Hints at the Onset of Strong Interactions in the Higgs Sector of the Standard Model

We calculate the two-loop corrections to a couple of processes which involve the Higgs boson, H, and longitudinally polarized gauge bosons, W_sp{L} {+/-}, Z_{L}, as a perturbation series in the quartic Higgs coupling lambda. In particular, we study the two-body scattering of the channels W_sp{L} {+}W_sp{L}{-}, Z_ {L}Z_{L}, HH and HZ _{L}, and examine the fermionic decay H to f| f. Both processes display a breakdown of the perturbation series in lambda, indicating the onset of strong interactions. The results are used to obtain upper limits on the mass M _{H} of a weakly interacting Higgs boson. The 2 to 2 processes involving W_sp{L}{+/-}, Z _{L}, H are calculated in the high -energy heavy-Higgs limit sqrt{s} gg M_{H} gg M_{W} using the equivalence theorem. Introducing the running Higgs coupling lambda_ {s}(s, M_{H}), the two -loop amplitudes are re-written as to satisfy the renormalization group equation. Using an Argand diagram analysis, we find that the perturbative eigen-amplitudes are approximately unitary for lambda_{s} _sp{~}{<} 2.3. For larger values, the perturbation series is not converging towards its unitary value. Assuming that the Standard Model is the correct theory to describe longitudinal gauge boson scattering up to energies of 5 TeV (10 ^{16} GeV), the bound lambda_{s} _sp{ ~}{<} 2.3 translates into an upper bound M_{H} _sp{~}{<} 380 (155) GeV for a weakly interacting Higgs boson. For larger M_{H}, the scattering amplitudes need to be unitarized in a phenomenological manner. We inspect the K-matrix and the Pade-resummation methods for unitarizing the two-loop amplitudes, and conclude that the Pade method is preferable. The decay H to f| f is calculated to two loops in lambda , again using the equivalence theorem. To one-loop we include the Yukawa couplings and show explicitly the validity of the equivalence-theorem approximation for the process H to t| t. We find that our result reproduces the full electroweak result reported by other authors to better than 97% for M_{H} >= 500 GeV. Neglecting the Yukawa couplings, we carry out a consistent two-loop calculation of the O( lambda^2) corrections. We find that the two-loop correction is in magnitude larger than the one-loop correction for M_{H} > 375 GeV, indicating the breakdown of the perturbation series in lambda. The derivation of this result is independent of any assumptions about the energy up to which the Standard Model is supposed to be valid, a completely new feature in the context of perturbative limits on the Higgs mass.