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 ΛMPP:=1017 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 λΨ of a newly introduced SM singlet
complex scalar Ψ must have a non-zero value at ΛMPP,
which means the breaking of the MPP. We find that the O(100)GeV
electroweak symmetry breaking can be achieved even if this breaking is very
small; λΨ(ΛMPP)≤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 λ is the Higgs
self coupling and vh is the Higgs expectation value. This is a remarkable
prediction of the (slightly broken) MPP. Furthermore, such a small
λΨ opens a new possibility: Ψ plays a roll of the inflaton
\cite{Okada:2011en}. Another purpose of this paper is to investigate the
λΨΨ4 inflation scenario with the non-minimal gravitational
coupling ξΨ2R based on the two-loop RGEs.Comment: 28 pages, 8 figures; some typos are corrected, Eq.(11) is corrected,
the predicted mass of M_{B-L} has changed, RGEs are corrected, footnotes
added, references added (v2); some typos are corrected, Eq.(26) is added,
references added, version to appear in PTEP(v3