Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts

TL;DR

This paper introduces symmetric reduction formulae for high-multiplicity one-loop integrals, simplifying calculations by expressing complex integrals in terms of lower-point ones with symmetric kinematic coefficients, aiding scattering amplitude computations.

Contribution

The paper presents novel symmetric reduction formulae for one-loop integrals with many external legs, applicable in 3+1 dimensions, and connects these to D-dimensional unitarity cuts for comprehensive scattering analysis.

Findings

Reduction formulae express high-multiplicity integrals in terms of lower-point integrals.

Formulas are symmetric in external momenta, simplifying calculations.

Applicable to arbitrary multiplicity in 3+1 dimensions, facilitating full epsilon-expansion.

Abstract

We derive useful reduction formulae which express one-loop Feynman integrals with a large number of external momenta in terms of lower-point integrals carrying easily derivable kinematic coefficients which are symmetric in the external momenta. These formulae apply for integrals with at least two more external legs than the dimension of the external momenta, and are presented in terms of two possible bases: one composed of a subset of descendant integrals with one fewer external legs, the other composed of the complete set of minimally-descendant integrals with just one more leg than the dimension of external momenta. In 3+1 dimensions, particularly compact representations of kinematic invariants can be computed, which easily lend themselves to spinor-helicity or trace representations. The reduction formulae have a close relationship with D-dimensional unitarity cuts, and thus provide a path towards computing full (all-epsilon) expressions for scattering at arbitrary multiplicity.