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

We derive the chiral kinetic theory under the presence of a gravitational Riemann curvature. It is well-known that in the chiral kinetic theory there inevitably appears a redundant ambiguous vector corresponding to the choice of the Lorentz frame. We reveal that on top of this conventional frame choosing vector, higher-order quantum correction to the chiral kinetic theory brings an additional degrees of freedom to specify the distribution function. Based on this framework, we derive new types of fermionic transport, that is, the charge current and energy-momentum tensor induced by the gravitational Riemann curvature. Such novel phenomena arise not only under genuine gravity but also in a (pseudo-)relativistic fluid, for which inhomogeneous vorticity or temperature are effectively represented by spacetime metric tensor. It is especially found that the charge and energy currents are antiparallelly induced by an inhomogeneous fluid vorticity (more generally, by the Ricci tensor R0i), as a consequence of the spin-curvature coupling. We also briefly discuss possible applications to Weyl/Dirac semimetals and heavy-ion collision experiments.


Introduction and summary
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) [1] can be regarded as the gravitational counterpart of the chiral magnetic effect (CME) [2][3][4], as is clear from the gravitoelectromagnetism [5]. The CVE is not only a theoretically interesting phenomenon in the sense that it is originated from quantum anomaly [5,6], but also an important experimental probe to study the rotation of quark-gluon plasmas JHEP05(2021)023 created in relativistic heavy-ion collision experiments [7]. Second, the mechanical strain plays a role of an effective U(1) or axial U(1) magnetic field, and accordingly yields a charge current [8][9][10]. Third, spacetime torsion is recently under active investigation, as it can bring novel currents, which are referred to as the chiral torsional effect [11][12][13][14][15][16].
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 [17][18][19]. 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 curvatureinduced 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) [20,21], which nicely reproduces the chiral anomaly, plays a pivotal role in the development of various studies of the chiral transport phenomena [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] 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 [37][38][39]. 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 [34]. 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( ) contribution to enter the distribution is anticipated to be the spin-vorticity coupling, if we recall the conservation of the total angular momentum [26]. On the other hand, this intuition is inapplicable to the O( 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 [27] even in the general coordinate system [34]. In this paper, thus we study the semiclassical transport theory with gravitational Riemann curvatures, based on the CKT derived with the Wigner function.

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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( 2 ). This is a contrast to the conventional works, where only the O( ) quantum correction is taken into account. It is intriguing that the CKT in curved spacetime is systematically solvable even with the O( 2 ) contributions, while the O( 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 [25]. As a result, a frame vector representing such an ambiguity is inevitably introduced [26]. From the Wigner function up to O( 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( 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 energymomentum tensor are given by the momentum integrals, as follows: and is the Wigner function of the right-handed Weyl fermions. Under a static weak gravity, the O( ) part of eq. (1.1) 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 [40], our calculation is its first verification based on the CKT.
The O( 2 ) contribution of eq. (1.1) 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: (1.2) Here C 0 = µ/(2π 2 ) and C 1 = µ 2 /(4π 2 ) + T 2 /12 are the metric-independent coefficients determined by temperature T and chemical potential for right-handed fermions µ. Also j µ and t µν are functions of Riemann curvature R µνρσ , Ricci tensor R µν and Ricci scalar R. In table 1, we summarize the analytic forms of j µ and t µν . The 'static' ('dynamical') implies that a time-independent (time-dependent) metric tensor is used. We denote ξ µ = δ µ 0 and η µν being the Minkowski metric tensor. It is demonstrated in appendix D that the same charge JHEP05(2021)023 charge current C 0 j µ energy-momentum tensor C 1 t µν 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: where h µν is the fluctuation around the flat spacetime. With this metric, temperature gradient and vorticity are given by 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 1 and 2 to the quantum anomaly, from different JHEP05(2021)023 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 1 and 2 for µ > 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 [41]. Also, the present work could be applicable to the physics of core-collapse supernova explosions and neutron star formations [42]. In this direction, we need to take the collisional effects into account [27,28,43], based on the Kadanoff-Baym equation in curved spacetime, which respects the diffeomorphism covariance [44].
This paper is organized as follows. In section 2, solving the constraint equations, we obtain the general solution of the Wigner function up to O( 2 ) under general curved spacetime. In section 3, we determine an equilibrium distribution function involving O( 2 ) corrections, based on the frame-independence of the Wigner function. In section 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 sections 5 and 6, we analyze the dynamical response from background gravitational fields. As a practical application, in section 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. [34].

Chiral kinetic theory at O( )
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 [45].
is the general relativistic extension of the one in gauge theory [45]. Here D µ is called the horizontal lift; for a function on (x µ , y µ ) and (x µ , p µ ), the horizontal lift is represented as where ∇ µ is the covariant derivative in terms of diffeomorphism and the local Lorentz transformation. The most beneficial property of D µ is that it commutes with both y µ and p µ , while ∇ µ 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 γ µ ∇ µ ψ(x) = 0, which brings the dynamical equation that the Wigner function W (x, p) obeys. After a long computation, the set of equations for R µ (x, p) is up to O( 2 ) given by [34] (D µ + 2 P µ )R µ = 0 , (2.4) where we introduce the following notations:

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This does not necessarily mean that R µ (1) itself vanishes for arbitrary p µ . Indeed if R µ (1) involves δ(p 2 ), it fulfils eq. (2.24). Therefore, the first-order correction is generally written as Here the undetermined part R We contract this with n ν /(2 p · n), where n µ (x) is a vector field independent of p µ . Then we get Thus the first-order correction is given by where we define (2.29) In the above R µ (1) , an arbitrary vector n µ emerges through Σ µν n . This ambiguity is related to the (local) Lorentz transformation [25], and thus Σ µν n is regarded as the spin tensor defined in the frame n µ [26]. In particular, at , 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 R µ (1) fulfils eqs. (2.14) and (2.17) as long as δ(p 2 )p 2 f (1) = 0 holds. Now we solve the second-order correction. By plugging the above R µ (0) and R µ (1) into eq. (2.15), we obtain where the derivative operator D µν is defined as The general form of the second-order correction then reads Here we again introduced the undetermined part R

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Similarly to eq. (2.26), we solve the above equation by introducing a vector u µ (this is in general different from n µ ), as follows: where we definedR αβµν = R αβ ρσ ε ρσµν /2 and In the second line of eq. (2.34), we utilized Therefore, the second-order correction reads We mention that eq. (2.15) is still fulfilled for the above R µ as long as holds. Indeed we can check In the second line, we utilized which follows from the Schouten identity: p µ ε νρσλ +p ν ε ρσλµ +p ρ ε σλµν +p σ ε λµνρ +p λ ε µνρσ = 0. Also the last line follows from [D µ , D ν ]f = −R αβµν p α ∂ β p f , and the classical kinetic equation (2.4), i.e., δ(p 2 ) p·Df (0) = 0. We stress that eq. (2.39) is a crucial constraint to f (2) , especially when we determine the equilibrium distribution function (see section 3).

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Eventually, the Wigner function up to O( 2 ) is derived as (2.42) As the above R µ 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 Here we used the expansion of the derivative; [34]). In section 4, we derive the equilibrium Wigner function and show that it yields the gravity-induced parts of J µ and T µν . We also demonstrate that the different approaches in appendix D lead to the same J µ [see eq.

Frame dependence and equilibrium
In the evaluation of physical quantities such as eq. (2.43), it is necessary to identify the explicit form of f (0),(1), (2) . For this purpose, the frame (i.e., n µ and u µ ) dependences of R µ 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 section 4. This section is devoted to the analysis of the frame dependence and the equilibrium solution found from it.
In the above derivation of R µ , the frame vectors n µ and u µ are algebraically introduced. It is valid to expect that the frame-dependence disappears in R µ , which generates physical quantities. Indeed, as is well-known in the O( ) CKT, the choice of the frame vector n µ 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( 2 ) CKT. That is, we determine the transformation law of f (2) under n µ → n µ and u µ → u µ so that the frame dependence vanishes in R µ .
Let us first take the Lorentz transformation in terms of n µ , namely, We also parametrize the transformation of f as Due to the Lorentz covariance of R µ , we have Contracting eq. (3.3) with n µ and picking up only the O( ) terms, we find Similarly, contracting eq. (3.3) with u µ , we obtain The above δ n f (1),(2) fulfills δ(p 2 )p 2 δ n f (1),(2) = 0. Also, we can show that they satisfy eq. (3.3). Let us also perform the Lorentz transformation with for which the Lorentz covariance of R µ requires From the O( ) part, we readily find By contracting eq. (3.7) with u µ , we find We can check that the above δ u f (2) fulfills δ(p 2 )p 2 δ u f (2) = 0 and eq. (3.7).

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In the Wigner function (2.42), the frame vectors n µ and u µ are in general chosen independently. As long as f (1) and f (2) obey the transformation laws (3.4), (3.5), (3.8) and (3.9), however, we can always set u µ = n µ by redefining f (2) . Then, eq. (2.42) is simplified as where we define The transformation laws under the change of the frames n µ and u µ are helpful to identify the equilibrium distribution function. Let us first start from the classical distribution f (0) , which is defined as a function of the collisional conserved quantities: For this f (0) , the transformation law (3.4) yields where we use eq. (2.41) and define f (0) = df (0) (g (0) )/dg (0) . Although the above relation identifies the frame-dependent part involved in f (1) at equilibrium, the frame-independent part is still undetermined. If we set such an ambiguous part in f (1) to be zero, however, we identify This is a plausible form in the sense that the spin-vorticity coupling term is correctly reproduced: . In this case, the first order Wigner function (2.28) is written as

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This R µ (1)eq fulfills the kinetic equation at O( ) [34]. Subsequently, with the above f (0) and f (1) , the transformation laws (3.5) and (3.9) lead to where we employ eq. (2.41). Therefore we find Here, unlike the first-order correction f (1) , we explicitly keep the frame-independent ambiguity φ (2) . Importantly, this ambiguous part should be taken into account for the realization of equilibrium, as we elaborate later. As shown in appendix A, inserting the above distribution functions f (0),(1),(2) into eq. (2.38), we reduce the second-order correction part to (3.19) The frame-dependence here vanishes totally, as it should. Plugging this into the chiral kinetic equation (2.4), and after a straightforward calculation in appendix B, we finally arrive at

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which is the equation to determine φ (2) . However, there is in general no solution, as is obvious from the constraint (2.39); φ (2) cannot have δ(p 2 )/p 2 terms. In other words, the collisionless CKT has no global equilibrium solution in general background curved geometry. We note that an equilibrium distribution function with φ (2) = 0 is realized in the flat spacetime limit g µν = η µν ; the Killing equation ∂ µ β ν + ∂ ν β µ = 0 leads to

Stationary weak gravity
Although the general curved spacetime does not realize an equilibrium, there may exist a special geometry having a solution for eq. (3.20). One of the simplest cases is the stationary and weak background gravitational field, where the metric tensor is given by In this case, the time-like Killing vector β µ ξ µ := δ µ 0 is admitted. Then, the kinetic equation (3.20) is drastically reduced as Therefore, for the metric tensor (4.1), we identify Hereafter, we call f = f (0) + f (1) + 2 f (2) with eqs. (3.12), (3.15), (3.18) and (4.3) an equilibrium distribution function. In this section, we focus on the geometry described by eq. (4.1). Let us evaluate the charge current and the symmetric energy-momentum tensor for the equilibrium distribution function. We employ the classical equilibrium state described by where β µ is a time-like Killing vector β µ =βξ µ withβ = β · β/g 00 . Alsoβ andμ are the global inverse temperature and chemical potential. In the flat spacetime limit, the classical charge density becomes From eq. (2.43), the equilibrium Wigner function R µ (1)eq yields the CVE [26]:

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where the vorticity vector is introduced as ω µ = ε µνρσ ξ ν ∇ ρ ξ σ /2. Here the coefficients are defined as and thus we find C 1 = µ 2 /(4π 2 ) + T 2 /12 and C 2 = µ 3 /6(π 2 ) + µT 2 /6 (see appendix C). The above charge current is conserved, namely, ∇ µ J µ (1)eq = 0. This reflects the absence of the gravitational contribution to the U(1) anomaly in the CKT up to O( ). Besides, the energy-momentum conservation holds, as we check ∇ µ T µν (1)eq = 0. From eq. (3.19) and (4.3), the second order equilibrium Wigner function reads where R = g µν R µν is the Ricci scalar. Here we dropped the terms including ∇ ρ β ν because they are of order O(h 2 µν ). In the momentum integral, the 1/p 2 terms can be rewritten as which follows from δ (x) = −δ(x)/x. The integral with 1/(p 2 ) 2 is also computed in a similar manner with δ (x) = 2δ(x)/x 2 . With the help of several formulas in appendix C, we eventually derive with C 0 = µ/(2π 2 ). Therefore, based on the CKT, we reveal that the nonvanishing fermion transport is induced by the background gravitational field even in equilibrium. This implies that these phenomena are nondissipative, as so are the CME, and CVE. This is one of the main findings of this paper. We can also derive the same current J µ (2)eq from the diagrammatic computation (see appendix D.1) and with the Riemann normal coordinate expansion (see appendix D.2). It is worthwhile to mention that g 0i enters in eqs. (4.6) and (4.10) only through the field strength f ij = ∂ i g 0j − ∂ j g 0i . This is a consequence of the Kaluza-Klein gauge symmetry [48].
For the left-handed Weyl fermion, J µ (2)eq and T µν (2)eq are written as the same form, while the sign of J µ (1)eq and T µν (1)eq flipped; the former does not involve ε µνρσ while the latter does.

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As a result, the vector and axial parts are written as with µ V and µ A being the vector and chiral chemical potential, respectively. In section 7, we argue the novelty and some implications of eq. (4.10) and (4.11).

Dynamical weak gravity
While so far we have focused on the equilibrium state, this section is dedicated to discuss the dynamical response from the time-dependent gravity. Specifically, we consider a planewave weak background gravitational field: where k µ = (k 0 , k) is the momentum of the gravitational field. Let us look for the perturbative distribution function represented as the following form: Expanding the above equation in terms of h µν and utilizing ∂ µ f flat = 0, we obtain where we denote β µ = βξ µ = βδ µ 0 . Note that after the weak gravitational field expansion, all indices are raised and lowered by η µν and the inner products are defined as A · B = η µν A µ B ν and A 2 = η µν A µ A ν . Especially, to get the above equation, we have taken the following replacement: with the linearized Christoffel symbol and Riemann tensor being Plugging the above distribution functions into eqs. (2.22) and (2.28), we get the Wigner function. It is here informative to decompose the Wigner function into the terms that involve the k-dependent pole in the denominator, and the others. As we show in section 6, the momentum integrals of the former vanishes in the static limit k 0 /|k| → 0, while those of the latter survives. In this sense, we denote such a (non)static part as R µ (non)st . We note that the static part R µ st reproduces the equilibrium Wigner function R µ eq in the previous section, as we show later.

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Applying the totally same manner to the O( 2 ) parts, we obtainf 2 and eventually the Wigner function as R µ for which the precise derivation is shown in appendix F. Again R µ (2)st is the same as R µ (2)eq in eq. (3.19) up to O(h µν ).

Dynamical response
In the following discussion, we evaluate the charge current J µ and the energy-momentum tensor T µν in eq. (2.43) with eqs. (5.15), and (5.16). As the Wigner function R µ is decomposed into the static (k-independent) and nonstatic (k-dependent) part, so are J µ and T µν , that is, J µ = J µ st + J µ nonst and T µν = T µν st + T µν nonst . The static part is calculated in the same manner as before. For instance, using the integral formulas in appendix C, the momentum integrals of the classical contribution (5.10) yield with C 3 = µ 4 /24π 2 + µ 2 T 2 /12. Here we use (−g) −1/2 1 − h µ µ /2, and perform the integration by part.
For the nonstatic part, it is helpful to additionally prepare the following tensor (scalar for n = 0) function: where we define x := k 0 /|k| and the integral is over the solid angle of p. The evaluations of I j 1 ···jn (x)'s are summarized in appendix G. Here x in the denominators is understood to involve the positive infinitesimal imaginary part +iη. The nonstatic part of the classical charge current is from eq. (5.11) evaluated as

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To obtain the above second line, we utilized I j 1 ···jn (−x) = (−1) n I j 1 ···jn (x) and 2k · β dp 0 2π 2πδ(p 2 ) with n F (y) = (e βy + 1) −1 and n (m) F (y) := d m n F (y)/dy m . Besides, the nonstatic part of the classical energy-momentum tensor is computed as It is worthwhile to notice several properties of the above I j 1 ···jn (x). First we find the following relations: From these, we can show the charge current and energy-momentum conservation for arbitrary k µ : Second, we check that I j 1 ···jn (x) fulfills another type of relations:

11)
I k + I j jk = 0 , (6.12) These bring the dilatation current conservation for arbitrary k µ :

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As a particular case, we consider the dynamical limit x = k 0 /|k| 1. We expand J µ (0)nonst and T µν (0)nonst in terms of 1/x, with the asymptotic forms of I j 1 ···jn (x)'s, which are derived in eqs. (G.9)-(G.13). For later convenience, we here define the total charge current and energy-momentum tensor in the dynamical limit, as follows: Their classical contributions hence become Let us also calculate quantum corrections. At O( ), the Wigner function (5.14) leads to where the vorticity is linearized as In particular, taking the dynamical limit, we find These results, combined with the static parts (4.6), yield the O( ) contributions of eq. (6.15), as follows: Therefore we conclude that the CVE vanishes in the dynamical limit. This is consistent with the diagrammatic calculation in ref. [40].

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At O( 2 ), from the Wigner function (5.16), the nonstatic parts are evaluated as In the dynamical limit x → ∞, we obtain Here we used 23) which follow from the second Bianchi identity. Combining these with the static contribution (4.10), we write the O( 2 ) contributions of eq. (6.15) as We note that eqs. (6.6)-(6.9) and eqs. (6.11)-(6.13) again result in the conservation laws In the next section, we discuss some implications of (6.24) in the fluid frame [see eq. (7.5)].

Fluid frame
The fermionic system under a background fluid is pedagogical and informative to show the novelty of the gravity-induced transport phenomena given by eqs. (4.10) and (6.24). In general, the effect of the fluid can be translated to that of an effective curved geometry with the following metric [48]: h 00 (t, x) , g 0i = h 0i (t, x) , g ij = η ij . (7.1)

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Adopting this metric, we represent the temperature gradient [49] and the fluid vorticity as withT being the global temperature. Alternatively, the present coordinate describes the system under the gravitoelectromagnetic fields E i = − 1 2 ∂ i g 00 and B i = − 1 2 ε ijk ∂ j g k0 . The nonvanishing components of the curvature tensors read . Inserting eq. (7.3) into eqs. (4.10) and (6.24), we readily obtain the transport phenomena under the temperature gradient and inhomogeneous vorticity. In the static limit (or equivalently, for the stationary metric ∂ 0 h µν = 0), eq. (4.10) is written explicitly as with C 0 = µ/2π 2 and C 1 = µ 2 /4π 2 +T 2 /12. Similarly, from the expression in the dynamical limit (6.24), we find where we introduce the shear tensor: The corresponding vector and axial-vector currents are obtained by replacing C 0 with C 0,V /A = µ V /A /π 2 , and C 1 with C 1,V = (µ 2 V + µ 2 A )/2π 2 + T 2 /6 and C 1,A = µ V µ A /π 2 respectively, as we have done to get eq. (4.11). Several comments are in order.
We again emphasize that eq. (7.4) represents equilibrium transport phenomena, like the CME and CVE. It is also intriguing to notice the difference between these and Fourier's law. For the former, the currents in eq. (7.4) come from the vorticity, namely, the magnetic part B of the gravity. This background source supplies no energy to particles, and thus the currents become finite even in equilibrium. On the other hand, the latter is the current generation by the temperature gradient. This electric part E of the gravitational field gives an energy to particles. Therefore the Fourier's law is dissipative and prohibited in equilibrium.
For an static and spatially inhomogeneous vorticity ω(x), there emerges the nonvanishing charge current J i (2)eq and the energy current T 0i (2)eq , on top of the contributions from the CVE (4.6). Unlike the vector part of the CVE, the curvature-induced currents (4.11) JHEP05(2021)023 or (7.4) does not require µ A = 0. In the system without the chiral imbalance, hence, J i (2)eq and T 0i (2)eq are the leading vortical contributions. In the dynamical limit, such second-order contributions become more important, since the CVE is washed out as shown in eq. (6.20) while the currents in eq. (7.5) are not.
Under the correspondence between magnetic field and vorticity, one would think that the charge current J i (2)eq/dyn is the gravitational analogue of Ampère's law: ∇ × B = ∇ × ω = J . The situation is however not so trivial since J i (2)eq/dyn is opposite-signed against the energy current T 0i (2)eq/dyn (for µ > 0). Namely, eqs. (7.4) and (7.5) cannot be explained based on the naive picture that a particle's momentum carries both charge and energy. This curious flow dynamics essentially comes from the quantum effects through the spin-curvature coupling. We should emphasize that such an antiparallel charge-energy flow is not restricted in the present coordinate, but more generally admitted in a lot of curved spacetime; this phenomenon always takes place as long as R 0 i = 0, as shown in eqs. (4.10) and (6.24). It is worthwhile to mention the feedback to the gravitational field from eq. (4.10). In our sign convention, the Einstein field equation is given by R µν − 1 2 g µν R = −8πGT µν with the gravitational constant G [50]. Following this, the induced Ricci tensor reads R ind 0i ∼ αR 0i with a positive coefficient α > 0. Hence, the initial gravitational field is enhanced, which evokes the possibility of instability. We will revisit and analyze more precisely the above brief argument in the future, including the existence of a novel collective dynamics [51] in a gravitational plasma [52,53].
One might think that eq. (7.4) is unrelated to an anomaly. Indeed, eq. (7.4) would be irrelevant to the chiral anomaly, according to the analysis of discrete symmetry [54]. Nevertheless, this fact is not sufficient to conclude the irrelevance to anomaly at all, as for the temperature dependent part of the CVE [55][56][57][58]. We also mention that the transport coefficients C 0 and C 1 are time-reversal even quantities, which could be associated with their nondissipative nature similarly to those of the CME and CVE [54]. It should be required to clarify the anomalous aspect of eq. (7.4) from different approaches, such as hydrodynamics. In the sense that they are of the higher-order of the derivative counting, usual hydrodynamics can neglect eqs. (7.4) and (7.5). This would not be the case, however, if these phenomena originate from quantum anomaly like the CME and CVE.
These novel contributions (7.4) lead to several implications in relativistic many-body systems where an inhomogeneous fluid vorticity is experimentally generated. In rotating quark-gluon plasma, there emerges the quadrupole configuration of the vorticity along the beam direction [59][60][61][62]. Thus, on the transverse plane to the beam direction, the inhomogeneous vorticities generate the charge current J ⊥ and the energy current J ⊥ , as depicted in figure 1. As a brief argument, we may estimate the scale of the vorticity gradient to be the inverse of the hot matter size. Indeed, at the collision energy √ s = 19.6 GeV, the gradient of the vorticity is estimated to be (∇ × ω)/ω ≈ 0.2 fm-0.5 fm ≈ 40 MeV-100 MeV [61]. Although the whole magnitudes of J ⊥ and J ⊥ are dependent on the scale of ω, hence, these are nonnegligible compared with the CVE.
On top of the charge and energy currents, the stress tensor T ij (2)eq is also induced. Let us consider a cylindrical system along the z direction with a spatially inhomogeneous temperature T (z). From the vector part of the energy-momentum tensor in eq. (7.4), the tempera-JHEP05(2021)023 Such a pressure correction is detectable in Weyl/Dirac semimetal experiments, similarly to the usual thermoelectric transport phenomena [63,64].
In table-top experiments, an inhomogeneous and dynamical vorticity can be generated by an acoustic surface wave. We consider a transverse wave propagating on the xy surface [65][66][67] of Weyl/Dirac semimetals. Also we prepare the wave propagating along the x direction, and its amplitude is normal to the surface, i.e., its displacement vector is given by u = (0, 0, u) with u =ūe −ik 0 t+ikx−κz . Here e −κz reflects unpenetrating into the material. Now the response to this surface wave can be evaluated in the coordinate space described by

C Integration formulas
Here, we present several Integration formulas. We first define

D.1 Diagrammatic computation
First, we derive the current of a chiral fluid under a gravitational field, based on the linear response theory. We consider a Weyl fermion, and the corresponding action is given by where we introduce σ a = (1, σ i ) with the Pauli matrices σ i (i = 1, 2, 3). Here e a µ (e µ a ) denotes (inverse) vierbein satisfying g µν = e a µ e b ν η ab , η ab = e a µ e b ν g µν with the spacetime curved metric g µν and Minkowski metric η ab = diag(1, −1, −1, −1), and e := det e a µ . The left and right covariant derivatives are defined as withσ a := (1, −σ i ), which satisfiesσ a σ b +σ b σ a = σ aσb + σ bσa = 2η ab . Furthermore, employing the torsionless condition, we can express the spin connection ω ab µ = −ω ba µ as (D. 3) The energy-momentum tensor T µν and U(1) covariant charge current J µ are defined as Note that T µν is not symmetric, so we introduce the symmetric energy-momentum tensor defined as T µν S := (T µν + T νµ )/2. In the following, we consider fluctuation around the flat metric g µν = η µν + h µν .
In the linear response theory, the current in momentum space can be expressed as where we define k µ = (0, k) and T τ denotes the imaginary time ordering. The two point correlator is computed with the Feynman rule in flat spacetime:

JHEP05(2021)023
In momentum space, at one-loop order, we get where we denote p = d 3 p (2π) 3 , p = p + k and p µ = (iπT (2n + 1) + µ, p), and the antisymmetric tensor ε µνρσ is normalized as ε 0123 = +1. Also we introduced In order to compute the liner response to the gravitational field, we expand I αβγ (k) in terms of k and define I (n) αβγ (k) to be the O(k n ) contribution of I αβγ (k). In particular, we find (D. 12) There are two steps to compute the momentum integrals. First, the radial integral is systematically evaluated with the following formulas: 16) JHEP05(2021)023 The above formulas are proved in appendix E. Second, for the angle integrals in eq. (D.12), we replace the momentum products p µ 1 · · · p µ j , as shown in eq. (C.6). Then we obtain where we denote ∆ µν = ξ µ ξ ν − η µν . As a result, the O(k) contribution in eq. (D.10) is written as which reproduces the CVE: with ω µ = ε µνρσ ξ ν ∂ ρ h ρλ ξ λ /2. Similarly, the O(k 2 ) parts are computed as For the stationary gravitational field (∂ 0 h µν = 0), we eventually derive where we employ (D.22) The above current J µ (2) is consistent with J µ (2)eq in eq. (4.10).

(E.11)
Here, (const) denotes the divergent term that is independent of T and µ.
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