Skip to main content

New developments in relativistic fluid dynamics with spin

Abstract

In this work, we briefly review the progress made in the formulation of hydrodynamics with spin with emphasis on the application to the relativistic heavy-ion collisions. In particular, we discuss the formulation of hydrodynamics with spin for perfect-fluid and the first order viscous corrections with some discussion on the calculation of spin kinetic coefficients. Finally, we apply relativistic hydrodynamics with spin to the relativistic heavy-ion collisions to calculate the spin polarization of \(\varLambda \)-particles.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    STAR Collaboration, L. Adamczyk et al., Global \(\varLambda \) hyperon polarization in nuclear collisions: evidence for the most vortical fluid. Nature 548, 62–65 (2017). https://doi.org/10.1038/nature23004. arXiv:1701.06657 [nucl-ex]

  2. 2.

    STAR Collaboration, J. Adam et al., Global polarization of \(\varLambda \) hyperons in Au+Au collisions at \(\sqrt{s_{{NN}}}\) = 200 GeV. Phys. Rev. C 98, 014910 (2018). https://doi.org/10.1103/PhysRevC.98.014910. arXiv:1805.04400 [nucl-ex]

  3. 3.

    ALICE Collaboration, S. Acharyaet al., Measurement of spin-orbital angular momentum interactions in relativistic heavy-ion collisions. Phys. Rev. Lett. 125(1), 012301 (2020). https://doi.org/10.1103/PhysRevLett.125.012301. arXiv:1910.14408 [nucl-ex]

  4. 4.

    Z.-T. Liang, X.-N. Wang, Globally polarized quark-gluon plasma in non-central A+A collisions. Phys. Rev. Lett. 94, 102301 (2005). https://doi.org/10.1103/PhysRevLett.94.102301. https://doi.org/10.1103/PhysRevLett.96.039901. arXiv:nucl-th/0410079 [Erratum: Phys. Rev. Lett. 96, 039901 (2006)]

  5. 5.

    Z.-T. Liang, X.-N. Wang, Spin alignment of vector mesons in non-central A+A collisions. Phys. Lett. B 629, 20–26 (2005). https://doi.org/10.1016/j.physletb.2005.09.060. arXiv:nucl-th/0411101

    ADS  Article  Google Scholar 

  6. 6.

    B. Betz, M. Gyulassy, G. Torrieri, Polarization probes of vorticity in heavy ion collisions. Phys. Rev. C 76, 044901 (2007). https://doi.org/10.1103/PhysRevC.76.044901. arXiv:0708.0035 [nucl-th]

    ADS  Article  Google Scholar 

  7. 7.

    S.A. Voloshin, Polarized secondary particles in unpolarized high energy hadron-hadron collisions? arXiv:nucl-th/0410089

  8. 8.

    F. Becattini, F. Piccinini, J. Rizzo, Angular momentum conservation in heavy ion collisions at very high energy. Phys. Rev. C 77, 024906 (2008). https://doi.org/10.1103/PhysRevC.77.024906. arXiv:0711.1253 [nucl-th]

    ADS  Article  Google Scholar 

  9. 9.

    F. Becattini, L. Csernai, D.J. Wang, \(\Lambda \) polarization in peripheral heavy ion collisions. Phys. Rev. C 88(3), 034905 (2013). https://doi.org/10.1103/PhysRevC.93.069901. https://doi.org/10.1103/PhysRevC.88.034905. arXiv:1304.4427 [nucl-th] [Erratum: Phys. Rev. C 93(6), 069901 (2016)]

  10. 10.

    F. Becattini, V. Chandra, L. Del Zanna, E. Grossi, Relativistic distribution function for particles with spin at local thermodynamical equilibrium. Ann. Phys. 338, 32–49 (2013). https://doi.org/10.1016/j.aop.2013.07.004. arXiv:1303.3431 [nucl-th]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    F. Becattini, F. Piccinini, The Ideal relativistic spinning gas: polarization and spectra. Ann. Phys. 323, 2452–2473 (2008). https://doi.org/10.1016/j.aop.2008.01.001. arXiv:0710.5694 [nucl-th]

    ADS  Article  MATH  Google Scholar 

  12. 12.

    F. Becattini, I. Karpenko, M. Lisa, I. Upsal, S. Voloshin, Global hyperon polarization at local thermodynamic equilibrium with vorticity, magnetic field and feed-down. Phys. Rev. C 95(5), 054902 (2017). https://doi.org/10.1103/PhysRevC.95.054902. arXiv:1610.02506 [nucl-th]

    ADS  Article  Google Scholar 

  13. 13.

    F. Becattini, G. Inghirami, V. Rolando, A. Beraudo, L. Del Zanna, A. De Pace, M. Nardi, G. Pagliara, V. Chandra, A study of vorticity formation in high energy nuclear collisions. Eur. Phys. J. C 75(9), 406 (2015). https://doi.org/10.1140/epjc/s10052-015-3624-1. https://doi.org/10.1140/epjc/s10052-018-5810-4. arXiv:1501.04468 [nucl-th] [Erratum: Eur. Phys. J. C 78(5), 354 (2018)]

  14. 14.

    I. Karpenko, F. Becattini, Study of \(\varLambda \) polarization in relativistic nuclear collisions at \(\sqrt{s_{\rm NN}}=7.7\)–200 GeV. Eur. Phys. J. C 77(4), 213 (2017). https://doi.org/10.1140/epjc/s10052-017-4765-1. arXiv:1610.04717 [nucl-th]

    ADS  Article  Google Scholar 

  15. 15.

    Y. Xie, D. Wang, L.P. Csernai, Global Lambda polarization in high energy collisions. Phys. Rev. C 95(3), 031901 (2017). https://doi.org/10.1103/PhysRevC.95.031901. arXiv:1703.03770 [nucl-th]

    ADS  Article  Google Scholar 

  16. 16.

    L.-G. Pang, H. Petersen, Q. Wang, X.-N. Wang, Vortical fluid and \(\varLambda \) spin correlations in high-energy heavy-ion collisions. Phys. Rev. Lett. 117(19), 192301 (2016). https://doi.org/10.1103/PhysRevLett.117.192301. arXiv:1605.04024 [hep-ph]

    ADS  Article  Google Scholar 

  17. 17.

    F. Becattini, I. Karpenko, Collective longitudinal polarization in relativistic heavy-ion collisions at very high energy. Phys. Rev. Lett. 120(1), 012302 (2018). https://doi.org/10.1103/PhysRevLett.120.012302. arXiv:1707.07984 [nucl-th]

    ADS  Article  Google Scholar 

  18. 18.

    F. Becattini, M.A. Lisa, Polarization and vorticity in the quark gluon plasma. arXiv:2003.03640 [nucl-ex]

  19. 19.

    STAR Collaboration, T. Niida, Global and local polarization of \(\lambda \) hyperons in au+au collisions at 200 GeV from star. In Global and Local Polarization of\(\Lambda \)Hyperons in Au+Au Collisions at 200 GeV from STAR, vol. 982, pp. 511–514 (2019). https://doi.org/10.1016/j.nuclphysa.2018.08.034. arXiv:1808.10482 [nucl-ex]

  20. 20.

    N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, D.H. Rischke, Generating spin polarization from vorticity through nonlocal collisions. arXiv:2005.01506 [hep-ph]

  21. 21.

    F. Becattini, W. Florkowski, E. Speranza, Spin tensor and its role in non-equilibrium thermodynamics. Phys. Lett. B 789, 419–425 (2019). https://doi.org/10.1016/j.physletb.2018.12.016. arXiv:1807.10994 [hep-th]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    W. Florkowski, B. Friman, A. Jaiswal, E. Speranza, Relativistic fluid dynamics with spin. Phys. Rev. C 97(4), 041901 (2018). https://doi.org/10.1103/PhysRevC.97.041901. arXiv:1705.00587 [nucl-th]

    ADS  Article  Google Scholar 

  23. 23.

    W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, E. Speranza, Spin-dependent distribution functions for relativistic hydrodynamics of spin-1/2 particles. Phys. Rev. D 97(11), 116017 (2018). https://doi.org/10.1103/PhysRevD.97.116017. arXiv:1712.07676 [nucl-th]

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    R. Singh, G. Sophys, R. Ryblewski, Spin polarization dynamics in the Gubser-expanding background (2020). arXiv:2011.14907 [hep-ph]

  25. 25.

    R. Singh, M. Shokri, R. Ryblewski, Spin polarization dynamics in the Bjorken-expanding resistive MHD background (2021). arXiv:2103.02592 [hep-ph]

  26. 26.

    A. Jaiswal, Dynamics of QCD matter—current status. Int. J. Mod. Phys. E 30(02), 2130001 (2021). https://doi.org/10.1142/S0218301321300010. arXiv:2007.14959 [hep-ph]

  27. 27.

    E. Speranza, N. Weickgenannt, Spin tensor and pseudo-gauges: from nuclear collisions to gravitational physics. arXiv:2007.00138 [nucl-th]

  28. 28.

    S. Shi, C. Gale, S. Jeon, Relativistic viscous spin hydrodynamics from chiral kinetic theory. arXiv:2008.08618 [nucl-th]

  29. 29.

    F. Haas, Quantum Plasmas: An Hydrodynamic Approach (2011)

  30. 30.

    G. Denicol, H. Niemi, E. Molnar, D. Rischke, Derivation of transient relativistic fluid dynamics from the Boltzmann equation. Phys. Rev. D 85, 114047 (2012). https://doi.org/10.1103/PhysRevD.85.114047. arXiv:1202.4551 [nucl-th] [Erratum: Phys. Rev. D 91, 039902 (2015)]

  31. 31.

    A. Jaiswal, Relativistic dissipative hydrodynamics from kinetic theory with relaxation time approximation. Phys. Rev. C 87(5), 051901 (2013). https://doi.org/10.1103/PhysRevC.87.051901. arXiv:1302.6311 [nucl-th]

    ADS  Article  Google Scholar 

  32. 32.

    A. Jaiswal, Relativistic third-order dissipative fluid dynamics from kinetic theory. Phys. Rev. C 88, 021903 (2013). https://doi.org/10.1103/PhysRevC.88.021903. arXiv:1305.3480 [nucl-th]

    ADS  Article  Google Scholar 

  33. 33.

    A. Dash, V. Roy, B. Mohanty, Magneto-vortical evolution of QGP in heavy ion collisions. J. Phys. G 46(1), 015103 (2019). https://doi.org/10.1088/1361-6471/aaeef2. arXiv:1705.05657 [nucl-th]

    ADS  Article  Google Scholar 

  34. 34.

    P. Mohanty, A. Dash, V. Roy, One particle distribution function and shear viscosity in magnetic field: a relaxation time approach. Eur. Phys. J. A 55, 35 (2019). https://doi.org/10.1140/epja/i2019-12705-7. arXiv:1804.01788 [nucl-th]

    ADS  Article  Google Scholar 

  35. 35.

    A. Dash, S. Samanta, J. Dey, U. Gangopadhyaya, S. Ghosh, V. Roy, Anisotropic transport properties of a hadron resonance gas in a magnetic field. Phys. Rev. D 102(1), 016016 (2020). https://doi.org/10.1103/PhysRevD.102.016016. arXiv:2002.08781 [nucl-th]

    ADS  MathSciNet  Article  Google Scholar 

  36. 36.

    D.E. Kharzeev, L.D. McLerran, H.J. Warringa, The effects of topological charge change in heavy ion collisions: ‘Event by event P and CP violation’. Nucl. Phys. A 803, 227–253 (2008). https://doi.org/10.1016/j.nuclphysa.2008.02.298. arXiv:0711.0950 [hep-ph]

  37. 37.

    K. Fukushima, D.E. Kharzeev, H.J. Warringa, The chiral magnetic effect. Phys. Rev. D 78, 074033 (2008). https://doi.org/10.1103/PhysRevD.78.074033. arXiv:0808.3382 [hep-ph]

    ADS  Article  Google Scholar 

  38. 38.

    D.E. Kharzeev, The chiral magnetic effect and anomaly-induced transport. Prog. Part. Nucl. Phys. 75, 133–151 (2014). https://doi.org/10.1016/j.ppnp.2014.01.002. arXiv:1312.3348 [hep-ph]

    ADS  Article  Google Scholar 

  39. 39.

    Y. Hirono, T. Hirano, D.E. Kharzeev, The chiral magnetic effect in heavy-ion collisions from event-by-event anomalous hydrodynamics. arXiv:1412.0311 [hep-ph]

  40. 40.

    Q. Li, D.E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic, A. Fedorov, R. Zhong, J. Schneeloch, G. Gu, T. Valla, Observation of the chiral magnetic effect in ZrTe5. Nat. Phys. 12, 550–554 (2016). https://doi.org/10.1038/nphys3648. arXiv:1412.6543 [cond-mat.str-el]

    Article  Google Scholar 

  41. 41.

    D. Kharzeev, J. Liao, S. Voloshin, G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions—a status report. Prog. Part. Nucl. Phys. 88, 1–28 (2016). https://doi.org/10.1016/j.ppnp.2016.01.001. arXiv:1511.04050 [hep-ph]

    ADS  Article  Google Scholar 

  42. 42.

    H. Li, X.-L. Sheng, Q. Wang, Electromagnetic fields with electric and chiral magnetic conductivities in heavy ion collisions. Phys. Rev. C 94(4), 044903 (2016). https://doi.org/10.1103/PhysRevC.94.044903. arXiv:1602.02223 [nucl-th]

    ADS  Article  Google Scholar 

  43. 43.

    S.Y. Liu, Y. Sun, C.M. Ko, Spin polarizations in a covariant angular-momentum-conserved chiral transport model. Phys. Rev. Lett. 125, 062301 (2020). https://doi.org/10.1103/PhysRevLett.125.062301. arXiv:1910.06774 [nucl-th]

    ADS  Article  Google Scholar 

  44. 44.

    J.-H. Gao, G.-L. Ma, S. Pu, Q. Wang, Recent developments in chiral and spin polarization effects in heavy-ion collisions. arXiv:2005.10432 [hep-ph]

  45. 45.

    Y.-C. Liu, X.-G. Huang, Anomalous chiral transports and spin polarization in heavy-ion collisions. Nucl. Sci. Tech. 31(6), 56 (2020). https://doi.org/10.1007/s41365-020-00764-z. arXiv:2003.12482 [nucl-th]

    Article  Google Scholar 

  46. 46.

    F. Li, S.Y. Liu, Anomalous Lorentz transformation and side jump of a massive fermion. arXiv:2004.08910 [nucl-th]

  47. 47.

    K. Hattori, Y. Hidaka, D.-L. Yang, Axial kinetic theory and spin transport for fermions with arbitrary mass. Phys. Rev. D 100(9), 096011 (2019). https://doi.org/10.1103/PhysRevD.100.096011. arXiv:1903.01653 [hep-ph]

    ADS  MathSciNet  Article  Google Scholar 

  48. 48.

    D.-L. Yang, K. Hattori, Y. Hidaka, Quantum kinetic theory for spin transport: general formalism for collisional effects. arXiv:2002.02612 [hep-ph]

  49. 49.

    D. Montenegro, L. Tinti, G. Torrieri, Sound waves and vortices in a polarized relativistic fluid. Phys. Rev. D 96(7), 076016 (2017). https://doi.org/10.1103/PhysRevD.96.076016. arXiv:1703.03079 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  50. 50.

    D. Montenegro, L. Tinti, G. Torrieri, The ideal relativistic fluid limit for a medium with polarization. Phys. Rev. D 96(5), 056012 (2017). https://doi.org/10.1103/PhysRevD.96.056012. arXiv:1701.08263 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  51. 51.

    D. Montenegro, G. Torrieri, Causality and dissipation in relativistic polarizeable fluids. arXiv:1807.02796 [hep-th]

  52. 52.

    D. Montenegro, R. Ryblewski, G. Torrieri, Relativistic fluid dynamics and its extensions as an effective field theory. In 25th Cracow Epiphany Conference on Advances in Heavy Flavour Physics (Epiphany 2019) Cracow, Poland, January 8–11, 2019 (2019). arXiv:1903.08729 [hep-th]

  53. 53.

    A.D. Gallegos, U. Gürsoy, Holographic spin liquids and Lovelock Chern-Simons gravity. JHEP 11, 151 (2020). https://doi.org/10.1007/JHEP11(2020)151. arXiv:2004.05148 [hep-th]

  54. 54.

    D. Gallegos, U. Gursoy, A. Yarom, Hydrodynamics of spin currents (2021). arXiv:2101.04759 [hep-th]

  55. 55.

    K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, H. Taya, Fate of spin polarization in a relativistic fluid: an entropy-current analysis. Phys. Lett. B 795, 100–106 (2019). https://doi.org/10.1016/j.physletb.2019.05.040. arXiv:1901.06615 [hep-th]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    F.A. Asenjo, V. Muñoz, J.A. Valdivia, S.M. Mahajan, A hydrodynamical model for relativistic spin quantum plasmas. Phys. Plasmas 18(1), 012107 (2011)

    ADS  Article  Google Scholar 

  57. 57.

    T. Takabayasi, Relativistic Hydrodynamics of the Dirac Matter. Part I. General Theory. Progress of Theoretical Physics Supplement 4, 1–80 (1957)

    ADS  MathSciNet  Article  Google Scholar 

  58. 58.

    J.R. Bhatt, M. George, Neutrino induced vorticity, Alfvén waves and the normal modes. Eur. Phys. J. C 77(8), 539 (2017). https://doi.org/10.1140/epjc/s10052-017-5100-6. arXiv:1608.05558 [hep-ph]

    ADS  Article  Google Scholar 

  59. 59.

    H.T. Elze, M. Gyulassy, D. Vasak, Transport equations for the QCD quark Wigner operator. Nucl. Phys. B 276, 706–728 (1986). https://doi.org/10.1016/0550-3213(86)90072-6

    ADS  Article  Google Scholar 

  60. 60.

    D. Vasak, M. Gyulassy, H.T. Elze, Quantum transport theory for Abelian plasmas. Ann. Phys. 173, 462–492 (1987). https://doi.org/10.1016/0003-4916(87)90169-2

    ADS  Article  Google Scholar 

  61. 61.

    H.-T. Elze, U.W. Heinz, Quark-gluon transport theory. Phys. Rep. 183(CERN–TH–5325–89), 81–135 (1989)

    ADS  Article  Google Scholar 

  62. 62.

    W. Florkowski, J. Hufner, S.P. Klevansky, L. Neise, Chirally invariant transport equations for quark matter. Ann. Phys. 245, 445–463 (1996). https://doi.org/10.1006/aphy.1996.0016. arXiv:hep-ph/9505407

    ADS  Article  Google Scholar 

  63. 63.

    P. Zhuang, U.W. Heinz, Relativistic quantum transport theory for electrodynamics. Ann. Phys. 245, 311–338 (1996). https://doi.org/10.1006/aphy.1996.0011. arXiv:nucl-th/9502034

    ADS  MathSciNet  Article  Google Scholar 

  64. 64.

    A. Alexandrov, P. Mitkin, Zilch vortical effect for fermions (2020). arXiv:2011.09429 [hep-th]

  65. 65.

    X.-L. Sheng, D.H. Rischke, D. Vasak, Q. Wang, Wigner functions for fermions in strong magnetic fields. Eur. Phys. J. A 54(2), 21 (2018). https://doi.org/10.1140/epja/i2018-12414-9. arXiv:1707.01388 [hep-ph]

    ADS  Article  Google Scholar 

  66. 66.

    X.-L. Sheng, R.-H. Fang, Q. Wang, D.H. Rischke, Wigner function and pair production in parallel electric and magnetic fields. Phys. Rev. D 99(5), 056004 (2019). https://doi.org/10.1103/PhysRevD.99.056004. arXiv:1812.01146 [hep-ph]

    ADS  MathSciNet  Article  Google Scholar 

  67. 67.

    X.-L. Sheng, Q. Wang, X.-G. Huang, Kinetic theory with spin: from massive to massless fermions. Phys. Rev. D 102(2), 025019 (2020). https://doi.org/10.1103/PhysRevD.102.025019. arXiv:2005.00204 [hep-ph]

    ADS  MathSciNet  Article  Google Scholar 

  68. 68.

    X.-L. Sheng, Wigner Function for Spin-1/2 Fermions in Electromagnetic Fields. PhD thesis, Frankfurt University (2019). arXiv:1912.01169 [nucl-th]

  69. 69.

    N. Weickgenannt, X.-L. Sheng, E. Speranza, Q. Wang, D.H. Rischke, Kinetic theory for massive spin-1/2 particles from the Wigner-function formalism. Phys. Rev. D 100(5), 056018 (2019). https://doi.org/10.1103/PhysRevD.100.056018. arXiv:1902.06513 [hep-ph]

  70. 70.

    S.R. De Groot, Relativistic Kinetic Theory. Principles and Applications (1980)

  71. 71.

    W. Florkowski, A. Kumar, R. Ryblewski, Thermodynamic versus kinetic approach to polarization-vorticity coupling. Phys. Rev. C 98, 044906 (2018). https://doi.org/10.1103/PhysRevC.98.044906. arXiv:1806.02616 [hep-ph]

    ADS  Article  Google Scholar 

  72. 72.

    W. Florkowski, R. Ryblewski, A. Kumar, Relativistic hydrodynamics for spin-polarized fluids. Prog. Part. Nucl. Phys. 108, 103709 (2019). https://doi.org/10.1016/j.ppnp.2019.07.001. arXiv:1811.04409 [nucl-th]

    Article  Google Scholar 

  73. 73.

    M. Mathisson, Neue mechanik materieller systemes. Acta Phys. Polon. 6, 163–2900 (1937)

    MATH  Google Scholar 

  74. 74.

    C. Itzykson, J.B. Zuber, Quantum Field Theory. International Series in Pure and Applied Physics. McGraw-Hill, New York (1980). https://doi.org/10.1063/1.2916419

  75. 75.

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, R. Ryblewski, Relativistic dissipative spin dynamics in the relaxation time approximation. Phys. Lett. B814, 136096 (2021). https://doi.org/10.1016/j.physletb.2021.136096arXiv:2002.03937 [hep-ph]

  76. 76.

    W. Florkowski, A. Kumar, R. Ryblewski, R. Singh, Spin polarization evolution in a boost invariant hydrodynamical background. Phys. Rev. C 99(4), 044910 (2019). https://doi.org/10.1103/PhysRevC.99.044910. arXiv:1901.09655 [hep-ph]

    ADS  Article  Google Scholar 

  77. 77.

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, R. Ryblewski, Dissipative spin dynamics in relativistic matter . Phys. Rev. D103, 014030 (2021). https://doi.org/10.1103/PhysRevD.103.014030. arXiv:2008.10976 [nucl-th]

  78. 78.

    J.D. Bjorken, Highly relativistic nucleus-nucleus collisions: the central rapidity region. Phys. Rev. D 27, 140–151 (1983). https://doi.org/10.1103/PhysRevD.27.140

    ADS  Article  Google Scholar 

  79. 79.

    F. Becattini, L. Tinti, The Ideal relativistic rotating gas as a perfect fluid with spin. Ann. Phys. 325, 1566–1594 (2010). https://doi.org/10.1016/j.aop.2010.03.007. arXiv:0911.0864 [gr-qc]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  80. 80.

    B. Singh, J.R. Bhatt, H. Mishra, Probing vorticity in heavy ion collision with dilepton production. Phys. Rev. D 100(1), 014016 (2019). https://doi.org/10.1103/PhysRevD.100.014016. arXiv:1811.08124 [hep-ph]

    ADS  Article  Google Scholar 

  81. 81.

    J.-Y. Ollitrault, Relativistic hydrodynamics for heavy-ion collisions. Eur. J. Phys. 29, 275–302 (2008). https://doi.org/10.1088/0143-0807/29/2/010. arXiv:0708.2433 [nucl-th]

    Article  Google Scholar 

  82. 82.

    W.-T. Deng, X.-G. Huang, Vorticity in heavy-ion collisions. Phys. Rev. C 93(6), 064907 (2016). https://doi.org/10.1103/PhysRevC.93.064907. arXiv:1603.06117 [nucl-th]

    ADS  Article  Google Scholar 

Download references

Acknowledgements

AK acknowledges the hospitality of National Institute of Science Education and Research where a part of this work was done. A. K. was supported in part by the department of Science and Technology Government of India under the SERB National Post Doctoral Fellowship Reference No. PDF/2020/000648. A.J. was supported in part by the DST-INSPIRE faculty award under Grant No. DST/INSPIRE/04/2017/000038.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jitesh Bhatt.

Appendices

List of integrals in spin space

Here we list various formula used to carry out integration in spin space. The detail calculations will be presented in our forthcoming paper [77]

$$\begin{aligned} \int \mathrm {d}S= & {} \frac{m}{\pi {\mathfrak {s}}} \int \mathrm {d}^{4}s\, \delta (s\cdot s + {{\mathfrak {s}}}^2) \delta (p\cdot s)=2\nonumber \\ \int \mathrm {d}S\,s^{\mu \nu }= & {} 0. \nonumber \\ \int \mathrm {d}S\, s_\sigma \, s_\delta= & {} -\frac{2 {\mathfrak {s}}^2}{3} \bigg (g_{\sigma \delta } - \frac{p_\sigma p_\delta }{m^2}\bigg )\nonumber \\ \int \mathrm {d}S s^{\mu \nu } s^{\alpha \beta }= & {} - \frac{2 {\mathfrak {s}}^2}{3m^2} \epsilon ^{\mu \nu \rho \sigma }\, \epsilon ^{\alpha \beta \gamma \delta }\, p_\rho \, p_\gamma \bigg (g_{\sigma \delta } \!-\! \frac{p_\sigma p_\delta }{m^2}\bigg )\nonumber \\ \end{aligned}$$
(114)

List of thermodynamic integrals \(I_{nq}^{(r)}\)

Thermodynamic integrals \(I_{nq}^{(r)}\) are obtained by

$$\begin{aligned} I_{nq}^{(r)}= & {} \frac{1}{(2q+1)!!} \int {\mathrm{d}P}\,(u\cdot p)^{n-2q-r} (\varDelta _{\alpha \beta } p^{\alpha } p^{\beta })^q e^{-\beta \cdot p}.\nonumber \\ \end{aligned}$$
(115)

From the above formula we can get,

$$\begin{aligned} I_{10}^{(0)}= & {} \frac{T^3 z^2 }{2 \pi ^2} K_2(z)\\ I_{20}^{(0)}= & {} \frac{T^4 z^2}{2 \pi ^2} \left[ 3 K_2(z) + zK_1(z)\right] \\ I_{21}^{(0)}= & {} - \frac{T^4 z^2}{2 \pi ^2} K_2(z)\\ I_{30}^{(0)}= & {} \frac{z^5 T^5}{2\pi ^2} \left[ K_5(z)+K_3(z)-2K_1(z)\right] \\ I_{31}^{(0)}= & {} - \frac{z^5T^5}{96\pi ^2} \left[ K_5(z)-3K_3(z)+2K_1(z)\right] \\ I_{40}^{(0)}= & {} \frac{T^6 z^6}{64 \pi ^2} \left[ K_{6}(z) + 2K_{4}(z) - K_{2}(z) - 2K_{0}(z)\right] \\ I_{41}^{(0)}= & {} - \frac{T^6 z^6}{192 \pi ^2} \left[ K_{6}(z) - 2K_{4}(z) - K_{2}(z) + 2K_{0}(z)\right] \\ I_{42}^{(0)}= & {} \frac{T^6 z^6}{960 \pi ^2} \left[ K_{6}(z) - 6K_{4}(z) + 15K_{2}(z) - 10K_{0}(z)\right] \\ I_{21}^{(1)}= & {} - \frac{T^3 z^3}{6 \pi ^2} \left[ \frac{1}{4} K_3(z) - \frac{5}{4} K_1(z) + K_{i,1}(z)\right] \\ I_{42}^{(1)}= & {} \frac{T^5 z^5}{480 \pi ^2} \Big [22K_{1}(z) - 7K_{3}(z) + K_5(z) - 16K_{i,1}(z)\Big ] \end{aligned}$$

In the above formulas, \(K_n(z)\) are the modified Bessel functions of the second kind while \(K_{i,1}(z)\) are the first order Bickley–Naylor function with the argument \(z = m/T\). The function \(K_n(z)\) and \(K_{i,1}(z)\) are expressed as

$$\begin{aligned} K_n(z)= & {} \int _0^{\infty } \mathrm{d}x\, \cosh {nx}\, e^{- z \cosh x}. \end{aligned}$$
(116)
$$\begin{aligned} K_{i,1}(z)= & {} \int _0^{\infty } \mathrm{d}x {{\,\mathrm{sech}\,}}{x}\, e^{-z \cosh x}\nonumber \\= & {} \frac{\pi }{2} \Big [ 1 - z\, K_{0}(z)\, L_{-1}(z) - z\, K_{1}(z)\, L_{0}(z)\Big ]\nonumber \\ \end{aligned}$$
(117)

In the expression for \(K_{i,1}(z)\), function \(L_i\) is the modified Struve function.

Note that here we have not listed the function \(I_{20}^{(1)}\), \(I_{30}^{(1)}\), \(I_{31}^{(1)}\), \(I_{40}^{(1)}\), \(I_{50}^{(1)}\), \(I_{51}^{(1)}\) and \(I_{52}^{(1)}\) as they all can be written in terms of above listed integrals using the recurrence relation as given below.

$$\begin{aligned} I_{\mathrm{n,q}}^{(r)}= & {} I_{\mathrm{n-1,q}}^{(r-1)};~~~ n\ge 2 q, \end{aligned}$$
(118)
$$\begin{aligned} I_{\mathrm{n,q}}^{(0)}= & {} \frac{1}{\beta } \left[ (n - 2 q) I_{\mathrm{n-1,q}}^{(0)} - I_{\mathrm{n-1, q-1}}^{(0)}\right] , \end{aligned}$$
(119)
$$\begin{aligned} {\dot{I}}_{\mathrm{n,q}}^{(0)}= & {} - {\dot{\beta }} I_{n+1, q}^{(0)} \end{aligned}$$
(120)

List of D-coefficients

Expressions for various D-coefficients are as follows

$$\begin{aligned}&D_{\varPi }^{\mu \nu } = D_{\varPi 1} \omega ^{\mu \nu } + D_{\varPi 2} u^{\alpha } u^{[\mu } \omega ^{\nu ]}{}_{\alpha } \end{aligned}$$
(121)
$$\begin{aligned}&D_n^{[\mu \nu ]}{}_{\alpha } = - D_{n1}\left( u^{[\mu } \omega ^{\nu ]}{}_{\alpha } + g^{[\mu }{}_{\alpha } u^{\kappa } \omega ^{\nu ]}{}_{\kappa }\right) \nonumber \\&\quad - D_{n2} u^{[\mu } \varDelta ^{\nu ]}{}_{\rho } \omega ^{\rho }{}_{\alpha } \end{aligned}$$
(122)
$$\begin{aligned}&D_{\pi }^{[\mu }{}_{\lambda } = - \omega ^{[\mu }{}_{\lambda } \frac{4 I_{31}^{(0)}}{(m^2\, I_{10}^{(0)} - 2 I_{31}^{(0)})} \nonumber \\&\quad - u^{[\mu } u^{\alpha } \omega _{\alpha \lambda } \frac{4 (I_{30}^{(0)} - I_{31}^{(0)}) I_{31}^{(0)}}{(m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}) \big [m^2\, I_{10}^{(0)} - (I_{30}^{(0)} + I_{31}^{(0)})\big ]}\nonumber \\ \end{aligned}$$
(123)
$$\begin{aligned}&D_{\varSigma 1}^{\alpha } = - u^{\alpha } \frac{2\, I_{31}^{(0)}}{(m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)})} \end{aligned}$$
(124)
$$\begin{aligned}&D_{\varSigma 2}^{[\mu \nu ] \alpha } = - u^{[\mu } g^{\nu ]\alpha } \frac{2\, I_{31}^{(0)}}{ \left( m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}\right) } \nonumber \\&\quad - u^{[\mu } \varDelta ^{\nu ]\alpha } \frac{2\, (I_{30}^{(0)} - I_{31}^{(0)}) I_{31}^{(0)}}{\left( m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}\right) \left[ m^2\, I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) \right] }\nonumber \\ \end{aligned}$$
(125)

where

$$\begin{aligned} D_{\varPi 1}= & {} - \frac{1}{\left( I_{10}^{(0)} - \frac{2}{m^2} I_{31}^{(0)}\right) } \left( \xi _{\theta } \tanh \xi \, I_{10}^{(0)} - \beta _{\theta } I_{20}^{(0)} + I_{10}^{(0)}\right. \nonumber \\&\left. - \frac{2}{m^2} \xi _{\theta }\, \tanh \xi \, I_{31}^{(0)} + \frac{2\, \beta _{\theta }\, I_{41}^{(0)}}{m^2} - \frac{10 I_{31}^{(0)}}{3\, m^2}\right) \end{aligned}$$
(126)
$$\begin{aligned} D_{\varPi 2}= & {} \frac{2}{m^2\, I_{10}^{(0)} \!-\! 2\, I_{31}^{(0)}} \Bigg [\beta _{\theta } \!\left( I_{40}^{(0)}\! -\! I_{41}^{(0)}\right) \!-\! \xi _{\theta }\! \left( I_{30}^{(0)} \!-\! I_{31}^{(0)}\right) \tanh \xi \nonumber \\&- \left( I_{30}^{(0)} - \frac{11}{3} I_{31}^{(0)}\right) + \frac{\left( I_{30}^{(0)} - I_{31}^{(0)}\right) }{m^2\, I_{10}^{(0)} - I_{30}^{(0)} - I_{31}^{(0)}}\nonumber \\&\!\times \bigg (\!m^2\, \xi _{\theta } \tanh \xi \, I_{10}^{(0)} - m^2 \beta _{\theta }\, I_{20}^{(0)} + m^2 I_{10}^{(0)} \nonumber \\&- \xi _{\theta }\! \left( \!I_{30}^{(0)} + I_{31}^{(0)}\!\right) \! \tanh \xi + \beta _\theta \! \left( \!I_{40}^{(0)} + I_{41}^{(0)}\!\right) \!\nonumber \\&+\beta I_{41}^{(0)} - \frac{5}{3} I_{31}^{(0)}\bigg )\! \Bigg ] \end{aligned}$$
(127)
$$\begin{aligned} D_{n1}= & {} \frac{2 \tanh \xi }{\left( m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}\right) } \left( I_{31}^{(0)} - \frac{n_0\, I_{41}^{(0)}}{\varepsilon _0 + P_0}\right) \end{aligned}$$
(128)
$$\begin{aligned} D_{n2}= & {} \frac{\tanh \xi }{m^2I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \left( I_{31}^{(0)} - \frac{n_0 I_{41}^{(0)}}{\varepsilon _0 + P_0}\right) \nonumber \\&\times \frac{2 \left( I_{30}^{(0)} - I_{31}^{(0)}\right) }{\left( m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}\right) } \end{aligned}$$
(129)

List of C-coefficients

Various C-coefficients are given by following expressions

$$\begin{aligned} C_{\varPi }= & {} - \frac{1}{m^2 I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \Bigg [m^2 \xi _{\theta } \tanh \xi I_{10}^{(0)} \nonumber \\&- m^2\, \beta _\theta \, I_{20}^{(0)} + m^2\, I_{10}^{(0)} - \tanh \xi \left( I_{30}^{(0)} + I_{31}^{(0)}\right) \xi _{\theta } \nonumber \\&+ \beta _{\theta } \left( I_{40}^{(0)} + I_{41}^{(0)}\right) + \beta \, I_{41}^{(0)} - \frac{5}{3} I_{31}^{(0)}\Bigg ] \end{aligned}$$
(130)
$$\begin{aligned} C_{n}= & {} \frac{\tanh \xi }{m^2\, I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \left( I_{31}^{(0)} - \frac{n_0 I_{41}^{(0)}}{\varepsilon _0 + P_0}\right) \end{aligned}$$
(131)
$$\begin{aligned} C_{\pi }= & {} - \frac{2 I_{31}^{(0)}}{m^2\, I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \end{aligned}$$
(132)
$$\begin{aligned} C_{\varSigma }= & {} \frac{I_{31}^{(0)}}{m^2\, I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \end{aligned}$$
(133)

List of \(\beta \)-coefficients

$$\begin{aligned} \beta _{\varPi }^{(1)}= & {} \frac{4\, {\mathfrak {s}}^2}{3}\Bigg (\!-\frac{2}{m^2} \xi _{\theta }\, \sinh \xi \, I_{41}^{(1)} + \frac{2}{m^2} I_{51}^{(1)} \beta _{\theta }\, \cosh \xi \nonumber \\&+ \frac{10}{3m^2} I_{52}^{(1)} \beta \, \cosh \xi - \frac{2}{m^2} I_{41}^{(1)} \cosh \xi \, D_{\varPi 1}\Bigg ) \end{aligned}$$
(134)
$$\begin{aligned} \beta _{\varPi }^{(2)}= & {} \frac{4\, {\mathfrak {s}}^2}{3} \Bigg [\!-\frac{2}{m^2} \xi _{\theta }\, \sinh \xi I_{40}^{(1)} + \frac{4}{m^2} \xi _{\theta }\, \sinh \xi \, I_{41}^{(1)} \nonumber \\&+ \frac{2}{m^2} I_{50}^{(1)} \beta _{\theta }\, \cosh \xi + \frac{2}{m^2} I_{51}^{(1)} \beta \, \cosh \xi - \frac{4}{m^2} I_{51}^{(1)} \beta _\theta \, \cosh \xi \nonumber \\&- \frac{20}{3m^2} I_{52}^{(1)} \beta \, \cosh \xi - \left( I_{20}^{(1)} - \frac{3}{m^2} I_{41}^{(1)}\right) \cosh \xi \, D_{\varPi 2} \nonumber \\&- \frac{2}{m^2} \left( I_{40}^{(1)} - 2\, I_{41}^{(1)}\right) \cosh \xi \, C_\varPi \Bigg ] \end{aligned}$$
(135)
$$\begin{aligned} \beta _{\varPi }^{(3)}= & {} \frac{4\, {\mathfrak {s}}^2}{3}\Bigg ( - \frac{2}{m^2} \xi _{\theta }\, \sinh \xi I_{41}^{(1)} + \frac{2}{m^2} I_{51}^{(1)} \beta _{\theta }\, \cosh \xi \nonumber \\&+ \frac{10}{3m^2} I_{52}^{(1)} \beta \, \cosh \xi - \frac{2}{m^2} I_{41}^{(1)} \cosh \xi \, C_\varPi \Bigg ) \end{aligned}$$
(136)
$$\begin{aligned} \beta _{\pi }^{(1)}= & {} \frac{16\, {\mathfrak {s}}^2}{3\, m^2} \beta \, \cosh \xi \, I_{42}^{(0)} \end{aligned}$$
(137)
$$\begin{aligned} \beta _{\pi }^{(2)}= & {} \frac{16\, {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left( \beta \, I_{42}^{(0)} - \frac{ I_{41}^{(1)}\, I_{31}^{(0)}}{m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}}\right) \end{aligned}$$
(138)
$$\begin{aligned} \beta _{\pi }^{(3)}= & {} \frac{16\, {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left( \frac{I_{41}^{(1)}\, I_{31}^{(0)}}{m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}}\right) \end{aligned}$$
(139)
$$\begin{aligned} \beta _{\pi }^{(4)}= & {} \frac{16\, {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left( \frac{I_{41}^{(1)}\, I_{31}^{(0)}}{m^2 I_{10}^{(0)}-\left( I^{(0)}_{30}+I_{31}^{(0)}\right) }\right) \end{aligned}$$
(140)
$$\begin{aligned} \beta _{n}^{(1)}= & {} \frac{4 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left[ -\tanh \xi \left( m^2\, I_{21}^{(1)} - 2\, I_{42}^{(1)}\right) \right. \nonumber \\&\left. + \bigg (\frac{n_0 \tanh (\xi )}{\varepsilon _0 + P_0} \bigg ) \left( m^2\, I_{31}^{(1)} - 2\, I_{52}^{(1)}\right) \right] \end{aligned}$$
(141)
$$\begin{aligned} \beta _{n}^{(2)}= & {} \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \Bigg [\!-\tanh \xi \!\left( I_{41}^{(1)} - I_{42}^{(1)}\right) \nonumber \\&+\bigg (\frac{n_0 \tanh \xi }{\varepsilon _0 + P_0} \bigg )\!\! \left( I_{51}^{(1)} - I_{52}^{(1)}\right) - \frac{I^{(1)}_{41}\, \tanh \xi }{\left( m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}\right) }\!\nonumber \\&\left( \!I_{31}^{(0)} - \frac{n_0 I_{41}^{(0)}}{\varepsilon _0 + P_0} \!\right) \!\! \Bigg ] \end{aligned}$$
(142)
$$\begin{aligned} \beta _{n}^{(3)}= & {} \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left[ - \tanh \xi \, I_{42}^{(1)} + \bigg (\frac{n_0 \tanh \xi }{\varepsilon _0 + P_0}\bigg ) I_{52}^{(1)}\right] \end{aligned}$$
(143)
$$\begin{aligned} \beta _{n}^{(4)}= & {} \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left[ \frac{I^{(1)}_{41}\, \tanh \xi }{\left( m^2 I_{10}^{(0)}-2 I_{31}^{(0)}\right) }\left( I_{31}^{(0)}-\frac{n_0 I_{41}^{(0)}}{\varepsilon _0+P_0}\right) \right] \nonumber \\ \end{aligned}$$
(144)
$$\begin{aligned} \beta _{n}^{(5)}= & {} \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \Bigg [ - \tanh \xi \, I_{42}^{(1)} + \bigg (\frac{n_0 \tanh \xi }{\varepsilon _0 + P_0} \bigg ) I_{52}^{(1)}\nonumber \\&- \frac{I_{41}^{(1)}\, \tanh \xi }{m^2\, I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \left( I_{31}^{(0)} - \frac{n_0 I_{41}^{(0)}}{\varepsilon _0 + P_0} \right) \Bigg ] \end{aligned}$$
(145)
$$\begin{aligned} \beta _{n}^{(6)}= & {} \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \Bigg [ \frac{I_{41}^{(1)}\, \tanh \xi }{m^2\, I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) } \left( I_{31}^{(0)} - \frac{n_0 I_{41}^{(0)}}{\varepsilon _0 + P_0} \right) \Bigg ] \nonumber \\ \end{aligned}$$
(146)
$$\begin{aligned} \beta _{\varSigma }^{(1)}= & {} - \frac{4 {\mathfrak {s}}^2}{3} \cosh \xi \, I_{21}^{(1)} \end{aligned}$$
(147)
$$\begin{aligned} \beta _{\varSigma }^{(2)}= & {} - \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left( I_{41}^{(1)} + \frac{I_{41}^{(1)}\, I_{31}^{(0)}}{m^2 I_{10}^{(0)}-2 I_{31}^{(0)}}\right) \end{aligned}$$
(148)
$$\begin{aligned} \beta _{\varSigma }^{(3)}= & {} - \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \, I_{42}^{(1)} \end{aligned}$$
(149)
$$\begin{aligned} \beta _{\varSigma }^{(4)}= & {} - \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left( \frac{I_{41}^{(1)}\, I_{31}^{(0)}}{m^2I_{10}^{(0)} - \left( I_{30}^{(0)} + I_{31}^{(0)}\right) }\right) \end{aligned}$$
(150)
$$\begin{aligned} B_{\varSigma }^{(5)}= & {} \frac{8 {\mathfrak {s}}^2}{3\, m^2} \cosh \xi \left( \frac{I_{41}^{(1)}\, I_{31}^{(0)}}{m^2\, I_{10}^{(0)} - 2\, I_{31}^{(0)}}\right) \end{aligned}$$
(151)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bhadury, S., Bhatt, J., Jaiswal, A. et al. New developments in relativistic fluid dynamics with spin. Eur. Phys. J. Spec. Top. 230, 655–672 (2021). https://doi.org/10.1140/epjs/s11734-021-00020-4

Download citation