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Scattering Amplitudes and BCFW Recursion in Twistor Space

Lionel Mason, David Skinner

TL;DR

This work develops explicit twistor-space formulations of scattering amplitudes by translating BCFW recursion into a twistor framework using seeds from three-point amplitudes and a real off-shell shift parameter. It delivers concrete constructions for tree-level amplitudes in $ ext{N}=4$ SYM and $ ext{N}=8$ supergravity (MHV, NMHV, and beyond), expressing them as Hilbert-transform operations on delta-function supports that localize on twistor-space geometric objects (lines and planes). The analysis reveals subtle conformal invariance breaking due to sign operators and split-signature topology, clarifies how even-point amplitudes can restore conformal invariance, and connects the twistor formulation to ambidextrous/ambitwistor approaches. The paper also discusses loop amplitudes in this setting and outlines avenues to relate these twistor constructions to twistor actions and complex/Lorentz-signature realizations, offering a practical toolkit for twistorial perturbation theory. Overall, it opens a concrete path to manifestly twistorial descriptions of gauge and gravity S-matrices while highlighting signature-dependent subtleties that require further refinement for a fully self-contained Lorentzian theory.

Abstract

Twistor ideas have led to a number of recent advances in our understanding of scattering amplitudes. Much of this work has been indirect, determining the twistor space support of scattering amplitudes by examining the amplitudes in momentum space. In this paper, we construct the actual twistor scattering amplitudes themselves. We show that the recursion relations of Britto, Cachazo, Feng and Witten have a natural twistor formulation that, together with the three-point seed amplitudes, allows us to recursively construct general tree amplitudes in twistor space. We obtain explicit formulae for $n$-particle MHV and NMHV super-amplitudes, their CPT conjugates (whose representations are distinct in our chiral framework), and the eight particle N^2MHV super-amplitude. We also give simple closed form formulae for the N=8 supergravity recursion and the MHV and conjugate MHV amplitudes. This gives a formulation of scattering amplitudes in maximally supersymmetric theories in which superconformal symmetry and its breaking is manifest. For N^kMHV, the amplitudes are given by 2n-4 integrals in the form of Hilbert transforms of a product of $n-k-2$ purely geometric, superconformally invariant twistor delta functions, dressed by certain sign operators. These sign operators subtly violate conformal invariance, even for tree-level amplitudes in N=4 super Yang-Mills, and we trace their origin to a topological property of split signature space-time. We develop the twistor transform to relate our work to the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed, Cachazo, Cheung and Kaplan.

Scattering Amplitudes and BCFW Recursion in Twistor Space

TL;DR

This work develops explicit twistor-space formulations of scattering amplitudes by translating BCFW recursion into a twistor framework using seeds from three-point amplitudes and a real off-shell shift parameter. It delivers concrete constructions for tree-level amplitudes in SYM and supergravity (MHV, NMHV, and beyond), expressing them as Hilbert-transform operations on delta-function supports that localize on twistor-space geometric objects (lines and planes). The analysis reveals subtle conformal invariance breaking due to sign operators and split-signature topology, clarifies how even-point amplitudes can restore conformal invariance, and connects the twistor formulation to ambidextrous/ambitwistor approaches. The paper also discusses loop amplitudes in this setting and outlines avenues to relate these twistor constructions to twistor actions and complex/Lorentz-signature realizations, offering a practical toolkit for twistorial perturbation theory. Overall, it opens a concrete path to manifestly twistorial descriptions of gauge and gravity S-matrices while highlighting signature-dependent subtleties that require further refinement for a fully self-contained Lorentzian theory.

Abstract

Twistor ideas have led to a number of recent advances in our understanding of scattering amplitudes. Much of this work has been indirect, determining the twistor space support of scattering amplitudes by examining the amplitudes in momentum space. In this paper, we construct the actual twistor scattering amplitudes themselves. We show that the recursion relations of Britto, Cachazo, Feng and Witten have a natural twistor formulation that, together with the three-point seed amplitudes, allows us to recursively construct general tree amplitudes in twistor space. We obtain explicit formulae for -particle MHV and NMHV super-amplitudes, their CPT conjugates (whose representations are distinct in our chiral framework), and the eight particle N^2MHV super-amplitude. We also give simple closed form formulae for the N=8 supergravity recursion and the MHV and conjugate MHV amplitudes. This gives a formulation of scattering amplitudes in maximally supersymmetric theories in which superconformal symmetry and its breaking is manifest. For N^kMHV, the amplitudes are given by 2n-4 integrals in the form of Hilbert transforms of a product of purely geometric, superconformally invariant twistor delta functions, dressed by certain sign operators. These sign operators subtly violate conformal invariance, even for tree-level amplitudes in N=4 super Yang-Mills, and we trace their origin to a topological property of split signature space-time. We develop the twistor transform to relate our work to the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed, Cachazo, Cheung and Kaplan.

Paper Structure

This paper contains 41 sections, 229 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Momentum Shift on Twistor Space
  3. Twistor Amplitudes and Conformal Invariance
  4. Distributions on projective twistor space
  5. The Hilbert transform and the $\mathrm{sgn}$ function
  6. Seed amplitudes in $\mathcal{N}=4$ SYM
  7. On conformal invariance
  8. BCFW Recursion in Twistor Space
  9. The $\mathrm{i}\epsilon$-prescription on $\mathbb{R}^{2,2}$
  10. Transform to twistor space
  11. SYM Twistor Amplitudes from BCFW Recursion
  12. On the general structure of $\mathcal{N}=4$ Amplitudes
  13. Solving the recursion relations
  14. The homogeneous term and MHV amplitudes
  15. The conjugate homogeneous term and the $\overline{\rm MHV}$ amplitudes
  16. ...and 26 more sections

Figures (2)

  • Figure 1: The MHV amplitude is supported on a line in twistor space.
  • Figure 2: The NMHV amplitude is supported on three coplanar lines in twistor space. Point 1 (distinguished by its role in the BCFW recursion relations) is located at the intersection of two of these lines.

Theorems & Definitions (1)

  • Definition 3.1