Imaginary Rotation Breaks Charge Conjugation in Hot QCD

Abstract

Starting from the identity $e^{2πi J_z}=(-1)^{F}=e^{iπQ}$ on the color-singlet physical state space, we derive a density-operator identity that ties imaginary rotation to an imaginary quark chemical potential, $ρ(Ω_I+2π,θ_q)=ρ(Ω_I,θ_q+π)$. Hence imaginarily rotating $SU(3)$ QCD at $(Ω_I,θ_q)=(2π,0)$ is mapped exactly onto the Roberge-Weiss (RW) point $(0,π)$, 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 $Ω_{I,C}/π=(22-2\sqrt{22})/9$, and a subsequent first-order transition at $Ω_{I,{\rm lock}}/π=2/3+2\sqrt{111}/27$. At smaller $Ω_I$, 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.