- 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

Late-time attractors in relativistic spin hydrodynamics in Gubser flow

Gen-Hui Li1,*, Xiang Ren1,†, Dong-Lin Wang1,2,‡, and Shi Pu1,3,§

  • *Contact author: genhuili@mail.ustc.edu.cn
  • †Contact author: renxiang0510@mail.ustc.edu.cn
  • ‡Contact author: donglinwang@mail.ustc.edu.cn
  • §Contact author: shipu@ustc.edu.cn
PDF

Phys. Rev. D 114, 074029 – Published 13 October, 2026

DOI: https://doi.org/10.1103/pr6f-t285

Abstract

We investigate the late-time asymptotic solutions and attractor structure of the spin density in minimal causal spin hydrodynamics in Gubser flow. After deriving the differential equation governing the spin density, we obtain its late-time asymptotic solutions and identify both attractors and repellers in the corresponding numerical solutions. We then map these solutions back to flat Minkowski space and find parameter regions where the spin density exhibits a power-law decay. We further show that, when the characteristic length scale of the system is much larger than the proper time, several components of the spin density can decay as slowly as conventional thermodynamic variables in relativistic hydrodynamics. In this regime, the spin density behaves as a hydrodynamic mode governed by the late-time scaling laws of the flow.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (149)

  1. Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, 102301 (2005); 96, 039901(E) (2006).
  2. Z.-T. Liang and X.-N. Wang, Phys. Lett. B 629, 20 (2005).
  3. STAR Collaboration, Nature (London) 548, 62 (2017).
  4. J. Adam et al. (STAR Collaboration), Phys. Rev. C 98, 014910 (2018).
  5. J. Adam et al. (STAR Collaboration), Phys. Rev. Lett. 126, 162301 (2021).
  6. M. S. Abdallah et al. (STAR Collaboration), Phys. Rev. C 104, L061901 (2021).
  7. M. I. Abdulhamid et al. (STAR Collaboration), Phys. Rev. C 108, 014910 (2023).
  8. I. Karpenko and F. Becattini, Eur. Phys. J. C 77, 213 (2017).
  9. H. Li, L.-G. Pang, Q. Wang, and X.-L. Xia, Phys. Rev. C 96, 054908 (2017).
  10. Y. Sun and C. M. Ko, Phys. Rev. C 96, 024906 (2017).
  11. D.-X. Wei, W.-T. Deng, and X.-G. Huang, Phys. Rev. C 99, 014905 (2019).
  12. O. Vitiuk, L. V. Bravina, and E. E. Zabrodin, Phys. Lett. B 803, 135298 (2020).
  13. B. Fu, K. Xu, X.-G. Huang, and H. Song, Phys. Rev. C 103, 024903 (2021).
  14. A. Lei, D. Wang, D.-M. Zhou, B.-H. Sa, and L. P. Csernai, Phys. Rev. C 104, 054903 (2021).
  15. S. Ryu, V. Jupic, and C. Shen, Phys. Rev. C 104, 054908 (2021).
  16. R. Abou Yassine et al. (HADES Collaboration), Phys. Lett. B 835, 137506 (2022).
  17. Y. Guo, J. Liao, E. Wang, H. Xing, and H. Zhang, Phys. Rev. C 104, L041902 (2021).
  18. X.-G. Deng, X.-G. Huang, Y.-G. Ma, and S. Zhang, Phys. Rev. C 101, 064908 (2020).
  19. X.-G. Deng, X.-G. Huang, and Y.-G. Ma, arXiv:2109.09956.
  20. K.-J. Sun, D.-N. Liu, Y.-P. Zheng, J.-H. Chen, C. Ming Ko, and Y.-G. Ma, Phys. Rev. Lett. 134, 022301 (2025).
  21. D.-N. Liu et al., arXiv:2508.12193.
  22. Y.-P. Zheng, D.-N. Liu, L.-W. Chen, J.-H. Chen, C. Ming Ko, Y.-G. Ma, K.-J. Sun, J. Xu, and B. Zhou, Phys. Rev. C 113, 054906 (2026).
  23. J. Xu, arXiv:2602.23793.
  24. S. Y. F. Liu and Y. Yin, J. High Energy Phys. 07 (2021) 188.
  25. B. Fu, S. Y. F. Liu, L. Pang, H. Song, and Y. Yin, Phys. Rev. Lett. 127, 142301 (2021).
  26. F. Becattini, M. Buzzegoli, and A. Palermo, Phys. Lett. B 820, 136519 (2021).
  27. F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, and A. Palermo, Phys. Rev. Lett. 127, 272302 (2021).
  28. Y. Hidaka, S. Pu, and D.-L. Yang, Phys. Rev. D 97, 016004 (2018).
  29. C. Yi, S. Pu, and D.-L. Yang, Phys. Rev. C 104, 064901 (2021).
  30. C. Yi, S. Pu, J.-H. Gao, and D.-L. Yang, Phys. Rev. C 105, 044911 (2022).
  31. W. Florkowski, A. Kumar, A. Mazeliauskas, and R. Ryblewski, Phys. Rev. C 105, 064901 (2022).
  32. S. Alzhrani, S. Ryu, and C. Shen, Phys. Rev. C 106, 014905 (2022).
  33. A. Palermo, F. Becattini, M. Buzzegoli, G. Inghirami, and I. Karpenko, EPJ Web Conf. 276, 01026 (2023).
  34. M. Buzzegoli, F. Becattini, G. Inghirami, I. Karpenko, and A. Palermo, Acta Phys. Pol. B Proc. Suppl. 16, 39 (2023).
  35. X.-Y. Wu, C. Yi, G.-Y. Qin, and S. Pu, Phys. Rev. C 105, 064909 (2022).
  36. C. Yi, X.-Y. Wu, D.-L. Yang, J.-H. Gao, S. Pu, and G.-Y. Qin, Phys. Rev. C 109, L011901 (2024).
  37. X.-Y. Wu, C. Yi, G.-Y. Qin, and S. Pu, Global and local polarization of Λ hyperons across RHIC-BES energies, in 30th International Conference on Ultrarelativstic Nucleus-Nucleus Collisions (2023), arXiv:2312.09068.
  38. A. Palermo, E. Grossi, I. Karpenko, and F. Becattini, Eur. Phys. J. C 84, 920 (2024).
  39. B. Fu, L. Pang, H. Song, and Y. Yin, arXiv:2201.12970.
  40. Y. B. Ivanov and A. A. Soldatov, JETP Lett. 116, 133 (2022).
  41. S. Fang and S. Pu, Phys. Rev. D 111, 034015 (2025).
  42. J.-R. Wang, S. Fang, D.-L. Yang, and S. Pu, Phys. Rev. D 113, 114026 (2026).
  43. S. Fang, S. Pu, and D.-L. Yang, Phys. Rev. D 106, 016002 (2022).
  44. S. Fang, S. Pu, and D.-L. Yang, Phys. Rev. D 112, 014038 (2025).
  45. C. Yi, X.-Y. Wu, J. Zhu, S. Pu, and G.-Y. Qin, Phys. Rev. C 111, 044901 (2025).
  46. A. Hayrapetyan et al. (CMS Collaboration), Phys. Rev. Lett. 135, 132301 (2025).
  47. 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.
  48. 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.
  49. J.-H. Gao, G.-L. Ma, S. Pu, and Q. Wang, Nucl. Sci. Tech. 31, 90 (2020).
  50. F. Becattini, Rep. Prog. Phys. 85, 122301 (2022).
  51. F. Becattini et al., arXiv:2402.04540.
  52. T. Niida and S. A. Voloshin, Int. J. Mod. Phys. E 33, 2430010 (2024).
  53. J.-H. Gao, Z.-T. Liang, S. Pu, Q. Wang, and X.-N. Wang, Phys. Rev. Lett. 109, 232301 (2012).
  54. J.-W. Chen, S. Pu, Q. Wang, and X.-N. Wang, Phys. Rev. Lett. 110, 262301 (2013).
  55. Y. Hidaka, S. Pu, and D.-L. Yang, Phys. Rev. D 95, 091901 (2017).
  56. J.-H. Gao and Z.-T. Liang, Phys. Rev. D 100, 056021 (2019).
  57. N. Weickgenannt, X.-L. Sheng, E. Speranza, Q. Wang, and D. H. Rischke, Phys. Rev. D 100, 056018 (2019).
  58. Y.-C. Liu, K. Mameda, and X.-G. Huang, Chin. Phys. C 44, 094101 (2020); 45, 089001(E) (2021).
  59. N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Phys. Rev. Lett. 127, 052301 (2021).
  60. N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Phys. Rev. D 104, 016022 (2021).
  61. X.-L. Sheng, N. Weickgenannt, E. Speranza, D. H. Rischke, and Q. Wang, Phys. Rev. D 104, 016029 (2021).
  62. Y. Hidaka, S. Pu, Q. Wang, and D.-L. Yang, Prog. Part. Nucl. Phys. 127, 103989 (2022).
  63. W.-B. Dong, Y.-L. Yin, and Q. Wang, Phys. Rev. C 106, 054909 (2022).
  64. W.-B. Dong, Y.-L. Yin, X.-L. Sheng, S.-Z. Yang, and Q. Wang, Phys. Rev. D 109, 056025 (2024).
  65. S. Fang, S. Pu, and D.-L. Yang, Phys. Rev. D 109, 034034 (2024).
  66. Y.-L. Yin, W.-B. Dong, J.-Y. Pang, S. Pu, and Q. Wang, Phys. Rev. C 110, 024905 (2024).
  67. S. Bhadury, Z. Drogosz, W. Florkowski, S. K. Kar, and V. Mykhaylova, arXiv:2505.02657.
  68. D. Montenegro, L. Tinti, and G. Torrieri, Phys. Rev. D 96, 056012 (2017); 96, 079901(A) (2017).
  69. W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, Phys. Rev. C 97, 041901 (2018).
  70. W. Florkowski, A. Kumar, and R. Ryblewski, Prog. Part. Nucl. Phys. 108, 103709 (2019).
  71. K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, and H. Taya, Phys. Lett. B 795, 100 (2019).
  72. K. Fukushima and S. Pu, Phys. Lett. B 817, 136346 (2021).
  73. F. Becattini, W. Florkowski, and E. Speranza, Phys. Lett. B 789, 419 (2019).
  74. J. Hu, Phys. Rev. C 107, 024915 (2023).
  75. S. Dey and A. Das, Phys. Rev. D 111, 074037 (2025).
  76. A. Tiwari and B. K. Patra, Phys. Rev. D 112, 036014 (2025).
  77. D. She, Y.-W. Qiu, and D. Hou, Phys. Rev. D 111, 036027 (2025).
  78. W. Florkowski, E. Speranza, and F. Becattini, Acta Phys. Pol. B 49, 1409 (2018).
  79. S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Lett. B 814, 136096 (2021).
  80. S. Shi, C. Gale, and S. Jeon, Phys. Rev. C 103, 044906 (2021).
  81. E. Speranza and N. Weickgenannt, Eur. Phys. J. A 57, 155 (2021).
  82. S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Rev. D 103, 014030 (2021).
  83. R. Singh, G. Sophys, and R. Ryblewski, Phys. Rev. D 103, 074024 (2021).
  84. H.-H. Peng, J.-J. Zhang, X.-L. Sheng, and Q. Wang, Chin. Phys. Lett. 38, 116701 (2021).
  85. N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Rischke, Phys. Rev. D 106, 096014 (2022).
  86. N. Weickgenannt, D. Wagner, and E. Speranza, Phys. Rev. D 105, 116026 (2022).
  87. N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Rischke, Phys. Rev. D 106, L091901 (2022).
  88. S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Rev. Lett. 129, 192301 (2022).
  89. D. Wagner, N. Weickgenannt, and E. Speranza, Phys. Rev. D 108, 116017 (2023).
  90. W. Florkowski and M. Hontarenko, Phys. Rev. Lett. 134, 082302 (2025).
  91. D. Wagner, M. Shokri, and D. H. Rischke, Phys. Rev. Res. 6, 043103 (2024).
  92. D. Wagner, Phys. Rev. D 111, 016008 (2025).
  93. A. Daher, X.-L. Sheng, D. Wagner, and F. Becattini, Phys. Rev. D 112, 094020 (2025).
  94. A. D. Gallegos and U. Gürsoy, J. High Energy Phys. 11 (2020) 151.
  95. S. Li, M. A. Stephanov, and H.-U. Yee, Phys. Rev. Lett. 127, 082302 (2021).
  96. A. D. Gallegos, U. Gürsoy, and A. Yarom, SciPost Phys. 11, 041 (2021).
  97. D. She, A. Huang, D. Hou, and J. Liao, Sci. Bull. 67, 2265 (2022).
  98. M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, and H.-U. Yee, J. High Energy Phys. 11 (2021) 150.
  99. R. Biswas, A. Daher, A. Das, W. Florkowski, and R. Ryblewski, Phys. Rev. D 108, 014024 (2023).
  100. S. Fang, K. Fukushima, S. Pu, and D.-L. Wang, Phys. Rev. Lett. 136, 182301 (2026).
  101. Z.-H. Zhang, X.-H. Lv, and X.-G. Huang, arXiv:2603.17794.
  102. D. Montenegro, L. Tinti, and G. Torrieri, Phys. Rev. D 96, 076016 (2017).
  103. P. Shi and H. Xu-Guang, Acta Phys. Sin. 72, 071202 (2023).
  104. X.-G. Huang, Nucl. Sci. Tech. 36, 208 (2025).
  105. X.-Q. Xie, D.-L. Wang, C. Yang, and S. Pu, Phys. Rev. D 108, 094031 (2023).
  106. G. Sarwar, M. Hasanujjaman, J. R. Bhatt, H. Mishra, and J.-e. Alam, Phys. Rev. D 107, 054031 (2023).
  107. A. Daher, A. Das, and R. Ryblewski, Phys. Rev. D 107, 054043 (2023).
  108. A. Daher, W. Florkowski, R. Ryblewski, and F. Taghinavaz, Phys. Rev. D 109, 114001 (2024).
  109. G. S. Denicol, T. Kodama, T. Koide, and P. Mota, J. Phys. G 35, 115102 (2008).
  110. S. Pu, T. Koide, and D. H. Rischke, Phys. Rev. D 81, 114039 (2010).
  111. P. Kovtun, J. High Energy Phys. 10 (2019) 034.
  112. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Phys. Rev. Lett. 130, 261601 (2023).
  113. L. Gavassino, Phys. Lett. B 840, 137854 (2023).
  114. L. Gavassino, M. M. Disconzi, and J. Noronha, Phys. Rev. Lett. 132, 162301 (2024).
  115. D.-L. Wang and S. Pu, Phys. Rev. D 109, L031504 (2024).
  116. R. E. Hoult and P. Kovtun, Phys. Rev. D 109, 046018 (2024).
  117. X. Ren, C. Yang, D.-L. Wang, and S. Pu, Phys. Rev. D 110, 034010 (2024).
  118. L. Gavassino, M. Antonelli, and B. Haskell, Phys. Rev. Lett. 128, 010606 (2022).
  119. S. Dey, W. Florkowski, A. Jaiswal, and R. Ryblewski, arXiv:2303.05271.
  120. M. Buzzegoli, Phys. Rev. C 105, 044907 (2022).
  121. M. Buzzegoli and A. Palermo, Phys. Rev. Lett. 133, 262301 (2024).
  122. F. Becattini and C. Hoyos, Phys. Rev. Lett. 137, 041602 (2026).
  123. F. Becattini, A. Daher, and X.-L. Sheng, Phys. Lett. B 850, 138533 (2024).
  124. F. Becattini and R. Singh, Eur. Phys. J. C 85, 1338 (2025).
  125. V. E. Ambrus and A. Gecić, arXiv:2509.17640.
  126. J. Armas and A. Jain, arXiv:2601.14421.
  127. D.-L. Wang, S. Fang, and S. Pu, Phys. Rev. D 104, 114043 (2021).
  128. D.-L. Wang, X.-Q. Xie, S. Fang, and S. Pu, Phys. Rev. D 105, 114050 (2022).
  129. D.-L. Wang, L. Yan, and S. Pu, Phys. Rev. D 111, 034033 (2025).
  130. M. P. Heller and M. Spalinski, Phys. Rev. Lett. 115, 072501 (2015).
  131. P. Romatschke, Phys. Rev. Lett. 120, 012301 (2018).
  132. J.-P. Blaizot and L. Yan, Phys. Lett. B 780, 283 (2018).
  133. G. S. Denicol and J. Noronha, Phys. Rev. D 97, 056021 (2018).
  134. M. Spaliński, Phys. Lett. B 776, 468 (2018).
  135. M. Strickland, J. High Energy Phys. 12 (2018) 128.
  136. J.-P. Blaizot and L. Yan, Ann. Phys. (N.Y.) 412, 167993 (2020).
  137. J. Brewer, L. Yan, and Y. Yin, Phys. Lett. B 816, 136189 (2021).
  138. D. Almaalol, A. Kurkela, and M. Strickland, Phys. Rev. Lett. 125, 122302 (2020).
  139. J.-P. Blaizot and L. Yan, Phys. Lett. B 820, 136478 (2021).
  140. M. P. Heller, R. Jefferson, M. Spaliński, and V. Svensson, Phys. Rev. Lett. 125, 132301 (2020).
  141. M. P. Heller, A. Serantes, M. Spaliński, V. Svensson, and B. Withers, Phys. Rev. D 104, 066002 (2021).
  142. M. P. Heller, A. Serantes, M. Spaliński, V. Svensson, and B. Withers, Phys. Rev. Lett. 128, 122302 (2022).
  143. J.-P. Blaizot and L. Yan, Phys. Rev. C 104, 055201 (2021).
  144. Z. Chen, D. Teaney, and L. Yan, Phys. Rev. C 108, 064911 (2023).
  145. S. S. Gubser, Phys. Rev. D 82, 085027 (2010).
  146. S. S. Gubser and A. Yarom, Nucl. Phys. B846, 469 (2011).
  147. M. Strickland, J. Noronha, and G. Denicol, Phys. Rev. D 97, 036020 (2018).
  148. G. S. Denicol and J. Noronha, Phys. Rev. D 99, 116004 (2019).
  149. Y.-C. Liu and X.-G. Huang, Sci. China Phys. Mech. Astron. 65, 272011 (2022).

Outline

Information

Phys. Rev. D 114, 074029– Published 13 October, 2026
  • Received 13 April 2026
  • Accepted 23 September 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