Precise bounds on the Higgs boson mass

Abstract

We study the renormalization group evolution of the Higgs quartic coupling ${\ensuremath{\lambda}}_{H}$. The one loop equation for ${\ensuremath{\lambda}}_{H}$ is nonlinear and it is of the Riccati type which we analytically and numerically solve in the energy range $[{m}_{t},{E}_{GU}]$ where ${m}_{t}$ is the mass of the top quark and ${E}_{GU}={10}^{14}\text{ }\text{ }\mathrm{GeV}$. We find that depending on the value of ${\ensuremath{\lambda}}_{H}({m}_{t})$ the solution for ${\ensuremath{\lambda}}_{H}(E)$ 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 ${\ensuremath{\lambda}}_{H}$ occurs. We then numerically solve the two loop renormalization group equation for ${\ensuremath{\lambda}}_{H}$ and compare it with the one loop solution. We find that the two loop running of ${\ensuremath{\lambda}}_{H}$ is very sensitive to the evolution of the top quark Yukawa coupling ${Y}_{t}$. This implies a strong dependence on the top quark mass ${m}_{t}$ and suggests that the choice of ${m}_{t}$ as the renormalization point, that we use, reduces theoretical errors. We find that in the approximation of one loop for $0.397\ensuremath{\le}{\ensuremath{\lambda}}_{H}({m}_{t})\ensuremath{\le}0.618$ the standard model is valid in the whole range $[{m}_{t},{E}_{GU}]$ while for two loops the bound is $0.368\ensuremath{\le}{\ensuremath{\lambda}}_{H}({m}_{t})\ensuremath{\le}0.621$. From the properties of ${\ensuremath{\lambda}}_{H}$ we then study the predictions for the Higgs mass. We use the effective potential to derive the relation between the Higgs mass and ${\ensuremath{\lambda}}_{H}$ 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 $[{m}_{t},{E}_{GU}]$ for the Higgs masses $153.5\ensuremath{\le}{M}_{H}\ensuremath{\le}191.1$ for one loop case and $148.5\ensuremath{\le}{M}_{H}\ensuremath{\le}193.1$ for two loops. The pattern of the behavior of ${\ensuremath{\lambda}}_{H}(E)$ for different values of ${\ensuremath{\lambda}}_{H}({m}_{t})$ indicates the existence of a phase transition in the standard model for ${\ensuremath{\lambda}}_{H}({m}_{t})=0.5$ which corresponds to the value of the Higgs mass ${M}_{H}={m}_{t}$.