Four-loop wave function renormalization in QCD and QED

TL;DR

The paper tackles the four‑loop on‑shell wave function renormalization constant $Z_2^{\rm OS}$ for heavy quarks in QCD, providing a complete set of numerical coefficients for all relevant color structures and establishing its role for high‑precision heavy‑quark physics. The authors employ a direct on‑shell self‑energy approach, expanding around $q^2=M^2$ and incorporating mass counterterm insertions, combined with extensive IBP reduction, master‑integral evaluation (including Mellin‑Barnes techniques and numerical methods like FIESTA), to obtain $\delta Z_2^{(4)}$ with quantified uncertainties. They further derive the four‑loop HQET anomalous dimension $\gamma_{\rm HQET}$ and perform nontrivial HQET/QCD consistency checks, including a QED limit, thereby providing robust cross‑validation of the four‑loop results. The work delivers a rigorous numerical benchmark for a challenging class of four‑loop on‑shell integrals and supplies essential data for future analytic progress and for building high‑precision predictions in heavy‑quark phenomenology and related QED applications.

Abstract

We compute the on-shell wave function renormalization constant to four-loop order in QCD and present numerical results for all coefficients of the SU$(N_c)$ colour factors. We extract the four-loop HQET anomalous dimension of the heavy quark field and also discuss the application of our result to QED.