One-loop Scattering Equations and Amplitudes from Forward Limit

TL;DR

The paper shows that the forward limit of tree-level scattering with two massive particles yields the SL(2,C) covariant form of one-loop scattering equations and provides a precise CHY-based formula for one-loop bi-adjoint scalar amplitudes. It derives the one-loop equations via the forward limit, analyzes the solution structure, and defines the corresponding measure. In the bi-adjoint theory, it proves a central identity relating one-loop double-partial amplitudes to forward-limited tree double-partials, producing a closed formula in Geyer’s one-loop representation. A detailed 4-point example establishes diagrammatic rules for assembling tree data into one-loop integrals, including cancellations of tadpoles and the handling of external-leg bubbles, thereby linking forward-limit tree amplitudes to bona fide one-loop amplitudes without invoking ambitwistor strings.

Abstract

We show that the forward limit of tree-level scattering equations with two massive particles yields the SL(2,C)-covariant form of the one-loop scattering equations recently proposed by Geyer et al. We clarify several properties about these equations and the formulas at one loop. We then argue that in the bi-adjoint scalar theory, such forward limit yields the correct one-loop massless amplitudes, which leads to a new formula for the latter.