Reduction of multi-leg loop integrals

TL;DR

This work tackles the reduction of multi-leg tensor one-loop integrals by focusing on the remaining rank-one contributions and addressing Gram determinant issues common in traditional methods. It merges spinor-helicity based trace techniques with dual-vector relations to produce a basic formula that expresses higher-point rank-one integrals in terms of scalar box and related integrals without introducing Gram determinant denominators. A practical three-step algorithm is presented: reduce to rank-one via Pittau's method when two external legs are massless, convert rank-one to scalars through trace relations, and show the correction term is perturbatively negligible for finite integrals. The paper also provides an explicit pentagon example and discusses extensions to massive external lines, thereby completing and operationalizing Pittau's tensor-reduction approach for efficient one-loop calculations.

Abstract

I give an efficient algorithm for the reduction of multi-leg one-loop integrals of rank one. The method combines the basic ideas of the spinor algebra approach with the dual vector approach and is applicable to box integrals or higher point integrals, if at least one external leg is massless. This method does not introduce Gram determinants in the denominator. It completes an algorithm recently given by R. Pittau.

Figures (4)

• Figure 1: A subgraph of a one-loop diagram. The loop momentum is flowing through $k$ and $k_4$, the $p_i$'s are assumed to be the external momenta.
• Figure 2: The labelling of a box integral with external momenta $l_i$.
• Figure 3: The labelling of a triangle integral.
• Figure 4: The Feynman diagram with the helicity assignment. All external momenta are taken to be outgoing.