Gravitational effects on vanishing Higgs potential at the Planck scale

We investigate gravitational effects on the so-called multiple point criticality principle (MPCP) at the Planck scale. The MPCP requires two degenerate vacua, whose necessary conditions are expressed by vanishing Higgs quartic coupling [lambda(M-P1) = 0] and vanishing its beta function [beta(lambda)(M-P1) = 0]. We discuss a case that a specific form of gravitational corrections are assumed to contribute to beta functions of coupling constants [although it is accepted that gravitational corrections do not alter the running of the standard model (SM) couplings]. To satisfy the above two boundary conditions at the Planck scale, we find that the top pole mass and the Higgs mass should be 170.8 GeV less than or similar to M-t less than or similar to 171.7 GeV and M-h = 125.7 +/- 0.4 GeV, respectively, as well as include suitable magnitude of gravitational effects (a coefficient of gravitational contribution as |a(lambda)| > 2). In this case, however, since the Higgs quartic coupling lambda becomes negative below the Planck scale, two vacua are not degenerate. We find that M-h greater than or similar to 131.5 GeV with M-t greater than or similar to 174 GeV is required by the realization of the MPCP. Therefore, the MPCP at the Planck scale cannot be realized in the SM, and also the SM with gravity since M-h greater than or similar to 131.5 GeV is experimentally ruled out.