The QCD equation of state in background magnetic fields

  • G. S. Bali
  • F. Bruckmann
  • G. EndrődiEmail author
  • S. D. Katz
  • A. Schäfer
Open Access


We determine the equation of state of 2+1-flavor QCD with physical quark masses, in the presence of a constant (electro)magnetic background field on the lattice. To determine the free energy at nonzero magnetic fields we develop a new method, which is based on an integral over the quark masses up to asymptotically large values where the effect of the magnetic field can be neglected. The method is compared to other approaches in the literature and found to be advantageous for the determination of the equation of state up to large magnetic fields. Thermodynamic observables including the longitudinal and transverse pressure, magnetization, energy density, entropy density and interaction measure are presented for a wide range of temperatures and magnetic fields, and provided in ancillary files. The behavior of these observables confirms our previous result that the transition temperature is reduced by the magnetic field. We calculate the magnetic susceptibility and permeability, verifying that the thermal QCD medium is paramagnetic around and above the transition temperature, while we also find evidence for weak diamagnetism at low temperatures.


Quark-Gluon Plasma Lattice QCD Phase Diagram of QCD 


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.

Supplementary material

13130_2014_8782_MOESM1_ESM.dat (286 kb)
ESM 1 (DAT 285 kb)
13130_2014_8782_MOESM2_ESM.dat (8 kb)
ESM 2 (DAT 8 kb)


  1. [1]
    Y. Aoki, G. Endrődi, Z. Fodor, S.D. Katz and K. Szabó, The order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443 (2006) 675 [hep-lat/0611014] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    T. Bhattacharya et al., The QCD phase transition with physical-mass, chiral quarks, arXiv:1402.5175 [INSPIRE].
  3. [3]
    D. Teaney, J. Lauret and E.V. Shuryak, A hydrodynamic description of heavy ion collisions at the SPS and RHIC, nucl-th/0110037 [INSPIRE].
  4. [4]
    P.F. Kolb and U.W. Heinz, Hydrodynamic description of ultrarelativistic heavy ion collisions, nucl-th/0305084 [INSPIRE].
  5. [5]
    J.M. Lattimer and M. Prakash, Neutron star structure and the equation of state, Astrophys. J. 550 (2001) 426 [astro-ph/0002232] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    M. Hindmarsh and O. Philipsen, WIMP dark matter and the QCD equation of state, Phys. Rev. D 71 (2005) 087302 [hep-ph/0501232] [INSPIRE].ADSGoogle Scholar
  7. [7]
    D. Grasso and H.R. Rubinstein, Magnetic fields in the early universe, Phys. Rept. 348 (2001) 163 [astro-ph/0009061] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    R.C. Duncan and C. Thompson, Formation of very strongly magnetized neutron starsimplications for gamma-ray bursts, Astrophys. J. 392 (1992) L9 [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    D. Kharzeev, K. Landsteiner, A. Schmitt and H.-U. Yee, Strongly interacting matter in magnetic fields, Lect. Notes Phys. 871 (2013) 1 [arXiv:1211.6245] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    E.S. Fraga, Thermal chiral and deconfining transitions in the presence of a magnetic background, Lect. Notes Phys. 871 (2013) 121 [arXiv:1208.0917] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    M. D’Elia, Lattice QCD simulations in external background fields, Lect. Notes Phys. 871 (2013) 181 [arXiv:1209.0374] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    G.S. Bali et al., Thermodynamic properties of QCD in external magnetic fields, PoS(Confinement X)197 [arXiv:1301.5826] [INSPIRE].
  13. [13]
    K. Szabó, QCD at non-zero temperature and magnetic field, PoS(LATTICE 2013) 014 [arXiv:1401.4192] [INSPIRE].
  14. [14]
    J. Engels, J. Fingberg, F. Karsch, D. Miller and M. Weber, Nonperturbative thermodynamics of SU(N ) gauge theories, Phys. Lett. B 252 (1990) 625 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    S. Borsányi et al., The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [arXiv:1007.2580] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    G.S. Bali et al., The QCD phase diagram for external magnetic fields, JHEP 02 (2012) 044 [arXiv:1111.4956] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    F. Bruckmann, G. Endrődi and T.G. Kovács, Inverse magnetic catalysis and the Polyakov loop, JHEP 04 (2013) 112 [arXiv:1303.3972] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    G.S. Bali, F. Bruckmann, G. Endrődi and A. Schäfer, Paramagnetic squeezing of QCD matter, Phys. Rev. Lett. 112 (2014) 042301 [arXiv:1311.2559] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of continuous media. Course of theoretical physics, Butterworth-Heinemann, Oxford U.K. (1995).Google Scholar
  20. [20]
    C. Kittel, Elementary statistical physics, Dover Books on Physics Series, Dover, New York U.S.A. (2004).Google Scholar
  21. [21]
    G.S. Bali, F. Bruckmann, G. Endrődi, F. Gruber and A. Schäfer, Magnetic field-induced gluonic (inverse) catalysis and pressure (an)isotropy in QCD, JHEP 04 (2013) 130 [arXiv:1303.1328] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    E.J. Ferrer, V. de la Incera, J.P. Keith, I. Portillo and P.L. Springsteen, Equation of state of a dense and magnetized fermion system, Phys. Rev. C 82 (2010) 065802 [arXiv:1009.3521] [INSPIRE].ADSGoogle Scholar
  23. [23]
    Y. Aoki, Z. Fodor, S.D. 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].ADSCrossRefGoogle Scholar
  24. [24]
    L. Levkova and C. DeTar, Quark-gluon plasma in an external magnetic field, Phys. Rev. Lett. 112 (2014) 012002 [arXiv:1309.1142] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    C. Bonati, M. D’Elia, M. Mariti, F. Negro and F. Sanfilippo, Magnetic susceptibility of strongly interacting matter across the deconfinement transition, Phys. Rev. Lett. 111 (2013) 182001 [arXiv:1307.8063] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    C. Bonati, M. D’Elia, M. Mariti, F. Negro and F. Sanfilippo, Magnetic susceptibility and equation of state of N f = 2 + 1 QCD with physical quark masses, Phys. Rev. D 89 (2014) 054506 [arXiv:1310.8656] [INSPIRE].ADSGoogle Scholar
  27. [27]
    G.S. Bali, F. Bruckmann, G. Endrődi and A. Schäfer, Magnetization and pressures at nonzero magnetic fields in QCD, arXiv:1310.8145 [INSPIRE].
  28. [28]
    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    L.F. Abbott, Introduction to the background field method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].MathSciNetGoogle Scholar
  30. [30]
    P. Elmfors, D. Persson and B.-S. Skagerstam, Real time thermal propagators and the QED effective action for an external magnetic field, Astropart. Phys. 2 (1994) 299 [hep-ph/9312226] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    G.V. Dunne, Heisenberg-Euler effective Lagrangians: basics and extensions, hep-th/0406216 [INSPIRE].
  32. [32]
    G. Endrődi, QCD equation of state at nonzero magnetic fields in the hadron resonance gas model, JHEP 04 (2013) 023 [arXiv:1301.1307] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    I.A. Shovkovy, Magnetic catalysis: a review, Lect. Notes Phys. 871 (2013) 13 [arXiv:1207.5081] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    V.P. Gusynin, V.A. Miransky and I.A. Shovkovy, Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field, Nucl. Phys. B 462 (1996) 249 [hep-ph/9509320] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    S.L. Adler, J.C. Collins and A. Duncan, Energy-momentum-tensor trace anomaly in spin 1/2 quantum electrodynamics, Phys. Rev. D 15 (1977) 1712 [INSPIRE].ADSGoogle Scholar
  36. [36]
    D.P. Menezes, M. Benghi Pinto, S.S. Avancini, A. Perez Martinez and C. Providência, Quark matter under strong magnetic fields in the Nambu-Jona-Lasinio model, Phys. Rev. C 79 (2009) 035807 [arXiv:0811.3361] [INSPIRE].ADSGoogle Scholar
  37. [37]
    E.S. Fraga and L.F. Palhares, Deconfinement in the presence of a strong magnetic background: an exercise within the MIT bag model, Phys. Rev. D 86 (2012) 016008 [arXiv:1201.5881] [INSPIRE].ADSGoogle Scholar
  38. [38]
    P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn and J. Rittinger, Vector correlator in massless QCD at order \( \mathcal{O}\left({\alpha}_s^4\right) \) and the QED β-function at five loop, JHEP 07 (2012) 017 [arXiv:1206.1284] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    G.S. Bali et al., QCD quark condensate in external magnetic fields, Phys. Rev. D 86 (2012) 071502 [arXiv:1206.4205] [INSPIRE].ADSGoogle Scholar
  40. [40]
    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].ADSCrossRefGoogle Scholar
  41. [41]
    A. Bazavov et al., The chiral and deconfinement aspects of the QCD transition, Phys. Rev. D 85 (2012) 054503 [arXiv:1111.1710] [INSPIRE].ADSGoogle Scholar
  42. [42]
    G. Boyd et al., Thermodynamics of SU(3) lattice gauge theory, Nucl. Phys. B 469 (1996) 419 [hep-lat/9602007] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    S. Borsányi et al., Full result for the QCD equation of state with 2 + 1 flavors, Phys. Lett. B 730 (2014) 99 [arXiv:1309.5258] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    T. Steinert and W. Cassing, Electric and magnetic response of hot QCD matter, Phys. Rev. C 89 (2014) 035203 [arXiv:1312.3189] [INSPIRE].ADSGoogle Scholar
  45. [45]
    V.D. Orlovsky and Y.A. Simonov, Magnetic susceptibility at zero and nonzero chemical potential in QCD and QED, arXiv:1406.1056 [INSPIRE].
  46. [46]
    P. Elmfors, D. Persson and B.-S. Skagerstam, QED effective action at finite temperature and density, Phys. Rev. Lett. 71 (1993) 480 [hep-th/9305004] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    D. Cangemi and G.V. Dunne, Temperature expansions for magnetic systems, Annals Phys. 249 (1996) 582 [hep-th/9601048] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    Particle Data Group collaboration, J. Beringer et al., Review of particle physics (RPP), Phys. Rev. D 86 (2012) 010001 [INSPIRE].ADSGoogle Scholar
  49. [49]
    E. Braaten and A. Nieto, On the convergence of perturbative QCD at high temperature, Phys. Rev. Lett. 76 (1996) 1417 [hep-ph/9508406] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    J.-P. Blaizot, E.S. Fraga and L.F. Palhares, Effect of quark masses on the QCD presssure in a strong magnetic background, Phys. Lett. B 722 (2013) 167 [arXiv:1211.6412] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Adler function, Bjorken sum rule and the Crewther relation to order α s4 in a general gauge theory, Phys. Rev. Lett. 104 (2010) 132004 [arXiv:1001.3606] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    V.G. Bornyakov, P.V. Buividovich, N. Cundy, O.A. Kochetkov and A. Schäfer, Deconfinement transition in two-flavour lattice QCD with dynamical overlap fermions in an external magnetic field, Phys. Rev. D 90 (2014) 034501 [arXiv:1312.5628] [INSPIRE].ADSGoogle Scholar
  53. [53]
    E.M. Ilgenfritz, M. Muller-Preussker, B. Petersson and A. Schreiber, Magnetic catalysis (and inverse catalysis) at finite temperature in two-color lattice QCD, Phys. Rev. D 89 (2014) 054512 [arXiv:1310.7876] [INSPIRE].ADSGoogle Scholar
  54. [54]
    E.S. Fraga, J. Noronha and L.F. Palhares, Large-N c deconfinement transition in the presence of a magnetic field, Phys. Rev. D 87 (2013) 114014 [arXiv:1207.7094] [INSPIRE].ADSGoogle Scholar
  55. [55]
    S. Fayazbakhsh and N. Sadooghi, Phase diagram of hot magnetized two-flavor color superconducting quark matter, Phys. Rev. D 83 (2011) 025026 [arXiv:1009.6125] [INSPIRE].ADSGoogle Scholar
  56. [56]
    E.S. Fraga, B.W. Mintz and J. Schaffner-Bielich, A search for inverse magnetic catalysis in thermal quark-meson models, Phys. Lett. B 731 (2014) 154 [arXiv:1311.3964] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    J.O. Andersen, W.R. Naylor and A. Tranberg, Chiral and deconfinement transitions in a magnetic background using the functional renormalization group with the Polyakov loop, JHEP 04 (2014) 187 [arXiv:1311.2093] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M. Ferreira, P. Costa, D.P. Menezes, C. Providência and N. Scoccola, Deconfinement and chiral restoration within the SU(3) Polyakov-Nambu-Jona-Lasinio and entangled Polyakov-Nambu-Jona-Lasinio models in an external magnetic field, Phys. Rev. D 89 (2014) 016002 [arXiv:1305.4751] [INSPIRE].ADSGoogle Scholar
  59. [59]
    R.L.S. Farias, K.P. Gomes, G.I. Krein and M.B. Pinto, The importance of asymptotic freedom for the pseudocritical temperature in magnetized quark matter, arXiv:1404.3931 [INSPIRE].
  60. [60]
    M. Ferreira, P. Costa, O. Lourenço, T. Frederico and C. Providência, Inverse magnetic catalysis in the (2 + 1)-flavor Nambu-Jona-Lasinio and Polyakov-Nambu-Jona-Lasinio models, Phys. Rev. D 89 (2014) 116011 [arXiv:1404.5577] [INSPIRE].ADSGoogle Scholar
  61. [61]
    A. Ayala, M. Loewe, A.J. Mizher and R. Zamora, Inverse magnetic catalysis for the chiral transition induced by thermo-magnetic effects on the coupling constant, Phys. Rev. D 90 (2014) 036001 [arXiv:1406.3885] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • G. S. Bali
    • 1
    • 2
  • F. Bruckmann
    • 1
  • G. Endrődi
    • 1
    Email author
  • S. D. Katz
    • 3
    • 4
  • A. Schäfer
    • 1
  1. 1.Institute for Theoretical PhysicsUniversität RegensburgRegensburgGermany
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia
  3. 3.Eötvös University, Theoretical PhysicsBudapestHungary
  4. 4.MTA-ELTE Lendület Lattice Gauge Theory Research GroupBudapestHungary

Personalised recommendations