New ideas for handling of loop and angular integrals in D-dimensions in QCD

TL;DR

This paper introduces new methods for simplifying loop tensor integrals and angular integrals in D-dimensional QCD calculations, enhancing computational efficiency and analytical clarity.

Contribution

It proposes a covariant tensor expansion formalism and a geometric partial fractioning algorithm, improving the handling of integrals in D-dimensional QCD.

Findings

Simplified tensor integral expansion into orthogonal basis.

Derived recursion relations for angular integrals.

Provided all-order epsilon expansion for basic angular integrals.

Abstract

We discuss new ideas for consideration of loop diagrams and angular integrals in $D$-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple presentation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order $\epsilon$-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.