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Spin alignment of vector mesons in local equilibrium by Zubarev’s approach

Shi-Zheng Yang1, Xin-Qing Xie1, Shi Pu1,2,3, Jian-Hua Gao4, and Qun Wang1,5

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Phys. Rev. D 112, 094040 – Published 20 November, 2025

DOI: https://doi.org/10.1103/x9mp-qthg

Abstract

We compute the 00 element of the spin density matrix, denoted as ρ00 and called the spin alignment, up to the second order of the gradient expansion in local equilibrium by Zubarev’s approach. In the first order, we obtain ρ00=1/3, meaning that the contributions from thermal vorticity and shear stress tensor are vanishing. The nonvanishing contributions to ρ00−1/3 appear in the second order of gradients in the Belinfante and canonical cases. We also discuss the properties of the spin density matrix under the time reversal transformation. The effective transport coefficient for the spin alignment induced by the thermal shear stress tensor is T odd in the first order, implying that the first order effect is dissipative.

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References (99)

  1. Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, 102301 (2005); 96, 039901(E) (2006).
  2. S. A. Voloshin, arXiv:nucl-th/0410089.
  3. Z.-T. Liang and X.-N. Wang, Phys. Lett. B 629, 20 (2005).
  4. J.-H. Gao, S.-W. Chen, W.-T. Deng, Z.-T. Liang, Q. Wang, and X.-N. Wang, Phys. Rev. C 77, 044902 (2008).
  5. F. Becattini, F. Piccinini, and J. Rizzo, Phys. Rev. C 77, 024906 (2008).
  6. B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 77, 061902 (2008).
  7. L. Adamczyk et al. (STAR Collaboration), Nature (London) 548, 62 (2017).
  8. J. Adam et al. (STAR Collaboration), Phys. Rev. Lett. 123, 132301 (2019).
  9. J. Adam et al. (STAR Collaboration), Phys. Rev. Lett. 126, 162301 (2021).
  10. M. S. Abdallah et al. (STAR Collaboration), Nature (London) 614, 244 (2023).
  11. S. Acharya et al. (ALICE Collaboration), Phys. Rev. Lett. 131, 042303 (2023).
  12. S. Acharya et al. (ALICE Collaboration), Phys. Rev. Lett. 125, 012301 (2020).
  13. R. Abou Yassine et al. (HADES Collaboration), Phys. Lett. B 835, 137506 (2022).
  14. I. Karpenko and F. Becattini, Eur. Phys. J. C 77, 213 (2017).
  15. Y. Xie, D. Wang, and L. P. Csernai, Phys. Rev. C 95, 031901 (2017).
  16. H. Li, L.-G. Pang, Q. Wang, and X.-L. Xia, Phys. Rev. C 96, 054908 (2017).
  17. Y. Sun and C. M. Ko, Phys. Rev. C 96, 024906 (2017).
  18. S. Shi, K. Li, and J. Liao, Phys. Lett. B 788, 409 (2019).
  19. D.-X. Wei, W.-T. Deng, and X.-G. Huang, Phys. Rev. C 99, 014905 (2019).
  20. X.-L. Xia, H. Li, Z.-B. Tang, and Q. Wang, Phys. Rev. C 98, 024905 (2018).
  21. O. Vitiuk, L. V. Bravina, and E. E. Zabrodin, Phys. Lett. B 803, 135298 (2020).
  22. B. Fu, K. Xu, X.-G. Huang, and H. Song, Phys. Rev. C 103, 024903 (2021).
  23. S. Ryu, V. Jupic, and C. Shen, Phys. Rev. C 104, 054908 (2021).
  24. A. Lei, D. Wang, D.-M. Zhou, B.-H. Sa, and L. P. Csernai, Phys. Rev. C 104, 054903 (2021).
  25. X.-Y. Wu, C. Yi, G.-Y. Qin, and S. Pu, Phys. Rev. C 105, 064909 (2022).
  26. T. Niida (STAR Collaboration), Nucl. Phys. A982, 511 (2019).
  27. F. Becattini, V. Chandra, L. Del Zanna, and E. Grossi, Ann. Phys. (Amsterdam) 338, 32 (2013).
  28. R.-h. Fang, L.-g. Pang, Q. Wang, and X.-n. Wang, Phys. Rev. C 94, 024904 (2016).
  29. H.-Z. Wu, L.-G. Pang, X.-G. Huang, and Q. Wang, Phys. Rev. Res. 1, 033058 (2019).
  30. S. Y. F. Liu and Y. Yin, J. High Energy Phys. 07 (2021) 188.
  31. Y.-C. Liu and X.-G. Huang, Sci. China Phys. Mech. Astron. 65, 272011 (2022).
  32. B. Fu, S. Y. F. Liu, L. Pang, H. Song, and Y. Yin, Phys. Rev. Lett. 127, 142301 (2021).
  33. F. Becattini, M. Buzzegoli, and A. Palermo, Phys. Lett. B 820, 136519 (2021).
  34. F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, and A. Palermo, Phys. Rev. Lett. 127, 272302 (2021).
  35. S. Y. F. Liu and Y. Yin, Phys. Rev. D 104, 054043 (2021).
  36. B. Fu, L. Pang, H. Song, and Y. Yin, arXiv:2201.12970.
  37. J.-A. Sun and L. Yan, Chin. Phys. C 49, 071001 (2025).
  38. S. Fang, S. Pu, and D.-L. Yang, Phys. Rev. D 109, 034034 (2024).
  39. A. Palermo, E. Grossi, I. Karpenko, and F. Becattini, Eur. Phys. J. C 84, 920 (2024).
  40. C. Yi, S. Fang, D.-L. Wang, and S. Pu, Local spin polarization of Λ hyperons and its interaction corrections, in 31st International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (2025), arXiv:2509.00377.
  41. A. Arslan, W.-B. Dong, G.-L. Ma, S. Pu, and Q. Wang, Phys. Rev. C 111, 044911 (2025).
  42. C. Yi, X.-Y. Wu, J. Zhu, S. Pu, and G.-Y. Qin, Phys. Rev. C 111, 044901 (2025).
  43. C. Yi, X.-Y. Wu, J. Zhu, S. Pu, and G.-Y. Qin, Hydrodynamic effects on spin polarization along the beam direction in Au+Au and p+Pb collisions, in 31st International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (2025), arXiv:2509.00380.
  44. S. Acharya et al. (ALICE Collaboration), Phys. Rev. Lett. 128, 172005 (2022).
  45. M. Abdulhamid et al. (STAR Collaboration), Phys. Rev. Lett. 131, 202301 (2023).
  46. Q. Wang, Nucl. Phys. A967, 225 (2017).
  47. W. Florkowski, A. Kumar, and R. Ryblewski, Prog. Part. Nucl. Phys. 108, 103709 (2019).
  48. Z.-T. Liang, M. A. Lisa, and X.-N. Wang, Nucl. Phys. News 30, 10 (2020).
  49. F. Becattini and M. A. Lisa, Annu. Rev. Nucl. Part. Sci. 70, 395 (2020).
  50. X.-G. Huang, J. Liao, Q. Wang, and X.-L. Xia, Lect. Notes Phys. 987, 281 (2021).
  51. J.-H. Gao, Z.-T. Liang, Q. Wang, and X.-N. Wang, Lect. Notes Phys. 987, 195 (2021).
  52. Y. Hidaka, S. Pu, Q. Wang, and D.-L. Yang, Prog. Part. Nucl. Phys. 127, 103989 (2022).
  53. F. Becattini, M. Buzzegoli, T. Niida, S. Pu, A.-H. Tang, and Q. Wang, Int. J. Mod. Phys. E 33, 2430006 (2024).
  54. T. Niida and S. A. Voloshin, Int. J. Mod. Phys. E 33, 2430010 (2024).
  55. P. Faccioli, C. Lourenco, J. Seixas, and H. K. Wohri, Eur. Phys. J. C 69, 657 (2010).
  56. X.-L. Sheng, L. Oliva, and Q. Wang, Phys. Rev. D 101, 096005 (2020).
  57. X.-L. Sheng, Q. Wang, and X.-N. Wang, Phys. Rev. D 102, 056013 (2020).
  58. X.-L. Sheng, L. Oliva, Z.-T. Liang, Q. Wang, and X.-N. Wang, Phys. Rev. Lett. 131, 042304 (2023).
  59. X.-L. Sheng, S. Pu, and Q. Wang, Phys. Rev. C 108, 054902 (2023).
  60. A. Kumar, B. Müller, and D.-L. Yang, Phys. Rev. D 107, 076025 (2023).
  61. A. Kumar, B. Müller, and D.-L. Yang, Phys. Rev. D 108, 016020 (2023).
  62. A. Kumar, D.-L. Yang, and P. Gubler, Phys. Rev. D 109, 054038 (2024).
  63. B. Fu, F. Gao, Y. Liu, and H. Song, Phys. Lett. B 855, 138821 (2024).
  64. Y.-L. Yin, W.-B. Dong, J.-Y. Pang, S. Pu, and Q. Wang, Phys. Rev. C 110, 024905 (2024).
  65. Y.-Q. Zhao, X.-L. Sheng, S.-W. Li, and D. Hou, J. High Energy Phys. 08 (2024) 070.
  66. X.-L. Sheng, Y.-Q. Zhao, S.-W. Li, F. Becattini, and D. Hou, Phys. Rev. D 110, 056047 (2024).
  67. F. Li and S. Y. F. Liu, arXiv:2206.11890.
  68. D. Wagner, N. Weickgenannt, and E. Speranza, Phys. Rev. D 108, 116017 (2023).
  69. W.-B. Dong, Y.-L. Yin, X.-L. Sheng, S.-Z. Yang, and Q. Wang, Phys. Rev. D 109, 056025 (2024).
  70. D. N. Zubarev, A. V. Prozorkevich, and S. A. Smolyanskii, Theor. Math. Phys. 40, 821 (1979).
  71. D. Zubarev, J. Sov. Math. 16, 1509 (1981).
  72. F. Becattini, M. Buzzegoli, and E. Grossi, Particles 2, 197 (2019).
  73. F. Becattini, G. Inghirami, V. Rolando, A. Beraudo, L. Del Zanna, A. De Pace, M. Nardi, G. Pagliara, and V. Chandra, Eur. Phys. J. C 75, 406 (2015); 78, 354(E) (2018).
  74. F. Becattini, I. Karpenko, M. Lisa, I. Upsal, and S. Voloshin, Phys. Rev. C 95, 054902 (2017).
  75. A. Palermo, M. Buzzegoli, and F. Becattini, J. High Energy Phys. 10 (2021) 077.
  76. A. Palermo and F. Becattini, Eur. Phys. J. Plus 138, 547 (2023).
  77. M. Buzzegoli, Phys. Rev. C 105, 044907 (2022).
  78. X.-L. Sheng, L. Oliva, Z.-T. Liang, Q. Wang, and X.-N. Wang, Phys. Rev. D 109, 036004 (2024).
  79. S. R. De Groot, W. A. van Leeuwen, and Ch. G. van Weert, Relativistic Kinetic Theory: Principles and Applications (North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1980).
  80. F. Becattini, J. Liao, and M. Lisa, Lect. Notes Phys. 987, 1 (2021).
  81. F. Becattini, Lect. Notes Phys. 987, 15 (2021).
  82. X.-L. Sheng, N. Weickgenannt, E. Speranza, D. H. Rischke, and Q. Wang, Phys. Rev. D 104, 016029 (2021).
  83. F. Becattini, W. Florkowski, and E. Speranza, Phys. Lett. B 789, 419 (2019).
  84. C. G. Van Weert, Ann. Phys. (N.Y.) 140, 133 (1982).
  85. K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, and H. Taya, Phys. Lett. B 795, 100 (2019).
  86. K. Fukushima and S. Pu, Phys. Lett. B 817, 136346 (2021).
  87. D.-L. Wang, S. Fang, and S. Pu, Phys. Rev. D 104, 114043 (2021).
  88. D.-L. Wang, X.-Q. Xie, S. Fang, and S. Pu, Phys. Rev. D 105, 114050 (2022).
  89. X.-Q. Xie, D.-L. Wang, C. Yang, and S. Pu, Phys. Rev. D 108, 094031 (2023).
  90. F. Becattini, L. Csernai, and D. J. Wang, Phys. Rev. C 88, 034905 (2013); 93, 069901(E) (2016).
  91. M. Buzzegoli, E. Grossi, and F. Becattini, J. High Energy Phys. 10 (2017) 091; 07 (2018) 119(E).
  92. J.-H. Gao and S.-Z. Yang, Chin. Phys. C 48, 053114 (2024).
  93. C. Yi, S. Pu, and D.-L. Yang, Phys. Rev. C 104, 064901 (2021).
  94. D. E. Kharzeev and H.-U. Yee, Phys. Rev. D 84, 045025 (2011).
  95. J.-W. Chen, T. Ishii, S. Pu, and N. Yamamoto, Phys. Rev. D 93, 125023 (2016).
  96. D. T. Son and P. Surowka, Phys. Rev. Lett. 103, 191601 (2009).
  97. S. Pu, J.-h. Gao, and Q. Wang, Phys. Rev. D 83, 094017 (2011).
  98. J.-R. Wang, S. Fang, D.-L. Yang, and S. Pu, arXiv:2507.15238.
  99. Z.-H. Zhang, X.-G. Huang, F. Becattini, and X.-L. Sheng, J. High Energy Phys. 07 (2025) 224.

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Phys. Rev. D 112, 094040– Published 20 November, 2025
  • Received 26 March 2025
  • Accepted 18 October 2025
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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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