Abstract
We present a systematic calculation of the corrections of the stress-energy tensor and currents of the free boson and Dirac fields up to second order in thermal vorticity, which is relevant for relativistic hydrodynamics. These corrections are non-dissipative because they survive at general thermodynamic equilibrium with non vanishing mean values of the conserved generators of the Lorentz group, i.e. angular momenta and boosts. Their equilibrium nature makes it possible to express the relevant coefficients by means of correlators of the angular-momentum and boost operators with stress-energy tensor and current, thus making simpler to determine their so-called “Kubo formulae”. We show that, at least for free fields, the corrections are of quantum origin and we study several limiting cases and compare our results with previous calculations. We find that the axial current of the free Dirac field receives corrections proportional to the vorticity independently of the anomalous term.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
18 July 2018
In section 3, the partition function is included in the definition of the statistical operator.
References
R. Baier, P. Romatschke and U.A. Wiedemann, Dissipative hydrodynamics and heavy ion collisions, Phys. Rev. C 73 (2006) 064903 [hep-ph/0602249] [INSPIRE].
P. Huovinen and P.V. Ruuskanen, Hydrodynamic Models for Heavy Ion Collisions, Ann. Rev. Nucl. Part. Sci. 56 (2006) 163 [nucl-th/0605008] [INSPIRE].
W. Florkowksi, Phenomenology of Ultra-Relativistic heavy ion collisions, World Scientific, Singapore (2010).
C. Gale, S. Jeon and B. Schenke, Hydrodynamic Modeling of Heavy-Ion Collisions, Int. J. Mod. Phys. A 28 (2013) 1340011 [arXiv:1301.5893] [INSPIRE].
U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Ann. Rev. Nucl. Part. Sci. 63 (2013) 123 [arXiv:1301.2826] [INSPIRE].
R. Derradi de Souza, T. Koide and T. Kodama, Hydrodynamic Approaches in Relativistic Heavy Ion Reactions, Prog. Part. Nucl. Phys. 86 (2016) 35 [arXiv:1506.03863] [INSPIRE].
A. Muronga, Relativistic Dynamics of Non-ideal Fluids: Viscous and heat-conducting fluids. II. Transport properties and microscopic description of relativistic nuclear matter, Phys. Rev. C 76 (2007) 014910 [nucl-th/0611091] [INSPIRE].
K. Tsumura, T. Kunihiro and K. Ohnishi, Derivation of covariant dissipative fluid dynamics in the renormalization-group method, Phys. Lett. B 656 (2007) 274 [Erratum ibid. B 656 (2007) 274] [hep-ph/0609056] [INSPIRE].
B. Betz, D. Henkel and D.H. Rischke, From kinetic theory to dissipative fluid dynamics, Prog. Part. Nucl. Phys. 62 (2009) 556 [arXiv:0812.1440] [INSPIRE].
M.A. York and G.D. Moore, Second order hydrodynamic coefficients from kinetic theory, Phys. Rev. D 79 (2009) 054011 [arXiv:0811.0729] [INSPIRE].
A. Monnai and T. Hirano, Relativistic Dissipative Hydrodynamic Equations at the Second Order for Multi-Component Systems with Multiple Conserved Currents, Nucl. Phys. A 847 (2010) 283 [arXiv:1003.3087] [INSPIRE].
G.S. Denicol, H. Niemi, E. Molnar and D.H. Rischke, Derivation of transient relativistic fluid dynamics from the Boltzmann equation, Phys. Rev. D 85 (2012) 114047 [Erratum ibid. D 91 (2015) 039902] [arXiv:1202.4551] [INSPIRE].
P. Van and T.S. Biro, First order and stable relativistic dissipative hydrodynamics, Phys. Lett. B 709 (2012) 106 [arXiv:1109.0985] [INSPIRE].
A. Jaiswal, R.S. Bhalerao and S. Pal, New relativistic dissipative fluid dynamics from kinetic theory, Phys. Lett. B 720 (2013) 347 [arXiv:1204.3779] [INSPIRE].
R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
M. Natsuume and T. Okamura, Causal hydrodynamics of gauge theory plasmas from AdS/CFT duality, Phys. Rev. D 77 (2008) 066014 [Erratum ibid. D 78 (2008) 089902] [arXiv:0712.2916] [INSPIRE].
V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, arXiv:1107.5780 [INSPIRE].
T. Koide, G.S. Denicol, P. Mota and T. Kodama, Relativistic dissipative hydrodynamics: A Minimal causal theory, Phys. Rev. C 75 (2007) 034909 [hep-ph/0609117] [INSPIRE].
M. Fukuma and Y. Sakatani, Relativistic viscoelastic fluid mechanics, Phys. Rev. E 84 (2011) 026316 [arXiv:1104.1416] [INSPIRE].
T. Koide and T. Kodama, Transport Coefficients of Non-Newtonian Fluid and Causal Dissipative Hydrodynamics, Phys. Rev. E 78 (2008) 051107 [arXiv:0806.3725] [INSPIRE].
Y. Minami and Y. Hidaka, Relativistic hydrodynamics from the projection operator method, Phys. Rev. E 87 (2013) 023007 [arXiv:1210.1313] [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].
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].
P. Romatschke, Do nuclear collisions create a locally equilibrated quark-gluon plasma?, Eur. Phys. J. C 77 (2017) 21 [arXiv:1609.02820] [INSPIRE].
P. Romatschke, Relativistic Viscous Fluid Dynamics and Non-Equilibrium Entropy, Class. Quant. Grav. 27 (2010) 025006 [arXiv:0906.4787] [INSPIRE].
G.D. Moore and K.A. Sohrabi, Kubo Formulae for Second-Order Hydrodynamic Coefficients, Phys. Rev. Lett. 106 (2011) 122302 [arXiv:1007.5333] [INSPIRE].
G.D. Moore and K.A. Sohrabi, Thermodynamical second-order hydrodynamic coefficients, JHEP 11 (2012) 148 [arXiv:1210.3340] [INSPIRE].
F. Becattini and E. Grossi, Quantum corrections to the stress-energy tensor in thermodynamic equilibrium with acceleration, Phys. Rev. D 92 (2015) 045037 [arXiv:1505.07760] [INSPIRE].
R. Panerai, Global equilibrium and local thermodynamics in stationary spacetimes, Phys. Rev. D 93 (2016) 104021.
L. Landau and L. Lifshitz, Statistical Physics, Pergamon Press, Oxford U.K. (1980).
A. Vilenkin, Quantum field theory at finite temperature in a rotating system, Phys. Rev. D 21 (1980) 2260 [INSPIRE].
STAR collaboration, L. Adamczyk et al., Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature 548 (2017) 62 [arXiv:1701.06657] [INSPIRE].
S. Bhattacharyya, Constraints on the second order transport coefficients of an uncharged fluid, JHEP 07 (2012) 104 [arXiv:1201.4654] [INSPIRE].
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.
Ch.G. Van Weert, Maximum entropy principle and relativistic hydrodynamics, Annals Phys. 140 (1982) 133.
H.A. Weldon, Covariant Calculations at Finite Temperature: The Relativistic Plasma, Phys. Rev. D 26 (1982) 1394 [INSPIRE].
F. Becattini, Covariant statistical mechanics and the stress-energy tensor, Phys. Rev. Lett. 108 (2012) 244502 [arXiv:1201.5278] [INSPIRE].
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].
M. Hongo, Path-integral formula for local thermal equilibrium, Annals Phys. 383 (2017) 1 [arXiv:1611.07074] [INSPIRE].
P. Ván and T.S. Biró, Dissipation flow-frames: particle, energy, thermometer, in Proceedings of the 12th Joint European Thermodynamics Conference, Brescia Italy (2013), pg. 546 [arXiv:1305.3190].
V.E. Ambrus and E. Winstanley, Rotating fermions inside a cylindrical boundary, Phys. Rev. D 93 (2016) 104014 [arXiv:1512.05239] [INSPIRE].
M.N. Chernodub and S. Gongyo, Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics, JHEP 01 (2017) 136 [arXiv:1611.02598] [INSPIRE].
C.G. Callan Jr., S.R. Coleman and R. Jackiw, A New improved energy-momentum tensor, Annals Phys. 59 (1970) 42 [INSPIRE].
J.I. Kapusta and C. Gale, Finite-temperature field theory: Principles and applications, Cambridge University Press, Cambridge U.K. (2006).
M. Laine and A. Vuorinen, Basics of Thermal Field Theory, Lect. Notes Phys. 925 (2016) 1 [arXiv:1701.01554].
N.P. Landsman and C.G. van Weert, Real and Imaginary Time Field Theory at Finite Temperature and Density, Phys. Rept. 145 (1987) 141 [INSPIRE].
F. Becattini and F. Piccinini, The Ideal relativistic spinning gas: Polarization and spectra, Annals Phys. 323 (2008) 2452 [arXiv:0710.5694] [INSPIRE].
D.E. Kharzeev, J. Liao, S.A. Voloshin and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions — A status report, Prog. Part. Nucl. Phys. 88 (2016) 1 [arXiv:1511.04050] [INSPIRE].
T. Kalaydzhyan, Temperature dependence of the chiral vortical effects, Phys. Rev. D 89 (2014) 105012 [arXiv:1403.1256] [INSPIRE].
A. Vilenkin, Macroscopic parity violating effects: neutrino fluxes from rotating black holes and in rotating thermal radiation, Phys. Rev. D 20 (1979) 1807 [INSPIRE].
D.T. Son and A.R. Zhitnitsky, Quantum anomalies in dense matter, Phys. Rev. D 70 (2004) 074018 [hep-ph/0405216] [INSPIRE].
D.T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].
A.V. Sadofyev and M.V. Isachenkov, The Chiral magnetic effect in hydrodynamical approach, Phys. Lett. B 697 (2011) 404 [arXiv:1010.1550] [INSPIRE].
M. Torabian and H.-U. Yee, Holographic nonlinear hydrodynamics from AdS/CFT with multiple/non-Abelian symmetries, JHEP 08 (2009) 020 [arXiv:0903.4894] [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].
I. Amado, K. Landsteiner and F. Pena-Benitez, Anomalous transport coefficients from Kubo formulas in Holography, JHEP 05 (2011) 081 [arXiv:1102.4577] [INSPIRE].
Y. Neiman and Y. Oz, Relativistic Hydrodynamics with General Anomalous Charges, JHEP 03 (2011) 023 [arXiv:1011.5107] [INSPIRE].
K. Landsteiner, E. Megias, L. Melgar and F. Pena-Benitez, Holographic Gravitational Anomaly and Chiral Vortical Effect, JHEP 09 (2011) 121 [arXiv:1107.0368] [INSPIRE].
J.-H. Gao, Z.-T. Liang, S. Pu, Q. Wang and X.-N. Wang, Chiral Anomaly and Local Polarization Effect from Quantum Kinetic Approach, Phys. Rev. Lett. 109 (2012) 232301 [arXiv:1203.0725] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].
K. Landsteiner, E. Megias and F. Pena-Benitez, Gravitational Anomaly and Transport, Phys. Rev. Lett. 107 (2011) 021601 [arXiv:1103.5006] [INSPIRE].
A. Avkhadiev and A.V. Sadofyev, Chiral Vortical Effect for Bosons, Phys. Rev. D 96 (2017) 045015 [arXiv:1702.07340] [INSPIRE].
S. Golkar and S. Sethi, Global Anomalies and Effective Field Theory, JHEP 05 (2016) 105 [arXiv:1512.02607] [INSPIRE].
S. Golkar and D.T. Son, (Non)-renormalization of the chiral vortical effect coefficient, JHEP 02 (2015) 169 [arXiv:1207.5806] [INSPIRE].
K. Jensen, P. Kovtun and A. Ritz, Chiral conductivities and effective field theory, JHEP 10 (2013) 186 [arXiv:1307.3234] [INSPIRE].
V. Braguta, M.N. Chernodub, V.A. Goy, K. Landsteiner, A.V. Molochkov and M.I. Polikarpov, Temperature dependence of the axial magnetic effect in two-color quenched QCD, Phys. Rev. D 89 (2014) 074510 [arXiv:1401.8095] [INSPIRE].
F. Becattini, V. Chandra, L. Del Zanna and E. Grossi, Relativistic distribution function for particles with spin at local thermodynamical equilibrium, Annals Phys. 338 (2013) 32 [arXiv:1303.3431].
A. Flachi and K. Fukushima, Chiral vortical effect in curved space and the Chern-Simons current, arXiv:1702.04753 [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series, Dover Publications, Mineola U.S.A. (1964).
E. Megias and M. Valle, Second-order partition function of a non-interacting chiral fluid in 3+1 dimensions, JHEP 11 (2014) 005 [arXiv:1408.0165] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, A Theory of first order dissipative superfluid dynamics, JHEP 05 (2014) 147 [arXiv:1105.3733] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1704.02808
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Buzzegoli, M., Grossi, E. & Becattini, F. General equilibrium second-order hydrodynamic coefficients for free quantum fields. J. High Energ. Phys. 2017, 91 (2017). https://doi.org/10.1007/JHEP10(2017)091
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)091