From 4d Ambitwistor Strings to On Shell Diagrams and Back

TL;DR

This work investigates the link between 4d ambitwistor string theory and on-shell diagrams for planar $N=4$ SYM and $N=8$ SUGRA, deriving new Grassmannian integral representations for tree-level amplitudes and constructing manifestly supersymmetric worldsheet formulae for 1-loop amplitudes refined by MHV degree. It shows that NMHV amplitudes in $N=4$ SYM can be captured by residues of a single top form in the Grassmannian, while decorated planar NMHV diagrams in $N=8$ SUGRA do not arise from a single top-form, indicating a more intricate contour structure. The paper also builds 1-loop worldsheet formulations that map to scalar box integrals in gauge theory and to Hodges-determinant structures in gravity, highlighting both the unifying framework and the theory-specific differences. Together, these results deepen the geometric understanding of perturbative amplitudes and point toward extensions to higher loops and non-planar sectors, with potential connections to underlying amplituhedron-like structures and new regularization schemes for loop integrals.

Abstract

We investigate the relation between 4d ambitwistor string theory and on-shell diagrams for planar N=4 super-Yang-Mills and N=8 supergravity, and deduce several new results about their scattering amplitudes at tree-level and 1-loop. In particular, we derive new Grassmannian integral formulae for tree-level amplitudes and obtain new worldsheet formulae for 1-loop amplitudes which are manifestly supersymmetric and supported on scattering equations refined by MHV degree.

Figures (29)

• Figure 1: Vertices and edges for on shell diagrams in $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ SUGRA
• Figure 2: Loop-level BCFW recursion for planar $\mathcal{N}=4$ SYM
• Figure 3: Square move equivalence relation SYM
• Figure 4: Merger equivalence relations for $\mathcal{N}=4$ SYM
• Figure 5: Tree-level BCFW recursion relations in $\mathcal{N}=8$ SUGRA