Optical theorem and probability conservation at O(λ⁴) in the CQ model

Establish conservation of total probability by verifying the open-systems analogue of the optical theorem for the classical–quantum Yukawa model at order λ⁴, including loop contributions, extending the O(λ²) tree-level check performed in the paper.

Background

The authors show that, to order λ², the CQ scattering framework conserves total probability (the open-systems version of the optical theorem), with forward-scattering contributions summing appropriately. However, they do not extend this proof to order λ⁴, where loop integrals become necessary, and explicitly defer such an analysis.

A rigorous O(λ⁴) verification would strengthen the consistency of the CQ framework beyond leading order and clarify how non-unitary effects interplay with probability conservation in higher-order processes.