Local Equilibrium Spin Distribution From Detailed Balance

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 Becattini:2013fla . 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 Becattini:2013fla . The same equilibrium distribution is also derived in Ref.Fang:2016vpj ; Gao:2018jsi by analyzing the free streaming spin transport equation. On the experimental side, the spin polarization effect in heavy ion collisions has attracted intense attention Liang:2004ph ; Voloshin:2004ha ; Betz:2007kg ; Becattini:2007sr . 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 momentumSTAR:2017ckg ; Adam:2018ivw ; Acharya:2019vpe . The magnitude of the global polarization of $\Lambda$ baryons can be very well described by models based on relativistic hydrodynamics and assuming local thermodynamic equilibrium of the spin degrees of freedomBecattini:2013vja ; Becattini:2015ska ; Becattini:2016gvu ; Karpenko:2016jyx ; Pang:2016igs ; Xie:2017upb . 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 productionBecattini:2014yxa or the collision terms for particles with spin. As one of the basic requirements, local thermodynamic equilibrium is defined by means of detailed balance of kinetic theory, namely the collisional integral of the Boltzmann equation vanishesDeGroot:1980dk . In this work, the equilibrium distribution functions for spin-$1/2$ particles are derived based on the detailed balance requirement.

The spin related anomalous transport phenomenon in heavy ion collisions, such as chiral magnetic effect (CME)Kharzeev:2004ey ; Fukushima:2008xe as well as chiral vortical effectNeiman:2010zi call for the spin related transport theory and hydrodynamic theory. The chiral kinetic theorySon:2012bg ; Son:2012wh ; Son:2012zy ; Stephanov:2012ki ; Pu:2010as ; Chen:2012ca ; Hidaka:2016yjf ; Huang:2018wdl ; Liu:2018xip ; Lin:2019ytz is developed to describe the anomalous transport of massless fermions, and is further extended to the spin transport theory of massive fermionsHattori:2019ahi ; Wang:2019moi ; Gao:2019znl ; Weickgenannt:2019dks ; Liu:2020flb . Recently, it is extended from the free streaming scenario to discussing the collisional effectsYang:2020hri ; Weickgenannt:2020aaf ; Carignano:2019zsh ; Li:2019qkf ; Hou:2020mqp . The general framework of spin transport with collision terms is derived based on the Keldysh theoryYang:2020hri . 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 orderYang:2020hri and weakly coupled quantum electrodynamics plasma in Ref.Hou:2020mqp . In this work, we investigate the collision term in spin transport theory based on the framework in Ref.Yang:2020hri . 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 Klevansky:1997wm . For massive fermions, spin is an independent degree of freedom, we take vector component $\mathcal{V}_{\mu}$ and axial-vector component $\mathcal{A}_{\mu}$ of the Wigner function as independent degrees of freedom and derive their kinetic equations at orders $\mathcal{O}(\hbar^{0})$ and $\mathcal{O}(\hbar^{1})$. The local equilibrium forms of vector and axial-vector components are then derived by requiring the detailed balance of the kinetic equations. Within such framework, one only needs to specify an interaction, and no more assumption is required.

The paper is organized as follows: In Section II, 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 Section III, 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. Eventually, we make concluding remarks and outlook in Section IV. For references, we present most of the details of computations and critical steps for derivations in the Appendix.

In this section, we review the basic steps of deriving the spin transport equation with collision term. Starting from the Wigner transformation applied to contour Green’s functionYang:2020hri ; Blaizot:2001nr

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

(1)

where $X=(x+y)/2$ and $Y=x-y$ are the center of mass coordinate and relative coordinate. Here, $\tilde{S}_{\alpha\beta}^{<}(x,y)=\langle\bar{\psi}_{\beta}(y)\psi_{\alpha}(x)\rangle$ and $\tilde{S}_{\alpha\beta}^{>}(x,y)=\langle\psi_{\alpha}(x)\bar{\psi}_{\beta}(y)\rangle$ are lessor and greater propagators, respectively. The Wigner transformation of the Dyson-Schwinger equation of the lessor and greater propagators gives the Kadanoff-Baym equationsYang:2020hri . The sum and difference of Kadanoff-Baym equations gives the constraint and transport equations. Hereafter, we focus only on $S_{\alpha\beta}^{<}(X,p)$,

	$\displaystyle\Big{\{}(\gamma^{\mu}p_{\mu}-m),S^{<}\Big{\}}+\frac{i\hbar}{2}\Big{[}\gamma^{\mu},\nabla_{\mu}S^{<}\Big{]}$	$\displaystyle=$	$\displaystyle\frac{i\hbar}{2}\Big{(}\big{[}\Sigma^{<},S^{>}\big{]}_{\star}-\big{[}\Sigma^{>},S^{<}\big{]}_{\star}\Big{)},$
	$\displaystyle\Big{[}(\gamma^{\mu}p_{\mu}-m),S^{<}\Big{]}+\frac{i\hbar}{2}\Big{\{}\gamma^{\mu},\nabla_{\mu}S^{<}\Big{\}}$	$\displaystyle=$	$\displaystyle\frac{i\hbar}{2}\Big{(}\{\Sigma^{<},S^{>}\}_{\star}-\{\Sigma^{>},S^{<}\}_{\star}\Big{)},$		(2)

where $m$ is mass of the fermion, $\Sigma^{<(>)}$ are the lessor and greater self-energy. 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

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

(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 get various components as,

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

(4)

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

	$\displaystyle C=\big{[}\Sigma^{<},S^{>}\big{]}_{\star}-~{}\big{[}\Sigma^{>},S^{<}\big{]}_{\star}$	$\displaystyle=$	$\displaystyle C_{S}+i\gamma^{5}C_{P}+\gamma^{\mu}C_{V_{\mu}}+\gamma^{5}\gamma^{\mu}C_{A_{\mu}}+\frac{1}{2}\sigma^{\mu\nu}C_{T\mu\nu},$
	$\displaystyle D=\{\Sigma^{<},S^{>}\}_{\star}-\{\Sigma^{>},S^{<}\}_{\star}$	$\displaystyle=$	$\displaystyle D_{S}+i\gamma^{5}D_{P}+\gamma^{\mu}D_{V_{\mu}}+\gamma^{5}\gamma^{\mu}D_{A_{\mu}}+\frac{1}{2}\sigma^{\mu\nu}D_{T\mu\nu}.$		(5)

Note that $C$ and $D$ contains both the loss term and the gain term, 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 spinor-basis decomposition for the self-energies is required to further derive the constraint and transport equation of each spin components,

$$\begin{split}\Sigma^{<}=\Sigma_{S}+i\Sigma_{P}\gamma^{5}+\Sigma_{V\mu}\gamma^{\mu}+\Sigma_{A\mu}\gamma^{5}\gamma^{\mu}+\frac{1}{2}\Sigma_{T\mu\nu}\sigma^{\mu\nu},\\ \Sigma^{>}=\bar{\Sigma}_{S}+i\bar{\Sigma}_{P}\gamma^{5}+\bar{\Sigma}_{V\mu}\gamma^{\mu}+\bar{\Sigma}_{A\mu}\gamma^{5}\gamma^{\mu}+\frac{1}{2}\bar{\Sigma}_{T\mu\nu}\sigma^{\mu\nu}.\end{split}$$

(6)

From the sum and difference of Kadanoff-Baym equations (II) as well as decomposition of the Wigner functions (4) and of the collision terms (II), one can derive the ten component functions

	$\displaystyle p_{\mu}\mathcal{V}^{\mu}-m\mathcal{S}=\frac{i\hbar}{4}C_{S},$
	$\displaystyle 2m\mathcal{P}+\hbar\nabla_{\mu}\mathcal{A}^{\mu}=-\frac{i\hbar}{2}C_{P},$
	$\displaystyle 2p_{\mu}\mathcal{S}-2m\mathcal{V}_{\mu}-\hbar\nabla^{\nu}\mathcal{S}_{\nu\mu}=\frac{i\hbar}{2}C_{V_{\mu}},$
	$\displaystyle\hbar\nabla_{\mu}\mathcal{P}-\epsilon_{\mu\nu\rho\sigma}p^{\sigma}\mathcal{S}^{\nu\rho}-2m\mathcal{A}^{\mu}=\frac{i\hbar}{2}C_{A_{\mu}},$
	$\displaystyle\hbar\nabla_{[\mu}\mathcal{V}_{\nu]}-2\epsilon_{\rho\sigma\mu\nu}p^{\rho}\mathcal{A}^{\sigma}-2m\mathcal{S}_{\mu\nu}=\frac{i\hbar}{2}C_{T\mu\nu},$		(7)

and

	$\displaystyle\nabla_{\mu}\mathcal{V}^{\mu}=\frac{1}{2}D_{S},$
	$\displaystyle 2p_{\mu}\mathcal{A}^{\mu}=\frac{\hbar}{2}D_{P},$
	$\displaystyle 2p^{\nu}\mathcal{S}_{\nu\mu}+\hbar\nabla_{\mu}\mathcal{S}=\frac{\hbar}{2}D_{V_{\mu}},$
	$\displaystyle 2p_{\mu}\mathcal{P}+\frac{\hbar}{2}\epsilon_{\mu\nu\rho\sigma}\nabla^{\sigma}\mathcal{S}^{\nu\rho}=-\frac{\hbar}{2}D_{A_{\mu}},$
	$\displaystyle 2p_{[\mu}\mathcal{V}_{\nu]}+\hbar\epsilon_{\mu\nu\rho\sigma}\nabla^{\rho}\mathcal{A}^{\sigma}=-\frac{\hbar}{2}D_{T\mu\nu}.$		(8)

Each component of the Wigner function and self-energies can be expanded by $\hbar$ and so as the constraint and transport equations Eq.(II) and Eq.(II). $\mathcal{V}$ and $\mathcal{A}$ give rise to the vector-charge and axial-charge currents through $J_{V}^{\mu}=\int q\mathcal{V}^{\mu}$ and $J_{5}^{\mu}=\int q\mathcal{A}^{\mu}$. The axial-charge currents can be regarded as a spin current of fermion. The 16 components given by the spin decomposition are not 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 $\mathcal{A}$, giving

	$\displaystyle\mathcal{S}^{(0)}$	$\displaystyle=$	$\displaystyle\frac{p_{\mu}}{m}\mathcal{V}^{(0)\mu},$
	$\displaystyle\mathcal{S}^{(1)}$	$\displaystyle=$	$\displaystyle\frac{p_{\mu}}{m}\mathcal{V}^{(1)\mu}-\frac{i}{4m}C_{S}^{(0)},$
	$\displaystyle\mathcal{P}^{(0)}$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle\mathcal{P}^{(1)}$	$\displaystyle=$	$\displaystyle-\frac{1}{2m}\nabla_{\mu}\mathcal{A}^{(0)\mu}-\frac{i}{4m}C_{P}^{(0)},$
	$\displaystyle\mathcal{S}_{\mu\nu}^{(0)}$	$\displaystyle=$	$\displaystyle-\frac{1}{m}\epsilon_{\rho\sigma\mu\nu}p^{\rho}\mathcal{A}^{(0)\sigma},$
	$\displaystyle\mathcal{S}^{(1)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{1}{2m}\nabla_{[\mu}\mathcal{V}^{(0)}_{\nu]}-\frac{1}{m}\epsilon_{\rho\sigma\mu\nu}p^{\rho}\mathcal{A}^{(1)\sigma}-\frac{i}{4m}C^{(0)}_{T\mu\nu}.$		(9)

Each of $\mathcal{V}_{\mu}$ and $\mathcal{A}_{\mu}$ contains 4 components, which are not all independent. $p_{\mu}\mathcal{A}^{(0)\mu}=0$ and $p_{[\mu}\mathcal{V}^{(0)}_{\nu]}=0$ indicates that $\mathcal{A}^{(0)}_{\mu}$ has three independent components, while $\mathcal{V}^{(0)}_{\mu}$ has only one independent component. Because of similar restrictions at $\mathcal{O}(\hbar)$, the number of independent components of $\mathcal{A}^{(1)}_{\mu}$ and $\mathcal{V}^{(1)}_{\mu}$ stays the same as at $\mathcal{O}(\hbar^{0})$. In order to keep the description covariant and symmetric, we in the following derive the transport equations of $\mathcal{V}_{\mu}$ 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 for number density and three for spin density.

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

	$\displaystyle(p\cdot\nabla)\mathcal{V}^{(0)}_{\mu}=\frac{m}{2}D_{V\mu}^{(0)}+\frac{i}{2}p^{\nu}C_{T\nu\mu}^{(0)},$
	$\displaystyle(p\cdot\nabla)\mathcal{A}^{(0)}_{\mu}=\frac{m}{2}D_{A\mu}^{(0)}-\frac{i}{2}p_{\mu}C_{P}^{(0)},$		(10)

With the spin decomposition of collision terms $C$ and $D$ given in Appendix.A, the transport equations of the vector and axial-vector components become,

	$\displaystyle p\cdot\nabla\mathcal{V}^{(0)}_{\mu}$	$\displaystyle=$	$\displaystyle m\widehat{\Sigma_{S}^{(0)}\mathcal{V}_{\mu}^{(0)}}+p^{\nu}\widehat{\Sigma_{V\nu}^{(0)}\mathcal{V}_{\mu}^{(0)}}+\frac{m}{2}\epsilon_{\alpha\beta\lambda\mu}\widehat{\Sigma_{T}^{(0)\alpha\beta}\mathcal{A}^{(0)\lambda}}-\frac{p_{\nu}}{m}\epsilon_{\alpha\mu\beta\lambda}p^{\beta}\widehat{\Sigma_{T}^{(0)\alpha\nu}\mathcal{A}^{(0)\lambda}}-p_{\mu}\widehat{\Sigma_{A}^{(0)\nu}\mathcal{A}_{\nu}^{(0)}},$
	$\displaystyle p\cdot\nabla\mathcal{A}^{(0)}_{\mu}$	$\displaystyle=$	$\displaystyle m\widehat{\Sigma_{S}^{(0)}\mathcal{A}_{\mu}^{(0)}}+p^{\nu}\widehat{\Sigma_{V\nu}^{(0)}\mathcal{A}_{\mu}^{(0)}}+\frac{m}{2}\epsilon_{\alpha\beta\lambda\mu}\widehat{\Sigma_{T}^{(0)\alpha\beta}\mathcal{V}^{(0)\lambda}}+\widehat{\Sigma_{A\mu}^{(0)}p^{\nu}\mathcal{V}_{\nu}^{(0)}}-p_{\mu}\widehat{\Sigma_{A\nu}^{(0)}\mathcal{V}^{(0)\nu}},$		(11)

where, same as in Ref.Yang:2020hri , 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 $\hbar$, especially the transport equation of $\mathcal{A}_{\mu}^{(1)}$. Taking the semiclassical expansion of Eq.(II) and Eq.(II), and considering the relation between various spin components Eq.(II), the on-shell conditions and transport equations at $\mathcal{O}(\hbar)$ are modified into

	$\displaystyle(p^{2}-m^{2})\mathcal{V}^{(1)}_{\mu}=\frac{ip_{\mu}}{4}C_{S}^{(0)}+\frac{im}{4}C_{V\mu}^{(0)},$
	$\displaystyle(p^{2}-m^{2})\mathcal{A}_{\mu}^{(1)}=\frac{1}{4}p^{\mu}D_{P}^{(0)}-\frac{i}{8}\epsilon_{\mu\alpha\beta\gamma}p^{\alpha}C_{T}^{(0)\beta\gamma}+\frac{i}{4}mC_{A\mu}^{(0)},$
	$\displaystyle(p\cdot\nabla)\mathcal{V}_{\mu}^{(1)}=\frac{m}{2}D_{V\mu}^{(1)}+\frac{i}{2}p^{\nu}C_{T\nu\mu}^{(1)}+\frac{i}{4}\nabla_{\mu}C_{S}^{(0)},$
	$\displaystyle(p\cdot\nabla)\mathcal{A}^{(1)}_{\mu}=\frac{m}{2}D_{A\mu}^{(1)}-\frac{i}{2}p_{\mu}C_{P}^{(1)}-\frac{i}{8}\epsilon_{\mu\sigma\nu\rho}\nabla^{\sigma}C_{T}^{(0)\nu\rho},$		(12)

together with the restrictions at first order of $\hbar$, $p_{\mu}\mathcal{A}^{(1)\mu}=\frac{1}{4}D_{P}^{(0)}$ and $p_{[\mu}\mathcal{V}^{(1)}_{\nu]}=-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\nabla^{\rho}\mathcal{A}^{(0)\sigma}-\frac{1}{4}D^{(0)}_{T\mu\nu}$. With the spin decomposition and semi-classical expansion of collision terms $C$ and $D$ in Appendix.A, as well as the relation between the spin components of the Wigner function Eq.(II), the on-shell relation become

	$\displaystyle(p^{2}-m^{2})\mathcal{V}^{(1)}_{\mu}$	$\displaystyle=$	$\displaystyle-\frac{m}{2}\widehat{\Sigma_{P}^{(0)}\mathcal{A}_{\mu}^{(0)}}-\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\widehat{\Sigma_{V}^{(0)\nu}\mathcal{A}^{(0)\beta}}-\frac{m}{2}\widehat{\Sigma_{T\mu\nu}^{(0)}\mathcal{V}^{(0)\nu}},$
	$\displaystyle(p^{2}-m^{2})\mathcal{A}_{\mu}^{(1)}$	$\displaystyle=$	$\displaystyle\frac{p^{\mu}}{2m}p^{\nu}\widehat{\Sigma_{P}^{(0)}\mathcal{V}_{\nu}^{(0)}}-\frac{m}{2}\widehat{\Sigma_{P}^{(0)}\mathcal{V}_{\mu}^{(0)}}+\epsilon_{\mu\alpha\beta\gamma}p^{\alpha}\widehat{\Sigma_{A}^{(0)\beta}\mathcal{A}^{(0)\gamma}},$		(13)

In general, at $\mathcal{O}(\hbar)$ order, $\mathcal{V}^{(1)}_{\mu}$ and $\mathcal{A}^{(1)}_{\mu}$ are off-shell because of the interaction. The restrictions at $\mathcal{O}(\hbar)$ again eliminate the redundant components in $\mathcal{A}^{(1)}_{\mu}$ and $\mathcal{V}^{(1)}_{\mu}$,

	$\displaystyle p_{[\mu}\mathcal{V}^{(1)}_{\nu]}$	$\displaystyle=$	$\displaystyle-\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}\nabla^{\alpha}\mathcal{A}^{(0)\beta}+\frac{1}{2m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\widehat{\Sigma_{S}^{(0)}\mathcal{A}^{(0)\beta}}+\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}\widehat{\Sigma_{V}^{(0)\alpha}\mathcal{A}^{(0)\beta}}+\frac{1}{2m}p_{[\mu}\widehat{\Sigma_{P}^{(0)}\mathcal{A}_{\nu]}^{(0)}}$
			$\displaystyle-\frac{1}{2m}p^{\rho}\widehat{\Sigma_{T\mu\nu}^{(0)}\mathcal{V}_{\rho}^{(0)}}-\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}\widehat{\Sigma_{A}^{(0)\alpha}\mathcal{V}^{(0)\beta}},$
	$\displaystyle p_{\mu}\mathcal{A}^{(1)\mu}$	$\displaystyle=$	$\displaystyle\frac{1}{2m}p^{\nu}\widehat{\Sigma_{P}^{(0)}\mathcal{V}_{\nu}^{(0)}}+\frac{1}{2m}p^{\rho}\widehat{\Sigma_{T\rho\nu}^{(0)}\mathcal{A}^{(0)\nu}}.$		(14)

Since the RHS of the restrictions contains only the $\mathcal{O}(\hbar^{0})$ components, $\mathcal{V}^{(1)}_{\mu}$ contains still only one independent component representing first order correction to number density, while $\mathcal{A}^{(1)}_{\mu}$ contains three independent components representing first order correction to spin density.

The transport equations of the first order components $\mathcal{V}^{(1)}_{\mu}$ and $\mathcal{A}^{(1)}_{\mu}$ are

	$\displaystyle(p\cdot\nabla)\mathcal{V}^{(1)}_{\mu}$	$\displaystyle=$	$\displaystyle+m\widehat{\Sigma_{S}^{(0)}\mathcal{V}_{\mu}^{(1)}}+p^{\nu}\widehat{\Sigma_{V\nu}^{(0)}\mathcal{V}_{\mu}^{(1)}}-p_{\mu}\widehat{\Sigma_{A}^{(0)\nu}\mathcal{A}^{(1)}_{\nu}}-\frac{p_{\nu}}{m}\epsilon_{\rho\sigma\alpha\mu}p^{\rho}\widehat{\Sigma_{T}^{(0)\alpha\nu}\mathcal{A}^{(1)\sigma}}+\frac{m}{2}\epsilon_{\sigma\nu\lambda\mu}\widehat{\Sigma_{T}^{(0)\sigma\nu}\mathcal{A}^{(1)\lambda}}$
			$\displaystyle+m\widehat{\Sigma_{S}^{(1)}\mathcal{V}_{\mu}^{(0)}}+p^{\nu}\widehat{\Sigma_{V\nu}^{(1)}\mathcal{V}_{\mu}^{(0)}}-p_{\mu}\widehat{\Sigma_{A}^{(1)\nu}\mathcal{A}_{\nu}^{(0)}}-\frac{p_{\nu}}{m}\epsilon_{\alpha\mu\beta\lambda}p^{\beta}\widehat{\Sigma_{T}^{(1)\alpha\nu}\mathcal{A}^{(0)\lambda}}+\frac{m}{2}\epsilon_{\sigma\nu\lambda\mu}\widehat{\Sigma_{T}^{(1)\sigma\nu}\mathcal{A}^{(0)\lambda}}$
			$\displaystyle+\frac{1}{2m}p^{\nu}[\widehat{\Sigma_{T\mu\nu}^{(0)}(p^{\alpha}\mathcal{V}_{\alpha}^{(0)}})]_{\text{P.B.}}-\frac{m}{2}[\widehat{\Sigma_{T\mu\nu}^{(0)}\mathcal{V}^{(0)\nu}}]_{\text{P.B.}}+\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}p^{\nu}[\widehat{\Sigma_{A}^{(0)\alpha}\mathcal{V}^{(0)\beta}}]_{\text{P.B.}}$
			$\displaystyle-\frac{1}{2m}p_{\nu}\widehat{\Sigma_{T\alpha\mu}^{(0)}\nabla^{[\alpha}\mathcal{V}^{(0)\nu]}}+\frac{1}{2m}p_{\nu}\widehat{\Sigma_{T}^{\alpha\nu(0)}\nabla_{[\alpha}\mathcal{V}^{(0)}_{\mu]}}+\frac{1}{2}\epsilon_{\beta\nu\lambda\mu}\widehat{\Sigma_{A}^{(0)\beta}\nabla^{\nu}\mathcal{V}^{(0)\lambda}}$
			$\displaystyle+\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}(\nabla^{\alpha}\widehat{\Sigma_{V}^{(0)\nu})\mathcal{A}^{(0)\beta}}-\frac{1}{2m}p_{\mu}(\nabla^{\nu}\widehat{\Sigma_{P}^{(0)})\mathcal{A}_{\nu}^{(0)}}-\frac{1}{2m}(p^{\nu}\nabla_{\nu}\widehat{\Sigma_{P}^{(0)})\mathcal{A}_{\mu}^{(0)}}+\frac{1}{2m}p^{\nu}\epsilon_{\mu\nu\alpha\beta}(\nabla^{\alpha}\widehat{\Sigma_{S}^{(0)})\mathcal{A}^{(0)\beta}},$

and

	$\displaystyle(p\cdot\nabla)\mathcal{A}_{\mu}^{(1)}$	$\displaystyle=$	$\displaystyle+m\widehat{\Sigma_{S}^{(0)}\mathcal{A}^{(1)}_{\mu}}+p^{\nu}\widehat{\Sigma_{V\nu}^{(0)}\mathcal{A}^{(1)}_{\mu}}+p^{\nu}\widehat{\Sigma_{A\mu}^{(0)}\mathcal{V}^{(1)}_{\nu}}+\frac{m}{2}\epsilon_{\alpha\beta\lambda\mu}\widehat{\Sigma_{T}^{(0)\alpha\beta}\mathcal{V}^{(1)\lambda}}-p_{\mu}\widehat{\Sigma_{A\nu}^{(0)}\mathcal{V}^{(1)\nu}}$
			$\displaystyle+m\widehat{\Sigma_{S}^{(1)}\mathcal{A}_{\mu}^{(0)}}+p^{\nu}\widehat{\Sigma_{V\nu}^{(1)}\mathcal{A}_{\mu}^{(0)}}+p^{\nu}\widehat{\Sigma_{A\mu}^{(1)}\mathcal{V}_{\nu}^{(0)}}+\frac{m}{2}\epsilon_{\alpha\beta\lambda\mu}\widehat{\Sigma_{T}^{(1)\alpha\beta}\mathcal{V}^{(0)\lambda}}-p_{\mu}\widehat{\Sigma_{A\nu}^{(1)}\mathcal{V}^{(0)\nu}}$
			$\displaystyle-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}(\nabla^{\sigma}\widehat{\Sigma_{V}^{(0)\nu})\mathcal{V}^{(0)\rho}}-\frac{m}{2}[\widehat{\Sigma_{P}^{(0)}\mathcal{V}_{\mu}^{(0)}}]_{\text{P.B.}}+\frac{1}{2m}p_{\mu}[\widehat{\Sigma_{P}^{(0)}(p^{\nu}\mathcal{V}_{\nu}^{(0)}})]_{\text{P.B.}}$
			$\displaystyle+\frac{1}{2}\epsilon_{\mu\sigma\nu\rho}\nabla^{\sigma}\widehat{\Sigma_{A}^{(0)\nu}\mathcal{A}^{(0)\rho}}+\frac{1}{2}\epsilon_{\nu\mu\alpha\beta}[\widehat{\Sigma_{A}^{(0)\nu}(p^{\alpha}\mathcal{A}^{(0)\beta}})]_{\text{P.B.}}-\frac{m}{2}[\widehat{\Sigma_{T\mu\nu}^{(0)}\mathcal{A}^{(0)\nu}}]_{\text{P.B.}}+\frac{1}{2m}p_{\mu}[\widehat{\Sigma_{T\rho\nu}^{(0)}(p^{\rho}\mathcal{A}^{(0)\nu}})]_{\text{P.B.}}$
			$\displaystyle-\frac{1}{2m}p_{\sigma}\nabla^{\sigma}(\widehat{\Sigma_{T\mu\nu}^{(0)}\mathcal{A}^{(0)\nu}})+\frac{1}{2m}p^{\nu}\nabla^{\sigma}(\widehat{\Sigma_{T\mu\nu}^{(0)}\mathcal{A}_{\sigma}^{(0)}})+\frac{1}{2m}p_{\mu}\nabla^{\sigma}(\widehat{\Sigma_{T\sigma\nu}^{(0)}\mathcal{A}^{(0)\nu}})-\frac{1}{2m}p^{\nu}\nabla^{\sigma}(\widehat{\Sigma_{T\sigma\nu}^{(0)}\mathcal{A}_{\mu}^{(0)}}).$

The first two lines in Eq.(II) and Eq.(II) 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 relate to the derivatives of self-energies and distribution functions, which are inhomogeneous effects. These terms are also quantum effects, which generate the coupled transport of vector charge and axial-vector charge. As we will see in the following, these inhomogeneous effects produce spin polarization from the thermal vorticity.

In this paper, we focus on deriving the local equilibrium distribution from the detailed balance principle. For this purpose, the interaction needs to be specified to calculate the explicit expression of off-diagonal self-energies $\Sigma^{<}$ and $\Sigma^{>}$. Considering the fact that, different interaction determines only how fast the system reaches equilibrium state, but not the equilibrium distribution function, therefore, we adopt the NJL-type model with scalar-channel of interaction and calculate the fermionic $2$ by $2$ scattering,

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

(17)

In general a large part of the light fermion mass comes from the chiral condensate, however, we here work in the chiral restored phase to simplify the calculation, and consider only the current mass. Due to the nature of the strong coupling theory, two expansions must be applied: one expansion in the inverse number of colors $1/N_{c}$, and one semiclassical expansion in powers of $\hbar$. Directly translating from the diagramsKlevansky:1997wm , and then perform the Wigner transformation, this results in the self-energy to the $1/N_{c}$ order (denoted by LO) and to the $1/N_{c}^{2}$ order (denoted by NL)

	$\displaystyle\Sigma^{>}_{\text{LO}}(X,p)$	$\displaystyle=$	$\displaystyle G^{2}\int dP~{}S^{>}(X,p_{1})\text{Tr}\Big{[}S^{<}(X,p_{2})S^{>}(X,p_{3})\Big{]},$
	$\displaystyle\Sigma^{>}_{\text{NL}}(X,p)$	$\displaystyle=$	$\displaystyle-G^{2}\int dP~{}S^{>}(X,p_{1})S^{<}(X,p_{2})S^{>}(X,p_{3}),$		(18)

with the momentum integral defined as $\int dP=\int\frac{d^{4}p_{1}d^{4}p_{2}d^{4}p_{3}}{(2\pi)^{4}(2\pi)^{4}(2\pi)^{4}}(2\pi)^{4}\delta(p-p_{1}+p_{2}-p_{3})$. The lesser self-energy $\Sigma^{<}_{\text{LO}}$ and $\Sigma^{<}_{\text{NL}}$ can be obtained by taking the exchange $S^{>}\leftrightarrow S^{<}$ from (III). As clarified in Klevansky:1997wm , the self-energy $\Sigma^{<(>)}_{\text{LO}}$ and $\Sigma^{<(>)}_{\text{NL}}$ correspond to different scattering channels. However, the spin decomposition of $\Sigma^{<(>)}_{\text{NL}}$ is much more complicated when the Green’s function involves Dirac structure. Besides, since the detailed balance requires that the gain term and the loss term cancel with each other in arbitrary collision channel, to simplify the calculation, we consider only the collisional self-energy at $\mathcal{O}(1/N_{c})$ order. The spin decomposition of $\Sigma_{\text{LO}}$ follows simply from that of $S^{<(>)}(X,p_{1})$, since the $\text{Tr}(S^{<}(X,p_{2})S^{>}(X,p_{3}))=\mathcal{S}^{2}\bar{\mathcal{S}}^{3}-\mathcal{P}^{2}\bar{\mathcal{P}}^{3}+\mathcal{V}^{2}_{\mu}\bar{\mathcal{V}}^{3\mu}-\mathcal{A}^{2}_{\mu}\bar{\mathcal{A}}^{3\mu}+\frac{1}{2}\mathcal{S}^{2}_{\mu\nu}\bar{\mathcal{S}}^{3\mu\nu}$ 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 $\bar{\mathcal{S}}$, $\bar{\Sigma}_{V\mu}$ corresponds to $\bar{\mathcal{V}}_{\mu}$, and $\bar{\Sigma}_{A\mu}$ corresponds to $\bar{\mathcal{A}}_{\mu}$ and so on.