Second order chiral kinetic theory under gravity and antiparallel charge-energy flow

Transport phenomena are a pivotal subject in modern quantum field theories. Similar to external electromagnetic fields, (effective) background gravitational fields are intriguing sources to generate various currents. First, the most widely well-known example is fluid vorticity; the fluid velocity can be described by a metric tensor of the comoving frame with fluid. The corresponding transport phenomenon, i.e., the chiral vortical effect (CVE) Vilenkin:1979ui can be regarded as the gravitational counterpart of the chiral magnetic effect (CME) Vilenkin:1980fu ; Nielsen:1983rb ; Fukushima:2008xe , as is clear from the gravitoelectromagnetism Landsteiner:2011cp . The CVE is not only a theoretically interesting phenomenon in the sense that it is originated from quantum anomaly Son:2009tf ; Landsteiner:2011cp , but also an important experimental probe to study the rotation of quark-gluon plasmas created in relativistic heavy-ion collision experiments STAR:2017ckg . Second, the mechanical strain plays a role of an effective $U(1)$ or axial $U(1)$ magnetic field, and accordingly yields a charge current PhysRevLett.115.177202 ; Cortijo:2016wnf ; PhysRevX.6.041046 . Third, spacetime torsion is recently under active investigation, as it can bring novel currents, which are referred to as the chiral torsional effect Sumiyoshi:2015eda ; Khaidukov:2018oat ; Ferreiros:2018udw ; Imaki:2019ite ; Imaki:2020csc ; Gao:2020gcf .

In contrast to the aforementioned effects from the spacetime geometry, we do not fully understand the effect of the gravitational Riemann curvature in the quantum transport theory. Even at the classical level, however, its importance has been known; the trajectory of a spinning particle is modified by the Riemann curvature Mathisson:1937zz ; Dixon:1970zza ; Papapetrou:1951pa . In the context of quantum transport theory, this knowledge suggests that the Riemann curvature can be the trigger of a characteristic transport of the fermion chirality (or spin, more generally). In cosmological systems, such spacetime distortions may become dominant contributions to determine the fermionic transport rather than background electromagnetic fields. In laboratory environments, a fluid motion and temperature gradient can be described by effective gravities leading to non-vanishing Riemann curvatures. Therefore, the curvature-induced transport phenomena could be relevant in a wide range of physics from table-top experiments to the Universe.

For the nonequilibrium dynamics in the weak interaction regime, one of the promising theoretical implements would be the kinetic theory. In particular, the so-called chiral kinetic theory (CKT) Stephanov:2012ki ; Son:2012wh , which nicely reproduces the chiral anomaly, plays a pivotal role in the development of various studies of the chiral transport phenomena Son:2012zy ; Chen:2012ca ; Manuel:2013zaa ; Chen:2014cla ; Chen:2015gta ; Hidaka:2016yjf ; Hidaka:2017auj ; Mueller:2017lzw ; Mueller:2017arw ; Huang:2018wdl ; Carignano:2018gqt ; Dayi:2018xdy ; Liu:2018xip ; Lin:2019ytz ; Carignano:2019zsh in the context of heavy-ion collision, condensed matter and neutrino physics; although the kinetic theory is inapplicable to strongly-coupled quark-gluon plasmas, the early stage of heavy-ion collisions is described well by the Boltzmann transport theory Baym:1984np ; Mueller:1999pi ; Baier:2000sb . The CKT conventionally involves only the leading order quantum correction so that the anomalous aspects can be taken into account as the Berry curvature. However, the leading order CKT is insufficient to capture the gravitational curvature contributions to the transport coefficients, although the kinetic equation involves the spin-curvature coupling Liu:2018xip . As is readily expected, higher-order corrections make the theory much more complicated, and an intuitive deduction would not avail. This fact can be found from the equilibrium distribution function. The $O(\hbar)$ contribution to enter the distribution is anticipated to be the spin-vorticity coupling, if we recall the conservation of the total angular momentum Chen:2015gta . On the other hand, this intuition is inapplicable to the $O(\hbar^{2})$ contribution, particularly, under a background gravitational field, as it is nontrivial to identify how the total angular momentum is modified at this order. Unlike the effective formalisms that relied on the Berry curvature, the derivation from quantum field theory works well against such a complication. In this case, the semiclassical (or weak coupling) dynamics is described by the Wigner transformation of the fermion propagator, which is a quantum-extended quantity of the classical distribution function. The Wigner function approach systematically involves quantum corrections appearing in the CKT, and definitely keeps the covariance of the fundamental theory Hidaka:2016yjf even in the general coordinate system Liu:2018xip . In this paper, thus we study the semiclassical transport theory with gravitational Riemann curvatures, based on the CKT derived with the Wigner function.

In the following, we present a summary of the findings in this paper. First, we solve the collisionless CKT in general coordinate and derive the Wigner function of Weyl fermions up to $O(\hbar^{2})$. This is a contrast to the conventional works, where only the $O(\hbar)$ quantum correction is taken into account. It is intriguing that the CKT in curved spacetime is systematically solvable even with the $O(\hbar^{2})$ contributions, while the $O(\hbar^{2})$ electromagnetic effect is not so tamable; the theory suffers from severe infrared divergence, for which so far no correct prescription is found.

The analysis of the higher-order quantum corrections reveals new aspects of the ambiguity underlying the CKT. It is well-known that due to degrees of freedom in terms of the Lorentz frame, the distribution function of chiral fermions cannot be uniquely determined Chen:2014cla . As a result, a frame vector representing such an ambiguity is inevitably introduced Chen:2015gta . From the Wigner function up to $O(\hbar^{2})$, we find that on top of the conventional frame vector, there emerges a different frame vector to define the distribution function. These extra degrees of freedom should be irrelevant to the Wigner function and thus physical quantities, as so is the conventional one. This is one of the guiding principles to identify an equilibrium distribution function. Indeed we find that there is no equilibrium solution for the kinetic equation in general curved spacetime. In other words, in general, the $O(\hbar^{2})$ CKT under gravity does not reach a global equilibrium without collisions. However, we elucidate that an equilibrium solution is admitted for several gravitational fields, such as the stationary weak one.

The remaining parts are devoted to the evaluation of physical quantities from the Wigner function that we derived before. For instance, the charge current and energy-momentum tensor are given by the momentum integrals, as follows:

$$\begin{split}J^{\mu}(x)=2\int_{p}\mathcal{R}^{\mu}(x,p)\,,\quad T^{\mu\nu}(x)=2\int_{p}p^{(\mu}\mathcal{R}^{\nu)}(x,p)+O(\hbar^{3})\end{split}$$

(1)

with $\int_{p}:=\int\frac{d^{4}p}{(2\pi)^{4}\sqrt{-g(x)}}$ and $X^{(\mu}Y^{\nu)}=\frac{1}{2}(X^{\mu}Y^{\nu}+Y^{\mu}X^{\nu})$. Here $\mathcal{R}^{\mu}(x,p)$ is the Wigner function of the right-handed Weyl fermions. Under a static weak gravity, the $O(\hbar)$ part of Eq. (1) correctly reproduces the CVE. We find that under a time-dependent gravity, the CVE totally vanishes in the dynamical limit of the background gravitational field. Although this fact is originally derived with the diagrammatic computation Landsteiner:2013aba , our calculation is its first verification based on the CKT.

The $O(\hbar^{2})$ contribution of Eq. (1) corresponds to the novel transport phenomena induced by Riemann curvatures, which is also our main finding. They are represented as the following charge current and energy-momentum tensor:

$$J^{\mu}=C_{1}j^{\mu}\,,\quad T^{\mu\nu}=C_{0}t^{\mu\nu}\,.$$

(2)

Here $C_{0}=\mu/(2\pi^{2})$ and $C_{1}=\mu^{2}/(4\pi^{2})+T^{2}/12$ are the metric-independent coefficients determined by temperature $T$ and chemical potential for right-handed fermions $\mu$. Also $j^{\mu}$ and $t^{\mu\nu}$ are functions of Riemann curvature $R_{\mu\nu\rho\sigma}$, Ricci tensor $R_{\mu\nu}$ and Ricci scalar $R$. In Table 2, we summarize the analytic forms of $j^{\mu}$ and $t^{\mu\nu}$. The ‘static’ (‘dynamical’) implies that a time-independent (time-dependent) metric tensor is used. We denote $\xi^{\mu}=\delta^{\mu}_{0}$ and $\eta_{\mu\nu}$ being the Minkowski metric tensor. It is demonstrated in Appendix D that the same charge current under a static gravity is also derived from different field-theoretical approaches. This consistency apparently guarantees the validity of our formalism in this paper.

The fermionic system under a background fluid is a pedagogical and informative environment to demonstrate the aforementioned novel phenomena. In general the fluid effect is translated into an effective curved spacetime described by the following metric tensor:

$$g_{00}=1+h_{00}(t,{\boldsymbol{x}})\,,\quad g_{0i}=h_{0i}(t,{\boldsymbol{x}})\,,\quad g_{ij}=\eta_{ij}\,,$$

(3)

where $h_{\mu\nu}$ is the fluctuation around the flat spacetime. With this metric, temperature gradient and vorticity are given by

$$\partial_{i}T/\bar{T}=-\frac{1}{2}\partial_{i}h_{00}\,,\quad\omega^{i}=-\frac{1}{2}\varepsilon^{0ijk}\partial_{j}h_{k0}\,.$$

(4)

Applying Eq. (3) to $j^{\mu}$ and $t^{\mu\nu}$ in Table 2, we get the transport phenomena induced by temperature gradient or the inhomogeneity of vorticity. The results are summarized in Table 2. Here we define the shear tensor as $\sigma^{ij}=\epsilon^{ij}-\frac{1}{3}\eta^{ij}{\epsilon_{k}}^{k}$ with $\epsilon_{ij}=\partial_{(i}h_{j)0}$.

There are two crucial features of the novel transport phenomena induced by gravity (or equivalently inhomogeneous fluid profiles). One is that even if the collisionless kinetic equation holds, these transport phenomena are induced, as so are the CME and CVE. Therefore these do not generate any entropy, and thus be nondissipative. This fact motivates us to analyze the relation of Tables 2 and 2 to the quantum anomaly, from different approaches, such as hydrodynamics. This is an interesting open question that will be revisited in the future. The other is the antiparallel flows of the charge current $j^{i}$ and energy current $t^{0i}$. Tables 2 and 2 for $\mu>0$ show the coefficients of these currents have opposite signs whether the metric is static or dynamical and whatever the metric tensor is. This is never explained by the classical particle motions; both charge and energy are carried along the classical particle momenta. The essential ingredient of the antiparallel flow is the spin-Riemann-curvature coupling. To find a more intuitive explanation of such curious flows is also a fascinating task.

In these respects, the novel gravity-induced transport phenomena should involve a lot of implications, e.g., in heavy-ion collisions or Dirac/Weyl semimetals (some of them are discussed in this paper). The latter systems may provide a good playground to study our novel phenomena and can be complementary environments to the former. For them, we need more detailed analysis based on the hydrodynamic model calculation, and quantitative comparison between theory and experiments. Besides, the CKT in curved spacetime and the resulting curvature-induced transport phenomena could play a more crucial role under genuine gravity, although we do not discuss it in this paper. For example, we can discuss the geodesics deviation of chiral fermions due to the spin-curvature coupling. Such a deviation may lead to some correction to the gravitational lensing of neutrinos liebes1964gravitational . Also, the present work could be applicable to the physics of core-collapse supernova explosions and neutron star formations Yamamoto:2015gzz . In this direction, we need to take the collisional effects into account Hidaka:2016yjf ; Hidaka:2017auj ; Yamamoto:2020zrs , based on the Kadanoff-Baym equation in curved spacetime, which respects the diffeomorphism covariance Hohenegger:2008zk .

This paper is organized as follows. In Sec. 2, solving the constraint equations, we obtain the general solution of the Wigner function up to $O(\hbar^{2})$ under general curved spacetime. In Sec. 3, we determine an equilibrium distribution function involving $O(\hbar^{2})$ corrections, based on the frame-independence of the Wigner function. In Sec. 4, we obtain curvature-induced charge current and energy-momentum tensor in equilibrium, which is consistent with different field-theoretical approaches in Appendix D. In Secs. 5 and 6, we analyze the dynamical response from background gravitational fields. As a practical application, in Sec. 7 we argue the transport phenomena in a fluid with inhomogeneous vorticity and temperature, which yields effective gravitational curvatures. In particular, we observe the antiparallel flows of charge and energy due to an inhomogeneous vorticity (or the Ricci tensor ${R_{0}}^{i}$). Through this paper, the convention follows from Ref. Liu:2018xip .

We first show the brief outline of the derivation of the CKT in the Wigner function approach. The kinetic theory is a perturbative effective theory in the infrared momentum regime. In quantum field theory, this perturbation is equivalent to the semiclassical truncation Elze:1986qd . The CKT is obtained from the semiclassical expansions of the equation of motions for the Wigner functions, that is, the Wigner transformed Dyson-Schwinger equation. The Wigner function of chiral fermions obeys three equations, which correspond to Eqs. (8)-(10) in this paper. Two of them are the constraints to the Wigner function, and the other becomes the kinetic equation eventually. In the classical limit, the equation for the Wigner function reduces to the Boltzmann equation (27). The Wigner function formalism gives the quantum generalization of the classical kinetic theory.

Let us start from the Wigner function for the right-handed Weyl fermions, which is defined as Fonarev:1993ht

		$\displaystyle\displaystyle\mathcal{R}^{\mu}(x,p)=\frac{1}{2}\text{tr}\biggl{[}\gamma^{\mu}\frac{1+\gamma^{5}}{2}W(x,p)\biggr{]}\,,$			(5)
		$\displaystyle\displaystyle W_{ab}(x,p)=\int d^{4}y\,\sqrt{-g(x)}\,e^{-ip\cdot y/\hbar}\langle\bar{\psi}_{b}(x,y/2)\psi_{a}(x,-y/2)\rangle\,,\ \ \$			(6)

with $g(x)=\det(g_{\mu\nu})$, $\bar{\psi}(x):=\psi^{\dagger}(x)\gamma^{\hat{0}}$, $\psi(x,y)=\exp(y\cdot D)\psi(x)$, $\bar{\psi}(x,y)=\bar{\psi}(x)\exp(y\cdot\overleftarrow{D})$, and $\bar{\psi}\overleftarrow{O}:=[O\psi]^{\dagger}\gamma^{\hat{0}}$. The above $W(x,p)$ is the general relativistic extension of the one in gauge theory Elze:1986qd . Here $D_{\mu}$ is called the horizontal lift; for a function on $(x^{\mu},y^{\mu})$ and $(x^{\mu},p_{\mu})$, the horizontal lift is represented as

$$D_{\mu}=\begin{cases}\nabla_{\mu}-\Gamma_{\mu\nu}^{\rho}y^{\nu}\partial_{\rho}^{y}\,,\\ \nabla_{\mu}+\Gamma_{\mu\nu}^{\rho}p_{\rho}\partial_{p}^{\nu}\,,\end{cases}$$

(7)

where $\nabla_{\mu}$ is the covariant derivative in terms of diffeomorphism and the local Lorentz transformation. The most beneficial property of $D_{\mu}$ is that it commutes with both $y^{\mu}$ and $p_{\mu}$, while $\nabla_{\mu}$ does not.

Hereafter we consider the Dirac theory under an external torsionless gravitational field. In this paper, we focus on the collisionless kinetic theory. The Dirac equation is thus given by $\gamma^{\mu}\nabla_{\mu}\psi(x)=0$, which brings the dynamical equation that the Wigner function $W(x,p)$ obeys. After a long computation, the set of equations for $\mathcal{R}^{\mu}(x,p)$ is up to $O(\hbar^{2})$ given by Liu:2018xip

		$\displaystyle(D_{\mu}+\hbar^{2}P_{\mu})\mathcal{R}^{\mu}=0\,,$			(8)
		$\displaystyle(p_{\mu}+\hbar^{2}Q_{\mu})\mathcal{R}^{\mu}=0\,,$			(9)
		$\displaystyle\displaystyle\hbar\varepsilon_{\mu\nu\rho\sigma}D^{\rho}\mathcal{R}^{\sigma}+4\Bigl{[}(p_{[\mu}+\hbar^{2}T_{[\mu})\mathcal{R}_{\nu]}+\hbar^{2}S_{\alpha\mu\nu}\mathcal{R}^{\alpha}\Bigr{]}=0\,,$			(10)

where we introduce the following notations:

		$\displaystyle\displaystyle P_{\mu}=-\frac{1}{8}\nabla_{\lambda}R_{\mu\nu}\partial_{p}^{\lambda}\partial_{p}^{\nu}-\frac{1}{24}\nabla_{\lambda}{R^{\rho}}_{\sigma\mu\nu}\partial_{p}^{\lambda}\partial_{p}^{\nu}\partial_{p}^{\sigma}p_{\rho}+\frac{1}{8}{R^{\rho}}_{\sigma\mu\nu}\partial_{p}^{\nu}\partial_{p}^{\sigma}D_{\rho}\,,$			(11)
		$\displaystyle\displaystyle Q_{\mu}=\frac{1}{8}R_{\mu\nu}\partial_{p}^{\nu}+\frac{1}{24}{R^{\rho}}_{\sigma\mu\nu}\partial^{\nu}_{p}\partial_{p}^{\sigma}p_{\rho}=3A_{\mu}+B_{\mu}\,,$			(12)
		$\displaystyle\displaystyle T_{\mu}=\frac{1}{4}R_{\mu\nu}\partial_{p}^{\nu}+\frac{1}{24}{R^{\rho}}_{\sigma\mu\nu}\partial^{\nu}_{p}\partial_{p}^{\sigma}p_{\rho}=6A_{\mu}+B_{\mu}\,,$			(13)
		$\displaystyle\displaystyle A_{\mu}=\frac{1}{24}R_{\mu\nu}\partial_{p}^{\nu}\,,\quad B_{\mu}=\frac{1}{24}{R^{\rho}}_{\sigma\mu\nu}\partial_{p}^{\nu}\partial_{p}^{\sigma}p_{\rho}\,,\quad S_{\alpha\mu\nu}=-\frac{1}{16}R_{\lambda\alpha\mu\nu}\partial^{\lambda}_{p}\,.$			(14)

In the above equations, we denote $X^{[\mu}Y^{\nu]}=(X^{\mu}Y^{\nu}-X^{\nu}Y^{\mu})/2$, the Riemann tensor is defined as ${R^{\rho}}_{\sigma\mu\nu}=2\bigl{(}\partial_{[\nu}\Gamma^{\rho}_{\mu]\sigma}+\Gamma_{\lambda[\nu}^{\rho}\Gamma_{\mu]\sigma}^{\lambda}\bigr{)}$ with $\Gamma^{\rho}_{\mu\nu}=g^{\rho\lambda}(\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu})/2$, and the Ricci tensor is $R_{\mu\nu}={R^{\lambda}}_{\mu\lambda\nu}$. For left-handed Weyl fermions, similar equations are derived, but only the sign in front of $\varepsilon_{\mu\nu\rho\sigma}$ is flipped, as is parity-odd. The first equation (8) corresponds to the kinetic equation while the others (9) and (10) are constraints that determine the functional form of $\mathcal{R}^{\mu}$. It is worthwhile to mention that Eqs. (8)-(10) are the Ward identities in terms of the symmetries that Weyl fermions respect in a given coordinate; the $U(1)$ gauge symmetry, the conformal symmetry, and the Lorentz symmetry, respectively Liu:2020flb .

Let us parametrize the solution for Eqs. (9) and (10) as

$$\mathcal{R}^{\mu}=\mathcal{R}^{\mu}_{(0)}+\hbar\mathcal{R}^{\mu}_{(1)}+\hbar^{2}\mathcal{R}_{(2)}^{\mu}\,.$$

(15)

Contracting Eq. (10) with $p^{\nu}$, we find

$$p^{2}\mathcal{R}_{\mu}=p_{\mu}p\cdot\mathcal{R}+\frac{\hbar}{2}\varepsilon_{\mu\nu\rho\sigma}p^{\nu}D^{\rho}\mathcal{R}^{\sigma}+2\hbar^{2}p^{\nu}\Bigl{(}T_{[\mu}\mathcal{R}_{\nu]}+S_{\alpha\mu\nu}\mathcal{R}^{\alpha}\Bigr{)}\,.$$

(16)

Combined with Eq. (9), this equation is decomposed into

	$\displaystyle p^{2}\mathcal{R}_{\mu}^{(0)}=0\,,$		(17)
	$\displaystyle\displaystyle p^{2}\mathcal{R}_{\mu}^{(1)}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}p^{\nu}D^{\rho}\mathcal{R}^{\sigma}_{(0)}\,,$		(18)
	$\displaystyle\displaystyle p^{2}\mathcal{R}_{\mu}^{(2)}=-p_{\mu}Q\cdot\mathcal{R}_{(0)}+\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}p^{\nu}D^{\rho}\mathcal{R}^{\sigma}_{(1)}+2p^{\nu}\Bigl{(}T_{[\mu}\mathcal{R}_{\nu]}^{(0)}+S_{\alpha\mu\nu}\mathcal{R}^{\alpha}_{(0)}\Bigr{)}\,.$		(19)

Also Eqs. (9) and (10) yield

	$\displaystyle p\cdot\mathcal{R}_{(0)}=0\,,$		(20)
	$\displaystyle p\cdot\mathcal{R}_{(1)}=0\,,$		(21)
	$\displaystyle p\cdot\mathcal{R}_{(2)}+Q\cdot\mathcal{R}_{(0)}=0\,,$		(22)
	$\displaystyle 4p_{[\mu}\mathcal{R}^{(0)}_{\nu]}=0\,,$		(23)
	$\displaystyle 4p_{[\mu}\mathcal{R}^{(1)}_{\nu]}+\varepsilon_{\mu\nu\rho\sigma}D^{\rho}\mathcal{R}^{\sigma}_{(0)}=0\,,$		(24)
	$\displaystyle 4p_{[\mu}\mathcal{R}_{\nu]}^{(2)}+\varepsilon_{\mu\nu\rho\sigma}D^{\rho}\mathcal{R}_{(1)}^{\sigma}+4\Bigl{(}T_{[\mu}\mathcal{R}_{\nu]}^{(0)}+S_{\alpha\mu\nu}\mathcal{R}_{(0)}^{\alpha}\Bigr{)}=0\,.$		(25)

In the following, we look for $\mathcal{R}^{\mu}_{(0)}$, $\mathcal{R}^{\mu}_{(1)}$ and $\mathcal{R}^{\mu}_{(2)}$ that satisfy Eqs. (17)-(25).

First, let us solve the zeroth and first-order parts. Equations (17) and (20) imply

$$\mathcal{R}^{\mu}_{(0)}=2\pi\delta(p^{2})p^{\mu}f_{(0)}\,,$$

(26)

where $f_{(0)}$ is a scalar function that satisfies $\delta(p^{2})p^{2}f_{(0)}=0$. From Eq. (23), we can check that there does not appear any other term in $\mathcal{R}_{\mu}^{(0)}$. Equation (8) in the zeroth order, $D_{\mu}\mathcal{R}^{\mu}_{(0)}=0$, gives the collisionless Boltzmann equation,

$$2\pi\delta(p^{2})(p^{\mu}\nabla_{\mu}+\Gamma_{\mu\nu}^{\rho}p^{\mu}p_{\rho}\partial_{p}^{\nu})f_{(0)}=0\,.$$

(27)

Higher-order terms give quantum corrections to the Boltzmann equation.

From Eq. (18) and the above $\mathcal{R}^{\mu}_{(0)}$, we find

$$p^{2}\mathcal{R}^{\mu}_{(1)}=0\,.$$

(28)

This does not necessarily mean that $\mathcal{R}^{\mu}_{(1)}$ itself vanishes for arbitrary $p_{\mu}$. Indeed if $\mathcal{R}^{\mu}_{(1)}$ involves $\delta(p^{2})$, it fulfils Eq. (28). Therefore, the first-order correction is generally written as

$$\mathcal{R}^{\mu}_{(1)}=2\pi\delta(p^{2})\widetilde{\mathcal{R}}^{\mu}_{(1)}\,.$$

(29)

Here the undetermined part $\widetilde{\mathcal{R}}_{\mu}^{(1)}$ satisfies $\delta(p^{2})p^{2}\widetilde{\mathcal{R}}^{\mu}_{(1)}=0$ so that Eq. (28) holds. Plugging this $\mathcal{R}^{\mu}_{(1)}$ and $\mathcal{R}^{\mu}_{(0)}$ into Eq. (24), we obtain

$$\delta(p^{2})\Bigl{[}\varepsilon_{\mu\nu\rho\sigma}p^{\rho}D^{\sigma}f_{(0)}-4p_{[\mu}\widetilde{\mathcal{R}}_{\nu]}^{(1)}\Bigr{]}=0\,.$$

(30)

We contract this with $n^{\nu}/(2\,p\cdot n)$, where $n^{\mu}(x)$ is a vector field independent of $p_{\mu}$. Then we get

$$\begin{split}\widetilde{\mathcal{R}}_{\mu}^{(1)}\delta(p^{2})=\delta(p^{2})\biggl{[}p_{\mu}\frac{n\cdot\widetilde{\mathcal{R}}_{(1)}}{p\cdot n}+\frac{\varepsilon_{\mu\nu\rho\sigma}p^{\rho}n^{\sigma}}{2p\cdot n}D^{\nu}f_{(0)}\biggr{]}\,.\end{split}$$

(31)

Thus the first-order correction is given by

$$\mathcal{R}^{\mu}_{(1)}=2\pi\delta(p^{2})\Bigl{[}p^{\mu}f_{(1)}+\Sigma_{n}^{\mu\nu}D_{\nu}f_{(0)}\Bigr{]}\,,$$

(32)

where we define

$$f_{(1)}=\frac{n\cdot\widetilde{\mathcal{R}}^{(1)}}{p\cdot n}\,,\quad\Sigma_{n}^{\mu\nu}=\frac{\varepsilon^{\mu\nu\rho\sigma}p_{\rho}n_{\sigma}}{2p\cdot n}\,.$$

(33)

In the above $\mathcal{R}^{\mu}_{{(1)}}$, an arbitrary vector $n^{\mu}$ emerges through $\Sigma^{\mu\nu}_{n}$. This ambiguity is related to the (local) Lorentz transformation Chen:2014cla , and thus $\Sigma^{\mu\nu}_{n}$ is regarded as the spin tensor defined in the frame $n^{\mu}$ Chen:2015gta . In particular, at $n^{\mu}=(1,{\boldsymbol{0}})$, we have $f_{(1)}=\widetilde{\mathcal{R}}^{(1)}_{0}/p_{0}$, i.e., the charge density divided by the particle energy. In this sense, $f_{(1)}$ can be regarded as the quantum correction to the distribution function. Note that the solution $\mathcal{R}^{\mu}_{(1)}$ fulfils Eqs. (18) and (21) as long as $\delta(p^{2})p^{2}f_{(1)}=0$ holds.

Now we solve the second-order correction. By plugging the above $\mathcal{R}_{(0)}^{\mu}$ and $\mathcal{R}_{(1)}^{\mu}$ into Eq. (19), we obtain

$$\begin{split}p^{2}\mathcal{R}_{\mu}^{(2)}&=2\pi\Bigl{(}-p_{\mu}Q\cdot p+p^{\nu}\mathcal{D}_{\mu\nu}\Bigr{)}\delta(p^{2})f_{(0)}\,,\end{split}$$

(34)

where the derivative operator $\mathcal{D}_{\mu\nu}$ is defined as

$$\mathcal{D}_{\mu\nu}=2\Bigl{(}T_{[\mu}p_{\nu]}+S_{\alpha\mu\nu}p^{\alpha}\Bigr{)}+\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}D^{\rho}\Sigma_{n}^{\sigma\lambda}D_{\lambda}\,.$$

(35)

The general form of the second-order correction then reads

$$\mathcal{R}_{\mu}^{(2)}=2\pi\delta(p^{2})\widetilde{\mathcal{R}}_{\mu}^{(2)}+\frac{2\pi}{p^{2}}\biggl{[}-p_{\mu}Q\cdot p+p^{\nu}\mathcal{D}_{\mu\nu}\biggr{]}\delta(p^{2})f_{(0)}\,.$$

(36)

Here we again introduced the undetermined part $\widetilde{\mathcal{R}}_{\mu}^{(2)}$, which satisfies $\delta(p^{2})p^{2}\widetilde{\mathcal{R}}^{\mu}_{(2)}=0$. Plugging $\mathcal{R}_{\mu}^{(0)}$, $\mathcal{R}_{\mu}^{(1)}$, and $\mathcal{R}_{\mu}^{(2)}$ into Eq. (25), we obtain

$$\begin{split}0&=4p_{[\mu}\mathcal{R}^{(2)}_{\nu]}+(2\pi)\varepsilon_{\mu\nu\rho\sigma}D^{\rho}\Bigl{(}p^{\sigma}f_{(1)}+\Sigma_{n}^{\sigma\lambda}D_{\lambda}f_{(0)}\Bigr{)}\delta(p^{2})+4(2\pi)\Bigl{(}T_{[\mu}p_{\nu]}+S_{\alpha\mu\nu}p^{\alpha}\Bigr{)}f_{(0)}\delta(p^{2})\\ &=2\pi\biggl{[}4p_{[\mu}\widetilde{\mathcal{R}}^{(2)}_{\nu]}+\varepsilon_{\mu\nu\rho\sigma}D^{\rho}p^{\sigma}f_{(1)}+\frac{4}{p^{2}}p_{[\mu}p^{\rho}\mathcal{D}_{\nu]\rho}f_{(0)}+2\mathcal{D}_{\mu\nu}f_{(0)}\Bigr{]}\delta(p^{2})\\ &=2\pi\biggl{[}4p_{[\mu}\widetilde{\mathcal{R}}^{(2)}_{\nu]}-\varepsilon_{\mu\nu\rho\sigma}p^{\rho}D^{\sigma}f_{(1)}-\frac{1}{p^{2}}\varepsilon_{\mu\nu\rho\sigma}p^{\rho}\varepsilon^{\alpha\beta\gamma\sigma}p_{\alpha}\mathcal{D}_{\beta\gamma}f_{(0)}\biggr{]}\delta(p^{2})\,.\end{split}$$

(37)

Similarly to Eq. (30), we solve the above equation by introducing a vector $u^{\mu}$ (this is in general different from $n^{\mu}$), as follows:

$$\begin{split}\widetilde{\mathcal{R}}_{\mu}^{(2)}\delta(p^{2})&=\delta(p^{2})\Bigl{[}p_{\mu}f_{(2)}+\Sigma_{\mu\nu}^{u}D^{\nu}f_{(1)}\Bigr{]}+\frac{1}{p^{2}}\varepsilon^{\alpha\beta\gamma\nu}\Sigma_{\mu\nu}^{u}p_{\alpha}\mathcal{D}_{\beta\gamma}\delta(p^{2})f_{(0)}\\ &=\delta(p^{2})\Bigl{[}p_{\mu}f_{(2)}+\Sigma_{\mu\nu}^{u}D^{\nu}f_{(1)}\Bigr{]}-\frac{\delta(p^{2})}{p^{2}}\Sigma_{\mu\nu}^{u}\biggl{[}\frac{1}{2}\tilde{R}^{\alpha\beta\nu\rho}p^{\rho}p^{\alpha}\partial_{p}^{\beta}+p\cdot D\Sigma_{n}^{\nu\rho}D_{\rho}\biggr{]}f_{(0)}\,,\end{split}$$

(38)

where we defined $\tilde{R}_{\alpha\beta\mu\nu}={R_{\alpha\beta}}^{\rho\sigma}\varepsilon_{\rho\sigma\mu\nu}/2$ and

$$f_{(2)}=\frac{u\cdot\widetilde{\mathcal{R}}_{(2)}}{p\cdot u}\,.$$

(39)

In the second line of Eq. (38), we utilized

$$\begin{split}[A_{\mu},p_{\nu}]=\frac{1}{24}R_{\mu\nu}\,,\qquad[B_{\mu},p_{\nu}]=-\frac{1}{24}R_{\mu\nu}+\frac{1}{24}\Bigl{(}{R^{\rho}}_{\nu\mu\sigma}+{R^{\rho}}_{\sigma\mu\nu}\Bigr{)}p_{\rho}\partial_{p}^{\sigma}\,,\end{split}$$

(40)

which yield

$$\begin{split}2\varepsilon^{\alpha\beta\gamma\nu}p_{\alpha}\Bigl{(}T_{\beta}p_{\gamma}+S_{\lambda\beta\gamma}p^{\lambda}\Bigr{)}&=-\frac{1}{2}\tilde{R}^{\alpha\beta\nu\rho}p_{\rho}p_{\alpha}\partial_{\beta}^{p}\,.\end{split}$$

(41)

Therefore, the second-order correction reads

$$\begin{split}\mathcal{R}_{\mu}^{(2)}&=2\pi\delta(p^{2})\biggl{[}p_{\mu}f_{(2)}+\Sigma_{\mu\nu}^{u}D^{\nu}f_{(1)}\biggr{]}+2\pi\frac{1}{p^{2}}\biggl{[}-p_{\mu}Q\cdot p+2p^{\nu}\Bigl{(}T_{[\mu}p_{\nu]}+S_{\alpha\mu\nu}p^{\alpha}\Bigr{)}\biggr{]}\delta(p^{2})f_{(0)}\\ &\quad+2\pi\frac{\delta(p^{2})}{p^{2}}\biggl{[}\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}p^{\nu}D^{\rho}\Sigma_{n}^{\sigma\lambda}D_{\lambda}-\Sigma_{\mu\nu}^{u}\biggl{(}\frac{1}{2}{\tilde{R}}^{\alpha\beta\nu\rho}p_{\rho}p_{\alpha}\partial_{\beta}^{p}+p\cdot D\Sigma_{n}^{\nu\rho}D_{\rho}\biggr{)}\biggr{]}f_{(0)}\,.\end{split}$$

(42)

We mention that Eq. (19) is still fulfilled for the above $\mathcal{R}_{\mu}^{(2)}$ as long as

$$\delta(p^{2})p^{2}f_{(2)}=0$$

(43)

holds. Indeed we can check

$$\begin{split}\delta(p^{2})p^{2}\widetilde{\mathcal{R}}^{(2)}_{\mu}&=-\delta(p^{2})\Sigma_{\mu\nu}^{u}\biggl{[}\frac{1}{2}\tilde{R}^{\alpha\beta\nu\rho}p_{\rho}p_{\alpha}\partial^{p}_{\beta}+p\cdot D\Sigma_{n}^{\nu\rho}D_{\rho}\biggr{]}f_{(0)}\\ &=-\delta(p^{2})\Sigma_{\mu\nu}^{u}\biggl{[}\frac{1}{2}\tilde{R}^{\alpha\beta\nu\rho}p_{\rho}p_{\alpha}\partial^{p}_{\beta}+D_{\lambda}\biggl{(}\Sigma_{n}^{\nu\lambda}p^{\rho}+\frac{1}{2}\varepsilon^{\nu\lambda\rho\sigma}p_{\sigma}-\frac{1}{2}\varepsilon^{\nu\lambda\rho\sigma}\frac{p^{2}n_{\sigma}}{p\cdot n}\biggr{)}D_{\rho}\biggr{]}f_{(0)}\\ &=-\delta(p^{2})\Sigma_{\mu\nu}^{u}D_{\lambda}\Sigma_{n}^{\nu\lambda}p\cdot Df_{(0)}=0\,.\end{split}$$

(44)

In the second line, we utilized

$$\Sigma_{\alpha[\mu}^{n}p_{\nu]}=-\frac{1}{2}\Sigma_{\mu\nu}^{n}p_{\alpha}-\frac{1}{4}\varepsilon_{\mu\nu\alpha\beta}\biggl{(}p^{\beta}-\frac{p^{2}n^{\beta}}{p\cdot n}\biggr{)}\,,$$

(45)

which follows from the Schouten identity: $p_{\mu}\varepsilon_{\nu\rho\sigma\lambda}+p_{\nu}\varepsilon_{\rho\sigma\lambda\mu}+p_{\rho}\varepsilon_{\sigma\lambda\mu\nu}+p_{\sigma}\varepsilon_{\lambda\mu\nu\rho}+p_{\lambda}\varepsilon_{\mu\nu\rho\sigma}=0$. Also the last line follows from $[D_{\mu},D_{\nu}]f=-R_{\alpha\beta\mu\nu}p^{\alpha}\partial^{\beta}_{p}f$, and the classical kinetic equation (8), i.e., $\delta(p^{2})\,p\cdot Df_{(0)}=0$. We stress that Eq. (43) is a crucial constraint to $f_{(2)}$, especially when we determine the equilibrium distribution function (see Sec. 3).

Eventually, the Wigner function up to $O(\hbar^{2})$ is derived as

$$\begin{split}\mathcal{R}_{\mu}&=2\pi\delta(p^{2})\Bigl{[}p_{\mu}\Bigl{(}f_{(0)}+\hbar f_{(1)}+\hbar^{2}f_{(2)}\Bigr{)}+\hbar\Sigma_{\mu\nu}^{n}D^{\nu}f_{(0)}+\hbar^{2}\Sigma_{\mu\nu}^{u}D^{\nu}f_{(1)}\Bigr{]}\\ &\quad+2\pi\hbar^{2}\frac{1}{p^{2}}\biggl{[}-p_{\mu}Q\cdot p+2p^{\nu}\Bigl{(}T_{[\mu}p_{\nu]}+S_{\alpha\mu\nu}p^{\alpha}\Bigr{)}\biggr{]}\delta(p^{2})f_{(0)}\\ &\quad+2\pi\hbar^{2}\frac{\delta(p^{2})}{p^{2}}\biggl{[}\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}p^{\nu}D^{\rho}\Sigma_{n}^{\sigma\lambda}D_{\lambda}-\Sigma_{\mu\nu}^{u}\biggl{(}\frac{1}{2}{\tilde{R}}^{\alpha\beta\nu\rho}p_{\rho}p_{\alpha}\partial_{\beta}^{p}+p\cdot D\Sigma_{n}^{\nu\rho}D_{\rho}\biggr{)}\biggr{]}f_{(0)}\,.\end{split}$$

(46)

As the above $\mathcal{R}^{\mu}$ is a fermion propagator, performing the momentum integration involving it we evaluate a corresponding quantity of Weyl fermions under a gravitational field. In particular, the charge current and the symmetric energy-momentum tensor are given by

$$\begin{split}J^{\mu}(x)&=\int_{p}\text{tr}\biggl{[}\gamma^{\mu}\frac{1+\gamma^{5}}{2}W(x,p)\biggr{]}=2\int_{p}\mathcal{R}^{\mu}(x,p)\,,\\ \quad T^{\mu\nu}(x)&=\int_{p}\text{tr}\biggl{[}\frac{i\hbar}{2}\gamma^{(\mu}\overleftrightarrow{D}^{\nu)}\frac{1+\gamma^{5}}{2}W(x,p)\biggr{]}=2\int_{p}p^{(\mu}\mathcal{R}^{\nu)}(x,p)+O(\hbar^{3})\,,\end{split}$$

(47)

where we define $\int_{p}:=\int\frac{d^{4}p}{(2\pi)^{4}\sqrt{-g(x)}}$, $X^{(\mu}Y^{\nu)}=(X^{\mu}Y^{\nu}+X^{\nu}Y^{\mu})/2$ and $\overleftrightarrow{D_{\mu}}(\bar{\psi}_{b}\psi_{a})=\bar{\psi}_{b}D_{\mu}\psi_{a}-\bar{\psi}_{b}\overleftarrow{D}_{\mu}\psi_{a}$. Here we used the expansion of the derivative; $D_{\mu}\psi(x,y)=\partial_{\mu}^{y}\psi(x,y)+O(\hbar^{2})$ (see Appendix C in Ref. Liu:2018xip ). In Sec. 4, we derive the equilibrium Wigner function and show that it yields the gravity-induced parts of $J^{\mu}$ and $T^{\mu\nu}$. We also demonstrate that the different approaches in Appendix D lead to the same $J^{\mu}$ [see Eq. (78) and Eqs. (169) and (186)].

In the evaluation of physical quantities such as Eq. (47), it is necessary to identify the explicit form of $f_{{(0)},{(1)},{(2)}}$. For this purpose, the frame (i.e., $n^{\mu}$ and $u^{\mu}$) dependences of $\mathcal{R}^{\mu}$ are a crucial concept; as shown below, we derive a constraint on the distribution function from the proper transformation law under the shift of the frames. Combined this with the kinetic equation, we determine $f_{{(0)},{(1)},{(2)}}$ at equilibrium, which are utilized to evaluate equilibrium transport phenomena induced by external gravity in Sec. 4. This section is devoted to the analysis of the frame dependence and the equilibrium solution found from it.

In the above derivation of $\mathcal{R}^{\mu}$, the frame vectors $n^{\mu}$ and $u^{\mu}$ are algebraically introduced. It is valid to expect that the frame-dependence disappears in $\mathcal{R}^{\mu}$, which generates physical quantities. Indeed, as is well-known in the $O(\hbar)$ CKT, the choice of the frame vector $n^{\mu}$ corresponds to the Lorentz transformation, and the frame-dependence is totally compensated in physical quantities, due to the shift of $f_{(1)}$. Hence we may plausibly require that the same is true in the $O(\hbar^{2})$ CKT. That is, we determine the transformation law of $f_{(2)}$ under $n^{\mu}\to n^{\prime\mu}$ and $u^{\mu}\to u^{\prime\mu}$ so that the frame dependence vanishes in $\mathcal{R}^{\mu}$.

Let us first take the Lorentz transformation in terms of $n^{\mu}$, namely, $(x^{\mu},p^{\mu})\to(x^{\prime\mu},p^{\prime\mu})={({\Lambda}_{n})^{\mu}}_{\nu}(x^{\nu},p^{\nu})$ and $u^{\mu}\to u^{\prime\mu}={({\Lambda}_{n})^{\mu}}_{\nu}u^{\nu}$, where ${({\Lambda}_{n})^{\mu}}_{\nu}$ is the matrix representation of the local Lorentz transformation. This transformation is equivalent to the one of the frame vector $n^{\mu}$ as

$$n^{\mu}\to n^{\prime\mu}={{({\Lambda}_{n}^{-1})}^{\mu}}_{\nu}n^{\nu}\,.$$

(48)

We also parametrize the transformation of $f$ as

$$f(x,p)\to f^{\prime}(x^{\prime},p^{\prime})=f(x,p)+\hbar\delta_{n}f_{(1)}(x,p)+\hbar^{2}\delta_{n}f_{(2)}(x,p)\,.$$

(49)

Due to the Lorentz covariance of $\mathcal{R}^{\mu}$, we have

$$\begin{split}0&={(\Lambda^{-1}_{n})_{\mu}}^{\nu}\mathcal{R}^{\prime}_{\nu}(x^{\prime},p^{\prime})-\mathcal{R}_{\mu}(x,p)\\ &=2\pi\delta(p^{2})\biggl{[}p_{\mu}\Bigl{(}\hbar\delta_{n}f_{(1)}+\hbar^{2}\delta_{n}f_{(2)}\Bigr{)}+\hbar\Bigl{(}\Sigma^{n^{\prime}}_{\mu\nu}-\Sigma^{n}_{\mu\nu}\Bigr{)}D^{\nu}f_{(0)}+\hbar^{2}\Sigma^{u}_{\mu\nu}D^{\nu}\delta_{n}f_{(1)}\\ &\qquad\qquad\qquad+\frac{\hbar^{2}}{p^{2}}\biggl{(}\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}p^{\nu}D^{\rho}\Bigl{(}\Sigma^{\sigma\lambda}_{n^{\prime}}-\Sigma^{\sigma\lambda}_{n}\Bigr{)}D_{\lambda}f_{(0)}-\Sigma_{\mu\nu}^{u}p\cdot D\Bigl{(}\Sigma_{n^{\prime}}^{\nu\rho}-\Sigma_{n}^{\nu\rho}\Bigr{)}D_{\rho}f_{(0)}\biggr{)}\biggr{]}\,.\end{split}$$

(50)

Contracting Eq. (50) with $n^{\mu}$ and picking up only the $O(\hbar)$ terms, we find

$$\begin{split}\delta_{n}f_{(1)}&=-\frac{n^{\mu}}{p\cdot n}\Sigma_{\mu\nu}^{n^{\prime}}D^{\nu}f_{(0)}\,.\end{split}$$

(51)

Similarly, contracting Eq. (50) with $u^{\mu}$, we obtain

$$\begin{split}\delta_{n}f_{(2)}&=\frac{1}{p^{2}}\Sigma_{\mu\nu}^{u}D^{\mu}\Bigl{(}\Sigma^{\nu\rho}_{n^{\prime}}-\Sigma^{\nu\rho}_{n}\Bigr{)}D_{\rho}f_{(0)}\,.\end{split}$$

(52)

The above $\delta_{n}f_{{(1)},{(2)}}$ fulfills $\delta(p^{2})p^{2}\delta_{n}f_{{(1)},{(2)}}=0$. Also, we can show that they satisfy Eq. (50).