Higgs sector renormalization group in the MS-bar and on-mass-shell scheme: The breakdown of perturbation theory for a heavy Higgs boson.

Abstract

We discuss different aspects of the Higgs self-interaction in the $\stackrel{-}{\mathrm{MS}}$ and the on-mass-shell (OMS) scheme. The running coupling $\ensuremath{\lambda}(\ensuremath{\mu})$ is investigated in great detail. The three-loop coefficient of the $\ensuremath{\beta}$ function in the OMS scheme is derived, and the three-loop running coupling is calculated. The breakdown of perturbation theory for large Higgs boson masses ${M}_{H}$ is analyzed in three physical observables for which two-loop results are known. Requiring the dependence on the renormalization scale to diminish order by order in $\ensuremath{\lambda}$, we find that perturbation theory breaks down for ${M}_{H}\ensuremath{\sim}700$ GeV in Higgs boson decays. Similarly, ${M}_{H}$ must be smaller than \ensuremath{\sim}400 GeV for perturbatively calculated cross sections to be trustworthy up to c.m. energies of \ensuremath{\sim}2 TeV. If the Higgs sector shall be perturbative up to the GUT scale, the Higgs boson must be lighter than \ensuremath{\sim}150 GeV. For the two-loop observables examined, the apparent convergence of the perturbation series is better in the OMS scheme than in the $\stackrel{-}{\mathrm{MS}}$ scheme.