SU(2) low energy quark effective couplings in weak external magnetic field

In this work corrections to the usual flavor SU(2) Nambu-Jona-Lasinio (NJL) coupling due to a weak external magnetic field are calculated by considering quark polarization in a (dressed) gluon exchange mechanism for quark interactions. The quark field is split into two components: one that condenses and another one that is a background field for interacting quarks, the former being integrated out. The resulting determinant is expanded for relatively large quark mass and small magnetic field (eB(0)/M*(2)) < 1 by resolving magnetic-field-dependent low energy quark effective interactions. Besides the corrections for the NJL and vector NJL effective couplings, different B-0-dependent effective couplings that break isospin and chiral symmetry emerge.