Scattering Amplitudes and Resonant Processes in QED with Chiral Chemical Potential and Chiral Magnetic Conductivity

Jonathan D. Kroth; Kirill Tuchin

arxiv: 2606.28118 · v1 · pith:CIUJM3UVnew · submitted 2026-06-26 · ✦ hep-ph · hep-th

Scattering Amplitudes and Resonant Processes in QED with Chiral Chemical Potential and Chiral Magnetic Conductivity

Jonathan D. Kroth , Kirill Tuchin This is my paper

Pith reviewed 2026-06-29 03:35 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords QED scattering amplitudeschiral chemical potentialchiral magnetic conductivityresonant processes1 to 2 processesquasi-stationary states

0 comments

The pith

QED scattering amplitudes in a chiral medium with constant μ5 and b0 exhibit resonant behavior in 1→2, 2→2, and 2→3 processes.

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

The paper analyzes the QED scattering amplitude in a medium with constant chiral chemical potential μ5 and chiral magnetic conductivity b0. It shows that resonant behavior emerges in one-to-two, two-to-two, and two-to-three particle processes. The rates of the basic 1→2 processes are computed to determine the widths of quasi-stationary fermion and photon states. A reader would care because these resonances arise from the medium properties and require specific regularization. The work elucidates the origin and conditions for these resonances.

Core claim

The QED scattering amplitude in a chiral medium characterized by constant μ5 and b0 develops resonances in 1→2, 2→2, and 2→3 processes. The rates of paradigm 1→2 processes determine the widths of quasi-stationary states, and the origin of resonances along with their regularization is explained.

What carries the argument

The modified QED scattering amplitude incorporating the effects of constant chiral chemical potential μ5 and chiral magnetic conductivity b0, which introduces resonant poles in the processes.

If this is right

The widths of quasi-stationary fermion and photon states are set by the computed 1→2 rates.
Resonant behavior appears in 1→2, 2→2, and 2→3 scattering channels under the medium conditions.
The resonances are regularized according to physical principles in the chiral medium.

Where Pith is reading between the lines

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

This framework could be extended to calculate cross sections in chiral plasmas relevant to heavy-ion collisions.
Similar resonances might appear in other gauge theories with chiral imbalances.
The regularization mechanism may connect to damping effects in real media.

Load-bearing premise

The analysis assumes the chiral medium has a constant chiral chemical potential μ5 and constant chiral magnetic conductivity b0.

What would settle it

An experimental observation of fermion or photon decay rates in a chiral medium that deviate significantly from the predicted 1→2 process rates would falsify the emergence of these resonances.

read the original abstract

The QED scattering amplitude in a chiral medium characterized by a constant chiral chemical potential $\mu_5$ and chiral magnetic conductivity $b_0$ is analyzed. We show the emergence of the resonant behavior in $1\to 2$, $2\to 2$, and $2\to 3$ processes. We compute the rates of paradigm $1\to 2$ processes that determine the widths of quasi-stationary fermion and photon states in the medium. We elucidate the origin of these resonances, the conditions of their emergence, and the physical principles of their regularization.

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.

Desk Editor's Note private letter to a colleague

The paper computes explicit rates for resonant 1→2 processes and state widths in an idealized chiral QED medium with constant μ5 and b0.

read the letter

The main takeaway is that the authors derive QED amplitudes in a medium with fixed chiral chemical potential and chiral magnetic conductivity, then show resonances appearing in 1→2, 2→2, and 2→3 channels while calculating the rates for the simplest 1→2 cases that fix the widths of quasi-stationary fermion and photon states.

They do the concrete work of writing down the amplitudes, identifying the resonance conditions, and spelling out the regularization rules that keep the rates finite. That supplies usable numbers for the widths, which is the part that could feed into transport models.

The setup treats μ5 and b0 as constants everywhere. That is a common starting point but restricts the results to a uniform medium; any real application would need to check how spatial or temporal variation changes the picture. The abstract gives no limit checks or comparison to the μ5 = b0 = 0 case, so it is not yet clear how large the corrections are or whether the resonances survive once those limits are taken.

The paper is aimed at people already working on chiral magnetic effects in heavy-ion collisions or topological materials. A reader who needs the resonance conditions or the width formulas might pull the rates directly. It is too specialized for a general audience.

I would send it to a referee. The calculations are in principle checkable, the claims are stated clearly, and the topic sits inside an active subfield even if the medium idealization keeps the scope narrow.

Referee Report

0 major / 1 minor

Summary. The manuscript analyzes the QED scattering amplitude in a chiral medium with constant chiral chemical potential μ₅ and chiral magnetic conductivity b₀. It claims to demonstrate the emergence of resonant behavior in 1→2, 2→2, and 2→3 processes, compute the rates of paradigm 1→2 processes that set the widths of quasi-stationary fermion and photon states, and elucidate the origin of the resonances along with their emergence conditions and regularization principles.

Significance. If the derivations and rate computations hold up under scrutiny, the results could contribute to understanding chiral effects in QED media, with potential relevance to systems like quark-gluon plasma. The explicit focus on 1→2 rates for state widths and the discussion of regularization would be concrete strengths, though the constant-μ₅/b₀ assumption limits generality.

minor comments (1)

[Abstract] Abstract: the statement that rates are computed and resonances emerge provides no derivation steps, explicit expressions, error estimates, or consistency checks against known limits (e.g., μ₅→0), preventing assessment of the central claims from the given information.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The report provides a concise summary of our analysis of QED scattering amplitudes in a chiral medium but lists no specific major comments. The recommendation of 'uncertain' appears to reflect a need for verification of the derivations. We are prepared to supply any additional details or clarifications requested.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper takes constant μ5 and b0 as the defining setup for the chiral medium and then derives scattering amplitudes, identifies resonant behavior in 1→2/2→2/2→3 processes, and explicitly computes 1→2 rates that set state widths. No quoted step reduces a claimed prediction or resonance condition to a fitted parameter, self-citation chain, or definitional tautology; the rates are presented as direct calculations from the amplitudes under the stated assumptions. The derivation chain is therefore self-contained and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating μ5 and b0 as constant external parameters supplied by the medium; the analysis assumes standard QED can be extended to this background without additional justification supplied in the abstract.

free parameters (2)

μ5
Chiral chemical potential introduced as a constant characterizing the medium.
b0
Chiral magnetic conductivity introduced as a constant characterizing the medium.

axioms (1)

domain assumption QED scattering amplitudes can be computed in a background with constant chiral chemical potential and conductivity
The entire analysis of resonances and rates presupposes this framework.

pith-pipeline@v0.9.1-grok · 5626 in / 1348 out tokens · 46819 ms · 2026-06-29T03:35:41.086443+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

77 extracted references · 42 linked inside Pith

[1]

This anomalous contribution to bremsstrahlung was computed in [39] under the conditionµ 5 = 0. B. Regularization of resonances The emergence of resonances indicates instability of the fermion and boson states in the chiral medium. Indeed this is precisely why the splitting processes discussed in Sec. IV are possible at all. Since the fermion and boson sta...
[2]

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

1969
[3]

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

1969
[4]

Kharzeev, Parity violation in hot QCD: Why it can happen, and how to look for it, Phys

D. Kharzeev, Parity violation in hot QCD: Why it can happen, and how to look for it, Phys. Lett. B633, 260 (2006), arXiv:hep-ph/0406125

Pith/arXiv arXiv 2006
[5]

Kharzeev and A

D. Kharzeev and A. Zhitnitsky, Charge separation induced by P-odd bubbles in QCD matter, Nucl. Phys. A797, 67 (2007), arXiv:0706.1026 [hep-ph]

Pith/arXiv arXiv 2007
[6]

Fukushima, D

K. Fukushima, D. E. Kharzeev, and H. J. Warringa, The Chiral Magnetic Effect, Phys. Rev. D78, 074033 (2008), arXiv:0808.3382 [hep-ph]

Pith/arXiv arXiv 2008
[7]

D. E. Kharzeev, Topologically induced local P and CP violation in QCD x QED, Annals Phys.325, 205 (2010), arXiv:0911.3715 [hep-ph]. 11

Pith/arXiv arXiv 2010
[8]

D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation’, Nucl. Phys. A803, 227 (2008), arXiv:0711.0950 [hep-ph]

Pith/arXiv arXiv 2008
[9]

Fujikawa and H

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

2004
[10]

Babaev, D

E. Babaev, D. Kharzeev, M. Larsson, A. Molochkov, and V. Zhaunerchyk,Chiral Matter: Proceedings of the Nobel Sym- posium 167(World Scientific Publishing Co., New Jersey, 2023)

2023
[11]

D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear colli- sions—A status report, Prog. Part. Nucl. Phys.88, 1 (2016), arXiv:1511.04050 [hep-ph]

Pith/arXiv arXiv 2016
[12]

D. E. Kharzeev, J. Liao, and P. Tribedy, Chiral magnetic effect in heavy ion collisions: The present and future, Int. J. Mod. Phys. E33, 2430007 (2024), arXiv:2405.05427 [nucl-th]

arXiv 2024
[13]

Sekine and K

A. Sekine and K. Nomura, Axion Electrodynamics in Topological Materials, J. Appl. Phys.129, 141101 (2021), arXiv:2011.13601 [cond-mat.mes-hall]

arXiv 2021
[14]

N. P. Ong and S. Liang, Review of experiments on the chiral anomaly in Dirac-Weyl semimetals, Nature Rev. Phys.3, 394 (2021), arXiv:2010.08564 [cond-mat.str-el]

arXiv 2021
[15]

Sikivie, Invisible Axion Search Methods, Rev

P. Sikivie, Invisible Axion Search Methods, Rev. Mod. Phys.93, 015004 (2021), arXiv:2003.02206 [hep-ph]

arXiv 2021
[16]

M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

1995
[17]

Z. Qiu, G. Cao, and X.-G. Huang, On electrodynamics of chiral matter, Phys. Rev. D95, 036002 (2017), arXiv:1612.06364 [cond-mat.mes-hall]

Pith/arXiv arXiv 2017
[18]

Colladay and V

D. Colladay and V. A. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D58, 116002 (1998), arXiv:hep-ph/9809521

Pith/arXiv arXiv 1998
[19]

Adam and F

C. Adam and F. R. Klinkhamer, Causality and CPT violation from an Abelian Chern-Simons like term, Nucl. Phys. B 607, 247 (2001), arXiv:hep-ph/0101087

Pith/arXiv arXiv 2001
[20]

S. M. Carroll, G. B. Field, and R. Jackiw, Limits on a Lorentz and Parity Violating Modification of Electrodynamics, Phys. Rev. D41, 1231 (1990)

1990
[21]

Lehnert and R

R. Lehnert and R. Potting, Vacuum Cerenkov radiation, Phys. Rev. Lett.93, 110402 (2004), arXiv:hep-ph/0406128

Pith/arXiv arXiv 2004
[22]

Joyce and M

M. Joyce and M. E. Shaposhnikov, Primordial magnetic fields, right-handed electrons, and the Abelian anomaly, Phys. Rev. Lett.79, 1193 (1997), arXiv:astro-ph/9703005

Pith/arXiv arXiv 1997
[23]

Colladay and V

D. Colladay and V. A. Kostelecky, CPT violation and the standard model, Phys. Rev. D55, 6760 (1997), arXiv:hep- ph/9703464

arXiv 1997
[24]

V. A. Kostelecky and A. G. M. Pickering, Vacuum photon splitting in Lorentz violating quantum electrodynamics, Phys. Rev. Lett.91, 031801 (2003), arXiv:hep-ph/0212382

Pith/arXiv arXiv 2003
[25]

P. O. Sukhachov, V. A. Miransky, I. A. Shovkovy, and E. V. Gorbar, Collective excitations in Weyl semimetals in the hydrodynamic regime, J. Phys. Condens. Matter30, 275601 (2018), arXiv:1802.10110 [cond-mat.str-el]

Pith/arXiv arXiv 2018
[26]

Sheng, D

X.-l. Sheng, D. H. Rischke, D. Vasak, and Q. Wang, Wigner functions for fermions in strong magnetic fields, Eur. Phys. J. A54, 21 (2018), arXiv:1707.01388 [hep-ph]

Pith/arXiv arXiv 2018
[27]

J. D. Kroth and K. Tuchin, Searching for missing direct photons in heavy-ion collisions with P and CP violation, arXiv:2602.02746 [hep-ph] (2026)

arXiv 2026
[28]

V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Phys. Rept.576, 1 (2015), arXiv:1503.00732 [hep-ph]

Pith/arXiv arXiv 2015
[29]

Lehnert and R

R. Lehnert and R. Potting, The Cerenkov effect in Lorentz-violating vacua, Phys. Rev. D70, 125010 (2004), [Erratum: Phys.Rev.D 70, 129906 (2004)], arXiv:hep-ph/0408285

Pith/arXiv arXiv 2004
[30]

F. R. Klinkhamer and G. E. Volovik, Emergent CPT violation from the splitting of Fermi points, Int. J. Mod. Phys. A20, 2795 (2005), arXiv:hep-th/0403037

Pith/arXiv arXiv 2005
[31]

Mattingly, Modern tests of Lorentz invariance, Living Rev

D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Rel.8, 5 (2005), arXiv:gr-qc/0502097

Pith/arXiv arXiv 2005
[32]

Jacobson, S

T. Jacobson, S. Liberati, and D. Mattingly, Lorentz violation at high energy: Concepts, phenomena and astrophysical constraints, Annals Phys.321, 150 (2006), arXiv:astro-ph/0505267

Pith/arXiv arXiv 2006
[33]

Altschul, Vacuum Cerenkov Radiation in Lorentz-Violating Theories Without CPT Violation, Phys

B. Altschul, Vacuum Cerenkov Radiation in Lorentz-Violating Theories Without CPT Violation, Phys. Rev. Lett.98, 041603 (2007), arXiv:hep-th/0609030

Pith/arXiv arXiv 2007
[34]

Altschul, Cerenkov Radiation in a Lorentz-Violating and Birefringent Vacuum, Phys

B. Altschul, Cerenkov Radiation in a Lorentz-Violating and Birefringent Vacuum, Phys. Rev. D75, 105003 (2007), arXiv:hep-th/0701270

Pith/arXiv arXiv 2007
[35]

J. R. Nascimento, E. Passos, A. Y. Petrov, and F. A. Brito, Lorentz-CPT violation, radiative corrections and finite temperature, JHEP06, 016, arXiv:0705.1338 [hep-th]

Pith/arXiv arXiv
[36]

Tuchin, Radiative instability of quantum electrodynamics in chiral matter, Phys

K. Tuchin, Radiative instability of quantum electrodynamics in chiral matter, Phys. Lett. B786, 249 (2018), arXiv:1806.07340 [hep-ph]

Pith/arXiv arXiv 2018
[37]

Huang and K

X.-G. Huang and K. Tuchin, Transition Radiation as a Probe of the Chiral Anomaly, Phys. Rev. Lett.121, 182301 (2018), arXiv:1808.00635 [hep-ph]

Pith/arXiv arXiv 2018
[38]

Tuchin, Chiral Cherenkov and chiral transition radiation in anisotropic matter, Phys

K. Tuchin, Chiral Cherenkov and chiral transition radiation in anisotropic matter, Phys. Rev. D98, 114026 (2018), arXiv:1809.08181 [hep-ph]

Pith/arXiv arXiv 2018
[39]

Hansen and K

J. Hansen and K. Tuchin, Collisional energy loss and the chiral magnetic effect, Phys. Rev. C104, 034903 (2021), arXiv:2012.06089 [hep-ph]

arXiv 2021
[40]

Hansen and K

J. Hansen and K. Tuchin, Bremsstrahlung in chiral medium: Anomalous magnetic contribution to the Bethe-Heitler formula, Phys. Rev. D105, 116008 (2022), arXiv:2203.13134 [hep-ph]

arXiv 2022
[41]

Hansen and K

J. Hansen and K. Tuchin, Electromagnetic bremsstrahlung and energy loss in chiral medium, Phys. Rev. D108, 076007 (2023), arXiv:2307.05761 [hep-ph]. 12

arXiv 2023
[42]

Hansen, K

J. Hansen, K. Ikeda, D. E. Kharzeev, Q. Li, and K. Tuchin, Magnetic Weyl semimetals as a source of circularly polarized THz radiation, Phys. Open23, 100268 (2025), arXiv:2405.11076 [cond-mat.mtrl-sci]

arXiv 2025
[43]

Hansen and K

J. Hansen and K. Tuchin, Chiral effects on radiation and energy loss in quark–gluon plasma, Int. J. Mod. Phys. E33, 2430011 (2024), arXiv:2409.16338 [hep-ph]

arXiv 2024
[44]

Hansen and K

J. Hansen and K. Tuchin, Time evolution of parity-odd cascades in homogeneous Abelian and non-Abelian media with chiral imbalance, Phys. Rev. D112, 014010 (2025), arXiv:2503.00933 [hep-ph]

arXiv 2025
[45]

Hansen and K

J. Hansen and K. Tuchin, Color chiral Cherenkov radiation and energy loss in the quark-gluon plasma, Phys. Rev. D110, 014027 (2024), arXiv:2405.08697 [hep-ph]

arXiv 2024
[46]

Akamatsu and N

Y. Akamatsu and N. Yamamoto, Chiral Plasma Instabilities, Phys. Rev. Lett.111, 052002 (2013), arXiv:1302.2125 [nucl- th]

Pith/arXiv arXiv 2013
[47]

Duari, N

S. Duari, N. Chaudhuri, P. Roy, and S. Sarkar, Dynamical color conductivity of a chiral quark-gluon plasma, Phys. Rev. D113, 014011 (2026), arXiv:2512.20050 [hep-ph]

arXiv 2026
[48]

Tuchin, Chiral Cherenkov radiation in the presence of a time-dependent chiral chemical potential, Phys

K. Tuchin, Chiral Cherenkov radiation in the presence of a time-dependent chiral chemical potential, Phys. Rev. C112, 044903 (2025), arXiv:2507.07324 [hep-ph]

arXiv 2025
[49]

Barredo-Alamilla, L

E. Barredo-Alamilla, L. F. Urrutia, and M. M. Ferreira, Jr., Electromagnetic radiation in chiral matter: The Cherenkov case, Phys. Rev. D107, 096024 (2023), arXiv:2305.07963 [hep-ph]

arXiv 2023
[50]

R. M. von Dossow, E. Barredo-Alamilla, M. A. Gorlach, and L. F. Urrutia, Cherenkov radiation in isotropic chiral matter: Unlocking threshold-free emission, Phys. Rev. D113, 016010 (2026), arXiv:2512.14676 [hep-ph]

arXiv 2026
[51]

Tuchin, Anomalous scattering and transport in chiral matter, Phys

K. Tuchin, Anomalous scattering and transport in chiral matter, Phys. Lett. B808, 135680 (2020), arXiv:2006.07950 [hep-ph]

arXiv 2020
[52]

Boyarsky, J

A. Boyarsky, J. Frohlich, and O. Ruchayskiy, Self-consistent evolution of magnetic fields and chiral asymmetry in the early Universe, Phys. Rev. Lett.108, 031301 (2012), arXiv:1109.3350 [astro-ph.CO]

Pith/arXiv arXiv 2012
[53]

D. E. Kharzeev, The Chiral Magnetic Effect and Anomaly-Induced Transport, Prog. Part. Nucl. Phys.75, 133 (2014), arXiv:1312.3348 [hep-ph]

Pith/arXiv arXiv 2014
[54]

Z. V. Khaidukov, V. P. Kirilin, A. V. Sadofyev, and V. I. Zakharov, On Magnetostatics of Chiral Media, Nucl. Phys. B 934, 521 (2018), arXiv:1307.0138 [hep-th]

Pith/arXiv arXiv 2018
[55]

V. P. Kirilin, A. V. Sadofyev, and V. I. Zakharov, Anomaly and long-range forces, in100th anniversary of the birth of I.Ya. Pomeranchuk(2014) pp. 272–286, arXiv:1312.0895 [hep-th]

Pith/arXiv arXiv 2014
[56]

Avdoshkin, V

A. Avdoshkin, V. P. Kirilin, A. V. Sadofyev, and V. I. Zakharov, On consistency of hydrodynamic approximation for chiral media, Phys. Lett. B755, 1 (2016), arXiv:1402.3587 [hep-th]

Pith/arXiv arXiv 2016
[57]

Dvornikov and V

M. Dvornikov and V. B. Semikoz, Magnetic field instability in a neutron star driven by the electroweak electron-nucleon interaction versus the chiral magnetic effect, Phys. Rev. D91, 061301 (2015), arXiv:1410.6676 [astro-ph.HE]

Pith/arXiv arXiv 2015
[58]

Tuchin, Electromagnetic field and the chiral magnetic effect in the quark-gluon plasma, Phys

K. Tuchin, Electromagnetic field and the chiral magnetic effect in the quark-gluon plasma, Phys. Rev. C91, 064902 (2015), arXiv:1411.1363 [hep-ph]

Pith/arXiv arXiv 2015
[59]

Manuel and J

C. Manuel and J. M. Torres-Rincon, Dynamical evolution of the chiral magnetic effect: Applications to the quark-gluon plasma, Phys. Rev. D92, 074018 (2015), arXiv:1501.07608 [hep-ph]

Pith/arXiv arXiv 2015
[60]

P. V. Buividovich and M. V. Ulybyshev, Numerical study of chiral plasma instability within the classical statistical field theory approach, Phys. Rev. D94, 025009 (2016), arXiv:1509.02076 [hep-th]

Pith/arXiv arXiv 2016
[61]

Sigl and N

G. Sigl and N. Leite, Chiral Magnetic Effect in Protoneutron Stars and Magnetic Field Spectral Evolution, JCAP01, 025, arXiv:1507.04983 [astro-ph.HE]

Pith/arXiv arXiv
[62]

X.-l. Xia, H. Qin, and Q. Wang, Approach to Chandrasekhar-Kendall-Woltjer State in a Chiral Plasma, Phys. Rev. D94, 054042 (2016), arXiv:1607.01126 [nucl-th]

Pith/arXiv arXiv 2016
[63]

D. B. Kaplan, S. Reddy, and S. Sen, Energy Conservation and the Chiral Magnetic Effect, Phys. Rev. D96, 016008 (2017), arXiv:1612.00032 [hep-ph]

Pith/arXiv arXiv 2017
[64]

V. P. Kirilin and A. V. Sadofyev, Anomalous Transport and Generalized Axial Charge, Phys. Rev. D96, 016019 (2017), arXiv:1703.02483 [hep-th]

Pith/arXiv arXiv 2017
[65]

M. Mace, N. Mueller, S. Schlichting, and S. Sharma, Chiral Instabilities and the Onset of Chiral Turbulence in QED Plasmas, Phys. Rev. Lett.124, 191604 (2020), arXiv:1910.01654 [hep-ph]

arXiv 2020
[66]

Tuchin, Polarized electromagnetic radiation by chiral media with time-dependent chiral chemical potential, Phys

K. Tuchin, Polarized electromagnetic radiation by chiral media with time-dependent chiral chemical potential, Phys. Lett. B872, 140080 (2026), arXiv:2508.12923 [hep-ph]

arXiv 2026
[67]

Tuchin, Quark and gluon production in the presence of the time-varying chiral magnetic current, arXiv:2604.21872 [hep-ph] (2026)

K. Tuchin, Quark and gluon production in the presence of the time-varying chiral magnetic current, arXiv:2604.21872 [hep-ph] (2026)

Pith/arXiv arXiv 2026
[68]

Stewart and K

E. Stewart and K. Tuchin, Optical manifestations of domains with constant topological charge density, Phys. Rev. Research. 1, 023005 (2019), arXiv:1906.04602 [hep-ph]

arXiv 2019
[69]

Tuchin, Photon radiation in hot nuclear matter by means of chiral anomalies, Phys

K. Tuchin, Photon radiation in hot nuclear matter by means of chiral anomalies, Phys. Rev. C99, 064907 (2019), arXiv:1903.02629 [hep-ph]

Pith/arXiv arXiv 2019
[70]

Weinberg,The Quantum theory of fields

S. Weinberg,The Quantum theory of fields. Vol. 1: Foundations(Cambridge University Press, 2005)

2005
[71]

D. T. Son and N. Yamamoto, Kinetic theory with Berry curvature from quantum field theories, Phys. Rev. D87, 085016 (2013), arXiv:1210.8158 [hep-th]. 13 Appendix A: Electromagnetic field at constantb 0

Pith/arXiv arXiv 2013
[72]

The complete energy-momentum tensor was obtained in [19]

Energy of electromagnetic field Using (4a) and (4c), we obtain: 1 2 ∂t Z E2 +B 2 d3x= I (B×E)·da−b 0 Z E·Bd 3x .(A1) Noting that ∂t Z A·Bd 3x=−2 Z E·Bd 3x+ I (A×E)·da(A2) we can rewrite the equation (A1) as: 1 2 ∂t Z E2 +B 2 −b 0A·B d3x= I (B×E+ 1 2 A×E)·da.(A3) This equation represents the conservation of electromagnetic field energy, which is given by: ...
[73]

We can similarly represent Dij(−x)θ(−x0) =i X λ Z d3k (2π)3 Z ∞ −∞ dk0 2π πij −k,λ 2ω−k,λ e−ik·x 1 −k0 −ω −k,λ +iϵ ,(A12) where nowk=−pandk 0 =−ω −k,λ −s

Photon propagator The Feynman propagator is defined as the following time-ordered commutator: Dij F (x) =D ij(x)θ(x0) +D ij(−x)θ(−x0),(A7) where Dij(x) =⟨0|A i(x)Aj(0)|0⟩= X λ Z d3p (2π)3 πij p,λ 2ωp,λ e−ip·x .(A8) The product of the polarization vectorsπ ij p,λ can be represented as a combination of symmetric and anti-symmetric matrices: πij p,λ =ϵ i p,λ...
[74]

Positive energy solutions of (18) The positive energy solution of (18) has a form: ψp,σ(x) =e −ip·xup,σ =e −ip·x φp,σ χp,σ ,(B1) 15 wherep µ = (Ep,σ,p),σis helicity,φandχare left and right two-component spinors respectively. Substituting (B1) into (18) and employing the chiral representation ofγ-matrices we obtain: −m E p,σ −p·σ+µ 5 Ep,σ +p·σ−µ 5 −m φp,σ ...
[75]

Negative energy solutions We seek the negative energy solutions to (18) in the form: ψp,σ(x) =e ip·xvp,σ =e ip·x φ′ p,σ χ′ p,σ ,(B11) wherep µ = (E′ p,σ,p). Substitution into (18) now yields: −m−E ′ p,σ +p·σ+µ 5 −E′ p,σ −p·σ−µ 5 −m φ′ p,σ χ′ p,σ = 0.(B12) The solution in terms of the normalized helicity eigenstates is: φ′ p,σ =N ′ σξp,σ ,(B13) χ′ p,σ =− N...
[76]

Using the Gordon identity, the current density is: jµ = 1 2Ep,σ ¯up,σγµup,σ = 1 2Ep,σ 1 2m ¯up,σ 2pµ −2iσ µνγ5aν up,σ = 1 Ep,σ (Ep,σ,p−µ 5⟨σ⟩).(B21) where⟨σ⟩=ξ † p,σσξp,σ

Gordon’s identity Consider the Dirac equation at finite chiral chemical potential (18) for the positive energy solutions: pµγµ −γ 5γµaµ −m up,σ = 0.(B17) The adjoint spinor obeys the equation ¯up,σ pµγµ −γ 5γµaµ −m = 0.(B18) It can be shown that the spinors satisfy the Gordon’s identity: ¯up′,σ′γµup,σ = 1 2m ¯up′,σ′ pµ +p ′µ +iσ µν(p′ ν −p ν)−2iσ µνγ5aν u...
[77]

It is worth noting that the two matrices enclosed within the two pairs of parentheses in the numerator of (25) commute

Fermion propagator The fermionic Green functionG(x) obeys the equation: i /∂−γ 5γ0µ5 −m G(x) =iδ(x).(B25) In the momentum space (B25) reads: /p−γ 5γ0µ5 −m ˜G(p) =i(B26) Using the properties ofγmatrices, we compute: /p−γ 5γ0µ5 +m /p−γ 5γ0µ5 −m =p 2 −µ 2 5 −m 2 + 2µ5γ5γ0γ·p(B27) and p2 −µ 2 5 −m 2 −2µ 5γ5γ0γ·p p2 −µ 2 5 −m 2 + 2µ5γ5γ0γ·p = p2 −µ 2 5 −m 2 2 ...

[1] [1]

This anomalous contribution to bremsstrahlung was computed in [39] under the conditionµ 5 = 0. B. Regularization of resonances The emergence of resonances indicates instability of the fermion and boson states in the chiral medium. Indeed this is precisely why the splitting processes discussed in Sec. IV are possible at all. Since the fermion and boson sta...

[2] [2]

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

1969

[3] [3]

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

1969

[4] [4]

Kharzeev, Parity violation in hot QCD: Why it can happen, and how to look for it, Phys

D. Kharzeev, Parity violation in hot QCD: Why it can happen, and how to look for it, Phys. Lett. B633, 260 (2006), arXiv:hep-ph/0406125

Pith/arXiv arXiv 2006

[5] [5]

Kharzeev and A

D. Kharzeev and A. Zhitnitsky, Charge separation induced by P-odd bubbles in QCD matter, Nucl. Phys. A797, 67 (2007), arXiv:0706.1026 [hep-ph]

Pith/arXiv arXiv 2007

[6] [6]

Fukushima, D

K. Fukushima, D. E. Kharzeev, and H. J. Warringa, The Chiral Magnetic Effect, Phys. Rev. D78, 074033 (2008), arXiv:0808.3382 [hep-ph]

Pith/arXiv arXiv 2008

[7] [7]

D. E. Kharzeev, Topologically induced local P and CP violation in QCD x QED, Annals Phys.325, 205 (2010), arXiv:0911.3715 [hep-ph]. 11

Pith/arXiv arXiv 2010

[8] [8]

D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation’, Nucl. Phys. A803, 227 (2008), arXiv:0711.0950 [hep-ph]

Pith/arXiv arXiv 2008

[9] [9]

Fujikawa and H

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

2004

[10] [10]

Babaev, D

E. Babaev, D. Kharzeev, M. Larsson, A. Molochkov, and V. Zhaunerchyk,Chiral Matter: Proceedings of the Nobel Sym- posium 167(World Scientific Publishing Co., New Jersey, 2023)

2023

[11] [11]

D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear colli- sions—A status report, Prog. Part. Nucl. Phys.88, 1 (2016), arXiv:1511.04050 [hep-ph]

Pith/arXiv arXiv 2016

[12] [12]

D. E. Kharzeev, J. Liao, and P. Tribedy, Chiral magnetic effect in heavy ion collisions: The present and future, Int. J. Mod. Phys. E33, 2430007 (2024), arXiv:2405.05427 [nucl-th]

arXiv 2024

[13] [13]

Sekine and K

A. Sekine and K. Nomura, Axion Electrodynamics in Topological Materials, J. Appl. Phys.129, 141101 (2021), arXiv:2011.13601 [cond-mat.mes-hall]

arXiv 2021

[14] [14]

N. P. Ong and S. Liang, Review of experiments on the chiral anomaly in Dirac-Weyl semimetals, Nature Rev. Phys.3, 394 (2021), arXiv:2010.08564 [cond-mat.str-el]

arXiv 2021

[15] [15]

Sikivie, Invisible Axion Search Methods, Rev

P. Sikivie, Invisible Axion Search Methods, Rev. Mod. Phys.93, 015004 (2021), arXiv:2003.02206 [hep-ph]

arXiv 2021

[16] [16]

M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

1995

[17] [17]

Z. Qiu, G. Cao, and X.-G. Huang, On electrodynamics of chiral matter, Phys. Rev. D95, 036002 (2017), arXiv:1612.06364 [cond-mat.mes-hall]

Pith/arXiv arXiv 2017

[18] [18]

Colladay and V

D. Colladay and V. A. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D58, 116002 (1998), arXiv:hep-ph/9809521

Pith/arXiv arXiv 1998

[19] [19]

Adam and F

C. Adam and F. R. Klinkhamer, Causality and CPT violation from an Abelian Chern-Simons like term, Nucl. Phys. B 607, 247 (2001), arXiv:hep-ph/0101087

Pith/arXiv arXiv 2001

[20] [20]

S. M. Carroll, G. B. Field, and R. Jackiw, Limits on a Lorentz and Parity Violating Modification of Electrodynamics, Phys. Rev. D41, 1231 (1990)

1990

[21] [21]

Lehnert and R

R. Lehnert and R. Potting, Vacuum Cerenkov radiation, Phys. Rev. Lett.93, 110402 (2004), arXiv:hep-ph/0406128

Pith/arXiv arXiv 2004

[22] [22]

Joyce and M

M. Joyce and M. E. Shaposhnikov, Primordial magnetic fields, right-handed electrons, and the Abelian anomaly, Phys. Rev. Lett.79, 1193 (1997), arXiv:astro-ph/9703005

Pith/arXiv arXiv 1997

[23] [23]

Colladay and V

D. Colladay and V. A. Kostelecky, CPT violation and the standard model, Phys. Rev. D55, 6760 (1997), arXiv:hep- ph/9703464

arXiv 1997

[24] [24]

V. A. Kostelecky and A. G. M. Pickering, Vacuum photon splitting in Lorentz violating quantum electrodynamics, Phys. Rev. Lett.91, 031801 (2003), arXiv:hep-ph/0212382

Pith/arXiv arXiv 2003

[25] [25]

P. O. Sukhachov, V. A. Miransky, I. A. Shovkovy, and E. V. Gorbar, Collective excitations in Weyl semimetals in the hydrodynamic regime, J. Phys. Condens. Matter30, 275601 (2018), arXiv:1802.10110 [cond-mat.str-el]

Pith/arXiv arXiv 2018

[26] [26]

Sheng, D

X.-l. Sheng, D. H. Rischke, D. Vasak, and Q. Wang, Wigner functions for fermions in strong magnetic fields, Eur. Phys. J. A54, 21 (2018), arXiv:1707.01388 [hep-ph]

Pith/arXiv arXiv 2018

[27] [27]

J. D. Kroth and K. Tuchin, Searching for missing direct photons in heavy-ion collisions with P and CP violation, arXiv:2602.02746 [hep-ph] (2026)

arXiv 2026

[28] [28]

V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Phys. Rept.576, 1 (2015), arXiv:1503.00732 [hep-ph]

Pith/arXiv arXiv 2015

[29] [29]

Lehnert and R

R. Lehnert and R. Potting, The Cerenkov effect in Lorentz-violating vacua, Phys. Rev. D70, 125010 (2004), [Erratum: Phys.Rev.D 70, 129906 (2004)], arXiv:hep-ph/0408285

Pith/arXiv arXiv 2004

[30] [30]

F. R. Klinkhamer and G. E. Volovik, Emergent CPT violation from the splitting of Fermi points, Int. J. Mod. Phys. A20, 2795 (2005), arXiv:hep-th/0403037

Pith/arXiv arXiv 2005

[31] [31]

Mattingly, Modern tests of Lorentz invariance, Living Rev

D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Rel.8, 5 (2005), arXiv:gr-qc/0502097

Pith/arXiv arXiv 2005

[32] [32]

Jacobson, S

T. Jacobson, S. Liberati, and D. Mattingly, Lorentz violation at high energy: Concepts, phenomena and astrophysical constraints, Annals Phys.321, 150 (2006), arXiv:astro-ph/0505267

Pith/arXiv arXiv 2006

[33] [33]

Altschul, Vacuum Cerenkov Radiation in Lorentz-Violating Theories Without CPT Violation, Phys

B. Altschul, Vacuum Cerenkov Radiation in Lorentz-Violating Theories Without CPT Violation, Phys. Rev. Lett.98, 041603 (2007), arXiv:hep-th/0609030

Pith/arXiv arXiv 2007

[34] [34]

Altschul, Cerenkov Radiation in a Lorentz-Violating and Birefringent Vacuum, Phys

B. Altschul, Cerenkov Radiation in a Lorentz-Violating and Birefringent Vacuum, Phys. Rev. D75, 105003 (2007), arXiv:hep-th/0701270

Pith/arXiv arXiv 2007

[35] [35]

J. R. Nascimento, E. Passos, A. Y. Petrov, and F. A. Brito, Lorentz-CPT violation, radiative corrections and finite temperature, JHEP06, 016, arXiv:0705.1338 [hep-th]

Pith/arXiv arXiv

[36] [36]

Tuchin, Radiative instability of quantum electrodynamics in chiral matter, Phys

K. Tuchin, Radiative instability of quantum electrodynamics in chiral matter, Phys. Lett. B786, 249 (2018), arXiv:1806.07340 [hep-ph]

Pith/arXiv arXiv 2018

[37] [37]

Huang and K

X.-G. Huang and K. Tuchin, Transition Radiation as a Probe of the Chiral Anomaly, Phys. Rev. Lett.121, 182301 (2018), arXiv:1808.00635 [hep-ph]

Pith/arXiv arXiv 2018

[38] [38]

Tuchin, Chiral Cherenkov and chiral transition radiation in anisotropic matter, Phys

K. Tuchin, Chiral Cherenkov and chiral transition radiation in anisotropic matter, Phys. Rev. D98, 114026 (2018), arXiv:1809.08181 [hep-ph]

Pith/arXiv arXiv 2018

[39] [39]

Hansen and K

J. Hansen and K. Tuchin, Collisional energy loss and the chiral magnetic effect, Phys. Rev. C104, 034903 (2021), arXiv:2012.06089 [hep-ph]

arXiv 2021

[40] [40]

Hansen and K

J. Hansen and K. Tuchin, Bremsstrahlung in chiral medium: Anomalous magnetic contribution to the Bethe-Heitler formula, Phys. Rev. D105, 116008 (2022), arXiv:2203.13134 [hep-ph]

arXiv 2022

[41] [41]

Hansen and K

J. Hansen and K. Tuchin, Electromagnetic bremsstrahlung and energy loss in chiral medium, Phys. Rev. D108, 076007 (2023), arXiv:2307.05761 [hep-ph]. 12

arXiv 2023

[42] [42]

Hansen, K

J. Hansen, K. Ikeda, D. E. Kharzeev, Q. Li, and K. Tuchin, Magnetic Weyl semimetals as a source of circularly polarized THz radiation, Phys. Open23, 100268 (2025), arXiv:2405.11076 [cond-mat.mtrl-sci]

arXiv 2025

[43] [43]

Hansen and K

J. Hansen and K. Tuchin, Chiral effects on radiation and energy loss in quark–gluon plasma, Int. J. Mod. Phys. E33, 2430011 (2024), arXiv:2409.16338 [hep-ph]

arXiv 2024

[44] [44]

Hansen and K

J. Hansen and K. Tuchin, Time evolution of parity-odd cascades in homogeneous Abelian and non-Abelian media with chiral imbalance, Phys. Rev. D112, 014010 (2025), arXiv:2503.00933 [hep-ph]

arXiv 2025

[45] [45]

Hansen and K

J. Hansen and K. Tuchin, Color chiral Cherenkov radiation and energy loss in the quark-gluon plasma, Phys. Rev. D110, 014027 (2024), arXiv:2405.08697 [hep-ph]

arXiv 2024

[46] [46]

Akamatsu and N

Y. Akamatsu and N. Yamamoto, Chiral Plasma Instabilities, Phys. Rev. Lett.111, 052002 (2013), arXiv:1302.2125 [nucl- th]

Pith/arXiv arXiv 2013

[47] [47]

Duari, N

S. Duari, N. Chaudhuri, P. Roy, and S. Sarkar, Dynamical color conductivity of a chiral quark-gluon plasma, Phys. Rev. D113, 014011 (2026), arXiv:2512.20050 [hep-ph]

arXiv 2026

[48] [48]

Tuchin, Chiral Cherenkov radiation in the presence of a time-dependent chiral chemical potential, Phys

K. Tuchin, Chiral Cherenkov radiation in the presence of a time-dependent chiral chemical potential, Phys. Rev. C112, 044903 (2025), arXiv:2507.07324 [hep-ph]

arXiv 2025

[49] [49]

Barredo-Alamilla, L

E. Barredo-Alamilla, L. F. Urrutia, and M. M. Ferreira, Jr., Electromagnetic radiation in chiral matter: The Cherenkov case, Phys. Rev. D107, 096024 (2023), arXiv:2305.07963 [hep-ph]

arXiv 2023

[50] [50]

R. M. von Dossow, E. Barredo-Alamilla, M. A. Gorlach, and L. F. Urrutia, Cherenkov radiation in isotropic chiral matter: Unlocking threshold-free emission, Phys. Rev. D113, 016010 (2026), arXiv:2512.14676 [hep-ph]

arXiv 2026

[51] [51]

Tuchin, Anomalous scattering and transport in chiral matter, Phys

K. Tuchin, Anomalous scattering and transport in chiral matter, Phys. Lett. B808, 135680 (2020), arXiv:2006.07950 [hep-ph]

arXiv 2020

[52] [52]

Boyarsky, J

A. Boyarsky, J. Frohlich, and O. Ruchayskiy, Self-consistent evolution of magnetic fields and chiral asymmetry in the early Universe, Phys. Rev. Lett.108, 031301 (2012), arXiv:1109.3350 [astro-ph.CO]

Pith/arXiv arXiv 2012

[53] [53]

D. E. Kharzeev, The Chiral Magnetic Effect and Anomaly-Induced Transport, Prog. Part. Nucl. Phys.75, 133 (2014), arXiv:1312.3348 [hep-ph]

Pith/arXiv arXiv 2014

[54] [54]

Z. V. Khaidukov, V. P. Kirilin, A. V. Sadofyev, and V. I. Zakharov, On Magnetostatics of Chiral Media, Nucl. Phys. B 934, 521 (2018), arXiv:1307.0138 [hep-th]

Pith/arXiv arXiv 2018

[55] [55]

V. P. Kirilin, A. V. Sadofyev, and V. I. Zakharov, Anomaly and long-range forces, in100th anniversary of the birth of I.Ya. Pomeranchuk(2014) pp. 272–286, arXiv:1312.0895 [hep-th]

Pith/arXiv arXiv 2014

[56] [56]

Avdoshkin, V

A. Avdoshkin, V. P. Kirilin, A. V. Sadofyev, and V. I. Zakharov, On consistency of hydrodynamic approximation for chiral media, Phys. Lett. B755, 1 (2016), arXiv:1402.3587 [hep-th]

Pith/arXiv arXiv 2016

[57] [57]

Dvornikov and V

M. Dvornikov and V. B. Semikoz, Magnetic field instability in a neutron star driven by the electroweak electron-nucleon interaction versus the chiral magnetic effect, Phys. Rev. D91, 061301 (2015), arXiv:1410.6676 [astro-ph.HE]

Pith/arXiv arXiv 2015

[58] [58]

Tuchin, Electromagnetic field and the chiral magnetic effect in the quark-gluon plasma, Phys

K. Tuchin, Electromagnetic field and the chiral magnetic effect in the quark-gluon plasma, Phys. Rev. C91, 064902 (2015), arXiv:1411.1363 [hep-ph]

Pith/arXiv arXiv 2015

[59] [59]

Manuel and J

C. Manuel and J. M. Torres-Rincon, Dynamical evolution of the chiral magnetic effect: Applications to the quark-gluon plasma, Phys. Rev. D92, 074018 (2015), arXiv:1501.07608 [hep-ph]

Pith/arXiv arXiv 2015

[60] [60]

P. V. Buividovich and M. V. Ulybyshev, Numerical study of chiral plasma instability within the classical statistical field theory approach, Phys. Rev. D94, 025009 (2016), arXiv:1509.02076 [hep-th]

Pith/arXiv arXiv 2016

[61] [61]

Sigl and N

G. Sigl and N. Leite, Chiral Magnetic Effect in Protoneutron Stars and Magnetic Field Spectral Evolution, JCAP01, 025, arXiv:1507.04983 [astro-ph.HE]

Pith/arXiv arXiv

[62] [62]

X.-l. Xia, H. Qin, and Q. Wang, Approach to Chandrasekhar-Kendall-Woltjer State in a Chiral Plasma, Phys. Rev. D94, 054042 (2016), arXiv:1607.01126 [nucl-th]

Pith/arXiv arXiv 2016

[63] [63]

D. B. Kaplan, S. Reddy, and S. Sen, Energy Conservation and the Chiral Magnetic Effect, Phys. Rev. D96, 016008 (2017), arXiv:1612.00032 [hep-ph]

Pith/arXiv arXiv 2017

[64] [64]

V. P. Kirilin and A. V. Sadofyev, Anomalous Transport and Generalized Axial Charge, Phys. Rev. D96, 016019 (2017), arXiv:1703.02483 [hep-th]

Pith/arXiv arXiv 2017

[65] [65]

M. Mace, N. Mueller, S. Schlichting, and S. Sharma, Chiral Instabilities and the Onset of Chiral Turbulence in QED Plasmas, Phys. Rev. Lett.124, 191604 (2020), arXiv:1910.01654 [hep-ph]

arXiv 2020

[66] [66]

Tuchin, Polarized electromagnetic radiation by chiral media with time-dependent chiral chemical potential, Phys

K. Tuchin, Polarized electromagnetic radiation by chiral media with time-dependent chiral chemical potential, Phys. Lett. B872, 140080 (2026), arXiv:2508.12923 [hep-ph]

arXiv 2026

[67] [67]

Tuchin, Quark and gluon production in the presence of the time-varying chiral magnetic current, arXiv:2604.21872 [hep-ph] (2026)

K. Tuchin, Quark and gluon production in the presence of the time-varying chiral magnetic current, arXiv:2604.21872 [hep-ph] (2026)

Pith/arXiv arXiv 2026

[68] [68]

Stewart and K

E. Stewart and K. Tuchin, Optical manifestations of domains with constant topological charge density, Phys. Rev. Research. 1, 023005 (2019), arXiv:1906.04602 [hep-ph]

arXiv 2019

[69] [69]

Tuchin, Photon radiation in hot nuclear matter by means of chiral anomalies, Phys

K. Tuchin, Photon radiation in hot nuclear matter by means of chiral anomalies, Phys. Rev. C99, 064907 (2019), arXiv:1903.02629 [hep-ph]

Pith/arXiv arXiv 2019

[70] [70]

Weinberg,The Quantum theory of fields

S. Weinberg,The Quantum theory of fields. Vol. 1: Foundations(Cambridge University Press, 2005)

2005

[71] [71]

D. T. Son and N. Yamamoto, Kinetic theory with Berry curvature from quantum field theories, Phys. Rev. D87, 085016 (2013), arXiv:1210.8158 [hep-th]. 13 Appendix A: Electromagnetic field at constantb 0

Pith/arXiv arXiv 2013

[72] [72]

The complete energy-momentum tensor was obtained in [19]

Energy of electromagnetic field Using (4a) and (4c), we obtain: 1 2 ∂t Z E2 +B 2 d3x= I (B×E)·da−b 0 Z E·Bd 3x .(A1) Noting that ∂t Z A·Bd 3x=−2 Z E·Bd 3x+ I (A×E)·da(A2) we can rewrite the equation (A1) as: 1 2 ∂t Z E2 +B 2 −b 0A·B d3x= I (B×E+ 1 2 A×E)·da.(A3) This equation represents the conservation of electromagnetic field energy, which is given by: ...

[73] [73]

We can similarly represent Dij(−x)θ(−x0) =i X λ Z d3k (2π)3 Z ∞ −∞ dk0 2π πij −k,λ 2ω−k,λ e−ik·x 1 −k0 −ω −k,λ +iϵ ,(A12) where nowk=−pandk 0 =−ω −k,λ −s

Photon propagator The Feynman propagator is defined as the following time-ordered commutator: Dij F (x) =D ij(x)θ(x0) +D ij(−x)θ(−x0),(A7) where Dij(x) =⟨0|A i(x)Aj(0)|0⟩= X λ Z d3p (2π)3 πij p,λ 2ωp,λ e−ip·x .(A8) The product of the polarization vectorsπ ij p,λ can be represented as a combination of symmetric and anti-symmetric matrices: πij p,λ =ϵ i p,λ...

[74] [74]

Positive energy solutions of (18) The positive energy solution of (18) has a form: ψp,σ(x) =e −ip·xup,σ =e −ip·x φp,σ χp,σ ,(B1) 15 wherep µ = (Ep,σ,p),σis helicity,φandχare left and right two-component spinors respectively. Substituting (B1) into (18) and employing the chiral representation ofγ-matrices we obtain: −m E p,σ −p·σ+µ 5 Ep,σ +p·σ−µ 5 −m φp,σ ...

[75] [75]

Negative energy solutions We seek the negative energy solutions to (18) in the form: ψp,σ(x) =e ip·xvp,σ =e ip·x φ′ p,σ χ′ p,σ ,(B11) wherep µ = (E′ p,σ,p). Substitution into (18) now yields: −m−E ′ p,σ +p·σ+µ 5 −E′ p,σ −p·σ−µ 5 −m φ′ p,σ χ′ p,σ = 0.(B12) The solution in terms of the normalized helicity eigenstates is: φ′ p,σ =N ′ σξp,σ ,(B13) χ′ p,σ =− N...

[76] [76]

Using the Gordon identity, the current density is: jµ = 1 2Ep,σ ¯up,σγµup,σ = 1 2Ep,σ 1 2m ¯up,σ 2pµ −2iσ µνγ5aν up,σ = 1 Ep,σ (Ep,σ,p−µ 5⟨σ⟩).(B21) where⟨σ⟩=ξ † p,σσξp,σ

Gordon’s identity Consider the Dirac equation at finite chiral chemical potential (18) for the positive energy solutions: pµγµ −γ 5γµaµ −m up,σ = 0.(B17) The adjoint spinor obeys the equation ¯up,σ pµγµ −γ 5γµaµ −m = 0.(B18) It can be shown that the spinors satisfy the Gordon’s identity: ¯up′,σ′γµup,σ = 1 2m ¯up′,σ′ pµ +p ′µ +iσ µν(p′ ν −p ν)−2iσ µνγ5aν u...

[77] [77]

It is worth noting that the two matrices enclosed within the two pairs of parentheses in the numerator of (25) commute

Fermion propagator The fermionic Green functionG(x) obeys the equation: i /∂−γ 5γ0µ5 −m G(x) =iδ(x).(B25) In the momentum space (B25) reads: /p−γ 5γ0µ5 −m ˜G(p) =i(B26) Using the properties ofγmatrices, we compute: /p−γ 5γ0µ5 +m /p−γ 5γ0µ5 −m =p 2 −µ 2 5 −m 2 + 2µ5γ5γ0γ·p(B27) and p2 −µ 2 5 −m 2 −2µ 5γ5γ0γ·p p2 −µ 2 5 −m 2 + 2µ5γ5γ0γ·p = p2 −µ 2 5 −m 2 2 ...