Gravitational effects on vanishing Higgs potential at the Planck scale

TL;DR

This paper examines how gravitational effects influence the conditions for multiple degenerate vacua at the Planck scale, concluding that the Standard Model cannot satisfy these conditions given current experimental Higgs and top quark masses.

Contribution

It introduces a specific gravitational correction to the Higgs potential boundary conditions at the Planck scale and analyzes their implications for the Standard Model.

Findings

Gravitational effects require a Higgs mass of about 125.7 GeV for boundary conditions.

The Higgs quartic coupling becomes negative below the Planck scale under these effects.

The multiple point criticality principle cannot be realized with current experimental Higgs and top masses.

Abstract

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 $\lambda(M_{\rm Pl})=0$ and vanishing its $\beta$ function $\beta_\lambda(M_{\rm Pl})=0$. 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 $170.8\,{\rm GeV} \lesssim M_t\lesssim 171.7\,{\rm GeV}$ and $M_h=125.7\pm0.4\,{\rm GeV}$, 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 $M_h \gtrsim 131.5\,{\rm GeV}$ with $M_t \gtrsim 174\,{\rm GeV}$ 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 $M_h \gtrsim 131.5\,{\rm GeV}$ is experimentally ruled out.