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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 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 splitted into two components, one that condenses and another one that is a background field for interacting quarks, being the former 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.