In this workshop we have presented the results obtained in the three-flavour ($N_f=3$) Nambu--Jona-Lasinio model Lagrangian which includes all non-derivative vertices at NLO in the $1/N_c$ expansion of spin zero multi-quark interactions. In particula
r the role played by the explicit chiral symmetry breaking interactions has been discussed in comparison with previous model Lagrangians.
We call attention to a class of current-quark mass dependent multi-quark interaction terms which break explicitly the chiral $SU(3)_Ltimes SU(3)_R$ and $U_A(1)$ symmetries. They complete the set of effective quark interactions that contribute at the
same order in $N_c$ as the t Hooft flavor determinant interaction and the eight quark interactions in the phase of spontaneously broken chiral symmetry. The $N_c$ classification scheme matches the counting rules based on arguments set by the scale of spontaneous chiral symmetry breaking. Together with the leading in $N_c$ four quark Nambu-Jona-Lasinio Lagrangian and current quark mass matrix, the model is apt to account for the correct empirical ordering and magnitude of the splitting of states in the low lying mass spectra of spin zero mesons. The new terms turn out to be essential for the ordering $m_K < m_eta$ in the pseudoscalar sector and $m_{kappa_0} < m_{a_0}sim m_{f0}$ for the scalars.
We present the results obtained in the three-flavour ($N_f=3$) Nambu--Jona-Lasinio model which is extended by the $U(1)_A$ breaking six-quark t Hooft interaction and eight-quark interactions. We address the problem of stability, and some phenomenolog
ical consequences of the models with multi-quark interactions.
The influence of a constant magnetic field on the order parameter of the four-dimensional Nambu and Jona-Lasinio model extended by the t Hooft six-quark term and eight-quark interactions is considered. It is shown that the multi-quark interactions ca
use the order parameter to increase sharply (secondary magnetic catalysis) with increasing strength of the field at the characteristic scale $Hsim 10^{14}Lambda^2$ G/MeV^2.
