An automatized algorithm to compute infrared divergent multi-loop integrals

Abstract

We describe a constructive procedure to separate overlapping infrared divergences in multiloop integrals. Working with a parametric representation in D = 4 − 2ǫ dimensions, adequate subtractions lead to a Laurent series in ǫ, where the coefficients of the pole-and finite terms are sums of regular parameter integrals which can be evaluated numerically. We fully automatized this algorithm by implementing it into algebraic manipulation programs and applied it to calculate numerically some nontrivial 2-loop 4-point and 3-loop 3-point Feynman diagrams. Finally, we discuss the applicability of our method to phenomenologically relevant multi-loop calculations such as the NNLO QCD corrections for e + e − → 3 jets.

Evaluating multiloop Feynman integrals by Mellin-Barnes representation

“…It is not clear at the moment whether these integrals can be written in terms of HPL or 2dHPL depending on special combinations of s, t and m 2 , or a new class of functions is needed. This result as well as many other results for double and triple boxes mentioned in this short review is confirmed by numerical evaluation by means of a method based on a sector decomposition in the space of alpha parameters [27].…”

Single scale tadpoles and O(GFmt2αs3) corrections to the ρ parameter

“…Although not all of the above numerical values could be checked with independent techniques, we were able to verify many of them, either with the sector decomposition method [27], or with Mellin-Barnes techniques [28,29] implemented in the MB package [30] (see also [31]). …”

Two-loop leading color corrections to heavy-quark pair production in the gluon fusion channel

“…The diagrams belonging to the sets ii) and iii) form part of the NLO corrections to top-pair-plus-jet production and have been calculated by several groups [49][50][51][52]. Aside from the calculation of the required Feynman diagrams, a full NNLO calculation of the top-quark pair production cross section is particularly challenging because it entails the development of a NNLO subtraction method, which could be used in processes with massive partons [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69].…”