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One-loop Amplitudes in Six-Dimensional (1,1) Theories from Generalised Unitarity

Andreas Brandhuber, Dimitrios Korres, Daniel Koschade, Gabriele Travaglini

TL;DR

This work develops a complete framework for six-dimensional (1,1) SYM amplitudes by combining six-dimensional spinor-helicity formalism with on-shell (1,1) superspace and generalized unitarity. The authors compute the one-loop four- and five-point superamplitudes: the four-point case is shown to factorize into the tree-level superamplitude times a scalar box integral via both two-particle and quadruple cuts, while the five-point case yields a linear pentagon integral that PV-reduces to a pentagon plus box contributions with well-defined coefficients. They further extract gluon components and perform consistency checks by dimensional reduction to four dimensions, recovering the Parke–Taylor structure and known ${\cal N}=4$ SYM results, as well as verifying soft limits. Overall, the paper provides a nontrivial test of unitarity-based techniques in higher dimensions and delivers explicit, finite one-loop results for 6D (1,1) amplitudes with up to five external legs, with clear 4D connections and potential for extending to higher-point and higher-loop analyses.

Abstract

Recently, the spinor helicity formalism and on-shell superspace were developed for six-dimensional gauge theories with (1,1) supersymmetry. We combine these two techniques with (generalised) unitarity, which is a powerful technique to calculate scattering amplitudes in any massless theory. As an application we calculate one-loop superamplitudes with four and five external particles in the (1,1) theory and perform several consistency checks on our results.

One-loop Amplitudes in Six-Dimensional (1,1) Theories from Generalised Unitarity

TL;DR

This work develops a complete framework for six-dimensional (1,1) SYM amplitudes by combining six-dimensional spinor-helicity formalism with on-shell (1,1) superspace and generalized unitarity. The authors compute the one-loop four- and five-point superamplitudes: the four-point case is shown to factorize into the tree-level superamplitude times a scalar box integral via both two-particle and quadruple cuts, while the five-point case yields a linear pentagon integral that PV-reduces to a pentagon plus box contributions with well-defined coefficients. They further extract gluon components and perform consistency checks by dimensional reduction to four dimensions, recovering the Parke–Taylor structure and known SYM results, as well as verifying soft limits. Overall, the paper provides a nontrivial test of unitarity-based techniques in higher dimensions and delivers explicit, finite one-loop results for 6D (1,1) amplitudes with up to five external legs, with clear 4D connections and potential for extending to higher-point and higher-loop analyses.

Abstract

Recently, the spinor helicity formalism and on-shell superspace were developed for six-dimensional gauge theories with (1,1) supersymmetry. We combine these two techniques with (generalised) unitarity, which is a powerful technique to calculate scattering amplitudes in any massless theory. As an application we calculate one-loop superamplitudes with four and five external particles in the (1,1) theory and perform several consistency checks on our results.

Paper Structure

This paper contains 27 sections, 161 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Spinor helicity formalism in six dimensions
  4. (1,1) on-shell superspace
  5. Tree-level amplitudes and their properties
  6. Three-point amplitude
  7. Four-point amplitude
  8. Five-point amplitude
  9. One-loop four-point amplitude
  10. The superamplitude from two-particle cuts
  11. The superamplitude from quadruple cuts
  12. Quadruple cut as reduced two-particle cuts
  13. Quadruple cut by Grassmann decomposition
  14. One-loop five-point superamplitude
  15. Quadruple cuts
  16. ...and 12 more sections

Figures (7)

  • Figure 1: The component fields of the (1,1) superfield given in \ref{['6d-superfield']}.
  • Figure 2: Double cut in the s-channel. The two internal cut-propagators, carrying momenta $l_1$ and $l_2$ set the two four-point subamplitudes on-shell. We identify $l_1 = l$ and $l_2 = l + p_1 + p_2$.
  • Figure 3: The quadruple cut of a four-point superamplitude. The primed momenta $l_i'$ are defined as $l_i' := - l_i$.
  • Figure 4: A specific quadruple cut of a five-point superamplitude. We choose to cut the legs such that we have the massive corner for momenta $p_3, p_4$.
  • Figure 5: A generic pentagon loop integral.
  • ...and 2 more figures