# Spin Boltzmann equation for non-relativistic spin-1/2 fermions

Wen-Bo Dong, Yi-Liang Yin, Qun Wang  Department of Modern Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

We derive the spin Boltzmann equations for spin-1/2 fermions in a non-
relativistic model with four-fermion contact interaction which conserves spin
degrees of freedom. A great advantage of the model is that the spin matrix
elements in collision terms can be completely worked out and be put into such
a compact form that one can clearly see how spins are coupled in particle
scatterings. A semi-classical expansion in the Planck constant has been made
and the on-shell part of the spin Boltzmann equation up to the next-to-leading
order is derived. At the leading order the equilibrium spin distribution can
be obtained from the vanishing of the collision term for the spin density. The
spin chemical potential emerges as a natural consequence of spin conservation.
The off-shell part of the spin Boltzmann equation is also discussed. The work
can be extended to more sophisticated interaction such as nuclear force in
order to apply to spin polarization phenomena in heavy-ion collisions at low
energies.

##  I Introduction

Very large orbital angular momenta (OAM) are generated in non-central heavy-
ion collisions which can be partially converted into the spin polarization of
hadrons along the direction of OAM or with respect to the reaction plane
(Liang:2004ph, ; Liang:2004xn, ; Gao:2007bc, ). This effect is called the
global spin polarization or global polarization for short. The global
polarization of $\Lambda$ hyperons (including $\overline{\Lambda}$) has been
measured for the first time by the STAR collaboration in Au+Au collision at
200 GeV and lower energies (STAR:2017ckg, ; STAR:2018gyt, ). The data show
that the global polarization is about 1.08$\pm$0.15% ($\Lambda$) and
1.38$\pm$0.30% ($\overline{\Lambda}$) with a decreasing behavior with the
collision energy.

Several theoretical methods have been developed for the global polarization.
These theoretical methods can be roughly put into three categories. One
category is related to the quantum statistical theory for particle systems
with spin degrees of freedom in equilibrium (Becattini:2007sr, ;
Becattini:2007nd, ; Becattini:2013fla, ; Becattini:2014yxa, ;
Becattini:2018duy, ; Becattini:2019dxo, ) [for a recent review, see, e.g.,
Ref. (Becattini:2020sww, )]. One category is the microscopic transport theory
based on kinetic or Boltzmann equations for spin degrees of freedom
(Fang:2016vpj, ; Zhang:2019xya, ; Weickgenannt:2019dks, ; Gao:2019znl, ;
Weickgenannt:2020aaf, ; Wang:2020pej, ; Sheng:2021kfc, ; Weickgenannt:2021cuo,
; Yang:2021fea, ; Sheng:2022ssd, ) in terms of covariant Wigner functions
(Gao:2012ix, ; Chen:2012ca, ; Hidaka:2016yjf, ; Gao:2017gfq, ; Gao:2018wmr, )
[see, e.g., Refs. (Gao:2020pfu, ; Hidaka:2022dmn, ) for recent reviews].
Another category is relativistic spin hydrodynamics (Montenegro:2017rbu, ;
Florkowski:2018fap, ; Montenegro:2018bcf, ; Becattini:2018duy, ;
Hattori:2019lfp, ; Li:2020eon, ; Fukushima:2020ucl, ; Wang:2021ngp, ;
Yi:2021unq, ; Weickgenannt:2022jes, ) [see Ref. (Florkowski:2018fap, ) for a
review], which incorporates spin degrees of freedom into conventional
relativistic hydrodynamics applied to the strong interaction matter in heavy-
ion collisions (Kolb:2003dz, ; Heinz:2013th, ; Florkowski:2017olj, ;
Romatschke:2017ejr, ). There are many phenomenological studies of the global
and local polarization using these theoretical methods to describe
experimental data (Karpenko:2016jyx, ; Li:2017slc, ; Xie:2017upb, ;
Sun:2017xhx, ; Xia:2018tes, ; Wei:2018zfb, ; Baznat:2017jfj, ;
Csernai:2018yok, ; Wu:2019eyi, ; Fu:2020oxj, ; Ivanov:2020qqe, ;
Ivanov:2020wak, ; Ryu:2021lnx, ; Fu:2021pok, ; Becattini:2021iol, ;
Yi:2021unq, ; Wu:2022mkr, ) [for recent reviews, see, e.g., Refs.
(Gao:2020lxh, ; Huang:2020dtn, ; Becattini:2020ngo, )].

Recently HADES collaboration measured the global $\Lambda$ polarization in
Ag+Ag collisions at 2.55 GeV and Au+Au collisions at 2.4 GeV (Kornas:2022cbl,
), while STAR collaboration measured the same observable in Au+Au collisions
at 3 GeV (PhysRevC.104.L061901, ). Combining all these low energy measurements
with the high energy ones, the $\Lambda$ polarization is observed to continue
the increasing trend with decreasing collision energy down to 2.4 GeV. At
these collision energies of $\mathcal{O}(m_{N})$ where $m_{N}$ is the nucleon
mass, the relativistic effect is small and non-relativistic theory can be a
proper approximation. Experiment data can be described by models such as UrQMD
and BUU (PhysRevC.103.L031903, ; Deng:2021miw, ). These models are based on
Boltzmann equations for hadrons which do not incorporate spin degrees of
freedom.

In this paper, we will derive the spin Boltzmann equation for spin-1/2
fermions in a non-relativistic model with four-fermion contact interaction
similar to the Nambu-Jona-Lasinio (NJL) model in relativistic theory
(PhysRev.122.345, ; Nambu:1961fr, ). The non-relativistic model has a feature
that the particle’s spin is decoupled from its momentum and is conserved in
the interaction. This is very different from a relativistic system in which
the particle’s spin and momentum are entangled. The method is based on a
previous work by one of us about the relativistic system of spin-1/2 fermions
(Sheng:2021kfc, ). A great advantage of the current non-relativistic model is
that the spin matrix elements in collision terms can be completely worked out
and be put into such a compact form that one can clearly see how spins are
coupled in two-to-two scatterings of particles. This is not the case in the
relativistic theory (Sheng:2021kfc, ). The current work can be extended to
nucleon-nucleon interaction via nuclear force and then can be applied to the
global polarization in heavy-ion collisions at low energies.

This paper is organized as follows. In Sect. II, we briefly introduce Green’s
functions in the CTP formalism. In Sect. III, we derive the KB equation from
the Schwinger-Dyson equation in quasi-particle approximation. In Sect. IV, we
derive the on-shell part of the spin Boltzmann equation at the leading and
next-to-leading order in $\hbar$. In Sect. V, the off-shell part of the KB
equation is discussed. A summary of the results is given in the final section.

##  II Green functions for fermions in CTP formalism

We consider a non-relativistic system of spin-1/2 fermions. A general form of
the Lagrangian with four-fermion interaction can be written as (2003Quantum, )

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\int
d^{3}\mathbf{x}\psi_{\alpha}^{\dagger}(t,\mathbf{x})\left(i\hbar\frac{\partial}{\partial
t}+\frac{\hbar^{2}}{2m}\nabla^{2}\right)\psi_{\alpha}(t,\mathbf{x})$ (1)
$\displaystyle-\frac{1}{2}\int
d^{3}\mathbf{x}d^{3}\mathbf{x}^{\prime}\psi_{\alpha}^{\dagger}(t,\mathbf{x})\psi_{\beta}^{\dagger}(t,\mathbf{x}^{\prime})V_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(\mathbf{x},\mathbf{x}^{\prime})\psi_{\beta^{\prime}}(t,\mathbf{x}^{\prime})\psi_{\alpha^{\prime}}(t,\mathbf{x}),$

where $\alpha,\beta=\pm$ denote spin states,
$V_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(\mathbf{x},\mathbf{x}^{\prime})$
is the spin-dependent potential, and repeated indices imply a summation if not
explicitly stated. Let us consider the contact interaction of the NJL type
(PhysRev.122.345, ; Nambu:1961fr, ),

$\mathcal{L}_{\mathrm{int}}=-g_{0}\int
d^{3}\mathbf{x}\left[\psi_{\alpha}^{\dagger}(t,\mathbf{x})\psi_{\alpha}(t,\mathbf{x})\right]^{2}-g_{\sigma}\sum_{i}\int
d^{3}\mathbf{x}\left[\psi_{\alpha}^{\dagger}(t,\mathbf{x})\sigma_{\alpha\beta}^{i}\psi_{\beta}(t,\mathbf{x})\right]^{2},$
(2)

which correpsonds to the potential in the form

$V_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(\mathbf{x},\mathbf{x}^{\prime})=2\delta^{(3)}(\mathbf{x}-\mathbf{x}^{\prime})\left(g_{0}\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}+g_{\sigma}\sum_{i}\sigma_{\alpha\alpha^{\prime}}^{i}\sigma_{\beta\beta^{\prime}}^{i}\right).$
(3)

The Lagrangian (1) with the interaction part (2) is invariant under the global
SU(2) transformation defined as

$\displaystyle U(\theta)$ $\displaystyle=$
$\displaystyle\exp\left(-i\frac{1}{2}\boldsymbol{\theta}\cdot\boldsymbol{\sigma}\right),$
$\displaystyle\psi_{\alpha}^{\prime}$ $\displaystyle=$ $\displaystyle
U_{\alpha\beta}(\theta)\psi_{\beta},$
$\displaystyle\psi_{\alpha}^{\prime\dagger}$ $\displaystyle=$
$\displaystyle\psi_{\beta}^{\dagger}U_{\beta\alpha}^{\dagger}(\theta),$ (4)

which means the spin is conserved. The kinetic term of the Lagrangian (1) and
$g_{0}$ term of the interaction Lagrangian (2) are obviously invariant under
the SU(2) transformation. Let us look at the $g_{\sigma}$ term

$\displaystyle\left[\psi_{\alpha}^{\prime\dagger}\sigma_{\alpha\beta}^{i}\psi_{\beta}^{\prime}\right]\left[\psi_{\alpha_{1}}^{\prime\dagger}\sigma_{\alpha_{1}\beta_{1}}^{i}\psi_{\beta_{1}}^{\prime}\right]$
$\displaystyle=$
$\displaystyle\left[\psi_{\gamma}^{\dagger}U_{\gamma\alpha}^{\dagger}\sigma_{\alpha\beta}^{i}U_{\beta\lambda}\psi_{\lambda}\right]\left[\psi_{\gamma_{1}}^{\dagger}U_{\gamma_{1}\alpha_{1}}^{\dagger}\sigma_{\alpha_{1}\beta_{1}}^{i}U_{\beta_{1}\lambda_{1}}\psi_{\lambda_{1}}\right]$
(5) $\displaystyle=$ $\displaystyle
V_{ij}V_{ik}\left[\psi_{\gamma}^{\dagger}\sigma_{\gamma\lambda}^{j}\psi_{\lambda}\right]\left[\psi_{\gamma_{1}}^{\dagger}\sigma_{\gamma_{1}\lambda_{1}}^{k}\psi_{\lambda_{1}}\right]$
$\displaystyle=$
$\displaystyle\left[\psi_{\gamma}^{\dagger}\sigma_{\gamma\lambda}^{j}\psi_{\lambda}\right]\left[\psi_{\gamma_{1}}^{\dagger}\sigma_{\gamma_{1}\lambda_{1}}^{j}\psi_{\lambda_{1}}\right],$

where we have used $U^{\dagger}\sigma_{i}U=V_{ij}\sigma_{j}$ with $V_{ij}$
denoting SO(3) matrices. Corresponding to the SU(2) invariance of the
Lagrangian, the Noether charge and current for spin are given by

$\displaystyle Q_{i}^{\mathrm{spin}}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\psi^{\dagger}\sigma_{i}\psi,$ $\displaystyle
J_{ij}^{\mathrm{spin}}$ $\displaystyle=$
$\displaystyle\frac{i\hbar^{2}}{4m}\left(\nabla_{j}\psi^{\dagger}\sigma_{i}\psi-\psi^{\dagger}\sigma_{i}\nabla_{j}\psi\right),$
(6)

which satisfy the conservation equation

$\frac{\partial}{\partial
t}Q_{i}^{\mathrm{spin}}+\nabla_{j}J_{ij}^{\mathrm{spin}}=0.$ (7)

Note that the Noether charge is a spin vector while the Noether current is a
tensor.

The fermion fields can be quantized as

$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\mathbf{p}}{(2\pi\hbar)^{3}}e^{-ip\cdot
x/\hbar}\sum_{s=\pm}a(s,\mathbf{p})\chi(s),$ $\displaystyle\psi^{\dagger}(x)$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}\mathbf{p}}{(2\pi\hbar)^{3}}e^{ip\cdot
x/\hbar}\sum_{s=\pm}a^{\dagger}(s,\mathbf{p})\chi^{\dagger}(s),$ (8)

where $x\equiv(x_{0},\mathbf{x})\equiv(t,\mathbf{x})$,
$p\equiv(p_{0},\mathbf{p})\equiv(\omega_{p},\mathbf{p})$ with
$\omega_{p}=\mathbf{p}^{2}/(2m)$, $p\cdot
x\equiv\omega_{p}t-\mathbf{p}\cdot\mathbf{x}$, $a(s,\mathbf{p})$ and
$a^{\dagger}(s,\mathbf{p})$ are annihilation and creation operators associated
with $\mathbf{p}$ and the spin state $s$ respectively, and $\chi(s)$ is the
spin state (Pauli spinor) which satisfies
$(\mathbf{n}\cdot\boldsymbol{\sigma})\chi(s)=s\chi(s)$ with $\mathbf{n}$ being
the spin quantum direction
$\mathbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The anti-
commutators of $a(s,\mathbf{p})$ and $a^{\dagger}(s,\mathbf{p})$ are given by

$\displaystyle\left\\{a(s_{1},\mathbf{p}_{1}),a^{\dagger}(s_{2},\mathbf{p}_{2})\right\\}$
$\displaystyle=$
$\displaystyle(2\pi\hbar)^{3}\delta_{s_{1}s_{2}}\delta^{(3)}(\mathbf{p}_{1}-\mathbf{p}_{2}),$
$\displaystyle\left\\{a(s_{1},\mathbf{p}_{1}),a(s_{2},\mathbf{p}_{2})\right\\}$
$\displaystyle=$
$\displaystyle\left\\{a^{\dagger}(s_{1},\mathbf{p}_{1}),a^{\dagger}(s_{2},\mathbf{p}_{2})\right\\}=0,$
(9)

which lead to equal-time anti-commutators for fermion fields

$\displaystyle\left\\{\psi_{\alpha}(t,\mathbf{x}),\psi_{\beta}^{\dagger}(t,\mathbf{x}^{\prime})\right\\}$
$\displaystyle=$
$\displaystyle\delta_{\alpha\beta}\delta^{(3)}(\mathbf{x}-\mathbf{x}^{\prime}),$
$\displaystyle\left\\{\psi_{\alpha}(t,\mathbf{x}),\psi_{\beta}(t,\mathbf{x}^{\prime})\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\psi_{\alpha}^{\dagger}(t,\mathbf{x}),\psi_{\beta}^{\dagger}(t,\mathbf{x}^{\prime})\right\\}=0.$
(10)

Now we define the two-point Green function in the closed-time-path (CTP)
formalism (Martin:1959jp, ; Keldysh:1964ud, ) (see, e.g., Refs. (Chou:1984es,
; Blaizot:2001nr, ; Berges:2004yj, ; Crossley:2015evo, ) for reviews) as

$G_{\alpha\beta}^{C}(x_{1},x_{2})=\left\langle
T_{C}\left[\psi_{\alpha}(x_{1})\psi_{\beta}^{\dagger}(x_{2})\right]\right\rangle,$
(11)

where $T_{C}$ denotes the time-ordered product on the CTP, and the angular
brackets denote averages weighted by the density operator at the initial time
$\rho(t_{0})$. Note that our definition for Green’s function is different from
Ref. (2003Quantum, ) without the additional factor $i=\sqrt{-1}$. Depending on
whether the two space-time points are on the positive or negative time branch,
there are four types of two-point functions

$\displaystyle G_{\alpha\beta}^{F}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle G_{\alpha\beta}^{++}(x_{1},x_{2})=\left\langle
T\psi_{\alpha}(x_{1})\psi_{\beta}^{\dagger}(x_{2})\right\rangle,$
$\displaystyle G_{\alpha\beta}^{\overline{F}}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle G_{\alpha\beta}^{--}(x_{1},x_{2})=\left\langle
T_{A}\psi_{\alpha}(x_{1})\psi_{\beta}^{\dagger}(x_{2})\right\rangle,$
$\displaystyle G_{\alpha\beta}^{<}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle
G_{\alpha\beta}^{+-}(x_{1},x_{2})=-\left\langle\psi_{\beta}^{\dagger}(x_{2})\psi_{\alpha}(x_{1})\right\rangle,$
$\displaystyle G_{\alpha\beta}^{>}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle
G_{\alpha\beta}^{-+}(x_{1},x_{2})=\left\langle\psi_{\alpha}(x_{1})\psi_{\beta}^{\dagger}(x_{2})\right\rangle,$
(12)

where $+/-$ stands for the positive/negative time branch respectively, and $T$
and $T_{A}$ denote the time-ordered and reverse-time-ordered product
respectively. Note that only three out of four types of two-point functions in
(12) are independent due to the identity

$G^{F}+G^{\overline{F}}=G^{<}+G^{>}.$ (13)

We can choose $(G^{F},G^{<},G^{>})$ as three independent two-point functions.
Equivalently we can also choose $(G^{R},G^{A},G^{S})$ as independent ones

$\displaystyle G^{R}$ $\displaystyle=$ $\displaystyle
G^{F}-G^{<}=\theta(t_{1}-t_{2})(G^{>}-G^{<}),$ $\displaystyle G^{A}$
$\displaystyle=$ $\displaystyle G^{F}-G^{>}=\theta(t_{2}-t_{1})(G^{<}-G^{>}),$
$\displaystyle G^{S}$ $\displaystyle=$ $\displaystyle G^{<}+G^{>},$ (14)

where $G^{R}$ and $G^{A}$ are retarded and advanced two-point functions.

The Wigner function is the building-block of the quantum transport theory
since it is the quantum analogue of the particle distribution in phase space.
The Wigner function can be obtained by taking the Fourier transformation with
respect to the distance of two space-time points in a two-point function. Then
the Wigner functions for $G_{\alpha\beta}^{<}(x_{1},x_{2})$ and
$G_{\alpha\beta}^{>}(x_{1},x_{2})$ are defined by

$\displaystyle G_{\alpha\beta}^{<}(x,p)$ $\displaystyle=$ $\displaystyle\int
d^{4}ye^{ip\cdot y/\hbar}G_{\alpha\beta}^{<}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\int d^{4}ye^{ip\cdot
y/\hbar}\left\langle\psi_{\beta}^{\dagger}\left(x-\frac{y}{2}\right)\psi_{\alpha}\left(x+\frac{y}{2}\right)\right\rangle,$
$\displaystyle G_{\alpha\beta}^{>}(x,p)$ $\displaystyle=$ $\displaystyle\int
d^{4}ye^{ip\cdot y/\hbar}G_{\alpha\beta}^{>}(x_{1},x_{2})$ (15)
$\displaystyle=$ $\displaystyle\int d^{4}ye^{ip\cdot
y/\hbar}\left\langle\psi_{\alpha}\left(x+\frac{y}{2}\right)\psi_{\beta}^{\dagger}\left(x-\frac{y}{2}\right)\right\rangle,$

where $p\cdot y\equiv\omega_{p}y_{0}-\mathbf{p}\cdot\mathbf{y}$, $x_{1}=x+y/2$
and $x_{2}=x-y/2$.

In non-equilibrium, we do not know $G_{\alpha\beta}^{<}(x,p)$ and
$G_{\alpha\beta}^{>}(x,p)$ exactly due to unknown ensemble averages. However,
using (8), we can make an ansatz for their forms in a power expansion of
$\hbar$. Inserting (8) to (15), the leading order contributions to the Wigner
functions have the form

$\displaystyle G_{\alpha\beta}^{<(0)}(x,p)$ $\displaystyle=$
$\displaystyle-(2\pi\hbar)\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\sum_{s_{1},s_{2}=\pm}\chi_{\alpha}(s_{1})\chi_{\beta}^{\dagger}(s_{2})f_{s_{1}s_{2}}^{(0)}(x,\mathbf{p}),$
$\displaystyle G_{\alpha\beta}^{>(0)}(x,p)$ $\displaystyle=$
$\displaystyle(2\pi\hbar)\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\sum_{s_{1},s_{2}=\pm}\chi_{\alpha}(s_{1})\chi_{\beta}^{\dagger}(s_{2})\left[\delta_{s_{1}s_{2}}-f_{s_{1}s_{2}}^{(0)}(x,\mathbf{p})\right],$
(16)

where $p\equiv(p_{0},\mathbf{p})\equiv(\omega,\mathbf{p})$ with $p_{0}$ or
$\omega$ being an independent variable, and the matrix valued spin-dependent
distribution (MVSD) at $\mathcal{O}(\hbar^{0})$ is defined as (Sheng:2021kfc,
)

$\displaystyle f_{s_{1}s_{2}}^{(0)}(x,\mathbf{p})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\mathbf{q}}{(2\pi\hbar)^{3}}\exp\left[i\frac{1}{\hbar}\left(-\frac{\mathbf{p}\cdot\mathbf{q}}{m}t+\mathbf{q}\cdot\mathbf{x}\right)\right]$
(17) $\displaystyle\times\left\langle
a^{\dagger}\left(s_{2},\mathbf{p}-\frac{\mathbf{q}}{2}\right)a\left(s_{1},\mathbf{p}+\frac{\mathbf{q}}{2}\right)\right\rangle.$

One can check that $G^{\lessgtr(0)}(x,p)$ are Hermitian matrices because
$f_{s_{1}s_{2}}^{(0)*}=f_{s_{2}s_{1}}^{(0)}$, i.e., $f^{(0)}$ is a Hermitian
matrix in spin space. The first order contributions at
$\mathcal{O}(\hbar^{1})$ are assumed to have the form

$\displaystyle\hbar G_{\alpha\beta}^{<(1)}(x,p)$ $\displaystyle=$
$\displaystyle\hbar G_{\alpha\beta}^{>(1)}(x,p)$ (18) $\displaystyle=$
$\displaystyle-\hbar(2\pi\hbar)\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\sum_{s_{1},s_{2}=\pm}\chi_{\alpha}(s_{1})\chi_{\beta}^{\dagger}(s_{2})f_{s_{1}s_{2}}^{(1)}(x,\mathbf{p}),$

where $f_{s_{1}s_{2}}^{(1)}(x,\mathbf{p})$ is unknown and can be determined by
solving the evolution equations. We assume
$f_{s_{1}s_{2}}^{(1)*}=f_{s_{2}s_{1}}^{(1)}$, which means $f^{(1)}$ and then
$G^{\lessgtr(1)}(x,p)$ are Hermitian matrices in spin space. Note that in Eq.
(18) we only include on-shell contributions to $G^{\lessgtr(1)}(x,p)$ which
are proportional to $\delta\left[p_{0}-\mathbf{p}^{2}/(2m)\right]$. In
principle there are off-shell contributions which are proportional to
$\delta^{\prime}\left[p_{0}-\mathbf{p}^{2}/(2m)\right]$. We will address off-
shell contributions in Section V separately.

Combining (16) and (18) we obtain $G^{\lessgtr}(x,p)$ up to
$\mathcal{O}(\hbar)$

$\displaystyle G^{<}(x,p)$ $\displaystyle=$ $\displaystyle G^{<(0)}(x,p)+\hbar
G^{<(1)}(x,p)+\mathcal{O}(\hbar^{2})$ $\displaystyle=$
$\displaystyle-(2\pi\hbar)\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\sum_{s_{1},s_{2}=\pm}\chi_{\alpha}(s_{1})\chi_{\beta}^{\dagger}(s_{2})f_{s_{1}s_{2}}(x,\mathbf{p}),$
$\displaystyle G^{>}(x,p)$ $\displaystyle=$ $\displaystyle G^{>(0)}(x,p)+\hbar
G^{>(1)}(x,p)+\mathcal{O}(\hbar^{2})$ (19) $\displaystyle=$
$\displaystyle(2\pi\hbar)\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\sum_{s_{1},s_{2}=\pm}\chi_{\alpha}(s_{1})\chi_{\beta}^{\dagger}(s_{2})\left[\delta_{s_{1}s_{2}}-f_{s_{1}s_{2}}(x,\mathbf{p})\right].$

Here the MVSD $f_{s_{1}s_{2}}(x,\mathbf{p})$ is given by

$\displaystyle f_{s_{1}s_{2}}(x,\mathbf{p})$ $\displaystyle=$ $\displaystyle
f_{s_{1}s_{2}}^{(0)}(x,\mathbf{p})+\hbar
f_{s_{1}s_{2}}^{(1)}(x,\mathbf{p})+\mathcal{O}(\hbar^{2})$ (20)
$\displaystyle=$
$\displaystyle\delta_{s_{1}s_{2}}\overline{f}(x,\mathbf{p})+\boldsymbol{\tau}_{s_{1}s_{2}}\cdot\mathbf{t}(x,\mathbf{p})+\mathcal{O}(\hbar^{2}),$

where $\boldsymbol{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ and
$\mathbf{t}(x,\mathbf{p})=[t_{1}(x,\mathbf{p}),t_{2}(x,\mathbf{p}),t_{3}(x,\mathbf{p})]$
denote Pauli matrices and the polarization vector in $(s_{1},s_{2})$ space
respectively. Note that $G^{\lessgtr}(x,p)$ are Hermitian. For convenience we
can write $G^{\lessgtr}(x,p)$ as

$G^{\lessgtr}(x,p)=-(2\pi\hbar)\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)g^{\lessgtr}(x,p),$
(21)

where $g^{\lessgtr}(x,p)$ denotes the part of $G^{\lessgtr}(x,p)$ without the
delta function and it can be decomposed in terms of 1 and Pauli matrices
$\boldsymbol{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ in spinor’s space,

$\displaystyle g^{<}(x,p)$ $\displaystyle\equiv$
$\displaystyle\sum_{s_{1},s_{2}=\pm}\chi(s_{1})\chi^{\dagger}(s_{2})f_{s_{1}s_{2}}(x,\mathbf{p})$
$\displaystyle=$
$\displaystyle\overline{f}(x,\mathbf{p})+\boldsymbol{\sigma}\cdot\mathbf{S}(x,\mathbf{p}),$
$\displaystyle g^{>}(x,p)$ $\displaystyle\equiv$
$\displaystyle-\sum_{s_{1},s_{2}=\pm}\chi(s_{1})\chi^{\dagger}(s_{2})\left[\delta_{s_{1}s_{2}}-f_{s_{1}s_{2}}(x,\mathbf{p})\right]$
(22) $\displaystyle=$
$\displaystyle\overline{f}(x,\mathbf{p})+\boldsymbol{\sigma}\cdot\mathbf{S}(x,\mathbf{p})-1.$

These components can be extracted by taking traces

$\displaystyle\overline{f}(x,\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{Tr}\left[g^{<}(x,p)\right],$
$\displaystyle\mathbf{S}(x,\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{Tr}\left[\boldsymbol{\sigma}g^{<}(x,p)\right].$
(23)

So $\mathbf{S}(x,\mathbf{p})$ is given by

$\displaystyle\mathbf{S}(x,\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s_{1},s_{2}=\pm}\chi^{\dagger}(s_{2})\boldsymbol{\sigma}\chi(s_{1})f_{s_{1}s_{2}}(x,\mathbf{p})$
(24) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s_{1},s_{2}=\pm}(\mathbf{n}_{1}\tau_{1}+\mathbf{n}_{2}\tau_{2}+\mathbf{n}_{3}\tau_{3})_{s_{2}s_{1}}f_{s_{1}s_{2}}(x,\mathbf{p})$
$\displaystyle=$ $\displaystyle\mathbf{n}_{i}t_{i}(x,\mathbf{p}),$

where we have used

$\displaystyle\chi^{\dagger}(s_{2})\boldsymbol{\sigma}\chi(s_{1})$
$\displaystyle=$
$\displaystyle(\mathbf{n}_{1}\tau_{1}+\mathbf{n}_{2}\tau_{2}+\mathbf{n}_{3}\tau_{3})_{s_{2}s_{1}}$
(27) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\mathbf{n}_{3}&\mathbf{n}_{1}-i\mathbf{n}_{2}\\\
\mathbf{n}_{1}+i\mathbf{n}_{2}&-\mathbf{n}_{3}\end{array}\right)_{s_{2}s_{1}}.$

Here $(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})$ are three basis vectors
given by

$\displaystyle\mathbf{n}_{1}$ $\displaystyle=$
$\displaystyle\left(\cos\phi\cos\theta,\sin\phi\cos\theta,-\sin\theta\right),$
$\displaystyle\mathbf{n}_{2}$ $\displaystyle=$
$\displaystyle\left(-\sin\phi,\cos\phi,0\right),$
$\displaystyle\mathbf{n}_{3}$ $\displaystyle=$
$\displaystyle\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right).$
(28)

Note that $\mathbf{n}_{3}=\mathbf{n}$ is the direction of the spin
quantization.

##  III Kadanoff-Baym equation

The time evolution of a many-body quantum system in non-equilibrium is
described by the Kadanoff-Baym (KB) equation (Kadanoff2018QuantumSM, ) [see,
e.g., Ref. (Cassing:2008nn, ) for a review]. The KB equations can be derived
from the Dyson-Schwinger equations for two-point Green functions on the CTP

$\displaystyle-i\left(i\hbar\frac{\partial}{\partial
t_{1}}+\frac{\hbar^{2}}{2m}\nabla_{x1}^{2}\right)G_{C}(x_{1},x_{2})$
$\displaystyle=$
$\displaystyle\hbar\delta_{C}^{(4)}(x_{1}-x_{2})+\hbar\int_{C}d^{4}x^{\prime}\Sigma_{C}(x_{1},x^{\prime})G_{C}(x^{\prime},x_{2}),$
$\displaystyle-
iG_{C}(x_{1},x_{2})\left(-i\hbar\frac{\overleftarrow{\partial}}{\partial
t_{2}}+\frac{\hbar^{2}}{2m}\overleftarrow{\nabla}_{x2}^{2}\right)$
$\displaystyle=$
$\displaystyle\hbar\delta_{C}^{(4)}(x_{1}-x_{2})+\hbar\int_{C}d^{4}x^{\prime}G_{C}(x_{1},x^{\prime})\Sigma_{C}(x^{\prime},x_{2}),$
(29)

where the index $C$ stands for the CTP, $\delta_{C}^{(4)}(x_{1}-x_{2})$ and
$\Sigma_{C}(x_{1},x^{\prime})$ are the delta-function and self-energy on the
CTP respectively. Note that $G_{C}$ and $\Sigma_{C}$ are $2\times 2$ matrices
in spinor space. In the case that $(t_{1},t_{2})$ are on ($+,-$) time branch,
Eq. (29) can be put into the conventional coordinate form

$\displaystyle-i\left(i\hbar\frac{\partial}{\partial
t_{1}}+\frac{\hbar^{2}}{2m}\nabla_{x1}^{2}\right)G^{<}(x_{1},x_{2})$
$\displaystyle=$ $\displaystyle\hbar\int
d^{4}x^{\prime}\left[\Sigma^{R}(x_{1},x^{\prime})G^{<}(x^{\prime},x_{2})\right.$
(30)
$\displaystyle\left.+\Sigma^{<}(x_{1},x^{\prime})G^{A}(x^{\prime},x_{2})\right],$
$\displaystyle-i\left(-i\hbar\frac{\partial}{\partial
t_{2}}+\frac{\hbar^{2}}{2m}\nabla_{x2}^{2}\right)G^{<}(x_{1},x_{2})$
$\displaystyle=$ $\displaystyle\hbar\int
d^{4}x^{\prime}\left[G^{R}(x_{1},x^{\prime})\Sigma^{<}(x^{\prime},x_{2})\right.$
(31)
$\displaystyle\left.+G^{<}(x_{1},x^{\prime})\Sigma^{A}(x^{\prime},x_{2})\right],$

where $\Sigma^{R}$ and $\Sigma^{A}$ are the retarded and advanced self-energy
respectively. Performing the Wigner transform for above equations, we obtain

$\left(i\hbar\frac{1}{2}\partial_{t}+i\frac{\hbar}{2m}\mathbf{p}\cdot\nabla_{x}+p_{0}-\frac{\mathbf{p}^{2}}{2m}+\frac{\hbar^{2}}{8m}\nabla_{x}^{2}\right)G^{<}(x,p)=I_{\mathrm{coll}},$
(32)

and

$\left(-i\hbar\frac{1}{2}\partial_{t}-i\frac{\hbar}{2m}\mathbf{p}\cdot\nabla_{x}+p_{0}-\frac{\mathbf{p}^{2}}{2m}+\frac{\hbar^{2}}{8m}\nabla_{x}^{2}\right)G^{<}(x,p)=I_{\mathrm{coll}}^{\dagger},$
(33)

where $I_{\mathrm{coll}}$ and $I_{\mathrm{coll}}^{\dagger}$ are given by

$\displaystyle I_{\mathrm{coll}}$ $\displaystyle=$ $\displaystyle
i\hbar\left[\Sigma^{R}(x,p)G^{<}(x,p)+\Sigma^{<}(x,p)G^{A}(x,p)\right]$ (34)
$\displaystyle+\frac{1}{2}\hbar^{2}\left[\left\\{\Sigma^{R}(x,p),G^{<}(x,p)\right\\}_{P.B.}+\left\\{\Sigma^{<}(x,p),G^{A}(x,p)\right\\}_{P.B.}\right],$
$\displaystyle I_{\mathrm{coll}}^{\dagger}$ $\displaystyle=$ $\displaystyle
i\hbar\left[G^{R}(x,p)\Sigma^{<}(x,p)+G^{<}(x,p)\Sigma^{A}(x,p)\right]$ (35)
$\displaystyle+\frac{1}{2}\hbar^{2}\left[\left\\{G^{R}(x,p),\Sigma^{<}(x,p)\right\\}_{P.B.}+\left\\{G^{<}(x,p),\Sigma^{A}(x,p)\right\\}_{P.B.}\right].$

Here the Poisson bracket of two matrices is defined as

$\left\\{A,B\right\\}_{\mathrm{PB}}\equiv\partial_{t}A\partial_{p_{0}}B-\partial_{p_{0}}A\partial_{t}B-\left(\nabla_{x}A\cdot\nabla_{p}B-\nabla_{p}A\cdot\nabla_{x}B\right).$
(36)

We see that $I_{\mathrm{coll}}^{\dagger}$ can be obtained by interchange of
$\Sigma$ and $G$ from $I_{\mathrm{coll}}$, and vice versa. With the relations
for $O=G,\Sigma$,

$[O^{R}]^{\dagger}=-O^{A},\;\;\left[O^{\lessgtr}\right]^{\dagger}=\left[O^{\lessgtr}\right],$
(37)

one can check that $I_{\mathrm{coll}}^{\dagger}$ is really the Hermitian
conjugate of $I_{\mathrm{coll}}$. Taking the sum and difference of Eq. (34)
and (35), we obtain an equation for the dispersion relation or the on-shell
equation

$\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}+\frac{\hbar^{2}}{8m}\nabla_{x}^{2}\right)G^{<}(x,p)=\frac{1}{2}\left(I_{\mathrm{coll}}+I_{\mathrm{coll}}^{\dagger}\right),$
(38)

and the evolution equation

$\hbar\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)G^{<}(x,p)=-i\left(I_{\mathrm{coll}}-I_{\mathrm{coll}}^{\dagger}\right).$
(39)

Equations (38,39) are one of the main results in this paper and the starting
point for the derivation of Boltzmann equations.

In the quasi-particle picture, the retarded and advanced self-energies and
two-point functions can be approximated as

$\displaystyle O^{R/A}(x,p)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int dk_{0}\frac{1}{k_{0}-p_{0}\mp
i\epsilon}\left[O^{>}(x,k_{0},\mathbf{p})-O^{<}(x,k_{0},\mathbf{p})\right]$
(40) $\displaystyle=$
$\displaystyle\pm\frac{1}{2}\left[O^{>}(x,p)-O^{<}(x,p)\right]+O_{\mathrm{pr}}^{R/A},$

where $O\equiv G,\Sigma$, and the principal part $O_{\mathrm{pr}}^{R/A}$ is
related to off-shell effects that modify the dispersion relation of the quasi-
particle which we will not consider in this paper. Using (40), the collision
terms (34) and (35) can be simplified as

$\displaystyle I_{\mathrm{coll}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\hbar\left[\Sigma^{>}(x,p)G^{<}(x,p)-\Sigma^{<}(x,p)G^{>}(x,p)\right]$
(41)
$\displaystyle+\frac{1}{4}\hbar^{2}\left[\left\\{\Sigma^{>}(x,p),G^{<}(x,p)\right\\}_{P.B.}-\left\\{\Sigma^{<}(x,p),G^{>}(x,p)\right\\}_{P.B.}\right],$

and

$\displaystyle I_{\mathrm{coll}}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\hbar\left[G^{>}(x,p)\Sigma^{<}(x,p)-G^{<}(x,p)\Sigma^{>}(x,p)\right]$
(42)
$\displaystyle+\frac{1}{4}\hbar^{2}\left[\left\\{G^{>}(x,p),\Sigma^{<}(x,p)\right\\}_{P.B.}-\left\\{G^{<}(x,p),\Sigma^{>}(x,p)\right\\}_{P.B.}\right].$

With collision terms (41) and (42), Eq. (39) is our starting point for the
derivation of Boltzmann equations.

The Boltzmann equations for $\overline{f}(x,\mathbf{p})$ and $\mathbf{S}(x,p)$
can be obtained by taking a trace of Eq. (39) and a trace of Eq. (39)
multiplied by $\boldsymbol{\sigma}$ as

$\displaystyle\hbar\left(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x}\right)\mathrm{Tr}\left(G^{<}\right)$
$\displaystyle=$ $\displaystyle
2\mathrm{Im}\mathrm{Tr}\left(I_{\mathrm{coll}}\right),$ (43)
$\displaystyle\hbar\left(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x}\right)\mathrm{Tr}\left(\boldsymbol{\sigma}G^{<}\right)$
$\displaystyle=$ $\displaystyle
2\mathrm{Im}\mathrm{Tr}\left(\boldsymbol{\sigma}I_{\mathrm{coll}}\right).$
(44)

From (19) we know that there is an on-shell delta-function
$\delta\left[p_{0}-\mathbf{p}^{2}/(2m)\right]$ in both sides of above
equations, therefore one can drop these delta-functions and derive equations
for $\overline{f}(x,\mathbf{p})$ and $\mathbf{S}(x,p)$ without delta-
functions.

##  IV Spin boltzmann equations: on-shell parts

In this section we consider the contact interaction of the NJL type, one of
the simplest cases for interaction. The interaction Lagrangian is given in
(2). The collision term $I_{\mathrm{coll}}$ in (41) and
$I_{\mathrm{coll}}^{\dagger}$ in (42) depend on the self-energy $\Sigma^{>}$
and $\Sigma^{<}$ whose Feynman diagrams are shown in Fig. 1. The self-energies
can be written as

$\displaystyle\Sigma^{>}(x,p)$ $\displaystyle=$ $\displaystyle
4\sum_{c1,c2}g_{c1}g_{c2}\int\frac{d^{4}p_{1}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{2}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{3}}{(2\pi\hbar)^{4}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Tr}\left[\Gamma^{(c2)}G^{>}(p_{1})\Gamma^{(c1)}G^{<}(p_{3})\right]\Gamma^{(c2)}G^{>}(p_{2})\Gamma^{(c1)}$
$\displaystyle-4\sum_{c1,c2}g_{c1}g_{c2}\int\frac{d^{4}p_{1}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{2}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{3}}{(2\pi\hbar)^{4}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\Gamma^{(c2)}G^{>}(p_{1})\Gamma^{(c1)}G^{<}(p_{3})\Gamma^{(c2)}G^{>}(p_{2})\Gamma^{(c1)},$
$\displaystyle\Sigma^{<}(x,p)$ $\displaystyle=$
$\displaystyle\Sigma^{>}[G^{>}\leftrightarrow G^{<}],$ (45)

where $g_{c1}$ and $g_{c2}$ can be $g_{0}$ or $g_{\sigma}$, and we have
suppressed the $x$ dependence of $G^{<}$ and $G^{>}$. If $g_{c}$ is
$g_{\sigma}$, a summation over $i$ for $\Gamma^{(\sigma)}=\sigma_{i}$ is
implied.

Figure 1: Feynman diagrams for $\Sigma^{>}(x,p)$ [figure (a) and (c)] and
$\Sigma^{<}(x,p)$ [figure (b) and (d)]. In the NJL model wavy lines attached
with two solid circles represent vertices $2g_{c}\Gamma^{(c)}$.

###  IV.1 Leading order

Using Eqs. (19), (45) and (41) in Eqs.(43) and (44) and performing an
integration of $p_{0}$ over 0 to $+\infty$, we obtain the Boltzmann equations
for the scalar and polarization part of the distribution at the leading order

$\displaystyle\hbar\left(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x}\right)\overline{f}_{p}^{(0)}$
$\displaystyle=$ $\displaystyle
4\hbar(g_{0}-3g_{\sigma})^{2}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{2}-p_{1})$
(46)
$\displaystyle\times\left[\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}(1-\overline{f}_{3}^{(0)})(1-\overline{f}_{p}^{(0)})-(1-\overline{f}_{1}^{(0)})(1-\overline{f}_{2}^{(0)})\overline{f}_{3}^{(0)}\overline{f}_{p}^{(0)}\right.$
$\displaystyle\left.-(1-\overline{f}_{3}^{(0)}-\overline{f}_{p}^{(0)})\mathbf{S}_{1}^{(0)}\cdot\mathbf{S}_{2}^{(0)}+(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)})\mathbf{S}_{3}^{(0)}\cdot\mathbf{S}_{p}^{(0)}\right],$
$\displaystyle\hbar\left(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x}\right)\mathbf{S}_{p}^{a,(0)}$
$\displaystyle=$ $\displaystyle
4\hbar(g_{0}-3g_{\sigma})^{2}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{2}-p_{1})$
(47)
$\displaystyle\times\left\\{\left[(1-\overline{f}_{p}^{(0)})\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}+\overline{f}_{p}^{(0)}(1-\overline{f}_{1}^{(0)})(1-\overline{f}_{2}^{(0)})\right]\mathbf{S}_{3}^{a,(0)}\right.$
$\displaystyle-\left[(1-\overline{f}_{3}^{(0)})\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}+\overline{f}_{3}^{(0)}(1-\overline{f}_{1}^{(0)})(1-\overline{f}_{2}^{(0)})\right]\mathbf{S}_{p}^{a,(0)}$
$\displaystyle\left.+\mathbf{S}_{1}^{(0)}\cdot\mathbf{S}_{2}^{(0)}(\mathbf{S}_{p}^{a,(0)}-\mathbf{S}_{3}^{a,(0)})\right\\},$

where $\overline{f}_{i}\equiv\overline{f}(x,\mathbf{p}_{i})$,
$\overline{f}_{p}\equiv\overline{f}(x,\mathbf{p})$,
$\mathbf{S}_{i}\equiv\mathbf{S}(x,\mathbf{p}_{i})$,
$\mathbf{S}_{p}\equiv\mathbf{S}(x,\mathbf{p})$, and the index ’(0)’ denotes
the leading order. If the system has no polarization at the leading order,
i.e. $\mathbf{S}_{i}^{(0)}=0$ for $i=1,2,3,p$, Eq. (46) is reduced to the
conventional Boltzmann equation for $\overline{f}$, with Eq. (47) for
$\mathbf{S}^{a}$ being trivially satisfied (both sides are vanishing). If we
assume polarized distributions are much smaller in magnitude than unpolarized
ones, i.e. $\left|\mathbf{S}_{i}^{(0)}\right|\ll\overline{f}_{j}$ for
$i,j=1,2,3,p$, then we can neglect quadratic terms of polarized distributions
in Eq. (46) relative to terms with only unpolarized distributions and neglect
cubic terms of polarized distributions in Eq. (47) relative to linear terms.
In this case the vanishing of the collision term in Eq. (46) gives the
equilibrium condition for unpolarized distributions

$\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}(1-\overline{f}_{3}^{(0)})(1-\overline{f}_{p}^{(0)})=(1-\overline{f}_{1}^{(0)})(1-\overline{f}_{2}^{(0)})\overline{f}_{3}^{(0)}\overline{f}_{p}^{(0)}.$
(48)

Similarly we can also obtain the equilibrium condition for unpolarized
distributions from the vanishing of the collision term in Eq. (47)

$\displaystyle\left[\left(1-\overline{f}_{p}^{(0)}\right)\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}+\overline{f}_{p}^{(0)}\left(1-\overline{f}_{1}^{(0)}\right)\left(1-\overline{f}_{2}^{(0)}\right)\right]\mathbf{S}_{3}^{a,(0)}$
$\displaystyle=\left[\left(1-\overline{f}_{3}^{(0)}\right)\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}+\overline{f}_{3}^{(0)}\left(1-\overline{f}_{1}^{(0)}\right)\left(1-\overline{f}_{2}^{(0)}\right)\right]\mathbf{S}_{p}^{a,(0)},$
(49)

which leads to

$\frac{\mathbf{S}_{3}^{(0)}}{\left(1-\overline{f}_{3}^{(0)}\right)\overline{f}_{3}^{(0)}}=\frac{\mathbf{S}_{p}^{(0)}}{\left(1-\overline{f}_{p}^{(0)}\right)\overline{f}_{p}^{(0)}},$
(50)

using the equilibrium condition (48). The equilibrium condition (48) implies
that $\overline{f}_{p}^{(0)}$ follows the Fermi-Dirac distribution

$\overline{f}_{p}^{(0)}=\frac{1}{\exp\left[\beta(\omega_{p}-\mu)\right]+1}.$
(51)

If we assume the ratio in Eq. (50) is a constant vector $\mathbf{c}$ which is
related to the spin potential $\boldsymbol{\mu}_{\mathrm{spin}}$, then at the
leading order the MVSD in Eq. (20) has the form

$f_{s_{1}s_{2}}^{(0)}(x,\mathbf{p})=\delta_{s_{1}s_{2}}\overline{f}_{p}^{(0)}+\overline{f}_{p}^{(0)}\left(1-\overline{f}_{p}^{(0)}\right)\boldsymbol{\tau}_{s_{1}s_{2}}\cdot\boldsymbol{\mu}_{\mathrm{spin}}+\mathcal{O}(\hbar),$
(52)

where the components of $\boldsymbol{\mu}_{\mathrm{spin}}$ are
$\mu_{\mathrm{spin}}^{i}=\mathbf{n}_{i}\cdot\mathbf{c}$ ($i=1,2,3$) with three
directions $(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})$ being given by Eq.
(28). Since we have assumed
$\left|\mathbf{S}_{i}^{(0)}\right|\ll\overline{f}_{j}$, i.e.
$|\boldsymbol{\mu}_{\mathrm{spin}}|\ll 1$, $f_{s_{1}s_{2}}^{(0)}$ can be put
into an approximated matrix form

$f^{(0)}(x,\mathbf{p})\approx\frac{1}{\exp\left[\beta(\omega_{p}-\mu)-\boldsymbol{\tau}\cdot\boldsymbol{\mu}_{\mathrm{spin}}\right]+1}.$
(53)

Here $\boldsymbol{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ are Pauli matrices in
spin space. The MVSD in (53) is the equilibrium distribution for fermions with
spin degrees of freedom.

###  IV.2 Next-to-leading order

At next to leading order, the Boltzmann equation for the unpolarized
distribution reads

$\hbar(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x})\overline{f}_{p}^{(1)}=I_{\mathrm{qc}}^{\mathrm{scalar}}\left[f^{(0)},f^{(1)}\right]+I_{\mathrm{PB}}^{\mathrm{scalar}}\left[f^{(0)}\right],$
(54)

where the subscript ’qc’ represents the quasi-classical contribution and ’PB’
represents the contribution from the Poisson bracket term. The quasi-classical
part can be obtained from Eq. (46) by replacing all zeroth order distributions
$\overline{f}_{i}^{(0)}$ and $\mathbf{S}_{i}^{(0)}$ by
$\overline{f}_{i}=\overline{f}_{i}^{(0)}+\overline{f}_{i}^{(1)}$ and
$\mathbf{S}_{i}=\mathbf{S}_{i}^{(0)}+\mathbf{S}_{i}^{(1)}$ respectively and
expanding it to the first order, the result is

$\displaystyle I_{\mathrm{qc}}^{\mathrm{scalar}}\left[f^{(0)},f^{(1)}\right]$
$\displaystyle=$ $\displaystyle
4\hbar(g_{0}-3g_{\sigma})^{2}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{2}-p_{1})$
(55)
$\displaystyle\times\left\\{\overline{f}_{1}^{(1)}\left[\overline{f}_{2}^{(0)}+\overline{f}_{3}^{(0)}\overline{f}_{p}^{(0)}-\overline{f}_{2}^{(0)}\overline{f}_{3}^{(0)}-\overline{f}_{2}^{(0)}\overline{f}_{p}^{(0)}\right]\right.$
$\displaystyle+\overline{f}_{2}^{(1)}\left[\overline{f}_{1}^{(0)}+\overline{f}_{3}^{(0)}\overline{f}_{p}^{(0)}-\overline{f}_{1}^{(0)}\overline{f}_{3}^{(0)}-\overline{f}_{1}^{(0)}\overline{f}_{p}^{(0)}\right]$
$\displaystyle-\overline{f}_{3}^{(1)}\left[\overline{f}_{p}^{(0)}+\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}-\overline{f}_{p}^{(0)}\overline{f}_{1}^{(0)}-\overline{f}_{p}^{(0)}\overline{f}_{2}^{(0)}\right]$
$\displaystyle-\overline{f}_{p}^{(1)}\left[\overline{f}_{3}^{(0)}+\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}-\overline{f}_{3}^{(0)}\overline{f}_{1}^{(0)}-\overline{f}_{3}^{(0)}\overline{f}_{2}^{(0)}\right]$
$\displaystyle-(1-\overline{f}_{3}^{(0)}-\overline{f}_{p}^{(0)})\mathbf{S}_{1}^{(0)}\cdot\mathbf{S}_{2}^{(1)}-(1-\overline{f}_{3}^{(0)}-\overline{f}_{p}^{(0)})\mathbf{S}_{1}^{(1)}\cdot\mathbf{S}_{2}^{(0)}$
$\displaystyle+(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)})\mathbf{S}_{3}^{(1)}\cdot\mathbf{S}_{p}^{(0)}+(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)})\mathbf{S}_{3}^{(0)}\cdot\mathbf{S}_{p}^{(1)}$
$\displaystyle\left.+\mathbf{S}_{1}^{(0)}\cdot\mathbf{S}_{2}^{(0)}(\overline{f}_{3}^{(1)}+\overline{f}_{p}^{(1)})-(\overline{f}_{1}^{(1)}+\overline{f}_{2}^{(1)})\mathbf{S}_{3}^{(0)}\cdot\mathbf{S}_{p}^{(0)}\right\\}.$

We see in each term there is only one first order distribution with all other
distributions being of zeroth order. One can prove

$I_{\mathrm{PB}}^{\mathrm{scalar}}\left[f^{(0)}\right]=0,$ (56)

for the scalar part of the Boltzmann equation.

In order to prove Eq. (56), we use the following property for the trace of
Pauli matrices

$\displaystyle\mathrm{Tr}\left[\sigma_{i}\sigma_{j}\cdots\sigma_{k}\right]$
$\displaystyle=$
$\displaystyle(-1)^{n}\mathrm{Tr}\left[\sigma_{k}\cdots\sigma_{j}\sigma_{i}\right],$
(57)

where the $n$ is number of Pauli matrices. To prove the above relation, we
insert $C^{2}=-1$ between Pauli matrices

$\displaystyle(-1)^{n}\mathrm{Tr}\left[C\sigma_{i}CC\sigma_{j}C\cdots
C\sigma_{k}C\right]$
$\displaystyle=(-1)^{n}\mathrm{Tr}[\sigma_{i}^{T}\sigma_{j}^{T}\cdots\sigma_{k}^{T}]=(-1)^{n}\mathrm{Tr}\left[\sigma_{k}\cdots\sigma_{j}\sigma_{i}\right],$
(58)

where $C=i\sigma^{2}$ and $C\boldsymbol{\sigma}C=\boldsymbol{\sigma}^{T}$.

With Eq. (58) we consider the quantities in the Poisson bracket collision term
$I_{\mathrm{PB}}^{\mathrm{scalar}}\left[f^{(0)}\right]$,

$\displaystyle I_{1}$
$\displaystyle=g_{c1}g_{c2}\mathrm{Im}\left[\mathrm{Tr}\left(\Gamma^{(c2)}G_{1}\Gamma^{(c1)}G_{3}\right)\mathrm{Tr}\left(\Gamma^{(c2)}G_{2}\Gamma^{(c1)}G_{p}\right)\right],$
$\displaystyle I_{2}$
$\displaystyle=g_{c1}g_{c2}\mathrm{Im}\mathrm{Tr}\left(\Gamma^{(c2)}G_{1}\Gamma^{(c1)}G_{3}\Gamma^{(c2)}G_{2}\Gamma^{(c1)}G_{p}\right).$
(59)

Note that $G_{i}$ $(i=1,2,3,p)$ can be either $G^{\lessgtr}(x,p_{i})$ or
derivatives of $G^{\lessgtr}(x,p_{i})$. We know that $G_{i}$ contain the
scalar and polarization parts, so we can express
$G_{i}=\sum_{b_{i}=1,0}G_{i}(b_{i})$, where $G_{i}(1)$ denotes its scalar part
and $G_{i}(0)$ denotes its polarization part. Let us work on $I_{1}$,

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\mathrm{Im}\left[\mathrm{Tr}\left(\Gamma^{(c2)}G_{1}\Gamma^{(c1)}G_{3}\right)\mathrm{Tr}\left(\Gamma^{(c2)}G_{2}\Gamma^{(c1)}G_{p}\right)\right]$
(60) $\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\sum_{b_{1},b_{2},b_{3},b}\mathrm{Im}\left\\{\mathrm{Tr}\left[\Gamma^{(c2)}G_{1}(b_{1})\Gamma^{(c1)}G_{3}(b_{3})\right]\mathrm{Tr}\left[\Gamma^{(c2)}G_{2}(b_{2})\Gamma^{(c1)}G_{p}(b)\right]\right\\}$
$\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\sum_{n=\mathrm{odd}}\mathrm{Im}\left\\{\mathrm{Tr}\left[\Gamma^{(c2)}G_{1}(b_{1})\Gamma^{(c1)}G_{3}(b_{3})\right]\mathrm{Tr}\left[\Gamma^{(c2)}G_{2}(b_{2})\Gamma^{(c1)}G_{p}(b)\right]\right\\}$
$\displaystyle\downarrow$ $\displaystyle\mathrm{insert}\;C^{2}$
$\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\sum_{n=\mathrm{odd}}(-1)^{n}\mathrm{Im}\left\\{\mathrm{Tr}\left[\Gamma^{(c1)}G_{1}(b_{1})\Gamma^{(c2)}G_{3}(b_{3})\right]\mathrm{Tr}\left[\Gamma^{(c1)}G_{2}(b_{2})\Gamma^{(c2)}G_{p}(b)\right]\right\\}$
$\displaystyle=$ $\displaystyle-
g_{c1}g_{c2}\sum_{n=\mathrm{odd}}\mathrm{Im}\left\\{\mathrm{Tr}\left[\Gamma^{(c2)}G_{1}(b_{1})\Gamma^{(c1)}G_{3}(b_{3})\right]\mathrm{Tr}\left[\Gamma^{(c2)}G_{2}(b_{2})\Gamma^{(c1)}G_{p}(b)\right]\right\\},$

where $n=b_{1}+b_{2}+b_{3}+b$ is the number of scalar parts in two traces, and
we have interchanged $c1\leftrightarrow c2$ in the last equality since a
summation over $c1$ and $c2$ is implied. Note that even or odd $n$ also
indicates even or odd number of Pauli matrices in two traces respectively.
Only when $n$ is odd does the product of two traces have an imaginary part. By
comparing the last equality with the second one of Eq. (60) we arrive at
$I_{1}=0$. For $I_{2}$, we have

$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\mathrm{Im}\mathrm{Tr}\left(\Gamma^{(c2)}G_{1}\Gamma^{(c1)}G_{3}\Gamma^{(c2)}G_{2}\Gamma^{(c1)}G_{p}\right)$
(61) $\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\sum_{b_{1},b_{2},b_{3},b}\mathrm{Im}\mathrm{Tr}\left[\Gamma^{(c2)}G_{1}(b_{1})\Gamma^{(c1)}G_{3}(b_{3})\Gamma^{(c2)}G_{2}(b_{2})\Gamma^{(c1)}G_{p}(b)\right]$
$\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\sum_{n=\mathrm{odd}}\mathrm{Im}\mathrm{Tr}\left[\Gamma^{(c2)}G_{1}(b_{1})\Gamma^{(c1)}G_{3}(b_{3})\Gamma^{(c2)}G_{2}(b_{2})\Gamma^{(c1)}G_{p}(b)\right]$
$\displaystyle\downarrow$ $\displaystyle\mathrm{insert}\;C^{2}$
$\displaystyle=$ $\displaystyle
g_{c1}g_{c2}\sum_{n=\mathrm{odd}}(-1)^{n}\mathrm{Im}\mathrm{Tr}\left[\Gamma^{(c1)}G_{2}(b_{2})\Gamma^{(c2)}G_{3}(b_{3})\Gamma^{(c1)}G_{1}(b_{1})\Gamma^{(c2)}G_{p}(b)\right]$
$\displaystyle=$ $\displaystyle-
g_{c1}g_{c2}\sum_{n=\mathrm{odd}}\mathrm{Im}\mathrm{Tr}\left[\Gamma^{(c2)}G_{1}(b_{1})\Gamma^{(c1)}G_{3}(b_{3})\Gamma^{(c2)}G_{2}(b_{2})\Gamma^{(c1)}G_{p}(b)\right],$

where $n=b_{1}+b_{2}+b_{3}+b$ is the number of scalar parts in the trace, in
the final equality we have interchanged $c1\leftrightarrow c2$ and
$G_{1}(b_{1})\leftrightarrow G_{2}(b_{2})$ since a summation over $c1$ and
$c2$ is implied and there is a symmetry in the labels 1 and 2 (in the
integration over $p_{1}$ and $p_{2}$ and the summation over $b_{1}$ and
$b_{2}$). The odd/even $n$ also corresponds to odd/even number of Pauli
matrices inside the trace. Only when $n$ is odd does the trace have an
imaginary part. By comparing the last equality with the second one of Eq.
(61), we obtain $I_{2}=0$.

We now derive the Boltzmann equation for the polarization distribution at the
next-to-leading order. The contributions can be grouped into the local (quasi-
classical) and nonlocal parts of the collision term. The local part contains
no space-time derivatives and can be obtained from Eq. (47) by the replacement
$\overline{f}_{i}^{(0)}\rightarrow\overline{f}_{i}=\overline{f}_{i}^{(0)}+\overline{f}_{i}^{(1)}$
and
$\mathbf{S}_{i}^{(0)}\rightarrow\mathbf{S}_{i}=\mathbf{S}_{i}^{(0)}+\mathbf{S}_{i}^{(1)}$
in the collision term and then by expanding the collision term to the next-to-
leading order. The nonlocal part comes from the Poisson bracket term with
space-time derivatives. Special care should be taken for the derivatives in
$\partial_{p_{0}}$ and $\nabla_{p}$ which act on the two-point function
$G^{\lessgtr}(x,p)$, giving terms with
$\delta^{\prime}(p_{0}-\mathbf{p}^{2}/2m)$. These terms belong to off-shell
contributions which we will neglect in this section and leave to the next
section for treatment. Substituting Eq. (19) into Eq. (44), we obtain the on-
shell Boltzmann equation for the polarization distribution at the next-to-
leading order

$\displaystyle\hbar\left(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x}\right)\mathbf{S}_{p}^{a,(1)}$
$\displaystyle=$ $\displaystyle
I_{\mathrm{qc}}^{\mathrm{pol}}\left[f^{(0)},f^{(1)}\right]+I_{\mathrm{PB}}^{\mathrm{pol}}\left[f^{(0)}\right].$
(62)

The explicit form of the local part (quasi-classical) of the collision term is
given by

$\displaystyle I_{\mathrm{qc}}^{\mathrm{pol}}\left[f^{(0)},f^{(1)}\right]$
$\displaystyle=$ $\displaystyle
4\hbar(g_{0}-3g_{\sigma})^{2}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{2}-p_{1})$
(63)
$\displaystyle\times\left\\{\mathbf{S}_{3}^{a,(0)}\left[\overline{f}_{p}^{(1)}(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)})+\overline{f}_{1}^{(1)}(\overline{f}_{2}^{(0)}-\overline{f}_{p}^{(0)})+\overline{f}_{2}^{(1)}(\overline{f}_{1}^{(0)}-\overline{f}_{p}^{(0)})\right]\right.$
$\displaystyle-\mathbf{S}_{p}^{a,(0)}\left[\overline{f}_{3}^{(1)}(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)})+\overline{f}_{1}^{(1)}(\overline{f}_{2}^{(0)}-\overline{f}_{3}^{(0)})+\overline{f}_{2}^{(1)}(\overline{f}_{1}^{(0)}-\overline{f}_{3}^{(0)})\right]$
$\displaystyle+\mathbf{S}_{3}^{a,(1)}\left(\overline{f}_{p}^{(0)}+\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}-\overline{f}_{p}^{(0)}\overline{f}_{1}^{(0)}-\overline{f}_{p}^{(0)}\overline{f}_{2}^{(0)}\right)$
$\displaystyle-\mathbf{S}_{p}^{a,(1)}\left(\overline{f}_{3}^{(0)}+\overline{f}_{1}^{(0)}\overline{f}_{2}^{(0)}-\overline{f}_{3}^{(0)}\overline{f}_{1}^{(0)}-\overline{f}_{3}^{(0)}\overline{f}_{2}^{(0)}\right)$
$\displaystyle+\left(\mathbf{S}_{1}^{(1)}\cdot\mathbf{S}_{2}^{(0)}+\mathbf{S}_{1}^{(0)}\cdot\mathbf{S}_{2}^{(1)}\right)\left(\mathbf{S}_{p}^{a,(0)}-\mathbf{S}_{3}^{a,(0)}\right)$
$\displaystyle\left.+\mathbf{S}_{1}^{(0)}\cdot\mathbf{S}_{2}^{(0)}\left(\mathbf{S}_{p}^{a,(1)}-\mathbf{S}_{3}^{a,(1)}\right)\right\\}.$

One can check that
$I_{\mathrm{qc}}^{\mathrm{pol}}\left[f^{(0)},f^{(1)}\right]$ is local and on-
shell.

The non-local part of the collision term contains on-shell and off-shell
contributions, as we have mentioned, in this section we focus on the on-shell
contribution and will treat the off-shell one in Sect. V. The explicit form of
the on-shell Poisson bracket term reads

$\displaystyle I_{\mathrm{PB}}^{\mathrm{pol}}\left[f^{(0)}\right]$
$\displaystyle\approx$ $\displaystyle
2\hbar^{2}\left(g_{0}-3g_{\sigma}\right)^{2}\epsilon_{aij}\int\frac{d^{3}p_{1}}{(2\pi\hbar)^{3}}\frac{d^{3}p_{2}}{(2\pi\hbar)^{3}}\frac{d^{3}p_{3}}{(2\pi\hbar)^{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
(64)
$\displaystyle\times\left\\{\mathbf{S}_{3}^{(0),i}\left[\nabla_{x}\left(\overline{f}_{1}^{(0)}+\overline{f}_{2}^{(0)}\right)\cdot\nabla_{p}\mathbf{S}_{p}^{(0),j}-\left(\nabla_{p_{1}}\overline{f}_{1}^{(0)}+\nabla_{p_{2}}\overline{f}_{2}^{(0)}\right)\cdot\nabla_{x}\mathbf{S}_{p}^{(0),j}\right]\right.$
$\displaystyle\left.+\left(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)}\right)\left(\nabla_{p_{3}}\mathbf{S}_{3}^{(0),i}\cdot\nabla_{x}\mathbf{S}_{p}^{(0),j}-\nabla_{x}\mathbf{S}_{3}^{(0),i}\cdot\nabla_{p}\mathbf{S}_{p}^{(0),j}\right)\right\\}.$

Here we have neglected a term with

$\partial_{p_{0}}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})\epsilon_{aij}\mathbf{S}_{3}^{(0),j}\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)\mathbf{S}_{p}^{j,(0)}$

in the integrand which is of order
$\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4})$ by Eq. (47), while the terms
in Eq. (64) are all of order $\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{2})$,
where $g_{\mathrm{couple}}=g_{0}$ or $g_{\sigma}$ denotes the coupling
constant. The collision term in Eq. (64) is non-local for it contains
derivatives of space-time. From Eq. (64), we see that the collision term
vanishes if there is no polarization density at the leading order, i.e.,
$\mathbf{S}_{p}^{a,(0)}=0$. This is the result of the non-relativistic
coupling of the NJL type. In contrast it has been proved in Ref.
(Sheng:2021kfc, ) that the polarization can be generated from the Poisson
bracket term even without polarization density at the leading order in a
relativistic NJL model.

##  V Spin boltzmann equations: off-shell parts

In this section we will investigate off-shell contributions. From Eq. (38) and
(39) $G^{<}(x,p)$ actually contains an off-shell part
$G_{\mathrm{off}}^{<}(x,p)$ besides the on-shell part in (19) which we denote
as $G_{\mathrm{on}}^{<}(x,p)$, so do $I_{\mathrm{coll}}$ and
$I_{\mathrm{coll}}^{\dagger}$.

Now we try to derive the connection between $G_{\mathrm{on}}^{<}(x,p)$ and
$G_{\mathrm{off}}^{<}(x,p)$. From Eq. (38) we obtain

$G_{\mathrm{off}}^{<}(x,p)=\frac{1}{2p_{0}-\mathbf{p}^{2}/m}\left(I_{\mathrm{coll}}+I_{\mathrm{coll}}^{\dagger}\right)+\mathcal{O}(\hbar^{2}).$
(65)

Note that the on-shell part $I_{\mathrm{coll}}^{\mathrm{on}}$ contains
$\delta\left[p_{0}-\mathbf{p}^{2}/(2m)\right]$ which, combining
$\left[p_{0}-\mathbf{p}^{2}/(2m)\right]^{-1}$, gives the derivative of the
delta-function, $\delta^{\prime}\left[p_{0}-\mathbf{p}^{2}/(2m)\right]$. The
explicit form of $G_{\mathrm{off}}^{<}(x,p)$ can be determined by the
collision term from Eq. (65) and is at least of the same order as $\hbar
G_{\mathrm{on}}^{\lessgtr(1)}$ in Eq. (18). We can express
$G_{\mathrm{off}}^{<}(x,p)$ as

$\displaystyle G_{\mathrm{off}}^{<}(x,p)$ $\displaystyle=$ $\displaystyle
G_{\mathrm{off}}^{>}(x,p)$ (66) $\displaystyle=$
$\displaystyle-\hbar(2\pi\hbar)\partial_{p_{0}}\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\sum_{s_{1},s_{2}=\pm}\chi(s_{1})\chi^{\dagger}(s_{2})f_{s_{1}s_{2}}^{\mathrm{off}}(x,\mathbf{p})$
$\displaystyle=$
$\displaystyle-\hbar(2\pi\hbar)\partial_{p_{0}}\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\left[\overline{f}_{\mathrm{off}}(x,\mathbf{p})+\boldsymbol{\sigma}\cdot\mathbf{S}_{\mathrm{off}}(x,\mathbf{p})\right]$
$\displaystyle=$
$\displaystyle-\hbar(2\pi\hbar)\partial_{p_{0}}\delta\left(p_{0}-\frac{\mathbf{p}^{2}}{2m}\right)\boldsymbol{\sigma}\cdot\mathbf{S}_{\mathrm{off}}(x,\mathbf{p}),$

where we used $\overline{f}_{\mathrm{off}}(x,\mathbf{p})=0$ and

$\displaystyle\mathbf{S}_{\mathrm{off}}^{a}(x,\mathbf{p})$ $\displaystyle=$
$\displaystyle
2(g_{0}-3g_{\sigma})^{2}\epsilon^{aij}\int\frac{d^{4}p_{1}}{(2\pi\hbar)^{3}}\frac{d^{4}p_{2}}{(2\pi\hbar)^{3}}\frac{d^{4}p_{3}}{(2\pi\hbar)^{3}}(2\pi\hbar)^{4}\delta(p+p_{3}-p_{2}-p_{1})$
(67)
$\displaystyle\times\delta\left(p_{1}^{0}-\frac{\mathbf{p}_{1}^{2}}{2m}\right)\delta\left(p_{2}^{0}-\frac{\mathbf{p}_{2}^{2}}{2m}\right)\delta\left(p_{3}^{0}-\frac{\mathbf{p}_{3}^{2}}{2m}\right)\left(1-\overline{f}_{1}^{(0)}-\overline{f}_{2}^{(0)}\right)\mathbf{S}_{3}^{i,(0)}\mathbf{S}_{p}^{j,(0)},$

following Eq. (65) at the leading order. Actually $G_{\mathrm{off}}^{<}$ in
the form of (66) would be added to the left-hand-side of Eq. (44) at the next-
to-leading order [since $\overline{f}_{\mathrm{off}}(x,\mathbf{p})=0$, there
is no off-shell correction to Eq. (43)], while there is also the off-shell
part from the Poisson bracket term in the right-hand-side. It can be shown
that the off-shell terms in both sides of Eq. (44) are equal at
$\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{2})$ and thus drop out from the
equation to leave Eq. (62) for the on-shell part.

From Eq. (39), we obtain

$\displaystyle\hbar\left(\partial_{t}+\frac{1}{m}\mathbf{p}\cdot\nabla_{x}\right)G_{\mathrm{on}}^{<}(x,p)$
$\displaystyle=$
$\displaystyle-i\left(I_{\mathrm{coll}}-I_{\mathrm{coll}}^{\dagger}\right)-\hbar\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)G_{\mathrm{off}}^{<}(x,p)$
(68) $\displaystyle=$
$\displaystyle-i\left(I_{\mathrm{coll}}^{\mathrm{on}}-I_{\mathrm{coll}}^{\mathrm{on}\dagger}\right)-i\left(I_{\mathrm{coll}}^{\mathrm{off}}-I_{\mathrm{coll}}^{\mathrm{off}\dagger}\right)$
$\displaystyle-\hbar\frac{1}{2p_{0}-\mathbf{p}^{2}/m}\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)\left(I_{\mathrm{coll}}+I_{\mathrm{coll}}^{\dagger}\right)+\mathcal{O}(\hbar^{3}).$

Now we act the operator in the left-hand side of Eq. (31) to the right-hand-
side of Eq. (30) and obtain

$\left(-i\hbar\frac{\partial}{\partial
t_{2}}+\frac{\hbar^{2}}{2m}\nabla_{x2}^{2}\right)I_{\mathrm{coll}}(x_{1},x_{2})=\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4}).$
(69)

Note that the operator only acts on Green functions in $I_{\mathrm{coll}}$
(34) instead of self-energies since only Green functions depend on $x_{2}$,
which gives a contribution of $\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4})$
using Eq. (31). Taking a Wigner transform of Eq. (69) we obtain an equation
for $I_{\mathrm{coll}}^{\mathrm{on}}$ and $I_{\mathrm{coll}}^{\mathrm{off}}$

$\displaystyle I_{\mathrm{coll}}^{\mathrm{off}}$ $\displaystyle=$
$\displaystyle\frac{1}{2p_{0}-\mathbf{p}^{2}/m}i\hbar\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)I_{\mathrm{coll}}+\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4})+\mathcal{O}(\hbar^{3}g_{\mathrm{couple}}^{2}),$
(70)

where the first term is at least of
$\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{2})$. Taking a Hermitian conjugate
of the above equation leads to

$\displaystyle I_{\mathrm{coll}}^{\mathrm{off}\dagger}$ $\displaystyle=$
$\displaystyle-\frac{1}{2p_{0}-\mathbf{p}^{2}/m}i\hbar\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)I_{\mathrm{coll}}^{\dagger}+\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4})+\mathcal{O}(\hbar^{3}g_{\mathrm{couple}}^{2}).$
(71)

Substituting Eqs. (70) and (71) into (68) we arrive at an equation for the on-
shell part of the two-point function

$\hbar\left(\partial_{t}+\frac{\mathbf{p}}{m}\cdot\nabla_{x}\right)G_{\mathrm{on}}^{<}(x,p)=-i\left(I_{\mathrm{coll}}^{\mathrm{on}}-I_{\mathrm{coll}}^{\mathrm{on}\dagger}\right)+\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4})+\mathcal{O}(\hbar^{3}g_{\mathrm{couple}}^{2}).$
(72)

The above equation for on-shell parts is the foundation for discussions in
Sect. IV, in which the index “on” has been suppressed for notational
simplicity. In this section we have resumed the use of index “on” to denote
$G_{\mathrm{on}}^{\lessgtr}(x,p)$ and $I_{\mathrm{coll}}^{\mathrm{on}}$.

Let us analyze $I_{\mathrm{coll}}^{\mathrm{off}}$ in Eq. (70). From Eq. (41),
$I_{\mathrm{coll}}$ has two parts: one is the local term or quasi-classical
term $I_{\mathrm{qc}}$ which is at least of $\mathcal{O}(\hbar)$, and the
other is the nonlocal term with Poisson brackets $I_{\mathrm{PB}}$ which is at
least of $\mathcal{O}(\hbar^{2})$. The leading contribution of
$I_{\mathrm{coll}}^{\mathrm{off}}$ is of
$\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{2})$ from $I_{\mathrm{PB}}$
containing derivatives $\partial_{p_{0}}$ and $\nabla_{p}$ acting on
$G^{\lessgtr}(x,p)$, while the contribution from $G_{\mathrm{off}}^{<}$ in
(66) to $I_{\mathrm{coll}}^{\mathrm{off}}$ through the expansion of
$I_{\mathrm{qc}}$ is of $\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{4})$ which
is in higher order. Note that the leading contribution in
$I_{\mathrm{coll}}^{\mathrm{off}}$ from $I_{\mathrm{PB}}$ is in the same order
as the first term in the right-hand-side of Eq. (70). So at
$\mathcal{O}(\hbar^{2}g_{\mathrm{couple}}^{2})$, Eq. (70) gives a constraint
for MVSD $f^{(0)}$ or equivalently $\overline{f}^{(0)}$ and $\mathbf{S}^{(0)}$
in addition to Eqs. (46) and (47).

##  VI Comments on nuclear force through OBEP

In order to apply our theory to a non-relativistic nucleon system in low
energy collisions, one has to go beyond the contact interaction of the NJL
type and consider nuclear force as interaction. The main features of nuclear
force can be effectively decribed by one boson exchange potential (OBEP)
(Bryan:1969mp, ; Machleidt:1989tm, ). The OBEPs through scalar, pseudoscalar,
and vector meson exchanges have terms with operators
$\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$,
$\mathbf{L}\cdot\mathbf{S}$ and $S_{12}$ defined as
$\mathbf{L}\equiv-i\mathbf{r}\times\nabla$,
$\mathbf{S}\equiv(\boldsymbol{\sigma}_{1}+\boldsymbol{\sigma}_{2})/2$, and

$S_{12}\equiv\frac{1}{r^{2}}\left[3(\boldsymbol{\sigma}_{1}\cdot\mathbf{r})(\boldsymbol{\sigma}_{2}\cdot\mathbf{r})-r^{2}\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\right],$
(73)

where $\mathbf{r}\equiv\mathbf{x}-\mathbf{x}^{\prime}$. So OBEPs are nolocal
and contain couplings between spin and coordinate (equivalently spin and
momentum). In this case the spin is not a conserved quantity as in the NJL-
like model. It may be converted from local orbital angular momentum or local
vorticity (Weickgenannt:2020aaf, ; Weickgenannt:2021cuo, ). The extension to
the OBEP is much more complicated and beyond the scope of this work. It will
be reserved for a future study.

##  VII Summary

We derive spin Boltzmann equations for non-relativistic spin-1/2 fermions from
the KB equation in the CTP formalism. The non-relativistic model is similar to
the NJL model with four-fermion contact interaction which conserves spins in
particle scatterings. The great merit of the model is that the spin matrix
element in the collision term can be completely worked out and be put into a
compact form. One can clearly see how spins are coupled in two-to-two
scatterings of particles. In contrast it is hard to envisage the structure of
the spin matrix element which is much more complicated in the relativistic
theory (Sheng:2021kfc, ).

Starting from the non-relativistic Lagrangian, the KB equation is derived from
the Dyson-Schwinger equation defined on the CTP. The spin Boltzmann equations
for the particle number and spin distribution are derived based on Wigner
functions and the KB equation. Since the spin polarization is a quantum
effect, we make an expansion in the Planck constant $\hbar$ for all quantities
in the spin Boltzmann equation. At the leading order, the equilibrium spin
distribution can be obtained under the condition of the vanishing collision
term for the spin phase space density. A spin chemical potential emerges in
the equilibrium spin distribution which is a natural consequence of spin
conservation. The off-shell parts of spin Boltzmann equations are also
discussed. The work can be extended to a system of nucleons which interact via
nuclear forces in low energy heavy-ion collisions.

Acknowledgement. This work is supported in part by the Strategic Priority
Research Program of the Chinese Academy of Sciences (CAS) under Grant No.
XDB34030102, and by the National Natural Science Foundation of China (NSFC)
under Grants No. 12135011, 11890713 (a subgrant of 11890710).

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