  In a recent work we have proposed a perturbative approach for the study of
the phase transition of pure Yang-Mills theories at finite temperature. This is
based on a simple massive extension of background field methods in the
Landau-DeWitt gauge, where the gluon mass term is related to the existence of
Gribov ambiguities. We have shown that a one-loop calculation of the background
field effective potential describes well the phase structure of the SU(2) and
SU(3) theories. Here, we present the calculation of the next-to-leading-order
contribution in perturbation theory for the SU(2) case. In particular, we
compute the background field effective potential at two-loop order and the
corresponding Polyakov loop, a gauge invariant order parameter of the
transition, at one-loop order. We show that the two-loop correction brings the
critical temperature closer to its actual value as compared to the previous
one-loop result. We also compute the thermodynamic pressure as a function of
the temperature and show that two-loop contributions play an important role in
the vicinity of the phase transition.
