[Pi]-D scattering using the Foldy-Walecka optical model

Text includes handwritten formulasWe discuss the possibility of producing a bound state of the pi-D system assuming that the only important interaction in the tr-N system below = 3 MeV is the 3-3 resonance. We find that the combined potentials of the two nucleons in the deuteron system are not sufficient to induce binding with the pion. However, the 3-3 resonance does produce a pi-D resonance at a lower energy and with a narrower half-width than the pi-N resonance. We use the approach of Foldy and Walecka^ to generate the equations describing the pi-D interaction. This approach neglects both the three body interactions in the pi-D system and the momentum exchanged between the pion and the deuteron. This enables us to describe the pi-D interaction in terms of partial waves knowing only the form of the pi-N scattering amplitude and the potential describing the Tr-N interaction. An S-wave resonance is forced to occur in the pi-N system so that the effects of the parity and A values of the Tr-D partial waves on the overall pi-D S-wave interaction can be clearly seen. The resonance in the pi-D interaction caused by the S-wave pi-N resonance indicates that the nucleons in the deuteron are weakly coupled together. Assuming this to be true in the P-wave pi-D interaction, we make a second order Born approximation of the P-wave matrix equations produced from the FoldyWalecka approach to simplify the numerical computations. We use both the rather simple pi-N scattering amplitudes and the TT-N scattering amplitudes derived from the Foldy-Walecka approach to describe the S-wave and P-wave pi-N resonances. Both forms of the pi-N scattering amplitudes give similar results when inserted into the Foldy-Walecka equations describing the pi-D interaction. The method used by Foldy and Walecka seems to be useful in describing the pi-D interaction, especially since the equations are easily expanded in a multiple scattering series. However, we believe their method would be too cumbersome as we have applied it for systems of larger size or higher angular momenta. We neglect spin and isospin, assume infinite masses for the nucleons in the pi-D system, and discuss forward scattering only. We find that if spin, isospin, or other refinements are included in our description of the pi-D interaction, the time required for the numerical computation of the Foldy-Walecka equations increases dramatically