A new algorithm towards quasi-Wigner solution of the gap equation beyond the chiral limit

We propose a new algorithm to solve the quasi-Wigner solution of the gap equation beyond chiral limit. Employing a Gaussian gluon model and rainbow truncation, we find that the quasi-Wigner solution exists in a limited region of current quark mass, $m<43.1$ MeV, at zero temperature $T$ and zero chemical potential $\mu$. The difference between Cornwall-Jackiw-Tomboulis (CJT) effective actions of quasi-Wigner and Nambu-Goldstone solutions shows that the Nambu-Goldstone solution is chosen by physics. Moreover, the quasi-Wigner solution is studied at finite temperature and chemical potential, the far infrared mass function of quasi-Wigner solution is negative and decrease along with $T$ at $\mu=0$. Its susceptibility is divergent at certain temperature with small $m$, and this temperature decreases along with $m$. Taking $T=80$ MeV as an example, the quasi-Wigner solution is shown at finite chemical potential upto $\mu=350$ MeV as well as Nambu solution, the coexistence of these two solutions indicates that the QCD system suffers the first order phase transition. The first order chiral phase transition line is determined by the difference of CJT effective actions.