Abstract
Using the second law of local thermodynamics and the first-order Palatini formalism, we formulate relativistic spin hydrodynamics for quantum field theories with Dirac fermions, such as QED and QCD, in a torsionful curved background. We work in a regime where spin density, which is assumed to relax much slower than other non-hydrodynamic modes, is treated as an independent degree of freedom in an extended hydrodynamic description. Spin hydrodynamics in our approach contains only three non-hydrodynamic modes corresponding to a spin vector, whose relaxation time is controlled by a new transport coefficient: the rotational viscosity. We study linear response theory and observe an interesting mode mixing phenomenon between the transverse shear and the spin density modes. We propose several field-theoretical ways to compute the spin relaxation time and the rotational viscosity, via the Green-Kubo formula based on retarded correlation functions.
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References
N. Isgur and M.B. Wise, Weak Decays of Heavy Mesons in the Static Quark Approximation, Phys. Lett. B 232 (1989) 113 [INSPIRE].
N. Isgur and M.B. Wise, Weak transition form-factors between heavy mesons, Phys. Lett. B 237 (1990) 527 [INSPIRE].
N. Isgur and M.B. Wise, Spectroscopy with heavy quark symmetry, Phys. Rev. Lett. 66 (1991) 1130 [INSPIRE].
M. Neubert, Heavy quark symmetry, Phys. Rept. 245 (1994) 259 [hep-ph/9306320] [INSPIRE].
A.V. Manohar and M.B. Wise, Heavy Quark Physics, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cambridge University Press, Cambridge U.K. (2000).
S. Maekawa, S.O. Valenzuela, T. Kimura and E. Saitoh, Series on Semiconductor Science and Technology. Vol. 22: Spin current, Oxford University Press, Oxford U.K. (2017).
STAR collaboration, Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature 548 (2017) 62 [arXiv:1701.06657] [INSPIRE].
STAR collaboration, Polarization of Λ (\( \overline{\Lambda} \)) hyperons along the beam direction in Au+Au collisions at \( \sqrt{s_{NN}} \) = 200 GeV, Phys. Rev. Lett. 123 (2019) 132301 [arXiv:1905.11917] [INSPIRE].
ALICE collaboration, Evidence of Spin-Orbital Angular Momentum Interactions in Relativistic Heavy-Ion Collisions, Phys. Rev. Lett. 125 (2020) 012301 [arXiv:1910.14408] [INSPIRE].
STAR collaboration, Global Polarization of Ξ and Ω Hyperons in Au+Au Collisions at \( \sqrt{s_{NN}} \) = 200 GeV, Phys. Rev. Lett. 126 (2021) 162301 [arXiv:2012.13601] [INSPIRE].
K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo and H. Taya, Fate of spin polarization in a relativistic fluid: An entropy-current analysis, Phys. Lett. B 795 (2019) 100 [arXiv:1901.06615] [INSPIRE].
K. Fukushima and S. Pu, Spin hydrodynamics and symmetric energy-momentum tensors – A current induced by the spin vorticity –, Phys. Lett. B 817 (2021) 136346 [arXiv:2010.01608] [INSPIRE].
S. Li, M.A. Stephanov and H.-U. Yee, Nondissipative Second-Order Transport, Spin, and Pseudogauge Transformations in Hydrodynamics, Phys. Rev. Lett. 127 (2021) 082302 [arXiv:2011.12318] [INSPIRE].
D. She, A. Huang, D. Hou and J. Liao, Relativistic Viscous Hydrodynamics with Angular Momentum, arXiv:2105.04060 [INSPIRE].
A.D. Gallegos, U. Gürsoy and A. Yarom, Hydrodynamics of spin currents, SciPost Phys. 11 (2021) 041 [arXiv:2101.04759] [INSPIRE].
W. Florkowski, B. Friman, A. Jaiswal and E. Speranza, Relativistic fluid dynamics with spin, Phys. Rev. C 97 (2018) 041901 [arXiv:1705.00587] [INSPIRE].
J.-H. Gao and Z.-T. Liang, Relativistic Quantum Kinetic Theory for Massive Fermions and Spin Effects, Phys. Rev. D 100 (2019) 056021 [arXiv:1902.06510] [INSPIRE].
K. Hattori, Y. Hidaka and D.-L. Yang, Axial Kinetic Theory and Spin Transport for Fermions with Arbitrary Mass, Phys. Rev. D 100 (2019) 096011 [arXiv:1903.01653] [INSPIRE].
S. Li and H.-U. Yee, Quantum Kinetic Theory of Spin Polarization of Massive Quarks in Perturbative QCD: Leading Log, Phys. Rev. D 100 (2019) 056022 [arXiv:1905.10463] [INSPIRE].
D.-L. Yang, K. Hattori and Y. Hidaka, Effective quantum kinetic theory for spin transport of fermions with collsional effects, JHEP 07 (2020) 070 [arXiv:2002.02612] [INSPIRE].
N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang and D.H. Rischke, Generating Spin Polarization from Vorticity through Nonlocal Collisions, Phys. Rev. Lett. 127 (2021) 052301 [arXiv:2005.01506] [INSPIRE].
Y.-C. Liu, K. Mameda and X.-G. Huang, Covariant Spin Kinetic Theory I: Collisionless Limit, Chin. Phys. C 44 (2020) 094101 [Erratum ibid. 45 (2021) 089001] [arXiv:2002.03753] [INSPIRE].
S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar and R. Ryblewski, Relativistic dissipative spin dynamics in the relaxation time approximation, Phys. Lett. B 814 (2021) 136096 [arXiv:2002.03937] [INSPIRE].
S. Shi, C. Gale and S. Jeon, From chiral kinetic theory to relativistic viscous spin hydrodynamics, Phys. Rev. C 103 (2021) 044906 [arXiv:2008.08618] [INSPIRE].
H.-H. Peng, J.-J. Zhang, X.-L. Sheng and Q. Wang, Ideal spin hydrodynamics from Wigner function approach, arXiv:2107.00448 [INSPIRE].
K. Hashimoto, N. Iizuka and T. Kimura, Towards Holographic Spintronics, Phys. Rev. D 91 (2015) 086003 [arXiv:1304.3126] [INSPIRE].
M. Garbiso and M. Kaminski, Hydrodynamics of simply spinning black holes & hydrodynamics for spinning quantum fluids, JHEP 12 (2020) 112 [arXiv:2007.04345] [INSPIRE].
A.D. Gallegos and U. Gürsoy, Holographic spin liquids and Lovelock Chern-Simons gravity, JHEP 11 (2020) 151 [arXiv:2004.05148] [INSPIRE].
D. Montenegro, L. Tinti and G. Torrieri, Ideal relativistic fluid limit for a medium with polarization, Phys. Rev. D 96 (2017) 056012 [Addendum ibid. 96 (2017) 079901] [arXiv:1701.08263] [INSPIRE].
D. Montenegro, L. Tinti and G. Torrieri, Sound waves and vortices in a polarized relativistic fluid, Phys. Rev. D 96 (2017) 076016 [arXiv:1703.03079] [INSPIRE].
D. Montenegro and G. Torrieri, Causality and dissipation in relativistic polarizable fluids, Phys. Rev. D 100 (2019) 056011 [arXiv:1807.02796] [INSPIRE].
D. Montenegro and G. Torrieri, Linear response theory and effective action of relativistic hydrodynamics with spin, Phys. Rev. D 102 (2020) 036007 [arXiv:2004.10195] [INSPIRE].
F. Becattini and F. Piccinini, The Ideal relativistic spinning gas: Polarization and spectra, Annals Phys. 323 (2008) 2452 [arXiv:0710.5694] [INSPIRE].
F. Becattini and L. Tinti, The Ideal relativistic rotating gas as a perfect fluid with spin, Annals Phys. 325 (2010) 1566 [arXiv:0911.0864] [INSPIRE].
F. Becattini and L. Tinti, Nonequilibrium Thermodynamical Inequivalence of Quantum Stress-energy and Spin Tensors, Phys. Rev. D 87 (2013) 025029 [arXiv:1209.6212] [INSPIRE].
F. Becattini, W. Florkowski and E. Speranza, Spin tensor and its role in non-equilibrium thermodynamics, Phys. Lett. B 789 (2019) 419 [arXiv:1807.10994] [INSPIRE].
J. Hu, Kubo formulae for first-order spin hydrodynamics, Phys. Rev. D 103 (2021) 116015 [arXiv:2101.08440] [INSPIRE].
W. Florkowski, A. Kumar and R. Ryblewski, Relativistic hydrodynamics for spin-polarized fluids, Prog. Part. Nucl. Phys. 108 (2019) 103709 [arXiv:1811.04409] [INSPIRE].
F. Becattini, Polarization in relativistic fluids: a quantum field theoretical derivation, arXiv:2004.04050 [INSPIRE].
E. Speranza and N. Weickgenannt, Spin tensor and pseudo-gauges: from nuclear collisions to gravitational physics, Eur. Phys. J. A 57 (2021) 155 [arXiv:2007.00138] [INSPIRE].
L.D. Landau and E.M. Lifshitz, Statistical Physics, third edition, Butterworth Heinemann, Oxford U.K. (1980).
J.I. Kapusta, E. Rrapaj and S. Rudaz, Relaxation Time for Strange Quark Spin in Rotating Quark-Gluon Plasma, Phys. Rev. C 101 (2020) 024907 [arXiv:1907.10750] [INSPIRE].
N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang and D.H. Rischke, Derivation of the nonlocal collision term in the relativistic Boltzmann equation for massive spin-1/2 particles from quantum field theory, Phys. Rev. D 104 (2021) 016022 [arXiv:2103.04896] [INSPIRE].
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, second edition, Butterworth Heinemann, Oxford U.K. (1987).
M. Stephanov and Y. Yin, Hydrodynamics with parametric slowing down and fluctuations near the critical point, Phys. Rev. D 98 (2018) 036006 [arXiv:1712.10305] [INSPIRE].
M.S. Green, Markoff random processes and the statistical mechanics of time-dependent phenomena. ii. irreversible processes in fluids, J. Chem. Phys. 22 (1954) 398.
R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12 (1957) 570.
H. Nakano, A method of calculation of electrical conductivity, Prog. Theor. Phys. 15 (1956) 77.
A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Matem. Palermo 43 (1919) 203.
F.W. Hehl, P. Von Der Heyde, G.D. Kerlick and J.M. Nester, General Relativity with Spin and Torsion: Foundations and Prospects, Rev. Mod. Phys. 48 (1976) 393 [INSPIRE].
J.C. Ward, An Identity in Quantum Electrodynamics, Phys. Rev. 78 (1950) 182 [INSPIRE].
Y. Takahashi, On the generalized Ward identity, Nuovo Cim. 6 (1957) 371 [INSPIRE].
E. Noether, Invariant variation problems, Transp. Theory Stat. Phys. 1 (1971) 186.
F. Belinfante, On the spin angular momentum of mesons, Physica 6 (1939) 887.
F. Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields, Physica 7 (1940) 449.
L. Rosenfeld, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields, Mem. Acad. Roy. Belg. Cl. Sc. 18 (1940) 1.
E. Leader and C. Lorcé, The angular momentum controversy: What’s it all about and does it matter?, Phys. Rept. 541 (2014) 163 [arXiv:1309.4235] [INSPIRE].
M. Wakamatsu, Is gauge-invariant complete decomposition of the nucleon spin possible?, Int. J. Mod. Phys. A 29 (2014) 1430012 [arXiv:1402.4193] [INSPIRE].
L.P. Kadanoff and P.C. Martin, Hydrodynamic equations and correlation functions, Annals Phys. 24 (1963) 419.
J.M. Luttinger, Theory of Thermal Transport Coefficients, Phys. Rev. 135 (1964) A1505 [INSPIRE].
P.C. Martin, O. Parodi and P.S. Pershan, Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids, Phys. Rev. A 6 (1972) 2401.
I. Muller, Zum Paradoxon der Warmeleitungstheorie, Z. Phys. 198 (1967) 329 [INSPIRE].
W. Israel, Nonstationary irreversible thermodynamics: A Causal relativistic theory, Annals Phys. 100 (1976) 310 [INSPIRE].
W. Israel and J. Stewart, Thermodynamics of nonstationary and transient effects in a relativistic gas, Phys. Lett. A 58 (1976) 213.
W. Israel and J.M. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals Phys. 118 (1979) 341 [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Transport coefficients in high temperature gauge theories. 1. Leading log results, JHEP 11 (2000) 001 [hep-ph/0010177] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Transport coefficients in high temperature gauge theories. 2. Beyond leading log, JHEP 05 (2003) 051 [hep-ph/0302165] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, The Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].
S. Nakajima, Thermal irreversible processes (in Japanese), Busseironkenkyu 2 (1957) 197.
H. Mori, Statistical-Mechanical Theory of Transport in Fluids, Phys. Rev. 112 (1958) 1829 [INSPIRE].
J.A. McLennan, Statistical mechanics of transport in fluids, Phys. Fluids 3 (1960) 493.
J.A. McLennan, Introduction to Non Equilibrium Statistical Mechanics, Prentice Hall Advanced Reference Series, Prentice Hall, Hoboken U.S.A. (1988).
D.N. Zubarev, A.V. Prozorkevich and S.A. Smolyanskii, Derivation of nonlinear generalized equations of quantum relativistic hydrodynamics, Theor. Math. Phys. 40 (1979) 821.
D.N. Zubarev, V. Morozov and G. Ropke, Statistical Mechanics of Nonequilibrium Processes. Volume 1: Basic Concepts, Kinetic Theory, first edition, Wiley-VCH, Weinheim Germany (1996).
D.N. Zubarev, V. Morozov and G. Ropke, Statistical Mechanics of Nonequilibrium Processes. Volume 2: Relaxation and Hydrodynamic Processes, Wiley-VCH, Weinheim Germany (1997).
K. Kawasaki and J.D. Gunton, Theory of nonlinear transport processes: Nonlinear shear viscosity and normal stress effects, Phys. Rev. A 8 (1973) 2048.
X.-G. Huang, A. Sedrakian and D.H. Rischke, Kubo formulae for relativistic fluids in strong magnetic fields, Annals Phys. 326 (2011) 3075 [arXiv:1108.0602] [INSPIRE].
S.-i. Sasa, Derivation of hydrodynamics from the hamiltonian description of particle systems, Phys. Rev. Lett. 112 (2014) 100602.
F. Becattini, L. Bucciantini, E. Grossi and L. Tinti, Local thermodynamical equilibrium and the beta frame for a quantum relativistic fluid, Eur. Phys. J. C 75 (2015) 191 [arXiv:1403.6265] [INSPIRE].
T. Hayata, Y. Hidaka, T. Noumi and M. Hongo, Relativistic hydrodynamics from quantum field theory on the basis of the generalized Gibbs ensemble method, Phys. Rev. D 92 (2015) 065008 [arXiv:1503.04535] [INSPIRE].
F. Becattini, M. Buzzegoli and E. Grossi, Reworking the Zubarev’s approach to non-equilibrium quantum statistical mechanics, Particles 2 (2019) 197 [arXiv:1902.01089] [INSPIRE].
M. Hongo and K. Hattori, Revisiting relativistic magnetohydrodynamics from quantum electrodynamics, JHEP 02 (2021) 011 [arXiv:2005.10239] [INSPIRE].
M. Hongo, Path-integral formula for local thermal equilibrium, Annals Phys. 383 (2017) 1 [arXiv:1611.07074] [INSPIRE].
K. Fukushima, D.E. Kharzeev and H.J. Warringa, The Chiral Magnetic Effect, Phys. Rev. D 78 (2008) 074033 [arXiv:0808.3382] [INSPIRE].
J. Erdmenger, M. Haack, M. Kaminski and A. Yarom, Fluid dynamics of R-charged black holes, JHEP 01 (2009) 055 [arXiv:0809.2488] [INSPIRE].
N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam and P. Surowka, Hydrodynamics from charged black branes, JHEP 01 (2011) 094 [arXiv:0809.2596] [INSPIRE].
D.T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].
K. Landsteiner, E. Megias and F. Pena-Benitez, Gravitational Anomaly and Transport, Phys. Rev. Lett. 107 (2011) 021601 [arXiv:1103.5006] [INSPIRE].
K. Landsteiner, Notes on Anomaly Induced Transport, Acta Phys. Polon. B 47 (2016) 2617 [arXiv:1610.04413] [INSPIRE].
D.E. Kharzeev and J. Liao, Chiral magnetic effect reveals the topology of gauge fields in heavy-ion collisions, Nature Rev. Phys. 3 (2021) 55 [arXiv:2102.06623] [INSPIRE].
N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, Constraints on Fluid Dynamics from Equilibrium Partition Functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
K. Jensen, Triangle Anomalies, Thermodynamics, and Hydrodynamics, Phys. Rev. D 85 (2012) 125017 [arXiv:1203.3599] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
S. Golkar and S. Sethi, Global Anomalies and Effective Field Theory, JHEP 05 (2016) 105 [arXiv:1512.02607] [INSPIRE].
S.D. Chowdhury and J.R. David, Global gravitational anomalies and transport, JHEP 12 (2016) 116 [arXiv:1604.05003] [INSPIRE].
P. Glorioso, H. Liu and S. Rajagopal, Global Anomalies, Discrete Symmetries, and Hydrodynamic Effective Actions, JHEP 01 (2019) 043 [arXiv:1710.03768] [INSPIRE].
J.L. Mañes, E. Megías, M. Valle and M.A. Vazquez-Mozo, Non-Abelian Anomalous (Super)Fluids in Thermal Equilibrium from Differential Geometry, JHEP 11 (2018) 076 [arXiv:1806.07647] [INSPIRE].
M. Hongo and Y. Hidaka, Anomaly-Induced Transport Phenomena from Imaginary-Time Formalism, Particles 2 (2019) 261 [arXiv:1902.09166] [INSPIRE].
J.L. Mañes, E. Megías, M. Valle and M.A. Vázquez-Mozo, Anomalous Currents and Constitutive Relations of a Chiral Hadronic Superfluid, JHEP 12 (2019) 018 [arXiv:1910.04013] [INSPIRE].
S.R. De Groot, W.A. van Leeuwen and C.G. van Weert, Relativistic Kinetic Theory: Principles and Applications, Elsevier Science Ltd., Amsterdam The Netherlands (1980).
F. Becattini, Covariant statistical mechanics and the stress-energy tensor, Phys. Rev. Lett. 108 (2012) 244502 [arXiv:1201.5278] [INSPIRE].
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Hongo, M., Huang, XG., Kaminski, M. et al. Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation. J. High Energ. Phys. 2021, 150 (2021). https://doi.org/10.1007/JHEP11(2021)150
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DOI: https://doi.org/10.1007/JHEP11(2021)150