Imaginary Rotation Breaks Charge Conjugation in Hot QCD

Abstract

Starting from the identity $e^{2\pi i J_z}=(-1)^{F}=e^{i\pi Q}$ on the color-singlet physical state space, we derive a density-operator identity that ties imaginary rotation to an imaginary quark chemical potential, $\rho(\Omega_I+2\pi,\theta_q)=\rho(\Omega_I,\theta_q+\pi)$. Hence imaginarily rotating $SU(3)$ QCD at $(\Omega_I,\theta_q)=(2\pi,0)$ is mapped exactly onto the Roberge-Weiss (RW) point $(0,\pi)$, where charge conjugation $C$ is spontaneously broken above the RW endpoint---a conclusion independent of any model or approximation. Minimizing the one-loop effective potential on the rotation axis, we obtain analytically, for three massless flavors, the second-order transition at $\Omega_{I,C}/\pi=(22-2\sqrt{22})/9$, and a subsequent first-order transition at $\Omega_{I,{\rm lock}}/\pi=2/3+2\sqrt{111}/27$. At smaller $\Omega_I$, a continuous degeneracy of the massless one-loop approximation leaves the realization of $C$ undecided; a finite strange-quark mass lifts it. We also state how the exact relations and the on-axis predictions can be tested in lattice QCD.