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 $(e{B}_0/{M^{*}}^2)<1$ 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.

Figure 1

In these diagrams, the wavy line with a full dot is a (dressed) nonperturbative gluon propagator, and the short bold line insertions for the vector potential, whereas the full triangle is for the magnetic field insertion ($F^{\mu \nu }$). Diagram (a) shows the effective mass due to the magnetic field contribution, whereas diagrams (b1)–(b3) represent all the quark-quark effective interactions shown above.