CoupledπNN−NNsystems in a Hamiltonian approach and in a relativistic off-mass-shell formalism

We discuss the scattering theory pertaining to the coupled $\pi NN-NN$ system and approach the problem in two independent ways. The first one starts from a Hamiltonian formalism and coupled Schrödinger equations, whereas the second one employs an off-mass-shell relativistic theory of classifying perturbation diagrams. Both ways lead to connected equations among transition operators in which $\pi \mathrm{NN}$ vertices, as well as nucleon propagators, are completely dressed and renormalized. Furthermore, the physical amplitudes obey two- and three-body unitarity relations. The resultant equations form a sound theoretical basis for subsequent numerical calculations leading to the evaluation of physical observables in the reactions $\pi +d\to \pi +d$, $\pi +d\leftrightarrow N+N$, and $N+N+\to N+N$.