Simplifying 5-point tensor reduction

TL;DR

This paper introduces a simplified method for 5-point tensor reduction that leverages properties of metric tensor insertions and Gram determinants to improve numerical efficiency in tensor calculations.

Contribution

The paper presents explicit formulas and a novel approach to tensor reduction that cancels metric tensor contributions and Gram determinants, enhancing computational efficiency.

Findings

Automatic cancellation of metric tensor contributions in 5-point tensors.

Efficient tensor reduction when the 4-point sub-Gram determinant is not small.

Explicit formulas provided for tensors of degree 2 and 3.

Abstract

The 5-point tensors have the property that after insertion of the metric tensor $g^{\mu \nu}$ in terms of external momenta, all $g^{\mu \nu}$-contributions in the tensor decomposition cancel. If furthermore the tensors are contracted with external momenta, the inverse 5-point Gram determinant $()_5$ cancels automatically. If the remaining 4-point sub-Gram determinant ${s\choose s}_5$ is not small then this approach appears to be particularly efficient in numerical calculations. We also indicate how to deal with small ${s\choose s}_5$. Explicit formulae for tensors of degree 2 and 3 are given for large and small (sub-) Gram determinants.