SU(2) higher-order effective quark interactions from polarization

Higher order quark effective interactions are found for SU(2) flavor by departing from a non local quark-quark interaction. By integrating out a component of the quark field, the determinant is expanded in chirally symmetric and symmetry breaking effective interactions up to the fifh order in the quark bilinears. The resulting coupling constants are resolved in the leading order of the longwavelength limit and exact numerical ratios between several of these coupling constants are obtained in the large quark mass limit. In this level, chiral invariant interactions only show up in even powers of the quark bilinears, i.e. ${\cal O}(\bar{\psi} \psi)^{2 n}$ ($n=1,2,3,..$), whereas (explicit) chiral symmetry breaking terms emerge as ${\cal O}(\bar{\psi} \psi)^{n}$ being always proportional to some power of the Lagrangian quark mass.