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Delineating neutral and charged mesons in magnetic fields

Toru Kojo* and Sakura Itatani†

  • *Contact author: torukojo@post.kek.jp
  • †Contact author: itasaku@post.kek.jp
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Phys. Rev. D 114, 034043 – Published 21 August, 2026

DOI: https://doi.org/10.1103/4jxs-zt5s

Abstract

We investigate the properties of neutral and charged mesons in magnetic fields, from weak-field to strong-field regimes. To develop analytic insights, we employ a nonrelativistic quark model with a confining potential of the harmonic oscillator type. Short-range correlations, such as Coulomb and color-magnetic interactions, are treated as perturbations. In particular, we focus on the magnetic field dependence of the relative and the center-of-mass motions. The qualitative trends differ significantly between neutral and charged mesons; for neutral mesons, the transverse momenta are conserved and continuous, while charged mesons exhibit quantized transverse dynamics. The Zeeman effects, arising from intrinsic spins and orbital angular momenta, are carefully examined. In particular, for charged mesons with spins s≥1, we discuss how the zero-point energy in the internal quark motion cancels the Zeeman energy from the orbital angular momentum, ensuring the energetic stability of mesons with high spins. The effectively reduced dimensionality of these mesons in the strong-field limit is also discussed.

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References (80)

  1. D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, The Effects of topological charge change in heavy ion collisions: ‘Event by event P and CP violation’, Nucl. Phys. A803, 227 (2008).
  2. V. Skokov, A. Y. Illarionov, and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A 24, 5925 (2009).
  3. W.-T. Deng and X.-G. Huang, Event-by-event generation of electromagnetic fields in heavy-ion collisions, Phys. Rev. C 85, 044907 (2012).
  4. K. Fukushima, D. E. Kharzeev, and H. J. Warringa, The Chiral magnetic effect, Phys. Rev. D 78, 074033 (2008).
  5. R. C. Duncan and C. Thompson, Formation of very strongly magnetized neutron stars—implications for gamma-ray bursts, Astrophys. J. Lett. 392, L9 (1992).
  6. S. Chakrabarty, Quark matter in strong magnetic field, Phys. Rev. D 54, 1306 (1996).
  7. E. J. Ferrer, V. de la Incera, and C. Manuel, Magnetic color flavor locking phase in high density QCD, Phys. Rev. Lett. 95, 152002 (2005).
  8. W. Gyory and V. de la Incera, Phase transitions and resilience of the magnetic dual chiral density wave phase at finite temperature and density, Phys. Rev. D 106, 016011 (2022).
  9. E. J. Ferrer and V. de la Incera, Magnetic dual Chiral density wave: A candidate quark matter phase for the interior of neutron stars, Universe 7, 458 (2021).
  10. E. J. Ferrer and V. de la Incera, Novel topological effects in dense QCD in a magnetic field, Nucl. Phys. B931, 192 (2018).
  11. P. Adhikari et al., Strongly interacting matter in extreme magnetic fields, Prog. Part. Nucl. Phys. 146, 104199 (2026).
  12. K. Hattori and X.-G. Huang, Novel quantum phenomena induced by strong magnetic fields in heavy-ion collisions, Nucl. Sci. Tech. 28, 26 (2017).
  13. D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions-A status report, Prog. Part. Nucl. Phys. 88, 1 (2016).
  14. V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Phys. Rep. 576, 1 (2015).
  15. J. O. Andersen, W. R. Naylor, and A. Tranberg, Phase diagram of QCD in a magnetic field: A review, Rev. Mod. Phys. 88, 025001 (2016).
  16. S. Iwasaki, M. Oka, and K. Suzuki, A review of quarkonia under strong magnetic fields, Eur. Phys. J. A 57, 222 (2021).
  17. V. A. Miransky and I. A. Shovkovy, Magnetic catalysis and anisotropic confinement in QCD, Phys. Rev. D 66, 045006 (2002).
  18. V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field, Nucl. Phys. B462, 249 (1996).
  19. G. Endrödi, QCD equation of state at nonzero magnetic fields in the Hadron Resonance Gas model, J. High Energy Phys. 04 (2013) 023.
  20. K. Fukushima and Y. Hidaka, Magnetic catalysis versus magnetic inhibition, Phys. Rev. Lett. 110, 031601 (2013).
  21. T. Kojo and N. Su, The quark mass gap in a magnetic field, Phys. Lett. B 720, 192 (2013).
  22. T. Kojo and N. Su, A renormalization group approach for QCD in a strong magnetic field, Phys. Lett. B 726, 839 (2013).
  23. T. Kojo and N. Su, The quark mass gap in strong magnetic fields, Nucl. Phys. A931, 763 (2014).
  24. K. Hattori, T. Kojo, and N. Su, Mesons in strong magnetic fields: (I) General analyses, Nucl. Phys. A951, 1 (2016).
  25. T. Kojo, Neutral and charged mesons in magnetic fields: A resonance gas in a non-relativistic quark model, Eur. Phys. J. A 57, 317 (2021).
  26. A. J. Arifi and K. Suzuki, Structure of heavy quarkonia in a strong magnetic field, Phys. Rev. D 112, 094013 (2025).
  27. J. Alford and M. Strickland, Charmonia and bottomonia in a magnetic field, Phys. Rev. D 88, 105017 (2013).
  28. E. S. Fraga, J. Noronha, and L. F. Palhares, Large Nc deconfinement transition in the presence of a magnetic field, Phys. Rev. D 87, 114014 (2013).
  29. E. S. Fraga and L. F. Palhares, Deconfinement in the presence of a strong magnetic background: An exercise within the MIT bag model, Phys. Rev. D 86, 016008 (2012).
  30. D. Ebert and K. G. Klimenko, Quark droplets stability induced by external magnetic field, Nucl. Phys. A728, 203 (2003).
  31. I. A. Shushpanov and A. V. Smilga, Quark condensate in a magnetic field, Phys. Lett. B 402, 351 (1997).
  32. G. Cao, Recent progresses on QCD phases in a strong magnetic field: Views from Nambu–Jona-Lasinio model, Eur. Phys. J. A 57, 264 (2021).
  33. J. Braun, W. A. Mian, and S. Rechenberger, Delayed magnetic catalysis, Phys. Lett. B 755, 265 (2016).
  34. N. Mueller and J. M. Pawlowski, Magnetic catalysis and inverse magnetic catalysis in QCD, Phys. Rev. D 91, 116010 (2015).
  35. N. Mueller, J. A. Bonnet, and C. S. Fischer, Dynamical quark mass generation in a strong external magnetic field, Phys. Rev. D 89, 094023 (2014).
  36. Z. Wang, Residue-enhanced Pion-Rho mixing as the origin of nonmonotonic charged pion mass in magnetic fields, arXiv:2602.15410.
  37. J. Mei, R. Wen, M. Zhou, S. Mao, and M. Huang, Spectral function for pions in magnetic field, arXiv:2601.22422.
  38. J. Mei, R. Wen, S. Mao, M. Huang, and K. Xu, Magnetic catalysis and diamagnetism from pion fluctuations, Phys. Rev. D 110, 034024 (2024).
  39. J. Mei, T. Xia, and S. Mao, Mass spectra of neutral mesons K0,π0,η,η’ at finite magnetic field, temperature and quark chemical potential, Phys. Rev. D 107, 074018 (2023); 110, 119901(E) (2024).
  40. S. Mao, Pions in magnetic field at finite temperature, Phys. Rev. D 99, 056005 (2019).
  41. M. A. Andreichikov and Y. A. Simonov, Chiral physics in the magnetic field with quark confinement contribution, Eur. Phys. J. C 78, 902 (2018).
  42. M. A. Andreichikov, B. O. Kerbikov, E. V. Luschevskaya, Y. A. Simonov, and O. E. Solovjeva, The evolution of meson masses in a strong magnetic field, J. High Energy Phys. 05 (2017) 007.
  43. Y. A. Simonov, Magnetic confinement and the Linde problem, Phys. Rev. D 96, 096002 (2017).
  44. T. H. Moreira and F. L. Braghin, Magnetic field induced corrections to the NJL model coupling constant from vacuum polarization, Phys. Rev. D 105, 114009 (2022).
  45. S. S. Avancini, R. L. S. Farias, M. Benghi Pinto, W. R. Tavares, and V. S. Timóteo, π0 pole mass calculation in a strong magnetic field and lattice constraints, Phys. Lett. B 767, 247 (2017).
  46. A. Ayala, R. L. S. Farias, S. Hernández-Ortiz, L. A. Hernández, D. M. Paret, and R. Zamora, Magnetic field-dependence of the neutral pion mass in the linear sigma model coupled to quarks: The weak field case, Phys. Rev. D 98, 114008 (2018).
  47. S. S. Avancini, R. L. S. Farias, and W. R. Tavares, Neutral meson properties in hot and magnetized quark matter: A new magnetic field independent regularization scheme applied to NJL-type model, Phys. Rev. D 99, 056009 (2019).
  48. A. Ayala, J. L. Hernández, L. A. Hernández, R. L. S. Farias, and R. Zamora, Magnetic field dependence of the neutral pion mass in the linear sigma model with quarks: The strong field case, Phys. Rev. D 103, 054038 (2021).
  49. A. Ayala, R. L. S. Farias, L. A. Hernández, A. J. Mizher, J. Rendón, C. Villavicencio, and R. Zamora, Magnetic field dependence of the neutral pion longitudinal screening mass in the linear sigma model with quarks, Phys. Rev. D 109, 074019 (2024).
  50. S. S. Avancini, R. L. S. Farias, W. R. Tavares, and V. S. Timóteo, Magnetized pole-mass of neutral ρ meson within full RPA evaluation, Nucl. Phys. B981, 115862 (2022).
  51. H.-T. Ding and D. Zhang, Chiral properties of (2+1)-flavor QCD in magnetic fields at zero temperature, Phys. Rev. D 113, 094503 (2026).
  52. H.-T. Ding, J.-B. Gu, A. Kumar, S.-T. Li, and J.-H. Liu, Baryon electric charge correlation as a magnetometer of QCD, Phys. Rev. Lett. 132, 201903 (2024).
  53. H. T. Ding, S. T. Li, Q. Shi, and X. D. Wang, Fluctuations and correlations of net baryon number, electric charge and strangeness in a background magnetic field, Eur. Phys. J. A 57, 202 (2021).
  54. H. T. Ding, S. T. Li, A. Tomiya, X. D. Wang, and Y. Zhang, Chiral properties of (2+1)-flavor QCD in strong magnetic fields at zero temperature, Phys. Rev. D 104, 014505 (2021).
  55. G. Endrodi, M. Giordano, S. D. Katz, T. G. Kovács, and F. Pittler, Magnetic catalysis and inverse catalysis for heavy pions, J. High Energy Phys. 07 (2019) 007.
  56. G. S. Bali, B. B. Brandt, G. Endrődi, and B. Gläßle, Meson masses in electromagnetic fields with Wilson fermions, Phys. Rev. D 97, 034505 (2018).
  57. G. S. Bali, F. Bruckmann, G. Endrödi, S. D. Katz, and A. Schäfer, The QCD equation of state in background magnetic fields, J. High Energy Phys. 08 (2014) 177.
  58. F. Bruckmann, G. Endrodi, and T. G. Kovacs, Inverse magnetic catalysis and the Polyakov loop, J. High Energy Phys. 04 (2013) 112.
  59. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, and A. Schafer, QCD quark condensate in external magnetic fields, Phys. Rev. D 86, 071502 (2012).
  60. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, The QCD phase diagram for external magnetic fields, J. High Energy Phys. 02 (2012) 044.
  61. G. Endrodi, Critical point in the QCD phase diagram for extremely strong background magnetic fields, J. High Energy Phys. 07 (2015) 173.
  62. M. D’Elia, L. Maio, F. Sanfilippo, and A. Stanzione, Confining and chiral properties of QCD in extremely strong magnetic fields, Phys. Rev. D 104, 114512 (2021).
  63. C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro, A. Rucci, and F. Sanfilippo, Magnetic field effects on the static quark potential at zero and finite temperature, Phys. Rev. D 94, 094007 (2016).
  64. V. G. Bornyakov, P. V. Buividovich, N. Cundy, O. A. Kochetkov, and A. Schäfer, Deconfinement transition in two-flavor lattice QCD with dynamical overlap fermions in an external magnetic field, Phys. Rev. D 90, 034501 (2014).
  65. Y. Hidaka and A. Yamamoto, Charged vector mesons in a strong magnetic field, Phys. Rev. D 87, 094502 (2013).
  66. K. Hattori and A. Yamamoto, Meson deformation by magnetic fields in lattice QCD, Prog. Theor. Exp. Phys. 2019, 043B04 (2019).
  67. M. Coppola, D. Gomez Dumm, S. Noguera, and N. N. Scoccola, Masses of magnetized pseudoscalar and vector mesons in an extended NJL model: The role of axial vector mesons, Phys. Rev. D 109, 054014 (2024).
  68. D. Gomez Dumm, S. Noguera, and N. N. Scoccola, Charged meson masses under strong magnetic fields: Gauge invariance and Schwinger phases, Phys. Rev. D 108, 016012 (2023).
  69. J. P. Carlomagno, D. Gomez Dumm, M. F. I. Villafañe, S. Noguera, and N. N. Scoccola, Charged pseudoscalar and vector meson masses in strong magnetic fields in an extended NJL model, Phys. Rev. D 106, 094035 (2022).
  70. J. P. Carlomagno, D. Gomez Dumm, S. Noguera, and N. N. Scoccola, Neutral pseudoscalar and vector meson masses under strong magnetic fields in an extended NJL model: Mixing effects, Phys. Rev. D 106, 074002 (2022).
  71. M. Coppola, D. Gomez Dumm, S. Noguera, and N. N. Scoccola, Neutral and charged pion properties under strong magnetic fields in the NJL model, Phys. Rev. D 100, 054014 (2019).
  72. N. Isgur and G. Karl, Ground state baryons in a quark model with hyperfine interactions, Phys. Rev. D 20, 1191 (1979).
  73. A. De Rujula, H. Georgi, and S. L. Glashow, Hadron masses in a gauge theory, Phys. Rev. D 12, 147 (1975).
  74. A. D. Sakharov, Mass formula for mesons and baryons, Sov. Phys. JETP 51, 1059 (1980).
  75. Y. B. Zeldovich and A. D. Sakharov, Mass formulas for mesons and baryons in the quark model, Acta Phys. Hung. 22, 153 (1967).
  76. Y. A. Simonov and M. A. Trusov, Confinement and αs in a strong magnetic field, Phys. Lett. B 747, 48 (2015).
  77. S. Godfrey and N. Isgur, Mesons in a relativized quark model with chromodynamics, Phys. Rev. D 32, 189 (1985).
  78. S. Capstick and N. Isgur, Baryons in a relativized quark model with chromodynamics, Phys. Rev. D 34, 2809 (1986).
  79. G. Cao and S. Wu, Single-flavor heavy baryons in a strong magnetic field, Phys. Rev. D 112, 074034 (2025).
  80. A. Deur, S. J. Brodsky, and G. F. de Teramond, The QCD running coupling, Prog. Part. Nucl. Phys. 90, 1 (2016).

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Phys. Rev. D 114, 034043– Published 21 August, 2026
  • Received 8 May 2026
  • Accepted 7 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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