Low energy constituent quark and pion effective couplings in a weak external magnetic field

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 is recovered. Some numerical estimates are provided for the effective coupling constants and parameters.