Integration Rules for Scattering Equations

TL;DR

The paper develops algorithmic CHY integration rules for evaluating tree-level amplitudes without solving the full set of scattering equations, by exploiting a Global Residue Theorem-based framework and parallels with the alpha' → 0 field-theory limit of string theory. It provides a concrete combinatorial rule that assigns propagator factors to compatible subsets of external legs, enabling direct evaluation of CHY integrals with simple poles, and extends these ideas to higher-order poles through Pfaffian identities. A complementary Pfaffian-centric approach yields reductions of double and triple poles, while a dedicated set of rules addresses phi^4-type CHY integrals with connected and disconnected perfect matchings. The work establishes a precise string-CHY correspondence and demonstrates that the resulting rules are algorithmic and broadly applicable to scalar, gauge, gravity, and phi^4 theories, with potential for further unification and extension in CHY formalisms.

Abstract

As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.