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Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes

Tristan Dennen, Yu-tin Huang

TL;DR

The paper proves that the delta-function-stripped tree-level amplitudes in six-dimensional maximal sYM covary under six-dimensional dual conformal inversion, with the delta functions carrying an extra weight. It extends this covariance to loop integrands using generalized unitarity, showing that all planar cuts invert consistently under the same weight pattern. Interpreting six-dimensional amplitudes as four-dimensional massively regulated amplitudes explains how the four-dimensional loop measure cancels the extra weight, yielding exact dual conformal symmetry for the regulated theory. The results motivate constructing six-dimensional dual-conformal covariants and momentum-twistor formulations to efficiently obtain higher-point amplitudes in this framework.

Abstract

We demonstrate that the tree-level amplitudes of maximal super-Yang-Mills theory in six dimensions, when stripped of their overall momentum and supermomentum delta functions, are covariant with respect to the six-dimensional dual conformal group. Using the generalized unitarity method, we demonstrate that this property is also present for loop amplitudes. Since the six-dimensional amplitudes can be interpreted as massive four-dimensional ones, this implies that the six-dimensional symmetry is also present in the massively regulated four-dimensional maximal super-Yang-Mills amplitudes.

Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes

TL;DR

The paper proves that the delta-function-stripped tree-level amplitudes in six-dimensional maximal sYM covary under six-dimensional dual conformal inversion, with the delta functions carrying an extra weight. It extends this covariance to loop integrands using generalized unitarity, showing that all planar cuts invert consistently under the same weight pattern. Interpreting six-dimensional amplitudes as four-dimensional massively regulated amplitudes explains how the four-dimensional loop measure cancels the extra weight, yielding exact dual conformal symmetry for the regulated theory. The results motivate constructing six-dimensional dual-conformal covariants and momentum-twistor formulations to efficiently obtain higher-point amplitudes in this framework.

Abstract

We demonstrate that the tree-level amplitudes of maximal super-Yang-Mills theory in six dimensions, when stripped of their overall momentum and supermomentum delta functions, are covariant with respect to the six-dimensional dual conformal group. Using the generalized unitarity method, we demonstrate that this property is also present for loop amplitudes. Since the six-dimensional amplitudes can be interpreted as massive four-dimensional ones, this implies that the six-dimensional symmetry is also present in the massively regulated four-dimensional maximal super-Yang-Mills amplitudes.

Paper Structure

This paper contains 12 sections, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of spinor helicity in six dimensions
  3. Dual conformal symmetry
  4. Dual conformal properties of tree-level amplitudes
  5. The BCFW shift in dual coordinates.
  6. BCFW diagrams without three-point subamplitudes
  7. BCFW diagrams with a three-point subamplitude
  8. Loop amplitudes through unitarity cuts
  9. Conclusion
  10. Acknowledgements
  11. Clifford algebra conventions
  12. Proof of variable inversion formulæ

Figures (3)

  • Figure 1: A BCFW diagram without three-point subamplitudes.
  • Figure 2: A BCFW diagram with a three-point subamplitude.
  • Figure 3: A cut of the two-loop four-point amplitude. (a) In the usual expression of the cut, this diagram is dressed with a tree-level amplitude for each blob and a state sum over each internal line. (b) As discussed in the text, for planar cuts this is equivalent to dressing the diagram with an $f$ function for each blob, introducing the dual variable constraints for every line, and integrating over the dual $\theta$ variables of every region. Finally, a state sum over each internal line is performed. One can check that the dressing of (b) contains $8\times 8=64$ fermionic delta functions and $5\times8=40$ integrations over $\theta$ (because one of the six regions is fixed by the shift symmetry), leaving $24$ unintegrated fermionic delta functions, which are exactly the supermomentum conservation of the subamplitudes in dressing (a).