From Twistor String Theory To Recursion Relations

TL;DR

The paper establishes a concrete bridge between Witten's twistor-string connected prescription and the BCFW/Arkani-Hamed dual formulations for tree-level N=4 SYM. By Fourier transforming to mixed twistor variables, it builds a link representation with variables $c_{iJ}$ and a tractable integration measure, enabling direct comparison of residues across representations. Through explicit 6- and 7-point examples, it shows how contour choices reproduce BCFW terms and uncover intermediate prescriptions, all tied together by the global residue theorem. This work clarifies how twistor-string frameworks encode multiple, contour-dependent representations of the same amplitudes and suggests a richer structure for more general amplitudes and loops.

Abstract

Witten's twistor string theory gives rise to an enigmatic formula [arXiv:hep-th/0403190] known as the "connected prescription" for tree-level Yang-Mills scattering amplitudes. We derive a link representation for the connected prescription by Fourier transforming it to mixed coordinates in terms of both twistor and dual twistor variables. We show that it can be related to other representations of amplitudes by applying the global residue theorem to deform the contour of integration. For six and seven particles we demonstrate explicitly that certain contour deformations rewrite the connected prescription as the BCFW representation, thereby establishing a concrete link between Witten's twistor string theory and the dual formulation for the S-matrix of N=4 SYM recently proposed by Arkani-Hamed et. al. Other choices of integration contour also give rise to "intermediate prescriptions". We expect a similar though more intricate structure for more general amplitudes.