CHY-Graphs on a Torus

TL;DR

The paper advances CHY methods to one-loop by formulating a torus-based (genus-1) construction on the moduli space ${\mathfrak M}_{1,n}$ using new connectors $H_{a:a}, T_{a:b}, G^{\pm}_{a:b}$. It provides a concrete, graphical, and rule-based translation between tree-level CHY integrands on ${\mathfrak M}_{0,n}$ and one-loop CHY integrands on ${\mathfrak M}_{1,n}$, with explicit application to bi-adjoint $\Phi^3$ theory and both forward- and inverse-constructions that map to and from one-loop Feynman diagrams. Central results include a proposition constraining valid one-loop integrands to products of chains with a fixed line/anti-line balance, and a construction rule that replaces sphere connectors with torus connectors while preserving loop topology. The Lambda-algorithm is employed to evaluate the resulting CHY graphs, yielding known one-loop topologies such as triangles and boxes, and clarifying the relationship between CHY integrands and standard Feynman diagrams. The work also discusses tadpole absence, potential higher-loop extensions, and a graphical blueprint for extending CHY to broader theories with loop-level CHY descriptions.

Abstract

Recently, we proposed a new approach using a punctured Elliptic curve in the CHY framework in order to compute one-loop scattering amplitudes. In this note, we further develop this approach by introducing a set of connectors, which become the main ingredient to build integrands on $\mathfrak{M}_{1,n}$, the moduli space of n-punctured Elliptic curves. As a particular application, we study the $Φ^3$ bi-adjoint scalar theory. We propose a set of rules to construct integrands on $\mathfrak{M}_{1,n}$ from $Φ^ 3$ integrands on $\mathfrak{M}_{0,n}$, the moduli space of n-punctured spheres. We illustrate these rules by computing a variety of $Φ^3$ one-loop Feynman diagrams. Conversely, we also provide another set of rules to compute the corresponding CHY-integrand on $\mathfrak{M}_{1,n}$ by starting instead from a given $Φ^ 3$ one-loop Feynman diagram. In addition, our results can easily be extended to higher loops.

Figures (28)

• Figure 1: The ${\cal I}^{\rm t}_5(1,2,3,4,5)$regular graph.
• Figure 2: Color Code.
• Figure 3: n-gon representation. (1) CHY-graph on a sphere (up to $\ell^2$ overall factor). (2) Feynman diagram.
• Figure 4: CHY Torus representation for the n-gon. After pinching the a-cycle one obtains the CHY-tree level graph. One can think that the anti-lines among $\sigma_\ell$ and $\sigma_{-\ell}$ arose in order to obtain $PSL(2,\mathbb{C})$ invariance on the CHY-tree level graph.
• Figure 5: Two different options to connect $\sigma_a$ with $\sigma_b$ on a Torus.