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One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

Humberto Gomez, Cristhiam Lopez-Arcos, Pedro Talavera

TL;DR

This work reformulates the one-loop CHY framework to produce quadratic propagators directly, by redefining the Parke-Taylor factors and systematically classifying one-loop CHY integrands. It introduces a new forward-limit construction with $n{+}4$ massless punctures and provides explicit proofs and examples for the bi-adjoint $\Phi^3$ theory, including three- and four-point amplitudes, showing precise mappings to Feynman diagrams and addressing regularization of forward-limit singularities. The approach yields a coherent, diagrammatic CHY dictionary that naturally reproduces the familiar loop content (boxes, triangles, bubbles) and connects with the Feynman $i\epsilon$ prescription via dimensional reduction. The results offer a stepping stone toward extensions to Yang-Mills theory, BCJ/KLT structures, and possibly higher-loop generalizations, with a clear pathway for incorporating non-planar sectors and regularization within the CHY formalism.

Abstract

In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint $Φ^3$ theory. The prescription directly reproduces the quadratic propagators from of the traditional Feynman approach.

One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

TL;DR

This work reformulates the one-loop CHY framework to produce quadratic propagators directly, by redefining the Parke-Taylor factors and systematically classifying one-loop CHY integrands. It introduces a new forward-limit construction with massless punctures and provides explicit proofs and examples for the bi-adjoint theory, including three- and four-point amplitudes, showing precise mappings to Feynman diagrams and addressing regularization of forward-limit singularities. The approach yields a coherent, diagrammatic CHY dictionary that naturally reproduces the familiar loop content (boxes, triangles, bubbles) and connects with the Feynman prescription via dimensional reduction. The results offer a stepping stone toward extensions to Yang-Mills theory, BCJ/KLT structures, and possibly higher-loop generalizations, with a clear pathway for incorporating non-planar sectors and regularization within the CHY formalism.

Abstract

In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint theory. The prescription directly reproduces the quadratic propagators from of the traditional Feynman approach.

Paper Structure

This paper contains 20 sections, 107 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Parke-Taylor at One-Loop
  3. Examples
  4. One-Loop Feynman Integrands Classification from CHY Approach
  5. One Loop Integrands Classification
  6. The bi-adjoint $\Phi^3$ scalar theory
  7. Bi-adjoint $\Phi^3$ at One-Loop
  8. A New Proposal
  9. Examples
  10. Three-point
  11. Obtaining $\mathfrak{M}_3^{\rm 1-loop}[1,2,3|1,2,3]$
  12. Obtaining $\mathfrak{M}_3^{\rm 1-loop}[1,2,3|3,2,1]$
  13. Four-point
  14. General Structure of $\mathfrak{M}_N^{\rm 1-loop}[\mathbb{1}_N|\mathbb{1}_N]$
  15. External-Leg Bubbles
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Vertex Color code in CHY-graphs for the $\Lambda-$algorithm.
  • Figure 2: Edges Color code in CHY-graphs for the $\Lambda-$algorithm.
  • Figure 3: Loop box and triangle definition .
  • Figure 4: Fundamental CHY-Integrands at one-loop.
  • Figure 5: Simplest cases of CHY-graphs from Fig. \ref{['3-CHY-graph']}.
  • ...and 2 more figures