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Quadratic Feynman Loop Integrands From Massless Scattering Equations

Humberto Gomez

TL;DR

This work presents a novel CHY-based prescription for obtaining quadratic Feynman loop propagators directly from massless scattering equations at one loop for ${\Phi}^3$ theory. By introducing two pairs of loop punctures and a forward-limit measure on an $(n{+}4)$-punctured sphere, the authors identify the loop momentum as a sum of massless momenta and show, via explicit examples (bubble, triangle, and four-point cases), that the resulting CHY integrand matches the standard quadratic Feynman integrand without requiring partial fraction identities or loop-momentum shifts. The construction has a geometric interpretation as a unitary cut on a genus-two Riemann surface and suggests a pathway to extend to other theories and higher-loop orders. The results offer a new route to connect CHY formulations with conventional Feynman diagrams and potentially illuminate loop-level relations (e.g., BCJ/KLT) within a massless, tree-level-like framework augmented by forward limits.

Abstract

Recently the Cachazo-He-Yuan (CHY) approach has been extended to loop level, but the resulting loop integrand has propagators that are linear in the loop momentum unlike Feynman's. In this note we present a new technique that directly produces quadratic propagators identical to Feynman's from the CHY approach. This paper focuses on $Φ^3$ theory but extensions to others theories are briefly discussed. In addition, our proposal has an interesting geometric meaning, we can interpret this new formula as a unitary cut on a higher genus Riemann surface.

Quadratic Feynman Loop Integrands From Massless Scattering Equations

TL;DR

This work presents a novel CHY-based prescription for obtaining quadratic Feynman loop propagators directly from massless scattering equations at one loop for theory. By introducing two pairs of loop punctures and a forward-limit measure on an -punctured sphere, the authors identify the loop momentum as a sum of massless momenta and show, via explicit examples (bubble, triangle, and four-point cases), that the resulting CHY integrand matches the standard quadratic Feynman integrand without requiring partial fraction identities or loop-momentum shifts. The construction has a geometric interpretation as a unitary cut on a genus-two Riemann surface and suggests a pathway to extend to other theories and higher-loop orders. The results offer a new route to connect CHY formulations with conventional Feynman diagrams and potentially illuminate loop-level relations (e.g., BCJ/KLT) within a massless, tree-level-like framework augmented by forward limits.

Abstract

Recently the Cachazo-He-Yuan (CHY) approach has been extended to loop level, but the resulting loop integrand has propagators that are linear in the loop momentum unlike Feynman's. In this note we present a new technique that directly produces quadratic propagators identical to Feynman's from the CHY approach. This paper focuses on theory but extensions to others theories are briefly discussed. In addition, our proposal has an interesting geometric meaning, we can interpret this new formula as a unitary cut on a higher genus Riemann surface.

Paper Structure

This paper contains 9 sections, 52 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Simple Review of CHY $\Phi^3$ Integrand at One-Loop
  3. Massless Scattering Equations and $\Phi^3$ Quadratic Feynman Loop Integrands
  4. A New CHY Proposal at One-Loop for $\Phi^3$
  5. Examples
  6. The Bubble
  7. The Triangle
  8. Four-particle
  9. Discussions

Figures (24)

  • Figure 1: $\Phi^3$ symmetrized Feynman diagram.
  • Figure 2: Schematic representation for the new CHY proposal.
  • Figure 3: Equality among CHY and Feynman integrands.
  • Figure 4: Color code in CHY-graphs.
  • Figure 5: Correspondence between the $\Phi^3$ Feynman diagrams ($n$-gon-symmetrized) and the CHY graphs, up to partial fraction identity and $\frac{1}{\ell^2}$ factor. $S_n$ is the permutation group and $\alpha_i : = \alpha(i)$.
  • ...and 19 more figures