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REVIEW 2 major objections 7 minor 71 references

Examining the Anomalous Nature of Chiral Effects in Thermodynamics

T0 review · 2 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A direct path-integral computation shows that the chiral anomaly in a fluid at local equilibrium depends on the local temperature and chemical potential, and that it vanishes exactly at global equilibrium.

desk verdict A careful top-down computation of the anomaly at local equilibrium that reproduces known transport coefficients, but the new thermodynamic anomaly terms still face an unresolved counter-term ambiguity that the paper itself flags. read the letter →

arxiv 2507.02079 v1 pith:S2VHIYVK submitted 2025-07-02 hep-th cond-mat.stat-mechhep-ph

Rémy Larue , Jérémie Quevillon , Diego Saviot This is my paper · ORCID
classification hep-thcond-mat.stat-mechhep-ph PACS 11.30.Rd11.10.Wx04.62.+v
keywords chiralanomalylocalthermalequilibriumvorticaleffectseparationfinitetemperaturefieldtheorycurvedspacetimecovariantderivativeexpansiondynamicalvorticity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper computes the chiral anomaly directly from the path integral for a massless Dirac fermion in a curved spacetime with background electromagnetic fields, in local thermal equilibrium with a position-dependent temperature $T(x)$ and electro-chemical potential $\mu_{ec}(x)$. Its main result is that the anomaly is not just the familiar topological density: it also contains new, finite terms built from the dynamical vorticity and the magnetic field, so the anomaly depends on the local thermodynamic state and its gradients. These terms are tied to the chiral vortical and chiral separation effects, and the paper argues they are physical rather than removable counter-term artifacts. The computation matters because vorticity, magnetic fields, and gradients of temperature and chemical potential coexist in heavy-ion collisions, condensed-matter experiments, and astrophysical plasmas, where the anomaly is often invoked to explain transport. As a corollary, the anomaly vanishes exactly at global equilibrium, $A = 0$, even though the axial current itself remains nonzero.

What carries the argument

The central object is the Jacobian $J[\theta]$ of the axial transformation, expressed as a ratio of functional determinants whose logarithm is $$\log J[\$\theta$] = \mathrm{Tr}\left[\left((\not{\partial}\$\theta$)\gamma_5 + 2im\$\theta$\gamma_5\right)\frac{1}{i\not{D} - \gamma^t\sqrt{g_{tt}}\mu_{ec} - m}\right],$$ evaluated in imaginary time with antiperiodic boundary conditions. At finite temperature the massless limit is safe because the temperature acts as an infrared regulator, so the anomaly reduces to the divergence of the axial-current expectation value. The trace is evaluated with a covariant derivative expansion organized as a weak-field expansion relative to $\mathrm{Max}\{\mu_{ec}, T\}$, with the electro-chemical potential shifted into the Matsubara frequencies; after a change of variables the local temperature $T(x) = T_0/\sqrt{g_{tt}}$ emerges naturally. The two structures that carry the new physics are the dynamical vorticity $\Theta^\mu = -\tfrac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_\nu u^\lambda \omega_{\lambda,\rho\sigma}$, which controls the chiral vortical effect contribution, and the magnetic field $B^\mu$, which controls the chiral separation effect contribution.

What would settle it

Recompute the same Jacobian in a solvable local-equilibrium model, such as a 2D massless fermion with a sharp step in temperature or chemical potential, and check whether the anomaly picks up the analogue of the new thermodynamic terms; alternatively, search explicitly for a local polynomial counter-term that removes Eq. (20) while leaving the chiral vortical and separation currents unchanged — if one exists, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is Eq. (20): the chiral anomaly obtained from the Jacobian of the axial transformation is $$A = \nabla_\mu\langle j_5^\mu\rangle = \left(\frac{\mu_{ec}\partial_\mu\mu_{ec}}{\$pi^{2}$} + \frac{T\partial_\mu T}{3}\right)\Theta^\mu + \left(\frac{\mu_{ec}^2}{2\$pi^{2}$} + \frac{$T^{2}$}{6}\right)\nabla_\mu\Theta^\mu + \frac{1}{2\$pi^{2}$}\left(\partial_\mu\mu_{ec} - \mu_{ec}a_\mu\right)B^\mu ,$$ up to third derivatives of the fields and thermodynamic variables, with $\Theta^\mu$ the dynamical vorticity, $B^\mu$ the magnetic field, and $a_\mu$ the fluid acceleration. The paper therefore claims that at local equilibrium the anomaly depends on the local temperature, the electro-chemical potential, and their gradients, and that the chiral vortical and chiral separation effects feed directly into the anomaly. When the global equilibrium conditions are imposed, all these terms cancel and the anomaly vanishes, Eq. (21). Along the way, the paper derives the local-equilibrium generalization of the chiral vortical effect, Eq. (12), and of the chiral separation effect, Eq. (16), which reduce to the known forms only when the temperature is at global equilibrium.

Load-bearing premise

The new finite, non-topological terms in the anomaly formula are assumed to be genuine physics that cannot be removed by local polynomial counter-terms; the paper explicitly notes this possibility has not been studied, so if such counter-terms exist, the central claim fails.

Editorial extensions

If this is right

  • The chiral anomaly in local equilibrium is no longer a purely topological quantity: it contains finite thermodynamic terms, so measurements of anomalous currents in vortical or magnetized fluids must be interpreted through Eq. (20) rather than the vacuum anomaly.
  • The temperature dependence of the chiral vortical effect is traced to a new mixed anomaly involving the spin connection, which resolves the power-counting mismatch between the two-derivative vortical current and the four-derivative axial-gravitational anomaly $R\tilde{R}$.
  • In flat spacetime with a background electric field and spatially varying chemical potential and temperature, the anomaly is $(1/2\pi^2)(E + T\,\partial(\mu/T))\cdot B$ rather than the commonly assumed $(1/2\pi^2)\,E\cdot B$.
  • At global equilibrium the anomaly vanishes exactly, $A = 0$, while the axial current remains nonzero, showing that the chiral effects studied here are inherently out-of-equilibrium phenomena.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the new thermodynamic terms are physical, then systems with strong gradients of temperature or chemical potential, such as the quark-gluon plasma in heavy-ion collisions or the early universe, should produce axial charge even without $E\cdot B$; that is a concrete and testable prediction.
  • The exact cancellation at global equilibrium suggests the local-equilibrium anomaly may be derivable from an effective action built only from the thermal data and fluid velocity, which would show whether the new terms are forced by thermodynamics or are specific to the microscopic computation.
  • Repeating the same path-integral computation with an axial chemical potential, which the authors postpone, would reveal whether the chiral magnetic effect also receives local thermodynamic corrections of the same kind.
  • A decisive check is scheme independence: recomputing the Jacobian with a different regularization of $\gamma_5$ would determine whether the finite non-topological terms are universal or artifacts of the particular regularization used here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper derives the chiral anomaly for a massless Dirac fermion in curved spacetime with a background electromagnetic field at local thermal equilibrium, using imaginary-time path integral and covariant derivative expansion. The main result is Eq. (20), which expresses the anomaly as a sum of terms involving the local electro-chemical potential μec, local temperature T, dynamical vorticity Θμ, and magnetic field Bμ, with coefficients that depend on μec and T. The authors also generalize the chiral vortical and separation effects and show that the anomaly vanishes under global equilibrium conditions. The computation is presented in detail in appendices, including Matsubara sums, dimensional regularization, and the BMHV scheme for γ5.

Significance. If Eq. (20) is scheme-independent, the result establishes a new connection between the chiral anomaly and local thermodynamic variables, with potential phenomenological consequences in heavy-ion physics, condensed matter, and cosmology. The paper is commendable for its fully explicit top-down derivation, with no free parameters and with the known CVE and CSE coefficients emerging from the computation. However, the physical significance of the new non-topological terms in Eq. (20) hinges on a counterterm analysis that the paper explicitly leaves open; moreover, part of the CSE derivation is restricted to flat spacetime. These points currently limit the strength of the central claim.

major comments (2)
  1. [Discussion and conclusions, paragraph following Eq. (20)] The central claim that the anomaly depends on local thermodynamic parameters is not yet established because the paper explicitly states that 'the possibility of canceling them with local polynomial counter-terms has not been studied here.' The new terms in Eq. (20) are finite, non-topological, and arise in the BMHV scheme, which breaks Lorentz invariance; the two arguments offered in defense—finiteness and Ref. [58]—do not rule out removal by finite local counter-terms. To make the claim load-bearing, the authors should compare Eq. (20) with the result obtained from a Lorentz-covariant regulator (e.g., Fujikawa/Leutwyler) and either show that the difference is not a local polynomial functional of T, μec, gμν, and Vμ, or identify the physical observable that fixes the scheme.
  2. [Appendix D, beginning of CSE computation] Eq. (16) in the main text presents the CSE current and Eq. (17) its contribution to the anomaly as generalizing to curved spacetime and background electric fields, but Appendix D explicitly restricts the k=2 computation to flat spacetime and drops higher-order terms in μec. The acceleration term −μec aμ Bμ /(2π^2) in Eq. (17) and Eq. (20) is a curved-space/fluid effect that is not derived in the flat-spacetime calculation. The authors should either provide the curved-spacetime computation or state clearly that Eqs. (16)-(20) are established only in flat spacetime, in which case the generalization claim in the abstract and introduction must be softened.
minor comments (7)
  1. [Fluid velocity within the metric] The sentence 'If ⃗ vhas vanishing material derivative' is repeated and the paragraph is garbled; please rewrite it for clarity.
  2. [Eq. (5)] The lower-right entry '−13' presumably denotes −I3, the negative 3×3 identity matrix, but the notation is ambiguous and should be made explicit.
  3. [Eq. (13)] The notation (⃗ u ∧ ∂t⃗ u)i is undefined; please clarify the definition of the wedge product and the index structure.
  4. [Eq. (9)] The denominator expression is missing parentheses; as written it could be misread as γtΩn + (γi qi)/(gttΩn^2 + ...). Please use full parentheses throughout.
  5. [Appendix E, Eq. (E5)] The Fermi-Dirac distribution uses β(x) while earlier in the paper the inverse temperature is β0; the relation between β(x) and β0 should be stated explicitly near Eq. (E5).
  6. [Introduction, Refs. [31]-[33]] Reference [33] is a textbook on electrochemistry; for the decomposition of the electro-chemical potential the authors might benefit from a more standard field-theory reference, though this is not required.
  7. [Global equilibrium] The sentence 'Note that we only apply these conditions at the end of our computations' could be misinterpreted; consider rephrasing to clarify that the equilibrium conditions are imposed only after evaluating the path integral.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: Eq. (20) is obtained by differentiating independently computed CDE currents, not by fitting or by self-referential input.

full rationale

The derivation chain runs from the path-integral Jacobian in Eq. (7), through the CDE expansion Eq. (8), to the k=1 and k=2 current expectation values Eqs. (12) and (16), and then by direct differentiation to the anomaly contributions Eqs. (15), (17), and finally Eq. (20). The numerical coefficients (µec^2/(2π^2)+T^2/6) and µec/(2π^2) are produced by Matsubara sums and traces in Apps. C-E, not fixed by fitting or by imposing the target result. The vanishing at global equilibrium, Eq. (21), is derived from the equilibrium conditions plus the identity ∇µΘµ+2Θµ∂µT/T=0 in App. C, not assumed. The self-citations [49] and [50] provide the functional-determinant representation and the curved-space covariant-derivative-expansion technique; these are methodological inputs with independent content. The one imported technical result, the BMHV vacuum cancellation in the k=1 channel cited to [50], concerns the T=µec=0 reference and is accompanied by a vector-symmetry rationale; it does not reduce the new thermodynamic terms in Eq. (20) to an input. The paper's own caveat that local polynomial counter-terms have not been studied is a legitimate scheme-dependence and correctness risk, not a circular step. No fitted parameter is relabeled as a prediction, and no ansatz or uniqueness theorem is smuggled in by citation to force the result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The computation rests on standard thermal field theory and several domain-specific assumptions (analytic continuation, massless limit with IR regulator, BMHV scheme, vacuum cancellation, and physicality of the new finite terms). No free parameters are fitted to data; no new entities are introduced.

assumptions (5)
  • domain assumption The imaginary-time path integral with the local equilibrium partition function Eq. (3) correctly describes a massless Dirac fermion at local temperature and chemical potential in a curved background.
    Invoked in the section 'Thermodynamics in Curved Spacetime' to set up Eq. (4); standard in thermal field theory but extended here to local equilibrium.
  • domain assumption Analytic continuation from imaginary to real time is valid for the local equilibrium results, including for the complex metric used to describe moving fluids.
    Stated under 'Global equilibrium': 'We however assume that we can analytically continue the imaginary-time to real-time to obtain the local equilibrium result.' Also footnote 8 assumes a complex momentum shift.
  • domain assumption Temperature acts as an infrared regulator so the massless limit can be taken by dropping the 2im theta gamma5 term.
    Section 'Chiral Anomaly and Chiral Effects': 'at finite temperature, the massless limit is well-defined since the temperature acts as an IR regulator'.
  • domain assumption The vacuum contribution to the k=1 spin-connection integral vanishes when regularized in the BMHV scheme, as shown in Ref. [50].
    App. C: 'when using the BMHV scheme it is showed in [50] that the vacuum contribution cancels in (C2)'. This cancellation is required for Eq. (12).
  • ad hoc to paper The new finite non-topological anomaly terms are physical and cannot be removed by local polynomial counter-terms.
    The Discussion assumes, following Ref. [58], that these terms should not be canceled, but explicitly states that the counter-term analysis 'has not been studied here'.

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Cite this review

Pith. "Pith review of Examining the Anomalous Nature of Chiral Effects in Thermodynamics." pith.science (2026). https://pith.science/paper/S2VHIYVK

@misc{pith2026250702079,
  author       = {Pith},
  title        = {Pith review of: Examining the Anomalous Nature of Chiral Effects in Thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2VHIYVK}},
  note         = {Machine review of arXiv:2507.02079}
}
read the original abstract

Quantum anomalies give rise to novel transport phenomena, including the generation of a current in a relativistic fluid due to the presence of magnetic field or vorticity. We present an exclusive and direct computation of the chiral anomaly within the path integral for a massless fermion on a generic electromagnetic and curved background, including local temperature and chemical potential. We identify new thermodynamical contributions to the anomaly which induce the Chiral Separation and Vortical Effects. Additionally, we show that the anomaly fully vanishes at global equilibrium.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

71 extracted references · 47 canonical work pages

  1. [58]

    Non-topological anomalies and Wess-Zumino effective action,

    J. Balog, “Non-topological anomalies and Wess-Zumino effective action,” Nucl. Phys. B 258 (1985) 361–372

    work page 1985

  2. [1]

    A PCAC puzzle: π0 → γγ in the σ model,

    J. S. Bell and R. Jackiw, “A PCAC puzzle: π0 → γγ in the σ model,” Nuovo Cim. A 60 (1969) 47–61

    1969

  3. [2]

    Axial vector vertex in spinor electrodynamics,

    S. L. Adler, “Axial vector vertex in spinor electrodynamics,” Phys. Rev. 177 (1969) 2426–2438

    1969

  4. [3]

    Chiral magnetic effect,

    K. Fukushima, D. E. Kharzeev, and H. J. Warringa, “Chiral magnetic effect,” Phys. Rev. D 78 (Oct, 2008) 074033. https: //link.aps.org/doi/10.1103/PhysRevD.78.074033

    doi:10.1103/physrevd.78.074033 2008

  5. [4]

    Anomalous axion interactions and topological currents in dense matter,

    M. A. Metlitski and A. R. Zhitnitsky, “Anomalous axion interactions and topological currents in dense matter,” Phys. Rev. D 72 (2005) 045011, arXiv:hep-ph/0505072

    arXiv 2005

  6. [5]

    Hydrodynamics with triangle anomalies,

    D. T. Son and P. Surowka, “Hydrodynamics with triangle anomalies,”Physical Review Letters 103 (Nov.,

    work page

  7. [6]

    Testing the chiral magnetic and chiral vortical effects in heavy ion collisions

    D. E. Kharzeev and D. T. Son, “Testing the chiral magnetic and chiral vortical effects in heavy ion collisions,” Phys. Rev. Lett. 106 (2011) 062301, arXiv:1010.0038 [hep-ph]

    work page Pith review arXiv 2011

  8. [7]

    Chiral magnetic effect in zrte5,

    Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosi´ c, 7 Let us recall that our conditions of global equilibrium are suffi- cient conditions. Less restrictive conditions can also lead to global equilibrium [59]. A. Fedorov, R. Zhong, J. Schneeloch, G. Gu, and T. Valla, “Chiral magnetic effect in zrte5,” Nature Physics 12 (June, 2016) 550–554

    work page 2016

Show all 71 references
  1. [8]

    Chiral magnetic effect in heavy ion collisions: The present and future,

    D. E. Kharzeev, J. Liao, and P. Tribedy, “Chiral magnetic effect in heavy ion collisions: The present and future,”International Journal of Modern Physics E 33 (Sept., 2024) . http://dx.doi.org/10.1142/S0218301324300078

    2024 doi

  2. [9]

    Chiral effects in astrophysics and cosmology,

    K. Kamada, N. Yamamoto, and D.-L. Yang, “Chiral effects in astrophysics and cosmology,” Prog. Part. Nucl. Phys. 129 (2023) 104016, arXiv:2207.09184 [astro-ph.CO]

    2023 arXiv

  3. [10]

    A Theory of first order dissipative superfluid dynamics,

    J. Bhattacharya, S. Bhattacharyya, S. Minwalla, and A. Yarom, “A Theory of first order dissipative superfluid dynamics,” JHEP 05 (2014) 147, arXiv:1105.3733 [hep-th]

    2014 arXiv

  4. [11]

    Constraints on Fluid Dynamics from Equilibrium Partition Functions,

    N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla, and T. Sharma, “Constraints on Fluid Dynamics from Equilibrium Partition Functions,” JHEP 09 (2012) 046, arXiv:1203.3544 [hep-th]

    2012 arXiv

  5. [12]

    Kubo formulas for thermodynamic transport coefficients,

    P. Kovtun and A. Shukla, “Kubo formulas for thermodynamic transport coefficients,” JHEP 10 (2018) 007, arXiv:1806.05774 [hep-th]

    2018 arXiv

  6. [13]

    Relativistic Hydrodynamics with General Anomalous Charges,

    Y. Neiman and Y. Oz, “Relativistic Hydrodynamics with General Anomalous Charges,” JHEP 03 (2011) 023, arXiv:1011.5107 [hep-th]

    2011 arXiv

  7. [14]

    Anomalous transport coefficients from kubo formulas in holography,

    I. Amado, K. Landsteiner, and F. Pena-Benitez, “Anomalous transport coefficients from kubo formulas in holography,”Journal of High Energy Physics 2011 (May,

    2011

  8. [15]

    Gravitational Anomaly and Transport,

    K. Landsteiner, E. Megias, and F. Pena-Benitez, “Gravitational Anomaly and Transport,” Phys. Rev. Lett. 107 (2011) 021601, arXiv:1103.5006 [hep-ph]

    2011 arXiv

  9. [16]

    Thermal transport, geometry, and anomalies,

    M. N. Chernodub, Y. Ferreiros, A. G. Grushin, K. Landsteiner, and M. A. H. Vozmediano, “Thermal transport, geometry, and anomalies,” Phys. Rept. 977 (2022) 1–58, arXiv:2110.05471 [cond-mat.mes-hall]

    2022 arXiv

  10. [17]

    Collisions in chiral kinetic theory,

    J.-Y. Chen, D. T. Son, and M. A. Stephanov, “Collisions in chiral kinetic theory,”Physical Review Letters 115 (July, 2015) 021601

    2015

  11. [18]

    Chiral kinetic theory in curved spacetime,

    Y.-C. Liu, L.-L. Gao, K. Mameda, and X.-G. Huang, “Chiral kinetic theory in curved spacetime,” Phys. Rev. D 99 (2019) no. 8, 085014, arXiv:1812.10127 [hep-th]

    2019 arXiv

  12. [19]

    Mixed Anomalies: Chiral Vortical Effect and the Sommerfeld Expansion,

    M. Stone and J. Kim, “Mixed Anomalies: Chiral Vortical Effect and the Sommerfeld Expansion,” Phys. Rev. D 98 (2018) no. 2, 025012, arXiv:1804.08668 [cond-mat.mes-hall]

    2018 arXiv

  13. [20]

    Anomalous transport from geometry,

    K. Landsteiner, S. Morales-Tejera, and P. Saura-Bastida, “Anomalous transport from geometry,”Physical Review D 107 (June, 2023) 125003

    2023

  14. [21]

    Hydrodynamic Manifestations of Gravitational Chiral Anomaly,

    G. Y. Prokhorov, O. V. Teryaev, and V. I. Zakharov, “Hydrodynamic Manifestations of Gravitational Chiral Anomaly,” Phys. Rev. Lett. 129 (2022) no. 15, 151601, arXiv:2207.04449 [hep-th]

    2022 arXiv

  15. [22]

    Chiral vortical effect with finite rotation, temperature, and curvature,

    A. Flachi and K. Fukushima, “Chiral vortical effect with finite rotation, temperature, and curvature,” Physical Review D 98 (Nov., 2018) . http://dx.doi.org/10.1103/PhysRevD.98.096011. 6

    2018 doi

  16. [23]

    The Axial Anomaly at Finite Temperature,

    H. Itoyama and A. H. Mueller, “The Axial Anomaly at Finite Temperature,” Nucl. Phys. B 218 (1983) 349–365

    1983

  17. [24]

    Chiral jacobians and two-dimensional qed at finite temperature,

    M. Reuter and W. Dittrich, “Chiral jacobians and two-dimensional qed at finite temperature,” Physical Review D 32 (July, 1985) 513–515

    1985

  18. [25]

    Axial anomaly at finite temperature,

    S. Chaturvedi, N. Gupte, and V. Srinivasan, “Axial anomaly at finite temperature,” Journal of Physics A: Mathematical and General 18 (Oct., 1985) L963–L967

    1985

  19. [26]

    The Derivative Expansion and the Anomaly at Finite Temperature,

    A. K. Das and A. Karev, “The Derivative Expansion and the Anomaly at Finite Temperature,” Phys. Rev. D 36 (1987) 623

    1987

  20. [27]

    Path integral approach to anomalies at finite temperature,

    T. F. Treml, “Path integral approach to anomalies at finite temperature,” Can. J. Phys. 68 (1990) 96–103

    1990

  21. [28]

    Anomalies in curved spacetime at finite temperature,

    H. Boschi-Filho and C. P. Natividade, “Anomalies in curved spacetime at finite temperature,” Physical Review D 46 (Dec., 1992) 5458–5466

    1992

  22. [29]

    Gravitational chiral anomaly at finite temperature and density,

    C. Corian` o, M. Cret ` ı, S. Lionetti, and R. Tommasi, “Gravitational chiral anomaly at finite temperature and density,”Physical Review D 110 (July, 2024) 025008

    2024

  23. [30]

    Chiral and parity anomalies at finite temperature and density,

    A. N. Sisakian, O. Y. Shevchenko, and S. B. Solganik, “Chiral and parity anomalies at finite temperature and density,” Nucl. Phys. B 518 (1998) 455–472, arXiv:hep-th/9710022

    1998 arXiv

  24. [31]

    Landau and E

    L. Landau and E. Lifshitz, Course of Theoretical Physics, Volume 6: Fluid Mechanics . 1987

    1987

  25. [32]

    Le Bellac, Thermal Field Theory

    M. Le Bellac, Thermal Field Theory . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1996

    1996

  26. [33]

    Newman and N

    J. Newman and N. P. Balsara, Electrochemical Systems. Wiley, 2021

    2021

  27. [34]

    Covariant Calculations at Finite Temperature: The Relativistic Plasma,

    H. A. Weldon, “Covariant Calculations at Finite Temperature: The Relativistic Plasma,” Phys. Rev. D 26 (1982) 1394

    1982

  28. [35]

    Covariant statistical mechanics and the stress-energy tensor,

    F. Becattini, “Covariant statistical mechanics and the stress-energy tensor,” Phys. Rev. Lett. 108 (2012) 244502, arXiv:1201.5278 [gr-qc]

    2012 arXiv

  29. [36]

    Path-integral formula for local thermal equilibrium,

    M. Hongo, “Path-integral formula for local thermal equilibrium,”Annals of Physics 383 (Aug., 2017) 1–32

    2017

  30. [37]

    Energy Momentum Tensor in Quantum Field Theory,

    K. Fujikawa, “Energy Momentum Tensor in Quantum Field Theory,” Phys. Rev. D 23 (1981) 2262

    1981

  31. [38]

    The Functional Measure for Quantum Field Theory in Curved Space-time,

    D. J. Toms, “The Functional Measure for Quantum Field Theory in Curved Space-time,” Phys. Rev. D 35 (1987) 3796

    1987

  32. [39]

    Fujikawa and H

    K. Fujikawa and H. Suzuki, Path integrals and quantum anomalies. 8, 2004

    2004

  33. [40]

    Transient relativistic thermodynamics and kinetic theory,

    W. Israel and J. M. Stewart, “Transient relativistic thermodynamics and kinetic theory,” Annals Phys. 118 (1979) 341–372

    1979

  34. [41]

    Thermodynamical equilibrium and spacetime geometry,

    T. Chrobok and H. H. von Borzeszkowski, “Thermodynamical equilibrium and spacetime geometry,” Gen. Rel. Grav. 38 (2006) 397–415

    2006

  35. [42]

    On the Weight of Heat and Thermal Equilibrium in General Relativity,

    R. C. Tolman, “On the Weight of Heat and Thermal Equilibrium in General Relativity,” Phys. Rev. 35 (1930) 904–924

    1930

  36. [43]

    Temperature Equilibrium in a Static Gravitational Field,

    R. Tolman and P. Ehrenfest, “Temperature Equilibrium in a Static Gravitational Field,” Phys. Rev. 36 (1930) no. 12, 1791–1798

    1930

  37. [44]

    Theory of Thermal Transport Coefficients,

    J. M. Luttinger, “Theory of Thermal Transport Coefficients,” Phys. Rev. 135 (1964) A1505–A1514

    1964

  38. [45]

    Perturbative confinement in thermal yang-mills theories induced by imaginary angular velocity,

    S. Chen, K. Fukushima, and Y. Shimada, “Perturbative confinement in thermal yang-mills theories induced by imaginary angular velocity,”Phys. Rev. Lett. 129 (Dec,

  39. [46]

    Inhomogeneous confinement and chiral symmetry breaking induced by imaginary angular velocity,

    S. Chen, K. Fukushima, and Y. Shimada, “Inhomogeneous confinement and chiral symmetry breaking induced by imaginary angular velocity,” Phys. Lett. B 859 (2024) 139107, arXiv:2404.00965 [hep-ph]

    2024 arXiv

  40. [47]

    Anomalous Transport from Kubo Formulae,

    K. Landsteiner, E. Megias, and F. Pena-Benitez, “Anomalous Transport from Kubo Formulae,” Lect. Notes Phys. 871 (2013) 433–468, arXiv:1207.5808 [hep-th]

    2013 arXiv

  41. [48]

    Path Integral Measure for Gauge Invariant Fermion Theories,

    K. Fujikawa, “Path Integral Measure for Gauge Invariant Fermion Theories,” Phys. Rev. Lett. 42 (1979) 1195–1198

    1979

  42. [49]

    Anomalies from an effective field theory perspective,

    B. Filoche, R. Larue, J. Quevillon, and P. N. H. Vuong, “Anomalies from an effective field theory perspective,” Phys. Rev. D 107 (2023) no. 2, 025017, arXiv:2205.02248 [hep-th]

    2023 arXiv

  43. [50]

    The universal one-loop effective action with gravity,

    R. Larue and J. Quevillon, “The universal one-loop effective action with gravity,” JHEP 11 (2023) 045, arXiv:2303.10203 [hep-th]

    2023 arXiv

  44. [51]

    Larue, A

    R. Larue, A. Marchon, J. Quevillon, and D. Saviot. Work in preparation

  45. [52]

    Regularization and Renormalization of Gauge Fields,

    G. ’t Hooft and M. Veltman, “Regularization and Renormalization of Gauge Fields,” Nucl. Phys. B 44 (1972) 189–213

    1972

  46. [53]

    Dimensional Renormalization and the Action Principle,

    P. Breitenlohner and D. Maison, “Dimensional Renormalization and the Action Principle,” Commun. Math. Phys. 52 (1977) 11–38

    1977

  47. [54]

    ON THE DETERMINANT OF THE WEYL OPERATOR,

    H. Leutwyler, “ON THE DETERMINANT OF THE WEYL OPERATOR,”

  48. [55]

    GRA VITATIONAL ANOMALIES,

    H. Leutwyler and S. Mallik, “GRA VITATIONAL ANOMALIES,” Z. Phys. C 33 (1986) 205

    1986

  49. [56]

    CHIRAL FERMION DETERMINANTS AND THEIR ANOMALIES,

    H. Leutwyler, “CHIRAL FERMION DETERMINANTS AND THEIR ANOMALIES,” Phys. Lett. B 152 (1985) 78–82

    1985

  50. [57]

    Consistent and Covariant Anomalies in Gauge and Gravitational Theories,

    W. A. Bardeen and B. Zumino, “Consistent and Covariant Anomalies in Gauge and Gravitational Theories,” Nucl. Phys. B 244 (1984) 421–453

    1984

  51. [59]

    Thermodynamic Equilibrium in General Relativity,

    J. A. S. Lima, A. Del Popolo, and A. R. Plastino, “Thermodynamic Equilibrium in General Relativity,” Phys. Rev. D 100 (2019) no. 10, 104042, arXiv:1911.09060 [gr-qc]

    2019 arXiv

  52. [60]

    Generalized Bloch theorem and chiral transport phenomena,

    N. Yamamoto, “Generalized Bloch theorem and chiral transport phenomena,” Phys. Rev. D 92 (2015) no. 8, 085011, arXiv:1502.01547 [cond-mat.mes-hall]

    2015 arXiv

  53. [61]

    Absence of equilibrium chiral magnetic effect,

    M. A. Zubkov, “Absence of equilibrium chiral magnetic effect,” Phys. Rev. D 93 (2016) no. 10, 105036, arXiv:1605.08724 [hep-ph]

    2016 arXiv

  54. [62]

    The Effective One Loop Lagrangian With Derivative Couplings,

    M. K. Gaillard, “The Effective One Loop Lagrangian With Derivative Couplings,” Nucl. Phys. B 268 (1986) 669–692

    1986

  55. [63]

    Effective Action for the Standard Model With Large Higgs Mass,

    O. Cheyette, “Effective Action for the Standard Model With Large Higgs Mass,” Nucl. Phys. B 297 (1988) 183–204. 7

    1988

  56. [64]

    The Leading Divergent Part of the Effective Action for the Nonlinear σ Model in n-dimensions,

    P. Binetruy and M. K. Gaillard, “The Leading Divergent Part of the Effective Action for the Nonlinear σ Model in n-dimensions,” Nucl. Phys. B 312 (1989) 341–401

    1989

  57. [65]

    On the effective action for scalars in a general manifold to any loop order,

    R. Alonso and M. West, “On the effective action for scalars in a general manifold to any loop order,” arXiv:2207.02050 [hep-th]

    arXiv

  58. [66]

    Dimensional regularization and mellin summation in high-temperature calculations,

    D. J. Bedingham, “Dimensional regularization and mellin summation in high-temperature calculations,” in Strong and Electroweak Matter 2000 , p. 226–230. July, 2001. http://arxiv.org/abs/hep-ph/0011012. arXiv:hep-ph/0011012. 8 In this Supplementary Material, we present details ...

    2000 arXiv

  59. [70]

    Matsubara sums The Matsubara sums are defined as Sn = 1 β X k∈Z 1 ( ˜Ω2 k + E2r )n ˜Ωk = (2k + 1)πT (x) − iµec(x) = Ωk/√gtt , (E5) with T (x) = 1/ p β2 = T0/√gtt. They can be computed using Sn+1 = (−1)n n!E2nr dn dzn S1(z) z=1 , where S1(z) = S1|Er→√zEr , S 1 = 1 − n(Er + µec)...

  60. [71]

    This integral is UV divergent and we compute in dimensional regualrisation by taking the spatial dimension to be d = 3 − ϵ

    (I 4[q4])αβγδ integral To keep the computation tractable, we compute this integral in flat spacetime. This integral is UV divergent and we compute in dimensional regualrisation by taking the spatial dimension to be d = 3 − ϵ. In the massless limit we find (I 4[q4])αβγδ = 1 β X...

  61. [2009]

    arXiv:0906.5044 [hep-ph, physics:hep-th, physics:nucl-th]

    191601. arXiv:0906.5044 [hep-ph, physics:hep-th, physics:nucl-th]

    arXiv

  62. [2011]

    arXiv:1102.4577 [hep-ph, physics:hep-th, physics:nucl-th]

    81. arXiv:1102.4577 [hep-ph, physics:hep-th, physics:nucl-th]

    arXiv

  63. [2022]

    https://link.aps.org/doi/10.1103/ PhysRevLett.129.242002

    242002. https://link.aps.org/doi/10.1103/ PhysRevLett.129.242002

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Reviewed August 6, 2026 · model on record in the stance chip above.