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

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 → ∞. The original integral is recovered from the limit y → 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 → 0. As a by-product, we obtain analytical results for some Feynman integrals typical for HQET.