Twistor transform of all tree amplitudes in N=4 SYM theory

TL;DR

This work provides a unified twistor-space framework for all tree-level ${\cal N}=4$ SYM amplitudes, showing that ${\rm N^kMHV}$ amplitudes are supported on $(2k+1)$ intersecting twistor lines with a corresponding lightlike $(2k+1)$-gon in moduli space. The authors introduce simple graphical rules to triangulate these polygons into two kinds of lightlike triangles, enabling direct construction of analytic expressions in both twistor and momentum space. They analyze conformal and dual conformal properties, address spurious singularities, and demonstrate how the full non-MHV sector can be assembled recursively from MHV via insertions that preserve the geometric line-polygon structure. The geometric picture suggests deep connections between twistor geometry, dual conformal symmetry, and potential extensions to Wilson-loop dualities and loop amplitudes, while highlighting sign-factor issues that break conformal invariance in certain kinematic regions. Overall, the paper provides a transparent, diagrammatic, and algebraic toolkit for representing and computing all tree-level ${\cal N}=4$ SYM amplitudes in twistor space.

Abstract

We perform the twistor (half-Fourier) transform of all tree n-particle superamplitudes in N=4 SYM and show that it has a transparent geometric interpretation. We find that the N^kMHV amplitude is supported on a set of (2k+1) intersecting lines in twistor space and demonstrate that the corresponding line moduli form a lightlike (2k+1)-gon in moduli space. This polygon is triangulated into two kinds of lightlike triangles lying in different planes. We formulate simple graphical rules for constructing the triangulated polygons, from which the analytic expressions of the N^kMHV amplitudes follow directly, both in twistor and in momentum space. We also discuss the ordinary and dual conformal properties and the cancellation of spurious singularities in twistor space.

Figures (17)

• Figure 1: A single line and two intersecting lines in twistor space. The arrowed line denotes the lightlike vector $X_{21}=\rho\tilde{\rho}$ in moduli space.
• Figure 2: The two basic three-line configurations, planar (right-hand side part) and three-dimensional (left-hand side part) . Blue lines are in twistor space, black lines are in moduli space.
• Figure 3: Transforming a single line into three lines.
• Figure 4: NMHV: The three twistor lines lie in a plane. The inscribed moduli space figure is a shaded triangle.
• Figure 5: N${}^2$MHV: Two-dimensional image of the pentagon corresponding to the term (A) in Eq. (\ref{['4-terms']}).