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Extensions of the NJL model which go beyond the original 4-quark interaction,which drives the dynamical mass generation, have proven to be quite successfulin describing low energy hadronic phenomenology. The inclusion of 8-quarkinteraction terms solved a metastability problem of the effective potentialintroduced by the inclusion of the 6-quark 't Hooft determinant term in the3-flavor version of the model (needed to eliminate the unwanted U(1) axialsymmetry) . 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 explicitchiral symmetry breaking, which are of the same order as the 't Hooft flavordeterminant in a 1/Nc expansion resulting in an unprecedented success inreproducing the low lying scalar and pseudoscalar meson spectra. This successcan be seen as a result of the inclusion of the full chiral symmetry breakingpattern. The two critical endpoints which are obtained in thetemperature/chemical potential phase diagram are shifted to lower chemicalpotential and higher temperature when the effect of magnetic field is takeninto account. For the studied magnetic field strengths (in the range$eH=0-0.4~GeV^2$) no significant extra transitions are seen to appear.

Strongly Interacting Matter Phase Diagram in the presence of Magnetic  Fields in an Extended Effective Lagrangian Approach with Explicit Chiral  Symmetry Breaking Interactions