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

Charge conjugation (C) and Parity (P) are exact symmetries at theta =pi and Theta = mu/(iT)=pi, where theta is the parameter of the so-called theta-vacuum, mu is the imaginary quark-number chemical potential and T is the temperature. Spontaneous breakings of these discrete symmetries are investigated by the the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) 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 theta =Theta =0 are remnants of the P restoration and the C breaking at theta =Theta =pi, respectively.