Phenomenological Hamiltonian for pions, nucleons, andΔisobars: Applications to the pion-deuteron system

A phenomenological model of the interactions between pions, nucleons, and $\Delta$ isobars is constructed. The mass operator is defined on a Hilbert space made up of $\mathrm{NN}$, $N\Delta$, and $\mathrm{NN}\pi$ states with the following interaction mechanism: (1) a $\Delta \rightleftarrows N\pi$ vertex in the $\pi N-P_{33}$ channel, (b) two-body $\pi N\to \pi N$ interactions in other $\pi N$ channels, and (c) two-body interactions for $\mathrm{NN}\to \mathrm{NN}$, $N\Delta \to N\Delta$, and $\mathrm{NN}\rightleftarrows N\Delta$ transitions. The model is Lorentz invariant and satisfies cluster separability. The interactions are parametrized by analytic separable forms with parameters determined by fitting the $\pi N$ scattering phase shifts for $l\le 1$ up to 300 MeV and $\mathrm{NN}$ scattering phase shifts for $l\le 4$ up to 800 MeV. In fitting parameters and in the applications, nonrelativistic approximations are used for the baryons in the $N\Delta$ and $\mathrm{NN}\pi$ channels. The model so determined gives a satisfactory description of pion absorption by deuterons and of elastic pion-deuteron scattering. Multiple rescattering of pions between $N$ and $\Delta$ as well as $\mathrm{NN}$ interactions in $\mathrm{NN}\pi$ intermediate states are found to be important in channels coupled to $N\Delta s$ waves. These effects enhance the cross sections for ${\pi }^{+}+d\to \mathrm{pp}$ by 40% in the resonance region. From the two-baryon Hamiltonian we construct a many-body Hamiltonian for nonrelativistic baryons.