The Six-Point NMHV amplitude in Maximally Supersymmetric Yang-Mills Theory

TL;DR

The paper computes the parity-even part of the two-loop six-point NMHV amplitude in planar ${ m N}=4$ SYM using a superspace generalized unitarity approach. It demonstrates that, after subtracting universal infrared divergences, the finite amplitude can be expressed as a sum of pseudo-conformal integrals with R-invariant spin factors, and it establishes dual conformal invariance for the NMHV ratio to this order. It further explores the all-loop structure via remainder-like functions dependent on conformal cross ratios and analyzes collinear and triple-collinear limits to constrain these functions. Numerical evaluations at multiple conformally related kinematic points corroborate the dual conformal invariance and illuminate the relationship between NMHV and MHV amplitudes, as well as potential connections to Wilson-loop formulations and strong-coupling expectations.

Abstract

We present an integral representation for the parity-even part of the two-loop six-point planar NMHV amplitude in terms of Feynman integrals which have simple transformation properties under the dual conformal symmetry. We probe the dual conformal properties of the amplitude numerically, subtracting the known infrared divergences. We find that the subtracted amplitude is invariant under dual conformal transformations, confirming existing conjectures through two-loop order. We also discuss the all-loop structure of the six-point NMHV amplitude and give a parametrization whose dual conformal invariant building blocks have a simple physical interpretation.

Figures (8)

• Figure 1: The integrals contributing to the six-point one-loop MHV and NMHV amplitudes. An arrow marks the leg with momentum $k_1$; the remaining momenta follow clockwise. The one-mass box $I^{\text{1m}}$ and two-mass easy $I^{\text{2me}}$ integrals contribute to the MHV amplitude and the one-mass box $I^{\text{1m}}$ and two-mass hard $I^{\text{2mh}}$ integrals contribute to the NMHV amplitude. The one-mass pentagon $I^{\text{1m,penta}}$ and the hexagon $I^{\text{hex}}$ have numerator factors of $\mu^2$ (the square of the $(-2 \epsilon)$-dimensional components of the loop momentum), and hence are finite. They contribute to both the MHV and NMHV amplitudes only at ${\cal O}(\epsilon)$ and higher ($I^{\text{hex}}$ contributes to the even parts while $I^{\text{1m,penta}}$ contributes to the odd parts).
• Figure 2: Generalized cuts required to determine the two-loop NMHV amplitude: (a) the 'double-pentagon' cut (b) the 'turtle' cut (c) the 'hexabox' cut (d) the 'flying-squirrel' cut (e) the 'rabbit-ears' cut. Unlike the MHV calculation, all permutations of the external legs must be considered.
• Figure 3: A cut of an $L$-loop six-point amplitude isolating an $(L-2)$-loop four-point amplitude with no external legs. The cut is proportional to a lone $R$ invariant.
• Figure 4: Two-loop topologies entering the 2-loop 6-point amplitudes. The arrow on the external line indicates leg number $1$.
• Figure 5: The two contributions to the 'double-pentagon' supercut \ref{['2loopcuts']}(a). The circled $+$ and $-$ denote MHV and ${\overline {\rm MHV}}$ superamplitudes, respectively. The middle amplitude may be chosen to be either of MHV or of ${\overline {\rm MHV}}$ type. Here we choose to present it as an MHV superamplitude.