Radial distribution of charge-conjugation breaking away from the rotation axis
Characterize the radial distribution of charge-conjugation breaking away from the rotation axis in imaginarily rotating hot QCD, beyond the one-loop analysis restricted to \(r=0\).
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The analysis is also restricted to the rotation axis---away from it the potential depends on \tilde r=rT---so, while eq:imaginary-rotation-C:RW-operator-mapping holds system-wide, the radial distribution of C breaking remains open.
The remaining question is how the order parameters vary between the axis and the bulk. On the axis the rotation leaves the point fixed, so the temperature remains $T$ and the rotation acts only on spin, and the minimum selected by the local effective potential there need not agree with the bulk state given by Eq.~eq:theorem. For instance, $\tilde\Omega_I=2\pi/3$ lies outside the on-axis $C$-breaking interval $1.40\pi\lesssim\tilde\Omega_I\lesssim2.60\pi$, so the potential on the axis selects a $C$-symmetric minimum, whereas the bulk breaks $C$ for $T>3T_{\rm RW}$; conversely, at $\tilde\Omega_I=3\pi/2$ the potential on the axis selects a $C$-broken minimum, whereas the bulk is the equivalent system at $\theta'=0$ and is $C$ symmetric. Likewise, the perturbative confinement found on the axis is a prediction of a one-loop calculation in the high-temperature limit, whereas at irrational $\tilde\Omega_I/2\pi$ the bulk is confined at every temperature, inheriting confinement from zero-temperature QCD. The one-loop calculation on the axis is justified as $T\to\infty$, yet at $\tilde\Omega_I=2\pi/3$ and $3\pi/2$ the potential on the axis and the bulk select different states even in that limit. The order parameter selected in the bulk may, however, reach the neighborhood of the axis through gradient terms and correlations; its actual value on the axis and its spatial distribution at finite radius $r$ require an analysis including gradient terms and are not addressed here.
At $T=qT_{\rm RW}$, with $r\to\infty$ taken first, the bulk inherits the criticality of the RW endpoint; how this criticality is approached at finite $r$, where the correlation length is not negligible compared with $r$, cannot be determined from Eq.~eq:theorem and remains, together with how the axis alters the critical fluctuations, a subject for future work.