Tensor decomposition for bosonic and fermionic scattering amplitudes

TL;DR

The authors address the combinatorial explosion in tensor bases for multiloop, multileg amplitudes by formulating a four-dimensional–external-state–consistent tensor decomposition. They show that, in the 't Hooft–Veltman scheme, a minimal set of relevant tensors exists, corresponding one-to-one with independent helicity amplitudes, while d-dimensional, irrelevant tensors decouple and need not be computed for helicity observables. The method is validated through explicit 4- and 5-point examples (gg → gg, q q̄ → gg, q q̄ → Q Q̄, and qq̄ → ggg, etc.), illustrating systematic reductions in the tensor basis and simple, stable projector constructions. The approach promises practical simplifications for higher-leg and higher-loop calculations in collider physics, with potential broad impact on complex amplitude computations.

Abstract

In this paper, we elaborate on a method to decompose multiloop multileg scattering amplitudes into Lorentz-invariant form factors, which exploits the simplifications that arise from considering four-dimensional external states. We propose a simple and general approach that applies to both fermionic and bosonic amplitudes and allows us to identify a minimal number of physically relevant form factors, which can be related one-to-one to the independent helicity amplitudes. We discuss explicitly its applicability to various four- and five-point scattering amplitudes relevant for LHC physics.