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arxiv: 2605.14372 · v1 · pith:OZEFJGVPnew · submitted 2026-05-14 · ✦ hep-ph · hep-th

Complete one-loop self-energies of the linear sigma model coupled to quarks at finite temperature and in a magnetic field

Adolfo Flores-Aguilar , Luis A. Hern\'andez , J. Carlos M\'arquez , R. Zamora This is my paper

Pith reviewed 2026-06-30 20:44 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords linear sigma modelone-loop self-energiesfinite temperaturemagnetic fieldquark couplingthermomagnetic correctionseffective QCD models
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The pith

One-loop self-energies for every field in the linear sigma model with quarks are derived at finite temperature and magnetic field strength.

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

The paper computes the one-loop self-energies for all fields in the linear sigma model coupled to quarks. It incorporates thermal effects and a uniform magnetic field for both neutral and charged degrees of freedom. The calculation remains valid for any temperature and any field strength. This produces expressions that separate vacuum contributions from matter contributions and isolate ultraviolet divergences.

Core claim

We present a complete calculation of the one-loop self-energies for all fields in the linear sigma model coupled to quarks at finite temperature and in the presence of a uniform magnetic field. The analysis consistently incorporates thermal and magnetic effects for both neutral and charged degrees of freedom, providing a unified framework valid for arbitrary values of the temperature and the field strength. The computation is performed using the Matsubara formalism to account for finite temperature effects and the Schwinger proper-time representation for charged propagators in a magnetic background. Special attention is given to loop contributions involving particles with different electric

What carries the argument

The full set of one-loop self-energies for scalar and quark fields, obtained by summing all diagrams while retaining the phase factors that arise when charged and neutral particles circulate in the same loop.

If this is right

  • Ultraviolet divergences in the thermomagnetic corrections can be identified term by term.
  • Vacuum pieces separate cleanly from temperature- and field-dependent pieces in every self-energy.
  • The expressions supply a consistent starting point for studying effective models of QCD that include both temperature and magnetic field.
  • Neutral and charged fields receive their corrections within the same calculational scheme.

Where Pith is reading between the lines

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

  • The self-energies could be inserted into gap equations to locate possible shifts in the chiral transition line as the magnetic field grows.
  • The separation of vacuum and matter pieces may simplify renormalization when these self-energies are used in higher-order calculations.
  • The same loop structure could be reused for related models that also contain both charged mesons and quarks.

Load-bearing premise

The phase factors that appear when a loop contains particles of unequal electric charge can be evaluated in position space and then converted to momentum space without loss of consistency.

What would settle it

An explicit numerical mismatch between the derived self-energy expression for a charged scalar and an independent evaluation of the same diagram at a chosen nonzero temperature and magnetic field value.

Figures

Figures reproduced from arXiv: 2605.14372 by Adolfo Flores-Aguilar, J. Carlos M\'arquez, Luis A. Hern\'andez, R. Zamora.

Figure 1
Figure 1. Figure 1: FIG. 1. Feynman diagrams corresponding to the nine inter [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. One-loop self-energy diagrams for the neutral pion [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. One-loop self-energy diagrams for the sigma meson [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. One-loop self-energy diagrams for the charged pion [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. One-loop self-energy diagrams for the quark fields (u [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

We present a complete calculation of the one-loop self-energies for all fields in the linear sigma model coupled to quarks at finite temperature and in the presence of a uniform magnetic field. The analysis consistently incorporates thermal and magnetic effects for both neutral and charged degrees of freedom, providing a unified framework valid for arbitrary values of the temperature and the field strength. The computation is performed using the Matsubara formalism to account for finite temperature effects and the Schwinger proper-time representation for charged propagators in a magnetic background. Special attention is given to loop contributions involving particles with different electric charges, for which the associated Schwinger phases do not cancel. We show that these terms can be systematically evaluated in coordinate space using the Ritus formalism, which provides the appropriate framework for treating external charged states in the presence of a magnetic background, and consistently expressed in momentum space. The resulting expressions exhibit a nontrivial interplay between thermal fluctuations and magnetic effects and allow for a clear separation between vacuum and matter contributions, providing a well-defined structure for the identification of ultraviolet divergences. Our results establish a consistent and systematic framework for the computation of thermomagnetic one-loop corrections in effective models of QCD, capturing the full interplay between thermal and magnetic effects for all dynamical degrees of freedom.

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

1 major / 0 minor

Summary. The manuscript claims to deliver a complete one-loop calculation of the self-energies for all fields (including neutral and charged mesons and quarks) in the linear sigma model coupled to quarks, at arbitrary finite temperature T and magnetic field strength B. The computation employs the Matsubara formalism for thermal effects and the Schwinger proper-time representation for charged propagators; special emphasis is placed on loops involving particles of unequal electric charges, where Schwinger phases do not cancel, and these are evaluated in coordinate space via the Ritus formalism before conversion to momentum space, yielding expressions that separate vacuum and matter contributions while isolating UV divergences.

Significance. If the central technical step is executed correctly, the work supplies a unified, gauge-consistent framework for thermomagnetic one-loop corrections in an effective QCD model. This would be a useful reference for subsequent studies of chiral symmetry restoration, meson masses, and transport coefficients in strong magnetic fields at finite temperature. The paper does not ship machine-checked proofs or reproducible code, but the explicit separation of vacuum/matter pieces and the claim of a parameter-free derivation (in the sense of no ad-hoc fitting) would constitute a concrete technical advance if verified.

major comments (1)
  1. [Abstract and method description of Ritus formalism application] The central claim that non-cancelling Schwinger phases from mixed-charge loops can be systematically evaluated in coordinate space with the Ritus formalism and then converted to momentum space while preserving a clean vacuum/matter separation and identifiable UV divergences is load-bearing for the entire framework. No explicit intermediate identities, cancellation checks after Matsubara summation, or verification that the conversion introduces neither gauge artifacts nor spurious infrared terms are supplied in the manuscript. This directly affects the validity of the resulting self-energy expressions for both neutral and charged degrees of freedom at arbitrary T and B.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim that non-cancelling Schwinger phases from mixed-charge loops can be systematically evaluated in coordinate space with the Ritus formalism and then converted to momentum space while preserving a clean vacuum/matter separation and identifiable UV divergences is load-bearing for the entire framework. No explicit intermediate identities, cancellation checks after Matsubara summation, or verification that the conversion introduces neither gauge artifacts nor spurious infrared terms are supplied in the manuscript. This directly affects the validity of the resulting self-energy expressions for both neutral and charged degrees of freedom at arbitrary T and B.

    Authors: We agree that the absence of explicit intermediate identities and verification steps limits the transparency of the derivation. The manuscript presents the final self-energy expressions after applying the Ritus formalism in coordinate space and converting to momentum space, with separation into vacuum and matter parts, but does not display the intermediate cancellation checks or explicit gauge-invariance verifications. In the revised version we will add these details in the sections describing the mixed-charge loops (currently Sections 3.3 and 4), including sample identities after Matsubara summation and explicit checks confirming the absence of gauge artifacts and spurious IR terms. These additions will not change the reported expressions or conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity: direct perturbative computation using established formalisms

full rationale

The paper performs an explicit one-loop calculation of self-energies via Matsubara sums and Schwinger proper-time propagators, with coordinate-space evaluation of non-cancelling Schwinger phases converted to momentum space via the Ritus formalism. No parameters are fitted to subsets of data and then relabeled as predictions; no result is defined in terms of itself; no uniqueness theorem or ansatz is imported solely via self-citation; and no known empirical pattern is merely renamed. The central expressions are obtained by direct integration of the loop integrals under the stated assumptions, making the derivation chain independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The calculation rests entirely on standard techniques of thermal field theory and QED in external magnetic fields; no new free parameters, axioms beyond textbook methods, or invented entities are introduced.

axioms (3)
  • standard math Matsubara formalism correctly sums over thermal modes at finite temperature
    Invoked to account for finite temperature effects.
  • standard math Schwinger proper-time representation gives the correct propagator for charged particles in a constant magnetic field
    Used for charged propagators in magnetic background.
  • domain assumption Ritus formalism provides the appropriate basis for treating external charged states when Schwinger phases do not cancel
    Required for loops involving particles with different electric charges.

pith-pipeline@v0.9.1-grok · 5771 in / 1414 out tokens · 30066 ms · 2026-06-30T20:44:04.423008+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

88 extracted references · 75 canonical work pages · 38 internal anchors

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    + eB 2 x1]2 |eB| } , ∫ dy1e { i ( − |eB| 4 (y1)2+y1[(p1 2− p1 1)− eB 2 x2] )} = √ 4π |eB|exp { − [ (p1 2 − p1

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    (105) We substitute the expressions in Eq

    − eB 2 x2]2 |eB| } , e− (p1 2 − p1 1)2 |eB| ∫ dx2e { − |eB| 2 (x2)2+x2 [ eB(p1 2− p1 1) |eB| +i(p2 1− p2 2) ]} = √ 2π |eB|e− (p1 2− p1 1)2 |eB| e { 2 |eB| [ eB(p1 2 − p1 1) |eB| +i(p2 1− p2 2) ]2 } , e− (p2 2 − p2 1)2 |eB| ∫ dx1e { − |eB| 2 (x1)2+x1 [ i(p1 1− p1 2)− eB(p2 2 − p2 1) |eB| ]} = √ 2π |eB|e− (p2 2− p2 1)2 |eB| e { 2 |eB| [ i(p1 1− p1 2)− eB(p2...

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    − 1 ) (−i) q+1e− iqϕ 0 q +n − (k1 1 − k2 1) ( ei2π (q+n− (k1 1− k2 1)) − 1 )∫ ∞ 0 dr r(k1 1 − k2 1)+1e− |qu B|r2 4 ×Lk1 1− k2 1 k2 1 (|quB|r 2 ) Jq ( r √ (p1 2 − p1 1)2 + (p2 2 − p2 1)2 ) ∫ ∞ 0 dr′r′(k1 2 − k2 2)+1e− |quB|r′2 4 Lk1 2− k2 2 k2 2 (|quB|r′ 2 ) ×Jm ( r′ √ (p1 2 − p1 1)2 + (p2 2 − p2 1)2 ) Jn ((qu − qd)Brr ′ 2 ) ×δ(2)(k1∥ +p1∥ − p2∥)δ(2)(p2∥ −...

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    +γ 3p3 1 ) + /p2⊥ cos(|qfB|s) ] . (166) For the integrals over p2⊥ , we use the following results ∫ d2p2⊥e − (p2⊥ )2 |qf B| [ i tan(|qf B|s)+1 ] e 2p2⊥ ·p1⊥ |qf B| = |qfB|π i tan(|qfB|s) + 1e p2 1⊥ |qf B|J, ∫ d2p2⊥ p2⊥e − (p2⊥ )2 |qf B| [i tan(|qf B|s)+1] e 2p2⊥ ·p1⊥ |qf B| = p1⊥ |qfB|π (i tan(|qfB|s) + 1)2e p2 1⊥ |qf B|J, (167) 27 and we obtain −iΠ (D. 6...

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    (168) We now rewrite the neutral pion propagator in the Schwinger prope r time representation and get −iΠ (D

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    6 becomes −iΠ (D

    For this purpose, we use ∫ dp3 1 e− i(p3 1)2(s+s′) = √ π i(s +s′), ∫ dp1⊥e − p2 1⊥ [ 1 |qf B| (1− 1 i tan(|qf B|s)+1 )+is′ ] = π 1 |qf B|(1 − 1 i tan(|qf B|s)+1 ) +is′, ∫ dp1⊥ p1⊥e − p2 1⊥ [ 1 |qf B| (1− 1 i tan(|qf B|s)+1 )+is′ ] = ∫ dp3 1 p3 1 e− (p3 1)2(s+s′) = 0, (170) and the contribution D. 6 becomes −iΠ (D. 6) f =g2 4π ∫ ∞ 0 ds cos(|qfB|s) ∫ ∞ 0 ds...

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