Integration Rules for Loop Scattering Equations

TL;DR

This paper introduces new integration rules for one-loop scattering equations, simplifying amplitude calculations in scalar theories within the CHY framework by avoiding explicit solution sums.

Contribution

It develops novel integration rules for one-loop scattering equations, extending tree-level methods to facilitate amplitude evaluations in scalar theories.

Findings

Validated the rules in non-trivial scalar amplitude cases

Reduced computational complexity in one-loop amplitude calculations

Enhanced the CHY formalism for loop-level processes

Abstract

We formulate new integration rules for one-loop scattering equations analogous to those at tree-level, and test them in a number of non-trivial cases for amplitudes in scalar $\phi^3$-theory. This formalism greatly facilitates the evaluation of amplitudes in the CHY representation at one-loop order, without the need to explicitly sum over the solutions to the loop-level scattering equations.