Tensor reduction of loop integrals

TL;DR

This paper introduces a closed-form solution for tensor integral reduction in loop calculations, significantly simplifying the diagonalisation process in high-energy physics Feynman diagram evaluations.

Contribution

It presents a novel, explicit formula for diagonalising tensor integrals using a basis of external momentum and metric tensors, applicable to multi-loop Feynman diagrams.

Findings

Efficient tensor reduction formula derived for arbitrary tensor integrals.

Application demonstrated on QCD 2→2 scattering processes up to three loops.

Reduces computational complexity in loop integral evaluations.

Abstract

The computational cost associated with reducing tensor integrals to scalar integrals using the Passarino-Veltman method is dominated by the diagonalisation of large systems of equations. These systems of equations are sized according to the number of independent tensor elements that can be constructed using the metric and external momenta. In this article, we present a closed-form solution of this diagonalisation problem in arbitrary tensor integrals. We employ a basis of tensors whose building blocks are the external momentum vectors and a metric tensor transverse to the space of external momenta. The scalar integral coefficients of the basis tensors are obtained by mapping the basis elements to the elements of an orthogonaldual basis. This mapping is succinctly expressed through a formula that resembles the ordering of operators in Wick's theorem. Finally, we provide examples demonstrating the application of our tensor reduction formula to Feynman diagrams in QCD $2 \to 2$ scattering processes, specifically up to three loops.