Toward an analytic determination of the deconfinement temperature in SU(2) L.G.T.

We consider the SU(2) lattice gauge theory at finite temperature in (d+1) dimensions, with different couplings $\beta_t$ and $\beta_s$ for timelike and spacelike plaquettes. By using the character expansion of the Wilson action and performing the integrals over space-like link variables, we find an effective action for the Polyakov loops which is exact to all orders in $\beta_t$ and to the first non-trivial order in $\beta_s$. The critical coupling for the deconfinement transition is determined in the (3+1) dimensional case, by the mean field method, for different values of the lattice size $N_t$ in the compactified time direction and of the asymmetry parameter $\rho = \sqrt{\beta_t/\beta_s}$. We find good agreement with Montecarlo simulations in the range $1\leq N_t \leq 5$, and good qualitative agreement in the same range with the logarithmic scaling law of QCD. Moreover the dependence of the results from the parameter $\rho$ is in excellent agreement with previous theoretical predictions.