CHY-Graphs on a Torus

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 ${\Phi }^3$ bi-adjoint scalar theory. We propose a set of rules to construct integrands on ${\mathfrak{M}}_{1,n}$ from ${\Phi }^3$ integrands on ${\mathfrak{M}}_{0,n}$, the moduli space of n-punctured spheres. We illustrate these rules by computing a variety of ${\Phi }^3$ 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 ${\Phi }^3$ one-loop Feynman diagram. In addition, our results can easily be extended to higher loops.

34 pages, 32 figures

References in corpus (6)

• New Relations for Gauge-Theory Amplitudes
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• On the solutions of the scattering equations
• Scattering equations, generating functions and all massless five point tree amplitudes
• Permutation Symmetry of the Scattering Equations

Cited by in corpus (4)

• Two-Loop Scattering Amplitudes from the Riemann Sphere
• Einstein-Yang-Mills from pure Yang-Mills amplitudes
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes