The Grassmannian and the twistor string: connecting all trees in $ \mathcal{N} = 4 $ SYM

Abstract

We present a new, explicit formula for all tree-level amplitudes in N=4 super Yang-Mills. The formula is written as a certain contour integral of the connected prescription of Witten's twistor string, expressed in link variables. A very simple deformation of the integrand gives directly the Grassmannian integrand proposed by Arkani-Hamed et al. together with the explicit contour of integration. The integral is derived by iteratively adding particles to the Grassmannian integral, one particle at a time, and makes manifest both parity and soft limits. The formula is shown to be related to those given by Dolan and Goddard, and generalizes the results of earlier work for NMHV and N^2MHV to all N^(k-2)MHV tree amplitudes in N=4 super Yang-Mills.