Electroweak phase transitions

Recent developments on the four dimensional (4d) lattice studies of the finite temperature electroweak phase transition (EWPT) are summarized. The phase diagram is given in the continuum limit. The finite temperature SU(2)-Higgs phase transition is of first order for Higgs-boson masses m(H) < 66.5 +/- 1.4 GeV. Above this endpoint only a rapid cross-over can be seen. The full 4d result agrees completely with that of the dimensional reduction approximation. The Higgs-boson endpoint mass in the Standard Model (SM) would be 72.1 +/- 1.4 GeV. Taking into account the LEP Higgs-boson mass lower bound excludes any EWPT in the SM. A one-loop calculation of the static potential in the SU(2)-Higgs model enables a precise comparison between lattice simulations and perturbative results. The most popular extension of the SM, the Minimal Supersymmetric SM (MSSM) is also studied on 4d lattices.