# Prescriptive Unitarity

## TL;DR

This paper presents a prescriptive unitarity method that simplifies the construction of loop integrands in quantum field theory, demonstrated through explicit multi-loop amplitude calculations in supersymmetric Yang-Mills theory.

## Contribution

It introduces a diagonal basis for loop integrands with coefficients as tailored residues, avoiding linear algebra in unitarity methods, applicable to various quantum field theories.

## Key findings

- Constructed all $n$-point N$^k$MHV amplitudes through three loops in planar maximally supersymmetric Yang-Mills.
- Developed a diagonal basis of integrands with coefficients as residues, simplifying calculations.
- Method applicable to theories beyond planarity, supersymmetry, or massless spectra.

## 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.