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MHV Diagrams in Momentum Twistor Space

Mathew Bullimore, Lionel Mason, David Skinner

TL;DR

This work recasts the MHV-diagram framework of planar N=4 SYM in momentum twistor space, replacing vertices by unity and associating dual superconformal R-invariants to propagators. The reformulation makes dual superconformal symmetry manifest (up to a reference twistor) and yields a direct all-loop algorithm for the planar integrand, with explicit tree- and loop-level examples. It connects to generalized unitarity at the integrand level and provides a coherent geometric interpretation via momentum twistors, including a natural regularisation approach for loops. The developed rules unify tree and loop amplitudes, enabling efficient computation of NMHV, N^2MHV, and higher MHV-degree amplitudes and their loop integrands within a single formalism. The work also lays groundwork for further links to Wilson loops and BCFW-type recursion in momentum-twistor space, highlighting the structural role of R-invariants in amplitude construction.

Abstract

We show that there are remarkable simplifications when the MHV diagram formalism for N=4 super Yang-Mills is reformulated in momentum twistor space. The vertices are replaced by unity while each propagator becomes a dual superconformal `R-invariant' whose arguments may be read off from the diagram. The momentum twistor MHV rules generate a formula for the full, all-loop planar integrand of the super Yang-Mills S-matrix that is manifestly dual superconformally invariant up to the choice of a reference twistor. We give a general proof of this reformulation and illustrate its use by computing the momentum twistor NMHV and NNMHV tree amplitudes and the integrands of the MHV and NMHV 1-loop and the MHV 2-loop planar amplitudes.

MHV Diagrams in Momentum Twistor Space

TL;DR

This work recasts the MHV-diagram framework of planar N=4 SYM in momentum twistor space, replacing vertices by unity and associating dual superconformal R-invariants to propagators. The reformulation makes dual superconformal symmetry manifest (up to a reference twistor) and yields a direct all-loop algorithm for the planar integrand, with explicit tree- and loop-level examples. It connects to generalized unitarity at the integrand level and provides a coherent geometric interpretation via momentum twistors, including a natural regularisation approach for loops. The developed rules unify tree and loop amplitudes, enabling efficient computation of NMHV, N^2MHV, and higher MHV-degree amplitudes and their loop integrands within a single formalism. The work also lays groundwork for further links to Wilson loops and BCFW-type recursion in momentum-twistor space, highlighting the structural role of R-invariants in amplitude construction.

Abstract

We show that there are remarkable simplifications when the MHV diagram formalism for N=4 super Yang-Mills is reformulated in momentum twistor space. The vertices are replaced by unity while each propagator becomes a dual superconformal `R-invariant' whose arguments may be read off from the diagram. The momentum twistor MHV rules generate a formula for the full, all-loop planar integrand of the super Yang-Mills S-matrix that is manifestly dual superconformally invariant up to the choice of a reference twistor. We give a general proof of this reformulation and illustrate its use by computing the momentum twistor NMHV and NNMHV tree amplitudes and the integrands of the MHV and NMHV 1-loop and the MHV 2-loop planar amplitudes.

Paper Structure

This paper contains 18 sections, 96 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Momentum Twistors for $\mathcal{N}=4$ super Yang-Mills
  3. The MHV formalism
  4. MHV tree diagrams in momentum twistor space
  5. Momentum twistor MHV rules for tree amplitudes
  6. NMHV tree amplitudes
  7. N$^2$MHV tree amplitudes
  8. A diagram for the N$^3$MHV tree
  9. General trees
  10. MHV loop diagrams in momentum twistor space
  11. Momentum twistor MHV rules at one loop
  12. MHV 1-loop
  13. NMHV 1-loop
  14. General 1-loop integrands
  15. The loop contour
  16. ...and 3 more sections

Figures (12)

  • Figure 1: The momentum twistor correspondence.
  • Figure 2: The shifted twistor $\widehat{Z_{i-1}}$ is the intersection of the line $(i\!-\!1,i)$ with the plane $(\,*\,,k\!-\!1,k)$.
  • Figure 3: An MHV diagram contributing $[*,i\!-\!1,i,j\!-\!1,j]$ to the tree-level NMHV amplitude.
  • Figure 4: An MHV diagram contributing to the tree-level N$^2$MHV amplitude, together with its associated product of dual superconformal invariants
  • Figure 5: MHV diagrams occurring in the tree-level $\mathrm{N}^2\mathrm{MHV}$ amplitude for which the two propagators are adjacent. Notice that $j\!-\!1$ is unshifted in the first R invariant, because the external leg $j\!-\!1$ is attached to the propagator on the left. Notice also that, with our choice of orientation, the external leg $j$ is never shifted.
  • ...and 7 more figures