Abstract.
An effective model with pions and constituent quarks in the presence of a weak external background electromagnetic field is derived by starting from a dressed one gluon exchange quark-quark interaction. By applying the auxiliary field and background field methods, the structureless pion limit is considered to extract effective pion and constituent quark couplings in the presence of a weak magnetic field. The leading terms of a large quark and gluon masses expansion are obtained by resolving effective coupling constants which turn out to depend on a weak magnetic field. Two pion field definitions are considered for that. Several relations between the effective coupling constants and parameters can be derived exactly or in the limit of very large quark mass at zero and weak constant magnetic field. Among these ratios, the Gell-Mann-Oakes-Renner and the quark level Goldberger-Treiman relations are obtained. In addition to that, in the pion sector, the leading terms of Chiral Perturbation Theory coupled to the electromagnetic field are recovered. Some numerical estimates are provided for the effective coupling constants and parameters.
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References
J. Gasser, H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984)
S. Scherer, Eur. Phys. J. A 28, 59 (2006)
S. Scherer, Prog. Part. Nucl. Phys. 64, 1 (2010)
J. Bijnens, G. Ecker, Annu. Rev. Nucl. Part. Sci. 64, 149 (2014)
S. Weinberg, Physica A 96, 327 (1979)
Yu.A. Simonov, Phys. Rev. D 65, 094018 (2002)
A.A. Osipov, B. Hiller, A.H. Blin, Eur. Phys. J. A 49, 14 (2013)
D. Ebert, H. Reinhardt, M.K. Volkov, Prog. Part. Nucl. Phys. 33, 1 (1994)
C.D. Roberts, R.T. Cahill, J. Praschifka, Ann. Phys. 188, 20 (1988)
B. Holdom, Phys. Rev. D 45, 2534 (1992)
Q. Wang, Yu-P. Kuang, X-Lei Wang, M. Xiao, Phys. Rev. D 61, 054011 (2000)
K. Ren, H.-F. Fu, Q. Wang, Phys. Rev. D 95, 074012 (2017)
S.P. Klevansky, Rev. Mod. Phys. 64, 649 (1992)
U. Vogl, W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991)
T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 1 (1994)
F.L. Braghin, Eur. Phys. J. A 52, 134 (2016)
E. de Rafael, Phys. Lett. B 703, 60 (2011)
M. Lavelle, D. McMullan, Phys. Rep. 279, 1 (1997)
A.W. Thomas, Nucl. Phys. B Proc. Suppl. 119, 50 (2003)
R.D. Young, D.B. Leinweber, A.W. Thomas, Prog. Part. Nucl. Phys. 50, 399 (2003) and references therein
S. Weinberg, The Quantum Theory of Fields, Vol. II (Cambridge University Press, 1996)
S.J. Brodsky, R. Shrock, Phys. Lett. B 666, 95 (2008)
S.J. Brodsky, R. Shrock, arXiv:0803.2541
S.J. Brodsky, R. Shrock, Proc. Natl. Acad. Sci. U.S.A. 108, 45 (2011)
S.J. Brodsky, R. Shrock, arXiv:0803.2554
H. Reinhardt, H. Weigel, Phys. Rev. D 85, 074029 (2012)
A. Manohar, G. Georgi, Nucl. Phys. B 233, 232 (1984)
S. Weinberg, Phys. Rev. Lett. 105, 261601 (2010)
K. Tuchin, Adv. High Energy Phys. 2013, 490495 (2013)
J.O. Andersen, W.R. Naylor, A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016) arXiv:1411.7176
V.A. Miransky, I.A. Shovkovy, Phys. Rep. 576, 1 (2015)
V.A. Miransky, I.A. Shovkovy, Phys. Rev. D 66, 045006 (2002)
V.P. Gusynin, V.A. Miransky, I.A. Shovkovy, Nucl. Phys. B 462, 249 (1996)
G.S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S.D. Katz, A. Schafer, Phys. Rev. D 86, 071502 (2012) arXiv:1206.4205
K. Fukushima, Y. Hidaka, Phys. Rev. Lett. 110, 031601 (2013)
F. Bruckmann, G. Endrodi, T.G. Kovacs, J. High Energy Phys. 04, 112 (2013)
B. Chatterjee, H. Mishra, A. Mishra, Phys. Rev. D 91, 034031 (2015)
M.N. Chernodub, Phys. Rev. Lett. 106, 142003 (2011)
K. Fukushima, D.E. Kharzeev, H.J. Warringa, Phys. Rev. D 78, 074033 (2008) arXiv:0808.3382 [hep-ph]
D.E. Kharzeev, Prog. Part. Nucl. Phys. 75, 133 (2014)
D.E. Kharzeev, Annu. Rev. Nucl. Part. Sci. 65, 193 (2015)
E.V. Gorbar, Phys. Lett. B 695, 354 (2011)
E.V. Gorbar, V.A. Miransky, I.A. Shovkovy, X. Wang, Phys. Rev. D 88, 025043 (2013)
L. Xia, Phys. Rev. D 90, 085011 (2014)
J. Chao, P. Chu, M. Huang, Phys. Rev. D 88, 054009 (2013) arXiv:1305.1100 [hep-ph]
L. Yu, H. Liu, M. Huang, arXiv:1404.6969 [hep-ph]
E.J. Ferrer, V. de la Incera, A. Sanchez, Phys. Rev. Lett. 107, 041602 (2011)
E.J. Ferrer, V. de la Incera, X.J. Wen, Phys. Rev. D 91, 054006 (2015)
E.J. Ferrer et al., Rev. D 89, 085034 (2014)
R.L.S. Farias, K.P. Gomes, G. Krein, M.B. Pinto, Phys. Rev. C 90, 025203 (2014)
A. Ayala,, arXiv:1510.09134 [hep-ph]
C.-F. Li, L. Yang, X.J. Wen, G.X. Peng, Phys. Rev. D 93, 054005 (2016)
M.A. Andreichikov, V.D. Orlovsky, Y.A. Simonov, Phys. Rev. Lett. 110, 162002 (2013)
F.L. Braghin, Phys. Rev. D 94, 074030 (2016)
F.L. Braghin, Phys. Rev. D 97, 014022 (2018)
D. Binosi, L. Chang, J. Papavassiliou, C.D. Roberts, Phys. Lett. B 742, 183 (2015) and references therein
K.-I. Kondo, Phys. Rev. D 57, 7467 (1998)
J.M. Cornwall, Phys. Rev. D 83, 076001 (2011)
P. Maris, C.D. Roberts, Int. J. Mod. Phys. E 12, 297 (2003)
P. Tandy, Prog. Part. Nucl. Phys. 39, 117 (1997)
K. Higashijima, Phys. Rev. D 29, 1228 (1984)
K.I. Aoki et al., Prog. Theor. Phys. 84, 683 (1990)
A.C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D 84, 085026 (2011)
M. Gell-Mann, R.J. Oakes, B. Renner, Phys. Rev. 175, 2195 (1968)
M.L. Goldberger, S. Treiman, Phys. Rev. 111, 354 (1966)
D. Barquilla-Cano, A.J. Buchmann, E. Hernández, Nucl. Phys. A 714, 611 (2003) arXiv:[nucl-th]/0204067
D. Ebert, H. Reinhardt, Nucl. Phys. B 271, 188 (1986)
S. Weinberg, Phys. Rev. 166, 1568 (1968)
L.F. Abbott, Acta Phys. Pol. B 13, 33 (1982)
M.O.C. Gomes, Teoria Quântica de Campos (EDUSP, São Paulo, Brazil, 2002)
F.L. Braghin, Phys. Rev. D 64, 125001 (2001)
H. Kleinert, in Erice Summer Institute 1976, Understanding the Fundamental Constituents of Matter, edited by A. Zichichi (Plenum Press, New York, 1978) p. 289
G. ’t Hooft, Nucl. Phys. B 72, 461 (1974)
W. Heisenberg, H. Euler, Z. Phys. 98, 714 (1936) translated into English: arXiv:[physics]/0605038
G.S. Bali et al., JHEP 02, 44 (2012) arXiv:1111.4956
G. Endrodi, JHEP 04, 023 (2013) arXiv:1301.1307
J. Schwinger, Phys. Rev. 82, 664 (1951)
J. Schwinger, Phys. Rev. 93, 615 (1953)
F.L. Braghin, Phys. Lett. B 761, 424 (2016)
E. Witten, Nucl. Phys. B 160, 57 (1979)
S. Weinberg, Phys. Rev. Lett. 18, 188 (1967)
J. Greensite, An Introduction to the Confinement Problem (Springer, Heildelberg, 2011)
P. Watson, H. Reinhardt, Phys. Rev. D 89, 045008 (2014)
G.S. Bali et al., J. High Energy Phys. 04, 130 (2013)
C. Itzikson, J.-B. Zuber, Quantum Field Theory (McGraw Hill, 1985)
J.O. Andersen, JHEP 10, 005 (2012)
A. Paulo jr., F.L. Braghin, Phys. Rev. D 90, 014049 (2014)
M. Benayoun, H.B. O’Connell, A.G. Williams, Phys. Rev. D 59, 074020 (1999)
G. Kalbermann, Phys. Rev. D 33, 1987 (1986)
T.-K. Chyi et al., Phys. Rev. D 62, 105014 (2000)
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Braghin, F.L. Low energy constituent quark and pion effective couplings in a weak external magnetic field. Eur. Phys. J. A 54, 45 (2018). https://doi.org/10.1140/epja/i2018-12485-6
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DOI: https://doi.org/10.1140/epja/i2018-12485-6