Covariant variational approach to Yang-Mills theory: Effective potential of the Polyakov loop

We compute the effective action of the Polyakov loop in $SU(2)$ and $SU(3)$ Yang-Mills theory using a previously developed covariant variational approach. The formalism is extended to background gauge and it is shown how to relate the low-order Green's functions to the ones in Landau gauge studied earlier. The renormalization procedure is discussed. The self-consistent effective action is derived and evaluated using the numerical solution of the gap equation. We find a clear signal for a deconfinement phase transition at finite temperatures, which is second order for $SU(2)$ and first order for $SU(3)$. The critical temperatures obtained are in reasonable agreement with high-precision lattice data.