How short is too short? Constraining zero-range interactions in nucleon-nucleon scattering

Abstract

We discuss a number of constraints on the effects of zero-range potentials in quantum mechanics. We show that for such a potential $p \cot(\delta)$, where $p$ is the momentum of the nucleon in the center of mass frame and $\delta$ is the S-wave phase shift, must be a monotonically decreasing function of energy. This implies that the effective range of the potential is non-positive. We also examine scattering from the sum of two potentials, one of which is a short-range interaction. We find that if the short-range interaction is of zero-range then it must be attractive, and the logarithmic derivative of the radial wave function at the origin must be a monotonically decreasing function of energy. If the short-range interaction is not of zero range then a constraint which gives the minimum possible range for it to fit the phase shifts exists. The implications of these results for effective field theory treatments of nucleon-nucleon interactions are discussed.