- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Spectral function for pions in a magnetic field

Jie Mei1,*, Rui Wen1,†, Min Zhou2,‡, Shijun Mao2,§, and Mei Huang1,∥

  • *Contact author: meijie22@mails.ucas.ac.cn
  • †Contact author: rwen@ucas.ac.cn
  • ‡Contact author: minzhou@stu.xjtu.edu.cn
  • §Contact author: maoshijun@mail.xjtu.edu.cn
  • ∥Contact author: huangmei@ucas.ac.cn
PDF

Phys. Rev. D 113, 074031 – Published 28 April, 2026

DOI: https://doi.org/10.1103/skmg-ql8c

Abstract

This study examines the spectral functions of neutral (π0) and charged (π±) pions under a uniform magnetic field using the SU(2) Nambu–Jona-Lasinio model with the Ritus method. The analysis highlights the complex interplay of magnetic field effects, thermal influences, and chiral symmetry on meson properties in extreme QCD environments. For π0, whose properties are governed by the behavior of its constituent quarks, magnetic field-induced Landau levels lead to a multipeak structure in its spectral function, reflecting stable and resonance solutions that evolve with temperature, showing shifts and critical enhancements near chiral restoration. For π±, cross terms that come from the asymmetry between the constituent quarks introduce Landau cuts alongside unitary cuts, indicating damping effects, with decay widths narrowing at higher temperatures, suggesting increased stability.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (83)

  1. F. Preis, A. Rebhan, and A. Schmitt, Inverse magnetic catalysis in field theory and gauge-gravity duality, Lect. Notes Phys. 871, 51 (2013).
  2. R. Gatto and M. Ruggieri, Quark matter in a strong magnetic background, Lect. Notes Phys. 871, 87 (2013).
  3. M. D’Elia, Lattice QCD simulations in external background fields, Lect. Notes Phys. 871, 181 (2013).
  4. 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).
  5. 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).
  6. 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).
  7. U. Gursoy, D. Kharzeev, and K. Rajagopal, Magnetohydrodynamics, charged currents and directed flow in heavy ion collisions, Phys. Rev. C 89, 054905 (2014).
  8. V. de la Incera, Nonperturbative physics in a magnetic field, AIP Conf. Proc. 1361, 74 (2011).
  9. R. C. Duncan and C. Thompson, Formation of very strongly magnetized neutron stars—Implications for gamma-ray bursts, Astrophys. J. Lett. 392, L9 (1992).
  10. H. Li, X.-L. Xia, X.-G. Huang, and H. Z. Huang, Dynamic calculations of magnetic field and implications on spin polarization and spin alignment in heavy ion collisions, Phys. Rev. C 108, 044902 (2023).
  11. K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Chiral magnetic effect, Phys. Rev. D 78, 074033 (2008).
  12. D. E. Kharzeev and H.-U. Yee, Chiral magnetic wave, Phys. Rev. D 83, 085007 (2011).
  13. S. F. Taghavi and U. A. Wiedemann, Chiral magnetic wave in an expanding QCD fluid, Phys. Rev. C 91, 024902 (2015).
  14. G. S. Bali et al., Thermodynamic properties of QCD in external magnetic fields, Proc. Sci. ConfinementX (2012) 197.
  15. 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).
  16. M. D’Elia, F. Manigrasso, F. Negro, and F. Sanfilippo, QCD phase diagram in a magnetic background for different values of the pion mass, Phys. Rev. D 98, 054509 (2018).
  17. G. Endrodi, Critical point in the QCD phase diagram for extremely strong background magnetic fields, J. High Energy Phys. 07 (2015) 173.
  18. 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.
  19. G. S. Bali, G. Endrődi, and S. Piemonte, Magnetic susceptibility of QCD matter and its decomposition from the lattice, J. High Energy Phys. 07 (2020) 183.
  20. J. Chao, P. Chu, and M. Huang, Inverse magnetic catalysis induced by sphalerons, Phys. Rev. D 88, 054009 (2013).
  21. L. Yu, H. Liu, and M. Huang, Spontaneous generation of local CP violation and inverse magnetic catalysis, Phys. Rev. D 90, 074009 (2014).
  22. S. Fayazbakhsh and N. Sadooghi, Anomalous magnetic moment of hot quarks, inverse magnetic catalysis, and reentrance of the chiral symmetry broken phase, Phys. Rev. D 90, 105030 (2014).
  23. J. Mei and S. Mao, Inverse catalysis effect of the quark anomalous magnetic moment to chiral restoration and deconfinement phase transitions, Phys. Rev. D 102, 114035 (2020).
  24. K. Xu, J. Chao, and M. Huang, Effect of the anomalous magnetic moment of quarks on magnetized QCD matter and meson spectra, Phys. Rev. D 103, 076015 (2021).
  25. F. Lin, K. Xu, and M. Huang, Magnetism of QCD matter and the pion mass from tensor-type spin polarization and the anomalous magnetic moment of quarks, Phys. Rev. D 106, 016005 (2022).
  26. M. Kawaguchi and M. Huang, Restriction on the form of the quark anomalous magnetic moment from lattice QCD results, Chin. Phys. C 47, 064103 (2023).
  27. W. R. Tavares, S. S. Avancini, R. L. S. Farias, and R. P. Cardoso, Artificial first-order phase transition in a magnetized Nambu–Jona-Lasinio model with a quark anomalous magnetic moment, Phys. Rev. D 109, 016011 (2024).
  28. N. Chaudhuri, S. Ghosh, P. Roy, and S. Sarkar, Collective modes of a massive fermion in a magnetized medium with finite anomalous magnetic moment, Phys. Rev. D 108, 116006 (2023).
  29. C. A. Islam, M. S. Ali, and M. Huang, Deciding on the anomalous magnetic moment of quarks in a framework of nonlocal NJL model, arXiv:2302.00696.
  30. M. Ferreira, P. Costa, O. Lourenço, T. Frederico, and C. Providência, Inverse magnetic catalysis in the (2+1)-flavor Nambu-Jona-Lasinio and Polyakov-Nambu-Jona-Lasinio models, Phys. Rev. D 89, 116011 (2014).
  31. H. Liu, L. Yu, M. Chernodub, and M. Huang, Possible formation of high temperature superconductor at an early stage of heavy-ion collisions, Phys. Rev. D 94, 113006 (2016).
  32. 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).
  33. A. Ayala, C. A. Dominguez, L. A. Hernandez, M. Loewe, A. Raya, J. C. Rojas, and C. Villavicencio, Thermomagnetic properties of the strong coupling in the local Nambu–Jona-Lasinio model, Phys. Rev. D 94, 054019 (2016).
  34. R. L. S. Farias, V. S. Timoteo, S. S. Avancini, M. B. Pinto, and G. Krein, Thermo-magnetic effects in quark matter: Nambu–Jona-Lasinio model constrained by lattice QCD, Eur. Phys. J. A 53, 101 (2017).
  35. X. Chen, L. Zhang, and D. Hou, Running coupling constant at finite chemical potential and magnetic field from holography, Chin. Phys. C 46, 073101 (2022).
  36. B.-k. Sheng, X. Wang, and L. Yu, Impacts of inverse magnetic catalysis on screening masses of neutral pions and sigma mesons in hot and magnetized quark matter, Phys. Rev. D 105, 034003 (2022).
  37. S. Mao and Y. Tian, Pion superfluid phase transition under an external magnetic field including the inverse magnetic catalysis effect, Phys. Rev. D 106, 094017 (2022).
  38. R. Wen, S. Yin, W.-j. Fu, and M. Huang, Functional renormalization group study of neutral and charged pions in magnetic fields in the quark-meson model, Phys. Rev. D 108, 076020 (2023).
  39. X. Li, W.-j. Fu, and Y.-x. Liu, Thermodynamics of 2+1 flavor Polyakov-loop quark-meson model under external magnetic field, Phys. Rev. D 99, 074029 (2019).
  40. K. Kamikado and T. Kanazawa, Chiral dynamics in a magnetic field from the functional renormalization group, J. High Energy Phys. 03 (2014) 009.
  41. K. Kamikado and T. Kanazawa, Magnetic susceptibility of a strongly interacting thermal medium with 2+1 quark flavors, J. High Energy Phys. 01 (2015) 129.
  42. K. Fukushima and J. M. Pawlowski, Magnetic catalysis in hot and dense quark matter and quantum fluctuations, Phys. Rev. D 86, 076013 (2012).
  43. W.-j. Fu and Y.-x. Liu, Four-fermion interactions and the chiral symmetry breaking in an external magnetic field, Phys. Rev. D 96, 074019 (2017).
  44. H. Bohra, D. Dudal, A. Hajilou, and S. Mahapatra, Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model, Phys. Lett. B 801, 135184 (2020).
  45. H. Bohra, D. Dudal, A. Hajilou, and S. Mahapatra, Chiral transition in the probe approximation from an Einstein-Maxwell-dilaton gravity model, Phys. Rev. D 103, 086021 (2021).
  46. N. Wen, X. Cao, J. Chao, and H. Liu, Neutral pion masses within a hot and magnetized medium in a lattice-improved soft-wall AdS/QCD model, Phys. Rev. D 109, 086021 (2024).
  47. R. Wen, C. Huang, and M. Huang, Functional renormalization group study of anomalous magnetic moment in a low energy effective theory, Phys. Rev. D 113, 054036 (2026).
  48. T. Hatsuda and T. Kunihiro, QCD phenomenology based on a chiral effective Lagrangian, Phys. Rep. 247, 221 (1994).
  49. M. A. Stephanov, QCD phase diagram and the critical point, Prog. Theor. Phys. Suppl. 153, 139 (2004).
  50. R. Gatto and M. Ruggieri, Deconfinement and chiral symmetry restoration in a strong magnetic background, Phys. Rev. D 83, 034016 (2011).
  51. B. Sheng, Y. Wang, X. Wang, and L. Yu, Pole and screening masses of neutral pions in a hot and magnetized medium: A comprehensive study in the Nambu–Jona-Lasinio model, Phys. Rev. D 103, 094001 (2021).
  52. J. Mei, R. Wen, S. Mao, M. Huang, and K. Xu, Magnetic catalysis and diamagnetism from pion fluctuations, Phys. Rev. D 110, 034024 (2024).
  53. A. Bazavov et al., Meson screening masses in (2+1)-flavor QCD, Phys. Rev. D 100, 094510 (2019).
  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. D. Laudicina, M. Dalla Brida, L. Giusti, T. Harris, and M. Pepe, Computation of QCD meson screening masses at high temperature, Proc. Sci. LATTICE2021 (2022) 190 [arXiv:2112.06662].
  56. M. Ishii, T. Sasaki, K. Kashiwa, H. Kouno, and M. Yahiro, Effective model approach to meson screening masses at finite temperature, Phys. Rev. D 89, 071901(R) (2014).
  57. P. Czerski and W. M. Alberico, Screening masses of scalar and pseudo-scalar excitations in quark-gluon plasma, Nucl. Phys. A918, 170 (2013).
  58. S. Mao, Inverse magnetic catalysis in Nambu–Jona-Lasinio model beyond mean field, Phys. Lett. B 758, 195 (2016).
  59. S. Mao, Chiral symmetry restoration and quark deconfinement beyond mean field in a magnetized PNJL model, Phys. Rev. D 97, 011501(R) (2018).
  60. 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).
  61. S. Mao, Chiral crossover characterized by Mott transition at finite temperature, Chin. Phys. C 45, 021004 (2021).
  62. 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).
  63. R.-A. Tripolt, N. Strodthoff, L. von Smekal, and J. Wambach, Finite-temperature spectral functions from the functional renormalization group, Proc. Sci. LATTICE2013 (2014) 457 [arXiv:1311.4304].
  64. R.-A. Tripolt, L. von Smekal, and J. Wambach, Flow equations for spectral functions at finite external momenta, Phys. Rev. D 90, 074031 (2014).
  65. J. Wambach, R.-A. Tripolt, N. Strodthoff, and L. von Smekal, Spectral functions from the functional renormalization group, Nucl. Phys. A928, 156 (2014).
  66. R.-A. Tripolt, L. von Smekal, and J. Wambach, Spectral functions and in-medium properties of hadrons, Int. J. Mod. Phys. E 26, 1740028 (2017).
  67. C. A. Islam, A. Bandyopadhyay, P. K. Roy, and S. Sarkar, Spectral function and dilepton rate from a strongly magnetized hot and dense medium in light of mean field models, Phys. Rev. D 99, 094028 (2019).
  68. M. Wei, C. A. Islam, and M. Huang, Production rate and ellipticity of lepton pairs from a rotating hot and dense QCD medium, Phys. Rev. D 105, 054014 (2022).
  69. M. Wei and M. Huang, Spin alignment of vector mesons from quark dynamics in a rotating medium, Chin. Phys. C 47, 104105 (2023).
  70. S. Ghosh, N. Chaudhuri, S. Sarkar, and P. Roy, Mass and spectral function of scalar and pseudoscalar mesons in a hot and chirally imbalanced medium using the two-flavor NJL model, Phys. Rev. D 109, 016021 (2024).
  71. M. L. Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2011).
  72. A. Patkos, Z. Szep, and P. Szepfalusy, Second sheet sigma pole and the threshold enhancement of the spectral function in the scalar isoscalar meson sector, Phys. Rev. D 66, 116004 (2002).
  73. A. Patkos, Z. Szep, and P. Szepfalusy, Universal threshold enhancement, Phys. Rev. D 68, 047701 (2003).
  74. S. Chiku and T. Hatsuda, Soft modes associated with chiral transition at finite temperature, Phys. Rev. D 57, 6 (1998).
  75. P. Chakraborty, Meson spectral function and screening masses in magnetized quark gluon plasma, Eur. Phys. J. Plus 134, 478 (2019).
  76. X.-L. Sheng, S.-Y. Yang, Y.-L. Zou, and D. Hou, Mass splitting and spin alignment for ϕ mesons in a magnetic field in NJL model, Eur. Phys. J. C 84, 299 (2024).
  77. X.-L. Sheng, Y.-Q. Zhao, S.-W. Li, F. Becattini, and D. Hou, Holographic spin alignment for vector mesons, Phys. Rev. D 110, 056047 (2024).
  78. 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).
  79. S. Mao and Y. Wang, Effect of discrete quark momenta on the Goldstone mode in a magnetic field, Phys. Rev. D 96, 034004 (2017).
  80. S. Mao, Pions in magnetic field at finite temperature, Phys. Rev. D 99, 056005 (2019).
  81. W.-j. Fu, J. M. Pawlowski, R. D. Pisarski, F. Rennecke, R. Wen, and S. Yin, QCD moat regime and its real-time properties, Phys. Rev. D 111, 094026 (2025).
  82. S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992).
  83. J. Mei, R. Wen, M. Zhou, S. Mao, and M. Huang, Spectral function for pions in magnetic field—Research data, https://github.com/ChiehMei/Spectral_Function_in_MFURL (2026).

Outline

Information

Phys. Rev. D 113, 074031– Published 28 April, 2026
  • Received 3 February 2026
  • Accepted 27 March 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation