New self-consistent mean field approximation and its application in strong interaction phase transition

In this letter, taking the Nambu--Jona--Lasinio model as an example, we propose a new self-consistent mean field approximation method by means of Fierz transformation. This new self-consistent mean field approximation introduces a new free parameter $\alpha$ to be determined by experiments and when $\alpha$ takes 0.5, it reduces to the mean field approximation that was commonly used in the past. Then based on this self-consistent mean field approximation, we study the influence of the undetermined parameter $\alpha$ on the phase diagram of the two-flavor strong interaction matter. It is found that the value of $\alpha$ plays an extremely important role in the study of strong interaction phase diagram. It not only changes the position of the phase transition point of strong interaction matter, but also affects the order of phase transition, for example, when $\alpha$ is greater than the critical value $\alpha_c = 0.71$, then the strong interaction matter phase diagram no longer exists critical end point. In addition, in the case of zero temperature and finite density, we also found that when $\alpha$ is greater than 1.044, the pseudo-critical chemical potential is about 4$\sim$5 times the saturation density of the nuclear matter, which agrees with the expected results from the image of hadrons degree of freedom. The resulting equations of state of strong interaction matter at low temperatures and high densities will have an important impact on the study of the mass radius relationship of neutron stars and the merging process of binary neutron stars.