Applying Mellin-Barnes technique and Groebner bases to the three-loop static potential

Abstract

The Mellin-Barnes technique to evaluate master integrals and the algorithm called FIRE to solve IBP relations with the help of Gröbner bases are briefly reviewed. In FIRE, an extension of the classical Buchberger algorithm to construct Gröbner bases is combined with the well-known Laporta algorithm. It is explained how both techniques are used when evaluating the three-loop correction to the static QCD quark potential. First results are presented: the coefficients of n 3 l and n 2 l , where n l is the number of light quarks.

Algorithm FIRE—Feynman Integral REduction

“…To illustrate the potential of FIRE let us describe a problem where FIRE was applied. It was used to calculate the fermionic contribution to the three-loop static quark potential [17]. The Feynman diagrams corresponding to this problem had 12 indices each (11 propagators and 1 irreducible numerator).…”

“…The beta-version of FIRE has been applied to a family of three-loop Feynman integrals necessary for the analysis of decoupling of c-quark loops in b-quark HQET [22]. Furthermore, the algorithm itself has been used to calculate the fermionic contribution to the three-loop static quark potential [17]. FIRE has also been used to crosscheck some of the integrals in a paper [23] where four-loop vacuum integrals have been evaluated.…”