# The linear mode analysis and spin relaxation

Jin Hu<EMAIL_ADDRESS>Department of Physics, Tsinghua
University, Beijing 100084, China

###### Abstract

In this paper, a detailed analysis on normal modes of the linearized Hermite
collision operator is presented, which follows from linearizing spin Boltzmann
equation for massive fermions proposed in [1] with the non-diagonal part of
the transition rate neglected and approximating what we got with a mutilated
operator. With the assumption of total angular momentum conservation, the
collision term is proved to well describe the equilibrium state and gives
proper interpretation for collisional invariants, thus is relevant for the
research on local spin polarization. Following the familiar fashion as used in
quantum mechanics, we treat the problem of solving normal modes as a
degenerate perturbation problem and calculate the dispersion relations for
intriguing eleven zero modes, which form one-to-one correspondence to all
collisional invariants. We find that the results of spinless modes appearing
in ordinary hydrodynamics are consistent with available conclusions in
textbooks. As for spin-related modes, we obtain the frequencies up to second
order in wave vector and relate them with the dissipation of spin density
fluctuation. In addition, the ratio of two relaxation time scales for spin and
momentum is shown as a function of reduced mass, which reads that based on
present framework spin equilibration is almost as slow as momentum
equilibration as far as the strange quark spins in quark gluon plasma (QGP)
are concerned.

##  I Introduction

Recent developments in the experiments of relativistic non-central heavy-ion
collisions have seen great progress in measuring the net spin polarization of
$\Lambda$ and $\bar{\Lambda}$ hyperons [2, 3]. In the early stage of evolution
of the hot, dense matter produced in non-central heavy-ion collisions, the
medium carries a huge total angular momentum that is converted into spin
angular momentum of the particles of final states via spin-orbit coupling.
Theoretical researches on the global polarization of $\Lambda$ hyperons have
long attracted extensive interests and correspondent numerical results satisfy
experimental data well [4, 5, 6, 7, 8, 9, 10, 11, 12]. However, they have
difficulties in providing satisfying explanations for the measurements of
differential spin polarization, i.e. the dependence of $\Lambda$ polarization
on the azimuthal angle and transverse momentum [13, 14], which is usually
called “spin sign problem” [15, 16]. Resolving this problem calls for new
theoretical frameworks and concerns one still unsettled question of how
strange quark spin comes to equilibrium. There are two possible mechanisms by
which strange quark spin could equilibrate. One is that fluctuations of the
vorticity will drive the spins towards equilibrium just as fluctuations around
an external magnetic field do, the other is dynamic mechanism originating from
the scatterings between strange quarks and other particles within the medium
[17]. If taking an analogy with transport equation, the proposed mechanisms
exactly refer to the external field term and the collision term. We here
address that the crucial point for answering these questions lies in the
appropriate extraction of spin equilibration time scale. In short, new
framework must take into account the influence of dissipation instead of
treating spin as a independent equilibrated quantity. Among all the
candidates, spin hydrodynamics and spin transport are thought to be promising.
Spin hydrodynamics extends the description of ordinary fluid theory by
including spin degree of freedom, based on which the relaxation of spin and
the evolution of vorticity become the focus of theoretical research, while
spin transport, namely, focuses on constructing a consistent theory for both
spin and momentum transport. Though, in principle, the latter possesses a
wider range of application, the constructed transport equation on the market
is too involved to be solved on account of the complexity and non-linearity.

There are many developments in the investigation of spin hydrodynamic. Among
them, “ideal” spin hydrodynamics was proposed in the context of the QGP [18]
and for massive spin-$1/2$ fermions [19]. Recently, first-order spin
hydrodynamics including non-equilibrium corrections has also been put into
efforts [20, 21, 22, 23, 24]. In a recent work, the authors construct a
second-order spin hydrodynamic theory based on the method of moment expansion
[25] from spin transport equation [1, 26], which is also our starting point
for linear mode analysis. As a reminder, one can also see [27, 28, 29, 30, 31,
32, 33] for continuous efforts on the research of spin transport theory.

In this paper, we present a detailed analysis on normal modes of the
linearized collision operator. To that end, we adopt the transport equation
along with a nonlocal collision kernel proposed in [1] as the start point of
our calculation. The linear mode analysis is closely associated with
hydrodynamics because the theory of fluids can be completely constructed from
these normal modes, which are nothing but collisional invariants, i.e, the
microscopic correspondence of macroscopic conserved laws. Moreover, one can
relate dispersion relations of normal modes with the dissipation of various
fluctuation amplitudes, from which various relaxation time scales can be
extracted. As is raised in previous paragraphs, the comparison for these time
scales is significant for investigating spin polarization and spin
equilibration. This paper is organized as follows. In Sec. II we present a
short review of spin Boltzmann equation with a nonlocal collision term. In
Sec. III the equilibrium distribution function is briefly discussed and the
conditions for global equilibrium are also obtained. After that, the mutilated
operator inheriting the important properties of full linearized collision
operator is proposed in Sec. IV. In Sec. V, we present the detailed analysis
on normal modes of the approximated linearized operator, i.e., mutilated
collision operator following the fashion as used in quantum mechanics [34,
35]. Summary and outlook are given in Sec. VI. Natural units $k_{B}=c=\hbar=1$
are utilized. The metric tensor here is given by
$g^{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ , while $\Delta^{\mu\nu}\equiv
g^{\mu\nu}-u^{\mu}u^{\nu}$ is the projection tensor orthogonal to the four-
vector fluid velocity $u^{\mu}$.

In addition, we employ the symmetric/antisymmetric shorthand notations:

$\displaystyle X^{(\mu\nu)}$ $\displaystyle\equiv$
$\displaystyle(X^{\mu\nu}+X^{\nu\mu})/2,$ (1) $\displaystyle X^{[\mu\nu]}$
$\displaystyle\equiv$ $\displaystyle(X^{\mu\nu}-X^{\nu\mu})/2,$ (2)
$\displaystyle X^{\langle\mu\nu\rangle}$ $\displaystyle\equiv$
$\displaystyle\bigg{(}\frac{\Delta^{\mu}_{\alpha}\Delta^{\nu}_{\beta}+\Delta^{\nu}_{\alpha}\Delta^{\mu}_{\beta}}{2}-\frac{\Delta^{\mu\nu}\Delta_{\alpha\beta}}{3}\bigg{)}X^{\alpha\beta}.$
(3)

Specially, we decompose the derivative $\partial$ according to

$\displaystyle\partial^{\mu}=u^{\mu}D+\nabla^{\mu},\quad D\equiv
u^{\mu}\partial_{\mu},\quad\nabla^{\mu}\equiv\Delta^{\mu\nu}\partial_{\nu}.$
(4)

##  II Review of the nonlocal transport equation

We start with the spin Boltzmann equation with a nonlocal collision term for
massive fermions proposed in [1]. Assuming that the evolution of the system of
our interest is governed by the proposed on-shell Boltzmann equation, which
extends the phase space to incorporate the variable $\bm{s}$ as a classical
description of spin degrees of freedom,

$\displaystyle p\cdot\partial f(x,p,\bm{s})=C[f],$ (5)

with the nonlocal collision term

$\displaystyle C[f]$ $\displaystyle\equiv$ $\displaystyle\int
d\Gamma_{1}d\Gamma_{2}d\Gamma^{\prime}\,\mathcal{W}\,[f(x+\Delta_{1},p_{1},\bm{s}_{1})f(x+\Delta_{2},p_{2},\bm{s}_{2})-f(x+\Delta,p,\bm{s})f(x+\Delta^{\prime},p^{\prime},\bm{s}^{\prime})],$
(6)

where the measure appearing in the collision kernel is defined as
$d\Gamma\equiv d^{4}p\,\delta(p^{2}-m^{2})dS(p)$ , the newly introduced
measure $dS(p)$ is given in Eq.(11) and the other collision term corresponding
to only spin changes in [1] is neglected. Here we note the transition rate
$\mathcal{W}$ is

$\displaystyle\mathcal{W}$ $\displaystyle\equiv$
$\displaystyle\delta^{(4)}(p+p^{\prime}-p_{1}-p_{2})\frac{1}{8}\sum_{s,r,s^{\prime},r^{\prime},s_{1},s_{2},r_{1},r_{2}}h_{sr}(p,\bm{s})h_{s^{\prime}r^{\prime}}(p^{\prime},\bm{s}^{\prime})\,h_{s_{1}r_{1}}(p_{1},\bm{s}_{1})\,h_{s_{2}r_{2}}(p_{2},\bm{s}_{2})$
(7)
$\displaystyle\times\langle{p,p^{\prime};r,r^{\prime}|t|p_{1},p_{2};s_{1},s_{2}}\rangle\langle{p_{1},p_{2};r_{1},r_{2}|t^{\dagger}|p,p^{\prime};s,s^{\prime}}\rangle\;$

with

$h_{sr}(p,\bm{s})\equiv\delta_{sr}+\frac{1}{2m}\,\bar{u}_{s}(p)\gamma^{5}\bm{s}\cdot\gamma
u_{r}(p)\;.$ (8)

As is shown by the two terms in Eq.(8), we split $\mathcal{W}$ into the
unpolarized part and polarized one. Namely, when neglecting the non-diagonal
part in Eq.(8), i.e., the term linearized to spin $\bm{s}$, the transition
rate takes exactly the unpolarized form (sum up the final states and average
the initial states),

$\displaystyle\bar{\mathcal{W}}$ $\displaystyle\equiv$
$\displaystyle\delta^{(4)}(p+p^{\prime}-p_{1}-p_{2})\,\frac{1}{8}\sum_{r,r^{\prime},r_{1},r_{2}}\langle{p,p^{\prime};r,r^{\prime}|t|p_{1},p_{2};r_{1},r_{2}}\rangle\langle{p_{1},p_{2};r_{1},r_{2}|t^{\dagger}|p,p^{\prime};r,r^{\prime}}\rangle\;,$
(9)

where the $\gamma$ matrices, spinor $u_{s}(p)$ and spin indices $r,s$
correspond to the spinor description for fermions as often used, and the
matrix element of $t$ is the conventional scattering amplitude defined in
quantum field theory. To proceed, we comment that the crucial point for the
nontrivial extension of the collision term lies in the spatial shift $\Delta$
manifesting the nonlocality of the collisions,

$\Delta^{\mu}\equiv-\frac{1}{2m(p\cdot\hat{t}+m)}\,\epsilon^{\mu\nu\alpha\beta}p_{\nu}\hat{t}_{\alpha}\bm{s}_{\beta}\;,$
(10)

where $\hat{t}^{\mu}$ is the time-like unit vector which is
$(1,\boldsymbol{0})$ in the frame where $p^{\mu}$ is measured. The collision
shift well captures the properties of spin-orbit coupling in nonlocal
collisions, thus is highly relevant for discussing the spin sign problem of
local polarization of $\Lambda$.

To proceed, we move on to the classical spin. Spin here is treated as an
additional variable in phase space [36, 37, 38, 39, 40, 26], which immediately
connects the first-principle quantum description to a “classical” description
of spin. In previous paragraphs, we have introduced the covariant integration
measure for spin

$\int dS(p)\equiv\sqrt{\frac{p^{2}}{3\pi^{2}}}\int
d^{4}\bm{s}\,\delta(\bm{s}\cdot\bm{s}+3)\delta(p\cdot\bm{s})\;.$ (11)

Then the following useful integrals can be easily obtained via rather
straightforward calculations,

$\displaystyle\int dS(p)$ $\displaystyle=$ $\displaystyle 2\;,$ (12a)
$\displaystyle\int dS(p)\,\bm{s}^{\mu}$ $\displaystyle=$ $\displaystyle 0\;,$
(12b) $\displaystyle\int dS(p)\,\bm{s}^{\mu}\bm{s}^{\nu}$ $\displaystyle=$
$\displaystyle-2\left(g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\;,$ (12c)
$\displaystyle\int dS(p)\,\bm{s}^{\lambda}\bm{s}^{\mu}\bm{s}^{\nu}$
$\displaystyle=$ $\displaystyle 0,$ (12d) $\displaystyle\int
dS(p)\Sigma^{\mu\nu}_{\bm{s}}\Sigma_{\bm{s}}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{2}{m^{2}}(g^{\mu\alpha}g^{\nu\beta}p^{2}+g^{\mu\beta}p^{\nu}p^{\alpha}+g^{\nu\alpha}p^{\mu}p^{\beta}-[\mu\leftrightarrow\nu])\;,$
(12e)

where the dipole tensor
$\Sigma_{\bm{s}}^{\mu\nu}\equiv-\frac{1}{m}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}\bm{s}_{\beta}$
is interpreted as the spin angular momentum of the particle [1].

With the extended phase space, our interesting tensors such as particle
current, energy-momentum tensor and spin tensor can be convenient written as

$\displaystyle N^{\mu}\equiv\int\,d\Gamma\,p^{\mu}f(x,p,\bm{s}),$ (13)
$\displaystyle
T_{\text{HW}}^{\mu\nu}\equiv\int\,d\Gamma\,p^{\mu}p^{\nu}f(x,p,\bm{s})+T_{\text{HW}}^{[\mu\nu]},$
(14) $\displaystyle
S_{\text{HW}}^{\lambda,\mu\nu}\equiv\int\,d\Gamma\,p^{\lambda}(\frac{1}{2}\Sigma^{\mu\nu}_{\bm{s}}-\frac{1}{2m^{2}}p^{[\mu}\partial^{\nu]})f(x,p,\bm{s}),$
(15)

where we have chosen the psudo-gauge proposed by Hilgevoord and Wouthuysen
(HW) [41, 42, 26], the spin tensor of which is proved to be conserved in
global equilibrium and non-conserved away from global equilibrium because of
mutual conversion between spin and orbital angular momentum and thus more
intuitively suitable for our discussion herein and we keep
$T_{\text{HW}}^{[\mu\nu]}$ to stress that $T^{\text{HW}}$ is not symmetric and
is in the second order in $\hbar$ expansion while its form is left unspecified
within our paper, see [25, 26] for more details.

By taking Landau choice of fluid velocity and Landau matching conditions,

$\displaystyle T_{\text{S}}^{\mu\nu}u_{\nu}=eu^{\mu},\quad
u_{\mu}N^{\mu}=u_{\mu}N^{\mu}_{\text{eq}},\quad
u_{\mu}T_{\text{S}}^{\mu\nu}u_{\nu}=u_{\mu}T^{\mu\nu}_{\text{eq}}u_{\nu},\quad
u_{\lambda}J^{\lambda,\mu\nu}=u_{\lambda}J_{\text{eq}}^{\lambda,\mu\nu},$ (16)

where $T_{\text{S}}^{\mu\nu}$ refers to the symmetric part of
$T^{\mu\nu}_{\text{HW}}$ and the total angular momentum tensor is defined as
$J^{\lambda,\mu\nu}\equiv
S_{\text{HW}}^{\lambda,\mu\nu}+x^{\mu}T_{\text{HW}}^{\lambda\nu}-x^{\nu}T_{\text{HW}}^{\lambda\mu}$,
we allow the following decompositions,

$\displaystyle N^{\mu}=nu^{\mu}+V^{\mu},$ (17) $\displaystyle
T_{\text{S}}^{\mu\nu}=eu^{\mu}u^{\nu}-P\Delta^{\mu\nu}+\pi^{\mu\nu}+\Pi\Delta^{\mu\nu},$
(18) $\displaystyle
S_{\text{HW}}^{\lambda,\mu\nu}=u^{\lambda}S^{\mu\nu}+\delta
S^{\lambda,\mu\nu},$ (19)

where $n,e,P,S^{\mu\nu}$ are the particle number density, energy density,
static pressure and spin density, and the dissipative quantities
$V^{\mu},\pi^{\mu\nu}$ and $\Pi$ are the diffusion current, shear stress
tensor, bulk viscous pressure respectively. Note that generally
$u_{\lambda}\delta S^{\lambda,\mu\nu}\neq 0$.

##  III Equilibrium

In this section, we will show that the collision term Eq.(6) is consistent
with the standard form of spin-dependent local equilibrium distribution
function [43, 18],

$\displaystyle
f_{\text{leq}}(x,p,\bm{s})=\frac{1}{(2\pi)^{3}}\exp[\xi-\beta\cdot
p+\frac{\Omega_{\mu\nu}\Sigma_{\bm{s}}^{\mu\nu}}{4}],$ (20)

where $\Omega_{\mu\nu}$ represents spin potential, while
$\beta^{\mu}\equiv\frac{u^{\mu}}{T},\xi\equiv\frac{\mu}{T},\beta\equiv\frac{1}{T}$
with the temperature $T$, and the chemical potential $\mu$ introduced for
conserved particle number (only elastic scatterings are considered). The
exponent in Eq.(20) is exactly the linear combination of all conserved
quantities, and $\xi,\beta$ and $\Omega^{\mu\nu}$ are the correspondent
Lagrangian multipliers maximizing the total entropy of the system. To prove
this, the substitution of Eq.(20) into Eq.(6) leads to

$\displaystyle C[f_{\text{leq}}]$ $\displaystyle=$
$\displaystyle-\frac{1}{(2\pi)^{6}}\int
d\Gamma^{\prime}d\Gamma_{1}d\Gamma_{2}\,\bar{\mathcal{W}}\,F[p,p^{\prime},p_{1},p_{2};\bm{s},\bm{s}^{\prime},\bm{s}_{1},\bm{s}_{2}]\exp(2\xi-\beta\cdot(p+p^{\prime}))$
(21)
$\displaystyle\times\big{[}-\partial_{\mu}\xi\frac{}{}\left(\Delta_{1}^{\mu}+\Delta_{2}^{\mu}-\Delta^{\mu}-\Delta^{\prime\mu}\right)+\partial_{\mu}\beta_{\nu}\frac{}{}\left(\Delta_{1}^{\mu}p_{1}^{\nu}+\Delta_{2}^{\mu}p_{2}^{\nu}-\Delta^{\mu}p^{\nu}-\Delta^{\prime\mu}p^{\prime\nu}\right)$
$\displaystyle\quad\,\,-\frac{1}{4}\Omega_{\mu\nu}\left(\Sigma_{\bm{s}_{1}}^{\mu\nu}+\Sigma_{\bm{s}_{2}}^{\mu\nu}-\Sigma_{\bm{s}}^{\mu\nu}-\Sigma_{\bm{s}^{\prime}}^{\mu\nu}\right)\big{]},$

where the local equilibrium distribution is Taylor expanded to first order in
$\Omega$ assuming small spin potential ( if the system in discussion is close
to the state of global equilibrium, $\Omega$ is about the order of the
gradient of $\beta$ field ).

When including spin, it is the global equilibrium distribution function that
makes the collision term vanish as long as the total angular momentum

$\displaystyle
J^{\mu\nu}=2\Delta^{[\mu}p^{\nu]}+\frac{1}{2}\Sigma^{\mu\nu}_{\bm{s}}$ (22)

is conserved in a collision, which is distinct from traditional definition for
local equilibrium. In that case, the conditions for vanishing Eq.(21) are

$\displaystyle\partial_{(\mu}\beta_{\nu)}=0,\quad\xi=\text{const},$
$\displaystyle\Omega_{\mu\nu}=-\partial_{[\mu}\beta_{\nu]}=\text{const},$ (23)

which implies that the spin potential $\Omega_{\mu\nu}$ is fixed to thermal
vorticity $\frac{1}{2}(\partial_{\nu}\beta_{\mu}-\partial_{\mu}\beta_{\nu})$.
$\beta^{\mu}$ can be further decomposed into a translation ($a^{\mu}$) and a
rigid rotation ($\Omega^{\mu\nu}x_{\nu}$) in global equilibrium,

$\displaystyle\beta^{\mu}=a^{\mu}+\Omega^{\mu\nu}x_{\nu},\quad
a^{\mu}=\text{const},$ (24)

which are consistent with the previous conclusions drawn in [43, 18]. One may
observe that Eq.(21) contains an extra dimensionless factor $F$ compared to
Eq.(6) and $\mathcal{W}$ is replaced by $\bar{\mathcal{W}}$. The reason why
these changes are necessary is elaborated in [44] and we want to emphasize
that the reformed collision operator or collision term respects hermity, non-
negative property and detailed balance as well as the collisional invariance
for total angular momentum, which is a necessary condition for a consistent
description of global equilibrium.

##  IV Mutilated collision operator

Following [44], we choose a quiescent (also unpolarized) background fluid
i.e., $u^{\mu}=(1,0,0,0),\,\Omega^{\mu\nu}=0$ with $\delta u^{\mu}$ and
$\delta\Omega^{\mu\nu}$ treated as perturbations. With more details given
therein, we start with the equation for $\tilde{\chi}$ (we want to find a
solution of the form $\chi\sim\tilde{\chi}e^{-ik\cdot x}$ to the linearized
spin Boltzmann equation)

$\displaystyle\tau\omega\tilde{\chi}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}+L_{2}[\tilde{\chi}]=-iL_{1}[\tilde{\chi}],$
(25)

with notations

$\displaystyle\tau\equiv\frac{p\cdot u}{T},\quad\omega\equiv\frac{u\cdot
k}{n\sigma(T)},\quad\hat{p}\equiv\frac{p}{T},\quad\kappa^{\alpha}\equiv\frac{\Delta^{\alpha\beta}k_{\beta}}{n\sigma(T)},\quad\kappa\equiv\sqrt{-\kappa\cdot\kappa},\quad
l^{\alpha}\equiv\frac{\kappa^{\alpha}}{\kappa},$ (26)

where $\sigma(T)$ is an arbitrary constant with the dimension of cross
sections [45], $\chi$ denotes the deviation from the global equilibrium
distribution, and $L_{1},L_{2}$ are the dimensionless collision operators
given by

$\displaystyle L_{1}[\phi]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(2\pi)^{3}n\sigma(T)T}\int
d\Gamma^{\prime}d\Gamma_{1}d\Gamma_{2}\,\bar{\mathcal{W}}\,F[p,p^{\prime},p_{1},p_{2};\bm{s},\bm{s}^{\prime},\bm{s}_{1},\bm{s}_{2}]\exp(\xi-\beta\cdot
p^{\prime}\,)$ (27) $\displaystyle\times$
$\displaystyle\Big{[}\phi(k,p,\bm{s})+\phi(k,p^{\prime},\bm{s}^{\prime})-\phi(k,p_{1},\bm{s}_{1})-\phi(k,p_{2},\bm{s}_{2})\,\Big{]},$
$\displaystyle L_{2}[\phi]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(2\pi)^{3}T}\int
d\Gamma^{\prime}d\Gamma_{1}d\Gamma_{2}\,\bar{\mathcal{W}}\,F[p,p^{\prime},p_{1},p_{2};\bm{s},\bm{s}^{\prime},\bm{s}_{1},\bm{s}_{2}]\exp(\xi-\beta\cdot
p^{\prime}\,)$ (28) $\displaystyle\times$
$\displaystyle\Big{[}\Delta\cdot\kappa\phi(k,p,\bm{s})+\Delta^{\prime}\cdot\kappa\phi(k,p^{\prime},\bm{s}^{\prime})-\Delta_{1}\cdot\kappa\phi(k,p_{1},\bm{s}_{1})-\Delta_{2}\cdot\kappa\phi(k,p_{2},\bm{s}_{2})\Big{]}.$

Here we remind that the inner product is defined in global equilibrium as

$\displaystyle(B,C)=\frac{1}{(2\pi)^{3}}\int d\Gamma\exp(\xi-\beta\cdot
p)B(p,\bm{s})C(p,\bm{s}).$ (29)

According to the calculations in [44], $L_{2}$ contributes nothing to the
dispersion law, which is known from the fact that $L_{2}$ is absent in the
formulas for $\omega$. Therefore, we neglect this term from now on but the
complexity of solving Eq.(25) does not decrease a lot. The linearized
collision operator has a very complicated structure even for the simplest
interaction. However, it seems reasonable that the qualitative features of the
normal modes depend only upon the universal properties independent of which
interaction to take, i.e., that the spectrum of the collision operator
$-L_{1}$ is composed of a eleven-fold degenerate zero, which are nothing but
all collisional invariants, and a sequence of negative eigenvalues. These zero
normal modes are protected by the conservation laws or essentially the
translational and Lorentz symmetries. In the linear analysis, a small
fluctuation $\chi$ on top of background distribution can always be expanded
with a set of eigenstates of the linearized collision operator formally [34]

$\displaystyle\chi=\sum_{n=0}^{\infty}a_{n}e^{\gamma_{n}t}|\gamma_{n}\rangle,$
(30)

where $\gamma_{n}$ and $|\gamma_{n}\rangle$ constitute the representation of
eigen spectrum of linearized collision operator, $a_{n}$ denotes the expansion
coefficient and $t$ is seen as a dimensionless time. As the system evolves in
time $t$, protected modes with $\gamma_{n}=0$ remain unchanged and other
negative modes become less important after a characteristic time scale. If
taking this idea as an ansatz, the full linearized operator can be
approximated as a mutilated collision operator in which all the negative
eigenvalues collapse into a single eigenvalue of infinite degeneracy [46],

$\displaystyle-L_{1}\approx n\sigma u\cdot
p\big{(}-\gamma+\gamma\sum_{n=1}^{11}|\lambda_{n}\rangle\langle\lambda_{n}|\big{)}$
(31)

with $|\lambda_{n}\rangle$ being the eleven degenerate orthonormal
eigenvectors of zero eigenvalue and $-\gamma$ is the remaining negative
eigenvalue (it is suggestive that $-\gamma$ is chosen to be the largest one of
all negative eigenvalues). To be concrete, we rewrite the Eq.(31) as

$\displaystyle-L_{1}\phi(k,p,\bm{s})\approx-n\sigma u\cdot
p\gamma\big{(}\phi-\sum_{n=1}^{11}(\tilde{\psi}_{n},\tau\phi)\tilde{\psi}_{n}\;\big{)},$
(32)

where the eigenfunction set is defined in Eq.(V) and there are no differences
between $\psi_{n}$ and $\tilde{\psi}_{n}$ for lack of the coordination $x$
dependence in both eigenfunction sets. It is easy to verify that this new
defined operator (the right hand side of Eq.(32)) has the basic features
belonging to the full operator with respect to the orthonormal condition
Eq.(37). One may have doubt why taking the orthonormal condition with a weight
function $\tau$. To answer this question, one should be informed that the
frequency $\omega$ of our interest is not the eigenvalue of the operator
$L_{1}$ but $\tau\omega$ is viewing Eq.(34). In addition, one may observe that
the proposed approximation is exactly reformed relaxation time approximation
(RTA) respecting $L_{1}|\lambda_{n}\rangle=0,n=1,2,\cdots 11$ and
$-L_{1}|\lambda_{n}\rangle=-\gamma|\lambda_{n}\rangle$ with $n>11$. This extra
factor $\tau$ is consistent with the form of relativistic relaxation time
approximation. Last but not the least, this novel version of RTA is proved to
reconcile the momentum dependence of the relaxation time with the conservation
laws. In fact, when the relaxation time has momentum dependence as it should
do in general, the argument via matching conditions used to resolve this
contradiction also fails. Here we do not go into details further and similar
results are also reported in a recent letter [47]. For simplicity, we require
that $\gamma$ is momentum independent hereafter and when nothing confusing
happens, $L_{1}$ represents the mutilated operator or RTA operator instead of
the full operator. Eventually the equation to be solved is

$\displaystyle\tau\omega\tilde{\chi}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}=-i\gamma\tau\big{(}\tilde{\chi}-\sum_{n=1}^{11}(\tilde{\psi}_{n},\tau\tilde{\chi})\tilde{\psi}_{n}\;\big{)}.$
(33)

##  V Degenerate perturbation theory and linear mode analysis

As a well-posed problem in the perturbation theory, the solutions to Eq.(25)
can be sought in the following steps. Firstly, treat the gradients of the
thermodynamic variables as a perturbation with respect to the linearized
collision operator $-iL_{1}$, then we obtain an eigenvalue problem for the
unperturbed equation,

$\displaystyle-iL_{1}\tilde{\chi}^{(0)}=\tau\omega^{(0)}\tilde{\chi}^{(0)}.$
(34)

Since the mutilated operator is constructed based on the priori knowledge
about the spectrum of the full collision operator, thus the conservation laws
for energy-momentum, particle number and total angular momentum have already
been contained and the eigenfunctions are all the linear combination of all
collisional invariants $1,p^{\mu}$ and $J^{\mu\nu}$. Compared to ordinary
hydrodynamic description, the six new modes arise from the nontrivial dynamics
of spin and we identify them with the spin modes. Due to the eleven-fold
degeneracy, the perturbation term should be taken into account to remove or
partly remove the degeneracy.

For the first-order perturbation of $p\cdot\kappa$, the evaluation of the
eigen spectrum can be done in a familiar fashion used in quantum mechanics. To
proceed, we denote the $n$-th order eigenvalues and eigenfunctions as
$\omega^{(n)}$ and $\tilde{\chi}^{(n)}$ for concreteness

$\displaystyle\tilde{\chi}=\tilde{\chi}^{(0)}+\tilde{\chi}^{(1)}+\cdots,$
$\displaystyle\omega=\omega^{(0)}+\omega^{(1)}+\omega^{(2)}+\cdots.$ (35)

Following the procedures in quantum mechanics, $\tilde{\chi}^{n}$ with $n>0$
is chosen as the combination of the eigenfunctions of Eq.(34) excluding the
zeroth-order eigenfunctions. In order to break the eleven-fold degeneracy, the
standard method of Schmidt orthogonalization is adopted and the eigenfunctions
for these eleven-fold degenerate zeros can be taken as

$\displaystyle\tilde{\psi}_{1}=\frac{1}{\sqrt{V_{1,1}}},\quad\tilde{\psi}_{2}=\beta\frac{u\cdot
p-\frac{e}{n}}{\sqrt{V_{2,2}}},\quad\tilde{\psi}_{3}=\frac{\beta l\cdot
p}{\sqrt{V_{3,3}}},\quad\tilde{\psi}_{4}=\frac{\beta j\cdot
p}{\sqrt{V_{3,3}}},\quad\tilde{\psi}_{5}=\frac{\beta v\cdot
p}{\sqrt{V_{3,3}}},$
$\displaystyle\tilde{\psi}_{6}=\frac{u_{\mu}J^{\mu\nu}l_{\nu}}{\sqrt{V_{6,6}}},\quad\tilde{\psi}_{7}=\frac{u_{\mu}J^{\mu\nu}j_{\nu}}{\sqrt{V_{6,6}}},\quad\tilde{\psi}_{8}=\frac{u_{\mu}J^{\mu\nu}v_{\nu}}{\sqrt{V_{6,6}}},\quad\tilde{\psi}_{9}=\frac{l_{\mu}J^{\mu\nu}j_{\nu}}{\sqrt{V_{9,9}}},$
$\displaystyle\tilde{\psi}_{10}=\frac{l_{\mu}J^{\mu\nu}v_{\nu}}{\sqrt{V_{9,9}}},\quad\tilde{\psi}_{11}=\frac{j_{\mu}J^{\mu\nu}v_{\nu}}{\sqrt{V_{9,9}}},$
(36)

to fulfill the orthonormal condition

$\displaystyle(\tilde{\psi}_{\alpha},\tau\tilde{\psi}_{\beta})=\delta_{\alpha\beta},$
(37)

where the definitions of two auxiliary vectors $j,v$ and the normalized factor
$V_{i,j}$ are all put in Appendix.(B). We now seek the solutions to the
inhomogeneous integral equation for $\tilde{\chi}^{(1)}$

$\displaystyle-
iL_{1}\tilde{\chi}_{\alpha}^{(1)}=\tau\omega_{\alpha}^{(1)}\tilde{\chi}_{\alpha}^{(0)}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}_{\alpha}^{(0)}.$
(38)

According to the fundamental theory of degenerate perturbation, when the
inhomogeneity is orthogonal to the solution of the associated homogeneous
equation Eq.(34), i.e.,

$\displaystyle(\tilde{\psi}_{\gamma},\,\tau\omega^{(1)}\tilde{\chi}^{(0)}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}^{(0)})=0,\quad\gamma=1\cdots,11$
(39)

with

$\displaystyle\tilde{\chi}_{\alpha}^{(0)}=\sum_{i=1}^{11}C_{\alpha\beta}\tilde{\psi}_{\beta},$
(40)

a unique solution to Eq.(38) exists. To ensure the existence of nontrivial
solutions of Eq.(39), the frequency $\omega$ has to obey the dispersion
relation, i.e., the secular equation,

$\displaystyle\text{Det}W_{\gamma\beta}=0,$ (41)

with the matrix elements taking the form,

$\displaystyle
W_{\gamma\beta}=\omega^{(1)}(\tilde{\psi}_{\gamma},\tau\tilde{\psi}_{\beta})+(\tilde{\psi}_{\gamma},\hat{p}\cdot\kappa\tilde{\psi}_{\beta}).$
(42)

With all these matrix elements calculated in Appendix.(C), the secular
equation is,

$\displaystyle\left|\begin{array}[]{cccccccccccc}\omega^{(1)}&0&H_{1,3}&0&0&0&0&0&0&0&0\\\
0&\omega^{(1)}&H_{2,3}&0&0&0&0&0&0&0&0\\\
H_{1,3}&H_{2,3}&\omega^{(1)}&0&0&0&0&0&0&0&0\\\
0&0&0&\omega^{(1)}&0&0&0&0&0&0&0\\\ 0&0&0&0&\omega^{(1)}&0&0&0&0&0&0\\\
0&0&0&0&0&\omega^{(1)}&0&0&0&0&0\\\ 0&0&0&0&0&0&\omega^{(1)}&0&H_{7,9}&0&0\\\
0&0&0&0&0&0&0&\omega^{(1)}&0&H_{7,9}&0\\\
0&0&0&0&0&0&H_{7,9}&0&\omega^{(1)}&0&0\\\
0&0&0&0&0&0&0&H_{7,9}&0&\omega^{(1)}&0\\\ 0&0&0&0&0&0&0&0&0&0&\omega^{(1)}\\\
\end{array}\right|=0\,,$ (54)

the roots of this equation are

$\displaystyle\omega^{(1)}_{1}=-\omega^{(1)}_{2}=\sqrt{H^{2}_{2,3}+H^{2}_{1,3}}{},\quad\omega^{(1)}_{3}=\omega^{(1)}_{4}=\omega^{(1)}_{5}=0,$
$\displaystyle\omega^{(1)}_{6}=\omega^{(1)}_{11}=0,\quad\omega^{(1)}_{7}=\omega^{(1)}_{8}=H_{7,9},\quad\omega^{(1)}_{9}=\omega^{(1)}_{10}=-H_{7,9},$
(55)

and one appropriate linear combination of $\tilde{\psi}$ satisfying the
solubility condition Eq.(39) can be chosen to be

$\displaystyle\tilde{\chi}^{(0)}_{1}=\frac{1}{\sqrt{2}}(-\frac{H_{1,3}}{\sqrt{H^{2}_{1,3}+H^{2}_{2,3}}}\tilde{\psi}_{1}-\frac{H_{2,3}}{\sqrt{H^{2}_{1,3}+H^{2}_{2,3}}}\tilde{\psi}_{2}+\tilde{\psi}_{3}),\quad\tilde{\chi}^{(0)}_{2}=\frac{1}{\sqrt{2}}(\frac{H_{1,3}}{\sqrt{H^{2}_{1,3}+H^{2}_{2,3}}}\tilde{\psi}_{1}+\frac{H_{2,3}}{\sqrt{H^{2}_{1,3}+H^{2}_{2,3}}}\tilde{\psi}_{2}+\tilde{\psi}_{3}),$
$\displaystyle\tilde{\chi}^{(0)}_{3}=\frac{1}{\sqrt{\frac{H_{2,3}^{2}}{H^{2}_{1,3}}+1}}(-\frac{H_{2,3}}{H_{1,3}}\tilde{\psi}_{1}+\tilde{\psi}_{2}),\quad\tilde{\chi}^{(0)}_{4}=\tilde{\psi}_{4},\quad\tilde{\chi}^{(0)}_{5}=\tilde{\psi}_{5},\quad\tilde{\chi}^{(0)}_{6}=\tilde{\psi}_{6},\quad$
$\displaystyle\tilde{\chi}^{(0)}_{7}=\frac{1}{\sqrt{2}}(\tilde{\psi}_{7}-\tilde{\psi}_{9}),\quad\tilde{\chi}^{(0)}_{8}=\frac{1}{\sqrt{2}}(\tilde{\psi}_{8}-\tilde{\psi}_{10}),\quad\tilde{\chi}^{(0)}_{9}=\frac{1}{\sqrt{2}}(\tilde{\psi}_{7}+\tilde{\psi}_{9}),\quad\tilde{\chi}^{(0)}_{10}=\frac{1}{\sqrt{2}}(\tilde{\psi}_{8}+\tilde{\psi}_{10}),\quad\tilde{\chi}^{(0)}_{11}=\tilde{\psi}_{11}.$
(56)

One can readily verify that the results of the first five modes are the same
as those in [45], which is independent of the details of interactions
involved. It is reasonable because we are solving the same problem as far as
the spinless modes are concerned. By substituting the zeroth order eigenvalues
and eigenfunctions into Eq.(38) the first-order eigenfunctions are

$\displaystyle\tilde{\chi}_{\alpha}^{(1)}=\frac{i}{\gamma\tau}(\tau\omega_{\alpha}^{(1)}\tilde{\chi}_{\alpha}^{(0)}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}_{\alpha}^{(0)}),\quad\alpha=1,2\cdots,11.$
(57)

Subsequently, the second order perturbation equation reads,

$\displaystyle-
iL_{1}\tilde{\chi}^{(2)}=\tau\omega^{(1)}\tilde{\chi}^{(1)}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}^{(1)}+\tau\omega^{(2)}\tilde{\chi}^{(0)}.$
(58)

At this time, the solubility condition turns into

$\displaystyle(\tilde{\chi}^{(0)}_{\alpha},\,\tau\omega_{\beta}^{(1)}\tilde{\chi}^{(1)}_{\beta}+\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}^{(1)}_{\beta}+\tau\omega_{\beta}^{(2)}\tilde{\chi}_{\beta}^{(0)})=0,$
(59)

equivalently, this can be written as with the assistance of bracket notation,

$\displaystyle\omega^{(2)}_{\alpha}=i[\tilde{\chi}^{(1)}_{\alpha},\tilde{\chi}^{(1)}_{\alpha}],\quad\alpha=1,2\cdots,11,$
(60)

where $\tilde{\chi}^{(1)}$ is given by Eq.(57) and the bracket is defined as
$[B,C]\equiv(L_{1}[B],C)$. Substitute the mutilated operator $L_{1}$ into
Eq.(60), and we get

$\displaystyle\omega^{(2)}_{\alpha}=$
$\displaystyle-i\big{(}(\hat{p}^{\mu}\kappa_{\mu}+\tau\omega^{(1)}_{\alpha})\tilde{\chi}^{(0)}_{\alpha},\,\frac{1}{\gamma\tau}(\hat{p}^{\mu}\kappa_{\mu}+\tau\omega^{(1)}_{\alpha})\tilde{\chi}^{(0)}_{\alpha}\big{)}$
$\displaystyle=$
$\displaystyle-\frac{i}{\gamma}\big{(}\,(\tilde{\chi}^{(0)}_{\alpha}\tilde{\chi}^{(0)}_{\alpha},\frac{1}{\tau}\hat{p}^{\mu}\kappa_{\mu}\hat{p}^{\nu}\kappa_{\nu})+\omega^{(1)}_{\alpha}\omega^{(1)}_{\alpha}(\tilde{\chi}^{(0)}_{\alpha},\tau\tilde{\chi}^{(0)}_{\alpha})+2\omega^{(1)}_{\alpha}(\tilde{\chi}^{(0)}_{\alpha},\hat{p}^{\mu}\kappa_{\mu}\tilde{\chi}^{(0)}_{\alpha})\big{)}.$
(61)

With a rather lengthy calculation, we obtain the frequencies in terms of
various thermodynamic integrals $I,L$ and $N$ displayed in Appendix.(A)

$\displaystyle\omega_{1}=\sqrt{H^{2}_{2,3}+H^{2}_{1,3}}-\frac{i}{\gamma}(Q_{1,1}-H^{2}_{2,3}-H^{2}_{1,3}),$
$\displaystyle\omega_{2}=-\sqrt{H^{2}_{2,3}+H^{2}_{1,3}}-\frac{i}{\gamma}(Q_{1,1}-H^{2}_{2,3}-H^{2}_{1,3}),\quad$
$\displaystyle\omega_{3}=-\frac{i}{\gamma}Q_{3,3},\quad\omega_{4}=-\frac{i}{\gamma}Q_{4,4},\quad\omega_{5}=-\frac{i}{\gamma}Q_{4,4},\quad\omega_{6}=-\frac{i}{\gamma}Q_{6,6},\quad$
$\displaystyle\omega_{7}=H_{7,9}-\frac{i}{\gamma}(Q_{7,7}-H^{2}_{7,9}),\quad\omega_{8}=H_{7,9}-\frac{i}{\gamma}(Q_{7,7}-H^{2}_{7,9}),\quad$
$\displaystyle\omega_{9}=-H_{7,9}-\frac{i}{\gamma}(Q_{9,9}-H^{2}_{7,9}),\quad\omega_{10}=-H_{7,9}-\frac{i}{\gamma}(Q_{9,9}-H^{2}_{7,9}),\quad\omega_{11}=-\frac{i}{\gamma}Q_{11,11},$
(62)

where the matrix elements $Q_{i,j}$ and $H_{i,j}$ are calculated and given in
Appendix.(C) and (D). By comparison with the results in [45], one can derive
the detailed results for all first-order transport coefficients of ordinary
hydrodynamics, but we here only concentrate on the spin modes. Here we comment
that these spin-related dispersion relations have no match with the remaining
six ones in [20], because in that work the linear analysis concentrates on the
non-conservative spin density. Among the spin modes there are four propagating
modes, two degenerate modes with the propagating speed $H_{7,9}/\kappa$ damp
according to the damping rate
$-[\tilde{\chi}^{(1)}_{7},\tilde{\chi}^{(1)}_{7}]$ while the other two travel
in the opposite direction with the damping rate
$-[\tilde{\chi}^{(1)}_{9},\tilde{\chi}^{(1)}_{9}]$. On the other hand, the
sixth and eleventh modes are purely decaying at their respective decaying
rates.

It is interesting to note that the imaginary parts of these frequencies encode
the information of the relaxation of related dissipative quantities, i.e, the
dissipation of a perturbation of conserved charges around the equilibrium
state. If the system of our interest is not too far away from the equilibrium
state, in that case the linear analysis suffice and we can get a quantitative
comparison of two typical relaxation times, one is the relaxation time for
energy momentum tensor, the other is for spin tensor, which is significantly
crucial in the investigation of spin polarization. Before that, we need to
clarify the definition of the spin relaxation time. As is shown in [44], we
propose that the relaxation time for spin density should be taken as the
longest one of all relaxation times for six spin modes,

$\displaystyle\tau_{s}=\max\\{\frac{1}{|\omega_{\alpha}^{(2)}|},\alpha=6,\cdots
11\\}$ (63)

with the sign $|A|$ representing the amplitude of a complex $A$. To clarify
the equilibrium picture, we compare it with another typical time scale, the
relaxation time for shear modes $\omega_{4,5}$, which describes the
dissipation of an initial disturbance through shear viscosity $\eta$. We
denote it with $\tau_{\eta}$ to reflect that this time scale is closely
connected with the relaxation of transverse momentum density via shear
viscosity $\eta$. If the former is far smaller than the latter, the initial
disturbance has minor effects on spin evolution because the spin density
relaxes monotonically and quickly to the equilibrium much earlier than other
spinless modes and almost independent. Reversely, the non-equilibrium effects
brought by the spin degrees of freedom play a big role and may be irrelevant
to heavy-ion collisions for much longer spin equilibration time. In an
intermediate case where these two scales are comparable, the status of spin
hydrodynamics is enhanced because this is exactly within the range of the
application for spin hydrodynamic theory.

To that end, we need to specify the parameters in our calculation. The spin
Boltzmann equation is specially derived for the spin transport for massive
quarks, i.e., strange quarks ($m=150\,\text{Mev}$) temporarily overlooking the
interactions with massless quarks and gluons. The relevant temperature of the
hot matter created in heavy-ion collisions ranges from about $T_{c}$ to
$5T_{c}$ (the critical temperature $T_{c}$ is $170\,\text{Mev}$). Since not
all thermodynamic integrals can be evaluated analytically, we calculate the
frequencies of normal modes numerically. The dependence of the relative
relaxation time on the reduced mass $z$ is exhibited in Fig.1.

Figure 1: The relative relaxation time defined by the ratio of two distinct
typical time scales $\tau_{s}/\tau_{\eta}$ as a function of $z(=\frac{m}{T})$.

From this figure, we can see it clearly that the spin relaxation time is a
little larger than shear relaxation time and their ratio grow with the
increasing reduced mass. Throughout temperature region we are interested in,
their departure from each other is rather small. Therefore, these two typical
times are comparable satisfying the intermediate scenario introduced in the
proceeding paragraphs, which calls for the tangled dynamic evolution of both
spin and momentum (their characteristic equilibration times are almost the
same), while identical conclusions are also reported in related works recently
[48, 25]. It seems that increasing temperature will lessen the separation of
two time scales. In the hot medium where strange quarks are generated, spin
and moment relax to equilibrium via the collisions between strange quarks at
almost the same time. In a different viewpoint, the dependence of
$\tau_{s}/\tau_{\eta}$ on $z$ can be also interpreted by holding $T$ fixed,
which reads that increase particle mass enhances the effect of separation for
two equilibration time scales. The slowness of spin dissipation may be
attributed to suppression of spin interaction by the mass of constitute
particles, which is consistent with the reason why spin rotation is suppressed
in microscopic collision process proposed in [20]. Nevertheless, present
results has dependence on the parameterization of $\gamma$. For the impacts of
the energy dependence of $\gamma$ on the separation of two time scales
discussed here, we leave it to a future work in preparation.

##  VI Summary and outlook

In this work, we present a detailed linear analysis of normal modes of
linearized collision operator based on the spin Boltzmann equation for massive
fermions proposed in [1]. When neglecting the non-diagonal part of the
transition rate, the linearized collision operator $L_{1}$ is proved to be
Hermite and solving normal modes can be done in the fashion of degenerate
perturbation theory. Moreover, the reformed collision kernel
phenomenologically incorporate the important ansatz of the conservation of
total angular momentum in a collision event. The eleven zero modes exactly
correspond to eleven conserved charges that are tightly associated with spin
hydrodynamics. Considering the complexity of solving integral equation, we
instead approximate the full linearized collision operator with a mutilated
operator, which inherits necessary properties of linearized collision
operator.

With the simplified collision kernel, solving normal modes completely reduces
to numerical integration order by order and we calculate the frequencies of
these normal modes growing from zero modes up to second order in wave vector.
We also show that our framework can be well applied to investigate the
relaxation of spin. Identifying spin equilibration time as the largest one of
all reciprocals of damping rates for spin modes, we compare it with typical
time scale for momentum equilibration. Our results manifest that these two
time scales are close over the temperature range of our interest, which also
show that increasing particle mass will lead to separation of scales while
high temperature puts them closer. Therefore, the evolution of spin can’t be
independent of that of momentum treating it as equilibrated. The clarification
of the hierarchy for relaxation times based on reliable quantum kinetic theory
is highly non-trivial in resolving the problem of discovering the local spin
polarization in the experiments of relativistic heavy-ion collisions.

There are still some improvements or possible extensions to our evaluation.
Firstly, the parameter $\gamma$ introduced to represent the eigenvalue with
infinite degeneracy is set to be spin and momentum independent, otherwise
$\gamma$ can’t be factorized out of the integral Eq.(V). In our framework,
$\gamma$ can be naturally parameterized as momentum dependent without
contradicting the conservation laws compared to traditional RTA. In principle,
$\gamma$ is not a free parameter and supposed to be determined by solving or
approximately solving eigen spectrum of the full linearized collision
operator, which is left as a further research in future. Secondly, the adopted
kinetic equation is derived only for massive quarks, although massive strange
quarks should contribute a lot as constitute components of $\Lambda$ hyperons.
However there are also other processes contributing to spin and momentum
relaxation such as collisions between strange quarks and massless gluons and
u,d quarks. They shall play a role but whether those processes are predominant
over collisions between massive quarks themselves calls for a further
investigation. For completeness and precision, the scattering of strange
quarks and massless quarks and gluons is also necessarily considered. Last but
not the least, we take one of the easiest equilibrium configuration, i.e.,
$\Omega=0$, on top of which the linear analysis is carried out. It is
generally believed that finite thermal vorticity can survive in global
equilibrium and so is the spin potential $\Omega$. If choosing the
configuration with finite vorticity, we are allowed with another power
counting scheme where vorticity field breaks rotation symmetry and the theory
then is anisotropic.

## Acknowledgments

This work was supported by the NSFC Grant No.11890710, No.11890712 and
No.12035006.

##  Appendix A Thermodynamic Integral

To proceed, we first introduce the following integration formula

$\displaystyle I^{(r)}_{\alpha_{1}\cdots\alpha_{n}}$ $\displaystyle\equiv
2\int\frac{\rm dP}{(2\pi)^{3}}\,p_{\alpha_{1}}p_{\alpha_{2}}\cdots
p_{\alpha_{n}}\frac{e^{\xi-\beta\cdot p}}{(u\cdot p)^{r}}$
$\displaystyle=I^{(r)}_{n0}u_{\alpha_{1}}\cdots
u_{\alpha_{n}}+I^{(r)}_{n1}(\Delta_{\alpha_{1}\alpha_{2}}u_{\alpha_{3}\cdots\alpha_{n}}+\text{permutations})+\cdots,$
(64)

where the abbreviation dP stands for $d^{4}p\,\delta(p^{2}-m^{2})$ and the
formal expression after the second equality comes from the analysis of Lorentz
covariance. Using the projection operator $u^{\alpha}$ and
$\Delta^{\alpha\beta}$, the scalar coefficients in the form of thermodynamic
integrals are given by

$\displaystyle I^{(r)}_{nq}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{(2q+1)!!}\int\frac{\rm dP}{(2\pi)^{3}}\,(u\cdot
p)^{n-2q-r}(\Delta_{\alpha\beta}p^{\alpha}p^{\beta})^{q}e^{\xi-\beta\cdot p},$
(65)

where $K_{n}(z)$ denotes the modified Bessel functions of the second kind
defined as

$\displaystyle K_{n}(z)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\,\cosh(nx)\,e^{-z\cosh x}.$ (66)

Specially, we note that $I^{(0)}_{10}=n$, $I^{(0)}_{20}=e,\,I^{(0)}_{21}=-P$ ,
$I^{(0)}_{31}=-hT$ and

$\displaystyle
I^{(0)}_{30}(z)=\frac{T^{5}z^{5}e^{\xi}}{32\pi^{2}}\big{(}K_{5}(z)+K_{3}(z)-2K_{1}(z)\big{)},$
(67)

with $z\equiv\frac{m}{T}$, $n,e,P,h$ are the number density, energy density,
static pressure and enthalpy respectively.

In addition, when handling the angular momentum integrations, the following
similar and useful formulas are also of our interest,

$\displaystyle L^{(r)}_{\alpha_{1}\cdots\alpha_{n}}$ $\displaystyle\equiv
2\int\frac{\rm dP}{(2\pi)^{3}(p^{0}+m)}\,p_{\alpha_{1}}p_{\alpha_{2}}\cdots
p_{\alpha_{n}}\frac{e^{\xi-\beta\cdot p}}{(u\cdot p)^{r}}$
$\displaystyle=L^{(r)}_{n0}u_{\alpha_{1}}\cdots
u_{\alpha_{n}}+L^{(r)}_{n1}(\Delta_{\alpha_{1}\alpha_{2}}u_{\alpha_{3}\cdots\alpha_{n}}+\text{permutations})+\cdots,$
$\displaystyle N^{(r)}_{\alpha_{1}\cdots\alpha_{n}}$ $\displaystyle\equiv
2\int\frac{\rm
dP}{(2\pi)^{3}(p^{0}+m)^{2}}\,p_{\alpha_{1}}p_{\alpha_{2}}\cdots
p_{\alpha_{n}}\frac{e^{\xi-\beta\cdot p}}{(u\cdot p)^{r}}$
$\displaystyle=N^{(r)}_{n0}u_{\alpha_{1}}\cdots
u_{\alpha_{n}}+N^{(r)}_{n1}(\Delta_{\alpha_{1}\alpha_{2}}u_{\alpha_{3}\cdots\alpha_{n}}+\text{permutations})+\cdots.$
(68)

The scalar functions $L^{(r)}_{nq}$ and $N^{(r)}_{nq}$ can be also defined
like Eq.(65). With extra factor appearing in the integrations, these integrals
can not be expressed with the modified Bessel functions of the second kind
$K_{n}(z)$.

##  Appendix B Normalized factors

First, we define two auxiliary unit vectors $j^{\mu}$ and $v^{\mu}$, which
satisfy

$\displaystyle u\cdot l=u\cdot j=u\cdot v=l\cdot j=l\cdot v=j\cdot v=0,$
$\displaystyle l^{2}=j^{2}=v^{2}=-1.$ (69)

Thus we can expand $p^{\mu}$ and $J^{\mu\nu}$ as

$\displaystyle p^{\mu}=u\cdot p\,u^{\mu}+l\cdot p\,l^{\mu}+j\cdot
p\,j^{\mu}+v\cdot p\,v^{\mu},$ $\displaystyle
J^{\mu\nu}=u_{\mu}J^{\mu\nu}l_{\nu}-l_{\mu}J^{\mu\nu}u_{\nu}+u_{\mu}J^{\mu\nu}j_{\nu}-j_{\mu}J^{\mu\nu}u_{\nu}+u_{\mu}J^{\mu\nu}v_{\nu}-v_{\mu}J^{\mu\nu}u_{\nu}$
$\displaystyle\quad\,\,\,\,+l_{\mu}J^{\mu\nu}j_{\nu}-j_{\mu}J^{\mu\nu}l_{\nu}+l_{\mu}J^{\mu\nu}v_{\nu}-v_{\mu}J^{\mu\nu}l_{\nu}+j_{\mu}J^{\mu\nu}v_{\nu}-v_{\mu}J^{\mu\nu}j_{\nu}.$
(70)

Accounting for the antisymmetric property of total angular momentum
$J^{\mu\nu}$, the effective degrees of freedom, or the effective basis can be
chosen as $(1,u\cdot p,l\cdot p,j\cdot p,v\cdot
p,u_{\mu}J^{\mu\nu}l_{\nu},u_{\mu}J^{\mu\nu}j_{\nu},u_{\mu}J^{\mu\nu}v_{\nu},l_{\mu}J^{\mu\nu}j_{\nu},l_{\mu}J^{\mu\nu}v_{\nu},j_{\mu}J^{\mu\nu}v_{\nu})$,
and they are label by the $i$-th basis respectively ($i=1,2,\cdots,11$).

The normalized factors for the zeroth eigenfunctions meeting the condition of
Eq.(37) are

$\displaystyle V_{1,1}$ $\displaystyle=$ $\displaystyle\exp(\xi)\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{u\cdot p}{T}\exp(-\beta\cdot p)=\frac{n}{T},$
$\displaystyle\quad V_{2,2}$ $\displaystyle=$
$\displaystyle\exp(\xi)\int\frac{\rm d\Gamma}{(2\pi)^{3}}\frac{(u\cdot
p-\frac{e}{n})^{2}(u\cdot p)}{T^{3}}\exp(-\beta\cdot
p)=\frac{I^{(0)}_{30}-\frac{e^{2}}{n}}{T^{3}},$ $\displaystyle V_{3,3}$
$\displaystyle=$ $\displaystyle V_{4,4}=V_{5,5}=\exp(\xi)\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{(u\cdot p)(l\cdot p)^{2}}{T^{3}}\exp(-\beta\cdot
p)=\frac{h}{T^{2}}.$ (71)

Because of the calibration settings mentioned before, we have
$\hat{t}=u=(1,0,0,0)$, then the other normalized factors are

$\displaystyle V_{6,6}=V_{7,7}=V_{8,8}$ $\displaystyle=\exp(\xi)\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{u\cdot
p}{T}u_{\mu}J^{\mu\nu}l_{\nu}u_{\rho}J^{\rho\sigma}l_{\sigma}\exp(-\beta\cdot
p)$
$\displaystyle=\frac{1}{2m^{2}T}(-I^{(0)}_{31}+2L^{(0)}_{41}-N^{(0)}_{51}),$
$\displaystyle V_{9,9}=V_{10,10}=V_{11,11}$
$\displaystyle=\exp(\xi)\int\frac{\rm
d\Gamma}{(2\pi)^{3}}l_{\mu}J^{\mu\nu}j_{\nu}(p\cdot
u)l_{\rho}J^{\rho\sigma}j_{\sigma}\exp(-\beta\cdot p)$
$\displaystyle=\frac{I^{(0)}_{30}+I^{(0)}_{31}+4L^{(0)}_{41}+10N^{(0)}_{52}}{4m^{2}T},$
(72)

where the thermodynamic integrals are given in Appendix.(A). We have checked
that the factors $V_{2,2},V_{6,6}$ and $V_{9,9}$ are all positive.

##  Appendix C Perturbation Matrix Elements I

In this section, we calculate the perturbation matrix elements which partly
breaks the degeneracy of normal modes. The matrix elements are given by
$H_{i,j}=\frac{\exp(\xi)}{T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\tilde{\psi}_{i}\kappa\cdot
p\,\tilde{\psi}_{j}\exp(-\beta\cdot p)$. When indices $i$ and $j$ range from
$1$ to $5$, only the following matrix elements are nonzero.

$\displaystyle H_{1,3}$ $\displaystyle=$
$\displaystyle\frac{\exp(\xi)}{\sqrt{V_{1,1}V_{3,3}}T^{2}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\kappa\cdot p\,l\cdot p\exp(-\beta\cdot
p)=\frac{P\kappa}{\sqrt{V_{1,1}V_{3,3}}T^{2}},$ $\displaystyle H_{2,3}$
$\displaystyle=$
$\displaystyle\frac{\exp(\xi)}{\sqrt{V_{2,2}V_{3,3}}T^{3}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}(u\cdot p-\frac{e}{n})p\cdot\kappa(l\cdot
p)\exp(-\beta\cdot p)=\frac{P\kappa}{\sqrt{V_{2,2}V_{3,3}}T^{2}},$ (73)

which break the five-fold degeneracy into three-fold. When it comes to spin
modes, we first introduce a four-indice tensor $H^{\mu\nu\rho\sigma}$, which
is defined as

$\displaystyle H^{\mu\nu\rho\sigma}=\frac{\exp(\xi)}{T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}J^{\mu\nu}(p\cdot\kappa)J^{\rho\sigma}\exp(-\beta\cdot p)$
$\displaystyle\quad\,\,\,\,=\frac{1}{4m^{2}}(g^{\mu\sigma}(\kappa^{\nu}u^{\rho}+\kappa^{\rho}u^{\nu})I^{(0)}_{31}+g^{\nu\rho}(\kappa^{\mu}u^{\sigma}+\kappa^{\sigma}u^{\mu})I^{(0)}_{31}-g^{\nu\sigma}(\kappa^{\mu}u^{\rho}+\kappa^{\rho}u^{\mu})I^{(0)}_{31}-g^{\mu\rho}(\kappa^{\nu}u^{\sigma}+\kappa^{\sigma}u^{\nu})I^{(0)}_{31})\,)$
$\displaystyle\quad\,\,\,\,+\frac{1}{4m^{2}}(L^{(0)}_{41}g^{\mu\rho}\kappa^{\sigma}u^{\nu}-5L^{(0)}_{42}g^{\mu\rho}u^{\sigma}\kappa^{\nu}-L^{(0)}_{41}g^{\mu\sigma}\kappa^{\rho}u^{\nu}+5L^{(0)}_{42}g^{\mu\sigma}u^{\rho}\kappa^{\nu})$
$\displaystyle\quad\,\,\,\,-\frac{1}{4m^{2}}(L^{(0)}_{41}g^{\nu\rho}\kappa^{\sigma}u^{\mu}-5L^{(0)}_{42}g^{\nu\rho}u^{\sigma}\kappa^{\mu}-L^{(0)}_{41}g^{\nu\sigma}\kappa^{\rho}u^{\mu}+5L^{(0)}_{42}g^{\nu\sigma}u^{\rho}\kappa^{\mu})$
$\displaystyle\quad\,\,\,\,+\frac{1}{4m^{2}}(L^{(0)}_{41}g^{\rho\mu}\kappa^{\nu}u^{\sigma}-5L^{(0)}_{42}g^{\rho\mu}u^{\nu}\kappa^{\sigma}-L^{(0)}_{41}g^{\rho\nu}\kappa^{\mu}u^{\sigma}+5L^{(0)}_{42}g^{\rho\nu}u^{\mu}\kappa^{\sigma})$
$\displaystyle\quad\,\,\,\,-\frac{1}{4m^{2}}(L^{(0)}_{41}g^{\sigma\mu}\kappa^{\nu}u^{\rho}-5L^{(0)}_{42}g^{\sigma\mu}u^{\nu}\kappa^{\rho}-L^{(0)}_{41}g^{\sigma\nu}\kappa^{\mu}u^{\rho}+5L^{(0)}_{42}g^{\sigma\nu}u^{\mu}\kappa^{\rho})$
$\displaystyle\quad\,\,\,\,+\frac{1}{4m^{2}}[g^{\mu\rho}\big{(}N^{(0)}_{51}(\kappa^{\nu}u^{\sigma}+u^{\nu}\kappa^{\sigma})+5N^{(0)}_{52}(u^{\nu}\kappa^{\sigma}+u^{\sigma}\kappa^{\nu}\,)\,\big{)}-g^{\nu\rho}\big{(}N^{(0)}_{51}(\kappa^{\mu}u^{\sigma}+u^{\mu}\kappa^{\sigma})+5N^{(0)}_{52}(u^{\mu}\kappa^{\sigma}+u^{\sigma}\kappa^{\mu}\,)\,\big{)}$
$\displaystyle\quad\quad\quad\;-(\rho\leftrightarrow\sigma)]$
$\displaystyle\quad\,\,\,\,-\frac{1}{4m^{2}}[N^{(0)}_{51}g^{\mu\rho}(\kappa^{\nu}u^{\sigma}+\kappa^{\sigma}u^{\nu}\,)-N^{(0)}_{51}g^{\nu\rho}(\kappa^{\mu}u^{\sigma}+\kappa^{\sigma}u^{\mu})-(\rho\leftrightarrow\sigma)],$
(74)

by projecting $H^{\mu\nu\rho\sigma}$ with four direction vectors $(u,l,j,v)$,
we get the following non-vanishing matrix elements,

$\displaystyle H_{7,9}=H_{8,10}$ $\displaystyle=$
$\displaystyle\frac{\exp(\xi)}{\sqrt{V_{6,6}V_{9,9}}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{p\cdot\kappa}{T}u_{\mu}J^{\mu\nu}j_{\nu}l_{\rho}J^{\rho\sigma}j_{\sigma}\exp(-\beta\cdot
p)=\frac{(-I^{(0)}_{31}+L^{(0)}_{41}-5L^{(0)}_{42}+5N^{(0)}_{52})\kappa}{4m^{2}T\sqrt{V_{6,6}V_{9,9}}}.$
(75)

It is easy to find that the perturbation matrix has a symmetry of
transposition. To be concrete, the matrix elements satisfy $H_{i,j}=H_{j,i}$.
We now see that there are no cross terms between the spinless part and the
spin part in the whole matrix $W$, which is clear from Eq.(54).

##  Appendix D Perturbation Matrix Elements II

In this section, we calculate the 2nd-order perturbation matrix elements
appearing in Eq.(V). The needed integrals are implemented with the assistance
of Eq.(A),

$\displaystyle q_{1,1}$ $\displaystyle=$
$\displaystyle\frac{\exp(\xi)}{V_{1,1}T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{(p\cdot\kappa)^{2}}{p\cdot u}\exp(-\beta\cdot
p)=-\frac{I^{(1)}_{21}}{V_{1,1}T}\kappa^{2},$ $\displaystyle q_{2,2}$
$\displaystyle=$ $\displaystyle\frac{\exp(\xi)}{V_{2,2}T^{3}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}(u\cdot p-\frac{e}{n})\frac{(p\cdot\kappa)^{2}}{u\cdot
p}(u\cdot p-\frac{e}{n})\exp(-\beta\cdot
p)=(-I^{(0)}_{31}-\frac{e^{2}I^{(1)}_{21}}{n^{2}}+\frac{2eI^{(0)}_{21}}{n})\frac{\kappa^{2}}{V_{2,2}T^{3}},$
$\displaystyle q_{1,2}$ $\displaystyle=$
$\displaystyle\frac{\exp(\xi)}{\sqrt{V_{1,1}V_{2,2}}T^{2}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{(p\cdot\kappa)^{2}}{u\cdot p}(u\cdot
p-\frac{e}{n})\exp(-\beta\cdot
p)=(-I^{(0)}_{21}+\frac{eI^{(1)}_{21}}{n})\frac{\kappa^{2}}{\sqrt{V_{1,1}V_{2,2}}T^{2}},$
$\displaystyle q_{3,3}$ $\displaystyle=$
$\displaystyle\frac{\exp(\xi)}{V_{3,3}T^{3}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{(p\cdot\kappa)^{2}(p\cdot l)^{2}}{p\cdot
u}\exp(-\beta\cdot p)=\frac{3I^{(1)}_{42}}{V_{3,3}T^{3}}\kappa^{2},$

and the final results for spinless modes are

$\displaystyle Q_{1,1}$ $\displaystyle=$ $\displaystyle
Q_{2,2}=\frac{H_{1,3}^{2}}{2(H^{2}_{1,3}+H^{2}_{2,3})}q_{1,1}+\frac{H_{2,3}^{2}}{2(H^{2}_{1,3}+H^{2}_{2,3})}q_{2,2}+\frac{1}{2}q_{3,3}+\frac{H_{1,3}H_{2,3}}{H^{2}_{1,3}+H^{2}_{2,3}}q_{1,2},$
$\displaystyle Q_{3,3}$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{H_{2,3}^{2}}{H^{2}_{1,3}}+1}(\frac{H_{2,3}^{2}}{H^{2}_{1,3}}q_{1,1}+q_{2,2}-\frac{2H_{2,3}}{H_{1,3}}q_{1,2}),$
$\displaystyle Q_{4,4}$ $\displaystyle=$ $\displaystyle
Q_{5,5}=\frac{\exp(\xi)}{V_{3,3}T^{3}}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}\frac{(p\cdot\kappa)^{2}(p\cdot j)^{2}}{p\cdot
u}\exp(-\beta\cdot p)=\frac{I^{(1)}_{42}}{V_{3,3}T^{3}}\kappa^{2}.$ (77)

It is rather cumbersome to carry out the rest of integrals associated with
spin modes. First, we introduce a new four-indice tensor
$Q^{\mu\nu\rho\sigma}$,

$\displaystyle Q^{\mu\nu\rho\sigma}=\frac{\exp(\xi)}{T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}J^{\mu\nu}\frac{(p\cdot\kappa)^{2}}{p\cdot
u}J^{\rho\sigma}\exp(-\beta\cdot p)$
$\displaystyle\quad\quad\,\,\,\,=\frac{1}{4m^{2}T}(-g^{\mu\rho}g^{\nu\sigma}(I^{(1)}_{41}+5I^{(1)}_{42})+g^{\mu\sigma}g^{\nu\rho}(I^{(1)}_{41}+5I^{(1)}_{42})\,)\kappa^{2}$
$\displaystyle\quad\quad\,\,\,\,+\frac{1}{4m^{2}T}\big{(}-g^{\mu\sigma}u^{\nu}u^{\rho}I^{(1)}_{41}-g^{\mu\sigma}\Delta^{\nu\rho}I^{(1)}_{42}+2g^{\mu\sigma}l^{\nu}l^{\rho}I^{(1)}_{42}-g^{\nu\rho}u^{\mu}u^{\sigma}I^{(1)}_{41}-g^{\nu\rho}\Delta^{\mu\sigma}I^{(1)}_{42}+2g^{\nu\rho}l^{\mu}l^{\sigma}I^{(1)}_{42}$
$\displaystyle\quad\quad\quad\quad\,\;\;+\;g^{\nu\sigma}u^{\mu}u^{\rho}I^{(1)}_{41}+g^{\nu\sigma}\Delta^{\mu\rho}I^{(1)}_{42}-2g^{\nu\sigma}l^{\mu}l^{\rho}I^{(1)}_{42}+g^{\mu\rho}u^{\nu}u^{\sigma}I^{(1)}_{41}+g^{\mu\rho}\Delta^{\nu\sigma}I^{(1)}_{42}-2g^{\mu\rho}l^{\nu}l^{\sigma}I^{(1)}_{42}\,\big{)}\kappa^{2}$
$\displaystyle\quad\quad\,\,\,\,-\frac{1}{2m^{2}T}\big{(}-g^{\mu\sigma}u^{\nu}u^{\rho}L^{(0)}_{41}-g^{\mu\sigma}\Delta^{\nu\rho}L^{(0)}_{42}+2g^{\mu\sigma}l^{\nu}l^{\rho}L^{(0)}_{42}-g^{\nu\rho}u^{\mu}u^{\sigma}L^{(0)}_{41}-g^{\nu\rho}\Delta^{\mu\sigma}L^{(0)}_{42}+2g^{\nu\rho}l^{\mu}l^{\sigma}L^{(0)}_{42}$
$\displaystyle\quad\quad\quad\quad\,\;\;+\;g^{\nu\sigma}u^{\mu}u^{\rho}L^{(0)}_{41}+g^{\nu\sigma}\Delta^{\mu\rho}L^{(0)}_{42}-2g^{\nu\sigma}l^{\mu}l^{\rho}L^{(0)}_{42}+g^{\mu\rho}u^{\nu}u^{\sigma}L^{(0)}_{41}+g^{\mu\rho}\Delta^{\nu\sigma}L^{(0)}_{42}-2g^{\mu\rho}l^{\nu}l^{\sigma}L^{(0)}_{42}\,\big{)}\kappa^{2}$
$\displaystyle\quad\quad\,\,\,\,+\frac{1}{2m^{2}T}[g^{\mu\rho}u^{\sigma}u^{\nu}(L^{(1)}_{51}+5L^{(1)}_{52})-g^{\mu\sigma}u^{\rho}u^{\nu}(L^{(1)}_{51}+5L^{(1)}_{52})-g^{\nu\rho}u^{\sigma}u^{\mu}(L^{(1)}_{51}+5L^{(1)}_{52})+g^{\nu\sigma}u^{\rho}u^{\mu}(L^{(1)}_{51}+5L^{(1)}_{52})\,]\kappa^{2}$
$\displaystyle\quad\quad\,\,\,\,-\frac{1}{4m^{2}T}[-\;g^{\mu\rho}\big{(}u^{\nu}u^{\sigma}N^{(0)}_{51}+(\Delta^{\nu\sigma}-2l^{\sigma}l^{\nu})N^{(0)}_{52}\,\big{)}+g^{\nu\rho}\big{(}u^{\mu}u^{\sigma}N^{(0)}_{51}+(\Delta^{\mu\sigma}-2l^{\mu}l^{\sigma})N^{(0)}_{52}\,\big{)}$
$\displaystyle\quad\quad\quad\;\quad+g^{\mu\sigma}\big{(}u^{\nu}u^{\rho}N^{(0)}_{51}+(\Delta^{\nu\rho}-2l^{\rho}l^{\nu})N^{(0)}_{52}\,\big{)}-g^{\nu\sigma}\big{(}u^{\mu}u^{\rho}N^{(0)}_{51}+(\Delta^{\mu\rho}-2l^{\mu}l^{\rho})N^{(0)}_{52}\big{)}]\kappa^{2}$
$\displaystyle\quad\quad\,\,\,\,+\frac{1}{4m^{2}T}[g^{\mu\rho}\big{(}-u^{\nu}u^{\sigma}N^{(1)}_{61}-(\Delta^{\nu\sigma}+5u^{\nu}u^{\sigma}-2l^{\nu}l^{\sigma})N^{(1)}_{62}+(-7\Delta^{\nu\sigma}+14l^{\nu}l^{\sigma})N^{(1)}_{63}\big{)}$
$\displaystyle\quad\quad\quad\,\,-g^{\nu\rho}\big{(}-u^{\mu}u^{\sigma}N^{(1)}_{61}-(\Delta^{\mu\sigma}+5u^{\mu}u^{\sigma}-2l^{\mu}l^{\sigma})N^{(1)}_{62}+(-7\Delta^{\mu\sigma}+14l^{\mu}l^{\sigma})N^{(1)}_{63}\big{)}$
$\displaystyle\quad\quad\quad\,\,-g^{\mu\sigma}\big{(}-u^{\nu}u^{\rho}N^{(1)}_{61}-(\Delta^{\nu\rho}+5u^{\nu}u^{\rho}-2l^{\nu}l^{\rho})N^{(1)}_{62}+(-7\Delta^{\nu\rho}+14l^{\nu}l^{\rho})N^{(1)}_{63}\big{)}$
$\displaystyle\quad\quad\quad\,\,+g^{\nu\sigma}\big{(}-u^{\mu}u^{\rho}N^{(1)}_{61}-(\Delta^{\mu\rho}+5u^{\mu}u^{\rho}-2l^{\mu}l^{\rho})N^{(1)}_{62}+(-7\Delta^{\mu\rho}+14l^{\mu}l^{\rho})N^{(1)}_{63}\big{)}]\kappa^{2}$
$\displaystyle\quad\quad\,\,\,\,-\frac{1}{4m^{2}T}[u^{\mu}u^{\rho}\big{(}-(\Delta^{\nu\sigma}-2l^{\nu}l^{\sigma})N^{(1)}_{62}+(-7\Delta^{\nu\sigma}+14l^{\nu}l^{\sigma})N^{(1)}_{63}\big{)}$
$\displaystyle\quad\quad\quad\,\,-u^{\nu}u^{\rho}\big{(}-(\Delta^{\mu\sigma}-2l^{\mu}l^{\sigma})N^{(1)}_{62}+(-7\Delta^{\mu\sigma}+14l^{\mu}l^{\sigma})N^{(1)}_{63}\big{)}$
$\displaystyle\quad\quad\quad\,\,-u^{\mu}u^{\sigma}\big{(}-(\Delta^{\nu\rho}-2l^{\nu}l^{\rho})N^{(1)}_{62}+(-7\Delta^{\nu\rho}+14l^{\nu}l^{\rho})N^{(1)}_{63}\big{)}$
$\displaystyle\quad\quad\quad\,\,+u^{\nu}u^{\sigma}\big{(}-(\Delta^{\mu\rho}-2l^{\mu}l^{\rho})N^{(1)}_{62}+(-7\Delta^{\mu\rho}+14l^{\mu}l^{\rho})N^{(1)}_{63}\big{)}]\kappa^{2},$
(78)

from which one can obtain the following indispensable integrals by projecting
onto various directions,

$\displaystyle q_{7,7}=q_{8,8}=\frac{\exp(\xi)}{V_{6,6}T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}u_{\mu}J^{\mu\nu}j_{\nu}\frac{(p\cdot\kappa)^{2}}{p\cdot
u}u_{\rho}J^{\rho\sigma}j_{\sigma}\exp(-\beta\cdot p)$
$\displaystyle\quad\,\,\,\,\,=\frac{1}{4m^{2}V_{6,6}T}(4I^{(1)}_{42}+2L^{(0)}_{41}+2L^{(0)}_{42}-2L^{(1)}_{51}-10L^{(1)}_{52}-N^{(0)}_{52}-N^{(0)}_{51}+N^{(1)}_{61}+5N^{(1)}_{62})\kappa^{2},$
$\displaystyle q_{9,9}=q_{10,10}=\frac{\exp(\xi)}{V_{9,9}T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}l_{\mu}J^{\mu\nu}j_{\nu}\frac{(p\cdot\kappa)^{2}}{p\cdot
u}l_{\rho}J^{\rho\sigma}j_{\sigma}\exp(-\beta\cdot p)$
$\displaystyle\quad\,\,\,\,\,=\frac{1}{4m^{2}V_{9,9}T}(-I^{(1)}_{41}-I^{(1)}_{42}-8L^{(0)}_{42}+4N^{(0)}_{52}-4N^{(1)}_{62}-28N^{(1)}_{63}\big{)}\kappa^{2},$
(79)

and finally,

$\displaystyle Q_{6,6}=\frac{\exp(\xi)}{V_{6,6}T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}u_{\mu}J^{\mu\nu}l_{\nu}\frac{(p\cdot\kappa)^{2}}{p\cdot
u}u_{\rho}J^{\rho\sigma}l_{\sigma}\exp(-\beta\cdot p)$
$\displaystyle\quad\,\,\,\,\,=\frac{1}{4m^{2}V_{6,6}T}(2I^{(1)}_{42}+2L^{(0)}_{41}+6L^{(0)}_{42}-2L^{(1)}_{51}-10L^{(1)}_{52}\,-N^{(0)}_{51}-3N^{(0)}_{52}+N^{(1)}_{61}+5N^{(1)}_{62})\kappa^{2},$
$\displaystyle
Q_{7,7}=Q_{8,8}=Q_{9,9}=Q_{10,10}=\frac{1}{2}(q_{7,7}+q_{9,9}),$
$\displaystyle Q_{11,11}=\frac{\exp(\xi)}{V_{9,9}T}\int\frac{\rm
d\Gamma}{(2\pi)^{3}}j_{\mu}J^{\mu\nu}v_{\nu}\frac{(p\cdot\kappa)^{2}}{p\cdot
u}j_{\rho}J^{\rho\sigma}v_{\sigma}\exp(-\beta\cdot p)$
$\displaystyle\quad\,\,\,\,\,=\frac{1}{4m^{2}V_{9,9}T}(-I^{(1)}_{41}-3I^{(1)}_{42}-4L^{(0)}_{42}+2N^{(0)}_{52}-2N^{(1)}_{62}-14N^{(1)}_{63}\big{)}\kappa^{2},$
(80)

where the diagonal matrix elements $Q_{i,i},\,i=1\cdots 11$, are exactly what
we present in Eq.(V).

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