Peace in the valley: Concordant approaches to distorted instantons

There are two competing technologies for computing the distorted instantons relevant to anomalous high-energy scattering, the ‘‘R-term method’’ and the ‘‘valley method’’, both designed for the calculation of final-state corrections. We show that, when restricted to final-state corrections, the two methods are formally equivalent at all orders. The proof makes use of Cutkosky’s cutting rules, appropriately extended to apply to perturbation theory in instanton backgrounds. However, the distinction that has come to be drawn between initial-state and final-state corrections to the total anomalous cross section ${\mathrm{\sigma }}_{\mathrm{anom}}$ is ambiguous, as it depends on one’s choice of Faddeev-Popov constraints (zero modes or otherwise) necessary to render perturbation theory around the instanton well defined. We discuss the relationship between the constraint choices in the two methods.