Extensions of the NJL model which go beyond the original 4-quark interaction, which drives the dynamical mass generation, have proven to be quite successful in describing low energy hadronic phenomenology. The inclusion of 8-quark interaction terms solved a metastability problem of the effective potential
introduced by the inclusion of the 6-quark 't Hooft determinant term in the 3-flavor version of the model (needed to eliminate the unwanted U(1) axial symmetry) . This model, that has proven to be quite powerful and feature-rich, has been expanded to include all the spin-0 terms, without and with explicit chiral symmetry breaking, which are of the same order as the 't Hooft flavor determinant in a 1/Nc expansion resulting in an unprecedented success in reproducing
the low lying scalar and pseudoscalar meson spectra. This success can be seen as a result of the inclusion of the full chiral symmetry breaking pattern. The two critical endpoints which are obtained in the temperature/chemical potential phase diagram are shifted to lower chemical potential and higher temperature when the effect of magnetic field is taken into account. For the studied magnetic field strengths (in the range $eH=0-0.4~GeV^2$) no significant extra transitions are seen to appear.