Fermionic contributions to the three-loop static potential

TL;DR

This work computes the fermionic three-loop corrections to the QCD static quark–antiquark potential, addressing IR safety and the role of a3 in heavy-quark phenomenology. The authors use a hybrid reduction approach (Gröbner/Laporta) via FIREFIRE and Mellin-Barnes techniques to evaluate ~100 master integrals, obtaining explicit color-structure–decomposed coefficients a1, a2, and a3. Numerically, the three-loop fermionic contributions can be sizable, notably around -27%, -8%, and -2% for charm, bottom, and top quark systems, indicating significant refinements over lower-order results. They also compare with Padé predictions and discuss QED limits, while noting that the nl-independent coefficient a3^(0) remains to be determined in future work.

Abstract

We consider the three-loop corrections to the static potential which are induced by a closed fermion loop. For the reduction of the occurring integrals a combination of the Gröbner and Laporta algorithm has been used and the evaluation of the master integrals has been performed with the help of the Mellin-Barnes technique. The fermionic three-loop corrections amount to 2% of the tree-level result for top quarks, 8% for bottom quarks and 27% for the charm quark system.

Figures (3)

• Figure 1: Sample diagrams contributing to the static potential at tree-level, one-, two- and three-loop order. In this paper only the fermionic corrections are considered at three-loop order which excludes diagrams of type (h).
• Figure 2: One-, two- and three-loop diagrams. The solid line stands for massless relativistic propagators and the zigzag line represents static propagators.
• Figure 3: Three-loop diagrams of "ladder" (first three diagrams), "non-planar" (forth diagram in upper row) and "mercedes" type (lower row) which have to be considered for the fermionic part of $a_3$.