Next-to-next-to-leading-order calculation of two-nucleon scattering in an effective field theory

We use effective field theory (EFT) to calculate the nucleon-nucleon scattering amplitude up to next-to-next-to-leading order (NNLO), where we include instantaneous pion exchange but ignore retardation effects. The scattering amplitude in the ¹S₀ channel is calculated using the Kaplan-Savage-Wise power counting scheme. An expansion of the amplitude about the pole at low (imaginary) momentum is used to derive matching conditions for the EFT couplings. After imposing constraints from a renormalization group flow analysis, there are no new free parameters in the amplitude at NNLO. We apply the calculation to a toy model in which there is only instantaneous pion exchange by construction, namely, the two-Yukawa model. We find that, through next-to-next-to-leading order, each successive term in the low energy expansion does systematically improve the approximation to the full amplitude.