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Effective potential of gluodynamics in background of Polyakov loop and colormagnetic field

In SU(N) gluodynamics, above the de-confinement temperature, the effective potential has minima at non-zero $A_0$-background fields in the two-loop approximation. Also, it has a minimum at non-zero chromomagnetic background field, known as 'Savvidy'-vacuum, which shows up on the one-loop level. In this paper, we join these two approaches. We formulate, at finite temperature, the effective action, or the free energy, in SU(2) gluodynamics on the two-loop level, with both, $A_0$ background and magnetic background present at the same time, which was not done so far. We provide the necessary representations for both, effective numerical calculation and high-temperature expansions. The results are represented as a 3D plot of the real part of the effective potential. Also, we reproduce for zero either, the $A_0$-background or the magnetic background, the known minima and compare them. The imaginary part is, on the two-loop level, still present. We mention that, as is known from literature for the case without $A_0$-background, the imaginary part is compensated by the ring ('daisy') diagrams. However, our results reveal an unnatural, singular behavior of the effective potential in the region, where the imaginary part sets in. Our conclusion is that one has to go beyond the two-loop approximation and its ring improved version, in order to investigate the minimum of the effective action as a function of $A_0$ and chromomagnetic field, and its stability, at least in the approximation of superdaisy diagrams, i.e., the Hartree approximation in the CJT formalism.