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CHY Loop Integrands from Holomorphic Forms

Humberto Gomez, Sebastian Mizera, Guojun Zhang

TL;DR

The paper develops a two-loop CHY formalism for $\Phi^3$ theory by deriving CHY integrands from holomorphic forms on genus-1 and genus-2 surfaces and by introducing explicit building blocks built from meromorphic forms. It defines a comprehensive gluing framework and extends the $\Lambda$-algorithm to two loops, enabling analytic evaluation of CHY integrals and verification against Feynman diagrams up to seven external legs. Key contributions include the introduction of three two-loop quadratic differentials $\mathsf{q}^1_a,\mathsf{q}^2_a,\mathsf{q}^3_a$, two-loop planar and non-planar CHY constructions, and a set of $\Lambda$-rules that control singularities and factorization. The framework provides a geometrical and scalable route to loop amplitudes in CHY language and offers a foundation for extending to higher loops and other theories such as YM and gravity, potentially enabling compact, double-copy compatible representations.

Abstract

Recently, the Cachazo-He-Yuan (CHY) approach for calculating scattering amplitudes has been extended beyond tree level. In this paper, we introduce a way of constructing CHY integrands for $Φ^3$ theory up to two loops from holomorphic forms on Riemann surfaces. We give simple rules for translating Feynman diagrams into the corresponding CHY integrands. As a complementary result, we extend the $Λ$-algorithm, originally introduced in arXiv:1604.05373, to two loops. Using this approach, we are able to analytically verify our prescription for the CHY integrands up to seven external particles at two loops. In addition, it gives a natural way of extending to higher-loop orders.

CHY Loop Integrands from Holomorphic Forms

TL;DR

The paper develops a two-loop CHY formalism for theory by deriving CHY integrands from holomorphic forms on genus-1 and genus-2 surfaces and by introducing explicit building blocks built from meromorphic forms. It defines a comprehensive gluing framework and extends the -algorithm to two loops, enabling analytic evaluation of CHY integrals and verification against Feynman diagrams up to seven external legs. Key contributions include the introduction of three two-loop quadratic differentials , two-loop planar and non-planar CHY constructions, and a set of -rules that control singularities and factorization. The framework provides a geometrical and scalable route to loop amplitudes in CHY language and offers a foundation for extending to higher loops and other theories such as YM and gravity, potentially enabling compact, double-copy compatible representations.

Abstract

Recently, the Cachazo-He-Yuan (CHY) approach for calculating scattering amplitudes has been extended beyond tree level. In this paper, we introduce a way of constructing CHY integrands for theory up to two loops from holomorphic forms on Riemann surfaces. We give simple rules for translating Feynman diagrams into the corresponding CHY integrands. As a complementary result, we extend the -algorithm, originally introduced in arXiv:1604.05373, to two loops. Using this approach, we are able to analytically verify our prescription for the CHY integrands up to seven external particles at two loops. In addition, it gives a natural way of extending to higher-loop orders.

Paper Structure

This paper contains 29 sections, 88 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Holomorphic Forms at One and Two Loops
  3. One--loop Holomorphic Form
  4. Geometric Interpretation
  5. Holomorphic Forms over the Double Torus
  6. Physical Requirements
  7. Building Blocks of CHY Integrands at One and Two Loops for $\Phi^3$
  8. One-Loop Building Block
  9. Two-loop Building Blocks
  10. Constructing CHY Graphs from Gluing Building Blocks
  11. Tree-level Gluing
  12. One-loop Level Gluing
  13. Two-loop Level Gluing
  14. $\Lambda-$Rules
  15. Examples in $\Phi^3$ Theory
  16. ...and 14 more sections

Figures (30)

  • Figure 1: Geometrical interpretation of reducing from the holomorphic form $\Omega(z)$ on a torus to the meromorphic form $\omega_z$ on a Riemann sphere.
  • Figure 2: Geometrical meaning of the two holomorphic forms on the double torus.
  • Figure 3: Three building blocks for two-loop Feynman diagrams for $\Phi^3$. We will use these to construct more complicated 1PI diagrams.
  • Figure 4: Correspondence between the $\Phi^3$ Feynman diagrams ($n$-gon symmetrized) and the $\mathbf{I}^{\rm n-gon-CHY}_{\rm sym}$ CHY-graphs. $S_n$ is the permutation group.
  • Figure 5: The $\Phi^3$ planar Feynman diagrams we want to compare with the CHY-graphs. $S_n$ is the permutation group.
  • ...and 25 more figures