Four-Loop Decoupling Relations for the Strong Coupling

TL;DR

The paper delivers a four-loop calculation of decoupling relations for the strong coupling α_s, enabling consistent five-loop running across heavy-quark thresholds once the five-loop β-function is available. Using a decoupling framework and four-loop vacuum integrals reduced by the Laporta method, it provides an analytic expression for the decoupling constant ζ_g^2 in the Nc=3, nh=1 case and explores the numerical impact on α_s evolution across the bottom threshold. A low-energy theorem ties ζ_g to the Higgs–gluon coupling, yielding a five-loop C1 coefficient and showing that its determination relies on the logarithmic pieces of ζ_g; the explicit C1 expansion for Nc=3, nh=1 is presented. The results enhance precision in QCD predictions by enabling higher-order matching across thresholds and offer building blocks for N^4LO Higgs production/decay in gluon fusion, with numerical studies indicating improved perturbative stability at higher orders and cross-validation with independent work.

Abstract

We compute the matching relation for the strong coupling constant within the framework of QCD up to four-loop order. This allows a consistent five-loop running (once the $β$ function is available to this order) taking into account threshold effects. As a side product we obtain the effective coupling of a Higgs boson to gluons with five-loop accuracy.