Precise bounds on the Higgs boson mass

We study the renormalization group evolution of the Higgs quartic coupling λH. The one loop equation for λH is nonlinear and it is of the Riccati type which we analytically and numerically solve in the energy range [mt,EGU] where mt is the mass of the top quark and EGU=1014GeV. We find that depending on the value of λH(mt) the solution for λH(E) may have a singularity or a zero and become negative in the former energy range so the ultraviolet cutoff of the standard model should be below or equal to the energy where the zero or singularity of λH occurs. We then numerically solve the two loop renormalization group equation for λH and compare it with the one loop solution. We find that the two loop running of λH is very sensitive to the evolution of the top quark Yukawa coupling Yt. This implies a strong dependence on the top quark mass mt and suggests that the choice of mt as the renormalization point, that we use, reduces theoretical errors. We find that in the approximation of one loop for 0.397≤λH(mt)≤0.618 the standard model is valid in the whole range [mt,EGU] while for two loops the bound is 0.368≤λH(mt)≤0.621. From the properties of λH we then study the predictions for the Higgs mass. We use the effective potential to derive the relation between the Higgs mass and λH and obtain that this relation is not very sensitive to the particular choice of the effective potential but for the large Higgs masses the two loop corrections are significant. We determine that the standard model is valid in the whole range [mt,EGU] for the Higgs masses 153.5≤MH≤191.1 for one loop case and 148.5≤MH≤193.1 for two loops. The pattern of the behavior of λH(E) for different values of λH(mt) indicates the existence of a phase transition in the standard model for λH(mt)=0.5 which corresponds to the value of the Higgs mass MH=mt.