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 (SYM), where we construct closed-form representations of all (n-point NkMHV) 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.
Highlights
This progress has been fueled by very concrete computational targets often guided by specific physical questions
We introduce a prescriptive approach to generalized unitarity, resulting in a strictly-diagonal basis of loop integrands with coefficients given by -tailored residues in field theory
(The size of this basis depends on the spacetime dimension and the power counting of the quantum field theory in question.) Given any complete basis {Ik}, the coefficients ck of any loop amplitude ALn in (1.1) can be determined by linear algebra via the criterion that residues match field theory
Summary
The basic idea of generalized unitarity is very simple: because Feynman diagrams are rational functions prior to loop integration, the loop integrands of arbitrary scattering amplitudes are rational functions of the external and internal momenta; being rational functions, they are expandable into a complete basis of functions, with coefficients determined by residues (or ‘poles’). Suppose {Ik} forms a complete basis of L loop integrands (appropriate for a particular field theory), an arbitrary scattering amplitude integrand, ALn , can be represented:. The coefficients in this expansion, ck, are determined by the criterion that the right hand side matches field theory on all residues (of arbitrary co-dimension). For related work trying to identify certain master integrands that are nonzero upon integration, see e.g. [116, 124,125,126]
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