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REVIEW 4 major objections 4 minor 39 references

Dynamical quark mass and finite volume effects in the Dyson-Schwinger Equations

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims that constituent quark masses in a magnetic field are strongly volume dependent, falling by about a third as the fireball radius shrinks from infinity to 2 fm.

desk verdict A new but not yet believable attempt to combine spherical MRE finite-volume effects with DSEs in a magnetic field; the headline mass drops rest on an unjustified phase-space replacement and an internal R>5 fm contradiction. read the letter →

arxiv 2508.11968 v1 pith:5MJXGGUG submitted 2025-08-16 hep-ph

Li-Juan Zhou , De-Xian Wei , Zhong-Yi Liu , Hong-Wei Zhong This is my paper
classification hep-ph PACS 12.38.Aw11.30.Rd
keywords Dyson-Schwingerequationsfinitevolumeeffectsconstituentquarkmassmagneticfieldrunningcouplingconstantmultiplereflectionexpansionchiralsymmetrybreakingheavy-ioncollisions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper targets a practical question for heavy-ion physics: is the quark-gluon fireball big enough that infinite-volume predictions apply? It argues it is not, once the strong coupling is allowed to depend on the magnetic field. Using Dyson-Schwinger equations with a multiple reflection expansion to impose a spherical volume, it finds the constituent quark mass depends on the fireball radius as well as on the field: at $eB=0.5$ GeV$^2$, $M_u$ drops from 392 to 240 MeV and $M_s$ from 519 to 377 MeV as the radius goes from infinity to 2 fm, with the sharpest volume sensitivity between 2 and 6 fm. The magnetic-field-dependent coupling also suppresses the familiar magnetic catalysis of the mass. If correct, the chiral transition in a realistic fireball is shifted away from infinite-volume expectations, and finite-size corrections matter for interpreting heavy-ion data.

What carries the argument

The multiple reflection expansion (MRE) density of states, $\rho_{\mathrm{MRE}}(q,m_f,R) = 1 + \frac{6\pi^2}{qR} f_S(q,m_f) + \frac{12\pi^2}{(qR)^2} f_C(q,m_f) + \cdots$, with an infrared cutoff where $\rho_{\mathrm{MRE}}=0$; it modifies the 3-momentum integration measure and carries the entire finite-volume effect in the quark self-energy. The second load-bearing piece is the magnetic-field-dependent running coupling $g_{II}=g_I/[1 + D_1 \ln(1 + D_2\, eB/\Lambda_{\mathrm{QCD}}^2)]$, whose parameters are fixed by matching the subtracted condensate to reference data. Together they turn 'smaller fireball' and 'stronger magnetic field' into concrete shifts in the dynamical quark mass.

What would settle it

Recompute the $u$-quark dressing functions at $eB=0.5$ GeV$^2$ with the same gluon and vertex inputs, but impose the finite volume through antiperiodic boundary conditions on a box whose momentum discretization corresponds to $R\approx2$–6 fm. The MRE claim requires the infrared value of $M_u$ to fall from about 392 MeV toward 240 MeV as the box shrinks; if it stays well above 350 MeV, the truncated spherical density of states is the step that fails.

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Extended reading notes

Core claim

With Dyson-Schwinger equations in a strong homogeneous magnetic field, the paper extracts the constituent quark mass $M = A_0/\sqrt{A_\parallel^2 + A_\perp^2}$ and changes the system from infinite volume to a sphere of radius $R$. Its central claim is that with a magnetic-field-dependent running coupling, the mass is substantially smaller in a finite fireball: at $eB=0.5$ GeV$^2$, $M_u$ falls from 392 MeV at $R=\infty$ to 240 MeV at $R=2$ fm, and $M_s$ from 519 to 377 MeV. The effect is concentrated in $R\approx2$–6 fm and negligible by $R\ge10$ fm. The paper also reports that the $eB$-dependent coupling suppresses naive magnetic catalysis, and that shrinking the volume acts on the mass like

Load-bearing premise

The whole finite-volume effect is assumed to enter through one spherical momentum-space substitution in the quark self-energy, and that substitution is assumed to stay accurate at radii of 2–6 fm inside a strong magnetic field.

Editorial extensions

If this is right

  • If the central claim is right, infinite-volume DSE calculations overestimate the dynamical quark mass for fireball radii below about 6 fm, so chiral symmetry restoration in heavy-ion collisions is expected to set in at smaller effective coupling than infinite-volume thermodynamics predicts.
  • Shrinking the system radius and reducing the magnetic field push the constituent mass in the same direction, so finite size and field strength combine to move the phase boundary.
  • At radii of 10 fm and larger the finite-volume correction is negligible, so only the smallest and most peripheral collision systems need this correction.
  • Stronger magnetic fields make the constituent mass vary more sharply with radius, widening the range of fireball sizes in which volume effects are visible.
  • Whether the magnetic field enhances or suppresses the dynamical mass depends on using the magnetic-field-dependent coupling $g_{II}$; switching from $g_I$ to $g_{II}$ changes the mass by about 30% at $eB=1.0$ GeV$^2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's volume measure is a sphere, while the magnetic field selects a longitudinal direction; a direction-dependent density of states would test whether the 2–6 fm window survives the loss of spherical symmetry.
  • The gluon propagator and vertex dressing are left volume-independent; allowing the volume to act there could shift the radius at which the mass reaches its infinite-volume limit.
  • Extending the zero-temperature calculation to finite temperature would convert the mass drop into a predicted shift of the chiral pseudocritical temperature with fireball size, a signal potentially visible in centrality-dependent heavy-ion data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript uses Dyson-Schwinger equations in a Ritus/Landau-level representation, combined with a Multiple Reflection Expansion density of states, to study the constituent quark mass in a finite spherical volume and an external magnetic field. A magnetic-field-dependent running coupling gII is introduced with parameters D1, D2, and a1 fitted to the lattice subtracted condensate. The numerical results show suppression of magnetic catalysis in the gII case and a strong finite-volume effect: at eB=0.5 GeV^2 the u-quark mass in the infrared falls from 392 MeV (R=infinity) to 240 MeV (R=2 fm), with a similar drop for the s quark; the main volume sensitivity is in R approximately 2-6 fm. The paper concludes that finite volume and the running coupling should be included when studying the chiral transition in heavy-ion fireballs.

Significance. If established, this work would provide a practical estimate of finite-size corrections to DSE chiral-condensate predictions in magnetized quark matter and could motivate more realistic fireball studies. Strengths: the quenched DSE formalism is spelled out, the MRE formulas are standard, and the condensate comparison to lattice QCD and NJL in Fig. 1 gives a concrete benchmark. However, the two load-bearing elements--the gII parametrization and the MRE replacement--are not independently established. The inverse magnetic catalysis is largely fitted, and the finite-volume implementation is not well defined after the Ritus reduction. The paper is therefore best read as a phenomenological exercise whose central numbers still require validation.

major comments (4)
  1. [Sec. II.C, Eqs. (14)-(15), Fig. 1] The parameters D1=2.39, D2=0.002515, and a1=0.04 are fixed by fitting the lattice (Sigma_u+Sigma_d)/2 curve. The gII results in Figs. 3-6 therefore inherit the inverse-magnetic-catalysis behavior of the lattice input; they do not independently predict it. The abstract and Sec. IV claim that the running coupling has considerable influence, but the supporting evidence is a fit. Please provide a goodness-of-fit measure, parameter sensitivity, and an out-of-sample check (e.g., reproduce an additional observable) before presenting this as a prediction.
  2. [Sec. II.B, Eqs. (3)-(5) and (12)] This is the central problem. After the Ritus transformation, the DSEs contain Landau-level sums and continuous longitudinal momentum integrals only; there is no 3D measure d^3q/(2*pi)^3 to replace with rho_MRE. Eq. (12) cannot be applied literally. The paper never defines how q in rho_MRE(q,m_f,R) is formed from q_parallel and the Landau-level transverse modes, nor whether m_f is the current or constituent mass. Since f_S and f_C in Eqs. (10)-(11) depend strongly on q/m_f and the IR cutoff is the zero of rho_MRE, the reported mass drops in Figs. 4-6 are not uniquely defined. The isotropy of the spherical MRE density is also incompatible with the strongly anisotropic Landau-level phase space. Without a derivation or a test against an alternative finite-volume prescription, the quantitative claims are unsupported.
  3. [Sec. III, text after Fig. 4, Figs. 5-6] The text states that 'within the framework of DSEs, thermodynamic quantities are expected to diverge when R>5 fm [41]', yet Figs. 4-6 report results at R=6 and R=10 fm, and the summary says R>=10 fm is safe. The manuscript must specify which quantity diverges, why the masses reported here are not affected, and why R>5 results are shown if the model is invalid. This internal inconsistency undermines the upper end of the claimed 2-6 fm range.
  4. [Sec. II.B, Eqs. (9)-(12), Sec. III] The MRE density becomes negative for small q and an IR cutoff is introduced as the largest root of rho_MRE=0, but the numerical value of Lambda_IR and its dependence on m_f and R are not given. The results will depend on which mass is used in f_S and f_C and on how the cutoff is applied in the Landau-level sums. Please report these choices and test the sensitivity of M(R) to a plausible range of Lambda_IR.
minor comments (4)
  1. [Throughout] There are several typos and presentation issues: 'cub' should be 'cube', 'i.g.' should be 'i.e.', and 'Comparision' in Fig. 3 should be 'Comparison'. Some references are incompletely formatted (e.g., Ref. [9]).
  2. [Eqs. (3)-(5)] The notation for the integral measure Z_q is nonstandard and appears to mix a two-dimensional and a one-dimensional integral. Please define all variables (q2, k1) explicitly.
  3. [Figs. 2-4] The dressing functions and masses are shown as functions of p_parallel, but the reader is not told at which fixed p_parallel the quoted mass values are taken. Clarify the 'small momentum region' and consider indicating p_parallel=0 or the quark pole definition.
  4. [Sec. III, paragraph after Fig. 4] The criterion 'V*m*<Psi-bar Psi> >> pi' is quoted without discussion of its provenance or applicability. A short explanation of how this condition relates to the present calculation would help.

Circularity Check

1 steps flagged · score 6.0 of 10

Partial circularity: gII is fitted to the very lattice condensate it then presents as a result; the finite-volume trend is not fitted and is independent.

  1. fitted input called prediction [Section II.C, Eq. (14); Section III, Fig. 1 discussion]
    "the free parameters D1 and D2 are fixed to obtain reasonable results of the lattice average(Σu + Σd)/2 [38] ... We obtain a good fit of the lattice QCD results [38] with D1 = 2.39, D2 = 0.002515 and a1 = 0.04 in Eq. (14). However, by including the eB-dependent running coupling constant (in gII case) leads to a suppression of the magnetic enhancement effect."

    The gII coupling constants D1 and D2 (and the condensate parameter a1) are fit to the same lattice (Σu+Σd)/2 curve that Fig. 1 then shows as a successful comparison. The subsequent claim that gII produces suppression/inverse magnetic catalysis in the condensate—and correspondingly in the constituent quark mass (Mu dropping from 687 to 483 MeV at eB=1.0 GeV²)—is not an independent prediction. It is the fitted target propagated through the DSE solution. The finite-volume dependence is separate, but the paper's central 'running coupling constant' effect is a fitted input described as a result.

full rationale

The only genuinely circular step is the magnetic-field-dependent running coupling: parameters are tuned to reproduce the lattice subtracted condensate, and the same condensate is then exhibited as output of the gII model. This fits the 'fitted input called prediction' pattern. The finite-volume part is not circular: the MRE replacement in Eq. (12) is a model assumption, and the R-dependent mass trend is not fitted to lattice data, so it counts as an independent model prediction. The self-citation for Eq. (15) (a definition of the subtracted condensate) is not load-bearing, and the external MRE references [22–25] are not author-self-citations. A separate internal inconsistency exists—the text says DSE thermodynamic quantities diverge for R>5 fm [41] while Figs. 4–6 use R=6 and 10 fm—but that is a correctness problem, not circularity, so it does not raise the circularity score. Overall: partial circularity because the headline running-coupling effect reduces to the fit, while the volume effect retains independent content.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

Central claim rests on a truncated MRE phase-space replacement, a phenomenological magnetic-field-dependent coupling whose parameters are fitted here, and the quenched ladder DSE model borrowed from [37]. No new particles, forces, or dimensions are introduced; the three fitted constants D1, D2, and a1 are the main extra load.

free parameters (3)
  • D1 = 2.39
    Free parameter in gII = gI / (1 + D1 ln(1 + D2 eB/Lambda_QCD^2)); fitted to reproduce the lattice (Sigma_u+Sigma_d)/2.
  • D2 = 0.002515
    Free parameter in the same magnetic-field-dependent coupling fit.
  • a1 = 0.04
    Coefficient in the subtracted quark condensate formula Eq. (15), fitted to lattice data and described as the anomalous magnetic moment factor.
assumptions (4)
  • domain assumption Quenched ladder approximation with the gluon propagator and vertex dressing of Eqs. (7)-(8), with parameters from Mueller et al. [37], provides a valid strong-field quark DSE at T=0.
    Invoked in Section II.A; the central mass calculation inherits this model and its parameter set.
  • ad hoc to paper The magnetic-field dependence of the running coupling is gII = gI / (1 + D1 ln(1 + D2 eB/Lambda_QCD^2)), with D1 and D2 fitted here to lattice data.
    Adopted in Section II.C, Eq. (14); the gII suppression that drives the magnetic-field results is built from this phenomenological ansatz.
  • domain assumption Finite-volume effects are fully captured by replacing the 3D momentum integral with the truncated MRE density of states, surface plus curvature terms only, plus an infrared cutoff where rho_MRE=0, while gluon and vertex dressing are left unchanged.
    Stated in Section II.B, Eqs. (9)-(13); neglected higher-order terms and the cutoff procedure are not quantitatively justified.
  • ad hoc to paper The isotropic spherical MRE density of states, designed for a free Fermi gas in a bag, can be applied to a Landau-level-resolved quark propagator in a magnetic field.
    Used in all finite-volume calculations; the paper does not reconcile the isotropic 3D phase-space density with the anisotropic momentum split into p_parallel and discrete Landau levels.

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Pith. "Pith review of Dynamical quark mass and finite volume effects in the Dyson-Schwinger Equations." pith.science (2026). https://pith.science/paper/5MJXGGUG

@misc{pith2026250811968,
  author       = {Pith},
  title        = {Pith review of: Dynamical quark mass and finite volume effects in the Dyson-Schwinger Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MJXGGUG}},
  note         = {Machine review of arXiv:2508.11968}
}
abstract

Within the framework of Dyson-Schwinger equations(DSEs) and by means of the Multiple Reflection Expansion approximation, we study the finite volume effects of the constituent quark mass in a strong external magnetic field. Since the magnetic field has influence on the coupling constant, the coupling constant controls the strength of strongly interaction in QCD, so we adopt the magnetic-field-dependent running coupling constant in simulation. The results show that in addition to the magnetic field, the masses of constituent quarks also have a significant dependence on the volume and the running coupling constant. The model behaves close to the infinite volume limit for large size, but the effect of the finite volume is significant when the system size $R$ is about $2-6$ fm.The finite volume effects and the magnetic-field-dependent running coupling constant have considerable influence on the phase transition.

Figures

Figures reproduced from arXiv: 2508.11968 by the authors.

Figure 1
Figure 1. FIG. 1: (Color online) The magnetic dependence of the subtracted quark condensate, comparing the results from DSEs, the NJL[ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (Color online) The quenched dressing functions [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (Color online) Comparision between [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) Constituent quark mass [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) The partial derivative respecting to the radius [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: (Color online) Constituent quark mass [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Works this paper leans on

39 extracted references · 21 canonical work pages

  1. [41]

    Y .-Z. Xu, C. Shi, X.-T. He, H.-S. Zong, Phys.Rev. D102, 114011 (2020), arXiv:2009.12035 [nucl-th]

    work page Pith review arXiv 2020

  2. [1]

    The Phase Diagram of Strongly-Interacting Matter

    P. B.Munzinger and J. Wambach, Rev. Mod. Phys. 81, 1031 (2009), arXiv:0801.4256[hep-ph]

    work page Pith review arXiv 2009

  3. [2]

    Philipsen, Prog

    O. Philipsen, Prog. Part. Nucl. Phys. 70, 55 (2013), arXiv:1207.5999[hep-lat]

    arXiv 2013

  4. [3]

    Gupta, X

    S. Gupta, X. Luo, B. Mohanty, H. G. Ritter, and N. Xu, Science 332, 1525 (2011), arXiv:1105.3934[hep-ph]

    arXiv 2011

  5. [4]

    Stephanov, K

    M. Stephanov, K. Rajagopal, and E. Shuryak, Phys.Rev. D 60, 114028 (1999), arXiv:hep-ph/9903292

    arXiv 1999

  6. [5]

    M. A. Stephanov, J. Phys. G 38, 124147 (2011)

    work page 2011

  7. [6]

    A.Bhattacharyya, P.Deb, S.K.Ghosh, R.Ray, S.Sur, Phys. Rev. D 87, 054009 (2013), arXiv:1212.5893[hep-ph]

    arXiv 2013

  8. [7]

    P.B.Munzinger, J.Stachel, Nature 448, 302-309 (2007)

    work page 2007

Show all 39 references
  1. [8]

    L.F.Palhares, E.S.Fraga, T.Kodama, J. Phys. G 38, 085101 (2011)

    2011

  2. [9]

    Nucl. Phys. A 757, 102 (2005) , Fisrt Three Years of Operation of RHIC

    2005

  3. [10]

    A.Bhattacharyya, R.Ray, S.Samanta, S.Sur, Phys. Rev. C 91, 041901 (2015), arXiv:1502.00889 [hep-ph]

    2015 arXiv

  4. [11]

    H.Kohyama, D.Kimura, T.Inagaki, Nucl. Phys. B 906, 524 (2016)

    2016

  5. [12]

    J.Braun, B.Klein, H.-J.Pirner, Phys. Rev. D 71, 014032 (2005), arXiv:hep-ph/0408116

    2005 arXiv

  6. [13]

    J.Braun, B.Klein, H.-J.Pirner, A.H.Rezaeian, Phys. Rev. D 73, 074010 (2006), arXiv:hep-ph/0512274

    2006 arXiv

  7. [14]

    H.T.Elze, W.Greiner, Phys. Lett. B 197, 385 (1986)

    1986

  8. [15]

    S.Samanta, S.Ghosh, B.Mohanty, J. Phys. G 45, 075101 (2018), arXiv:1706.07709 [hep-ph]

    2018 arXiv

  9. [16]

    L.M.Abreu, E.S.Nery, Phys. Rev. C 96, 055204 (2017), arXiv:1711.07934 [nucl-th]

    2017 arXiv

  10. [17]

    C.S.Fischer, R.Alkofer, H.Reinhardt, Phys. Rev. D 65, 094008 (2002), arXiv:hep-ph/0202195

    2002 arXiv

  11. [18]

    B.-L.Li, Z.-F.Cui, B.-W.Zhou, S.An, L.-P.Zhang, H.-S.Zong, Nucl. Phys. B 938, 298 (2019), arXiv:1711.04914 [hep-ph]

    2019 arXiv

  12. [21]

    L.Jiang, F.Gao, Y .-X.Liu, arXiv:2310.15770 [nucl-th]

    arXiv

  13. [22]

    Balian and C

    R. Balian and C. Bloch, Annals Phys. 60, 401 (1970)

    1970

  14. [23]

    Madsen, Phys

    J. Madsen, Phys. Rev. D 50, 3328 (1994), arXiv:hep-ph/9407314. 9

    1994 arXiv

  15. [24]

    Kiriyama and A

    O. Kiriyama and A. Hosaka, Phys. Rev. D 67, 085010 (2003), arXiv:hep-ph/0211235

    2003 arXiv

  16. [25]

    Kiriyama, Phys

    O. Kiriyama, Phys. Rev. D 72, 054009(2005), arXiv:hep-ph/0507188

    2005 arXiv

  17. [26]

    Sarkar, P

    A. Sarkar, P. Deb, B. Mandal, R. Varma, arXiv:2507.21744 [hep-ph]

    arXiv

  18. [27]

    N.Chahal, S.Dutt, A.Kumar, arXiv:2303.16840 [hep-ph]

    arXiv

  19. [28]

    P.Adhikari, B.C.Tiburzi, arXiv:2302.09179 [hep-lat]

    arXiv

  20. [29]

    A.Ayala,S.B.Langarica, C.Villavicencio, arXiv:2111.05951 [hep-ph]

    arXiv

  21. [30]

    Miransky and I.A.Shovkovy, Phys.Rev.D66, 045006 (2002), arXiv:hep-ph/0205348

    V .A. Miransky and I.A.Shovkovy, Phys.Rev.D66, 045006 (2002), arXiv:hep-ph/0205348

    2002 arXiv

  22. [31]

    R.L.S.Farias, K.P.Gomes, G.I.Krein, and M.B.Pinto,Phys.Rev.C 90, 025203 (2014), arXiv:1404.3931 [hep-ph]

    2014 arXiv

  23. [32]

    E.J.Ferrer, V .Incera, and X.J.Wen,Phys.Rev.D91, 054006 (2015), arXiv:1407.3503 [nucl-th]

    2015 arXiv

  24. [33]

    R.L.S.Farias, V .S.Tim´oteo, S.S.Avancini, M.B.Pinto, and G.Krein, Eur. Phys. J. A53, 101 (2017), arXiv:1603.03847 [hep-ph]

    2017 arXiv

  25. [34]

    W.R.Tavares, R.L.S.Farias, S.S.Avancini, V .S.Tim´oteo, and M.B.Pinto et al, Eur. Phys. J. A 57, 278 (2021), arXiv:2104.11117 [hep-ph]

    2021 arXiv

  26. [35]

    D.-X.Wei, L.-J.Zhou, Int. J. Mod. Phys. E, 32(05), 2350024(2023), arXiv:2212.03508[hep-ph]

    2023 arXiv

  27. [36]

    L.-J.Zhou, B.Zheng, H.-w.Zhong, W.-X.Ma, Chin. Phys. C 39, 033101 (2014), arXiv:1403.6965 [hep-ph]

    2014 arXiv

  28. [37]

    N.Mueller, J.A.Bonnet, and C.S.Fischer,Phys.Rev.D 89, 094023 (2014), arXiv:1401.1647 [hep-ph]

    2014 arXiv

  29. [38]

    D105, 034514 (2022), arXiv:2201.02349 [hep-lat]

    H.-T.Ding, S.-T.Li, J.-H.Liu, and X.-D.Wang, Phys.Rev. D105, 034514 (2022), arXiv:2201.02349 [hep-lat]

    2022 arXiv

  30. [39]

    C 47, 064103 (2023), arXiv: 2205.08169 [hep-ph]

    M.Kawaguchi, and M.Huang, Chin.Phys. C 47, 064103 (2023), arXiv: 2205.08169 [hep-ph]

    2023 arXiv

  31. [40]

    Bernhardt, C

    J. Bernhardt, C. S. Fischer, P. Isserstedt, B.-J. Schaefer, Phys.Rev. D 104, 074035 (2021), arXiv:2107.05504 [hep-ph]

    2021 arXiv

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