On the analysis of the pion--nucleon σ--term: The size of the remainder at the Cheng--Dashen point

We calculate the one--loop contributions of order $M_\pi^4$ to the difference $\Delta_R$ between the on--shell pion--nucleon scattering amplitude $\bar{D}^+(0,2M_\pi^2)$ at the Cheng--Dashen point $\nu =0$, $t=2M_\pi^2$ and the scalar form factor $\sigma (2M_\pi^2)$ in the framework of heavy baryon chiral perturbation theory. We proof that to this order $\Delta_R$ contains $no$ chiral logarithms and therefore it vanishes simply as $M_\pi^4$ in the chiral limit. Numerically, we find as an upper limit $\Delta_R \simeq 2\,$MeV.