Pion Conspiracy and Pion Residue from the Finite-Energy Photoproduction Sum Rule

The pion residue is obtained as an integral over the relevant photoproduction amplitude following the usual prescription of the finite-energy sum rule. The latter, studied as a function of t, reveals the following interesting features of the pion residue. It is nonzero at $t=0$ (thus showing pion conspiracy), but vanishes around -1.5μ², as required by high-energy data. The magnitude of the residue at $t=0$ also agrees extremely well with that obtained from the high-energy forward photoproduction cross section.