Dispersion Theoretic Calculation of Two-Pion-Exchange Contribution in Nucleon-Nucleon Scattering. I

Abstract

The two-pion-exchange contribution without pion-pion correlation in nucleon-nucleon scattering is investigated in detail using the partial-wave dispersion relation. With the aim to clarify the contribution we devide the amplitude into the contributions corresponding to the fourth order perturbation (N-part) and the additional effect from the pion-nucleon resonance in the (33) state (33 part), and each part is further divided with respect to the angular momentum of the exchanged two pions, L, so that the characteristics of quantities appearing in the intermediate steps of calculation such as the effective mass distribution functions may be presented in a somewhat concrete way. Through the investigation of the effective mass distribution functions, we found (i) the 33 part contributions mostly to the L=0 and 1 parts, which are the lowest L contributions to h^(a) and h^(b) respectively, where we describe the whole amplitude as h^(a)+τ⁽¹⁾τ⁽²⁾h^(b) for the isospin freedom; (ii) the relative order of magnitude of 33 part to N part shows that the (33) effect is considerably large for L=0 part at the large effective mass region. This means that the effects of (33) πN resonance in the two-pion-exchange contribution for the nucleon-nucleon scattering have a very similar behaviour to the one-scalar-boson exchange contributions, where relatively large effective masses are assumed. Consequently, such features of the 33 part is likely to specify the characteristics of the two-pion-exchange contributions. In fact, the (33) effect is generally very large for lower partial wave amplitudes than the F wave, in the form of “proper” 2π contribution. Comparison with other calculations and discussions of related problem are also given.