Violations of parity and charge conjugation in the θ vacuum with imaginary chemical potential

Charge conjugation (C) and parity (P) are exact symmetries at θ=π and Θμ/(iT)=π, where θ is the parameter of the so-called θ vacuum, μ is the imaginary quark-number chemical potential and T is the temperature. Spontaneous breakings of these discrete symmetries are investigated by the Polyakov-loop extended Nambu-Jona-Lasinio model. At zero T, P symmetry is spontaneously broken while C symmetry is conserved. As T increases, P symmetry is restored just after C symmetry is spontaneously broken, so that either P or C symmetry or both the symmetries are spontaneously broken for any T. The chiral-symmetry restoration and the deconfinement transition at θ=Θ=0 are remnants of the P restoration and the C breaking at θ=Θ=π, respectively.