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

  • Roman N. LeeEmail author
  • Vladimir A. Smirnov
Open Access
Regular Article - Theoretical Physics

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 → ∞. 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.

Keywords

NLO Computations QCD Phenomenology 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Budker Institute of Nuclear PhysicsNovosibirskRussia
  2. 2.Skobeltsyn Institute of Nuclear PhysicsMoscow State UniversityMoscowRussia

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