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 $t=0\mathrm{}$ (thus showing pion conspiracy), but vanishes around $-1.5\mathrm{}{\mu }^2$, as required by high-energy data. The magnitude of the residue at $t=0\mathrm{}$ also agrees extremely well with that obtained from the high-energy forward photoproduction cross section.