The Electroweak Phase Transition through the Renormalization Group

We study the high-temperature phase transitions for the Abelian and $SU(2)$ Higgs models, using the exact renormalization group. The evolution equation for a properly-defined coarse-grained free energy is solved. The phase diagram of the Abelian Higgs model has a region of second-order phase transitions governed by fixed points, for which we calculate critical exponents and crossover curves. It also has a region of first-order phase transitions, which we discuss in terms of the coarse-grained free energy. The $SU(2)$ Higgs model for small Higgs field masses has only first-order phase transitions. We derive their properties and compare with lattice and perturbative results. For larger Higgs field masses the strength of the first-order phase transitions diminishes. For masses larger than 80-100 GeV we find evidence that they are replaced by analytical crossovers.