Journal of High Energy Physics

, 2012:44 | Cite as

The QCD phase diagram for external magnetic fields

  • G. S. Bali
  • F. Bruckmann
  • G. Endrődi
  • Z. Fodor
  • S. D. Katz
  • S. Krieg
  • A. Schäfer
  • K. K. Szabó


The effect of an external (electro)magnetic field on the finite temperature transition of QCD is studied. We generate configurations at various values of the quantized magnetic flux with N f  = 2 + 1 flavors of stout smeared staggered quarks, with physical masses. Thermodynamic observables including the chiral condensate and susceptibility, and the strange quark number susceptibility are measured as functions of the field strength. We perform the renormalization of the studied observables and extrapolate the results to the continuum limit using N t  = 6, 8 and 10 lattices. We also check for finite volume effects using various lattice volumes. We find from all of our observables that the transition temperature T c significantly decreases with increasing magnetic field. This is in conflict with various model calculations that predict an increasing T c (B). From a finite volume scaling analysis we find that the analytic crossover that is present at B = 0 persists up to our largest magnetic fields eB ≈ 1 GeV2, and that the transition strength increases mildly up to this eB ≈ 1 GeV2.


Lattice QCD Lattice Quantum Field Theory 


  1. [1]
    T. Vachaspati, Magnetic fields from cosmological phase transitions, Phys. Lett. B 265 (1991) 258 [INSPIRE].ADSGoogle Scholar
  2. [2]
    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
  3. [3]
    V. Skokov, A. Illarionov and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A 24 (2009) 5925 [arXiv:0907.1396] [INSPIRE].ADSGoogle Scholar
  4. [4]
    D.E. Kharzeev, L.D. McLerran and H.J. Warringa, The effects of topological charge change in heavy ion collisions:Event by event P and CP-violation’, Nucl. Phys. A 803 (2008) 227 [arXiv:0711.0950] [INSPIRE].ADSGoogle Scholar
  5. [5]
    K. Fukushima, D.E. Kharzeev and H.J. Warringa, The chiral magnetic effect, Phys. Rev. D 78 (2008) 074033 [arXiv:0808.3382] [INSPIRE].ADSGoogle Scholar
  6. [6]
    D. Kharzeev, Parity violation in hot QCD: Why it can happen and how to look for it, Phys. Lett. B 633 (2006) 260 [hep-ph/0406125] [INSPIRE].ADSGoogle Scholar
  7. [7]
    STAR collaboration, S.A. Voloshin, Probe for the strong parity violation effects at RHIC with three particle correlations, Indian J. Phys. 85 (2011) 1103 [arXiv:0806.0029] [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    STAR collaboration, B. Abelev et al., Azimuthal charged-particle correlations and possible local strong parity violation, Phys. Rev. Lett. 103 (2009) 251601 [arXiv:0909.1739] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    ALICE collaboration, I. Selyuzhenkov, Anisotropic flow and other collective phenomena measured in Pb-Pb collisions with ALICE at the LHC, arXiv:1111.1875 [INSPIRE].
  10. [10]
    F. Wang, Effects of cluster particle correlations on local parity violation observables, Phys. Rev. C 81 (2010) 064902 [arXiv:0911.1482] [INSPIRE].ADSGoogle Scholar
  11. [11]
    B. Müller and A. Schäfer, Charge fluctuations from the chiral magnetic effect in nuclear collisions, Phys. Rev. C 82 (2010) 057902 [arXiv:1009.1053] [INSPIRE].Google Scholar
  12. [12]
    V. Voronyuk et al., (Electro-)Magnetic field evolution in relativistic heavy-ion collisions, Phys. Rev. C 83 (2011) 054911 [arXiv:1103.4239] [INSPIRE].ADSGoogle Scholar
  13. [13]
    A.J. Mizher, M. Chernodub and E.S. Fraga, Phase diagram of hot QCD in an external magnetic field: possible splitting of deconfinement and chiral transitions, Phys. Rev. D 82 (2010) 105016 [arXiv:1004.2712] [INSPIRE].ADSGoogle Scholar
  14. [14]
    E.S. Fraga and A.J. Mizher, Can a strong magnetic background modify the nature of the chiral transition in QCD?, Nucl. Phys. A 820 (2009) 103 C-106 C [arXiv:0810.3693] [INSPIRE].ADSGoogle Scholar
  15. [15]
    R. Gatto and M. Ruggieri, Deconfinement and chiral symmetry restoration in a strong magnetic background, Phys. Rev. D 83 (2011) 034016 [arXiv:1012.1291] [INSPIRE].ADSGoogle Scholar
  16. [16]
    R. Gatto and M. Ruggieri, Dressed Polyakov loop and phase diagram of hot quark matter under magnetic field, Phys. Rev. D 82 (2010) 054027 [arXiv:1007.0790] [INSPIRE].ADSGoogle Scholar
  17. [17]
    A. Osipov, B. Hiller, A. Blin and J. da Providencia, Dynamical chiral symmetry breaking by a magnetic field and multi-quark interactions, Phys. Lett. B 650 (2007) 262 [hep-ph/0701090] [INSPIRE].ADSGoogle Scholar
  18. [18]
    K. Kashiwa, Entanglement between chiral and deconfinement transitions under strong uniform magnetic background field, Phys. Rev. D 83 (2011) 117901 [arXiv:1104.5167] [INSPIRE].Google Scholar
  19. [19]
    C.V. Johnson and A. Kundu, External fields and chiral symmetry breaking in the Sakai-Sugimoto model, JHEP 12 (2008) 053 [arXiv:0803.0038] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    S. Kanemura, H.-T. Sato and H. Tochimura, Thermodynamic Gross-Neveu model under constant electromagnetic field, Nucl. Phys. B 517 (1998) 567 [hep-ph/9707285] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    K. Klimenko, Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Theor. Math. Phys. 90 (1992) 1 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    J. Alexandre, K. Farakos and G. Koutsoumbas, Magnetic catalysis in QED(3) at finite temperature: beyond the constant mass approximation, Phys. Rev. D 63 (2001) 065015 [hep-th/0010211] [INSPIRE].ADSGoogle Scholar
  23. [23]
    N. Evans, T. Kalaydzhyan, K.-y. Kim and I. Kirsch, Non-equilibrium physics at a holographic chiral phase transition, JHEP 01 (2011) 050 [arXiv:1011.2519] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    N. Agasian and S. Fedorov, Quark-hadron phase transition in a magnetic field, Phys. Lett. B 663 (2008) 445 [arXiv:0803.3156] [INSPIRE].ADSGoogle Scholar
  25. [25]
    F. Preis, A. Rebhan and A. Schmitt, Inverse magnetic catalysis in dense holographic matter, JHEP 03 (2011) 033 [arXiv:1012.4785] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    P. Cea and L. Cosmai, Color dynamics in external fields, JHEP 08 (2005) 079 [hep-lat/0505007] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    P. Cea and L. Cosmai, Abelian chromomagnetic fields and confinement, JHEP 02 (2003) 031 [hep-lat/0204023] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    P. Cea, L. Cosmai and M. D’Elia, QCD dynamics in a constant chromomagnetic field, JHEP 12 (2007) 097 [arXiv:0707.1149] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    V. Gusynin, V. Miransky and I. 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
  30. [30]
    S.-i. Nam and C.-W. Kao, Chiral restoration at finite T under the magnetic field with the meson-loop corrections, Phys. Rev. D 83 (2011) 096009 [arXiv:1103.6057] [INSPIRE].ADSGoogle Scholar
  31. [31]
    J.K. Boomsma and D. Boer, The influence of strong magnetic fields and instantons on the phase structure of the two-flavor NJLS model, Phys. Rev. D 81 (2010) 074005 [arXiv:0911.2164] [INSPIRE].ADSGoogle Scholar
  32. [32]
    I. Shushpanov and A.V. Smilga, Quark condensate in a magnetic field, Phys. Lett. B 402 (1997) 351 [hep-ph/9703201] [INSPIRE].ADSGoogle Scholar
  33. [33]
    T.D. Cohen, D.A. McGady and E.S. Werbos, The chiral condensate in a constant electromagnetic field, Phys. Rev. C 76 (2007) 055201 [arXiv:0706.3208] [INSPIRE].ADSGoogle Scholar
  34. [34]
    N.O. Agasian, Chiral thermodynamics in a magnetic field, Phys. Atom. Nucl. 64 (2001) 554 [hep-ph/0112341] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A. Zayakin, QCD vacuum properties in a magnetic field from AdS/CFT: chiral condensate and Goldstone mass, JHEP 07 (2008) 116 [arXiv:0807.2917] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    V. Miransky and I. Shovkovy, Magnetic catalysis and anisotropic confinement in QCD, Phys. Rev. D 66 (2002) 045006 [hep-ph/0205348] [INSPIRE].ADSGoogle Scholar
  37. [37]
    M. D’Elia, S. Mukherjee and F. Sanfilippo, QCD phase transition in a strong magnetic background, Phys. Rev. D 82 (2010) 051501 [arXiv:1005.5365] [INSPIRE].ADSGoogle Scholar
  38. [38]
    G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    M. Al-Hashimi and U.-J. Wiese, Discrete accidental symmetry for a particle in a constant magnetic field on a torus, Annals Phys. 324 (2009) 343 [arXiv:0807.0630] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  40. [40]
    G. Martinelli, G. Parisi, R. Petronzio and F. Rapuano, The proton and neutron magnetic moments in lattice QCD, Phys. Lett. B 116 (1982) 434 [INSPIRE].ADSGoogle Scholar
  41. [41]
    C.W. Bernard, T. Draper, K. Olynyk and M. Rushton, Lattice QCD calculation of some baryon magnetic moments, Phys. Rev. Lett. 49 (1982) 1076 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    L. Zhou, F. Lee, W. Wilcox and J.C. Christensen, Magnetic polarizability of hadrons from lattice QCD, Nucl. Phys. Proc. Suppl. 119 (2003) 272 [hep-lat/0209128] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    D.S. Roberts, P.O. Bowman, W. Kamleh and D.B. Leinweber, Wave functions of the proton ground state in the presence of a uniform background magnetic field in lattice QCD, Phys. Rev. D 83 (2011) 094504 [arXiv:1011.1975] [INSPIRE].ADSGoogle Scholar
  44. [44]
    M. D’Elia and F. Negro, Chiral properties of strong interactions in a magnetic background, Phys. Rev. D 83 (2011) 114028 [arXiv:1103.2080] [INSPIRE].ADSGoogle Scholar
  45. [45]
    F. Bruckmann and G. Endrődi, Dressed Wilson loops as dual condensates in response to magnetic and electric fields, Phys. Rev. D 84 (2011) 074506 [arXiv:1104.5664] [INSPIRE].ADSGoogle Scholar
  46. [46]
    S. Dürr, Theoretical issues with staggered fermion simulations, PoS(LAT2005)021 [hep-lat/0509026] [INSPIRE].
  47. [47]
    H. Leutwyler and A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev. D 46 (1992) 5607 [INSPIRE].MathSciNetADSGoogle Scholar
  48. [48]
    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].ADSCrossRefGoogle Scholar
  49. [49]
    A. Salam and J. Strathdee, Transition electromagnetic fields in particle physics, Nucl. Phys. B 90 (1975) 203 .MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison Wesley, Reading U.S.A. (1995).Google Scholar
  51. [51]
    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].ADSCrossRefGoogle Scholar
  52. [52]
    M. Chernodub, Superconductivity of QCD vacuum in strong magnetic field, Phys. Rev. D 82 (2010) 085011 [arXiv:1008.1055] [INSPIRE].ADSGoogle Scholar
  53. [53]
    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].ADSCrossRefGoogle Scholar
  54. [54]
    S. Borsányi et al., The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [arXiv:1007.2580] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    N. Ishizuka, M. Fukugita, H. Mino, M. Okawa and A. Ukawa, Operator dependence of hadron masses for Kogut-Susskind quarks on the lattice, Nucl. Phys. B 411 (1994) 875 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    G. Endrődi, Multidimensional spline integration of scattered data, Comput. Phys. Commun. 182 (2011) 1307 [arXiv:1010.2952] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    F. Karsch, E. Laermann and A. Peikert, Quark mass and flavor dependence of the QCD phase transition, Nucl. Phys. B 605 (2001) 579 [hep-lat/0012023] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    Y. Aoki, G. Endrődi, Z. Fodor, S. 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
  59. [59]
    Y. Aoki, Z. Fodor, S. Katz and K. Szabó, The QCD transition temperature: results with physical masses in the continuum limit, Phys. Lett. B 643 (2006) 46 [hep-lat/0609068] [INSPIRE].ADSGoogle Scholar
  60. [60]
    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

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • G. S. Bali
    • 1
  • F. Bruckmann
    • 1
  • G. Endrődi
    • 1
  • Z. Fodor
    • 2
    • 3
    • 4
  • S. D. Katz
    • 3
  • S. Krieg
    • 2
    • 4
  • A. Schäfer
    • 1
  • K. K. Szabó
    • 2
  1. 1.Institute for Theoretical PhysicsUniversität RegensburgRegensburgGermany
  2. 2.Department of PhysicsBergische Universität WuppertalWuppertalGermany
  3. 3.Institute for Theoretical PhysicsEötvös UniversityBudapestHungary
  4. 4.Jülich Supercomputing CentreForschungszentrum JülichJülichGermany

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