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The ideal hydrodynamic limit and non-Abelian gauge symmetries

  • Regular Article -Theoretical Physics
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Abstract

We show that the ideal fluid local equilibrium limit, defined as the existence of a flow frame \(u_\mu \) which characterises the direction of both a conserved entropy current and conserved charge currents is incompatible with non-Abelian gauge theory if local color charge density is non-zero. Instead, the equation of state becomes dependent on \(u_\mu \) via modes which are roughly equivalent to ghost modes in the hydrodynamic limit. These modes can be physically imagined as a field of “purcell swimmers” whose “arms and legs” are outstretched in Gauge space. Also, vorticity should couple to the Wilson loop via the chromo-electro-magnetic field tensor, which in local equilibrium is not a “force” but instead represents the polarization tensor of the gluons. We show that because of this coupling vorticity also acquires swirling non-hydrodynamic modes. We then argue that these swirling and swimming non-hydrodynamic modes are the manifestation of gauge redundancy within local equilibrium, and speculate on their role in quark-gluon plasma thermalization.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theory paper, with analytical calculations. There is no data associated with it beyond the equations in the paper.]

Notes

  1. A globally equilibrated color-neutral state in the Grand Canonical ensemble is of course well-defined and understood using lattice techniques. The relation between such a system and the locally equilibrated ideal fluid examined here is the same as the relation between a hydrostatic bath in global equilibrium, and the same bath with sound-waves bouncing around it. Even without Gauge theories the relationship between the two setups can be extraordinarily subtle, something the last section will discuss further.

  2. Equation 29, with both \(T_{\mu \nu },n_\mu \) and \(J_{\mu }\) becoming gauge dependent but the whole exponent gauge-invariant.

References

  1. R. Derradi de Souza, T. Koide, T. Kodama, Prog. Part. Nucl. Phys. 86, 35 (2016). https://doi.org/10.1016/j.ppnp.2015.09.002. arXiv:1506.03863 [nucl-th]

    Article  ADS  Google Scholar 

  2. W. Florkowski, M.P. Heller, M. Spalinski, Rep. Prog. Phys. 81(4), 046001 (2018). https://doi.org/10.1088/1361-6633/aaa091. arXiv:1707.02282 [hep-ph]

    Article  ADS  Google Scholar 

  3. S. Endlich, A. Nicolis, R. Rattazzi, J. Wang, JHEP 1104, 102 (2011). arXiv:1011.6396 [hep-th]

    Article  ADS  Google Scholar 

  4. S. Dubovsky, L. Hui, A. Nicolis, D.T. Son, Phys. Rev. D 85, 085029 (2012). https://doi.org/10.1103/PhysRevD.85.085029. arXiv:1107.0731 [hep-th]

    Article  ADS  Google Scholar 

  5. D. Montenegro, G. Torrieri, Phys. Rev. D 94(6), 065042 (2016). https://doi.org/10.1103/PhysRevD.94.065042. arXiv:1604.05291 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  6. D. Montenegro, L. Tinti, G. Torrieri, Phys. Rev. D 96(5), 056012 (2017). Addendum: [Phys. Rev. D 96, no. 7, 079901 (2017)] https://doi.org/10.1103/PhysRevD.96.079901, https://doi.org/10.1103/PhysRevD.96.056012 arXiv:1701.08263 [hep-th]

  7. D. Montenegro, L. Tinti, G. Torrieri, Phys. Rev. D 96(7), 076016 (2017). https://doi.org/10.1103/PhysRevD.96.076016. arXiv:1703.03079 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  8. D. Montenegro, G. Torrieri, arXiv:1807.02796 [hep-th]

  9. B. Bistrovic, R. Jackiw, H. Li, V.P. Nair, S.Y. Pi, Phys. Rev. D 67, 025013 (2003). https://doi.org/10.1103/PhysRevD.67.025013. arXiv:hep-th/0210143

    Article  ADS  MathSciNet  Google Scholar 

  10. S.K. Wong, Nuovo Cim. A 65, 689 (1970). https://doi.org/10.1007/BF02892134

    Article  ADS  Google Scholar 

  11. U.W. Heinz, Phys. Rev. Lett. 51, 351 (1983). https://doi.org/10.1103/PhysRevLett.51.351

    Article  ADS  Google Scholar 

  12. J.J. Fernandez-Melgarejo, S.J. Rey, P. Surwka, JHEP 1702, 122 (2017). https://doi.org/10.1007/JHEP02(2017)122. arXiv:1605.06080 [hep-th]

    Article  ADS  Google Scholar 

  13. C. Manuel, S. Mrowczynski, Phys. Rev. D 68, 094010 (2003). https://doi.org/10.1103/PhysRevD.68.094010. arXiv:hep-ph/0306209

    Article  ADS  Google Scholar 

  14. C. Manuel, S. Mrowczynski, Phys. Rev. D 70, 094019 (2004). https://doi.org/10.1103/PhysRevD.70.094019. arXiv:hep-ph/0403024

    Article  ADS  Google Scholar 

  15. M. Asakawa, S.A. Bass, B. Muller, Phys. Rev. Lett. 96, 252301 (2006). https://doi.org/10.1103/PhysRevLett.96.252301. arXiv:hep-ph/0603092

    Article  ADS  Google Scholar 

  16. S. Mrowczynski, B. Schenke, M. Strickland, Phys. Rep. 682, 1 (2017). https://doi.org/10.1016/j.physrep.2017.03.003. arXiv:1603.08946 [hep-ph]

    Article  ADS  MathSciNet  Google Scholar 

  17. S. Mrowczynski, E.V. Shuryak, Acta Phys. Polon. B 34, 4241 (2003). arXiv:nucl-th/0208052

    ADS  Google Scholar 

  18. S. Vogel, G. Torrieri, M. Bleicher, Phys. Rev. C 82, 024908 (2010). https://doi.org/10.1103/PhysRevC.82.024908. arXiv:nucl-th/0703031

    Article  ADS  Google Scholar 

  19. V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 765, 193 (2017) https://doi.org/10.1016/j.physletb.2016.12.009. arXiv:1606.06198 [nucl-ex]

    Article  ADS  Google Scholar 

  20. J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, U.A. Wiedemann, Gauge/string duality, Hot QCD and heavy ion collisions (Cambridge University Press, Cambridge, 2014). https://doi.org/10.1017/CBO9781139136747. arXiv:1101.0618 [hep-th]

    Book  MATH  Google Scholar 

  21. D. Bak, A. Karch, L.G. Yaffe, JHEP 0708, 049 (2007). https://doi.org/10.1088/1126-6708/2007/08/049. arXiv:0705.0994 [hep-th]

    Article  ADS  Google Scholar 

  22. H.W. Lin, Chin. J. Phys. 49, 827 (2011). arXiv:1106.1608 [hep-lat]

    Google Scholar 

  23. H. Mntysaari, B. Schenke, Phys. Rev. Lett. 117(5), 052301 (2016). https://doi.org/10.1103/PhysRevLett.117.052301. arXiv:1603.04349 [hep-ph]

    Article  ADS  Google Scholar 

  24. J.L. Albacete, P. Guerrero-Rodrguez, C. Marquet, JHEP 1901, 073 (2019). https://doi.org/10.1007/JHEP01(2019)073. arXiv:1808.00795 [hep-ph]

    Article  ADS  Google Scholar 

  25. S. Banerjee, A. Bhattacharyya, S.K. Ghosh, E.M. Ilgenfritz, S. Raha, B. Sinha, E. Takasugi, H. Toki, Nucl. Phys. A 774, 769 (2006). https://doi.org/10.1016/j.nuclphysa.2006.06.133. arXiv:hep-ph/0602202

    Article  ADS  Google Scholar 

  26. D. Miskowiec, PoS CPOD 07, 020 (2007). https://doi.org/10.22323/1.047.0020. arXiv:0707.0923 [hep-ph]

    Article  Google Scholar 

  27. A.D. Linde, Phys. Lett. 96B, 289 (1980). https://doi.org/10.1016/0370-2693(80)90769-8

    Article  ADS  Google Scholar 

  28. P.B. Arnold, Int. J. Mod. Phys. E 16, 2555 (2007). https://doi.org/10.1142/S021830130700832X. arXiv:0708.0812 [hep-ph]

    Article  ADS  Google Scholar 

  29. R. Alkofer, L. von Smekal, Phys. Rept. 353, 281 (2001). https://doi.org/10.1016/S0370-1573(01)00010-2. arXiv:hep-ph/0007355

    Article  ADS  Google Scholar 

  30. B. Muller, AIP Conf. Proc. 482(1), 1 (1999). https://doi.org/10.1063/1.59571. arXiv:nucl-th/9902065

    Article  ADS  Google Scholar 

  31. F. Becattini, L. Tinti, Ann. Phys. 325, 1566 (2010). https://doi.org/10.1016/j.aop.2010.03.007. arXiv:0911.0864 [gr-qc]

    Article  ADS  Google Scholar 

  32. S.J. Van Enk, G. Nienhuis, J. Modern Opt. 41(5), 963–977 (1994). https://doi.org/10.1080/09500349414550911

    Article  ADS  Google Scholar 

  33. X.S. Chen, X.F. Lu, W.M. Sun, F. Wang, T. Goldman, Phys. Rev. Lett. 100, 232002 (2008). https://doi.org/10.1103/PhysRevLett.100.232002. arXiv:0806.3166 [hep-ph]

    Article  ADS  Google Scholar 

  34. F.W. Hehl, Rept. Math. Phys. 9, 55 (1976). https://doi.org/10.1016/0034-4877(76)90016-1

    Article  ADS  Google Scholar 

  35. W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, E. Speranza, arXiv:1807.04946 [nucl-th]

  36. for a worked out example, see J.D.Jackson, “classical electrodynamics”, third edition, Chapter 12.10 Of course there the gauge theory is electromagnetism, where photons never reach local equilibrium, so local equilibration in relation to the regularisation of \(T_{\mu \nu }\) is not discussed

  37. T. Brauner, arXiv:1910.12224 [hep-th]

  38. S. Elitzur, Phys. Rev. D 12, 3978 (1975). https://doi.org/10.1103/PhysRevD.12.3978

    Article  ADS  Google Scholar 

  39. J. Kapusta, C. Gale, thermal field theory

  40. G. Torrieri, Phys. Rev. D 85, 065006 (2012). https://doi.org/10.1103/PhysRevD.85.065006. [arXiv:1112.4086 [hep-th]]

    Article  ADS  Google Scholar 

  41. R.F. Sobreiro, S.P. Sorella, arXiv:hep-th/0504095

  42. M.A.L. Capri, M.S. Guimaraes, I.F. Justo, L.F. Palhares, S.P. Sorella, Phys. Lett. B 735, 277 (2014). https://doi.org/10.1016/j.physletb.2014.06.035. arXiv:1404.7163 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  43. R.D. Pisarski, Phys. Rev. D 74, 121703 (2006). https://doi.org/10.1103/PhysRevD.74.121703. arXiv:hep-ph/0608242

    Article  ADS  MathSciNet  Google Scholar 

  44. J. Liao, E.V. Shuryak, Nucl. Phys. A 775, 224 (2006). https://doi.org/10.1016/j.nuclphysa.2006.06.169. arXiv:hep-ph/0508035

    Article  ADS  Google Scholar 

  45. A. Nicolis, F. Piazza, Phys. Rev. Lett. 110(1), 011602 (2013) Addendum: [Phys. Rev. Lett. 110, 039901 (2013)] https://doi.org/10.1103/PhysRevLett.110.011602, https://doi.org/10.1103/PhysRevLett.110.039901 arXiv:1204.1570 [hep-th]

  46. H. Watanabe, T. Brauner, H. Murayama, Phys. Rev. Lett. 111(2), 021601 (2013). https://doi.org/10.1103/PhysRevLett.111.021601. arXiv:1303.1527 [hep-th]

    Article  ADS  Google Scholar 

  47. M. Buballa, I.A. Shovkovy, Phys. Rev. D 72, 097501 (2005). https://doi.org/10.1103/PhysRevD.72.097501

    Article  ADS  Google Scholar 

  48. D. Blaschke, D. Gomez Dumm, A.G. Grunfeld, N.N. Scoccola, arXiv:hep-ph/0507271

  49. H. Abuki, T. Brauner, Phys. Rev. D 85, 116004 (2012). https://doi.org/10.1103/PhysRevD.85.116004. arXiv:1203.1705 [hep-ph]

    Article  ADS  Google Scholar 

  50. X.G. Huang, A. Sedrakian, Phys. Rev. D 82, 045029 (2010). https://doi.org/10.1103/PhysRevD.82.045029. arXiv:1006.0147 [nucl-th]

    Article  ADS  Google Scholar 

  51. A.W. Steiner, S. Reddy, M. Prakash, Phys. Rev. D 66, 094007 (2002). https://doi.org/10.1103/PhysRevD.66.094007. arXiv:hep-ph/0205201

    Article  ADS  Google Scholar 

  52. P.D. Powell, G. Baym, Phys. Rev. D 88(1), 014012 (2013). https://doi.org/10.1103/PhysRevD.88.014012. arXiv:1302.0416 [hep-ph]

    Article  ADS  Google Scholar 

  53. A. Shapere, F. Wilczek, Self-propulsion at low reynolds number. Phys. Rev. Lett. 58, 2051–2054 (1987)

    Article  ADS  Google Scholar 

  54. J. Avron, O. Raz, New J. Phys. 10, 063016 (2008). 0712.2047

    Article  ADS  Google Scholar 

  55. R. Montgomery, “Gauge theory of the falling cat”, Dynamics and control of mechanical systems, pp, 218 (1993)

  56. D.E. Kharzeev, J. Liao, S.A. Voloshin, G. Wang, Prog. Part. Nucl. Phys. 88, 1 (2016). https://doi.org/10.1016/j.ppnp.2016.01.001. arXiv:1511.04050 [hep-ph]

    Article  ADS  Google Scholar 

  57. Camillo De Lellis, Laszlo Szekelyhidi Jr., Arch. Ration. Mech. Anal. 195(1), 225–260 (2010). 0712.3288

    Article  MathSciNet  Google Scholar 

  58. T. Burch, G. Torrieri, Phys. Rev. D 92(1), 016009 (2015). https://doi.org/10.1103/PhysRevD.92.016009. arXiv:1502.05421 [hep-lat]

    Article  ADS  Google Scholar 

  59. T. Nishioka, Rev. Mod. Phys. 90(3), 035007 (2018). https://doi.org/10.1103/RevModPhys.90.035007. arXiv:1801.10352 [hep-th]

    Article  ADS  Google Scholar 

  60. D.N. Zubarev, V.G. Morozov, Statistical mechanics of nonlinear hydrodynamics fluctuations. Phys. A 120(3), 411 (1983)

    Article  MathSciNet  Google Scholar 

  61. F. Becattini, L. Bucciantini, E. Grossi, L. Tinti, Eur. Phys. J. C 75(5), 191 (2015). https://doi.org/10.1140/epjc/s10052-015-3384-y. arXiv:1403.6265 [hep-th]

    Article  ADS  Google Scholar 

  62. Y. Kats, P. Petrov, JHEP 0901, 044 (2009). https://doi.org/10.1088/1126-6708/2009/01/044. arXiv:0712.0743 [hep-th]

    Article  ADS  Google Scholar 

  63. L. Fabbri, Phys. Lett. B 707, 415 (2012). https://doi.org/10.1016/j.physletb.2012.01.008. arXiv:1101.1761 [gr-qc]. And discussion with the author

    Article  ADS  MathSciNet  Google Scholar 

  64. W. Li, Nucl. Phys. A 967, 59 (2017). https://doi.org/10.1016/j.nuclphysa.2017.05.011. arXiv:1704.03576 [nucl-ex]

    Article  ADS  Google Scholar 

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Acknowledgements

GT acknowledges support from FAPESP proc. 2017/06508-7, partecipation in FAPESP tematico 2017/05685-2 and CNPQ bolsa de produtividade 301996/2014-8. DM was supported by CNPQ graduate fellowship n. 147435/2014-5. This work is a part of the project INCT-FNA Proc. No. 464898/2014-5. We wish to thank David Montenegro, Radoslaw Ryblewski, Saso Grozdanov, Luca Fabbri and Henrique Sa Earp for fruitful discussions.

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    Giorgio Torrieri

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Torrieri, G. The ideal hydrodynamic limit and non-Abelian gauge symmetries. Eur. Phys. J. A 56, 121 (2020). https://doi.org/10.1140/epja/s10050-020-00121-z

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