Sharpening The Leading Singularity

TL;DR

This paper introduces a new residue-based method for computing multi-loop scattering amplitudes in N=4 super Yang-Mills theory, simplifying calculations by solving linear equations and analyzing leading singularities with complex momenta.

Contribution

The authors develop a novel technique using leading singularities and residues to efficiently compute multi-loop amplitudes, extending quadruple cut methods to more complex scenarios.

Findings

Successfully computed the five-particle two-loop amplitude.

Demonstrated the universality of the homogeneous linear equations.

Applied the method to MHV and next-to-MHV six-particle amplitudes.

Abstract

We show how studying leading singularities of Feynman diagrams, when all momenta are complex, gives a simple way of writing multi-loop and multi-particle scattering amplitudes in N=4 super Yang-Mills. The simplicity of the method is equivalent to that of the quadruple cut technique introduced in hep-th/0412103 at one-loop. The new technique only involves the computation of residues and the solution of linear equations. In our technique both parity even and parity odd pieces of a coefficient are computed simultaneously and it is only at the end that a separation can be made if desired. We explain the procedure via examples. The main example, which we compute in detail, is the five-particle two-loop amplitude first given in hep-th/0604074. Another feature of our method is that the helicity structure of the amplitude only enters in the inhomogeneous part of the linear equations. In other words, the homogeneous part is universal. We illustrate this feature by presenting the linear equations which determine a large class of terms for MHV and next-to-MHV six-particle two-loop amplitudes.