The 2–Loop Effective Potential of the 3d SU(2)–Higgs Model in a General Covariant Gauge

To study the convergence of the loop expansion at the high–temperature electroweak phase transition, we calculate the 2–loop effective potential of the 3d SU(2)–Higgs model in a general covariant gauge. We find that the loop expansion definitely breaks down for large ξ, but converges rather well for smaller values, deep in the broken phase. 1 Electronic address: [email protected] Recently, both perturbative [1, 2, 3] and non–perturbative [4, 5] studies of the high–T electroweak phase transition have been performed, with the indication that perturbation theory works poorly in the symmetric phase, due to non–perturbative effects. The purpose of this note is to investigate the convergence of loop expansion in the broken phase. The method is to calculate the high–T asymptotic of the 2–loop effective potential in a general covariant gauge, and to extract from it a quantity which is gauge–independent in the full theory. The degree of gauge–dependence at the 2– loop level is then expected to tell something about the convergence of the expansion. In accordance with this philosophy, we do not convert the loop expansion into an expansion in the coupling constants at any stage. The convergence of loop expansion in the broken phase has also been studied in ref. [5], by comparing lattice data and perturbation theory. Our starting point is the action