is of specific relevance for relativistic heavy ion collisions. Of fundamental importance in the study of the phase diagram are chiral symmetry and confinement, however the finite size ($m_u, m_d, m_s\neq 0$) as well as the difference in the bare quark masses ($m_u\neq m_d\neq m_s$) pose major problems both from the calculational point of view and in the implications due to deviations from the ideal situations, where the chiral condensate and the Polyakov loop are known to be the appropriate order parameters to characterize the phase state of the quark-gluon system.

The present study focuses on the chiral symmetry breaking aspects related to non-zero current quark mass values. Our arguments will be based on the successful model of Nambu–Jona-Lasinio (NJL) [@Nambu:1961], combined with the $U_A(1)$ breaking $2N_f$ flavor determinant of ’t Hooft [@Hooft:1976]-[@Reinhardt:1988] (NJLH). Moreover the most general $U(3)_L\otimes U(3)_R$ chiral invariant non derivative eight-quark ($8q$) interactions [@Osipov:2005b] are included. These terms were proven to render the effective potential of the NJLH model globally stable, and their effect has been thoroughly studied in low energy characteristics of pseudoscalar and scalar mesons [@Osipov:2006a], at finite temperature [@Osipov:2007b; @Osipov:2008] and in presence of a constant magnetic field [@Osipov:2007c]. These studies have lead 