Abstract
We present new formulas for n-particle tree-level scattering amplitudes of six-dimensional mathcal{N}=left(1,1right) super Yang-Mills (SYM) and mathcal{N}=left(2,2right) 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 mathcal{N}=left(1,1right) 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 mathcal{N}=left(2,2right) 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 mathcal{N}=4 SYM on the Coulomb branch.
Highlights
Scattering amplitudes have been the subject of great interest especially since the introduction of Witten’s twistor string theory in 2003 [1]
We present new formulas for n-particle tree-level scattering amplitudes of sixdimensional N = (1, 1) super Yang-Mills (SYM) and N = (2, 2) supergravity (SUGRA)
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
Summary
Scattering amplitudes have been the subject of great interest especially since the introduction of Witten’s twistor string theory in 2003 [1]. Having explicit integrands for the complete 6D N = (1, 1) SYM tree amplitudes allows the construction of the 6D N = (2, 2) SUGRA integrand by the standard replacement of the left-integrand Parke-Taylor factor by a copy of the right integrand, which contains the necessary new supersymmetric information. We end with various applications to other theories in four, five, and six dimensions These include mixed superamplitudes of 6D N = (1, 1) SYM coupled to a single D5-brane, 5D SYM and SUGRA, and 4D scattering amplitudes involving massive particles of N = 4 SYM on the Coulomb branch of its moduli space. In appendix A we present the algebra of the new T-shift, and in appendix B we give details of the soft-limit calculations
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