Interplay between quark-antiquark and diquark condensates in vacuum in a two-flavor Nambu-Jona-Lasinio model

Abstract

By means of a relativistic effective potential, we have analytically researched competition between the quark-antiquark condensates $<\bar{q}q>$ and the diquark condensates $<qq>$ in vacuum in ground state of a two-flavor Nambu-Jona-Lasinio (NJL) model and obtained the $G_S-H_S$ phase diagram, where $G_S$ and $H_S$ are the respective four-fermion coupling constants in scalar quark-antiquark channel and scalar color anti-triplet diquark channel. The results show that, in the chiral limit, there is only the pure $<\bar{q}q>$ phase when $G_S/H_S>2/3$, and as $G_S/H_S$ decreases to $2/3>G_S/H_S\geq 0$ one will first have a coexistence phase of the condensates $<\bar{q}q>$ and $<qq>$ and then a pure $<qq>$ phase. In non-zero bare quark mass case, the critical value of $G_S/H_S$ at which the pure $<\bar{q}q>$ phase will transfer to the coexistence phase of the condensates $<\bar{q}q>$ and $<qq>$ will be less than 2/3. Our theoretical results, combined with present phenomenological fact that there is no diquark condensates in the vacuum of QCD, will also impose a real restriction to any given two-flavor NJL model which is intended to simulate QCD, i.e. in such model the resulting smallest ratio $G_S/H_S$ after the Fierz transformations in the Hartree approximation must be larger than 2/3. A few phenomenological QCD-like NJL models are checked and analyzed.