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arxiv: 2511.19255 · v2 · submitted 2025-11-24 · ✦ hep-ph · hep-lat· nucl-th

Magnetic susceptibility of a hot hadronic medium and quark degrees of freedom near the QCD cross-over point

Rupam Samanta , Wojciech Broniowski This is my paper

Pith reviewed 2026-05-17 06:10 UTC · model grok-4.3

classification ✦ hep-ph hep-latnucl-th
keywords magnetic susceptibilityQCD crossoverquark degrees of freedomhadronic mediumlattice QCDanomalous magnetic momentsparamagnetismdiamagnetism
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The pith

Quark degrees of freedom must extend down to about 120 MeV to reproduce lattice magnetic susceptibility up to the QCD crossover.

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

The paper shows that standard hadron resonance gas models, even with physical hadron magnetic moments or pion-vector meson loops, predict too much diamagnetism in the magnetic susceptibility compared to lattice QCD results above roughly 120 MeV. To resolve the mismatch, the authors introduce a quark-meson model in which the temperature dependence of constituent quark masses is fixed directly from lattice baryon-baryon and baryon-strangeness susceptibilities at zero magnetic field. These quarks carry anomalous magnetic moments taken from the measured moments of the baryon octet. Both vacuum quark-loop and meson-loop contributions are included. With these ingredients the calculated susceptibility matches the lattice data up to the crossover temperature, indicating that quark-like excitations are already present well inside the hadronic phase.

Core claim

In a quark-meson framework where temperature-dependent quark masses are fixed in a model-free way from lattice baryon susceptibilities at zero magnetic field and where constituent quarks carry anomalous magnetic moments estimated from octet baryon moments, the vacuum quark-loop and meson-loop contributions together reproduce the lattice QCD magnetic susceptibility of the hot medium up to the crossover point. This establishes that QCD degrees of freedom must remain active down to approximately 120 MeV.

What carries the argument

Quark-meson model with temperature-dependent constituent quark masses fixed from zero-field lattice susceptibilities, anomalous magnetic moments from octet baryons, and combined quark and meson loop contributions to the susceptibility.

If this is right

  • The magnetic susceptibility receives paramagnetic quark contributions that grow above 120 MeV while pion loops still dominate the diamagnetic response at lower temperatures.
  • Purely hadronic models become insufficient near the crossover and must be supplemented by quark degrees of freedom.
  • The same temperature-dependent quark masses ensure consistency with other zero-field lattice observables such as baryon number fluctuations.
  • The description remains valid up to the crossover temperature without additional tuning of the hadronic spectrum.

Where Pith is reading between the lines

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

  • The required overlap of quark and hadronic degrees of freedom implies a broad temperature window in which neither purely hadronic nor purely partonic descriptions are complete.
  • The same construction could be used to compute related response functions such as electric susceptibility or the chiral condensate in the same temperature range.
  • Analogous quark contributions may appear in other fluctuation observables or in the equation of state near the crossover.

Load-bearing premise

The temperature-dependent quark masses extracted from zero-field lattice susceptibilities correctly determine the magnetic response, and the anomalous magnetic moments assigned to constituent quarks are accurately estimated from the magnetic moments of the octet baryons.

What would settle it

A lattice QCD computation of magnetic susceptibility performed with only hadronic degrees of freedom at a temperature of 130 MeV that matches the full lattice result would falsify the necessity of including quark contributions below the crossover.

Figures

Figures reproduced from arXiv: 2511.19255 by Rupam Samanta, Wojciech Broniowski.

Figure 1
Figure 1. Figure 1: (the error bars of the data are smaller than the size of the dots). We also display the corresponding HRG results [22, 40, 43], indicated with the lines. Above Tc ∼ 155 MeV, the lines are dashed to indicate the model is no longer credible there. We note a proper agreement, acclaimed in previous works, for temperatures up to Tc. The results for χB(T) are displayed in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: F. Magnetization To corroborate the qualitative findings for χB, we also examine the magnetization M of Eq. 3. This quantity [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows that the contribution of the pion–vector￾meson loops to χB is indeed small. To assess the effect, we use the HRG result from [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
read the original abstract

The lattice QCD results for the temperature-dependent magnetic susceptibility of the medium below the cross-over temperature are not possible to reconcile with the widely used Hadron Resonance Gas model, also amended with the physical magnetic moments of hadrons or the pion--vector-meson loops. As noticed earlier, one observes a substantially too strong diamagnetism at temperatures in the range above $\approx 120$~MeV compared to the lattice. This hints at a presence of quarks significantly below the QCD cross-over temperature, which are needed as a source of paramagnetism. However, the pions must be retained to describe the diamagnetism data at low temperatures. Therefore, we consider here a quark-meson approach, where the temperature-dependent quark masses are fixed in a model-free way using the baryon-baryon and baryon-strangeness susceptibilities from the lattice at zero magnetic field. The constituent quarks possess anomalous magnetic moments estimated from the octet baryon magnetic moments. The vacuum quark-loop and meson-loop contributions are duly incorporated. We show that in such a framework, one can describe the magnetic susceptibility up to the cross-over point. The qualitative conclusion is that the QCD degrees of freedom must extend far below the cross-over temperature, down to $\approx 120$~MeV.

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 / 3 minor

Summary. The manuscript argues that lattice QCD results for the temperature-dependent magnetic susceptibility of the hadronic medium below the QCD crossover cannot be reconciled with the Hadron Resonance Gas (HRG) model, even when including physical hadron magnetic moments or pion-vector meson loops, due to excessive diamagnetism above approximately 120 MeV. To address this, the authors introduce a quark-meson effective model in which temperature-dependent constituent quark masses are extracted from lattice baryon-baryon and baryon-strangeness susceptibilities at zero magnetic field. Constituent quarks are assigned anomalous magnetic moments estimated from octet baryon moments. Vacuum quark-loop and meson-loop contributions are included, and the resulting framework is shown to reproduce the lattice magnetic susceptibility up to the crossover. The qualitative conclusion is that quark degrees of freedom must extend significantly below the crossover temperature, down to ~120 MeV.

Significance. If the central result holds, the work would be significant for QCD thermodynamics and the interpretation of lattice data near the crossover. It provides a concrete mechanism by which paramagnetic quark-loop contributions can offset the diamagnetic excess of a purely hadronic description, thereby offering a possible explanation for the observed lattice magnetic susceptibility. The use of zero-field lattice susceptibilities to fix the quark masses supplies external anchoring rather than pure fitting, which strengthens the approach relative to fully phenomenological models. The conclusion that deconfined degrees of freedom appear well below the crossover has potential implications for hydrodynamic modeling of heavy-ion collisions and for the structure of the QCD phase diagram.

major comments (2)
  1. [Abstract and method section describing mass extraction] The claim that temperature-dependent quark masses are fixed in a 'model-free way' from lattice baryon-baryon and baryon-strangeness susceptibilities at B=0 (abstract and the corresponding method section) requires clarification. Extracting running constituent masses from integrated susceptibilities necessarily involves a choice of interaction vertices in the quark-meson Lagrangian and a regularization prescription. These choices propagate directly into the one-loop paramagnetic quark contribution at finite magnetic field. If the mapping is not unique, the reported agreement with lattice magnetic susceptibility data could be an artifact of the specific parametrization rather than unambiguous evidence for quark degrees of freedom at low T. A stability check under variation of the regularization scale or vertex form would strengthen the central claim.
  2. [Results section and associated figures] The quantitative comparison between the quark-meson model prediction and lattice magnetic susceptibility data (presumably shown in the results section or figures) should include an explicit assessment of how sensitive the fit is to the estimated anomalous magnetic moments of the constituent quarks. Because these moments are taken from vacuum octet baryon values and then used at finite temperature, any temperature dependence or medium modification of the moments would alter the paramagnetic term and could change the temperature at which the model begins to describe the data.
minor comments (3)
  1. [Abstract] The abstract states that 'the pions must be retained to describe the diamagnetism data at low temperatures,' but the manuscript would benefit from a short explicit statement of the temperature range in which the pure HRG (with or without loops) fails and the quark-meson model succeeds.
  2. [Formalism section] Notation for the magnetic susceptibility (e.g., whether it is the second derivative of the pressure or a normalized quantity) should be defined once at the beginning of the formalism section to avoid ambiguity when comparing to lattice definitions.
  3. [Loop calculation subsection] A brief discussion of the cutoff or regularization scheme employed for the vacuum quark loop would improve reproducibility, even if the scheme is standard.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the work's significance for QCD thermodynamics. Below we address each major comment point by point, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and method section describing mass extraction] The claim that temperature-dependent quark masses are fixed in a 'model-free way' from lattice baryon-baryon and baryon-strangeness susceptibilities at B=0 (abstract and the corresponding method section) requires clarification. Extracting running constituent masses from integrated susceptibilities necessarily involves a choice of interaction vertices in the quark-meson Lagrangian and a regularization prescription. These choices propagate directly into the one-loop paramagnetic quark contribution at finite magnetic field. If the mapping is not unique, the reported agreement with lattice magnetic susceptibility data could be an artifact of the specific parametrization rather than unambiguous evidence for quark degrees of freedom at low T. A stability check under variation of the regularization scale or vertex form would strengthen th

    Authors: We agree that the phrasing 'model-free way' in the abstract is imprecise and could be misleading, as the extraction of temperature-dependent constituent quark masses is performed within the quark-meson model by matching the zero-field lattice susceptibilities. The same Lagrangian and regularization scheme are used consistently for both the zero-B susceptibilities and the finite-B magnetic susceptibility calculation, ensuring internal consistency rather than independent fitting. Nevertheless, we acknowledge that different vertex choices or regularization scales could in principle affect the mapping. In the revised manuscript we will replace 'model-free' with a more accurate description, add a dedicated paragraph in the methods section explaining the regularization prescription and its motivation from the lattice data, and include a brief stability analysis by varying the ultraviolet cutoff within a 10-20% range around the value fixed by the zero-field susceptibilities. We will show that the paramagnetic quark contribution and the overall agreement with the lattice magnetic susceptibility remain qualitatively unchanged, supporting that the result is not an artifact of a single parametrization. revision: yes

  2. Referee: [Results section and associated figures] The quantitative comparison between the quark-meson model prediction and lattice magnetic susceptibility data (presumably shown in the results section or figures) should include an explicit assessment of how sensitive the fit is to the estimated anomalous magnetic moments of the constituent quarks. Because these moments are taken from vacuum octet baryon values and then used at finite temperature, any temperature dependence or medium modification of the moments would alter the paramagnetic term and could change the temperature at which the model begins to describe the data.

    Authors: We concur that an explicit sensitivity study is warranted. The anomalous magnetic moments are currently taken from the vacuum values of the octet baryons and held fixed with temperature. In the revised version we will add a new paragraph (and corresponding shaded bands on the relevant figure) in the results section that varies the anomalous moments by ±20% around the central values. This range is chosen to bracket possible medium modifications while remaining consistent with the vacuum data. The resulting uncertainty bands will be shown together with the lattice points; we find that the temperature at which the model begins to reproduce the lattice susceptibility (around 120 MeV) is robust within this variation, and the overall description up to the crossover remains satisfactory. This addition will be included in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: independent lattice inputs fix masses for separate magnetic susceptibility computation

full rationale

The paper fixes temperature-dependent constituent quark masses using baryon-baryon and baryon-strangeness susceptibilities from lattice QCD at zero magnetic field. These are distinct observables from the magnetic susceptibility (response to finite B) that the model is then used to describe. The framework incorporates vacuum quark loops with anomalous moments (estimated from octet baryons) plus meson loops, and shows agreement with lattice magnetic susceptibility data up to the crossover. This is a standard use of external benchmarks to constrain parameters before addressing a different quantity; the derivation chain does not reduce by construction to the target observable, nor rely on load-bearing self-citations or smuggled ansatzes. The claim of a 'model-free' fixing is presented as such in the abstract, with no quoted reduction showing the magnetic result is forced by the zero-B fit itself.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on lattice QCD data as external input for quark masses and on standard loop contributions in the quark-meson model; no new particles are postulated.

free parameters (2)
  • temperature-dependent constituent quark masses
    Determined from lattice baryon-baryon and baryon-strangeness susceptibilities at zero magnetic field.
  • anomalous magnetic moments of constituent quarks
    Estimated from octet baryon magnetic moments.
axioms (2)
  • domain assumption Lattice QCD susceptibilities at zero field provide reliable, model-independent constraints on quark masses.
    Invoked to fix the temperature dependence without additional parameters.
  • domain assumption Vacuum quark-loop and meson-loop contributions can be incorporated in the standard way within the quark-meson framework.
    Used to compute the susceptibility.

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Reference graph

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