Elliptic scattering equations

TL;DR

This work develops an algebraic, elliptic-curve-based generalization of the CHY framework for one-loop amplitudes, avoiding elliptic theta-function machinery by embedding the elliptic curve in $\mathbb{C}P^2$ and constructing a meromorphic differential $\Omega^\mu$. By integrating over the torus modulus with a global residue theorem and applying a $\Lambda$-algorithm, the authors derive a tree-level off-shell recurrence for the $n$-gon integrand, relate the elliptic setup to the forward-limit CHY picture, and show that at the nodal degenerations the problem reduces to a sum of tree-level subamplitudes with a loop momentum $\ell$ identified as the flux through the a-cycle. They provide explicit lower-point checks (3-, 4-, 5-particle) and conjecture a permutation-based partial fraction form for the recurrence, supported numerically up to nine points. The framework naturally extends to higher genus (hyperelliptic curves) and offers a practical route toward one-loop amplitudes in other theories, with discussion of dimensional constraints and potential extensions to YM and gravity.

Abstract

Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a $\mathbb{C}P^2$ space. We show that for the simplest integrand, namely the ${\rm n-gon}$, our proposal indeed reproduces the expected result. By using the recently formulated $Λ-$algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.