Abstract
We investigate the quark-meson model in a magnetic field using the functional renormalization group equation beyond the local-potential approximation. Our truncation of the effective action involves anisotropic wave function renormalization for mesons, which allows us to investigate how the magnetic field distorts the propagation of neutral mesons. Solving the flow equation numerically, we find that the transverse velocity of mesons decreases with the magnetic field at all temperatures, which is most prominent at zero temperature. The meson screening masses and the pion decay constants are also computed. The constituent quark mass is found to increase with magnetic field at all temperatures, resulting in the crossover temperature that increases monotonically with the magnetic field. This tendency is consistent with most model calculations but not with the lattice simulation performed at the physical point. Our work suggests that the strong anisotropy of meson propagation may not be the fundamental origin of the inverse magnetic catalysis.
Article PDF
Similar content being viewed by others
References
K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rept. Prog. Phys. 74 (2011) 014001 [arXiv:1005.4814] [INSPIRE].
R.C. Duncan and C. Thompson, Formation of very strongly magnetized neutron stars - implications for gamma-ray bursts, Astrophys. J. 392 (1992) L9 [INSPIRE].
D. Grasso and H.R. Rubinstein, Magnetic fields in the early universe, Phys. Rept. 348 (2001) 163 [astro-ph/0009061] [INSPIRE].
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].
H. Suganuma and T. Tatsumi, On the Behavior of Symmetry and Phase Transitions in a Strong Electromagnetic Field, Annals Phys. 208 (1991) 470 [INSPIRE].
K. Klimenko, Three-dimensional Gross-Neveu model in an external magnetic field, Theor. Math. Phys. 89 (1992) 1161 [INSPIRE].
K. Klimenko, Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Z. Phys. C 54 (1992) 323 [INSPIRE].
V. Gusynin, V. Miransky and I. Shovkovy, Catalysis of dynamical flavor symmetry breaking by a magnetic field in (2+1)-dimensions, Phys. Rev. Lett. 73 (1994) 3499 [Erratum ibid. 76 (1996) 1005] [hep-ph/9405262] [INSPIRE].
V. Gusynin, V. Miransky and I. Shovkovy, Dynamical flavor symmetry breaking by a magnetic field in (2+1)-dimensions, Phys. Rev. D 52 (1995) 4718 [hep-th/9407168] [INSPIRE].
V. Gusynin, V. Miransky and I. Shovkovy, Dimensional reduction and dynamical chiral symmetry breaking by a magnetic field in (3+1)-dimensions, Phys. Lett. B 349 (1995) 477 [hep-ph/9412257] [INSPIRE].
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].
D. Ebert and K. Klimenko, Quark droplets stability induced by external magnetic field, Nucl. Phys. A 728 (2003) 203 [hep-ph/0305149] [INSPIRE].
T. Inagaki, D. Kimura and T. Murata, Four fermion interaction model in a constant magnetic field at finite temperature and chemical potential, Prog. Theor. Phys. 111 (2004) 371 [hep-ph/0312005] [INSPIRE].
E.S. Fraga and A.J. Mizher, Chiral transition in a strong magnetic background, Phys. Rev. D 78 (2008) 025016 [arXiv:0804.1452] [INSPIRE].
D. Menezes, M. Benghi Pinto, S. Avancini, A. Perez Martinez and C. Providencia, Quark matter under strong magnetic fields in the Nambu-Jona-Lasinio Model, Phys. Rev. C 79 (2009) 035807 [arXiv:0811.3361] [INSPIRE].
D. Menezes, M. Benghi Pinto, S. Avancini and C. Providencia, Quark matter under strong magnetic fields in the SU(3) Nambu-Jona-Lasinio Model, Phys. Rev. C 80 (2009) 065805 [arXiv:0907.2607] [INSPIRE].
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].
K. Fukushima, M. Ruggieri and R. Gatto, Chiral magnetic effect in the PNJL model, Phys. Rev. D 81 (2010) 114031 [arXiv:1003.0047] [INSPIRE].
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].
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].
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].
B. Chatterjee, H. Mishra and A. Mishra, Vacuum structure and chiral symmetry breaking in strong magnetic fields for hot and dense quark matter, Phys. Rev. D 84 (2011) 014016 [arXiv:1101.0498] [INSPIRE].
M. Frasca and M. Ruggieri, Magnetic Susceptibility of the Quark Condensate and Polarization from Chiral Models, Phys. Rev. D 83 (2011) 094024 [arXiv:1103.1194] [INSPIRE].
A. Rabhi and C. Providencia, Quark matter under strong magnetic field in chiral models, Phys. Rev. C 83 (2011) 055801 [arXiv:1104.1512] [INSPIRE].
K. Kashiwa, Entanglement between chiral and deconfinement transitions under strong uniform magnetic background field, Phys. Rev. D 83 (2011) 117901 [arXiv:1104.5167] [INSPIRE].
J.O. Andersen and R. Khan, Chiral transition in a magnetic field and at finite baryon density, Phys. Rev. D 85 (2012) 065026 [arXiv:1105.1290] [INSPIRE].
V. Skokov, Phase diagram in an external magnetic field beyond a mean-field approximation, Phys. Rev. D 85 (2012) 034026 [arXiv:1112.5137] [INSPIRE].
D.D. Scherer and H. Gies, Renormalization Group Study of Magnetic Catalysis in the 3d Gross-Neveu Model, Phys. Rev. B 85 (2012) 195417 [arXiv:1201.3746] [INSPIRE].
K. Fukushima and J.M. Pawlowski, Magnetic catalysis in hot and dense quark matter and quantum fluctuations, Phys. Rev. D 86 (2012) 076013 [arXiv:1203.4330] [INSPIRE].
J.O. Andersen and A. Tranberg, The Chiral transition in a magnetic background: Finite density effects and the functional renormalization group, JHEP 08 (2012) 002 [arXiv:1204.3360] [INSPIRE].
S. Fayazbakhsh, S. Sadeghian and N. Sadooghi, Properties of neutral mesons in a hot and magnetized quark matter, Phys. Rev. D 86 (2012) 085042 [arXiv:1206.6051] [INSPIRE].
S. Fayazbakhsh and N. Sadooghi, Weak decay constant of neutral pions in a hot and magnetized quark matter, Phys. Rev. D 88 (2013) 065030 [arXiv:1306.2098] [INSPIRE].
G.N. Ferrari, A.F. Garcia and M.B. Pinto, Chiral Transition Within Effective Quark Models Under Magnetic Fields, Phys. Rev. D 86 (2012) 096005 [arXiv:1207.3714] [INSPIRE].
F. Preis, A. Rebhan and A. Schmitt, Inverse magnetic catalysis in field theory and gauge-gravity duality, Lect. Notes Phys. 871 (2013) 51 [arXiv:1208.0536] [INSPIRE].
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, arXiv:1311.2093 [INSPIRE].
E.J. Ferrer, V. de la Incera, I. Portillo and M. Quiroz, A new look at the QCD ground state in a magnetic field, arXiv:1311.3400 [INSPIRE].
R. Gatto and M. Ruggieri, Quark Matter in a Strong Magnetic Background, Lect. Notes Phys. 871 (2013) 87 [arXiv:1207.3190] [INSPIRE].
I.A. Shovkovy, Magnetic Catalysis: A Review, Lect. Notes Phys. 871 (2013) 13 [arXiv:1207.5081] [INSPIRE].
P. Buividovich, M. Chernodub, E. Luschevskaya and M. Polikarpov, Numerical study of chiral symmetry breaking in non-Abelian gauge theory with background magnetic field, Phys. Lett. B 682 (2010) 484 [arXiv:0812.1740] [INSPIRE].
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].
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].
V. Braguta, P. Buividovich, M. Chernodub, A.Y. Kotov and M. Polikarpov, Electromagnetic superconductivity of vacuum induced by strong magnetic field: numerical evidence in lattice gauge theory, Phys. Lett. B 718 (2012) 667 [arXiv:1104.3767] [INSPIRE].
G. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. Katz et al., The QCD phase diagram for external magnetic fields, JHEP 02 (2012) 044 [arXiv:1111.4956] [INSPIRE].
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].
E. Luschevskaya and O. Larina, The ρ and a mesons in a strong abelian magnetic field in SU(2) lattice gauge theory, arXiv:1203.5699 [INSPIRE].
G. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. Katz et al., QCD quark condensate in external magnetic fields, Phys. Rev. D 86 (2012) 071502 [arXiv:1206.4205] [INSPIRE].
G. Bali, F. Bruckmann, M. Constantinou, M. Costa, G. Endrodi 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].
G. Bali, F. Bruckmann, G. Endrodi, F. Gruber and A. Schaefer, Magnetic field-induced gluonic (inverse) catalysis and pressure (an)isotropy in QCD, JHEP 04 (2013) 130 [arXiv:1303.1328] [INSPIRE].
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].
L. Levkova and C. DeTar, quark-gluon plasma in an external magnetic field, Phys. Rev. Lett. 112 (2014) 012002 [arXiv:1309.1142] [INSPIRE].
E.M. Ilgenfritz, M. Muller-Preussker, B. Petersson and A. Schreiber, Magnetic catalysis (and inverse catalysis) at finite temperature in two-color lattice QCD, arXiv:1310.7876 [INSPIRE].
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, arXiv:1310.8656 [INSPIRE].
M. D’Elia, Lattice QCD Simulations in External Background Fields, Lect. Notes Phys. 871 (2013) 181 [arXiv:1209.0374] [INSPIRE].
V. Bornyakov, P. Buividovich, N. Cundy, O. Kochetkov and A. Schäfer, Deconfinement transition in two-flavour lattice QCD with dynamical overlap fermions in an external magnetic field, arXiv:1312.5628 [INSPIRE].
K. Fukushima and Y. Hidaka, Magnetic Catalysis vs Magnetic Inhibition, Phys. Rev. Lett. 110 (2013) 031601 [arXiv:1209.1319] [INSPIRE].
T. Kojo and N. Su, The quark mass gap in a magnetic field, Phys. Lett. B 720 (2013) 192 [arXiv:1211.7318] [INSPIRE].
F. Bruckmann, G. Endrodi and T.G. Kovacs, Inverse magnetic catalysis and the Polyakov loop, JHEP 04 (2013) 112 [arXiv:1303.3972] [INSPIRE].
J. Chao, P. Chu and M. Huang, Inverse magnetic catalysis induced by sphalerons, Phys. Rev. D 88 (2013) 054009 [arXiv:1305.1100] [INSPIRE].
C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90 [INSPIRE].
J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363 (2002) 223 [hep-ph/0005122] [INSPIRE].
J.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831 [hep-th/0512261] [INSPIRE].
B. Delamotte, An Introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852 (2012) 49 [cond-mat/0702365] [INSPIRE].
J. Braun, Fermion Interactions and Universal Behavior in Strongly Interacting Theories, J. Phys. G 39 (2012) 033001 [arXiv:1108.4449] [INSPIRE].
R.D. Pisarski and M. Tytgat, Propagation of cool pions, Phys. Rev. D 54 (1996) 2989 [hep-ph/9604404] [INSPIRE].
D. Son and M.A. Stephanov, Pion propagation near the QCD chiral phase transition, Phys. Rev. Lett. 88 (2002) 202302 [hep-ph/0111100] [INSPIRE].
D. Son and M.A. Stephanov, Real time pion propagation in finite temperature QCD, Phys. Rev. D 66 (2002) 076011 [hep-ph/0204226] [INSPIRE].
D. Jungnickel and C. Wetterich, Effective action for the chiral quark-meson model, Phys. Rev. D 53 (1996) 5142 [hep-ph/9505267] [INSPIRE].
J. Berges, D. Jungnickel and C. Wetterich, Two flavor chiral phase transition from nonperturbative flow equations, Phys. Rev. D 59 (1999) 034010 [hep-ph/9705474] [INSPIRE].
T.R. Morris, Equivalence of local potential approximations, JHEP 07 (2005) 027 [hep-th/0503161] [INSPIRE].
T.R. Morris and J.F. Tighe, Convergence of derivative expansions of the renormalization group, JHEP 08 (1999) 007 [hep-th/9906166] [INSPIRE].
J. Braun, Thermodynamics of QCD low-energy models and the derivative expansion of the effective action, Phys. Rev. D 81 (2010) 016008 [arXiv:0908.1543] [INSPIRE].
B.-J. Schaefer and J. Wambach, The Phase diagram of the quark meson model, Nucl. Phys. A 757 (2005) 479 [nucl-th/0403039] [INSPIRE].
V. Skokov, B. Stokic, B. Friman and K. Redlich, Meson fluctuations and thermodynamics of the Polyakov loop extended quark-meson model, Phys. Rev. C 82 (2010) 015206 [arXiv:1004.2665] [INSPIRE].
T.K. Herbst, J.M. Pawlowski and B.-J. Schaefer, The phase structure of the Polyakov-quark-meson model beyond mean field, Phys. Lett. B 696 (2011) 58 [arXiv:1008.0081] [INSPIRE].
B. Stokic, B. Friman and K. Redlich, The Functional Renormalization Group and O(4) scaling, Eur. Phys. J. C 67 (2010) 425 [arXiv:0904.0466] [INSPIRE].
Y. Hidaka and A. Yamamoto, Charged vector mesons in a strong magnetic field, Phys. Rev. D 87 (2013) 094502 [arXiv:1209.0007] [INSPIRE].
I. Shushpanov and A.V. Smilga, Quark condensate in a magnetic field, Phys. Lett. B 402 (1997) 351 [hep-ph/9703201] [INSPIRE].
N.O. Agasian and I. Shushpanov, Gell-Mann-Oakes-Renner relation in a magnetic field at finite temperature, JHEP 10 (2001) 006 [hep-ph/0107128] [INSPIRE].
J.O. Andersen, Thermal pions in a magnetic background, Phys. Rev. D 86 (2012) 025020 [arXiv:1202.2051] [INSPIRE].
J.O. Andersen, Chiral perturbation theory in a magnetic background - finite-temperature effects, JHEP 10 (2012) 005 [arXiv:1205.6978] [INSPIRE].
V. Orlovsky and Y. Simonov, Nambu-Goldstone mesons in strong magnetic field, JHEP 09 (2013) 136 [arXiv:1306.2232] [INSPIRE].
J.O. Andersen and A.A. Cruz, Two-color QCD in a strong magnetic field: The role of the Polyakov loop, Phys. Rev. D 88 (2013) 025016 [arXiv:1211.7293] [INSPIRE].
J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [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: 1312.3124
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Kamikado, K., Kanazawa, T. Chiral dynamics in a magnetic field from the functional renormalization group. J. High Energ. Phys. 2014, 9 (2014). https://doi.org/10.1007/JHEP03(2014)009
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2014)009