CHY-graphs on a torus

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 M1,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathfrak{M}}_{1,n} $$\end{document}, the moduli space of n-punctured Elliptic curves. As a particular application, we study the Φ3 bi-adjoint scalar theory. We propose a set of rules to construct integrands on M1,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathfrak{M}}_{1,n} $$\end{document} from Φ3 integrands on M0,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathfrak{M}}_{0,n} $$\end{document}, the moduli space of n-punctured spheres. We illustrate these rules by computing a variety of Φ3 one-loop Feynman diagrams. Conversely, we also provide another set of rules to compute the corresponding CHY-integrand on M1,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathfrak{M}}_{1,n} $$\end{document} by starting instead from a given Φ3 one-loop Feynman diagram. In addition, our results can easily be extended to higher loops.