The Finite temperature SU(2) Savvidy model with a nontrivial Polyakov loop

We calculate the complete one-loop effective potential for $\mathrm{SU}(2)$ gauge bosons at temperature T as a function of two variables: $\ensuremath{\varphi},$ the angle associated with a nontrivial Polyakov loop, and H, a constant background chromomagnetic field. These two variables are indicators for confinement and scale symmetry breaking, respectively. Using techniques broadly applicable to finite temperature field theories, we develop both low and high temperature expansions. At low temperatures, the real part of the effective potential ${V}_{R}$ indicates a rich phase structure, with a discontinuous alternation between confined $(\ensuremath{\varphi}=\ensuremath{\pi})$ and deconfined phases $(\ensuremath{\varphi}=0).$ The background field H moves slowly upward from its zero temperature value as T increases, in such a way that $\sqrt{\mathrm{gH}}/\ensuremath{\pi}T$ is approximately an integer. This behavior stops at ${T}_{c}=0.722(1){\ensuremath{\mu}}_{0},$ where ${\ensuremath{\mu}}_{0}$ is a zero-temperature renormalization group invariant scale; beyond this temperature, the deconfined phase is always preferred. At high temperatures, where perturbation theory should be reliable as a consequence of asymptotic freedom, the deconfined phase $(\ensuremath{\varphi}=0)$ is always preferred, and $\sqrt{\mathrm{gH}}$ is of order ${g}^{2}(T)T.$ The imaginary part of the effective potential ${V}_{I},$ which originates in a tachyonic mode associated with the lowest Landau level, is nonzero at the global minimum of ${V}_{R}$ for all temperatures. A nonperturbative magnetic screening mass of the form ${M}_{m}{=cg}^{2}(T)T$ with a sufficiently large coefficient c removes this instability at high temperature, leading to a stable high-temperature phase with $\ensuremath{\varphi}=0$ and $H=0,$ characteristic of a weakly interacting gas of gauge particles. The value of ${M}_{m}$ obtained is comparable with lattice estimates.