CHY-construction of Planar Loop Integrands of Cubic Scalar Theory

TL;DR

This paper introduces a novel method for constructing planar loop integrands in cubic scalar theories using CHY formalism, simplifying calculations by relating loop momenta to higher-dimensional massless momenta.

Contribution

It presents a new approach to derive all-loop scattering equations as tree-level equations and provides explicit two-loop planar integrands for bi-adjoint phi_3 theory.

Findings

Explicit two-loop planar integrand formula matches Feynman diagram results

Method effectively handles forward singularities in loop integrands

Unified treatment of all-loop scattering equations as tree-level equations

Abstract

In this paper, by treating massive loop momenta to massless momenta in higher dimension, we are able to treat all-loop scattering equations as tree ones. As an application of the new aspect, we consider the CHY-construction of bi-adjoint phi_3 theory. We present the explicit formula for two-loop planar integrands. We discuss carefully how to subtract various forward singularities in the construction. We count the number of terms obtained by our formula and by direct Feynman diagram calculation and find the perfect match, thus provide a strong support for our results.