Recursion for Differential Cross-Section from the Optical Theorem

Abstract: We present a novel framework for computing differential cross-sections in quantum field theory using the optical theorem and loop amplitudes, circumventing the traditional method of squaring scattering amplitudes. This approach addresses two major computational challenges in high-multiplicity processes: complexity from amplitude squaring and the extensive summations over color and helicity. Our method employs quantum off-shell recursion, a loop-level generalization of Berends--Giele recursion, combined with Veltman's largest time equation (LTE) through a doubling prescription of fields. By deriving Dyson--Schwinger equations within this doubled framework and constructing quantum perturbiner expansions, we develop recursive relations for generating LTEs. We validate our method by successfully reproducing the differential cross-section for tree-level $2 \to 2$ and $2 \to 4$ scalar scattering for $\phi<sup>{4}$ theory through one-loop and three-loop amplitude calculation respectively. This framework offers an efficient alternative to conventional methods and can be broadly applied to theories with color charges, such as QCD and the Standard Model.