A self consistent study of the phase transition in the scalar electroweak theory at finite temperature

We propose the study of the phase transition in the scalar electroweak theory at finite temperature by a two - step method. It combines i) dimensional reduction to a 3-dimensional {\it lattice\/} theory via perturbative blockspin transformation, and ii) either further real space renormalization group transformations, or solution of gap equations, for the 3d lattice theory. A gap equation can be obtained by using the Peierls inequality to find the best quadratic approximation to the 3d action. % This method avoids the lack of self consistency of the usual treatments which do not separate infrared and UV-problems by introduction of a lattice cutoff. The effective 3d lattice action could also be used in computer simulations