All tree-level amplitudes in N=4 SYM

TL;DR

This work provides an explicit, manifestly supersymmetric formula for all tree-level amplitudes in N=4 SYM by solving supersymmetric on-shell recursion relations in on-shell superspace. The amplitudes are expressed as the MHV prefactor times a universal structure P_n built from dual superconformal invariants (R-invariants), ensuring both conventional and dual superconformal covariance. The authors systematically derive NMHV, NNMHV, and higher N^pMHV cases, introducing generalized R-invariants and a path-based, Catalan-number organized tree representation that unifies all tree amplitudes. The framework enables direct extraction of any component amplitude, including gluon amplitudes, and suggests potential extensions to gravity via KLT and links to AdS/CFT definitively tying symmetry considerations to explicit formulae.

Abstract

We give an explicit formula for all tree amplitudes in N=4 SYM, derived by solving the recently presented supersymmetric tree-level recursion relations. The result is given in a compact, manifestly supersymmetric form and we show how to extract from it all possible component amplitudes for an arbitrary number of external particles and any arrangement of external particles and helicities. We focus particularly on extracting gluon amplitudes which are valid for any gauge theory. The formula for all tree-level amplitudes is given in terms of nested sums of dual superconformal invariants and it therefore manifestly respects both conventional and dual superconformal symmetry.

Figures (6)

• Figure 1: Illustration of the r.h.s of the on-shell recursion relations (\ref{['BCF']}),(\ref{['BCF-super']}). The picture on the right illustrates the transition to dual variables.
• Figure 2: The two contributions to the supersymmetric recursion relation for NMHV amplitudes. We call term $B$ inhomogeneous and $A$ homogeneous. $B$ can be easily computed since it is built from MHV amplitudes only. $\hat{1}$ means that $\lambda_{1}$ is shifted, and $\bar{n}$ means that $\tilde{\lambda}_{n}$ is shifted.
• Figure 3: The three contributions to the supersymmetric recursion relation for NNMHV amplitudes.
• Figure 4: Graphical representation of the formula for tree-level amplitudes in $\mathcal{N}=4$ SYM.
• Figure 5: The rule for going from line $p-1$ to line $p$ (for $p>1$) in Fig. \ref{['fig-rec-solution']}. For every vertex in line $p-1$ of the form given at the top of the diagram, there are $r+2$ vertices in the lower line (line $p$). The labels in these vertices start with $v_{1}u_{1};\ldots v_{r}u_{r};b_{p-1}a_{p-1};a_{p}b_{p}$ and they get sequentially shorter, with each step to the right removing the pair of labels adjacent to the last pair $a_p,b_p$ until only the last pair is left. The summation limits between each line are also derived from the labels of the vertex above. The right superscripts associated to each vertex are obtained by deleting the final pair of labels $a_pb_p$ and reversing the order the last pair which remain. The left superscript of a given vertex coincides with the right superscript of the vertex to its left.