A differential operator for integrating one-loop scattering equations
Gang Chen, Yeuk-Kwan E. Cheung, Tianheng Wang, Feng Xu
TL;DR
The paper introduces a differential operator framework for evaluating multivariable residues in CHY representations of scattering amplitudes, with the operator uniquely fixed by the local duality theorem and divisor-intersection numbers. It demonstrates the method on a 5-point tree-level $\phi^3$ example and a four-point one-loop Yang-Mills case, showing how homogenization and residue-constrained operators can replace solving scattering-equation systems. The results reproduce known amplitudes and reveal a clear link to Q-cut forward-channel structures, suggesting practical reductions in computational complexity and a path to higher-point and higher-loop CHY analyses. The approach offers a new algebraic-geometry-based tool for CHY calculations and may illuminate symmetries and relations in gauge and gravity theories beyond leading order.
Abstract
We propose a differential operator for computing the residues associated with a class of meromorphic $n$-forms that frequently appear in the Cachazo-He-Yuan form of the scattering amplitudes. This differential operator is conjectured to be uniquely determined by the local duality theorem and the intersection number of the divisors in the $n$-form. We use the operator to evaluate the tree-level amplitude of $φ^3$ theory and the one-loop integrand of Yang-Mills theory from their CHY forms. The method can reduce the complexity of the calculation. In addition, the expression for the 1-loop four-point Yang-Mills integrand obtained in our approach has a clear correspondence with the Q-cut results.