$S$-matrix calculation of $BQ$ correlation at finite baryon density

Boris Tomasik; Pok Man Lo; Vojtech Honek

arxiv: 2604.09951 · v1 · submitted 2026-04-10 · ⚛️ nucl-th · hep-ph

S-matrix calculation of BQ correlation at finite baryon density

Vojtech Honek , Pok Man Lo , Boris Tomasik This is my paper

Pith reviewed 2026-05-10 16:00 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph

keywords baryon susceptibilitycharge susceptibilityS-matrix formalismhadron gas modelfinite baryon densitychemical freeze-outpartial chemical equilibrium

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The pith

Hadron gas model with S-matrix pion-nucleon interactions predicts sharp rise in baryon-electric charge susceptibility at higher chemical potential.

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

The paper computes the baryon number-electric charge susceptibility in a hadron gas model that includes pion-nucleon interactions through the S-matrix formalism at non-zero baryon chemical potential. It finds that the susceptibility increases substantially as the chemical potential grows in ranges relevant to heavy-ion collisions. The results are evaluated along the chemical freeze-out line and during the cooling evolution of a fireball under the partial chemical equilibrium model. This matters because susceptibilities connect directly to measurable fluctuation observables that probe the equation of state of dense QCD matter.

Core claim

Accounting for pion-nucleon interactions via the S-matrix formalism inside the hadron resonance gas model produces a large increase in the baryon-electric charge susceptibility when the baryon chemical potential is raised within a phenomenologically relevant interval. The enhanced values are tracked along the chemical freeze-out line and followed through the expansion and cooling of the fireball in partial chemical equilibrium.

What carries the argument

The S-matrix formalism for pion-nucleon scattering embedded in the hadron gas model, which incorporates interaction phase shifts to adjust the thermodynamic susceptibilities beyond the basic resonance counting.

If this is right

The susceptibility grows significantly with rising baryon chemical potential in the relevant interval.
Along the chemical freeze-out line the susceptibility takes higher values than in the non-interacting resonance gas.
In the cooling fireball the susceptibility evolves according to the partial chemical equilibrium prescription.
This changes the expected size of baryon-charge correlations accessible in fluctuation measurements.

Where Pith is reading between the lines

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

The rise could shift baseline predictions for cumulant ratios used in searches for a critical point at finite density.
Direct comparison with fluctuation data from lower-energy collisions would provide a concrete test of the model.
The same S-matrix approach could be applied to other mixed susceptibilities for consistency checks.

Load-bearing premise

The S-matrix formalism for pion-nucleon interactions remains valid and the hadron gas plus partial chemical equilibrium assumptions hold at the finite baryon densities and temperatures considered.

What would settle it

Precise experimental extraction of the baryon-electric charge susceptibility or its ratio to other susceptibilities in heavy-ion collisions at beam energies spanning a range of baryon chemical potentials would directly test whether the predicted increase appears.

read the original abstract

We calculate the baryon number--electric charge susceptibility at non-vanishing baryo-chemical potential within the model of hadron gas where pion-nucleon interaction is accounted for by the $S$-matrix formalism. The susceptibility is largely increased when the chemical potential grows within a phenomenologically relevant interval. The results are then evaluated along the chemical freeze-out line. We also calculate the evolution of the susceptibility in a cooling fireball by making use of the Partial Chemical Equilibrium model.

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 extends the S-matrix hadron-gas model to finite mu_B and reports a clear rise in baryon-charge susceptibility, but the vacuum phase-shift inputs are used without medium corrections that likely matter at the densities reached.

read the letter

The main takeaway is that the susceptibility grows noticeably with baryon chemical potential in the range probed by current heavy-ion runs. They apply the established S-matrix treatment of pion-nucleon scattering inside the hadron resonance gas, evaluate the result along the chemical freeze-out line, and also follow its evolution in a cooling fireball under partial chemical equilibrium. That gives a concrete baseline number for fluctuation observables at finite density, which is the part that could be directly useful to people analyzing data from BES or future facilities. The calculation itself follows the standard virial expansion route, so once the phase shifts are inserted the numerics are straightforward and the partial-equilibrium extension adds a dynamical layer that prior static calculations lacked. Credit is due for producing a specific, usable result rather than stopping at the formalism. The soft spot is the reliance on vacuum S-matrix elements. At mu_B values of a few hundred MeV the nucleon density approaches nuclear saturation, where Pauli blocking, modified dispersions, and self-energies would change the interaction contribution to the pressure. Since the susceptibility is a second derivative, any density dependence in that term scales directly into the reported increase. The work gives no indication that these corrections were bounded or included, so the size of the growth remains sensitive to an assumption whose validity shrinks exactly where the effect is claimed to be largest. The paper is aimed at the heavy-ion fluctuation community and at modelers who need a reference curve at finite density. A reader already working with S-matrix or virial approaches would find the numbers and the freeze-out-line plot worth checking, even if they later add medium modifications themselves. It deserves a serious referee. The method is applied consistently and the question is timely, but the referees should press on the range of validity of the vacuum inputs and ask for explicit checks or error bands on that point.

Referee Report

3 major / 3 minor

Summary. The paper computes the baryon-electric charge susceptibility χ_BQ in a hadron-resonance gas at finite baryon chemical potential μ_B, with pion-nucleon interactions incorporated via the S-matrix formalism applied to vacuum phase shifts. It reports a substantial increase in χ_BQ as μ_B grows in the phenomenologically relevant range, evaluates the quantity along the chemical freeze-out line, and follows its evolution in a cooling fireball under the partial chemical equilibrium (PCE) assumption.

Significance. If the central result holds, the work supplies a parameter-free estimate (using external scattering data) of a fluctuation observable directly relevant to the RHIC beam-energy scan and future FAIR/NICA experiments. The S-matrix treatment of πN interactions is a clear strength, as it incorporates both resonant and non-resonant contributions without additional model parameters. The extension to PCE evolution along an isentropic trajectory further connects the static susceptibility to dynamical freeze-out scenarios.

major comments (3)

[§2] §2 (S-matrix formalism for the pressure): the virial/S-matrix correction to the hadron-gas pressure is constructed exclusively from vacuum πN phase shifts. At μ_B ≳ 300 MeV the nucleon density becomes comparable to 0.05–0.1 fm^{-3}; the manuscript provides no medium-modified dispersion relations, self-energies, or Pauli-blocking factors in the two-body kernel. Because χ_BQ is obtained by second derivatives of this pressure with respect to μ_B and μ_Q, any density dependence in the interaction term directly scales the reported growth.
[§3] §3 (numerical results): the increase in χ_BQ is shown as a function of μ_B, yet no uncertainty band arising from the choice of phase-shift parametrization or from the omission of medium effects is displayed. A quantitative sensitivity study (e.g., varying the cutoff in the S-matrix integral or comparing to a resonance-gas baseline) is required to establish that the enhancement is robust rather than an artifact of the vacuum approximation.
[§4] §4 (PCE evolution): the time-dependent susceptibility is obtained by evolving the system along an isentropic trajectory while keeping the S-matrix correction fixed. The manuscript does not address whether the vacuum phase shifts remain appropriate once the system cools through the region where baryon density is still appreciable; this assumption is load-bearing for the final dynamical prediction.

minor comments (3)

[Abstract] The abstract states that the susceptibility is 'largely increased' but supplies no numerical factor; the text should quote the ratio χ_BQ(μ_B=400 MeV)/χ_BQ(μ_B=0) at a representative temperature.
Notation for the susceptibility (χ_BQ versus χ_{BQ}) and for the chemical potentials should be made uniform between equations and figure labels.
[Introduction] A short paragraph comparing the present S-matrix result to existing lattice QCD or HRG calculations at μ_B=0 would help place the finite-density enhancement in context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below, indicating the revisions we will implement.

read point-by-point responses

Referee: §2: The virial/S-matrix correction uses exclusively vacuum πN phase shifts. No medium-modified dispersion relations, self-energies, or Pauli-blocking factors are included at μ_B ≳ 300 MeV, where nucleon density reaches 0.05–0.1 fm^{-3}. Any density dependence in the interaction term would scale the reported growth in χ_BQ.

Authors: We acknowledge that the calculation employs vacuum phase shifts without medium modifications. This is the standard implementation of the S-matrix formalism in hadron resonance gas models, relying on experimental scattering data. Incorporating finite-density effects would require additional model inputs (e.g., self-energies from effective theories) that are outside the scope of this work. In the revised manuscript we will add an explicit discussion in §2 on the range of validity, noting that the dominant enhancement in χ_BQ stems from the rising nucleon density itself rather than details of the interaction kernel. revision: partial
Referee: §3: No uncertainty band from phase-shift parametrization or omission of medium effects is shown. A quantitative sensitivity study (varying cutoff or comparing to resonance-gas baseline) is required to confirm the enhancement is robust.

Authors: We agree that a sensitivity analysis would improve the presentation. In the revised version we will add a direct comparison of χ_BQ with and without the S-matrix correction to quantify the πN contribution. We will also examine the dependence on the integration cutoff in the S-matrix formula and reference the uncertainties in the input phase-shift parametrizations. A complete uncertainty band incorporating medium effects is not feasible within the present vacuum-based framework and will be noted as a limitation. revision: yes
Referee: §4: The PCE evolution keeps the S-matrix correction fixed along the isentropic trajectory without addressing whether vacuum phase shifts remain appropriate as the system cools while baryon density is still appreciable.

Authors: Within the PCE framework the S-matrix term is evaluated at chemical freeze-out and held fixed during the subsequent isentropic expansion, consistent with the frozen chemical composition assumption. We will clarify this point in the revised §4 and note that, as temperature decreases and baryon density drops along the trajectory, the vacuum phase shifts become progressively more suitable. A fully dynamical treatment with temperature- and density-dependent interactions lies beyond the present study. revision: partial

Circularity Check

0 steps flagged

No circularity: external S-matrix inputs drive independent thermodynamic output

full rationale

The derivation begins from the standard hadron-resonance gas pressure with the S-matrix correction for πN interactions taken from measured vacuum phase shifts. The baryon-electric charge susceptibility is then obtained strictly as the second mixed derivative of that pressure with respect to the chemical potentials. Because the phase shifts enter as fixed external data and are not adjusted to reproduce the susceptibility itself, the reported growth with μ_B follows directly from the density dependence in the virial term without any definitional or fitting closure. No self-citation chain is invoked to justify the central formula, and the partial chemical equilibrium evolution is a standard auxiliary assumption unrelated to the susceptibility calculation.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The calculation rests on the standard assumptions of thermal equilibrium in a hadron resonance gas, the validity of the S-matrix representation of pion-nucleon scattering, and the partial-chemical-equilibrium ansatz for the cooling stage. No new free parameters or invented entities are introduced in the abstract.

axioms (3)

domain assumption Hadron resonance gas in thermal equilibrium at finite baryon chemical potential
Invoked to define the baseline susceptibility before S-matrix corrections.
domain assumption S-matrix formalism accurately encodes pion-nucleon interactions at the relevant energies and densities
Central modeling choice stated in the abstract.
domain assumption Partial chemical equilibrium governs particle number evolution during fireball cooling
Used for the time-dependent calculation.

pith-pipeline@v0.9.0 · 5368 in / 1344 out tokens · 31302 ms · 2026-05-10T16:00:14.961117+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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[1] [1]

S Matrix formulat ion of statistical mechanics,

R. Dashen, S. K. Ma and H. J. Bernstein, Phys. Rev.187(1969), 345-370 doi:10.1103/PhysRev.187.345

work page doi:10.1103/physrev.187.345 1969

[2] [2]

Weinhold, B

W. Weinhold, B. Friman and W. Norenberg, Phys. Lett. B433(1998), 236-242 doi:10.1016/S0370-2693(98)00639-X [arXiv:nucl-th/9710014 [nucl-th]]

work page doi:10.1016/s0370-2693(98)00639-x 1998

[3] [3]

P. M. Lo, B. Friman, M. Marczenko, K. Redlich and C. Sasaki, Phys. Rev. C 96(2017) no.1, 015207 doi:10.1103/PhysRevC.96.015207 [arXiv:1703.00306 [nucl- th]]

work page doi:10.1103/physrevc.96.015207 2017

[4] [4]

Bazavov, T

A. Bazavovet al.[HotQCD], Phys. Rev. D86, 034509 (2012) doi:10.1103/PhysRevD.86.034509 [arXiv:1203.0784 [hep-lat]]

work page doi:10.1103/physrevd.86.034509 2012

[5] [5]

P. M. Lo, B. Friman, K. Redlich and C. Sasaki, Phys. Lett. B778(2018), 454-458 doi:10.1016/j.physletb.2018.01.016 [arXiv:1710.02711 [hep-ph]]

work page doi:10.1016/j.physletb.2018.01.016 2018

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Stachel, A

J. Stachel, A. Andronic, P. Braun-Munzinger and K. Redlich, J. Phys. Conf. Ser.509(2014), 012019 doi:10.1088/1742-6596/509/1/012019 [arXiv:1311.4662 [nucl-th]]

work page doi:10.1088/1742-6596/509/1/012019 2014

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Andronic, P

A. Andronic, P. Braun-Munzinger, B. Friman, P. M. Lo, K. Redlich and J. Stachel, Phys. Lett. B792(2019), 304-309 doi:10.1016/j.physletb.2019.03.052 [arXiv:1808.03102 [hep-ph]]

work page doi:10.1016/j.physletb.2019.03.052 2019

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P. M. Lo and F. Giacosa, Eur. Phys. J. C79, no.4, 336 (2019) doi:10.1140/epjc/s10052-019-6844-y 13

work page doi:10.1140/epjc/s10052-019-6844-y 2019

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R. L. Workman, R. A. Arndt, W. J. Briscoe, M. W. Paris and I. I. Strakovsky, Phys. Rev. C86(2012), 035202 doi:10.1103/PhysRevC.86.035202 [arXiv:1204.2277 [hep-ph]]

work page doi:10.1103/physrevc.86.035202 2012

[10] [10]

P. M. Lo, PoSCPOD2021, 037 (2022) doi:10.22323/1.400.0037

work page doi:10.22323/1.400.0037 2022

[11] [11]

Bollweg, J

D. Bollweget al.[HotQCD], Phys. Rev. D104, no.7, 074512 (2021) doi:10.1103/PhysRevD.104.074512 [arXiv:2107.10011 [hep-lat]]

work page doi:10.1103/physrevd.104.074512 2021

[12] [12]

Vovchenko, V

V. Vovchenko, V. V. Begun and M. I. Gorenstein, Phys. Rev. C93(2016) no.6, 064906 doi:10.1103/PhysRevC.93.064906 [arXiv:1512.08025 [nucl-th]]

work page doi:10.1103/physrevc.93.064906 2016

[13] [13]

B. I. Abelevet al.[STAR], Phys. Rev. C79(2009), 034909 doi:10.1103/PhysRevC.79.034909 [arXiv:0808.2041 [nucl-ex]]

work page doi:10.1103/physrevc.79.034909 2009

[14] [14]

Abelev, et al., Centrality dependence ofπ, K, p production in Pb-Pb collisions at √sNN = 2.76 TeV, Phys

B. Abelevet al.[ALICE], Phys. Rev. C88(2013), 044910 doi:10.1103/PhysRevC.88.044910 [arXiv:1303.0737 [hep-ex]]

work page doi:10.1103/physrevc.88.044910 2013

[15] [15]

Adamczyk, et al., Bulk Properties of the Medium Produced in Rel- ativistic Heavy-Ion Collisions from the Beam Energy Scan Program, Phys

L. Adamczyket al.[STAR], Phys. Rev. C96(2017) no.4, 044904 doi:10.1103/PhysRevC.96.044904 [arXiv:1701.07065 [nucl-ex]]

work page doi:10.1103/physrevc.96.044904 2017

[16] [16]

Figueroa

I. Melo and B. Tom´ aˇ sik, J. Phys. G47(2020) no.4, 045107 doi:10.1088/1361- 6471/ab5f03 [arXiv:1908.03023 [nucl-th]]

work page doi:10.1088/1361- 2020

[17] [17]

Bebie, P

H. Bebie, P. Gerber, J. L. Goity and H. Leutwyler, Nucl. Phys. B378(1992), 95-128 doi:10.1016/0550-3213(92)90005-V 14

work page doi:10.1016/0550-3213(92)90005-v 1992