Sum Rules for Almost Backward Pion-Nucleon Elastic Scattering

Sum rules are derived, assuming Regge asymptotic behavior, for the $\pi N$ elastic scattering amplitudes at $u=0\mathrm{}$. The four sum rules obtained are found to be consistent with each other and with the forward-scattering sum rule of Sakita and Wali if $\rho$ and $f^0\mathrm{}\left(1250\right)\mathrm{}$ are included in the $t$ channel. Without the $f^0$, the sum rules are inconsistent. Predictions are made for the nucleon-$\rho$-meson coupling constants.