Validity of on-shell distortion factors for NN →πd

We test the validity of calculating the process NN →πd in lowest order, with initial- and final-state interactions being included via square roots of s matrices for the respective elastic processes (so-called Sopkovich approximation). This is done by constructing a coupled-channels model with effective two-body potentials between the channels NN, πd, and NΔ. These potentials are taken to be separable, and are fitted to partial-wave amplitudes coming from a three-body calculation. We obtain reasonable fits in those partial waves where coupling to the NΔ channel is strongest; in particular, the dominant J=2⁺ channel is well described. Within such a two-body potential model, it is known that the Sopkovich approximation should be valid for small, short-range transition potentials in the limit of high energies. Using this coupled-channels model, we find that the Sopkovich approximation in the J=2⁺ channel works well only above 200-MeV total center-of-mass kinetic energy. Comparison with an approximation which assumes only a small transition potential leads us to observe that (i) the 2⁺ NN →πd transition potential is indeed small everywhere except at the resonance peak (160 MeV), and (ii) away from this peak, the Sopkovich approximation breaks down below 200 MeV mainly due to the neglect of off-shell effects. For the smaller J=2^-, 3^-, and 4⁺ partial waves, we find the Sopkovich approximation to work at least as well as for the 2⁺. No conclusions are made about the J=0⁺ and 1^- partial waves; these obtain large contributions from s-state πN rescattering which is not easily included in our model.