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A note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix

Andreas Brandhuber, Paul Heslop, Gabriele Travaglini

TL;DR

The paper develops a supersymmetric BCFW-like recursion for ${\cal N}=4$ SYM in dual superspace and proves that the tree-level S-matrix is covariant under dual superconformal transformations, with explicit inversion weights tied to region momenta. It then shows that the coefficients of one-loop amplitudes expanded in a basis of scalar box functions transform covariantly in the same way as the tree-level amplitudes, using four-dimensional quadruple cuts to avoid dimensional regularization artifacts. Together, these results reinforce the duality between scattering amplitudes and Wilson loops and connect to the DHKS framework by demonstrating uniform covariance of both tree and loop structures. The work provides a concrete, section-by-section mechanism to relate on-shell recursion, dual conformal symmetry, and loop integrals in planar ${\cal N}=4$ SYM, offering a robust, symmetry-based understanding of the S-matrix.

Abstract

We present a supersymmetric recursion relation for tree-level scattering amplitudes in N=4 super Yang-Mills. Using this recursion relation, we prove that the tree-level S-matrix of the maximally supersymmetric theory is covariant under dual superconformal transformations. We further analyse the consequences that the transformation properties of the trees under this symmetry have on those of the loops. In particular, we show that the coefficients of the expansion of generic one-loop amplitudes in a basis of pseudo-conformally invariant scalar box functions transform covariantly under dual superconformal symmetry, and in exactly the same way as the corresponding tree-level amplitudes.

A note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix

TL;DR

The paper develops a supersymmetric BCFW-like recursion for SYM in dual superspace and proves that the tree-level S-matrix is covariant under dual superconformal transformations, with explicit inversion weights tied to region momenta. It then shows that the coefficients of one-loop amplitudes expanded in a basis of scalar box functions transform covariantly in the same way as the tree-level amplitudes, using four-dimensional quadruple cuts to avoid dimensional regularization artifacts. Together, these results reinforce the duality between scattering amplitudes and Wilson loops and connect to the DHKS framework by demonstrating uniform covariance of both tree and loop structures. The work provides a concrete, section-by-section mechanism to relate on-shell recursion, dual conformal symmetry, and loop integrals in planar SYM, offering a robust, symmetry-based understanding of the S-matrix.

Abstract

We present a supersymmetric recursion relation for tree-level scattering amplitudes in N=4 super Yang-Mills. Using this recursion relation, we prove that the tree-level S-matrix of the maximally supersymmetric theory is covariant under dual superconformal transformations. We further analyse the consequences that the transformation properties of the trees under this symmetry have on those of the loops. In particular, we show that the coefficients of the expansion of generic one-loop amplitudes in a basis of pseudo-conformally invariant scalar box functions transform covariantly under dual superconformal symmetry, and in exactly the same way as the corresponding tree-level amplitudes.

Paper Structure

This paper contains 9 sections, 84 equations, 4 figures.

Table of Contents

  1. Introduction and background
  2. $\mathcal{N}=4$ supersymmetric recursion relations
  3. Supersymmetric anti-MHV three-point amplitudes
  4. Large $z$ behaviour of the supersymmetric amplitudes ${{\cal A}}(z)$
  5. Examples
  6. First example, supersymmetric MHV amplitudes
  7. Second example, five-point $\overline{\mathrm{MHV}}$ amplitudes
  8. Proof of tree-level covariance
  9. Covariance of the coefficients of one-loop amplitudes

Figures (4)

  • Figure 1: Recursive diagram for the MHV four-point amplitude. Given the $[12\rangle$ shift we have chosen, the amplitude on the left must be $\mathrm{MHV}$, and that on the right $\overline{\mathrm{MHV}}$.
  • Figure 2: Recursive diagram for the five-point $\overline{\mathrm{MHV}}$ amplitude.
  • Figure 3: Generic recursion diagram used in the proof of covariance.
  • Figure 4: Quadruple cut of a one-loop superamplitude in $\mathcal{N}=4$ SYM. The four blobs represent tree-level $\mathcal{N}=4$ superamplitudes. The $K_{1\ldots4}$ correspond to sums of momenta $p_i$ of the external particles.