Prescriptive Unitarity

TL;DR

The paper introduces prescriptive unitarity, a method to build a strictly diagonal, integrand-level basis for perturbative amplitudes where each coefficient is a single on-shell residue. It applies the framework to planar $\mathcal{N}=4$ SYM up to three loops, producing closed-form, all-multiplicity representations for $n$-point $N^kMHV$ amplitudes and highlighting the method's potential generality to nonplanar and massive theories. By carefully selecting a basis and exploiting on-shell functions and leading singularities, the authors demonstrate how higher-loop integrands can be fixed without large linear-algebra solves, and they discuss the interplay with infrared structure and composite residues. The work reveals a path toward simpler, more modular amplitude representations and provides a concrete, testable blueprint for extending prescriptive unitarity to broader quantum field theories and higher perturbative orders.

Abstract

We introduce a prescriptive approach to generalized unitarity, resulting in a strictly-diagonal basis of loop integrands with coefficients given by specifically-tailored residues in field theory. We illustrate the power of this strategy in the case of planar, maximally supersymmetric Yang-Mills theory, where we construct closed-form representations of all ($n$-point N$^k$MHV) scattering amplitudes through three loops. The prescriptive approach contrasts with the ordinary description of unitarity-based methods by avoiding any need for linear algebra to determine integrand coefficients. We describe this approach in general terms as it should have applications to many quantum field theories, including those without planarity, supersymmetry, or massless spectra defined in any number of dimensions.