On the Short Range Interaction between Pion and Nucleon

Asymptotic behaviors of the so-called force functions, dispersion integrals over left-hand cuts of partial-wave amplitudes, in pion-nucleon scattering are investigated. By virtue of the exponential decrease of fixed cos 8 amplitudes for s-?+oo with Ieos 81<1, the left-hand cut contributions to these amplitudes, from which we obtain the force function by doing the partial-wave projection, can be evaluated in terms of the right-hand cut integrals. The empirical formulae for high-energy scattering are used for evaluating these right-hand cut integrals. The two cases are considered, in which Krisch's formula and Orear's one a:re used for the large-angle scattering, respectively. For the case of Krisch's formula, it is found that the asymptotic behavior of nN force function for any partial wave is dominated by the contribution from a diffraction-scattering term and this contribution gives rise to an attractive short-range force, and that for the P11 state, this attractive force seems to be somewhat weaker than that required in the previous N/D bootstrap calculation. For the case of Orear's formula, no definite conclusion is obtained because of the possibility that large angle regions give rise to large contributions.