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

Abstract

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