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Polarization-dependent mass modifications of ϕ meson with finite momentum in nuclear matter

Ahmad Jafar Arifi1,2,*, Philipp Gubler1,†, and Kazuo Tsushima3,‡

  • *Contact author: aj.arifi01@gmail.com
  • †Contact author: gubler.philipp@jaea.go.jp; philipp.gubler1@gmail.com
  • ‡Contact author: kazuo.tsushima@cruzeirodosul.edu.br; kazuo.tsushima@gmail.com
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Phys. Rev. D 114, 034041 – Published 21 August, 2026

DOI: https://doi.org/10.1103/wlvt-szn4

Abstract

We investigate the in-medium properties of the ϕ meson with finite momentum, going beyond the commonly studied case at rest. In a nuclear medium, Lorentz invariance is broken, leading to distinct longitudinal and transverse polarization modes that evolve differently with density and momentum. Within an effective Lagrangian approach, we calculate the polarization-dependent mass shifts and width modifications of the ϕ meson arising from KK¯ loops and mean-field interactions. The divergent loop integrals are regulated using two different schemes: a covariant form factor and dimensional regularization. Our results show that the mass shift of the transverse polarization is independent of the ϕ-meson momentum, whereas that of the longitudinal polarization decreases quadratically with momentum. This difference originates from the coupling of the longitudinal mode to the vector mean field and derivative-type interactions in the self-energy. These effects have direct implications for experimental observables, especially for upcoming measurements at J-PARC, and provide a new prediction for experiments studying hadron dynamics in dense matter.

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References (72)

  1. G. E. Brown and M. Rho, Chiral restoration in hot and/or dense matter, Phys. Rep. 269, 333 (1996).
  2. S. Leupold, V. Metag, and U. Mosel, Hadrons in strongly interacting matter, Int. J. Mod. Phys. E 19, 147 (2010).
  3. R. S. Hayano and T. Hatsuda, Hadron properties in the nuclear medium, Rev. Mod. Phys. 82, 2949 (2010).
  4. G. E. Brown and M. Rho, Scaling effective Lagrangians in a dense medium, Phys. Rev. Lett. 66, 2720 (1991).
  5. T. Hatsuda and S. H. Lee, QCD sum rules for vector mesons in the nuclear medium, Phys. Rev. C 46, R34 (1992).
  6. P. Gubler and K. Ohtani, Constraining the strangeness content of the nucleon by measuring the ϕ meson mass shift in nuclear matter, Phys. Rev. D 90, 094002 (2014).
  7. P. Gubler and W. Weise, Phi meson spectral moments and QCD condensates in nuclear matter, Nucl. Phys. A954, 125 (2016).
  8. J. Kim, P. Gubler, and S. H. Lee, ϕ meson properties in nuclear matter from QCD sum rules with chirally separated four-quark condensates, Phys. Rev. D 105, 114053 (2022).
  9. S. Durr et al., Lattice computation of the nucleon scalar quark contents at the physical point, Phys. Rev. Lett. 116, 172001 (2016).
  10. Y.-B. Yang, A. Alexandru, T. Draper, J. Liang, and K.-F. Liu (xQCD Collaboration), πN and strangeness sigma terms at the physical point with chiral fermions, Phys. Rev. D 94, 054503 (2016).
  11. A. Abdel-Rehim, C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco (ETM Collaboration), Direct evaluation of the quark content of nucleons from lattice QCD at the physical point, Phys. Rev. Lett. 116, 252001 (2016).
  12. S. Borsanyi, Z. Fodor, C. Hoelbling, L. Lellouch, K. K. Szabo, C. Torrero, and L. Varnhorst, Ab-initio calculation of the proton and the neutron’s scalar couplings for new physics searches, arXiv:2007.03319.
  13. G. S. Bali, S. Collins, P. Georg, D. Jenkins, P. Korcyl, A. Schäfer, E. E. Scholz, J. Simeth, W. Söldner, and S. Weishäupl (RQCD Collaboration), Scale setting and the light baryon spectrum in Nf=2+1 QCD with Wilson fermions, J. High Energy Phys. 05 (2022) 035.
  14. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG) Collaboration), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
  15. Y.-S. Oh and H. C. Bhang, Asymmetries in phi photoproduction and the OZI violation, Phys. Rev. C 64, 055207 (2001).
  16. S.-H. Kim, T. S. H. Lee, S.-i. Nam, and Y. Oh, Dynamical model of ϕ meson photoproduction on the nucleon and He4, Phys. Rev. C 104, 045202 (2021).
  17. Y. Lyu, T. Doi, T. Hatsuda, Y. Ikeda, J. Meng, K. Sasaki, and T. Sugiura, Attractive N−ϕ interaction and two-pion tail from lattice QCD near physical point, Phys. Rev. D 106, 074507 (2022).
  18. L. M. Abreu, P. Gubler, K. P. Khemchandani, A. Martinez Torres, and A. Hosaka, A study of the ϕN correlation function, Phys. Lett. B 860, 139175 (2025).
  19. V. Metag, M. Nanova, and E. Y. Paryev, Meson–nucleus potentials and the search for meson–nucleus bound states, Prog. Part. Nucl. Phys. 97, 199 (2017).
  20. L. Tolos and L. Fabbietti, Strangeness in nuclei and neutron stars, Prog. Part. Nucl. Phys. 112, 103770 (2020).
  21. W. Cassing and E. L. Bratkovskaya, Parton-hadron-string dynamics: An off-shell transport approach for relativistic energies, Nucl. Phys. A831, 215 (2009).
  22. P. Muhlich, T. Falter, C. Greiner, J. Lehr, M. Post, and U. Mosel, Photoproduction of phi mesons from nuclei, Phys. Rev. C 67, 024605 (2003).
  23. P. Gubler, M. Ichikawa, T. Song, and E. Bratkovskaya, Production and in-medium modification of φ mesons in proton-nucleus reactions from a transport approach, Phys. Rev. C 111, 034908 (2025).
  24. G. Balassa, K. Aoki, P. Gubler, S. H. Lee, H. Sako, and G. Wolf, Studying the in-medium φ meson spectrum through kaons in proton–nucleus reactions, Prog. Theor. Exp. Phys. 2025, 113C01 (2025).
  25. M. Ichikawa et al. (KEK-PS E325 Collaboration), Analysis of spectral modification of φ mesons at finite density using a transport approach in 12 GeV pA reactions, Prog. Theor. Exp. Phys. 2025, 093D01 (2025).
  26. C. M. Ko and B. H. Sa, Phi meson production in hadronic matter, Phys. Lett. B 258, 6 (1991).
  27. W. S. Chung, G.-Q. Li, and C. M. Ko, Phi meson production in heavy ion collisions at SIS energies, Nucl. Phys. A625, 347 (1997).
  28. W. S. Chung, C. M. Ko, and G.-Q. Li, Seeing phi meson through the dilepton spectra in heavy ion collisions, Nucl. Phys. A641, 357 (1998).
  29. S. Pal, C. M. Ko, and Z.-w. Lin, Phi meson production in relativistic heavy ion collisions, Nucl. Phys. A707, 525 (2002).
  30. B. B. Back et al. (E917 Collaboration), Production of phi mesons in Au+Au collisions at 11.7-A-GeV/c, Phys. Rev. C 69, 054901 (2004).
  31. B. I. Abelev et al. (STAR Collaboration), Measurements of phi meson production in relativistic heavy-ion collisions at RHIC, Phys. Rev. C 79, 064903 (2009).
  32. T. Song, J. Aichelin, and E. Bratkovskaya, In-medium effects in ϕ meson production in heavy-ion collisions from subthreshold to relativistic energies, Phys. Rev. C 106, 024903 (2022).
  33. J. Steinheimer, T. Reichert, and M. Bleicher, Determination of the ϕ-meson production process and its absorption cross section via directed flow, Phys. Lett. B 869, 139870 (2025).
  34. T. Ishikawa et al., ϕ photo-production from Li, C, Al, and Cu nuclei at E(γ)=1.5  GeV to 2.4 GeV, Phys. Lett. B 608, 215 (2005).
  35. R. Muto et al. (KEK-PS-E325 Collaboration), Evidence for in-medium modification of the phi meson at normal nuclear density, Phys. Rev. Lett. 98, 042501 (2007).
  36. F. Sakuma et al. (E325 Collaboration), Study of nuclear matter modification of decay widths in ϕ→e+e− and ϕ→K+K− channels, Phys. Rev. Lett. 98, 152302 (2007).
  37. M. H. Wood et al. (CLAS Collaboration), Absorption of the ω and ϕ mesons in nuclei, Phys. Rev. Lett. 105, 112301 (2010).
  38. A. Polyanskiy et al., Measurement of the in-medium phi-meson width in proton-nucleus collisions, Phys. Lett. B 695, 74 (2011).
  39. M. Hartmann et al., Momentum dependence of the phi-meson nuclear transparency, Phys. Rev. C 85, 035206 (2012).
  40. J. Adamczewski-Musch et al. (HADES Collaboration), Strong absorption of hadrons with hidden and open strangeness in nuclear matter, Phys. Rev. Lett. 123, 022002 (2019).
  41. S. Yokkaichi et al., Electron pair spectrometer at the J-PARC 50-GeV PS to explore the chiral symmetry in QCD, https://ribf.riken.jp/~yokkaich/E16/pub/proposal.pdf, (2007).
  42. M. Naruki, Hadron physics at J-PARC, Prog. Theor. Exp. Phys. 2012, 02B013 (2012).
  43. K. Aoki et al., Experimental study of In-medium spectral change of vector mesons at J-PARC, Few Body Syst. 64, 63 (2023).
  44. K. Aoki et al., Experimental investigation of vector mesons in medium through dielectron decay at J-PARC, J. Subatomic Part. Cosmol. 3, 100019 (2025).
  45. H. Sako et al., Experimental studies of in-medium modification of ϕ meson mass through ϕ→K+K− decays, J. Subatomic Part. Cosmol. 1–2, 100012 (2024).
  46. J. G. Messchendorp et al., Hadron physics opportunities at FAIR, arXiv:2512.15986.
  47. C. M. Ko, P. Levai, X. J. Qiu, and C. T. Li, Phi meson in dense matter, Phys. Rev. C 45, 1400 (1992).
  48. M. Asakawa and C. M. Ko, Phi meson mass in hot and dense matter, Nucl. Phys. A572, 732 (1994).
  49. F. Klingl, T. Waas, and W. Weise, Modification of the phi meson spectrum in nuclear matter, Phys. Lett. B 431, 254 (1998).
  50. S. Zschocke, O. P. Pavlenko, and B. Kampfer, Evaluation of QCD sum rules for light vector mesons at finite density and temperature, Eur. Phys. J. A 15, 529 (2002).
  51. D. Cabrera and M. J. Vicente Vacas, Phi meson mass and decay width in nuclear matter, Phys. Rev. C 67, 045203 (2003).
  52. D. Cabrera, A. N. Hiller Blin, and M. J. Vicente Vacas, ϕ meson self-energy in nuclear matter from ϕN resonant interactions, Phys. Rev. C 95, 015201 (2017).
  53. J. J. Cobos-Martínez, K. Tsushima, G. Krein, and A. W. Thomas, ϕ meson mass and decay width in nuclear matter and nuclei, Phys. Lett. B 771, 113 (2017).
  54. Z. Ahmad, N. Chahal, A. Kumar, and S. Dutt, Impact of finite volume on kaon, antikaon, and ϕ meson masses and decay widths in asymmetric strange hadronic matter, Prog. Theor. Exp. Phys. 2025, 013B03 (2025).
  55. A. Mondal and A. Mishra, ϕ meson in nuclear matter and atomic nuclei, Phys. Rev. D 111, 094037 (2025).
  56. M. Kaur and A. Kumar, ϕ meson properties in dense resonance matter at finite temperature, Phys. Rev. D 112, 014030 (2025).
  57. S. H. Lee, Vector mesons in-medium with finite three momentum, Phys. Rev. C 57, 927 (1998); 58, 3771(E) (1998).
  58. S. Leupold and U. Mosel, On QCD sum rules for vector mesons in nuclear medium, Phys. Rev. C 58, 2939 (1998).
  59. H. Kim and P. Gubler, The ϕ meson with finite momentum in a dense medium, Phys. Lett. B 805, 135412 (2020).
  60. K. Tsushima, K. Saito, A. W. Thomas, and S. V. Wright, In-medium kaon and antikaon properties in the quark meson coupling model, Phys. Lett. B 429, 239 (1998); 436, 453(E) (1998).
  61. I. W. Park, H. Sako, K. Aoki, P. Gubler, and S. H. Lee, Disentangling longitudinal and transverse modes of the ϕ meson through dilepton and kaon decays, Phys. Rev. D 107, 074033 (2023).
  62. C. Gale and J. I. Kapusta, Vector dominance model at finite temperature, Nucl. Phys. B357, 65 (1991).
  63. P. A. M. Guichon, A possible quark mechanism for the saturation of nuclear matter, Phys. Lett. B 200, 235 (1988).
  64. K. Saito, K. Tsushima, and A. W. Thomas, Nucleon and hadron structure changes in the nuclear medium and impact on observables, Prog. Part. Nucl. Phys. 58, 1 (2007).
  65. P. A. M. Guichon, J. R. Stone, and A. W. Thomas, Quark–Meson-Coupling (QMC) model for finite nuclei, nuclear matter and beyond, Prog. Part. Nucl. Phys. 100, 262 (2018).
  66. G. E. Brown, C. M. Ko, and K. Kubodera, Strangeness production in relativistic heavy ion collisions, Z. Phys. A 341, 301 (1992).
  67. S. Navas et al. (Particle Data Group Collaboration), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  68. C. Fuchs, Kaon production in heavy ion reactions at intermediate energies, Prog. Part. Nucl. Phys. 56, 1 (2006).
  69. A. J. Arifi, P. T. P. Hutauruk, and K. Tsushima, In-medium properties of the light and heavy-light mesons in a light-front quark model, Phys. Rev. D 107, 114010 (2023).
  70. W. Peters, M. Post, H. Lenske, S. Leupold, and U. Mosel, The spectral function of the rho meson in nuclear matter, Nucl. Phys. A632, 109 (1998).
  71. I. W. Park, H. Sako, K. Aoki, P. Gubler, and S. H. Lee, Identifying the transverse and longitudinal modes of the K* and K1 mesons through their angular-dependent decay modes, Phys. Rev. D 109, 114042 (2024).
  72. S. Yeo, H. Kim, and S. H. Lee, K1± mesons moving in nuclear matter, Phys. Rev. D 110, 014013 (2024).

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Phys. Rev. D 114, 034041– Published 21 August, 2026
  • Received 25 March 2026
  • Accepted 23 July 2026
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