Leading QCD-induced four-loop contributions to the $\beta$-function of the Higgs self-coupling in the SM and vacuum stability

TL;DR

This paper provides the leading four-loop QCD and top-Yukawa contributions to the Higgs self-coupling beta function in the Standard Model, with implications for vacuum stability up to the Planck scale.

Contribution

It presents the first analytical four-loop results for the Higgs self-coupling beta function, including new relations between QCD anomalous dimensions and vacuum stability analysis.

Findings

Four-loop beta function contributions are numerically estimated at the top-quark scale.

Results impact the understanding of Higgs self-coupling evolution and vacuum stability.

New relation between QCD vacuum energy anomalous dimension and scalar current correlator is found.

Abstract

We present analytical results for the leading top-Yukawa and QCD contribution to the $\beta$-function for the Higgs self-coupling $\lambda$ of the Standard Model at four-loop level, namely the part $\propto y_t^4 g_s^6$ independently confirming a result given in [1]. We also give the contribution $\propto y_t^2 g_s^6$ of the anomalous dimension of the Higgs field as well as the terms $\propto y_t g_s^8$ to the top-Yukawa $\beta$-function which can also be derived from the anomalous dimension of the top quark mass. We compare the results with the RG functions of the correlators of two and four scalar currents in pure QCD and find a new relation between the anomalous dimension $\gamma_0$ of the QCD vacuum energy and the anomalous dimension $\gamma_m^{SS}$ appearing in the RG equation of the correlator of two scalar currents. Together with the recently computed top-Yukawa and QCD contributions to $\beta_{g_s}$ [2,3] the $\beta$-functions presented here constitute the leading four-loop contributions to the evolution of the Higgs self-coupling. A numerical estimate of these terms at the scale of the top-quark mass is presented as well as an analysis of the impact on the evolution of $\lambda$ up to the Planck scale and the vacuum stability problem.