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REVIEW 3 major objections 4 minor 1 cited by

Including the charm quark in QCD shifts the chiral critical endpoint to lower baryon chemical potential by about 3 percent, while leaving the crossover line almost unchanged.

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2026-08-02 18:46 UTC pith:6R7K3KXV

load-bearing objection A same-truncation 2+1 vs 2+1+1 miniDSE comparison showing the charm loop shifts the CEP by ~3% in μ_B — plausible but the number sits right at the size of the truncation error the authors themselves quote. the 3 major comments →

arxiv 2603.04728 v3 pith:6R7K3KXV submitted 2026-03-05 hep-ph hep-th

The effect of charm quark on the QCD chiral phase diagram

Fei Gao , Yuepeng Guan , Shinya Matsuzaki This is my paper
classification hep-ph hep-th
keywords QCD phase diagramcritical endpointcharm quarkDyson-Schwinger equationsminiDSEchiral crossoverheavy-flavor effectsgluon propagator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether the charm quark, usually ignored in QCD phase-transition studies because of its large mass, leaves a measurable imprint on the phase diagram. Using the miniDSE truncation of the Dyson-Schwinger equations, the authors compare 2+1 and 2+1+1 flavor QCD with all parameters fixed in vacuum. They find that including the charm quark barely moves the chiral crossover line, but shifts the critical endpoint—the point where the smooth crossover would become a sharp transition—from (102.9, 618.8) MeV to (104.3, 600.1) MeV in temperature and baryon chemical potential. That is about 1.4% hotter and 3% lower in baryon chemical potential. The result matters because experimental searches for the QCD critical endpoint aim at percent-level precision, and heavy-flavor loops are a correction that has usually been dropped.

Core claim

The central claim is that the charm-quark loop acts through the gluon propagator rather than directly on the light quarks: it suppresses the gluon dressing function's peak from 1.93 to 1.85, alters the effective gluon mass scale, and thereby makes chiral symmetry restoration occur at a slightly lower baryon chemical potential. Within the same truncation, the 2+1+1 phase diagram shows a crossover line that coincides with the 2+1 case, and a critical endpoint at (104.3, 600.1) MeV versus (102.9, 618.8) MeV. The authors present this as a controlled estimate of the charm-loop effect and note that the shift's size, about 3%, is comparable to the spread among existing functional-QCD predictions fo

What carries the argument

The machinery is the miniDSE difference scheme: the quark gap equation is solved self-consistently, while the gluon propagator is treated as a 2+1-flavor hard-thermal-loop background with an O(4)-symmetric dressing; the charm contribution enters through a difference DSE for the gluon self-energy (their Eq. 20), essentially the charm-quark vacuum polarization evaluated with a full quark propagator and a vertex constrained by the Slavnov-Taylor identity. A Brown-Pennington projection removes the quadratic ultraviolet divergence, leaving a logarithmic piece absorbed by renormalization. The scheme isolates the charm loop as the only new ingredient between two otherwise identical calculations, so

Load-bearing premise

The calculation inherits a 2+1-flavor gluon background whose thermal mass comes from a hard-thermal-loop formula that drops the strange-quark loop and treats chromoelectric and chromomagnetic gluons as degenerate; if that background is not accurate at temperatures around 100–150 MeV and baryon chemical potential around 600 MeV, the 3% shift attributed to charm could be an artifact.

What would settle it

Recompute the phase diagram with a chromoelectric/chromomagnetic-split gluon propagator, or with the strange-quark loop restored in the hard-thermal-loop mass, and compare the 2+1 versus 2+1+1 endpoint shift; if the shift changes sign or grows beyond about 5%, the claim that charm moves the endpoint by 3% is not robust. Independently, a lattice or functional calculation that finds no suppression of the gluon dressing peak (1.93 to 1.85) would falsify the proposed mechanism.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the claim holds, precision determinations of the QCD critical endpoint should use at least 2+1+1 flavors; neglecting charm biases the endpoint's baryon chemical potential by roughly 19 MeV in this scheme.
  • The crossover line's insensitivity to charm means earlier 2+1-flavor estimates of the transition temperature at zero baryon density remain valid at about the 1 MeV level; the charm effect is concentrated near the endpoint.
  • The mechanism—suppression of the gluon dressing by the charm loop—is systematic, so including even heavier flavors (bottom) should continue to move the endpoint, though by smaller amounts.
  • The vacuum comparison ties the charm effect to an observable quantity, the peak height of the gluon dressing function, which can be checked against independent nonperturbative computations.

Where Pith is reading between the lines

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

  • The 3% shift is well inside the spread of published functional-QCD endpoint predictions (roughly 567–672 MeV in baryon chemical potential), so the charm effect is not yet distinguishable from truncation uncertainty; a sharper claim would require also controlling the O(4)-symmetric gluon approximation and the dropped strange-quark thermal loop.
  • The same difference logic yields a testable bottom-quark prediction: with a mass around 4.2 GeV the shift should shrink, and the gluon dressing peak should move back toward the 2+1-flavor value.
  • Because the crossover curvature is unaffected at the level shown, lattice QCD at imaginary chemical potential could look for the 2+1 versus 2+1+1 difference in the curvature; if the curvature is truly unchanged, the charm effect would have to appear in higher-order cumulants or in the endpoint region itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This manuscript uses the miniDSE framework to compare QCD chiral phase diagrams for 2+1 and 2+1+1 flavors. The charm quark enters through a difference DSE for the gluon propagator, as a vacuum-polarization correction to an externally fitted 2+1-flavor gluon background. After fixing parameters to vacuum pion properties and quark masses, the authors solve the quark gap equations at finite temperature and baryon chemical potential. The main result is Table I: including charm shifts the critical endpoint from (T, mu_B) = (102.9, 618.8) MeV to (104.3, 600.1) MeV, i.e. about +1.4% in T and -3.0% in mu_B, while the crossover line is said to be essentially unchanged. The paper concludes that charm-loop effects are small but non-negligible for precision CEP predictions.

Significance. If the result holds, this is a useful quantitative estimate that goes beyond the common 2+1-flavor treatment and exploits a same-truncation comparison that cancels many shared approximations. The vacuum charm self-energy is checked against dimensional regularization in the UV (Fig. 2), parameters are calibrated to pion observables and quark masses, and the CEP results are benchmarked against several functional QCD works (Table I). However, the claimed effect is of the same nominal size as the paper's own estimate of truncation-related CEP uncertainties, and no error budget is attached to Table I. This limits the strength of the central claim; the result is better described as a first estimate than as a fully controlled precision statement.

major comments (3)
  1. [Sec. III B / Table I] The central claim — a 3.0% downward shift in mu_B^CEP and a 1.4% upward shift in T^CEP — is presented without any uncertainty estimate. In the same section the authors state that higher-order correlations (hadron-resonance channels, Polyakov loop, heavier quark loops) move the CEP by 'around 3% deviations'. The charm effect is therefore of the same nominal size as the known truncation uncertainty. Shared approximations do cancel in the difference, but the charm loop is inserted into an externally fitted 2+1-flavor background (Eqs. 14-15), so not all errors are common. Please quantify the stability of the 3% shift under at least: variation of alpha_HTL^S in Eq. (15), inclusion of the strange-quark thermal loop, and inclusion of the T(4) vertex contribution in Eq. (20). Without this, the word 'controlled' in the abstract and Sec. IV is not supported.
  2. [Eq. (15) / Sec. II B] The 2+1-flavor gluon background is O(4)-symmetric and its HTL thermal mass includes only light-quark loops; the strange-quark thermal contribution is dropped. At the relevant conditions (T about 150 MeV, mu_B about 600 MeV), the strange quark mass is not much larger than T, so this term need not be negligible. Since the claimed charm effect is a small difference computed against this background, a moderate error in the strange/thermal sector could masquerade as a charm-induced shift. The authors should either estimate the strange-loop contribution to the thermal mass or show that varying it changes the CEP difference by much less than 3%.
  3. [Eq. (20)] The charm vacuum polarization is evaluated keeping only the Dirac tensor in the quark-gluon vertex; the T(4) Pauli term is dropped with a reference to Refs. [42,46]. Because the signal is only a few percent, the claim that T(4) is negligible should be demonstrated in the present setup, especially in the momentum region relevant for chiral symmetry breaking (roughly 0.5-2 GeV in Fig. 3). A numerical estimate of the T(4) contribution to Pi_2(k) in Eq. (22) would directly test whether this truncation can change the CEP shift at the advertised level.
minor comments (4)
  1. [Abstract / Sec. IV] The abstract quotes 'approximately 3%' and then 'approximately 2-3%' for the CEP shift; Sec. IV describes the effect as 'sizable' and 'noticeable' while Sec. III B states the crossover line is 'essentially unchanged.' These statements should be reconciled and made quantitative.
  2. [Table I] Adding an uncertainty column, or at least a parenthetical spread, would make the comparison with previous functional QCD results more informative and would directly address the resolution of the 3% shift.
  3. [Fig. 3 caption] Typo: 'solving the coupled DSEa' should read 'DSEs'. Also in Sec. III A, 'formual' should be 'formula'.
  4. [Eqs. (4), (20)] The coupling notation changes between g_s in Eq. (4) and g_HTL^s in Eq. (20); please clarify the relation and the separate roles of the renormalization constants Z_1^f and the HTL coupling.

Circularity Check

0 steps flagged

No significant circularity: the charm-loop CEP shift is an output of a self-consistent DSE calculation, not a fitted target.

full rationale

The paper's central claim is a ~2–3% downward shift of μ_B^CEP when the charm quark is added to the 2+1-flavor system. This shift is obtained by comparing two gap-equation solutions that differ only by the charm vacuum-polarization term ΔΠ in Eq. (20) entering the gluon difference-DSE (12). The CEP is not used as an input: the coupling α_s and current quark masses are fixed in vacuum by the condensate, pion mass, and decay constant (Eqs. (26)–(30)), and the gluon inputs are either external fits (gluon dressing Eq. (16)–(19), ghost dressing Eq. (10)–(11)) or standard HTL approximations (Eq. (15)). None of these inputs contain the CEP location or the charm-induced shift. The 2+1 baseline itself is benchmarked in Table I against independent fRG and DSE calculations, so the paper is not predicting a pre-determined number. The self-citations to [42] and [44] supply the miniDSE truncation and the 2+1 gluon baseline, but these are methodological/published inputs with stated assumptions; they do not assume the charm effect. The authors' statement that higher-order correlations move the CEP by 'around 3%' (Sec. III B) is a precision limitation rather than a circularity — it does not show that the charm shift equals the truncation uncertainty by construction. No equation reduces the predicted shift to an input definition, and no parameter is fitted to the CEP. Therefore no significant circularity is present.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The central comparison is anchored by many external fit parameters; the only new physics entered in this paper is the charm vacuum polarization, but the conclusion is sensitive to untested truncations such as the O(4)-symmetric gluon, the vacuum ghost dressing, and the alpha_s re-tuning between the two flavored runs.

free parameters (7)
  • Strong coupling alpha_s per flavor setup = 0.257 (2+1), 0.255 (2+1+1)
    Re-calibrated to reproduce pion mass, f_pi and condensate; the CEP shift is between two differently calibrated runs.
  • Light current quark mass m_l = 1.8 MeV at mu=10 GeV
    Set by the Gell-Mann-Oakes-Renner relation and the empirical pion mass.
  • Strange/light quark mass ratio m_s/m_l = 27
    Empirical input from prior literature.
  • Charm current quark mass m_c = 1270 MeV
    PDG value used for the charm loop.
  • HTL gauge coupling alpha_HTL^S = 0.115
    Chosen for the thermal gluon mass; not varied in an error estimate.
  • Vacuum gluon fit parameters = a=1 GeV, b=0.735 GeV, c=0.12, d=0.0257 GeV^-1, e=0.081 GeV^-1, f=0.65 GeV, g=0.87 GeV
    Fitted to 2+1 functional QCD gluon data; the charm-induced change to this dressing is the driver of the CEP shift.
  • Ghost dressing fit parameters = a1=3.169 GeV^2, b1=4.744 GeV, c1=6.045, d1=1.558 GeV^2, e1=6.385 GeV
    External lattice/functional input; T and mu dependence are dropped.
axioms (6)
  • domain assumption Vertex truncation: only Dirac (T1) and Pauli (T4) tensor structures, with STI-based dressing (Eqs. 5-8).
    Closes the gap equation; higher-rank vertices are neglected without an error estimate.
  • ad hoc to paper Ghost dressing F(k^2) is taken from vacuum and its temperature/chemical-potential dependence is dropped.
    Stated in Sec. II A; no sensitivity study is presented.
  • domain assumption Gluon propagator is O(4)-symmetric; chromoelectric and chromomagnetic sectors do not split; HTL thermal mass includes only light-quark loops (Eq. 15).
    Underpins the gluon background against which the charm loop is measured; this is the weakest load-bearing modeling choice.
  • domain assumption Strange quark contribution to the gluon thermal mass is negligible.
    Dropped due to its heaviness in Sec. II B; questionable at T around 150 MeV.
  • domain assumption Brown-Pennington projection removes the quadratic UV divergence and leaves IR physics intact.
    Used in Eqs. (21)-(22); no independent check that the projection does not distort the small effect.
  • ad hoc to paper T(4) tensor contribution to the gluon self-energy is negligible.
    Stated after Eq. (20), citing Refs. [42,46]; accepted without re-evaluation.

pith-pipeline@v1.3.0-alltime-deepseek · 9604 in / 16843 out tokens · 152223 ms · 2026-08-02T18:46:04.849064+00:00 · methodology

0 comments
read the original abstract

We study the influence of charm-quark dynamics on the chiral phase structure of Quantum Chromodynamics (QCD) using the recently developed miniDSE scheme for the Dyson-Schwinger equations. By comparing the quark and gluon propagators in the $2+1$- and $2+1+1$-flavor setups within the same truncation scheme, we qualify the impact of the charm-quark loop on the QCD phase diagram. Our results show that the charm quark has only a mild effect on the crossover boundary, which remains almost unchanged within the present setup. The most visible effect is a small shift of the critical endpoint (CEP) toward lower baryon chemical potential, by approximately $3\%$. The present result provides a controlled estimate of the charm-loop effect within the same miniDSE truncation. It indicates that heavy-flavor contributions may become relevant when aiming at precision studies of the CEP location.

Figures

Figures reproduced from arXiv: 2603.04728 by Fei Gao, Shinya Matsuzaki, Yuepeng Guan.

Figure 1
Figure 1. Figure 1: FIG. 1. Feynman diagrams of the difference DSE ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dimensionless gluon self-energies Π [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Top panel: the dressing function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. QCD chiral phase diagram obtained within the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Reference graph

Works this paper leans on

49 extracted references · 45 linked inside Pith · cited by 1 Pith paper

  1. [1]

    M. M. Aggarwalet al.(STAR), (2010), arXiv:1007.2613 [nucl-ex]

    Pith/arXiv arXiv 2010

  2. [2]

    Mohanty (STAR), J

    B. Mohanty (STAR), J. Phys. G38, 124023 (2011), arXiv:1106.5902 [nucl-ex]

    Pith/arXiv arXiv 2011

  3. [3]

    Luo and N

    X. Luo and N. Xu, Nucl. Sci. Tech.28, 112 (2017), arXiv:1701.02105 [nucl-ex]

    Pith/arXiv arXiv 2017

  4. [4]

    Nonaka, JPS Conf

    T. Nonaka, JPS Conf. Proc.26, 024007 (2019)

    2019

  5. [5]

    Chenet al., Nucl

    J. Chenet al., Nucl. Sci. Tech.35, 214 (2024), arXiv:2407.02935 [nucl-ex]

    Pith/arXiv arXiv 2024

  6. [6]

    Buballa, Phys

    M. Buballa, Phys. Rept.407, 205 (2005), arXiv:hep- ph/0402234

    arXiv 2005

  7. [7]

    Fukushima and C

    K. Fukushima and C. Sasaki, Prog. Part. Nucl. Phys.72, 99 (2013), arXiv:1301.6377 [hep-ph]

    Pith/arXiv arXiv 2013

  8. [8]

    Oertel, M

    M. Oertel, M. Hempel, T. Kl¨ ahn, and S. Typel, Rev. Mod. Phys.89, 015007 (2017), arXiv:1610.03361 [astro- ph.HE]

    Pith/arXiv arXiv 2017

  9. [9]

    Aghanimet al.(Planck), Astron

    N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  10. [10]

    Gao and I

    F. Gao and I. M. Oldengott, Phys. Rev. Lett.128, 131301 (2022), arXiv:2106.11991 [hep-ph]

    Pith/arXiv arXiv 2022

  11. [11]

    Zheng, F

    H.-w. Zheng, F. Gao, L. Bian, S.-x. Qin, and Y.-x. Liu, Phys. Rev. D111, L021303 (2025), arXiv:2407.03795 [hep-ph]

    Pith/arXiv arXiv 2025

  12. [12]

    F. Gao, J. Harz, C. Hati, Y. Lu, I. M. Oldengott, and G. White, JHEP06, 247 (2025), arXiv:2407.17549 [hep- ph]

    Pith/arXiv arXiv 2025

  13. [13]

    G. Y. Shao, M. Di Toro, V. Greco, M. Colonna, S. Plumari, B. Liu, and Y. X. Liu, Phys. Rev. D84, 034028 (2011), arXiv:1105.4528 [nucl-th]

    Pith/arXiv arXiv 2011

  14. [14]

    X. Y. Xin, S. X. Qin, and Y. X. Liu, Phys. Rev. D90, 076006 (2014), arXiv:2109.09935 [hep-ph]

    Pith/arXiv arXiv 2014

  15. [15]

    He, S.-Y

    S. He, S.-Y. Wu, Y. Yang, and P.-H. Yuan, JHEP04, 093 (2013), arXiv:1301.0385 [hep-th]

    Pith/arXiv arXiv 2013

  16. [16]

    Chelabi, Z

    K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, JHEP04, 036 (2016), arXiv:1512.06493 [hep-ph]

    Pith/arXiv arXiv 2016

  17. [17]

    T. Kojo, D. Hou, J. Okafor, and H. Togashi, Phys. Rev. D104, 063036 (2021), arXiv:2012.01650 [astro-ph.HE]

    Pith/arXiv arXiv 2021

  18. [18]

    X. Chen, L. Zhang, D. Li, D. Hou, and M. Huang, JHEP 07, 132 (2021), arXiv:2010.14478 [hep-ph]. 7

    Pith/arXiv arXiv 2021

  19. [19]

    Hippert, J

    M. Hippert, J. Grefa, T. A. Manning, J. Noronha, J. Noronha-Hostler, I. Portillo Vazquez, C. Ratti, R. Rougemont, and M. Trujillo, (2023), arXiv:2309.00579 [nucl-th]

    Pith/arXiv arXiv 2023

  20. [20]

    R. G. Cai, S. He, L. Li, and Y. X. Wang, Phys. Rev. D 106, L121902 (2022), arXiv:2201.02004 [hep-th]

    Pith/arXiv arXiv 2022

  21. [21]

    X. Chen, D. Li, and M. Huang, Chin. Phys. C43, 023105 (2019), arXiv:1810.02136 [hep-ph]

    Pith/arXiv arXiv 2019

  22. [22]

    Basar, Phys

    G. Basar, Phys. Rev. C110, 015203 (2024), arXiv:2312.06952 [hep-th]

    Pith/arXiv arXiv 2024

  23. [23]

    A. Adam, S. Bors´ anyi, Z. Fodor, J. N. Guenther, P. Parotto, A. P´ asztor, D. Peszny´ ak, L. Pirelli, and C. H. Wong, PoSLA TTICE2024, 178 (2025), arXiv:2502.03211 [hep-lat]

    Pith/arXiv arXiv 2025

  24. [24]

    Ecker, N

    C. Ecker, N. Jokela, and M. J¨ arvinen, (2025), arXiv:2506.10065 [astro-ph.HE]

    Pith/arXiv arXiv 2025

  25. [25]

    Borsanyi, Z

    S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabo, Phys. Rev. Lett.125, 052001 (2020), arXiv:2002.02821 [hep-lat]

    Pith/arXiv arXiv 2020

  26. [26]

    Bazavovet al.(HotQCD), Phys

    A. Bazavovet al.(HotQCD), Phys. Lett. B795, 15 (2019), arXiv:1812.08235 [hep-lat]

    Pith/arXiv arXiv 2019

  27. [27]

    Bonati, M

    C. Bonati, M. D’Elia, F. Negro, F. Sanfilippo, and K. Zambello, Phys. Rev. D98, 054510 (2018), arXiv:1805.02960 [hep-lat]

    Pith/arXiv arXiv 2018

  28. [28]

    S. X. Qin, L. Chang, H. Chen, Y. X. Liu, and C. D. Roberts, Phys. Rev. Lett.106, 172301 (2011), arXiv:1011.2876 [nucl-th]

    Pith/arXiv arXiv 2011

  29. [29]

    C. S. Fischer, J. Luecker, and C. A. Welzbacher, Phys. Rev. D90, 034022 (2014), arXiv:1405.4762 [hep-ph]

    Pith/arXiv arXiv 2014

  30. [30]

    Gao and Y

    F. Gao and Y. X. Liu, Phys. Rev. D94, 076009 (2016), arXiv:1607.01675 [hep-ph]

    Pith/arXiv arXiv 2016

  31. [31]

    C. S. Fischer, Prog. Part. Nucl. Phys.105, 1 (2019), arXiv:1810.12938 [hep-ph]

    Pith/arXiv arXiv 2019

  32. [32]

    Gao and J

    F. Gao and J. M. Pawlowski, Phys. Rev. D102, 034027 (2020), arXiv:2002.07500 [hep-ph]

    Pith/arXiv arXiv 2020

  33. [33]

    Gao and J

    F. Gao and J. M. Pawlowski, Phys. Lett. B820, 136584 (2021), arXiv:2010.13705 [hep-ph]

    Pith/arXiv arXiv 2021

  34. [34]

    P. J. Gunkel and C. S. Fischer, Phys. Rev. D104, 054022 (2021), arXiv:2106.08356 [hep-ph]

    Pith/arXiv arXiv 2021

  35. [35]

    W. J. Fu, J. M. Pawlowski, and F. Rennecke, Phys. Rev. D101, 054032 (2020), arXiv:1909.02991 [hep-ph]

    Pith/arXiv arXiv 2020

  36. [36]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, Phys. Rept. 910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2021

  37. [37]

    W. J. Fu, Commun. Theor. Phys.74, 097304 (2022), arXiv:2205.00468 [hep-ph]

    Pith/arXiv arXiv 2022

  38. [38]

    B. E. Aboonaet al.(STAR), Phys. Rev. Lett.135, 142301 (2025), arXiv:2504.00817 [nucl-ex]

    Pith/arXiv arXiv 2025

  39. [39]

    van Hees, V

    H. van Hees, V. Greco, and R. Rapp, Phys. Rev. C73, 034913 (2006), arXiv:nucl-th/0508055

    Pith/arXiv arXiv 2006

  40. [40]

    Goswami, K

    K. Goswami, K. K. Pradhan, D. Sahu, J. Dey, and R. Sa- hoo, Phys. Rev. D111, 014029 (2025), arXiv:2409.13255 [hep-ph]

    Pith/arXiv arXiv 2025

  41. [41]

    C. A. Welzbacher, C. S. Fischer, and J. Luecker, J. Phys. Conf. Ser.599, 012015 (2015), arXiv:1412.3650 [hep-ph]

    Pith/arXiv arXiv 2015

  42. [42]

    Y. Lu, F. Gao, Y.-X. Liu, and J. M. Pawlowski, Phys. Rev. D110, 014036 (2024), arXiv:2310.18383 [hep-ph]

    Pith/arXiv arXiv 2024

  43. [43]

    C. Tang, F. Gao, and Y.-X. Liu, Phys. Rev. D100, 056001 (2019), arXiv:1902.01679 [hep-ph]

    Pith/arXiv arXiv 2019

  44. [44]

    F. Gao, J. Papavassiliou, and J. M. Pawlowski, Phys. Rev. D103, 094013 (2021), arXiv:2102.13053 [hep-ph]

    Pith/arXiv arXiv 2021

  45. [45]

    A. C. Aguilar, C. O. Ambr´ osio, F. De Soto, M. N. Ferreira, B. M. Oliveira, J. Papavassiliou, and J. Rodr ´ ıguez-Quintero, Phys. Rev. D104, 054028 (2021), arXiv:2107.00768 [hep-ph]

    Pith/arXiv arXiv 2021

  46. [46]

    Y. Lu, F. Gao, Y.-x. Liu, and J. M. Pawlowski, (2025), arXiv:2504.05099 [hep-ph]

    Pith/arXiv arXiv 2025

  47. [47]

    Brown and M

    N. Brown and M. R. Pennington, Phys. Rev. D38, 2266 (1988)

    1988

  48. [48]

    C. S. Fischer and J. Luecker, Phys. Lett. B718, 1036 (2013), arXiv:1206.5191 [hep-ph]

    Pith/arXiv arXiv 2013

  49. [49]

    M. D. Schwartz,Quantum Field Theory and the Standard Model(Cambridge University Press, 2014)

    2014