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.
Several relevant thermodynamic observables obtained within the (2+1) flavor and spin zero NJL and PNJL models with inclusion of the t Hooft determinant and $8q$ interactions are compared with lattice-QCD (lQCD) results. In the case that a small ratio
$R=frac{mu_B}{T_c}sim 3$ at the critical end point (CEP) associated with the hadron gas to quark-gluon plasma transition is considered, combined with fits to the lQCD data of the trace anomaly[1], subtracted light quark condensate[1] and continuum extrapolated data of the light quark chiral condensate[2], a reasonable description for the PNJL model is obtained with a strength $g_1sim 5...6 times 10^3$ GeV$^{-8}$ of the $8q$ interactions. The dependence on the further model parameters is discussed.
It is shown that the endpoint of the first order transition line which merges into a crossover regime in the phase diagram of the Nambu--Jona-Lasinio model, extended to include the six-quark t Hooft and eight-quark interaction Lagrangians, is pushed
towards vanishing chemical potential and higher temperatures with increasing strength of the OZI-violating eight-quark interactions. We clarify the connection between the location of the endpoint in the phase diagram and the mechanism of chiral symmetry breaking at the quark level. We show how the 8q interactions affect the number of effective quark degrees of freedom. We are able to obtain the correct asymptotics for this number at large temperatures by using the Pauli-Villars regularization.
It is shown that the endpoint of the first order transition line which merges into a crossover regime in the phase diagram of the Nambu--Jona-Lasinio model, extended to include the six-quark t Hooft and eight-quark interaction Lagrangians, is pushed
towards vanishing chemical potential and higher temperatures with increasing strength of the OZI-violating eight-quark interactions. We clarify a connection between the location of the endpoint in the phase diagram and the mechanism of chiral symmetry breaking at the quark level. Constraints on the coupling strengths based on groundstate stability and physical considerations are explained.
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.
In this talk attention is drawn to thermal properties due to the addition of eight quark interactions in the standard $SU(3)times SU(3)$ chiral Nambu-Jona-Lasinio model (NJL) with t Hooft interaction (NJLH). The schematic SU(3) flavor limit with mass
less current quarks as well as the realistic case $m_u=m_d e m_s$ are discussed.
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.
A NJL Lagrangian extended to six and eight quark interactions is applied to study temperature effects (SU(3) flavor limit, massless case), and (realistic massive case). The transition temperature can be considerably reduced as compared to the standar
d approach, in accordance with recent lattice calculations. The mesonic spectra built on the spontaneously broken vacuum induced by the t Hooft interaction strength, as opposed to the commonly considered case driven by the four-quark coupling, undergoes a rapid crossover to the unbroken phase, with a slope and at a temperature which is regulated by the strength of the OZI violating eight-quark interactions. This strength can be adjusted in consonance with the four-quark coupling and leaves the spectra unchanged, except for the sigma meson mass, which decreases. A first order transition behavior is also a possible solution within the present approach.
The thermodynamic potential and thermal dependence of low lying mass spectra of scalars and pseudoscalars are evaluated in a generalized Nambu -- Jona-Lasinio model, which incorporates eight-quark interactions. These are necessary to stabilize the sc
alar effective potential for the light and strange quark flavors, which would be otherwise unbounded from below. In addition it turns out that they are also crucial to i) lower the temperature of the chiral transition, in conformity with lattice calculations, ii) sharpen the temperature interval in which the crossover occurs, iii) or even allow for first order transitions to occur with realistic quark mass values, from certain critical values of the parameters. These are unprecedented results which cannot be obtained within the NJL approaches restricted to quartic and six-quark interactions.
