Abstract
We study the ABDK relation using maximal cuts of one- and two-loop integrals with up to five external legs. We show how to find a special combination of integrals that allows the relation to exist, and how to reconstruct the terms with one-loop integrals squared. The reconstruction relies on the observation that integrals across different loop orders can have support on the same generalized unitarity cuts and can share global poles. We discuss the appearance of nonhomologous integration contours in multivariate residues. Their origin can be understood in simple terms, and their existence enables us to distinguish contributions from different integrals. Our analysis suggests that maximal and near-maximal cuts can be used to infer the existence of integral identities more generally.
Highlights
The development of on-shell methods [1,2,3,4,5,6,7] for computing scattering amplitudes in quantum field theory has led to rapid progress in numerous directions in recent years, including higher-loop computations in the maximally supersymmetric (N 1⁄4 4) Yang-Mills theory (MSYM) [8,9,10,11,12,13,14,15,16,17,18,19,20,21], the understanding of dual conformal [22] and Yangian [23] symmetries, the development of alternate viewpoints on amplitudes such as twistor strings [24] and Grassmannians [25,26,27], as well as the development of numerical one-loop libraries [28,29,30,31,32] applied to next-toleading order (NLO) calculations for phenomenology at CERN’s Large Hadron Collider
Bern, Dixon, and Smirnov (BDS) wrote down a remarkable conjecture [40], namely, that the planar part of all maximally helicity-violating (MHV) amplitudes in N 1⁄4 4 supersymmetric Yang-Mills theory can be written in a certain sense as exponentials of the oneloop amplitude, 1 þ X ∞ aLMðnLÞðfsijg; εÞ
We must first work out the relevant maximal cuts of the five-point two-loop integrals that appear on the left-hand side of the five-point relation (8.1)
Summary
The development of on-shell methods [1,2,3,4,5,6,7] for computing scattering amplitudes in quantum field theory has led to rapid progress in numerous directions in recent years, including higher-loop computations in the maximally supersymmetric (N 1⁄4 4) Yang-Mills theory (MSYM) [8,9,10,11,12,13,14,15,16,17,18,19,20,21], the understanding of dual conformal [22] and Yangian [23] symmetries, the development of alternate viewpoints on amplitudes such as twistor strings [24] and Grassmannians [25,26,27], as well as the development of numerical one-loop libraries [28,29,30,31,32] applied to next-toleading order (NLO) calculations for phenomenology at CERN’s Large Hadron Collider. In its numerical form, the formalism underlies recent software libraries and programs used for LHC phenomenology In this approach, the one-loop amplitude in a quantum field theory is written as a sum over a set of basis integrals, with coefficients that are rational in external spinor variables, X. We will work within a maximal unitarity approach, cutting all propagators in a given integral, and further seeking to localize integrands onto global poles to the extent possible. In principle, this allows one to isolate individual integrals on the right-hand side of the higherloop analog of Eq (1.3).
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