the parametrization $F_2\approx x^\lambda$, the “effective power” $\lambda$ rises on average from about 0.15 around $Q^2\approx 1$2 to 0.4 at $Q^2\approx 1000$2 \[3\]. This exponent cannot be identified with the intercept - 1 of a simple Pomeron pole since by factorization it cannot depend on the virtuality of the external particle. A $Q^2-$ dependent intercept, compatible with the data, may arise from unitarization. However such a model \[4\] leaves much flexibility since neither the input (Born) value of the intercept is known for sure, nor a reliable unitarization procedure exists (for a recent attempt see however \[5\]). Moreover, claims exist that the HERA data are compatible with a softer, namely logarithmic behavior in $x$ (obeying the Froissart bound) with a factorized $Q^2$ dependence \[6, 7\].
On the other hand the $Q^2$, or GLAP, evolution in the “leading-log” approximation, has the following asymptotic solution for the singlet SF, valid for low $x$ and high $Q^2$ \[1, 8\] $$F_2\approx \sqrt{\gamma_1\ell n(1/x)\ \ell n\ell n{Q^2}} \ , \eqno(1.1)$$ with $\gamma_1={16N_c\over(11-2f/3)}$. For 4 flavours $(f=4)$ and three colours $(N_c=3)$, one gets $\gamma_1=5.76$.
The asymptotic solution of the BFKL evolution equation is the so-called “Lipatov Pomeron” \[2\]. The numerical value of its intercept was calculated \[2\] to be between 1.3 and 1.5. This large value gave rise to speculations that the “Lipatov Pomeron” has been seen at HERA, where the large - $Q^2$ data seemed to be compatible with a steep rise $\approx x^{-0.4}$ (for an alternative interpretation of the relation between the “Lipatov Pomeron” and the “HERA effect” see for example \[9\]). However, according to the results of a recent calculation \[10\], the sub-asymptotic corrections to the Pomeron pole in perturbative QCD are larger than expected and they contribute distructively to the intercept, thus lowering its value and making it compatible with the intercept of the soft Pomeron $\footnote {One of us (F.P.) thanks B.I. Ermolaev for a
discussion of this issue.}$.
The technical difficulties of the purely perturbative calculations are aggravated by the unpredictable non-perturbative contributions, both in the BFKL and GLAP evolutions, thus reducing the precision of the theoretical calculations and their predictive power. All these difficulties are redoubled by the unknown unitarity corrections to be included in the final result.
Attempts to extrapolate the Pomeron-dominated “soft” SF by applying GLAP evolution towards higher $Q^2$ are known in the literature (see ). They differ in some details, namely in the choice of the model for the Pomeron, its range, the value of $Q^2$ from which the evolution starts, and in the details of the evaluation (explicit in \[11\] or numerical in \[4a\]) of this evolution. We are not aware of any results of an “inverse extrapolation”.
The situation has been recently summarized \[3\] in a figure (see also Sec. 6) showing the $x$ - and $Q^2$ - dependence of the derivative $dF_2/d\ell n Q^2$. The philosophy behind this figure is that the turning point (located at $Q^2\sim2$ 2 ) divides “soft” and “hard” dynamics. As shown in \[3\], one of the most successful approaches to the GLAP evolution, that by Ref. \[12\], fails to follow the soft dynamics. A phenomenological model (called “ALLM”) for the structure function and cross-section, applicable in a wide range of their kinematical variables is well known in the literature \[13\]. Recently \[14\] it was updated to fit the data and shown to exhibit both - the rising and falling - parts of the derivative versus $x$ (or $Q^2$). We will comment more the behavior of this derivative in Sec. 6.
Below we pursue a pragmatic approach to the problem. We seek for an interpolation formula between the known asymptotic solutions imposed as boundary conditions. Clearly, such an interpolation is not unique, but it seems to be among the simplest. Moreover, it fits the data remarkably well, thus indicating that the interpolation is not far from reality.
[**2 KINEMATICS**]{}
We use the standard kinematic variables to describe deep inelastic scattering: $$e(k) \ +\ p(P) \ \to\ e(k')\ +\ X\ , \eqno(2.1)$$ where $ k, k', P$ are the four-momenta of the incident electron, scattered electron and incident proton. $ Q^2$ is the negative squared four-momentum transfer carried by the virtual exchanged boson (photon) $$Q^2 \ =\ -q^2\ =\ -(k-k')^2\ , \eqno(2.2)$$ $x$ is the Björken variable $$x\ =\ {Q^2\over 2P.q} \ , \eqno(2.3)$$ $y$ (the inelasticity parameter) describes the energy transfer to the final hadronic state $$y\ =\ {q.P\over k.P} \ ,
\eqno(2.4)$$ $W$ is the center of mass energy of the $\gamma^* p$ system $$W^2 \ =\ Q^2{1-x\over x}+m_p^2 \ , \eqno(2.5)$$ with $m_p$, being the proton mass. Note that only two of these variables are independent and that, at high energies for a virtual photon with $x\ll 1$, one has $W^2\ \sim {Q^2\over x}$.
[**3 STRUCTURE FUNCTION FOR SMALL $x$ AND ALL $Q^2$**]{}
Following the strategy outlined in the Introduction, we suggest the following ansatz for the small-$x$ singlet part (labelled by the upper index $S,0$) of the proton structure function, interpolating between the soft (VMD, Pomeron) and hard (GLAP evolution) regimes:
$$F_{2}^{(S,0)}(x,Q^2) =
A\left({Q^2\over Q^2+a}\right)^{1+\widetilde{\Delta} (Q^2)}
e^{\Delta (x,Q^2)} , \eqno(3.1)$$ with the “effective power” $$\widetilde{\Delta }(Q^2) =\epsilon+\gamma_1\ell n {
\left(1+\gamma_2\ell n{\left[1+{Q^2\over Q^2_0}\right]}\right)} ,
\eqno(3.2)$$ and $$\Delta (x,Q^2) = \left(\widetilde{\Delta } (Q^2) \ell n{x_0\over x}\right)
^{f(Q^2)}, \eqno(3.3)$$ where $$f(Q^2) = {1\over 2}\left( {1+e^{-{Q^2/Q_1^2}}}\right) . \eqno(3.4)$$
At small and moderate values of $Q^2$ (to be specified from the fits, see below), the exponent $\widetilde{\Delta}(Q^2)$ (3.2) may be interpreted as a $Q^2$-dependent “effective Pomeron intercept”.
The function $f(Q^2)$ has been introduced in order to provide for the transition from the Regge behavior, where $f(Q^2)=1$, to the asymptotic solution of the GLAP evolution equation, where $f(Q^2)=1/2$.
By construction, the model has the following asymptotic limits:
a\) Large $Q^2$, fixed $x$: $$F_{2}^{(S,0)}(x,Q^2\to \infty)\to A\
\exp^{\sqrt{\gamma_1\ell n\ell n{Q^2\over Q_0^2}\ \ell n{x_0\over x}}}\ ,
\eqno(3.5)$$ which is the asymptotic solution of the GLAP evolution equation (see Sec. 1).
b\) Low $Q^2$, fixed $x$: $$F_{2}^{(S,0)}(x,Q^2\to 0) \to A\
e^{\Delta (x,Q^2\to 0)} \
\left({Q^2\over a}\right)^{1+\widetilde{\Delta }(Q^2\to 0)} \eqno(3.6)$$ with $$\widetilde{\Delta }(Q^2\to 0) \to
\epsilon+\gamma_1 \gamma_2 { \left({{Q^2\over Q^2_0} }\right)}\
\to\ \epsilon , \eqno(3.7)$$ $$f(Q^2\to 0) \to 1 , \eqno(3.8)$$ whence $$F_{2}^{(S,0)}(x,Q^2\to 0) \to A\ \left( {x_0\over x}
\right)^\epsilon \ \left({Q^2\over a}\right)^{1+\epsilon} \ \propto
(Q^2)^{1+\epsilon} \ \to 0\ , \eqno(3.9)$$ as required by gauge invariance.
c\) Low $x$, fixed $Q^2$:
$$F_{2}^{(S,0)}(x\to 0,Q^2) \ =\
A\left({Q^2\over Q^2 + a}\right)^{1+\widetilde{\Delta }(Q^2)}
e^{\Delta (x\to 0,Q^2)} . \eqno(3.10)$$ If $$f(Q^2)\sim 1\ , \eqno(3.11)$$ when $Q^2\ll Q_1^2$, we get the standard (Pomeron-dominated) Regge behavior (with a $Q^2$ dependence in the effective Pomeron intercept) $$F_{2}^{(S,0)}(x\to 0,Q^2) \to A\ \left({Q^2\over Q^2 +
a}\right)^{1+\widetilde{\Delta}(Q^2)}\ \left({x_0\over x}
\right)^{\widetilde{\Delta}(Q^2)} \ \propto
x^{-\widetilde{\Delta}(Q^2)} .
\eqno(3.12)$$
Within this approximation, the total cross-section for $(\gamma,p)$ scattering as a function of the center of mass energy $W$ is $$\sigma ^{tot,(0)}_{\gamma,p} (W)=
4\pi^2\alpha\ \left[{F_{2}^{(S,0)}(x,Q^2)\over
Q^2}\right]_{Q^2\to 0} =\ 4\pi^2\alpha\ A\ a^{-1-\epsilon}\
x_0^\epsilon\ W^{2\epsilon} . \eqno(3.13)$$
[**4 EXTENSION TO LARGE $x$**]{}
In this section we complete our model by including the large-$x$ domain, extending to $x=1$, and for all kinematically allowed $Q^2$. Since we are essentially concerned with the small-$x$ dynamics (transition between the GLAP and BFKL evolution), the present extension serves merely to have as good fits as possible with a minimal number of extra parameters. To this end we rely on the existing successful phenomenological models, in particular on that of \[4a\] (CKMT).
Following CKMT, we multiply the singlet part of the above structure function $F_{2}^{(S,0)}$ (defined in (3.1-3.4)) by a standard large-$x$ factor to get $$F_{2}^{(S)}(x,Q^2) = F_{2}^{(S,0)}(x,Q^2)\
(1-x)^{n(Q^2)}, \eqno(4.1)$$ with $$n(Q^2) ={3\over 2} \left(1+{Q^2\over Q^2+c}\right), \eqno(4.2)$$ where $c=3.5489 $ GeV$^2$ \[4a\].
Next we add the nonsinglet $(NS)$ part of the structure function, also borrowed from CKMT $$F_{2}^{(NS)}(x,Q^2) =
B\ (1-x)^{n(Q^2)}\ x^{1-\alpha_r}\ \left({Q^2\over Q^2+b}\right)^{\alpha_r} \,
. \eqno(4.3)$$ The free parameters that appear with this addendum are $c, B, b$ and $\alpha_r$. The final and complete expression for the proton structure function thus becomes $$F_{2}(x,Q^2) = F_{2}^{(S)}(x,Q^2) + F_{2}^{(NS)}(x,Q^2)\ . \eqno(4.4)$$
The total cross-section for $(\gamma,p)$ scattering is $$\sigma ^{tot}_{(\gamma,p)} (W)=\
4\pi^2\alpha\ \left( A\ a^{-1-\epsilon}\ x_0^\epsilon\
W^{2\epsilon} + B\ b^{-\alpha_r}\ W^{2(\alpha_r-1)}\ \right) \ .
\eqno(4.5)$$
[**5 FITTING TO THE DATA**]{}
In fitting to the data, the complete experimental “H1” data set (which encloses 237 points: 193 from \[15\] and 44 from \[16\]) for the proton structure function $F_{2}(x,Q^2)$ was used as well as, 76 data points \[17\] on the $(\gamma,p)$ total cross-section $ \sigma ^{tot}_{(\gamma,p)} (W)$.
We note that among a total of 12 parameters, 8 are free, the resting 4 being fixed in the following way:
1\. $\epsilon=0.08$ is a “canonical” value \[18\], leaving little room for variations (although, in principle, it can be also subject to the fitting procedure);
2\. when left free in the fitting procedure, $x_0$ takes a value slightly beyond 1. Thus, we can safely fix $x_0=1$ without practically affecting the resulting fits;
3\. as already mentioned, we have set $c=3.5489$ GeV$^2$ relying on CKMT. This parameter is responsible for the large-$x$ and small-$Q^2$ region, outside the domain of our present interest;
4\. as argued above, we may estimate from QCD the parameter $\gamma_1=16N_c(11-2f/3)$ with four flavours $(f=4)$ and three colors $(N_c=3)$, it equals 5.76. It corresponds to the asymptotic regime (when $Q^2\rightarrow\infty$, or $f(Q^2)\rightarrow 1/2))$, far away from the region of the fits, where $f=1$ is more appropriate, hence the value ${\gamma_1}=\sqrt{5.76}=2.4$ is more appropriate in the domain under consideration. Remarkably, this value comes also independently from the fits if $\gamma_1$ is let free.
To compare with, the CKMT model \[4a\] depends on 8 adjustable parameters in the “soft” region, to be completed by QCD evolution at higher values of $Q^2$, and with a higher twist term added. On the other hand, the proton structure function and $(\gamma^*,p)$ cross section in the ALLM model \[13,14\] are given explicitly in the whole range of the kinematical variables, and the fits to the data are good with a total of 23 adjustable parameters.
When limiting the fitted data to the structure function only \[15,16\] with $x<0.1$ (all $Q^2$), the singlet contribution alone, as approximated in Sec. 3, gives a very good fit (2 $\sim 0.59$), shown in Figs. 1a, 2a. We mention that this result is obtained with an economical set of 8 parameters (5 free), listed in Table 1.
The complete model of Sec. 4 gives very good fits in the whole ranges in $x$, $Q^2$ and $W$ covered by measurements. To be specific, we find 2 $\sim$ 0.69. We show the contributions to the $\chi^2$ of the 3 data sets we used in Table 2, the numerical values for the 12 parameters (8 free) are presented in Table 3.
The results of our fits for the structure function versus $Q^2$ for fixed $x$ are shown in Fig. 1 b and for fixed $Q^2$ as a function of $x$ are in Figs. 2b, 3. The total cross section for real photons on protons as function of $W$ is displayed in Fig. 4.
[**6 INTERFACE BETWEEN SOFT AND HARD DYNAMICS AND TRANSITION FROM BFKL TO GLAP EVOLUTION**]{}
[**6.1 ${\partial F_{2}\over \partial (\ell n Q^2) }$ as a function of $x$ and $Q^2$.**]{}
The derivative of the SF with respect to $\ell n Q^2$ (slope for brevity) measures the amount of the scaling violation and eventually shows the transition from soft to hard dynamics. This derivative depends on two variables ($x$ and $Q^2$). It was recently calculated from the HERA data \[3\]; in Figs. 5, 6a we have quoted the corresponding results. In those calculations the variables $x$ and $Q^2$ are strongly correlated, it is implied that, for a limited acceptance (as it is the case in the HERA experiments) and for a fixed energy, one always has a limited band in $Q^2$ at any given $x$, with average $Q^2$ becoming smaller for smaller $x$. From a theoretical point of view, however, $x$ and $Q^2$ are quite independent and one is not restricted to follow a particular path on the surface representing the slope. Therefore we plot in Fig. 5 the slope calculated from our model (4.4) with the parameters fitted to the data, in one more dimension than usual, as a function of the two independent variables - $x$ and $Q^2$. The two slopes on the hill of ${\partial F_{2}\over \partial (\ell n Q^2) }$ in Fig. 5 correspond to soft and hard dynamics. The division line is only symbolic since there is a wide interface region where both dynamics mix, each tending to dominate on the lower side of its own slope. Remarkably, the division line - or line of maxima of this surface - turns out to be almost $Q^2$-independent ($\sim 40$ 2 ). The difference with the maximum at 2 2 exhibited in \[3\] is due to the special experimental set of ($x,Q^2$) chosen in \[3\], discussed above and shown in Fig. 5.
Notice that the slope becomes negative in a region between $Q^2\sim
200$ and $\sim 4000$ 2 , at small $x$ ; this region tends to narrow when $x$ increases beyond $x=$ 0.0005 and finally disappears when $x$ exceeds 0.05.
The same results are exposed on families of 2-dimensional figures as well (Figs. 6a, 6b) showing the $x$ - (and $Q^2$ -) dependence of the slope when the other variable takes fixed values. Fig. 6a shows that our predictions are quite in agreement with the data from \[3\]; also shown is the failure of the approach of the GLAP evolution equation \[12\] to follow the low $x$ ($Q^2$) dynamics as reported in \[3\]. Fig. 6b shows the variation with $x$ of the region with negative slope. Notice that the rising part to large extent is a threshold effect due to the increasing phase space (see \[19\]).
[**6.2 ${\partial \ell n F_{2}\over \partial (\ell n (1/x))} $ as a function of $Q^2$ for some $x$ values.**]{}
The derivative of the logarithm of the SF with respect to $\ell n 1/x$, when measured in the Regge region, can be related (for low $x$) to the Pomeron intercept. In Fig. 7 the $Q^2$-dependence of this derivative is shown for some low $x$ - values, together with the “effective power” $\tilde\Delta$ (3.2). On the same figure, the behavior of the function $f(Q^2)$ (3.4) is also shown. In our model, Regge behavior is equivalent to the condition that $f(Q^2)$ is close to unity. This lower limit, marked on Fig. 7 (tentatively approximated within a 2 % accuracy for the function $f(Q^2)$), is located near $40$ 2 . Until this landmark, the effective power $\tilde\Delta$ indeed remains very close to ${\partial \ell n F_{2}\over \partial (\ell n (1/x))} $, beyond Regge behavior is not valid (since $f\ \ne 1$) and $\tilde\Delta$ cannot be considered as the effective slope any more. On the other hand, ${\partial \ell n F_{2}\over \partial (\ell n (1/x))} $ turns down as $Q^2$ increases, approaching its “initial value” of $\approx 0.1$ at largest $Q^2$ and coming closer to the unitarity bound. Notably, at large $Q^2$ the derivative gets smaller as $x$ decreases, contrary to the general belief that dynamics becomes harder for smaller $x$, but in accord with an observation made in \[20\]. Care should be however taken in interpreting the “hardness” of the effective power outside the Regge region.
According to our model, the change from the BFKL (Pomeron) to the GLAP evolution occurs when $f(Q^2)$ changes from 1 to 1/2. This variation happens in a band in $Q^2$, namely between $\sim 40$ 2 and $\sim 4000$ 2 .
Let us remind once more that our interpolating formula (3.1) between Regge behavior and GLAP evolution was suggested for small $x (\ x\leq0.1)$. The larger $x$ part was introduced for completeness and better fits only, without any care of its correspondence to the GLAP evolution equation. It does not affect however the kinematical domain of the present and future HERA measurements and Pomeron dominance (BFKL evolution) we are interested in.
[**7 CONCLUSIONS**]{}
Once the “boundary conditions” (at low and high $Q^2$) are satisfied, the interpolation may be considered as an approximate solution valid for all $Q^2$. Clearly, our interpolation is not unique. For example, the choice of $f(Q^2)$, satisfying the boundary conditions $f(0)=1$ and $f(\infty)=1/2$, may be different from ours. However, there is little freedom in the choice of the asymptotic forms, different from those we have used, namely (3.5) and (3.12). The utilization of a soft Pomeron input different from (3.12) is credible. For example, a dipole Pomeron was shown \[7, 11\] to have the required formal properties and to fit the data at small and moderate $Q^2$. Moreover, the dipole Pomeron does not violate the Froissart bound, so it does not need to be unitarized. Attempts \[6, 7\] to fit the high-$Q^2$ HERA data without a power in $x$, i.e. with logarithmic functions, attributing the whole $Q^2$ dependence to the (factorized) “residue function”, are disputable. What is even more important from the point of view of the present interpolation, a power in $x$ must be introduced anyway to match the high $Q^2$ GLAP evolution solution (3.5). This discussion brings us back to the interesting but complicated problem of unitarity.
As it is well known, the power increase of the total cross sections, or of the SF towards small $x$ cannot continue indefinitely. It will be slowed down by unitarity, or shadowing corrections, whose calculation or even recipe - especially for high virtualities $Q^2$ - is a delicate and complicated problem, beyond the scope of the present paper. Here we only mention, that once the model fits the data, it cannot be far from the “unitarized” one in the fitted range, since the data “obey” unitarity.
To conclude:
1\. Strong interaction dynamics is continuous, hence the relevant solutions should be described by continuous solutions as well;
2\. The formal solutions of the GLAP equations, even in their most advanced forms, ultimately contain some freedom (e.g. “higher twists”, or non-perturbative corrections) or approximations;
3\. However so elaborated or “precise” the existing solutions are, unitarity corrections will modify their form anyway;
The above remarks justify the use for practical purposes of an explicit solution that satisfies the formal theoretical requirements and yet fits the data. Its simplicity and flexibility make possible its further improvement and its use as a laboratory in studying complicated and yet little understood transition phenomena.
We thank M. Bertini for discussions and L.A. Bauerdick for a useful correspondence.
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193 values of the structure function of the proton, for $Q^2$, between 1.52 and 50002, and $x$, between 3 10$^{-5}$ to 0.32 (of which 169 data correspond to $x < 0.1$).
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44 values of the structure function of the proton, at low $x$, down to 6 10$^{-6}$, and low $Q^2$, between 0.352 and 3.52.
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2.cm
**Tables captions**
[**Table 1.**]{}
Parameters used in our “first approximation fit” ($x<0.1$).
[**Table 2.**]{}
$\chi^2$ - contributions of each set of data used in our fit with the parameters listed in Table 3.
Parameters used in our fit in the whole kinematical range (see the text).
1.cm
**Figures captions**
[**Fig. 1 a**]{} Proton structure function $F_{2}(x,Q^2)$ as a function of $Q^2$ at various values of fixed $x$. For a better display, the structure function values have been scaled at each $x$ by the factor shown in brackets on the same line as the $x$ values. The shown H1 - data are from \[15,16\], the error bars represent the statistical and systematic errors added in quadrature, the curves are the results of our first parametrization fitted on $x<0.1$ data (“low-$x$, Pomeron dominated” approximation, the parameters being listed in Table 1).
[**Fig. 1 b**]{} Proton structure function $F_{2}(x,Q^2)$ as a function of $Q^2$ at various values of fixed $x$ as in Fig. 1 a but the curves being the results of our second parametrization fitted to all H1 data \[15,16\] of the proton structure function and to the total cross-sections of the $(\gamma ,p)$ process \[17\] (the parameters are listed in Table 3).
[**Fig. 2 a**]{} Proton structure function $F_{2}(x,Q^2)$ as a function of $x$ at various values of fixed $Q^2$. Results of our first approximation, see also Fig. 1 a.
[**Fig. 2 b**]{} Proton structure function $F_{2}(x,Q^2)$ as a function of $x$ at various values of fixed $Q^2$. Results of our second parametrization, see also Fig. 1 b.
[**Fig. 3**]{} Proton structure function $F_{2}(x,Q^2)$ as a function of $x$ at various low $Q^2$ values. See also Fig. 1 b.
[**Fig. 4**]{} Total cross-section of the reaction $(\gamma,p)$ $ \sigma ^{tot}_{(\gamma,p)} $ as a function of $W$, center of mass energy. (see also Fig. 1 b).
[**Fig. 5**]{} Two-dimensional projection of the three dimensional “slope” of the proton structure function. The surface represents ${\partial F_{2}(x,Q^2) \over \partial (\ln Q^2)}\ $ as a function of $x$ and $Q^2$ as following from the present parametrization with its line of maximum (open squares). The crosses are the points calculated from the HERA data in \[3\], located on an experimental ($x,Q^2$) path.
[**Fig. 6a**]{} Derivative of the proton structure function ${\partial F_{2}(x,Q^2) \over \partial (\ln Q^2)} $ as a function of $x$, for some $Q^2$ values as indicated. The round dots are the HERA data, the open squares the results from \[12\] taken from \[3\] and the hollow triangles are the results of the present parametrization.
[**Fig. 6b**]{} Same derivative as in Fig. 6a ${\partial F_{2}(x,Q^2)\over \partial (\ln Q^2)}\ $ as a function of $Q^2$, for some $x$ values as indicated. The solid curves are the results of the present parametrization.
[**Fig. 7**]{} Derivative of the logarithm of the proton structure function ${\partial \ell n F_{2}(x,Q^2)\over \partial (\ell n (1/x))} $ versus $Q^2$ for some $x$ values as indicated. Also plotted on the same (left) scale is the effective exponent $\widetilde{\Delta}$ (3.2), representing the Pomeron intercept $-1.$ only when $f(Q^2)\approx 1$. The function $f(Q^2)$ (3.4) is also shown as a dashed line (right scale); the transition between the Regge behavior ($ f=1.$) and the GLAP evolution ($f=0.5$) occurs within an estimated band located between vertical landmarks (see the text).
$$\vbox{\init\halign to 12 cm{
\strut#&\vrule#\tabskip=1em plus 2em&
\hfil$#$&
\vrule#&
%\vrule$\,$\vrule#&
\hfil$#$\hfil&
\vrule#\tabskip 0pt\crr
&&{\bf A} &&0.1612&\cr
&&{\bf a{\rm \ (GeV)}^2} &&0.2133&\cr
&&{\bf \gamma_2 }&&0.02086 &\cr
&&{\bf Q_0^2 {\rm \ (GeV)}^2} &&0.2502 &\cr
&&{\bf Q_1^2 {\rm \ (GeV)}^2} &&676.9 &\crr
&&{\bf x_0 }&&1.0 \ {\rm (fixed)} &\cr
&&{\bf \epsilon}&&0.08\ {(\rm fixed\ [18])} &\cr
&&{\bf \gamma_1 }&&2.4 \ {\rm (fixed\ QCD)} &\crr}}$$
**Table 1.**
Parameters used in our “first approximation fit” ($x<0.1$).
$$\vbox{\init\halign to 12 cm{
\strut#&\vrule#\tabskip=1em plus 2em&
\hfil$#$&
\vrule$\,$\vrule#&
\hfil$#$\hfil&
\vrule#&
\hfil$#$\hfil&
\vrule#\tabskip 0pt\crr
&&{\rm Data\ set} && {\rm N.\ of\ points}&& \chi^2&\crr
&& \sigma ^{tot}_{(\gamma,p)}\ (W>3\ GeV^2)\ {\rm \ [17] } &&73 &&73 &\cr
&&F_{2},\ {\rm H1\ [15] } &&193 &&116 &\cr
&&F_{2},\ {\rm H1\ (low }\ x {\rm )\ [16] } &&44 &&20 &\crr
}}$$
**Table 2.**
$\chi^2$ - contributions of each set of data used in our fit with the parameters listed in Table 3.
$$\vbox{\init\halign to 12 cm{
\strut#&\vrule#\tabskip=1em plus 2em&
\hfil$#$&
\vrule#&
%\vrule$\,$\vrule#&
\hfil$#$\hfil&
\vrule#\tabskip 0pt\crr
&&{\bf A} &&0.1623&\cr
&&{\bf a{\rm \ (GeV)}^2} &&0.2919&\cr
&&{\bf \gamma_2 }&&0.01936 &\cr
&&{\bf Q_0^2 {\rm \ (GeV)}^2} &&0.1887 &\cr
&&{\bf Q_1^2 {\rm \ (GeV)}^2} &&916.1 &\cr
&&{\bf B} &&0.3079&\cr
&&{\bf b{\rm \ (GeV)}^2} &&0.06716&\cr
&&{\bf \alpha_r}&&0.5135\ &\crr
&&{\bf x_0 }&&1.0 \ {\rm (fixed)} &\cr
&&{\bf \epsilon}&&0.08\ {\rm (fixed\ [18])} &\cr
&&{\bf \gamma_1 }&&2.4 {\rm (fixed\ QCD)} &\cr
&&{\bf c{\rm \ (GeV)}^2} &&3.549\ {\rm (fixed\ [4a])} &\crr
}}$$
**Table 3.**
Parameters used in our fit in the whole kinematical range (see the text).
Fig. 1a
Fig. 1b
Fig. 2a
Fig. 2b
Fig. 3
Fig. 4
Fig. 5
Fig. 6a
Fig. 6b
Fig. 7
[^1]: E-mail: desgrolard@ipnl.in2p3.fr
[^2]: E-mail: jenk@gluk.apc.org
[^3]: E-mail: paccanoni@pd.infn.it
|
---
author:
- 'A.V. Ugryumov'
- 'D. Engels'
- 'A.Y. Kniazev'
- 'R.F. Green'
- 'Y.I. Izotov'
- 'U. Hopp'
- 'S.A. Pustilnik'
- 'A.G. Pramsky'
- 'T.F. Kniazeva'
- 'N. Brosch'
- 'H.-J. Hagen'
- 'V.A. Lipovetsky[^1]'
- 'J. Masegosa'
- 'I. Márquez'
- 'J.-M. Martin'
date: 'Received ; Accepted'
subtitle: ' V. The Fifth List of 161 Galaxies '
title: ' The Hamburg/SAO Survey for Emission–Line Galaxies '
---
[llccccccc]{} [2]{}[c]{}[Candidate Sample]{} & N & BCG & Other & QSO & Galaxies & Stars & Not\
& & & & & ELGs & & without & & Classified\
& & & BCG? & & SA & & emission & &\
First priority & new & 141 & 64 & 43 & 5 & 3 & 18 & 8\
& already known & 28 & 21 & 7 & – & – & – & –\
& total & 169 & 85 & 50 & 5 & 3 & 18 & 8\
Random sample & new & 16 & 3 & 1 & – & 1 & 2 & 9\
& already known & 1 & – & 1 & – & – & – & –\
APM selected sample & new & 22 & 10 & 10 & – & – & 1 & 1\
& already known & 1 & – & 1 & – & – & – & –\
Second priority & total & 40 & 13 & 13 & – & 1 & 3 & 10\
[2]{}[l]{}[Objects presented in this paper]{} & 209 & 98 & 63 & 5 & 4 & 21 & 18\
[ccccccc]{}\
[1]{}[c]{}[ Date ]{} & [1]{}[c]{}[ Telescope ]{} & [1]{}[c]{}[ Instrument ]{} & [1]{}[c]{}[ Grating, ]{} & [1]{}[c]{}[ Wavelength ]{} & [1]{}[c]{}[ Dispersion ]{} & [1]{}[c]{}[ Observed ]{}\
[1]{}[c]{}[ ]{} & & & [1]{}[c]{}[ grism ]{} & [1]{}[c]{}[ range \[Å\] ]{} & [1]{}[c]{}[ \[Å/pixel\] ]{} & [1]{}[c]{}[ number ]{}\
[1]{}[c]{}[ (1) ]{} & [1]{}[c]{}[ (2) ]{} & [1]{}[c]{}[ (3) ]{} & [1]{}[c]{}[ (4) ]{} & [1]{}[c]{}[ (5) ]{} & [1]{}[c]{}[ (6) ]{} & [1]{}[c]{}[ (7) ]{}\
\
09.06-18.06.1999 & 2.2 m CAHA & CAFOS & G-200 & 3700–9500 & 4.5 & 117\
— & — & — & B-200 & 3500–7400 & 4.7 & 18$^*$\
17.06-20.06.1999 & [1]{}[r]{}[4 m KPNO]{} & R.C.Sp. & KPC-10A & 3700–8300 & 2.8 & 46\
08.12-11.12.1999 & 2.2 m CAHA & CAFOS & G-200 & 3700–9500 & 9.0 & 46\
\
[7]{}[l]{}[$^*$ – objects reobserved ]{}
Introduction
============
The problem of creating large, homogeneous and deep samples of actively star-forming low-mass galaxies is very important for several applications in studies of galaxy evolution and spatial distribution. Several earlier projects, like the Second Byurakan Survey (SBS) (Markarian et al. [@Markarian83], Stepanian [@Stepanian94]), the University of Michigan (UM) survey (e.g., Salzer et al. [@Salzer89]), and the Case survey (Pesch et al. [@Pesch95], Salzer et al. [@Salzer95], Ugryumov et al. [@Ugryumov98]), as well as some others, identified on objective prism plates many hundreds of emission-line galaxies. The Hamburg/SAO survey (HSS) is intended to create a new very large homogeneous sample of such galaxies in the region of the Northern sky with an area of some 1700 square degrees. The basic outline of the HSS and first results are described in Paper I (Ugryumov et al. [@Ugryumov99]), while additional results from follow-up spectroscopy are given in papers II, III and IV (Pustilnik et al. [@Pustilnik99], Hopp et al. [@Hopp00], Kniazev et al. [@Kniazev01]). In this paper we present the results of follow-up spectroscopy of another 209 objects selected on the Hamburg Quasar Survey (HQS) prism spectral plates as ELG candidates.
The article is organized as follows. In section \[Obs\_red\] we give the details of the spectroscopic observations and of the data reduction. In section \[Res\_follow\] the results of the observations are presented in several tables. In section \[Discussion\] we briefly discuss the new data and summarize the current state of the Hamburg/SAO survey. Throughout this paper a Hubble constant H$_0$ = 75 km$\,$s$^{-1}$ Mpc$^{-1}$ is used.
Spectral observations and data reduction {#Obs_red}
========================================
Observations
------------
The results presented here were obtained in a snap-shot observing mode during two runs with the Calar Alto 2.2m and one run with the KPNO 4m telescopes (see Table \[Tab2\]).
Observations with the KPNO 4m telescope
---------------------------------------
The observations were carried out with the Ritchey-Chretien Spectrograph attached to a Tektronix 2K$\times$2K CCD detector. We used a 2$''$$\times$205$''$ slit with a KPC-10A grating (316 grooves mm$^{-1}$) in its first order, and a GG 375 order separation filter cutting off second-order contamination for wavelengths blueward of 7400Å. This instrumental setup allows a spatial scale along the slit of $0\farcs69$ pixel$^{-1}$, a scale perpendicular to the slit of 2.77Åpixel$^{-1}$, a spectral range of $3700-8300$Åand a spectral resolution of $\sim$ 7Å (FWHM). Short exposures ($3-5$ minutes) were used in order to detect strong emission lines to allow measurement of redshifts and a crude classification. No orientation of the slit along the parallactic angle was done because of the snap-shot observing mode. Reference spectra of an Ar-Ne-He lamp were recorded to provide wavelength calibration. Spectrophotometric standard stars from Oke ([@Oke90]) and Bohlin ([@Bohlin96]) were observed at the beginning and at the end of each night for flux calibration. The dome flats, bias, dark and twilight sky frames were accumulated each night. The weather conditions were photometric, with seeing variations between 25 and 3 (FWHM).
Calar Alto 2.2m telescope observations
--------------------------------------
Follow-up spectroscopy with the CAHA 2.2m telescope was carried out during two runs (June and December, 1999, see Table \[Tab2\]), using the Calar Alto Faint Object Spectrograph (CAFOS) and Cassegrain focal reducer. During these runs a long slit of $300\arcsec \times (2\arcsec-3\arcsec)$ and a G-200 grism (187Åmm$^{-1}$, first order) were used. The B-200 grism (185Åmm$^{-1}$, first order) was also used to reobserve 18 objects in order to improve the \[O[ii]{}\]$\lambda$3727Å value. There were no order separation filters applied. The spatial scale along the slit was $0\farcs53$ pixel$^{-1}$. A SITE 15 2K$\times$2K CCD was operated in a 2$\times$1 binned mode during the December run (binning only along the dispersion direction), while in the June run there was no binning applied. The wavelength coverages were $\lambda$3700 – $\lambda$9500Åwith maximum sensivity at $\sim$ 6000Åfor the G-200 grism and $\lambda$3500 – $\lambda$7400Åwith maximum sensivity at $\sim$ 4000Å for the B-200 grism. The spectral resolution was $\sim$ 10Å (FWHM) in the June run and due to CCD binning $\sim$ 20Å (FWHM) in the December run. The slit orientation was not aligned with the parallactic angle because of the snap-shot observing mode. The exposure times varied within $2-15$ minutes depending on the object brightness. The observations were complemented by standard star flux measurements (Oke [@Oke90], Bohlin [@Bohlin96]), reference spectra (Hg-Cd lamp) for wavelength calibration, dome flat, bias and dark frames. Most of the time the weather conditions were photometric with a seeing $\approx$1.5 (FWHM). Only during two nights the weather conditions were variable with a seeing of 3 – 4(FWHM). The measurements of these nights are marked by “$*$” in Table 4.
Data reduction
--------------
Reduction of the CAHA and KPNO spectral data was performed at the SAO using the standard reduction systems MIDAS[^2] and IRAF.
The MIDAS command FILTER/COSMIC was found to be a quite successful way to remove automatically all cosmic ray hits from the images. After that we applied the IRAF package CCDRED for bad pixel removal, trimming, bias-dark subtraction, slit profile and flat-field corrections.
To do accurate wavelength calibration, correction for distortion and tilt for each frame, sky substraction and correction for atmospheric extinction, the IRAF package LONGSLIT was used with invoking the IDENTIFY, REIDENTIFY, FITCOORD, TRANSFORM, BACKGROUND and EXTINCTION tasks.
To obtain an instrumental response function from observed spectrophotometric flux standards, the APSUM procedure from the APEXTRACT package was used first to extract apertures of standard stars. Then the sensivity curve determined by the STANDARD and SENSFUNC procedures was applied by the CALIBRATE task to perform flux calibration for all object images. Finally the APSUM task was used to extract one-dimensional spectra from the flux calibrated images. In case that more than one exposure was obtained with the same setup for an object, the extracted spectra were co-added and a mean vector was calculated. In case of several observations with different setups (telescopes or grisms) for the same object, the data were reduced and measured independently and the more reliable values were taken.
To speed-up and facilitate the line measurements we employed dedicated command files created at the SAO using the FIT context and MIDAS command language. The procedures for the measurement of line parameters and redshifts applied were described in detail in Papers III and IV.
Results of follow–up spectroscopy {#Res_follow}
=================================
In Table \[summary\] we present the results of the observations. The 209 candidates were selected from our first and second priority samples introduced in Paper IV.
Of 169 first priority candidates, 141 objects appeared in our list as new ones. 28 objects were listed in the NED as galaxies or objects from various catalogs with known redshifts and some of them already had information on emission lines in earlier publications. All objects were included in our observing program in order to improve spectral information. Comparison of our velocities with those of galaxies with already known redshift shows acceptable consistency within the uncertainties given.
Another 40 candidates observed were taken from the list of second priority candidates. As described in Paper IV two samples were created from this list. A “random selected sample” containing randomly selected objects from the list and the “APM selected sample” which uses additional information for selection. The “random selected sample” was created to access the fraction of BCGs in the second priority list and contains 43 objects. For 26 of them the spectral data were presented in Paper IV, while for the remaining 17 candidates observed with the CAHA 2.2m telescope, the spectral information is presented here. The results of the analysis for this sample were presented in Paper IV. We found that the second priority list contains at most 10% BCG/H[ii]{}-galaxies.
The second, “APM selected” sample comprises second priority candidates which are classified as non-stellar on Palomar Sky Survey plates (PSS) in the APM database, and have blue color according to the APM color system ($(B-R) <$ 1.0). Here we give spectral data for 23 objects from this sample. Except one all are ELGs confirming the efficiency of this selection criterium to pick up the BCG/H[ii]{}-galaxies from the second priority list (cf. Paper IV).
Emission-line galaxies
----------------------
The observed emission line galaxies are listed in Table 3 containing the following information:\
[*column 1:*]{} The object’s IAU-type name with the prefix HS.\
[*column 2:*]{} Right ascension for equinox B1950.\
[*column 3:*]{} Declination for equinox B1950. The coordinates were measured on direct plates of the HQS and are accurate to $\sim$ 2$\arcsec$ (Hagen et al. [@Hagen95]).\
[*column 4:*]{} Heliocentric velocity and its r.m.s. uncertainty in km s$^{-1}$.\
[*column 5:*]{} Apparent B-magnitude obtained by calibration of the digitized photoplates with photometric standard stars (Engels et al. [@Engels94]), having an r.m.s. accuracy of $\sim$ $0\fm5$ for objects fainter than m$_{\rm B}$ = $16\fm0$ (Popescu et al. [@Popescu96]). Since the algorithm to calibrate the objective prism spectra is optimized for point sources the brightnesses of extended galaxies are underestimated. The resulting systematic uncertainties are expected to be as large as 2 mag (Popescu et al. [@Popescu96]). For about 20% of our objects, B-magnitudes are unavailable at the moment. We present for them blue magnitudes obtained from the APM database. They are marked by a “+” before the value in the corresponding column. According to our estimate they are systematically brighter by $0\fm92$ than the B-magnitudes obtained by calibration of the digitized photoplates (r.m.s. $1\fm02$). Objects referred to Popescu & Hopp ([@Popescu00]) have precise B-magnitude in Vennik, Hopp & Popescu ([@Vennik2000]). We do not list them here for the sake of homogeneity. The B-magnitude for HS 1213+3636A was determined by eye estimate as 175 and marked by a “:” as less confident.\
[*column 6:*]{} Absolute B-magnitude, calculated from the apparent B-magnitude and the heliocentric velocity. The only exception is made for HS 1213+3636B, a super-association in the nearby galaxy NGC 4214 for which the distance is known from stellar photometry (see comments below). No correction for galactic extinction is made because all objects are located at high galactic latitudes and the corrections are significantly smaller than the uncertainties of the magnitudes.\
[*column 7:*]{} Preliminary spectral classification type according to the spectral data presented in this article. BCG means a galaxy possessing a characteristic H[ii]{}-region spectrum with low enough luminosity (M$_B \geq -$20$^m$). SBN and DANS are galaxies of lower excitation with a corresponding position in line ratio diagrams, as discussed in Paper I. SBN are the brighter fraction of this type. Here we follow the notation of Salzer et al. ([@Salzer89]). Three objects were recognized as Seyfert galaxies. Two of them (HS 1317+4521B and HS 1616+3627) are Sy1 galaxies due to the presence of broad Balmer lines and of broad \[Fe[ii]{}\] emission. Our spectrum of HS 1616+3627 has insufficient quality to show this, but independent spectroscopy data of Grupe et al. ([@Grupe99]) clearly classify this object as a narrow-line Sy1 galaxy. The third one (HS 1220+3845) is a narrow-line ELG, which is classified as a Sy2 galaxy in diagnostic diagrams. Typical spectra of low-ionisation nuclear emission-line regions (LINERs) are identified for 3 galaxies. SA stands for two probable super-associations in two dwarf spiral galaxies. Six ELGs are difficult to classify. They are coded as NON.\
[*column 8:*]{} One or more alternative names, according to the information from NED. References are given to the other sources of the redshift-spectral information indicating that a galaxy is an ELG.
The spectra of all emission-line galaxies are shown in Appendix A, which is available only in the electronic version of the journal.
The results of line flux measurements are given in Table 4. It contains the following information:\
[*column 1:*]{} The object’s IAU-type name with the prefix HS. By asterisk we note the objects observed during non-photometric conditions.\
[*column 2:*]{} Observed flux (in 10$^{-16}$ergs$^{-1}$cm$^{-2}$) of the H$\beta$ line. For several objects without H$\beta$ emission line the fluxes are given for H$\alpha$ and marked by a “+”.\
[*columns 3,4,5:*]{} The observed flux ratios \[O[ii]{}\]/H$\beta$, \[O[iii]{}\]/H$\beta$ and H$\alpha$/H$\beta$.\
[*columns 6,7:*]{} The observed flux ratios \[N[ii]{}\]$\lambda$6583Å/H$\alpha$, and (\[S[ii]{}\]$\lambda$6716Å + $\lambda$6731Å)/H$\alpha$.\
[*columns 8,9,10:*]{} Equivalent widths of the lines \[O[ii]{}\]$\lambda$3727Å, H$\beta$ and \[O[iii]{}\]$\lambda$5007Å. For several objects without detected H$\beta$ emission line the equivalent widths are given for H$\alpha$ and marked by a “+”.\
Below we give comments on some specific cases:\
[*HS 0948+3723*]{} and [*HS 1430+4040*]{}: H$\alpha$-emission in these galaxies is affected by cosmic ray hits. The intensity ratio of H$\gamma$/H$\beta$ is close to the recombination one with c$_{H\beta}=$0. So we accepted for them the recombination H$\alpha$/H$\beta$ flux ratio, and corrected respectively the ratios of \[N[ii]{}\]/H$\alpha$ and \[S[ii]{}\]/H$\alpha$ in Table 4.\
[*HS 1213+3636A*]{}: this is a compact slightly elongated ($\sim$9) object at $\sim$60 to NW from the center of the star-bursting barred Magellanic type galaxy (SBm) NGC 4214 (with M$_B=-$170). Its radial velocity of 522$\pm$35 [kms$^{-1}$]{} is higher by $\sim$220 [kms$^{-1}$]{} than the systemic velocity of NGC 4214 and higher by $\sim$260 [kms$^{-1}$]{} than the velocity of H[i]{} gas in this place (McIntyre [@McIntyre97]). This implies that this compact emission region is a kinematically detached object, either a background dwarf, or more probably, a tiny “satellite” galaxy, passing by its much more massive neighbouring galaxy. For the former case its distance from the Hubble flow is 7 Mpc, M$_B=-$117: and the size $\sim$0.3 kpc. For the latter case, if HS 1213+3636A is located at the same distance as NGC 4214 (2.64 Mpc, as found through stellar photometry with HST by Drozdovsky et al. [@Drozd01]), its M$_B=-$96: and the size $\sim$0.1 kpc. The nature of this tiny galaxy can be checked in principle by the use of well known luminosity-abundance relationship for gas-rich galaxies. If it is situated far from NGC 4214 one should expect that this galaxy should have metallicity Z $\leq$ 1/30 Z$_{\odot}$. On the other hand, if HS 1213+3636A is connected to NGC 4214, one can speculate that this companion formed of gas pulled out of the parent SBm galaxy as a result of its strong interaction with other galaxy. In this case the metallicity of this companion should be close to that of NGC 4214, that is known to be $\sim$1/4–1/5 Z$_{\odot}$ (e.g., Kobulnitsky & Skillman [@KS96]). Thus, metallicity determination of this dwarf with strong starburst will probably resolve the dilemma.\
[*HS 1213+3636B*]{}: this well elongated object ($\sim$12) is projected onto the middle part ($\sim$46, or $\sim$0.6 kpc to NWW from the center) of the same galaxy NGC 4214. It is at $\sim$25 to the South from HS 1213+3636A. Our velocity differs by only $\sim$60 [kms$^{-1}$]{} from the NGC 4214 systemic velocity and is well consistent with the H[i]{} velocity of 260 [kms$^{-1}$]{} at this place from McIntyre ([@McIntyre97]). From our data it follows that HS 1213+3636B is a super-association (SA) in a spiral arm of NGC 4214. Its M$_B=-$98: and the size $\sim$0.15 kpc. The spectra of both A and B objects were acquired in one slit, so their large velocity difference is out of doubt.\
[*HS 1311+3628*]{}: this elongated object ($\sim$11) is projected onto the southern edge ($\sim$50 to the South from the center) of Im galaxy UGC 8303 (Holmberg VIII). A large difference of its radial velocity (1094$\pm$12 [kms$^{-1}$]{}) with that of the systemic velocity of the dwarf irregular UGC 8303 (944$\pm$5 [kms$^{-1}$]{}from Huchra et al. [@Huchra95]) and the local H[i]{} velocity in this region ($\sim$945 [kms$^{-1}$]{}) from the H[i]{} map by Thean et al. ([@Thean97]) (where this galaxy by error is called UGC 8314), implies that this object probably does not belongs to UGC 8303 and is a nearby separate star-forming dwarf. We observed this object twice on different nights and with different setups. Both spectra have the same redshifts within the observational uncertainties. HS 1311+3628 is therefore not an H[ii]{}-region in UGC 8303 as suggested by Popescu et al. ([@Popescu96]).\
[*HS 1311+3924*]{}: this is the irregular (probably merging) galaxy UGC 8315 and member of a group, with a radial velocity of 1215 [kms$^{-1}$]{} according to Garcia et al. ([@Garcia93]). We got the spectrum of only one of two bright knots in the galaxy elongated body, namely the NE one. While its velocity 1401$\pm$43 differs by 186 [kms$^{-1}$]{}from that given by Garcia et al. ([@Garcia93]) the combined uncertainty of latter value is not well constrained. A long-slit spectrum along the major axis will be helpful for understanding the nature of this object.\
[*HS 1423+3945*]{}: the radial velocity of this object is very close to that of UGC 9242 (our 1480$\pm$35 versus 1440$\pm$11 [kms$^{-1}$]{} in the RC3 catalog) and therefore the object is very probably an SA in this edge-on dwarf spiral.\
[*HS 1242+4058A*]{}: The nearby BCG HS 1242+4058B (cf. Paper II) is separated by $\sim$4 from HS 1242+4058A and the radial velocities differ by $\approx$300 km/s. They are probably not associated with each other.\
[*HS 1317+4521B*]{}: The nearby BCG-candidate ($\sim$10 to SW) HS 1317+4521A is not observed yet.\
[*HS 1646+4003B*]{}: its faint companion ($\sim$30 to SSW), HS 1646+4003A was earlier identified as NON (cf. Paper II).\
[*HS 0935+4135, HS 0958+3654, HS 1025+3707B, HS 1027+4728, HS 1206+4012, HS 1236+4532, HS 1433+4245, HS 1514+4602*]{}: In all these galaxies the only emission line suitable for redshift measurement is H$\alpha$. A conservative estimate of their radial velocity uncertainty is 300 kms$^{-1}$.\
For some of the galaxies either the snap-shot spectra had strong enough emission lines and a well detected \[O[iii]{}\]$\lambda$4363Å or they were reobserved later with better S/N ratio. For these galaxies we determined metallicities and the following galaxies appeared to have O/H $\le$ 1/10 (O/H)$_{\odot}$: HS 1313+4521, HS 1330+3651, HS 1334+3957, HS 1408+4227, HS 1417+4433, HS 1545+4055, HS 1650+3706, HS 1655+3845.\
For two BCGs (HS 0940+4052 and HS 1311+3628) the blue WR-bump is more or less evident from the snap-shot spectra, and for 8 more BCGs this feature is probably present.\
Several “BCG?” were listed in Table 3 with absolute magnitudes brighter than $-20$0. In fact, these M$_B$ are uncertain, since they come from the estimates from the Hamburg Quasar Survey plates. Independent CCD measurements or estimates from alternative sources show that usually they are fainter by $0\fm5-1\fm0$, and should fulfill therefore the criterion M$_B \geq -$200 (Kniazev et al. [@Kn-HS37]).
Quasars
-------
The main criteria applied to search for BCGs are a blue continuum near $\lambda$4000Å and a strong emission line — expected is \[O[iii]{}\]$\lambda$5007Å — in the wavelength region between 5000Å and the sensitivity break of the Kodak IIIa-J photoemulsion near 5400Å (see Paper I). For this reason faint QSOs with Ly$\alpha$$\lambda$1216Åredshifted to z $\sim$ 3 or with Mg[ii]{} $\lambda$2798Åredshifted to z $\sim$ 0.8 could be selected as BCG candidates. In Papers I–IV we already reported on the discovery of a number of such faint QSOs. They have been missed by the proper Hamburg Quasar Survey since the latter is restricted to bright QSOs (B $\leq 17-17.5$). Here we report on the discovery of five high-redshift faint (B $> 17.5$) QSOs. For all but one of them, we identified Ly$\alpha$$\lambda$1216Åredshifted to z $\sim$ 3 as the line responsible for its selection. One object (HS 1301+4233) shows a broad emission line tentatively identified with Mg[ii]{}$\lambda$2798Å. However this identification remains somewhat uncertain due to low signal-to-noise of the spectrum. Two of our new QSOs: HS 1224+4410 and HS 1541+4452, have recently appeared in the NED as radio sources with no redshift determination (reported as \[HVG99\] R17 in Hunter et al. ([@Hunter99]) and B3 1541+448A in Douglas et al. ([@Douglas96])). The data for all five quasars are presented in Table 5. Finding charts and plots of their spectra can be found on the www-site of the Hamburg Quasar Survey (http://www.hs.uni-hamburg.de/hqs.html).
Non-emission-line objects
-------------------------
In total, for 43 candidates no (trustworthy) emission lines were detected. We divided them into three categories.
### Absorption-line galaxies
For four non-ELG galaxies the signal-to-noise ratio of our spectra was sufficient to detect absorption lines, allowing the determination of redshifts. The data are presented in Table 6.
### Stellar objects
To separate the stars among the objects missing detectable emission lines we cross-correlated a list of the most common stellar features with the observed spectra. In total, 21 objects with definite stellar spectra and redshifts close to zero were identified. All of them were crudely classified in categories from definite A-stars to G-stars, with most of them intermediate between A and F. The data for these stars are presented in Table 7.
### Non-classified objects
There was no possibility of classifying 18 objects without emission lines. Their spectra have too low signal-to-noise ratio to detect trustworthy absorption features, or the EWs of their emission lines are too small.
Discussion {#Discussion}
==========
The fifth list
--------------
As result we have 209 observed candidates preselected on HQS objective prism plates, of which 169 were first priority candidates and 40 were second priority ones. 166 objects (79 % from 209 objects) are found to be either ELGs (161), or quasars (5).
Of 161 ELGs 98 galaxies (61 %) were classified based on the character of their spectra and their luminosity as H[ii]{}/BCGs or probable BCGs.
Two of them (HS 1213+3636A and HS 1311+3628) are low-mass neighbours/satellites of the dwarf spiral NGC 4214 and of the Im galaxy UGC 8303. As the discussion of the local environment of BCGs is out of the scope of this paper, we refer to a recent analysis of this issue by Pustilnik et al. ([@Pustilnik01]).
HS 1213+3636B and HS 1423+3945 are SAs in the dwarf spirals NGC 4214 and UGC 9242.
Six ELGs are difficult to classify at all due to their poor signal-to-noise spectra. Six more ELGs were classified as Active Galactic Nuclei (AGNs): 3 as Seyfert galaxies and 3 as LINERs. The remaining 49 ELGs are objects with low excitation: either starburst nuclei galaxies (SBN and probable SBN) or their lower mass analogs – dwarf amorphous nuclear starburst galaxies (DANS or probable DANS).
By keeping a high fraction of BCGs ($\sim$ 62 % among the first priority candidates) we continue to have a high efficiency of discovery new BCGs, which is the main goal of the HSS. Since the completeness of the BCG sample under construction is an important parameter for many follow-up statistical studies, we observed a randomly selected sample of candidates from our list of second priority candidates. As discussed already in Paper IV at most 10 % of them turned out to be BCGs. Using additional information from the APM is an efficient means to pick up these BCGs among the second priority candidates: among 23 objects observed 10 BCGs (43 %) were discovered (Table \[summary\]).
Summary of the present status of the survey
-------------------------------------------
Summarizing the results of the Hamburg/SAO survey presented in Papers I through V, we discovered altogether from the 1-st priority candidates 433 new emission-line objects (25 of them are QSOs), and for 85 known ELGs we got quantitative data for their emission lines. At the moment the total number of confident or probable blue compact/low-mass H[ii]{}-galaxies reaches 360. Relative to all observed 493 ELGs the fraction of BCGs (360/493 or 73 %) demonstrates the high efficiency of the survey to find this type of galaxies. 21 more new BCGs and 20 other type ELGs are found among the second priority candidates. To estimate the total number of BCGs in the HSS zone we should count new BCGs expected from the remaining candidates and those selected in the HSS, but not observed by us since they already were known from other surveys. Thus we expect the total number of BCGs in this sky region to be $\sim$500. This will be the largest homogeneous BCG sample in both hemispheres.
Conclusions
===========
We made follow-up spectroscopy of the fifth list of candidates from the Hamburg/SAO Survey for ELGs. Summarizing the results of the spectroscopy, the analysis of spectral information and the discussion above we draw the following conclusions:
- The intended methods to detect ELG candidates on the plates of the Hamburg Quasar Survey give a reasonably high detection rate of emission-line objects. In total, out of both priority categories, we observed 209 objects among which we found 166 emission-line objects corresponding to a detection rate of $\sim$ 79 %.
- Besides the ELGs we found also 5 new quasars, mostly with Ly$\alpha$ in the wavelength region $4950-5100$Å (z $\sim$ 3), near the red boundary of the IIIa-J photoplates.
- The high fraction of BCG/H[ii]{} galaxies among all observed ELGs (about 61 % in this paper) is in line with our main goal – to pick up efficiently a statistically well selected and deep BCG sample in the sky region under analysis.
This work was supported by the grant of the Deutsche Forschungsgemeinschaft No. 436 RUS 17/77/94. U.A.V. is very grateful to the staff of the Hamburg Observatory for their hospitality and kind assistance. Support by the INTAS grant No. 96-0500 was crucial to proceed with the Hamburg/SAO survey declination band centered on +37.5$^{\circ}$. SAO authors acknowledge also partial support from the INTAS grant No. 97-0033. We note that the use of APM facility was extremely valuable for selection methodology of additional candidates to BCGs from the 2-nd priority list. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We have also used the Digitized Sky Survey, produced at the Space Telescope Science Institute under government grant NAG W-2166.
[99]{}
Bohlin, R. C. 1996, AJ, 111, 1743 Douglas, J. N., Bash, F. N., Bozyan, F. A., Torrence, G. W., & Wolfe, C. 1996, AJ, 111, 1945 Drozdovsky, I., et al. 2001, in Bad Honnef conference “Dwarf galaxies and their environment”, Jan. 23–27, 2001 Engels, D., Cordis, L., & Köhler, T. 1994, Proc. IAU Symp. 161, ed. H. T. MacGillivray (Kluwer: Dordrecht), 317 Garcia, A. M., Bottinelli, L., Garnier, R., Gouguenheim, L., & Paturel, G. 1993, A&AS, 97, 801 Grupe, D., Beuermann, K., Mannheim, K., & Thomas H.-C. 1999, A&A, 350, 895 Hagen, H.-J., Groote, D., Engels, D., & Reimers, D. 1995, A&AS, 111, 195 Hopp, U., Engels, D., Green, R., et al. 2000, A&AS, 142, 417 ([**Paper III**]{}) Huchra, J. P., Geller, M. J., & Corwin, H. G. Jr. 1995, ApJS, 99, 391 Hunter, D., van Woerden, H., & Gallagher, J. 1999, AJ, 118, 2184 Kniazev, A. Y., Engels, D., Pustilnik, S. A. et al. 2001a, A&A, 366, 771 ([**Paper IV**]{}) Kniazev, A. Y., Pustilnik, S. A., Ugryumov A.V. et al. 2001b, in preparation Kobulnicky, H.A., Skillman, E.D. 1996, ApJ, 471, 211 Markarian, B. E., Lipovetsky, V. A., & Stepanian, J. A. 1983, Afz, 19, 29 McIntyre, V. J. 1997, Publ. Astron. Soc. Australia, 14, 122 Oke, J. B. 1990, AJ, 99, 1621 Pesch, P., Stephenson, C. B., & MacConnell, D. J. 1995, ApJS, 98, 41 Popescu, C. C., & Hopp, U. 2000, A&AS, 142, 247 Popescu, C. C., Hopp, U., Hagen, H.-J., & Elsässer, H. 1996, A&AS, 116, 43 Pustilnik, S. A., Engels, D., Ugryumov, A. V., et al. 1999, A&AS, 135, 299 ([**Paper II**]{}) Pustilnik, S. A., Kniazev, A. Y., Lipovetsky, V. A., & Ugryumov, A. V. 2001, A&A, in press = astro-ph/0104334 Salzer, J. J., MacAlpine, G. M., & Boroson, T. A. 1989, ApJS, 70, 479 Salzer, J. J., Moody, J. W., Rosenberg, J. L., Gregory, S. A., & Newberry, M. V. 1995, AJ, 109, 2376 Stepanian, J. A. 1994, Proc. IAU Symp. 161, ed. H. T. MacGillivray (Kluwer: Dordrecht), 731 Thean, A. H. C., Mundell, C. G., Pedlar, A., & Nicholson, R. A. 1997, MNRAS, 290, 15 Ugryumov, A. V., Pustilnik, S. A., Lipovetsky, V. A., Izotov, Yu. I., & Richter, G. M. 1998, A&AS, 131, 295 Ugryumov, A. V., Engels, D., Lipovetsky, V. A., et al. 1999, A&AS, 135, 511 ([**Paper I**]{}) Vennik, J., Hopp, U., & Popescu, C. C. 2000, A&AS, 142, 399
HSSV\_Tab3.tex HSSV\_Tab4.tex HSSV\_Tab5.tex HSSV\_Tab6.tex HSSV\_Tab7.tex
[^1]: Deceased 1996 September 22.
[^2]: MIDAS is an acronym for the European Southern Observatory package – Munich Image Data Analysis System.
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---
abstract: 'We investigate the fermions of the standard model without a Higgs scalar. Instead, we consider a non-local four-quark interaction in the tensor channel which is characterized by a single dimensionless coupling $f$. Quantization leads to a consistent perturbative expansion for small $f$. The running of $f$ is asymptotically free and therefore induces a non-perturbative scale $\Lambda_{ch}$, in analogy to the strong interactions. We argue that spontaneous electroweak symmetry breaking is triggered at a scale where $f$ grows large and find the top quark mass of the order of $\Lambda_{ch}$. We also present a first estimate of the effective Yukawa coupling of a composite Higgs scalar to the top quark, as well as the associated mass ratio between the top quark and the W boson.'
author:
- 'Jan-Markus Schwindt, Christof Wetterich'
title: 'Asymptotically free four-fermion interactions and electroweak symmetry breaking'
---
*Institut für Theoretische Physik, Philosophenweg 16, 69120 Heidelberg, Germany*
*E-mail: Schwindt@thphys.uni-heidelberg.de, C.Wetterich@thphys.uni-heidelberg.de*
Introduction
============
The large hadron collider (LHC) will soon test the mechanism of spontaneous electroweak symmetry breaking. It is widely agreed that this phenomenon is associated to the expectation value of a scalar field which transforms as a doublet with respect to the weak SU(2)-symmetry. The origin and status of this order parameter wait, however, for experimental clarification. In particular, an effective description in terms of a scalar field does not tell us if this scalar is “fundamental" in the sense that it constitutes a dynamical degree of freedom in a microscopic theory which is formulated at momentum scales much larger than the Fermi scale. Alternatively, no fundamental Higgs scalar may be present, and the order parameter rather involves an effective “composite field". In this note we investigate the second possibility and therefore consider the fermions of the standard model without a fundamental Higgs scalar.
As long as we include only the gauge interactions of the standard model, the microscopic or classical action of such a theory does not involve any mass scale. An effective mass scale $\Lambda_{QCD}$ will only be generated by the running of the strong gauge coupling, which is asymptotically free [@asfr]. Confinement removes then the gluons and quarks from the massless spectrum. In addition, the spontaneous chiral symmetry breaking by quark-antiquark condensates would also imply electroweak symmerty breaking by a composite order parameter, in this case the chiral condensate. In this setting all particle masses would be of the order $\Lambda_{QCD}$ or zero, such that this scenario cannot explain why the top quark mass or the $W$,$Z$-boson masses are much larger than 1GeV. Furthermore, all leptons would remain massless. One concludes that any realistic model needs further interactions beyond the standard model gauge interactions. Since the strong and electroweak gauge couplings remain actually quite small at the Fermi scale of electroweak symmetry breaking, they are expected to produce only some quantitative corrections to the dominant mechanism of electroweak symmetry breaking. We will therefore neglect the gauge couplings in this paper.
Our knowledge about the effective interactions between the quarks and leptons at some microscopic or “ultraviolet" scale $\Lambda_{UV}$ is very limited. Furthermore, it is not known which scale $\Lambda_{UV}$ has to be taken. Typically, one may associate $\Lambda_{UV}$ with the scale where further unification takes place, as a grand unified scale or the Planck scale for the unification with gravity. This would suggest a very high scale , $\Lambda_{UV}\geq 10^{16}$ GeV and we will have this scenario in mind for our discussion. However, much smaller values of $\Lambda_{UV}$ are also possible. In practice, we will only assume here that $\Lambda_{UV}$ is sufficiently above the Fermi scale (say $\Lambda_{UV}>100$ TeV), such that an effective description involving only the fermions of the standard model becomes possible in the momentum range $\Lambda_{QCD}\ll |q| \ll \Lambda_{UV}$.
We will formulate our model in terms of effective fermion interactions at the scale $\Lambda_{UV}$ and restrict the discussion to a four-fermion interaction involving only the right-handed top quark and the left-handed top and bottom quarks. This is motivated by the observation that only the top quark has a mass comparable to the Fermi scale. Interactions with the other quarks and leptons are assumed to be much smaller than the top quark interactions - typically their relative suppression is reflected in the much smaller masses of the other fermions. For the discussion in this paper we omit all “light" fermions and the gauge bosons.
Since we do not know the effective degrees of freedom at the scale $\Lambda_{UV}$, the effective interaction is not necessarily local. Non-localities involving inverse powers of the exchanged momenta are typically generated by the propagators of exchanged massless particles. With the inclusion of possibly non-local interactions the limitation to an effective four-fermion interaction poses no severe restriction. Many models with additional degrees of freedom can be effectively described in this way.
Local four-fermion interactions have already been investigated earlier, for example in the models of “top quark condensation" [@TC]. By simple dimensional analysis a local interaction involves a coupling $\sim$(mass)$^{-2}$. Such models therefore exhibit an explicit mass scale in the microscopic action. It is indeed possible to obtain spontaneous electroweak symmetry breaking in this way - the prototype is the Nambu-Jona-Lasinio model [@njl]. Without a tuning of parameters the top quark mass $m_t$ turns out, however, to be of the same order as $\Lambda_{UV}$, in contradiction to the assumed separation of scales. By a tuning of parameters it is possible to obtain $m_t \ll \Lambda_{UV}$, but the issue is now similar to the “gauge hierarchy problem" in presence of a fundamental scalar field. In order to obtain $m_t \ll \Lambda_{UV}$ the microscopic effective action must be close to an ultraviolet fixed point. The necessity of tuning arises from a “relevant parameter" in the vicinity of the fixed point (in the sense of statistical physics for critical phenomena) which has a dimension not much smaller than one. A rather extensive search for possible ultraviolet fixed points for pointlike four-fermion interactions has been performed in [@gjw]. Many fixed points have been found, but all show a relevant direction with substantial dimension, and therefore the need for a parameter tuning for $m_t \ll \Lambda_{UV}$.
We will therefore concentrate in this paper on non-local effective interactions. For such interactions the coupling $\sim M^{-2}$ is replaced by $f^2/q^2$, with $q^2$ the square of some appropriate exchanged momentum and $f$ a dimensionless coupling. Interactions of this type do not involve a mass scale on the level of the microscopic action - the classical action exhibits dilatation symmetry. Still, the quantum fluctuations typically induce an anomaly for the scale symmetry, associated to the running of the dimensionless coupling $f$. This may be responsible for the generation of the Fermi scale, in analogy to the “confinement scale" $\Lambda_{QCD}$ for QCD. Since the running of dimensionless couplings is only logarithmic, this offers a chance for a large natural hierarchy $\Lambda_{UV} \gg m_t$, without tuning of parameters. We will present a model of this type.
Let us first discuss the possible tensor structures for a non-local four-fermion interaction $\sim (\bar{\psi}A\psi)^2$. The basic building block is a color singlet fermion bilinear $\bar{\psi}A\psi$, where the color indices are contracted. The tensor structure with respect to the Lorentz symmetry is determined by $A$ such that $\bar{\psi}A\psi$ is a scalar or pseudoscalar, a vector or pseudovector, or a second rank antisymmetric tensor. Interactions in the vector or pseudovector channels involve bilinears with two left-handed or two right-handed fermions, $\bar{\psi}_L \gamma^\mu \psi_L$ or $\bar{\psi}_R \gamma^\mu \psi_R$. They conserve chiral flavour symmetries which act separately on $\psi_L$ and $\psi_R$ and therefore forbid mass terms for the fermions $\sim \bar{\psi}_L \psi_R$. Interactions capable of producing masses for the top quark and W/Z-bosons of comparable magnitude must therefore involve the (pseudo)scalar or tensor channels. A non-local scalar interaction $\sim(\bar{\psi}_L \psi_R)(\bar{\psi}_R \psi_L)q^{-2}$, with $q^2$ the squared momentum in the scalar exchange channel, has very similar properties as a model with a fundamental Higgs scalar which is massless at the scale $\Lambda_{UV}$. We therefore expect the usual necessity of parameter tuning if we want to achieve a small ratio $m_t/\Lambda_{UV}$. A local coupling $(\bar{\psi}_L \psi_R)(\bar{\psi}_R \psi_L)m^{-2}$ is allowed by the symmetries and will be generated by quantum fluctuations. The interesting remaining candidate is a tensor interaction, with $A\sim [\gamma^\mu , \gamma^\nu]$. For chiral tensors no local interaction in this channel is consistent with the SU(3)$\times$SU(2)$\times$U(1) symmetries of the standard model as well as Lorentz symmetry. We will therefore investigate a model with a non-local interaction of this type.
We define the microscopic or classical action for a Lorentz invariant theory of massless interacting fermions by $S=S_2 + S_4$, defined in momentum space as $$\label{a1}
-S_2=-\int \frac{d^4 q}{(2 \pi)^4} \bigg( \bar{t}(q)\gamma^\mu q_\mu t(q)
+ \bar{b}(q)\gamma^\mu q_\mu b(q) \bigg)$$ and $$\begin{aligned}
\label{a2}
-S_4 = 4 f^2 \int \frac{d^4 q \, d^4 p \, d^4 p'}{(2 \pi)^{12}}\;
\frac{P_{kl}^*(q)}{q^4}
& \bigg\{ & \left[ \bar{t}(q+p)\sigma_+^k t(p)\right] \;
\left[ \bar{t}(p')\sigma_-^l t(p'+q)\right] \\ \nonumber
&+& \left[ \bar{t}(q+p)\sigma_+^k b(p)\right] \;
\left[ \bar{b}(p')\sigma_-^l t(p'+q)\right] \bigg\}. \end{aligned}$$ Here $t$ and $b$ are Dirac spinor fields describing the top and bottom quark, respectively. (The theory can be easily extended to all three generations of quarks and also to leptons.) Contracted indices are summed. The $3 \times 3$ matrix $P(q)$ involves the spacelike indices $k$,$l$ and is defined by $$\label{a3}
P_{kl}(q)=-(q_0^2 + q_j q_j)\delta_{kl} + 2q_k q_l - 2i \epsilon_{klj}q_0 q_j$$ It has the properties $$\label{a4}
P_{kl}P_{lj}^* = q^4 \delta_{kj}, \qquad P_{kl}^*(q)=P_{lk}(q).$$ The non-local character of the interaction arises from the factor $1/q^4$.
In a standard spinor basis in which $\psi=\left( \begin{array}{c}\psi_L \\ \psi_R
\end{array}\right)$, $\bar{\psi}=\psi^\dagger \gamma^0=(\bar{\psi}_R,\bar{\psi}_L)$, the $4 \times 4$ matrices $\sigma_\pm^k$ are defined in terms of the Pauli matrices $\tau^k$, $$\label{a5}
\sigma_+^k = \left( \begin{array}{cc} \tau^k & 0 \\ 0 & 0 \end{array} \right), \qquad
\sigma_-^k = \left( \begin{array}{cc} 0 & 0 \\ 0 & \tau^k \end{array} \right).$$ The fermion bilinears $\bar{\psi}\sigma_+ \psi \sim \bar{\psi}_R \sigma_+ \psi_L$ and $\bar{\psi}\sigma_- \psi \sim \bar{\psi}_L \sigma_+ \psi_R$ therefore mix left- and right-handed spinors and violate the chiral symmetry that would protect the top quark from acquiring a mass. The matrices $\sigma_\pm$ correspond to appropriate projections of the commutator $[\gamma^\mu, \gamma^\nu]$ on left (right) handed spinors, such that eq. (\[a2\]) indeed describes a tensor exchange interaction (cf. the appendix A). The Lorentz-invariance of the interaction can be checked explicitly.
Furthermore, the action (\[a1\]), (\[a2\]) has a global SU(2)$\times$U(1) symmetry - the remnant of the electroweak gauge symmetry in the limit of neglected gauge couplings. This symmetry forbids mass terms for the quarks, such that a mass term can be generated only by spontaneous symmetry breaking. Similarly, the interaction is invariant under the color symmetry SU(3), with implicitly summed color indices in the bilinears. We note that only the left-handed bottom quarks are involved in the interaction and we will therefore omit the right-handed bottom quark together with the other light quarks and the leptons. Similar to the photon-exchange description of the non-local Coulomb interaction we may obtain $S_4$ from the exchange of chiral tensor fields [@cw1], according to the Feynman diagrams in fig. 1. The Lorentz symmetry of $S_4$ becomes more apparent in this formulation. A summary of the properties of the associated tensor fields and a proof of equivalence of a local theory with massless chiral tensor fields with our non-local fermion interaction (\[a2\]) is provided in appendix A. In this paper we will not use tensor fields and concentrate on the purely fermionic action (\[a1\]), (\[a2\]). In particular, this avoids the delicate issues of consistency of the quantization of chiral antisymmetric tensors [@cw2]. One can write a consistent functional integral and therefore a consistent quantum field theory based on the action (\[a1\]), (\[a2\]). (Anomaly cancellation involves the omitted light quarks and leptons.)
(400,120)(0,0)
(50,20)(70,60) (70,60)(50,100) (150,20)(130,60) (130,60)(150,100) (70,60)(130,60)[3]{} (55,100)\[tl\][$t_L$]{} (55,20)\[bl\][$t_R$]{} (145,100)\[tr\][$t_R$]{} (145,20)\[br\][$t_L$]{} (100,65)\[b\][$B^{+0}$]{}
(250,20)(270,60) (270,60)(250,100) (350,20)(330,60) (330,60)(350,100) (270,60)(330,60)[3]{} (255,100)\[tl\][$b_L$]{} (255,20)\[bl\][$t_R$]{} (345,100)\[tr\][$t_R$]{} (345,20)\[br\][$b_L$]{} (300,65)\[b\][$B^{++}$]{}
This paper is organized as follows. In sect. 2 we compute the running of the dimensionless coupling $f$ and demonstrate that it is asymptotically free. In consequence, a non-perturbative “chiral" scale $\Lambda_{ch}$ is generated where the running coupling $f$ grows large. Sect. 3 evaluates the effective interactions in the scalar and vector channels which are induced in one loop-order. We find that the induced interaction in the scalar channel $\sim f^4$ also gets large once the chiral scale is approached. As a consequence of this large interaction spontaneous electroweak symmetry breaking can be generated similar to the NJL-model [@njl]. We compute the resulting top quark mass $m_t$ in sect. 4, using a Schwinger-Dyson [@sd] or gap equation. We find $m_t \approx \Lambda_{ch}$. In sect. 5 we introduce a composite Higgs field by “partial bosonization" of the interaction in the scalar channel. We compute the running of its Yukawa coupling $h$ to the top quark. The latter is directly related to the mass ratio $m_W/m_t=g_W/(\sqrt{2}h)$, with $m_W$ and $g_W$ the mass and gauge coupling of the $W$ boson. In sect. 6 we present our conclusions.
Asymptotic Freedom
==================
The dimensionless coupling constant $f$ is the only free parameter in our model. We will show in this section that the corresponding renormalized running coupling is asymptotically free in the ultraviolet. On the other side, there is a characteristic infrared scale $\Lambda_{ch}$ where the coupling $f$ grows large. This is in complete analogy to QCD, where $\Lambda_{QCD}$ corresponds to the scale where the strong gauge coupling grows large. By “dimensional transmutation" we may therefore trade $f$ for the mass scale $\Lambda_{ch}$. Besides this mass scale the model has no free dimensionless parameter. In particular, we will argue in sect. 4 that our model leads to spontaneous breaking of the SU(2)$\times$U(1) symmetry. The Fermi scale $\varphi$ characterizing the electroweak symmetry breaking must be proportional to $\Lambda_{ch}$, $\varphi = c_w \Lambda_{ch}$, with a dimensionless proportionality coefficient $c_w$ that is, in principle, calculable without involving a further free parameter. If the model leads to a composite Higgs scalar, its mass in units of $\varphi$, $M_H/\varphi$, as well as its effective Yukawa coupling to the top quark, $h=m_t/\varphi$, are calculable quantities in our model.
In order to investigate the running of $f$, we wish to compute the effective action $\Gamma$ corresponding to the classical action (\[a1\]),(\[a2\]), to one-loop order. The one-loop correction reads $$\label{c1}
\Delta\Gamma^{(1l)}= \frac{i}{2} \; {\rm Tr} \ln S^{(2)},$$ where the field dependent inverse propagator $S^{(2)}$ is defined as the second functional derivative of the action with respect to the quark fields, $$\label{c2}
(S^{(2)})^{cc'}_{AB,\alpha\beta}(p,p')=-\frac{\delta}{\delta\Psi^{c'}_{B\beta}(p')}
\frac{\delta}{\delta\Psi^{c}_{A\alpha}(p)}\;(S_2+S_4).$$ Here $\Psi$ are the quark fields, with color indices $c,c'$, flavor indices $A,B$ (taking values $t,b,\bar{t},\bar{b}$), spinor indices $\alpha,\beta$ and momenta $p,p'$. The trace Tr involves an integral over momentum and summation over all kinds of indices. We write $S^{(2)}=P_0 + F$, where $P_0$ is the “free" part of the propagator, derived from $S_2$, $$\label{c3}
(P_0)^{cc'}_{AB,\alpha\beta}(p,p') = \left[
(-\gamma^\mu p_\mu)_{\alpha\beta}(\delta_{A\bar{t}}\delta_{Bt}
+\delta_{A\bar{b}}\delta_{Bb})
+ (\gamma^\mu p_\mu)_{\beta\alpha}(\delta_{At}\delta_{B\bar{t}}
+\delta_{Ab}\delta_{B\bar{b}}) \right]\;
\delta_{cc'}\delta(p-p'),$$ and $\delta(p-p')=(2 \pi)^4 \delta^4(p_\mu - p'_\mu)$. We treat $F \sim f^2$ as a perturbative correction due to $S_4$. Then $\Delta \Gamma ^{(1l)}$ reads, up to an “infinite constant", $$\label{c4}
\Delta\Gamma^{(1l)}=
\frac{i}{2} \; {\rm Tr} \left( P_0^{-1}\,F \right) - \frac{i}{4}\; {\rm Tr}
\left( P_0^{-1}\,F \,P_0^{-1}\,F \right) + \cdots ,$$ where the dots stand for neglected six-quark and higher interactions. We display the explicit expressions for $F$, as well as the formal expressions for the first two terms on the r.h.s. of eq. (\[c4\]), in appendix B.
(400,120)(0,0)
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(340,10)(340,35) (340,35)(340,85) (340,85)(340,110) (340,60)(25,90,270)[3]{} (335,20)\[r\][$b_L$]{} (335,60)\[r\][$t_R$]{} (335,100)\[r\][$b_L$]{} (370,60)\[l\][$B^{++}$]{}
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The nonlocal factors $\sigma(P^*(q)/q^4)\sigma$ appearing in the interaction (\[a2\]) are attached in different ways to the fermion lines. We represent them as dashed lines in the corresponding Feynman diagrams in figs. 2-4. Our notation recalls the one-to-one correspondence with the exchange of the associated chiral tensor fields. The diagrams in fig. 2 correspond to the first term $\sim F$ in eq. (\[c4\]), while figs. 3,4 reflect the terms $\sim F^2$.\
Our one-loop calculation results in an effective action $$\label{eff1}
\Gamma = \Gamma_2 + \Gamma_4^{(T)} + \Gamma_4^{(V)} + \Gamma_4^{(S)},$$ with a kinetic term $$\label{eff2}
\Gamma_2 = -\int \frac{d^4q}{(2 \pi)^4}\bigg( Z_L (\bar{t}_L \gamma^\mu q_\mu t_L
+ \bar{b}_L \gamma^\mu q_\mu b_L) + Z_R \bar{t}_R \gamma^\mu q_\mu t_R \bigg)$$ and three different types of quartic interactions $\Gamma_4$. (Note the relative minus sign between $\Gamma$ and the classical action $S$ which is chosen in order to make analytic continuation to the euclidean effective action straightforward by replacing $q_0 \rightarrow i q_0$.) The “tensorial part", $\Gamma_4^{(T)}$, corresponds to the exchange of a tensor field and has the form of the classical interaction $S_4$. The other two parts, $\Gamma_4^{(V)}$ and $\Gamma_4^{(S)}$, correspond to the exchange of vectors and scalars, respectively, and will be further discussed in sect. 3.
The momentum intergals in the loop expansion (\[c4\]) are logarithmically divergent, both in the ultraviolet and the infrared. We regularize our model in the ultraviolet by a suitable momentum cutoff $\Lambda_{UV}$. In order to investigate the flow of effective couplings we also introduce an effective infrared cutoff $k$. The effective action $\Gamma_k$ depends then on the scale $k$, resulting in an effective coupling $f(k)$. Instead of the infrared scale induced by non-vanishing external momenta for the vertices (as most common for perturbative renormalization) we introduce the cutoff by modifying the quark propagators\
$\slash\hspace{-0.2cm} q \,^{-1}
\rightarrow \slash\hspace{-0.2cm} q \,(q^2+k^2)^{-1}$. This is a procedure known from functional renormalization. Indeed, $\Gamma_k$ may be considered as the “average action" or “flowing action" [@cwfr]. The precise implementation of the infrared cutoff is not important and does not influence the one-loop beta function for the running coupling $f(k)$ that we will derive next.
The fermion anomalous dimensions arise from eq. (\[c9\]) or fig. 2. Our computation in the purely fermionic model yields the same result as in ref. [@cw1], where it was computed in the equivalent model with chiral tensors, namely $$\label{c11}
\eta_R \equiv -k \frac{\partial}{\partial k} Z_R =-\frac{3}{2 \pi^2}f^2 \; ,
\qquad \eta_L \equiv -k \frac{\partial}{\partial k} Z_L =-\frac{3}{4 \pi^2}f^2.$$
Only the terms visualized diagramatically in fig. 3, which are $\sim A_2$ in eq. (\[c10\]), generate the same tensor structure as the classical interaction (\[a2\]). They provide the one-loop correction to the inverse chiron propagator, i.e. to the matrix $P_{kl}$, and one obtains $$\Gamma_4^{(T)}= -S_4 (P_{kl}^* \rightarrow Z_+ P_{kl}^*).$$ Again our fermionic computation reproduces the computation in ref. [@cw1] with chiral tensors. The correction results in an anomalous dimension for the chiron, $$\label{c12}
\eta_+ \equiv -k \frac{\partial}{\partial k} Z_+ = \frac{f^2}{2 \pi^2}.$$ There are no further one loop corrections in the tensor exchange channel. Among the quartic corrections shown in fig. 4, the first four terms in eq. (\[c10\]) contribute to an interaction channel which is equivalent to the exchange of a vector particle. These diagrams will be evaluated in the next section. Similarly, the fifth and sixth term in eq. (\[c10\]) contribute to an interaction with a tensor structure different from the classical action (\[a2\]). It can be interpreted as an effective scalar exchange and will also be discussed in the next section.
The running of the renormalized coupling $f^2$ (to one-loop order) is therefore given by the anomalous dimensions of the fermions and the correction to $P_{kl}$, $$\label{c13}
k \frac{\partial}{\partial k} f^2 = (\eta_R + \eta_L + \eta_+)f^2
= - \frac{7}{4 \pi^2} f^4.$$ This implies that the interaction is asymptotically free. The solution to the renormalization group equation (\[c13\]) is $$\label{c14}
f^2(k)=\frac{4 \pi^2}{7 \ln(k/\Lambda_{ch})},$$ where the “chiral scale" $\Lambda_{ch}$ is the scale at which $f^2/4\pi$ becomes much larger than one. This is completely equivalent to the result of ref. [@cw1]. The asymptotic freedom of the chiral fermion-tensor interaction is simply taken over to the non-local four-fermion interaction.
Induced scalar and vector interactions
======================================
(450,240)(0,0)
(30,130)(30,155) (30,155)(30,205) (30,205)(30,230) (110,130)(110,155) (110,155)(110,205) (110,205)(110,230) (30,205)(70,180)[3]{} (110,205)(70,180)[3]{} (70,180)(110,155)[3]{} (70,180)(30,155)[3]{} (25,140)\[r\][$t_L$]{} (25,180)\[r\][$t_R$]{} (25,220)\[r\][$t_L$]{} (115,140)\[l\][$t_L$]{} (115,180)\[l\][$t_R$]{} (115,220)\[l\][$t_L$]{} (40,202)\[bl\][$B^{+0}$]{} (100,202)\[br\][$B^{+0}$]{}
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In this section we evaluate the diagrams in fig. 4, representing the terms $\sim A_1$ in eq. (\[c10\]). We begin with the fifth diagram with all external momenta set to zero. (The contribution of the sixth diagram is equivalent, with $t_L$ substituted by $b_L$.) The contribution to $\Gamma$ is $$\label{d1}
\Delta\Gamma^{(1)} = 16 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{P_{kl}^*(q)}{q^4}
\frac{P_{mn}^*(q)}{q^4} \left[ \bar{t}\sigma^m_+ \frac{(-\slash\hspace{-0.2cm} q)}{q^2}
\sigma^l_- t \right] \left[\bar{t}\sigma^n_- \frac{\slash\hspace{-0.2cm} q}{q^2}
\sigma^k_+ t \right].$$ This may be rewritten in terms of Weyl spinors $$\label{d2}
\Delta\Gamma^{(1)} = 16 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{P_{kl}^*(q)}{q^4}
\frac{P_{mn}^*(q)}{q^4} \left[ t^\dagger_R \tau^m \frac{(-\slash\hspace{-0.2cm} q)}{q^2}
\tau^l t_R \right] \left[ t^\dagger_L \tau^n \frac{\slash\hspace{-0.2cm} \bar{q}}{q^2}
\tau^k t_L \right],$$ where now $$\label{d3}
\slash\hspace{-0.2cm} q = q_0 + q_i \tau^i, \qquad
\slash\hspace{-0.2cm} \bar{q} = q_0 - q_i \tau^i.$$ With the identity $$\label{d4}
P_{kl}^*(q) P_{mn}^*(q) \left[ \tau^m \slash\hspace{-0.2cm} q \tau^l \right]_{\alpha\beta}
\left[ \tau^n \slash\hspace{-0.2cm} \bar{q} \tau^k \right]_{\lambda\eta}
=5 q^4 \left[ \slash\hspace{-0.2cm} q \right]_{\alpha\beta}
\left[ \slash\hspace{-0.2cm} \bar{q} \right]_{\lambda\eta}
+ 4 q^6 \delta_{\alpha\eta}\delta_{\beta\lambda}$$ this simplifies to $$\label{d5}
\Delta\Gamma^{(1)} = 16 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{1}{q^8}
\left( -5 [t^\dagger_R \slash\hspace{-0.2cm} q t_R]
[t^\dagger_L \slash\hspace{-0.2cm} \bar{q} t_L]
+4 q^2 [t^{c \dagger}_R t^{c'}_L][t^{c' \dagger}_L t^c_R] \right).$$ The relative minus sign is due to an exchange of Grassmann variables. Note the different color structure of the second term. The momentum integral can be evaluated by analytic continuation to Euclidean space, $q_0=iq_{E0}$. Introducing the infrared cutoff in the quark propagators (i.e. substituting $q^{-4}\rightarrow(q^2+k^2)^{-2}$), one obtains $$\label{d6}
\Delta\Gamma^{(1)} = \frac{f^4}{4 \pi^2 k^2} \left( 5 g_{\mu\nu}[t^\dagger_R \tau^\mu t_R]
[t^\dagger_L \bar{\tau}^\nu t_L] -
16 [t^{c \dagger}_R t^{c'}_L][t^{c' \dagger}_L t^c_R] \right).$$ Using the identities $$\label{d7}
(\tau^\mu)_{\alpha\beta}(\bar{\tau}_\mu)_{\lambda\eta}
= -2 \delta_{\alpha\eta}\delta_{\beta\lambda}$$ and $$\label{d7a}
\delta_{cd'}\delta_{c'd} = \frac{1}{3}\delta_{cd}\delta_{c'd'}
+ \frac{1}{2} T^z_{cd}T^z_{c'd'}$$ (where $T^z$ are the eight Gell-Mann matrices) this can be further simplified to $$\label{d8}
\Delta\Gamma^{(1)} = \frac{f^4}{2 \pi^2 k^2}[\bar{t}_R t_L][\bar{t}_L t_R]
+\frac{3 f^4}{4 \pi^2 k^2}[\bar{t}^c_R T^z_{cd} t^d_L]
[\bar{t}^{c'}_L T^z_{c'd'} t^{d'}_R].$$ The total contribution to $\Gamma$ from the fifth and sixth diagram in fig. 4 reads $$\label{d10}
\Gamma_4^{(S)}= \Delta\Gamma^{(1)} + \Delta\Gamma^{(1)}(t_L \rightarrow b_L).$$
The first term in eq. (\[d8\]) is equivalent to the four-fermion interaction generated at tree level by a Yukawa interaction with a scalar field $\phi$, which has a mass $k$ and a Yukawa coupling $\bar{h}$ given by $$\label{d9}
\bar{h}^2= \frac{f^4}{2 \pi^2}.$$ This scalar transforms as a singlet under color and a doublet with respect to the electroweak interactions. It therefore has the same quantum numbers as a (composite) Higgs doublet. The second term in eq. (\[d8\]) corresponds to the exchange of a second scalar which is an octet with respect to color.
Finally we evaluate the first four diagrams of fig. 4, which generate an interaction equivalent to the exchange of a vector particle. The expression for the first diagram is $$\label{e1}
\Delta\Gamma^{(2)} = 8 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{P_{kl}^*(q)}{q^4}
\frac{P_{mn}^*(q)}{q^4} \left[ \bar{t}\sigma^l_- \frac{\slash\hspace{-0.2cm} q}{q^2}
\sigma^m_+ t \right] \left[\bar{t}\sigma^n_- \frac{\slash\hspace{-0.2cm} q}{q^2}
\sigma^k_+ t \right].$$ This can again be rewritten in terms of Weyl spinors $$\label{e2}
\Delta\Gamma^{(2)} = 8 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{P_{kl}^*(q)}{q^4}
\frac{P_{mn}^*(q)}{q^4} \left[ t^\dagger_L \tau^l \frac{\slash\hspace{-0.2cm} \bar{q}}{q^2}
\tau^m t_L \right] \left[ t^\dagger_L \tau^n \frac{\slash\hspace{-0.2cm} \bar{q}}{q^2}
\tau^k t_L \right].$$ With the identities $$\label{e4}
P_{kl}^*(q) P_{mn}^*(q)\left[ \tau^l \slash\hspace{-0.2cm}\bar{q} \tau^m \right]_{\alpha\beta}
\left[ \tau^n \slash\hspace{-0.2cm} \bar{q} \tau^k \right]_{\lambda\eta}
=5 q^4 \left[ \slash\hspace{-0.2cm} \bar{q} \right]_{\alpha\beta}
\left[ \slash\hspace{-0.2cm} \bar{q} \right]_{\lambda\eta}
+ 4 q^6 (\delta_{\alpha\beta}\delta_{\eta\lambda}-\delta_{\alpha\eta}\delta_{\beta\lambda})$$ and $$\label{e7}
(\bar{\tau}^\mu)_{\alpha\beta}(\bar{\tau}_\mu)_{\lambda\eta}
= -2 (\delta_{\alpha\beta}\delta_{\eta\lambda}-\delta_{\alpha\eta}\delta_{\beta\lambda})$$ the expression (\[e2\]) simplifies to $$\label{e5}
\Delta\Gamma^{(2)} = 8 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{1}{q^8}
\left( 5 [t^\dagger_L \slash\hspace{-0.2cm} \bar{q} t_L]
[t^\dagger_L \slash\hspace{-0.2cm} \bar{q} t_L]
-2 q^2 [t^\dagger_L \bar{\tau}^\mu t_L][t^\dagger_L \bar{\tau}_\mu t_L] \right).$$ With an infrared cutoff $k$ in the quark propagator the momentum integral yields $$\label{e8}
\Delta\Gamma^{(2)} = \frac{3f^4}{8 \pi^2 k^2}[\bar{t}_L \gamma^\mu t_L]
[\bar{t}_L \gamma_\mu t_L].$$ This is equivalent to the four-fermion interaction generated at tree level by the exchange of a vector field with mass $k$ and coupling $\tilde{g}$ given by $$\tilde{g}^2 = \frac{3f^4}{8 \pi^2}.$$ The evaluation of the diagrams 3 and 4 is equivalent, with two or four external $t_L$ fields substituted by $b_L$.
The expression for the second diagram, in terms of Weyl spinors, is $$\label{f2}
\Delta\Gamma^{(3)} = 16 i f^4 \int\frac{d^4 q}{(2 \pi)^4} \frac{P_{kl}^*(q)}{q^4}
\frac{P_{mn}^*(q)}{q^4} \left[ t^\dagger_R \tau^m \frac{\slash\hspace{-0.2cm} q}{q^2}
\tau^l t_R \right] \left[ t^\dagger_R \tau^k \frac{\slash\hspace{-0.2cm} q}{q^2}
\tau^n t_R \right].$$ Identities similar to eqs. (\[e4\]) and (\[e7\]) result in $$\label{f8}
\Delta\Gamma^{(3)} = \frac{3f^4}{4 \pi^2 k^2}[\bar{t}_R \gamma^\mu t_R]
[\bar{t}_R \gamma_\mu t_R].$$ The “vector exchange" interaction generated at one loop level can be summarized as $$\begin{aligned}
\label{f10}
\Gamma_4^{(V)} = \frac{3f^4}{8 \pi^2 k^2} & \bigg( [\bar{t}_L \gamma^\mu t_L]
[\bar{t}_L \gamma_\mu t_L] + 2 [\bar{b}_L \gamma^\mu t_L]
[\bar{t}_L \gamma_\mu b_L] \qquad
\\ \nonumber & + [\bar{b}_L \gamma^\mu b_L]
[\bar{b}_L \gamma_\mu b_L] + 2 [\bar{t}_R \gamma^\mu t_R]
[\bar{t}_R \gamma_\mu t_R]\bigg).\end{aligned}$$
Electroweak symmetry breaking
=============================
The presence of the effective Yukawa interaction in eq. (\[d8\]) indicates the possibility of spontaneous symmetry breaking and an analogue of the Higgs mechanism. If we neglect for $k$ close to $\Lambda_{ch}$ all interactions except for the scalar singlet exchange channel in the first term of eq. (\[d8\]), we may characterize the effective action by a “scalar four-fermion coupling" $\lambda(k)$, $$\label{g1a}
\Gamma_k^{(S)}=\frac{\lambda}{2}\int d^4 x
\bigg[ (\bar{\psi}\psi)^2 - (\bar{\psi}\gamma^5 \psi)^2
\bigg] = 2 \lambda (\bar{\psi}_L \psi_R)(\bar{\psi}_R \psi_L)$$ with $$\label{g1b}
\lambda(k)=\frac{f^4(k)}{4 \pi^2 k^2}.$$ In the limit where the momentum dependence of the scalar four-fermion interactions can be neglected this can be interpreted as an effective NJL model. Here the role of the UV cutoff in the effective NJL model is played by the scale $k$, since the computation of $\Gamma_k^{(S)}$ has involved only fluctuations with momenta $q^2>k^2$ (due to the infrared cutoff) such that the remaining fluctuations with $q^2<k^2$ still have to be included. It is well known that for $\lambda k^2$ exceeding a critical value the NJL model leads to spontaneous symmetry breaking.
For a rough estimate of the top quark mass $m_t$ induced by spontaneous electroweak symmetry breaking we consider the Schwinger Dyson equation $$\label{g1}
-\gamma^\mu q_\mu + m_k \gamma^5 = -\gamma^\mu q_\mu +
2i \lambda(k) \int_{p^2<k^2}\frac{d^4 p}{(2 \pi)^4}
\frac{\gamma^\mu p_\mu - 6 m_k \gamma^5}{p^2 + m_k^2}.$$ Setting $q=0$ and performing the momentum integral yields the gap equation for the top quark mass $m_k$ $$\label{g2}
m_k = \frac{3\lambda(k)}{4 \pi^2 } \bigg( k^2 -
m_k^2 \ln \frac{k^2 + m_k^2}{m_k^2} \bigg) m_k .$$ We have indicated the scale $k$ for $m_k$ in order to recall that the solution of the gap equation will depend on the choice of the scale $k$ which we use as an UV cutoff for the effective NJL model. Of course, for an exact treatment the physical top quark mass should no longer depend on $k$.
Let us next discuss the solution of eq. (\[g2\]). Obviously, $m_k=0$ is always one possible solution. The expression in brackets is always $\leq k^2$. If we write $$\label{g3}
\alpha(k)=\lambda(k)k^2,$$ we see that non-zero solutions for $m_k$ (indicating spontaneous symmetry breaking) occur for $$\label{g4}
\alpha(k) > \frac{4 \pi^2}{3}.$$ Once the condition (\[g4\]) is obeyed, one finds indeed a non-zero $m_k$ obeying $$\label{xx}
\frac{m_k^2}{k^2} \ln \bigg( 1+\frac{k^2}{m_k^2} \bigg) = 1-\frac{4\pi^2}{3\alpha(k)}.$$ Since $\alpha(k)\sim f^4(k)$ grows arbitrarily large as $k$ approaches $\Lambda_{ch}$ and $f(k)$ diverges, the condition (\[g4\]) is always fulfilled for $k$ sufficiently close to $\Lambda_{ch}$.
For a qualitative investigation we first use the one loop result $\alpha = f^4 / 4 \pi^2$ and replace in eq. (\[c14\]) $k \rightarrow (k^2 + c^2 m_k^2)^{1/2}$ with $c$ of order 1. This is motivated by the effective infrared cutoff $\sim m_k$ which stops or slows down the running of $f^2$. In fig. 5 we show $m_k / \Lambda_{ch}$ as a function of $k / \Lambda_{ch}$ for $c=1$. After the onset of a nonzero $m_k$ for $k \approx \Lambda_{ch}$ we find first a very rapid increase until $m_k$ settles at $m_k=\Lambda_{ch}/c$ for $k \rightarrow 0$. It is obvious that $\Lambda_{ch}$ sets the scale for the top quark mass. This coincides with the result of a two-loop Schwinger-Dyson equation in a formulation with chiral tensor fields in [@cw1]. Indeed, since the generation of $\alpha$ is a one-loop effect, and the gap equation (\[g1\]) involves a further loop, the generation of the top quark mass consists of two nested one-loop integrals, which are equivalent to a two-loop integral.
The solution of eq. (\[xx\]) for $k \rightarrow 0$, $m_k \neq 0$ corresponds to an asymptotic value which is obtained from the condition $\alpha(m_k , k \rightarrow 0)\rightarrow\infty$.
As an alternative to the ad hoc insertion of the quark mass cutoff in eq. (\[c14\]) we may take into account the additional infrared cutoff due to $m_k$ by replacing in the quark propagator $\slash\hspace{-0.2cm} q \,^{-1} \rightarrow \slash\hspace{-0.2cm} q \,
(q^2+k^2+m_k^2)^{-1}$. As a consequence, the anomalous dimensions involve a threshold function $s(m_k^2 / k^2)$, $$\label{xz}
\eta_L = - \frac{3}{4 \pi^2} f^2 s(m_k^2 / k^2), \qquad
\eta_R = - \frac{3}{2 \pi^2} f^2 s(m_k^2 / k^2), \qquad
\eta_+ = \frac{1}{2 \pi^2} f^2 s(m_k^2 / k^2) ,$$ given by $$\label{xa}
s(m_k^2 / k^2) = \frac{k^2}{k^2 + m_k^2}.$$ The one loop expression for $\lambda$ (\[g1b\]) can be replaced by a flow equation $$\label{x1}
k \frac{\partial}{\partial k}\lambda = - \frac{f^4}{2 \pi^2}\frac{k^2}{(k^2+m_k^2)^2}
+(\eta_L + \eta_R)\lambda .$$ (For $m_k=0$, $\eta_{L,R}=0$ and constant $f$ this reproduces eq. (\[g1b\]).) Nonzero $m_k$ results in a threshold function for the running of $\alpha$, $$\nonumber
k \frac{\partial}{\partial k} \alpha =
(2+\eta_L +\eta_R) \alpha - \frac{f^4}{2 \pi^2} \tilde{s}(m_k^2/k^2),$$ $$\label{x2}
\tilde{s}(m_k^2/k^2) = \frac{k^4}{(k^2+m_k^2)^2}.$$ We show the running of $f^2$ and $\alpha$ in fig. 6, for different values of $m_k^2/\Lambda_{ch}^2$. Again we stop the flow at some value of $k=\Lambda_{UV}^{SDE}$ and solve the Schwinger-Dyson equation with this UV cutoff. The value of $k$ for which the SDE yields the given $m_k^2/\Lambda_{ch}^2$ is indicated in fig. 6 by a dot. The dots in fig. 5 show the corresponding $k$ dependence of $m_k / \Lambda_{ch}$.
 
We are aware that our treatment of the infrared cutoff is only qualitative. While the one-loop form of the flow equation can be maintained if we interpret the flow as an approximation to the exact functional renormalization group equation for the average action [@cwfr], the approximation of the exact inverse quark and chiron propagators by $Z_{L,R}\slash\hspace{-0.2cm} q$ and $Z_+ P_{kl}(q)$ with momentum independent $Z$-factors becomes questionable in the presence of large anomalous dimensions.
Composite Higgs scalar
======================
An elegant method for the description of composite particles in the context of functional renormalization uses partial bosonization [@giesw]. It is based on the observation that an interaction of the type (\[g1a\]) can be described by the exchange of a composite scalar field. Indeed, one may add to the flowing action $\Gamma_k$ a scalar piece, with $\bar{\varphi}$ a complex doublet scalar field $$\label{s1}
\Gamma_k^{(S)} = \int_x \left\{ Z_\varphi \partial^\mu \bar{\varphi}^\dagger \partial_\mu
\bar{\varphi} + \bar{m}_\varphi^2 \bar{\varphi}^\dagger \bar{\varphi}
+ \bar{h}(\bar{\psi}_R \bar{\varphi}^\dagger \psi_L - \bar{\psi}_L \bar{\varphi}\psi_R)
\right\}$$ “Integrating out" the scalar field by solving its field equation as a functional of $\psi$ and reinserting the solution into eq. (\[s1\]) yields eq. (\[g1a\]), with $\lambda$ dependent on the squared exchanged momentum in the scalar channel $$\label{s2}
\lambda(q) = \frac{\bar{h}^2}{2 (Z_\varphi q^2 + \bar{m}_\varphi^2)}.$$ As long as the momentum dependence of the effective scalar exchange vertex is not resolved (as in our computation where the vertex is only evaluated for $q^2=0$), one may take arbitrary $Z_\varphi$. We will therefore replace $\Gamma_k^{(S)}$ in eq. (\[g1a\]) by eq. (\[s1\]), and replace the flow of $\lambda$ in eq. (\[x1\]) by $$\label{s3}
k \frac{\partial}{\partial k} \bar{h}^2 =
-\frac{f^4}{\pi^2} \tilde{s}(m_t^2/k^2)\frac{\bar{m}_\varphi^2}{k^2}
+ 2 \bar{h}^2 .$$
At this stage our reformulation precisely reproduces the results in sect. 4. The rule for the replacement of the flow of $\lambda(q)$ by a running of the renormalized Yukawa coupling and dimensionless mass term $$\label{s5}
h^2 = \frac{\bar{h}^2}{Z_\varphi} , \qquad
\tilde{m}_\varphi^2=\bar{m}_\varphi^2/(Z_\varphi k^2)$$ is to adjust the flow of $h^2$ and $\tilde{m}_\varphi^2$ such that the flow of $\lambda(q)$ is reproduced, with $$\label{s3a}
\lambda(q) = \frac{h^2}{2 k^2} \left( \frac{q^2}{k^2} + \tilde{m}_\varphi^2 \right)^{-1}.$$ For the approximately scale invariant situation for small coupling one expects that the relative split into $q^2$-dependent and $q^2$-independent parts does not depend much on $k$. We therefore make the approximation that the flow of $\tilde{m}_\varphi^2$ receives no contribution from bosonization, such that $$\partial_k \lambda = \partial_k (h^2/k^2)/(2 \tilde{m}_\varphi^2)$$ or $$k \partial_k h^2 - 2 h^2 = 2 \tilde{m}_\varphi^2 k^3 \partial_k \lambda
= -\frac{f^4}{\pi^2} \tilde{s}(m_t^2/k^2)\tilde{m}_\varphi^2 + (\eta_L+\eta_R)h^2.$$ The effective initial value $\tilde{m}_\varphi^2(\Lambda)$ can be computed by evaluating the momentum dependence of the four-fermion interaction in the scalar channel. At present, it remains a free parameter. In a more complete calculation one should also evaluate the diagrams in fig. 4 for non-zero external momenta and choose the flow of $h^2$ and $\tilde{m}_\varphi^2$ such that the flow of the vertex $\lambda(q)$ is well approximated by eq. (\[s3a\]). We expect that the resulting flow for $\tilde{m}_\varphi^2$ will be attracted to an approximate fixed point. We note that for a fixed (point) value of $\tilde{m}_\varphi^2$ the mass term $\bar{m}_\varphi^2$ decreases roughly $\sim k^2$.
(320,120)(0,0)
(20,40)(45,40) (45,40)(95,40) (95,40)(120,40) (45,40)[2]{} (95,40)[2]{} (70,40)(26,180,0) (70,40)(24,180,0) (20,37)\[tl\][$t_L$]{} (45,37)\[t\][$\bar{h}$]{} (70,37)\[t\][$t_R$]{} (95,37)\[t\][$\bar{h}$]{} (120,37)\[tr\][$t_L$]{} (70,70)\[b\][$\varphi$]{}
(300,10)(300,35) (300,35)(300,85) (300,85)(300,110) (300,35)(250,35)[3]{} (250,85)(300,85)[3]{} (250,60)(25,90,270) (250,60)(25,270,90) (304,22)\[tl\][$t_L$]{} (304,60)\[l\][$t_R$]{} (304,98)\[bl\][$t_L$]{} (275,32)\[t\][$B^{+0}$]{} (275,88)\[b\][$B^{+0}$]{} (279,60)\[l\][$t_L$]{} (222,60)\[r\][$t_R$]{}
(10,115)\[tl\][(a)]{} (190,115)\[tl\][(b)]{} (160,60)\[t\][$\hat{=}$]{}
The formulation in terms of a composite scalar field allows an inclusion of the scalar field fluctuations as well. This adds new diagrams, which can be viewed as a partial resummation of the box diagrams in fig. 4. In particular, the quark wave function renormalizations $Z_{L,R}$ get additional contributions from the scalar exchange diagrams in fig. 7 (with scalars represented as double lines), as well known from computations in the standard model. In consequence, the anomalous dimensions acquire a scalar contribution with opposite sign to the tensor contribution. $$\nonumber
\eta_L = -\frac{3 f^2}{4 \pi^2} s(\tilde{m}_t^2)
+ \frac{h^2}{16 \pi^2} s_\varphi(\tilde{m}_\varphi^2,\tilde{m}_t^2),$$ $$\label{s4}
\eta_R = -\frac{3 f^2}{2 \pi^2} s(\tilde{m}_t^2)
+ \frac{h^2}{8 \pi^2} s_\varphi(\tilde{m}_\varphi^2,\tilde{m}_t^2),$$ with $\tilde{m}_t^2=m_t^2/k^2$ and $$\label{s6}
s_\varphi(\tilde{m}_\varphi^2,\tilde{m}_t^2)=
\frac{1+\ln[(1+\tilde{m}_t^2+\tilde{m}_\varphi^2)/(1+\tilde{m}_t^2)]}
{1+\tilde{m}_t^2+\tilde{m}_\varphi^2} .$$ For $\tilde{m}_t=0$ the scalar correction to $\eta_{L,R}$ becomes approximately $\Delta\eta_{L,R}\sim h^2/\tilde{m}_\varphi^2\sim f^4$. As long as $f^2$ remains small, the effective two-loop contribution corresponding to scalar exchange (cf. fig. 7b) remains subleading. However, for large $f^2$ the scalar contribution to $\eta_{L,R}$ may become important and modify the anomalous dimensions towards positive values. This could contribute to a final stop of the increase of $f^2$ which obeys now $$k \frac{\partial}{\partial k} f^2 = (\eta_R + \eta_L + \eta_+) f^2,$$ with $\eta_{L,R}$ given by eq. (\[s4\]).
(200,100)(0,0)
(75,50)[2]{} (125,50)[2]{} (100,50)(25,180,0) (100,50)(25,0,180) (35,51)(75,51) (35,49)(75,49) (125,51)(165,51) (125,49)(165,49) (145,55)\[b\][$\varphi$]{} (55,55)\[b\][$\varphi$]{} (72,47)\[tr\][$\bar{h}$]{} (128,47)\[tl\][$\bar{h}$]{} (100,22)\[t\][$t_R$]{} (100,78)\[b\][$t_L , b_L$]{}
We next turn to the scalar contributions to the running of $\bar{m}_\varphi^2$ and $Z_\varphi$. The fermion loop correction to the inverse scalar propagator, as depicted in fig. 8, results in the standard result for the anomalous dimension of the scalar field, $$\label{s7}
k \frac{\partial}{\partial k}Z_\varphi = -\frac{3}{8 \pi^2}\bar{h}^2 s(\tilde{m}_t^2),
\qquad \eta_\varphi = - k \frac{\partial}{\partial k}\ln Z_\varphi
= \frac{3}{8 \pi^2}h^2 s(\tilde{m}_t^2) .$$ Due to the Yukawa coupling $\bar{h}^2$ a positive non-vanishing $Z_\varphi$ is generated, even if it is absent at some microscopic scale. This produces a pole in the scalar propagator for $q^2 = -m_\varphi^2 =
-\bar{m}_\varphi^2 / Z_\varphi$, such that the scalar behaves as a dynamical particle. The contribution of the quark loop shown in fig. 8 to the flow of the scalar mass term is $\sim h^2$. Within functional renormalization it has been investigated in [@cwtop] and one finds with our cutoff in the fermion propagator $$\label{s8}
k \frac{\partial}{\partial k} \tilde{m}_\varphi^2 = \frac{3}{2 \pi^2}h^2
\hat{s}(\tilde{m}_t^2),$$ $$\nonumber
\hat{s}(\tilde{m}_t^2)=\ln\frac{\tilde{\Lambda}^2}{k^2} -1-\ln(1+\tilde{m}_t^2).$$ The momentum integral for the contribution of fig. 8 to the flow of $\bar{m}_\varphi^2$ has been cut at some effective scale $\tilde{\Lambda}$. Here $\tilde{\Lambda}$ is a characteristic scale below which the description of the flow in terms of scalar fluctuations becomes a reasonable approximation. This should be somewhat above $\Lambda_{ch}$, but the precise value remains somewhat arbitrary without additional computations. (In any case the use of an improved infrared cutoff within functional renormalization would remove this spurious dependence on a scale.)
For sufficiently large $h^2$ the flow (\[s8\]) drives $\tilde{m}_\varphi^2$ to negative values, indicating the onset of spontaneous symmetry breaking. This is the same physics that is responsible for the nontrivial solutions of the Schwinger-Dyson equation for $m_t$ in the preceding section. Indeed, for a qualitative picture we can take for $k<k_0$ a constant $\bar{h}^2$ and $\hat{s}$, and neglect $\eta_\varphi$ as well as the contribution from bosonization. This replaces eq. (\[s8\]) by $$k \frac{\partial}{\partial k} \bar{m}_\varphi^2 =
\frac{3 \hat{s} \bar{h}^2}{2 \pi^2 k^2}.$$ For the solution one finds the critical value $\bar{h}_c^2=(4 \pi^2/3 \hat{s}) \bar{m}_\varphi^2(k_0)/k_0^2$ for which $\bar{m}_\varphi^2$ reaches zero for $k \rightarrow 0$. Inserting $\hat{s}=\frac{1}{2}$ and using $$\label{s8a}
\alpha = \frac{\bar{h}^2 k_0^2}{2 \bar{m}_\varphi^2(k_0)},$$ this indeed corresponds to $\alpha_c = 4 \pi^2/3$. The vacuum expectation value $\varphi_0 = Z_\varphi^{1/2} \bar{\varphi}_0$, with $\bar{\varphi}_0$ the location of the minimum of the scalar effective potential, differs from zero if $\bar{m}_\varphi^2$ gets negative. For the computation of its value, which should be $\varphi_0=$175 GeV in a realistic model, one further needs to compute the quartic scalar self interaction $\lambda_\varphi$. We can adjust the value of $\Lambda_{ch}$ (or the ultraviolet value $f^2(\Lambda_{UV})$) such that the Fermi scale $\varphi_0$ has the correct value.
A particularly interesting quantity is the renormalized Yukawa coupling $h$ (\[s5\]). Its value for $k=0$ determines the top quark mass in terms of the Fermi scale $$\label{s9}
m_t = h(k=0)\varphi_0 = h_t \varphi_0.$$ In other words, the knowledge of $h_t=h(k=0)$ is equivalent to a determination of the mass ratio $m_t/m_W$, where we use $m_W=(g_W/ \sqrt{2})\varphi_0$, with $g_W$ the known weak gauge coupling ($g_W^2/4\pi \approx 0.033$). The observational value is $h_t=0.98$. While the scale $\varphi_0$ is set by dimensional transmutation and therefore a free parameter, a computation of $h_t$ is equivalent to a parameter-free “pre"-diction for the mass ratio $m_t/m_W$.
In our approximation we can infer the flow equation for $h^2$ from eq. (\[s3\]), $$\begin{aligned}
\label{s10}
k \frac{\partial}{\partial k} h^2 &=&
( 2+ \eta_L + \eta_R + \eta_\varphi ) h^2 - \frac{f^4}{\pi^2}
\tilde{m}_\varphi^2 \tilde{s}(\tilde{m}_t^2) \\ \nonumber
&=& 2 h^2 -\frac{9}{4 \pi^2} f^2 h^2 s(\tilde{m}_t^2) + \frac{3}{16 \pi^2}h^4
\left[ 2s(\tilde{m}_t^2) + s_\varphi(\tilde{m}_\varphi^2,\tilde{m}_t^2)\right]
-\frac{f^4}{\pi^2} \tilde{m}_\varphi^2 \tilde{s}(\tilde{m}_t^2) .\end{aligned}$$
 
We have solved the flow equations numerically until the onset of spontaneous symmetry breaking at $k_{SSB}$ where $\bar{m}_\varphi^2(k_{SSB})=0$. For $k>k_{SSB}$ one has $\tilde{m}_t=0$ such that many threshold functions equal one. We display the running of $h$ and $f$ in fig. 9. For $k<k_{SSB}$ one has to continue the flow in the regime with spontaneous symmetry breaking and non-zero $\varphi_0(k)$, as well known from many studies of functional flow equations [@cwtop; @btw]. We will not do so here.
In the present approximation to the flow equations we observe an increase of $f$ and $h$ to very large values as $m_\varphi^2$ approaches zero. We do not expect our approximations to remain valid for large couplings, even though the one-loop form of the functional flow equations is exact [@cwfr]. One of the main shortcomings is the inaccurate truncation of the general form of momentum dependence for the fermion propagators and non-local interactions in a region where the anomalous dimensions $\eta_{L,R,+}$ are of the order one. For example, the quartic interactions for the bare quarks in the tensor channel replaces in our approximation $P^*_{kl}/q^4\rightarrow
P^*_{kl}/(Z_+ q^4)$ in eq. (\[a2\]). We may consider an effective momentum dependence of $Z_+$ given by the anomalous dimension $\eta_+(q^2)$ evaluated for $k=\sqrt{q^2}$, i.e. $Z_+\sim (q^2)^{-2\eta_+(q^2)}$ - this would indeed be a valid approximation for small $|\eta_+|\ll 1$ and $q^2 \gg k^2$. However, for $\eta_+=1$ the momentum dependence of $Z_+$ would effectively cancel the nonlocality $\sim 1/q^2$ of the interaction. For a quasi-local interaction in the tensor channel we expect strong modifications of the flow of $f$ - for example, it could reach a fixed point $f_*$, replacing in the interaction $f^2/q^2 \rightarrow f_*^2/(c \Lambda_{ch}^2)$. (In the language of chiral tensor fields this would correspond to the generation of a non-local mass for the “chirons" [@cw2].)
For a realistic model of electroweak interactions the increase of the Yukawa coupling should stop or be substantially slowed down in the vicinity of its final value at $k=0$, say for $1 \apprle h \apprle 2$. For the corresponding region in the scale $k$, $1.1 < k/\Lambda_{ch} < 1.4$ we find values $0.8 < f^2/(2 \pi^2)=\eta_+ < 2.4$. (We use $\tilde{m}_\varphi^2(\Lambda)=0.1$ for the quantitative estimates.) In view of our discussion, it seems not unreasonable that the non-perturbative infrared physics stops the further increase of $f$ and $h$ in this range. If this happens, the ratio $m_t/m_W$ may come out in a reasonable range. A typical value of the scale for non-perturbative physics where the increase of $f$ and $h$ stops, may be $k_{np}= 1.3 \Lambda_{ch}$. At this scale the quantity $\alpha$ (cf. eq. (\[s8a\]) with $k_0=k_{np})$ has reached a value $\alpha(k_{np})\approx 10$, not too far from the critical value in the Schwinger-Dyson approach. It is well conceavable that the top quark mass is substantially below the scale $k_{np}$, such that the region of strong interactions may correspond to the multi-TeV-scales and not disturb too much the LEP precision tests of the electroweak theory.
Our computation of the flow of $f$ and the Yukawa coupling $h$ has further substantial quantitative uncertainties. For example, we have neglected effects from the interactions in sect. 3 that correspond to the exchange of color-octet scalars or vectors. This may be motivated for the region of $k$ where $m_\varphi^2$ is already small, since a resonance type behavior and spontaneous symmetry breaking is only expected in the scalar singlet sector. On the other hand, these contributions may still play a role in the interesting region where the increase of $f$ may stop.
Conclusions
===========
We conclude that models for quarks and leptons with a non-local four-fermion interaction in the tensor channel appear to be promising candidates for an understanding of the electroweak symmetry breaking. No fundamental Higgs scalar is needed. The theory is asymptotically free and generates an exponentially small “chiral scale" $\Lambda_{ch}$ where the dimensionless coupling $f$ grows large. Furthermore, a strong interaction in the scalar channel is generated at scales where $f$ is large. A solution of Schwinger-Dyson equations suggests that this interaction triggers the spontaneous breaking of the electroweak symmetry at a scale determined by $\Lambda_{ch}$. This would solve the gauge hierarchy problem.
We have also introduced a composite Higgs scalar and investigated the flow of its mass. We find indeed spontaneous symmetry breaking with a Fermi scale somewhat below the chiral scale. Our first attempt of an estimate of the Yukawa coupling of the top quark is encouraging, yielding a reasonable range for the ratio $m_t/m_W$.
The model has also other interesting features. It was shown [@cw1] that the flavor and CP-violation is completely described by the CKM matrix [@ckm]. Masses of the light quarks and leptons arise from appropriate four-fermion couplings [@fm]. First phenomenological constraints from LEP precision tests and the anomalous magnetic moment of the muon have been computed [@cw1].
At the present stage the understanding of the strong interactions around the scale $\Lambda_{ch}$, which would be a few TeV in a realistic model, is still in its infancy. For this reason our estimate of $m_t/m_W$ is only very crude. Nevertheless, no free parameter enters in the determination of this ratio in our model. If the strong interactions can be understood quantitatively, our model leads to a unique “pre"-diction of $m_t/m_W$. It would also predict the masses and interactions of the composite scalar fields.
[**Appendix A: Non-local four-fermion interaction and chiral tensor fields**]{}\
In this appendix we show the equivalence of the microscopic action with a theory of chiral tensor fields [@cw1; @cw2; @cwjs; @cht; @cht2]. This is similar to the equivalence of quantum electrodynamics to a theory with a non-local four-fermion interaction, generated by integrating out the photon. We may start from the non-local four-fermion interaction and obtain the chiral tensor theory by a Hubbard-Stratonovich transformation. We proceed here in the opposite way, starting with a chiral tensor theory and integrating out the chiral tensors.
Starting point is the theory of chiral antisymmetric tensor fields investigated in ref. [@cw1]. We consider a complex antisymmetric tensor field $\beta_{\mu\nu}=-\beta_{\nu\mu}$ which is a doublet of weak isospin and carries hypercharge $Y=1$. The field can be decomposed into two parts which correspond to irreducible representations of the Lorentz group. The two parts are the “chiral" components of $\beta_{\mu\nu}$: $$\label{b1}
\beta_{\mu\nu}^\pm = \frac{1}{2}\beta_{\mu\nu}\pm \frac{i}{4}\epsilon_{\mu\nu}
\,^{\rho\sigma}\beta_{\rho\sigma}.$$ These components can be written as $4 \times 4$ matrices acting in the space of Dirac spinors via $$\label{b2}
\beta_\pm = \frac{1}{2}\beta_{\mu\nu}^\pm \sigma^{\mu\nu},$$ where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu , \gamma^\nu]$. The matrix $\beta_+$ ($\beta_-$) acts only on left-handed (right-handed) fermions, i.e. $$\label{b3}
\beta_\pm=\beta_\pm \frac{1 \pm \gamma^5}{2}.$$ One may introduce an interaction between the fermions and the chiral tensors: $$\label{b4}
-\mathcal{L}_{ch}= \bar{u}_R F_U \tilde{\beta}_+ q_L
-\bar{q}_L F_U^\dagger \bar{\tilde{\beta}}_+ u_R
+\bar{d}_R F_D \bar{\beta}_- q_L -\bar{q}_L F_D^\dagger \beta_- d_R
+\bar{e}_R F_L \bar{\beta}_- l_L -\bar{l}_L F_L^\dagger \beta_- e_R .$$ Here the chiral couplings $F_{U,D,L}$ are $3 \times 3$ matrices in generation space, $q_L$ are the left-handed quark doublets, $u_R$ ($d_R$) are the right-handed up-type (down-type) quarks, $l_L$ are the left-handed lepton doublets, $e_R$ are the right-handed electron-type leptons, and we defined $$\label{b5}
\bar{\beta}_\pm = \frac{1}{2} (\beta_{\mu\nu}^\pm)^* \sigma^{\mu\nu}
=\bar{\beta}_\pm \frac{1 \mp \gamma^5}{2}, \qquad
\tilde{\beta}_+ = -i \beta^T_+ \tau_2, \qquad
\bar{\tilde{\beta}}_+ = - \tau_2 \bar{\beta}_+,$$ where the transposition $\beta^T$ and the Pauli matrix $\tau_2$ act in weak isospin space, i.e. on the two components of the weak doublet $\beta_{\mu\nu}^+$.
The fields $\beta_{\mu\nu}^\pm$ can be represented by three-vectors $B_k^\pm$, $$\label{b6}
\beta_{jk}^+ = \epsilon_{jkl}B_l^+ , \qquad \beta_{0k}^+ = iB_k^+ , \qquad
\beta_{jk}^- = \epsilon_{jkl}B_l^- , \qquad \beta_{0k}^- = -iB_k^-.$$ Rewriting the kinetic term $$\label{b7}
-\mathcal{L}_{kin}= \frac{1}{4}\bigg( (\partial^\rho \beta^{\mu\nu})^*
\partial_\rho \beta_{\mu\nu} - 4 (\partial_\mu \beta^{\mu\nu})^*
\partial_\rho \beta^\rho \,_\nu \bigg)$$ in terms of the $B$-fields and in Fourier space gives $$\label{b8}
-S_{kin}= \int \frac{d^4 q}{(2 \pi)^4}\bigg[ (B_k^+)^*(q) P_{kl}(q)B_l^+(q)
+ (B_k^-)^*(q)P_{kl}^*(q)B_l^-(q)\bigg].$$ The propagator $$\label{b9}
P_{kl}=-(q_0^2 + q_j q_j)\delta_{kl} + 2q_k q_l - 2i \epsilon_{klj}q_0 q_j$$ has the properties $$\label{b10}
P_{kl}P_{lj}^* = q^4 \delta_{kj}, \qquad P_{kl}^*(q)=P_{lk}(q).$$
In the following we ignore for simplicity all couplings except $f \equiv f_t$, i.e. we assume $$\label{b11}
F_U = \left( \begin{array}{ccc} f & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0
\end{array}\right) , \quad F_D = F_L =0 .$$ In this case, only the $B^+$ fields couple to top quarks $t$ and left-handed bottom quarks $b$. All other fields are free if we ignore gauge couplings. We denote the two weak isospin components of $B^+$ as $B^{++}$ and $B^{+0}$, since they correspond to electric charge $+1$ and $0$ after electroweak symmetry breaking. The action for the chiral interactions is then $$\label{b12}
-S_{ch}= 2f \; \int d^4 x \,
\left[ -B_k^{+0}\,\bar{t}\,\sigma_+^k t + B_k^{++}\,\bar{t}\,\sigma_+^k b
+(B_k^{+0})^*\, \bar{t}\,\sigma_-^k \, t - (B_k^{++})^*\, \bar{b} \,\sigma_-^k \, t \right],$$ In a spinor basis in which $$\label{b14}
\gamma^0 = -i \left( \begin{array}{cc} 0 & 1_2 \\ 1_2 & 0 \end{array}\right), \quad
\gamma^i = -i \left( \begin{array}{cc} 0 & \tau^i \\ -\tau^i & 0 \end{array}\right), \quad
\gamma^5 = -i \gamma^0 \gamma^1 \gamma^2 \gamma^3
= \left( \begin{array}{cc} 1_2 & 0 \\ 0 & -1_2 \end{array}\right)$$ the matrices $\sigma_\pm^k$ are defined in terms of the Pauli matrices $\tau^k$ as $$\label{b13}
\sigma_+^k = \left( \begin{array}{cc} \tau^k & 0 \\ 0 & 0 \end{array} \right), \qquad
\sigma_-^k = \left( \begin{array}{cc} 0 & 0 \\ 0 & \tau^k \end{array} \right).$$ In the action $-S_{ch}$, a summation over quark color is understood. The classical field equations are obtained by varying $S_B \equiv S_{kin}+S_{ch}$ with respect to the $B^+$ fields. They determine these fields as functionals of the quark fields. In momentum space, these relations are $$\begin{aligned}
B_k^{+0}(q) &=& -2f \frac{P_{kl}^*(q)}{q^4} \int \frac{d^4 k}{(2 \pi)^4}\;
\bar{t}(k)\sigma_-^l \, t(k+q), \\ \nonumber
(B_k^{+0})^*(q) &=& 2f \frac{P_{kl}(q)}{q^4} \int \frac{d^4 k}{(2 \pi)^4}\;
\bar{t}(k)\sigma_+^l \, t(k-q), \\ \nonumber
B_k^{++}(q) &=& 2f \frac{P_{kl}^*(q)}{q^4} \int \frac{d^4 k}{(2 \pi)^4}\;
\bar{b}(k)\sigma_-^l \, t(k+q), \\ \nonumber
(B_k^{++})^*(q) &=& -2f \frac{P_{kl}(q)}{q^4} \int \frac{d^4 k}{(2 \pi)^4}\;
\bar{t}(k)\sigma_+^l \, t(k-q). \end{aligned}$$ We then insert these relations into $S_B$ and obtain the action $S_4$ of the non-local four-fermion interaction given in eq. (\[a2\]). In a functional integral formulation this procedure is equivalent to performing the Gaussian integral over the $B^+$ fields.\
[**Appendix B: One loop expressions for the effective action**]{}\
In this appendix we evaluate the interaction contribution $F$ to the inverse propagator $S^{(2)}$ in eq. (\[c2\]) for the different quark types separately, ($S^{(2)}=P_0+F$). One obtains $$\begin{aligned}
\nonumber
F^{cc'}_{\bar{t}t,\alpha\beta}(p,p')=& 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
\bigg\{ \bar{t}^{c'}_\gamma(q+p')t^c_\eta(q+p) \left[ \sigma^k_{+ \alpha\eta}
\sigma^l_{- \gamma\beta} + \sigma^l_{- \alpha\eta} \sigma^k_{+ \gamma\beta}\right]
\left( -\frac{P_{kl}^*(q)}{q^4}\right)\\ \nonumber
& + \delta_{cc'}\bar{t}^{c''}_\gamma(q+p')t^{c''}_\eta(q+p) \left[ \sigma^k_{+ \alpha\beta}
\sigma^l_{- \gamma\eta} + \sigma^l_{- \alpha\beta} \sigma^k_{+ \gamma\eta}\right]
\frac{P_{kl}^*(p'-p)}{(p'-p)^4}\\ \nonumber
& + \bar{b}^{c'}_\gamma(q+p')b^c_\eta(q+p) \sigma^k_{+ \alpha\eta}
\sigma^l_{- \gamma\beta} \left( -\frac{P_{kl}^*(q)}{q^4}\right)\bigg\},\end{aligned}$$ $$\nonumber
F^{cc'}_{\bar{b}b,\alpha\beta}(p,p')= - 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
\bar{t}^{c'}_\gamma(q+p')t^c_\eta(q+p)
\sigma^l_{- \alpha\eta} \sigma^k_{+ \gamma\beta} \frac{P_{kl}^*(q)}{q^4},$$ $$\nonumber
F^{cc'}_{\bar{t}b,\alpha\beta}(p,p')= 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
\delta_{cc'} \left[ \bar{b}(q+p')\sigma_-^l t(q+p) \right]
\sigma^k_{+\alpha\beta} \frac{P_{kl}^*(p'-p)}{(p'-p)^4},$$ $$\nonumber
F^{cc'}_{\bar{b}t,\alpha\beta}(p,p')= 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
\delta_{cc'} \left[ \bar{t}(q+p')\sigma_-^l b(q+p) \right]
\sigma^k_{-\alpha\beta} \frac{P_{kl}^*(p'-p)}{(p'-p)^4},$$ $$\nonumber
F^{cc'}_{\bar{t}\bar{t},\alpha\beta}(p,p')= 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
t^{c'}_\gamma(p'-q) t^c_\eta(p+q) \left[ \sigma^k_{+ \beta\gamma}
\sigma^l_{-\alpha\eta} - \sigma^k_{+\alpha\eta} \sigma^l_{-\beta\gamma}\right]
\frac{P_{kl}^*(q)}{q^4},$$ $$\nonumber
F^{cc'}_{tt,\alpha\beta}(p,p')= 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
\bar{t}^{c'}_\gamma(p'-q) \bar{t}^c_\eta(p+q) \left[ \sigma^k_{+ \gamma\beta}
\sigma^l_{-\eta\alpha} - \sigma^k_{+\eta\alpha} \sigma^l_{-\gamma\beta}\right]
\frac{P_{kl}^*(q)}{q^4},$$ $$\nonumber
F^{cc'}_{\bar{t}\bar{b},\alpha\beta}(p,p')= - 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
t^{c'}_\gamma(p'-q) t^c_\eta(p+q) \sigma^k_{+\alpha\eta} \sigma^l_{-\beta\gamma}
\frac{P_{kl}^*(q)}{q^4},$$ $$\nonumber
F^{cc'}_{tb,\alpha\beta}(p,p')= 4f^2 \int \frac{d^4 q}{(2 \pi)^4}
\bar{t}^{c'}_\gamma(p'-q) \bar{t}^c_\eta(p+q) \sigma^k_{+ \gamma\beta}
\sigma^l_{-\eta\alpha} \frac{P_{kl}^*(q)}{q^4},$$ $$\label{c8}
F^{cc'}_{BA,\alpha\beta}(p,p')= - F^{c'c}_{AB,\beta\alpha}(p',p).$$ Inserting this into the expression (\[c4\]), we find an anomalous dimension term for $t$ and for the left-handed $b$ $$\begin{aligned}
\label{c9}
\frac{i}{2} \, {\rm Tr} \left( \frac{1}{P_0} F \right) = &
4i f^2 \int \frac{d^4 p \; d^4 q}{(2 \pi)^8}
\frac{p_\mu}{p} \frac{P_{kl}^*(q-p)}{(q-p)^4} \; \bigg\{
\bar{t}(q)\sigma_-^l \gamma^\mu \sigma_+^k t(q) \\ \nonumber
& + 2\bar{t}(q)\sigma_+^k \gamma^\mu \sigma_-^l t(q)
+\bar{b}(q)\sigma_-^l \gamma^\mu \sigma_+^k b(q)\bigg\}.\end{aligned}$$ The term $\sim F^2$ in eq. (\[c4\]) produces an effective four fermion interaction $$\begin{aligned}
\label{c10}
-\frac{i}{4}\, {\rm Tr} \left( \frac{1}{P_0} F \frac{1}{P_0} F \right) = &
8i\, f^4 \int \frac{d^4 p \; d^4 p' \; d^4 q \; d^4 q'}{(2 \pi)^{16}}
\frac{p_\mu}{p^2}\frac{p'_\mu}{p'^2} \bigg\{
\frac{P_{kl}^*(q)}{q^4}\frac{P_{mn}^*(q')}{q'^4} A_1\\ \nonumber
& -6 \frac{P_{kl}^*(p-p')}{(p-p')^4}\frac{P_{mn}^*(p-p')}{(p-p')^4}
\; {\rm tr}(\gamma^\mu \sigma_+^k \gamma^\nu \sigma_-^n) A_2 \bigg\},\end{aligned}$$ with $$\begin{aligned}
\label{c10a}
A_1 = & \bigg( \bar{t}(q'+p)\sigma_-^n \gamma^\mu \sigma_+^k t(q+p) \bigg)
\bigg( \bar{t}(q+p')\sigma_-^l \gamma^\nu \sigma_+^m t(q'+p') \bigg) \\ \nonumber
& +2 \bigg( \bar{t}(q'+p)\sigma_+^m \gamma^\mu \sigma_-^l t(q+p) \bigg)
\bigg( \bar{t}(q+p')\sigma_+^k \gamma^\nu \sigma_-^n t(q'+p') \bigg) \\ \nonumber
& +2 \bigg( \bar{b}(q'+p)\sigma_-^n \gamma^\mu \sigma_+^k t(q+p) \bigg)
\bigg( \bar{t}(q+p')\sigma_-^l \gamma^\nu \sigma_+^m b(q'+p') \bigg) \\ \nonumber
& + \bigg( \bar{b}(q'+p)\sigma_-^n \gamma^\mu \sigma_+^k b(q+p) \bigg)
\bigg( \bar{b}(q+p')\sigma_-^l \gamma^\nu \sigma_+^m b(q'+p') \bigg) \\ \nonumber
& +2 \bigg( \bar{t}(p+q)\sigma_+^k \gamma^\mu \sigma_-^n t(p+q') \bigg)
\bigg( \bar{t}(p'-q)\sigma_-^l \gamma^\nu \sigma_+^m t(p'-q') \bigg) \\ \nonumber
& +2 \bigg( \bar{t}(p+q)\sigma_+^k \gamma^\mu \sigma_-^n t(p+q') \bigg)
\bigg( \bar{b}(p'-q)\sigma_-^l \gamma^\nu \sigma_+^m b(p'-q') \bigg)\end{aligned}$$ and $$\begin{aligned}
\label{c10b}
A_2 = & \bigg( \bar{t}(q+p')\sigma_-^l t(q+p) \bigg)
\bigg( \bar{t}(q'+p)\sigma_+^m t(q'+p') \bigg) \\ \nonumber
& + \bigg( \bar{b}(q+p')\sigma_-^l t(q+p) \bigg)
\bigg( \bar{t}(q'+p)\sigma_+^m b(q'+p') \bigg). \end{aligned}$$ Color indices are suppressed, since all pairs of fermions in large brackets are color singlets.
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|
---
abstract: 'In this paper, we consider a one dimensional pursuit law with delay which is derived from traffic flow modelling. It takes the form of an infinite system of first order coupled delayed equations. Each equation describes the motion of a driver who interacts with the preceding one, taking into account his reaction time. We derive a macroscopic model, namely a Hamilton-Jacobi equation, by a homogenization process for reaction times that are below an explicit threshold. The key idea is to show, that below this threshold, a strict comparison principle holds for the infinite system. In a second time, for well-chosen dynamics and higher reaction times, we show that there exist some microscopic pursuit laws that do not lead to the previous macroscopic model.'
author:
- 'Jérémy FIROZALY, Université Paris-Est, Cermics (ENPC), F-77455 Marne-la-Vallée'
bibliography:
- 'biblio.bib'
title: 'Homogenization of a 1D pursuit law with delay and a counter-example'
---
\[section\] \[theorem\][Definition]{} \[theorem\][Lemma]{} \[theorem\][Remark]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Example]{}
**Keywords:** Hamilton-Jacobi equations, infinite system, delay time, strict comparison principle, homogenization.
Introduction
============
In the present paper we consider an infinite system of delay differential equations (DDEs). This form can especially be derived from a one dimensional pursuit law on a straight road with a very simple model. Such models, where we follow the time position of every driver are called microscopic models. The effect of drivers’ reaction times on the stability of the corresponding systems of DDEs is studied in [@atay2010complex] and [@nagatani2002physics]. Presentations of some classical microscopic traffic flow models of first order or second order are made in [@chand], [@gipps] or [@costeseque]. Conversely, macroscopic models describe the time evolution of densities in global traffic flow. We want to study if we can recover a macroscopic model from our microscopic one. Formally, it corresponds to making the interdistance between drivers go to zero or to observe the pursuit law from a height and time that go to infinity. When possible, the passage can be made rigorous by an homogenization process that will be done here in the framework of viscosity solutions of Hamilton-Jacobi equations. The interested reader is referred to the pionnering works [@LPV] or [@Evans] about homogenization of Hamilton-Jacobi equations. He can see the works of [@light] or [@richards] for the historical approach used to define and analyse macroscopic traffic flow models and links with fluid mechanics. Several examples (and counter-examples) will be exhibited throughout this paper to show that both the initial dynamics and reaction time values will have a huge influence on the homogenization process.
This paper derives from Régis Monneau’s work [@monne] (see also [@costeseque]).
Description of the model and main results
-----------------------------------------
The common velocity of each driver is supposed to be a Lipschitz continuous, bounded, nondecreasing function, $F$ say, of the distance that separates each driver from the preceding one (with $F(0)=0$). Each driver has its own reaction time and we assume that the reaction times are uniformly bounded from above and from below; we denote $\tau$ the upper bound and $\xi>0$ the lower bound. Let us study the evolution of the system during a time $T\in(0,+\infty]$ with $T>\tau$.
Given a sequence $(X_i)_{i\in \mathbb{Z}}$ of drivers’ positions on the road, the microscopic model then takes the form of an infinite system of DDEs: $$\begin{aligned}
\label{microdelay}
\frac{dX_i}{dt}(t)=F(X_{i+1}(t-\tau_i)-X_{i}(t-\tau_i)) \hspace{3 cm} (i,t)\in \mathbb{Z}\times (0,T),\end{aligned}$$ where $\tau_i$ denotes the individual reaction time.
If we suppose that we know the dynamics of the cars during the initial time interval, we can define time functions $(x_i^0)_{i\in\mathbb{Z}}$ such that: $$\begin{aligned}
\label{microinitial}
X_i(t)=x_i^0(t) \qquad \forall t\in [-2\tau,0].\end{aligned}$$
The first step is to embed the microscopic system dynamics into a single PDE. In order to go from the microscopic scale to the macroscopic one, we have to introduce a small parameter, $\varepsilon>0$ say, so as to rescale properly the function and to take into account the microscopic space and time oscillations with high frequency.
Hence, by hyperbolic change of variables, we define the function $u^\varepsilon$ as follows (the space variable will be denoted as $x$): $$\begin{aligned}
u^\varepsilon(x,t)=\varepsilon X_{\lfloor \frac{x}{\varepsilon} \rfloor}(\frac{t}{\varepsilon}).\end{aligned}$$ We suppose that there exists a Lipschitz continuous function $u_0: \mathbb{R} \times [-2\tau,0] \rightarrow \mathbb{R}$ such that: $$x_i^0(t)=\frac{u_0(i\varepsilon, t \varepsilon)}{\varepsilon} \hspace{3cm} (i,t)\in \mathbb{Z}\times [-2\tau,0],$$ and a function $\tau_0: \mathbb{R} \rightarrow [\xi,\tau]$ such that: $$\tau_i=\tau_0(i\varepsilon) \qquad \forall i\in\mathbb{Z}.$$ The rescaled model can thus be embedded into the following equation: $$\begin{aligned}
\left\{
\begin{array}{l}
\partial_t u^\varepsilon (x,t) =F\left(\frac{u^\varepsilon (x+\varepsilon,t-\varepsilon\tau_0(x))-u^\varepsilon (x,t-\varepsilon \tau_0(x))}{\varepsilon}\right) \hspace{1 cm} (x,t)\in \mathbb{R}\times (0,T), \label{exo} \\
u^\varepsilon (x,s)=u_0(x,s) \hspace{4.8 cm} (x,s) \in \mathbb{R} \times [-2\varepsilon \tau,0],
\end{array}
\right.\end{aligned}$$ with the following assumptions on $F$, $\tau_0$ and $u_0$: $$\begin{aligned}
\left\{
\begin{array}{l}
F \mbox{ is a non-decreasing, bounded and } C_F-\mbox{Lipschitz continuous function on }\mathbb{R}, \\
\overline{\tau_0(\mathbb{R})}= [\xi, \tau], \\
u_0 \mbox{ is a globally } L-\mbox{Lipschitz continuous function on }\mathbb{R} \times [-2\tau,0] . \label{assumptions}
\end{array}
\right.\end{aligned}$$ As the argument of $F$ in converges towards a space derivative for vanishing $\varepsilon$, it is natural to introduce the following dynamics in the form of a first-order Hamilton-Jacobi equation: $$\begin{aligned}
\left\{
\begin{array}{l}
\partial_t u^0 (x,t) =F(\partial_x u^0 (x,t)) \hspace{1 cm} (x,t)\in \mathbb{R}\times (0,T), \label{exo2} \\
u^0(x,0)=u_0(x,0) \hspace{2.6 cm} x \in \mathbb{R}.
\end{array}
\right.\end{aligned}$$ The first goal of the article is to prove that $u^\varepsilon$ tends locally uniformly towards $u^0$:
\[convergenceth\]$\,$\
For $\tau\in \bigg(0;\frac{1}{e C_F}\bigg)$ and under assumptions , the solution $u^\varepsilon$ to converges locally uniformly towards the (unique viscosity) solution of .
The key idea of the convergence proof relies on a strict comparison principle. This one is stated for the microscopic model for $\varepsilon=1$: $$\begin{aligned}
\left\{
\begin{array}{l}
\partial_t u (x,t) =F({u (x+1,t-\tau_0(x))-u (x,t- \tau_0(x))}) \hspace{1 cm} (x,t)\in \mathbb{R}\times (0,T), \label{exo3} \\
u (x,s)=u_0(x,s) \hspace{5 cm} (x,s) \in \mathbb{R} \times [-2 \tau;0].
\end{array}
\right.\end{aligned}$$ In contrast with the classical cases where it is sufficient to keep the initial order of the solutions, here, because of the reaction time, those solutions (namely the vehicles in traffic flow modelling) must be sufficiently spaced out at initial times in the sense made precise in the following theorem:
\[compa\]$\,$\
Under the assumptions , let $v$ and $u$ be respectively a supersolution and a subsolution of . Let us assume that there exist $\delta>0$, $t_0\in [0,T)$, $R\in[0,+\infty)$, $x_0\in\mathbb{R}$, and $\rho$ a positive, non decreasing function on $\mathbb{R}$ such that: $$\begin{aligned}
v(x,t)\geq u(x,t) \hspace{1 cm} \mbox{ for } R\leq |x-x_0|\leq R+1,\hspace{0.3cm} t\in [t_0-\tau,T) \label{hypborne}\end{aligned}$$ $$\begin{aligned}
\delta\leq (v-u)(x,t-\tau ')\leq \rho(\tau ') (v-u)(x,t) \hspace{0.8 cm} \mbox{ for } \tau '\in [0,\tau], (x,t) \in [x_0-R,x_0+R] \times [t_0-\tau,t_0], \label{eqini1}\end{aligned}$$ where $\rho$ is such that: $$\begin{aligned}
1+C_F \rho(\tau)\int_0^{\tau '}\rho(s)ds < \rho(\tau ') \hspace{1 cm} \tau ' \in [0,\tau]. \label{condrho}\end{aligned}$$ Then we have: $$\begin{aligned}
0<\delta e^{-C_F\rho(\tau)(t-t_0)}\leq v(x,t)-u(x,t) \hspace{1 cm} \mbox{ for } (x,t) \in [x_0-R,x_0+R]\times [t_0,T). \label{finalres}\end{aligned}$$
The theorem is still valid when $R=+\infty$ (substituting $[x_0-R,x_0+R]$ by $\mathbb{R}$), is defined on $\emptyset$.
All the proofs involved in this paper remain the same when dealing with stochastic reaction times provided those are uniformly bounded from above and from below.
The homogenization proof is different from the one performed in [@forca] on generalized Frenkel-Kontorova models, concerning an infinite system of particles that interact with a finite number of neighbours and that are subject to a periodic potential (this system takes the form of non-linear ODEs). The convergence proof in [@forca] relies on the construction of hull functions in a periodic setting. The case where particles interact with an infinite number of other particles is covered in [@disloc]. Other homogenization results in traffic flow modelling without considering the reaction time can be found in [@salaz1] when adding a junction condition in the microscopic model and in [@salaz2] for a second order microscopic model.
The second goal of the article is to provide a “counter-example to homogenization” in the following sense: we will display an explicit model for which $u^\varepsilon$ does not converge locally uniformly towards $u^0$ in the special case where all the drivers have a common reaction time larger than the threshold:
\[counterexample\]$\,$\
In the special case $\tau_0\equiv\tau$, there exist a Lipschitz continuous $F$, an initial data, and $\tau>\frac{1}{e C_F}$ such that the solution $u^\varepsilon$ to does not converge locally uniformly towards the solution of .
We derive such a counter-example by a small perturbation of a trivial but unstable case where the homogenization holds for all reaction times. This trivial case is a stationnary one where all the vehicles are initially equally spaced. This also highlights the fact that the initial dynamics is at least as important as the value of the reaction time from the homogenization perspective.
Organisation of the article
---------------------------
To get the homogenization result, we will first show the existence and uniqueness of solutions to and in Section \[exun\]. We will then prove the strict comparison principle for in Section \[proofcompa\] and we will show that the restriction on $\tau$ in Theorem \[convergenceth\] is equivalent to the existence of functions $\rho$ that verify . We will also state and prove a direct corollary of Theorem \[compa\] on drivers’ positions and give a counter-example to Theorem \[compa\] when the vehicles are not suitably spaced out at initial times, regardless of the threshold on $\tau$. We will give the explicit convergence proof in Section \[proofconv\] and give the unstable example where homogenization holds for all reaction times, provided the vehicles are perfectly spaced at initial times. In section \[counter\], we will give the explicit model and will prove Theorem \[counterexample\].
Existence and uniqueness {#exun}
=========================
This section is devoted to the study of the existence and uniqueness of solutions to and . Studying existence and uniqueness of solutions to is stricly equivalent to studying .
Under the assumptions , there exists a unique classical solution to .\[uniqeps1\]
\
The proof is based on an explicit and incremental construction. Indeed, on the time interval $(0;\xi]$ the equation can be written as follows: $$\partial_t u (x,t) =F({u_0 (x+1,t-\tau_0(x))-u_0 (x,t-\tau_0(x))}) \hspace{2 cm} (x,t)\in \mathbb{R}\times (0,\xi].$$ Hence, it can be explicitly integrated and we get: $$u(x,t)=u_0(x,0)+\int_0^t F({u_0 (x+1,s-\tau_0(x))-u_0 (x,s- \tau_0(x))}) ds \hspace{1 cm} \hspace{2 cm} (x,t)\in \mathbb{R}\times (0,\xi].$$ This enables to define the function $u_1$ which coincides with $u_0$ at initial times and that is defined by the previous expression in $[0,\xi]$.
Hence, in the time interval $(\xi;2\xi]$, the equation can be written as follows: $$\partial_t u (x,t) =F({u_1 (x+1,t-\tau_0(x))-u_1 (x,t- \tau_0(x))}) \hspace{0.7 cm} (x,t)\in \mathbb{R}\times (\xi;2\xi].$$ Here again, it can be explicitly integrated: $$u(x,t)=u_1(x,\xi)+\int_\xi^t F({u_1 (x+1,s-\tau_0(x))-u_1 (x,s- \tau_0(x))}) ds \hspace{1 cm} (x,t)\in \mathbb{R}\times [\xi,2\xi].$$ The process can be iterated to obtain a function $u$ defined piecewise which is, by construction, continuous on $\mathbb{R}\times [0,\infty)$ and which solves equation in each $\mathbb{R}\times(k\xi,(k+1)\xi)$, $k= 0,.., \lfloor \frac{T}{\xi}\rfloor -1$. It remains to prove that it solves the equation globally and so, that $u$ is $C^1$ in time on $(0,T)$. Indeed, at each $k\xi$, the left and right limits of the derivative coincide and verify the equation.
The uniqueness also comes from the process as we see that on each time interval, the solution is completely determined by its expression on the previous interval which implies global uniqueness for a given initial condition $u_0$.
Equation is an Hamilton-Jacobi equation and is therefore studied in the viscosity solutions’ framework. The existence and uniqueness are classical and are respectively given by Perron’s method and the usual comparison principle, see [@barles] for instance.
Strict comparison principle {#proofcompa}
===========================
This section is first devoted to the proof of Theorem \[compa\]. We will then explain the restriction on $\tau$. We will also prove the conservation of initial order of the vehicles as a corollary when those are well spaced out at initial times. Finally, we will finally present a counter-example to comparison principle for any reaction time $\tau>0$. Indeed, the existence of $\rho$ functions that verify is equivalent to consider $\tau$ under the threshold but both and are necessary for the strict comparison principle to hold and hence, the restriction on $\tau$ is not a sufficient condition to ensure the result. The existence of $\rho$ functions is not enough, it has to be linked with the initial dynamics of the vehicles. Namely, we will show that if the vehicles are not suitably spaced out at initial times (in the sense made precise in ), their initial order can be disturbed even for reaction times that are below the threshold.
Proof of Theorem \[compa\]
--------------------------
\
Let us consider $d:=v-u$ and define $T^*$ as: $$T^*=\sup \{S\in [0,+\infty)/\forall \tau'\in[0,\tau], \forall (x,t) \in [x_0-R,x_0+R] \times [t_0-\tau,S], \hspace{0.2 cm} d(x,t-\tau')\leq \rho (\tau') d(x,t) \}.$$ The set is not empty as it contains $t_0$ so $T^*$ is well-defined.
- We first claim that to establish , it is sufficient to prove that $T^* \geq T$.
Thanks to we have $\rho(0)>1$. Taking $\tau '=0$ in the definition of $T^*$, with $T^*\geq T$, implies that: $$\begin{aligned}
d(x,t)\geq 0 \hspace{1cm}(x,t)\in [x_0-R,x_0+R]\times [t_0-\tau,T). \label{posi}\end{aligned}$$ By combining with , we get: $$\begin{aligned}
d(x,t)\geq 0 \hspace{1cm}(x,t)\in [x_0-R-1,x_0+R+1]\times [t_0-\tau,T). \label{posi2}\end{aligned}$$ Let us consider $(x,t)\in [x_0-R,x_0+R]\times (t_0,T)$. By definition of $d$, we have: $$\begin{aligned}
\partial_t d(x,t) \geq F({v (x+1,t-\tau_0(x))-v (x,t- \tau_0(x))})- F({u (x+1,t-\tau_0(x))-u (x,t- \tau_0(x))}). \label{lipdessous}\end{aligned}$$ By combining with the fact that $F$ is non-decreasing gives: $$\begin{aligned}
\partial_t d(x,t) \geq F({u (x+1,t-\tau_0(x))-v (x,t- \tau_0(x))})- F({u (x+1,t-\tau_0(x))-u (x,t- \tau_0(x))}).\end{aligned}$$ Using the fact that $F$ is $C_F-$Lipschitz, we get: $$\begin{aligned}
\partial_t d(x,t) \geq -C_F d(x,t-\tau_0(x)).\end{aligned}$$ If $T^* \geq T$, then we have: $$\begin{aligned}
\partial_t d(x,t) \geq -C_F\rho(\tau_0(x)) d(x,t).\end{aligned}$$ By using the fact that $\rho$ is non-decreasing, we get: $$\begin{aligned}
\partial_t d(x,t) \geq -C_F\rho(\tau) d(x,t). \label{decroi}\end{aligned}$$ Hence, $d$ is a supersolution of the problem: $$\begin{aligned}
\left\{
\begin{array}{l}
\partial_t w(x,t)=-C_F \rho(\tau)w(x,t), \\
w(x,t_0)=\delta. \label{sys2}
\end{array}
\right.\end{aligned}$$ We conclude by a standard comparison principle for this ODE.
- Let us now show that $T^*\geq T$.
By contradiction, if we suppose that $T^* < T$, then for all $\beta\in \left(0,\inf\left(\frac{T-T^*}{2},1 \right) \right)$, there exists $\tau_\beta\in [0,\tau]$, $(x_\beta, t_\beta)\in [x_0-R,x_0+R]\times (T^*,T^*+\beta]$ such that: $$\begin{aligned}
d(x_\beta, t_\beta-\tau_\beta)> \rho (\tau_\beta) d(x_\beta, t_\beta). \label{absurde}\end{aligned}$$ Let us notice that holds true for all $t\in [t_0,T^*]$ (and so does ).
As a preliminary result, let us first show that: $$\begin{aligned}
d(x,t-\tau')\leq \bar{\rho}(\tau')d(x,t) \hspace{1 cm} \tau ' \in [0,\tau], (x,t) \in [x_0-R,x_0+R] \times [t_0,T^*], \label{controstrict}\end{aligned}$$ with $\bar{\rho}$ given by: $$\bar{\rho}( \tau')= 1+C_F \rho(\tau)\int_0^{\tau '}\rho(s)ds.$$ By remarking that $\bar{\rho}(0)=1$, we notice the equality for $\tau'=0$. For $\tau'\in [0,\tau], (x,t)\in [x_0-R,x_0+R] \times [t_0,T^*]$, we have: $$\partial_{\tau'}(\bar{\rho}(\tau')d(x,t))=C_F\rho(\tau)\rho(\tau')d(x,t).$$ Let us remark that $\tau' \mapsto d(x,t-\tau')$ is a subsolution of this equation. Indeed: $$\begin{aligned}
\partial_{\tau'} \left(d(x,t-\tau')\right)= -\partial_t d(x,t-\tau').\end{aligned}$$ We can use as $t-\tau'\in [t_0-\tau,T^*]$: $$\begin{aligned}
\partial_{\tau'} \left(d(x,t-\tau')\right) \leq C_F \rho(\tau) d(x,t-\tau ').\end{aligned}$$ Finally, by definition of $T^*$, we get: $$\begin{aligned}
\partial_{\tau'} \left(d(x,t-\tau')\right) \leq C_F \rho(\tau) \rho(\tau')d(x,t).\end{aligned}$$ The preliminary result is then obtained by a comparison principle on the equation $\partial_{\tau'} W=C_F\rho(\tau)\rho(\tau')W$. Let us notice that and imply that: $$\begin{aligned}
\partial_t d(x,t)\geq -2 ||F||_\infty.\end{aligned}$$ By integrating it, we get: $$\begin{aligned}
d(x,t)\geq d(x,s)-2||F||_\infty(t-s) \hspace{1cm} (x,t) \in [x_0-R,x_0+R]\times [t_0,T], s\in [t_0,t].
\label{lipdessous2}\end{aligned}$$ **Case 1: $t_\beta-\tau_\beta\leq T^*$**.
In this case, by defining $\tau'_\beta:=T^*-t_\beta+\tau_\beta\in [0,\tau]$ we get: $$d(x_\beta, t_\beta-\tau_\beta)=d(x_\beta, T^*-\tau'_\beta).$$ Hence, by we get: $$d(x_\beta, t_\beta-\tau_\beta)\leq \bar{\rho}(\tau'_\beta)d(x_\beta,T^*).$$ Thanks to , we have: $$\begin{aligned}
d(x_\beta, t_\beta-\tau_\beta)\leq \bar{\rho}(\tau'_\beta)(d(x_\beta,t_\beta)+2||F||_\infty \beta). \label{intermediaire}\end{aligned}$$ In order to get a contradiction with , it is sufficient to consider $\beta<\min (\beta_1,\beta_2)$ with: $$\begin{aligned}
\left\{
\begin{array}{l}
\beta_1:=\frac{\delta}{2||F||_\infty+1} e^{-C_F\rho(\tau)(T^*-t_0)}>0, \\
\beta_2:= \frac{\beta_1}{2||F||_\infty} \inf_{[0,\tau]} (\frac{\rho}{\bar{\rho}}-1)>0.
\end{array}
\right.\end{aligned}$$
Indeed, for $\beta<\beta_1$, thanks to (which is still valid in $[x_0-R,x_0+R]\times [t_0,T^*]$) and to , we have: $$\begin{aligned}
d(x_\beta, t_\beta)\geq d(x_\beta,T^*)-2||F||_\infty\beta\geq \delta e^{-C_F\rho(\tau)(T^*-t_0)}-2||F||_\infty\beta_1=\beta_1, \end{aligned}$$ and for $\beta<\min (\beta_1,\beta_2)$ we have: $$\begin{aligned}
\bar{\rho}(\tau'_\beta)(d(x_\beta,t_\beta)+2||F||_\infty\beta)\leq \rho (\tau'_\beta) d(x_\beta,t_\beta).\end{aligned}$$ Together with , this implies: $$d(x_\beta, t_\beta-\tau_\beta)\leq \rho (\tau'_\beta) d(x_\beta,t_\beta).$$ Finally, as $\rho$ is non decreasing and $\tau'_\beta\leq \tau_\beta$, we get a contradiction with .
**Case 2: $t_\beta-\tau_\beta> T^*$**.
As we have $t_\beta<T^*+\beta$, this implies that $\tau_\beta\in[0,\beta]$.
Then, thanks to and we have: $$\begin{aligned}
d(x_\beta,t_\beta)+2||F||_\infty\beta\geq d(x_\beta,t_\beta)+ 2||F||_\infty\tau_\beta\geq d(x_\beta, t_\beta-\tau_\beta)>\rho (\tau_\beta) d(x_\beta,t_\beta).\end{aligned}$$ In particular, we have: $$\begin{aligned}
(\rho(\tau_\beta)-1)d(x_\beta,t_\beta)<2||F||_\infty\beta.\end{aligned}$$ Thanks to , this implies: $$\begin{aligned}
(\rho(\tau_\beta)-1)d(x_\beta,T^*)<2||F||_\infty\rho(\tau_\beta)\beta.\end{aligned}$$ Thus, $$\begin{aligned}
0< \delta e^{-C_F\rho(\tau)(T^*-t_0)} <\frac{\rho(\tau_\beta)}{\rho(\tau_\beta)-1}2||F||_\infty\beta.\end{aligned}$$ For vanishing $\beta$ this implies $\delta e^{-C_F\rho(\tau)(T^*-t_0)}=0 $ which is false.
Restriction on $\tau$ and existence of solutions to
----------------------------------------------------
Condition explains why we chose $\tau<\frac{1}{eC_F}$. In fact, we have the following proposition:
\[A\] admits solutions if and only if $\tau<\frac{1}{eC_F}$.
We now give a proof of Proposition \[A\].
We want to show that condition implies $\tau<\frac{1}{eC_F}$.
Let us define: $$R(\tau ')=\int_0^{\tau '} \rho(s)ds.$$ We have $R(0)=0$. By defining $\alpha=R'(\tau)=\rho(\tau)$, condition is equivalent for $R$ to be a strict supersolution of: $$\begin{aligned}
\left\{
\begin{array}{l}
y'=C_F\alpha y+1, \mbox { in } [0,\tau], \\
y(0)=0. \label{sys}
\end{array}
\right.\end{aligned}$$ As $S: \tau ' \mapsto \frac{e^{C_F\alpha\tau '}-1}{C_F\alpha}$ is a solution of this Cauchy problem on $[0,\tau]$, by Gronwall’s lemma or a standard comparison principle, we deduce that $R\geq S$ or equivalently that: $$1+C_F R(\tau ')R'(\tau)\geq e^{C_F R'(\tau)\tau '} \hspace{1 cm} \tau ' \in [0,\tau].$$
Moreover, as $R$ is a strict supersolution of , we deduce that: $R'(\tau ') > e^{C_F R'(\tau)\tau '}$ for all $\tau ' \in [0,\tau]$. By considering this inequality for $\tau '= \tau$ and taking the logarithm, we get: $$C_F \tau < \frac{ \ln R'(\tau)}{R'(\tau)}\leq \max_{x>0} \frac{\ln x}{x}=\frac{1}{e}.$$
Let us now show that admits solutions as soon as $\tau<\frac{1}{e C_F}$.
It is equivalent to find positive and non-decreasing solutions $R$ of: $$\begin{aligned}
\left\{
\begin{array}{l}
R'(\tau')>C_F R'(\tau)R(\tau')+1 \hspace{2cm} \tau'\in[0,\tau], \\
R(0)=0. \label{sys2}
\end{array}
\right.\end{aligned}$$ Let us consider $\lambda\in\left(1,\sqrt{\frac{1}{e\tau C_F}}\right)$. The function $h:\gamma \mapsto \ln \gamma - \lambda \gamma \tau C_F$ is smooth on $(0,+\infty)$ and admits a maximum at $\gamma_{max}=\frac{1}{\lambda\tau C_F}$. As $\lambda<\sqrt{\frac{1}{e\tau C_F}}$, it is straightforward to check that $\ln \lambda < h(\gamma_{max})$ and so, there exists $\gamma_0>0$ such that $h(\gamma_0)=\ln \lambda$ or equivalently that $\gamma_0=\lambda e^{\lambda\gamma_0 C_F \tau}$.
Let us define $U_\lambda: \tau' \mapsto \frac{1}{\gamma_0 C_F} (e^{\lambda\gamma_0 C_F \tau'}-1).$ We have: $U_\lambda'(\tau)=\lambda e^{\lambda\gamma_0 C_F \tau}=\gamma_0$. As $\lambda>1$, $U_\lambda$ verifies: $$\begin{aligned}
\left\{
\begin{array}{l}
U_\lambda '(\tau')=\lambda \gamma_0 C_F U_\lambda(\tau')+\lambda>\gamma_0 C_F U_\lambda(\tau')+1=C_F U_\lambda'(\tau) U_\lambda(\tau')+1\hspace{2cm} \tau'\in[0,\tau], \\
U_\lambda(0)=0.
\end{array}
\right.\end{aligned}$$ For all $\lambda\in\left(1,\sqrt{\frac{1}{e\tau C_F}}\right)$, we can construct a solution $U_\lambda$ to . Therefore, as soon as $\tau<\frac{1}{e C_F}$, there exist infinitely many solutions.
Condition is reduced to: $$\rho C_F \tau ' <1-\frac{1}{\rho },$$ if we consider $\rho$ as a constant. If it is considered for $\tau'=\tau$, we get $ \tau<\frac{1}{4C_F}$ because the condition tells that the function $x\mapsto C_F \tau x^2-x+1$ has a negative part. Conversely, for all fixed $\tau<\frac{1}{4 C_F}$, we can always find constant solutions.
Consequence on drivers’ positions and a counter-example
-------------------------------------------------------
A very practical consequence of this strict comparison principle for the microscopic model derived from traffic flow is the conservation of initial order for vehicles, provided they are suitably spaced out at initial times.
\[Conservation of initial order\] Let us consider $(X_i)_{i\in \mathbb{Z}}$ a sequence of drivers’ positions that evolve under the dynamics with initial conditions given by and for $\tau_0\equiv\tau$. We consider the solution $u$ to such that: $$u_0(i,t)=x_i^0(t) \hspace{0.5cm} (i,t)\in \mathbb{Z}\times [-\tau,0].$$ We suppose that there exist $\sigma>0$, a time function $\rho_{p}$ which verifies such that: $$\begin{aligned}
\sigma \leq u_0(x+1,t-\tau')-u_0(x,t-\tau') \leq \rho_{p}(\tau') (u_0(x+1,t)-u_0(x,t)),\mbox{ } \tau'\in[0,\tau], (x,t)\in \mathbb{R}\times [-\tau,0]. \label{goodhyp}\end{aligned}$$ Then, $$X_{i+1}(t) > X_{i}(t) \hspace{0.5cm} (i,t)\in \mathbb{Z}\times [0,T).$$
\
By uniqueness, we have: $$\begin{aligned}
X_i(t)=u(i,t) \hspace{0.5cm} (i,t)\in \mathbb{Z}\times [0,T). \label{edoedp}\end{aligned}$$ We consider $v:(x,t)\mapsto u(x+1,t)$. In particular, we have: $$\begin{aligned}
X_{i+1}(t)=v(i,t) \hspace{0.5cm} (i,t)\in \mathbb{Z}\times [0,T). \label{edoedp2}\end{aligned}$$ For $d:=v-u$, thanks to , the conditions of Theorem \[compa\] (and its remark) are fulfilled with $\delta = \sigma$, $t_0=0$, $R=+\infty$ and $\rho=\rho_{p}$. Therefore we have $v>u$ on $\mathbb{R}\times [0,T)$ and in particular on $\mathbb{Z}\times [0,T)$. Using and this is equivalent to have: $$X_{i+1}(t) > X_{i}(t) \hspace{2cm} (i,t)\in \mathbb{Z}\times [0,T).$$
Our microscopic model is valid when the drivers always stay in the same order, otherwise it is not adapted anymore to the physical situation. More explicitly, the behaviour of the driver at position $X_i$ does not depend anymore on the position $X_{i+1}$ if at some time the car $i+1$ is not in front of the car $i$.
In contrast with most Hamilton-Jacobi equations, it is not possible to state a classical comparison principle because of the delay time. More accurately, it is compulsory to ensure that the vehicles are suitably spaced out (this space is represented by the function $\rho$).
\[nonconserv\] We consider the case: $\tau_0\equiv\tau$. For any delay time $\tau>0$, it is straightforward to show that the initial order of solutions is sometimes not conserved when the vehicles are not suitably spaced in the sense made precise in Theorem \[compa\]. Let us consider $n_0\in\mathbb{N}^*$ such that $\tau>\frac{2}{n_0}$. Let us consider the particular case for $F$ being identically equal to the identity on $[0;1]$ (and that verifies in $\mathbb{R}$). Let us consider two sets of drivers $(X_i)_{i\in \mathbb{Z}}$ and $(Y_i)_{i\in \mathbb{Z}}$ that evolve under the same dynamics and such that: $$y_i^0(t)< x_i^0(t) \hspace{2cm} (i,t)\in \mathbb{Z}\times [-\tau,0],$$ and there exists $j\in\mathbb{Z}$ such that for $t\in [-\tau,0]$: $$\begin{aligned}
\left\{
\begin{array}{l}
y_j^0(t)=j-1+\frac{n_0}{n_0+1} e^{n_0 t}< x_j^0(t)=j, \\
y_{j+1}^0(t)=j+\frac{n_0}{n_0+1}e^{n_0 t}, \\
x_{j+1}^0(t)=y_{j+1}^0(t) +\frac{1}{n_0+1}
\end{array}
\right.\end{aligned}$$ Those two sets of drivers do not see each other, that means that they do not evolve on the same physical road but on two identical copies of the real line. We can integrate on $[0,\tau]$ for the two sets and we find that: $$(Y_j-X_j)(\tau)=\frac{n_0}{n_0+1}\tau-\frac{1}{n_0+1}-\int_{-\tau}^0 e^{n_0 u} du=\frac{n_0}{n_0+1}\tau-\frac{1}{n_0+1}-\frac{1-e^{-n_0\tau}}{n_0+1}>0.$$ Therefore, the initial order is disrupted even if: $$\frac{1}{n_0+1}\leq x_i^0(t)- y_i^0(t)\hspace{2cm} (i,t)\in \mathbb{Z}\times [-\tau,0].$$
Convergence {#proofconv}
===========
This section is mainly devoted to the proof of Theorem \[convergenceth\]. The unstable stationnary case will be given as the final remark of this section.
Proof of Theorem \[convergenceth\]
----------------------------------
\
Let us first show that $u^\varepsilon$ is globally Lipschitz continuous in space and time uniformly in $\varepsilon$.
In time: By looking at , we remark that $u^\varepsilon$ $||F||_\infty-$Lipschitz continuous in time.
In space: The equation is translation invariant in space and invariant by addition of constants to the solutions. Let $h>0$.
The solution corresponding to the initial condition $u_1: (x,t)\mapsto u_0(x+h,t)$ is the function $w: (x,t)\mapsto u^\varepsilon(x+h,t)$.
The one associated to $u_2: (x,t)\mapsto u_0(x,t)+2 L h$ is $v: (x,t)\mapsto u^\varepsilon(x,t)+2 L h$. We define $\psi_1: (x,t)\mapsto \frac{1}{\varepsilon} w(\varepsilon x, \varepsilon t)$ and $\psi_2: (x,t)\mapsto \frac{1}{\varepsilon} v(\varepsilon x, \varepsilon t) $.
$\psi_1$ solves: $$\begin{aligned}
\left\{
\begin{array}{l}
\partial_t u (x,t) =F({u (x+1,t-\tau_0(\varepsilon x))-u (x,t- \tau_0(\varepsilon x))}) \hspace{1 cm} (x,t)\in \mathbb{R}\times (0,T), \label{psi1} \\
u (x,t)=\frac{1}{\varepsilon} u_0(\varepsilon x+h,\varepsilon t) \hspace{4 cm} (x,t) \in \mathbb{R} \times [-2 \tau;0].
\end{array}
\right.\end{aligned}$$ $\psi_2$ solves: $$\begin{aligned}
\left\{
\begin{array}{l}
\partial_t u (x,t) =F({u (x+1,t-\tau_0(\varepsilon x))-u (x,t- \tau_0(\varepsilon x))}) \hspace{1 cm} (x,t)\in \mathbb{R}\times (0,T), \label{psi2} \\
u (x,t)=\frac{1}{\varepsilon} u_0(\varepsilon x,\varepsilon t)+\frac{2Lh}{\varepsilon} \hspace{3.6 cm} (x,t) \in \mathbb{R} \times [-2 \tau;0].
\end{array}
\right.\end{aligned}$$ By , $\psi_1$ and $\psi_2$ are $L-$ Lipschitz continuous functions on $\mathbb{R}\times [-2\tau,0]$.
Hence, $d:=\psi_2-\psi_1$ is a $2L-$ Lipschitz continuous function on $\mathbb{R}\times [-2\tau,0]$. Otherwise, we have: $$\begin{aligned}
d(x,t)= \frac{1}{\varepsilon} \left( u_0(\varepsilon x,\varepsilon t)-u_0(\varepsilon x+h,\varepsilon t)\right)+\frac{2Lh}{\varepsilon}\geq -\frac{Lh}{\varepsilon}+\frac{2Lh}{\varepsilon}=\frac{Lh}{\varepsilon} \hspace{0.5cm} (x,t) \in \mathbb{R} \times [- \tau;0]\label{dgood}.\end{aligned}$$ Let us now consider $\rho_m$ any solution to for $\tau<\frac{1}{eC_F}$ (see Proposition \[A\]). We recall that implies $\rho_m\geq \rho_m(0)>1$.
We have: $$d(x,t-\tau')\leq d(x,t)+2L\tau \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in \mathbb{R}\times [-\tau,0].$$ Therefore, to get: $$d(x,t-\tau')\leq \rho_m(\tau')d(x,t) \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in \mathbb{R}\times [-\tau,0],$$ it is sufficient to verify: $$2L\tau\leq (\rho_m(\tau')-1)d(x,t) \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in \mathbb{R}\times [-\tau,0].$$ As $\rho_m$ is a non-decreasing function, and as $d\geq 0$, it is finally sufficient to verify: $$\frac{2L\tau}{\rho_m(0)-1}\leq d(x,t) \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in \mathbb{R}\times [-\tau,0].$$ Thanks to , this will be the case for $\varepsilon$ small enough. The conditions of the strict comparison principle are fulfilled with $\delta = \frac{Lh}{\varepsilon}$, $t_0=0$, $R=+\infty$ and $\rho=\rho_m$. Then, by Theorem \[compa\] we have $\psi_2-\psi_1>0$ in $\mathbb{R}\times [0,T)$ and this implies for all $(y,s)\in \mathbb{R}\times [0,T)$: $$u^\varepsilon(y+h,s)\leq u^\varepsilon(y,s)+2 L h.$$ Analogously, we can show that: $$u^\varepsilon(y+h,s)\geq u^\varepsilon(y,s)-2 L h.$$
Hence, $u^\varepsilon$ is $2L$-Lipschitz continuous in space. Therefore, we can define the relaxed upper and lower semi-limits $\bar u(x,t)=\lim_{\varepsilon \rightarrow 0} \sup^* u^\varepsilon(x,t)$ and $ \underline {u}(x,t)=\lim_{\varepsilon \rightarrow 0} \inf_* u^\varepsilon(x,t)$. By definition, we have $\underline {u} \leq \bar u$.
Given $\nu>0$, for $\varepsilon$ small enough, we can apply Theorem \[compa\] on $v_b: (x,t)\mapsto \frac{1}{\varepsilon} \left(u_0(\varepsilon x,\varepsilon t)+\nu+(||F||_\infty+L) \varepsilon t \right)$ and on $u_b: (x,t)\mapsto \frac{1}{\varepsilon} u^\varepsilon (\varepsilon x,\varepsilon t) $ and we get: $$u^\varepsilon(x,t) \leq u_0(x,t)+\nu+(||F||_\infty+L) t \qquad (x,t)\in \mathbb{R}\times [-2\varepsilon \tau,T),$$ which gives for vanishing $\nu$: $$u^\varepsilon(x,t) \leq u_0(x,t)+(||F||_\infty+L) t \qquad (x,t)\in \mathbb{R}\times [-2\varepsilon \tau,T).$$ Similarly, we get: $$u^\varepsilon(x,t) \geq u_0(x,t)-(||F||_\infty+L) t \qquad (x,t)\in \mathbb{R}\times [-2\varepsilon \tau,T),$$ and so: $$|u^\varepsilon(x,t) -u_0(x,t)|\leq (||F||_\infty+L) t \qquad (x,t)\in \mathbb{R}\times [-2\varepsilon \tau,T).$$ This implies: $$\bar u(0,x)= \underline{u}(0,x)=u_0(x,0) \qquad x\in\mathbb{R}.$$ To get the convergence, it is sufficient to show that $\bar u$ and $\underline{u}$ are respectively a subsolution and a supersolution of . Indeed, by using a classical comparison principle we will then get: $$\underline{u}\leq \bar{u}\leq u^0\leq\underline {u}$$ Let us only show by contradiction that $\bar u$ is a subsolution of (the other one being very similar). Assume that there exist $(\bar x, \bar t)$, $\varphi\in C^1_{x,t}$, $(r,\theta, \eta)\in (0,+\infty)^3$ such that:
$$\begin{aligned}
\left\{ \label{testfun}
\begin{array}{l}
\bar u(\bar x, \bar t)=\varphi (\bar x, \bar t), \\
\bar u < \varphi \mbox{ in } B_{2r}(\bar x, \bar t)\setminus\{(\bar x, \bar t)\}, \\
\bar u \leq \varphi -2 \eta \mbox{ in } B_{2r}(\bar x, \bar t)\setminus B_{r}(\bar x, \bar t),\\
\partial_t\varphi (\bar x, \bar t)= 2 \theta + F(\partial_x \varphi (\bar x, \bar t)), \\
\partial_t\varphi ( x, t)\geq \theta + F(\partial_x \varphi (x,t)) \mbox{ in } B_{r}(\bar x, \bar t),
\end{array}
\right.\end{aligned}$$
where we set $B_s(p,q):=(p-s,p+s)\times (q-s,q+s)$.
As $u^\varepsilon$ and $\varphi$ are continuous, we can define: $$\begin{aligned}
M_\varepsilon:= \max_{B_{2r}(\bar x, \bar t)} (u^\varepsilon- \varphi):=(u^\varepsilon- \varphi)(x_\varepsilon, t_\varepsilon).\end{aligned}$$
$M_\varepsilon\geq -\eta$ for $\varepsilon$ small enough and hence $(x_\varepsilon, t_\varepsilon)\in B_{r}(\bar x, \bar t)$.
Let us define: $\varphi^\varepsilon:=\varphi+M_\varepsilon$. This function satisfies: $$\begin{aligned}
\left\{ \label{testfun2}
\begin{array}{l}
\varphi^\varepsilon(x_\varepsilon, t_\varepsilon)=u^\varepsilon(x_\varepsilon, t_\varepsilon)\\
\partial_t\varphi^\varepsilon ( x, t)\geq \theta + F(\partial_x \varphi^\varepsilon (x,t)) \mbox{ in } B_{r}(\bar x, \bar t).
\end{array}
\right.\end{aligned}$$ By regularity of $\varphi^\varepsilon$ and $F$, for $\varepsilon$ small enough, we have in $B_{r}(\bar x, \bar t) $: $$\begin{aligned}
\partial_t \varphi^\varepsilon(x,t) \geq F\bigg(\frac{\varphi^\varepsilon (x+\varepsilon,t-\varepsilon\tau_0( x))-\varphi^\varepsilon (x,t-\varepsilon \tau_0( x))}{\varepsilon}\bigg)+\frac{\theta}{2}.\end{aligned}$$ Let us define: $d^\varepsilon(x,t)=\varphi^\varepsilon ( x, t)-u^\varepsilon ( x, t)$. In $B_{2r}(\bar x, \bar t)\setminus B_{r}(\bar x, \bar t)$ we have: $$\begin{aligned}
d^\varepsilon(x,t)=\varphi ( x, t)-u^\varepsilon ( x, t)+M_\varepsilon\geq \frac{3\eta}{2} - \eta= \frac{\eta}{2}. \label{minordeps}\end{aligned}$$ Using the fact that $u^\varepsilon$ is $||F||_\infty-$Lipschitz continuous in time and that $\varphi$ is smooth, we remark that $d^\varepsilon$ is Lipschitz continuous in $B_{2r}(\bar x, \bar t)$. Let $K$ denote its Lipschitz constant. Let us define: $$d_1: (x,t)\mapsto \frac{1}{\varepsilon} d^\varepsilon (\varepsilon x, \varepsilon t).$$ $d_1$ is $K-$ Lipschitz continuous in $B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)$. Thanks to , we have in $B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)$: $$\begin{aligned}
d_1(x,t)\geq \frac{\eta}{2\varepsilon}\label{d1good}.\end{aligned}$$ Let us now consider $\rho_1$ any solution to for $\tau<\frac{1}{eC_F}$ (see Proposition \[A\]). We recall that implies $\rho_1\geq \rho_1(0)>1$.
We have: $$d_1(x,t-\tau')\leq d_1(x,t)+K\tau \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right).$$ Therefore, to get: $$d_1(x,t-\tau')\leq \rho_1(\tau')d_1(x,t) \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right),$$ it is sufficient to verify: $$K\tau\leq (\rho_1(\tau')-1)d_1(x,t) \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right).$$ As $\rho_1$ is a non-decreasing function, and as $d_1\geq 0$, it is finally sufficient to verify: $$\frac{K\tau}{\rho_1(0)-1}\leq d_1(x,t) \hspace{0.5cm}\tau'\in[0,\tau], (x,t)\in B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right).$$ Thanks to , this will be the case for $\varepsilon$ small enough. We choose $\varepsilon$ such that $\max(1,\tau)<\frac{r}{8\varepsilon}$. The previous inequalities enable us to apply the strict comparison principle with $\delta=\frac{\eta}{2\varepsilon}$, $R=\frac{3r}{2\varepsilon}$, $x_0=\frac{\bar{x}}{\varepsilon}$, $T=\frac{\bar{t}}{\varepsilon}+R$ and $t_0=\frac{\bar{t}}{\varepsilon}-R$. Indeed, we have: $$\begin{aligned}
\left\{ \label{inclu}
\begin{array}{l}
[x_0-R- 1 , x_0+R+1 ] \setminus ]x_0-R,x_0+R[\times [t_0-\tau,T)\subset B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right) \\
\mbox {}[x_0-R,x_0+R] \times [t_0-\tau,t_0] \subset B_{\frac{2r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\setminus B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right).
\end{array}
\right.\end{aligned}$$ Hence, by Theorem \[compa\], we have $d_1>0$ in $[x_0-R,x_0+R]\times [t_0,T)$ which is in contradiction with the definition of $(x_\varepsilon, t_\varepsilon)$ as we have: $$\begin{aligned}
\left\{
\begin{array}{l}
\left(\frac{x_\varepsilon}{\varepsilon}, \frac{t_\varepsilon}{\varepsilon}\right)\in B_{\frac{r}{\varepsilon}}\left(\frac{\bar x}{\varepsilon}, \frac{\bar t}{\varepsilon}\right)\subset [x_0-R,x_0+R]\times [t_0,T) , \\
d_1\left(\frac{x_\varepsilon}{\varepsilon}, \frac{t_\varepsilon}{\varepsilon}\right)=0.
\end{array}
\right.\end{aligned}$$
A special case: homogenization for any reaction time
----------------------------------------------------
A natural question arises. What happens for higher reaction times? Example \[nonconserv\] highlights the fact that the initial dynamics is very important and that reaction times cannot be considered separately. The answer to this question is not invariable and the following example shows that the expected macrosopic model can be derived for any reaction time for a special initial condition.
\[unstable\] For $T=+\infty$, let us consider the case where all drivers have the same reaction time $\tau\in (0,+\infty)$. Let $L>0$ be the common interdistance between all the vehicles. We consider that the vehicles do not move at initial times: $$x_i^0(t)=L i, \qquad t\in[-\tau,0].$$
This corresponds to: $$u_0(x,t)=L x, \qquad (x,t)\in\mathbb{R}\times [-\tau,0].$$
By incremental construction (or directly by uniqueness), we see that the solution $u^\varepsilon$ to for this initial condition does not depend on $\varepsilon$ and is given by: $$u^\varepsilon(x,t)=Lx+F(L)t, \qquad (x,t)\in\mathbb{R}\times [0,+\infty).$$
The unique solution $u^0$ of corresponding to this initial linear data is also given by the same expression. Therefore we have: $$u^\varepsilon(x,t)=u^0(x,t)=Lx+F(L)t, \qquad (x,t)\in\mathbb{R}\times [0,+\infty).$$
A counter-example to homogenization {#counter}
===================================
The goal of this section is to exhibit a counter-example derived from Example \[unstable\]. We will first give the explicit expressions of $F$, of the vehicles’ initial positions and the value of $\tau$ and give a list of lemmas that will be useful to finally prove Theorem \[counterexample\]. For simplicity, we fix $T=+\infty$. Let us consider the same $L>0$ and $(k,\beta, \alpha)\in (0,+\infty)^3$ with the condition $\alpha>4 \beta L$. We consider the following $F$: $$F(x)=k+\beta(x-L)^2+\alpha (x-L) \qquad x\in [0,2L],$$ and we continuously extend $F$ to $\mathbb{R}$ by constants; hence $F$ satisfies .
We here choose the common reaction time: $\tau=\frac{\pi}{4\alpha}$.
Let us consider $A\in (0,\frac{L}{2})$. We now introduce the initial positions of the vehicles: $$\begin{aligned}
\label{initi}
x_{i}^0(t)=i L+(-1)^i\frac{A}{2} \sin (2\alpha t) \qquad (i,t)\in \mathbb{Z}\times [-\tau,0].\end{aligned}$$
\[ini\] We have $x_{i+2}^0-x_{i}^0=2L $ but $x_{i+1}^0-x_{i}^0 \neq L$. We recover the initial data of Example \[unstable\] for $A=0$. This means that in this case, the vehicles alternatively oscillate around the previous stationnary positions at initial times. Those oscillations will be essential in the construction of the counter-example: they will last for all time and will raise the time velocity of the vehicles that will become striclty superior to the value of the velocity function of the corresponding macroscopic space gradient.
The key relationship between $\tau$ and $\alpha$ will allow the periodic oscillations to remain for all times as expressed in the following lemma:
\[osci\] Under the previous initial conditions, we have: $$X_{i+1}(t)-X_i(t)=L+A (-1)^{i+1} \sin (2\alpha t) \qquad t\geq 0.$$
We have $X_{i+2}-X_i=2L$ for all time.
The proof is based on the incremental construction of the solutions and is thus an induction proof. We show the result for each $[n\tau,(n+1)\tau], n\in\mathbb{N}$. We only perform the first step as the next ones are identical.
Let us define: $d_i:=X_{i+1}-X_i$.
For $t\in[0,\tau]$, we have: $$\begin{aligned}
d_i'(t)=F(X_{i+2}(t-\tau)-X_{i+1}(t-\tau))-F(d_i(t-\tau)).\end{aligned}$$ As $t-\tau\in[-\tau,0]$: $$\begin{aligned}
d_i'(t)=F(x_{i+2}^0(t-\tau)-x_{i+1}^0(t-\tau))-F(d_i(t-\tau)).\end{aligned}$$ Thanks to Remark \[ini\], we get: $$\begin{aligned}
d_i'(t)=F(2L-d_i(t-\tau))-F(d_i(t-\tau)).\end{aligned}$$ With the expression of $F$, this equivalently gives for $\bar{d_i}:=d_i-L$: $$\begin{aligned}
\bar{d_i}'(t)=-2\alpha \bar{d_i}(t-\tau).\end{aligned}$$ Thanks to , we have: $$\begin{aligned}
\bar{d_i}'(t)=-2\alpha A (-1)^{i+1} \sin (2\alpha t-2\alpha\tau).\end{aligned}$$ We remind that we have chosen $\tau=\frac{\pi}{4\alpha}$ to get: $$\begin{aligned}
\bar{d_i}'(t)=2\alpha A (-1)^{i+1} \cos (2\alpha t).\end{aligned}$$ We then integrate: $$\begin{aligned}
\bar{d_i}(t)=A (-1)^{i+1} \sin (2\alpha t).\end{aligned}$$ This gives the result for $n=0$.
For $\varepsilon>0$, we now consider $u^\varepsilon$ associated to this initial data and we recall the relationship: $$\begin{aligned}
\label{relationmicro}
X_i(t)=\frac{u^\varepsilon(i\varepsilon,t\varepsilon)}{\varepsilon} \qquad(i,t)\in\mathbb{Z}\times [-\tau,+\infty).\end{aligned}$$ Thanks to the previous remark, we have: $$\begin{aligned}
\frac{u^\varepsilon((i+2)\varepsilon,t\varepsilon)-u^\varepsilon(i\varepsilon,t\varepsilon)}{2\varepsilon}=L \qquad (i,t)\in\mathbb{Z}\times [-\tau,+\infty),\end{aligned}$$ or equivalently: $$\begin{aligned}
\frac{u^\varepsilon((i+2)\varepsilon,s)-u^\varepsilon(i\varepsilon,s)}{2\varepsilon}=L \qquad (i,s)\in\mathbb{Z}\times [-\varepsilon\tau,+\infty).\end{aligned}$$ For $s=0$, this gives for the initial condition: $$\begin{aligned}
\label{gradini}
\frac{u_0((i+2)\varepsilon,0)-u_0(i\varepsilon,0)}{2\varepsilon}=L \qquad (i,\varepsilon)\in\mathbb{Z}\times (0,+\infty).\end{aligned}$$ Moreover, thanks to , we have: $$x_0^0(0)=0.$$ Thus, we also have: $$\begin{aligned}
\label{zeroini}
u_0(0,0)=0.\end{aligned}$$ This leads to the following lemma:
The initial condition is given on the real line by: $u_0(x,0)=L x$.
Let us consider $\varepsilon>0$, $h>0$ and $x\in\mathbb{R}$. We define: $$\begin{aligned}
\left\{
\begin{array}{l}
n:=\lfloor \frac{h}{2\varepsilon} \rfloor, \\
m:=\lfloor \frac{x}{2\varepsilon} \rfloor.
\end{array}
\right.\end{aligned}$$ Let us now compute the rate of change: $$\begin{gathered}
\label{rateini}
\frac{u_0(x+h,0)-u_0(x,0)}{h}=\\
\frac{u_0(x+h,0)-u_0(2(m+n)\varepsilon,0)}{h}+\frac{u_0(2(m+n)\varepsilon,0)-u_0(2m\varepsilon,0)}{h}+\frac{u_0(2m\varepsilon,0)-u_0(x,0)}{h}.\end{gathered}$$ We introduce a telescopic sum for the second term: $$\begin{aligned}
\frac{u_0(2(m+n)\varepsilon,0)-u_0(2m\varepsilon,0)}{h}=\sum_{i=0}^{n-1} \frac{u_0(2m\varepsilon+2(i+1)\varepsilon,0)-u_0(2m\varepsilon+2i\varepsilon,0)}{h},\end{aligned}$$ or equivalently: $$\begin{aligned}
\frac{u_0(2(m+n)\varepsilon,0)-u_0(2m\varepsilon,0)}{h}=\frac{2\varepsilon}{h}\sum_{i=0}^{n-1} \frac{u_0(2m\varepsilon+2i\varepsilon+2\varepsilon,0)-u_0(2m\varepsilon+2i\varepsilon,0)}{2\varepsilon}.\end{aligned}$$ Thanks to , we get: $$\begin{aligned}
\frac{u_0(2(m+n)\varepsilon,0)-u_0(2m\varepsilon,0)}{h}=\frac{2n\varepsilon}{h}L.\end{aligned}$$ Thus, becomes: $$\begin{aligned}
\label{rateini2}
\frac{u_0(x+h,0)-u_0(x,0)}{h}=\frac{u_0(x+h,0)-u_0(2(m+n)\varepsilon,0)}{h}+\frac{2n\varepsilon}{h}L+\frac{u_0(2m\varepsilon,0)-u_0(x,0)}{h}.\end{aligned}$$ Let us remind that: $$\begin{aligned}
\label{estim}
\left\{
\begin{array}{l}
|h-2n\varepsilon|\leq 2\varepsilon, \\
|x-2m\varepsilon|\leq 2\varepsilon.
\end{array}
\right.\end{aligned}$$ The left-hand side of does not depend on $\varepsilon$. By taking the limit when $\varepsilon\rightarrow 0$, using and the fact that $u_0$ is a $L-$Lipschitz continuous function, we get: $$\begin{aligned}
\frac{u_0(x+h,0)-u_0(x,0)}{h}=L.\end{aligned}$$ Thanks to , we obtain the desired result on $[0,+\infty)$: $$\begin{aligned}
u_0(h,0)=Lh \qquad h\geq 0.\end{aligned}$$ For $h<0$, we define $h':=-h>0$, and we have for $x=h$: $$\begin{aligned}
\frac{u_0(h+h',0)-u_0(h,0)}{h'}=L,\end{aligned}$$ and thus: $$\begin{aligned}
u_0(h,0)=-Lh'=Lh.\end{aligned}$$
\[solumacro\] The solution of the macroscopic equation corresponding to this initial data is: $$u^0(x,t)=Lx+F(L)t \qquad (x,t)\in\mathbb{R}\times[0,+\infty).$$
This is a classical solution and the solution is unique.
We see that we have: $$\partial_t u^0(0,t)=k=F(L) \qquad t>0.$$
That was also the case for $u^\varepsilon$ in Example \[unstable\] but it is not true anymore for $A>0$.
Indeed, we have for $s>0$: $$\begin{aligned}
\partial_t u^\varepsilon (0,s)=F\left(\frac{u^\varepsilon (\varepsilon,s-\varepsilon\tau)-u^\varepsilon (0,s-\varepsilon\tau)}{\varepsilon}\right)\end{aligned}$$ Thanks to , we get: $$\begin{aligned}
\partial_t u^\varepsilon (0,s)=F\left(X_1\left(\frac{s}{\varepsilon}-\tau\right)-X_0\left(\frac{s}{\varepsilon}-\tau\right)\right).\end{aligned}$$ From Lemma \[osci\], we get for the velocity function $F$ considered: $$\begin{aligned}
\partial_t u^\varepsilon (0,s)=F(L)+\beta A^2 \sin^2\left(2\alpha\left(\frac{s}{\varepsilon}-\tau\right)\right)-\alpha A \sin\left(2\alpha\left(\frac{s}{\varepsilon}-\tau\right)\right),\end{aligned}$$ or equivalently, for $\tau=\frac{\pi}{4\alpha}$: $$\begin{aligned}
\label{derivative}
\partial_t u^\varepsilon (0,s)=F(L)+\beta A^2 \cos^2\left(2\alpha \frac{s}{\varepsilon}\right)+\alpha A \cos\left(2\alpha\frac{s}{\varepsilon}\right).\end{aligned}$$
We are now ready to prove Theorem \[counterexample\].
\[Proof of Theorem \[counterexample\]\] By contradiction, we suppose that $u^\varepsilon$ converges locally uniformly towards the solution $u^0$ of whose expression is given in corollary \[solumacro\].
Let us consider $s>0$ and $h>0$. We then have: $$\begin{aligned}
\label{limitmacro1}
\lim_{\varepsilon\rightarrow 0} \frac{u^\varepsilon(0,s+h)-u^\varepsilon(0,s)}{h}=\frac{u^0(0,s+h)-u^0(0,s)}{h}=F(L).\end{aligned}$$ Otherwise, we have for $\varepsilon>0$: $$\begin{aligned}
\frac{u^\varepsilon(0,s+h)-u^\varepsilon(0,s)}{h}=\frac{1}{h}\int_s^{s+h} \partial_t u^\varepsilon (0,s') ds'.\end{aligned}$$ From , we get: $$\begin{aligned}
\frac{u^\varepsilon(0,s+h)-u^\varepsilon(0,s)}{h}=F(L)+\beta \frac{A^2}{2}+\frac{\beta\varepsilon A^2}{8\alpha h} \left [\sin \left(\frac{4\alpha s'}{\varepsilon}\right)\right]_s^{s+h}+\frac{\varepsilon A}{2h} \left [\sin \left(\frac{2\alpha s'}{\varepsilon}\right)\right]_s^{s+h}\end{aligned}$$ As $|\sin|\leq 1$, we immediately get: $$\begin{aligned}
\lim_{\varepsilon\rightarrow 0} \frac{u^\varepsilon(0,s+h)-u^\varepsilon(0,s)}{h}=F(L)+\beta \frac{A^2}{2},\end{aligned}$$ which is in contradiction with .
**Acknowledgements.** I would like to thank Cyril Imbert for all his support, help and remarks on this article. I would also like to thank Régis Monneau for all his contribution. This work was partly supported by Labex Bézout, (ANR-10-LABX-58-01) and ANR HJnet, (ANR-12-BS01-0008-01).
|
---
abstract: 'We find a five-parameter family of partial differential systems in two variables with two polynomial Hamiltonians. We give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new.'
author:
- Yusuke Sasano
title: 'Five-parameter family of partial differential systems in two variables\'
---
Introduction
============
In this paper, we present a 5-parameter family of partial differential systems in two variables explicitly given by $$\begin{aligned}
\label{1}
\begin{split}
dq_1&=\frac{\partial H_1}{\partial p_1}dt+\frac{\partial H_2}{\partial p_1}ds, \quad dp_1=-\frac{\partial H_1}{\partial q_1}dt-\frac{\partial H_2}{\partial q_1}ds,\\
dq_2&=\frac{\partial H_1}{\partial p_2}dt+\frac{\partial H_2}{\partial p_2}ds, \quad dp_2=-\frac{\partial H_1}{\partial q_2}dt-\frac{\partial H_2}{\partial q_2}ds
\end{split}\end{aligned}$$ with the polynomial Hamiltonians: $$\begin{aligned}
\label{2}
\begin{split}
H_1 &=H_{VI}(q_1,p_1,t;\alpha_1,\alpha_2,\alpha_3,\alpha_4)\\
&+\alpha_2 p_2\left\{\frac{-(t-1)sq_1+t(s-1)q_2+(t-s)q_1q_2}{t(t-1)(t-s)}+\frac{(q_1-t)q_2(q_2-1)}{t(t-1)(t-\eta)}\right\}\\
&+\alpha_5 p_1\left\{\frac{(t-s)q_1(q_1-1)+t(t-1)(q_1-q_2)}{t(t-1)(t-s)}+\frac{(q_1-t)((t-1)q_1+(q_1-t)q_2)}{t(t-1)(t-\eta)}\right\}\\
&- p_1p_2\left\{\frac{(t-1)(sq_1^2+tq_2^2)-(t-s)q_2(q_1^2+t)-2t(s-1)q_1q_2}{t(t-1)(t-s)}-\frac{(q_1-t)^2q_2(q_2-1)}{t(t-1)(t-\eta)}\right\}\\
&+\frac{\alpha_2\alpha_5(2tq_1-q_1-tq_2+q_1q_2-\eta q_1)}{t(t-1)(t-\eta)},\\
H_2&=\pi(H_1),
\end{split}\end{aligned}$$ where the transformation $\pi$ is explicitly given by $$\begin{aligned}
\begin{split}
\pi:&(q_1,p_1,q_2,p_2,t,s;\alpha_0,\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)\\
&\rightarrow(q_2,p_2,q_1,p_1,s,t;\alpha_0,\alpha_1,\alpha_5,\alpha_3,\alpha_4,\alpha_2).
\end{split}\end{aligned}$$ Here $q_1,p_1,q_2$ and $p_2$ denote unknown complex variables, and $\alpha_0,\alpha_1,\ldots,\alpha_5$ are complex parameters satisfying the relation: $$\label{3}
\alpha_0+\alpha_1+2\alpha_2+\alpha_3+\alpha_4+2\alpha_5=1.$$ This parameter’s relation can be obtained by holomorphy conditions in Theorem \[th;holoG\].
The symbol $H_{VI}(q,p,t;\beta_1,\beta_2,\beta_3,\beta_4)$ denotes the Hamiltonian of the second-order Painlevé VI equations (see [@Sasa5]) given by $$\begin{aligned}
\label{6}
\begin{split}
&t(t-1)(t-\eta)H_{VI}(q,p,t;\beta_1,\beta_2,\beta_3,\beta_4)\\
&=q(q-1)(q-\eta)(q-t)p^2\\
&+\{\beta_1(t-\eta)q(q-1)+2\beta_2q(q-1)(q-\eta)\\
&+\beta_3(t-1)q(q-\eta)+\beta_4 t(q-1)(q-\eta)\}p\\
&+\beta_2\{(\beta_1+\beta_2)(t-\eta)+\beta_2(q-1)\\
&+\beta_3(t-1)+t \beta_4\}q \quad (\beta_0+\beta_1+2\beta_2+\beta_3+\beta_4=1, \quad \eta \in {\Bbb C}-\{0,1\}).
\end{split}\end{aligned}$$
We give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new.
After we review the notion of accessible singularity and local index, we make its holomorphy conditions by resolving the accessible singularities.
Symmetry and holomorphy conditions
==================================
In this section, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new.
\[th;holoG\] Let us consider a polynomial Hamiltonian system with Hamiltonians $H_i \in {\Bbb C}(t,s)[q_1,p_1,q_2,p_2] \ (i=1,2)$. We assume that
$(A1)$ $deg(H_i)=6$ with respect to $q_1,p_1,q_2,p_2$.
$(A2)$ This system becomes again a polynomial Hamiltonian system in each coordinate $r_i, \ i=0,1,\dots,5$[:]{} $$\begin{aligned}
\begin{split}
&r_0:x_0=-p_1((q_1-t)p_1+(q_2-s)p_2-\alpha_0), \ y_0=\frac{1}{p_1}, \ z_0=(q_2-s)p_1, \ w_0=\frac{p_2}{p_1}, \\
&r_1:x_1=-p_1((q_1-\eta)p_1+(q_2-\eta)p_2-\alpha_1), \ y_1=\frac{1}{p_1}, \ z_1=(q_2-\eta)p_1, \ w_1=\frac{p_2}{p_1}, \\
&r_2:x_2=\frac{1}{q_1}, \ y_2=-q_1(q_1p_1+\alpha_2), \ z_2=q_2, \ w_2=p_2, \\
&r_3:x_3=-p_1((q_1-1)p_1+(q_2-1)p_2-\alpha_3), \ y_3=\frac{1}{p_1}, \ z_3=(q_2-1)p_1, \ w_3=\frac{p_2}{p_1}, \\
&r_4:x_4=-p_1(q_1p_1+q_2p_2-\alpha_4), \ y_4=\frac{1}{p_1}, \ z_4=q_2p_1, \ w_4=\frac{p_2}{p_1}, \\
&r_5:x_5=q_1, \ y_5=p_1, \ z_5=\frac{1}{q_2}, \ w_5=-(q_2p_2+\alpha_5)q_2.
\end{split}\end{aligned}$$ Then such a system coincides with the system with two polynomial Hamiltonians .
In each coordinate $r_i, \ i=0,1,\dots,5$, the Hamiltonians $H_{j1}$ and $H_{j2}$ on $U_j \times B$ are expressed as a polynomial in $x_j,y_j,z_j,w_j$ and a rational function in $t$ and $s$, and satisfy the following conditions[: ]{} $$\begin{aligned}
\label{symplectic}
\begin{split}
&dq_1 \wedge dp_1 +dz \wedge dp_2 - dH_1 \wedge dt- dH_2 \wedge ds\\
&=dx_j \wedge dy_j +dz_j \wedge dw_j - dH_{j1} \wedge dt- dH_{j2} \wedge ds \quad (j=1,2,\dots,5),\\
&dq_1 \wedge dp_1 +dq_2 \wedge dp_2 - d(H_1-p_1) \wedge dt- d(H_2-p_2) \wedge ds\\
&=dx_0 \wedge dy_0 +dz_0 \wedge dw_0 - dH_{01} \wedge dt- dH_{02} \wedge ds.
\end{split}\end{aligned}$$
0.1in
(35.87,16.56)(17.90,-24.16)
(35.0200,-9.4700)[(0,0)\[lb\][$p_1$]{}]{}(35.2600,-23.4400)[(0,0)\[lb\][$p_2$]{}]{}(18.5400,-15.2300)[(0,0)\[lb\][$q_1-\eta$]{}]{}(18.6200,-17.2200)[(0,0)\[lb\][$q_2-\eta$]{}]{}(29.7400,-15.3400)[(0,0)\[lb\][$q_1$]{}]{}(29.7400,-17.2200)[(0,0)\[lb\][$q_2$]{}]{}(39.4200,-17.1600)[(0,0)\[lb\][$q_2-s$]{}]{}(49.2600,-15.5100)[(0,0)\[lb\][$q_1-1$]{}]{}(49.3400,-17.1600)[(0,0)\[lb\][$q_2-1$]{}]{}(39.4200,-15.4500)[(0,0)\[lb\][$q_1-t$]{}]{}
\[UraDynkin1\]
\[inv\]
codimension invariant cycles parameter’s relation
------------- ---------------------------------------------- ----------------------
1 $f_2:=p_1$ $\alpha_2=0$
1 $f_5:=p_2$ $\alpha_5=0$
2 $f_0^{(1)}:=q_1-t, \ f_0^{(2)}:=q_2-s$ $\alpha_0=0$
2 $f_1^{(1)}:=q_1-\eta, \ f_1^{(2)}:=q_2-\eta$ $\alpha_1=0$
2 $f_3^{(1)}:=q_1-1, \ f_3^{(2)}:=q_2-1$ $\alpha_3=0$
2 $f_4^{(1)}:=q_1, \ f_4^{(2)}:=q_2$ $\alpha_4=0$
We note that when $\alpha_2=0$, we see that the system admits a particular solution $f_2=0$, and when $\alpha_0=0$, we see that the system admits a particular solution $f_0^{(1)}=f_0^{(2)}=0$.
B[ä]{}cklund transformations
============================
\[th:in3\] The system admits the following transformations as its B[ä]{}ckl-\
und transformations[:]{} with the notation $(*)=(q_1,p_1,q_2,p_2,\eta,t,s;\alpha_0,\alpha_1,\dots,\alpha_5),$ $$\begin{aligned}
\begin{split}
s_1: (*) \rightarrow &\left(q_1+\frac{\alpha_2}{p_1},p_1,q_2,p_2,\eta,t,s;\alpha_0+\alpha_2,\alpha_1+\alpha_2,-\alpha_2,\alpha_3+\alpha_2,\alpha_4+\alpha_2,\alpha_5 \right), \\
s_2: (*) \rightarrow &\left(q_1,p_1,q_2+\frac{\alpha_5}{p_2},p_2,\eta,t,s;\alpha_0+\alpha_5,\alpha_1+\alpha_5,\alpha_2,\alpha_3+\alpha_5,\alpha_4+\alpha_5,-\alpha_5 \right),\\
\pi_1: (*) \rightarrow &(\frac{\eta-q_1}{\eta-1},-(\eta-1)p_1,\frac{\eta-q_2}{\eta-1},-(\eta-1)p_2,\frac{\eta}{\eta-1},\frac{\eta-t}{\eta-1},\frac{\eta-s}{\eta-1};\\
&\alpha_0,\alpha_4,\alpha_2,\alpha_3,\alpha_1,\alpha_5),\\
\pi_2: (*) \rightarrow &(\frac{q_1(t-\eta)}{t-q_1+tq_1-\eta t},-\frac{(t-q_1+tq_1-\eta t)\{(t-q_1+tq_1-\eta t)p_1+\alpha_2(t-1)\}}{t(t-\eta)(\eta-1)},\\
&\frac{q_2(s-\eta)}{s-q_2+sq_2-\eta s},-\frac{(s-q_2+sq_2-\eta s)\{(s-q_2+sq_2-\eta s)p_2+\alpha_5(s-1)\}}{s(s-\eta)(\eta-1)},\\
&\eta,\frac{\eta-t}{1-2t+\eta t},\frac{\eta-s}{1-2s+\eta s};\alpha_3,\alpha_1,\alpha_2,\alpha_0,\alpha_4,\alpha_5),\\
\pi_3: (*) \rightarrow &(\frac{(t-1)q_1}{t-q_1-\eta t+\eta tq_1},\frac{(t-q_1+\eta t(q_1-1))\{(q_1-t)p_1+\alpha_2-\eta t((q_1-1)p_1+\alpha_2)\}}{t(t-1)(\eta-1)},\\
&\frac{(s-1)q_2}{s-q_2-\eta s+\eta sq_2},\frac{(s-q_2+\eta s(q_2-1))\{(q_2-s)p_2+\alpha_5-\eta s((q_2-1)p_2+\alpha_5)\}}{s(s-1)(\eta-1)},\\
&\frac{1}{\eta},\frac{\eta(t-1)}{t-\eta-\eta t+\eta^2 t},\frac{\eta(s-1)}{s-\eta-\eta s+\eta^2 s};\alpha_1,\alpha_0,\alpha_2,\alpha_3,\alpha_4,\alpha_5),\\
\pi_4: (*) \rightarrow &(1-q_1,-p_1,1-q_2,-p_2,1-\eta,1-t,1-s;\alpha_0,\alpha_1,\alpha_2,\alpha_4,\alpha_3,\alpha_5),\\
\pi_5: (*) \rightarrow &(q_2,p_2,q_1,p_1,\eta,s,t;\alpha_0,\alpha_1,\alpha_5,\alpha_3,\alpha_4,\alpha_2).
\end{split}
\end{aligned}$$
The B[ä]{}cklund transformations $s_1,s_2$ are determined by the invariant divisors .
Accessible singularity and local index
======================================
Let us review the notion of [*accessible singularity*]{}. Let $B$ be a connected open domain in $\Bbb C$ and $\pi : {\mathcal W} \longrightarrow B$ a smooth proper holomorphic map. We assume that ${\mathcal H} \subset {\mathcal W}$ is a normal crossing divisor which is flat over $B$. Let us consider a rational vector field $\tilde v$ on $\mathcal W$ satisfying the condition $$\tilde v \in H^0({\mathcal W},\Theta_{\mathcal W}(-\log{\mathcal H})({\mathcal H})).$$ Fixing $t_0 \in B$ and $P \in {\mathcal W}_{t_0}$, we can take a local coordinate system $(x_1,\ldots ,x_n)$ of ${\mathcal W}_{t_0}$ centered at $P$ such that ${\mathcal H}_{\rm smooth \rm}$ can be defined by the local equation $x_1=0$. Since $\tilde v \in H^0({\mathcal W},\Theta_{\mathcal W}(-\log{\mathcal H})({\mathcal H}))$, we can write down the vector field $\tilde v$ near $P=(0,\ldots ,0,t_0)$ as follows: $$\tilde v= \frac{\partial}{\partial t}+g_1
\frac{\partial}{\partial x_1}+\frac{g_2}{x_1}
\frac{\partial}{\partial x_2}+\cdots+\frac{g_n}{x_1}
\frac{\partial}{\partial x_n}.$$ This vector field defines the following system of differential equations $$\label{39}
\frac{dx_1}{dt}=g_1(x_1,\ldots,x_n,t),\ \frac{dx_2}{dt}=\frac{g_2(x_1,\ldots,x_n,t)}{x_1},\cdots, \frac{dx_n}{dt}=\frac{g_n(x_1,\ldots,x_n,t)}{x_1}.$$ Here $g_i(x_1,\ldots,x_n,t), \ i=1,2,\dots ,n,$ are holomorphic functions defined near $P=(0,\dots ,0,t_0).$
\[Def1\] With the above notation, assume that the rational vector field $\tilde v$ on $\mathcal W$ satisfies the condition $$(A) \quad \tilde v \in H^0({\mathcal W},\Theta_{\mathcal W}(-\log{\mathcal H})({\mathcal H})).$$ We say that $\tilde v$ has an [*accessible singularity*]{} at $P=(0,\dots ,0,t_0)$ if $$x_1=0 \ {\rm and \rm} \ g_i(0,\ldots,0,t_0)=0 \ {\rm for \rm} \ {\rm every \rm} \ i, \ 2 \leq i \leq n.$$
If $P \in {\mathcal H}_{{\rm smooth \rm}}$ is not an accessible singularity, all solutions of the ordinary differential equation passing through $P$ are vertical solutions, that is, the solutions are contained in the fiber ${\mathcal W}_{t_0}$ over $t=t_0$. If $P \in {\mathcal H}_{\rm smooth \rm}$ is an accessible singularity, there may be a solution of which passes through $P$ and goes into the interior ${\mathcal W}-{\mathcal H}$ of ${\mathcal W}$.
Here we review the notion of [*local index*]{}. Let $v$ be an algebraic vector field with an accessible singular point $\overrightarrow{p}=(0,\ldots,0)$ and $(x_1,\ldots,x_n)$ be a coordinate system in a neighborhood centered at $\overrightarrow{p}$. Assume that the system associated with $v$ near $\overrightarrow{p}$ can be written as $$\begin{aligned}
\label{b}
\begin{split}
\frac{d}{dt}\begin{pmatrix}
x_1 \\
x_2 \\
\vdots\\
x_{n-1} \\
x_n
\end{pmatrix}=\frac{1}{x_1}\left\{\begin{bmatrix}
a_{11} & 0 & 0 & \hdots & 0 \\
a_{21} & a_{22} & 0 & \hdots & 0 \\
\vdots & \vdots & \ddots & 0 & 0 \\
a_{(n-1)1} & a_{(n-1)2} & \hdots & a_{(n-1)(n-1)} & 0 \\
a_{n1} & a_{n2} & \hdots & a_{n(n-1)} & a_{nn}
\end{bmatrix}\begin{pmatrix}
x_1 \\
x_2 \\
\vdots\\
x_{n-1} \\
x_n
\end{pmatrix}+\begin{pmatrix}
x_1h_1(x_1,\ldots,x_n,t) \\
h_2(x_1,\ldots,x_n,t) \\
\vdots\\
h_{n-1}(x_1,\ldots,x_n,t) \\
h_n(x_1,\ldots,x_n,t)
\end{pmatrix}\right\},\\
(h_i \in {\Bbb C}(t)[x_1,\ldots,x_n], \ a_{ij} \in {\Bbb C}(t))
\end{split}
\end{aligned}$$ where $h_1$ is a polynomial which vanishes at $\overrightarrow{p}$ and $h_i$, $i=2,3,\ldots,n$ are polynomials of order at least 2 in $x_1,x_2,\ldots,x_n$, We call ordered set of the eigenvalues $(a_{11},a_{22},\cdots,a_{nn})$ [*local index*]{} at $\overrightarrow{p}$.
We are interested in the case with local index $$\label{integer}
(1,a_{22}/a_{11},\ldots,a_{nn}/a_{11}) \in {\Bbb Z}^{n}.$$ These properties suggest the possibilities that $a_1$ is the residue of the formal Laurent series: $$y_1(t)=\frac{a_{11}}{(t-t_0)}+b_1+b_2(t-t_0)+\cdots+b_n(t-t_0)^{n-1}+\cdots \quad (b_i \in {\Bbb C}),$$ and the ratio $(1,a_{22}/a_{11},\ldots,a_{nn}/a_{11})$ is resonance data of the formal Laurent series of each $y_i(t) \ (i=2,\ldots,n)$, where $(y_1,\ldots,y_n)$ is original coordinate system satisfying $(x_1,\ldots,x_n)=(f_1(y_1,\ldots,y_n),\ldots,f_n(y_1,\ldots,y_n)), \ f_i(y_1,\ldots,y_n) \in {\Bbb C}(t)(y_1,\ldots,y_n)$.
If each component of $(1,a_{22}/a_{11},\ldots,a_{nn}/a_{11})$ has the same sign, we may resolve the accessible singularity by blowing-up finitely many times. However, when different signs appear, we may need to both blow up and blow down.
The $\alpha$-test, $$\label{poiuy}
t=t_0+\alpha T, \quad x_i=\alpha X_i, \quad \alpha \rightarrow 0,$$ yields the following reduced system: $$\begin{aligned}
\label{ppppppp}
\begin{split}
\frac{d}{dT}\begin{pmatrix}
X_1 \\
X_2 \\
\vdots\\
X_{n-1} \\
X_n
\end{pmatrix}=\frac{1}{X_1}\begin{bmatrix}
a_{11}(t_0) & 0 & 0 & \hdots & 0 \\
a_{21}(t_0) & a_{22}(t_0) & 0 & \hdots & 0 \\
\vdots & \vdots & \ddots & 0 & 0 \\
a_{(n-1)1}(t_0) & a_{(n-1)2}(t_0) & \hdots & a_{(n-1)(n-1)}(t_0) & 0 \\
a_{n1}(t_0) & a_{n2}(t_0) & \hdots & a_{n(n-1)}(t_0) & a_{nn}(t_0)
\end{bmatrix}\begin{pmatrix}
X_1 \\
X_2 \\
\vdots\\
X_{n-1} \\
X_n
\end{pmatrix},
\end{split}
\end{aligned}$$ where $a_{ij}(t_0) \in {\Bbb C}$. Fixing $t=t_0$, this system is the system of the first order ordinary differential equation with constant coefficient. Let us solve this system. At first, we solve the first equation: $$X_1(T)=a_{11}(t_0)T+C_1 \quad (C_1 \in {\Bbb C}).$$ Substituting this into the second equation in , we can obtain the first order linear ordinary differential equation: $$\frac{dX_2}{dT}=\frac{a_{22}(t_0) X_2}{a_{11}(t_0)T+C_1}+a_{21}(t_0).$$ By variation of constant, in the case of $a_{11}(t_0) \not= a_{22}(t_0)$ we can solve explicitly: $$X_2(T)=C_2(a_{11}(t_0)T+C_1)^{\frac{a_{22}(t_0)}{a_{11}(t_0)}}+\frac{a_{21}(t_0)(a_{11}(t_0)T+C_1)}{a_{11}(t_0)-a_{22}(t_0)} \quad (C_2 \in {\Bbb C}).$$ This solution is a single-valued solution if and only if $$\frac{a_{22}(t_0)}{a_{11}(t_0)} \in {\Bbb Z}.$$ In the case of $a_{11}(t_0)=a_{22}(t_0)$ we can solve explicitly: $$X_2(T)=C_2(a_{11}(t_0)T+C_1)+\frac{a_{21}(t_0)(a_{11}(t_0)T+C_1){\rm Log}(a_{11}(t_0)T+C_1)}{a_{11}(t_0)} \quad (C_2 \in {\Bbb C}).$$ This solution is a single-valued solution if and only if $$a_{21}(t_0)=0.$$ Of course, $\frac{a_{22}(t_0)}{a_{11}(t_0)}=1 \in {\Bbb Z}$. In the same way, we can obtain the solutions for each variables $(X_3,\ldots,X_n)$. The conditions $\frac{a_{jj}(t)}{a_{11}(t)} \in {\Bbb Z}, \ (j=2,3,\ldots,n)$ are necessary condition in order to have the Painlevé property.
Construction of the holomorphy conditions
=========================================
In this section, we will give the holomorphy conditions $r_i \ (i=0,1,\ldots,5)$ by resolving some accessible singular loci of the system .
In order to consider the singularity analysis for the system , as a compactification of ${\Bbb C}^4$ which is the phase space of the system , we take 4-dimensional complex manifold $\mathcal S$ given in the paper [@Sasa5]. This manifold can be considered as a generalization of the Hirzebruch surface.
We easily see that the rational vector field $\tilde v$ associated with the system satisfies the condition: $$\tilde v \in H^0({\mathcal S},\Theta_{\mathcal S}(-\log{\mathcal H})({\mathcal H})).$$
0.1in
( 55.1000, 18.0000)( 15.7000,-28.0000)
(16.7000,-29.2000)[(0,0)\[lb\][$Y_3$]{}]{}(67.4000,-29.0000)[(0,0)\[lb\][$W_4$]{}]{}
(31.3000,-28.2000)[(0,0)\[lb\][$C_0$]{}]{}(33.1000,-23.9000)[(0,0)\[lb\][$C_1$]{}]{}(34.2000,-20.1000)[(0,0)\[lb\][$C_2$]{}]{}(41.3000,-18.4000)[(0,0)\[lb\][$C_3$]{}]{}
\[Garnierfig1\]
\[lem1\] The rational vector field $\tilde v$ has the following accessible singular loci [(see figure 2)]{}$:$ $$\left\{
\begin{aligned}
C_0=&\{(X_3,Y_3,Z_3,W_3)|X_3=t,Z_3=s,Y_3=0\}\\
&\cup \{(X_4,Y_4,Z_4,W_4)|X_4=t,Z_4=s,W_4=0\} \cong {\Bbb P}^1,\\
C_1=&\{(X_3,Y_3,Z_3,W_3)|X_3=\eta,Z_3=\eta,Y_3=0\}\\
&\cup \{(X_4,Y_4,Z_4,W_4)|X_4=\eta,Z_4=\eta,W_4=0\} \cong {\Bbb P}^1,\\
C_2=&\{(X_3,Y_3,Z_3,W_3)|X_3=1,Z_3=1,Y_3=0\}\\
&\cup \{(X_4,Y_4,Z_4,W_4)|X_4=1,Z_4=1,W_4=0\} \cong {\Bbb P}^1,\\
C_3=&\{(X_3,Y_3,Z_3,W_3)|X_3=Z_3=Y_3=0\}\\
&\cup \{(X_4,Y_4,Z_4,W_4)|X_4=Z_4=W_4=0\} \cong {\Bbb P}^1.
\end{aligned}
\right.$$
Here, the coordinate systems $(X_i,Y_i,Z_i,W_i) \ (i=3,4)$ (see [@Sasa5]) are explicitly given by $$\begin{aligned}
\begin{split}
(X_3,Y_3,Z_3,W_3)=\left(q_1,\frac{1}{p_1},q_2,\frac{p_2}{p_1} \right),\\
(X_4,Y_4,Z_4,W_4)=\left(q_1,\frac{p_1}{p_2},q_2,\frac{1}{p_2} \right).
\end{split}\end{aligned}$$
This lemma can be proven by a direct calculation.
Next, we calculate its local index at the point $P:=\{(X_3,Y_3,Z_3,W_3)|X_3=t,Z_3=s,Y_3=W_3=0\}$.
[**Step 0:**]{} We make a change of variables. $$X_3^{(1)}=X_3-t, \quad Y_3^{(1)}=Y_3, \quad Z_3^{(1)}=Z_3-s, \quad W_3^{(1)}=W_3.$$ Around the point $P$, we rewrite the system as follows: $$\begin{aligned}
\frac{d}{dt}\begin{pmatrix}
X_3^{(1)} \\
Y_3^{(1)} \\
Z_3^{(1)} \\
W_3^{(1)}
\end{pmatrix}&=\frac{1}{Y_3^{(1)}}\left\{\begin{pmatrix}
2 & -\alpha_0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & \frac{\alpha_5}{t-s} & 0 & 0
\end{pmatrix}\begin{pmatrix}
X_3^{(1)} \\
Y_3^{(1)} \\
Z_3^{(1)} \\
W_3^{(1)}
\end{pmatrix}+\cdots\right\}.
\end{aligned}$$ We see that this system has its local index $(2,1,1,0)$ at the point $P$.
For the remaining accessible singular loci, the local index is same.
\[prop3\] If we resolve the accessible singular loci given in Lemma \[lem1\] by blowing-ups, then we can obtain the canonical coordinate systems $r_i \ (i=0,1,3,4)$.
[*Proof.*]{} By the following steps, we can resolve the accessible singular locus $C_0$.
[**Step 1:**]{} We blow up along the curve $C_0$. $$X_3^{(2)}=\frac{X_3^{(1)}}{Y_3^{(1)}}, \quad Y_3^{(2)}=Y_3^{(1)}, \quad Z_3^{(2)}=\frac{Z_3^{(1)}}{Y_3^{(1)}}, \quad W_3^{(2)}=W_3^{(1)}.$$
[**Step 2:**]{} We blow up along the surface $\{(X_3^{(2)},Y_3^{(2)},Z_3^{(2)},W_3^{(2)})|X_3^{(2)}=-Z_3^{(2)} W_3^{(2)}+\alpha_0\}$ $$X_3^{(3)}=\frac{X_3^{(2)}+Z_3^{(2)} W_3^{(2)}-\alpha_0}{Y_3^{(2)}}, \quad Y_3^{(3)}=Y_3^{(2)}, \quad Z_3^{(3)}=Z_3^{(2)}, \quad W_3^{(3)}=W_3^{(2)}.$$ By choosing a new coordinate system as $$(x_0,y_0,z_0,w_0)=(-X_3^{(3)},Y_3^{(3)},Z_3^{(3)},W_3^{(3)}),$$ we can obtain the coordinate system $r_0$.
For the remaining accessible singular loci, the proof is similar. Thus, we have completed the proof of Proposition \[prop3\].
After a series of explicit blowing-ups given in Proposition \[prop3\], we obtain the smooth projetive $4$-fold $\tilde{\mathcal S}$ and a birational morphism $\varphi:\tilde{\mathcal S} \rightarrow {\mathcal S}$. Its canonical divisor $K_{\tilde{\mathcal S}}$ is given by $$K_{\tilde{\mathcal S}}=-3\tilde{\mathcal H}-\sum_{i=0}^3{\mathcal E}_i,$$ where the symbol $\tilde{\mathcal H}$ denotes the proper transform of ${\mathcal H}$ by $\varphi$ and ${\mathcal E}_i$ denote the exceptional divisors obtained by Step $1$ [(see Proof of Proposition \[prop3\])]{}.
[99]{}
H. Kimura, [*Uniform foliation associated with the Hamiltonian system ${\mathcal H}_{n}$*]{}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) [**20**]{} (1993), no. 1, 1–60.
H. Kimura and K. Okamoto, [*On the polynomial Hamiltonian structure of the Garnier systems*]{}, J. Math. Pures Appl. [**63**]{} (1984), 129–146.
Y. Sasano, [*Coupled Painlevé V systems in dimension [4 ]{}*]{}, Funkcial. Ekvac. [**49**]{} (2006), 133–161.
Y. Sasano, [*The phase space of coupled Painlevé III system in dimension four*]{}, to appear in Kyoto Journal.
Y. Sasano, [*Coupled Painlevé II systems in dimension four and the systems of type ${A_4}^{(1)}$*]{}, to appear in Tohoku Journal.
Y. Sasano, [*Higher order Painlevé equations of type ${D_l}^{(1)}$*]{}, RIMS Kokyuroku [**1473**]{} (2006), 143–163.
Y. Sasano, [*Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D_6^{(1)}$, II*]{}, RIMS Kokyuroku Bessatsu. [**B5**]{} (2008), 137–152.
|
---
author:
- ROBERTO SEBASTIANI and SILVIA TOMASI
bibliography:
- 'sathanbook.bib'
- 'rs\_ownrefs.bib'
- 'rs\_refs.bib'
- 'st\_refs.bib'
- 'mrg\_a-l.bib'
- 'mrg\_m-z.bib'
- 'rs\_specific\_refs.bib'
title: '[Optimization Modulo Theories with Linear Rational Costs]{}'
---
Authors’ addresses: Roberto Sebastiani ([roberto.sebastiani@unitn.it]{}), Silvia Tomasi ([silvia.tomasi@disi.unitn.it]{}), <span style="font-variant:small-caps;">DISI</span>, Università di Trento, via Sommarive 9, I-38123 Povo, Trento, Italy. Roberto Sebastiani is supported by Semiconductor Research Corporation (SRC) under GRC Research Project 2012-TJ-2266 WOLF.
Introduction {#sec:intro}
============
Optimization in {#sec:optsmt}
================
Procedures for and {#sec:algorithms}
===================
An offline schema for {#sec:algorithms_offline}
----------------------
An inline schema for {#sec:algorithms_inline}
---------------------
Extensions to {#sec:algorithms_omlaratplus}
--------------
Experimental evaluation {#sec:expeval}
=======================
Encodings. {#sec:expeval_enc}
----------
Comparison on LGDP problems {#sec:expeval_lgdp}
---------------------------
### The strip-packing problem. {#sec:expeval_sp}
### The zero-wait jobshop problem. {#sec:expeval_js}
### Discussion {#sec:expeval_disc}
Comparison on SMT-LIB problems {#sec:expeval_smtlib}
------------------------------
Comparison on SAL problems {#sec:expeval_sal}
--------------------------
Comparison on pseudo-Boolean SMT problems {#sec:expeval_pb}
-----------------------------------------
Related work. {#sec:related}
=============
Conclusions and Future Work {#sec:concl}
===========================
Appendix: Proof of the Theorems {#sec:appendix}
===============================
|
---
author:
- 'F. Brimioulle, M. Lerchster, S. Seitz, R. Bender, and J. Snigula'
date: 'Received; accepted'
title: 'Photometric redshifts for the CFHTLS-Wide [^1]'
---
[We want to derive bias free, accurate photometric redshifts for those fields of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) Wide Data which are covered in the $u^*, g', r', i'$ and $z'$ filters and are public on January 2008. These are 21, 5 and 11 square degrees in the `W1`, `W3` and `W4` fields with photometric data for $1.397.545$ (`W1`), $366.190$ (`W3`) and $833.504$ (`W4`) galaxies i.e. for a total of $2.597.239$ galaxies.]{} [We use the photometric redshift code PHOTO-z of Bender et al. (2001).]{} [To study the reliability of the photometric redshifts for the CFHTLS broad band filter set we first derive redshifts for the CFHTLS-Deep field `D1`, and compare the results to the spectroscopic and photometric redshifts presented in Ilbert et al. (2006). After that we compare our redshifts for the `W1`, `W3` and `W4` fields to about 7500 spectroscopic redshifts from the VVDS therein. For galaxies with $17.5 \leq i'_{AB} \leq 22.5$ the accuracies and outlier rates become $\sigma_{\rm \Delta z/(1+z)} = 0.033 $, $\eta\sim$ 2 % for `W1`, $\sigma_{\rm \Delta z/(1+z)} = 0.037 $, $\eta \sim $ 2% for `W3` and $\sigma_{\rm \Delta z/(1+z)} = 0.035 $, $\eta=\sim$ 2.5 % outliers for `W4` fields.\
Finally we consider the photometric redshifts of Erben et al. (2008) which were obtained with exactly the same photometric catalog using the BPZ-redshift code and compare them with our computed redshifts. For the total galaxy sample with about 9000 spectroscopic redshifts from VVDS, DEEP2 or SDSS we obtain a $\sigma_{\rm \Delta z/(1+z)}=0.04$ and $\eta=5.7\%$ for the PHOTO-z redshifts. We also merge the subsample with good photometric redshifts from PHOTO-z with that one from BPZ to obtain a sample which then contains ‘secure’ redshifts according to both the PHOTO-z and the BPZ codes. This sample contains about 6100 spectra and the photometric redshift qualities become $\sigma_{\rm \Delta z/(1+z)}=0.037$ and $\eta=1.0\%$ for our PHOTO-z redshifts. ]{} [We conclude that this work provides a bias free, low dispersion photometric redshift catalog (given the depth and filter set of the data), that we have criteria at hand to select a ‘robust’ subsample with fewer outliers. Such a subsample is very useful to study the redshift dependent growth of the dark matter fluctuations with weak lensing cosmic shear analyses or to investigate the redshift dependent weak lensing signal behind clusters of galaxies in the framework of dark energy equation of state constraints.\
The PHOTO-z photometric redshift catalog is provided on request. Send emails to fabrice@usm.lmu.de]{}
Introduction {#sec:intro}
============
The CFHTLS Wide survey plans to image 170 square degrees in four patches of 25 to 72 square degrees through the whole filter set ($u^*
g' r' i' z'$) down to $i'=24.5$. This survey will (among other goals) allow to study the evolution of galaxies, the large scale structures as traced by galaxies, groups and clusters of galaxies. Due to its superb PSF-quality one can also directly study the line of sight matter distribution through weak lensing analysis. Full exploitation of the data requires the redshift of galaxies to be known in order to obtain the 3 dimensional arrangement of galaxies and to turn the observed galaxy colors into restframe properties. Obtaining spectra for millions of galaxies is impossible at the moment. The photometric redshift technique, however, can provide redshifts for large numbers of faint galaxies with an accuracy that eg. allows galaxy evolution studies or 3D lensing analysis. For high quality photometric redshifts the photometry should cover a wide wavelength range. The necessary wavelength range has to be adapted to the depth of the survey. For a survey as shallow as SDSS, NIR data are not essential since basically all redshifts are low ($z<1$), and there are hardly any SED-z degeneracies. This holds as long as U-band data are available, which locate the Balmer or $4000$ Angstroem break; therefore, the central wavelength of the U-band (or in general, the bluest) filter determines the redshift above which photometric redshifts are trustable (see Gabasch et al. 2007 and Niemack et al. 2008 for the impact of GALEX data on photoz-accuracies). The CFHTLS-Wide (`W1`-`W4`) and Deep surveys are deeper than SDSS, which implies that larger redshifts but also smaller absolute luminosities and thus different SED-types are traced. In this situation SED-z degeneracies can occur (eg. degeneracies between a redshift $z=0.7$ emission line galaxy and a ’normal’ $z=1.2$ galaxy) which can be cured either with bluer U-band (from space) or with NIR data. The classical ‘catastrophic’ failures become really relevant only in data as deep or deeper than the CFHTLS-Deep fields. Only in data as deep as this there is a significant number of galaxies with sufficiently high redshifts where Lyman break and $4000$ Angstrom might be misidentified. This effect could be supressed with absolute luminosity priors, and (almost) avoided with NIR data.\
The principle disadvantage of photometric redshifts is the relatively low redshift resolution (due to the width of filters) compared to spectroscopic redshifts. On the other hand, photometric redshifts have turned into a vital tool in resolving redshift ambiguities where spectra show single (emission) line features only (Lilly et al. 2006).\
\
In the Erben et al. [@erben08] we combined publicly available[^2] $u^*g'r'i'z'$ data, remapped and coadded the frames, derived an $i'$-band detected photometric catalog; we also made pixel based data and the photometric catalog public. This catalog also contains a photometric redshift estimate obtained with the BPZ-code (Benítez et al. 2000), using the original CWW-templates and redshift priors developed from Benítez for the HDF. The redshifts suffer a bias for $z<1.0$, where low redshift galaxies are at too high redshift and high redshift galaxies are at too low redshifts. Also, it was emphasized in Erben et al. [@erben08] that photometric redshifts in this catalog are not trustable above redshifts of $z=1.4$. Providing redshifts with strongly reduced bias is the main goal of this paper: In section \[sec:data\] we briefly summarize our previous work. We then describe the spectroscopic data that we use for calibration and for redshift accuracy tests in section \[sec:spec\]. We shortly describe the photometric redshift method of Bender et al. [@bender01] and our relative zeropoint recalibration method in section \[sec:photozmethod\]. In section \[sec:photoztest\] we demonstrate that this redshift code is bias free and works as good as the method of Ilbert et al. [@ilbert06] by comparing to spectroscopic and photometric redshift results of Ilbert et al. for the CFHTLS `D1` field. We then present photometric redshifts for the `W1`, `W3` and `W4` fields, and infer their quality (as a function of SED type and brightness) from spectroscopic data in these fields. Our redshifts are then compared to the previous ones from Erben et al. [@erben08]. We also empirically correct for their redshift bias. We end up with two sets of photometric redshifts, which can also be used in combination, to select a subsample with most reliable redshifts.
Data Acquisition, Reduction and Photometric Catalogs {#sec:data}
====================================================
---------- ---------- ----------- -- -- --
ID RA Dec
(J2000) (J2000)
`W1` 02:18:00 -07:00:00
`W2`[^3] 08:54:00 -04:15:00
`W3` 14:17:54 +54:30:31
`W4` 22:13:18 +01:19:00
---------- ---------- ----------- -- -- --
: CFHTLS Wide Fields: location[]{data-label="tab:fieldpos"}
{width="8.5cm"} {width="8.5cm"}
Here we briefly review the data acquisition, the data reduction steps and the creation of the multicolor catalogs; more details can be found in Erben et al. [@erben08].\
The data used in this analysis are taken in the framework of the synoptic CFHTLS-Wide observations with the MegaPrime instrument mounted at the Canada-France-Hawaii Telescope (CFHT). See <http://www.cfht.hawaii.edu/Science/CFHTLS/> and <http://terapix.iap.fr/cplt/oldSite/Descart/summarycfhtlswide.html> for further information on survey goals and survey implementation. We consider all [`Elixir`]{}processed CFHTLS-Wide fields with observations in all five optical bands $u^*g'r'i'z'$ which are publicly available. After downloading all data from CADC we further process them with our GaBoDS/THELI pipeline (astrometric solution, remapping, stacks). The stacked data have a pixel size of 0.186”, a typical PSF of 0.8” and limiting AB-magniudes of about 24.5 ($5\sigma$ within a 2” aperture for a point source) in the $i'$-band.\
For the creation of the multicolor catalogs we first cut all images in the filter $u^*g'r'i'z'$ of a given field to the same size. We then measure the seeing in each band and convolve all images with a Gaussian to degrade the seeing to that of the worst band.
For object detection we use [`SExtractor`]{}in dual-image mode with the unconvolved $i'$-band image as the detection image. We measure the fluxes in apertures in the convolved images and obtain aperture colors. The aperture we use for photoz estimates has a diameter of $1.86''$. It is important to keep track of locations which have increased photometric errors that are not accounted for in the [`SExtractor`]{}flux errors. These are: halos of very bright stars, defraction spikes of stars, areas around large and extended galaxies and various kinds of image reflections. Masks are automatically generated but then finalized by human eye. These masks can also be used as masks where shape estimates are unreliable, and they can be obtained from Erben et al. [@erben08] on request. We generate photometric redshifts for all objects. They have a non-zero-flag (equal to the MASK value in Erben et al. 2008) if photometry and thus redshifts (and possibly also shape estimates) could not be trustable. The fraction of flagged objects/area is about 20 percent. This is in line with conservative flagging, e.g., in previous work (compare Ilbert et al. 2006).
Spectroscopic Redshifts {#sec:spec}
=======================
The CFHTLS Wide fields `W1` and `W4` have a good spectroscopic coverage: Le Fèvre et al. [@lefevre04] and [@lefevre05] have released a catalog of 8981 spectra of galaxies, stars and QSOs with 17.5 $<$$i_{AB}$$ < $ 24.0 in the VVDS 0226-04 field (which is located within the W1-field). The spectroscopic redshifts are within 0 $<$ z $<$ 5, with a median redshift of about 0.76. The sample covers 0.5 $deg^2$ of sky area. For the CFHTLS `W4` there are 17928 spectra of galaxies, stars and QSOs (Le Fèvre et al. 2004, 2005, Garilli et al. 07) located in the VVDS-F22 field with a magnitude limit of $i_{AB} < 22.5$. This sample covers 4 ${\rm deg}^2$ of sky area.\
The online database gives access to the redshifts and quality flags, to the multi-wavelength photometric information, as well as to the images and VIMOS spectra. The data can be accessed via the CENCOS[^4]web tool.\
The CFHTLS `W3` has public spectroscopic data from the DEEP survey[^5] (Davis et al. 2003, 2007; Vogt et al. 2005; Weiner et al. 2005). The DEEP1 redshift catalog [^6] contains 658 objects with a median redshift of z = 0.65. The DEEP2 DR3 redshift catalog [^7] contains 47700 unique objects with redshifts $>$ 0.7 and covers 4 regions, each $120'\times30'$ large. The targets were selected from the CFHT12K BRI imaging, eligible DEEP2 targets have $18.5\leq
R_{AB}\leq24.1$. The region on the Groth Survey Strip, with $120'\times15'$ has an overlap to the CFHTLS `W3` field.\
For the comparison to photometric redshifts we consider galaxies with trustworthy (for the `W1` and `W4` $\geq$ 95%, for the `W3` $\equiv 100\%$) spectroscopic redshifts only. Due to low S/N (at high redshift) and the limited wavelength ranges of the spectra only 2933 objects on the CFHTLS `W1`, 410 objects on the CFHTLS `W3` and 3688 objects on the CFHTLS `W4` could be considered. In Fig. \[FigHistPhotSpecComp\_1\_4\] we compare the photometric redshift distribution with the VVDS spectroscopic redshift distribution. The agreement is very good.\
Our data from patches `W3` and `W4` have complete SDSS coverage [^8], from patch `W1` only southern pointings `W1p4m0`, `W1p3m0` and `W1p1m1` have SDSS overlap. Via the flexible web-interface `SkyServer` of the Catalog Archive Server (CAS), we get access to the spectra catalog.\
For our purpose we only consider objects clearly classified as galaxy and a redshift trustworthy $\geq$ 95%. By matching the SDSS catalog with our photometric catalog we end up with 39 objects on the CFHTLS `W1`, 180 objects on the CFHTLS `W3` and 309 objects on the CFHTLS `W4`. In total there are 528 objects with the spectroscopic redshift from the SDSS (Adelman-McCarthy et al. 2007).
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Photometric redshift method and photometric calibration of CFHTLS Wide and Deep data {#sec:photozmethod}
====================================================================================
We use the PHOTO-z code of Bender et al. [@bender01] (see also Gabasch et al. 2004 about the construction of SED templates). The code calculates for each SED the full redshift likelihood function including priors for redshift and absolute luminosity. The stepsize for the redshift grid is equal to $0.01$. Our aim in this paper is to estimate photometric redshifts from the optical bands of the CFHTLS-Wide data based on approved SED-templates and prior settings from earlier publications. The SEDs and priors used for this work are the same as in Bender et al. [@bender01] and Gabasch et al. 2004. Development of new SED-templates adapted to the CFHTLS data, application of better adjusted priors and further investigations, data from near infrared bands will be part of an upcoming paper (Lerchster et al. in prep.). Generally, our priors are weak and do hardly influence the result. The luminosity prior deals the fact that the absolute luminosity of galaxies are limited and do not exceed a certain value. It is flat over a broad range of restframe luminosities with a supression of absolute Magnitudes brighter than -25 and fainter than -13 by a factor of 2 in probablilty at these luminosity values. The relatively strongest prior is the redshift prior for old stellar populations. The redshift prior considers that certain SED-Types do not exist at higher redshifts, it supresses e.g. the probability for elliptical galaxies at a redshift of 0.8 by a factor of 2. We choose a prior which makes red SED types at $z=0.6$ and S0-like galaxies at $z=1$ one fifth as likely as at $ z=0.1$. The redshift priors for other SED types are almost flat. The redshift of the highest probability among all SEDs becomes the ‘photometric redshift’ of the object. The redshift ‘error’ of the object is obtained as $$\triangle z_{\rm phot}=\sqrt {\sum_{i,j} (z_i -
z_{max})^2 \cdot P_{ij}},$$ where the sum runs over all (discrete) redshifts $z_i$, and all SEDs and $P_{ij}$ is the contribution of the j-th SED to the total (normalized) probability function at redshift $z_i$. Hence, the meaning of the ‘error’ is how well the galaxy is locatable in redshift space around the ‘best’ redshift. Sometimes, the redshift probabilities have a maximum at another distinct redshift (where so called ‘catastrophic outliers’ could arise). The potential redshift-SED-degeneracy can be read off in the chi-squares of the most likely and the second most likely SED (which in general is different from the redshift likelihood ratios).
They could be improved with the now available, much larger spectroscopic and photometric datasets. This is subject of future work and will allow to further improve the photometric redshift accuracy.\
The photometric calibration of the data is very important. Small errors in the photometric zero-points or false assumptions on the wavelength dependent transmission of the system (sky, telescope optics, filters, CCDs) have to be avoided or corrected. As throughput of the system we use the filter curves in <http://www1.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/CFHTLS-SG/docs/extra/filters.html>. These include optics (wide field corrector, image stabilizing plate, camera window), the mirror (approximated with the reflectivity of freshly alumium coated glass) and the CCDs (QE is given only between 350 and 1000 nm). For the atmospheric extinction we used [http://www.cfht.hawaii.edu/Instruments/ObservatoryManual/
om-extinction.gif](http://www.cfht.hawaii.edu/Instruments/ObservatoryManual/
om-extinction.gif) (for the blue optical part) where extinction is featureless. This wavelength dependent extinction shifts the effective wavelength of the filters to the red (relevant for the $u^*$-band, where it implies a shift of about 20 Angstroems.\
With these ingredients we calculate the location of the Pickles stellar libary stars (Pickles 1998) in color-color diagrams and compare them with colors of observed stars. The observed comparison stars are selected in the central region of the frames using their [`SExtractor`]{}CLASS\_STAR and SExtractor flag parameters. We derive their aperture colors (after seeing matched convolution). An example is shown in Fig.\[FigW1p2p3color\]: red dots are used for measured stellar colors, and blue dots are used for the Pickles libary stars. Stellar sequences are well located and can evidently be used to measure relative zeropoint offsets.\
Our $u^*-r'$ vs $r'-i'$ diagram looks very different from that of Erben et al. [@erben08], where the stars show a huge scatter in $u^*-r'$ colors for large values of $r'-i'$. We show only the brightest (unsaturated) stars with highest [`SExtractor`]{}CLASS\_STAR values, which implies that photometric errors are very small for stars in our diagrams. This then allows, to measure relative zeropoint offsets and to see whether the spread of stellar colors and the shape of the color-colors diagrams also agrees with expectations from stellar libaries for the assumed system throughput.\
The $r'-i'$ vs $g'-r'$ diagram of observed and libary stars has a very strong curvature at $g'-r'=1.2$. This implies, that one can adjust the zeropoint offsets of the $g'$ [*and*]{} $i'$-bands relative to the $r'$-band very well. If this is done one can proceed with the $z'$-band, using, e.g. the $g'-z'$ colors. In this way one gets zeropoints, for which stars have colors consistent to the Pickles libary. After these adjustments g’, r’ and i’ band data usually match the expected curve very well. For $z'$-band data one might expect larger ‘scatter’ around the Pickles points, because some fields do show considerable amount of fringes, which can lead to systematic (relative to the fringe pattern) magnitude offsets for some of the stars.
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We finally also show all examples which involve $u^*$-band data. They don’t match the Pickles stars, since they occupy a larger interval in the $u^*-g'$ color (which can’t be fixed by a zeropoint offset, obviously). Since the $g'-r'$ color is corrected for zeropoint offsets already, it means, that stars, which are blue in $g'-r'$ are bluer than predicted in $u^*-g'$ for $u^*-g'<1.7$ than expected. This could be explained by the fact that stars targetted by CFHTLS are more metal poor than those in the solor neighbourhood and therefore bluer in the UV . Alternatively, if throughput is the explanation, it would surprisingly imply that the throughput in the blue parts of the $u^*$-band has been underestimated; this does not appear very likely. On the other hand, if the mismatch of colors is due to metal poor stars (UV-excess), then the effect might have shown up more strongly in the deeper FORS Deep Field (Gabasch et al. 2004), where more of the halo is traced, and where the U-band filter curve is considerably bluer. As long as one cannot firmly identify the reason for this strech in the $u^*-g'$ color, one could claim that it is not obvious, whether a ‘good’ stellar color-color diagram involving $u^*$-band data should match the Pickles colors at the blue or red end (or somewhere else). Furtheron difficulties in measuring the transmission function of the CFHT-$u^*$ are well known, as several versions of the transmission curve can be found in the web which additionally complicates the analysis. The effect described above shows up in any CFHT $u^*$-band data we investigated (DEEP and WIDE fields), independent of the applied reduction pipeline but not in other fields we previously investigated (e.g. ESO DPS, in particular GOODS-S, compare Gabasch et al. 2004). Nevertheless, any remaining systematic effects leading to systematic errors in the zeropoint offset determination can be detected and corrected by calibrating the zeropoints using spectroscopic redshifts. We therefore looked into subfields with spectroscopic data (we took `W1p2p2, W1p2p3, W4m0m0, W4m0m1, W4m0m2, W4p1m0, W4p1m1, W4p1m2, W4p2m0, W4p2m1, W4p2m2`) and investigated how the color-color diagrams (involving $u^*$-band) looked like when photometric and spectroscopic redshifts matched well. It turned out, that a slight shift to the color-color diagrams of stars relative to that which matches the stars at the red end and which is shown in Fig. \[FigW1p2p3color\] was necessary, for all the fields, and that this shift was consistent in size from field to field. We took that as a description how observed stars have to look relative to the Pickles libary, and calibrated other fields without spectroscopic data like that. We have tested how good this empiric photometric ZP-calibration works in the fields where we predicted the photometric redshifts based on the ‘ideal empirical color-color diagram’ and compared to available spectroscopic data (this was done for the fields `W3m1m2, W3m1m3` using DEEP2 spectra and for many subfields of `W1`, `W3` and `W4` using SDSS redshifts).
{width="14cm"}
Photometric Redshifts: Accuracy tests with the CFHTLS `D1` data {#sec:photoztest}
===============================================================
Results for the complete sample
-------------------------------
We had been testing our photoz-method previously on the FDF, GOODS, MUNICS and COSMOS fields (Gabasch et al. 2007, 2006, 2004, Feulner et al. 2006, 2005; Gabasch et al. 2004 and Drory et al. 2001). Since any photometric redshift method is likely to fail if the system throughput is misunderstood, we first test our method on the CFHTLS `D1` subfield, which has many spectra from the VVDS 0226-04 field, and where photometric redshifts have been derived by Ilbert et al. [@ilbert06] before. For the photometry of this deep field we use the $u^*g'r'i'z'$ science data from the T0003-release (including their weight and mask frames)[^9]. We then obtain photometric catalogs in the same way as described before. For the spectroscopic sample we use all matchable spectra which fulfill the redshift quality criteria defined in Section 3. The match of spectroscopic and photometric redshifts is quantified the same way as in Ilbert et al. [@ilbert06], i.e., we define the outlier fraction and redshift accuracy of non-outliers as: $$\eta= {\rm fraction \; of \; outliers \; with } \;
|z_{spec}-z_{phot}|/(1+z_{spec})> 0.15$$ $$\sigma_{\rm \Delta z/(1+z)} = 1.48 \times median (|z_{\rm spec}-z_{\rm phot}
|/(1+z_{\rm spec}))_{\rm non-outliers}.
\label{sigclip}$$ This definition of $\sigma_{\rm \Delta z/(1+z)} $ is quite different from the ’true’ dispersion $$\sigma =
\sqrt {\sum_{i}^{N_{\rm spec}}
(z_{i, {\rm photz}} - z_{i, {\rm spec}})^2 / (N_{\rm spec}-1) }
\quad,
\label{sigtrue}$$ since it describes only the typical redshift deviation within a narrow range around the true value. We recommend to also consider $\sigma$ in parallel to $\sigma_{\rm \Delta z/(1+z)}$ and $\eta$, since this tells how ‘off’ outliers typical are. Also, if one compares the performance of photometric redshifts with optical data alone to the case where eg. NIR data are added, it usually happens that $\eta$ and $\sigma$ decrease, whereas $\sigma_{\rm \Delta z/(1+z)}$ can even increase, since more data points end in the ‘almost true’ section, but the median deviation within this $|z_{spec}-z_{phot}|/(1+z_{spec})< 0.15$ interval can increase. Nevertheless, one would, for many application prefer the situation with reduced $\eta$ and $\sigma$, even if $\sigma_{\rm
clip}$ is slightly increased. Using these definitions from above, our fraction of outliers for the `D1` field is $\eta \sim 5\%$ and the accuracy becomes $\sigma_{\rm \Delta z/(1+z)}=0.033$. This accuracy has to be compared to that of Ilbert et al. [@ilbert06]. They have obtained redshifts with the LePHARE code using about 70 template SEDs, which have been optimized with spectroscopic-photometric data in the `D1` field. The photometry comes from the CFHTLS Deep1 with integration times of about 11h, 7h, 17h, 37h and 17hours in the $u^*g'r' i'z'$-filters and PSFs between 1.1 and 0.9 arcseconds (as released in the Terapix T0003 release) and the BVRI-VVDS with integration times of 3 to 7 hours in the CFH12K BVRI filters and median PSF of 0.8” -0.9” (as described in McCracken et al. 2003). The 50 percent point source completeness is at AB magnitudes of 26.5, 26.4, 26.1, 25.9 and 25.0 for the $u^*g'r'i'z'$-filters and 26.5, 26.2, 25.9 and 25.0 for the $BVRI$-filters according to Ilbert et al. [@ilbert06]. So, their VVDS data are fairly deep in the B and V filters, and we therefore expect a gain in the photometric redshift accuracies, when these data are used: The VVDS B- and V-filters can sample breaks that are within the very broad CFHTLS-g’ filter, and the R-band helps to locate breaks that are within either the CFHTLS-r’ and i’-filters. In addition to the optical data Ilbert et al. [@ilbert06] could use deep J and K band data for 13$ \% $ of their objects. Their results for `D1` are released at at <http://terapix.iap.fr/rubrique.php?id_rubrique=227>. They reached values of $\sigma_{\rm \Delta z/(1+z)}=0.029$ and $\eta= 3.8\% $ using all their photometric data and galaxies with $i<24$ according to their paper. This result is surprisingly close to our result if one accounts for the denser wavelength coverage and the fractional coverage with NIR data.
Creating subsamples with higher photo-z precision
-------------------------------------------------
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We now explore whether we can identify a subsample of galaxies with more precise photometric redshifts. One expects that the quality of photometric redshifts varies with the SED properties of the galaxies. Using optical bands ($u^*g'r'i'z'$) only, there is a redshift range where strongly starforming galaxies cannot be discriminated from ‘more normal’ galaxies at other redshifts. In Fig. \[FigSEDzdeg\] we show the photometric redshift probability distribution for two spectroscopic objects, and the SED-fit for the most likely and second most likely SED. One can see that without near infrared data (e.g., J or H-band) the SED-photoz degeneracy cannot be broken. In both cases the true redshift is that of a ‘normal’ galaxies, without strong emsission lines. Since at the depth of the survey we do not expect very many strongly starforming galaxies anyhow, we decided to reject galaxies for which the strongly starforming SEDs is formally most likely. In this way we get rid of the ‘systematic arms’ in the spec-z photoz plots; the accuracy becomes $\sigma_{\rm \Delta z/(1+z)} \approx 0.031$ and the outlier rate is $\eta =2.7\%$. If we further exclude objects with photometric redshift errrors $\triangle_z{\rm photoz} >0.25 *
(1+z_{\rm photoz})$ the remaining accuracy and fraction of outliers become $\sigma_{\rm \Delta z/(1+z)}=0.031$ and $\eta \sim 2\%$. The fraction of galaxies that we loose with these selection criteria (small width of the most likely redshift, and excluding galaxies with SEDs degenerate to strongly starforming ones) is about 20 percent. We nevertheless provide photozs for all redshifts and flag galaxies with likely imprecise redshifts. The meaning of the photometric redshift flag values can be looked up in the Table \[tab:keys\] in the Appendix.\
Finally, if we limit the magnitudes of catalog to $i'_{AB}\leq 22.5$ mag, which corresponds to the limiting magnitude of the primary targets of the spectroscopic survey on the VVDS-F22 field, this leads to an accuracy of $\sigma_{\rm \Delta z/(1+z)}
=0.030$ and the fraction of catastrophic outliers is $\eta \sim1\%$.\
We now more directly compare the performence of our photometric redshifts with that of Ilbert et al. [@ilbert06]. We retrieve their photometric redshift catalog [^10], and merge our spectro-photometric-redshift sample with their photometric redshift. This ‘merged’ sample is now smaller than our spectro-photometric-redshift sample alone, since we skip sources that have more than one photometric-redshift counterpart within the search radius in the Ilbert et al. [@ilbert06] catalog.
-------------------------------------------------------------------------------------------------------------------------------------
Code Sample: CFHTLS-D1 $N_{zspec}$ [^11] Median error Mean error $\sigma$ $\sigma_{\rm \Delta z/(1+z)}$ $\eta
[\%]$
--------- ---------------------- ------------------- -------------- ------------ ---------- ------------------------------- ---------
PHOTO-z all PHOTO-z objects 3035 -0.011 -0.006 0.138 0.033 4.6$\%$
PHOTO-z good PHOTO-z objects 2477 -0.010 -0.006 0.082 0.031 1.8$\%$
PHOTO-z common good objects 2468 -0.010 -0.006 0.082 0.031 1.7$\%$
LePHARE all LePHARE objects 3035 -0.003 0.015 0.187 0.028 4.3$\%$
LePHARE good LePHARE objects 3012 -0.003 0.013 0.173 0.028 3.9$\%$
LePHARE common good objects 2468 -0.003 0.004 0.099 0.029 2.6$\%$
-------------------------------------------------------------------------------------------------------------------------------------
The definitions of the samples are:\
all PHOTO-z objects: $z_{\rm spec}>0,z_{\rm PHOTO-z}>0$ good PHOTO-z objects: $z_{\rm spec}>0,z_{\rm PHOTO-z}>0$, SED-type filtering, photometric redshift error filtering $\Delta z _{\rm photoz}< 0.25~(1+z_{\rm photoz})$\
common good objects: $z_{\rm spec}>0,z_{\rm PHOTO-z}>0$, SED-type filtering, photometric redshift error filtering for both codes\
all LePHARE objects: $z_{\rm spec}>0,z_{\rm LePHARE}>0$\
good LePHARE objects: $z_{\rm spec}>0,z_{\rm LePHARE}>0 $, photometric redshift error filtering $ (\rm (z_{\rm sup-{68}}- z_{\rm inf_{68}}) ) \le 0.25~(1+z_{phot}) $\
\[allz\_table\_D1\]
Table \[allz\_table\_D1\] and Fig. \[FigLePHARE\_PHOTO-z\] show, that if one takes all spectroscopic objects in the merged sample of the PHOTO-z and LePHARE catalogs, the clipped dispersions and the outlier rates are similar ($\sigma_{\rm \Delta z/(1+z)}=0.033$, $\eta=4.6\%$ and $\sigma_{\rm
clip}=0.028$, $\eta=4.3\%$ for the PHOTO-z and LePHARE code respectively), whereas the true dispersion of PHOTO-z is significantly smaller than that of the LePHARE catalog ($\sigma=0.138$ vs $\sigma=0.187$). It also shows, that using SED-filtering and photometric redshift error filtering (for the PHOTO-z case) and photometric redshift error filtering (for the LePHARE case –note, that we don’t make use of the full probability function but just of the redshift range, including $68\%$ of the redshift probability when defining ‘good objects’ as $
(\rm (z_{\rm sup-{68}}- z_{\rm inf_{68}}) \le 0.25~(1+z_{phot}) $ for the LePHARE catalog) reduces the outlier rate for both cases. The stronger decrease of outliers in the PHOTO-z case ($1.8\%$ vs $3.9\%$) is also caused by the effect that more objects are filtered out when defining a ‘good object catalog’, relative to the LePHARE catalog, as can be seen from the sample size. To see how photometric redshifts compare for objects which are considered as ‘good’ objects in both catalogs, we define a catalog of ‘common good objects’ and compare their photometric redshift quality in Table \[allz\_table\_D1\] as well. For common good objects, the outlier rate and the true dispersion is smaller using PHOTO-z redshift, whereas the clipped sigma is slightly smaller for LePHARE photometric redshifts. However the PHOTO-z redshifts show a small systematic effect compared to the LePHARE ones, as the redshifts seem to slightly oscillate around the 45 degree line. Whether this effect might be cured by optimizing the SED-templates for the CFHT filters will be investigated in an upcoming paper (Lerchster et al. in prep.), but one should keep in mind that the LePHARE redshifts are more immune against these oscillations since they do not use only the CFHT-$u^*g'r'i'z'$ data but also include the CFH12K $BVRI$ data and in addition partly NIR data (J and K). If one derives photometric redshifts for large area surveys, one will not have a spectroscopic sample to compare with; however comparing the redshift results and the assignemnts of ‘good or secure photometric redshift objects’ will improve the selection of a robust sample with few outliers.
Photometric Redshifts in the CFHTLS “Wide Fields” `W1`, `W3` and `W4`
=====================================================================
![Histogram of the reduced $\chi^2$ for all galaxies in the CFHTLS Wide field `W1` as obtained for the best fitting template and redshift. The dotted vertical line indicates the median reduced $\chi^2$ of 0.5. []{data-label="FigHistoChiError"}](figures/AA_2008_10733_09.ps){width="6.cm"}
![Histogram of the photometric redshift errors in the CFHTLS Wide field `W1`. The median redshift error is -0.004. The values for the formal dispersion and the clipped dispersion are $\sigma=0.08$ and $\sigma_{\rm \Delta z/(1+z)}=0.038$ for the W1 field. The central distribution nearly is gaussian (with a width of $0.038$) whereas the wings beyond $ |z_{\rm phot}-z_{\rm spec}|> 0.1$ cannot be described with the same gaussian at all. To illustrate that, we have added two gaussians with width of $0.038$ (in red) and $0.08$ (dashed blue) and amplitudes matching the true error distribution at zero. The high formal dispersion of $\sigma=0.08$ comes from the outliers; there are 27 objects with $|z_{\rm phot}-z_{\rm spec}|> 0.3$ and 16 objects with $|z_{\rm phot}-z_{\rm spec}|> 0.4$. Considering both $\sigma_{\rm \Delta z/(1+z)}$ and $\sigma $ tells, how ‘severe’ the outliers are.[]{data-label="FigW1zError"}](figures/AA_2008_10733_10.ps){width="6.cm"}
{width="8.5cm"} {width="8.5cm"}
We obtain photometric redshifts after zeropoint calibration as described before. A comparison between spectroscopic (VVDS sample only) and photometric redshifts for the CHTLS Wide Field `W1` and `W4` is shown in Fig. \[FigZZ\]. The outlier fraction is $\eta \leq 4\%$. In Fig. \[FigW1zError\] we show the distribution of the redshift errors on CFHTLS Wide field `W1` using the VVDS spectra. The error distribution is only gaussian in its center, which has a width of $0.038$ equal to the ‘clipped width’, $\sigma_{\rm \Delta z/(1+z)}$ defined by Ilbert et al. [@ilbert06]. The outliers are too many to be compatible with such a narrow gaussian; the true dispersion is equal to $0.08$ (for galaxies with VVDS spectra in the `W1`-field)
Fig. \[FigHistoChiError\] presents the $\chi^2$ distribution of the best fitting templates and photometric redshifts for all the objects. The median value of the reduced $\chi^2$ is 0.5 which implies that the galaxy templates describe the galaxies rather well.
Tests of the zeropoint calibration method
-----------------------------------------
We now investigate how well our empirical calibration of zeropoints using the color-color diagram of stars works. We derive zeropoints offset from matching the color-color diagram of stars as learned in the `W1` and `W4` subfields fields with VIMOS spectroscopic data. This offsets are used in the photoz code. We then compare our photometric redshifts to 410 galaxies from the Deep survey (Weiner et al. 2005; Davis et al. 2007) in the CFHTLS `W3` field (which cover areas different from the VIMOS galaxies). The photometric redshift prediction for these Deep galaxies is accurate to $\sigma_{\rm \Delta z/(1+z)} \sim 0.041$ with an outlier rate of $\eta \sim 6\%$.\
Since our data (partly) overlap with the Sloan Digital Sky Survey (SDSS), we can compare the photozs to spectroscopic ones for further 528 galaxies from the SDSS DR6. These low redshift objects give an accuracy of $\sigma_{\rm \Delta z/(1+z)} \sim 0.036 $ and a outlier rate of only $1/528$. When we finally combine the DEEP2 and the SDSS DR6 spectra we determine an accuracy of $\sigma_{\rm \Delta z/(1+z)} \sim 0.045$ and an outlier rate of $\eta \sim 1.5\%$ for the combined sample, see Fig. \[FigCompSDSS\].\
We think, that this shows, that our photometric calibration method can be applied to all CFHTLS Wide fields.\
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![Photometric redshifts in the CFHTLS Wide fields against the spectroscopic redshifts of the SDSS DR6 (blue dots) and DEEP2 (red dots). The dotted lines are for $z_{phot}=z_{spec}\pm
0.15~(1+z_{spec})$. These spectroscopic data have not been used for the calibration of zeropoint shifts, and thus provide an independent estimate of the redshift accuray.[]{data-label="FigCompSDSS"}](figures/AA_2008_10733_15.ps){width="8.5cm"}
{width="8.5cm"} {width="8.5cm"}
We show in Fig. \[FigPhotSpecCompW1\] how the photometric redshifts change if we use for the zeropoint calibrations stars only (instead of galaxy redshifts, see Fig. \[FigZZ\] for that case). Results are shown for the CFHTLS `W1` field and the CFHTLS `W4` field. The accuracy deteriorates from $\sigma_{\rm \Delta z/(1+z)}=0.035$ ($\sigma_{\rm \Delta z/(1+z)}=0.038$) to $\sigma_{\rm \Delta z/(1+z)}=0.04$ ($\sigma_{\rm
clip}=0.044$) for the field `W4` (`W1`), the outlier rate roughly stays the same.
Redshift accuracies as a function of type & brightness
------------------------------------------------------
In Fig. \[FigPhotSpecCompMag\] we compare the photometric redshifts to the spectroscopic redshifts for different apparent magnitude intervals for the CFHTLS `W1` and `W4` fields. The fraction of catastrophic outliers $\eta$ increases from 2.7 % to 9.7 %, going from $17.5 \leq i'_{AB} \leq 21.5$ up to $23.5 \leq i'_{AB}
\leq 24.5$, this has been also seen by Ilbert et al. [@ilbert06].\
Fig. \[FigPhotSpecCompSED\] shows the photometric redshifts versus the spectroscopic redshifts for different spectral types for the CFHTLS `W1` and `W4` field. We sort the SEDs we use to describe galaxies into 4 groups. The first one contains SEDs that describe ellipticals and S0s and is colored red in plots, the fourth contains very blue, strongly starforming SEDs (colored blue in plots) and the third (magenta) and fourth (green) group form a continuous sequence of SEDs in color and star forming activity between the first and fourth group of galaxy-SEDs. The accuracies of the photometric redshifts become $\sigma_{\rm \Delta z/(1+z)} \sim 0.037$ for group one and two (red and magenta), and $\sigma_{\rm \Delta z/(1+z)} \sim 0.044$ for group three and four (green and blue). The catastrophic outliers increase by a factor of about three from group old (old SEDs) to the other groups (younger SEDs) which has also been found by Ilbert et al. [@ilbert06].\
It is worth to note, that the integration time eg. of the i’-band on the Deep Field `D1` is $>$ 35 times longer then integration time of the i’-band data in the Wide Field `W1`. Nevertheless, the outlier rate is only slightly larger and the accuracy is almost the same in the shallower sample (if one limits the comparison sample to $I<22.5$). In other words: in order to obtain photometric redshifts for $I<22.5$ galaxies, it does not play a role whether the data are ‘rather deep’ (`W1` , with $I<24.5$) or ‘very deep’ (`D1` -T0003, with $I<25.9$). For these depths the photon noise is not relevant any longer, but solely how well the templates can reproduce the true galaxy SEDs.
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Field $\sigma_{\rm \Delta z/(1+z)}$ $\eta$ \[%\]
---------- ------------------------------- --------------
`W1p2p2` 0.042 4.0
`W1p2p3` 0.038 4.0
`W4m0m0` 0.034 1.3
`W4m0m1` 0.030 2.8
`W4m0m2` 0.033 1.9
`W4p1m0` 0.034 0.7
`W4p1m1` 0.036 3.4
`W4p1m2` 0.043 2.1
`W4p2m0` 0.032 2.4
`W4p2m1` 0.037 3.0
`W4p2m2` 0.031 2.4
: Photometric redshift accuracy seperated for each subfield (this makes use of the VVDS spectra only)[]{data-label="tab:subfieldcheck"}
Redshift distributions
----------------------
![Photometric redshift distributions of ‘good’ objects in the three CFHTLS Wide fields, `W1` (red line), `W3` (blue line) and `W4` (green line). The redshift distributions are shown from bright to faint selected samples.[]{data-label="FigHistoGalFields"}](figures/AA_2008_10733_22.ps){width="8.cm"}
![Same as Fig. \[FigHistoGalFields\] , but each panel shows a different selection of spectral types, according to the best-fitting template. From top to the bottom: The first group contains SEDs that describe ellipticals and S0s and the fourth group contains very blue, strongly starforming SEDs; the second and third group form a continuous sequence of SEDs in color and star forming activity between the first and fourth group.[]{data-label="FigHistoGalFieldsSED"}](figures/AA_2008_10733_23.ps){width="8.cm"}
Fig. \[FigHistoGalFields\] shows the galaxy redshift histogram of all objects in the four CFHTLS Wide Fields. The median redshift (see Tab. \[tab:meanred\]) is in good agreement in the four fields although the redshift distribution in the `W1` field is shifted to higher redshift. In Fig. \[FigHistoGalFieldsSED\] the SED redshift distribution of all galaxies in the CFHTLS Wide Field `W1` and `W4` is shown.
![Redshift number distributions for galaxies in the three CFHTLS Wide fields. The distributions are red, blue and green for the `W1`, `W3` and `W4` fields, the median redshifts in these fields are $z_{median, W1}=0.84$, $z_{median,W3}=0.79$ and $z_{median, W4}=0.80$. The red solid curve is a fit to the mean galaxy distribution for the `W1`-field using the parametric description given, e.g., in Van Warbeke et al. [@waerbeke01].[]{data-label="FigHistoGal"}](figures/AA_2008_10733_24.ps){width="8.cm"}
Magnitude interval $z_{\rm median}[\texttt{W1}]$ $z_{\rm median}[\texttt{W3}]$ $z_{\rm median}[\texttt{W4}]$
------------------------------ ------------------------------- ------------------------------- ------------------------------- -- --
$17.5\leq i'_{AB} \leq 22.0$ 0.53 0.53 0.54
$22.0\leq i'_{AB} \leq 23.0$ 0.70 0.68 0.68
$23.0\leq i'_{AB} \leq 24.0$ 0.77 0.75 0.74
$24.0\leq i'_{AB} \leq 25.0$ 0.79 0.76 0.76
: Median redshifts in the three CFHTLS Wide Fields (columns) for samples selected according to $17.5\leq i'_{AB} \leq 22$, $22\leq i'_{AB} \leq 23$, $23\leq i'_{AB} \leq 24$, and $24\leq i'_{AB} \leq 25$ from the top to the bottom. Only galaxies with ‘good’ photometric redshifts are considered.[]{data-label="tab:meanred"}
Field $z_0$ $\alpha$ $\beta$
------- ------- ---------- --------- -- --
`W1` 0.84 2.2 2.4
`W3` 0.79 2.2 2.4
`W4` 0.80 2.2 2.4
: Galaxy redshift distribution for objects with ‘good’ photometric redshifts in the CFHTLS Wide fields, using the parametrisation of Van Waerbeke et al. [@waerbeke01].
\[tab:reddist\]
In Fig. \[FigHistoGal\] the galaxy redshift histogram of all objects in the CFHTLS Wide is shown. The galaxy redshift distribution can be parameterized, following Van Waerbeke et al. [@waerbeke01]: $$n(z_s)=\frac{\beta}{z_0
\Gamma\bigl(\frac{1+\alpha}{\beta}\bigr)}\bigl(\frac{z_s}{z_0}\bigr)^{\alpha}
exp\bigl[-\bigl(\frac{z_s}{z_0}\bigr)^{\beta}\bigr],$$ where $(z_0, \alpha, \beta)$ are free parameters. The best fitting values for the 3 fields (`W1;W3;W4`) are show in Table \[tab:reddist\]. It can be seen from median redshift that the three fields were observed to different depths.
Comparing CFHTLS-W photozs from different methods
==================================================
{width="12.5cm"}
{width="12.5cm"}
In Erben et al. [@erben08] we had derived photometric redshifts with the BPZ algorithm. In this method the ‘ODDS’ output parameter of the BPZ method provides a very efficient way to dissentangle likely accurate from likely inaccurate photometric redshift. We used an odds parameter of ODDS=$0.9$ to select reliable redshifts and displayed the results for the reliable ones in Erben et al. [@erben08]. The photo-$z$’s were systematically overestimated at low spectroscopic redshifts and underestimated at higher $z$. We have discussed potential origins of this bias already in Erben et al. [@erben08].
This trend has turned out to be very stable over all investigated fields with VVDS overlap (13 fields in total) and can be described as $$z_{\rm BPZ-corr} \approx 1.29 ~z_{\rm BPZ} -0.13 \quad
.$$ \[correct-z\] Using this empirical relation (obtained from 13 fields) between the true redshift and the BPZ-photometric redshifts we can provide an alternative redshift estimate, which we call $z_{\rm BPZ corr}$.\
We finally take all ’reliable’ spectroscopic redshifts (VVDS, Deep and SDSS) and merge them with the photometric catalogs. Requiring a match in position within $0.56$ arcseconds distances yields a combined catalog with photometric and spectroscopic redshifts for 9079 objects, 9130 objects and 9118 objects for the PHOTO-z, the BPZ- (from Erben et al. 2008) and the BPZ-corrected catalogs. Note, that the number of objects is not the same, because eg. PHOTO-z identifies likely stars (which are then removed from the sample) and some objects wich have small redshifts in BPZ-catalog can obtain negative redshifts after applying the correction estimate from equation \[correct-z\]. We show the comparison of the spectroscopic and photometric redshifts in the `W1`, `W3` and `W4` fields in Fig. \[specz\_photoz\_all\_objects\_1\]. Black points denote all objects in the sample, and red symbols are used, for objects, which a method flags as having ‘good’ or ‘reliable’ photometric redshifts. In the overall sample, one sees a lot of systematics, most severly for the BPZ method (which is in the upper left panel). The relation for bias corrected BPZ redshifts and PHOTO-z redshifts are shown in the upper right and and lower left panels. The PHOTO-z method has fewer systematics in the total galaxy sample shown (black points) than the BPZ method. Fig. \[specz\_photoz\_all\_objects\_1\] demonstrates that the [ODDS]{} parameter is very helpful in sorting out outliers (see red points in the upper panels of the same Figure). Fig. \[specz\_photoz\_all\_objects\_2\] shows the density distributions of points in the $z_{\rm true}-z_{\rm
photoz}$-space. The contours are isodensity contours, and their levels are choosen such that they contain 99 (black), 90 (blue) and 50 (red) percent of all objects which have true redshifts between zero and 1.5. Compared to the BPZ and BPZ-corrected redshifts, the PHOTO-z method shows hardly any bias. We also build that sample of objects which has ‘good’ redshifts with both the BPZ and PHOTO-z method, and show the comparison of true and PHOTO-z redhifts in the upper right subpanel of Fig. \[specz\_photoz\_all\_objects\_2\]. These redshifts are bias free, have an outlier rate of only $1\%$, and in quality exceed the BPZ and BPZ-corrected redshifts according to Table \[allz\_table\].\
This shows, that one can construct a subsample of objects with very robust photometric redshifts, which is useful for weak lensing studies. One should however try to keep the ‘good’ subsample as large as possible in order to have enough galaxies to measure the shapes. We therefore now compare the yield of ‘good’ objects with the PHOTO-z and BPZ-methods as a function of object magnitude for the photometric and spectroscopic sample. We use only objects from the `W1p2p3`-field (which are located in an area not flagged having potentially unreliable photometry, i.e. objects having a flag of zero in the photometric catalog) and show results in Fig. \[good\_objects\_vs\_mag\]. The histogram for the number of objects with a given magnitude in the `W1`-field is shown in black in the upper two panels of this figure. The same histogram for objects which have spectroscopic data (and are not stars) in the `W1`-field is shown in black in the lower two panels. All objects (including stars) are shown as yellow histogram. One can see, that there are fairly many stars at the bright end. The histograms for those objects for which photometric redshifts could be derived and which are not classified as stars in terms of morphology or SED are shown in red, and those which have ‘good’ photometric redshifts are shown in green. The left panels are for PHOTO-z redshifts, the right panels are for BPZ-redshifts. One can see, that the yield of ‘good’ objects is quite complete for the spectroscopic and photometric sample for both the PHOTO-z and BPZ-method up to $i'=22$. For fainter magnitudes the BPZ-method is less complete: one obtains good redshifts for objects brighter than $i'$=24.5 only for 54 percent of all objects using BPZ. For the PHOTO-z method, this ratio equals 70 percent.
We are aware that the fraction of galaxies which have ‘good’ photometric redshifts (low clipped dispersion and low outlier rate) should be increased, or become ‘identical’ to the original sample. This can be achieved by adding NIR data, by improving our SED-templates, and probably more important by making our photometry (including the convolution to the same PSF) more accurate. This will be subject of a further study.
Code Sample: CFHTLS-W1, $N_{zspec}$ Median-error Mean-error $\sigma$ $\sigma_{clip}$ $\eta$ \[%\]
------------------------ ---------------------- ------------- -------------- ------------ ---------- ----------------- --------------
PHOTO-z all objects 9079 -0.0036 0.0065 0.158 0.040 5.7$\%$
PHOTO-z good PHOTO-z objects 7469 -0.0025 0.0019 0.071 0.038 2.8$\%$
PHOTO-z common good objects 6088 -0.0025 0.0001 0.053 0.037 1.0$\%$
BPZ all objects 9130 -0.0005 0.0257 0.194 0.056 10.5$\%$
BPZ good BPZ objects 7108 -0.0017 0.0013 0.066 0.054 2.7$\%$
BPZ common good objects 6088 -0.0050 0.0000 0.062 0.056 1.5$\%$
${\rm BPZ_{\rm corr}}$ all objects 9118 +0.0076 0.0414 0.241 0.051 11.0$\%$
${\rm BPZ_{\rm corr}}$ good BPZ objects 7104 +0.0024 0.0055 0.065 0.047 2.7$\%$
${\rm BPZ_{\rm corr}}$ common good objects 6088 +0.0063 0.0098 0.060 0.048 1.8$\%$
The samples are defined as follows:\
all PHOTO-z objects: [$ z_{\rm spec}>0,z_{\rm PHOTO-z}>0 $]{}\
good PHOTO-z objects: [$z_{\rm spec}>0,z_{\rm PHOTO-z}>0,flag_{\rm PHOTO-z}<=1$]{}\
common good objects: [$z_{\rm spec}>0,z_{\rm PHOTO-z}>0,z_{\rm BPZ}>0,z_{\rm BPZ-corr}>0,flag_{\rm PHOTO-z}<=1, ODDS_{\rm BPZ}>0.9$]{}\
all BPZ- objects: [ $z_{\rm spec}>0,z_{\rm BPZ}>0$]{}\
good BPZ objects: [$z_{\rm spec}>0,z_{\rm BPZ}>0,ODDS_{\rm BPZ}>0.9$]{}\
all BPZ-corr objects: [ $z_{\rm spec}>0,z_{\rm BPZ-corr}>0$]{}\
good BPZ-corr objects: [$z_{\rm spec}>0,z_{\rm BPZ-corr}>0,ODDS_{\rm BPZ}>0.9$]{}\
\[allz\_table\]
![This figure shows normalized histograms for the photometric redshift error, $\Delta z = z_{\rm true}-z_{\rm photoz}$ for the PHOTO-z method (left panel), for the BPZ-method (middle panel) and for the BPZ-(bias) corrected catalogs (right panel), using the `W1`, `W2`, and `W3` field and all reliable spectroscopic VVDS, DEEP2 and SDSS redshift data. Redshift biases as seen in the BPZ catalog don’t show up in this histogram (which combines all spectroscopic redshifts up to $z=4.5$) since only the histograms within small redshift slices are shifted relative to $\Delta z=0$. One can see, however, that the PHOTO-z redshifts have more objects with redshifts very close to the true redshift relatively to the other two methods (note, that these histograms are normalized). In addition to that, the PHOTO-z method provides the largest amount of galaxies in its ‘good’ sample. []{data-label="specz_photoz_all_objects_3"}](figures/AA_2008_10733_27.ps){width="8.5cm"}
{width="18.cm"}
Summary {#sec:sum}
=======
We tested the performance of the photometric redshift code of Bender et al. (2001) with the CFHT-MegaCam filter system using the CFHTLS `D1` data. The comparison of our photometric redshift results with spectroscopic data and photometric redshift results from Ilbert et al. [@ilbert06] shows: our performance is very close to that of Ilbert et al. [@ilbert06], although we use only about half of the optical bands (and no NIR data). This makes us confident that results we then derived for the CFHTLS-Wide fields are reliable.
We analyzed the CFHTLS-Wide data and showed that the colors of stars can be measured accurately enough and that the throughput of the system is known well enough to allow relative zero point calibration for the $g'r'i'z'$ bands using the colors predicted from Pickles libary stars. We could not match the color-color diagrams on the Pickles-libary-star colors when the $u^*$-band is involved. We nevertheless found out, that calibrating zeropoints such that the $u^*-g'$ colors approximately match at the red end, gives results which describe the galaxy colors correctly. In this case the likely reason for the mismatch would be that Pickles stars are metal enriched and do not show the UV excess of metal poor halo stars.
After the improved relative zeropoint calibration we derived photometric redshifts for 2.5 million of galaxies. We identified galaxies with likely inaccurate redshifts as those which have formally a large photometric redshift error (as provided by the code) and which have SEDs which are likely to be mismatched with another SED at different redshifts. By flagging those objects we end up with a sample of galaxies with fairly precise photometric redshifts. We investigated the redshift accuracy as a function of brightness and SED-type, and find similar results (in numbers and in trend) as Ilbert et al. [@ilbert06] for CFHTLS-Deep data set.
The overall photometric redshift precision was quantified by comparing all `W1`, `W3` and `W4` photometric redshifts to VVDS, DEEP2 and SDSS spectra. We then also investigated the BPZ-redshifts from Erben et al. [@erben08]. These redshifts are biased, and can be corrected according to $z_{\rm BPZ-corr}= 1.29 ~z_{\rm BPZ}-0.13$. This correction slightly overcorrects at redshifts larger than 1. We then analyzed all three photometric redshifts samples (PHOTO-z, BPZ, BPZ-corr) in more detail: Taking all objects (irrespective of photoz quality flags) the outlier-rate varies between 10 percent (BPZ/BPZ-corr) and 6 percent (PHOTO-z). If we select only good objects (using photometric redshift errors and SED types for the PHOTO-z method and using the ODDS parameter for the BPZ method) we can reduce the outlier rate to about 2.7 percent for all three catalogs. The width of the distributions (after clipping) then becomes 0.038, 0.054 and 0.057 for the PHOTO-z, the BPZ and the BPZ-corr catalogs. The width of the (unclipped) distribution is 0.071 (PHOTO-z) and 0.066 (BPZ/BPZ-corr).
Finally we consider only galaxies which are classified as ‘reliable’ objects in all three catalogs, and investigate the photometric redshift quality for this ‘common sample’. We indeed can reduce the outlier rate to 1 percent (PHOTO-z) and 1.5 to 1.8 percent for the BPZ-versions. The width of the distributions (after clipping) then becomes 0.037, 0.056 and 0.058 for the PHOTO-z, the BPZ and the BPZ-corr catalogs. The width of the (unclipped) distribution is 0.05 (PHOTO-z) and 0.06 (BPZ/BPZ-corr).
We conclude that this common sample defines a high quality redshift sample, which has (in the case of PHOTO-z) no bias and a very low outlier rate. This sample is ideally suited for weak lensing analysis like growth of cosmic shear and in particular the shear ratio test behind clusters of galaxies. Since these ‘good’ redshift samples include several selections steps, and are fairly incomplete at faint magnitudes we don’t recommend this sample to be taken for galaxy evolution studies in general.
The original sample (all galaxies) can be taken for that, which however requires to understand the impact of outliers (eg. on derived luminosity functions, galaxy colors as a function of redshift and environment density);
We are currently working on increasing the ‘good’ sample, or decreasing the outlier rate of the ‘remaining’ sample, to finally unite them to one again. Goal is to obtain a ‘complete sample’ with outlier rates as low as 2 percent. This requires a more detailed study of photometric calibration, improved convolution for more precise aperture colors, improved SEDs, potentially varying priors, and including further colors where possible.
We provide these catalogs (with future updates and extensions) on request.
We are greatful to the Terapix consortium for developing and providing tools for the handling of large CCD images in general, and for the image processing and pipeline software for MegaCam, and finally for the production of photometrically and astrometically corrected images within the CFHTLS survey. This work uses images which have been compared to the CFHTL T0003 release when deriving the photometric and astrometric accuracy.
We acknowledge use of the Canadian Astronomy Data Centre, which is operated by the Dominion Astrophysical Observatory for the National Research Council of Canada’s Herzberg Institute of Astrophysics.
We thank Y. Mellier and J. Coupon for the friendly spirit in which the Munich and Paris teams independently worked on their photometric redshifts.\
This work was supported by the DFG Sonderforschungsbereich 375 “Astro-Teilchenphysik”, the DFG priority program 1177 (Se1038), TRR33 “The Dark Universe” and the DFG ‘Cluster of Excellence on the Origin and Structure of the Universe’’.\
We all, in particular M. L., thank the European Community for the support by the Marie Curie research training network “DUEL”. M.L. further thanks the University of Bonn and the University of British Columbia for hospitality.
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Details on the photometric redshift redshift catalog
====================================================
Using the notation introduced in Erben et al. [@erben08], we briefly explain the most important `FITS` keys in the multi-color catalogs in the Table \[tab:keys\] .\
key name description measured on
------------------ -------------------------------------------- ----------------------------
`SeqNr` Running object number -
`ALPHA_J2000` Right ascension unconvolved $i$-band image
`DELTA_J2000` Declination unconvolved $i$-band image
`Xpos` x pixel position unconvolved $i$-band image
`Ypos` y pixel position unconvolved $i$-band image
`FWHM_WORLD` FWHM assuming a Gaussian core unconvolved $i$-band image
`FLUX_RADIUS` half-light-radius unconvolved $i$-band image
`A_WORLD` profile RMS along major axis unconvolved $i$-band image
`B_WORLD` profile RMS along minor axis unconvolved $i$-band image
`THETA_WORLD` position angle unconvolved $i$-band image
`Flag` [`SExtractor`]{}extraction flags unconvolved $i$-band image
`CLASS_STAR` star-galaxy classifier unconvolved $i$-band image
`MAG_AUTO` total $i$-band magnitude unconvolved $i$-band image
`MAGERR_AUTO` total $i$-band magnitude error unconvolved $i$-band image
`MAG_ISO_x` isophotal magnitude in x-band PSF-equalised x-band image
`MAGERR_ISO_x` isophotal magnitude error in x-band PSF-equalised x-band image
`MAG_APER_x` aperture magnitude vector in x-band PSF-equalised x-band image
`MAGERR_APER_x` aperture magnitude error vector in x-band PSF-equalised x-band image
`MAG_LIM_x` limiting magnitude in x-band unconvolved x-band image
`FLUX_ISO_x` isophotal flux in x-band PSF-equalised x-band image
`FLUXERR_ISO_x` isophotal flux error in x-band PSF-equalised x-band image
`FLUX_APER_x` aperture flux vector in x-band PSF-equalised x-band image
`FLUXERR_APER_x` aperture flux error vector in x-band PSF-equalised x-band image
`Z1_PHOT` photometric redshift best-fit SED -
`ERR_Z1_PHOT` error of photometric redshift best-fit SED -
`SED_TYPE ` SED type -
`Flag_PHOT` global photometric redshift flag key [^12] -
`MASK` global mask key[^13] -
[^1]: Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based on data products produced at TERAPIX and the Canadian Astronomy Data Centre (CADC) as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.
[^2]: We are very greatful for the CFHTLS survey team to conduct the survey, and for the Terapix team (http://terapix.iap.fr/) for developing software, for preprocessing the images and carrying out the numerous data control steps. A description of the CFHTLS survey can be found at <http://www.cfht.hawaii.edu/Science/CFHTLS>
[^3]: We will provide photozs for this field later
[^4]: <http://cencosw.oamp.fr/EN/index.en.html>
[^5]: Data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.
[^6]: <http://mingus.as.arizona.edo/~bjw/papers/>
[^7]: <http://deep.berkeley.edu/DR3/zcat.dr3.v1_0.uniq.dat>
[^8]: Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is <http://www.sdss.org/>;
[^9]: We would like to thank the terapix team for reducing and releasing these data.
[^10]: We would like to thank the authors for providing their photometric redshift catalog to the public.
[^11]: Number of objects in the photometric redshift sample with spectroscopic redshifts
[^12]: $1=$ object FHWM$<$PSF, $2=\triangle z > 0.25 * (1+z)$, $4=$ ext. flag ([`SExtractor`]{}, absolute photometry) for object, $8=$ SED rejected,\
$16=$ star SED, $32=$ no photometric redshift, and combinations.
[^13]: 0 for objects inside masks and 1 otherwise
|
---
abstract: 'The discovery of cosmic acceleration is one of the most important developments in modern cosmology. The observation, thirteen years ago, that type Ia supernovae appear dimmer that they would have been in a decelerating universe followed by a series of independent observations involving galaxies and cluster of galaxies as well as the cosmic microwave background, all point in the same direction: we seem to be living in a flat universe whose expansion is currently undergoing an acceleration phase. In this paper, we review the various observational evidences, most of them gathered in the last decade, and the improvements expected from projects currently collecting data or in preparation.'
address: 'Laboratoire de Physique Nucléaire et de Hautes Energies, Université Pierre et Marie Curie, Université Paris Diderot and CNRS/IN2P3, 4 place Jussieu, 75005 Paris, France'
author:
- Pierre Astier and Reynald Pain
bibliography:
- 'biblio.bib'
- 'biblio-reynald.bib'
title: Observational evidence of the accelerated expansion of the universe
---
Introduction
============
Soon after the expansion of the universe was firmly established, were observational cosmologists already trying to detect a modification of the expansion speed as a function of redshift. So confident were they that the expansion had to decelerate due to gravitational interaction of galaxies that they introduced the so-called deceleration parameter $q_0$, thought to be positive[@Sandage61]. Together with $H_0$, the deceleration parameter remained, for some time, the main cosmological parameters accessible to measurement. Nowadays, one prefers to describes the variation of the expansion of the universe in terms of the energy density of its constituents and their equation of state.
These first “classical tests” of the expansion involved measuring brightnesses of galaxies, but questions concerning galaxy brightness evolution with redshift rapidly surfaced and astronomers started looking for a better standard candle. The accelerated expansion was finally discovered at the very end of the last century and came as a surprise [@Riess98b; @Perlmutter99]. The two teams of discoverers were aiming at measuring the matter density parameter through the distance-redshift relation of Type Ia supernovae, and faced a paradox: when they fitted a matter-dominated cosmology to their data, the matter density parameter had to be significantly negative. Relying on the reproducibility of Type Ia supernova explosion, the two projects were, by the end of the 1990, gaining access for the first time to a precise distance-redshift relation extending to $z\sim 0.7$ (about half of the age of the universe), and the observed relation favored an accelerated expansion. This was surprising because there is no room for an accelerated expansion in a matter-dominated universe. However, a cosmological model mixing matter and a cosmological constant could describe well these observations [@Riess98b; @Perlmutter99]. The cosmological constant enters in such a model as a source term with static density, while the matter density decreases with expansion.
Before the discovery of accelerated expansion, there had been earlier hints that matter might not constitute the dominant component of the universe at late times. In 1975, assuming that the brightest galaxy in galaxy clusters can be used as a standard candle (in a way very similar to type Ia supernovae), Gunn and Tinsley [@1975Natur.257..454G] boldly suggested that the universe was accelerating, but also (wrongly) concluded that the total energy density exceeds the critical density (the density for which the universe is spatially flat, see §\[sec:theory\]), and that deuterium cannot be formed in the early universe. In 1984, Peebles [@Peebles84] gathered the arguments, mostly based on matter clustering (we discuss the physics at §\[sec:growth-theory\]), in favour of a low matter density universe, and explicitly considered what has become our standard cosmological model. In 1990, the data from the new APM galaxy survey provided stronger evidence that the matter density is at most 1/3 of the critical density (see [@Maddox90; @Efstathiou90] and §\[sec:BAO\]). And, in 1993, the measurement of the baryon fraction in galaxy clusters (i.e. the ratio of visible to total mass, see §\[sec:clusters\]), associated to the baryon density from big bang nucleosynthesis[^1] also challenged the matter-dominated flat universe model [@White93], favouring as well about 1/3 of the critical density in matter. These indications favouring a low matter density universe call for some other content, when associated to the theoretical prejudice of a flat (hence at critical density) universe[^2]. In this context, a low matter density calls for some sort of “complement”, although not necessarily causing acceleration. Note that observational evidence in favour of a critical matter-dominated universe was also produced concurrently (e.g. [@Loh86; @NusserDekel93]). So, around 1997, observational cosmologists were mostly considering two possible models for the late-time universe: a low matter density (sub-critical) universe, and a critical matter-dominated universe. Our current paradigm is both low matter density and flat. The discovery of accelerated expansion then reconciled the measurements of matter density with the theoretical inclination for a critical universe.
The accelerated expansion raises deep issues with likely connections to general relativity and particle physics. In the framework of general relativity a fluid with a static or almost static density may cause the acceleration of the expansion. The expression “dark energy” used nowadays refers to such a hypothetical fluid.
Although a cosmological constant still accurately describes all available large scale cosmological observations, phenomenologists have been studying a very vast range of possible models to incarnate dark energy. These models often involve scalar fields of some nature inspired by a range of particle physics or quantum gravity theories. We will not discuss these models here but refer to other papers of this special issue [@kunz2012; @martin2012; @clarkson2012; @derham2012]. The distinguishing feature of these models (or at least of classes of model) is the way the density of dark energy evolves with the expansion, commonly described using the “equation of state” parameter $w$ relating pressure and density $p= w \rho$. Non relativistic matter has $w=0$, while a fluid of constant density (e. g. the cosmological constant) follows $w=-1$. We will report, in this review, on recent constraints obtained on $w$ and discuss prospects for future improvements.
The discovery of an accelerated expansion was initially relying only on the distance-redshift relation of type Ia supernovae and the results were questioned. Could there be dust in the distant universe making distant supernovae appear dimmer? Were the supernovae brightnesses evolving with redshift ? But independent observational evidence of an acceleration of the expansion grew rapidly. First, early ground based cosmic microwave background (CMB, discussed in §\[sec:CMB\]) measurement pointing to a flat universe [@Bernardis00] which was hard to reconcile with observed low mass density without involving a non zero cosmological constant or something alike; then, the detection of the baryon acoustic oscillations (BAO, discussed in §\[sec:BAO\]) in the galaxies two-point correlation function measured by the Sloan Digital Sky Survey [@Eisenstein05]. We now have strong evidence for an accelerated expansion without invoking at all SNe Ia (e.g. §4.1 in [@Spergel07], [@Blanchard06]), and, in a matter of a few years, a new model of the universe has emerged, the “concordance model”. In this model, the energy density content consists now of about a quarter of matter and three quarters of dark energy, often assumed to be of constant density, as observations indicate more and more tightly [@Sullivan11]. The cosmological model where dark energy is assumed to be the cosmological constant, $\Lambda$ is called $\Lambda$CDM.
In this paper, we review the evidence for cosmic acceleration. In §\[sec:theory\], we briefly describe the cosmological framework and introduce the observables for which currents constraints are reported in §\[sec:sne\] to \[sec:age-of-the-universe\], from observations obtained using a number of different techniques. An example of combined constraints on $w$ and $\Omega_M$ is shown in §\[sec:combination\]. In §\[sec:future\], we briefly describe future projects that will help better constraint the acceleration and possibly shade new light on what could be the source of it. We conclude in §\[sec:conclusion\].
This review is part of a 5 paper special issue on Dark Energy with the companion papers being: The Phenomenological Approach to Modeling Dark Energy[@kunz2012], Everything You always Wanted to Know about the Cosmological Constant (but Were Afraid to Ask)[@martin2012], Establishing Homogeneity of the Universe in the Shadow of Dark Energy[@clarkson2012] and Galileons in the Sky[@derham2012].
Cosmic acceleration and dark energy \[sec:theory\]
==================================================
The cosmological principle states that the universe is homogeneous and isotropic, and the Friedman-Lemaitre-Robertson-Walker (FLRW) metric encodes this principle into its symmetries: $$ds^2 = dt^2-R^2(t)\left (\frac{dr^2}{1-kr^2}
+r^2(d\theta^2+\sin^2\theta d\phi^2) \right )$$ $R(t)$ is called the scale factor, and $k = -1$, $0$ or $1$, is the sign of the spatial curvature[^3]. Rather than $R(t)$, one often referes to $a(t) \equiv R(t)/T_{\mathrm now}$. Objects with constant coordinates $(r,\theta,\phi)$ are called comoving. In the FLRW framework, it is easy to show that photons emitted by comoving sources and detected by comoving observers see their wavelength scale with $R(t)$: $$\frac{\lambda_\mathrm{reception}}{\lambda_\mathrm{emission}} =
\frac{R(t_\mathrm{reception})}{R(t_\mathrm{emission})} \equiv 1+z$$ where $z$ is the redshift of the (comoving) source. General relativity postulates a relation between sources and the metric, which for the FLRW metric are called the Friedman equations [@Friedmann24]: $$\begin{aligned}
H^2(t) \equiv \left( \frac{\dot R}{R} \right)^2 & = &
\frac{8 \pi G}{3} \rho - \frac{k}{R^2(t)} + \Lambda/3 \label{eq:friedmann_1}\\
\frac{\ddot R}{R} & = & - \frac{4\pi G}{3} (\rho+3p) + \Lambda/3 \label{eq:friedmann_2}\end{aligned}$$ where $\Lambda$ is the cosmological constant, $\rho$ stands for the energy density, and $p$ for the pressure. The second equation is often called the Raychaudhuri equation. The energy conservation equation $$\label{eq:energy-conservation}
\frac{d}{dt}(\rho R^3) = -3 p R^2 \dot{R}$$ relates pressure to density evolution and applies also separately to the various fluids in the universe. For non-relativistic matter, $\rho
R^3$ is constant and hence $p=0$. A fluid with static density ($\dot
\rho = 0$) has $p = -\rho$. In both Friedman equations, $\Lambda$ could be summed into the density and pressure terms: $\rho_\Lambda=-p_\Lambda\equiv \Lambda/8\pi G$. Relation (\[eq:energy-conservation\]) can be obtained by eliminating $\ddot R$ between Eq. \[eq:friedmann\_1\] and \[eq:friedmann\_2\].
Fluids can be characterised by a relation between $p$ and $\rho$. The equation of state of each fluid $w_X$ is defined by $p_X=w_X\rho_X$, and for a constant $w_X$, we have $\rho_X(t) \propto R(t)^{-3(1+w_X)}$. For matter $w=0$, while $w=-1$ for $\Lambda$, and $w=1/3$ for radiation. Given the densities at one epoch (e.g. now) and the equations of state of the fluids of the universe, one can solve the first Friedman equation (Eq. \[eq:friedmann\_1\]) for $R(t)$. One can define the current critical density i.e. the density for which $k=0$, and the universe is flat: $$\rho_c = \frac{3 H_0^2}{8\pi G}$$ where $H_0 = (\dot R/R)_{now}$ is the Hubble constant. One conveniently expresses current densities in units of the current critical density: $$\Omega_M = \frac{\rho_M}{\rho_c} = \frac{8 \pi G \rho_M}{3 H_0^2}, \hspace{1cm}
\Omega_\Lambda = \frac{\Lambda}{3 H_0^2}, \hspace{1cm} \Omega_k = - \frac{k}{R_0^2 H_0^2}$$ and for, e.g., a universe of matter and a cosmological constant, the first Friedman equation simplifies to $1=\Omega_M+\Omega_\Lambda+\Omega_k$. The quantity $H_0$ is often introduced into expressions under the form $h \equiv H_0/100 km/s/Mpc$. For example, the matter physical density today is usually expressed as $\Omega_M h^2$, and turns out to be better determined than $\Omega_M$.
From Eq. \[eq:friedmann\_2\], one notes that a matter-dominated ($p
\simeq 0$) universe or a radiation-dominated ($p>0$) universe sees its expansion decelerate ($\ddot R<0$). More generally, once one integrates $\Lambda$ into density and pressure, the deceleration parameter $q(z)$ can then be expressed as: $$q (z) \equiv -{\ddot R\over R H^2} = {1\over 2} \sum_i \Omega_i(z) \left[ 1+ 3w_i(z) \right]
\label{eq:q}$$ where $\Omega_i(z)\equiv \rho_i(z)/\rho_{\rm crit}(z)$ is the fraction of critical density of component $i$ at redshift $z$, and $w_i(z) \equiv p_i(z)/\rho_i(z)$, the equation of state of component $i$ at redshift $z$. The sign of $\ddot R$ is the one of the resulting $(\rho +
3p)$[^4]. Therefore a fluid with $p <-\rho/3$ (i.e. $w<-1/3$) will cause the expansion to accelerate (${\ddot R}>0$) when it comes to dominate; its pressure is negative and pressure sources gravity in general relativity. By definition, such a component will be called dark energy. Note also that a mixture of matter and $\Lambda$ (a $w=-1$ dark energy) sees its expansion accelerate as soon as $\rho_m<2\rho_{\Lambda}$.
Following Frieman [*et al*]{} , we identify three possible classes of explanations for the acceleration of the expansion:
1. a source term in Friedman equations with a negative enough equation of state, for which various forms have been proposed from the simple vacuum energy $\Lambda$ to more complicated time-variable scalar fields
2. Einstein equations of relativity need to be modified such as the acceleration is a manifestation of gravitational physics. This requires a modification of geometric part of the Einstein equation rather than of the stress-energy part (“left side as opposed to right side of the equations”). For this to work the modifications have to apply on large scales only.
3. A third explanation involves dropping the assumption that the universe if spatially homogeneous on large scales. The idea is that non linear gravitational effects of spatial density fluctuations should alter the distance-redshift relation (see below) in such a way that it would explain its apparent departure from a dark energy free universe.
It is not the purpose of this review to discuss the possible sources of acceleration. Here, we will rather concentrate on discussing the evidence for cosmic acceleration through constraints obtained the values of $\Omega_i$ and $w_i$ as measured today. For an in-depth review of the phenomenology associated with the specific case of a cosmological constant, we refer the reader to [@martin2012].
The first constraints, which lead to the discovery of dark energy were obtained using type Ia supernovae to measure the distance-redshift relation
Cosmological distances and comoving volume \[sec:distances\]
------------------------------------------------------------
With the Friedman equation, one can integrate the photon path equation $ds=0$ for r(t), and compute various distances relevant to describe cosmological observations [@CPT92], as a function of redshift of the emitter and the parameters describing the source terms in the Friedman equation. For a source emitting a (rest frame) power L and with a measured energy flux L, the luminosity distance is defined by $d_L(z) \equiv \sqrt{L/4\pi F}$. Its expression as a function of cosmological parameters reads [@CPT92]:
$$\begin{aligned}
\hspace{-15mm}
d_L(z) &= & (1+z)H_0^{-1} |\Omega_k|^{-1/2}Sin \left\{ |\Omega_k|^{1/2} r(z) \right\} \label{eq:r_of_z} \\
r(z) &\equiv& \int_0^z \frac{dz'}{H(z')} = \int_0^z[ \Omega_M(1+z')^3+\Omega_\Lambda+\Omega_k(1+z')^2]^{-1/2} dz' \nonumber\end{aligned}$$
where $Sin(x) = sin(x),x,sinh(x)$ for $k=1,0,-1$; note that the expression is continuous in $\Omega_k=0$. This expression indicates that the distance-redshift relation probes the source terms of the Friedman equation. A Taylor expansion around $z=0$ reads $d_L(z) = z/H_0 + O(z^2)$, which shows that densities only enter the expression beyond the first order in redshift. $H_0$ only enters as a global factor in the distance expression, so that $H_0d_L$ only depends on redshift and reduced densities. One generalises Eq. \[eq:r\_of\_z\] to alternatives to $\Lambda$ (where dark energy might have a time-variable density) by replacing the $\Omega_{\Lambda}$ term by the (reduced) density of the considered fluid. For a constant equation of state, $\Omega_\Lambda \rightarrow
\Omega_X(1+z)^{3(1+w)}$, and for a time-variable equation of state w(z), $\Omega_\Lambda \rightarrow\Omega_X \ {\rm exp} [ 3 \int_0^z\frac{1+w(z')}{1+z'} dz']$. If one considers epochs when radiation was important, one should add $\Omega_r(1+z)^4$ to the sum of densities.
Fig. \[fig:plotdl\] displays the luminosity distance for a few cosmologies with varying admixtures of matter and cosmological constant, corresponding to a range of acceleration values now. One can note that the curve corresponding to our present $\Lambda$CDM paradigm cannot be mimicked with matter-dominated distance-redshift relations.
The angular distance $d_A$ is defined via the apparent angular size $\theta$ of an object of comoving physical size D : $d_A \equiv
D/\theta$. Because photons follow null-geodesics of the metric [@Etherington33], we have $d_L = (1+z)^2d_A$ and hence $d_L$ and $d_A$ convey the same cosmological information. $d_L$ measurements rely on “standard candles” while $d_A$ measurements rely on “standard rulers”.
Comoving volumes can be used to constrain cosmological parameters from e.g. counts of objects of known comoving densities. For convenience, the comoving volume should be indexed by redshift : $$\frac{d^2V}{dz\ d\Omega} = \frac{d_M^2(z)}{H(z)}
\label{eq:dvdz}$$ where $d_M = (1+z) d_A$.
Growth of structures \[sec:growth-theory\]
------------------------------------------
The expansion of the universe is accompanied by the increase in density contrast, essentially on all scales. This is called growth of structures. The subject is considerably more complex than homogeneous cosmologies and we will concentrate here on the salient features for what follows and point the interested readers to, e.g., chapter 15 of [@Peacock-book-99] (and references therein).
Density perturbations $\delta$ are defined by $1+\delta(x) =
\rho(x)/<\rho>$. In the late universe, matter density perturbations follow the following differential equation: $$\ddot \delta + 2 H \dot \delta = 4\pi G \rho_M \delta.
\label{eq:perturbations}$$ where we have assumed that matter is pressure-less, radiation is negligible and we are considering scales (well) below the Hubble radius. This equation is perturbative in the sense that it results from a first order expansion, and requires in particular that $\delta\ll 1$. Dark energy impacts the growth of structures through its contribution to $H$ and the evolution of $\rho_M(t)$. This equation has two solutions, and as a rule, at most one is growing. For a critical matter-dominated universe (an excellent approximation at $1000>z>1$), the growing mode follows $\delta(t) \propto a(t)$, which is called the “linear growth of structures”. Fig. \[fig:plotgrowth\] displays the growth factor at late times of the growing solution for a set of chosen cosmologies.
![*Linear growth factor (growing mode) for various cosmologies. Note that the ordering is quite different from distances of Fig. \[fig:plotdl\]. \[fig:plotgrowth\]*](plotdl){width="\textwidth"}
![*Linear growth factor (growing mode) for various cosmologies. Note that the ordering is quite different from distances of Fig. \[fig:plotdl\]. \[fig:plotgrowth\]*](plotgrowth){width="\textwidth"}
In the early universe, matter perturbations still follow an equation similar to Eq. \[eq:perturbations\] with however two differences : the expansion rate is faster in a radiation-dominated universe, and when applicable, one should consider the couplings of radiation to charged matter, which allow sound waves to propagate in the primordial plasma. In a radiation-dominated universe, matter density perturbations (both for charged and collision-free matter) grow logarithmically with time (i.e. very slowly), and hence, the horizon size, when matter and radiation densities are equal (the equality epoch), is imprinted in the matter power spectrum[^5]. After equality, matter density perturbations grow, and the charged matter perturbations propagate as sound waves, until the temperature is small enough to allow atoms to form, at an epoch called “recombination”. The travel length of these sound waves is also imprinted in the matter correlation function as a (slight) excess at the comoving sound horizon size at recombination (see Fig. \[fig:plotps\]).
So, the clustering of matter contains two distinctive features: the horizon size at equivalence (which scales as $\rho_{M,0}^{-1}$) and the “sound horizon at recombination” (sometimes called the acoustic scale), which is a function of the matter and baryonic matter densities. Fig. \[fig:plotps\] displays the canonical matter power spectrum and correlation function (they are related by a Fourier transform) at low redshift, where both features are visible.
![ *Left: dark matter power spectrum at z=0, computed using CMBEasy [@CMBEasy] for a flat standard $\Lambda$CDM, as a function of comoving wave number. The two key features are the maximum (which indicates the horizon size at matter-radiation equality, and depends on the matter density), and the series of wiggles which indicate the size of the sound horizon at recombination (and depends on matter and baryon densities). The latter corresponds to a single peak in direct space, as shown on the right. These features can be observed in Cosmic Microwave Background (CMB) anisotropies and galaxy redshift surveys, and their angular size constrains the expansion history. Right : the matter correlation function (times $s^2$) as a function of comoving separation $s$, for a universe without baryons (dashed line), or with baryons (full line). The acoustic peak (causing the wiggles in the power spectrum) is clearly visible (figure adapted from [@Blake11BAO]).\[fig:plotps\]* ](plotps){width="\textwidth"}
![ *Left: dark matter power spectrum at z=0, computed using CMBEasy [@CMBEasy] for a flat standard $\Lambda$CDM, as a function of comoving wave number. The two key features are the maximum (which indicates the horizon size at matter-radiation equality, and depends on the matter density), and the series of wiggles which indicate the size of the sound horizon at recombination (and depends on matter and baryon densities). The latter corresponds to a single peak in direct space, as shown on the right. These features can be observed in Cosmic Microwave Background (CMB) anisotropies and galaxy redshift surveys, and their angular size constrains the expansion history. Right : the matter correlation function (times $s^2$) as a function of comoving separation $s$, for a universe without baryons (dashed line), or with baryons (full line). The acoustic peak (causing the wiggles in the power spectrum) is clearly visible (figure adapted from [@Blake11BAO]).\[fig:plotps\]* ](corr-function){width="\textwidth"}
Cosmic variance \[cosmic-variance\]
-----------------------------------
Cosmological models do not predict the actual observed patterns, but rather their statistical properties, such as the average correlation function, or higher order statistics. Unavoidably, the comparison of observations with the model is limited by the ensemble variance of the observations, whatever their quality. This variance floor is referred to as “cosmic variance”, to be understood as cosmic sampling variance. For example, the theory predicts the average angular power spectrum of cosmic microwave background anisotropies (see §\[sec:CMB\]), but we only observe a single map and cannot practically average over the observer location. The importance of cosmic variance in a given data set for measuring fluctuations at a given spatial scale can be appreciated in practice by evaluating the number of cells of this scale that the data set contains. The cosmic variance hence goes down when going to smaller spatial scales. The cosmic (sampling) variance has practical consequences: it sets a limit beyond which measuring more accurately the fluctuations will not improve significantly the cosmological constraints. For example, some surveys have mapped the three-dimensional positions of galaxies in some parts of the sky, up to a certain redshift, and the galaxy counts can be used as a proxy for the matter density field. Once the surveyed galaxy density is large enough for the Poisson noise to go below the cosmic variance, surveying the same volume by other means will not improve significantly our knowledge of e.g. the average power spectrum of the density field. Cosmic variance sets hard limits about the possible measurements of matter fluctuations on large scales in the nearby universe, and on the cosmic microwave background fluctuations on large angular scales. For the latter, the only practical approach consists in surveying the whole sky.
A brief survey of dark energy probes
------------------------------------
Quantifying the merits of dark energy probes and predicting future dark energy constraints has been attempted by (at least) two working groups and the interested reader should consult their detailed reports [@DETF06; @ESO-ESA]. Studying dark energy consists first in constraining its equation of state $w$, because the cosmological constant has $w=-1$ while other implementations give, in general, different values. This is constrained by measuring the expansion history, in practise the distance-redshift relation or more directly $H(z)$. The growth rate of structures is another handle on dark energy, because it can deliver constraints on its own. More interestingly, measuring both the growth rate and the expansion history will allow us to test General Relativity on the largest spatial scales. All present evidence for dark energy rely on general relativity (more precisely the Friedman equations) properly describing gravitation on the largest spatial scales.
The expansion history is in practice constrained through the distance-redshift relation. Distances can be either luminosity distances from “standard candles”, or angular distances from “standard rods”. Today, the best known approximation to a standard candle is provided by type Ia supernovae, observable at redshifts beyond 1. For standard rods, baryon acoustic oscillations (BAO) provide the “acoustic peak” in the correlation function of matter, which requires large volume surveys to be detected, and we now have measurements over a few redshift bins. Standard rods measured across the line of sight constrain the angular distance and may directly constrain $H(z)$ when measured along the line of sight.
Gravitational lensing provides a handle on the matter distribution between distant galaxies and us. The “cosmic shear” phenomenon refers to weak lensing by large scale structures and allows one to constrain both distances and the matter clustering on large scales (and hence the growth of structures).
Studying the evolution of galaxy cluster counts with redshift allows one to constrain both the growth of structures (in a non-linear regime), and the expansion history through the comoving volume (Eq. \[eq:dvdz\]).
Supernovae, baryon acoustic oscillations, weak lensing, and clusters were the four canonical dark energy probes studied at length in [@DETF06; @ESO-ESA]. They remain today the main probes of cosmic acceleration. The next section is devoted to the particular role of CMB in dark energy constraints. The following sections then describe the principle behind the measurements and current achievements of each probe.
The role of the cosmic microwave background (CMB) \[sec:CMB\]
=============================================================
The cosmic microwave background (CMB) plays a particular role in constraining the acceleration of the expansion. Studying anisotropies (including polarisation) of CMB has become the key handle on cosmological parameters. The current best determination of cosmological parameters comes from seven years of observations with the WMAP satellite [@Komatsu11], and these results should be superseded by the Planck satellite in early 2013.
If dark energy density evolves slowly or not at all with redshift, it is negligible in the early universe, in particular before recombination, the epoch of CMB emission. However, dark energy impacts the angular scale of anisotropy correlations through our distance to the CMB emission (called last scattering surface or LSS), because dark energy contributes to the expansion rate at late times. Cosmological parameters hence contribute in two ways to the geometrical aspects of the observed anisotropies: directly in determining the detailed correlations of CMB anisotropies, and indirectly through our distance to LSS (see e.g. [@Hu05]). The physics of the acoustic waves is driven by the matter and baryon densities. Our distance to LSS depends on matter density, dark energy parameters, and the Hubble constant. At constant matter and baryon densities, and constant distance to LSS, the observed power spectrum of fluctuations is essentially unchanged (see e.g. [@Bond97]). CMB anisotropies thus provide a single constraint on dark energy and $H_0$, which can be turned into a constraint on dark energy alone by using either a local measurement of $H_0$, the flatness assumption (see e.g. [@Riess11] and references therein), or some other cosmological constraint. Assuming flatness and that dark energy is a cosmological constant, [@Komatsu11] obtains $\Omega_\Lambda=0.727\pm 0.030$. CMB constraints can of course be integrated into fits involving more general dark energy parameters, e.g. [@Sullivan11].
The CMB “geometrical degeneracy” is simply due to the fact that observations depend on a single distance: our distance to LSS. Secondary CMB anisotropies are the ones that build up as CMB travels to us, and hence break this degeneracy by introducing other distances. However, secondary anisotropies are subtle (e.g. [@Aghanim08]).
CMB is affected by the gravitational deflections of foreground structures: the image we obtain is remapped with respect to an homogeneous universe. This phenomenon has been studied in detail [@Seljak96; @Stompor99; @Hu00], and turns out to be a small correction to most of the observables. Recently a ground-based high resolution imager reported the detection of gravitational lensing of the CMB [@Das11]. By introducing intermediate distances between us and LSS, the phenomenon breaks the geometrical degeneracy and allowed the same team to obtain a 3.2 $\sigma$ evidence for dark energy [*from CMB alone*]{} [@Sherwin11].
The integrated Sachs-Wolfe effect [@SachsWolfe67] is due to the time evolution of gravitational wells when photons traverse those, and the growth of structures is sensitive to dark energy. The Sachs-Wolfe effect only affects large scales, which are also the most affected by cosmic variance. Following a suggestion by [@CrittendenTurok95], mild evidence for dark energy was found in [@Scranton03] through the correlation of CMB temperature maps and galaxy distributions. Recent analyses [@Giannantonio08; @Ho08] typically reach a 4 $\sigma$ detection of dark energy through this cross-correlation technique. Note that this detection is indeed independent of lifting the geometrical degeneracy of CMB.
Hence, the CMB [*on its own*]{} is of limited use when constraining dark energy. However since CMB constitutes the best probe to constrain our cosmological model (see e.g. [@Komatsu11]), its indirect contribution to dark energy constraints is essential (e.g. [@Percival10; @Amanullah10; @Sullivan11]). Most if not all recent studies forecasting dark energy constrains (see e.g. [@DETF06; @ESO-ESA; @EuclidRedBook11]) now assume they will eventually use “Planck priors” (e.g. [@Mukherjee08]).
Hubble diagram of SNe Ia \[sec:sne\]
====================================
Hubble originally published a distance-velocity diagram[@Hubble29], which was the first indication of the expansion. We now call Hubble diagrams flux-redshift relations (or more commonly magnitude-redshift relations), for objects of similar intrinsic luminosity. Hubble initially reported distances to galaxies, but the chemical evolution of galaxies seems too fast to extend the galaxy “Hubble diagram” to high enough redshifts [@OstrikerTremaine75].
The Hubble diagram of type I supernovae was proposed to measure distances [@Kowal68; @KirshnerKwan74] and soon after to constrain the evolution of the expansion rate [@Wagoner77]. Supernovae are explosions of stars, and their taxonomy has been refined over the last 70 years. The current classification is detailed in e.g. [@DaSilva93; @Filippenko97]. Type Ia supernovae (SNe Ia) constitute an homogeneous subclass of supernovae, believed to be a complete thermonuclear combustion of a white dwarf reaching the Chandrasekhar mass (1.4 $M_\odot$), or the fusion of two white dwarfs (for a review of our current knowledge of progenitors, see [@HowellReview11]). These events are homogeneous in the sense that they show a reproducible luminosity (see e.g. [@Hamuy96c]), and can hence be used to infer luminosity distances. The luminosity rises in about 20 rest frame days and fades over months (e.g. [@LeibundgutPhD; @Contreras10]) in visible bands. One usually uses the peak brightness as the primary distance indicator, which requires to measure the luminosity as a function of time (called “light curve”) in several spectral bands (see below). These events are bright enough to be measurable up to $z\simeq 1$ using ground-based 4-m class telescopes.
The Calan-Tololo survey delivered the first set of precise measurements of SNe Ia in 1996 [@Hamuy96b], 29 events up to $z\sim 0.1$, which allowed the authors to get residuals to the Hubble diagram (magnitude-redshift) of 0.17 mag r.m.s or better[^6] (i.e. relative distance uncertainties of $\sim$8%). These distances make use of two empirical luminosity indicators, the decline rate of the light curves ([@Pskovskii84; @Phillips93]), and the color[^7] of the event (measured e.g. at peak luminosity). SNe Ia are sometimes called “standardisable candles”, and are the best known distance indicator to date.
In order to efficiently constrain cosmology from distances, one should use as long a redshift lever arm as possible: this is why cosmological constraints obtained from supernovae make use of nearby (typically $z<0.1$) events, which, on their own, do not bring any interesting constraint on the expansion evolution. Those are however vital to almost all supernova cosmology analyses.
The discovery of the accelerated expansion
------------------------------------------
At the time the Calan-Tololo was succeeding at measuring distances, other teams were trying to discover and measure SNe Ia at $z\sim 0.5$, in order to probe the distance-redshift relation beyond the linear regime. Finding distant supernovae required using image subtraction techniques [@Hansen87; @Norgaard89; @Alard98], which consist in digitally subtracting images of the same areas of the sky taken a few weeks apart in order to locate light excesses. These light excesses can then be spectroscopically identified (and their redshift measured), and eventually their light curves measured. Both methods and instruments for this demanding program started to be available at the beginning of the 1990’s, and by the end of the decade, the Supernova Cosmology project and the High-Z teams had collected large enough high redshift samples to start probing the cosmic acceleration. The two groups compared their high redshift ($z\sim 0.5$) sample to the Calan-Tololo nearby sample ($z<0.1$) and reached the striking conclusion that no matter-dominated universe could describe their magnitude-redshift relation [@Riess98b; @Perlmutter99]. The measured high redshift distances are too large, compared to nearby ones, for a decelerating universe and all matter-only universes decelerate (see Eq. \[eq:friedmann\_2\] and Fig. \[fig:plotdl\]). All together the two projects had gathered $\sim$50 distant events in total, and the shot noise affecting their photometry was significantly degrading the distance scatter compared to nearby events. A few years later, a sample of 11 events measured with the Hubble space telescope (HST), and featuring a photometric quality comparable to nearby events confirmed the picture [@Knop03].
Based on this success, second generation supernovae surveys were then launched, which aimed at constraining a constant equation of state of dark energy to 0.1 or better. This was carried out in two complementary ways: measuring distances to very high redshift supernovae using the HST, and running large dedicated ground-based surveys.
Second generation supernova surveys
-----------------------------------
Very high redshift supernovae see their light shifted in the near IR, which is very difficult to observe from the ground because of the atmosphere large absorption bands, and emits a bright glow. Observing from space is then essentially mandatory for SNe Ia at $z>1$. A large HST program was devoted to measuring distances to high redshift supernovae and delivered 37 events among which 18 were at $z>1$ [@Riess04; @Riess07]. This sample extends deep enough in redshift to find evidence for a past deceleration era. Nowadays, the impact of this sample is however limited by the modest sampling of light curves and the photometric calibration uncertainties (discussed in e.g. §5.1.3 of [@Conley11]). More recently a higher quality sample of 10 HST $z>1$ events was published [@Suzuki12], and confirms the picture.
Ground-based wide-field imagers can efficiently tackle the $z<1$ regime by repeatedly imaging the same area of the sky, thus building light curves of variable objects. By tailoring the exposure time for a $z=1$ supernova, a 1 deg$^2$ image delivers about 10 useful measurements of SN Ia light curves. The advent of wide-field imagers allowed observers to propose efficient second generation SN surveys relying on this multiplex advantage, with the promise of bringing new constraints on the equation of state of dark energy. Three surveys, which benefited from large observing time allocations, are listed here in the order of their median redshift:
- The SDSS SN search used the 1.4 deg$^2$ imager on the SDSS 2.5-m telescope to monitor 300 deg$^2$ in 5 bands every second night for 3 months a year during 3 years (2005-2007). The survey delivered light curves of $\sim 500$ spectroscopically identified SNe Ia to $z=0.4$.
- The Essence project used the 0.36 ${\rm deg^2}$ Mosaic-II imager on the CTIO-4m to monitor in 2 bands $\sim$ 10 ${\rm deg^2}$ for 3 months over 5 years (2003-2008), and has measured light curves of 228 identified SN Ia.
- The SNLS project relied on the 1 deg$^2$ Megacam imager on the 3.6-m Canada-France-Hawaii Telescope. It monitored 4 pointings in 4 bands for 5 years and measured the light curves of 450 spectroscopically identified SN Ia events to $z\sim1$.
The three projects acquired as many spectra as they could to identify spectroscopically the candidates detected in the imaging program, and were limited by the amount of spectroscopic observing time. They however were able to increase the statistics of distant events by more than a factor 10, and these events have distance accuracies that compare well with nearby ones. During the last decade, second generation nearby SN surveys have also been run, and we now have about 120 high quality nearby events (see Tab 2. of [@Conley11]), and more to come.
![*Cosmological constraints obtained from the Hubble diagram of Fig. \[fig:snls3-hd\] ([@Conley11]), from CMB anisotropies ([@Komatsu11]) and the matter power spectrum ([@Percival10]), and combined ([@Sullivan11]). The SN contours account for systematic uncertainties dominated by photometric calibration. Fig. from [@Sullivan11]. \[fig:snls3-w0-om\]*](snls3-hd){width="\textwidth"}
![*Cosmological constraints obtained from the Hubble diagram of Fig. \[fig:snls3-hd\] ([@Conley11]), from CMB anisotropies ([@Komatsu11]) and the matter power spectrum ([@Percival10]), and combined ([@Sullivan11]). The SN contours account for systematic uncertainties dominated by photometric calibration. Fig. from [@Sullivan11]. \[fig:snls3-w0-om\]*](snls3-w0-om){width="\textwidth"}
So far, these 3 surveys have published partial analyses [@Astier06; @WoodVasey07; @Kessler09; @Conley11], and none has delivered its final sample yet. The latest compilation can be found in [@Conley11] which collects 3 years of SNLS, 1 year of SDSS, and the nearby samples, reaching a total of about 500 events passing stringent quality cuts, including spectroscopic identification. The resulting Hubble diagram is shown in Fig. \[fig:snls3-hd\]. With a thorough accounting of systematic uncertainties, the cosmological fit of a flat universe where dark energy has a constant equation of state yields $w=-1.07
\pm 0.07$ [@Sullivan11] where the uncertainty accounts for both statistics and systematics, which contribute almost equally. This constraint, among the tightest to date, is illustrated on Fig. \[fig:snls3-w0-om\]. It is highly compatible with the cosmological constant hypothesis.
Baryon acoustic oscillations (BAO) \[sec:BAO\]
==============================================
The acoustic signatures observed in the CMB anisotropies survive recombination and leave their imprint on the matter distribution. Namely, the correlation function of matter density shows a peak at a comoving separation around 150 Mpc. This feature can be used as a standard ruler and can in principle be detected both along and across the line of sight, and yields constraints on $H(z)$ and the angular distance $d_A(z)$ respectively. Baryon acoustic oscillations are a small signal: the probability to find a galaxy pair at 150 Mpc separation is less than 1% larger than at 100 or 200 Mpc; detecting the signal require to survey at least a volume of the order of 1 $h^{-3}Gpc^3$ (e.g. [@Tegmark97]).
So far, all detections made use of the galaxy distribution and merged the longitudinal and transverse directions. The first detections were reported in 2005 by the SDSS [@Eisenstein05] and the 2dF [@Cole05] from the three-dimensional distribution of galaxies. Both surveys have made use of multi-object spectrographs, which allow one to collect hundreds of spectra at a time. Their samples of a few 100,000 galaxies used hundreds to thousands of observing nights. Both were redshift (i.e. 3 dimensional) galaxy surveys, limited to $z<0.3$ for the 2dF and $0.16<z<0.47$ for the SDSS. The significance of BAO detections is modest ($\sim$ 2.5 to 3.5 $\sigma$) but they add up since the samples map distinct volumes. Despite this modest significance, the SDSS measures the distance to the median redshift of the survey ($z=0.35$) to better than 5% using the whole correlation function. Multi-band imaging data from the SDSS allows one to derive “photometric redshifts” of galaxies, and the volume thus covered extends to $0.2<z<0.6$ where the BAO signal is detected to 2.5 $\sigma$ [@Padmanabhan07]. Compared to spectroscopic redshifts, the noise of photometric redshifts blurs the BAO peak across the line of sight, and destroys the whole signal along the line of sight.
The SDSS spectroscopic sample has been doubled since the first detection[@Percival10], and the new WiggleZ survey has published its first results [@Blake11BAO]; The measured correlation function sn shown in Fig. \[fig:bao\]. All these spectroscopic studies of BAO provide measurements of the acoustic scale at various redshifts and are usually expressed using a hybrid distance (proposed in [@Eisenstein05]) : $$D_V = \left[ (1+z)^2 d_A^2(z) cz/H(z) \right ]^{1/3}
\label{eq:dv}$$ which expresses that the measurement relies on two transverse and one longitudinal directions. The obtained constraints, independent of the acoustic scale and the growth of structure, are displayed in Fig. \[fig:bao\]: they convey essentially the same information as the SN Hubble diagram, and reach very compatible conclusions, but are not as precise yet.
![ *Right: correlation function from the WiggleZ redshift survey (borrowed from [@Blake11BAO]) as a function of comoving separation, with the expectations from the best fitting model, and a baryon-free reference model. The acoustic peak is clearly visible, but note that the measured points are heavily correlated. Left: Distances measured in redshift slices from the acoustic feature in BAO surveys. The two low-redshift point come from [@Percival10] (which makes use of [@Cole05]), and the high redshift point is from [@Blake11BAO]. The solid curve is the expectation for $D_V$ (defined in Eq. \[eq:dv\]), for a flat $\Lambda$CDM cosmology, where the overall scale was adjusted to the data. The dashed curve, does not much the data, even adjusting a global factor. In [@Blake11-baodistances], a similar figure is proposed where the WiggleZ data is split into 3 overlapping (hence correlated) redshift bins.* []{data-label="fig:bao"}](BAO-peak-wigglez){width="\textwidth"}
![ *Right: correlation function from the WiggleZ redshift survey (borrowed from [@Blake11BAO]) as a function of comoving separation, with the expectations from the best fitting model, and a baryon-free reference model. The acoustic peak is clearly visible, but note that the measured points are heavily correlated. Left: Distances measured in redshift slices from the acoustic feature in BAO surveys. The two low-redshift point come from [@Percival10] (which makes use of [@Cole05]), and the high redshift point is from [@Blake11BAO]. The solid curve is the expectation for $D_V$ (defined in Eq. \[eq:dv\]), for a flat $\Lambda$CDM cosmology, where the overall scale was adjusted to the data. The dashed curve, does not much the data, even adjusting a global factor. In [@Blake11-baodistances], a similar figure is proposed where the WiggleZ data is split into 3 overlapping (hence correlated) redshift bins.* []{data-label="fig:bao"}](dv_bao){width="\textwidth"}
The redshifts surveys that deliver BAO constraints can also efficiently constrain the matter density from the shape of the matter power spectrum. In most of the analyses discussed above [@Cole05; @Eisenstein05; @Percival10; @Blake11BAO], a global fit to the matter correlation function or power spectrum yields essentially a constraint on $\Omega_M$, which very efficiently complements distance measurements from SN to constrain dark energy. The clustering of matter is sensitive to $\Omega_M$, through the “horizon at equality” turnover discussed in § \[sec:growth-theory\] and displayed in Fig. \[fig:plotps\]. Indeed, more than 20 years ago, one of the first measurements of the angular correlation of galaxies was found to be incompatible with an Einstein-De Sitter (flat $\Omega_M=1$) universe[@Maddox90; @Efstathiou90], preferring $\Omega_M \simeq 0.3$, because this delays equality and shifts the turnover to larger spatial scales. With the prejudice of flatness, these results could be regarded as the first evidence for the presence of “something more than just matter”.
Cosmic variance really limits the reach of BAOs at low redshift: since the SDSS 2005 result [@Eisenstein05] makes use of $\sim$10 % of the sky, is almost cosmic variance limited (sampling variance is about 1/3 of cosmic variance), one should not expect more than a 10 $\sigma$ whole-sky detection of BAOs at $z<\sim 0.4$, at least using similar strategies. Similarly, cosmic variance limits the accuracy of a distance measurement to $z\sim 0.3$ using galaxy clustering over the whole sky to a few percent. This limitation rapidly vanishes as redshift rises, and is totally irrelevant at $z>1$.
An interesting avenue for BAO surveys consists in accounting for the displacement of tracers with respect to their Hubble flow positions, due to their motions towards surrounding mass excesses. This technique, called “reconstruction”, was proposed[@Eisenstein07], and applied on the SDSS data recently [@Padmanabhan12], where the variance of $D_V(z=0.35)$ is remarkably improved by about a factor of 3. This technique is expected to be mostly effective in the low redshift universe, where it is very welcome because cosmic variance severely limits the ultimately achievable precision.
Direct measure of the growth rate \[sec:growth-rate-measurements\]
==================================================================
Three-dimensional galaxy redshift surveys allow one to probe the growth rate of fluctuations, assuming the expansion history is known. One relies on the distortion of redshift from velocity due to relative attraction of close-by galaxies which compresses their redshift difference[@Kaiser87]. Assuming one knows $d_A(z)$ and $H(z)$, and that galaxy clustering is isotropic on average, one can compare the clustering across and along the line of sight and detect these redshift distortions. For that, one defines the nuisance bias parameter $b$, assuming the relation $$\left. \frac{\delta \rho}{\rho} \right|_{galaxies} = b \left. \frac{\delta \rho}{\rho} \right|_{mass}$$ and measurable e.g. by comparing the fluctuations of the CMB (evolved to current epoch using Eq. \[eq:perturbations\]) with those of galaxy density, on spatial scales where perturbation theory holds. One thus measures a combination of parameters $\beta \equiv f/b$ ([@Kaiser87]), where $f \equiv d \log \delta/d \log R$ describes the growth rate, and approximately reads $\Omega_M^{0.6}$ for standard gravity in a wide class of cosmologies around $\Lambda$CDM (e.g. p. 378 of [@KolbTurnerBook]).
The first evidence for redshift distortions were proposed in 2000 [@Hamilton00; @Taylor00; @Outram01]. A more precise measurement from the two-degree-field galaxy redshift survey (2dfGRS) data (mostly at $z<0.2$) [@Peacock01] (using the amplitude of CMB fluctuations to derive bias) concludes that $\Omega_M \simeq 0.3$. Similar conclusions are reached using a smaller sample at $z \simeq 0.55$ [@Ross07].
Going to higher redshifts allows one in principle to probe the evolution of growth rate between then and now. In [@Guzzo08], the measurement at $z\sim 0.8$ is limited by the sample size of about 10,000 galaxies of the VIPERS survey, and measures the growth rate to 2.5 $\sigma$. On a much larger volume, the WiggleZ survey measures the growth rate all the way to $z=0.9$ at high significance [@Blake11Growth] and discusses in detail the uncertainties in the way to account for non-linear effects, but does not venture into a fit of cosmological parameters.
To conclude, growth rate measurements from the redshift distortions are highly compatible with the current cosmological paradigm, not yet at a level to significantly contribute to cosmological constrains, and the way to overcome systematic uncertainties when measurements get more precise is still unclear. One may note that galaxy redshift surveys primarily aimed at measuring BAO over very large volume and redshift intervals will deliver high quality redshift distortion measurements in the same data sets.
Clusters of galaxies \[sec:clusters\]
=====================================
The use of clusters of galaxies samples to study the acceleration of the universe started in the mid 1970’s. At that time, brightest galaxies of clusters were used as standard candles to build Hubble diagrams extending to high enough redshifts that deviations from a straight Hubble line started to be detectable (see for example [@1975ApJ...195..255G; @1978ApJ...221..383K; @1976ApJ...205..688S] ). Interestingly, Gunn and Tinsley published in 1975 a letter titled “An accelerating Universe” (cautiously) reporting evidence that we live in an accelerating universe [@1975Natur.257..454G]. The conclusion was largely based on constraints obtained from measuring the brightness of galaxies in clusters [@1975ApJ...195..255G].
These results, however, were marginally significant and possibly subject to large systematic errors as pointed by the authors themselves, in particular galaxy evolutionary corrections.
Most of the constraints derived from clusters nowadays are not obtained from a fit to a Hubble diagram but rather from the variation of the number density of clusters as a function of redshift as described below.
Clusters as cosmological probes
-------------------------------
In the framework of the cold dark matter model, the number density of dark matter halos as a function of redshift can be calculated and compared to numbers obtained in large area cluster surveys that nowadays extend to high enough redshift.
Galaxy clusters are the largest virialized[^8] objects in the universe and are therefore expected to trace dark matter halos. The difficulty arises from the fact that clusters are, in practice, selected according to some observable $O$, such as X-ray luminosity or temperature. Other observables often used are cluster galaxy richness, weak lensing shear or Sunayaev-Zeldovich effect on the cosmic microwave background flux. The relation of these observable $O$ selected cluster distributions with cluster mass distributions can be written as $$\frac{d^{2}N(z)}{dzd\Omega} = \frac{d_M(z)}{H(z)}
\int^{\infty}_{0}f(O,z)dO\int^{\infty}_{0}p(O|M,z)\frac{dn(z)}{dM}dM ~,
\label{eq:clustercount}$$ where $f(O,z)$ is the observable redshift dependent selection function, $dn(z)/dM$ is the comoving density of dark halos, and $p(O|M,z)$ is the probability that a halo of mass $M$ at redshift $z$ is observed as a cluster with observable $O$.
Eq. (\[eq:clustercount\]) is sensitive to cosmology through the comoving volume element $ d_M^2(z)/H(z)$ (Eq \[eq:dvdz\]) and the growth of structure term, $dn(z)/dM$ which depends on the primordial spectrum and the evolution of density perturbations. As mentioned above, several techniques are used to detect clusters as well as for estimating their masses. Systematic uncertainties, however, can greatly affect the determination of the mass-observable relation $p(O|M,z)$ and of the selection function $f(O,z)$ (see for example [@2009ApJ...691.1307H]).
Multi-band imaging, for example, allows clusters to be efficiently detected as excesses in the surface density of galaxies, and observed colors provide reliable enough redshift estimates that accidental projections can be greatly reduced. Moreover, various other effects such as weak lensing (e.g. [@Mandelbaum10]), can be used to calibrate the mass-observable relations. Although weak lensing seems a safe approach to evaluate cluster masses, unrelated large scale structures along the line of sight are a serious source of bias in this calibration, see e.g. [@2002ApJ...575..640W; @Hoekstra11].
Clusters are also detected in X-ray emitted by the hot baryon gas trapped in the dark matter potential well, and their mass derived from X-ray luminosity or gas temperature (see for example ).
Since it does not, in principle, depend on the source distance, the Sunayaev-Zeldovich effect can also be used to detect clusters out to higher redshift. (see for example ), and weak lensing can also provide cluster detections (e.g. [@Miyazaki07]).
Another approach is to measure the baryonic gas mass from X-ray or SZ measurements and compare it with the virial mass estimates. The ratio of the two should be independent of redshift, which can only be achieved with the correct cosmology.
Current cosmological constraints
--------------------------------
![ *Measured mass functions of clusters at low and high redshifts compared with predictions of a flat accelerating model and an open model without dark energy (from [@2009ApJ...692.1060V])* []{data-label="fig:mfcn"}](Vikhlinin09_mfun.pdf){width="\textwidth"}
![ *Measured mass functions of clusters at low and high redshifts compared with predictions of a flat accelerating model and an open model without dark energy (from [@2009ApJ...692.1060V])* []{data-label="fig:mfcn"}](Vikhlinin09_mfun_nolambda.pdf){width="\textwidth"}
Cosmological constraints obtained from clusters have greatly improved during the last decade. X-ray observations of clusters obtained by “Chandra”, for example, have confirmed the acceleration of the expansion (see [@2010MNRAS.406.1759M] and more recently [@2012arXiv1202.2889B]). Fig. \[fig:mfcn\] illustrates the power of cluster measurements to constraints acceleration. It shows that the measured mass function of clusters is correctly described provided a non zero amount of dark energy is accounted for in the model.
Taken as a whole, results obtained by most of the groups and using different techniques now agree at the $\sim 20\%$ precision level on the measurements of $w$ and $\Omega_{M}$ (see Table 2 of ), and are in agreement with constraints obtained with other techniques. Clusters are now playing an important role in constraining the cosmic acceleration.
Gravitational lensing and cosmic shear \[sec:cosmic-shear\]
===========================================================
Gravitational lensing refers to the deflection of light by masses, or in a cosmological context by mass contrasts (see e.g. §2.4 of [@Peacock-book-99]). The transverse gravitational potential between sources and observers bends light bundles, thus remapping the image plane, which leaves observational signatures. Strong lensing refers to singular mappings, due to steep mass contrasts, and leads to spectacular signatures such as giant arcs and multiple images.
Weak lensing refers to non singular mappings and has tenuous observational signatures. The image displacement of distant galaxies is not observable but its gradient will coherently shear their images: locally, galaxies will display a coherent average elongation on top of natural randomly oriented ellipticities. This “cosmic shear” probes the density gradients, and the angular correlations of the cosmic shear probe the correlations of density perturbations (see e.g. [@Mellier99; @BartelmannSchneider01; @Refregier03Review]). The cosmic shear signal is weak : induced ellipticities are of order 0.01, when galaxies have natural ellipticities of $\sim$0.3 [@Mellier99], and one has to beat down this noise by brute force averaging. Shape distortions from the telescope (and atmosphere, when applicable) are commonly larger than the signal, and foreground stars are vital to map those distortions.
Rather than the mass power spectrum itself, its evolution with redshift (see Eq. \[eq:perturbations\]) is sensitive to dark energy properties, and splitting the shear signal in source redshift slices improves cosmological constraints [@Hu99]. This technique called “lensing tomography”, is expected to deliver strong dark energy constraints in the future (e.g. [@Amara07]). For this application, redshift of galaxies can be approximate (however unbiased) and one uses “photometric redshifts” derived from multi-band photometry (considered for this purpose as coarse spectroscopy).
First evidence of cosmic shear were found in 2000 [@VanWaerbeke00; @Bacon00; @Wittman00] from a few deg$^2$ surveys. In the following decade, two complementary paths were followed: improved galaxy shape measurements from the Hubble Space Telescope (HST), and much larger surveys from the ground. The COSMOS field covers 1.64 deg$^2$ imaged with the HST [@Scoville07], which delivers an image quality (quantified by the size of star images) far better than from the ground. Multi-band photometry from UV to IR has been collected from ground and space to estimate the photometric redshifts. Using shear tomography, a $\sim$90% CL evidence for acceleration was obtained from this data set [@Schrabback10].
The wide CFHT legacy survey (CFHTLS, 2003-2008) has collected images on 170 deg$^2$ in five bands from the ground. Preliminary results do not use the tomography ([@Fu08] and references therein), find a signal amplitude compatible with $\Lambda$CDM, measure the signal on large angular scales where perturbation theory applies, and detect the expected signal rise with redshift of sources. Shear tomography results from this survey are expected very soon (see cfhtlens.org), but given the survey area, dark energy constraints cannot yet outperform the current supernova and cluster counts results.
Age of the universe \[sec:age-of-the-universe\]
===============================================
Given a current cosmological model, one can integrate the Friedman equation and derive the time elapsed since the Big Bang. The uncertainties associated to the exact nature of the “beginning” are totally negligible in this context. In a matter-dominated universe, the age of the universe tends to $t_0 = H_0^{-1}$ for low matter density, reads $t_0 = 2/3 H_0^{-1}$ for a flat universe, and has even shorter values for closed universes. For $H_0 = 70 km/s/Mpc$, $H_0^{-1} \simeq 13$ Gy.
An observational lower limit on the age of the universe can be derived from the confrontation of star models with real stars. The constraint is cosmologically relevant since the oldest stars in globular clusters have an age $12 < t_0 < 15$ Gy [@KraussChaboyer03]. For a matter-dominated universe, all densities but $\Omega_M\lesssim 0.1$ yield shorter ages, which is incompatible with constraints from galaxy clustering (§\[sec:BAO\]). With dark energy, we can have both a $\sim$13 Gy age and a low matter density ($\Omega_M \sim 0.3$) : cosmological models accelerating now had a slower expansion in the past and hence predict a larger age than the same models without dark energy.
The CMB anisotropies measure the distance to last scattering surface (LSS) (i.e. when hydrogen atoms combine and light no longer scatters), which is an increasing function of age. In a flat $\Lambda$CDM model, the CMB anisotropies constrain $t_0 = 13.77 \pm 0.13$ [@Komatsu11]. The constrain involves either the flatness assumption or some other measurement, such as a local $H_0$ measurement.
Current constraints on dark energy \[sec:combination\]
======================================================
Figure \[fig:constraints:Mantz\_w\] displays a recent combination of constraints on a constant dark energy equation if state in a spatially flat universe (extracted from ). Over-plotted are constraints obtained from WMAP [@2009ApJS..180..306D], SNIa [@2008ApJ...686..749K], BAO [@2010MNRAS.401.2148P], abundance and growth of RASS clusters at $z<0.5$ (labeled XLF; [@2010MNRAS.406.1759M]) and gas fraction measurements at $z<1.1$ [@2008MNRAS.383..879A], as well as the combined result shown by the orange (95%) and yellow (68%) confidence ellipses. Current precisions on these parameters are at the level of 10% both statistically and for systematics.
![ *Joint 68.3% and 95.4% confidence regions for the dark energy equation of state and mean matter density (from )* []{data-label="fig:constraints:Mantz_w"}](MantzI_Wm_w){width="80.00000%"}
So far in this review, we have mainly discussed the achievements of observational programs in measuring the equation of state parameter $w$, assuming that $w$ does not vary with cosmic time. $w$ characterises the evolution with redshift of the dark energy density: for a constant $w$, $\rho_{DE} \propto (1+z)^{3(1+w)}$, and more generally, $\dot \rho_{DE} = -3\rho_{DE} H (1+w)$. When challenging the cosmological constant paradigm, or simply aiming at a finer characterisation of dark energy, one may consider a first order variation of the equation of state such as [@ChevallierPolarski01]: $w(z)\equiv w_0+w_a(1-a)=w_0+w_a
z/(1+z)$, where $w_a$ characterises the variation. Since observations span at most $0<a \leq 1$, only confidence intervals significantly smaller than 1 are really constraining. Unsurprisingly, this has not happened yet, even when fits gather essentially all available data: [@Komatsu11] reports $w_a = -0.38 \pm 0.66$ (flat universe), and with more supernovae but a careful accounting of systematic uncertainties [@Sullivan11] finds $w_a = -0.984\pm 1.09$. So, the current limits on a varying equation of state are of limited interest. Future large projects such as Euclid, WFIRST or LSST could bring the uncertainty of $w_a$ down to about 0.2 provided their level of systematics uncertainties are kept low.
Future and prospects \[sec:future\]
===================================
Dark energy science relies today almost entirely on imaging and spectroscopy in the visible and near infrared. Several new wide-field imaging projects are starting to take data or soon will. The Table below summarizes the key figures of some present and future instruments and the size of anticipated (or executed) observing programs. Only currently approved programs are listed.
-------------- ----------------- ----------------- ------- --------------
Project Mirror Area First Large survey
$\diameter$ (m) (${\rm deg^2}$) light (nights)
CFHT/Megacam 3.6 1. 2002 500
Pan-STARRS 1.8 7. 2009 $>$1000
Blanco/DEC 4.0 3. 2012 500
Subaru/HSC 8.2 1.8 2013 $\sim$500
LSST 6.5 10. 2019 3500
Euclid 1.2 0.5 2019 5 years
-------------- ----------------- ----------------- ------- --------------
: *Key figures for the major past and future wide-field imaging facilities. SNLS was part of the CFHTLS, a cosmology-oriented 500-night survey, executed at CFHT. Pan-STARRS, is currently constructing a second telescope and aims at eventually operating 4 of them. The 500-night survey to be run at the CTIO-Blanco is called Dark Energy Survey (DES). LSST main mirror diameter is 8.4 m but suffers from a 5 m central occultation. This facility will almost entirely observe in survey mode during its anticipated 10-year lifetime. Euclid is an approved ESA space project meant to observe in visible and near IR. \[tab:wide-field-projects\]*
They are imaging programs at the exception of the Euclid project, which will carry out both imaging and spectroscopy for high redshift galaxy survey. There are very few wide-field spectroscopy projects and Euclid’s galaxy redshift survey is the main approved program in this field.
Regarding dark energy constraints these projects might deliver in the future, the main forecasts [@DETF06; @ESO-ESA] show that systematic uncertainties are at play for all dark energy probes, and insist on a multi-probe approach. One should also note that all these telescopes will deliver data sets of major interest besides dark energy.
Second generation supernova surveys are expected to deliver their final samples in the next two years. This will amount to about 1000 distant events (at $z>0.1$) and a growing set of nearby supernovae. Pan-STARSS is running a supernova program, DES is expected to[@Bernstein11], as well as LSST. Euclid does not currently have a supernovae program, although such a program could significantly contribute to dark energy constraints [@Astier11]. The accuracy of cosmological constrains will primarily depend on the accuracy of the relative photometric calibration of the various samples.
Although weak lensing has not delivered strong dark energy constraints yet, it concentrates hopes for the future: forecasts place the shear correlations ahead of all other dark energy probes (e.g. [@DETF06; @ESO-ESA; @EuclidRedBook11]). However, measuring the shear field from galaxy shapes is a difficult problem (see e.g. [@Zhang11] and references therein), and relies on a very precise knowledge of the imaging system response (e.g. [@Paulin-Henriksson09] and references therein). Current methodologies [@Great08results] are improving rapidly but still behind the required measurement accuracy [@Amara08]. One can now distinguish two observational complementary approaches: high signal to noise repeated measurements from the ground (pursued by LSST), and high resolution images from space (pursued by Euclid). Besides the difficult shear measurement, the comparison with expectations is not straightforward on small spatial scales, because of non linearity (e.g. [@Sato09]) and poorly known “baryon” physics [@Semboloni11]. The observations required for tomographic shear measurements also allow one to measure cosmic magnification statistics [@VanWaerbeke10], which are free of all issues associated to shear measurements. Both approaches probe the same density field but with unrelated systematics, and can hence be compared without cosmic variance.
Galaxy redshift surveys have been dominated lately by WiggleZ and the SDSS. The SDSS telescope is currently running the BOSS program, to deliver galaxy redshifts up to $z\sim 0.7$, from which results are expected very soon (see [@BOSS12]). Measuring BAO at large redshifts might come from the Fastsound redshift survey on Subaru, and also from the currently observing BOSS Quasar survey on SDSS-III [@LeGoff11]. Beyond these existing instruments, BigBOSS [@BigBossProposal11] constitutes a natural far-reaching ground-based follow-up project, but is not approved yet. Euclid’s core program [@EuclidRedBook11] includes a galaxy redshift survey (mainly at $z>\sim 0.8$) mostly for BAO and redshift distortions.
Over the next few years, the completion of the South Pole Telescope (SPT), the Atacama Cosmology telescope (ACT) and of the Planck SZ surveys will result in a large increase of the number of known clusters up to redshifts $z>1$: about 1000 new clusters are expected to be discovered. Used in combination with existing optical and X-ray catalogs they should lead to significant improvement of our understanding of cluster growth and therefore further help constraint the acceleration of the expansion. However, calibration of mass proxies is likely to remain a limitation for these surveys and will require improvement of spatial resolution and sensitivity planned for the next generation of surveys such as the CCAT project. On the optical and near-infrared front, the large number of new ground-based surveys (see list above) will also result in a significant increase of the number of clusters and provide needed additional data such as the photometric redshifts and lensing data. Here again, the difficulty will be to define improved mass proxies in the redshift range of interest. In space, the planned Euclid and WFIRST near-infrared missions will allow cluster studies and measurements to be extended to higher redshifts and larger volumes to be probed. X-ray clusters samples detected with Chandra and XMM-Newton will help, in the near future, improve cosmological constraints from clusters. They will pave the way to the use of high statistics from the eROSITA X-ray telescope, which is expected to detect more than 50000 clusters with unprecedented purity and completeness.
Summary and conclusions \[sec:conclusion\]
==========================================
Thirteen years after its discovery the acceleration of the expansion is now firmly established and the concordance model constitutes the frame of a standard model of cosmology. Several techniques are now used at telescopes around the word to probe dark energy following, and sometimes driving, the fast development of wide-field imaging and multi-object spectroscopy, and making use of increased precision X-ray and CMB measurements. Of those techniques, SN and BAO are the most developed today, closely followed by the use of galaxy clusters, which has made considerable progress in the last few years and the emerging use of weak shear which promises to become one of the best tools to measure dark energy. All these techniques make use of the impressive precision on cosmological parameters obtained by WMAP, soon to be improved by Planck results. One of the key features and power of using several techniques is that these techniques do not always probe the same domain of the cosmological model. Supernovae are a pure geometrical test, BAO are mostly geometrical too, while Weak Lensing and Clusters probe both geometry and growth of structure. Put together, they not only help break cosmological parameter degeneracies, but more importantly are subject to unrelated systematics. Mixing probes could also help finding out whether the acceleration of the expansion requires changing gravity on large scale or not.
To date, constraints on the dark energy equation of state require combining different techniques and are at the level of 10% both statistically and for systematics. In the future, with the coming next generation of experiments, each technique, combined with CMB precision measurements from the Planck satellite, will provide individual constraints on the dark energy equation of state, and combined they should be able to reach percent level precision. These projects may or may not see departure from $w=-1$ but if they do or if $w$ is found to vary with time, they would rule out a cosmological constant or vacuum energy as the source of acceleration and open the way to new physics. Likewise, if the values of $w$ determined from geometry or growth of structure methods are not equal, it would point toward a modification of gravity as the cause of acceleration. The next decade could bring important new observational clues on the origin of the acceleration of the expansion.
å[A&A]{}
[^1]: Big-Bang Nucleosynthesis (BBN) refers to the synthesis of nuclei via the fusion of light elements in the first minutes of the universe (see e.g. , §4 of [@KolbTurnerBook] & [@Wagoner73]). The primordial abundance of light elements (and in particular He) depend on the baryon-to-photon density ratio then.
[^2]: Flatness is an inevitable consequence of the inflation theory, meant to solve some observation-based puzzles of hot big bang cosmology (see e.g. §8 of [@KolbTurnerBook] and references therein).
[^3]: For textbooks covering these matters, we suggest [@KolbTurnerBook; @Peacock-book-99].
[^4]: In Newtonian gravity, since only masses source gravity, we would find $\rho$ instead of $\rho+3p$ there.
[^5]: Note that the electromagnetic radiation density is precisely known from FIRAS (aboard COBE) measurements [@Mather94] of Cosmic Microwave Background (CMB) temperature : $T_0=2.275 \pm 0.001 K$ [@Fixsen02], and hence the CMB radiation density is known to 1.5 $10^{-3}$. This uncertainty has a negligible impact on our cosmological model [@Hamman08]. Since neutrinos contribute to radiation density at equality, their number density relative to photons has to be assumed for the above arguments to hold. In particular, we [*assume*]{} that there are 3 neutrino species in the universe, i.e. that only the neutrinos with standard interactions (such as counted at the LEP collider) exist in the universe. CMB precision experiments now enter in a precision regime that might allow this hypothesis to be tested (e.g. [@Keisler11]).
[^6]: Astronomical magnitudes are a logarithmic scale used to describe relative fluxes: $m=-2.5 log_{10}(Flux/Flux_{ref})$
[^7]: Colors are defined in astronomy as the ratio of fluxes measured in two different bands, or in practise the difference of magnitudes in two different bands. A typical color indicator, widely used for distances to supernovae and elsewhere is B-V, where B is a (blue) filter covering \[390,480\]nm, and V covers the green region \[500,600\]nm.
[^8]: In an expanding universe, strong over-densities no longer follow the expansion, and are bound. They approximately respect the virial relation between kinetic and potential energy, hence their name.
|
---
abstract: 'The kinetic Brownian motion on the cosphere bundle of a Riemannian manifold ${\mathbb{M}}$ is a stochastic process that models the geodesic equation perturbed by a random white force of size ${\varepsilon}$. When ${\mathbb{M}}$ is compact and negatively curved, we show that the $L^2$-spectrum of the infinitesimal generator of this process converges to the Pollicott–Ruelle resonances of ${\mathbb{M}}$ as ${\varepsilon}$ goes to $0$.'
author:
- Alexis Drouot
title: 'Pollicott–Ruelle resonances via kinetic Brownian motion'
---
Introduction {#sec:1}
============
We consider a smooth compact Riemannian manifold ${\mathbb{M}}$ with negative sectional curvatures and cosphere bundle $S^*{\mathbb{M}}$. The generator of the geodesic flow $H_1 \in TS^*{\mathbb{M}}$ has the Anosov property and, on suitable spaces, $P_0 {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\frac{1}{i} H_1$ has a discrete spectrum with eigenvalues called Pollicott–Ruelle resonances. We denote it by ${\mathrm{Res}}(P_0)$ These complex numbers appear in expansions of classical correlations – see Tsuji [@Ts] and Nonnemacher–Zworski [@NZ]. We refer to §\[sec:2.3\] for precise definitions.
Recently, several authors studied a stochastic process on $S^*{\mathbb{M}}$ called the kinetic Brownian motion – see Franchi–Le Jan [@LF], Grothaus–Stilgenbauer [@GS2], Angst–Bailleul–Tardif [@ABT] and Li [@Li]. In contrast with the Langevin process, the kinetic Brownian motion models diffusive phenomena with finite speed of propagation. Its infinitesimal generator is $i P_{\varepsilon}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}H_1+{\varepsilon}\Delta_{\mathbb{S}}$, where $\Delta_{\mathbb{S}}\geq 0$ is the vertical spherical Laplacian – see §\[sec:2.1\].
In this paper, we investigate the convergence of the $L^2$-spectrum $\Sigma(P_{\varepsilon})$ of $P_{\varepsilon}$, as ${\varepsilon}$ goes to $0^+$. Although the $L^2$-spectrum of $P_0$ is absolutely continuous and equal to ${\mathbb{R}}$, we have:
\[thm:0\] The set of accumulation points of $\Sigma(P_{\varepsilon})$ as ${\varepsilon}\rightarrow 0^+$ is equal to ${\mathrm{Res}}(P_0)$.
Theorem \[thm:6\] below is a finer statement: the spectral projections of $P_{\varepsilon}$ depend smoothly on ${\varepsilon}$; and if each Pollicott–Ruelle resonance of $P_0$ is simple, the $L^2$-eigenvalues of $P_{\varepsilon}$ admit a full expansion in powers of ${\varepsilon}$. Remark \[rem:1\] analyzes the convergence as ${\varepsilon}\rightarrow 0^-$.
We proved Theorem \[thm:0\] when ${\mathbb{M}}$ is an orientable surface in an earlier version [@D2] of this paper.
Motivation and outline of proof {#motivation-and-outline-of-proof .unnumbered}
-------------------------------
Dyatlov–Zworski [@DZ2] showed that the Pollicott–Ruelle resonances of an Anosov vector field $X$ on a Riemannian manifold are the limits as ${\varepsilon}\rightarrow 0^+$ of the $L^2$-eigenvalues of $\frac{1}{i}(X + {\varepsilon}\Delta)$. From the point of view of partial differential equations, this realizes resonances as viscosity limits. From the point of view of probability theory, this indicates stochastic stability of Pollicott–Ruelle resonances, because the operator $\frac{1}{i}X + i {\varepsilon}\Delta$ generates the stochastic differential equation $$\label{Eq:0a}
{{\partial}}_t\Phi_t = - X(\Phi_t) - \sqrt{2{\varepsilon}} B(t), \ \ \ \ \Phi_0 = {{\operatorname{Id}}}_{\mathcal{M}},$$ where $B(t)$ is a Brownian motion on ${\mathcal{M}}$. Their approach also shows that the $L^2$-eigenvalues of $\frac{1}{i}X + i {\varepsilon}\Delta$ converge to complex conjugates of Pollicott–Ruelle resonances as ${\varepsilon}\rightarrow 0^-$. This fact also holds here, see Remark \[rem:1\].
The geodesic flow on the cosphere bundle $S^*{\mathbb{M}}$ of a Riemannian manifold ${\mathbb{M}}$ is a fundamental example of Anosov flow. If $X$ denotes the generator of the geodesic flow, is a random perturbation of the geodesic equation. The perturbative term in acts on both momenta and positions. As was first modeled by Langevin’s equation [@L08], a physical random perturbation created by collisions should only act on the momentum variables. A generalization of Langevin’s equation to cotangent bundles $T^*{\mathbb{M}}$ was studied in J$\o$rgensen [@J], Soloveitchik [@So] and Kolokoltsov [@K].
In this paper, we remain on the cosphere bundle $S^*{\mathbb{M}}$ and we consider the *kinetic Brownian motion*. This stochastic process is a random perturbation in the momentum random of the geodesic equation on $S^*{\mathbb{M}}$. It models diffusions with constant speed of propagation, and has generator $H_1 + {\varepsilon}\Delta_{\mathbb{S}}$. The kinetic Brownian motion was first introduced in Franchi–Le Jan [@LF], as an extension of Langevin’s equation in general relativity: it models the relativistic motion of random particles, whose speed has to be bounded by the speed of light. Grothaus–Stilgenbauer [@GS2] extended the construction to cosphere bundles of Riemannian manifolds, with applications to industry. Li [@Li] showed the first perturbative results in the small-and-large white force limit (respectively, ${\varepsilon}\rightarrow 0$ and ${\varepsilon}\rightarrow \infty$). Angst–Bailleul–Tardif [@ABT] improved upon Li’s result and derived asymptotic in the context of rotationally invariant manifolds. We refer to §\[sec:2.1\] for precise definitions.
Dolgopyat–Liverani [@DL] studied another perturbation of the geodesic equation. They considered the geodesic motion of particles, coupled with an interaction of size ${\varepsilon}$. When the initial data is random and ${\varepsilon}$ goes to $0$, they showed that a suitable rescaling of the energy at time $t$ solves an explicit stochastic differential equation. Bernadin et al. [@BeL] obtained a formal expansion of the heat conductivity for systems of weakly coupled random particles. Conceptually, both results can be seen as a step towards deriving macroscopic equations from principles of microscopic dynamics.
This paper aims to generalize the main result of Dyatlov–Zworski [@DZ2] to the kinetic Brownian motion. In contrast with [@DZ2], the operator $P_{\varepsilon}= \frac{1}{i}(H_1+{\varepsilon}\Delta_{\mathbb{S}})$ is hypoelliptic instead of being elliptic. An earlier version [@D2] contains a proof of Theorem \[thm:0\] when ${\mathbb{M}}$ is an orientable surface. It can be seen as an introduction to the present paper. The technical details are simpler there, because in that case $\Delta_{\mathbb{S}}= -V^2$, with $V$ the generator of the circle action on the fibers of $S^*{\mathbb{M}}$.
The lack of ellipticity of $P_{\varepsilon}$ creates serious new difficulties that we overcome by showing that the operator $P_{\varepsilon}$ is maximally hypoelliptic in the regime ${\varepsilon}\rightarrow 0$, see Theorem \[thm:1\]. For technical reasons, we will lift $P_{\varepsilon}$ to an operator ${\widetilde{P}}_{\varepsilon}$ acting on functions on the orthonormal coframe bundle of ${\mathbb{M}}$. The proof continues with the Lebeau [@L], where the maximal hypoellipticity of Bismut’s hypoelliptic Laplacian [@B] is shown. Lebeau ingeniously uses certain commutation relations to reduce his study to the case of the model operator $x_1^2 D_{x'}^2 + D_{x_1}$, microlocally near $(0, x', 0, \xi')$, $\xi' \neq 0$. In our approach, we bypass the microlocal reduction and we work directly with $P_{\varepsilon}$. We replace Lebeau’s main step with a positive commutator argument. This yields a maximal hypoellipticity result for ${\widetilde{P}}_{\varepsilon}$, that descends to an estimate for $P_{\varepsilon}$. Lifting geometric equations to the orthonormal frame bundle has been an efficient technique in probability theory, starting with the pioneering constructions of stochastic processes on manifolds by Elworthy [@El]. It was used in both Li [@Li] and Angst–Bailleul–Tardif [@ABT] to show asymptotic results for the kinetic Brownian motion.
The remainder of the proof of Theorem \[thm:0\] is similar to [@DZ2]. We will decompose the operator $P_{\varepsilon}$ in two parts $P^\sharp_{\varepsilon}+ P^\flat_{\varepsilon}$. The first part acts on momentum frequencies greater than ${\varepsilon}^{-1}$, and the maximal hypoelliptic estimate will take care of it. For the second part, we will use the anisotropic Sobolev spaces designed in Faure–Sjöstrand [@FS] in a modified form due to Dyatlov–Zworski [@DZ1]. Their construction relies on Melrose’s propagation estimate at radial points [@Me], in the improved version of [@DZ1 Propositions 2.6-2.7]. For the original version of anisotropic spaces used in Anosov dynamics, see Baladi [@Ba], Liverani [@Liv], Gouëzel–Liverani [@GoLi] and Baladi–Tsuji [@BaTs]. We also mention Vasy [@Va] for application of similar anisotropic Sobolev spaces in the context of asymptotically hyperbolic manifolds and general relativity.
The operator $P_{\varepsilon}$ can be realized as the restriction of the hypoelliptic Laplacian of Bismut [@B] to the cosphere bundle. This connection provides another motivation for the study of $P_{\varepsilon}$. Li [@Li] and Angst–Bailleul–Tardif [@ABT] showed that the kinetic Brownian motion interpolates between geodesic trajectories as ${\varepsilon}\rightarrow 0$ and the Brownian motion on ${\mathbb{M}}$ as ${\varepsilon}\rightarrow \infty$ (after projection and rescaling). This dramatically echoes Bismut–Lebeau’s motivation to study the hypoelliptic Laplacian, obtained in [@BL1] as an operator interpolating between the generator of the geodesic flow and the Laplacian on ${\mathbb{M}}$ (after rescaling and projection). For the corresponding interpretation in probability theory, see Bismut [@B2]. Improving upon work of Bismut [@BL2], Shen [@S] recently obtained far-reaching applications of the hypoelliptic Laplacian, including a proof of Fried’s conjecture [@Fr] for maximally symmetric spaces.
Baudoin–Tardif [@BT] showed exponential convergence of the heat operator $e^{-itP_{\varepsilon}}$ to equilibrium: there exists $\nu_{\varepsilon}> 0$ such that for every $u \in S^\infty(S^*{\mathbb{M}})$, $$\left|e^{-itP_{\varepsilon}} u-\int_{S^*{\mathbb{M}}} u\right| \leq Ce^{-\nu_{\varepsilon}t}\left|u-\int_{S^*{\mathbb{M}}} u\right|.$$ Because of the connection of $P_{\varepsilon}$ with the Laplacian on ${\mathbb{M}}$, Baudoin and Tardif expected that the optimal value of $\nu_{\varepsilon}$ converges as ${\varepsilon}\rightarrow \infty$ to the first eigenvalue of the *non-negative* Laplacian on ${\mathbb{M}}$. Though the explicit value of $\nu_{\varepsilon}$ derived there converges to $0$ as ${\varepsilon}\rightarrow \infty$. When ${\mathbb{M}}$ is negatively curved, we conjecture that the optimal value of $\nu_{\varepsilon}$ converges as ${\varepsilon}\rightarrow 0$ to the largest imaginary parts of Pollicott–Ruelle resonances of $\frac{1}{i}H_1$.
When ${\mathbb{M}}$ is not negatively curved, we can still study the accumulation points of the $L^2$-eigenvalues of $P_{\varepsilon}$ as ${\varepsilon}\rightarrow 0$. Already in the case of the $2$-torus, the behavior of this spectrum is quite mysterious. See (in a slightly different context) [@DZ2 Figure 3] and the discussion following it, originating from Galtsev–Shafarevich [@GS06]. The general case is far from being understood. Recently, Dyatlov–Zworski [@DZ3] showed a deep connection between Pollicott–Ruelle resonances and topology: the order of vanishing of the Ruelle zeta function at $0$ determines the genus of a negatively curved surface. We believe that the spectrum of $P_{\varepsilon}$ relates closed geodesics and topology, even when ${\mathbb{M}}$ is not negatively curved. The maximal hypoelliptic estimate holds with no restrictions on the sign of the curvature. However, the methods of §\[sec:5\] are strictly restricted to the negative curvature case.
**Acknowledgment.** We are very grateful to Maciej Zworski for suggesting the problem and for his invaluable guidance. We would also like to thank Semyon Dyatlov and Gilles Lebeau for various fruitful discussions. This research was partially supported by the Fondation CFM and the National Science Foundation grant DMS-1500852.
Preliminaries
=============
The generator of the kinetic Brownian motion {#sec:2.1}
--------------------------------------------
Let ${\mathbb{M}}$ be a smooth compact Riemannian manifold of dimension $d \geq 2$ and $S^*{\mathbb{M}}= \{(z,\zeta^1) \in T^* {\mathbb{M}}: |\zeta^1|=1\}$ be its cosphere bundle – the notation $\zeta^1$ instead of $\zeta$ for the dual variable of $z$ will be clear later. The restriction of the Liouville $1$-form $z \cdot d\zeta^1$ to $S^*{\mathbb{M}}$ is a contact form on $S^*{\mathbb{M}}$, denoted ${\alpha}$. Its Reeb vector field $H_1$ generates the geodesic flow. In particular, ${\alpha}(H_1) = 1$, $d{\alpha}(H_1,\cdot) = 0$, and $H_1$ is divergence-free with respect to the Liouville measure $\mu = {\alpha}\wedge (d{\alpha})^{d-1}$. The $L^2$-norm with respect to this measure will be denoted by $|\cdot|$. Alternatively, $H_1$ is the restriction to $T^*{\mathbb{M}}$ of the Hamiltonian vector field of the function $\frac{1}{2}|\zeta^1|^2$, with respect to the canonical symplectic form $dz \wedge d\zeta^1$ on $T^*{\mathbb{M}}$.
For every $z \in {\mathbb{M}}$, the fiber $T_z^*{\mathbb{M}}$ admits a Euclidean structure and the fiber $S^*_z{\mathbb{M}}$, provided with the induced metric, is a Riemannian submanifold of $T_z^*{\mathbb{M}}$. The *non-negative* Laplacian on $S_z^*{\mathbb{M}}$ is a differential operator $\Delta_{\mathbb{S}}(z) : C^\infty(S^*_z{\mathbb{M}}) \rightarrow C^\infty(S^*_z{\mathbb{M}})$. Varying $z$ we obtain a differential operator $\Delta_{\mathbb{S}}: C^\infty(S^*_z{\mathbb{M}}) \rightarrow C^\infty(S^*_z{\mathbb{M}})$ called the spherical vertical Laplacian. Similarly there is a spherical vertical gradient operator $\nabla_{\mathbb{S}}: C^\infty(S^*{\mathbb{M}}) \rightarrow C^\infty(TS^*{\mathbb{M}})$, defined on each fiber $S_z^*{\mathbb{M}}$ as the standard gradient.
Let $P_{\varepsilon}$ be the operator $$P_{\varepsilon}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\dfrac{1}{i}(H_1 + {\varepsilon}\Delta_{\mathbb{S}}) = \dfrac{1}{i}H_1 - i{\varepsilon}\Delta_{\mathbb{S}},$$ with $L^2$-domain $D(P_{\varepsilon}) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\{ u \in L^2 : \ P_{\varepsilon}u \in L^2\}$ – here $P_{\varepsilon}u$ is seen as a distribution. Angst–Bailleul–Tardif [@ABT] call $P_{\varepsilon}$ the generator of the *kinetic Brownian motion*. In §\[sec:2.2.4\] below we compute certain Lie brackets, showing that $P_{\varepsilon}$ satisfies Hörmander’s condition [@Ho] for hypoellipticity. The Rothschild–Stein theory of hypoelliptic operators [@RS §18] yields the subelliptic estimate : there exists a constant $c_{\varepsilon}> 0$ such that $|u|_{H^{2/3}} \leq c_{\varepsilon}(|P_{\varepsilon}u|+|u|)$. A significant part of this paper, §\[sec:3\], studies the behavior of $c_{\varepsilon}$ as ${\varepsilon}\rightarrow 0$ when $H^{2/3}$ is replaced by its semiclassical version $H^{2/3}_{\varepsilon}$.
The operator $P_{\varepsilon}$ is semibounded: ${\operatorname{Re}}({\langle iP_{\varepsilon}u,u \rangle}) \geq 0$. Combined with the hypoellipticity of $P_{\varepsilon}$ and the compactness of $S^*{\mathbb{M}}$, this shows that $P_{\varepsilon}$ has a discrete spectrum on $L^2$. This paper studies the accumulation points as ${\varepsilon}\rightarrow 0$ of the $L^2$-eigenvalues of $P_{\varepsilon}$ when ${\mathbb{M}}$ has negative curvature.
Operators on frame bundles
--------------------------
This section reviews Cartan’s lifting process from the cosphere bundle $S^*{\mathbb{M}}$ to the bundle of orthonormal frames $O^*{\mathbb{M}}$. Angst–Bailleul–Tardif [@ABT] and Li [@Li] previously used it to show asymptotic of the kinetic Brownian motion in the limits ${\varepsilon}\rightarrow 0, \infty$. We mention that when ${\mathbb{M}}$ is an orientable surface, $O^*{\mathbb{M}}\equiv S^*{\mathbb{M}}\times \{\pm 1\}$ and this lifting process is unnecessary. This simplifies the technical aspects in the earlier version [@D2] of this paper.
### Horizontal and vertical vector fields.
The space of frames at $z \in {\mathbb{M}}$ – denoted ${\mathcal{F}}_z^*{\mathbb{M}}$ – is the vector space of linear maps $\zeta : {\mathbb{R}}^d \rightarrow T_z^*{\mathbb{M}}$. At this point $\zeta$ is not required to be orthogonal nor an invertible. The space ${\mathcal{F}}_z^*{\mathbb{M}}$ is a Euclidean when provided with the scalar product $(\zeta,\zeta') \mapsto {{\operatorname{Tr}}}(\zeta^* \zeta')$. Varying the base point $z$ we obtain a vector bundle ${\mathcal{F}}^*{\mathbb{M}}$ over ${\mathbb{M}}$ which admits a Riemannian structure.
For $(z_0,\zeta_0) \in {\mathcal{F}}^*{\mathbb{M}}$, a vector $X_0 \in T_{z_0,\zeta_0}{\mathcal{F}}^*{\mathbb{M}}$ is said to be vertical if $X_0$ is tangent to the fiber ${\mathcal{F}}_{z_0}^*{\mathbb{M}}$. A smooth vector field $X \in T{\mathcal{F}}^*{\mathbb{M}}$ is vertical if $X(z_0,\zeta_0)$ is vertical for all $(z_0,\zeta_0) \in {\mathcal{F}}^*{\mathbb{M}}$. A curve $t \mapsto (z_t,\zeta_t) \in {\mathcal{F}}^*{\mathbb{M}}$ is said to be horizontal if for all $e \in {\mathbb{R}}^d$, $\zeta_t(e)$ (which belongs to $T_{z_t}{\mathbb{M}}$) is parallel along $z_t$ with respect to the Levi–Civita connection. A vector $X_0 \in T_{z_0,\zeta_0}{\mathcal{F}}^*{\mathbb{M}}$ is horizontal if there exists a horizontal curve $(z_t,\zeta_t)$ with ${{\partial}}_t (z_t,\zeta_t)(0) = X_0$; a smooth vector field $X \in T{\mathcal{F}}^*{\mathbb{M}}$ is horizontal if $X(z,\zeta)$ is horizontal for every $(z,\zeta) \in {\mathcal{F}}^*{\mathbb{M}}$.
The bundle of orthonormal frames $O^*{\mathbb{M}}$ is the subbundle of ${\mathcal{F}}^*{\mathbb{M}}$ with fibers formed of orthogonal maps $\zeta : {\mathbb{R}}^d \rightarrow T_z^*{\mathbb{M}}$. Since parallel transport preserves angles, the Levi–Civita derivative of an orthogonal frame along a curve is still an orthogonal frame. Vertical and horizontal vector fields in $TO^*{\mathbb{M}}$ are defined similarly as before. We also observe that $O^*{\mathbb{M}}$ is a bundle over $S^*{\mathbb{M}}$, provided with the projection $\pi_{\mathbb{S}}: (z,\zeta) \mapsto (z,\zeta(e_1))$, where $e_1=(1,0,...,0) \in {\mathbb{R}}^d$.
Geodesics on ${\mathbb{M}}$ are identified with integral curves of the vector field $H_1$ defined in \[sec:2.1\]; the geodesic flow is then $\exp(tH_1)$. The vector field $H_1$ lifts to a horizontal vector field ${{\widetilde{H}}}_1$ on $O^*{\mathbb{M}}$ defined as follows. Fix $(z_0,\zeta_0) \in O^*{\mathbb{M}}$ and let $(z_0,\zeta_0^1) = (z_0,\zeta_0(e_1))$ be its projection on $S^*{\mathbb{M}}$; let $(z_t,\zeta_t^1) = \exp(tH_1)(z_0,\zeta_0(e_1))$ be the geodesic starting at $(z_0,\zeta_0^1)$. Parallel transport of $\zeta_0$ along $z_t$ yields a flow $(z_t,\zeta_t) = \Phi_t(z_0,\zeta_0)$ on ${\mathcal{F}}^*{\mathbb{M}}$. Since the parallel transport preserves angles this flow actually takes values in $O^*{\mathbb{M}}$. As $(z_t,\zeta_t^1)$ is a geodesic, $\zeta_t^1 = \zeta_t(e_1)$ is the parallel transport of $\zeta_0^1$ along $z_t$ hence $\zeta_t(e_1) = \zeta_t^1$. This shows that $(z_t,\zeta_t)$ is a lift of $(z_t,\zeta_t^1)$ to the orthogonal frame bundle. The vector field ${{\widetilde{H}}}_1 \in T O^*{\mathbb{M}}$ is the generator of $\Phi_t$: $${{\widetilde{H}}}_1(z_0,\zeta_0) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\left.\dfrac{d}{dt}\right|_{t=0} \Phi_t(z_0,\zeta_0).$$ The integral curves of ${{\widetilde{H}}}_1$ are horizontal, which shows that ${{\widetilde{H}}}_1$ is horizontal.
Let $E_{k\ell}$ be the matrix $E_{k\ell} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}(\delta_{ki} \delta_{j\ell})_{ij}$ and $A_{k \ell}$ be the anti-symmetric matrix $A_{k\ell} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}E_{k \ell}-E_{\ell k}$. The matrix $e^{tA_{k\ell}}$ is orthogonal and $V_{k \ell}$ is the vector field on $O^*{\mathbb{M}}$ given by $$V_{k \ell}(z,\zeta) = \left. \dfrac{d}{dt} \right|_{t=0} \left(z,\zeta \circ e^{tA_{k \ell}}\right).$$ Since the projection of $(z,\zeta \circ e^{tA_{k\ell}})$ on ${\mathbb{M}}$ does not depend on $t$ the vector fiels $V_{k\ell}$ are vertical. The brackets of ${{\widetilde{H}}}_1$ with $V_{1k}$ define new vector fields on $O^*{\mathbb{M}}$: ${{\widetilde{H}}}_k {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}[{{\widetilde{H}}}_1, V_{1k}]$.
### Expression in coordinates
A system of coordinates $z_m \in {\mathbb{R}}^d$ on ${\mathbb{M}}$ lifts canonically to a system of coordinates $(z_m, \zeta_j^1)$ on $T^*{\mathbb{M}}$. If $(z,\zeta) \in {\mathcal{F}}^*{\mathbb{M}}$ then $\zeta(e_i) \in T_z^*{\mathbb{M}}$ and we denote by $\zeta^i_j$ its coordinates. This defines a system of coordinates on ${\mathcal{F}}^*{\mathbb{M}}$.
Unless precised otherwise, all the sums appearing below are run through indices from $1$ to $d$. Let $(z,\zeta) \in O^*{\mathbb{M}}\subset {\mathcal{F}}^*{\mathbb{M}}$ with coordinates $(z_m,\zeta_i^j)$. Then $$\zeta \circ e^{tA_{k\ell}} (e_i) = \zeta + t\zeta(A_{k\ell}e_i) + O(t^2) = \zeta + t\delta_{i\ell} \zeta(e_k)-t\delta_{ik}\zeta(e_\ell) + O(t^2).$$ Hence $\zeta \circ e^{tA_{k\ell}}$ has coordinates $\zeta^i_j + t \delta_{i\ell} \zeta^k_j -t \delta_{ik}\zeta^\ell_j + O(t^2)$ and $$\label{eq:0yab}
V_{k\ell} = \sum_{i,j} \left(\delta_{i\ell} \zeta^k_j - \delta_{ik}\zeta^\ell_j \right){\dfrac{\partial }{\partial \zeta^i_j}} = \sum_j \zeta^k_j {\dfrac{\partial }{\partial \zeta^\ell_j}} - \zeta_j^\ell {\dfrac{\partial }{\partial \zeta^k_j}}.$$
Geodesic trajectories $(z,\zeta^1) \in T^*{\mathbb{M}}$ satisfy the equation $$\dot{z}_m = \zeta_m^1, \ \ \dot{\zeta}_m^1 = \sum_{i,j} \Gamma_{ij}^m(z) \zeta_i^1 \zeta_j^1$$ while covectors $\eta \in T^*{\mathbb{M}}$ that are parallely transported along $(z,\zeta^1)$ satisfy $$\dot{\eta}_m = - \sum_{i,j} \Gamma_{ij}^m \zeta_j^1 \eta_i.$$ This yields the coordinate expression of ${{\widetilde{H}}}_1$, ${{\widetilde{H}}}_m$: $${{\widetilde{H}}}_1 = \sum_i \zeta_i^1 {\dfrac{\partial }{\partial z_i}} - \sum_{i,j,k,\ell} \Gamma_{ij}^\ell \zeta^1_i\zeta_j^k {\dfrac{\partial }{\partial \zeta_\ell^k}}, \ \ \ \
{{\widetilde{H}}}_m = \sum_i \zeta_i^m {\dfrac{\partial }{\partial z_i}} - \sum_{i,j,k,\ell} \Gamma_{ij}^\ell \zeta^m_i\zeta_j^k {\dfrac{\partial }{\partial \zeta^k_\ell}}.$$
### Some differential operators {#sec:2.2.3}
Recall that $\Delta_{\mathbb{S}}$ is the operator defined in §\[sec:2.1\] and let $\Delta_{\mathbb{M}}$ the *non-negative* Laplacian operator of ${\mathbb{M}}$. The operator $\Delta {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\Delta_{\mathbb{M}}+\Delta_{\mathbb{S}}$ is an elliptic operator acting on $C^\infty(S^*{\mathbb{M}})$.
The operators $\Delta_O^V, \Delta_O^H$ acting on $C^\infty(O^*{\mathbb{M}})$ are defined by $\Delta_O^V {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}-\sum_{i, j} V_{ij}^2$, $\Delta_O^H {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}-\sum_{i} {{\widetilde{H}}}_i^2$. The operator $\Delta_O {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\Delta_O^H+\Delta_O^V$ is an elliptic operator on $O^*{\mathbb{M}}$. Let $\pi_{\mathbb{S}}: (z,\zeta) \in O^*{\mathbb{M}}\mapsto (z,\zeta(e_1)) \in S^*{\mathbb{M}}$ be the bundle projection of $O^*{\mathbb{M}}$ to $S^*{\mathbb{M}}$. It lifts the operators $\Delta_O^V, \Delta_O^H, {{\widetilde{H}}}_1$ as follows: $$\label{eq:1b}
\Delta_O^V \pi^*_{\mathbb{S}}= \pi^*_{\mathbb{S}}\Delta_{\mathbb{S}}, \ \ \ \ \Delta_O^H \pi^*_{\mathbb{S}}= \pi_{\mathbb{S}}^* \Delta_{\mathbb{M}}, \ \ \ \ \pi_{\mathbb{S}}^* H_1 = {{\widetilde{H}}}_1 \pi_{\mathbb{S}}^*.$$
In order to prove the first identity of it is enough to show that for every $z \in {\mathbb{M}}$, $\pi_{\mathbb{S}}(z)^* \Delta_{\mathbb{S}}(z) = -\pi_{\mathbb{S}}(z)^*\sum_{i,j} V_{ij}^2(z)$, where $\pi_{\mathbb{S}}(z)$ is the canonical projection $\zeta \in O_z^*{\mathbb{M}}\rightarrow \zeta(e_1) \in S_z^*{\mathbb{M}}$ and $V_{ij}(z) = V_{ij}|_{C^\infty(O_z^*{\mathbb{M}})}$. Normal coordinates centered at $z$ on ${\mathbb{M}}$ induce coordinates $\zeta_i^1$ on $T_z^*{\mathbb{M}}$ (and $\zeta_i^j$ on ${\mathcal{F}}_z^*{\mathbb{M}}$). In these coordinates the Euclidean metric on $T_z^*{\mathbb{M}}$ takes the form $\sum_i (d\zeta_i^1)^2$; hence they provide an isometric identification of $S_z^*{\mathbb{M}}$ with ${\mathbb{S}}^{d-1}$, $O_z^*{\mathbb{M}}$ with $O(d)$, and ${\mathcal{F}}_z^*{\mathbb{M}}$ with ${\mathbb{R}}^{d \times d}$. Therefore, it suffices to show that if $\pi_{{\mathbb{S}}^{d-1}} : O(d) \rightarrow {\mathbb{S}}^{d-1}$ is the canonical projection, if $\Delta_{{\mathbb{S}}^{d-1}}$ and $\Delta_{O(d)}$ are respectively the Laplacians on ${\mathbb{S}}^{d-1}$ and $O(d)$ (with respect to the metric induce by the Euclidean structure of ${\mathbb{R}}^{d\times d}$), then $$\label{eq:5l}
\Delta_{O(d)} \pi^*_{{\mathbb{S}}^{d-1}}= \pi^*_{{\mathbb{S}}^{d-1}}\Delta_{{\mathbb{S}}^{d-1}}.$$ This identity should be available in the literature, though we have found no reference. We prove it below.
Since ${\mathbb{S}}^{d-1} \subset {\mathbb{R}}^d \subset {\mathbb{R}}^{d \times d}$, $\Delta_{{\mathbb{S}}^{d-1}}$ can be written as $-\sum_j X_j^2$, where the $X_j$ are the projections of ${{\partial}}_{\zeta_j^1}$ on ${\mathbb{S}}^{d-1}$ – see [@Hsu Theorem 3.1.4]. In coordinates, $$\label{eq:5m}
X_j = {{\partial}}_{\zeta_j^1} - \sum_k \zeta_j^1 \zeta_j^k {{\partial}}_{\zeta_j^k}.$$ A direct computation combining with $\sum_j (\zeta_j^1)^2 = 1$ on ${\mathbb{S}}^{d-1}$ shows that if $u$ is a function on ${\mathbb{R}}^{d \times d} $ depending only on $(\zeta^1_1, ..., \zeta^1_d)$, $$\Delta_{{\mathbb{S}}^{d-1}} u|_{{\mathbb{S}}^{d-1}} = -\sum_j {\dfrac{\partial ^2 u}{\partial {\zeta_j^1}^2}} + \sum_{j,k} \zeta_j^1 \zeta_k^1 {\dfrac{\partial ^2u}{\partial \zeta_k^1 {{\partial}}\zeta_j^1}}
+ (d-1) \sum_k \zeta_k^1 {\dfrac{\partial u}{\partial \zeta_k^1}}.$$ We similarly compute $\Delta_{O(d)} u|_{O(d)}$. Using and that $u$ depends only on $(\zeta^1_1, ..., \zeta^1_d)$,
\_[O(d)]{}u|\_[O(d)]{} = - \_[k,]{} ( \_j \^k\_j - \_j\^ )\^2 u =\_[i > 1]{} \_[j,k]{} (\_j\^1 - \_j\^i ) \_k\^i\
= -\_[i > 1]{} \_[j,k]{} \_j\^i \_k\^i +\_[i > 1]{} \_[j,k]{} \_j\^1 \_[jk]{} = -\_[i > 1]{} \_[j,k]{} \_j\^i \_k\^i +(d-1) \_j \_j\^1 .
Because of these formula, proving amounts to show that for $\zeta \in O(d)$, $$\label{eq:0u}
\sum_j {\dfrac{\partial ^2 u}{\partial {\zeta_j^1}^2}} = \sum_{i,j,k} \zeta_j^i \zeta_k^i {\dfrac{\partial ^2u}{\partial \zeta^1_k {{\partial}}\zeta^1_j}}.$$ Since $\zeta \in O(d)$, $\zeta^* \in O(d)$ which implies that $\sum_i \zeta_j^i \zeta_k^i = \delta_{jk}$. This relation shows that holds on $O(d)$, which proves and the first identity of .
The second identity in is [@Hsu Proposition 3.1.2].
If $(z_t,\zeta_t) = \exp(t{{\widetilde{H}}}_1)(z_0,\zeta_0)$ with $(z_0,\zeta_0) \in O^*{\mathbb{M}}$ then $\pi_{\mathbb{S}}(z_t,\zeta_t)$ is the geodesic starting at $\pi_{\mathbb{S}}(z_0,\zeta_0)$: $\pi_{\mathbb{S}}(z_t,\zeta_t) = \exp(tH_1) \pi_{\mathbb{S}}(z_0,\zeta_0)$. The identity $\pi_{\mathbb{S}}^* H_1 = {{\widetilde{H}}}_1\pi_{\mathbb{S}}^*$ follows.
We define ${\widetilde{P}}_{\varepsilon}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\frac{1}{i}({{\widetilde{H}}}_1 + {\varepsilon}\Delta_O^V)$. Because of , the operator ${\widetilde{P}}_{\varepsilon}$ is the lift of $P_{\varepsilon}$ to the orthogonal coframe bundle: ${\widetilde{P}}_{\varepsilon}\pi_{\mathbb{S}}^* = \pi_{\mathbb{S}}^* P_{\varepsilon}$.
### Commutation identities {#sec:2.2.4}
A computation using yields the commutation relation $$\label{eq:1d}
[V_{k\ell},V_{mn}] = \delta_{\ell m}V_{k n} + \delta_{n k} V_{\ell m} + \delta_{km} V_{n\ell} + \delta_{\ell n} V_{mk}.$$
We next study the commutation relations between the $V_{k\ell}$ and ${{\widetilde{H}}}_m$. Fix $z \in {\mathbb{M}}$ together with normal coordinates centered at $z$. In particular, $\Gamma_{ij}^\ell(z) = 0$ and $$[V_{k\ell},{{\widetilde{H}}}_m](z) = \sum_{i,j} [\zeta^k_j {{\partial}}_{\zeta_j^\ell} - \zeta^\ell_j {{\partial}}_{\zeta^k_j} ,\zeta_i^m {{\partial}}_{z_i}] = \sum_{i} \delta_{\ell m} \zeta^k_i {{\partial}}_{z_i} - \delta_{km} \zeta^\ell_i {{\partial}}_{z_i} = \delta_{\ell m}{{\widetilde{H}}}_k(z) - \delta_{km} {{\widetilde{H}}}_\ell(z).$$ Since $z$ was arbitrary, this shows that $$\label{eq:1e}
[V_{k\ell},{{\widetilde{H}}}_m] = \delta_{\ell m}{{\widetilde{H}}}_k - \delta_{km} {{\widetilde{H}}}_\ell.$$
We conclude this section by proving that the operators $\Delta_O^V, \Delta_O^H$ enjoy some important commutation properties: $$\label{eq:1c}
[\Delta_O^V, V_{mn}] = 0, \ \ \ \ [\Delta_O^H, \Delta_O^V] = 0, \ \ \ \ [\Delta_{\mathbb{M}}, \Delta_{\mathbb{S}}] = 0.$$
We start with the first identity. By , $[\Delta_O^V, V_{mn}]$
=-\_[k, ]{} (\_[m]{} V\_[k n]{} + \_[n k]{} V\_[m]{} + \_[km]{} V\_[n]{} + \_[n]{} V\_[mk]{}) V\_[k]{} + V\_[k]{} (\_[m]{} V\_[k n]{} + \_[n k]{} V\_[m]{} + \_[km]{} V\_[n]{} + \_[n]{} V\_[mk]{})\
= -\_k V\_[kn]{} (V\_[km]{}+V\_[mk]{}) + (V\_[km]{}+V\_[mk]{}) V\_[kn]{} + \_V\_[n]{} (V\_[m]{}+V\_[m]{}) + (V\_[m]{}+V\_[m]{}) V\_[k]{} = 0,
where we used that $V_{ij} + V_{ji}=0$.
For the second identity, we first observe that implies
\[V\_[k]{},\_O\^H\] = -\_m (\_[m]{}\_k - \_[km]{} \_) \_m + \_m (\_[m]{}\_k - \_[km]{} \_)\
= -\_k \_+ \_\_k - \_\_k + \_k \_= 0.
Therefore $\Delta_O^H$ commutes with the $V_{k\ell}$ and a fortiori with $\Delta_O^V$.
The third identity is equivalent to $\pi_{\mathbb{S}}^*[\Delta_{\mathbb{M}},\Delta_{\mathbb{S}}] = 0$. This is automatically satisfied since $[\Delta_O^H, \Delta_O^V] = 0$ and $\pi_{\mathbb{S}}^*$ intertwines $\Delta_{\mathbb{M}}$ with $\Delta_O^H$ and $\Delta_{\mathbb{S}}$ with $\Delta_O^V$ – see .
### Sobolev equivalence {#subsec:1}
Recall that $\mu$ is the Liouville measure on $S^*{\mathbb{M}}$, that $\pi_{\mathbb{S}}$ denotes the bundle projection $O^*{\mathbb{M}}\rightarrow S^*{\mathbb{M}}$ and that $\pi_{\mathbb{S}}$ intertwines $\Delta_O$ with $\Delta$ – see . Let $\mu_O$ be a measure on $O^*{\mathbb{M}}$ with $$\label{eq:0o}
v \in C^\infty(S^*{\mathbb{M}}) \ \Rightarrow \ \int_{S^*{\mathbb{M}}} v d\mu = \int_{O^*{\mathbb{M}}} \pi_{\mathbb{S}}^*v d\mu_O.$$ Let $\Lambda_s = ({{\operatorname{Id}}}+ {\varepsilon}^2 \Delta)^{s/2}$, ${{\widetilde{\Lambda}}}_s = ({{\operatorname{Id}}}+ {\varepsilon}^2 \Delta_O)^{s/2}$. We *define* the semiclassical Sobolev space $H^s_{\varepsilon}$ on $S^*{\mathbb{M}}$ (resp. ${{\widetilde{H}}}^s_{\varepsilon}$ on $O^*{\mathbb{M}}$) by $H^s_{\varepsilon}= \Lambda_{-s} L^2$ (resp. ${{\widetilde{\Lambda}}}_{-s} L^2$) with the corresponding norm with respect to $\mu$ (resp. $\mu_O$). The identity implies $$\label{eq:7f}
|\pi_{\mathbb{S}}^*u|_{{{\widetilde{H}}}^s_{\varepsilon}}^2 = \int_{O^*{\mathbb{M}}} \left|{{\widetilde{\Lambda}}}_s \pi_{\mathbb{S}}^* u\right|^2 d\mu_O = \int_{S^*{\mathbb{M}}} \left|\Lambda _s u\right|^2 \mu = |u|_{H^s_{\varepsilon}}^2.$$ The commutation relation shows that the vector fields $V_{k\ell}, [V_{1m},{{\widetilde{H}}}_1]$ span the whole tangent bundle $TO^*{\mathbb{M}}$. The operator ${\widetilde{P}}_{\varepsilon}$ satisfies Hörmander’s condition [@Ho] for hypoellipicity, with only one commutator needed. The Rothschild–Stein theory [@RS §18] shows that there exists a constant $C_{\varepsilon}> 0$ such that $$\label{eq:0p}
v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \
|v|_{{{\widetilde{H}}}^{2/3}_{\varepsilon}} \leq C_{\varepsilon}(|{\widetilde{P}}_{\varepsilon}v| + |v|).$$ Thanks to , this subelliptic estimate for ${\widetilde{P}}_{\varepsilon}$ transfers to a subelliptic estimate on $P_{\varepsilon}$: it suffices to plug $v = \pi_{\mathbb{S}}^* u$ in to obtain $$\label{eq:0pa}
u \in C^\infty(S^*{\mathbb{M}}) \ \Rightarrow \ |u|_{H^{2/3}_{\varepsilon}} \leq C_{\varepsilon}(|P_{\varepsilon}u| + |u|).$$
### Spherical vertical Laplacian as a sum of squares
We will need the following result: there exist $n > 0$ and $X_1, ..., X_n$ smooth vector fields on $S^*{\mathbb{M}}$ such that $$\label{eq:7e}
\Delta_{\mathbb{S}}= -\sum_{j=1}^n X_j^2, \ \ \ \ {{\operatorname{div}}}(X_j) = 0.$$ Indeed, Nash’s theorem shows there exist $n > 0$ and an isometric embedding $\iota : {\mathbb{M}}\hookrightarrow {\mathbb{R}}^n$. The manifold $S^*{\mathbb{M}}$ can be seen as a submanifold of $T^* {\mathbb{R}}^n$ thanks to the embedding $$\left(z,\zeta^1\right) \mapsto \left(\iota(z), \ (d\iota(z)^*)^{-1} \cdot \zeta^1\right),$$ which in addition preserves the bundle structure. Let $X_1, ..., X_n$ be the orthogonal projections of ${{\partial}}_{n+1}, ..., {{\partial}}_{2n}$ on $S^*{\mathbb{M}}$. Following the proof of [@Hsu Theorem 3.1.4], the $X_j$’s are divergence-free vector fields hence holds.
Dynamical systems and microlocal analysis {#sec:2.3}
-----------------------------------------
The material here is mostly taken from [@DZ1 §2.1] and [@DZ Appendix E.5.2].
When ${\mathbb{M}}$ has negative curvature, $H_1$ generates an Anosov flow on $S^*{\mathbb{M}}$: there exists a decomposition of $TS^*{\mathbb{M}}$, invariant under the geodesic flow $e^{tH_1}$, of the form $$T_xS^*{\mathbb{M}}= E_0(x) \oplus E_u(x) \oplus E_s(x),$$ where $E_0(x) = {\mathbb{R}}\cdot H_1(x)$ and $E_u(x), E_s(x)$ satisfy:
v E\_u(x) |de\^[tH\_1]{}(x) v| Ce\^[ct]{} |v|, t < 0,\
v E\_s(x) |de\^[tH\_1]{}(x) v| Ce\^[-ct]{} |v|, t > 0.
For $(x,\xi) \in TS^*{\mathbb{M}}$, let $\sigma_{H_1}(x,\xi) = {\langle \xi,H_1(x) \rangle}$ – a smooth function on $TS^*{\mathbb{M}}$. The Hamiltonian vector field $H_{\sigma_{H_1}}$ of $\sigma_{H_1}$ generates the flow $\exp(tH_{\sigma_{H_1}})$ given by $$\label{eq:9f}
\exp(tH_{\sigma_{H_1}})(x,\xi) = \left(e^{tH_1}(x), ^Tde^{tH_1}(x)^{-1}\xi\right).$$ Since $\sigma_{H_1}(x,\xi) = \frac{1}{i}{\langle \xi,H_1(x) \rangle}$ is homogeneous of degree $1$ in $\xi$, $\exp(tH_{\sigma_{H_1}})$ extends to a map ${{\overline{T}}}^*S^*{\mathbb{M}}\rightarrow {{\overline{T}}}^*S^*{\mathbb{M}}$, see [@DZ1 Proposition E.5]. A radial sink (with respect to $H_{\sigma_{H_1}}$) is a $\exp(tH_{\sigma_{H_1}})$-invariant closed conic set $L \subset T^* S^*{\mathbb{M}}\setminus 0$ with a conical neighborhood $U$ satisfying
\[Eq:7k\] t + (((tH\_[\_[H\_1]{}]{})(U)), (L)) 0,\
(x,) U |\_(tH\_[\_[H\_1]{}]{})(x,)| C\^[-1]{} e\^[c t]{} ||.
Here $\pi_\xi(x,\xi) = \xi$. A radial source is defined by reversing the flow direction in .
The decomposition $T_xS^*{\mathbb{M}}= E_u(x) \oplus E_0(x) \oplus E_s(x)$ induces a dual decomposition $T_x^* S^*{\mathbb{M}}= E_s^*(x) \oplus E_0^*(x) \oplus E_u^*(x)$. *Note that in this notation, $E_s^*(x)$ is the dual of $E_u(x)$ and $E_u^*(x)$ is the dual of $E_s^*(x)$.* The stable and unstable foliations of Anosov flows are related to the radial source and sinks as follows: $E_s^* \setminus 0$ is a radial source and $E_u^*\setminus 0$ is a radial sink, see [@DZ1 §2.3].
Pollicott–Ruelle resonances are dynamical quantities associated to ${\mathbb{M}}$, that quantify the decay of classical correlations, see [@Ts Corollary 1.2] and [@NZ Corollary 5]. These numbers can also be realized as eigenvalues of $\frac{1}{i} H_1$ on specifically designed Sobolev spaces. They are the poles of the meromorphic continuation of the Fredholm family of operators $(P_0-\lambda)^{-1} = (\frac{1}{i}H_1-\lambda)^{-1} : C^\infty \rightarrow {{\mathcal{D}}}'$, where ${{\mathcal{D}}}'$ is the set of distributions on $S^*{\mathbb{M}}$. The poles of $(P_0-\lambda)^{-1}$ have finite rank; the multiplicity of a pole $\lambda_0 \in {\mathbb{C}}$ is $\text{rank}(\Pi_{\lambda_0})$, where $$\label{eq:5n}
\Pi_{\lambda_0} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\dfrac{1}{2\pi i} \oint_{{{\partial}}{\mathbb{D}}(\lambda_0,r_0)} (P_0-\lambda)^{-1} d\lambda$$ and $r_0$ is small enough so that $\lambda_0$ is the unique pole of $(P_0-\lambda)^{-1}$ on ${\mathbb{D}}(\lambda_0,r_0)$. In order to investigate further the residues of $(P_0-\lambda)^{-1}$, we recall that one can associate to each $u \in {{\mathcal{D}}}'$ a conical set ${{\operatorname{WF}}}(u)$, called the classical wavefront set, which measures in phase space where $u$ is not smooth. We refer to [@GS §7] for precise definitions. For $\Gamma \subset T^*S^*{\mathbb{M}}$ a conical set, let ${{\mathcal{D}}}'_\Gamma$ be the set of distributions with classical wavefront set contained in $\Gamma$.
\[lem:3\] If $\lambda_0$ is a simple Pollicott–Ruelle resonance of $P_0 = \frac{1}{i}H_1$, there exist $u \in {{\mathcal{D}}}'_{E_u^*}, v \in {{\mathcal{D}}}'_{E_s^*}$ and a holomorphic family of operators $A(\lambda)$ defined near $\lambda_0$, with $$(P_0 - \lambda)^{-1} = \dfrac{u \otimes v}{\lambda-\lambda_0} + A(\lambda).$$
According to [@DZ1 Proposition 3.3], the operator $\Pi_{\lambda_0}$ defined in is equal to $u \otimes v$, where ${{\operatorname{WF}}}(u) \subset E_u^*$, ${{\operatorname{WF}}}(v) \subset E_s^*$; and there exist $J > 0$ and a family of operators $A(\lambda) : C^\infty \rightarrow {{\mathcal{D}}}'$ holomorphic near $\lambda_0$ such that $$\label{eq:0k}
(P_0 - \lambda)^{-1} = A(\lambda) + \sum_{j=1}^J \dfrac{(P_0-\lambda_0)^{j-1} \Pi_{\lambda_0}}{(\lambda-\lambda_0)^j}.$$ By the same argument as in the proof of [@DZ Theorem 2.4] the operator $P_0 - \lambda_0$ maps ${{\mathrm{Range}}}(\Pi_{\lambda_0})$ to itself and $(P_0 - \lambda_0)|_{{{\mathrm{Range}}}(\Pi_{\lambda_0})}$ is nilpotent. Since ${{\mathrm{Range}}}(\Pi_{\lambda_0})$ has dimension $1$, $(P_0-\lambda)|_{{{\mathrm{Range}}}(\Pi_{\lambda_0})}$ is equal to $0$ and the index $J$ in is equal to $1$.
In [@DZ1] the meromorphic continuation of $(P_0-\lambda)^{-1}$ is realized via analytic Fredholm theory. Therefore, Pollicott–Ruelle resonances of $P_0$ are identified with the roots of a suitable Fredholm determinant, see [@DZ2 Proposition 3.2].
Semiclassical analysis {#subsec:2.2}
----------------------
We recall some facts about the semiclassical calculus on $S^*{\mathbb{M}}$ (or $O^*{\mathbb{M}}$). Unless specified otherwise, our basic reference is [@DZ Appendix E]. *In the rest of the paper, $h$ is a parameter satisfying $0 < h < 1$.*
For $m \in {\mathbb{R}}$, a function $a \in C^\infty(T^*S^*{\mathbb{M}})$ depending on $h$ lies in the symbol class $S^m$ if $$\forall {\alpha}, \beta, \ \exists C_{{\alpha}\beta} >0, \ \forall 0 < h < 1, \ \sup_{(x,\xi) \in T^* S^*{\mathbb{M}}} {\langle \xi \rangle}^{m-|\beta|} |{{\partial}}_x^{\alpha}{{\partial}}_\xi^\beta a(x,\xi)| \leq C_{{\alpha}\beta}.$$ Semiclassical pseudodifferential operators on $S^*{\mathbb{M}}$ are $h$-quantization of symbols in $S^m$ and form an algebra denoted $\Psi_h^m$, see [@DZ Appendix E.1]. Conversely, to each $A \in \Psi^m_h$, we can associate a principal symbol $\sigma(A) \in S^m/hS^{m-1}$. For example if $X$ is a smooth vector field on $S^*{\mathbb{M}}$, the principal symbols of $\frac{h}{i}X$ is $$\label{EQ:0a}
\sigma_X(x,\xi) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}{\langle \xi,X(x) \rangle} \mod hS^0.$$ The principal symbol of $h^2\Delta = h^2\Delta_{\mathbb{S}}+ h^2 \Delta_O$ induces a positive definite quadratic form on the fibers of $T^*S^*{\mathbb{M}}$, thus a metric on $S^*{\mathbb{M}}$. We denote it by $g$, so that the principal symbol of $h^2 \Delta$ is $|\xi|^2_g$ modulo $hS^1$. We refer to [@DZ Appendix E.1] for additional properties of operators in $\Psi^m_h$.
Let ${{\overline{T}}}^*S^*{\mathbb{M}}$ be the radial compactification of $T^*S^*{\mathbb{M}}$: it is a compact manifold with interior $T^*S^*{\mathbb{M}}$ and boundary $S^*S^*{\mathbb{M}}$ associated with a map $\kappa : T^*S^*{\mathbb{M}}\setminus 0 \rightarrow {{\partial}}{{\overline{T}}}^*S^*{\mathbb{M}}$, see [@DZ Appendix E.1]. To each operator $A \in \Psi^m_h$, we associate an invariant closed set ${{\operatorname{WF}}}_h(A) \subset {{\overline{T}}}^* S^*{\mathbb{M}}$ called the wavefront set of $A$, which measures where $A$ is not semiclassically negligible. We also associate to $A$ an invariant open set ${{\operatorname{Ell}}}_h(A) \subset {{\overline{T}}}^*S^*{\mathbb{M}}$ called the elliptic set, which measures where $A$ is semiclassically invertible. See [@DZ Appendix E.2] for precise definitions. The main interest of the elliptic set is the elliptic estimate [@DZ1 Proposition 2.4]. Among classical results we record the sharp G$\mathring{\text{a}}$rding inequality [@DZ Proposition E.34] and the Duistermaat–Hörmander propagation of singularities theorem [@DZ1 Proposition 2.5].
A less classical result needed here is the radial source (resp. radial sink) estimate, first introduced by Melrose [@Me] and developed further in [@DZ1 Propositions 2.6-2.7]. This estimate applies microlocally near a radial source (resp. near a radial sink); it enables us to control certain semiclassical quantities provided that the regularity index is high (resp. low) enough. This motivates the definition of semiclassical anisotropic Sobolev spaces that have high microlocal regularity near radial sources and low microlocal regularity near radial sinks. See [@Z Chapter 8] for a general theory of semiclassical anisotropic Sobolev spaces and [@DZ1 §3.1] for the specific scale of Sobolev space we will use in this paper.
We can also consider operators on ${\mathbb{R}}^n, n > 0$, that belong to a more general class than $\Psi_h^0$. These are realized as quantization of symbols $a$ satisfying $$\forall {\alpha}, \beta, \ \exists C_{{\alpha}\beta} >0, \forall 0 < h < 1 , \ \sup_{(x,\xi) \in T^* {\mathbb{R}}^n} |{{\partial}}_x^{\alpha}{{\partial}}_\xi^\beta a(x,\xi)| \leq C_{{\alpha}\beta}.$$ The space of resulting symbols (resp. resulting operators) is denoted by $S$ (resp. $\Psi_h$). This space is *not* invariant under change of variables. In the class $\Psi_h$, the remainders in the composition formula are smaller than the leading part, but they are not mor smoothing – in contrast with $\Psi_h^0$. We will use this class exclusively in §\[subsec:3.3\]. Our basic reference for such operators is [@Z Chapter 4].
Maximal hypoelliptic estimates {#sec:3}
==============================
Statement of the result
-----------------------
Recall that the operator $P_{\varepsilon}$ is given by $\frac{1}{i}(H_1 + {\varepsilon}\Delta_{\mathbb{S}})$, that the semiclassical Sobolev spaces $H^s_{\varepsilon}$ were defined in §\[subsec:1\], and that there exist $X_1, ..., X_n \in T S^*{\mathbb{M}}$ such that such that $\Delta_{\mathbb{S}}= - \sum_{j=1}^n X_j^2$. Here we prove an estimate for $P_{\varepsilon}$ similar to [@RS Theorem 18], but uniform in the *semiclassical regime* ${\varepsilon}\rightarrow 0$. Let $\rho_1, \rho_2$ be two smooth functions satisfying $$\label{eq:0g}
{\mathrm{supp}}(\rho_1, \rho_2) \subset {\mathbb{R}}\setminus 0, \ \ \ \ 1-\rho_1, \ 1-\rho_2 \in C^\infty_0({\mathbb{R}},[0,1]), \ \ \ \ \rho_2 = 1 \text{ on } {\mathrm{supp}}(\rho_1).$$
\[thm:1\] Let $R > 0$ and $\rho_1, \rho_2$ two functions satisfying . For any $N > 0$, there exists $C_{N,R} > 0$ such that for every $|\lambda| \leq R$, $u \in C^\infty(S^*{\mathbb{M}})$, and $0 < {\varepsilon}< 1$,
\[eq:5k\] \^[2/3]{}|\_1(\^2 ) u|\_[H\^[2/3]{}\_]{} + \^[1/3]{} \_[j=1]{}\^n| X\_j \_1(\^2 ) u|\_[H\^[1/3]{}\_]{} + |\_1(\^2 ) \^2 \_u|\
C\_[N,R]{} |\_2(\^2) (P\_-)u| + O(\^N)|u|.
This Theorem applies to any smooth compact Riemannian manifold ${\mathbb{M}}$, *with no restriction on the sign of its sectional curvatures, and with no change in the proof.*
The paper [@RS] shows that for every ${\varepsilon}> 0$ there exists $C_{\varepsilon}> 0$ such that $$|\rho_1({\varepsilon}^2\Delta) u|_{H^{2/3}_{\varepsilon}} \leq C_{\varepsilon}(|\rho_2({\varepsilon}^2\Delta){\varepsilon}(P_{\varepsilon}-\lambda)u| + |u|).$$ Theorem \[thm:1\] shows that $C_{\varepsilon}= O({\varepsilon}^{-2/3})$. Because of related estimates in [@DSZ] and [@L §3] we believe that this upper bound is optimal. This is the subject of a work in progress of Smith [@Sm].
We proved Theorem \[thm:1\] in [@D2], when ${\mathbb{M}}$ is an orientable surface. In this case, $\Delta_{\mathbb{S}}= -V^2$ where $V\in TS^*{\mathbb{M}}$ generates the circle action on the fibers of $S^*{\mathbb{M}}$. Thus, $\Delta_{\mathbb{S}}$ is a sum of squares of vector fields that commute with $\Delta_{\mathbb{S}}$, a fact used in a crucial manner in the proof of [@D2 Proposition 3.1]. This no longer holds when $d \geq 3$ or ${\mathbb{M}}$ is not orientable. In order to apply nevertheless the main idea of [@D2] we observe that $\Delta_O^V$ – the lift of $\Delta_{\mathbb{S}}$ to the orthonormal coframe bundle $O^*{\mathbb{M}}$ – is the sum of squares of vector fields which all commute with $\Delta_O^V$: $$\label{eq:0v}
\Delta_O^V = -\sum_{i,j} V_{ij}^2, \ \ \ \ [\Delta_O^V, V_{ij}] = 0,$$ see §\[sec:2.2.3\]-\[sec:2.2.4\]. The operator $P_{\varepsilon}= \frac{1}{i}(H_1+{\varepsilon}\Delta_{\mathbb{S}})$ on $C^\infty(S^*{\mathbb{M}})$ lifts to ${\widetilde{P}}_{\varepsilon}= \frac{1}{i}({{\widetilde{H}}}_1+{\varepsilon}\Delta_O^V)$ on $C^\infty(O^*{\mathbb{M}})$. Because of , we can modify the techniques of [@D2] to apply them to the operator ${\widetilde{P}}_{\varepsilon}$. This will yield estimates for functions on $O^*{\mathbb{M}}$, which we will descend to function on $S^*{\mathbb{M}}$.
We will use semiclassical analysis to show Theorem \[thm:1\]. To conform with standard notations, we define $$h {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}{\varepsilon}, \ \ \ \ P {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}ihP_h = h^2 \Delta_{\mathbb{S}}+ hH_1, \ \ \ \ {\widetilde{P}}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}ih{\widetilde{P}}_h = h^2 \Delta_O^V+h{{\widetilde{H}}}_1,$$ *for use in §\[sec:3.2\]-\[subsec:3.3\] only*. We see $h$ as a small parameter and $P$ as a $h$-semiclassical operator in $\Psi_h^2$. As in [@D2], we base our investigation on ideas of Lebeau [@L], where a subelliptic estimate for the Bismutian is shown, for ${\varepsilon}= 1$. The strategy starts to differ when Lebeau uses a microlocal reduction to a toy model. Instead, we continue to work with $P_{\varepsilon}$ and we replace the microlocal reduction by a positive commutator estimate. This avoids to use semiclassical Fourier integral operators.
Reduction to a subelliptic estimate {#sec:3.2}
-----------------------------------
The first lemma shows that Theorem \[thm:1\] is a consequence of a subelliptic estimate.
\[lem:1a\] Let ${{\mathcal{S}}}_1, ..., {{\mathcal{S}}}_q, {\mathcal{T}}\subset \Psi_h^1$ be a collection of selfadjoint semiclassical operators on $S^*{\mathbb{M}}$ or $O^*{\mathbb{M}}$ and ${{\mathcal{P}}}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\sum_{j=1}^q {{\mathcal{S}}}_j^2 + i{\mathcal{T}}$. There exist $C, h_0>0$ such that $$0 < h < h_0 \ \Rightarrow \ |{\mathcal{T}}u| + \left|\sum_{j=1}^q {{\mathcal{S}}}_j^2 u \right| + \sum_{j=1}^q h^{1/3} |{{\mathcal{S}}}_j u|_{H_h^{1/3}} \leq C |{{\mathcal{P}}}u| + O(h^{2/3})|u|_{H^{2/3}_h}.$$
We prove the result only in the case of $S^*{\mathbb{M}}$; the proof is identical when considering operators on $O^*{\mathbb{M}}$. We first show the estimate $$\label{eq:1l}
|{{\mathcal{S}}}_j u|_{H^{1/3}_h}^2 \leq C|{{\mathcal{P}}}u| |u|_{H^{2/3}_h} + O(h) |u|_{H_h^{2/3}}^2.$$ Recall that $\Delta {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\Delta_{\mathbb{M}}+ \Delta_{\mathbb{S}}$, where $\Delta_{\mathbb{M}}$ is the *non-negative* standard Laplacian on ${\mathbb{M}}$ (lifted to $S^*{\mathbb{M}}$) and $\Delta_{\mathbb{S}}$ is the spherical Laplacian on $S^*{\mathbb{M}}$. The $H^s_h$-norm was defined in §\[subsec:1\] by $|u|_{H^s_h} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}|\Lambda_s u|$, where $\Lambda_s = ({{\operatorname{Id}}}+h^2 \Delta)^{s/2}$. Thus, $$\label{eq:7a}
|{{\mathcal{S}}}_j u|_{H^{1/3}_h}^2 = |\Lambda_{1/3} {{\mathcal{S}}}_j u|^2 \leq 2|{{\mathcal{S}}}_j \Lambda_{1/3} u|^2 + 2|[\Lambda_{1/3},{{\mathcal{S}}}_j]u|^2 \leq 2|{{\mathcal{S}}}_j \Lambda_{1/3} u|^2 + O(h^2)|u|_{H^{1/3}_h}^2$$ because $[\Lambda_{1/3},{{\mathcal{S}}}_j] \in h \Psi^{1/3}_h$. Next we study $|{{\mathcal{S}}}_j \Lambda_{1/3} u|$: using $\sum_{j=1}^q {{\mathcal{S}}}_j^2 = {\operatorname{Re}}({{\mathcal{P}}})$,
\[eq:7b\] |\_j \_[1/3]{} u|\^2 = [\_j\^2 \_[1/3]{} u, \_[1/3]{} u ]{} [\_[j=1]{}\^q \_j\^2 \_[1/3]{} u, \_[1/3]{} u ]{}\
([\_[1/3]{} u, \_[1/3]{} u ]{}) = ([u, \_[2/3]{} u ]{}) + ([u,\_[1/3]{} u ]{}).
We can estimate ${\langle {{\mathcal{P}}}u, \Lambda_{2/3} u \rangle}$ by $|{{\mathcal{P}}}u| |u|_{H^{2/3}_h}$. The identity ${{\mathcal{P}}}=\sum_{j=1}^q {{\mathcal{S}}}_j^2+i{\mathcal{T}}$ yields $${\operatorname{Re}}({\langle [{{\mathcal{P}}},\Lambda_{1/3}]u,\Lambda_{1/3} u \rangle}) = \sum_{j=1}^q {\operatorname{Re}}({\langle [{{\mathcal{S}}}_j^2,\Lambda_{1/3}]u,\Lambda_{1/3} u \rangle}) + {\operatorname{Re}}({\langle [i{\mathcal{T}}, \Lambda_{1/3}] u, \Lambda_{1/3} u \rangle}).$$ The operator $[i{\mathcal{T}}, \Lambda_{1/3}]$ belongs to $h\Psi^{1/3}_h$ therefore $|{\langle [i{\mathcal{T}}, \Lambda_{1/3}] u, \Lambda_{1/3} u \rangle}| = O(h)|u|_{H^{1/3}_h}^2$. Using the relation $[{{\mathcal{S}}}_j^2,\Lambda_{1/3}] = {{\mathcal{S}}}_j [{{\mathcal{S}}}_j,\Lambda_{1/3}] + [{{\mathcal{S}}}_j,\Lambda_{1/3}] {{\mathcal{S}}}_j$ and the fact that $[{{\mathcal{S}}}_j,\Lambda_{1/3}]$ is anti-selfadjoint we obtain
[u,\_[1/3]{} u ]{} = -[\_j u,\[\_j,\_[1/3]{}\]\_[1/3]{} u ]{} + [u,\_j\_[1/3]{} u ]{}\
= -[\_j u,\[\_j,\_[1/3]{}\]\_[1/3]{} u ]{} + [\_[1/3]{}\[\_j,\_[1/3]{}\]u, \_ju ]{} + [u, \[\_j,\_[1/3]{}\] u ]{}.
The operators $\Lambda_{1/3}[{{\mathcal{S}}}_j,\Lambda_{1/3}]$ and $[{{\mathcal{S}}}_j,\Lambda_{1/3}]$ belong to $h\Psi^{2/3}_h$ and $h\Psi^{1/3}_h$, respectively. Moreover ${{\mathcal{S}}}_j^2 \leq {\operatorname{Re}}({{\mathcal{P}}})$, hence $|{{\mathcal{S}}}_ju| \leq |{{\mathcal{P}}}u|^{1/2} |u|^{1/2}$. It follows that
|[u,\_[1/3]{} u ]{}| |\_j u| |\_[1/3]{}\[\_j,\_[1/3]{}\] u| + |\[\_j,\_[1/3]{}\] u|\^2\
O(h)|u|\^[1/2]{} |u|\^[1/2]{} |u|\_[H\^[2/3]{}\_h]{} + O(h\^2)|u|\_[H\^[1/3]{}\_h]{}\^2.
Gluing this estimate with , , we get the bound
|\_ju|\_[H\_h\^[1/3]{}]{}\^2 C|u| |u|\_[H\^[2/3]{}\_h]{} + O(h) |u|\_[H\_h\^[1/3]{}]{}\^2 + O(h) |u|\^[1/2]{} |u|\^[1/2]{} |u|\_[H\^[2/3]{}\_h]{}\
C|u| |u|\_[H\^[2/3]{}\_h]{} + O(h) |u|\_[H\_h\^[2/3]{}]{}\^2 + O(h) |u|\^[1/2]{} |u|\_[H\^[2/3]{}\_h]{}\^[3/2]{} C|u| |u|\_[H\^[2/3]{}\_h]{} +O(h) |u|\_[H\_h\^[2/3]{}]{}\^2.
In the last inequality we used $a b \leq a^2 + b^2$ with $a = |{{\mathcal{P}}}u|^{1/2} |u|_{H^{2/3}_h}^{1/2}$ and $b = h |u|_{H^{2/3}_h}$. This proves . We observe that gives the estimate on $|{{\mathcal{S}}}_j u|_{H^{1/3}_h}$ provided by the lemma: $$\label{eq:7c}
h^{2/3} |{{\mathcal{S}}}_j u|^2 \leq C h^{2/3} |{{\mathcal{P}}}u| |u|_{H_h^{2/3}} + O(h^{5/3})|u|_{H^{2/3}_h}^2 \leq C|Pu|^2 + O(h^{4/3})|u|_{H^{2/3}_h}^2.$$
Next we observe that $$|{{\mathcal{P}}}u|^2 = \left|\sum_{j=1}^q {{\mathcal{S}}}_j^2 u\right|^2+|{\mathcal{T}}u|^2 + \sum_{j=1}^q {\langle [{{\mathcal{S}}}_j^2,i{\mathcal{T}}]u,u \rangle}.$$ To conclude the proof of the lemma it suffices to control the commutator terms ${\langle [{{\mathcal{S}}}_j^2,i{\mathcal{T}}]u,u \rangle}$. We have $${\langle [{{\mathcal{S}}}_j^2,i{\mathcal{T}}]u,u \rangle} = {\langle [{{\mathcal{S}}}_j,i{\mathcal{T}}]u,{{\mathcal{S}}}_ju \rangle}+ {\langle {{\mathcal{S}}}_ju,[{{\mathcal{S}}}_j,i{\mathcal{T}}]u \rangle} = 2 {\operatorname{Re}}({\langle {{\mathcal{S}}}_ju,[{{\mathcal{S}}}_j,i{\mathcal{T}}]u \rangle}).$$ By interpolation, $|{\langle [{{\mathcal{S}}}_j^2,i{\mathcal{T}}]u,u \rangle}| \leq |{{\mathcal{S}}}_j u|_{H^{1/3}_h} |[{{\mathcal{S}}}_j,i{\mathcal{T}}]u|_{H^{-1/3}_h}$. Since $[{{\mathcal{S}}}_j,i{\mathcal{T}}] \in h\Psi^1_h$ it is bounded from $H^{2/3}_h$ to $H^{-1/3}_h$ with norm $O(h)$. By , $$|{\langle [{{\mathcal{S}}}_j^2,i{\mathcal{T}}]u,u \rangle}| \leq Ch \left( |{{\mathcal{P}}}u|^{1/2} |u|_{H^{2/3}_h}^{1/2} + h^{1/2} |u|_{H_h^{2/3}} \right) |u|_{H^{2/3}_h}.$$ Hence we obtain
\[eq:7d\] |\_[j=1]{}\^q \_j\^2 u|\^2+|u|\^2 C |u|\^2 + O(h) |u|\^[1/2]{} |u|\_[H\^[2/3]{}\_h]{}\^[3/2]{} + O(h\^[3/2]{}) |u|\_[H\^[2/3]{}\_h]{}\^2\
C |u|\^2 + O(h\^[4/3]{}) |u|\_[H\^[2/3]{}\_h]{}\^2.
In the second line we used Young’s inequality: $ab \leq a^4 + b^{4/3}$ with $a = |{{\mathcal{P}}}u|^{1/2}$, $b = h |u|_{H^{2/3}_h}^{3/2}$. The estimates , conclude the proof.
Roughly speaking, this lemma reduces the proof of to an estimate of the form $$\label{eq:7g}
u \in C^\infty(S^*{\mathbb{M}}) \ \Rightarrow \ h^{2/3}|\rho_1(h^2 \Delta) u|_{H^{2/3}_h} \leq C |\rho_2(h^2 \Delta) P u| + O(h^\infty) |u|.$$ Because of the reasons detailed above, we will work with the lift of $P$ to $O^*{\mathbb{M}}$ rather than directly with $P$. We will show the estimate $$\label{eq:7h}
v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ h^{2/3}|\rho_1(h^2 \Delta_O) v|_{H^{2/3}_h} \leq C |\rho_2(h^2 \Delta_O) {\widetilde{P}}v| + O(h^\infty) |v|.$$ To see that implies we plug $v=\pi_{\mathbb{S}}^*u$ in , then we use the identity between ${\widetilde{P}}$ and $P$, $\Delta$ and $\Delta_O$, and finally the relation between Sobolev spaces on $S^*{\mathbb{M}}$ and $O^*{\mathbb{M}}$. The bound will be implied by microlocal estimates on ${\widetilde{P}}$:
\[prop:1\] For every $(x_0,\xi_0) \in {{\overline{T}}}^*O^*{\mathbb{M}}\setminus 0$ there exists an open neighborhood $W_{x_0,\xi_0}$ of $(x_0,\xi_0)$ in ${{\overline{T}}}^*O^*{\mathbb{M}}\setminus 0$ with the following property. For every $A \in \Psi_h^0$ with ${{\operatorname{WF}}}_h(A) \subset W_{x_0,\xi_0}$, there exists $B$ with ${{\operatorname{WF}}}_h(B) \subset W_{x_0,\xi_0}$ such that $$v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ h^{2/3}|Av|_{{{\widetilde{H}}}^{2/3}_h} \leq C|{\widetilde{P}}Bv|+O(h)|v|_{{{\widetilde{H}}}^{3/5}_h}.$$
It suffices to prove the Theorem when $h$ is sufficiently small. We first fix $N, R > 0$ and $\rho_1, \rho_2$ two functions satisfying . Recall that we can write $P = -h^2 \sum_{j=1}^n X_j^2 + h H_1$, where $\frac{h}{i} X_j, \frac{h}{i} H_1$ are selfadjoint semiclassical operators in $\Psi_h^1$.
**Step 1.** By Lemma \[lem:1a\] applied to $P$ instead of ${{\mathcal{P}}}$ and $\rho_1(h^2\Delta) u$ instead of $u$,
|h\^2 \_\_1(h\^2) u| + h\^[1/3]{} \_[j=1]{}\^n |hX\_j \_1(h\^2) u|\_[H\^[1/3]{}\_h]{} + h\^[2/3]{}|\_1(h\^2) u|\_[H\^[2/3]{}\_h]{}\
C |P \_1(h\^2) u| + O(h\^[2/3]{}) |\_1(h\^2) u|\_[H\^[2/3]{}\_h]{}.
Let ${\tilde{\rho}}_1 \in C_0^\infty$, be equal to $1$ on ${\mathrm{supp}}(\rho_1)$ and $0$ where $\rho_2 \neq 1$. Since $\Delta$ and $\Delta_{\mathbb{S}}$ commute, we have $P \rho_1(h^2\Delta) = \rho_1(h^2\Delta) (P-\lambda h) + \lambda h \rho_1(h^2\Delta)+[\frac{h}{i}H_1,\rho_1(h^2\Delta)]$. Both $\lambda h \rho_1(h^2\Delta)$ and $[\frac{h}{i}H_1,\rho_1(h^2\Delta)]$ have wavefront set contained in the elliptic set of ${\tilde{\rho}}_1(h^2\Delta)$. Therefore,
C |P \_1(h\^2) u| + O(h\^[2/3]{}) |\_1(h\^2) u|\_[H\^[2/3]{}\_h]{}\
C |\_2(h\^2) (P-h) u| + O(h\^[2/3]{}) |\_1(h\^2) u|\_[H\^[2/3]{}\_h]{} + O(h\^)|u|.
Hence the theorem follows from a bound on $h^{2/3} |{\tilde{\rho}}_1(h^2\Delta) u|_{H^{2/3}_h}$. After lifting to $O^*{\mathbb{M}}$ and using and it suffices to show that $$\label{eq:7i}
v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ h^{2/3} |{\tilde{\rho}}_1(h^2\Delta_O) v|_{{{\widetilde{H}}}^{2/3}_h} \leq C|\rho_2(h^2\Delta_O)({\widetilde{P}}-\lambda h)v| + O(h^N)|v|.$$
**Step 2.** Since ${{\operatorname{WF}}}_h({\tilde{\rho}}_1(h^2 \Delta))$ is a compact subset of ${{\overline{T}}}^*O^*{\mathbb{M}}\setminus 0$, there exists a finite collection of points $(x_1,\xi_1), ..., (x_\nu,\xi_\nu) \in {{\overline{T}}}^*O^*{\mathbb{M}}$ and open sets $W_{x_1, \xi_1}, ..., W_{x_\nu,\xi_\nu}$ given by Proposition \[prop:1\] such that $$\label{eq:9z}
{{\operatorname{WF}}}_h({\tilde{\rho}}_1(h^2 \Delta)) \subset \bigcup_{k=1}^\nu W_{x_k,\xi_k}.$$ Let $\Psi_{h,k}^m$ be the set of operators in $\Psi^m_h$ with wavefront set contained in $W_{x_k,\xi_k}$. Using and a microlocal partition of unity, we can construct operators $E_k \in \Psi_{h,k}^{2/3}$ with $$\label{eq:0q}
v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ |{\tilde{\rho}}_1(h^2 \Delta_O) v|_{{{\widetilde{H}}}^{2/3}_h} \leq \sum_{k=1}^\nu |E_k v| + O(h^\infty)|v|.$$ Below we obtain bounds on the terms $|E_k v|$.
**Step 3.** Let $\delta = 1/15$ and $m \leq 2/3$. We first claim that for every $A \in \Psi_{h,k}^m$, there exist $B_1 \in \Psi_{h,k}^{m-2/3}$ and $A' \in \Psi_{h,k}^{m-\delta}$ with $$\label{eq:0e}
h^{2/3}|A v| \leq C|{\widetilde{P}}B_1 v|+O(h)|A' v| + O(h^\infty)|v|.$$ The operator $\Lambda_{-2/3}A \Lambda_{-m+2/3}$ belongs to $\Psi^{0}_{h,k}$. Proposition \[prop:1\] gives an operator $B \in \Psi_{h,k}^0$ such that $$h^{2/3}|\Lambda_{-2/3}A \Lambda_{-m+2/3} v|_{{{\widetilde{H}}}^{2/3}_h} \leq |{\widetilde{P}}Bv| + O(h) |v|_{{{\widetilde{H}}}^{3/5}_h}.$$ Pick $B' \in \Psi_{h,k}^0$ with ${{\operatorname{WF}}}_h(B' - {{\operatorname{Id}}}) \cap {{\operatorname{WF}}}_h(A) = \emptyset$ and replace $v$ by $\Lambda_{m-2/3} B'v$: $$h^{2/3}|AB'v| \leq C|{\widetilde{P}}B\Lambda_{m-2/3} B'v| + O(h) |\Lambda_{m-2/3} B' v|_{{{\widetilde{H}}}^{3/5}_h}.$$ Since $h^{2/3}|A ({{\operatorname{Id}}}- B') v| = O(h^\infty) |v|$, holds with $B_1 {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}B \Lambda_{m-2/3} B' \in \Psi^{m-2/3}_{h,k}$ and $A' {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\Lambda_{3/5} \Lambda_{m-2/3} B' \in \Psi^{m-\delta}_{h,k}$.
**Step 4.** The goal is now to iterate . We first need a commutator-like estimate. For $B_1$ belongs to $\Psi_{h,k}^{m-2/3}$,
B\_1 = B\_1 + \[,B\_1\] = B\_1 (-h) + 2h \_[k,]{} V\_[k]{}\[hV\_[k]{},B\_1\] + h \_[h,k]{}\^[m-2/3]{}\
= B\_1 (-h) + 2h \_[k,]{} \_[1/3]{}hV\_[k]{} \_[h,k]{}\^[m-1]{} + h \_[h,k]{}\^[m-2/3]{}.
Hence there exist operators $B_2 \in \Psi_{h,k}^{m-1}$ and $C_0 \in \Psi_{h,k}^{m-2/3}$ such that
|B\_1 v| C|B\_1 (-h)v| + h \_[k,]{} |hV\_[k]{} B\_2 v|\_[\^[1/3]{}\_h]{}+ h |C\_0 v|\
C|\_2(h\^2\_O) (-h)v| + h\^[4/3]{}|B\_2 v|\_[\^[2/3]{}\_h]{}+ h\^[2/3]{} |B\_2 v| + h |C\_0 v| + O(h\^)|v|.
In the second line we used Lemma \[lem:1a\] and the elliptic estimate. The slightly weaker bound holds: there exist $B_2 \in \Psi_{h,k}^{m-1}$ and $C_1 \in \Psi_{h,k}^{m-1/3}$ such that $$\label{eq:0s}
|{\widetilde{P}}B_1 v| \leq C|\rho_2(h^2\Delta_O) ({\widetilde{P}}-\lambda h)v| + h^{2/3} |{\widetilde{P}}B_2 v| + h |C_1 v| + O(h^\infty)|v|.$$ Iterate to obtain $B_N \in \Psi^{m-2/3-N/3}$ and $C_N \in \Psi_{h,k}^{m-1/3}$ such that $$|{\widetilde{P}}B_1 v| \leq C|\rho_2(h^2\Delta_O) ({\widetilde{P}}-\lambda h) v| + h^{2N/3} |{\widetilde{P}}B_N v| + h |C_N v| + O(h^\infty)|v|.$$ For $N \geq 6$ the operator ${\widetilde{P}}B_N$ belongs to $\Psi_h^0$ and $|{\widetilde{P}}B_N v| = O(|v|)$. It follows that for $N$ large enough, $$\label{eq:9w}
|{\widetilde{P}}B_1 v| \leq C|\rho_2(h^2\Delta_O) ({\widetilde{P}}-\lambda h) v| + h |C_{2N} v| + O(h^N)|v|.$$
**Step 5.** The estimate combined with show that for every $A_1 \in \Psi^m_{h,k}$ there exists $A_2 \in \Psi^{m-\delta}_{h,k}$ with $$h^{2/3}|A_1 v| \leq C |\rho_2(h^2\Delta_O) ({\widetilde{P}}-\lambda h) v| + h |A_2 v| + O(h^N)|v|.$$ Here again we can iterate this inequality sufficiently many times to obtain $$\label{eq:0r}
h^{2/3}|A_1 v| \leq C |\rho_2(h^2\Delta_O) ({\widetilde{P}}-\lambda h) v| + O(h^N)|v|.$$ Recall that ${\tilde{\rho}}_1$ is controlled by operators microlocalized inside $W_{x_k,\xi_k}$ thanks to . Apply with $A_1 = E_k$, $k=1, ..., \nu$ and sum over $k$ to get : $$h^{2/3}|{\tilde{\rho}}_1(h^2 \Delta_O) v|_{{{\widetilde{H}}}^{2/3}_h} \leq C |\rho_2(h^2\Delta_O) ({\widetilde{P}}-\lambda h) v| + O(h^N)|v|.$$ This ends the proof of the theorem.
Proof of the subelliptic estimate {#subsec:3.3}
---------------------------------
In this subsection we show Proposition \[prop:1\]. We fix $(x_0,\xi_0) \in {{\overline{T}}}^*O^*{\mathbb{M}}\setminus 0$. We distinguish three cases: whether $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h(h^2\Delta^V_O)$ – in this case ${\widetilde{P}}$ is elliptic at $(x_0,\xi_0)$ – or $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h(h{{\widetilde{H}}}_1)$ – in this case ${\operatorname{Im}}({\widetilde{P}})$ is elliptic at $(x_0,\xi_0)$ – or $(x_0,\xi_0) \notin {{\operatorname{Ell}}}_h(h{{\widetilde{H}}}_1) \cup {{\operatorname{Ell}}}_h(h{{\widetilde{H}}}_1)$. The latter is the hardest; we will use that one of the commutators $[hV_{1\ell},h{{\widetilde{H}}}_1]$ is elliptic at $(x_0,\xi_0)$.
In this case $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h({\widetilde{P}})$. Let $W_{x_0,\xi_0}$ be an open neighborhood of $(x_0,\xi_0)$ contained in ${{\operatorname{Ell}}}_h({\widetilde{P}})$, and $A \in \Psi_h^0$ with wavefront set contained in $W_{x_0,\xi_0}$. Let $B \in \Psi_h^0$ elliptic on ${{\operatorname{WF}}}_h(A)$ and with wavefront set contained in $W_{x_0,\xi_0}$. The operator ${\widetilde{P}}B$ is elliptic on the wavefront set of $A$. The elliptic estimate [@DZ Theorem E.32] shows that for $h$ small enough, $$v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ h^{2/3}|Av|_{{{\widetilde{H}}}^{2/3}_h} \leq C |{\widetilde{P}}Bv|+O(h^\infty)|v|.$$ This shows the proposition in this case.
Without loss of generalities $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h(h {{\widetilde{H}}}_1) \setminus {{\operatorname{Ell}}}_h(h^2 \Delta_O^V)$. In particular $V_{1\ell}$ is characteristic at $(x_0,\xi_0)$ for any $\ell$. Let $\sigma_{{{\widetilde{H}}}_m}, \sigma_{V_{k\ell}}$ be the principal symbols of $\frac{h}{i}{{\widetilde{H}}}_m, \frac{h}{i}V_{k\ell}$ given accordingly by . We can find an open neighborhood $W_{x_0,\xi_0} \subset {{\operatorname{Ell}}}_h(h{{\widetilde{H}}}_1)$ of $(x_0, \xi_0)$ in ${{\overline{T}}}^*O^*{\mathbb{M}}$ such that on $W_{x_0,\xi_0} \cap T^*O^*{\mathbb{M}}$, $\sigma_{{{\widetilde{H}}}_1}^2 - 2\sigma_{V_{1\ell}} \sigma_{{{\widetilde{H}}}_\ell} \geq 0$. Let $A \in \Psi_h^0$ with wavefront set contained in $W_{x_0,\xi_0}$ and $B \in \Psi_h^0$ elliptic on ${{\operatorname{WF}}}_h(A)$, with wavefront set contained in $W_{x_0,\xi_0}$, and principal symbol $\sigma_B$. The operator $h {{\widetilde{H}}}_1 B$ is elliptic on ${{\operatorname{WF}}}_h(A)$ and [@DZ Theorem E.32] shows that $$|Av|_{{{\widetilde{H}}}^{2/3}_h} \leq C|h {{\widetilde{H}}}_1 B v| + O(h^\infty)|v|.$$ It remains to control $|h {{\widetilde{H}}}_1 B v|$. Using that ${\widetilde{P}}$ is equal to $h^2 \Delta_O^V + h {{\widetilde{H}}}_1$ with $\Delta_O^V$ selfadjoint and ${{\widetilde{H}}}_1$ anti-selfadjoint,
\[eq:0t\] |Bv|\^2 = |h\^2 \_O\^V B v|\^2 + |h \_1 B v|\^2 + [Bv,Bv ]{}\
= |h\^2 \_O\^V B v|\^2 + |h \_1 B v|\^2 + 2h ([B\^\*(h\^2 V\_[1]{} \_)Bv,v ]{}),
where we used that ${\operatorname{Re}}([h^2 \Delta_O^V,h{{\widetilde{H}}}_1]) = 2h\sum_\ell {\operatorname{Re}}(h^2V_{1\ell} {{\widetilde{H}}}_\ell)$. On $W_{x_0,\xi_0} \cap T^*O^*{\mathbb{M}}$, $\sigma_{{{\widetilde{H}}}_1}^2 - 2\sigma_{V_{1\ell}} \sigma_{{{\widetilde{H}}}_\ell} \geq 0$; hence $2|\sigma(B)|^2 i\sigma_{V_{1\ell}} i\sigma_{{{\widetilde{H}}}_\ell} \geq |\sigma_B|^2 (i\sigma_{{{\widetilde{H}}}_1})^2$. The sharp G$\mathring{\text{a}}$rding inequality (see [@DZ Proposition E.34]) shows that
2h ([B\^\*(h\^2 V\_[1]{} \_)Bv,v ]{}) [B\^\* h\^2 \_1\^2 B v, v ]{} - O(h) |v|\^2\_[\^[1/2]{}\_h]{} = -|h \_1 B v|\^2 - O(h) |v|\^2\_[\^[1/2]{}\_h]{}.
Plug this inequality in to obtain $$v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ |Av|^2_{{{\widetilde{H}}}^{2/3}_h} \leq C|h{{\widetilde{H}}}_1 B v|^2 + O(h^\infty)|v|^2 \leq C|{\widetilde{P}}Bv|^2 + O(h^2) |v|^2_{{{\widetilde{H}}}^{1/2}_h}.$$ This shows the proposition in the case $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h(h{{\widetilde{H}}}_1)$.
In this case $(x_0,\xi_0) \notin {{\operatorname{Ell}}}_h(hV_{k\ell})$ for any $k,\ell$. Since $\{ V_{k\ell}, {{\widetilde{H}}}_m\}$ span $TO^*{\mathbb{M}}$, there exists $m$ such that $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h(h{{\widetilde{H}}}_m)$; and for $(x,\xi) \in T^*O^*{\mathbb{M}}$ in a neighborhood of $(x_0,\xi_0)$, $\sigma_{{{\widetilde{H}}}_m}(x,\xi) \neq 0$. Changing $V_{1m}$ to $-V_{1m}$ does not change ${\widetilde{P}}$; and under this change ${{\widetilde{H}}}_m = [V_{1m},{{\widetilde{H}}}_1]$ becomes $-{{\widetilde{H}}}_m$. Hence we can assume without of generalities that $\sigma_{{{\widetilde{H}}}_m}(x,\xi) > 0$ for $(x,\xi) \in T^*O^*{\mathbb{M}}$ in a neighborhood of $(x_0,\xi_0)$.
We subdivide the proof it in 7 short steps. In the first step we localize the functions and operators involved in a small neighborhood of $x_0$, diffeomorphic to ${\mathbb{R}}^{d(d+1)/2}$. It allows us to use the class $\Psi_h$ introduced in §\[subsec:2.2\] and to perform a second microlocalization in the steps 2 and 3. Step 4 is the main argument. Instead of using an energy estimate obtained after a microlocal reduction as in [@L] we apply a positive commutator estimate. This allows us to control microlocally $u$ over certain small frequencies. In step 5 we use the spectral theorem to control microlocally $u$ over the remaining frequencies. In step 6 we combine the results of steps 4,5 to conclude the proof modulo an error term which is shown to be negligible in step 7.
**Step 1.** The first step in the proof is a localization process. We fix $W_{x_0,\xi_0}$ an open neighborhood of $(x_0,\xi_0)$ in ${{\overline{T}}}^*O^*{\mathbb{M}}$. We assume that $W_{x_0,\xi_0}$ is small enough, so that for all $(x,\xi) \in W_{x_0,\xi_0} \cap T^*O^*{\mathbb{M}}$, $\sigma_{{{\widetilde{H}}}_m}(x,\xi) > c|\xi|_g$, $c >0$; and so that there exists a smooth diffeomorphism $\gamma : {{\mathcal{U}}}\subset {\mathbb{R}}^{d(d+1)/2-1}_y \times {\mathbb{R}}_{\theta}\rightarrow U {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\{x \in O^*{\mathbb{M}}: \exists \xi, (x,\xi) \in W_{x_0,\xi_0}\}$ such that $d\gamma({{\partial}}_{\theta}|_{{\mathcal{U}}}) = V_{1m}|_U$. Let $\Gamma : T^*{{\mathcal{U}}}\rightarrow T^*U$ be the symplectic lift of $\gamma$.
Let ${{\mathcal{V}}}_{k\ell} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\frac{1}{2}(d\gamma^{-1} V_{k\ell}|_U - (d\gamma^{-1} V_{k\ell}|_U)^*)$. This is an anti-selfadjoint differential operator on ${{\mathcal{U}}}$ which has the same principal symbol as $d\gamma^{-1} V_{k\ell}|_U$. In particular there exists a function $f_{k\ell} \in C^\infty(O^*{\mathbb{M}})$ such that $$\label{eq:0a}
{{\mathcal{V}}}_{k\ell} = d\gamma^{-1}V_{k\ell}|_U + \gamma^* f_{k\ell}|_U.$$ Extend ${{\mathcal{V}}}_{k\ell}$ to an anti-selfadjoint differential operator of order $1$ on ${\mathbb{R}}^{d(d+1)/2}$ with coefficients in $C^\infty_b({\mathbb{R}}^{d(d+1)/2})$ – with ${{\mathcal{V}}}_{1m}$ specifically continued by ${{\partial}}_{\theta}$ – and define ${{\mathcal{L}}}_O^V {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}-\sum_{k,\ell} {{\mathcal{V}}}_{k\ell}^2$. Since $\Delta_O^V$ commutes with $V_{1m}$, $[{{\mathcal{L}}}_O^V,D_{\theta}] w = 0$ for each $w \in C^\infty({\mathbb{R}}^{d(d+1)/2})$ supported on ${{\mathcal{U}}}$.
Similarly, we define ${\mathcal{H}}_1 {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\frac{1}{2}(d\gamma^{-1} {{\widetilde{H}}}_1 |_U - (d\gamma^{-1} {{\widetilde{H}}}_1 |_U)^*)$, which is an anti-selfadjoint differential operator on ${{\mathcal{U}}}$. It satisfies $$\label{eq:0b}
{\mathcal{H}}_1|_{{\mathcal{U}}}= d\gamma^{-1} {{\widetilde{H}}}_1 |_U + \gamma^*f|_U,$$ for a certain function $f \in C^\infty(O^*{\mathbb{M}})$. It extends to an anti-selfadjoint differential operator of order $1$ on ${\mathbb{R}}^{d(d+1)/2}$ with coefficients in $C^\infty_b({\mathbb{R}}^{d(d+1)/2})$. We define ${{\mathcal{P}}}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}h^2{{\mathcal{L}}}_O^V + h {\mathcal{H}}_1$.
Let $A \in \Psi_h^0$ with ${{\operatorname{WF}}}_h(A) \subset W_{x_0,\xi_0}$ and $\psi \in C^\infty(O^*{\mathbb{M}})$ be equal to $1$ on the set $\{x \in O^*{\mathbb{M}}, \exists \xi, (x,\xi) \in {{\operatorname{WF}}}_h(A) \}$ and $0$ outside $U$. The function $1-\psi$ can be seen as a pseudodifferential operator in $\Psi_h^0$ with ${{\operatorname{WF}}}_h(1-\psi) \cap W_{x_0,\xi_0} = \emptyset$. In particular $A(1-\psi) \in h^\infty \Psi_h^{-\infty}, (1-\psi) A \in h^\infty \Psi_h^{-\infty}$ and to prove the proposition it suffices to show that $$\label{eq:7l}
v \in C^\infty(O^*{\mathbb{M}}) \ \Rightarrow \ h^{2/3}|\psi A \psi^2 v|_{{{\widetilde{H}}}^{2/3}_h} \leq C|{\widetilde{P}}\psi A \psi^2 v|+O(h) |v|_{{{\widetilde{H}}}^{3/5}_h}.$$ We define $(\gamma^*)^{-1}$ (resp. $\gamma^*$) the operator defined on functions on ${{\mathcal{U}}}$ (resp. $O^*{\mathbb{M}}$) by $$(\gamma^*)^{-1}w (x) = \systeme{ w(\gamma^{-1}(x)) \text{ if } x \in U \\ 0 \ \ \ \ \text{ otherwise}} \ \ \ \left(\text{resp. } \gamma^*v (z) = \systeme{ v(\gamma(z)) \text{ if } z \in {{\mathcal{U}}}\\ 0 \ \ \ \ \text{ otherwise}}\right).$$ The function $\psi A \psi^2 u$ has support in $U$; the operator ${\mathcal{A}}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\gamma^* \psi A \psi (\gamma^*)^{-1}$ is a pseudodifferential operator in $\Psi^0_h$ on ${\mathbb{R}}^3$ with wavefront set in $\Gamma^{-1} (W_{x_0,\xi_0})$; and $$\label{eq:7k}
|\psi A \psi^2 v|_{{{\widetilde{H}}}^{2/3}_h} \leq C |\gamma^* \psi A \psi^2 v|_{{{\widetilde{H}}}^{2/3}_h} = C |{\mathcal{A}}v|_{{{\widetilde{H}}}^{2/3}_h}, \ \ w {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\gamma^* \psi v.$$ Thanks to , ,
\^\* = -\^\* \_[k,]{} h\^2(V\_[k]{} + f\_[k]{})\^2 + \^\*h(\_1 + f) = \^\* - 2h \^\* \_[k,]{} f\_[k]{} h V\_[k]{} + h\^\* g ,
where $g {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}f- h \sum_{k,\ell} f_{k\ell}^2 + (V_{k\ell} f_{k\ell})$ belongs to $C^\infty(O^*{\mathbb{M}})$. It follows that $|{{\mathcal{P}}}{\mathcal{A}}v|$
\[eq:7j\] |A \^2 v| + O(h)\_[k,]{} |h V\_[k]{} A \^2 v| + O(h) |v| 2 |A \^2 u| + O(h) |v|.
In the last inequality we used that ${\operatorname{Re}}({\widetilde{P}}) = h^2\Delta_O^V = -\sum_{k,\ell} (hV_{k\ell})^2$ hence $|hV_{k\ell} v|^2 \leq |{\widetilde{P}}v| |v|$. Finally we observe that since $w = \gamma^*\psi v$, $|w|_{{{\widetilde{H}}}^{3/5}_h} = |\gamma^* \psi v|_{{{\widetilde{H}}}^{3/5}_h} \leq C |v|_{{{\widetilde{H}}}^{3/5}_h}$. Thanks to and the bound will follow from the estimate $$\label{eq:7}
w \in C^\infty_0({\mathbb{R}}^{d(d+1)/2}), \ {\mathrm{supp}}(w) \subset {{\mathcal{U}}}\ \Rightarrow \ h^{2/3} |{\mathcal{A}}w|_{{{\widetilde{H}}}^{2/3}_h} \leq C |{{\mathcal{P}}}{\mathcal{A}}w| + O(h) |w|_{{{\widetilde{H}}}_h^{3/5}}.$$ We have reduced the estimate on $O^*{\mathbb{M}}$ to an estimate on ${\mathbb{R}}^{d(d+1)/2}$. In the following steps we prove .
**Step 2.** Let $\chi, \chi_0 \in C_0^\infty({\mathbb{R}}^{d(d+1)/2})$ be two functions such that $\chi$ is supported away from $0$, ${{\operatorname{WF}}}_h({\mathcal{A}}) \cap {{\operatorname{WF}}}_h(\chi_0(hD)) = \emptyset$, and $$1 = \sum_{j=0}^\infty \chi_j(\xi), \ \ \chi_j(\xi) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\chi(2^{-j}\xi) \text{ for } j \geq 1.$$ Write a Littlewood-Paley decomposition of ${\mathcal{A}}$: $${\mathcal{A}}= \sum_{j=0}^\infty {\mathcal{A}}_j, \ \ {\mathcal{A}}_j {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\chi_j(hD) {\mathcal{A}}.$$ Given $a$ a symbol on ${\mathbb{R}}^{d(d+1)/2} \times {\mathbb{R}}^{d(d+1)/2}$ we denote by ${{\operatorname{Op}}}_h(a)$ the standard quantization of $a$ – see [@Z §4]. The following lemma studies the composition of a pseudodifferential operator with symbol in $S^m$ with a dyadic decomposition:
\[lem:1\] If $a \in S^m$, both the operators $2^{-jm} {{\operatorname{Op}}}_h(a) \chi(2^{-j} h D)$ and $2^{-jm} \chi(2^{-j} h D) {{\operatorname{Op}}}_h (a)$ belong to $\Psi_{2^{-j} h}$, with semiclassical symbol $a_j \chi + 2^{-j}h \cdot S$.
We first note that if $a_j(x,\xi) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}2^{-jm} a(x,2^j \xi)$ then $$2^{-jm} {{\operatorname{Op}}}_h(a) \chi(2^{-j}h D) = {{\operatorname{Op}}}_{2^{-j}h} (a_j \# \chi)= {{\operatorname{Op}}}_{2^{-j}h} (a_j \chi).$$ It suffices to show that the $S$-seminorms of $a_j \chi$ are uniformly bounded in $j$. We have $$\label{Eq:6}
\left|{{\partial}}_x^{\alpha}{{\partial}}_{\xi}^\beta a_j(x,\xi) \chi(\xi)\right| \leq C_{{\alpha}\beta} \sup_{2^j \xi \in {\mathrm{supp}}(\chi)} 2^{-jm+j|\beta|} {\langle 2^j \xi \rangle}^{m-|\beta|}.$$ Since ${\mathrm{supp}}(\chi)$ is a compact subset of ${\mathbb{R}}^3 \setminus 0$, the right hand side of is uniformly bounded in $j$. This shows that $a_j \# \chi = a_j \chi \in S$, hence $2^{-jm} {{\operatorname{Op}}}_h(a) \chi(2^{-j}h D)$ belongs to $\Psi_{2^{-j}h}$ with symbol $a_j \chi$. The operator $2^{-jm} \chi(2^{-j}h D){{\operatorname{Op}}}_h(a)$ is the adjoint of $2^{-jm} {{\operatorname{Op}}}_h(a^*)\chi(2^{-j}h D)$, thus it also belongs to $\Psi_{2^{-j}h}$. By the composition formula for symbols of semiclassical operators, its semiclassical symbol is equal to $a_j \chi + 2^{-j}h \cdot S$.
A direct application of this result shows that ${\mathcal{A}}_j$ belongs to $\Psi_{2^{-j}h}$. In addition, ${\mathcal{A}}_0 \in h^\infty \Psi^{-\infty}_h$, which implies immediately $|{\mathcal{A}}_0 w| \leq O(h) |w|_{{{\widetilde{H}}}^{3/5}_h}$. We obtain in the next steps estimates on $|{\mathcal{A}}_j w|$ for $j \geq 1$.
**Step 3.** We start with a simple result:
\[lem:2\] There exist functions $\Phi \in C^\infty_b({\mathbb{R}})$ and $\phi \in C_0^\infty({\mathbb{R}})$ such that $\phi(0) = 1$, $\phi \geq 0$ and $\phi^2 = (\Phi^2)'$.
It is enough to construct $\phi$ with $\phi(0) > 0$ then to multiply $\Phi, \phi$ by a suitable multiplicative constant. Let $\Phi$ be a smooth non-decreasing function with $$\Phi(x) = \systeme{0 \ \ \ \ \ \ \ \text{ if } x \leq -1, \\
e^{-(x+1)^{-1}} \ \text{ if } x \in [-1,0], \\
1 \ \ \ \ \ \text{ if } x \geq 1.}$$ If $\phi$ is the non-negative root of $(\Phi^2)'$ then $\phi$ has compact support and $\phi(0) > 0$. Since the $s \in [0,\infty) \mapsto \sqrt{s}$ is smooth everywhere but at $0$, $\phi$ is smooth everywhere but possibly at $-1$. But $$\phi(x) = \systeme{0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ if } x \leq -1, \\
2^{1/2} (x+1)^{-1}e^{-(x+1)^{-1}} \text{ if } x \in [-1,0],}$$ which is smooth at $x=-1$.
Let $\Phi, \phi$ be given by Lemma \[lem:2\]. Let $h_j {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}h^{2/3} 2^{-j/3}$ and consider the operator $\Phi(h_j D_{\theta})$. This operator belongs to $\Psi_{h_j}$ with semiclassical symbol $\Phi(\xi_{\theta})$. Below we show an estimate on $|{\mathcal{A}}_j w|$, by splitting it into two parts, $|\phi(h_j D_{\theta}) w|$ and $|({{\operatorname{Id}}}-\phi(h_jD_{\theta}))w|$.
**Step 4.** In order to estimate $|\phi(h_j D_{\theta}) w|$ we use a positive commutator argument and the sharp G$\mathring{\text{a}}$rding inequality. Observing that $\sigma_{{{\widetilde{H}}}_m}(x,\xi) > c|\xi|_g$ on $W_{x_0,\xi_0} \cap T^*O^*{\mathbb{M}}$, the principal symbol $\sigma_{{\mathcal{H}}_1}$ of $\frac{1}{i} {\mathcal{H}}_1$ satisfies
{ \_, \_[\_1]{} }(x,) > c ||\_g \^[-1]{}(W\_[x\_0,\_0]{}) T\^\*O\^\*.
Recall that ${{\mathcal{P}}}= h^2 {{\mathcal{L}}}_O^V + h {\mathcal{H}}_1$. Similarly to [@L Equation (2.47)],
([\_jw,(h\_jD\_)\^2 \_jw ]{})\
= ([h\^2 (h\_jD\_) \_O\^V \_j w, (h\_jD\_) \_j w ]{}) + ([h\_1 \_jw, (h\_jD\_)\^2 \_jw ]{}).
We study the first term. We observe that ${{\mathcal{L}}}_O^V {\mathcal{A}}_j = {{\mathcal{L}}}_O^V {\tilde{\chi}}_j(hD) \cdot {\mathcal{A}}_j$. Lemma \[lem:1\] shows that both the operators $2^{-2j} h^2 {{\mathcal{L}}}_O^V {\tilde{\chi}}_j(hD)$ and ${\mathcal{A}}_j$ belong to $\Psi_{2^{-j} h}$. Since $2^{-j} h \leq h_j = h^{2/3} 2^{-j/3}$, they *a fortiori* belong to $\Psi_{h_j}$. In addition, $D_{\theta}$ and ${{\mathcal{L}}}_O^V$ commute on ${{\mathcal{U}}}$ and ${\mathcal{A}}$ has wavefront set contained in $T{{\mathcal{U}}}$. The asymptotic expansion formula for composition of pseudodifferential operators [@Z Theorem 4.14] show that $$h^2 \Phi(h_jD_{\theta}) {{\mathcal{L}}}_O^V {\mathcal{A}}_j = h^2 {{\mathcal{L}}}_O^V \Phi(h_jD_{\theta}) {\mathcal{A}}_j + h_j^\infty \Psi_{h_j}.$$ Using that ${{\mathcal{L}}}_V^O = -\sum_{k,\ell} {{\mathcal{V}}}_{k\ell}^2 \geq 0$ we get ${\operatorname{Re}}({\langle h^2 \Phi(h_jD_{\theta}) {{\mathcal{L}}}_O^V {\mathcal{A}}_j w, \Phi(h_jD_{\theta}) {\mathcal{A}}_j w \rangle}) =$ $${\langle h^2 {{\mathcal{L}}}_O^V \Phi(h_jD_{\theta}) {\mathcal{A}}_j w, \Phi(h_jD_{\theta}) {\mathcal{A}}_j w \rangle} + O(h_j^\infty)|w| \geq O(h_j^\infty)|w|.$$ We next focus on the term ${\operatorname{Re}}({\langle h{\mathcal{H}}_1 {\mathcal{A}}_jw, \Phi(h_jD_{\theta})^2 {\mathcal{A}}_jw \rangle})$. Since $h{\mathcal{H}}_1$ is anti-selfadjoint, it is equal to ${\operatorname{Re}}({\langle [h{\mathcal{H}}_1,\Phi(h_jD_{\theta})]{\mathcal{A}}_jw,\Phi(h_jD_{\theta}) {\mathcal{A}}_jw \rangle})$. The real part of the operator $\Phi(h_jD_{\theta})[h{\mathcal{H}}_1,\Phi(h_jD_{\theta})]$ is equal to $\frac{1}{2}[h{\mathcal{H}}_1,\Phi(h_jD_{\theta})^2]$. We obtain $$\label{eq:2}
{\operatorname{Re}}({\langle {{\mathcal{P}}}{\mathcal{A}}_jw,\Phi(h_jD_{\theta})^2 {\mathcal{A}}_jw \rangle}) \geq \frac{1}{2}{\langle {\mathcal{A}}_j^*[h{\mathcal{H}}_1,\Phi(h_jD_{\theta})^2]{\mathcal{A}}_j w, w \rangle} + O(h_j^\infty)|w|.$$
We now study the commutator term ${\mathcal{E}}_j {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}2^{-j} {\mathcal{A}}_j^*[h{\mathcal{H}}_1,\Phi(h_jD_{\theta})^2]{\mathcal{A}}_j$. We claim that it belongs to $\Psi_{h_j}$. To show this claim we fix ${\tilde{\chi}}\in C_0^\infty({\mathbb{R}}^{d(d+1)/2} \setminus 0)$ equal to $1$ near ${\mathrm{supp}}(\chi)$ and we write
\_j = 2 ( (\_j(hD)) (2\^[-j]{}\_j(hD) h \_1) (h\_j D\_)\^2 \_j ).
By Lemma \[lem:1\], the operators ${\mathcal{A}}{\tilde{\chi}}_j(hD)$, $2^{-j}\chi_j(hD) h {\mathcal{H}}_1$ and ${\mathcal{A}}_j$ belong to $\Psi_{2^{-j}h}$. Since $2^{-j} h \leq h_j = h^{2/3} 2^{-j/3}$, they also belong to $\Psi_{h_j}$. The operator $\Phi(h_j D_{\theta})$ has symbol equal to $\Phi(\xi_{\theta})$ in the $h_j$-quantization and the composition theorem for semiclassical operators shows that ${\mathcal{E}}_j \in \Psi_{h_j}$.
The semiclassical symbols of ${\mathcal{A}}_j$ and $2^{-j} h {\mathcal{H}}_1$ are given modulo $O(h_j) S$ by $$a(x,h^{1/3} 2^{j/3} \xi) \chi(h^{1/3} 2^{-2j/3} \xi), \ \ \ \ 2^{-2j/3}h^{1/3} \sigma_{{\mathcal{H}}_1},$$ where $a$ is the semiclassical symbol of ${\mathcal{A}}$ in the $h$-quantization. By the composition formula for symbols of semiclassical operators [@Z Theorem 4.14], the semiclassical symbol $\sigma_{{\mathcal{E}}_j}$ of ${\mathcal{E}}_j$ in the $h_j$-quantization is given modulo $O(h_j^2)S$ by
\[eq:1\] (h\^[1/3]{} 2\^[-2j/3]{} )\^2 |a(x,h\^[1/3]{} 2\^[j/3]{} )|\^2 { (\_)\^2, 2\^[-2j/3]{}h\^[1/3]{} \_[\_1]{}}\
= (h\^[1/3]{} 2\^[-2j/3]{} )\^2 |a(x,h\^[1/3]{} 2\^[j/3]{} )|\^2 (\_)\^2 2\^[-j]{}h { \_, \_[\_1]{} }.
The wavefront set of ${\mathcal{A}}$ (hence the support of $a$) is contained in $\Gamma^{-1}(W_{x_0,\xi_0})$ itself contained in the conical set $\overline{\{ \{ \xi_{\theta}, \sigma_{{\mathcal{H}}_1} \} \geq c|\xi| \}}$, and $|\xi| \geq c h^{-1/3} 2^{2j/3}$ whenever $\chi(h^{1/3} 2^{-2j/3} \xi) \neq 0$. It follows that $$\sigma_{{\mathcal{E}}_j} \geq \chi(h^{1/3} 2^{-2j/3} \xi)^2 |a(x,h^{1/3} 2^{j/3} \xi)|^2 \phi(\xi_{\theta})^2 \cdot c h^{2/3} 2^{-j/3}.$$ The sharp G$\mathring{\text{a}}$rding inequality [@Z Theorem 4.32] implies $${\langle {\mathcal{E}}_j w, w \rangle} \geq c h^{2/3} 2^{-j/3} |\phi(h_jD_{\theta}) {\mathcal{A}}_j w|^2 + O(h_j^2) |w|^2.$$ Since $h_j = h^{2/3} 2^{-j/3}$ and ${\mathcal{E}}_j = 2^{-j}{\mathcal{A}}_j^*[h{\mathcal{H}}_1,\Phi(h_jD_{\theta})^2]{\mathcal{A}}_j$ this yields $${\langle {\mathcal{A}}_j^*[h{\mathcal{H}}_1,\Phi(h_jD_{\theta})^2]{\mathcal{A}}_j w, w \rangle} \geq c h^{2/3} 2^{2j/3} |\phi(h_jD_{\theta}) {\mathcal{A}}_j w|^2 + O(h^{4/3} 2^{j/3}) |w|^2.$$ Therefore we can come back to and obtain $${\operatorname{Re}}({\langle {{\mathcal{P}}}{\mathcal{A}}_jw,\Phi(h_jD_{\theta})^2 {\mathcal{A}}_jw \rangle}) \geq c h^{2/3} 2^{2j/3} |\phi(h_jD_{\theta}) {\mathcal{A}}_j w|^2 + O(h^{4/3} 2^{j/3}) |w|^2.$$ Since $\Phi$ is uniformly bounded, the operator $\Phi(h_jD_{\theta})^2$ is bounded on $L^2$. This gives the estimate on $|\phi(h_jD_{\theta}) {\mathcal{A}}_j w|$: $$h^{2/3} 2^{2j/3} |\phi(h_jD_{\theta}) {\mathcal{A}}_j w|^2 \leq C|{{\mathcal{P}}}{\mathcal{A}}_jw||{\mathcal{A}}_jw| + O(h^{4/3} 2^{j/3}) |w|^2.$$
**Step 5.** The estimate on $|({{\operatorname{Id}}}-\phi(h_j D_{\theta})) w|$ follows from the spectral theorem. Since $\phi(0) = 1$ there exists a smooth bounded function $\varphi$ such that $1-\phi(t) = t\varphi(t)$. The operator $\varphi(h_jD_{\theta})$ is uniformly bounded on $L^2$ hence $$\label{eq:7n}
h^{2/3} 2^{2j/3} |({{\operatorname{Id}}}-\phi(h_j D_{\theta})) {\mathcal{A}}_j w|^2 = h^{2/3} 2^{2j/3} |\varphi(h_jD_{\theta}) h_jD_{\theta}{\mathcal{A}}_jw|^2 \leq C |h D_{\theta}{\mathcal{A}}_jw|^2.$$ We recall that ${{\partial}}_{\theta}= {{\mathcal{V}}}_{1m}$ and that ${{\mathcal{L}}}_O^V = -\sum_{k,\ell} {{\mathcal{V}}}_{k\ell}^2 \geq -{{\mathcal{V}}}_{1m}^2$; hence
h\^[2/3]{} 2\^[2j/3]{} |(-(h\_j D\_)) \_j w|\^2 C [\_O\^V \_jw, \_j w ]{} C|\_jw| |\_jw|.
**Step 6.** Combining the results of the steps 4 and 5, we obtain the estimate $$h^{2/3} 2^{2j/3}|{\mathcal{A}}_j w|^2 \leq C|{{\mathcal{P}}}{\mathcal{A}}_j w| |{\mathcal{A}}_j w|+O(h^{4/3}2^{j/3}) |w|^2.$$ Let ${\tilde{\chi}}\in C_0^\infty({\mathbb{R}}^{d(d+1)/2} \setminus 0)$ equal to $1$ on ${\mathrm{supp}}(\chi)$. We apply the above estimate to ${\tilde{\chi}}_j(hD) w$ and we observe that both ${\mathcal{A}}_j$ and ${{\operatorname{Id}}}- {\tilde{\chi}}_j(hD)$ belong to $\Psi_{2^{-j}h}$ and that their symbols have disjoint supports; therefore $|{\mathcal{A}}_j ({{\operatorname{Id}}}- {\tilde{\chi}}_j(hD))w| = O(h^\infty 2^{-j\infty})|w|$ by the composition theorem. Similarly by Lemma \[lem:1\], $2^{-2j}{{\mathcal{P}}}{\mathcal{A}}_j$ belongs to $\Psi_{2^{-j}h}$ and its symbol has disjoint support from the one of ${{\operatorname{Id}}}- {\tilde{\chi}}_j(hD)$; therefore $2^{-2j}|{{\mathcal{P}}}{\mathcal{A}}_j ({{\operatorname{Id}}}- {\tilde{\chi}}_j(hD))w| = O(h^\infty2^{-j\infty})|w|$. It follows that $$h^{2/3} 2^{2j/3}|{\mathcal{A}}_j w|^2 \leq C|{{\mathcal{P}}}{\mathcal{A}}_j w| |{\mathcal{A}}_jw|+O(h^{4/3}2^{j/3}) |{\tilde{\chi}}_j(hD) w|^2 + O(h^\infty 2^{-j\infty})|w|^2.$$ The inequality $ab \leq a^2 + b^2$ and the identity ${\mathcal{A}}_j = \chi_j(hD) {\mathcal{A}}$ shows that
\[eq:3\] h\^[4/3]{} 2\^[4j/3]{}|\_j(hD) w|\^2 C|\_j w|\^2+ O(h\^2 2\^j) |\_j(hD) w|\^2 + O(h\^2\^[-j]{})|w|\^2\
C|\_j(hD) w|\^2+ C |\[,\_j(hD)\]w|\^2 + O(h\^2 2\^j) |\_j(hD) w|\^2 + O(2\^[-j]{}h\^2)|w|\^2.
**Step 7.** To conclude we show the commutator term $|[{{\mathcal{P}}},\chi_j(hD)]{\mathcal{A}}w|$ in the right hand side of is negligible. Recall that ${{\mathcal{P}}}= -h^2 \sum_{k,\ell} {{\mathcal{V}}}_{k\ell}^2 + h {\mathcal{H}}_1$ and write $$[{{\mathcal{P}}},\chi_j(hD)] = [h{\mathcal{H}}_1,\chi_j(hD)]-\sum_{k,\ell} 2h {{\mathcal{V}}}_{k\ell} [h{{\mathcal{V}}}_{k\ell}, \chi_j(hD)] + [h{{\mathcal{V}}}_{k\ell}, [h{{\mathcal{V}}}_{k\ell}, \chi_j(hD)]].$$
We first control the term $|[h{\mathcal{H}}_1,\chi_j(hD)]{\mathcal{A}}w|$. We can write
2\^[-j/2]{}\[h\_1,\_j(hD)\][hD ]{}\^[-1/2]{}\
= 2\^[-j]{} h\_1 \_j(hD)2\^[j/2]{} \_j(hD) [hD ]{}\^[-1/2]{} - 2\^[j/2]{} \_j(hD)[hD ]{}\^[-1/2]{} 2\^[-j]{} h\_1 \_j(hD).
By Lemma \[lem:1\], both $2^{-j} h{\mathcal{H}}_1 {\tilde{\chi}}_j(hD)$ and $2^{j/2} \chi_j(hD){\langle hD \rangle}^{-1/2}$ belong to $\Psi_{2^{-j}h}$. It follows that the operator $2^{-j/2} [h{\mathcal{H}}_1, \chi_j(hD)] {\langle hD \rangle}^{-1/2}$ belongs to $\Psi_{2^{-j} h}$. Its symbol in the $2^{-j}h$-quantization is given by the asymptotic formula and has vanishing leading term; therefore $2^{-j/2} [h{\mathcal{H}}_1, \chi_j(hD)] {\langle hD \rangle}^{-1/2}$ belongs to $2^{-j} h \Psi_{2^{-j}h}$. As such it is bounded on $L^2$ with norm $O(2^{-j} h)$. This yields $$\label{eq:0d}
|[h{\mathcal{H}}_1, \chi_j(hD)] {\mathcal{A}}w| = O(2^{-j/2} h) |w|_{{{\widetilde{H}}}^{1/2}_h} = O(2^{-j/2} h) |w|_{{{\widetilde{H}}}^{3/5}_h}.$$ By arguments similar to the one needed to show , $[h{{\mathcal{V}}}_{k\ell}, [h{{\mathcal{V}}}_{k\ell}, \chi_j(hD)]] {\langle hD \rangle}^{-1/2}$ belongs to $2^{-j/2} h^2 \Psi_{2^{-j} h}$ and $$\label{eq:0y}
|[h{{\mathcal{V}}}_{k\ell}, [h{{\mathcal{V}}}_{k\ell}, \chi_j(hD)]] {\mathcal{A}}w| = O(2^{-j/2} h^2) |w|_{{{\widetilde{H}}}^{1/2}_h}= O(2^{-j/2} h^2) |w|_{{{\widetilde{H}}}^{3/5}_h}.$$
The term $h {{\mathcal{V}}}_{k\ell} [h{{\mathcal{V}}}_{k\ell}, \chi_j(hD)]$ requires some extra work. Fix $j,k,\ell$ and define $B = [h{{\mathcal{V}}}_{k\ell}, \chi_j(hD)]$. Then, $$|h {{\mathcal{V}}}_{k\ell} B {\mathcal{A}}w|^2 \leq {\operatorname{Re}}({\langle {{\mathcal{P}}}B {\mathcal{A}}w, B {\mathcal{A}}w \rangle}) = {\operatorname{Re}}({\langle {{\mathcal{P}}}{\mathcal{A}}w, B^*B {\mathcal{A}}w \rangle}) + {\operatorname{Re}}({\langle [P,B]{\mathcal{A}}w,B{\mathcal{A}}w \rangle}).$$ By the same arguments as needed to show , the operator $B^*B {\langle hD \rangle}^{-1/2}$ belongs to $2^{-j/2} h^2 \Psi_{2^{-j}h}$. Hence ${\operatorname{Re}}({\langle {{\mathcal{P}}}{\mathcal{A}}w, B^*B {\mathcal{A}}w \rangle}) = O(2^{-j/2}h^2) |{{\mathcal{P}}}{\mathcal{A}}w| |w|_{{{\widetilde{H}}}^{1/2}_h}$. On the other hand the operator ${\langle hD \rangle}^{-3/5}[P,B] {\langle hD \rangle}^{-3/5}$ belongs to $2^{-j/5} h^2 \Psi_{h_j}$ and this implies that ${\operatorname{Re}}({\langle [P,B]{\mathcal{A}}w,B {\mathcal{A}}w \rangle}) = O(2^{-j/5} h^3) |w|_{{{\widetilde{H}}}^{3/5}_h}^2$. Combining all these estimates together we obtain that $$\label{eq:0x}
|h {{\mathcal{V}}}_{k\ell} B {\mathcal{A}}w|^2 = O(2^{-j/5}h^3) |w|^2_{{{\widetilde{H}}}^{3/5}_h} + O(2^{-j/2}h^2) |{{\mathcal{P}}}{\mathcal{A}}w| |w|_{{{\widetilde{H}}}^{3/5}_h}.$$
We plug , , in to obtain the estimate $h^{4/3} 2^{4j/3}|\chi_j(hD) {\mathcal{A}}w|^2 $ $$\leq C|\chi_j(hD) {{\mathcal{P}}}{\mathcal{A}}w|^2+ O(h^2 2^j) |{\tilde{\chi}}_j(hD) w|^2 + O(2^{-j/5} h^3) |w|_{{{\widetilde{H}}}^{3/5}_h}^2 + O(2^{-j/2}h^2) |{{\mathcal{P}}}{\mathcal{A}}w| |w|_{{{\widetilde{H}}}^{3/5}_h}.$$ Summation over $j$ allows us to conclude thanks to [@Z Equation (9.3.29)]: $$h^{4/3} |{\mathcal{A}}w|_{{{\widetilde{H}}}^{2/3}_h}^2 \leq C |{{\mathcal{P}}}{\mathcal{A}}w|^2 + O(h^2)|w|_{{{\widetilde{H}}}^{1/2}_h}^2 + O(h^3)|w|_{{{\widetilde{H}}}^{3/5}_h} + O(h^2) |{{\mathcal{P}}}{\mathcal{A}}w||w|_{{{\widetilde{H}}}^{3/5}_h}.$$ This implies , hence the proof is over.
Subelliptic estimates in Anisotropic Sobolev spaces
===================================================
Anisotropic Sobolev spaces
--------------------------
To define Pollicott–Ruelle resonances as eigenvalues we need to change the spaces on which $ H_1 $ acts. These spaces originally appeared as anisotropic Sobolev spaces in Baladi [@Ba], Liverani [@Liv], Gouëzel–Liverani [@GoLi], Baladi–Tsuji [@BaTs]. We follow a microlocal approach due to Faure–Sjöstrand [@FS] in a version given by Dyatlov–Zworski [@DZ1]. It allows the use of PDE methods in the study of the Pollicott–Ruelle spectrum.
For $s,r \in {\mathbb{R}}$, let $G_{r,s}(h) \in \Psi_h^{0+}$ with principal symbol $\sigma_{G_{r,s}}$ given by $$\label{Eq:3c}
\sigma_{G_{r,s}}(x,\xi) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}(sm(x,\xi) + r)\rho_0(|\xi|_g) \log(|\xi|_g),$$ where $\rho_0 \in C^\infty({\mathbb{R}},[0,1])$ vanishes on $[-1,1]$ and is equal to $1$ on ${\mathbb{R}}\setminus [-2,2]$ and $m \in C^\infty(T^*S^*{\mathbb{M}}\setminus 0,[-1,1])$ is homogeneous of degree $0$ with $$\systeme{ m(x,\xi) = 1 \ \ \text{ near } E_s^* \\ m(x,\xi) = -1 \text{ near } E_u^*
} \ \ \text{and } \{m,\sigma_{H_1}\} \geq 0.$$ The existence of $m$ is proved in [@DZ1 Lemma 3.1]. For every $s, r \geq 0$, the operator $e^{G_{r,s}(h)}$ belongs to $\Psi_h^{s+r+}$ and the semiclassical spaces of [@DZ1] are defined as $H_h^{r,s} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}e^{-G_{r,s}(h)} L^2$. In particular functions in $H_h^{r,s}$ are in $H^{r+s}_h$ microlocally near $E_s^*$ and in $H^{r-s}_h$ microlocally near $E_u^*$:
\[Eq:6d\] A \_h\^0, \_h(A) E\_s\^\* |Au|\_[H\^[r+s]{}\_h]{} C|u|\_[H\_h\^[r,s]{}]{},\
A \_h\^0, \_h(A) E\_u\^\* |Au|\_[H\^[r-s]{}\_h]{} C|u|\_[H\_h\^[r,s]{}]{}.
In addition if $r,s \in {\mathbb{R}}$ are fixed and $h > 0$ varies the spaces $H_h^{r,s}$ are equal and there exists a constant $C$ such that $$\label{eq:0j}
C^{-1} h^{|s|+|r|} |u|_{H_1^{r,s}} \leq |u|_{H_h^{r,s}} \leq C h^{-|s|-|r|}|u|_{H_1^{r,s}}.$$
High frequency estimate in $H_1^{r,s}$
--------------------------------------
The first result of this section extends the $L^2$-based hypoelliptic estimate of Theorem \[thm:1\] to anisotropic Sobolev spaces:
\[prop:1d\] For every $R,N \geq 0$ and $r,s \in {\mathbb{R}}$, $\rho_1, \rho_2$ satisfying , there exist $C_{R,N,r,s} > 0$ and ${\varepsilon}_0 >0$ such that for every $0 < {\varepsilon}\leq {\varepsilon}_0$ and $|\lambda| \leq R$, $$\label{eq:1w}
|\rho_1({\varepsilon}^2\Delta) {\varepsilon}^2 \Delta_{\mathbb{S}}u|_{H_1^{r,s}} \leq C_{R,N,r,s} |\rho_2({\varepsilon}^2\Delta) {\varepsilon}(P_{\varepsilon}-\lambda) u|_{H_1^{r,s}}+ O({\varepsilon}^N) |u|_{H_1^{r,s}}.$$
First observe that as in [@DZ2 Equation (4.4)], if $B$ is a semiclassical pseudodifferential operator then $${{\operatorname{WF}}}_{\varepsilon}(B) \subset {{\overline{T}}}^*S^*{\mathbb{M}}\setminus 0 \ \Rightarrow \ (e^{G_{r,s}(1)} - e^{G_{r,s}({\varepsilon})}) B \in {\varepsilon}^\infty \Psi^{-\infty}_{\varepsilon}.$$ Since $\rho_1({\varepsilon}^2 \Delta), \rho_2({\varepsilon}^2 \Delta)$ are microlocalized away from the zero section and because of , the proposition will follow from the bound $$\label{eq:3a}
|\rho_1({\varepsilon}^2\Delta) {\varepsilon}^2 \Delta_{\mathbb{S}}u|_{H_{\varepsilon}^{r,s}} \leq C |\rho_2({\varepsilon}^2\Delta) {\varepsilon}(P_{\varepsilon}-\lambda) u|_{H_{\varepsilon}^{r,s}}+ O({\varepsilon}^N) |u|_{H_{\varepsilon}^{r,s}}.$$ Below we conjugate the operators involved in with $e^{G_{r,s}({\varepsilon})}$ and show a $L^2$-based estimate equivalent to .
For $A \in \Psi_{\varepsilon}^m$, let $[A]_{r,s}$ be the operator $e^{G_{r,s}({\varepsilon})} A e^{-G_{r,s}({\varepsilon})}$. We have $$\label{eq:3b}
[A]_{r,s} = A + [G_{r,s}({\varepsilon}),A] + {\varepsilon}^2 \Psi_{\varepsilon}^{m-2+},$$ see the equation [@DDZ (3.11)] and the discussion following it. For $\rho_1, \rho_2$ satisfying , let ${\tilde{\rho}}_1, {\tilde{\rho}}_2$ be smooth functions satisfying , with ${\tilde{\rho}}_1 = 1$ on ${\mathrm{supp}}(\rho_1)$ and ${\tilde{\rho}}_2 = 0$ on $\{\rho_2 \neq 1\}$. We use the identity to prove that: $$\label{eq:7da}
\left|\left(\rho_1({\varepsilon}^2 \Delta) {\varepsilon}^2 \Delta_{\mathbb{S}}-[\rho_1({\varepsilon}^2 \Delta) {\varepsilon}^2 \Delta_{\mathbb{S}}]_{r,s} \right)v\right| \leq C |{\tilde{\rho}}_2({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda) v| + O({\varepsilon}^N) |v|.$$ Since $\Delta_{\mathbb{S}}= -\sum_{j=1}^n X_j^2$, we have
\[eq:3c\] \_1(\^2) \^2 \_- \[\_1(\^2) \^2 \_\]\_[r,s]{} \_[j=1]{}\^n \_\^[0+]{} X\_j + \^2 \_\^[0+]{},
where the terms in $\Psi_{\varepsilon}^{0+}$ have wavefront sets contained in ${{\operatorname{WF}}}_{\varepsilon}(\rho_1({\varepsilon}^2 \Delta))$, itself contained in ${{\operatorname{Ell}}}_{\varepsilon}({\tilde{\rho}}_1({\varepsilon}^2 \Delta))$. Thus,
|(\_1(\^2 ) \^2 \_-\[(\_1(\^2 ) \^2 \_\]\_[r,s]{} )v|\
O() \_[j=1]{}\^n |\_1(\^2 ) X\_j v|\_[H\^[1/3]{}\_]{} + O(\^2) |\_1(\^2 ) v|\_[H\^[2/3]{}\_]{} + O(\^)|v|.
Theorem \[thm:1\] applied with the pair $({\tilde{\rho}}_1, {\tilde{\rho}}_2)$ estimates the right hand side by $C |{\tilde{\rho}}_2({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda) v|+O({\varepsilon}^N)|v|$. This gives .
Thanks to ,
\[eq:3e\] |\[\_1(\^2) \^2 \_\]\_[r,s]{} v| |\_1(\^2) \^2 \_v| + C|\_2(\^2 ) (P\_-) v| + O(\^N) |v|\
C|\_2(\^2 ) (P\_-) v| + O(\^N) |v|.
In the second line we used Theorem \[thm:1\] with the pair $(\rho_1, {\tilde{\rho}}_2)$. To show , it remains to control $|{\tilde{\rho}}_2({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda) v|$ by $|[\rho_2({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda)]_{r,s} v|$. We will need the following lemma:
\[lem:8\] Let $m \leq 0$ and $B_0 \in \Psi_{\varepsilon}^m$ such that ${{\operatorname{WF}}}_{\varepsilon}(B_0) \subset {{\operatorname{Ell}}}_{\varepsilon}(\rho_2({\varepsilon}^2\Delta))$. For every $N > 0$ there exists $B_1 \in \Psi_{\varepsilon}^{m-1/4}$ with ${{\operatorname{WF}}}_h(B_1) \subset {{\operatorname{Ell}}}_{\varepsilon}(\rho_2({\varepsilon}^2\Delta))$ such that $$\label{eq:0w}
|B_0 {\varepsilon}(P_{\varepsilon}-\lambda) v| \leq |B_0{\varepsilon}[P_{\varepsilon}- \lambda]_{r,s} v| + O({\varepsilon}^{1/3}) |B_1 {\varepsilon}(P_{\varepsilon}-\lambda) v| + O({\varepsilon}^N) |v|.$$
The idea is similar to the second part of the proof of Theorem \[thm:1\]. We have
\[eq:7o\] B\_0 (P\_-) = B\_0 \[P\_-\]\_[r,s]{} + \_[j=1]{}\^n X\_j \^[m+]{}\_+ \^[m+]{}\_\
= B\_0 \[P\_-\]\_[r,s]{} + \_[1/3]{} \_[j=1]{}\^n X\_j \^[m-1/4]{}\_+ \_[2/3]{} \^[m-1/4]{}\_.
Let ${\tilde{\rho}}_3, {\tilde{\rho}}_4 \in C^\infty({\mathbb{R}}^3)$ satisfying and such that ${{\operatorname{WF}}}_{\varepsilon}(B_0) \cap {{\operatorname{WF}}}_{\varepsilon}({\tilde{\rho}}_3({\varepsilon}^2\Delta)-{{\operatorname{Id}}}) = \emptyset$. Equivalently, ${\tilde{\rho}}_3({\varepsilon}^2 \Delta) B_0 = B_0 + {\varepsilon}^\infty \Psi^{-\infty}_{\varepsilon}$. We multiply both sides of by ${\tilde{\rho}}_3({\varepsilon}^2 \Delta)$ to obtain $B_0 {\varepsilon}(P_{{\varepsilon}}-\lambda) - B_0 {\varepsilon}[P_{{\varepsilon}}-\lambda]_{r,s}$ $$\label{eq:8c}
= {\varepsilon}\Lambda_{1/3} \cdot {\tilde{\rho}}_3({\varepsilon}^2 \Delta) \sum_{j=1}^n {\varepsilon}X_j \cdot {\varepsilon}\Psi^{m-1/4}_{\varepsilon}+ {\varepsilon}\Lambda_{2/3} \cdot {\tilde{\rho}}_3({\varepsilon}^2 \Delta) \cdot \Psi^{m-1/4}_{\varepsilon}+ {\varepsilon}^\infty \Psi^{-\infty}_{\varepsilon}.$$ Thus there exist operators ${\widetilde{B}}_1^j \in \Psi^{m-1/3+}_{\varepsilon}\subset \Psi^{m-1/4}_{\varepsilon}$ and ${\widetilde{B}}_2^j \in \Psi^{m-2/3+} \subset \Psi^{m-1/4}_{\varepsilon}$ with wavefront sets contained in ${{\operatorname{WF}}}_{\varepsilon}(B_0)$ such that $|B_0 {\varepsilon}(P_{\varepsilon}-\lambda) v - B_0 {\varepsilon}[P_{{\varepsilon}}-\lambda]_{r,s} v|$ $$\label{eq:0c}
\leq {\varepsilon}\sum_{j=1}^n \sum_{k=1,2} |{\tilde{\rho}}_3({\varepsilon}^2 \Delta) {\varepsilon}X_j {\widetilde{B}}_k^j v|_{H^{1/3}_{\varepsilon}} + {\varepsilon}|{\tilde{\rho}}_3({\varepsilon}^2 \Delta) {\widetilde{B}}_k^j v|_{H^{2/3}_{\varepsilon}} + O({\varepsilon}^\infty) |v|.$$ Theorem \[thm:1\] applied to $({\tilde{\rho}}_3, {\tilde{\rho}}_4)$ estimates the right hand side of : $$|B_0 {\varepsilon}(P_{\varepsilon}-\lambda) v| \leq |B_0 {\varepsilon}[P_{{\varepsilon}}-\lambda]_{r,s} v| + O({\varepsilon}^{1/3}) \sum_{j,k} |{\tilde{\rho}}_4({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_k^j v| + O({\varepsilon}^N)|v|.$$ Since ${{\operatorname{WF}}}_{\varepsilon}({\widetilde{B}}_k^j) \cap {{\operatorname{WF}}}_{\varepsilon}({\tilde{\rho}}_3({\varepsilon}^2 \Delta) - {{\operatorname{Id}}})$ is empty and ${\tilde{\rho}}_4=1$ on ${\mathrm{supp}}({\tilde{\rho}}_3)$, ${\tilde{\rho}}_4({\varepsilon}^2\Delta) {\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_k^j = {\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_k^j+ {\varepsilon}^\infty \Psi^{-\infty}_{\varepsilon}$. It follows that $$\label{eq:6k}
|B_0 {\varepsilon}(P_{\varepsilon}-\lambda) v| \leq |B_0 {\varepsilon}[P_{{\varepsilon}}-\lambda]_{r,s} v| + O({\varepsilon}^{1/3}) \sum_{j,k} |{\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_k^j v| + O({\varepsilon}^N)|v|.$$
Fix $1 \leq j \leq n$ and $1 \leq k \leq 2$. We recall that ${\widetilde{B}}_k^j \in \Psi^{m-1/4}_{\varepsilon}$ with wavefront sets contained in ${{\operatorname{WF}}}_{\varepsilon}(B_0)$. Similarly to , we write $$[{\varepsilon}(P_{\varepsilon}-\lambda), {\widetilde{B}}_k^j] = {\varepsilon}\Lambda_{1/3} \cdot {\tilde{\rho}}_3({\varepsilon}^2 \Delta) \sum_{\ell=1}^n {\varepsilon}X_\ell {\widetilde{B}}_{k,1}^{j,\ell} + {\varepsilon}\Lambda_{2/3} \cdot {\tilde{\rho}}_3({\varepsilon}^2\Delta) \cdot {\varepsilon}{\widetilde{B}}_{k,2}^{j,\ell} + {\varepsilon}^\infty \Psi_{\varepsilon}^{\infty},$$ for some operators ${\widetilde{B}}_{k,1}^{j,\ell}, {\widetilde{B}}_{k,2}^{j,\ell} \in \Psi^{m-1/2}_{\varepsilon}$ with wavefront sets contained in ${{\operatorname{WF}}}_h(B_0)$. And similarly to , we obtain the estimate $$\label{eq:3d}
|{\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_k^j v| \leq |{\widetilde{B}}_k^j {\varepsilon}(P_{\varepsilon}-\lambda) v| + O({\varepsilon}^{1/3}) \sum_{\ell, k, k'} |{\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_{k,k'}^{j,\ell} v| + O({\varepsilon}^N)|v|.$$ We observe that the terms $O({\varepsilon}^{1/3}) |{\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_{k,k'}^{j,\ell} v|$ above involve a factor ${\varepsilon}^{1/3}$ and an operator ${\widetilde{B}}_{k,k'}^{j,\ell}$ that is $1/4$-smoother than ${\widetilde{B}}_k^j$. Since ${\varepsilon}(P_{\varepsilon}- \lambda) \cdot \Psi_{\varepsilon}^{-2} \subset \Psi_{\varepsilon}^0$, we can then iterate sufficiently many times to get an operator $B_1 \in \Psi^{m-1/4}_h$ with wavefront set contained in ${{\operatorname{WF}}}_h(B_0)$, such that $$\sum_{j,k} |{\varepsilon}(P_{\varepsilon}-\lambda) {\widetilde{B}}_k^j v| \leq |B_1 {\varepsilon}(P_{\varepsilon}-\lambda) v| + O({\varepsilon}^N)|v|.$$ We combine this bound with to conclude the proof.
The right hand side of involves the term $O({\varepsilon}^{1/3}) |B_1 {\varepsilon}(P_{\varepsilon}-\lambda) v|$ which comes with the factor ${\varepsilon}^{1/3}$, and the operator $B_1$. This operator is $1/4$-smoother than $B_0$. We can then iterate sufficiently many times starting from $B_0 = {\tilde{\rho}}_2({\varepsilon}^2 \Delta) \in \Psi_{\varepsilon}^{0}$ to obtain operators $B_1 \in \Psi_{\varepsilon}^{-1/4}, ..., B_{3N} \in \Psi^{-3N/4}_{\varepsilon}$ with wavefront sets contained in ${{\operatorname{Ell}}}_{\varepsilon}(\rho_2({\varepsilon}^2 \Delta))$ and such that $$|{\tilde{\rho}}_2({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda) v| \leq \sum_{k=0}^{3N-1} {\varepsilon}^{k/3} |B_k {\varepsilon}[P_{{\varepsilon}}-\lambda]_{r,s} v| + O({\varepsilon}^N)|{\varepsilon}(P_{\varepsilon}-\lambda) B_{3N} v|.$$ For $N$ large enough, ${\varepsilon}(P_{\varepsilon}-\lambda) B_{3N} \in \Psi^0_{\varepsilon}$ and $O({\varepsilon}^N)|{\varepsilon}(P_{\varepsilon}-\lambda) B_{3N} v| = O({\varepsilon}^N)|v|$. In addition the operator $[\rho_2({\varepsilon}^2 \Delta)]_{r,s}$ is elliptic on the wavefront set of the $B_k$ thus $$|{\tilde{\rho}}_2({\varepsilon}^2 \Delta) {\varepsilon}(P_{\varepsilon}-\lambda) v| \leq |[\rho_2({\varepsilon}^2 \Delta){\varepsilon}(P_{\varepsilon}-\lambda)]_{r,s} v| + O({\varepsilon}^N)|v|.$$ Plug this estimate back in to conclude the proof of the proposition.
*Starting now we consider $R,N,r,s$ fixed, ${\varepsilon}_0$ given by Proposition \[prop:1d\] and ${\varepsilon}, h$ satisfying $0 < {\varepsilon}\leq h \leq {\varepsilon}_0$.* Fix $\rho_1, \rho_2$ satisfying , $\chi_1 {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}1-\rho_1$ and $\chi$ be equal to $1$ near $0$ and such that $\chi \rho_2 = 0$. Define $Q{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\chi(h^2\Delta)$ and
\[Eq:2i\] P\_() h(P\_-)-iQ = -ih\_-ih H\_1-h-iQ ,\
\_() -i h\_1(\^2 ) \_-i h H\_1-h-iQ.
If $P_{{\varepsilon}}(\lambda) u {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}f$ then ${\widetilde{P}}_{{\varepsilon}}(\lambda)u = f + ih {\varepsilon}\rho_1({\varepsilon}^2 \Delta)\Delta_{\mathbb{S}}u {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}F$. We use to go from the space $H^{r,s}_h$ to the space $H^{r,s}_1$ and we bound $F$ by Proposition \[prop:1d\]:
|F|\_[H\^[r,s]{}\_h]{} |f|\_[H\^[r,s]{}\_h]{} + h |\_1(\^2 )\_u|\_[H\^[r,s]{}\_h]{} |f|\_[H\^[r,s]{}\_h]{} + h\^[-|s|-|r|+1]{} \^[-1]{} |\_1(\^2 ) \^2 \_u|\_[H\^[r,s]{}\_1]{}\
|f|\_[H\^[r,s]{}\_1]{}+C h\^[-|s|-|r|+1]{} \^[-1]{}|\_2(\^2) (P\_- )u|\_[H\^[r,s]{}\_1]{}+O(h\^[-|s|-|r|]{}\^N)|u|\_[H\^[r,s]{}\_1]{}.
We note that $\rho_2({\varepsilon}^2\Delta) Q = 0$ because ${\varepsilon}\leq h$, hence
h \^[-1]{} \_2(\^2) (P\_-)u = \_2(\^2) (h( P\_-) - iQ )u =\_2(\^2) P\_()u = \_2(\^2 ) f.
It follows that $$\label{eq:3g}
|F|_{H^{r,s}_h} \leq |f|_{H^{r,s}_h}+C h^{-|s|-|r|}|\rho_2({\varepsilon}^2 \Delta) f|_{H^{r,s}_1}+O(h^{-|s|-|r|}{\varepsilon}^N)|u|_{H^{r,s}_1}.$$ The operator $\rho_2({\varepsilon}^2 \Delta)$ is bounded on $H^{r,s}_1$ since $\rho_2({\varepsilon}^2 \Delta) \in \Psi_{\varepsilon}^0 \subset \Psi_1^0$ and by , $$e^{G_{r,s}(1)} \rho_2({\varepsilon}^2 \Delta) e^{-G_{r,s}(1)} = \rho_2({\varepsilon}^2 \Delta) + \Psi_1^{-1+} \in \Psi^0_1.$$ Therefore $|\rho_2({\varepsilon}^2 \Delta) f|_{H^{r,s}_1} \leq C|f|_{H^{r,s}_1}$; and $|f|_{H^{r,s}_1}$ is controlled by $h^{-|s|-|r|} |f|_{H^{r,s}_h}$ because of . The estimate yields $$|F|_{H^{r,s}_h} \leq C h^{-2|s|-2|r|} |f|_{H^{r,s}_h} + O(h^{-2|s|-2|r|}{\varepsilon}^N)|u|_{H^{r,s}_h}.$$ Recalling that $f=P_{{\varepsilon}}(\lambda) u$ and $F={\widetilde{P}}_{{\varepsilon}}(\lambda)u$ we obtain the main result of this section:
\[thm:2\] For every $R,N \geq 0$, and $r, s \in {\mathbb{R}}$ there exist $C_{R,N,r,s} > 0$ and ${\varepsilon}_0 >0$ such that if $P_{\varepsilon}(\lambda)$ and ${\widetilde{P}}_{\varepsilon}(\lambda)$ are defined in ,
(0,R), 0 < h \_0\
|\_()u|\_[H\^[r,s]{}\_h]{} C\_[R,N,r,s]{} h\^[-2|s|-2|r|]{} |P\_()u|\_[H\^[r,s]{}\_h]{} + O(h\^[-2|s|-2|r|]{}\^N)|u|\_[H\^[r,s]{}\_h]{}.
Stochastic stability of Pollicott–Ruelle resonances {#sec:5}
===================================================
Invertibility of $P_{{\varepsilon}}(\lambda)$
---------------------------------------------
Recall that $P_{\varepsilon}(\lambda)$ is given by $P_{\varepsilon}(\lambda) = h(P_{\varepsilon}-\lambda)-iQ$ on $H^{r,s}_h$, and let $D^{r,s}_h$ be its domain on $H^{r,s}_h$: $$D^{r,s}_h {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\{ u \in H^{r,s}_h, \ H_1 u \in H^{r,s}_h, \Delta_{\mathbb{S}}u \in H^{r,s}_h \},$$ where $H_1 u, \Delta_{\mathbb{S}}u$ are first seen as distributions. We prove here that the operator $P_{\varepsilon}(\lambda)$ is invertible from $D^{r,s}_h$ to $H^{r,s}_h$, provided that $\lambda$ is in a compact set and that $h$ is small enough, $s$ is large enough.
\[thm:5\] Let $R > 0$ and $r \in {\mathbb{R}}$. There exists $s_0 > 0$ such that for every $s \geq s_0$, there exists $h_0 > 0$ with $${\varepsilon}\leq h \leq h_0, \ \ |\lambda| \leq R \ \ \Rightarrow \ \ P_{\varepsilon}(\lambda) : D^{r,s}_h \rightarrow H^{r,s}_h \text{ is invertible. }$$
A necessary step to prove this result is a bound of the form $|u|_{H^{r,s}_h} \leq C_h|P_{{\varepsilon}}(\lambda)|_{H^{r,s}_h}$. In view of Theorem \[thm:2\] applied with $N=2|s|+2|r|+1$ it suffices to show that $|u|_{H^{r,s}_h} \leq Ch^{-1} |{\widetilde{P}}_{{\varepsilon}}(\lambda)|_{H^{r,s}_h}$ where we recall that ${\widetilde{P}}_{{\varepsilon}}(\lambda)$ is given by $${\widetilde{P}}_{{\varepsilon}}(\lambda) = -i h{\varepsilon}\chi_1({\varepsilon}^2 \Delta) \Delta_{\mathbb{S}}-i h H_1-\lambda h-iQ.$$ See ${\widetilde{P}}_{\varepsilon}(\lambda)$ as a pseudodifferential operator in the semiclassical parameter $h$. Its semiclassical principal symbol is $p_{\varepsilon}- i q_{\varepsilon}$, where $p_{\varepsilon}= \sigma_{H_1}$ and $$q_{\varepsilon}(x,\xi) = \chi_1\left(\frac{{\varepsilon}^2}{h^2} |\xi|_g^2\right) \frac{{\varepsilon}}{h} \sigma_{\Delta_{\mathbb{S}}}(x,\xi)+\chi(|\xi|^2_g).$$ It is clear that $p_{\varepsilon}$ belongs to $S^{1}/hS^0$. We claim that $q_{\varepsilon}$ also belong to $S^1/hS^0$ or equivalently that $$\label{eq:0f}
\chi_1\left(\frac{{\varepsilon}^2}{h^2} |\xi|_g^2\right) \frac{{\varepsilon}}{h} \sigma_{\Delta_{\mathbb{S}}}(x,\xi) \in S^1/hS^0.$$ Recall that $\Delta_{\mathbb{S}}= - \sum_{j=1}^n X_j^2$, write $\sigma_{X_j}$ for the principal symbol of $\frac{h}{i} X_j$ and note that $$q_{\varepsilon}(x,\xi) = \sum_{j=1}^n \sigma_{X_j}(x,\xi)\chi_1\left(|\xi'|_g^2\right) \sigma_{X_j}(x,\xi')+\chi(|\xi|_g^2), \ \ \ \ \ \xi' {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}{\varepsilon}h^{-1} \xi.$$ It suffices to show that each term in the above sum belongs to $S^1/hS_0$, thus that $(x,\xi) \mapsto \chi_1(|\xi'|_g^2) \sigma_{X_j}(x,\xi')$ belongs to $S^0/hS^{-1}$. When $|{\alpha}|+|\beta|>0$,
\^[||]{} |\_x\^\_\^( \_1(|’|\_g\^2) \_[X\_j]{}(x,’) )| = \^[||]{}(h\^[-1]{})\^[||]{} |\_x\^\_[’]{}\^\_1(|’|\_g\^2) \_[X\_j]{}(x,’)|\
[’ ]{}\^[||]{} |\_x\^\_[’]{}\^\_1(|’|\_g\^2) \_[X\_j]{}(x,’)| C\_,
where in the last inequality we used that $\chi'$ vanishes in a neighborhood of $0$ and that $\chi_1(|\xi'|_g^2) \sigma_{X_j}(x,\xi')$ belongs to $S^0$ as a symbol in $\xi'$. Since for ${\alpha}=\beta=0$ there is nothing to prove, we obtain and $q_{\varepsilon}\in S^1/hS^0$.
Hence the operator ${\widetilde{P}}_{{\varepsilon}}(\lambda)$ belongs to $\Psi^1_h$. We next compute the principal symbol of the operator $[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}e^{G_{r,s}(h)} P_{\varepsilon}(\lambda) e^{-G_{r,s}(h)}$. We write $p_{{\varepsilon},r,s} - i q_{{\varepsilon},r,s}$ for the principal symbol of $[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s}$, where $p_{{\varepsilon},r,s}, q_{{\varepsilon},r,s}$ are real-valued. The symbol $p_{{\varepsilon},r,s}$ is given by:
\[Eq:2za\] p\_[,r,s]{} = \_[H\_1]{} - { \_[G\_[r,s]{}]{}, \_1 ( ||\_g\^2 ) \_[\_]{} }\
= \_[H\_1]{} - sh { m, \_1 ( ||\_g\^2 ) \_[\_]{} } \_0(||\_g\^2) ||\_g hS\^0.
Here we used that $\sigma_{G_{r,s}} = \log(|\xi|_g) \rho_0(|\xi|_g^2) m \mod h S^{-1}$ by , and that $\{\sigma_{\Delta_{\mathbb{S}}}, |\xi|_g^2 \} = 0$ because $\Delta_{\mathbb{S}}$ commutes with $\Delta$, see . Since $m$ is homogeneous of degree $0$, we deduce from and that $$\label{Eq:2z}
p_{{\varepsilon},r,s} = \sigma_{H_1} + sh \log |\xi|_g \cdot S^0 \mod hS^0.$$ Similarly the symbol $q_{{\varepsilon},r,s}$ is given by:
\[Eq:2k\] q\_[,r,s]{} = Q(||\_g\^2) + ( ||\_g\^2 ) \_[\_]{} + h { \_[G\_[r,s]{}]{},\_[H\_1]{}}\
= Q(||\_g\^2) + ( ||\_g\^2 ) \_[\_]{} + sh {m,\_[H\_1]{}} \_0(||\_g\^2) ||\_g h S\^0,
where we used that $h\rho_0 m \{ \sigma_{H_1}, \log |\xi|_g \} \in hS^0$ and that $h \{ \sigma_{H_1}, \rho_0(|\xi|_g^2) \} \log |\xi|_g \in hS^0$. We remark that since $\{m, \sigma_{H_1}\} \geq 0$, $q_{{\varepsilon},r,s}$ is nonnegative when $s \geq 0$.
The key step to prove Theorem \[thm:5\] is the following Proposition, whose proof is largely inspired from [@DZ1 Proposition 3.1] and [@DZ2 Lemma 4.2]:
\[prop:3\] Let $R > 0$, $r \in {\mathbb{R}}$. There exists $s_0$ such that for $s \geq s_0$, there exist $h_0 > 0$ and $C_{R,r,s} > 0$ with $$0 < {\varepsilon}\leq h \leq h_0, \ |\lambda| \leq R \ \ \Rightarrow \ \ |u|_{H^{r,s}_h} \leq C_{R,r,s} h^{-1} |{\widetilde{P}}_{\varepsilon}(\lambda) u|_{H^{r,s}_h}.$$
We define $v {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}e^{G_{r,s}(h)} u \in L^2$ and we recall that $[A]_{r,s} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}e^{G_{r,s}(h)} A e^{-G_{r,s}(h)}$ when $A \in \Psi^m_h$. Using a microlocal partition of unity it is sufficient to show the inequality $$|[A]_{r,s} v| \leq C h^{-1} |[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|,$$ when ${{\operatorname{WF}}}_h(A)$ is supported in a small neighborhood of $(x_0,\xi_0) \in {{\overline{T}}}^*S^*{\mathbb{M}}$ in each of the following cases:
**Case I: $(x_0,\xi_0) \in {{\operatorname{Ell}}}_h(Q)$.** Since $\{m,\sigma_{H_1}\} \geq 0$ by construction of $m$, shows that $q_{{\varepsilon},r,s}(x_0,\xi_0) > 0$ when $s \geq 0$. In particular, $[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s}$ is elliptic at $(x_0,\xi_0)$. By the elliptic estimate, $|A_{r,s}v| \leq C |[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty) |v|$.
**Case II: $(x_0,\xi_0) \in \kappa(E_s^*)$.** Here $\kappa : T^*S^*{\mathbb{M}}\rightarrow {{\partial}}{{\overline{T}}}^*S^*{\mathbb{M}}$ is the projection map defined in [@DZ Appendix E.1]. The operator ${\widetilde{P}}_{\varepsilon}(\lambda)$ has semiclassical principal symbol $p_{\varepsilon}-iq_{\varepsilon}$. We note that $q_{\varepsilon}\geq 0$ everywhere and that $p_{\varepsilon}= \sigma_{H_1}$ is homogeneous of degree $1$ and independent of $h$. Hence we can apply the radial source estimate [@DZ1 Proposition 2.6]. Fix $B_1 \in \Psi_h^0$ with wavefront set contained in the set $\{ \rho_0 m = 1\}$ so that on $WF_h(B_1)$ the space $H^{r,s}_h$ and $H^{r+s}_h$ are microlocally equivalent, see . There exist $s_0 > 0$ and $U_0$ neighborhood of $\kappa(E_s^*)$ in ${{\overline{T}}}^*S^*{\mathbb{M}}$ such that $$s \geq s_0, \ \ {{\operatorname{WF}}}_h(A) \subset U_0 \ \Rightarrow \ |Au|_{H^{r+s}_h} \leq Ch^{-1}|B_1 {\widetilde{P}}_{\varepsilon}(\lambda) u|_{H^{r+s}_h} + O(h^\infty)|u|_{H^{-|r|-s}_h}.$$ After possibly shrinking the size of ${{\operatorname{WF}}}_h(A)$ we can use that $H^{r,s}_h$ and $H^s_h$ are microlocally equivalent near ${{\operatorname{WF}}}_h(A)$, ${{\operatorname{WF}}}_h(B_1)$ to conclude that $$|Au|_{H^{r,s}_h} \leq Ch^{-1}|{\widetilde{P}}_{\varepsilon}(\lambda) v|_{H^{r,s}_h} + O(h^\infty)|u|_{H^{-|r|-s}_h}.$$ Since $H^{r,s}_h$ embeds in $H^{-|r|-s}_h$, we deduce that for $v {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}e^{G_{r,s}(h)} u$, $$|[A]_{r,s}v| \leq Ch^{-1}|[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|.$$
**Case III: $(x_0,\xi_0) \in {{\overline{T}}}^*S^*{\mathbb{M}}$, $(x_0,\xi_0) \notin \overline{E_0^*} \oplus \overline{E_u^*}$.** In this case $(x_0,\xi_0)$ admits a neighborhood $U$ in ${{\overline{T}}}^*S^*{\mathbb{M}}$ such that $$\mathrm{d}( \exp(-tH_{\sigma_{H_1}})(U), \kappa(E_s^*)) \rightarrow 0 \text{ as } t \rightarrow -\infty,$$ see [@DZ1 Equation (3.2)]. Hence for $T$ large enough, $\exp(-TH_{\sigma_{H_1}})(U) \subset U_0$ where $U_0$ is the open set defined in Case II. We recall that $p_{{\varepsilon},r,s} - i q_{{\varepsilon},r,s}$ is the principal symbol of $[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s}$, and that $p_{{\varepsilon},r,s} = \sigma_{H_1} + h S^{1/2}$ and $q_{{\varepsilon},r,s} \geq 0$. Since $\sigma_{H_1}$ is homogeneous of degree $1$ we can apply [@DZ2 Proposition 2.2]. It shows that if $B \in \Psi_h^0$ has wavefront set contained in $U_0$ then $|[A]_{r,s} v| \leq C |Bv| + Ch^{-1} |[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|$. Combined with the result of Case II, we get $$|[A]_{r,s} v| \leq Ch^{-1}|[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|.$$
**Case IV: $(x_0,\xi_0) \in E_u^* \setminus 0$.** We recall that the lifted geodesic flow $\exp(-tH_{\sigma_{H_1}})(x_0,\xi_0)$ is equal to $\left(e^{-tH_1}(x_0), ^Tde^{-tH_1}(x_0)^{-1}\xi_0\right)$. We observe that $\exp(-tH_{\sigma_{H_1}})(x_0,\xi_0)$ converges to the zero section as $t \rightarrow +\infty$: because of $\xi_0 \in E_u^*(x_0) = E_s(x_0)$ and of ,
|\^Tde\^[-tH\_1]{}(x\_0)\^[-1]{} \_0|\_g = |\^Tde\^[tH\_1]{}(e\^[-tH\_1]{}(x\_0)) \_0|\_g Ce\^[-ct]{}.
Since ${{\operatorname{Ell}}}_h(Q)$ contains the zero section, there exists $T > 0$ such that $\exp(-TH_{\sigma_{H_1}})(x_0,\xi_0)$ belongs to ${{\operatorname{Ell}}}_h(Q)$. We apply again [@DZ2 Proposition 2.2]: if ${{\operatorname{WF}}}_h(A)$ is supported sufficiently close to $E_u^*$, there exists $B \in \Psi_h^0$ with wavefront set contained in the elliptic set of $Q$ such that $|[A]_{r,s} v| \leq C |Bv| + Ch^{-1} |[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|$. Together with Case I, it implies $$|[A]_{r,s} v| \leq Ch^{-1} |[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|.$$
**Case V: $(x_0,\xi_0) \in \kappa(E_u^*)$.** We recall that $q_{\varepsilon}\geq 0$ everywhere and that $p_{\varepsilon}= \sigma_{H_1}$ is homogeneous of degree $1$ and independent of $h$. Hence we can apply [@DZ1 Proposition 2.7]. Fix $B_1 \in \Psi_h^0$ elliptic on $\kappa(E_u^*)$, such that ${{\operatorname{WF}}}_h(B_1) \cap \overline{E_0^*} = \emptyset$ and such that $\rho_0 m = -1$ on ${{\operatorname{WF}}}_h(B_1)$. Then (after possibly increasing the value of $s_0$ given in Case II) there exist a neighborhood $U_1$ of $(x_0,\xi_0)$ and $B \in \Psi_h^0$ with ${{\operatorname{WF}}}_h(B) \subset {{\operatorname{WF}}}_h(B_1) \setminus \kappa(E_u^*)$ such that if ${{\operatorname{WF}}}_h(A) \subset U_1$ and $s \geq s_0$, $$\label{Eq:2l}
|Au|_{H^{r-s}_h} \leq C|Bu|_{H^{r-s}_h}+ C h^{-1}|B_1 {\widetilde{P}}_{\varepsilon}(\lambda)u|_{H^{r-s}_h} + O(h^\infty)|u|_{H^{-|r|-s}_h}.$$ Without loss of generality $U_1$ is small enough so that the spaces $H^{r-s}_h$, $H^{r,s}_h$ are microlocally equivalent on ${{\operatorname{WF}}}_h(A), {{\operatorname{WF}}}_h(B_1), {{\operatorname{WF}}}_h(B)$. Hence we can replace $H_h^{r-s}$ by $H^{r,s}_h$ in . In addition since ${{\operatorname{WF}}}_h(B_1)$ is supported away from $\kappa(E_0^*)$, it can be written as a finite sum of operators in $\Psi_h^0$ whose wavefront sets are supported near points $(x_0,\xi_0)$ satisfying Cases I-IV. Finally, since $H_h^{r,s}$ embedds in $H^{-s-|r|}_h$, the term $O(h^\infty)|u|_{H^{-|r|-s}_h}$ in the right hand side of is bounded by $O(h^\infty)|u|_{H^{r,s}_h}$. It follows that $$|Au|_{H^{r,s}_h} \leq Ch^{-1}|{\widetilde{P}}_{\varepsilon}(\lambda) u|_{H^{r,s}_h} + O(h^\infty)|u|_{H^{r,s}_h}.$$ Since $v = e^{G_{r,s}(h)} u$ we deduce that $$|[A]_{r,s}v| \leq Ch^{-1}|[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|.$$
**Case VI: $(x_0,\xi_0) \in \overline{E_0^*} \setminus {{\operatorname{Ell}}}_h(Q)$.** In particular, $\xi_0 \neq 0$ and $\sigma_{H_1}(x_0,\xi_0) \neq 0$. By , we have $p_{{\varepsilon},r,s} = \sigma_{H_1} + sh\log|\xi|_g \cdot S^0$. This shows that the operator $[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s}$ is elliptic at $(x_0,\xi_0)$. Therefore if $A$ has wavefront set contained in a small neighborhood of $(x_0,\xi_0)$ the elliptic estimate shows that $|[A]_{r,s}v| \leq C |[{\widetilde{P}}_{\varepsilon}(\lambda)]_{r,s} v| + O(h^\infty)|v|$.
Since Cases I-VI cover the whole ${{\overline{T}}}^*S^*{\mathbb{M}}$ this ends the proof of the theorem.
It is very similar to the end of the proof of [@DZ1 Proposition 3.1]. Fix $R > 0$ and $r \in {\mathbb{R}}$. Proposition \[prop:3\] shows that $|u|_{H^{r,s}_h} \leq C_R h^{-1} |{\widetilde{P}}_{\varepsilon}(\lambda) u|_{H^{r,s}_h}$ as long as $0 < {\varepsilon}\leq h \leq h_0$ and $s$ is large enough. Theorem \[thm:2\] applied with $N=2|s|+2|r|+1$ yields the estimate $$|u|_{H^{r,s}_h} \leq C_{R} h^{-2r-2s-1} |P_{{\varepsilon}}(\lambda)u|_{H^{r,s}_h} + O(h)|u|_{H^{r,s}_h}.$$ After possibly decreasing the value of $h_0$ we can absorb the term $O(h)|u|_{H^{r,s}_h}$ by the left hand side. We get $|u|_{H^{r,s}_h} \leq C_{R} h^{-2r-2s-1} |P_{{\varepsilon}}(\lambda)u|_{H^{r,s}_h}$. This estimate implies that the operator $P_{{\varepsilon}}(\lambda) : D^{r,s}_h \rightarrow H^{r,s}_h$ is injective.
To show the surjectivity of $P_{\varepsilon}(\lambda)$ we first note that the range of $P_{{\varepsilon}}(\lambda)$ is closed in $H^{r,s}_h$. Indeed, let $u_j \in D^{r,s}_h$ such that $P_{{\varepsilon}}(\lambda)u_j$ converges in $H^{r,s}_h$. Then $u_j$ is a Cauchy sequence in $H^{r,s}_h$ and it converges to some $u \in H^{r,s}_h$. We must show that $u \in D^{r,s}_h$. The sequence $P_{{\varepsilon}}(\lambda)u_j$ is bounded in $H^{r,s}_h$ hence it converges weakly; it follows that $P_{{\varepsilon}}(\lambda)u \in H^{r,s}_h$. By Proposition \[prop:1d\], $\rho_1({\varepsilon}^2\Delta) \Delta_{\mathbb{S}}u \in H^{r,s}_h$. In addition for any ${\varepsilon}> 0$, $\chi_1({\varepsilon}^2 \Delta) \Delta_{\mathbb{S}}u \in C^\infty$. It follows that $\Delta_{\mathbb{S}}u \in H^{r,s}_h$ hence $H_1 u \in H^{r,s}_h$. Therefore $u$ belongs to the domain of $P_{\varepsilon}(\lambda)$ and the range of $P_{\varepsilon}(\lambda)$ is closed.
To conclude we show that the range of $P_{\varepsilon}(\lambda)$ is dense in $H_h^{r,s}$. The dual of $H^{r,s}_h$ is $H^{-r,-s}_h$. Thus it suffices to prove that if $f \in H^{-r,-s}_h$ is such that ${\langle f,P_{\varepsilon}(\lambda)u \rangle} = 0$ for every $u \in H^{r,s}_h$ then $f=0$, or equivalently that $P_{\varepsilon}(\lambda)$ is injective. We have $$P_{{\varepsilon}}(\lambda) = -ih{\varepsilon}\Delta_{\mathbb{S}}-ih H_1-\lambda h-iQ, \ \ \
-P_{\varepsilon}(-{{\overline{\lambda}}})^* = -ih{\varepsilon}\Delta_{\mathbb{S}}+ih H_1-\lambda h-iQ.$$ Therefore $-P_{\varepsilon}(-{{\overline{\lambda}}})^*$ is equal to $P_{\varepsilon}(\lambda)$ except for $H_1$ which is replaced by $-H_1$. For the dynamics of $-H_1$, $E_u^*$ is a radial source and $E_s^*$ a radial sink. Moreover the imaginary part of $-P_{\varepsilon}(\lambda)^*$ is non-positive. The space $H^{-r,-s}_h$ has low regularity near $E_s^*$ (the radial sink for $-H_1$) since it is microlocally equivalent to $H^{-r-s}_h$ near $E_s^*$. Similarly $H^{-r,-s}_h$ has high regularity near $E_s^*$ (the radial source for $-H_1$) since it is microlocally equivalent to $H^{-r-s}_h$ near $E_s^*$. Hence the same analysis as in the proof of Proposition \[prop:3\] can be applied to $-P_{\varepsilon}(\lambda)^*$. It shows that for $s$ large enough and $0 < {\varepsilon}\leq h$ small enough, $\lambda \in {\mathbb{D}}(0,R)$, $$|f|_{H_h^{-r,-s}} \leq C_R h^{-2r-2s-1}|P_{\varepsilon}(-{{\overline{\lambda}}})^* f|_{H_h^{-r,-s}}.$$ This shows that $P_{\varepsilon}(\lambda)^*$ is injective. Hence the range of $P_{\varepsilon}(\lambda)$ is dense and $P_{\varepsilon}(\lambda)$ is surjective. This ends the proof of the theorem.
Proof of Theorem \[thm:0\] {#subsec:5.2}
--------------------------
We conclude the paper with a more precise version of Theorem \[thm:0\]. A function ${\varepsilon}\in [0,{\varepsilon}_0) \mapsto f({\varepsilon})$ is said to be $C^1([0,{\varepsilon}_0))$ if $f$ is $C^1$ on $(0,{\varepsilon}_0)$ and $f'({\varepsilon})$ has a limit when ${\varepsilon}\rightarrow 0$. By induction we define the class $C^k([0,{\varepsilon}_0))$. In the following, we shall say that $f$ is smooth at $0$ if for every $k > 0$, there exists ${\varepsilon}_k > 0$ such that $f \in C^k([0,{\varepsilon}_k))$. The set $\Sigma(P_{\varepsilon})$ (resp. ${\mathrm{Res}}(P_0)$) is defined as the $L^2$-spectrum of $P_{\varepsilon}= \frac{1}{i}(H_1+{\varepsilon}\Delta_{\mathbb{S}})$ (resp. Pollicott–Ruelle resonances of $P_0 = \frac{1}{i}H_1$), with inclusion according to multiplicity.
\[thm:6\] The set of accumulation points of $\Sigma(P_{\varepsilon})$, as ${\varepsilon}\rightarrow 0$, is contained in ${\mathrm{Res}}(P_0)$. Conversely, if $\lambda_0 \in {\mathrm{Res}}(P_0)$ has multiplicity $m$, there exist $r_0 > 0, {\varepsilon}_0 > 0$ such that for every ${\varepsilon}\in (0,{\varepsilon}_0)$, $\Sigma(P_{\varepsilon}) \cap {\mathbb{D}}(\lambda_0,r_0) = \{\lambda_j({\varepsilon})\}_{j=1}^m$. Moreover,
1. If $m=1$, then ${\varepsilon}\mapsto \lambda_1({\varepsilon})$ is smooth at ${\varepsilon}= 0$ and $$\label{eq:0m}
\lambda_1({\varepsilon}) = \lambda_0 + i {\varepsilon}\int_{S^*{\mathbb{M}}} {\langle \nabla_{\mathbb{S}}u, \nabla_{\mathbb{S}}v \rangle} d\mu + O({\varepsilon}^2),$$ where $u, v$ are the left and right resonant states defined in Lemma \[lem:3\].
2. The finite-rank operators $$\label{eq:0h}
\Pi_{\varepsilon}{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\dfrac{1}{2\pi i} \oint_{{{\partial}}{\mathbb{D}}(\lambda_0,r_0)} (P_{\varepsilon}-\lambda)^{-1} d\lambda \ : \ C^\infty(S^*{\mathbb{M}}) \rightarrow {{\mathcal{D}}}'(S^*{\mathbb{M}})$$ form a smooth trace-class family of operators at ${\varepsilon}=0$.
\[rem:1\] Theorem \[thm:6\] shows that as ${\varepsilon}\rightarrow 0^-$, the spectrum of $P_{-{\varepsilon}}^*$ converges to complex conjugates of Pollicott–Ruelle resonances. Because of the identity $P_{\varepsilon}= P_{-{\varepsilon}}^*$, we deduce that the spectrum of $P_{\varepsilon}$ converges to complex conjugates of Pollicott–Ruelle resonances as ${\varepsilon}\rightarrow 0^-$.
Fix $R > 0$ and $k_0$ a positive integer. For $1 \leq k \leq k_0$, let $r_k {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}2^{k+1}-2$. By Theorem \[thm:5\] and [@DZ1 Proposition 3.4] there are $s_0, h_0 > 0$ such that for every $0 \leq {\varepsilon}\leq h_0$, $r \in [\![0,r_{k_0}]\!]$ and $\lambda \in {\mathbb{D}}(0,R)$ the operator $$P_{\varepsilon}(\lambda) = -i{\varepsilon}h_0 \Delta_{\mathbb{S}}-ih_0 H_1 - \lambda h_0 - iQ$$ admits a right inverse on ${\mathcal{H}}^{-r} {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}H_{h_0}^{-r,s_0}$: there exists a bounded operator $P_{\varepsilon}(\lambda)^{-1} : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r}$ with range contained in the domain of $P_{\varepsilon}(\lambda)$ such that $P_{\varepsilon}(\lambda) P_{\varepsilon}(\lambda)^{-1} = {{\operatorname{Id}}}_{{\mathcal{H}}^{-r}}$. We show below that for every $r \in [\![0, r_{k_0}-r_k]\!]$, the operator $P_{\varepsilon}(\lambda)^{-1} : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-r_k}$ is $C^k([\![0,h_0))$. We proceed by induction on $k$.
We start with $k=1$. For every $r \in [\![0, r_{k_0}]\!] \cap [\![-2, r_{k_0}-2]\!] = [\![0,r_{k_0}-r_1]\!]$, the operator $P_{\varepsilon}(\lambda)^{-1}$ maps ${\mathcal{H}}^{-r}$ to itself and ${\mathcal{H}}^{-r-2}$ to itself. This fact, together with the identity $$\label{eq:0n}
P_{{\varepsilon}}(\lambda)^{-1} - P_{{\varepsilon}'}(\lambda)^{-1} = -i ({\varepsilon}-{\varepsilon}') P_{{\varepsilon}'}(\lambda)^{-1} h_0 \Delta_{\mathbb{S}}P_{{\varepsilon}}(\lambda)^{-1}$$ shows that ${\varepsilon}\in [0,h_0) \mapsto P_{{\varepsilon}}(\lambda)^{-1} : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-2}$ is differentiable (in particular continuous) with $$\label{eq:0l}
{{\partial}}_{\varepsilon}P_{\varepsilon}(\lambda) = -i P_{{\varepsilon}}(\lambda)^{-1} h_0 \Delta_{\mathbb{S}}P_{{\varepsilon}}(\lambda)^{-1}.$$ The right hand side of is continuous, hence ${\varepsilon}\in [0,h_0) \mapsto P_{{\varepsilon}}(\lambda)^{-1} : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-2}$ is $C^1([0,h_0))$.
Assume now that $k \leq {k_0}-1$ and that for every $r \in [\![0, r_{k_0}-r_k]\!]$, $P_{\varepsilon}(\lambda)^{-1} : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-r_k}$ is $C^k([\![0,h_0))$. The identity shows that ${{\partial}}_{\varepsilon}P_{\varepsilon}(\lambda) : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-2r_k-2}$ is also $C^k([0,h_0))$ as long as $r \in [\![0,r_{k_0}-r_k]\!] \cap [\![-r_k - 2, r_{k_0}-2r_k-2]\!]$. Since $r_{k+1} = 2r_k-2$, the operator ${{\partial}}_{\varepsilon}P_{\varepsilon}(\lambda) : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-r_{k+1}}$ is $C^k([\![0,h_0))$ as long as $r \in [0, r_{k_0}-r_{k+1}]\!]$. This implies that $P_{\varepsilon}(\lambda) : {\mathcal{H}}^{-r} \rightarrow {\mathcal{H}}^{-r-r_{k+1}}$ is $C^{k+1}([0,h_0))$ for the above range of $r$. This completes the induction process.
It follows that the operator $P_{\varepsilon}(\lambda)^{-1} : {\mathcal{H}}^0 \rightarrow {\mathcal{H}}^{-r_{k_0}}$ is $C^{k_0}([0,h_0))$. We recall that $Q$ is a smoothing operator. In particular, $Q$ maps ${\mathcal{H}}^{-r_{k_0}}$ to the Sobolev space $H^N$ for any $N$. It follows that $Q P_{\varepsilon}(\lambda)$ is a trace-class operator with holomorphic dependence in $\lambda \in {\mathbb{D}}(0,R)$ and $C^{k_0}$ dependence in ${\varepsilon}\in [0,h_0)$. Since ${k_0}$ was arbitrary, $Q P_{\varepsilon}(\lambda)$ is smooth at ${\varepsilon}= 0$. For ${\varepsilon}\in [0,h_0)$ and $\lambda \in {\mathbb{D}}(0,R)$, we define the Fredholm determinant $$D_{\varepsilon}(\lambda) = {{\text{Det}}}_{{\mathcal{H}}^0}({{\operatorname{Id}}}+ i Q P_{\varepsilon}(\lambda)^{-1}),$$ which depends holomorphically in $\lambda$, and which is smooth at ${\varepsilon}=0$.
The operator $h_0(P_{\varepsilon}- \lambda) = P_{\varepsilon}(\lambda) + iQ$ is Fredholm, because where $P_{\varepsilon}(\lambda)$ admits a right inverse on ${\mathcal{H}}^0$ and $Q$ is compact. Hence, the ${\mathcal{H}}^0$-spectrum of $P_{\varepsilon}$ in ${\mathbb{D}}(0,R)$ is discrete and equal to the zero set of $D_{\varepsilon}(\lambda)$. When ${\varepsilon}\neq 0$ the operator $P_{\varepsilon}$ is subelliptic. Consequently, ${\mathcal{H}}^0$-eigenvectors of $P_{\varepsilon}$ must belong to the (standard) Sobolev space $H^2$, thus to the domain of $P_{\varepsilon}$ on $L^2$. Conversely, $L^2$-eigenvectors of $P_{\varepsilon}$ must belong to the (standard) Sobolev space $H^{s_0}$, thus to ${\mathcal{H}}^0$. This shows that for ${\varepsilon}\neq 0$, the $L^2$-spectrum and ${\mathcal{H}}^0$-spectrum of $P_{\varepsilon}$ in ${\mathbb{D}}(0,R)$ are equal, and the $L^2$-eigenvalues of $P_{\varepsilon}$ in ${\mathbb{D}}(0,R)$ are exactly the zeroes of $D_{\varepsilon}(\lambda)$.
For ${\varepsilon}> 0$, $D_{\varepsilon}(\lambda)$ is a holomorphic function of $\lambda$ whose zero set is the $L^2$-spectrum of $P_{\varepsilon}$ in ${\mathbb{D}}(0,R)$, and the zero set of $D_0(\lambda)$ is the Pollicott–Ruelle spectrum of $P_0$ in ${\mathbb{D}}(0,R)$ – see [@DZ Proposition 3.2]. Since $D_{\varepsilon}(\lambda)$ is smooth at ${\varepsilon}= 0$, the first part of the theorem follows from an application of Hurwitz’s theorem.
If $\lambda_0$ is a Pollicott–Ruelle resonance of $P_0$ and $\lambda_1({\varepsilon})$ is the unique eigenvalue of $P_{\varepsilon}$ converging to $\lambda_0$, the implicit function theorem shows that ${\varepsilon}\mapsto \lambda_1({\varepsilon})$ is smooth. We compute now the leading terms in the expansion , inspired by the method of [@D §3.1]. Denote by ${\mathrm{Res}}(P_0)$ the set of Pollicott–Ruelle resonances of $P_0 = \frac{1}{i}H_1$ and fix $K$ be a compact subset of ${\mathbb{D}}(0,R) \setminus {\mathrm{Res}}(P_0)$. For every $\lambda \in K$, $D_0(\lambda) \neq 0$ and the operator ${{\operatorname{Id}}}+ iQ P_0(\lambda)^{-1} : {\mathcal{H}}^0 \rightarrow {\mathcal{H}}^0$ is invertible. Therefore, for every $0 < {\varepsilon}\leq h_0$ and $\lambda \in K$, $${{\operatorname{Id}}}+ iQ P_{\varepsilon}(\lambda)^{-1} = \left({{\operatorname{Id}}}+ iQP_0(\lambda)^{-1}\right) \cdot \left( {{\operatorname{Id}}}+ \left( {{\operatorname{Id}}}+ iQP_0(\lambda)^{-1}\right)^{-1} iQ \left(P_{\varepsilon}(\lambda)^{-1} - P_0(\lambda)^{-1}\right) \right).$$ Uniformly for $\lambda \in K$, the operator $\left( {{\operatorname{Id}}}+ iQP_0(\lambda)^{-1} \right)^{-1}$ is bounded on ${\mathcal{H}}^0$ and by , $Q \left(P_{\varepsilon}(\lambda)^{-1} - P_0(\lambda)^{-1} \right)$ has trace-class norm $O({\varepsilon})$. The identity implies for $\lambda \in K$,
D\_() = D\_0() \_[\^0]{} ( + i( + iQP\_0()\^[-1]{})\^[-1]{} Q (P\_()\^[-1]{} - P\_0()\^[-1]{}) )\
= D\_0() ( 1 + h\_0 \_[\^0]{}(( + iQP\_0()\^[-1]{})\^[-1]{} Q P\_0()\^[-1]{} \_P\_()\^[-1]{} ) + O(\^2) ).
The operator $Q P_0(\lambda)^{-1} \Delta_{\mathbb{S}}$ extends to a trace-class operator in ${\mathcal{H}}^0$. Because of the identity , we have uniformly for $\lambda \in K$,
h\_0 \_[\^0]{}(( + iQP\_0()\^[-1]{})\^[-1]{} Q P\_0()\^[-1]{} \_P\_()\^[-1]{} ) = f\_1() + O(),\
f\_1() h\_0 \_[\^0]{}(( + iQP\_0()\^[-1]{})\^[-1]{} Q P\_0()\^[-1]{} \_P\_0()\^[-1]{} ).
It follows that uniformly for $\lambda \in K$, $$\label{eq:0i}
D_{\varepsilon}(\lambda) = D_0(\lambda) \cdot \left( 1 + f_1(\lambda) {\varepsilon}+ O({\varepsilon}^2) \right).$$
In , the function $D_{\varepsilon}$ is holomorphic on ${\mathbb{D}}(0,R)$ and $f_1(\lambda)$ is meromorphic in ${\mathbb{D}}(0,R)$, with poles in ${\mathbb{D}}(0,R) \cap {\mathrm{Res}}(P_0)$. Therefore we can apply [@D Lemma 4.4] with $E = {\mathbb{D}}(0,R)$, $S_0 = {\mathrm{Res}}(P_0)$, $D_{\varepsilon}(\lambda)/D_0(\lambda) = 1 + f_1(\lambda) {\varepsilon}+ O({\varepsilon}^2)$ and $g(\lambda,{\varepsilon}) = D_0(\lambda)$ (strictly speaking, [@D Lemma 4.4] is stated there with $E = {\mathbb{C}}$ or ${\mathbb{C}}\setminus 0$; but it also holds without change in the proof when $E = {\mathbb{D}}(0,R)$). It shows that is valid uniformly for $\lambda \in {\mathbb{D}}(0,R) \setminus {\mathrm{Res}}(P_0)$ and that the function $D_0(\lambda) f_1(\lambda)$ is holomorphic on ${\mathbb{D}}(0,R)$.
Let $\lambda_0 \in {\mathbb{D}}(0,R)$ be a simple resonance of ${\mathrm{Res}}(P_0)$. We now work with $f_1(\lambda)$ for $\lambda$ in a small punctured disk ${{\mathcal{D}}}\setminus \lambda_0 \subset {\mathbb{D}}(0,R)$, so that $\lambda_0$ is the only resonance of $P_0$ in ${{\mathcal{D}}}$. We have
f\_1() = h\_0 \_[\^0]{}(( + iQP\_0()\^[-1]{})\^[-1]{} Q P\_0()\^[-1]{} \_P\_0()\^[-1]{} )\
= h\_0 \_[\^0]{}(P\_0()\^[-1]{} ( + iQP\_0()\^[-1]{})\^[-1]{} Q P\_0()\^[-1]{} \_) = \_[\^0]{}( (P\_0-)\^[-1]{} Q P\_0()\^[-1]{} \_).
In the above we used the cyclicity of the trace and the identity $$P_0(\lambda)^{-1} \left( {{\operatorname{Id}}}+ iQP_0(\lambda)^{-1} \right)^{-1} = (P_0(\lambda) + iQ)^{-1} = h_0^{-1} (P_0 - \lambda )^{-1}.$$
Because of and since $P_0(\lambda)^{-1}$ is holomorphic near $\lambda_0$, we can write $$\label{eq:0ya}
(P_0-\lambda)^{-1} Q P_0(\lambda)^{-1} \Delta_{\mathbb{S}}= (i(P_0-\lambda)^{-1} - h_0 P_0(\lambda)^{-1}) \Delta_{\mathbb{S}}= \dfrac{i u \otimes v \Delta_{\mathbb{S}}}{\lambda-\lambda_0} + B(\lambda),$$ where $B(\lambda)$ denotes a holomorphic family of operators near $\lambda_0$. The right hand side of is trace-class on ${\mathcal{H}}^0$ and the operator $u \otimes v \Delta_{\mathbb{S}}$ is of rank $1$. Therefore $B(\lambda)$ is trace-class on ${\mathcal{H}}^0$ and $f_0(\lambda) {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}{{\operatorname{Tr}}}_{{\mathcal{H}}^0}(B(\lambda))$ is holomorphic. It follows that $$f_1(\lambda)-f_0(\lambda) = \dfrac{i{{\operatorname{Tr}}}_{{\mathcal{H}}^0}\left(u \otimes v \Delta_{\mathbb{S}}\right)}{\lambda-\lambda_0} = \dfrac{i {{\operatorname{Tr}}}_{{\mathcal{H}}^0}\left(\Delta_{\mathbb{S}}u \otimes v \right)}{\lambda-\lambda_0} = \dfrac{i}{\lambda_0-\lambda}\int_{S^*{\mathbb{M}}} {\langle \nabla_{\mathbb{S}}u, \nabla_{\mathbb{S}}v \rangle}.$$ In the last equality we used that $\Delta_{\mathbb{S}}u$ and $v$ have wavefront sets contained in $E_u^*$ and $E_s^*$, respectively. Hence the trace of the operator $\Delta_{\mathbb{S}}u \otimes v$ is given by integrating the kernel $\Delta_{\mathbb{S}}u(x) v(y)$ along the diagonal $\{x=y\}$ according to [@GS Proposition 7.6]. The operator $\nabla_{\mathbb{S}}$ was defined in §\[sec:2.1\] and the scalar product ${\langle \cdot, \cdot \rangle}$ is inherited from the Euclidean structure on the fibers of $T^*{\mathbb{M}}$.
Combining the above, we obtain that uniformly in ${\varepsilon}$ small enough and $\lambda \in {{\mathcal{D}}}\setminus \lambda_0$,
D\_() = D\_0() - i \_[S\^\*]{} [\_u, \_v ]{} + D\_0() f\_0() + O(\^2)\
= D’\_0(\_0) ( -\_0 - i \_[S\^\*]{} [\_u, \_v ]{} + O((-\_0)) + O(\^2) ).
Recall that $\lambda_1({\varepsilon})$ is the unique eigenvalue of $P_{\varepsilon}$ near $\lambda_0$. In particular $D_{\varepsilon}(\lambda_1({\varepsilon})) = 0$. Since ${\varepsilon}\mapsto \lambda_1({\varepsilon})$ is smooth, $\lambda_1({\varepsilon}) = \lambda_0 + O({\varepsilon})$. This yields $$\lambda_1({\varepsilon}) = \lambda_0 + i {\varepsilon}\int_{S^*{\mathbb{M}}} {\langle \nabla_{\mathbb{S}}u, \nabla_{\mathbb{S}}v \rangle} + O({\varepsilon}^2).$$ This concludes the proof of $(i)$.
For $(ii)$, we fix $k_0 > 0$ and we recall that $P_{\varepsilon}(\lambda)^{-1} : {\mathcal{H}}^0 \rightarrow {\mathcal{H}}^{-r_{k_0}}$ is $C^{k_0}([0,h_0))$. Since $h_0(P_{\varepsilon}-\lambda) = P_{\varepsilon}(\lambda)+Q$, where $Q$ is smoothing, the family $P_{\varepsilon}-\lambda : {\mathcal{H}}^{-r_{k_0}} \rightarrow {\mathcal{H}}^0$ is Fredholm with $C^{k_0}$ dependence in ${\varepsilon}$. Hence, $(P_{\varepsilon}-\lambda)^{-1}$ is a meromorphic family of operators with poles of finite rank, with $C^{k_0}$ dependence in ${\varepsilon}$. This shows that the family of operators ${\varepsilon}\rightarrow \Pi_{\varepsilon}: {\mathcal{H}}^0 \rightarrow {\mathcal{H}}^{-r_{k_0}}$ given by is $C^{k_0}([0,h_0))$. A fortiori, ${\varepsilon}\mapsto \Pi_{\varepsilon}: C^\infty(S^*{\mathbb{M}}) \rightarrow {{\mathcal{D}}}'(S^*{\mathbb{M}})$ is also $C^{k_0}([0,h_0))$, hence smooth at ${\varepsilon}= 0$.
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abstract: 'Intermolecular correlations in liquid water at ambient conditions have generally been characterized through short range density fluctuations described through the atomic pair distribution functions (PDF). Recent numerical and experimental results have suggested that such a description of order or structure in liquid water is incomplete and there exists considerably longer ranged orientational correlations in water that can be studied through dipolar correlations. In this study, using large scale classical, atomistic molecular dynamics (MD) simulations using TIP4P-Ew and TIP3P models of water, we show that salts such as sodium chloride (NaCl), potassium chloride (KCl), caesium chloride (CsCl) and magnesium chloride (MgCl$_2$) have a long range effect on the dipolar correlations, which can not be explained by the notion of structure making and breaking by dissolved ions. The relative effects of cations on dipolar correlations are observed to be consistent with the well-known Hofmeister series. Observed effects are explained through orientational stratification of water molecules around ions, and their long range coupling to the global hydrogen bond network by virtue of the sum rule for water. The observations for single hydrophilic solutes are contrasted with the same for a single methane (CH$_4$) molecule. We observe that even a single small hydrophobe can result in enhancement of long range orientational correlations in liquid water,- contrary to the case of dissolved ions, which have been observed to have a reducing effect. The observations from this study are discussed in the context of hydrophobic effect.'
author:
- Upayan Baul
- 'J. Maruthi Pradeep Kanth'
- Ramesh Anishetty
- Satyavani Vemparala
title: Effect of simple solutes on the long range dipolar correlations in liquid water
---
Introduction
============
Liquid water and aqueous solutions, ubiquitous in nature, provide the native environment for numerous processes taking place in nature. Detailed understanding of structural and dynamical behavior of water and the effect of simple salts (or ion combinations) on the same are thus critical to the understanding of further complex and interesting processes of biological [@kunz04; @nostro; @ball15] and environmental [@buszek; @jung06] relevance. Many of the unique properties of water can be attributed to the ability of water molecules to form an extensively connected network of hydrogen bonds in which the four nearest neighbors of a water molecule arrange themselves in a nearly tetrahedral geometry [@eisen69; @ziel05; @stan00; @kumar09]. This local structuring, in addition to the generally observed short range structure in liquids, owing to the excluded volume effects, has led to consensus in treating water as a highly structured liquid [@marcus09chemrev; @str1; @str2; @str3; @str4; @str5]. However, the precise definition of microscopic structure in water is not without ambiguity [@marcus09chemrev; @ball15].
The structure of liquid water has been extensively probed using pair distribution functions (PDFs) among its atomic constituents using experimental as well as numerical techniques both in its bulk liquid phase and in presence of ionic impurities [@gordon02; @marcus09chemrev]. Observed PDFs have suggested [@t4pew; @5p; @t3p; @gr1; @kanth10] that at ambient conditions, the structure of liquid water is short range and the spatial distribution of water molecules are uncorrelated beyond a distance of $\sim(8-10)\mathring{A}$. However, PDFs yield a measure for density fluctuations alone and are devoid of information on orientational correlation, especially at longer separations. The mutual orientations of water molecules separated in space can, however, be envisaged through correlations involving dipolar and higher order multipole moments of the constituent molecules [@mat04; @kanth10; @dipzhang14; @elton14]. Interestingly, while quadrupolar fluctuations have been observed to disappear within a distance of $\sim3\mathring{A}$, correlations involving the dipolar degree of freedom of water molecules have been observed to be considerably longer ranged [@kanth10]. These dipole - dipole correlations were decomposed in the same report into transverse or the trace part, involving the alignment of the dipoles with respect to themselves and longitudinal or the traceless part, measuring the alignment of the dipoles with respect to the radial vector separating them [@kanth10]. Both of these correlations show oscillatory solvation structure and are longer ranged compared to density correlations in water. The former vanishes beyond $\sim14\mathring{A}$ in liquid water in compliance with rotational symmetry. The latter, oscillatory in nature and always positive, was observed [@kanth10] to be non-vanishing even at $\sim75\mathring{A}$ separations and decays exponentially beyond solvation region ($14\mathring{A}$) with largest correlation length of $\sim24\mathring{A}$. A non-vanishing alignment of dipole vectors with respect to each other over separations comparable to system dimensions would indicate the existence of spontaneous polarization in the medium. These results have been observed to be in good agreement with further numerical observations of long range angular correlations in liquid water [@angliu13; @dipzhang14]. Another dipolar correlation of interest for bulk liquid water is the oxygen - dipole correlation which represents the propensity for alignment of a dipole vector to the radially outward direction when observed from the position of another water molecule. This correlation also shows oscillatory solvation structure at radial separations below $14\mathring{A}$ and vanishes beyond the same [@kanth10].
The experimental observations of such long range dipolar or angular correlation are not accessible to conventional spectroscopic techniques owing to the lack of positional ordering at longer length scales. However, recent hyper-Rayleigh light scattering (HRS) experiments [@hrs1; @hrs2; @hrs3; @hrs5] have revealed the existence of long range orientational correlations longer than 230$\mathring{A}$ in liquid water [@angliu13]. Dipolar correlations in liquid water have been studied for a variety of non-polarizable as well as polarizable models of water, including (TIP5P, TIP3P) [@kanth10] and (SPC/E, swm4-DP) [@dipzhang14]. Within the limitations of system sizes studied, the results for the various water models have been consistent. With the SPC/E and swm4-DP water models, the longitudinal part of the correlations was qualitatively observed to be in agreement with the results for TIP5P and TIP3P water models [@dipzhang14; @kanth10] for radial separations $\leq12\mathring{A}$. The longer correlation length of $\sim24\mathring{A}$, however, requires considerably larger system sizes than reported [@dipzhang14]. In a recent work [@hrs5], correlation functions computed from HRS measurements were directly compared with the longitudinal and transverse correlations from MD simulations [@dipzhang14]. While the experimentally observed correlations did not reproduce the oscillatory solvation structure, the asymptotic behavior was observed to be in good qualitative agreement. The long range correlations from HRS experiments were stated to have $r^{-3}$ dependence, which indicative of the importance of dipole - dipole interactions.
It is important to note that the long range behavior of the longitudinal correlation is not a consequence of long range electrostatic interactions, but has its origin in the fluctuations of the underlying hydrogen bond network of water. The same was established in our previous work, where truncating the electrostatic interactions beyond $12\mathring{A}$ was observed to have no effect on the asymptotic behavior of the longitudinal correlation [@kanth10]. While hydrogen bonding interactions are intrinsically short range, the fluctuations of the hydrogen bond network are restricted by topological constraints. The entropy of hydrogen bond network thus has contributions from both single-water and collective dynamics [@netz]. In a statistical description of liquid water, the hydrogen bond fluctuations need to be consistent with a sum rule, which states that sum of density of dangling bonds (i.e., a water molecules bond arms which are not hydrogen bonded to any other molecule) and twice the density of hydrogen bonds should be equal to four times the density of water molecules [@kanth12]. The dynamics of the hydrogen bond network of water, governed by orientational fluctuations of water molecules, thus results in orientational correlations that are substantially long range compared to the length scale ($\sim 2 \mathring{A}$) of hydrogen bonding interactions [@kanth10; @kanth12; @kanth13].
Temperature and density have been observed to have small but monotonic effects on the transverse and longitudinal dipolar correlations [@kanth10; @dipzhang14]. Presence of salt (CaCl$_2$), however, was observed to have a considerably larger effect at reported [@dipzhang14] length scales of $\leq15\mathring{A}$. The system sizes reported therein, with $\sim432$ water molecules for CaCl$_2$ - water systems, are however, unlikely to capture the quantitative aspects of the effects of salts on the long range correlations.
Over decades, effort has been made to characterize the effect of salts on the structure of water based on the ability of the dissolved ions to enhance or reduce the same [@marcus09chemrev]. Consequently, ions have often been characterized as *structure makers* and *structure breakers*. This putative notion has been recently subjected to considerable criticism, primarily owing to the absence of an unambiguous definition of structure in water [@ball15; @makbr1; @makbr2].
In the current study, we investigate the effect of multiple salts (CsCl, KCl, NaCl and MgCl$_2$) on the dipolar correlations for large system sizes using TIP4P-Ew and TIP3P water models. Our results clearly demonstrate that structure making (and breaking) is not a generic concept, but crucially depends on the particular correlation in water. Without invoking a definition of structure, we discuss the consistency of our observations in a general framework motivated by the sum rule and associated orientational entropy. For the same, we invoke electrostatics driven orientational stratification of water molecules situated close to the ions. To quantify the same, we define a solute - dipole correlation which is a trivial extension of the oxygen - dipole correlation. Finally, as a first extension to the case of non - ionic solutes, we contrast the effects observed for single salt molecules dissolved in water with the same for a methane molecule and show that hydrophilic and hydrophobic impurities of comparable sizes differ characteristically in their effects on the long range orientational correlations in water. We discuss the implications of this observation in the context of hydrophobic effect.
The rest of the article is arranged as follows: in section II, we define the dipolar correlations used in the current study (elaborated in reference [@kanth10]) followed by a description of systems studied and simulation details. The results obtained are discussed in section III, followed by discussions and future directions in section IV.
Methods
=======
Dipolar correlations
--------------------
### dipole - dipole correlations
Let $\hat{\mu}$ denote the normalized dipole vector of a water molecule. Statistical correlations among any two water molecules, whose oxygen atoms are situated at $\bf{r_1}$ and $\bf{r_2}$, involving the dipolar degree of freedom can be formulated [@kanth10] as $$\langle \hat{\mu}^i (\mathbf{r_1}) \hat{\mu}^j (\mathbf{r_2}) \rangle = \frac{1}{2} \left( \delta^{ij} - \frac{r^i r^j}{r^2} \right)t(r) - \frac{1}{2} \left( \delta^{ij} - 3\frac{r^i r^j}{r^2} \right)l(r)$$ where indices $i,j$ denote directions in three-dimensional space, $\mathbf{r}=(\mathbf{r_1} - \mathbf{r_2})$, $r=|\mathbf{r}|$ and the angular brackets denote ensemble averages. The decomposed scalar functions $t(r)$ and $l(r)$, defined as $$\begin{aligned}
t(r) & \,=\, & \langle \hat{\mu}(\mathbf{r_1}) . \hat{\mu}(\mathbf{r_2}) \rangle \\
l(r) & \,=\, & \langle \hat{\mu}(\mathbf{r_1}) . \hat{\mathbf{r}} \, \hat{\mu}(\mathbf{r_2}) . \hat{\mathbf{r}} \rangle\end{aligned}$$ describe the transverse (trace) part and the longitudinal (traceless) part of the tensorial correlations $\langle \hat{\mu}^i (\mathbf{r_1}) \hat{\mu}^j (\mathbf{r_2}) \rangle$. Physically, $t(r)$ and $l(r)$ are measures for the statistical alignment of water dipole vectors spaced $r$ distance apart with respect to themselves and with respect to the radial vector separating them respectively.
### position - dipole correlations
The oxygen - dipole correlation ($d(r)$) is defined as [@kanth10] $$d(r)\,=\,\langle \hat{\mu}(\mathbf{r_1}) \ldotp \mathbf{\hat{r}} \rho (\mathbf{r_2})\rangle$$ where $\rho(\mathbf{r})$ takes a value of 1 or 0 depending on presence or absence of oxygen atom at $\mathbf{r}$. Similarly, the solute - dipole correlation can be defined through the same expression, where the density field for oxygen atoms ($\rho$(${\mathbf{r}}$)) is replaced by the same for a solute species. The oxygen - dipole and solute - dipole correlations represent the propensity for alignment of the dipole vector of a water molecule to the radially outward direction, when observed from the position of another water molecule or a solute respectively.
System setup
------------
Classical atomistic molecular dynamics simulations were performed using simulation package NAMD 2.9 [@namd] using TIP4P-Ew [@t4pew] and TIP3P [@t3p] water models. Recently developed parameters that correctly reproduce the right coordination number were used for simulating divalent Mg$^{2+}$ ions with TIP4P-Ew water model [@divt4p]. Rest of the monovalent ion parameters for simulating TIP4P-Ew systems were taken from extensively used halide and alkali ion parameters available in literature [@monot4p]. For simulating aqueous solutions using TIP3P water model, standard CHARMM parameters [@iont3p], used extensively in biomolecular systems, were used for all ions. Methane parameters were derived from aliphatic *sp3* carbon and nonpolar hydrogen parameters in CHARMM [@charmmProt].
Given the longest correlation length is $\sim24\mathring{A}$ for $l(r)$ dipolar correlations [@kanth10], cubic box lengths of $\gtrsim50\mathring{A}$ need to be simulated for quantitative estimates of the same. As to enable the identification of any enhancement of the same, or the appearances of even longer correlation lengths, if any, considerably larger cubic boxes of dimensions $\approx 100\mathring{A}\times100\mathring{A}\times100\mathring{A}$ were simulated in the investigations of $t(r)$ and $l(r)$ correlations. Solute free water systems comprising of 36054 water molecules for TIP4P-Ew and 34194 water molecules for TIP3P were initially equilibrated for 5ns under isothermal, isobaric (NPT) conditions under a pressure of 1atm. The temperature was maintained at 298K for TIP4P-Ew and at 305K for TIP3P. The system configurations at the end of the NPT equilibration were used to generate 0.15M and 1.0M aqueous solutions of CsCl, NaCl, KCl and MgCl$_2$ using *solvate* plugin of VMD [@vmd]. The salt-water systems were further equilibrated for 5ns under NPT conditions (pressure 1atm; temperature 298K and 305K for TIP4P-Ew and TIP3P respectively). All systems, including solute free water systems were equilibrated for further 5ns under NVE conditions resulting in total equilibration time of 10ns for each system. Following equilibration, production runs for all systems were carried out for additional 10ns, over which the system configurations were written every 4ps. The analyses for each system were performed on the 2500 uncorrelated system configurations thus generated.
For comparative study of effects of single hydrophilic versus hydrophobic solutes, and the analyses of $d(r)$ correlations, a pre-equilibrated water box of dimensions $\approx 50\mathring{A}\times50\mathring{A}\times50\mathring{A}$ was extracted from the 10ns equilibrated configuration of the $\approx 100\mathring{A}\times100\mathring{A}\times100\mathring{A}$ water box for TIP4P-Ew water model. Single MgCl$_2$, NaCl and CH$_4$ molecules were added to the same to construct extremely dilute (1 solute molecule in $\sim$4100 water molecules) solutions. All four systems were further equilibrated for 2ns under NPT conditions (pressure 1atm; temperature 298K) followed by 3ns under NVE. Production runs for 10ns were then carried out for each of the three systems under NVE conditions, generating 5000 configurations for analyses. Ion pairing effects were not observed for the salt solutions studied here.
For the NPT equilibration stages, constant pressure and temperature were maintained using Langevin Piston [@pres] and temperature coupling to external reservoir respectively. Timesteps of 1fs and 2fs were used for simulating systems with TIP4P-Ew and TIP3P water model respectively. For all simulations, long range electrostatic interactions were computed using particle mesh Ewald (PME) and Lennard-Jones interactions were smoothly truncated beyond $12\mathring{A}$ through the use of switching functions between $10\mathring{A}$ and $12\mathring{A}$.
Results
=======
In the following subsections we report the observations of the effects of studied solutes on the dipolar correlations $t(r)$, $l(r)$ and $d(r)$. All the results in the following subsections are for TIP4P-Ew water model and the corresponding results for TIP3P water model are given in *Supp. Info* [@supplement]. In subsections A and B we describe the effects of salts on the $t(r)$ and $l(r)$ correlations at salt concentrations of 0.15M and 1.0M. In subsection C we describe $d(r)$ correlation, both in absence of salts and around cationic species. We conclude by comparing results for the presence of single molecules of NaCl, MgCl$_2$ and CH$_4$ in subsection D.
Effect of salts on transverse correlations
------------------------------------------
Computed $t(r)$ correlation functions for systems with TIP4P-Ew water model are shown in FIG. 1 (see FIG. SI of *Supp. Info* [@supplement] for similar plots with TIP3P water model). For clarity of comparison, the plots have been shown for $r \geq 5\mathring{A}$. Plots for the full range of $r$ are shown in FIG. SIV of *Supp. Info* [@supplement]. In good agreement with prior results for solute free liquid water [@kanth10; @dipzhang14], $t(r)$ shows oscillatory solvation structure over radial separations below $14\mathring{A}$ and vanishes beyond the same. All salts at low concentration (0.15M) enhance $t(r)$ in comparison with solute free water at both smaller ($\leq10\mathring{A}$) and larger ($\geq10\mathring{A}$) separations (FIG. 1(a,b)). At smaller separations, the enhancement is observed to be most pronounced in the presence of MgCl$_2$. The other salts studied (NaCl, KCl and CsCl) are observed to induce smaller, roughly equal enhancements in $t(r)$. Beyond $10\mathring{A}$, however, the effects are reversed with NaCl, KCl and CsCl resulting in a greater enhancement in $t(r)$ over MgCl$_2$. Further, all salts at 0.15M concentration can clearly be observed to lead to an enhancement in the range of $t(r)$, being non-vanishing till $\sim 18 \mathring{A}$ for MgCl$_2$ and $\sim 24 \mathring{A}$ for the rest. No appreciable shift in peak positions is observed at 0.15M concentration for all salts.
At 1.0M concentration , the effect of MgCl$_2$ can be observed to be substantially different from that of the other salts, as can be seen in FIG. 1(c). While NaCl, KCl and CsCl still result in an enhancement of $t(r)$ over that for solute free water for smaller separations, MgCl$_2$ leads to a reduction of the same beyond the first solvation shell. The effect is most prominently observable beyond the second solvation peak, where presence of MgCl$_2$ results in an anticorrelation in the transverse part of dipolar correlations. Effects of NaCl, KCl and CsCl are also distinguishable at the higher concentration, with CsCl resulting in a greater enhancement in peak heights for $t(r)$ over NaCl and KCl. The positions of the second and third solvation peaks are visibly shifted to left for all salts, the shift being maximum for MgCl$_2$. Further, the appearance of a quasi long range nature in $t(r)$ at the lower concentration (0.15M) is washed off at the higher (1.0M) (see FIG. 1(b,d)). The concentration dependent enhancing / reducing effect observed for MgCl$_2$ is in good agreement with reported results [@dipzhang14] for CaCl$_2$ which was also observed to affect enhancement in $t(r)$ at 0.25M concentration followed by similar anticorrelations at the concentration of 1.56M. The trait thus appears to be a general characteristics for salts with strongly solvated cations capable of inducing strong perturbations in the orientation of water molecules.
Effect of salts on longitudinal correlations
--------------------------------------------
Longitudinal correlations $l(r)$ are of greater interest over $t(r)$ in the study of long range dipolar correlations in water since $l(r)$ can be described as a truly long range correlation with an exponential decay [@kanth10]. In solute free liquid water at ambient conditions, $l(r)$ has been shown [@kanth10] to exhibit solvation peaks till $14 \mathring{A}$, beyond which it decays exponentially with longest correlation length of $\sim 24 \mathring{A}$ and is non-vanishing even at $75 \mathring{A}$. It is always positive in solute free water and decays in an oscillatory manner. The characteristics of this function are closely shared by another long range angular correlation function computable using classical density functional theory calculations [@angliu13], further strengthening the likely existence of long range orientationally correlated domains of water molecules. It has been suggested from HRS observations that long range orientational correlations in molecular liquids are expressed through propagating waves in molecular reorientation instead of diffusional orientation of molecules [@hrs4]. $l(r)$ as well as the stated angular correlation function [@angliu13] are thus in qualitative agreement with HRS results. Recently, correlations from HRS measurements have been directly compared to $l(r)$ and a linear combination of it with $t(r)$, computed using MD simulations [@hrs5]. Results have been in qualitative agreement with similar asymptotic behavior. However, the oscillatory nature of the correlations computed using molecular simulations have not been reflected in the HRS results. A direct quantitative comparison with the experimental results is still not possible since (a) system dimensions studied using HRS are well beyond the scope of atomistic simulations [@hrs3] and (b) HRS does not provide explicit information of microscopic correlations among molecular dipoles [@angliu13].
[@|Y|Y|Y|Y|Y|Y|@]{} water model & salt (concentration) & $a_1$ & $r_1$ & $a_2$ & $r_2$\
& none & 0.31 ($\pm$ 0.01) & 4.82 ($\pm$ 0.08) & 0.0207 ($\pm$ 0.0008) & 24.0 ($\pm$ 0.6)\
& CsCl (0.15M) & 0.321 ($\pm$ 0.004) & 4.93 ($\pm$ 0.02) & &\
& KCl (O.15 M) & 0.34 ($\pm$ 0.01) & 4.82 ($\pm$ 0.03) & &\
& NaCl (O.15 M) & 0.34 ($\pm$ 0.01) & 4.77 ($\pm$ 0.03) & &\
& MgCl$_2$ (O.15 M) & 0.74 ($\pm$ 0.04) & 2.94 ($\pm$ 0.04) & &\
& none & 0.34 ($\pm$ 0.01) & 5.24 ($\pm$ 0.12) & 0.026 ($\pm$ 0.001) & 24.8 ($\pm$ 0.9)\
& CsCl (0.15M) & 0.344 ($\pm$ 0.006) & 5.95 ($\pm$ 0.04) & &\
& KCl (O.15 M) & 0.359 ($\pm$ 0.006) & 5.93 ($\pm$ 0.04) & &\
& NaCl (O.15 M) & 0.323 ($\pm$ 0.005) & 6.10 ($\pm$ 0.04) & &\
& MgCl$_2$ (O.15 M) & 0.82 ($\pm$ 0.04) & 3.01 ($\pm$ 0.03) & &\
Computed $l(r)$ correlations for systems with TIP4P-Ew water model are shown in FIG. 2 and the same for systems with TIP3P model of water are included in the *Supp. Info* [@supplement] (FIG. SII). As with $t(r)$, the plots in the article have been shown for $r \geq 5\mathring{A}$. Plots for the full range of $r$ are shown in FIG. SIV of *Supp. Info* [@supplement]. Given the system sizes ($\sim 100 \times 100 \times 100 \mathring{A}^3$) studied in the current work, the correlations have been computed upto $48 \mathring{A}$ separations. The results for solute free water has been observed to be consistent with prior results [@kanth10; @dipzhang14]. All salts have been observed to induce reduction in both the strength and range of $l(r)$ correlations at both the concentrations studied. The reduction is more prominent at the higher salt concentration (1.0M) studied. At 0.15M salt concentrations, the $l(r)$ correlations for salt - water systems retain the essential solvation structures of solute free $l(r)$, such as the positions of solvation peaks and the absence of anticorrelation. However, the long range the decay of the correlations, which are bi-exponential for solute free water, are severely effected even at 0.15M salt concentration (FIG. 2(a,b)). The longest correlation length $\sim 24 \mathring{A}$ for solute free water is not observed for any of the salts, even at 0.15M concentrations. The correlation lengths have been obtained by fitting $l(r)$ for $r > 12\mathring{A}$ to a bi-exponential or a mono-exponential ($a_2 \,=\,0$, for salt - water systems) function of Ornstein-Zernike kind [@OZmarch]. $$l(r) = \frac{a_{1}}{r}\exp(-r/r_{1}) + \frac{a_{2}}{r}\exp(-r/r_{2})$$ The fit results are shown in TABLE I for both TIP4P-Ew and TIP3P water models. Within the limitations of model dependence, the results from both water models are in good agreement. The comparable effects of NaCl, KCl and CsCl are reflected in the correlation lengths too.
While the correlation lengths of $l(r)$, computed for individual salt - water systems, can not distinguish the relative effects of NaCl, KCl and CsCl, their relative influences are discernible even at 0.15M concentrations through their solvation peaks (as shown in the inset of FIG. 2(a) for the third solvation peak). Owing to the common choice of anion (Cl$^-$), the cations studied can be ordered based on their relative effect on the longitudinal correlation $l(r)$. The order, in terms of increasing effect on $l(r)$ is observed to be Cs$^+ <$K$^+ <$Na$^+ <$Mg$^{2+}$ which is consistent with the ordering of the same ions in the cationic Hofmeister series [@tiel]. The same trend is also observed to be retained at 1.0M salt concentrations.
At the higher concentration of 1.0M, salts are observed to severely reduce the $l(r)$ correlations. The absence of anticorrelations in $l(r)$ is observably lost for MgCl$_2$ and NaCl as seen from FIG. 2(c). Closer scrutiny of curves for KCl and CsCl also indicate anticorrelated regions. The trends ensure that at higher concentrations, all studied salts would result in negatively correlated regions in $l(r)$ profile. Salts studied have not been observed to have any appreciable effect on the positions of observable solvation peaks in $l(r)$, except for MgCl$_2$, which is observed to induce small shift in peak positions to smaller separations. Most notably, however, the long range nature of $l(r)$ visibly disappears within 1.0M concentrations for all salts studied (FIG. 2(d)), fit results in TABLE I show that the long range component vanishes even at 0.15M). The long range behavior of $l(r)$ correlations in liquid water was shown to be predominantly governed by local fluctuations in the underlying hydrogen bond network of water [@kanth10]. Observed effects of salts on $l(r)$ thus clearly indicate that the salts studied induce perturbations in the hydrogen bond network of water,- especially MgCl$_2$, for which the magnitude of effect indicates at highly non-local effects owing to the presence of strongly solvated Mg$^{2+}$ ions.
Effect of salts on position - dipolar orientational correlations
----------------------------------------------------------------
The position - dipole correlations have been computed using TIP4P-Ew water model alone. The oxygen - dipole correlation $d(r)$ for solute free liquid water is observed to display oscillatory solvation structure at small radial separations below $14\mathring{A}$ and vanish beyond the same (FIG. 3 (a)). The result is in good agreement with prior observations of the same correlation [@kanth10]. Dissolved ions have strong electrostatic interactions with water molecules in their first few solvation shells. The solute - dipole correlations for ionic species are thus expected to deviate widely from the oxygen - dipole correlations in solute free water. Further, the nature of the correlations is expected to vary strongly between cationic and anionic species owing to their preferential interactions with the oxygen and hydrogen atoms of a water molecule respectively. The charge density of the ions can also be envisaged to play a critical role in the patterning of water molecules around ions. Thus, the solute - dipole correlations for ions are likely to be highly ion specific, and the same can have important consequences on the connectivity, as well as the dynamics of the hydrogen bond network both within and beyond the first few solvation shells. Rich literature exists on ion specific effects on the structure and dynamics of water molecules, encompassing both local and non-local effects [@makbr2; @tiel; @tiel2; @obr1; @obr2; @UBpre; @yang; @iru]. In the following, we investigate the effects of the presence of a single divalent (Mg$^{2+}$) or monovalent (Na$^+$) cation (with requisite Cl$^-$ anion(s) for charge neutrality) on the orientational behavior of water around them and compare the same with solute free water $d(r)$ correlations. To retain similar statistics for solute free water and salt - water systems, the $d(r)$ calculation for solute free water was carried out by considering a single *tagged* water molecule at the center of the simulation box (since the same for salt - water systems can be carried out only over a single cation per frame). The results of the analyses are shown in FIG. 3(a,b). $d(r)$ for solute free water with TIP4P-Ew water model is observed to reproduce all qualitative trends of the same, previously reported with TIP5P water model [@kanth10].
As can be clearly seen from FIG. 3 (b), the presence of a cation has a significant effect on the orientation of water dipoles, causing them to align along the radial vector. The effect is strongest for the first solvation shell waters and gradually falls off with distance, with solvation peak structure. The correlation is long range, being non-vanishing at $>14\mathring{A}$, at least when single salt molecules are present in a box of water. Interestingly, solute - dipole $d(r)$ for ionic solutes falls off faster than the charge oscillations within spheres of increasing radii around ions (FIG. SIII of *Supp. Info* [@supplement]), indicating that the neighboring water molecules (hence hydrogen bonding) also have considerable effects on the structuring of water molecules around ions. The divalent cation (Mg$^{2+}$) has a stronger effect compared to the monovalent one (Na$^+$), while both result in $\sim$2 orders of magnitude enhancement over $d(r)$ correlations in solute free water. Further, anticorrelated regions present for solute free water are absent when observed from ion sites.
Such preferential patterning of water molecules’ dipoles, or equivalently of hydrogen bonding arms, around each ion can have important consequences on the long range dipolar correlations. As a consequence of the patterning, orientation fluctuations of neighboring water molecules are restricted as compared to those in solute free water. Such local perturbations to the underlying dynamics of hydrogen bond network is coupled to the long range orientational correlations, and more generally to the orientational entropy of the liquid, through the sum rule [@kanth12; @kanth13]. The ions can thus be envisaged as de-correlating centers, which along with their regions of influence, effectively screen bulk like water molecules from one another,- thus causing a decrease in the range of (ensemble averaged) long range correlations. The spatial extent of region of influence, as well as the strength of the patterning are critically related to the charge density of the ions [@iru]. Thus, strongly solvated ions can be expected to affect the long range orientational correlations to a greater extent than weakly solvated ones. With increasing salt concentration, there is likely to be a two-fold effect. Trivially, there is an enhanced fraction of water in the orientationally restricted regions of influence, whose orientational fluctuations are suppressed. Further, an enhancement in the number of such de-correlation centers with increasing number of ions can be expected to result in a more effective screening. Our observations are generally consistent with such effects, with the exception of enhancement in $t(r)$ at lower salt concentrations.
Interestingly, the decay of the solute - dipole correlation around both Na$^+$ and Mg$^{2+}$ ions show similar long range behavior. Fitting the $d(r)$ correlations for them for $r \geq10 \mathring{A}$ with the fit function defined in equation 5 ($a_2$ = 0), yields correlation lengths of (9.45$\pm$0.42) for Na$^+$ and (8.81$\pm$0.28) for Mg$^{2+}$. This indicates that, while the effects of ions on the orientations of first few solvation shell waters is strongly governed by the valency and charge density of the ions, the approach to pure water-like orientation can be independent of such parameters.
Comparison of effects : hydrophilic and hydrophobic solute
----------------------------------------------------------
At the very low concentration of solutes studied, with only one solute molecule among $\sim$4100 water molecules, the effects of the solutes on the dipolar correlations $t(r)$ and $l(r)$ are expected to be extremely small. As shown in the inset plots of FIG. 3 (c,d), the correlations only differ marginally from that in absence of solutes [@foot]. However, the long range parts of the correlations $t(r)$ and $l(r)$ can still be seen to be capable of distinguishing the effects of individual solute molecules, further emphasizing the applicability of the correlations in investigating orientational correlations among water molecules (FIG. 3 (c,d)). The $t(r)$ correlations for NaCl and MgCl$_2$ can be seen to have the same qualitative differences as observed at 0.15M concentration, both among themselves and with water in absence of solutes. $t(r)$ correlation for CH$_4$ is observed to be almost identical to that for solute free water, with indications of enhanced structuring, especially at larger separations. $l(r)$ correlation for CH$_4$, however shows clear differences, being enhanced compared to solute free water. $l(r)$ correlation can be observed to be reduced for MgCl$_2$, and comparable for NaCl when compared to solute free water. The results clearly show that small hydrophilic and hydrophobic solutes have contrasting effects on the long range orientational correlations among water molecules,- the former causing reduction and the latter leading to enhancement. $d(r)$ correlation around CH$_4$ shows no long range preferential orientation, as expected from its charge neutrality and size.
Discussion
==========
The results in this study indicate that the presence of dissolved salts (ions) leads to considerable changes in the dipolar correlations. Such changes are both salt species and concentration dependent. The effects on the transverse and longitudinal parts of the dipolar correlations are considerably varied. The transverse part of the correlation is observed to be enhanced at lower (0.15M) concentrations for all salts with CsCl, NaCl and KCl leading to enhancement of statistically comparable magnitude. The presence MgCl$_2$ at 0.15M concentration is observed to lead to a greater enhancement in the transverse part for smaller separations, which is compensated at larger separations in comparison to the other salts. At higher salt concentration of 1.0M, the presence of MgCl$_2$ leads to strong reduction in the transverse alignment of water dipoles resulting in anticorrelations over shorter ($\sim12\mathring{A}$) separations, while the other salts studied retain the enhancement in correlation over solute free water. The longitudinal part of the dipolar correlation shows uniform reduction with salt concentration for all salts studied. The magnitude of reduction is further observed to be consistent with the relative positions of the cations in the Hofmeister cationic series for both salt concentrations studied. Interestingly, the long range exponential decay of the longitudinal correlation observed for solute free water is greatly reduced in the presence of salts, completely vanishing at the higher concentration studied. The reduction in the long range longitudinal correlation in presence of ions can be explained by considering ion locations as de-correlation centers. Within their regions of influence, ions result in orientational stratification of water molecules and a consequent restriction in their orientational fluctuations. Such local perturbations are coupled to the long range dynamics of the underlying hydrogen bond network through the sum rule. As a result, long range orientational correlations, whose molecular origin lies in the inherent fluctuations of the hydrogen bond network, get suppressed.
Further, the varied responses of the transverse and longitudinal parts of the dipolar correlations to the presence of salts at lower concentration (0.15M), as well as the concentration dependent enhancement or reduction of the transverse part for the presence of the same salt (MgCl$_2$) further bolsters the criticism of the prevalent classification of ions along the lines of structure making and breaking. The contrasting effects of hydrophilic and hydrophobic solutes on long range $l(r)$ correlations can have interesting implications in understanding the long range ($10 - 100\,\mathring{A}$) component of hydrophobic force. It was suggested in our earlier publication that $l(r)$ correlations can lead to a shape dependent attraction between two hydrophobic surfaces at large distances of separation [@kanth10], decaying with a correlation length of $\sim 12\mathring{A}$. Experimental evidences have shown that the force between hydrophobic surfaces, acting in spatial separation between ($10 - 100\,\mathring{A}$), is exponential in nature, and the range, as well as the magnitude of the same is reduced in presence of ionic impurities [@meyer]. Our observations of reduction in the range of $l(r)$ correlations is consistent with the same. The observations for methane can not be extended to extended hydrophobic surfaces in a straightforward manner. However, sum rule for water dictates that long range $l(r)$ correlations in the presence of extended hydrophobic surfaces would be considerably altered compared to the same for water in absence of impurities, owing to the presence of dangling bonds at hydrophobic surfaces. Future research would be directed at studying the effects of larger hydrophobes as well as surfactants and osmolytes on the discussed correlations.
Acknowledgments
===============
All the simulations in this work have been carried out on 1024-cpu Annapurna cluster at The Institute of Mathematical Sciences.
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---
abstract: 'Given a Kripke structure $M$ and CTL formula $\phi$, where $M$ does not satisfy $\phi$, the problem of *Model Repair* is to obtain a new model $M''$ such that $M''$ satisfies $\phi$. Moreover, the changes made to $M$ to derive $M''$ should be minimum with respect to all such $M''$. As in model checking, *state explosion* can make it virtually impossible to carry out model repair on models with infinite or even large state spaces. In this paper, we present a framework for model repair that uses *abstraction refinement* to tackle state explosion. Our framework aims to repair Kripke Structure models based on a Kripke Modal Transition System abstraction and a 3-valued semantics for CTL. We introduce an abstract-model-repair algorithm for which we prove soundness and semi-completeness, and we study its complexity class. Moreover, a prototype implementation is presented to illustrate the practical utility of abstract-model-repair on an Automatic Door Opener system model and a model of the Andrew File System 1 protocol.'
address:
- '$^a$Department of Informatics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece'
- '$^b$Department of Computing and Software, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L7, Canada'
- '$^c$Department of Informatics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece'
- '$^d$Department of Computer Science, Stony Brook University, Stony Brook, NY 11794-4400, USA'
author:
- George Chatzieleftheriou
- Borzoo Bonakdarpour
- Panagiotis Katsaros
- 'Scott A. Smolka'
bibliography:
- 'amr.bib'
title: Abstract Model Repair
---
Introduction {#sec:intro}
============
Given a model $M$ and temporal-logic formula $\phi$, *model checking* [@CES09] is the problem of determining whether or not $M \models \phi$. When this is not the case, a model checker will typically provide a *counterexample* in the form of an execution path along which $\phi$ is violated. The user should then process the counterexample manually to correct $M$.
An extended version of the model-checking problem is that of *model repair*: given a model $M$ and temporal-logic formula $\phi$, where $M \not\models \phi$, obtain a new model $M'$, such that $M' \models \phi$. The problem of Model Repair for Kripke structures and Computation Tree Logic (CTL) [@EH85] properties was first introduced in [@BEGL99].
*State explosion* is a well known limitation of automated formal methods, such as model checking and model repair, which impedes their application to systems having large or even infinite state spaces. Different techniques have been developed to cope with this problem. In the case of model checking, *abstraction* [@CGL94; @LGSBBP95; @GS97; @DGG97; @GHJ01] is used to create a smaller, more abstract version $\hat{M}$ of the initial concrete model $M$, and model checking is performed on this smaller model. For this technique to work as advertised, it should be the case that if $\hat{M} \models \phi$ then $M \models \phi$.
Motivated by the success of abstraction-based model checking, we present in this paper a new framework for Model Repair that uses *abstraction refinement* to tackle state explosion. The resulting *Abstract Model Repair* (AMR) methodology makes it possible to repair models with large state spaces, and to speed-up the repair process through the use of smaller abstract models. The major contributions of our work are as follows:
- We provide an AMR framework that uses Kripke structures (KSs) for the concrete model $M$, Kripke Modal Transition Systems (KMTSs) for the abstract model $\hat{M}$, and a 3-valued semantics for interpreting CTL over KMTSs [@HJS01]. An iterative refinement of the abstract KMTS model takes place whenever the result of the 3-valued CTL model-checking problem is undefined. If the refinement process terminates with a KMTS that violates the CTL property, this property is also falsified by the concrete KS $M$. Then, the repair process for the refined KMTS is initiated.
- We strengthen the Model Repair problem by additionally taking into account the following *minimality* criterion (refer to the definition of Model Repair above): the changes made to $M$ to derive $M'$ should be minimum with respect to all $M'$ satisfying $\phi$. To handle the minimality constraint, we define a metric space over KSs that quantifies the structural differences between them.
- We introduce an Abstract Model Repair algorithm for KMTSs, which takes into account the aforementioned minimality criterion.
- We prove the soundness of the Abstract Model Repair algorithm for the full CTL and the completeness for a major fragment of it. Moreover, the algorithm’s complexity is analyzed with respect to the abstract KMTS model size, which can be much smaller than the concrete KS.
- We illustrate the utility of our approach through a prototype implementation used to repair a flawed Automatic Door Opener system [@BK08] and the Andrew File System 1 protocol. Our experimental results show significant improvement in efficiency compared to a concrete model repair solution.
*Organization.* The rest of this paper is organized as follows. Sections \[sec:mc\] and \[sec:abstr\] introduce KSs, KMTSs, as well as abstraction and refinement based on a 3-valued semantics for CTL. Section \[sec:mrp\] defines a metric space for KSs and formally defines the problem of Model Repair. Section \[sec:absmrp\] presents our framework for Abstract Model Repair, while Section \[sec:alg\] introduces the abstract-model-repair algorithm for KMTSs and discusses its soundness, completeness and complexity properties. Section \[sec:exp\] presents the experimental evaluation of our method through its application to the Andrew File System 1 protocol (AFS1). Section \[sec:relwork\] considers related work, while Section \[sec:concl\] concludes with a review of the overall approach and pinpoints directions for future work.
Kripke Modal Transition Systems {#sec:mc}
===============================
Let $AP$ be a set of [*atomic propositions*]{}. Also, let $Lit$ be the set of [*literals*]{}: $$Lit = AP \; \cup \; \{ \neg p \mid p \in AP\}$$
\[def:ks\] A [*Kripke Structure*]{} (KS) is a quadruple $M = (S, S_{0}, R, L)$, where:
1. $S$ is a finite set of [*states*]{}.
2. $S_{0}\subseteq S$ is the set of [*initial states*]{}.
3. $R\subseteq S \times S$ is a [*transition relation*]{} that must be total, i.e., $$\forall s \in S: \exists s' \in S:R(s,s').$$
4. $L: S \rightarrow 2^{Lit}$ is a state [*labeling function*]{}, such that $$\forall s \in S: \forall p \in AP: p \in L(s)
\Leftrightarrow \neg p \notin L(s).\eqno{\qEd}$$
The fourth condition in Def. \[def:ks\] ensures that any atomic proposition $p \in AP$ has one and only one truth value at any state.\
*Example.* We use the Automatic Door Opener system (ADO) of [@BK08] as a running example throughout the paper. The system, given as a KS in Fig \[fig:ado\_system\], requires a three-digit code $(p_{0},p_{1},p_{2})$ to open a door, allowing for one and only one wrong digit to be entered at most twice. Variable $\mathit{err}$ counts the number of errors, and an alarm is rung if its value exceeds two. For the purposes of our paper, we use a simpler version of the ADO system, given as the KS $M$ in Fig. \[fig:ado\_initial\], where the set of atomic propositions is $AP = \{q\}$ and $q \equiv (open = true)$.
\[def:kmts\] A [*Kripke Modal Transition System*]{} (KMTS) is a 5-tuple $\hat{M} = (\hat{S}, \hat{S_{0}},$ $R_{must}, R_{may}, \hat{L})$, where:
1. $\hat{S}$ is a finite set of *states*.
2. $\hat{S_{0}}\subseteq \hat{S}$ is the set of *initial states*.
3. $R_{must} \subseteq \hat{S} \times \hat{S}$ and $R_{may} \subseteq
\hat{S} \times \hat{S}$ are *transition relations* such that $R_{must} \subseteq R_{may}$.
4. $\hat{L}: \hat{S} \rightarrow 2^{Lit}$ is a state-labeling such that $\forall \hat{s} \in \hat{S}$, $\forall p \in AP$, $\hat{s}$ is labeled by [*at most*]{} one of $p$ and $\neg p$.
A KMTS has two types of transitions: *must-transitions*, which exhibit *necessary* behavior, and *may-transitions*, which exhibit *possible* behavior. Must-transitions are also may-transitions. The “at most one” condition in the fourth part of Def. \[def:kmts\] makes it possible for the truth value of an atomic proposition at a given state to be [*unknown*]{}. This relaxation of truth values in conjunction with the existence of may-transitions in a KMTS constitutes a *partial modeling* formalism.
Verifying a CTL formula $\phi$ over a KMTS may result in an undefined outcome ($\bot$). We use the *3-valued semantics* [@HJS01] of a CTL formula $\phi$ at a state $\hat{s}$ of KMTS $\hat{M}$.
\[def:ctl3\_semantics\] [**[@HJS01]**]{} Let $\hat{M} = (\hat{S}, \hat{S_{0}}, R_{must}, R_{may},
\hat{L})$ be a KMTS. The 3-valued semantics of a CTL formula $\phi$ at a state $\hat{s}$ of $\hat{M}$, denoted as $(\hat{M},\hat{s}) \models^{3} \phi$, is defined inductively as follows:
- If $\phi = \mathit{false}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$
- If $\phi = \mathit{true}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$
- If $\phi = p$ where $p \in AP$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff $p \in \hat{L}(\hat{s})$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff $\neg p \in \hat{L}(\hat{s})$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = \neg\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff $[(\hat{M},\hat{s}) \models^{3} \phi_{1} ] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff $[(\hat{M},\hat{s}) \models^{3} \phi_{1} ] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = \phi_{1} \, \vee \, \phi_{2}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff $[(\hat{M},\hat{s}) \models^{3} \phi_{1}] = \mathit{true}$ or $[(\hat{M},\hat{s}) \models^{3} \phi_{2}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff $[(\hat{M},\hat{s}) \models^{3} \phi_{1}] = \mathit{false}$ and $[(\hat{M},\hat{s}) \models^{3} \phi_{2}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = \phi_{1} \, \wedge \, \phi_{2}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff $[(\hat{M},\hat{s}) \models^{3} \phi_{1}] = \mathit{true}$ and $[(\hat{M},\hat{s}) \models^{3} \phi_{2}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff $[(\hat{M},\hat{s}) \models^{3} \phi_{1}] = \mathit{false}$ or $[(\hat{M},\hat{s}) \models^{3} \phi_{2}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = AX\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff for all $\hat{s}_{i}$ such that $(\hat{s},\hat{s}_{i}) \in R_{may}$, $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff there exists some $\hat{s}_{i}$ such that $(\hat{s},\hat{s}_{i}) \in R_{must}$ and $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = EX\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff there exists $\hat{s}_{i}$ such that $(\hat{s},\hat{s}_{i}) \in R_{must}$ and $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff for all $\hat{s}_{i}$ such that $(\hat{s},\hat{s}_{i}) \in R_{may}$, $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = AG\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff for all may-paths $\pi_{may} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$ and for all $\hat{s}_{i} \in \pi_{may}$ it holds that $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff there exists some must-path $\pi_{must} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, such that for some $\hat{s}_{i} \in \pi_{must}$, $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = EG\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff there exists some must-path $\pi_{must} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, such that for all $\hat{s}_{i} \in \pi_{must}$, $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff for all may-paths $\pi_{may} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, there is some $\hat{s}_{i} \in \pi_{may}$ such that $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = AF\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff for all may-paths $\pi_{may} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, there is a $\hat{s}_{i} \in \pi_{may}$ such that $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff there exists some must-path $\pi_{must} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, such that for all $\hat{s}_{i} \in \pi_{must}$, $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = EF\phi_{1}$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff there exists some must-path $\pi_{must} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, such that there is some $\hat{s}_{i} \in \pi_{must}$ for which $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff for all may-paths $\pi_{may} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$ and for all $\hat{s}_{i} \in \pi_{may}$, $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{1}] = \mathit{false}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = A(\phi_{1} \, U \, \phi_{2})$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff for all may-paths $\pi_{may} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, there is $\hat{s}_{i} \in \pi_{may}$ such that $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{2}] = \mathit{true}$ and $\forall j < i: [(\hat{M},\hat{s}_{j}) \models^{3} \phi_{1}]
= true$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff there exists some must-path $\pi_{must} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$, such that
- for all $0\leq k< |\pi_{must}|:$\
$(\forall j < k : [(\hat{M},\hat{s}_{j}) \models^{3} \phi_{1}] \neq \mathit{false})
\Rightarrow ([(\hat{M},\hat{s}_{k}) \models^{3} \phi_{2}] = \mathit{false})$
- $(\text{for all } 0 \leq k < |\pi_{must}|:[(\hat{M},\hat{s}_{k}) \models^{3} \phi_{2}] \neq \mathit{false})
\Rightarrow |\pi_{must}| = \infty$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
- If $\phi = E(\phi_{1}U\phi_{2})$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{true}$, iff there exists some must-path $\pi_{must} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$ such that there is a $\hat{s}_{i} \in \pi_{must}$ with $[(\hat{M},\hat{s}_{i}) \models^{3} \phi_{2}] = \mathit{true}$ and for all $j < i, [(\hat{M},\hat{s}_{j}) \models^{3}
\phi_{1}] = \mathit{true}$.
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \mathit{false}$, iff for all may-paths $\pi_{may} = [\hat{s},\hat{s}_{1},\hat{s}_{2},...]$
- for all $0 \leq k < |\pi_{may}|:$\
$(\forall j < k :
[(\hat{M},\hat{s}_{j}) \models^{3} \phi_{1}] \neq \mathit{false})
\Rightarrow ([(\hat{M},\hat{s}_{k}) \models^{3} \phi_{2}] = \mathit{false})$
- $(\text{for all } 0 \leq k < |\pi_{may}| :
[(\hat{M},\hat{s}_{k}) \models^{3} \phi_{2}] \neq \mathit{false})
\Rightarrow |\pi_{may}| = \infty$
- $[(\hat{M},\hat{s}) \models^{3} \phi ] = \bot$, otherwise.
From the 3-valued CTL semantics, it follows that must-transitions are used to check the truth of existential CTL properties, while may-transitions are used to check the truth of universal CTL properties. This works inversely for checking the refutation of CTL properties. In what follows, we use $\models$ instead of $\models^{3}$ in order to refer to the 3-valued satisfaction relation.
Abstraction and Refinement for 3-Valued CTL {#sec:abstr}
===========================================
Abstraction
-----------
*Abstraction* is a state-space reduction technique that produces a smaller abstract model from an initial [*concrete*]{} model, so that the result of model checking a property $\phi$ in the abstract model is preserved in the concrete model. This can be achieved if the abstract model is built with certain requirements [@CGL94; @GHJ01].
\[def:abs\_kmts\] Given a KS $M = (S, S_{0}, R, L)$ and a pair of total functions $(\alpha : S \rightarrow \hat{S}, \gamma : \hat{S}
\rightarrow 2^{S})$ such that $$\forall s \in S:
\forall\hat{s} \in \hat{S}: (\alpha(s) = \hat{s} \Leftrightarrow s \in
\gamma(\hat{s}))$$ the KMTS $\alpha(M) = (\hat{S}, \hat{S_{0}}, R_{must}, R_{may},
\hat{L})$ is defined as follows:
1. $\hat{s} \in \hat{S_{0}}$ iff $\exists s \in
\gamma(\hat{s})$ such that $s \in S_{0}$
2. $lit \in \hat{L}(\hat{s})$ only if $\forall s \in
\gamma(\hat{s}): lit \in L(s)$
3. $R_{must} = \left\{(\hat{s_{1}},\hat{s_{2}}) \mid
\forall s_{1} \in \gamma(\hat{s_{1}}): \exists s_{2} \in
\gamma(\hat{s_{2}}): (s_{1},s_{2}) \in R\right\}$
4. $R_{may} = \left\{(\hat{s_{1}},\hat{s_{2}}) \mid
\exists s_{1} \in \gamma(\hat{s_{1}}): \exists s_{2} \in
\gamma(\hat{s_{2}}): (s_{1},s_{2}) \in R\right\}$
For a given KS $M$ and a pair of abstraction and concretization functions $\alpha$ and $\gamma$, Def. \[def:abs\_kmts\] introduces the KMTS $\alpha(M)$ defined over the set $\hat{S}$ of *abstract states*. In our AMR framework, we view $M$ as the *concrete model* and the KMTS $\alpha(M)$ as the *abstract model*. Any two concrete states $s_{1}$ and $s_{2}$ of $M$ are abstracted by $\alpha$ to a state $\hat{s}$ of $\alpha(M)$ if and only if $s_{1}$, $s_{2}$ are elements of the set $\gamma(\hat{s})$ (see Fig \[fig:abstract\_concrete\]). A state of $\alpha(M)$ is initial *if and only if* at least one of its concrete states is initial as well. An atomic proposition in an abstract state is true (respectively, false), *only if* it is also true (respectively, false) in all of its concrete states. This means that the value of an atomic proposition may be unknown at a state of $\alpha(M)$. A must-transition from $\hat{s_{1}}$ to $\hat{s_{2}}$ of $\alpha(M)$ exists, if and only if there are transitions from all states of $\gamma(\hat{s_{1}})$ to at least one state of $\gamma(\hat{s_{2}})$ $(\forall\exists-condition)$. Respectively, a may-transition from $\hat{s_{1}}$ to $\hat{s_{2}}$ of $\alpha(M)$ exists, if and only if there is at least one transition from some state of $\gamma(\hat{s_{1}})$ to some state of $\gamma(\hat{s_{2}})$ $(\exists\exists-condition)$.
![Abstraction and Concretization.[]{data-label="fig:abstract_concrete"}](abscon){width="38mm"}
\[def:concretize\_kmts\] Given a pair of total functions $(\alpha : S \rightarrow \hat{S}, \gamma :
\hat{S} \rightarrow 2^{S})$ such that $$\forall s \in S: \forall
\hat{s} \in \hat{S}: (\alpha(s) = \hat{s} \Leftrightarrow s \in
\gamma(\hat{s}))$$ and a KMTS $\hat{M} = (\hat{S}, \hat{S_{0}}, R_{must}, R_{may}, \hat{L})$, the set of KSs $\gamma(\hat{M}) = \{M \mid M = (S, S_{0}, R, L)\}$ is defined such that for all $M \in \gamma(\hat{M})$ the following conditions hold:
1. $s \in S_{0}$ iff $\alpha(s) \in \hat{S_{0}}$
2. $lit \in L(s)$ if $lit \in \hat{L}(\alpha(s))$
3. $(s_{1},s_{2}) \in R$ iff
- $\exists s_{1}^{\prime} \in \gamma(\alpha(s_{1})):
\exists s_{2}^{\prime} \in \gamma(\alpha(s_{2})) :
(\alpha(s_{1}),\alpha(s_{2})) \in R_{may}$ and,
- $\forall s_{1}^{\prime} \in \gamma(\alpha(s_{1})):
\exists s_{2}^{\prime} \in \gamma(\alpha(s_{2})) :
(\alpha(s_{1}),\alpha(s_{2})) \in R_{must}$
For a given KMTS $\hat{M}$ and a pair of abstraction and concretization functions $\alpha$ and $\gamma$, Def. \[def:concretize\_kmts\] introduces a set $\gamma(\hat{M})$ of *concrete* KSs. A state $s$ of a KS $M \in \gamma(\hat{M})$ is initial if its abstract state $\alpha(s)$ is also initial. An atomic proposition in a concrete state $s$ is true (respectively, false) if it is also true (respectively, false) in its abstract state $\alpha(s)$. A transition from a concrete state $s_{1}$ to another concrete state $s_{2}$ exists, if and only if
- there are concrete states $s_{1}^{\prime} \in \gamma(\alpha(s_{1}))$ and $s_{2}^{\prime} \in \gamma(\alpha(s_{2}))$, where $(\alpha(s_{1}),\alpha(s_{2})) \in R_{may}$, and
- there is at least one concrete state $s_{2}^{\prime} \in \gamma(\alpha(s_{2}))$ such that for all $s_{1}^{\prime} \in \gamma(\alpha(s_{1}))$ it holds that $(\alpha(s_{1}),\alpha(s_{2})) \in R_{must}$.
#### Abstract Interpretation.
A pair of abstraction and concretization functions can be defined within an *Abstract Interpretation* [@CC77; @CC79] framework. Abstract interpretation is a theory for a set of abstraction techniques, for which important properties for the model checking problem have been proved [@DGG97; @D96].
\[def:mixsimul\] * [@DGG97; @GJ02]* Let $M = (S, S_{0}, R, L)$ be a concrete KS and $\hat{M}$ = $(\hat{S}, \hat{S_{0}}, R_{must},$ $R_{may}, \hat{L})$ be an abstract KMTS. A relation $H \subseteq S \times \hat{S}$ for $M$ and $\hat{M}$ is called a *mixed simulation*, when $H(s,\hat{s})$ implies:
- $\hat{L}(\hat{s}) \subseteq L(s)$
- if $r = (s,s^{\prime}) \in R$, then there is exists $\hat{s}^{\prime} \in
\hat{S}$ such that $r_{may} = (\hat{s},\hat{s}^{\prime}) \in R_{may}$ and $(s^{\prime},\hat{s}^{\prime}) \in H$.
- if $r_{must} = (\hat{s},\hat{s}^{\prime}) \in R_{must}$, then there exists $s^{\prime} \in S$ such that $r = (s,s^{\prime}) \in R$ and $(s^{\prime},\hat{s}^{\prime})
\in H$.
The abstraction function $\alpha$ of Def. \[def:abs\_kmts\] is a mixed simulation for the KS $M$ and its abstract KMTS $\alpha(M)$.
\[theor:preserv\] *[@GJ02]* Let $H \subseteq S \times \hat{S}$ be a mixed simulation from a KS $M = (S, S_{0}, R, L)$ to a KMTS $\hat{M} =
(\hat{S}, \hat{S_{0}}, R_{must}, R_{may}, \hat{L})$. Then, for every CTL formula $\phi$ and every $(s,\hat{s})
\in H$ it holds that $$[(\hat{M},\hat{s}) \models \phi] \neq \bot \Rightarrow
[(M,s) \models \phi] = [(\hat{M},\hat{s}) \models \phi]$$
Theorem \[theor:preserv\] ensures that if a CTL formula $\phi$ has a definite truth value (i.e., true or false) in the abstract KMTS, then it has the same truth value in the concrete KS. When we get $\bot$ from the 3-valued model checking of a CTL formula $\phi$, the result of model checking property $\phi$ on the corresponding KS can be either true or false.\
*Example.* An abstract KMTS $\hat{M}$ is presented in Fig. \[fig:ado\_initial\], where all the states labeled by $q$ are grouped together, as are all states labeled by $\neg q$.
Refinement
----------
When the outcome of verifying a CTL formula $\phi$ on an abstract model using the 3-valued semantics is $\bot$, then a *refinement* step is needed to acquire a more *precise* abstract model. In the literature, there are refinement approaches for the 2-valued CTL semantics [@CGJLV00; @CPR05; @CGR07], as well as a number of techniques for the 3-valued CTL model checking [@GHJ01; @SG04; @SG07; @GLLS07]. The refinement technique that we adopt is an automated two-step process based on [@CGJLV00; @SG04]:
1. Identify a *failure state* in $\alpha(M)$ using the algorithms in [@CGJLV00; @SG04]; the cause of failure for a state $\hat{s}$ stems from an atomic proposition having an undefined value in $\hat{s}$, or from an outgoing may-transition from $\hat{s}$.
2. Produce the abstract KMTS $\alpha_{\mathit{Refined}}(M)$, where $\alpha_{\mathit{Refined}}$ is a new abstraction function as in Def. \[def:abs\_kmts\], such that the identified failure state is refined into two states. If the cause of failure is an undefined value of an atomic proposition in $\hat{s}$, then $\hat{s}$ is split into states $\hat{s}_{1}$ and $\hat{s}_{2}$, such that the atomic proposition is true in $\hat{s}_{1}$ and false in $\hat{s}_{2}$. Otherwise, if the cause of failure is an outgoing may-transition from $\hat{s}$, then $\hat{s}$ is split into states $\hat{s}_{1}$ and $\hat{s}_{2}$, such that there is an outgoing must-transition from $\hat{s}_{1}$ and no outgoing may- or must-transition from $\hat{s}_{2}$.
The described refinement technique does not necessarily converge to an abstract KMTS with a definite model checking result. A promising approach in order to overcome this restriction is by using a different type of abstract model, as in [@SG04], where the authors propose the use of Generalized KMTSs, which ensure monotonicity of the refinement process.\
*Example.* Consider the case where the ADO system requires a mechanism for opening the door from any state with a direct action. This could be an action done by an expert if an immediate opening of the door is required. This property can be expressed in CTL as $\phi = AGEXq$. Observe that in $\alpha(M)$ of Fig. \[fig:ado\_initial\], the absence of a must-transition from $\hat{s}_{0}$ to $\hat{s}_{1}$, where $[(\alpha(M),\hat{s}_{1}) \models q]
= true$, in conjunction with the existence of a may-transition from $\hat{s}_{0}$ to $\hat{s}_{1}$, i.e. to a state where $[(\alpha(M),\hat{s}_{1}) \models q] = true$, results in an undefined model-checking outcome for $[(\alpha(M),\hat{s}_{0}) \models \phi]$. Notice that state $\hat{s}_{0}$ is the failure state, and the may-transition from $\hat{s}_{0}$ to $\hat{s}_{1}$ is the cause of the failure. Consequently, $\hat{s}_{0}$ is refined into two states, $\hat{s}_{01}$ and $\hat{s}_{02}$, such that the former has no transition to $\hat{s}_{1}$ and the latter has an outgoing must-transition to $\hat{s}_{1}$. Thus, the may-transition which caused the undefined outcome is eliminated and for the refined KMTS $\alpha_{\mathit{Refined}}(M)$ it holds that $[\alpha_{\mathit{Refined}}(M),\hat{s}_{1}) \models \phi] =
\mathit{false}$. The initial KS and the refined KMTS $\alpha_{\mathit{Refined}}(M)$ are shown in Fig. \[fig:ado\_refined\].
The Model Repair Problem {#sec:mrp}
========================
In this section, we formulate the problem of Model Repair. A metric space over Kripke structures is defined to quantify their structural differences. This allows us taking into account the *minimality of changes* criterion in Model Repair.
Let $\pi$ be a function on the set of all functions $f: X \rightarrow Y$ such that: $$\pi(f) = \{(x, f(x)) \mid x \in X\}$$
A *restriction operator* (denoted by $\upharpoonright$) for the domain of function $f$ is defined such that for $X_{1} \subseteq X$, $$f\upharpoonright_{X_{1}} = \{(x, f(x)) \mid x \in X_{1}\}$$ By $S^C$, we denote the complement of a set $S$.
\[def:metric\_space\] For any two $M = (S,S_{0},R,L)$ and $M^{\prime} = (S^{\prime},S^{\prime}_{0},R^{\prime},L^{\prime})$ in the set $K_{M}$ of all KSs, where
- $S^{\prime} = (S \cup S_{\mathit{IN}}) - S_{\mathit{OUT}}$ for some $S_{\mathit{IN}} \subseteq S^{C}$, $S_{\mathit{OUT}} \subseteq S$,
- $R^{\prime} = (R \cup R_{\mathit{IN}}) - R_{\mathit{OUT}}$ for some $R_{\mathit{IN}} \subseteq R^{C}$, $R_{\mathit{OUT}} \subseteq R$,
- $L^{\prime} = S^{\prime} \rightarrow 2^{LIT}$,
the [*distance function*]{} $d$ over $K_{M}$ is defined as follows: $$d(M,M^{\prime}) = |S\,\Delta \, S^{\prime}| + |R \, \Delta \, R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}}) \,\Delta \,
\pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$$ with $A \, \Delta \, B$ representing the symmetric difference $(A-B)\cup(B-A)$.
For any two KSs defined over the same set of atomic propositions $AP$, function $d$ counts the number of differences $|S\,\Delta\, S^{\prime}|$ in the state spaces, the number of differences $|R\,\Delta\, R^{\prime}|$ in their transition relation and the number of common states with altered labeling.
\[prop:metric\_space\] The ordered pair $(K_{M},d)$ is a metric space.
We use the fact that the cardinality of the symmetric difference between any two sets is a distance metric. It holds that:
1. $|S\Delta S^{\prime}| \geq 0$, $|R\Delta R^{\prime}| \geq 0$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta
\pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| \geq 0$ (non-negativity)
2. $|S\Delta S^{\prime}| = 0$ iff $S = S^{\prime}$, $|R\Delta R^{\prime}| = 0$ iff $R = R^{\prime}$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})|\Delta |\pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$ iff $\pi(L\upharpoonright_{S\cap S^{\prime}}) = \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})$ (identity of indiscernibles)
3. $|S\Delta S^{\prime}| = |S^{\prime}\Delta S|$, $|R\Delta R^{\prime}| = |R^{\prime}\Delta R|$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| =\\
|\pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}}) \Delta \pi(L\upharpoonright_{S\cap S^{\prime}})|$(symmetry)
4. $|S^{\prime}\Delta S^{\prime\prime}| \leq |S^{\prime}\Delta S| + |S \Delta S^{\prime\prime}|$, $|R^{\prime}\Delta R^{\prime\prime}| \leq |R^{\prime}\Delta R| + |R \Delta R^{\prime\prime}|$,\
$|\pi(L^{\prime}\upharpoonright_{S^{\prime}\cap S^{\prime\prime}})\Delta \pi(L^{\prime\prime}|_{S^{\prime}\cap S^{\prime\prime}})| \leq
|\pi(L^{\prime}\upharpoonright_{S^{\prime}\cap S})\Delta \pi(L\upharpoonright_{S^{\prime}\cap S})| + \\
|\pi(L\upharpoonright_{S\cap S^{\prime\prime}})\Delta \pi(L^{\prime\prime}|_{S\cap S^{\prime\prime}})|$\
(triangle inequality)
We will prove that $d$ is a metric on $K_{M}$. Suppose $M, M^{\prime}, M^{\prime\prime} \in K_{M}$
- It easily follows from (1) that $d(M,M^{\prime}) \geq 0$ (non-negativity)
- From (2), $d(M,M^{\prime}) = 0$ iff $M = M^{\prime}$ (identity of indiscernibles)
- Adding the equations in (3), results in $d(M,M^{\prime}) = d(M^{\prime},M)$ (symmetry)
- If we add the inequalities in (4), then we get $d(M^{\prime},M^{\prime\prime}) \leq d(M^{\prime},M) + d(M,M^{\prime\prime})$ (triangle inequality)
So, the proposition is true.
\[def:metric\_space\_kmts\] For any two $\hat{M}$ = $(\hat{S}, \hat{S_{0}}, R_{must}, R_{may}, \hat{L})$ and $\hat{M}^{\prime}$ = $(\hat{S}^{\prime}, \hat{S_{0}}^{\prime}, R_{must}^{\prime},$ $R_{may}^{\prime}, \hat{L}^{\prime})$ in the set $K_{\hat{M}}$ of all KMTSs, where
- $\hat{S}^{\prime} = (\hat{S} \cup \hat{S}_{\mathit{IN}}) - \hat{S}_{\mathit{OUT}}$ for some $\hat{S}_{\mathit{IN}} \subseteq \hat{S}^{C}$, $\hat{S}_{\mathit{OUT}} \subseteq \hat{S}$,
- $\hat{R}_{must}^{\prime} = (\hat{R}_{must} \cup \hat{R}_{\mathit{IN}}) - \hat{R}_{\mathit{OUT}}$ for some $\hat{R}_{\mathit{IN}} \subseteq \hat{R}_{must}^{C}$, $\hat{R}_{\mathit{OUT}} \subseteq \hat{R}_{must}$,
- $\hat{R}_{may}^{\prime} = (\hat{R}_{may} \cup \hat{R}_{\mathit{IN}}^{\prime}) - \hat{R}_{\mathit{OUT}}^{\prime}$ for some $\hat{R}_{\mathit{IN}}^{\prime} \subseteq \hat{R}_{may}^{C}$, $\hat{R}_{\mathit{OUT}}^{\prime} \subseteq \hat{R}_{may}$,
- $\hat{L}^{\prime} = \hat{S}^{\prime} \rightarrow 2^{LIT}$,
the [*distance function*]{} $\hat{d}$ over $K_{\hat{M}}$ is defined as follows: $$\begin{split}
\hat{d}(M,M^{\prime}) = |\hat{S} \, \Delta \, \hat{S}^{\prime}| +
|\hat{R}_{must} \, \Delta \, \hat{R}_{must}^{\prime}| +
|(\hat{R}_{may} - \hat{R}_{must}) \, \Delta \, (\hat{R}_{may}^{\prime} -
\hat{R}_{must}^{\prime})| + \\
\frac{|\pi(\hat{L}\upharpoonright_{\hat{S}\cap \hat{S}^{\prime}}) \, \Delta \,
\pi(\hat{L}^{\prime}\upharpoonright_{\hat{S}\cap \hat{S}^{\prime}})|}{2}
\end{split}$$ with $A \Delta B$ representing the symmetric difference $(A-B)\cup(B-A)$.
We note that $\hat{d}$ counts the differences between $\hat{R}_{may}^{\prime}$ and $\hat{R}_{may}$, and those between $\hat{R}_{must}^{\prime}$ and $\hat{R}_{must}$ separately, while avoiding to count the differences in the latter case twice (we remind that must-transitions are also included in $\hat{R}_{may}$).
\[prop:kmts\_metric\_space\] The ordered pair $(K_{\hat{M}},\hat{d})$ is a metric space.
The proof is done in the same way as in Prop. \[prop:metric\_space\].
Given a KS $M$ and a CTL formula $\phi$ where $M \not\models \phi$, the Model Repair problem is to find a KS $M^{\prime}$, such that $M^{\prime} \models
\phi$ and $d(M,M^{\prime})$ is minimum with respect to all such $M^{\prime}$.
The Model Repair problem aims at modifying a KS such that the resulting KS satisfies a CTL formula that was violated before. The distance function $d$ of Def. \[def:metric\_space\] features all the attractive properties of a distance metric. Given that no quantitative interpretation exists for predicates and logical operators in CTL, $d$ can be used in a model repair solution towards selecting minimum changes to the modified KS.
The Abstract Model Repair Framework {#sec:absmrp}
===================================
Our AMR framework integrates 3-valued model checking, model refinement, and a new algorithm for selecting the repair operations applied to the abstract model. The goal of this algorithm is to apply the repair operations in a way, such that the number of structural changes to the corresponding concrete model is minimized. The algorithm works based on a partial order relation over a set of basic repair operations for KMTSs. This section describes the steps involved in our AMR framework, the basic repair operations, and the algorithm.
The Abstract Model Repair Process {#subsec:absmrp_approach}
---------------------------------
![Abstract Model Repair Framework.[]{data-label="fig:abs_repair"}](AMR){width="120mm"}
The process steps shown in Fig. \[fig:abs\_repair\] rely on the KMTS abstraction of Def. \[def:abs\_kmts\]. These are the following:
Step 1.
: Given a KS $M$, a state $s$ of $M$, and a CTL property $\phi$, let us call $\hat{M}$ the KMTS obtained as in Def. \[def:abs\_kmts\].
Step 2.
: For state $\hat{s} = \alpha(s)$ of $\hat{M}$, we check whether $(\hat{M},\hat{s}) \models \phi$ by 3-valued model checking.
Case 1.
: If the result is *true*, then, according to Theorem \[theor:preserv\], $(M,s) \models \phi$ and there is no need to repair $M$.
Case 2.
: If the result is *undefined*, then a refinement of $\hat{M}$ takes place, and:
Case 2.1.
: If an $\hat{M}_{Refined}$ is found, the control is transferred to Step 2.
Case 2.2.
: If a refined KMTS cannot be retrieved, the repair process terminates with a failure.
Case 3.
: If the result is *false*, then, from Theorem \[theor:preserv\], $(M,s) \not\models \phi$ and the repair process is enacted; the control is transferred to Step 3.
Step 3.
: The *AbstractRepair* algorithm is called for the abstract KMTS ($\hat{M}_{Refined}$ or $\hat{M}$ if no refinement has occurred), the state $\hat{s}$ and the property $\phi$.
Case 1.
: *AbstractRepair* returns an $\hat{M}^{\prime}$ for which $(\hat{M}^{\prime},\hat{s}) \models \phi$.
Case 2.
: *AbstractRepair* fails to find an $\hat{M}^{\prime}$ for which the property holds true.
Step 4.
: If *AbstractRepair* returns an $\hat{M}^{\prime}$, then the process ends with selecting the subset of KSs from $\gamma(\hat{M}^{\prime})$, with elements whose distance $d$ from the KS $M$ is minimum with respect to all the KSs in $\gamma(\hat{M}^{\prime})$.
Basic Repair Operations {#subsec:basic_ops}
-----------------------
We decompose the KMTS repair process into seven basic repair operations:
AddMust
: Adding a must-transition
AddMay
: Adding a may-transition
RemoveMust
: Removing a must-transition
RemoveMay
: Removing a may-transition
ChangeLabel
: Changing the labeling of a KMTS state
AddState
: Adding a new KMTS state
RemoveState
: Removing a disconnected KMTS state
### Adding a must-transition
\[def:AddMust\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$ and $\hat{r}_{n} = (\hat{s}_{1},\hat{s}_{2}) \notin R_{must}$, $AddMust(\hat{M},\hat{r}_{n})$ is the KMTS $\hat{M^{\prime}} = (\hat{S}, \hat{S_{0}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L})$ such that $R_{must}^{\prime} = R_{must} \cup \{\hat{r}_{n}\}$ and $R_{may}^{\prime} = R_{may} \cup \{\hat{r}_{n}\}$.
Since $R_{must} \subseteq R_{may}$, $\hat{r}_{n}$ must also be added to $R_{may}$, resulting in a new may-transition if $\hat{r}_{n} \notin R_{may}$. Fig. \[fig:AddMust\] shows how the basic repair operation *AddMust* modifies a given KMTS. The newly added transitions are in bold.
\[prop:AddMust\] For any $\hat{M}^{\prime} = AddMust(\hat{M},\hat{r}_{n})$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:add\_must\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = AddMust(\alpha(M),\hat{r}_{n})$ for some $\hat{r}_{n} = (\hat{s}_{1},\hat{s}_{2}) \notin R_{must}$. The set $K_{min} \subseteq \gamma(\hat{M}^{\prime})$ with all KSs, whose distance $d$ from $M$ is minimized is: $$K_{min} = \{M^{\prime} \mid M^{\prime} = (S, S_{0}, R \cup R_{n}, L)\}$$ where $R_{n}$ is given for one $s_{2} \in \gamma(\hat{s}_{2})$ as follows: $$\nonumber
R_{n} = \bigcup_{s_{1} \in \gamma(\hat{s}_{1})} \{(s_{1}, s_{2}) \mid \nexists s \in \gamma(\hat{s}_{2}): (s_{1},s) \in R\} \eqno{\qEd}$$
Def. \[def:add\_must\_ks\] implies that when the *AbstractRepair* algorithm applies *AddMust* on the abstract KMTS $\hat{M}$, then a set of KSs is retrieved from the concretization of $\hat{M}^{\prime}$. The same holds for all other basic repair operations and consequently, when *AbstractRepair* finds a repaired KMTS, one or more KSs can be obtained for which property $\phi$ holds.
\[prop:add\_must\] For all $M^{\prime} \in K_{min}$, it holds that $1 \leq d(M,M^{\prime}) \leq \left|S\right|$.
Recall that $$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta
\pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$$ Since $|S\Delta S^{\prime}| = 0$ and $|\pi(L\upharpoonright_{S\cap
S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$, $d(M,M^{\prime}) = |R\Delta R^{\prime}| =
|R - R^{\prime}| + |R^{\prime} - R| = 0 + |R_{n}|$. Since $|R_{n}| \geq 1$ and $|R_{n}| \leq |S|$, it is proved that $1 \leq d(M,M^{\prime}) \leq \left|S\right|$.
From Prop. \[prop:add\_must\], we conclude that a lower and upper bound exists for the distance between $M$ and any $M^{\prime} \in K_{min}$.
### Adding a may-transition
\[def:AddMay\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$ and $\hat{r}_{n} = (\hat{s}_{1},\hat{s}_{2}) \notin R_{may}$, $AddMay(\hat{M},\hat{r}_{n})$ is the KMTS $\hat{M^{\prime}} = (\hat{S}, \hat{S_{0}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L})$ such that $R_{must}^{\prime} = R_{must} \cup \{\hat{r}_{n}\}$ if $\left|S_{1}\right| = 1$ or $R_{must}^{\prime} = R_{must}$ if $\left|S_{1}\right| > 1$ for $S_{1} = \{s_{1} \mid s_{1} \in \gamma(\hat{s}_{1})\}$ and $R_{may}^{\prime} = R_{may} \cup \{\hat{r}_{n}\}$.
From Def. \[def:AddMay\], we conclude that there are two different cases in adding a new may-transition $\hat{r}_{n}$; adding also a must-transition or not. In fact, $\hat{r}_{n}$ is also a must-transition if and only if the set of the corresponding concrete states of $\hat{s}_{1}$ is a singleton. Fig. \[fig:AddMay\] displays the two different cases of applying basic repair operation *AddMay* to a KMTS.
\[prop:AddMay\] For any $\hat{M}^{\prime} = AddMay(\hat{M},\hat{r}_{n})$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:add\_may\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = AddMay(\alpha(M),\hat{r}_{n})$ for some $\hat{r}_{n} = (\hat{s}_{1},\hat{s}_{2}) \notin R_{may}$. The set $K_{min} \subseteq \gamma(\hat{M}^{\prime})$ with all KSs, whose structural distance $d$ from $M$ is minimized is given by: $$K_{min} = \{M^{\prime} \mid M^{\prime} = (S, S_{0}, R \cup \{r_{n}\}, L)\}$$ where $r_{n} \in R_{n}$ and $R_{n} = \{r_{n}=(s_{1},s_{2}) \mid
s_{1} \in \gamma(\hat{s}_{1}), s_{2} \in \gamma(\hat{s}_{2})$ and $r_{n} \notin R\}$.
\[prop:add\_may\] For all $M^{\prime} \in K_{min}$, it holds that $d(M,M^{\prime}) = 1$.
$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$. Because $|S\Delta S^{\prime}| = 0$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$, $d(M,M^{\prime}) = |R\Delta R^{\prime}| =
|R - R^{\prime}| + |R^{\prime} - R| = 0 + |\{r_{n}\}| = 1$. So, we prove that $d(M,M^{\prime}) = 1$.
### Removing a must-transition
\[def:RemoveMust\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$ and $\hat{r}_{m} = (\hat{s}_{1},\hat{s}_{2}) \in R_{must}$, $RemoveMust(\hat{M},\hat{r}_{m})$ is the KMTS $\hat{M^{\prime}} = (\hat{S}, \hat{S_{0}},
R_{must}^{\prime},$ $R_{may}^{\prime}, \hat{L})$ such that $R_{must}^{\prime} = R_{must} - \{\hat{r}_{m}\}$ and $R_{may}^{\prime} = R_{may} - \{\hat{r}_{m}\}$ if $\left|S_{1}\right| = 1$ or $R_{may}^{\prime} = R_{may}$ if $\left|S_{1}\right| > 1$ for $S_{1} = \{s_{1} \mid s_{1} \in \gamma(\hat{s}_{1})\}$.
Removing a must-transition $\hat{r}_{m}$, in some special and maybe rare cases, could also result in the deletion of the may-transition $\hat{r}_{m}$ as well. In fact, this occurs if transitions to the concrete states of $\hat{s}_{2}$ exist only from one concrete state of the corresponding ones of $\hat{s}_{1}$. These two cases for function *RemoveMust* are presented graphically in Fig. \[fig:RemoveMust\].
\[prop:RemoveMust\] For any $\hat{M}^{\prime} = RemoveMust(\hat{M},\hat{r}_{m})$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:remove\_must\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = RemoveMust(\alpha(M),\hat{r}_{m})$ for some $\hat{r}_{m} = (\hat{s}_{1},\hat{s}_{2}) \in R_{must}$. The set $K_{min} \subseteq \gamma(\hat{M}^{\prime})$ with all KSs, whose structural distance $d$ from $M$ is minimized is given by: $$K_{min} = \{M^{\prime} \mid M^{\prime} = (S, S_{0}, R - \{R_{m}\}, L)\}$$ where $R_{m}$ is given for one $s_{1} \in \gamma(\hat{s}_{1})$ as follows: $$\nonumber
R_{m} = \bigcup_{s_{2} \in \gamma(\hat{s}_{2})} \{(s_{1}, s_{2}) \in R\} \eqno{\qEd}$$
\[prop:remove\_must\] For $M^{\prime}$, it holds that $1 \leq d(M,M^{\prime}) \leq \left|S\right|$.
$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$. Because $|S\Delta S^{\prime}| = 0$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$, $d(M,M^{\prime}) = |R\Delta R^{\prime}| =
|R - R^{\prime}| + |R^{\prime} - R| = |R_{m}| + 0 = |R_{m}|$. It holds that $|R_{m}| \geq 1$ and $|R_{m}| \leq |S|$. So, we proved that $1 \leq d(M,M^{\prime}) \leq \left|S\right|$.
### Removing a may-transition
\[def:RemoveMay\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$ and $\hat{r}_{m} = (\hat{s}_{1},\hat{s}_{2}) \in R_{may}$, $RemoveMay(\hat{M},\hat{r}_{m})$ is the KMTS $\hat{M^{\prime}} = (\hat{S}, \hat{S_{0}},
R_{must}^{\prime}, R_{may}^{\prime},$ $\hat{L})$ such that $R_{must}^{\prime} = R_{must} - \{\hat{r}_{m}\}$ and $R_{may}^{\prime} = R_{may} - \{\hat{r}_{m}\}$.
Def. \[def:RemoveMay\] ensures that removing a may-transition $\hat{r}_{m}$ implies the removal of a must-transition, if $\hat{r}_{m}$ is also a must-transition. Otherwise, there are not any changes in the set of must-transitions $R_{must}$. Fig. \[fig:RemoveMay\] shows how function *RemoveMay* works in both cases.
\[prop:RemoveMay\] For any $\hat{M}^{\prime} = RemoveMay(\hat{M},\hat{r}_{m})$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:remove\_may\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = RemoveMay(\alpha(M),\hat{r}_{m})$ for some $\hat{r}_{m} = (\hat{s}_{1},\hat{s}_{2}) \in R_{may}$ with $\hat{s}_{1},\hat{s}_{2} \in \hat{S}$. The KS $M^{\prime} \in \gamma(\hat{M}^{\prime})$, whose structural distance $d$ from $M$ is minimized is given by: $$M^{\prime} = (S, S_{0}, R - R_{m}, L\}$$ where $R_{m} = \{r_{m}=(s_{1},s_{2}) \mid
s_{1} \in \gamma(\hat{s}_{1}),
s_{2} \in \gamma(\hat{s}_{2})$ and $r_{m} \in R\}$.
\[prop:remove\_may\] For $M^{\prime}$, it holds that $1 \leq d(M,M^{\prime}) \leq \left|S\right|^{2}$.
$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$. Because $|S\Delta S^{\prime}| = 0$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$, $d(M,M^{\prime}) = |R\Delta R^{\prime}| =
|R - R^{\prime}| + |R^{\prime} - R| = 0 + |R_{m}| = |R_{m}|$. It holds that $|R_{m}| \geq 1$ and $|R_{m}| \leq |S|^{2}$. So, we proved that $1 \leq d(M,M^{\prime}) \leq \left|S\right|^{2}$.
### Changing the labeling of a KMTS state
\[def:ChangeLabel\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$, a state $\hat{s} \in \hat{S}$ and an atomic CTL formula $\phi$ with $\phi \in 2^{LIT}$, $ChangeLabel(\hat{M},\hat{s},\phi)$ is the KMTS $\hat{M^{\prime}} = (\hat{S}, \hat{S_{0}},
R_{must}, R_{may}, \hat{L^{\prime}})$ such that $\hat{L^{\prime}} = ( \hat{L} - \{\hat{l}_{old}\} )
\cup \{\hat{l}_{new}\}$ for $\hat{l}_{old} = (\hat{s},lit_{old})$ and $\hat{l}_{new} = (\hat{s},lit_{new})$ where $lit_{new} = \hat{L}(\hat{s}) \cup \{ lit \mid lit \in \phi \}
- \{ \neg lit \mid lit \in \phi \}$.
Basic repair operation *ChangeLabel* gives the possibility of repairing a model by changing the labeling of a state, thus without inducing any changes in the structure of the model (number of states or transitions). Fig. \[fig:ChangeLabel\] presents the application of *ChangeLabel* in a graphical manner.
\[prop:ChangeLabel\] For any $\hat{M}^{\prime} = ChangeLabel(\hat{M},\hat{s},\phi)$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:change\_label\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = ChangeLabel(\alpha(M),\hat{s},\phi)$ for some $\hat{s} \in \hat{S}$ and $\phi \in 2^{LIT}$. The KS $M^{\prime} \in \gamma(\hat{M}^{\prime})$, whose structural distance $d$ from $M$ is minimized, is given by: $$M^{\prime} = (S, S_{0}, R, L - L_{old} \cup L_{new}\}$$ where $$\nonumber
L_{old} = \{ l_{old} = (s,lit_{old}) \mid s \in \gamma(\hat{s}),
s \in S, \neg lit_{old} \not\in \phi \; \text{and} \; l_{old} \in L \}$$ $$\nonumber
L_{new} = \{ l_{new} = (s,lit_{new}) \mid s \in \gamma(\hat{s}),
s \in S, lit_{new} \in \phi \; \text{and} \; l_{new} \notin L \}$$
\[prop:change\_label\] For $M^{\prime}$, it holds that $1 \leq d(M,M^{\prime}) \leq |S|$.
$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$. Because $|R\Delta R^{\prime}| = 0$ and $|R\Delta R^{\prime}| = 0$, $d(M,M^{\prime}) =
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}=
\frac{|L_{old}| + |L_{new}|}{2} = |L_{old}| = |L_{new}|$. It holds that $L_{new} \geq 1$ and $L_{new} \leq |S|$. So, we prove that $1 \leq d(M,M^{\prime}) \leq |S|$.
### Adding a new KMTS state
\[def:AddState\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$ and a state $\hat{s}_{n} \notin \hat{S}$, $AddState(\hat{M},\hat{s}_{n})$ is the KMTS $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}},
R_{must}, R_{may}, \hat{L^{\prime}})$ such that $\hat{S^{\prime}} = \hat{S} \cup \{\hat{s}_{n}\}$ and $\hat{L^{\prime}} = \hat{L} \cup \{\hat{l}_{n}\}$, where $\hat{l}_{n} = (\hat{s}_{n},\bot)$.
The most important issues for function $AddState$ is that the newly created abstract state $\hat{s}_{n}$ is isolated, thus there are no ingoing or outgoing transitions for this state, and additionally, the labeling of this new state is $\bot$. Another conclusion from Def. \[def:AddState\] is the fact that the inserted stated is not permitted to be initial. Application of function $AddState$ is presented graphically in Fig. \[fig:AddState\].
\[prop:AddState\] For any $\hat{M}^{\prime} = AddState(\hat{M},\hat{s}_{n})$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:add\_state\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = AddState(\alpha(M),\hat{s}_{n})$ for some $\hat{s}_{n} \notin \hat{S}$. The KS $M^{\prime} \in \gamma(\hat{M}^{\prime})$, whose structural distance $d$ from $M$ is minimized is given by: $$M^{\prime} = (S \cup \{s_{n}\}, S_{0}, R, L \cup \{l_{n}\})$$ where $s_{n} \in \gamma(\hat{s}_{n})$ and $l_{n} = (s_{n},\bot)$.
\[prop:add\_state\] For $M^{\prime}$, it holds that $d(M,M^{\prime}) = 1$.
$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$. Because $|R\Delta R^{\prime}| = 0$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$, $d(M,M^{\prime}) = |S\Delta S^{\prime}| =
|S - S^{\prime}| + |S^{\prime} - S| = 0 + |\{s_{n}\}| = 1$. So, we proved that $d(M,M^{\prime}) = 1$.
### Removing a disconnected KMTS state
\[def:RemoveState\] For a given KMTS $\hat{M} = (\hat{S},\hat{S_{0}}, R_{must},
R_{may}, \hat{L})$ and a state $\hat{s}_{r} \in \hat{S}$ such that $\forall \hat{s} \in \hat{S} :
(\hat{s},\hat{s}_{r}) \not\in R_{may} \, \wedge \, (\hat{s}_{r},\hat{s})
\not\in R_{may}$, $RemoveState(\hat{M},\hat{s}_{r})$ is the KMTS $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}, R_{may}, \hat{L^{\prime}})$ such that $\hat{S^{\prime}} = \hat{S} - \{\hat{s}_{r}\}$, $\hat{S_{0}^{\prime}} = \hat{S_{0}} - \{\hat{s}_{r}\}$ and $\hat{L^{\prime}} = \hat{L} - \{\hat{l}_{r}\}$, where $\hat{l}_{r} = (\hat{s}_{r},lit) \in \hat{L}$.
From Def. \[def:RemoveState\], it is clear that the state being removed should be isolated, thus there are not any may- or must-transitions from and to this state. This means that before using *RemoveState* to an abstract state, all its ingoing or outgoing must have been removed by using other basic repair operations. *RemoveState* are also used for the elimination of dead-end states, when such states arise during the repair process. Fig. \[fig:RemoveState\] presents the application of *RemoveState* in a graphical manner.
\[prop:RemoveState\] For any $\hat{M}^{\prime} = RemoveState(\hat{M},\hat{s}_{r})$, it holds that $\hat{d}(\hat{M},\hat{M}^{\prime}) = 1$.
\[def:remove\_state\_ks\] Let $M = (S,S_{0},R,L)$ be a KS and let $\alpha(M) = (\hat{S},\hat{S_{0}},
R_{must}, R_{may}, \hat{L})$ be the abstract KMTS derived from $M$ as in Def. \[def:abs\_kmts\]. Also, let $\hat{M}^{\prime} = RemoveState(\alpha(M),\hat{s}_{r})$ for some $\hat{s}_{r} \in \hat{S}$ with $\hat{l}_{r} = (\hat{s}_{r},lit) \in \hat{L}$. The KS $M^{\prime} \in \gamma(\hat{M}^{\prime})$, whose structural distance $d$ from $M$ is minimized, is given by: $$M^{\prime} = (S^{\prime}, S_{0}^{\prime},
R^{\prime}, L^{\prime}) \mbox{ s.t. }
S^{\prime} = S - S_{r},
S_{0}^{\prime} = S_{0} - S_{r}, R^{\prime} = R,
L^{\prime} = L - L_{r}$$ where $S_{r} = \{ s_{r} \mid s_{r} \in S \mbox{ and } s_{r}
\in \gamma(\hat{s}_{r}) \}$ and $L_{r} = \{ l_{r} = (s_{r},lit) \mid l_{r} \in L \}$.
\[prop:remove\_state\] For $M^{\prime}$, it holds that $1 \leq d(M,M^{\prime}) \leq |S|$.
$d(M,M^{\prime}) = |S\Delta S^{\prime}| + |R\Delta R^{\prime}| +
\frac{|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})|}{2}$. Because $|R\Delta R^{\prime}| = 0$ and $|\pi(L\upharpoonright_{S\cap S^{\prime}})\Delta \pi(L^{\prime}\upharpoonright_{S\cap S^{\prime}})| = 0$, $d(M,M^{\prime}) = |S\Delta S^{\prime}| =
|S - S^{\prime}| + |S^{\prime} - S| = |S_{r}| + 0 = |S_{r}|$. It holds that $|S_{r}| \geq 1$ and $|S_{r}| \leq |S|$. So, we proved that $1 \leq d(M,M^{\prime}) \leq |S|$.
### Minimality Of Changes Ordering For Basic Repair Operations {#subsec:minimal_basic_ops}
The distance metric $d$ of Def. \[def:metric\_space\] reflects the need to quantify structural changes in the concrete model that are attributed to model repair steps applied to the abstract KMTS. Every such repair step implies multiple structural changes in the concrete KSs, due to the use of abstraction. In this context, our distance metric is an essential means for the effective application of the abstraction in the repair process.
Based on the upper bound given by Prop. \[prop:add\_must\] and all the respective results for the other basic repair operations, we introduce the partial ordering shown in Fig. \[fig:order\_basic\_ops\]. This ordering is used in our *AbstractRepair* algorithm to *heuristically* select at each step the basic repair operation that *generates the KSs with the least changes*. When it is possible to apply more than one basic repair operation with the same upper bound, our algorithm successively uses them until a repair solution is found, in an order based on the computational complexity of their application.
If instead of our approach, all possible repaired KSs were checked to identify the basic repair operation with the minimum changes, this would defeat the purpose of using abstraction. The reason is that such a check inevitably would depend on the size of concrete KSs.
![Minimality of changes ordering of the set of basic repair operations[]{data-label="fig:order_basic_ops"}](MinimalityOrderingBasicOperations)
The Abstract Model Repair Algorithm {#sec:alg}
===================================
The *AbstractRepair* algorithm used in Step 3 of our repair process is a recursive, syntax-directed algorithm, where the syntax for the property $\phi$ in question is that of CTL. The same approach is followed by the SAT model checking algorithm in [@HR04] and a number of model repair solutions applied to concrete KSs [@ZD08; @CR09]. In our case, we aim to the repair of an abstract KMTS by successively calling primitive repair functions that handle atomic formulas, logical connectives and CTL operators. At each step, the repair with the least changes for the concrete model among all the possible repairs is applied first.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi$ in PNF for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. FAILURE $AbstractRepair_{ATOMIC}(\hat{M},\hat{s},\phi,C)$ $AbstractRepair_{AND}(\hat{M},\hat{s},\phi,C)$ $AbstractRepair_{OR}(\hat{M},\hat{s},\phi,C)$ $AbstractRepair_{OPER}(\hat{M},\hat{s},\phi,C)$ where $OPER \in \{AX,EX,AU,EU,AF,EF,AG,EG\}$
The main routine of *AbstractRepair* is presented in Algorithm \[alg:main\]. If the property $\phi$ is not in Positive Normal Form, i.e. negations are applied only to atomic propositions, then we transform it into such a form before applying Algorithm \[alg:main\].
An initially empty set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ is passed as an argument in the successive recursive calls of *AbstractRepair*. We note that these constraints can also specify *existing* properties that should be preserved during repair. If $C$ is not empty, then for the returned KMTS $\hat{M}^{\prime}$, it holds that $(\hat{M^{\prime}},\hat{s}_{c_{i}}) \models \phi_{c_{i}}$ for all $(\hat{s}_{c_{i}},\phi_{c_{i}}) \in C$. For brevity, we denote this with $\hat{M}^{\prime} \models C$. We use $C$ in order to handle conjunctive formulas of the form $\phi = \phi_{1} \wedge \phi_{2}$ for some state $\hat{s}$. In this case, *AbstractRepair* is called for the KMTS $\hat{M}$ and property $\phi_{1}$ with $C = \{ (\hat{s},\phi_{2}) \}$. The same is repeated for property $\phi_{2}$ with $C = \{ (\hat{s},\phi_{1}) \}$ and the two results are combined appropriately.
For any CTL formula $\phi$ and KMTS state $\hat{s}$, *AbstractRepair* either outputs a KMTS $\hat{M}^{\prime}$ for which $(\hat{M^{\prime}},\hat{s}) \models \phi$ or else returns FAILURE, if such a model cannot be found. This is the case when the algorithm handles conjunctive formulas and a KMTS that simultaneously satisfies all conjuncts cannot be found.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi$ where $\phi$ is an atomic formula for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{M^{\prime}} := ChangeLabel(\hat{M},\hat{s},\phi)$ $\hat{M^{\prime}}$ FAILURE
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = \phi_{1} \vee \phi_{2}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = ( (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) )$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$, $\hat{s} \in \hat{S^{\prime}}$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $RET_{1} := AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ $RET_{2} := AbstractRepair(\hat{M},\hat{s},\phi_{2},C)$ $\hat{M}_{1} := RET_{1}$ $\hat{M}_{2} := RET_{2}$ $\hat{M^{\prime}} := MinimallyChanged(\hat{M},\hat{M_{1}},\hat{M_{2}})$ $\hat{M^{\prime}} := RET_{1}$ $\hat{M^{\prime}} := RET_{2}$ FAILURE $\hat{M}^{\prime}$
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = \phi_{1} \wedge \phi_{2}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = ( (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) )$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$, $\hat{s} \in \hat{S^{\prime}}$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $RET_{1} := AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ $RET_{2} := AbstractRepair(\hat{M},\hat{s},\phi_{2},C)$ $C_{1} := C \cup \{ (\hat{s},\phi_{1}) \}$, $C_{2} := C \cup \{(\hat{s},\phi_{2})\}$ $RET_{1}^{\prime} := FAIURE$, $RET_{2}^{\prime} := FAIURE$ $\hat{M}_{1} := RET_{1}$ $RET_{1}^{\prime} := AbstractRepair(\hat{M}_{1},\hat{s},\phi_{2},C_{1})$ $\hat{M}_{1}^{\prime} := RET_{1}^{\prime}$ $\hat{M}_{2} := RET_{2}$ $RET_{2}^{\prime} := AbstractRepair(\hat{M}_{2},\hat{s},\phi_{1},C_{2})$ $\hat{M}_{2}^{\prime} := RET_{2}^{\prime}$ $\hat{M^{\prime}} := MinimallyChanged(\hat{M},\hat{M}_{1}^{\prime},\hat{M}_{2}^{\prime})$ $\hat{M^{\prime}} := RET_{1}^{\prime}$ $\hat{M^{\prime}} := RET_{2}^{\prime}$ FAILURE $\hat{M}^{\prime}$
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = AG\phi_{1}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $RET := AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ FAILURE $\hat{M^{\prime}} := RET$ $\hat{M^{\prime}} := \hat{M}$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{k},\phi_{1},C)$ FAILURE $\hat{M^{\prime}} := RET$ $\hat{M^{\prime}}$ FAILURE
Primitive Functions {#subsec:alg_prim_func}
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Algorithm \[alg:ATOMIC\] describes $AbstractRepair_{ATOMIC}$, which for a simple atomic formula, updates the labeling of the input state with the given atomic proposition. Disjunctive formulas are handled by repairing the disjunct leading to the minimum change (Algorithm \[alg:OR\]), while conjunctive formulas are handled by the algorithm with the use of constraints (Algorithm \[alg:AND\]).
Algorithm \[alg:AG\] describes the primitive function $AbstractRepair_{AG}$ which is called when $\phi = AG\phi_{1}$. If $AbstractRepair_{AG}$ is called for a state $\hat{s}$, it recursively calls *AbstractRepair* for $\hat{s}$ and for all reachable states through may-transitions from $\hat{s}$ which do not satisfy $\phi_{1}$. The resulting KMTS $\hat{M}^{\prime}$ is returned, if it does not violate any constraint in $C$.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = EX\phi_{1}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{M}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{r}_{i} := (\hat{s},\hat{s}_{i})$, $\hat{M^{\prime}} := AddMust(\hat{M},\hat{r}_{i})$ $\hat{M^{\prime}}$ $RET := AbstractRepair(\hat{M},\hat{s}_{i},\phi_{1},C)$ $\hat{M^{\prime}} := RET$ $\hat{M^{\prime}}$ $\hat{M^{\prime}} := AddState(\hat{M},\hat{s}_{n})$, $\hat{r}_{n} := (\hat{s},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMust(\hat{M^{\prime}},\hat{r}_{n})$ $\hat{r}_{n} := (\hat{s}_{n},\hat{s}_{n})$ $\hat{M^{\prime}} := AddMay(\hat{M^{\prime}},\hat{r}_{n})$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{n},\phi_{1},C)$ $\hat{M^{\prime}} := RET$ $\hat{M^{\prime}}$ FAILURE
$AbstractRepair_{EX}$ presented in Algorithm \[alg:EX\] is the primitive function for handling properties of the form $EX\phi_{1}$ for some state $\hat{s}$. At first, $AbstractRepair_{EX}$ attempts to repair the KMTS by adding a must-transition from $\hat{s}$ to a state that satisfies property $\phi_{1}$. If a repaired KMTS is not found, then *AbstractRepair* is recursively called for an immediate successor of $\hat{s}$ through a must-transition, such that $\phi_{1}$ is not satisfied. If a constraint in $C$ is violated, then (i) a new state is added, (ii) *AbstractRepair* is called for the new state and (iii) a must-transition from $\hat{s}$ to the new state is added. The resulting KMTS is returned by the algorithm if all constraints of $C$ are satisfied.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = AX\phi_{1}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{M}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{M^{\prime}} := \hat{M}$ $RET := FAILURE$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{i},\phi_{1},C)$ BREAK $\hat{M^{\prime}} := RET$ $\hat{M^{\prime}}$ $\hat{M^{\prime}} := \hat{M}$ $\hat{M^{\prime}} := RemoveMay(\hat{M^{\prime}},\hat{r}_{i})$ $\hat{M^{\prime}}$ $\hat{r}_{j} := (\hat{s},\hat{s}_{j})$, $\hat{M^{\prime}} := AddMay(\hat{M^{\prime}},\hat{r}_{j})$ $\hat{M^{\prime}}$ $\hat{M^{\prime}} := AddState(\hat{M},\hat{s}_{n})$ $\hat{r}_{n} := (\hat{s}_{n},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMay(\hat{M^{\prime}},\hat{r}_{n})$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{n},\phi_{1},C)$ $\hat{M^{\prime}} := RET$, $\hat{r}_{n} := (\hat{s},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMay(\hat{M^{\prime}},\hat{r}_{n})$ $\hat{M^{\prime}}$ FAILURE
Algorithm \[alg:AX\] presents primitive function $AbstractRepair_{AX}$ which is used when $\phi = AX\phi_{1}$. Firstly, $AbstractRepair_{AX}$ tries to repair the KMTS by applying $AbstractRepair$ for all direct may-successors $\hat{s}_{i}$ of $\hat{s}$ which do not satisfy property $\phi_{1}$, and in the case that all the constraints are satisfied the new KMTS is returned by the function. If such states do not exist or a constraint is violated, all may-transitions $(\hat{s},\hat{s}_{i})$ for which $(\hat{M},\hat{s}_{i}) \not\models \phi_{1}$, are removed. If there are states $\hat{s}_{i}$ such that $r_{m} := (\hat{s},\hat{s}_{i}) \in R_{may}$ and all constraints are satisfied then a repaired KMTS has been produced and it is returned by the function. Otherwise, a repaired KMTS results by the application of $AddMay$ from $\hat{s}$ to all states $\hat{s}_{j}$ which satisfy $\phi_{1}$. If any constraint is violated, then the KMTS is repaired by adding a new state, applying $AbstractRepair$ to this state for property $\phi_{1}$ and adding a may-transition from $\hat{s}$ to this state. If all constraints are satisfied, the repaired KMTS is returned.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = EG\phi_{1}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{M}_{1} := \hat{M}$ $RET := AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ FAILURE $\hat{M}_{1} := RET$ $\hat{r}_{1} := (\hat{s},\hat{s}_{1})$, $\hat{M^{\prime}} := AddMust(\hat{M}_{1},\hat{r}_{1})$ $\hat{M^{\prime}}$ $\hat{M^{\prime}} := \hat{M}_{1}$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{i},\phi_{1},C)$ $\hat{M^{\prime}} := RET$ continue to next path $\hat{M^{\prime}}$ $\hat{M^{\prime}} := AddState(\hat{M}_{1},\hat{s}_{n})$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{n},\phi_{1},C)$ $\hat{M^{\prime}} := RET$ $\hat{r}_{n} := (\hat{s},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMust(\hat{M^{\prime}},\hat{r}_{n})$ $\hat{r}_{n} := (\hat{s}_{n},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMust(\hat{M^{\prime}},\hat{r}_{n})$ $\hat{M^{\prime}}$ FAILURE
$AbstractRepair_{EG}$ which is presented in Algorithm \[alg:EG\] is the primitive function which is called when input CTL property is in the form of $EG\phi_{1}$. Initially, if $\phi_{1}$ is not satisfied at $\hat{s}$ $AbstractRepair$ is called for $\hat{s}$ and $\phi_{1}$, and a KMTS $\hat{M}_{1}$ is produced. At first, a must-transition is added from $\hat{s}$ to a state $\hat{s}_{1}$ of a maximal must-path (i.e. a must-path in which each transition appears at most once) $\pi_{must} := [\hat{s}_{1},\hat{s}_{2},...]$ such that $\forall \hat{s}_{i} \in \pi_{must}$, $(\hat{M}_{1},\hat{s}_{i}) \models \phi_{1}$. If all constraints are satisfied, then the repaired KMTS is returned. Otherwise, a KMTS is produced by recursively calling $AbstractRepair$ to all states $\hat{s}_{i} \neq \hat{s}$ of any maximal must-path $\pi_{must} := [\hat{s}_{1},\hat{s}_{2},...]$ with $\forall \hat{s}_{i} \in \pi_{must}$, $(\hat{M}_{1},\hat{s}_{i}) \not\models \phi_{1}$. If there are violated constraints in $C$, then a repaired KMTS is produced by adding a new state, calling $AbstractRepair$ for this state and property $\phi_{1}$ and calling $AddMust$ to insert a must-transition from $\hat{s}$ to the new state. The resulting KMTS is returned by the algorithm, if all constraints in $C$ are satisfied.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = AF\phi_{1}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{M^{\prime}} := \hat{M}$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{i},\phi_{1},C)$ $\hat{M^{\prime}} := RET$ continue to next path FAILURE $\hat{M}^{\prime}$
$AbstractRepair_{AF}$ shown in Algorithm \[alg:AF\] is called when the CTL formula $\phi$ is in the form of $AF\phi_{1}$. While there is maximal may-path $\pi_{may} := [\hat{s},\hat{s}_{1},...]$ such that $\forall \hat{s}_{i} \in \pi_{may}$, $(\hat{M^{\prime}},\hat{s}_{i}) \not\models \phi_{1}$, $AbstractRepair_{AF}$ tries to obtain a repaired KMTS by recursively calling $AbstractRepair$ to some state $\hat{s}_{i} \in \pi_{may}$. If all constraints are satisfied to the new KMTS, then it is returned as the repaired model.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = EF\phi_{1}$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{r}_{k} := (\hat{s}_{i},\hat{s}_{k})$, $\hat{M^{\prime}} := AddMust(\hat{M},\hat{r}_{k})$ $\hat{M^{\prime}}$ $RET := AbstractRepair(\hat{M},\hat{s}_{i},\phi_{1},C)$ $\hat{M^{\prime}} := RET$ $\hat{M^{\prime}}$ $\hat{M}_{1} := AddState(\hat{M^{\prime}},\hat{s}_{n})$, $RET := AbstractRepair(\hat{M}_{1},\hat{s}_{n},\phi_{1},C)$ $\hat{M}_{1} := RET$ $\hat{r}_{i} := (\hat{s}_{i},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMust(\hat{M}_{1},\hat{r}_{i})$ $\hat{r}_{n} := (\hat{s}_{n},\hat{s}_{n})$, $\hat{M^{\prime}} := AddMust(\hat{M^{\prime}},\hat{r}_{n})$ $\hat{M^{\prime}}$ FAILURE
$AbstractRepair_{EF}$ shown in Algorithm \[alg:EF\] is called when the CTL property $\phi$ is in the form $EF\phi_{1}$. Initially, a KMTS is acquired by adding a must-transition from a must-reachable state $\hat{s}_{i}$ from $\hat{s}$ to a state $\hat{s}_{k} \in \hat{S}$ such that $(\hat{M},\hat{s}_{k}) \models \phi_{1}$. If all constraints are satisfied then this KMTS is returned. Otherwise, a KMTS is produced by applying $AbstractRepair$ to a must-reachable state $\hat{s}_{i}$ from $\hat{s}$ for $\phi_{1}$. If none of the constraints is violated then this KMTS is returned. At any other case, a new KMTS is produced by adding a new state $\hat{s}_{n}$, recursively calling $AbstractRepair$ for this state and $\phi_{1}$ and adding a must-transition from $\hat{s}$ or from a must-reachable $\hat{s}_{i}$ from $\hat{s}$ to $\hat{s}_{n}$. If all constraints are satisfied, then this KMTS is returned as a repaired model by the algorithm.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = A(\phi_{1}U\phi_{2})$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{M}_{1} := \hat{M}$ $RET := AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ FAILURE $\hat{M}_{1} := RET$ $RET := AbstractRepair(\hat{M}_{1},\hat{s}_{j},\phi_{2},C)$ $\hat{M^{\prime}} := RET$ continue to next path FAILURE $\hat{M^{\prime}}$
$AbstractRepair_{AU}$ is presented in Algorithm \[alg:AU\] and is called when $\phi = A(\phi_{1}U\phi_{2})$. If $\phi_{1}$ is not satisfied at $\hat{s}$, then a KMTS $\hat{M}_{1}$ is produced by applying $AbstractRepair$ to $\hat{s}$ for $\phi_{1}$. Otherwise, $\hat{M}_{1}$ is same to $\hat{M}$. A new KMTS is produced as follows: for all may-paths $\pi_{may} := [\hat{s}_{1},...,\hat{s}_{m}]$ such that $\forall \hat{s}_{i} \in \pi_{may}$, $(\hat{M}_{1},\hat{s}_{i}) \models \phi_{1}$ and for which there does not $\hat{r}_{m} := (\hat{s}_{m},\hat{s}_{n}) \in R_{may}$ with $(\hat{M}_{1},\hat{s}_{n}) \models \phi_{2}$, $AbstractRepair$ is called for property $\phi_{2}$ for some state $\hat{s}_{j} \in \pi_{may}$ with $(\hat{M}_{1},\hat{s}_{j}) \not\models \phi_{2}$. If the resulting KMTS satisfies all constraints, then it is returned as a repair solution.
$\hat{M} = (\hat{S}, \hat{S}_{0}, R_{must}, R_{may},
\hat{L})$, $\hat{s} \in \hat{S}$, a CTL property $\phi = E(\phi_{1}U\phi_{2})$ for which $(\hat{M},\hat{s}) \not\models \phi$, and a set of constraints $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}),
(\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ where $\hat{s}_{c_{i}} \in \hat{S}$ and $\phi_{c_{i}}$ is a CTL formula. $\hat{M^{\prime}} = (\hat{S^{\prime}}, \hat{S_{0}^{\prime}},
R_{must}^{\prime}, R_{may}^{\prime}, \hat{L^{\prime}})$ and $(\hat{M^{\prime}},\hat{s}) \models \phi$ or FAILURE. $\hat{M}_{1} := \hat{M}$ $RET := AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ FAILURE $\hat{M}_{1} := RET$ $\hat{r}_{j} := (\hat{s}_{m},\hat{s}_{j})$, $\hat{M}^{\prime} := AddMust(\hat{M}_{1},\hat{r}_{j})$ $\hat{M^{\prime}}$ $\hat{M^{\prime}} := AddState(\hat{M}_{1},\hat{s}_{k})$ $RET := AbstractRepair(\hat{M^{\prime}},\hat{s}_{k},\phi_{2},C)$ $\hat{M^{\prime}} := RET$ $\hat{r}_{n} := (\hat{s},\hat{s}_{k})$, $\hat{M^{\prime}} := AddMust(\hat{M^{\prime}},\hat{r}_{n})$ $\hat{r}_{k} := (\hat{s}_{k},\hat{s}_{k})$, $\hat{M^{\prime}} := AddMust(\hat{M^{\prime}},\hat{r}_{k})$ $\hat{M^{\prime}}$ FAILURE
$AbstractRepair_{EU}$ is called if for input CTL formula $\phi$ it holds that $\phi = E(\phi_{1}U\phi_{2})$. $AbstractRepair_{EU}$ is presented in Algorithm \[alg:EU\]. Firstly, if $\phi_{1}$ is not satisfied at $\hat{s}$, then $AbstractRepair$ is called for $\hat{s}$ and $\phi_{1}$ and a KMTS $\hat{M}_{1}$ is produced for which $(\hat{M}_{1},\hat{s}) \models \phi_{1}$. Otherwise, $\hat{M}_{1}$ is same to $\hat{M}$. A new KMTS is produced as follows: for a must-path $\pi_{must} := [\hat{s}_{1},...,\hat{s}_{m}]$ such that $\forall \hat{s}_{i} \in \pi_{must}$, $(\hat{M}_{1},\hat{s}_{i}) \models \phi_{1}$ and for a $\hat{s}_{j} \in \hat{S}$ with $(\hat{M}_{1},\hat{s}_{j}) \models \phi_{2}$, a must-transition is added from $\hat{s}_{m}$ to $\hat{s}_{j}$. If all constraints are satisfied then the new KMTS is returned. Alternatively, a KMTS is produced by adding a new state $\hat{s}_{n}$, recursively calling $AbstractRepair$ for $\phi_{2}$ and $\hat{s}_{n}$ and adding a must-transition from $\hat{s}$ to $\hat{s}_{n}$. In the case that no constraint is violated then this is a repaired KMTS and it is returned from the function.
Properties of the Algorithm {#subsec:alg_props}
---------------------------
*AbstractRepair* is *well-defined* [@BGS07], in the sense that the algorithm always proceeds and eventually returns a result $\hat{M}^{\prime}$ or FAILURE such that $(\hat{M}^\prime,\hat{s}) \models \phi$, for any input $\hat{M}$, $\phi$ and $C$, with $(\hat{M},\hat{s}) \not\models \phi$. Moreover, the algorithm steps are well-ordered, as opposed to existing concrete model repair solutions [@CR11; @ZD08] that entail nondeterministic behavior.
### Soundness {#subsubsec:alg_soundness}
\[theor:sound\_help\] Let a KMTS $\hat{M}$, a CTL formula $\phi$ with $(\hat{M},\hat{s}) \not\models \phi$ for some $\hat{s}$ of $\hat{M}$, and a set $C = \{ (\hat{s}_{c_{1}},\phi_{c_{1}}), (\hat{s}_{c_{2}},\phi_{c_{2}}), ..., (\hat{s}_{c_{n}},\phi_{c_{n}}) \}$ with $(\hat{M},\hat{s}_{c_{i}}) \models \phi_{c_{i}}$ for all $(\hat{s}_{c_{n}},\phi_{c_{n}}) \in C$. If $AbstractRepair(\hat{M},\hat{s},\phi,C)$ returns a KMTS $\hat{M}^{\prime}$, then $(\hat{M}^{\prime},\hat{s}) \models \phi$ and $(\hat{M}^{\prime},\hat{s}_{c_{i}}) \models \phi_{c_{i}}$ for all $(\hat{s}_{c_{i}},\phi_{c_{i}}) \in C$.
We use structural induction on $\phi$. For brevity, we write $\hat{M} \models C$ to denote that $(\hat{M},\hat{s}_{c_{i}}) \models \phi_{c_{i}}$, for all $(\hat{s}_{c_{i}},\phi_{c_{i}}) \in C$.
#### Base Case:
- if $\phi = \top$, the lemma is trivially true, because $(\hat{M},\hat{s}) \models \phi$
- if $\phi = \bot$, then $AbstractRepair(\hat{M},\hat{s},\phi,C)$ returns FAILURE at line 2 of Algorithm \[alg:main\] and the lemma is also trivially true.
- if $\phi = p \in AP$, $AbstractRepair_{ATOMIC}(\hat{M},\hat{s},p,C)$ is called at line 4 of Algorithm \[alg:main\] and an $\hat{M^{\prime}} = ChangeLabel(\hat{M},\hat{s},p)$ is computed at line 1 of Algorithm \[alg:ATOMIC\]. Since $p \in \hat{L}^{\prime}(\hat{s})$ in $\hat{M^{\prime}}$, from 3-valued semantics of CTL over KMTSs we have $(\hat{M^{\prime}},\hat{s}) \models \phi$. Algorithm \[alg:ATOMIC\] returns $\hat{M^{\prime}}$ at line 3, if and only if $\hat{M}^{\prime} \models C$ and the lemma is true.
#### Induction Hypothesis:
For CTL formulae $\phi_{1}, \phi_{2}$, the lemma is true. Thus, for $\phi_{1}$ (resp. $\phi_{2}$), if $AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ returns a KMTS $\hat{M}^{\prime}$, then $(\hat{M^{\prime}},\hat{s}) \models \phi_{1}$ and $\hat{M}^{\prime} \models C$.
#### Inductive Step:
- if $\phi = \phi_{1} \vee \phi_{2}$, then $AbstractRepair(\hat{M},\hat{s},\phi,C)$ calls $AbstractRepair_{OR}(\hat{M},\hat{s},\phi_{1} \vee \phi_{2},C)$ at line 8 of Algorithm \[alg:main\]. From the induction hypothesis, if a KMTS $\hat{M}_{1}$ is returned by $AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ at line 1 of Algorithm \[alg:OR\] and a KMTS $\hat{M}_{2}$ is returned by $AbstractRepair(\hat{M},\hat{s},\phi_{2},C)$ respectively, then $(\hat{M}_{1},\hat{s}) \models \phi_{1}$, $\hat{M}_{1} \models C$ and $(\hat{M}_{2},\hat{s}) \models \phi_{1}$, $\hat{M}_{2} \models C$. $AbstractRepair_{OR}(\hat{M},\hat{s},\phi_{1} \vee \phi_{2},C)$ returns at line 8 of Algorithm \[alg:main\] the KMTS $\hat{M^{\prime}}$, which can be either $\hat{M}_{1}$ or $\hat{M}_{2}$. Therefore, $(\hat{M^{\prime}},\hat{s}) \models \phi_{1}$ or $(\hat{M^{\prime}},\hat{s}) \models \phi_{2}$ and $\hat{M^{\prime}} \models C$ in both cases. From 3-valued semantics of CTL, $(\hat{M^{\prime}},\hat{s}) \models \phi_{1} \vee \phi_{2}$ and the lemma is true.
- if $\phi = \phi_{1} \wedge \phi_{2}$, then $AbstractRepair(\hat{M},\hat{s},\phi,C)$ calls $AbstractRepair_{AND}(\hat{M},\hat{s},\phi_{1} \wedge \phi_{2},C)$ at line 6 of Algorithm \[alg:main\]. From the induction hypothesis, if at line 1 of Algorithm \[alg:AND\] $AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ returns a KMTS $\hat{M}_{1}$, then $(\hat{M}_{1},\hat{s}) \models \phi_{1}$ and $\hat{M}_{1} \models C$. Consequently, $\hat{M}_{1} \models C_{1}$, where $C_{1} = C \cup {(\hat{s},\phi_{1})}$. At line 7, if $AbstractRepair(\hat{M}_{1},\hat{s},\phi_{2},C_{1})$ returns a KMTS $\hat{M}_{1}^{\prime}$, then from the induction hypothesis $(\hat{M}_{1}^{\prime},\hat{s}) \models \phi_{2}$ and $\hat{M}_{1}^{\prime} \models C_{1}$.
In the same manner, if the calls at lines 2 and 12 of Algorithm \[alg:AND\] return the KMTSs $\hat{M}_{2}$ and $\hat{M}_{2}^{\prime}$, then from the induction hypothesis $(\hat{M}_{2},\hat{s}) \models \phi_{2}$, $\hat{M}_{2} \models C$ and $(\hat{M}_{2}^{\prime},\hat{s}) \models \phi_{1}$, $\hat{M}_{2}^{\prime} \models C_{2}$ with $C_{2} = C \cup {(\hat{s},\phi_{2})}$.
The KMTS $\hat{M^{\prime}}$ at line 6 of Algorithm \[alg:main\] can be either $\hat{M}_{1}^{\prime}$ or $\hat{M}_{2}^{\prime}$ and therefore, $(\hat{M^{\prime}},\hat{s}) \models \phi_{1}$, $(\hat{M^{\prime}},\hat{s}) \models \phi_{2}$ and $\hat{M^{\prime}} \models C$. From 3-valued semantics of CTL it holds that $(\hat{M^{\prime}},\hat{s}) \models \phi_{1} \wedge \phi_{2}$ and the lemma is true.
- if $\phi = EX\phi_{1}$, $AbstractRepair(\hat{M},\hat{s},\phi,C)$ calls $AbstractRepair_{EX}(\hat{M},\hat{s},EX\phi_{1},C)$ at line 10 of Algorithm \[alg:main\].
If a KMTS $\hat{M}^{\prime}$ is returned at line 5 of Algorithm \[alg:EX\], there is a state $\hat{s}_{1}$ with $(\hat{M},\hat{s}_{1}) \models \phi_{1}$ such that $\hat{M}^{\prime} = AddMust(\hat{M},(\hat{s},\hat{s}_{1}))$ and $\hat{M^{\prime}} \models C$. From 3-valued semantics of CTL, we conclude that $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
If a $\hat{M}^{\prime}$ is returned at line 11, there is $(\hat{s},\hat{s}_{1}) \in R_{must}$ such that $(\hat{M^{\prime}},\hat{s}_{1}) \models \phi_{1}$ and $\hat{M^{\prime}} \models C$ from the induction hypothesis, since $\hat{M^{\prime}}=AbstractRepair(\hat{M},\hat{s}_{1},\phi_{1},C)$. From 3-valued semantics of CTL, we conclude that $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
If a $\hat{M}^{\prime}$ is returned at line 18, a must transition $(\hat{s},\hat{s}_{n})$ to a new state has been added and $\hat{M}^{\prime}=AbstractRepair(AddMust(\hat{M},(\hat{s},\hat{s}_{n})),\hat{s}_{n},\phi_{1},C)$. Then, from the induction hypothesis $(\hat{M^{\prime}},\hat{s}_{n}) \models \phi_{1}$, $\hat{M^{\prime}} \models C$ and from 3-valued semantics of CTL, we also conclude that $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
- if $\phi = AG\phi_{1}$, $AbstractRepair(\hat{M},\hat{s},\phi,C)$ calls $AbstractRepair_{AG}(\hat{M},\hat{s},AG\phi_{1},C)$ at line 10 of Algorithm \[alg:main\]. If $(\hat{M},\hat{s}) \not\models \phi_{1}$ and $AbstractRepair(\hat{M},\hat{s},\phi_{1},C)$ returns a KMTS $\hat{M}_{0}$ at line 2 of Algorithm \[alg:AG\], then from the induction hypothesis $(\hat{M}_{0},\hat{s}) \models \phi_{1}$ and $\hat{M}_{0} \models C$. Otherwise, $\hat{M}_{0} = \hat{M}$ and $(\hat{M}_{0},\hat{s}) \models \phi_{1}$ also hold true.
If Algorithm \[alg:AG\] returns a $\hat{M}^{\prime}$ at line 16, then $\hat{M}^{\prime} \models C$ and $\hat{M}^{\prime}$ is the result of successive $AbstractRepair(\hat{M_{i}},\hat{s}_{k},\phi_{1},C)$ calls with $\hat{M_{i}}=AbstractRepair(\hat{M}_{i-1},\hat{s}_{k},\phi_{1},C)$ and $i=1, . . .$, for all may-reachable states $\hat{s}_{k}$ from $\hat{s}$ such that $(\hat{M}_{0},\hat{s}_{k}) \not\models \phi_{1}$. From the induction hypothesis, $(\hat{M}^{\prime},\hat{s}_{k}) \models \phi_{1}$ and $\hat{M^{\prime}} \models C$ for all such $\hat{s}_{k}$ and from 3-valued semantics of CTL we conclude that $(\hat{M^{\prime}},\hat{s}) \models AG\phi_{1}$.
We prove the lemma for all other cases in a similar manner.
\[theor:sound\] Let a KMTS $\hat{M}$, a CTL formula $\phi$ with $(\hat{M},\hat{s}) \not\models \phi$, for some $\hat{s}$ of $\hat{M}$. If $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ returns a KMTS $\hat{M}^{\prime}$, then $(\hat{M}^{\prime},\hat{s}) \models \phi$.
We use structural induction on $\phi$ and Lemma \[theor:sound\_help\] in the inductive step for $\phi_{1} \wedge \phi_{2}$.
#### Base Case:
- if $\phi = \top$, Theorem \[theor:sound\] is trivially true, because $(\hat{M},\hat{s}) \models \phi$.
- if $\phi = \bot$, then $AbstractRepair(\hat{M},\hat{s},\bot,\emptyset)$ returns FAILURE at line 2 of Algorithm \[alg:main\] and the theorem is also trivially true.
- if $\phi = p \in AP$, $AbstractRepair_{ATOMIC}(\hat{M},\hat{s},p,\emptyset)$ is called at line 4 of Algorithm \[alg:main\] and an $\hat{M^{\prime}} =
ChangeLabel(\hat{M},\hat{s},p)$ is computed at line 1. Because of the fact that $p \in \hat{L}^{\prime}(\hat{s})$ in $\hat{M^{\prime}}$, from 3-valued semantics of CTL over KMTSs we have $(\hat{M^{\prime}},\hat{s}) \models \phi$. Algorithm \[alg:ATOMIC\] returns $\hat{M^{\prime}}$ at line 3 because $C$ is empty, and the theorem is true.
#### Induction Hypothesis:
For CTL formulae $\phi_{1}$, $\phi_{2}$, the theorem is true. Thus, for $\phi_{1}$ (resp. $\phi_{2}$), if $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ returns a KMTS $\hat{M}^{\prime}$, then $(\hat{M^{\prime}},\hat{s}) \models \phi_{1}$.
#### Inductive Step:
- if $\phi = \phi_{1} \vee \phi_{2}$, then $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ calls $AbstractRepair_{OR}(\hat{M},\hat{s},\phi_{1} \vee \phi_{2},\emptyset)$ at line 8 of Algorithm \[alg:main\].
From the induction hypothesis, if $AbstractRepair(\hat{M},\hat{s},\phi_{1},\emptyset)$ returns a KMTS $\hat{M}_{1}$ at line 1 of Algorithm \[alg:OR\] and $AbstractRepair(\hat{M},\hat{s},\phi_{2},\emptyset)$ returns a KMTS $\hat{M}_{2}$ respectively, then $(\hat{M}_{1},\hat{s}) \models \phi_{1}$ and $(\hat{M}_{2},\hat{s}) \models \phi_{1}$. $AbstractRepair_{OR}(\hat{M},\hat{s},\phi_{1} \vee \phi_{2},\emptyset)$ returns at line 8 of Algorithm \[alg:main\] the KMTS $\hat{M^{\prime}}$, which can be either $\hat{M}_{1}$ or $\hat{M}_{2}$. Therefore, $(\hat{M^{\prime}},\hat{s}) \models \phi_{1}$ or $(\hat{M^{\prime}},\hat{s}) \models \phi_{2}$. From 3-valued semantics of CTL, $(\hat{M^{\prime}},\hat{s}) \models \phi_{1} \vee \phi_{2}$ and the theorem is true.
- if $\phi = \phi_{1} \wedge \phi_{2}$, then $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ calls $AbstractRepair_{AND}(\hat{M},\hat{s},\phi_{1} \wedge \phi_{2},\emptyset)$ at line 6 of Algorithm \[alg:main\]. From the induction hypothesis, if at line 1 of Algorithm \[alg:AND\] $AbstractRepair(\hat{M},\hat{s},\phi_{1},\emptyset)$ returns a KMTS $\hat{M}_{1}$, then $(\hat{M}_{1},\hat{s}) \models \phi_{1}$. Consequently, $\hat{M}_{1} \models C_{1}$, where $C_{1} = \emptyset \cup {(\hat{s},\phi_{1})}$. At line 7, if $AbstractRepair(\hat{M}_{1},\hat{s},\phi_{2},C_{1})$ returns a KMTS $\hat{M}_{1}^{\prime}$, then from Lemma \[theor:sound\_help\] $(\hat{M}_{1}^{\prime},\hat{s}) \models \phi_{2}$ and $\hat{M}_{1}^{\prime} \models C_{1}$.
Likewise, if the calls at lines 2 and 12 of Algorithm \[alg:AND\] return the KMTSs $\hat{M}_{2}$ and $\hat{M}_{2}^{\prime}$, then from the induction hypothesis $(\hat{M}_{2},\hat{s}) \models \phi_{2}$ and from Lemma \[theor:sound\_help\] $(\hat{M}_{2}^{\prime},\hat{s}) \models \phi_{1}$, $\hat{M}_{2}^{\prime} \models C_{2}$ with $C_{2} = \emptyset \cup {(\hat{s},\phi_{2})}$.
The KMTS $\hat{M^{\prime}}$ at line 7 of Algorithm \[alg:main\] can be either $\hat{M}_{1}^{\prime}$ or $\hat{M}_{2}^{\prime}$ and therefore, $(\hat{M^{\prime}},\hat{s}) \models \phi_{1}$ and $(\hat{M^{\prime}},\hat{s}) \models \phi_{2}$. From 3-valued semantics of CTL it holds that $(\hat{M^{\prime}},\hat{s}) \models \phi_{1} \wedge \phi_{2}$ and the lemma is true.
- if $\phi = EX\phi_{1}$, $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ calls $AbstractRepair_{EX}(\hat{M},\hat{s},EX\phi_{1},\emptyset)$ at line 10 of Algorithm \[alg:main\].
If a KMTS $\hat{M}^{\prime}$ is returned at line 5 of Algorithm \[alg:EX\], there is a state $\hat{s}_{1}$ with $(\hat{M},\hat{s}_{1}) \models \phi_{1}$ such that $\hat{M}^{\prime} = AddMust(\hat{M},(\hat{s},\hat{s}_{1}))$. From 3-valued semantics of CTL, we conclude that $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
If a $\hat{M}^{\prime}$ is returned at line 11, there is $(\hat{s},\hat{s}_{1}) \in R_{must}$ such that $(\hat{M^{\prime}},\hat{s}_{1}) \models \phi_{1}$ from the induction hypothesis, since $\hat{M^{\prime}} = AbstractRepair(\hat{M},\hat{s}_{1},\phi_{1},\emptyset)$. From 3-valued semantics of CTL, we conclude that $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
If a $\hat{M}^{\prime}$ is returned at line 18, a must transition $(\hat{s},\hat{s}_{n})$ to a new state has been added and $\hat{M}^{\prime} = AbstractRepair(AddMust(\hat{M},(\hat{s},\hat{s}_{n})),\hat{s}_{n},\phi_{1},\emptyset)$. Then, from the induction hypothesis $(\hat{M^{\prime}},\hat{s}_{n}) \models \phi_{1}$ and from 3-valued semantics of CTL, we also conclude that $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
- if $\phi = AG\phi_{1}$, $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ calls $AbstractRepair_{AG}(\hat{M},\hat{s},AG\phi_{1},\emptyset)$ at line 10 of Algorithm \[alg:main\]. If $(\hat{M},\hat{s}) \not\models \phi_{1}$ and $AbstractRepair(\hat{M},\hat{s},\phi_{1},\emptyset)$ returns a KMTS $\hat{M}_{0}$ at line 2 of Algorithm \[alg:AG\], then from the induction hypothesis $(\hat{M}_{0},\hat{s}) \models \phi_{1}$. Otherwise, $\hat{M}_{0} = \hat{M}$ and $(\hat{M}_{0},\hat{s}) \models \phi_{1}$, $\hat{M}_{0} \models C$ also hold true.
If Algorithm \[alg:AG\] returns a $\hat{M}^{\prime}$ at line 16, this KMTS is the result of successive calls of $AbstractRepair(\hat{M_{i}},\hat{s}_{k},\phi_{1},\emptyset)$ with $\hat{M_{i}}=AbstractRepair(\hat{M}_{i-1},\hat{s}_{k},\phi_{1},\emptyset)$ and $i=1, . . .$, for all may-reachable states $\hat{s}_{k}$ from $\hat{s}$ such that $(\hat{M}_{0},\hat{s}_{k}) \not\models \phi_{1}$. From the induction hypothesis, $(\hat{M}^{\prime},\hat{s}_{k}) \models \phi_{1}$ for all such $\hat{s}_{k}$ and from 3-valued semantics of CTL we conclude that $(\hat{M^{\prime}},\hat{s}) \models AG\phi_{1}$.
We prove the theorem for all other cases in the same way.
Theorem \[theor:sound\] shows that *AbstractRepair* is *sound* in the sense that if it returns a KMTS $\hat{M}^{\prime}$, then $\hat{M}^{\prime}$ satisfies property $\phi$. In this case, from the definitions of the basic repair operations, it follows that one or more KSs can be obtained for which $\phi$ holds true.
### Semi-completeness {#subsubsec:alg_completeness}
Given a set $AP$ of atomic propositions, we define the syntax of a CTL fragment inductively via a Backus Naur Form: $$\begin{aligned}
\phi ::== &\bot \, | \, \top \, | \, p \, | \, (\neg \phi) \, | \, (\phi \vee \phi) \, | \, AXp \, | \, EXp \, | \, AFp \\
& | \, EFp \, | \, AGp \, | \, EGp \, | \, A[p \, U \, p] \, | \, E[p \, U \, p]\end{aligned}$$ where $p$ ranges over $AP$.
*mr*-CTL includes most of the CTL formulae apart from those with nested path quantifiers or conjunction.
\[theor:complete\] Given a KMTS $\hat{M}$, an *mr*-CTL formula $\phi$ with $(\hat{M},\hat{s}) \not\models \phi$, for some $\hat{s}$ of $\hat{M}$, if there exists a KMTS $\hat{M}^{\prime\prime}$ over the same set $AP$ of atomic propositions with $(\hat{M}^{\prime\prime},\hat{s}) \models \phi$, $AbstractRepair(\hat{M},\hat{s},\phi,\emptyset)$ returns a KMTS $\hat{M}^{\prime}$ such that $(\hat{M}^{\prime},\hat{s}) \models \phi$.
We prove the theorem using structural induction on $\phi$.
#### Base Case:
- if $\phi = \top$, Theorem \[theor:complete\] is trivially true, because for any KMTS $\hat{M}$ it holds that $(\hat{M},\hat{s}) \models \phi$.
- if $\phi = \bot$, then the theorem is trivially true, because there does not exist a KMTS $\hat{M}^{\prime\prime}$ such that $(\hat{M}^{\prime\prime},\hat{s}) \models \phi$.
- if $\phi = p \in AP$, there is a KMTS $\hat{M}^{\prime\prime}$ with $p \in \hat{L}^{\prime\prime}(\hat{s})$ and therefore $(\hat{M}^{\prime\prime},\hat{s}) \models \phi$. Algorithm \[alg:main\] calls $AbstractRepair_{ATOMIC}(\hat{M},\hat{s},p,\emptyset)$ at line 4 and an $\hat{M^{\prime}} = ChangeLabel(\hat{M},\hat{s},p)$ is computed at line 1 of Algorithm \[alg:ATOMIC\]. Since $C$ is empty, $\hat{M^{\prime}}$ is returned at line 3 and $(\hat{M}^{\prime},\hat{s}) \models \phi$ from 3-valued semantics of CTL. Therefore, the theorem is true.
#### Induction Hypothesis:
For *mr*-CTL formulae $\phi_{1}$, $\phi_{2}$, the theorem is true. Thus, for $\phi_{1}$ (resp. $\phi_{2}$), if there is a KMTS $\hat{M}^{\prime\prime}$ over the same set $AP$ of atomic propositions with $(\hat{M}^{\prime\prime},\hat{s}) \models \phi_{1}$, $AbstractRepair(\hat{M},\hat{s},\phi_{1},\emptyset)$ returns a KMTS $\hat{M}^{\prime}$ such that $(\hat{M}^{\prime},\hat{s}) \models \phi_{1}$.
#### Inductive Step:
- if $\phi = \phi_{1} \vee \phi_{2}$, from the 3-valued semantics of CTL a KMTS that satisfies $\phi$ exists if and only if there is a KMTS satisfying any of the $\phi_{1}$, $\phi_{2}$. From the induction hypothesis, if there is a KMTS $\hat{M}_{1}^{\prime\prime}$ with $(\hat{M}_{1}^{\prime\prime},\hat{s}) \models \phi_{1}$, $AbstractRepair(\hat{M},\hat{s},\phi_{1},\emptyset)$ at line 1 of Algorithm \[alg:OR\] returns a KMTS $\hat{M}_{1}^{\prime}$ such that $(\hat{M}_{1}^{\prime},\hat{s}) \models \phi_{1}$. Respectively, $AbstractRepair(\hat{M},\hat{s},\phi_{2},\emptyset)$ at line 2 of Algorithm \[alg:OR\] can return a KMTS $\hat{M}_{2}^{\prime}$ with $(\hat{M}_{2}^{\prime},\hat{s}) \models \phi_{2}$. In any case, if either $\hat{M}_{1}^{\prime}$ or $\hat{M}_{2}^{\prime}$ exists, for the KMTS $\hat{M}^{\prime}$ that is returned at line 13 of Algorithm \[alg:OR\] we have $(\hat{M}^{\prime},\hat{s}) \models \phi_{1}$ or $(\hat{M}^{\prime},\hat{s}) \models \phi_{2}$ and therefore $(\hat{M}^{\prime},\hat{s}) \models \phi$.
- if $\phi = EX\phi_{1}$, from the 3-valued semantics of CTL a KMTS that satisfies $\phi$ at $\hat{s}$ exists if and only if there is KMTS satisfying $\phi_{1}$ at some direct must-successor of $\hat{s}$.
If in the KMTS $\hat{M}$ there is a state $\hat{s}_{1}$ with $(\hat{M},\hat{s}_{1}) \models \phi_{1}$, then the new KMTS $\hat{M}^{\prime} = AddMust(\hat{M},(\hat{s},\hat{s}_{1}))$ is computed at line 3 of Algorithm \[alg:EX\]. Since $C$ is empty $\hat{M}^{\prime}$ is returned at line 5 and $(\hat{M^{\prime}},\hat{s}) \models EX\phi_{1}$.
Otherwise, if there is a direct must-successor $\hat{s}_{i}$ of $\hat{s}$, $AbstractRepair(\hat{M},\hat{s}_{i},\phi_{1},\emptyset)$ is called at line 8. From the induction hypothesis, if there is a KMTS $\hat{M}^{\prime\prime}$ with $(\hat{M}^{\prime\prime},\hat{s}_{i}) \models \phi_{1}$, then a KMTS $\hat{M}^{\prime}$ is computed such that $(\hat{M}^{\prime},\hat{s}_{i}) \models \phi_{1}$ and therefore the theorem is true.
If there are no must-successors of $\hat{s}$, a new state $\hat{s}_{n}$ is added and subsequently connected with a must-transition from $\hat{s}$. $AbstractRepair$ is then called for $\phi_{1}$ and $\hat{s}_{n}$ as previously and the theorem holds also true.
- if $\phi = AG\phi_{1}$, from the 3-valued semantics of CTL a KMTS that satisfies $\phi$ at $\hat{s}$ exists, if and only if there is KMTS satisfying $\phi_{1}$ at $\hat{s}$ and at each may-reachable state from $\hat{s}$.
$AbstractRepair(\hat{M},\hat{s},\phi_{1},\emptyset)$ is called at line 2 of Algorithm \[alg:AG\] and from the induction hypothesis if there is KMTS $\hat{M}_{0}^{\prime}$ with $(\hat{M}_{0}^{\prime},\hat{s}) \models \phi_{1}$, then a KMTS $\hat{M}_{0}$ is computed such that $(\hat{M}_{0},\hat{s}) \models \phi_{1}$. $AbstractRepair$ is subsequently called for $\phi_{1}$ and for all may-reachable $\hat{s}_{k}$ from $\hat{s}$ with $(\hat{M}_{0},\hat{s}_{k}) \not\models \phi_{1}$ one-by-one. From the induction hypothesis, if there is KMTS $\hat{M}_{i}^{\prime}$ that satisfies $\phi_{1}$ at each such $\hat{s}_{k}$, then all $\hat{M}_{i}=AbstractRepair(\hat{M}_{i-1},\hat{s}_{k},\phi_{1},\emptyset), \, i=1, . . .,$ satisfy $\phi_{1}$ at $\hat{s}_{k}$ and the theorem holds true.
We prove the theorem for all other cases in the same way.
Theorem \[theor:complete\] shows that *AbstractRepair* is *semi-complete* with respect to full CTL: if there is a KMTS that satisfies a *mr*-CTL formula $\phi$, then the algorithm finds one such KMTS.
Complexity Issues {#subsec:alg_complex}
-----------------
AMR’s complexity analysis is restricted to *mr*-CTL, for which the algorithm has been proved complete. For these formulas, we show that AMR is upper bounded by a polynomial expression in the state space size and the number of may-transitions of the abstract KMTS, and depends also on the length of the *mr*-CTL formula.
For CTL formulas with nested path quantifiers and/or conjunction, AMR is looking for a repaired model satisfying all conjunctives (constraints), which increases the worst-case execution time exponentially to the state space size of the abstract KMTS. In general, as shown in [@BK12], the complexity of all model repair algorithms gets worse when raising the level of their completeness, but AMR has the advantage of working exclusively over an abstract model with a reduced state space compared to its concrete counterpart.
Our complexity analysis for *mr*-CTL is based on the following results. For an abstract KMTS $\hat{M} = (\hat{S}, \hat{S_{0}},$ $R_{must}, R_{may},
\hat{L})$ and a *mr*-CTL property $\phi$, (i) 3-valued CTL model checking is performed in $O(|\phi| \cdot (|\hat{S}|+|R_{may}|))$ [@GHJ01], (ii) Depth First Search (DFS) of states reachable from $\hat{s} \in \hat{S}$ is performed in $O(|\hat{S}|+|R_{may}|)$ in the worst case or in $O(|\hat{S}|+|R_{must}|)$ when only must-transitions are accessed, (iii) finding a maximal path from $\hat{s} \in \hat{S}$ using Breadth First Search (BFS) is performed in $O(|\hat{S}|+|R_{may}|)$ for may-paths and in $O(|\hat{S}|+|R_{must}|)$ for must-paths.
We analyze the computational cost for each of the AMR’s primitive functions:
- if $\phi = p \in AP$, $AbstractRepair_{ATOMIC}$ is called and the operation $ChangeLabel$ is applied, which is in $O(1)$.
- if $\phi = EX\phi_{1}$, then $AbstractRepair_{EX}$ is called and the applied operations with the highest cost are: (1) finding a state satisfying $\phi_{1}$, which depends on the cost of 3-valued CTL model checking and is in $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$, (2) finding a must-reachable state, which is in $O(|\hat{S}| + |R_{must}|)$. These operations are called at most once and the overall complexity for this primitive functions is therefore in $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$.
- if $\phi = AX\phi_{1}$, then $AbstractRepair_{AX}$ is called and the most costly operations are: (1) finding a may-reachable state, which is in $O(|\hat{S}| + |R_{may}|)$, and (2) checking if a state satisfies $\phi_{1}$, which is in $O(|\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$. These operations are called at most $|\hat{S}|$ times and the overall bound class is $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$.
- if $\phi = EF\phi_{1}$, $AbstractRepair_{EF}$ is called and the operations with the highest cost are: (1) finding a must-reachable state, which is in $O(|\hat{S}| + |R_{must}|)$, (2) checking if a state satisfies $\phi_{1}$ with its bound class being $O(|\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$ and (3) finding a state that satisfies $\phi_{1}$, which is in $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$. These three operations are called at most $|\hat{S}|$ times and consequently, the overall bound class is $O(|\hat{S}|^{2} \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$.
- if $\phi = AF\phi_{1}$, $AbstractRepair_{AF}$ is called and the most costly operation is: finding a maximal may-path violating $\phi_{1}$ in all states, which is in $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|)$. This operation is called at most $|\hat{S}|$ times and therefore, the overall bound class is $O(|\hat{S}|^2 \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$.
In the same way, it is easy to show that: (i) if $\phi = EG\phi_{1}$, then $AbstractRepair_{EG}$ is in $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{must}|)$, (ii) if $\phi = AG\phi_{1}$, then $AbstractRepair_{AG}$ is in $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$, (iii) if $\phi = E(\phi_{1}U\phi_{2})$, then the bound class of $AbstractRepair_{EU}$ is $O(|\hat{S}| \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{must}|)$, (iv) if $\phi = A(\phi_{1}U\phi_{2})$ then $AbstractRepair_{AU}$ is in $O(|\hat{S}|^2 \cdot |\phi_{1}| \cdot (|\hat{S}|+|R_{may}|))$.
For a *mr*-CTL property $\phi$, the main body of the algorithm is called at most $|\phi|$ times and the overall bound class of the AMR algorithm is $O(|\hat{S}|^2 \cdot |\phi|^{2} \cdot (|\hat{S}|+|R_{may}|))$.
Application {#sec:app}
-----------
We present the application of *AbstractRepair* on the ADO system from Section \[sec:mc\]. After the first two steps of our repair process, *AbstractRepair* is called for the KMTS $\alpha_{\mathit{Refined}}(M)$ that is shown in Fig. \[fig:ado\_refined\], the state $\hat{s}_{01}$ and the CTL property $\phi = AGEXq$.
*AbstractRepair* calls $AbstractRepair_{AG}$ with arguments $\alpha_{\mathit{Refined}}(M)$, $\hat{s}_{01}$ and $AGEXq$. The $AbstractRepair_{AG}$ algorithm at line 10 triggers a recursive call of *AbstractRepair* with the same arguments. Eventually, $AbstractRepair_{EX}$ is called with arguments $\alpha_{\mathit{Refined}}(M)$, $\hat{s}_{01}$ and $EXq$, that in turn calls *AddMust* at line 3, thus adding a must-transition from $\hat{s}_{01}$ to $\hat{s}_{1}$. *AbstractRepair* terminates by returning a KMTS $\hat{M^{\prime}}$ that satisfies $\phi = AGEXq$. The repaired KS $M^{\prime}$ is the single element in the set of KSs derived by the concretization of $\hat{M^{\prime}}$ (cf. Def. \[def:add\_must\_ks\]). The execution steps of *AbstractRepair* and the obtained repaired KMTS and KS are shown in Fig. \[fig:ado\_repair\_process\] and Fig. \[fig:ado\_repaired\] respectively.
Although the ADO is not a system with a large state space, it is shown that the repair process is accelerated by the proposed use of abstraction. If on the other hand model repair was applied directly to the concrete model, new transitions would have have been inserted from all the states labeled with $\neg open$ to the one labeled with *open*. In the ADO, we have seven such states, but in a system with a large state space this number can be significantly higher. The repair of such a model without the use of abstraction would be impractical.
Experimental Results: The Andrew File System 1 (AFS1) Protocol {#sec:exp}
==============================================================
In this section, we provide experimental results for the relative performance of a prototype implementation of our AMR algorithm in comparison with a prototype implementation of a concrete model repair solution [@ZD08]. The results serve as a proof of concept for the use of abstraction in model repair and demonstrate the practical utility of our approach.
As a model we use a KS for the Andrew File System Protocol 1 (AFS1) [@WV95], which has been repaired for a specific property in [@ZD08]. AFS1 is a client-server cache coherence protocol for a distributed file system. Four values are used for the client’s belief about a file (nofile, valid, invalid, suspect) and three values for the server’s belief (valid, invalid, none).
A property which is not satisfied in the AFS1 protocol in the form of CTL is: $$AG((Server.belief = valid) \rightarrow (Client.belief = valid))$$
We define the atomic proposition $p$ as $Server.belief = valid$ and $q$ as $Client.belief = valid$, and the property is thus written as $AG(p \rightarrow
q)$. The KS for the AFS1 protocol is depicted in Fig. \[fig:afs1\_refined2\_ks\]. State colors show how they are abstracted in the KMTS of Fig. \[fig:afs1\_refined2\_kmts\], which is derived after the 2nd refinement step of our AMR framework (Fig. \[fig:abs\_repair\]). The shown KMTS and the CTL property of interest are given as input in our prototype AMR implementation.
To obtain larger models of AFS1 we have extended the original model by adding one more possible value for three model variables. Three new models are obtained with gradually increasing size of state space.
The results of our experiments are presented in Table \[table:exp\_results\]. The time needed for the AMR prototype to repair the original AFS1 model and its extensions is from 124 to even 836 times less than the needed time for concrete model repair. The repaired KMTS and KS for the original AFS1 model are shown in Fig. \[fig:afs1\_repaired\_ks\_kmts\].
An interesting observation from the application of the AMR algorithm on the repair of the AFS1 KS is that the distance $d$ (cf. Def. \[def:metric\_space\]) of the repaired KS from the original KS is less than the corresponding distance obtained from the concrete model repair algorithm in [@ZD08]. This result demonstrates in practice the effect of the minimality of changes ordering, on which the AMR algorithm is based on (cf. Fig. \[fig:order\_basic\_ops\]).
Models Concrete States Concr. Repair (Time in sec.) AMR (Time in sec.) Improvement (%)
---------------------- ----------------- ------------------------------ -------------------- -----------------
$AFS1$ $26$ $17.4$ $0.14$ $124$
$AFS1 (Extension 1)$ $30$ $24.9$ $0.14$ $178$
$AFS1 (Extension 2)$ $34$ $35.0$ $0.14$ $250$
$AFS1 (Extension 3)$ $38$ $117.0$ $0.14$ $836$
: Experimental results of AMR with respect to concrete repair[]{data-label="table:exp_results"}
Related Work {#sec:relwork}
============
To the best of our knowledge this is the first work that suggests the use of abstraction as a means to counter the state space explosion in search of a Model Repair solution. However, abstraction and in particular abstract interpretation has been used in *program synthesis* [@VYY2010], a different but related problem to the Model Repair. Program synthesis refers to the automatic generation of a program based on a given specification. Another related problem where abstraction has been used is that of *trigger querying* [@AK14]: given a system $M$ and a formula $\phi$, find the set of scenarios that trigger $\phi$ in $M$.
The related work in the area of *program repair* do not consider KSs as the program model. In this context, abstraction has been previously used in the repair of data structures [@ZMK13]. The problem of repairing a Boolean program has been formulated in [@SJB05; @JGB07; @GBC06; @EJ12] as the finding of a winning strategy for a game between two players. The only exception is the work reported in [@SDE08].
Another line of research on program repair treats the repair as a search problem and applies innovative evolutionary algorithms [@A11], *behavioral programming* techniques [@HKMW12] or other informal heuristics [@WC08; @AAG11; @WPFSBMZ10].
Focusing exclusively on the area of Model Repair without the use of abstraction, it is worth to mention the following approaches. The first work on Model Repair with respect to CTL formulas was presented in [@A95]. The authors used only the removal of transitions and showed that the problem is NP-complete. Another interesting early attempt to introduce the Model Repair problem for CTL properties is the work in [@BEGL99]. The authors are based on the AI techniques of abductive reasoning and theory revision and propose a repair algorithm with relatively high computational cost. A formal algorithm for Model Repair in the context of KSs and CTL is presented in [@ZD08]. The authors admit that their repair process strongly depends on the model’s size and they do not attempt to provide a solution for handling conjunctive CTL formulas.
In [@CR09], the authors try to render model repair applicable to large KSs by using “table systems”, a concise representation of KSs that is implemented in the NuSMV model checker. A limitation of their approach is that table systems cannot represent all possible KSs. In [@ZKZ10], tree-like local model updates are introduced with the aim of making the repair process applicable to large-scale domains. However, the proposed approach is only applicable to the universal fragment of the CTL.
A number of works attempt to ensure completeness for increasingly larger fragments of the CTL by introducing ways of handling the constraints associated with conjunctive formulas. In [@KPYZ10], the authors propose the use of constraint automata for ACTL formulas, while in [@CR11] the authors introduce the use of protected models for an extension of the CTL. Both of the two methods are not directly applicable to formulas of the full CTL.
The Model Repair problem has been also addressed in many other contexts. In [@E12], the author uses a distributed algorithm and the processing power of computing clusters to fight the time and space complexity of the repair process. In [@MLB11], an extension of the Model Repair problem has been studied for Labeled Transition Systems. In [@BGKRS11], we have provided a solution for the Model Repair problem in probabilistic systems. Another recent effort for repairing discrete-time probabilistic models has been proposed in [@PAJTK15]. In [@BBG11], model repair is applied to the *fault recovery* of component-based models. Finally, a slightly different but also related problem is that of Model Revision, which has been studied for UNITY properties in [@BEK09; @BK08-OPODIS] and for CTL in [@GW10]. Other methods in the area of fault-tolerance include the work in [@gr09], which uses discrete controller synthesis and [@fb15], which employs SMT solving. Another interesting work in this direction is in [@df09], where the authors present a repair algorithm for fault-tolerance in a fully connected topology, with respect to a temporal specification.
Conclusions {#sec:concl}
===========
In this paper, we have shown how abstraction can be used to cope with the state explosion problem in Model Repair. Our model-repair framework is based on Kripke Structures, a 3-valued semantics for CTL, and Kripke Modal Transition Systems, and features an abstract-model-repair algorithm for KMTSs. We have proved that our AMR algorithm is sound for the full CTL and complete for a subset of CTL. We have also proved that our AMR algorithm is upper bounded by a polynomial expression in the size of the abstract model for a major fragment of CTL. To demonstrate its practical utility, we applied our framework to an Automatic Door Opener system and to the Andrew File System 1 protocol.
As future work, we plan to apply our method to case studies with larger state spaces, and investigate how abstract model repair can be used in different contexts and domains. A model repair application of high interest is in the design of fault-tolerant systems. In [@bka12], the authors present an approach for the repair of a distributed algorithm such that the repaired one features fault-tolerance. The input to this model repair problem includes a set of uncontrollable transitions such as the faults in the system. The model repair algorithm used works on concrete models and it can therefore solve the problem only for a limited number of processes. With this respect, we believe that this application could be benefited from the use of abstraction in our AMR framework.
At the level of extending our AMR framework, we aim to search for “better" abstract models, in order to either restrict failures due to refinement or ensure completeness for a larger fragment of the CTL. We will also investigate different notions of minimality in the changes introduced by model repair and the applicability of abstraction-based model repair to probabilistic, hybrid and other types of models.
Acknowledgment
==============
This work was partially sponsored by Canada NSERC Discovery Grant 418396-2012 and NSERC Strategic Grants 430575-2012 and 463324-2014. The research was also co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: Thalis Athens University of Economics and Business - SOFTWARE ENGINEERING RESEARCH PLATFORM.
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---
abstract: 'This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We prove that the problem is locally well-posed. Our method relies on energy methods in Sobolev spaces for a collection of related linear and nonlinear problems.'
address:
- |
Department of Mathematics\
University of California, Riverside\
Riverside, CA 92521, USA
- |
Department of Mathematical Sciences\
Carnegie Mellon University\
Pittsburgh, PA 15213, USA
- |
School of Mathematical Sciences\
Xiamen University\
Xiamen, Fujian 361005, China
author:
- Juhi Jang
- Ian Tice
- Yanjin Wang
title: 'The compressible viscous surface-internal wave problem: local well-posedness '
---
Introduction
============
Formulation in Eulerian coordinates
-----------------------------------
We consider two distinct, immiscible, viscous, compressible, barotropic fluids evolving in a moving domain $\Omega(t)=\Omega_+(t)\cup \Omega_-(t)$ for time $t\ge0$. One fluid $(+)$, called the “upper fluid,” fills the upper domain $$\label{omega_plus}
\Omega_+(t)=\{y\in \mathrm{T}^2\times \mathbb{R}\mid \eta_-(y_1,y_2,t)<y_3< \ell +\eta_+(y_1,y_2,t)\},$$ and the other fluid $(-)$, called the “lower fluid,” fills the lower domain $$\label{omega_minus}
\Omega_-(t)=\{y\in \mathrm{T}^2\times \mathbb{R}\mid -b <y_3<\eta_-(y_1,y_2,t)\}.$$ Here we assume the domains are horizontal periodic by setting $\mathrm{T}^2=(2\pi L_1\mathbb{T}) \times (2\pi L_2\mathbb{T})$ for $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ the usual 1–torus and $L_1,L_2>0$ the periodicity lengths. We assume that $\ell,b >0$ are two fixed and given constants, but the two surface functions $\eta_\pm$ are free and unknown. The surface $\Gamma_+(t) = \{y_3= \ell + \eta_+(y_1,y_2,t)\}$ is the moving upper boundary of $\Omega_+(t)$ where the upper fluid is in contact with the atmosphere, $\Gamma_-(t) = \{y_3=\eta_-(y_1,y_2,t)\}$ is the moving internal interface between the two fluids, and $\Sigma_b = \{y_3=-b \}$ is the fixed lower boundary of $\Omega_-(t)$.
The two fluids are described by their density and velocity functions, which are given for each $t\ge0$ by $\tilde{\rho}_\pm
(\cdot,t):\Omega_\pm (t)\rightarrow \mathbb{R}^+$ and $\tilde{u}_\pm (\cdot,t):\Omega_\pm (t)\rightarrow \mathbb{R}^3$, respectively. In each fluid the pressure is a function of density: $P_\pm =P_\pm(\tilde{\rho}_\pm)>0$, and the pressure function is assumed to be smooth, positive, and strictly increasing. For a vector function $u\in {\mathbb{R}^{3}}$ we define the symmetric gradient by $(\sg u)_{ij} = \p_i u_j + \p_j u_i$ for $i,j=1,2,3$; its deviatoric (trace-free) part is then $$\label{deviatoric_def}
\sgz u = \sg u - \frac{2}{3} \operatorname{div}{u} I,$$ where $I$ is the $3 \times 3$ identity matrix. The viscous stress tensor in each fluid is then given by $$\S_\pm(\tilde{u}_\pm) := \mu_\pm \sgz \tilde u_\pm +\mu'_\pm \operatorname{div}\tilde{u}_\pm I,$$ where $\mu_\pm$ is the shear viscosity and $\mu'_\pm$ is the bulk viscosity; we assume these satisfy the usual physical conditions $$\label{viscosity}
\mu_\pm>0,\quad \mu_\pm'\ge 0.$$ The tensor $P_\pm(\tilde{\rho}_\pm) I-\S_\pm(\tilde{u}_\pm)$ is known as the stress tensor. The divergence of a symmetric tensor $\mathbb{M}$ is defined to be the vector with components $(\operatorname{div}\mathbb{M})_i = \p_j \mathbb{M}_{ij}$. Note then that $$\operatorname{div}\left( P_\pm(\tilde{\rho}_\pm) I-\S_\pm(\tilde{u}_\pm) \right) = \nab P_\pm(\tilde{\rho}_\pm) - \mu_\pm \Delta \tilde{u}_\pm - \left(\frac{\mu_\pm}{3} + \mu_\pm' \right)\nab \operatorname{div}{\tilde{u}_\pm}.$$
For each $t>0$ we require that $(\tilde{u}_\pm, \tilde{\rho}_\pm,\eta_\pm)$ satisfy the following equations: $$\label{ns_euler}
\begin{cases}
\partial_t\tilde{\rho}_\pm+\operatorname{div}(\tilde{\rho}_\pm \tilde{u}_\pm)=0 & \text{in }\Omega_\pm(t)
\\\tilde{\rho}_\pm (\partial_t\tilde{u}_\pm + \tilde{u}_\pm \cdot \nabla \tilde{u}_\pm ) +\nab P_\pm(\tilde{\rho}_\pm) - \operatorname{div}\S_\pm(\tilde{u}_\pm) =-g\tilde{\rho}_\pm e_3 & \text{in } \Omega_\pm(t)
\\\partial_t\eta_\pm=\tilde{u}_{3,\pm}-\tilde{u}_{1,\pm}\partial_{y_1}\eta_\pm-\tilde{u}_{2,\pm}\partial_{y_2}\eta_\pm &\hbox{on } \Gamma_\pm(t)
\\(P_+(\tilde{\rho}_+)I-\S_+(\tilde{u}_+))n_+=p_{atm}n_+-\sigma_+ \mathcal{H}_+ n_+ &\hbox{on }\Gamma_+(t)
\\ (P_+(\tilde{\rho}_+)I-\S_+( \tilde{u}_+))n_-=(P_-(\tilde{\rho}_-)I-\S_-( \tilde{u}_-))n_-+ \sigma_- \mathcal{H}_-n_- &\hbox{on }\Gamma_-(t) \\\tilde{u}_+=\tilde{u}_- &\hbox{on }\Gamma_-(t) \\\tilde{u}_-=0 &\hbox{on }\Sigma_b.
\end{cases}$$ In the equations $-g \tilde{\rho}_\pm e_3$ is the gravitational force with the constant $g>0$ the acceleration of gravity and $e_3$ the vertical unit vector. The constant $p_{atm}>0$ is the atmospheric pressure, and we take $\sigma_\pm\ge 0$ to be the constant coefficients of surface tension. In this paper, we let $\nabla_\ast$ denote the horizontal gradient, $\operatorname{div}_\ast$ denote the horizontal divergence and $\Delta_\ast$ denote the horizontal Laplace operator. Then the upward-pointing unit normal of $\Gamma_\pm(t)$, $n_\pm$, is given by $$n_\pm=\frac{(-\nabla_\ast\eta_\pm,1)}
{\sqrt{1+|\nabla_\ast\eta_\pm|^2}},$$ and $\mathcal{H}_\pm$, twice the mean curvature of the surface $\Gamma_\pm(t)$, is given by the formula $$\mathcal{H}_\pm=\operatorname{div}_\ast\left(\frac{\nabla_\ast\eta_\pm}
{\sqrt{1+|\nabla_\ast\eta_\pm|^2}}\right).$$ The third equation in is called the kinematic boundary condition since it implies that the free surfaces are advected with the fluids. The boundary equations in involving the stress tensor are called the dynamic boundary conditions. Notice that on $\Gamma_-(t)$, the continuity of velocity, $\tilde{u}_+ = \tilde{u}_-$, means that it is the common value of $\tilde{u}_\pm$ that advects the interface. For a more physical description of the equations and the boundary conditions in , we refer to [@3WL].
To complete the statement of the problem, we must specify the initial conditions. We suppose that the initial surfaces $\Gamma_\pm(0)$ are given by the graphs of the functions $\eta_\pm(0)$, which yield the open sets $\Omega_\pm(0)$ on which we specify the initial data for the density, $\tilde{\rho}_\pm(0): \Omega_\pm(0) \rightarrow \mathbb{R}^+$, and the velocity, $\tilde{u}_\pm(0): \Omega_\pm(0) \rightarrow \mathbb{R}^3$. We will assume that $\ell+\eta_+(0)>\eta_-(0)>-b $ on $\mathrm{T}^2$, which means that at the initial time the boundaries do not intersect with each other.
Equilibria
----------
Here we seek a steady-state equilibrium solution to with $\tilde{u}_\pm=0, \eta_\pm =0$, and the equilibrium domains given by $$\Omega_+=\{y\in \mathrm{T}^2\times \mathbb{R}\mid 0 < y_3< \ell \} \text{ and }
\Omega_-=\{y\in \mathrm{T}^2\times \mathbb{R}\mid -b <y_3< 0 \}.$$ Then reduces to an ODE for the equilibrium densities $\tilde\rho_\pm = \bar{\rho}_\pm(y_3)$: $$\label{steady}
\begin{cases}
\displaystyle\frac{d(P_+ (\bar{\rho}_+ ))}{dy_3} = -g\bar{\rho}_+, & \text{for }y_3 \in (0,\ell), \\
\displaystyle\frac{d(P_- (\bar{\rho}_- ))}{dy_3} = -g\bar{\rho}_-, & \text{for } y_3 \in (-b,0), \\
P_+(\bar{\rho}_+(\ell)) = p_{atm}, \\
P_+(\bar{\rho}_+(0)) =P_-(\bar{\rho}_-(0)).
\end{cases}$$ The system admits a solution $\bar{\rho}_\pm >0$ if and only if we assume that the equilibrium heights $b,\ell>0$, the pressure laws $P_\pm$, and the atmospheric pressure $p_{atm}$ satisfy a collection of admissibility conditions. These are enumerated in detail in our companion paper [@JTW_GWP]. For the sake of brevity we will not mention these here, but we will assume they are satisfied so that an equilibrium exists. The resulting function $\bar{\rho}$ is strictly positive and smooth when restricted to to $[-b,0]$ and $[0,\ell]$.
We give special names to the equilibrium density at the fluid interfaces: $$\bar{\rho}_1 = \bar{\rho}_+(\ell), \; \bar{\rho}^+ = \bar{\rho}_+(0),\; \bar{\rho}^- = \bar{\rho}_-(0).$$ Notice in particular that the equilibrium density can jump across the internal interface. The jump in the equilibrium density, which we denote by $$\label{rho+-}
\rj := \bar{\rho}_+(0)-\bar{\rho}_-(0)= \bar{\rho}^+ - \bar{\rho}^-,$$ is of fundamental importance in the the analysis of solutions to near equilibrium. Indeed, if $\rj > 0$ then the upper fluid is heavier than the lower fluid along the equilibrium interface, and the fluid is susceptible to the well-known Rayleigh-Taylor gravitational instability. This is not particularly important for the local theory developed in this paper, but of fundamental importance in the stability theory.
In studying perturbations of the equilibrium density it will be useful throughout the paper to employ the enthalpy functions. These are defined in terms of the pressure laws $P_\pm$ and the equilibrium density values via $$\label{h'}
h_+(z) = \int_{\bar{\rho}_1}^z \frac{P'_+(r)}{r}dr \text{ and } h_-(z) = \int_{\bar{\rho}^-}^z \frac{P'_+(r)}{r}dr.$$
Reformulation in flattened coordinates
--------------------------------------
The movement of the free surfaces $\Gamma_\pm(t)$ and the subsequent change of the domains $\Omega_\pm(t)$ create numerous mathematical difficulties. To circumvent these, we will switch to a coordinate system in which the boundaries and the domains stay fixed in time. In order to be consistent with our study of the nonlinear stability of the equilibrium state in [@JTW_GWP], we will use the equilibrium domain. We will not use a Lagrangian coordinate transformation, but rather utilize a special flattening coordinate transformation motivated by Beale [@B2].
To this end, we define the fixed domain $$\Omega = \Omega_+\cup\Omega_-\text{ with }\Omega_+:=\{0<x_3<\ell \} \text{ and } \Omega_-:=\{-b<x_3<0\},$$ for which we have written the coordinates as $x\in \Omega$. We shall write $\Sigma_+:=\{x_3= \ell\}$ for the upper boundary, $\Sigma_-:=\{x_3=0\}$ for the internal interface and $\Sigma_b:=\{x_3=-b\}$ for the lower boundary. Throughout the paper we will write $\Sigma = \Sigma_+ \cup \Sigma_-$. We think of $\eta_\pm$ as a function on $\Sigma_\pm$ according to $\eta_+: (\mathrm{T}^2\times\{\ell\}) \times \mathbb{R}^{+} \rightarrow\mathbb{R}$ and $\eta_-:(\mathrm{T}^2\times\{0\}) \times \mathbb{R}^{+} \rightarrow \mathbb{R}$, respectively. We will transform the free boundary problem in $\Omega(t)$ to one in the fixed domain $\Omega $ by using the unknown free surface functions $\eta_\pm$. For this we define $$\bar{\eta}_+:=\mathcal{P}_+\eta_+=\text{Poisson extension of }\eta_+ \text{ into }\mathrm{T}^2 \times \{x_3\le \ell\}$$ and $$\bar{\eta}_-:=\mathcal{P}_-\eta_-=\text{specialized Poisson extension of }\eta_-\text{ into }\mathrm{T}^2 \times \mathbb{R},$$ where $\mathcal{P}_\pm$ are defined by and . The Poisson extensions $\bar{\eta}_\pm$ allow us to flatten the coordinate domains via the following special coordinate transformation: $$\label{cotr}
\Omega_\pm \ni x\mapsto(x_1,x_2, x_3+ \tilde{b}_1\bar{\eta}_++\tilde{b}_2\bar{\eta}_-):=\Theta (t,x)=(y_1,y_2,y_3)\in\Omega_\pm(t),$$ where we have chosen $\tilde{b}_1=\tilde{b}_1(x_3), \tilde{b}_2=\tilde{b}_2(x_3)$ to be two smooth functions in $\mathbb{R}$ that satisfy $$\label{b function}
\tilde{b}_1(0)=\tilde{b}_1(-b)=0, \tilde{b}_1(\ell)=1\text{ and }\tilde{b}_2(\ell)=\tilde{b}_2(-b)=0, \tilde{b}_2(0)=1.$$ Note that $\Theta(\Sigma_+,t)=\Gamma_+(t),\ \Theta (\Sigma_-,t)=\Gamma_-(t)$ and $\Theta(\cdot,t) \mid_{\Sigma_b} = Id \mid_{\Sigma_b}$.
Note that if $\eta $ is sufficiently small (in an appropriate Sobolev space), then the mapping $\Theta $ is a diffeomorphism. This allows us to transform the problem to one in the fixed spatial domain $\Omega$ for each $t\ge 0$. In order to write down the equations in the new coordinate system, we compute $$\label{A_def}
\begin{array}{ll} \nabla\Theta =\left(\begin{array}{ccc}1&0&0\\0&1&0\\A &B &J \end{array}\right)
\text{ and }\mathcal{A} := \left(\nabla\Theta
^{-1}\right)^T=\left(\begin{array}{ccc}1&0&-A K \\0&1&-B K \\0&0&K
\end{array}\right)\end{array}.$$ Here the components in the matrix are $$\label{ABJ_def}
A =\p_1\theta ,\
B =\p_2\theta,\
J = 1 + \p_3\theta,\ K =J^{-1},$$ where we have written $$\label{theta}
\theta:=\tilde{b}_1\bar{\eta}_++\tilde{b}_2\bar{\eta}_-.$$ Notice that $J={\rm det}\, \nabla\Theta $ is the Jacobian of the coordinate transformation. It is straightforward to check that, because of how we have defined $\bar{\eta}_-$ and $\Theta $, the matrix $\mathcal{A}$ is regular across the interface $\Sigma_-$.
We now define the density $\rho_\pm$ and the velocity $u_\pm$ on $\Omega_\pm$ by the compositions $\rho_\pm(x,t)=\tilde \rho_\pm(\Theta_\pm(x,t),t)$ and $ u_\pm(x,t)=\tilde u_\pm(\Theta_\pm(x,t),t)$. Since the domains $\Omega_\pm$ and the boundaries $\Sigma_\pm$ are now fixed, we henceforth consolidate notation by writing $f$ to refer to $f_\pm$ except when necessary to distinguish the two; when we write an equation for $f$ we assume that the equation holds with the subscripts added on the domains $\Omega_\pm$ or $\Sigma_\pm$. To write the jump conditions on $\Sigma_-$, for a quantity $f=f_\pm$, we define the interfacial jump as $${\left\llbracket f \right\rrbracket } := f_+ \vert_{\{x_3=0\}} - f_- \vert_{\{x_3=0\}}.$$ Then in the new coordinates, the PDE becomes the following system for $(u,\rho,\eta)$: $$\label{ns_geometric}
\begin{cases}
\partial_t \rho-K\p_t\theta\p_3\rho +\operatorname{div}_\a ( {\rho} u)=0 & \text{in }
\Omega \\
\rho (\partial_t u -K\p_t\theta\p_3 u+u\cdot\nabla_\a u ) + \nabla_\a P ( {\rho} ) -\diva \S_\a (u) =- g\rho e_3 & \text{in }
\Omega
\\ \partial_t \eta = u\cdot \n &
\text{on }\Sigma
\\ (P ( {\rho} ) I- \S_{\a}(u))\n
=p_{atm}\n -\sigma_+ \mathcal{H} \n &\hbox{on }\Sigma_+
\\ {\left\llbracket P ( {\rho} ) I- \S_\a(u) \right\rrbracket }\n
= \sigma_- \mathcal{H} \n &\hbox{on }\Sigma_-
\\{\left\llbracket u \right\rrbracket }=0 &\hbox{on }\Sigma_-\\ {u}_- =0 &\text{on }\Sigma_b.
\end{cases}$$ Here we have written the differential operators $\naba$, $\diva$, and $\da$ with their actions given by $$(\naba f)_i := \a_{ij} \p_j f,\; \diva X := \a_{ij}\p_j X_i, \text{ and }\da f := \diva \naba f$$ for appropriate $f$ and $X$. We have also written $$\label{n_def}
\n := (-\p_1 \eta, - \p_2 \eta,1)$$ for the non-unit normal to $\Sigma(t)$, and we have written $$\begin{gathered}
\label{deviatoric_a_def}
(\sg_{\a} u)_{ij} = \a_{ik} \p_k u_j + \a_{jk} \p_k u_i, \qquad \sgz_{\a} u = \sg_{\a} u - \frac{2}{3} \diva u I,\\
\text{and } \S_{\a,\pm}(u): =\mu_\pm \sgz_{\a} u+ \mu_\pm' \diva u I.\end{gathered}$$ Note that if we extend $\diva$ to act on symmetric tensors in the natural way, then $\diva \S_{\a} u =\mu\Delta_\a u+(\mu/3+\mu')\nabla_\a \operatorname{div}_\a u$. Recall that $\a$ is determined by $\eta$ through . This means that all of the differential operators in are connected to $\eta$, and hence to the geometry of the free surfaces.
The equilibrium state given by corresponds to the static solution $(u,\rho, \eta)=(0,\bar{\rho},0)$ of .
Perturbation equations
----------------------
We will now rephrase the PDE in a perturbation formulation around the steady state solution $(0,\bar\rho,0)$. We define a special density perturbation by $$\label{q_def}
\q=\rho-\bar\rho- \p_3\bar\rho\theta.$$ In order to deal with the pressure term $P(\rho)=P(\bar\rho+\q+ \p_3\bar\rho\theta)$ we introduce the Taylor expansion: by we have $$\label{R1}
P (\bar\rho+\q+\p_3\bar\rho\theta)=P (\bar{\rho} )+P '(\bar{\rho} )(\q+\p_3\bar\rho\theta)+\mathcal{R}=P (\bar{\rho} )+P '(\bar{\rho} ) \q -g\bar\rho\theta+\mathcal{R},$$ where the remainder term is defined via $$\label{R_def}
\mathcal{R} =\int_{\bar{\rho} }^{\bar{\rho} +\q+\p_3\bar\rho\theta}(\bar{\rho} +\q+\p_3\bar\rho\theta-z) P ^{\prime\prime}(z)\,dz.$$ The advantage of defining the perturbation in this manner is seen in the following cancellation: $$\begin{split}
&\a_{ij}\p_j P ( {\rho} )+g\rho \delta_{i3}=\a_{ij}\p_j P ( {\rho} )+g\rho \a_{ij}\p_j\Theta_3
\\&\quad=\a_{ij}\p_j (P (\bar{\rho} )+P '(\bar{\rho} ) \q -g\bar\rho\theta+\mathcal{R})+g(\bar\rho+\q+\p_3\bar\rho\theta) \a_{ij}\p_j(x_3+\theta)
\\&\quad=\a_{ij}\p_j ( P '(\bar{\rho} ) \q )-g\a_{ij}\p_j (\bar\rho \theta )+\a_{ij}\p_j\mathcal{R}+g \bar\rho \a_{ij}\p_j\theta +g(\q+\p_3\bar\rho\theta) \a_{i3} +g( \q+\p_3\bar\rho\theta) \a_{ij}\p_j\theta
\\&\quad=\a_{ij}\p_j ( P '(\bar{\rho} ) \q ) +\a_{ij}\p_j\mathcal{R}+g \q \a_{i3}+g( \q+\p_3\bar\rho\theta) \a_{ij}\p_j\theta
\\&\quad=\bar\rho\a_{ij}\p_j ( h '(\bar{\rho} ) \q ) +\a_{ij}\p_j\mathcal{R}+g( \q+\p_3\bar\rho\theta) \a_{ij}\p_j\theta,
\end{split}$$ where we have used and . Recalling also , , and , we find that $$-g\bar\rho_+\theta=-\bar{\rho}_1 g\eta_+\text{ on }\Sigma_+,\text{ and } {\left\llbracket -g\bar\rho\theta \right\rrbracket } =-\rj g \eta_-\text{ on }\Sigma_-.$$
The equations become the following system when perturbed around the equilibrium $(0,\bar{\rho},0)$: $$\label{geometric}
\begin{cases}
\partial_t \q +\operatorname{div}_\a((\bar{\rho} + \q+\p_3\bar\rho\theta) u ) - \p_3^2\bar\rho K \theta \p_t\theta - K\p_t\theta \pa_3 \q =0 & \text{in } \Omega
\\
( \bar{\rho} + \q+\p_3\bar\rho\theta)\partial_t u
+( \bar{\rho} + \q+\p_3\bar\rho\theta) (-K\p_t\theta \pa_3 u + u \cdot \nab_\a u )
+ \bar{\rho}\nabla_\a \left(h'(\bar{\rho})\q\right) \\
\quad -\diva \S_{\a} u =- \nabla_\a\mathcal{R}-g( \q+\p_3\bar\rho\theta ) \nabla_\a \theta & \text{in }
\Omega
\\
\partial_t \eta = u\cdot \n & \text{on }\Sigma
\\
( P'(\bar\rho)\q I- \S_{\a}( u))\n = \bar{\rho}_1 g \eta \n-\sigma_+ \mathcal{H} \n
- \mathcal{R}_+ \n
& \text{on } \Sigma_+
\\
{\left\llbracket P'(\bar\rho)\q I- \S_\a(u) \right\rrbracket }\n = \rj g\eta\n +\sigma_- \mathcal{H} \n - {\left\llbracket \mathcal{R} \right\rrbracket }\n
&\text{on }\Sigma_-
\\
{\left\llbracket u \right\rrbracket }=0 &\text{on } \Sigma_-
\\
u_-=0 &\hbox{on }\Sigma_b.
\end{cases}$$
The introduction of the special density perturbation $q$ given by and the subsequent perturbation equations of form is crucial for our study of the nonlinear stability in [@JTW_GWP]; it is not essential for the local theory developed in this paper. Indeed, we could consider $\rho$ or $\rho-\bar\rho$ directly. We choose here to consider $q$ in order to be consistent with the study in [@JTW_GWP].
Main results and discussion
===========================
Previous work
-------------
Free boundary problems in fluid mechanics have attracted much interest in the mathematical community. A thorough survey of the literature would prove impossible here, so we will primarily mention the work most relevant to our present setting, namely that related to layers of viscous fluid. We refer to the review of Shibata and Shimizu [@ShSh] for a more proper survey of the literature.
The dynamics of a single layer of viscous incompressible fluid lying above a rigid bottom, i.e. the incompressible viscous surface wave problem, have attracted the attention of many mathematicians since the pioneering work of Beale [@B1]. For the case without surface tension, Beale [@B1] proved the local well-posedness in the Sobolev spaces. Hataya [@H] obtained the global existence of small, horizontally periodic solutions with an algebraic decay rate in time. Guo and Tice [@GT_per; @GT_inf; @GT_lwp] developed a two-tier energy method to prove global well-posedness and decay of this problem. They proved that if the free boundary is horizontally infinite, then the solution decays to equilibrium at an algebraic rate; on the other hand, if the free boundary is horizontally periodic, then the solution decays at an almost exponential rate. The proofs were subsequently refined by the work of Wu [@Wu]. For the case with surface tension, Beale [@B2] proved global well-posedness of the problem, while Allain [@A] obtained a local existence theorem in two dimension using a different method. Bae [@B] showed the global solvability in Sobolev spaces via energy methods. Beale and Nishida [@BN] showed that the solution obtained in [@B2] decays in time with an optimal algebraic decay rate. Nishida, Teramoto and Yoshihara [@NTY] showed the global existence of periodic solutions with an exponential decay rate in the case of a domain with a flat fixed lower boundary. Tani [@Ta] and Tani and Tanaka [@TT] also discussed the solvability of the problem with or without surface tension by using methods developed by Solonnikov in [@So; @So_2; @So_3]. Tan and Wang [@TW] studied the vanishing surface tension limit of the problem.
There are fewer results on two-phase incompressible problems, i.e. the incompressible viscous surface-internal wave or internal wave problems. Hataya [@H2] proved an existence result for a periodic free interface problem with surface tension, perturbed around Couette flow; he showed the local existence of small solution for any physical constants, and the existence of exponentially decaying small solution if the viscosities of the two fluids are sufficiently large and their difference is small. Prüss and Simonett [@PS] proved the local well-posedness of a free interface problem with surface tension in which two layers of viscous fluids fill the whole space and are separated by a horizontal interface. For two horizontal fluids of finite depth with surface tension, Xu and Zhang [@XZ] proved the local solvability for general data and global solvability for data near the equilibrium state using Tani and Tanaka’s method. Wang and Tice [@WT] and Wang, Tice and Kim [@WTK] adapted the two-tier energy methods of [@GT_per; @GT_inf; @GT_lwp] to develop the nonlinear Rayleigh-Taylor instability theory for the problem, proving the existence of a sharp stability criterion given in terms of the surface tension coefficient, gravity, periodicity lengths, and $\rj$.
The free boundary problems corresponding to a single horizontally periodic layer of compressible viscous fluid with surface tension have been studied by several authors. Jin [@jin] and Jin-Padula [@jin_padula] produced global-in-time solutions using Lagrangian coordinates, and Tanaka and Tani [@tanaka_tani] produced global solutions with temperature dependence. However, to the best of our knowledge, even the local existence problem for two layers of compressible viscous fluids remains unsolved.
The two-layer problem is important because it allows for the development of the classical Rayleigh-Taylor instability [@3R; @3T], at least when the equilibrium has a heavier fluid on top and a lighter one below and there is a downward gravitational force. In our companion paper [@JTW_GWP] we identify a stability criterion and prove the existence of global solutions that decay to equilibrium. In our companion paper [@JTW_nrt] we show that the stability criterion is sharp, as in the incompressible case [@WT; @WTK], and that the Rayleigh-Taylor instability persists at the nonlinear level (the linear analysis was developed by Guo and Tice in [@GT_RT]). The Rayleigh-Taylor instability is a long time phenomenon; for the local-in-time theory developed in this paper it plays no essential role.
Local existence
---------------
In our companion paper [@JTW_GWP] we deal with questions of global existence and asymptotic stability of solutions to the perturbed system . The analysis there is carried out in a high-regularity functional framework that is indexed by an integer $N\ge 3$ related to the decay properties of solutions. Consequently, we must first guarantee the local existence of solutions in this framework for every $N \ge 3$.
Our main result guarantees the existence of high-regularity solutions to under smallness conditions on the initial free surface as well as the temporal interval. Notice, though, that there are no smallness conditions placed on the initial velocity or density. We refer to Appendix \[sec\_en\_dis\] for the definitions of the terms ${\EEE}$, $\E$, $\D$, $\hat{\E}^\sigma$, $\hat{\D}^\sigma$, and $\Lf$ (which all depend on $N$) appearing in the statement of the theorem.
\[local\_existence\_intro\] Let $N \ge 3$ be an integer. Assume that either $\sigma_\pm> 0$ or else $\sigma_\pm=0$. Suppose that $(u_0,q_0,\eta_0)$ satisfy ${\EEE}<\infty$ in addition to the compatibility conditions , and that $$\label{q_0_assump}
\rho_\ast\le \rho_0:=\bar\rho+q_0+\p_3\bar\rho\theta_0\le \rho^\ast$$ for two constants $0<\rho_\ast,\rho^\ast<\infty$. There exists a universal constant $\delta_\eta >0$ and a $T_{loc} = T_{loc}({\EEE})$ such that if $$\Lf[\eta_0] \le \frac{\delta_\eta}{2} \text{ and } 0 < T \le T_{loc},$$ then there exists a triple $(u,q,\eta)$ defined on the temporal interval $[0,T]$ satisfying the following three properties. First, $(u,q,\eta)$ achieve the initial data at $t=0$. Second, the triple uniquely solve . Third, the triple obey the estimates $$\begin{gathered}
\label{le_intro_01}
\sup_{0\le t \le T} \left( \E[u(t)] + \E[q(t)] + \hat{\E}^\sigma[\eta(t)] + {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \right)
\\
+ \int_0^T \left( \D[u(t)] + {{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D[q(t)] + \hat{\D}^\sigma[\eta(t)] \right) dt \le P({\EEE})\end{gathered}$$ for a universal positive polynomial $P$ with $P(0)=0$. Also, $$\frac{1}{2}\rho_\ast\le \rho(x,t) :=\bar\rho(x)+q(x,t)+\p_3\bar{\rho}(x)\theta(x,t)\le \frac{3}{2}\rho^\ast \text{ for all } x \in \Omega \text{ and } t \in [0,T]$$ and $$\Lf[\eta](T) \le \delta_\eta.$$ Moreover, the mapping $\Theta(\cdot,t)$ defined by is a $C^{4N-2}$ diffeomorphism for each $t \in [0,T]$.
The diffeomorphism condition in Theorem \[local\_existence\_intro\] guarantees that the solution to also gives rise to a solution to in the original coordinate system.
The temporal existence interval of Theorem \[local\_existence\_intro\] with $\sigma_\pm>0$ is independent of $\sigma_\pm$. As such, we can use a standard limiting argument to produce a solution to when $\sigma_+ >0$ and $\sigma_- =0$ or $\sigma_+ =0$ and $\sigma_- >0$ (the case $\sigma_\pm=0$ is already covered by the theorem). While this produces a solution, it does not yield a satisfactory local well-posedness theory. Indeed, if we are given data $(u_0,q_0,\eta_0)$ satisfying the compatibility conditions with, say $\sigma_+ >0$ and $\sigma_- =0$, we would need to produce a sequence of approximate data $(u_0^\sigma,q_0^\sigma,\eta_0^\sigma)$ satisfying with $\sigma_\pm >0$ that converge to $(u_0,q_0,\eta_0)$ as $\sigma_- \to 0$. While this construction might be possible, the compatibility conditions are sufficiently complicated that we have not pursued it in this paper.
It is noteworthy, though, that if we send both $\sigma_\pm \to 0$, then the same sort of limiting argument can provide a sort of continuity result, connecting our result with $\sigma_\pm >0$ to our result with $\sigma_\pm=0$. See [@JTW_GWP] for more details.
Theorem \[local\_existence\_intro\] is proved in a somewhat more general form (the smallness conditions on the data are more general) in Theorems \[local\_existence\] and \[local\_existence\_no\_ST\]. The former handles the case $\sigma_\pm >0$, while the latter handles the case $\sigma_\pm =0$. The proof of Theorem \[local\_existence\] is more complicated than that of Theorem \[local\_existence\_no\_ST\] because of the regularity requirements for dealing with the surface tension terms in . Therefore, most of the paper is devoted to the proof of Theorem \[local\_existence\]. We sketch below some of the main ideas of the proof.
[ **Picard iteration** ]{}
We construct solutions to with $\sigma_\pm >0$ by way of a Picard iteration scheme, which is developed in Section \[sec\_lwp\_st\]. This scheme is built on two sub-problems: the transport problem for $q$, and the two-phase free boundary Lamé problem for $(u,\eta)$. The sequence of Picard iterates $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ is constructed in Theorem \[approx\_solns\] under some smallness assumptions on $\eta_0$ and time. The theorem also provides estimates of various high-regularity norms in terms of polynomials of ${\EEE}$.
The uniform bounds are sufficient for extracting weak and weak-$\ast$ limits from the sequence of Picard iterates. Space-time compactness results then provide strong convergence results. However, the nature of our iteration scheme does not allow us to immediately deduce from the strong subsequential limits that a solution to exists. Instead, we show that the sequence of iterates actually contracts in certain low-regularity norms, as long as further smallness conditions are imposed on $\eta_0$ and the time interval. This is the content of Theorem \[contraction\_thm\].
Theorems \[approx\_solns\] and \[contraction\_thm\] are then combined in Theorem \[local\_existence\] to produce the desired high-regularity solutions to . Indeed, the low-regularity strong convergence and the high-regularity bounds combine to show that the sequence of iterates actually converges in a sufficiently high-regularity context to pass to the limit in the sequence (without extracting a subsequence) and produce a solution to satisfying the bounds . The high regularity of our solutions requires that we impose the compatibility conditions mentioned in Theorem \[local\_existence\_intro\].
[ **The transport problem for $q$** ]{}
We study the problem in Section \[sec\_xport\]; the pair $(u,\eta)$ are assumed to be given, and $q$ is solved for. The key feature of the transport equation in a bounded domain is the behavior of the normal component of the transport velocity on the boundary: its vanishing allows for the construction of solutions without imposing boundary conditions. This vanishing occurs for precisely because the pair $(u,\eta)$ satisfy the equations $\dt \eta = u \cdot \n$ on $\Sigma$, ${\left\llbracket u \right\rrbracket }=0$ on $\Sigma_-$, and $u_-=0$ on $\Sigma_b$.
Using this condition and the high regularity of $(u,\eta)$, we can produce a solution to using the method of characteristics. In a $C^k$-smoothness context we could then simply apply standard a priori estimates (which again depend critically on the vanishing of the normal transport velocity) to deduce the regularity estimates for $q$ desired in our iteration scheme. However, due to the Sobolev regularity $(u,\eta)$, it is not a priori clear that those estimates can be rigorously applied to the solution given by characteristics. To get around this technical obstacle, we show that the problem can be “lifted” to a corresponding problem on the spatial domain $\mathrm{T}^2\times \mathbb{R}$, and that the restriction of the lifted solution agrees with the solution given by characteristics. The advantage of working with the lifted problem is that it is amenable to solution by the Friedrichs mollification method, which employs delicate properties of mollification operators that are unavailable in bounded domains. The Friedrichs method allows for the rigorous derivation of the desired a priori Sobolev estimates, and we deduce in Theorem \[xport\_well-posed\] that solutions to exist and obey the estimates needed for our iteration scheme.
[ **The two-phase free boundary Lamé problem** ]{}
The sub-problem takes $\rho$ as a given function in addition to various forcing terms, and produces a solution pair $(u,\eta)$. Notice that is not a linear problem: the $\a$ coefficients of the differential operators and the non-unit normal $\n$ are both determined by $\eta$. We need this nonlinearity for two crucial reasons. First, as mentioned above, the equation $\dt \eta = u \cdot \n$ is essential in using $(u,\eta)$ to generate the transport velocity in . Second, it allows us to take advantage of the regularity gain offered by the surface tension terms in . Without it, our iteration scheme would fail to control $\eta$ at the highest level of regularity.
We produce a solution to in Section \[sec\_lame\_free\]. It is simple to see as two coupled linear problems: a parabolic problem for $u$ with $\eta$ given, and a transport problem for $\eta$ with $u$ given. This perspective works well without surface tension, and indeed this is how we proceed in Theorem \[local\_existence\_no\_ST\]. However, with surface tension the regularity demands of the $\sigma_\pm \Delta_\ast \eta_\pm$ terms are higher than what can be recovered from the transport equation for $\eta$, given the target regularity for $u$. Our way around this difficulty is to recover a solution to as a limit as $\kappa \to 0$ of the $\kappa-$approximation problem . This is accomplished in Theorems \[lame\_local\] and \[lame\_exist\]. The former establishes the existence of solutions to on $\kappa-$independent intervals, while the latter passes to the limit $\kappa \to 0$ to solve and refines some of the estimates from the former.
[ **The $\kappa$-approximation problem** ]{}
Theorems \[lame\_local\] and \[lame\_exist\] are predicated on the analysis in Section \[sec\_kappa\_exist\], where solutions to are constructed, and Section \[sec\_kappa\_est\], where $\kappa-$independent a priori estimates for solutions to are derived.
Solutions to are produced in Theorem \[kappa\_contract\] by way of the contraction mapping principle in an appropriate space. The contracting map is constructed by solving the linear sub-problems and . The problem is a heat equation; its well-posedness is standard, but we record (without proof) the precise form of the estimates of solutions in terms of $\kappa$ in Theorem \[heat\_estimates\] of Section \[sec\_heat\]. The only subtlety in is the appearance of the term $\Xi$. This is inserted as a “corrector function” in order to force the compatibility conditions for to match those of for every $\kappa \in (0,1)$. The problem is studied in Section \[sec\_lame\], where solutions are produced via a Galerkin scheme and an iteration argument in Theorems \[lame\_strong\] and \[lame\_high\].
The analysis in Section \[sec\_kappa\_est\] develops a priori estimates for that are strong enough to allow us to bound the existence time from below in a $\kappa-$independent manner and to obtain $\kappa-$independent energy and dissipation estimates. Here again the nonlinear structure of is essential in closing the estimates (see for example the handling of the $F^{3,\alpha}$ terms in Proposition \[k\_hor\_est\]). It is noteworthy that our main estimate, Theorem \[kappa\_apriori\], actually fails to provide an estimate of ${{\left\Vertu\right\Vert}^2}_{L^2_T H^{4N+1}}$, though this term is still guaranteed to be finite. This is due to the fact that this estimate depends on being able to simultaneously estimate ${{\left\Vert\eta\right\Vert}^2}_{L_T^\infty H^{4N+1/2}}$. The structure of only permits $\kappa-$dependent estimates of this term, and so we fail to estimate ${{\left\Vertu\right\Vert}^2}_{L^2_T H^{4N+1}}$ independently of $\kappa$. It is only later, in Theorem \[lame\_exist\], after passing to the limit $\kappa \to 0$ that we are able to estimate ${{\left\Vert\eta\right\Vert}^2}_{L_T^\infty H^{4N+1/2}}$ and thereby recover an estimate of ${{\left\Vertu\right\Vert}^2}_{L^2_T H^{4N+1}}$.
[ **Smallness condition**]{}
Throughout the paper we impose smallness conditions on $\eta_0$ and the size of the temporal interval on which solutions exist. No smallness condition is imposed on $u_0$ or $q_0$. The smallness of $\eta_0$ is used with a small-time condition to control various norms of $\eta(\cdot,t)$, which in turn provides control of the coefficients of the differential operators in (see Lemma \[eta\_small\]), a Korn inequality (see Proposition \[korn\]), and elliptic regularity estimates (see Proposition \[lame\_elliptic\]). The smallness of time is used as above and to guarantee that various absorption arguments work throughout the paper.
Definitions and terminology {#def-ter}
---------------------------
We now mention some of the definitions, bits of notation, and conventions that we will use throughout the paper.
[ **Universal constants and polynomials**]{}
Throughout the paper we will refer to generic constants as “universal” if they depend on $N$, $\Omega_\pm$, the various parameters of the problem (e.g. $g$, $\mu_\pm$, $\sigma_\pm$), $\rho_\ast,\;\rho^\ast$ and the functions $\bar{\rho}_\pm$, with the caveat that if the constant depends on $\sigma_\pm$, then it remains bounded above as either $\sigma_\pm$ tend to $0$. For example this allows constants of the form $g\mu_+ + 3 \sigma_-^2 + \sigma_+$ but forbids constants of the form $3 + 1/\sigma_-$. Whenever behavior of the latter type appears we will keep track of the dependence on $\sigma_\pm$ in our estimates. We make this choice in order to handle the vanishing surface tension limit.
We will employ the notation $a \ls b$ to mean that $a \le C b$ for a universal constant $C>0$. We also employ the convention of saying that a polynomial $P$ is a “universal positive polynomial” if $P(x) = \sum_{j=0}^m C_j x^j$ for some $m \ge 0$, where each $C_j >0$ is a universal constant. Universal constants and polynomials are allowed to change from one inequality to the next as needed.
[ **Norms** ]{}
We write $H^k(\Omega_\pm)$ with $k\ge 0$ and and $H^s(\Sigma_\pm)$ with $s \in {\mathbb{R}^{}}$ for the usual Sobolev spaces. We will typically write $H^0 = L^2$. If we write $f \in H^k(\Omega)$, the understanding is that $f$ represents the pair $f_\pm$ defined on $\Omega_\pm$ respectively, and that $f_\pm \in H^k(\Omega_\pm)$. We employ the same convention on $\Sigma_\pm$. Throughout the paper we will refer to the space $\H(\Omega)$ defined as follows: $$\H(\Omega) = \{ v \in H^1(\Omega) \; \vert \; {\left\llbracket v \right\rrbracket }=0 \text{ on } \Sigma_- \text{ and } v_- = 0 \text{ on } \Sigma_b\}.$$
To avoid notational clutter, we will avoid writing $H^k(\Omega)$ or $H^k(\Sigma)$ in our norms and typically write only ${\left\Vert\cdot\right\Vert}_{k}$, which we actually use to refer to sums $${{\left\Vertf\right\Vert}^2}_k = {{\left\Vertf_+\right\Vert}^2}_{H^k(\Omega_+)} + {{\left\Vertf_-\right\Vert}^2}_{H^k(\Omega_-)} \text{ or } {{\left\Vertf\right\Vert}^2}_k = {{\left\Vertf_+\right\Vert}^2}_{H^k(\Sigma_+)} + {{\left\Vertf_-\right\Vert}^2}_{H^k(\Sigma_-)}.$$ Since we will do this for functions defined on both $\Omega$ and $\Sigma$, this presents some ambiguity. We avoid this by adopting two conventions. First, we assume that functions have natural spaces on which they “live.” For example, the functions $u$, $\rho$, $q$, and $\bar{\eta}$ live on $\Omega$, while $\eta$ lives on $\Sigma$. As we proceed in our analysis, we will introduce various auxiliary functions; the spaces they live on will always be clear from the context. Second, whenever the norm of a function is computed on a space different from the one in which it lives, we will explicitly write the space. This typically arises when computing norms of traces onto $\Sigma_\pm$ of functions that live on $\Omega$. We use ${\left\Vert\cdot\right\Vert}_{L^p_TX}$ to denote the norm of the space $L^p([0,T];X)$.
Occasionally we will need to refer to the product of a norm of $\eta$ and a constant that depends on $\pm$. To denote this we will write $$\gamma {{\left\Vert\eta\right\Vert}^2}_{k} = \gamma_+ {{\left\Vert\eta_+\right\Vert}^2}_{H^k(\Sigma_+)} + \gamma_- {{\left\Vert\eta_-\right\Vert}^2}_{H^k(\Sigma_-)}.$$
[ **Derivatives** ]{}
We write $\mathbb{N} = \{ 0,1,2,\dotsc\}$ for the collection of non-negative integers. When using space-time differential multi-indices, we will write $\mathbb{N}^{1+m} = \{ \alpha = (\alpha_0,\alpha_1,\dotsc,\alpha_m) \}$ to emphasize that the $0-$index term is related to temporal derivatives. For just spatial derivatives we write $\mathbb{N}^m$. For $\alpha \in \mathbb{N}^{1+m}$ we write $\pal = \dt^{\alpha_0} \p_1^{\alpha_1}\cdots \p_m^{\alpha_m}.$ We define the parabolic counting of such multi-indices by writing ${\left\vert\alpha\right\vert} = 2 \alpha_0 + \alpha_1 + \cdots + \alpha_m.$ We will write $\nab_{\ast}f$ for the horizontal gradient of $f$, i.e. $\nab_{\ast}f = \p_1 f e_1 + \p_2 f e_2$, while $\nab f$ will denote the usual full gradient.
The two-phase Lamé problem {#sec_lame}
==========================
In this section we are concerned with solving the two-phase Lamé problem for the velocity field $u$: $$\label{lame}
\begin{cases}
\rho \dt u - \diva \Sa u = F^1 & \text{in }\Omega \\
-\Sa u \n = F^2_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa u \right\rrbracket } \n = -F^2_- &\text{on } \Sigma_- \\
{\left\llbracket u \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b \\
u(\cdot,0) = u_0 &\text{in } \Omega.
\end{cases}$$ Here we assume that $\eta$ is given and determines $\a$ and $\n$ via and , respectively, and that $\rho$ is given as well. We also assume that $\rho$ obeys the estimate $$\label{rho_assump_1}
\frac{1}{2} \rho_\ast\le \rho(x,t) \le \frac{3}{2} \rho^\ast \text{ for all } x \in \Omega \text{ and } t \in [0,T].$$
In our analysis of we will need to make various assumption about energies associated to $\eta$ and $\rho$. We record these now, first defining a low regularity term that will be used in controlling the coefficients of the equations in : $$\mathcal{K}[\eta,\rho](T) = \sup_{0\le t\le T} \left( {{\left\Vert\eta(t)\right\Vert}^2}_{9/2} + {{\left\Vert\dt \eta(t)\right\Vert}^2}_{7/2} + {{\left\Vert\dt^2 \eta(t)\right\Vert}^2}_{5/2} + {{\left\Vert\rho(t)\right\Vert}^2}_{3} + {{\left\Vert\dt \rho(t)\right\Vert}^2}_3 \right).$$ We also define some high regularity terms: $$\begin{gathered}
\Kf[\eta,\rho](T) = \int_0^T \left( {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} + \sum_{j=1}^{2N+1} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+3/2} \right)dt \\
+ \sup_{0\le t\le T} \left( {{\left\Vert\eta(t)\right\Vert}^2}_{4N} + \sum_{j=1}^{2N} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+1/2} \right)
+ \sup_{0\le t\le T} \sum_{j=0}^{2N} {{\left\Vert\dt^j \rho(t)\right\Vert}^2}_{4N-2j} ,\end{gathered}$$ and $$\Ef[\eta,\rho](T) = \sup_{0\le t\le T} \left( \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j} \right) +
\sup_{0\le t\le T} \left( {{\left\Vert\rho(t)\right\Vert}^2}_{4N} +\sum_{j=1}^{2N} {{\left\Vert\dt^j \rho(t)\right\Vert}^2}_{4N-2j+1} \right).$$
We will also need to make certain assumptions on the forcing terms. To describe these we first define $$\label{F2_def}
\f_2(t) := \sum_{j=0}^{2N-1} {{\left\Vert\dt^j F^1(t)\right\Vert}^2}_{4N-2j-1} + {{\left\Vert\dt^{2N} F^1(t)\right\Vert}^2}_{\Hd} + \sum_{j=0}^{2N} {{\left\Vert\dt^j F^2(t)\right\Vert}^2}_{4N-2j-1/2}$$ and $$\label{Finf_def}
\f_\infty(t) := \sum_{j=0}^{2N-1} \left[ {{\left\Vert\dt^j F^1(t)\right\Vert}^2}_{4N-2j-2} + {{\left\Vert\dt^j F^2(t)\right\Vert}^2}_{4N-2j-3/2} \right].$$ We will assume throughout this section that the forcing terms obey the estimate $$\label{F12_assump}
\sup_{0 \le t \le T} \f_\infty(t) + \int_0^T \f_2(t) dt < \infty.$$
We define $\dt u(\cdot,0)$ via $$\label{u_data_1}
(\rho \dt u) \vert_{t=0} = (\diva \Sa u + F^1)\vert_{t=0},$$ and then for $j=0,\dotsc,2N-1$ we inductively define $$\label{u_data_2}
(\rho \dt^{j+1} u)\vert_{t=0} = - \sum_{k=0}^{j-1} C_{j}^k (\dt^{j-k} \rho \dt^{k+1} u)\vert_{t=0} + \dt^j (\diva \Sa u + F^1)\vert_{t=0}.$$ We must assume that $u_0$, $F^1(\cdot,0)$, and $F^2(\cdot,0)$ satisfy the following $2N$ systems of compatibility conditions: $$\label{lame_ccs}
\begin{cases}
-\dt^j(\Sa u \n) \vert_{t=0} = \dt^j F^2_+\vert_{t=0} &\text{on } \Sigma_+ \\
-{\left\llbracket \dt^j(\Sa u\n) \right\rrbracket } \vert_{t=0} = -\dt^j F^2_- \vert_{t=0} &\text{on } \Sigma_- \\
{\left\llbracket \dt^j u \right\rrbracket }\vert_{t=0} =0 &\text{on } \Sigma_- \\
\dt^j u_-\vert_{t=0} = 0 &\text{on } \Sigma_b
\end{cases}$$ for $j=0,\dotsc,2N-1$.
Weak solutions
--------------
Weak solutions will only arise in our analysis in the context of the highest-order time derivatives of solutions to , where they will arise as a byproduct of the regularity assumptions on the forcing terms. As such, we will only discuss the meaning of weak solution and provide a simple uniqueness result, neglecting a construction of weak solutions.
We define the time-dependent inner-product on $\H$ according to $${\left( \mspace{-2.5mu} \left(u,v\right) \mspace{-2.5mu} \right) } = \int_\Omega J(t) \left( \frac{\mu}{2} \sg^0_{\a(t)} u : \sg^0 _{\a(t)} v + \mu' \operatorname{div}_{\a(t)} u \operatorname{div}_{\a(t)} v \right),$$ where $J(t)>0$. Notice that for the sake of notational brevity we have neglected to include $t$ in ${\left( \mspace{-2.5mu} \left(u,v\right) \mspace{-2.5mu} \right) }$ even though the inner-product actually changes in time. With this definition in hand we can give the meaning of weak solutions.
We say $u$ is a weak solution to if $$\label{weak_def_1}
u \in L^\infty([0,T]; H^0) \cap L^2([0,T]; \H) \text{ and }\rho J \dt u \in L^2([0,T]; \Hd),$$ $u(\cdot,0) = u_0$, and $$\label{weak_def_2}
{\left\langle \rho J \dt u,v \right\rangle}_{\ast} + {\left( \mspace{-2.5mu} \left(u,v\right) \mspace{-2.5mu} \right) } = {\left\langle J F^1,v \right\rangle}_{\ast} - {\left\langle F^2,v \right\rangle}_{-1/2,\Sigma}$$ for almost every $t \in [0,T]$ and for every $v \in \H$. Here ${\left\langle \cdot,\cdot \right\rangle}_\ast$ denotes the dual pairing for $\H$, while ${\left\langle \cdot,\cdot \right\rangle}_{-1/2,\Sigma}$ denotes the dual-pairing on $H^{1/2}(\Sigma)$.
Next we show that weak solutions are unique.
\[weak\_unique\] Weak solutions to are unique.
If $u$ is a weak solution, then we choose $v = u$ as a test function in . Employing Proposition \[korn\], we may easily derive the differential inequality $$\ddt \int_\Omega \rho J \frac{{\left\vertu\right\vert}^2}{2} + C {{\left\Vertu\right\Vert}^2}_1 \ls \int_\Omega \frac{\dt(\rho J)}{\rho J} \rho J \frac{{\left\vertu\right\vert}^2}{2} + {{\left\VertJ F^1\right\Vert}^2}_{\ast} + {{\left\VertF^2\right\Vert}^2}_{-1/2}$$ for some universal $C>0$. Then Gronwall implies that $$\label{w_u_1}
\int_\Omega \rho J \frac{{\left\vertu\right\vert}^2}{2}(t) \le e^{\alpha t} \left( \int_\Omega \rho(0) J(0) \frac{{\left\vertu_0\right\vert}^2}{2} + \int_0^t {{\left\VertJ F^1\right\Vert}^2}_{\ast} + {{\left\VertF^2\right\Vert}^2}_{-1/2}\right)$$ for $\alpha = C + {\left\Vert\dt(\rho J)/(\rho J) \right\Vert}_{L^\infty}$, with $C$ universal.
Then if $u_1, u_2$ are two weak solutions, then $u = u_1 - u_2$ is a weak solution with $u(0) =0$, $F^1=0$, and $F^3=0$. The equality $u_1 =u_2$ then follows from and the fact that $\min{\rho J} > 0$.
Strong solutions
----------------
Now we turn to the construction of strong solutions to . We say that $u$ is a strong solution to on $[0,T]$ if $$\begin{cases}
u \in L^\infty([0,T];H^2\cap \H) \cap L^2([0,T]; H^3) \\
\dt u \in L^\infty([0,T]; H^0)\cap L^2 ([0,T];\H),
\end{cases}$$ $u(\cdot, 0) = u_0$, and $u$ satisfies almost everywhere (with respect to Lebesgue measure on $\Omega$ and Hausdorff measure on $\Sigma_\pm$ and $\Sigma_b$).
Our next result establishes the existence of strong solutions.
\[lame\_strong\] Suppose that $\rho$ satisfies and that $\Lf[\eta](T) < \delta_0$, where $\delta_0$ is given by Proposition \[lame\_elliptic\]. Suppose that $u_0 \in H^2$, $F^1 \in L^\infty_T H^0 \cap L^2_T H^1$, $\dt F^1 \in L^2_T (\H)^\ast$, $F^2 \in L^\infty_T H^{1/2} \cap L^2_T H^{3/2}$, $\dt F^2 \in L^2_T H^{-1/2}$, and that $u_0$ and $F^2(\cdot,0)$ satisfy the system of compatibility conditions $$\label{ls_01}
\begin{cases}
- \Sa u \n \vert_{t=0} = F^2_+ \vert_{t=0} &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa u \right\rrbracket } \n \vert_{t=0} = -F^2_- \vert_{t=0} &\text{on } \Sigma_- \\
{\left\llbracket u \right\rrbracket }\vert_{t=0} =0 &\text{on } \Sigma_- \\
u_-\vert_{t=0} = 0 &\text{on } \Sigma_b.
\end{cases}$$ Then there exists a unique $u$ that is a strong solution to . Additionally, $\dt u$ is a weak solution to $$\label{ls_02}
\begin{cases}
\rho \dt(\dt u) - \diva \Sa (\dt u) = F^{1,1} & \text{in }\Omega \\
-\Sa (\dt u) \n = F^{2,1}_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa (\dt u) \right\rrbracket } \n = -F^{2,1}_- &\text{on } \Sigma_- \\
{\left\llbracket \dt u \right\rrbracket } =0 &\text{on } \Sigma_- \\
\dt u_- = 0 &\text{on } \Sigma_b \\
\dt u(\cdot,0) = [\rho^{-1}(\diva \Sa u + F^1)] \vert_{t=0} &\text{in } \Omega,
\end{cases}$$ where $$\begin{split}
F^{1,1} &= \dt F^1 -\dt \rho \dt u + \operatorname{div}_{\dt \a}\Sa u + \diva{\S_{\dt \a} u} \\
F^{2,1}_+& = \dt F^{2}_+ + \S_{\dt \a} u \n + \Sa u \dt \n \\
F^{2,1}_- &= \dt F^{2}_+ - {\left\llbracket \S_{\dt \a} u \right\rrbracket } \n - {\left\llbracket \Sa u \right\rrbracket } \dt \n.
\end{split}$$ The solution obeys the estimate $$\begin{gathered}
\label{ls_03}
{{\left\Vertu\right\Vert}^2}_{L^\infty_T H^2} + {{\left\Vertu\right\Vert}^2}_{L^2_T H^3} +{{\left\Vert\dt u\right\Vert}^2}_{L^\infty_T H^0} + {{\left\Vert\dt u\right\Vert}^2}_{L^2_T H^1} + {{\left\Vert\rho J \dt^2 u\right\Vert}^2}_{L^2_T \Hd} \\
\ls (1+ P(\mathcal{K}[\eta,\rho](T))) \exp(T(1+ P(\mathcal{K}[\eta,\rho](T))))
\left( {{\left\Vertu_0\right\Vert}^2}_2 + {{\left\VertF^1\right\Vert}^2}_{L^\infty_T H^0} \right. \\
\left. + {{\left\VertF^1\right\Vert}^2}_{L^2_T H^1} + {{\left\Vert\dt F^1\right\Vert}^2}_{L^2_T (\H)^\ast}
+ {{\left\VertF^2\right\Vert}^2}_{L^\infty_T H^{1/2}} + {{\left\VertF^2\right\Vert}^2}_{L^2_T H^{3/2}} + {{\left\Vert\dt F^2\right\Vert}^2}_{L^2_T H^{-1/2}}\right),\end{gathered}$$ where $P$ is a positive universal polynomial such that $P(0)=0$.
The proof is a fairly standard application of the Galerkin method. For the sake of brevity we will provide only a terse sketch. For more details we refer to the incompressible case (where the analysis is actually somewhat more complicated): see for example [@GT_lwp], [@WTK], or [@TW].
Step 1 – The Galerkin approximation
Let $\{w_k\}_{k=1}^\infty$ be a basis of $\H$ and for $m \ge 1$ set $u^m(x,t) = a^m_k(t) w_k(x)$, where the $k$ summation runs over $k=1,\dotsc,m$. Using the theory of linear ODEs we can choose the $a^m_k$ such that $$\label{ls_1}
{\left(\rho J \dt u^m,w_k\right)}_0 + {\left( \mspace{-2.5mu} \left(u^m,w_k\right) \mspace{-2.5mu} \right) } = {\left(J F^1,w_k\right)}_0 - {\left(F^2,w_k\right)}_{0,\Sigma}$$ and ${\left( \mspace{-2.5mu} \left(u^m(0),w_k\right) \mspace{-2.5mu} \right) } = {\left( \mspace{-2.5mu} \left(u_0,w_k\right) \mspace{-2.5mu} \right) }$ for $k=1,\dotsc,m$. Here ${\left(\cdot,\cdot\right)}_0$ denotes the $L^2$ inner product on $\Omega$, while ${\left(\cdot,\cdot\right)}_{0,\Sigma}$ denotes the $L^2$ inner product on $\Sigma$. We then take linear combinations of so that $w_k$ may be replaced by $u^m$. This results in the basic energy equality $$\label{ls_2}
\frac{d}{dt} \int_\Omega \rho J \frac{{\left\vertu^m\right\vert}^2}{2} + {\left( \mspace{-2.5mu} \left(u^m,u^m\right) \mspace{-2.5mu} \right) } = \int_\Omega \dt(\rho J) \frac{{\left\vertu^m\right\vert}^2}{2} + {\left(J F^1,u^m\right)}_0 - {\left(F^2,u^m\right)}_{0,\Sigma}.$$ Notice that the assumption on $\Lf[\eta](T)$ guarantees that that the smallness conditions of Lemma \[eta\_small\] and Proposition \[korn\] are satisfied. Korn’s inequality, Proposition \[korn\], allows us to bound $${{\left\Vertu^m\right\Vert}^2}_{1} \ls {\left( \mspace{-2.5mu} \left(u^m,u^m\right) \mspace{-2.5mu} \right) },$$ while Lemma \[eta\_small\] and provide the estimates $${{\left\Vertu^m\right\Vert}^2}_0 \ls \int_\Omega \rho J \frac{{\left\vertu^m\right\vert}^2}{2} \ls {{\left\Vertu^m\right\Vert}^2}_0 \text{ and } {\left(J F^1,u^m\right)}_0 \ls {\left\VertF^1\right\Vert}_0 \int_\Omega \rho J \frac{{\left\vertu^m\right\vert}^2}{2},$$ and trace theory provides the estimate $$\label{ls_3}
- {\left(F^2,u^m\right)}_{0,\Sigma} \ls {\left\VertF^2\right\Vert}_0 {\left\Vertu^m\right\Vert}_1.$$ We combine – with Cauchy’s inequality to absorb the ${\left\Vertu^m\right\Vert}_1$ term onto the left; applying Gronwall’s lemma to the resulting inequality results in the bound $$\label{ls_4}
{{\left\Vertu^m\right\Vert}^2}_{L^\infty_T H^0} + {{\left\Vertu^m\right\Vert}^2}_{L^2_T H^1} \ls \exp(T(1+ P(\mathcal{K}[\eta,\rho](T))) )
\left( {{\left\Vertu_0\right\Vert}^2}_0 + {{\left\VertF^1\right\Vert}^2}_{L^2_T H^0} + {{\left\VertF^2\right\Vert}^2}_{L^2_T H^{0}} \right).$$
Next we differentiate in time and use a similar argument to get estimates of the sum ${{\left\Vert\dt u^m\right\Vert}^2}_{L^\infty_T H^0} + {{\left\Vert\dt u^m\right\Vert}^2}_{L^2_T H^1}$. Two complications arise. The first is that differentiating introduces a number of commutators. These may be dealt with via the estimates and the usual Sobolev embeddings. The second is that we must estimate ${{\left\Vert\dt u^m(0)\right\Vert}^2}_0$ in terms of $u_0$. This can be accomplished by using , evaluated at $t=0$, in conjunction with the choice of initial condition for $u^m(0)$ and the compatibility conditions . The resulting estimate bounds ${{\left\Vert\dt u^m\right\Vert}^2}_{L^\infty_T H^0} + {{\left\Vert\dt u^m\right\Vert}^2}_{L^2_T H^1}$ by the right side of .
Step 2 – Passing to the limit and higher regularity.
The estimates from Step 1 are uniform in $m$ and hence allow for the extraction of weak and weak-$\ast$ limits. Passing to the limit yields a function such that $u, \dt u \in L^\infty_T H^0 \cap L^2 \H$ and $$\label{ls_5}
{\left(\rho J \dt u,w\right)}_0 + {\left( \mspace{-2.5mu} \left(u,w\right) \mspace{-2.5mu} \right) } = {\left(J F^1,w\right)}_0 - {\left(F^2,w\right)}_{0,\Sigma}$$ for a.e. $t \in [0,T]$ and every $w \in \H$. The $L^\infty_T H^0$ and $L^2_T H^1$ norms of $u$ and $\dt u$ are controlled by the right side of by virtue of lower semicontinuity. We then apply the elliptic regularity result for the Lamé problem (a variant of Proposition \[lame\_elliptic\] adapted to the boundary conditions involving $\S$, which can be proved in a similar manner by appealing to Section 3 of [@WTK]) to deduce that ${{\left\Vertu\right\Vert}^2}_{L^\infty_T H^2} + {{\left\Vert u\right\Vert}^2}_{L^2_T H^3}$ is also bounded by the right side of . The only term remaining to estimate in is ${{\left\Vert\rho J \dt^2 u\right\Vert}^2}_{L^2_T (\H)^\ast}$. This can be accomplished by using and the existing estimates. The equation is seen to hold by differentiating in time.
Higher regularity
-----------------
Our aim now is to show that the strong solutions of Theorem \[lame\_strong\] are of higher regularity when the forcing terms are more regular. We first define the forcing terms that appear for the time differentiated versions of . We set $$F^{1,0} = F^1, \; F^{2,0}_+ = F^2_+, \;\text{ and }F^{2,0}_- = F^2_-,$$ and then for $j=1,\dotsc,2N$ we iteratively define $$\begin{aligned}
F^{1,j} &= \dt F^{1,j-1} -\dt \rho \dt^j u + \operatorname{div}_{\dt \a}\Sa \dt^{j-1} u + \diva{\S_{\dt \a} \dt^{j-1}u}
\label{F1j_def} \\
F^{2,j}_+ &= \dt F^{2,j-1}_+ + \S_{\dt \a} \dt^{j-1} u \n + \Sa \dt^{j-1} u \dt \n
\label{F2j_def} \\
F^{2,j}_- &= \dt F^{2,j-1}_+ - {\left\llbracket \S_{\dt \a} \dt^{j-1} u \right\rrbracket } \n - {\left\llbracket \Sa \dt^{j-1} u \right\rrbracket } \dt \n.
\label{F3j_def}\end{aligned}$$
The following lemma will be used in conjunction with an iteration argument to improve the regularity of strong solutions to when the forcing terms are of high regularity. It is a simple variant of Lemma 3.2 of [@TW], which is a technical refinement of Lemma 4.5 of [@GT_lwp]. As such, we omit the proof.
\[lame\_forcing\_est\] For $m=1,\dotsc,2N-1$ and $j=1,\dotsc,m$, we have the following estimates, where $P$ is a positive universal polynomial such that $P(0)=0$: $$\begin{gathered}
{{\left\Vert F^{1,j}\right\Vert}^2}_{L^2_T H^{2m-2j+1}} + {{\left\Vert F^{2,j}\right\Vert}^2}_{L^2_T H^{4N-2j+3/2}}
\ls (1+ P(\Kf[\eta,\rho](T))) \\
\times \left( \int_0^T \f_2(t)dt + {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{2m-2j+1}} +\sum_{\ell=0}^{j-1} {{\left\Vert\dt^\ell u\right\Vert}^2}_{L^2_T H^{2m-2j+3}} + {{\left\Vert\dt^\ell u\right\Vert}^2}_{L^\infty_T H^{2m-2j+2}}
\right),\end{gathered}$$ $$\begin{gathered}
{{\left\VertF^{1,j}\right\Vert}^2}_{L^\infty_T H^{2m-2j}} + {{\left\VertF^{2,j}\right\Vert}^2}_{L^\infty_T H^{2m-2j+1/2}}
\\
\ls (1+ P(\Kf[\eta,\rho](T))) \left( \sup_{0 \le t \le T} \f_\infty(t) + {{\left\Vert\dt^j u\right\Vert}^2}_{L^\infty_T H^{2m-2j}} + \sum_{\ell=0}^{j-1} {{\left\Vert\dt^\ell u\right\Vert}^2}_{L^\infty_T H^{2m-2j+2}} \right),\end{gathered}$$ and if we denote the functional $v \mapsto {\left(J F^{1,m},v\right)}_0 - {\left(F^{3,m},v\right)}_{\Sigma}$ in $\Hd$ by $\hat{F}^m$, then $$\begin{gathered}
{{\left\Vert\dt \hat{F}^{m} \right\Vert}^2}_{L^2_T \Hd}
\ls(1+ P(\Kf[\eta,\rho](T))) \left( \int_0^T \f_2(t)dt + \sum_{\ell=0}^{m-1} {{\left\Vert\dt^\ell u\right\Vert}^2}_{L^\infty_T H^{2}} + {{\left\Vert\dt^\ell u\right\Vert}^2}_{L^2_T H^{3}} \right. \\
\left. +{{\left\Vert\dt^m u\right\Vert}^2}_{L^2_T H^{2}} + {{\left\Vert\dt^{m+1} u\right\Vert}^2}_{L^2_T H^0}
\right).\end{gathered}$$
We can now state our higher regularity result.
\[lame\_high\] Assume that $u_0 \in H^{4N}$, that $F^1,F^2$ satisfy , and that the compatibility conditions are satisfied. Assume that $\rho$ satisfies , and that $\Lf[\eta](T) < \delta_0$, where $\delta_0$ is given by Proposition \[lame\_elliptic\]. Then there exists a unique $u$ satisfying $\dt^j u \in L^\infty([0,T];H^{4N-2j}) \cap L^2([0,T];H^{4N-2j+1})$ for $j=0,\dotsc,2N$ and $\rho J \dt^{2N+1}u \in L^2([0,T];\Hd)$ so that $\dt^j u$ solves $$\label{lh_01}
\begin{cases}
\rho \dt(\dt^j u) - \diva \Sa (\dt^j u) = F^{1,j} & \text{in }\Omega \\
-\Sa (\dt^j u) \n = F^{2,j}_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa (\dt^j u) \right\rrbracket } \n = -F^{2,j}_- &\text{on } \Sigma_- \\
{\left\llbracket \dt^j u \right\rrbracket } =0 &\text{on } \Sigma_- \\
\dt^j u_- = 0 &\text{on } \Sigma_b \\
\end{cases}$$ strongly for $j=0,\dotsc,2N-1$ and weakly for $j=2N$. The solution obeys the estimates $$\begin{gathered}
\label{lh_02}
\sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{4N-2j+1}} + {{\left\Vert\rho J \dt^{2N+1} u\right\Vert}^2}_{L^2_T \Hd}
\\
\ls (1+P(\Kf[\eta,\rho](T))) \exp\left((1+P(\Ef[\eta,\rho](T)))T \right) \left( \sum_{j=0}^{2N} {{\left\Vert\dt^j u(\cdot,0)\right\Vert}^2}_{4N-2j} + \int_0^T \f_2(t) dt \right)\end{gathered}$$ and $$\begin{gathered}
\label{lh_03}
\sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^\infty_T H^{4N-2j}}
\ls (1+P(\Kf[\eta,\rho](T))) \exp\left((1+P(\Ef[\eta,\rho](T)))T \right) \\
\times \left( \sum_{j=0}^{2N} {{\left\Vert\dt^j u(\cdot,0)\right\Vert}^2}_{4N-2j} + \sup_{0\le t \le T} \f_\infty(t) + \int_0^T \f_2(t) dt \right),\end{gathered}$$ where $P$ is a universal positive polynomial such that $P(0)=0$.
The proof follows the same basic outline (though the arguments here are somewhat simpler), as that of Theorem 4.8 of [@GT_lwp]: we iteratively apply Theorem \[lame\_strong\], Lemma \[lame\_forcing\_est\] to estimate the forcing terms appearing for each $j$, and the elliptic regularity result. We omit further details for the sake of brevity.
Heat estimates {#sec_heat}
==============
Suppose that $0 < \kappa < 1$. In this section we are concerned with the problem $$\label{heat}
\begin{cases}
\dt \eta - \kappa \Delta_\ast \eta = f & \text{in }\Sigma \times (0,T) \\
\eta(\cdot,0) = \eta_0 & \text{in }\Sigma
\end{cases}$$ where $\eta_0 \in H^{4N+1}(\Sigma)$ and $f$ satisfies $$\label{eta_f_assump}
\int_0^T \sum_{j=0}^{2N} {{\left\Vert\dt^j f(t)\right\Vert}^2}_{4N-2j } dt < \infty.$$ A simple interpolation argument (see for example Lemma A.4 of [@GT_lwp]) allows us to deduce from that $\dt^j f \in C^0([0,T];H^{4N-2j-1}(\Sigma))$ for $j=0,\dotsc,2N-1$. We may use this in to inductively define $\dt^j \eta(\cdot,0)\in H^{4N-2j+1}(\Sigma)$ for $j=1,\dotsc,2N$
We now state a result on the well-posedness of and estimates of its solutions. The proof is standard and thus omitted.
\[heat\_estimates\] Suppose that $\eta_0 \in H^{4N+1}(\Sigma)$, that $f$ satisfies , and that the data $\dt \eta(\cdot,0) \in H^{4N-2j+1}(\Sigma)$ are determined as above for $j=1,\dotsc,2N$. The problem admits a unique solution that achieves the initial data $\dt^j\eta(\cdot,0)$ for $j=0,\dotsc,2N$ and that satisfies the estimate $$\begin{gathered}
\label{he_01}
\sum_{j=0}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} + \sum_{j=1}^{2N+1} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^2_T H^{4N-2j+2}} \\
+ \kappa \left[ \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j+1}} + \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^2_T H^{4N-2j+1}} \right] + \kappa^2 {{\left\Vert\eta\right\Vert}^2}_{L^2_T H^{4N+2}}
\\
\ls e^T \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(\cdot,0)\right\Vert}^2}_{4N-2j+1} + \int_0^T e^{T-t} \sum_{j=0}^{2N} {{\left\Vert\dt^j f(t)\right\Vert}^2}_{4N-2j } dt.\end{gathered}$$
The $\kappa-$approximation problem {#sec_kappa_exist}
==================================
Our goal in this section is to produce a solution pair $(u,\eta)$ to the problem $$\label{kappa_problem}
\begin{cases}
\rho \dt u - \diva \Sa u = F^1 & \text{in }\Omega \\
\dt \eta - \kappa \Delta_\ast \eta = u \cdot \n - \kappa \Xi &\text{on }\Sigma \\
-\Sa u \n = - \sigma_+ \Delta_\ast \eta \n + F^2_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa u \right\rrbracket } \n = \sigma_- \Delta_\ast \eta \n - F^2_- &\text{on } \Sigma_- \\
{\left\llbracket u \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b \\
u(\cdot,0) = u_0, \eta(\cdot, 0) = \eta_0.
\end{cases}$$ Here we assume that $u_0 \in H^{4N}(\Omega)$, $\eta_0 \in H^{4N+1}(\Sigma)$, $F^1$, $F^2$ satisfy . Notice that the problem is nonlinear: the terms $\a$ and $\n$ are viewed as being generated by $\eta$ itself. We will again assume that $\rho$ is a given function on the given time interval $[0,T_\ast]$ and satisfies (with $T$ replaced by $T_\ast$) in addition to the bound $$\label{rho_assump_2}
\Ef[\rho](T_*) := \sup_{0 \le t \le T_\ast} \Ef_\infty[\rho(t)] := \sup_{0\le t \le T_\ast} \left({{\left\Vert\rho(t)\right\Vert}^2}_{4N} + \sum_{j=1}^{2N} {{\left\Vert\dt^j \rho(t)\right\Vert}^2}_{4N-2j+1} \right) \ls 1+ P({\EEE}).$$
We now show how to construct the data $(\dt^j u(\cdot,0),\dt^j\eta(\cdot,0))$ for $j=1,\dotsc,2N$ from the pair $(u_0,\eta_0)$ in a manner that is consistent with the compatibility conditions for the problem . Given $(\dt^j u(\cdot,0),\dt^j\eta(\cdot,0))$ for some $j=0,\dotsc,2N-1$ we first use the first equation in to solve for $\dt^{j+1} u(\cdot,0)$. Then we set $$\label{eta_data}
\dt^{j+1} \eta(\cdot,0) = \dt^j(u \cdot \n)\vert_{t=0}.$$ In this way we iteratively define all the time-differentiated data for $j=1,\dotsc,2N$. However, the only way this can be consistent with the natural compatibility conditions needed to study is if $\Xi:\Sigma \times [0,\infty)$ is chosen so that $$\label{Xi_prop}
\begin{split}
\Xi(\cdot,0) &= \Delta_\ast \eta_0 \in H^{4N-1}(\Sigma), \text{ and }\\
\dt^j \Xi(\cdot,0) &= \left. \Delta_\ast \left(\dt^{j-1}(u\cdot \n) \right) \right\vert_{t=0} \in H^{4N-2j-1/2}(\Sigma) \text{ for }j=1,\dotsc,2N.
\end{split}$$ According to Proposition \[extension\_Xi\], there exists a $\Xi$ satisfying as well as the bounds $$\begin{gathered}
\label{Xi_bound}
\sum_{j=0}^{2N-1} \sup_{t>0} {{\left\Vert\dt^j \Xi(t)\right\Vert}^2}_{4N-2j-1} + \sum_{j=0}^{2N} \int_0^\infty {{\left\Vert\dt^j \Xi(t)\right\Vert}^2}_{ 4N-2j}dt
\\
\ls {{\left\Vert\Delta_\ast \eta_0\right\Vert}^2}_{4N-1} + \sum_{j=1}^{2N} {{\left\Vert\Delta_\ast \left(\dt^{j-1}(u\cdot \n) \right) (0)\right\Vert}^2}_{4N-2j-1}\end{gathered}$$
In order to produce a high regularity solution to we must assume that the $u_0$, $\eta_0$, and the forcing terms $F^1$, $F^2$ satisfy the compatibility conditions $$\label{kappa_ccs}
\begin{cases}
-\dt^j(\Sa u \n) \vert_{t=0} = \dt^j\left( - \sigma_+ \Delta_\ast \eta \n + F^2_+ \right) \vert_{t=0} &\text{on } \Sigma_+ \\
-{\left\llbracket \dt^j(\Sa u\n) \right\rrbracket } \vert_{t=0} = \dt^j \left( \sigma_- \Delta_\ast \eta \n - F^2_- \right) \vert_{t=0} &\text{on } \Sigma_- \\
{\left\llbracket \dt^j u \right\rrbracket }\vert_{t=0} =0 &\text{on } \Sigma_- \\
\dt^j u_-\vert_{t=0} = 0 &\text{on } \Sigma_b
\end{cases}$$ for $j=0,\dotsc,2N-1$.
For $0<T$ we define $$\begin{split}
\j_2[u](T) &= \int_0^T \left( \sum_{j=0}^{2N} {{\left\Vert\dt^j u(t)\right\Vert}^2}_{4N-2j+1} + {{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{\Hd} \right) dt \\
\j_\infty[u](T) &= \sup_{0 \le t \le T} \sum_{j=0}^{2N} {{\left\Vert\dt^j u(t)\right\Vert}^2}_{4N-2j}.
\end{split}$$ Similarly, we write $$\i_2[\eta](T) = \int_0^T \sum_{j=0}^{2N+1} {{\left\Vert\dt^j \eta (t) \right\Vert}^2}_{4N-2j+2} dt \text{ and }
\i_\infty[\eta](T) = \sup_{0\le t \le T} \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+1}.$$ We then define $$\label{ab_def}
\j[u](T) = \j_2[u](T) + \j_\infty[u](T) \text{ and } \i[\eta](T) = \i_2[\eta](T) + \i_\infty[\eta](T).$$ Finally, we also define $$\label{ij_def}
\Jf[u](T) = \sup_{0\le t \le T} \sum_{j=0}^{2N-1} {{\left\Vert\dt^j u(t)\right\Vert}^2}_{4N-2j-1} \text{ and } \If[\eta](T) = \sup_{0\le t \le T} \sum_{j=0}^{2N-1} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j-1}.$$
Constructing the mapping
------------------------
We will employ a fixed point argument in order to produce a solution to the nonlinear $\kappa-$approximation problem . We thus begin with the development of an appropriate metric space in which to work.
Notice that if $\j_2[v](T) + \i_2[\zeta](T)< \infty$, then the maps $[0,T]\ni t \mapsto \dt^j v(\cdot,t) \in H^{4N-2j}(\Omega)$ and $[0,T]\ni t \mapsto \dt^j \zeta(\cdot,t) \in H^{4N-2j}(\Sigma)$ are absolutely continuous for $j=0,\dotsc,2N$. In particular, $\dt^j v(\cdot,0)$ and $\dt^j \zeta(\cdot,0)$ are well-defined for $j=0,\dotsc,2N$. This then allows us to define the metric space, for $0 < T \le T_\ast$, $M_1,M_2>0$ and $r\ge 2$ an integer, $$\begin{gathered}
\label{metric_space_def}
\XT = \{ (v,\zeta) \;\vert\; \j[v](T) \le M_1 \kappa^{-r}, \i[\zeta](T) \le M_2 \kappa^{-2}, \\
\text{ and } \dt^j v(\cdot, 0) = \dt^j u(\cdot,0), \dt^j \zeta(\cdot,0) = \dt^j \eta(\cdot,0) \text{ for }j=0,\dots,2N\}.\end{gathered}$$ The space $\XT$ is complete due to the fact that $\{ (v,\zeta) \;\vert\; \j[v](T) + \i[\zeta](T)< \infty\}$ is a Banach space.
We now define a mapping $\M : \XT \to \XT$ as $\M(v,\zeta) = (u,\eta)$, where $\eta$ and $u$ are determined through the following two steps. In these steps we will assume that $0<T<1$ and that $\Lf[\eta_0] < \delta_0/2$, where $\delta_0$ is given by Proposition \[lame\_elliptic\].
*Step 1 – The $\eta$ equation*
Note that $$\label{s1_1}
\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j (v\cdot \n[\zeta])(t)\right\Vert}^2}_{4N-2j} dt \ls (1+ \i_\infty[\zeta](T)) \sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N-2j+1/2}dt.$$ The usual Sobolev interpolation allows us to estimate $${{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N-2j+1/2} \ls {\left\Vert\dt^j v(t)\right\Vert}_{4N-2j} {\left\Vert\dt^j v(t)\right\Vert}_{4N-2j+1}$$ when $j =0,\dotsc,2N$, and hence $$\begin{gathered}
\label{s1_1_5}
\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N-2j+1/2}dt \ls \sum_{j=0}^{2N} \int_0^T {\left\Vert\dt^j v(t)\right\Vert}_{4N-2j} {\left\Vert\dt^j v(t)\right\Vert}_{4N-2j+1}dt \\
\ls \left(\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N-2j} dt \right)^{1/2} \left(\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N-2j+1} dt \right)^{1/2} \\
\ls \sqrt{T} \sqrt{\j_\infty[v](T)} \sqrt{\j_2[v](T)}.\end{gathered}$$ Combining the estimates and , we find that $$\label{s1_1_6}
\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j (v\cdot \n[\zeta])(t)\right\Vert}^2}_{4N-2j} dt \ls (1+ \i_\infty[\zeta](T)) \sqrt{T} \sqrt{\j_\infty[v](T)} \sqrt{\j_2[v](T)}.$$
Estimate and allow us to use Theorem \[heat\_estimates\] with $f = v\cdot \n[\zeta] - \kappa \Xi$ and $\Xi$ defined as in – to produce a unique solution $\eta$ to $$\label{kappa_heat}
\begin{cases}
\dt \eta - \kappa \Delta_\ast \eta = v \cdot \n[\zeta] - \kappa \Xi & \text{in }\Sigma \times (0,T) \\
\eta(\cdot,0) = \eta_0 & \text{in }\Sigma.
\end{cases}$$ The theorem guarantees that $\dt^{j+1} \eta(\cdot,0) = \dt^j (v \cdot \n)\vert_{t=0} = \dt^j (u\cdot \n)\vert_{t=0}$ for $j=0,\dotsc,2N-1$, with the understanding that this last quantity is the initial data computed previously in , and that $\eta$ obeys the estimate (utilizing ) $$\begin{gathered}
\label{s1_2}
\kappa^2 \left(\i_2[\eta](T) + \i_\infty[\eta](T) \right)\ls e^T \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(\cdot,0)\right\Vert}^2}_{4N-2j+1}
\\
+ \int_0^T e^{T-t} \sum_{j=0}^{2N} {{\left\Vert\dt^j (v \cdot \n[\zeta] -\kappa \Xi)(t)\right\Vert}^2}_{4N-2j } dt
\\
\ls e^T \left( {{\left\Vert\eta_0\right\Vert}^2}_{4N+1} + \sum_{j=0}^{2N-1} {{\left\Vert\dt^j (u \cdot \n)\vert_{t=0}\right\Vert}^2}_{4N-2j-1} \right)
+ (1+ \i_\infty[\zeta](T)) \sqrt{T} \sqrt{\j_\infty[v](T)} \sqrt{\j_2[v](T)}
\\
\ls {{\left\Vert\eta_0\right\Vert}^2}_{4N+1} + \sum_{j=0}^{2N-1} {{\left\Vert\dt^j (u \cdot \n)\vert_{t=0}\right\Vert}^2}_{4N-2j-1} + \left(1+\frac{M_2}{\kappa^2}\right) \frac{M_1}{\kappa^r} \sqrt{T} ,\end{gathered}$$ where in the last line we have used the bounds built into the definition of $\XT$ and the assumption that $T<1$.
From we see that if $M_2$ is sufficiently large (depending on the data, $\sigma$, etc) and $T$ is sufficiently small (depending on $\kappa$ and $M_1,M_2$), then $$\label{s1_3}
\i[\eta](T) \le \frac{M_2}{\kappa^2},$$ which is the requirement for $\eta$ to be part of a pair belonging in $\XT$. On the other hand, from Lemma \[time\_interp\] $$\label{s1_4}
\Lf[\eta](T) \le \Lf[\eta_0] + \int_0^T \ih_\infty[\eta(t)]dt \le \Lf[\eta_0] + T \i_\infty[\eta](T) \le \frac{\delta_0}{2} + \frac{TM_2}{\kappa^2} < \delta_0$$ if $T$ is further restricted.
*Step 2 – The $u$ equation*
Now we seek to solve the problem $$\label{kappa_lame}
\begin{cases}
\rho \dt u - \operatorname{div}_{\a[\eta]} \S_{\a[\eta]} u = F^1 & \text{in }\Omega \\
-\S_{\a[\eta]} u \n[\eta] = -\sigma_+ \Delta_\ast \eta \n[\eta] + F^2_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \S_{\a[\eta]} u \right\rrbracket } \n[\eta] = \sigma_- \Delta_\ast \eta \n[\eta] -F^2_- &\text{on } \Sigma_- \\
{\left\llbracket u \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b \\
u(\cdot,0) = u_0,
\end{cases}$$ where $\eta$ is the function constructed in Step 1 and $\n[\eta], \a[\eta]$ are determined in terms of $\eta$ as in and . Let us write $\tilde{F}^1 = F^1$, $\tilde{F}^2_+ =-\sigma_+ \Delta_\ast \eta\n[\eta] + F^2_+$, and $\tilde{F}^2_- = \sigma_- \Delta_\ast \eta\n[\eta] -F^2_-$. Then we write $\tilde{\f}_2(t)$ and $\tilde{\f}_\infty(t)$ as in and , except with $\tilde{F}^1$ and $\tilde{F}^2$ in place of $F^1,F^2$. Then clearly $$\label{s2_1}
\begin{split}
\tilde{\f}_\infty(t) &\ls \f_\infty(t) +
\left(1+\sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+1} \right) \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+1} , \\
\tilde{\f}_2(t) &\ls \f_2(t) + (1+\sigma^2) \left(1 + \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+1} \right) \sum_{j=0}^{2N}{{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+3/2}.
\end{split}$$
A simple interpolation argument, in conjunction with the Cauchy-Schwarz inequality, provides us with the estimate $$\begin{gathered}
\label{s2_1_5}
\int_0^T \sum_{j=0}^{2N}{{\left\Vert\dt^j \eta(t)\right\Vert}^2}_{4N-2j+3/2} dt \ls \int_0^T \sum_{j=0}^{2N}{\left\Vert\dt^j \eta(t)\right\Vert}_{4N-2j+1} {\left\Vert\dt^j \eta(t)\right\Vert}_{4N-2j+2}dt \\
\ls \sqrt{T} \sqrt{ \i_\infty[\eta](T)} \sqrt{\i_2[\eta](T) } \ls \sqrt{T} \left( \i_2[\eta](T) + \i_\infty[\eta](T)\right).\end{gathered}$$ We then deduce from and that $$\label{s2_2_1}
\sup_{0\le t \le T} \tilde{\f}_\infty(t) \ls \sup_{0\le t \le T} \f_\infty(t) + (1+\i_\infty[\eta](T)) \i_\infty[\eta](T)$$ and $$\begin{gathered}
\label{s2_2_2}
\int_0^T \tilde{\f}_2(t) dt \ls \int_0^T \f_2(t)dt +(1+\sigma^2)(1+\i_\infty[\eta](T)) \sqrt{T} \left( \i_2[\eta](T) + \i_\infty[\eta](T)\right) .\end{gathered}$$
Now, owing to the bounds –, the compatibility conditions , and the estimate , we may apply Theorem \[lame\_high\] with $\tilde{F}^1$ and $\tilde{F}^2$ in place of $F^1,F^2$. This yields a unique solution $u$ to satisfying (after combining – with –) $$\begin{gathered}
\label{s2_3}
\j_2[u](T) \ls (1+ P(\i[\eta](T) +\Ef[\rho](T) )) \exp\left(T(1+ P(\i_\infty[\eta](T) +\Ef[\rho](T))) \right) \\
\times \left( \sum_{j=0}^{2N} {{\left\Vert\dt^j u(\cdot,0)\right\Vert}^2}_{4N-2j} + \int_0^T \f_2(t)dt
+(1+\sigma^2)\sqrt{T} \i[\eta](T) \right),\end{gathered}$$ and $$\begin{gathered}
\label{s2_4}
\j_\infty[u](T) \ls (1+ P(\i[\eta](T) +\Ef[\rho](T)))\exp\left(T(1+P(\i_\infty[\eta](T) +\Ef[\rho](T))) \right)
\\
\times
\left( \sum_{j=0}^{2N} {{\left\Vert\dt^j u(\cdot,0)\right\Vert}^2}_{4N-2j} + \sup_{0\le t \le T} \f_\infty(t) + \i_\infty[\eta](T) + \int_0^T \f_2(t)dt +(1+\sigma^2)\sqrt{T} \i[\eta](T) \right)\end{gathered}$$ for some universal positive polynomial $P(\cdot)$ with $P(0)=0$.
Recall that we assume the bound . Then by restricting $T$ further and employing the estimate , we may deduce from and that there exists a natural number $\ell$ such that $$\label{s2_5}
\j[u](T) \ls \left( \frac{1+P({\EEE})+M_2^\ell}{\kappa^\ell} \right)
\left( \sum_{j=0}^{2N} {{\left\Vert\dt^j u(\cdot,0)\right\Vert}^2}_{4N-2j} + \sup_{0\le t \le T} \f_\infty(t) + \int_0^T \f_2(t)dt
+1 \right).$$ If $M_1$ is sufficiently large and $r$ is set to $\ell$, then $$\label{s2_6}
\j[u](T) \le \frac{M_1}{\kappa^r},$$ which means that $u$ satisfies the estimate required to belong to a pair in $\XT$.
*Step 3 – Inclusion in $\XT$.*
In particular, and imply that if $M_1,M_2>0$ are taken to be sufficiently large, and $0<T<1$ is taken to be sufficiently small, then $(u,\eta) \in \XT$. This allows us to set $\M(v,\zeta) = (u,\eta) \in \XT$, and so the mapping $\M: \XT \to \XT$ is well-defined.
Fixed point
-----------
Now we show that the mapping constructed above has a fixed point.
\[kappa\_contract\] Assume that $\rho$ is given on the time interval $[0,T_\ast]$ and satisfies and . Let $\delta_0$ be given by Proposition \[lame\_elliptic\], and assume that $\delta \in (0,\delta_0)$. Suppose that that $\Lf[\eta_0] \le \delta/2$, where $\Lf$ is given by . There exist $M_1,M_2 >0$, $r \in \mathbb{N}$, and $T_0 = T_0(\kappa,\sigma,\delta,\TE) \in (0,T_\ast]$ such that if $0 < T \le T_0$ then $\M : \XT \to \XT$ is a contraction, and therefore admits a unique fixed point $(u,\eta) = \M(u,\eta)$. In particular, there exists a unique pair $(u,\eta)$ satisfying $$\label{kc_01}
\j[u](T) \le M_1 \kappa^{-r} \text{ and } \i[\eta](T) \le M_2 \kappa^{-2}$$ that solve and achieve the initial data $\dt^j u(\cdot,0)$ and $\dt^j \eta(\cdot,0)$ for $j=0,\dotsc,2N$. Moreover, $$\label{kc_02}
\Lf[\eta](T) \le \delta \text{ and } \If[\eta](T) + \Jf[u](T) \ls \tilde{\E}^0[u_0,\eta_0] + 1,$$ where $\tilde{\E}^0[u_0,\eta_0]$ is as defined in .
Let $(v_i,\zeta_i) \in \XT$, $(u_i,\eta_i) = \M(v_i,\zeta_i) \in \XT$ for $i=1,2$. Then $u:= u_1 - u_2$ and $\eta := \eta_1 -\eta_2$ satisfy $$\label{kc_1}
\begin{cases}
\rho \dt u - \operatorname{div}_{\a[\eta_1]} \S_{\a[\eta_1]} u = H^1 & \text{in }\Omega \\
-\S_{\a[\eta_1]} u \n[\eta_1] = -\sigma_+ \Delta_\ast \eta \n[\eta_1] + H^2_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \S_{\a[\eta_1]} u \right\rrbracket } \n[\eta_1] = \sigma_- \Delta_\ast \eta \n[\eta_1] + H^2_- &\text{on } \Sigma_- \\
{\left\llbracket u \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b \\
u(\cdot,0) = 0
\end{cases}$$ and $$\label{kc_2}
\begin{cases}
\dt \eta - \kappa \Delta_\ast \eta = (v_1 - v_2)\cdot \n[\zeta_1] + v_2 \cdot (\n[\zeta_1] - \n[\zeta_2]) \\
\eta(\cdot,0) = 0,
\end{cases}$$ where $$\begin{split}
H^1 &= \operatorname{div}_{\a[\eta_1] - \a[\eta_2]} \S_{\a[\eta_2]} u_2 + \operatorname{div}_{\a[\eta_1]} \S_{\a[\eta_1] - \a[\eta_2]} u_2 \\
H^2_+ &= -\sigma_+ \Delta_\ast \eta_2 (\n[\eta_1] - \n[\eta_2]) -\S_{\a[\eta_1]}u_2 (\n[\eta_1] - \n[\eta_2])
- \S_{\a[\eta_1] - \a[\eta_2]} u_2 \n[\eta_2] \\
H^2_- &= \sigma_- \Delta_\ast \eta_2 (\n[\eta_1] - \n[\eta_2]) -{\left\llbracket \S_{\a[\eta_1]}u_2 \right\rrbracket } \cdot (\n[\eta_1] - \n[\eta_2]) - {\left\llbracket \S_{\a[\eta_1] - \a[\eta_2]} u_2 \right\rrbracket } \n[\eta_2].
\end{split}$$ Notice that the $\Xi$ terms cancel in since they are both computed from the same initial data.
From in Step 1 above, we know that $$\begin{gathered}
\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j ((v_1-v_2)\cdot \n[\zeta_1])(t)\right\Vert}^2}_{4N-2j} dt
\\
\ls (1+ \i_\infty[\zeta_1](T)) \sqrt{T} \sqrt{\j_\infty[v_1-v_2](T)} \sqrt{\j_2[v_1-v_2](T)}
\\
\ls \left(1 + \frac{M_2}{\kappa^2} \right) \sqrt{T} \j[v_1-v_2](T).\end{gathered}$$ A similar argument shows that $$\begin{gathered}
\sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j (v_2\cdot (\n[\zeta_1]) -\n[\zeta_2])(t)\right\Vert}^2}_{4N-2j} dt
\\
\ls \i_\infty[\zeta_1-\zeta_2](T) \sqrt{T} \sqrt{\j_\infty[v_2](T)} \sqrt{\j_2[v_2](T)}
\ls \frac{M_1}{\kappa^r} \sqrt{T} \i_\infty[\zeta_1-\zeta_2](T).\end{gathered}$$ Then, as in , we have the estimate $$\label{kc_3}
\i_2[\eta](T) + \i_\infty[\eta](T) \ls \left(\frac{1}{\kappa^2} + \frac{M_2}{\kappa^4} \right) \sqrt{T} \j[v_1-v_2](T)
+ \frac{M_1}{\kappa^{r+2}} \sqrt{T} \i_\infty[\zeta_1-\zeta_2](T).$$
From in Step 2 above, we know that $$\begin{gathered}
\label{kc_4}
\j_2[u](T) \ls (1+ P(\i[\eta](T) +\Ef[\rho](T))) \exp\left(T(1+P(\i_\infty[\eta](T) +\Ef[\rho](T))) \right) \\
\times \left( \int_0^T \mathcal{H}_2(t)dt
+(1+\sigma^2)\sqrt{T} \i[\eta](T) \right)\end{gathered}$$ for some universal positive polynomial $P$ with $P(0)=0$, where we have written $$\mathcal{H}_2(t) = \sum_{j=0}^{2N-1} {{\left\Vert\dt^j H^1(t)\right\Vert}^2}_{4N-2j-1} + {{\left\Vert\dt^{2N} H^1(t)\right\Vert}^2}_{\Hd} + \sum_{j=0}^{2N} {{\left\Vert\dt^j H^2(t)\right\Vert}^2}_{4N-2j-1/2}.$$ Similarly, in Step 2 yields $$\begin{gathered}
\label{kc_5}
\j_\infty[u](T) \ls ( 1+P(\i[\eta](T) +\Ef[\rho](T) ))\exp\left(T(1+P(\i_\infty[\eta](T) +\Ef[\rho](T))) \right)
\\
\times
\left( \sup_{0\le t \le T} \mathcal{H}_\infty(t) + \int_0^T \mathcal{H}_2(t)dt + \i_\infty[\eta](T) +(1+\sigma^2)\sqrt{T} \i[\eta](T) \right),\end{gathered}$$ where we have written $$\
\mathcal{H}_\infty(t) := \sum_{j=0}^{2N-1} \left[ {{\left\Vert\dt^j H^1(t)\right\Vert}^2}_{4N-2j-2} + {{\left\Vert\dt^j H^2(t)\right\Vert}^2}_{4N-2j-3/2} \right].$$
We now seek to estimate the $\mathcal{H}$ terms appearing on the right side of and . Standard nonlinear estimates lead us to the bounds $$\begin{gathered}
\label{kc_7}
\int_0^T \mathcal{H}_2(t)dt \ls \i_\infty[\eta](T) (1+P(\i_\infty[\eta_1](T) +\i_\infty[\eta_2](T) ) ) \j_2[u_2](T)
+ \i_\infty[\eta](T) \i_2[\eta_2](T)
\\ \ls
P\left(\frac{M_2}{\kappa^r} +\frac{M_1 M_2}{\kappa^{r+2}} + \frac{M_1}{\kappa^2}\right)\i_\infty[\eta](T)\end{gathered}$$ and $$\begin{gathered}
\label{kc_8}
\sup_{0\le t \le T} \mathcal{H}_\infty(t) \ls \i_\infty[\eta](T) (1+\i_\infty[\eta_1](T) +\i_\infty[\eta_2](T) ) \j_\infty[u_2](T) + \i_\infty[\eta](T) \i_\infty[\eta_2](T)
\\
\ls P\left(\frac{M_2}{\kappa^r} +\frac{M_1 M_2}{\kappa^{r+2}} + \frac{M_1}{\kappa^2}\right)\i_\infty[\eta](T).\end{gathered}$$
Now we sum the estimates and and then combine the resulting estimate with and to deduce that $$\begin{gathered}
\label{kc_9}
\j[u](T) \ls \left( 1+ P\left(\frac{M_2}{\kappa^r} \right)\right) \exp\left(T \left(1+ P\left(\frac{M_2}{\kappa^2}\right) \right) \right)
\\
\times \left[
P \left(\frac{M_2}{\kappa^r} +\frac{M_1 M_2}{\kappa^{r+2}} + \frac{M_1}{\kappa^2}\right)\i_\infty[\eta](T)
+(1+\sigma^2)\sqrt{T} \i[\eta](T)
\right]\end{gathered}$$ for some universal positive polynomial with $P(0)=0$. Combining and then yields the estimate $$\begin{gathered}
\label{kc_10}
\j[u](T) + \i[\eta](T) \ls \left( 1+ P\left(\frac{M_2}{\kappa^r} \right)\right) \exp\left(T \left(1+ P\left(\frac{M_2}{\kappa^2}\right)\right) \right) \\
\times
\left( P\left( \frac{M_2}{\kappa^r} +\frac{M_1 M_2}{\kappa^{r+2}} + \frac{M_1}{\kappa^2} \right) + (1+\sigma^2)\sqrt{T} \right)
\\
\times
\left[
\left(\frac{1}{\kappa^2} + \frac{M_2}{\kappa^4} \right) \sqrt{T} \j[v_1-v_2](T)
+ \frac{M_1}{\kappa^{r+2}} \sqrt{T} \i_\infty[\zeta_1-\zeta_2](T)
\right].\end{gathered}$$ Finally, from we see that if we further restrict $T$ in terms of $M_1,$ $M_2,$ $\ell,$ $\sigma,$ and $\kappa$, then $$\j[u_1-u_2](T) + \i[\eta_1-\eta_2](T) \le \hal \left( \j[v_1-v_2](T) + \i[\zeta_1-\zeta_2](T) \right),$$ which implies that the mapping $\M : \XT \to \XT$ is a contraction.
The existence of a fixed point satisfying is then an easy consequence. The estimates follow from , Lemma \[time\_interp\], and standard estimates of the data.
Estimates for the $\kappa-$problem {#sec_kappa_est}
===================================
Our goal in this section is to derive $\kappa-$independent estimates for the problem . These will eventually allow us to pass to the limit $\kappa \to 0$. In this section and the next we must work with a slightly weaker form of the dissipation for $u$, which is defined as $$\label{dissipation_weak_u}
\check{\D}[u] = {{\left\Vertu\right\Vert}^2}_{4N} + {{\left\Vert\nab_{\ast,0}^{4N-1} u\right\Vert}^2}_{2} + \sum_{j=1}^{2N} {{\left\Vert\dt^j u \right\Vert}^2}_{4N-2j+1}.$$ Here and at several points in this section we employ the notational convention $$\label{horiz_sum}
{{\left\Vert\nab_{\ast,0}^{m} u\right\Vert}^2} = \sum_{\substack{ \alpha \in \mathbb{N}^2 \\ {\left\vert\alpha\right\vert} \le m }} {{\left\Vert\pal u\right\Vert}^2}$$ for any $m\ge 0$ and any norm ${\left\Vert\cdot\right\Vert}$.
Recall that $\Xi$ is the function defined in – in terms of the data. We will need to refer to the following functional: $$\label{Xi_fnal}
\zf := \sum_{j=0}^{2N-1} \sup_{t>0} {{\left\Vert\dt^j \Xi(t)\right\Vert}^2}_{4N-2j-1} + \sum_{j=0}^{2N} \int_0^\infty {{\left\Vert\dt^j \Xi(t)\right\Vert}^2}_{ 4N-2j}dt.$$ It’s easy to see from that $$\label{Xi_data_est}
\zf \ls \frac{1}{\sigma} {{\left\Vert\sqrt{\sigma} \nab_\ast \eta_0\right\Vert}^2}_{4N} + P(\TE[u_0,\eta_0])$$ for a positive universal polynomial $P$ such that $P(0)=0$.
Preliminaries
-------------
Rather than work directly with the solutions from Theorem \[kappa\_contract\] we will prove our estimates in a somewhat more general context. We assume the following for some parameter $\delta >0$ and time interval $[0,T_\ast]$. $$\begin{aligned}
&\text{The parameter } \delta>0 \text{ satisfies } \delta \le \delta_0 \text{ where }\delta_0 \text{ is given by Proposition } \ref{lame_elliptic}. \label{kap_assump_1}\\
&\text{The initial data } \dt^j u(\cdot,0) \text{ and }\dt^j \eta(\cdot,0) \text{ for }j=0,\dotsc,2N \text{ are given as before and} \\
& \quad\text{ satisfy the compatibility conditions } \eqref{kappa_ccs}. \nonumber \\
& \text{The data satisfy }\Lf[\eta_0] \le \delta/2, \text{ where }\Lf[\eta_0] \text{ is given by } \eqref{l0_def}. \\
& \text{A solution }(u,\eta) \text{ to }\eqref{kappa_problem} \text{ exists on the interval } [0,T_\ast] \text{ and achieves the initial data}. \\
& \text{The solution satisfies the estimate }\j[u](T_\ast) + \i[\eta](T_\ast) < \infty, \text{ where } \j[u] \text{ and } \i[\eta] \nonumber \\
&\quad\text{are as defined in }\eqref{ab_def}. \label{kap_assump_2} \\
& \text{We have the estimates } \Lf[\eta](T_\ast) < \delta \text{ and }\If[\eta](T_\ast) + \Jf[u](T_\ast) \ls P(\TE) + 1 \nonumber \\
&\quad \text{for a universal positive polynomial }P\text{ such that }P(0)=0. \text{ Here } \If[\eta], \Jf[u] \nonumber \\
&\quad\text{are defined by } \eqref{ij_def}. \\
& \text{The forcing terms satisfy } \sup_{0\le t\le T_\ast} \f_\infty(t) + \int_0^{T_\ast} \f_2(t)dt \ls 1+ P(\TE) \nonumber \\
& \quad \text{for a universal positive polynomial }P\text{ such that }P(0)=0. \label{kap_assump_4}\\
& \text{The function } \rho \text{ is defined on }[0,T_\ast] \text{ and satisfies } \eqref{rho_assump_1} \text{ and } \eqref{rho_assump_2}. \label{kap_assump_3}\end{aligned}$$
The assumption guarantees that $(u,\eta)$ are regular enough for us to apply $\pal$ to for $\alpha \in \mathbb{N}^{1+2}$ with ${\left\vert\alpha\right\vert} \le 4N$. This results in
$$\label{kappa_deriv}
\begin{cases}
\rho \dt \pal u - \diva \Sa \pal u = \pal F^1 + F^{1,\alpha} & \text{in }\Omega \\
\dt \pal \eta - \kappa \Delta_\ast \pal \eta = (\pal u) \cdot \n - \kappa \pal \Xi + F^{3,\alpha} &\text{on }\Sigma \\
-\Sa \pal u \n = - \sigma_+ \Delta_\ast \pal\eta \n + \pal F^2_+ + F^{2,\alpha}_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa \pal u \right\rrbracket } \n = \sigma_- \Delta_\ast \pal \eta \n - \pal F^2_- - F^{2,\alpha}_- &\text{on } \Sigma_- \\
{\left\llbracket \pal u \right\rrbracket } =0 &\text{on } \Sigma_- \\
\pal u_- = 0 &\text{on } \Sigma_b,
\end{cases}$$
where for $i=1,2,3$, $$\begin{gathered}
\label{kF1_def}
F^{1,\alpha}_i = -\sum_{\beta < \alpha} C_{\alpha,\beta} \p^{\alpha-\beta} \rho \dt \p^\beta u_i + \sum_{\beta < \alpha} C_{\alpha,\beta} \p^{\alpha-\beta} \a_{\ell m} \p^\beta \p_m(\mu \a_{\ell k} \p_k u_i)
\\
+ \sum_{\beta < \alpha} C_{\alpha,\beta} \left( \mu' + \frac{\mu}{3}\right) \left( \a_{ik} \p_k (\p^{\alpha-\beta} \a_{\ell m} \p^\beta \p_m u_\ell ) + \p^{\alpha-\beta} \a_{ik} \p^{\beta} \p_k(\a_{\ell m} \p_m u_\ell) \right),\end{gathered}$$ $$\begin{gathered}
F^{2,\alpha}_{+,i} = \sum_{\beta < \alpha} C_{\alpha,\beta} \left( \mu \p^{\alpha-\beta} (\n_\ell \a_{ik}) \p^\beta \p_k u_\ell + \mu \p^{\alpha-\beta}(\n_\ell \a_{\ell k} )\p^\beta \p_k u_i \right)
\\
+ \sum_{\beta < \alpha} C_{\alpha,\beta} \left(\left( \mu' - \frac{2\mu}{3}\right) \p^{\alpha-\beta}(\n_i \a_{\ell k}) \p^\beta \p_k u_\ell - \sigma_+ \p^{\alpha-\beta}\n_i \p^\beta \Delta_\ast \eta \right)\end{gathered}$$ and $$\begin{gathered}
-F^{2,\alpha}_{-,i} = \sum_{\beta < \alpha} C_{\alpha,\beta} \left( \ \p^{\alpha-\beta} (\n_\ell \a_{ik}) {\left\llbracket \mu \p^\beta \p_k u_\ell \right\rrbracket } + \p^{\alpha-\beta}(\n_\ell \a_{\ell k} ) {\left\llbracket \mu \p^\beta \p_k u_i \right\rrbracket } \right)
\\
+ \sum_{\beta < \alpha} C_{\alpha,\beta} \left(\p^{\alpha-\beta}(\n_i \a_{\ell k}) {\left\llbracket \left( \mu' - \frac{2\mu}{3}\right) \p^\beta \p_k u_\ell \right\rrbracket } +\sigma_- \p^{\alpha-\beta}\n_i \p^\beta \Delta_\ast \eta \right).\end{gathered}$$ Also, $$\label{kF3_def}
F^{3,\alpha} = \sum_{\beta < \alpha} C_{\alpha,\beta} \p^{\alpha-\beta} \n \p^\beta u.$$
The estimates
-------------
We begin with a basic energy identity.
\[kappa\_en\_ident\] Assume –. Let $\alpha \in \mathbb{N}^{1+2}$ satisfy ${\left\vert\alpha\right\vert} \le 4N$. Then $$\begin{gathered}
\label{kei_01}
\ddt \left( \int_\Omega \frac{\rho J}{2} {\left\vert\pal u\right\vert}^2 + \int_{\Sigma} \frac{1}{2} {\left\vert\pal \eta\right\vert}^2 + \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2 \right)
+ \int_\Omega \frac{\mu J}{2} {\left\vert\sgz_{\a} \pal u\right\vert}^2 + J \mu' {\left\vert\diva \pal u\right\vert}^2
\\
+ \kappa \int_{\Sigma} {\left\vert\nab_\ast \pal \eta\right\vert}^2 + \sigma {\left\vert\Delta_\ast \pal \eta\right\vert}^2
= \int_\Omega \frac{\dt(\rho J)}{\rho J} \frac{\rho J}{2} {\left\vert\pal u\right\vert}^2 + J \pal F^1 \cdot \pal u - \int_\Sigma \pal F^2 \cdot \pal u \\
+ \int_\Omega J F^{1,\alpha} \cdot \pal u - \int_\Sigma F^{2,\alpha} \cdot \pal u + \int_{\Sigma} \pal \eta \pal u \cdot \n
+ \int_{\Sigma} (\kappa \pal \Xi - F^{3,\alpha} ) (- \pal \eta + \sigma \Delta_\ast \pal \eta).\end{gathered}$$
We multiply the first equality in by $J\pal u$ and integrate over $\Omega$. After integrating by parts and using the boundary conditions, we find that $$\begin{gathered}
\label{kei_1}
\ddt \left( \int_\Omega \frac{\rho J}{2} {\left\vert\pal u\right\vert}^2 + \int_{\Sigma} \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2 \right)
+ \int_\Omega \frac{\mu J}{2} {\left\vert\sgz_{\a} \pal u\right\vert}^2 + J \mu' {\left\vert\diva \pal u\right\vert}^2
+ \kappa \int_{\Sigma} \sigma {\left\vert\Delta_\ast \pal\eta\right\vert}^2
\\
= \int_\Omega \frac{\dt(\rho J)}{\rho J} \frac{\rho J}{2} {\left\vert\pal u\right\vert}^2 + J \pal F^1 \cdot \pal u - \int_\Sigma \pal F^2 \cdot \pal u
\\+ \int_\Omega J F^{1,\alpha} \cdot \pal u - \int_\Sigma F^{2,\alpha} \cdot \pal u
+ \int_{\Sigma} (\kappa \pal \Xi - F^{3,\alpha} ) \sigma \Delta_\ast \pal \eta.\end{gathered}$$ Also, we multiply the second equality in by $\pal \eta$ and integrate over $\Sigma$. After again integrating by parts, we find that $$\label{kei_2}
\ddt \int_{\Sigma} \hal {\left\vert\pal \eta\right\vert}^2 + \kappa \int_{\Sigma} {\left\vert\nab_\ast \pal \eta\right\vert}^2 = \int_{\Sigma} \pal \eta (\pal u \cdot \n) + (-\kappa \pal \Xi + F^{3,\alpha}) \pal \eta.$$ Summing and yields .
Our next result provides some estimates of the forcing terms that appear in the equations as a result of commutators with $\p^\alpha$. The proof is similar to that of Lemma \[lame\_forcing\_est\] and the $H$ forcing estimates of Theorem \[kappa\_contract\], and is similarly omitted. We recall that $\Ef_\infty[\rho]$ is defined in .
\[k\_F\_ests\] Let $\alpha \in \mathbb{N}^{1+2}$ satisfy ${\left\vert\alpha\right\vert} \le 4N$ and let $F^{i,\alpha}$ be given by –. Then there exists a polynomial $P$ with universal positive coefficients and $P(0)=0$ such that $$\label{kfE_01}
{{\left\VertF^{1,\alpha}\right\Vert}^2}_0 + {{\left\VertF^{2,\alpha}\right\Vert}^2}_{-1/2}
\ls ( P(\E^0[\eta]) + \Ef_\infty[\rho]) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \check{\D}[u].$$ Additionally, $$\label{kfE_02}
{{\left\Vert(F^{3,\alpha} + (\pal \nab_\ast \eta) \cdot u ) \right\Vert}^2}_{0} \ls \E^0[\eta] \E[u] + \Lf[\eta]\check{\D}[u],$$ and $$\label{kfE_03}
{{\left\Vert\nab_\ast (F^{3,\alpha} + (\pal \nab_\ast \eta) \cdot u ) \right\Vert}^2}_0 \ls \sigma^{-1}\E^\sigma[\eta] \E[u] + \Lf[\eta]\check{\D}[u].$$
With this lemma in hand, we can turn the energy identity of Lemma \[kappa\_en\_ident\] into a useful estimate. Note that in this proposition we employ the notation defined in .
\[k\_hor\_est\] Assume –. Then for $0 \le T \le T_\ast$ we have the estimate $$\begin{gathered}
\label{khe_01}
\sum_{j=0}^{2N} {{\left\Vert\dt^j \nab_{\ast,0}^{4N-2j} u\right\Vert}^2}_{L^\infty_T H^0} + {{\left\Vert\dt^j \nab_{\ast,0}^{4N-2j} u\right\Vert}^2}_{L^2_T H^1} + {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} + \sigma {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} \\
+ \kappa \sum_{j=0}^{2N} {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{L^2_T H^{4N-2j}} + \sigma {{\left\Vert\dt^j \Delta_\ast \eta\right\Vert}^2}_{L^2_T H^{4N-2j}}
\ls e^{\gamma T} \TE[u_0,\eta_0]
\\
+ e^{\gamma T} \int_0^T \left( \f_2(t) + ( P(\E^0[\eta]) + \Ef_\infty[\rho]) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \check{\D}[u]
\right) dt,\end{gathered}$$ where $ \gamma = C \left( 1 + \If[\eta](T) + \Ef[\rho](T) + \Jf[u](T) ) \right)$, and $P$ is a polynomial with positive universal coefficients satisfying $P(0)=0$.
Let $\alpha \in \mathbb{N}^{1+2}$ satisfy ${\left\vert\alpha\right\vert} \le 4N$. Taking of Lemma \[kappa\_en\_ident\] as our starting point, we seek to estimate each term on the right hand side and to estimate non-time-derivative term on the left. In our subsequent analysis we will rewrite $F^{3,\alpha} = (F^{3,\alpha} + (\pal \nab_\ast \eta)\cdot u) - (\pal \nab_\ast \eta)\cdot u$.
First note that we may use Proposition \[korn\] to bound $$\label{khe_1}
{{\left\Vert\pal u\right\Vert}^2}_{1} \ls \int_\Omega \frac{\mu J}{2} {\left\vert\sgz_{\a} \pal u\right\vert}^2 + J \mu' {\left\vert\diva \pal u\right\vert}^2.$$ This will allow us to absorb most of the $\pal u$ terms appearing on the right side of . Using the definition of $\f_2$, given in , in conjunction with the estimates and of Lemma \[k\_F\_ests\] and the bound $0 < \kappa <1$, we may bound $$\begin{gathered}
\label{khe_3_5}
{{\left\Vert\pal F^1\right\Vert}^2}_0 + {{\left\Vert\pal F^2\right\Vert}^2}_0 + {{\left\VertF^{1,\alpha}\right\Vert}^2}_0 + {{\left\VertF^{2,\alpha}\right\Vert}^2}_{-1/2}
+ {{\left\Vert(F^{3,\alpha} + (\pal \nab_\ast \eta) \cdot u ) \right\Vert}^2}_{0} \ + \kappa^2 {{\left\Vert\pal \Xi\right\Vert}^2}_0
\\
\ls \f_2(t)
+ ( P(\E^0[\eta]) + \Ef_\infty[\rho]) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \check{\D}[u]
+ \kappa \sum_{j=0}^{2N} {{\left\Vert\dt^j \Xi\right\Vert}^2}_{4N-2j},\end{gathered}$$ where $P$ is a universal positive polynomial with $P(0)=0$. Next, we use Cauchy’s inequality, Proposition \[korn\], trace theory, and the estimates ${{\left\VertJ\right\Vert}^2}_{L^\infty} + {{\left\Vert\n\right\Vert}^2}_{L^\infty} \ls 1$ (which follow from Lemma \[eta\_small\] and the bounds on $\Lf[\eta](T)$) and to bound, for any $0 < \ep_1 < 1$, $$\begin{gathered}
\label{khe_2}
\int_\Omega J \pal F^1 \cdot \pal u - \int_\Sigma \pal F^2 \cdot \pal u + \int_\Omega J F^{1,\alpha} \cdot \pal u - \int_\Sigma F^{2,\alpha} \cdot \pal u \\
+ \int_{\Sigma} \pal \eta (\pal u \cdot \n)
+ \int_{\Sigma} (\kappa \pal \Xi - (F^{3,\alpha} + (\pal \nab_\ast \eta) \cdot u )) \pal \eta
\\
\ls \ep_1 {{\left\Vert\pal u\right\Vert}^2}_1
+\frac{1}{\ep_1} \left( {{\left\Vert \pal \eta\right\Vert}^2}_0 + {{\left\Vert\pal F^1\right\Vert}^2}_0 + {{\left\Vert\pal F^2\right\Vert}^2}_0 + {{\left\VertF^{1,\alpha}\right\Vert}^2}_0 + {{\left\VertF^{2,\alpha}\right\Vert}^2}_{-1/2} \right)
\\
+ {{\left\VertF^{3,\alpha} + (\pal \nab_\ast \eta) \cdot u \right\Vert}^2}_0 + \kappa^2 {{\left\Vert\pal \Xi\right\Vert}^2}_0
\ls \ep_1 {{\left\Vert\pal u\right\Vert}^2}_1
+\frac{1}{\ep_1} {{\left\Vert \pal \eta\right\Vert}^2}_0
\\
+\frac{1}{\ep_1} \left(
\f_2(t) + ( P(\E^0[\eta]) + \Ef_\infty[\rho]) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \check{\D}[u]
+ \kappa\sum_{j=0}^{2N} {{\left\Vert\dt^j \Xi\right\Vert}^2}_{4N-2j}
\right).\end{gathered}$$
For the remaining $\Xi$ terms on the right side of we estimate $$\label{khe_3}
\int_{\Sigma} \kappa \pal \Xi \sigma \Delta_\ast \pal \eta \le \hal \int_\Sigma \kappa \sigma {\left\vert\Delta_\ast \pal \eta\right\vert}^2 +
\frac{\kappa\sigma}{2}\sum_{j=0}^{2N} {{\left\Vert\dt^j \Xi\right\Vert}^2}_{4N-2j}$$ with the aim of absorbing the $\Delta_\ast \pal \eta$ term onto the left of . Next we handle the $(F^{3,\alpha} + (\pal \nab_\ast \eta)\cdot u) ( \sigma \Delta_\ast \pal \eta)$ terms on the right side of . We integrate by parts and then use of Lemma \[k\_F\_ests\] to see that $$\begin{gathered}
\label{khe_4}
\int_\Sigma -(F^{3,\alpha} + (\pal \nab_\ast \eta)\cdot u) \sigma \Delta_\ast \pal \eta = \int_\Sigma \nab_\ast (F^{3,\alpha} + (\pal \nab_\ast \eta)\cdot u) \cdot \sigma \nab_\ast \pal \eta
\\
\le \frac{\sigma}{2} {{\left\Vert\nab_\ast (F^{3,\alpha} + (\pal \nab_\ast \eta)\cdot u)\right\Vert}^2}_0 + \int_\Sigma \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2 \ls \E^0[\eta] \E[u] + \Lf[\eta]\check{\D}[u]+ \int_\Sigma \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2.\end{gathered}$$
It remains to handle the term $$\label{khe_5}
\int_{\Sigma} (\pal \nab_\ast \eta)\cdot u ( - \pal \eta + \sigma \Delta_\ast \pal \eta) .$$ We have that $$\begin{gathered}
\label{khe_6}
-\int_{\Sigma } (\pal \nab_\ast \eta)\cdot u \pal \eta
=- \int_{\Sigma } \frac{1}{2} u\cdot \nab_\ast {\left\vert\pal \eta\right\vert}^2
= \int_{\Sigma } \frac{1}{2} \operatorname{div}_\ast u {\left\vert\pal \eta\right\vert}^2
\\
\ls {\left\Vert\operatorname{div}_\ast u\right\Vert}_{L^\infty} \int_\Sigma \frac{1}{2} {\left\vert\pal \eta\right\vert}^2 \ls \Jf[u](T) \int_\Sigma \frac{1}{2} {\left\vert\pal \eta\right\vert}^2.\end{gathered}$$ Similarly, $$\begin{gathered}
\label{khe_7}
\int_{\Sigma} (\pal \nab_\ast \eta)\cdot u \sigma \Delta_\ast \pal \eta = - \sum_{i,j=1}^2\int_{\Sigma} \sigma \p_i \pal \eta (\p_i u_j \p_j \pal \eta + u_j \p_i \p_j \pal \eta)
\\ = - \int_{\Sigma} \sigma u\cdot \nab_\ast \frac{{\left\vert\nab_\ast \pal \eta\right\vert}^2 }{2} + \sum_{i,j=1}^2 \sigma \p_i \pal \eta \p_i u_j \p_j \pal \eta
\\
\ls {\left\Vert\nab_\ast u\right\Vert}_{L^\infty} \int_\Sigma \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2 \ls \Jf[u](T) \ \int_\Sigma \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2.\end{gathered}$$ Combining –, we find that $$\label{khe_8}
\int_{\Sigma} (\pal \nab_\ast \eta)\cdot u (- \pal \eta + \sigma \Delta_\ast \pal \eta)
\ls \Jf[u](T) \ \int_\Sigma \frac{1}{2} {\left\vert\pal \eta\right\vert}^2 + \frac{\sigma}{2} {\left\vert \nab_\ast \pal \eta\right\vert}^2.$$
Now we employ the estimates , , , , and in . By choosing $\ep_1$ small enough (but universal), we may absorb the $\ep_1 {{\left\Vert\pal u\right\Vert}^2}_1$ term in onto the left. This results in the differential inequality $$\begin{gathered}
\label{khe_9}
\ddt \left( \int_\Omega \frac{\rho J}{2} {\left\vert\pal u\right\vert}^2 + \int_{\Sigma} \frac{1}{2} {\left\vert\pal \eta\right\vert}^2 + \frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2 \right)
+ C {{\left\Vert\pal u\right\Vert}^2}_{1}
+ \frac{\kappa}{2} \int_{\Sigma} {\left\vert\nab_\ast \pal \eta\right\vert}^2 + \sigma {\left\vert\Delta_\ast \pal \eta\right\vert}^2
\\
\ls \left(1 + {\left\Vert\dt(\rho J)(\rho J)^{-1}\right\Vert}_{L^\infty} + \Jf[u](T) \right) \left( \int_\Omega \frac{\rho J}{2} {\left\vert\pal u\right\vert}^2 + \int_\Sigma \frac{1}{2} {\left\vert\pal \eta\right\vert}^2 +\frac{\sigma}{2} {\left\vert\nab_\ast \pal \eta\right\vert}^2 \right)
\\
+\f_2(t) + ( P(\E^0[\eta]) + \Ef_\infty[\rho]) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \check{\D}[u]
+\kappa \sum_{j=0}^{2N} {{\left\Vert\dt^j \Xi\right\Vert}^2}_{4N-2j}.\end{gathered}$$ Notice that $$\sup_{0\le t \le T} {\left\Vert\dt(\rho J)(\rho J)^{-1}\right\Vert}_{L^\infty} \ls \If[\eta](T) + \Ef[\rho](T).$$ We may then apply Gronwall’s inequality to and sum over $\alpha$ to deduce the bound $$\begin{gathered}
\label{khe_10}
\sum_{j=0}^{2N} {{\left\Vert\dt^j \nab_{\ast,0}^{4N-2j} u\right\Vert}^2}_{L^\infty_T H^0} + {{\left\Vert\dt^j \nab_{\ast,0}^{4N-2j} u\right\Vert}^2}_{L^2_T H^1} + {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} + \sigma {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} \\
+ \kappa \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^2_T H^{4N-2j}} + \sigma {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{L^2_T H^{4N-2j}}
\ls e^{\gamma T}\TE[u_0,\eta_0]
\\
+ e^{\gamma T} \int_0^T \left( \f_2(t) + ( P(\E^0[\eta]) + \Ef_\infty[\rho]) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \check{\D}[u] +\kappa \sum_{j=0}^{2N} {{\left\Vert\dt^j \Xi\right\Vert}^2}_{4N-2j} \right) dt,\end{gathered}$$ where $ \gamma = C \left( 1 + \If[\eta](T) + \Ef[\rho](T) + \Jf[u](T) ) \right). $ Then follows from and .
Next we employ various elliptic estimates to gain control of all derivatives of $(u,\eta)$.
\[kappa\_improved\] Assume –. Then for $0 \le T \le T_\ast$ we have the estimate $$\begin{gathered}
\label{ki_01}
\sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^\infty_T H^{4N-2j}} + {{\left\Vertu\right\Vert}^2}_{L^2_T H^{4N}} + {{\left\Vert\nab_{\ast,0}^{4N-1} u \right\Vert}^2}_{L^2_T H^{2}} +\sum_{j=1}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{4N-2j+1}}
\\
+ {{\left\Vert\rho J \dt^{2N+1} u\right\Vert}^2}_{L^2_T \Hd} +
\sum_{j=0}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} + \sum_{j=0}^{2N} \sigma {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{L^\infty_T H^{4N-2j}} \\
+ \sigma^2 {{\left\Vert\eta\right\Vert}^2}_{L^2_T H^{4N+3/2}}
+ {{\left\Vert\dt \eta\right\Vert}^2}_{L^2_T H^{4N-1/2}} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^2_T H^{4N-2j+2}}
\\+ \kappa \sum_{j=0}^{2N} {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{L^2_T H^{4N-2j}}
+ \sigma {{\left\Vert\dt^j \Delta_\ast \eta\right\Vert}^2}_{L^2_T H^{4N-2j}}
\ls e^{\gamma T}\TE[u_0,\eta_0]
+\sup_{0\le t \le T} \f_\infty(t) \\
+ T \sup_{0 \le t \le T} \E^0[\eta(t)] + e^{\gamma T} \int_0^T \left( \f_2(t) + ( P(\E^0[\eta]) + P(\Ef_\infty[\rho])) (\E[u] + \E^0[\eta]) + P(\Lf[\eta]) \D[u]
\right) dt \\
+\int_0^T \E[u] \frac{1}{\sigma} \E^\sigma[\eta] dt + \kappa^2 \zf,\end{gathered}$$ where $P$ is a polynomial with positive universal coefficients such that $P(0)=0$ and $\gamma = C \left( 1 + \If[\eta](T)+ \Ef[\rho](T) + \Jf[u](T) ) \right).$
The argument is very similar to one used in [@JTW_GWP], so we will only provide a sketch. Let us write $\z$ to denote a term of the same form as the right hand side of . From line to line we will let the polynomials vary as well as the universal constant $C>0$ appearing in $\gamma$.
First we use trace estimates to bound $$\label{ki_20}
\sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{4N-2j+1/2}(\Sigma)} \ls \sum_{j=0}^{2N} {{\left\Vert\dt^j\nab_{\ast,0}^{4N-2j} u\right\Vert}^2}_{L^2_T H^1} \ls \z.$$ Then we use elliptic regularity for the Lamé system with Dirichlet boundary conditions, Proposition \[lame\_elliptic\], to deduce that $$\label{ki_11}
{{\left\Vertu\right\Vert}^2}_{L^2_T H^{4N}} + \sum_{j=1}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{4N-2j+1}} \ls \z.$$ Notice here that we have only used the estimate in order to avoid the introduction of the term ${{\left\Vert\eta\right\Vert}^2}_{4N+1/2}$. This causes no difficulty in getting $4N-2j+1$ estimates when $j=1,\dotsc,2N$ but prevents us from obtaining an estimate of ${{\left\Vertu\right\Vert}^2}_{4N+1}$ at this point. Instead we sum the elliptic estimate for $\p^\alpha u$ obtained from with $\alpha \in \mathbb{N}^2$ and ${\left\vert\alpha\right\vert}\le 4N-1$ to obtain the estimate $$\label{ki_19}
{{\left\Vert\nab_{\ast,0}^{4N-1} u\right\Vert}^2}_2 \ls \z .$$
Now we derive the $L^\infty$ in time estimate for $u$ and its time derivatives. Lemma \[time\_interp\] provides us with $L^\infty$ in time estimates at lower regularity: $$\sum_{j=1}^{2N-1} {{\left\Vert\dt^j u\right\Vert}^2}_{L^\infty_T H^{4N-2j}} \ls \TE[u_0] + \sum_{j=1}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{4N-2j+1}},$$ and hence and the bound of ${{\left\Vert\dt^{2N} u \right\Vert}^2}_{L^\infty_T H^0}$ provided by imply that $$\label{ki_5}
\sum_{j=1}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{L^\infty_T H^{4N-2j}} \ls \z.$$ Similarly, Lemma \[time\_interp\] and imply that $$\label{ki_21}
{{\left\Vertu\right\Vert}^2}_{L^\infty_T H^{4N-1/2}(\Sigma)} \ls \TE[u_0] + \sum_{j=0}^{1} {{\left\Vert\dt^j u\right\Vert}^2}_{L^2_T H^{4N-2j+1/2}(\Sigma)} \ls \z.$$ We may then use and the elliptic estimate of Proposition \[lame\_elliptic\] to bound $$\begin{gathered}
\label{ki_22}
{{\left\Vertu\right\Vert}^2}_{L^\infty_T H^{4N}} \ls {{\left\Vert\rho \dt u\right\Vert}^2}_{L^\infty_T H^{4N-2}} + {{\left\VertF^1\right\Vert}^2}_{L^\infty_T H^{4N-2}} + {{\left\Vertu\right\Vert}^2}_{L^\infty_T H^{4N-1/2}(\Sigma)} \\
\ls \Ef[\rho](T) {{\left\Vert \dt u\right\Vert}^2}_{L^\infty_T H^{4N-2}} +\sup_{0\le t \le T} \f_\infty(t) + {{\left\Vertu\right\Vert}^2}_{L^\infty_T H^{4N-1/2}(\Sigma)} \\
\ls (1+\Ef[\rho](T)) \z +\sup_{0\le t \le T} \f_\infty(t) \ls \z +\sup_{0\le t \le T} \f_\infty(t).\end{gathered}$$ Note that in the last inequality we have used the bound $\Ef[\rho](T) \z \ls \z$, which holds since we may increase the constant $C>0$ appearing in $\gamma$.
Next we derive the $L^2$ in time estimates for $\eta$ and its derivatives. We first use the dynamic boundary condition and to estimate $$\begin{gathered}
\label{ki_1}
\sigma^2 {{\left\Vert \eta\right\Vert}^2}_{L^2_T H^{4N+3/2}} \ls \sigma^2 {{\left\Vert\eta\right\Vert}^2}_{L^2_T H^0} + \sigma^2 {{\left\Vert\Delta_\ast \eta\right\Vert}^2}_{L^2_T H^{4N-1/2}} \\
\ls T \sup_{0\le t \le T} \E^0[\eta(t)] + \z + \int_0^T \left((1+P(\E^0[\eta]) {{\left\Vert\nab_\ast \eta\right\Vert}^2}_{4N-1/2} \E[u]\right) dt \\
\ls T \sup_{0\le t \le T} \E^0[\eta(t)] + \z + \int_0^T \left((1+P(\E^0[\eta]) \frac{1}{\sigma} \E^\sigma[\eta] \E[u]\right) dt.\end{gathered}$$ Then we use the parabolic modification of the kinematic boundary condition to get improved $L^2$ in time estimates for $\dt^j \eta$ for $j\ge 1$. Standard heat equation estimates yield the bounds $$\label{ki_12}
{{\left\Vert\dt \eta\right\Vert}^2}_{L^2_T H^{4N-1}} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j \eta\right\Vert}^2}_{L^2_T H^{4N-2j+2}} \ls \z + \int_0^T \E[u] \frac{1}{\sigma} \E^\sigma[\eta] dt + \kappa^2 \zf.$$
We then combine the estimates of Proposition \[k\_hor\_est\] with , , , , , and to deduce all estimates in except that of ${{\left\Vert\rho J \dt^{2N+1} u\right\Vert}^2}_{L^2_T \Hd}$. To recover this estimate we simply appeal to with $\pa = \dt^{2N}$. A standard duality argument then allows us to estimate ${{\left\Vert\rho J \dt^{2N+1} u\right\Vert}^2}_{L^2_T \Hd}$ in terms of all of the existing terms in ; this yields in the form written.
Finally, we combine Propositions \[k\_hor\_est\] and \[kappa\_improved\] in order to obtain $\kappa-$independent estimates.
\[kappa\_apriori\] Assume –. Assume also that $0<\kappa < \min\{1,\sigma_+,\sigma_-\}$. There exists a universal constant $0 < \delta_1 < \delta_0 /2$ (where $\delta_0$ is from Proposition \[lame\_elliptic\]) and a $0 < T_1 = T_1(\TE,\sigma) \le 1$ such that if $0<\delta < \delta_1$ in – and $0\le T \le \min\{T_1,T_\ast\}$, then $$\begin{gathered}
\label{ka_01}
\sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \int_0^T (\check{\D}[u(t)] +{{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D^\sigma[\eta(t)] )dt
\\
+ \kappa \sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j \nab_\ast \eta(t)\right\Vert}^2}_{4N-2j} +\sigma {{\left\Vert\dt^j \Delta_\ast \eta(t) \right\Vert}^2}_{4N-2j} dt
\ls P(\TE[u_0,\eta_0]) +\sup_{0\le t \le T} \f_\infty(t) + \int_0^T \f_2(t) dt\end{gathered}$$ for a positive universal polynomial $P$ such that $P(0)=0$.
We first notice that $$\gamma = C \left( 1 + \If[\eta](T)+ \Ef[\rho](T) + \Jf[u](T) ) \right) \le C(1 + P(\TE)),$$ and so we can make $T_2$ small in terms of $\TE$, $\sigma$, and a universal constant so that $e^{\gamma T} \le 2$, $\sqrt{T} P(\Ef[\rho](T)) \le 1$, and $\sqrt{T} \le \sigma$ whenever $T \le \min\{T_2,T_\ast\}$. Then Proposition \[kappa\_improved\] yields the estimate $$\begin{gathered}
\label{ka_1}
\sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \int_0^T (\check{\D}[u(t)] +{{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D^\sigma[\eta(t)] )dt
\\
+ \kappa \sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j \nab_\ast \eta(t)\right\Vert}^2}_{4N-2j} + \sigma {{\left\Vert\dt^j \Delta_\ast \eta(t) \right\Vert}^2}_{4N-2j} dt
\le C\TE[u_0,\eta_0] + C \sup_{0\le t\le T} \f_\infty(t)
\\
+ C\int_0^T \f_2(t) dt
+ C\kappa^2 \zf + \sqrt{T} P\left( \sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)] \right) + C\sup_{0\le t \le T} \Lf[\eta(t)] \int_0^T \check{\D}[u(t)]dt\end{gathered}$$ for every $T \le \min\{T_2, T_\ast\}$, where $P$ is a universal positive polynomial such that $P(0)=0$, and $C\ge 1$ is a universal constant.
Since $$\sup_{0\le t \le T} \Lf[\eta(t)] \le \delta \le \delta_1,$$ we may choose a universal $\delta_1>0$ such that $C \delta_1 =1/2$. We may then absorb the last term on the right side of onto the left. This yields the estimate $$\begin{gathered}
\label{ka_2}
\sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \hal \int_0^T (\check{\D}[u(t)] +{{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D^\sigma[\eta(t)] )dt
\\
+ \kappa \sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j \nab_\ast \eta(t)\right\Vert}^2}_{4N-2j} + \sigma{{\left\Vert\dt^j \Delta_\ast \eta(t) \right\Vert}^2}_{4N-2j} dt
\\
\le C\TE[u_0,\eta_0] + C \sup_{0\le t\le T} \f_\infty(t) + C\int_0^T \f_2(t) dt + C \kappa^2 \zf
\\
+ \sqrt{T} P\left( \sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) \right)\end{gathered}$$ for every $0 \le T \le \min\{T_2,T_\ast\}$.
We may view abstractly as an inequality of the form $$\label{ka_3}
X(T) \le C Z(T) + \sqrt{T} P(X(T))$$ for $X,Z:[0,\min\{T_2,T_\ast\}] \to [0, \infty)$ continuous functions such that $Z$ is non-decreasing and $X(0) \le C Z(0)$. By continuity we know that either $X(T) \le 2C Z(T)$ for all $0\le T \le \min\{T_2,T_\ast\}$, or else there exists a first time $0 < T_3 < \min\{T_2,T_\ast\}$ such that $X(T_3) = 2C Z(T_3).$ Plugging this equality into implies that $$2C Z(T_3) = X(T_3) \le C Z(T_3) + \sqrt{T_3} P( X(T_3)) = C Z(T_3) + \sqrt{T_3} P( 2C Z(T_3) ),$$ and hence $$C Z(T_3) \le \sqrt{T_3} P( 2C Z(T_3) ).$$ From this we deduce that $$\sqrt{T_3} \ge \hal \frac{2 C Z(T_3)}{ P( 2C Z(T_3) )} \ge \hal \min_{z \in [0,2C Z_{max}]} \frac{z}{P(z)},$$ where $$Z_{max} = \max_{0 \le T \le \min\{T_2,T_\ast\}} Z(T) = Z(\min\{T_2,T_\ast\}) \le P(\TE) + \tilde{C}.$$ The last estimate follows because of assumption , from which the universal constant $\tilde{C} >0$ comes, and combined with the bound $\kappa < \min\{1,\sigma_+,\sigma_-\}$. Since $P(0)=0$, we then find that $$\sqrt{T_3} \ge \hal \min_{z \in [0,2C(P({\TE}) + \tilde{C} ) ]} \frac{z}{P(z)} =: \sqrt{T_4} > 0.$$ This leads us to define $$T_1 =T_1(\TE,\sigma) = \min\{T_2,T_4 \}$$ so that if $0 \le T \le \min\{T_1,T_\ast\}$ we have the estimate $$X(T) \le 2C Z(T).$$ Removing the abstraction, this implies that $$\begin{gathered}
\label{ka_4}
\sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \hal \int_0^T (\check{\D}[u(t)] +{{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D^\sigma[\eta(t)] )dt
\\
+ \kappa \sum_{j=0}^{2N} \int_0^T {{\left\Vert\dt^j \nab_\ast \eta(t)\right\Vert}^2}_{4N-2j} + \sigma {{\left\Vert\dt^j \Delta_\ast \eta(t) \right\Vert}^2}_{4N-2j} dt
\\
\le 2C\TE[u_0,\eta_0] + 2C \sup_{0\le t\le T} \f_\infty(t) + 2C\int_0^T \f_2(t) dt + 2C\kappa^2 \zf\end{gathered}$$ for $0 \le T \le \min\{T_1,T_\ast\}$. This and yield .
The two-phase free boundary Lamé problem {#sec_lame_free}
=========================================
Our goal in this section is to produce a solution to $$\label{lame_free_bndry}
\begin{cases}
\rho \dt u - \diva \Sa u = F^1 & \text{in }\Omega\\
\dt \eta = u \cdot \n &\text{on }\Sigma \\
-\Sa u \n = - \sigma_+ \Delta_\ast \eta \n + F^2_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \Sa u \right\rrbracket } \n = \sigma_- \Delta_\ast \eta \n - F^2_- &\text{on } \Sigma_- \\
{\left\llbracket u \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b \\
u(\cdot,0) = u_0, \eta(\cdot, 0) = \eta_0
\end{cases}$$ by passing to the limit $\kappa \to 0$ in the $\kappa-$approximation problem .
Our strategy is as follows. First, we will combine the local existence result of Theorem \[kappa\_contract\] with the a priori estimates of Theorem \[kappa\_apriori\] to produce solutions on a time interval that is independent of $\kappa$ and that satisfy $\kappa-$independent estimates of the form . Second, we will pass to the limit $\kappa \to 0$ to recover a solution to .
Local existence on $\kappa-$independent intervals
-------------------------------------------------
Our goal now is to combine Theorems \[kappa\_contract\] and \[kappa\_apriori\]. We first describe some assumptions on the data and forcing terms that will be needed.
We say the data and forcing satisfy $\mathfrak{P}(\delta)$ on the time interval $[0,T_\ast]$ if the following hold. $$\begin{aligned}
&\text{The initial data } \dt^j u(\cdot,0) \text{ and }\dt^j \eta(\cdot,0) \text{ for }j=0,\dotsc,2N \text{ are given as before and} \label{lame_assump_1}\\
& \quad\text{ satisfy the compatibility conditions } \eqref{kappa_ccs}. \nonumber \\
& \text{The data satisfy }\Lf[\eta_0] < \delta/2, \text{ where }\Lf[\eta_0] \text{ is given by }\eqref{l0_def}. \label{lame_assump_2}\\
& \text{The forcing terms satisfy } \sup_{0\le t\le T_\ast} \f_\infty(t) + \int_0^{T_\ast} \f_2(t)dt \le P(\TE) \nonumber \\
& \quad \text{for a universal positive polynomial }P\text{ such that }P(0)=0. \label{lame_assump_3}\\
& \text{The function } \rho \text{ is defined on }[0,T_\ast] \text{ and satisfies } \eqref{rho_assump_1} \text{ and } \eqref{rho_assump_2}. \label{lame_assump_4}\end{aligned}$$
\[lame\_local\] Assume that $0<\kappa < \min\{1,\sigma_+,\sigma_-\}$. Let $0 < \delta_1$ be the universal constant and $0 < T_1 = T_1(\TE,\sigma)$ be from Theorem \[kappa\_apriori\]. Assume that the data and forcing satisfy $\mathfrak{P}(\delta_1)$ on the time interval $[0,T_\ast]$. Then there exists $0 < T_2 = T_2(\TE,\sigma) \le T_1(\TE,\sigma)$ such that if $0 < T \le \min\{T_\ast,T_2\}$, then the following hold.
1. A unique solution $(u,\eta)$ to exists on the interval $[0,T]$ and achieves the initial data.
2. The solution satisfies the estimate $$\label{lam_loc_01}
\j[u](T) + \i[\eta](T) < \infty,$$ where $\j[u]$ and $\i[\eta]$ are as defined in .
3. We have the estimate $$\begin{gathered}
\label{lam_loc_02}
\sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \int_0^{T}(\check{\D}[u(t)] +{{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D^\sigma[\eta(t)] )dt
\\
+ \kappa \sum_{j=0}^{2N} \int_0^{T} {{\left\Vert\dt^j \nab_\ast \eta(t)\right\Vert}^2}_{4N-2j} + \sigma{{\left\Vert\dt^j \Delta_\ast \eta(t) \right\Vert}^2}_{4N-2j} dt
\ls P(\TE[u_0,\eta_0]) +\sup_{0\le t \le T} \f_\infty(t) + \int_0^{T} \f_2(t) dt\end{gathered}$$ for a universal positive polynomial $P$ such that $P(0)=0$. Here we recall the notation $\check{\D}$ defined in .
4. We also have the estimates $$\label{lam_loc_03}
\Lf[\eta](T) < \delta_1$$ and $$\label{lam_loc_04}
\If[\eta](T) + \Jf[u](T) \ls P(\TE) + 1$$ for a universal positive polynomial $P$ such that $P(0)=0$, where $\Lf$ is defined by and $\If$ and $\Jf$ are defined by .
5. One of the following is true. Either $T_2 = T_1$, or else there exists a universal positive polynomial $P$ satisfying $P(0)=0$ and a universal constant $C>0$ such that $$\label{lam_loc_05}
\frac{\sigma \delta_1}{ C (1+ P({\EEE}))} \le T_2.$$
For $r >0$ let $\mathfrak{S}(r)$ denote the proposition that the following three conditions hold. First, a unique solution to exists on $[0,r]$ and achieves the initial data. Second, the solution satisfies with $T$ replaced by $r$. Third, the solution satisfies and with $T$ replaced by $r$. Define the set $$\mathfrak{R} = \{ r \in [0,\min\{T_\ast,T_1\}] \;\vert \; \mathfrak{S}(r) \text{ holds} \}.$$ Theorem \[kappa\_apriori\] guarantees that $\delta_1 \le \delta_0/2$, where $\delta_0>0$ is defined by Proposition \[lame\_elliptic\]. We may then apply Theorem \[kappa\_contract\] to see that $\mathfrak{R} \neq \varnothing$. Let $T_{\mathfrak{R}} = \sup \mathfrak{R} \in (0, T_\ast]$.
If $T_{\mathfrak{R}} = \min\{T_\ast,T_1\}$ then we set $T_2 = T_1$, and we are done. Indeed, the hypotheses of Theorem \[kappa\_apriori\] are satisfied, and so follows. We may assume then that $T_{\mathfrak{R}} < \min\{T_\ast,T_1\}$ throughout the rest of the proof.
If $T_{\mathfrak{R}}= \max \mathfrak{R}$, then a standard continuation argument, employing Theorem \[kappa\_contract\] to extend the solutions, yields a contradiction, and so we deduce that $T_{\mathfrak{R}} \notin \mathfrak{R}$. This means that $\mathfrak{S}(T_{\mathfrak{R}})$ fails. We claim that in fact it is only the third condition in $\mathfrak{S}(T_{\mathfrak{R}})$ that can fail, i.e. the first two conditions remain true at $T_{\mathfrak{R}}$.
We know that $\mathfrak{S}(T_{\mathfrak{R}} -\ep)$ is true for $\ep$ sufficiently small. The a priori estimates of Theorem \[kappa\_apriori\] are then valid and provide $\ep-$independent control of $\E(T_{\mathfrak{R}} -\ep)$ in terms of $\TE$. We may then use Theorem \[kappa\_contract\] to extend the solutions to $T_3 = T_{\mathfrak{R}} - \ep + T_0(\kappa,\sigma,\delta,\TE)$. When $\ep$ is sufficiently small we have that $T_3 > T_{\mathfrak{R}}$, and so the first condition of $\mathfrak{S}(T_{\mathfrak{R}})$ must hold. Additionally, the functional setting of Theorem \[kappa\_contract\] guarantees that the second condition of $\mathfrak{S}(T_{\mathfrak{R}})$ must also hold. We deduce then that it is the third condition of $\mathfrak{S}(T_{\mathfrak{R}})$ that fails at $T_{\mathfrak{R}}$, proving the claim.
From Lemma \[time\_interp\], Theorem \[kappa\_apriori\], and we may estimate $$\If[\eta](T_{\mathfrak{R}}) + \Jf[u](T_{\mathfrak{R}}) \ls \TE[u_0,\eta_0] + \int_0^{T_{\mathfrak{R}}} (\E[u(t)] + \E^0[\eta(t)] )dt \ls \TE[u_0,\eta_0] + \int_0^{T_{\mathfrak{R}}} P(\TE) dt,$$ and since $T_{\mathfrak{R}} \le T_1 \le 1$, we then have that $\If[\eta](T_2) + \Jf[u](T_2) \ls P(\TE) + 1.$ This means that estimate remains true at $T_{\mathfrak{R}}$, so it is actually estimate that fails at time $T_{\mathfrak{R}}$. Arguing similarly, we deduce that $$\begin{gathered}
\delta_1 = \Lf[\eta(T_{\mathfrak{R}})] = {{\left\Vert\eta(T_{\mathfrak{R}})\right\Vert}^2}_{4N-1/2} \le \Lf[\eta_0] + C\int_0^{T_{\mathfrak{R}}} ({{\left\Vert\eta(t)\right\Vert}^2}_{4N} + {{\left\Vert\dt \eta(t)\right\Vert}^2}_{4N-1}) dt \\
\le \frac{\delta_1}{2} + \frac{C}{\sigma} \int_0^{T_{\mathfrak{R}}} \E^\sigma[\eta(t)] dt \le \frac{\delta_1}{2} + \frac{C T_{\mathfrak{R}} }{\sigma}(1 + P({\EEE})),\end{gathered}$$ and hence that $$\label{lam_loc_1}
\frac{\sigma \delta_1}{2 C (1+ P({\EEE}))} \le T_{\mathfrak{R}}.$$ To conclude the proof we then set $T_2 = T_{\mathfrak{R}} /2$, apply Theorem \[kappa\_apriori\], and use to produce .
Sending $\kappa \to 0$
----------------------
Our aim now is to use Theorem \[lame\_local\] to send $\kappa \to 0$ in in order to produce a solution to .
\[lame\_exist\] Let $\delta_1 >0$ be the universal constant from Theorem \[kappa\_apriori\], and assume that the data and forcing satisfy $\mathfrak{P}(\delta_1)$ on the time interval $[0,T_\ast].$ There exists a $0 < T_3 = T_3(\TE)$ such that if $0 < T \le \min\{T_\ast,T_3\}$, then the following hold.
1. A unique solution $(u,\eta)$ to exists on the interval $[0,T]$ and achieves the initial data.
2. We have the estimate $$\begin{gathered}
\label{lamex_01}
\sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \int_0^{T}(\D[u(t)] +{{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} +\D^\sigma[\eta(t)] )dt
\\
\ls P(\TE[u_0,\eta_0]) +\sup_{0\le t \le T} \f_\infty(t) + \int_0^{T} \f_2(t) dt\end{gathered}$$ for a universal positive polynomial $P$ such that $P(0)=0$.
3. We have the estimates $$\label{lamex_02}
\Lf[\eta](T) < \delta_1
\text{ and }
\If[\eta](T) + \Jf[u](T) \ls P(\TE) + 1$$ for a universal positive polynomial $P$ such that $P(0)=0$, where $\Lf[\eta]$ is defined by and $\If[\eta]$ and $\Jf[u]$ are defined by .
4. We have the estimate $$\begin{gathered}
\label{lamex_04}
\sup_{0\le t \le T} \sum_{j=1}^{2N} {{\left\Vert\dt^j \eta(t) \right\Vert}^2}_{4N-2j+3/2} + \int_0^T \left( {{\left\Vert\dt \eta\right\Vert}^2}_{4N-1/2} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j \eta(t) \right\Vert}^2}_{4N-2j+5/2} \right)dt\\
\ls P(\TE) +\sup_{0\le t \le T} \f_\infty(t) + \int_0^{T} \f_2(t) dt\end{gathered}$$ for a universal positive polynomial $P$ such that $P(0)=0$.
5. We also have the estimate $$\begin{gathered}
\label{lamex_05}
\sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \le \exp\left(C T \int_0^T {{\left\Vertu(t)\right\Vert}^2}_{H^{4N+1/2}(\Sigma)} dt \right) \\
\times \left({{\left\Vert\eta_0\right\Vert}^2}_{4N+1/2} + T \int_0^T {{\left\Vertu(t)\right\Vert}^2}_{H^{4N+1/2}(\Sigma)} dt \right).\end{gathered}$$
We divide the proof into two steps. In the first we will initially prove the theorem with the weaker condition that $T_3 = T_3(\TE,\sigma)$, i.e. that $T_3$ is allowed to depend on $\sigma$ as well as the data, but not on $\kappa$. In the second we will use the first step and some auxiliary arguments to remove the $\sigma$ dependence.
Step 1 – $T_3 = T_3(\TE,\sigma)$.
For each $0 < \kappa < \min\{1,\sigma_+,\sigma_-\}$ Theorem \[lame\_local\] provides us with a pair $(u_\kappa,\eta_\kappa)$ solving on $(0,T_2)$ and satisfying the conclusions of the theorem. We shall consider $\kappa$ to index a sequence of values chosen in $(0,\min\{1,\sigma_+,\sigma_-\})$ that decrease to $0$. The $\kappa-$independent estimates of show that the sequence $\{(u_\kappa,\eta_\kappa)\}_\kappa$ is bounded uniformly in the function space determined by the first line of (i.e. the left hand side of the inequality with $\kappa=0$). These uniform bounds allow us to argue as in Theorem 6.3 of [@GT_lwp], using weak and weak-$\ast$ compactness arguments together with interpolation and lower semi-continuity arguments, to extract a subsequence (still denoted by $\kappa$) such that $$\label{lamex_1}
\begin{split}
\dt^j u_\kappa &\to \dt^j u \text{ in } C^0([0,T];H^{4N-2j-2}(\Omega)) \text{ for }j=0,1,2 \\
\dt^j \eta_\kappa &\to \dt^j \eta \text{ in } C^0([0,T];H^{4N-2j-1}(\Sigma)) \text{ for }j=0,1,2,
\end{split}$$ where $(u,\eta)$ satisfies and achieves the initial data. Note here that the convergence can be improved to a larger range of $j$ and to higher regularity for the various $\eta$ terms. We state as is because it is more than sufficient to pass to the limit in and deduce that $(u,\eta)$ are a strong solution to . The estimates follow from and using similar lower semi-continuity arguments.
It remains to derive the improved estimates. This entails improving to by obtaining an estimate for ${{\left\Vertu\right\Vert}^2}_{L^2_T H^{4N+1}}$ and also proving . To accomplish the first we will need . This estimate is proved in Step 1 of Theorem 5.4 of [@GT_lwp] by using estimates developed in [@danchin] for solutions to the kinematic transport equation .
With in hand, we use and in conjunction with the simple estimate $${{\left\Vertu\right\Vert}^2}_{H^{4N+1/2}(\Sigma)} \ls {{\left\Vert\nab_{\ast,0}^{4N-1} u\right\Vert}^2}_{H^{3/2}(\Sigma)} \ls {{\left\Vert\nab_{\ast,0}^{4N-1} u\right\Vert}^2}_{2}$$ in order to bound $$\sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \ls \exp\left(C T P(\TE) \right)
\left(\TE + T P(\TE) \right).$$ Then if $T_3 \le T_2$ is taken sufficiently small, we may bound $$\sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \ls P(\TE).$$ We then appeal to the elliptic estimate of Proposition \[lame\_elliptic\] to bound $$\begin{gathered}
\int_0^T {{\left\Vertu(t)\right\Vert}^2}_{4N+1} dt \ls \int_0^T \left( {{\left\Vert\rho \dt u(t)\right\Vert}^2}_{4N-1} + {{\left\VertF^1(t)\right\Vert}^2}_{4N-1} + {{\left\Vertu(t)\right\Vert}^2}_{H^{4N+1/2}(\Sigma)} \right) dt \\
+ \sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \int_0^T \left( {{\left\Vert\rho \dt u(t)\right\Vert}^2}_{2} + {{\left\VertF^1(t)\right\Vert}^2}_{2} + {{\left\Vertu(t)\right\Vert}^2}_{H^{7/2}(\Sigma)} \right) dt.\end{gathered}$$ Hence , , and allow us to bound $$\begin{gathered}
\label{lamex_3}
\int_0^T {{\left\Vertu(t)\right\Vert}^2}_{4N+1} dt \ls \int_0^T \left(\check{D}[u(t)] + \f_2(t) \right) dt + T(1+P(\TE) ) \sup_{0 \le t \le T} \left( \E[u(t)] + \f_\infty(t) \right)
\\ \ls P(\TE) +\sup_{0\le t \le T} \f_\infty(t) + \int_0^{T} \f_2(t) dt\end{gathered}$$ once we further restrict $T_3$ so that $T_3 (1+P(\TE) )\le 1$. Summing and then yields .
Finally, we prove the improved estimates . For this we will use the fact that $(u,\eta)$ satisfy the kinematic equation $$\label{lame_xport}
\dt \eta = u_3 + \nabla_\ast \eta\cdot u\text{ on }\Sigma.$$ We may use this equality as in Theorem 5.4 of [@GT_lwp], using the usual estimates of products in Sobolev spaces, to deduce that $$\begin{gathered}
\label{lamex_2}
\sup_{0\le t \le T} \sum_{j=1}^{2N} {{\left\Vert\dt^j \eta(t) \right\Vert}^2}_{4N-2j+3/2} + \int_0^T \left( {{\left\Vert\dt \eta\right\Vert}^2}_{4N-1/2} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j \eta(t) \right\Vert}^2}_{4N-2j+5/2} \right)dt \\
\le P\left( \sup_{0\le t \le T} (\E[u(t)] + \E^\sigma[\eta(t)]) + \int_0^{T}(\D[u(t)] +\D^\sigma[\eta(t)] )dt \right)\end{gathered}$$ for some universal positive polynomial with $P(0)=0$. Then from , , and we deduce that holds.
Step 2 – Improvement to $T_3 = T_3(\TE)$
With the Theorem in hand for a $T_3 = T_3(\TE,\sigma)$, we can employ a continuation argument in order to remove the dependence on $\sigma$. The argument is similar to the one used in Theorem \[lame\_local\], so in the interest of brevity we will only point out the key point. This lies in the fact that our result from Step 1 requires $\mathfrak{P}(\delta_1)$ to hold on $[0,T_\ast]$, which in turn demands that ${{\left\Vert\eta_0\right\Vert}^2}_{4N-1/2} < \delta_1/2$. The estimate provides us with an estimate of $\sup_{0 \le t \le T} {{\left\Vert \dt \eta(t)\right\Vert}^2}_{4N-1/2}$, which when coupled to the fundamental theorem of calculus, allows us to estimate $\mathfrak{L}[\eta](T)$. Using this, we can prove that the theorem remains true on a time interval $[0, \min\{T_\ast, T_3\}]$, where either $T_3 = T_\ast$ or else $T_3$ is when ${{\left\Vert\eta(T_3)\right\Vert}^2}_{4N-1/2} \ge \frac{\delta_1}{2}.$ We may then argue as in , using the estimate for ${{\left\Vert\dt \eta\right\Vert}^2}_{4N-1/2}$ in place of Lemma \[time\_interp\], to show that $T_3$ is bounded below by a positive function of $\TE$ that is independent of $\sigma$. Hence $T_3 = T_3(\TE)$.
The improved $\eta$ estimates of are the key to eliminating the dependence of the existence interval on $\sigma$. These in turn depend on the structure of the equation $\dt \eta = u \cdot \n$ in place of the $\kappa$ approximation. We thus see the importance of passing to the limit $\kappa \to 0$ before attempting to remove the dependence on $\sigma$.
The transport problem {#sec_xport}
=====================
Our goal in this section is to study the transport problem $$\label{xport_fundamental}
\begin{cases}
\dt q +(\a^T u - K \dt \theta e_3) \cdot \nab q + \diva(u) q = f & \text{in } \Omega \times (0,T) \\
q(\cdot, 0) = q_0 &\text{in }\Omega,
\end{cases}$$ where $u$, $\eta$ (and hence $\a$, etc), and $f$ are given. To simplify the structure of we will initially study the more general problem $$\label{xport_proto}
\dt q + v \cdot \nab q + c q = f,$$ where $v$, $c$, and $f$ are given.
The key to the analysis of a transport problem on a domain with boundaries is the behavior of $v$ at the boundary. For the $v$ arising in with $(u,\eta)$ satisfying we have the following crucial identities: on $\Sigma_\pm$ we have $$\label{xport_v1}
v \cdot e_3 = (\a^T u - K \dt \theta e_3)\cdot e_3 = K(u \cdot J\a e_3 - \dt \eta) = K (u_\pm \cdot\n_\pm - \dt \eta_\pm) =0,$$ while on $\Sigma_b$ we have $$\label{xport_v2}
v \cdot e_3 = (\a^T u - K \dt \theta e_3)\cdot e_3 = K(u_- \cdot J \a e_3 - \tilde{b}_1 \dt \bar{\eta}_+ - \tilde{b}_2 \dt \bar{\eta}_1) =0$$ since $\tilde{b}_1 = \tilde{b}_2 =0$ and $u_- =0$ on $\Sigma_b$. Note that the condition $u_- =0$ on $\Sigma_b$ could be weakened to $u_{3,-}=0$ on $\Sigma_b$ since $J \a e_3=e_3$ on $\Sigma_b$. The upshot of these identities is that we may assume that the vector field $v$ satisfies $$\label{xport_v}
v \cdot e_3 =0 \text{ on } \p \Omega.$$
Notice that we do not couple the transport equation to boundary conditions, and in fact the equations in $\Omega_+$ and $\Omega_-$ are decoupled from one another. This allows us to solve the equations in each domain separately. To this end we will let $\Upsilon = \Omega_+$ or $\Upsilon = \Omega_-$ and discuss the transport equation in $\Upsilon$. Let us now define various functionals that will appear in our analysis of the transport problem.
For a function $g$ we define $$\label{rqe_def}
\Qf_e[g] = \sum_{j=1}^{2N} {{\left\Vert\dt^j g\right\Vert}^2}_{4N-2j+1} \text{ and } \Rf_e[g] = \sum_{j=0}^{2N-1} {{\left\Vert\dt^j g\right\Vert}^2}_{4N-2j-1}.$$ and $$\label{rqd_def}
\Qf_d[g] = {{\left\Vert\dt g\right\Vert}^2}_{4N-1} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j g\right\Vert}^2}_{4N-2j+2} \text{ and } \Rf_d[g] = {{\left\Vertg\right\Vert}^2}_{4N-1} + \sum_{j=1}^{2N} {{\left\Vert\dt^j g\right\Vert}^2}_{4N-2j}.$$
We will assume that $v$, $c$, and $f$ satisfy $$\label{xport_reg}
\sup_{0 \le t \le T} \left( \Rf_e[v(t)] + \Rf_e[c(t)] + \Rf_e[f(t)] \right) + \int_0^T \left( \Rf_d[v(t)] + \Rf_d[c(t)] + \Rf_d[f(t)] \right) dt <\infty.$$
Solution by characteristics
---------------------------
Consider the transport problem $$\label{xport_eqn}
\begin{cases}
\dt q + v \cdot \nab q + c q = f & \text{in } \Upsilon \times (0,T) \\
q(\cdot,0) = q_0 &\text{in }\Upsilon.
\end{cases}$$ Here we assume that $v$, $c$, and $f$ satisfy . In particular the usual Sobolev embeddings require that $v$, $c$, and $f$ are all $C^1(\Upsilon \times [0,T])$.
To produce a solution to we use the method of characteristics. We let $\zeta_t(x)$ denote the solution to the ODE $$\label{xport_char_1}
\begin{cases}
\dt \zeta_t(x) = v(\zeta_t(x),t) \\
\zeta_0(x) = x
\end{cases}$$ for $x \in \Upsilon$ and $t \in [0,T]$. The identity is essential here, since it guarantees that for each $t \in [0,T]$, $\zeta_t : \Upsilon \to \Upsilon$ is a diffeomorphism. Let us define the map $\omega : [0,T]^2 \times \Upsilon \to \Upsilon$ via $$\label{xport_char_2}
\omega(s,t,x) = \zeta_s(\zeta^{-1}_t(x)).$$
Then the solution to is $$\begin{gathered}
\label{xport_soln}
q(x,t) = q_0(\omega(0,t,x)) \exp\left[ - \int_0^t c(\omega(s,t,x),s)ds \right] \\
+ \int_0^t f(\omega(s,t,x),s) \exp\left[ - \int_s^t c(\omega(r,t,x),r)dr \right]ds.\end{gathered}$$ While this explicit form of the solution is nice, it is not convenient for making higher regularity estimates. In particular it is not immediately obvious from that $q$ belongs to the space defined by and , and it is not clear that we can justify applying $\p^\alpha$ to and performing a priori estimates. The usual solution to this difficulty is the Friedrichs mollification method: we first solve a mollified version of so that the solution is smooth enough to justify the a priori estimates, then we derive the a priori estimates in a manner independent of the mollification parameter, and then finally we pass to the limit. This works well when $\p \Upsilon=\varnothing$, but the mollification procedure runs into technical obstructions when $\p \Upsilon \neq \varnothing$, which is the case here.
Two options then present themselves. The first is to modify the mollification procedure in a manner that makes sense in our $\Upsilon$ but does not destroy the structure of the a priori estimates. The second is to transfer the problem to a new problem on a set without boundary in such a way that the estimates from Friedrichs’ method carry over to . We have chosen to go for the second option since Friedrichs’ method is so well-known. Our goal then is to justify the transfer of the problem and the estimates.
Transfer
--------
Consider the transport problem where we only assume for now that $q_0 \in L^2(\Upsilon)$ and $f \in L^2([0,T];L^2(\Upsilon))$. We say that $q \in L^2([0,T];L^2(\Upsilon))$ is a weak solution to if $$\label{xport_weak}
\int_0^T \int_\Upsilon -q \left( \dt \varphi + \operatorname{div}(v \varphi ) - c\varphi\right) = \int_0^T \int_\Upsilon \varphi f + \int_\Upsilon q_0 \varphi(\cdot,0)$$ for all $\varphi \in C_c^1(\Upsilon \times [0,T))$. The identity is clearly satisfied by any regular solution to , and in particular by the solution given by .
We have the following simple lemma on the uniqueness of weak solutions, which we state without proof.
\[xport\_unique\_char\] The following are equivalent.
1. For every $f \in L^2([0,T];L^2(\Upsilon))$ and $q_0 \in L^2(\Upsilon)$, there exists at most one function $q \in L^2([0,T];L^2(\Upsilon))$ that is a weak solution to .
2. If $q \in L^2([0,T];L^2(\Upsilon))$ satisfies $$\label{xuc_01}
\int_0^T \int_\Upsilon -q \left( \dt \varphi + \operatorname{div}(v \varphi ) - c\varphi\right) =0$$ for every $\varphi \in C_c^1(\Upsilon \times [0,T))$, then $q=0$.
Our next goal is to verify that the second item of Lemma \[xport\_unique\_char\] holds, which means that weak solutions to are unique. To this end we first study the adjoint problem determined by what appears in parentheses in .
Let $\psi \in C_c^\infty(\Upsilon \times (0,T))$. We want to find $\varphi \in C_c^1(\Upsilon \times [0,T))$ satisfying the adjoint problem $$\label{xport_adjoint}
\begin{cases}
\dt \varphi + \operatorname{div}(v \varphi) - c \varphi = \psi &\text{in }\Upsilon \times (0,T) \\
\varphi(\cdot,T) = 0 &\text{in }\Upsilon.
\end{cases}$$ Note that this is a terminal value problem; we seek to solve this so that $\varphi$ can be taken to be compactly supported in $\Upsilon \times [0,T)$.
To solve the adjoint problem we use $\zeta_t$ given by to reduce to an ODE along the characteristics. Recall the function $\omega :[0,T]^2 \times \Upsilon \to \Upsilon$ given by ; we may use it to derive the solution $$\varphi(x,t) = - \int_t^T \psi(\omega(s,t,x),s) \exp\left[ \int_t^r \left( \operatorname{div}{v}(\omega(r,t,x),r) - c(\omega(r,t,x),r) \right)dr \right] ds.$$ From this formula and the inclusion $\psi \in C_c^\infty(\Upsilon \times (0,T))$ it is easy to see that $\varphi \in C_c^1(\Upsilon \times [0,T))$, as desired. We have thus proved the following lemma.
\[xport\_adjoint\_soln\] Let $\psi \in C_c^\infty(\Upsilon \times (0,T))$. Then there exists $\varphi \in C_c^1(\Upsilon \times [0,T))$ solving .
With Lemma \[xport\_adjoint\_soln\] in hand we can prove that weak solutions to are unique.
\[xport\_weak\_unique\] Let $f \in L^2([0,T];L^2(\Upsilon))$ and $q_0 \in L^2(\Upsilon)$. Then there exists at most one function $q \in L^2([0,T];L^2(\Upsilon))$ that is a weak solution to in the sense of .
To prove uniqueness we will show that the second item of Lemma \[xport\_unique\_char\] holds. Suppose that $q \in L^2([0,T];L^2(\Upsilon))$ satisfies for every $\varphi \in C_c^1(\Upsilon \times [0,T))$. For any $\psi \in C_c^\infty(\Upsilon \times (0,T))$ we may use Lemma \[xport\_adjoint\_soln\] to find $\varphi \in C_c^1(\Upsilon \times [0,T))$ solving . Using this $\varphi$ in yields the equality $$\int_0^T \int_\Upsilon q \psi =0.$$ Since $\psi$ was arbitrary, we deduce that $q =0$. Hence the second item of Lemma \[xport\_unique\_char\] holds.
Next we define the extended problem, which is easier to handle with Friedrichs’ method. Let $\Gamma = \mathrm{T}^2\times \mathbb{R}$ denote the extended domain in which we will pose the extended problem. Let $E$ denote a Sobolev extension operator such that $E : H^m(\Upsilon) \to H^m(\Gamma)$ for every $m=0,\dotsc,M$, where $M >0$ is large enough to surpass every regularity index in the energy and dissipation defined in and . We then define $$\bar{v} = E v, \; \bar{c} = Ec, \;\bar{q}_0 = Eq_0, \text{ and } \bar{f} = E f.$$
We say that $\bar{q} \in L^2([0,T];L^2(\Gamma))$ is a weak solution to $$\label{xport_extended}
\begin{cases}
\dt \bar{q} + \bar{v} \cdot \nab \bar{q} + \bar{c} \bar{q} = \bar{f} & \text{in } \Gamma \times (0,T) \\
\bar{q}(\cdot,0) = \bar{q}_0 &\text{in }\Gamma
\end{cases}$$ if $$\label{xport_weak_extended}
\int_0^T \int_\Gamma -\bar{q} \left( \dt \varphi + \operatorname{div}(\bar{v} \varphi ) - \bar{c}\varphi\right) = \int_0^T \int_\Gamma \varphi \bar{f} + \int_\Gamma \bar{q}_0 \varphi(\cdot,0)$$ for all $\varphi \in C_c^1(\Gamma \times [0,T))$.
\[xport\_transfer\] Suppose that $\bar{q} \in L^2([0,T];L^2(\Gamma))$ is a weak solution to in the sense of . Let $q$ denote the restriction of $\bar{q}$ to $\Upsilon \times (0,T)$. Then $q$ is the unique weak solution to in the sense of .
Since holds for all $\varphi \in C_c^1(\Gamma \times [0,T))$ it must also hold for all $\varphi \in C_c^1(\Upsilon \times [0,T))$. For such $\varphi$ the equality is identical to because $\bar{v},$ $\bar{c}$,$\bar{q}_0$, and $\bar{f}$ are extensions of $v$, $c$, $\bar{q}_0$, and $f$ to $\Gamma$. Hence $q$ is a weak solution to to in the sense of . Uniqueness follows from Proposition \[xport\_weak\_unique\].
The upshot of Lemma \[xport\_transfer\] is that if we can produce a solution to the extended problem that obeys the estimates we seek for the original problem , then we know that those estimates are also valid for the solution given by characteristics on $\Upsilon$. This leads us to study the extended problem .
The extended problem
---------------------
### The mollified problem
We begin by defining space and time mollification operators. Since the horizontal directions in $\Gamma$ are periodic, it is convenient to decompose our spatial mollifiers into horizontal and vertical parts. Let $\vartheta \in C_c^\infty({\mathbb{R}^{}})$ be a standard mollifier. For $n \in \mathbb{N}$ and $i=1,2$, let $F_n^i \in C^\infty(2\pi L_i \mathbb{T})$ denote the Fejér kernel. Then we define the spatial mollifier via $$K_\ep^{sp} g(x) = \int_\Gamma g(x-y) F_{\lfloor 1/\ep\rfloor}^1(y_1) F_{\lfloor 1/\ep\rfloor}^2(y_2) \frac{1}{\ep} \vartheta\left(\frac{y_3}{\ep}\right) dy,$$ where $\lfloor z \rfloor$ denotes the integer part of $z$. To define the temporal mollification operator we must first define a temporal extension. For a function $g$ defined on $\Gamma \times (0,T)$ we first extend to $\tilde{g}$ defined on $\Gamma \times {\mathbb{R}^{}}$ via $$\tilde{g}(x,t) =
\begin{cases}
g(x,t) & \text{if }t \in [0,T] \\
0 & \text{if }t \notin [0,T].
\end{cases}$$ Then the temporal mollification is $$K_\ep^{te} g(x,t) = \int_{{\mathbb{R}^{}}} \tilde{g}(x,t-s) \frac{1}{\ep}\vartheta\left(\frac{s}{\ep}\right) ds.$$ The operators $K_\ep^{sp}$ and $K_\ep^{te}$ satisfy all the usual properties of a mollification operators.
The mollified problem studied in Friedrichs’ method is $$\label{xport_mollified}
\begin{cases}
\dt \bar{q}_\ep + K_\ep^{sp}[ (K_\ep^{te}\bar{v}) \cdot \nab (K_\ep^{sp} \bar{q}_\ep) ] + (K_\ep^{te} \bar{c}) \bar{q}_\ep = K_\ep^{te} \bar{f} &\text{in } \Gamma \times (0,\infty) \\
\bar{q}_\ep(\cdot,0) =\bar{q}_0 &\text{in } \Gamma.
\end{cases}$$ Notice that $K_\ep^{te}\bar{v}$, $K_\ep^{te}\bar{c}$ and $K_\ep^{te}\bar{f}$ belong to $C^\infty({\mathbb{R}^{}};H^{4N}(\Gamma))$. Then the theory of linear ODEs in Banach spaces provides us with a unique solution $\bar{q}_\ep \in C^\infty([0,\infty);H^{4N}(\Gamma))$ to .
Our next goal is to produce $\ep-$independent estimates to the solutions to . To this end we note that for $\alpha \in \mathbb{N}^{3}$ we may apply $\p^\alpha$ to to find that $\p^\alpha \bar{q}_\ep$ solves $$\label{xport_mollified_diff}
\begin{cases}
\dt \p^\alpha \bar{q}_\ep + K_\ep^{sp}[ (K_\ep^{te}\bar{v}) \cdot \nab (K_\ep^{sp} \p^\alpha \bar{q}_\ep) ] + (K_\ep^{te} \bar{c}) \p^\alpha\bar{q}_\ep = f^\alpha_\ep &\text{in } \Gamma \times (0,\infty) \\
\p^\alpha\bar{q}_\ep(\cdot,0) =\p^\alpha \bar{q}_0 &\text{in } \Gamma,
\end{cases}$$ where $$f^\alpha_\ep = \p^\alpha K_\ep^{te} \bar{f} - \sum_{0 < \beta \le \alpha }C_{\alpha, \beta} \left( K_\ep^{sp}[ (K_\ep^{te} \p^\beta \bar{v}) \cdot \nab (K_\ep^{sp} \p^{\alpha-\beta} \bar{q}_\ep) ] + (K_\ep^{te} \p^\beta \bar{c}) \p^{\alpha-\beta}\bar{q}_\ep \right).$$
### Estimates
By multiplying the first equation in by $\p^\alpha \bar{q}_\ep$ and integrating by parts over $\Gamma$ we may derive the basic energy identity $$\frac{d}{dt} \int_{\Gamma} \hal {\left\vert\p^\alpha \bar{q}_\ep\right\vert}^2 + \int_\Gamma (K_\ep^{te}\bar{c}) {\left\vert\p^\alpha \bar{q}_\ep\right\vert}^2- (K_\ep^{te} \operatorname{div}\bar{v} ){\left\vertK_\ep^{sp} \p^\alpha \bar{q}_\ep\right\vert}^2 = \int_\Gamma f_\ep^\alpha \p^\alpha \bar{q}_\ep.$$ We may use standard Sobolev embeddings and properties of mollifiers to estimate $$\int_\Gamma f_\ep^\alpha \p^\alpha \bar{q}_\ep \ls {\left\Vert\bar{f}\right\Vert}_{4N} {\left\Vert\bar{q}_\ep\right\Vert}_{4N} + \left( {\left\Vert\bar{v}\right\Vert}_{4N} + {\left\Vert\bar{c}\right\Vert}_{4N} \right) {\left\Vert\bar{q}_\ep\right\Vert}_{4N}^2.$$ Then $$\frac{d}{dt} {\left\Vert\bar{q}_\ep\right\Vert}_{4N}^2 \ls {\left\Vert\bar{f}\right\Vert}_{4N} {\left\Vert\bar{q}_\ep\right\Vert}_{4N} + \left( {\left\Vert\bar{v}\right\Vert}_{4N} + {\left\Vert\bar{c}\right\Vert}_{4N} \right) {\left\Vert\bar{q}_\ep\right\Vert}_{4N}^2,$$ which leads us to the fundamental estimate $$\label{xport_main_1}
\sup_{0\le t \le T} {\left\Vert\bar{q}_\ep(t)\right\Vert}_{4N} \ls \exp\left( C\int_0^T \left( {\left\Vert\bar{v}(t)\right\Vert}_{4N} + {\left\Vert\bar{c}(t)\right\Vert}_{4N} \right)dt \right) \left({\left\Vert\bar{q}_0\right\Vert}_{4N} + \int_0^T {\left\Vert\bar{f}(t)\right\Vert}_{4N} dt \right).$$ Here the constants on the right-hand side do not depend on $\ep$.
In order to record other estimates we recall the functionals $\Qf_e$, $\Rf_e$ given by and $\Qf_d$, $\Rf_d$ given by . Here the norms are understood to be computed over $\Gamma$. With the estimate in hand we may use the equation to directly estimate ${\left\Vert\dt \bar{q}_\ep\right\Vert}_{4N-1}^2$: $${\left\Vert\dt \bar{q}_\ep\right\Vert}_{4N-1}^2 \ls {{\left\Vert\bar{f}\right\Vert}^2}_{4N-1} + \left( {{\left\Vert\bar{v}\right\Vert}^2}_{4N-1} + {{\left\Vert\bar{c}\right\Vert}^2}_{4N-1} \right){{\left\Vert\bar{q}_\ep\right\Vert}^2}_{4N}.$$ We may iteratively apply $\dt$ to to estimate higher-order temporal derivatives; this leads us to the estimate $$\Qf_e[\bar{q}_\ep] \ls (1+ P(\Rf_e[\bar{v}] + \Rf_e[\bar{c}])) \Rf_e[\bar{f}] + P(\Rf_e[\bar{v}] + \Rf_e[\bar{c}]) {{\left\Vert\bar{q}_\ep\right\Vert}^2}_{4N}$$ for some universal positive polynomial $P$ with $P(0)=0$. Hence $$\begin{gathered}
\label{xport_main_2}
\sup_{0\le t\le T} \E[\bar{q}_\ep(t)] \ls \sup_{0\le t\le T} \left(1+ P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)])) \Rf_e[\bar{f}(t)] \right) \\
+ \left(\sup_{0\le t\le T} \left(1+P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \right) \\
\times \exp\left( C T \int_0^T \left( {{\left\Vert\bar{v}(t)\right\Vert}^2}_{4N} + {{\left\Vert\bar{c}(t)\right\Vert}^2}_{4N} \right)dt \right) \left({{\left\Vert\bar{q}_0\right\Vert}^2}_{4N} + T\int_0^T {{\left\Vert\bar{f}(t)\right\Vert}^2}_{4N} dt \right).\end{gathered}$$
Similarly, we may derive the bound $$\Qf_d[\bar{q}_\ep] \ls (1+ P(\Rf_e[\bar{v}] + \Rf_e[\bar{c}])) \Rf_d[\bar{f}] + P(\Rf_e[\bar{v}] + \Rf_e[\bar{c}]) \left( {{\left\Vert\bar{q}_\ep\right\Vert}^2}_{4N} + (\Rf_d[\bar{v}] + \Rf_d[\bar{c}])\E[\bar{q}_\ep] \right).$$ This leads us to the estimate $$\begin{gathered}
\int_0^T \D[\bar{q}_\ep(t)]dt \le \sup_{0\le t\le T} \left(1+ P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \int_0^T \Rf_d[\bar{f}(t)]dt \\
+ \sup_{0\le t\le T} \left( 1+ P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \int_0^T {{\left\Vert\bar{q}_\ep(t)\right\Vert}^2}_{4N} dt \\
+ \left(\sup_{0\le t\le T} P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \left( \sup_{0\le t \le T} \E[\bar{q}_\ep(t)] \right) \int_0^T \left( \Rf_d[\bar{v}(t)] + \Rf_d[\bar{c}(t)]\right) dt.\end{gathered}$$ Combining with and then leads to the bound $$\begin{gathered}
\label{xport_main_3}
\int_0^T \D[\bar{q}_\ep(t)]dt \le \sup_{0\le t\le T} \left(1+ P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \int_0^T \Rf_d[\bar{f}(t)]dt \\
+ \sup_{0\le t\le T} \left( P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \left( \int_0^T \left( \Rf_d[\bar{v}(t)] + \Rf_d[\bar{c}(t)]\right) dt \right) \sup_{0\le t\le T} \left( \Rf_e[\bar{f}(t)]dt \right) \\
+ \left[ T \sup_{0\le t\le T} \left(1+ P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \right. \\
\left. + \sup_{0\le t\le T} \left( P(\Rf_e[\bar{v}(t)] + \Rf_e[\bar{c}(t)]) \right) \int_0^T \left( \Rf_d[\bar{v}(t)] + \Rf_d[\bar{c}(t)]\right) dt \right]
\\
\times \exp\left( CT \int_0^T \left( {{\left\Vert\bar{v}(t)\right\Vert}^2}_{4N} + {{\left\Vert\bar{c}(t)\right\Vert}^2}_{4N} \right)dt \right) \left({{\left\Vert\bar{q}_0\right\Vert}^2}_{4N} + T\int_0^T {{\left\Vert\bar{f}(t)\right\Vert}^2}_{4N} dt \right).\end{gathered}$$
### Passing to the limit
The estimates and provide us with estimates of $$\sup_{0\le t \le T} \E[\bar{q}_\ep(t)] + \int_0^T \D[\bar{q}_\ep(t)]dt$$ that are independent of $\ep$. We may then extract a subsequence of $\ep$ values such that $\bar{q}_\ep$ converges to some $\bar{q}$, which by virtue of lower semicontinuity, must also obey the estimates and . Multiplying by $\varphi \in C_c^1(\Gamma \times [0,T))$ and integrating by parts leads to the identity $$\int_0^T \int_\Gamma -\bar{q}_\ep \left( \dt \varphi + K_\ep^{sp} \operatorname{div}((K_\ep^{te}\bar{v}) \varphi ) - (K_\ep^{te}\bar{c})\varphi\right) = \int_0^T \int_\Gamma \varphi K_\ep^{te}\bar{f} + \int_\Gamma \bar{q}_0 \varphi(\cdot,0).$$ We may then send $\ep \to 0$ to deduce that $\bar{q}$ is weak solution to in the sense of . This proves the following proposition.
\[xport\_ext\_exist\] There exists a weak solution $\bar{q}$ to obeying the estimates and .
Estimates for the solution to
------------------------------
We are now ready to return to the problem .
We begin with estimates of the $v$ and $c$ terms that arise in . They are simple variants of previous nonlinear estimates (for example Lemmas \[lame\_forcing\_est\] and \[k\_F\_ests\], Theorem \[kappa\_contract\], and Proposition \[kappa\_improved\]), so we omit the proof.
\[xport\_nonlins\] Let $v$ and $c$ be given on $\Omega$ as in and let $\bar{v}$, $\bar{c}$ denote their bounded extensions from $\Upsilon = \Omega_\pm$ to $\Gamma$. Then we have the following estimates, where $P$ is a universal positive polynomial with $P(0)=0$: $$\Rf_e[\bar{v}] + \Rf_e[\bar{c}] \le P( \E^0[\eta] ) \left( \E[u] + \hat{\E}^0[\eta] \right),$$ $$\label{xp_n_01}
\Rf_d[\bar{v}] + \Rf_d[\bar{c}] \le P( \hat{\E}^0[\eta] ) \left( \D[u] + \hat{\E}^0[\eta] + \hat{\D}^0[\eta] \right),$$ and $$\label{xp_n_02}
{{\left\Vert\bar{v}\right\Vert}^2}_{4N} + {{\left\Vert\bar{c}\right\Vert}^2}_{4N} \le P( \E[u] + \hat{\E}^0[\eta] ) \left( \D[u] + \hat{\D}^0[\eta] + {{\left\Vert\eta\right\Vert}^2}_{4N+1/2} \right).$$
The term $\hat{\E}^0[\eta]$ is added onto the right side of only because $\hat{\D}^0[\eta]$ provides no control of $\eta$ itself. The $\eta$ term in $\hat{\E}^0[\eta]$ is enough to make up for this deficit. The term ${{\left\Vert\eta\right\Vert}^2}_{4N+1/2}$ is added on the right side of for a similar reason, but in this case the regularity demands require more than $\hat{\E}^0[\eta]$.
Now we prove that the solution produced by the method of characteristics in obeys various useful estimates.
\[xport\_well-posed\] Suppose that $u$ and $\eta$ are given and satisfy $$\sup_{0 \le t \le T} \left( \E[u(t)] + \hat{\E}^0[\eta(t)] \right) + \int_0^T \left(\D[u(t)] + \hat{\D}^0[\eta(t)]\right)dt < \infty$$ and $$\label{xwp_03}
\dt \eta = u \cdot \n \text{ on } \Sigma, {\left\llbracket u \right\rrbracket }=0 \text{ on } \Sigma_-, \text{ and } u_- =0 \text{ on } \Sigma_b.$$ Let $q$ be given by . Then $q$ is the unique solution to . Moreover, the solution obeys the following estimates for some universal positive polynomial $P$ with $P(0)=0$: $$\begin{gathered}
\label{xwp_01}
\sup_{0\le t\le T} \E[q(t)] \ls \sup_{0\le t\le T} \left(1+ P(\E[u(t)] + \hat{\E}^0[\eta(t)]) \right) \sup_{0\le t\le T}\left( \Rf_e[f(t)] \right) \\
+ \left(\sup_{0\le t\le T} \left(1 + P(\E[u(t)] + \hat{\E}^0[\eta(t)]) \right) \right) \Xi(T)
\left({{\left\Vertq_0\right\Vert}^2}_{4N} + T\int_0^T {{\left\Vertf(t)\right\Vert}^2}_{4N} dt \right),\end{gathered}$$ and $$\begin{gathered}
\label{xwp_02}
\int_0^T \D[q(t)]dt \le \sup_{0\le t\le T} \left(1+ P(\E[u(t)] + \hat{\E}^0[\eta(t)]) \right) \int_0^T \Rf_d[f(t)]dt \\
+ \sup_{0\le t\le T} \left( P(\E[u(t)] + \hat{\E}^0[\eta(t)]) \right) \left( \int_0^T \left( \D[u(t)] + \hat{D}^0[\eta(t)] + \hat{\E}^0[\eta(t)] \right) dt \right) \sup_{0\le t\le T} \left( \Rf_e[f(t)]dt \right) \\
+ \Xi(T) \left({{\left\Vertq_0\right\Vert}^2}_{4N} + T\int_0^T {{\left\Vertf(t)\right\Vert}^2}_{4N} dt \right) \left[ T \sup_{0\le t\le T} \left(1+ P(\E[u(t)] + \hat{\E}^0[\eta(t)]) \right) \right. \\
\left. + \sup_{0\le t\le T} \left( P(\E[u(t)] + \hat{\E}^0[\eta(t)]) \right) \int_0^T \left( \D[u(t)] + \hat{D}^0[\eta(t)] +\hat{\E}^0[\eta(t)] \right) dt \right],\end{gathered}$$ where we have written $$\Xi(T) := \exp\left( C T \int_0^T P( \E[u(t)] + \hat{\E}^0[\eta(t)] ) \left( \D[u(t)] + \hat{\D}^0[\eta(t)] + {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \right) dt \right).$$
The equations in guarantee that the computations in and are valid, and so $v$ satisfies . First consider $\Upsilon = \Omega_+$. Proposition \[xport\_ext\_exist\] yields a weak solution $\bar{q}$ to on $\Gamma$ obeying the estimates and on all of $\Gamma$. We call $q$ the restriction of $\bar{q}$ to $\Upsilon$; Lemma \[xport\_transfer\] then guarantees that $q$ is the unique weak solution to on $\Upsilon$, but Proposition \[weak\_unique\] guarantees that $q$ coincides with the solution produced by characteristics in . The estimates and follow easily from and and Proposition \[xport\_nonlins\]. A similar argument works for $\Upsilon = \Omega_-$.
Some more useful estimates for
-------------------------------
To conclude our analysis of the transport problem we record another a priori estimate for solutions to that will be useful in the next section.
\[xport\_low\_reg\_est\] Suppose that $q$ is a solution to satisfying $$\sup_{0 \le t \le T} \E[q(t)] + \int_0^T \D[q(t)]dt < \infty.$$ Assume also that $$\gamma = \sup_{0 \le t \le T} \left( {\left\Vertc(t)\right\Vert}_{C^k} + {\left\Vertv(t)\right\Vert}_{C^k} \right) < \infty$$ for some $1 \le k \le 4N$. Then there exists a universal constant $C>0$ such that $$\label{xlre_01}
{\left\Vertq(t)\right\Vert}_k \le e^{C \gamma t} {\left\Vertq_0\right\Vert}_k + \int_0^t e^{C \gamma (t-s)} {\left\Vertf(s)\right\Vert}_k ds$$ for $t \in [0,T]$. In particular, if $q_0 =0$ then $$\label{xlre_02}
\sup_{0\le t \le T} {{\left\Vertq(t)\right\Vert}^2}_k \le T e^{2 C \gamma T} \int_0^T {{\left\Vertf(s)\right\Vert}^2}_k ds.$$
Let $\alpha \in \mathbb{N}^3$ with ${\left\vert\alpha \right\vert} \le k$. Applying $\p^\alpha$ to leads to the equation $$\begin{cases}
\dt \p^\alpha q + v \cdot \nab \p^\alpha q + c \p^\alpha q = \p^\alpha f - f^\alpha, \\
\p^\alpha q(t=0) = \p^\alpha q_0
\end{cases}$$ where $$f^\alpha = \sum_{\beta < \alpha} C_{\alpha,\beta} \left( \p^{\alpha- \beta} v \cdot \nab \p^\beta q + \p^{\alpha- \beta} c \p^\beta q \right).$$ Because of the condition we may multiply by $\p^\alpha q$ and integrate to deduce the standard energy identity $$\frac{d}{dt} \frac{{{\left\Vert\p^\alpha q\right\Vert}^2}_0 }{2} = \int_\Upsilon (\operatorname{div}v - 2c) \frac{{\left\vert\p^\alpha q\right\vert}}{2} + \int_\Upsilon (\p^\alpha f - f^\alpha)\p^\alpha q.$$ From this and the structure of $f^\alpha$ we may easily deduce the estimate $$\frac{d}{dt} \frac{{{\left\Vert\p^\alpha q\right\Vert}^2}_0 }{2} \le \left( {\left\Vertc\right\Vert}_{C^0} + \hal {\left\Vertv\right\Vert}_{C^1} \right) {{\left\Vert\p^\alpha q\right\Vert}^2}_0 + C\left( {\left\Vertc\right\Vert}_{C^k} + {\left\Vertv\right\Vert}_{C^k}\right) {\left\Vertq\right\Vert}_k {\left\Vert\p^\alpha q\right\Vert}_0 + \int_\Upsilon \p^\alpha f \p^\alpha q$$ for some universal $C>0$. Summing this inequality over all ${\left\vert\alpha\right\vert} \le k$, we find (since $k \ge 1$) that $$\label{xlre_1}
\frac{d}{dt} \frac{ {{\left\Vertq\right\Vert}^2}_{k} }{2} \le C \gamma {{\left\Vertq\right\Vert}^2}_k + {\left\Vertf\right\Vert}_k {\left\Vertq\right\Vert}_k,$$ where $C>0$ is another universal constant. The differential inequality and a standard Gronwall argument then imply . The bound follows from and the Cauchy-Schwarz inequality: $$\begin{gathered}
\left(\int_0^t e^{C \gamma (t-s)} {\left\Vertf(s)\right\Vert}_k ds \right)^2 \le e^{2C \gamma t} \left(\int_0^t {{\left\Vertf(s)\right\Vert}^2}_k ds\right) \left( \int_0^t e^{-2C \gamma s} ds \right) \\
\le t e^{2C \gamma t} \int_0^t {{\left\Vertf(s)\right\Vert}^2}_k ds \le T e^{2C \gamma T} \int_0^T {{\left\Vertf(s)\right\Vert}^2}_k ds.\end{gathered}$$
Local well-posedness of with $\sigma_\pm >0$ {#sec_lwp_st}
============================================
Data: construction and estimates
--------------------------------
Our goal now is to deal with the initial data. We assume initially that $$u_0 \in H^{4N}, \eta_0 \in H^{4N+1/2}, \sqrt{\sigma} \nab_\ast \eta_0 \in H^{4N}, \text{ and } q_0 \in H^{4N}.$$ We may then construct the data for the higher temporal derivatives as in Appendix \[app\_ccs\]. This process leads us to the following estimate.
\[data\_estimate\] Suppose that $\Lf[\eta_0] \le \delta_1/2$, where $\Lf[\eta_0]$ is given by and $\delta_1>0$ is from Theorem \[kappa\_apriori\]. Then $$\sum_{j=1}^{2N} {{\left\Vert\dt^j u(0)\right\Vert}^2}_{4N-2j} + {{\left\Vert\dt^j q(0)\right\Vert}^2}_{4N-2j+1}+ {{\left\Vert\dt^j \eta(0)\right\Vert}^2}_{4N-2j+3/2} \ls {{\left\Vertu_0\right\Vert}^2}_{4N} + {{\left\Vert\eta_0\right\Vert}^2}_{4N+1/2} + {{\left\Vertq_0\right\Vert}^2}_{4N}.$$ Consequently, $$\TE \ls \EEE \le \TE.$$
The proof is essentially the same as that of Proposition 5.3 of [@GT_lwp] and is thus omitted.
Approximate solutions
---------------------
Suppose that $u, q, \eta$ are given. We then define the forcing terms $F^1$ on $\Omega$ and $F^2_\pm$ on $\Sigma_\pm$ according to $$\label{forcing_def_1}
F^1[u,q,\eta] = -(\bar{\rho} + q + \p_3 \bar{\rho} \theta)(-K \dt \theta \p_3 u + u \cdot \naba u)
- \bar{\rho} \naba (h'(\bar{\rho}) q) - \naba \mathcal{R}
- g(q + \p_3 \bar{\rho} \theta) \naba \theta,$$ $$\label{forcing_def_2}
F^{2}_+[q,\eta] = -P_+'(\bar{\rho}_+) q_+ \n_++g\eta_+ \n_+ - \mathcal{R}_+ \n_+ - \sigma_+ \operatorname{div}_\ast\left((1+ {\left\vert\nab_\ast \eta_+\right\vert}^2)^{-1/2} -1) \nab_\ast \eta_+ \right) \n_+,$$ and $$\label{forcing_def_3}
-F^2_-[q,\eta] = -{\left\llbracket P'(\bar{\rho}) q \right\rrbracket }\n_- +\rj g\eta_-\n_- - {\left\llbracket \mathcal{R} \right\rrbracket } \n_- + \sigma_- \operatorname{div}_\ast\left((1+ {\left\vert\nab_\ast \eta_-\right\vert}^2)^{-1/2} -1) \nab_\ast \eta_- \right) \n_-.$$ Here the function $\mathcal{R}$ is determined by . We also define the forcing term $f$ on $\Omega$ via $$\label{forcing_def_4}
f[u,\eta] = -\diva(\bar{\rho} u) + K \dt \theta \p_3^2 \bar{\rho} \theta - \diva( \p_3 \bar{\rho} \theta u).$$ Finally we define the density function $$\label{forcing_def_5}
\rho[q,\eta] = \bar{\rho} +q + \p_3 \bar{\rho} \theta .$$
We define the following functionals for use in the forcing estimates: $$\begin{gathered}
\mathfrak{Y}_\infty[u] := \sum_{j=0}^{2N-1} {{\left\Vert\dt^j u\right\Vert}^2}_{4N-2j-1}, \mathfrak{Y}_\infty[q] := \sum_{j=0}^{2N-1} {{\left\Vert\dt^j q\right\Vert}^2}_{4N-2j-1}, \\
\mathfrak{Y}_\infty[\eta] := {{\left\Vert\eta\right\Vert}^2}_{4N-1/2} + \sum_{j=1}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{4N-2j+1/2}\end{gathered}$$ and $$\begin{gathered}
\mathfrak{Y}_2[u] := \sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{4N-2j}, \mathfrak{Y}_2[q] := \sum_{j=0}^{2N} {{\left\Vert\dt^j q\right\Vert}^2}_{4N-2j}, \\
\mathfrak{Y}_2[\eta] := {{\left\Vert\eta \right\Vert}^2}_{4N} + \sum_{j=1}^{2N+1} {{\left\Vert\dt^j \eta\right\Vert}^2}_{4N-2j+3/2}.\end{gathered}$$ For the sake of brevity we will write multiple arguments within brackets to indicate sums; for example, $\mathfrak{Y}_\infty[u,q,\eta] = \mathfrak{Y}_\infty[u] + \mathfrak{Y}_\infty[q]+ \mathfrak{Y}_\infty[\eta]$.
Our next two results contain the estimates of the forcing terms used in the problems and \[xport\_fundamental\]. The proofs are again standard nonlinear estimates and are thus omitted.
\[force\_est\_1\] Let $F^1$ and $F^2 $ be determined by $(u,q,\eta)$ as in –. Let $\f_2$ and $\f_\infty$ be given by this choice of $F^1, F^2$ as in and . Then we have the estimates $$\f_\infty \ls \mathfrak{Y}_\infty[q,\eta]( 1 + P(\mathfrak{Y}_\infty[\eta] )) + P(\mathfrak{Y}_\infty[u,q,\eta])\mathfrak{Y}_\infty[u,q,\eta] + \sigma^2 {{\left\Vert\nab_\ast \eta\right\Vert}^2}_{4N-1/2} P(\mathfrak{Y}_\infty[\eta] )$$ and $$\f_2 \ls \mathfrak{Y}_2[q,\eta]( 1 + P(\mathfrak{Y}_\infty[\eta] )) + P(\mathfrak{Y}_\infty[u,q,\eta]) \mathfrak{Y}_2[u,q,\eta] + \sigma^2 {{\left\Vert\nab_\ast \eta\right\Vert}^2}_{4N+1/2} P(\mathfrak{Y}_\infty[\eta] )$$ for some universal positive polynomial $P$ such that $P(0)=0$.
\[force\_est\_2\] Let $f$ be determined by $(u,\eta)$ as in . Let $\mathfrak{R}_e$ be defined by . Then we have the estimates $$\mathfrak{R}_e[f] \le \mathfrak{Y}_\infty[u] ( 1 + P(\mathfrak{Y}_\infty[\eta] )) + P(\mathfrak{Y}_\infty[\eta] )\mathfrak{Y}_\infty[\eta]$$ and $$\mathfrak{R}_d[f] + {{\left\Vertf\right\Vert}^2}_{4N} \le \D[u] ( 1 + P(\hat{\E}^0[\eta] )) + P(\hat{\E}^0[\eta] )\left({{\left\Vert\eta\right\Vert}^2}_{4N+1/2} + \hat{\D}^0[\eta]\right)$$ for some universal positive polynomial $P$ such that $P(0)=0$.
The following proposition allows us to estimate $\mathfrak{Y}_\infty[u,q,\eta]$ in terms of $\E$ and $\D$ with the benefit of introducing time factors.
\[Y\_est\] We have the following estimates: $$\sup_{0 \le t \le T} \mathfrak{Y}_\infty[u(t)] \ls \TE + T \int_0^T \D[u(t)]dt,$$ $$\sup_{0 \le t \le T} \mathfrak{Y}_\infty[q(t)] \ls \TE + T^2 \sup_{0 \le t \le T} \E[q(t)],$$ and $$\sup_{0 \le t \le T} \mathfrak{Y}_\infty[\eta(t)] \ls \TE + T \int_0^T \hat{\D}^0[\eta(t)]dt.$$
The estimates follow easily from the fundamental theorem of calculus and the Cauchy-Schwarz inequality.
Now we present the construction of a sequence of approximate solutions.
\[approx\_solns\] Suppose that $(u_0,q_0,\eta_0)$ satisfy the compatibility conditions as well as the bound . Further assume that $$\Lf[\eta_0] \le \frac{\delta_1}{2} ,$$ where $\Lf[\eta_0]$ is given by and $\delta_1$ is given by Theorem \[kappa\_apriori\]. Then there exists a $T_4 = T_4({\EEE})$ such that if $0 < T \le T_4$ then there exists a sequence $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ defined on the temporal interval $[0,T]$ satisfying the following three properties. First, $(u^n,q^n,\eta^n)$ achieve the initial data at $t=0$. Second, for $n \ge 1$ we have that $$\label{aps_01}
\begin{cases}
\rho^{n-1} \dt u^n - \operatorname{div}_{\a^{n}} \mathbb{S}_{\a^{n}} u^n = F^1[u^{n-1},q^{n-1},\eta^{n-1}] & \text{in }\Omega\\
\dt \eta^{n} = u^{n} \cdot \n^{n} &\text{on }\Sigma \\
-\mathbb{S}_{\a^{n}} u^n \n^{n} = - \sigma_+ \Delta_\ast \eta^{n} \n^{n} + F^2_+[q^{n-1},\eta^{n-1}] &\text{on } \Sigma_+ \\
-{\left\llbracket \mathbb{S}_{\a^{n}} u^{n} \right\rrbracket } \n^{n} = \sigma_- \Delta_\ast \eta^{n} \n^{n} - F^2_-[q^{n-1},\eta^{n-1}] &\text{on } \Sigma_- \\
{\left\llbracket u^{n} \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_-^{n} = 0 &\text{on } \Sigma_b \\
u^{n}(\cdot,0) = u_0, \eta^{n}(\cdot, 0) = \eta_0,
\end{cases}$$ and $$\label{aps_02}
\begin{cases}
\dt q^{n} - K^{n} \dt \theta^{n} \p_3 q^{n} + \operatorname{div}_{\a^{n}}(q^{n}u^{n})
= f[u^{n},\eta^{n}] & \text{in } \Omega \times (0,T) \\
q^{n}(\cdot, 0) = q_0 &\text{in }\Omega,
\end{cases}$$ where $F^1$, $F^2$, and $f$ are defined by – and $\rho^{n-1} = \rho[q^{n-1},\eta^{n-1}]$ is given by . Third, we have the estimates $$\begin{gathered}
\label{aps_03}
\sup_{0\le t \le T} \left( \E[u^{n}(t)] + \hat{\E}^\sigma[\eta^{n}(t)] \right) \\
+ \int_0^T \left( \D[u^{n}(t)] + {{\left\Vert\rho^{n-1} J^{n} \dt^{2N+1} u^{n}(t)\right\Vert}^2}_{ \Hd} + \hat{\D}^\sigma[\eta^{n}(t)] \right) dt \le P_1( {\EEE}),\end{gathered}$$ $$\label{aps_04}
\sup_{0\le t \le T} {{\left\Vert\eta^{n}(t)\right\Vert}^2}_{4N+1/2} \le P_2( {\EEE}),$$ $$\label{aps_05}
\sup_{0\le t \le T} \E[q^{n}(t)] + \int_0^T \D[q^{n}(t)] dt \le P_3( {\EEE}) ,$$ $$\label{aps_06}
\Lf[\eta^n](T) \le \delta_1 \text{ and }\frac{1}{2}\rho_\ast\le \rho^n=\rho[q^{n},\eta^{n}]\le \frac{3}{2}\rho^\ast$$ for all $n \ge 1$, where $P_i$ for $i=1,2,3$ is a universal positive polynomial with $P_i(0)=0$.
We divide the proof into three steps.
Step 1 - Seeding the sequence
To begin, we extend the initial data to a triple that belongs to the function spaces necessary for the construction of solutions. We combine the data estimates of Proposition \[data\_estimate\] with the extension results of Propositions \[extension\_u\], \[extension\_eta\], and \[extension\_q\] in order to produce a triple $(u^0,q^0,\eta^0)$ defined on the temporal interval $[0,\infty)$, achieving the initial data, and satisfying the estimates $$\label{aps_1}
\sup_{t \ge 0} \E[u^0(t)] + \int_0^\infty \D[u^0(t)] dt \le P_0(\EEE),$$ $$\sup_{t \ge 0} \E[q^0(t)] + \int_0^\infty \D[q^0(t)] dt \le P_0(\EEE),$$ and $$\label{aps_2}
\sup_{t \ge 0} \hat{\E}^\sigma[\eta^0(t)] + \sup_{t \ge 0} {{\left\Vert\eta^0(t)\right\Vert}^2}_{4N+1/2} + \int_0^\infty \hat{\D}^\sigma[\eta^0(t)]dt \le P_0(\EEE)$$ for some universal polynomial $P_0>0$ with $P_0(0)=0$. Step 2 - The iteration procedure
We claim that there exist universal positive polynomials $P_i$ for $i=1,2,3$ such that $P_i(0)=0$, and $\alpha >0$ (depending on ${\EEE}$) with the following two properties. First, $$\label{aps_3}
\min\{ P_1(z),P_2(z), P_3(z)\} \ge P_0(z) \text{ for all } z \ge 0.$$ That is, each of the coefficients of $P_i(z)$, $i=1,2,3$, is bounded below by the corresponding coefficient of $P_0(Z)$. Second, if $T \le \min\{\alpha,T_3\}$ (where $T_3= T_3(\TE)>0$ is given by Theorem \[lame\_exist\]), and the triple $(u^{n-1},q^{n-1},\eta^{n-1})$ is given, achieves the initial data, and obeys the estimates $$\label{aps_4}
\sup_{0\le t \le T} \left( \E[u^{n-1}(t)] + \hat{\E}^\sigma[\eta^{n-1}(t)] \right) + \int_0^T \left( \D[u^{n-1}(t)] + \hat{\D}^\sigma[\eta^{n-1}(t)] \right) dt \le P_1( {\EEE}),$$ $$\label{aps_5}
\sup_{0\le t \le T} {{\left\Vert\eta^{n-1}(t)\right\Vert}^2}_{4N+1/2} \le P_2( {\EEE}),$$ and $$\label{aps_6}
\sup_{0\le t \le T} \E[q^{n-1}(t)] \le P_3({\EEE}),$$ then there exists a triple $(u^{n},q^{n},\eta^{n})$ that solves and , achieves the initial data, and obeys the estimates $$\begin{gathered}
\label{aps_7}
\sup_{0\le t \le T} \left( \E[u^{n}(t)] + \hat{\E}^\sigma[\eta^{n}(t)] \right) \\
+ \int_0^T \left( \D[u^{n}(t)] + {{\left\Vert\rho^{n-1} J^{n} \dt^{2N+1} u^{n}(t)\right\Vert}^2}_{ \Hd} + \hat{\D}^\sigma[\eta^{n}(t)] \right) dt \le P_1( {\EEE}),\end{gathered}$$ $$\label{aps_8}
\sup_{0\le t \le T} {{\left\Vert\eta^{n}(t)\right\Vert}^2}_{4N+1/2} \le P_2( {\EEE}),$$ and $$\label{aps_9}
\sup_{0\le t \le T} \E[q^{n}(t)] + \int_0^T \D[q^{n}(t)] dt \le P_3( {\EEE}).$$
The proof of the claim is very similar to Step 2 of Theorem 6.1 in [@GT_lwp], so we will only provide a sketch of the idea. It suffices to show that if – hold for some choice of $P_i$, $i=1,2,3$, satisfying , then $(u^n,q^n,\eta^n)$ can be constructed and must satisfy – so long as the constants and degree of the $P_i$ are sufficiently large and $T \le \alpha$ for some small $\alpha$.
The first step is to use Theorem \[lame\_exist\] to produce a $(u^n,\eta^n)$ solving . For this we must verify that the hypotheses $\mathfrak{P}(\delta_1)$ are satisfied. The hypotheses and are satisfied by assumption. The hypothesis follows by combining the estimates of Propositions \[data\_estimate\], \[force\_est\_1\], and \[Y\_est\] with the bounds –. The hypothesis requires that $\rho^{n-1} = \rho[q^{n-1},\eta^{n-1}]$ satisfies and . The condition follows trivially from and . To verify we first estimate $$\begin{gathered}
\label{aps_10}
\sup_{0\le t \le T}{\left\Vert\rho^{n-1}(t)-\rho_0 \right\Vert}_{L^\infty} \le \int_0^T {\left\Vert\dt \rho^{n-1}(t)\right\Vert}_{L^\infty} dt
\ls \int_0^T \left({\left\Vert\dt q^{n-1}(t)\right\Vert}_{L^\infty}+{\left\Vert\dt \eta^{n-1}(t)\right\Vert}_{L^\infty}\right) dt \\
\le T \sup_{0 \le t \le T}\sqrt{\E[q^{n-1}(t)]+ \hat{\E}^0[\eta^{n-1}(t)]}
\le T \sqrt{P_1({\EEE})+P_3({\EEE})} \le \alpha\sqrt{P_1({\EEE})+P_3({\EEE})}.\end{gathered}$$ Then we find that $$\label{aps_12}
\sup_{0\le t \le T}{\left\Vert\rho^{n-1}(t)-\rho_0 \right\Vert}_{L^\infty} \le \frac{\rho_\ast}{2}$$ if $\alpha$ is chosen sufficiently small with respect to the $P_1$, $P_3$, ${\EEE}$ and $\rho_\ast$. Hence is satisfied by the assumption on $\rho_0$ in .
We may thus apply Theorem \[lame\_exist\] to produce the solution pair $(u^n,\eta^n)$ solving . To derive the estimate we sum and and again employ Propositions \[data\_estimate\], \[force\_est\_1\], and \[Y\_est\] and – to estimate the forcing terms. The actual derivation of is tedious and will be omitted, but we will point out the key observation. The estimates of Proposition \[Y\_est\] guarantee that any appearance of $P_i$ in the resulting estimates is multiplied by at least one factor of $T$ and so by choosing $\alpha$ small enough (in particular a bound like $\alpha \le \alpha_0(1+{\EEE})^{-m}$ for $\alpha_0$ small and $m$ large is needed to reduce the degrees of various polynomials appearing in the estimates) and the constants and degrees of $P_1$ large enough, we can show that holds. The estimate follows from via a similar argument.
The second step is to use the newly-constructed pair $(u^n,\eta^n)$ to construct $q^n$, the solution to , through an application of Theorem \[xport\_well-posed\]. The hypotheses of the theorem are satisfied due to , , and . The estimates and then lead to the estimate by employing Propositions \[force\_est\_2\] and \[Y\_est\] and arguing as above, except that we use and since the forcing terms are generated by $(u^n,\eta^n)$.
Step 3 - Constructing the sequence
To conclude the proof we combine the previous two steps as follows. We set $(u^0,q^0,\eta^0)$ to be the triple constructed in Step 1. The bounds – imply – with $n=1$ due to . We then set $T_3 = \min\{T_2,\alpha\}$ and use Step 2 to construct $(u^1,q^1,\eta^1)$ satisfying – and solving and . The bounds allow us to iteratively apply Step 2 to produce $(u^n,q^n,\eta^n)$ for $n \ge 2$. This produces the sequence $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ satisfying – for $n \ge 1$. It remains only to prove . The estimates of $ \rho^n=\rho[q^{n},\eta^{n}]$ can be derived exactly as in –. The estimate of $\Lf[\eta^n](T)$ can be derived similarly: $$\begin{gathered}
\Lf[\eta^n](T) \le \frac{3}{2} \Lf[\eta_0] + 3 T \int_0^T {{\left\Vert\dt \eta^n(t)\right\Vert}^2}_{4N-1/2} dt \le \frac{3 \delta_1}{4} + 3T^2 \sup_{0 \le t \le T} \hat{\E}^0[\eta^{n}(t)] \\
\le \frac{3 \delta_1 }{4} + 3T^2 P_1( {\EEE}) \le \frac{3\delta_1}{4} + 3\alpha^2 P_1( {\EEE}) \le \delta_1\end{gathered}$$ if $\alpha$ is further restricted.
Contraction
-----------
We wish to ultimately show that the sequence $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ contracts in some lower-order regularity space than that given by –. Our goal now is to prove such a contraction result. We will prove the result in a somewhat more general context than within the sequence $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ in order for the result to be applicable in proving uniqueness of solutions to .
Before stating the result we define the low-regularity norms in which contraction occurs. We define $$\label{w_def_start}
\Wf_\infty[u] = {{\left\Vertu\right\Vert}^2}_{2} + {{\left\Vert\dt u\right\Vert}^2}_{0}
\text{ and }
\Wf_2[u] = {{\left\Vertu\right\Vert}^2}_{3} + {{\left\Vert\dt u\right\Vert}^2}_{1},$$ $$\Wf_\infty[\eta] = {{\left\Vert\eta\right\Vert}^2}_{5/2} + {{\left\Vert\dt \eta\right\Vert}^2}_{3/2} + \sigma {{\left\Vert\nab_\ast \eta\right\Vert}^2}_{2} + \sigma {{\left\Vert\nab_\ast \dt \eta\right\Vert}^2}_{0} \text{ and }\Wf_2[\eta] = \sigma^2 {{\left\Vert\eta\right\Vert}^2}_{7/2} + {{\left\Vert\dt^2 \eta\right\Vert}^2}_{1/2},$$ and $$\label{w_def_end}
\Wf_\infty [q] = {{\left\Vertq\right\Vert}^2}_{2} + {{\left\Vert\dt q\right\Vert}^2}_{1}.$$
Now we state our contraction result.
\[contraction\_thm\]
Suppose that the triples $(u^i,q^i,\eta^i)$ and $(v^i,p^i,\zeta^i)$ for $i=1,2$ satisfy $$\label{cot_01}
\begin{cases}
\rho^{i} \dt u^i - \operatorname{div}_{\a^{i}} \mathbb{S}_{\a^{i}} u^i = F^1[v^{i},p^{i},\zeta^{i}] & \text{in }\Omega \\
\dt \eta^{i} = u^{i} \cdot \n^{i} &\text{on }\Sigma \\
-\mathbb{S}_{\a^{i}} u^i \n^{i} = - \sigma_+ \Delta_\ast \eta^{i} \n^{i} + F^2_+[p^{i},\zeta^{i}] &\text{on } \Sigma_+ \\
-{\left\llbracket \mathbb{S}_{\a^{i}} u^{i} \right\rrbracket } \n^{i} = \sigma_- \Delta_\ast \eta^{i} \n^{i} - F^2_-[p^{i},\zeta^{i}] &\text{on } \Sigma_- \\
{\left\llbracket u^{i} \right\rrbracket } =0 &\text{on } \Sigma_- \\
u_-^{i} = 0 &\text{on } \Sigma_b \\
u^{i}(\cdot,0) = u_0, \eta^{i}(\cdot, 0) = \eta_0,
\end{cases}$$ and $$\label{cot_02}
\begin{cases}
\dt q^{i} - K^{i} \dt \theta^{i} \p_3 q^{i} + \operatorname{div}_{\a^{i}}(q^{i}u^{i})
= f[u^{i},\eta^{i}] & \text{in } \Omega \times (0,T) \\
q^{i}(\cdot, 0) = q_0 &\text{in }\Omega,
\end{cases}$$ where $\a^i,$ $\n^i,$ $\theta^i$, and $K^i$ are given by $\eta^i$, and $F^1$, $F^2_\pm$, and $f$ are defined by – and $\rho^{i} = \rho[p^{i},\zeta^{i}]$ is given by . Further suppose that $$\max\left\{ \sup_{i} \sup_{0 \le t \le T} \E[u^i,q^i,\eta^i] , \sup_{i} \sup_{0 \le t \le T} \E[v^i,p^i,\zeta^i] \right\} \le M^2,$$ and that $$\label{cot_04}
\Lf[\eta^1](T) \le \delta_1 \text{ and }\frac{1}{2}\rho_\ast\le \rho^1=\rho[p^{1},\zeta^{1}]\le \frac{3}{2}\rho^\ast,$$ where $\Lf$ is given by and $\delta_1$ is given by Theorem \[kappa\_apriori\].
There exist universal constants $\gamma >0$ and a constant $T_5 = T_5(M) \in (0,1)$ such that if $0 < T \le T_5$ and $$\label{cot_05}
\max\left\{ \sup_{i} \sup_{0 \le t \le T} {{\left\Vert\eta^i(t)\right\Vert}^2}_7, \sup_{i} \sup_{0 \le t \le T} {{\left\Vert\zeta^i(t)\right\Vert}^2}_7 \right\} \le \gamma^2,$$ then $$\begin{gathered}
\label{cot_03}
\sup_{0\le t \le T} \Wf_\infty[u^1(t) - u^2(t),q^1(t) - q^2(t),\eta^1(t) - \eta^2(t)] + \int_0^T \Wf_2[u^1(t) - u^2(t),\eta^1(t) - \eta^2(t)] dt \\
\le \hal \sup_{0\le t \le T} \Wf_\infty[v^1(t) - v^2(t),p^1(t) - p^2(t) ,\zeta^1(t) - \zeta^2(t)] + \hal \int_0^T \Wf_2[v^1(t) - v^2(t),\zeta^1(t)-\zeta^2(t)] dt .\end{gathered}$$
We divide the proof into several steps.
Step 1 – Differences
To begin we define $u= u^1 - u^2$, $q =q^1 - q^2$, $\eta = \eta^1 - \eta^2$, $v = v^1 - v^2$, $p = p^1 - p^2$, and $\zeta = \zeta^1 - \zeta^2$. Then we subtract the equations with $i=2$ from the same equations with $i=1$ to deduce equations for $(u,\eta)$. We then apply $\pal$ for $\alpha \in \mathbb{N}^{1+2}$ with ${\left\vert\alpha\right\vert}\le 2$. This results in the equations $$\label{cot_1}
\begin{cases}
\rho^{1} \dt \pal u - \operatorname{div}_{\a^{1}} \mathbb{S}_{\a^{1}} \pal u = \operatorname{div}_{\a^{1}} \mathbb{S}_{\pal (\a^{1}-\a^{2})} u^2 + \pal H^1 + H^{1,\alpha} & \text{in }\Omega \\
\dt \pal \eta + u^2 \cdot \nab_\ast \pal \eta = \pal u \cdot \n^{1} + H^{3,\alpha} &\text{on }\Sigma\\
-\mathbb{S}_{\a^{1}} \pal u \n^{1} = - \sigma_+ \Delta_\ast \pal \eta \n^{1} + \mathbb{S}_{\pal (\a^{1}-\a^{2})} u^2 \n^1 + \pal H^2_+ + H^{2,\alpha}_+ &\text{on } \Sigma_+ \\
-{\left\llbracket \mathbb{S}_{\a^{1}} \pal u \right\rrbracket } \n^{1} = \sigma_- \Delta_\ast \pal \eta \n^{1} + {\left\llbracket \mathbb{S}_{\pal (\a^{1}-\a^{2})} u^2 \right\rrbracket } \n^1 - \pal H^2_- - H^{2,\alpha}_- &\text{on } \Sigma_- \\
{\left\llbracket \pal u \right\rrbracket } =0 &\text{on } \Sigma_- \\
\pal u_- = 0 &\text{on } \Sigma_b \\
\pal u(\cdot,0) = 0, \pal \eta(\cdot, 0) = 0.
\end{cases}$$ A similar argument with but not employing derivatives yields an equation for $q$: $$\label{cot_2}
\begin{cases}
\dt q - K^{1} \dt \theta^{1} \p_3 q + \operatorname{div}_{\a^{1}}(q u^{1})
= H^4 & \text{in } \Omega \times (0,T) \\
q(\cdot, 0) =0 &\text{in }\Omega.
\end{cases}$$ Here we have written the forcing terms as follows: $$\label{cot_3}
\begin{split}
H^1 & := F^1[v^1,p^1,\zeta^1] - F^1[v^2,p^2,\zeta^2] - (p + \p_3 \bar{\rho} \theta[\zeta]) \dt u^2 + G^1 \\
H^2_\pm & := F^2_\pm[p^1,\zeta^1] - F^2_\pm[p^2,\zeta^2] + G^2_\pm \\
H^4 &:= f[u^1,\eta^1] - f[u^2,\eta^2] + G^4,
\end{split}$$ where $\theta[\zeta]$ is determined by $\zeta$, $$\label{cot_4}
\begin{split}
G^1 &:= \operatorname{div}_{(\a^{1}-\a^{2})} \mathbb{S}_{\a^{2}} u^2 \\
G^2_+ &:= - \sigma_+ \Delta_\ast \eta^2 (\n^1 - \n^2) + \mathbb{S}_{\a^{2}} u^2 (\n^1 - \n^2) \\
G^2_- &:= - \sigma_- \Delta_\ast \eta^2 (\n^1 - \n^2) + {\left\llbracket \mathbb{S}_{\a^{2}} u^2 \right\rrbracket } (\n^1 - \n^2) \\
G^4 &:= ((K^1-K^2) \dt \theta^1 + K^2 (\dt \theta^1 -\dt \theta^2))\p_3 q^2 - \operatorname{div}_{(\a^1-\a^{2})}(q^2 u^1) - \operatorname{div}_{\a^{2}}(q^2 u),
\end{split}$$ and $$\begin{split}
H^{1,\alpha} & := \left(\pal(\operatorname{div}_{\a^{1}} \mathbb{S}_{\a^{1}} u) - \operatorname{div}_{\a^{1}} \mathbb{S}_{\a^{1}} \pal u \right) + \left( \pal (\operatorname{div}_{\a^{1}} \mathbb{S}_{(\a^{1}-\a^{2})} u^2) - \operatorname{div}_{\a^{1}} \mathbb{S}_{\pal (\a^{1}-\a^{2})} u^2 \right)\\
H^{2,\alpha}_+ & := \left( \pal(\mathbb{S}_{\a^{1}} u \n^{1}) -\mathbb{S}_{\a^{1}} \pal u \n^{1} \right) + \left( \pal ( \mathbb{S}_{ (\a^{1}-\a^{2})} u^2 \n^1)- \mathbb{S}_{\pal (\a^{1}-\a^{2})} u^2 \n^1 \right) \\
& \quad - \sigma_+\left( \pal [ \Delta_\ast \eta \n^{1}] - \Delta_\ast \pal \eta \n^{1} \right)\\
H^{2,\alpha}_- & := \left( -\pal{\left\llbracket \mathbb{S}_{\a^{1}} u \n^{1} \right\rrbracket } +{\left\llbracket \mathbb{S}_{\a^{1}} \pal u \right\rrbracket } \n^{1} \right) + \left( -\pal {\left\llbracket \mathbb{S}_{ (\a^{1}-\a^{2})} u^2 \n^1 \right\rrbracket }+ {\left\llbracket \mathbb{S}_{\pal (\a^{1}-\a^{2})} u^2 \right\rrbracket } \n^1 \right) \\
& \quad - \sigma_-\left( \pal [ \Delta_\ast \eta ) \n^{1}] -\Delta_\ast \pal \eta \n^{1} \right)\\
H^{3,\alpha} & := \left( - \pal (u^2 \cdot \nab_\ast \eta) + u^2 \cdot \nab_\ast \pal \eta \right) +
\left( \pal (u \cdot \n^1) - \pal u \cdot \n^1 \right). \\
\end{split}$$ We have written the forcing terms in this manner in order to isolate those terms depending on $(v^i,p^i,\zeta^i)$ from those depending on $(u^i,q^i,\eta^i)$ and to single out some special delicate terms.
Step 2 – Energy estimate
The starting point for the contraction analysis is a basic energy estimate for . Arguing as in Lemma \[kappa\_en\_ident\] leads us to the equality $$\begin{gathered}
\label{cot_5}
\frac{d}{dt} \left( \int_\Omega \rho^1J^1 \frac{{\left\vert\pal u\right\vert}^2}{2} + \int_{\Sigma} \frac{{\left\vert\pal \eta\right\vert}^2}{2} + \sigma \frac{{\left\vert\pal\nab_\ast \eta\right\vert}^2}{2} \right)
+ \int_\Omega \frac{\mu J^1}{2} {\left\vert\sgz_{\a^1} \pal u\right\vert}^2 + J^1 \mu' {\left\vert\operatorname{div}_{\a^1} \pal u\right\vert}^2
\\
= \int_\Omega J^1 \pal u \cdot ( \pal H^1 + H^{1,\alpha}) - \int_\Sigma \pal u \cdot ( \pal H^2 + H^{2,\alpha}) \\
+ \int_{\Sigma} (- \pal \eta + \sigma \Delta_\ast \pal \eta) (u^2 \cdot \nab_\ast \pal \eta- H^{3,\alpha})
+ \int_{\Sigma} \pal \eta (\pal u \cdot \n^1) \\
+ \int_\Omega \dt (J^1 \rho^1) \frac{{\left\vert\pal u\right\vert}^2}{2}
- \int_\Omega \frac{\mu J^1}{2} \sgz_{\a^1} \pal u : \sgz_{\pal(\a^1-\a^2)} u^2 + J^1 \mu' (\operatorname{div}_{\a^1} \pal u )( \operatorname{div}_{\pal(\a^1 - \a^2)}u^2).\end{gathered}$$
Let us define $$\mathfrak{U}(t) = \sum_{ \substack{\alpha \in \mathbb{N}^{1+2} \\ {\left\vert\alpha\right\vert} \le 2} } \left( \int_\Omega \rho^1J^1 \frac{{\left\vert\pal u(t)\right\vert}^2}{2} + \int_{\Sigma} \frac{{\left\vert\pal \eta(t)\right\vert}^2}{2} + \sigma \frac{{\left\vert\pal\nab_\ast \eta(t)\right\vert}^2}{2} \right)$$ and $$\mathfrak{V}(t) = \sum_{ \substack{\alpha \in \mathbb{N}^{1+2} \\ {\left\vert\alpha\right\vert} \le 2} } {{\left\Vert\pal u(t)\right\Vert}^2}_{1}.$$
We sum over $\alpha \in \mathbb{N}^{1+2}$ with ${\left\vert\alpha\right\vert} \le 2$; applying Proposition \[korn\] and arguing as per usual (as in Lemmas \[lame\_forcing\_est\] and \[k\_F\_ests\], Theorem \[kappa\_contract\], and Propositions \[kappa\_improved\] and \[force\_est\_1\]) to estimate the various nonlinearities, we derive the differential inequality $$\begin{gathered}
\label{cot_6}
\frac{d}{dt} \mathfrak{U}(t) + C \mathfrak{V}(t) \ls P(\gamma) \sqrt{\Wf_2[u]} \sqrt{\Wf_2[\zeta]} \\
+ (1 + P(M)) \sqrt{\Wf_\infty[u]} \left( \sqrt{\Wf_\infty[v,p,\zeta]} + \sqrt{\Wf_\infty[v,\zeta]} \right)
+ (1 + P(M)) \sqrt{\Wf_2[u]} \sqrt{\Wf_\infty[p,\zeta]} \\
+ (1+ P(M)) \Wf_\infty[u,\eta] + (1+ P(M)) \sqrt{\Wf_\infty[u,\eta]} \sqrt{\Wf_2[u,\eta]}\end{gathered}$$ for some universal positive polynomial with $P(0)=0$ and a universal constant $C>0$. We should note that two of the terms appearing on the right side of require some delicate treatment. The first are terms involving $\nab p$ in $H^1$. In order to handle these when two horizontal spatial derivatives are applied, we must integrate by parts to move one horizontal derivative onto $J^1 \p^\alpha u$ and reduce to only two derivatives on $p$, which is all that is controlled by $\Wf_\infty[p]$. The second are terms involving $\sigma_\pm$ multiplying two spatial derivatives of $\zeta_\pm$ in $H^2_\pm$; these give rise to the term $P(\gamma) \sqrt{\Wf_2[u]} \sqrt{\Wf_2[\zeta]}$. The key part of this is $P(\gamma)$, which appears because the nonlinear terms only involve spatial derivatives of $\zeta^i$.
Integrating in time, using the fact that $\mathfrak{U}(0)=0$, and applying the Cauchy-Schwarz inequality then yields the bound $$\begin{gathered}
\label{cot_7}
\sup_{0\le t \le T} \mathfrak{U}(t) + \int_0^T \mathfrak{V}(t) dt \ls P(\gamma) \int_0^T \left(\Wf_2[\zeta] + \Wf_2[u] \right) dt \\
+ \sqrt{T}(1 + P(M)) \int_0^T \left( \Wf_2[v,\zeta] + \Wf_2[u,\eta] \right) dt
\\
+ (\sqrt{T} +T) (1 + P(M)) \sup_{0 \le t \le T} \left( \Wf_\infty[v,p,\zeta] + \Wf_\infty[u,\eta] \right),\end{gathered}$$ where again $P$ is a universal positive polynomial such that $P(0)=0$.
Throughout the rest of the proof we will let $\z$ denote a quantity of the form $$\z \simeq (1+ P(\gamma)) \left( \text{RHS of } \eqref{cot_7} \right),$$ where $P$ is some universal positive polynomial such that $P(0)=0$. From one estimate to another the polynomials and constants may change, but the structure of $\z$ does not.
Step 3 – Improved $u$ estimates
The usual trace theory allows us to estimate $$\int_0^T {{\left\Vertu\right\Vert}^2}_{H^{5/2}(\Sigma)} \ls \int_0^T \sum_{\substack{\alpha \in \mathbb{N}^2 \\ {\left\vert\alpha\right\vert}\le 2}} {{\left\Vert\pal u\right\Vert}^2}_{1} \ls \int_0^T \mathfrak{V}(t)dt \ls \z.$$ We may apply the elliptic estimate of Proposition \[lame\_elliptic\], which is applicable due to the first estimate in , to bound $$\begin{gathered}
{{\left\Vertu\right\Vert}^2}_3 \ls {{\left\Vert\rho^1 \dt u\right\Vert}^2}_1 + {{\left\Vert\operatorname{div}_{\a^{1}} \mathbb{S}_{ (\a^{1}-\a^{2})} u^2 + H^1\right\Vert}^2}_1 + {{\left\Vertu\right\Vert}^2}_{H^{5/2}(\Sigma)} \\
\ls {{\left\Vert\rho^1\right\Vert}^2}_{L^\infty}{{\left\Vert\dt u\right\Vert}^2}_1+ {{\left\Vert\nab \rho^1\right\Vert}^2}_{L^\infty} {{\left\Vert\dt u\right\Vert}^2}_0 + (1+P(M)) \left( \Wf_\infty[v,p,\zeta] + \Wf_\infty[\eta] \right) + {{\left\Vertu\right\Vert}^2}_{H^{5/2}(\Sigma)} \\
\ls \mathfrak{V} + (1+P(M)) \Wf_\infty[u] + (1+P(M)) \left( \Wf_\infty[v,p,\zeta] + \Wf_\infty[\eta] \right) + {{\left\Vertu\right\Vert}^2}_{H^{5/2}(\Sigma)}.\end{gathered}$$ Here in the third inequality we have used the second estimate in . Hence $$\label{cot_8}
\int_0^T {{\left\Vertu(t)\right\Vert}^2}_3 dt \ls \int_0^T \mathfrak{V}(t) dt + \z \ls \z.$$ We improve the $L^\infty$ in time estimate for $u$ by employing Lemma \[time\_interp\]: $$\label{cot_9}
\sup_{0\le t \le T} {{\left\Vertu(t)\right\Vert}^2}_2 \ls \int_0^T ({{\left\Vertu(t)\right\Vert}^2}_3 + {{\left\Vert\dt u(t)\right\Vert}^2}_1 )dt \ls \int_0^T \left( {{\left\Vertu(t)\right\Vert}^2}_3 +\mathfrak{V}(t)\right) dt \ls \z.$$ Combining , , and then provides us with the bound $$\label{cot_30}
\sup_{0 \le t \le T} \Wf_\infty[u] + \int_0^T \Wf_2[u] dt \ls \z.$$
Step 4 – Improved $\eta$ estimates
Now we improve the estimates for $\eta$. Note first that we already have the bound $$\label{cot_10}
\sup_{0\le t \le T} \left( {{\left\Vert\eta(t)\right\Vert}^2}_2 + {{\left\Vert\dt \eta (t) \right\Vert}^2}_0 + \sigma {{\left\Vert\nab_\ast \eta(t)\right\Vert}^2}_2 + \sigma {{\left\Vert\nab_\ast \dt \eta(t) \right\Vert}^2}_0 \right) \le \sup_{0\le t \le T} \mathfrak{U}(t) \ls \z.$$ By solving the second and third equations in for $\sigma \Delta \eta$ and employing , , and , we may then bound $$\begin{gathered}
\label{cot_11}
\int_0^T \sigma^2 {{\left\Vert\eta(t)\right\Vert}^2}_{7/2}dt \ls \int_0^T \left( {{\left\Vert\eta(t)\right\Vert}^2}_{0} + {{\left\Vert\sigma \Delta_\ast \eta(t) \right\Vert}^2}_{3/2} \right)dt \\
\ls T \sup_{0 \le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_0 + \int_0^T \left( (1+P(\gamma)) \Wf_2[u] + P(\gamma) \Wf_2[\zeta] \right)dt \\
+ T (1+P(M)) \sup_{0 \le t \le T} \left( \Wf_\infty[p,\zeta] + \Wf_\infty[\eta] \right)
\ls \z.\end{gathered}$$ Next we employ the kinematic equation in along with the transport estimates of Proposition 2.1 of [@danchin] to bound $$\sup_{0\le t\le T} {{\left\Vert\eta(t)\right\Vert}^2}_{5/2} \ls \exp\left(CT \int_0^T {{\left\Vertu^2(t)\right\Vert}^2}_3 dt\right) T \int_0^T {{\left\Vertu\cdot \n^1 (t)\right\Vert}^2}_{5/2} dt.$$ Hence, if we assume that $T_5 M \le 1$ we may bound the exponential term above by a universal constant and then estimate $$\label{cot_12}
\sup_{0\le t\le T} {{\left\Vert\eta(t)\right\Vert}^2}_{5/2} \ls T(1+P(\gamma)) \int_0^T {{\left\Vertu(t)\right\Vert}^2}_3 dt \ls T \z.$$ Then we solve for $\dt \eta$ in to bound $$\label{cot_13}
\sup_{0\le t \le T} {{\left\Vert\dt \eta(t)\right\Vert}^2}_{3/2} \ls M \sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{5/2} + (1+P(\gamma)) \sup_{0\le t \le T} {{\left\Vertu(t)\right\Vert}^2}_{2}
\ls MT \z + \z \ls \z$$ since $M T \le M T_5 \le 1$. Similarly, we solve for $\dt^2 \eta$ to estimate $$\begin{gathered}
\label{cot_14}
\int_0^T {{\left\Vert\dt^2 \eta(t)\right\Vert}^2}_{1/2} dt \ls MT \sup_{0 \le t \le T} {{\left\Vert\dt \eta(t)\right\Vert}^2}_{3/2} + (1+\gamma) \int_0^T {{\left\Vert\dt u(t)\right\Vert}^2}_1 dt \\
+ MT \sup_{0\le t\le T} \left( {{\left\Vertu(t)\right\Vert}^2}_2 + {{\left\Vert\eta(t)\right\Vert}^2}_{5/2} \right) \ls \z.
\end{gathered}$$ Summing , , and – then yields the bound $$\label{cot_31}
\sup_{0 \le t \le T} \Wf_\infty[\eta] + \int_0^T \Wf_2[\eta] dt \ls \z.$$
Step 5 – Estimates of $q$
Next we employ Proposition \[xport\_low\_reg\_est\] to get estimates for $q$. First we find that $$\sup_{0 \le t \le T} {{\left\Vertq(t)\right\Vert}^2}_2 \ls \exp(C (1+ P(M)) T) T \int_0^T {{\left\VertH^4(t)\right\Vert}^2}_{2} dt.$$ If we further restrict $T_5$ so that $(1+P(M))T \le 1$ we can again treat the exponential as a universal constant. Then $$\begin{gathered}
\label{cot_15}
\sup_{0 \le t \le T} {{\left\Vertq(t)\right\Vert}^2}_2 \ls T \int_0^T {{\left\VertH^4(t)\right\Vert}^2}_{2} dt \ls T (1+P(M))\int_0^T \left( \Wf_2[u] + \Wf_\infty[u,\eta] \right)dt \\
\ls T(1+P(M)) \z.\end{gathered}$$
Next we use to solve for $\dt q$ and estimate $$\label{cot_16}
\sup_{0 \le t \le T} {{\left\Vert\dt q(t)\right\Vert}^2}_1 \ls P(M) \sup_{0 \le t \le T} {{\left\Vertq(t)\right\Vert}^2}_2 + \sup_{0 \le t \le T} {{\left\VertH^4\right\Vert}^2}_1$$ The term $H^4$ may be estimated as follows. First we use the fact that $u(t=0)=0$ and $\eta(t=0)=0$ to estimate $$\label{cot_33}
\sup_{0\le t \le T} {{\left\Vert\dt \eta(t)\right\Vert}^2}_{1/2} + \sup_{0\le t \le T} {{\left\Vertu(t)\right\Vert}^2}_{1} \le T \int_0^T \left( {{\left\Vert\dt^2 \eta(t)\right\Vert}^2}_{1/2} + {{\left\Vert\dt u(t)\right\Vert}^2}_1\right) dt \le T \z.$$ Then we bound $$\begin{gathered}
\label{cot_17}
\sup_{0 \le t \le T} {{\left\VertH^4(t)\right\Vert}^2}_1 \ls P(M) \sup_{0 \le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{3/2} + P(M) \sup_{0 \le t \le T} {{\left\Vert\dt \eta(t)\right\Vert}^2}_{1/2} \\
+ P(M) \sup_{0\le t \le T} {{\left\Vertu(t)\right\Vert}^2}_1 + (1+ P(\gamma)) \sup_{0 \le t \le T} {{\left\Vertu(t)\right\Vert}^2}_2
\\
\ls T P(M) \z + (1+ P(\gamma)) \z \ls \z\end{gathered}$$ if $T_5$ is further restricted so that $T_5 P(M) \le 1$. Note here that we have crucially employed the $T$ factor appearing on the right side of and . We can now combine , , and to deduce that $$\label{cot_32}
\sup_{0 \le t \le T} \Wf_\infty[q] \ls T(1+P(M)) \z +\z\ls \z$$ if $T_5$ is further restricted.
Step 6 – Synthesis
Now we sum the estimates , , and to deduce that $$\sup_{0\le t \le T} \Wf_\infty[u,q,\eta](t) + \int_0^T \Wf_2[u,\eta](t) dt \ls \z.$$ Assuming that $\gamma$ and $T_5$ are sufficiently small and using the previous inequality, we may absorb the terms involving $(u,q,\eta)$ from the right side to the left; this results in the estimate $$\begin{gathered}
\sup_{0\le t \le T} \Wf_\infty[u,q,\eta](t) + \int_0^T \Wf_2[u,\eta](t) dt \ls P(\gamma) \int_0^T \Wf_2[\zeta] dt \\
+ \sqrt{T}(1 + P(M)) \int_0^T \Wf_2[v,\zeta] dt
+ (\sqrt{T} +T) (1 + P(M)) \sup_{0 \le t \le T} \Wf_\infty[v,p,\zeta].\end{gathered}$$ By further restricting $\gamma$ and $T_5$ we deduce that holds.
Local well-posedness
--------------------
We now have all of the ingredients necessary to prove a more general version of Theorem \[local\_existence\_intro\] in the case $\sigma_\pm >0$.
\[local\_existence\] Assume that $\sigma_\pm >0$. Suppose that $(u_0,q_0,\eta_0)$ satisfy the compatibility conditions as well as the bound , and that $$\label{le_00}
\Lf[\eta_0] \le \frac{\delta_1}{2},$$ where $\Lf[\eta_0]$ is given by and $\delta_1$ is given by Theorem \[kappa\_apriori\]. Further assume that $$\label{le_00_2}
{{\left\Vert\eta_0\right\Vert}^2}_7 \le \frac{\gamma^2}{2},$$ where $\gamma$ is as given in Theorem \[contraction\_thm\]. Set $T_6 = T_6({\EEE}) = \min\{T_4({\EEE}),T_5(P_1({\EEE}) + P_3({\EEE})\}$, where $T_4$ and $P_1, P_3$ are given by Theorem \[approx\_solns\] and $T_5$ is given by Theorem \[contraction\_thm\].
If $0 < T \le T_6$ then there exists a triple $(u,q,\eta)$ defined on the temporal interval $[0,T]$ satisfying the following three properties. First, $(u,q,\eta)$ achieve the initial data at $t=0$. Second, the triple uniquely solve . Third, the triple obey the estimates $$\label{le_01}
\sup_{0\le t \le T} \left( \E[u(t)] + \hat{\E}^\sigma[\eta(t)] \right)
+ \int_0^T \left( \D[u(t)] + {{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} + \hat{\D}^\sigma[\eta(t)] \right) dt \le P_1({\EEE}),$$ $$\label{le_02}
\sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \le P_2({\EEE}),$$ $$\label{le_03}
\sup_{0\le t \le T} \E[q(t)] + \int_0^T \D[q(t)] dt \le P_3({\EEE}),$$ and $$\label{le_04}
\Lf[\eta](T) \le \delta_1 \text{ and }\frac{1}{2}\rho_\ast\le \rho=\rho[q,\eta]\le \frac{3}{2}\rho^\ast.$$
The proof is very similar to that of Theorem 6.2 in [@GT_lwp], so we will only provide a quick sketch.
First we use Theorem \[approx\_solns\] to produce a sequence of approximate solutions $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ on the temporal interval $[0,T]$. The uniform estimates – along with standard compactness and weak compactness arguments yield a subsequence converging to a limiting triple $(u,q,\eta)$ that achieves the initial data and satisfies the estimates –.
Because we only know the convergence of a subsequence, we cannot immediately pass to the limit in and . Instead we first use Theorem \[contraction\_thm\] to deduce that the sequence $\{(u^n,q^n,\eta^n)\}_{n=0}^\infty$ actually contracts in the lower regularity norm defined by . In order to apply the theorem we must verify that and are satisfied; these follow from and , an argument like that used in –, and a further restriction of time. This low regularity convergence, when combined with the bounds – and various interpolation arguments, shows that the original sequence actually converges to $(u,q,\eta)$ in a regularity class slightly larger than that defined by – but more than sufficient for passing to the limit in –. We deduce then that $(u,q,\eta)$ satisfy . The uniqueness claim follows from another application of Theorem \[contraction\_thm\].
Local well-posedness of with $\sigma_\pm =0$ {#local-well-posedness-of-with-sigma_pm-0}
============================================
We now state a result on the local existence of solutions to without surface tension.
\[local\_existence\_no\_ST\] Assume that $\sigma_\pm =0$. Suppose that $(u_0,q_0,\eta_0)$ satisfy the compatibility conditions as well as the bounds , and that $$\Lf[\eta_0] \le \frac{\delta_1}{2},$$ where $\Lf[\eta_0]$ is given by and $\delta_1$ is given by Theorem \[kappa\_apriori\]. Further assume that $${{\left\Vert\eta_0\right\Vert}^2}_7 \le \frac{\gamma^2}{2},$$ where $\gamma$ are as given in Theorem \[contraction\_thm\]. Set $T_6 = T_6({\EEE}) = \min\{T_4({\EEE}),T_5(P_1({\EEE}) + P_3({\EEE})\}$, where $T_4$ and $P_1, P_3$ are given by Theorem \[approx\_solns\] and $T_5$ is given by Theorem \[contraction\_thm\].
If $0 < T \le T_6$ then there exists a triple $(u,q,\eta)$ defined on the temporal interval $[0,T]$ satisfying the following three properties. First, $(u,q,\eta)$ achieve the initial data at $t=0$. Second, the triple uniquely solve . Third, the triple obey the estimates $$\label{le_st_01}
\sup_{0\le t \le T} \left( \E[u(t)] + \hat{\E}^0[\eta(t)] \right)
+ \int_0^T \left( \D[u(t)] + {{\left\Vert\rho J \dt^{2N+1} u(t)\right\Vert}^2}_{ \Hd} + \hat{\D}^0[\eta(t)] \right) dt \le P_1({\EEE}),$$ $$\sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N+1/2} \le P_2({\EEE}),$$ $$\sup_{0\le t \le T} \E[q(t)] + \int_0^T \D[q(t)] dt \le P_3({\EEE}),$$ and $$\label{le_st_02}
\Lf[\eta](T) \le \delta_1\text{ and }\frac{1}{2}\rho_\ast\le \rho=\rho[q,\eta]\le \frac{3}{2}\rho^\ast.$$
The proof follows from an argument similar to that used in the proof of Theorem \[local\_existence\], but actually somewhat easier and more akin to that used in Section 6 of [@GT_lwp]. We will provide only a sketch of the ideas.
The main difference between the method to produce solutions to with $\sigma_\pm >0$ and the method used with $\sigma_\pm =0$ lies in the use of the $\kappa$ approximation, which replaces the kinematic transport equation for $\eta$ with a parabolic problem. This is essential in studying the problem with surface tension, as it leads to a regularity gain for $\eta$ that enables us to treat $\sigma_\pm \Delta_\ast \eta_\pm$ as a forcing term when solving for $u$. However, when $\sigma_\pm =0$, this regularity gain is unnecessary, as only $\eta$ appears as a forcing term in the $u$ equation. In place of the parabolic problem we simply study the kinematic transport problem directly, using Theorem 5.4 of [@GT_lwp] to produce solutions and derive estimates.
We then proceed essentially as in Section \[sec\_lwp\_st\]. First we prove that Theorem \[approx\_solns\] holds with $\sigma_\pm =0$. The iteration scheme begins with a triple $(u^{n-1},q^{n-1},\eta^{n-1})$ and then uses Theorem \[lame\_high\] to produce $u^n$. Then the equation $\dt \eta^n = u^n \cdot \n^n$ is solved using Theorem 5.4 of [@GT_lwp]. Then $(u^n,\eta^n)$ are used to solve for $q^n$ in by way of Theorem \[xport\_well-posed\]. Next we observe that Theorem \[contraction\_thm\] remains true as stated with $\sigma_\pm =0$. Finally, we combine these two theorems to produce a sequence $\{(u^n,q^n,\eta^n\}_{n=1}^\infty$ of approximate solutions that remain uniformly bounded at high regularity and contract in a lower-regularity norm.
Energy and dissipation functionals {#sec_en_dis}
==================================
Here we collect the definitions of various functionals that are used throughout the paper. We define the energies associated to $(u,q,\eta)$ via $$\label{energy_def_u}
\E[u] = \sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{4N-2j},$$ $$\label{energy_def_q}
\E[q] = {{\left\Vertq\right\Vert}^2}_{4N} +\sum_{j=1}^{2N} {{\left\Vert\dt^j q\right\Vert}^2}_{4N-2j+1},$$ and $$\label{energy_def_eta}
\E^\sigma[\eta] = \sum_{j=0}^{2N} {{\left\Vert\dt^j \eta\right\Vert}^2}_{ 4N-2j} + \sigma {{\left\Vert\dt^j \nab_\ast \eta\right\Vert}^2}_{ 4N-2j}.$$
We define the corresponding dissipation functionals via $$\label{dissipation_def_u}
\D[u] = \sum_{j=0}^{2N} {{\left\Vert\dt^j u\right\Vert}^2}_{ 4N-2j+1}$$ $$\label{dissipation_def_q}
\D[q] = {{\left\Vertq\right\Vert}^2}_{4N} + {{\left\Vert\dt q\right\Vert}^2}_{4N-1} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j q\right\Vert}^2}_{4N-2j+2},$$ and $$\label{dissipation_def_eta}
\D^\sigma[\eta] = \sigma^2 {{\left\Vert\eta\right\Vert}^2}_{ 4N+3/2} + {{\left\Vert\dt \eta\right\Vert}^2}_{ 4N-1} + \sum_{j=2}^{2N+1} {{\left\Vert\dt^j \eta\right\Vert}^2}_{ 4N-2j+2}.$$
For $\eta$ we also need to define some improved terms: $$\label{energy_eta_improved}
\hat{\E}^\sigma[\eta] = \E^\sigma[\eta] + \sum_{j=1}^{2N} {{\left\Vert\dt^j \eta \right\Vert}^2}_{4N-2j+3/2}$$ and $$\label{dissipation_eta_improved}
\hat{\D}^\sigma[\eta] = \D^\sigma[\eta]+ {{\left\Vert\dt \eta\right\Vert}^2}_{4N-1/2} +\sum_{j=2}^{2N} {{\left\Vert\dt^j \eta \right\Vert}^2}_{4N-2j+5/2}.$$ We must also define the term $$\label{l_def}
\Lf[\eta](T) = \sup_{0\le t \le T} {{\left\Vert\eta(t)\right\Vert}^2}_{4N-1/2}.$$ In estimating this term we often refer to the following term associated with the data: $$\label{l0_def}
\Lf[\eta_0] = {{\left\Vert\eta_0\right\Vert}^2}_{4N-1/2}.$$
For the data $(u_0,q_0,\eta_0)$ we define $${\EEE} = {{\left\Vertu_0\right\Vert}^2}_{4N} +{{\left\Vertq_0\right\Vert}^2}_{4N} + {{\left\Vert\eta_0\right\Vert}^2}_{4N+1/2} + \sigma {{\left\Vert\nab_\ast \eta_0\right\Vert}^2}_{4N},$$ and when $\{(\dt^j u(0),\dt^j q(0),\dt^j\eta(0))\}_{j=0}^{2N}$ are known we write $$\begin{gathered}
\label{TE_def1}
\TE[u_0] = \sum_{j=0}^{2N} {{\left\Vert\dt^j u(0)\right\Vert}^2}_{4N-2j}, \; \TE[q_0] = {{\left\Vertq(0)\right\Vert}^2}_{4N} +\sum_{j=1}^{2N} {{\left\Vert\dt^j q(0)\right\Vert}^2}_{4N-2j+1}, \\
\text{ and }
\TE[\eta_0] = {{\left\Vert\eta(0)\right\Vert}^2}_{4N+1/2} + \sigma {{\left\Vert\nab_\ast \eta(0)\right\Vert}^2}_{ 4N} + \sum_{j=1}^{2N} {{\left\Vert\dt^j \eta(0) \right\Vert}^2}_{4N-2j+3/2} $$ We will often abbreviate $$\label{TE_def2}
\TE = \TE[u_0] + \TE[q_0] + \TE[\eta_0] \text{ and } \TE[u_0,\eta_0] = \TE[u_0] + \TE[\eta_0].$$
Compatibility conditions {#app_ccs}
=========================
Here we record the system of compatibility conditions that the initial data $(u_0,q_0,\eta_0)$ must satisfy in order to produce high-regularity solutions to . To state the compatibility conditions we must first show how to construct $(\dt^j u(\cdot,0), \dt^j q(\cdot,0), \dt^j \eta(\cdot,0))$ for $j=1,\dotsc,2N$ from the triple $(u_0,q_0,\eta_0)$.
For the purposes of constructing the temporal-derivative data we rewrite the first, second and third equations in in the form $$\label{cc_data}
\begin{split}
\dt \eta &= F_1(u,\eta) \\
\dt q &= F_2(u,q,\eta,\dt \eta) \\
\dt u &= F_3(u,q,\eta,\dt \eta).
\end{split}$$ Assuming that we are given $\{(\dt^k u(\cdot,0), \dt^k q(\cdot,0), \dt^k \eta(\cdot,0))\}_{k=0}^j$ for some $j\in \{0,\dotsc,2N-1\}$, we construct $(\dt^{j+1} u(\cdot,0), \dt^{j+1} q(\cdot,0), \dt^{j+1} \eta(\cdot,0))$ as follows. First we apply $\dt^j$ to the first equation in and define $$\left.\dt^{j+1} \eta(\cdot,0) = \dt^j F_1(u,\eta) \right\vert_{t=0},$$ which is possible because all terms appearing on the right are already known. We can now perform a similar operation on the second and third equations in , setting $$\left.\dt^{j+1} u(\cdot,0) = \dt^j F_2(u,q,\eta,\dt \eta) \right\vert_{t=0} \text{ and } \left.\dt^{j+1} q(\cdot,0) = \dt^j F_3(u,q,\eta,\dt \eta) \right\vert_{t=0},$$ both of which can be computed in terms of known quantities since we have already computed $\dt^{j+1}\eta(\cdot,0)$. Using this argument, we may inductively define $\{(\dt^j u(\cdot,0), \dt^j q(\cdot,0), \dt^j \eta(\cdot,0))\}_{j=1}^{2N}$ as desired.
We may now state the compatibility conditions. We say that $(u_0,q_0,\eta_0)$ satisfy the compatibility conditions at level $2N$ if $$\label{ccs}
\begin{cases}
\dt^j\left( P'(\bar\rho)\q \n - \S_{\a}( u)\n \right) \vert_{t=0} = \dt^j\left( \bar{\rho}_1 g \eta \n-\sigma_+ \mathcal{H}_+ \n
- \mathcal{R} \n\right)\vert_{t=0} &\text{on } \Sigma_+ \\
{\left\llbracket \dt^j\left( P'(\bar\rho)\q \n- \S_\a(u)\n \right) \right\rrbracket } \vert_{t=0} = \dt^j \left( \rj g\eta\n+\sigma_- \mathcal{H}_- \n - {\left\llbracket \mathcal{R} \right\rrbracket }\n \right) \vert_{t=0} &\text{on } \Sigma_- \\
{\left\llbracket \dt^j u \right\rrbracket }\vert_{t=0} =0 &\text{on } \Sigma_- \\
\dt^j u_-\vert_{t=0} = 0 &\text{on } \Sigma_b
\end{cases}$$ for $j=0,\dotsc,2N-1$.
Poisson extension
=================
We will now define the appropriate Poisson integrals that allow us to extend $\eta_\pm$, defined on the surfaces $\Sigma_\pm$, to functions defined on $\Omega$, with “good” boundedness.
Suppose that $\Sigma_+ = \mathrm{T}^2\times \{\ell\}$, where $\mathrm{T}^2:=(2\pi L_1 \mathbb{T}) \times (2\pi L_2 \mathbb{T})$. We define the Poisson integral in $\mathrm{T}^2 \times (-\infty,\ell)$ by $$\label{P-1def}
\mathcal{P}_{-,\ell}f(x) = \sum_{\xi \in (L_1^{-1} \mathbb{Z}) \times
(L_2^{-1} \mathbb{Z}) } \frac{e^{i \xi \cdot x' }}{2\pi \sqrt{L_1 L_2}} e^{|\xi|(x_3-\ell)} \hat{f}(\xi),$$ where for $\xi \in (L_1^{-1} \mathbb{Z}) \times (L_2^{-1} \mathbb{Z})$ we have written $$\label{horiz_ft_def}
\hat{f}(\xi) = \int_{\mathrm{T}^2} f(x') \frac{e^{- i \xi \cdot x' }}{2\pi \sqrt{L_1 L_2}} dx'.$$ Here “$-$” stands for extending downward and “$\ell$” stands for extending at $x_3=\ell$, etc. It is well-known that $\mathcal{P}_{-,\ell}:H^{s}(\Sigma_+) \rightarrow H^{s+1/2}(\mathrm{T}^2 \times (-\infty,\ell))$ is a bounded linear operator for $s>0$. Certain improvements of this are available when we restrict to $\Omega$; we refer to the appendix of [@WTK] for details.
We extend $\eta_+$ to be defined on $\Omega$ by $$\label{P+def}
\bar{\eta}_+(x',x_3)=\mathcal{P}_+\eta_+(x',x_3):=\mathcal{P}_{-,\ell}\eta_+(x',x_3),\text{ for } x_3\le \ell.$$ If $\eta_+\in H^{s-1/2}(\Sigma_+)$ for $s\ge 0$, then $\bar{\eta}_+\in H^{s}(\Omega)$.
Similarly, for $\Sigma_- = \mathrm{T}^2\times \{0\}$ we define the Poisson integral in $\mathrm{T}^2 \times (-\infty,0)$ by $$\label{P-0def}
\mathcal{P}_{-,0}f(x) = \sum_{\xi \in (L_1^{-1} \mathbb{Z}) \times (L_2^{-1} \mathbb{Z}) } \frac{e^{ i \xi \cdot x' }}{2\pi \sqrt{L_1 L_2}} e^{ |\xi|x_3} \hat{f}(\xi).$$ It is clear that $\mathcal{P}_{-,0}$ has the same regularity properties as $\mathcal{P}_{-,\ell}$. This allows us to extend $\eta_-$ to be defined on $\Omega_-$. However, we do not extend $\eta_-$ to the upper domain $\Omega_+$ by the reflection since this will result in the discontinuity of the partial derivatives in $x_3$ of the extension. For our purposes, we instead to do the extension through the following. Let $0<\lambda_0<\lambda_1<\cdots<\lambda_m<\infty$ for $m\in \mathbb{N}$ and define the $(m+1) \times (m+1)$ Vandermonde matrix $V(\lambda_0,\lambda_1,\dots,\lambda_m)$ by $V(\lambda_0,\lambda_1,\dots,\lambda_m)_{ij} = (-\lambda_j)^i$ for $i,j=0,\dotsc,m$. It is well-known that the Vandermonde matrices are invertible, so we are free to let $\alpha=(\alpha_0,\alpha_1,\dots,\alpha_m)^T$ be the solution to $$\label{Veq}
V(\lambda_0,\lambda_1,\dots,\lambda_m)\,\alpha=q_m,\ q_m=(1,1,\dots,1)^T.$$ Now we define the specialized Poisson integral in $\mathrm{T}^2 \times (0,\infty)$ by $$\label{P+0def}
\mathcal{P}_{+,0}f(x) = \sum_{\xi \in (L_1^{-1} \mathbb{Z}) \times
(L_2^{-1} \mathbb{Z}) } \frac{e^{ i \xi \cdot x' }}{2\pi \sqrt{L_1 L_2}} \sum_{j=0}^m\alpha_j
e^{- |\xi|\lambda_jx_3} \hat{f}(\xi).$$ It is easy to check that, due to , $\partial_3^l\mathcal{P}_{+,0}f(x',0)=
\partial_3^l\mathcal{P}_{-,0}f(x',0)$ for all $0\le l\le m$ and hence $$\partial^\alpha\mathcal{P}_{+,0}f(x',0)=
\partial^\alpha\mathcal{P}_{-,0}f(x',0), \ \forall\, \alpha\in \mathbb{N}^3 \text{ with }0\le |\alpha|\le m.$$ These facts allow us to extend $\eta_-$ to be defined on $\Omega$ by $$\bar{\eta}_-(x',x_3)=
\mathcal{P}_-\eta_-(x',x_3):=\left\{\begin{array}{lll}\mathcal{P}_{+,0}\eta_-(x',x_3),\quad
x_3> 0 \\
\mathcal{P}_{-,0}\eta_-(x',x_3),\quad x_3\le
0.\end{array}\right.\label{P-def}$$ It is clear now that if $\eta_-\in H^{s-1/2}(\Sigma_-)$ for $ 0\le s\le m$, then $\bar{\eta}_-\in H^{s}(\Omega)$. Since we will only work with $s$ lying in a finite interval, we may assume that $m$ is sufficiently large in for $\bar{\eta}_- \in H^s(\Omega)$ for all $s$ in the interval.
Estimates of Sobolev norms
==========================
Here we record an estimate involving space-time norms.
\[time\_interp\] Let $\Gamma$ denote either $\Sigma$ or $\Omega$. Suppose that $\zeta \in L^2([0,T]; H^{s_1}(\Gamma))$ and $\dt \zeta \in L^2([0,T]; H^{s_2}(\Gamma))$ for $s_1 \ge s_2 \ge 0$. Let $s = (s_1+s_2)/2$. Then $\zeta \in C^0([0,T]; H^{s}(\Gamma))$ (after possibly being redefined on a set of measure $0$), and $$\label{l_sobi_01}
{{\left\Vert\zeta\right\Vert}^2}_{L^\infty H^{s}} \le {{\left\Vert\zeta(0)\right\Vert}^2}_{H^s} + C {{\left\Vert \zeta\right\Vert}^2}_{L^2 H^{s_1}} + C {{\left\Vert\dt \zeta\right\Vert}^2}_{L^2 H^{s_2}}$$ for some universal constant $C>0$.
This is a slight variant of Lemma A.4 of [@GT_lwp] that follows from the same argument.
Some estimates involving the geometric terms
============================================
Coefficient estimates
---------------------
Here we are concerned with how the size of $\eta$ can control the “geometric” terms that appear in the equations.
\[eta\_small\] There exists a universal $0 < \delta < 1$ so that if ${{\left\Vert\eta\right\Vert}^2}_{5/2} \le \delta,$ then $$\label{es_01}
\begin{split}
& {\left\VertJ-1\right\Vert}_{L^\infty(\Omega)} +{\left\VertA\right\Vert}_{L^\infty(\Omega)} + {\left\VertB\right\Vert}_{L^\infty(\Omega)} \le \hal, \\
& {\left\Vert\n-1\right\Vert}_{L^\infty(\Gamma)} + {\left\VertK-1\right\Vert}_{L^\infty(\Gamma)} \le \hal, \text{ and } \\
& {\left\VertK\right\Vert}_{L^\infty(\Omega)} + {\left\Vert\mathcal{A}\right\Vert}_{L^\infty(\Omega)} \ls 1.
\end{split}$$ Also, the map $\Theta$ defined by is a diffeomorphism, and $$\label{es_02}
\hal \int_\Omega {\left\vert\varphi\right\vert}^2 \le \int_\Omega J {\left\vert \varphi\right\vert}^2 \le 2 \int_\Omega {\left\vert\varphi\right\vert}^2$$ for all $\varphi \in L^2(\Omega)$.
The estimate is guaranteed by Lemma 2.4 of [@GT_per]. The estimate then follows trivially from .
Korn’s inequality
-----------------
Here we record a version of Korn’s inequality that is needed throughout our analysis. First we record a version involving only the deviatoric part of the symmetric gradient, $\sgz$, defined by .
\[layer\_korn\] There exists a universal constant $C>0$ so that $${{\left\Vertu\right\Vert}^2}_1 = {{\left\Vertu_+\right\Vert}^2}_{1} + {{\left\Vertu_-\right\Vert}^2}_{1} \le C({{\left\Vert\sgz{u_+}\right\Vert}^2}_0 + {{\left\Vert\sgz{u_-}\right\Vert}^2}_0)$$ for all $u_\pm \in H^1(\Omega_\pm)$ with ${\left\llbracket u \right\rrbracket }=0$ along $\Sigma$ and $u_- =0$ on $\Sigma_b$.
The proof is based on the “deviatoric Korn inequality” of Dain, Theorem 1.1 of [@dain]. For details of the proof see the appendix of [@JTW_GWP].
Next we extend Proposition \[layer\_korn\] to $\sgz_{\a}$.
\[korn\] Assume that $\mu >0$ and $\mu' \ge 0$. There exists a universal $0 < \delta < 1$, smaller than the $\delta$ appearing in Lemma \[eta\_small\], such that if ${{\left\Vert\eta\right\Vert}^2}_{5/2} \le \delta,$ then $$\label{ko_01}
{{\left\Vertu\right\Vert}^2}_1 \ls \int_\Omega \frac{\mu}{2} J {\left\vert\sgz_\a u\right\vert}^2 + \mu' J {\left\vert\diva u\right\vert}^2 \ls {{\left\Vertu\right\Vert}^2}_1$$ for all $u_\pm \in H^1(\Omega_\pm)$ with ${\left\llbracket u \right\rrbracket }=0$ along $\Sigma$ and $u_- =0$ on $\Sigma_b$.
Let $\delta$ be as small as in Lemma \[eta\_small\]. With Proposition \[layer\_korn\] in hand we may then argue as in Lemma 2.1 of [@GT_lwp], further restricting $\delta$ as needed, to derive .
Elliptic estimates
------------------
Here we record elliptic estimates for the two-phase geometric Lamé problem with Dirichlet boundary conditions: $$\label{lam_dir}
\begin{cases}
- \diva \Sa u = G & \text{in }\Omega \\
u_+ = h_+ &\text{on } \Sigma_+ \\
u_+ = u_- = h_- &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b.
\end{cases}$$
Our elliptic regularity result is contained in the following.
\[lame\_elliptic\] Let $k\ge 4$ be an integer and suppose that $\eta \in H^{k+1/2}$. There exists $0 < \delta_0 \le \delta$ (where $\delta$ is given by Proposition \[korn\]) so that if ${{\left\Vert\eta\right\Vert}^2}_{k-1/2} \le \delta_0$, then solutions to satisfy $$\label{ld_01}
{\left\Vertu\right\Vert}_{r} \ls \left( {\left\VertG\right\Vert}_{r-2} + {\left\Verth\right\Vert}_{r-1/2} \right)$$ for $r=2,\dotsc,k$, whenever the right side is finite.
In the case $r=k+1$, solutions to satisfy $$\label{ld_02}
{\left\Vertu\right\Vert}_{k+1} \ls \left( {\left\VertG\right\Vert}_{k-1} + {\left\Verth \right\Vert}_{k+1/2} \right)
+ {\left\Vert\eta\right\Vert}_{k+1/2} \left( {\left\VertF^1\right\Vert}_{2} + {\left\Verth\right\Vert}_{7/2} \right)$$ whenever the right side is finite.
Estimates of the form for all $r \ge 2$ follow from the standard elliptic regularity theory for the problem $$\begin{cases}
- \operatorname{div}\S u = \tilde{G} & \text{in }\Omega \\
u_+ = h_+ &\text{on } \Sigma_+ \\
u_+ = u_- = h_- &\text{on } \Sigma_- \\
u_- = 0 &\text{on } \Sigma_b.
\end{cases}$$ With these estimates in hand, we may argue as in Section 3 of [@WTK] to deduce and under the smallness assumption ${{\left\Vert\eta\right\Vert}^2}_{k-1/2} \le \delta_0$.
Data extension results
======================
Next we need some extension results. Our first one allows us to take data with parabolic scaling and extend them to space-time functions with the same scaling. The proof can be found in Lemma A.5 of [@GT_lwp] and is thus omitted.
\[extension\_u\] Suppose that for $j=0,\dotsc,2N$ we have that $\dt^j v(0) \in H^{4N-2j}(\Omega)$. Then there exists an extension $u$, achieving the initial data, such that $$\dt^j u \in L^\infty([0,\infty); H^{4N-2j}(\Omega)) \cap L^2([0,\infty);H^{4N-2j+1}(\Omega)) \text{ for }j=0,\dotsc,2N.$$ Moreover, $$\sup_{t \ge 0} \sum_{j=0}^{2N} {{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N} + \int_0^\infty \sum_{j=0}^{2N}{{\left\Vert\dt^j v(t)\right\Vert}^2}_{4N-2j+1} dt \ls \sum_{j=0}^{2N} {{\left\Vert\dt^j v(0)\right\Vert}^2}_{4N-2j}$$
We also need a version of Proposition \[extension\_u\] that works to extend $\eta$. The difference between $\eta$ and $u$ is that Proposition \[data\_estimate\] provides higher regularity for the time derivatives that we would get from strict parabolic scaling. Nevertheless, Lemma A.5 of [@GT_lwp] may be readily modified to construct an extension.
\[extension\_eta\] Suppose that $\zeta(0) \in H^{4N+1/2}(\Sigma)$ and that $\sqrt{\sigma} \nab_\ast \zeta(0) \in H^{4N}(\Sigma)$. Further suppose that for $j=1,\dotsc,2N$ we have that $\dt^j \zeta(0) \in H^{4N-2j+3/2}(\Sigma)$. Then there exists an extension $\zeta$, achieving the initial data, such that $$\begin{split}
\zeta &\in L^\infty([0,\infty); H^{4N+1/2}(\Sigma)) \cap L^2([0,\infty);H^{4N+1}(\Sigma)), \\
\sqrt{\sigma} \nab_\ast \zeta &\in L^\infty ([0,\infty); H^{4N}(\Sigma)) \cap L^2([0,\infty);H^{4N+1/2}(\Sigma)) , \\
\dt \zeta &\in L^\infty ([0,\infty); H^{4N-1/2}(\Sigma)) \cap L^2([0,\infty);H^{4N}(\Sigma)) , \\
\dt^j \zeta & \in L^\infty([0,\infty); H^{4N-2j+3/2}(\Sigma)) \cap L^2([0,\infty);H^{4N-2j+5/2}(\Sigma)) \text{ for }j=2,\dotsc,2N,\\
\dt^{2N+1}\zeta &\in L^2([0,\infty);H^{1/2}(\Sigma)).
\end{split}$$ Moreover, we have the estimates $$\begin{gathered}
\label{ex_e01}
\sup_{t \ge 0} \left( {{\left\Vert\zeta(t)\right\Vert}^2}_{4N+1/2} + \sum_{j=1}^{2N} {{\left\Vert\dt^j \zeta(t)\right\Vert}^2}_{4N-2j+3/2} \right) \\
+ \int_0^\infty \left( {{\left\Vert \zeta(t)\right\Vert}^2}_{4N+1} +{{\left\Vert\dt \zeta(t)\right\Vert}^2}_{4N}+ \sum_{j=2}^{2N+1}{{\left\Vert \dt^j \zeta(t)\right\Vert}^2}_{4N-2j +5/2} \right) dt \\
\ls {{\left\Vert\zeta(0)\right\Vert}^2}_{4N+1/2} +\sum_{j=1}^{2N} {{\left\Vert\dt^j \zeta(0)\right\Vert}^2}_{4N-2j+3/2}\end{gathered}$$ and $$\label{ex_e02}
\sup_{t\ge 0} \sigma {{\left\Vert\nab_\ast \zeta\right\Vert}^2}_{4N} + \int_0^\infty \sigma {{\left\Vert\nab_\ast \zeta(\cdot,t)\right\Vert}^2}_{4N+1/2} dt \ls \sigma {{\left\Vert\nab_\ast \zeta(0)\right\Vert}^2}_{4N} + \sigma \sum_{j=1}^{2N} {{\left\Vert\dt^j \zeta(0)\right\Vert}^2}_{4N-2j+3/2}.$$
For each $j=0,\dotsc,2N$ let $\varphi_j \in C_c^\infty({\mathbb{R}^{}})$ be such that $\varphi_j^{(k)}(0) = \delta_{j,k}$ for $k=0,\dotsc,2N$ (here $(k)$ is the number of derivatives and $\delta_{j,k}$ is the Kronecker delta). For the same $j$ let $f_j = \dt^j \zeta(0)$. Then $f_0 \in H^{4N+1/2}(\Sigma)$, $\sqrt{\sigma} \nab_\ast f_0 \in H^{4N}(\Sigma)$, and $f_j \in H^{4N-2j+3/2}(\Sigma)$ for $j=1,\dotsc,2N$.
We will construct $\zeta$ as a sum $\zeta = \sum_{j=0}^{2N} F_j$. To construct the $F_j$ we must break to cases, considering $j=0$ and $j\ge 1$ separately. We begin with the latter case, in which case we use the Fourier transform given by for the construction. We define $F_j$ via its Fourier coefficients: $$\hat{F}_j(\xi,t) = \varphi_j(t {\left\langle \xi \right\rangle}^2) \hat{f}_j(\xi) {\left\langle \xi \right\rangle}^{-2j},$$ where ${\left\langle \xi \right\rangle} = \sqrt{1+ {\left\vert\xi\right\vert}^2}$.
By construction, $\dt^k \hat{F}_j(\xi,t) = \varphi_j^{(k)}(t {\left\langle \xi \right\rangle}^2) \hat{f}_j(\xi) {\left\langle \xi \right\rangle}^{2(k-j)}$, and hence $\dt^k F(\cdot,0) = \delta_{j,k} f_j$. We estimate $$\begin{split}
{{\left\Vert\dt^k F_j(\cdot,t)\right\Vert}^2}_{4N-2k+3/2} & = \sum_{\xi} {\left\langle \xi \right\rangle}^{2(4N-2k+3/2)} {\left\vert\varphi_j^{(k)}(t {\left\langle \xi \right\rangle}^2)\right\vert}^2 {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(2k-2j)} d\xi \\
& = \sum_\xi {\left\vert\varphi_j^{(k)}(t {\left\langle \xi \right\rangle}^2)\right\vert}^2 {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(4N-2j+3/2)} d\xi \\
&\ls {{\left\Vert\varphi_j^{(k)}\right\Vert}^2}_{L^\infty({\mathbb{R}^{}})} {{\left\Vertf_j\right\Vert}^2}_{4N-2j+3/2},
\end{split}$$ so $$\label{ex_e1}
\sup_{t\ge 0} {{\left\Vert\dt^k F_j(\cdot,t)\right\Vert}^2}_{ 4N-2k+3/2} \ls {{\left\Vertf_j\right\Vert}^2}_{4N-2j+3/2} \text{ for }k=0,\dotsc,2N.$$ Similarly, $$\begin{split}
\int_0^\infty{{\left\Vert\dt^k F_j(\cdot,t)\right\Vert}^2}_{4N-2k+5/2}dt &= \int_0^\infty \sum_\xi {\left\langle \xi \right\rangle}^{2(4N-2k+5/2)} {\left\vert\varphi_j^{(k)}(t {\left\langle \xi \right\rangle}^2)\right\vert}^2 {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(2k-2j)} dt\\
& = \int_0^\infty \sum_\xi {\left\vert\varphi_j^{(k)}(t {\left\langle \xi \right\rangle}^2)\right\vert}^2 {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(4N-2j+5/2)} dt \\
& = \sum_\xi {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(4N-2j+5/2)} \left( \int_0^\infty {\left\vert\varphi_j^{(k)}(t {\left\langle \xi \right\rangle}^2)\right\vert}^2 dt\right) \\
& = \sum_\xi {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(4N-2j+5/2)} \left( \frac{1}{{\left\langle \xi \right\rangle}^2}\int_0^\infty {\left\vert\varphi_j^{(k)}(r)\right\vert}^2 dr\right) \\
&= {{\left\Vert\varphi_j^{(k)}\right\Vert}^2}_{L^2({\mathbb{R}^{}})} \sum_\xi {\left\vert\hat{f}_j(\xi)\right\vert}^2 {\left\langle \xi \right\rangle}^{2(4N-2j+3/2)} \\
&\ls {{\left\Vert\varphi_j^{(k)}\right\Vert}^2}_{L^2({\mathbb{R}^{}})}{{\left\Vertf_j\right\Vert}^2}_{4N-2j+3/2},
\end{split}$$ so $$\label{ex_e2}
\int_0^\infty{{\left\Vert\dt^k F_j(\cdot,t)\right\Vert}^2}_{4N-2k+5/2}dt \ls {{\left\Vertf_j\right\Vert}^2}_{4N-2j+3/2} \text{ for }k=0,\dotsc,2N+1.$$
It remains to handle the case $j=0$. We define $\hat{F}_0(\xi,t) = \varphi_0(t{\left\langle \xi \right\rangle}) \hat{f}_0(\xi)$. Then we may argue as above to deduce the bounds $$\label{ex_e3}
\sup_{t\ge 0} \left( {{\left\VertF_0(\cdot,t)\right\Vert}^2}_{4N+1/2} + \sum_{k=1}^{2N} {{\left\Vert \dt^k F_0(\cdot,t)\right\Vert}^2}_{4N-2k+3/2} \right) \ls {{\left\Vertf_0\right\Vert}^2}_{4N+1/2},$$ and $$\label{ex_e4}
\int_0^\infty\left({{\left\VertF_0(\cdot,t)\right\Vert}^2}_{4N+1}+ {{\left\Vert\dt F_0(\cdot,t)\right\Vert}^2}_{4N} +\sum_{k=2}^{2N+1}{{\left\Vert\dt^k F_0(\cdot,t)\right\Vert}^2}_{4N-2k+5/2} \right)dt \ls {{\left\Vertf_0\right\Vert}^2}_{4N+1/2}$$ for all $k \ge 0$. We may also estimate $$\label{ex_e5}
\sup_{t\ge 0} {{\left\Vert \nab_\ast F_0(\cdot,t)\right\Vert}^2}_{4N} + \int_0^\infty {{\left\Vert\nab_\ast F_0(\cdot,t)\right\Vert}^2}_{4N+1/2} dt \ls {{\left\Vert\nab_\ast f_0\right\Vert}^2}_{4N}.$$
Now, since we set $\zeta = \sum_{j=0}^{2N} F_j$ we may sum , , , and to deduce that holds. To prove we sum and with $k=0$ and : $$\begin{gathered}
\sup_{t\ge 0} \sigma {{\left\Vert\nab_\ast \zeta(\cdot,t)\right\Vert}^2}_{4N} + \int_0^\infty \sigma {{\left\Vert\nab_\ast \zeta(\cdot,t)\right\Vert}^2}_{4N+1/2} dt
\ls \sup_{t\ge 0}\sigma {{\left\Vert\nab_\ast F_1(\cdot,t)\right\Vert}^2}_{4N} \\
+ \int_0^\infty \sigma {{\left\Vert\nab_\ast F_0(\cdot,t)\right\Vert}^2}_{4N+1/2} dt
+ \sup_{t\ge 0} \sigma \sum_{j=1}^{2N}{{\left\Vert \nab_\ast F_j(\cdot,t) \right\Vert}^2}_{4N}
+ \int_0^\infty \sigma \sum_{j=1}^{2N}{{\left\Vert\nab_\ast F_j(\cdot,t)\right\Vert}^2}_{4N+1/2}
\\
\ls \sigma {{\left\Vert\nab_\ast \zeta(0)\right\Vert}^2}_{4N} + \sigma \sum_{j=1}^{2N}{{\left\Vert \dt^j \zeta(0) \right\Vert}^2}_{4N-2j+3/2}.\end{gathered}$$
We will also need the following simple variant of Proposition \[extension\_eta\]. We omit the proof for the sake of brevity.
\[extension\_Xi\] Suppose that $\Xi(0) \in H^{4N-1}(\Sigma)$ and that for $j=1,\dotsc,2N$ we have that $\dt^j \Xi(0) \in H^{4N-2j-1/2}(\Sigma)$. Then there exists an extension $\Xi$, achieving the initial data, such that $$\dt^j \Xi \in L^\infty([0,\infty); H^{4N-2j-1}(\Sigma)) \cap L^2([0,\infty);H^{4N-2j}(\Sigma)) \text{ for }j=1,\dotsc,2N.$$ Moreover, we have the estimates $$\begin{gathered}
\sum_{j=0}^{2N-1} \sup_{t>0} {{\left\Vert\dt^j \Xi(t)\right\Vert}^2}_{4N-2j-1} + \sum_{j=0}^{2N} \int_0^\infty {{\left\Vert\dt^j \Xi(t)\right\Vert}^2}_{ 4N-2j}dt
\\
\ls {{\left\Vert\Xi(0)\right\Vert}^2}_{4N-1} + \sum_{j=1}^{2N} {{\left\Vert\dt^j \Xi(0)\right\Vert}^2}_{4N-2j-1}\end{gathered}$$
Finally, we record a similar extension for use on $q$. The proof follows from an argument like that used in Proposition \[extension\_eta\] and is thus omitted.
\[extension\_q\] Suppose that $p(0) \in H^{4N}(\Omega)$ and that for $j=1,\dotsc,2N$ we have that $\dt^j p(0) \in H^{4N-2j+1}(\Omega)$. Then there exists an extension $p$, achieving the initial data, such that $$\begin{split}
p &\in L^\infty([0,\infty); H^{4N}(\Omega)) \cap L^2([0,\infty);H^{4N+1/2}(\Omega)), \\
\dt p & \in L^\infty([0,\infty); H^{4N-1}(\Omega)) \cap L^2([0,\infty);H^{4N-1/2}(\Omega)), \\
\dt^j p & \in L^\infty([0,\infty); H^{4N-2j+1}(\Omega)) \cap L^2([0,\infty);H^{4N-2j+2}(\Omega)) \text{ for }j=2,\dotsc,2N, \\
\dt^{2N+1} p & \in L^2([0,\infty);H^{0}(\Omega))
\end{split}$$ Moreover, we have the estimate $$\begin{gathered}
\sup_{t \ge 0} \left( {{\left\Vertp(t)\right\Vert}^2}_{4N} + \sum_{j=1}^{2N} {{\left\Vert\dt^j p(t)\right\Vert}^2}_{4N-2j+1} \right) \\
+ \int_0^\infty \left( {{\left\Vert p(t)\right\Vert}^2}_{4N+1/2} +{{\left\Vert\dt p(t)\right\Vert}^2}_{4N-1/2}+ \sum_{j=2}^{2N+1}{{\left\Vert \dt^j p(t)\right\Vert}^2}_{4N-2j +2} \right) dt \\
\ls {{\left\Vertp(0)\right\Vert}^2}_{4N} +\sum_{j=1}^{2N} {{\left\Vert\dt^j p(0)\right\Vert}^2}_{4N-2j+1}.\end{gathered}$$
Acknowledgements {#acknowledgements .unnumbered}
================
I. Tice would like to thank the Laboraroire Jacques-Louis Lions at Université Pierre et Marie Curie for their hospitality during his visit.
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abstract: 'The construction of the non-logarithmic conformal field theory based on ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ is revisited. Without resorting to free-field methods, the determination of the spectrum and fusion rules is streamlined and the $\beta \gamma$ ghost system is carefully derived as the extended algebra generated by the unique finite-order simple current. A brief discussion of modular invariance is given and the Verlinde formula is explicitly verified.'
address: |
Theory Group, DESY\
Notkestraße 85\
D-22603, Hamburg, Germany
author:
- David Ridout
title: '${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$: A Case Study'
---
[^1]
Introduction {#secIntro}
============
Fractional level [Wess-Zumino-Witten]{} models were posited long ago as a tool to construct the non-unitary minimal models. Their introduction was facilitated by the discovery of Kac and Wakimoto [@KacMod88; @KacMod88b; @KacCla89] of a class of irreducible representations of affine algebras whose (normalised) characters carry a representation of the modular group ${{\mathsf{SL}} \left( 2 ; {\mathbb{Z}}\right)}$. These so-called admissible representations include, but are not limited to, the integrable representations from which the rational [Wess-Zumino-Witten]{} models are constructed. The integrable representations necessarily have non-negative integer levels, so the fractional level models must be constructed from non-integrable admissible representations.
Whereas the rational models have a well-known geometric description as non-linear sigma models on compact (simple) group manifolds [@WitNon84], this cannot be generalised to fractional level models. Indeed, the action defining such a sigma model is ambiguous unless the level is an integer[^2] [@NovMul81]. Of course an action is not a prerequisite for constructing a [conformal field theory]{}, especially a non-unitary one, and one can proceed in a purely algebraic manner from the representation theory of the appropriate affine algebra.
At each level there are only finitely many admissible representations. Indeed, this number is almost always zero, and levels for which this is not the case are sometimes referred to as being admissible themselves. This finiteness property led to the conjecture that such algebraically-defined fractional level [Wess-Zumino-Witten]{} models were also rational [conformal field theories]{}. Indeed, the characters of the admissible representations close under the modular group action and this action is unitary, as in the integer level case. However, it was quickly realised that the Verlinde formula, which relates the fusion coefficients of the theory to the modular $S$-matrix [@VerFus88], gives negative fusion coefficients in general [@KohFus88; @BerFoc90; @MatFra90]. Moreover, the matrix representing conjugation, $S^2$, was also observed to contain negative entries. Even worse, subsequent investigations determining the fusion rules from the decoupling of the null vectors of the representations (in correlation functions) gave different results [@AwaFus92; @FeiFus94; @AndOpe95; @PetFus96; @FurAdm97]. Whilst there have been some proposals for how to interpret these negative coefficients [@BerFoc90; @RamNew93], this resulted in a general feeling that the fractional level models suffered from an “intrinsic sickness” [@DiFCon97] and that only their coset theories were well-defined.
All of these efforts were hampered by the seemingly natural assumption that the fusion of admissible modules decomposed into direct sums of admissible modules. This was pointed out by Gaberdiel [@GabFus01], who studied the “smallest” fractional level model corresponding to the affine algebra ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ at level $\tfrac{-4}{3}$ (smallest in the sense of having the minimal number of admissible representations). Using a purely algebraic algorithm to compute the fusion rules of the admissible representations [@NahQua94; @GabInd96], rather than the Verlinde formula or correlation functions, he was able to show that fusing admissible representations sometimes resulted in reducible but indecomposable representations of the type found in [logarithmic conformal field theory]{}. Furthermore, he also gave strong evidence that these fusions sometimes produced representations for which the conformal dimensions of the states were *not* bounded below.
This may seem like a textbook definition of “intrinsic sickness”, but there is a very natural way to understand these unbounded-below representations. The fusion rules of the rational [Wess-Zumino-Witten]{} models respect, in a natural way, the automorphisms of the underlying affine algebra. It is therefore natural to expect that the fusion rules of the fractional level models will too, and explicit computations completely support this expectation (however, we mention that no proof of this property has yet been advanced). Whereas these automorphisms transform integrable representations into one another, the same is not true for the admissible representations. There, one finds that the infinite group of affine algebra automorphisms leads to an infinite number of distinct transformed representations, only a finite number of which have conformal dimensions which are bounded below.
This ruins the hope that fractional level [Wess-Zumino-Witten]{} models would be rational [conformal field theories]{}, but in a manner which is easy to control. The inherent irrationality seems to be restricted to these automorphic copies (in modern parlance, the images under *spectral flow*) of the admissible representations. Of course, there is still the realisation that these models are logarithmic — work on understanding the nature of the indecomposable representations that arise in these models is still in its infancy. Nevertheless, this provides a convenient handle with which one can try to understand the true nature of fractional level models. It is no longer appropriate to regard these models as poorly-defined curiosities. Rather, it is natural to regard these models as fundamental building blocks for irrational and logarithmic [conformal field theories]{}, much as their integer level cousins are for rational theories.
With this in mind, another fractional level model was studied in [@LesSU202], ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ at level $\tfrac{-1}{2}$. This model is particularly interesting to field theorists as it has been known for some time (see [@GurRel98] for a statement to this effect) that the $\beta \gamma$ system of ghost fields exhibits the same symmetry. In other words, this fractional level model is equivalent to a free field theory. Somewhat perversely, the authors of [@LesSU202] did not analyse ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ using this equivalence, but instead realised it in terms of a different ghost system and a lorentzian boson. The advantage of this approach was that they were also able to divine the existence of unbounded-below representations in terms of “multiple-twist” fields, albeit at a formidable computation cost. More interestingly, the theory they explored was not logarithmic, in contrast to the $k = \tfrac{-4}{3}$ theory of [@GabFus01].
In this note, we revisit the construction of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ [conformal field theory]{}. Our aim is threefold. First, we emphasise that this theory is in fact extremely easy to analyse if one abandons free-field constructions. We do so here in an expository fashion which makes it clear how to generalise to other admissible levels. Indeed, $k = \tfrac{-1}{2}$ is the first of an infinite series of admissible levels $k = \tfrac{1}{2} {\left( 2m-1 \right)}$ ($m \in {\mathbb{N}}$) which give rise to non-[logarithmic conformal field theories]{}. We expect that all other admissible levels give rise to logarithmic theories. Our second aim is to make precise the relation between the algebra ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ and the ghost algebra considered in [@LesSU202]. This provides an excellent example of the extended algebra formalism of [@RidSU206; @RidMin07], in which all the subtleties uncovered there are present. Our last aim is to point out that there is nothing mysterious or “sick” about the modular properties of this theory. The partition functions, conjugation matrices and Verlinde formula all work *exactly* as expected.
The organisation is as follows. After first introducing our notations and conventions ([Section \[secAlg\]]{}), we derive the structure of the irreducible vacuum module in [Section \[secVacMod\]]{}. It is worthwhile seeing explicitly in at least one case that admissibility just means that the corresponding Verma module has the same “braided” singular vector structure as the integrable modules. This gives us the “null-vector constraints” on the other representations of the theory, thence the other admissible [highest weight modules]{} ([Section \[secAdmiss\]]{}). Character formulae for all these are derived.
We then proceed to the computation of the fusion rules of the theory ([Section \[secFusion\]]{}). This involves considering the purely algebraic algorithm of Nahm [@NahQua94] and Gaberdiel-Kausch [@GabInd96]. Whilst this algorithm is computationally intensive, we note that by making two very plausible assumptions, we do not actually have to perform any computations and can proceed using only logical consequences of the algorithm. First, we assume that the irreducible vacuum module acts as the fusion identity. We could of course prove this easily using the fusion algorithm, but prefer to note that this assumption is consistent unless we uncover a logarithmic partner state to the vacuum (which we do not). Logic alone then allows us to compute the fusion rules of the admissible modules. In particular, we prove that certain fusions of admissible modules lead to modules whose conformal dimensions are unbounded below. The second assumption then allows us to identify these modules. This is the assumption that the fusion rules respect the *spectral flow* automorphisms of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$.
This then gives us a complete infinite spectrum of irreducible modules which closes under fusion. In [Section \[secChar\]]{}, we determine the full set of characters of the theory, noting that they are not all independent as one might expect from rational theories. Instead, there are only four linearly independent characters. We argue, following [@LesSU202], that a module is determined by its character *and* a prescription of how to expand it. The latter is implicit in rational theories, but the presence of unbounded-below modules (and non-integrable modules in general) forces its explicit acknowledgement here. The consequent lack of a bijection between the modules and the characters therefore leads us to introduce a Grothendieck ring of characters.
[Sections \[secGhost\] – \[secGhostReps\]]{} are devoted to a detailed study of the extended algebra of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ algebra, which is the $\beta \gamma$ ghost system. In [Section \[secGhost\]]{}, we show that the (chiral) primary fields defining this extension cannot be taken to be mutually bosonic with the affine currents, and that associativity of the operator product algebra forces the introduction of an additional operator into the theory. The bosonic ghost fields are defined, but they are not mutually bosonic with respect to the affine currents either. At issue here is the definition of the adjoint, an integral part of any symmetry algebra. In [Section \[secGhost2\]]{}, we change the adjoint and repeat the analysis of the previous section finding satisfying simplifications — all fields are mutually bosonic and the operator product algebra is associative without need of additional operators.
We then briefly discuss ([Section \[secGhostReps\]]{}) the representation theory of this extended algebra, remarking upon the consistency of the monodromy charge and the lifted extended algebra spectral flow. The Verma modules of the extended algebra are verified to be irreducible — in this sense the $\beta \gamma$ ghost system may be said to be free — and fermionic character formulae for them and their ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ counterparts are derived. These formulae give simple expressions for the string functions of all the modules of the theory.
Finally, we conclude by reconsidering the modular properties of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ theory in [Section \[secMod\]]{}. We derive the $S$ and $T$-matrices of the theory, verify that they are symmetric and unitary, and write down a complete set of modular invariants. Moreover, we check that $S^2$ represents conjugation and the Verlinde formula recovers the fusion coefficients in the Grothendieck ring of characters.
There are also two appendices, the second of which ([Appendix \[appTheta\]]{}) is just a summary of our notations and conventions for Jacobi theta functions. The first, [Appendix \[appSpecFlow\]]{}, gives a detailed description of the spectral flow automorphisms as affine Weyl group translations (by elements of the coroot lattice) and affine outer automorphisms (as translations by elements of the dual root lattice). We are not aware of a comprehensive discussion of this viewpoint in the literature, so we hope that this will be of independent use in the future.
Algebraic Preliminaries {#secAlg}
=======================
Let ${{\mathfrak{sl}} \left( 2 \right)}$ be the complex Lie algebra spanned by three generators $E$, $H$ and $F$ subject to the commutation relations $${\bigl[ H , E \bigr]} = 2 E, \qquad {\bigl[ E , F \bigr]} = H \qquad \text{and} \qquad {\bigl[ H , F \bigr]} = -2 F.$$ We define the Killing form to be the trace of the product in the defining (fundamental) two-dimensional representation (equivalently, $1/4$ of the trace of the product in the adjoint representation). This gives $$\label{eqnKilling}
{\kappa \bigl( H , H \bigr)} = 2 \qquad \text{and} \qquad {\kappa \bigl( E , F \bigr)} = 1,$$ with all other combinations vanishing. The affine Kac-Moody algebra ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ is then the vector space $${{\mathfrak{sl}} \left( 2 \right)} \otimes {\mathbb{C}}{\left[ t ; t^{-1} \right]} \oplus \operatorname{span}_{{\mathbb{C}}} {\left\{ K , L_0 \right\}}$$ equipped with the commutation relations
\[eqnCommRels\] $$\begin{gathered}
{\bigl[ J^a_m , J^b_n \bigr]} = {\bigl[ J^a , J^b \bigr]}_{m+n} + m {\kappa \bigl( J^a , J^b \bigr)} \delta_{m+n,0} K, \qquad {\bigl[ J^a_m , K \bigr]} = 0, \\
{\bigl[ L_0 , J^a_m \bigr]} = -m J^a_m \qquad \text{and} \qquad {\bigl[ L_0 , K \bigr]} = 0.\end{gathered}$$
Here, $J^a_m$ denotes $J^a \otimes t^m$, where $J^a$ can represent $H$, $E$ or $F$. We are generally interested in representations of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ on which the central element $K$ acts as $k$ times the identity, for some common scalar $k$ called the level. In what follows, we will be principally interested in the case where $k = \tfrac{-1}{2}$.
As is well known, the [universal enveloping algebra]{} of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ contains a subalgebra isomorphic to the ([universal enveloping algebra]{} of the) Virasoro algebra (when $k \neq -2$). This is the Sugawara construction. Here, the Virasoro elements are realised as quadratic elements normally ordered in the standard way: $$\label{eqnDefVir}
L_n = \frac{1}{2 {\left( k+2 \right)}} \sum_{r \in {\mathbb{Z}}} {{} : \frac{1}{2} H_r H_{n-r} + E_r F_{n-r} + F_r E_{n-r} : {}}.$$ As usual, we will identify $L_0 \in {{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ with the quadratic element $L_0$ constructed in [Equation (\[eqnDefVir\])]{}. The central charge defined by the Sugawara construction is $c = 3k / {\left( k+2 \right)}$.
We define a triangular decomposition of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ as follows: The span of $H_0$, $K$ and $L_0$ defines the Cartan subalgebra, the raising operators are $E_{n-1}$, $H_n$ and $F_n$ for $n \geqslant 1$, and the adjoint is defined by $$\label{eqnDefAdj}
E_n^{\dag} = F_{-n}, \qquad H_n^{\dag} = H_{-n}, \qquad K^{\dag} = K \qquad \text{and} \qquad L_n^{\dag} = L_{-n}.$$ We can now talk about [highest weight states]{} and Verma modules. It is easy to check from [Equation (\[eqnDefVir\])]{} that an affine [highest weight state]{} with ${{\mathfrak{sl}} \left( 2 \right)}$-weight ($H_0$-eigenvalue) $\lambda$ has conformal dimension ($L_0$-eigenvalue) $$h_{\lambda} = \frac{\lambda {\left( \lambda + 2 \right)}}{4 {\left( k+2 \right)}}.$$ The ${{\mathfrak{sl}} \left( 2 \right)}$-weight $\lambda$, conformal dimension $h$ and the level $k$ completely determine an ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-weight ${\widehat{\lambda}} = {\left( \lambda , k , h \right)}$. As the level is given and the conformal dimension of a [highest weight state]{} is determined by its ${{\mathfrak{sl}} \left( 2 \right)}$-weight, it follows that an ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-Verma module is characterised solely by the latter. We therefore denote Verma modules by ${{\widehat{\mathcal{V}}}_{\lambda}}$.
The fundamental question to ask about Verma modules concerns their reducibility. If a Verma module contains a proper submodule, then this submodule is generated by singular vectors, non-trivial descendant [highest weight states]{}. Quotienting ${{\widehat{\mathcal{V}}}_{\lambda}}$ by its maximal proper submodule gives the corresponding irreducible module ${{\widehat{\mathcal{L}}}_{\lambda}}$. To find singular vectors, we can use the fact that Verma modules come equipped with a unique (up to normalisation) invariant inner product defined by the adjoint (\[eqnDefAdj\]), the Shapovalov form. With respect to this form, the (non-trivial) singular vectors and their descendants are all null, meaning that their norm is zero. The presence of such null states can be detected by computing the determinant of the Shapovalov form in each affine weight space.
Happily, there is an explicit form for this determinant, given by the Kac-Kazhdan formula [@KacStr79]: The Shapovalov determinant of ${{\widehat{\mathcal{V}}}_{\lambda}}$ in the weight space ${\left( \lambda - \mu , k , h_{\lambda} + m \right)}$ is $$\begin{gathered}
\label{eqnKacKazhdan}
{{\textstyle \det_{\lambda}} \left( \mu , m \right)} = \prod_{\ell=1}^{\infty} \Biggl\{ {\left( \lambda + 1 - \ell \right)}^{{P \left( -\mu + 2 \ell , m \right)}} \prod_{n=1}^{\infty} \bigl( \lambda + 1 + n {\left( k+2 \right)} - \ell \bigr)^{{P \left( -\mu + 2 \ell , m - n \ell \right)}} \Biggr. \\
\Biggl. \cdot \bigl( -\lambda - 1 + n {\left( k+2 \right)} - \ell \bigr)^{{P \left( -\mu - 2 \ell , m - n \ell \right)}} \bigl( n {\left( k+2 \right)} \bigr)^{{P \left( -\mu , m - n \ell \right)}} \Biggr\},\end{gathered}$$ where ${P \left( \mu , m \right)}$ denotes the multiplicity with which the weight ${\left( \mu , 0 , m \right)}$ appears in the module ${{\widehat{\mathcal{V}}}_{0}}$ (this is independent of $k$). The presence of a singular vector in ${{\widehat{\mathcal{V}}}_{\lambda}}$ is signalled by the vanishing of one of the factors appearing in this formula and the vanishing of the arguments of the function $P$ occurring in the corresponding exponent (non-vanishing arguments of this $P$ in general correspond to descendants of the singular vector). We will refer to weights which admit a singular vector as singular weights.
Vacuum Module Structure {#secVacMod}
=======================
We now specialise to $k = \tfrac{-1}{2}$, with the aim of constructing a [conformal field theory]{}. This theory will therefore have central charge $c = -1$. The first step is to determine a vacuum module. By definition, the vacuum ${\bigl\lvert 0 \bigr\rangle}$ is an ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-[highest weight state]{} which is also annihilated by all the zero-modes, in particular by $H_0$ and $F_0$. The vacuum module is therefore a quotient module of ${{\widehat{\mathcal{V}}}_{0}}$. We can analyse these quotients by determining the singular vector structure of the Verma module, and to do this we use the Kac-Kazhdan formula (\[eqnKacKazhdan\]).
Setting $\lambda = 0$ in this formula, we see that the determinant vanishes when $$\ell = 1, \qquad \ell = \frac{3n}{2} + 1 \qquad \text{or} \qquad \ell = \frac{3n}{2} - 1 \qquad \text{($n \in 2 {\mathbb{Z}}_+$).}$$ In the first case, the arguments of $P$ in the corresponding exponent vanish if $\mu = 2 \ell = 2$ and $m = 0$, indicating that the singular vector has weight ${\left( -2,\tfrac{-1}{2},0 \right)}$. This is clearly the singular vector $F_0 {\bigl\lvert 0 \bigr\rangle}$ which is set to zero by definition. The other two cases are more interesting and the weights of the corresponding singular vectors are found to be $$\label{eqnSW1}
{\left( -6m-2,\frac{-1}{2},2m {\left( 3m+1 \right)} \right)} \qquad \text{and} \qquad {\left( 6m-2,\frac{-1}{2},2m {\left( 3m-1 \right)} \right)} \qquad \text{($m = \frac{n}{2} \in {\mathbb{Z}}_+$)},$$ respectively. The first few singular weights are therefore $${\left( 4,\frac{-1}{2},4 \right)}, \quad {\left( -8,\frac{-1}{2},8 \right)}, \quad {\left( 10,\frac{-1}{2},20 \right)}, \quad {\left( -14,\frac{-1}{2},28 \right)}, \quad \ldots$$ These weights do not determine the singular vector itself, but it can be shown that every weight space of a Verma module admits at most one singular vector.
Unfortunately, these are not the only singular weights of ${{\widehat{\mathcal{V}}}_{0}}$. We also have to check for singular vectors which are descended from those we have already found. In other words, we should check the submodules which the known singular vectors generate for further singular vectors. Repeating the above Kac-Kazhdan analysis for the submodule generated by the singular weight ${\left( -2,\tfrac{-1}{2},0 \right)}$, we find further singular weights of the form $$\label{eqnSW2}
{\left( -6m,\frac{-1}{2},2m {\left( 3m-1 \right)} \right)} \qquad \text{and} \qquad {\left( 6m,\frac{-1}{2},2m {\left( 3m+1 \right)} \right)} \qquad \text{($m \in {\mathbb{Z}}_+$)}.$$ These describe two series of singular vectors which are completely disjoint from those found above. The first few weights are $${\left( -6,\frac{-1}{2},4 \right)}, \quad {\left( 6,\frac{-1}{2},8 \right)}, \quad {\left( -12,\frac{-1}{2},20 \right)}, \quad {\left( 12,\frac{-1}{2},28 \right)}, \quad \ldots$$
However, the weight ${\left( -2,\tfrac{-1}{2},0 \right)}$ and those given in (\[eqnSW1\]) and (\[eqnSW2\]) exhaust the singular weights of ${{\widehat{\mathcal{V}}}_{0}}$. This is not hard to check explicitly: For example, the singular weights descended from that of weight ${\left( -6m-2,\frac{-1}{2},2m {\left( 3m+1 \right)} \right)}$ of (\[eqnSW1\]) all have the form $$\begin{split}
{\left( -6 {\left( m'-m \right)},\frac{-1}{2},2 {\left( m'-m \right)} {\left( 3 {\left( m'-m \right)} - 1 \right)} \right)} \qquad \text{($m' > 2m$)} \\
\text{or} \qquad {\left( 6 {\left( m'+m \right)},\frac{-1}{2},2 {\left( m'+m \right)} {\left( 3 {\left( m'+m \right)} + 1 \right)} \right)} \qquad \text{($m' \in {\mathbb{Z}}_+$),}
\end{split}$$ which are both of the form given in (\[eqnSW2\]). It follows now that the singular vector structure of ${{\widehat{\mathcal{V}}}_{0}}$ is as shown in [Figure \[figVacVerMod\]]{}. Note the braiding pattern familiar from the integrable modules (and the Virasoro algebra).
\[\]\[\][$\scriptstyle {\left( 0,0 \right)}$]{} \[\]\[\][$\scriptstyle {\left( -2,0 \right)}$]{} \[\]\[\][$\scriptstyle {\left( 4,4 \right)}$]{} \[\]\[\][$\scriptstyle {\left( -6,4 \right)}$]{} \[\]\[\][$\scriptstyle {\left( 6,8 \right)}$]{} \[\]\[\][$\scriptstyle {\left( -8,8 \right)}$]{} \[\]\[\][$\scriptstyle {\left( 10,20 \right)}$]{} \[\]\[\][$\scriptstyle {\left( -12,20 \right)}$]{} \[\]\[\][$\scriptstyle {\left( 12,28 \right)}$]{} \[\]\[\][$\scriptstyle {\left( -6m+4,2 {\left( m-1 \right)} {\left( 3m-2 \right)} \right)}$]{} \[\]\[\][$\scriptstyle {\left( 6m-2,2m {\left( 3m-1 \right)} \right)}$]{} \[\]\[\][$\scriptstyle {\left( -6m,2m {\left( 3m-1 \right)} \right)}$]{} \[\]\[\][$\scriptstyle {\left( 6m,2m {\left( 3m+1 \right)} \right)}$]{} \[\]\[\][${{\widehat{\mathcal{V}}}_{0}} \qquad k = \frac{-1}{2}$]{}
![The singular vector structure of the vacuum Verma module at level $\tfrac{-1}{2}$. Each singular vector is labelled by its ${{\mathfrak{sl}} \left( 2 \right)}$-weight and conformal dimension (respectively).[]{data-label="figVacVerMod"}](Affsl2VacMod){width="13cm"}
It follows that the descendant singular vectors of ${{\widehat{\mathcal{V}}}_{0}}$ are generated by the two singular vectors of weights ${\left( -2,\tfrac{-1}{2},0 \right)}$ and ${\left( 4,\tfrac{-1}{2},4 \right)}$. The former is the vector $F_0 {\bigl\lvert 0 \bigr\rangle}$ which we have already set to zero, so we see that there are only two possible choices for the vacuum module. Either we set the dimension $4$ singular vector to zero, or we do not. We choose to set this singular vector to zero, thereby taking the vacuum module to be the irreducible quotient ${{\widehat{\mathcal{L}}}_{0}}$. The alternative, in which this singular vector is not set to zero, will undoubtedly lead to a logarithmic [conformal field theory]{} [@GabAlg03] (assuming it can be defined), which we do not want to consider here[^3].
The character for the vacuum Verma module is easily computed from the standard Poincaré-Birkhoff-Witt basis and has the form $$\label{eqnChV0}
{{\chi_{{{\widehat{\mathcal{V}}}_{0}}} \left( z ; q \right)}} = {\textstyle \operatorname{tr}_{{{\widehat{\mathcal{V}}}_{0}}}} z^{H_0} q^{L_0} = \frac{1}{\displaystyle \prod_{i=1}^{\infty} {\left( 1 - z^{-2} q^{i-1} \right)} {\left( 1 - q^i \right)} {\left( 1 - z^2 q^i \right)}} = \frac{1}{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^{2n} q^{n {\left( n+1 \right)} / 2}},$$ where we have used Jacobi’s triple product identity, [Equation (\[eqnJacobiTriple\])]{}, in the last step. It now follows from the embedding pattern of [Figure \[figVacVerMod\]]{} that the character of the (irreducible) vacuum module takes the form $$\begin{aligned}
\label{eqnCh0}
{{\chi_{{{\widehat{\mathcal{L}}}_{0}}} \left( z ; q \right)}} &= {\left[ 1 - \sum_{n=1}^{\infty} {\left( z^{6n-2} q^{2n {\left( 3n-1 \right)}} + z^{-6n+4} q^{2 {\left( n-1 \right)} {\left( 3n-2 \right)}} \right)} + \sum_{n=1}^{\infty} {\left( z^{6n} q^{2n {\left( 3n+1 \right)}} + z^{-6n} q^{2n {\left( 3n-1 \right)}} \right)} \right]} {{\chi_{{{\widehat{\mathcal{V}}}_{0}}} \left( z ; q \right)}} \notag \\
&= \frac{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( z^{-6n} - z^{6n-2} \right)} q^{2n {\left( 3n-1 \right)}}}{\displaystyle \prod_{i=1}^{\infty} {\left( 1 - z^{-2} q^{i-1} \right)} {\left( 1 - q^i \right)} {\left( 1 - z^2 q^i \right)}} = \frac{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( z^{-6n} - z^{6n-2} \right)} q^{2n {\left( 3n-1 \right)}}}{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^{2n} q^{n {\left( n+1 \right)} / 2}}.\end{aligned}$$
Admissible Representations {#secAdmiss}
==========================
Now that we have a vacuum module, we can ask if it constrains the spectrum of the theory. Since the vacuum module has a vanishing singular vector at ${{\mathfrak{sl}} \left( 2 \right)}$-weight $4$ and grade $4$, the answer is “yes” (setting $F_0 {\bigl\lvert 0 \bigr\rangle}$ to zero does not affect the spectrum as it is a part of the definition of the vacuum). To derive the constraints, we need the explicit form of this singular vector. There exist semi-explicit formulae for such singular vectors in the literature [@MalSin86; @BauFus93; @MatPri99], but it is not hard to compute it directly in this case. It turns out to be $$\label{eqnVacSV}
{\left( 156 E_{-3} E_{-1} - 71 E_{-2}^2 + 44 E_{-2} H_{-1} E_{-1} - 52 H_{-2} E_{-1}^2 - 16 F_{-1} E_{-1}^3 - 4 H_{-1}^2 E_{-1}^2 \right)} {\bigl\lvert 0 \bigr\rangle} = 0,$$ up to normalisation. By the state-field correspondence of [conformal field theory]{}, this vanishing singular vector gives rise to a vanishing chiral field whose modes must therefore annihilate any physical state [@FeiAnn92]. These are the constraints we seek.
Instead of considering this singular vector itself, it is convenient to consider its ${{\mathfrak{sl}} \left( 2 \right)}$-weight $0$ descendant obtained by acting with $F_0^2$. This descendant field is (up to normalisation) $$\begin{gathered}
\Lambda = 64 {{} : EEFF : {}} - 16 {{} : EHHF : {}} + 136 {{} : EH \partial F : {}} - 128 {{} : E \partial HF : {}} + 12 {{} : E \partial^2 F : {}} - 8 {{} : HHHH : {}} \\
- 200 {{} : \partial EHF : {}} +108 {{} : \partial E \partial F : {}} + 8 {{} : \partial HHH : {}} - 38 {{} : \partial H \partial H : {}} - 156 {{} : \partial^2 EF : {}} + 24 {{} : \partial^2 HH : {}} - \partial^3 H.\end{gathered}$$ Let ${\bigl\lvert \lambda \bigr\rangle}$ be a [highest weight state]{} with ${{\mathfrak{sl}} \left( 2 \right)}$-weight $\lambda$. Since the modes of $\Lambda$ must annihilate any physical state, $$0 = \Lambda_0 {\bigl\lvert \lambda \bigr\rangle} = {\left( -8 H_0^4 - 8 H_0^3 - 38 H_0^2 + 48 H_0^2 + 6 H_0 \right)} {\bigl\lvert \lambda \bigr\rangle} = -2 \lambda {\left( \lambda - 1 \right)} {\left( 2 \lambda + 1 \right)} {\left( 2 \lambda + 3 \right)} {\bigl\lvert \lambda \bigr\rangle},$$ implying that $\lambda = 0, 1, \tfrac{-1}{2}, \tfrac{-3}{2}$. These are the only allowed [highest weight states]{} of the theory. Their conformal dimensions are $0$, $\tfrac{1}{2}$, $\tfrac{-1}{8}$ and $\tfrac{-1}{8}$, respectively.
Now that we know the possible [highest weight states]{}, we can ask about the possible [highest weight modules]{}. For example, repeating the analysis of [Section \[secVacMod\]]{} shows that the Verma module ${{\widehat{\mathcal{V}}}_{1}}$ has singular vectors of weights ${\left( -3,\frac{-1}{2},\frac{1}{2} \right)}$, $${\left( \pm 6m-3,\frac{-1}{2},\frac{1}{2} + 2m {\left( 3m \mp 2 \right)} \right)} \qquad \text{and} \qquad {\left( \pm 6m+1,\frac{-1}{2},\frac{1}{2} + 2m {\left( 3m \pm 2 \right)} \right)} \qquad \text{($m \in {\mathbb{Z}}_+$).}$$ The embedding pattern is again of braided type and the two generating singular vectors are those with weights $$\label{eqnV1SVs}
{\left( -3,\frac{-1}{2},\frac{1}{2} \right)} \qquad \text{and} \qquad {\left( 3,\frac{-1}{2},\frac{5}{2} \right)}.$$ The ${{\mathfrak{sl}} \left( 2 \right)}$-weights of these singular vectors do not belong to the allowed set, hence we can conclude that all descendant singular vectors vanish in the physical module. It follows that the [highest weight state]{} with $\lambda = 1$ generates the irreducible module ${{\widehat{\mathcal{L}}}_{1}}$. Its character is $$\label{eqnCh1}
{{\chi_{{{\widehat{\mathcal{L}}}_{1}}} \left( z ; q \right)}} = z q^{1/2} \ \frac{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( z^{-6n} - z^{6n-4} \right)} q^{2n {\left( 3n-2 \right)}}}{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^{2n} q^{n {\left( n+1 \right)} / 2}}.$$ Similarly, one finds that the modules corresponding to $\lambda = \tfrac{-1}{2}$ and $\tfrac{-3}{2}$ are also irreducible with characters $$\begin{aligned}
{{\chi_{{{\widehat{\mathcal{L}}}_{-1/2}}} \left( z ; q \right)}} &= z^{-1/2} q^{-1/8} \ \frac{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( z^{6n} q^{n {\left( 6n+1 \right)}} - z^{-6n+2} q^{{\left( 2n-1 \right)} {\left( 3n-1 \right)}} \right)}}{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^{2n} q^{n {\left( n+1 \right)} / 2}} \label{eqnCh-1/2} \\
\text{and} \qquad {{\chi_{{{\widehat{\mathcal{L}}}_{-3/2}}} \left( z ; q \right)}} &= z^{-3/2} q^{-1/8} \ \frac{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( z^{6n} q^{n {\left( 6n-1 \right)}} - z^{-6n+4} q^{{\left( 2n-1 \right)} {\left( 3n-2 \right)}} \right)}}{\displaystyle \sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^{2n} q^{n {\left( n+1 \right)} / 2}}, \label{eqnCh-3/2}\end{aligned}$$ respectively. These are the *admissible* [highest weight modules]{} of Kac and Wakimoto [@KacMod88b]. We illustrate them in [Figure \[figAdmissReps\]]{}.
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Fusion and the Spectrum {#secFusion}
=======================
We turn now to the derivation of the fusion rules of the admissible modules. Normally, we could investigate this by computing $3$-point correlation functions of the primary fields. However, doing this requires making a number of non-trivial assumptions. In particular, we must assume that every field has a conjugate so that the matrix of $2$-point functions is non-degenerate. Moreover, we would also be implicitly assuming that we have already identified every field of the theory. Since there are no candidates within the admissible representations for the conjugate fields to the dimension $\tfrac{-1}{8}$ primaries, we must conclude that there are further fields to discover. But if we admit to not knowing the field content of the theory, then it follows that we cannot be sure that the $2$-point functions we will use in our fusion computations are non-degenerate. For example, it seems reasonable to declare that the vacuum is self-conjugate, so that the $2$-point function of the identity is constant. However, if subsequent fusion computations revealed that the vacuum had a logarithmic partner state, then it would follow from general principles [@FloBit03] that the conjugate field to the identity would be this logarithmic partner (and the $2$-point function of the identity would vanish identically), contradicting our original declaration.
For this reason, we will be careful and compute fusion using a purely algebraic method that makes no reference to correlation functions nor non-degeneracy. This is the algorithm of Nahm and Gaberdiel-Kausch [@NahQua94; @GabInd96]. Happily, the situation here is sufficiently simple that we will not have to make any explicit computations with this algorithm; we will be able to proceed with a few logical consequences which are easy to state (and hopefully understand).
In general, this algorithm constructs a representation $\Delta$ of the symmetry algebra on the fusion product of two modules ${\mathcal{M}_{1}}$ and ${\mathcal{M}_{2}}$. Decomposing this representation gives the fusion rule ${\mathcal{M}_{1}} {\times_{\! f}}{\mathcal{M}_{2}}$. In practice, one only constructs this representation to a chosen finite grade $g$ — all the “deeper” structure of the fused module is thrown away. The idea is to choose $g$ large enough that one obtains as much information as is required. The representation $\Delta$ is constructed within the *working space*, which for affine symmetry algebras consists of the tensor product of the zero-grade subspace of ${\mathcal{M}_{1}}$ and the subspace of ${\mathcal{M}_{2}}$ consisting of elements with grade at most $g$. The working space is then reduced by removing the so-called *spurious states* which reflect the vanishing of certain singular vectors of ${\mathcal{M}_{1}}$ and ${\mathcal{M}_{2}}$. This is achieved by employing three master equations [@GabInd96 Eqs. 2.2–2.4] iteratively on expressions formed from these singular vectors. The fusion representation space is thereby constructed (to grade $g$) when all spurious states are removed. The master equations then define the action of the symmetry algebra (that is $\Delta$) upon what remains.
It is extremely important to note that we need to assume that the conformal dimensions of the states composing each module are bounded from below[^4]. Then, we can define the *grade* of an arbitrary state in this module to be the difference between the dimension of the state and the minimal dimension. It should be clear that we require this bounded-below property for the modules we are fusing *and* the modules we generate via fusion. Indeed, when we say that the fusion representation space is constructed to grade $g$, we mean that upon decomposition, the structure of each component module is determined to grade $g$ in the above sense.
We will not need to enter into the details of this algorithm. Computing to grade $0$ turns out to suffice for our purposes, and so we will only determine the action of the ${{\mathfrak{sl}} \left( 2 \right)}$-subalgebra spanned by the zero-modes $E_0$, $H_0$ and $F_0$. The master equations give this action as $${\Delta \left( J_0 \right)} = J_0 \otimes \operatorname{id}+ \operatorname{id}\otimes J_0 \qquad \text{($J = E, H, F$),}$$ which is identical to the tensor product ${{\mathfrak{sl}} \left( 2 \right)}$-action (on ${{\mathfrak{sl}} \left( 2 \right)}$-modules). The fusion of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-modules to grade $0$ therefore only differs from the tensor product of the corresponding grade $0$ ${{\mathfrak{sl}} \left( 2 \right)}$-modules if there are non-trivial spurious states. We point out that if we find a spurious state, then acting upon it with any ${\Delta \left( J_0 \right)}$ must give another spurious state. The spurious states therefore form a representation of the zero-mode ${{\mathfrak{sl}} \left( 2 \right)}$-subalgebra.
Consider first fusing the vacuum module ${{\widehat{\mathcal{L}}}_{0}}$ with some other module ${\mathcal{M}_{}}$. We require only that the zero-grade states of ${\mathcal{M}_{}}$ form an irreducible ${{\mathfrak{sl}} \left( 2 \right)}$-module. The working space is then the tensor product of the trivial ${{\mathfrak{sl}} \left( 2 \right)}$-module with this irreducible ${{\mathfrak{sl}} \left( 2 \right)}$-module. There are vanishing singular vectors in at least one of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-modules, so there could be spurious states. But, the working space is isomorphic to a single irreducible ${{\mathfrak{sl}} \left( 2 \right)}$-module, so the existence of spurious states would mean that the fusion product is empty! The possibilities are therefore that fusing a module with the vacuum gives the module back again or nothing.
Suppose therefore that ${{\widehat{\mathcal{L}}}_{0}} {\times_{\! f}}{\mathcal{M}_{}}$ is empty for some module ${\mathcal{M}_{}}$ in our theory. As ${\mathcal{M}_{}}$ must have a conjugate representation ${\mathcal{M}_{}}^+$ in the theory, ${\mathcal{M}_{}} {\times_{\! f}}{\mathcal{M}_{}}^+ = {{\widehat{\mathcal{L}}}_{0}} + \ldots$. By hypothesis, the result of fusing the left hand side of this rule with ${{\widehat{\mathcal{L}}}_{0}}$ is empty, hence ${{\widehat{\mathcal{L}}}_{0}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{0}}$ must also be empty. But this implies that the vacuum is a null state, which requires the existence of a logarithmic partner (as we noted above). We may therefore proceed under the assumption that ${{\widehat{\mathcal{L}}}_{0}} {\times_{\! f}}{\mathcal{M}_{}}$ is not empty for any module in our theory — this will only be invalidated if we find that it leads to a logarithmic partner to the vacuum. As we will see, we do not find this outcome, hence it is consistent to insist that the irreducible vacuum module ${{\widehat{\mathcal{L}}}_{0}}$ acts as the fusion identity on every module in the theory[^5]. Note that it follows from this that the vacuum module is self-conjugate.
A more interesting computation is to determine the fusion of the module ${{\widehat{\mathcal{L}}}_{1}}$ with itself. Computing to grade $0$ again, we may regard the working space as the tensor product of the fundamental representation of ${{\mathfrak{sl}} \left( 2 \right)}$ with itself. This decomposes as the direct sum of the trivial and adjoint representations, so the working space contains a ${{\mathfrak{sl}} \left( 2 \right)}$-[highest weight state]{} of weight $2$. In the absence of any spurious states, this would imply that the fused module contains a ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-[highest weight state]{} of weight ${\left( 2, \tfrac{-1}{2}, \tfrac{4}{3} \right)}$. But this is forbidden by the vacuum singular vector ([Section \[secAdmiss\]]{}), so the weight $2$ [highest weight state]{} must be spurious. It then follows that the entire adjoint representation must also be spurious, so we are left with the trivial ${{\mathfrak{sl}} \left( 2 \right)}$-representation. If this were also spurious, then the fusion product would be empty. The requirement of a conjugate for ${{\widehat{\mathcal{L}}}_{1}}$ would then force ${{\widehat{\mathcal{L}}}_{0}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{1}}$ to be empty, contradicting the fact that ${{\widehat{\mathcal{L}}}_{0}}$ is the fusion identity. The ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-module corresponding to the trivial ${{\mathfrak{sl}} \left( 2 \right)}$-representation is clearly the vacuum module[^6], so we have derived the following fusion rule: $$\label{eqnFR1x1}
{{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{1}} = {{\widehat{\mathcal{L}}}_{0}}.$$ The module ${{\widehat{\mathcal{L}}}_{1}}$ is therefore also self-conjugate.
We can continue in a similar fashion to determine the fusion rules of the other admissible modules. In particular, we obtain $$\label{eqnFR1xAdmiss}
{{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-1/2}} = {{\widehat{\mathcal{L}}}_{-3/2}} \qquad \text{and} \qquad {{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-3/2}} = {{\widehat{\mathcal{L}}}_{-1/2}}$$ without fuss. The rest of the fusion rules are more delicate to analyse however. For example, considering ${{\widehat{\mathcal{L}}}_{-1/2}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-1/2}}$ to grade $0$ as above, the working space decomposes as an ${{\mathfrak{sl}} \left( 2 \right)}$-representation into an infinite direct sum[^7] of irreducibles whose highest weights are $-1, -3, -5, \ldots$. Proceeding as above, we would conclude that none of these highest weights are allowed, hence that all states are spurious and the fusion product is empty.
But as with ${{\widehat{\mathcal{L}}}_{1}}$, insisting on a conjugate for ${{\widehat{\mathcal{L}}}_{-1/2}}$, even if we have not yet identified it, again leads to a contradiction. The loophole is in trusting that a non-spurious ${{\mathfrak{sl}} \left( 2 \right)}$-[highest weight state]{} corresponds to an ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-[highest weight state]{}. We could trust this correspondence in the previous examples because the Nahm-Gaberdiel-Kausch algorithm gives us, grade by grade, the affine structure of the fused module. However, this algorithm does not make sense if the conformal dimensions of the states of the fused module are not bounded from below (recall that in this situation, the concept of grade is not defined). Before, this boundedness property was guaranteed because we only had finitely many irreducible representations of the zero-mode ${{\mathfrak{sl}} \left( 2 \right)}$-subalgebra. In the case at hand however, there are infinitely many such representations, so the conformal dimension need not be bounded from below. Indeed, we cannot even compute the conformal dimensions of the ${{\mathfrak{sl}} \left( 2 \right)}$-[highest weight states]{} without assuming something about the fused module structure. The correct conclusion to draw from our analysis is that either ${{\widehat{\mathcal{L}}}_{-1/2}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-1/2}}$ is empty, which leads to a contradiction, or that it gives a module whose states have arbitrarily negative conformal dimension.
In fact, it is not too difficult to justify directly that the second option is what actually occurs. To do this, we make use of the automorphisms of our symmetry algebra, in particular the *spectral flow* automorphisms (described in detail in [Appendix \[appSpecFlow\]]{}). For ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$, the spectral flow is freely generated by a single automorphism $\gamma$ which may be taken to act by (see [Equation (\[eqnSpecFlowA1\])]{})
\[eqnSF\] $$\begin{gathered}
{\gamma \left( E_n \right)} = E_{n-1}, \qquad {\gamma \left( H_n \right)} = H_n - \delta_{n,0} K, \qquad {\gamma \left( F_n \right)} = F_{n+1}, \\
{\gamma \left( K \right)} = K \qquad \text{and} \qquad {\gamma \left( L_0 \right)} = L_0 - \frac{1}{2} H_0 + \frac{1}{4} K.\end{gathered}$$
We consider the induced action of $\gamma$ on the vacuum. Specifically, we determine the ${{\mathfrak{sl}} \left( 2 \right)}$-weight and conformal dimension of ${\gamma \bigl( {\bigl\lvert 0 \bigr\rangle} \bigr)}$: $$\begin{aligned}
H_0 {\gamma \left( {\bigl\lvert 0 \bigr\rangle} \right)} &= {\gamma \left( {\gamma^{-1} \left( H_0 \right)} {\bigl\lvert 0 \bigr\rangle} \right)} = {\gamma \left( {\left( H_0 + K \right)} {\bigl\lvert 0 \bigr\rangle} \right)} = \frac{-1}{2} {\gamma \left( {\bigl\lvert 0 \bigr\rangle} \right)}, \\
L_0 {\gamma \left( {\bigl\lvert 0 \bigr\rangle} \right)} &= {\gamma \left( {\gamma^{-1} \left( L_0 \right)} {\bigl\lvert 0 \bigr\rangle} \right)} = {\gamma \left( \Bigl( L_0 + \frac{1}{2} H_0 + \frac{1}{4} K \Bigr) {\bigl\lvert 0 \bigr\rangle} \right)} = \frac{-1}{8} {\gamma \left( {\bigl\lvert 0 \bigr\rangle} \right)}.\end{aligned}$$ Similarly, one can check that ${\gamma \left( {\bigl\lvert 0 \bigr\rangle} \right)}$ is a [highest weight state]{}. This therefore suggests that $$\label{eqnMapI}
{\gamma \left( {{\widehat{\mathcal{L}}}_{0}} \right)} = {{\widehat{\mathcal{L}}}_{-1/2}}.$$ This is to be interpreted as $\gamma \circ \pi_0 \circ \gamma^{-1} = \pi_{-1/2}$, where $\pi_{\lambda}$ denotes the representation (map) of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ on ${{\widehat{\mathcal{L}}}_{\lambda}}$. Note that here $\gamma^{-1}$ is acting as an automorphism on ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$, whereas $\gamma$ denotes the induced isomorphism of vector spaces acting on the states (as in ${\gamma \bigl( {\bigl\lvert 0 \bigr\rangle} \bigr)}$ above).
We can prove (\[eqnMapI\]) in many ways. First, we can note that what we have proven is the corresponding equality of Verma modules. We should therefore explicitly check that the expressions for the two vanishing singular vectors of each module are mapped to zero by $\gamma$ and $\gamma^{-1}$ (as appropriate). A second proof involves verifying that the characters satisfy $$\begin{aligned}
\label{eqnCh1stSF}
{{\chi_{{{\widehat{\mathcal{L}}}_{-1/2}}} \left( z ; q \right)}} &= {{\chi_{{\gamma \left( {{\widehat{\mathcal{L}}}_{0}} \right)}} \left( z ; q \right)}} = \sum_{\text{basis } {\gamma \left( {\left\lvert \psi \right\rangle} \right)}} z^{\lambda_{{\gamma \left( {\left\lvert \psi \right\rangle} \right)}}} q^{\Delta_{{\gamma \left( {\left\lvert \psi \right\rangle} \right)}}} \notag \\
&= \sum_{\text{basis } {\left\lvert \psi \right\rangle}} z^{\lambda_{{\left\lvert \psi \right\rangle}} + k} q^{\Delta_{{\left\lvert \psi \right\rangle}} + \frac{1}{2} \lambda_{{\left\lvert \psi \right\rangle}} + \frac{1}{4} k} = z^{-1/2} q^{-1/8} {{\chi_{{{\widehat{\mathcal{L}}}_{0}}} \left( z q^{1/2} ; q \right)}},\end{aligned}$$ where $\lambda_{{\left\lvert \psi \right\rangle}}$ and $\Delta_{{\left\lvert \psi \right\rangle}}$ denote the ${{\mathfrak{sl}} \left( 2 \right)}$-weight and conformal dimension of ${\bigl\lvert \psi \bigr\rangle}$ (respectively). This can be done in a straight-forward fashion using the character formulae given in [Equations (\[eqnCh0\]) and (\[eqnCh-1/2\])]{}. However, it is far more elegant to simply observe that twisting a representation by an algebra automorphism clearly preserves irreducibility. To this third proof, we add the practical method of looking at the picture of ${{\widehat{\mathcal{L}}}_{0}}$ in [Figure \[figAdmissReps\]]{}, turning one’s head $45^{\circ}$ to the right, and comparing with the picture of ${{\widehat{\mathcal{L}}}_{-1/2}}$ there. In this way, we observe that the multiplicities of the appropriate weight spaces precisely match. This actually constitutes a rigorous proof in itself because the pictures in [Figure \[figAdmissReps\]]{} show the multiplicities to sufficiently deep grades (in general, both pictures must show that the generating singular vectors vanish).
We can similarly study the spectral flow of ${{\widehat{\mathcal{L}}}_{1}}$. Proceeding as above, we compute that the image of the [highest weight state]{} ${\bigl\lvert 1 \bigr\rangle}$ under $\gamma$ has ${{\mathfrak{sl}} \left( 2 \right)}$-weight $\tfrac{1}{2}$ and conformal dimension $\tfrac{7}{8}$. However, it is not a [highest weight state]{}: $$F_1 {\gamma \left( {\bigl\lvert 1 \bigr\rangle} \right)} = {\gamma \left( {\gamma^{-1} \left( F_1 \right)} {\bigl\lvert 1 \bigr\rangle} \right)} = {\gamma \left( F_0 {\bigl\lvert 1 \bigr\rangle} \right)} \neq 0.$$ Instead, it is the image of $F_0 {\bigl\lvert 1 \bigr\rangle}$ which becomes the [highest weight state]{} of the flowed module ${\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$. This can be checked explicitly, but is most easily seen from [Figure \[figAdmissReps\]]{}. Since ${\gamma \left( F_0 {\bigl\lvert 1 \bigr\rangle} \right)}$ has ${{\mathfrak{sl}} \left( 2 \right)}$-weight $\tfrac{-1}{2}$ and conformal dimension $\tfrac{-1}{8}$, it now follows that $${\gamma \left( {{\widehat{\mathcal{L}}}_{1}} \right)} = {{\widehat{\mathcal{L}}}_{-3/2}}.$$
We can apply our new-found knowledge regarding the action of the spectral flow automorphisms to the computation of fusion rules. This relies upon the principle that the fusion rules respect these automorphisms in the following manner[^8]: $${\mathcal{M}_{}} {\times_{\! f}}{\mathcal{M}_{}}' = {\mathcal{M}_{}}'' \qquad \Rightarrow \qquad {\Omega \left( {\mathcal{M}_{}} \right)} {\times_{\! f}}{\Omega' \left( {\mathcal{M}_{}}' \right)} = {\Omega \Omega' \left( {\mathcal{M}_{}}'' \right)}.$$ Here, $\Omega$ and $\Omega'$ are automorphisms. This principle is well-known from studies of rational [conformal field theories]{} with Lie algebra symmetries. Despite its natural appearance, we are not aware of any formal general proof. It has however been checked explicitly in many non-trivial cases (see [@GabFus01] in particular). For example, we can determine ${{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-1/2}}$ by applying $\operatorname{id}{\times_{\! f}}\gamma$ (in hopefully obvious notation) to ${{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{0}}$. The result reproduces the first fusion rule of (\[eqnFR1xAdmiss\]).
More importantly, we can apply this principle to compute the fusion of ${{\widehat{\mathcal{L}}}_{-1/2}}$ with itself. The result is therefore that this gives the module ${\gamma^2 \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)} = {\gamma \bigl( {{\widehat{\mathcal{L}}}_{-1/2}} \bigr)}$ (and indeed we see that the fusion is not empty). This module is not one of the admissibles that we have considered. Indeed, it is not even a [highest weight module]{}, as can be seen by looking at the picture of ${{\widehat{\mathcal{L}}}_{-1/2}}$ in [Figure \[figAdmissReps\]]{} and turning one’s head $45^{\circ}$ to the right. This is perhaps physically distasteful, but is an unavoidable feature of the theory. We remark that the conformal dimensions of the states in this module with a given ${{\mathfrak{sl}} \left( 2 \right)}$-weight *are* bounded below. It follows from this that operator products of the corresponding fields may still be expanded as a Laurent series (with poles of finite order). The standard field-theoretic machinery of [conformal field theory]{} may therefore be carried across to these modules without difficulty.
It is now trivial to determine the remaining fusion rules of the admissible modules: $${{\widehat{\mathcal{L}}}_{-1/2}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-1/2}} = {{\widehat{\mathcal{L}}}_{-3/2}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-3/2}} = {\gamma^2 \left( {{\widehat{\mathcal{L}}}_{0}} \right)} \qquad \text{and} \qquad {{\widehat{\mathcal{L}}}_{-1/2}} {\times_{\! f}}{{\widehat{\mathcal{L}}}_{-3/2}} = {\gamma^2 \left( {{\widehat{\mathcal{L}}}_{1}} \right)}.$$ Moreover, we now see that the spectrum contains the modules ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ for all $\ell$. Extending this to $\ell$ negative also makes sense, and is in fact necessary for physical consistency. Otherwise (for example), ${{\widehat{\mathcal{L}}}_{-1/2}}$ would have no conjugate within the spectrum, so correlation functions of its fields with any other fields would vanish, leading to the effective decoupling (and removal) of ${{\widehat{\mathcal{L}}}_{-1/2}}$ from the theory. The conjugate of ${{\widehat{\mathcal{L}}}_{-1/2}} = {\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ is of course ${\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$. This is not a [highest weight module]{} — one pictures it by looking at ${{\widehat{\mathcal{L}}}_{0}}$ in [Figure \[figAdmissReps\]]{} and turning one’s head $45^{\circ}$ to the left — as its zero-grade states form a *lowest* weight representation of ${{\mathfrak{sl}} \left( 2 \right)}$. We indicate this module (and other flowed modules) schematically in [Figure \[figSpecFlow\]]{}. Note that even the Weyl group of ${{\mathfrak{sl}} \left( 2 \right)}$ does not preserve the modules in the spectrum: The non-trivial reflection induces a (grade-preserving) map between ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{\lambda}} \bigr)}$ and ${\gamma^{-\ell} \bigl( {{\widehat{\mathcal{L}}}_{\lambda}} \bigr)}$ (for $\lambda = 0 , 1$). Of course, this map is nothing but conjugation (as usual for ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ theories).
\[\]\[\][${{\widehat{\mathcal{L}}}_{0}}$]{} \[\]\[\][${{\widehat{\mathcal{L}}}_{1}}$]{} \[\]\[\][${{\widehat{\mathcal{L}}}_{-1/2}}$]{} \[\]\[\][${{\widehat{\mathcal{L}}}_{-3/2}}$]{} \[\]\[\][$\gamma$]{} \[\]\[\][$\scriptstyle {\left( 0,0 \right)}$]{} \[\]\[\][$\scriptstyle {\left( \tfrac{-1}{2},\tfrac{-1}{8} \right)}$]{} \[\]\[\][$\scriptstyle {\left( \tfrac{1}{2},\tfrac{-1}{8} \right)}$]{} \[\]\[\][$\scriptstyle {\left( -1,\tfrac{-1}{2} \right)}$]{} \[\]\[\][$\scriptstyle {\left( 1,\tfrac{-1}{2} \right)}$]{} \[\]\[\][$\scriptstyle {\left( 1,\tfrac{1}{2} \right)}$]{} \[\]\[\][$\scriptstyle {\left( -1,\tfrac{1}{2} \right)}$]{} \[\]\[\][$\scriptstyle {\left( \tfrac{-3}{2},\tfrac{-1}{8} \right)}$]{} \[\]\[\][$\scriptstyle {\left( \tfrac{1}{2},\tfrac{7}{8} \right)}$]{} \[\]\[\][$\scriptstyle {\left( \tfrac{3}{2},\tfrac{-1}{8} \right)}$]{} \[\]\[\][$\scriptstyle {\left( \tfrac{-1}{2},\tfrac{7}{8} \right)}$]{} \[\]\[\][$\scriptstyle {\left( -2,-1 \right)}$]{} \[\]\[\][$\scriptstyle {\left( 0,1 \right)}$]{} \[\]\[\][$\scriptstyle {\left( 2,-1 \right)}$]{}
![Depictions of the modules appearing in the spectrum and the action of the spectral flow automorphism $\gamma$. Each “corner state” is labelled by its ${{\mathfrak{sl}} \left( 2 \right)}$-weight and conformal dimension (in that order).[]{data-label="figSpecFlow"}](Affsl2SpecFlow){width="15cm"}
We have therefore shown that the the spectrum of our ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ theory consists of two infinite series of modules, ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ ($\ell \in {\mathbb{Z}}$). The fusion rules may be summarised by $$\label{eqnFR}
{\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{\lambda}} \bigr)} {\times_{\! f}}{\gamma^{m} \bigl( {{\widehat{\mathcal{L}}}_{\mu}} \bigr)} = {\gamma^{\ell + m} \bigl( {{\widehat{\mathcal{L}}}_{\lambda + \mu}} \bigr)},$$ where $\lambda$ and $\mu$ take value $0$ or $1$ and their sum is taken *modulo* $2$. It should be clear that all the modules in the spectrum are mutually distinct (there are no module isomorphisms between them).
Characters and Modular Invariants {#secChar}
=================================
Consider now the characters of the modules comprising our theory. We have already determined the characters of ${{\widehat{\mathcal{L}}}_{0}}$ and ${{\widehat{\mathcal{L}}}_{1}}$ and their images under $\gamma$, and it is easy to use the explicit spectral flow action to determine expressions for those which remain. But let us first take this opportunity to rewrite the known characters in a more standard form [@KacMod88b]. Recall the explicit form of the vacuum character, given in [Equation (\[eqnCh0\])]{}. We split the denominator (as an infinite sum over $n$) into sums over $n$ even and $n$ odd. Completing the square in the $q$-exponents of both the numerator and denominator then gives $$\label{eqnChar0}
{{\chi_{{{\widehat{\mathcal{L}}}_{0}}} \left( z ; q \right)}} = q^{-1/24} \frac{\displaystyle \sum_{r \in {\mathbb{Z}}+ 1/6} z^{6r} q^{6r^2} - \sum_{r \in {\mathbb{Z}}- 1/6} z^{6r} q^{6r^2}}{\displaystyle \sum_{r \in {\mathbb{Z}}+ 1/4} z^{4r} q^{2r^2} - \sum_{r \in {\mathbb{Z}}- 1/4} z^{4r} q^{2r^2}}.$$ The reader will no doubt recognise that the numerator and denominator are differences of classical theta functions [@KacInf90]. The factor $q^{-1/24}$ is the standard modular anomaly $q^{c/24}$. A similar massaging of [Equation (\[eqnCh1\])]{} gives $$\label{eqnChar1}
{{\chi_{{{\widehat{\mathcal{L}}}_{1}}} \left( z ; q \right)}} = q^{-1/24} \frac{\displaystyle \sum_{r \in {\mathbb{Z}}+ 1/3} z^{6r} q^{6r^2} - \sum_{r \in {\mathbb{Z}}- 1/3} z^{6r} q^{6r^2}}{\displaystyle \sum_{r \in {\mathbb{Z}}+ 1/4} z^{4r} q^{2r^2} - \sum_{r \in {\mathbb{Z}}- 1/4} z^{4r} q^{2r^2}}.$$ Apply now the spectral flow automorphism $\gamma^{\ell}$. Generalising [Equation (\[eqnCh1stSF\])]{}, we quickly derive that $$\label{eqnCharSF}
{{\chi_{{\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{\lambda}} \bigr)}} \left( z ; q \right)}} = z^{-\ell / 2} q^{-\ell^2 / 8} {{\chi_{{{\widehat{\mathcal{L}}}_{\lambda}}} \left( z q^{\ell / 2} ; q \right)}}.$$ From [Equations (\[eqnChar0\]) and (\[eqnChar1\])]{} we therefore obtain $$\begin{aligned}
{{\chi_{{\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \left( z ; q \right)}} &= q^{-1/24} \frac{\displaystyle \sum_{r \in {\mathbb{Z}}+ {\left( 3 \ell + 2 \right)} / 12} z^{6r} q^{6r^2} - \sum_{r \in {\mathbb{Z}}+ {\left( 3 \ell - 2 \right)} / 12} z^{6r} q^{6r^2}}{\displaystyle \sum_{r \in {\mathbb{Z}}+ {\left( 2 \ell + 1 \right)} / 4} z^{4r} q^{2r^2} - \sum_{r \in {\mathbb{Z}}+ {\left( 2 \ell - 1 \right)} / 4} z^{4r} q^{2r^2}} \label{eqnCharSF0} \\
\text{and} \qquad {{\chi_{{\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \left( z ; q \right)}} &= q^{-1/24} \frac{\displaystyle \sum_{r \in {\mathbb{Z}}+ {\left( 3 \ell + 4 \right)} / 12} z^{6r} q^{6r^2} - \sum_{r \in {\mathbb{Z}}+ {\left( 3 \ell - 4 \right)} / 12} z^{6r} q^{6r^2}}{\displaystyle \sum_{r \in {\mathbb{Z}}+ {\left( 2 \ell + 1 \right)} / 4} z^{4r} q^{2r^2} - \sum_{r \in {\mathbb{Z}}+ {\left( 2 \ell - 1 \right)} / 4} z^{4r} q^{2r^2}}. \label{eqnCharSF1}\end{aligned}$$
This appears to provide a satisfying answer to the determination of the characters of our theory. However, it is easy to check from [Equations (\[eqnCharSF0\]) and (\[eqnCharSF1\])]{} that the common denominator is antiperiodic under $\ell \rightarrow \ell + 1$, hence periodic under $\ell \rightarrow \ell + 2$. Moreover, the numerators of the spectrally-flowed characters of ${{\widehat{\mathcal{L}}}_{0}}$ and ${{\widehat{\mathcal{L}}}_{1}}$ are interchanged (with an additional factor of $-1$) under $\ell \rightarrow \ell + 2$, and are thus periodic under $\ell \rightarrow \ell + 4$. It therefore follows that these expressions for the spectrally-flowed characters are periodic in $\ell$ with period $4$, and that there are only four linearly independent characters (there are actually eight distinct characters, but four are just the negatives of the other four). We can take these to be the characters of the admissible [highest weight modules]{} ${{\widehat{\mathcal{L}}}_{0}}$, ${{\widehat{\mathcal{L}}}_{1}}$, ${{\widehat{\mathcal{L}}}_{-1/2}} = {\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${{\widehat{\mathcal{L}}}_{-3/2}} = {\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$. The spectral flow action on the characters may then be summarised as $$\label{eqnLinIndChars}
\begin{split}
\cdots \overset{\gamma}{\longrightarrow} -\chi_{{{\widehat{\mathcal{L}}}_{1}}} \overset{\gamma}{\longrightarrow} -\chi_{{{\widehat{\mathcal{L}}}_{-3/2}}} \overset{\gamma}{\longrightarrow} \chi_{{{\widehat{\mathcal{L}}}_{0}}} \overset{\gamma}{\longrightarrow} \chi_{{{\widehat{\mathcal{L}}}_{-1/2}}} \overset{\gamma}{\longrightarrow} -\chi_{{{\widehat{\mathcal{L}}}_{1}}} \overset{\gamma}{\longrightarrow} \cdots \\
\cdots \overset{\gamma}{\longrightarrow} -\chi_{{{\widehat{\mathcal{L}}}_{0}}} \overset{\gamma}{\longrightarrow} -\chi_{{{\widehat{\mathcal{L}}}_{-1/2}}} \overset{\gamma}{\longrightarrow} \chi_{{{\widehat{\mathcal{L}}}_{1}}} \overset{\gamma}{\longrightarrow} \chi_{{{\widehat{\mathcal{L}}}_{-3/2}}} \overset{\gamma}{\longrightarrow} -\chi_{{{\widehat{\mathcal{L}}}_{0}}} \overset{\gamma}{\longrightarrow} \cdots
\end{split}$$ This seems to contradict the fact that the corresponding modules are all distinct. There are no isomorphisms between the spectrally-flowed modules, but nevertheless there is an infinite degeneracy of the characters.
A resolution to this seeming contradiction was proposed in [@LesSU202], where it was noted that one has to pay close attention to the regions of convergence of such character formulae. The problem is very much related to the more transparent example of ${{\mathfrak{sl}} \left( 2 \right)}$ characters. Here, a highest weight Verma module with highest weight $\lambda$ has character $$z^{\lambda} + z^{\lambda - 2} + z^{\lambda - 4} + \ldots = \frac{z^{\lambda}}{1 - z^{-2}} \qquad \text{(${\left| z \right|} > 1$).}$$ Similarly, a lowest weight Verma module with lowest weight $\lambda + 2$ has character $$z^{\lambda + 2} + z^{\lambda + 4} + z^{\lambda + 6} + \ldots = \frac{z^{\lambda + 2}}{1 - z^2} \qquad \text{(${\left| z \right|} < 1$).}$$ Formally, these characters give the same function (up to a conspicuous factor of $-1$), but the notion that the modules are (almost) the same is patently absurd. The point is that in general the physical module is determined by its character *and* the given region of convergence. It is the latter which dictates the formal expansion, here in powers of $z^2$ or $z^{-2}$. Note that finite-dimensional ${{\mathfrak{sl}} \left( 2 \right)}$-modules have characters that are polynomial in $z$ and $z^{-1}$, hence converge when ${\left| z \right|} = 1$ (indeed, everywhere).
The ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ character formulae we have derived have to be understood in a similar way. Specifically, the infinite sums appearing in the numerators and denominators of these formulae are easily checked to converge for all $z \in {\mathbb{C}}$, provided that ${\left| q \right|} < 1$. However, the common denominator of these expressions vanishes whenever $z^2 = q^i$ ($i \in {\mathbb{Z}}$). This is obvious from its product form (displayed in [Equation (\[eqnChV0\])]{} for example), but it is also useful to check this from the above sum form: $$\begin{aligned}
\sum_{r \in {\mathbb{Z}}+ \tfrac{2 \ell + 1}{4}} q^{2r^2 + 2ir} - \sum_{r \in {\mathbb{Z}}+ \tfrac{2 \ell - 1}{4}} q^{2r^2 + 2ir} &= q^{-i^2 / 2} {\left[ \sum_{s \in {\mathbb{Z}}+ \tfrac{\ell + i}{2} + \tfrac{1}{4}} q^{2s^2} - \sum_{s \in {\mathbb{Z}}+ \tfrac{\ell + i}{2} - \tfrac{1}{4}} q^{2s^2} \right]} \notag \\
&= q^{-i^2 / 2} {\left[ \sum_{s \in {\mathbb{Z}}- \tfrac{\ell + i}{2} - \tfrac{1}{4}} q^{2s^2} - \sum_{s \in {\mathbb{Z}}+ \tfrac{\ell + i}{2} - \tfrac{1}{4}} q^{2s^2} \right]} = 0,\end{aligned}$$ as $\ell + i \in {\mathbb{Z}}$. The character formulae will therefore have poles at $z^2 = q^i$ ($i \in {\mathbb{Z}}$) unless the zeroes of the denominator are cancelled by zeroes in the numerator (this is what happens in the integrable module case). But, analysing the numerators of [Equations (\[eqnCharSF0\]) and (\[eqnCharSF1\])]{} as above, we find that zeroes occur only at $z^2 = q^{\ell + i}$ with $\ell + i \in 2 {\mathbb{Z}}$. It follows that the character formulae we have given for the modules ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ have poles at $z^2 = q^i$ for all $i \in 2 {\mathbb{Z}}- 1 - \ell$.
We have argued above with the example of ${{\mathfrak{sl}} \left( 2 \right)}$ that the relationship between a character and the module it is supposed to describe is determined by the region in which the character is to be expanded. It is now clear how this applies to the present case. The characters given for ${{\widehat{\mathcal{L}}}_{0}}$ and ${{\widehat{\mathcal{L}}}_{1}}$ should be expanded on the annulus ${\left| q \right|}^{1/2} < {\left| z \right|} < {\left| q \right|}^{-1/2}$. Note that as ${\left| q \right|} < 1$, this covers the case ${\left| z \right|} = 1$. Accordingly, when $q$-expanding [Equations (\[eqnChar0\]) and (\[eqnChar1\])]{}, the coefficients simplify to give (Laurent) polynomials in $z$ as the constituent ${{\mathfrak{sl}} \left( 2 \right)}$-modules are all finite-dimensional. The spectral flow action now implies that the appropriate region on which to properly expand the characters (\[eqnCharSF0\]) and (\[eqnCharSF1\]) with $\ell \neq 0$ is the annulus $$\label{eqnAnnulus}
{\left| q \right|}^{{\left( -\ell + 1 \right)}/2} < {\left| z \right|} < {\left| q \right|}^{{\left( -\ell - 1 \right)}/2}.$$ Note that when $\ell > 0$, we have ${\left| z \right|} > 1$, appropriate for highest weight ${{\mathfrak{sl}} \left( 2 \right)}$-modules, and for $\ell < 0$, we have ${\left| z \right|} < 1$, appropriate for lowest weight ${{\mathfrak{sl}} \left( 2 \right)}$-modules. This accords with the pictures we have drawn in [Figure \[figSpecFlow\]]{}.
For $\ell = \pm 1$, $q$-expanding the character gives coefficients which are rational functions of $z$. These coefficients may then be expanded for either ${\left| z \right|} > 1$ or ${\left| z \right|} < 1$, as appropriate, in order to recover the correct weight multiplicities of the module. However, for ${\left| \ell \right|} > 1$, this procedure fails. For example, $q$-expanding [Equation (\[eqnCharSF0\])]{} with $\ell = 2$ gives polynomial coefficients in $z$ because this character coincides with that of ${{\widehat{\mathcal{L}}}_{1}}$ up to an overall factor of $-1$ ([Equation (\[eqnLinIndChars\])]{}). The expansion annulus (\[eqnAnnulus\]) for $\ell = 2$ is disjoint from that for $\ell = 0$, so a naïve $q$-expansion[^9] is no longer appropriate. Indeed, for this module the conformal dimension is not bounded below, so the correct expansion would have to include arbitrarily negative powers of $q$ as well as the usual positive powers. To obtain such an expansion, we would have to change variables to $u = {\left( z q^{\ell / 2} \right)}^{-1}$, effectively undoing the spectral flow, then $q$-expand and change $u$ back again.
It is clear therefore that the explicit expressions we have given for the modules ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ with ${\left| \ell \right|} > 1$ are not actually particularly useful. The point however is that any other expression we might cook up will be equivalent to these because the classical theta functions from which they are constructed are entire in the $z$-plane (when ${\left| q \right|} < 1$). The conclusion is that by expressing the characters in terms of these functions, instead of as formal power series, we lose the equivalence between modules and characters. We have an infinite collection of distinct modules, but only four linearly independent characters.
More precisely, the ${\mathbb{Z}}$-linear map which assigns to each module in the fusion ring its character is not one-to-one. It is not hard to check that the kernel of this map is generated by the modules ${\gamma^{\ell \pm 1} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)} \oplus {\gamma^{\ell \mp 1} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ and that these modules are closed under fusion. It follows that we can consistently define fusion at the level of characters. We call the resulting ring over ${\mathbb{Z}}$ the *Grothendieck ring* of characters[^10]. Assigning modules their characters therefore defines a projection (more precisely, an onto ring homomorphism) from the fusion ring onto the Grothendieck ring.
This has a peculiar effect when considering modular invariance. Specifically, one expects from rational theories that pairing each module with itself under the holomorphic and antiholomorphic ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$-actions leads to a modular invariant partition function. But in our case, the coincidence of characters means that there are infinitely many modules all contributing the same amount to the partition function, which therefore diverges. One can of course regularise this divergence by only allowing the linearly independent characters to contribute, effectively dividing the modular invariant by the infinite multiplicity of each independent character, and in this way one recovers the modular invariant of Kac and Wakimoto [@KacMod88b] (we postpone a proper discussion of modularity until [Section \[secMod\]]{}). This is indeed invariant under the usual action of ${{\mathsf{SL}} \left( 2 ; {\mathbb{Z}}\right)}$, but we should be uneasy about its status as a physical partition function. It does not, strictly speaking, refer to a complete set of modules of the theory. In particular, there is no set of modules corresponding to this partition function which is closed under fusion.
In essence however, what this does is determine a modular invariant partition function in the Grothendieck ring of characters. This is no different to what one does in rational theories, and evidence is steadily mounting that this is what one should do in logarithmic theories as well. However, it is clear that determining a modular invariant in this way does not answer the fundamental question of how the holomorphic and antiholomorphic sectors of the theory are glued together. For this reason, we advise caution in treating such modular invariants as physical. Applications require a justification of why such a partition function is appropriate.
We briefly compare this conclusion with that of [@LesSU202]. Their proposal for making sense of Kac and Wakimoto’s invariant is to regard the character of ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{\lambda}} \bigr)}$ as only being defined on the annulus (\[eqnAnnulus\]). Summing to get a partition function is therefore viewed as summing over the different annuli in order to have a finite meromorphic partition function on the $z$-plane (with ${\left| q \right|} < 1$). Presumably this means each character should take value zero outside its given annulus, in defiance of analytic continuation. Evidence for this proposal is quoted in the claim that a particular modular transformation maps the annuli into one another. This claim is not true. Even if it were, the other transformations do not preserve this annulus structure, hence one is *forced* to analytically continue the characters into the rest of the $z$-plane.
We agree that it would be better to extend the definition of partition function so that every module contributes, but the interpretation of [@LesSU202] does not achieve this goal. What is needed in our opinion is an additional quantum number to distinguish representations with the same character. It is not clear however that such a quantum number need exist. It seems plausible that modular invariants for fractional level models can only be defined at the level of Grothendieck rings.
Extended Algebras and Ghosts {#secGhost}
============================
Note that the fusion rules of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ ([Equation (\[eqnFR\])]{}) show that every module in the spectrum is a simple current — these are distinguished in general by the property that fusing them with any irreducible module gives a single irreducible factor [@SchSim90]. This is clear for the modules ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ as they are automorphic images of the vacuum module. For the other modules, this follows because ${{\widehat{\mathcal{L}}}_{1}}$ happens to be a simple current. This is somewhat mysterious as ${{\widehat{\mathcal{L}}}_{1}}$ is not related to the vacuum by any automorphism of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$. Nevertheless, this is the only simple current which has finite order, and its order is $2$ ([Equation (\[eqnFR1x1\])]{}).
It is therefore interesting to study the *extended algebra* defined by this simple current, more precisely, by the zero-grade fields of the module ${{\widehat{\mathcal{L}}}_{1}}$. Their conformal dimension is $\tfrac{1}{2}$ which suggests some sort of fermionic behaviour. Correctly determining an extended algebra can be a somewhat subtle business, and we shall proceed carefully in an elementary fashion. The final answer may not be particularly surprising, but there are several pitfalls to avoid during the derivation which we would like to draw attention to.
Let us begin by introducing some convenient notation for the zero-grade fields of the simple current. We will denote the field corresponding to the [highest weight state]{} ${\bigl\lvert 1 \bigr\rangle} \in {{\widehat{\mathcal{L}}}_{1}}$ by $\phi$ and that corresponding to the descendant $F_0 {\bigl\lvert 1 \bigr\rangle}$ by $\psi$. We recall [@RidSU206] that in general such zero-grade fields are mutually bosonic with respect to the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ current field $H$, but mutually fermionic with respect to $E$ and $F$. This is true even for admissible levels, and the proof is easy in this special case. Suppose therefore that $$\label{eqnMutLoc}
{J \left( z \right)} {\phi \left( w \right)} = \mu_{J,\phi} {\phi \left( w \right)} {J \left( z \right)}, \qquad \text{($J = E, H, F$),}$$ for some $\mu_{J,\phi} \in {\mathbb{C}}$. Then, we can write $${J \left( z \right)} {J' \left( x \right)} {\phi \left( w \right)} = \mu_{J',\phi} {J \left( z \right)} {\phi \left( w \right)} {J' \left( x \right)} = \mu_{J,\phi} \mu_{J',\phi} {\phi \left( w \right)} {J \left( z \right)} {J' \left( x \right)}.$$ Alternatively, we can make use of the [operator product expansion]{} to get $${J \left( z \right)} {J' \left( x \right)} {\phi \left( w \right)} = {\left( \frac{{\kappa \bigl( J , J' \bigr)} K}{{\left( z-x \right)}^2} + \frac{{{\bigl[ J , J' \bigr]} \left( x \right)}}{z-x} + \ldots \right)} {\phi \left( w \right)} = \mu_{{\left[ J,J' \right]},\phi} {\phi \left( w \right)} {J \left( z \right)} {J' \left( x \right)},$$ if ${\bigl[ J , J' \bigr]} \neq 0$. Thus we have $$\mu_{J,\phi} \mu_{J',\phi} = \mu_{{\left[ J,J' \right]},\phi} \qquad \text{if ${\bigl[ J , J' \bigr]} \neq 0$.}$$ In addition, $K$ remains central in the extended algebra so we can also conclude that $$\mu_{J,\phi} \mu_{J',\phi} = 1 \qquad \text{if ${\kappa \bigl( J , J' \bigr)} \neq 0$.}$$
These constraints (which are equivalent to the generalised Jacobi identity at the level of modes) fix $\mu_{H , \phi} = 1$ and $\mu_{E , \phi} \mu_{F , \phi} = 1$. $H$ and $\phi$ are therefore mutually bosonic, but the case of $E$ or $F$ and $\phi$ is not decided. The corresponding conclusion for the zero-grade descendant field $\psi$ is identical. To settle the remaining ambiguity, we need to extend the adjoint (\[eqnDefAdj\]) to the simple current fields. Needless to say, the adjoint must define an (antilinear) antiautomorphism on the extended algebra, and it is this requirement that we shall exploit.
Since ${{\widehat{\mathcal{L}}}_{1}}$ is self-conjugate, the extended adjoint must take the form $$\phi_n^{\dag} = {\varepsilon}\psi_{-n} \qquad \Rightarrow \qquad \psi_n^{\dag} = \overline{{\varepsilon}}^{-1} \phi_{-n},$$ where the bar denotes complex conjugation. We now translate the primary field [operator product expansions]{} into modes using [Equation (\[eqnMutLoc\])]{}. For example, we have
$$\begin{aligned}
{F \left( z \right)} {\phi \left( w \right)} &= \frac{{\psi \left( w \right)}}{z-w} + \ldots & &\Rightarrow & F_m \phi_n - \mu_{F , \phi} \phi_n F_m &= \psi_{m+n} \label{eqnI} \\
\text{and} \qquad {E \left( z \right)} {\psi \left( w \right)} &= \frac{{\phi \left( w \right)}}{z-w} + \ldots & &\Rightarrow & E_m \psi_n - \mu_{E , \psi} \psi_n E_m &= \phi_{m+n}. \label{eqnII} \end{aligned}$$
Taking the adjoint of [Equation (\[eqnI\])]{} and using $\mu_{E , \phi} \mu_{F , \phi} = 1$, we find that $$- {\left| {\varepsilon}\right|}^2 \overline{\mu}_{F , \phi} {\left( E_m \psi_n - \overline{\mu}_{E , \phi} \psi_n E_m \right)} = \phi_{m+n}.$$ Comparing with [Equation (\[eqnII\])]{}, we finally conclude that $\overline{\mu}_{F , \phi} = - {\left| {\varepsilon}\right|}^{-2}$ and $\overline{\mu}_{E , \phi} = \mu_{E , \psi}$, hence that $\mu_{E , \phi} = \mu_{E , \psi}$ and $\mu_{F , \phi} = \mu_{F , \psi}$ are real and negative. There are no further constraints to be found, so we are free to choose the most symmetric consistent solution: $${\varepsilon}= 1 \qquad \text{hence} \qquad \mu_{E , \phi} = \mu_{F , \phi} = \mu_{E , \psi} = \mu_{F , \psi} = -1.$$ It follows that $E$ and $F$ are both mutually fermionic with respect to $\phi$ and $\psi$, as claimed.
We can now turn to the [operator product expansions]{} of the simple current fields. From [Equation (\[eqnFR1x1\])]{} and conservation of ${{\mathfrak{sl}} \left( 2 \right)}$-weight, we know that these must take the form
$$\begin{aligned}
{\phi \left( z \right)} {\phi \left( w \right)} &= \alpha {E \left( w \right)} + \ldots & {\psi \left( z \right)} {\psi \left( w \right)} &= \gamma {F \left( w \right)} + \ldots \\
{\phi \left( z \right)} {\psi \left( w \right)} &= \frac{1}{z-w} + \beta {H \left( w \right)} + \ldots & {\psi \left( z \right)} {\phi \left( w \right)} &= \frac{1}{z-w} + \beta' {H \left( w \right)} + \ldots, \label{eqnOPEsphipsi}\end{aligned}$$
for some constants $\alpha$, $\beta$, $\beta'$ and $\gamma$. These are easily computed. For example, the first expansion implies that $\phi_{-1/2} {\bigl\lvert \phi \bigr\rangle} = \alpha E_{-1} {\bigl\lvert 0 \bigr\rangle}$. Comparing $${\bigl\langle 0 \bigr\rvert F_1 \phi_{-1/2} \bigl\lvert \phi \bigr\rangle} = {\bigl\langle 0 \bigr\rvert \psi_{1/2} \phi_{-1/2} \bigl\lvert 0 \bigr\rangle} = 1 \qquad \text{and} \qquad {\bigl\langle 0 \bigr\rvert F_1 E_{-1} \bigl\lvert 0 \bigr\rangle} = {\bigl\langle 0 \bigr\rvert K - H_0 \bigl\lvert 0 \bigr\rangle} = \frac{-1}{2}$$ immediately yields $\alpha = k^{-1} = -2$. Similarly, $\beta = -1$, $\beta' = 1$ and $\gamma = -2$. Note that we have normalised the zero-grade state ${\bigl\lvert \phi \bigr\rangle}$ to have norm $1$. It follows that ${\bigl\lvert \psi \bigr\rangle} = F_0 {\bigl\lvert \phi \bigr\rangle}$ also has norm $1$ (these norms are the respective constants in the singular term of the [operator product expansions]{} (\[eqnOPEsphipsi\])).
We also need to determine the mutual locality of the simple current fields with one another, and this follows easily from the above [operator product expansions]{}. For example, if ${\phi \left( z \right)} {\psi \left( w \right)} = \mu {\psi \left( w \right)} {\phi \left( z \right)}$, then inserting the [operator product expansions]{} (\[eqnOPEsphipsi\]) gives $$\frac{1}{z-w} - {H \left( w \right)} + \ldots = \mu {\left( \frac{1}{w-z} + {H \left( z \right)} + \ldots \right)}.$$ Taylor-expanding ${H \left( z \right)}$ about $w$, we see that $\mu = -1$. Thus, $\phi$ and $\psi$ are mutually fermionic with respect to each other. Similarly, we can prove that both simple current fields are mutually bosonic with respect to themselves.
Finally, we have to verify that the [operator product expansions]{} we have derived are *associative*. For this we need to consider operator products of three fields. The associativity when at least one of the fields is an affine current is built into the above derivations, so we only need to check the case where all three fields are simple current fields. For example, since $E$ and $\phi$ are mutually fermionic, we see that $${\phi \left( z \right)} {\phi \left( w \right)} {\phi \left( x \right)} = {\left[ -2 {E \left( w \right)} + \ldots \right]} {\phi \left( x \right)} = - {\phi \left( x \right)} {\left[ -2 {E \left( w \right)} + \ldots \right]} = - {\phi \left( x \right)} {\phi \left( z \right)} {\phi \left( w \right)}.$$ However, this contradicts the fact that $\phi$ is mutually bosonic with respect to itself. In fact, further computation shows that *every* combination of three simple current fields exhibits the same contradiction — there is always a lone factor of $-1$ unaccounted for by the mutual locality.
This problem has been observed before in the algebra defining graded parafermions [@JacQua06] and certain minimal model extended algebras [@RidMin07]. The remedy is to introduce an auxiliary operator $\mathcal{S}$ which commutes with the affine generators, leaves the vacuum invariant, but *anticommutes* with the simple current fields. The defining [operator product expansions]{} of the extended algebra are thereby modified to be
\[eqnExtAlg\] $$\begin{aligned}
{\phi \left( z \right)} {\phi \left( w \right)} &= \mathcal{S} {\left[ -2 {E \left( w \right)} + \ldots \right]} & {\psi \left( z \right)} {\psi \left( w \right)} &= \mathcal{S} {\left[ -2 {F \left( w \right)} + \ldots \right]} \\
{\phi \left( z \right)} {\psi \left( w \right)} &= \mathcal{S} {\left[ \frac{1}{z-w} - {H \left( w \right)} + \ldots \right]} & {\psi \left( z \right)} {\phi \left( w \right)} &= \mathcal{S} {\left[ \frac{1}{z-w} + {H \left( w \right)} + \ldots \right]}\end{aligned}$$
One can check that introducing such an $\mathcal{S}$ precisely accounts for the factor of $-1$ observed above, restoring associativity. The mutual localities then give the corresponding mode algebra as $$\label{eqnModeAlg}
{\bigl[ \phi_m , \phi_n \bigr]} = 0, \qquad {\bigl\{ \phi_m , \psi_n \bigr\}} = \delta_{m+n,0} \mathcal{S} \qquad \text{and} \qquad {\bigl[ \psi_m , \psi_n \bigr]} = 0.$$ It is easy to check that $\mathcal{S}$ acts as the identity on each ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$, but as *minus* the identity on each ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$. $\mathcal{S}$ is therefore self-inverse, and by [Equation (\[eqnModeAlg\])]{}, self-adjoint.
Finally, the singular terms of the [operator product expansions]{} derived here suggest, together with the conformal dimensions of the simple current fields, that what we have constructed is nothing but a complex fermion, or equivalently, a system of fermionic ghosts. However, this system has central charge $1$ whereas ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ has central charge $-1$. The correct identification is that our extended algebra realises a system of *bosonic* ghosts, given by $${\beta \left( z \right)} = {\phi \left( z \right)} \qquad \text{and} \qquad {\gamma \left( z \right)} = \mathcal{S}^{-1} {\psi \left( z \right)}.$$ The defining [operator product expansions]{} then become
\[eqnGhost\] $$\begin{aligned}
{\beta \left( z \right)} {\beta \left( w \right)} &= \mathcal{S} {\left[ -2 {E \left( w \right)} + \ldots \right]} & {\gamma \left( z \right)} {\gamma \left( w \right)} &= \mathcal{S}^{-1} {\left[ 2 {F \left( w \right)} + \ldots \right]} \\
{\beta \left( z \right)} {\gamma \left( w \right)} &= \frac{-1}{z-w} + {H \left( w \right)} + \ldots & {\gamma \left( z \right)} {\beta \left( w \right)} &= \frac{1}{z-w} + {H \left( w \right)} + \ldots\end{aligned}$$
It is easy to check that $\beta$ and $\gamma$ are mutually bosonic with respect to themselves and each other.
By making the redefinitions $\widetilde{E} = \mathcal{S} E$, $\widetilde{H} = H$ and $\widetilde{F} = \mathcal{S}^{-1} F$ (which do not affect the affine algebra structure), we recover the standard $\beta \gamma$ ghost [operator product expansions]{}. Whilst this trick allows us to remove any trace of $\mathcal{S}$ from the defining equations, and even makes all the fields mutually bosonic with respect to one another, the ghost adjoint still requires the $\mathcal{S}$ operator: $$\beta^{\dag} = \mathcal{S} \gamma \qquad \text{and} \qquad \gamma^{\dag} = \beta \mathcal{S}^{-1}.$$ As the adjoint is vital for computations, we see therefore that we cannot do without $\mathcal{S}$ completely!
A Simplification {#secGhost2}
================
It is worth emphasising once again the fundamental rôle played by the adjoint (\[eqnDefAdj\]) in deriving the extended algebra in the previous section. This is the adjoint corresponding to the real form ${{\mathfrak{su}} \left( 2 \right)}$ of ${{\mathfrak{sl}} \left( 2 \right)} = {{\mathfrak{sl}} \left( 2 ; {\mathbb{C}}\right)}$. We could also consider the adjoint corresponding to the real form ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$: $$\label{eqnDefOtherAdj}
J_n^{\ddag} = -J_{-n}, \qquad K^{\ddag} = K \qquad \text{and} \qquad L_n^{\ddag} = L_{-n} \qquad \text{($J = E,H,F$).}$$ When we wish to emphasise that the chiral algebra ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ comes equipped with one of these adjoints, we will denote it by ${{\widehat{{\mathfrak{su}}}} \left( 2 \right)}$ or ${{\widehat{{\mathfrak{sl}}}} \left( 2 ; {\mathbb{R}}\right)}$, as appropriate. We stress that these are still complex Lie algebras. In general, every order-$2$ automorphism of a complex simple Lie algebra ${\mathfrak{g}}$ induces[^11] an adjoint on ${\mathfrak{g}}$ and its untwisted affinisation ${\widehat{{\mathfrak{g}}}}$. For ${\mathfrak{g}} = {{\mathfrak{sl}} \left( 2 ; {\mathbb{C}}\right)}$, the adjoint given in [Equation (\[eqnDefAdj\])]{} corresponds to the non-trivial Weyl reflection whereas that of [Equation (\[eqnDefOtherAdj\])]{} corresponds to the trivial automorphism.
We want to repeat the derivation of the extended algebra using the ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$ adjoint. The result will be slightly different, but the derivation is significantly simpler. The point here is that the choice of adjoint makes a real difference to simple current extensions of a chiral algebra. In the theory we are constructing, we have no physical intuition to support either choice, so it is interesting and valid to consider both possibilities. However, in concrete applications one generally does have a given adjoint, so it is extremely important to be sure that the extended algebra one derives and works with is the correct one.
To proceed, we have to change our basis to something appropriate for ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$. The problem here is that the eigenvectors $E$ and $F$ of ${\operatorname{ad}\left( H \right)}$ are not raising and lowering operators with respect to the adjoint (\[eqnDefOtherAdj\]). Instead, we introduce the linear combinations $$h = {\mathfrak{i}}{\left( E - F \right)}, \qquad e = \frac{1}{2} {\left( {\mathfrak{i}}E + {\mathfrak{i}}F - H \right)} \qquad \text{and} \qquad f = \frac{1}{2} {\left( {\mathfrak{i}}E + {\mathfrak{i}}F + H \right)}$$ (of ${{\mathfrak{sl}} \left( 2 ; {\mathbb{C}}\right)}$). These can be quickly checked to satisfy $$e^{\ddag} = f, \qquad h^{\ddag} = h, \qquad {\bigl[ h , e \bigr]} = 2e \qquad \text{and} \qquad {\bigl[ h , f \bigr]} = -2f,$$ so we have recovered the formalism of raising and lowering operators. The subtle but important difference between these operators and those considered in [Section \[secAlg\]]{} is that $${\bigl[ e , f \bigr]} = -h.$$ This difference is mirrored in the Killing form which is given in this basis by (compare [Equation (\[eqnKilling\])]{}) $${\kappa \bigl( h , h \bigr)} = 2 \qquad \text{and} \qquad {\kappa \bigl( e , f \bigr)} = -1,$$ with all other entries vanishing. The basis ${\left\{ e,h,f \right\}}$ can now be affinised in the usual manner to define a new basis ${\left\{ e_n,h_n,f_n,K,L_0 \right\}}$ of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$. We will *not* change the normalisation of the central extension $K$ and derivation $L_0$, as compared with [Equation (\[eqnCommRels\])]{} (we mention this as many articles implicitly replace $K$ by $-K$ which changes the prefactor of $L_0$ in the Sugawara construction).
Consider now the zero-grade states ${\bigl\lvert \phi \bigr\rangle}$ and ${\bigl\lvert \psi \bigr\rangle}$ of the simple current module ${{\widehat{\mathcal{L}}}_{1}}$. Just as we have had to change the basis of algebra generators to account for the ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$ adjoint, so we need to change this basis. The problem now is that ${\bigl\lvert \phi \bigr\rangle}$ is not a [highest weight state]{} with respect to the triangular decomposition afforded by $e_n$, $h_n$ and $f_n$. Indeed, it is not even an eigenstate of $h_0$. A better basis of zero-grade states is given by $${\bigl\lvert \Phi \bigr\rangle} = {\bigl\lvert \phi \bigr\rangle} - {\mathfrak{i}}{\bigl\lvert \psi \bigr\rangle} \qquad \text{and} \qquad {\bigl\lvert \Psi \bigr\rangle} = {\bigl\lvert \phi \bigr\rangle} + {\mathfrak{i}}{\bigl\lvert \psi \bigr\rangle}.$$ One can easily check that these are $h_0$-eigenstates with eigenvalues $1$ and $-1$, respectively, and that $e_0 {\bigl\lvert \Phi \bigr\rangle} = f_0 {\bigl\lvert \Psi \bigr\rangle} = 0$. Again, there is a subtle difference in the structure: $$f_0 {\bigl\lvert \Phi \bigr\rangle} = {\bigl\lvert \Psi \bigr\rangle} \qquad \text{but} \qquad e_0 {\bigl\lvert \Psi \bigr\rangle} = - {\bigl\lvert \Phi \bigr\rangle}.$$ This is reflected in the norms: If ${\bigl\lvert \Phi \bigr\rangle}$ has norm $1$, then ${\bigl\lvert \Psi \bigr\rangle}$ has norm $-1$ (the fundamental representation of ${{\mathfrak{sl}} \left( 2 \right)}$ is not unitarisable with respect to the ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$ adjoint).
We can now repeat the computations of [Section \[secGhost\]]{}. First, it is clear that the [operator product expansions]{} of $e$, $h$ or $f$ with the simple current fields $\Phi$ or $\Psi$ will lead to the same constraints on the mutual localities, namely $$\mu_{h,\Phi} = \mu_{h,\Psi} = 1 \qquad \text{and} \qquad \mu_{e,\Phi} \mu_{f,\Phi} = \mu_{e,\Psi} \mu_{f,\Psi} = 1.$$ We therefore determine when the adjoint extends to an antiautomorphism of the extended algebra. Defining $\Phi_n^{\ddag} = {\varepsilon}\Psi_{-n}$, hence $\Psi_n^{\ddag} = \overline{{\varepsilon}}^{-1} \Phi_{-n}$, we take the adjoint of the algebra relation $$f_m \Phi_n - \mu_{f,\Phi} \Phi_n f_m = \Psi_{m+n}$$ and compare it to the dual relation $$e_m \Psi_n - \mu_{e,\Psi} \Psi_n e_m = -\Phi_{m+n}$$ (note the minus sign!). This time we find that $\overline{\mu}_{f , \Phi} = {\left| {\varepsilon}\right|}^{-2}$ and $\overline{\mu}_{e , \Phi} = \mu_{e , \Psi}$, hence that $\mu_{e , \Phi} = \mu_{e , \Psi}$ and $\mu_{f , \Phi} = \mu_{f , \Psi}$ are real and *positive*. The simplest solution is therefore $${\varepsilon}= 1 \qquad \text{hence} \qquad \mu_{e , \Phi} = \mu_{f , \Phi} = \mu_{e , \Psi} = \mu_{f , \Psi} = 1.$$ The algebra generators are therefore mutually bosonic with respect to the simple current fields in this picture!
This is evidently a more familiar situation than that which we found in [Section \[secGhost\]]{} with the ${{\mathfrak{su}} \left( 2 \right)}$ adjoint. Continuing with the extended algebra derivation, we can determine the defining [operator product expansions]{}:
\[eqnNewOPEs\] $$\begin{aligned}
{\Phi \left( z \right)} {\Phi \left( w \right)} &= 2 {e \left( w \right)} + \ldots & {\Psi \left( z \right)} {\Psi \left( w \right)} &= 2 {f \left( w \right)} + \ldots \\
{\Phi \left( z \right)} {\Psi \left( w \right)} &= \frac{-1}{z-w} + {h \left( w \right)} + \ldots & {\Psi \left( z \right)} {\Phi \left( w \right)} &= \frac{1}{z-w} + {h \left( w \right)} + \ldots\end{aligned}$$
(note that the constants appearing in the singular term of these [operator product expansions]{} correspond to the respective norms of ${\bigl\lvert \Psi \bigr\rangle}$ and ${\bigl\lvert \Phi \bigr\rangle}$). It follows immediately from these expansions that the simple current fields are mutually bosonic with respect to themselves and each other, and it is simple to show that these expansions determine an associative operator product algebra (without any additional $\mathcal{S}$-type operators). The correspondence with the ghost fields is therefore as natural as it could be: $$\label{eqnNewGhosts}
{\beta \left( z \right)} = {\Phi \left( z \right)} \qquad \text{and} \qquad {\gamma \left( z \right)} = {\Psi \left( z \right)}.$$ Moreover, the adjoint on the ghost fields is just $\beta^{\ddag} = \gamma$ and $\gamma^{\ddag} = \beta$.
It is appropriate now to discuss the reverse procedure, obtaining the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ symmetry from studying the $\beta \gamma$ ghost system, for this was how these theories were first related (see [@GurRel98] for example). From [Equations (\[eqnNewOPEs\]) and (\[eqnNewGhosts\])]{}, we see that the composite fields $$e = \frac{1}{2} {{} : \beta \beta : {}}, \qquad h = {{} : \beta \gamma : {}} \qquad \text{and} \qquad f = \frac{1}{2} {{} : \gamma \gamma : {}}$$ together reconstitute the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ generators. Moreover, explicit calculation confirms that these are the ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$-type generators of this section. Furthermore, the ghost adjoint $\beta^{\ddag} = \gamma$ now implies the adjoint (\[eqnDefOtherAdj\]).
To summarise, the last two sections prove that the $\beta \gamma$ ghost system is naturally a simple current extension of ${{\widehat{{\mathfrak{sl}}}} \left( 2 ; {\mathbb{R}}\right)}_{-1/2}$ (we remind the reader again that we do not negate the level here). In order to realise these ghosts as an extension of ${{\widehat{{\mathfrak{su}}}} \left( 2 \right)}_{-1/2}$, it is *necessary* to augment the ghost algebra by the operator $\mathcal{S}$. It is not hard to find examples in the literature where this subtlety has been overlooked, so we want to emphasise the precise results derived here. Ignoring this leads to contradictions in the algebra when delving deeper into the module structure [@RidSU206].
Extended Algebra Representation Theory {#secGhostReps}
======================================
We now turn to a discussion of the representations of our extended algebra (\[eqnNewOPEs\]), or equivalently, of the ghost system (\[eqnGhost\]) (we work with ${{\widehat{{\mathfrak{sl}}}} \left( 2 ; {\mathbb{R}}\right)}$ for simplicity). The corresponding algebra relations are $$\label{eqnExtAlgComm}
{\bigl[ \Phi_m , \Phi_n \bigr]} = 0, \qquad {\bigl[ \Psi_m , \Phi_n \bigr]} = \delta_{m+n,0} \qquad \text{and} \qquad {\bigl[ \Psi_m , \Psi_n \bigr]} = 0,$$ where we write ${\Phi \left( z \right)} = \sum_n \Phi_n z^{-n-1/2}$ and ${\Psi \left( z \right)} = \sum_n \Psi_n z^{-n-1/2}$ as usual. Since $$\label{eqnFRSC}
{{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{\gamma^{\ell} \left( {{\widehat{\mathcal{L}}}_{0}} \right)} = {\gamma^{\ell} \left( {{\widehat{\mathcal{L}}}_{1}} \right)} \qquad \text{and} \qquad {{\widehat{\mathcal{L}}}_{1}} {\times_{\! f}}{\gamma^{\ell} \left( {{\widehat{\mathcal{L}}}_{1}} \right)} = {\gamma^{\ell} \left( {{\widehat{\mathcal{L}}}_{0}} \right)},$$ we see that each (irreducible) extended module will be labelled by a single integer $\ell$, and be composed of two ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$-modules, ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$. We denote this extended module by ${\mathbb{L}_{\ell}}$ ($\ell \in {\mathbb{Z}}$).
To understand the structure of these extended modules, we must first determine their monodromy charges $\theta_{\ell}$. These determine whether the extended algebra modes $\Phi_n$ and $\Psi_n$ have indices which are integers, half-integers, or something else entirely. The monodromy charge may be defined [@SchSim90; @RidSU206] in terms of the powers of $z-w$ appearing in the [operator product expansions]{} of the simple current fields ${\Phi \left( z \right)}$ and ${\Psi \left( z \right)}$ with a field ${\xi^{{\left( \ell \right)}} \left( z \right)}$ associated to a state of the extended module ${\mathbb{L}_{\ell}}$. The simple current property means that the powers of $z-w$ which appear only differ by an integer, and their common value *modulo* ${\mathbb{Z}}$ defines the monodromy charge (strictly speaking, this is the negative of the monodromy charge). This is also obviously independent of the choice of $\xi^{{\left( \ell \right)}}$ and simple current field.
From [Equation (\[eqnTransAuts\])]{}, we can compute (as in [Section \[secFusion\]]{}) that ${\gamma^{\ell} \bigl( {\bigl\lvert 0 \bigr\rangle} \bigr)}$ has conformal dimension $-\ell^2 / 8$ whereas that of ${\gamma^{\ell} \bigl( {\bigl\lvert 1 \bigr\rangle} \bigr)}$ is $-{\left( \ell^2 - 4 \ell - 4 \right)} / 8$. The fusion rules (\[eqnFRSC\]) then imply that the monodromy charges of ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{\ell} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ are $$\tfrac{1}{2} - \tfrac{1}{8} \ell^2 + \tfrac{1}{8} {\left( \ell^2 - 4 \ell - 4 \right)} = \tfrac{-1}{2} \ell \qquad \text{and} \qquad \tfrac{1}{2} - \tfrac{1}{8} {\left( \ell^2 - 4 \ell - 4 \right)} + \tfrac{1}{8} \ell^2 = \tfrac{1}{2} \ell + 1,$$ respectively. The monodromy charge of the extended module ${\mathbb{L}_{\ell}}$ is therefore well-defined (as claimed) and is simply $$\theta_{\ell} = \frac{\ell}{2} \pmod{{\mathbb{Z}}}.$$ It now follows that when ${\Phi \left( z \right)}$ and ${\Psi \left( z \right)}$ act upon a state ${\bigl\lvert \xi^{{\left( \ell \right)}} \bigr\rangle} \in {\mathbb{L}_{\ell}}$ (with monodromy charge $\theta_{\ell}$), they must be expanded in the forms $${\Phi \left( z \right)} {\bigl\lvert \xi^{{\left( \ell \right)}} \bigr\rangle} = \sum_{n \in {\mathbb{Z}}+ \theta_{\ell} - 1/2} \Phi_n z^{-n - 1/2} {\bigl\lvert \xi^{{\left( \ell \right)}} \bigr\rangle} \qquad \text{and} \qquad {\Psi \left( z \right)} {\bigl\lvert \xi^{{\left( \ell \right)}} \bigr\rangle} = \sum_{n \in {\mathbb{Z}}+ \theta_{\ell} - 1/2} \Psi_n z^{-n - 1/2} {\bigl\lvert \xi^{{\left( \ell \right)}} \bigr\rangle}.$$ In other words, the modes $\Phi_n$ and $\Psi_n$ with $n \in {\mathbb{Z}}+ \tfrac{1}{2}$ act on the extended algebra modules ${\mathbb{L}_{\ell}}$ with $\ell \in 2 {\mathbb{Z}}$, and the modes $\Phi_n$ and $\Psi_n$ with $n \in {\mathbb{Z}}$ act on the extended algebra modules ${\mathbb{L}_{\ell}}$ with $\ell \in 2 {\mathbb{Z}}+ 1$.
For example, the extended vacuum module ${\mathbb{L}_{0}} \sim {{\widehat{\mathcal{L}}}_{0}} \oplus {{\widehat{\mathcal{L}}}_{1}}$ has monodromy charge $0$, so $\Phi_n$ and $\Psi_n$ act upon it with half-integer indices. In particular, $\Phi_{-1/2}$ and $\Psi_{-1/2}$ act on the vacuum to create the zero-grade states ${\bigl\lvert \Phi \bigr\rangle}$ and ${\bigl\lvert \Psi \bigr\rangle}$ of the simple current module ${{\widehat{\mathcal{L}}}_{1}}$ (these are not the same as ${\bigl\lvert 1 \bigr\rangle} = {\bigl\lvert \phi \bigr\rangle}$ and $F_0 {\bigl\lvert 1 \bigr\rangle} = {\bigl\lvert \psi \bigr\rangle}$ as we changed basis in [Section \[secGhost2\]]{}). We recall from [Section \[secFusion\]]{} that every module in our ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ theory could be regarded as the image under a spectral flow automorphism of either ${{\widehat{\mathcal{L}}}_{0}}$ or ${{\widehat{\mathcal{L}}}_{1}}$. It is reasonable therefore to expect that the same conclusion will hold for the extended algebra modules.
This is indeed the case. The spectral flow[^12]
\[eqnSFsl2R\] $$\begin{gathered}
{\widetilde{\gamma} \left( e_n \right)} = e_{n-1} \qquad {\widetilde{\gamma} \left( h_n \right)} = h_n - \delta_{n,0} K \qquad {\widetilde{\gamma} \left( f_n \right)} = f_{n+1} \\
{\widetilde{\gamma} \left( K \right)} = K \qquad {\widetilde{\gamma} \left( L_0 \right)} = L_0 - \frac{1}{2} h_0 + \frac{1}{4} K\end{gathered}$$
may be derived from the following extended algebra automorphism (which we also denote by $\widetilde{\gamma}$) $$\label{eqnExtSF}
{\widetilde{\gamma} \left( \Phi_n \right)} = \Phi_{n - 1/2} \qquad {\widetilde{\gamma} \left( \Psi_n \right)} = \Psi_{n + 1/2}.$$ Glancing at [Equation (\[eqnExtAlgComm\])]{}, this is obviously an extended algebra automorphism, and the change of mode indices from integer to half-integer and vice-versa precisely accounts for the fact that the monodromy charge changes in this way when applying the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ spectral flow.
To show that [Equation (\[eqnExtSF\])]{} implies [Equation (\[eqnSFsl2R\])]{}, we derive certain generalised commutation relations relating extended algebra modes and affine modes. These are obtained by evaluating $$\oint_0 \oint_w {\Phi \left( z \right)} {\Phi \left( w \right)} z^{m+1/2} w^{n-1/2} {\left( z-w \right)}^{-1} \frac{{\mathrm{d}}z}{2 \pi {\mathfrak{i}}} \frac{{\mathrm{d}}w}{2 \pi {\mathfrak{i}}}$$ in two different ways (and by replacing one or both of the fields $\Phi$ by $\Psi$). We can expand the operator product directly, using [Equation (\[eqnNewOPEs\])]{}, or we can break the $z$-contour around $w$ into the difference of two contours around the origin, one with ${\left| z \right|} > {\left| w \right|}$ and the other with ${\left| z \right|} < {\left| w \right|}$. The results of this procedure are the following generalised commutation relations:
\[eqnGCRs\] $$\begin{aligned}
\sum_{j=0}^{\infty} {\left[ \Phi_{m-j} \Phi_{n+j} + \Phi_{n-j-1} \Phi_{m+j+1} \right]} &= 2 e_{m+n}, \label{eqnGCRe} \\
\sum_{j=0}^{\infty} {\left[ \Psi_{m-j} \Phi_{n+j} + \Phi_{n-j-1} \Psi_{m+j+1} \right]} &= h_{m+n} + {\left( m + \frac{1}{2} \right)} \delta_{m+n,0}, \label{eqnGCRh} \\
\sum_{j=0}^{\infty} {\left[ \Psi_{m-j} \Psi_{n+j} + \Psi_{n-j-1} \Psi_{m+j+1} \right]} &= 2 f_{m+n}. \label{eqnGCRf}\end{aligned}$$
Regarding these as defining relations for the affine modes[^13], it is easy to check that applying [Equation (\[eqnExtSF\])]{} recovers the affine spectral flow (with the implicit replacement of $K$ by $k = \tfrac{-1}{2}$).
Now consider the singular vectors of the extended algebra module ${\mathbb{L}_{0}}$, or rather of the corresponding Verma module ${\mathbb{V}_{0}}$. Since one expects this module to be composed of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ Verma modules ${{\widehat{\mathcal{V}}}_{0}}$ and ${{\widehat{\mathcal{V}}}_{1}}$, there are four non-trivial singular vector combinations to consider:
$$\begin{gathered}
f_0 {\bigl\lvert 0 \bigr\rangle} \in {{\widehat{\mathcal{V}}}_{0}}, \qquad f_0 {\bigl\lvert \Psi \bigr\rangle} \in {{\widehat{\mathcal{V}}}_{1}}, \\
{\left( 156 e_{-3} e_{-1} - 71 e_{-2}^2 + 44 e_{-2} h_{-1} e_{-1} - 52 h_{-2} e_{-1}^2 + 16 f_{-1} e_{-1}^3 - 4 h_{-1}^2 e_{-1}^2 \right)} {\bigl\lvert 0 \bigr\rangle} \in {{\widehat{\mathcal{V}}}_{0}} \label{eqnSV0} \\
\text{and} \qquad {\left( 7 e_{-2} - 2 h_{-1} e_{-1} \right)} {\bigl\lvert \Phi \bigr\rangle} + 4 e_{-1}^2 {\bigl\lvert \Psi \bigr\rangle} \in {{\widehat{\mathcal{V}}}_{1}}. \label{eqnSV1}\end{gathered}$$
Note the slight sign change in (\[eqnSV0\]) as compared to (\[eqnVacSV\]) due to our change of basis. Note also that (\[eqnSV1\]) has the correct dimension and ${{\mathfrak{sl}} \left( 2 \right)}$-weight as given in [Equation (\[eqnV1SVs\])]{}.
But applying [Equation (\[eqnGCRf\])]{} with $m = \tfrac{-1}{2}$ to $f_0 {\bigl\lvert 0 \bigr\rangle}$ gives $$f_0 {\bigl\lvert 0 \bigr\rangle} = \sum_{j=0}^{\infty} \Psi_{-j - 1/2} \Psi_{j + 1/2} {\bigl\lvert 0 \bigr\rangle} = 0,$$ since there are no states in ${\mathbb{V}_{0}}$ with conformal dimension less than $0$. Similarly, $$f_0 {\bigl\lvert \Psi \bigr\rangle} = f_0 \Psi_{-1/2} {\bigl\lvert 0 \bigr\rangle} = \Psi_{-1/2} f_0 {\bigl\lvert 0 \bigr\rangle} = 0.$$ We therefore see that $f_0 {\bigl\lvert 0 \bigr\rangle}$ and $f_0 {\bigl\lvert \Psi \bigr\rangle}$ are not (non-trivial) singular vectors in ${\mathbb{V}_{0}}$, rather they vanish *identically*. It is somewhat more surprising that the same is true for the vectors (\[eqnSV0\]) and (\[eqnSV1\]). We will detail this computation for the latter vector leaving the former as a simple if tedious exercise.
Consider therefore the first term of (\[eqnSV1\]), $e_{-2} {\bigl\lvert \Phi \bigr\rangle} = e_{-2} \Phi_{-1/2} {\bigl\lvert 0 \bigr\rangle}$. Commuting the affine mode to the right and using [Equation (\[eqnGCRe\])]{} with $m = \tfrac{-1}{2}$ gives $$e_{-2} {\bigl\lvert \Phi \bigr\rangle} = \frac{1}{2} \Phi_{-1/2} {\left( \Phi_{-1/2} \Phi_{-3/2} + \Phi_{-3/2} \Phi_{-1/2} \right)} {\bigl\lvert 0 \bigr\rangle} = \Phi_{-3/2} \Phi_{-1/2}^2 {\bigl\lvert 0 \bigr\rangle}.$$ Repeating this process with $e_{-1} {\bigl\lvert \Phi \bigr\rangle}$ and then $h_{-1} e_{-1} {\bigl\lvert \Phi \bigr\rangle}$ (using [Equation (\[eqnGCRh\])]{}) yields $$h_{-1} e_{-1} {\bigl\lvert \Phi \bigr\rangle} = {\left( \frac{3}{2} \Phi_{-3/2} \Phi_{-1/2}^2 + \frac{1}{2} \Psi_{-1/2} \Phi_{-1/2}^4 \right)} {\bigl\lvert 0 \bigr\rangle}.$$ Finally, recalling that ${\bigl[ e_m , \Psi_n \bigr]} = -\Phi_{m+n}$, we derive that $$e_{-1}^2 {\bigl\lvert \Psi \bigr\rangle} = {\left( \frac{1}{4} \Psi_{-1/2} \Phi_{-1/2}^4 - \Phi_{-3/2} \Phi_{-1/2}^2 \right)} {\bigl\lvert 0 \bigr\rangle}.$$ We therefore see that all the terms of (\[eqnSV1\]) explicitly cancel, hence that this singular vector also vanishes identically in ${\mathbb{V}_{0}}$.
It follows from the identical vanishing of these singular vectors that the extended algebra Verma module ${\mathbb{V}_{0}}$ is irreducible and is therefore composed of irreducible ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$-modules: ${\mathbb{V}_{0}} = {\mathbb{L}_{0}} \sim {{\widehat{\mathcal{L}}}_{0}} \oplus {{\widehat{\mathcal{L}}}_{1}}$. Because the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ spectral flow lifts to a spectral flow automorphism (\[eqnExtSF\]) on the extended algebra, we may immediately deduce that the other extended algebra modules ${\mathbb{V}_{\ell}} = {\widetilde{\gamma}^{\ell} \bigl( {\mathbb{V}_{0}} \bigr)}$ ($\ell \in {\mathbb{Z}}$), which will not be Verma modules in general, are likewise irreducible. We mention that the irreducibility of extended algebra Verma modules is generic for (finite) simple current extensions [@RidSU206; @RidMin07], although the extended algebra will usually have to be defined by generalised commutation relations.
The extended algebra characters are therefore easily deduced from the obvious Verma module (Poincaré-Birkhoff-Witt) bases. Indeed, the character of the extended vacuum module is just $$\label{eqnExtChar0}
{{\chi_{{\mathbb{L}_{0}}} \left( z ; q \right)}} = \prod_{i=1}^{\infty} \frac{1}{{\left( 1 - z^{-1} q^{i-1/2} \right)} {\left( 1 - z q^{i-1/2} \right)}} = \sum_{n \in {\mathbb{Z}}/ 2} \sum_{m = {\left| n \right|}}^{\infty} \frac{q^m}{{\left( q \right)_{m-n}} {\left( q \right)_{m+n}}} z^{2n},$$ where ${\left( q \right)_{m}} = \prod_{i=1}^m {\left( 1 - q^i \right)}$ as usual, and we have used the well-known partition identity [@AndThe76 Eq. 2.2.5] $$\prod_{i=1}^{\infty} \frac{1}{1 - z q^i} = \sum_{j=0}^{\infty} \frac{q^j}{{\left( q \right)_{j}}} z^j.$$ This is an example of a so-called *fermionic character formula* — upon expanding the ${\left( q \right)_{m}}$ factors in the denominator, we find that all the contributions to the sums come with positive signs. Splitting the sum over $n$ into $n \in {\mathbb{Z}}$ and $n \in {\mathbb{Z}}+ \tfrac{1}{2}$ gives fermionic character formulae for the affine modules ${{\widehat{\mathcal{L}}}_{0}}$ and ${{\widehat{\mathcal{L}}}_{1}}$, respectively. This is to be contrasted with the *bosonic* character formulae given for these modules in [Equations (\[eqnCh0\]) and (\[eqnCh1\])]{} which are not manifestly positive in this sense. The difference is that before we had to subtract and add contributions corresponding to the braiding pattern of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ singular vectors ([Figure \[figVacVerMod\]]{}). In the extended algebra picture, these singular vectors all vanish identically, leading to far nicer, manifestly positive character formulae.
Applying the spectral flow one more time, we get expressions for the characters of the extended modules ${\mathbb{L}_{\ell}}$: $${{\chi_{{\mathbb{L}_{\ell}}} \left( z ; q \right)}} = \frac{z^{-\ell / 2} q^{-\ell^2 / 8}}{\displaystyle \prod_{i=1}^{\infty} {\left( 1 - z^{-1} q^{i - {\left( \ell + 1 \right)}/2} \right)} {\left( 1 - z q^{i + {\left( \ell - 1 \right)}/2} \right)}} = z^{-\ell / 2} q^{-\ell^2 / 8} \sum_{n \in {\mathbb{Z}}/ 2} \sum_{m = {\left| n \right|}}^{\infty} \frac{q^{m + \ell n}}{{\left( q \right)_{m-n}} {\left( q \right)_{m+n}}} z^{2n}.$$ The product forms tell us directly (compare [Section \[secChar\]]{}) that these characters have simple poles when $z^2 = q^i$ for all $i \in 2 {\mathbb{Z}}- 1 - \ell$. The fermionic sum form is even nicer. It gives the decomposition of the character into so-called string functions of constant ${{\mathfrak{sl}} \left( 2 \right)}$-weight. Unlike the $q$-expansions of [Section \[secChar\]]{}, these string functions have $q$-expansions which always give the multiplicities of the weights of the modules correctly. For example, when $\ell = 2$ the terms with ${{\mathfrak{sl}} \left( 2 \right)}$-weight $2n$ have $q$-expansion $q^{{\left| n \right|} + 2n} + \ldots$, so the lowest power of $q$ is $3n > 0$ when $n$ is positive, but is $n < 0$ when $n$ is negative (compare with the depictions of the affine modules in [Figure \[figSpecFlow\]]{}). Again, restricting the sum to $n$ integer or half-integer recovers fermionic character formulae for the constituent ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$-modules.
Modular Invariance {#secMod}
==================
Finally, we consider the modular properties of the ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$-characters. Whereas the bosonic character formulae (\[eqnCharSF0\]) and (\[eqnCharSF1\]) for the affine modules were naturally expressed in terms of classical theta functions, the characters of the extended algebra may be expressed in terms of ordinary Jacobi theta functions (our conventions for these are summarised in [Appendix \[appTheta\]]{}). Before giving these expressions, it is convenient to redefine the characters (in the standard manner) by $${{\widetilde{\chi}_{{\mathcal{M}_{}}} \left( y ; z ; q \right)}} = \operatorname{tr}_{{\mathcal{M}_{}}} y^K z^{H_0} q^{L_0 - C/24}.$$ Since $C$ and $K$ are central, the only effect of this redefinition is to multiply the characters by the factors $q^{-c/24} = q^{1/24}$ and $y^k = y^{-1/2}$. This may seem trivial, especially the inclusion of the new variable $y$, but is in fact essential for constructing representations of the modular group ${{\mathsf{SL}} \left( 2 ; {\mathbb{Z}}\right)}$ [@KacInf84].
To begin, let us compare [Equations (\[eqnThIdPF\]) and (\[eqnDefeta\])]{} with the product form of the character formula (\[eqnExtChar0\]). We find that $${{\widetilde{\chi}_{{\mathbb{L}_{0}}} \left( y ; z ; q \right)}} = y^{-1/2} \frac{{\eta \left( q \right)}}{{\vartheta_{4} \bigl( z ; q \bigr)}}.$$ As the ${{\mathfrak{sl}} \left( 2 \right)}$-weights of ${{\widehat{\mathcal{L}}}_{0}}$ are all even whereas those of ${{\widehat{\mathcal{L}}}_{1}}$ are all odd, we can project onto the affine characters using the known behaviour of the theta functions under $z \rightarrow e^{{\mathfrak{i}}\pi} z$ ([Equation (\[eqnThId-z\])]{}):
$$\begin{aligned}
{{\widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \left( y ; z ; q \right)}} &= \frac{y^{-1/2}}{2} {\left[ \frac{{\eta \left( q \right)}}{{\vartheta_{4} \bigl( z ; q \bigr)}} + \frac{{\eta \left( q \right)}}{{\vartheta_{3} \bigl( z ; q \bigr)}} \right]} &
{{\widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{1}}} \left( y ; z ; q \right)}} &= \frac{y^{-1/2}}{2} {\left[ \frac{{\eta \left( q \right)}}{{\vartheta_{4} \bigl( z ; q \bigr)}} - \frac{{\eta \left( q \right)}}{{\vartheta_{3} \bigl( z ; q \bigr)}} \right]}.\end{aligned}$$
Spectral flow and [Equation (\[eqnThIdSF\])]{} then give $$\begin{aligned}
{{\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \left( y ; z ; q \right)}} &= \frac{y^{-1/2}}{2} {\left[ \frac{-{\mathfrak{i}}{\eta \left( q \right)}}{{\vartheta_{1} \bigl( z ; q \bigr)}} + \frac{{\eta \left( q \right)}}{{\vartheta_{2} \bigl( z ; q \bigr)}} \right]} &
{{\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \left( y ; z ; q \right)}} &= \frac{y^{-1/2}}{2} {\left[ \frac{-{\mathfrak{i}}{\eta \left( q \right)}}{{\vartheta_{1} \bigl( z ; q \bigr)}} - \frac{{\eta \left( q \right)}}{{\vartheta_{2} \bigl( z ; q \bigr)}} \right]}.\end{aligned}$$
These are the four linearly independent (admissible) characters of our theory.
It is now clear from [Equations (\[eqnThIdS\]) and (\[eqnetaS\])]{} that the action of the modular transformation $S$ on the ratios $\eta / \vartheta_i$ appearing in the admissible characters will be to recover such a ratio, but multiplied by the factor ${\exp \left( -{\mathfrak{i}}\pi \zeta^2 / \tau \right)}$, where $z = {\exp \left( 2 \pi {\mathfrak{i}}\zeta \right)}$ and $q = {\exp \left( 2 \pi {\mathfrak{i}}\tau \right)}$. Cancelling this unwanted factor is the reason why we must include the variable $y$ in the normalised characters. Specifically, if $y = {\exp \left( 2 \pi {\mathfrak{i}}t \right)}$, then we can extend the action (\[eqnModGenAct\]) of the modular group generators as follows: $$S \colon {\left( t , \zeta , \tau \right)} \longmapsto {\left( t - \zeta^2 / \tau , \zeta / \tau , -1 / \tau \right)} \qquad T \colon {\left( t , \zeta , \tau \right)} \longmapsto {\left( t , \zeta , \tau + 1 \right)}.$$ One can easily check that $S^4 = {\left( ST \right)}^6 = \operatorname{id}$ as before. With this extended action, we can now compute (in hopefully obvious notation) $$\begin{aligned}
{{\widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \left( t - \zeta^2 / \tau \mid \zeta / \tau \mid -1 / \tau \right)}} &= \frac{e^{-{\mathfrak{i}}\pi t}}{2} {\left[ \frac{{\eta \left( \tau \right)}}{{\vartheta_{2} \bigl( \zeta \mid \tau \bigr)}} + \frac{{\eta \left( \tau \right)}}{{\vartheta_{3} \bigl( \zeta \mid \tau \bigr)}} \right]} \notag \\
&= \frac{1}{2} {\left[ \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} - \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} + \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} - \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{1}}} \right]} {\left( t \mid \zeta \mid \tau \right)} \\[1mm]
{{\widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \left( t \mid \zeta \mid \tau + 1 \right)}} &= e^{{\mathfrak{i}}\pi / 12} {{\widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \left( t \mid \zeta \mid \tau \right)}}.\end{aligned}$$
Repeating these computations for the other admissible characters, we obtain the $S$-matrix and $T$-matrix representing these modular transformations on the vector space spanned by the admissible characters. With respect to the ordered basis $$\label{eqnAdmChBasis}
{\left\{ \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}}, \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{1}}}, \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}}, \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \right\}}$$ (which corresponds to the admissible [highest weight modules]{}), these matrices are $$S = \frac{1}{2}
\begin{pmatrix}
1 & -1 & 1 & -1 \\
-1 & 1 & 1 & -1 \\
1 & 1 & {\mathfrak{i}}& {\mathfrak{i}}\\
-1 & -1 & {\mathfrak{i}}& {\mathfrak{i}}\end{pmatrix}
\qquad \text{and} \qquad
T =
\begin{pmatrix}
e^{{\mathfrak{i}}\pi / 12} & 0 & 0 & 0 \\
0 & -e^{{\mathfrak{i}}\pi / 12} & 0 & 0 \\
0 & 0 & e^{-{\mathfrak{i}}\pi / 6} & 0 \\
0 & 0 & 0 & e^{-{\mathfrak{i}}\pi / 6}
\end{pmatrix}
.$$ Both matrices are symmetric and unitary. We note that $S^2 \colon {\left( t , \zeta , \tau \right)} \longmapsto {\left( t , -\zeta , \tau \right)}$ represents conjugation, but that $$\label{eqnS^2=C}
S^2 =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & -1 & 0
\end{pmatrix}
.$$ This indicates that ${{\widehat{\mathcal{L}}}_{0}}$ and ${{\widehat{\mathcal{L}}}_{1}}$ are self-conjugate, as we know, but the appearance of the negative entries in the last two rows deserves comment. These negative entries may be explained by noting that the conjugates of the [highest weight modules]{} ${\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ are the non-[highest weight modules]{} ${\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}$ and ${\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$ (respectively). The latter modules do not appear in the list of admissible modules, but their characters satisfy ([Section \[secChar\]]{}) $${{\widetilde{\chi}_{{\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \left( y ; z ; q \right)}} = -{{\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \left( y ; z ; q \right)}} \qquad \text{and} \qquad {{\widetilde{\chi}_{{\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \left( y ; z ; q \right)}} = -{{\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \left( y ; z ; q \right)}}.$$ This precisely accounts for the negative off-diagonal entries in $S^2$. Put differently, this shows that $S^2$ represents conjugation on the Grothendieck ring of characters ([Section \[secChar\]]{}).
The diagonal modular invariant therefore takes the form $$\begin{aligned}
{\mathcal{Z}_{\text{diag.}} \left( y ; z ; q \right)} &= {\bigl| \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \bigr|}^2 + {\bigl| \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{1}}} \bigr|}^2 + {\bigl| \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \bigr|}^2 + {\bigl| \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \bigr|}^2 \notag \\
&= \frac{1}{2 {\left| y \right|}} {\left[ \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{4} \bigl( z ; q \bigr)} \right|}^2} + \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{3} \bigl( z ; q \bigr)} \right|}^2} + \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{2} \bigl( z ; q \bigr)} \right|}^2} + \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{1} \bigl( z ; q \bigr)} \right|}^2} \right]}.\end{aligned}$$ Furthermore, [Equation (\[eqnS\^2=C\])]{} specifies that the charge-conjugate modular invariant takes the form $$\begin{aligned}
{\mathcal{Z}_{\text{cc.}} \left( y ; z ; q \right)} &= {\bigl| \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \bigr|}^2 + {\bigl| \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{1}}} \bigr|}^2 - \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}}^* - \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}}^* \notag \\
&= {\bigl| \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}} \bigr|}^2 + {\bigl| \widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{1}}} \bigr|}^2 + \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \widetilde{\chi}_{{\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}}^* + \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}}
\widetilde{\chi}_{{\gamma^{-1} \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}}^* \notag \\
&= \frac{1}{2 {\left| y \right|}} {\left[ \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{4} \bigl( z ; q \bigr)} \right|}^2} + \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{3} \bigl( z ; q \bigr)} \right|}^2} + \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{2} \bigl( z ; q \bigr)} \right|}^2} - \frac{{\left| {\eta \left( q \right)} \right|}^2}{{\left| {\vartheta_{1} \bigl( z ; q \bigr)} \right|}^2} \right]},\end{aligned}$$ where the asterisks denote complex conjugation. We emphasise the negative coefficients appearing with respect to the basis (\[eqnAdmChBasis\]). If one neglects these signs (as in [@LesSU202]), then the “invariant” transforms non-trivially under the modular $S$ transformation. Indeed, it is not hard to show that every modular invariant must have the form $${\mathcal{Z}_m \left( y ; z ; q \right)} = {\mathcal{Z}_{\text{diag.}} \left( y ; z ; q \right)} + m {\bigl| \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} + \widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \bigr|}^2, \qquad m \in {\mathbb{Z}}.$$ (In this classification, $\mathcal{Z}_{\text{diag.}} = \mathcal{Z}_0$ and $\mathcal{Z}_{\text{cc.}} = \mathcal{Z}_{-1}$.) This reflects the simple observation that $${{\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} \left( y ; z ; q \right)}} + {{\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} \left( y ; z ; q \right)}} = {{\widetilde{\chi}_{{\mathbb{L}_{1}}} \left( y ; z ; q \right)}} = -{\mathfrak{i}}y^{-1/2} \frac{{\eta \left( q \right)}}{{\vartheta_{1} \bigl( z ; q \bigr)}}$$ is itself ${{\mathsf{SL}} \left( 2 ; {\mathbb{Z}}\right)}$-invariant, up to a factor of ${\mathfrak{i}}$.
Finally, it is appropriate to discuss the Verlinde formula. In rational theories, this summarises a remarkable connection between the modular properties of the characters and the fusion ring. If $\mathcal{N}_{\lambda \mu}^{\hphantom{\lambda \mu} \nu}$ denotes the multiplicity with which ${{\widehat{\mathcal{L}}}_{\nu}}$ appears in the fusion decomposition of ${{\widehat{\mathcal{L}}}_{\lambda}}$ and ${{\widehat{\mathcal{L}}}_{\mu}}$, then the Verlinde formula relates these fusion multiplicities to the modular $S$-matrix via $$\mathcal{N}_{\lambda \mu}^{\hphantom{\lambda \mu} \nu} = \sum_{\sigma} \frac{S_{\lambda \sigma} S_{\mu \sigma} S_{\nu \sigma}^*}{S_{0 \sigma}}.$$ Here the sum runs over all irreducible modules ${{\widehat{\mathcal{L}}}_{\sigma}}$ in the fusion ring, and the index $0$ refers to the vacuum module ${{\widehat{\mathcal{L}}}_{0}}$.
In our fractional level theory, we no longer have a bijective correspondence between the modules of the theory and the characters, so it is pointless to expect a direct relation between the fusion ring of our theory and the $S$-matrix. However, we can compute the “fusion multiplicities” obtained from the Verlinde formula by restricting the sum to the linearly independent admissible characters (\[eqnAdmChBasis\]). Collecting these multiplicities in fusion matrices, ${\left( N_{\lambda} \right)}_{\mu \nu} = \mathcal{N}_{\lambda \mu}^{\hphantom{\lambda \mu} \nu}$, the results are
$$\begin{gathered}
N_{{{\widehat{\mathcal{L}}}_{0}}} =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\qquad N_{{{\widehat{\mathcal{L}}}_{1}}} =
\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{pmatrix}
\\
N_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} =
\begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0
\end{pmatrix}
\qquad N_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} =
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{pmatrix}
.\end{gathered}$$
Whilst the negative “fusion” multiplicities might seem alarming at first sight, it is easy to check that these are precisely the structure constants of the Grothendieck ring of characters. For example, the Verlinde formula gives $$\mathcal{N}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)} {\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}}^{\hphantom{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)} {\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} {{\widehat{\mathcal{L}}}_{0}}} = -1,$$ which reflects the Grothendieck fusion rule $$\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}} {\times_{\! f}}\widetilde{\chi}_{{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)}} = -\widetilde{\chi}_{{{\widehat{\mathcal{L}}}_{0}}}.$$ This is of course the projection of the fusion rule $${\gamma \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)} {\times_{\! f}}{\gamma \bigl( {{\widehat{\mathcal{L}}}_{0}} \bigr)} = {\gamma^2 \bigl( {{\widehat{\mathcal{L}}}_{1}} \bigr)}$$ onto the characters, by [Equation (\[eqnLinIndChars\])]{}. There is no mystery here — the modular $S$-matrix only sees the Grothendieck ring of characters, so it is no surprise that the Verlinde formula reconstructs the structure constants of this ring, rather than that of the full fusion ring. And as we have seen, these structure constants are quite often negative.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Vladimir Mitev for initiating these thoughts by asking me what the extended algebra of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ would be. I also thank Volker Schomerus for introducing me to spectral flow, Thomas Creutzig for enlightening discussions on what this actually means, and Pierre Mathieu for explaining [@LesSU202] to me. This work was partially supported by the Galileo Galilei Institute for Theoretical Physics, the INFN, and the Marie Curie Excellence Grant MEXT-CT-2006-042.
Spectral Flow {#appSpecFlow}
=============
In this appendix, we detail the construction of spectral flow automorphisms. Spectral flow has a long history in the [conformal field theory]{} literature, and can be traced back at least as far as [@SchCom87]. The name refers to the fact that these automorphisms do not preserve the conformal dimension, hence the spectrum “flows” (discretely in this case) under their action. We are actually only interested in the case where ${\widehat{{\mathfrak{g}}}} = {{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$, but it is not much harder to develop the theory for general (untwisted) affine Kac-moody algebras ${\widehat{{\mathfrak{g}}}}$ (and it is very beautiful).
Affine Weyl Group Translations {#appAffW}
------------------------------
Let ${\mathfrak{g}}$ be the horizontal subalgebra of ${\widehat{{\mathfrak{g}}}}$, let $\alpha$ denote a root of ${\mathfrak{g}}$ with root vector $e^{\alpha}$ and coroot $\alpha^{\vee}$, and let ${\mathsf{W}}$ be the Weyl group of ${\mathfrak{g}}$. Then, each $w \in {\mathsf{W}}$ permutes the roots and thereby induces an automorphism of ${\mathfrak{g}}$ via $${w \left( e^{\alpha} \right)} = e^{{w \left( \alpha \right)}}, \qquad \text{hence} \qquad {w \left( \alpha^{\vee} \right)} = {w \left( \alpha \right)}^{\vee}.$$ This generalises to ${\widehat{{\mathfrak{g}}}}$ as follows. The real roots now take the form $\alpha + n {\widehat{\delta}}$ ($n \in {\mathbb{Z}}$), where $\alpha$ is a root of ${\mathfrak{g}}$ and ${\widehat{\delta}}$ is the generating imaginary root. The corresponding root vector is $e^{\alpha}_n$. The root vectors corresponding to the imaginary root $n {\widehat{\delta}}$ ($n \neq 0$) are denoted by $h^i_n$, $i = 1, 2, \ldots, \operatorname{rank}{\mathfrak{g}}$, and we will associate the $h^i$ with the simple coroots of ${\mathfrak{g}}$: $h^i = \alpha_i^{\vee}$. The affine Weyl group decomposes as ${\widehat{{\mathsf{W}}}} = {\mathsf{W}} \ltimes {\mathsf{Q}}^{\vee}$, where ${\mathsf{Q}}^{\vee}$ is the coroot lattice of ${\mathfrak{g}}$. The coroot lattice acts on the roots of ${\widehat{{\mathfrak{g}}}}$ by translations in the imaginary direction: $$\alpha^{\vee} \colon \beta + n {\widehat{\delta}} \longmapsto \beta + {\left( n - {\left\langle \beta , \alpha^{\vee} \right\rangle} \right)} {\widehat{\delta}}.$$ This is nothing but the usual affine Weyl group action obtained by embedding the roots into the weight space of ${\widehat{{\mathfrak{g}}}}$.
It follows that the simple coroots $\alpha_i^{\vee}$ ($i=1, 2, \ldots, \operatorname{rank}{\mathfrak{g}}$) of ${\mathfrak{g}}$ each define an independent transformation $\tau_i$ on the root vectors of ${\widehat{{\mathfrak{g}}}}$ via $$\label{eqnStartingPoint}
{\tau_i \left( e^{\alpha}_n \right)} = e^{\alpha}_{n - {\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} \quad \text{($n \in {\mathbb{Z}}$)} \qquad \text{and} \qquad {\tau_i \left( h^j_n \right)} = h^j_n \quad \text{($n \neq 0$).}$$ We extend these transformations to automorphisms of ${\widehat{{\mathfrak{g}}}}$. First we compute $$\begin{aligned}
{\tau_i \left( h^j_0 \right)} &= {\tau_i \left( {\bigl[ e^{\alpha_j}_n , e^{-\alpha_j}_{-n} \bigr]} - n {\kappa \bigl( e^{\alpha_j} , e^{-\alpha_j} \bigr)} K \right)} = {\bigl[ e^{\alpha_j}_{n - {\left\langle \alpha_j , \alpha_i^{\vee} \right\rangle}} , e^{-\alpha_j}_{-n + {\left\langle \alpha_j , \alpha_i^{\vee} \right\rangle}} \bigr]} - \frac{2n}{{\left\| \alpha_j \right\|}^2} {\tau_i \left( K \right)} \notag \\
&= h^j_0 - \frac{2 {\left\langle \alpha_j , \alpha_i^{\vee} \right\rangle}}{{\left\| \alpha_j \right\|}^2} K + \frac{2n}{{\left\| \alpha_j \right\|}^2} {\left( K - {\tau_i \left( K \right)} \right)}.\end{aligned}$$ Here, ${\kappa \bigl( \cdot , \cdot \bigr)}$ denotes the Killing form of ${\mathfrak{g}}$. Since this computation holds for all $n \in {\mathbb{Z}}$, we must have $${\tau_i \left( h^j_0 \right)} = h^j_0 - {\kappa \bigl( \alpha_i^{\vee} , \alpha_j^{\vee} \bigr)} K \qquad \text{and} \qquad {\tau_i \left( K \right)} = K.$$
It remains to determine the action of the $\tau_i$ on $L_0$. This is fixed by the Sugawara construction, but requires a little work. The normal-ordering appearing in this construction turns out to cause some difficulties and we will treat these by working in the (equivalent) field-theoretic framework, rather than at the level of the algebra itself. Note that the automorphisms $\tau_i$ act on the fields ${e^{\alpha} \left( z \right)} = \sum_n e^{\alpha}_n z^{-n-1}$ and ${h^j \left( z \right)} = \sum_n h^j_n z^{-n-1}$ by $${\tau_i \left( {e^{\alpha} \left( z \right)} \right)} = z^{-{\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} {e^{\alpha} \left( z \right)} \qquad \text{and} \qquad {\tau_i \left( {h^j \left( z \right)} \right)} = {h^j \left( z \right)} - {\kappa \bigl( \alpha_i^{\vee} , \alpha_j^{\vee} \bigr)} K z^{-1}.$$ Our goal is therefore to determine the corresponding action on $$\label{eqnDefT}
{T \left( z \right)} = \frac{1}{2 {\left( K + {\mathsf{h}^{\vee}}\right)}} {\left[ \sum_{m,n=1}^{\operatorname{rank}{\mathfrak{g}}} {\kappa^{-1} \bigl( h^m , h^n \bigr)} {{} : {h^m \left( z \right)} {h^n \left( z \right)} : {}} + \sum_{\alpha \in \Delta} {\kappa^{-1} \bigl( e^{\alpha} , e^{-\alpha} \bigr)} {{} : {e^{\alpha} \left( z \right)} {e^{-\alpha} \left( z \right)} : {}} \right]},$$ where ${\mathsf{h}^{\vee}}$ is the dual Coxeter number of ${\mathfrak{g}}$ and $\Delta$ is the set of roots of ${\mathfrak{g}}$.
We first note that $${\tau_i \left( {{} : {h^m \left( z \right)} {h^n \left( z \right)} : {}} \right)} = {{} : {h^m \left( z \right)} {h^n \left( z \right)} : {}} - \kappa_{im} {h^n \left( z \right)} K z^{-1} - \kappa_{in} {h^m \left( z \right)} K z^{-1} + \kappa_{im} \kappa_{in} K^2 z^{-2},$$ where $\kappa_{ab} = \kappa_{ba}$ denotes ${\kappa \bigl( h^a , h^b \bigr)}$. Under $\tau_i$, the sum over $m$ and $n$ in [Equation (\[eqnDefT\])]{} therefore gives $$\sum_{m,n=1}^{\operatorname{rank}{\mathfrak{g}}} {\kappa^{-1} \bigl( h^m , h^n \bigr)} {{} : {h^m \left( z \right)} {h^n \left( z \right)} : {}} - 2 {h^i \left( z \right)} K z^{-1} + \frac{4}{{\left\| \alpha_i \right\|}^2} K^2 z^{-2}.$$ Since $\tau_i$ changes the dimension of the ${e^{\alpha} \left( z \right)}$, it affects the normal-ordering in the corresponding terms in a non-trivial way. Using the standard definition of normal-ordering in [conformal field theory]{}, we compute $$\begin{aligned}
{\tau_i \left( {{} : {e^{\alpha} \left( w \right)} {e^{-\alpha} \left( w \right)} : {}} \right)} &= \oint_w {e^{\alpha} \left( z \right)} {e^{-\alpha} \left( w \right)} z^{-{\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} w^{{\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} {\left( z-w \right)}^{-1} \frac{{\mathrm{d}}z}{2 \pi {\mathfrak{i}}} \notag \\
&= w^{{\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} \oint_w z^{-{\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} {\left[ \frac{2K / {\left\| \alpha \right\|}^2}{{\left( z-w \right)}^3} + \frac{{\alpha^{\vee} \left( w \right)}}{{\left( z-w \right)}^2} + \frac{{{} : {e^{\alpha} \left( w \right)} {e^{-\alpha} \left( w \right)} : {}}}{z-w} \right]} \frac{{\mathrm{d}}z}{2 \pi {\mathfrak{i}}} \notag \\
&= {{} : {e^{\alpha} \left( w \right)} {e^{-\alpha} \left( w \right)} : {}} - {\left\langle \alpha , \alpha_i^{\vee} \right\rangle} w^{-1} {\alpha^{\vee} \left( w \right)} + \frac{{\left\langle \alpha , \alpha_i^{\vee} \right\rangle} {\left( {\left\langle \alpha , \alpha_i^{\vee} \right\rangle} + 1 \right)}}{{\left\| \alpha \right\|}^2} K w^{-2}.\end{aligned}$$ Under $\tau_i$, the sum over the roots in [Equation (\[eqnDefT\])]{} gives $$\begin{aligned}
\sum_{\alpha \in \Delta} &{\left[ {\kappa^{-1} \bigl( e^{\alpha} , e^{-\alpha} \bigr)} {{} : {e^{\alpha} \left( z \right)} {e^{-\alpha} \left( z \right)} : {}} - \frac{{\left\| \alpha \right\|}^2}{2} {\left\langle \alpha , \alpha_i^{\vee} \right\rangle} z^{-1} {\alpha^{\vee} \left( z \right)} + \frac{{\left\langle \alpha , \alpha_i^{\vee} \right\rangle} {\left( {\left\langle \alpha , \alpha_i^{\vee} \right\rangle} + 1 \right)}}{2} K z^{-2} \right]} \notag \\
&= \sum_{\alpha \in \Delta} {\left[ {\kappa^{-1} \bigl( e^{\alpha} , e^{-\alpha} \bigr)} {{} : {e^{\alpha} \left( z \right)} {e^{-\alpha} \left( z \right)} : {}} - \frac{2}{{\left\| \alpha_i \right\|}^2} {\left( \alpha , \alpha_i \right)} z^{-1} {\alpha \left( z \right)} + \frac{2}{{\left\| \alpha_i \right\|}^4} {\left( \alpha_i , \alpha \right)} {\left( \alpha , \alpha_i \right)} K z^{-2} \right]} \notag \\
&= \sum_{\alpha \in \Delta} {\kappa^{-1} \bigl( e^{\alpha} , e^{-\alpha} \bigr)} {{} : {e^{\alpha} \left( z \right)} {e^{-\alpha} \left( z \right)} : {}} - 2 {\mathsf{h}^{\vee}}z^{-1} {\alpha_i^{\vee} \left( z \right)} + \frac{4 {\mathsf{h}^{\vee}}}{{\left\| \alpha_i \right\|}^2} K z^{-2}.\end{aligned}$$ Here in the first step, we have used the fact that summands over $\Delta$ which are odd under $\alpha \rightarrow - \alpha$ give vanishing sums. In the second step, we use (twice) the fact that $$\sum_{\alpha \in \Delta} {\left( \lambda , \alpha \right)} {\left( \alpha , \mu \right)} = 2 {\mathsf{h}^{\vee}}{\left( \lambda , \mu \right)}$$ for all weights $\lambda$ and $\mu$.
Putting this all together (and remembering that $h^i = \alpha_i^{\vee}$), we finally obtain $$\begin{gathered}
{\tau_i \left( {T \left( z \right)} \right)} = {T \left( z \right)} - z^{-1} {h^i \left( z \right)} + \frac{2}{{\left\| \alpha_i \right\|}^2} K z^{-2} \\
\Rightarrow \qquad {\tau_i \left( L_0 \right)} = L_0 - h^i_0 + \frac{2}{{\left\| \alpha_i \right\|}^2} K.\end{gathered}$$ This then completes the description of the automorphisms of ${\widehat{{\mathfrak{g}}}}$ induced by the translation subgroup of the affine Weyl group. It is not hard to check now that powers of $\tau_i$ act as follows:
\[eqnTransAuts\] $$\begin{gathered}
{\tau_i^{\ell} \left( e^{\alpha}_n \right)} = e^{\alpha}_{n - \ell {\left\langle \alpha , \alpha_i^{\vee} \right\rangle}} \qquad {\tau_i^{\ell} \left( h^j_n \right)} = h^j_n - \ell {\kappa \bigl( \alpha_i^{\vee} , \alpha_j^{\vee} \bigr)} \delta_{n,0} K \\
{\tau_i^{\ell} \left( K \right)} = K \qquad {\tau_i^{\ell} \left( L_0 \right)} = L_0 - \ell h^i_0 + \ell {\left( \ell + \frac{2}{{\left\| \alpha_i \right\|}^2} - 1 \right)} K.\end{gathered}$$
These automorphisms are examples of *spectral flow* automorphisms. However, they do not usually exhaust the latter in general, as we shall see.
Outer Automorphisms {#appAffOut}
-------------------
Having determined the explicit action of the algebra automorphisms induced by the affine Weyl group, we can turn to the remaining automorphisms of ${\widehat{{\mathfrak{g}}}}$, the outer automorphisms induced by the symmetries of the Dynkin diagram. Unlike the (non-trivial) affine Weyl transformations, these preserve a given set of a simple roots. Indeed, an outer automorphism is completely determined by the permutation it induces on the (chosen set of) simple roots.
The outer automorphisms of ${\mathfrak{g}}$ therefore just permute the root vectors $e^{\alpha}_n$ and $h^j_n$ of ${\widehat{{\mathfrak{g}}}}$ without changing the grade $n$. But, by analogy with the results of the previous section, we would like to understand the general case. Happily, this is a simple endeavour. The automorphisms of ${\widehat{{\mathfrak{g}}}}$ which preserve the chosen Cartan subalgebra can be decomposed into $${{\mathsf{Aut}} \ }{\widehat{{\mathfrak{g}}}} = {{\mathsf{Out}} \ }{\widehat{{\mathfrak{g}}}} \ltimes {\widehat{{\mathsf{W}}}} = {{\mathsf{Aut}} \ }{\mathfrak{g}} \ltimes {\mathsf{Q}}^*,$$ where ${\mathsf{Q}}^*$ denotes the dual of the root lattice. Thus, our endeavour corresponds to generalising the results of [Appendix \[appAffW\]]{} to the outer automorphisms of ${\mathfrak{g}}$ (which is trivial) and replacing coroot lattice translations by dual root lattice translations. It is these dual root translations which generate the complete set of spectral flow automorphisms.
In fact, it is easy to understand these latter translations. Recall from [Equation (\[eqnStartingPoint\])]{} that our starting point for constructing the automorphisms corresponding to a translation by the simple coroot $\alpha_i^{\vee}$ was the effect on $e^{\alpha}_n$. Everything else follows from this effect, which was to lower $n$ by ${\left\langle \alpha , \alpha_i^{\vee} \right\rangle}$. However, this index will still be an integer (for all roots $\alpha$) if we replace $\alpha_i^{\vee}$ by an element of the dual root lattice ${\mathsf{Q}}^*$, so it follows that such a replacement will still lead to a well-defined automorphism of ${\widehat{{\mathfrak{g}}}}$.
In fact, we can always choose a basis of ${\mathsf{Q}}^*$ whose $\operatorname{rank}{\mathfrak{g}}$ elements are of the form $q_i^{\vee} / m_i$ for some $q_i^{\vee} \in {\mathsf{Q}}^{\vee}$ and $m_i \in {\mathbb{Z}}$ (the fact that ${\mathsf{Q}}^*$ contains ${\mathsf{Q}}^{\vee}$ follows from the integrality of the Cartan matrix). We may therefore determine generators of the automorphism group corresponding to dual root lattice translations by finding such a basis and applying [Equation (\[eqnTransAuts\])]{} with $\ell$ fractional. Note however that scaling $\alpha^{\vee}$ by some factor $t$ corresponds to scaling $\alpha$ by $t^{-1}$.
For example, the coroot lattice of ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}$ is generated by $\alpha_1^{\vee}$, so the coroot spectral flow automorphisms are generated by $\tau_1$: $${\tau_1 \left( e^{\alpha}_n \right)} = e^{\alpha}_{n - 2}, \qquad {\tau_1 \left( h^1_n \right)} = h^1_n - 2 \delta_{n,0} K, \qquad {\tau_1 \left( K \right)} = K, \qquad {\tau_1 \left( L_0 \right)} = L_0 - h^1_0 + K.$$ The dual root lattice is however generated by $\alpha_1^{\vee} / 2$. It follows that the spectral flow automorphisms are generated by $\gamma = \tau_1^{1/2}$. By [Equation (\[eqnTransAuts\])]{}, the action of $\gamma$ is given by $$\label{eqnSpecFlowA1}
{\gamma \left( e^{\alpha}_n \right)} = e^{\alpha}_{n - 1}, \qquad {\gamma \left( h^1_n \right)} = h^1_n - \delta_{n,0} K, \qquad {\gamma \left( K \right)} = K, \qquad {\gamma \left( L_0 \right)} = L_0 - \frac{1}{2} h^i_0 + \frac{1}{4} K.$$ It should be clear from these formulae why $\tau_1$ has a square root.
As a second example, the dual root lattice of ${{\widehat{{\mathfrak{sl}}}} \left( 3 \right)}$ is generated by $\tfrac{2}{3} \alpha_1^{\vee} + \tfrac{1}{3} \alpha_2^{\vee}$ and $\tfrac{1}{3} \alpha_1^{\vee} + \tfrac{2}{3} \alpha_2^{\vee}$. We therefore have the spectral flow generators $\gamma_1 = \tau_1^{2/3} \tau_2^{1/3}$ and $\gamma_2 = \tau_1^{1/3} \tau_2^{2/3}$, which act on ${{\widehat{{\mathfrak{sl}}}} \left( 3 \right)}$ via
$$\begin{gathered}
{\gamma_i \left( e^{\alpha_j}_n \right)} = e^{\alpha_j}_{n - \delta_{i,j}}, \qquad {\gamma_i \left( e^{\theta}_n \right)} = e^{\theta}_{n - 2}, \qquad {\gamma_i \left( h^j_n \right)} = h^j_n - \delta_{n,0} \delta_{i,j} K, \\
{\gamma_i \left( K \right)} = K, \qquad {\gamma_i \left( L_0 \right)} = L_0 - \frac{1}{3} {\left( h^1_0 + h^2_0 \right)} - \frac{1}{3} h^i_0 + \frac{1}{3} K.\end{gathered}$$
Finally, note that composing any representation of ${\widehat{{\mathfrak{g}}}}$ with an automorphism gives another representation. Hence, spectral flow automorphisms induce maps (vector space isomorphisms) between ${\widehat{{\mathfrak{g}}}}$-modules. Since such maps must preserve integrability, the set of integrable ${\widehat{{\mathfrak{g}}}}$-modules must close under the induced spectral flow. In fact, integrable modules are mapped to themselves when the spectral flow corresponds to a translation by a coroot lattice element. More general translations induce maps between integrable modules whose highest weights are related by an outer automorphism. In both cases, these maps are non-trivial and provide a wealth of information about the integrable modules. When the modules are not integrable, the spectral flow generally does not map any module to itself, even if the flow corresponds to a coroot translation. In this case, spectral flow automorphisms are useful for understanding the spectrum and for investigating the structure of the unfamiliar modules which arise.
Jacobi Theta Functions {#appTheta}
======================
We collect here for convenience our notation for the Jacobi theta functions and some of their important properties. First we define
$$\begin{aligned}
{\vartheta_{1} \bigl( z ; q \bigr)} &= -{\mathfrak{i}}\sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^{n+1/2} q^{{\left( n+1/2 \right)}^2 / 2} & {\vartheta_{3} \bigl( z ; q \bigr)} &= \sum_{n \in {\mathbb{Z}}} z^n q^{n^2 / 2} \\
{\vartheta_{2} \bigl( z ; q \bigr)} &= \sum_{n \in {\mathbb{Z}}} z^{n+1/2} q^{{\left( n+1/2 \right)}^2 / 2} & {\vartheta_{4} \bigl( z ; q \bigr)} &= \sum_{n \in {\mathbb{Z}}} {\left( -1 \right)}^n z^n q^{n^2 / 2}.\end{aligned}$$
From these definitions follow a number of simple relations:
\[eqnThId-z\] $$\begin{aligned}
{\vartheta_{1} \bigl( e^{{\mathfrak{i}}\pi} z ; q \bigr)} &= {\vartheta_{2} \bigl( z ; q \bigr)} & {\vartheta_{3} \bigl( e^{{\mathfrak{i}}\pi} z ; q \bigr)} &= {\vartheta_{4} \bigl( z ; q \bigr)} \\
{\vartheta_{2} \bigl( e^{{\mathfrak{i}}\pi} z ; q \bigr)} &= -{\vartheta_{1} \bigl( z ; q \bigr)} & {\vartheta_{4} \bigl( e^{{\mathfrak{i}}\pi} z ; q \bigr)} &= {\vartheta_{3} \bigl( z ; q \bigr)}\end{aligned}$$
\[eqnThIdSF\] $$\begin{aligned}
{\vartheta_{1} \bigl( zq^{1/2} ; q \bigr)} &= \frac{{\mathfrak{i}}}{z^{1/2} q^{1/8}} {\vartheta_{4} \bigl( z ; q \bigr)} & {\vartheta_{3} \bigl( zq^{1/2} ; q \bigr)} &= \frac{1}{z^{1/2} q^{1/8}} {\vartheta_{2} \bigl( z ; q \bigr)} \\
{\vartheta_{2} \bigl( zq^{1/2} ; q \bigr)} &= \frac{1}{z^{1/2} q^{1/8}} {\vartheta_{3} \bigl( z ; q \bigr)} & {\vartheta_{4} \bigl( zq^{1/2} ; q \bigr)} &= \frac{{\mathfrak{i}}}{z^{1/2} q^{1/8}} {\vartheta_{1} \bigl( z ; q \bigr)}\end{aligned}$$
By making use of Jacobi’s triple product identity [@AndThe76 Eq. 2.2.10], $$\label{eqnJacobiTriple}
\prod_{i=1}^{\infty} {\left( 1 + z q^{i-1/2} \right)} {\left( 1 - q^i \right)} {\left( 1 + z^{-1} q^{i-1/2} \right)} = \sum_{n \in {\mathbb{Z}}} z^n q^{n^2 / 2},$$ each of the theta functions may be written in product form:
\[eqnThIdPF\] $$\begin{aligned}
{\vartheta_{1} \bigl( z ; q \bigr)} &= -{\mathfrak{i}}z^{1/2} q^{1/8} \prod_{i=1}^{\infty} {\left( 1 - z q^i \right)} {\left( 1 - q^i \right)} {\left( 1 - z^{-1} q^{i-1} \right)} \\
{\vartheta_{2} \bigl( z ; q \bigr)} &= z^{1/2} q^{1/8} \prod_{i=1}^{\infty} {\left( 1 + z q^i \right)} {\left( 1 - q^i \right)} {\left( 1 + z^{-1} q^{i-1} \right)} \\
{\vartheta_{3} \bigl( z ; q \bigr)} &= \prod_{i=1}^{\infty} {\left( 1 + z q^{i-1/2} \right)} {\left( 1 - q^i \right)} {\left( 1 + z^{-1} q^{i-1/2} \right)} \\
{\vartheta_{4} \bigl( z ; q \bigr)} &= \prod_{i=1}^{\infty} {\left( 1 - z q^{i-1/2} \right)} {\left( 1 - q^i \right)} {\left( 1 - z^{-1} q^{i-1/2} \right)}.\end{aligned}$$
This also gives us the identity $$\label{eqnIdeta}
{\vartheta_{2} \bigl( 1 ; q \bigr)} {\vartheta_{3} \bigl( 1 ; q \bigr)} {\vartheta_{4} \bigl( 1 ; q \bigr)} = 2 {\eta \left( q \right)}^3,$$ where $\eta$ is Dedekind’s eta function $$\label{eqnDefeta}
{\eta \left( q \right)} = q^{1/24} \prod_{i=1}^{\infty} {\left( 1 - q^i \right)}.$$
The most important property of these functions is their behaviour under modular transformations. Setting $z = {\exp \left( 2 \pi {\mathfrak{i}}\zeta \right)}$ and $q = {\exp \left( 2 \pi {\mathfrak{i}}\tau \right)}$, the modular group ${{\mathsf{SL}} \left( 2 ; {\mathbb{Z}}\right)}$ is generated by two transformations $S$ and $T$ which act via $$\label{eqnModGenAct}
S \colon {\left( \zeta , \tau \right)} \longmapsto {\left( \zeta / \tau , -1 / \tau \right)} \qquad T \colon {\left( \zeta , \tau \right)} \longmapsto {\left( \zeta , \tau + 1 \right)}.$$ One can check that $S^4 = {\left( ST \right)}^6 = \operatorname{id}$. Writing ${\vartheta_{i} \bigl( \zeta \mid \tau \bigr)}$ for ${\vartheta_{i} \bigl( e^{2 \pi {\mathfrak{i}}\zeta} ; e^{2 \pi {\mathfrak{i}}\tau} \bigr)}$, $T$ is therefore represented on the space of theta functions by
\[eqnThIdT\] $$\begin{aligned}
{\vartheta_{1} \bigl( \zeta \mid \tau + 1 \bigr)} &= e^{{\mathfrak{i}}\pi / 4} {\vartheta_{1} \bigl( \zeta \mid \tau \bigr)} & {\vartheta_{3} \bigl( \zeta \mid \tau + 1 \bigr)} &= {\vartheta_{4} \bigl( \zeta \mid \tau \bigr)} \\
{\vartheta_{2} \bigl( \zeta \mid \tau + 1 \bigr)} &= e^{{\mathfrak{i}}\pi / 4} {\vartheta_{2} \bigl( \zeta \mid \tau \bigr)} & {\vartheta_{4} \bigl( \zeta \mid \tau + 1 \bigr)} &= {\vartheta_{3} \bigl( \zeta \mid \tau \bigr)}.\end{aligned}$$
[Equation (\[eqnDefeta\])]{} gives (in hopefully obvious notation) $$\label{eqnetaT}
{\eta \left( \tau + 1 \right)} = e^{{\mathfrak{i}}\pi / 12} {\eta \left( \tau \right)}.$$ Determining the corresponding transformations under $S$ requires a specialisation of the Poisson resummation formula from Fourier analysis. With this tool, we derive
\[eqnThIdS\] $$\begin{aligned}
{\vartheta_{1} \bigl( \zeta / \tau , -1 / \tau \bigr)} &= -{\mathfrak{i}}\sqrt{-{\mathfrak{i}}\tau} \ e^{{\mathfrak{i}}\pi \zeta^2 / \tau} {\vartheta_{1} \bigl( \zeta \mid \tau \bigr)} & {\vartheta_{3} \bigl( \zeta / \tau , -1 / \tau \bigr)} &= \sqrt{-{\mathfrak{i}}\tau} \ e^{{\mathfrak{i}}\pi \zeta^2 / \tau} {\vartheta_{3} \bigl( \zeta \mid \tau \bigr)} \\
{\vartheta_{2} \bigl( \zeta / \tau , -1 / \tau \bigr)} &= \sqrt{-{\mathfrak{i}}\tau} \ e^{{\mathfrak{i}}\pi \zeta^2 / \tau} {\vartheta_{4} \bigl( \zeta \mid \tau \bigr)} & {\vartheta_{4} \bigl( \zeta / \tau , -1 / \tau \bigr)} &= \sqrt{-{\mathfrak{i}}\tau} \ e^{{\mathfrak{i}}\pi \zeta^2 / \tau} {\vartheta_{2} \bigl( \zeta \mid \tau \bigr)}.\end{aligned}$$
The additional factor of $-{\mathfrak{i}}$ for $\vartheta_1$ reflects the fact that this theta function is antisymmetric under $z \rightarrow z^{-1}$ whereas the others are symmetric. [Equation (\[eqnIdeta\])]{} now gives $$\label{eqnetaS}
{\eta \left( -1 / \tau \right)} = \sqrt{-{\mathfrak{i}}\tau} \ {\eta \left( \tau \right)}.$$
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[^1]:
[^2]: This is not necessarily true if one drops the requirement of compactness. However, investigations of conformally invariant sigma models on non-compact group manifolds have not yet revealed any clear relation to the fractional level models.
[^3]: We remark that a logarithmic [conformal field theory]{} with ${{\widehat{{\mathfrak{sl}}}} \left( 2 \right)}_{-1/2}$ symmetry was proposed in [@LesLog04], based on free field constructions. We do not expect that keeping the singular vector in the vacuum module will lead to this theory. We intend to return to a detailed discussion of how the theory discussed here can be extended to something similar to that of [@LesLog04] in a future publication.
[^4]: In fact, one can sometimes bypass this requirement [@GabFus01], but it adds significantly to the complexity of the computations.
[^5]: Of course, we can explicitly show that ${{\widehat{\mathcal{L}}}_{0}} {\times_{\! f}}{\mathcal{M}_{}} = {\mathcal{M}_{}}$ for each of our admissible modules using the Nahm-Gaberdiel-Kausch algorithm. But the above argument is much more elementary, and has the additional advantage of drawing attention to the subtleties possible when one does find logarithmic structure. It does assume the existence of conjugates, however this is physically necessary in all (quasirational) theories, even logarithmic ones (with a suitable interpretation) — fields without a conjugate decouple within correlation functions.
[^6]: Matthias Gaberdiel points out that this assumes that the result of the fusion is a module whose conformal dimensions are bounded below. I believe that this assumption is warranted because of the finite-dimensionality of the working space, but I have no proof of this at present. In any case, the conclusion of the above argument has been confirmed by explicitly calculating the fusion structure to grade $1$ using the full Nahm-Gaberdiel-Kausch algorithm.
[^7]: We know that this is a direct sum because these ${{\mathfrak{sl}} \left( 2 \right)}$-representations are unitary with respect to the ${{\mathfrak{sl}} \left( 2 ; {\mathbb{R}}\right)}$ adjoint $J^{\ddag} = -J$, $J = E,H,F$.
[^8]: This can only apply when the automorphisms acting commute. It is not clear what should replace this principle in general.
[^9]: Here, we mean an expansion in which the powers of $q$ are bounded from below, such as one obtains from computer algebra packages.
[^10]: We should mention that the notion of Grothendieck ring which we have defined here is not quite the same as that used in [logarithmic conformal field theory]{} (and in category theory in general). There, the Grothendieck ring makes precise the notion of forgetting the indecomposable structure of the modules in the fusion ring, essentially regarding these objects as graded vector spaces (for a precise definition, see [@GabFro08 App. C]). Since this is exactly what the characters do, we see that the spirit of the two definitions is the same, and so it is reasonable to call the ring of characters a Grothendieck ring (despite the absence of indecomposable structure in the fusion ring).
[^11]: If $\Omega$ is the automorphism, the induced adjoint is given by $x^{\dag} = -{\Omega \left( x \right)}$, where $x$ is either an element of the Cartan subalgebra or a root vector. This is then extended antilinearly to the entire complex Lie algebra.
[^12]: We denote this spectral flow by $\widetilde{\gamma}$ because this automorphism is not the same as the spectral flow automorphism $\gamma$ which was introduced in [Equation (\[eqnSF\])]{}. Whereas the latter denotes a spectral flow naturally defined on the ${{\widehat{{\mathfrak{su}}}} \left( 2 \right)}$ basis, the flow $\widetilde{\gamma}$ is naturally defined on the ${{\widehat{{\mathfrak{sl}}}} \left( 2 ; {\mathbb{R}}\right)}$ basis. We can see that these are different by determining the action of $\widetilde{\gamma}$ on the ${{\widehat{{\mathfrak{su}}}} \left( 2 \right)}$ basis: $$\begin{aligned}
{\widetilde{\gamma} \left( E_n \right)} &= \frac{1}{4} {\left( E_{n-1} + 2 E_n + E_{n+1} + {\mathfrak{i}}H_{n-1} + 2 {\mathfrak{i}}\delta_{n,0} K - {\mathfrak{i}}H_{n+1} + F_{n-1} - 2 F_n + F_{n+1} \right)} \\
{\widetilde{\gamma} \left( H_n \right)} &= \frac{1}{2} {\left( -{\mathfrak{i}}E_{n-1} + {\mathfrak{i}}E_{n+1} + H_{n-1} + H_{n+1} - {\mathfrak{i}}F_{n-1} + {\mathfrak{i}}F_{n+1} \right)} \\
{\widetilde{\gamma} \left( F_n \right)} &= \frac{1}{4} {\left( E_{n-1} - 2 E_n + E_{n+1} + {\mathfrak{i}}H_{n-1} - 2 {\mathfrak{i}}\delta_{n,0} K - {\mathfrak{i}}H_{n+1} + F_{n-1} + 2 F_n + F_{n+1} \right)}.\end{aligned}$$ While this can be checked to indeed provide a non-trivial automorphism (of ${{\widehat{{\mathfrak{sl}}}} \left( 2 ; {\mathbb{C}}\right)}$), it is not clear whether it is of any use in further analysing our theory. Note that it preserves the adjoint (\[eqnDefOtherAdj\]) but not (\[eqnDefAdj\]).
[^13]: We mention that this is the correct way of defining these modes given the [operator product expansions]{} (\[eqnNewOPEs\]), despite the fact that $m$ can be chosen arbitrarily (up to monodromy charge considerations). Naïvely defining the affine modes as the obvious normally-ordered products of the extended algebra modes gives equivalent results, *except* for $h_0$ when $m \in {\mathbb{Z}}$. Then the naïve result is incorrect, and must be adjusted by the appropriate multiple of the identity. This correction phenomenon should be familiar from the computation of the Virasoro zero-mode in the Ramond sector of the free fermion.
|
---
abstract: 'In this note we obtain a unique continuation result for the differential inequality $|\overline{\partial}u|\leq|Vu|$, where $\overline{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function in $L^2(\mathbb{R}^2)$.'
address: 'Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea'
author:
- Ihyeok Seo
title: 'A remark on unique continuation for the Cauchy-Riemann operator'
---
Introduction
============
The unique continuation property is one of the most interesting properties of holomorphic functions $f\in H(\mathbb{C})$. This property says that if $f$ vanishes in a non-empty open subset of $\mathbb{C}$ then it must be identically zero. Note that $u\in C^1(\mathbb{R}^2)$ satisfies the Cauchy-Riemann equation $(i\partial_y+\partial_x)u=0$ if and only if it defines a holomorphic function $f(x+iy)\equiv u(x,y)$ on $\mathbb{C}$. From this point of view, one can see that a $C^1$ function satisfying the equation has the unique continuation property.
In this note we consider a class of non-holomorphic functions $u$ which satisfy the differential inequality $$\label{inequality}
|\overline{\partial}u|\leq|Vu|,$$ where $\overline{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function on $\mathbb{R}^2$.
The best positive result for is due to Wolff [@W] (see Theorem 4 there) who proved the property for $V\in L^p$ with $p>2$. On the other hand, there is a counterexample [@M] to unique continuation for with $V\in L^p$ for $p<2$. The remaining case $p=2$ seems to be unknown for the differential inequality , and note that $L^2$ is a scale-invariant space of $V$ for the equation $\overline{\partial}u=Vu$. Here we shall handle this problem. Our unique continuation result is the following theorem which is based on bounds for a Fourier multiplier from $L^p$ to $L^q$.
\[thm\] Let $1<p<2<q<\infty$ and $1/p-1/q=1/2$. Assume that $u\in L^p\cap L^q$ satisfies the inequality with $V\in L^2$ and vanishes in a non-empty open subset of $\mathbb{R}^2$. Then it must be identically zero.
The unique continuation property also holds for harmonic functions, which satisfy the Laplace equation $\Delta u=0$, since they are real parts of holomorphic functions. This was first extended by Carleman [@C] to a class of non-harmonic functions satisfying the inequality $|\Delta u|\leq|Vu|$ with $V\in L^\infty(\mathbb{R}^2)$. There is an extensive literature on later developments in this subject. In particular, the problem of finding all the possible $L^p$ functions $V$, for which $|\Delta u|\leq|Vu|$ has the unique continuation, is completely solved (see [@JK; @KN; @KT]). See also the survey papers of Kenig [@K] and Wolff [@W2] for more details, and the recent paper of Kenig and Wang [@KW] for a stronger result which gives a quantitative form of the unique continuation.
Throughout the paper, the letter $C$ stands for positive constants possibly different at each occurrence. Also, the notations $\widehat{f}$ and $\mathcal{F}^{-1}(f)$ denote the Fourier and the inverse Fourier transforms of $f$, respectively.
A preliminary lemma
===================
The standard method to study the unique continuation property is to obtain a suitable Carleman inequality for relevant differential operator. This method originated from Carleman’s classical work [@C] for elliptic operators. In our case we need to obtain the following inequality for the Cauchy-Riemann operator $\overline{\partial}=(i\partial_y+\partial_x)/2$, which will be used in the next section for the proof of Theorem \[thm\]:
\[lem\] Let $f\in C_0^\infty(\mathbb{R}^2\setminus\{0\})$. For all $t>0$, we have $$\label{Sobo}
\big\||z|^{-t}f\big\|_{L^q}
\leq C\big\||z|^{-t}\overline{\partial}f\big\|_{L^p}$$ if $1<p<2<q<\infty$ and $1/p-1/q=1/2$. Here, $z=x+iy\in\mathbb{C}$ and $C$ is a constant independent of $t$.
First we note that $$\overline{\partial}(z^{-t}f)=z^{-t}\overline{\partial}f+f\overline{\partial}(z^{-t})
=z^{-t}\overline{\partial}f$$ for $z\in\mathbb{C}\setminus\{0\}$. Then the inequality is equivalent to $$\big\|z^{-t}f\big\|_{L^q}
\leq C\big\|\overline{\partial}(z^{-t}f)\big\|_{L^p}.$$ By setting $g=z^{-t}f$, we are reduced to showing that $$\|g\|_{L^q}
\leq C\|(i\partial_y+\partial_x)g\|_{L^p}$$ for $g\in C_0^\infty(\mathbb{R}^2\setminus\{0\})$. To show this, let us first set $$\label{0}
(i\partial_y+\partial_x)g=h,$$ and let $\psi_\delta:\mathbb{R}^2\rightarrow[0,1]$ be a smooth function such that $\psi_\delta=0$ in the ball $B(0,\delta)$ and $\psi_\delta=1$ in $\mathbb{R}^2\setminus B(0,2\delta)$. Then, using the Fourier transform in , we see that $$(-\eta+i\xi)\widehat{g}(\xi,\eta)=\widehat{h}(\xi,\eta).$$ Thus, by Fatou’s lemma we are finally reduced to showing the following uniform boundedness for a multiplier operator having the multiplier $m(\xi,\eta)=\psi_\delta(\xi,\eta)/(-\eta+i\xi)$: $$\label{multi}
\bigg\|\mathcal{F}^{-1}\bigg(\frac{\psi_\delta(\xi,\eta)
\widehat{h}(\xi,\eta)}{-\eta+i\xi}\bigg)\bigg\|_{L^q}\leq C\|h\|_{L^p}$$ uniformly in $\delta>0$.
From now on, we will show using Young’s inequality for convolutions and Littlewood-Paley theorem ([@G]). Let us first set for $k\in\mathbb{Z}$ $$\widehat{Th}(\xi,\eta)=m(\xi,\eta)\widehat{h}(\xi,\eta)\quad\text{and}\quad
\widehat{T_kh}(\xi,\eta)=m(\xi,\eta)\chi_k(\xi,\eta)\widehat{h}(\xi,\eta),$$ where $\chi_k(\cdot)=\chi(2^k\cdot)$ for $\chi\in C_0^\infty(\mathbb{R}^2)$ which is such that $\chi(\xi,\eta)=1$ if $|(\xi,\eta)|\sim1$, and zero otherwise. Also, $\sum_k\chi_k=1$. Now we claim that $$\label{multi2}
\|T_kh\|_{L^q}\leq C\|h\|_{L^p}$$ uniformly in $k\in\mathbb{Z}$. Then, since $1<p<2<q<\infty$, by the Littlewood-Paley theorem together with Minkowski’s inequality, we get the desired inequality as follows: $$\begin{aligned}
\big\|\sum_kT_kh\big\|_{L^q}&\leq C\big\|\big(\sum_k|T_kh|^2\big)^{1/2}\big\|_{L^q}\\
&\leq C\big(\sum_k\|T_kh\|_{L^q}^2\big)^{1/2}\\
&\leq C\big(\sum_k\|h_k\|_{L^p}^2\big)^{1/2}\\
&\leq C\big\|\big(\sum_k|h_k|^2\big)^{1/2}\big\|_{L^p}\\
&\leq C\big\|\sum_kh_k\big\|_{L^p},\end{aligned}$$ where $h_k$ is given by $\widehat{h_k}(\xi,\eta)=\chi_k(\xi,\eta)\widehat{h}(\xi,\eta)$. Now it remains to show the claim . But, this follows easily from Young’s inequality. Indeed, note that $$T_kh=\mathcal{F}^{-1}\bigg(\frac{\psi_\delta(\xi,\eta)\chi_k(\xi,\eta)}{-\eta+i\xi}\bigg)
\ast h$$ and by Plancherel’s theorem $$\begin{aligned}
\bigg\|\mathcal{F}^{-1}\bigg(\frac{\psi_\delta(\xi,\eta)\chi_k(\xi,\eta)}{-\eta+i\xi}\bigg)\bigg\|_{L^2}
&=\bigg\|\frac{\psi_\delta(\xi,\eta)\chi_k(\xi,\eta)}{-\eta+i\xi}\bigg\|_{L^2}\\
&\leq C\bigg(\int_{|(\xi,\eta)|\sim2^{-k}}\frac1{\eta^2+\xi^2}d\xi d\eta\bigg)^{1/2}\\
&\leq C.\end{aligned}$$ Since we are assuming the gap condition $1/p-1/q=1/2$, by Young’s inequality for convolutions, this readily implies that $$\|T_kh\|_{L^q}\leq \bigg\|\mathcal{F}^{-1}\bigg(\frac{\psi_\delta(\xi,\eta)\chi_k(\xi,\eta)}{-\eta+i\xi}\bigg)\bigg\|_{L^2}\|h\|_{L^p}
\leq C\|h\|_{L^p}$$ as desired.
Proof of Theorem \[thm\]
========================
The proof is standard once one has the Carleman inequality in Lemma \[lem\].
Without loss of generality, we may show that $u$ must vanish identically if it vanishes in a sufficiently small neighborhood of zero. Then, since we are assuming that $u\in L^p\cap L^q$ vanishes near zero, from with a standard limiting argument involving a $C_0^\infty$ approximate identity, it follows that $$\big\||z|^{-t}u\big\|_{L^q}
\leq C\big\||z|^{-t}\overline{\partial}u\big\|_{L^p}.$$ Thus by we see that $$\begin{aligned}
\big\||z|^{-t}u\big\|_{L^q(B(0,r))}&\leq C\big\||z|^{-t}Vu\big\|_{L^p(B(0,r))}\\
&+C\big\||z|^{-t}\overline{\partial}u\big\|_{L^p(\mathbb{R}^2\setminus B(0,r))},\end{aligned}$$ where $B(0,r)$ is the ball of radius $r>0$ centered at $0$. Then, using Hölder’s inequality with $1/p-1/q=1/2$, the first term on the right-hand side in the above can be absorbed into the left-hand side as follows: $$\begin{aligned}
C\big\||z|^{-t}Vu\big\|_{L^p(B(0,r))}&\leq
C\|V\|_{L^2(B(0,r))}\big\||z|^{-t}u\big\|_{L^q(B(0,r))}\\
&\leq \frac12\big\||z|^{-t}u\big\|_{L^q(B(0,r))}\end{aligned}$$ if we choose $r$ small enough. Here, $\||z|^{-t}u\|_{L^q(B(0,r))}$ is finite since $u\in L^q$ vanishes near zero. Hence we get $$\begin{aligned}
\|(r/|z|)^{t}u\|_{L^q(B(0,r))}
&\leq2C\|\overline{\partial}u\|_{L^p(\mathbb{R}^2\setminus B(0,r))}\\
&\leq2C\|V\|_{L^2}\|u\|_{L^q}\\
&<\infty.\end{aligned}$$ By letting $t\rightarrow\infty$, we now conclude that $u=0$ on $B(0,r)$. This implies $u\equiv0$ by a standard connectedness argument.
Acknowledgment {#acknowledgment .unnumbered}
==============
I would like to thank Jenn-Nan Wang for pointing out a preprint ([@KW]) and for some comments.
[ho]{}
T. Carleman, *Sur un problème d’unicité pour les systèmes d’équations aux derivées partielles à deux variables indépendantes*, Ark. Mat., Astr. Fys., 26 (1939), 1-9.
L. Grafakos, *Classical Fourier Analysis*, Springer, New York, 2008.
D. Jerison and C. E. Kenig, *Unique continuation and absence of positive eigenvalues for Schrödinger operators*, Ann. of Math. 121 (1985), 463-494.
C. E. Kenig, *Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuations*, Harmonic analysis and partial differential equations (El Escorial, 1987), 69-90, Lecture Notes in Math. 1384, Springer, Berlin, 1989.
C. E. Kenig and N. Nadirashvili, *A counterexample in unique continuation*, Math. Res. Lett. 7 (2000), 625-630.
C. E. Kenig and J-N Wang, *Quantitative uniqueness estimates for second order elliptic equations with unbounded drift*, Preprint, arXiv:1407.1536.
H. Koch and D. Tataru, *Sharp counterexamples in unique continuation for second order elliptic equations*, J. Reine Angew. Math. 542 (2002), 133-146.
N. Mandache, *A counterexample to unique continuation in dimension two*, Commun. Anal. Geom., 10 (2002), 1-10.
T. H. Wolff, *A property of measures in $\mathbb{R}^n$ and an application to unique continuation*, Geom. Funct. Anal., 2 (1992), 225-284.
T. H. Wolff, *Recent work on sharp estimates in second-order elliptic unique continuation problems*, J. Geom. Anal., 3 (1993), 621-650.
|
---
abstract: 'We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin $\sigma$-$\P$-porous sets where $\P$ can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Suslin cone small set in Asplund spaces. As an application we prove a theorem stating that a continuous approximately convex function on an Asplund space is Fréchet differentiable up to a cone small set.'
address: 'Charles University, Faculty of Mathematics and Physics, Department of MathematicalAnalysis, Sokolovská 83, 186 75 Prague 8, Czech Republic'
author:
- Marek Cúth
- Martin Rmoutil
- Miroslav Zelený
title: |
On Separable Determination of $\sigma$-$\P$-Porous Sets\
in Banach Spaces
---
[^1]
Introduction
============
The present paper could be considered as a sequel to the articles [@Cuth] and [@Cuth-Rmoutil]. Our main aim is to further investigate separable determination of various properties of sets and functions in metric spaces (especially Banach spaces). That is, given a nonseparable metric space $X$ and a property of sets (or functions etc.) in $X$, we are interested whether certain statements about the property hold, provided that they hold in (some) separable subspaces of $X$. In particular, if $X$ is a metric space, we are interested in $\sigma$-$\P$-porous sets in $X$, where $\P$ is a porosity-like relation on $X$ (see the definition below).
The key method we use to obtain separable determination results uses countable elementary structures which we call elementary submodels. This method is described in Section \[S:submodels\]. Further details and examples of use can be found in [@Cuth] and [@Cuth-Rmoutil]. However, the reader ought to note that there are other ways to go about this topic. An example is the use of rich families of Banach spaces, which is described in detail, e.g., in [@LPT]. Sometimes one can also opt to prove this sort of results in the “elementary way”, in some sense replicating parts of the proof of the Löwenheim-Skolem theorem. This approach would in many cases be very complicated, but can give a deeper insight, when possible.
In the rather technical Section \[S:Foran\] we prove essential auxiliary results which we use in Section \[S:porosity-like\] to prove our general separable determination result, Proposition \[P:reduction-porosity-like\]. The general scheme of our proof is rather similar to that of separable determination of $\sigma$-upper porous sets in [@Cuth-Rmoutil] and involves the Foran lemma and it’s partial converse. The difference is that here we need no inscribing theorems as in [@Cuth-Rmoutil] to prove the statement for all Suslin sets.
Section \[S:cone\] contains the main result of this article, the separable determination of the notion of cone small sets in Asplund spaces (Theorem \[T:cone\]). Arguably the deeper part of this statement is a consequence of Proposition \[P:reduction-porosity-like\]. In the last section we provide several applications of our results, most notably Theorem \[T:appl-approx-convexity\].
Let us recall the most relevant notions, definitions, and notations. We denote by $\omega$ the set of all natural numbers (including $0$), by $\en$ the set $\omega \setminus \{0\}$, by $\er_+$ the interval $(0,\infty)$, and $\qe_+$ stands for $\er_+ \cap \qe$. We denote by ${\omega}^{<\omega}$ the set of all finite sequences of elements of $\omega$, and by $\omega^\omega$ the set of all infinite sequences. The notion of a countable set includes also finite sets. If $f$ is a mapping then we denote by $\rng f$ the range of $f$ and by $\dom f$ the domain of $f$. By writing $f\colon X \to Y$ we mean that $f$ is a mapping with $\dom f = X$ and $\rng f \subset Y$. By the symbol $f\rest_{Z}$ we denote the restriction of the mapping $f$ to the set $Z$.
If $(X,\rho)$ is a metric space, we denote by $U_X(x,r)$ the open ball, i.e., the set $\{y \in X{;\ }\rho(x,y) < r\}$. We often write $U(x,r)$ instead of $U_X(x,r)$. If $A,B \subset X$ are nonempty sets in a metric space $X$, we denote by $\d(A,B)$ the distance between the sets $A$ and $B$, i.e., $\d(A,B) = \inf\{\rho(a,b){;\ }a \in A, b \in B\}$. We shall consider normed linear spaces over the field of real numbers. If $X$ is a normed linear space, $X^*$ stands for the dual space of $X$.
We say that $\P$ is a *porosity-like relation* on the space $X$ if $\P$ is a relation between points $x \in X$ and sets $A \subset X$ (i.e. $\P \subset X \times 2^X$) satisfying the following conditions:
- if $A\subset B$ and $\P(x,B)$, then $\P(x,A)$,
- $\P(x,A)$ if and only if there is $r > 0$ such that $\P(x,A \cap U(x,r))$,
- $\P(x,A)$ if and only if $\P(x,\overline{A})$.
A set is $\P$-porous, if $\P(x,A)$ for every $x \in A$. A set is $\sigma$-$\P$-porous, if it is a countable union of $\P$-porous sets.
Elementary submodels {#S:submodels}
====================
In this section we recall some basic notions concerning the method of elementary submodels. A brief description of this method can be found in [@Cuth-Rmoutil], for a more detailed description see [@Cuth].
First, let us recall some definitions. Let $N$ be a fixed set and $\phi$ a formula in the language of $ZFC$. Then the *relativization of $\phi$ to $N$* is the formula $\phi^N$ which is obtained from $\phi$ by replacing each quantifier of the form “$\forall x$” by “$\forall x \in N$” and each quantifier of the form “$\exists x$” by “$\exists x \in N$”. If $\phi(x_1,\ldots,x_n)$ is a formula with all free variables shown (i.e. a formula whose free variables are exactly $x_1,\ldots,x_n$) then *$\phi$ is absolute for $N$* if and only if $$\forall a_1, \ldots, a_n \in N\colon \bigl(\phi^N(a_1,\ldots,a_n) \leftrightarrow \phi(a_1,\ldots,a_n)\bigr).$$ The method is based mainly on the following theorem (a proof can be found in [@Kunen Chapter IV, Theorem 7.8]).
\[T:countable-model\] Let $\phi_1, \ldots, \phi_n$ be any formulas and $X$ any set. Then there exists a set $M \supset X$ such that $\phi_1, \ldots, \phi_n \text{ are absolute for } M$ and $|M| \leq \max(\omega,|X|)$.
Since the set from Theorem \[T:countable-model\] will often be used, the following notation is useful.
Let $\phi_1, \ldots, \phi_n$ be any formulas and $X$ be any countable set. Let $M \supset X$ be a countable set such that $\phi_1, \ldots, \phi_n$ are absolute for $M$. Then we say that $M$ is an *elementary submodel for $\phi_1,\ldots,\phi_n$ containing $X$*. This is denoted by $M \prec (\phi_1,\ldots,\phi_n; X)$.
The fact that certain formula is absolute for $M$ will always be used in order to satisfy the assumption of the following lemma. It is a similar statement to [@Cuth Lemma 2.6]. Using this lemma we can force the elementary submodel $M$ to contain all the needed objects created (uniquely) from elements of $M$.
\[L:unique-M\] Let $\phi(y,x_1,\ldots,x_n)$ be a formula with all free variables shown and $Y$ be a countable set. Let $M$ be a fixed set, $M \prec (\phi, \exists y \colon \phi(y,x_1,\ldots,x_n);\; Y)$, and $a_1,\ldots,a_n \in M$ be such that there exists a set $u$ satisfying $\phi(u,a_1,\ldots,a_n)$. Then there exists $u \in M$ such that $\phi(u,a_1,\ldots,a_n)$.
Using the absoluteness of the formula $\exists u\colon \phi(u,x_1,\ldots,x_n)$ there exists $u\in M$ satisfying $\phi^M(u,a_1,\ldots,a_n)$. Using the absoluteness of $\phi$ we get, that for this $u\in M$ the formula $\phi(u,a_1,\ldots,a_n)$ holds.
It would be very laborious and pointless to use only the basic language of the set theory. For example, having a Banach space $X$ and $x, y \in X$ we often write $x + y$ and we know that this is a shortcut for a formula with free variables $x$, $y$, and $+$ (note that the symbol $+$ is here considered to be a variable as it depends on the Banach space). Therefore, in the following text we use this extended language of the set theory as we are used to. We shall also use the following convention.
Whenever we say “*for any suitable elementary submodel $M$ (the following holds …)*” we mean that “*there exists a list of formulas $\phi_1,\ldots,\phi_n$ and a countable set $Y$ such that for every $M \prec (\phi_1,\ldots,\phi_n;Y)$ (the following holds …)*”.
By using this new terminology we lose the information about the formulas $\phi_1,\ldots,\phi_n$ and the set $Y$. However, this is not important in applications.
Let us recall several further results about suitable elementary submodels (all the proofs are based on Lemma \[L:unique-M\] and they can be found in [@Cuth Chapters 2 and 3]).
For any suitable elementary submodel $M$ the following holds.
- If $A,B \in M$, then $A \cap B \in M$, $B \setminus A \in M$ and $A \cup B \in M$.
- Let $f$ be a function such that $f\in M$. Then $\dom{f} \in M$, $\rng{f} \in M$ and for every $x \in \dom{f} \cap M$ we have $f(x) \in M$.
- Let $S$ be a finite set. Then $S \in M$ if and only if $S \subset M$.
- Let $S \in M$ be a countable set. Then $S \subset M$.
- For every natural number $n > 0$ and for arbitrary $n+1$ sets $a_0, \ldots, a_n$ it is true that $$a_0, \ldots, a_n \in M \iff (a_0, \ldots, a_n) \in M.$$
<!-- -->
- If $A$ is a set, then by saying that an elementary model $M$ contains $A$ we mean that $A\in M$.
- If $(X,\rho)$ is a metric space (resp. $(X, +, \cdot, \| \cdot \|)$ is a normed linear space) and $M$ is an elementary submodel, then by saying [*$M$ contains $X$*]{} (or by writing $X\in M$) we mean that $(X,\rho) \in M$ (resp. $(X, +, \cdot, \| \cdot \|) \in M$).
- If $X$ is a topological space and $M$ is an elementary submodel, then we denote by $X_M$ the set $\ov{X\cap M}$.
For any suitable elementary submodel $M$ the following holds.
- If $X$ is a metric space then whenever $M$ contains $X$, it is true that $$\forall r \in \er_+ \cap M\; \forall x \in X \cap M\colon U(x,r) \in M.$$
- If $X$ is a normed linear space then whenever $M$ contains $X$, it is true that $X_M$ is a closed separable subspace of $X$.
The proofs in the following text often begin in the same way. To avoid unnecessary repetitions, by saying “*Let us fix a $(*)$-elementary submodel $M$ (containing $A_1,\ldots,A_n$)*” we will understand the following.
*Let us have formulas $\varphi_1,\ldots,\varphi_m$ and a countable set $Y$ such that the elementary submodel $M \prec(\varphi_1,\ldots,\varphi_m; Y)$ is suitable for all the propositions from [@Cuth] and [@Cuth-Rmoutil]. Add to them formulas marked with $(*)$ in all the preceding proofs from this paper and formulas marked with $(*)$ in the proof below and all their subformulas. Denote such a list of formulas by $\psi_1,\ldots,\psi_k$. Let us fix a countable set $X$ containing the sets $Y$, $\omega$, $\omega^\omega$, $\omega^{<\omega}$, $\zet$, $\qe$, $\qe_+$, $\er$, $\er_+$, and all the common operations and relations on real numbers ($+$, $-$, $\cdot$, $:$, $<$). Fix an elementary submodel $M$ for formulas $\psi_1,\ldots,\psi_k$ containing $X$ (such that $A_1, \ldots, A_n \in M$).*
Thus, any $(*)$-elementary submodel $M$ is suitable for the results from [@Cuth], [@Cuth-Rmoutil] and all the preceding theorems and propositions from this paper, making it possible to use all of these results for $M$.
In order to demonstrate how this technique works, we prove the following lemma which we use later.
\[L:density\] For any suitable elementary submodel $M$ the following holds. Let $(X,\rho)$ be a metric space and $\F$ be a countable collection of subsets of $X$. Then whenever $M$ contains $X$ and $\F \subset M$, it is true that $$\bigcup \F \text{ is dense in } X \quad \Rightarrow \quad \bigcup \F \cap X_M \text{ is dense in } X_M.$$
Let us fix a $(*)$-elementary submodel $M$ containing $X$ such that $\F \subset \mathcal{P}(X) \cap M$ and $\bigcup \F$ is dense in $X$. In order to see that $\bigcup\F\cap X_M$ is dense in $X_M$, it is sufficient to prove that, for every $x\in X\cap M$ and $r \in \qe_+$, there exists $F \in \F$ such that $U(x,r) \cap X_M \cap F \neq \emptyset$. Fix some $x \in X \cap M$ and $r \in \qe_+$. Then there exists $F\in\F$ such that the following formula is satisfied $$\tag{$*$}
\exists y \colon (y \in F \; \wedge \; \rho(x,y) < r).$$ The preceding formula has free variables $F$, $\rho$, $<$, $x$, and $r$. Those are in $M$; hence, by Lemma \[L:unique-M\], there exists $y\in M$ such that $y \in F$ and $\rho(x,y) < r$. Consequently, $U(x,r) \cap X_M \cap F \neq \emptyset$.
Foran scheme {#S:Foran}
============
We employ the following notation. Given $s,t \in \S$, we write $s \prec t$ if $t$ is an extension of $s$ (not necessarily proper). We denote the concatenation of $s \in \S$ and $t \in \S$ by $s \h t$. If $s \in \S$ and $i \in \omega$, we write $s\h i$ instead of $s\h (i)$. If $\nu = (\nu_0, \nu_1,\nu_2, \dots ) \in \omega^{\omega}$ and $n \in \omega$, then the symbol $\nu|n$ means the finite sequence $(\nu_0,\nu_1,\dots,\nu_{n-1})$. If $t \in \S$, then the symbol $|t|$ denotes the length of $t$. By a *tree* we mean any subset $T$ of $\S$ such that for every $s \in \S$ and $t \in T$ with $s \prec t$, we have $s \in T$. We say that a tree $T$ is *pruned* if for every $t \in T$ there exists $n \in \omega$ such that $t \h n \in T$.
Let us recall that any family $\mathcal A = \{A(s){;\ }s \in \S\}$ of sets is called a *Suslin scheme* and *Suslin operation* $\mathcal S$ is defined by $$\mathcal S(\mathcal A) = \mathcal S_s(A(s)) = \bigcup_{\nu \in \omega^{\omega}}\bigcap_{n \in \omega} A(\nu|n).$$ A Suslin scheme $\{A(s){;\ }s \in \S\}$ is called *monotone* if $A(s) \supset A(t)$ whenever $s,t \in \S, s \prec t$. Finally, a subset $A$ of a topological space $X$ is a *Suslin* set (in $X$) if there exists a Suslin scheme $\mathcal A$ consisting of closed subsets of $X$ with $\mathcal S(\mathcal A) = A$.
Let $X$ be a complete metric space, $\P$ be a porosity-like relation on $X$, and let $\mathcal B$ be a basis of open sets in $X$.
For any $A \subset X$ we define the following set operators: $$\begin{aligned}
\ker_{\P}(A) &= A \setminus \bigcup \{U \subset X {;\ }U \text{ is open and $U \cap A$ is $\sigma$-$\P$-porous}\}, \\
N_{\P}(A) &= \{x \in A{;\ }\neg \P(x,A)\}.\end{aligned}$$
The following lemma is easy to prove. Its assertions (i) and (ii) can be found, e.g., in [@Rmoutil].
\[L:basic\] Let $A \subset X$. Then we have
- $A \setminus \ker_{\P}(A)$ is $\sigma$-$\P$-porous,
- $\ker_{\P}(\ker_{\P}(A)) = \ker_{\P}(A)$,
- if $A \subset X$ is a Suslin set then $\ker_{\P}(A)$ is a Suslin set,
- if $A \subset B \subset X$, $\ker_{\P}(B) = B$, and $B \setminus A$ is $\sigma$-$\P$-porous, then $\ker_{\P}(A) = A$,
- $A \setminus N_{\P}(A)$ is $\P$-porous.
\[D:Foran-scheme\] Let $\F = \{S(t){;\ }\ t \in \omega^{<\omega}\}$ be a system of nonempty subsets of $X$ such that for each $t\in\omega^{<\omega}$ and each $k \in \omega$ we have
- $\bigcup_{j\in\omega} S(t \h j)$ is a dense subset of $S(t)$,
- $S(t)$ is $\P$-porous at no point of $S(t \h k)$,
- for any $\nu\in \omega^\omega$ and any sequence $\{G_n\}_{n\in\omega}$ of sets from $\mathcal B$ satisfying:
- $\lim_{n \to \infty} \diam(G_n) = 0$,
- $\overline{G_{n+1}} \subset G_n$ for every $n \in \omega$,
- $S(\nu |n) \cap G_n \neq \emptyset$ for every $n \in \omega$,
we have $$\bigcap_{n\in\omega} \bigl(S(\nu |n) \cap G_n\bigr) \neq \emptyset.$$
Then we say that $\F$ is a $(\mathcal B,\P)$*-Foran scheme in* $X$. If there is no danger of confusion we will say just *Foran scheme*.
\[L:Foran-lemma\] Let $\F$ be a Foran scheme in $X$. Then each element of $\F$ is a non-$\sigma$-$\P$-porous set.
We mimic the standard proof which works for Foran systems, see [@Zajicek-3 Lemma 4.3]. It is sufficient to prove that $S(\emptyset)$ is non-$\sigma$-$\P$-porous. Suppose on the contrary that $S(\emptyset) = \bigcup_{n=1}^{\infty} A_n$, where each $A_n$ is $\P$-porous. We set $A_0 = \emptyset$. We will construct $\nu = (\nu_0,\nu_1,\dots) \in \omega^{\omega}$ and a sequence of open sets $\{G_n\}_{n \in \omega}$ such that for every $n \in \omega$ we have
- $\diam(G_n) < 2^{-n}$,
- $\overline{G_{n+1}} \subset G_n$,
- $G_n \cap S(\nu |n) \neq \emptyset$,
- $S(\nu|n) \cap G_n \cap A_n = \emptyset$,
- $G_n \in \mathcal B$.
We will construct inductively $\nu_n$’s and $G_n$’s. If $n = 0$, then we find an open set $G_0 \in \mathcal B$ intersecting $S(\emptyset)$ with $\diam G_0 < 1$. Then conditions (a)–(e) are clearly satisfied. Now suppose that we have already constructed $G_n$ and $s = (\nu_0,\dots,\nu_{n-1})$ for $n \in \omega$. We distinguish two cases.
If $A_{n+1}$ is not dense in $S(s) \cap G_n$, then we find a nonempty open set $G_{n+1} \in \mathcal B$ such that $G_{n+1} \cap S(s) \neq \emptyset$, $\overline{G_{n+1}} \subset G_n \setminus A_{n+1}$, and $\diam G_{n+1} < 2^{-(n+1)}$. Further, using condition (i) from Definition \[D:Foran-scheme\] we find $\nu_n \in \omega$ such that $S(s \h \nu_n) \cap S(s) \cap G_{n+1} \neq \emptyset$.
Now suppose that $A_{n+1}$ is dense in $S(s) \cap G_n$. Then we find $\nu_n$ such that $S(s \h \nu_n) \cap S(s) \cap G_n \neq \emptyset$ by condition (i) from Definition \[D:Foran-scheme\]. Suppose that $x \in A_{n+1} \cap S(s) \cap G_n$. Then we have $\P(x,A_{n+1})$ and by density of $A_{n+1}$ in $S(s) \cap G_n$ we get $\P(x,S(s) \cap G_n)$ since $\P$ is a porosity-like relation. It implies $x \notin S(s \h \nu_n)$ by condition (ii) from Definition \[D:Foran-scheme\]. Thus we get that $S(s \h \nu_n) \cap G_n \cap A_{n+1} = \emptyset$. We find an open set $G_{n+1} \in \mathcal B$ such that $\overline{G_{n+1}} \subset G_n$, $G_{n+1} \cap S(s \h \nu_n) \neq \emptyset$, and $\diam G_{n+1} < 2^{-(n+1)}$. This finishes the construction of $\nu$ and $G_n$’s.
Since $\F$ is a Foran scheme there exists $x \in \bigcap_{n \in \omega} \bigl(S(\nu|n) \cap G_n\bigr)$. By (b) and (d) we have $x \in S(\emptyset) \setminus \bigcup_{n=1}^{\infty} A_n = \emptyset$, a contradiction.
A Suslin scheme $\mathcal C = \{C(s) {;\ }s \in \S\}$ is *subordinate* to a Suslin scheme $\mathcal A = \{A(s){;\ }s \in \S\}$ (notation $\mathcal C \sqsubseteq \mathcal A$) if there exists a mapping $\varphi\colon \S \to \S$ such that for each $s \in \S$ we have
- $|\varphi(s)| = |s|$,
- if $t \in \S, s \prec t$, then $\varphi(s) \prec \varphi(t)$,
- $C(s) \subset A(\varphi(s))$.
\[D:regular\] Let $\mathcal A = \{A(s){;\ }s \in \S\}$ be a Suslin scheme. Denote $C(s) = \mathcal S_t(A (s \h t))$, $s \in \S$. We say that $\mathcal A$ is *$\P$-regular* if $\mathcal A$ is monotone and for every $s \in \S$ we have $\ker_{\P} (C(s)) = C(s) \neq \emptyset$.
\[L:sub\] Let $\mathcal A$ be a Suslin scheme consisting of closed sets and $C \subset \mathcal S(\mathcal A)$ be a Suslin set. Then there exists a Suslin scheme $\mathcal C$ consisting of closed sets which is subordinate to $\mathcal A$ and $C = \mathcal S(\mathcal C)$.
Let $\mathcal L = \{L(s){;\ }s \in \S\}$ be a Suslin scheme consisting of closed sets with $\mathcal S(\mathcal L) = C$. Let $\mathcal A = \{A(s){;\ }s \in \S \}$. Fix a bijection $\psi = (\psi_1,\psi_2) \colon \omega \to \omega^2$. We define mappings $\varphi\colon \S \to \S$, $\rho\colon \S \to \S$ by $\varphi(\emptyset) = \rho(\emptyset) = \emptyset$ and $$\begin{aligned}
\varphi(s) &= \psi_1(s_0) \h \psi_1(s_1) \h \dots \h \psi_1(s_{|s|-1}), \\
\rho(s) &= \psi_2(s_0) \h \psi_2(s_1) \h \dots \h \psi_2(s_{|s|-1}),\end{aligned}$$ where $s = (s_0, \dots, s_{|s|-1}) \in \S \setminus \{\emptyset\}$. We define the desired scheme $\mathcal C$ by $C(s) = A(\varphi(s)) \cap L(\rho(s))$. The scheme $\mathcal C = \{C(s){;\ }s \in \S\}$ consists of closed sets and is clearly subordinate to $\mathcal A$ via the mapping $\varphi$.
We verify the equality $C = \mathcal S(\mathcal C)$. If $x \in C$, then there exist $\nu,\mu \in \omega^{\omega}$ such that $x \in L(\nu|k)$ and $x \in A(\mu|k)$ for every $k \in \omega$. Since $\psi$ is a bijection of $\omega$ onto $\omega^2$ we can find $\tau \in \omega^{\omega}$ such that $\varphi(\tau|k) = \nu|k$ and $\rho(\tau|k) = \mu|k$ for every $k \in \omega$. Then we have $x \in C(\tau|k)$ for every $k \in \omega$. Consequently, $x \in \mathcal S(\mathcal C)$.
If $x \in \mathcal S(\mathcal C)$, then there exists $\tau \in \omega^{\omega}$ such that $x \in C(\tau|k)$ for every $k \in \omega$. We find $\mu,\nu \in \omega^{\omega}$ such that $\varphi(\tau|k) = \mu|k$ and $\rho(\tau|k) = \nu|k$ for every $k \in \omega$. Then we have $x \in A(\mu|k) \cap L(\nu|k)$. Consequently, $x \in C$. Thus we have proved $C = \mathcal S(C)$.
\[L:regular\] Let $\mathcal A$ be a Suslin scheme consisting of closed subsets of $X$ and $C \subset \mathcal S(\mathcal A)$ be a Suslin set with $\ker_{\P}(C) = C \neq \emptyset$. Then there exists a $\P$-regular Suslin scheme $\mathcal L = \{L(s) {;\ }s \in \S\}$ consisting of closed subsets of $X$ such that $\mathcal L$ is subordinate to $\mathcal A$ and $\mathcal S(\mathcal L)$ is a dense subset of $C$.
Let $\mathcal A = \{A(s){;\ }s \in \S\}$. Using Lemma \[L:sub\] we find a Suslin scheme $\mathcal D = \{D(s){;\ }s \in \S\}$ consisting of closed subsets of $X$ which is subordinate to $\mathcal A$ and $\mathcal S(\mathcal D) = C$. For $s \in \S$ we set $$E(s) = \mathcal S_t(D(s \h t)), \qquad
H(s) = \ker_{\P}(E(s)), \qquad
P(s) = \overline{H(s)}, \qquad
\text{and} \qquad Q(s) = \mathcal S_t(P(s \h t)).$$ We verify that $\ker_{\P}(Q(s)) = Q(s)$ for every $s \in \S$. For every $u \in \S$ we have $$E(u) = \bigcup_{j \in \omega} E(u \h j)$$ and $$H(u) \setminus \bigcup_{j \in \omega} H(u \h j) \subset \Bigl(E(u) \setminus \bigcup_{j \in \omega} E(u \h j)\Bigr) \cup
\bigcup_{j \in \omega} \bigl(E(u \h j)\setminus H(u \h j)\bigr)
= \bigcup_{j \in \omega} \bigl(E(u \h j)\setminus H(u \h j)\bigr).$$ Since $E(u \h j) \setminus H(u \h j)$ is $\sigma$-$\P$-porous for every $j \in \omega$ (Lemma \[L:basic\](i)), we conclude that the set $H(u) \setminus \bigcup_{j \in \omega} H(u \h j)$ is $\sigma$-$\P$-porous. Since for every $s \in \S$ we have $$H(s) \setminus \mathcal S_t(H(s\h t)) \subset \bigcup_{t\in\S} \Bigl(H(s\h t) \setminus \bigcup_{j\in \omega} H(s\h t \h j)\Bigr),$$ we get that $H(s) \setminus \mathcal S_t(H(s\h t))$ is $\sigma$-$\P$-porous. Therefore $$\label{E:ker-S-t}
\ker_{\P}(\mathcal S_t(H(s\h t))) = \mathcal S_t(H(s\h t))$$ by Lemma \[L:basic\](iv). Further, we get that $\mathcal S_t(H(s\h t))$ is a dense subset of $H(s)$, thus $\overline{\mathcal S_t(H(s\h t))} = \overline{H(s)}$. Observing $$\label{E:inclusions}
\mathcal S_t(H(s\h t)) \subset \mathcal S_t(P(s\h t)) = Q(s) \subset P(s) = \overline{H(s)} = \overline{\mathcal S_t(H(s\h t))}$$ and using we get $\ker_{\P}(Q(s)) = Q(s)$. Indeed, fix an open set $U$ with $U\cap Q(s)\neq\emptyset$. Using , $U\cap S_t(H(s\h t))\neq\emptyset$ and, by , $U\cap S_t(H(s\h t))$ is not $\sigma$-$\P$-porous set. Hence, $U\cap Q(s)$ is not $\sigma$-$\P$-porous set and, as $U$ was an arbitrary open set intersecting $Q(s)$, $\ker_{\P}(Q(s)) = Q(s)$.
Further, we set $T = \{s \in \S{;\ }P(s) \neq \emptyset\}$. The set $T$ is obviously a nonempty tree. Moreover, $T$ is pruned. Indeed, if $s \in T$, then $H(s) \neq \emptyset$ and thus $E(s)$ is non-$\sigma$-$\P$-porous. We have $E(s) = \bigcup_{n \in \omega} E(s \h n)$ and therefore there exists $m \in \omega$ such that $E(s \h m)$ is non-$\sigma$-$\P$-porous. Thus $P(s \h m) \neq \emptyset$ and $s\h m \in T$.
We find a mapping $\varphi \colon \S \to \S$ such that for every $s \in \S$ we have
- $|\varphi(s)| = |s|$,
- if $t \in \S, s \prec t$, then $\varphi(s) \prec \varphi(t)$,
- $\{\varphi(s \h n){;\ }n \in \omega\} = \{\varphi(s) \h k {;\ }k \in \omega\} \cap T$.
We have $\emptyset \in T$ since $T$ is nonempty. We set $\varphi(\emptyset) = \emptyset$. Suppose that $\varphi(s) \in T$ has been already defined for some $s \in \S$. The set $W := \{k \in \omega {;\ }\varphi(s) \h k \in T\}$ is nonempty since $T$ is pruned. Thus we can find a mapping $\psi\colon \omega \to \omega$ such that $\psi(\omega) = W$. We define $\varphi(s \h n) = \varphi(s) \h \psi(n)$. This finishes the construction of $\varphi$. It is easy to check that the mapping $\varphi$ has all the required properties.
We set $L(s) = P(\varphi(s))$ and $\mathcal L = \{L(s) {;\ }s \in \S\}$. The scheme $\{E(s){;\ }s \in \S\}$ is monotone by definition. This easily gives that the scheme $\mathcal L$ is also monotone.
By the properties of $\varphi$ and the definition of $T$ we have $\mathcal S_t(L(s \h t)) = Q(\varphi(s)) \neq \emptyset$ for every $s \in \S$. Indeed, if $x \in Q(\varphi(s))$ for some $s \in \S$, then there exists $\nu \in \omega^{\omega}$ such that $x \in P(\varphi(s) \h \nu|n)$ for every $n \in \omega$. Thus $P(\varphi(s) \h \nu|n) \neq \emptyset$ for every $n \in \omega$. This means that $\varphi(s) \h \nu|n \in T$ for every $n \in \omega$. Using the properties of $\varphi$ we find $\mu \in \omega^{\omega}$ such that $\varphi(s \h \mu|n) = \varphi(s) \h \nu|n$ for every $n \in \omega$. Thus we have $x \in \bigcap_{n\in\omega}P(\varphi(s) \h \nu|n) = \bigcap_{n\in\omega} L(s \h \mu|n) \subset \mathcal S_t(L(s \h t))$. If $x \in \mathcal S_t(L(s \h t))$, then there exists $\nu \in \omega^{\omega}$ such that $x \in L(s \h \nu|n) = P(\varphi(s \h \nu|n))$ for every $n \in \omega$. Using the properties of $\varphi$ again, we get $x \in \bigcap_{n \in \omega} P(\varphi(s \h \nu|n)) \subset Q(\varphi(s))$. Finally, if $s \in \S$ we have $H(\varphi(s)) \neq \emptyset$ and $H(\varphi(s)) \setminus \mathcal S_t(H(\varphi(s) \h t))$ is $\sigma$-$\P$-porous. Thus $\mathcal S_t(H(\varphi(s) \h t)) \neq \emptyset$ and by we get $Q(\varphi(s)) \neq \emptyset$. Thus $\mathcal L$ is $\P$-regular.
Clearly $\mathcal L \sqsubseteq \mathcal D$. Using the fact that $\mathcal D \sqsubseteq \mathcal A$, we get $\mathcal L \sqsubseteq \mathcal A$. It remains to verify that $\mathcal S(\mathcal L)$ is dense in $C$. Since by definition we have $P(s) \subset D(s)$ for every $s \in \S$, we get $Q(\emptyset) \subset E(\emptyset) = C$. The set $\mathcal S_t(H(t))$ is a dense subset of $H(\emptyset)$. We get by that $Q(\emptyset)$ is a dense subset of $H(\emptyset) = C$. It concludes the proof since $\mathcal S(\mathcal L) = Q(\emptyset)$.
\[P:Foran-converse\] Suppose that $N_{\P}(A)$ is a Suslin set whenever $A \subset X$ is Suslin. If $S \subset X$ is a Suslin non-$\sigma$-$\P$-porous set, then there exists a $(\mathcal B,\P)$-Foran scheme $\F$ in $X$ such that each element of $\F$ is a subset of $S$.
We will construct a sequence $(\mathcal A_n)_{n \in \omega}$ of $\P$-regular Suslin schemes consisting of closed sets, where $\mathcal A_n = \{A^n(s){;\ }s \in \S\}$. For $s \in \S$ we denote $C^n(s) = \mathcal S_t (A^n(s \h t))$ and, for every $n \in \omega$, we require
- $\mathcal S (\mathcal A_0) \subset S$,
- $\mathcal A_{n+1} \sqsubseteq \mathcal A_n$ and this fact is witnessed by a mapping $\varphi_{n+1}\colon \S \to \S$,
- $\varphi_{n+1}(s) = s$ for every $s \in \S, |s| \leq n$.
Using Lemma \[L:regular\] we find a $\P$-regular Suslin scheme $\mathcal A_0$ consisting of closed sets with $\mathcal S (\mathcal A_0) \subset \ker_{\P}(S) \subset S$. Suppose that $n \in \omega, n > 0$, and we have already constructed the desired schemes $\mathcal A_j$, $j < n$. Fix $s \in \omega^{n-1}$. The set $\ker_{\P}(N_{\P}(C^{n-1}(s)))$ is nonempty by $\P$-regularity of $\mathcal A_{n-1}$. This set is also Suslin by the assumption and Lemma \[L:basic\](iii). By Lemma \[L:regular\] we find a $\P$-regular Suslin scheme $\mathcal L_s = \{L^s(t) {;\ }t \in \S\}$ such that $\mathcal S(\mathcal L_s)$ is a dense subset of $\ker_{\P}(N_{\P}(C^{n-1}(s)))$ and $\mathcal L_s \sqsubseteq \{A^{n-1}(s \h t){;\ }t \in \S\}$. The last fact is witnessed by a mapping $\varphi^s_n \colon \S \to \S$.
For $t = (t_0,\dots,t_{|t|-1}) \in \S$, we set $$\begin{aligned}
A^{n}(t) &=
\begin{cases}
A^{n-1}(t), & |t| < n, \\
L^{t|(n-1)}(t_{n-1},\dots,t_{|t|-1}), & |t| \geq n;
\end{cases}
\\
\varphi_n(t) &=
\begin{cases}
t, & |t| < n, \\
t|(n-1) \h \varphi_n^{t|(n-1)}(t_{n-1},\dots,t_{|t|-1}), & |t| \geq n.
\end{cases}\end{aligned}$$ Further, we set $\mathcal A_n = \{A^n(t){;\ }t \in \S\}$.
We define $S(s) = C^{|s|}(s)$, $s \in \S$, and $\F = \{S(s){;\ }s \in \S\}$. We verify the conditions (i)–(iii) from Definition \[D:Foran-scheme\].
\(i) Let $n \in \omega$ and $s \in \omega^n$. Since $\ker_{\P}(C^n(s)) = C^n(s)$ we have that the set $\ker_{\P}(N_{\P}(C^n(s)))$ is dense in $C^n(s)$. Since $\mathcal S(\mathcal L_s)$ is dense in $\ker_{\P}(N_{\P}(C^n(s)))$, we get that $\mathcal S(\mathcal L_s)$ is dense in $C^n(s)$. Thus we have that $\bigcup_{j \in \omega} S(s \h j) = \bigcup_{j \in \omega} C^{n+1}(s \h j) = C^{n+1}(s) = \mathcal S(\mathcal L_s)$ is a dense subset of $S(s) = C^n(s)$.
\(ii) We have $S(t \h k) = C^{|t|+1}(t \h k) \subset N_{\P}(C^{|t|}(t)) = N_{\P}(S(t))$ for every $t \in \S$ and $k \in \omega$.
\(iii) Suppose that we have $\nu \in \omega^{\omega}$ and a sequence $\{G_n\}_{n\in\omega}$ of open sets such that
- $\lim_{n \to \infty} \diam(G_n) = 0$,
- $\overline{G_{n+1}} \subset G_n$ for every $n \in \omega$,
- $S(\nu |n) \cap G_n \neq \emptyset$ for every $n \in \omega$.
We have that $\bigcap_{n \in \omega} G_n = \{x\}$ for some $x \in X$, since $X$ is complete. Our task is to show that $x \in S(\nu|m)$ for every $m \in \omega$. Fix $m \in \omega$. For each $k \in \omega$ there exists $y_k \in S(\nu|k) \cap G_k$. We have $\lim y_k = x$. Fix $k \in \omega$, $k \geq m$. For every $n \in \omega, n > k$, we have $$\label{E:inclusions2}
y_n \in S(\nu|n) = C^n(\nu|n) \subset A^n(\nu|n) \subset A^{n-1}(\varphi_n(\nu|n)) \subset \dots \subset A^m(\varphi_{m+1} \circ \dots \circ \varphi_n(\nu|n)).$$ Since $\mathcal A_m$ is a $\P$-regular scheme, it is monotone. Using this fact and we get $$\label{E:inclusions3}
y_n \in A^m(\varphi_{m+1} \circ \dots \circ \varphi_n(\nu|n))
\subset A^m(\varphi_{m+1} \circ \dots \circ \varphi_n(\nu|k)).$$ Since $\varphi_j(\nu|k) = \nu|k$ for every $j \in \omega, j > k$, we get $$A^m(\varphi_{m+1} \circ \dots \circ \varphi_n(\nu|k)) = A^m(\varphi_{m+1} \circ \dots \circ \varphi_{k+1}(\nu|k)).$$ Using this and we get $x \in A^m(\varphi_{m+1} \circ \dots \circ \varphi_{k+1}(\nu|k))$ since the latter set is closed. Since $\nu|m \prec \varphi_{m+1} \circ \dots \circ \varphi_{k+1}(\nu|k)$ we can conclude $x \in S(\nu|m)$.
Porosity-like relations {#S:porosity-like}
=======================
Let $X$ be a metric space and ${\bf R}$ be a point-set relation on $X$ (i.e. ${\bf R}\subseteq X\times \mathcal{P}(X)$). Let $M$ be a set and ${\bf R}'$ be a point-set relation on $X_M = \ov{X\cap M}$. We say that the set $M$ is a *pointwise $({\bf R}\to {\bf R}')$-model* if $$\forall A \in \mathcal{P}(X) \cap M \ \forall x \in X_M \colon \bigl({\bf R}(x,A)\;\to\; {\bf R}'(x,A\cap X_M)\bigr).$$ Similarly, we define the notion of a *pointwise $({\bf R}\leftrightarrow {\bf R}')$-model*.
Let $X$ be a metric space and $\P$ be a porosity-like relation on $X$. Let $M$ be a set and $\P'$ be a porosity-like relation on $X_M = \ov{X\cap M}$. We say that the set $M$ is a *$(\P \to \P')$-model* if for every set $A\in \mathcal{P}(X)\cap M$ $$A \text{ is $\P$-porous in the space } X\;\to\;A \cap X_M \text{ is $\P'$-porous in the space }X_M.$$ We say that the set $M$ is a *$(\sigma$-$\P \to \sigma$-$\P')$-model* if for every set $A\in \mathcal{P}(X)\cap M$ $$A \text{ is $\sigma$-$\P$-porous in the space } X\;\to\;A \cap X_M\text{ is $\sigma$-$\P'$-porous in the space }X_M.$$ Similarly, we define the notion of *$(\P \leftarrow \P')$-model*, *$(\P \leftrightarrow \P')$-model*, and *$(\sigma$-$\P \leftrightarrow \sigma$-$\P')$-model*.
\[P:porosity-like\] For any suitable elementary submodel $M$ the following holds. Let $X$ be a metric space, $\P$ be a porosity-like relation on $X$ and $\P'$ be a porosity-like relation on $X_M$. Assume $M$ contains $X$ and $\P$.
- If $M$ is a pointwise $(\P \to \P')$-model, then $M$ is a $(\P \rightarrow \P')$-model.
- If $M$ is a pointwise $(\neg \P \to \neg \P')$-model, then $M$ is a $(\P \leftarrow \P')$-model.
- If $M$ is a $(\P \to \P')$-model, then $M$ is a $(\sigma$-$\P \to \sigma$-$\P')$-model.
In particular, if $M$ is a pointwise $(\P \leftrightarrow \P')$-model, then $M$ is a $(\P \leftrightarrow \P')$-model and a $(\sigma$-$\P \to \sigma$-$\P')$-model.
Let us fix a $(*)$-elementary submodel $M$ containing $X$ and $\P$.
\(i) The statement follows immediately from definitions.
\(ii) Let us suppose $M$ is a pointwise $(\neg \P \to \neg \P')$-model and let us fix a non-$\P$-porous set $A\in\mathcal{P}(X)\cap M$. Consider the formula $$\tag{$*$}
\exists x\in A \colon (x,A) \notin \P,$$ with free variables $A$ and $\P$. Since $A \in M$, $\P \in M$, and the above formula is absolute for $M$, there exists $x \in A \cap M$ such that $(x,A) \notin \P$, i.e., $A$ is not $\P$-porous at $x$. Hence $A\cap X_M$ is not $\P'$-porous at $x$. Thus, $A\cap X_M$ is not $\P'$-porous in the space $X_M$ and (ii) holds.
\(iii) Suppose that $A \in M \cap \mathcal P(X)$ is $\sigma$-$\P$-porous. Then the next formula is satisfied $$\tag{$*$}
\begin{split}
\exists D \colon \bigl(& D \text{ is a function with }\dom D = \en, D(n) \subset X \text{ is $\P$-porous set}\\
& \text{for every }n \in \en,\; A \subset \bigcup_{n \in \en} D(n)\bigr).
\end{split}$$ Now by Lemma \[L:unique-M\] we find $D \in M$ such that $$\begin{split}
& D \text{ is a function with }\dom D = \en, D(n) \subset X \text{ is $\P$-porous set}\\
& \text{for every }n \in \en,\; A \subset \bigcup_{n \in \en} D(n).
\end{split}$$ We have $D(n) \in M$ for every $n \in \en$. Since $M$ is a $(\P \to \P')$-model, we obtain that $D(n) \cap X_M$ is $\P'$-porous in $X_M$ for every $n \in \en$, hence, $A \cap X_M$ is $\sigma$-$\P'$-porous in $X_M$.
\[L:metric-lemma\] Let $(X,\varrho)$ be a metric space and $(Y,\varrho)$ be its complete subspace. Assume we have a sequence $U_Y^n = U_Y(x_n,r_n) \subseteq Y$, $n \in \en$, of open balls in $Y$ and assume the following conditions hold:
- $\lim_{n\to \infty}r_n = 0$,
- for each $n\in \en$ we have $\ov{U_Y^{n+1}\;}^Y \subseteq U_Y^n$.
Denote $U_X^n := U_X(x_n,r_n)$. Then there exists an increasing sequence of integers $\{ n(k) \}_{k=1}^\infty$ such that for each $k\in \en$ we have $$\ov{U_X^{n(k+1)}\;}^X \subseteq U_X^{n(k)}.$$
We shall prove the following statement which implies the conclusion of the lemma: For each $k\in\en$ there is $l\in\en$, $l>k$ such that $\ov{U_X^l\;}^X\subseteq U_X^k$.
Assume this is not the case, i.e., there is a natural number $n_0$ such that $\ov{U_X^n\;}^X \setminus U_X^{n_0} \neq \emptyset$ for each natural number $n>n_0$. Choose a sequence $\{y_n\}_{n=n_0+1}^\infty$ such that $y_n\in\ov{U_X^n\;}^X \setminus U_X^{n_0}$ for each $n>n_0$.
From the assumptions it is obvious that the sequence $\{x_n\}_{n=1}^\infty$ is Cauchy and hence it has a limit $x\in Y$ (as $Y$ is complete). Since $\varrho (x_n,y_n)\leq r_n$, it also follows from (i) that $\lim_{n\to\infty}y_n=x$. Consequently, $x\notin U_X^{n_0}$ as $U_X^{n_0}$ is open and $y_n \notin U_X^{n_0}$ for any $n > n_0$.
On the other hand, the assumption (ii) gives that $\{ x_n{;\ }n>n_0\}\subseteq U_Y^{n_0+1}$ and so, again by (ii), $x = \lim_{n\to\infty} x_n \in \ov{U_Y^{n_0+1}\;}^Y \subseteq U_Y^{n_0}$. This is a contradiction.
\[P:reduction-Foran\] For any suitable elementary submodel $M$ the following holds. Let $X$ be a complete metric space, $A\subset X$ and $\P$ be a porosity-like relation on $X$. Let there exist a $(\B, \P)$-Foran scheme $\F$ in $X$, where $\B = \{U(x,r){;\ }x \in X,\; r \in \er_+\}$, such that each element of $\F$ is a subset of $A$. Assume that $M$ contains $X$, $A$, $\P$ and $M$ is a pointwise $(\neg \P \to \neg \P')$-model for some porosity-like relation $\P'$ on $X_M$.
Then there exists a $(\B', \P')$-Foran scheme $\F'$ in $X_M$, where $\B' = \{U(x,r)\cap X_M {;\ }x \in X \cap M,\; r \in \qe_+\}$, such that each element of $\F'$ is a subset of $A \cap X_M$.
By the assumption, the following formula is true: $$\tag{$*$}
\begin{split}
\exists S\;(S:\omega^{<\omega}\to\mathcal{P}(X)\text{ is such that } \{S(t){;\ }t\in\omega^{<\omega}\}&\text{ is a }(\B,\P)\text{-Foran scheme in }X\\
& \text{and, for every $t\in\omega^{<\omega}$, $S(t)\subset A$}).
\end{split}$$ Using Lemma \[L:unique-M\] and absoluteness of the preceding formula and its subformula for $M$, we find the corresponding $S \in M$. Consequently, for every $t \in \omega^{<\omega}$, $S(t) \in M$. Now it is sufficient to prove, that $$\F' = \{S(t)\cap X_M {;\ }t \in \omega^{<\omega}\}$$ is a $(\B',\P')$-Foran scheme in $X_M$. By [@Cuth-Rmoutil Lemma 2.10], $S(t)\cap M$ is a dense subset of $S(t)\cap X_M$ for every $t \in \omega^{<\omega}$. Hence, by Lemma \[L:density\], the condition (i) from Definition \[D:Foran-scheme\] is satisfied. Since $M$ is a pointwise $(\neg \P \to \neg \P')$-model for $\P'$ on $X_M$ we get by Proposition \[P:porosity-like\](ii) that the condition (ii) is satisfied.
In order to prove that (iii) holds, let us take some $\nu\in \omega^\omega$, a sequence $\{x_n\}_{n\in\omega}$ of elements of $X\cap M$ and a sequence $\{r_n\}_{n\in\omega}$ of numbers from $\qe_+$ such that the open balls $G_n = U(x_n,r_n)\cap X_M$ satisfy conditions (a), (b), and (c) in the space $X_M$. It is easy to see that the radii $r_n$ can be chosen in such a way that $r_n<2\diam(U(x_n,r_n)\cap X_M)=2\diam(G_n)$, and hence $r_n\to 0$. Then Lemma \[L:metric-lemma\] gives the existence of an increasing sequence of integers $\{n(k)\}_{k=1}^\infty$ such that $\ov{U(x_{n(k+1)},r_{n(k+1)})\;}^X \subseteq U(x_{n(k)},r_{n(k)})$ for each $k$. Hence we have that the sequence $\left\{U(x_{n(k)},r_{n(k)}) \right\}_{k=1}^\infty$ satisfies condition (b) from the definition of $(\B, \P)$-Foran scheme in $X$ and the condition (a) follows from our assumption that $r_n\to 0$. Now we verify the condition (c). From the assumptions on $G_n$ we know that $U(x_{n(k)},r_{n(k)}) \cap S(\nu \rest k) \supseteq U(x_{n(k)},r_{n(k)}) \cap S(\nu\rest n(k)) \cap X_M \neq \emptyset$ and so (as $\F$ is a $(\B, \P)$-Foran scheme in $X$) we have that there exists $x \in \bigcap_{k=1}^\infty U(x_{n(k)},r_{n(k)}) \cap S(\nu\rest k)$. Since $\lim x_n = x$ by (a) and (b), we have $x \in X_M$. Consequently, $$x \in \bigcap_{n=1}^\infty(G_n \cap S(\nu\rest n)\cap X_M).$$ This verifies (iii) from Definition \[D:Foran-scheme\].
\[P:reduction-porosity-like\] For any suitable elementary submodel $M$ the following holds. Let $X$ be a complete metric space, $\P$ be a porosity-like relation on $X$, and ${\P'}$ be a porosity-like relation on $X_M$. Suppose that $N_{\P}(S)$ is a Suslin set whenever $S\subset X$ is Suslin. Assume $M$ contains $X$, $\P$, and a Suslin set $A\subset X$. Then whenever $M$ is a pointwise $(\P \leftrightarrow \P')$-model, then the following holds: $$\begin{aligned}
A \text{ is $\P$-porous in the space }X &\iff A \cap X_M \text{ is $\P'$-porous in the space } X_M, \\
A \text{ is $\sigma$-$\P$-porous in the space }X &\iff A \cap X_M \text{ is $\sigma$-$\P'$-porous in the space } X_M.\end{aligned}$$
By Proposition \[P:porosity-like\], it is sufficient to prove the implication from the right to the left in the second equivalence. Let us fix a $(*)$-elementary submodel $M$ containing $X$, $\P$, and a non-$\sigma$-$\P$-porous Suslin set $A\subset X$. We would like to verify that $A$ is not $\sigma$-$\P'$-porous in the space $X_M$. Let $\B$, $\B'$ be as in Proposition \[P:reduction-Foran\]. By Proposition \[P:Foran-converse\], there exists a $(\B,\P)$-Foran scheme $\F$ in $X$ such that each element of $\F$ is a subset of $A$. Using Proposition \[P:reduction-Foran\], there exists a $(\B',\P')$-Foran scheme $\F'$ in $X_M$ such that each element of $\F'$ is a subset of $A \cap X_M$. Hence, by Lemma \[L:Foran-lemma\], $A \cap X_M$ is non-$\sigma$-$\P'$-porous in the space $X_M$.
Let $X$ be a complete metric space and $\P_{up}$ be the porosity-like relation defined by (for the definition of the upper porosity, see for example [@Cuth]) $${\P_{up}} = \{(x,A) \in X \times \mathcal{P}(X) {;\ }A \text{ is upper porous at the point } x\}.$$ Let us fix a $(*)$-elementary submodel $M$ containing $X$ and $\P_{up}$. Denote by ${\P'_{up}}$ the porosity-like relation defined by $${\P'_{up}} = \{(x,A) \in X_M \times \mathcal{P}(X_M) {;\ }A \text{ is upper porous at the point } x \text{ in the space } X_M\}.$$ Then, by results from [@Cuth] and [@Cuth-Rmoutil], $M$ is a pointwise $({\P_{up}} \leftrightarrow {\P'_{up}})$-model. It is easy to see that $N_{\P_{up}}(S)$ is a Suslin set whenever $S\subset X$ is Suslin. Thus, by Proposition \[P:reduction-porosity-like\], $\sigma$-upper porosity is separably determined property. This result has already been proved in [@Cuth-Rmoutil]. However, a nontrivial inscribing theorem ([@Zeleny-Pelant Theorem 3.1]) is needed in the proof there. The above mentioned method enables us to avoid the usage of this result.
It is known to the authors that also the notions of lower porosity, $\langle g\rangle$-porosity, and $(g)$-porosity satisfy the assumptions of Proposition \[P:reduction-porosity-like\] (for definitions see [@Zajicek-1]). Consequently, those porosities (and corresponding $\sigma$-porosities) are separably determined when taking Suslin sets in complete metric spaces. We do not present proofs of those results here since now we see no interesting applications.
Note that in the following section we prove that also the notion of $\alpha$-cone porosity in Asplund spaces satisfies the assumptions of Proposition \[P:reduction-porosity-like\] and, therefore, cone smallness is separably determined.
It is an open problem whether the notion of $\sigma$-directional porosity (see [@Zajicek-4] for the definition) is separably determined in the sense of Corollary \[C:cone-small\].
Note that the notion of $\sigma$-directional porosity is defined also in [@LPT], but in a slightly different way which is equivalent to the definition from [@Zajicek-4] only in separable Banach spaces.
Cone porosity {#S:cone}
=============
In the following section we prove that the notion of $\alpha$-cone porosity in Asplund spaces satisfies the assumptions of Proposition \[P:reduction-porosity-like\] and, therefore, cone smallness is separably determined. First, let us give the definition.
Let $X$ be a Banach space. For $x^*\in X^*\setminus\{0\}$ and $\alpha\in[0,1)$ we define the *$\alpha$-cone* $$C(x^*,\alpha) = \{x\in X {;\ }\alpha\|x\|\cdot\|x^*\| < x^*(x)\}.$$ A set $A \subset X$ is said to be *$\alpha$-cone porous* at $x \in X$ in the space $X$ if there exists $R>0$ such that for each $\varepsilon > 0$ there exists $z \in U(x,\varepsilon)$ and $x^* \in X^* \setminus \{0\}$ such that $$U(x,R) \cap \bigl(z+C(x^*,\alpha)\bigr) \cap A = \emptyset.$$ The corresponding porosity-like relation is denoted by ${\P_X^{\alpha\text{-}cone}}$. A set is said to be *cone small* if it is $\sigma$-${\P_X^{\alpha\text{-}cone}}$-porous for each $\alpha\in(0,1)$.
The following lemma comes from [[@Cuth Lemma 4.14]]{}.
\[L:sup-fin-M\] For any suitable elementary submodel $M$ the following holds. Let $(X,\rho)$ be a metric space and $f \colon X \to \er$ be a function. Then whenever $M$ contains $X$ and $f$, it is true that for every $R > 0$ and $x \in X_M$ we have $$\sup_{u\in U(x,R)}f(u) = \sup_{u\in U(x,R) \cap X_M} f(u).$$
\[P:pointwise-cone-first\] For any suitable elementary submodel $M$ the following holds. Let $X$ be a Banach space and $\alpha \in [0,1)$. Then whenever $M$ contains $X$ and $\alpha$, $M$ is a pointwise $({\P_X^{\alpha\text{-}cone}}\to {\P_{X_M}^{\alpha\text{-}cone}})$-model.
Let us fix a $(*)$-elementary submodel $M$ containing $X$, $\alpha$ and a set $A \in \mathcal P(X) \cap M$. Fix some $x\in X_M$ such that $A$ is $\alpha$-cone porous at $x$. Then there exists a rational number $R > 0$ such that $$\forall \varepsilon > 0\; \exists z\in U(x,\varepsilon)\; \exists x^*\in X^*\setminus\{0\}\colon U(x,R) \cap \bigl(z+C(x^*,\alpha)\bigr) \cap A = \emptyset.$$ We will show that this formula is true in the space $X_M$ with the constant $\frac14 R$ instead of $R$. Let us fix a rational $\varepsilon > 0$. Fix a number $0 < \delta < \min\{\frac13 \varepsilon,\frac14 R\}$ and a point $x' \in U(x,\delta) \cap M$. Then it is easy to observe that the following formula is true $$\tag{$*$}
\exists z'\in U(x',\tfrac23 \varepsilon)\;\exists x^*\in X^*\setminus\{0\} \colon U(x',\tfrac12 R) \cap \bigl(z'+C(x^*,\alpha)\bigr) \cap A = \emptyset.$$ (Indeed, it is enough to take a point $z \in U(x,\tfrac13 \varepsilon) \subset U(x',\tfrac23 \varepsilon)$ and $x^* \in X^* \setminus \{0\}$ satisfying $U(x,R) \cap \bigl(z+C(x^*,\alpha)\bigr) \cap A = \emptyset$ and to observe that $U(x',\tfrac12 R) \subset U(x,R)$.) Using the absoluteness of this formula (and its subformulas) we find $z' \in U(x',\frac23 \varepsilon) \cap M \subset U(x,\varepsilon) \cap M$ and $x^*\in X^* \cap M \setminus \{0\}$ such that $$\label{E:cone-empty}
U(x',\tfrac{R}{2}) \cap \bigl(z'+C(x^*,\alpha)\bigr) \cap A = \emptyset.$$ By Lemma \[L:sup-fin-M\] we have $\|x^*\| = \|{x^*\rest_{X_M}}\|$. Hence, the cone $C({x^*\rest_{X_M}},\alpha)$ in the space $X_M$ equals to $C(x^*,\alpha) \cap X_M$. We need to verify that $$U(x,\tfrac{R}{4}) \cap \bigl(z'+C(x^*,\alpha)\bigr) \cap A \cap X_M = \emptyset.$$ Fix some $a \in A \cap X_M$ such that $\|x-a\| < \tfrac14 R$. Then $a$ is an element of $U(x',\frac12 R)$. By we conclude $a \notin z'+C(x^*,\alpha)$ and the proof is finished.
In order to show existence of a pointwise $({\P_X^{\alpha\text{-}cone}} \leftarrow {\P_{X_M}^{\alpha\text{-}cone}})$-models we restrict our attention to Asplund spaces. First, we need to prove that “functionals from the model $M$ are dense in $(X_M)^*$ when $X$ is an Asplund space”. This seems to be a nontrivial and very useful result and it might be useful in other separable reduction theorems as well. The proof can be done using the existence of a “projectional generator with domain $X$” in the dual space of an Asplund space $X$. In fact, it is sufficient to use only the first part of the proof of this statement from [@Fabian].
\[T:Asplund-model\] For any suitable elementary submodel $M$ the following holds. Let $X$ be an Asplund space. Then whenever $M$ contains $X$, it is true that $\ov{\{x^*\rest_{X_M} {;\ }x^*\in X^*\cap M\}} = (X_M)^*$.
It is obvious that the inclusion “$\subset$” holds. Let us show that the opposite one holds as well. It is proved in the second step of the proof of [[@Fabian Theorem 1]]{} that there exists continuous mappings $D(n) \colon X \to X^*$, $n \in \en$, such that that, for every closed separable subspace $V \subset X$, we have $$\ov{\sspan}\{D(n)(x) \rest_V {;\ }n \in \en, x \in V\} = V^*.$$ Using the absoluteness of the formula (and its subformula) $$\tag{$*$}
\begin{split}
\exists D \colon \bigl(& D \text{ is a function, } \dom D = \en, \ D(n) \text{ are $\|\cdot\|$-$\|\cdot\|$ continuous mappings} \\
&\text{from $X$ into $X^*$ and for every closed separable subspace $V$ of $X$ we have } \\
&\ov{\sspan}\{D(n)(x) \rest_V {;\ }n \in \en, x \in V\} = V^* \bigr),\\
\end{split}$$ we may without loss of generality assume that $D(n) \in M$ for every $n \in \en$. Thus, for every $n \in \en$ and $x \in X \cap M$, we have $D(n)(x) \in M$. Using the continuity of $D(n)$ for every $n \in \en$, we get $$\{D(n)(x) {;\ }n \in \en, x \in X_M\} \subset \ov{\{D(n)(x) {;\ }n \in \en, x \in X \cap M\}} \subset \ov{X^* \cap M}.$$ Finally, $$\begin{aligned}
(X_M)^* &= \ov{\sspan}\{{D(n)(x)\rest_{X_M}} {;\ }n \in \en, x \in X_M\}
\subset \ov{\{x^*\rest_{X_M} {;\ }x^* \in \ov{X^* \cap M}\}} \\
&\subset \ov{\{x^*\rest_{X_M} {;\ }x^* \in X^* \cap M\}}.\end{aligned}$$
Now, we need to observe that it is enough to consider functionals from a dense subset of $X^*$ in the definition of $\alpha$-cone porosity.
\[L:equivalent-def-cone\] Let $X$ be a Banach space and let $E\subset X$ and $D\subset X^*$ be norm-dense subsets. Let $A\subset X$ and $x\in X$ and $\alpha\in [0,1)$. Then $A$ is $\alpha$-cone porous at $x$ if and only if the following is true: $$\exists R\in \qe_+ \ \forall \varepsilon\in \qe_+ \ \exists y^* \in D \ \exists w \in B(x,\varepsilon) \cap E \colon B(x,R) \cap (w + C(y^*,\alpha)) \cap A = \emptyset.$$
The sufficiency of our condition is easy to see. Let us, therefore, assume that a given set $A$ is $\alpha$-cone porous at a given point $x\in X$, and deduce from it the desired condition.
Since $A$ is $\alpha$-cone porous at $x$, it is easy to see that there exists $R\in \qe_+$ such that $$\label{E:cone-porous}
\forall \varepsilon > 0 \ \exists x^* \in X \ \exists z \in B(x,\varepsilon) \colon B(x,R) \cap \bigl(z + C(x^*,\alpha)\bigr) \cap A = \emptyset.$$ Let $\varepsilon\in \qe_+$. Using we find $x^* \in X^*$ and $z \in B(x,\min\{\varepsilon,R\})$ such that $$B(x,R) \cap (z + C(x^*,\alpha)) \cap A = \emptyset.$$ Choose $w \in B(x,\min\{\varepsilon,R\}) \cap (z + C(x^*,\alpha))\cap E$. Since $w-z \in C(x^*,\alpha)$, we have $$x^*(w-z) - \alpha \|x^*\| \|w-z\| > 0.$$ Using the last inequality and density of $D$ we find $y^* \in D$ such that
- $\|y^*\| \geq \|x^*\|$,
- $\|x^*-y^*\| < \frac{1}{2R}\bigl(x^*(w-z)-\alpha\|x^*\|\|w-z\|\bigr)$.
Now it is sufficient to prove that $$\label{E:inclusion-of-cones}
B(x,R) \cap (w + C(y^*,\alpha)) \subset z + C(x^*,\alpha).$$ Indeed, since then we have $$B(x,R) \cap (w + C(y^*,\alpha)) \cap A \subset B(x,R) \cap (z + C(x^*,\alpha)) \cap A = \emptyset.$$ To verify take $u \in C(y^*,\alpha)$ with $w + u \in B(x,R)$. Then we have $$\label{E:estimate-2R}
\|u\| = \|(u+w-x)+(x-w)\| \leq \|u+w-x\| + \|x-w\| \leq 2R.$$ We compute $$\begin{split}
x^*(w+u-z) &= x^*(w-z) + x^*(u) \\
&= x^*(w-z) + y^*(u) + (x^*-y^*)(u) \\
&> x^*(w-z) + \alpha\|y^*\| \cdot \|u\| - \|x^*-y^*\|\cdot \|u\| \\
&\geq x^*(w-z) + \alpha \|x^*\| \cdot \|u\| - \|x^*-y^*\|\cdot 2R \qquad (\text{by} \eqref{E:estimate-2R}) \\
&\geq \alpha \|x^*\| \cdot \|u\| + \alpha \|x^*\| \cdot \|w-z\| \qquad (\text{by (b)}) \\
&\geq \alpha \|x^*\| \cdot \|u+w-z\|.
\end{split}$$ This shows that $w+u-z \in C(x^*,\alpha)$. Consequently, we get $w + u \in z + C(x^*,\alpha)$.
Now, we are ready to see the existence of a pointwise $({\P_X^{\alpha\text{-}cone}} \leftrightarrow {\P_{X_M}^{\alpha\text{-}cone}})$-models in Asplund spaces.
\[P:point-cone\] For any suitable elementary submodel $M$ the following holds. Let $X$ be an Asplund space and $\alpha \in [0,1)$. Then whenever $M$ contains $X$ and $\alpha$, $M$ is a pointwise $({\P_X^{\alpha\text{-}cone}}\leftrightarrow {\P_{X_M}^{\alpha\text{-}cone}})$-model.
Let us fix a $(*)$-elementary submodel $M$ containing $X$, $\alpha$, and a set $A \subset X$. By Proposition \[P:pointwise-cone-first\], $M$ is a pointwise $({\P_X^{\alpha\text{-}cone}}\to {\P_{X_M}^{\alpha\text{-}cone}})$-model. Fix some $x\in X_M$ such that $A$ is not $\alpha$-cone porous at $x$. We will show that $A\cap X_M$ is not $\alpha$-cone porous at $x$ in the space $X_M$. Notice that, by Lemma \[L:sup-fin-M\], $\|{x^*\rest_{X_M}}\| = \|x^*\|$ for every $x^*\in X^* \cap M$. Hence, the cone $C({x^*\rest_{X_M}},\alpha)$ in the space $X_M$ equals $C(x^*,\alpha)\cap X_M$. Thus, by Lemma \[L:equivalent-def-cone\] and Theorem \[T:Asplund-model\], it is sufficient to prove that the following formula is true $$\begin{aligned}
\forall R\in\qe_+\;\exists\varepsilon\in\qe_+\;\forall z\in U(x,\varepsilon)\cap M\;
& \forall x^* \in (X^*\cap M) \setminus\{0\}\colon \\
&A \cap U(x,R) \cap X_M \cap \bigl(z+C(x^*,\alpha)\bigr) \neq \emptyset.\end{aligned}$$ Fix $R \in \qe_+$. As $A$ is not $\alpha$-cone porous at $x$, there exists $\varepsilon \in \qe_+$ such that $$\label{E:cone-in-X}
\forall z \in U(x,\varepsilon)\; \forall x^* \in X^* \setminus \{0\} \colon A \cap U(x,\tfrac13 R) \cap \bigl(z+C(x^*,\alpha)\bigr) \neq \emptyset.$$ Let us fix $z\in U(x,\varepsilon)\cap M$ and $x^*\in (X^* \cap M) \setminus \{0\}$. Find some $x' \in U(x,\tfrac13 R) \cap M$. Then $U(x,\tfrac13 R)\subset U(x',\tfrac23 R)$. By , the following formula is true $$\tag{$*$}
\exists a \in A \colon a \in \bigl(z+C(x^*,\alpha)\bigr) \cap U(x',\tfrac23 R).$$ Using the absoluteness of the formula (and its subformula) above, there exists $a \in A \cap M$ satisfying the formula above. It is easy to verify that $a\in U(x,R) \cap \bigl(z+C(x^*,\alpha)\bigr)$. Hence, $$A \cap U(x,R) \cap X_M \cap \bigr(z+C(x^*,\alpha)\bigl) \neq \emptyset.$$ Thus, $A \cap X_M$ is not $\alpha$-cone porous at $x$ in the space $X_M$. This finishes the proof.
In the remainder of the section we prove that the assumption on descriptive quality of $N_{\P}(S)$ from Proposition \[P:reduction-porosity-like\] is satisfied for cone porosity. We begin with the following lemma.
\[L:dist-cone\] Let $X$ be a Banach space, $x^* \in X^*$, and $\alpha \in [0,1)$. Then for each $x\in C(x^*,\alpha)$ it is true that $\d(X\setminus C(x^*,\alpha), x+C(x^*,\alpha)) > 0$.
It is easy to verify that $C(x^*,\alpha)$ is an open set and that it is a convex cone in the sense that for any two points $y,z$ from $C(x^*,\alpha)$ and any $c > 0$ the points $cy$ and $y+z$ also belong to $C(x^*,\alpha)$. Let $x \in C(x^*,\alpha)$. Set $\delta := \d(x,X \setminus C(x^*,\alpha))$. The number $\delta$ is positive, since $C(x^*,\alpha)$ is open. Take any point $y \in x+C(x^*,\alpha)$ (then $y-x\in C(x^*,\alpha)$). Hence, $U(x,\delta)\subset C(x^*,\alpha)$, and so $U(y,\delta) = (y-x)+U(x,\delta)\subset C(x^*,\alpha)$. Since $y\in x+C(x^*,\alpha)$ was chosen arbitrarily, we conclude that $\d(X\setminus C(x^*,\alpha), x+C(x^*,\alpha))\geq\delta>0$.
\[L:points-cone-porous\] Let $X$ be a Banach space, $\alpha \in [0,1)$, and $A \subset X$ be any set. Then the set $S$ of all points $x \in X$ at which $A$ is $\alpha$-cone porous is Borel (of the type $G_{\delta\sigma}$).
For $x,z \in X$, $R > 0$, $x^* \in X^* \setminus \{0\}$, and $\alpha \in [0,1)$ we set $$T(x,R,z,x^*,\alpha) = U(x,R) \cap \bigl(z+C(x^*,\alpha)\bigr).$$ First we show that $$\label{E:cone-porosity}
S = \bigcup_{R\in\qe_+} \bigcap_{\varepsilon\in\qe_+} \bigcup_{x^*\in X^*\setminus \{0\}} G(R,\varepsilon,x^*),$$ where $$G(R,\varepsilon,x^*) = \{x \in X {;\ }\exists z \in U(x,\varepsilon)\colon \d(T(x,R,z,x^*,\alpha),A) > 0 \}.$$ It is easy to see that the inclusion “$\supset$” holds. To prove the opposite one consider $x \in S$. Then we can find $R' > 0$ such that for every $\varepsilon > 0$ there is a $z' \in U(x,\varepsilon)$ and $x^* \in X^*\setminus \{0\}$ such that $T(x,R,z',x^*,\alpha) \cap A = \emptyset$. Fix $R \in (0,R') \cap \qe$ and take any $\varepsilon \in \qe_+$. Then we find $z' \in U(x,\varepsilon)$ and $x^* \in X^* \setminus \{0\}$ such that $T(x,R,z',x^*,\alpha) \cap A = \emptyset$. For $z \in \bigl(z' + C(x^*,\alpha)\bigr) \cap U(x,\varepsilon)$ we have $\d(T(x,R,z,x^*,\alpha),X \setminus U(x,R')) > 0$ and by Lemma \[L:dist-cone\] we have that $$\d\bigl(T(x,R,z,x^*,\alpha),X \setminus (z'+C(x^*,\alpha))\bigr) > 0.$$ Since $$A \subset \bigl(X \setminus U(x,R')\bigr) \cup \bigl(X \setminus (z' + C(x^*,\alpha)\bigr),$$ we get $x \in G(R,\varepsilon,x^*)$ and the equality is proved.
Now it is sufficient to prove that the set $G(R,\varepsilon,x^*)$ is open. To this end fix $R > 0$, $\varepsilon > 0$, $x^* \in X^* \setminus \{0\}$ and consider $x \in G(R,\varepsilon,x^*)$. There exists $z \in U(x,\varepsilon)$ with $\d(T(x,R,z,x^*,\alpha),A) > 0$. Denote $\eta = \d(T(x,R,z,x^*,\alpha),A)$. We have $\eta > 0$. For $x' \in U(x,\frac12 \eta)$ we have $$T(x',R,z+x'-x,x^*,\alpha) = (x'-x) + T(x,R,z,x^*,\alpha).$$ This gives $\d(T(x',R,z+x'-x,x^*,\alpha),A) \geq \frac12 \eta > 0$. Since $z+x'-x \in U(x',\varepsilon)$ we have $x' \in G(R,\varepsilon,x^*)$. This implies $U(x,\frac12 \eta) \subset G(R,\varepsilon,x^*)$ and we are done.
\[C:cone-por-suslin\] Let $X$ be a Banach space, $\alpha\in [0,1)$, and $A \subset X$ be a Suslin set. Then the set $N_{\P_X^{\alpha\text{-}cone}}(A)$ is Suslin.
\[T:cone\] For any suitable elementary submodel $M$ the following holds. Let $X$ be an Asplund space, $A \subset X$ be Suslin, and $\alpha\in [0,1)$. Then whenever $M$ contains $X$, $A$, and $\alpha$, the following are true: $$\begin{aligned}
A \text{ is $\alpha$-cone porous in the space } X &\iff A \cap X_M \text{ is $\alpha$-cone porous in the space } X_M, \\
A \text{ is $\sigma$-$\alpha$-cone porous in the space } X &\iff A \cap X_M \text{ is $\sigma$-$\alpha$-cone porous in the space } X_M, \\
A \text{ is cone small in the space } X &\iff A \cap X_M \text{ is cone small in the space } X_M.\end{aligned}$$
Let us fix a $(*)$-elementary submodel $M$ containing $X$, $A$, and $\alpha$. Then the following formula is clearly true $$\tag{$*$}
\exists {\bf R} \text{ point-set relation on } X \; \forall x \in X \; \forall B \subset X \colon
(B \text{ is $\alpha$-cone porous at } x \iff (x,B)\in {\bf R}).$$ The absoluteness of this formula and its subformula implies that ${\P_X^{\alpha\text{-}cone}} \in M$. The first two parts of the theorem now follow using Propositions \[P:reduction-porosity-like\] and \[P:point-cone\] and Corollary \[C:cone-por-suslin\]. To prove the last part one just needs to observe that any set in any Banach space is cone small if and only if it is $\sigma$-$\beta$-cone porous for each $\beta \in (0,1) \cap \qe$.
\[C:cone-small\] Let $X$ be an Asplund space and $A \subset X$ be a Suslin set. Then for every separable space $V_0\subset X$ there exists a closed separable space $V \subset X$ such that $V_0 \subset V$ and $$A \text{ is cone small in } X \iff A \cap V \text{ is cone small in } V.$$
Applications
============
Let $X$ be a real Banach space, $G \subset X$ be open. A function $f\colon G \to \er$ is called *approximately convex at $x_0 \in G$* if for every $\varepsilon >0$ there exists $\delta>0$ such that $$f(\lambda x + (1-\lambda)y)\leq \lambda f(x) + (1-\lambda) f(y) + \varepsilon \lambda(1-\lambda) \| x-y \|$$ whenever $\lambda \in [0,1]$ and $x,y \in U(x_0,\delta)$. We say $f$ is *approximately convex on $G$* if it is approximately convex at each $x_0 \in G$.
The class of approximately convex functions includes semiconvex functions and strongly paraconvex functions (for definitions see, e.g., [@Zajicek-5]).
We shall apply our result about cone small sets to prove the following generalization of [@Zajicek-5 Theorem 5.5] to nonseparable Asplund spaces. Note that the following theorem is also a strengthening of [@Zajicek-5 Theorem 5.9] which states that a continuous approximately convex function on an Asplund space is Fréchet differentiable except for points from a union of a cone small set and a $\sigma$-cone supported set. Note also that unlike [@Zajicek-5 Theorem 5.5], our Theorem \[T:appl-approx-convexity\] states that the exceptional set is cone small and not angle small. However, these two notions are equivalent if $X$ is separable.
\[T:appl-approx-convexity\] Let $X$ be an Asplund space and $G \subset X$ be open. Let $f\colon G \to \er$ be a continuous and approximately convex function. Then the set $N_F(f)$ of all points of $G$ at which $f$ is not Fréchet differentiable is cone small.
To prove the theorem we will need several notions and a lemma. The notion of LAN mapping is defined and studied in [@Zajicek-2].
Let $X$ be a Banach space and $G \subset X$ be open. We say a (singlevalued) mapping $g\colon G \to X^*$ is *LAN* (*locally almost nonincreasing*) if for any $a \in G$ and $\varepsilon > 0$ there exists $\delta > 0$ such that for any $x_1, x_2 \in U(a,\delta)$ we have $$\langle g(x_1)-g(x_2), x_1-x_2 \rangle \leq \varepsilon \| x_1 - x_2 \|.$$
We say a multivalued mapping $T \colon G \to X^*$ is *submonotone on $G$* if for any $a\in G$ and $\varepsilon > 0$ there exists $\delta > 0$ such that for any $x_1,x_2 \in U(a,\delta)$, $x_1^* \in T(x_1)$, and $x_2^* \in T(x_2)$ we have $$\langle x_1^*-x_2^*, x_1-x_2 \rangle \geq - \varepsilon \| x_1 - x_2 \|.$$
\[R:LAN-monot\] Clearly $T$ is LAN if and only if $-T$ is singlevalued and submonotone.
The following lemma generalizes [@Zajicek-2 Lemma 3] to general Asplund spaces.
\[L:red-monot\] Let $X$ be Asplund, $G \subset X$ be open and $g\colon G \to X^*$ be LAN. Then $g$ is continuous at all points of $G$ except those which belong to a cone small set.
Denote by $A$ the set of all points of $G$ at which $g$ is not continuous (then $A$ is Borel) and let us fix a $(*)$-elementary submodel $M$ containing $X$ and $g$. Then $X_M$ is a Banach space with separable dual and $g\rest_{X_M}$ is clearly LAN. Denote by $B$ the set of all points of $G\cap X_M$ (the intersection is nonempty) at which $g\rest_{X_M}$ is not continuous. By [@Zajicek-2 Lemma 3], $B$ is angle small in $X_M$. But [@Cuth Theorem 5.1] gives that $B = A \cap X_M$ and that $A\in M$. Hence, by Theorem \[T:cone\], $A$ is cone small.
Let $X$ be a Banach space, $G\subset X$ be open, and $f \colon G \to \er$. The Fréchet subdifferential of $f$ at $a$ is defined by $$\partial^F f(a) = \Bigl\{ x^*\in X^*{;\ }\liminf_{h\to 0} \frac{f(a+h)-f(a)-\langle x^*, h \rangle}{\| h \|}\geq 0\Bigr\}.$$
By [@Zajicek-5 Lemma 2.5 (ii) and (iii)] the multivalued mapping $x\mapsto \partial^F f(x)$ is submonotone on $G$. Choose any selection $g$ of $\partial^F f$ on $G$; then $g$ is also submonotone. Lemma \[L:red-monot\] (together with Remark \[R:LAN-monot\]) implies that $g$ is continuous on $G$ up to a cone small set. Now, [@Zajicek-5 Lemma 5.4] says that $f$ is Fréchet differentiable at points of continuity of $g$, concluding the proof.
Another possible application of Theorem \[T:cone\] is the following strengthening of [@Vesely Proposition 4.2] (for definitions see [@Vesely]).
Let $Y$ be a countably Daniell ordered Banach space with the Radon-Nikodým property. Assume that
- either $X$ is a closed subspace of $c_0(\Delta)$, where $\Delta$ is an uncountable set,
- or $X=C(K)$, where $K$ is scattered compact topological space.
Let $A \subset X$ be an open convex set and $f \colon A \to Y$ be a continuous convex operator. Then $f$ is Fréchet differentiable on $A$ except for a cone small $\Gamma$-null set.
The only difference from the original assertion is that, instead of $\sigma$-lower porous, we have the exceptional set cone small which is a stronger assertion. We shall, however, omit the proof, as there is no difference from the proof in [@Vesely]; one just needs to use our Theorem \[T:cone\] instead of [@Cuth-Rmoutil Theorem 5.4] which is an analogue of \[T:cone\] for $\sigma$-lower porosity.
Note that we also obtain an analogue of [@Vesely Proposition CR] for cone smallness. Since this could be of some independent interest, it is, perhaps, worth stating (see also [@Cuth-Rmoutil Theorem 1.2]).
Let $X, Y$ be Banach spaces, $G \subset X$ be an open set, and $f \colon G \to Y$ an arbitrary mapping. Then for every separable space $V_0 \subset X$ there exists a closed separable space $V \subset X$ such that $V_0 \subset V$ and that the following are equivalent:
- the set of all points where $f$ is not Fréchet differentiable is cone small in $X$,
- the set of all points where $f\rest _{V \cap G}$ is not Fréchet differentiable is cone small in $V$.
[10]{}
Marek C[ú]{}th. Separable reduction theorems by the method of elementary submodels. , 219(3):191–222, 2012.
Marek C[ú]{}th and Martin Rmoutil. -porosity is separably determined. , 63(138)(1):219–234, 2013.
Mari[á]{}n Fabi[á]{}n and Gilles Godefroy. The dual of every [A]{}splund space admits a projectional resolution of the identity. , 91(2):141–151, 1988.
Kenneth Kunen. , volume 102 of [*Studies in Logic and the Foundations of Mathematics*]{}. North-Holland Publishing Co., Amsterdam, 1980. An introduction to independence proofs.
Joram Lindenstrauss, David Preiss, and Jaroslav Ti[š]{}er. , volume 179 of [*Annals of Mathematics Studies*]{}. Princeton University Press, Princeton, NJ, 2012.
Huynh Van Ngai, Dinh The Luc, and Michel Th[é]{}ra. Approximate convex functions. , 1(2):155–176, 2000.
Martin Rmoutil. Products of non-[$\sigma$]{}-lower porous sets. , 63(138)(1):205–217, 2013.
Libor Vesel[ý]{} and Lud[ě]{}k Zaj[í]{}[č]{}ek. On differentiability of convex operators. , 402(1):12–22, 2013.
Lud[ě]{}k Zaj[í]{}[č]{}ek. Sets of [$\sigma $]{}-porosity and sets of [$\sigma $]{}-porosity [$(q)$]{}. , 101(4):350–359, 1976.
Lud[ě]{}k Zaj[í]{}[č]{}ek. On the [F]{}réchet differentiability of distance functions. In [*Proceedings of the 12th winter school on abstract analysis ([S]{}rní, 1984)*]{}, number Suppl. 5, pages 161–165, 1984.
Lud[ě]{}k Zaj[í]{}[č]{}ek. Porosity and [$\sigma$]{}-porosity. , 13(2):314–350, 1987/88.
Lud[ě]{}k Zaj[í]{}[č]{}ek. On [$\sigma$]{}-porous sets in abstract spaces. , (5):509–534, 2005.
Lud[ě]{}k Zaj[í]{}[č]{}ek. Differentiability of approximately convex, semiconcave and strongly paraconvex functions. , 15(1):1–15, 2008.
Miroslav Zelen[ý]{} and Jan Pelant. The structure of the [$\sigma$]{}-ideal of [$\sigma$]{}-porous sets. , 45(1):37–72, 2004.
[^1]: M.Cúth was supported by the Grant No. 282511/B-MAT/MFF of Grant Agency of Charles University in Prague. M.Rmoutil was supported by the Grant No. 710812/B-MAT/MFF of Grant Agency of Charles University in Prague. M.Zelený was supported by the grant P201/12/0436.
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---
abstract:
- |
Based on the three parametric Lorenz system, a model was developed that permits to describe the behavior of the plasma-condensate system near phase equilibrium in a self-consistent way. Considering the effect of fluctuations of the growth surface temperature, the evolution equation and the corresponding Fokker-Planck equation were obtained. The phase diagram is built which determines the system parameters corresponding to the regime of the porous structure formation.
self-organization, supersaturation, phase equilibrium, condensation, phase diagram
64.70.fm, 68.43.Hn, 05.10.Gg
- '=3000На основі трипараметричної системи Лоренца була розвинена модель, що дозволяє самоузгодженим чином описати поведінку системи плазма-конденсат поблизу фазової рівноваги. Враховуючи вплив флуктуацій температури ростової поверхні, були знайдені рівняння еволюції та відповідне рівняння Фоккера-Планка. Побудована фазова діаграма, на основі якої визначені параметри системи, які відповідають режимові утворення пористих структур. самоорганізація, пересичення, фазова рівновага, конденсація, фазова діаграма'
address:
- ' Sumy State University, 2 Rimskii-Korsakov St., 40007 Sumy, Ukraine'
- 'Сумський державний університет, вул. Римського-Корсакова, 2, 40007 Суми, Україна'
author:
- 'O.V. Yushchenko, T.I. Zhylenko'
- 'О.В. Ющенко, Т.І. Жиленко'
date: 'Received October 16, 2012, in final form December 5, 2012'
title: Дослідження нанопористих матеріалів за умов квазирівноваги
---
\[sec:level1\]Introduction
==========================
Nowadays, modern nanotechnologies are developed by using a variety of methods, one of which is the condensation process in the steady state close to phase equilibrium. This method makes it possible to obtain various structures of a condensate, fractal surfaces, porous structures, etc. [@POKK-3; @FTT]. The principal feature of this condensation process is the plasma-condensate system being close to phase equilibrium. Consequently, the adsorbed atoms are arranged on the active centers of crystallization forming structures with different architectures. In quasi-equilibrium conditions for continuous copper condensation of about 7 hours, the formation of highly porous structures, whiskers, and some intermediate structures (fibrous structures with alternating crystalline and porous parts) was observed. Of particular interest is the structure shown in figure \[copper\], which unlike ordinary single crystals, is realized under unstable temperature regime.
[i]{}[0.49]{}
{width="48.00000%"}
These structures may be of great practical interest. For example, they can be used as a molecular sieve.
However, a question arises regarding the reasons of holding the plasma-condensate system near phase equilibrium. Taking into account a universal nature of the condensation process, we assumed earlier [@FTT; @MNT; @UFZh; @OYBZKP_18_3] that it is caused by self-organization of a multi-phase plasma-condensate system. From the physical point of view, the mentioned self-organization is explained by an increase of the energy of the adsorbed atoms which results in the temperature increase of the growth surface under the effect of the plasma within the condensation process. On the other hand, an increase of the growth surface temperature is compensated by the desorption flow of the adsorbed atoms, which are responsible for supersaturation. As a result, within the framework of synergetic ideology [@Haken], our consideration is based on the three-parameter Lorenz system [@Ol_13; @Ol_14].
The paper is organized as follows. In section \[sec:level2\], the self-organized system that forms the basis of our consideration is described. Section \[sec:level3\] discusses the statistical analysis of the motion equation. The stationary solution of the Fokker-Planck equation is considered in section \[sec:level4\]. General conclusions are presented in section \[sec:level5\].
\[sec:level2\]Basic equations
=============================
Considering that a quasi-equilibrium condensation is provided on the growth surface due to a self-consistent development of the processes in the plasma volume, we will further use a three-dimensional (volume) concentration of the condensate $N$ and a two-dimensional (surface) concentration $n\equiv Na$. Here $a$ is a scale factor, which plays the role of the lattice parameter and its value will be determined below.
For a given value of the equilibrium concentration $n_{\mathrm{e}}$, the increasing supersaturation $n-n_{\mathrm{e}}$ is provided by the diffusion component defined by the Onsager relation [@Lifshic_15; @Landau_16] for the adsorption flow $$J_{\mathrm{ad}}\equiv D|\pmb{\nabla} N|\simeq \frac{D}{\lambda}(N_{\mathrm{ac}}-N).\label{1}$$ It takes into account that the main decrease in the concentration value takes place near the cathode layer, whose thickness is determined by the screening length $\lambda$. The latter and the diffusion coefficient $D$ are given by the equations [@Lifshic_15; @Landau_16] $$\begin{aligned}
\lambda^{2}=\frac{\varepsilon T_{\mathrm{p}}}{4\pi e^{2} N_{\mathrm{i}}}\,, \qquad
D=\frac{\sigma T_{\mathrm{p}}}{e^{2}N_{\mathrm{i}}}\,, \label{2}\end{aligned}$$ where $\varepsilon$, $\sigma$ are the dielectric permittivity and the conductivity of the plasma, respectively, $T_{\mathrm{p}}$ is its temperature measured in the energy units; $e$, $N_{\mathrm{i}}$ are the charge and the total concentration of the ions of the deposited substance and the inert gas.
In the second relation (\[1\]), it is considered that at the upper boundary layer of the cathode the volume concentration of the deposited atoms is reduced to the accumulated value $N_{\mathrm{ac}}$, and the lower boundary of this layer presents the growth surface, near which the concentration of atoms is $N$.
A decrease of supersaturation $n-n_{\mathrm{e}}$ is ensured by the desorption flow $\textbf{J}$, which is directed up from the growth surface, so that $J<0$, while the value of the adsorption flow $J_{\mathrm{ad}}>0$. In case there is no condensate (when all the adsorbed atoms have evaporated from the substrate), the condition $J=-J_{\mathrm{ad}}$ is performed for the desorption component. Here, the accumulated flow $J_{\mathrm{ad}}$ is defined by the equation (\[1\]), where $N=N_{\mathrm{e}}$. The diffusion changes of the concentration $N$ of the deposited atoms are presented by the continuity equation $\dot{n}/a+{\pmb{\nabla}}{\textbf{J}}_{\mathrm{ad}}=0$. Here, the point over $n$ denotes the differentiation with respect to time and the source effect is given by the estimate $$|{\pmb{\nabla}} \textbf{J}_{\mathrm{ad}}|\simeq \frac{J_{\mathrm{ad}}}{\lambda} \simeq
\frac{(D/\lambda^{2})(n-n_{\mathrm{e}})}{a}\,.\label{3}$$ Thus, the diffusion dissipation of concentration is expressed by the equation $\dot{n}\simeq(D/\lambda^{2})(n-n_{\mathrm{e}})/a$.
On the other hand, the velocity of desorption of atoms $\int_{v}\dot{N}\rd v$ in volume $v$, based on the growth surface $s$, is as follows: $$\int_{\nu} \dot{N}\rd v=-\int_{\nu} (\pmb{\nabla} \textbf{J})\rd v =
-\int_{\bar{S}} \textbf{J} \rd \textbf{s},\label{4}$$ where the first equation takes into account the continuity condition, while the second equation considers the Gauss theorem. As a result, the total change of concentration $n=n(t)$ near the growth surface is described by the equation $$\dot{n}=\frac{n_{\mathrm{e}}-n}{\tau_{n}} -J.\label{5}$$ At the same time, the characteristic relaxation time of the supersaturation is determined by the equalities $$\tau_{n} \equiv \frac{\lambda^{2}}{D}=\frac{\varepsilon}{4\pi\sigma}\,,\label{6}$$ the second equality being in agreement with the second relation (\[2\]).
Within the framework of the synergetic picture [@Ol_14], the quasi-equilibrium condensation process is caused by the fact, that along with an increase of the supersaturation $n-n_{\mathrm{e}}$, the condensed atoms transfer the excess of their energy to the growth surface. As a result, its temperature $T$ (measured from the ambient temperature) increases as well. This enhances the evaporation of the deposited atoms due to an increase of the absolute value of the desorption flow $J<0$, which compensates the initial supersaturation.
Thus, an appropriate representation of the sequential picture of quasi-equilibrium condensation process requires a self-consistent description of the time dependence of the concentration $n(t)$ of adsorbed atoms, the growth surface temperature $T(t)$ and the desorption flow $J(t)$. According to [@Ol_14], the evolution equations of these values contain dissipative components and the terms presenting positive and negative feedbacks, the balance of which provides a self-organization process. Thereby, in the equation (\[5\]), the first term on the right hand side represents the dissipation contribution, and the second term presents a linear relation between the rate of the concentration changes and the desorption flow.
The evolution equation for the temperature of the growth surface is presented in a similar way $$\tau_{\mathrm{T}}\dot{T}=-T-a_{\mathrm{T}}nJ+\zeta(t), \label{7}$$ where $\tau_{\mathrm{T}}$ is a corresponding relaxation time, $a_{\mathrm{T}}>0$ is the coupling constant. In contrast to the equation (\[5\]), it is assumed that dissipation leads to the relaxation of the growth surface temperature to the value $T=0$[^1]. The second term represents the nonlinear relationship of $\dot T$ with concentration and flow. Since the structure shown in figure [\[copper\]]{}, was obtained at an unstable temperature regime, the third term on the right hand side of (\[7\]) is a stochastic source of temperature changes representing the Ornstein-Uhlenbeck process[^2]: $$\langle\zeta(t)\rangle=0, \qquad
\langle\zeta(t)\zeta(t')\rangle=\frac{I}{\tau_{\zeta}}
\exp\left\{-\frac{|t-t'|}{\tau_{\zeta}}\right\}.
\label{8}$$ Here, $I$ is the intensity of temperature fluctuations, $\tau_{\zeta}$ is the time of their correlation.
To ensure self-organization, it is required to compensate the negative relationship in the expression (\[7\]) by a positive component in the evolution equation of the flow: $$\tau_{J}\dot{J}=-(J_{\mathrm{ac}}+J)+a_{J}nT, \label{80}$$ where $\tau_{J}$ is a corresponding relaxation time, $J_{\mathrm{ac}}$ is the accumulation flow, $a_{J}>0$ is a constant of a positive feedback, allowing the growth of the $\dot J$ due to the mutual effect of the concentration of the adsorbed atoms and the growth surface temperature.
Thus, equations (\[5\]), (\[7\]), (\[80\]) present a synergetic system, where the supersaturation $n-n_{\mathrm{e}}$ is reduced to the order parameter, the temperature $T$ of the growth surface – to the conjugate field, and the desorption flow $J$ – to the control parameter [@Ol_14]. As a result, the task is to investigate the possible stationary regimes in a stochastic plasma-condensate system, in particular, to consider the regime of the formation of porous structures.
The most simple investigation of the system (\[5\]), (\[7\]), (\[80\]) is possible within a dimensionless form using the characteristic scales for the time $t$, the concentration $n$, the temperature of the growth surface $T$, the flow $J$, and for the intensity of the temperature fluctuations $I$: $$t_{\mathrm{s}}\equiv \tau_{n},\qquad n_{\mathrm{s}}\equiv a^{-2}, \qquad T_{\mathrm{s}}\equiv
\varepsilon, \qquad J_{\mathrm{s}}\equiv \tau^{-1}_{n}a^{-2},\qquad I_{\mathrm{s}}\equiv
\tau^{-1}_{n}a_J^{-2}, \label{9}$$ where the above-mentioned length $a=(a_{\mathrm{T}}a_{J})^{1/4}$ and energy $\varepsilon=(\tau_{n}a_{J})^{-1}$ were used.
Thus, the dimensionless system of equations describing the fluctuational transition in a plasma-condensate system takes the form $$\begin{aligned}
\dot{n}&=&-(n-n_{\mathrm{e}})-J, \nonumber\\
\epsilon\dot{T}&=&-T-n J+\zeta(t), \nonumber\\
\sigma\dot{J}&=&-(J_{\mathrm{ac}}+J)+n T, \label{10}\end{aligned}$$ where we introduced the relations for the relaxation times $$\epsilon=\frac{\tau_{\mathrm{T}}}{\tau_{n}}\,,\qquad
\sigma=\frac{\tau_{J}}{\tau_{n}}\,.\label{11}$$
\[sec:level3\] Statistical analysis
====================================
While this system has no analytical solution, we will use the approximation $\tau_{n}\simeq \tau_{J}\gg \tau_{\mathrm{T}}$, which means that the temperature varies most rapidly. This situation is realized in the experiment very rarely, but the structure, presented in figure \[copper\], is obtained exactly under unstable temperature regime (unstable cooling).
Then, on the left hand side of the equation (\[80\]), we can assume $\epsilon\dot{T}\simeq 0$, and the conjugate field is expressed by the equation $T=-nJ+\zeta(t)$.
After some simple mathematical operations [@Ol_14; @OK-18_1; @Kharch] the system (\[10\]) reduces to the evolution equation having a canonical form of the nonlinear stochastic Van der Pol oscillator [@sinchroniz] $$\sigma \ddot{n}+\gamma(n)\dot{n}=f(n)+g(n)\zeta(t).\label{14_1}$$ Here, the friction coefficient $\gamma(n)$, the force $f(n)$ and the noise amplitude $g(n)$ are presented by the equations $$\begin{aligned}
\gamma(n)&=&1+\sigma+n^2,\nonumber\\
f(n)&=& J_{\mathrm{ac}}-(n-n_{\mathrm{e}})(1+n^2),\nonumber\\
g(n)&=& n.\label{gfg}\end{aligned}$$
Then, the task is to find a distribution function of the system in the phase space formed by the concentration $n$ and the rate of its change $p=\sigma \dot n$ depending on time $t$.
To this end, the Euler equation (\[14\_1\]) is conveniently represented by the Hamilton formalism $$\begin{aligned}
\dot n&=&\sigma^{-1}p,\nonumber\\
\dot p&=&-\sigma^{-1}\gamma(n)p+f(n)+g(n)\zeta(t).\label{Ham}\end{aligned}$$ Thus, the above-mentioned probability density $P(n,p,t)$ is reduced to the distribution function $\rho(n,p,t)$ for the solutions of the system (\[Ham\]): $$P(n,p,t)=\langle\rho(n,p,t)\rangle_{\zeta}\,,\label{49}$$ where $\langle\ldots\rangle_{\zeta}$ means the averaging over noise $\zeta$.
We will proceed from the continuity equation $$\frac{\partial }{\partial
t}\rho(n,p,t)+\left\{\frac{\partial}{\partial n}\left[\dot n
\rho(n,p,t)\right]+\frac{\partial}{\partial p}\left[\dot p
\rho(n,p,t)\right] \right\}=0.\label{50}$$
Further, the substitution of the equalities (\[Ham\]) leads to the Liouville equation $$\left[\frac{\partial}{\partial{t}}+\hat{\mathcal{L}}(n,p)\right]\rho{(n,p,t)}=
-g(n)\zeta(t)\frac{\partial}{\partial p}\rho{(n,p,t)},\label{51}$$ where the operator $$\hat{\mathcal{L}}(n,p)=\frac{p}{\sigma}\frac{\partial}{\partial n}+
\frac{\partial}{\partial
p}\left[f(n)-\frac{\gamma(n)}{\sigma}\right].\label{52}$$
Turning to the interaction representation [@Shapiro] $$\varrho(n,p,t)=\re^{\hat{\mathcal{L}}(n,p)t}\rho(n,p,t),\label{53}$$ the equation (\[51\]) takes the form $$\frac{\partial \varrho(n,p,t)}{\partial
t}=-\re^{\hat{\mathcal{L}}(n,p)t}g(n)\zeta(t)\frac{\partial}{\partial
p}\re^{-\hat{\mathcal{L}}(n,p)t}
\varrho(n,p,t)\equiv\varepsilon\mathcal{R}(n,p,t)\varrho(n,p,t),\label{54}$$ where $\varepsilon$ is a dimensionless small parameter [@Shapiro]. Then, using the cumulant expansion method [@VanK], one can obtain the kinetic equation[^3] $$\frac{\partial \varrho(n,p,t)}{\partial t}=\varepsilon^2
\int_0^t\left\langle\mathcal{R}(n,p,t)\mathcal{R}(n,p,t')
\right\rangle\langle\varrho(n,p,t')\rangle
\rd t',\label{55}$$ neglecting the terms of $\varepsilon^3$ order [@Ol_14].
Since a physical time $t$ is usually much longer than the noise correlation time $\tau_\zeta$, the upper limit of the integration can be set equal to infinity. Then, returning from the interaction presentation to the original presentation, for the distribution function (\[49\]) we obtain $$\left[\frac{\partial}{\partial{t}}+\hat{\mathcal{L}}(n,p)\right]P(n,p,t)=
\varepsilon^{-2}\hat{\mathcal{N}}P(n,p,t). \label{56}$$ Here, $\hat{\mathcal N}$ is a scattering operator, which is given by the expression $$\hat{\mathcal N}= \left[M_0(t)-\gamma(n)M_1(t)\right]g^2(n)
\frac{\partial^2}{\partial p^2}+\varepsilon M_1(t)g^2(n)
\left[-\frac{1}{g(n)}\frac{\partial g(n)}{\partial
n}\left(\frac{\partial}{\partial p}+p \frac{\partial^2}{\partial
p^2} \right)+\frac{\partial^2}{\partial n\partial p}
\right]+\mathcal{O}(\varepsilon^2),\label{57}$$ where $M_0(t)$ and $M_1(t)$ are the moments of the correlation function (\[8\]) $$M_i(t)=\frac{1}{i!}\int^{\infty}_{0}t^{i}\langle\zeta(t)\zeta(0)\rangle
\rd t.\label{17}$$ From equation (\[17\]) one can obtain $$M_0(t)=I, \qquad M_1(t)= I\tau_\zeta\,. \label{17_1}$$
Since, for this task, a complete distribution function $P(n,p,t)$ has a lower practical interest than its integral $$\mathcal P(n,t)=\int P(n,p,t)\rd p,\label{19_1}$$ it makes sense to consider the moments of the initial distribution function $$\mathcal P_i(n,t)=\int p^i P(n,p,t)\rd p.\label{19}$$ Then, the zero moment is reduced to the required integral (\[19\_1\]).
Multiplying equation (\[56\]) by $p^i$ and integrating over all $p$, we arrive at the relation that can be written as a Fokker-Planck equation presented in the Kramers-Moyal form [@Risken] $$\frac{\partial \mathcal
P(n,t)}{\partial{t}}=-\frac{\partial}{\partial{n}}\left[D_1(n)\mathcal
P(n,t)\right]+ \frac{\partial^{2}}{\partial n^2}\left[D_2(n)\mathcal
P(n,t)\right],\label{20}$$ where the drift coefficient $$D_1(n)=\frac{1}{\gamma(n)}\left[f(n)-M_0(t)\frac{g^2(n)}{\gamma^{2}(n)}\frac{\partial
\gamma(n)}{\partial n}+M_1(t)g(n)\frac{\partial g(n)}{\partial n}
\right]\label{21}$$ and the diffusion coefficient $$D_2(n)=M_0(t) \frac{g^2(n)}{\gamma^{2}(n)}\label{22}$$ are presented by the functions (\[gfg\]).
\[sec:level4\]Stationary solution
=================================
A stationary solution of the Fokker-Planck equation [@Risken] yields a stationary distribution [@Ol_14; @OK-18_1] $$\mathcal{P}(n)=\frac{Z^{-1}}{D_2(n)}{\exp{\int_0^n\frac{D_1(n')}{D_2(n')}\rd n'}},\label{42}$$ where the partition function $Z$ is presented by the equation $$Z={\int_0^\infty\frac{\rd n}{D_2(n)}\exp{\int_0^n\frac{D_1(n')}{D_2(n')}\rd n'}}.
\label{30}$$
The extremum condition for the distribution (\[42\]) $$D_1(n)-\frac{\partial }{\partial n}D_2(n)=0\label{31}$$ defines the stationary states of the plasma-condensate system.
[o]{}[0.48]{}
{width="48.00000%"}
Substituting expressions (\[21\]), (\[22\]), (\[gfg\]), and (\[17\_1\]) into the equation (\[31\]) we obtain the equation defining the stationary concentration dependence $$J_{\mathrm{ac}}=\frac{2I(1+\sigma)n}{\left[(1+\sigma)+n^{2}\right]^{2}}+(n-n_{\mathrm{e}})(1+n^{2})-I
\tau_{\zeta}n. \label{31_1}$$
Then, the condition that restricts the domain of the existence of the solution $n = 0$ corresponding to the complete evaporation of the condensate from the growth surface, has the form $$J_{\mathrm{ac}}=-n_{\mathrm{e}}\,. \label{12}$$
The corresponding phase diagram of the system is shown in figure \[diagr\].
While figure \[diagr\] (a) shows the effect of the system parameters (the equilibrium concentration $n_{\mathrm{e}}$ and the correlation time of fluctuations $\tau_\zeta$), figure \[diagr\] (b) considers in detail the domains of the phase diagram. In particular, the domain $C$ corresponds to the condensation process, the domain $S$ is characterized by the formation of porous structures, and a complete evaporation of the condensed matter takes place at the domain $O$. The domains, which are indicated by two letters, meet the coexistence of the above mentioned regimes.
It is more convenient to understand the processes occurring in each domain considering the example of the stationary concentration dependence presented in figure \[St\].
Each point on the phase diagram \[figure \[diagr\] (b)\] corresponds to the ray in figure \[St\] (a), (b).
For example, for the ray **1** \[figure \[St\] (a)\] only the condensation process is realized. It corresponds to the point $C'$ characterized by a sufficient stationary concentration $n$. With a decrease of the accumulated flow (ray **2**), there is observed a gradual disassembly of the previously formed condensate. This situation corresponds to the existence of two steady states with different concentrations (points $C$ and $S'$).
![ The dependence of the stationary concentration $n$ on the accumulated flow $J_{\mathrm{ac}}$ at $n_{\mathrm{e}}=0.25$, $\tau_\zeta=0.5$, (a) $I=8$, (b) $I=14$.[]{data-label="St"}](fig3){width="150mm"}
It should be noted that the additional intersection point of the main curve and the ray **2** (which is located between $C$ and $S'$) applies to the non-physical plot and, therefore, it is not considered[^4]. Turning to the state of the surface disassembly (point $S'$), it is worth noting that in this case the usual evaporation of the upper layer of the condensate does not take place. First, the atoms which are less connected with the crystallization centers, are detached from the condensate surface. With a further decrease of the accumulated flow (ray **3**), the only state of the surface disassembly remains (point $S$). This is the situation that characterizes the pattern shown in figure \[copper\] and is of great interest to us. Going to the ray **4**, the disassembly is replaced by the usual evaporation process (point $O$).
Analyzing the relationship (\[31\_1\]) for higher intensity of fluctuations \[figure \[St\] (b)\], one can see that some changes occur. As previously, only condensation process (point $C''$) is realized for ray **5**, while ray **6** is characterized by the coexistence of disassembly (point $S$) and condensation (point $C'$) processes. The main difference is found for the ray 7, when together with the condensation process (point $C$), evaporation (point $O'$) takes place. At the parameters specified for the ray **8**, only evaporation occurs.
\[sec:level5\]Conclusion
========================
Based on the above analysis, we can conclude that processes occurring in the plasma-condensate system can be represented within the system (\[5\]), (\[7\]) and (\[80\]) describing the self-consistent behavior of concentration, temperature of the growth surface and desorption flow. Taking into account the fluctuations of the growth surface temperature with the correlation function (\[8\]) makes it possible to describe the most specific state (disassembly of the surface), when porous nanostructures may be formed. In addition, as shown in figure \[diagr\] (a), the system parameters have a significant effect on the domain of the formation of such structures. With an increase of the correlation time of fluctuations, this domain significantly decreases and shifts towards the lower values of the fluctuation intensity, while an increase in the equilibrium concentration results in a less significant decrease, as well as causes a shift along two axis (fluctuation intensity and accumulated flow). As a real experiment [@POKK-3; @FTT], our theoretical approach has shown that the state of the surface disassembly is rarely realized. However, controlling the parameters of a system, we can reach the regime under which porous nanostructures are formed.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to professor V.I. Perekrestov for placing the experimental material at their disposal.
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[^1]: It should be noted that the condition $T=0$ does not correspond to the absolute zero since the temperature $T$ is measured from the ambient temperature.
[^2]: Investigation of the temperature fluctuation in the form of white noise was carried out in [[@FTT]]{}.
[^3]: It takes into account that the time derivatives in equations (\[51\]), (\[54\]) were treated according to the Stratonovich rule.
[^4]: This also applies to the similar cases in figure \[St\] (b).
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abstract: 'We propose a new formulation of the equivariant Tamagawa number conjecture (ETNC) for non-commutative coefficients. We remove Picard groupoids, determinant functors, virtual objects and relative $K$-groups. Our Tamagawa numbers lie in an idèle group instead of any kind of $K$-group. Our formulation is proven equivalent to the one of Burns–Flach.'
address: 'Mathematical Institute, University of Freiburg, Ernst-Zermelo-Strasse 1, 79104 Freiburg im Breisgau, Germany'
author:
- Oliver Braunling
bibliography:
- 'ollinewbib.bib'
title: An alternative construction of equivariantTamagawa numbers
---
[^1]
In this paper we give a new approach to constructing the equivariant Tamagawa numbers of Burns–Flach [@MR1884523]. We do *not* wish to claim that this method is in any way better or worse than their original one. We merely have a different kind of perspective, focussing on adèles and local compactness, and we just want to set up Tamagawa numbers in the way which appears most natural from this slightly different angle.
As in our previous paper [@etnclca] we restrict to regular orders $\mathfrak{A}\subset A$ in a semisimple algebra $A$. Given the current state of our foundations, this presently cannot be avoided. However, we will remove this assumption in a future paper and our formulation will remain intact almost verbatim.
Let us explain our approach: Suppose $A$ is a finite-dimensional semisimple $\mathbb{Q}$-algebra and $\mathfrak{A}\subset A$ a regular order, e.g. a hereditary or maximal one. Usually (following [@MR1884523]) the equivariant Tamagawa number is an element $T\Omega$ in $K_{0}(\mathfrak{A},\mathbb{R})$, a relative $K$-group, whose elements have the shape$$\lbrack P,\varphi,Q]$$ in the so-called Swan presentation. Already here, we shall proceed a little differently. The relative $K$-group sits in an exact sequence$$\cdots\longrightarrow K_{1}(\mathfrak{A})\longrightarrow K_{1}(A_{\mathbb{R}})\longrightarrow K_{0}(\mathfrak{A},\mathbb{R})\overset{\operatorname*{cl}}{\longrightarrow}\operatorname*{Cl}(\mathfrak{A})\longrightarrow0\text{,}
\label{l_Rado_1}$$ where $\operatorname*{Cl}(\mathfrak{A})$ is the locally free class group. In [@MR0376619] Fröhlich has proven a formula for the latter, namely$$\operatorname*{Cl}(\mathfrak{A})\cong\frac{J(A)}{J^{1}(A)\cdot A^{\times}\cdot
U\mathfrak{A}\cdot A_{\mathbb{R}}^{\times}}\text{,}$$ where $J(A)$ denotes the non-commutative idèles of $A$, $J^{1}(A)$ the reduced norm one idèles, and $U\mathfrak{A}$ the unit finite idèles of the order. Our first idea is to extend this formula of Fröhlich to $K_{0}(\mathfrak{A},\mathbb{R})$:
There is a canonical isomorphism$$K_{0}(\mathfrak{A},\mathbb{R})\cong\frac{J(A)}{J^{1}(A)\cdot A^{\times}\cdot
U\mathfrak{A}}\text{,} \label{l_Rado_3}$$ and under this identification the map ‘$\operatorname*{cl}$’ in Equation \[l\_Rado\_1\] amounts to quotienting out $A_{\mathbb{R}}^{\times}$ in the infinite places of $J(A)$.[^2]
So, in our picture, we will most naturally view an equivariant Tamagawa number as an element of the idèle-style group on the right. Next, from a previous article, we already know that $K_{0}(\mathfrak{A},\mathbb{R})\cong
K_{1}(\mathsf{LCA}_{\mathfrak{A}})$, where $\mathsf{LCA}_{\mathfrak{A}}$ denotes the category of locally compact topological $\mathfrak{A}$-modules. We refer to [@etnclca] for both the philosophy why this should hold (keyword: equivariant Haar measures), as well as an actual proof.
Now suppose we are given a pure motive $M$ with an action by the semisimple algebra $A$. As in Burns–Flach [@MR1884523] some further assumptions and data, like lattices $T_{v}$, are needed[^3]. It would unreasonably inflate this introduction to carefully go through the rather involved setup, so we shall assume that the reader is familiar with [@MR1884523 §2-3]. We use exactly the same notation. As in Burns–Flach, we begin with the objects$$R\Gamma_{c}(\mathcal{O}_{F,S_{p}},T_{p})\qquad\text{and}\qquad\Xi(M)$$ of a certain nature, defined over $\mathfrak{A}_{p}$ and $A$. In [@MR1884523] the former object is regarded as a bounded complex and the latter, the *fundamental line*, as an object in a certain Picard groupoid.
We drop this and shall look at them differently: We work in a special model of algebraic $K$-theory due to [@MR1167575]. It provides us with a space such that (a) bounded complexes define points in it, (b) generalized lines like $\Xi(M)$ also define points in it, and (c), quasi-isomorphisms between bounded complexes define paths between points.
Our Tamagawa numbers are now defined as follows: We prove that there is a fibration of pointed spaces$$K(\widehat{\mathfrak{A}})\times K(A)\longrightarrow K(\widehat{A})\times
K(A_{\mathbb{R}})\longrightarrow K(\mathsf{LCA}_{\mathfrak{A}})\text{.}
\label{l_Rado_2}$$ The reader should really think of this in the sense of topology. A base space on the right, the total space in the middle, and on the left the fiber over the base point. Because we use the aforementioned versatile model of $K$-theory, $R\Gamma_{c}(\mathcal{O}_{F,S_{p}},T_{p})$ and $\Xi(M)$ simply define points in the fiber, i.e. the leftmost space in Equation \[l\_Rado\_2\]. $${\includegraphics[
height=1.2142in,
width=4.7818in
]{fibration_2a-eps-converted-to.pdf}}
\label{l_Rado_4}$$ After base change from $\mathfrak{A}_{p}$ to $A_{p}$, and from $A$ to $A_{\mathbb{R}}$, there exist comparison isomorphisms $\vartheta_{p}$ resp. $\vartheta_{\infty}$ (this is exactly as in [@MR1884523 §3.4]). However, the underlying quasi-isomorphisms then just define paths between the points in the middle term, i.e. the total space $K(\widehat{A})\times
K(A_{\mathbb{R}})$. But wait: The start point and end point objects of these paths all came from the fiber, i.e. once we go all to the right to the base space of the fibration, all these points collapse to the base point. However, this means that the paths all get mapped to closed loops. So they define an element in the fundamental group $\pi_{1}K(\mathsf{LCA}_{\mathfrak{A}})=:K_{1}(\mathsf{LCA}_{\mathfrak{A}})$.
This is it. As in Burns–Flach, we call this element $R\Omega$, and after adding the term coming from the $L$-function, $L\Omega$, **we have now constructed our Tamagawa number**. Returning to Equation \[l\_Rado\_3\], we may, if we want, get rid of $K$-groups and regard this as an element of$$\frac{J(A)}{J^{1}(A)\cdot A^{\times}\cdot U\mathfrak{A}}\text{,} \label{lsol1}$$ leading to a formulation in which not a single $K$-group is present anymore. Or, at least not literally.
In the end, we have just obtained the same concept as Burns–Flach. Using $K_{0}(\mathfrak{A},\mathbb{R})\cong K_{1}(\mathsf{LCA}_{\mathfrak{A}})$ we prove the following.
\[thm2\_Intro\]Our construction of the equivariant Tamagawa number $T\Omega$ is equivalent to the one of Burns–Flach in [@MR1884523].
See §\[sect\_CompatProofSection\]. The reader might worry that the identification of our closed loops with idèles in Equation \[lsol1\] is something elusive. Not at all. The non-commutative idèles $J(A)$ act as automorphisms on the non-commutative adèles$$A_{\mathbb{A}}:=\left\{ \left. (x_{\mathfrak{p}})_{\mathfrak{p}}\in
\prod_{\mathfrak{p}}A_{\mathfrak{p}}\right\vert a_{\mathfrak{p}}\in\mathfrak{A}_{\mathfrak{p}}\text{ for all but finitely many places
}\mathfrak{p}\right\} \text{,}$$ where $\mathfrak{p}$ runs through the finite and infinite places. As we had already explained above, in our model of $K$-theory any isomorphism determines a path. Hence, any $\alpha\in J(A)$ defines a loop$${\includegraphics[
height=0.4987in,
width=0.894in
]{alphact-eps-converted-to.pdf}}
$$ and thus we get a map $\varphi:J(A)\rightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})$. When phrasing Equation \[l\_Rado\_3\] in terms of $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$ instead of $K_{0}(\mathfrak{A},\mathbb{R})$, this map $\varphi$ is the one inducing the isomorphism. This will be Theorem \[thm\_GlobalLocal\_Nenashev\].
In Figure \[l\_Rado\_4\] we had equipped $R\Gamma_{c}(\mathcal{O}_{F,S_{p}},T_{p})\otimes A_{p}$ and $\Xi(M)\otimes A_{\mathbb{R}}$ with their natural topologies, giving objects which carry $p$-adic and real topologies, just like the adèles $A_{\mathbb{A}}$. Our theorem says that the closed loops made from $\vartheta_{p}$, $\vartheta_{\infty}$ on the right in Figure \[l\_Rado\_4\] are equivalent[^4] to a closed loop coming from an automorphism of the non-commutative adèles, and in fact one coming from plainly multiplying with an idèle. This idèle (class) *is* the Tamagawa number.
$-$ *Some further results *$-\medskip$
We prove Equation \[l\_Rado\_3\] in terms of the Swan presentation of $K_{0}(\mathfrak{A},\mathbb{R})$, but we can also spell out an explicit map from the idèle quotient to the Nenashev presentation of the $K$-group $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$. Running those isomorphisms back to back, we obtain a new isomorphism$$K_{0}(\mathfrak{A},\mathbb{R})\overset{\sim}{\longrightarrow}K_{1}(\mathsf{LCA}_{\mathfrak{A}})\text{.}$$ Hence, we now have *three* such isomorphisms: the inexplicit one of the first paper [@etnclca], a rather enigmatic one with the special property to be ‘universal in Swan generators’ in [@etnclca2], as well as the one of this paper. *We do not know* whether any two of them agree. Given how reluctant the Nenashev presentation is towards Swan generators, we wish to propose the following analogy:$$\begin{aligned}
\text{ideal class group} & \leftrightarrow\text{id\`{e}le class group}\\
\text{Swan presentation} & \leftrightarrow\text{Nenashev presentation.}$$ Why the ideal class group allusion makes sense is surely clear from the map ‘$\operatorname*{cl}$’ in Equation \[l\_Rado\_1\]: It sends $[P,\varphi,Q]$ to $[P]-[Q]$. To justify the right side, consider the explicit formulas for the maps in Theorem \[thm\_GlobalLocal\_Swan\] and Theorem \[thm\_GlobalLocal\_Nenashev\]. Both essentially rely on Fröhlich’s idèle classification of projective $\mathfrak{A}$-modules, [@MR0376619].
We heartily thank B. Chow, B. Drew, A. Huber, M. Wendt for discussions and help. The delicate role of signs in the proof of Theorem \[thm\_PrincipalIdeleFibration\] only became clear after some very valuable remarks by Brad Drew. Most of this project was carried out at FRIAS and I heartily thank them for providing perfect working conditions. I still cannot imagine a better place for inspiration and creativity.
\[sect\_Overview\]Detailed Overview {#marker_OverviewTNCBegin}
===================================
We first recall some basics of $K$-theory, but rather differently from the material of many surveys. We allow ourselves some imprecisions, for pedagogical reasons, and provide rigorous details only later in §\[sect\_GettingPrecise\].
Suppose $\mathsf{C}$ is an exact category, e.g., finitely generated projective modules over a ring, for which we write $\operatorname*{PMod}(R)$, or any abelian category. In many situations people are only interested in the $K$-groups $K_{i}(\mathsf{C})$ themselves. However, being all honest, $K$-theory is a pointed space $K(\mathsf{C})$ and then the $K$-groups arise as its homotopy groups $K_{i}(\mathsf{C}):=\pi_{i}K(\mathsf{C})$. The space $K(\mathsf{C})$ is practically never the kind of space one could draw on a sheet of paper. And really, what kind of space it is, depends on our concrete approach to $K$-theory, e.g. Quillen’s $Q$-construction, Waldhausen’s $S$-construction, etc. These spaces are all distinct, but have the same homotopy type. What we describe in this section is the Gillet–Grayson model in a version due to [@MR1167575]. A textbook explanation of the basic method can be found in Weibel’s book [@MR3076731 Chapter IV, §9]. This space has a number of properties making it a lot more convenient than other spaces giving $K$-theory.
Before giving a precise description, let us just summarize the most important principles:$\left. \qquad\text{\textbf{(a)}}\right. $ Every object in $\mathsf{C}$ determines a point in the space $K(\mathsf{C})$.$\left. \qquad\text{\textbf{(b)}}\right. $ Every isomorphism $X\overset
{\sim}{\rightarrow}Y$ determines a path from the point of $X$ to $Y$ in $K(\mathsf{C})$.Let us quickly connect this with $K$-groups: The zero-th $K$-group $K_{0}(\mathsf{C})=\pi_{0}K(\mathsf{C})$ corresponds to the connected components of the space. Usually one writes $[X]$ for the $K_{0}$-class determined by an object $X\in\mathsf{C}$. And indeed, if two objects $X,Y$ are isomorphic, say $\varphi:X\overset{\sim}{\rightarrow}Y$, then there is a path between them by principle (b), so they lie in the same connected component and correspondingly $[X]=[Y]$ in $K_{0}(\mathsf{C})$.$${\includegraphics[
height=0.8294in,
width=3.1548in
]{principleA_B-eps-converted-to.pdf}}
$$ Further, if $\mathsf{C}=\operatorname*{PMod}(R)$, then it is well-known that$$K_{1}(R)=\operatorname*{GL}(R)/[\operatorname*{GL}(R),\operatorname*{GL}(R)]\text{,} \label{lww1}$$ the abelianization of $\operatorname*{GL}(R)$. And indeed, given any element $\alpha\in\operatorname*{GL}(R)$, it determines an automorphism $R^{n}\overset{\sim}{\rightarrow}R^{n}$ for some sufficiently large $n$, and by principle (b) this determines a path from the point of $R^{n}$ to itself, i.e. a closed loop, and thus an element in $\pi_{1}$. This leads us back to the fact that $K_{1}$, beyond the description in Equation \[lww1\], is also the fundamental group $\pi_{1}K(\mathsf{C})$.
At this point, the reader might wish for a more precise formulation of (a) and (b) and a concrete rigorous justification of these principles. We will do this, but only later, see §\[sect\_GettingPrecise\].
There is also an addition operation on $K(\mathsf{C})$ and a negation map:$$\left. +\right. :K(\mathsf{C})\times K(\mathsf{C})\longrightarrow
K(\mathsf{C})\qquad\text{and}\qquad\left. -\right. :K(\mathsf{C})\longrightarrow K(\mathsf{C})\text{.} \label{lww_AddSubtract}$$ If $X,Y$ are objects, the point of the direct sum $X\oplus Y$ is the sum of the points of $X$ and $Y$ under this map $+$. We write $-X$ for the negation of the point of $X$. While this exists as a point in $K(\mathsf{C})$, there is usually no object producing this point under principle (a).
Care is needed here: The above maps do *not* give the space $K(\mathsf{C})$ a group structure. The problem is that while for example $X\oplus Y$ and $Y\oplus X$ are canonically isomorphic, according to principle (b) this canonical isomorphism merely defines a path from the point of $X\oplus Y$ to the point of $Y\oplus X$, but they will usually be different points. Similarly, the associativity isomorphism $(X\oplus Y)\oplus
Z\overset{\sim}{\rightarrow}X\oplus(Y\oplus Z)$ only yields a path between the corresponding points. Once going to the homotopy groups $\pi_{i}K(\mathsf{C})$ these problems all disappear and the above two maps induce honest group structures. For example, $\pi_{0}K(\mathsf{C})$ only sees connected components, and since the above remarks mean that paths exist between these points, they lie in the same component and thus $\pi_{0}K(\mathsf{C})$ gets an honest group structure.
In general the weaker type of structure given by the maps in Equation \[lww\_AddSubtract\] is sometimes called a ‘homotopy commutative and homotopy associative $H$-space’[^5]. All these phenomena are well-understood and a big and active field of investigation. However, for our purposes, *they do not matter* and we will be fine not digging deeper into this. We merely wanted to point out that this is an issue where some caution is appropriate.
A second elaboration: For $K$-theory it does not matter whether we work with the genuine category $\mathsf{C}$ or with bounded complexes in $\mathsf{C}$. Thus, simultaneously to the above principles (a) and (b), the following two are also true:$\left. \qquad\text{\textbf{(a')}}\right. $ Every bounded complex $X_{\bullet}$ of objects in $\mathsf{C}$ determines a point $X_{\bullet}$ in the space $K(\mathsf{C})$, and also determines a point $-X_{\bullet}$ using negation.$\left. \qquad\text{\textbf{(b')}}\right. $ Every quasi-isomorphism $X_{\bullet}\overset{\sim}{\rightarrow
}Y_{\bullet}$ determines a path from the point of $X_{\bullet}$ to $Y_{\bullet}$ in $K(\mathsf{C})$.Finally, if $F:\mathsf{C}\longrightarrow\mathsf{C}^{\prime}$ is an exact functor of exact categories, then there is an induced map of spaces $K(\mathsf{C})\rightarrow
K(\mathsf{C}^{\prime})$.
In §\[sect\_GettingPrecise\] we will give a fully rigorous and precise justification for principles (a) and (b), as well as (a’) and (b’).
One may think about the principles (a), (b), (a’), (b’) as follows, in analogy: Both a single point as well as the real line $\mathbb{R}$ have the same homotopy type. However, $\mathbb{R}$ has a lot more points and a lot more paths. Similarly, when we choose whether we use Quillen’s plus construction, the $Q$-construction, Waldhausen’s $S$-construction, etc. to model the $K$-theory space, we always get the same homotopy type having the same homotopy groups and therefore same $K$-groups. However, the wealth of points or paths in these spaces varies a lot. Thus, our choice to use the Gillet–Grayson type model from [@MR1167575] is special. One may think of it as a sufficiently fattened up incarnation of the homotopy type of $K$-theory such that all our principles (a), (b), (a’), (b’) hold.[^6]
We are ready to state our construction of the Tamagawa number. Let $F$ be a number field, $S_{\infty}$ its set of infinite places, and fix a separable closure $F^{\operatorname*{sep}}$ (i.e. an algebraic closure). Let $M\in\operatorname*{CHM}(F,\mathbb{Q})$ be a Chow motive over $F$ in the category of Chow motives with $\mathbb{Q}$-coefficients, [@MR2115000], [@MR1265529]. If the reader does not like motives, we may take $M$ to be a smooth proper $F$-variety, or even something as concrete as an elliptic curve. Let $A$ be a finite-dimensional semisimple $\mathbb{Q}$-algebra and suppose $M$ carries a right action by $A$ as a Chow motive. This just means that we provide a $\mathbb{Q}$-algebra homomorphism$$A\longrightarrow\operatorname*{End}\nolimits_{\operatorname*{CHM}(F,\mathbb{Q})}(X)\text{.}\label{lww_Action}$$
If $\mathfrak{A}\subset A$ is an order in the algebra, we use the standard notation$$\widehat{\mathfrak{A}}:=\mathfrak{A}\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}}\text{,}\qquad\widehat{A}:=A\otimes_{\mathbb{Q}}\mathbb{A}_{fin}\text{,}\qquad A_{\mathbb{R}}:=A\otimes_{\mathbb{Q}}\mathbb{R}\text{,}$$$$\mathfrak{A}_{p}:=\mathfrak{A}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\qquad\text{and}\qquad A_{p}:=A\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\text{,}\qquad\qquad\text{(for any prime number }p\text{)}$$ where $\mathbb{A}_{fin}$ denotes the ring of finite adèles of the rationals (Example: $\widehat{F}$ are the finite adèles of $F$). We follow Burns and Flach and shall use the same notation, so in particular
1. for any infinite place $v$ we write $H_{v}(M):=H^{\ast}(M_{v}(\mathbb{C}),(2\pi i)^{\ast}\mathbb{Q})$ for the Betti realization $M_{v}$ along $v:F\hookrightarrow\mathbb{C}$, and for $(\ast,\ast)$ picked appropriately,
2. and for each prime $p$, we write $H_{p}(M):=H_{\acute{e}t}^{\ast
}(M\times_{F}F^{\operatorname*{sep}},\mathbb{Q}_{p}(\ast))$ for the étale realization, for $(\ast,\ast)$ picked appropriately,
3. $H_{dR}$ for the de Rham realization with suitably shifted Hodge filtration.
We will not go into details. This setting is roughly the same in all papers about the ETNC or its historical ancestors.
\[elab\_Motives1\]How to attach realizations to the motive is explained in many places. Omitting a lot of details, the story is as follows: If $X$ is a smooth proper $F$-variety, a typical choice of $M$ would be$$M:=h^{i}(X)(r)\text{.}$$ In general, the splitting of $X$ into a direct sum of cohomology pieces $h^{i}$ in the category of Chow motives is conjectural. But let us assume this direct summand exists. Then$$M=(X,q,r)\text{,}$$ where $X$ is the variety as before, $q$ an idempotent self-correspondence which has the property to cut out the direct summand $h^{i}$, $r$ a formal Tate twist parameter. One may write $M=q_{\ast}(X,\Gamma_{\operatorname*{id}:X\rightarrow X},0)(r)$, where $\Gamma_{\operatorname*{id}:X\rightarrow X}$ is the graph of the identity map. Then, just as $M$ is cut out from an idempotent inside the motive of all of $X$, for an infinite place $v$ we would define the Betti realization as the corresponding image of the idempotent in the Betti cohomology of $X$,$$H_{v}(M)=q_{\ast}H^{\ast}(X_{v}(\mathbb{C}),(2\pi i)^{r}\mathbb{Q})\text{.}$$ This story then is analogous for the other realizations. The realizations all come with extra structures (e.g. a pure $\mathbb{Q}$-Hodge structure on the Betti cohomology groups), which we also tacitly keep as data, and which need to be shifted according to $r$.
The key point is that correspondences act on all Weil cohomology theories and thus one can define these cohomology groups for all Chow motives, [@MR2115000]. The same is possible for mixed motives, albeit technically much harder [@MR1775312; @MR2008720].
\[[[@MR1884523 §3.3, Definition 1]]{}\]For every infinite place $v\in
S_{\infty}$ pick a choice of $T_{v}$ of a projective $\mathfrak{A}$-lattice in the Betti realization $H_{v}(M)$ (which is a right $A$-module by Equation \[lww\_Action\]).
1. Let $p$ be any prime. Suppose for all $v\in S_{\infty}$ the Betti-to-étale comparison isomorphism$$H_{v}(M)\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\overset{\sim}{\longrightarrow}H_{p}(M)$$ sends the $\mathfrak{A}_{p}$-submodule $T_{v}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$ to the same image $T_{p}$ on the right-hand side.
2. Suppose further that the $T_{p}$ of (1) is stable under the Galois action $G_{F}$ on the right side.
Any such choice $(T_{v})_{v\in S_{\infty}}$ is called a *projective* $\mathfrak{A}$*-structure* on the motive $M$.
As explained in loc. cit., a projective $\mathfrak{A}$-structure need not exist in general. We write $\mathsf{LCA}_{\mathfrak{A}}$ for the exact category of locally compact topological right $\mathfrak{A}$-modules, as introduced in [@etnclca].
In this paper, we shall establish that the following is a fibration:
Suppose $\mathfrak{A}\subset A$ is a regular order. Then there is a canonical fibration of pointed spaces$$K(\widehat{\mathfrak{A}})\times K(A)\longrightarrow K(\widehat{A})\times
K(A_{\mathbb{R}})\longrightarrow K(\mathsf{LCA}_{\mathfrak{A}})\text{,}
\label{lcixi1}$$ which we call the *principal idèle fibration*.
This will be Theorem \[thm\_PrincipalIdeleFibration\]. We will shortly see how the spaces in this sequence relate to the $p$-adic, de Rham and Betti realization. The word fibration is meant in the sense of topology (but we shall not need to know anything technical about it right now): The zero object $0$ of any of the involved categories defines, by principle (a), a point in the $K$-theory spaces. We use this canonical point as the base point for each of the involved spaces. The statement means that the space $K(\widehat{A})\times K(A_{\mathbb{R}})$ is fibered over the base space $K(\mathsf{LCA}_{\mathfrak{A}})$, and each fiber looks like (that is: has the homotopy type of) $K(\widehat{\mathfrak{A}})\times K(A)$. For example, the Möbius band is fibered over the circle,$${\includegraphics[
height=0.8441in,
width=2.1958in
]{fibration-eps-converted-to.pdf}}
\label{l_Mobius}$$ and the above three constituents, fiber (left), total space (middle), and base space (right) correspond to the three terms in Equation \[lcixi1\]. Of course, the spaces in Equation \[lcixi1\] are a lot more complicated and it would be impossible to draw a picture.
Actually, our construction of the Tamagawa number does not even need the full strength of the above theorem. We shall only use that the composition of both maps is zero. More geometrically: Once the fiber is mapped all to the right in Equation \[lcixi1\], there exists a homotopy contracting this image to the base point.
Using exactly the same notation as in Burns–Flach [@MR1386106], [@MR1884523] for the individual groups, we define $$\begin{aligned}
\Xi(M) & :=H_{f}^{0}(F,M)-H_{f}^{1}(F,M)+H_{f}^{1}(F,M^{\ast}(1))^{\ast
}-H_{f}^{0}(F,M^{\ast}(1))^{\ast}\label{ltiops1}\\
& -\sum_{v\in S_{\infty}}H_{v}(M)^{G_{v}}+\sum_{v\in S_{\infty}}\left(
H_{dR}(M)/F^{0}\right) \nonumber\end{aligned}$$ as a point in $K(A)$. To clarify: (a) The meaning of $+$ and $-$ is as in Equation \[lww\_AddSubtract\] and unravelled from left to right.
Exactly as in Burns–Flach, we write $H^{i}(F,M)$ for (what is usually shortened to be called) the motivic cohomology of $M$ with $\mathbb{Q}$-coefficients. See [@MR1884523 §3.1]. This notation is chosen to be suggestive for$$H^{i}(F,M)=\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{i}(\mathbb{Q},M)\text{,}\label{l_lemo2}$$ which is how these motivic cohomology groups can be defined in terms of the category of mixed motives over $F$ (e.g. using Voevodsky’s $DM$) and using the natural contravariant functor sending a Chow motive into mixed motives. Analogously, we write $H_{f}^{i}(F,M)$ for what is called the ‘finite part’ (and also known as the ‘unramified part’ sometimes); $H_{v}(M)$ denotes Betti cohomology under the complex realization of the base change $M\times
_{F}\mathbb{C}$ along $\sigma:F\hookrightarrow\mathbb{C}$, $G_{v}$ the decomposition group, so that $H_{v}(M)^{G_{v}}$ is the piece fixed under complex conjugation for real places, $H_{dR}(M)/F^{0}$ is de Rham cohomology (over $F$) modulo $F^{0}$, where $F^{\bullet}$ is the standard decreasing filtration.
Let us continue Elaboration \[elab\_Motives1\]. The picture is as follows: In a lot of literature by the motivic cohomology of a motive $M$ one would mean $H^{i}(M,\mathbb{Q}(j))=\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{i}(M,\mathbb{Q}(j))$, where $\mathbb{Q}(j)$ are the motivic coefficient sheaves as for example introduced in the book [@MR2242284 Lecture 3]. As we restrict to rational coefficients, the motivic cohomology groups $\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{i}(X,\mathbb{Q}(r))$ can also be expressed as eigenspaces of the Adams operations on $K$-theory, which is historicially the pioneering approach to define them at all. This is also called *absolute* motivic cohomology and is simply called motivic cohomology in [@MR2242284]. The connection between this usage of the term and the one here is as follows: if we again consider a motive of the particular form$$M:=h^{i}(X)(r)$$ as in Elaboration \[elab\_Motives1\], then in a 6-functor formalism of mixed motivic sheaves[^7] there is a (Leray-type) spectral sequence$$\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{j}(\mathbb{Q},h^{i}(X)(r))\Longrightarrow\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{j+i}(X,\mathbb{Q}(r))\text{,}$$ where one has $\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{j}(\mathbb{Q},-)=0$ for $j\neq0,1$ thanks to $F$ being a number field ([@MR1775312 Corollary 1.1.13]). Having only two possibly non-zero columns, one gets a supply of short exact sequences$$0\rightarrow\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{1}(\mathbb{Q},h^{i}(X)(r))\rightarrow\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{i+1}(X,\mathbb{Q}(r))\rightarrow\operatorname*{Ext}\nolimits_{\mathcal{MM}_{F}}^{0}(\mathbb{Q},h^{i+1}(X)(r))\rightarrow0\text{.}\label{l_lemo3}$$ Instead of following this picture coming from the Beilinson conjectures, a lot of literature (like [@MR1386106], [@MR1884523], [@MR2882695],…) takes this as an implicit axiom, and presents the $A$-modules $H^{i}(F,M)$ of Equation \[l\_lemo2\] as being plainly defined as the output of what the sequence in Equation \[l\_lemo3\] would give. The $H^{i}(F,M)$ would be called *geometric* motivic cohomology; see for example the introduction of [@MR1439046] or [@MR1265544].
All these objects$$H^{i}(F,M)\qquad H_{f}^{i}(F,M)\qquad H_{v}(M)\qquad H_{dR}(M)$$ as well as their counterparts for the Tate twist $M(1)$, carry a canonical right $A$-module structure coming from the right action of $A$ on the motive, Equation \[lww\_Action\]. Hence, by principle (a) they each define a point in $K(A)$ and via the operations of Equation \[lww\_AddSubtract\] we can form $\Xi(M)$. Yes, it is true, this object *depends* on how we bracket it to evaluate the sums and negations, but we just once and for all choose to unravel it from left to right. In fact, a posteriori the choice turns out not to matter, so this aspect is not very important. Any choice is good enough.
Now we have a point $\Xi(M)$ in the space $K(A)$. Next, for any prime number $p$ there is a comparison quasi-isomorphism leading via principle (b’) to a path$$\vartheta_{p}:\Xi(M)\otimes_{A}A_{p}\overset{\sim}{\longrightarrow}R\Gamma
_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right) \otimes_{\mathfrak{A}_{p}}A_{p} \label{lww_Z3}$$ i.e. a path from $\Xi(M)\otimes_{A}A_{p}$ to $R\Gamma_{c}\left(
\mathcal{O}_{F,S_{p}},T_{p}\right) \otimes_{\mathfrak{A}_{p}}A_{p}$.
\[elab\_ThetaPConstructedAsInBF\]We explain where this comes from: The bounded complex $R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right)
\otimes_{\mathfrak{A}_{p}}A_{p}$ defines a point in $K(\widehat{A})$ by principle (a’). Next, $\Xi(M)$ is a point in $K(A)$ by construction, and under the exact functor $(-)\mapsto(-)\otimes_{A}A_{p}$ inducing a map $K(A)\rightarrow K(\widehat{A})$ it gets sent to a point which we may reasonably call $\Xi(M)\otimes_{A}A_{p}$ (it can be spelled out explicitly as the result of tensoring each summand in Equation \[ltiops1\] with $A_{p}$). Now use the quasi-isomorphisms of (and here we quote Burns–Flach [@MR1884523 §3.4, middle of page 526] directly) (27), (28), (23), the isomorphisms (24), (19) or the triangle (22) for all $v\in
S_{p,f}$ and finally \[to\] the triangle (26)$_{\text{vert}}$ (exactly the input used in the framework of virtual objects loc. cit.) as input for principle (b’) to turn a quasi-isomorphism into a path. By directly quoting this from Burns–Flach, we do not only save ourselves from repeating the setup of [@MR1884523 §3.2], it will also help us in §\[sect\_CompatProofSection\] to prove the comparison to the Burns–Flach approach, because our path $\vartheta_{p}$ is literally made from the same quasi-isomorphisms as the analogous isomorphism of virtual objects in their paper (and also called $\vartheta_{p}$ there).
Thus, we learn: Once we send $\Xi(M)$ along the first arrow to $K(\widehat
{A})$, then in this space we have a canonical path from $\Xi(M)\otimes
_{A}A_{p}$ to $R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right)
\otimes_{\mathfrak{A}_{p}}A_{p}$. On the other hand, it is conjectured that:
\[[e.g., [@MR1206069], [@MR1884523 Conjecture 1]]{}\]There is the basic exact sequence$$\begin{aligned}
& 0\longrightarrow H^{0}(F,M)_{\mathbb{R}}\overset{\epsilon}{\longrightarrow
}\ker\left( \alpha_{M}\right) \overset{r_{B}^{\ast}}{\longrightarrow}\left(
H_{f}^{1}(F,M^{\ast}(1))_{\mathbb{R}}\right) ^{\ast}\overset{\delta
}{\longrightarrow}\\
& \qquad\qquad\qquad H_{f}^{1}(F,M)_{\mathbb{R}}\overset{r_{B}}{\longrightarrow}\operatorname*{coker}\left( \alpha_{M}\right)
\overset{\epsilon^{\ast}}{\longrightarrow}\left( H^{0}(F,M^{\ast
}(1))_{\mathbb{R}}\right) ^{\ast}\longrightarrow0\text{.}$$ in the category $\operatorname*{PMod}(A_{\mathbb{R}})$.
However, a sequence being exact is of course the same as saying that it is quasi-isomorphic to the zero complex. The map to a zero object is canonical, so we obtain a canonical quasi-isomorphism$$\vartheta_{\infty}:\Xi(M)\otimes_{A}A_{\mathbb{R}}\overset{\sim}{\longrightarrow}\mathbf{0}\text{.} \label{lww_x5}$$ Hence, by principle (b’) we learn: Once we send $\Xi(M)$ along the first arrow in Equation \[lcixi1\] to $K(A_{\mathbb{R}})$, then in this space we have a canonical path from $\Xi(M)\otimes_{A}A_{\mathbb{R}}$ to zero.
Let us summarize this: Under the first arrow in the principal idèle fibration, the point $\Xi(M)$ gets send to a point in $K(\widehat{A})\times
K(A_{\mathbb{R}})$ and here for all primes and at infinity we get a canonical collection of paths$$\vartheta_{(-)}:\Xi(M)\otimes_{A}(-)\qquad\longrightarrow\qquad\left( \left.
\text{zero/some object from }K(\mathfrak{A}_{p})\right. \right) \text{.}$$ Now, let us use the second arrow in Equation \[lcixi1\]. Since the composition of both arrows in a fibration is zero (or more precisely: can be contracted to the constant zero map), we obtain the following: The object $\Xi(M)$ goes to the zero base point in $K(\mathsf{LCA}_{\mathfrak{A}})$ since it comes all from the left, namely $K(A)$. Moreover, all the objects $R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right) \otimes_{\mathfrak{A}_{p}}A_{p}$ on the right in Equation \[lww\_Z3\] go to zero in $K(\mathsf{LCA}_{\mathfrak{A}})$ because they come all from the left, namely $K(\widehat{\mathfrak{A}})$ (the $\mathfrak{A}_{p}$-modules are $\widehat
{\mathfrak{A}}$-modules as well). But this just means that the endpoints of the paths $\vartheta_{p}$ resp. $\vartheta_{\infty}$ all go to zero in $K(\mathsf{LCA}_{\mathfrak{A}})$. That is: *they all define closed loops* around the zero object. Hence, we get elements$$\vartheta_{p},\vartheta_{\infty}\in\pi_{1}K(\mathsf{LCA}_{\mathfrak{A}})\text{,}$$ and this in turn is just the $K$-group $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$.
We illustrate this construction by drawing the principal idèle fibration in a similar style as our example of the Möbius band in Equation \[l\_Mobius\]. For simplicity, we only include a single finite prime $p$ and $\infty$.$${\includegraphics[
height=1.2635in,
width=4.9744in
]{fibration_2a-eps-converted-to.pdf}}
$$ On the left, we have the fiber. Besides the zero object $\mathbf{0}$, we have $\Xi(M)$ as a point in $K(A)$ and $R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right) $ as a point in $K(\widehat{\mathfrak{A}})$, by principle (a) and its variations. These are the various points depicted on the left. Once we map them to the total space, depicted in the middle, we can construct paths between these points. These come from principle (b) and its variations: Quasi-isomorphisms like $\vartheta_{p}$ and $\vartheta_{\infty}$ give rise to paths. These paths do not need to exist in the fiber on the left, because in general they are not $\widehat{\mathfrak{A}}$- or $A$-module isomorphisms. On the right, we have the base space. The three points in the fiber now all get mapped to the base point. Thus, the paths we have drawn in the total space now become closed loops. Thus, they define an element in the fundamental group $\pi_{1}K(\mathsf{LCA}_{\mathfrak{A}})$.
We also construct a class $L\Omega(M,\mathfrak{A}):=\hat{\delta}_{\mathfrak{A},\mathbb{R}}^{1}(L^{\ast}(\left. M_{A}\right. ,s))$ attached to the equivariant special $L$-value. To this end, we construct an extended boundary map $\hat{\delta}_{\mathfrak{A},\mathbb{R}}^{1}$ as in [@MR1884523]. The construction of $L\Omega$ is essentially the same as Burns and Flach give, so nothing new happens here. It turns out that all but finitely many of the loops $\vartheta_{(-)}$ are trivial, so:
We define$$R\Omega(M,\mathfrak{A}):=\prod_{v}\vartheta_{v}\qquad\in\qquad K_{1}(\mathsf{LCA}_{\mathfrak{A}})\text{,} \label{lww_Z4}$$ where $v$ runs through all places of $\mathbb{Q}$. We call$$T\Omega(M,\mathfrak{A}):=L\Omega(M,\mathfrak{A})+R\Omega(M,\mathfrak{A})$$ the *equivariant Tamagawa number* of the motive $M$ with respect to the order $\mathfrak{A}$.
We have swept the dependency on $S$ and the projective $\mathfrak{A}$-structure under the rug. However, an argument analogous to [@MR1884523 Lemma 5] removes this apparent dependency in a way completely analogous to how Burns–Flach establish this.
\[sect\_GettingPrecise\]Getting precise
=======================================
In the previous section we have explained the construction of our equivariant Tamagawa number $T\Omega$ along what we have called principles (a) and (b). We had focussed on explaining the geometry of our construction, but had neglected justifying these principles rigorously. We do this in this section.
Spaces\[subsect\_Spaces\]
-------------------------
In §\[sect\_Overview\] we were talking about $K$-theory as a *space*. What do we mean? Basically, there are two fundamentally equivalent ways to do homotopy theory. Close to intuition is the following one: By space we refer to a topological space $X$. A point $x$ is really an element $x\in X$ of this space, a path is a continuous map $p:[0,1]\rightarrow X$, $p(0)$ is the starting point and $p(1)$ the endpoint, and so on.
The reader has surely seen this. The category $\mathsf{Top}_{\bullet}$ has all topological spaces as objects along with a chosen point, called the base point. Morphisms are the continuous maps preserving the pointing. The term fibration is defined as a Serre fibration. Homotopy groups are defined as the based homotopy classes of pointed maps$$S^{n}\longrightarrow X\text{,}$$ where $S^{n}$ denotes the $n$-sphere, pointed at $(1,0,\ldots,0)$.
This setup is very intuitive since it connects well with how we usually do geometry. However, one can also do homotopy theory entirely combinatorially without ever touching a topology: Then, by space we refer to a simplicial set $X_{\bullet}$. A point $x$ is a $0$-simplex, i.e. an element $x\in X_{0}$. An elementary path is a $1$-simplex, i.e. an element $p\in X_{1}$, $\partial_{1}p\in X_{0}$ is the starting point and $\partial_{0}p\in X_{0}$ the endpoint, paths are finite concatenations of elementary paths[^8], and so on.
The category $\mathsf{sSet}_{\bullet}$ has simplicial sets as objects along with a chosen point, called the base point. Morphisms are maps of simplicial sets preserving the pointing. The term fibration is defined as a Kan fibration. Homotopy groups are defined as the simplicial homotopy classes of pointed maps$$\Delta(n)\longrightarrow\operatorname*{Ex}\nolimits^{\infty}X\text{,}
\label{l_Kan}$$ where $\Delta(n)$ denotes the standard simplicial $n$-simplex and $\operatorname*{Ex}\nolimits^{\infty}$ is Kan’s functorial fibrant replacement functor (a technical device which is of no importance to what we do in this paper). General references for simplicial homotopy theory are [@MR1206474], [@MR1650134], or [@MR2840650].
Both approaches are very parallel. And indeed Quillen proved both $\mathsf{Top}_{\bullet}$ and $\mathsf{sSet}_{\bullet}$ are so-called ‘model categories’, which one can think of as saying that they both possess all the structure to do homotopy theory. A few references: $\mathsf{Top}_{\bullet}$ and its model category structure is very carefully set up and discussed in [@MR1650134 §2.4]; $\mathsf{sSet}_{\bullet}$ and its model category structure is set up in [@MR1650134 §3.2].
Indeed, there is an adjunction$$\mathsf{sSet}_{\bullet}\rightleftarrows\mathsf{Top}_{\bullet}\text{,}
\label{lwaa1}$$ the left adjoint sending a simplicial set $X_{\bullet}$ to its geometric realization $\left\vert X_{\bullet}\right\vert $ (which basically glues topological $i$-cells according to the glueing rules prescribed by the simplicial set structure) and reversely the right adjoint sending a space to its simplicial set of maps $\operatorname*{Sing}_{\bullet}(X):=\{f:S^{n}\rightarrow X$, $f$ continuous$\}$, [@MR1650134 p. 77]. Sweeping some technicalities under the rug, this adjunction can be promoted to a so-called Quillen equivalence, which roughly speaking means that the concepts of fibration, homotopy groups, etc. of both model categories are compatible[^9]. We do not need to understand any of that for this paper, only the following consequence: There is no difference between whether we do homotopy theory in $\mathsf{sSet}_{\bullet}$ or $\mathsf{Top}_{\bullet}$.
As a convention: From now on, we work in the setting of $\mathsf{sSet}_{\bullet}$, i.e. the word ‘space’ means a simplicial set. Keeping the equivalence of $\mathsf{sSet}_{\bullet}$ and $\mathsf{Top}_{\bullet}$ in mind, we may however always use $\mathsf{Top}_{\bullet}$ whenever we feel in need to get some geometric intuition.
Algebraic $K$-theory\[sect\_AlgKThy\]
-------------------------------------
### Definition as a space
As a motivation, recall the definition of $K_{0}$ (we ask the reader for forgiveness if this appears too elementary, but there is a good reason to go through this): If $R$ is a ring, let $Pr(R)$ denote the set of isomorphism classes of finitely generated projective right $R$-modules. This is an abelian monoid under the direct sum $[X]+[Y]:=[X\oplus Y]$. However, there is no reason why additive inverses, like some $-[X]$ would have to exist. Then define$$K_{0}(R):=GC(Pr(R))\text{,}$$ where $GC(-)$ denotes the group completion: This is a general operation turning abelian monoids into abelian groups. It can be defined as follows: If $M$ is an abelian monoid, consider the quotient set$$GC(M):=\left\{ \frac{\text{pairs }(P,Q)\in M\times M}{(P,Q)\sim(P\underset
{M}{+}S,Q\underset{M}{+}S)\text{ for all }S\in M}\right\} \text{.}$$ It is easy to show that defining$$(P,Q)+(P^{\prime},Q^{\prime}):=(P\underset{M}{+}P^{\prime},Q\underset{M}{+}Q^{\prime})\qquad\text{and}\qquad-(P,Q):=(Q,P)$$ renders $GC(M)$ into an abelian group. Define$$M\longrightarrow GC(M)\qquad\text{by}\qquad P\mapsto(0,P)\text{.}$$ One can show that any monoid morphism from $M$ to an abelian group $A$ factors uniquely over $GC(M)$, so $GC(-)$ is the universal construction transforming abelian monoids into abelian groups.[^10] This construction of $K_{0}$ extends to split exact categories $\mathsf{C}$, define $K_{0}(\mathsf{C}):=GC(Iso(\mathsf{C}))$, where $Iso(\mathsf{C})$ is the set of isomorphism classes of objects, turned into a monoid using the direct sum.
\[rmk\_MinusOneMapGC\]Every element $P\in M$ maps to $(0,P)$ in $GC(M)$. Correspondingly, we observe $-P=(P,0)$, and in particular the automorphism of $GC(M)$ exchanging $(P,Q)$ with $(Q,P)$ corresponds to multiplication by $-1$.
There is a way to define also all higher $K$-groups and in particular the $K$-theory space $K(\mathsf{C})$ in a rather similar way: If $\mathsf{C}$ is a category, we write $s\mathsf{C}$ for the category of simplicial objects in $\mathsf{C}$. A *Waldhausen category* is a pointed[^11] category with a choice of cofibrations and a choice of weak equivalences, satisfying the usual axioms. A detailed definition is given in Weibel [@MR3076731 Ch. II, Definition 9.1.1]. We write $X^{\prime}\hookrightarrow X$ to denote cofibrations. In a category with cofibrations every cofibration admits a cokernel (as follows from the axioms), and any sequence$$X^{\prime}\hookrightarrow X\longrightarrow C\text{,}$$ where $C$ is a cokernel object, is called a *cofibration sequence*. We will write any cofibration sequence as$$X^{\prime}\hookrightarrow X\twoheadrightarrow C\text{.}$$
\[Coproducts exist\]\[rmk\_Coproducts\]Every Waldhausen category has finite coproducts. By the axioms for any object $X\in\mathsf{C}$ there is a canonical cofibration $0\hookrightarrow C$ and taking the pushout of this arrow along a second copy of itself, which exists by the axioms, we get an object which we denote by $C\vee C$ (following the notation of [@MR1167575]). This object is unique up to unique isomorphism.
As is customary, we usually write $w\mathsf{C}$ for the category whose objects are the same as those of the Waldhausen category $\mathsf{C}$, but we only keep the weak equivalences as morphisms. Similarly, we write $i\mathsf{C}$ if we use the same objects, but only keep the isomorphisms. Let $\mathsf{Wald}$ denote the category[^12] whose objects are Waldhausen categories and morphisms are exact functors. In [@MR1167575] this category is called $w\mathcal{C}of$. Waldhausen’s $S$-construction is a functor$$S_{\bullet}:\mathsf{Wald}\longrightarrow s\mathsf{Wald}$$ from Waldhausen categories to simplicial Waldhausen categories. Note that the latter is a simplicial object in categories (and not, as one could think, a category enriched in simplicial sets. Unfortunately, both are frequently called a ‘simplicial category’). We write $PX_{\bullet}$ for the simplicial path space (where customarily, we use the right path space. There is also a left path space, see [@MR3782417 §2.2.3] for a comparison. In the end, this choice does not matter). On simplices, $PX_{n}:=X_{n+1}$, in both cases.
Following [@MR1167575], define a functor$$G_{\bullet}:\mathsf{Wald}\longrightarrow s\mathsf{Wald}$$ by forming the Cartesian square$$\xymatrix{
G_{\bullet}\mathsf{C} \ar[r] \ar[d] & PS_{\bullet}\mathsf{C} \ar[d] \\
PS_{\bullet}\mathsf{C} \ar[r] & S_{\bullet}\mathsf{C}.
}
\label{lkiops1}$$ While this a priori only defines an object $G\in s\mathsf{Cat}$, define the cofibrations (resp. weak equivalences) to be the Cartesian product, too. This means that a morphism in $G_{n}$ is a cofibration (resp. weak equivalence) if it is given by a pair $(f_{1},f_{2})$ of cofibrations (resp. weak equivalences) of $PS_{\bullet}\mathsf{C}\times PS_{\bullet}\mathsf{C}$.
\[[@MR909784], [@MR1167575]\]If $\mathsf{C}$ is a Waldhausen category, the construction $G_{\bullet}\mathsf{C}$ is called the *Gillet–Grayson model* of $\mathsf{C}$.
\[Exact categories as Waldhausen categories\]\[rmk\_ExactToWald\]Gillet and Grayson [@MR909784; @MR2007234] originally introduced $G_{\bullet
}\mathsf{C}$, but they only considered it for exact categories. This amounts to taking a pointed exact category, taking its admissible monics as the class of cofibrations and its isomorphisms as the class of weak equivalences.
Only in [@MR1167575] the construction was extended to (fairly) arbitrary Waldhausen categories. We will have crucial need for this broader variant, so [@MR1167575] will be our principal foundation for the Gillet–Grayson model.
\[example\_GG\]We can unravel the definition of $G_{\bullet}(\mathsf{C})$ in concrete terms. Its $q$-simplices $G_{q}(\mathsf{C})$ is given by the following Waldhausen category. Its objects are pairs of diagrams $$\xymatrix@!=0.157in{
& & & & X_{n/(n-1)}
\\
& & & \cdots\ar@{^{(}.>}[r] & \vdots\ar@{.>>}[u] \\
& & X_{2/1} \ar@{^{(}.>}[r] & \cdots\ar@{^{(}.>}[r] & X_{n/1} \ar@{.>>}[u] \\
& X_{1/0} \ar@{^{(}.>}[r] & X_{2/0} \ar@{^{(}.>}[r] \ar@{.>>}[u] & \cdots
\ar@{^{(}.>}[r] & X_{n/0} \ar@{.>>}[u] \\
X_0 \ar@{^{(}.>}[r] & X_1 \ar@{^{(}.>}[r] \ar@{.>>}[u] & X_2 \ar@{^{(}.>}[r] \ar@{.>>}[u] & \cdots\ar@{^{(}.>}[r] & X_n \ar@{.>>}[u]
}\qquad\xymatrix@!=0.157in{
& & & & X_{n/(n-1)}
\\
& & & \cdots\ar@{^{(}->}[r] & \vdots\ar@{->>}[u] \\
& & X_{2/1} \ar@{^{(}->}[r] & \cdots\ar@{^{(}->}[r] & X_{n/1} \ar@{->>}[u] \\
& X_{1/0} \ar@{^{(}->}[r] & X_{2/0} \ar@{^{(}->}[r] \ar@{->>}[u] & \cdots
\ar@{^{(}->}[r] & X_{n/0} \ar@{->>}[u] \\
X^{\prime}_0 \ar@{^{(}->}[r] & X^{\prime}_1 \ar@{^{(}->}[r] \ar@
{->>}[u] & X^{\prime}_2 \ar@{^{(}->}[r] \ar@{->>}[u] & \cdots\ar@{^{(}->}[r] & X^{\prime}_n \ar@{->>}[u]
}\text{,} \label{lkiops4}$$ such that (1) the diagrams commute and agree above the bottom row, (2) every sequence $X_{i}\hookrightarrow X_{j}\twoheadrightarrow X_{j/i}$ is a cofibration sequence, (2’) every sequence $X_{i}^{\prime}\hookrightarrow
X_{j}^{\prime}\twoheadrightarrow X_{j/i}^{\prime}$ is a cofibration sequence, (3) every sequence $X_{i/j}\hookrightarrow X_{m/j}\twoheadrightarrow X_{m/i}$ is a cofibration sequence. The face and degeneracy maps amount to duplicating the $i$-th row and column or deleting them.We only use the solid vs. dotted arrows to distinguish the two pieces of the otherwise fully symmetric pairs. We call the side with solid arrows the *Yin side*, and the one with dotted arrows the *Yang side*. The morphisms in $G_{q}(\mathsf{C})$ are all morphisms between such diagrams (i.e. such that all arrows commute). The cofibrations (resp. weak equivalences) in $G_{q}(\mathsf{C})$ are those morphisms such that for all objects in the Diagram \[lkiops4\] entry-wise it is a cofibration (resp. weak equivalence) in $\mathsf{C}$.
The main theorem of Gillet and Grayson is that the simplicial set $G_{\bullet
}(\mathsf{C})$ has the same homotopy type as the $K$-theory space $K(\mathsf{C})$ as defined by Quillen. In particular, we may simply use the Gillet–Grayson model as *the* definition of the $K$-theory space in this paper.
Working in terms of simplicial homotopy theory, for us, the term ‘geometric realization $\left\vert X_{\bullet}\right\vert $’ of a simplicial set $X_{\bullet}$ denotes a functorial fibrant replacement functor. To fix matters, let us use Kan’s $\operatorname*{Ex}^{\infty}$-functor (as we had already done in Equation \[l\_Kan\]), although the precise nature of the fibrant replacement will be fully irrelevant for what is to come.
We may now define the $K$-theory space of a (pseudo-additive[^13]) Waldhausen category $\mathsf{C}$ with weak equivalences $w$ by$$K(\mathsf{C})=\left\vert wG_{\bullet}\mathsf{C}\right\vert \text{.}
\label{latix5}$$ While writing this in this way is standard accepted practice, this notation sweeps a bunch of things under the rug, so let us instead give a precise definition:
If $\mathsf{C}$ denotes any category, we write $N_{\bullet}\mathsf{C}$ to denote the nerve of the category.
If we write a bisimplicial set $X_{\bullet,\bullet}\in ss\mathsf{Set}$ as a functor$$X_{\bullet,\bullet}:\triangle^{op}\times\triangle^{op}\longrightarrow
\mathsf{Set}\text{,}$$ where $\triangle$ is the ordinal number category, then the diagonal simplicial set $(\operatorname*{diag}X)_{\bullet}$ is defined as the composite functor$$\triangle^{op}\overset{d}{\longrightarrow}\triangle^{op}\times\triangle
^{op}\longrightarrow\mathsf{Set}\text{,}$$ where $d$ is the diagonal functor $q\mapsto(q,q)$ for ordinal numbers $q$. We can make this more concrete: For the simplices we have$$(\operatorname*{diag}X_{\bullet,\bullet})_{q}:=X_{q,q}$$ and if we write $\partial_{\ast}^{h}$, $\partial_{\ast}^{v}$ to denote the horizontal (resp. vertical) face maps of $X_{\bullet,\bullet}$, then$$\partial_{i}^{\operatorname*{diag}X}:=\partial_{i}^{h}\circ\partial_{i}^{v}\text{.} \label{lkiops3}$$
\[def\_KThyViaWaldGG\]For every pseudo-additive Waldhausen category $\mathsf{C}$ with weak equivalences $w$, we call$$K(\mathsf{C}):=\left\vert \operatorname*{diag}N_{\bullet}wG_{\bullet
}(\mathsf{C})\right\vert$$ the $K$*-theory space* of $\mathsf{C}$. Take $(0,0)$ as the base point. Equation \[latix5\] is just a shorthand for the same thing.
This includes the case where $\mathsf{C}$ is an exact category by using the Waldhausen category structure of Remark \[rmk\_ExactToWald\]. The above is the simplicial geometric realization of the diagonal of a bisimplicial set (one simplicial direction comes from the Gillet–Grayson construction, the other from taking the nerve of the categories $wG_{q}\mathsf{C}$ for any $q$).
If in §\[subsect\_Spaces\] you prefer doing homotopy theory in $\mathsf{Top}_{\bullet}$, you need to take the true geometric realization (i.e. in the original meaning of this term) of this simplicial set to obtain an object in $\mathsf{Top}_{\bullet}$ as in Equation \[lwaa1\]. On the other hand, if you prefer simplicial sets, Definition \[def\_KThyViaWaldGG\] *is* the space on the nose.
### Explicit structure for $K_{0}$\[subsect\_ExplicitK0\]
We extract from Example \[example\_GG\] that a $0$-simplex in $G_{\bullet
}(\mathsf{C})$ corresponds to a pair of objects $(P,Q)$ with $P,Q\in
\mathsf{C}$. Indeed, the concrete isomorphism$$\pi_{0}\left\vert G_{\bullet}(\mathsf{C})\right\vert \longrightarrow
K_{0}(\mathsf{C}) \label{lmexi1}$$ is given as follows: Given a connected component on the left, let $(P,Q)$ be any point in this component and then send it to $[Q]-[P]$ in $K_{0}(\mathsf{C})$, i.e. the difference of the isomorphism classes of these objects. See [@MR3076731 Ch. IV, Lemma 9.2] for a proof.
### Justification of principles (a) and (b)\[subsect\_JusitfyPrinciplesAB\]
In §\[sect\_Overview\] our construction of the Tamagawa number rested on the following basic principles: Suppose $\mathsf{C}$ is an exact category.$\left. \qquad\text{\textbf{(a)}}\right. $ Every object in $\mathsf{C}$ determines a point in the space $K(\mathsf{C})$.$\left.
\qquad\text{\textbf{(b)}}\right. $ Every isomorphism $X\overset{\sim
}{\rightarrow}Y$ determines a path from the point of $X$ to $Y$ in $K(\mathsf{C})$.
We can fully justify them now: For (a) if $P\in\mathsf{C}$ is an object, simply take the $0$-simplex $(0,P)$ as the point in $K(\mathsf{C})$. Using the map in Equation \[lmexi1\] we see that it lies in the connected component of $[P]\in K_{0}(\mathsf{C})$, so this is in line with our descriptions given in §\[sect\_Overview\]. For (b), let us look at the Gillet–Grayson model again. Unravelling the definition of $G_{1}(\mathsf{C})$ explicitly, the $1$-simplices turn out to be given by pairs of exact sequences$$\xymatrix{
P_0 \ar@{^{(}.>}[r] & P_1 \ar@{.>>}[r] & P_{1/0} & \qquad& P^{\prime}_0 \ar@{^{(}->}[r] & P^{\prime}_1 \ar@{->>}[r] & P_{1/0}
}
\label{lmexid1}$$ having the same cokernel. Using the description of paths in simplicial sets, such a $1$-simplex is a path from $(P_{0},P_{0}^{\prime})$ to $(P_{1},P_{1}^{\prime})$.
Hence, if $\varphi:X\rightarrow Y$ is an isomorphism in $\mathsf{C}$, attach the $1$-simplex of the pair of exact sequences $$\xymatrix{
0 \ar@{^{(}.>}[r] & 0 \ar@{.>>}[r] & 0 & \qquad& X \ar@{^{(}->}[r]^{\varphi}
& Y \ar@{->>}[r] & 0
}
\label{lmexi4}$$ with matching cokernel zero to it. As discussed above, this is a path from $(0,X)$ to $(0,Y)$, i.e. a path between the points associated to the objects $X$ and $Y$ by principle (a).
Thus, principles (a) and (b) are set up rigorously now.
### Justification of principles (a’) and (b’)\[subsect\_JusitfyPrinciplesABprime\]
We had also claimed that the same principles hold on the derived level, essentially. Let $\mathsf{C}$ be any exact category. Let $\mathbf{Ch}^{b}(\mathsf{C})$ be the exact category of bounded chain complexes in $\mathsf{C}$, [@MR2606234 §10]. We write $q\mathbf{Ch}^{b}(\mathsf{C})$ to denote the subcategory where we only keep quasi-isomorphisms as morphisms, [@MR2606234 §10.3]. Now make $\mathbf{Ch}^{b}(\mathsf{C})$ a Waldhausen category as in Remark \[rmk\_ExactToWald\], but use the class of morphisms $q$ (i.e. the quasi-isomorphisms) as weak equivalences instead. We note that this is a pseudo-additive Waldhausen category in the sense of [@MR1167575].
Write $K(\mathsf{C},w)$ if we wish to stress that we use the class of weak equivalences $w$. By the Gillet–Waldhausen theorem ([@MR3076731 Chapter V, Theorem 2.2]), we have the equivalence$$K(\mathsf{C},i)\overset{\sim}{\longrightarrow}K(\mathbf{Ch}^{b}(\mathsf{C}),q)\text{,} \label{lseip1}$$ which is usually proven in terms of the Waldhausen $S$-construction, but since the Gillet–Grayson model of [@MR1167575] can also handle Waldhausen categories with non-trivial weak equivalences, we obtain$$K(\mathsf{C},i)\overset{\sim}{\longrightarrow}K(\mathbf{Ch}^{b}(\mathsf{C}),q)=\left\vert \operatorname*{diag}N_{\bullet}qG_{\bullet}(\mathbf{Ch}^{b}\mathsf{C})\right\vert \text{,}$$ using Definition \[def\_KThyViaWaldGG\] for the right-hand side of Equation \[lseip1\]. Now repeat the constructions of §\[subsect\_JusitfyPrinciplesAB\]. We obtain principle (a’) since $0$-simplices in $G_{\bullet}(\mathbf{Ch}^{b}\mathsf{C})$ are pairs $(-,-)$ of bounded complexes in $\mathsf{C}$ now, and principle (b’) since we can now plug in quasi-isomorphisms for $\varphi$ in Equation \[lmexi4\].
### Justification of sum and negation\[subsect\_JustifySumAndNegation\]
Next, let us set up the maps$$\left. +\right. :K(\mathsf{C})\times K(\mathsf{C})\longrightarrow
K(\mathsf{C})\qquad\text{and}\qquad\left. -\right. :K(\mathsf{C})\longrightarrow K(\mathsf{C}) \label{lwimo1}$$ of Equation \[lww\_AddSubtract\]. For the addition in Equation \[lwimo1\] let$$\left. \oplus\right. :\mathsf{C}\times\mathsf{C}\longrightarrow\mathsf{C}
\label{lexit1}$$ be a symmetric monoidal structure giving the coproduct $\vee
$ (see Remark \[rmk\_Coproducts\]). Then define$$(P,Q)+(P^{\prime},Q^{\prime}):=(P\oplus P^{\prime},Q\oplus Q^{\prime})\text{.}
\label{l_maxiu_1}$$ Since $\left. \oplus\right. $ is a functor, one can naturally extend this to a map$$G_{\bullet}\mathsf{C}\times G_{\bullet}\mathsf{C}\longrightarrow G_{\bullet
}\mathsf{C}\text{.}$$ As we had pointed out, this map is neither associative nor commutative, and in fact depends on choosing a concrete bifunctor as in Equation \[lexit1\] (since in general coproducts are only well-defined up to unique isomorphism). However, the above definition is good enough for the moment. We shall later set up a homotopy correct addition, see Definition \[def\_SegalNerve\], but the above operation is one possible representative. In particular, we defer justifying that this map induces addition to Corollary \[cor\_SegalNerveSum\] much later. On $K_{0}$ it is easy to check directly, of course.
We define the negation$$\left. -\right. :K(\mathsf{C})\longrightarrow K(\mathsf{C})$$ by simply swapping the Yin and Yang side, i.e. on $0$-simplices this is $(P,Q)\mapsto(Q,P)$. These maps have all the properties we had discussed in §\[sect\_Overview\], and more concretely:
\[[@MR909784]\]\[prop\_NegationOnGGModel\]With these definitions, $K(\mathsf{C})$ is an $H$-space,
1. on all homotopy groups $\pi_{i}K(\mathsf{C})$ this addition map induces the genuine addition of the homotopy group,
2. on all homotopy groups $\pi_{i}K(\mathsf{C})$ this negation map induces multiplication with $-1$,
and in particular on the level of homotopy groups both operations define an abelian group structure.
This is proven by Gillet and Grayson in [@MR909784 Theorem 3.1]. Note that negation is given by swapping $(P,Q)\mapsto(Q,P)$, fully analogous to what happens for the group completion $GC(-)$, see Remark \[rmk\_MinusOneMapGC\].
\[Rigorous interpretation of §\[sect\_Overview\]\]\[convention\_KThySpace\]Use the space $K(\mathbf{Ch}^{b}(\mathsf{C}),q)$ of §\[subsect\_JusitfyPrinciplesABprime\] as the meaning of the $K$-theory space for the constructions in §\[sect\_Overview\]. As we have just explained, principles (a’) and (b’) are available, and so are (a) and (b) by viewing objects as complexes concentrated in degree zero. Along with the sum and negation, now all the operations employed in §\[sect\_Overview\] have a rigorous foundation.
We have just set up $K$-theory as a space and $H$-space here. There is also a canonical infinite loop space structure. We will not discuss this yet because it only becomes relevant later, but the reader may jump ahead to Lemma \[lemma\_InfLoopSpaceStructures\] to see how this structure arises.
### Explicit structure of $K_{1}$
The explicit description of $K_{0}$ in §\[subsect\_ExplicitK0\] can be complemented by a description of $K_{1}$. Suppose $\mathsf{C}$ is a pointed exact category. We write $0$ for the designated zero object. A *double (short) exact sequence* (in the sense of Nenashev) consists of two short exact sequences$$\mathrm{Yin}:A\overset{p}{\hookrightarrow}B\overset{r}{\twoheadrightarrow
}C\qquad\quad\text{and}\quad\qquad\mathrm{Yang}:A\overset{q}{\hookrightarrow
}B\overset{s}{\twoheadrightarrow}C\text{,}$$ whose three objects are the same for Yin and Yang, but the morphisms $p,r$ resp. $q,s$ need not agree. We denote this datum in the format$$l=\left[
\xymatrix{
A \ar@<1ex>@{^{(}->}[r]^{p} \ar@<-1ex>@{^{(}.>}[r]_{q} & B \ar@<1ex>@{->>}[r]^{r} \ar@<-1ex>@{.>>}[r]_{s} & C,
}\right]$$ as a shorthand. Given any such $l$, it describes a closed loop around the base point $(0,0)$ in the Gillet–Grayson model $G_{\bullet}(\mathsf{C})$ which is made up from the concatenation of three elementary paths (i.e. three $1$-simplices), namely$${\includegraphics[
height=1.2246in,
width=1.6873in
]{gfx2-eps-converted-to.pdf}}
\label{lmixi3b}$$ where $e(l)$ denotes the $1$-simplex from $(A,A)$ to $(B,B)$ which comes from interpreting $l$ as a pair of exact sequences with the same cokernel (see Equation \[lmexid1\], where we had already talked about the $1$-simplices). Moreover, for any object $A\in\mathsf{C}$, $e(A)$ denotes the $1$-simplex from the pair $0\hookrightarrow A\overset{1}{\twoheadrightarrow}A$, taken both for Yin and Yang, which also defines a pair of exact sequences with the same cokernel. See also [@etnclca2 Definition 3.1] or [@MR1409623; @MR1621690; @MR1637539].
\[Nenashev\]Suppose $\mathsf{C}$ is an arbitrary exact category. Then the abelian group $K_{1}(\mathsf{C})$ has the following explicit presentation:
1. Attach an abstract generator to each double exact sequence$$\xymatrix{
A \ar@<1ex>@{^{(}->}[r]^{p} \ar@<-1ex>@{^{(}.>}[r]_{q} & B \ar@<1ex>@{->>}[r]^{r} \ar@<-1ex>@{.>>}[r]_{s} & C.
}$$
2. Whenever the Yin and Yang sides happen to agree, i.e.,$$\xymatrix{
A \ar@<1ex>@{^{(}->}[r]^{p}_{=} \ar@<-1ex>@{^{(}.>}[r]_{p} & B \ar@
<1ex>@{->>}[r]^{r}_{=} \ar@<-1ex>@{.>>}[r]_{r} & C\text{,}
}$$ declare the class of this generator to vanish.
3. Suppose there is a (not necessarily commutative) $(3\times3)$-diagram$$\xymatrix@W=0.3in@H=0.3in{
A \ar@<1ex>@{^{(}->}[r] \ar@<1ex>@{^{(}.>}[d] \ar@<-1ex>@{^{(}->}[d] \ar@<-1ex>@{^{(}.>}[r] & B \ar@<1ex>@{->>}[r] \ar@<-1ex>@{.>>}[r] \ar@<1ex>@{^{(}.>}[d] \ar@<-1ex>@{^{(}->}[d] & C \ar@<1ex>@{^{(}.>}[d] \ar@<-1ex>@{^{(}->}[d] \\
D \ar@<1ex>@{^{(}->}[r] \ar@<-1ex>@{^{(}.>}[r] \ar@<1ex>@{.>>}[d] \ar@
<-1ex>@{->>}[d] & E \ar@<1ex>@{->>}[r] \ar@<-1ex>@{.>>}[r] \ar@<1ex>@{.>>}[d] \ar@<-1ex>@{->>}[d] & F \ar@<1ex>@{.>>}[d] \ar@<-1ex>@{->>}[d] \\
G \ar@<1ex>@{^{(}->}[r] \ar@<-1ex>@{^{(}.>}[r] & H \ar@<1ex>@{->>}[r] \ar@<-1ex>@{.>>}[r] & I, \\
}$$ whose rows $Row_{i}$ and columns $Col_{j}$ are each a double exact sequence. Suppose after removing all Yin (resp. all Yang) exact sequences, the remaining diagram commutes. Whenever this holds, impose the relation$$Row_{1}-Row_{2}+Row_{3}=Col_{1}-Col_{2}+Col_{3}\text{.} \label{l_C_Nenashev}$$
\[example\_MakeAutToNenashevRepresentative\]We use the same notation as in the paper [@etnclca2], which is a little different from the one in Nenashev’s papers. If $\varphi:X\rightarrow X$ is an automorphism of an object in $\mathsf{C}$, the canonical map $\operatorname*{Aut}(X)\rightarrow
K_{1}(\mathsf{C})$ sends it to the Nenashev representative$$\xymatrix{
0 \ar@<1ex>@{^{(}->}[r] \ar@<-1ex>@{^{(}.>}[r] & X \ar@<1ex>@{->>}[r]^{\varphi} \ar@<-1ex>@{.>>}[r]_{1} & X
}\text{.} \label{lmixi3a}$$
\[sect\_Setup\]Equivariance setup
---------------------------------
Let $F$ be a number field and $\mathcal{O}_{F}$ its ring of integers. Let $A$ be a finite-dimensional semisimple $F$-algebra and $\mathfrak{A}$ an $\mathcal{O}_{F}$-order. This assumption implies that $\mathcal{O}_{F}$ lies in the center of $\mathfrak{A}$.
These assumptions in place, $A$ is also a finite-dimensional semisimple $\mathbb{Q}$-algebra and $\mathfrak{A}$ a $\mathbb{Z}$-order, i.e. an order in the usual sense.
For any place $\mathfrak{p}$ of $\mathcal{O}_{F}$, we write $A_{\mathfrak{p}}:=A\otimes_{F}F_{\mathfrak{p}}$, where $F_{\mathfrak{p}}$ is the local field at $\mathfrak{p}$. If $\mathfrak{p}$ is a finite place, define $\mathfrak{A}_{\mathfrak{p}}:=\mathfrak{A}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{F_{\mathfrak{p}}}$, where $\mathcal{O}_{F_{\mathfrak{p}}}$ is the ring of integers of $F_{\mathfrak{p}}$. For any finitely generated projective right $\mathfrak{A}$-module $\mathfrak{X}$, we introduce the shorthands$$\mathfrak{X}_{\mathfrak{p}}:=\mathfrak{X}\otimes_{\mathfrak{A}}\mathfrak{A}_{\mathfrak{p}}\text{,}\qquad\qquad X_{\mathfrak{p}}:=\mathfrak{X}\otimes_{\mathfrak{A}}A_{\mathfrak{p}}\text{,}\qquad\qquad X:=\mathfrak{X}\otimes_{\mathfrak{A}}A\text{.}$$ Here we tacitly equip $X$ with the discrete topology, i.e. its natural topology as a finite-dimensional $\mathbb{Q}$-vector space. We equip $X_{\mathfrak{p}}$ with its natural topology as a finite-dimensional $F_{\mathfrak{p}}$-vector space. We also write$$X_{\mathbb{R}}:=X\otimes_{A}A_{\mathbb{R}}=X\otimes_{\mathbb{Q}}\mathbb{R}\text{,}$$ equipped with the real vector space topology.
In the special case $F=\mathbb{Q}$ this is compatible with the notation used by Burns and Flach, see [@MR1884523 §2.7]. In particular, it is compatible with the notation in [@etnclca] and [@etnclca2].
We also use the notation$$\widehat{A}:=A\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}}=\mathfrak{A}\otimes_{\mathbb{Z}}\mathbb{A}_{fin}\text{.}$$ Here $\mathbb{A}_{fin}$ denotes the *finite part* of the adèles of the rational number field $\mathbb{Q}$, i.e. the restricted product $\left.
\prod\nolimits_{p}^{\prime}\right. (\mathbb{Q}_{p},\mathbb{Z}_{p})$, where $p$ only runs over the primes. We equip $\widehat{A}$ with the locally compact topology coming from the adèles. Note that$$\widehat{F}=F\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}}$$ agrees with the finite part of the adèles of the number field $F$.
Next, we discuss idèles in the non-commutative setting, following Fröhlich [@MR0376619 §2]. Define the *idèle group* by$$J(A):=\left\{ (a_{\mathfrak{p}})_{\mathfrak{p}}\in\left. \prod
_{\mathfrak{p}}A_{\mathfrak{p}}^{\times}\right\vert a_{\mathfrak{p}}\in\mathfrak{A}_{\mathfrak{p}}^{\times}\text{ for all but finitely many places
}\mathfrak{p}\right\} \text{.} \label{lmixi1}$$ This group is independent of the choice of the order $\mathfrak{A}$ (because if $\mathfrak{A}^{\prime}$ is a further $\mathcal{O}_{F}$-order in $A$, we have $\mathfrak{A}_{\mathfrak{p}}=(\mathfrak{A}^{\prime})_{\mathfrak{p}}$ for all but finitely many places $\mathfrak{p}$). If $\mathfrak{A}\subset A$ is a fixed order, we also get the group of *unit finite idèles* $U(\mathfrak{A})\subset J(A)$ defined by$$U(\mathfrak{A}):=\left\{ (a_{\mathfrak{p}})_{\mathfrak{p}}\in\prod
_{\mathfrak{p}\text{ finite}}\mathfrak{A}_{\mathfrak{p}}^{\times}\right\}
\qquad\text{(also denoted }U^{\operatorname*{fin}}(\mathfrak{A})\text{)}
\label{lcixiDex1}$$ This group depends crucially on the choice of the order $\mathfrak{A}$. We view this as a subgroup of $J(A)$ by letting $a_{\mathfrak{p}}=1$ for all infinite places $\mathfrak{p}$.
\[rmk\_FroehlichU\_DifferentDef\]This differs from [@MR0376619], because Fröhlich’s definition of $U(\mathfrak{A})$ includes the infinite places, so what he calls $U$ is $U(\mathfrak{A})\cdot A_{\mathbb{R}}^{\times}$ in our notation.
The *ideal class group* of $\mathfrak{A}$ can be defined as$$\operatorname*{Cl}(\mathfrak{A}):=\ker\left( \operatorname*{rk}:K_{0}(\mathfrak{A})\longrightarrow\mathbb{Z}\right) \text{,} \label{lmixi7}$$ where $\operatorname*{rk}$ denotes the rank map. If $A$ is a number field and $\mathfrak{A}$ any order, $\operatorname*{Cl}(\mathfrak{A})$ agrees with the usual ideal class group. To see this, use [@MR3076731 Ch. II, Corollary 2.6.3].
There is a reduced norm$$\operatorname*{nr}:A^{\times}\longrightarrow\zeta(A)^{\times}\text{,}$$ where $\zeta(A)$ denotes the center of $A$, see [@MR0376619 §2] (if $A$ is not simple, define it by taking the direct sum of the reduced norms of each simple summand). We define the *reduced norm one subgroup*$$J^{1}(A):=\ker\left( \operatorname*{nr}:J(A)\longrightarrow\zeta(A)^{\times
}\right) \text{.}$$ As $\zeta(A)^{\times}$ is abelian, it follows that $[J(A),J(A)]\subseteq
J^{1}(A)$, i.e. the commutator subgroup is contained in $J^{1}(A)$. Next, we recall the classification of finitely generated projective right $\mathfrak{A}$-modules. Given any $a=(a_{\mathfrak{p}})_{\mathfrak{p}}\in
J(A)$, there exists a unique $\mathcal{O}_{F}$-lattice, denoted $a\mathfrak{A}$, inside $A$ such that$$(a\mathfrak{A})_{\mathfrak{p}}=a_{\mathfrak{p}}\mathfrak{A}_{\mathfrak{p}}
\label{lciops1}$$ for all finite places $\mathfrak{p}$ of $F$. By a result of Fröhlich [@MR0376619 Theorem 1], every finitely generated projective right $\mathfrak{A}$-module $\mathfrak{X}$ of rank $m\geq1$ is isomorphic to$$\mathfrak{X}\cong a_{1}\mathfrak{A}\oplus\cdots\oplus a_{m}\mathfrak{A}
\label{lmixi4}$$ for some $a_{1},\ldots,a_{m}\in J(A)$, and further any two such are isomorphic if and only if they have (a) the same rank and, (b) moreover $a_{1}\cdots
a_{m}\equiv a_{1}^{\prime}\cdots a_{m}^{\prime}$ holds in the double coset set $A^{\times}\backslash J(A)/(U(\mathfrak{A})\cdot A_{\mathbb{R}}^{\times})$ for $m=1$, resp. in the quotient group$$\frac{J(A)}{J^{1}(A)\cdot A^{\times}\cdot U(\mathfrak{A})\cdot A_{\mathbb{R}}^{\times}} \label{lmixiDex1}$$ in the case of $m\geq2$. (Keep in mind Remark \[rmk\_FroehlichU\_DifferentDef\] when comparing this with [@MR0376619].)
\[example\_CancellationRankTwo\]For idèles $a=(a_{\mathfrak{p}})_{\mathfrak{p}}$ and $b=(b_{\mathfrak{p}})_{\mathfrak{p}}$, there exists an isomorphism $a\mathfrak{A}\oplus b\mathfrak{A}\overset{\sim}{\longrightarrow
}ab\mathfrak{A}\oplus\mathfrak{A}$.
\[example\_IdeleNontrivOnlyAtInfinity\]If an idèle $a=(a_{\mathfrak{p}})_{\mathfrak{p}}$ satisfies $a_{\mathfrak{p}}=1$ for all finite places, then $a\mathfrak{A}=\mathfrak{A}$.
Since $J^{1}(A)$ contains the commutator subgroup, Equation \[lmixiDex1\] describes an abelian group. The classification result generalizes Steinitz’s classification of vector bundles over affine Dedekind schemes. Based on this result, Fröhlich obtains a second characterization of the ideal class group of Equation \[lmixi7\].
\[Fröhlich\]\[thm\_FrohlichTheory\]There exists an isomorphism$$\begin{aligned}
\operatorname*{Cl}(\mathfrak{A}) & \longrightarrow\frac{J(A)}{J^{1}(A)\cdot
A^{\times}\cdot U(\mathfrak{A})\cdot A_{\mathbb{R}}^{\times}}\label{lmixi5}\\
\lbrack\mathfrak{X}]-[\mathfrak{A}^{n}] & \longmapsto\lbrack a_{1}\cdots
a_{m}]\text{,}\nonumber\end{aligned}$$ where we take any presentation of the module $\mathfrak{X}$ as in Equation \[lmixi4\] for any $m\geq2$ and $n:=\operatorname*{rk}(\mathfrak{A})$. Since all classes on the left-hand side are represented by rank one modules, the map is uniquely determined by declaring $[a\mathfrak{A}]-[\mathfrak{A}]\mapsto\lbrack a]$, using the notation $a\mathfrak{A}$ of Equation \[lciops1\].
See [@MR0376619 Consequence II of Theorem 1] (and again keep in mind Remark \[rmk\_FroehlichU\_DifferentDef\]).
Additivity and its consequences
===============================
Let us recall the Additivity Theorem of algebraic $K$-theory. Suppose $\mathsf{C}$, $\mathsf{D}$ are exact categories and $f_{i}:\mathsf{C}\rightarrow\mathsf{D}$ for $i=1,2,3$ are exact functors such that for each object $C\in\mathsf{C}$ we get a short exact sequence $f_{1}(C)\hookrightarrow
f_{2}(C)\twoheadrightarrow f_{3}(C)$, and this short exact sequence is functorial in $C$. A more elegant and precise way to set this up is to write $\mathcal{E}\mathsf{D}$ for the exact category of exact sequences in $\mathsf{D}$, [@MR2606234 Exercise 3.9] and consider $(f_{i})_{i=1,2,3}$ a single exact functor $\mathsf{C}\rightarrow\mathcal{E}\mathsf{D}$.$$\xymatrix{
& C \ar[dl]_{f_1} \ar[d]^{f_2} \ar[dr]^{f_3} \\
f_1(C) \ar@{^{(}->}[r] & f_2(C) \ar@{->>}[r] & f_3(C)
}
\label{lmisu6}$$
\[Additivity\]For every exact functor $(f_{i})_{i=1,2,3}:\mathsf{C}\rightarrow\mathcal{E}\mathsf{D}$ we have $f_{2\ast}=f_{1\ast}+f_{3\ast}$, where $f_{i\ast}:K(\mathsf{C})\rightarrow K(\mathsf{D})$ denotes the map induced from the exact functor $f_{i}$.
See [@MR3076731 Ch. V, Theorem 1.2]. First of all, we deduce the following standard vanishing theorem.
\[Eilenberg swindle\]\[lemma\_EilenbergSwindle\]If an exact category $\mathsf{C}$ is closed under countable products (or under countable coproducts), then $K(\mathsf{C})=0$.
For example given in [@obloc Lemma 4.2], but since the proof illustrates how to use Additivity in a powerful way, we repeat the full argument here: Suppose $\mathsf{C}$ is closed under coproducts. Use the exact functor $\mathsf{C}\rightarrow\mathcal{E}\mathsf{C}$ sending any object $X$ to the exact sequence $X\hookrightarrow\bigoplus_{i\in\mathbb{N}}X\overset
{s}{\twoheadrightarrow}\bigoplus_{i\in\mathbb{N}}X$, where the map $s$ sends the $i$-th factor to the $(i-1)$-th for $i\geq1$. Naming these functors $f_{1},f_{2},f_{3}$ as in Diagram \[lmisu6\], we obtain $\operatorname*{id}_{\mathsf{C}\ast}=f_{2\ast}-f_{3\ast}$, but $f_{2}=f_{3}$, showing that the identity map agrees with the zero map, forcing our claim to hold. If $\mathsf{C}$ is closed under products instead, use the same sequence, but with products instead.
We can now prove several fundamental theorems solely on the basis of Additivity and topological considerations in the category $\mathsf{LCA}_{\mathfrak{A}}$. In particular, at this point we will do a few things which hinge mostly on topology, and far less on the underlying algebraic right $\mathfrak{A}$-module structure of objects.
\[Local Triviality\]\[thm\_LocalTriviality\]Let $F$ be a number field and $\mathfrak{A}$ an order in a finite-dimensional semisimple $F$-algebra $A$. Suppose $\mathfrak{p}$ is a finite place of $F$. Let $\mathfrak{O}$ be any order in $A_{\mathfrak{p}}$ (for example $\mathfrak{A}_{p}$ or the maximal order). Then the composition$$K(\mathfrak{O})\longrightarrow K(A_{\mathfrak{p}})\longrightarrow
K(\mathsf{LCA}_{\mathfrak{A}})$$ is zero. Here the first arrow is induced from the ring inclusion $\mathfrak{O}\subset A_{\mathfrak{p}}$, while the latter sends $A_{\mathfrak{p}}$ to itself, but equipped with the locally compact topology.
We define an exact functor $p:\operatorname*{PMod}(\mathfrak{O})\rightarrow
\mathcal{E}\mathsf{LCA}_{\mathfrak{A}}$. We define it on the projective generator $\mathfrak{O}$ of the category $\operatorname*{PMod}(\mathfrak{O})$ by sending it to$$\mathfrak{O}\hookrightarrow A_{\mathfrak{p}}\twoheadrightarrow A_{\mathfrak{p}}/\mathfrak{O}$$ in $\mathsf{LCA}_{\mathfrak{A}}$. Let $p$ be the residual characteristic of the local field $Z_{\mathfrak{p}}$. Then $\mathfrak{O}$ carries the topology of a finite rank free $\mathcal{O}_{\mathfrak{p}}$-module, where $\mathcal{O}_{\mathfrak{p}}$ is the ring of integers of $Z_{\mathfrak{p}}$, $A_{\mathfrak{p}}$ carries the topology of a finite-dimensional $Z_{\mathfrak{p}}$-vector space and the quotient $A_{\mathfrak{p}}/\mathfrak{O}$ is seen to necessarily carry the discrete topology. Note that $\mathfrak{O}$ is compact. The Additivity Theorem implies that $p_{2\ast
}=p_{1\ast}+p_{3\ast}$, where $p_{1},p_{2},p_{3}$ denote the exact functors to the left (resp. middle, resp. right) entry of the short exact sequence. Since $p_{1\ast}$ and $p_{3\ast}$ factor over $\mathsf{LCA}_{\mathfrak{A},D}$ resp. $\mathsf{LCA}_{\mathfrak{A},C}$, both of which have zero $K$-theory (Lemma \[lemma\_EilenbergSwindle\], use that arbitrary direct sums of discrete groups are discrete, and arbitrary products of compact groups are compact by Tychonoff’s Theorem), it follows that $p_{2\ast}=0+0$.
Of course it would have been sufficient to prove this with $\mathfrak{O}$ the unique maximal order and use that every order is contained in it. However, the way we present the proof above it is particularly clear that all which is really used is the compactness of $\mathfrak{O}$ and the discreteness of the respective cokernel, so the above proof is in a way simpler since it does not even use the algebraic theory of orders in semisimple algebras.
In degree one this has the following important consequence.
\[cor\_LocalTrivInDegreeOne\]Let $F$ be a number field and $\mathfrak{A}$ an order in a finite-dimensional semisimple $F$-algebra $A$. Suppose $\mathfrak{p}$ is a finite place of $F$. Then the composition$$\mathfrak{A}_{\mathfrak{p}}^{\times}\longrightarrow A_{\mathfrak{p}}^{\times
}\longrightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})$$ is zero.
We recall the reciprocity law, [@etnclca Theorem 13.1]:
\[Reciprocity Law\]\[thm\_reciprocity\_law\]Let $F$ be a number field and $\mathfrak{A}$ an order in a finite-dimensional semisimple $F$-algebra $A$. Then the composition$$K(A)\longrightarrow K(\widehat{A})\oplus K(A_{\mathbb{R}})\overset
{\operatorname*{sum}}{\longrightarrow}K(\mathsf{LCA}_{\mathfrak{A}})
\label{lmits1}$$ is zero.
Here the first map stems from the exact functor $X\mapsto(X\otimes_{A}\widehat{A}\,,\,X\otimes_{\mathbb{Z}}\mathbb{R)}$. The second map sends an $\widehat{A}$-module to itself, but equipped with the adelic topology, and maps a free right $A_{\mathbb{R}}$-module to itself, equipped with the real vector space topology.
\[Signs\]\[rem\_SignInReciprocityLaw\]In Theorem \[thm\_reciprocity\_law\] we really mean the sum map on the right, and not the difference.
\[cor\_RecipLawInDegreeOne\]Let $F$ be a number field and $\mathfrak{A}$ an order in a finite-dimensional semisimple $F$-algebra $A$. Then the composition$$A^{\times}\longrightarrow J(A)\longrightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})$$ is zero.
Noncommutative idèles I
=======================
In this section we shall establish an idèle presentation of the group $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$. We first prove an analogous result using $K_{1}$-idèles and then use reduced norms to translate this into the claim which we want to prove. This is analogous to the proof of Fröhlich’s idèle presentation, Equation \[lmixi5\] in Curtis–Reiner [MR892316]{}. It originates from ideas of Wilson [@MR0447211].
For auxiliary purposes, we define the $K_{1}$-analogue of the idèle group,$$JK_{1}(A):=\left\{ (a_{\mathfrak{p}})_{\mathfrak{p}}\in\left. \prod
_{\mathfrak{p}}K_{1}(A_{\mathfrak{p}})\right\vert a_{\mathfrak{p}}\in\operatorname*{im}K_{1}(\mathfrak{A}_{\mathfrak{p}})\text{ for all but
finitely many places }\mathfrak{p}\right\} \text{,} \label{lzal1}$$ where $\mathfrak{p}$ runs over all places, finite and infinite, and the condition on $a_{\mathfrak{p}}$ is considered satisfied if $\mathfrak{p}$ is an infinite place. The image in $\operatorname*{im}K_{1}(\mathfrak{A}_{\mathfrak{p}})$ refers to the natural map $K_{1}(\mathfrak{A}_{p})\longrightarrow K_{1}(A_{\mathfrak{p}})$, which in general need not be injective. The definition of $JK_{1}(A)$ does not depend on the choice of the order $\mathfrak{A}$, for the same reason as in the definition of $J(A)$. If $\mathfrak{A}^{\prime}$ is a further order, we have $\mathfrak{A}_{\mathfrak{p}}=(\mathfrak{A}^{\prime})_{\mathfrak{p}}$ for all but finitely many places. Next, we define$$UK_{1}^{\operatorname{fin}}(\mathfrak{A}):=\left\{ (a_{\mathfrak{p}})_{\mathfrak{p}}\in\prod_{\mathfrak{p}\text{ finite}}K_{1}(\mathfrak{A}_{\mathfrak{p}})\right\} \text{.} \label{lmixi5a}$$ These definitions roughly match the ones in Curtis–Reiner [@MR892316 (49.16) Proposition] and Wilson [@MR0447211], except that we also include the infinite places.
Suppose $\mathfrak{p}$ is any place of $F$. Define$$\tilde{\xi}_{\mathfrak{p}}:K_{1}(A_{\mathfrak{p}})\longrightarrow
K_{1}(\mathsf{LCA}_{\mathfrak{A}})\text{,} \label{lmixi6}$$ based on the exact functor sending a finitely generated projective right $A_{\mathfrak{p}}$-module to itself, equipped with its natural locally compact topology (i.e. the $\mathbb{Q}_{p}$-vector space topology if $\mathfrak{p}$ is a finite place over the prime $p$, or the $\mathbb{R}$-vector space topology if $\mathfrak{p}$ is an infinite place).
We can make this map explicit in the Nenashev presentation: By Example \[example\_MakeAutToNenashevRepresentative\] the natural morphism below on the left$$A_{\mathfrak{p}}^{\times}\rightarrow K_{1}(A_{\mathfrak{p}})\text{,}\qquad
a\mapsto\left[
\xymatrix{
0 \ar@<1ex>@{^{(}->}[r]^-{0} \ar@<-1ex>@{^{(}.>}[r]_-{0} & {A_{\mathfrak{p}}}
\ar@<1ex>@{->>}[r]^{\cdot a} \ar@<-1ex>@{.>>}[r]_{1} & {{A_{\mathfrak{p}}}}
}\right] \label{lmixi10_1}$$ is given in terms of the Nenashev presentation by the double exact sequence above on the right, and moreover this map is an isomorphism since $A_{\mathfrak{p}}$ is semisimple. Use the same Nenashev representative for its image in $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$, just additionally equipped with the natural topology.
\[Prop\_IdentifyK1LCA\_Use\_K1Ideles\]Let $\mathfrak{A}$ be a regular order in a finite-dimensional semisimple $\mathbb{Q}$-algebra $A$. The map$$\tilde{\xi}:\frac{JK_{1}(A)}{\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})}\overset{\sim}{\longrightarrow}K_{1}(\mathsf{LCA}_{\mathfrak{A}})$$ given by $\tilde{\xi}_{\mathfrak{p}}$ on all factors in the restricted product in Equation \[lzal1\], induces an isomorphism.
We shall see how the possibility to quotient out the image of $K_{1}(A)$ comes precisely from the Reciprocity Law, Theorem \[thm\_reciprocity\_law\], while quotienting out the image of $UK_{1}^{\operatorname{fin}}(\mathfrak{A})$ stems from Local Triviality, Theorem \[thm\_LocalTriviality\]. The key point in the proof of the proposition is to show that these account for the entire kernel of $\tilde{\xi}$.
We split the proof into a series of individual verifications.
\[mz1\]The map $\tilde{\xi}$ is well-defined.
Observe that$$JK_{1}(A)=\underset{S}{\underrightarrow{\operatorname*{colim}}}\left(
\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})+\bigoplus
_{\mathfrak{p}\in S}K_{1}(A_{\mathfrak{p}})\right) \text{,}$$ where $S$ runs over all finite subsets of places of $Z$, partially ordered by inclusion, and we understand the sum in the big round brackets as the subgroup generated inside $JK_{1}(A)$ by these subgroups. Hence, in order to define $\tilde{\xi}$ it suffices to define it on each $\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})+\bigoplus_{\mathfrak{p}\in S}K_{1}(A_{\mathfrak{p}})$ in a way compatible with replacing $S$ by a bigger finite set. We define$$\tilde{\xi}_{S}:u\cdot\prod_{\mathfrak{p}\in S}a_{\mathfrak{p}}\mapsto
\prod_{\mathfrak{p}\in S}\tilde{\xi}_{\mathfrak{p}}(a_{\mathfrak{p}})\qquad\text{for}\qquad u\in\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})\text{, }a_{\mathfrak{p}}\in K_{1}(A_{\mathfrak{p}})\text{.}$$ We claim that $\tilde{\xi}_{S}$ is well-defined: We only need to show that the intersection$$\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})\cap\left(
\bigoplus_{\mathfrak{p}\in S}K_{1}(A_{\mathfrak{p}})\right) =\bigoplus
_{\mathfrak{p}\in S}\operatorname*{im}K_{1}(\mathfrak{A}_{\mathfrak{p}})$$ gets sent to zero. However, this follows from Local Triviality, Theorem \[thm\_LocalTriviality\]. Thus, $\tilde{\xi}:=$ $\operatorname*{colim}_{S}\tilde{\xi}_{S}$, verifying that we get a well-defined map$$\frac{JK_{1}(A)}{\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})}\longrightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})\text{.} \label{lmixi2}$$ Given any $a\in K_{1}(A)$, we obtain that $\tilde{\xi}(a)=\tilde{\xi}_{S}(a)$ for $S$ sufficiently big. Hence, by the fundamental Reciprocity Law, Theorem \[thm\_reciprocity\_law\], we have $\xi(a)=0$. Thus, the morphism set up in Equation \[lmixi2\] descends to the quotient modulo $\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})$.
Next, we set up a commutative diagram$$\xymatrix{
& K_2(\mathsf{LCA}_{\mathfrak{A} }) \ar[d] \\
K_1(\mathfrak{A}) \ar[d]_{\gamma} \ar[r]^{1} \ar@{}[dr]|{X}
& K_1(\mathfrak{A}) \ar[d] \\
K_1(A) \oplus K_1(A_{\mathbb{R}}) \ar[r]^{\operatorname{pr}_2 } \ar@
{->}[d]_{\alpha} \ar@{}[dr]|{Y} & K_1(A_{\mathbb{R}}) \ar[d] \\
\frac{JK_{1}(A)}{\operatorname*{im}UK^{\operatorname{fin}}_{1}(\mathfrak{A})}
\ar[r]^{\tilde{\xi} }
\ar@{->>}[d]_{j} \ar@{}[dr]|{Z} & K_1(\mathsf{LCA}_{\mathfrak{A} }) \ar@
{->>}[d] \\
\frac{JK_{1}(A)}{\operatorname*{im}K_1(A) + \operatorname*{im}UK^{\operatorname{fin}}_{1}(\mathfrak{A}) + \operatorname*{im}K_1(A_{\mathbb
{R}})} \ar[r]_-{w}
& \operatorname{Cl}(\mathfrak{A}),
}
\label{lfigA1}$$ also in several steps. The morphism $\alpha$ is the difference of the natural maps (we elaborate on the precise definition in the proof), $\operatorname*{pr}_{2}$ denotes the projection to the second summand. The bottom horizontal map $w$ amounts to Fröhlich’s idèle description of the class group, Equation \[lmixi5\].
\[mz2\]The bottom horizontal map $w$ is an isomorphism.
This is [@MR892316 (49.16) Proposition]. Loc. cit. Curtis and Reiner define $JK_{1}(A)$ without the infinite places. However, since we additionally quotient out by the infinite place contribution $\operatorname*{im}K_{1}(A_{\mathbb{R}})$, this difference gets remedied.
\[mz3\]The columns in Figure \[lfigA1\] are exact.
(Step 1) In the right column, we use the long exact sequence of [@etnclca Theorem 11.3]. This is the only input of the proof which uses the assumption that $\mathfrak{A}$ is a regular order. This sequence terminates in$$\cdots\longrightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})\overset
{c}{\longrightarrow}K_{0}(\mathfrak{A})\overset{a}{\longrightarrow}K_{0}(A_{\mathbb{R}})\longrightarrow K_{0}(\mathsf{LCA}_{\mathfrak{A}})\longrightarrow0\text{.} \label{lmixi8}$$ We know that $K_{0}(A_{\mathbb{R}})\cong\mathbb{Z}^{n}$ for some $n$ since $A_{\mathbb{R}}$ is semisimple (and concretely $n$ is the number of factors in the Artin–Wedderburn decomposition), and $\operatorname*{Cl}(\mathfrak{A})\hookrightarrow K_{0}(\mathfrak{A})$ is the kernel of the rank map. It follows that $\ker(a)$, in the notation introduced in Equation \[lmixi8\], contains at most the subgroup $\operatorname*{Cl}(\mathfrak{A})$. On the other hand, by the Jordan–Zassenhaus theorem [@MR1972204 (26.4) Theorem] the class group $\operatorname*{Cl}(\mathfrak{A})$ is finite, but $K_{0}(A_{\mathbb{R}})\cong\mathbb{Z}^{n}$ is torsion-free, so $\operatorname*{Cl}(\mathfrak{A})$ is contained in the kernel. We deduce $\ker
(a)=\operatorname*{Cl}(\mathfrak{A})$, and by the exactness of Equation \[lmixi8\], we have $\operatorname*{im}(c)=\operatorname*{Cl}(\mathfrak{A})$. This yields the truncation of the exact sequence which we use as the right column.(Step 2) The left column is set up as follows: The map $\gamma$ is just the sum of the natural maps coming from the ring homomorphisms $\mathfrak{A}\rightarrow A$ and $\mathfrak{A}\rightarrow A_{\mathbb{R}}$. Analogously, $\alpha$ is the difference of the identity map $K_{1}(A_{\mathbb{R}})\rightarrow K_{1}(A_{\mathbb{R}})$ for the infinite places, minus the diagonal map$$K_{1}(A)\longrightarrow JK_{1}(A)\text{,}\qquad\qquad a\longmapsto
(a,a,\ldots)$$ (involving all places of $F$, even the infinite ones). The composition is zero: Given any $\alpha\in K_{1}(\mathfrak{A})$, in the factors of $JK_{1}(A)$ corresponding to infinite places we subtract the same values, so it is zero at the infinite places. At each finite place $\mathfrak{p}$, we have the factorization $\mathfrak{A}\longrightarrow\mathfrak{A}_{\mathfrak{p}}\longrightarrow A_{p}$, showing that the image of this contribution comes from $UK_{1}^{\operatorname{fin}}(\mathfrak{A})$.(Step 3) It is also exact at this position. Assume we are given $(x,y)\in K_{1}(A)\oplus
K_{1}(A_{\mathbb{R}})$ such that $\alpha(x,y)=0$. Firstly, this means that for every finite place $\mathfrak{p}$ the image of $x$ under $K_{1}(A)\rightarrow
K_{1}(A_{\mathfrak{p}})$ lies in the image $\operatorname*{im}K_{1}(\mathfrak{A}_{\mathfrak{p}})$. Collecting this data for all finite places, we find $x^{\prime}\in K_{1}(\widehat{\mathfrak{A}})$ such that $(x,x^{\prime})$ maps to zero in the Wall exact sequence$$K_{1}(\mathfrak{A})\overset{\operatorname*{diag}}{\longrightarrow}\underset{(x,x^{\prime})}{K_{1}(A)\oplus K_{1}(\widehat{\mathfrak{A}})}\overset{\operatorname*{diff}}{\longrightarrow}K_{1}(\widehat{A})\longrightarrow K_{0}(\mathfrak{A})\longrightarrow\cdots\text{,}$$ see [@MR892316 (42.19) Theorem]. By the exactness of this sequence, we learn that $x=x^{\prime}\in K_{1}(\mathfrak{A})$. For the infinite places, $\alpha(x,y)=0$ now just means that $y\in K_{1}(A_{\mathbb{R}})$ also agrees with the image of $x$ under the map $\mathfrak{A}\rightarrow A_{\mathbb{R}}$. However, this means that $(x,y)$ is diagonal coming from $K_{1}(\mathfrak{A})$, settling exactness at this point of the column.(Step 4) The composition $j\circ\alpha$ is visibly zero, we just quotient out exactly the image of this map; and for the same reason we have exactness here. Finally, being a quotient, the last arrow is clearly surjective.
The square $X$ in Figure \[lfigA1\] commutes.
Obvious.
\[sw\_SqY\]The square $Y$ in Figure \[lfigA1\] commutes.
(Step 1) Suppose $x\in K_{1}(A)$. Then $\alpha$ sends it to the diagonal element $(-x,-x,\ldots)$. By the Reciprocity Law, Theorem \[thm\_reciprocity\_law\], this gets mapped to zero in $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$. Correspondingly, the projection on the second factor, $\operatorname*{pr}_{2}$, also sends it to zero. (Step 2) Suppose $y\in
K_{1}(A_{\mathbb{R}})$. Then $\alpha$ sends it to the $K_{1}$-idèle $(z_{\mathfrak{p}})_{\mathfrak{p}}$ with $z_{\mathfrak{p}}=y$ (under the natural map) for $\mathfrak{p}$ an infinite place, and $z_{\mathfrak{p}}=1$ for a finite place. The map $K_{1}(A_{\mathbb{R}})\longrightarrow
K_{1}(\mathsf{LCA}_{\mathfrak{A}})$ is induced from sending a right $A_{\mathbb{R}}$-module to itself, equipped with the real topology, see [@etnclca Theorem 11.2]. However, this is the same as what $\tilde{\xi
}_{\mathfrak{p}}$ (of Equation \[lmixi6\]) does in the case of $\mathfrak{p}$ infinite.
It remains to check that the square $Z$ commutes. This is a little more difficult than the previous steps in the proof. Let us first recall a few useful facts about the structure of division algebras over the $p$-adic numbers.
\[rmk\_LocalStructureAtFinitePlace\]Let $F$ be a number field and suppose $\mathfrak{p}$ is a finite place. Suppose $D$ is a division algebra over $F$ (this means: its center comes with a given inclusion $F\hookrightarrow
\zeta(D)$) and whose center is a finite field extension of $F_{\mathfrak{p}}$. Let$$v:F_{\mathfrak{p}}^{\times}\longrightarrow\mathbb{R}$$ be the normalized $\mathfrak{p}$-adic valuation, i.e. its image (usually called the ‘value group’) is $\alpha\mathbb{Z}\subset\mathbb{R}$ for some $\alpha\in\mathbb{R}$. Then there is a unique extension $\tilde{v}:D^{\times
}\rightarrow\mathbb{R}$ to a discrete valuation on the division algebra. It is still a discrete valuation with value group $\frac{\alpha}{e}\mathbb{Z}\subset\mathbb{R}$ for some integer $e\geq1$. Define$$\Delta:=\{x\in D\mid\tilde{v}(x)\geq0\}\text{.}$$ Then $\Delta$ is an $\mathcal{O}_{\mathfrak{p}}$-order in $D$ and more generally (a) it is the unique maximal $\mathcal{O}_{F}$-order inside $D_{\mathfrak{p}}$, (b) it can alternatively be characterized as the integral closure of $\mathcal{O}_{\mathfrak{p}}$ inside $D$. Upon normalization to have integer values, the valuation $\tilde{v}$ gives rise to an exact sequence$$\Delta^{\times}\hookrightarrow D^{\times}\twoheadrightarrow\mathbb{Z}\text{.}
\label{lmixi11}$$ A uniformizer $\pi$ is any element $\pi\in D^{\times}$ which gets mapped to $+1$ in this sequence, as in the commutative case. These results are found as an overview in [@MR1278263 §1.4], or with complete proofs in [@MR1972204 Ch. 3, §12].
The square $Z$ in Figure \[lfigA1\] commutes.
(Step 1) It suffices to check this for an arbitrary place $\mathfrak{p}$ and an arbitrary $a\in K_{1}(A_{\mathfrak{p}})$, because if the claim is settled in all these cases, it follows from all maps being group homomorphisms. So, let us assume $\mathfrak{p}$ is chosen and fixed.(Step 2) Let us get the case of $\mathfrak{p}$ an infinite place out of the way. In this case, we need to check it for an $a$ coming from a summand of $K_{1}(A_{\mathbb{R}})$. However, any such $a$ lies in the image of $\alpha$ and hence by the commutativity of the square $Y$ (Lemma \[sw\_SqY\]), any such element goes to zero in the bottom row of Figure \[lfigA1\]. In particular, the square $Z$ commutes for this input.(Step 3) It remains to deal with $\mathfrak{p}$ a finite place. By the Artin–Wedderburn Theorem, we can split $A_{\mathfrak{p}}$ into a direct sum of matrix algebras $M_{n}(D)$ over division algebras over $F_{\mathfrak{p}}$. By Morita invariance of $K$-theory, we have the equivalence of $K$-theory spaces$$K(D)\overset{\sim}{\longrightarrow}K(M_{n}(D))\qquad\text{under}\qquad
D\hookrightarrow M_{n}(D)$$ as a top left $(1\times1)$-minor in an $(n\times n)$-matrix, so it suffices to check it for arbitrary $a\in K_{1}(D)$, where $D$ is a division algebra over $F_{\mathfrak{p}}$. The natural map $A_{\mathfrak{p}}^{\times}\rightarrow
K_{1}(A_{\mathfrak{p}})$ is surjective, see Equation \[lmixi10\_1\]. Hence, we can start with an arbitrary $a\in A_{\mathfrak{p}}^{\times}$. Now, we use a little bit of structure theory: Since $D$ is a division algebra over $F$ and $\mathfrak{p}$ a finite place, and we are in the setting which we had recalled in Elaboration \[rmk\_LocalStructureAtFinitePlace\]. Let us use the notation loc. cit. Then by Equation \[lmixi11\] we may write $a=u\pi^{n}$ for $u\in\Delta^{\times}$ a unit of the maximal order, $\pi$ a uniformizer and $n\in\mathbb{Z}$. Since our maps are group homomorphisms, it suffices to check commutativity in the two cases$$\text{(a) }a:=u\qquad\qquad\text{and}\qquad\qquad\text{(b) }a:=\pi
\label{lmixi15}$$ separately. Doing this, the first steps of the computation agree in the cases (a) and (b), so henceforth assume we are in one (but any) of these cases.(Step 4) We begin by considering first the map $\tilde{\xi}$, followed by the right downward arrow to $\operatorname*{Cl}(\mathfrak{A})$. From here onward, the proof uses the same strategy as [@etnclca2 Lemma 3.5]. We recall from loc. cit. that the right downward sequence comes from the long exact sequence$$\cdots\longrightarrow K_{1}(A_{\mathbb{R}})\longrightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})\overset{\partial}{\longrightarrow}K_{0}(\mathsf{Mod}_{\mathfrak{A},fg})\longrightarrow K_{0}(A_{\mathbb{R}})\longrightarrow\cdots$$ and more specifically the right downward arrow in Figure \[lfigA1\] corresponds to $\partial$ in the above sequence. Thus, in order to compute $\partial$, we need to go through the construction of this long exact sequence. As was explained loc. cit., this differential agrees with$$\partial=\partial^{\ast}\circ\Phi^{-1}\circ q\text{,} \label{lmixi12}$$ where these maps come from the diagram$$\xymatrix{
\cdots\ar[r] & {{\pi}_1}K({\mathsf{Mod}_{\mathfrak{A}}}) \ar[r] \ar
[d] & {{\pi}_1}K({{\mathsf{Mod}_{\mathfrak{A}}}}/{{\mathsf{Mod}_{{\mathfrak
{A}},fg}}}) \ar@{=}[d]^{\Phi} \ar[r]^-{{\partial}^{\ast}} & {{\pi}_0}K({\mathsf{Mod}_{{\mathfrak{A}},fg}}) \ar[d] \\
{{\pi}_1}K(\mathsf{LCA}_{\mathfrak{A},cg}) \ar[r] & {{\pi}_1}K(\mathsf
{LCA}_{\mathfrak{A}}) \ar[r]_-{q} & {{\pi}_1}K({\mathsf{LCA}_{\mathfrak{A}}}/{\mathsf{LCA}_{\mathfrak{A},cg}}) \ar[r] & {{\pi}_0}K(\mathsf{LCA}_{\mathfrak{A},cg}),
}
\label{lmixi14}$$ where in turn $\partial^{\ast}$ arises as the boundary map of the long exact sequence of homotopy groups coming from the localization sequence$$\mathsf{Mod}_{\mathfrak{A},fg}\longrightarrow\mathsf{Mod}_{\mathfrak{A}}\longrightarrow\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}\text{.}$$ We refer to the proof of [@etnclca2 Lemma 3.5] for a little more background how these facts are proven. Following Equation \[lmixi10\_1\], the $K_{1}$-class we consider can explicitly be spelled out as$$\tilde{\xi}(a)=\left[
\xymatrix{
0 \ar@<1ex>@{^{(}->}[r]^-{0} \ar@<-1ex>@{^{(}.>}[r]_-{0} & D \ar@
<1ex>@{->>}[r]^{a} \ar@<-1ex>@{.>>}[r]_{1} & D
}\right]$$ in the Nenashev presentation. Now, we simply follow the formula in Equation \[lmixi12\] step by step. Applying $q$, we still can use the same representative in $K_{1}(\mathsf{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg})$. However, we also have the exact sequence $\Delta
\hookrightarrow D\twoheadrightarrow D/\Delta$, in $\mathsf{LCA}_{\mathfrak{A}}$, where $\Delta$ is the maximal order of $D$. Now, $\Delta$ is a free finite rank $\mathbb{Z}_{p}$-module and a compact clopen subgroup of $D$. Hence, the quotient $D/\Delta$ carries the discrete topology. Since $\Delta$ is compact, it is in particular compactly generated, and thus a zero object in the quotient exact category $\mathsf{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg}$. Thus, we get an isomorphism $D\cong D/\Delta$ in $\mathsf{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg}$. Hence,$$\left[
\xymatrix{
0 \ar@<1ex>@{^{(}->}[r]^-{0} \ar@<-1ex>@{^{(}.>}[r]_-{0} & D/{\Delta}
\ar@<1ex>@{->>}[r]^{a} \ar@<-1ex>@{.>>}[r]_{1} & D/{\Delta}
}\right] \label{lmixi14b}$$ represents that *same* class in $K_{1}(\mathsf{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg})$. Now, simply read the above as a Nenashev representative in $K_{1}(\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg})$. The exact functor $\Phi$ as in Diagram \[lmixi14\] sends this to itself, equipped with the discrete topology, but since $D/\Delta$ has carried the discrete topology anyway, we see that we have found a preimage of the element in Equation \[lmixi14b\] under $\Phi$. Hence, in view of Equation \[lmixi12\], it suffices to compute $\partial^{\ast}$ of the element, regarded in $\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}$. As was explained in the proof of [@etnclca2 Lemma 3.5], this reduces to a homotopical problem: The boundary map $\partial^{\ast}$ in$$K_{1}(\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg})\longrightarrow K_{0}(\mathsf{Mod}_{\mathfrak{A},fg})$$ does the following: Starting with a closed loop around the zero object in the $K$-theory space of $\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}$, it lifts it to a non-closed path from zero to some object $(P,Q)$ under the fibration$$K(\mathsf{Mod}_{\mathfrak{A}})\longrightarrow K(\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg})\text{,}$$ and then the output is the $\pi_{0}$-element corresponding to the connected component of $(P,Q)$ in which this path ends. Hence, in order to compute $\partial^{\ast}$, we need to produce an explicit lift of the closed loop in question.(Step 5) Following the concrete properties of the Gillet–Grayson model as summarized in §\[sect\_AlgKThy\] and Figure \[lmixi3b\], the element in Equation \[lmixi14b\] corresponds to a loop, depicted below on the left:$${\includegraphics[
height=1.2168in,
width=4.9415in
]{gfx1-eps-converted-to.pdf}}
\label{lmixivio}$$ where the top horizontal arrow $e(l)$ comes from the $1$-simplex$$\xymatrix@!=0.5in{
0 \ar@{^{(}.>}[r] & D/{\Delta} \ar@{.>>}[r]^{1} & D/{\Delta} & \qquad&
0 \ar@{^{(}->}[r] & D/{\Delta} \ar@{->>}[r]^{a} & D/{\Delta}
}$$ in the Gillet–Grayson model of $K(\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg})$. Consider the $1$-simplex $T$ given by$$\xymatrix@!=0.5in{
0 \ar@{^{(}.>}[r] & D/{\Delta} \ar@{.>>}[r]^{1} & D/{\Delta} & \qquad&
{a^{-1}\Delta/ \Delta} \ar@{^{(}->}[r] & D/{\Delta} \ar@{->>}[r]^{a}
& D/{\Delta}
}$$ in the Gillet–Grayson model of $K(\mathsf{Mod}_{\mathfrak{A}})$. Next, note that $a^{-1}\Delta/\Delta$ is a finitely generated $\mathfrak{p}$-torsion right $\mathfrak{A}_{\mathfrak{p}}$-module (note: this is true for both $a$ a unit since then it is zero, or if $a=\pi$, for $\pi$ is a uniformizer of $D$ and since the valuation of $D$ extends the one of $F_{\mathfrak{p}}$, the $\mathfrak{p}$-torsion property follows, see Elaboration \[rmk\_LocalStructureAtFinitePlace\]). But being $\mathfrak{p}$-torsion, it follows that $a^{-1}\Delta/\Delta$ is also a finitely generated right ($\mathfrak{p}$-torsion) $\mathfrak{A}$-module. Thus, $a^{-1}\Delta/\Delta
\in\mathsf{Mod}_{\mathfrak{A},fg}$ and we conclude that the quotient exact functor $\mathsf{Mod}_{\mathfrak{A}}\longrightarrow\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}$ sends the $1$-simplex $T$ to the top horizontal arrow in Figure \[lmixivio\]. Thus, we have found a candidate for the desired lift of the closed loop. It is depicted above in Figure \[lmixivio\] on the right. Analogous to the argument in the proof of [@etnclca2 Lemma 3.5], we obtain that the endpoint of the path is the vertex $(0,a^{-1}\Delta/\Delta)$ in the Gillet–Grayson model. In summary,$$\begin{aligned}
\partial(a) & =\left( \partial^{\ast}\circ\Phi^{-1}\circ q\right) \left[
\xymatrix{
0 \ar@<1ex>@{^{(}->}[r]^-{0} \ar@<-1ex>@{^{(}.>}[r]_-{0} & D \ar@
<1ex>@{->>}[r]^{\cdot a} \ar@<-1ex>@{.>>}[r]_{1} & D
}\right] \\
& =\text{connected component of }(0,a^{-1}\Delta/\Delta)\end{aligned}$$ and by §\[subsect\_ExplicitK0\], this is $[a^{-1}\Delta/\Delta]-[0]\in
K_{0}(\mathsf{Mod}_{\mathfrak{A},fg})$.(Step 6) We follow the element $a$ through the square $Z$ in the other way. The map $j$ sends it simply to itself, merely under a further quotient operation. Finally, the bottom horizontal map $w$ in Figure \[lfigA1\] needs to be unravelled. To this end, we refer to [@MR892316 proof of (49.14) Corollary, p. 223]. In the notation loc. cit. we consider the idèle formed with $a$ in the $\mathfrak{p}$-component and $1$ as the component of all other places. Since $a\in\Delta$ in both cases (a) and (b) of Equation \[lmixi15\] we get$$\lbrack\Delta/a\Delta]\in K_{0}(\mathfrak{A})\text{.}$$ Finally, we have the isomorphism$$\frac{a^{-1}\Delta}{\Delta}\underset{\cdot a}{\overset{\sim}{\longrightarrow}}\frac{\Delta}{a\Delta}\text{,}$$ and hence $[\Delta/a\Delta]=[a^{-1}\Delta/\Delta]$ agree in $K_{0}(\mathfrak{A})$. This confirms that in both cases (a) and (b) we get the same element, whichever way we follow the square $Z$. This finishes the proof.
Now we can finally prove Proposition \[Prop\_IdentifyK1LCA\_Use\_K1Ideles\].
\[Proof of Proposition \[Prop\_IdentifyK1LCA\_Use\_K1Ideles\]\]Consider the commutative diagram in Figure \[lfigA1\]. (Step 1) We truncate it to three rows by quotienting out the images of the top vertical arrows, so that the top horizontal arrow now reads$$\operatorname*{pr}\nolimits_{2}:\left( K_{1}(A)\oplus K_{1}(A_{\mathbb{R}})\right) /\operatorname*{im}K_{1}(\mathfrak{A})\longrightarrow
K_{1}(A_{\mathbb{R}})/\operatorname*{im}K_{1}(\mathfrak{A})\text{.}$$ Since the map is projection to the second factor, the image of $K_{1}(A)$ in the left-hand side quotient is the kernel of this map. On the other hand, the map is obviously surjective. (Step 2) Now apply the Snake Lemma to the remaining diagram. Since $w$ is an isomorphism by Lemma \[mz2\], the resulting snake long exact sequence is$$0\rightarrow\operatorname*{im}K_{1}(A)\overset{\alpha}{\rightarrow}\ker
(\tilde{\xi})\rightarrow0\rightarrow0\rightarrow\operatorname*{coker}(\tilde{\xi})\rightarrow0\text{,}$$ where by $\operatorname*{im}K_{1}(A)$ we mean the kernel discussed in Step 1. We deduce that $\tilde{\xi}$ is surjective, and that once we additionally quotient out by $\operatorname*{im}K_{1}(A)$ on the left-hand side (now with image taken under $\alpha$), $\tilde{\xi}$ will be additionally injective on the quotient. However, this is precisely the claim of Proposition \[Prop\_IdentifyK1LCA\_Use\_K1Ideles\].
Noncommutative idèles II
========================
We work under the standing assumptions of §\[sect\_Setup\]. In particular, $F$ denotes a number field and $\mathfrak{A}\subset A$ an $\mathcal{O}_{F}$-order in a finite-dimensional semisimple $F$-algebra $A$. We shall write $K_{0}(\mathfrak{A},\mathbb{R})_{\operatorname{Swan}}$ whenever we want to stress that we think of the relative $K$-group $K_{0}(\mathfrak{A},\mathbb{R})$ in terms of the explicit Swan presentation. Concretely, in the case at hand this means that generators have the shape $[P,\varphi,Q]$, where $P,Q$ are finitely generated projective right $\mathfrak{A}$-modules and$$\varphi:P_{\mathbb{R}}\overset{\sim}{\longrightarrow}Q_{\mathbb{R}}$$ an isomorphism of right $A_{\mathbb{R}}$-modules. Then $K_{0}(\mathfrak{A},\mathbb{R})_{\operatorname{Swan}}$ is the free abelian group generated by these formal elements modulo Swan’s Relation A and Relation B. We will not recall these in full, see [@etnclca2 §1.1], or [@MR3076731 Chapter II, Definition 2.10], where they are called relation (a) and (b).
\[example\_OneMiddleIsZero\]In $K_{0}(\mathfrak{A},\mathbb{R})_{\operatorname{Swan}}$ the identity $[\mathfrak{X},1,\mathfrak{X}]=0$ holds for any finitely generated projective right $\mathfrak{A}$-module $\mathfrak{X}$. To see this, use Swan’s Relation B to obtain $[\mathfrak{X},1,\mathfrak{X}]+[\mathfrak{X},1,\mathfrak{X}]=[\mathfrak{X},1\cdot
1,\mathfrak{X}]$.
\[def\_FrohlichTheory\]Let an idèle $a=(a_{\mathfrak{p}})_{\mathfrak{p}}\in J(A)$ be given.
1. Then there is a unique $\mathcal{O}_{F}$-lattice $a\mathfrak{A}$ inside $A$ such that$$(a\mathfrak{A})_{\mathfrak{p}}=a_{\mathfrak{p}}\mathfrak{A}_{\mathfrak{p}}
\label{lciops1E}$$ holds for all finite places $\mathfrak{p}$ of $F$. See [@MR0376619 §2, Equation 2.2 and Theorem 1] for background.
2. Secondly, we write $a_{\infty}$ for the map$$a_{\infty}:A_{\mathbb{R}}\overset{\sim}{\longrightarrow}A_{\mathbb{R}}$$ coming from those components $a_{\mathfrak{p}}$ alone for which $\mathfrak{p}$ runs through the infinite places of $F$.
Given any $a\mathfrak{A}$ as in part (1) of the definition, tensoring with the rationals yields an injection$$a\mathfrak{A}\subset\mathbb{Q}\cdot(a\mathfrak{A})=\mathbb{Q}\cdot
\mathfrak{A}=A$$ since each $a\mathfrak{A}$ is a torsion-free right $\mathfrak{A}$-module. We we view $a\mathfrak{A}\subset A\subset A_{\mathbb{R}}$ as a full rank $\mathbb{Z}$-lattice inside the real vector space $A_{\mathbb{R}}$. Thus, we may alternatively regard $a_{\infty}$ as a map $a_{\infty}:\mathfrak{A}_{\mathbb{R}}\overset{\sim}{\rightarrow}(a\mathfrak{A})_{\mathbb{R}}$. We use this for the following definition.
\[def\_MapTheta\]Define$$\theta:J(A)\longrightarrow K_{0}(\mathfrak{A},\mathbb{R})\text{,}\qquad
\qquad(a_{\mathfrak{p}})_{\mathfrak{p}}\longmapsto\lbrack\mathfrak{A},a_{\infty},a\mathfrak{A}]$$ with $a\mathfrak{A}$ and $a_{\infty}$ as in Definition \[def\_FrohlichTheory\].
We first need to check that this definition makes sense at all.
\[lemma\_ThetaWelldef\]The map $\theta$ is well-defined.
We check that $\theta$ is a group homomorphism. Suppose $a:=(a_{\mathfrak{p}})_{\mathfrak{p}}$ and $b:=(b_{\mathfrak{p}})_{\mathfrak{p}}$ are idèles. We compute$$\theta(a)+\theta(b)=[\mathfrak{A},a_{\infty},a\mathfrak{A}]+[\mathfrak{A},b_{\infty},b\mathfrak{A}]=[\mathfrak{A}\oplus\mathfrak{A},a_{\infty}\oplus
b_{\infty},a\mathfrak{A}\oplus b\mathfrak{A}]$$ by using Swan’s Relation A for the split exact sequence of the direct sum. Now we use the classification of projective modules, in the following concrete form: There exists an isomorphism of right $\mathfrak{A}$-modules,$$\varphi:a\mathfrak{A}\oplus b\mathfrak{A}\overset{\sim}{\longrightarrow
}ab\mathfrak{A}\oplus\mathfrak{A}\text{.}$$ by Example \[example\_CancellationRankTwo\] (see [@MR0376619 Theorem 1, (ii)] for the proof this example is based on), and this isomorphism sits in a commutative diagram of projective right $\mathfrak{A}$-modules comprising the solid arrows in$$\xymatrix{
\mathfrak{A} \oplus\mathfrak{A} \ar@{..>}[d]_{a_{\infty} \oplus b_{\infty}}
\ar@{^{(}->}[r]^{1 \oplus1} & \mathfrak{A} \oplus\mathfrak{A} \ar@
{->>}[r] \ar@{..>}[d]_{a_{\infty} \cdot b_{\infty} \oplus1} & 0 \ar@
{..>}[d]^{1} \\
a\mathfrak{A} \oplus b\mathfrak{A} \ar@{^{(}->}[r]_{\varphi} & ab\mathfrak{A}
\oplus\mathfrak{A} \ar@{->>}[r] & 0, \\
}$$ while the dotted downward arrows only exist (and commute alongside the solid arrows) after tensoring everything over $\mathfrak{A}$ with $A_{\mathbb{R}}$. This data can serve as the input for Swan’s Relation A, and implies that$$\begin{aligned}
\lbrack\mathfrak{A}\oplus\mathfrak{A},a_{\infty}\oplus b_{\infty
},a\mathfrak{A}\oplus b\mathfrak{A}] & =[\mathfrak{A}\oplus\mathfrak{A},(a_{\infty}\cdot b_{\infty})\oplus1,ab\mathfrak{A}\oplus\mathfrak{A}]-[0,1,0]\\
& =[\mathfrak{A},a_{\infty}b_{\infty},ab\mathfrak{A}]+[\mathfrak{A},1,\mathfrak{A}]-[0,1,0]\end{aligned}$$ and by Example \[example\_OneMiddleIsZero\] we obtain equality to $[\mathfrak{A},a_{\infty}b_{\infty},ab\mathfrak{A}]=\theta(a\cdot b)$, proving our claim.
\[lemma\_mixi1\]The map $\theta$ sends $\operatorname*{im}UK^{\operatorname{fin}}_{1}(\mathfrak{A})\subseteq JK_{1}(A)$ to zero.
Suppose the idèle $a:=(a_{\mathfrak{p}})_{\mathfrak{p}}$ comes from $UK_{1}^{\operatorname{fin}}(\mathfrak{A})$. Then it sits entirely in the components of the finite places, so $a_{\infty}=1$. Next, by Equation \[lciops1E\] (or Equation \[lciops1\]) the lattice $a\mathfrak{A}$ is uniquely determined by the equation $(a\mathfrak{A})_{\mathfrak{p}}=a_{\mathfrak{p}}\mathfrak{A}_{\mathfrak{p}}$ for all finite places $\mathfrak{p}$. Since $\mathfrak{A}_{p}^{\times}\rightarrow K_{1}(\mathfrak{A}_{p})$ is surjective, we have $a_{\mathfrak{p}}\cdot
\mathfrak{A}_{\mathfrak{p}}=\mathfrak{A}_{\mathfrak{p}}$, and thus $(a\mathfrak{A})_{\mathfrak{p}}=\mathfrak{A}_{\mathfrak{p}}$ holds for all finite $\mathfrak{p}$, uniquely characterizing $a\mathfrak{A}$ as $a\mathfrak{A}=\mathfrak{A}$. Hence, $\theta(a)=[\mathfrak{A},a_{\infty
},a\mathfrak{A}]=[\mathfrak{A},1,\mathfrak{A}]=0$ by Example \[example\_OneMiddleIsZero\].
\[lemma\_mixi2\]The map $\theta$ sends $\operatorname*{im}K_{1}(A)\subseteq
JK_{1}(A)$ to zero.
The map $A^{\times}\rightarrow K_{1}(A)$ is an isomorphism. Thus, the image in question consists of the diagonally constant idèles $(a)_{\mathfrak{p}}$, with $a\in A^{\times}$. In particular, $a_{\infty}=a$. Since $A=\mathfrak{A}\otimes_{\mathbb{Z}}\mathbb{Q}$, we can write $a=\frac{1}{n}a_{0}$ for some $a_{0}\in\mathfrak{A}\setminus\{0\}$ and $n\in\mathbb{Z}_{\geq1}\subset\mathfrak{A}\setminus\{0\}$. We have $\mathfrak{A}\setminus\{0\}\subset
A^{\times}$ and since $\theta$ is a group homomorphism, without loss of generality it suffices to prove our claim for all elements $a\in
\mathfrak{A}\setminus\{0\}$. For such an element, we find a commutative diagram$$\xymatrix{
\mathfrak{A} \ar@{..>}[d]_{1} \ar@{^{(}->}[r]^{1} & \mathfrak{A} \ar@
{->>}[r] \ar@{..>}[d]_{a} & 0 \ar@{..>}[d]^{1} \\
\mathfrak{A} \ar@{^{(}->}[r]_{a} & a\mathfrak{A} \ar@{->>}[r] & 0, \\
}$$ where the solid arrows exist on the level of projective right $\mathfrak{A}$-modules and the dotted arrows only after tensoring with $A_{\mathbb{R}}$ (the same notation which we had used in the proof of Lemma \[lemma\_ThetaWelldef\]). Having such a diagram, Swan’s Relation A yields $[\mathfrak{A},a,a\mathfrak{A}]=[\mathfrak{A},1,\mathfrak{A}]+[0,1,0]$ and by Example \[example\_OneMiddleIsZero\] both terms on the right vanish.
We will construct a commutative diagram$$\xymatrix{
K_1(\mathfrak{A}) \ar[d] \ar[r]^{1} \ar@{}[dr]|{X}
& K_1(\mathfrak{A}) \ar[d] \\
K_1(A_{\mathbb{R}}) \ar[r]^{1} \ar@{->}[d]_{\beta} \ar@{}[dr]|{Y}
& K_1(A_{\mathbb{R}}) \ar[d] \\
\frac{JK_{1}(A)}{\operatorname*{im}K_1(A) + \operatorname*{im}UK^{\operatorname{fin}}_{1}(\mathfrak{A})} \ar[r]^{{\theta} }
\ar@{->>}[d]_{j} \ar@{}[dr]|{Z} & K_0({\mathfrak{A} },\mathbb{R}) \ar@
{->>}[d] \\
\frac{JK_{1}(A)}{\operatorname*{im}K_1(A) + \operatorname*{im}UK^{\operatorname{fin}}_{1}(\mathfrak{A}) + \operatorname*{im}K_1(A_{\mathbb
{R}})} \ar[r]_-{-{\sigma}_{\mathfrak{A}}}
& \operatorname{Cl}(\mathfrak{A}),
}
\label{lbixi1}$$ using the following maps: (a) The map $\beta$ sends $a\in A_{\mathbb{R}}^{\times}$ to the idèle $(a_{\mathfrak{p}})_{\mathfrak{p}}$ with $a_{\mathfrak{p}}=1$ for all finite places $\mathfrak{p}$, while $a_{\mathfrak{p}}$ agrees with the corresponding component of $a$ for all infinite places. We may suggestively write$$(1,1,\ldots,1,\underset{a_{\infty}}{\underbrace{a_{v},\ldots,a_{v^{\prime}}}})\text{,}$$ where $v,\ldots,v^{\prime}$ are the infinite places. (b) The map $\sigma_{\mathfrak{A}}$ sends an idèle to its associated ideal class. This construction comes from Fröhlich’s idèle classification of projective modules, see [@MR0376619 §2, Theorem 1, especially Consequence II]. This is the inverse map to the one in Theorem \[thm\_FrohlichTheory\].
Diagram \[lbixi1\] commutes.
The commutativity of the square $X$ is obvious. Square $Y$ is not much harder: Suppose we are given $a\in A_{\mathbb{R}}$. Then $\beta$ maps it to the idèle$$\hat{a}:=(1,1,\ldots,1,\underset{a_{\infty}}{\underbrace{a_{v},\ldots
,a_{v^{\prime}}}})$$ with $a_{\infty}=a$. Next, $\theta$ sends this to $[\mathfrak{A},a,\hat
{a}\mathfrak{A}]$. However, the idèle $\hat{a}$ differs from the neutral element $(1,1,\ldots,1)$ only in the infinite places. Thus, $\hat
{a}\mathfrak{A}=\mathfrak{A}$ by Example \[example\_IdeleNontrivOnlyAtInfinity\], i.e. we get $[\mathfrak{A},a,\mathfrak{A}]$. On the other hand, the map $K_{1}(A_{\mathbb{R}})\rightarrow K_{0}(\mathfrak{A},\mathbb{R})$ sends $a$ to the same class, by definition (see [@etnclca2 Theorem 3.2], the map $\delta$ in the diagram loc. cit., or [@MR0245634 p. 215]). Hence, Square $Y$ commutes. The map $\theta$ is well-defined and vanishes on $\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})+\operatorname*{im}K_{1}(A)$ by Lemma \[lemma\_mixi1\] and \[lemma\_mixi2\]. It remains to check Square $Z$. Let $a:=(a_{\mathfrak{p}})_{\mathfrak{p}}$ be some idèle in $JK_{1}(A)$. Then $\theta(a)=[\mathfrak{A},a_{\infty},a\mathfrak{A}]$ and going down on the right sends this to $[\mathfrak{A}]-[a\mathfrak{A}]$ (see [@etnclca2 Theorem 3.2], and concretely the proof of Lemma 3.5 loc. cit.). On the other hand, the idèle $a$ under Fröhlich’s idèle classification of projective right $\mathfrak{A}$-modules corresponds to $\sigma_{\mathfrak{A}}(a)=[a\mathfrak{A}]-[\mathfrak{A}]$, see [@MR0376619 §2, Consequence II of Theorem 1]. The map $\sigma_{\mathfrak{A}}$ loc. cit. specifically describes the class group as the kernel from $K_{0}(\mathfrak{A})$ under the rank, as in Equation \[lmixi7\]. We use the same notation as in Fröhlich’s article. Thanks to the negative sign, Square $Z$ commutes.
\[Lemma\_ExactColumns\]The columns in Diagram \[lbixi1\] are exact.
(Step 1) The right column comes from the standard long exact sequence of relative $K$-groups,$$\cdots\longrightarrow K_{1}(\mathfrak{A})\longrightarrow K_{1}(A_{\mathbb{R}})\longrightarrow K_{0}(\mathfrak{A},\mathbb{R})\longrightarrow K_{0}(\mathfrak{A})\longrightarrow K_{0}(A_{\mathbb{R}})\longrightarrow
\cdots\text{.}$$ From this sequence we obtain that$$K_{1}(\mathfrak{A})\longrightarrow K_{1}(A_{\mathbb{R}})\longrightarrow
K_{0}(\mathfrak{A},\mathbb{R})\longrightarrow\ker\left( K_{0}(\mathfrak{A})\longrightarrow K_{0}(A_{\mathbb{R}})\right) \longrightarrow0$$ is exact, but one can show that the kernel on the right consists precisely of those classes in $K_{0}(\mathfrak{A})$ with vanishing rank, i.e. this kernel agrees with the one in Equation \[lmixi7\]. This settles the right column. (Step 2) For the left column it is clear that $j$ is just the quotient map under the image coming from $\beta$. Thus, exactness is clear, except perhaps at $K_{1}(A_{\mathbb{R}})$. We check this now: Firstly, we have $j\circ
\beta=0$ because the image of $K_{1}(\mathfrak{A})$ in $JK_{1}(A)$ is contained in the image of $K_{1}(A)$, which we had quotiented out. Thus, we only need to show that every $a\in K_{1}(A_{\mathbb{R}})$ such that $\beta(a)\equiv0$ comes from $K_{1}(\mathfrak{A})$. Suppose such an $a\in
K_{1}(A_{\mathbb{R}})$ is given. As an idèle, we may write a representative of its image in $JK_{1}(A)$ suggestively as$$\beta(a)=(1,1,\ldots,1,\underset{a_{\infty}}{\underbrace{a_{v},\ldots
,a_{v^{\prime}}}})\text{.} \label{lmixi20}$$ However, since we had assumed that $\beta(a)\equiv0$, we also know that $\beta(a)=x\cdot y$ with $x\in UK_{1}^{\operatorname{fin}}(\mathfrak{A})$ and $y\in K_{1}(A)$. We have tacitly dropped distinguishing between these elements are their images in the idèles. Since the image of $UK_{1}^{\operatorname{fin}}(\mathfrak{A})$ is supported in the finite places alone by Equation \[lmixi5a\], we learn that $y_{\mathfrak{p}}=a_{\mathfrak{p}}$ holds for all infinite places. Further, since $\operatorname*{im}K_{1}(A)$ is diagonal, this means that we can assume that $a=y\otimes1_{\mathbb{R}}$. However, Equation \[lmixi20\] also implies that$$1=x_{\mathfrak{p}}\cdot y_{\mathfrak{p}}\qquad\text{for all finite places
}\mathfrak{p}\text{.}$$ Thus, in Wall’s exact sequence for $K$-theory, [MR892316]{},$$K_{1}(\mathfrak{A})\overset{\operatorname*{diag}}{\longrightarrow}\underset{(y,x)}{K_{1}(A)\oplus K_{1}(\widehat{\mathfrak{A}})}\overset
{\operatorname*{diff}}{\longrightarrow}K_{1}(\widehat{A})\longrightarrow
K_{0}(\mathfrak{A})\longrightarrow\cdots$$ we learn that the pair $(y,x)$ goes to zero, and therefore there exists some $z\in K_{1}(\mathfrak{A})$ such that $(y,x)=(z\otimes1_{A},z\otimes
1_{\widehat{\mathfrak{A}}})$ under the diagonal map. Hence, $a=y\otimes
1_{\mathbb{R}}=(z\otimes1_{A})\otimes1_{\mathbb{R}}$, proving that $a$ lies in the image from $K_{1}(\mathfrak{A})$ as desired.
Main results regarding Fröhlich’s idèle perspective
---------------------------------------------------
We write $U^{\operatorname*{fin}}(\mathfrak{A})$ with the same meaning as $U(\mathfrak{A})$ in this section in order to stress the analogy with $UK_{1}^{\operatorname*{fin}}$. In case the reader has forgotten the definition, see Equation \[lcixiDex1\].
\[Global–Local Formula, Swan presentation\]\[thm\_GlobalLocal\_Swan\]Let $F$ be a number field, $\mathcal{O}_{F}$ its ring of integers. Suppose $\mathfrak{A}\subset A$ is an $\mathcal{O}_{F}$-order in a finite-dimensional semisimple $F$-algebra $A$. Then the following diagrams, whose rows are isomorphisms, commute:
1. (Classical idèle formulation)$$\xymatrix{
\cfrac{J(A)}{J^{1}(A)+\operatorname*{im}(A^{\times})+\operatorname
*{im}U^{\operatorname{fin}}(\mathfrak{A})} \ar[r]^-{\theta}_-{\sim} \ar@
{->>}[d] &
K_{0}(\mathfrak{A},\mathbb{R}) \ar@{->>}[d] \\
\cfrac{J(A)}{J^{1}(A)+\operatorname*{im}(A^{\times})+\operatorname
*{im}U^{\operatorname{fin}}(\mathfrak{A})+\operatorname*{im}(A_{\mathbb{R}
}^{\times})}
\ar[r]_-{\sim} & \operatorname{Cl}(\mathfrak{A})
}$$
2. ($K_{1}$-idèle formulation)$$\xymatrix{
\cfrac{JK_{1}(A)}{\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})} \ar[r]^-{\theta}_-{\sim} \ar@{->>}[d] &
K_{0}(\mathfrak{A},\mathbb{R}) \ar@{->>}[d] \\
\cfrac{JK_{1}(A)}{\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A}) + \operatorname*{im}K_1(A_{\mathbb{R} })}
\ar[r]_-{\sim} & \operatorname{Cl}(\mathfrak{A})
}$$
3. (Formulation in terms of the center)$$\xymatrix{
\cfrac{J(\zeta(A))}{\operatorname*{im}(\zeta(A)^{+,\times})+\prod
_{\mathfrak{p},\text{fin.}}\operatorname*{im}(\operatorname{nr}(\mathfrak
{A}^{\times}_{\mathfrak{p} }))} \ar[r]^-{\vartheta}_-{\sim} \ar@{->>}[d] &
K_{0}(\mathfrak{A},\mathbb{R}) \ar@{->>}[d] \\
\cfrac{J(\zeta(A))}{\operatorname*{im}(\zeta(A)^{+,\times})+\prod
_{\mathfrak{p},\text{fin.}}\operatorname*{im}(\operatorname{nr}(\mathfrak
{A}^{\times}_{\mathfrak{p} }))+\operatorname*{im}(A_{\mathbb{R} }^{\times})}
\ar[r]_-{\sim} & \operatorname{Cl}(\mathfrak{A})
}$$ Here $\zeta(-)$ denotes the center, and $()^{+}$ means: We restrict to $a\in\zeta(A)$ such that $a_{\mathfrak{p}}>0$ for all real places of $F$ which ramify in $A$. The products run only over the finite places of $F$.
Recall the notation $a\mathfrak{A}$ of Fröhlich’s idèle classification of projective $\mathfrak{A}$-modules (Equation \[lciops1\]). In terms of the Swan presentation, the maps are given by$$\theta:(a_{\mathfrak{p}})_{\mathfrak{p}}\longmapsto\lbrack\mathfrak{A},a_{\infty},a\mathfrak{A}]\qquad\vartheta:(a_{\mathfrak{p}})_{\mathfrak{p}}\longmapsto\lbrack\mathfrak{A},\operatorname*{nr}\nolimits^{-1}(a_{\infty
}),\operatorname*{nr}\nolimits^{-1}(a)\mathfrak{A}]\text{.}$$ Here the reduced norm maps $\operatorname*{nr}\nolimits^{-1}(-)$ are understood component-wise for each place $\mathfrak{p}$. See the proof for further clarification.
The notation $(-)^{+}$ is standard, and for example also used by Curtis and Reiner [@MR1038525], [@MR892316]. Note that its meaning depends on $A$ and not just on $\zeta(A)$. Let us point out that some readers might prefer to think of the idèles as a multiplicative gadget and would prefer writing $\cdot$ in the quotients rather than $+$. This is a matter of taste and we hope it does not lead to confusion.
The bottom horizontal map in all formulations (1)-(3) of the theorem give well-known variations on a theme due to Fröhlich. The idèle formulation (1) of the bottom horizontal map appears as Consequence II in Fröhlich [@MR0376619]. For the other formulations we refer the reader to [@MR892316 (49.16)-(49.23)] or Wilson’s paper [@MR0447211]. Thus, in a sense, the above theorem generalizes these results from the locally free class group $\operatorname*{Cl}(\mathfrak{A})$ to all of $K_{0}(\mathfrak{A},\mathbb{R})$.
The theorem is not really new. Agboola and Burns [@MR2192383] basically give the $\operatorname*{Hom}$-description version of a more general result, also for more general relative $K$-groups $K(\mathfrak{A},-)$, see [@MR2192383 Theorem 3.5]. See also [@MR2192383 Example 3.9, (2)] for an idèle description derived from it, especially Equation (10) loc. cit., which agrees literally with our formula (in this particular example $\mathfrak{A}$ is commutative, so there is no $J^{1}$-quotient in their paper since it is zero on the nose).
(Step 1) The core of the proof is the verification of (2), the $K_{1}$-idèle formulation. We claim that the map $\theta$ of Definition \[def\_MapTheta\] induces the isomorphism. Diagram \[lbixi1\] is commutative with exact columns by Lemma \[Lemma\_ExactColumns\]. The two top horizontal maps are the identity and thus isomorphisms. The bottom horizontal map $-\sigma_{\mathfrak{A}}$ is an isomorphism by Fröhlich’s description of the locally free class group, in the format of [MR892316]{}. A crucial remark on notation: Loc. cit. the group Curtis and Reiner denote by $JK_{1}(A)$ does not contain the infinite places. Correspondingly, their $UK_{1}(A)$ is precisely the group $UK_{1}^{\operatorname{fin}}(\mathfrak{A})$ in this paper. Since we also quotient out by $\operatorname*{im}K_{1}(A_{\mathbb{R}})$ in the bottom row of Diagram \[lbixi1\], the quotient on the left is the same as the one discussed in Curtis and Reiner [@MR892316 (49.16) Proposition] on the left. Finally, the Five Lemma implies that $\theta$ is also an isomorphism. Now the commutativity of Square $Z$ in Diagram \[lbixi1\], as well as both horizontal maps being isomorphisms, is the same as claim (2) in the above theorem.(Step 2) The rest of the proof follows the pattern of [@MR892316 (49.17)-(49.23)]. We first prove (3). As in the proof of [@MR892316 (49.17) Theorem], the reduced norm induces isomorphisms$$\operatorname*{nr}:K_{1}(A_{\mathfrak{p}})\overset{\sim}{\longrightarrow}\zeta(A_{\mathfrak{p}})^{\times} \label{lfixi1}$$ for all places $\mathfrak{p}$. Hence, we obtain an isomorphism$$\operatorname*{nr}:JK_{1}(A)\overset{\sim}{\longrightarrow}J(\zeta(A))\text{.}
\label{lfixi2}$$ Next, by the Hasse–Schilling–Maass theorem [MR1972204]{}, the image of the reduced norm of $\operatorname*{im}K_{1}(A)$ inside $JK_{1}(A)$ under this map is $\zeta(A)^{+,\times}$. Similarly, the image of the group $K_{1}(\mathfrak{A}_{\mathfrak{p}})$ gets sent to $\operatorname*{im}(\operatorname*{nr}K_{1}(\mathfrak{A}_{\mathfrak{p}}))$. However, since $\mathfrak{A}_{\mathfrak{p}}$ is local, the natural map $\mathfrak{A}_{\mathfrak{p}}^{\times}\rightarrow K_{1}(\mathfrak{A}_{\mathfrak{p}})$ is surjective, so this image agrees with the image $\operatorname*{im}(\operatorname*{nr}\mathfrak{A}_{\mathfrak{p}}^{\times})$ under the composition$$\mathfrak{A}_{\mathfrak{p}}^{\times}\longrightarrow K_{1}(\mathfrak{A}_{\mathfrak{p}})\longrightarrow K_{1}(A_{\mathfrak{p}})\overset
{\operatorname*{nr}}{\longrightarrow}\zeta(A_{\mathfrak{p}})^{\times}\text{.}$$ All we have just done was transporting the subgroups appearing in the denominator in (2) under the isomorphism of Equation \[lfixi2\]. Thus, we obtain an isomorphism$$\frac{JK_{1}(A)}{\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})}\overset{\sim}{\longrightarrow}\frac{J(\zeta(A))}{\operatorname*{im}\zeta(A)^{+,\times}+{\textstyle\prod\nolimits_{\mathfrak{p}}}
\operatorname*{im}(\operatorname*{nr}\mathfrak{A}_{\mathfrak{p}}^{\times})}\text{.}$$ Since the map is still described by the reduced norm in all components of the idèles, formulation (2) implies formulation (3) in our claim. Finally, use that the natural map $A_{\mathbb{R}}^{\times}\rightarrow K_{1}(A_{\mathbb{R}})$ is also surjective.(Step 3) Finally, we prove formulation (1). Define $c:J(A)\longrightarrow JK_{1}(A)$ by using the maps $A_{\mathfrak{p}}^{\times}\rightarrow K_{1}(A_{\mathfrak{p}})$ for all places $\mathfrak{p}$. Since both $A_{\mathfrak{p}}^{\times}\rightarrow K_{1}(A_{\mathfrak{p}})$ as well as $\mathfrak{A}_{\mathfrak{p}}^{\times}\rightarrow K_{1}(\mathfrak{A}_{\mathfrak{p}})$ are surjective, it is clear that the morphism $c$ is surjective. Let $J^{1}(A)$ denote the kernel of this map. Consider the commutative diagram$$\xymatrix{
J(A) \ar@{->>}[r]^-{c} \ar[dr]_{\operatorname{nr}} & JK_1(A) \ar
[d]^{\operatorname{nr}}_{\cong} \\
& J(\zeta(A)).
}$$ This yields the alternative characterization$$J^{1}(A)=\left\{ (a_{\mathfrak{p}})_{\mathfrak{p}}\in J(A)\mid
\operatorname*{nr}\nolimits_{A_{\mathfrak{p}}}(a_{\mathfrak{p}})=1\right\}$$ as the idèles of reduced norm one. We obtain the isomorphism of groups$$\frac{J(A)}{J^{1}(A)}\underset{c}{\overset{\sim}{\longrightarrow}}JK_{1}(A)\text{.}$$ Moreover, under this isomorphism, the image $\operatorname*{im}A^{\times}$ inside $J(A)$ gets identified with $\operatorname*{im}K_{1}(A)$, and the image $\operatorname*{im}U^{\operatorname{fin}}(\mathfrak{A})$ with $\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})$. This finishes the proof.
\[Global–Local Formula, Nenashev presentation\]\[thm\_GlobalLocal\_Nenashev\]Let $F$ be a number field, $\mathcal{O}_{F}$ its ring of integers. Suppose $\mathfrak{A}\subset A$ is a regular $\mathcal{O}_{F}$-order in a finite-dimensional semisimple $F$-algebra $A$. Then the following diagrams, whose rows are isomorphisms, commute:
1. (Classical idèle formulation)$$\xymatrix{
\cfrac{J(A)}{J^{1}(A)+\operatorname*{im}(A^{\times})+\operatorname
*{im}U^{\operatorname{fin}}(\mathfrak{A})} \ar[r]^-{\theta}_-{\sim} \ar@
{->>}[d] &
{K_{1}(\mathsf{LCA}_{\mathfrak{A} })} \ar@{->>}[d] \\
\cfrac{J(A)}{J^{1}(A)+\operatorname*{im}(A^{\times})+\operatorname
*{im}U^{\operatorname{fin}}(\mathfrak{A})+\operatorname*{im}(A_{\mathbb{R}
}^{\times})}
\ar[r]_-{\sim} & \operatorname{Cl}(\mathfrak{A})
}$$
2. ($K_{1}$-idèle formulation)$$\xymatrix{
\cfrac{JK_{1}(A)}{\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A})} \ar[r]^-{\theta}_-{\sim} \ar@{->>}[d] &
{K_{1}(\mathsf{LCA}_{\mathfrak{A} })} \ar@{->>}[d] \\
\cfrac{JK_{1}(A)}{\operatorname*{im}K_{1}(A)+\operatorname*{im}UK_{1}^{\operatorname{fin}}(\mathfrak{A}) + \operatorname*{im}K_1(A_{\mathbb{R} })}
\ar[r]_-{\sim} & \operatorname{Cl}(\mathfrak{A})
}$$
3. (Formulation in terms of the center)$$\xymatrix{
\cfrac{J(\zeta(A))}{\operatorname*{im}(\zeta(A)^{+,\times})+\prod
_{\mathfrak{p},\text{fin.}}\operatorname*{im}(\operatorname{nr}(\mathfrak
{A}^{\times}_{\mathfrak{p} }))} \ar[r]^-{\vartheta}_-{\sim} \ar@{->>}[d] &
{K_{1}(\mathsf{LCA}_{\mathfrak{A} })} \ar@{->>}[d] \\
\cfrac{J(\zeta(A))}{\operatorname*{im}(\zeta(A)^{+,\times})+\prod
_{\mathfrak{p},\text{fin.}}\operatorname*{im}(\operatorname{nr}(\mathfrak
{A}^{\times}_{\mathfrak{p} }))+\operatorname*{im}(A_{\mathbb{R} }^{\times})}
\ar[r]_-{\sim} & \operatorname{Cl}(\mathfrak{A})
}$$ Here $\zeta(-)$ denotes the center, and $()^{+}$ means: We restrict to $a\in\zeta(A)$ such that $a_{\mathfrak{p}}>0$ for all real places of $F$ which ramify in $A$. The products run only over the finite places of $F$.
In terms of the Nenashev presentation, the maps are given by$$\theta:(a_{\mathfrak{p}})_{\mathfrak{p}}\mapsto\left[
\xymatrix{
0 \ar@<1ex>@{^{(}->}[r]^-{0} \ar@<-1ex>@{^{(}.>}[r]_-{0} & {A_{\mathbb{A}}}
\ar@<1ex>@{->>}[r]^{\cdot(\ldots,a_{\mathfrak{p}},\ldots)} \ar@<-1ex>@{.>>}[r]_{1}
& {A_{\mathbb{A}}}
}\right] \qquad\vartheta:(a_{\mathfrak{p}})_{\mathfrak{p}}\mapsto\left[
\xymatrix{
0 \ar@<1ex>@{^{(}->}[r]^-{0} \ar@<-1ex>@{^{(}.>}[r]_-{0} & {A_{\mathbb{A}}}
\ar@<1ex>@{->>}[r]^{\cdot(\ldots,{\operatorname{nr}\nolimits^{-1}}a_{\mathfrak{p}},\ldots)} \ar@<-1ex>@{.>>}[r]_{1}
& {A_{\mathbb{A}}}
}\right] \text{.}$$
We use the same proof as for Theorem \[thm\_GlobalLocal\_Swan\]. Simply replace Step 1 loc. cit. by Proposition \[Prop\_IdentifyK1LCA\_Use\_K1Ideles\]. Step 2 and Step 3 then follow analogously.
Extended boundary map
---------------------
We define the *relative free class group* as$$\operatorname*{Cl}(\mathfrak{A},\mathbb{R}):=\ker\left( K_{1}(\mathsf{LCA}_{\mathfrak{A}})\overset{\partial}{\longrightarrow}K_{0}(\mathfrak{A})\longrightarrow\prod_{p}K_{0}(\mathfrak{A}_{p})\right) \text{,}$$ where $\partial$ is the boundary map in the long exact sequence of [@etnclca Theorem 11.3]. This theorem also implies that this definition is equivalent to the one in Burns–Flach [@MR1884523 §2.9]. We follow the notation of *loc. cit.*: For an associative algebra $R$, we write $\zeta(R)$ for its center, and $\operatorname*{nr}_{R}$ denotes the reduced norm (see also [@MR1038525 §7D]).
We define the *extended boundary map* $\hat{\delta}_{\mathfrak{A},\mathbb{R}}^{1}:\zeta(A_{\mathbb{R}})^{\times}\rightarrow\operatorname*{Cl}(\mathfrak{A},\mathbb{R})$ as follows: Given $y\in\zeta(A_{\mathbb{R}})^{\times}$, pick some $\lambda\in\zeta(A)^{\times}$ such that $\lambda
y\in\operatorname*{im}(\operatorname*{nr}\nolimits_{A_{\mathbb{R}}})$. Then define$$\psi_{y,\lambda}:=\left( \prod_{p}\operatorname*{nr}\nolimits_{A_{p}}^{-1}(\lambda),\operatorname*{nr}\nolimits_{A_{\mathbb{R}}}^{-1}(\lambda
y)\right) \in K_{1}(\widehat{A})\oplus K_{1}(A_{\mathbb{R}})\text{.}$$ Then $\hat{\delta}_{\mathfrak{A},\mathbb{R}}^{1}(y):=\operatorname*{sum}(\psi_{y,\lambda})$, where the sum map is the one from Theorem \[thm\_reciprocity\_law\].$$\xymatrix{
\cdots\ar[r] & K_1(\mathfrak{A}) \ar[r] & K_1(A_{\mathbb{R}}) \ar[r] \ar@
{^{(}->}[d]_{\operatorname{nr}_{A_{\mathbb{R}}}} & \operatorname
*{Cl}(\mathfrak{A},\mathbb{R}) \\
& & \zeta(A_{\mathbb{R}})^{\times} \ar@{-->}[ur]_{\hat{\delta}_{\mathfrak
{A},\mathbb{R}}^{1}}
}$$
This definition is very close to the one given in Burns–Flach, albeit with a sum instead of a difference (and this is for the same reason as in Remark \[rem\_SignInReciprocityLaw\]).
The map $\hat{\delta}_{\mathfrak{A},\mathbb{R}}^{1}$ is well-defined.
We adapt [@MR1884523 Lemma 9] to our locally compact setting. We shall use the structure of the image of the reduced norm map in both the local as well as the global situation, see [@MR892316 (45.3)] for a summary sufficient for our purposes. As in loc. cit., given $y$, by Weak Approximation we find a (highly non-unique) $\lambda\in\zeta(A)^{\times}$ such that $y\lambda\in\operatorname*{im}(\operatorname*{nr}\nolimits_{A_{\mathbb{R}}})$. This is possible by the description of the image of the reduced norm of units over the reals, [@MR1972204 (33.4) Theorem], i.e. we just need to make $y\lambda$ positive at real places. Then $\operatorname*{nr}\nolimits_{A_{\mathbb{R}}}^{-1}(\lambda y)$ is a unique element, because the reduced norm is injective when restricted to $K_{1}(A_{\mathbb{R}})$ by [@MR1972204 (33.1) Theorem, (ii)]. For all but finitely many primes $p$, we have that the image of $\lambda$ in $\zeta(A_{p})^{\times}$ lies even in $\operatorname*{nr}\nolimits_{A_{p}}^{-1}\zeta(\mathfrak{A}_{p})^{\times}$ and that the latter lies in the image of $K_{1}(\mathfrak{A}_{p})$. If $\lambda^{\prime}$ is an alternative choice, we find$$\psi_{y,\lambda}\psi_{y,\lambda^{\prime}}^{-1}=\left( \prod_{p}\operatorname*{nr}\nolimits_{A_{p}}^{-1}(\lambda\lambda^{\prime-1}),\operatorname*{nr}\nolimits_{A_{\mathbb{R}}}^{-1}(\lambda
y)\operatorname*{nr}\nolimits_{A_{\mathbb{R}}}^{-1}(\lambda^{\prime}y)^{-1}\right) =\left( \prod_{p}\operatorname*{nr}\nolimits_{A_{p}}^{-1}(\lambda\lambda^{\prime-1}),\operatorname*{nr}\nolimits_{A_{\mathbb{R}}}^{-1}(\lambda\lambda^{\prime-1})\right) \text{.}$$ However, for elements $x\in A$ we have $\operatorname*{nr}\nolimits_{A}(x)=\operatorname*{nr}\nolimits_{A_{p}}(x)=\operatorname*{nr}\nolimits_{A_{\mathbb{R}}}(x)$ by [@MR1972204 (33.3) Theorem]. Thus, we get$$=\left( \prod_{p}\operatorname*{nr}\nolimits_{A}^{-1}(\lambda\lambda
^{\prime-1}),\operatorname*{nr}\nolimits_{A}^{-1}(\lambda\lambda^{\prime
-1})\right)$$ and then $\lambda\lambda^{\prime-1}\in\operatorname*{im}(\operatorname*{nr}_{A})$ by the Hasse–Schilling–Maass norm theorem, see [@MR1972204 (33.15) Theorem]. Thus, $\psi_{y,\lambda}\psi_{y,\lambda
^{\prime}}^{-1}$ is the image of $\operatorname*{nr}_{A}^{-1}(\lambda
\lambda^{\prime-1})$ in $K_{1}(\widehat{A})\oplus K_{1}(A_{\mathbb{R}})$ in Equation \[lmits1\]. But then $\operatorname*{sum}(\psi_{y,\lambda}\psi_{y,\lambda^{\prime}}^{-1})=0$ by the reciprocity law, Theorem \[thm\_reciprocity\_law\].
Similarly to the discussion in [@MR1884523 §2.9], the exact sequence$$\cdots\longrightarrow K_{1}(A_{\mathbb{R}})\longrightarrow K_{1}(\mathsf{LCA}_{\mathfrak{A}})\longrightarrow K_{0}(\mathfrak{A})\longrightarrow\cdots$$ can be truncated on the right and re-spliced to $$\cdots\longrightarrow K_{1}(A_{\mathbb{R}})\longrightarrow\operatorname*{Cl}(\mathfrak{A},\mathbb{R})\longrightarrow\operatorname*{Cl}(\mathfrak{A})\longrightarrow0\text{.}$$
Proof of the principal idèle fibration
======================================
This section is fairly independent of the rest of the text. It is entirely devoted to proving that$$K(\widehat{\mathfrak{A}})\times K(A)\longrightarrow K(\widehat{A})\times
K(A_{\mathbb{R}})\longrightarrow K(\mathsf{LCA}_{\mathfrak{A}})$$ in Equation \[lcixi1\] is indeed a fibration. While loc. cit. it is stated as a fibration of pointed simplicial sets having our conventions of §\[sect\_GettingPrecise\] in mind, we work on the level of spectra in this section, relying on the results and language of the previous article [@etnclca]. As the $K$-theory spaces in question are infinite loop spaces, this amounts to the same and is just a change of language.
A certain sign switch will play an important rôle in the proof, so let us begin with some careful considerations around signs:
\[elab\_SignsHtpyCartesianSquares\]Choose some $\varepsilon\in\{-1,+1\}$. Suppose $\mathsf{C}$ is a stable $\infty$-category and $h\mathsf{C}$ its homotopy category. We write $\Sigma$ and $\Omega=\Sigma^{-1}$ for the translation functors of $h\mathsf{C}$. Then a square$$\xymatrix{
A \ar[r]^{a} \ar[d]_{f} & B \ar[d]^{g} \\
A^{\prime} \ar[r]_{a^{\prime}} & B^{\prime}
}
\label{lmisu2}$$ in $\mathsf{C}$ is called *(homotopy) Cartesian* if there exists a morphism $\partial_{\square}:B^{\prime}\rightarrow\Sigma A$ in $h\mathsf{C}$ such that$$\xymatrix{
A \ar[r]^-{f+a} & A^{\prime} \oplus B \ar[r]^-{\varepsilon({a^{\prime}} - g)}
& B^{\prime} \ar[r]^-{\partial_{\square}} & \Sigma A
}
\label{lmisu1}$$ is a distinguished triangle in the category $h\mathsf{C}$. See Neeman [@MR1812507 §1.4] for a careful discussion purely on the level of $h\mathsf{C}$. For both choices of $\varepsilon$ this definition makes sense and one obtains the full theory. This choice of orientation is also discussed by Lurie, from a slightly different angle [@LurieHA Lemma 1.1.2.10]. In this paper we use the convention $\varepsilon:=1$ (which is compatible to [@obloc], [@etnclca], [@etnclca2]), but the other option would also work. Nothing would change, except a few signs here and there. Nonetheless, the following is important: Suppose we are given the commutative diagram$$\xymatrix{
A \ar[r]^{a} \ar@{}[dr]|{\square} \ar[d]_{f} & B \ar[d]^{g} \ar[r]^{b}
& C \ar[r]^-{\partial_{F_1}} \ar[d]^{h}_{\cong} & \Sigma A \ar[d]^{\Sigma f}
\\
A^{\prime} \ar[r]_{a^{\prime}} & B^{\prime} \ar[r]_{b^{\prime}} & C^{\prime}
\ar[r]_-{\partial_{F_2}} & \Sigma A^{\prime}
}
\label{lmisu4}$$ in $h\mathsf{C}$ (with the left two squares lifted to $\mathsf{C}$) and with $h$ an isomorphism in $h\mathsf{C}$. Then there is an attached distinguished triangle as in Equation \[lmisu1\] with $\partial_{\square}$ given as the composition$$B^{\prime}\overset{b^{\prime}}{\longrightarrow}C^{\prime}\underset{\sim
}{\overset{h}{\longleftarrow}}C\overset{\partial_{F_{1}}}{\longrightarrow
}\Sigma A\text{.}$$ in $h\mathsf{C}$. This is a variation of [@MR1812507 Lemma 1.4.3]. By the above definition, this means that the square on the left (marked by the central ‘$\square$’) is homotopy Cartesian in the stable $\infty$-category. Now, what if $f$ instead of $h$ is an equivalence? To figure this out, we rotate both distinguished triangles, giving the commutative diagram$$\xymatrix{
\Omega C \ar[r]^-{-\Omega\partial_{F_1}} \ar[d]_{\Omega h}^{\cong}
& A \ar[r]^{-a} \ar[d]_{f} & B \ar[d]^{g} \ar[r]^{-b} & C \ar[d]^{h}_{\cong}
\\
\Omega C^{\prime} \ar[r]_-{-\Omega\partial_{F_2}} & A^{\prime} \ar
[r]_{-a^{\prime}} & B^{\prime} \ar[r]_{-b^{\prime}} & C^{\prime}
}$$ in $h\mathsf{C}$ so that upon renaming $A,B,C$ we are in the desired situation. Next, check that$$\xymatrix@L=2.6mm{
A \ar[r]^-{f-a} & A^{\prime} \oplus B \ar[r]^-{\varepsilon({-a^{\prime}} - g)}
& B^{\prime} \ar[r]^-{-\partial_{\square}} & \Sigma A
}$$ is isomorphic to the triangle in Equation \[lmisu1\] (to see this: Map $A$ and $A^{\prime}$ to themselves via the identity, on $B$ and $B^{\prime}$ use the negative of the identity; all resulting squares commute. Note that this is only true because we use $-\partial_{\square}$; it is not possible to make this work without changing the sign there, or at some other point). It follows that this triangle is also distinguished. Now rename $\tilde{a}:=-a$, $\tilde{a}^{\prime}:=-a^{\prime}$ (same for $b,b^{\prime}$) and $C:=\Sigma D$. Then$$\xymatrix{
A \ar[r]^-{f+\tilde{a}} & A^{\prime} \oplus B \ar[r]^-{\varepsilon({\tilde
{a}^{\prime}} - g)} & B^{\prime} \ar[r]^-{-\partial_{\square}} & \Sigma A
}
\label{lmisu3}$$ is distinguished, and our input diagram reads$$\xymatrix{
D \ar[r] \ar[d]_{\Omega h}^{\cong} & A \ar[r]^{\tilde{a}} \ar[d]_{f} \ar@
{}[dr]|{\square} & B \ar[d]^{g} \ar[r]^{\tilde{b}} & \Sigma D \ar
[d]^{h}_{\cong} \\
D^{\prime} \ar[r] & A^{\prime} \ar[r]_{{\tilde{a}}^{\prime}} & B^{\prime}
\ar[r]_{{\tilde{b}}^{\prime}} & \Sigma D^{\prime}.
}
\label{lmisu3b}$$ Note that the distinguished triangle in Equation \[lmisu3\] has exactly the same shape as the one in Equation \[lmisu1\] except for the different sign of $\partial_{\square}$. We may summarize this as follows: Depending on whether the first or third vertical arrow of a commutative diagram of the shape of Equation \[lmisu4\] is an isomorphism (that is: $f$ or $h$), the other square will be homotopy Cartesian, and both variants only differ by the sign of $\partial_{\square}$ (which by an extension of [@LurieHA Lemma 1.1.2.10] is equivalent to mirroring the diagram along the diagonal from the upper left to the lower right). This is true independently of which sign $\varepsilon$ we use in the first place. We repeat that we use the convention $\varepsilon:=+1$ in this paper.
With this preparation on signs in place, we can begin the proof. Firstly, we elaborate on a theme due to Wall.
\[lemma\_WallSeqForGTheory\]Suppose $A$ is a finite-dimensional semisimple $\mathbb{Q}$-algebra and $\mathfrak{A}\subset A$ an order. Then there is a canonical fiber sequence$$G(\mathfrak{A})\overset{\iota}{\longrightarrow}G(\widehat{\mathfrak{A}})\oplus
G(A)\overset{\operatorname*{diff}}{\longrightarrow}G(\widehat{A})$$ in spectra. Here $\iota$ is the induced map on $K$-theory coming from the exact functors of tensoring with $\widehat{\mathfrak{A}}$ resp. $A$ on the right. Moreover, $\operatorname*{diff}=\rho-\tau$, where $\rho$ and $\tau$ are induced from the exact functors of tensoring with $\widehat{A}$ in both cases.
For the algebraic $K$-theory of projective modules and restricted to low degrees, this result was originally established by Wall. It was originally proven using a different method based on excision squares. A more general version is due to Swan, [@MR892316 (42.22) Remark, (ii)]. We give a quick self-contained account in contemporary language, if only to set up notation and signs.
Let $R$ be a unital associative ring, finite as a $\mathbb{Z}$-module. We write $\mathsf{Mod}_{R,fg}^{tor}$ for the abelian category of finitely generated right $R$-modules which are torsion over $\mathbb{Z}$, that is: The support of each modules over $\mathbb{Z}$ is supposed to be of codimension $\geq1$ in $\operatorname*{Spec}\mathbb{Z}$. Then $\mathsf{Mod}_{R,fg}^{tor}$ is a Serre subcategory of $\mathsf{Mod}_{R,fg}$ and the quotient abelian category is $\mathsf{Mod}_{R\otimes\mathbb{Q},fg}$. Now, applying Quillen’s Localization Theorem for Serre subcategories [@MR3076731 Ch. V, Theorem 5.1] both for $R=\mathfrak{A}$ as well as $R=\widehat{\mathfrak{A}}$, we obtain that the two rows in the diagram $$\xymatrix@L=2.6mm@!C=0.9in{
K({\mathsf{Mod}_{\mathfrak{A},fg}^{tor}}) \ar[r] \ar[d]_{i}^{\cong}
& K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar@{}[dr]|{\Diamond} \ar[r] \ar[d]_{j}
& K({\mathsf{Mod}_{A,fg}}) \ar[d]^{k} \ar[r]^{\partial_{\mathfrak{A}}^{tor \hookrightarrow fg}} & \Sigma K({\mathsf{Mod}_{\mathfrak{A},fg}^{tor}}) \ar[d]^{\Sigma i}_{\cong} \\
K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}}) \ar[r] & K({\mathsf
{Mod}_{\widehat{\mathfrak{A}},fg}}) \ar[r] & K({\mathsf{Mod}_{\widehat{A},fg}}) \ar[r]_{\partial_{\widehat{\mathfrak{A}}}^{tor \hookrightarrow fg}}
& \Sigma K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}})
}
\label{ltixi1}$$ are distinguished. The downward arrows are$$j:\mathsf{Mod}_{\mathfrak{A},fg}\longrightarrow\mathsf{Mod}_{\widehat
{\mathfrak{A}},fg}\text{,}\qquad M\mapsto M\otimes_{\mathfrak{A}}\widehat{\mathfrak{A}}$$$$k:\mathsf{Mod}_{A,fg}\longrightarrow\mathsf{Mod}_{\widehat{A},fg}\text{,}\qquad M\mapsto M\otimes_{A}\widehat{A}$$ and $i$ is the restriction of $j$ to torsion modules. Note that since completions are flat, the functors $j$ and $k$ are exact. The functor $i$ is not just exact; it induces an equivalence of categories. By functoriality of localization, Diagram \[ltixi1\] commutes. Thus, we are in the situation of Diagram \[lmisu3b\] in Elaboration \[elab\_SignsHtpyCartesianSquares\]. Hence, Equation \[lmisu3\] gives a corresponding distinguished square in the homotopy category of spectra $h\mathsf{Sp}$. Concretely, this means that$$\xymatrix@L=2.6mm{
K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar[r]^-{(-)\otimes\widehat{\mathbb{Z}}+(-)\otimes\mathbb{Q}} & K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}}) \oplus
K({\mathsf{Mod}_{A,fg}}) \ar[r]^-{{(-)\otimes\mathbb{Q}} - {(-)\otimes
\widehat{\mathbb{Q}}}} & K({\mathsf{Mod}_{\widehat{A},fg}}) \ar[r]^-{-\partial
_{\Diamond}} & \Sigma K({\mathsf{Mod}_{\mathfrak{A},fg}})
}
\label{l_WallDistTriangle}$$ is distinguished, where ‘$\Diamond$’ refers to the respective square Diagram \[ltixi1\]. Note the negative sign in front of $\partial_{\Diamond}$.
\[thm\_PrincipalIdeleFibration\]Let $A$ be a finite-dimensional semisimple $\mathbb{Q}$-algebra. Suppose $\mathfrak{A}\subset A$ is a regular order. Then there is a canonical fibration of pointed spaces$$K(\widehat{\mathfrak{A}})\times K(A)\longrightarrow K(\widehat{A})\times
K(A_{\mathbb{R}})\longrightarrow K(\mathsf{LCA}_{\mathfrak{A}})\text{,}$$ which we call the *principal idèle fibration*.
1. The first arrow is induced from the exact functors$$\begin{aligned}
\operatorname*{PMod}(\widehat{\mathfrak{A}}) & \longrightarrow
\operatorname*{PMod}(\widehat{A})\text{,}\qquad X\mapsto X\otimes
_{\widehat{\mathfrak{A}}}\widehat{A}\\
\operatorname*{PMod}(A) & \longrightarrow\operatorname*{PMod}(\widehat
{A})\times\operatorname*{PMod}(A_{\mathbb{R}})\text{,}\qquad X\mapsto
(X\otimes_{A}\widehat{A},X\otimes_{A}A_{\mathbb{R}})\text{.}$$
2. The second arrow is induced from the exact functor sending a right $\widehat{A}$-module to itself, but equipped with the natural adèle topology. Similarly, a right $A_{\mathbb{R}}$-module gets sent to itself, equipped with the natural real vector space topology.
(Step 1) The commutative diagram$$\xymatrix@L=2.6mm@!C=1.2in{
K({\mathsf{Mod}_{{\mathfrak{A}},fg}}) \ar[r] \ar[d]_{l} \ar@{}[dr]|{\ddag}
& K({\mathsf{Mod}_{\mathfrak{A}}}) \ar[r] \ar[d]_{m} & K({{\mathsf
{Mod}_{\mathfrak{A}}}}/{{\mathsf{Mod}_{{\mathfrak{A}},fg}}}) \ar[d]^{\Phi
}_{\cong} \ar[r]^-{\partial_{\mathfrak{A}}^{fg \hookrightarrow all}} & \Sigma
K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar[d]^{\Sigma l} \\
K(\mathsf{LCA}_{\mathfrak{A},cg}) \ar[r] & K(\mathsf{LCA}_{\mathfrak{A}}) \ar[r] & K({\mathsf{LCA}_{\mathfrak{A}}}/{\mathsf{LCA}_{\mathfrak{A},cg}}) \ar[r]_-{\partial_{\mathsf{LCA}}^{cg \hookrightarrow all}} & \Sigma
K(\mathsf{LCA}_{\mathfrak{A},cg})
}
\label{lmisu5}$$ was set up in [@etnclca Proposition 11.1], using the same notation. Loc. cit. we have only spelled out a commutative diagram of fiber sequences in $\mathsf{Sp}$, whereas here we have expanded the entire datum including the maps $\partial$ belonging to the underlying homotopy Cartesian squares. The maps $l$ and $m$ come from reading the discrete $\mathfrak{A}$-modules and locally compact $\mathfrak{A}$-modules, equipped with the discrete topology. This clearly defines an exact functor. Since $\Phi$ (in the notation of the reference) stems from an exact equivalence of exact categories, it induces an isomorphism in $h\mathsf{Sp}$. Hence, we are in the situation of Diagram \[lmisu4\]. Thus, the square denoted by ‘$\ddag$’ is homotopy Cartesian. Unravelling the meaning of this along Elaboration \[elab\_SignsHtpyCartesianSquares\], we obtain the distinguished triangle$$\xymatrix@L=2.6mm{
K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar[r]^-{\text{incl}+\operatorname{incl}}
& K(\mathsf{LCA}_{\mathfrak{A},cg}) \oplus K({\mathsf{Mod}_{\mathbb{A}}}) \ar[r]^-{\operatorname{incl}_1 - \operatorname{incl}_2} & K(\mathsf
{LCA}_{\mathfrak{A}}) \ar[r]^-{\partial_{\ddag}} & \Sigma K({\mathsf
{Mod}_{\mathfrak{A},fg}})
}$$ Let us stress that this time the map $\partial$ carries a positive sign, as carefully discussed in Elaboration \[elab\_SignsHtpyCartesianSquares\] on the basis of the equivalence $\Phi$ in Diagram \[lmisu5\] sitting on a different position as in Diagram \[ltixi1\]. Note that the signs we get here are exactly the ones as in [@etnclca Proposition 11.1], justifying our choice of $\varepsilon=+1$ in Elaboration \[elab\_SignsHtpyCartesianSquares\].(Step 2) Next, the category of all right $\mathfrak{A}$-modules $\mathsf{Mod}_{\mathfrak{A}}$ is closed under coproducts, so by the Eilenberg swindle, Lemma \[lemma\_EilenbergSwindle\], we have $K(\mathsf{Mod}_{\mathfrak{A}})=0$.(Step 3) Now we shall set up the following diagram:$$\xymatrix@L=2.6mm{
K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar@{}[dr]|{\sharp} \ar[d]_{1}
\ar[r]^-{(-)\otimes\widehat{\mathbb{Z}}+(-)\otimes\mathbb{Q}} & K({\mathsf
{Mod}_{\widehat{\mathfrak{A}},fg}}) \oplus K({\mathsf{Mod}_{A,fg}}) \ar@
{}[dr]|{\square} \ar[d]^{(0,(-)\otimes\mathbb{R})} \ar[r]^-{{(-)\otimes
\mathbb{Q}} - {(-)\otimes\widehat{\mathbb{Q}}}} & K({\mathsf{Mod}_{\widehat
{A},fg}}) \ar@{}[dr]|{\natural} \ar[d]_{\text{top. realiz.}}
\ar[r]^-{-\partial_{\Diamond}} & \Sigma K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar[d]^{1} \\
K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar[r]_-{\operatorname{incl}}
& K(\mathsf{LCA}_{\mathfrak{A},cg}) \ar[r]_-{\operatorname{incl}}
& K(\mathsf{LCA}_{\mathfrak{A}}) \ar[r]_-{\partial_{\ddag}} & \Sigma
K({\mathsf{Mod}_{\mathfrak{A},fg}})
}$$ Both rows are the distinguished triangles which we had produced in Lemma \[lemma\_WallSeqForGTheory\] (and more specifically given in detail in Equation \[l\_WallDistTriangle\]), and the one coming from Step 1 and Step 2. Thus, it remains to describe the downward arrows and prove the commutativity of the three squares.(Square ‘$\sharp$’) We compose the underlying exact functors, first going down and then right resp. the other way round. We obtain the exact functors$$h_{i}:\mathsf{Mod}_{\mathfrak{A},fg}\longrightarrow\mathsf{LCA}_{\mathfrak{A},cg}\qquad\text{(for }i=1,2\text{),}$$ where $h_{1}$ sends a right $\mathfrak{A}$-module $X$ to itself, equipped with the discrete topology, while $h_{2}$ sends it to $X_{\mathbb{R}}:=X\otimes_{\mathfrak{A}}A_{\mathbb{R}}$, and regards this as a topological right $\mathfrak{A}$-module, equipped with the real topology. Clearly, $h_{1}\neq h_{2}$ as exact functors. However, we only need to show that the induced square commutes in $h\mathsf{Sp}$ after taking $K$-theory. To this end, consider the exact functor $\operatorname*{PMod}(\mathfrak{A})\rightarrow\mathcal{E}\mathsf{LCA}_{\mathfrak{A},cg}$ sending a finitely generated projective right $\mathfrak{A}$-module $X$ to the exact sequence$$X\hookrightarrow X_{\mathbb{R}}\twoheadrightarrow X_{\mathbb{R}}/X$$ in $\mathsf{LCA}_{\mathfrak{A},cg}$. Here $X$ carries the discrete topology, $X_{\mathbb{R}}$ the real vector space topology and $X_{\mathbb{R}}/X$ the torus topology (topologically it stems from quotienting a real vector space by a full rank $\mathbb{Z}$-lattice). Denote the individual functors $f_{i}$ for $i=1,2,3$ for the left (resp. middle, resp. right) individual exact functor. Since $\mathfrak{A}$ is regular, $\operatorname*{PMod}(\mathfrak{A})$ and $\mathsf{Mod}_{\mathfrak{A},fg}$ have the same $K$-theory by resolution. Thus, it suffices to define this exact functor on $\operatorname*{PMod}(\mathfrak{A})$. By Additivity we get $f_{2\ast}=f_{1\ast}+f_{3\ast}$. Next, note that $f_{3}$ can be factored as$$\operatorname*{PMod}(\mathfrak{A})\longrightarrow\mathsf{LCA}_{\mathfrak{A},C}\longrightarrow\mathsf{LCA}_{\mathfrak{A},cg}\text{,} \label{lmisu7}$$ where $\mathsf{LCA}_{\mathfrak{A},C}$ denotes the exact category of compact right $\mathfrak{A}$-modules. Since products of compact spaces are compact, the latter category is closed under products, so $K(\mathsf{LCA}_{\mathfrak{A},C})$ by the Eilenberg swindle, Lemma \[lemma\_EilenbergSwindle\]. Thus, we necessarily have $f_{3\ast}=0$ since it can be factored over a zero object. Hence, $f_{2\ast}=f_{1\ast}$, but $f_{1}=h_{1}$ and $f_{2}=h_{2}$, proving $h_{1\ast}=h_{2\ast}$, and thus proving the commutativity of the square ‘$\sharp$’. Let us point out that a factorization of $f_{2}$ as in$$\text{\textquotedblleft}\operatorname*{PMod}(\mathfrak{A})\longrightarrow
\mathsf{LCA}_{\mathfrak{A},D}\longrightarrow\mathsf{LCA}_{\mathfrak{A},cg}\text{\textquotedblright} \label{lmisu7a}$$ with $\mathsf{LCA}_{\mathfrak{A},D}$ the discrete right $\mathfrak{A}$-modules *does not exist*. The point is that while all compact right $\mathfrak{A}$-modules are compactly generated, leading to Equation \[lmisu7\], a discrete right $\mathfrak{A}$-module is compactly generated if and only if it is finitely generated, so we cannot define the second arrow in Equation \[lmisu7a\] on all of $\mathsf{LCA}_{\mathfrak{A},D}$. We could only define it on the finitely generated ones at best, but this category then is not closed under countable coproducts, so the Eilenberg swindle cannot be applied.(Square ‘$\square$’) This square commutes if and only if the following two squares commute:$$\xymatrix{
K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}}) \ar[d]^{0} \ar[r]^-{{(-)\otimes
\mathbb{Q}}} & K({\mathsf{Mod}_{\widehat{A},fg}}) \ar[d]_{\text{top. realiz.}}
\\
K(\mathsf{LCA}_{\mathfrak{A},cg}) \ar[r]_-{\operatorname{incl}} & K(\mathsf
{LCA}_{\mathfrak{A}})
}\qquad\xymatrix{
K({\mathsf{Mod}_{A,fg}}) \ar[d]^{(-)\otimes\mathbb{R}} \ar[r]^-{-{(-)\otimes
\widehat{\mathbb{Q}}}} & K({\mathsf{Mod}_{\widehat{A},fg}}) \ar[d]_{\text
{top. realiz.}} \\
K(\mathsf{LCA}_{\mathfrak{A},cg}) \ar[r]_-{\operatorname{incl}} & K(\mathsf
{LCA}_{\mathfrak{A}})
}$$ In the left square, we compare the zero map with the composition$$K(\mathsf{Mod}_{\widehat{\mathfrak{A}},fg})\longrightarrow K(\mathsf{Mod}_{\widehat{A},fg})\longrightarrow K(\mathsf{LCA}_{\mathfrak{A}})\text{,}$$ but the latter is also zero by Local Triviality, Theorem \[thm\_LocalTriviality\] (either give a precise argument using the isomorphism of rings $\widehat{\mathfrak{A}}\cong\prod\widehat{\mathfrak{A}}_{p}$ and an approximation argument, or much more elegantly: Copy the proof of Theorem \[thm\_LocalTriviality\] and use that $\widehat{\mathfrak{A}}$ is a compact clopen in $\widehat{A}$ with discrete quotient $\widehat{A}/\widehat{\mathfrak{A}}$ if we equip $\widehat{\mathfrak{A}}$ with its natural profinite topology, and $\widehat{A}/\widehat{\mathfrak{A}}$ with the natural product as a restricted product of $\mathbb{Q}_{p}$-vector spaces. This way, one can avoid any approximation argument). The right square can be done very similarly: The two functors are induced from$$v_{i}:\mathsf{Mod}_{A,fg}\longrightarrow\mathsf{LCA}_{\mathfrak{A}}\qquad\text{(for }i=1,2\text{)}$$$$v_{1}(X):=X_{\mathbb{R}}\text{ (}=X\otimes_{A}A_{\mathbb{R}}\text{)}\qquad\text{and}\qquad v_{2}(X):=\widehat{X}\text{ (}=X\otimes_{A}\widehat
{A}\text{),}$$ where instead of $v_{2}$ we take the negative of what is induced by this functor. Because of this sign switch, the two induced maps on $K$-theory agree if and only if *their sum* is the zero map $K(\mathsf{Mod}_{A,fg})\rightarrow K(\mathsf{LCA}_{\mathfrak{A}})$. However, this is precisely the statement of the fundamental Reciprocity Law, Theorem \[thm\_reciprocity\_law\].(Square ‘$\natural$’) The commutativity of this square is the most delicate part of the proof.$$\xymatrix@L=2.6mm{
K({\mathsf{Mod}_{\widehat{A},fg}}) \ar@{}[dr]|{\natural} \ar[d]_{\text
{top. realiz.}}
\ar[r]^-{-\partial_{\Diamond}} & \Sigma K({\mathsf{Mod}_{\mathfrak{A},fg}}) \ar[d]^{1} \\
K(\mathsf{LCA}_{\mathfrak{A}}) \ar[r]_-{\partial_{\ddag}} & \Sigma
K({\mathsf{Mod}_{\mathfrak{A},fg}})
}
\label{ltixi8b}$$ Note that, from the point of view of regarding $\mathsf{Sp}$ as a stable $\infty$-category, checking the commutativity of this square amounts to checking that the fiber sequences attached to the two rows have compatible nullhomotopies.We first follow the top horizontal arrow and then go down. We unravel the definition of $\partial_{\Diamond}$. It comes from the homotopy Cartesian square in Diagram \[ltixi1\]. We have recalled how to set up the attached distinguished triangle in Elaboration \[elab\_SignsHtpyCartesianSquares\], namely$$\partial_{\Diamond}:\xymatrix{
K(\mathsf{Mod}_{\widehat{A},fg}) \ar[r]^-{\partial_{\widehat{\mathfrak{A}}}^{tor \hookrightarrow fg}} & \Sigma K(\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}) & \Sigma K(\mathsf{Mod}_{\mathfrak{A},fg}^{tor}) \ar
[r] \ar[l]^{\sim} & \Sigma K(\mathsf{Mod}_{\mathfrak{A},fg})
}\text{.} \label{ltixi8a}$$ On the other hand, going around the square ‘$\natural$’ the other way, we unravel$$\partial_{\ddag}:\xymatrix@L=1.4mm{
K(\mathsf{Mod}_{\widehat{A},fg}) \ar[r]^-{\text{top. realiz.}} \ar@
/^2pc/[rr]^{f_2} & K(\mathsf{LCA}_{\mathfrak{A}}) \ar[r] & K(\mathsf
{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg}) \\
& K(\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}) \ar
[ur]^-{\sim}_-{\Phi} \ar[r]_-{\partial_{{\mathfrak{A}}}^{fg \hookrightarrow
all}} & \Sigma K(\mathsf{Mod}_{\mathfrak{A},fg})
}
\label{ltixi4}$$ Ignore the arrow with the label $f_{2}$ temporarily. Let us first focus on $\partial_{\ddag}$. We have an exact equivalence of exact categories$$\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}/\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}\overset{\sim}{\longrightarrow}\mathsf{Mod}_{\widehat{A}}\text{,}\qquad M\mapsto M\otimes\mathbb{Q}\text{.}$$ This is the same equivalence which underlies the fiber sequences in Diagram \[ltixi1\]. Consider the exact functor$$\operatorname*{PMod}(\widehat{\mathfrak{A}})\longrightarrow\mathcal{E}\mathsf{LCA}_{\mathfrak{A}} \label{ltixi3}$$ sending $\mathfrak{A}$ to the exact sequence$$\widehat{\mathfrak{A}}\hookrightarrow\widehat{A}\twoheadrightarrow\widehat
{A}/\widehat{\mathfrak{A}} \label{ltixi3a}$$ in $\mathsf{LCA}_{\mathfrak{A}}$, where (a) $\widehat{\mathfrak{A}}$ is equipped with its natural compact topology. Its underlying LCA group is a product $\prod\mathbb{Z}_{p}$; (b) $\widehat{A}$ is equipped with its natural locally compact topology. Its underlying LCA group is a restricted product $\left. \prod\nolimits^{\prime}\right. (\mathbb{Q}_{p}:\mathbb{Z}_{p})$; (c) and $\widehat{A}/\widehat{\mathfrak{A}}$ is equipped with the quotient topology. This just amounts to the discrete topology since by the construction of the restricted product topology, $\widehat{\mathfrak{A}}$ sits as a clopen subgroup in it. We want to use the Additivity Theorem. Write $f_{i}$, $i=1,2,3$ for the three exact functors $f_{i}:\mathsf{Mod}_{\widehat
{\mathfrak{A}},fg}\longrightarrow\mathsf{LCA}_{\mathfrak{A}}$ pinned down by the functor in Equation \[ltixi3\]. We get an induced exact functor$$\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}/\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}\longrightarrow\mathcal{E}\left( \mathsf{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg}\right) \text{.} \label{ltixi5}$$ By the Additivity Theorem, $f_{2\ast}=f_{1\ast}+f_{3\ast}$. However, since $\widehat{\mathfrak{A}}$ is compact, it is compactly generated, so $f_{1}$ sends all objects to zero objects in the quotient exact category $\mathsf{LCA}_{\mathfrak{A}}/\mathsf{LCA}_{\mathfrak{A},cg}$. Thus, $f_{2\ast
}=f_{3\ast}$. However, note that $f_{2}$ agrees with the functor, suggestively denoted by $f_{2}$, in Diagram \[ltixi4\]: The straight arrows just equip $\widehat{A}$ with its natural locally compact topology. This is the same as using the identification of Equation \[ltixi5\] first, and then equipping the outcome with the topology as discussed above in (b). Thus, by Additivity, we may work with the functor underlying $f_{3}$ instead, since it induces the same map on the level of $K$-theory.Now, we repeat the same trick in a similar fashion. Consider the exact functor$$\operatorname*{PMod}(\widehat{\mathfrak{A}})\longrightarrow\mathcal{E}\left(
\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}\right) \text{,}$$ sending $\mathfrak{A}$ again to the exact sequence $\widehat{\mathfrak{A}}\hookrightarrow\widehat{A}\twoheadrightarrow\widehat{A}/\widehat
{\mathfrak{A}}$, but now regarded in the category $\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}$ (this is a precise statement already, but note that philosophically it corresponds to considering the same functor, but this time equipping all terms in the exact sequence with the discrete topology instead. Of course this is still exact). Again, we get an induced functor from $\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}/\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}$ since the torsion modules go to zero objects in $\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}$. Write $g_{i}$, $i=1,2,3$ for the three exact functors$$g_{i}:\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}/\mathsf{Mod}_{\widehat
{\mathfrak{A}},fg}^{tor}\longrightarrow\mathsf{Mod}_{\mathfrak{A}}/\mathsf{Mod}_{\mathfrak{A},fg}\text{.}$$ The key point is the following: Running the equivalence $\Phi$ in Diagram \[ltixi4\] backwards, we get$$\Phi_{\ast}^{-1}\circ f_{3\ast}=g_{3\ast}\text{.}$$ The point behind this is that the quotient $\widehat{A}/\widehat{\mathfrak{A}}$ in Equation \[ltixi3a\] carries the discrete topology. However, by Additivity we have $g_{2\ast}=g_{1\ast}+g_{3\ast}$, so combining these two equations, and remembering $f_{2\ast}=f_{3\ast}$, we get$$\Phi_{\ast}^{-1}\circ f_{2\ast}=g_{2\ast}-g_{1\ast}\text{.} \label{ltixi5a}$$ Finally, consider the diagram$$\xymatrix@L=2.6mm@!C=1.2in{
& & K({\mathsf{Mod}_{{\widehat{\mathfrak{A}}},fg}}/{{\mathsf{Mod}^{tor}_{{\widehat{\mathfrak{A}}},fg}}}) \ar[d]^{\otimes\mathbb{Q}}_{g_2}
\ar@{-->}[dl] \ar[dr]^{0} \\
\cdots\ar[r]
& K({\mathsf{Mod}_{\mathfrak{A}}}) \ar[r] & K({{\mathsf{Mod}_{\mathfrak{A}}}}/{{\mathsf{Mod}_{{\mathfrak{A}},fg}}}) \ar[r]_-{\partial_{\mathfrak{A}}^{fg \hookrightarrow all}} & \Sigma K({\mathsf{Mod}_{\mathfrak{A},fg}}).
}$$ The bottom row stems from the localization sequence. The exact functor $g_{2}$ admits a lift to $\mathsf{Mod}_{\mathfrak{A}}$. This would not work for $g_{1}$ for example since $g_{1}$ would send the torsion modules $\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}$ would go to non-zero objects in $\mathsf{Mod}_{\mathfrak{A}}$. However, since $g_{2}$ sends torsion modules to zero anyway, this lift exists. We deduce that$$\partial_{\mathfrak{A}}^{fg\hookrightarrow all}\circ g_{2\ast}=0\text{.}$$ Thus, Equation \[ltixi5a\] leads to$$\partial_{\mathfrak{A}}^{fg\hookrightarrow all}\circ\Phi_{\ast}^{-1}\circ
f_{2\ast}=\partial_{\mathfrak{A}}^{fg\hookrightarrow all}\circ g_{2\ast
}-\partial_{\mathfrak{A}}^{fg\hookrightarrow all}\circ g_{1\ast}=-\partial_{\mathfrak{A}}^{fg\hookrightarrow all}\circ g_{1\ast}\text{.}
\label{ltixi5b}$$ Returning to Diagram \[ltixi4\], we have shown that the morphism $\partial_{\ddag}$ is the same morphism as$$\xymatrix@L=2.6mm{
K(\mathsf{Mod}_{\widehat{A},fg}) & K(\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}/\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor})
\ar[l]_-{\sim} \ar[r]^-{-\partial_{\mathfrak{A}}^{fg\hookrightarrow all}}
& K(\mathsf{Mod}_{\mathfrak{A},fg})
}$$ since the functor $g_{1}$ just sends $\widehat{\mathfrak{A}}$ to itself, treated as a right $\mathfrak{A}$-module. We swallow this rather naïve operation into the notation. Consider the commutative diagram$$\xymatrix@L=2.6mm@!C=0.9in{
K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}}) \ar[r] \ar[d]_{s}
& K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}}) \ar[r] \ar[d] & K({\mathsf
{Mod}_{\widehat{A},fg}}) \ar[d]_{g_1} \ar[r]^-{\partial_{\widehat{\mathfrak
{A}}}^{tor \hookrightarrow fg}}
& \Sigma K({\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}}) \ar[d]^{\Sigma s}
\\
K(\mathsf{Mod}_{\mathfrak{A},fg}) \ar[r] & K(\mathsf{Mod}_{\mathfrak{A}}) \ar[r] &
K({\mathsf{Mod}_{\mathfrak{A}}}/\mathsf{Mod}_{\mathfrak{A},fg}) \ar
[r]_-{\partial_{{\mathfrak{A}}}^{fg \hookrightarrow all}}
& \Sigma K(\mathsf{Mod}_{\mathfrak{A},fg}).
}$$ Both rows are distinguished triangles coming from the respective localization sequences of $\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}$ as a Serre subcategory of $\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}$, resp. $\mathsf{Mod}_{\mathfrak{A},fg}$ as a Serre subcategory of $\mathsf{Mod}_{\mathfrak{A}}$. The functor underlying $s$ sends a torsion right $\widehat{\mathfrak{A}}$-module to itself, regarded as a right $\mathfrak{A}$-module. Note that just like $\mathsf{Mod}_{\widehat{\mathfrak{A}},fg}^{tor}\cong\mathsf{Mod}_{\mathfrak{A},fg}^{tor}$ are equivalent categories, finitely generated torsion $\widehat{\mathfrak{A}}$-modules are indeed finitely generated right $\widehat{\mathfrak{A}}$-modules. The commutativity of the right square implies that we can continue the computation in Equation \[ltixi5b\] as$$-\partial_{\mathfrak{A}}^{fg\hookrightarrow all}\circ g_{1\ast}=-(\Sigma
s)\circ\partial_{\widehat{\mathfrak{A}}}^{tor\hookrightarrow fg}\text{.}$$ Thus, in total, the map $\partial_{\ddag}$ of Diagram \[ltixi4\] can be expressed as follows.$$\partial_{\ddag}=-(\Sigma s)\circ\partial_{\widehat{\mathfrak{A}}}^{tor\hookrightarrow fg}\text{.}$$ Finally, compare this to the map $\partial_{\Diamond}$ of Equation \[ltixi8a\]. The part $\partial_{\widehat{\mathfrak{A}}}^{tor\hookrightarrow
fg}$ agrees for both maps, and the following maps in Equation \[ltixi8a\] merely amount to regarding a finitely generated torsion right $\widehat
{\mathfrak{A}}$-module as a finitely generated right $\mathfrak{A}$-module. This is the same functor as $s$. Thus, in total the only difference is the sign, $\partial_{\ddag}=-\partial_{\Diamond}$. However, this is exactly what we had to show, see Diagram \[ltixi8b\]. Note that the appearance of this sign is quite subtle. In our computations above it arose from a sign when using the Additivity theorem, while in general it is needed for the right compatibility because of rôle of signs as explained in Elaboration \[elab\_SignsHtpyCartesianSquares\].
Proof of compatibility\[sect\_CompatProofSection\]
==================================================
In this section we will prove that our approach is equivalent to the original construction of Burns and Flach in [@MR1884523]. To this end, let us go through their construction, so roughly from [@MR1884523] §2.1 to §4.3 (although we can jump over certain parts).
Regarding our approach on the other hand, we use the construction of $T\Omega$ of §\[sect\_Overview\], using the rigorous justification of all construction steps from §\[sect\_GettingPrecise\], especially Convention \[convention\_KThySpace\].
We recall the concept of a determinant functor from [@MR902592 §4.3]. Given any category $\mathsf{C}$, we write $\mathsf{C}^{\times}$ for its internal groupoid, i.e. we delete all morphisms which are not isomorphisms.
\[def\_DeterminantFunctor\]Suppose $\mathsf{C}$ is an exact category and let $(\mathsf{P},\otimes)$ be a Picard groupoid. A *determinant functor* on $\mathsf{C}$ is a functor$$\mathcal{D}:\mathsf{C}^{\times}\longrightarrow\mathsf{P}$$ along with the following extra structure and axioms:
1. For any exact sequence $\Sigma:G^{\prime}\hookrightarrow
G\twoheadrightarrow G^{\prime\prime}$ in $\mathsf{C}$, we are given an isomorphism$$\mathcal{D}(\Sigma):\mathcal{D}(G)\overset{\sim}{\longrightarrow}\mathcal{D}(G^{\prime})\underset{\mathsf{P}}{\otimes}\mathcal{D}(G^{\prime\prime})$$ in $\mathsf{P}$. This isomorphism is required to be functorial in morphisms of exact sequences.
2. For every zero object $Z$ of $\mathsf{C}$, we are given an isomorphism $z:\mathcal{D}(Z)\overset{\sim}{\rightarrow}1_{\mathsf{P}}$ to the neutral object of the Picard groupoid.
3. Suppose $f:G\rightarrow G^{\prime}$ is an isomorphism in $\mathsf{C}$. We write$$\Sigma_{l}:0\hookrightarrow G\twoheadrightarrow G^{\prime}\qquad
\text{and}\qquad\Sigma_{r}:G\hookrightarrow G^{\prime}\twoheadrightarrow0$$ for the depicted exact sequences. We demand that the composition$$\mathcal{D}(G)\underset{\mathcal{D}(\Sigma_{l})}{\overset{\sim}{\longrightarrow}}\mathcal{D}(0)\underset{\mathsf{P}}{\otimes}\mathcal{D}(G^{\prime})\underset{z\otimes1}{\overset{\sim}{\longrightarrow}}1_{\mathsf{P}}\underset{\mathsf{P}}{\otimes}\mathcal{D}(G^{\prime})\underset{\mathsf{P}}{\overset{\sim}{\longrightarrow}}\mathcal{D}(G^{\prime})
\label{l_CDetFunc1}$$ and the natural map $\mathcal{D}(f):\mathcal{D}(G)\overset{\sim}{\rightarrow
}\mathcal{D}(G^{\prime})$ agree. We further require that $\mathcal{D}(f^{-1})$ agrees with a variant of Equation \[l\_CDetFunc1\] using $\Sigma_{r}$ instead of $\Sigma_{l}$.
4. If a two-step filtration $G_{1}\hookrightarrow G_{2}\hookrightarrow
G_{3}$ is given, we demand that the diagram$$\xymatrix{
\mathcal{D}(G_3) \ar[r]^-{\sim} \ar[d]_{\sim} & \mathcal{D}(G_1) \otimes
\mathcal{D}(G_3/G_1) \ar[d]^{\sim} \\
\mathcal{D}(G_2) \otimes\mathcal{D}(G_3/G_2) \ar[r]_-{\sim} & \mathcal
{D}(G_1) \otimes\mathcal{D}(G_2/G_1) \otimes\mathcal{D}(G_3/G_2)
}$$ commutes.
5. Given objects $G,G^{\prime}\in\mathsf{C}$ consider the exact sequences$$\Sigma_{1}:G\hookrightarrow G\oplus G^{\prime}\twoheadrightarrow G^{\prime
}\qquad\text{and}\qquad\Sigma_{2}:G^{\prime}\hookrightarrow G\oplus G^{\prime
}\twoheadrightarrow G$$ with the natural inclusion and projection morphisms. Then the diagram$$\xymatrix{
& \mathcal{D}(G \oplus G^{\prime}) \ar[dl]_{\mathcal{D}(\Sigma_1)}
\ar[dr]^{\mathcal{D}(\Sigma_2)} \\
\mathcal{D}(G) \otimes\mathcal{D}(G^{\prime}) \ar[rr]_{s_{G,G^{\prime}}}
& & \mathcal{D}(G^{\prime}) \otimes\mathcal{D}(G)
}$$ commutes, where $s_{G,G^{\prime}}$ denotes the symmetry constraint of $\mathsf{P}$.
As usual, suppose $A$ is a finite-dimensional semisimple $\mathbb{Q}$-algebra, $\mathfrak{A}\subset A$ an order. Also, let $F$ be a number field, $S_{\infty
}$ the set of infinite places of $F$, and $M\in\operatorname*{CHM}(F,\mathbb{Q})$ a Chow motive over $F$ (where we take the category of Chow motives to have $\mathbb{Q}$-coefficients). Let$$A\longrightarrow\operatorname*{End}\nolimits_{\operatorname*{CHM}(F,\mathbb{Q})}(X)\text{.}$$ be the action of $A$ on the motive. Pick a projective $\mathfrak{A}$-structure $\{T_{v},v\in S_{\infty}\}$ and assume the *Coherence Hypothesis* (as defined and discussed in detail in [@MR1884523 §3.3]).
In the construction of Burns and Flach, they work with the framework of Picard groupoids[^14]. The connection to our approach is as follows: in our picture the Tamagawa number $T\Omega$ lives in $\pi_{1}K(\mathsf{LCA}_{\mathfrak{A}})$, so instead of working with the full $K$-theory space, it is sufficient to work with a $1$-skeleton of that space (i.e. it does not matter if we kill all homotopy groups $\pi_{i}$ for $i\geq2$). Viewed from the angle of homotopy theory, this $1$-skeleton is a stable $(0,1)$-type. A priori it would only be an unstable $(0,1)$-type, but since $K$-theory really comes from a spectrum (or: when being viewed as a simplicial set in our setting of §\[subsect\_Spaces\], it comes equipped with a $\Gamma$-space structure), it is a stable homotopy type. However, the category of stable $(0,1)$-types can alternatively be modelled in a somewhat more concrete fashion through Picard groupoids.
The precise relation is as follows:
\[thm\_PicardGrpdsAnd01Types\]There is an equivalence of homotopy categories,$$\Psi:\operatorname*{Ho}(\mathsf{Picard})\overset{\sim}{\longrightarrow
}\operatorname*{Ho}(\mathsf{Sp}^{0,1})\text{,} \label{lcimez20}$$ where $\mathsf{Picard}$ denotes the $\infty$-category of Picard groupoids, and $\mathsf{Sp}^{0,1}$ denotes the $\infty$-category of spectra such that $\pi_{i}X=0$ for $i\neq0,1$, also known as *stable* $(0,1)$*-types*. The functor $\Psi^{-1}$ can be described as follows: If $E\in\mathsf{Sp}^{0,1}$ is the input spectrum, let $\Omega^{\infty}E$ denote its infinite loop space. Define$$\Psi^{-1}(E):=GP(\left\vert \Omega^{\infty}E\right\vert )\text{,}$$ i.e. where $GP$ denotes the fundamental groupoid (in the setting of simplicial homotopy theory, see [@MR2840650 Ch. I, p. 42] for the Gabriel–Zisman fundamental groupoid). The infinite loop space structure equips this groupoid with a symmetric monoidal structure, which gives rise to the Picard groupoid structure in question.
We refer to [@MR2981817 §5.1, Theorem 5.3] or alternatively [@MR2981952 1.5 Theorem] for detailed proofs.
\[Prop\_CompatPatel\]This description of $\Psi$ is equivalent to the one given by Patel [@MR2981817].
Just follow Patel’s description of his construction, [@MR2981817 §5.1]. Starting from a very special $\Gamma$-space $X$, we attaches to it the topological fundamental groupoid (which he calls Poincaré groupoid. Objects are points in $X(\mathbf{1})$ and morphisms are homotopy classes of paths in $\left\vert X(\mathbf{1})\right\vert $). This is equivalent to what we do; we just take the Gabriel–Zisman fundamental groupoid instead. The equivalence of these two ways to form the fundamental groupoid is proven in [@MR2840650 Chapter III, §1, Theorem 1.1].
It is also equivalent to the one given by Johnson and Osorno [@MR2981952]. Instead of using $\Gamma$-spaces to model the connective spectrum, they use operads. However, the basic link is also a (topological) fundamental groupoid, just as in Patel.
We write $(V(\mathsf{C}),\boxtimes)$ for Deligne’s Picard groupoid of virtual objects of an exact category $\mathsf{C}$, [@MR902592]. Deligne proved in this paper that there is a determinant functor$$\mathcal{D}:\mathsf{C}^{\times}\longrightarrow(V(\mathsf{C}),\boxtimes
)\text{,}$$ which is actually (2-)universal, and in particular for any other determinant functor $\mathcal{D}^{\prime}:\mathsf{C}^{\times}\longrightarrow
(\mathsf{P},\otimes)$ to some Picard groupoid $(\mathsf{P},\otimes)$, there exists a factorization$$\mathsf{C}^{\times}\overset{\mathcal{D}}{\longrightarrow}(V(\mathsf{C}),\boxtimes)\longrightarrow(\mathsf{P},\otimes)$$ such that the composition is the given $\mathcal{D}^{\prime}$. The precise notion of (2-)universality is actually rather subtle, see for example [@MR2842932 §4.1], because it needs to take the entire symmetric monoidal structure into consideration. Even better, there is also a map of spaces$$\mathsf{C}^{\times}\longrightarrow K(\mathsf{C})$$ (where $\mathsf{C}^{\times}$ is regarded as its nerve) and under the truncation to the $1$-skeleton,$$\mathsf{C}^{\times}\longrightarrow K(\mathsf{C})\longrightarrow\tau_{\leq
1}K(\mathsf{C})\text{,}$$ if we apply $\Psi^{-1}$, this map transforms into the universal determinant functor $\mathcal{D}$ above. In particular, it follows that there is a canonical equivalence of stable $(0,1)$-types $\tau_{\leq1}K(\mathsf{C})\overset{\sim}{\rightarrow}\Psi(V(\mathsf{C}),\boxtimes)$. Thus, pre-composing this with the $1$-truncation, we obtain a map of spectra$$J:K(\mathsf{C})\longrightarrow\tau_{\leq1}K(\mathsf{C})\overset{\sim
}{\rightarrow}\Psi(V(\mathsf{C}),\boxtimes)\text{.} \label{llh}$$ Next, let us show that our concept of fundamental line is compatible with Burns–Flach.
\[thm\_LinesAgree\]The fundamental line point $\Xi(M)$ in $K(A)$ of Equation \[ltiops1\] under $J$ gets sent to the fundamental line virtual object $\Xi(M)^{\operatorname*{BF}}$ of Burns–Flach [@MR1884523 §3.4].
We will split the proof into several parts. In general, even when coproducts exist in a category, they are only well-defined up to unique isomorphism, so technically the expression $P\oplus P^{\prime}$ does not define a point in the nerve. We circumvented this problem by picking a concrete bifunctor in Equation \[lexit1\], but we shall now see that this was merely an ad hoc choice compatible with a fully homotopy coherent solution of the issue, which we shall recall now:
\[Segal [@MR0353298], 2$^{\text{nd}}$ page\]\[def\_SegalNerve\]Suppose $\mathsf{C}$ is a pointed category (we write $0$ for the base point object) which admits finite coproducts. Write $\oplus$ for the coproduct[^15]. Then we write $N_{\bullet}^{\oplus}$ to denote its categorical *Segal nerve*. That is: $N_{\bullet}^{\oplus}$ is a simplicial category and the objects in $N_{n}^{\oplus}$ are $n$-tuples$$(X_{1},\ldots,X_{n})$$ of objects $X_{i}\in\mathsf{C}$ along with a choice of a coproduct $X_{i_{1}}\oplus\cdots\oplus X_{i_{r}}$, where $\{i_{1},\ldots,i_{r}\}$ runs through all finite subsets of $\{1,\ldots,n\}$. We demand additionally that
1. for the empty subset the choice of the (empty) coproduct is the base point object $0$;
2. for the singleton subsets $\{i\}$, we pick $X_{i}$ itself as the (one-element) coproduct.
The simplicial structure comes from deleting (resp. duplicating) the $i$-th entry. A detailed definition and discussion is given in [MR802796]{}.
Each category $N_{n}^{\oplus}$ is equivalent to the $n$-fold product category $\mathsf{C}\times\cdots\times\mathsf{C}$. The geometric realization $\left\vert N_{\bullet}^{\oplus}\mathsf{C}\right\vert $ of the Segal nerve carries a canonical structure as a $\Gamma$-space, see [@MR0353298 §2].
\[[[@MR1167575 Observation 3.2]]{}\]\[lemma\_InfLoopSpaceStructures\]Suppose $\mathsf{C}$ is a pseudo-additive[^16] Waldhausen category. The $K$-theory space $K(\mathsf{C})$ carries a canonical infinite loop space structure. This infinite loop space structure equivalently comes from
1. iterates of the Waldhausen $S$-construction:$$w\mathsf{C}\longrightarrow\Omega\left\vert wS_{\bullet}\mathsf{C}\right\vert
\overset{\sim}{\longrightarrow}\Omega^{2}\left\vert wS_{\bullet}S_{\bullet
}\mathsf{C}\right\vert \overset{\sim}{\longrightarrow}\Omega^{3}\left\vert
wS_{\bullet}S_{\bullet}S_{\bullet}\mathsf{C}\right\vert \overset{\sim
}{\longrightarrow}\cdots\text{,}$$
2. the Waldhausen $S$-construction, and iterates of the Segal nerve with respect to the composition law $\vee$ of Remark \[rmk\_Coproducts\]:$$w\mathsf{C}\longrightarrow\Omega\left\vert wS_{\bullet}\mathsf{C}\right\vert
\overset{\sim}{\longrightarrow}\Omega^{2}\left\vert wS_{\bullet}N_{\bullet
}^{\vee}\mathsf{C}\right\vert \overset{\sim}{\longrightarrow}\Omega
^{3}\left\vert wS_{\bullet}N_{\bullet}^{\vee}N_{\bullet}^{\vee}\mathsf{C}\right\vert \overset{\sim}{\longrightarrow}\cdots\text{,}$$
3. the $G$-construction, and iterates of the Segal nerve with respect to the composition law of $\vee$:$$w\mathsf{C}\longrightarrow\left\vert wG_{\bullet}\mathsf{C}\right\vert
\overset{\sim}{\longrightarrow}\Omega\left\vert wG_{\bullet}N_{\bullet}^{\vee
}\mathsf{C}\right\vert \overset{\sim}{\longrightarrow}\Omega^{2}\left\vert
wG_{\bullet}N_{\bullet}^{\vee}N_{\bullet}^{\vee}\mathsf{C}\right\vert
\overset{\sim}{\longrightarrow}\cdots\text{,}$$
and the stabilized terms (anywhere starting from the second term) are equivalent to $\left\vert D_{\bullet}\right\vert $.
Regarding (1) and (2), this is already mentioned in Waldhausen’s classic [@MR802796 §1.3, the paragraphs after the definition], and in more precise form in [@MR802796 Lemma 1.8.6], applied to the identity functor $\mathsf{C}\longrightarrow\mathsf{C}$. Or, as mentioned, see [@MR1167575 Observation 3.2]. The cited lemma yields the equivalence$$\left\vert wS_{\bullet}N_{\bullet}^{\vee}\mathsf{C}\right\vert \overset{\sim
}{\longrightarrow}\left\vert wS_{\bullet}S_{\bullet}\mathsf{C}\right\vert
\text{.} \label{lapix2}$$ The claim (3) is only a mild variation: We have$$\left\vert wG_{\bullet}\mathsf{C}\right\vert \overset{\sim}{\longrightarrow
}\Omega\left\vert wS_{\bullet}\mathsf{C}\right\vert$$ by [@MR1167575 Theorem 2.6] and we can functorially apply this to the Segal nerve $N_{\bullet}^{\vee}\mathsf{C}$, getting$$\left\vert wG_{\bullet}N_{\bullet}^{\vee}\mathsf{C}\right\vert \overset{\sim
}{\longrightarrow}\Omega\left\vert wS_{\bullet}N_{\bullet}^{\vee}\mathsf{C}\right\vert \text{.}$$ Now use Equation \[lapix2\] and composing these equivalences, we obtain$$\Omega\left\vert wG_{\bullet}N_{\bullet}^{\vee}\mathsf{C}\right\vert
\overset{\sim}{\longrightarrow}\Omega^{2}\left\vert wS_{\bullet}N_{\bullet
}^{\vee}\mathsf{C}\right\vert \overset{\sim}{\longrightarrow}\Omega
^{2}\left\vert wS_{\bullet}S_{\bullet}\mathsf{C}\right\vert \overset{\sim
}{\longrightarrow}\Omega\left\vert wS_{\bullet}\mathsf{C}\right\vert \text{.}$$ Replacing $\mathsf{C}$ inductively by $N_{\bullet}^{\vee}\mathsf{C}$ in this entire equivalence, and using Equation \[lapix2\] repeatedly on the right, we obtain (3).
\[cor\_SegalNerveSum\]Using the Segal nerve $N_{\bullet}^{\vee}$, the above observation equips $K(\mathsf{C})$ with a concrete structure as a $\Gamma
$-space with $\vee$ as the underlying composition law. The resulting infinite loop space structure is the same one as coming from the $S$-construction in (1) of the Lemma.
This corollary is the homotopy correct replacement for Equation \[l\_maxiu\_1\].
\[Proof of Theorem \[thm\_LinesAgree\]\]We consider the map $J:K(\mathsf{C})\rightarrow\tau_{\leq1}K(\mathsf{C})\overset{\sim}{\rightarrow}\Psi(V(\mathsf{C}),\boxtimes)$ of Equation \[llh\]. Since the target is only a stable $(0,1)$-type, we can check this statement by truncating to a stable $(0,1)$-type all along. By Theorem \[thm\_PicardGrpdsAnd01Types\] we may equivalently perform this verification in the framework of Picard groupoids. Now observe that$$\begin{aligned}
\Xi(M) & =H_{f}^{0}(F,M)-H_{f}^{1}(F,M)+H_{f}^{1}(F,M^{\ast}(1))^{\ast
}-H_{f}^{0}(F,M^{\ast}(1))^{\ast}\label{lef3}\\
& -\sum_{v\in S_{\infty}}H_{v}(M)^{G_{v}}+\sum_{v\in S_{\infty}}\left(
H_{dR}(M)/F^{0}\right) \nonumber\end{aligned}$$ of Equation \[ltiops1\] is formed using the sum and negation map of §\[subsect\_JustifySumAndNegation\]. We had picked a concrete choice for the coproduct $\left. \oplus\right. :\mathsf{C}\times\mathsf{C}\longrightarrow\mathsf{C}$ as in Equation \[lexit1\]. Now by Corollary \[cor\_SegalNerveSum\] the $\Gamma$-space structure of $K$-theory is compatible with any such choice, and further with the $\Gamma$-space structure coming from the infinite loop space structure of the $S$-construction. As Equation \[llh\] comes from a map of spectra, it induces (as spaces) a map of $\Gamma$-spaces. However, by Theorem \[thm\_PicardGrpdsAnd01Types\] and Proposition \[Prop\_CompatPatel\] the symmetric monoidal structure on the Picard groupoids$$\Psi^{-1}\tau_{\leq1}K(\mathsf{C})\overset{\sim}{\rightarrow}(V(\mathsf{C}),\boxtimes)$$ stems from this $\Gamma$-space structure. Finally, the universal determinant functor $[-]$ used by Burns–Flach in [@MR1884523 §2.3-2.4] is taken exactly with respect to this symmetric monoidal structure. Thus, Equation \[lef3\] gets mapped to [@MR1884523 Equation (29) in §3.4], i.e. $\Xi(M)^{\operatorname*{BF}}$. This proves the claim.
By Theorem \[thm\_PrincipalIdeleFibration\] and shifting (this corresponds to rotating the attached distinguished triangle on the level of the homotopy category), we have the fiber sequence of spectra$$\Omega K(\mathsf{LCA}_{\mathfrak{A}})\longrightarrow K(\widehat{\mathfrak{A}})\times K(A)\longrightarrow K(\widehat{A})\times K(A_{\mathbb{R}})\text{.}$$ But this just means that$$\Omega K(\mathsf{LCA}_{\mathfrak{A}})\overset{\sim}{\longrightarrow
}\operatorname*{fib}\left( K(\widehat{\mathfrak{A}})\times
K(A)\longrightarrow K(\widehat{A})\times K(A_{\mathbb{R}})\right) \text{.}
\label{lxmc3}$$ We will shortly use this below. We can now compare the construction of our $R\Omega$ versus the one in [@MR1884523]. Burns and Flach consider the diagram of exact functors (tensoring) between exact categories$$\xymatrix{
\operatorname{PMod}(\mathfrak{A}) \ar[r] \ar[d] & \operatorname{PMod}(A) \ar[d] \\
\operatorname{PMod}(\widehat{\mathfrak{A}}) \ar[r] & \operatorname
{PMod}(\widehat{\mathfrak{A}})
}$$ and using these exact functors, one $2$-functorially gets induced morphisms between the attached Picard groupoids of virtual objects $V(-)$. From the resulting diagram, they define $$\mathbb{V}(\mathfrak{A}):=V(\widehat{\mathfrak{A}})\times_{V(\widehat{A})}V(A)
\label{llh2}$$ as a fiber product in Picard groupoids. They show [@MR1884523 Proposition 2.3],$$\pi_{0}\mathbb{V}(\mathfrak{A})\cong\pi_{0}V(\mathfrak{A})=K_{0}(\mathfrak{A})\text{.} \label{llh3}$$ Further, they define $\mathbb{V}(\mathfrak{A},\mathbb{R}):=\mathbb{V}(\mathfrak{A})\times_{V(A_{\mathbb{R}})}0$, where we write $0$ for the trivial Picard groupoid (this is $\mathcal{P}_{0}$ loc. cit.), so there is another Cartesian diagram$$\xymatrix{
\mathbb{V}(\mathfrak{A},\mathbb{R}) \ar[r] \ar[d] & 0 \ar[d] \\
\mathbb{V}(\mathfrak{A}) \ar[r] & V(A_{\mathbb{R}})
}
\label{lww_Z1}$$ of Picard groupoids. Hence, by Equation \[llh3\] it follows that$$\pi_{0}\mathbb{V}(\mathfrak{A},\mathbb{R})\cong K_{0}(\mathfrak{A},\mathbb{R})
\label{lww_Z2}$$ since both are merely the groups $\pi_{0}$ of the fiber along maps induced from the same functor, namely tensoring to $A_{\mathbb{R}}$. Now [@MR1884523 §3.4] define$$\Xi(M,T_{p},S)^{\operatorname*{BF}}:=([R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right) ],\Xi(M)^{\operatorname*{BF}},\vartheta_{p})\in
V(\mathfrak{A}_{p})\times_{V(A_{p})}V(A) \label{lix1}$$ (see loc. cit. for the meaning of $S$, $S_{p}$; $T_{p}$ stems from the projective $\mathfrak{A}$-structure picked above), where they use the notation of their concrete model of fiber products of Picard groupoids and a $p$-local slight variant of Equation \[llh2\]. The map $\vartheta_{p}$ is the same as we use in Equation \[lww\_Z3\]. They go on to prove that there is no actual dependency on $T_{p}$ or $S$, [@MR1884523 Lemma 5]. Next, they glue from this $p$-local data a virtual object$$\Xi(M,T,S)_{\mathbb{Z}}^{\operatorname*{BF}}\in\mathbb{V}(\mathfrak{A})$$ encompassing all finite primes $p$. For this, see [@MR1884523 Lemma 6]. Finally, they use $\vartheta_{\infty}$ (same as in our Equation \[lww\_x5\]) to get a further trivialization, moving this virtual object into the fiber in Diagram \[lww\_Z1\].
What has happened here: We have twice constructed an object by using the defining property of the fiber product Picard groupoid: (1) first we used (modulo some details around [@MR1884523 §3.4] and [@MR1884523 Lemma 6]) the fiber $\mathbb{V}(\mathfrak{A})$, i.e.$$V(\widehat{\mathfrak{A}})\times V(A)\longrightarrow V(\widehat{A})
\label{lepsi1}$$ and then (2) secondly the fiber of Diagram \[lww\_Z1\], i.e.$$\mathbb{V}(\mathfrak{A})\longrightarrow V(A_{\mathbb{R}})\text{.}
\label{lepsi2}$$ Taking the fiber twice consecutively can equivalently be described as taking the fiber of$$\operatorname*{fib}\left( V(\widehat{\mathfrak{A}})\times V(A)\longrightarrow
V(\widehat{A})\times V(A_{\mathbb{R}})\right) \text{.} \label{lgg1}$$ We have not spelled out the maps here, but they just stem from tensoring. Now we may truncate Equation \[lxmc3\] to the attached stable $(0,1)$-type, giving$$\tau_{\leq1}\Omega K(\mathsf{LCA}_{\mathfrak{A}})\overset{\sim}{\longrightarrow}\operatorname*{fib}\left( \tau_{\leq1}K(\widehat
{\mathfrak{A}})\times\tau_{\leq1}K(A)\longrightarrow\tau_{\leq1}K(\widehat
{A})\times\tau_{\leq1}K(A_{\mathbb{R}})\right) \text{,}$$ where we now mean the fiber in $\mathsf{Sp}^{0,1}$. However, Equation \[llh\] is ($2$-)functorial in exact functors between exact categories, so firstly the truncations of the $K$-theory spaces can all be identified with the stable $(0,1)$-types of their virtual objects, and the middle arrow is functorially induced. Finally, since $\Psi$ is an equivalence of homotopy categories, the notions of fiber are compatible. Thus,$$\begin{aligned}
\Psi^{-1}\tau_{\leq1}\Omega K(\mathsf{LCA}_{\mathfrak{A}}) & \cong\operatorname*{fib}\left( \Psi^{-1}\tau_{\leq1}K(\widehat{\mathfrak{A}})\times\Psi^{-1}\tau_{\leq1}K(A)\longrightarrow\Psi^{-1}\tau_{\leq
1}K(\widehat{A})\times\Psi^{-1}\tau_{\leq1}K(A_{\mathbb{R}})\right)
\label{lgg2}\\
& \cong\operatorname*{fib}\left( V(\widehat{\mathfrak{A}})\times
V(A)\longrightarrow V(\widehat{A})\times V(A_{\mathbb{R}})\right)
\text{,}\nonumber\end{aligned}$$ which agrees with the fiber which Burns and Flach take, see Equation \[lgg1\]. Here we have tacitly used that the maps in the fibration sequence of Theorem \[thm\_PrincipalIdeleFibration\] are induced from the same functors (tensoring).
Thus, in Equation \[lgg2\] we have produced an isomorphism between the object$$\Psi^{-1}\tau_{\leq1}\Omega K(\mathsf{LCA}_{\mathfrak{A}})$$ in which our construction of the Tamagawa number is formulated (modulo truncating to the $1$-skeleton and $\Psi^{-1}$, but as discussed above truncating does not affect $\pi_{1}$, where our $T\Omega$ lies, and $\Psi
^{-1}$ preserves $\pi_{1}$, transforming it into the notion of $\pi_{1}$ for Picard groupoids); and the object$$\operatorname*{fib}\left( V(\widehat{\mathfrak{A}})\times V(A)\longrightarrow
V(\widehat{A})\times V(A_{\mathbb{R}})\right)$$ in which Burns and Flach run their construction of their Tamagawa number, which we shall call $T\Omega^{\operatorname*{BF}}$.
This isomorphism being set up, we need to compare the actual constructions: The object $\Xi(M,T_{p},S)^{\operatorname*{BF}}$ of Equation \[lix1\] stems from the input$$\lbrack R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right) ]\text{, }\Xi(M)^{\operatorname*{BF}}\text{, }\vartheta_{p}$$ and we had used the same object $R\Gamma_{c}\left( \mathcal{O}_{F,S_{p}},T_{p}\right) $ for our construction, the same map $\vartheta_{p}$, and $\Xi(M)^{\operatorname*{BF}}$ was already shown to be the image of our $\Xi(M)$ in Theorem \[thm\_LinesAgree\]. Similarly for $\vartheta_{\infty}$ in the fiber of $\mathbb{V}(\mathfrak{A})\rightarrow V(A_{\mathbb{R}})$. Finally, take $\pi_{0}$ of Equation \[lgg2\]. We get$$\pi_{0}\Psi^{-1}\tau_{\leq1}\Omega K(\mathsf{LCA}_{\mathfrak{A}})=\pi
_{0}\Omega K(\mathsf{LCA}_{\mathfrak{A}})=\pi_{1}K(\mathsf{LCA}_{\mathfrak{A}})=K_{1}(\mathsf{LCA}_{\mathfrak{A}})\text{,}$$ while$$\pi_{0}\operatorname*{fib}\left( V(\widehat{\mathfrak{A}})\times
V(A)\longrightarrow V(\widehat{A})\times V(A_{\mathbb{R}})\right) =\pi
_{0}\mathbb{V}(\mathfrak{A},\mathbb{R})=K_{0}(\mathfrak{A},\mathbb{R})$$ by Equation \[lww\_Z2\]. This gives an identification of the groups in question, coming from the identification of the separate fibers of Equations \[lepsi1\] and \[lepsi2\] with the composite fiber in Equation \[lgg1\]. The two elements $\Xi(M,T_{p},S)^{\operatorname*{BF}}$ in (essentially) $V(\widehat{\mathfrak{A}})\times_{V(\widehat{A})}V(A)$ of [@MR1884523 §3.4, page 526] (plus the independence lemma proven loc. cit.) and $(\Xi(M)_{\mathbb{Z}},\vartheta_{\infty})^{\operatorname*{BF}}$ in $\mathbb{V}(\mathfrak{A},\mathbb{R})$, given in terms of the explicit structure of relative Picard groupoids, then topologically can be unravelled to give paths after the respective base change of the relative Picard groupoid. They correspond to the path we define in Equations \[lww\_Z3\] (and see Elaboration \[elab\_ThetaPConstructedAsInBF\] why it is clear that they match) and the path of Equation \[lww\_x5\] respectively.
This construction gives a more concrete formulation of Theorem \[thm2\_Intro\] and proves the equivalence.
Complements
===========
\[Arakelov interpretation\]\[example\_Arakelov\]If the semisimple algebra $A$ is merely a number field, i.e. $A:=F$ and $\mathfrak{A}:=\mathcal{O}_{F}$ its ring of integers, then one can interpret the idèle group of Equation \[l\_Rado\_3\] as an extension of the Arakelov–Picard group, i.e. a group classifying metrized line bundles. Write $s$ for the number of real places of $F$, and $r$ for the number of complex places. Consider the following commutative diagram with exact rows and columns:$$\xymatrix{
0 \ar[r] & \mu_{F} \ar@{^{(}->}[d] \ar[r] & {{\textstyle\prod_{\mathfrak
{p}\text{ fin.}}}\mathcal{O}_{\mathfrak{p}}^{\times}\oplus(S^{1})^{r}\oplus\{\pm1\}^{s}} \ar[r] \ar@{^{(}->}[d] & T \ar[r] \ar@{^{(}->}[d] & 0 \\
0 \ar[r] & F^{\times} \ar@{->>}[d] \ar[r] & {{\textstyle\prod_{\mathfrak
{p}\text{ fin.}}^{\prime}}F_{\mathfrak{p}}^{\times}\oplus{\textstyle
\bigoplus_{\sigma}}\mathbb{R}_{\sigma}^{\times}} \ar[r] \ar@{->>}[d] & C_{F}
\ar@{->>}[d] \ar[r] & 0 \\
0 \ar[r] & F^{\times}/\mu_{F} \ar[r] & {{\textstyle\bigoplus_{\mathfrak
{p}\text{ fin.}}}\mathbb{Z}\oplus{\textstyle\bigoplus_{\sigma}}\mathbb{R}}
\ar[r] & \widehat{\operatorname*{Pic}}_{F} \ar[r] & 0
}
\label{l_Fig_E1}$$ We write $\mu_{F}$ for the roots of unity in $F$, $\mathbb{R}_{\sigma}$ to denote the closure of $F$ in the image of the infinite place $\sigma$, i.e. this can be either $\mathbb{R}$ or $\mathbb{C}$. We write $C_{F}$ for the idèle class group and $\widehat{\operatorname*{Pic}}_{F}$ for the Arakelov–Picard group, in the sense of [@MR1847381]. We explain how to construct Figure \[l\_Fig\_E1\]: Take the two bottom rows as the input for the snake lemma to get the top row. As all the downward arrows of the bottom rows are surjective, the exactness of the top row follows. The bottom two rows stem from the map (a) $F^{\times}$ being sent along the embeddings along all the places in the middle row, and (b) $F^{\times}$ being sent to its valuation at the finite places and $x\mapsto\log\left\vert \sigma(x)\right\vert $ for each infinite place $\sigma:F\hookrightarrow\mathbb{R}_{\sigma}$. We wrote $T$ merely to denote the cokernel in the top row. The middle downward surjection sends each element in $F_{\mathfrak{p}}^{\times}$ to its $\mathfrak{p}$-adic valuation for finite places, and $x\mapsto\log\left\vert x\right\vert $ for infinite places.Now quotient out the image of $U(\mathfrak{A})=_{def}{\textstyle\prod_{\mathfrak{p}\text{ fin.}}}
\mathcal{O}_{\mathfrak{p}}^{\times}$ in $T$, transforming the right downward column into$$\frac{(S^{1})^{r}\times\{\pm1\}^{s}}{\mu_{F}}\hookrightarrow\frac{JA}{J^{1}(A)\cdot A^{\times}\cdot U(\mathfrak{A})}\twoheadrightarrow
\widehat{\operatorname*{Pic}}_{F}\text{.} \label{l_h_Arak}$$ We summarize this as follows.
\[prop\_XArak1\]If $F$ is a number field, pick $A:=F$ and $\mathfrak{A}:=\mathcal{O}_{F}$. Then there is a canonical extension of abelian groups,$$\frac{(S^{1})^{r}\times\{\pm1\}^{s}}{\mu_{F}}\hookrightarrow K_{0}(\mathfrak{A},\mathbb{R})\twoheadrightarrow\widehat{\operatorname*{Pic}}_{F}\text{;}$$ and of course we could also write $K_{1}(\mathsf{LCA}_{\mathfrak{A}})$ for the middle group.
Given an Arakelov divisor $\sum x_{\mathfrak{p}}+\sum x_{\sigma}$, i.e. an element of ${\textstyle\bigoplus_{\mathfrak{p}\text{ fin.}}}
\mathbb{Z}\oplus{\textstyle\bigoplus_{\sigma}}
\mathbb{R}$, representing a class in $\widehat{\operatorname*{Pic}}_{F}$, one can attach to this an Arakelov line bundle, by equipping a genuine line bundle $L$ within the isomorphism class of the image under $\widehat
{\operatorname*{Pic}}_{F}\twoheadrightarrow\operatorname*{Pic}(\mathcal{O}_{F})$ with the metric such that on $L\otimes_{\mathbb{Z}}\mathbb{R}$ we have$$\begin{tabular}
[c]{ll}$\left\Vert 1\right\Vert _{\sigma}^{2}=e^{-2x_{\sigma}}$ & $\text{for }\sigma\text{ real}$\\
$\left\Vert 1\right\Vert _{\sigma}^{2}=2e^{-2x_{\sigma}}$ & $\text{for }\sigma\text{ complex}$\end{tabular}
\ $$ in terms of the norm of the image of $1\in F^{\times}$ under the embedding $\sigma$. Relating this to our constructions, this means that $(z_{\sigma
})_{\sigma\in S_{\infty}}\in\prod\mathbb{R}_{\sigma}^{\times}$ goes to$$\left\Vert 1\right\Vert _{\sigma}^{2}=c_{\sigma}e^{-2\log\left\vert \sigma
z_{\sigma}\right\vert }=c_{\sigma}\left\vert \sigma z_{\sigma}\right\vert
^{-2}\qquad\text{with}\qquad c_{\sigma}\in\{1,2\}\text{.} \label{l_h_Arak2}$$ The group on the left in Equation \[l\_h\_Arak\] thus corresponds precisely to the kernel of the absolute values occuring in Equation \[l\_h\_Arak2\]. Thus, if one insisted on giving the middle group of Equation \[l\_h\_Arak\] a geometric interpretation, it would be (angularly) decorated metrized line bundles. This could be extended to the non-commutative setting, where now real, complex and quaternion embeddings as in the Artin–Wedderburn decomposition of $A\otimes_{\mathbb{Q}}\mathbb{R}$ would play a rôle.
If $A$ is a finite-dimensional semisimple $\mathbb{Q}$-algebra and $\mathfrak{A}\subset A$ an arbitrary order, Proposition \[prop\_XArak1\] should have analogues in a suitably formulated theory of $\mathfrak{A}$-equivariant Arakelov modules. The papers [@MR1914000], [MR2192383]{} give possible answers to this.
[^1]: The author was supported by DFG GK1821 Cohomological Methods in Geometry and a Junior Fellowship at the Freiburg Institute for Advanced Studies (FRIAS)
[^2]: This is not really a new result. Agboola and Burns have given a Hom-description formulation of a much more general result in [@MR2192383].
[^3]: a projective $\mathfrak{A}$-structure plus the Coherence Hypothesis, [@MR1884523 §3.3]
[^4]: in a complicated sense: firstly derived, i.e. up to quasi-isomorphisms, and secondly $K$-theoretically, i.e. after transforming exact sequences into alternating sums.
[^5]: The book [@MR0270372] might be a bit out of date, but it carefully develops and explores such structures along a lot of examples.
[^6]: and to be fully honest we do not just need that the model has the correct homotopy type; it also needs to have the correct infinite loop space structure, which is true for all of the well-known models of $K$-theory (except for the plus construction approach).
[^7]: as exists by work of Ayoub
[^8]: Some people prefer only working with elementary paths as the notion of path. This is also reasonable (and simpler), but then one only gets a well-behaved concept of paths for fibrant simplicial sets.
[^9]: Really $\mathsf{Top}_{\bullet}$ should first be replaced by $k$-spaces $\mathsf{K}_{\bullet}$. Doing this, the said adjunction with the same functors gives a Quillen equivalence, [@MR1650134 Theorem 3.6.7]. Then, there is a further Quillen equivalence between $\mathsf{K}_{\bullet}$ and $\mathsf{Top}_{\bullet}$, [@MR1650134 Corollary 2.4.24]. It follows that both approaches are connected by a zig-zag of Quillen equivalences.
[^10]: more precise: it is the left adjoint of the forgetful functor from abelian groups to abelian monoids.
[^11]: a pointed category is a category along with a choice of a fixed zero object
[^12]: treat this as a $2$-category if you prefer, but it is not really necessary for our purposes
[^13]: in the sense of [@MR1167575 Definition 2.3]
[^14]: So this is inspired from the fact that before the introduction of non-commutative coefficients, this would have been phrased in terms of determinant lines (e.g. Fontaine, Perrin–Riou,…), and these form a Picard groupoid.
[^15]: So, in the context of this definition, we do *not* (yet) demand that we have picked a bifunctor $\mathsf{C}\times\mathsf{C}\rightarrow\mathsf{C}$ which exhibits these coproducts as a monoidal structure. Thus, (at this point) for these coproducts it suffices to be well-defined up to unique isomorphism.
[^16]: in the sense of [@MR1167575 Definition 2.3].
|
---
abstract: 'At the Relativistic Heavy Ion Collider, jets have been a useful tool to probe the properties of the hot, dense matter created. At the Large Hadron Collider, collisions of Pb+Pb at $\sqrt{s_{NN}}$ = 5.5 TeV will provide a large cross section of jets at high $E_T$ above the minimum bias heavy ion background. Simulations of the Compact Muon Solenoid (CMS) experiment’s capability to measure jets in heavy ion collisions are presented. In particular, $\gamma$-jet measurements can estimate the amount of energy lost by a jet interacting strongly with the medium, since the tagged photon passes through unaffected.'
author:
- 'M.B. Tonjes (for the CMS collaboration)'
title: Jet Analysis in Heavy Ion Collisions in CMS
---
INTRODUCTION
============
In relativistic heavy ion collisions, it is theorized that a hot, dense medium known as the quark-gluon plasma is formed. The energy loss of hard partonic jets propagating through the medium can serve as a useful probe. At the Relativistic Heavy Ion Collider (RHIC), a suppression of hadron yield at high transverse momentum ($p_T$) was observed in 200 GeV Au+Au collisions when compared to p+p collisions (scaled by the number of binary collisions, $N_{coll}$) [@WhitePaper]. This jet quenching observation is believed to be from energy loss of fast partons traversing the medium produced. However, high $p_T$ hadrons can also be emitted from the surface of the collision, which makes the quenching measurement harder to quantify [@Eskola]. A direct measurement of jet production, as opposed to leading high $p_T$ hadrons, would be illuminating.
At the Large Hadron Collider (LHC), there will be a much higher rate of particle production than at RHIC [@Vitev]. In one year of Pb+Pb collisions at $\sqrt{s_{NN}} = 5.5$ TeV, $0.5$ $nb^{-1}$ integrated luminosity will be obtained, with an estimated 7.8 b inelastic Pb+Pb cross section. This gives a total of $3.9$ x $10^9$ Pb+Pb collisions. It is expected that fully formed high $E_T$ jets will be made at a rate of more than 10 pairs per second.
The Compact Muon Solenoid (CMS) detector is well suited to measure high $p_T$ jets and photons in heavy ion collisions [@HIPTDR]. CMS has high precision tracking over $|\eta|<2.5$, calorimetry in $|\eta|<5$, as well as muon identification over $|\eta|<2.5$, with a large bandwidth data acquisition and high level trigger. In events in which a photon and jet are created together, the initial transverse energy of the fragmenting parton can be determined from the photon $E_T$. The modification of the parton fragmentation function by the dense medium in heavy ion collisions can be studied in comparison to p+p collisions.
ANALYSIS DETAILS
================
Simulations
-----------
For this analysis [@GammaJet], the $\gamma$-jet channel was simulated within a Pb+Pb environment. Simulated events were created by generating p+p events that include high $p_T$ $\gamma$-jet interactions, as well as QCD background. The p+p generators used were PYTHIA [@Pythia] and PYQUEN [@Pyquen], where PYQUEN includes a jet quenching scenario. In addition, heavy ion background events were created using HYDJET [@Hydjet] at $\sqrt{s_{NN}}$ = 5.5 TeV using either the unquenched or quenched scenario (quenched includes parton energy loss due to the medium). A number of 0-10% central events equivalent to one year of heavy ion running were created. The $\gamma$-jet signal and Pb+Pb background events were mixed for either the unquenched or quenched scenarios. Data taking conditions were simulated by GEANT-4 with a full CMS detector, followed by full CMS reconstruction.
Tracking
--------
Charged particle reconstruction was performed using the CMS tracker. The algorithm used is based on seeding from hits in the silicon pixel detector [@Tracking]. This algorithm is an extension of that used in p+p collisions with quality cuts optimized for heavy ion collisions. At midrapidity in a heavy ion environment, the algorithmic efficiency is about 70% near midrapidity for charged particles of $p_T > 1$ GeV/c, with a fake rate of a few percent. For reconstructed tracks of $p_T < 100$ GeV/c, the momentum resolution is $\Delta p_T/p_T$ $<$ 1.5%.
Jet Finding
-----------
Jet finding was performed with the pileup jet finding algorithm using calorimeter energy deposition [@PileupJet]. The algorithm is a standard p+p iterative cone jet finding algorithm with a noise/pedestal subtraction. Calorimeter energy from both the electromagnetic (ECAL) and hadronic (HCAL) calorimeters are combined to make towers. Average tower transverse energy and dispersion are calculated for rings in pseudorapidity for each event. Then the tower energy is recalculated by subtracting the mean and dispersion, dropping any towers with negative energy after the subtraction. The first set of jets are found with an iterative cone algorithm (cone radius = 0.5) and the pedestal subtracted tower energy. Then, using the original tower energy, mean and dispersion are found for towers outside of the first set of found jets. This second set of jets have their energy adjusted again with the pedestal subtraction, dropping any negative towers. With these final background subtracted energies, the iterative cone algorithm is used and jets are found. The jet finder has good performance in heavy ion events, with an efficiency above 80% for Monte Carlo (MC) jets of $E_T$ above 100 GeV. Reconstructed jets with transverse energy below 30 GeV are cut to reduce the high rate of fakes. Jet energies are not corrected for variations due to pseudorapidity or energy dependent particle response. However, this analysis does not use jet energy except for the $E_T$ cut to reduce the number of fake jets found.
Photon Isolation and Multivariate Analysis
------------------------------------------
Photon reconstruction is performed using several parts of the CMS detector. Superclusters of energy deposits in the ECAL that have $E_T > 70$ GeV are found using a standard p+p clustering algorithm. Ten cluster shape variables from the ECAL are combined with isolation variables based on both the ECAL and HCAL, and tracking information. These variables are processed in a multivariate analysis, using the TMVA package of ROOT [@TMVA]. The TMVA package is used to determine an optimal cut to divide the candidates into isolated photons (signal) and background. [Figure \[fig:PhotonEff\]]{} (left panel) shows the $E_T$ distribution comparing isolated photons (red squares) to non-isolated particles (black circles), which has a signal to background ratio of 0.3 before the TMVA analysis. After the TMVA analysis, the signal to background ratio has improved to 4.5, as can be seen in [Figure \[fig:PhotonEff\]]{} (right panel). For photons above a transverse energy of 70 GeV, the average $E_T$ resolution is about 4.5%. This analysis includes a fully simulated QCD background with a number of possible false photons: from fragmentation, pions, or even mis-identified photons. However, the signal to background improvement in the heavy ion environment is significant.
RESULTS
=======
A fragmentation function is created by correlating isolated photons ($E_T > 70$ GeV) with jets that are back-to-back, that is with an angle of separation between the photon and away side jet greater than $172^{\circ}$. The parton $E_T$ is estimated from the photon $E_T$. The jet $p_T$ is found from reconstructed tracks which are within the jet cone (R=0.5). The variable $\xi$ is then defined as the natural logarithm of the ratio of the photon $E_T$ over the jet $p_T$. The fragmentation function characterizes the process by which high $p_T$ partons become final state hadrons.
To understand the contribution of the underlying heavy ion event, tracks which are outside of the jet cone are studied. The momentum distribution of tracks within a radius of 0.5 cone that is $90^{\circ}$ away from the jet is used to estimate the underlying event contribution. This underlying event fragmentation function is then subtracted from the measured fragmentation function. The results are shown in [Figure \[fig:FragFunc\]]{} (left panel) for unquenched events, and in [Figure \[fig:FragFunc\]]{} (right panel) for quenched events. The fragmentation function created from the simulated (MC) truth data before reconstruction is represented by the solid line. There are four main contributions to the systematic errors (grey band) which are added in quadrature. One is QCD jet fragmentation products which pass ECAL cluster cuts and are misidentified as photons. Another is the association of a wrongly paired or fake jet on the away side of the isolated photon. Uncertainty in charged particle reconstruction efficiency also contributes to the final measurement. The largest systematic uncertainty is from the low jet reconstruction efficiency for low $E_T$ jets.
[Figure \[fig:QuenUnquenRatio\]]{} shows the ratio of the underlying event subtracted fragmentation functions of quenched events to unquenched events, with the systematic errors shown by the grey band. The ratio of the fragmentation functions for simulation truth is shown by the solid line. The agreement between the truth and reconstructed fragmentation functions is quite good. The reconstructed fragmentation function reproduces the MC truth over the full $\xi$ range within uncertainties. The change in the fragmentation function between quenched and unquenched scenarios is larger than the estimated uncertainty.
CONCLUSION
==========
CMS will be able to quantitatively study high $p_T$ parton fragmentation within the medium by using $\gamma$-jet events. Simulations of one year of Pb+Pb collisions show that this measurement is sensitive to anticipated changes in the fragmentation function within expected statistical and systematic uncertainties.
[9]{} RHIC White Papers, Nucl. Phys. A[**757**]{} (2005) 28. K. J. Eskola, H. Honkanen, C. A. Salgado, and U. A. Wiedemann, Nucl. Phys. A[**747**]{} (2005) 511. I. Vitev, arXiv:hep-ph/0212109. The CMS Collaboration, J. Phys. G: Nucl. Part. Phys. 34 (2007) 2307-2455. The CMS Collaboration, [**CMS Note PAS HIN-07-002**]{}; C. Loizides for the CMS Collaboration, [*[proceedings for Quark Matter 2008]{}*]{}, arXiv:0804.3679 \[nucl-ex\]. T. Sjostrand, S. Mrenna, and P. Skands, JHEP [**[0605]{}**]{} (2006) 026 \[PYTHIA v6.411 is used\]. I. P. Lokhtin and A. M. Snigirev, Eur. Phys. J. C [**[45]{}**]{} (2006) 211 \[PYQUEN v1.2 is used\]. I. P. Lokhtin and A. M. Snigirev, arXiv:hep-ph/0312204 \[HYDJET v1.2 is used\]. C. Roland, Nucl. Instrum. Meth. A[**[566]{}**]{} (2006) 123-126. O. Kodolova, I. Vardanyan, A. Nikitenko, and A. Oulianov, Eur. Phys. J. C [**[50]{}**]{} (2007) 117. Toolkit for Multivariate Data Analysis with ROOT, v3.8.11 <http://tmva.sourceforge.net>.
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---
abstract: 'Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.'
author:
- Shai Sarussi
title: 'Quasi-valuations - topology and the weak approximation theorem'
---
\[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{}
\[thm\][Definition]{} \[thm\][Remark]{}\[thm\][Example]{}
Introduction
============
Recall that a valuation on a field $F$ is a function $v : F
\rightarrow \Gamma \cup \{ \infty \}$, where $\Gamma$ is a totally ordered abelian group and $v$ satisfies the following conditions:
(A1) $v(x) = \infty$ iff $x=0$;
(A2) $v(xy) = v(x)+v(y)$ for all $x,y \in F$;
(A3) $v(x+y) \geq \min \{ v(x),v(y) \}$ for all $x,y \in F$.
There has been considerable interest in recent years in generalizations of valuations, in order to treat rings that are not integral domains, and also to handle several valuations simultaneously. For example, pseudo-valuations ([*see*]{} \[Co\],\[Hu\], and \[MH\]), Manis-valuations and PM-valuations ([*see*]{} \[KZ\]), value functions ([*see*]{} \[Mor\]), and gauges ([*see*]{} \[TW\]). These related theories are discussed briefly in the introduction of \[Sa\].
In this paper we continue our study from \[Sa\] of quasi-valuations. Recall that a [*quasi-valuation*]{} on a ring $A$ is a function $w
: A \rightarrow ~M \cup \{ \infty \}$, where $M$ is a totally ordered abelian monoid, to which we adjoin an element $\infty$ greater than all elements of $M$, and $w$ satisfies the following properties:
(B1) $w(0) = \infty$;
(B2) $w(xy) \geq w(x) + w(y)$ for all $x,y \in A$;
(B3) $w(x+y) \geq \min \{ w(x), w(y)\}$ for all $x,y \in A$.
The minimum of a finite number of valuations with the same value group is a quasi-valuation. For example, the $n$-adic quasi-valuation on $\Bbb Q$ (for any positive $n \in \Bbb Z$) already has been studied in \[Ste\]. (Stein calls it the $n$-adic valuation.) It is defined as follows: for any $0 \neq \frac{c}{d}
\in \Bbb Q$ there exists a unique $e \in \Bbb Z$ and integers $a,b
\in \Bbb Z$, with $b$ positive, such that $\frac{c}{d}=n^{e}\frac{a}{b}$ with $n \nmid a$, $(n,b)=1$ and $(a,b)=1$. Define $w_{n}(\frac{c}{d})=e$ and $w_{n}(0)=\infty$.
In \[Sa\] we develop the theory of quasi-valuations on finite dimensional field extensions that extend a given valuation. For the reader’s convenience we briefly overview some of the results from \[Sa\]. Let $F$ be a field with valuation $v$ and valuation ring $O_{v}$, let $E$ be a finite field extension and let $w$ be a quasi-valuation on $E$ extending $v$ with a corresponding quasi-valuation ring $O_{w}$. We prove that $O_{w}$ satisfies INC (incomparability), LO (lying over), and GD (going down) over $O_{v}$; in particular, $O_{w}$ and $O_{v}$ have the same Krull Dimension. We also prove that every such quasi-valuation is dominated by some valuation extending $v$. Namely, there exists a valuation $u$ extending $v$ on $E$ so that $\forall x \in E$, $w(x) \leq u(x)$.
Under the assumption that the value monoid of the quasi-valuation is a group we prove that $O_{w}$ satisfies GU (going up) over $O_{v}$, and a bound on the size of the prime spectrum is given. In addition, a 1:1 correspondence is obtained between exponential quasi-valuations and integrally closed quasi-valuation rings.
Given $R$, an algebra over $O_{v}$, we construct a quasi-valuation on $R$; we also construct a quasi-valuation on $R \otimes _{O_{v}}
F$, which helps us prove our main Theorem. The main Theorem states that if $R \subseteq E$ satisfies $R \cap F=O_{v}$ and $E$ is the field of fractions of $R$, then $R$ and $v$ induce a quasi-valuation $w$ on $E$ such that $R=O_{w}$ and $w$ extends $v$; thus $R$ satisfies the properties of a quasi-valuation ring.
In this paper, we extend some fundamental results from valuation theory. For example, we prove that the topology induced by a quasi-valuation is Hausdorff and totally disconnected. We also prove a weak version of the approximation theorem for quasi-valuations.
The topology induced by a quasi-valuation
=========================================
In this section we introduce the topology induced by a quasi-valuation. We show that this topology is close to the topology induced by a valuation in the sense that they share some basic topological properties such as being both Hausdorff and totally disconnected. In the main theorem of this section we prove that the topology induced by the quasi-valuation is determined by the corresponding quasi-valuation ring.
In this section $F$ denotes a field with a nontrivial valuation $v$, a value group $\Gamma$, and a valuation ring $O_{v}$. $E$ denotes a finite dimensional field extension with $n=[E:F]$, $w:E \rightarrow M \cup \{ \infty \}$ is a quasi-valuation on $E$ with quasi-valuation ring $O_{w}$ (namely, $O_{w}=\{x\in E \ | \ w(x)\geq0\}$), such that $w|_{F}=v$, and $M$ is a totally ordered abelian monoid containing $\Gamma$.
We note that w(-1)=0 because $w$ extends $v$. Thus, by \[Sa, Lemma 1.3\], we have $w(x)=w(-x)$ for all $x \in E$. Moreover, by \[Sa, Lemma 1.4\], for all $x,y \in E$ such that $w(x) \neq w(y)$, we have $w(x+y)= min \{w(x),w(y)\}$. Recall from \[Sa, Definition 1.5\] that an element $c \in E$ is called stable with respect to $w$ if $w(cx)=w(c)+w(x)$ for every $x \in E$. Thus, by \[Sa, Lemma 1.6\], every $a\in F$ is stable with respect to $w$. We shall freely use these facts throughout the paper.
Let $x \in E$ and $m \in M$; we denote $$U_{m}^{w}(x)=\{ y \in E \mid
w(y-x)>m \};$$we suppress $w$ when it is understood.
\[x in umx\] Let $x \in E$ and $m \in M$; then $x \in U_{m}(x)$. Indeed, $$w(x-x)=w(0)=\infty>m.$$
We shall repeatedly use Remark \[x in umx\] without reference.
If $y \in U_{m_{1}}(x_{1}) \cap
U_{m_{2}}(x_{2})$ and $m_{1} \leq m_{2}$, then $$y \in U_{m_{2}}(y) \subseteq U_{m_{1}}(x_{1}) \cap
U_{m_{2}}(x_{2}).$$
Since $y \in U_{m_{1}}(x_{1}) \cap U_{m_{2}}(x_{2})$, we have $w(y-x_{1}) > m_{1}$ and $w(y-x_{2}) > m_{2}
\geq m_{1}$. Let $z \in U_{m_{2}}(y)$; then $w(z-y) > m_{2}$. Thus, $$w(z-x_{1}) = w(z-y+y-x_{1})$$ $$\geq min \{ w(z-y),
w(y-x_{1})\} > m_{1}$$ and $$w(z-x_{2}) = w(z-y+y-x_{2})$$ $$\geq
min \{ w(z-y), w(y-x_{2})\} > m_{2}.$$ Thus $z \in U_{m_{1}}(x_{1}) \cap
U_{m_{2}}(x_{2})$
We denote $B=\{ U_{m}(x) \mid x \in E$, $m \in M\}$.
The set $B$ is a base for a topology on $E$.
In view of Corollary 2.3 we define,
The topology whose base is $B$ will be denoted by $T_{w}$. We call $T_{w}$ the topology induced by the quasi-valuation $w$.
We recall the following lemma from \[Sa, Lemma 2.8\]:
\[wx alpha\] Let $E/F$ be a finite field extension and let $w$ be a quasi-valuation on $E$ extending a valuation $v$ on $F$. Then $w(x) \neq \infty$ for all $0\neq x \in E$. In fact, for all $0\neq x \in E$, there exists $\alpha \in \Gamma$ such that $w(x) < \alpha$.
We denote $$M^{G}=\{ m \in M \mid \text{\ there exists\ } 0 \neq y\in E
\text{\ such that\ } m \leq w(y) \}.$$
\[MG is a submonoid and contains Gamma\] $M^{G}$ is a submonoid of $M$ containing $\Gamma$.
Let $m_1,m_2 \in M^{G}$; then there exist nonzero $y_1, y_2 \in E$ such that $m_1 \leq w(y_1)$ and $m_2 \leq w(y_2)$. Thus, $$m_1 +m_2 \leq w(y_1)+w(y_2) \leq w(y_1y_2).$$ Note that $y_1y_2 \neq 0$ and thus $m_1 +m_2 \in M^{G}$. It is easy to see that $\Gamma \subseteq M^{G}$; indeed, for every $\alpha \in \Gamma$ there exists a nonzero $a \in F$ such that $w(a)=v(a)=\alpha$.
\[MG does not have a maximal element\] $M^{G}$ does not have a maximal element; in fact, for all $m \in M^G$ there exists $\alpha \in \Gamma \cap M^G$ such that $m < \alpha$.
Let $m \in M^G$. Then there exists $0 \neq y \in E$ such that $m \leq w(y)$. By Lemma \[wx alpha\] there exists $\alpha \in \Gamma$ such that $w(y) < \alpha$. So, $m < \alpha $. By Remark \[MG is a submonoid and contains Gamma\], $\alpha \in \Gamma \subseteq M^G$.
\[T\_w is discrete\] $T_{w}$ is discrete iff there exists an element $m \in M \setminus M^{G}$.
($\Leftarrow$) Let $m \in M \setminus M^{G}$ and let $x \in E$. Then for every $y \neq x$ we have $w(y-x)<m$; thus $U_{m}(x)=\{ x \}$.
($\Rightarrow$) We assume $M=M^{G}$ and we show that $T_{w}$ is not discrete. It is enough to show that every open set has infinitely many elements. Now, since every open set contains some $U_{m}(x)$ (for $m \in M$, $x \in E$), it is enough to show that every $U_{m}(x)$ has infinitely many elements. By our assumption $M=M^{G}$ and thus for every $m \in M$ there exists $0 \neq z
\in E$ such that $m \leq w(z)$; also, by Lemma \[wx alpha\], for every such $z$ there exists $0 < \alpha \in \Gamma$ such that $w(z) < \alpha $. Take $a \in O_{v}$ with $v(a)=\alpha$. Then $x+a^{n} \in U_{m}(x)$ for each $ n \in \Bbb N$, proving the set $U_{m}(x)$ has infinitely many elements.
In view of Proposition \[T\_w is discrete\], we restrict our discussion to $M^{G}$; namely we denote $B=\{ U_{m}(x) \mid x\in E,\ m \in M^{G} \}$ as a base for $T_{w}$.
\[w y-x >m\] Let $x,y,z \in E$ and $m \in M^{G}$. If $z \in U_{m}(x) \cap
U_{m}(y)$ then $w(y-x) >m$.
By definition, $z \in U_{m}(x) \cap
U_{m}(y)$ implies $w(z-x)>m$ and $w(y-z)>m$. Thus, $$w(y-x) \geq \min \{w(y-z),w(z-x) \}>m.$$
$T_{w}$ is Hausdorff.
Let $x, y \in E$ with $x \neq y$, and write $w(y-x)=m \in M^{G}$. By Lemma \[w y-x >m\] we have, $$U_{m}(x) \cap U_{m}(y) =
\emptyset .$$
\[U\_mx is clopen\] Let $x \in E$ and $m \in M^{G}$. Then $U_{m}(x)$ is closed as well as open.
Let $y \notin U_{m}(x)$; then $w(y-x) \leq m$. By Lemma \[w y-x >m\] we have, $$U_{m}(y)\cap U_{m}(x)= \emptyset;$$ obviously $y \in U_{m}(y)$. So, $U_{m}(y)$ is an open set containing $y$ disjoint from $U_{m}(x)$.
The following lemma shows that $E$ is totally disconnected, in the following sense.
The only nonempty connected subsets of $E$ are the singleton sets $\{ x \}$ for $x \in E$.
Let $S \subseteq E$ be a nonempty set containing at least two elements, $x \neq y$. Write $w(x-y)=m$ and $U_{1}=U_{m}(x)$. Let $U_{2}$ denote the complement of $U_{1}$ in $E$, which is open by Lemma \[U\_mx is clopen\]. Note that $x \in U_{1}$ and $y \in U_{2}$ (since $w(y-x)=m \ngtr m$), and thus by definition $S$ is disconnected.
Let $x\in E$ and $m \in M^{G}$; we denote $\widetilde{U}_{m}(x)=\{ y \in E \mid w(y-x) \geq m \}$. Obviously, $ U_{m}(x) \subseteq \widetilde{U}_{m}(x)$. Thus, as in Remark \[x in umx\], we have $x \in
\widetilde{U}_{m}(x)$.
\[U\_my subset of U\_mx\] Let $x,y \in E$ and $m \in M^G$. If $y \in \widetilde{U}_{m}(x)$ then $\widetilde{U}_{m}(y) \subseteq \widetilde{U}_{m}(x).$ If $y \notin \widetilde{U}_{m}(x)$ then $ \widetilde{U}_{m}(y) \subseteq (\widetilde{U}_{m}(x))^{c}.$
Suppose $y \in \widetilde{U}_{m}(x)$; then $w(y-x) \geq m$. Let $z \in \widetilde{U}_{m}(y)$; then $w(z-y) \geq m$. Hence, $w(z-x) \geq min \{w(z-y), w(y-x) \} \geq m$. Thus, $$\widetilde{U}_{m}(y) \subseteq \widetilde{U}_{m}(x).$$
Suppose $y \notin \widetilde{U}_{m}(x)$; then $w(y-x) < m$. Let $z \in \widetilde{U}_{m}(y)$; then $w(z-y) \geq m$. Hence, $w(z-x) = min \{w(z-y), w(y-x) \} = w(y-x)<m$. Thus, $$\widetilde{U}_{m}(y) \subseteq (\widetilde{U}_{m}(x))^{c}.$$
\[widetildeU\_mx is clopen\] $\widetilde{U}_{m}(x)$ is both open and closed, for any $x \in E$ and $m \in M^{G}$.
Let $y \in \widetilde{U}_{m}(x)$; then by Lemma \[U\_my subset of U\_mx\],$$y \in
U_{m}(y) \subseteq \widetilde{U}_{m}(y) \subseteq \widetilde{U}_{m}(x).$$ Let $y \notin \widetilde{U}_{m}(x)$; then by Lemma \[U\_my subset of U\_mx\], $$y \in U_{m}(y) \subseteq \widetilde{U}_{m}(y) \subseteq (\widetilde{U}_{m}(x))^{c}.$$
\[formula\] Let $x \in E$ and let $m,m' \in
M^{G}$ such that $m<m'$. Then $$U_{m}(x)=\bigcup_{y \in U_{m}(x)} \widetilde{U}_{m'}(y).$$
$(\subseteq)$ holds because every $y \in U_{m}(x)$ is obviously in $\widetilde{U}_{m'}(y)$. To prove $(\supseteq)$, we need to show that $\widetilde{U}_{m'}(y) \subseteq U_{m}(x)$ for all $y \in U_{m}(x)$. So, let $y \in U_{m}(x)$ and let $z \in \widetilde{U}_{m'}(y)$. Then, $w(y-x)>m$ and $w(z-y) \geq m'$. Thus, since $m<m'$, $$w(z-x) \geq min \{ w(z-y),w(y-x)\}>m.$$
We denote $B_{1}=\{ \widetilde{U}_{m}(x) \mid x\in
E, \ m \in M^{G}\}$.
The set $B_{1}$ is a base for $T_{w}$.
First, by Corollary \[widetildeU\_mx is clopen\], $\widetilde{U}_{m}(x)$ is open for all $x \in E$ and $ m \in M^{G}$. Now, let $x \in E$ and $m \in M^{G}$. By Remark \[MG does not have a maximal element\] there exists $m<m' \in M^{G}$. By Lemma \[formula\] we have, $U_{m}(x)=\bigcup_{y \in U_{m}(x)} \widetilde{U}_{m'}(y).$ Thus, every open set in $T_{w}$ is a union of elements of $B_{1}$.
In fact, we can describe the topology in terms of $\Gamma$ (the value group of the valuation) as the following proposition shows.
First, we denote $B_{2}=\{ \widetilde{U}_{\alpha}(x) \mid x\in
E, \ \alpha \in \Gamma\}$.
\[B\_2 is a base for thetopology\] The set $B_{2}$ is a base for $T_{w}$.
First, by Remark \[MG is a submonoid and contains Gamma\], $B_{2} \subseteq B_{1}$. So every element of $B_{2}$ is open in $T_{w}$. Now, let $x \in E$ and $m \in M^{G}$. By Remark \[MG does not have a maximal element\], there exists $\alpha \in \Gamma \cap M^G$ such that $m < \alpha$. By Lemma \[formula\], $$U_{m}(x)=\bigcup_{y \in U_{m}(x)} \widetilde{U}_{\alpha}(y).$$
Recall from \[Sa, Section 10\] that for every ring $R \subseteq E$ satisfying $R \cap F=O_{v}$, we denote $$\mathcal W_{R}=\{ w \mid~
w \text{\ is a quasi-valuation on\ } E \text{\ extending\ } v
\text{\ with\ } O_{w}=R\}.$$
Also recall that the class $\mathcal W_{R}$ is not empty, by \[Sa, Theorem 9.35\].
\[wx geq va iff xa-1>0\] Let $w \in \mathcal W_{R}$, $ x \in E$ and $0 \neq a \in F$. The following are equivalent:
\(a) $w(x) \geq v(a);$
\(b) $w(x)-v(a) \geq 0;$
\(c) $w(xa^{-1}) \geq 0;$
\(d) $xa^{-1} \in R.$
(a)$\Leftrightarrow$(b). Because $v(a) \in \Gamma$. (b)$\Leftrightarrow$(c). $v$ is a valuation and $0 \neq a \in F$; thus $-v(a)=v(a^{-1})$. Therefore, $w(x) - v(a)=w(x) + v(a^{-1})$. Since $w$ extends $v$ and $a$ is stable with respect to $w$, we get $$w(x) + v(a^{-1})=w(x) + w(a^{-1})=w(xa^{-1}).$$ (c)$\Leftrightarrow$(d). By assumption, $w \in \mathcal W_{R}$; thus $O_{w}=R$.
\[w\_1x iff w\_2x\] Let $w_{1},w_{2} \in \mathcal W_{R}$ and let $\alpha \in \Gamma$; then $$w_{1}(x)\geq \alpha \text{\ iff\
}w_{2}(x) \geq \alpha.$$
By assumption, $O_{w_1}=O_{w_2}=R$. Let $a \in F$ such that $v(a)=\alpha$; clearly, $a \neq 0$ (since $\alpha \in \Gamma$). Thus, using Lemma \[wx geq va iff xa-1>0\] twice, we get $$w_{1}(x)\geq \alpha \text{\ iff\ } xa^{-1} \in O_{w_1}=O_{w_2} \text{\ iff\ } w_{2}(x)\geq \alpha.$$
We are ready to prove the main theorem of this section.
If $w_{1},w_{2} \in \mathcal W_{R}$, then $T_{w_{1}}=T_{w_{2}}$. In other words, the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation.
By Proposition \[B\_2 is a base for thetopology\], the set $C=\{ \widetilde{U}_{\alpha}^{w_{1}}(x) \mid x\in E, \ \alpha \in \Gamma\}$ is a base for $T_{w_{1}}$ and the set $D=\{ \widetilde{U}_{\alpha}^{w_{2}}(x) \mid x\in E, \ \alpha \in \Gamma\}$ is a base for $T_{w_{2}}$. However, for every $x\in
E$ and $ \alpha \in \Gamma$ we have, by Lemma \[w\_1x iff w\_2x\], $\widetilde{U}_{\alpha}^{w_{1}}(x)=\widetilde{U}_{\alpha}^{w_{2}}(x)$. Thus, $C=D$ and the theorem is proved.
Weak approximation theorem
==========================
In this section we prove a weak version of the approximation theorem for quasi-valuations. We call it the weak approximation theorem since it relies on the independence of the valuation rings in $F$ and not on the independence of the quasi-valuations in $E$. (The independence of the valuation rings in $F$ implies the independence of the quasi-valuation rings in $E$ but not vice versa).
In this section $F$ denotes a field and $E$ denotes a finite dimensional field extension with $[E:F]=n$.
Let $A$ and $B$ be two subrings of $F$. $A$ and $B$ are called independent if $AB=F$. Two valuations are called independent, if their rings are independent; likewise, two valuations are called dependent, if their rings are dependent.
We recall the Approximation Theorem for valuations.
(Approximation Theorem for valuations) (\[Bo, Section 7.2, Thm. 1\]) Let $\{ v_{i}\}_{1 \leq i \leq k}$ be a set of valuations on a field $F$ which are independent in pairs and let $\Gamma_{i}$ be the value group of $v_{i}$. Let $x_{i} \in
F$ and $\alpha_{i} \in \Gamma_{i}$ for $1 \leq i \leq k$. Then there exists an $x \in F$ such that $v_{i}(x-x_{i}) \geq \alpha
_{i}$ for all $i$.
Let $\{ O_{v_{i}} \}_{1 \leq i \leq k}$ be a finite set of valuation rings of $F$. We denote by $B$ their intersection, i.e., $$B= \bigcap _{1 \leq i \leq k} O_{v_{i}}.$$ Let $\{ R_{i} \}_{1
\leq i \leq k}$ be a finite set of subrings of $E$ such that $E$ is the field of fractions of each $R_{i}$ and $R_{i} \cap F=
O_{v_{i}}$ for every $1 \leq i \leq k$. Recall that by \[Sa, Theorem 9.35\], for every $1 \leq i \leq k$ there exists a filter quasi-valuation $w_{i}$ on $E$ corresponding to $R_{i}$ and such that $w_{i}$ extends $v_{i}$ (so the collections of quasi-valuations corresponding to the $R_{i}$’s are not empty.) We shall prove our theorem for every quasi-valuation $w_{i}$ on $E$ corresponding to $R_{i}$ (not necessarily filter quasi-valuations). Note that for every $1 \leq i \leq k$, $R_{i}F$ is an integral domain finite dimensional over $F$ and thus a field containing $R_{i}$; hence $R_{i}F=E$. Moreover, by \[Bo, Section 7, Proposition 1\], the field of fractions of $B$ is $F$.
We denote $R= \bigcap _{1 \leq i \leq k} R_{i}$.
The next observation is well known.
Let $C$ be an integral domain, $S$ a multiplicative closed subset of $C$ with $0 \notin S$, and $R$ an algebra over $C$. We claim that every $x \in R \otimes_{C}CS^{-1}$ is of the form $r \otimes \frac{1}{\beta}$ for $r \in R$ and $\beta \in S$. Indeed, write $x=\sum_{i=1}^{t}(r_{i} \otimes
\frac{\alpha_{i}}{\beta_{i}})$ where $r_{i} \in R$, $\alpha_{i}
\in C$ and $\beta_{i} \in S$. Let $\beta=\Pi_{i=1}^{t}\beta_{i}$ and $\alpha_{i}'=\alpha_{i}\beta \beta_{i}^{-1} \in C$. Thus, $$\sum_{i=1}^{t}(r_{i} \otimes \frac{\alpha_{i}}{\beta_{i}})=
\sum_{i=1}^{t}(r_{i} \otimes \frac{\alpha_{i}'}{\beta})=
\sum_{i=1}^{t} (\alpha_{i}' r_{i} \otimes \frac{1}{\beta})=r
\otimes \frac{1}{\beta}.$$
Where $r=\sum_{i=1}^{t} \alpha_{i}' r_{i}$.
$E=S^{-1}R \cong R \otimes_{B} F$, where $S=B \setminus \{ 0\}$.
$S^{-1}R$ is an integral domain finite dimensional over $F$, so is a field. It remains to show that any $x \in E$ has the form $r/b$ where $r\in R$ and $b \in S$. By the previous Remark, $x$ can be written in the form $r_{i}/b_{i}$ where $r_{i}\in
R_{i}$ and $b_{i}\in B \setminus \{ 0\}$. Write $b= \prod_{1 \leq i \leq k} b_{i}$ and get $$bx=r_{i} \prod _{l \neq i}b_{l} \in R_{i}B=R_{i}$$ for every $1 \leq i \leq k$, and thus $x=bx/b$ has the desired form.
We are ready for the main theorem of this section: the weak approximation theorem for quasi-valuations.
Let $E/F$ be a finite field extension and let $\{ O_{v_{i}} \}_{1 \leq i \leq k}$ be a finite set of valuation rings of $F$ which are pairwise independent. Let $\{
R_{i} \}_{1 \leq i \leq k}$ be a finite set of subrings of $E$ such that the field of fractions of each $R_{i}$ is $E$ and $R_{i}
\cap F= O_{v_{i}}$ for every $1 \leq i \leq k$. Let $\{ w_{i} \mid w_i:E \rightarrow M_i \cup \{ \infty\}, \ 1 \leq i \leq k \}$ be a set of quasi-valuations on $E$ such that for every $1 \leq i \leq
k$, $w_{i} \in \mathcal W_{R_i}$. Let $\{ x_{i}
\}_{1 \leq i \leq k} \subseteq E$ and let $\{ m_{i} \}_{1 \leq i
\leq k}$ be a set of elements such that, for every $1 \leq i \leq
k$, $m_{i} \in M^{G}_{i}$. Then there exists an element $x \in E$ such that $$w_{i}(x-x_{i}) \geq m_{i}.$$ for all $1 \leq i \leq k$.
Since $m_{i} \in M^{G}_{i}$ for every $1 \leq i \leq
k$, we get by Remark \[MG does not have a maximal element\] that there exist $\alpha_{i} \in \Gamma_{i}$ such that $m_{i}
< \alpha_{i}$ for all $1 \leq i \leq
k$. We shall prove that $w_{i}(x-x_{i}) \geq
\alpha_{i}$ for every $1 \leq i \leq k$. Let $R= \bigcap
_{1 \leq i \leq k} R_{i}$; by Proposition 3.4, $R$ contains a basis $\{ r_{1}, r_{2},...,r_{n}\} $ of $E$ over $F$. Write, for every $1 \leq i \leq k$, $$x_{i}=\sum_{1 \leq j \leq n} c_{ij}r_{j}$$ where $c_{ij} \in F$. The approximation theorem for valuations gives $d_{1},...,d_{n}
\in ~F$ such that $$v_{i}(d_{j}-c_{ij}) \geq \alpha_{i},$$ for $1
\leq i \leq k$, $1 \leq j \leq n$.
Define $x= \sum _{1 \leq j \leq n} d_{j}r_{j}$ and get, for every $1 \leq i \leq k$, $$w_{i}(x-x_{i})=w_{i}(\sum _{1 \leq j \leq n} d_{j}r_{j}-\sum_{1
\leq j \leq n} c_{ij}r_{j})$$ $$=w_{i}(\sum _{1
\leq j \leq n} (d_{j}- c_{ij})r_{j}).$$
Note that, for every $1 \leq j \leq n$ and $1 \leq i \leq k$, $$w_{i}(d_{j}-
c_{ij})=v_{i}(d_{j}- c_{ij}) \geq \alpha_{i}$$ and $w_{i}(r_{j})
\geq 0$ (since $r_{j} \in R$). Thus, $$w_{i}(\sum _{1
\leq j \leq n} (d_{j}- c_{ij})r_{j}) \geq \min_{1
\leq j \leq n} w_{i}( (d_{j}- c_{ij})r_{j}) \geq \alpha_i.$$
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Department of Mathematics, Sce College, Ashdod, 77245, Israel.
[*E-mail address: sarusss1@gmail.com*]{}
|
---
abstract: 'We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of ${{\mathbb Z}}^d$ with CPE.'
address: '-Center of Mathematics, Chongqing University, Chongqing 401331, China. Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, U.S.A.'
author:
- Hanfeng Li
date: 'January 5, 2018'
title: Garden of Eden and Specification
---
Introduction {#S-introduction}
============
Let a countable discrete group $\Gamma$ act on a compact metrizable space $X$ continuously and let $T$ be a $\Gamma$-equivariant continuous map $X\rightarrow X$. In this paper we consider the relation between the surjectivity of $T$ and a weak form of injectivity of $T$.
In the case $X=A^\Gamma$ for some finite set $A$ and $\Gamma$ acts on $A^\Gamma$ by shifts, $T$ is called a cellular automaton [@CC10] and can be thought of as an evolution determined by a local rule. In this case $T$ is not surjective exactly when there is a $w\in A^F$ for some finite set $F\subseteq \Gamma$ such that $w$ is not equal to the restriction of any element of $T(A^\Gamma)$ on $F$. Such a $w$ is called a Garden of Eden (GOE) pattern, meaning that it could appear only in the first of the sequence $A^\Gamma, T(A^\Gamma), T^2(A^\Gamma), \dots$. A pair of mutually erasable patterns is a pair $(w_1, w_2)$ of distinct elements of $A^K$ for some finite set $K\subseteq \Gamma$ such that whenever $x_1$ and $x_2$ are elements of $A^\Gamma$ coinciding on $\Gamma\setminus K$ and extending $w_1$ and $w_2$ respectively, one has $T(x_1)=T(x_2)$. In 1963, Moore [@Moore] showed that when $\Gamma={{\mathbb Z}}^d$, if there is a pair of mutually erasable patterns, then there is a GOE pattern. Soon afterwards Myhill [@Myhill] proved the converse. The results of Moore and Myhill were extended to finitely generated groups with subexponential growth by Machì and Mignosi [@MM], and then to all amenable groups by Ceccherini-Silberstein, Machì, and Scarabotti [@CMS]. On the other hand, Bartholdi [@Bartholdi10; @Bartholdi16] showed that the results of Moore and Myhill fail for every nonamenable group.
For general actions of $\Gamma$ on a compact metrizable space $X$, one has the homoclinic equivalence relation defined on $X$ (see Section \[SS-homoclinic\]) and $T$ is called [*pre-injective*]{} if it is injective on each homoclinic equivalence class [@Gromov]. For the shift action $\Gamma\curvearrowright A^\Gamma$ with finite $A$, the map $T$ is not pre-injective exactly when there is a pair of mutually erasable patterns. Thus we say the action $\Gamma\curvearrowright X$ has [*the Moore property*]{} if surjectivity implies pre-injectivity for every $\Gamma$-equivariant continuous map $T:X\rightarrow X$, the action has [*the Myhill property*]{} if pre-injectivity implies surjectivity for every such $T$, and the action has the [*Moore-Myhill property*]{} if surjectivity is equivalent to pre-injectivity for every such $T$. There has been quite some work trying to establish these properties for various actions [@CC12; @CC15; @CC16; @CC17a; @Fiorenzi00; @Fiorenzi03; @Gromov]; see also the book [@CC10] and the recent survey [@CC17b].
We say an action $\Gamma\curvearrowright X$ is [*surjunctive*]{} if injectivity implies surjectivity for every $\Gamma$-equivariant continuous map $T:X\rightarrow X$. Thus the Myhill property implies surjunctivity. Gottschalk’s surjunctivity conjecture says that the shift action $\Gamma\curvearrowright A^\Gamma$ for finite $A$ is surjunctive for every group $\Gamma$ [@Got]. This was proved for sofic groups by Gromov [@Gromov; @Weiss; @CC10; @KL].
Specification plays a vital role in our work. It is a strong orbit tracing property introduced by Bowen for ${{\mathbb Z}}$-actions [@Bowen] and extended to ${{\mathbb Z}}^d$-actions by Ruelle [@Ruelle]. There are a few versions of the specification property [@LS Definition 5.1] [@CL Definition 6.1]. For our purpose, the weak specification property (see Section \[SS-specification\]) suffices. Our result regarding the Myhill property is the following.
\[T-Myhill1\] Every expansive continuous action of a countable amenable group on a compact metrizable space with the weak specification property has the Myhill property. In particular, such actions are surjunctive.
Theorem \[T-Myhill1\] strengthens a few known results and is new even for the case $\Gamma={{\mathbb Z}}$. Fiorenzi [@Fiorenzi03 Corollary 4.8] proved Theorem \[T-Myhill1\] under the further assumption of subshifts of finite type. (Actually Fiorenzi stated her result for strongly irreducible subshifts of finite type, but strongly irreducible subshifts are exactly subshifts with the weak specification property, see Appendix \[S-subshifts\].) Ceccherini-Silberstein and Coornaert proved Theorem \[T-Myhill1\] under the further assumption of subshifts [@CC12 Theorem 1.1] and under the further assumption of a uniformly bounded-to-one factor of a weak specification subshift [@CC15 Theorem 1.1].
Algebraic actions are actions of $\Gamma$ on compact metrizable abelian groups by automorphisms. The algebraic actions of ${{\mathbb Z}}^d$ were studied extensively in 1990s, and the last decade has seen much progress towards understanding the algebraic actions of nonabelian groups; see [@Schmidt; @KL] and the references therein. Our result concerning the Moore property is the following.
\[T-Moore1\] Every expansive algebraic action of a countable amenable group with completely positive entropy with respect to the normalized Haar measure has the Moore property.
If an algebraic action of a countable amenable group has the weak specification property, then it has completely positive entropy with respect to the normalized Haar measure (see Corollary \[C-CPE\]). For expansive algebraic actions, the converse is also true in the case $\Gamma={{\mathbb Z}}^d$ [@LS Theorem 5.2], and is conjectured to hold for polycyclic-by-finite groups [@CL Conjecture 6.5, Theorem 1.2, Corollary 8.4]. The principal algebraic actions are the actions of the form $\Gamma\curvearrowright \widehat{{{\mathbb Z}}\Gamma/{{\mathbb Z}}\Gamma f}$ for $f$ in the integral group ring ${{\mathbb Z}}\Gamma$ (see Section \[SS-algebraic\]). By Lemma \[L-principal\] every expansive principal algebraic action has the weak specification property. Thus combining Theorems \[T-Myhill1\] and \[T-Moore1\] we have
\[T-MM\] Every expansive algebraic action of a countable amenable group with the weak specification property has the Moore-Myhill property. In particular, every expansive principal algebraic action of a countable amenable group and every expansive algebraic action of ${{\mathbb Z}}^d$ with completely positive entropy with respect to the normalized Haar measure have the Moore-Myhill property.
Theorem \[T-MM\] applies to the shift actions $\Gamma\curvearrowright A^\Gamma$ for finite $A$, since one can identify $A^\Gamma$ with $\widehat{{{\mathbb Z}}\Gamma/{{\mathbb Z}}\Gamma f}$ for $f=|A|$. Previously, Ceccherini-Silberstein and Coornaert established the Moore-Myhill property for hyperbolic toral automorphisms [@CC16] and expansive principal algebraic actions of countable abelian groups with connected underlying space [@CC17a].
We remark that even for $\Gamma={{\mathbb Z}}$, not every expansive action with the weak specification property has the Moore property. Indeed, Fiorenzi showed that the even shift in $\{0, 1\}^{{\mathbb Z}}$ consisting of all elements with an even number of $0$’s between any two $1$’s does not have the Moore property [@Fiorenzi00].
This paper is organized as follows. In Section \[S-preliminaries\] we recall some definitions. Theorems \[T-Myhill1\] and \[T-Moore1\] are proved in Sections \[S-Myhill\] and \[S-Moore\] respectively. We study the implications of weak specification for combinatorial independence in Section \[S-ind\]. The equivalence of weak specification and strong irreducibility for subshifts is proved in Appendix \[S-subshifts\].
[*Acknowledgements.*]{} The author was partially supported by NSF and NSFC grants. He is grateful to Michel Coornaert for helpful discussion, and to Tullio Ceccherini-Silberstein, David Kerr, Douglas Lind and Chung Nhân Phú for comments. He also thanks the referee for remarks.
preliminaries {#S-preliminaries}
=============
In this section we recall some definitions and set up some notations. Throughout this paper $\Gamma$ will be a countable discrete group with identity element $e_\Gamma$.
Expansiveness and weak specification {#SS-specification}
------------------------------------
Let $\Gamma$ act on a compact metrizable space $X$ continuously, and let $\rho$ be a compatible metric on $X$. The action is called [*expansive*]{} if there is some $\kappa>0$ such that $\sup_{s\in \Gamma}\rho(sx, sy)>\kappa$ for all distinct $x, y\in X$. Such a $\kappa$ is called an [*expansive constant*]{} with respect to $\rho$.
The action $\Gamma\curvearrowright X$ is said to have [*the weak specification property*]{} [@CL Definition 6.1] if for any $\varepsilon>0$ there exists a nonempty symmetric finite subset $F$ of $\Gamma$ such that for any finite collection $\{F_j\}_{j\in J}$ of finite subsets of $\Gamma$ satisfying $$F F_i\cap F_j=\emptyset \mbox{ for all distinct } i, j\in J,$$ and any collection of points $\{x_j\}_{j\in J}$ in $X$, there exists a point $x\in X$ such that $$\rho(sx, sx_j)\le \varepsilon \mbox{ for all } j\in J, s\in F_j.$$ Using the compactness of $X$, it is easy to see that when the action has the weak specification property, one can actually allow $J$ and $F_j$ to be infinite. It is also easy to see that weak specification passes to factors.
Group rings and algebraic actions {#SS-algebraic}
---------------------------------
We refer the reader to [@KL; @Schmidt] for detail on group rings and algebraic actions. The [*integral group ring*]{} ${{\mathbb Z}}\Gamma$ of $\Gamma$ is defined as the set of all finitely supported functions $\Gamma\rightarrow {{\mathbb Z}}$, written as $\sum_{s\in \Gamma} f_s s$ with $f_s\in {{\mathbb Z}}$ for all $s\in \Gamma$ and $f_s=0$ for all but finitely many $s$, with addition and multiplication given by $$\begin{aligned}
\sum_{s\in \Gamma} f_s s+\sum_{s\in \Gamma} g_s s&=\sum_{s\in \Gamma} (f_s+g_s)s,\\
(\sum_{s\in \Gamma}f_ss)(\sum_{t\in \Gamma}g_tt)&=\sum_{s, t\in \Gamma}f_sg_t(st).\end{aligned}$$ The group algebra $\ell^1(\Gamma)$ is the set of all functions $f: \Gamma\rightarrow {{\mathbb R}}$ satisfying $\sum_{s\in \Gamma}|f_s|<+\infty$, with addition and multiplication defined in the same way.
An action of $\Gamma$ on a compact metrizable abelian group by (continuous) automorphisms is called [*an algebraic action*]{}. Up to isomorphism, there is a natural one-to-one correspondence between algebraic actions of $\Gamma$ and countable left ${{\mathbb Z}}\Gamma$-modules as follows. For any algebraic action $\Gamma\curvearrowright X$, the Pontrjagin dual $\widehat{X}$ consisting of all continuous group homomorphisms $X\rightarrow {{\mathbb R}}/{{\mathbb Z}}$ is a countable abelian group and the action of $\Gamma$ on $X$ induces an action of $\Gamma$ on $\widehat{X}$ which makes $\widehat{X}$ into a left ${{\mathbb Z}}\Gamma$-module. Conversely, for any countable left ${{\mathbb Z}}\Gamma$-module ${{\mathcal M}}$, the Pontrjagin dual $\widehat{{{\mathcal M}}}$ consisting of all group homomorphisms ${{\mathcal M}}\rightarrow {{\mathbb R}}/{{\mathbb Z}}$ under the pointwise convergence topology forms a compact metrizable abelian group and the ${{\mathbb Z}}\Gamma$-module structure of ${{\mathcal M}}$ gives rise to an action of $\Gamma$ on ${{\mathcal M}}$ which induces an action of $\Gamma$ on $\widehat{{{\mathcal M}}}$ in turn.
For each $f\in {{\mathbb Z}}\Gamma$, the associated algebraic action $\Gamma\curvearrowright \widehat{{{\mathbb Z}}\Gamma/{{\mathbb Z}}\Gamma f}$ is called a [*principal algebraic action*]{}.
\[L-principal\] Every expansive principal algebraic action has the weak specification property.
For any $f\in {{\mathbb Z}}\Gamma$, the principal algebraic action $\Gamma\curvearrowright \widehat{{{\mathbb Z}}\Gamma/{{\mathbb Z}}\Gamma f}$ is expansive exactly when $f$ is invertible in $\ell^1(\Gamma)$ [@DS Theorem 3.2]. If $f\in {{\mathbb Z}}\Gamma$ is invertible in $\ell^1(\Gamma)$, then the principal algebraic action $\Gamma\curvearrowright \widehat{{{\mathbb Z}}\Gamma/{{\mathbb Z}}\Gamma f}$ has the weak specification property [@Ren Theorem 1.2].
Homoclinic pairs {#SS-homoclinic}
----------------
Let $\Gamma$ act on a compact metrizable space $X$ continuously, and let $\rho$ be a compatible metric on $X$. We say a pair $(x, y)\in X^2$ is [*homoclinic*]{} or [*asymptotic*]{} if $\rho(sx, sy)\to 0$ as $\Gamma \ni s\to \infty$. The set of all homoclinic pairs is an equivalence relation on $X$, and does not depend on the choice of $\rho$. A map from $X$ to another space is called [*pre-injective*]{} if it is injective on every homoclinic equivalence class.
Now assume that $\Gamma\curvearrowright X$ is an algebraic action. We can always choose $\rho$ to be translation-invariant. It follows that the homoclinic equivalence class of the identity element $0_X$ is a $\Gamma$-invariant subgroup of $X$, which we shall denote by $\Delta(X)$. Furthermore, for any $x\in X$, its homoclinic equivalence class is exactly $x+\Delta(X)$. The following lemma will be crucial for the proof of Theorem \[T-Moore\].
\[L-metric\] Let $\Gamma\curvearrowright X$ be an expansive algebraic action. Then there is a translation-invariant compatible metric $\rho$ on $X$ such that $\sum_{s\in \Gamma}\rho(sx, 0_X)<+\infty$ for all $x\in \Delta(X)$.
Since the algebraic action $\Gamma\curvearrowright X$ is expansive, by [@CL Theorem 5.6] [@KL Theorem 13.31] we have $\Delta(X)=\Delta^1(X)$, where $\Delta^1(X)$ is the $1$-homoclinic group of $X$ defined in [@CL Definition 5.1] [@KL Definition 13.26].
Again using the expansiveness of the algebraic action $\Gamma\curvearrowright X$, by [@Schmidt Proposition 2.2, Corollary 2.16] or [@KL Lemma 13.6] we know that $\widehat{X}$ is a finitely generated ${{\mathbb Z}}\Gamma$-module. Then by [@CL Proposition 5.7] [@KL Lemma 13.33] there is a translation-invariant compatible metric $\rho$ on $X$ such that $\sum_{s\in \Gamma}\rho(sx, 0_X)<+\infty$ for all $x\in \Delta^1(X)$. Consequently, $\sum_{s\in \Gamma}\rho(sx, 0_X)<+\infty$ for all $x\in \Delta(X)$.
Amenable groups and entropy {#SS-entropy}
---------------------------
We refer the reader to [@MO; @CC10; @KL] for details on amenable groups and the entropy theory of their actions. A countable group $\Gamma$ is called [*amenable*]{} if it has a left F[ø]{}lner sequence, i.e. a sequence $\{F_n\}_{n\in {{\mathbb N}}}$ of nonempty finite subsets of $\Gamma$ satisfying $$\lim_{n\to \infty}\frac{|KF_n\Delta F_n|}{|F_n|}=0$$ for all nonempty finite subsets $K$ of $\Gamma$.
Let $\Gamma$ act on a compact metrizable space $X$ continuously. For a finite open cover ${{\mathcal U}}$ of $X$, we denote by $N({{\mathcal U}})$ the minimal number of elements of ${{\mathcal U}}$ needed to cover $X$. Then the limit $\lim_{n\to \infty}\frac{1}{|F_n|}\log N(\bigvee_{s\in F_n}s^{-1}{{\mathcal U}})$ exists and does not depend on the choice of the F[ø]{}lner sequence $\{F_n\}_{n\in {{\mathbb N}}}$. We denote this limit by ${{\text{\rm h}}_{\text{\rm top}}}({{\mathcal U}})$. The [*topological entropy of the action $\Gamma\curvearrowright X$*]{} is defined as $${{\text{\rm h}}_{\text{\rm top}}}(X):=\sup_{{{\mathcal U}}}{{\text{\rm h}}_{\text{\rm top}}}({{\mathcal U}}),$$ where ${{\mathcal U}}$ runs over all finite open covers of $X$.
Let $\rho$ be a compatible metric on $X$, and let $\varepsilon>0$. A set $Z\subseteq X$ is called [*$(\rho, \varepsilon)$-separated*]{} if $\rho(x, z)\ge \varepsilon$ for all distinct $x, z\in Z$. Denote by ${{\rm sep}}(X, \rho, \varepsilon)$ the maximal cardinality of $(\rho, \varepsilon)$-separated subsets of $X$. A set $Z\subseteq X$ is called [*$(\rho, \varepsilon)$-spanning*]{} if for any $x\in X$ there exists some $z\in Z$ with $\rho(x, z)< \varepsilon$. Denote by ${{\rm span}}(X, \rho, \varepsilon)$ the minimal cardinality of $(\rho, \varepsilon)$-spanning subsets of $X$. For any nonempty finite subset $F$ of $\Gamma$, we define a new metric $\rho_F$ on $X$ by $\rho_F(x, y)=\max_{s\in F}\rho(sx, sy)$.
The case $\Gamma={{\mathbb Z}}$ of the following lemma is [@Walters Theorem 7.11], whose proof extends to amenable group case easily.
\[L-expansive\] Suppose that the action $\Gamma\curvearrowright X$ is expansive, and let $\kappa$ be an expansive constant with respect to a compatible metric $\rho$ on $X$. Then the following holds.
1. For any finite open cover ${{\mathcal U}}$ of $X$ such that each item of ${{\mathcal U}}$ has $\rho$-diameter at most $\kappa$, one has ${{\text{\rm h}}_{\text{\rm top}}}(X)={{\text{\rm h}}_{\text{\rm top}}}({{\mathcal U}})$.
2. For any $0<\varepsilon<\kappa/4$, one has $${{\text{\rm h}}_{\text{\rm top}}}(X)=\lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm sep}}(X, \rho_{F_n}, \varepsilon)=\lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm span}}(X, \rho_{F_n}, \varepsilon).$$
Let $\mu$ be a $\Gamma$-invariant Borel probability measure on $X$. For each finite Borel partition ${{\mathcal P}}$ of $X$, one defines the Shannon entropy $$H_\mu({{\mathcal P}})=\sum_{P\in {{\mathcal P}}}-\mu(P)\log \mu(P),$$ where the convention is $0\log 0=0$, and the dynamical entropy $$h_\mu({{\mathcal P}})=\lim_{n\to \infty}\frac{1}{|F_n|}H_\mu(\bigvee_{s\in F_n}s^{-1}{{\mathcal P}}).$$ The [*measure entropy of the action $\Gamma\curvearrowright (X, \mu)$*]{} is defined as $$h_\mu(X):=\sup_{{\mathcal P}}h_\mu({{\mathcal P}})$$ for ${{\mathcal P}}$ ranging over all finite Borel partitions of $X$. The action $\Gamma\curvearrowright (X, \mu)$ is said to have [*completely positive entropy*]{} (CPE) if $h_\mu({{\mathcal P}})>0$ for every finite Borel partition ${{\mathcal P}}$ of $X$ with $H_\mu({{\mathcal P}})>0$.
The variational principle says that ${{\text{\rm h}}_{\text{\rm top}}}(X)=\sup_\mu h_\mu(X)$ for $\mu$ ranging over all $\Gamma$-invariant Borel probability measures of $X$.
Myhill property {#S-Myhill}
===============
In this section we prove Theorem \[T-Myhill1\]. Throughout this section, we let a countable amenable group $\Gamma$ act on compact metrizable spaces $X$ and $Y$ continuously, and fix a left F[ø]{}lner sequence $\{F_n\}_{n\in {{\mathbb N}}}$ for $\Gamma$.
\[P-subaction\] Assume that the action $\Gamma\curvearrowright Y$ is expansive and has the weak specification property. For any nonempty closed $\Gamma$-invariant subset $Z$ of $Y$ with $Z\neq Y$, we have ${{\text{\rm h}}_{\text{\rm top}}}(Z)<{{\text{\rm h}}_{\text{\rm top}}}(Y)$.
Take a compatible metric $\rho$ on $Y$. Let $\kappa>0$ be an expansive constant of $\Gamma \curvearrowright Y$ with respect to $\rho$. Take $y_0\in Y\setminus Z$ and set $\eta=\min(\kappa/10, \rho(y_0, Z))>0$. By the weak specification property, there exists a symmetric finite set $F\subseteq \Gamma$ containing $e_\Gamma$ such that for any finite collection of finite subsets $\{K_j\}_{j\in J}$ of $\Gamma$ satisfying $FK_i\cap K_j=\emptyset$ for all distinct $i, j\in J$ and any collection $\{y_j\}_{j\in J}$ of points in $Y$, there exists $y\in Y$ such that $\rho(sy, sy_j)\le \eta/4$ for all $j\in J$ and $s\in K_j$.
Let $n\in {{\mathbb N}}$. Take a maximal set $K_n\subseteq F_n$ subject to the condition that for any distinct $s, t\in K_n$, one has $s\not \in Ft$. Then $FK_n\supseteq F_n$, and hence $$|K_n|\ge |F_n|/|F|.$$ Let $A\subseteq K_n$. Take a $(\rho_{F_n\setminus (FA)}, \eta)$-spanning subset $W_{F_n\setminus (FA)}$ of $Z$ with cardinality ${{\rm span}}(Z, \rho_{F_n\setminus (FA)}, \eta)$, and for each $s\in A$ take a $(\rho_{Fs}, \eta)$-spanning subset $W_{Fs}$ of $Z$ with cardinality ${{\rm span}}(Z, \rho_{Fs}, \eta)={{\rm span}}(Z, \rho_F, \eta)$. For each $z\in Z$, we can take $z_{F_n\setminus (FA)}\in W_{F_n\setminus (FA)}$ with $\rho_{F_n\setminus (FA)}(z, z_{F_n\setminus (FA)})<\eta$ and $z_{Fs}\in W_{Fs}$ with $\rho_{Fs}(z, z_{Fs})<\eta$ for each $s\in A$. For any $z, z'\in Z$, if $z_{F_n\setminus (FA)}=z_{F_n\setminus (FA)}'$ and $z_{Fs}=z_{Fs}'$ for all $s\in A$, then $\rho_{F_n}(z, z')<2\eta$. It follows that $$\begin{aligned}
{{\rm sep}}(Z, \rho_{F_n}, 2\eta)&\le |W_{F_n\setminus (FA)}|\cdot \prod_{s\in A}|W_{Fs}|\\
&= {{\rm span}}(Z, \rho_{F_n\setminus (FA)}, \eta){{\rm span}}(Z, \rho_F, \eta)^{|A|}\\
&\le {{\rm sep}}(Z, \rho_{F_n\setminus (FA)}, \eta){{\rm sep}}(Z, \rho_F, \eta)^{|A|}.\end{aligned}$$ Take a $(\rho_{F_n\setminus (FA)}, \eta)$-separated subset $\Omega_A$ of $Z$ with maximal cardinality. For each $\omega\in \Omega_A$, take $\omega_A\in Y$ such that $\rho(s\omega_A, y_0)\le \eta/4$ for all $s\in A$ and $\rho(t\omega_A, t\omega)\le \eta/4$ for all $t\in F_n\setminus (FA)$. For any distinct $\omega, \omega'\in \Omega_A$, we have $\rho(t\omega, t\omega')\ge \eta$ for some $t\in F_n\setminus (FA)$, and hence $$\rho(t\omega_A, t\omega'_A)\ge \rho(t\omega, t\omega')-\rho(t\omega, t\omega_A)-\rho(t\omega', t\omega'_A)\ge \eta-\eta/4-\eta/4=\eta/2.$$ For any distinct $A, B\subseteq K_n$ and any $\omega\in \Omega_A, \omega'\in \Omega_B$, say $s\in A\setminus B$, we have $$\rho(s\omega_A, s\omega'_B)\ge \rho(y_0, s\omega')-\rho(s\omega_A, y_0)-\rho(s\omega'_B, s\omega')\ge \eta-\eta/4-\eta/4=\eta/2.$$ Thus the set $\{\omega_A: A\subseteq K_n, \omega\in \Omega_A\}$ is a $(\rho_{F_n}, \eta/2)$-separated subset of $Y$ with cardinality $\sum_{A\subseteq K_n}{{\rm sep}}(Z, \rho_{F_n\setminus (FA)}, \eta)$. Therefore $$\begin{aligned}
{{\rm sep}}(Y, \rho_{F_n}, \eta/2)&\ge \sum_{A\subseteq K_n}{{\rm sep}}(Z, \rho_{F_n\setminus (FA)}, \eta)\\
&\ge \sum_{A\subseteq K_n}{{\rm sep}}(Z, \rho_{F_n}, 2\eta){{\rm sep}}(Z, \rho_F, \eta)^{-|A|}\\
&={{\rm sep}}(Z, \rho_{F_n}, 2\eta)(1+{{\rm sep}}(Z, \rho_F, \eta)^{-1})^{|K_n|}.\end{aligned}$$ Thus by Lemma \[L-expansive\] we have $$\begin{aligned}
{{\text{\rm h}}_{\text{\rm top}}}(Y)&=\lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm sep}}(Y, \rho_{F_n}, \eta/2)\\
&\ge \lim_{n\to \infty}\frac{1}{|F_n|}\log{{\rm sep}}(Z, \rho_{F_n}, 2\eta)+ \limsup_{n\to \infty}\frac{1}{|F_n|}\log (1+{{\rm sep}}(Z, \rho_F, \eta)^{-1})^{|K_n|}\\
&\ge {{\text{\rm h}}_{\text{\rm top}}}(Z)+\frac{1}{|F|}\log (1+{{\rm sep}}(Z, \rho_F, \eta)^{-1})\\
&> {{\text{\rm h}}_{\text{\rm top}}}(Z)\end{aligned}$$ as desired.
Proposition \[P-subaction\] was proved before under the further assumption that $Y$ is a subshift of finite type by Fiorenzi in the proof of [@Fiorenzi03 Proposition 4.6], and under the further assumption that $Y$ is a subshift by Ceccherini-Silberstein and Coornaert [@CC12 Proposition 4.2].
\[P-factor\] Assume that the action $\Gamma\curvearrowright X$ is expansive and has the weak specification property. Let $\Gamma \curvearrowright Y$ be a factor of $\Gamma\curvearrowright X$ such that $\Gamma \curvearrowright Y$ is expansive and ${{\text{\rm h}}_{\text{\rm top}}}(Y)<{{\text{\rm h}}_{\text{\rm top}}}(X)$. Then for any homoclinic equivalence class $\Xi$ of $X$, the factor map $T:X\rightarrow Y$ fails to be injective on $\Xi$.
Take compatible metrics $\rho_X$ and $\rho_Y$ on $X$ and $Y$ respectively. Take a common expansive constant $\kappa>0$ for the action $\Gamma\curvearrowright X$ with respect to $\rho_X$ and the action $\Gamma\curvearrowright Y$ with respect to $\rho_Y$. Write $\varepsilon=\kappa/15$. As $T$ is continuous and $X$ is compact, there exists $0<\delta<\varepsilon$ such that for any $x, x'\in X$ with $\rho_X(x, x')\le \delta$ one has $\rho_Y(Tx, Tx')\le \varepsilon$.
Fix a point $z\in \Xi$. Since the action $\Gamma\curvearrowright X$ has the weak specification property, there exists a symmetric finite set $F\subseteq \Gamma$ containing $e_\Gamma$ such that for any $x\in X$ and any finite set $K\subseteq \Gamma$ there exists $x'\in X$ with $\max_{s\in K}\rho_X(sx', sx)\le \delta$ and $\sup_{s\in \Gamma\setminus (FK)}\rho_X(sx', sz)\le \delta$. Set $C={{\rm span}}(Y, \rho_Y, \varepsilon)$.
By Lemma \[L-expansive\] we have $$\begin{aligned}
{{\text{\rm h}}_{\text{\rm top}}}(X)=\lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm sep}}(X, \rho_{X, F_n}, 3\varepsilon),\end{aligned}$$ and $$\begin{aligned}
{{\text{\rm h}}_{\text{\rm top}}}(Y)=
\lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm span}}(Y, \rho_{Y, F_n}, \varepsilon).\end{aligned}$$
Take $\eta>0$ with ${{\text{\rm h}}_{\text{\rm top}}}(X)>3\eta+{{\text{\rm h}}_{\text{\rm top}}}(Y)$. When $n$ is large enough, we have ${{\rm span}}(Y, \rho_{Y, F_n}, \varepsilon)\le e^{|F_n|({{\text{\rm h}}_{\text{\rm top}}}(Y)+\eta)}$ and $${{\rm sep}}(X, \rho_{X, F_n}, 3\varepsilon)\ge e^{|F_n|({{\text{\rm h}}_{\text{\rm top}}}(X)-\eta)}\ge e^{\eta |F_n|}{{\rm span}}(Y, \rho_{Y, F_n}, \varepsilon).$$ Let $\Omega_n$ be a $(\rho_{X, F_n}, 3\varepsilon)$-separated subset of $X$ with maximum cardinality, and $\Lambda_n$ be a $(\rho_{Y, F_n}, \varepsilon)$-spanning subset of $Y$ with minimal cardinality. Then there exist a set $\Omega'_n\subseteq \Omega_n$ with $|\Omega'_n|\ge e^{\eta |F_n|}$ and $y\in \Lambda_n$ such that $\max_{s\in F_n}\rho_Y(sTx, sy)\le \varepsilon$ for all $x\in \Omega'_n$. For each $x\in \Omega'_n$, take $x'\in X$ with $\max_{s\in F_n}\rho_X(sx', sx)\le \delta<\varepsilon$ and $\sup_{s\in \Gamma\setminus (FF_n)}\rho_X(sx', sz)\le \delta$. Then $\sup_{s\in \Gamma\setminus (FF_n)}\rho_X(sx', sz)\le \kappa$, and hence by [@CL Lemma 6.2] the pair $(x', z)$ is homoclinic. Thus $x'\in \Xi$. By our choice of $\delta$, we get $$\begin{aligned}
\max_{s\in F_n}\rho_Y(sTx', sy)&\le \max_{s\in F_n}\rho_Y(sTx', sTx)+\max_{s\in F_n}\rho_Y(sTx, sy)\\
&\le \max_{s\in F_n}\rho_Y(T(sx'), T(sx))+\varepsilon\\
&\le \varepsilon+\varepsilon=2\varepsilon,\end{aligned}$$ and $$\sup_{s\in \Gamma\setminus (FF_n)}\rho_Y(sTx', sTz)=\sup_{s\in \Gamma\setminus (FF_n)}\rho_Y(T(sx'), T(sz))\le \varepsilon.$$ Now take a set $\Omega''_n\subseteq \Omega'_n$ with $|\Omega''_n|\ge |\Omega'_n|/C^{|FF_n\setminus F_n|}$ such that for any $x, \omega\in \Omega''_n$ we have $\max_{s\in FF_n\setminus F_n}\rho_Y(sTx', sT\omega')<2\varepsilon$. Then for any $x, \omega\in \Omega''_n$ we have $\sup_{s\in \Gamma}\rho_Y(sTx', sT\omega')\le 4\varepsilon$, whence $Tx'=T\omega'$. If $x\neq \omega$, then $$\max_{s\in F_n}\rho_X(sx', s\omega')\ge \max_{s\in F_n}\rho_X(sx, s\omega)-\max_{s\in F_n}\rho_X(sx, sx')-\max_{s\in F_n}\rho_X(s\omega, s\omega')>\varepsilon,$$ and hence $x'\neq \omega'$. When $n$ is large enough, we have $C^{|FF_n\setminus F_n|}<e^{\eta |F_n|}$, and hence $|\Omega''_n|>1$.
Proposition \[P-factor\] was proved before under the further assumption that $X$ is a subshift of finite type and $Y$ is a subshift by Fiorenzi [@Fiorenzi03 Proposition 4.5], and under the further assumption that $X$ and $Y$ are subshifts by Ceccherini-Silberstein and Coornaert [@CC12 Theorem 5.1].
From Propositions \[P-subaction\] and \[P-factor\] we obtain:
\[T-Myhill\] Let $\Gamma\curvearrowright X$ and $\Gamma\curvearrowright Y$ be expansive actions with the weak specification property. Assume that ${{\text{\rm h}}_{\text{\rm top}}}(X)={{\text{\rm h}}_{\text{\rm top}}}(Y)$. Then every pre-injective continuous $\Gamma$-equivariant map $X\rightarrow Y$ is surjective.
Theorem \[T-Myhill\] was proved before under the further assumption that $X$ is a subshift of finite type and $Y$ is a subshift by Fiorenzi [@Fiorenzi03 Theorem 4.7], and under the further assumption that $X$ and $Y$ are subshifts by Ceccherini-Silberstein and Coornaert [@CC12 Corollary 5.2].
Theorem \[T-Myhill1\] follows from Theorem \[T-Myhill\] directly.
Moore Property {#S-Moore}
==============
In this section we prove Theorem \[T-Moore1\]. Throughout this section $\Gamma$ will be a countable amenable group and $\{F_n\}_{n\in {{\mathbb N}}}$ will be a left F[ø]{}lner sequence of $\Gamma$.
\[L-measure\] Let $\Gamma$ act on compact metrizable groups $X$ and $Y$ by automorphisms. Denote by $\mu_X$ and $\mu_Y$ the normalized Haar measure of $X$ and $Y$ respectively. Suppose that the action $\Gamma\curvearrowright (X, \mu_X)$ has CPE. Also assume that ${{\text{\rm h}}_{\text{\rm top}}}(X)={{\text{\rm h}}_{\text{\rm top}}}(Y)<+\infty$. Let $T: X\rightarrow Y$ be a $\Gamma$-equivariant continuous surjective map. Then $T_*\mu_X=\mu_Y$.
Since $\Gamma$ is amenable and $T$ is surjective, there is a $\Gamma$-invariant Borel probability measure $\nu$ on $X$ satisfying $T_*\nu=\mu_Y$. Then $${{\text{\rm h}}_{\text{\rm top}}}(X)\ge h_\nu(X)\ge h_{\mu_Y}(Y)={{\text{\rm h}}_{\text{\rm top}}}(Y),$$ where the equality is from [@Den Theorem 2.2] [@KL Proposition 13.2]. Thus $h_\nu(X)={{\text{\rm h}}_{\text{\rm top}}}(X)$. Since the action $\Gamma\curvearrowright (X, \mu_X)$ has CPE and $h_{\mu_X}(X)\le {{\text{\rm h}}_{\text{\rm top}}}(X)<+\infty$, by [@CL Theorem 8.6] we have $h_{\nu'}(X)<h_{\mu_X}(X)$ for every $\Gamma$-invariant Borel probability measure $\nu'$ on $X$ different from $\mu_X$. Therefore $\nu=\mu_X$. Thus $T_*\mu_X=\mu_Y$.
\[T-Moore\] Let $\Gamma\curvearrowright X$ be an expansive algebraic action with CPE with respect to the normalized Haar measure and $\Gamma\curvearrowright Y$ be an expansive action on a compact metrizable group by automorphisms. Assume that ${{\text{\rm h}}_{\text{\rm top}}}(X)={{\text{\rm h}}_{\text{\rm top}}}(Y)$. Then every surjective continuous $\Gamma$-equivariant map $T:X\rightarrow Y$ is pre-injective.
When $\Gamma$ is finite, we have $|X|=|\Gamma|{{\text{\rm h}}_{\text{\rm top}}}(X)=|\Gamma|{{\text{\rm h}}_{\text{\rm top}}}(Y)=|Y|<+\infty$. Then $T$ is actually injective. Thus we may assume that $\Gamma$ is infinite.
Assume that $T$ is not pre-injective. Then there is a homoclinic pair $(x, \omega)\in X^2$ such that $x\neq \omega$ and $Tx=T\omega$. We get $\omega-x\in \Delta(X)$.
By Lemma \[L-metric\] we can find a compatible translation-invariant metric $\rho_X$ on $X$ such that $\sum_{s\in \Gamma}\rho_X(sx', 0_X)<+\infty$ for all $x'\in \Delta(X)$. We also take a compatible translation-invariant metric $\rho_Y$ on $Y$.
Denote by $\mu_X$ and $\mu_Y$ the normalized Haar measure of $X$ and $Y$ respectively.
Take a common expansive constant $\kappa>0$ for the action $\Gamma\curvearrowright X$ with respect to $\rho_X$ and the action $\Gamma\curvearrowright Y$ with respect to $\rho_Y$. Let $0<\varepsilon<\kappa/4$. By Lemma \[L-expansive\] we have $${{\text{\rm h}}_{\text{\rm top}}}(Y)=\lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm sep}}(Y, \rho_{Y, F_n}, \varepsilon).$$ For each $n\in {{\mathbb N}}$, denote by $D_n$ the set of $y\in Y$ satisfying $\max_{s\in F_n}\rho_Y(sy, e_Y)<\varepsilon/2$, where $e_Y$ denotes the identity element of $Y$, and take a $(\rho_{Y, F_n}, \varepsilon)$-separated subset $W_n$ of $Y$ with maximal cardinality. For any distinct $y, z\in W_n$, one has $(yD_n)\cap (zD_n)=\emptyset$. Thus $\mu_Y(D_n)|W_n|\le 1$, and hence $$1/\mu_Y(D_n)\ge |W_n|.$$
Since $T$ is continuous and $X$ is compact, we can find $0<\delta<\min(\kappa/4, \rho_X(\omega-x, 0_X)/2)$ such that for any $x_1, x_2\in X$ with $\rho_X(x_1, x_2)\le 2\delta$, one has $\rho_Y(Tx_1, Tx_2)<\varepsilon/16$. Since $\sum_{s\in \Gamma}\rho_X(s(\omega-x), 0_X)<+\infty$, we can find a symmetric finite set $F\subseteq \Gamma$ containing $e_\Gamma$ such that $$\sum_{s\in \Gamma\setminus F}\rho_X(s(\omega-x), 0_X)<\delta.$$ Take $0<\tau<\delta$ such that for any $x_1, x_2\in X$ with $\rho_X(x_1, x_2)\le \tau$, one has $\max_{s\in F}\rho_X(sx_1, sx_2)\le \delta$.
Denote by $B$ the set of all $x'\in X$ satisfying $\rho_X(x', x)\le \tau$. Then $\mu_X(B)>0$. Since the action $\Gamma\curvearrowright (X, \mu_X)$ has CPE, $\mu_X$ is ergodic. By the mean ergodic theorem [@KL Theorem 4.22], we have $\|\frac{1}{|F_n|}\sum_{s\in F_n}s^{-1}1_{B}-\mu_X(B)\|_2\to 0$ as $n\to \infty$, where $1_B$ denotes the characteristic function of $B$. For each $n\in {{\mathbb N}}$, denote by $X_n$ the set of $x'\in X$ satisfying $|\{s\in F_n: sx'\in B\}|\ge |F_n|\mu_X(B)/2$. Then $X_n$ is closed and $\mu_X(X_n)\to 1$ as $n\to \infty$.
Since the action $\Gamma\curvearrowright X$ is expansive, by Lemma \[L-expansive\] we have ${{\text{\rm h}}_{\text{\rm top}}}(X)<+\infty$. By Lemma \[L-measure\] we have $T_*\mu_X=\mu_Y$. Then $\mu_Y(T(X_n))\ge \mu_X(X_n)$, and hence $\mu_Y(T(X_n))\to 1$ as $n\to \infty$. Take a maximal $(\rho_{Y, F_n}, \varepsilon/2)$-separated subset $W_n'$ of $T(X_n)$. Then $yD_n$ for $y\in W_n'$ covers $T(X_n)$. Thus $\mu_Y(T(X_n))\le \mu_Y(D_n)|W_n'|$, whence $$|W'_n|\ge \mu_Y(T(X_n))/\mu_Y(D_n)\ge \mu_Y(T(X_n))|W_n|.$$ Take a subset $V_n$ of $X_n$ with $|V_n|=|W'_n|$ and $T(V_n)=W'_n$. Then $$\begin{aligned}
\liminf_{n\to \infty}\frac{1}{|F_n|}\log |V_n|&=\liminf_{n\to \infty}\frac{1}{|F_n|}\log |W'_n|\\
&\ge \liminf_{n\to \infty}\frac{1}{|F_n|}\log \mu_Y(T(X_n))+\liminf_{n\to \infty}\frac{1}{|F_n|}\log |W_n|\\
&={{\text{\rm h}}_{\text{\rm top}}}(Y).\end{aligned}$$
For any $n\in {{\mathbb N}}$ and $v\in V_n$, write $E_{n, v}=\{s\in F_n: sv\in B\}$ and take a maximal set $E'_{n, v}\subseteq E_{n, v}$ subject to the condition $Ft\cap Fs=\emptyset$ for all distinct $s, t\in E'_{n, v}$. Then $F^2E'_{n, v}\supseteq E_{n, v}$, and hence $$|E'_{n, v}|\ge |E_{n, v}|/|F|^2\ge |F_n|\mu_X(B)/(2|F|^2).$$ For each set $A\subseteq E'_{n, v}$, define $$v_A=v+\sum_{s\in A}s^{-1}(\omega-x)\in X.$$ We claim that $Tv_A=Tv$. It suffices to show $\rho_Y(tTv_A, tTv)=\rho_Y(T(tv_A), T(tv))<\varepsilon/8$ for all $t\in \Gamma$. Let $t\in \Gamma$. If $t\not \in FA$, then $$\rho_X(tv_A, tv)=\rho_X(\sum_{s\in A}ts^{-1}(\omega-x), 0_X)\le \sum_{s\in \Gamma\setminus F}\rho_X(s(\omega-x), 0_X)<\delta,$$ and hence $\rho_Y(T(tv_A), T(tv))<\varepsilon/16$. Now consider the case $t\in FA$. Say $t=\gamma s'$ for some $\gamma \in F$ and $s'\in A$. Then $$\begin{aligned}
\rho_X(tv_A, \gamma \omega)&=\rho_X(\gamma s'v+\sum_{s\in A\setminus \{s'\}}ts^{-1}(\omega-x), \gamma x)\\
&\le \rho_X(\gamma s'v, \gamma x)+\sum_{s\in \Gamma\setminus F}\rho_X(s(\omega-x), 0_X)<2\delta,\end{aligned}$$ and $$\rho_X(tv, \gamma x)=\rho_X(\gamma s'v, \gamma x)\le \delta.$$ Therefore $$\begin{aligned}
\rho_Y(T(tv_A), \gamma Tx)=\rho_Y(T(tv_A), T(\gamma \omega))<\varepsilon/16,\end{aligned}$$ and $$\rho_Y(T(tv), \gamma Tx)=\rho_Y(T(tv), T(\gamma x))<\varepsilon/16.$$ Consequently, $$\rho_Y(T(tv_A), T(tv))\le \rho_Y(T(tv_A), \gamma Tx)+\rho_Y(T(tv), \gamma Tx)<\varepsilon/8.$$ This proves our claim.
Write $V^\dag_n:=\{v_A: v\in V_n, A\subseteq E'_{n, v}\}$. For any $v\in V_n$ and distinct $A, A'\subseteq E'_{n, v}$, say $t\in A\setminus A'$, we have $$\begin{aligned}
\rho_X(tv_A, tv_{A'})&\ge
\rho_X(\omega-x, 0_X)-\sum_{s\in (A\setminus \{t\})\Delta A'}\rho_X(ts^{-1}(\omega-x), 0_X)\\
&\ge \rho_X(\omega-x, 0_X)-\sum_{s\in \Gamma\setminus F}\rho_X(s(\omega-x), 0_X)\\
&\ge \delta.\end{aligned}$$ For any distinct $v, z\in V_n$, and $A\subseteq E'_{n, v}$ and $A'\subseteq E'_{n, z}$, we have $$\max_{s\in F_n}\rho_Y(Tsv_A, Tsz_{A'})=\max_{s\in F_n}\rho_Y(sTv_A, sTz_{A'})=\max_{s\in F_n}\rho_Y(sTv, sTz)\ge \varepsilon/2,$$ whence $\max_{s\in F_n}\rho_X(sv_A, sz_{A'})> 2\delta$. Thus $V^\dag_n$ is $(\rho_{X, F_n}, \delta)$-separated, and $$|V^\dag_n|\ge |V_n|2^{|F_n|\mu_X(B)/(2|F|^2)}.$$ Therefore by Lemma \[L-expansive\] we have $$\begin{aligned}
{{\text{\rm h}}_{\text{\rm top}}}(X)&= \lim_{n\to \infty}\frac{1}{|F_n|}\log {{\rm sep}}(X, \rho_{X, F_n}, \delta)\\
&\ge \liminf_{n\to \infty}\frac{1}{|F_n|}\log |V_n|+\frac{\mu_X(B)\log 2}{2|F|^2}\\
&\ge {{\text{\rm h}}_{\text{\rm top}}}(Y)+\frac{\mu_X(B)\log 2}{2|F|^2},\end{aligned}$$ which is a contradiction to the hypothesis ${{\text{\rm h}}_{\text{\rm top}}}(X)={{\text{\rm h}}_{\text{\rm top}}}(Y)$. Thus $T$ is pre-injective.
Now Theorem \[T-Moore1\] follows from Theorem \[T-Moore\] directly.
Weak specification and independence {#S-ind}
===================================
In this section we discuss implications of weak specification to combinatorial independence and prove Corollary \[C-CPE\].
Let a countable (not necessarily amenable) group $\Gamma$ act on a compact metrizable space $X$ continuously. Let ${\bf A}=(A_1, \dots, A_k)$ be a tuple of subsets of $X$. A nonempty finite set $K'\subseteq \Gamma$ is called [*an independence set*]{} for ${\bf A}$ if $\bigcap_{s\in K'}s^{-1}A_{\omega(s)}\neq \emptyset$ for all maps $\omega: K'\rightarrow \{1, \dots, k\}$ [@KL Definition 8.7]. For any nonempty finite set $K\subseteq \Gamma$ write $\varphi_{\bf A}(K)$ for the maximal cardinality of independence sets $K'$ of $\bf A$ satisfying $K'\subseteq K$. The [*independence density*]{} of $\bf A$ is defined as $$I({\bf A}):=\inf_K\frac{\varphi_{\bf A}(K)}{|K|},$$ where $K$ ranges over all nonempty finite subsets of $\Gamma$ [@KL13b Definition 3.1]. A tuple ${\bf x}=(x_1, \dots, x_k)\in X^k$ is called [*an orbit IE-tuple*]{} if for every product neighborhood $U_1\times \cdots \times U_k$ of $\bf x$, the tuple $(U_1, \dots, U_k)$ has positive independence density [@KL13b Definition 3.2].
\[P-UPE\] For any action $\Gamma\curvearrowright X$ with the weak specification property, every tuple is an orbit IE-tuple.
Let $k\in {{\mathbb N}}$ and $x_1, \dots, x_k\in X$. We shall show that $(x_1, \dots, x_k)\in X^k$ is an orbit IE-tuple.
Let $\rho$ be a compatible metric on $X$, and let $\varepsilon>0$. Denote by $D_i$ the set of all $x\in X$ satisfying $\rho(x, x_i)\le \varepsilon$. By the weak specification property there is some symmetric finite subset $F$ of $\Gamma$ containing $e_\Gamma$ such that for any finite collection $\{F_j\}_{j\in J}$ of finite subsets of $\Gamma$ satisfying $FF_i\cap F_j=\emptyset$ for all distinct $i, j\in J$ and any collection $\{y_j\}_{j\in J}$ of points in $X$, there is some $y\in X$ such that $\rho(y, sy_j)\le \varepsilon$ for all $j\in J$ and $s\in F_j$.
Let $K$ be a nonempty finite subset of $\Gamma$. Take a maximal subset $K'$ of $K$ subject to the condition that $s\not\in Ft$ for all distinct $s, t\in K'$. Then $FK'\supseteq K$, and hence $|K'|\ge |K|/|F|$. Let $\omega$ be a map $K'\rightarrow \{1, \dots, k\}$. Then there is some $x\in X$ such that $\rho(sx, x_{\omega(s)})\le \varepsilon$ for all $s\in K'$. Thus $K'$ is an independence set for the tuple $(D_1, \dots, D_k)$. It follows that $(D_1, \dots, D_k)$ has independence density at least $1/|F|$. Therefore $(x_1, \dots, x_k)$ is an orbit IE-tuple.
Now we consider the case $\Gamma$ is amenable. An action $\Gamma\curvearrowright X$ has positive entropy if and only if there is at least one non-diagonal orbit IE-pair in $X^2$ [@KL Definition 12.5, Theorem 12.19]. Thus we get
\[C-factor\] For any countable amenable group, every continuous action on a compact metrizable space with the weak specification property and more than one point has positive entropy.
Corollary \[C-factor\] was proved before under the further assumption of subshifts by Ceccherini-Silberstein and Coornaert [@CC12 Proposition 4.5].
For any action of a countable amenable group $\Gamma$ on a compact metrizable group $X$ by automorphisms, every pair in $X^2$ is an orbit IE-pair if and only if the action has CPE with respect to the normalized Haar measure [@KL Definition 12.5] [@CL Theorem 7.3, Corollary 8.4]. Thus we get
\[C-CPE\] Every weak specification action of a countable amenable group on a compact metrizable group by automorphisms has CPE with respect to the normalized Haar measure.
Subshifts with weak specification {#S-subshifts}
=================================
Let $\Gamma$ be a countable (not necessarily amenable) group, and let $A$ be a finite set. We consider the shift action $\Gamma\curvearrowright A^\Gamma$ given by $(sx)_t=x_{s^{-1}t}$ for all $x\in A^G$ and $s, t\in \Gamma$. A closed $\Gamma$-invariant subset $X$ of $A^\Gamma$ is called [*strongly irreducible*]{} [@Fiorenzi03 Definition 4.1] if there exists a nonempty symmetric finite set $F\subseteq \Gamma$ such that for any finite sets $F_1, F_2\subseteq \Gamma$ with $F_1F\cap F_2=\emptyset$ and any $x_1, x_2\in X$, there exists $x\in X$ such that $x=x_1$ on $F_1$ and $x=x_2$ on $F_2$.
\[P-subshifts\] For any closed $\Gamma$-invariant subset $X$ of $A^\Gamma$, $X$ is strongly irreducible if and only if it has the weak specification property.
Take a compatible metric $\rho$ of $X$.
Suppose that $X$ is strongly irreducible. Let $F\subseteq \Gamma$ witness the strong irreducibility of $X$. By induction it is easy to see that for any finite collection $\{F_j\}_{j\in J}$ of finite subsets of $\Gamma$ satisfying $F_iF\cap F_j=\emptyset$ for all distinct $i, j\in J$ and any collection $\{x_j\}_{j\in J}$ of points in $X$, there is some $x\in X$ such that $x=x_j$ on $F_j$ for all $j\in J$. Let $\varepsilon>0$. Then there is some nonempty finite subset $K$ of $\Gamma$ such that for any $y, z\in X$ with $y=z$ on $K$ one has $\rho(y, z)\le \varepsilon$. Now let $\{F_j\}_{j\in J}$ be a finite collection of finite subsets of $\Gamma$ satisfying $(KFK^{-1})F_i\cap F_j=\emptyset$ for all distinct $i, j\in J$ and $\{x_j\}_{j\in J}$ be a collection of points in $X$. Then $(F_i^{-1}K)F\cap (F_j^{-1}K)=\emptyset$ for all distinct $i, j\in J$. Thus there is some $x\in X$ satisfying $x=x_j$ on $F_j^{-1}K$ for all $j\in J$. For any $j\in J$ and $s\in F_j$, we get $sx=sx_j$ on $K$, and hence $\rho(sx, sx_j)\le \varepsilon$. Therefore $X$ has the weak specification property.
Conversely suppose that $X$ has the weak specification property. Take $\varepsilon>0$ such that any two points $y, z\in X$ satisfying $\rho(y, z)\le \varepsilon$ must coincide at $e_\Gamma$. Then there is some nonempty symmetric finite subset $F$ of $\Gamma$ such that for any finite subsets $F_1$ and $F_2$ of $\Gamma$ satisfying $FF_1\cap F_2=\emptyset$ and any points $x_1, x_2\in X$, there is some $x\in X$ such that $\rho(sx, sx_j)\le \varepsilon$ for all $j=1, 2$ and $s\in F_j$. Now let $F_1$ and $F_2$ be finite subsets of $\Gamma$ satisfying $F_1F\cap F_2=\emptyset$ and $x_1, x_2\in X$. Then $FF_1^{-1}\cap F_2^{-1}=\emptyset$. Thus there is some $x\in X$ such that $\rho(sx, sx_j)\le \varepsilon$ for all $j=1, 2$ and $s\in F_j^{-1}$. Then $sx=sx_j$ at $e_\Gamma$, which means $x=x_j$ at $s^{-1}$. Therefore $X$ is strongly irreducible.
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abstract: 'Near-infrared spectroscopy from APOGEE and wide-field optical photometry from Pan-STARRS1 have recently made possible precise measurements of the shape of the extinction curve for tens of thousands of stars, parameterized by $R(V)$. These measurements revealed structures in $R(V)$ with large angular scales, which are challenging to explain in existing dust paradigms. In this work, we combine three-dimensional maps of dust column density with $R(V)$ measurements to constrain the three-dimensional distribution of $R(V)$ in the Milky Way. We find that variations in $R(V)$ are correlated on kiloparsec scales. In particular, most of the dust within one kiloparsec in the outer Galaxy, including many local molecular clouds (Orion, Taurus, Perseus, California, Cepheus), has a significantly lower $R(V)$ than more distant dust in the Milky Way. These results provide new input to models of dust evolution and processing, and complicate application of locally derived extinction curves to more distant regions of the Milky Way and to other galaxies.'
author:
- 'E. F. Schlafly, J. E. G. Peek, D. P. Finkbeiner, G. M. Green'
bibliography:
- '2dmap.bib'
title: 'Mapping the Extinction Curve in 3D: Structure on Kiloparsec Scales'
---
Introduction {#sec:intro}
============
Dust is a key component of galaxies. It is an important source of cooling in the interstellar medium, and it shields and catalyzes the formation of molecular hydrogen, allowing the formation of stars. Additionally, dust dramatically reshapes the interstellar radiation field of galaxies, extinguishing blue light preferentially to red light, and reradiating absorbed starlight at long wavelengths [for a review, see @Draine:2003]. The extinction curve describes the wavelength dependence of dust absorption and scattering of light. The shape of this curve is an important diagnostic of the properties of dust.
Since the 1950s, significant work has focused on the ratio $R(V) = A(V)/E(B-V)$, the total-to-selective extinction ratio. This parameter is especially important because it allows reddenings $E(B-V)$, which can be easily measured, to be transformed into total extinctions $A(V)$, which are necessary to derive distances to stars from their observed and absolute magnitudes. It also plays a principal role in the parameterization of extinction curves.
A mean value of $R(V) = 3.0 \pm 0.2$, similar to the commonly adopted present value of 3.1, was found early on [e.g., @Morgan:1953; @Whitford:1958]. Further investigation revealed substantial variation in $R(V)$, in both the Milky Way [e.g. @Whittet:1976], and in external galaxies like the Large Magellanic Cloud and Small Magellanic Cloud [@Nandy:1984; @Gordon:1998; @Gordon:2003]. The work of @Fitzpatrick:1986 [@Fitzpatrick:1988; @Fitzpatrick:1990] developed an empirical six-parameter description of the ultraviolet extinction curve which appears to account for nearly all variation at those wavelengths. The work of @Cardelli:1989 [CCM] showed that much of the variation found by @Fitzpatrick:1988 could be described by a single parameter, usually taken to be $R(V)$.
The shape of the extinction curve is set by the properties of the dust grains: their size, shape, and chemical composition. Work by @vandeHulst:1946 was among the first to begin to put all of these pieces together. The work of @Mathis:1977 employed more realistic grain compositions and found that the observed extinction curve was relatively insensitive to the material used, but constrained the grain size distribution to be a rough power law. Through changes to the grain size distribution, the shape of the extinction curve could be varied to reproduce observed variations in $R(V)$ [@Kim:1994; @Weingartner:2001; @Hirashita:2012].
While the grain size distribution is important for determining the dust extinction curve, dust chemical composition is important as well. The work of @Mulas:2013 fit extinction curves with a realistic set of interstellar molecules, using over 200 free parameters, emphasizing the importance of grains of different chemical species. The work of @Jones:2013 focuses instead on the role of processing of carbonaceous grains in the interstellar medium (ISM) by ultraviolet light, which can aromatize the grains, altering their extinction properties. Both the grain size distribution and the composition and types of dust grains affect the extinction curve [e.g. @Siebenmorgen:2014], a remarkable fact given the apparent single-parameter nature of the extinction curve shape.
The work of @Whittet:1988 [@Whittet:2001] found that dust properties and the extinction curve varied systematically with dust column density, with ice mantle formation and $R(V)$ increasing from its nominal, diffuse value at $A_V \approx 3.2$. A number of works have focused on the extinction curves of stars in dense star-forming regions, likewise finding elevated $R(V)$ (flatter extinction curves) in these regions [e.g. @Herbst:1976b; @Chini:1981; @Chini:1983; @Flaherty:2007]. This effect is often associated with grain growth (accretion, agglomeration) in dense regions, though star-formation in these regions also alters the radiation field and ISM environment more generally, complicating the interpretation.
In the recent past, the ever-increasing scale of astronomical surveys has allowed the extinction curve to be probed over larger regions. Many seek higher precision measurements of the extinction curve in diffuse regions to allow precise dereddening [e.g. @Schlafly:2011; @JonesD:2011; @Yuan:2013; @Wang:2014; @Xue:2016]. Others map out variations in the extinction curve, in the Milky Way [e.g. @Zasowski:2009; @Schlafly:2016; @Gontcharov:2012; @Nataf:2013; @Nataf:2015; @Schultheis:2014; @Schultheis:2015], and in other nearby galaxies like the Large Magellanic Cloud [@MaizApellaniz:2014]. These works have been valuable in refining measurements of the extinction curve and the character of its variation, but have not yet been able to identify what underlying physical mechanisms lead to extinction curve variations.
These surveys have, however, enabled exploration of a new observable constraining the origin of extinction curve variations: the spatial structure of the variations. Spatial structure can be a critical tool for understanding the origin of $R(V)$ variation. For instance, if elevated $R(V)$ is tied to dense regions in the ISM, cloud boundaries should tightly track $R(V)$ boundaries. In principle, adequately large samples of extinction curves could be correlated with supernovae and superbubble catalogs, to attempt to track the destruction and shattering of dust grains through a steepening of the extinction curve.
Unfortunately, until recently, catalogs of $R(V)$ measurements have been too sparse to address these questions. However, the catalog of @Schlafly:2016 [S16] is sufficiently dense to track large-scale 3D variation in the extinction curve over the nearest $\approx 4~\mathrm{kpc}$ of the Galaxy, which we investigate in this work.
The work of S16 uses near-infrared spectroscopy from the APOGEE survey [@Majewski:2015] in concert with broadband photometry from Pan-STARRS1 [@Magnier:2013] to measure the extinction curve to tens of thousands of reddened stars in the Galactic plane. We combine these data with distance estimates from @Ness:2016 and the 3D extinction map of @Green:2015 [G15] to infer the 3D distribution of $R(V)$. In this work, we find that roughly half of the observed variation in $R(V)$ in the S16 catalog is explained by large, kiloparsec-scale structures in $R(V)$. In particular, many of the nearby, outer Galaxy molecular clouds (Orion, Taurus, Perseus, California, Cepheus) share a lower typical $R(V)$ than clouds farther away in the Galaxy, though the existing data do not allow us to track detailed variations within those clouds. It is not presently clear whether these large scale features can be produced by any existing evolutionary models of dust grains.
We begin this work by first discussing in \[sec:data\] the adopted data sets. Second, in \[sec:rv3d\], we show the clear relationship between structures in the projected $R(V)$ map and the 3D distribution of dust, and present our mapping technique. In \[sec:results\] and \[sec:discussion\], we present our $R(V)$ map, and discuss its implications. Finally, in \[sec:conclusion\], we conclude.
Data {#sec:data}
====
Our analysis relies on three data sets. First, we employ the $R(V)$ measurements of S16. Second, we assign distances to the stars in S16 from the work of @Ness:2016, which determines distances based on a star’s magnitude and spectroscopically determined temperature, metallicity, and gravity. Finally, we use the three-dimensional dust map of G15 to determine where the dust in the Galaxy resides.
$R(V)$ Catalog {#subsec:rvcatalog}
--------------
We use the $R(V)$ catalog of S16 to provide the basic measurements of $R(V)$. That work measures $R(V)$ through determination of the broad band optical-infrared extinction curve, as derived from the combination of APOGEE spectroscopy [@Wilson:2010; @Nidever:2015; @GarciaPerez:2015; @Zasowski:2013; @Majewski:2015] and Pan-STARRS1 [@PS1_Optics; @PS1_GPCA; @PS1_GPCB; @Magnier:2013; @JTphoto; @Schlafly:2012], 2MASS [@Skrutskie:2006], and WISE [@Wright:2010] photometry. This work uses the $R(V)$ proxy $$\label{eq:rpv}
{\ensuremath{R^\prime(V)}}= 1.2 E({\ensuremath{g_{\rm P1}}}-\mathrm{W2})/E({\ensuremath{g_{\rm P1}}}-{\ensuremath{r_{\rm P1}}}) - 1.18$$ introduced in S16 as an estimate of $R(V)$. Accordingly, we also only use catalog [$R^\prime(V)$]{} measurements when a star is detected in all of the Pan-STARRS1 [$g_{\rm P1}$]{}, [$r_{\rm P1}$]{} and the WISE $\mathrm{W2}$ bands. We further use only stars where the total reddening $E(B-V)$ is greater than $0.3~\mathrm{mag}$, for which the uncertainty in [$R^\prime(V)$]{} is typically smaller than 0.2. Finally, we require $2 < R(V) < 6$ to remove 12 serious outliers. This leaves us with 15003 $R(V)$ measurements.
Distance Catalog {#subsec:distancecatalog}
----------------
We adopt the distance estimates of @Ness:2016 to the APOGEE stars. These estimates are made by using the stars’ temperatures, metallicities, and gravities to predict the absolute magnitude of the star. The distance modulus to a star is then found by comparing the absolute H-band magnitude to the apparent H-band magnitude of the star, after correction for extinction following @Zasowski:2013. The work of @Ness:2016 reports a distance uncertainty of about 30%.
3D Dust Map {#subsec:3ddust}
-----------
We employ the 3D dust column density map of G15. The map was developed by estimating the distances and reddening to roughly a billion stars, as inferred from their Pan-STARRS1 and 2MASS photometry [@Green:2014]. The technique has been shown to well reproduce the distances to the dust clouds [@Schlafly:2014] and the true dust column density [@Schlafly:2014b], until saturating at $E(B-V) \approx 2$. The map is pixelized so that pixels grow longer when they are located farther away from the sun; they have a length equal to roughly 25% of their distance from the sun. This places a limit of 25% on the distance accuracy obtainable from the 3D dust map, comparable to the uncertainty in the distances to the APOGEE stars from @Ness:2016.
$R(V)$ in 3D {#sec:rv3d}
============
We measure the 3D spatial variation in $R(V)$ throughout the Galactic plane. This measurement is motivated by strong correlations between the 3D morphology of the dust column and the projected $R(V)$ map, which we investigate in \[subsec:simplerv3d\]. Emboldened by this signal, we model the variation in $R(V)$ throughout the Galactic plane in \[subsec:modelrv3d\].
Observations of $R(V)$ in 3D {#subsec:simplerv3d}
----------------------------
Angular maps of $R(V)$ from the catalog of S16 show significant variations in $R(V)$, which we reproduce in Figure \[fig:rvangpanels\]. The first panel of Figure \[fig:rvangpanels\] shows the mean $R(V)$ of the S16 stars in different locations in the sky. The second panel compares this with $\beta$, the emissivity spectral index of the dust, as measured by @Planck:2014. The third panel shows the column density of dust within one kiloparsec from G15. The fourth panel shows the predicted $R(V)$ map from our 3D $R(V)$ model (\[subsec:modelrv3d\]), and the fifth panel shows the residuals between the data and the model.
The most dramatic large scale features in the $R(V)$ map in the first panel are: an extended low latitude region from $180{\ensuremath{^\circ}}> l > 240{\ensuremath{^\circ}}$ with higher-than-average $R(V)$; a curving low $R(V)$ region extending from $(l, b) = (200{\ensuremath{^\circ}}, -15{\ensuremath{^\circ}})$ to $(130{\ensuremath{^\circ}}, 10{\ensuremath{^\circ}})$; a high $R(V)$ region centered at $(l, b) = (110{\ensuremath{^\circ}}, 0{\ensuremath{^\circ}})$; and an extended low $R(V)$ region with $90{\ensuremath{^\circ}}> l > 0{\ensuremath{^\circ}}$. These structures are highly correlated with the Planck measurements of the emissivity spectral index, which show the same general features (second panel). Comparison with the 3D dust map (third panel) shows that the extended, curving, low $R(V)$, high $\beta$ feature in the outer Galaxy is associated with dust within one kiloparsec, including the Orion, Taurus, Perseus, California, and Cepheus molecular clouds, which are often associated with the Gould Belt.
Indeed, the 3D model of $R(V)$ we construct in \[subsec:modelrv3d\] does a good job of reproducing these general features (fourth panel), qualitatively reproducing all of the features we identified in the first panel. The residuals of the 3D-pixelized fit (fifth panel) show much reduced large scale spatial structure, though in some regions it seems we have not captured all of the $R(V)$ variation. Notably, in the low $R(V)$, high $\beta$ structure centered at $(l, b) = (125{\ensuremath{^\circ}}, 7.5{\ensuremath{^\circ}})$, our model predicts significantly higher $R(V)$ than present in the data.
[{width="\linewidth"}]{}
The strong correlation in the outer Galaxy between the presence of significant columns of nearby dust and low $R(V)$ suggests that nearby dust has a steeper extinction curve than more distant dust. To make this point more clearly, we show in Figure \[fig:rvskewerstack\] the distribution of dust along the line of sight toward each of the S16 stars (including any dust behind the stars), as a function of the $R(V)$ of the star. There is a clear correlation between $R(V)$ and the spatial distribution of dust along a line of sight: most of the column toward low-$R(V)$ stars is within about 1 kpc, while most of the column toward high-$R(V)$ stars lies beyond 1 kpc.
[![ \[fig:rvskewerstack\] Distribution of dust density along the line of sight toward S16 stars, as a function of the stars’ $R(V)$. Low $R(V)$ sight lines are dominated by nearby dust, while high $R(V)$ sight lines are dominated by more distant dust. ](rvskewerstack.eps "fig:"){width="\linewidth"}]{}
This is strong evidence that changes in $R(V)$ along typical lines of sight are driven by underlying 3D structures that have kiloparsec scales, and motivates the development of a 3D $R(V)$ map.
3D $R(V)$ Modeling {#subsec:modelrv3d}
------------------
Existing photometric dust mapping techniques allow billions of stars to be used to constrain the detailed 3D structure of the Milky Way’s dust. However, these techniques currently lack the sensitivity to simultaneously map variations in the dust extinction curve. Meanwhile, extinction mapping programs using spectroscopy are capable of sensitive measurements of changes in the extinction curve, but are limited to samples of hundreds of thousands of stars, rather than billions. We take advantage of the fact that $R(V)$ seems to typically vary on much larger spatial scales than $E(B-V)$ to map $R(V)$ at low resolution while combining with higher resolution 3D extinction maps to track the detail in the 3D dust distribution.
We assume in this work that the 3D dust map and the distances to the S16 stars are exactly correct and have no uncertainty (though this assumption is false; see \[subsec:limitations\]). The total dust column $E(g-r)$ to a star is then given by $$E(g-r) = \int_0^D ds\, \rho_{g-r}(l, b, s)$$ where $l$ and $b$ are the Galactic longitude and latitude of the star, $D$ is the distance to the star from @Ness:2016, and $\rho_{g-r}(l, b, s)$ is the 3D dust map from G15, giving the dust density in $\mathrm{mag}\ E(g-r)\ \mathrm{kpc}^{-1}$ in a particular direction at a distance $s$.
In the work of S16, $R(V)$ is inferred as a linear function $L$ of $E(g-W2)/E(g-r)$. Insofar as the 3D map of G15 is primarily sensitive to optical reddenings, we can treat it as a map of the density of $E(g-r)$ reddening dust. Then $$\begin{aligned}
\label{eq:linearrv1}
R &=& &L\left(\frac{E(g-W2)}{E(g-r)}\right) \\
\label{eq:linearrv2}
&=& &L\left(\frac{\int_0^D ds\, \rho_{g-r}(l, b, s) E(g-W2)/E(g-r)}{\int_0^D ds\, \rho_{g-r}(l, b, s)}\right) \\
\label{eq:linearrv3}
&=& &\frac{\int_0^D ds\, \rho_{g-r}(l, b, s) R(l, b, s)}{\int_0^D ds\, \rho_{g-r}(l, b, s)}\end{aligned}$$ where Equation \[eq:linearrv3\] follows from Equation \[eq:linearrv2\] since the argument of $L$ is equivalent to the expectation value of $E(g-\mathrm{W2})/E(g-r)$ along the line of sight, and the expectation value is a linear operator. Accordingly, $R(V)$ to any star is simply the average $R(V)$ along the line of sight to the star, weighted by the amount of dust ($\rho_{g-r}$) at each point.
This procedure relies on G15 being a map of $E(g-r)$ reddening. The work of G15 assumes that all dust is described by a single extinction curve, so in that context, this assumption is fine. This work, however, studies variations in the extinction curve, so it is not clear how we should treat the G15 3D reddening map. Our assumption that it most closely maps $E(g-r)$ is motivated by the fact that [$g_{\rm P1}$]{} and [$r_{\rm P1}$]{} are the most reddening sensitive bands in PS1, and because this assumption makes for the simplest analysis. Future 3D maps will need to better account for variability in the extinction curve; see \[subsec:limitations\].
We choose to parameterize $R(l, b, s)$ as $R(X, Y, Z)$, centered on the Galactic center, with $Z=0$ corresponds to $b=0$, the Galactic plane. We take the position of the sun to be $(X, Y) = (8~\mathrm{kpc}, 0)$, and choose the positive $Z$ axis to point toward the North Galactic pole. The $Y$ axis is fixed so that the coordinate system is right-handed; $l=90{\ensuremath{^\circ}}$ points in the negative $Y$ direction.
We use two different schemes to pixelize the Galactic plane: a 2D and a 3D pixelization. In our 2D scheme, we use pixels $62.5 \mathrm{pc}$ on a side within 5 kpc in the $X$ or $Y$ direction from the sun. Beyond 5 kpc, we extend the edges of the pixels to infinity. All pixels extend to infinity in the $Z$ direction. Alternatively, in our 3D scheme, we use pixels $200 \mathrm{pc}$ on a side in the $X$ and $Y$ directions, and slice the Galactic plane into three pixels in the $Z$ direction: a midplane pixel extending from $-50 \mathrm{pc}$ to $+50 \mathrm{pc}$, and above-plane and below-plane pixels extending to infinity outside of the midplane. We adopt these two schemes to balance having a huge number of parameters in the fit routines with wanting to explore potential variation in the extinction curve with $Z$.
Most of the stars for which we have good $R(V)$ measurements lie within 5 kpc, and the 3D dust map loses reliability beyond that distance, so the treatment of pixels beyond this distance is largely irrelevant. In the 3D scheme, finer resolution is required in the $Z$ direction than in the $X$ and $Y$ directions, since the scale height of the ISM disk is only about $100 \mathrm{pc}$. In both pixelization schemes, $R(V)$ is constant within a pixel in the model.
We seek a linear model for the $R(V)$ observed to each star, $$Ap \approx R \, ,$$ where $R_i$ is the observed $R(V)$ of star $i$, $p$ is a vector with the $R(V)$ we find for each pixel of the 3D map, and $A$ is the design matrix. Each row of $A$ gives the fraction of dust in front of star $i$ contained in pixel $j$, so each row of $A$ sums to unity. More explicitly, according to Equation \[eq:linearrv3\], $$\begin{aligned}
A_{ij} &= \frac{\int_{N(i,j)}^{F(i,j)} ds\, \rho_{g-r}(l, b, s)}{\int_0^{D_i} ds\, \rho_{g-r}(l, b, s)} \\
N(i,j) &= \min(D_i, \widetilde{N}(l, b, j)) \\
F(i,j) &= \min(D_i, \widetilde{F}(l, b, j))\end{aligned}$$ where $D_i$ is the distance to star $i$, and $\widetilde{N}(l, b, j)$ and $\widetilde{F}(l, b, j)$ are the nearest and farthest distances where the sight line toward the coordinates $(l, b)$ intersects pixel $j$ (or 0 when there is no intersection).
Many pixels in the $R(V)$ map have no 3D map data or no $R(V)$ data from APOGEE. Most of these cases are due to the pixel’s lying in the southern hemisphere, inaccessible to observations from the north. Some further cases are caused by the patchy coverage of the APOGEE data. Another limitation of the method is the low resolution of the 3D dust map and APOGEE distances, which are both worse than 25%, which is larger than 1 kpc at the 5 kpc boundary. The $R(V)$ values we fit at different distances in these regions are highly degenerate with one another; increasing $R(V)$ nearby can be largely compensated by decreasing it farther away. For these reasons, the problem is underdetermined, and to obtain a stable solution we need to add some regularization to the design matrix $A$. We choose to demand that the eight nearest neighbors to any pixel have $R(V)$ not far from one another, by appending to $A$ $$\begin{aligned}
A_{N+q,i(q)} &= \phantom{-}1 \\
A_{N+q,j(q)} &= -1 \, ,\end{aligned}$$ where $q$ indexes over all pairs of nearest neighbor pixels $i(q)$, $j(q)$ with $i < j$, and $N$ is the number of $R(V)$ stars. We correspondingly append $0$ to the vector $R$ for each pair; this enforces an L2 penalty on the $R(V)$ differences between nearest neighbors, i.e., Tikhonov regularization of $p$ via nearest neighbor differences.
The usual least-squares solution is to determine $p$ by minimization of $$\label{eq:chi2}
\chi^2 = \sum_i \chi_i^2 = \sum_{i} ((A p - R)_i/\sigma_i)^2 \, ,
$$ which produces a 3D $R(V)$ map encoded in $p$. For $\sigma_i$ we currently adopt a diagonal matrix, with $\sigma_i = 0.2$ for $i \leq N$. Here we have chosen not to adopt the actual measurement uncertainties, since we believe that the residuals are dominated by model imperfections (for example, in the 3D map and stellar distances) rather than in the $R(V)$ uncertainties.
We set $\sigma_i = \lambda$ for $i > N$; $\lambda$ controls the strength of the nearest-neighbor regularization. We use a cross-validation technique to determine $\lambda$. The general concept is to split the data into two sets, a training set and a test set. The training set is used to determine the model, and the test set is used to determine the accuracy of the model as a function of $\lambda$. The value of $\lambda$ is chosen to minimize the prediction error. We explored two cross-validation techniques. In the first, we made a test set from a random 10% of the data, and found that $1 < \lambda < 2$ minimized the prediction error, depending on the 10% of the data removed. In the second, we select 10 stars at random, and construct a test set from all stars within 2 degrees of these stars. This technique obtains $0.05 < \lambda < 0.4$—a roughly 10$\times$ smoother model. The first technique can be thought of exploring how well the data predict other data spatially distributed like the APOGEE data. The second technique, on the other hand, better describes the prediction error obtained by extrapolating somewhat outside the existing APOGEE data. Since we seek to interpolate over the sparse, non-contiguous APOGEE coverage to create a relatively uniform map of $R(V)$, we adopt $\lambda = 0.2$, consistent with the results of the second cross-validation technique.
We refine the usual least-squares solution of Equation \[eq:chi2\] in order to reduce the influence of outliers, though these have little influence in these data. We accomplish this by replacing $\chi_i$ in Equation \[eq:chi2\] with $$\label{eq:chidamp}
\chi^\prime_i = 2 D \operatorname{sgn}(\chi)(\sqrt{1+|\chi|/D}-1) \, .$$ This function has $\lim_{x\to 0} \chi^\prime(x) = x$ and $\lim_{x\to\infty} \chi^\prime(x) = 2\sqrt{Dx}$, effectively giving points with small $\chi$ full weight, while reducing the weight of points with $x \gg D$. We choose $D = 3$.
3D $R(V)$ Modeling Limitations {#subsec:limitations}
------------------------------
Our technique is subject to a number of limitations:
1. distance resolution,
2. treatment of distance as known,
3. patchy coverage,
4. limited coverage in dense regions, and
5. an [*ad hoc*]{} regularization scheme.
Two of these points are relatively simple to address. First, the uncertainties in distance, especially for nearby stars, will be dramatically reduced when Gaia data for fainter stars become available. Second, the patchy coverage will be substantially resolved with upcoming data from APOGEE-II and future surveys with the APOGEE instrument.
The remaining limitations are difficult to address. Our coverage in dense regions is limited because of our reliance on $g$-band photometry. Obtaining this photometry through the densest clouds is simply expensive. Upcoming surveys of the Galactic plane from DECam and LSST will help, but we know of no planned deeper northern Galactic plane surveys. Proxies for $R(V)$ that do not use $g$ are available, but S16 makes clear that the widest possible baseline is ideal for simultaneous determination of $E(B-V)$ and $R(V)$.
We treat the distances to dust clouds and to APOGEE stars as known in this work, ignoring their distance uncertainty. In principle, we should fit these distances simultaneously with the $R(V)$ map. For the distances to the stars, we might adopt a constant 30% uncertainty in distance, as recommended by @Ness:2016, though this neglects correlations in the uncertainties due to an unknown bias, which may vary as a function of stellar type. Adequate treatment of the 3D dust map uncertainties is more problematic. These are highly non-uniform and correlated along the line of sight. The work of G15 does provide Markov chains describing the uncertainty in the 3D dust map, but incorporating these into the existing 3D $R(V)$ framework would make the analysis vastly more computationally expensive. Moreover, the 3D map was created in the first place assuming that $R(V)$ was constant. The correct procedure is then to go back to the photometry of a billion PS1 and 2MASS stars and simultaneously fit the 3D column and $R(V)$ maps: a worthy goal, but beyond the scope of this work.
Finally, we adopt a simple regularization scheme, choosing to minimize the differences between adjacent pixels in the $R(V)$ map to enhance smoothness. If we had an underlying model for how $R(V)$ variations should be spatially correlated throughout the Galaxy, we could use it to better model the $R(V)$ map. This initial exploration is in part intended to spur the development of such models.
Results {#sec:results}
=======
Figure \[fig:rvmap\] shows our resulting maps, for the 2D pixelization. The top-left panel shows the measured $E(B-V)$ to the APOGEE stars, projected face-on into the Milky Way plane. The top-right panel shows the projected total amount of dust in the plane, as determined by G15, for $|Z| < 0.2\,\mathrm{kpc}$. The lower-left panel shows the measured [$R^\prime(V)$]{} of the APOGEE stars, and the lower-right panel shows our inferred $R(V)$ map. In each panel, the $\times$ symbol indicates the adopted location of the sun at $(X, Y) = (8~\mathrm{kpc}, 0)$, and the red circle shows a circle of 1 kpc centered at the sun. The blue contours in the left two panels show where most of the APOGEE data lie.
[{width="\linewidth"}]{}
The $R(V)$ map is colored light blue in regions where the result is particularly uncertain (estimated uncertainty $0.16$), though the value is somewhat arbitrary because of the [*ad hoc*]{} regularization we have employed. This masks the Southern Galactic plane, where we have no APOGEE or 3D dust map information, as well as most of the sky beyond 5 kpc, where likewise few observations are available.
The clearest structure we see in the lower right panel of Figure \[fig:rvmap\] is the region of relatively low $R(V)$ within about 1 kpc, anticipated in \[subsec:simplerv3d\]. Toward $l=180{\ensuremath{^\circ}}$ and $l=100{\ensuremath{^\circ}}$, there are clear transitions between nearby low $R(V)$ dust and more distant high $R(V)$ dust at a distance of about 1 kpc. Many features of the map, however, are strongly heliocentrically radial. For example, there is a narrow radial wedge of low $R(V)$ at $l = 160{\ensuremath{^\circ}}$, and a wide wedge of intermediate $R(V)$ at $30{\ensuremath{^\circ}}< l < 90{\ensuremath{^\circ}}$. Unfortunately, given the significant uncertainty in distance in both the 3D dust map and the distances to the individual APOGEE stars, as well as our rudimentary treatment of this uncertainty, such “fingers of god” are expected.
We show in Figure \[fig:ebvrvmap\] an attempt to combine the $R(V)$ and $E(B-V)$ maps. The darkness of a region in Figure \[fig:ebvrvmap\] is set by the amount of dust column $E(B-V)$ in that region from G15, while the color of the pixel is determined by our $R(V)$ map. Since regions with little dust are not useful for constraining $R(V)$, this visualization makes it easier to focus on the $R(V)$ map where it is actually constrained by the data. The features reproduce those in Figure \[fig:rvmap\], however.
[![ \[fig:ebvrvmap\] A map of $E(B-V)$ and $R(V)$ in the Galactic plane. Color shows $R(V)$, with deep red indicating $R(V) < 3.1$, gray indicating $R(V) = 3.4$, and deep blue indicating $R(V) > 3.7$. The darkness of a region shows how much dust is present in that region, using $E(B-V)$ per unit distance as a proxy. ](rvebvsimul.eps "fig:"){width="\linewidth"}]{}
Our 3D-pixelized model contains three dust slices at different heights above the Galactic plane, which we show in Figure \[fig:rvmap3slice\]. The slices show reasonable consistency with one another, though this is partially enforced by the regularization scheme. The nearest kiloparsec has reduced $R(V)$ in each slice, though nearby, above the plane, toward the Galactic center the map prefers a high $R(V)$, presumably due to the influence of the Ophiuchus molecular cloud. The very lowest $R(V)$ is found above the plane near $l = 125{\ensuremath{^\circ}}$, corresponding to the low $R(V)$ region at $(l, b) = (125{\ensuremath{^\circ}}, 7.5{\ensuremath{^\circ}})$ identified in Figure \[fig:rvangpanels\].
[{width="\linewidth"}]{}
We can test how good of a description Figure \[fig:rvmap\] is of the $R(V)$ measurements by comparing predictions of $R(V)$ from the map and the stellar distances with the measurements. We make this comparison for the 3D-pixelized map in Figure \[fig:rvvsrvpred\] (second panel; the results for the 2D-pixelized map are similar). Unsurprisingly, given that we are fitting the $R(V)$ measurements, the correlation between our model and the measurements is good. The standard deviation of all S16 [$R^\prime(V)$]{} measurements is 0.19, while the model residuals have a standard deviation of 0.13; the model explains roughly half of the variance in $R(V)$. The median uncertainty in [$R^\prime(V)$]{} is 0.1, so the dispersion in the residuals is roughly half uncertainty in the data and half inadequacies in the model.
[{width="\linewidth"}]{}
The $R(V)$ map is limited in its accuracy by the accuracy of the underlying 3D dust map and stellar distance catalog. We test one important aspect of these underlying catalogs in the first panel of Figure \[fig:rvvsrvpred\]. The Figure compares the measured reddenings of the APOGEE sources with the reddenings we expect from integrating through the 3D extinction map of G15 to the stars, with distances given by @Ness:2016. We find excellent agreement in the mean.
Nevertheless, the differences between the two $E(B-V)$ measurements are significant. The rms disagreement is 0.1 mag $E(B-V)$, but the APOGEE $E(B-V)$ should be accurate to better than 0.03 mag, while the extinction map likewise claims $\approx 0.02$ mag accuracy from comparisons with @Schlegel:1998 [hereafter SFD] at high latitudes. Some part of this discrepancy can be explained by the uncertain distances, which are not important at high latitudes where the SFD comparison was performed. We find it likely that angular differential extinction at the low latitudes and high reddenings where the APOGEE stars reside is also a significant source of error in the APOGEE-3D map comparison. These $0.1$ mag errors correspond to 15% errors at the median extinction of the sample, and will translate to error in the $R(V)$ map.
$R(V)$ Input Catalog Limitations {#subsec:rvlimitations}
--------------------------------
Our technique transforms a set of $R(V)$ measurements to objects at known distances into a 3D map of $R(V)$. The resulting 3D map naturally inherits all of the limitations and systematics of the $R(V)$ catalog on which it is based. In the case of the S16 catalog used in this work, we consider two possible systematic errors: an overall offset in the $R(V)$ proxy adopted in S16, $R(V) \approx 1.2 E(g-W2)/E(g-r) - 1.18$ (Equation \[eq:rpv\]), and the dependence of this proxy on stellar type in the absence of true variation in the extinction curve.
We first consider the effect of a systematic offset in the input $R(V)$ catalog. The work of S16 uses the linear function of Equation \[eq:rpv\] to determine $R(V)$ from the color excess ratio $E(g-W2)/E(g-r)$. This color excess ratio is a decent proxy for $R(V) = A(V)/E(B-V)$ because $g-r$ and $B-V$ are similar optical colors and because $A(W2)$ is much smaller than $A(g)$, so $E(g-W2)$ roughly equals $A(g)$. Unsurprisingly therefore, typical extinction curves predict strong, nearly linear correlations between $E(g-W2)/E(g-r)$ and $R(V)$ with slopes not far from unity. Unfortunately, different extinction curves predict significantly different constant offsets in Equation \[eq:rpv\], so that for a given $E(g-W2)/E(g-r)$, the $R(V)$ from @Fitzpatrick:1999 and from CCM may be different by 0.5 or more. Given this significant disagreement between standard extinction curves, it is possible that the $R(V)$ measurements in the S16 catalog are systematically off by a few tenths. Fortunately, the problem we solve in Section \[subsec:modelrv3d\] is linear, so any constant offset in the $R(V)$ catalog can be accommodated by subtracting the offset from our derived 3D map. Moreover, in this work we are primarily interested in how $R(V)$ varies throughout the Galaxy, and this variation is unaffected by the addition of a constant term.
We next consider the dependence of the S16 $R(V)$ proxy (Equation \[eq:rpv\]) on the stellar type of the target. This proxy relies on broadband photometric magnitudes, whose effective wavelengths depend modestly on stellar type, dust column, and the extinction curve. This leads to variation in color excess ratios like that in Equation \[eq:rpv\] with stellar type, even in the absence of true variation in the extinction curve [e.g. @Fernie:1963; @Sale:2015]. We simulate this effect for the particular $R(V)$ proxy of Equation \[eq:rpv\], and show the results in Figure \[fig:rvvsstellartype\]. Because S16 compares stars of different reddenings with one another, rather than comparing unreddened stars to reddened stars, we compute $$R(V) \approx 1.2 \frac{E(g-W2)_{2.5} - E(g-W2)_{1.5}}{E(g-r)_{2.5}-E(g-r)_{1.5}} - 1.18 \, ,$$ where the subscripts $1.5$ and $2.5$ indicate the amount of reddening $A(V)$ at which the color excess was computed. We note that the obvious approach using $A_V = 0$ and $A_V = 2$ changes the result only by a roughly constant offset of 0.05. The blue line shows the derived $R(V)$ as a function of temperature for solar metallicity stars with $\log g = 2.5$, and the green histogram shows the distribution of temperatures for stars considered in this work. Changes in the star’s temperature alone can lead to variations in the $R(V)$ we derive by up to 0.15, but due to the strong clustering of the stellar temperatures near 5000 K, the root-mean-square induced variation in $R(V)$ is only 0.02. We could in principle correct this effect, but given its small amplitude relative to the $R(V) \approx 0.2$ variations we observe, we choose to neglect it.
[![ \[fig:rvvsstellartype\] Derived $R(V)$ vs. effective temperature $T$ for solar metallicity, $\log g = 2.5$ stars, for an input F99 extinction curve with $R(V) = 3.1$ (blue line). The distribution of stellar temperatures used in this work is shown by the green histogram. The full range of variation in $R(V)$ induced by temperature variation is 0.15 within the sample used in this work, though due to the clustering of the temperatures near 5000 K, the root-mean-square variation in $R(V)$ is only 0.02. ](rvvsteffmodhist.eps "fig:"){width="\linewidth"}]{}
In principle, this effect could lead to small variations in $R(V)$ over the sky as the stellar populations in the sample vary. We consider this effect in Figure \[fig:tlb\], which shows how the mean temperature of the sample varies with Galactic latitude and longitude. The dominant signal is a smooth variation from hotter stars ($\sim 4800 \mathrm{K}$) in the outer Galaxy to cooler stars in the inner Galaxy ($\sim 4000 \mathrm{K}$), inducing an artificial variation in $R(V)$ of 0.05. If we were to apply a correction, we would slightly increase $R(V)$ in the outer Galaxy while decreasing it in the inner Galaxy, but the effect is not large enough to influence our conclusions.
[{width="\linewidth"}]{}
Discussion {#sec:discussion}
==========
We present measurements of the spatial variation of $R(V)$ across the sky, and find that variations on kiloparsec scales explain roughly half of all variation in the $R(V)$ measurements. Ideally we would now compare this rough spatial morphology with the predictions from evolutionary models of dust grains in the interstellar medium, but we are unaware of sufficiently detailed models. The problem is complicated by the fact that it is not even certain what physically is different about dust grains in high $R(V)$ versus low $R(V)$ regions. The most common explanation is that grains in high $R(V)$ regions are larger, though compositional variations in the dust grains have also been proposed [@Mulas:2013; @Jones:2013].
Given the uncertainty in the underlying physical framework, we can make only broad, qualitative statements about the distribution of the dust. First, critically, the features in the $R(V)$ map are much larger in scale than the features in the $E(B-V)$ map. At some level, this is due to the regularization in the $R(V)$ map, but the fact that all of the clouds beyond 1 kpc in the outer Galaxy show up as clearly “blue” and high $R(V)$ in Figure \[fig:ebvrvmap\] is significant, and that the more nearby clouds within 1 kpc have lower, red-gray $R(V)$. Moreover, the $R(V)$ measurements to different stars are independent, and one can already see large scale correlations in Figure \[fig:rvangpanels\], top panel.
The large-scale structure of the $R(V)$ variation seems to suggest interpretation in terms of large-scale features of the Galaxy. Figure \[fig:ebvrvmap\] could be interpreted as a gradient with Galactocentric radius, where $R(V)$ is higher in the outer Galaxy than it is in the inner Galaxy. Such an effect could be generated by the interstellar radiation field, star formation history (and hence age of the dust grains and their processing history), or the chemical composition of the interstellar medium.
It is also possible that $R(V)$ is connected to the Galaxy’s spiral structure. We explore this possibility in Figure \[fig:maser\], which shows our $R(V)$ map in the context of the @Reid:2014 masers and spiral arm identifications. The most striking feature is that the Perseus spiral arm neatly matches onto a high $R(V)$ region, and the transition from low to high $R(V)$ occurs at the Local-Perseus Arm boundary. It is difficult to understand how different arms give rise to different $R(V)$, however.
[![ \[fig:maser\] $R(V)$ map and Galactic spiral structure, as traced by the @Reid:2014 maser sample. The Outer Arm (Out), Perseus Arm (Per), Local Arm (Loc), Sagittarius Arm (Sgr), and Scutum Arm (Sct) are labeled, together with masers which could not be reliably associated with a spiral arm (labeled ‘?’). The transition from low to high $R(V)$ at about $X = 8.5 \mathrm{kpc}$ corresponds surprisingly well with the Local to Perseus Arm transition. ](masers.eps "fig:"){width="\linewidth"}]{}
We can also try to exclude mechanisms for generating $R(V)$ variations using the spatial structure we observe. The large scale region of elevated $R(V)$ in the outer Galaxy seems to be at best indirectly connected with grain growth in dense regions of the interstellar medium. In particular, we are fortunate that the APOGEE footprint overlaps the California molecular cloud at $(l, b) = (165{\ensuremath{^\circ}}, -8{\ensuremath{^\circ}})$. There is little evidence of increased $R(V)$ with increasing $E(B-V)$ in this cloud, and the entire cloud has lower than average $R(V)$. This seems inconsistent with a picture in which $R(V)$ variations are driven due to grain growth in dense regions. Moreover, the California molecular cloud has had a relatively quiescent star formation history [@Lada:2009], which is somewhat inconsistent with the picture of low $R(V)$ dust having been shattered in supernovae.
An additional challenge to dust modeling is the overall small amplitude of changes in $R(V)$. The dispersion in the projected $R(V)$ measurements from S16 is only 0.18, and our deprojected, 3D $R(V)$ map has a dispersion of only 0.2, with a full range of only about 0.8 (considering only regions of the map with estimated uncertainties less than 0.2). Existing dust models, motivated by observations of individual stars with $R(V) > 5$ [@Cardelli:1989], can accommodate a substantially larger range of $R(V)$ variations. It is not yet understood why the extinction curve varies as little as it does, given its sensitivity to a wide range of model parameters.
Future programs surveying $R(V)$ variations will address these questions. Ongoing programs within APOGEE-II seek to systematically chart the variation of $R(V)$ within nearby molecular clouds, to better characterize if and when column density leads to grain growth. Future large-scale studies in the Magellanic clouds and Andromeda could track $R(V)$ variations across entire Galactic disks, clarifying questions of whether $R(V)$ varies with Galactic radius or star-formation activity. In principle, the $R(V)$-morphology may unveil the signatures of grain destruction and shattering through correlation with past supernovae and superbubbles. In conjunction with improved modeling of the evolution of dust in the Galaxy, such programs may be capable of identifying the underlying physical mechanisms determining dust properties and their variation.
Conclusion {#sec:conclusion}
==========
We have made three-dimensional maps of the shape of the dust extinction curve, as parameterized by $R(V)$. Roughly half of the variance in $R(V)$ over the APOGEE footprint can be explained by an $R(V)$ map containing information only at scales larger than $200~\text{pc}$ in the $X$ and $Y$ directions and $100~\text{pc}$ in the $Z$ direction.
Our map features structures on kiloparsec scales. This result complements findings in the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC), which have different extinction curves than the Milky Way [@Gordon:2003], emphasizing that galaxy-scale mechanisms can be dominant determinants of extinction curve shape.
In particular, our map shows that much of the dust within 1 kpc of the sun has systematically lower $R(V)$ than more distant dust in the Galactic plane, especially in the outer Galaxy, including the Orion, Taurus, Perseus, California, and Cepheus molecular clouds. This result argues that the physical processes that set dust properties act on scales larger than these individual clouds. In particular, this result is in tension with the usual picture that $R(V)$ growth is driven by grain growth in individual dense regions.
Existing dust models can explain variations in the extinction curve via the dust grain size distribution, via the chemical composition of the dust, as well as via the chemical and physical processing of the dust grains. Studies of the spatial morphology of the dust extinction curve can shed light on which of these factors are at work, and to which extent. These measurements will be especially valuable in combination with simulations of the evolution of dust in the interstellar medium, tracking its life through molecular clouds and supernovae remnants, its growth and destruction, and the morphology these processes imprint on it.
Future studies of the extinction curve will be able to provide much greater detail than has been possible in this work. Gaia parallaxes and spectrophotometry will improve the 3D resolution and accuracy of the extinction curve maps. Ongoing spectroscopic surveys of the Galactic plane like APOGEE-II will dramatically extend the coverage of the Galactic plane, enabling complete maps of the dust in the nearby Galaxy. Extension to other galaxies like the LMC [@MaizApellaniz:2014] and M31 [@Clayton:2015] will become increasingly effective with massively multiplexed spectrographs, allowing determination of the processes that set the properties of dust grains.
We thank the referee for helpful comments which improved the manuscript. ES acknowledges support for this work provided by NASA through Hubble Fellowship grant HST-HF2-51367.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.
|
---
abstract: 'For some centuries, first order chemical rate constants were determined mainly by a linear logarithmic plot of reagent concentration terms against time where the initial concentration was required, which is experimentally often a challenging task to derive accurate estimates. By definition, the rate constant was deemed to be invariant and the kinetic equations were developed with this assumption. A reason for these developments was the ease in which linear graphs could be plotted. Here, different methods are discussed that does not require exact knowledge of initial concentrations and which require elementary nonlinear analysis and the ensuing results are compared with those derived from the standard methodology from an actual chemical reaction, with its experimental determination of the initial concentration with a degree of uncertain. We verify experimentally our previous theoretical conclusion based on simulation data \[ J. Math . Chem [**43**]{} (2008) 976–1023\] that the so called rate constant is never constant even for elementary reactions and that all the rate laws and experimental determinations to date are actually averaged quantities over the reaction pathway. We conclude that nonlinear methods in conjunction with experiments could in the future play a crucial role in extracting information of various kinetic parameters.'
author:
- |
Christopher G. Jesudason\
[Dept. of Chemistry and Center for Theoretical and Computational Physics,]{}\
[University of Malaya, 50603 Kuala Lumpur]{}\
[Malaysia]{}\
[jesuum.edu.my, chrysostomggmail.com]{}
title: Determination of Rate Constant of Chemical Reactions by Simple Numerical Nonlinear Analysis
---
[ **Keywords**]{}: \[1\] elementary reaction rate constant , \[2\] activity and reactivity coefficients, \[3\] elementary and ionic reactions without pre-equilibrium. : 80A10, 80A30, 81T80, 82B05, 92C45, 92E10, 92E20.
1. INTRODUCTION AND METHODS
===========================
As alluded in the abstract, most kinetic determinations use logarithmic plots with known initial concentrations, although there have been attempts [@moore1; @gug1 and refs. therein]. (There are possible ambiguities in [@moore1] concerning choice of variables that will not be discussed.) However all these publications hitherto assume constancy of the rate constant $k$ and do not focus on nonlinear analysis (NLA), as will be attempted here in preliminary form. We analyze kinetic data of the tert butyl chloride hydrolysis reaction in ethanol solvent (80%v/v) derived from the Year III teaching laboratory of this University (UM); 0.3mL of the reactant was dissolved in 50mL of ethanol initially. The reaction was conducted at $30^oC$ and monitored over time (minutes) by measuring conductivity ($\mu\text{S cm}^{-1}$) due to the release of $\text{H}^+$ and $\text{Cl}^-$ ions as shown below (\[eq:1\]), $$\label{eq:1}
\mbox{C$_4$H$_9$Cl}+\mbox{H$_2$O}\,\, \stackrel{k}{\longrightarrow}\,\, \mbox{C$_4$H$_9$OH} + \mbox{H}^{+} + \mbox{Cl}^{-}$$ and $\lambda_\infty=2050 \mu\text{S cm}^{-1}$ was determined by heating the reaction vessel at the end of the monitoring to $60^o$C until there was no apparent change in the conductivity when equilibrated back at $30^o$C. “Units” in the figures and text refers to $\mu\text{S cm}^{-1}$. It would be inferred here that either because of evaporation or the temperatures not equilibrating after heating, the measured $\lambda_\infty$ is larger than the actual one. Linear proportionality is assumed in $\lambda$ and the extent of reaction $x$, where the first order law (c being the instantaneous concentration and $a$ the initial concentration) is $\frac{dc}{dt}=-kc=-k(a-x)$; with $\lambda_{\infty}=\alpha a,\lambda_{t}=\alpha x$ and $\lambda(0)=\lambda_{0}=\alpha x_0$, integration yields for assumed constant $k$ $$\label{eq:2}
\ln\frac{(\lambda_\infty- \lambda_0)}{(\lambda_\infty-\lambda(t))}=kt$$ Eqn.(\[eq:2\]) determines $k$ if $\lambda_{0}$ and $\lambda_{\infty}$ are known.
The plot of (\[eq:2\]) was made for the same experimental values with different $\lambda_\infty$’s, both higher and lower than the experimental value. We find that the rate constant for the NLA was higher, leading to a lower value of $\lambda_\infty$ which is consonant with evaporation of solvent or the non-equilibration of temperature prior to measurement to determine $\lambda_{\infty}$.Except for the last subsection, we shall do a NLA based on constant $k$ assumption.
1.1 Method 1
------------
Under linearity argument and constant $k$, the rate equation $\frac{dc}{dt}=-kc=-k(a-x)$ reduces to $$\label{eq:2b}
\frac{\lambda(t)}{dt}= -k\lambda(t)+\lambda_{\infty}.k$$ Hence a plot of $\frac{\lambda(t)}{dt}$ vs $t$ would be linear. We find this to be the case for polynomial order $npoly\leq 3$ as in Fig.(\[fig:2\]) below for all data values; higher polynomial orders can be used in selected data points of the curve below, especially in the central region. Thus criteria must be set up to determine the appropriate regime of datapoints in the NLA.
![Method 1 graph showing linearity lower order polynomial fits[]{data-label="fig:2"}](m1all.eps){width="7cm"}
1.2 Method 2
------------
Let $\alpha'=\lambda_\infty-\lambda_0$, then $\ln \alpha' -\ln(\lambda_\infty-\lambda)=kt$, then noting this and differentiating yields $$\label{eq:3}
\underbrace{\ln\left(\frac{d\lambda}{dt}\right)}_{Y}=\underbrace{-kt}_{Mt}+\underbrace{\ln[k(\lambda_\infty- \lambda_0)]}_{C}$$ A typical plot that can extract $k$ as a linear plot of $\ln(d\lambda/dt)$ vs $t$ is given in Fig.(\[fig:3\]). Linearity is observed for $npoly=2$ and smooth curves without oscillations for at least $npoly\leq3$.
![Method 2 where smooth curves are obtained for at least $npoly<4$ []{data-label="fig:3"}](m2p235.eps){width="7cm"}
1.3 Other methods and considerations
------------------------------------
A variant method similar to the Guggenheim method [@gug1] of elimination is given below but where gradients to the conductivity curve is required, and where the average over all pairs is required. $$\label{eq:4}
\left\langle k\right\rangle=\frac{-2}{N(N-1)}\sum_{i}^{N}\sum_{j>i}^N\ln\left(\lambda'(t_i)/\lambda'(t_j)\right)/
(t_i-t_j)$$ It was discovered that the normal least squares polynomial method using Gaussian elimination [@yak1 Sec.6.2.4,p.318 ]to derive the coefficients of the polynomial was highly unstable for $npoly>4$ and so for this work, we used a variant of the Orthogonal method modified for determination of differentials. The normal method defines the nth order polynomial $p_n(t)$ which is then expressed as a sum of square terms over the domain of measurement to yield $Q$ in eqns(\[eq:5\]). $$\label{eq:5}
\begin{array}{rcl}
p_n(t) &=& \sum_{j=0}^{n}h_it^j\\
Q(f,p_n) &=& \sum_{i=1}^N\left[f_i-p_n(t_i)\right]^2
\end{array}$$ The $Q$ function is minimized over the polynomial coefficient space. In the Orthogonal method adopted here, we express our polynomial expression $p_m(t)$ linearly in coefficients $a_j$ of $\varphi_j$ functions that are orthogonal with respect to an [*inner*]{} product definition. For arbitrary functions $f,g$, the inner product $(f,g)$ is defined below, together with properties of the $\varphi_j$ orthogonal polynomials.
$$\label{eq:6}
\begin{array}{rcl}
(f,g) &=& \sum_{k=1}^{N} f(t_k).g(t_k)\\
(\varphi_i,\varphi_j)=0 &(i\neq j);& \,\, \mbox{and} \,\, (\varphi_i,\varphi_i)\neq 0.
\end{array}$$
$$\label{eq:7}
\begin{array}{rll}
\varphi_i(t)&=&(t-b_i)\varphi_{i-1}(t)-c_i\varphi_{i-2}(t)\,(i\geq1)\\
\varphi_0(t)&=&1,\mbox{and}\,\, \varphi_j=0 \,\,\, j<1,\\
b_i &=& (t\varphi_{i-1},\varphi_{i-1})/( \varphi_{i-1}, \varphi_{i-1}) \,\, (i\geq1) \\
c_i &=&(t\varphi_{i-1},\varphi_{i-2})/( \varphi_{i-2}, \varphi_{i-2}) \,\, (i\geq2), c_i=0.
\end{array}$$
We define the $m^{th}$ order polynomial and associated $a_j$ coefficients as:
$$\label{eq:8}
\begin{array}{rll}
p_m(t)&=& \sum_{j=0}^m a_j\varphi_{j}(t)\\[0.5cm]
a_j &=&(f, \varphi_{j})/(\varphi_{j}, \varphi_{j}), (j=0,1,\ldots m) \\
\end{array}$$
The recursive definitions for the first and second derivatives are given respectively as: $$\label{eq:9}
\begin{array}{rll}
\varphi^{\prime}_i(t)&=&\varphi^{\prime}_{i-1}(t) (t-b_i)+\varphi_{i-1}(t)-c_i\varphi^{\prime}_{i-2}(t)\,(i\geq1)\\
\varphi^{\prime\prime}_i(t)&=&\varphi^{\prime\prime}_{i-1}(t) (t-b_i)+2\varphi_{i-1}^{\prime}(t)-c_i\varphi^{\prime\prime}_{i-2}(t)\,(i\geq2)
\end{array}$$ Here the codes were developed in C/C++ which provides for recursive functions which we exploited for the evaluation of all the terms. The experimental data were fitted to an $m^{th}$ order expression $\lambda_m(t)$ defined below $$\label{eq:10}
\lambda_m(t) = \sum_{j=0}^{n}h_it^j\\$$ Figure(\[fig:4\])are plots for the different polynomial orders n. The orthogonal polynomial method is stable and the mean square error decreases with higher polynomial order (for the 36 data points) monotonically, but the differentials are not so stable, as shown in the previous figures.
![Plot using orthogonal polynomials for various orders $n$[]{data-label="fig:4"}](phimv2.eps){width="7cm"}
Differentiating (\[eq:3\]) for constant $k$ leads to (\[eq:11\]) expressed in two ways $$\label{eq:11}
\frac{d^2 \lambda}{dt^2}=-k\left(\frac{d\lambda}{dt}\right)\,(a) \,\mbox{or}\,
k=-\frac{d^2 \lambda}{dt^2}/\left(\frac{d\lambda}{dt}\right)\,(b)$$
Eq.(\[eq:11\](b)) suggests another way of computing $k$ for “well-behaved” values of the differentials, meaning regions where $k$ would appear to be a reasonable constant. The (a) form suggests an exponential solution. Define $\frac{d\lambda}{dt}\equiv dl$ and $\frac{d^2\lambda}{dt^2}\equiv d2l$. Then $dl(t)=A\exp(-kt)$ and $dl(0)=A=h_2$ from (\[eq:10\]).Furthermore, as $t\rightarrow 0$, $k=\left(-2h_2/h_1\right)$ and a global definition of the rate constant becomes possible based on the total system $\lambda(t)$ curve.
With a slight change of notation, we now define $dl$ and $d2l$ as referring to the continuous functions $dl(t)=A\exp(-kt)$ and $d2l(t)=-kA\exp(-kt)$ and we consider $(d\lambda/dt)$ and $d^2\lambda/dt^2 $ to belong to the values (\[eq:10\]) derived from ls fitting where $(d\lambda/dt)=\lambda_m^{\prime}$, $(d^2\lambda/dt^2)=\lambda_m^{\prime\prime}$ which are the experimental values for a curve fit of order $m$. From the experimentally derived gradients and differentials, we can define two non-negative functions $R_a(k)$ and $R_b(k)$ as below: $$\label{eq:12}
\begin{array}{rll}
R_a(k)&=&\sum^N_{i=1}\left(\frac{d^2\lambda(t_i)}{dt^2}+kdl(t_i)\right)^2\\[.3cm]
R_b(k)&=&\sum^N_{i=1}\left(\frac{d\lambda(t_i)}{dt}-dl(t_i)\right)^2\\
&\mbox{where}& \\
f_a(k)=R^{\prime}_a(k) &\mbox{and}& f_b(k)=R^{\prime}_b(k)
\end{array}$$ and a minimum exists at $f_a(k)=f_b(k)=0.$ We solve the equations $f_a$ , $f_b$ for their roots in k using the Newton-Raphson method and compute the rate constant $k$. The error threshold in the Newton-Raphson method was set at $\epsilon=1.0\times10^{-7}$ We provide a series of data of the form $\left[n,A,k_a,k_b,\lambda_{a,\infty},\lambda_{b,\infty}\right]$ where $n$ refers to the polynomial degree, $A$ the initial value constant as above, $k_a$ and $k_b$ is the rate constant for function $f_a$ and $f_b$ (solved when the functions are zero respectively ) and likewise for $\lambda_{a,\infty}$ and $\lambda_{b,\infty}$. The $e$ symbol refers to base $10$ (decimal) exponents. The $\lambda_{\infty}$ values are averaged over all the (36) data points from the equation $$\label{eq:12a}
\lambda_{\infty}=\frac{d\lambda(t)}{dt}\frac{1}{k}+ \lambda(t)$$ The results are as follows:\
$\left[2, 3.7634e0, 3.2876e-3, 3.2967e-3, 1.1506e3, 1.1477e3 \right]$,\
$\left[3, 3.6745e0, 2.7537e-3, 2.7849e-3, 1.34756e3,1.3334e3\right]$,\
$\left[4, 3.6380e0, 2.0973e-3, 2.4716e-3, 1.7408e3, 1.4900e3\right]$,\
$\left[5, 4.0210e0, 9.7622e-3, 4.9932e-3, 4.4709e2, 7.9328e2,\right]$,\
$\left[6, 4.5260e0, 4.1270e-2, 8.9257e-3, 1.7101e2, 4.8403e2\right]$.\
We noticed as in the previous cases that the most linear values occur for $1<n<4$. In this approach, we can use the $f_a$ and $f_b$ function similarity of solution for $k$ to determine the appropriate regime for a reasonable solution. Here, we notice a sudden departure of similar value between $k_a$ and $k_b$ (about 0.4 difference ) at $n=4$ and so we conclude that the probable “rate constant” is about the range given by the values spanning $n=2$ and $n=3$. Interestingly, the $\lambda_{\infty}$ values are approximately similar to the ones for method 1 and 2 for polynomial evaluation 2 and 3 for those methods. More study with reliable data needs to be done in order to discern and select appropriate criteria that can be applied to these non-linear methods.
1.4 Evidence of varying kinetic coefficient $k$
-----------------------------------------------
Finally, what of direct methods that do not assume the constancy of $k$ which was the case in the above subsections? Under the linearity assumption $x=\alpha\lambda(t)$, the rate law has the form $dc/dt=-k(t)(a-x)$ where $k(t)$ is the instantaneous rate constant and this form implies $$\label{eq:13}
k(t)=\frac{d\lambda/dt}{\lambda_{\infty}-\lambda(t)}$$ If $\lambda_\infty$ is known from accurate experiments or from our computed estimates, then $k(t)$ is determined; the variation of $k(t)$ provides crucial information concerning reaction kinetic mechanism and energetics, from at least one theory recently developed for elementary reactions [@cgj1] and for such theories and developments, it may be anticipated that nonlinear methods would be used to accurately determine $k(t)$ that would yield the so-called “reactivity coefficients” [@cgj1] that account for variations in $k$ that would provide fundamental information concerning activation and free energy changes.
![Variation of $k$ with time or concentration changes based on the experimental value $\lambda_{\infty}=2050 \text{units}$ and the computations based on different polynomial degrees $n=2,3,4$ and the computed $\lambda_{\infty}$ values for Method 1 and Method 2 for fixed polynomial degree $n=3$.[]{data-label="fig:5"}](m3ratevarytot.eps){width="7cm"}
Figure(\[fig:5\]) refers to the computations under the assumption of first order linearity of concentration and the conductivity. Whilst very preliminary, non-constancy of the rate constants are evident, and one can therefore expect that another area of fruitful experimental and theoretical development can be expected from these results.
2. CONCLUSIONS
==============
The results presented here provides alternative developments based on NLA that is able to probe into the finer details of kinetic phenomena than what the standard representations allow for, especially in the the areas of changes of the rate constant with the reaction environment. Such studies would involve building up another set of axioms that is consistent with a varying $k(t)$ kinetic coefficient. Even with the assumption of invariance of $k$, one can always choose the best type of polynomial order that is consistent with the assumption, and it appears that the initial concentration as well as the rate constant seems be be predicted as global properties based on the polynomial expansion.
ACKNOWLEDGMENTS
===============
This work was supported by Science Faculty conference allocation and grants UMRG(RG077/09AFR), FRGS(FP037/2008C) and PJP(FP037/2007C) of the Malaysian Government.\
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E. A. Guggenheim, On the determination of the velocity constant of a unimolecular reaction, [*Philos Mag J Sci*]{} [**2**]{} (1926) 538–543 .
C. G. Jesudason, The form of the rate constant for elementary reactions at equilibrium from MD: framework and proposals for thermokinetics, [*J. Math . Chem*]{} [**43**]{} (2008) 976–1023.
S. Yakowitz and F. Szidarovsky,[*An Introduction to Numerical Computations*]{}, Maxwell Macmillan, New York, 1990
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---
abstract: 'Supernova explosions and their remnants (SNRs) drive important feedback mechanisms that impact considerably the galaxies that host them. Then, the knowledge of the SNRs evolution is of paramount importance in the understanding of the structure of the interstellar medium (ISM) and the formation and evolution of galaxies. Here we study the evolution of SNRs in homogeneous ambient media from the initial, ejecta-dominated phase, to the final, momentum-dominated stage. The numerical model is based on the Thin-Shell approximation and takes into account the configuration of the ejected gas and radiative cooling. It accurately reproduces well known analytic and numerical results and allows one to study the SNR evolution in ambient media with a wide range of densities $n_{0}$. It is shown that in the high density cases, strong radiative cooling alters noticeably the shock dynamics and inhibits the Sedov-Taylor stage, thus limiting significantly the feedback that SNRs provide to such environments. For $n_{0}>5 \times 10^{5}$ cm$^{-3}$, the reverse shock does not reach the center of the explosion due to the rapid fall of the thermal pressure in the shocked gas caused by strong radiative cooling.'
author:
- |
Santiago Jiménez,[^1] Guillermo Tenorio-Tagle, Sergiy Silich\
Instituto Nacional de Astrofísica, Óptica y Electrónica, AP 51, 72000 Puebla, México\
bibliography:
- 'ref.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: The full evolution of supernova remnants in low and high density ambient media
---
\[firstpage\]
Shock waves – ISM: evolution – ISM: Supernova Remnants–ISM: Kinematics and Dynamics.
Introduction
============
Supernova Remnants (SNRs) are powerful sources of mass, momentum and energy. They shape the interstellar medium (ISM) of their host galaxies, determine the evolution of the ISM chemical composition and are sources of cosmic rays, radio and X-ray emission [e.g. @mckee1977theory; @tenorio2015supernovae; @2015MNRAS.451.2757W; @elmegreen2017globular]. It has also been suggested that SNRs are effective dust producers (e.g. @2001Todini, @dust1, @Bianchi2007, @dust2, @Micelotta2016).\
{width="1.55\columnwidth"}
The SNRs undergo several evolutionary stages when the progenitor star explodes in an uniform density media [@1977Chevalier]. The first stage, known as the ejecta-dominated (ED) or thermalization phase, begins when the high velocity supernova ejecta collides with the ambient gas and forms a leading shock. The large thermal pressure behind this shock leads to the formation of another, reverse shock, which decelerates and thermalizes the ejected matter. At this stage, the velocity and density structure of the ejecta strongly affects the dynamics of the SNR (e.g. @draine1993theory, @2017Tang).\
In low density media, after the reverse shock reaches the center of the explosion and the thermalization process is terminated, the adiabatic Sedov-Taylor (ST) stage begins (e.g @sedov1993similarity, @astrowaves, @bisno1). This stage is described by a self-similar hydrodynamic solution: the shock radius and velocity are given by power-law functions of time ($R\propto t^{2/5}$, $V\propto t^{-3/5}$ ) and the kinetic ($E_{k}$) and thermal ($E_{th}$) energies are conserved ($E_{k}\approx0.3 E_{0}$ and $E_{th}\approx0.7 E_{0}$, where $E_{0}$ is the explosion energy, see @sedov1993similarity). This solution has been widely used as initial condition in many SNR evolution models [e.g. @1975MNRAS.172...55F; @KimOstriker], neglecting thus the thermalization phase and assuming that all remnants enter the ST stage. This, as shown here, is not always the case.\
In homogeneous media, the leading shock slows down with time during the ST stage such that the post-shock temperature reaches values close to the maximum of the cooling function (e.g. @Raymond1976, @Wiersma2009, @Schure2009). Therefore, at late times radiative cooling becomes important. When this occurs, the remnant enters the snowplough (SP) phase (e.g. @CioffiMckee, @Blondin1998, @mihalas2013foundations). At this stage, a very thin, cold and dense shell is formed at the outer edge of the SNR. In order to preserve pressure, the density increases in response to the sudden fall of the post-shock temperature.\
Throughout the course of the SP stage, a SNR loses most of its thermal energy. The remnant then moves for a while in the momentum conserving stage (MCS) and finally merges with the surrounding medium when its expansion velocity becomes comparable to the sound speed in the ambient gas.\
The evolutionary tracks described above reproduce the physical conditions in many SNRs (e.g @2000ApJ...528L.109H, @slane2000chandra, @borkowski2001supernova, @Laming2003). However, the standard theory does not consider the radiative losses of energy at the early ED-phase which, as we show bellow, dominate the SNR evolution when the explosion occurs in a high density medium. Indeed, [@terlevich1992starburst] considered a particular case of an ambient gas with density $n_{0}=10^{7}$ cm$^{-3}$, and found that the SNR does not reach the ST stage in such a case.\
Here we present a model based on the Thin-shell approximation [e.g. @astrowaves; @silich1992; @bisno1] that allows one to follow the full evolution of SNRs, i.e., the thermalization of the SN ejecta, the Sedov-Taylor and the Snowplough stages. The numerical scheme includes both the initial distribution of density and velocity in the ejecta and radiative cooling of the shocked gas. Our results are tested against well known numerical and analytic results. This model allows us to study the SNRs evolution for a wide range of ambient gas densities ($1$ cm$^{-3}$ $\leq n_{0} \leq$ $10^{7}$ cm$^{-3}$).\
The paper is organized as follows. In section \[sec2\], we introduce the numerical model: the equations of motion and energy conservation and the set of initial conditions. In Section \[sec3\], we compare our results with previous numerical and analytic results. Section \[sec4\] discusses the impact of the ambient gas density on the evolution of SNRs. It is shown that the Sedov-Taylor stage does not occur for SNRs that evolve in densities larger than $5 \times 10^{5}$ cm$^{-3}$ and that scaling relations for the SNRs evolution are not applicable for ambient gas densities above this value. The main differences with the standard evolutionary tracks are addressed including those from high density runs with different metallicities. Finally, Section \[sec5\] summarizes and discusses our main findings.
Model Set-up {#sec2}
============
We model the evolution of a SNR in a homogeneous medium with a number density $n_{0}$ from the Ejecta-Dominated to the Snowplough stages. Fig. \[fig1\] presents a schematic illustration of the initial condition (left panel) and the resultant SNR structure (right panel), which inside-out presents: the free ejecta, the shocked ejecta and the shocked ambient gas, with kinetic energies $E_{k,free}$, $E_{k,ej}$ and $E_{k,ism}$, respectively. The two outer zones of shocked gas are separated by a contact discontinuity $R_{CD}$. The shocked gas (A and B) loses its thermal energy $E_{th}$ due to radiative cooling. The instabilities of the gas flow are not considered here and therefore no mass traverses the contact discontinuity.
The mass and momentum conservation equations {#massConserva}
--------------------------------------------
The evolution of the leading shock is determined by the mass and momentum conservation equations (see @bisno1 and references therein for a discussion on the Thin-Shell approximation), which in the adiabatic case are: $$\label{eq:1}
\frac{dM_{s1}}{dt}= \rho_{0} U_{LS} 4 \pi R_{LS}^{2},$$ $$\label{eq:2}
\frac{d}{dt}\left(M_{s1}U_{s1} \right)=4 \pi P R_{LS}^{2},$$ $$\label{eq:3}
\frac{d R_{LS}}{dt}=U_{LS},$$ where: $$\label{eq:3a}
U_{LS}=\frac{\gamma+1}{2}U_{s1}.$$ In these equations, $R_{LS}$ and $U_{LS}$ are the leading shock radius and velocity, $M_{s1}$ and $U_{s1}$ are the mass and velocity of the swept-up ambient gas, $\gamma=5/3$ is the specific heats ratio, $\rho_{0}=\mu n_{0}$ is the ambient gas density, $\mu=14/11 m_{H}$ is the mean mass per particle in the neutral gas with 10 hydrogen atoms per helium atom and $P$ is the thermal pressure of the shocked ambient gas.\
Note that as the swept-up gas cools down, the remnant enters to the SP stage, and in such case equation (\[eq:3a\]) becomes $U_{LS}=U_{S1}$. The transition to this phase occurs at the *thin shell-formation time* $t_{sf}$, which is the time when the swept-up gas begins to collapse into a cold, dense shell (e.g. @CioffiMckee). If an element of gas is shocked at time $t$, it cools at: $$t_{c}=t+\Delta t_{cool} \left(t \right),$$ where $\Delta t_{cool} \left(t \right) $ is the gas cooling time (e.g. @Petruk2006 [@KimOstriker]): $$\Delta t_{cool} \left(t \right) =\frac{1}{\gamma+1}\frac{k_{B}T_{LS}}{n_{0}\Lambda\left(T_{LS}\right)}.
\label{coolA}$$ In this expression, $k_{B}$ is the Boltzmann constant, $T_{LS}$ is the post-shock temperature at the leading shock (calculated by means of the Rankine-Hugoniot relations) and $\Lambda$ is the cooling function for a gas in collisional ionization equilibrium (CIE). In our simulations, $t_{c}$ is calculated at each time-step and the minimum $t_{min}$ is determined: $$t_{min}=min \left(t_{c}\left(t \right), t_{c}\left(t+\Delta t \right), \ldots \right).
\label{shellftime}$$ The transition time is $t_{sf}=t_{min}$.\
The evolution of the reverse shock position $R_{RS}$ is calculated as: $$\label{eq:4}
\frac{d R_{RS}}{dt}=\frac{R_{RS}}{t}-\tilde{V}_{RS},$$ where $\tilde{V}_{RS}$ is the reverse shock velocity in the frame of the unshocked ejecta. From the Rankine-Hugoniot relations (e.g. @1995PhRMckee): $$\label{eq:5}
\tilde{V}_{RS}^{2}=\frac{\gamma+1}{2}\frac{P_{RS}\left(R_{RS},t \right)}{\rho_{ej}\left(R_{RS},t \right)},$$ where $P_{RS}\left(R_{RS},t \right)$ is the gas pressure just behind the reverse shock in zone B (Fig. \[fig1\]) and $\rho_{ej}\left(R_{RS},t \right)$ is the density of the unshocked ejecta in front of the reverse shock (zone C in Fig. \[fig1\]).\
Several calculations (e.g. @gull1973numerical, @gull1975x, @chevalier1982radio, @hamilton1984new, @Silich2018) have shown that the thermal pressure of the shocked ejecta and the shocked ambient gas in zones B and A rapidly becomes almost homogeneous but presents a sharp fall just behind the reverse shock. Therefore, $P_{RS}$ is smaller than the average pressure of the shocked gas in zones A and B: $P_{RS}<P$. Thus, the pressure ratio $\phi=P_{RS}/P<1$ [@1999Mckee hereafter ]. For example, $\phi=0.3$ for both steep power-law ejecta [@chevalier1982self] and uniform ejecta [@hamilton1984new]. As it is shown in Appendix \[Ap2\], $\phi \left(t_{0} \right)\approx 0.3$ also for the fiducial initial conditions adopted here. Moreover, as the reverse shock approaches the center of the explosion, $\phi$ also reaches values close to 0.3 [@Gaffet1]. Numerical simulations also show that $\phi$ slowly changes with time (e.g. @Fabian). Therefore, hereafter the thermal pressure $P$ between $R_{LS}$ and $R_{RS}$ is assumed to be uniform but drops rapidly nears the reverse shock such that $\phi=0.3$.
The evolution of the remnant energies
-------------------------------------
In order to solve equations (\[eq:1\]-\[eq:4\]), one needs to know the thermal pressure $P$, which is calculated here by means of the energy conservation equation. It is assumed that the ejecta density distribution is given by a power-law of index $n<3$ and that the ejected gas freely expands. Hence, the kinetic energy of the free ejecta is readily integrated (see Appendix \[Ap1\]): $$\label{eq:7}
E_{k,free}=\frac{1}{2}M_{ej}V_{ej}^{2} \left(\frac{3-n}{5-n} \right)\left(\frac{R_{RS}}{t V_{ej}} \right)^{5-n},$$ where $M_{ej}$ and $V_{ej}$ are the ejecta mass and the maximum expansion velocity, respectively. The kinetic energy of the swept-up ambient gas is: $$\label{eq:8}
E_{k,ism}=\frac{1}{2}M_{s1}U_{s1}^{2}.$$ It is assumed that the shocked ejecta moves with the same velocity as the swept-up ambient gas, therefore: $$\label{eq:9}
E_{k,ej}=\frac{1}{2}M_{s2}U_{s1}^{2},$$ where $M_{s2}$ is the mass of the thermalized ejecta (see Appendix \[Ap1\]): $$\label{eq:10}
M_{s2}=M_{ej}\left[1-\left(\frac{R_{RS}}{V_{ej}t} \right)^{3-n} \right].$$ The energy conservation equation reads as: $$\label{eq:11a}
E_{0}=E_{th}+E_{k,free}+E_{k,ej}+E_{k,ism}+E_{rad1}+E_{rad2},$$ where $E_{rad1}$ and $E_{rad2}$ are energies lost by radiation at the outer and inner shells, respectively. Equation (\[eq:11a\]) can be written as a differential equation for the thermal energy: $$\label{eq:11}
\frac{d E_{th}}{dt}=-\frac{d E_{k,free}}{dt}-\frac{d E_{k,ej}}{dt}-\frac{d E_{k,ism}}{dt}-Q_{1}-Q_{2},$$ where $Q_{1}$ and $Q_{2}$ are the cooling rates in the shocked ambient gas and the shocked ejecta, respectively: $$\label{eq:12}
Q_{1} = \left\lbrace
\begin{array}{ll}
n_{s1}^{2}\Lambda \left(T_{LS} \right)\Omega_{s1}, &\text{if} \hspace{0.3cm} t \leq t_{sf}, \\
4 \pi R_{LS}^{2}U_{LS}P-\frac{dE_{k,sim}}{dt}, & \text{if} \hspace{0.3cm} t >t_{sf},
\end{array}
\right.$$ $$\label{eq:15}
Q_{2}= n_{s2}^{2}\Lambda \left(T_{RS} \right) \Omega_{s2}.$$ The terms $n_{s1}$, $n_{s2}$ and $\Omega_{s1}$, $\Omega_{s2}$ in equations (\[eq:12\]-\[eq:15\]) are the densities and the volumes occupied by the shocked ambient gas and the shocked ejecta: $$\label{eq:13}
n_{s1} = \frac{\rho_{s1}}{\mu m_{H}}=\frac{P}{k_{B}T_{LS}m_{H}},$$ $$\label{eq:14}
\Omega_{s1} = \frac{M_{s1}}{\rho_{s1}}.$$ $$\label{eq:16}
n_{s2}=\frac{P_{RS}}{k_{B}T_{RS}m_{H}},$$
$$\label{eq:17}
\Omega_{s2}=\frac{M_{s2}}{\rho_{s2}}, \hspace{0.5cm} \rho_{s2}=\mu m_{H} n_{s2}.$$
In these equations, $T_{RS}$ is the post-shock temperature at the reverse shock.\
The time derivatives of the kinetic energies are calculated with equations (\[eq:7\]-\[eq:10\]). Indeed, from equations (\[eq:8\]-\[eq:9\]), and by making use of equations (\[eq:2\]) and (\[eq:4\]) one can obtain: $$\frac{d E_{k,ism}}{dt}=4 \pi P R_{LS}^{2}U_{s1}-\frac{1}{2}U_{s1}^{2}\frac{d M_{s1}}{dt},$$ $$\frac{d E_{k,ej}}{dt}=\frac{U_{s1}^{2}}{2}\frac{dM_{s2}}{dt}+\frac{M_{s2}}{M_{s1}}U_{s1}\left[\frac{d E_{k,ism}}{dt}-\frac{U_{s1}^{2}}{2}\frac{dM_{s1}}{dt} \right],$$ where: $$\frac{dM_{s2}}{dt}=\left(3-n \right) \frac{M_{ej}}{V_{ej}t} \left( \frac{R_{RS}}{V_{ej}t}\right)^{2-n}\tilde{V}_{RS}.$$ From equations (\[eq:7\]) and (\[eq:4\]): $$\frac{d E_{k,free}}{dt}=-\frac{\left(3-n\right)}{2v_{ej}t}M_{ej}V_{ej}^{2}\tilde{V}_{RS}\left(\frac{R_{RS}}{V_{ej}t} \right)^{4-n}.$$ The thermal pressure is then calculated as: $$\label{eq:18}
P=\left(\gamma-1 \right) \frac{E_{th}}{\Omega_{LS}-\Omega_{RS}},$$ where $\Omega_{LS}$ and $\Omega_{RS}$ are the volumes encompassed by the leading and the reverse shock, respectively.
The Initial Conditions {#initialc}
----------------------
The values of $E_{0}$, $M_{ej}$ and $n$ are set at the initial time $t_{0}$. Then, $V_{ej}$ is obtained from equation (\[eq:A4\]). At $t_{0}$, a small fraction $\beta$ (usually $\beta <5 \%$) of the energy $E_{0}$ is assumed to be already transformed into $E_{k,ism}$, $E_{th}$ and $E_{k,ej}$, i.e: $$\label{eq:19a}
E_{k,free}\left(R_{RS}\left(t_{0} \right), t_{0} \right)=\left(1-\beta \right)E_{0}.$$
![The evolution of the shocks radii for a SNR with $E_{0}=10^{51}$ erg, $n_{0}=1$ cm$^{-3}$, $M_{ej}=3 M_{\odot}$ and index $n=2$. The top panel presents the case of the leading shock $R_{LS}$ and the bottom panel the reverse shock $R_{RS}$. The solid lines show our results and the dashed and dotted lines are the analytic and numerical radii obtained by , respectively. The starred variables at the axis are dimensionless variables as defined in . []{data-label="fig2"}](fig2){width="\columnwidth"}
Here, we show that the parameters $E_{0}$, $M_{ej}$, $n$, and $\beta$ define the initial conditions for the further remnant evolution.\
At the initial position of the reverse shock $R_{RS}\left(t_{0} \right)$, the free ejecta velocity is: $$V_{0}=\frac{R_{RS}\left( t_{0}\right)}{t_{0}}=V_{RS}\left(t_{0} \right).$$ In order to determine $V_{0}$, one can make use of the equations (\[eq:7\]) and (\[eq:19a\]): $$\label{eq:19}
\left(1-\beta \right)E_{0}=\frac{1}{2}M_{ej}V_{ej}^{2} \left(\frac{3-n}{5-n} \right)\left(\frac{V_{0}}{V_{ej}} \right)^{5-n}.$$ This equation together with equation (\[eq:A4\]) from Appendix \[Ap1\] yield: $$\label{eq:20}
V_{0}=V_{ej}\left(1-\beta \right)^{1/\left(5-n \right)},$$ where $V_{ej}$ is the maximum expansion velocity of the ejecta. Following [@chevalier1982self], [@hamilton1984new], [@Hwang2012a], we define the leading factor $l_{ED}$ at $t_{0}$ as: $$\label{eq:21}
l_{ED}= \frac{R_{LS}\left( t_{0} \right)}{R_{RS}\left( t_{0} \right)}.$$ The relation between the initial leading and reverse shock radii then is: $$\label{eq:22}
R_{LS}\left( t_{0} \right)=l_{ED}R_{RS}\left( t_{0} \right).$$
{width="1.8\columnwidth"}
The position of the reverse shock is assumed to be coincident with the contact discontinuity (, see left panel of Fig. \[fig1\]). One can obtain then the value of $l_{ED}=1.1$ from the mass conservation equation. From equation (\[eq:22\]): $$\label{eq:23}
V_{LS}\left( t_{0} \right)=l_{ED}V_{0},$$ where $V_{LS}\left( t_{0} \right)$ is the velocity of the leading shock at $t_{0}$. The initial velocity of the shocked ambient gas then is: $$\label{eq:24}
U_{s1}\left(t_{0} \right)=\frac{2}{\gamma+1}l_{ED}V_{0}.$$ Finally, the shock positions are determined from equation (\[eq:22\]) and the energy conservation equation. Indeed, as $\beta$ is the fraction of the ejecta kinetic energy converted into other energies, then: $$\label{eq:25}
\beta E_{0}= E^{0}_{th}+E^{0}_{k,ej}+E^{0}_{k,ism},$$ where the terms on the right-hand side of this expression are the thermal and kinetic energies of the shocked ejecta and the shocked ambient gas at $t_{0}$: $$\label{eq:26}
E^{0}_{k,ism}= \frac{1}{2}\rho_{0}\frac{4 \pi}{3}R_{LS}^{3}\left( t_{0} \right)\left( \frac{2}{\gamma+1}l_{ED}V_{0}\right)^{2},$$ $$\label{eq:27}
E^{0}_{k,ej}=\frac{1}{2}M_{ej}\left[1-\left(\frac{V_{0}}{V_{ej}}\right)^{3-n} \right]\left( \frac{2}{\gamma+1}l_{ED}V_{0}\right)^{2},$$
$$\label{eq:28}
E^{0}_{th}=\frac{4}{\gamma-1}\frac{k_{B}\rho_{0}}{\mu} T_{LS} \frac{4\pi}{3}\left(1-\frac{1}{l_{ED}^{3}} \right)R_{LS}^{3}\left( t_{0} \right).$$
Equation (\[eq:28\]) was derived under the assumption that the post-shock density is $4 n_{0}$. The initial position for the leading shock radius $R_{LS}\left( t_{0} \right)$ is calculated from equation (\[eq:25\]) as this is the only unknown parameter in equations (\[eq:26\]-\[eq:28\]). Finally, the initial time $t_{0}$ is: $$t_{0}=\frac{R_{RS}\left(t_{0} \right)}{V_{0}},$$ where $R_{RS}\left( t_{0} \right)$ is determined by means of equation (\[eq:22\]).
{width="2\columnwidth"}
. \[fig4\]
Comparison to numerical and analytic models {#sec3}
===========================================
In order to integrate equations (\[eq:1\]-\[eq:4\]) and (\[eq:11\]), we have used the Dormand-Prince method for the eight order Runge-Kutta-Fehlberg integrator, coupled with a PI step-size control [@press2007numerical]. The cooling was included trough a table lookup/interpolation method. In all calculations, we have used the cooling function from [@Raymond1976] for a solar metallicity, unless otherwise stated.
SNR evolution in a low density ambient medium
---------------------------------------------
Fig. \[fig2\] presents the evolution of the leading (upper panel) and reverse (bottom panel) shock radii predicted by our model (solid lines) and compare them to the analytic and numerical solutions obtained by (dashed and dotted lines) in the case when $E_{0}=10^{51}$ erg, $M_{ej}=3 M\odot$, $n_{0}= 1$ cm$^{-3}$ and $n=2$. Following , the time and shock radii in Fig. \[fig2\] are presented in the dimensionless form $t^{*}=t/t_{ch}$, $R_{RS}^{*}=R_{RS}/R_{ch}$, $R_{LS}^{*}=R_{LS}/R_{ch}$, where:
$$\label{char2}
R_{ch}=M_{ej}^{1/3}\rho_{0}^{-1/3},$$
$$\label{char3}
t_{ch}=E^{-1/2}M_{ej}^{5/6}\rho_{0}^{-1/3}.$$
studied analytically and numerically the ED and the ST stages for low density media. Cooling was assumed to be negligible and therefore their model is adiabatic.\
Our results are in excellent agreement with the numerical results obtained by both for the leading (see the upper panel in Fig. \[fig2\]) and the reverse (see lower panel in Fig. \[fig2\]) shocks. Small differences between ours and numerical calculations are likely to be produced because in our calculations $\phi$ was assumed to have a constant value while in numerical calculations $\phi$ slightly changes with time. Note that the reverse shock positions obtained in all cases coincide very well at the ED stage. However, after that the analytic solution (dashed line) is not able to reproduce the correct position of the reverse shock which, as noticed by , results from a significant error in the reverse shock velocity around the Sedov-Taylor transition time $t_{ST}$.\
Fig. \[fig3\] presents the evolution of the remnant energies. The three evolutionary stages are separated by vertical lines. As one can see, during the ED phase, the kinetic energy of the free ejecta is converted into kinetic and thermal energies of the shocked gas. The dashed vertical line marks the beginning of the ST stage. At this time, the ratio of the ambient swept up mass to the ejecta mass is about 38, which is in good agreement with the results of [@gull1973numerical] and [@1990MNRASTenorio]. At this moment, the energy lost by radiation, $E_{rad}=\int \left(Q_{1}+Q_{2} \right)dt$ (dash-dotted line), is negligible, thus allowing the total kinetic and thermal energies to approximately reach constant values ($E_{k} = E_{k,free}+E_{k,ej}+E_{k,ism} \approx 0.33 E_{0}$ and $E_{th}\approx 0.66 E_{0}$). The leading shock radius then evolves as $R_{LS} \propto t^{0.39}$, which is close to the analytic solution (@sedov1993similarity).\
Note that $E_{rad}$ steadily grows several orders of magnitude to finally make an impact on the evolution terminating the ST stage. The vertical solid line in Fig. \[fig3\] marks the transition to the SP phase at $t_{sf}$ (see \[massConserva\]). For the case considered here, $t_{sf} \approx 5 \times 10^{4}$ yr, which is in agreement with recent results (e.g. @LiOstriker [@KimOstriker; @Haid2016]).\
The left panel in Fig. \[fig4\] presents the energy loss rate behind the leading ($Q_{1}$) and the reverse ($Q_{2}$) shocks. The right panel shows the reverse shock velocity $\tilde{V}_{RS}$ in the frame of the unshocked ejecta (see equation \[eq:4\]) and the velocity of the leading shock $V_{LS}$ in the rest frame, as a function of time. The vertical dashed lines on both plots indicate the thin-shell formation time. At the beginning of the evolution, the reverse shock is radiative as the ejecta density is large and the reverse shock velocity $\tilde{V}_{RS}$ is small (solid line on the right panel). However, this period is short because the shock velocity $\tilde{V}_{RS}$ grows and the ejecta density drops. Hence, the radiative losses $Q_{2}$ become negligible for most of the evolution. On the other hand, the leading shock velocity is high at early times (dotted line on the right panel). This implies that $Q_{1}$ is initially small but continuously increases as the shock slows down. When the post-shock temperature drops to values close to the maximum of the cooling function, $Q_{1}$ reaches the maximum value. Note how close this luminosity peak is to $t_{sf}$. This is in agreement with previous results by [@Thornton1998], who used the luminosity peak as the definition of $t_{sf}$.\
![The kinetic (dashed line) and thermal (solid line) energies of a SNR evolving in an ambient medium with density $n_{0}=10^{7}$ cm$^{-3}$. []{data-label="fig5"}](fig5){width="\columnwidth"}
{width="2\columnwidth"}
. \[fig6b\]
SNR evolution in a high density medium
--------------------------------------
A test case for the SNR evolution in a high density medium was presented in [@terlevich1992starburst], who discussed the SNR evolution in a $n_{0}=10^{7}$ cm$^{-3}$ ambient medium. To compare our model with these results, the velocity and density distributions of the ejected gas and the initial conditions were modified (see Appendix \[Ap3\]) to account for the initial values used by [@terlevich1992starburst]. Fig. \[fig5\] presents the evolution of the remnant energies for this case. The strong radiative cooling begins to be a dominant factor around $\approx 1 $ yr after the explosion, which is the time when the leading shock becomes radiative. This leads to the rapid remnant evolution as the thermal energy dramatically decreases in a short time interval. The fact that the remnant energies are decaying for most of the time covered by our calculations implies that the energy of the newly shocked gas is radiated very efficiently. The total kinetic and thermal energies never reach the values of $E_{k} \approx 0.33 E_{0}$ and $E_{th} \approx 0.66 E_{0}$ and therefore the Sedov-Taylor stage is inhibited and radiative cooling sets in before the thermalization is completed, in agreement with the numerical results presented by [@terlevich1992starburst].
![The thermal energy of SNRs evolving into an ISM with different densities (listed in the legend in units of cm$^{-3}$). []{data-label="fig6"}](fig7){width="\columnwidth"}
SNR Evolution in different ambient media {#sec4}
========================================
The impact of the ambient gas density {#sec4-1}
-------------------------------------
Our numerical model allows one to study the evolution of SNRs in a wide range of ambient gas densities. This is problematic if one uses full hydrodynamical codes, because such calculations require high spatial and temporal resolutions and therefore are time-consuming (e.g. @leveque2006computational). This section presents the results of simulations which were provided for ambient gas densities: $n_{0}$ $[$cm$^{-3}]=10^{2},$ $10^{3},$ $5 \times 10^{3},$ $10^{4},$ $10^{5},$ $5 \times 10^{5},$ $10^{6},$ and $10^{7}$. All calculations assume that the ejecta mass is $M_{ej}=3$ $M_{\odot}$, the total energy is $E_{0}=10^{51}$ erg and the ejecta density distribution is a power-law with index $n=2$.\
![Top panel: Fraction of the thermalized ejecta $M_{s2}/M_{ej}$ as a function of time for different values of the ambient gas density (shown in the legend). The open squares indicate the time when the leading shock becomes radiative (i.e., when $t=t_{sf}$) for each case. For low density models this occur after full thermalization of the ejecta ($M_{s2}/M_{ej}=1$), in contrast with the large density models when $M_{s2}/M_{ej}<1$ at $t_{sf}$. Bottom panel: Same as the top panel but the time coordinate is now expressed in dimensionless units (see Appendix \[Ap1\]). Unified solutions must lie over the same curve in this plot. This is the case for densities $n_{0}<5 \times 10^{5}$ cm$^{-3}$ but higher densities cases depart from this curve. []{data-label="fig6a"}](fig8){width="\columnwidth"}
{width="2\columnwidth"}
Fig. \[fig6b\] presents the shocked ejecta and the shocked ambient gas densities ($n_{s2}$ and $n_{s1}$) and cooling rates ($Q_{2}$, $Q_{1}$) for different ambient gas densities as functions of time. One can note that the initial value of $n_{s2}$ is large in all cases. However, it rapidly drops with time, slightly increasing at the end of the ED stage as the ejecta density is large near the center of the explosion. The shocked ejecta density $n_{s2}$ reaches smaller values in calculations with smaller ambient gas densities as in these cases the ejecta gas passes through the reverse shock at larger distances from the center of the explosion. The density of the shocked ambient gas $n_{s1}$ (right upper panel) is initially close to the adiabatic strong shock limit $4 n_{0}$. However, it increases orders of magnitude upon strong radiative cooling at the transition time $t_{sf}$. At final stages of the SNRs evolution, $n_{s1}$ falls because radiative cooling becomes inefficient as the leading shock decelerates.\
The left bottom panel in Fig. \[fig6b\] presents the evolution of $Q_{2}$. In large ambient gas densities strong radiative cooling sets in earlier and $Q_{2}$ reaches larger maximum values than in low ambient gas densities. This occurs because in these models the density of the shocked ejecta is larger. $Q_{1}$ presents a similar trend: it reaches larger maximum values at earlier evolutionary times in models with larger ambient gas densities (see the right bottom panel in Fig. \[fig6b\]). This is due to larger shocked ambient gas densities $n_{s1}$ (see right upper panel). Note that in the high-density models, the cooling rates $Q_{1}$ and $Q_{2}$ reach their maximum values at similar times whereas in the low density models this does not occur.\
Fig. \[fig6\] shows the evolution of the thermal energy for various values of $n_{0}$. As discussed before, in the large ambient density cases radiative cooling sets in at early stages of the SNR evolution and therefore $E_{th}$ never reaches values predicted by the ST solution.\
In low density media, thermalization is well separated from the radiative stage by the Sedov-Taylor regime. However, one can notice from our calculations that in high density media the leading shock becomes radiative before the thermalization is completed. This implies that in these cases both processes (the ejecta thermalization and the shocked gas cooling) proceed simultaneously. Indeed, the top panel of Fig. \[fig6a\] presents the fraction of thermalized ejecta $M_{s2}/M_{ej}$ as a function of time. The open squares mark this fraction at the time when the leading shock becomes radiative (i.e, $t=t_{sf}$). For densities $n_{0}<5 \times 10^{5}$ cm$^{-3}$, practically all the ejected gas is thermalized by this time, while in the larger ambient gas densities this fraction becomes progressively smaller.\
As has been shown by [@1999Mckee], , [@2009MNRAS.397.2106T], in low density media the leading and reverse shock radii and velocities scale with the input parameters $E_{0}$, $n_{0}$ and $M_{ej}$. This implies that the radii and velocities are determined by a unified dimensionless solution. Equation (\[Ap:terma\]) shows that for a given power-law index $n$, the thermalized ejecta mass $M_{s2}/M_{ej}$ is also a function of the dimensionless variables $t^{*}$ and $R^{*}_{RS}$ (). The bottom panel on Fig. \[fig6a\] presents the ratio $M_{s2}/M_{ej}$ as a function of $t^{*}$. As both axes are now dimensionless, the ratio $M_{s2}/M_{ej}$ should not change with the ambient gas density $n_{0}$ if a unified solution exists. Indeed, this is the case for low densities ($n_{0}<5 \times 10^{5}$ cm$^{-3}$) but is not true in denser ambient media. Therefore, the early radiative cooling in high density media breaks the scaling properties of the SNRs evolution.\
The radiative cooling impact the reverse shock dynamics. For low density cases, the reverse shock promptly reaches the center of the explosion (see upper panels of Fig. \[fig8\]) as the thermal pressure of the shocked gas is always larger than the ejecta ram pressure, thus allowing the SNRs to thermalize all of its ejected mass before the onset of the SP stage. However, in the high density cases, the thermal pressure drops drastically and becomes smaller than the ejecta ram pressure. Our calculations show that for densities larger than the critical density $n_{0,cri}=5 \times 10^{5}$ cm$^{-3}$, the reverse shock never reaches the center of the explosion and instead moves outwards (see bottom panels of the Fig. \[fig8\]). In these cases the leading shock decelerates rapidly and this leads to the merging of the shells of shocked ejecta and ambient gas, shortly after the explosion (see the bottom right panel of \[fig8\], where the dotted line indicates the leading shock position).\
{width="1.75\columnwidth"}
Fig. \[fig7\] presents the momentum carried by the shocked gas (i.e. the shocked ejecta and the shocked ambient gas) for several values of the ambient gas density. The horizontal line is the initial momentum of the ejecta $p_{ej}\approx 3.0\times 10^{42}$ g cm s$^{-1}$ (see Appendix \[Ap1\]). Note that for the same values of the explosion energy and the ambient gas density $n_{0}=1$ cm$^{-3}$, [@LiOstriker] obtained numerically a final momentum of $p_{s} \approx 5.0 \times 10^{43}$ g cm s$^{-1}$ while our calculations lead to $p_{s} \approx 8.8 \times 10^{43}$ g cm s$^{-1}$. Note also that in all cases the remnants momentum asymptotically approaches a constant final value $p_{f}$ at the end of the calculations. This is because the thermal pressure of the shocked gas becomes negligible due to cooling and therefore at late stages the SNR evolves in a momentum conservation regime (see equation \[eq:2\]).\
Fig. \[fig7\] also shows that the work done by the shocked gas cannot boost the momentum injected by the explosion too much ($1$ $\leq p_{s}/p_{ej} \leq$ $29$) and that the boosting factor becomes negligible for SNRs evolving in high density media as in these cases the shocked gas cools down rapidly and the SNRs evolve practically in the momentum-dominated regime. This agrees with recent results by [@2013ApJ...770...25A], [@2015MNRAS.450..504M] and [@2015MNRAS.451.2757W], who studied the SNRs evolution for the density range $1$ cm$^{-3}$ $\leq n_{0} \leq$ $100$ cm$^{-3}$.
The impact of the gas metallicity
---------------------------------
{width="2\columnwidth"}
The calculations presented in the previous sections assumed a solar metellicity for the ambient gas and for the SN ejecta. Here, models with $M_{ej}=3 M_{\odot}$, $E_{0}=10^{51}$ erg , $n=2$ and lower metallicities are discussed (see Table \[TableMetal\]). Note that model $M1$ is the one discussed in previous sections.\
Fig. \[fig7a\] presents the evolution of the reverse shock for all these models in different ambient gas densities. In the case of $n_{0}=10^{4}$ cm$^{-3}$ shown in the upper left panel, and in lower density cases, the thermalization of the ejecta gas occurs before radiative cooling becomes important and therefore the reverse shock position practically does not depend on the gas metallicity. The right upper panel shows that in the case $n_{0}=5 \times 10^{5}$ cm$^{-3}$, the reverse shock reaches the center of the explosion only in the low-metallicity models M2 and M4. However, a slight increase in $n_{0}$ to $3.5 \times 10^{6}$ cm$^{-3}$ leads in all cases, to reverse shocks unable to reach the center (see the bottom left panel in Fig. \[fig7a\]).\
Model Reference
----------------- ----------- -----------
Ejecta ISM
M1 1 1
M2 $10^{-1}$ $10^{-1}$
M3 1 $10^{-1}$
M4 $10^{-2}$ $10^{-2}$
M5 1 $10^{-2}$
: Set of models with different gas compositions. Left panel: model identifier. Second and third columns: the gas metallicity for the ejecta and the ambient gas, respectively.
\[TableMetal\]
The bottom panels in Fig. \[fig7a\] show that the variation on the gas composition may alter the shock dynamics as the cooling rates change accordingly, but the impact of the ambient gas metallicity is less important than the ambient gas density (see models M1, M3 and M5 in Fig. \[fig7a\]). This is due to the fact that free-free cooling dominates the energy losses in SNRs that evolve in large ambient gas densities. To clarify this point, Fig. \[fig7b\] shows the evolution of the leading shock velocity for a low ($n_{0}=1$ cm$^{-3}$) and a high ($n_{0}=10^{5}$ cm$^{-3}$) density media for models M1 and M4, which are the cases with the highest and lowest metallicities. Fig. \[fig7b\] also presents the post-shock temperature $T_{LS}$ at the right $y$-axis. The vertical lines indicate $t_{sf}$ for each case. In the low density $n_{0}=1$ cm$^{-3}$, the transition to the SP stage occurs when the post-shock temperatures are $4.3 \times 10^{5}$ K and $ 1.2 \times 10^{5}$ K for M1 and M4, respectively. These temperatures are well within the line-cooling regime. Note that $t_{sf}$ is considerably larger in M4 than in M1 because the shocked ambient gas requires a larger time to cool down in the lower metallicity case. For $n_{0}=10^{5}$ cm$^{-3}$, $t_{sf}$ is similar in both, M1 and M4 models. In this case, the temperatures at $t_{sf}$ are still high ($T_{LS}\approx 4.6 \times 10^{7}$ K and $T_{LS}\approx 4.0 \times 10^{7}$ K for M1 and M4, respectively). At these temperatures, the gas cools mostly due to free-free emission, which is less sensitive to the gas metallicity. Therefore SNRs in low density media cool down as a consequence of the post-shock temperatures reaching values close to the maximum of the cooling function while in high density media, SNRs cool mostly due to the $n_{shock}^{2}$ dependence.
{width="2\columnwidth"}
Summary and Discussion {#sec5}
======================
A numerical scheme based on the Thin-Shell approximation, which allows one to study the full evolution of SNRs, from the early Ejecta-dominated to the Snowplough stages, has been developed and confronted with a number of previous results. The scheme accounts for the ejecta density and velocity distributions, and for the radiative cooling of the shocked gas. The initial shock radii and velocities are obtained from the explosion parameters: ejecta mass $M_{ej}$, total energy $E_{0}$, power-law index $n$ of the ejecta density distribution and the ambient gas density $n_{0}$, once the fraction of the ejecta kinetic energy $\beta$ that has been thermalized at the initial time $t_{0}$, is selected.\
Our model was compared with several previous simulations and reproduces well the evolution of the expansion radii, the remnant energetics, and the momentum deposited by SNRs both in low and high density cases.\
Our calculations show that radiative cooling speeds up drastically the evolution of SNRs in high density media. In these cases, the thermal energy of the remnants reaches lower maximum values as one considers larger densities. This limits the SNR lifetime and the feedback that SNe provide to the ambient gas.\
It was shown that in high density cases the leading shock becomes radiative long before the thermalization of the ejecta is completed. Therefore in these cases the SNRs never reach the Sedov-Taylor stage, in contrast with the predictions of the standard theory. This implies that the Sedov-Taylor solution cannot be used as the initial condition for numerical simulations of the SNR evolution in high density cases.\
Strong radiative cooling also impacts the reverse shock dynamics. For densities $n_{0}>10^{5}$ cm$^{-3}$, the thermal pressure falls faster than the ejecta ram pressure and therefore, the reverse shock does not reach the center of the explosion and it is weaker compared to low-density cases. As a consequence, we have shown that scaling relations for the SNRs dynamical evolution are only applicable for $n_{0}<10^{5}$ cm$^{-3}$.\
The work done by the hot shocked gas increases the momentum deposited by a SN explosion into the ambient medium, although radiative cooling constrains the boosting factor. The lowest boosting factor ($p_{s}/p_{ej} \approx 1$), which corresponds to the highest density medium, implies that in this case the SNR evolves in a momentum conservation regime during most of its evolution, and that the Sedov-Taylor stage is inhibited.\
The impact of the gas metallicity on the evolution of SNRs was also addressed. Several models with sub-solar compositions for the ejecta and the ambient gas were discussed. In low-density media, the SP stage begins when the post-shock temperatures reach values close to the maximum of the cooling function and therefore the onset of the SP stage depends on the gas metallicity. In high density models, however, the impact of the ambient gas metallicity is small as in these cases most of the energy is radiated away in the free-free emission regime. Hence, regardless of the metallicity, the density $n_{0,cri} \approx 5 \times 10^{5}$ cm$^{-3}$ is still the approximate value that inhibits the Sedov-Taylor phase.\
The results obtained here can contribute to the understanding of dust formation and evolution at early stages of the Universe, when most of the dust grains are expected to be formed in the supernovae ejecta (e.g. @2001Todini, @2006Marchenko, @Bianchi2007, @2013ApJ...778..159T). Indeed, recent calculations suggest that in low ambient gas densities ($n_{0}<100$ cm$^{-3}$), just a small fraction (about $10\%-20 \% $) of dust grains could survive crossing the reverse shock (e.g. @Bianchi2007, @Micelotta2016) due to thermal sputtering (e.g. @1979ApJ...231...77D, @2017MNRAS.468.1505M). Our results suggest that a larger fraction of dust grains could survive in SNRs which evolve in high density media as in these cases the reverse shock is weak and the post-shock temperature drops in short timescales, limiting the window of opportunity for thermal sputtering. Therefore it is likely that SN explosions in high density media may explain high redshift objects with such a large amount of dust (e.g. @2006Hines, @2014MNRAS.441.1040R, , @2016ApJ...816...39M).
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to our anonymous referee for multiple comments and suggestions which have greatly improved the presentation of our results. This study was supported by CONACYT-México research grant A1-S-28458. SJ acknowledge the support by CONACYT-México (scholarship registration number 613136) and by the Sistema Nacional de Investigadores (SNI), through its program of research assistants (grant number 620/2018).
The initial configuration of the ejected gas {#Ap1}
============================================
The ejecta is considered to have a negligible thermal pressure and to be freely expanding. Its velocity then is (): $$\label{eq:A1}
V\left(R,t\right)=\left\{
\begin{array}{ll}
\frac{R}{t} & \mbox{if } R \leq R_{ej},\\
0 & \mbox{if } R > R_{ej},
\end{array}
\right.$$ where $R_{ej}=V_{ej}t$. The initial ejecta mass density is assumed to be: $$\label{eq:A2}
\rho_{ej}\left(R,t \right)=\frac{M_{ej}}{V_{ej}^{3}}f_{n} \left(\frac{V}{V_{ej}} \right)^{-n} t^{-3},$$ where $M_{ej}$ and $V_{ej}$ are the ejecta mass and the free-expansion velocity, respectively. $f_{n}$ is a parameter determined by continuity and mass normalization (). As we are considering cases with $n<3$: $$\label{eq:A3}
f_{n}=\frac{3-n}{4 \pi}, \hspace{0.5cm} n<3.$$ The kinetic energy of the ejecta enclosed by the reverse shock, i.e., the energy of the free ejecta, is: $$\label{eq:A3a}
E_{k,free}=2 \pi \int_{0}^{R_{RS}} \rho_{ej}\left(R,t \right) V^{2} R^{2} dR.$$ Substituting equation (\[eq:A2\]) into (\[eq:A3a\]): $$E_{k,free}=\frac{1}{2}M_{ej}v_{ej}^{2} \left(\frac{3-n}{5-n} \right)\left(\frac{R_{RS}}{t V_{ej}} \right)^{5-n}.$$ The explosion releases a total energy $E_{0}$, which is assumed to be all as kinetic energy of the ejected gas, hence: $$\label{eq:A4}
\frac{E_{0}}{\left(1/2 \right)M_{ej}V_{ej}^{2}}=\frac{3-n}{5-n}, \hspace{0.5cm} n<3.$$ The independent parameters are $E_{0}$ and $M_{ej} $. The velocity $V_{ej}$ is calculated from equation (\[eq:A4\]).\
The ejecta mass enclosed by the reverse shock is: $$M_{free}=4 \pi \int_{0}^{R_{RS}} \rho_{ej}\left(R,t \right)R^{2} dR=M_{ej}\left(\frac{R_{RS}}{V_{ej}t} \right)^{3-n}.$$ Therefore, the thermalized ejecta mass is: $$M_{s2}=M_{ej}\left[1- \left(\frac{R_{RS}}{V_{ej}t} \right)^{3-n}\right].$$ This equation can be written in a dimensionless form by making use of equations (\[char1\]-\[char3\]) and equation (\[eq:A4\]): $$\label{Ap:terma}
M^{*}_{s2}=1-\left[2 \left(\frac{5-n}{3-n} \right) \right]^{\frac{3-n}{2} }\left(\frac{R^{*}_{RS}}{t^{*}} \right)^{3-n}.$$ Finally, the momentum carried by the free ejecta is: $$p_{free}=4 \pi \int_{0}^{R_{RS}} \rho_{ej}\left(R,t \right) V R^{2} dR,$$ hence, the total momentum $p_{ej}$ injected by the SN explosion is: $$p_{ej}=\left( \frac{3-n}{4-n}\right) M_{ej}V_{ej}.$$
The pressure ratio at the beginning of the ED stage {#Ap2}
===================================================
Here, the pressure ratio $\phi$ is estimated at the early ejecta-dominated phase. The pressure gradient between the leading shock $P_{LS}$ and the reverse shock $P_{RS}$ can be estimated from the stationary Euler equation: $$\frac{dP}{dr}= -\rho u \frac{du}{dr}.$$ At the initial time $t_{0}$: $$P_{RS}- P_{LS}\approx -\rho_{LS}U_{LS} \left( U_{RS}-U_{LS}\right),
\label{ap1a}$$ where $\rho_{LS}$ is the density behind the leading shock, $U_{LS}$ and $U_{RS}$ are the gas velocities behind the leading and the reverse shock in the rest frame, respectively. The ratio of the gas pressures $\phi$ then is: $$\phi=\frac{P_{RS}}{P_{LS}}= 1-\frac{\rho_{LS}U_{LS}}{P_{LS}}\left(U_{RS}-U_{LS} \right).
\label{ap2a}$$ From the Rankine-Hugoniot conditions: $$P_{LS}=\frac{\gamma+1}{2}\rho_{0} U_{LS}^{2}, \hspace{0.5cm}\rho_{LS}=\frac{\gamma+1}{\gamma-1}\rho_{0},
\label{ap3a}$$ where $\rho_{0}$ is the density of the ambient medium. Substituting equation (\[ap3a\]) into (\[ap2a\]): $$\phi= 1-\frac{2}{\gamma-1}\left(\frac{U_{RS}}{U_{LS}}-1 \right).
\label{ap4a}$$ The post-shock velocities are: $$U_{RS}=\frac{2}{\gamma+1}V_{RS}+\frac{\gamma-1}{\gamma+1}\frac{R_{RS}}{t},
\label{apen1a}$$ where $V_{RS}$ and $R_{RS}$ are the velocity and position of the reverse shock. At $t_{0}$: $$V_{RS}\left(t_{0} \right)=\frac{R_{RS}\left(t_{0} \right)}{t_{0}}=V_{0},
\label{apen2}$$ Hence: $$U_{RS}\left(t_{0}\right)=V_{0}.
\label{ap01}$$ The gas velocity behind the leading shock $U_{LS}$ is (see section \[initialc\]): $$U_{LS}\left(t_{0} \right)=\frac{2}{\gamma+1}l_{ED}V_{0},
\label{ap02}$$ where $l_{ED}=1.1$ is the leading factor. Substituting equations (\[ap01\]) and (\[ap02\]) into equation (\[ap4a\]): $$\phi\left(t_{0} \right)=1-\frac{2}{\gamma-1}\left(\frac{\gamma+1}{2l_{ED}}-1 \right)=0.3636.$$
The initial conditions for the High density test {#Ap3}
================================================
The ejecta density and its velocity structure presented at [@terlevich1992starburst] and related works (e.g, @1991Franco [@1991Tenorio]) are here discussed.
The density and velocity structure
----------------------------------
The ejected gas is assumed to have a velocity given by: $$v\left(r,t\right)=\left\{
\begin{array}{ll}
\frac{r-R_{c}}{R_{ej}\left( t\right)-R_{c}} & \mbox{if } r \geq R_{c},\\
0 & \mbox{if } r < R_{c},
\end{array}
\right.$$ where: $$R_{ej}\left( t \right)=R^{0}_{ej}+v_{ej}t,
\label{apend2}$$ is the free-expansion radius of the ejecta, $v_{ej}$ its maximum velocity and $R_{c}$ is the inner surface of the ejected mass, i.e., the boundary defining the size of the stellar remnant. The term $R^{0}_{ej}$ is the initial outer boundary of the ejected matter. The mass $M_{ej}$ expelled by the explosion is assumed to be located between $R_{c}$ and $R_{ej}\left(t \right)$: $$\rho_{ej}\left(r,t\right)=\left\{
\begin{array}{ll}
\frac{M_{ej}}{4 \pi \ln \left(R_{ej}\left(t \right) /R_{c}\right)}r^{-3} & \mbox{if } r \geq R_{c},\\
0 & \mbox{if } r < R_{c},
\end{array}
\right.
\label{apend3}$$ The fraction of thermalized ejecta mass is now given by: $$M_{th}=M_{ej}(1-\frac{\ln\left(R_{RS}/R_{c} \right)}{\ln\left(R_{ej}\left(t \right)/R_{c} \right)})
\label{apend5}$$ The parameters $R_{ej}^{0}$ and $R_{c}$ are calculated from the initial conditions fulfilling the data from [@terlevich1992starburst]. Indeed, The authors set $M_{ej}=2.5$ M$_{\odot}$, and state that an initial energy $E=10^{51}$ erg and momentum $p_{0}= 2.44 \times 10^{42}$ g cm s$^{-1}$ were deposited into an ambient medium of $n_{0}=10^{7}$ cm$^{-3}$. Table \[table1\] presents the set of parameters that satisfy these initial conditions.
\[table1\]
--------------------------- ---------------------
$M_{ej}[$M$_{\odot}]$ 2.5
$v_{ej}[$km s$^{-1}]$ $1.4 \times 10^{4}$
$E_{0}$ \[erg\] $10^{51}$
$R_{ej}^{0}[10^{-2}$ pc\] 0.70
$R_{c}[10^{-2}$ pc\] 0.11
--------------------------- ---------------------
: Initial conditions for a SNR evolving into a medium of density $n_{0}=10^{7}$ cm$^{-3}$.
\[lastpage\]
[^1]: E-mail: sjimenez@inaoep.mx
|
---
abstract: 'Electric charge detection by atomic force microscopy (AFM) with single-electron resolution ($e$-EFM) is a promising way to investigate the electronic level structure of individual quantum dots (QD). The oscillating AFM tip modulates the energy of the QDs, causing single electrons to tunnel between QDs and an electrode. The resulting oscillating electrostatic force changes the resonant frequency and damping of the AFM cantilever, enabling electrometry with a single-electron sensitivity. Quantitative electronic level spectroscopy is possible by sweeping the bias voltage. Charge stability diagram can be obtained by scanning the AFM tip around the QD. $e$-EFM technique enables to investigate individual colloidal nanoparticles and self-assembled QDs without nanoscale electrodes. $e$-EFM is a quantum electromechanical system where the back-action of a tunneling electron is detected by AFM; it can also be considered as a mechanical analog of admittance spectroscopy with a radio frequency resonator, which is emerging as a promising tool for quantum state readout for quantum computing. In combination with the topography imaging capability of the AFM, $e$-EFM is a powerful tool for investigating new nanoscale material systems which can be used as quantum bits.'
address: 'Department of Physics, McGill University, 3600 rue University, Montreal, H3A 2T8, Quebec, Canada'
author:
- 'Yoichi Miyahara, Antoine Roy-Gobeil and Peter Grutter'
title: Quantum state readout of individual quantum dots by electrostatic force detection
---
Introduction
============
Quantum dots (QD) are one of the most interesting and important entities in nanoscience and nanotechnology. QDs are often called artificial atoms as their electronic states become discrete just like those in atoms because of quantum confinement in three dimensions. The emergence of atom-like discrete energy levels leads to particular optical and electronic properties of QDs. Unlike real atoms, one can engineer the size and shape of the QDs, leading to their tunable optical and electronic properties. Therefore, understanding the energy level structure and its relation to the shape and size of the QD is a key to understand the properties of QDs, which remains to be elucidated. For this end, spectroscopic measurement on individual QDs is essential. QDs have been attracting much attention in the field of quantum information processing as their atom-like discrete energy levels can host charge or spin qubit and their quantum state can be read out by probing the charge state of the QD [@Elzerman2004]. Many studies have already been reported [@Elzerman2005; @Petta09302005; @Gorman2005].
Current transport spectroscopy performed in single-electron transistor (SET) structures [@Fulton1987] has been instrumental to understanding the electronic levels of individual QDs. In a SET structure, the energy level of the QD can be shifted by the gate voltage with respect to the source and drain electrodes. When one of the energy levels of the QD lies in the bias window set by the source-drain bias, a single electron can tunnel through two tunnel barriers at the source-QD and the drain-QD. When the source-drain current is measured as a function of gate bias voltage, peaks appear in the conductance versus gate voltage curves. This is known as a Coulomb-blockade oscillation peak. The energy of these peaks is the sum of the energy of the discrete levels and the Coulomb charging energy of the QD. A detailed analysis of these spacings thus allows the spectroscopy of the electronic energy levels of the QD [@Kouwenhoven01]. Although current transport spectroscopy is very powerful, its application has been limited mainly to gate-defined QDs where a QD is formed in a two dimensional electron gas using surface gate electrodes to create a confinement potential. Other interesting classes of QDs such as self-assembled QDs and colloidal nanoparticles have not been studied as much by current transport spectroscopy because attaching electrodes to these QDs by lithography techniques is challenging due to their smaller sizes. Recently, nanoparticles with more complex shape and structure have been developed for more functionalities such as biochemical sensing [@Milliron2004a; @Aldaye2007a; @Chen2012]. As attaching electrodes to these nanoparticles and their complexes is even more challenging, current transport spectroscopy has not been done in most of these complex structures.
In order to overcome the difficulty, we have developed an alternative experimental technique ($e$-EFM). The charge state of the QD can be detected by electrostatic force microscopy with sensitivity much better than a single electron charge [@Schonenberger1990]. The AFM tip also acts as a scannable gate, thus enabling spectroscopic measurements of single electron charging. To enable these experiments, the QD needs to be tunnel-coupled to a back-electrode. Oscillating the AFM tip modulates the energy of the QD, causing a single electron back and forth to tunnel between the QD and back-electrode. The resulting oscillating electrostatic force changes the resonant frequency and damping of the AFM cantilever, enabling the sensitive electrometry with single-electron sensitivity.
As $e$-EFM requires no nanometer scale electrode to be attached to the QD, it makes it possible to investigate individual colloidal nanoparticles and self-assembled QDs which pose major challenges for transport spectroscopy. This technique is equivalent to admittance spectroscopy with a radio frequency resonator, which is emerging as a promising tool for quantum state readout [@Petersson2010; @Petersson2012; @Colless2013]. In combination with the topography imaging capability of the AFM, $e$-EFM can be a powerful tool for investigating new nanoscale material systems which can be used as quantum bits. The fundamental physics of $e$-EFM is also of great interest as the measured interaction between the AFM tip and QD can be described as a back-action of a measurements on quantum electromechanical system [@Clerk2005].
In this review, we will first describe the basic principle of operation of the technique, followed by its application to several experimental systems. We will then discuss more detailed theoretical aspects which lead to physical insights which we can gain by this technique. Finally, we will discuss prospect of the technique, including possible application to other physical systems.
Single-electron detection by force detection
============================================
Overview of technique
---------------------
The experimental technique we describe here is essentially based on electrostatic force microscopy (EFM) which is a variant of atomic force microscopy (AFM). While AFM was shown to be capable of detecting a single-electron charge soon after its invention [@Schonenberger1990], this single-electron charge sensitivity has not been widely exploited.
Figure \[fig:setup\](a) depicts the experimental setup of our technique. QDs are separated from a conductive substrate (the back-electrode) by a tunnel barrier that allows electrons to tunnel between the QD and substrate. An oscillating conductive AFM tip is used as a sensitive charge detector as well as a movable (scannable) gate. A dc bias voltage is applied between the AFM tip and conductive substrate (back-electrode) to control the electron tunneling. The tip-QD distance is set to the order of 10 nm so that no tunneling is allowed across the vacuum gap between the tip and QD, making the system a so-called “single-electron box” [@Wasshuber2001]. The system with only a single tunnel barrier has several important advantages both experimentally and theoretically as will be shown later.
Choice of an appropriate dc bias enables a single electron to tunnel back and forth in response to the oscillating energy detuning across the barrier which is induced by the tip oscillation. The tunneling single-electron thus produces an oscillating electrostatic force which causes peaks both in the resonance frequency and the damping (dissipation) of the AFM cantilever. As is shown later, these peaks are essentially equivalent to the Coulomb peaks usually observed in dc transport spectroscopy and capacitance/impedance spectroscopy on single-electron transistors [@Kouwenhoven01; @Ashoori1993].
The frequency modulation (FM) mode operation [@Albrecht1991] of AFM is used to detect the oscillating electrostatic force caused by the single-electron tunneling. Because of the finite tunneling rate, there is a time delay for the oscillating force with respect to the tip motion. The in-phase component of the oscillating electrostatic force gives rise to the frequency shift signal and its quadrature component to the damping (dissipation) signal. The effect can also be described as a result of the quantum back-action of the tunneling electrons [@Clerk2005].
Another unique feature of this technique is the scanning capability which enables the spatial mapping of single-electron tunneling events. We will see that the spatial maps of the resonance frequency shift and dissipation are equivalent to the so-called charge stability diagram [@Wiel2003; @Hanson07] which provides a wealth of information particularly on the systems containing multiple QDs. The described technique, termed single-electron electrostatic force microscopy ($e$-EFM), was first demonstrated on QDs formed in carbon nanotube SETs [@Woodside2002; @Zhu05; @Zhu08]. As the present technique requires no patterned electrode to be defined around the QDs, it can open up the spectroscopy on individual QDs of various kinds that have been very difficult to investigate, such as colloidal nanoparticle dots and those having complex structures [@Milliron2004a; @Chen2012; @Swart2016].
Electrostatic force in AFM tip - QD system
------------------------------------------
![(a) Schematic of experimental setup. (b) Equivalent circuit of the system shown in (a). (c) Energy diagram of the system. \[fig:setup\]](Figure1.pdf){width="150mm"}
Following the equivalent circuit model commonly used in the QD transport studies [@Wasshuber2001], we model the system as shown in figure \[fig:setup\](b). The electrostatic force acting on an AFM tip shown in figure \[fig:setup\](a) can be calculated by considering electrostatic free energy of the system, $W$, which is shown below [@Stomp05]:
$$\label{eq:Free_energy}
W=\frac{\qdot^{2}}{2\Ctot}
+\frac{\Ctip}{\Ctot}\qdot\Vbias
-\frac{1}{2}\frac{\Csub \Ctip}{\Ctot}\Vbias ^{2}$$
where $\qdot$ is the electrical charge in the QD, $\Ctip$ the tip-QD capacitance, $\Csub$ the QD-back-electrode capacitance and $\Ctot \equiv \Ctip + \Csub$.
Differentiating the electrostatic free energy with respect to the tip position, $z$, yields the expression for the electrostatic force as follows:
$$\label{eq:elec_force}
\Felec = - \frac{\partial W}{\partial z }
= -\frac{1}{\Ctot^2}\frac{\partial \Ctip}{\partial z}\left\{\frac{\qdot^2}{2}
- \Csub \, \qdot \, \Vbias + \frac{1}{2}\Csub^2 \Vbias ^2\right\}$$
The third term on the right hand side represents the electrostatic interaction between the charges in the tip and conducting substrate (back-electrode) which is known as the capacitive force. This term is responsible for the parabolic background commonly observed in frequency shift versus bias voltage curves. The second term is proportional to the charge in the QD and responsible for the detection of single-electron tunneling. In the system shown in figure \[fig:setup\](a), the charge in the QD, $\qdot=-ne$, ($e$:elementary charge) is determined by the number of electrons in the QD, $n$, and can be varied solely by the electron tunneling through the tunnel barrier to the substrate because of the much larger tunnel barrier height of the vacuum gap prohibiting the tunneling between tip-QD.
Coulomb blockade effect in a single quantum dot system probed by electrostatic force detection {#CB_effect}
----------------------------------------------------------------------------------------------
In order for a single electron to tunnel into the QD from the back-electrode, the final state must be more energetically favorable than the initial one. Considering the electron tunneling process between $n$ electron state and $n+1$ electron state, the free energy of two states needs to be equal to $W(n+1) = W(n)$. This condition sets the threshold bias voltages for the tunneling to be
$$\label{eq:threshold_bias}
\Vbias ^{n+1}= \frac{e}{\Ctip}\left(n+\frac{1}{2}\right)$$
The separation of two successive peaks, $\Delta \Vbias = e/\Ctip$, is proportional to so-called addition energy which represents the energy required to add an extra electron to the QD. In general, the addition energy, $\Eadd$, is determined not only by the Coulomb charging energy of the QD, $2\Ec = e^2/\Ctot$, but also the energy difference between consecutive quantum states, $\delta$. $\Eadd$ is related to $\Delta \Vbias$ through the relation $\Eadd = e \alpha \Delta \Vbias $ where $\alpha$ is the fraction of $\Vbias$ applied across the tunnel barrier and given as $\alpha = \Ctip/\Ctot$. $\alpha$ is often called the lever-arm. In order to get the true energy scale in the QD, $\alpha$ needs to be determined. As we will see later, $\alpha$ can be determined experimentally from the peak shape of the frequency shift or dissipation peak.
As $\alpha$ is determined by $\Ctip$ and $\Csub$, the oscillating tip leads to the oscillation of the QD energy levels through the oscillation of $\alpha$. In other words, the oscillating tip causes the modulation of the energy level detuning, $\Delta E$, across the tunnel barrier. At each $\Vbias^{n+1}$, a single electron can tunnel back and forth between the QD and back-electrode in response to the oscillation in the QD energy. It is this oscillating single electron that induces the peaks in frequency shift and dissipation. At low temperature, $T$, ($\kB T \ll \Eadd$, $\kB$: Boltzmann constant), the number of electron is fixed due to the large addition energy of the QDs between two adjacent peaks (Coulomb blockade).
The shape of the peaks can be derived by considering the equation of motion of the AFM cantilever that is subject to the oscillating electrostatic force caused by the tunneling single electron, $\Fdot$ as described below:
$$\label{eq:eom}
m\ddot{z} + m\gamma_{0} \dot{z} + k(z-\zO)=\Fext (z,t) = \Fdot (t) + \Fexc(t)$$
where $m$, $\gamma_0$ and $k$ are the effective mass, intrinsic damping and spring constant of the AFM cantilever, respectively. $\Fexc$ is the external driving force that is controlled by the self-oscillation electronics [@Albrecht1991].
The second term of Eq. \[eq:elec\_force\], $\Fdot$, is the back-action of the tunneling process on the detector (*i.e.* AFM cantilever) and is given by: $$\label{eq:Fdot}
\Fdot(t) = \frac{\Csub}{\Ctot^2}(\qdot \, \Vbias)\frac{\partial \Ctip}{\partial z}
= -\frac{2\Ec \Vbias}{e}(1-\alpha) \frac{\partial \Ctip}{\partial z}n = -An$$ where $A = -(2\Ec \Vbias/e)(1-\alpha) \partial \Ctip/\partial z$. $A$ describes the magnitude of the force due to the capacitive tip-QD coupling and is dependent on the geometry of the tip and QD and bias voltage. In practice, the tip-QD distance is a very important parameter and $A$ increases with decreasing tip-QD distance.
Dynamics of the charge in the QD can be described by the master equation, ${\partial \langle n \rangle}/\partial t = -\Gamma_-(z)\langle n \rangle
+ \Gamma_+(z)(1 - \langle n \rangle)$ where $\Gamma_+$ ($\Gamma_-$ ) are $z$-dependent tunneling rates to add (remove) an electron to the QD and $\langle n \rangle$ denotes the average number of electrons on the QD. The widely accepted treatment of single-electron tunneling implies that the operation of single-electron devices that is governed by stochastic tunneling events, can be described well by time-evolution of the average value [@Wasshuber2001].
For a small tip oscillation amplitude, considering a single non-degenerate level in the QD and the linear response of the average charge on the QD, the changes in frequency shift, $\dw$, and dissipation, $\Delta \gamma$, due to the single-electron tunneling are given as follows [@Cockins2010a]:
$$\dw= -\frac{\omega_0^2 A^2}{2 k \kB T} \frac{1}{1+(\ogO)^2}f(\dE)[1-f(\dE)]
\label{eq:single_level_deltaf}$$
$$\Delta \gamma
= \frac{\omega_0^2 A^2}{k \kB T}\frac{\ogO}{1+(\ogO)^2}f(\dE)[1-f(\dE)]
\label{eq:single_level_gamma}$$
where $\omega_0$ is the angular resonance frequency of the cantilever, $\Gamma$ the tunneling rate between the QD and back-electrode and $f$ the Fermi-Dirac distribution function. In this single non-degenerate level case, the tunneling rate is independent of the energy and equal to tunnel coupling constant. In general, however, the tunneling rate is energy-dependent and determined by the energy-level structure of the QD and back-electrode. We will see this aspect in the section \[section:dos\_effect\]. Note that each expression contains a prefactor containing the ratio between the oscillation frequency of the cantilever and the tunneling rate. The tunneling rate can thus be extracted directly by taking these ratio such as $$\label{eq:tunnel_rate}
\Gamma = -2\omega_0 \frac{\Delta \omega}{\Delta \gamma}$$ In other words, the magnitude of $\Delta \omega$ and $\Delta \gamma$ depends on the ratio of the cantilever oscillation frequency to the tunneling rate, $\omega_0/\Gamma$, as shown in figure \[fig:diss\_peak\_and\_ring\](c). The magnitude of $\Delta \gamma$ takes its maximum when $\Gamma \approx \omega_0$. Note that this determines the condition for which both frequency shift and damping signals are observable [@Woodside2002; @Zhu08; @Brink2007; @Tekiel2013]. Specifically, the tunneling rate between the QD and the back-electrode needs to be engineered to be similar to the cantilever resonance frequency (to within a factor of 10). Note that the resonance frequency of AFM cantilevers is typically a few hundred kHz. As we will see later, this tunneling rate can be engineered by selecting a barrier of suitable thickness. In the later sections, we will see that the detailed analysis of the peak shape provide such useful information as shell-filling of the energy levels in the QD and the density-of-states of the QD levels.
![(a) Single-electron tunneling peaks in dissipation versus bias voltage curve. (b) Single-electron tunneling rings in the dissipation image taken at a constant bias voltage. Each peak in (a) corresponds to a ring in (b). (c) Normalized amplitude of Dissipation and frequency shift as a function of tunneling rate. \[fig:diss\_peak\_and\_ring\]](Figure2.jpg){height="42mm"}
Spatial mapping of charge state {#sec:spatial mapping}
-------------------------------
Changing the distance between tip and QD changes the lever-arm and thus the energy between QD and back electrode. Since the AFM tip is a scannable gate and charge sensor, one can spatially map the charging events, a unique feature, not available in conventional transport measurements. The dissipation image shown in figure \[fig:diss\_peak\_and\_ring\](b) demonstrates the unique capability of this technique. This image is taken by scanning the tip over a QD at a constant tip-height with a constant bias voltage while acquiring the frequency shift and dissipation signals. Each of the circular concentric rings in figure \[fig:diss\_peak\_and\_ring\](b) correspond to a peak in the dissipation versus bias voltage spectrum obtained at a fixed tip position $(x,y,z)$.
The concentric rings can be understood by considering the energy diagram of the system. The energy detuning, $\Delta E$, which governs the electron tunneling is dependent on the lever-arm, $\alpha$, as well as $\Vbias$. In fact, $\alpha$ is a function of the tip position such as $\alpha(x,y,z)$ because the tip-sample capacitance depends on the tip position. If $\alpha$ depends only on the distance between a QD and tip such as $\alpha(x,y,z)=\alpha(r)$ where $r=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$ ($(x_0, y_0, z_0)$ denotes the position of the QD) [@Tekiel_thesis], a specific $\Delta E$ allowing tunneling is mapped to a circular ring around the center of the QD. Although the real effect of the biased tip on the QD confinement potential can be more complex, this simple model has so far been able to explain most of the important features observed in experiments.
The image indicates that the number of electrons in the QD can be controlled by the tip position without changing $\Vbias$. It is true even for multiple QD complex as shown in figure \[fig:multiQD\_image\](a) because even though there is only one gate (AFM tip), changing tip position change the lever-arm, $\alpha$, for each QD, thus the energy of each QD (figure \[fig:multiQD\_image\](c)). Figure \[fig:multiQD\_image\](a) and (b) show the charging of three QDs located close to each other. We can see that the number of electrons in each QD can be controlled by the tip position. Furthermore, as expected, we observe avoided-crossings of the rings, indicative of the coupling between the QDs. In fact, these dissipation images are equivalent to the charge stability diagram [@Wiel2003] of multiple QDs as we will see more detail in the later section \[sec:charge stability diagram\]. Characterizing coupled-QDs is of great interest because if the inter-dot tunneling coupling is coherent, the QD complex should behave as an artificial molecule which can host a charge or spin qubit [@Petta09302005; @Gorman2005].
![(a) and (b) Dissipation images of the same area containing three QDs taken at two different bias voltages, $\Vbias = -7.6$ V (a) and $\Vbias = -9.0$ V (b). The numbers in parenthesis in (a) indicate the number of electrons in the left and right QDs. Scalebar is 20 nm [@Cockins2010a]. (c) Schematic showing different coupling capacitances to each QD. \[fig:multiQD\_image\]](Figure3.jpg){width="15.5cm"}
Experimental
============
The experiments are performed with a home-built cryogenic atomic force microscope [@Roseman00], in which a fiber optic interferometer is used for cantilever deflection sensing [@Rugar89; @Albrecht1992]. A fiber-pigtailed laser diode operating at a wavelength of 1550 nm is used for the interferometer. The laser diode current is modulated by a radio frequency signal through a bias-tee to reduce the coherence length of the laser to reduce phase noise and suppress undesirable interferences. A typical optical power of $100~\mu$W is emitted from a cleaved optical fiber end. We use commercially available AFM cantilevers (NCLR, Nanosensors) with typical resonance frequency and spring constant of 160 kHz and 20 N/m, respectively. The tip side of the cantilevers are coated with 20 nm thick Pt/10 nm thick Ti by sputtering to ensure a good electrical conductivity even at liquid helium temperature. A dilute helium exchange gas ($\sim 10^{-3}$ mbar) is introduced in the vacuum can for the experiments at low temperature for good thermalization. The typical quality factor of the cantilevers ranges from 30,000 to 50,000 at 77 K and 100,000 to 200,000 at 4.5 K, which is unaffected by the exchange gas.
The fundamental flexural-mode oscillation is controlled either by a self-oscillation feedback electronics which consists of a phase shifter and an amplitude controller (Nanosurf EasyPLLplus) or by a digital phase-locked loop oscillation controller (Nanonis OC4). The resonance frequency shift is measured by a phase-locked loop frequency detector (Nanosurf EasyPLLplus or Nanonis OC4). The amplitude of the excitation signal is measured as a dissipation signal. Care must be taken to correct for the background dissipation signal caused by the non-flat frequency response of piezoacoustic excitation systems [@Labuda2011].
Epitaxially grown self-assembled InAs QD on InP
===============================================
In our first experiment, we used epitaxially grown self-assembled InAs QDs [@Poole01]. The schematic of the structure and the energy level diagram of the sample are shown in figure \[fig:InAsQD\](a) and (c). The two-dimensional electron gas (2DEG) layer formed in a InGaAs quantum well serves as a back-electrode which works at liquid helium temperature and the 20 nm thick undoped InP layer serves as a tunnel barrier. In this system, InAs islands are spontaneously formed after the growth of monolayer thick InAs wetting layer due to the lattice mismatch between InAs and InP. Figure \[fig:InAsQD\](b) shows an AFM topography image of the sample surface. The variation in size and shape of the InAs QDs can be clearly seen, indicating the spectroscopy of individual QDs is highly desired.
![(a) Schematic of InAs QD on InP sample structure. (b) Tapping mode AFM topogaphy images of the InAs QD sample. Scale bar is 500 nm. (c) Energy diagram of the system.[]{data-label="fig:InAsQD"}](Figure4.jpg){height="4.2cm"}
Figure \[fig:InAsQD\_spectra\] shows typical bias-spectroscopy curves taken over one of the InAs QDs by sweeping the tip-sample bias voltage, $\Vbias$. While sharp dips appear in the frequency shift versus $\Vbias$ curve on top of the parabolic background, sharp peaks appear in the dissipation versus $\Vbias$ curve at the same bias voltages. When the sample is negatively biased, the electronic levels in the InAs QD are brought down with respect to the back-electrode (InGaAs quantum well) as shown in figure \[fig:InAsQD\](c). At a sufficiently high negative sample bias voltage, one of these levels is lined up with the back-electrode states and a single electron can tunnel between the QD and back-electrode.
We can obtain the lever-arm, $\alpha$, experimentally by fitting the observed spectrum with the theoretical one. Figure \[fig:Effect\_of\_degeneracy\](b) and (c) show the results of the fitting. By assuming the temperature, we can extract the lever-arm, $\alpha=0.04$. This means only 4 % of the applied bias voltage is applied across the QD-back electrode tunnel barrier and 96 % falls off across the tip-QD spacing. The converted energy in the QD is indicated as a scale bar in figure \[fig:InAsQD\_spectra\]. Notice that as the temperature broadening becomes smaller at lower temperature, the oscillation amplitude needs to be reduced accordingly so that the peak shape is determined not by the tip oscillation but by the temperature.
We notice the larger peak separation between $n=2$ and $n=3$ peaks and between $n=6$ and $n=7$. It is indicative of the shell structure expected for 2D circularly symmetric potential that consists of lowest two-fold degenerate levels ($s$-shell, 2-spin $\times$ 1-orbital degeneracy) and the next lowest four-fold degenerate levels ($p$-shell, 2-spin $\times$ 2-orbital degeneracy). This type of shell structure has previously been observed for an ensemble of self-assembled InAs QD [@Drexler94; @Miller97]. Assuming this shell structure, we can obtain the charging energy from this peak separation between peak $n=1$ and $n=2$ as 30 meV using $\alpha=0.04$ obtained before.
However, this simple argument to identify the shell structure is far from convincing. The more compelling identification usually requires the measurements under high magnetic field to acquire the evolution of energy level structure (Fock-Darwin spectrum) [@Kouwenhoven01; @Tarucha1996]. In the next section, we will show an alternative way for the shell structure identification based on the asymmetry of the tunneling in and out processes.
![(a) Frequency shift and (b) Dissipation versus bias voltage spectra measured on an InAs QD shown in figure \[fig:InAsQD\] at 4.5 K. The energy scale in the QD calculated from the lever-arm, $\alpha =0.04$, is indicated as a scale bar in each figure.[]{data-label="fig:InAsQD_spectra"}](Figure5.jpg){width="14cm"}
Effect of degenerate levels on the dissipation peak {#effect_of_degeneracy}
---------------------------------------------------
When the degeneracy of the energy levels in the QD is taken into account, the tunneling in and out process is no longer symmetric. Figure \[fig:Effect\_of\_degeneracy\](a) illustrates the asymmetric tunneling process. When we consider the $n=1$ peak in the spectrum shown in figure \[fig:Effect\_of\_degeneracy\](b), the peak arises from the transition between $n=0$ and $n=1$ states. Assuming twofold spin degeneracy under no magnetic field, there are two ways for an electron to tunnel into the QD from the back-electrode, with either a spin-up or a spin-down, whereas there is only one way for the already occupied electron to tunnel out of the QD. For the $n=2$ peak, the situation is opposite (one way to tunnel in, two ways to tunnel out). The same argument for the peaks in $p$-shell reveal the similar but more pronounced asymmetry for $n=3$ and $n=6$ peaks as illustrated in figure \[fig:Effect\_of\_degeneracy\](a).
The effect of the asymmetric tunneling process manifests itself in two different ways depending on the tip oscillation amplitude. In the small oscillation amplitude case, the positions of the dissipation/frequency shift peaks shift with increasing temperature in the way that the peaks that belong to the same shell repel each other as indicated by arrows in figure \[fig:Effect\_of\_degeneracy\](b). In the large oscillation amplitude case, the peaks that belong to the same shell get skewed away from each other as shown in figure \[fig:large\_oscillation\].
![ (a) Schematic representation of asymmetric tunneling processes. Solid horizontal lines depict electronic levels on the QD and the grayed area in the right hand side depicts Fermi distribution of electrons in the back-electrode. The fine dashed line is where the chemical potentials line up and where a dissipation peak would occur for a nondegenerate level. (b) The measured dissipation and frequency shift versus bias curves measured at 30 K. The parabolic background is subtracted from the frequency shift spectrum. The energy scale in the QD is indicated as a scale bar (20 meV). The arrows indicate the directions of the peak shifts with increasing temperature. (c) The measured and fitted dissipation spectra as a function of temperature. The curves are fitted with Eq. \[eq:diss\_small\]. The energy scale in the QD is indicated as a scale bar (10 meV). Adapted from [@Cockins2010a].\[fig:Effect\_of\_degeneracy\]](Figure6.pdf){width="16cm"}
### Small tip oscillation amplitude case (Weak coupling) {#section:weak_coupling}
In this case, we can obtain the analytical expression for the frequency shift and dissipation by considering the master equation and linear response theory [@Cockins2010a; @Roy-Gobeil2015].
$$\dw= -\frac{\omega_0}{2k}\frac{A^2 \Gamma^2}{\kB T}
\frac{(\nshell + 1)(\nu - \nshell)}{\omega ^2 +(\phi \Gamma)^2}f(\dE)[1-f(\dE)]$$
$$\Delta \gamma
= \frac{\omega_0^2 A^2 \Gamma}{k \kB T}
\frac{(\nshell + 1)(\nu - \nshell)}{\omega ^2 +(\phi \Gamma)^2}f(\dE)[1-f(\dE)]
\label{eq:diss_small}$$
where $\nshell$ is the number of electrons already in the shell before adding a new electron, $\nu$ level degeneracy and $\phi = (\nu-\nshell)f + (\nshell + 1)(1-f)$. While these results indicate the deviation of the peak shape from the simple form of $4\cosh^{-2}(\dE/2) = f(\dE)(1-f(\dE))$, which reflects the asymmetry of the tunneling process, the deviation is too small to be discernible. Instead, each peak position changes with temperature in its own unique way from which the level degeneracy can be identified.
Figure \[fig:Effect\_of\_degeneracy\](b) and (c) show the direction of the peak shift of each peak for increasing temperature and the temperature dependence of $n=1$ and $n=2$ peaks, respectively. The peak shift is linearly proportional to the temperature and the level degeneracy can be inferred from the slope of the peak shift versus temperature line. Although a similar effect has been predicated for the current transport measurements in single-electron transistors [@Beenakker91; @Tews2004], it has rarely been observed experimentally [@Deshpande2000; @Bonet2002]. The peak shift observed in the dissipation measurement is much more clear because the effect appearing in the dissipation peak is much more pronounced compared with the conductance peak as predicted by theory [@Cockins2010a].
### Large tip oscillation amplitude case (Strong coupling) {#section:strong_coupling}
![(a) Theoretical dissipation peak for $n=1$ transition with different oscillation amplitudes. $\mathcal{N}=-\Vbias \Ctip/e$ is the dimensionless bias voltage. Dotted (solid) lines are calculated from simulation (semianalytic theory). The green dashed line is from linear response. (b) Adiabatic approximation (dash-dotted), semianalytic theory (solid), and simulation (dot) for oscillation amplitude $a=1$ nm. (c)-(e) Average dot charge versus time (solid) for $a=1$ nm, at voltages marked in (a). Tip position is shown (thin dashed) as a reference. (f) Experimental (solid) and theoretical (dashed) dissipation peaks for $n=1$ transition with three different oscillation amplitudes. (g) Dissipation peak for $n=2$ transition. (h) Experimental dissipation peaks for the transitions in $p$-shell with three oscillation amplitudes. (i) Theoretical dissipation peaks calculated for the peaks in $p$-shell. Adapted with permission from [@Bennett2010]. Copyright (2010) by the American Physical Society.[]{data-label="fig:large_oscillation"}](Figure7.jpg){width="15cm"}
In the large tip oscillation case, the effect of degeneracy appears as an asymmetric peak shape that is much more pronounced and thus clearly observable than in the small oscillation amplitude case. This makes it possible to identify the shell structure by sweeping $\Vbias$ with a large tip oscillation amplitude than measuring the more challenging temperature dependence of the peak position described above. The large oscillation amplitude refers to the case where the maximum change in the energy detuning due to the oscillator, $\Delta E_\mathrm{osc} = Aa$ ($a$: oscillation amplitude), is comparable to thermal energy, $\kB T$. In this case, the tunneling rates depend nonlineary on the tip position, $z$ and the master equation needs to be numerically solved to obtain $\langle n(t) \rangle$. The expressions for frequency shift and dissipation can be obtained using the theory of FM-AFM [@Holscher2001; @Kantorovich2004] as shown below:
$$\Delta \omega = -\frac{\omega_0^2}{2 \pi k a} \int_0^{2 \pi/\omega_0} \Fdot (t) \cos(\omega_0 t)dt
=- \frac{\omega_0^2 A}{2 \pi k a} \int_0^{2 \pi/\omega_0} \langle n(t) \rangle\ \cos(\omega_0 t) dt
% \Delta \omega = - \frac{\omega_0 A}{2 \pi k a} \int_0^{2 \pi/\omega_0} dt \cos(\omega_0 t)\langle P(t) \rangle\
\label{strong_fshift}$$
$$\Delta \gamma = \frac{\omega_0^2 }{\pi k a} \int_0^{2 \pi/\omega_0} \Fdot (t) \sin(\omega_0 t) dt
=\frac{\omega_0^2 A}{\pi k a} \int_0^{2 \pi/\omega_0} \langle n(t) \rangle\ \sin(\omega_0 t) dt
\label{strong_diss}$$
Eqs. \[strong\_fshift\] and \[strong\_diss\] can also be derived from a more quantum mechanical approach [@Rodrigues2007].
Figure \[fig:large\_oscillation\] shows the theoretical ((a)-(e)) and experimental ((f)-(g)) dissipation versus bias voltage spectra for the $n=1$ transition with different tip oscillation amplitudes. The theoretical curves are obtained from Eq. \[strong\_diss\] using $\langle n \rangle$ calculated from the master equation numerically. Figure \[fig:large\_oscillation\](c)-(e) shows the time evolution of $\langle n \rangle$ calculated for different bias positions. We notice that $\langle n (t)\rangle$ is no longer sinusoidal and that the dissipation peaks become wider and get more skewed towards the direction of $n=0$ state for larger amplitude. The skewed peak stems from the asymmetric tunneling process we described above (figure \[fig:Effect\_of\_degeneracy\](a)). Figure \[fig:large\_oscillation\](f)-(i) show the excellent agreement between the theory and experiments. All the theoretical curves are calculated using the following parameters, $2\Ec=31$ meV, $\alpha=0.04$, and $\Gamma =70, 90$ kHz for peaks $n=1$ (figure \[fig:large\_oscillation\](f)) and $n=2$ (figure \[fig:large\_oscillation\](g)), all of which are obtained from the experimental results taken at weak coupling conditions (small oscillation amplitude), only $A$ being a fitting parameter. The tip oscillation amplitude, $a$, is obtained from the calibrated interferometer signal. The peaks shown in figure \[fig:large\_oscillation\](g) corresponds to the $n=2$ transition and is skewed in the opposite direction as $n=1$ peak because of the opposite asymmetry of the tunneling process. Figure \[fig:large\_oscillation\](h) and (i) show the experimental and theoretical dissipation peaks versus voltage for the peaks belonging to the $p$-shell, respectively. The excellent agreement between the theory and the experiment also demonstrates the validity of the theoretical interpretation. In summary, by finding a pair of peaks which are skewed away from each other, the shell-filling can be identified without measuring magnetic field or temperature dependence of the spectra.
Excited-states spectroscopy with a large oscillation amplitude {#section:excited_states}
--------------------------------------------------------------
The analysis in the previous section suggests the possibility of probing the excited states of the QDs by taking the dissipation spectra with large oscillation amplitudes. When the tip motion is large enough, the energy detuning (bias window) modulated by the tip motion becomes large enough to allow the tunneling to the excited states (figure \[fig:ESS\_principle\](b)). This situation is illustrated in figure \[fig:ESS\_principle\]. This additional tunneling paths to the excited states show up as additional features in damping versus voltage spectra. Figure \[fig:ESS\_results\] shows such experimental results [@Cockins2010a]. The figure consists of the derivatives of 67 dissipation versus voltage spectra taken with different tip oscillation amplitudes.
![(a) Derivative of dissipation versus bias voltage curves taken at different oscillation amplitudes. (b) Zoom-up of the region around $A$ and $A'$ indicated in (a). Reprinted with permission from [@Cockins2011]. Copyright (2012) American Chemical Society. \[fig:ESS\_results\]](Figure9.jpg){width="12cm"}
The down-pointing triangles in the figure are equivalent to the Coulomb diamonds commonly observed in the current transport spectroscopy experiments where the vertical axis is the source-drain bias voltage of a single electron transistor (SET) in place of the oscillation amplitude. The lines that run parallel to the rightmost triangle edges (e.g.,$B$-$B'$) in figure \[fig:ESS\_results\](a) result from the opening of additional tunneling paths which become available in the tunnel bias window. In this case, the bias window is set by the lowest and highest energy detuning each of which is determined by the closest and furthest position of the tip. The technique allows us to probe directly the excitation spectrum of the QDs for a fixed number of electron without optical measurement and can thus be applied to measure spin-excitation energy under magnetic field which is important for single electronic spin manipulations [@Elzerman2005]. The additional tunneling paths can also be due to inelastic tunneling or features in the density of states of the back-electrode [@Escott2010]. Ability to probe excited states directly can be applied to such interesting applications as inelastic tunneling spectroscopy [@Natelson2006; @Leturcq2009] or single-electron spin detection [@Elzerman2005].
Spatial mapping of charging on strongly interacting QD - Charge stability diagram {#sec:charge stability diagram}
---------------------------------------------------------------------------------
As we have already seen in the section \[sec:spatial mapping\], $e$-EFM is capable of obtaining spatial maps of single-electron charging events in the QD. Figure \[fig:Strongly\_coupled\_QD\] shows such an example. Figure \[fig:Strongly\_coupled\_QD\](a) is the topography image of an InAs island and (b) is the dissipation image taken at the same location with $\Vbias=-8.0$ V. While the topography shows one connected island, the dissipation image shows at least two sets of concentric rings, indicating the existence of multiple QDs in the island.
Although the current transport spectroscopy on a single QD has been performed on InAs QDs by attaching a pair of electrodes with a nanogap [@Jung05], it would not be straightforward to identify such multiple dots just with the conventional conductance versus bias voltage spectra. The similar spatial maps showing concentric rings have been observed by a related technique, scanning gate microscopy (SGM) [@Woodside2002; @Pioda04; @Bleszynski2007; @Boyd2011; @Zhou2014]. In SGM, the AFM tip is used just as a movable gate and the conductance of QD devices is measured as a function of the tip position. While SGM is more widely used to investigate the properties of individual QDs, it still requires the fabrication of the wired QD devices.
![(a) AFM topography of an InAs QD island. (b) dissipation images of InAs QD taken at $\Vbias = -8.0$ V. (c) Zoom up of the lower left corner of the image (b) showing many avoided crossings [@Cockins2010a].[]{data-label="fig:Strongly_coupled_QD"}](Figure10.jpg){width="16cm"}
Figure \[fig:Strongly\_coupled\_QD\](c) is the zoom-up of the lower left corner of figure \[fig:Strongly\_coupled\_QD\](b). Overall, the image resembles the charge stability diagram of double QD systems. The image shows two set of rings which avoid each other. These avoided crossings are indicative of the coupling between two QDs. More detailed analysis of the avoided crossing can quantitatively reveal the nature of the coupling, either capacitive or quantum mechanical (tunnel) [@Gardner2011], which is of critical importance for quantum information processing application.
Figure \[fig:converted\_charge\_diagram\] shows that a set of concentric rings shown in figure \[fig:converted\_charge\_diagram\](b) can be converted to a conventional charge stability diagram. Figure \[fig:converted\_charge\_diagram\](c) can be obtained by the coordinate transformation, $(x, y) \rightarrow (V_\mathrm{U}, V_\mathrm{L})$ where $V_\mathrm{U}$ and $V_\mathrm{L}$ are equivalent bias for the upper and left QDs. In this particular case, we use the experimentally obtained relations, $V_\mathrm{U}=\beta_\mathrm{U} \sqrt{(x-x_\mathrm{U})^2 + (y-y_\mathrm{U})^2 }$ and $V_\mathrm{L}=\beta_\mathrm{L} \sqrt{(x-x_\mathrm{L})^2 + (y-y_\mathrm{L})^2 }$ where $(x_\mathrm{U}, y_\mathrm{U})$ and $(x_\mathrm{L}, y_\mathrm{L})$ are the center of each QD, $\beta_\mathrm{U}$ and $\beta_\mathrm{L}$ are constants which can be obtained by measuring the radius of each ring at different bias voltages, $\Vbias$.
![(a) Dissipation image of the same QD as that in figure \[fig:Strongly\_coupled\_QD\] taken with $\Vbias = -8.0$ V at a larger tip-QD distance. (b) Zoom-up of (a). (c) Charge stability diagram obtained from (b) by coordinate transformation. Vertical axis and horizontal axis are effective bias voltage applied to the upper and left QD, respectively. The two numbers in parenthesis in (b) and (c) indicate the number of electrons in the upper and left QDs. \[fig:converted\_charge\_diagram\]](Figure11.jpg){width="16cm"}
Imaging of Capped QD
--------------------
For practical device applications such as quantum dot lasers, epitaxially grown semiconductor quantum dots are usually covered with a protecting layer [@Dalacu09] or even buried in the host matrix to prevent them from being oxidized. As $e$-EFM technique relies on the detection of long-range electrostatic force, it can be used to probe the capped or buried QDs. Figure \[fig:cappedQD\] shows the topography, frequency shift and dissipation images of the InAs QD capped with a thin layer of InP. The topography image is taken with frequency modulation mode and the frequency shift and dissipation images are simultaneously taken in constant height mode. While the topography of the QDs is much less obvious than uncapped QDs, the charging rings appear very clearly in frequency shift and dissipation images. The background in the frequency shift image originates from the topography through the third term of the expression of the electrostatic force in Eq. \[eq:elec\_force\]. This demonstrates a clear advantage of the $e$-EFM technique over tunneling spectroscopy by STM which requires a clean surface to ensure a reliable vacuum tunnel gap [@Maltezopoulos03].
![(a) Frequency modulation mode AFM topography, (b) frequency shift and (c) dissipation images of of the same location of a capped InAs QD grown on InP taken at $T=4.5$ K. Scale bar is 1 $\mu$m. The topography image was taken separately. The frequency shift and dissipation images were taken simultaneously in constant height mode.[]{data-label="fig:cappedQD"}](Figure12.jpg){height="4.5cm"}
Colloidal Au nanoparticles on alkane-dithiol self-assembled monolayer
=====================================================================
![(a) AFM topography images of 5 nm diameter Au nanoparticles (five bright spots) on C16S2 SAM. Scale bar is 500 nm. (b) Schematic of the cross sectional view of the sample. (c) Schematic of Au NP -C16S2 - Au substrate structure.[]{data-label="fig:AuNP"}](Figure13.jpg){width="14cm"}
Quantum dots based on colloidal nanoparticles (NP) have been attracting considerable attention because of their tunable optoelectronic properties by their size and constituent material. The capability of producing a wide variety of nanoparticles with different materials, shapes and sizes together with the possibility of arranging them and integrating them with other structures has a huge potential for various applications [@Kim2013]. Investigating the electronic structure of individual colloidal QDs have been of great importance in order to uncover the relationship between their electronic and geometric structures and also to engineer more complex structure based on colloidal QDs. Scanning tunneling microscopy and spectroscopy (STM and STS) have been so far used to this end [@Swart2016]. Although the conductance spectroscopy in SET with NP as an island have also been performed [@Kuemmeth08; @Nishino2010; @Guttman2011], the device fabrication remains challenging.
![(a) Dissipation Spectrum on Au NP taken at $T=4.5$ K. The energy scale in the QD is indicated as a scale bar (10 meV). (Inset) Blue solid curve is a fitted curve with the theory. (b) and (e) Topography, (c) and (f) frequency shift, (d) and (g) dissipation images on Au NP taken at 77 K and 4.5 K, respectively. \[fig:AuNP\_results\]](Figure14.jpg){width="15cm"}
We apply $e$-EFM technique to study the single-electron charging in colloidal bare gold nanoparticles (Au NP). For this study, the sample as shown in figure \[fig:AuNP\](b) has been prepared. The self-assembled monolayer (SAM) of 1,16-hexadecanedithiol (C16S2) is formed on a template strip gold surface and is used as a tunnel barrier. Bare Au NPs with 5 nm diameter are deposited on the C16S2 SAM layer to form a Au NP-C16S2-Au junction (figure \[fig:AuNP\](c)). Figure \[fig:AuNP\](a) shows the AFM topography image of the prepared sample. Covalent Au-S bonds form stable links between Au NPs and Au substrate through alkyl chains that act as a tunnel barrier [@Akkerman2007; @Song2010]. The alkanedithiol SAM is ideally suited to optimize the tunnel barrier so that the tunneling rate matches the oscillation frequency of the AFM cantilever.
Figure \[fig:AuNP\_results\](a) shows a typical dissipation versus bias voltage curve taken above a 5 nm Au NP on the sample at the temperature of 4.5 K. A set of very sharp single-electron charging peaks are clearly seen in the figure. The equal separations between the neighboring peaks indicate that the Au NP is in the classical Coulomb blockade regime ($\delta \ll \kB T \ll \Ec$), which is consistent with the expected $\delta \approx 1$ meV estimated for 5 nm Au NPs [@VonDelft2001]. The inset shows the zoom up of the rightmost peak together with the fitted curve (blue line) with the function, $\cosh^{-2}[e\alpha(\Vbias-V_0)/2\kB T]$ from which $\alpha=0.0054$ is extracted. Using the extracted $\alpha$ value, the charging energy of the Au NP is obtained as $\Ec = 16$ meV.
Figure \[fig:AuNP\_results\](b-g) shows the spatial mapping results on a Au NP. The top and bottom panel shows the result at 77 K and 4.5 K, respectively. The concentric rings due to the single-electron charging can be clearly seen even at 77 K but the contrast of the ring to the background is much higher at 4.5 K. As is shown here, $e$-EFM technique can be used to investigate the nanoparticle complex as those demonstrated in Ref. [@Milliron2004a; @Aldaye2007a; @Chen2012].
Revealing density-of-states from tunneling-rate spectrum {#section:dos_effect}
--------------------------------------------------------
The expressions for $\Delta \omega$ and $\Delta \gamma$ given in Eq. \[eq:single\_level\_deltaf\] and \[eq:single\_level\_gamma\] are derived for a single non-degenerate level, in which case the total tunneling rate, $ \Gamma_{\Sigma} = \Gammap+\Gamman$, is constant and equal to the tunnel coupling constant. In general, the tunneling rate depends on the energy level structure and is thus energy dependent. The corresponding results for generic energy-dependent tunneling rate in the limit of weak coupling (small oscillation amplitude) are obtained as follows [@Roy-Gobeil2015]:
$$\label{fshift}
\Delta \omega = -\frac{\omega_0 A^2}{2 k} \frac{(\Gammap' \Gamma_{\Sigma} - \Gammap \Gamma_{\Sigma}')}{(\Gamma_{\Sigma}^2 + \omega^2)}$$
$$\label{diss}
\Delta \gamma = \frac{\omega_0^2 A^2}{k \Gamma_{\Sigma}} \frac{(\Gammap' \Gamma_{\Sigma} - \Gammap \Gamma_{\Sigma}')}{(\Gamma_{\Sigma}^2 + \omega^2)}$$
where $'$ denotes derivative with respect to energy. The total tunneling rate between the QD and the back-electrode, $\Gamma_{\Sigma}$, can be directly measured as a function of the electrochemical potential detuning by noting that:
$$\label{total_trate}
\Gamma_{\Sigma}(\Delta E) = -2 \omega_0 \frac{\Delta \omega}{\Delta \gamma}$$
This points to the interesting possibility of tunneling-rate spectroscopy from which the energy level structure (density-of-states) of the QD can be extracted. It should be noted that the measuring the tunneling rate with current spectroscopy measurement in SET is not as straightforward because two tunnel barriers are involved. Alternative single-electron counting techniques require fast charge sensors [@Muller2012].
Figure \[fig:DOS\_effect\] shows three examples of such tunneling-rate spectroscopy using $e$-EFM. Figure \[fig:DOS\_effect\](b) shows energy dependent tunneling rate for a Au NP which is obtained from the $\Delta \omega$ and $\Delta \gamma$ spectra shown in figure \[fig:DOS\_effect\](a). A parabola like energy dependent tunneling rate (blue curve) is clearly seen in the spectrum which is fitted with the energy dependent tunneling rate expected for the continuous density of states (dashed line) [@Brink2007; @Beenakker91]. Figure \[fig:DOS\_effect\](c) shows the experimental frequency shift and dissipation spectra measured on InAs QD at 4.5 K. (similar to the result in figure \[fig:InAsQD\_spectra\] but the bias axis is reversed). Three peaks for $n=1$, $n=2$ and $n=3$ are seen from left to right. Figure \[fig:DOS\_effect\](d) shows the corresponding tunneling rate spectra for each peak (blue line). The green and red lines are fits with the the theoretical tunneling rate expression expected for 2-fold degenerate levels with the shell-filling, $\nshell=0$ and $\nshell=1$, respectively. The yellow circle is a fit to the theory for 4-fold degenerate levels with $\nshell=0$. We can clearly see the asymmetric tunneling processes which is discussed in section \[effect\_of\_degeneracy\] in the tunneling rate spectra although it is not obvious at all to identify them in the charging peaks in either frequency shift or dissipation spectrum. In other words, it is possible to identify the shell filling from the slope of the tunneling rate spectra. This approach can be extended to identify more general energy level structures [@Roy-Gobeil2015].
![Effect of density-of-states on tunneling rate. (a) Experimental frequency shift and dissipation versus bias voltage curves measured on Au NP at 77 K. The energy scale in the Au NP is indicated as a scale bar (100 meV) determined from the lever-arm, $alpha=0.064$. (b) Extracted tunneling rate from the data above using Eq. \[total\_trate\] (blue solid line). Dashed line is a fit to the analytical expression for a continuous density of states. (c) Experimental frequency shift and dissipation versus bias voltage curves measured on InAs QD at 4 K. The energy scale in the QD is indicated as a scale bar (10 meV) determined from the lever-arm, $alpha=0.04$. (d) Corresponding tunneling rate data (blue) obtained from the data above. Green and blue solid lines are fits to the tunneling rate expected for 2-fold degenerate levels. Circles represent a best fit solution assuming tunneling into an empty 4-fold degenerate level accounting for the strong coupling effect. Adapted with permission from [@Roy-Gobeil2015]. Copyright (2015) American Chemical Society.[]{data-label="fig:DOS_effect"}](Figure15.jpg){width="15.5cm"}
Au island on a few monolayer thick NaCl grown on Fe(001) surface
================================================================
The charge transfer process to metallic islands on insulator surfaces have been the subject of active research particularly in the context of supported model catalysts [@Henry2015]. STM has been a main tool for investigating the relationship between the size and shape of individual islands and their electronic properties including charge transfer [@Giordano2011]. However, as STM requires a dc current typically higher than 1 pA, it can only be applied to systems of a few monolayers of insulators. Here, we demonstrate that $e$-EFM technique can be applied to the gold islands deposited onto thicker insulating films.
Figure \[fig:Au\_island\_schematic\](a) shows the AFM topography image of the sample with a 7 ML thick NaCl grown on a Fe(001) surface with a submonolayer coverage of Au. The NaCl layer grows in a nearly perfect layer-by-layer growth mode [@Tekiel2012], allowing for the tunneling rate to be controlled. Figure \[fig:Au\_island\_schematic\](b) and (c) show the schematic and the energy level diagram of the experimental setup. The size of the studied Au nanoislands is 3.5 nm in height and 10 nm in apparent diameter which is likely to be larger than the actual dimension of the QDs due to tip convolution effects.
Figure \[fig:Au\_island\_results\](b) shows the dissipation image taken over the three Au islands shown in figure \[fig:Au\_island\_results\](a) with the tip height of 7.5 nm and $\Vbias=10$ V. The measurements were performed at room temperature under ultra-high vacuum condition. A circular ring with a dot at its center appear for each of the islands, indicating single-electron tunneling between the Au islands and Fe(100) back-electrode through 7 ML NaCl film [@Tekiel2013]. Figure \[fig:Au\_island\_results\](c) is the dissipation versus bias voltage spectrum taken at the center of one of the three islands (indicated by $\times$). We can see three single-electron charging peaks whose separations are roughly equal. Fitting the peaks with $\cosh^{-2}[e\alpha(\Vbias-V_0)/2\kB T]$ provides $\alpha=0.04$ by which the addition energy of $137\pm27$ meV can be obtained. By taking this value as the charging energy of the Au island (assuming the negligible contribution of the quantum confinement energy), we can determine the shape of the Au island as a truncated sphere with the base diameter of 4.2 nm from the finite-element electrostatic modeling of the system using the measured height of the island. Similar charging peaks are observed in the frequency shift versus bias voltage spectrum as well, enabling the determination of the electron tunneling rate through NaCl layer ranging from 61 kHz to 285 kHz for the three peaks.
The approach presented in this section can be applied to a variety of interesting systems such as metallic islands on oxide films that are widely studied model catalysts. The interaction of even smaller islands with molecules could be investigated by monitoring the electronic level structure of these islands with the technique discussed above.
![(a) Frequency modulation mode AFM topography of Au islands on 7 ML thick NaCl. (b) Schematic diagram of the experimental setup. (c) Energy level diagram of the system. Adapted with permission from [@Tekiel2013]. Copyright (2012) American Chemical Society.[]{data-label="fig:Au_island_schematic"}](Figure16.jpg){width="15cm"}
![(a) Frequency modulation mode AFM topography image of Au islands on 7 ML thick NaCl. The height of the islands is 3.5 nm. (b) Dissipation image on the same Au islands taken at $\Vbias=10$ V and $a=1$ nm. (c) Dissipation versus $\Vbias$ curve taken on the Au island indicated as $\times$ in (a). The energy scale in the Au NP is indicated as a scale bar (50 meV) determined from the lever-arm, $\alpha=0.04$. Adapted with permission from [@Tekiel2013]. Copyright (2012) American Chemical Society.[]{data-label="fig:Au_island_results"}](Figure17.jpg){width="15.5cm"}
Possible application to other systems
=====================================
The application of $e$-EFM technique is not limited to QDs. Single-electron transistors incorporating individual molecules have been studied actively in recent years [@Natelson2006], and different charge states of single molecules have been observed in conductance spectroscopy [@Park2000; @Park2002; @Kubatkin2003]. We can therefore imagine that $e$-EFM can be applied to investigate the properties of single molecules. As the fabrication of single-molecule SETs is inherently challenging [@Yu2004] because a single molecule cannot be placed in the nanogap electrodes in a well controlled manner, $e$-EFM could be used as a new technique to investigate the electronic energy levels of individual molecules that interact with a substrate [@Kubatkin2003].
Another interesting system for $e$-EFM technique is individual dopant atoms in semiconductors. While the behavior of individual dopant atoms is becoming increasingly important in ever shrinking semiconductor electronic devices, these individual dopants are emerging as a new class of quantum mechanical entities whose discrete energy levels can be used for applications in quantum information processing such as qubits and non-classical light sources [@Koenraad2011]. Although spectroscopy of individual dopants can be performed in a single-electron transistor geometry [@Sellier2006; @Fuechsle2012], STM has also been applied to detect the charge state of individual dopant by detecting the electron tunneling through bound states due to the dopant atoms [@Teichmann2008a]. In addition to the advantage of STM based experiments which do not require the fabrication of device, $e$-EFM can be applied to the dopants that are much more weakly coupled to the bulk states as the required tunneling rate to observe a signal is much lower than that for STM (recall that 1 nA corresponds to $10^{10}$ electrons per second).
Relation to other techniques
============================
The $e$-EFM technique shares several common features with capacitance/admittance spectroscopy. Single-electron tunneling spectroscopy has been performed with capacitance spectroscopy technique [@Drexler94; @Ashoori92]. In capacitance spectroscopy, a QD is connected to a lead through a tunnel barrier and the charge in the QD is controlled by a gate voltage. Single-electron tunneling is induced by applying an ac voltage to the gate and the resulting ac current is measured by a lock-in amplifier. Although the capacitance spectroscopy has successfully been applied to gate-defined QDs [@Ashoori1993; @Ashoori92; @Ashoori93], it has not been widely adapted for individual QDs because of the difficulty in detecting the weak electrical signal from single-electron tunneling. Recently, similar measurements using radio frequency (RF) resonators have emerged as an alternative technique to current transport spectroscopy for probing the quantum states of QDs. In this scheme, an RF resonator is directly coupled to the QD through either a source/drain electrode [@Petersson2010; @Villis2011a; @Chorley2012; @Frey2012; @Hile2015] or a gate electrode [@Colless2013; @Verduijn2014; @GonzalezZalba2015; @Gonzalez-Zalba2016; @Frake2015] and the changes in the phase and amplitude of the reflected RF signal due to the single-electron tunneling are detected (RF reflectometry). The changes in the phase and dissipated power of the microwave signal are expressed for small RF excitation limit as follows: $$\label{eq:phase_response}
\Delta \phi \simeq -\frac{\pi Q_\mathrm{res}}{C_\mathrm{p}} \frac{(e\alpha)^2}{2\kB T}
\frac{1}{1+(\omega_0/\Gamma)^2} f(\dE)[1-f(\dE)]$$
$$\label{eq:capacitance}
\Delta P \simeq \frac{(e\alpha V_\mathrm{g}^\mathrm{rf})^2}{2\kB T}
\frac{\ogO}{1+(\ogO)^2}f(\dE)[1-f(\dE)]$$
where $Q_\mathrm{res}$ is the quality factor of the RF resonator, $C_\mathrm{p}$ the paracitic capacitance, $\alpha$ is the lever-arm, $\alpha = C_\mathrm{g}/C_\Sigma$, $V_\mathrm{g}^\mathrm{rf}$ the RF signal excitation amplitude [@GonzalezZalba2015].
This technique is emerging as a promising way to detect the charge state of QDs which is indispensable for readout of the qubits based on QDs. $e$-EFM can be considered as a mechanical analog of this technique. In $e$-EFM, a mechanical resonator (AFM cantilever) is capacitively coupled to the QDs in place of an RF resonator and its response to the change in charge state is detected as the changes in resonance frequency and dissipation, similarly to the RF admittance measurements. Therefore, admittance spectroscopy should be able to measure shell-filling (\[section:weak\_coupling\], \[section:strong\_coupling\]), perform excited-states spectroscopy (\[section:excited\_states\]) and determine density-of-states (\[section:dos\_effect\]) in complete analogy to our $e$-EFM measurements. Although RF reflectometry techniques have been applied for other systems than gate-defined QDs such as single dopants [@Verduijn2014; @Hile2015] or nanoparticles [@Frake2015], the devices which incorporate dopants or nanoparticles still need to be fabricated. Along with the topographic imaging capability, $e$-EFM can be a powerful tool for exploring new nanoscale material systems such as single molecules, nanoparticle complexes as possible quantum bits.
Conclusion
==========
We demonstrated that charge sensing by electrostatic force microscopy is capable of quantitative electronic energy level spectroscopy of quantum dots in three distinct systems, epitaxially grown InAs self-assembled QDs on InP, colloidal Au nanoparticles on alkanedithiol self-assembled monolayer and Au nanoislands deposited on NaCl.
Three different methods for probing the energy level structure of QDs are demonstrated. The direction of the temperature-dependent shift of the charging peaks in the small tip oscillation case (section \[section:weak\_coupling\]) and the direction of the skewness of the peaks in the large tip oscillation case (section \[section:strong\_coupling\]) enable straightforward identification of shell-filling of QDs. The energy-dependent tunneling rate which can easily be obtained by $e$-EFM offers more general approach to probe energy level structure of the QDs. Excited-states spectroscopy is also possible by measuring the tip oscillation amplitude dependence of the single-electron charging spectra. All the features described above makes $e$-EFM a powerful technique for readout of the charge state of QDs.
The spatial mapping capability of the $e$-EFM technique provides a new route for the research on unconventional QDs such as multi-QDs based on nanoparticles, individual molecules, dopants and defects. In combination with topographic imaging and structural characterization capabilities of AFM, $e$-EFM will play an important role in searching for new device elements for nanoelectronics including qubits.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank P. Poole, S. Studenikin, and A. Sachrajda at the National Research Council of Canada for providing the InAs QD sample and for fruitful discussion, L. Cockins and A. Tekiel for their experimental contributions, A. A. Clerk and S. D. Bennette for their theoretical contributions. This work was supported by funding from NSERC and FQRNT.
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