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QCD $θ$-vacua from the chiral limit to the quenched limit

We investigate the dependence of the QCD vacuum structure on the $θ$-angle and quark mass, using the Di-Vecchia--Veneziano model. Although the Di-Vecchia--Veneziano model is a chiral effective model, it contains the topological properties of the pure Yang--Mills theory. It is shown that within this model, the ground state energies for all $θ$ are continuous functions of quark mass from the chiral limit to the quenched limit, even including the first order phase transition at $θ= π$. Based on this effective model, we discuss (i) how the ground state depends on quark mass, and (ii) why the phase transition at $θ= π$ is present both in the chiral and quenched limit. In order to analyze the relation between quark mass and the $θ$-vacua, we calculate the chiral condensate as a function of quark mass. Also, considering the presence of the innate metastable states included in the QCD $θ$-vacuum, we also give a unified understanding of the phase transitions at $θ= π$ in the chiral and quenched limit.