Short-range interactions in an effective field theory approach for nucleon-nucleon scattering

We investigate in detail the effect of making the range of the “contact” interaction used in effective field theory (EFT) calculations of $\mathrm{NN}$ scattering finite. This is done in both an effective field theory with explicit pions, and one where the pions have been integrated out. In both cases we calculate $\mathrm{NN}$ scattering in the ${}^1{S}_0$ channel using potentials which are second order in the EFT expansion. The contact interactions present in the EFT Lagrangian are made finite by use of a square-well regulator. We find that there is an optimal radius for this regulator, at which second-order corrections to the EFT are identically zero; for radii near optimal these second-order corrections are small. The cutoff EFT’s which result from this procedure appear to be valid for momenta up to about $100–150\mathrm{MeV}/c.\mathrm{}$ We also find that the radius of the square well cannot be reduced to zero if the theory is to reproduce both the experimental scattering length and effective range. Indeed, we show that, if the $\mathrm{NN}$ potential is the sum of a one-pion-exchange piece and a short-range interaction, then the short-range piece must extend out beyond 1.05 fm, regardless of its particular form.