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Eliminating spurious poles from gauge-theoretic amplitudes

Andrew Hodges

TL;DR

Spurious poles hinder straightforward BCFW-based decompositions of gauge-theory amplitudes. The author introduces momentum-twistor coordinates and dual conformal symmetry to recast amplitudes as geometric integrals over momentum-space polytopes, yielding spurious-pole-free NMHV results and revealing a dihedral-symmetric supersymmetric structure. By interpreting NMHV amplitudes as volumes of polytopes H_n (and P_n in the bosonic case), the framework unifies different BCFW representations as equivalent polytope decompositions and makes hidden symmetries explicit. The approach also suggests a duality between these momentum-twistor integrals and traditional twistor diagrams, with promising extensions to all helicities and potentially to loop amplitudes through region-space and polytope geometry.

Abstract

This note addresses the problem of spurious poles in gauge-theoretic scattering amplitudes. New twistor coordinates for the momenta are introduced, based on the concept of dual conformal invariance. The cancellation of spurious poles for a class of NMHV amplitudes is greatly simplified in these coordinates. The poles are eliminated altogether by defining a new type of twistor integral, dual to twistor diagrams as previously studied, and considerably simpler. The geometric features indicate a supersymmetric extension of the formalism at least to all NMHV amplitudes, allowing the dihedral symmetry of the super-amplitude to be made manifest. More generally, the definition of `momentum-twistor' coordinates suggests a powerful new approach to the study of scattering amplitudes.

Eliminating spurious poles from gauge-theoretic amplitudes

TL;DR

Spurious poles hinder straightforward BCFW-based decompositions of gauge-theory amplitudes. The author introduces momentum-twistor coordinates and dual conformal symmetry to recast amplitudes as geometric integrals over momentum-space polytopes, yielding spurious-pole-free NMHV results and revealing a dihedral-symmetric supersymmetric structure. By interpreting NMHV amplitudes as volumes of polytopes H_n (and P_n in the bosonic case), the framework unifies different BCFW representations as equivalent polytope decompositions and makes hidden symmetries explicit. The approach also suggests a duality between these momentum-twistor integrals and traditional twistor diagrams, with promising extensions to all helicities and potentially to loop amplitudes through region-space and polytope geometry.

Abstract

This note addresses the problem of spurious poles in gauge-theoretic scattering amplitudes. New twistor coordinates for the momenta are introduced, based on the concept of dual conformal invariance. The cancellation of spurious poles for a class of NMHV amplitudes is greatly simplified in these coordinates. The poles are eliminated altogether by defining a new type of twistor integral, dual to twistor diagrams as previously studied, and considerably simpler. The geometric features indicate a supersymmetric extension of the formalism at least to all NMHV amplitudes, allowing the dihedral symmetry of the super-amplitude to be made manifest. More generally, the definition of `momentum-twistor' coordinates suggests a powerful new approach to the study of scattering amplitudes.

Paper Structure

This paper contains 20 sections, 50 equations, 5 figures.

Table of Contents

  1. The problem of spurious poles
  2. The split-helicity 6-field NMHV amplitude
  3. Super-amplitudes and dihedral symmetry
  4. Dual conformal symmetry
  5. Momentum-twistor coordinates
  6. Translating the amplitude
  7. New integrals in momentum-twistor space
  8. Why spurious poles cancel: spurious boundaries
  9. Generalisation to more than six fields
  10. NMHV split-helicity amplitudes for seven fields
  11. NMHV split-helicity amplitudes for $n$ fields
  12. Supersymmetric extension to all helicities
  13. Polytopes with dihedral symmetry
  14. The combinatorics of BCFW
  15. Summary: super-amplitudes as super-volumes
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Regions for a process with outgoing null momenta, with $p_i = x_i - x_{i-1}$.
  • Figure 2: The polyhedron $P_6 = T_{13[46]5}$, with 6 vertices, 9 edges and 5 faces.
  • Figure 3: In the dual representation, $P_6$ appears as the join of two tetrahedra, with 5 vertices, 9 edges and 6 faces.
  • Figure 4: Twistor diagrams for the two-term representation of the split-helicity amplitude $A(1^- 2^- 3^- 4^+ 5^+ 6^+)$
  • Figure 5: Polyhedron $P_9$