Production and Scattering in Simple Models

Some properties of the $V\theta$ scattering amplitude in the Lee model are discussed with reference to the influence of production ($N\theta \theta$) on scattering. As long as the coupling is not too weak, production has important effects on scattering even at low energies. For a wide class of well-behaved cutoff functions there can be no resonance in $V\theta$ scattering unless production is taken into account. Static model calculations for meson-baryon scattering neglecting production can therefore be quite misleading. There is in general no simple relation between the locations of resonances in $V\theta$ scattering and the production threshold, even in the strong coupling limit $Z\approx 0$. A modified version of the Lee model with odd-$\mathrm{NV}\theta$ parity is discussed. Two different sets of integral equations for the $V\theta \to V\theta$ and $V\theta \to N\theta \theta$ amplitudes are derived, in which the influence of spin enters in a nontrivial way.