Charge conjugation ($C$) and parity ($P$) are exact symmetries at $\theta =\pi$ and $\Theta \equiv \mu /(iT)=\pi$, 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 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 $\theta =\Theta =0$ are remnants of the $P$ restoration and the $C$ breaking at $\theta =\Theta =\pi$, respectively.

• Received 3 February 2011

DOI:https://doi.org/10.1103/PhysRevD.83.076009