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

Abstract

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.

Comments: 7 pp, LaTeX, uses epsf, 1 figure in separate PS file