Renormalization Group Effects on the Mass Relation Predicted by the Standard Model with Generalized Covariant Derivatives

Renormalization group analysis is made on the relation $m_{\rm H} \simeq \sqrt{2}m_t$ for masses of the top quark and the Higgs boson, which is predicted by the standard model based on generalized covariant derivatives with gauge and Higgs fields. This relation is a low energy manifestation of a tree level constraint which holds among the quartic Higgs self-coupling constant and the Yukawa coupling constants at a certain high energy scale $\mu_0$. With the renormalization group equation at one-loop level, the evolution of the constraint is calculated from $\mu_0$ down to the low energy region around the observed top quark mass. The result of analysis shows that the Higgs boson mass is in $m_t \lesssim m_{\rm H} \lesssim \sqrt{2}m_t$ for a wide range of the energy scale $\mu_0 \gtrsim m_t$ and it approaches to 177 GeV ($\approx m_t$) for large values of $\mu_0$.