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How gluon leading singularities discover curves on surfaces

Sérgio Carrôlo, Carolina Figueiredo

TL;DR

This work develops a graphical, curve-based interpretation of leading singularities for gluon amplitudes by recasting on-shell spin contractions as curves on fatgraphs. The authors show that LS correspond to maximal residues in a surface-integral formulation and map these residues to simple combinatorial coverings of the fatgraph by non-overlapping curves, with cores and extensions encoding the Lorentz contractions. Loop-level corrections arise from ghost-related spin sums, self-intersecting curves, and closed curves whose exponents Δ_J depend on whether the curve is purely left-turning (Δ_J = 1−D) or not (Δ_J = −D); this framework also yields a V-rule that explains cancellations among competing contraction patterns. The approach extends from tree to all-loop orders, clarifying the exponents of closed curves and the role of mapping-class group symmetries, and it opens the way to incorporating fermions and other vertices into the surface-integral picture. Overall, the paper provides a unified, graphical method to compute and interpret gluon leading singularities across dimensions and loop levels, linking field-theory gluing to geometric curve coverings on surfaces.

Abstract

We study the leading singularities for pure gluon amplitudes obtained by on-shell gluing of three-particle amplitudes for an arbitrary graph in any number of dimensions. By encoding the polarization vector contractions in a graphical way, on-shell gluing "discovers" curves on surfaces, and we find that the leading singularity is determined by a simple combinatorial question: what are all ways of covering the graph with non-overlapping curves such that each edge is covered exactly once? This precisely matches the formula from the surfaceology formulation of gluons, where the leading singularities are given by maximal residues, with the combinatorial problem arising from the linearized form of the $u$ variables. At loop-level we describe how the novelties associated with spin sums (related with the need for ghosts when working off-shell using Lagrangians) can be easily encoded in this combinatorial picture. Matching the leading singularities also lets us settle an open question in the surface formulation of gluons, determining the exponents of the closed curves at any loop order.

How gluon leading singularities discover curves on surfaces

TL;DR

This work develops a graphical, curve-based interpretation of leading singularities for gluon amplitudes by recasting on-shell spin contractions as curves on fatgraphs. The authors show that LS correspond to maximal residues in a surface-integral formulation and map these residues to simple combinatorial coverings of the fatgraph by non-overlapping curves, with cores and extensions encoding the Lorentz contractions. Loop-level corrections arise from ghost-related spin sums, self-intersecting curves, and closed curves whose exponents Δ_J depend on whether the curve is purely left-turning (Δ_J = 1−D) or not (Δ_J = −D); this framework also yields a V-rule that explains cancellations among competing contraction patterns. The approach extends from tree to all-loop orders, clarifying the exponents of closed curves and the role of mapping-class group symmetries, and it opens the way to incorporating fermions and other vertices into the surface-integral picture. Overall, the paper provides a unified, graphical method to compute and interpret gluon leading singularities across dimensions and loop levels, linking field-theory gluing to geometric curve coverings on surfaces.

Abstract

We study the leading singularities for pure gluon amplitudes obtained by on-shell gluing of three-particle amplitudes for an arbitrary graph in any number of dimensions. By encoding the polarization vector contractions in a graphical way, on-shell gluing "discovers" curves on surfaces, and we find that the leading singularity is determined by a simple combinatorial question: what are all ways of covering the graph with non-overlapping curves such that each edge is covered exactly once? This precisely matches the formula from the surfaceology formulation of gluons, where the leading singularities are given by maximal residues, with the combinatorial problem arising from the linearized form of the variables. At loop-level we describe how the novelties associated with spin sums (related with the need for ghosts when working off-shell using Lagrangians) can be easily encoded in this combinatorial picture. Matching the leading singularities also lets us settle an open question in the surface formulation of gluons, determining the exponents of the closed curves at any loop order.

Paper Structure

This paper contains 28 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scalar-scaffolded gluon leading singularities and surfaces
  3. Leading singularities from linearized $u$'s
  4. Tree and loop examples
  5. $3$-point interaction
  6. One-loop bubble
  7. Tree-level leading singularities from explicit gluing
  8. Building blocks: 3-point amplitudes
  9. 4-point s-channel LS
  10. From contractions to curves and cores
  11. Example
  12. From cores to contractions
  13. Tr $F^3$ leading singularities
  14. Leading singularities at one-loop
  15. Corrections to loop-level gluing
  16. ...and 13 more sections

Figures (2)

  • Figure 1: (Left) $s$-channel fatgraph and respective dual $\mathcal{T}_{\text{int}} = \{(1,5)\}$ for the $4$-point gluon amplitude. (Right) Correspondence between curves on the surface and paths on the fat-graph. The curve $(3,7)$ in blue LHS maps to the path in the RHS. The path on the fat-graph can be further truncated into a core piece (solid) plus its extensions (dashed).
  • Figure 2: (Left) Triangulation of the punctured disk containing curves $\mathcal{T}=\{(1,p),(3,p),(1,3),(3,1)\}$, where $p$ is labelling the puncture. (Right) Dual fatgraph with the internal edges labelled accordingly. In blue we represent the core of curve $X_{2,4}$ and in red the one for the tadpole $X_{3,3}$.