Higgs mass and grand unification

Abstract

The knowledge of the solution of the Renormalization Group Equation (RGE) for the quartic self-interaction of the Higgs scalar $\lambda _{H}$ is crucial for the determination of the energy limits on the Higgs mass. $\lambda _{H}$ is also important in order to solve the RGEs at the two loop level for other observables like the quark masses or the Cabibbo-Kobayashi-Maskawa (CKM) matrix. We obtain an analytical and numerical (considering Grand Unification) solution to the one loop renormalization group evolution of the Higgs quartic coupling $\lambda _{H}$ in the energy range $[m_{t},E_{GU}]$, where $m_{t}$ is the mass of the top quark and $E_{GU}=10^{14}$~GeV. We find that depending on the value of $\lambda _{H}(m_{t})$ the solution for $ \lambda _{H}(E)$ may have singularities or zeros and become negative, in which case the Standard Model (SM) becomes inconsistent. We obtain that for $0.65\leq \lambda _{H}(m_{t})\leq 0.69$ the SM is valid in the whole range $[m_{t},E_{GU}]$. These values of $\lambda _{H}(m_{t})$ correspond to the following Higgs mass $198\leq m_{H}\leq 205\,$GeV.