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Published September 2018 | Published + Submitted
Journal Article Open

The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps


We present new formulas for n-particle tree-level scattering amplitudes of six-dimensional N=(1,1) super Yang-Mills (SYM) and N=(2,2) supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the (n − 3)! solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N=(1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2,ℂ) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N=(2,2) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N=4 SYM on the Coulomb branch.

Additional Information

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received: June 13, 2018; Accepted: September 14, 2018; Published: September 20, 2018. We thank Yvonne Geyer, Song He, Yu-tin Huang, Lionel Mason, and Kai Roehrig for useful discussions, and we are grateful to Song He, Yu-tin Huang, Zhengwen Liu, and Yong Zhang for comments on early drafts. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. A.G. thanks CONICYT for financial support. C.W. is supported by a Royal Society University Research Fellowship no. UF160350. This work was supported in part by the Walter Burke Institute for Theoretical Physics at Caltech and by U.S. DOE Grant DE-SC0011632. M.H. would like to thank Perimeter Institute for their hospitality.

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Published - Cachazo2018_Article_TheSMatrixOf6DSuperYang-MillsA.pdf

Submitted - 1805.11111


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August 21, 2023
October 18, 2023