Generalized unitarity for N=4 super-amplitudes

TL;DR

This paper develops a manifestly supersymmetric generalized unitarity framework for N=4 SYM amplitudes and applies it to compute one-loop NMHV amplitudes, yielding a compact, dual superconformally covariant tree-level NMHV form and a dual conformal invariant NMHV/MHV ratio at one loop. By working in on-shell N=4 superspace and using quadruple cuts, the authors systematically derive coefficients for all scalar box integrals, organize them into R-invariants, and demonstrate infrared consistency to fix tree-level NMHV amplitudes. They provide explicit N=6 and N=7 results, establishing finite, dual-conformal combinations V_{rst} that govern the finite parts and show dual superconformal symmetry extends to NMHV amplitudes. The work reinforces the amplitude/Wilson loop duality and clarifies how dual conformal symmetry emerges from a supersymmetric unitarity approach, with implications for higher-point amplitudes and potential extensions to N=8 supergravity.

Abstract

We develop a manifestly supersymmetric version of the generalized unitarity cut method for calculating scattering amplitudes in N=4 SYM theory. We illustrate the power of this method by computing the one-loop n-point NMHV super-amplitudes. The result confirms two conjectures which we made in arXiv:0807.1095 [hep-th]. Firstly, we derive the compact, manifestly dual superconformally covariant form of the NMHV tree amplitudes for arbitrary number and types of external particles. Secondly, we show that the ratio of the one-loop NMHV to the MHV amplitude is dual conformal invariant.

Figures (9)

• Figure 1: Quadruple cut of the scalar box integral $I(K_1,K_2,K_3,K_4)$. The four cut conditions $l_i^2=0$ and momentum conservation at each corner leave precisely two solutions for the $l_i$.
• Figure 2: Two adjacent three-point $\overline{\rm MHV}$ or two adjacent three-point MHV vertices. In either case the on-shell momentum conservation conditions imply that $(p_1+p_2)^2=0$ so the configuration does not exist for general kinematics.
• Figure 3: The only allowed configuration contributing to the one-loop MHV super-amplitude. It corresponds to a cut two-mass easy integral in the general case. If $s=3$ or $s=n-1$ then it is a one-mass integral and if $s=3=n-1$ then it is a massless box.
• Figure 4: The configuration contributing to the three-mass coefficient $\mathcal{C}^{\rm 3m}_{r,r+1,s,t}$. The empty vertices are MHV super-amplitudes and the shaded vertex is a three-particle $\overline{\rm MHV}$ super-amplitude.
• Figure 5: The two configurations contributing to the two-mass-hard coefficient $\mathcal{C}^{\rm 2mh}_{r-1,r,r+1,s}$. They are simply related to the three-mass coefficients $\mathcal{C}^{\rm 3m}_{r,r+1,s,r-1}$ (left) and $\mathcal{C}^{\rm 3m}_{r-1,r,r+1,s}$ (right).