The three loop on-shell renormalization of QCD and QED

TL;DR

This paper computes the on-shell renormalization constants for heavy quarks in QCD and QED at three loops, providing explicit results for $Z_m$ and $Z_2$ and showing that $Z_2$ in QCD becomes gauge dependent at $O(\alpha_s^3)$, thereby refuting the conjecture of all-orders gauge independence. The calculation relies on dimensional regularization, IBP reduction to 18 master integrals, and new master integrals $I_{10}$, $I_{5}$, and $I_{18}$ evaluated via large-mass expansions, dispersion methods, and Mellin-Barnes techniques. The authors also derive the three-loop HQET heavy-quark anomalous dimension from the OS/MSbar ratio and verify HQET decoupling constraints, while confirming gauge-independence of $Z_2$ in QED. Together, these results advance three-loop on-shell computations for heavy quarks and provide robust checks across QCD, QED, and HQET frameworks.

Abstract

We describe a calculation of the on-shell renormalization factors in QCD and QED at the three loop level. Explicit results for the fermion mass renormalization factor Zm and the on-shell fermion wave function renormalization constant Z2 are given. We find that at O(alpha_s^3) the wave function renormalization constant Z2 in QCD becomes gauge dependent also in the on-shell scheme, thereby disproving the ``gauge-independence'' conjecture based on an earlier two-loop result. As a byproduct, we derive an O(alpha_s^3) contribution to the anomalous dimension of the heavy quark field in HQET.

Figures (2)

• Figure 1: Examples of three-loop quark propagator diagrams corresponding to eleven integration topologies.
• Figure 2: Pictorial representation of the complete set of primitive 3-loop on-shell integrals $I_1$ -- $I_{18}$. Thick lines are used to indicate massive scalar propagators and thin lines indicate massless propagators.