SpontaneousCPviolation in the strong interaction atθ=π

Abstract

Spontaneous $CP$ violation in the strong interaction is analyzed at $\ensuremath{\theta}=\ensuremath{\pi}$ within the framework of the two-flavor Nambu-Jona-Lasinio model. It is found that the occurrence of spontaneous $CP$ violation at $\ensuremath{\theta}=\ensuremath{\pi}$ depends on the strength of the 't Hooft determinant interaction, which describes the effect of instanton interactions. The dependence of the phase structure, and, in particular, of the $CP$-violating phase, on the quark masses, temperature, baryon and isospin chemical potential is examined in detail. When available a comparison to earlier results from chiral perturbation theory is made. From our results we conclude that spontaneous $CP$ violation in the strong interaction is an inherently low-energy phenomenon. In all cases we find agreement with the Vafa-Witten theorem, also at nonzero density and temperature. Meson masses and mixing in the $CP$-violating phase display some unusual features as a function of instanton interaction strength. A modification of the condition for charged pion condensation at nonzero isospin chemical potential and a novel phase of charged ${a}_{0}$ mesons are discussed.