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 ${\Omega }_{I,\mathrm{lock}}/\pi =2/3+2\sqrt{111}/27$. 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.

Comments: 6 pages, 4 figures