Higgs boson mass bounds in the Standard Model with type III and type I seesaw

In type III seesaw utilized to explain the observed solar and atmospheric neutrino oscillations the Standard Model (SM) particle spectrum is extended by introducing three SU(2)_L triplet fermion fields. This can have important implications for the SM Higgs boson mass (MH$M_H$) bounds based on vacuum stability and perturbativity arguments. We compute the appropriate renormalization group equations for type III seesaw, and then proceed to identify regions of the parameter space such that the SM Higgs boson mass window is enlarged to 125 GeV?MH?174 GeV$125\text{GeV}?M_H?174\text{GeV}$, with the type III seesaw scale close to TeV. We also display regions of the parameter space for which the vacuum stability and perturbativity bounds merge together for large neutrino Yukawa couplings. Comparison with type I seesaw is also presented.