Effect of chiral imbalance on the electrical conductivity of hot and dense quark matter using Green-Kubo Method within the 2-flavour gauged NJL model

Nilanjan Chaudhuri; Pradip Roy; Snigdha Ghosh; Sourav Sarkar

REVIEW 3 major objections 4 minor 113 references

Effect of chiral imbalance on the electrical conductivity of hot and dense quark matter using Green-Kubo Method within the 2-flavour gauged NJL model

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A chiral imbalance, quantified by a chiral chemical potential $\mu_5$, significantly suppresses the ratio $\sigma_{\rm el}/T$ of hot quark matter; in a two-flavor NJL model the suppression is strongest in the low-temperature chirally…

desk verdict First NJL calculation of σel at finite μ5, with a believable qualitative suppression but a quantitatively insecure width identification. read the letter →

arxiv 2411.17117 v2 pith:IV37PJKR submitted 2024-11-26 hep-ph nucl-th

Snigdha Ghosh , Nilanjan Chaudhuri , Sourav Sarkar , Pradip Roy This is my paper · ORCID

classification hep-phnucl-th MSC 81T2881V05 PACS 12.38.-t12.39.-x25.75.-q

keywords electricalconductivitychiralchemicalpotentialimbalanceNJLmodelGreen-Kuborelationthermalwidthquarkmatterfinite-temperaturefieldtheory

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 tries to establish that a chiral imbalance, an excess of right-handed over left-handed quarks parametrized by a chiral chemical potential $\mu_5$, acts as a strong brake on electrical conduction in hot quark matter, and that the effect is largest exactly where chiral symmetry is still broken. Using the two-flavor Nambu-Jona-Lasinio model and the Green-Kubo relation, the authors compute $\sigma_{\rm el}/T$ from the vector-current spectral function, feeding in quark thermal widths obtained from $2\to 2$ scattering. The central numerical finding is a monotonic decrease of $\sigma_{\rm el}/T$ with both temperature and $\mu_5$, with the $\mu_5$ effect most pronounced at low temperature and nearly washed out above about 200 MeV. The authors present this as the first NJL-model determination of electrical conductivity with a chiral imbalance, and the matter matters because the same chirality imbalance is expected in the quark-gluon plasma created in heavy-ion collisions and in certain astrophysical settings.

What carries the argument

The load-bearing identity is the Green-Kubo expression for the electrical conductivity, given here as Eq. (22): $\sigma_{\rm el}$ is proportional to $1/T$ times an integral over quark momenta of occupation factors $f_\pm^r(1-f_\pm^r)$ divided by the thermal width $\Gamma^r$, with a helicity-dependent factor $(|\mathbf{k}|+r\mu_5)^2/(\omega_{\mathbf{k}}^r)^2$. The machinery that makes the one-loop formula finite is the replacement of the Dirac delta function by a Breit-Wigner form with width $\Gamma$, which is then identified with the momentum-dependent thermal width of quarks and antiquarks obtained from $2\to 2$ scattering. The scattering amplitudes are built from mesonic propagators $D_h=2G/(1-2G\Pi_h)$ in the scalar ($\sigma$) and pseudoscalar ($\pi$) channels, with polarization functions $\Pi_h$ evaluated at finite temperature, baryon chemical potential, and $\mu_5$; the polarization functions develop Landau cuts in the time-like region only when $\mu_5\neq 0$.

What would settle it

Compute $\sigma_{\rm el}/T$ in the same NJL model with the vector correlator evaluated from a self-consistent quark spectral function, for example by resumming the leading ladder diagrams, and compare with the one-loop Breit-Wigner result; if the $\mu_5$ suppression disappears or changes sign, the central claim is falsified. A complementary check is to extract the vector spectral function from lattice simulations at imaginary $\mu_5$ and analytically continue to real $\mu_5$ to compare $\sigma_{\rm el}/T$ directly.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dimensionless electrical conductivity $\sigma_{\rm el}/T$ of hot, dense quark matter is significantly reduced by a finite chiral chemical potential $\mu_5$, with the reduction concentrated in the chirally broken phase. In the NJL model, turning on $\mu_5$ increases the constituent quark mass at low temperature (chiral catalysis), lowers the chiral crossover temperature (inverse chiral catalysis), and makes the quark and antiquark occupation factors in Eq. (22) helicity dependent. The authors evaluate the one-loop vector spectral function in the real-time thermal field theory formalism, regulate the pinch singularity by a finite Breit-Wigner width $\Gamma$, identify that width with the thermal width computed from $2\to 2$ scattering, and find that $\sigma_{\rm el}/T$ falls monotonically as both $T$ and $\mu_5$ grow; at $T\simeq 120$ MeV the drop caused by $\mu_5=250$ MeV is large relative to $\mu_5=0$, while at $T\simeq 220$ MeV the $\mu_5$ dependence is weak and slightly non-monotonic.

Load-bearing premise

The central bet is that the small artificial width used to smooth the delta-function in the one-loop formula can be treated as the real scattering width of a quark in the medium; if that identification fails, the absolute values and the detailed temperature and $\mu_5$ dependence of $\sigma_{\rm el}$ change, even if the direction of the effect could survive.

Editorial extensions

If this is right

A finite chiral chemical potential lowers $\sigma_{\rm el}/T$ monotonically, and in the low-temperature chirally broken phase the suppression is large: at $T\simeq 120$ MeV, raising $\mu_5$ to 250 MeV substantially reduces the ratio relative to $\mu_5=0$.
The $\mu_5$ dependence nearly disappears at high temperature; around $T\simeq 220$ MeV the curves become weakly non-monotonic, so chiral imbalance affects transport mainly when chiral symmetry is still broken or only partially restored.
A nonzero $\mu_5$ opens Landau cuts in the scalar and pseudoscalar polarization functions, which modifies the $2\to 2$ cross sections that set the thermal width; the conductivity inherits the chiral-imbalance dependence through that width.
Because the vector-current response controls the time evolution of electromagnetic fields and the emission of photons and dileptons from the plasma, a lower conductivity in chirally imbalanced quark matter would feed into those observables if the central claim is correct.

Reading between the lines

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

A test the paper does not perform is replacing the $2\to 2$ scattering width by a self-consistently computed quark self-energy and checking whether the $\mu_5$ suppression survives; that calculation would separate the physics of the width prescription from a genuine medium effect.
The same vector spectral function determines photon and dilepton emission rates, so the finite-$\mu_5$ machinery developed here immediately implies altered emission rates in a chirally imbalanced plasma, although the paper does not compute them.
A lattice simulation of the vector spectral function at imaginary $\mu_5$, followed by analytic continuation, would be a direct benchmark for this conductivity prediction; given that the paper's $T_c(\mu_5)$ curve disagrees with current lattice results, any eventual retuning of the model parameters could shift the predicted magnitude of the effect.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: This is a workmanlike first calculation of electrical conductivity in the two-flavour NJL model at finite chiral chemical potential. The central qualitative claim—σel/T decreases as μ5 grows, most sharply in the chirally broken phase—is credible. The framework is inherited from the authors' earlier papers on vector spectral functions and polarization functions, but the application to conductivity is new, and the authors are candid about the model's limitations.

What the paper does well: it gives the full chain—gap equation, polarization functions, 2→2 cross sections, thermal widths, Kubo formula—with enough analytic detail to reproduce the results. The μ5→0 limit of Eq. (22) correctly reduces to the known expression. They also correct a typo from their earlier work, and they openly compare their inverse chiral catalysis with the lattice chiral catalysis, discussing possible fixes instead of hiding the discrepancy. No code or data is shipped, but the expressions are explicit.

The soft spots are real, though not fatal. First, the title promises 'hot and dense,' but all numerics are at μ=0; the authors justify this as parameter-space reduction, but it is a mismatch with the framing. Second, and more load-bearing: the width Γ that regularizes the Kubo pinch singularity is identified with the total 2→2 scattering rate, not a transport relaxation rate. Electrical conductivity in kinetic theory weights collisions by (1−cosθ); using σ_total instead of σ_tr can change both the magnitude and the μ5-dependence of Γ. The paper itself notes that a rigorous treatment requires ladder resummation and is only justified at large N_c. So the absolute values, and the detailed T/μ5 shape of σel/T, are not secure—although the qualitative suppression is robust because both the larger constituent mass and the larger scattering-partner density at finite μ5 push σel down. Third, the strong cutoff sensitivity of M and fπ shown in Fig. 2 is not propagated to σel; that is a missing check.

The citation pattern is largely self-referential, but this is a direct continuation of the authors' own formalism, and they engage the lattice and effective-model literature that disagrees with them.

This paper is for people doing NJL transport in chirally imbalanced matter. It deserves a serious referee. I would send it to review, with the expectation of a conditional acceptance after the μ=0 framing is fixed or softened, the transport-width issue is addressed quantitatively (or at least honestly caveated), and a cutoff-sensitivity estimate for σel is added. I would not use the numbers quantitatively until then.

Referee Report

3 major / 4 minor

Summary. The paper computes the electrical conductivity σel of two-flavour quark matter in the NJL model at finite temperature and chiral chemical potential μ5, using the one-loop Green-Kubo formula with the vector-current spectral function evaluated in the real-time formalism. The pinch singularity in the Kubo expression is regulated by a Breit-Wigner width Γ, which is then identified with the thermal width obtained from 2→2 quark/antiquark scattering with meson-exchange amplitudes. The central numerical result is that σel/T decreases with increasing μ5, especially in the low-temperature chirally broken phase (Fig. 8), and the paper also reports the temperature/μ5 dependence of the constituent mass, polarization functions, cross sections, and relaxation times.

Significance. If the result holds, it provides a new model-based estimate of how chiral imbalance affects charge transport in the quark-gluon plasma, a quantity relevant for electromagnetic-field evolution and charge fluctuations in heavy-ion collisions. The paper is transparent: all analytic expressions for the spectral function, polarization functions, and scattering amplitudes are given, and the numerical procedure is reproducible in principle. The authors also explicitly acknowledge known limitations, including the need for ladder resummation and the discrepancy between their inverse chiral catalysis and the LQCD trend. The qualitative prediction that σel/T is suppressed by μ5 in the broken phase is an interesting and falsifiable statement, but its quantitative support depends on several unverified identifications discussed below.

major comments (3)

[Secs. III and IV, Eqs. (16), (22), (24), (25)] The central quantitative claim is that the Breit-Wigner width Γ introduced in Eq. (16) can be identified with the thermal width computed from total 2→2 cross sections in Eqs. (24)-(25). The paper itself concedes (Sec. III, after Eq. (22)) that a rigorous Kubo calculation requires ladder resummation and that the one-loop result is only justified in the large-N_c limit. Even granting this, the width entering the current-current correlation function is not generally the total collision rate: for electrical conductivity the relevant relaxation rate is weighted by (1−cosθ), i.e. by the transport cross section, not by σ_total. Since the NJL amplitudes in Eqs. (27)-(28) depend on the Mandelstam variables t and u through the meson propagators, the angular weighting is non-trivial and can affect both the magnitude and the μ5 dependence of the effective width. The paper does not address this distinction, so the absolute values of σel/T in Fig. 8 and the detailed temperature/μ5 dependence are not quantitatively secured, although the qualitative suppression trend may be robust.
[Sec. VI, Fig. 2] The sensitivity of the input constituent mass M to the smooth cutoff Λ is very large: a 10% change in Λ changes M by up to 60% at μ5=0 and the authors state that results with Λ changed by more than 5% cannot be trusted. However, this cutoff sensitivity is not propagated to σel. Since σel depends on M through the dispersion relations, the thermal distributions, and the cross sections entering Γ, the quantitative values of σel/T could shift significantly under a different regularization or a different μ5-dependent cutoff. A robustness test varying Λ within the allowed 5% window (or using a second regulator) should be reported before the numbers in Fig. 8 can be taken as a quantitative prediction.
[Abstract, Title, Sec. VI] The title and abstract advertise 'hot and dense quark matter', but all numerical results are presented at vanishing quark chemical potential μ=0, as explicitly stated in Sec. VI ('we have chosen to set μ to zero'). While the analytic expressions in Eqs. (8), (22), (24)-(25) contain μ, no numerical dependence on μ is shown. This makes the 'dense' part of the central claim unsubstantiated and creates a mismatch between the advertised scope and the actual content. The authors should either present representative finite-μ results (even a limited scan) or revise the title and abstract to reflect that only the μ=0 case is studied numerically.

minor comments (4)

[Fig. 6 caption] The caption states that the plots correspond to the same nine representative combinations of temperature and CCP chosen in Fig. 4, but Fig. 6 contains only six panels; the three μ5 values are shown as curves within each panel. This wording is confusing and should be corrected.
[Sec. III, after Eq. (22)] There is a typo: 'aniquarks' should read 'antiquarks'.
[Sec. II, Eq. (5)] The authors note that the sign of the σμν term in Eq. (5) was corrected from a previous work; it would be helpful to comment briefly on whether this correction affects any of the earlier published results, if at all.
[Eqs. (24)-(25)] The degeneracy factor g=N_cN_f in the width formulas is stated without derivation; clarifying whether this accounts for the summed helicity and flavor indices as shown would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the μ5 dependence of σel emerges from the NJL gap equation, propagator, and scattering widths rather than from fitting or constructed definitions.

full rationale

The derivation chain is self-contained. The electrical conductivity is obtained from the Kubo formula (Eqs. 10-22) using an explicit μ5-dependent quark propagator (Eq. 4) and the gap equation (Eq. 8). The Breit-Wigner width Γ that regulates the pinch singularity is not fitted to conductivity data; it is computed from 2→2 NJL scattering cross sections (Eqs. 24-28), with polarization functions Π_h evaluated at finite μ5 in Sec. V. The μ5 dependence of σel/T therefore emerges from the model dynamics (quark densities, constituent mass, and scattering amplitudes) rather than being imposed by construction. The paper's reliance on its earlier Refs. [65,69] for the spectral-function and polarization-function formalism is not load-bearing: the relevant equations are re-derived in this paper (Eqs. 14-18 and 31-37), and no conclusion is supported solely by a self-citation chain or by a uniqueness theorem imported from the authors' prior work. The paper's own caveat that a rigorous treatment would require ladder resummation (Sec. III) concerns the quantitative accuracy of the one-loop approximation and the identification of Γ with the physical thermal width, not a circular reduction of the result to its inputs. Thus no specific circular step can be exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the NJL model's three fitted parameters, the choice of a particular regularization, and the identification of the thermal width. No new particles or forces are introduced. The chiral chemical potential is an external control parameter, not a fitted quantity.

free parameters (3)

NJL current quark mass m = 5.6 MeV
Fixed by reproducing vacuum pion properties; used in the gap equation (Eq. 8).
NJL scalar coupling G_s = 5.742 GeV^-2
Fixed to vacuum fπ and mπ; controls meson propagators (Eq. 29) and the gap equation.
Smooth cutoff Λ = 568.69 MeV
Fixed to vacuum quark condensate and pion decay constant; regulates UV divergences. The paper shows strong sensitivity of M to Λ (Fig. 2).

assumptions (5)

domain assumption The 2-flavour gauged NJL Lagrangian with scalar and pseudoscalar interactions and a chiral chemical potential is a valid low-energy description of QCD for this transport calculation.
The model replaces gluon dynamics with point-like four-fermion interactions; this is the standard NJL approximation, not derived from QCD.
standard math Real-time finite-temperature field theory with the Schwinger-Keldysh contour is used for all spectral and polarization functions.
This is a standard, well-established formalism for thermal field theory.
domain assumption The large-N_c expansion justifies the mean-field gap equation and the neglect of ladder resummation in the Kubo formula.
The paper argues that ladder diagrams are subleading in 1/N_c because each rung costs a factor G_s N_c ~ 1/N_c (Sec. III). This is an approximation, not an exact result.
ad hoc to paper The Breit-Wigner width Γ that regularizes the pinch singularity can be identified with the thermal width computed from 2→2 scattering.
This identification is the central dynamical input linking the one-loop spectral function to σel (Eqs. 16-22, Sec. IV). It is not derived from first principles.
ad hoc to paper The μ5-independent smooth three-momentum cutoff does not qualitatively alter the results.
The paper discusses that μ5-dependent regulators could be chosen and that the regularization is non-unique (Eq. 9 discussion, Sec. II), so conclusions carry regulator dependence.

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Cite this review

Pith. "Pith review of Effect of chiral imbalance on the electrical conductivity of hot and dense quark matter using Green-Kubo Method within the 2-flavour gauged NJL model." pith.science (2026). https://pith.science/paper/IV37PJKR

@misc{pith2026241117117,
  author       = {Pith},
  title        = {Pith review of: Effect of chiral imbalance on the electrical conductivity of hot and dense quark matter using Green-Kubo Method within the 2-flavour gauged NJL model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IV37PJKR}},
  note         = {Machine review of arXiv:2411.17117}
}

abstract

The electrical conductivity of hot and dense quark matter is calculated using the 2-flavour gauged Nambu-Jona--Lasinio (NJL) model in the presence of a chiral imbalance quantified in terms of a chiral chemical potential (CCP). To this end, the in-medium spectral function corresponding to the vector current correlator is evaluated employing the real time formulation of finite temperature field theory. Taking the long wavelength limit of the spectral function we extract the electrical conductivity using the Green-Kubo relation. The thermal widths of the quarks/antiquarks that appear in the expression of electrical conductivity are calculated by considering the $2\to2$ scattering in the NJL model. The scattering amplitudes containing the polarization functions of the mesonic modes in the scalar and pseudoscalar channels are also evaluated by considering finite value of CCP. We find that the ratio of electrical conductivity to temperature has significant dependence on CCP especially in the low temperature region.

Figures

Figures reproduced from arXiv: 2411.17117 by the authors.

**Figure 2.** (Color Online) Relative cutoff parameter [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** (Color Online) The variation of the (a) constituent quark mass [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: (Color Online) The variation of polarization functions of the scalar ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: (Color Online) The variation of polarization functions of the scalar ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: (Color Online) The isospin-averaged total cross sections for the processes [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: (Color Online) The variation of the average relaxation time [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: (Color Online) The variation of 𝜎el/𝑇 as a function of (a) temperature, (b) CCP approximating ⟨𝜏⟩± ∼ 1 𝑛𝜎 from Eqs. (24) and (25) where 𝑛 is the quark/antiquark density (they are same at 𝜇 = 0) at finite CCP and 𝜎 is the value of typical 2 → 2 cross section in the medi…

**Figure 9.** Figure 9: (Color Online) The Feynman diagrams for 𝑞(𝑘)𝑞(𝑝) → 𝑞(𝑘 ′ )𝑞(𝑝 ′ ) scattering via (a) 𝔱-channel and (b) 𝔲-channel in RPA. The arrows represent the direction of momentum. Phase Approximation (RPA) come out to be M(1) = M(6) = [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: (Color Online) The Feynman diagrams for 𝑞(𝑘)𝑞¯(𝑝) → 𝑞(𝑘 ′ )𝑞¯(𝑝 ′ ) scattering via (a) 𝔰-channel and (b) 𝔱-channel in RPA. The arrows represent the direction of momentum. to be M˜ (1) = M˜ (6) = −2M˜𝜋 𝔰 + M˜𝜋 𝔱 − M˜𝜎 𝔱 , (A29) M˜ (2) = M˜ (5) = −M˜𝜋 𝔰 − M˜𝜋 𝔱 − …

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