Skip to main content
SpringerLink
Log in

QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order μ 2

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We determine the equation of state of QCD for nonzero chemical potentials via a Taylor expansion of the pressure. The results are obtained for N f  = 2 + 1 flavors of quarks with physical masses, on various lattice spacings. We present results for the pressure, interaction measure, energy density, entropy density, and the speed of sound for small chemical potentials. At low temperatures we compare our results with the Hadron Resonance Gas model. We also express our observables along trajectories of constant entropy over particle number. A simple parameterization is given (the Matlab/Octave script parameterization.m, submitted to the arXiv along with the paper), which can be used to reconstruct the observables as functions of T and μ, or as functions of T and S/N.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Teaney, J. Lauret and E.V. Shuryak, Flow at the SPS and RHIC as a quark gluon plasma signature, Phys. Rev. Lett. 86 (2001) 4783 [nucl-th/0011058] [INSPIRE].

    Article  ADS  Google Scholar 

  2. D. Teaney, J. Lauret and E. Shuryak, A Hydrodynamic description of heavy ion collisions at the SPS and RHIC, nucl-th/0110037 [INSPIRE].

  3. P.F. Kolb and U.W. Heinz, Hydrodynamic description of ultrarelativistic heavy ion collisions, nucl-th/0305084 [INSPIRE].

  4. P. Jacobs and X.-N. Wang, Matter in extremis: ultrarelativistic nuclear collisions at RHIC, Prog. Part. Nucl. Phys. 54 (2005) 443 [hep-ph/0405125] [INSPIRE].

    Article  ADS  Google Scholar 

  5. P. Romatschke, New developments in relativistic viscous hydrodynamics, Int. J. Mod. Phys. E 19 (2010) 1 [arXiv:0902.3663] [INSPIRE].

    Article  ADS  Google Scholar 

  6. C. Hung and E.V. Shuryak, Equation of state, radial flow and freezeout in high-energy heavy ion collisions, Phys. Rev. C 57 (1998) 1891 [hep-ph/9709264] [INSPIRE].

    ADS  Google Scholar 

  7. V. Toneev, J. Cleymans, E. Nikonov, K. Redlich and A. Shanenko, Dynamical interpretation of chemical freezeout in heavy ion collisions, J. Phys. G G 27 (2001) 827 [nucl-th/0011029] [INSPIRE].

    Article  ADS  Google Scholar 

  8. M. Bluhm, B. Kampfer, R. Schulze, D. Seipt and U. Heinz, A family of equations of state based on lattice QCD: Impact on flow in ultrarelativistic heavy-ion collisions, Phys. Rev. C 76 (2007) 034901 [arXiv:0705.0397] [INSPIRE].

    ADS  Google Scholar 

  9. Y. Aoki, G. Endrodi, Z. Fodor, S. Katz and K. Szabo, The order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443 (2006) 675 [hep-lat/0611014] [INSPIRE].

    Article  ADS  Google Scholar 

  10. M. Cheng et al., The transition temperature in QCD, Phys. Rev. D 74 (2006) 054507 [hep-lat/0608013] [INSPIRE].

    ADS  Google Scholar 

  11. Y. Aoki, Z. Fodor, S. Katz and K. Szabo, The QCD transition temperature: results with physical masses in the continuum limit, Phys. Lett. B 643 (2006) 46 [hep-lat/0609068] [INSPIRE].

    Article  ADS  Google Scholar 

  12. A. Bazavov et al., The chiral and deconfinement aspects of the QCD transition, Phys. Rev. D 85 (2012) 054503 [arXiv:1111.1710] [INSPIRE].

    ADS  Google Scholar 

  13. Y. Aoki et al., The QCD transition temperature: results with physical masses in the continuum limit II, JHEP 06 (2009) 088 [arXiv:0903.4155] [INSPIRE].

    Article  ADS  Google Scholar 

  14. Wuppertal-Budapest collaboration, S. Borsányi et al., Is there still any T c mystery in lattice QCD? Results with physical masses in the continuum limit III, JHEP 09 (2010) 073 [arXiv:1005.3508] [INSPIRE].

    Article  ADS  Google Scholar 

  15. HotQCD collaboration, A. Bazavov and P. Petreczky, Deconfinement and chiral transition with the Highly Improved Staggered Quark (HISQ) action, J. Phys. Conf. Ser. 230 (2010) 012014 [arXiv:1005.1131] [INSPIRE].

    Article  ADS  Google Scholar 

  16. S. Ejiri, F. Karsch, E. Laermann and C. Schmidt, The isentropic equation of state of 2-flavor QCD, Phys. Rev. D 73 (2006) 054506 [hep-lat/0512040] [INSPIRE].

    ADS  Google Scholar 

  17. C. Bernard et al., QCD equation of state with 2 + 1 flavors of improved staggered quarks, Phys. Rev. D 75 (2007) 094505 [hep-lat/0611031] [INSPIRE].

    ADS  Google Scholar 

  18. M. Cheng et al., Equation of state for physical quark masses, Phys. Rev. D 81 (2010) 054504 [arXiv:0911.2215] [INSPIRE].

    ADS  Google Scholar 

  19. S. Borsányi et al., The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [arXiv:1007.2580] [INSPIRE].

    Article  ADS  Google Scholar 

  20. M.P. Lombardo, High temperature QCD, talk given at Lattice 2012, June 24–29, Cairns, Australia (2012).

  21. S. Borsányi et al., The QCD equation of state and the effects of the charm, PoS(LATTICE2011)201 [arXiv:1204.0995] [INSPIRE].

  22. Z. Fodor and S. Katz, A new method to study lattice QCD at finite temperature and chemical potential, Phys. Lett. B 534 (2002) 87 [hep-lat/0104001] [INSPIRE].

    Article  ADS  Google Scholar 

  23. Z. Fodor and S. Katz, The phase diagram of quantum chromodynamics, arXiv:0908.3341 [INSPIRE].

  24. F. Csikor et al., Equation of state at finite temperature and chemical potential, lattice QCD results, JHEP 05 (2004) 046 [hep-lat/0401016] [INSPIRE].

    Article  ADS  Google Scholar 

  25. Z. Fodor, S. Katz and K. Szabo, The QCD equation of state at nonzero densities: Lattice result, Phys. Lett. B 568 (2003) 73 [hep-lat/0208078] [INSPIRE].

    Article  ADS  Google Scholar 

  26. C. Allton et al., The equation of state for two flavor QCD at nonzero chemical potential, Phys. Rev. D 68 (2003) 014507 [hep-lat/0305007] [INSPIRE].

    ADS  Google Scholar 

  27. C. Allton et al., Thermodynamics of two flavor QCD to sixth order in quark chemical potential, Phys. Rev. D 71 (2005) 054508 [hep-lat/0501030] [INSPIRE].

    ADS  Google Scholar 

  28. C. Bernard et al., QCD thermodynamics with 2 + 1 flavors at nonzero chemical potential, Phys. Rev. D 77 (2008) 014503 [arXiv:0710.1330] [INSPIRE].

    ADS  Google Scholar 

  29. MILC collaboration, S. Basak et al., QCD equation of state at non-zero chemical potential, PoS(LATTICE 2008)171 [arXiv:0910.0276] [INSPIRE].

  30. C. DeTar et al., QCD thermodynamics with nonzero chemical potential at N t  = 6 and effects from heavy quarks, Phys. Rev. D 81 (2010) 114504 [arXiv:1003.5682] [INSPIRE].

    ADS  Google Scholar 

  31. T. Takaishi, P. de Forcrand and A. Nakamura, Equation of state at finite density from imaginary chemical potential, PoS(LAT2009)198 [arXiv:1002.0890] [INSPIRE].

  32. Y. Aoki, Z. Fodor, S. Katz and K. Szabó, The equation of state in lattice QCD: With physical quark masses towards the continuum limit, JHEP 01 (2006) 089 [hep-lat/0510084] [INSPIRE].

    Article  ADS  Google Scholar 

  33. S. Borsányi et al., Fluctuations of conserved charges at finite temperature from lattice QCD, JHEP 01 (2012) 138 [arXiv:1112.4416] [INSPIRE].

    Article  ADS  Google Scholar 

  34. G. Endrődi, Z. Fodor, S. Katz and K. Szabó, The QCD phase diagram at nonzero quark density, JHEP 04 (2011) 001 [arXiv:1102.1356] [INSPIRE].

    Article  ADS  Google Scholar 

  35. G.I. Egri et al., Lattice QCD as a video game, Comput. Phys. Commun. 177 (2007) 631 [hep-lat/0611022] [INSPIRE].

    Article  ADS  Google Scholar 

  36. J. Kapusta, Finite-temperature field theory, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge U.K. (1989).

Download references

Author information

Authors and Affiliations

  1. Bergische Universität Wuppertal, Theoretical Physics, 42119, Wuppertal, Germany

    Sz. Borsányi, Z. Fodor, S. Krieg & K. K. Szabó

  2. Institute for Theoretical Physics, Universität Regensburg, D-93040, Regensburg, Germany

    G. Endrődi

  3. Eötvös University, Theoretical Physics, Pázmány P. S 1/A, H-1117, Budapest, Hungary

    Z. Fodor & S. D. Katz

  4. Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52425, Jülich, Germany

    Z. Fodor & S. Krieg

  5. Dipartimento di Fisica, Universita‘ degli Studi di Torino and INFN — Sezione di Torino, I-10125, Torino, Italy

    C. Ratti

Authors
  1. Sz. Borsányi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Endrődi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Fodor
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. D. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. S. Krieg
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. C. Ratti
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. K. K. Szabó
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Endrődi.

Additional information

ArXiv ePrint: 1204.6710

Electronic supplementary material

Below is the link to the electronic supplementary material.

ESM 1

(M 6 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borsányi, S., Endrődi, G., Fodor, Z. et al. QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order μ 2 . J. High Energ. Phys. 2012, 53 (2012). https://doi.org/10.1007/JHEP08(2012)053

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP08(2012)053

Keywords

Search

Navigation