Leading Singularities of the Two-Loop Six-Particle MHV Amplitude

TL;DR

The paper applies the leading singularity method to compute the planar two-loop six-particle MHV amplitude in N=4 SYM, expressing it in a geometric-integral basis and determining coefficients via residue-based linear equations. It reveals an overcomplete basis (177 geometric integrals) with 18 reduction identities, yet shows the amplitude is uniquely fixed by leading singularities after accounting for these relations. The parity-even portion agrees with previous unitarity-based results, while the parity-odd portion is new and, through numerical ABDK/BDS checks, appears to satisfy the ABDK/BDS relation despite the parity-even violation. The work demonstrates the efficacy of leading singularities in generating both even and odd contributions and clarifies the role of integral reductions and basis choices in fully determining multi-loop amplitudes.

Abstract

We use the leading singularity technique to determine the planar six-particle two-loop MHV amplitude in N=4 super Yang-Mills in terms of a simple basis of integrals. Our result for the parity even part of the amplitude agrees with the one recently presented in arXiv:0803.1465. The parity-odd part of the amplitude is a new result. The leading singularity technique reduces the determination of the amplitude to finding the solution to a system of linear equations. The system of equations is easily found by computing residues. We present the complete system of equations which determines the whole amplitude, and solve the two-by-two blocks analytically. Larger blocks are solved numerically in order to test the ABDK/BDS iterative structure.

Figures (4)

• Figure 1: The five indepedent 8-propagator topologies. Each diagram represents a sum of those Feynman diagrams in which all of the 8 indicated propagators are present. In each diagram the external momenta are labeled clockwise beginning with $k_1$ at the position of the arrow. Also in each diagram $p$ is the loop momentum in the left loop and $q$ is the momentum in the right loop.
• Figure 2: The eight independent 7-propagator topologies. See fig. \ref{['EightPropagatorTopologies']} for details.
• Figure 3: The geometric integrals for the two-loop six-particle amplitude in ${\cal N}=4$ SYM. In each diagram the external momenta are labeled clockwise beginning with $k_1$ at the position of the arrow and the number in parentheses denotes the number of independent permutations of the diagrams. As discussed in section IV, this is an overcomplete set: the 177 integrals here obey 18 linear relations.
• Figure 4: An extra 12 integrals we add to the set of geometric integrals in order to facilitate comparison with Bern:2008ap. These integrals are dual conformally invariant but not geometric. Nevertheless they can be expressed as linear combinations of the geometric integrals in fig. \ref{['GeometricBasis']} using first class identities (see section IV).