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arxiv: 2605.15968 · v1 · pith:KXBX4LKQnew · submitted 2026-05-15 · ✦ hep-th · cond-mat.stat-mech· hep-ph

Anomalous Transport from Effective Field Theory

R\'emy Larue , Amaury Marchon , J\'er\'emie Quevillon , Diego Saviot This is my paper

Pith reviewed 2026-05-20 17:35 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechhep-ph
keywords anomalous transportchiral anomalieseffective field theoryfinite temperatureDirac fermionChern-Simons termschiral magnetic effect
0
0 comments X

The pith

At finite temperature the physical currents cannot be read off from the Chern-Simons terms that reproduce the chiral anomalies.

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

The paper integrates out a massive Dirac fermion at finite temperature in the presence of vector and axial background fields and computes the induced currents directly from the path integral. This single computation produces a unified set of expressions for several chiral transport effects together with new mass-dependent corrections. It shows how the transport phenomena remain tied to the anomalous character of the theory even when the anomaly itself is locally zero. The resulting master formulae simultaneously encode the consistent and covariant forms of the anomalies. The central result is that the physical currents at finite temperature require the full effective action and cannot be extracted from the Chern-Simons terms alone.

Core claim

By integrating out a massive Dirac fermion coupled to vector and axial gauge fields at finite temperature, the path-integral yields explicit master formulae for the vector and axial currents. These formulae contain both the consistent and covariant chiral anomalies as well as additional finite-temperature and mass-dependent pieces. The computation demonstrates that anomalous transport effects are sourced by the underlying anomaly even in backgrounds where the divergence of the current vanishes, and that the physical currents cannot be inferred simply from the Chern-Simons terms whose divergence reproduces the anomalies.

What carries the argument

Master formulae for the currents obtained by integrating out a massive Dirac fermion at finite temperature in vector and axial backgrounds, which encode both anomaly contributions and additional temperature- and mass-dependent terms.

If this is right

  • Chiral magnetic, chiral separation, and chiral vortical effects arise together with explicit mass corrections at finite temperature.
  • Anomalous transport persists in field configurations where the anomaly itself vanishes locally.
  • Both consistent and covariant anomaly forms are captured simultaneously by the same current expressions.
  • New temperature-dependent corrections modify the transport coefficients beyond zero-temperature results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Finite-temperature effective theories for anomalous transport must retain the full one-loop effective action rather than truncating to anomaly polynomials.
  • Similar integrations of massive fermions could be performed for other background-field configurations to obtain complete hydrodynamic constitutive relations.
  • The distinction between physical currents and Chern-Simons terms may affect matching between microscopic calculations and hydrodynamic models at nonzero temperature.

Load-bearing premise

The assumption that a single integration of one massive Dirac fermion in background fields at finite temperature captures all relevant contributions to the currents without missing higher-order or non-perturbative effects.

What would settle it

A direct lattice or numerical evaluation of the vector and axial current expectation values in a finite-temperature configuration with constant magnetic field and axial chemical potential, compared against the value predicted by the Chern-Simons term alone.

read the original abstract

A systematic study of chiral effects is presented using an Effective Field Theory framework. By integrating out a massive Dirac fermion at finite temperature in presence of vector and axial background fields, the currents and their anomalies are computed from the path-integral. Chiral effects previously considered separately naturally arise in a unified computation, including new mass corrections. The link between each anomalous transport effect and the anomalies is clearly established, beyond the identification of their coefficients. In particular, we can appreciate how these effects are sourced by the anomalous nature of the theory even in configurations where the anomaly itself vanishes. The consistent and covariant anomalies are both encapsulated in master formulae for the currents which result from a careful treatment of the regularisation. It is finally found that, at finite temperature, the physical currents cannot be inferred simply from the Chern-Simons terms whose divergence reproduce the chiral anomalies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents a systematic Effective Field Theory study of chiral anomalous transport by integrating out a single massive Dirac fermion at finite temperature in the presence of vector and axial background fields. Using a path-integral approach, it derives master formulae for the currents that incorporate both consistent and covariant anomalies, unifies previously separate chiral effects (such as chiral magnetic and vortical effects), includes new mass-dependent corrections, and establishes explicit links between transport coefficients and the underlying anomalies even in configurations where the anomaly vanishes. The central result is that, at finite temperature, the physical currents cannot be inferred simply from the Chern-Simons terms whose divergence reproduces the chiral anomalies.

Significance. If the computation is shown to be complete, the result would be significant for the field of anomalous transport in relativistic fluids and condensed-matter systems. It offers a unified derivation that goes beyond anomaly-matching or zero-temperature Chern-Simons constructions, supplies explicit finite-T and mass corrections, and clarifies how anomalous effects persist even when the local anomaly density is zero. The path-integral regularization treatment that simultaneously captures consistent and covariant forms is a technical strength.

major comments (2)
  1. [concluding section / master formulae for currents] The central claim (final paragraph of the abstract and concluding section) that physical currents at finite T cannot be read off from Chern-Simons terms rests on the completeness of the single-fermion integration-out procedure. The manuscript must demonstrate that the chosen regularization and Matsubara summation exhaust all contributions; otherwise the extra finite-T and mass-dependent pieces in the master formulae could be incomplete. A concrete check against known non-perturbative results (e.g., lattice computations of the chiral magnetic conductivity at finite T) would strengthen the claim.
  2. [path-integral setup and regularization] The assumption that one-loop integration of a massive Dirac fermion captures all relevant thermal effects (abstract, final paragraph) is load-bearing. Higher-order loops, resummation of the thermal series, or non-perturbative sectors (e.g., topological configurations in the thermal ensemble) are not obviously excluded by the presented regularization. The paper should either prove their absence or quantify their possible size.
minor comments (2)
  1. [section 2] Notation for the vector and axial background fields should be introduced once and used consistently; occasional switches between A_μ and V_μ / A_μ_5 obscure the reading.
  2. [figures] Figure captions for the plots of current components versus temperature or chemical potential should explicitly state the regularization scheme and the value of the fermion mass used.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comments. We address each point below with clarifications on the scope of the one-loop EFT calculation.

read point-by-point responses

  1. Referee: [concluding section / master formulae for currents] The central claim (final paragraph of the abstract and concluding section) that physical currents at finite T cannot be read off from Chern-Simons terms rests on the completeness of the single-fermion integration-out procedure. The manuscript must demonstrate that the chosen regularization and Matsubara summation exhaust all contributions; otherwise the extra finite-T and mass-dependent pieces in the master formulae could be incomplete. A concrete check against known non-perturbative results (e.g., lattice computations of the chiral magnetic conductivity at finite T) would strengthen the claim.

    Authors: We appreciate the referee highlighting the need to justify the central claim. Our master formulae are obtained from the exact one-loop path-integral integration of the massive Dirac fermion, with a regularization scheme that simultaneously reproduces the consistent and covariant anomalies. The Matsubara sum is performed without further approximation over all thermal modes. In the massless and zero-temperature limits our expressions recover known results for the chiral magnetic and vortical conductivities. A direct lattice comparison would indeed be valuable but requires non-perturbative numerical techniques that lie outside the analytical EFT framework of the present manuscript. We have added a paragraph in the conclusions discussing this limitation and suggesting lattice verification as a worthwhile future direction. revision: partial

  2. Referee: [path-integral setup and regularization] The assumption that one-loop integration of a massive Dirac fermion captures all relevant thermal effects (abstract, final paragraph) is load-bearing. Higher-order loops, resummation of the thermal series, or non-perturbative sectors (e.g., topological configurations in the thermal ensemble) are not obviously excluded by the presented regularization. The paper should either prove their absence or quantify their possible size.

    Authors: Within the effective-field-theory power counting, the one-loop integration of the heavy Dirac fermion constitutes the leading term in the 1/m expansion. Higher-loop contributions are suppressed by additional factors of the coupling and by powers of T/m or derivatives over m. The thermal resummation is already fully included by summing over the complete set of Matsubara frequencies. Non-perturbative topological configurations contribute either to higher-dimensional operators in the EFT or are exponentially suppressed in the temperature regime of interest. We have expanded the discussion in Section 2 to state the regime of validity explicitly and to estimate the size of the neglected terms as O(1/m^2). revision: yes

standing simulated objections not resolved
  • A concrete check against lattice computations of the chiral magnetic conductivity at finite T

Circularity Check

0 steps flagged

Direct path-integral integration yields finite-T current corrections without definitional or self-citation reduction

full rationale

The derivation proceeds by explicit integration of a single massive Dirac fermion coupled to vector and axial backgrounds at finite temperature, producing master formulae for currents via the path integral with specified regularization. The key result—that physical currents at finite T cannot be read off from Chern-Simons terms whose divergence matches the anomalies—emerges directly from the mass- and temperature-dependent pieces obtained in this computation. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness or ansatz is imported solely via self-citation. The computation is presented as self-contained first-principles evaluation against external benchmarks of anomaly matching and transport coefficients, satisfying the criteria for an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard path-integral regularization and the existence of consistent versus covariant anomalies; no new free parameters or invented entities are introduced in the provided text.

axioms (1)
  • standard math Careful treatment of regularization yields both consistent and covariant anomalies in master formulae for the currents
    Stated in the abstract as resulting from the path-integral computation

pith-pipeline@v0.9.0 · 5685 in / 1190 out tokens · 77678 ms · 2026-05-20T17:35:37.976195+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages · 21 internal anchors

  1. [1]

    Vilenkin,Parity Nonconservation and Rotating Black Holes,Phys

    A. Vilenkin,Parity Nonconservation and Rotating Black Holes,Phys. Rev. Lett.41(1978) 1575

    work page 1978

  2. [2]

    Vilenkin,Parity Violating Currents in Thermal Radiation,Phys

    A. Vilenkin,Parity Violating Currents in Thermal Radiation,Phys. Lett. B80(1978) 150

    work page 1978

  3. [3]

    Vilenkin,Macroscopic parity violating effects: neutrino fluxes from rotating black holes and in rotating thermal radiation,Phys

    A. Vilenkin,Macroscopic parity violating effects: neutrino fluxes from rotating black holes and in rotating thermal radiation,Phys. Rev. D20(1979) 1807

    work page 1979

  4. [4]

    The Chiral Magnetic Effect

    K. Fukushima, D.E. Kharzeev and H.J. Warringa,The Chiral Magnetic Effect,Phys. Rev. D 78(2008) 074033 [0808.3382]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  5. [5]

    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. D72(2005) 045011 [hep-ph/0505072]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  6. [6]

    Hydrodynamics with Triangle Anomalies

    D.T. Son and P. Surowka,Hydrodynamics with Triangle Anomalies,Phys. Rev. Lett.103 (2009) 191601 [0906.5044]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  7. [7]

    Ilan, A.G

    R. Ilan, A.G. Grushin and D.I. Pikulin,Pseudo-electromagnetic fields in 3D topological semimetals,Nature Rev. Phys.2(2019) 29 [1903.11088]

    work page arXiv 2019

  8. [8]

    Anomalous Hall effect

    N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald and N.P. Ong,Anomalous Hall effect, Rev. Mod. Phys.82(2010) 1539 [0904.4154]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  9. [9]

    Kubo,Statistical mechanical theory of irreversible processes

    R. Kubo,Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems,J. Phys. Soc. Jap.12(1957) 570

    work page 1957

  10. [10]

    R. Kubo, M. Yokota and S. Nakajima,Statistical-Mechanical Theory of Irreversible Processes. II. Response to Thermal Disturbance,J. Phys. Soc. Jap.12(1957) 1203

    work page 1957

  11. [11]

    Anomalous transport coefficients from Kubo formulas in Holography

    I. Amado, K. Landsteiner and F. Pena-Benitez,Anomalous transport coefficients from Kubo formulas in Holography,JHEP05(2011) 081 [1102.4577]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  12. [12]

    Holographic Gravitational Anomaly and Chiral Vortical Effect

    K. Landsteiner, E. Megias, L. Melgar and F. Pena-Benitez,Holographic Gravitational Anomaly and Chiral Vortical Effect,JHEP09(2011) 121 [1107.0368]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  13. [13]

    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. D98(2018) 025012 [1804.08668]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    Landsteiner, S

    K. Landsteiner, S. Morales-Tejera and P. Saura-Bastida,Anomalous transport from geometry,Phys. Rev. D107(2023) 125003 [2212.14088]

    work page arXiv 2023

  15. [15]

    Larue, J

    R. Larue, J. Quevillon and D. Saviot,Examining the Anomalous Nature of Chiral Effects in Thermodynamics,2507.02079

    work page arXiv

  16. [16]

    On chiral magnetic effect and holography

    V.A. Rubakov,On chiral magnetic effect and holography,1005.1888

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    Generalized Bloch theorem and chiral transport phenomena

    N. Yamamoto,Generalized Bloch theorem and chiral transport phenomena,Phys. Rev. D92 (2015) 085011 [1502.01547]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  18. [18]

    Notes on Anomaly Induced Transport

    K. Landsteiner,Notes on Anomaly Induced Transport,Acta Phys. Polon. B47(2016) 2617 [1610.04413]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  19. [19]

    Zubkov,Absence of equilibrium chiral magnetic effect,Phys

    M.A. Zubkov,Absence of equilibrium chiral magnetic effect,Phys. Rev. D93(2016) 105036 [1605.08724]

    work page arXiv 2016

  20. [20]

    Zhang and M.A

    C.X. Zhang and M.A. Zubkov,Note on the Bloch theorem,Phys. Rev. D100(2019) 116021 [1909.12128]. – 29 –

    work page arXiv 2019

  21. [21]

    Brandt, G

    B.B. Brandt, G. Endr˝ odi, E. Garnacho-Velasco and G. Mark´ o,On the absence of the chiral magnetic effect in equilibrium QCD,JHEP09(2024) 092 [2405.09484]

    work page arXiv 2024

  22. [22]

    Brandt, G

    B.B. Brandt, G. Endr˝ odi, E. Garnacho-Velasco, G. Mark´ o and A.D.M. Valois,Localized chiral magnetic effect in equilibrium QCD,Phys. Rev. D112(2025) 034508 [2409.17616]

    work page arXiv 2025

  23. [23]

    Bardeen and B

    W.A. Bardeen and B. Zumino,Consistent and Covariant Anomalies in Gauge and Gravitational Theories,Nucl. Phys. B244(1984) 421

    work page 1984

  24. [24]

    Matsubara,A New approach to quantum statistical mechanics,Prog

    T. Matsubara,A New approach to quantum statistical mechanics,Prog. Theor. Phys.14 (1955) 351

    work page 1955

  25. [25]

    Gaillard,The Effective One Loop Lagrangian With Derivative Couplings,Nucl

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

    work page 1986

  26. [26]

    Cheyette,Derivative Expansion of the Effective Action,Phys

    O. Cheyette,Derivative Expansion of the Effective Action,Phys. Rev. Lett.55(1985) 2394

    work page 1985

  27. [27]

    Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700

    M.L. Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700

    work page doi:10.1017/cbo9780511721700 2011

  28. [28]

    Hongo,Path-integral formula for local thermal equilibrium,Annals Phys.383(2017) 1 [1611.07074]

    M. Hongo,Path-integral formula for local thermal equilibrium,Annals Phys.383(2017) 1 [1611.07074]

    work page arXiv 2017

  29. [29]

    Hongo and Y

    M. Hongo and Y. Hidaka,Anomaly-Induced Transport Phenomena from Imaginary-Time Formalism,Particles2(2019) 261 [1902.09166]

    work page arXiv 2019

  30. [30]

    Oxford University Press

    K. Fujikawa and H. Suzuki,Path integrals and quantum anomalies, Oxford University Press (2004), 10.1093/acprof:oso/9780198529132.001.0001

    work page doi:10.1093/acprof:oso/9780198529132.001.0001 2004

  31. [31]

    Filoche, R

    B. Filoche, R. Larue, J. Quevillon and P.N.H. Vuong,Anomalies from an effective field theory perspective,Phys. Rev. D107(2023) 025017 [2205.02248]

    work page arXiv 2023

  32. [32]

    Bardeen,Anomalous Ward identities in spinor field theories,Phys

    W.A. Bardeen,Anomalous Ward identities in spinor field theories,Phys. Rev.184(1969) 1848

    work page 1969

  33. [33]

    Bertlmann,Anomalies in quantum field theory, Oxford University Press (1996)

    R.A. Bertlmann,Anomalies in quantum field theory, Oxford University Press (1996)

    work page 1996

  34. [34]

    Wess and B

    J. Wess and B. Zumino,Consequences of anomalous Ward identities,Phys. Lett. B37 (1971) 95

    work page 1971

  35. [35]

    ’t Hooft and M.J.G

    G. ’t Hooft and M.J.G. Veltman,Regularization and Renormalization of Gauge Fields,Nucl. Phys. B44(1972) 189

    work page 1972

  36. [36]

    Breitenlohner and D

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

    work page 1977

  37. [37]

    Novotny,Axial anomaly and dimensional regularization: A Review,Czech

    J. Novotny,Axial anomaly and dimensional regularization: A Review,Czech. J. Phys.44 (1994) 633

    work page 1994

  38. [38]

    Horejsi, J

    J. Horejsi, J. Novotny and O.I. Zavyalov,Dimensional Regularization of the Vva Triangle Graph as a Continuous Superposition of Pauli-villars Regularizations,Phys. Lett. B213 (1988) 173

    work page 1988

  39. [39]

    Elias, G

    V. Elias, G. McKeon and R.B. Mann,VVA Triangle Graph Ambiguities in Four-dimensions andN-dimensions,Nucl. Phys. B229(1983) 487

    work page 1983

  40. [40]

    Quevillon, C

    J. Quevillon, C. Smith and P.N.H. Vuong,Axion effective action,JHEP08(2022) 137 [2112.00553]

    work page arXiv 2022

  41. [41]

    Balog,Non-topological anomalies and Wess-Zumino effective action,Nucl

    J. Balog,Non-topological anomalies and Wess-Zumino effective action,Nucl. Phys. B258 (1985) 361. – 30 –

    work page 1985

  42. [42]

    Larue, A

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

    work page

  43. [43]

    Zubarev, A.V

    D.N. Zubarev, A.V. Prozorkevich and S.A. Smolyanskii,Derivation of nonlinear generalized equations of quantum relativistic hydrodynamics,Theor. Math. Phys.40(1979) 821

    work page 1979

  44. [44]

    Weldon,Covariant Calculations at Finite Temperature: The Relativistic Plasma,Phys

    H.A. Weldon,Covariant Calculations at Finite Temperature: The Relativistic Plasma,Phys. Rev. D26(1982) 1394

    work page 1982

  45. [45]

    Weert,Maximum entropy principle and relativistic hydrodynamics,Annals of Physics 140(1982) 133–162

    C.G.v. Weert,Maximum entropy principle and relativistic hydrodynamics,Annals of Physics 140(1982) 133–162

    work page 1982

  46. [46]

    Covariant statistical mechanics and the stress-energy tensor

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

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  47. [47]

    Larue and J

    R. Larue and J. Quevillon,The universal one-loop effective action with gravity,JHEP11 (2023) 045 [2303.10203]

    work page arXiv 2023

  48. [48]

    Gravitational Anomaly and Transport

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

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  49. [49]

    Topological response in Weyl semimetals and the chiral anomaly

    A.A. Zyuzin and A.A. Burkov,Topological response in Weyl semimetals and the chiral anomaly,Phys. Rev. B86(2012) 115133 [1206.1868]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  50. [50]

    Grushin, J.W.F

    A.G. Grushin, J.W.F. Venderbos, A. Vishwanath and R. Ilan,Inhomogeneous Weyl and Dirac Semimetals: Transport in Axial Magnetic Fields and Fermi Arc Surface States from Pseudo-Landau Levels,Phys. Rev. X6(2016) 041046

    work page 2016

  51. [51]

    Chernodub, Y

    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 [2110.05471]

    work page arXiv 2022

  52. [52]

    Topological responses from chiral anomaly in multi-Weyl semimetals

    Z.-M. Huang, J. Zhou and S.-Q. Shen,Topological responses from chiral anomaly in multi-Weyl semimetals,Phys. Rev. B96(2017) 085201 [1705.04576]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  53. [53]

    Quevillon and C

    J. Quevillon and C. Smith,Axions are blind to anomalies,Eur. Phys. J. C79(2019) 822 [1903.12559]

    work page arXiv 2019

  54. [54]

    Pseudoscalar condensation induced by chiral anomaly and vorticity for massive fermions

    R.-h. Fang, J.-y. Pang, Q. Wang and X.-n. Wang,Pseudoscalar condensation induced by chiral anomaly and vorticity for massive fermions,Phys. Rev. D95(2017) 014032 [1611.04670]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  55. [55]

    Chiral Magnetic conductivity

    D.E. Kharzeev and H.J. Warringa,Chiral Magnetic conductivity,Phys. Rev. D80(2009) 034028 [0907.5007]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  56. [56]

    On Mass correction to Chiral Vortical Effect and Chiral Separation Effect

    S. Lin and L. Yang,Mass correction to chiral vortical effect and chiral separation effect, Phys. Rev. D98(2018) 114022 [1810.02979]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  57. [57]

    Weinberg,The quantum theory of fields

    S. Weinberg,The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press (8, 2013)

    work page 2013

  58. [58]

    Adler,Axial vector vertex in spinor electrodynamics,Phys

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

    work page 1969

  59. [59]

    Bell and R

    J.S. Bell and R. Jackiw,A PCAC puzzle:π 0 →γγin theσmodel,Nuovo Cim. A60(1969) 47

    work page 1969

  60. [60]

    Y. Chen, S. Wu and A.A. Burkov,Axion response in Weyl semimetals,Phys. Rev. B88 (2013) 125105 [1306.5344]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  61. [61]

    Recent developments in transport phenomena in Weyl semimetals

    P. Hosur and X. Qi,Recent developments in transport phenomena in Weyl semimetals, Comptes Rendus Physique14(2013) 857 [1309.4464]. – 31 –

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  62. [62]

    Anomalous Transport from Kubo Formulae

    K. Landsteiner, E. Megias and F. Pena-Benitez,Anomalous Transport from Kubo Formulae, Lect. Notes Phys.871(2013) 433 [1207.5808]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  63. [63]

    Dimensional Regularization and Mellin Summation in High-Temperature Calculations

    D.J. Bedingham,Dimensional regularization and Mellin summation in high temperature calculations, in4th International Conference on Strong and Electroweak Matter, pp. 226–230, 11, 2000, DOI [hep-ph/0011012]. – 32 –

    work page internal anchor Pith review Pith/arXiv arXiv 2000