Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the Standard Model

TL;DR

We address how the top‑Yukawa coupling $y_t$, the strong coupling $g_s$, and the Higgs self‑coupling $λ$ run at three‑loop order in the unbroken Standard Model by computing the dominant QCD and top‑Yukawa contributions to the $β$‑functions and the anomalous dimensions. The authors provide complete analytic expressions for $β_{λ}$, $β_{y_t}$, and $β_{g_s}$ with general gauge factors, including $ζ_3$ terms, and assess three‑loop effects on the evolution of $λ$ and vacuum stability. They perform the calculation in the MSbar scheme using massless propagator diagrams, an auxiliary‑mass method to handle subdivergences, and cross‑checks with multiple computational tools, along with careful treatment of $γ_5$. Their phenomenological analysis shows that three‑loop corrections are numerically small for $β_{λ}$ at realistic scales due to cancellations, but the running of $λ$ remains highly sensitive to $M_H$, $M_t$ and $α_s$, affecting electroweak vacuum stability up to the Planck scale.

Abstract

We analytically compute the dominant contributions to the β-functions for the top-Yukawa coupling, the strong coupling and the Higgs self-coupling as well as the anomalous dimensions of the scalar, gluon and quark fields in the unbroken phase of the Standard Model at three-loop level. These are mainly the QCD and top-Yukawa corrections. The contributions from the Higgs self-interaction which are negligible for the running of the top-Yukawa and the strong coupling but important for the running of the Higgs self-coupling are also evaluated.

Figures (7)

• Figure 1: Some diagrams contributing to $Z^{(tb\Phi)}_1$
• Figure 2: Some diagrams contributing to $Z^{(4\Phi)}_1$
• Figure 3: Some diagrams contributing to $Z^{(2g)}_3$
• Figure 4: Some diagrams contributing to $Z^{(2\Phi)}_2$
• Figure 5: Some diagrams contributing to $Z^{(2t)}_{2,L/R}$