Chiral Prediction for the π N Scattering Length a^- to Order {cal O}(M_π⁴)

We evaluate the S-wave pion--nucleon scattering length $a^-$ in the framework of heavy baryon chiral perturbation theory up--to--and--including terms of order $M_\pi^4$. We show that the order $M_\pi^4$ piece of the isovector amplitude at threshold, $T^-_{\rm thr}$, vanishes exactly. We predict for the isovector scattering length, $0.088 \, M_{\pi^+}^{-1} \le a^- \le 0.096 \, M_{\pi^+}^{-1}$.