Superconvergence Relations for Pion-Baryon Scattering

Superconvergence relations are discussed for the spin-flip amplitudes in pion-baryon scattering at $t=0\mathrm{}$ and at $u=0\mathrm{}$. These sum rules are evaluated in the resonance saturation approximation, and in this case an extension of the derivation is given. This allows the sum rules to be discussed even if the relevant leading Regge trajectory lies above zero at $t=0\mathrm{}$ for mesons or above -½ at $u=0\mathrm{}$ for baryons. We consider all 14 possible sum rules, as well as the moment sum rules, which require stronger assumptions. The spin-flip sum rules are well satisfied for $\pi N\to \pi N$, and in the other cases the $\rho$ and pion couplings to $\Sigma$, $\Lambda$, and $\Xi$ required to satisfy the relations are in agreement with other models. The moment sum rules are less convergent and so more difficult to test, but for $\pi N$ the moment sum rule for $B^{\mathrm{}(-)\mathrm{}}$ at $t=0\mathrm{}$ is not satisfied with resonance saturation, and this fact can be understood.