Landau pole in the Standard Model with weakly interacting scalar fields

Abstract

We consider the Standard Model with a new scalar field $X$ which is a $n_X^{}$ representation of the $SU(2)_L$ with a hypercharge $Y_X$. The renormalization group running effects on the new scalar quartic coupling constants are evaluated. Even if we set the scalar quartic coupling constants to be zero at the scale of the new scalar field, the coupling constants are induced by the one-loop effect of the weak gauge bosons. Once non-vanishing couplings are generated, the couplings rapidly increase by renormalization group effect of the quartic coupling constant itself. As a result, the Landau pole appears below Planck scale if $n_X^{}\geq 4$. We find that the scale of the obtained Landau pole is much lower than that evaluated by solving the one-loop beta function of the gauge coupling constants.