HIGGS SECTOR RENORMALIZATION-GROUP IN THE (MS)OVER-BAR AND ON-MASS-SHELL SCHEME - THE BREAKDOWN OF PERTURBATION-THEORY FOR A HEAVY HIGGS-BOSON

We discuss different aspects of the Higgs self-interaction in the <(MS )over bar> and the on-mass-shell (OMS) scheme. The running coupling la mbda(mu) is investigated in great detail. The three-loop coefficient o f the beta function in the OMS scheme is derived, and the three-loop r unning coupling is calculated. The breakdown of perturbation theory fo r large Higgs boson masses M(H) is analyzed in three physical observab les for which two-loop results are known. Requiring the dependence on the renormalization scale to diminish order by order in lambda, we fin d that perturbation theory breaks down for M(H) similar to 700 GeV in Higgs boson decays. Similarly, M(H) must be smaller than similar to 40 0 GeV for perturbatively calculated cross sections to be trustworthy u p to c.m. energies of similar to 2 TeV. If the Higgs sector shall be p erturbative up to the GUT scale, the Higgs boson must be lighter than similar to 150 GeV, For the two-loop observables examined, the apparen t convergence of the perturbation series is better in the OMS scheme t han in the <(MS)over bar> scheme.