Systematic power counting in cutoff effective field theories for nucleon-nucleon interactions and the equivalence with PDS

An analytic expression for the 1S0 phase shifts in nucleon-nucleon scattering is derived in the context of the Schrödinger equation in configuration space with a short distance cutoff and with a consistent power counting scheme including pionic effects. The scheme treats the pion mass and the inverse scattering length over the intrinsic short distance scale as small parameters. Working at next-to-leading order in this scheme, we show that the expression obtained is identical to one obtained using the recently introduced PDS approach which is based on dimensional regularization with a novel subtraction scheme. This strongly supports the conjecture that the schemes are equivalent provided one works to the same order in the power counting.