Bounds on the Higgs mass in the standard model and the minimal supersymmetric standard model

Depending on the Higgs-boson and top-quark masses, M_H and M_t, the effective potential of the {\bf Standard Model} can develop a non-standard minimum for values of the field much larger than the weak scale. In those cases the standard minimum becomes metastable and the possibility of decay to the non-standard one arises. Comparison of the decay rate to the non-standard minimum at finite (and zero) temperature with the corresponding expansion rate of the Universe allows to identify the region, in the (M_H, M_t) plane, where the Higgs field is sitting at the standard electroweak minimum. In the {\bf Minimal Supersymmetric Standard Model}, approximate analytical expressions for the Higgs mass spectrum and couplings are worked out, providing an excellent approximation to the numerical results which include all next-to-leading-log corrections. An appropriate treatment of squark decoupling allows to consider large values of the stop and/or sbottom mixing parameters and thus fix a reliable upper bound on the mass of the lightest CP-even Higgs boson mass. The discovery of the Higgs boson at LEP~2 might put an upper bound (below the Planck scale) on the scale of new physics \Lambda and eventually disentangle between the Standard Model and the Minimal Supersymmetric Standard Model.