Interacting hadron resonance gas model in magnetic field and the fluctuations of conserved charges

Abhijit Bhattacharyya; Guruprasad Kadam; Somenath Pal

REVIEW 4 major objections 5 minor 2 cited by

Interacting hadron resonance gas model in magnetic field and the fluctuations of conserved charges

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the low-temperature hadronic phase of QCD remains paramagnetic even when short-range repulsive interactions between hadrons are included, and that excluded-volume repulsion barely changes the magnetization.

desk verdict A useful hadronic EoS baseline in a magnetic field, but the vacuum-pressure sign that drives the paramagnetic claim is inconsistent as printed. read the letter →

arxiv 1908.10618 v2 pith:VXTC3KJZ submitted 2019-08-28 hep-ph nucl-th

Guruprasad Kadam , Somenath Pal , Abhijit Bhattacharyya This is my paper · ORCID

classification hep-phnucl-th PACS 12.38.Mh12.39.-x11.30.Rd11.30.Er

keywords hadronresonancegasexcludedvolumemagneticfieldmagnetizationparamagneticmatterconservedchargesusceptibilitiesLandaulevelsQCDphasediagram

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 extends the hadron resonance gas model to a constant external magnetic field with short-range repulsive interactions between hadrons, implemented through an excluded-volume correction to the pressure. It asks whether the magnetic response of low-temperature hadronic matter survives once repulsion is included. The authors find that the magnetization of hadronic matter is positive, so the hadronic phase is paramagnetic, and that repulsive interactions have only a negligible effect on the magnetization. They also find that both the magnetic field and excluded-volume repulsion suppress the baryon and electric-charge susceptibilities, with the suppression stronger for higher-order fluctuations. If the paper is right, the hadronic phase below roughly 160 MeV is paramagnetic even when hadron sizes are taken into account.

What carries the argument

The load-bearing object is the excluded-volume pressure equation $P_{\mathrm{EV}}(T,\mu,B) = P_{\mathrm{id}}(T, \mu - v P_{\mathrm{EV}}, B)$, a self-consistent transcendental equation that shifts the chemical potential by the excluded-volume pressure. The ideal pressure for charged species is a sum over Landau levels and spin projections, while neutral species use the free-particle dispersion. Vacuum divergences are removed by magnetic-field-independent regularization, which separates the magnetic-field-dependent vacuum pressure from the zero-field vacuum pressure and yields renormalized expressions for spin-0, spin-1/2, and spin-1 hadrons. This regularized vacuum term supplies the dominant positive magnetization, while the thermal term carries the temperature dependence and the excluded-volume correction carries the repulsive interaction.

What would settle it

Compute the excluded-volume pressure using the parallel and perpendicular pressures in place of the isotropic ideal pressure in the self-consistent equation and check whether the magnetization remains positive; alternatively, measure the magnetization of QCD matter on the lattice at $eB\approx 0.2$ GeV$^2$ and temperatures below 160 MeV and see whether it is positive and nearly independent of hadron hard-core radii.

Watch

Extended reading notes

Core claim

The central claim is that an interacting hadron resonance gas in a constant magnetic field produces positive magnetization for low-temperature hadronic matter. The vacuum contribution, regularized by a magnetic-field-independent scheme, dominates and fixes the sign; the thermal contribution is negative at low temperature once pions populate but turns positive when spin-1 and spin-1/2 hadrons contribute. Adding excluded-volume repulsion suppresses the pressure, energy density, entropy density, and all studied baryon and electric-charge susceptibilities, but changes the magnetization by only a negligible amount, leaving the paramagnetic character intact. At finite baryon chemical potential, $\mu_B = 300$ MeV, the same qualitative picture holds.

Load-bearing premise

The argument hinges on assuming that the excluded-volume equation, built from the isotropic ideal pressure, remains valid inside a magnetic field with a single magnetic-field-independent excluded-volume parameter, even though the pressure in a magnetic field is anisotropic.

Editorial extensions

If this is right

Below about 160 MeV, the hadronic phase should be paramagnetic, with positive magnetization, even when hadron hard-core repulsion is included.
Excluded-volume repulsion suppresses pressure, energy density, and entropy density, and the suppression grows with temperature and becomes stronger at finite baryon chemical potential.
Baryon and electric-charge susceptibilities of all studied orders are reduced by both the magnetic field and repulsive interactions, with higher-order susceptibilities suppressed more strongly.
Ratios such as $\chi^6_B/\chi^4_B$ drop sharply in the interacting model, making higher-order fluctuation ratios a sensitive probe of repulsion.
The magnetic field suppresses the thermal population of spin-0 hadrons by increasing their effective mass, while spin-1 hadrons become lighter, and this shapes the thermal contribution to magnetization.

Reading between the lines

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

If the positive vacuum magnetization is dominant, hadron-gas magnetization is largely fixed by the vacuum sector, so a lattice QCD measurement of magnetization below the transition temperature could directly test whether the vacuum and thermal balance is as the model predicts.
A testable extension would replace the isotropic pressure in the excluded-volume equation with the parallel or perpendicular pressure of the magnetized system; this could change the size of the repulsive correction, though it would likely leave the sign of the magnetization unchanged.
The model's susceptibility ratios suggest an experimental route: comparing higher-order fluctuation ratios between central and peripheral heavy-ion collisions, where the magnetic field differs, could probe magnetic-field effects if freeze-out parameters are controlled.
The assumption that the excluded-volume parameter is independent of the magnetic field could be relaxed, and a field-dependent hard-core radius would introduce a new source of suppression that could be checked against future lattice results.

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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

Short version: this is the first combination I know of the excluded-volume HRG with a magnetic-field HRG, and the qualitative claims—positive magnetization that is insensitive to excluded volume, magnetic suppression of baryon and charge susceptibilities—are the sort of baseline the field can use. It is not a breakthrough, and the quantitative results should not be used until a sign problem at the center of the derivation is resolved.

What's good: the model has no fitted parameters aimed at the target observables. The hard-core radii come from an earlier yield fit, g=2 is the tree-level value, and eB=0.2 GeV^2 is representative. The MFIR regularization follows Endrődi and earlier work, and the particle list is standard. The paper is also clear that no experimental comparison is possible yet because there is no parametrization of B with collision energy, and lattice results for the susceptibilities in a field are not available. That is honest.

The soft spots are real. The printed derivation of the vacuum pressure, Eqs. (10)–(11), contains a sign inconsistency: the dispersion in Eq. (5) for the lowest spin-up Landau level is p_z^2 + m^2 + 2eBn, but Eq. (10) writes sqrt(p_z^2 + m^2 − 2eBn). The Hurwitz-zeta form in Eq. (11) implicitly uses the plus sign, since x = m^2/(2eB) would not arise from the minus sign. So as printed the derivation cannot be followed. This matters because Fig. 3 shows the total magnetization is almost entirely the vacuum contribution; the thermal part is two orders of magnitude smaller. If the sign of ΔP_vac is wrong, the paramagnetic claim goes with it. My own quick evaluation of the printed Eq. (22) at x=2 gives a negative vacuum pressure, which conflicts with the text's claim that it is positive for a wide range and with the log plot in Fig. 1(a). This might be a typo or a sign convention issue, but it has to be fixed and checked against Endrődi's result before the physics claim can be trusted.

The second soft spot is the excluded-volume extension in Eq. (23), which uses the isotropic ideal pressure in a magnetic field where the pressure is anisotropic. The paper does not justify this or explain why the excluded-volume parameter v should be B-independent. That concern mostly affects the thermal part and hence the quantitative susceptibility suppression; it does not rescue or destroy the vacuum part.

Bottom line: this is a useful paper for people building hadronic equations of state in magnetic fields, but as written the key derivation is inconsistent. A serious referee could sort it out; I would send it out rather than desk reject. If the sign check fails, the paper is mainly a null/incomplete result; if it passes, the model is a solid baseline.

Referee Report

4 major / 5 minor

Summary. The paper extends the hadron resonance gas (HRG) model to a constant magnetic field, regularizes the charged-hadron vacuum pressure with the magnetic-field-independent regularization (MFIR) scheme, and adds van der Waals excluded-volume interactions with hard-core radii taken from an earlier fit to hadron yields. It computes the pressure, energy and entropy densities, magnetization, and baryon- and electric-charge susceptibilities of hadronic matter up to T ≈ 0.16 GeV at zero and finite baryon chemical potential, comparing the equation of state with lattice data of Ref. [77] up to T ≈ 140 MeV. The central claims are: (i) low-temperature hadronic matter is paramagnetic, with the sign of the magnetization set by the regularized vacuum contribution; (ii) repulsive excluded-volume interactions have negligible effect on the total magnetization; and (iii) both the magnetic field and the excluded-volume repulsion suppress the conserved-charge susceptibilities and modify the susceptibility ratios.

Significance. If the claims hold, the paramagnetism of the low-temperature hadronic phase and its insensitivity to repulsive interactions are clean, falsifiable predictions that can be confronted with lattice QCD results for the equation of state in magnetic fields; the same model reproduces the available lattice pressure data up to T ≈ 140 MeV both with and without B. The paper deserves credit for not fitting any parameter to its target observables: the hard-core radii are taken from Ref. [44], the gyromagnetic ratios are the tree-level values g_i = 2, and eB = 0.2 GeV^2 is a representative choice, so the results are genuine model predictions rather than fits. The vacuum-pressure formulas for spins 0, 1/2 and 1 are given explicitly in the appendices, and the susceptibility ratios χ4/χ2 and χ6/χ4 are concrete outputs that can be compared with future lattice data. The main caveats are that the printed vacuum-pressure derivation contains internal sign inconsistencies that must be corrected before the central claim can be trusted, and that the susceptibility conclusions rest on an assumed, rather than derived, extension of the excluded-volume scheme to magnetic fields.

major comments (4)

[Sec. 3, Eqs. (9)-(11)] The printed derivation of the vacuum pressure is internally inconsistent. Eq. (10) contains sqrt(p_z^2 + m^2 - 2eBn), which becomes imaginary for n > m^2/(2eB) and is inconsistent both with the Landau dispersion in Eq. (5), which has a plus sign for the lowest spin-up level, and with the Hurwitz-zeta evaluation in Eq. (11), whose argument x = m^2/(2eB) can only arise from the plus-sign form sqrt(p_z^2 + m^2 + 2eBn). In addition, Eq. (9) subtracts E_{p,0}/2 inside the sum while Eq. (10) subtracts E_0 once, so the lowest-Landau-level subtraction differs by a factor of two between the two equations. Since Eq. (22) determines the sign of the vacuum magnetization, which is the basis of the paper's headline paramagnetic claim, this step must be corrected and re-derived, or explicitly identified as typographical with the corrected intermediate steps supplied.
[Sec. 3, Eq. (22) and Fig. 1(a)] The sign of the spin-1/2 vacuum pressure given by the paper's own formula appears to contradict the text. Evaluating Eq. (22) at x = 2 gives Delta P_vac ≈ -0.468 (eB)^2/(2π^2) ≈ -0.024 (eB)^2, i.e., a negative value, and the value at the proton-relevant x ≈ 2.2 is likewise negative, whereas Sec. 5 states that the vacuum pressure is positive for a wide range of magnetic fields for all three spin channels. Because Fig. 1(a) uses a logarithmic vertical scale it cannot display negative values, so the figure should state what is actually plotted, and the sign of Delta P_vac, and of M_vac = ∂(Delta P_vac)/∂B, should be given explicitly over the plotted range of x; this is the quantity that supports the paramagnetic conclusion.
[Sec. 4, Eq. (23)] The extension of the excluded-volume scheme to finite magnetic fields is assumed rather than derived. In a magnetic field the ideal-gas pressure is anisotropic, with P_parallel = -Ω and P_perpendicular = -Ω - MB, whereas Eq. (23) is a scalar transcendental equation using the isotropic P_id, and the excluded-volume parameter v is taken to be B-independent. This equation is the sole input generating the interaction effects on the susceptibilities in Figs. 5-8, so the paper should either justify the scalar form from a microscopic starting point or clearly state that Eq. (23) is an ansatz and estimate the resulting uncertainty; a concrete diagnostic would be to compare the solution of Eq. (23) obtained with P_parallel and with P_perpendicular in place of P_id.
[Sec. 2, Eqs. (2)-(6)] The Landau spectrum and the degeneracy factor are written with the signed charge e_i throughout, e.g., E_{i,c}^2 = p_z^2 + m_i^2 + 2 e_i B (n + 1/2 - S_z) and the prefactor eB/(2π)^2. For the negatively charged hadrons in Table 1 (π^-, K^-, ρ^-, K*^-, Σ^- and antiprotons) this would give imaginary energies at large n and a negative degeneracy factor; the physical spectrum and degeneracy involve |e_i|B. The text should state explicitly that |e_i|B is used throughout; as printed, the model equations are only defined for positively charged species.

minor comments (5)

[Sec. 2, Eq. (5)] The text says n is 'any positive integer', but the lowest Landau level n = 0 is used throughout, for instance in Eq. (9); n should be described as a non-negative integer.
[Eq. (25)] The Boltzmann factors contain unmatched parentheses, with '(exp [Ej - µi)/T ]))' appearing twice; the intended expression is exp[(E_j - μ_i)/T], and the sign convention for fermions and bosons should be stated explicitly.
[Fig. 4 caption] The caption contains a duplicated '(b)' label in the sub-caption for the energy-density panel.
[Sec. 6] The summary states that 'all the thermodynamical quantities are strongly suppressed due to non-zero background magnetic field', which is not consistent with Fig. 1(b), where the vacuum contribution makes the total pressure at low T larger for eB = 0.2 GeV^2 than for eB = 0; the statement should be restricted to the thermal parts of the thermodynamic quantities.
[Sec. 5] Describing the positive sign of the magnetization as 'a fundamental characteristic of thermal QCD vacuum' overstates what an HRG model calculation can establish; a more cautious formulation is advised, since the result is a property of the MFIR-regularized hadron resonance gas model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported magnetization and susceptibilities are model outputs, and the model inputs come from external fits and standard regularization, not from the target quantities.

full rationale

The derivation chain is self-contained in the sense required for this circularity pass. The hard-core radii used in the excluded-volume equation (Eq. 23) are taken from Ref. [44], an external fit to hadron yields, and they are not tuned to the magnetization or charge susceptibilities that the paper reports. The gyromagnetic factors gi = 2|ei/e| are fixed by the tree-level argument of Ref. [65], and eB = 0.2 GeV^2 is a representative input. The vacuum pressure is obtained by the MFIR regularization formulas of Sec. 3, with divergences removed by charge and field renormalization; the sign and magnitude of Delta P_vac then determine the magnetization as a derived consequence, not as an imposed constraint. The excluded-volume extension is an explicit model assumption cited to Ref. [73] and is not advertised as a first-principles derivation of the target claims. No parameter is fitted to the predicted quantities, no uniqueness theorem from the same authors is invoked to force a choice, and the self-citations in the paper are contextual rather than load-bearing. The paper also checks its pressure results against external lattice QCD data from Ref. [77] and explicitly notes where quantitative comparison is not yet possible. Even if the sign issue in Eq. (10) raised by the skeptic were correct, it would be a correctness or consistency problem, not a circularity problem, because it would not make the derivation equivalent to its inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The calculation relies on standard HRG and EVHRG assumptions, a set of hard core radii from a prior fit, and a representative magnetic field value. No new particles or forces are introduced.

free parameters (2)

hadron hard core radii (r_pi, r_K, r_m, r_b) = r_pi=0 fm, r_K=0.35 fm, r_m=0.3 fm, r_b=0.5 fm
Multi-component excluded volume radii taken from Bugaev et al. (Ref 44), where they were fitted to hadron yield ratios; they are inputs here, not fitted in this paper.
magnetic field strength eB = 0.2 GeV^2
Representative value for off central heavy-ion collisions, chosen by hand for the plots; the results depend on this value.

assumptions (5)

domain assumption Dashen-Ma-Bernstein theorem: the partition function of a hadronic system is a sum of non-interacting hadrons and narrow resonances.
Basis of the HRG model, invoked in Section 2 before Eq. (2).
ad hoc to paper The thermodynamically consistent excluded volume equation P_EV = P_id(T, mu - v P_EV, B) remains valid in a magnetic field.
Eq. (23) introduces this extension without derivation or discussion of pressure anisotropy in a magnetic field.
domain assumption Gyromagnetic ratio gi = 2|ei/e| for all charged hadrons.
Section 2 states this tree level value; experimental values are known for few hadrons and have large uncertainties.
domain assumption Only hadrons with spin S < 3/2 are included for stability reasons.
Section 5 and Table 1: heavier spin states are excluded following Ref 63 to avoid unphysical vacuum behavior.
standard math The vacuum divergence is cancelled using magnetic field independent regularization with charge and field renormalization, with M* = m.
Section 3, Eqs. (19)-(22); standard MFIR scheme from Refs 67, 68, 63.

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Pith. "Pith review of Interacting hadron resonance gas model in magnetic field and the fluctuations of conserved charges." pith.science (2026). https://pith.science/paper/VXTC3KJZ

@misc{pith2026190810618,
  author       = {Pith},
  title        = {Pith review of: Interacting hadron resonance gas model in magnetic field and the fluctuations of conserved charges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXTC3KJZ}},
  note         = {Machine review of arXiv:1908.10618}
}

abstract

In this paper we discuss the interacting hadron resonance gas model in presence of a constant external magnetic field. The short range repulsive interaction between hadrons are accounted through van der Waals excluded volume correction to the ideal gas pressure. Here we take the sizes of hadrons as $r_\pi$ (pion radius) $= 0$ fm, $r_K$ (kaon radius) $= 0.35$ fm, $r_m$ (all other meson radii) $= 0.3$ fm and $r_b$ (baryon radii) $= 0.5$ fm. We analyse the effect of uniform background magnetic field on the thermodynamic properties of interacting hadron gas. We especially discuss the effect of interactions on the behaviour of magnetization of low temperature hadronic matter. The vacuum terms have been regularized using magnetic field independent regularization scheme. We find that the magnetization of hadronic matter is positive which implies that the low temperature hadronic matter is paramagnetic. We further find that the repulsive interactions have very negligible effect on the overall magnetization of the hadronic matter and the paramagnetic property of the hadronic phase remains unchanged. We have also investigated the effects of short range repulsive interactions as well as the magnetic field on the baryon and electric charge number susceptibilities of hadronic matter within the ambit of excluded volume hadron resonance gas model.

Figures

Figures reproduced from arXiv: 1908.10618 by the authors.

**Figure 2.** (Color Online) Energy density (left panel) and entropy den [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** (Color Online) Magnetization of hadronic matter estimated [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: (Color Online) Thermodynamic variables at [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: (Color Online) Baryon susceptibilities of the hadronic matte [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: (Color Online) Electric charge susceptibilities of the hadron [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: (Color Online) Ratios of baryon susceptibilities of the hadro [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: (Color Online) Ratios of electric charge susceptibilities of th [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reviewed August 14, 2026 · model on record in the stance chip above.

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Introduction The phase diagram of strongly interacting matter described by qua ntum chromody- namics (QCD) is one of the active topic of research today. At low tem perature ( T ) and low baryon chemical potential ( µB) strongly interacting matter consists of colorless hadrons, while at high temperature and/or high baryon density the fundamental degrees of...

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Hadron resonance gas model in magnetic ﬁeld The free energy density in the presence of constant external ma gnetic ﬁeld B is written in terms of partition function as F = −T lnZ = Fvac + Fth (1) where Fvac and Fth are vacuum and thermal parts respectively. In the ideal HRG mode l the free energy at low temperature and in the dilute gas approximat ion is g...

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It has been recently shown that the appropriate regularization sche me is necessary to avoid certain unphysical results

Regularization of vacuum pressure The vacuum pressure is divergent and it needs to be properly regula rized ﬁrst. It has been recently shown that the appropriate regularization sche me is necessary to avoid certain unphysical results. For instance, some studies have found the oscillations in the magnetization while others have found imaginary meson masses...

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Interacting Hadron resonance gas model in magnetic ﬁeld The hadron resonance gas model, deﬁned by Eqs.(

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We would like to ment ion at this stage that the results have been obtained around the point µQ = 0 and µS = 0 and both zero and ﬁnite µB cases have been explored

Results and discussion Let us now discuss the results we have obtained. We would like to ment ion at this stage that the results have been obtained around the point µQ = 0 and µS = 0 and both zero and ﬁnite µB cases have been explored. Table 1 shows all the hadrons and resonances included in the HRG descriptio n. For the stability reasons, as discussed in...

work page

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One can extend this ideal HRG model by taking in to account the repulsive interactions between hadrons via Van der Waals (VDW) excluded-volume correction to the partition function

and ( 7), corresponds to non- interacting gas of hadrons. One can extend this ideal HRG model by taking in to account the repulsive interactions between hadrons via Van der Waals (VDW) excluded-volume correction to the partition function. In the thermodynamically con sistent excluded volume formulation one can obtain the transcendental equation fo r the (...