Magnetic field-induced gluonic (inverse) catalysis and pressure (an)isotropy in QCD

  • G. S. Bali
  • F. Bruckmann
  • G. Endrődi
  • F. Gruber
  • A. Schäfer


We study the influence of strong external magnetic fields on gluonic and fermionic observables in the QCD vacuum at zero and nonzero temperatures, via lattice simulations with N f  = 1 + 1 + 1 staggered quarks of physical masses. The gluonic action density is found to undergo magnetic catalysis at low temperatures and inverse magnetic catalysis near and above the transition temperature, similar to the quark condensate. Moreover, the gluonic action develops an anisotropy: the chromo-magnetic field parallel to the external field is enhanced, while the chromo-electric field in this direction is suppressed. We demonstrate that the same hierarchy is obtained using the Euler-Heisenberg effective action. Conversely, the topological charge density correlator does not reveal a significant anisotropy up to magnetic fields eB ≈ 1 GeV2. Furthermore, we show that the pressure remains isotropic even for nonzero magnetic fields, if it is defined through a compression of the system at fixed external field. In contrast, if the flux of the field is kept fixed during the compression — which is the situation realized in the lattice simulation — the pressure develops an anisotropy. We estimate the quark and gluonic contributions to this anisotropy, and relate them to the magnetization of the QCD vacuum. After performing electric charge renormalization, we obtain an estimate for the magnetization, which indicates that QCD is paramagnetic.


Lattice QCD Phase Diagram of QCD 


  1. [1]
    T. Vachaspati, Magnetic fields from cosmological phase transitions, Phys. Lett. B 265 (1991) 258 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    R.C. Duncan and C. Thompson, Formation of very strongly magnetized neutron starsImplications for γ-ray bursts, Astrophys. J. 392 (1992) L9 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    V. Skokov, A.Y. 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].ADSCrossRefGoogle 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].ADSCrossRefGoogle 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]
    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
  7. [7]
    L. Landau, Diamagnetismus der Metalle, Z. Physik A 64 (1930) 629.ADSCrossRefGoogle Scholar
  8. [8]
    T. Banks and A. Casher, Chiral symmetry breaking in confining theories, Nucl. Phys. B 169 (1980) 103 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    G. Endrodi, QCD equation of state at nonzero magnetic fields in the hadron resonance gas model, JHEP 04 (2013) 023 [arXiv:1301.1307] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    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
  11. [11]
    F. Bruckmann and G. Endrodi, 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
  12. [12]
    G. Bali et al., The QCD phase diagram for external magnetic fields, JHEP 02 (2012) 044 [arXiv:1111.4956] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    G. Bali et al., QCD quark condensate in external magnetic fields, Phys. Rev. D 86 (2012) 071502 [arXiv:1206.4205] [INSPIRE].ADSGoogle Scholar
  14. [14]
    E.-M. Ilgenfritz, M. Kalinowski, M. Muller-Preussker, B. Petersson and A. Schreiber, Two-color QCD with staggered fermions at finite temperature under the influence of a magnetic field, Phys. Rev. D 85 (2012) 114504 [arXiv:1203.3360] [INSPIRE].ADSGoogle Scholar
  15. [15]
    I.A. Shovkovy, Magnetic catalysis: a review, arXiv:1207.5081 [INSPIRE].
  16. [16]
    F. Bruckmann, G. Endrődi and T. Kovács, Inverse magnetic catalysis and the Polyakov loop, JHEP 04 (2013) 112.ADSCrossRefGoogle Scholar
  17. [17]
    G. Basar, G.V. Dunne and D.E. Kharzeev, Electric dipole moment induced by a QCD instanton in an external magnetic field, Phys. Rev. D 85 (2012) 045026 [arXiv:1112.0532] [INSPIRE].ADSGoogle Scholar
  18. [18]
    B. Ioffe and A.V. Smilga, Nucleon magnetic moments and magnetic properties of vacuum in QCD, Nucl. Phys. B 232 (1984) 109 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    G. Bali et al., Magnetic susceptibility of QCD at zero and at finite temperature from the lattice, Phys. Rev. D 86 (2012) 094512 [arXiv:1209.6015] [INSPIRE].ADSGoogle Scholar
  20. [20]
    P. Buividovich, M. Chernodub, E. Luschevskaya and M. Polikarpov, Chiral magnetization of non-abelian vacuum: a lattice study, Nucl. Phys. B 826 (2010) 313 [arXiv:0906.0488] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    V. Braguta, P. Buividovich, T. Kalaydzhyan, S. Kuznetsov and M. Polikarpov, The chiral magnetic effect and chiral symmetry breaking in SU(3) quenched lattice gauge theory, Phys. Atom. Nucl. 75 (2012) 488 [arXiv:1011.3795] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    R. Blandford and L. Hernquist, Magnetic susceptibility of a neutron star crust, J. Phys C 15 (1982) 6233.ADSGoogle Scholar
  23. [23]
    E.J. Ferrer, V. de la Incera, J.P. Keith, I. Portillo and P.P. Springsteen, Equation of state of a dense and magnetized fermion system, Phys. Rev. C 82 (2010) 065802 [arXiv:1009.3521] [INSPIRE].ADSGoogle Scholar
  24. [24]
    A. Potekhin and D. Yakovlev, Comment onEquation of state of dense and magnetized fermion system’, Phys. Rev. C 85 (2012) 039801 [arXiv:1109.3783] [INSPIRE].ADSGoogle Scholar
  25. [25]
    E.J. Ferrer and V. de la Incera, Reply to comment onEquation of state of dense and magnetized fermion system’, Phys. Rev. C 85 (2012) 039802 [arXiv:1110.0420] [INSPIRE].ADSGoogle Scholar
  26. [26]
    M. Strickland, V. Dexheimer and D. Menezes, Bulk properties of a Fermi gas in a magnetic field, Phys. Rev. D 86 (2012) 125032 [arXiv:1209.3276] [INSPIRE].ADSGoogle Scholar
  27. [27]
    A. Kandus, K.E. Kunze and C.G. Tsagas, Primordial magnetogenesis, Phys. Rept. 505 (2011) 1 [arXiv:1007.3891] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    B. Müller, J. Schukraft and B. Wyslouch, First results from Pb+Pb collisions at the LHC, Ann. Rev. Nucl. Part. Sci. 62 (2012) 361 [arXiv:1202.3233] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    B. Schenke, P. Tribedy and R. Venugopalan, Fluctuating glasma initial conditions and flow in heavy ion collisions, Phys. Rev. Lett. 108 (2012) 252301 [arXiv:1202.6646] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    U.W. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, arXiv:1301.2826 [INSPIRE].
  31. [31]
    S. Borsányi et al., The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [arXiv:1007.2580] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    P. Weisz, Continuum limit improved lattice action for pure Yang-Mills theory. 1, Nucl. Phys. B 212 (1983) 1 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    P.J. Moran and D.B. Leinweber, Over-improved stout-link smearing, Phys. Rev. D 77 (2008) 094501 [arXiv:0801.1165] [INSPIRE].ADSGoogle Scholar
  34. [34]
    S.O. Bilson-Thompson, D.B. Leinweber and A.G. Williams, Highly improved lattice field strength tensor, Annals Phys. 304 (2003) 1 [hep-lat/0203008] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  35. [35]
    E.-M. Ilgenfritz et al., Vacuum structure revealed by over-improved stout-link smearing compared with the overlap analysis for quenched QCD, Phys. Rev. D 77 (2008) 074502 [Erratum ibid. D 77 (2008) 099902] [arXiv:0801.1725] [INSPIRE].Google Scholar
  36. [36]
    F. Bruckmann, F. Gruber, N. Cundy, A. Schafer and T. Lippert, Topology of dynamical lattice configurations including results from dynamical overlap fermions, Phys. Lett. B 707 (2012)278 [arXiv:1107.0897] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    F. Niedermayer, Exact chiral symmetry, topological charge and related topics, Nucl. Phys. Proc. Suppl. 73 (1999) 105 [hep-lat/9810026] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  38. [38]
    I. Horvath et al., On the local structure of topological charge fluctuations in QCD, Phys. Rev. D 67 (2003) 011501 [hep-lat/0203027] [INSPIRE].ADSGoogle Scholar
  39. [39]
    J. Rafelski, Electromagnetic fields in the QCD vacuum, hep-ph/9806389 [INSPIRE].
  40. [40]
    H.T. Elze, B. Müller and J. Rafelski, Interfering QCD/QED vacuum polarization, hep-ph/9811372 [INSPIRE].
  41. [41]
    M. D’Elia, M. Mariti and F. Negro, Susceptibility of the QCD vacuum to CP-odd electromagnetic background fields, Phys. Rev. Lett. 110 (2013) 082002 [arXiv:1209.0722] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    E. Seiler and I. Stamatescu, Some remarks on the Witten-Venziano formula for the eta-prime mass, MPI-PAE/PTh 10/87 (1987).Google Scholar
  43. [43]
    B. Ioffe and K. Zyablyuk, Gluon condensate in charmonium sum rules with three loop corrections, Eur. Phys. J. C 27 (2003) 229 [hep-ph/0207183] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    L. Landau and E. Lifshits, The classical theory of fields, Course on Theoretical Physics volume 2, Pergamon Press, U.K. (1971).Google Scholar
  45. [45]
    V. Canuto and H. Chiu, Quantum theory of an electron gas in intense magnetic fields, Phys. Rev. 173 (1968) 1210 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    A.P. Martinez, H.P. Rojas and H.J. Mosquera Cuesta, Magnetic collapse of a neutron gas: can magnetars indeed be formed?, Eur. Phys. J. C 29 (2003) 111 [astro-ph/0303213] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    J. Jackson, Classical electrodynamics, Wiley, U.S.A. (1975).MATHGoogle Scholar
  48. [48]
    F. Rohrlich, Classical charged particles, World Scientific, Singapore (2007).CrossRefMATHGoogle Scholar
  49. [49]
    R.D. Blandford and L. Hernquist, Magnetic susceptibility of a neutron star crust, J. Phys. C 15 (1982) 6233.ADSGoogle Scholar
  50. [50]
    L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of continuous media, Course of theoretical physics, Butterworth-Heinemann, U.K. (1995).Google Scholar
  51. [51]
    F. Karsch, SU(N) gauge theory couplings on asymmetric lattices, Nucl. Phys. B 205 (1982) 285 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    F. Karsch and I. Stamatescu, QCD thermodynamics with light quarks: quantum corrections to the fermionic anisotropy parameter, Phys. Lett. B 227 (1989) 153 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    E.S. Fraga, J. Noronha and L.F. Palhares, Large N c deconfinement transition in the presence of a magnetic field, arXiv:1207.7094 [INSPIRE].
  54. [54]
    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
  55. [55]
    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  56. [56]
    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
  57. [57]
    G.V. Dunne, Heisenberg-Euler effective lagrangians: basics and extensions, hep-th/0406216 [INSPIRE].
  58. [58]
    M. Cheng et al., The QCD equation of state with almost physical quark masses, Phys. Rev. D 77 (2008) 014511 [arXiv:0710.0354] [INSPIRE].ADSGoogle Scholar
  59. [59]
    W.-J. Lee, Quark mass renormalization on the lattice with staggered fermions, Phys. Rev. D 49 (1994) 3563 [hep-lat/9310018] [INSPIRE].ADSGoogle Scholar
  60. [60]
    J. Engels, F. Karsch, H. Satz and I. Montvay, Gauge field thermodynamics for the SU(2) Yang-Mills system, Nucl. Phys. B 205 (1982) 545 [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    G.S. Bali, Casimir scaling of SU(3) static potentials, Phys. Rev. D 62 (2000) 114503 [hep-lat/0006022] [INSPIRE].ADSGoogle Scholar
  62. [62]
    S. Borsányi et al., Anisotropy tuning with the Wilson flow, arXiv:1205.0781 [INSPIRE].
  63. [63]
    L. Levkova, T. Manke and R. Mawhinney, Two-flavor QCD thermodynamics using anisotropic lattices, Phys. Rev. D 73 (2006) 074504 [hep-lat/0603031] [INSPIRE].ADSGoogle Scholar
  64. [64]
    S. Sakai, T. Saito and A. Nakamura, Anisotropic lattice with improved gauge actions. 1. Study of fundamental parameters in weak coupling regions, Nucl. Phys. B 584 (2000) 528 [hep-lat/0002029] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    W. Heisenberg and H. Euler, Consequences of Diracs theory of positrons, Z. Phys. 98 (1936) 714 [physics/0605038] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    V. Novikov, M.A. Shifman, A. Vainshtein and V.I. Zakharov, Calculations in external fields in quantum chromodynamics. Technical review, Fortsch. Phys. 32 (1984) 585 [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • G. S. Bali
    • 1
    • 2
  • F. Bruckmann
    • 1
  • G. Endrődi
    • 1
  • F. Gruber
    • 1
  • A. Schäfer
    • 1
  1. 1.Institute for Theoretical PhysicsUniversität RegensburgRegensburgGermany
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations