Quark and pion effective couplings from polarization effects

A flavor SU(2) effective model for pions and quarks is derived by considering polarization effects departing from the usual quark-quark effective interaction induced by dressed gluon exchange, i.e. a global color model for QCD. For that, the quark field is decomposed into a component that yields light mesons and the quark-antiquark condensate, being integrated out by means of the auxiliary field method, and another component which yields constituent quarks, which is basically a background quark field. Within a longwavelength and weak quark field expansion (or large quark effective mass expansion) of a quark determinant, the leading terms are found up to the second order in a zero order derivative expansion, by neglecting vector mesons that are considerably heavier than the pion. Pions are considered in the structureless limit and, besides the chiral invariant terms that reproduce previously derived expressions, symmetry breaking terms are also presented. The leading chiral quark-quark effective couplings are also found corresponding to a NJL and a vector-NJL couplings. All the resulting effective coupling constants and parameters are expressed in terms of the current and constituent quark masses and of the coupling $g$.