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

TL;DR

The paper addresses the problem of obtaining an analytic value for the last missing constant in the three-loop quark static potential by focusing on the master integral $I_1$. It introduces a parametric family of Feynman integrals with an auxiliary variable $y$ and derives a system of 109 master integrals satisfying $\partial_y \mathbf{F}=\mathrm{A}(y,\epsilon)\mathbf{F}$, aiming for a canonical $\epsilon$-form. Although a full $\epsilon$-form is not achieved for the entire matrix, a block-wise $\epsilon$-form is obtained, and the solution is constructed in an $\epsilon$-expansion after a change of variables to $x=1/y$, with results expressed in terms of multiple polylogarithms up to weight six. Boundary conditions are fixed from the large-$y$ asymptotics via expansion by regions, including HQET-like hard-region integrals and DRA-computed contributions, allowing the extraction of $I_1(0)$ in analytic form. The final result yields a compact analytic expression for $I_1(0)$ in terms of polylogarithms and zeta values, enabling the fully analytic three-loop quark static potential in the companion work, and providing analytic HQET integrals as by-products; the master-integral data are publicly available.

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 $ε$-form of the differential system. Though this step appears to be, strictly speaking, not possible, we succeed to find an $ε$-form of all irreducible diagonal blocks, which is sufficient for solving the differential system in terms of an $ε$ 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.