NNLO Calculation of Two-Nucleon Scattering in EFT for a Two Yukawa Toy Model

The effective field theory (EFT) for a toy model of nucleons interacting via a short range and a long range Yukawa potential is presented. The scattering amplitude in the 1S0 channel is calculated up to NNLO using KSW power counting scheme. A particular expansion of the amplitude about the pole at low (imaginary) momentum is used to derive matching conditions for the EFT couplings. In addition, after imposing constraints from a Renormalization Group analysis, there are no new free parameters in the amplitude at NNLO. Comparing the NNLO phase shifts to the full model we show that the EFT expansion is converging.