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arxiv: 2606.28111 · v1 · pith:O5KPO4ZNnew · submitted 2026-06-26 · ✦ hep-ph · hep-lat· hep-th· nucl-th

Dense and Cold Magnetized Quark Matter: A Review of Magnetic-Field-Independent Regularization and the Medium Separation Scheme

Francisco X. Azeredo , Dyana C. Duarte , Ricardo L. S. Farias , Bruno S. Lopes , Jo\~ao A. R. S. Prado , William R. Tavares This is my paper

Pith reviewed 2026-06-29 03:40 UTC · model grok-4.3

classification ✦ hep-ph hep-lathep-thnucl-th
keywords magnetized quark matterregularization schemescolor superconductivityNambu-Jona-Lasinio modeldense matterzero temperaturemagnetic fields
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The pith

The superconducting gap in cold dense magnetized quark matter stays finite at all chemical potentials under proper vacuum-medium separation.

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

This review examines regularization of ultraviolet divergences in nonrenormalizable models of quark matter placed in strong magnetic fields at finite density. It presents Magnetic-Field-Independent Regularization, which isolates vacuum divergences from magnetic-field-dependent terms, and the Medium Separation Scheme, which further separates vacuum quantities from all medium contributions so that only the former are regularized. When these methods are applied to the thermodynamics of color-superconducting phases at zero temperature, the gap equation yields a nonzero gap that persists to arbitrarily large chemical potentials even for strong fields. Traditional regularization schemes instead produce an artificial closing of the gap and a transition to the normal phase at zero temperature because they mix vacuum and medium terms. The separation removes unphysical oscillations and yields a consistent thermodynamic description relevant to compact stars.

Core claim

Within the unified MFIR and MSS framework the superconducting gap remains finite at large chemical potentials even in the presence of strong magnetic fields, with no evidence for a transition to a normal phase at zero temperature.

What carries the argument

The Medium Separation Scheme (MSS), which isolates only vacuum quantities for regularization while leaving all medium contributions unregularized.

If this is right

  • Superconducting phases persist to arbitrarily high densities without closing.
  • No zero-temperature transition to the normal phase occurs under strong magnetic fields.
  • Spurious oscillations in thermodynamic quantities disappear.
  • The equation of state for magnetized quark matter becomes free of regularization artifacts.

Where Pith is reading between the lines

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

  • Astrophysical models of magnetars may need to include persistent color superconductivity at all densities.
  • Phase diagrams obtained from other effective models should be recomputed with the same vacuum-medium separation.
  • Finite-temperature extensions could reveal whether thermal effects restore a normal phase at lower densities than previously thought.

Load-bearing premise

The Medium Separation Scheme correctly isolates only vacuum quantities for regularization while leaving all medium contributions unregularized.

What would settle it

A direct solution of the gap equation under the MSS showing that the superconducting gap vanishes at some finite chemical potential for nonzero magnetic field strength.

Figures

Figures reproduced from arXiv: 2606.28111 by Bruno S. Lopes, Dyana C. Duarte, Francisco X. Azeredo, Jo\~ao A. R. S. Prado, Ricardo L. S. Farias, William R. Tavares.

Figure 1
Figure 1. Figure 1: Effective quark mass M as a function of the magnetic field eB for µ = 0 (left panel), and diquark condensate ∆ as a function of the chemical potential µ for different values of eB (right panel). requiring one fewer iteration than in the previous two cases, since the last term is already finite when integrated over the entire momentum space. The integration over a can be performed analytically, while the la… view at source ↗
Figure 2
Figure 2. Figure 2: Constituent quark mass M (left panels) and diquark condensate ∆ (right panels) as a function of the chemical potential µ within the MSS + MFIR method at finite magnetic field. The results are presented for three different values of the ratio between couplings η [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Constituent quark mass M (left panels) and diquark condensate ∆ (right panels) as a function of the magnetic field eB within the MSS + MFIR method at finite chemical potential. The results are presented for three different values of the ratio between couplings η [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Phase diagram in the eB × µ plane within the MSS + MFIR method. The results are presented for three different values of the coupling ratio η. These diagrams display the chiral symmetry breaking phase (χSB) and the color superconducting phase (CSC). The phase structure in the eB × µc plane, obtained within the MSS + MFIR scheme, is presented in [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normalized baryon number density nB/n0 (left panels) and magnetization M (right panels) as functions of the magnetic field strength eB (upper panels) and the chemical potential µ (lower panels), calculated within the MSS+MFIR scheme. The results shown in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Equation of State (EoS) (upper panels), and the squared speed of sound [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Phase diagram in the η × µ plane, comparing the MSS and TRS schemes at zero magnetic field (top panels) and at a finite magnetic field using the MSS+MFIR and TRS+MFIR methods (bottom panels). The diagrams display the chiral symmetry breaking (χSB), color superconducting phase (CSC), and a phase with restored chiral symmetry (normal). The red dot indicates the critical end point (CEP). Finally, [PITH_FULL_… view at source ↗
read the original abstract

We present a comprehensive review of regularization schemes for magnetized dense quark matter within effective models of quantum chromodynamics, focusing on the Magnetic-Field-Independent Regularization (MFIR) and the Medium Separation Scheme (MSS) at finite chemical potential and magnetic field. In nonrenormalizable frameworks such as the Nambu-Jona-Lasinio model, the treatment of ultraviolet divergences is crucial, particularly in magnetized and dense environments where conventional regularization procedures may introduce unphysical artifacts. We show that MFIR consistently isolates divergent vacuum contributions from finite magnetic-field-dependent terms, while MSS extends this separation to the medium sector, ensuring that only vacuum quantities are regularized. Within this unified framework, we analyze the thermodynamics of cold and dense quark matter, including color-superconducting phases, and demonstrate that the superconducting gap remains finite at large chemical potentials, even in the presence of strong magnetic fields. In contrast to results obtained with traditional regularization schemes, we find no evidence for a transition to a normal phase at zero temperature, highlighting the importance of a proper separation between vacuum and medium contributions. These results eliminate spurious oscillations and other nonphysical artifacts, leading to a more robust and physically consistent description of strongly interacting matter under extreme conditions relevant to compact stars and heavy-ion collisions.

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 reviews regularization schemes for the Nambu-Jona-Lasinio model in magnetized dense quark matter, focusing on Magnetic-Field-Independent Regularization (MFIR) and the Medium Separation Scheme (MSS). It claims that these approaches isolate vacuum divergences while leaving medium contributions unregularized, leading to a finite color-superconducting gap at large chemical potentials even under strong magnetic fields, with no transition to a normal phase at zero temperature and the elimination of spurious oscillations seen in conventional schemes.

Significance. If the central claims hold, the work offers a more consistent thermodynamic description of cold dense quark matter relevant to compact stars and heavy-ion collisions by avoiding unphysical artifacts from traditional regularizations. As a review, it synthesizes prior results on vacuum-medium separation but does not introduce new derivations or numerical validations within the manuscript itself.

major comments (1)
  1. [Abstract] Abstract and the discussion of MSS: the central claim that the gap remains finite at large mu with no T=0 normal-phase transition rests on the assumption that MSS unambiguously isolates only vacuum divergences for regularization while leaving all medium contributions finite. When Landau levels mix vacuum and Fermi-sea pieces (particularly when mu ~ sqrt(eB)), the split is not obviously unique, which could render the gap equation scheme-dependent and undermine the contrast with traditional regularizations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the MSS. We respond point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the discussion of MSS: the central claim that the gap remains finite at large mu with no T=0 normal-phase transition rests on the assumption that MSS unambiguously isolates only vacuum divergences for regularization while leaving all medium contributions finite. When Landau levels mix vacuum and Fermi-sea pieces (particularly when mu ~ sqrt(eB)), the split is not obviously unique, which could render the gap equation scheme-dependent and undermine the contrast with traditional regularizations.

    Authors: The MSS defines the vacuum-medium separation uniquely by subtracting the mu-independent vacuum contribution (containing all ultraviolet divergences) from the full thermodynamic potential at the integrand level, prior to the sum over Landau levels. This subtraction criterion fixes the split unambiguously for any relation between mu and sqrt(eB), with only the vacuum term regularized and all mu-dependent medium terms left finite. The resulting gap equation therefore yields a finite superconducting gap at large mu with no T=0 transition to the normal phase. This construction follows directly from the definitions in the works reviewed in the manuscript and eliminates the artifacts of conventional schemes without introducing scheme dependence within the MSS framework. revision: no

Circularity Check

0 steps flagged

No circularity; review applies external regularization schemes without self-referential reduction

full rationale

The manuscript is explicitly a review of MFIR and MSS schemes developed in prior literature. The central claim (finite superconducting gap at large mu, no T=0 normal-phase transition) is presented as a consequence of applying the MSS vacuum-medium separation to the NJL thermodynamic potential, not as a new derivation that reduces to fitted parameters or self-citations within this paper. No equations are shown that equate a 'prediction' to an input by construction, and no load-bearing uniqueness theorem or ansatz is imported solely via overlapping-author citations. The separation procedure is the definitional content of MSS itself and is treated as an established input rather than derived here. This is the expected outcome for a review paper whose results remain externally falsifiable against traditional regularization benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The review rests on the standard assumptions of the Nambu-Jona-Lasinio effective model and the validity of the MFIR and MSS schemes as developed in earlier papers; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The Nambu-Jona-Lasinio model provides a suitable effective description of QCD at finite density and magnetic field.
    The entire analysis is performed inside this framework.

pith-pipeline@v0.9.1-grok · 5802 in / 1217 out tokens · 53982 ms · 2026-06-29T03:40:33.680242+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

203 extracted references · 114 linked inside Pith

  1. [1]

    Rafelski and B

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

    1976

  2. [2]

    D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A803, 227 (2008), arXiv:0711.0950 [hep-ph]

    Pith/arXiv arXiv 2008

  3. [3]

    Skokov, A

    V. Skokov, A. Y. Illarionov, and V. Toneev, Int. J. Mod. Phys. A24, 5925 (2009), arXiv:0907.1396 [nucl-th]

    Pith/arXiv arXiv 2009

  4. [4]

    Voronyuk, V

    V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Konchakovski, and S. A. Voloshin, Phys. Rev. C83, 054911 (2011), arXiv:1103.4239 [nucl-th]

    Pith/arXiv arXiv 2011

  5. [5]

    Bzdak and V

    A. Bzdak and V. Skokov, Phys. Lett. B710, 171 (2012), arXiv:1111.1949 [hep-ph]

    Pith/arXiv arXiv 2012

  6. [6]

    Ou and B.-A

    L. Ou and B.-A. Li, Phys. Rev. C84, 064605 (2011), arXiv:1107.3192 [nucl-th]

    Pith/arXiv arXiv 2011

  7. [7]

    Deng and X.-G

    W.-T. Deng and X.-G. Huang, Phys. Rev. C85, 044907 (2012), arXiv:1201.5108 [nucl-th]

    Pith/arXiv arXiv 2012

  8. [8]

    Bloczynski, X.-G

    J. Bloczynski, X.-G. Huang, X. Zhang, and J. Liao, Phys. Lett. B718, 1529 (2013), arXiv:1209.6594 [nucl-th]

    Pith/arXiv arXiv 2013

  9. [9]

    Bloczynski, X.-G

    J. Bloczynski, X.-G. Huang, X. Zhang, and J. Liao, Nucl. Phys. A939, 85 (2015), arXiv:1311.5451 [nucl-th]

    Pith/arXiv arXiv 2015

  10. [10]

    Zhong, C.-B

    Y. Zhong, C.-B. Yang, X. Cai, and S.-Q. Feng, Adv. High Energy Phys.2014, 193039 (2014), arXiv:1408.5694 [hep-ph]

    Pith/arXiv arXiv 2014

  11. [11]

    Zhong, C.-B

    Y. Zhong, C.-B. Yang, X. Cai, and S.-Q. Feng, Chin. Phys. C39, 104105 (2015), arXiv:1410.6349 [hep-ph]. 19

    Pith/arXiv arXiv 2015

  12. [12]

    Adhikari et al., Prog

    P. Adhikari et al., Prog. Part. Nucl. Phys.146, 104199 (2026), arXiv:2412.18632 [nucl-th]

    Pith/arXiv arXiv 2026

  13. [13]

    Endrodi, Prog

    G. Endrodi, Prog. Part. Nucl. Phys.141, 104153 (2025), arXiv:2406.19780 [hep-lat]

    arXiv 2025

  14. [14]

    M. G. Mustafa, A. Bandyopadhyay, and C. A. Islam, Prog. Part. Nucl. Phys.148, 104234 (2026), arXiv:2503.00075 [nucl-th]

    arXiv 2026

  15. [15]

    Fukushima, Prog

    K. Fukushima, Prog. Part. Nucl. Phys.107, 167 (2019), arXiv:1812.08886 [hep-ph]

    Pith/arXiv arXiv 2019

  16. [16]

    Hattori, K

    K. Hattori, K. Itakura, and S. Ozaki, Prog. Part. Nucl. Phys.133, 104068 (2023), arXiv:2305.03865 [hep-ph]

    arXiv 2023

  17. [17]

    Huang, Rept

    X.-G. Huang, Rept. Prog. Phys.79, 076302 (2016), arXiv:1509.04073 [nucl-th]

    Pith/arXiv arXiv 2016

  18. [18]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP05, 051 (2003), arXiv:hep-ph/0302165

    Pith/arXiv arXiv 2003

  19. [19]

    Aarts, C

    G. Aarts, C. Allton, J. Foley, S. Hands, and S. Kim, Phys. Rev. Lett.99, 022002 (2007), arXiv:hep-lat/0703008

    Pith/arXiv arXiv 2007

  20. [20]

    H. T. Ding, A. Francis, O. Kaczmarek, F. Karsch, E. Laermann, and W. Soeldner, Phys. Rev. D83, 034504 (2011), arXiv:1012.4963 [hep-lat]

    Pith/arXiv arXiv 2011

  21. [21]

    Francis and O

    A. Francis and O. Kaczmarek, Prog. Part. Nucl. Phys.67, 212 (2012), arXiv:1112.4802 [hep-lat]

    Pith/arXiv arXiv 2012

  22. [22]

    H.-T. Ding, O. Kaczmarek, and F. Meyer, PoSLA TTICE2014, 216 (2015), arXiv:1412.5869 [hep-lat]

    Pith/arXiv arXiv 2015

  23. [23]

    B. B. Brandt, A. Francis, H. B. Meyer, and H. Wittig, JHEP03, 100 (2013), arXiv:1212.4200 [hep-lat]

    Pith/arXiv arXiv 2013

  24. [24]

    Amato, G

    A. Amato, G. Aarts, C. Allton, P. Giudice, S. Hands, and J.-I. Skullerud, Phys. Rev. Lett.111, 172001 (2013), arXiv:1307.6763 [hep-lat]

    Pith/arXiv arXiv 2013

  25. [25]

    Aarts, C

    G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands, and J.-I. Skullerud, JHEP02, 186 (2015), arXiv:1412.6411 [hep-lat]

    Pith/arXiv arXiv 2015

  26. [26]

    Fukushima, D

    K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D78, 074033 (2008), arXiv:0808.3382 [hep-ph]

    Pith/arXiv arXiv 2008

  27. [27]

    Ali Hassan Abdallah et al

    D. Ali Hassan Abdallah et al. (ALICE), (2026), arXiv:2602.22900 [nucl-ex]

    arXiv 2026

  28. [28]

    Acharya et al

    S. Acharya et al. (ALICE), Phys. Lett. B856, 138862 (2024), arXiv:2210.15383 [nucl-ex]

    arXiv 2024

  29. [29]

    B. E. Aboona et al. (STAR), Phys. Rev. C113, 014912 (2026), arXiv:2506.00278 [nucl-ex]

    arXiv 2026

  30. [30]

    Abdallah et al

    M. Abdallah et al. (STAR), Phys. Rev. C105, 014901 (2022), arXiv:2109.00131 [nucl-ex]

    arXiv 2022

  31. [31]

    G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, and A. Schafer, Phys. Rev. D86, 071502 (2012), arXiv:1206.4205 [hep-lat]

    Pith/arXiv arXiv 2012

  32. [32]

    Endrodi, M

    G. Endrodi, M. Giordano, S. D. Katz, T. G. Kov´ acs, and F. Pittler, JHEP07, 007 (2019), arXiv:1904.10296 [hep-lat]

    arXiv 2019

  33. [33]

    G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber, and A. Schaefer, JHEP04, 130 (2013), arXiv:1303.1328 [hep-lat]

    Pith/arXiv arXiv 2013

  34. [34]

    G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, JHEP02, 044 (2012), arXiv:1111.4956 [hep-lat]

    Pith/arXiv arXiv 2012

  35. [35]

    Hidaka and A

    Y. Hidaka and A. Yamamoto, Phys. Rev. D87, 094502 (2013), arXiv:1209.0007 [hep-ph]

    Pith/arXiv arXiv 2013

  36. [36]

    E. V. Luschevskaya and O. V. Larina, Nucl. Phys. B884, 1 (2014), arXiv:1203.5699 [hep-lat]

    Pith/arXiv arXiv 2014

  37. [37]

    M. A. Andreichikov, B. O. Kerbikov, E. V. Luschevskaya, Y. A. Simonov, and O. E. Solovjeva, JHEP05, 007 (2017), arXiv:1610.06887 [hep-ph]

    Pith/arXiv arXiv 2017

  38. [38]

    Endr¨ odi, JHEP04, 023 (2013), arXiv:1301.1307 [hep-ph]

    G. Endr¨ odi, JHEP04, 023 (2013), arXiv:1301.1307 [hep-ph]

    Pith/arXiv arXiv 2013

  39. [39]

    G. S. Bali, B. B. Brandt, G. Endr˝ odi, and B. Gl¨ aßle, Phys. Rev. D97, 034505 (2018), arXiv:1707.05600 [hep-lat]

    Pith/arXiv arXiv 2018

  40. [40]

    D’Elia, L

    M. D’Elia, L. Maio, F. Sanfilippo, and A. Stanzione, Phys. Rev. D105, 034511 (2022), arXiv:2111.11237 [hep-lat]

    arXiv 2022

  41. [41]

    J. N. Guenther, Eur. Phys. J. A57, 136 (2021), arXiv:2010.15503 [hep-lat]

    arXiv 2021

  42. [42]

    D’Elia, F

    M. D’Elia, F. Manigrasso, F. Negro, and F. Sanfilippo, Phys. Rev. D98, 054509 (2018), arXiv:1808.07008 [hep-lat]

    Pith/arXiv arXiv 2018

  43. [43]

    H.-T. Ding, C. Schmidt, A. Tomiya, and X.-D. Wang, Phys. Rev. D102, 054505 (2020), arXiv:2006.13422 [hep-lat]

    arXiv 2020

  44. [44]

    V. V. Braguta, M. N. Chernodub, A. Y. Kotov, A. V. Molochkov, and A. A. Nikolaev, Phys. Rev. D100, 114503 (2019), arXiv:1909.09547 [hep-lat]

    arXiv 2019

  45. [45]

    G. S. Bali, F. Bruckmann, G. Endrodi, and A. Schafer, Phys. Rev. Lett.112, 042301 (2014), arXiv:1311.2559 [hep-lat]

    Pith/arXiv arXiv 2014

  46. [46]

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

    arXiv 2022

  47. [47]

    J. O. Andersen, W. R. Naylor, and A. Tranberg, Rev. Mod. Phys.88, 025001 (2016), arXiv:1411.7176 [hep-ph]

    Pith/arXiv arXiv 2016

  48. [48]

    J. O. Andersen, JHEP10, 005 (2012), arXiv:1205.6978 [hep-ph]

    Pith/arXiv arXiv 2012

  49. [49]

    Adhikari, T

    P. Adhikari, T. D. Cohen, and J. Sakowitz, Phys. Rev. C91, 045202 (2015), arXiv:1501.02737 [nucl-th]

    Pith/arXiv arXiv 2015

  50. [50]

    Adhikari and B

    P. Adhikari and B. C. Tiburzi, Phys. Rev. D107, 094504 (2023), arXiv:2302.09179 [hep-lat]

    arXiv 2023

  51. [51]

    C. P. Hofmann, Phys. Rev. D104, 014025 (2021), arXiv:2012.06461 [hep-ph]

    arXiv 2021

  52. [52]

    C. P. Hofmann, Phys. Rev. D101, 114031 (2020), arXiv:2004.01247 [hep-ph]

    arXiv 2020

  53. [53]

    C. P. Hofmann, Phys. Rev. D99, 014030 (2019), arXiv:1710.05820 [hep-ph]

    Pith/arXiv arXiv 2019

  54. [54]

    Endr˝ odi and G

    G. Endr˝ odi and G. Mark´ o, JHEP08, 036 (2019), arXiv:1905.02103 [hep-lat]

    arXiv 2019

  55. [55]

    S. P. Klevansky, Rev. Mod. Phys.64, 649 (1992)

    1992

  56. [56]

    Skokov, Phys

    V. Skokov, Phys. Rev. D85, 034026 (2012), arXiv:1112.5137 [hep-ph]

    Pith/arXiv arXiv 2012

  57. [57]

    Ayala, C

    A. Ayala, C. A. Dominguez, L. A. Hernandez, M. Loewe, and R. Zamora, Phys. Rev. D92, 096011 (2015), [Addendum: Phys.Rev.D 92, 119905 (2015)], arXiv:1509.03345 [hep-ph]

    Pith/arXiv arXiv 2015

  58. [58]

    Magdy, M

    N. Magdy, M. Csan´ ad, and R. A. Lacey, J. Phys. G44, 025101 (2017), arXiv:1510.04380 [nucl-th]

    Pith/arXiv arXiv 2017

  59. [59]

    Fang and L

    Z. Fang and L. Zhang, (2019), arXiv:1910.02269 [hep-ph]

    arXiv 2019

  60. [60]

    J. O. Andersen, Eur. Phys. J. A57, 189 (2021), arXiv:2102.13165 [hep-ph]

    arXiv 2021

  61. [61]

    S. S. Avancini, D. P. Menezes, M. B. Pinto, and C. Providencia, Phys. Rev. D85, 091901 (2012), arXiv:1202.5641 [hep-ph]

    Pith/arXiv arXiv 2012

  62. [62]

    Gatto and M

    R. Gatto and M. Ruggieri, Phys. Rev. D82, 054027 (2010), arXiv:1007.0790 [hep-ph]

    Pith/arXiv arXiv 2010

  63. [63]

    M. S. Ali, C. A. Islam, and R. Sharma, Phys. Rev. D110, 096011 (2024), arXiv:2407.14449 [hep-ph]

    arXiv 2024

  64. [64]

    J. O. Andersen and A. A. Cruz, Phys. Rev. D88, 025016 (2013), arXiv:1211.7293 [hep-ph]

    Pith/arXiv arXiv 2013

  65. [65]

    R. L. S. Farias, V. S. Timoteo, S. S. Avancini, M. B. Pinto, and G. Krein, Eur. Phys. J. A53, 101 (2017), arXiv:1603.03847 [hep-ph]. 20

    Pith/arXiv arXiv 2017

  66. [66]

    D. C. Duarte, R. L. S. Farias, and R. O. Ramos, Phys. Rev. D84, 083525 (2011), arXiv:1108.4428 [hep-ph]

    Pith/arXiv arXiv 2011

  67. [67]

    Krein and C

    G. Krein and C. Miller, Symmetry13, 551 (2021), arXiv:2103.15665 [hep-ph]

    arXiv 2021

  68. [68]

    F. L. Braghin, Eur. Phys. J. A54, 45 (2018), arXiv:1705.05926 [hep-ph]

    Pith/arXiv arXiv 2018

  69. [69]

    Ayala, L

    A. Ayala, L. A. Hern´ andez, A. J. Mizher, J. C. Rojas, and C. Villavicencio, Phys. Rev. D89, 116017 (2014), arXiv:1404.6568 [hep-ph]

    Pith/arXiv arXiv 2014

  70. [70]

    Ferreira, P

    M. Ferreira, P. Costa, and C. Providˆ encia, Phys. Rev. D90, 016012 (2014), arXiv:1406.3608 [hep-ph]

    Pith/arXiv arXiv 2014

  71. [71]

    Ferreira, P

    M. Ferreira, P. Costa, O. Louren¸ co, T. Frederico, and C. Providˆ encia, Phys. Rev. D89, 116011 (2014), arXiv:1404.5577 [hep-ph]

    Pith/arXiv arXiv 2014

  72. [72]

    Bandyopadhyay, R

    A. Bandyopadhyay, R. L. S. Farias, B. S. Lopes, and R. O. Ramos, Phys. Rev. D100, 076021 (2019), arXiv:1906.09250 [hep-ph]

    arXiv 2019

  73. [73]

    H. T. Ding, S. T. Li, A. Tomiya, X. D. Wang, and Y. Zhang, Phys. Rev. D104, 014505 (2021), arXiv:2008.00493 [hep-lat]

    arXiv 2021

  74. [74]

    S. S. Avancini, R. L. S. Farias, M. Benghi Pinto, W. R. Tavares, and V. S. Tim´ oteo, Phys. Lett. B767, 247 (2017), arXiv:1606.05754 [hep-ph]

    Pith/arXiv arXiv 2017

  75. [75]

    S. S. Avancini, R. L. S. Farias, and W. R. Tavares, Phys. Rev. D99, 056009 (2019), arXiv:1812.00945 [hep-ph]

    Pith/arXiv arXiv 2019

  76. [76]

    S. S. Avancini, R. L. S. Farias, W. R. Tavares, and V. S. Tim´ oteo, Nucl. Phys. B981, 115862 (2022), arXiv:2202.03328 [hep-ph]

    arXiv 2022

  77. [77]

    Coppola, W

    M. Coppola, W. R. Tavares, S. S. Avancini, J. C. Sodr´ e, and N. N. Scoccola, Phys. Rev. D110, 114036 (2024), arXiv:2410.05568 [hep-ph]

    arXiv 2024

  78. [78]

    Fayazbakhsh, S

    S. Fayazbakhsh, S. Sadeghian, and N. Sadooghi, Phys. Rev. D86, 085042 (2012), arXiv:1206.6051 [hep-ph]

    Pith/arXiv arXiv 2012

  79. [79]

    Ayala, J

    A. Ayala, J. L. Hern´ andez, L. A. Hern´ andez, R. L. S. Farias, and R. Zamora, Phys. Rev. D103, 054038 (2021), arXiv:2011.03673 [hep-ph]

    arXiv 2021

  80. [80]

    Ayala, R

    A. Ayala, R. L. S. Farias, S. Hern´ andez-Ortiz, L. A. Hern´ andez, D. M. Paret, and R. Zamora, Phys. Rev. D98, 114008 (2018), arXiv:1809.08312 [hep-ph]

    Pith/arXiv arXiv 2018

Showing first 80 references.