Efficient integrand reduction for particles with spin

TL;DR

This paper develops a $d$-dimensional, multi-copy tensor basis for scattering amplitudes with spinning particles that factorizes amplitudes into copies of simple building blocks, enabling reduction to sums over scalar master integrals. By expressing amplitudes in terms of the building blocks $A_i(j,k)$ and $C_{i,j}$ and constructing dual projectors $A^i(k)$ and $D^{s,i}$, it reveals consistency relations among master integral coefficients via differential equations and the absence of residues at unphysical poles. The framework is applied to planar gluon amplitudes, yielding analytic results for the planar five-gluon two-loop and planar four-gluon three-loop cases, and it demonstrates leading singularities up to four loops. The method has potential to relieve the IBP bottleneck, provides internal checks, and opens avenues for extensions to massive matter and gravity, with important implications for analytic QFT understanding and collider predictions.

Abstract

Scattering amplitudes with spinning particles are shown to decompose into multiple copies of simple building blocks to all loop orders, which can be used to efficiently reduce these amplitudes to sums over scalar integrals. Absence of unphysical kinematic singularities cleanly exposed by the method uncover novel consistency relations among master integrals and their coefficients. Analytic results are obtained for the five gluon, two loop, and four gluon, three loop planar scattering amplitudes in pure Yang-Mills theory as well as for leading singularities to even higher orders.

Figures (2)

• Figure 1: P-matrix for five gluons for the basis constructed from $A$ and $D$ tensors, evaluated on random integer values.
• Figure 2: Deriving a two loop integral parametrisation of a five point scattering problem from a one loop topology.