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Article

Anomaly-induced transport phenomenafrom the imaginary-time formalism

Masaru Hongo 1,†, and Yoshimasa Hidaka 2,1

1 RIKEN iTHEMS, RIKEN, Wako, Saitama 351-0198, Japan; [email protected] Quantum Hadron Physics Laboratory, RIKEN Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan;

[email protected]* Correspondence: [email protected]; Tel.: +81-48-462-1226

Received: date; Accepted: date; Published: date

Abstract: A derivation of the anomaly-induced transport phenomena—the chiral magnetic/vorticaleffect—is revisited based on the imaginary-time formalism of quantum field theory. Consideringthe simplest anomalous system composed of a single Weyl fermion, we provide two derivations:perturbative (one-loop) evaluation of the anomalous transport coefficient, and the anomaly matchingfor the local thermodynamic functional.

Keywords: Finite temperature field theory; Path integrals; Quantum fields in curved spacetime;Quantum statistical mechanics; Symmetries; Quantum anomalies; Hydrodynamics;

1. Introduction

Quantum anomaly is one of the most fundamental properties of quantum systems, whichkeeps staying in the low-energy regime once it appears in an underlying UV theory [1,2]. As aconsequence, the low-energy dynamics is strongly influenced by the existence of the quantum anomaly.A well-known example is the chiral anomaly in QCD, which gives rise to the Wess-Zumino term in thelow-energy effective theory of QCD (the chiral perturbation theory) describing the neutral pion decayinto two photons (π0 → γγ) [3–5]. The notion of anomaly can be generalized to discrete symmetriesof systems such as time-reversal symmetry. The anomaly matching argument [6,7] is actively appliedto restrict the possible nontrivial ground states (See Refs. [8–19] for recent applications).

It has been recently noticed that quantum anomaly also appears even in the effective theorydescribing the real-time dynamics of nonequilibrium systems, e.g., hydrodynamics and the kinetictheory, and it affects the macroscopic transport properties in the hydrodynamic regime [20–58] (Seealso pioneering works by Vilenkin [59,60]). For example, the simplest anomalous system composed ofa single right-handed Weyl fermion coupled to a background electromagnetic field shows interestingtransport . When this system is put into an environment with a temperature T and a chemical potentialµR, the chiral anomaly induces the dissipationless current along the magnetic field Bi given by

〈 JiR〉ano = σBBi + σωωi with σB =

µR

4π2 , σω =µ2

R4π2 +

T2

12, (1)

where 〈 JµR〉ano denotes the anomalous part of the expectation value of the right-handed current, and σB

(σω) is regarded as the chiral magnetic (vortical) conductivity. The first and second terms in Eq. (1) arecalled the chiral magnetic effect (CME) and chiral vortical effect (CVE), respectively (See Fig. 1). It isworth pointing out that even in the weak coupling limit, σB and σω do not diverge unlike the usualconductivity because their existence is protected by the quantum anomaly.

These anomalous transports are believed to be universally present when the system underconsideration contains the chiral anomaly. For example, they are expected to take place in the

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(a) Chiral Magnetic Effect (b) Chiral Vortical Effect

µR 6= 0<latexit sha1_base64="wFxHs0cTycPtqcYMQ2Inpu05IJI=">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</latexit>

~JR / ~B<latexit sha1_base64="tp2nOsT/3USWx7hCpYdBINqxEg4=">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</latexit> µR, T 6= 0

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Figure 1. The schematic picture of the anomaly-induced transport phenomena: (a) Chiral magneticeffect. (b) Chiral vortical effect.

quark-gluon plasma created in high-energy heavy-ion collisions [61–71], astrophysical plasmaincluding neutrino process [72–77], and Weyl semimetals realized in condensed matter physics [78–87].While we have not observed clear experimental signal of the anomaly-induced transport in the firsttwo systems, it has been recently reported that the experimental signal of the CME are achieved in theWeyl semimetal [88–90].

The theoretical derivation of the anomaly-induced transport phenomena has been remarkablydeveloped in the past ten years, e.g., the direct field theoretical evaluation [20], the fluid/gravitycorrespondence [21–23,25], the phenomenological entropy-current analysis [24], the linear responsetheory [26,31,34], the kinetic theory [27,28,33,36,37,41,42,44,48–52,54–58], and the hydrostatic partitionfunction method and extensions [29,30,32,35,38–40,43,45,46,53]. In this paper, we review the derivationof the anomaly-induced transport phenomena from the statistical mechanical viewpoint with thehelp of the imaginary-time (Matsubara) formalism of quantum field theory [91–94]. In particular, wedemonstrate two derivations, which are basically on the same line as the last two derivations raisedabove. For that purpose, we consider the simplest anomalous system composed of a single Weylfermion coupled to an external electromagnetic field. Although most results given in this paper hasbeen already known, we gives the clear rigorous justification of the hydrostatic partition functionmethod for the anomalous system based on the statistical ensemble describing systems in general localthermal equilibrium. This shows that the hydrostatic partition function method is indeed not restrictedto the real hydrostatic situation, but applicable to systems in general local thermal equilibrium.

The paper is organized as follows: In Sec. 2, we review the basic setup and formulation includingthe Zubarev’s nonequilibrium statistical operator methods [95–97] (See also Refs. [98–101] for a recentsophisticated revival of a similar idea). In Sec. 3, we then provide the perturbative evaluation of thechiral magnetic/vortical conductivity with the help of the (equilibrium) linear response theory, fromwhich we can read off the constitutive relation for the anomalous current. In Sec. 4, we give anothernonperturbative derivation based on the anomaly matching for the local thermodynamic functional.Sec. 5 is devoted to the summary and discussion.

2. Preliminaries for the anomaly-induced transport phenomena

In this section, we briefly summarize the formulation to derive the anomaly-induced transportphenomena based on the imaginary-time formalism [95–101].

2.1. Anomalous (non-)conservation laws for a single Weyl fermion

Let us consider the system consisting of a right-handed Weyl fermions ξ under an external U(1)gauge field Aµ in a (3 + 1) dimensional curved spacetime, whose action has the form:

S [ξ, ξ†; Aµ, e aµ ] =

∫d4xe

[i2

ξ†(

e µa σa−→D µ −

←−D µσae µ

a

)ξ

]with e ≡ det(e a

µ ), (2)


where we introduced σa = (1, σi) with the Pauli matrices σi (i = 1, 2, 3). Here e aµ (e

µa) denotes

(inverse) vierbein satisfying gµν = e aµ e b

ν ηab, ηab = e aµ e b

ν gµν with the spacetime curved metric gµν andMinkowski metric ηab = diag(−1,+1,+1,+1). The left and right covariant derivatives are defined as

−→D µξ ≡ ∂µξ − i(Aµ + Aµ)ξ, ξ†←−D µ ≡ ∂µξ† + iξ†(Aµ + Aµ) with Aµ ≡

12

ω abµ Σab, (3)

where we introduced Σab ≡ i(σaσb − σbσa)/4 with σa ≡ (−1, σi), which satisfies σaσb + σbσa = 2ηab.Furthermore, employing the torsionless condition, we can express the spin connection ω ab

µ = −ω baµ as

ω abµ ≡

12

eaνebρ(Cνρµ − Cρνµ − Cµνρ) with Cµνρ ≡ e cµ (∂νeρc − ∂ρeνc). (4)

Although the classical action (2) is invariant under a set of infinitesimal diffeomorphism, local Lorentz,and U(1) gauge transformations with parameters χ ≡ {ζµ, αab, θ}:

δχe aµ = ζν∇νe a

µ + e aν ∇µζν + αa

be bµ ,

δχ Aµ = ζν∇ν Aµ + Aν∇µζν + ∂µθ,

δχξ = ζν∂νξ − i2

αabΣabξ + iθξ,

(5)

we encounter with the quantum anomaly attached to the Weyl fermion. As a consequence, theanomalous Ward-Takahashi identities results in the following operator identities corresponding to the(non-)conservation laws:∇µTµ

ν = Fνµ Jµ,

∇µ Jµ = −18

CεµνρσFµνFρσ − λεµνρσRαβµνRβ

αρσ,with Tab − Tba = 0, (6)

where we introduced the energy-momentum tensor Tµν, U(1) covariant charge current Jµ, a field

strength tensor for the background electromagnetic field Fµν ≡ ∂µ Aν − ∂ν Aµ, and the Riemanncurvature tensor Rµ

νρσ with the totally antisymmetric tensor εµνρσ satisfying ε0123 = 1/e. For notationalsimplicity, we drop the subscript R for the U(1) current. Here C = 1/(4π2) and λ = 1/(768π2)

denote the anomaly coefficients coming from gauge and gravitational sectors, respectively. SinceλεµνρσRα

βµνRβαρσ contains four derivatives, it does not contribute to the first order hydrodynamics that

we are interested in. Therefore, we will omit the gravitational part in the following discussion. Notethat while the gauge and diffeomorphism invariance provides two (non-)conservation laws, the localLorentz invariance results in the symmetric property of the energy-momentum tensor operator. It isworth emphasizing that Jµ in Eq. (6) is the covariant current which can be related to the consistentcurrent Jµ

con by

Jµ = Jµcon −

16

Cεµνρσ AνFρσ. (7)

An analogue of this relation in local thermal equilibrium will appear in Sec. 4, and it plays an importantrole to see how the anomaly matching is realized for the local thermodynamic functional.

2.2. Zubarev’s formula: Decomposing dissipative and nondissipative transport

We then briefly review the Zubarev’s nonequilibrium statistical operator method from the modernviewpoint (See e.g., Refs. [95–101] for recent discussions) and specify from where the anomaly-inducedtransport arises. Assuming that the system is initially in local thermal equilibrium, the Zubarev’sformula provides us the expectation values of conserved current operators J µ

a ≡ {Tµν, Jµ} over the

initial density operator in the following compact form:

〈J µa(t, x)〉 = 〈J µ

a(t, x)〉LGt + Lµν

ab (t, x)∇νλb(t, x) + O((∇λ)2), (8)


where we introduced the intensive local thermodynamic parameters λa ≡ {βµ, ν}, which are relatedto the local fluid temperature T = 1/β, four-velocity uµ, and the chemical potential µ throughβµ ≡ βuµ, ν ≡ βµ. We also defined the average over the local Gibbs distribution ρLG[λ; t], whichdescribes systems in local thermal equilibrium, for an arbitrary operator O as

〈O〉LGt ≡ Tr

(ρLG[λ; t]O

)with ρLG[λ; t] ≡ exp

[−S[λ; t]

], S[λ; t] = K[λ; t] + Ψ[λ; t], (9)

where the entropy operator S[λ; t] is composed of the part including operators J µa and normalization

part for the density operator:

K[λ; t] ≡ −∫

dΣtµ

[βν(t, x)Tµ

ν(t, x) + ν(t, x) Jµ(t, x)]

, (10)

Ψ[λ; t] ≡ log Tr exp[−K[λ; t]

]. (11)

We here employed the fully covariant notion by introducing the constant time (spacelike) hypersurfacedefined by its perpendicular surface vector dΣtµ ≡ −d3x

√γnµ. Choosing a certain globally defined

time-coordinate function t(x), the unit normal vector nµ can be expressed as

nµ(x) = −N(x)∂µ t(x) with N(x) ≡(−∂µ t(x)∂µ t(x)

)−1/2 , (12)

where N(x) is a so-called Lapse function. In addition, introducing the spatial coordinate on the x,we have the induced metric γµν = gµν + nµnν whose spatial part gives γ ≡ detγi j (See e.g., Refs. [99,100] for a detailed geometric setup). The introduction of the covariantized notion looks a little bitcomplicated, but one can always take the flat limit by setting

(t(x), x(x)

)= (t, x), which results in e. g.

dΣtµ|flat = d3xδ0µ. Although it might be desirable to distinguish two coordinate systems defined by

(t, x) and (t, x), we will basically omit overline for the later one for notational simplicity since only(t, x)-coordinate system is mainly used. The normalization part Ψ[λ; t] is the local thermodynamicfunctional called the Massieu-Planck functional, and plays a central role in Sec. 4.

The crucial point here is that, by construction, we can identify the first term in the right-hand-sideof Eq. (8) as the nondissipative transport taking place in locally thermalized system, whereas thesecond term as the dissipative correction coming from the deviation from local thermal equilibrium.In other words, the formula (8) gives a way to decompose the non-dissipative and dissipativetransport at least in the leading-order derivative expansion. The second term is proportional tothe (local) thermodynamic forces ∇νλb, and coefficients in front of them are indeed specified astransport coefficients such as the bulk/shear viscosity, and conductivity. They are expressed by thetwo-point (Kubo) correlation function, which is nothing but the Green-Kubo formula for the transportcoefficient [95–101]. On the other hand, nondissipative part is often assumed to be simply givenby the usual constitutive relation for a perfect fluid. This is the case for parity-invariant systems,since the nondissipative derivative corrections are accompanied with higher-order derivatives forparity-invariant systems. Nevertheless, if we consider a system without parity symmetry—likethe Weyl fermion system given in Eq. (2)—we generally encounter with first-order nondissipativederivative corrections in 〈J µ

a(t, x)〉LGt . This is the origin of the anomaly-induced transport, and we

will focus on how we can evaluate 〈J µa(t, x)〉LG

t in the remaining part of this paper.Before closing this section, we put a short comment on the absence of the anomalous contribution

to the entropy production. To see this, using the conservation laws (6), we express the entropyproduction operator Σ[t, t0; λ] ≡ S[λ; t]− S[λ; t0] as

Σ[t, t0; λ] =∫ t

t0

d4xe∇µ sµ with ∇µ sµ ≡ −(∇νβµ)δTµν − (∇µν + βνFµν)δ Jµ, (13)

where we defined the local entropy production rate ∇µ sµ with δO(t) ≡ O(t) − 〈O(t)〉LGt . We

thus find that the local equilibrium part of the constitutive relation 〈J µa〉LG

t which also contains the


anomaly-induced transport as first-order derivative corrections, does not contribute to the local entropyproduction. This is perfectly consistent with the phenomenological derivation of the anomaly-inducedtransport based on the entropy-current analysis given in Ref. [24].

3. Perturbative evaluation of anomalous transport coefficients

In this section, we provide a simple perturbative derivation of the anomaly-induced transportgiven in Eq. (1), and calculate anomalous transport coefficients σB and σω at the one-loop level.

3.1. Derivative expansion of the local Gibbs distribution

First of all, we note that the local equilibrium part of the constitutive relation, or 〈J µa(t, x)〉LG

t , is afunctional of local thermodynamic parameters λa = {βµ, ν} and external fields j ≡ {Aµ, e a

µ } at a fixedconstant time t since the local Gibbs distribution ρLG[λ; t] depends on the configuration of them. Thus,〈J µ

a(t, x)〉LGt inherently contains the derivative correction coming from the local Gibbs distribution

itself.Suppose that our system is described by the local Gibbs distribution slightly deviated from the

global equilibrium (Gibbs) distribution only with the magnetic field and fluid vorticity. We also turnoff the external fields and take the flat limit. In that situation, approximating the fluid velocity and themagnetic field as uj(x) = (xi − xi

0)∂iuj|x=x0 = (xi − xi0)εijkωk,

Aj(x) = (xi − xi0)∂i Aj|x=x0 =

12(xi − xi

0)εijkBk,(14)

we can expand the local Gibbs distribution on the top of the global Gibbs distribution as

ρLG[λ; t] =1Z

e−β(H−µN)

[1 + Tτ

∫ β

0dτ∆S(t− iτ)

]with ∆S ≡ 1

2

∫d3xεijk(xi− xi

0)(

J jBk + 2T0jωk)

,

(15)where we defined O(t− iτ) ≡ eτ(H−µN)O(t)e−τ(H−µN). Here Z ≡ Tr e−β(H−µN) denotes the partitionfunction for the globally thermalized system, and we use 〈O〉eq ≡ Tr(e−β(H−µN)O)/Z. Then, notingthat the averaged current in global thermal equilibrium vanishes 〈 Ji(t, x0)〉eq = 0, we can evaluate〈 Ji(t, x0)〉LG

t as

〈 Ji(t, x0)〉LGt =

12

∫ β

0dτ∫

d3xεjkl(xj − xj0)

×[〈 Jk(t− iτ, x) Ji(t, x0)〉eqBl(t, x0) + 2〈T0k(t− iτ, x) Ji(t, x0)〉eqωl(t, x0)

]=

i2

εjkl

[∂qj ∆Jk Ji (ωn, q)

∣∣ωn=0, q=0Bl(t, x0) + 2∂qj ∆T0k Ji (ωn, q)

∣∣ωn=0, q=0ωl(t, x0)

],

(16)

where we performed the Fourier transformation to proceed the second line. It is now clear thatwe only need to evaluate two-point imaginary-time—not real-time—correlation functions, namely〈 Jk(t− iτ, x) Ji(t, x0)〉eq and 〈T0k(t− iτ, x) Ji(t, x0)〉eq, or their low-frequency and wave-number in theFourier space.

3.2. One-loop evaluation of anomalous transport coefficients

We then evaluate the anomalous transport coefficients with the help of the Matsubara formalism.Since we expand the local Gibbs distribution on the top of global Gibbs distribution, the Euclideanaction SE[ξ, ξ†, µ] for the right-handed Weyl fermion is simply given by

SE[ξ, ξ†] = −∑P

ξ†a(P)

(G−1

0 (P))

abξb(P) with G−1

0 (P) ≡ σµPµ, G0(P) =σµPµ

P2 , (17)


where a, b(= 1, 2) denote the spinor indices, and G0(P) the free propagator for the Weyl fermion.We also defined Pµ ≡ (−iωn − µ, p) with the Matsubara frequency ωn ≡ (2n + 1)πT and chemicalpotential µ. As usual, we introduced the Fourier transformation

ξ(τ, x) = T ∑n

∫ d3 p(2π)3 e−iωnτ+ip·xξ(ωn, p), (18)

with the temperature T ≡ 1/β. Note that the argument of the propagator in Eq. (17) is not P but P,and, thus, it represents the propagator fully dressed by the chemical potential µ. By using these, weneed to evaluate the following diagrams:

P

Q Q

P + QQ Q

A� �gµ⌫<latexit sha1_base64="aQ3680/b8rTItLpZQzC4xAGptac=">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</latexit>

P

Aµ

P + Q

A�

and

P

Q Q

P + QQ Q


P

Aµ

P + Q

A�

, (19)

where we will take the long-wave-length limit Q ∼ 0.First, let us evaluate the two-point current-current correlation function given by

P

Q Q

P + QQ Q


P

Aµ

P + Q

A�

= −T0 ∑n

∫ d3 p(2π)3 tr

((Qσ + Pσ)Pρσρσµσσσν

(Q + P)2P2

), (20)

where we used the free propagator defined in Eq. (17). Here “tr” denotes the trace over the spinorindices. With the help of the trace formula for the Pauli matrices

tr σµσνσασβ = −2iεµναβ + 2ηµνηαβ − 2ηµαηνβ + 2ηµβηνα, (21)

we can decompose the two-point functions into the antisymmetric part and other parts. Since we areinterested in the anomalous term which results from the antisymmetric part, we only focus on thatpart:

P

Q Q

P + QQ Q


P

Aµ

P + Q

A�

= − iµ4π2 ε0µνρQρ + (symmetric terms) + O(Q2), (22)

Next, let us evaluate the two-point momentum-current correlation function. Then, the same calculusbrings about the following result

P

Q Q

P + QQ Q


P

Aµ

P + Q

A�

= −14

T0 ∑n

∫ d3 p(2π)3 (2Pγ + Qγ)(δ

µβ δν

γ + δνβδ

µγ) tr

((Qσ + Pσ)Pρσρσβσσσα

(Q + P)2P2

)

= iQρ

(ην0ερµ0α + ηµ0ερν0α + δν

j ερµjα + δµj ερνjα

)( µ2

16π2 +T2

048

)+ (symmetric terms) +O(Q2). (23)


Putting these results all together, Eq. (16) results in

〈 Ji(t, x0)〉LGt =

µ

4π2 Bi(t, x0) +

(µ2

4π2 +T2

12

)ωi(t, x0), (24)

which is nothing but Eq. (1). To summarize the above analysis, we have derived the anomaly-inducedtransport—chiral magnetic/vortical effect—for the Weyl fermion by expanding the local Gibbsdistribution. This clearly shows that information on the anomaly-induced transport is fully containedin 〈J µ

a(t, x)〉LGt . Although we performed the direct expansion of the local Gibbs distribution in this

section, there is another way to systematically evaluate 〈J µa(t, x)〉LG

t as we will see in the next section.

4. Anomaly matching for local thermodynamic functional

In the previous section, we have explicitly shown that the local equilibrium part of constitutiverelations 〈J µ

a(t, x)〉LGt indeed contains the information on the anomaly-induced transport. Although

it is the one-loop perturbative calculation, we expect the result, or the value of anomalous transportcoefficients, is protected by the underlying chiral anomaly, and remain the same even if we take intoaccount the effect of interactions nonperturbatively. In this section, we provide another way to see theanomaly-induced transport putting the emphasis on the nonperturbative aspect of the anomaly. Thekey quantity is the local thermodynamic functional Ψ[λ, j; t] already defined in Eq. (11).

4.1. Basic properties of local thermodynamic functional

We here summarize basic properties of the Massieu-Planck functional Ψ[λ, j; t]: the exactpath-integral expression of Ψ[λ, j; t] and resulting symmetry properties together with the variationalformula.

4.1.1. Path-integral formula and resulting symmetry

We will first summarize the key result for the Massieu-Planck functional (See Refs. [99,100] forthe derivation). Using the energy-momentum tensor operator Tµ

ν and covariant current operator Jµ

resulting from (2), we can express the Massieu-Planck functional by the imaginary-time path integralin the same way with the usual Matsubara formalism for global thermal equilibrium. After a little bittedious calculation (See Ref. [100]), we eventually obtain

Ψ[λ, j; t] =∫DξDξ† exp

(S [ξ, ξ†; A, e]

), (25)

with the manifestly covariant action S [ξ, ξ†; Aµ, e aµ ] given by

S [ξ, ξ†; Aµ, e aµ ] =

∫ β0

0dτd3xe

[i2

ξ†(

e µa σa−→D µ −

←−D µσa e µ

a

)ξ

]with e ≡ det(e a

µ ). (26)

Here we introduced the thermal (inverse) vierbein e aµ (e µ

a ) and the external U(1) gauge field Aµ inthermally emergent curved spacetime as

e a0 = eσua, e a

i = e ai and A0 = eσµ, Ai = Ai, (27)

where, recalling βµ(x) ≡ β(x)uµ(x) and ν(x) = β(x)µ(x), we used

eσ(x) ≡ β(x)/β0, µ(x) ≡ ν(x)/β(x), β(x) ≡√−gµν(x)βµ(x)βν(x), (28)


with a constant reference inverse temperature β0. We also introduced e ≡ det e aµ and the covariant

derivative in thermal spacetime as−→D µξ ≡ ∂µξ − i(Aµ + Aµ)ξ,

ξ†←−D µ ≡ ∂µξ† + iξ†(Aµ + Aµ),

with ∂µ ≡ (i∂τ , ∂i), Aµ ≡12

ω abµ Σab, (29)

where the thermal spin connection is expressed by the thermal vierbein e aµ through the same relation

in the original spacetime (4).As is shown in these, we can say that the Massieu-Planck functional is expressed as the path

integral in the presence of the emergent background curved spacetime and U(1) gauge field. Notethat this background structure is completely determined by configurations of the local thermodynamicvariables λa (and external fields j) on the constant time hypersurface in the original spacetime. Thecrucial point here is that all these quantities do not depend on the imaginary-time coordinate τ,which leads to the Kaluza-Klein gauge symmetry. To see this clearly, we express the line elementds2 ≡ e a

µ e bν ηabdxµ ⊗ dxν and U(1) gauge connection A ≡ Aµdxµ in thermal spacetime as

ds2 = −e2σ(dt + aidxi)2 + γ′ijdxidxj, (30)

A = A0(dt + aidxi) + A′idxi, (31)

with dt ≡ −idτ. Here we defined the following quantities

ai ≡ −e−σui, γ′ij ≡ γij+e2σaiaj, A′i ≡= Ai − A0ai. (32)

Then, in addition to the spatial diffeomorphism invariance—invariance under spatial coordinatetransformation x→ x′(x)— we now see the background (30)-(31) is invariant under the transformationgiven by

t→ t + χ(x),

x→ x,

ai(x)→ ai(x)− ∂iχ(x).

(33)

This is nothing but Kaluza-Klein gauge transformation, and ai is identified as the Kaluza-Klein gaugefield. Note that γij and Ai = Ai do transform under the Kaluza-Klein gauge transformation sothat γ′ij and A′i do not. Therefore, it is useful to employ Kaluza-Klein gauge invariant quantities γ′ijand A′i rather than γij and Ai as basic building blocks to construct the Massieu-Planck functional.Furthermore, since the system is composed of the Weyl fermion, the apparent U(1) gauge invariancefor A′i is anomalously broken. These spatial diffeomorphism, Kaluza-Klein gauge, and anomalousU(1) gauge symmetries provide a basic restriction to the Massieu-Planck functional.

4.1.2. Variational formula in the presence of quantum anomaly

We then provide the variational formula for the Massieu-Planck functional Ψ[λ, j; t], and showall information on 〈J µ

a(t, x)〉LGt is fully installed in it. To show this, let us consider the variation of K

defined in Eq. (10) under the infinitesimal general coordinate and gauge transformation with a set ofparameters ζµ = εβµ and θ = ε(ν− β · A). (ε denotes an infinitesimal constant.) As a result of thecombination of diffeomorphism and U(1) gauge transformations, the variation of the backgroundU(1) gauge field δλ Aµ has the simple expression:

δλ Aµ = £β Aµ +∇µ(ν− β · A) = ∇µν + βνFνµ. (34)


The crucial point here is that K remains invariant under the simultaneous transformation acting onboth operators and external fields: δλK ≡ δ

paraλ K + δ

opeλ K = 0. This invariance can be shown by

recalling all operators in K are U(1) gauge invariant, and, furthermore, rewriting K as

K[t, λa, e aµ , Aµ] =

∫d4x√

γδ(t− t(x))nµ(x)λa(x)J µa(x), (35)

from which we can clearly see diffeomorphism (reparametrization) invariance. Moreover, δopeλ K will

also trivially vanish just because δopeλ K = [iK, K] = 0. As a result, we have the operator identity

δparaλ K = 0.

Then, let us investigate δparaλ K in detail, whose explicit definition is given by

δparaλ K ≡

∫d4x

[δK

δt(x)£βt(x) +

δKδλa(x)

£βλa(x) +δK

δe aµ (x)

£βe aµ (x) +

δKδAµ(x)

δλ Aµ(x)

]. (36)

To rewrite the first term of this equation, noting δ(t− t(x))nµ = −Nδ(t− t(x))∂µt = N∂µθ(t− t(x))following from the definition of nµ, and performing the integration by parts, we rewrite K in Eq. (35)as

K[t, λa, e aµ , Aµ] = −

∫d4xeθ(t− t(x))∇µ(λ

a(x)J µa (x))

= −∫

d4xeθ(t− t(x))(

Tµν∇µβν + Jµ(∇µν + βνFνµ)−

18

CνεµνρσFµνFρσ

),

(37)

where we used e = N√

γ and employed the operator identity for current operators (6) to proceedthe second line. With the help of Eq. (34) together with ∇µβν = e ν

a £βe aµ + βρω ν

ρ µ followed from theso-called (torsionless) tetrad postulate ∇µea

ν + ω aµ be b

ν = 0, Eq. (37) enables us to obtain

∫d4x

δKδt(x)

£βt(x) =∫

d4x√

γδ(t− t(x))[

Tµa£βe a

µ + Jµδλ Aµ −18

CνεµνρσFµνFρσ

]β′, (38)

where we defined β′ ≡ −βµnµ and used the operator identity Tab − Tba = 0. By using the identity

nαεµνρσFµνFρσ = −4εµνρσnνFρσFαµ. (39)

the last term in the second line of Eq. (38) can be further simplified as

∫d4x√

γδ(t− t(x))[

18

CνβαnαεµνρσFµνFρσ

]= −

∫d4x√

γδ(t− t(x))[

12

CνβαnνεµνρσFρσFαµ

]= −

∫d4x√

γδ(t− t(x))CνBµδλ Aµ.(40)

Here we defined the four-magnetic field as Bµ ≡ Fµνnν = εµνρσnνFρσ/2, and neglected the surfaceterm accompanied by the integration by parts. We thus obtain the following compact result:

∫d4x

δKδt(x)

£βt(x) =∫

d4xβ′√

γδ(t− t(x))[

Tµa£βe a

µ +(

Jµ − Cβ′−1νBµ)δλ Aµ

]. (41)

Equipped with this formula together with £ββµ = 0, and £βν = £β(ν− β · A) + βµ£β Aµ = βµδλ Aµ, weare now ready to express δ

paraλ K in Eq. (36) by the use of the variation of the vierbein and gauge field:

δparaλ K =

∫d3x

[(β′√

γTµa +

δKδe a

µ

)£βe a

µ +

(β′√

γ[

Jµ − Cβ′−1νBµ]+

δKδν

βµ +δK

δAµ

)δλ Aµ

]. (42)


Let us then take the average of this operator identity over the local Gibbs distribution ρLG[λ; t]. In theabsence of the quantum anomaly, we can simply replace the averaged variation of K with the variationof the Massieu-Planck functional: 〈δK/δj〉LG

t = −δΨ/δj. Nevertheless, since we are considering thesystem with the chiral anomaly, we need to be careful when we take the variation of the charge densitycoupled to the local chemical potential. Using the relation ∂(e J0)/∂Aµ =

√γCBµ resulting from the

covariant anomaly, we can show

δKδAµ

= e J0βµ +√

γνCBµ − β′√

γ∂L

∂Aµ. (43)

We can then identify the local Gibbs average of the last term in this equation as the covariant current inthermal spacetime, which results in the sum of the consistent current and the Bardeen-Zumino currentcomposed of Aµ:

β′√

γ

⟨∂L

∂Aµ

⟩LG

t

= N∫DξDξ†eS [ξ,ξ† ;A,e] δS

δAµ

=δΨ

δAµ

− C6

εµνρσ AνFρσ, (44)

where N denotes a normalization constant, and we introduced a field strength tensor in thermalspacetime Fµν ≡ ∂µ Aν − ∂ν Aµ together with the totally antisymmetric tensor εµνρσ ≡ N(β0/β′)εµνρσ.Using this together with 〈δK/δe a

µ 〉LGt = −δΨ/δe a

µ , we eventually obtain the following identity:

〈δparaλ K〉LG

t =∫

d3x

[(β′√

γ〈Tµa〉LG

t −δΨδe a

µ

)£βe a

µ +

(β′√

γ〈 Jµ〉LGt −

δΨδAµ

+C6

εµνρσ AνFρσ

)δλ Aµ

].

(45)Therefore, noting that that this identity holds for an arbitrary variation of the background vierbeinand gauge field, the identity 〈δpara

λ K〉LGt = 0 provides the variational formula for the Masseiu-Planck

functional

〈Tµa(t, x)〉LG

t =1

β′√

γ

δΨ[λ, j, t]δe a

µ (x), (46)

〈 Jµ(t, x)〉LGt =

1β′√

γ

δΨ[λ, j, t]δAµ(x)

− C6

εµνρσ AνFρσ. (47)

We thus conclude that the average values of any conserved current operator over local thermalequilibrium is fully captured by the single (local thermodynamic) functional known as theMasseiu-Planck functional. It is worth pointing out that because we deal with the average of thecovariant current 〈 J(x)〉LG

t , we have the last term in Eq. (47) analogous to the Bardeen-Zuminocurrent [102] (See also Refs. [29,43,47]). In summary, we can identify the Massieu-Planck functionalΨ[λ, j; t] as a generating functional for a (nondissipative) local equilibrium part of hydrodynamics, or〈J µ

a(t, x)〉LGt .

Before moving to the path-integral formula for the Massieu-Planck functional, we put a shortcomment on the useful “gauge and coordinate choice”, which we call hydrostatic gauge. Since we havea freedom to choose the local time-direction and time-component of the external gauge field, we canemploy the hydrostatic gauge fixing condition

tµ(x) = βµ(x)/β0, tµ(x)Aµ(x) = ν(x)/β0, (48)

with a constant reference temperature β0. In this special choice of the gauge, the above transformationdoes not induce the gauge transformation because θ = ε(ν− β · A) = 0, and furthermore, thanks tothe refined choice of our local time-direction, the fluid looks like entirely at rest. This is the originof the name hydrostatic. Nevertheless, note that this does not means the system is in a stationary


hydrostatic state since we do not assume βµ is a killing vector: £βgµν 6= 0. The main reason why thehydrostatic gauge gives the most useful gauge is that we can equate the background field in original(real) spacetime with that in (imaginary) thermal spacetime: e a

µ |hs = e aµ and Aµ|hs = Aµ. As a result,

the above variational formula results in (46)-(47) as

〈Tµa(t, x)〉LG

t =1

β0eδΨ[λ, j; t]

δe aµ (x)

∣∣∣∣∣hs

, (49)

〈 Jµ(t, x)〉LGt =

1β0e

δΨ[λ, j; t]δAµ(x)

∣∣∣∣hs− C

6εµνρσ AνFρσ

∣∣hs , (50)

which enable us to regard the Massieu-Planck functional as a usual generating functional.

4.2. Anomaly matching for local thermodynamic functional

Based on the obtained formulae, we now discuss the anomaly-induced transport from the pointof view of the anomaly matching for the Massieu-Planck functional.

Before moving to the anomaly-induced transport, let us briefly see how we can derive theconstitutive relation for a perfect fluid. Employing the simplest power counting scheme λ =

O(∇0), j = O(∇0), we perform the derivative expansion of the Massieu-Planck function as follows:

Ψ[λ, j; t] = Ψ(0)[λ, j; t] + Ψ(1)[λ, j; t] + O(∇2), (51)

where the superscript represents the number of spatial derivatives acting on parameters λ and j.Then, the symmetry argument reviewed in the previous subsection tells us that we cannot use theKaluza-Klein and U(1) gauge fields in the leading-order derivative expansion. As a result, the generalform of the leading-order Massieu-Planck functional Ψ(0)[λ, j; t] is expressed as

Ψ(0)[λ, j; t] =∫ β0

0dτd3xep(β, ν) =

∫d3xβ′

√γp(β, ν), (52)

where p(β, ν) is a certain function depending on β and ν. By taking the variation with respect to thevierbein and gauge field, we are able to obtain the leading-order constitutive relation as

〈Tµν(t, x)〉LG(0) = (e + p)uµuν + pgµν + O(∇1), 〈 Jµ(t, x)〉LG

(0) = nuµ + O(∇1). (53)

This is nothing but the constitutive relation for the perfect fluid with e, n, p being the energy density,charge density, and fluid pressure, respectively.

Then, the next problem is to specify the first-order derivative correction of the Massieu-Planckfunctional Ψ(1)[λ, j; t], which is present (absent) in the absence (presence) of the parity symmetry. Sinceour system is composed of the right-handed Weyl fermion, and thus, there is no parity symmetry, thefirst-order correction is not prohibited. In this case, two (anomalous) gauge symmetries again plays acentral role to extract information on the anomaly-induced transport contained in Ψ(1)[λ, j; t]. In thefollowing, after giving a bottom up view relying on the one-loop result in the previous section, weswitch to a top down view of the anomaly matching, from which we can derive the anomaly-inducedtransport beyond the one-loop level.

4.2.1. Chiral anomaly in thermal spacetime

At one-loop level, we have already derived the anomaly-induced transport given in Eq. (24). Onthe other hand, we also have the variational formula (47) in a general gauge, or (50) in the hydrostatic


gauge. Let us take the hydrostatic gauge. Then, the combination of the above results enables us toobtain the following functional differential equation for Ψ(1):

1β0

δΨ(1)ano[λ, j; t]δAi(x)

∣∣∣∣∣hs

+µ

12π2 Bi − 112π2 ε0ijk Ak∂jµ =

µ

4π2 Bi +

(µ2

4π2 +T2

12

)ωi, (54)

where we take the flat limit and assume global thermal equilibrium with a constant temperature β0 inthe variational formula. This equation can be easily solved as

Ψ(1)ano[λ, j; t]

∣∣∣eq

hs=

β0

12π2

∫d3xµAiBi + β0

∫d3x

(1

4π2 µ2 +1

12T2)

Aiωi

=β0

12π2

∫d3xε0ijkµAi∂j Ak +

β0

2

∫d3xε0ijk

(1

4π2 µ2 +1

12T2)

Ai∂juk

(55)

up to irrelevant constants. On the other hand, we have already clarified that the Massieu-Planckfunctional need to respect both U(1) and Kaluza-Klein gauge invariance. This constraint then enablesus to guess the full result on Ψ(1) for general local thermal equilibrium though Eq. (55) is obtained bymatching with the one-loop result for linear perturbations on the top of global thermal equilibrium. Byusing the U(1) and Kaluza-Klein gauge covariant quantities—A′i and ai, respectively—together withA0 = eσµ, we specify the first-order derivative correction as

Ψ(1)ano[λ, j; t] =

Cβ0

3

∫d3xeε0ijk A0 A′i∂j A′k +

Cβ0

6

∫d3xeε0ijk A2

0 A′i∂jak,− C1

2β0

∫d3xeε0ijk A′i∂jak, (56)

with C1 ≡ 1/12. Note that A0 and A′i defined in Eqs. (27) and (32) are manifestly Kaluza-Klein gaugeinvariant quantities.

Let us then confirm the consistency for this result based on the anomaly matching for theMassieu-Planck functional itself. For that purpose, we consider the time-independent gaugetransformation given by δθ A0 = 0, δθ Ai = ∂iθ(x). Under this gauge transformation, the Fujikawamethod [2] says that the anomalous shift of the Massieu-Planck functional is given by the consistentanomaly:

δθΨ[λ, j; t] = −Cβ0

3

∫d3xθeε0ijk∂i A0∂j Ak. (57)

On the other hand, one can directly show that the first two term of Ψ(1)ano[λ, j; t] in Eq. (56) correctly

reproduces this anomalous shift as

δθΨ(1)ano[λ, j; t] =

Cβ0

3

∫d3xeε0ijk A0∂iθ∂j A′k +

Cβ0

6

∫d3xeε0ijk A2

0∂iθ∂jak

= −Cβ0

3

∫d3xθeε0ijk∂i A0∂j(Ak − A0ak)−

Cβ0

3

∫d3xθeε0ijk A0∂i A0∂jak + (surface terms)

= −Cβ0

3

∫d3xθε0ijk∂i A0∂j Ak + (surface terms). (58)

Therefore, we see that the anomalous transport coefficients C proportional to the chemical potential µ

is indeed related to the anomaly coefficient attached to the Weyl fermion.Nevertheless, the last term in Eq. (56), which brings about the chiral vortical effect proportional

to T2, is not restricted by the chiral anomaly. From the symmetry point of view, this is just becausethe last term in Eq. (56) remains invariant under the U(1) gauge transformation. This corresponds thefact that the entropy production argument with chiral anomaly leads to the existence of both chiralmagnetic and vortical effect [24], in which only the anomalous transport coefficients proportionalto the chemical potential are determined. Then, the natural question is ”Does the chiral vortical effectproportional to T2 have any relation with the quantum anomaly?”


4.2.2. Global anomaly for Kaluza-Klein gauge transformation

It was pointed out the T2 term of the chiral vortical coefficient is related to the gravitationalcontribution to the chiral anomaly [26]. However, unlike the chiral magnetic coefficient discussedin this section, it is not clear that how the chiral vortical effect relates to the εµνρσRα

βµνRβαρσ,

because the number of derivative in εµνρσRαβµνRβ

αρσ is higher than that in εµνρσFµνFρσ. In other

words, εµνρσRαβµνRβ

αρσ does not directly contribute to the first order hydrodynamics. An alternative

explanation of T2 term is that the chiral vortical coefficient is related to a global anomaly [45,46,103].Here, we show the relation between the global anomaly and chiral vortical effect.

As a warm up exercise, let us first consider the global anomaly attached to the Weyl fermion in1 + 1 dimensions, which possesses the chiral anomaly given by

∂µ Jµ = −12

C2DεµνFµν with C2D ≡1

2π, (59)

where Jµ again denotes the covariant current in 1 + 1 dimensional system. In this case, there are nochiral magnetic and vortical effects because there is no transverse direction, and thus, no magnetic fieldand vorticity. However, there exist nonvanishing 〈 Jz〉 and 〈T0

z〉 caused by chiral and global anomalies.The direct calculation at equilibrium shows

〈T0z〉eq =

∫ ∞

0

dpz

2πpz[nF(|pz| − µ) + nF(|pz|+ µ)

]=

µ2

4π+

π

12T2,

〈 Jz〉eq =∫ ∞

0

dpz

2π

pz

|pz|[nF(|pz| − µ)− nF(|pz|+ µ)

]=

µ

2π.

(60)

On the other hand, the same procedure given above leads to the variational formula in (1 + 1)dimensions:

〈Tµa〉LG

t =1

β′√

γ

δΨ[t; λ]

δe aµ (x)

,

〈 Jµ〉LGt =

1β′√

γ

δΨ[t; λ]

δAµ(x)− 1

2C2D εµν Aν,

(61)

where εµν = N(β0/β′)εµν. Then, the matching condition for the momentum density and currentresults in

1β0

δΨano

δe z0

= − 1β0

δΨano

δaz=

C2D2

µ2 + πC1T2, (62)

1β0

δΨano

δAz+

C2D2

µ = C2Dµ. (63)

Solving Eqs. (62) and (63), we find

Ψano =C2Dβ0

2

∫dzA0 A′z − π

C1

β0

∫dzaz. (64)

This gives the anomalous part of the Masseiu-Planck functional. In order to detect anomalies, wecompactify the spatial direction with the length L. Here we will show Ψano has two types of anomalies.One is the chiral anomaly: Under U(1) gauge transformation Az → Az + ∂zθ(z), the anomalous shiftof Ψ arises:

δθΨano = −C2Dβ0

2

∫dzθ∂z A0, (65)


which correctly reproduces the consistent anomaly in thermal spacetime. The other is the globalanomaly associated with the Kaluza-Klein gauge transformation:{

t→ t + χ(z),

az → az − ∂zχ(z),(66)

where A′z remains invariant. Under this transformation, Ψano also acquires the anomalous shift givenby

δχΨano = πC1

β0

∫dz∂zχ(z), (67)

which is just a boundary term, so that Ψano is invariant under local transformation with χ(0) = χ(L).However, if we consider global transformation, χ(z) = −2iβ0z/L, which corresponds to the imaginarytime shift τ → τ + 2zβ0/L that keep the boundary condition, we have an additional phase

Ψano → Ψano − 2πiC1, (68)

which can be understood as the global anomaly associated with the large diffeomorphism. Thisanomalous phase is related to the three dimensional gravitational Chern-Simons term through theanomaly inflow mechanism, which is also related to the gravitational contribution to chiral anomaly in3 + 1 dimensions [104,105].

This argument can be generalized to higher dimensions. In (3 + 1) dimensions, Ψano is given inEq. (56). In order to detect the global anomaly, we compactify the space to S1 × S2, where we choose zas the coordinate on S1. Under the large diffeomorphism, τ → τ + 2zβ0/L, the term contributing tothe T2 part of chiral vortical effect transforms as

Ψano → Ψano − 2πiC1

∫ d2x2π

eε0ijz∂i A′j. (69)

This is the global mixed anomaly between U(1) gauge and large diffeomorphism. Therefore, we seethat the chiral vortical coefficient proportional to T2, which is nothing but C1, is related to the mixedglobal anomaly.

5. Summary and discussion

In this paper, we have discussed two approaches to derive the anomaly-induced transportphenomena for the system composed of a Weyl fermion: perturbative evaluation of the chiralmagnetic/vortical conductivity with the help of the (equilibrium) linear response theory, and thenonperturbative determination of anomalous parts of the local thermodynamic functional on thebasis of the anomaly matching. Both derivations are based on the imaginary-time formalism of thequantum field theory, and we have seen that the obtained anomalous constitutive relations correctlydescribe the chiral magnetic/vortical effect. Although it is not so clear in the first derivation, thesecond derivation shows that the chiral magnetic/vortical effect results from the first-order derivativecorrections of the local thermodynamic functional, and thus, they are clearly nondissipative in nature.This is perfectly consistent with the known result obtained from the hydrostatic partition functionmethod [29–32,35,38–40,43,45,46], and we rigorously clarify why that method works well. This localequilibrium part of the constitutive relation also complete the application of Zubarev’s nonequilibriumstatistical operator method to derive the hydrodynamic equation for the parity-violating (anomalous)fluid.

There are several interesting questions related to the current work. It has been already pointedout that the coefficient in front of the T2-term of the chiral vortical effect will be renormalized in the


presence of dynamical gauge fields such as the gluon in the QCD plasma [106]. It may be interestingto examine which part of the anomaly matching argument associated with the large diffeomorphism(Kaluza-Klein gauge) transformation should be modified due to the existence of the dynamical gaugefield. Another important issue associated with the inclusion of dynamical electromagnetic field isits dynamics. When we consider the dynamics of the electromagnetic field rather than treating it asthe background one, we encounter with several interesting phenomena such as the chiral plasmainstability [107–111], and mixing of some hydrodynamic modes (chiral magnetic wave) to be themassive collective excitation (chiral plasmon) [48,62,112,113]. It is desirable to systematically describethem based on the generalization of magnetohydrodynamics for the chiral plasma by formulatingchiral magnetohydrodynamics. Chiral magnetohydrodynamics is just recently formulated basedon e.g., the phenomenological entropy-current analysis [114] (See also Refs. [115–119]), but less isclarified from the underlying quantum field theory. Combined with the recent development of themagnetohydrodynamics itself from the field theoretical viewpoint [120–124], it may be interesting toformulate chiral magnetohydrodynamics based on the Zubarev’s nonequilibrium statistical operatormethod equipped with the path-integral formula for the local thermodynamic functional reviewed inthis paper.

Funding: This research was funded by Japan Society of Promotion of Science (JSPS) Grant-in-Aid for ScientificResearch grant number JP16J02240, 16K17716, 17H06462, and 18H01211.

Acknowledgments: M. H. was supported by the Special Postdoctoral Researchers Program at RIKEN. This workwas partially supported by the RIKEN iTHES/iTHEMS Program, in particular, iTHEMS STAMP working group.

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