One-loop tensor Feynman integral reduction with signed minors

TL;DR

This paper introduces an algebraic method for reducing one-loop tensor Feynman integrals to scalar functions, implemented in the PJFry C++ package, with efficient evaluation and special handling for small Gram determinants.

Contribution

It provides a systematic algebraic reduction approach for tensor integrals up to five points and ranks five, along with a practical C++ implementation for numerical evaluation.

Findings

Efficient reduction of tensor integrals to scalar functions.

Handling of small or vanishing Gram determinants.

Implementation of the PJFry package for practical computations.

Abstract

We present an algebraic approach to one-loop tensor integral reduction. The integrals are presented in terms of scalar one- to four-point functions. The reduction is worked out explicitly until five-point functions of rank five. The numerical C++ package PJFry evaluates tensor coefficients in terms of a basis of scalar integrals, which is provided by an external library, e.g. QCDLoop. We shortly describe installation and use of PJFry. Examples for numerical results are shown, including a special treatment for small or vanishing inverse four-point Gram determinants. An extremely efficient application of the formalism is the immediate evaluation of complete contractions of the tensor integrals with external momenta. This leads to the problem of evaluating sums over products of signed minors with scalar products of chords. Chords are differences of external momenta. These sums may be evaluated analytically in a systematic way. The final expressions for the numerical evaluation are then compact combinations of the contributing basic scalar functions.