Scalar one-loop integrals for QCD

TL;DR

This paper develops a comprehensive set of analytic formulas for divergent scalar one-loop integrals in QCD, including new results for certain box integrals, and provides a publicly available code for their calculation.

Contribution

It introduces a basis set of divergent scalar one-loop integrals for QCD and supplies analytic formulas and a calculation code, including previously missing results.

Findings

Analytic formulas for 6 divergent triangle integrals.

Analytic formulas for 16 divergent box integrals.

A publicly available code for calculating divergent and finite integrals.

Abstract

We construct a basis set of infra-red and/or collinearly divergent scalar one-loop integrals and give analytic formulas, for tadpole, bubble, triangle and box integrals, regulating the divergences (ultra-violet, infra-red or collinear) by regularization in $D=4-2\epsilon$ dimensions. For scalar triangle integrals we give results for our basis set containing 6 divergent integrals. For scalar box integrals we give results for our basis set containing 16 divergent integrals. We provide analytic results for the 5 divergent box integrals in the basis set which are missing in the literature. Building on the work of van Oldenborgh, a general, publicly available code has been constructed, which calculates both finite and divergent one-loop integrals. The code returns the coefficients of $1/\epsilon^2,1/\epsilon^1$ and $1/\epsilon^0$ as complex numbers for an arbitrary tadpole, bubble, triangle or box integral.