Higgs mass and grand unification

The knowledge of the solution of the Renormalization Group Equation (RGE) for the quartic self-interaction of the Higgs scalar λ_H is crucial for the determination of the energy limits on the Higgs mass. λ_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 λ_H in the energy range [m_t, E_GU], where m_t is the mass of the top quark and E_GU = 10¹⁴ GeV. We find that depending on the value of λ_H(m_t) the solution for λ_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 ≤ λ_H (m_t) ≤ 0.69 the SM is valid in the whole range [m_t, E_GU]- These values of λ_H(m_t) correspond to the following Higgs mass 198 ≤ m_H ≤ 205 GeV.