  We consider a complex scalar $\Phi^4 $ theory with spontaneously broken
global U(1) symmetry, minimally coupling to perturbatively quantized Einstein
gravity which is treated as an effective theory at the energy well below the
Planck scale. Both the lowest order pure real scalar correction and the
gravitational correction to the renormalization of the Higgs sector in this
model have been investigated. Our results show that the gravitational
correction renders the renormalization of the Higgs sector in this model
inconsistent while the pure real scalar correction to it leads to a compatible
renormalization.
