1 Introduction

The single-particle distribution function is of fundamental significance in off-equilibrium kinetic theory as well as many-body physics in equilibrium state. It has long been well known that for a system consisting of identical particles in thermodynamic equilibrium, the average number of particles in a single-particle state is described by Boltzmann distribution for non-relativistic system, Bose-Einstein distribution for bosons, and Fermi-Dirac distribution for fermions. However, in general, when spin of fermions is an independent degree of freedom, the distribution for spin-1/2 particles needs to be extended to describe the thermodynamical equilibrium of spin degrees of freedom [1]. By analyzing the density matrix for spin-1/2 particles, it is found that the non-even population of the polarization states arises from a steady gradient of temperature, and is orthogonal to particle momentum [1]. The same equilibrium distribution is also derived in Refs. [2, 3] by analyzing the free streaming spin transport equation. On the experimental side, the spin polarization effect in heavy ion collisions has attracted intense attention [4,5,6,7]. A large global angular momentum is produced in non-central heavy ion collisions and the spin of hadrons emitted is aligned with the direction of the global angular momentum [8,9,10]. The magnitude of the global polarization of \(\Lambda \) baryons can be very well described by models based on relativistic hydrodynamics with thermodynamic equilibrium of spin degrees of freedom [11,12,13,14,15,16]. The distribution function of a system of spin-1/2 particles is thus not only of significant importance for theoretical interest, but also required to explain the experimental data. As a matter of fact, different forms of equilibrium distribution functions are proposed based on different arguments. The most optimal situation would be to derive an equilibrium form from the entropy production [17] or the collision terms for particles with spin. In a classical kinetic theory, as one of the basic requirements, the local equilibrium is defined by means of detailed balance, namely the vanishing of the collision kernel in the Boltzmann equation [18]. In this work, we investigate the detailed balance principle in a quantum kinetic theory. For spin transport at the leading order in \(\hbar \), the vanishing of the collision terms together with the Killing condition lead to global equilibrium spin distribution function. Different from the classical kinetic theory where the collision terms are eliminated by the local equilibrium distribution, in a quantum kinetic theory, the collision terms vanish only in global equilibrium.

The spin related anomalous transport phenomena in heavy ion collisions, such as chiral magnetic effect [19, 20] and chiral vortical effect [21], call for the spin related transport theory and hydrodynamic theory. The chiral kinetic theory [22,23,24,25,26,27,28,29,30,31] is developed to describe these anomalous transport of massless fermions and is further extended to the spin transport of massive fermions [32,33,34,35,36]. Recently, it is also extended from the free streaming scenario to including collisional effects [37,38,39,40,41,42]. The general framework of spin transport with collision terms is derived based on the Keldysh formalism [38]. This framework is then applied to the weakly coupled quark-gluon plasma at high temperature to compute the spin-diffusion term for massive quarks up to the leading logarithmic order [38] and weakly coupled quantum electrodynamics plasma [42]. In this work, we investigate the collision terms in spin transport theory in the framework developed in Ref. [38]. In order to include fermionic 2-by-2 scattering, we consider the interaction among fermions by adopting the Nambu–Jona-Lasinio (NJL) model and calculate the collisional self-energy by taking semi-classical (\(\hbar \)) expansion and non-perturbative (\(1/N_c\)) expansion [43]. For massive fermions, spin is an independent degree of freedom, we take vector and axial-vector components of the Wigner function as independent degrees of freedom and derive their kinetic equations at classical level and the leading order in \(\hbar \). The vector and axial-vector components in equilibrium state are derived by requiring the detailed balance of the kinetic equations. Since our goal is to derive the spin distribution in equilibrium state where the system has forgotten the history of its evolution, the distribution derived from detailed balance should be independent of the details of the interaction.

The paper is organized as follows: In Sect. 2, we briefly review the Wigner-function approach and derive the kinetic equations for vector and axial-vector components to the first order of \(\hbar \). In Sect. 3, after specifying the scalar four-fermion interaction and reviewing the free fermion solution of the classical Wigner function, we derive the local equilibrium formulae of vector and axial-vector components under the requirement of detailed balance. The spin is found to be polarized by the local vorticity. The conservation of angular momentum is verified in Sect. 4. Eventually, we make concluding remarks and outlook in Sect. 5. For references, we present most of the details of computations and critical steps for derivations in the Appendix.

2 Constraint and transport equation

In this section, we review the basic steps of deriving the spin transport equation with collision terms. Starting from the Wigner transformation applied to contour Green’s function [38, 44]

$$\begin{aligned} S_{\alpha \beta }^{<(>)}(X,p)=\int d^4 Y e^{ip\cdot Y/\hbar }{\tilde{S}}_{\alpha \beta }^{<(>)}(x,y), \end{aligned}$$
(1)

where \(X=(x+y)/2\) and \(Y=x-y\) are the center-of-mass and relative coordinates, and \({\tilde{S}}_{\alpha \beta }^{<}(x,y)=\langle {\bar{\psi }}_\beta (y)\psi _\alpha (x)\rangle \) and \({\tilde{S}}_{\alpha \beta }^{>}(x,y)=\langle \psi _\alpha (x){\bar{\psi }}_\beta (y)\rangle \) are lessor and greater propagators. The Wigner transformation of the Dyson-Schwinger equation of the lessor and greater propagators gives the Kadanoff–Baym equations [38], and the sum and difference of the Kadanoff–Baym equations give the constraint and transport equations. Hereafter, we focus only on the equations for \(S_{\alpha \beta }^{<}(X,p)\),

$$\begin{aligned} \begin{aligned}&\big \{(\gamma ^\mu p_\mu -m),S^<\big \}+\frac{i\hbar }{2}\big [\gamma ^\mu ,\nabla _\mu S^<\big ]\\&\quad =\frac{i\hbar }{2}\big (\big [\Sigma ^<,S^>\big ]_\star -\big [\Sigma ^>,S^<\big ]_\star \big ),\\&\big [(\gamma ^\mu p_\mu -m),S^<\big ]+\frac{i\hbar }{2}\big \{\gamma ^\mu ,\nabla _\mu S^<\big \}\\&\quad =\frac{i\hbar }{2}\big (\{\Sigma ^<,S^>\}_\star -\{\Sigma ^>,S^<\}_\star \big ), \end{aligned} \end{aligned}$$
(2)

where m is the fermion mass and \(\Sigma ^{<(>)}\) are the lessor and greater self-energies. The scattering process involves only \(\Sigma ^{<(>)}\), thus we have dropped the real parts of the retarded and advanced self-energies and of the retarded propagators. The star product of two functions A(q, X) and B(q, X) is generated from the Wigner transformation and stands for the shorthand notation of the following calculations

$$\begin{aligned} A\star B=AB+\frac{i\hbar }{2}[AB{]}_{\text {P.B.}}+{\mathcal {O}}(\hbar ^2), \end{aligned}$$
(3)

where the Poisson bracket is \([AB{]}_{\text {P.B.}}\equiv (\partial _q^\mu A)(\partial _\mu B)-(\partial _\mu A)(\partial _q^\mu B)\). The commutators are \(\{F,G\}\equiv FG+GF\), \([F,G{]}\equiv FG-GF\), \(\{F,G\}_\star \equiv F\star G+G\star F\) and \([F,G{]}_\star \equiv F\star G-G\star F\), with F and G being arbitrary matrix-valued functions.

Different Dirac components of the Wigner function have different physical meanings. Performing the spin decomposition of the Wigner function, one obtains various components,

$$\begin{aligned} \begin{aligned} S^<={\mathcal {S}}+i{\mathcal {P}}\gamma ^5+{\mathcal {V}}_\mu \gamma ^\mu +{\mathcal {A}}_\mu \gamma ^5\gamma ^\mu +\frac{1}{2}{\mathcal {S}}_{\mu \nu }\sigma ^{\mu \nu },\\ S^>=\bar{{\mathcal {S}}}+i\bar{{\mathcal {P}}}\gamma ^5+\bar{{\mathcal {V}}}_\mu \gamma ^\mu +\bar{{\mathcal {A}}}_\mu \gamma ^5\gamma ^\mu +\frac{1}{2}\bar{{\mathcal {S}}}_{\mu \nu }\sigma ^{\mu \nu } \end{aligned} \end{aligned}$$
(4)

with \(\sigma ^{\mu \nu }=i[\gamma ^\mu ,\gamma ^\nu ]/2\) and \(\gamma ^5=i\gamma ^0\gamma ^1\gamma ^2\gamma ^3\). Similarly, the collisions terms in (2) are also decomposed by the Clifford algebra,

$$\begin{aligned} \begin{aligned} C&=\big [\Sigma ^<,S^>\big ]_\star -~\big [\Sigma ^>,S^<\big ]_\star \\&=C_S+i\gamma ^5 C_P+\gamma ^\mu C_{V_\mu }+\gamma ^5\gamma ^\mu C_{A_\mu }+\frac{1}{2}\sigma ^{\mu \nu }C_{T\mu \nu },\\ D&=\{\Sigma ^<,S^>\}_\star -\{\Sigma ^>,S^<\}_\star \\&=D_S+i\gamma ^5 D_P+\gamma ^\mu D_{V_\mu }+\gamma ^5\gamma ^\mu D_{A_\mu }+\frac{1}{2}\sigma ^{\mu \nu }D_{T\mu \nu }. \end{aligned} \end{aligned}$$
(5)

Note that C and D contain both the loss and gain terms, they can be recognized as \(I_{\text {gain}}^c=\big [\Sigma ^<,S^>\big ]\), \(I^c_{\text {loss}}=\big [\Sigma ^>,S^<\big ]\), \(I_{\text {gain}}^a=\{\Sigma ^<,S^>\}\) and \(I_{\text {loss}}^a=\{\Sigma ^>,S^<\}\), with c and a denoting commutator and anti-commutator respectively. Since \(\Sigma \) and S are both \(4\times 4\) matrices, their multiplication is not commutative. The same spin decomposition for the self-energies is required to further derive the constraint and transport equations for the spin components,

$$\begin{aligned} \begin{aligned} \Sigma ^<\!=\!\Sigma _S+i\Sigma _P\gamma ^5+\Sigma _{V\mu }\gamma ^\mu +\Sigma _{A\mu }\gamma ^5\gamma ^\mu +\frac{1}{2}\Sigma _{T\mu \nu }\sigma ^{\mu \nu },\\ \Sigma ^>\!=\!{\bar{\Sigma }}_S+i{\bar{\Sigma }}_P\gamma ^5+{\bar{\Sigma }}_{V\mu }\gamma ^\mu +{\bar{\Sigma }}_{A\mu }\gamma ^5\gamma ^\mu +\frac{1}{2}{\bar{\Sigma }}_{T\mu \nu }\sigma ^{\mu \nu }.\nonumber \end{aligned}\!\!\!\!\!\!\\ \end{aligned}$$
(6)

From the sum and difference of the Kadanoff–Baym equations (2) as well as the decomposition of the Wigner functions (4) and of the collision terms (5), one derives ten equations for the components,

$$\begin{aligned} \begin{aligned}&p_\mu {\mathcal {V}}^\mu -m{\mathcal {S}}=\frac{i\hbar }{4}C_S,\\&2m{\mathcal {P}}+ \hbar \nabla _\mu {\mathcal {A}}^\mu =-\frac{i\hbar }{2}C_P,\\&2p_\mu {\mathcal {S}}-2m{\mathcal {V}}_\mu -\hbar \nabla ^\nu {\mathcal {S}}_{\nu \mu }=\frac{i\hbar }{2}C_{V\mu },\\&\hbar \nabla _\mu {\mathcal {P}}-\epsilon _{\mu \nu \rho \sigma }p^\sigma {\mathcal {S}}^{\nu \rho }-2m{\mathcal {A}}^\mu =\frac{i\hbar }{2}C_{A\mu },\\&\hbar \nabla _{[\mu }{\mathcal {V}}_{\nu ]}-2\epsilon _{\rho \sigma \mu \nu }p^\rho {\mathcal {A}}^\sigma -2m{\mathcal {S}}_{\mu \nu }=\frac{i\hbar }{2}C_{T\mu \nu } \end{aligned} \end{aligned}$$
(7)

and

$$\begin{aligned} \begin{aligned}&\nabla _\mu {\mathcal {V}}^\mu =\frac{1}{2}D_S,\\&2p_\mu {\mathcal {A}}^\mu =\frac{\hbar }{2}D_P,\\&2p^\nu {\mathcal {S}}_{\nu \mu }+ \hbar \nabla _\mu {\mathcal {S}}=\frac{\hbar }{2}D_{V\mu },\\&2p_\mu {\mathcal {P}}+\frac{\hbar }{2}\epsilon _{\mu \nu \rho \sigma }\nabla ^\sigma {\mathcal {S}}^{\nu \rho }=-\frac{\hbar }{2}D_{A\mu },\\&2p_{[\mu }{\mathcal {V}}_{\nu ]}+\hbar \epsilon _{\mu \nu \rho \sigma }\nabla ^\rho {\mathcal {A}}^\sigma =-\frac{\hbar }{2}D_{T\mu \nu }. \end{aligned} \end{aligned}$$
(8)

The Wigner function and self-energies in equations (7) and (8) can be expanded in terms of \(\hbar \). \({\mathcal {V}}\) and \({\mathcal {A}}\) give rise to the vector-charge and axial-charge currents through \(J_V^\mu (x)=\int d^4p{\mathcal {V}}^\mu (x,p)\) and \(J_5^\mu (x)=\int d^4p{\mathcal {A}}^\mu (x,p)\). The axial-charge currents can be regarded as the spin current of fermion.

The 16 components given by the spin decomposition are not all independent. Up to the first order of \(\hbar \), the scalar component \({\mathcal {S}}\), pseudo-scalar component \({\mathcal {P}}\) and tensor component \({\mathcal {S}}_{\mu \nu }\) can be expressed in terms of \({\mathcal {V}}\) and \({\mathcal {A}}\),

$$\begin{aligned} \begin{aligned} {\mathcal {S}}^{(0)}&=\frac{p_\mu }{m}{\mathcal {V}}^{(0)\mu },\\ {\mathcal {S}}^{(1)}&=\frac{p_\mu }{m}{\mathcal {V}}^{(1)\mu }-\frac{i}{4m}C_S^{(0)}, \\ {\mathcal {S}}_{\mu \nu }^{(0)}&=-\frac{1}{m}\epsilon _{\rho \sigma \mu \nu }p^\rho {\mathcal {A}}^{(0)\sigma }, \\ {\mathcal {S}}^{(1)}_{\mu \nu }&=\frac{1}{2m}\nabla _{[\mu }{\mathcal {V}}^{(0)}_{\nu ]}-\frac{1}{m}\epsilon _{\rho \sigma \mu \nu }p^\rho {\mathcal {A}}^{(1)\sigma }-\frac{i}{4m}C^{(0)}_{T\mu \nu }, \\ {\mathcal {P}}^{(0)}&=0,\\ {\mathcal {P}}^{(1)}&=-\frac{1}{2m}\nabla _\mu {\mathcal {A}}^{(0)\mu }-\frac{i}{4m}C_P^{(0)}. \end{aligned} \end{aligned}$$
(9)

The four components of \({\mathcal {V}}_\mu \) and \({\mathcal {A}}_\mu \) are not all independent. The constraints \(p_\mu {\mathcal {A}}^{(0)\mu }=0\) and \(p_{[\mu }{\mathcal {V}}^{(0)}_{\nu ]}=0\) indicate that \({\mathcal {A}}^{(0)}_\mu \) has three independent components and \({\mathcal {V}}^{(0)}_{\mu }\) has only one independent component. Considering the similar restrictions at \({\mathcal {O}}(\hbar )\), there are the same number of independent components for \({\mathcal {A}}^{(1)}_\mu \) and \({\mathcal {V}}^{(1)}_{\mu }\). In order to keep the description covariant and symmetric, we derive in the following the transport equations for \({\mathcal {V}}_\mu \) and \({\mathcal {A}}_\mu \), but keep in mind that \({\mathcal {V}}_\mu \) and \({\mathcal {A}}_\mu \) has redundant components and that the system has 4 independent degrees of freedom in total: one is number density and the other three are spin density.

The classical components are on the mass shell \((p^2-m^2){\mathcal {V}}^{(0)}_\mu =0\) and \((p^2-m^2){\mathcal {A}}^{(0)}_\mu =0\). Their transport equations are

$$\begin{aligned} \begin{aligned}&(p\cdot \nabla ){\mathcal {V}}^{(0)}_\mu =\frac{m}{2}D_{V\mu }^{(0)}+\frac{i}{2}p^\nu C_{T\nu \mu }^{(0)},\\&(p\cdot \nabla ){\mathcal {A}}^{(0)}_\mu =\frac{m}{2}D_{A\mu }^{(0)}-\frac{i}{2}p_\mu C_P^{(0)}. \end{aligned} \end{aligned}$$
(10)

With the spin decomposition of the collision terms C and D given in Appendix A, the transport equations become

$$\begin{aligned} p\cdot \nabla {\mathcal {V}}^{(0)}_\mu =&-m\widehat{\Sigma _S^{(0)}{\mathcal {V}}_{\mu }^{(0)}} -p^\nu \widehat{\Sigma _{V\nu }^{(0)}{\mathcal {V}}_{\mu }^{(0)}} +p_\mu \widehat{\Sigma _{A}^{(0)\nu }{\mathcal {A}}_{\nu }^{(0)}}\nonumber \\&-\frac{m}{2}\epsilon _{\alpha \beta \lambda \mu }\widehat{\Sigma _{T}^{(0)\alpha \beta }{\mathcal {A}}^{(0)\lambda }} +\frac{p_\beta }{m}\epsilon _{\mu \nu \alpha \lambda }p^\nu \widehat{\Sigma _T^{(0)\alpha \beta }{\mathcal {A}}^{(0)\lambda }},\nonumber \\ p\cdot \nabla {\mathcal {A}}^{(0)}_\mu =&-m\widehat{\Sigma _S^{(0)}{\mathcal {A}}_\mu ^{(0)}} -p^\nu \widehat{\Sigma _{V\nu }^{(0)}{\mathcal {A}}_\mu ^{(0)}} +p_\mu \widehat{\Sigma _{A\nu }^{(0)}{\mathcal {V}}^{(0)\nu }}\nonumber \\&-\frac{m}{2}\epsilon _{\alpha \beta \lambda \mu }\widehat{\Sigma _T^{(0)\alpha \beta }{\mathcal {V}}^{(0)\lambda }} -\widehat{\Sigma _{A\mu }^{(0)}p^\nu {\mathcal {V}}_\nu ^{(0)}}, \end{aligned}$$
(11)

where, the same as in Ref. [38], the hat operator is defined as \({\widehat{FG}}={\bar{F}}G-F{\bar{G}}\). Since the spin polarization is in general a quantum effect, it is crucial to investigate the transport equation at the first order of \(\hbar \). Taking the semiclassical expansion for equations (7) and (8) and considering the relations among different spin components shown in equation (9), the on-shell conditions and transport equations at \({\mathcal {O}}(\hbar )\) are modified as

$$\begin{aligned} \begin{aligned}&(p^2-m^2){\mathcal {V}}^{(1)}_\mu =\frac{ip_\mu }{4}C_S^{(0)}+\frac{im}{4}C_{V\mu }^{(0)},\\&(p^2-m^2){\mathcal {A}}_\mu ^{(1)}=\frac{1}{4}p^\mu D_P^{(0)}-\frac{i}{8}\epsilon _{\mu \alpha \beta \gamma }p^\alpha C_{T}^{(0)\beta \gamma }+\frac{i}{4}mC_{A\mu }^{(0)},\\&(p\cdot \nabla ){\mathcal {V}}_\mu ^{(1)}=\frac{m}{2}D_{V\mu }^{(1)}+\frac{i}{2}p^\nu C_{T\nu \mu }^{(1)}+\frac{i}{4}\nabla _\mu C_S^{(0)},\\&(p\cdot \nabla ){\mathcal {A}}^{(1)}_\mu =\frac{m}{2}D_{A\mu }^{(1)}-\frac{i}{2}p_\mu C_P^{(1)}-\frac{i}{8}\epsilon _{\mu \sigma \nu \rho }\nabla ^\sigma C_{T}^{(0)\nu \rho }\nonumber \end{aligned}\!\!\!\!\!\!\\ \end{aligned}$$
(12)

and the restrictions \(p_{[\mu }{\mathcal {V}}^{(1)}_{\nu ]}=-\frac{1}{2}\epsilon _{\mu \nu \rho \sigma }\nabla ^\rho {\mathcal {A}}^{(0)\sigma }-\frac{1}{4}D^{(0)}_{T\mu \nu }\) and \(p_\mu {\mathcal {A}}^{(1)\mu }=\frac{1}{4}D_P^{(0)}\). With the spin decomposition and semi-classical expansion for the collision terms C and D in Appendix A and the relations among the spin components of the Wigner function, the restrictions at \({\mathcal {O}}(\hbar )\) again reduce the number of independent components of \({\mathcal {A}}^{(1)}_\mu \) and \({\mathcal {V}}^{(1)}_\mu \),

$$\begin{aligned} p_{[\mu }{\mathcal {V}}^{(1)}_{\nu ]} =&-\frac{1}{2m}\epsilon _{\mu \nu \alpha \beta }p^\alpha \widehat{\Sigma _S^{(0)}{\mathcal {A}}^{(0)\beta }} +\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }\nabla ^\alpha {\mathcal {A}}^{(0)\beta }\nonumber \\&-\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }\widehat{\Sigma _{V}^{(0)\alpha }{\mathcal {A}}^{(0)\beta }}-\frac{1}{2m}p_{[\mu }\widehat{\Sigma _P^{(0)}{\mathcal {A}}_{\nu ]}^{(0)}}\nonumber \\&+\frac{1}{2m}p^\rho \widehat{\Sigma _{T\mu \nu }^{(0)}{\mathcal {V}}_\rho ^{(0)}} +\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }\widehat{\Sigma _{A}^{(0)\alpha }{\mathcal {V}}^{(0)\beta }},\nonumber \\ p_\mu {\mathcal {A}}^{(1)\mu }=&-\frac{1}{2m}p^\nu \widehat{\Sigma _P^{(0)}{\mathcal {V}}_\nu ^{(0)}}-\frac{1}{2m}p^\rho \widehat{\Sigma _{T\rho \nu }^{(0)}{\mathcal {A}}^{(0)\nu }}. \end{aligned}$$
(13)

Since the right hand side of the constraint and transport equations (14) and (13) contain only the \({\mathcal {O}}(\hbar ^0)\) components, \({\mathcal {V}}^{(1)}_\mu \) contains only one independent component representing the first order correction to the number density, and \({\mathcal {A}}^{(1)}_\mu \) contains only three independent components representing the first order correction to the spin density. The on-shell relations become

$$\begin{aligned} (p^2-m^2){\mathcal {V}}^{(1)}_\mu =&\frac{m}{2}\widehat{\Sigma _P{\mathcal {A}}_\mu }+\frac{m}{2}\widehat{\Sigma _{T\mu \nu }{\mathcal {V}}^{\nu }}\nonumber \\&+\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }p^\alpha \widehat{\Sigma _{V}^{\nu }{\mathcal {A}}^{\beta }},\nonumber \\ (p^2-m^2){\mathcal {A}}_\mu ^{(1)} =&-\frac{p^\mu }{2m}p^\nu \widehat{\Sigma _P{\mathcal {V}}_\nu }+\frac{m}{2}\widehat{\Sigma _P{\mathcal {V}}_\mu }\nonumber \\&-\epsilon _{\mu \alpha \beta \gamma }p^\alpha \widehat{\Sigma _{A}^{\beta }{\mathcal {A}}^{\gamma }}, \end{aligned}$$
(14)

with components of self-energy and Wigner function at order \({\mathcal {O}}(\hbar ^0)\). The transport equations for the first order components \({\mathcal {V}}^{(1)}_\mu \) and \({\mathcal {A}}^{(1)}_\mu \) are

$$\begin{aligned}&(p\cdot \nabla ){\mathcal {V}}^{(1)}_\mu \nonumber \\&\quad = -m\widehat{\Sigma _S^{(0)}{\mathcal {V}}_{\mu }^{(1)}} -p^\nu \widehat{\Sigma _{V\nu }^{(0)}{\mathcal {V}}_{\mu }^{(1)}}\nonumber \\&\qquad +p_\mu \widehat{\Sigma _{A}^{(0)\nu } {\mathcal {A}}^{(1)}_{\nu }} +\frac{p_\nu }{m}\epsilon _{\rho \sigma \alpha \mu }p^\rho \widehat{\Sigma _{T}^{(0)\alpha \nu }{\mathcal {A}}^{(1)\sigma }}\nonumber \\&\qquad -\frac{m}{2}\epsilon _{\sigma \nu \lambda \mu }\widehat{\Sigma _{T}^{(0)\sigma \nu }{\mathcal {A}}^{(1)\lambda }}\nonumber \\&\qquad -m\widehat{\Sigma _S^{(1)}{\mathcal {V}}_{\mu }^{(0)}} -p^\nu \widehat{\Sigma _{V\nu }^{(1)}{\mathcal {V}}_{\mu }^{(0)}} +p_\mu \widehat{\Sigma _{A}^{(1)\nu }{\mathcal {A}}_\nu ^{(0)}}\nonumber \\&\qquad +\frac{p_\nu }{m}\epsilon _{\alpha \mu \beta \lambda }p^\beta \widehat{\Sigma _{T}^{(1)\alpha \nu }{\mathcal {A}}^{(0)\lambda }}\nonumber \\&\qquad -\frac{m}{2}\epsilon _{\sigma \nu \lambda \mu }\widehat{\Sigma _{T}^{(1)\sigma \nu }{\mathcal {A}}^{(0)\lambda }}\nonumber \\&\qquad -\frac{1}{2m}p^\nu [\widehat{\Sigma _{T\mu \nu }^{(0)}(p^\alpha {\mathcal {V}}_\alpha ^{(0)}})]_{\text {P.B.}}\nonumber \\&\qquad +\frac{m}{2}[\widehat{\Sigma _{T\mu \nu }^{(0)}{\mathcal {V}}^{(0)\nu }}]_{\text {P.B.}} -\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }p^\nu [\widehat{\Sigma _{A}^{(0)\alpha }{\mathcal {V}}^{(0)\beta }}]_{\text {P.B.}}\nonumber \\&\qquad -\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }(\nabla ^\alpha \widehat{\Sigma _{V}^{(0)\nu }){\mathcal {A}}^{(0)\beta }} +\frac{1}{2m}p_\mu (\nabla ^\nu \widehat{\Sigma _P^{(0)}){\mathcal {A}}_\nu ^{(0)}}\nonumber \\&\qquad +\frac{1}{2m}(p^\nu \nabla _\nu \widehat{\Sigma _P^{(0)}){\mathcal {A}}_\mu ^{(0)}} -\frac{1}{2m}p^\nu \epsilon _{\mu \nu \alpha \beta }(\nabla ^\alpha \widehat{\Sigma _S^{(0)}){\mathcal {A}}^{(0)\beta }}\nonumber \\&\qquad +\frac{1}{2m}p_\nu \widehat{\Sigma _{T\alpha \mu }^{(0)}\nabla ^{[\alpha }{\mathcal {V}}^{(0)\nu ]}} -\frac{1}{2m}p_\nu \widehat{\Sigma _{T}^{\alpha \nu (0)}\nabla _{[\alpha }{\mathcal {V}}^{(0)}_{\mu ]}}\nonumber \\&\qquad -\frac{1}{2}\epsilon _{\beta \nu \lambda \mu }\widehat{\Sigma _{A}^{(0)\beta }\nabla ^{\nu }{\mathcal {V}}^{(0)\lambda }} \end{aligned}$$
(15)

and

$$\begin{aligned}&(p\cdot \nabla ){\mathcal {A}}_\mu ^{(1)} \nonumber \\&\quad = -m\widehat{\Sigma _S^{(0)}{\mathcal {A}}^{(1)}_{\mu }} -p^\nu \widehat{\Sigma _{V\nu }^{(0)}{\mathcal {A}}^{(1)}_{\mu }}\nonumber \\&\qquad -p^\nu \widehat{\Sigma _{A\mu }^{(0)}{\mathcal {V}}^{(1)}_{\nu }} -\frac{m}{2}\epsilon _{\alpha \beta \lambda \mu }\widehat{\Sigma _T^{(0)\alpha \beta }{\mathcal {V}}^{(1)\lambda }}\nonumber \\&\qquad +p_\mu \widehat{\Sigma _{A\nu }^{(0)}{\mathcal {V}}^{(1)\nu }}\nonumber \\&\qquad -m\widehat{\Sigma _S^{(1)}{\mathcal {A}}_\mu ^{(0)}} -p^\nu \widehat{\Sigma _{V\nu }^{(1)}{\mathcal {A}}_\mu ^{(0)}}\nonumber \\&\qquad -p^\nu \widehat{\Sigma _{A\mu }^{(1)}{\mathcal {V}}_\nu ^{(0)}} -\frac{m}{2}\epsilon _{\alpha \beta \lambda \mu }\widehat{\Sigma _T^{(1)\alpha \beta }{\mathcal {V}}^{(0)\lambda }}\nonumber \\&\qquad +p_\mu \widehat{\Sigma _{A\nu }^{(1)}{\mathcal {V}}^{(0)\nu }}\nonumber \\&\qquad +\frac{1}{2}\epsilon _{\mu \nu \rho \sigma }(\nabla ^\sigma \widehat{\Sigma _{V}^{(0)\nu }){\mathcal {V}}^{(0)\rho }} +\frac{m}{2}[\widehat{\Sigma _P^{(0)}{\mathcal {V}}_\mu ^{(0)}}]_{\text {P.B.}}\nonumber \\&\qquad -\frac{1}{2m}p_\mu [\widehat{\Sigma _P^{(0)}(p^\nu {\mathcal {V}}_\nu ^{(0)}})]_{\text {P.B.}} -\frac{1}{2}\epsilon _{\mu \sigma \nu \rho }\nabla ^\sigma \widehat{\Sigma _{A}^{(0)\nu }{\mathcal {A}}^{(0)\rho }}\nonumber \\&\qquad -\frac{1}{2}\epsilon _{\nu \mu \alpha \beta }[\widehat{\Sigma _{A}^{(0)\nu }(p^\alpha {\mathcal {A}}^{(0)\beta }})]_{\text {P.B.}} +\frac{m}{2}[\widehat{\Sigma _{T\mu \nu }^{(0)}{\mathcal {A}}^{(0)\nu }}]_{\text {P.B.}}\nonumber \\&\qquad -\frac{1}{2m}p_\mu [\widehat{\Sigma _{T\rho \nu }^{(0)}(p^\rho {\mathcal {A}}^{(0)\nu }})]_{\text {P.B.}}\nonumber \\&\qquad +\frac{1}{2m}p_\sigma \nabla ^\sigma (\widehat{\Sigma _{T\mu \nu }^{(0)}{\mathcal {A}}^{(0)\nu }}) -\frac{1}{2m}p^\nu \nabla ^\sigma (\widehat{\Sigma _{T\mu \nu }^{(0)}{\mathcal {A}}_\sigma ^{(0)}})\nonumber \\&\qquad -\frac{1}{2m}p_\mu \nabla ^\sigma (\widehat{\Sigma _{T\sigma \nu }^{(0)}{\mathcal {A}}^{(0)\nu }}) +\frac{1}{2m}p^\nu \nabla ^\sigma (\widehat{\Sigma _{T\sigma \nu }^{(0)}{\mathcal {A}}_\mu ^{(0)}}). \end{aligned}$$
(16)

The first two lines in equations (15) and (16) are dynamical effects which contain for instance the diffusion effect. These terms have the same structure as the collision terms in the classical limit (11). The last three lines in both transport equations are related to the derivatives of self-energies and distribution functions, which are inhomogeneous effects. As we will see in the following, these inhomogeneous effects produce spin polarization from the thermal vorticity.

3 Fermionic 2 by 2 scattering

In this paper, we focus on deriving the equilibrium distribution from the detailed balance principle. To this end, the interaction needs to be specified to obtain the explicit expression of the off-diagonal self-energies \(\Sigma ^<\) and \(\Sigma ^>\). Considering the fact that, while the process from non-equilibrium to equilibrium depends strongly on the interaction among particles of the system, the equilibrium distribution itself should be independent of the details of the interaction. Therefore, we adopt a NJL-type model with only scalar interaction and calculate the fermionic 2 by 2 scattering,

$$\begin{aligned} {\mathcal {L}}={{\bar{\psi }}}(i\hbar \partial \!\!\!/-m)\psi +G({{\bar{\psi }}}\psi )^2. \end{aligned}$$
(17)

In general a large part of the light fermion mass comes from the chiral condensate. To simplify the calculation, we work here in the chiral restored phase and consider only the current mass. Due to the nature of the strong coupling, we take two expansions, one is in the inverse number of colors \(1/N_c\) and the other in \(\hbar \). Directly translating from the diagrams [43] and performing the Wigner transformation, the self-energies to the leading order (LO) and next to the leading order (NL) of the \(1/N_c\) expansion can be explicitly expressed as

$$\begin{aligned} \begin{aligned}&\Sigma ^{>}_{\text {LO}}(X,p)\,=\,G^2\int dP~S^{>}(X,p_1)\text {Tr}\Big [S^{<}(X,p_2)S^{>}(X,p_3)\Big ],\\&\Sigma ^{>}_{\text {NL}}(X,p)\,=\,-G^2\int dP~S^{>}(X,p_1)S^{<}(X,p_2)S^{>}(X,p_3)\nonumber \end{aligned}\\ \end{aligned}$$
(18)

with \(\int dP=\int \frac{d^4p_1d^4p_2d^4p_3}{(2\pi )^4(2\pi )^4(2\pi )^4}(2\pi )^4\delta (p-p_1+p_2-p_3)\) for the momentum integral. The lesser self-energy \(\Sigma ^{<}_{\text {LO}}\) and \(\Sigma ^{<}_{\text {NL}}\) can be obtained by taking the exchange \(S^>\leftrightarrow S^<\) in (18). As clarified in [43], the self-energies \(\Sigma ^{<(>)}_{\text {LO}}\) and \(\Sigma ^{<(>)}_{\text {NL}}\) correspond to different scattering channels. Since the detailed balance requires that the gain term and the loss term cancel with each other in arbitrary collision channel, we consider, to simplify the calculation, only the collisional self-energy at leading order. The spin decomposition of \(\Sigma _{\text {LO}}\) follows simply from that of \(S^{<(>)}(X,p_1)\), since the factor \(\text {Tr}(S^<(X,p_2)S^>(X,p_3))={\mathcal {S}}^2\bar{{\mathcal {S}}}^3-{\mathcal {P}}^2\bar{{\mathcal {P}}}^3+{\mathcal {V}}^2_\mu \bar{{\mathcal {V}}}^{3\mu }-{\mathcal {A}}^2_\mu \bar{{\mathcal {A}}}^{3\mu }+\frac{1}{2}{\mathcal {S}}^2_{\mu \nu }\bar{{\mathcal {S}}}^{3\mu \nu }\) is a number. The self-energy \(\Sigma ^>_{\text {LO}}\) can be decomposed as \({{\bar{\Sigma }}}^{\text {LO}}_i=G^2\int dP~\text {Tr}\big (S^{<}(X,p_2)S^{>}(X,p_3)\big ){S}^<_i(X,p_1)\). For instance, \({\bar{\Sigma }}_{S}\) corresponds to \(\bar{{\mathcal {S}}}\), \({\bar{\Sigma }}_{V\mu }\) to \(\bar{{\mathcal {V}}}_\mu \) and \({\bar{\Sigma }}_{A\mu }\) to \(\bar{{\mathcal {A}}}_\mu \).

3.1 Classical limit

Taking the spin components of the self-energy, one obtains the transport equations for \({\mathcal {V}}_\mu ^{(0)}\) and \({\mathcal {A}}_\mu ^{(0)}\) including collision terms. Considering the relation between \({\mathcal {S}}^{(0)}\) and \({\mathcal {V}}^{(0)}_\mu \) and the fact that \({\mathcal {V}}^{(0)}_\mu \) can be decomposed to \({\mathcal {V}}^{(0)}_\mu \propto \delta (p^2-m^2)p^\mu f_V\), it would be convenient to derive the collision terms in transport equations for the vector charge distribution \(f_V\) and the axial-vector charge distribution \(f_A\) from the following two equations,

$$\begin{aligned}&p\cdot \nabla {\mathcal {S}}^{(0)}\nonumber \\&\quad =-G^2\int _p\Big [\Big (1+\frac{p_2\cdot p_3}{m^2}\Big )\left( \widehat{{\mathcal {S}}^2{{\mathcal {S}}}^3}\right. \nonumber \\&\qquad \left. -\widehat{{\mathcal {A}}^{2\mu }{\mathcal {A}}^{3}_\mu }\right) +\frac{p_3^\mu p_2^\nu }{m^2}\widehat{{{\mathcal {A}}^2_\mu {\mathcal {A}}}^{3}_\nu }\Big ]\nonumber \\&\qquad \Big [\Big (m+\frac{p\cdot p_1}{m}\Big )\left( \widehat{{\mathcal {S}}{{\mathcal {S}}}^1}\right. \nonumber \\&\qquad \left. - \widehat{{\mathcal {A}}^\nu {{\mathcal {A}}}^{1}_\nu }\right) +\frac{p_{1}^\mu p^\nu }{m} \widehat{{\mathcal {A}}^\mu {{\mathcal {A}}}^{1}_\nu }\Big ],\;\;\;\; \end{aligned}$$
(19)
$$\begin{aligned}&p\cdot \nabla {\mathcal {A}}^{(0)}_\mu \nonumber \\&\quad =-G^2\int _p\Big [\Big (1+\frac{p_2\cdot p_3}{m^2}\Big )\left( \widehat{{\mathcal {S}}^2{{\mathcal {S}}}^3}\right. \nonumber \\&\qquad \left. -\widehat{{\mathcal {A}}^{2\mu }{\mathcal {A}}^{3}_\mu }\right) +\frac{p_3^\mu p_2^\nu }{m^2}\widehat{{{\mathcal {A}}^2_\mu {\mathcal {A}}}^{3}_\nu }\Big ]\nonumber \\&\qquad \Big [\Big (m+\frac{p\cdot p_1}{m}\Big )(\widehat{{\mathcal {A}}_\mu {{\mathcal {S}}}^1}\nonumber \\&\qquad +\widehat{{\mathcal {S}}{\mathcal {A}}_\mu ^1})-\frac{p_\mu +p_{1\mu }}{m}p^\nu \widehat{{\mathcal {S}}{\mathcal {A}}^1_\nu }\Big ]. \end{aligned}$$
(20)

Note that all the components on the right hand side are at leading order of \(\hbar \). Before moving on to analyzing the scattering channels, we first recall the classical free fermion solution of the Wigner function [35]. From the definition of the Wigner function as well as the contour green’s function, the classical Wigner function for a free fermion system is given by

$$\begin{aligned} \begin{aligned} S^<(X,p)=&\frac{\delta (p^2-m^2)}{(2\pi )^3} \sum _{sr}\Big \{\theta (p^0){\bar{u}}_s({\mathbf {p}}) {u}_r({\mathbf {p}})f_q^{sr}(X,p)\\&+\theta (-p^0){\bar{v}}_s(-{\mathbf {p}}) {v}_r(-{\mathbf {p}}){\bar{f}}_{{\bar{q}}}^{sr}(X,-p)\Big \},\\ S^>(X,p)=&\frac{\delta (p^2-m^2)}{(2\pi )^3}\sum _{sr}\Big \{\theta (p^0){\bar{u}}_s({\mathbf {p}}) {u}_r({\mathbf {p}}){\bar{f}}_q^{sr}(X,p)\\&+\theta (-p^0){\bar{v}}_s(-{\mathbf {p}}) {v}_r(-{\mathbf {p}}){f}_{{\bar{q}}}^{sr}(X,-p)\Big \}, \end{aligned} \end{aligned}$$
(21)

where \(s,r=\pm 1\) denote the spin up and down along the direction set by the unit vector \(n^{\pm \mu }\), \({\bar{f}}_{{q}}^{sr}(X,p)=\delta _{sr}-{f}_{{q}}^{sr}(X,p)\) and \({\bar{f}}_{{\bar{q}}}^{sr}(X,-p)=\delta _{sr}-{f}_{{\bar{q}}}^{sr}(X,-p)\) can be obtained from the ensemble average of the creation and annihilation operators, and \(f_q^{sr}\) is an element of a \(2\times 2\) Hermitian matrix which can be diagonalized to give \(f_q^s\). One can easily find that \({\bar{f}}_q^{sr}\) gives \(1-f_q^s\) after diagonalization. The mean polarization vector is defined as \(n^{(0)\mu }(X,p)=\theta (p^0)n^{+\mu }(X,{\mathbf {p}})-\theta (-p^0)n^{-\mu }(X,-{\mathbf {p}})\) with

$$\begin{aligned} n^{\pm \mu }(X,{\mathbf {p}})=\pm \left( \frac{{\mathbf {n}}_*^\pm \cdot {\mathbf {p}}}{m}~,~{\mathbf {n}}_*^\pm +\frac{{\mathbf {n}}_*^\pm \cdot {\mathbf {p}}}{m(E_{\mathbf {p}}+m)}{\mathbf {p}}\right) , \end{aligned}$$
(22)

where \({\mathbf {n}}_*^\pm \) is the direction of the mean polarization for particles (+) and anti-particles (-) satisfying \({\mathbf {n}}_*^\pm \cdot {\mathbf {n}}_*^\pm =-1\), and the momentum \({\mathbf {p}}\) is measured in the Particle Rest Frame. The mean polarization is a unit time-like vector satisfying \(n^{(0)\mu }(X,p)n^{(0)}_{\mu }(X,p)=-1\). The spin decomposition of \(S^>\) can be obtained by simply taking the exchange \(f_q\leftrightarrow {\bar{f}}_q\) and \(f_{{\bar{q}}}\leftrightarrow {\bar{f}}_{{\bar{q}}}\). Note that \(f_q^s\) is the particle distribution parallel (\(s=+\)) and anti-parallel (\(s=-\)) to the unit vector \(n^{\pm \mu }\). The vector charge distribution \(f_V\) and axial charge distribution \(f_A\) are combinations of \(f_q^s\), \(f_{Vq}=f_q^+ + f_q^-\) and \(f_{Aq}=f_q^+ - f_q^-\). The magnitude of polarization can be defined through the positive quantity \(\zeta _{q/{\bar{q}}}(X,{\mathbf {p}})\), \(f_{Aq}=f_{Vq}(X,{\mathbf {p}})\zeta _q(X,{\mathbf {p}})\) and \(f_q^s=f_{Vq}(1+s\zeta _q)\). \(\zeta _{q/{\bar{q}}}(X,{\mathbf {p}})=1\) corresponds to a pure state, while \(\zeta _{q/{\bar{q}}}(X,{\mathbf {p}})<1\) describes a mixed state [45]. It is worth noticing that, for outgoing particles there is \({\bar{f}}_q^s=1-f_q^s\) which leads to \({\bar{f}}_V=1-f_V\) and \({\bar{f}}_A=-f_A\). This can be understood from the fact that, the axial distribution function \(f_A\) comes from the off-diagonal component of \(f^{sr}\), and the relation \({\bar{f}}^{sr}=\delta _{rs}-{f}^{sr}\) results in \({\bar{f}}_A=-f_A\). With \(f_V\) and \(f_A\), the classical components can be rewritten as

$$\begin{aligned} \begin{aligned} {\mathcal {V}}^{(0)}_\mu (X,p)&=\frac{2p_\mu }{(2\pi )^32E_{\mathbf {p}}}\Big \{\delta (p^0-E_{\mathbf {p}})f_{Vq}(X,{\mathbf {p}})\\&\quad +\delta (p^0+E_{\mathbf {p}}){\bar{f}}_{V{\bar{q}}}(X,-{\mathbf {p}})\Big \},\\ {\mathcal {A}}^{(0)}_\mu (X,p)=&\frac{2m}{(2\pi )^32E_{\mathbf {p}}}\Big \{\delta (p^0-E_{\mathbf {p}})n^{+}_{\mu }(X,{\mathbf {p}})f_{Aq}(X,{\mathbf {p}})\\&\quad -\delta (p^0+E_{\mathbf {p}})n^{-}_{\mu }(X,-{\mathbf {p}}){\bar{f}}_{A{\bar{q}}}(X,-{\mathbf {p}})\Big \}. \end{aligned} \end{aligned}$$
(23)

The similar relations for \(\bar{{\mathcal {V}}}^{(0)}_\mu \) and \(\bar{{\mathcal {A}}}_\mu ^{(0)}\) can be obtained by taking the exchange \(f_{Vq/{\bar{q}}}\leftrightarrow {\bar{f}}_{Vq/{\bar{q}}}\) and \(f_{Aq/{\bar{q}}}\leftrightarrow {\bar{f}}_{Aq/{\bar{q}}}\).

Let’s consider the particle sector of the transport equations for \({\mathcal {V}}^{(0)}_\mu \) and \({\mathcal {A}}_\mu ^{(0)}\), namely the terms with the delta function \(\delta (p^0-E_{{\mathbf {p}}})\). While both particles and anti-particles exist in \(S^<(p_2)\), \(S^>(p_3)\) and \(S^<(p_1)\), corresponding to different scattering process, only three of the eight channels are allowed by the energy-momentum conservation. Each channel contains a product of four quark and anti-quark distribution functions \(f_{Vq}\) and \({{\bar{f}}}_{V{{\bar{q}}}}\). One may attribute a diagram to each of these processes in a loose sense, by assigning \(f_{Vq}\) to an incoming quark, \(f_{V{{\bar{q}}}}\) to an incoming anti-quark, \({\bar{f}}_{Vq}\) to a outgoing quark, and \({\bar{f}}_{V{{\bar{q}}}}\) to a outgoing anti-quark. The allowed channels correspond to the quark–quark scattering and quark–antiquark scattering. The other channels involving particle and antiparticle creation and annihilation can be categorized as off-shell processes. Together with the gain term, one can obtain the transport equation for the vector charge. In the following, when considering detailed balance, we only focus on the first channel, namely the quark–quark scattering.

In kinetic theory, the local equilibrium state is specified by the distribution functions that eliminate the collision kernel. This implies that the distribution functions must depend only on the linear combination of the conserved quantities, namely the particle number, energy and momentum, and angular momentum.

We first focus on the collision terms for \(p\cdot \nabla {\mathcal {S}}^{(0)}\) in equation (19). The collision terms include three parts: the term with only vector charge distribution \(f_V\), the term with only axial charge distribution \(f_A\), and the term with both \(f_V\) and \(f_A\). The term involving only \(f_V\) is

$$\begin{aligned}&\!\!\!m(m^2+{p_2\cdot p_3})(m^2+p\cdot p_1)\nonumber \\&\!\!\!\qquad \big (f_{V}^{{\mathbf {p}}_1}{\bar{f}}_{V}^{{\mathbf {p}}}{\bar{f}}_{V}^{{\mathbf {p}}_2}{f}_{V}^{{\mathbf {p}}_3}-f_{V}^{{\mathbf {p}}}{\bar{f}}_{V}^{{\mathbf {p}}_1} {f}_{V}^{{\mathbf {p}}_2}{\bar{f}}_{V}^{{\mathbf {p}}_3}\big ), \end{aligned}$$
(24)

where we have neglected the subscript q in \(f_{Vq}\) and \(f_{Aq}\). The detailed balance requires all the collision terms to vanish. This implies that the local equilibrium distribution \(f_V\) is the Fermi-Dirac function. Requiring that \({\mathcal {A}}_\mu ^{(0)}\) has the structure \({\mathcal {A}}_\mu ^{(0)}=m n_\mu f_A\), one can easily show that the term containing only \({\mathcal {A}}_\mu \) is

$$\begin{aligned}&m\big [(p_2\cdot n_3)(p_3\cdot n_2)(p\cdot n_1)(p_1\cdot n)\nonumber \\&\quad +(m^2+p_2\cdot p_3)(m^2+p\cdot p_1)(n\cdot n_1)(n_2\cdot n_3)\nonumber \\&\quad -(m^2+p_2\cdot p_3)(p\cdot n_1)(p_1\cdot n)(n_3\cdot n_2)\nonumber \\&\quad -(m^2+p\cdot p_1)(n\cdot n_1)(p_2\cdot n_3)(p_3\cdot n_2)\big ]\nonumber \\&\quad \times ({\bar{f}}_{A}^{{\mathbf {p}}}f_{A}^{{\mathbf {p}}_1}{\bar{f}}_{A}^{{\mathbf {p}}_2}f_{A}^{{\mathbf {p}}_3}-{f}_{A}^{{\mathbf {p}}}{\bar{f}}_{A}^{{\mathbf {p}}_1}{f}_{A}^{{\mathbf {p}}_2}{\bar{f}}_{A}^{{\mathbf {p}}_3}). \end{aligned}$$
(25)

Considering \({\bar{f}}_{A}=-{f}_{A}\), the detailed balance in local equilibrium state requires \({\bar{f}}_{A}^{{\mathbf {p}}}f_{A}^{{\mathbf {p}}_1}{\bar{f}}_{A}^{{\mathbf {p}}_2}f_{A}^{{\mathbf {p}}_3}-{f}_{A}^{{\mathbf {p}}}{\bar{f}}_{A}^{{\mathbf {p}}_1}{f}_{A}^{{\mathbf {p}}_2}{\bar{f}}_{A}^{{\mathbf {p}}_3}=0\). This does not have any restriction on the equilibrium distribution function. Finally, the term involving mixture of \(f_V\) and \(f_A\) is

$$\begin{aligned}&m(m^2+p_2\cdot p_3)((p\cdot n^{1}) (p_{1}\cdot n)-(m^2+{p\cdot p_1}) n_1\cdot n)\nonumber \\&\quad \times ({\bar{f}}_{V}^{{\mathbf {p}}_2}{f}_{V}^{{\mathbf {p}}_3}-{f}_{V}^{{\mathbf {p}}_2}{\bar{f}}_{V}^{{\mathbf {p}}_3}) {f}_{A}^{{\mathbf {p}}} {f}_{A}^{{\mathbf {p}}_1}\nonumber \\&\quad +m(m^2+p\cdot p_1)((p_2\cdot n^{3})(p_3\cdot n^2)\nonumber \\&\quad -(m^2+{p_2\cdot p_3})n_2\cdot n_3)\nonumber \\&\quad \times ({\bar{f}}_{V}^{{\mathbf {p}}_1} {f}_{V}^{{\mathbf {p}}}-{f}_{V}^{{\mathbf {p}}_1}{\bar{f}}_{V}^{{\mathbf {p}}}){f}_{A}^{{\mathbf {p}}_2}{f}_{A}^{{\mathbf {p}}_3}. \end{aligned}$$
(26)

A trivial solution of the detailed balance for this term is \(f_A=0\).

We then consider the collision terms for \(p\cdot \nabla {\mathcal {A}}^{(0)}_\mu \) in equation (19). The terms in the momentum integral can be simplified as

$$\begin{aligned}&-m(m^2+p_2\cdot p_3)(m^2+p_1\cdot p)n_\mu {f}_{A}^{{\mathbf {p}}}(f_V^{{\mathbf {p}}_1} {\bar{f}}_{V}^{{\mathbf {p}}_2}{f}_{V}^{{\mathbf {p}}_3}\nonumber \\&\quad +{\bar{f}}_{V}^{{\mathbf {p}}_1} {f}_{V}^{{\mathbf {p}}_2}{\bar{f}}_{V}^{{\mathbf {p}}_3})\nonumber \\&\quad +\left[ (m^2+p_1\cdot p)n_\mu ^1-(p\cdot n^1)(p_\mu +p_{1\mu })\right] \nonumber \\&\quad \times m(m^2+p_2 \cdot p_3)f_{A}^{{\mathbf {p}}_1}({\bar{f}}_{V}^{{\mathbf {p}}}{\bar{f}}_{V}^{{\mathbf {p}}_2}{f}_{V}^{{\mathbf {p}}_3}+{f}_{V}^{{\mathbf {p}}}{f}_{V}^{{\mathbf {p}}_2}{\bar{f}}_{V}^{{\mathbf {p}}_3})\nonumber \\&\quad -m\big [(m^2+p_1\cdot p)n^1_\mu -(p_\mu +p_{1\mu })p\cdot n^1\big ]\nonumber \\&\quad \times \big [(p_2\cdot n^{3})(p_3\cdot n^2)-(m^2+p_2\cdot p_3)n^2\cdot n^{3}\big ] f_{A}^{{\mathbf {p}}_1}{f}_{A}^{{\mathbf {p}}_2}f_{A}^{{\mathbf {p}}_3},\nonumber \\&\quad +n_\mu \left[ (p_2\cdot n^{3})(p_3\cdot n^2)-(m^2+{p_2\cdot p_3}) n_2\cdot n_3\right] \nonumber \\&\quad \times m(m^2+p_1\cdot p){f}_{A}^{{\mathbf {p}}} {f}_{A}^{{\mathbf {p}}_2}{f}_{A}^{{\mathbf {p}}_3} \end{aligned}$$
(27)

where we have used the relations \({\bar{f}}_V+f_V=1\) and \({\bar{f}}_A=-f_A\). Again the detailed balance is satisfied by the trivial equilibrium solution \(f_A=0\) at classical level. Since the couplings between spin and external background field, vorticity, and orbital angular momentum are all at first order, and the strong interaction under consideration does not introduce any preferable direction, there cannot be any physical mechanism that affects the rotational symmetry of the system as well as the spin at classical level. Therefore, if the system is not initially polarized, the spin distribution will keep zero during the evolution of the system.

Considering the relations between spin components of the Wigner function shown in equation (9), the above solution of \(f_V\) and \(f_A\) in local equilibrium state leads to a nonzero scalar component \({\mathcal {S}}^{(0)}\ne 0\) and a vanishing tensor component \({\mathcal {S}}_{\mu \nu }^{(0)}=0\). The pseudo-scalar component at order \({\mathcal {O}}(\hbar )\) is directly shown to be vanished by the constraint equations. Since the axial charge current appears at order \({\mathcal {O}}(\hbar )\), it is necessary to analyze the transport equations at the leading order in \(\hbar \). In an initially unpolarized system, when considering the fermionic 2 by 2 collisions, the spin polarization can be produced as a quantum effect. The equilibrium spin polarization is then derived from the detailed balance of the collision terms in the transport equations at order \({\mathcal {O}}(\hbar )\).

3.2 Collision term at quantum level

The classical solution \({\mathcal {A}}_\mu ^{(0)}=0\) in equilibrium state greatly simplifies the transport equations (15) and (16). With the NJL model one can further verify that the vanishing \({\mathcal {A}}_\mu ^{(0)}\) leads to vanishing \(\Sigma _A^{(0)}\) and \(\Sigma _P^{(0)}\). At the leading order in \(1/N_c\) expansion, \({\bar{\Sigma }}_T^{(0)\alpha \beta }\) can also be shown to vanish. With these conditions, the vector component and axial-vector component are still on-shell at order \({\mathcal {O}}(\hbar )\), \((p^2-m^2){\mathcal {V}}_\mu ^{(1)}=0\) and \((p^2-m^2){\mathcal {A}}_\mu ^{(1)}=0\), and the orthogonal relations are unchanged as well, \(p_{[\mu }{\mathcal {V}}_{\nu ]}^{(1)}=0\) and \(p^\mu {\mathcal {A}}_\mu ^{(1)}=0\). The transport equations for \({\mathcal {V}}^{(1)}_\mu \) to the first order in \(\hbar \) can be simplified to be

$$\begin{aligned} \begin{aligned} (p\cdot \nabla ){\mathcal {V}}^{(1)}_\mu =&-m\big (\widehat{\Sigma _S{\mathcal {V}}_{\mu }}\big )^{(1)}-p^\nu \big (\widehat{\Sigma _{V\nu }{\mathcal {V}}_{\mu }}\big )^{(1)}, \end{aligned} \end{aligned}$$
(28)

with \((\widehat{\Sigma _S{\mathcal {V}}_{\mu }}\big )^{(1)}=\widehat{\Sigma _S^{(1)}{\mathcal {V}}_{\mu }^{(0)}}+\widehat{\Sigma _S^{(0)}{\mathcal {V}}_{\mu }^{(1)}}\) and the similar expression for the second term. It is obvious that, at order \({\mathcal {O}}(\hbar )\) the transport equation for \({\mathcal {V}}^{(1)}_\mu \) contains only diffusion terms. The transport equation for \({\mathcal {A}}^{(1)}_\mu \) at order \({\mathcal {O}}(\hbar )\) can be simplified to be

$$\begin{aligned} (p\cdot \nabla ){\mathcal {A}}^{(1)}_\mu&=-m\widehat{{\Sigma }_S^{(0)}{\mathcal {A}}^{(1)}_{\mu }}-p_\nu \widehat{{\Sigma }_{V}^{(0)\nu }{\mathcal {A}}^{(1)}_{\mu }}\nonumber \\&\quad -p_\nu \widehat{{\Sigma }_{A\mu }^{(1)}{\mathcal {V}}^{(0)\nu }}+p_\mu \widehat{{\Sigma }_{A\nu }^{(1)}{\mathcal {V}}^{(0)\nu }} \nonumber \\&\quad +\frac{1}{2}\epsilon _{\mu \nu \rho \sigma }(\nabla ^\sigma \widehat{{\Sigma }_V^{(0)\nu }){\mathcal {V}}^{(0)\rho }}\nonumber \\&\quad -\frac{m}{2}\epsilon _{\rho \nu \lambda \mu }\widehat{{\Sigma }_T^{(1)\rho \nu }{\mathcal {V}}^{(0)\lambda }}. \end{aligned}$$
(29)

The first two terms on the right hand side can be understood as spin diffusion, like the terms appeared in the transport equation for \({\mathcal {A}}_\mu ^{(0)}\) at classical level. The third term contains only the vector component, indicating that in an initially unpolarized system, a non-zero spin polarization can be generated through interaction with the medium. Considering that at zeroth order any vector can be decomposed by \(p^\mu \) and \(\beta ^\mu =u^\mu /T\), it would be possible to generate thermal vorticity from the third term. At the leading order in \(1/N_c\) expansion, the self-energy \(\Sigma ^>_{\text {LO}}\) is defined in Eq. (18). After making the substitution \(p_2\rightarrow k+q\) and \(p_3\rightarrow k\), and carrying out the integration over \(p_1\) using the \(\delta -\)function, the zeroth- and first-order self-energy becomes

$$\begin{aligned} {\Sigma }^{>(0)}(p)&=G^2\int _{qk}{S}^{>(0)}(p+q)\text {Tr}\big [S^{<}(k+q){S}^{>}(k)\big ]^{(0)},\nonumber \\ {\Sigma }^{>(1)}(p)&=G^2\int _{qk}{S}^{>(1)}(p+q)\text {Tr}\big [S^{<}(k+q){S}^{>}(k)\big ]^{(0)}\nonumber \\&\quad +G^2\int _{qk}{S}^{>(0)}(p+q)\text {Tr}\big [S^{<}(k+q){S}^{>}(k)\big ]^{(1)}\nonumber \\ \end{aligned}$$
(30)

with \(\int _{qk}=\int \frac{d^4q d^4k}{(2\pi )^8}\), where \(S^<\) and \(S^>\) in \([S^{<}(k+q){S}^{>}(k)]^{(0)}\) are both at the order \({\mathcal {O}}(\hbar ^0)\), and \([S^< S^>]^{(1)}\) is defined as \([S^{<}(k+q){S}^{>}(k)]^{(1)}=S^{<(0)}(k+q){S}^{>(1)}(k)+S^{<(1)}(k+q){S}^{>(0)}(k)\). In the above two transport equations for \({{\mathcal {V}}}_\mu ^{(1)}\) and \({{\mathcal {A}}}_\mu ^{(1)}\), the components \(\Sigma _S^{(0)}\), \(\Sigma _V^{(0)}\), \(\Sigma _S^{(1)}\), \(\Sigma _V^{(1)}\), \(\Sigma _A^{(1)}\) and \(\Sigma _T^{(1)}\) are all involved. At the leading order in \(1/N_c\), taking \({\mathcal {S}}\) and \({\mathcal {A}}_\mu \) as independent components, and considering the relations between the spin components shown in (9), the components of the self-energy can be evaluated and are presented in Appendix B. With the known self-energy, the loss term on the right hand side of the transport equations (28) and (29) can be evaluated explicitly. For instance, the loss term in (28) is given by

$$\begin{aligned} I_{V,\text {loss}}^{(1)}&=G^2p^\mu \int _{qk}~\frac{[m^2+(k+q)\cdot k][m^2+p\cdot (p+q)]}{m^3}\nonumber \\&\quad \times \Big \{{\mathcal {S}}^{(0)}(k+q)\bar{{\mathcal {S}}}^{(0)}(k)\bar{{\mathcal {S}}}^{(0)}(p+q){{\mathcal {S}}}^{(1)}(p)\nonumber \\&\quad +{\mathcal {S}}^{(1)}(k+q)\bar{{\mathcal {S}}}^{(0)}(k)\bar{{\mathcal {S}}}^{(0)}(p+q){{\mathcal {S}}}^{(0)}(p)\nonumber \\&\quad +{\mathcal {S}}^{(0)}(k+q)\bar{{\mathcal {S}}}^{(1)}(k)\bar{{\mathcal {S}}}^{(0)}(p+q){{\mathcal {S}}}^{(0)}(p)\nonumber \\&\quad +{\mathcal {S}}^{(0)}(k+q)\bar{{\mathcal {S}}}^{(0)}(k)\bar{{\mathcal {S}}}^{(1)}(p+q){{\mathcal {S}}}^{(0)}(p)\Big \}. \end{aligned}$$
(31)

The loss term on the right hand side of the transport equation (29) can be similarly expressed as

$$\begin{aligned} I_{A,\text {loss}}^{(1)}= & {} G^2\int _{qk}\Bigg \{\Big (1+\frac{(k+q)\cdot k}{m^2}\Big )\Big (m+\frac{p\cdot (p+q)}{m}\Big )\nonumber \\&\times {\mathcal {S}}^{(0)}(k+q)\bar{{\mathcal {S}}}^{(0)}(k)\bar{{\mathcal {S}}}^{(0)}(p+q){\mathcal {A}}^{(1)}_{\mu }(p)\nonumber \\&+\Big (1+\frac{(k+q)\cdot k}{m^2}\Big )\Big (m+\frac{p\cdot (p+q)}{m}\Big ){\mathcal {S}}^{(0)}(k+q)\nonumber \\&\times \bar{{\mathcal {S}}}^{(0)}(k)\bar{{\mathcal {A}}}_\mu ^{(1)}(p+q){\mathcal {S}}^{(0)}(p)\nonumber \\&-\Big (1+\frac{(k+q)\cdot k}{m^2}\Big )\frac{(2p+q)_\mu p^\nu }{m} {\mathcal {S}}^{(0)}(k+q)\nonumber \\&\times \bar{{\mathcal {S}}}^{(0)}(k){\mathcal {S}}^{(0)}(p)\bar{{\mathcal {A}}}^{(1)}_{\nu }(p+q)\nonumber \\&+\Big (1+\frac{(k+q)\cdot k}{m^2}\Big )\epsilon _{\mu \nu \rho \sigma }\frac{(p+q)^\nu p^\rho }{2m^2}\nonumber \\&\times {\mathcal {S}}^{(0)}(k+q)\bar{{\mathcal {S}}}^{(0)}(k)\bar{{\mathcal {S}}}^{(0)}(p+q)\big [\nabla ^\sigma {\mathcal {S}}^{(0)}(p)\big ]\nonumber \\&-\Big (1+\frac{(k+q)\cdot k}{m^2}\Big )\epsilon _{\mu \nu \rho \sigma }\frac{(p+q)^{\nu }p^\rho }{2m^2}{\mathcal {S}}^{(0)}(k+q)\nonumber \\&\times \bar{{\mathcal {S}}}^{(0)}(k){\mathcal {S}}^{(0)}(p)\big [\nabla ^{\sigma }\bar{{\mathcal {S}}}^{(0)}(p+q)\big ] \Bigg \}. \end{aligned}$$
(32)

The first order component \({\mathcal {A}}^{(1)}_\mu \) appears in the first three terms, and the last two terms contain only the classical scalar component \({\mathcal {S}}^{(0)}\) and its derivative. The appearance of the terms involving purely classical components indicates that the spin polarization \({\mathcal {A}}^{(1)}_\mu \) can be generated by collisions. Since such terms involve also spatial derivatives of \({\mathcal {S}}^{(0)}\), the spin polarization does not appear in homogeneous systems. When the system achieves local equilibrium, the detailed balance requires that the gain term and loss term cancel to each other. In the following, we show that the terms containing purely classical components would contribute to the equilibrium distribution \({\mathcal {A}}^{(1)}_\mu \). With the classical expression \({\mathcal {S}}^{(0)}\) in equation (23), considering only the particle part in \({\mathcal {S}}^{(0)}\) and \({\mathcal {A}}^{(1)}_\mu \) (namely the terms with \(\delta (p^0-E_p)\)), and using the identity \(p^\lambda \epsilon _{\mu \nu \rho \sigma }+p^\mu \epsilon _{\nu \rho \sigma \lambda }+p^\nu \epsilon _{\rho \sigma \lambda \mu }+p^\rho \epsilon _{\sigma \lambda \mu \nu }+p^\sigma \epsilon _{\lambda \mu \nu \rho }=0\), the terms with spatial derivatives in (32) are largely simplified. Considering \(\bar{{\mathcal {A}}}^{(1)}_\mu =-{{\mathcal {A}}}^{(1)}_\mu \) and \({\bar{f}}_V=1-f_V\), and using the detailed balance condition for the classical transport equation of \(f_V^{(0)}\), one has \(\frac{d}{d(\beta \cdot p)}\Big (f_V^{k+q}{\bar{f}}_V^{k}{\bar{f}}_V^{p+q}f_V^{p}-{\bar{f}}_V^{k+q}{f}_V^{k}{f}_V^{p+q}{\bar{f}}_V^{p}\Big )=0\) which then leads to

$$\begin{aligned} \begin{aligned}&\Big ({\bar{f}}_V^{k+q}{f}_V^{k}{f}_V^{p+q}+{f}_V^{k+q}{\bar{f}}_V^{k}{\bar{f}}_V^{p+q}\Big ) f'_V(p)\\&\quad -\Big (f_V^{k+q}{\bar{f}}_V^{k}f_V^{p}+{\bar{f}}_V^{k+q}{f}_V^{k}{\bar{f}}_V^{p}\Big )f'_V(p+q)=0. \end{aligned} \end{aligned}$$
(33)

The subtraction between the gain and loss terms can be expressed as

$$\begin{aligned} I_{A,\text {gain}}^{(1)}-I_{A,\text {loss}}^{(1)}=G^2\int _{qk} F(X;p,q,k), \end{aligned}$$
(34)

where F contains contributions from the lose and gain terms,

$$\begin{aligned} F&=-\frac{[m^2+(k+q)\cdot k][m^2+p\cdot (p+q)]}{[(2\pi )^3]^3E_{k+q}E_{k}E_{q+p}}\nonumber \\&\quad \times \Big ({\bar{f}}_V^{k+q}{f}_V^{k}{f}_V^{p+q}+{f}_V^{k+q}{\bar{f}}_V^{k}{\bar{f}}_V^{p+q}\Big ) \nonumber \\&\quad \times \bigg \{{{\mathcal {A}}}^{(1)}_{\mu }(p)+\frac{1}{(2\pi )^34E_{p}}\epsilon _{\mu \nu \sigma \lambda }p^\nu \nabla ^\sigma \beta ^\lambda f'_V(p)\nonumber \\&\quad + \frac{\epsilon _{\mu \nu \rho \lambda }p^\rho (p+q)^{\nu } p_\sigma (\nabla ^{\sigma }\beta ^\lambda +\nabla ^{\lambda }\beta ^\sigma ) f'_V(p)}{(2\pi )^3[m^2+p\cdot (p+q)]4E_{p}}\bigg \}\nonumber \\&\quad +\frac{[m^2+(k+q)\cdot k][m^2+p\cdot (p+q)]}{[(2\pi )^3]^3E_{k+q}E_{k}E_{p}}\nonumber \\&\quad \times \Big ({\bar{f}}_V^{k+q}{f}_V^{k}{\bar{f}}_V^{p} +{f}_V^{k+q}{\bar{f}}_V^{k}{f}_V^{p}\Big )\nonumber \\&\quad \times \bigg \{{{\mathcal {A}}}_\mu ^{(1)}(p+q)+\frac{1}{(2\pi )^34E_{q+p}}\epsilon _{\mu \sigma \lambda \nu }(p+q)^{\nu }\nabla ^{\sigma }\beta ^\lambda f'_V(p+q)\nonumber \\&\quad +\frac{\epsilon _{\mu \nu \rho \lambda }(p+q)^{\rho }p^\nu (p+q)_\sigma (\nabla ^{\sigma }\beta ^\lambda +\nabla ^{\lambda }\beta ^\sigma ) f'_V(p+q)}{(2\pi )^3[m^2+p\cdot (p+q)]4E_{q+p}}\bigg \}\nonumber \\&\quad -\frac{[m^2+(k+q)\cdot k]}{[(2\pi )^3]^3E_{k+q}E_{k}E_{p}} \Big ({\bar{f}}_V^{k+q}{f}_V^{k}{\bar{f}}_V^{p}+{f}_V^{k+q}{\bar{f}}_V^{k}{f}_V^{p}\Big )(2p+q)_\mu p^\nu \nonumber \\&\quad \times \bigg \{{{\mathcal {A}}}^{(1)}_{\nu }(p+q) +\frac{1}{(2\pi )^34E_{p+q}}\epsilon _{\nu \rho \sigma \lambda }(p+q)^{\rho }\nabla ^{\sigma }\beta ^\lambda f'_V(p+q)\bigg \}. \end{aligned}$$
(35)

A solution of \({{\mathcal {A}}}_\mu ^{(1)}\) that eliminates the collision term is

$$\begin{aligned} {{\mathcal {A}}}^{(1)}_{\mu } =&-\frac{1}{(2\pi )^34E_{p}}\epsilon _{\mu \nu \sigma \lambda }p^\nu \nabla ^\sigma \beta ^\lambda f'_V(p)\nonumber \\&-\frac{\epsilon _{\mu \nu \rho \lambda }(p+q)^\nu p^\rho p_\sigma (\nabla ^\sigma \beta ^\lambda +\nabla ^\lambda \beta ^\sigma )f'_V(p)}{ (2\pi )^3[m^2+p\cdot (p+q)]4E_p}.\nonumber \\ \end{aligned}$$
(36)

For a system in non-equilibrium state, particles can have different momentum. When the system is in equilibrium state, however, all the particles are thermalized and have only one momentum scale (here the momentum p). Therefore, the second term in above solution which depends on two momenta p and q should vanish. This requires the Killing condition \(\nabla ^\sigma \beta ^\lambda +\nabla ^\lambda \beta ^\sigma =0\). This is to say that the detailed balance in a quantum kinetic theory is fulfilled only in global equilibrium. In this sense the solution \({{\mathcal {A}}}^{(1)}_{\mu } =-\frac{1}{(2\pi )^34E_{p}}\epsilon _{\mu \nu \sigma \lambda }p^\nu \nabla ^\sigma \beta ^\lambda f'_V(p)\) can be called as the global equilibrium distribution. The result here is consistent with the conclusion in previous studies on non-local collisions [39] and relaxation from spin chemical potential to thermal vorticity in global equilibrium [46,47,48,49]. The solution indicates that in an initially unpolarized system, non-zero spin polarization can be generated from the collision terms, especially the coupling between vector and axial-vector charges. Different from the classical transport theory where the collision terms can be eliminated by the local equilibrium distribution, for the spin transport at order \({{\mathcal {O}}}(\hbar )\), the collision terms vanish only in global equilibrium.

In the above calculations we have used a simple NJL model with only scalar interaction. We believe that the equilibrium distribution should be independent of the interaction model, while the process from non-equilibrium to equilibrium depends strongly on the interaction itself. However, as far as we know, this is not yet strictly proven in quantum kinetic theory. We checked this problem in the NJL model and found that, 1) the above conclusions do not depend on the coupling constant G, namely the strength of the interaction, and 2) with other interaction channels such as the pseudo-scalar channel we obtain the same equilibrium distribution.

4 Angular momentum conservation

In the last section, we have verified that the distribution function in global equilibrium eliminates the collision terms, as required by the detailed balance principle. For an initially unpolarized system, the spin can get polarized by collisions, indicating the conversion between orbital and spin angular momentum. In this section, we check the total angular momentum conservation of the system.

The energy-momentum tensor and spin tensor are related to the Wigner function through the vector component and axial-vector component,

$$\begin{aligned}&T^{\mu \nu } = \int d^4p {\mathcal {V}}^\mu p^\nu \nonumber \\&S^{\rho ,\mu \nu } = -\frac{1}{2}\int d^4 p \epsilon ^{\rho \mu \nu \lambda }{\mathcal {A}}_\lambda , \end{aligned}$$
(37)

and the total angular momentum contains the orbital part and spin part,

$$\begin{aligned} M_{\rho ,\mu \nu }=x_\mu T_{\rho \nu }-x_\nu T_{\rho \mu }+\hbar S_{\rho ,\mu \nu }. \end{aligned}$$
(38)

Taking momentum integral over the last equation of (8) leads to

$$\begin{aligned}&\qquad \partial ^\rho M_{\rho ,\mu \nu } = \frac{\hbar }{4}\int d^4p\ D_{T\mu \nu }.\\ \nonumber \end{aligned}$$
(39)

At classical level with \(\hbar =0\), there is clearly

$$\begin{aligned} \partial ^\rho M_{\rho ,\mu \nu }^{(0)} = 0, \end{aligned}$$
(40)

which means orbital angular momentum conservation.

At first order in \(\hbar \), there is

$$\begin{aligned} \partial ^\rho M_{\rho ,\mu \nu }^{(1)} = \frac{1}{4}\int d^4p\ D_{T\mu \nu }^{(0)} \end{aligned}$$
(41)

with the classical collision term

$$\begin{aligned} D_{T\mu \nu }^{(0)}= & {} -2\Big [\widehat{\Sigma _S{\mathcal {S}}_{\mu \nu }} + \widehat{\Sigma _{T\mu \nu }{\mathcal {S}}} + \epsilon _{\mu \nu \alpha \beta }\Big (\widehat{\Sigma _{A}^\alpha {\mathcal {V}}^\beta }\nonumber \\&-\widehat{\Sigma _{V}^\alpha {\mathcal {A}}^\beta }-\frac{1}{2}\widehat{\Sigma _{T}^{\alpha \beta }{\mathcal {P}}}-\frac{1}{2}\widehat{\Sigma _P{\mathcal {S}}^{\alpha \beta }}\Big )\Big ]^{(0)}. \end{aligned}$$
(42)

Considering the classical constraints \({\mathcal {P}}^{(0)}, {\mathcal {A}}_\mu ^{(0)}, {\mathcal {S}}_{\mu \nu }^{(0)}=0\) and \(\Sigma _{T\mu \nu }^{(0)}, \Sigma _{A\mu }^{(0)}=0\), we have again the orbital angular momentum conservation,

$$\begin{aligned} \begin{aligned}&D_{T\mu \nu }^{(0)} = 0,\\&\partial ^\rho M_{\rho ,\mu \nu }^{(1)} = 0. \end{aligned} \end{aligned}$$
(43)

At second order in \(\hbar \), the transfer from orbital angular momentum to spin angular momentum starts. In this case, the collision term controlling the total angular momentum conservation becomes

$$\begin{aligned} D_{T\mu \nu }^{(1)}= & {} -2\Big [\widehat{\Sigma _S{\mathcal {S}}_{\mu \nu }} + \widehat{\Sigma _{T\mu \nu }{\mathcal {S}}} + \epsilon _{\mu \nu \alpha \beta }\Big (\widehat{\Sigma _{A}^\alpha {\mathcal {V}}^\beta }\nonumber \\&-\widehat{\Sigma _{V}^\alpha {\mathcal {A}}^\beta }\Big )\Big ]^{(1)}+\left[ \widehat{\Sigma _{V[\mu }{\mathcal {V}}_{\nu ]}}\right] _{\text {P.B.}}^{(0)}. \end{aligned}$$
(44)

Taking its momentum integration and considering the asymmetry under momentum exchange \(p\leftrightarrow p_1\) and \(p_2\leftrightarrow p_3\), the two integrals of the above first-order components disappear,

$$\begin{aligned}&\frac{(-2)}{4}\int d^4p \left[ \widehat{\Sigma _S{\mathcal {S}}_{\mu \nu }} +\widehat{\Sigma _{T\mu \nu }{\mathcal {S}}}\right] ^{(1)}\nonumber \\&\quad = (-2)\int d^4pd^4p_1d^4p_2d^4p_3(m^2+p_2\cdot p_3)\nonumber \\&\qquad \times \delta (p-p_1+p_2-p_3)\delta (p^2-m^2)\delta (p_1^2-m^2)\nonumber \\&\qquad \times \delta (p_2^2-m^2)\delta (p_3^2-m^2)\times \left[ f(p_2){\bar{f}}(p_3)\bar{{\mathcal {S}}}(p_1)\right. \nonumber \\&\qquad \times {\mathcal {S}}_{\mu \nu }(p)-{\bar{f}}(p_2){f}(p_3){{\mathcal {S}}}(p_1)\bar{{\mathcal {S}}}_{\mu \nu }(p)+f(p_2){\bar{f}}(p_3)\nonumber \\&\qquad \times \left. \bar{{\mathcal {S}}}_{\mu \nu }(p_1){\mathcal {S}}(p)-{\bar{f}}(p_2){f}(p_3){{\mathcal {S}}}_{\mu \nu }(p_1)\bar{{\mathcal {S}}}(p)\right] = 0,\nonumber \\&\qquad \times \frac{(-2)}{4}\int d^4p\ \epsilon _{\mu \nu \alpha \beta }\left( \widehat{\Sigma _{A}^\alpha {\mathcal {V}}^\beta }-\widehat{\Sigma _{V}^\alpha {\mathcal {A}}^\beta }\right) ^{(1)}\nonumber \\&\quad =(-2)\epsilon _{\mu \nu \alpha \beta }\int d^4pd^4p_1d^4p_2d^4p_3(m^2+p_2\cdot p_3)\nonumber \\&\qquad \times \delta (p-p_1+p_2-p_3)\delta (p^2-m^2)\delta (p_1^2-m^2)\nonumber \\&\qquad \times \delta (p_2^2-m^2)\delta (p_3^2-m^2)\times \left[ f(p_2){\bar{f}}(p_3)\bar{{\mathcal {A}}}^\alpha (p_1){\mathcal {V}}^\beta (p)\right. \nonumber \\&\qquad -{\bar{f}}(p_2){f}(p_3){{\mathcal {A}}}^\alpha (p_1)\bar{{\mathcal {V}}}^\beta (p)-f(p_2){\bar{f}}(p_3)\nonumber \\&\qquad \times \left. \bar{{\mathcal {V}}}^\alpha (p_1){\mathcal {A}}^\beta (p)+{\bar{f}}(p_2){f}(p_3){{\mathcal {V}}}^\alpha (p_1)\bar{{\mathcal {A}}}^\beta (p)\right] = 0.\nonumber \\ \end{aligned}$$
(45)

It is easy to see that the momentum integral of the Poisson bracket is also asymmetric under the momentum exchange, which results in

$$\begin{aligned} \frac{1}{4}\int d^4p\left[ \widehat{\Sigma _{V[\mu }{\mathcal {V}}_{\nu ]}}\right] _{\text {P.B.}}^{(0)} = 0. \end{aligned}$$
(46)

Therefore, the total angular momentum is conserved at the second order in \(\hbar \),

$$\begin{aligned} \partial ^\rho M_{\rho ,\mu \nu }^{(2)} = \frac{\hbar }{4}\int d^4p\ D_{T\mu \nu }^{(1)}=0. \end{aligned}$$
(47)

5 Conclusion

Spin is a quantum effect and is normally neglected in a classical transport theory. In this work, we addressed the problem of spin polarization in the Wigner function formalism of quantum kinetic theory. While non-equilibrium distributions are related to the details of the interaction of the system, namely the collision terms, the corresponding equilibrium distributions are determined only by the detailed balance between the loss and gain terms, namely the disappearing of the total collision terms. We obtained the equilibrium spin distribution by requiring the detailed balance for the Kadanoff–Baym equations. To be specific, we take a NJL model as an example to calculate the collision terms in the constraint and transport equations at classical level and to the leading order in \(\hbar \). We found that, for an initially non-polarized system without external electromagnetic fields, while the equilibrium spin distribution is trivial at classical level, the quantum correction internally generated by the inhomogeneous vorticity of the system leads to a non-trivial spin distribution. Different from the classical transport theory where the collision terms are eliminated by the local equilibrium distribution, for the spin transport in a quantum kinetic theory, the collision terms vanish only in global equilibrium.