One-Loop Gluon Scattering Amplitudes in Theories with $N < 4$ Supersymmetries

TL;DR

One-loop gluon scattering amplitudes in theories with $N<4$ supersymmetries are studied using generalized unitarity. The authors decompose amplitudes into box, triangle, and bubble functions, compute box coefficients via quadruple cuts, relate $N=1$ and $N=4$ (and $N=0$) contributions with rho factors, and determine triangle coefficients from triple cuts, yielding an explicit $A^{N=1}$ NMHV amplitude. The results reveal that box coefficients obey the same twistor-space coplanarity/collinearity constraints as in $N=4$ theories, and demonstrate the feasibility of reconstructing amplitudes with fewer supersymmetries from unitarity data. The work extends the unitarity program to non-supersymmetric theories and hints at broader twistor-like structures, with potential relevance for gravity amplitudes.

Abstract

Generalised unitarity techniques are used to calculate the coefficients of box and triangle integral functions of one-loop gluon scattering amplitudes in gauge theories with $N < 4$ supersymmetries. We show that the box coefficients in N=1 and N=0 theories inherit the same coplanar and collinear constraints as the corresponding N=4 coefficients. We use triple cuts to determine the coefficients of the triangle integral functions and present, as an example, the full expression for the one-loop amplitude $A^{N=1}(1^-,2^-,3^-,4^+,..,n^+)$.