Vacuum stability in the $U(1)_\chi$ extended model with vanishing scalar potential at the Planck scale

We investigate the vacuum stability in a scale invariant local $U(1)_\chi$ model with vanishing scalar potential at the Planck scale. We find that it is impossible to realize the Higgs mass of 125\,GeV while keeping the Higgs quartic coupling $\lambda_H$ to be positive in all energy scale, that is the same as the standard model. Once one allows $\lambda_H<0$, the lower bounds of the $Z'$ boson mass are obtained through the positive definiteness of the scalar mass squared eigenvalues, while the bounds are smaller than the LHC bounds. On the other hand, the upper bounds strongly depend on the number of relevant Majorana Yukawa couplings of the right-handed neutrinos $N_\nu$. Considering decoupling effects of the $Z'$ boson and the right-handed neutrinos, the condition of the singlet scalar quartic coupling $\lambda_\phi>0$ gives the upper bound in $N_\nu=1$ case, while it does not constrain $N_\nu=2$ and 3 cases. Especially, we find that $Z'$ boson mass is tightly restricted for $N_\nu=1$ case as $M_{Z'} \lesssim 3.7\,{\rm TeV}$.