pith. sign in

arxiv: 2605.14674 · v1 · pith:I6QJI77Jnew · submitted 2026-05-14 · ⚛️ nucl-th

Fourth order correlation of baryon number and electric charge as a better magnetometer of QCD

Shijun Mao , Shuai Yang , Sicheng Lin , Xinran Yang , Guoyun Shao , Wen-Chao Zhang This is my paper

Pith reviewed 2026-06-30 20:06 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords QCD phase transitionmagnetic fieldbaryon number correlationselectric charge fluctuationschiral restorationPNJL modelhigher-order susceptibilities
0
0 comments X

The pith

The fourth-order baryon-electric charge correlation χ^{BQ}_{31} is more sensitive to magnetic fields near the QCD chiral transition than other correlations.

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

The paper investigates fourth-order correlations involving baryon number, electric charge, and strangeness using a three-flavor PNJL model at finite temperature and magnetic field with zero chemical potential. It identifies that χ^{BQ}_{31} at the chiral restoration phase transition responds more strongly to changes in the magnetic field than other second- and fourth-order quantities. This matters because such correlations could be measured in heavy-ion collisions to probe the magnetic fields generated in those experiments. The results hold whether or not the model includes inverse magnetic catalysis.

Core claim

Within the three-flavor PNJL model, the fourth order correlation χ^{BQ}_{31} demonstrates higher sensitivity to the magnetic field at the chiral restoration phase transition than other considered correlations and fluctuations, positioning it as a more effective magnetometer of QCD.

What carries the argument

The fourth-order mixed susceptibility χ^{BQ}_{31} computed in the PNJL model, which measures the correlated fluctuations of baryon number and electric charge under varying magnetic fields.

Load-bearing premise

The three-flavor PNJL model accurately reproduces the dependence of QCD thermodynamics on magnetic fields near the chiral transition.

What would settle it

A lattice QCD calculation or heavy-ion collision measurement showing that the magnetic field dependence of χ^{BQ}_{31} is comparable to or weaker than that of quantities such as χ^{BQ}_{22} or second-order fluctuations would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.14674 by Guoyun Shao, Shijun Mao, Shuai Yang, Sicheng Lin, Wen-Chao Zhang, Xinran Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: The fourth order correlations [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The fourth order correlations [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The fourth order correlations [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fig.5. Such ratios are widely adopted in heavy-ion ex [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Double ratio [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (upper panel) Magnetic field dependent param [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The scaled fourth and second order correlations of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The scaled fourth and second order correlations [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The scaled fourth order correlations of baryon [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
read the original abstract

This work focuses on the fourth order correlations $\chi^{BQ}_{31}$, $\chi^{QB}_{31}$, $\chi^{BQ}_{22}$, $\chi^{BS}_{31}$, $\chi^{SB}_{31}$, $\chi^{BS}_{22}$, $\chi^{QS}_{31}$, $\chi^{SQ}_{31}$, $\chi^{QS}_{22}$, $\chi^{BQS}_{211}$, $\chi^{QBS}_{211}$, $\chi^{SBQ}_{211}$ of baryon number $B$, electric charge $Q$ and strangeness $S$ at finite temperature, magnetic field and vanishing quark chemical potential. The study is carried out in the framework of a three-flavor PNJL model, considering both cases with and without inverse magnetic catalysis effect. We find that, fourth order correlations $\chi^{BQ}_{31}$ at chiral restoration phase transition is more sensitive to the magnetic field than other second order and fourth order correlations and fluctuations, and can be served as a more effective magnetometer of QCD.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript computes a suite of second- and fourth-order mixed susceptibilities (χ^{BQ}_{31}, χ^{BQ}_{22}, χ^{BS}_{31}, χ^{QS}_{31}, χ^{BQS}_{211} and their permutations) of baryon number, electric charge and strangeness in a three-flavor PNJL model at μ=0, both with and without an inverse-magnetic-catalysis term. It reports that χ^{BQ}_{31} displays the largest relative change with eB near the pseudocritical temperature and therefore proposes this correlator as a superior magnetometer for QCD.

Significance. A model-independent identification of a higher-order mixed correlator that is especially sensitive to magnetic fields would be valuable for heavy-ion phenomenology. The present work systematically surveys many correlators inside one effective model and isolates one candidate; this computational effort is useful, but the significance remains provisional until the model dependence is quantified or the result is corroborated by lattice QCD.

major comments (2)
  1. [Abstract and §IV] Abstract and §IV (results): the central claim that χ^{BQ}_{31} is 'more sensitive' and 'a more effective magnetometer' rests entirely on the ordering of |∂χ/∂(eB)| extracted from the PNJL model; no error bands, parameter-variation tests, or comparison to other models or lattice data are shown to establish that the ordering is robust rather than an artifact of the fitted couplings.
  2. [§II] §II (model): the PNJL parameters (G, K, T0, etc.) are fixed exclusively to vacuum meson masses and decay constants at B=0; the inverse-magnetic-catalysis term is inserted by hand. Because the magnetic response of the fourth-order BQ correlators is generated by these same parameters, the reported ranking reduces to an output of the vacuum fit rather than an independent prediction of QCD thermodynamics.
minor comments (2)
  1. [§II] The notation χ^{BQ}_{31} versus χ^{QB}_{31} is introduced without an explicit definition of the ordering of derivatives; a short clarifying sentence in §II would remove ambiguity.
  2. [Figures] Figure captions should state whether the curves include statistical or systematic uncertainties from the numerical solution of the gap equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and §IV] Abstract and §IV (results): the central claim that χ^{BQ}_{31} is 'more sensitive' and 'a more effective magnetometer' rests entirely on the ordering of |∂χ/∂(eB)| extracted from the PNJL model; no error bands, parameter-variation tests, or comparison to other models or lattice data are shown to establish that the ordering is robust rather than an artifact of the fitted couplings.

    Authors: The manuscript presents a systematic computation of multiple mixed susceptibilities within the three-flavor PNJL model and identifies χ^{BQ}_{31} as the correlator exhibiting the largest relative variation with eB near the pseudocritical temperature. This ordering is indeed a result obtained inside the model. We will revise the abstract and the concluding section to state explicitly that the proposed magnetometer role is a finding of the PNJL framework and that independent confirmation in other models or lattice QCD is required to assess robustness. No parameter scans or error bands were included because the couplings are fixed by the standard vacuum fit; we will add a brief remark on this point in §IV. revision: partial

  2. Referee: [§II] §II (model): the PNJL parameters (G, K, T0, etc.) are fixed exclusively to vacuum meson masses and decay constants at B=0; the inverse-magnetic-catalysis term is inserted by hand. Because the magnetic response of the fourth-order BQ correlators is generated by these same parameters, the reported ranking reduces to an output of the vacuum fit rather than an independent prediction of QCD thermodynamics.

    Authors: The parameters are determined from vacuum phenomenology at B=0, which is the conventional procedure for PNJL studies. The inverse-magnetic-catalysis term is introduced phenomenologically to reproduce the lattice trend of decreasing T_c with eB. While the magnetic-field dependence of the susceptibilities therefore inherits the model’s vacuum calibration, the explicit evaluation of the fourth-order mixed correlators at finite T and B constitutes a dynamical prediction of the effective theory. We will expand the model-description paragraph in §II to clarify the phenomenological status of the IMC term and the consequent model dependence of the reported ordering. revision: partial

Circularity Check

0 steps flagged

No significant circularity; explicit model computation with transparent assumptions.

full rationale

The paper performs a numerical study of susceptibilities inside the three-flavor PNJL model (with and without inverse magnetic catalysis) at μ=0. The central claim is a comparative statement about which correlator shows the largest variation with eB at the pseudocritical temperature, obtained directly from the model's partition function and its derivatives. No step equates a fitted parameter to a predicted observable by construction, renames a known result, or invokes a self-citation chain whose only support is the present work. The model parameters and the inverse-catalysis implementation are stated as inputs; the ordering of sensitivities is an output of those inputs rather than a tautology. Because the derivation chain consists of explicit thermodynamic derivatives inside a specified effective model, it is self-contained and does not reduce to its own fitted values.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on the validity of the PNJL model and on the numerical extraction of susceptibilities from it; no independent first-principles derivation or experimental benchmark is supplied in the abstract.

free parameters (1)
  • PNJL coupling constants and Polyakov-loop parameters
    Standard vacuum fits to meson masses and decay constants that are required to run the model at finite T and B.
axioms (1)
  • domain assumption The PNJL model provides a reliable approximation to QCD thermodynamics in a magnetic field near the chiral transition.
    Invoked as the sole computational framework.

pith-pipeline@v0.9.1-grok · 5725 in / 1281 out tokens · 26975 ms · 2026-06-30T20:06:37.420265+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Isospin-Driven Splitting of Chemical Potentials in Isobar Collisions from Lattice QCD

    hep-lat 2026-06 unverdicted novelty 6.0

    Lattice QCD yields first-principles splitting ratios for chemical potentials in Ru+Ru vs Zr+Zr collisions that are comparable in size to Bayesian STAR extractions, with Δμ_Q negative, Δμ_S positive, and only moderate ...

Reference graph

Works this paper leans on

100 extracted references · 2 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Notes Phys

    F.Preis, A.Rebhan and A.Schmitt, Lect. Notes Phys. 871, 51(2013)

    2013

  2. [2]

    Notes Phys

    R.Gatto and M.Ruggieri, Lect. Notes Phys. 871, 87(2013)

    2013

  3. [3]

    V.A.Miransky and I.A.Shovkovy, Phys. Rep. 576, 1(2015)

    2015

  4. [4]

    J.O.Anderson and W.R.Naylor, Rev. Mod. Phys. 88, 025001(2016)

    2016

  5. [5]

    G.Q.Cao, Eur. Phys. J. A57, 264(2021)

    2021

  6. [6]

    Notes Phys

    M.D’Elia, Lect. Notes Phys. 871, 181(2013)

    2013

  7. [7]

    High Energy Phys

    G.S.Bali, F.Bruckmann, G.Endr¨ odi, Z.Fodor, S.D.Katz, S.Krieg, A.Schaefer and K.K.Szabo, J. High Energy Phys. 02, 044(2012)

    2012

  8. [8]

    G.S.Bali, F.Bruckmann, G.Endr¨ odi, Z.Fodor, S.D.Katz and A.Schaefer, Phys. Rev. D86, 071502(2012). 11

    2012

  9. [9]

    High Energy Phys

    G.S.Bali, F.Bruckmann, G.Endr¨ odi, Z.Fodor, S.D.Katz and A.Schaefer, J. High Energy Phys. 08, 177(2014)

    2014

  10. [10]

    High Energy Phys

    F.Bruckmann, G.Endr¨odi and T.G.Kovacs, J. High Energy Phys. 04, 112(2013)

    2013

  11. [11]

    V.G.Bornyakov, P.V.Buividovich, N.Cundy, O.A.Kochetkov and A.Schaefer, Phys. Rev. D90, 034501(2014)

    2014

  12. [12]

    High Energy Phys

    G.Endr¨odi, J. High Energy Phys. 07, 173(2015)

    2015

  13. [13]

    High Energy Phys

    G.Endr¨odi, M.Giordano, S.D.Katz, T.G.Kovacs and F.Pittler, J. High Energy Phys. 07, 009(2019)

    2019

  14. [14]

    H.T.Ding, S.T.Li, J.H.Liu and X.D.Wang, Phys. Rev. D 105 , 034514(2022)

    2022

  15. [15]

    H.T.Ding, J.B.Gu, A.Kumar and S.T.Li, Phys. Rev. D 111 , 114522 (2025)

    2025

  16. [16]

    M.D’Elia, F.Manigrasso, F.Negro and F.Sanfilippo, Phys. Rev. D98, 054509(2018)

    2018

  17. [17]

    K.Fukushima and Y.Hidaka, Phys. Rev. Lett 110, 031601(2013)

    2013

  18. [18]

    S.J.Mao, Phys. Lett. B758, 195(2016)

    2016

  19. [19]

    S.J.Mao, Phys. Rev. D94, 036007(2016)

    2016

  20. [20]

    S.J.Mao, Phys. Rev. D97, 011501(R)(2018)

    2018

  21. [21]

    S.J.Mao, Chin. Phys. C45, 021004(2021)

    2021

  22. [22]

    S.J.Mao, Phys. Rev. D106, 034018(2022)

    2022

  23. [23]

    High Energy Phys

    K.Kamikado and T.Kanazawa, J. High Energy Phys. 03, 009(2014)

    2014

  24. [24]

    J.Y.Chao, P.C.Chu and M.Huang, Phys. Rev. D88, 054009(2013)

    2013

  25. [25]

    J.Braun, W.A.Mian and S.Rechenberger, Phys. Lett. B755, 265(2016)

    2016

  26. [26]

    N.Mueller and J.M.Pawlowski, Phys. Rev. D91, 116010(2015)

    2015

  27. [27]

    T.Kojo and N.Su, Phys. Lett. B720, 192(2013)

    2013

  28. [28]

    A.Ayala, M.Loewe, A.J.Mizher and R.Zamora, Phys. Rev. D90, 036001(2014)

    2014

  29. [29]

    A.Ayala, L.A.Hernandez, A.J.Mizher, J.C.Rojas and C.Villavicencio, Phys. Rev. D89, 116017(2014)

    2014

  30. [30]

    A.Ayala, C.A.Dominguez, L.A.Hernandez, M.Loewe and R.Zamora, Phys. Rev. D92, 096011(2015)

    2015

  31. [31]

    R.L.S.Farias, K.P.Gomes, G.Krein, and M.B.Pinto, Phys. Rev. C90, 025203(2014)

    2014

  32. [32]

    M.Ferreira, P.Costa, O.Lourenco, T.Frederico, and C.Providˆencia, Phys. Rev. D89, 116011(2014)

    2014

  33. [33]

    High Energy Phys

    F.Preis, A.Rebhan and A.Schmitt, J. High Energy Phys. 1103, 033(2011)

    2011

  34. [34]

    E.S.Fraga and A.J.Mizher, Phys. Rev. D78, 025016(2008)

    2008

  35. [35]

    E.S.Fraga and A.J.Mizher, Nucl. Phys. A820, 103C(2009)

    2009

  36. [36]

    K.Fukushima, M.Ruggieri and R.Gatto, Phys. Rev. D81, 114031(2010)

    2010

  37. [37]

    High Energy Phys

    C.V.Johnson and A.Kundu, J. High Energy Phys. 12, 053(2008)

    2008

  38. [38]

    V.Skokov, Phys. Rev. D85, 034026(2012)

    2012

  39. [39]

    E.S.Fraga, J.Noronha and L.F.Palhares, Phys. Rev. D87, 114014(2013)

    2013

  40. [40]

    R.Gatto and M.Ruggieri, Phys. Rev. D82, 054027(2010)

    2010

  41. [41]

    R.Gatto and M.Ruggieri, Phys. Rev. D83, 034016(2011)

    2011

  42. [42]

    M.Ferreira, P.Costa and C.Providˆ encia, Phys. Rev. D89, 036006(2014)

    2014

  43. [43]

    M.Ferreira, P.Costa, D.P.Menezes, C.Providˆencia and N.N.Scoccola, Phys. Rev. D89, 016002(2014)

    2014

  44. [44]

    P.Costa, M.Ferreira, H.Hansen, D.P.Menezes and C.Providˆencia, Phys. Rev. D89, 056013(2014)

    2014

  45. [45]

    A.J.Mizher, M.N.Chernodub and E.S.Fraga, Phys. Rev. D82, 105016(2010)

    2010

  46. [46]

    Mintz and J.Schaffner-Bielich, Phys

    E.S.Fraga, B.W. Mintz and J.Schaffner-Bielich, Phys. Lett B731, 154-158(2014)

    2014

  47. [47]

    E.J.Ferrer, V.de la Incera, I.Portillo and M.Quiroz, Phys. Rev. D89, 085034(2014)

    2014

  48. [48]

    E.J.Ferrer, V.de la Incera, and X.J.Wen, Phys. Rev. D91, 054006(2015)

    2015

  49. [49]

    J.Mei and S.J.Mao, Phys. Rev. D102, 114035(2020)

    2020

  50. [50]

    K.Xu, J.Y.Chao and M.Huang, Phys. Rev. D103, 076015(2021)

    2021

  51. [51]

    J.Mei, R.Wen, S.J.Mao, M.Huang and K.Xu, Phys. Rev. D110, 034024(2024)

    2024

  52. [52]

    S.J.Mao, Phys. Rev. D110, 054002(2024)

    2024

  53. [53]

    G.Cao and X.G.Huang, Phys. Rev. D93, 016007(2016)

    2016

  54. [54]

    W.R.Tavares, R.L.S.Farias and S.S.Avancini, Phys. Rev. D101, 016017(2020)

    2020

  55. [55]

    W.R.Tavares, S.S.Avancini and R.L.S.Farias, Phys. Rev. D108, 016017(2023)

    2023

  56. [56]

    G.Endr¨odi and G.Marko, Phys. Rev. D109, 034506(2024)

    2024

  57. [57]

    S.J.Mao and S.Yang, Phys. Rev. D112, 014026(2025)

    2025

  58. [58]

    Z.Y.Qin, B.Feng, Y.H.Hou, H.Y.Song, W.C.Zhang, H.Zheng, S.J.Mao, arXiv:2602.11713

    work page internal anchor Pith review Pith/arXiv arXiv

  59. [59]

    H.T.Ding, S.T.Li, Q.Shi, and X.D.Wang, Eur. Phys. J. A57, 202(2021)

    2021

  60. [60]

    H.T.Ding, S.T.Li, J.H.Liu and X.D.Wang, arXiv:2208.07285

    work page arXiv

  61. [61]

    H.T.Ding, J.B.Gu, A.Kumar, S.T.Li and J.H.Liu, Phys. Rev. Lett. 132, 201903(2024)

    2024

  62. [62]

    K.Fukushima and Y.Hidaka, Phys. Rev. Lett. 117, 102301(2016)

    2016

  63. [63]

    A.Bhattacharyya, S.K.Ghosh, R.Ray, and S.Samanta, Eur. Phys. Lett. 115, 62003(2016)

    2016

  64. [64]

    G.Kadam, S.Pal, and A.Bhattacharyya, J. Phys. G 47, 125106(2020)

    2020

  65. [65]

    W.J.Fu, Phys. Rev. D88, 014009(2013)

    2013

  66. [66]

    N.Chahal, S.Dutt, and A.Kumar, Phys. Rev. C107, 045203(2023)

    2023

  67. [67]

    Mao, Chin

    S.J. Mao, Chin. Phys. C49, 063106(2025)

    2025

  68. [68]

    Rafelski and B

    J. Rafelski and B. Muller, Phys. Rev. Lett. 36, 517 (1976)

    1976

  69. [69]

    D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A803, 227 (2008)

    2008

  70. [70]

    Skokov, A

    V. Skokov, A. Y. Illarionov and T. Toneev, Int. J. Mod. Phys. A24, 5925 (2009)

    2009

  71. [71]

    W. T. Deng and X. G. Huang, Phys. Rev. C85, (2012) 044907; Phys. Lett. B742, 296 (2015)

    2012

  72. [72]

    Tuchin, Adv

    K. Tuchin, Adv. High Energy Phys. 490495 (2013)

    2013

  73. [73]

    H.T.Ding, F.Karsch and S.Mukherjee, Int. J. Mod. Phys. E24, 1530007(2015)

    2015

  74. [74]

    W.J.Fu, Commun. Theor. Phys. 74, 097304(2022)

    2022

  75. [75]

    X.Luo and N.Xu, Nucl. Sci. Tech. 28, 112(2017)

    2017

  76. [76]

    A.Pandav, D.Mallic and B.Mohanty, Prog. Part. Nucl. Phys. 125, 103960(2022)

    2022

  77. [77]

    276, 01007(2023)

    A.Rustanmov, EPJ Web Conf. 276, 01007(2023)

    2023

  78. [78]

    Nonaka, Acta Phys

    T. Nonaka, Acta Phys. Pol. B Proc. Suppl. 16, 1- A14(2023)

    2023

  79. [79]

    H.S.Ko(STAR Collaboration), Acta Phys. Pol. B Proc. Suppl. 16, 1-A87(2023)

    2023

  80. [80]

    STAR Collaboration, Phys. Rev. C100, 12 014902(2019)

    2019

Showing first 80 references.