Criticality and Inflation of the Gauged B-L Model

We consider the multiple point principle (MPP) and the inflation of the gauged B-L extension of the Standard Model (SM) with a classical conformality. We examine whether the scalar couplings and their beta functions can become simultaneously zero at ${\mathrm{\Lambda }}_{\text{MPP}}:={10}^{17}$ GeV by using the two-loop renormalization group equations (RGEs). We find that we can actually realize such a situation and that the parameters of the model are uniquely determined by the MPP. However, as discussed in \cite{Iso:2012jn}, if we want to realize the electroweak symmetry breaking by the radiative B-L symmetry breaking, the self coupling ${\lambda }_{\mathrm{\Psi }}$ of a newly introduced SM singlet complex scalar $\mathrm{\Psi }$ must have a non-zero value at ${\mathrm{\Lambda }}_{\text{MPP}}$, which means the breaking of the MPP. We find that the $\mathcal{O}(100)$GeV electroweak symmetry breaking can be achieved even if this breaking is very small; ${\lambda }_{\mathrm{\Psi }}({\mathrm{\Lambda }}_{\text{MPP}})\le {10}^{-10}$. Within this situation, the mass of the B-L gauge boson is predicted to be \begin{equation} M_{B-L}=2\sqrt{2}\times\sqrt{\frac{\lambda(v_{h})}{0.10}}\times v_{h}\simeq 696\hspace{1mm}$\text{GeV}$,\nonumber\end{equation} where $\lambda$ is the Higgs self coupling and $v_h$ is the Higgs expectation value. This is a remarkable prediction of the (slightly broken) MPP. Furthermore, such a small ${\lambda }_{\mathrm{\Psi }}$ opens a new possibility: $\mathrm{\Psi }$ plays a roll of the inflaton \cite{Okada:2011en}. Another purpose of this paper is to investigate the ${\lambda }_{\mathrm{\Psi }}{\mathrm{\Psi }}^4$ inflation scenario with the non-minimal gravitational coupling $\xi {\mathrm{\Psi }}^2\mathcal{R}$ based on the two-loop RGEs.