Evaluating the last missing ingredient for the three-loop quark static potential by differential equations

TL;DR

This paper analytically evaluates a complex three-loop Feynman integral crucial for the three-loop quark static potential using differential equations and auxiliary parameters, providing explicit solutions in terms of polylogarithms.

Contribution

Introduces a novel differential equation approach with an auxiliary parameter to evaluate the last missing three-loop Feynman integral for the quark static potential.

Findings

Analytical expression for the three-loop Feynman integral obtained.

Solution expressed in terms of multiple polylogarithms up to weight six.

Additional analytical results for HQET-related integrals.

Abstract

We analytically evaluate the three-loop Feynman integral which was the last missing ingredient for the analytical evaluation of the three-loop quark static potential. To evaluate the integral we introduce an auxiliary parameter $y$, which corresponds to the residual energy in some of the HQET propagators. We construct a differential system for 109 master integrals depending on $y$ and fix boundary conditions from the asymptotic behaviour in the limit $y\to \infty$. The original integral is recovered from the limit $y\to 0$. To solve these linear differential equations we try to find an $\epsilon$-form of the differential system. Though this step appears to be, strictly speaking, not possible, we succeed to find an $\epsilon$-form of all irreducible diagonal blocks, which is sufficient for solving the differential system in terms of an $\epsilon$ expansion. We find a solution up to weight six in terms of multiple polylogarithms and obtain an analytical result for the required three-loop Feynman integral by taking the limit $y\to 0$. As a by-product, we obtain analytical results for some Feynman integrals typical for HQET.