$Λ$ Scattering Equations

TL;DR

The paper develops a double-cover reformulation of CHY scattering equations by embedding the punctured sphere in the curve $y^2=\sigma^2-\Lambda^2$, introducing a branch-cut parameter $\Lambda$ and an extra $\mathbb{C}^*$ redundancy. By promoting $\Lambda$ to a variable and using a GL(2,$\mathbb{C}$) gauge, the authors derive a $\Lambda$-prescription and a residue-based $\Lambda$-algorithm that expands CHY integrals into sums of products of off-shell subamplitudes linked by propagators. They construct a diagrammatic language with $\Lambda$-diagrams and building blocks, outline a systematic procedure (the $\Lambda$-algorithm) to evaluate generic rational CHY integrals, and illustrate the method with six- and eight-point examples, including a comparison with BBBD rules. The formalism naturally encodes factorization and crossing, accommodates off-shell and numerator-containing integrands, and offers a potential route toward loop-level CHY computations via iterative factorization. Overall, the work provides a powerful, gauge-guided framework to compute CHY integrals more efficiently and to understand their factorization structure through a new algebraic-geometry perspective.

Abstract

The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter $Λ$ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting $Λ$ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the $Λ$ algorithm.