Chiral Prediction for the $\pi N$ Scattering Length $a^{-}$ to Order $\mathcal{O}(M_{\pi }^4)$

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, $T_{\mathrm{t}\mathrm{h}\mathrm{r}}^{-}$, vanishes exactly. We predict for the isovector scattering length, $0.088 , M_{\pi^+}^{-1} \le a^- \le 0.096 , M_{\pi^+}^{-1}$.