Kinematical Considerations on Three-Pion-Exchange Contributions in Nucleon-Nucleon Scattering

Recent theoretical investigations of nucleonnucleon scattering have clarified the detailed features of the nuclear forces in the intermediate region.1l These analyses have shown that the nuclear forces in this region (this roughly corresponds to the elastic region except for the S waves) can be explained by contributions from the one-boson exchange of various types, the one-boson-exchange (OBE) model, which at least include the pion, the p meson, the m meson and an I= 0 scalar meson. 2) On the other hand, from the standpoint of the current field theory, the two-pionexchange effects are also considered to be important in this region and extensive studies3l-sl have shown that, as f~r as we are concerned with the nucleon-nucleon partial wave amplitudes, the two-pion-exchange contributions are very similar in their behaviour and magnitude to the OBE amplitudes which fit the experiment. An immediate question arising from these results is how much the three-pion-exchange contributions in this region are and what significance can be given to an analysis of the two-pion exchange without considering the three-pion exchange. In this respect Amati, Leader and Vitale6l have proposed to cut off the two-pion-exchange contributions from the large effective mass region of the exchanged two pion (t:Zl216m!) for which the three-pion exchange could be significant, when we make the analysis without taking the contributions from the three-pion exchange. Static three-pion-exchange potentials have been calculated by Machida and Senba7l and these are shown to be comparable to or even larger than the two-pion-exchange ones at r<0.7 pion compton wave length. It is, however, generally quite difficult to evaluate ' the three-pion-exchange effects with the accuracy which has been attained for the twopion exchange. In this paper we give some preliminary discussion about the three-pion effects, namely how these could compete with the two-pion effects, along the line of investigations made in reference 3). First we summarize the analyses made about the twopion exchange. We can classify the contents of the two-pion-exchange contributions kinematically by the states of exchanged two pions as the contributions from the I= 0, L =even states and those from the I= 1, L =odd states. *l The contributions from each two-pion state have been calculated and it has been shown