Physical projectors for multi-leg helicity amplitudes

TL;DR

The paper addresses the exponential growth of tensor structures in multi-leg amplitude computations by introducing physical projector operators that target four-dimensional helicity amplitudes. By exploiting the four-dimensional external state space and spinor-helicity/momentum-twistor formalisms, the authors construct a compact, d-independent set of helicity projectors, reducing the required tensor basis from the full d-dimensional set to a minimal $h_\\lambda\times h_\\lambda$ system with rational coefficients. They explicitly build the complete set of five-point, helicity-specific projectors (32 in total) for five-gluon scattering and demonstrate analytic one-loop results that reproduce known amplitudes, highlighting substantial computational savings. The approach promises significant simplifications for higher-point processes and offers a practical path to incorporate projector-based methods alongside existing techniques like integrand reduction and unitarity.

Abstract

We present a method for building physical projector operators for multi-leg helicity amplitudes. For any helicity configuration of the external particles, we define a physical projector which singles out the corresponding helicity amplitude. For processes with more than four external legs, these physical projectors depend on significantly fewer tensor structures and exhibit a remarkable simplicity compared with projector operators defined with traditional approaches. As an example, we present analytic formulas for a complete set of projectors for five-gluon scattering. These have been validated by reproducing known results for five-gluon amplitudes up to one-loop.