  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.

High Energy Physics - Theory