# Wilsonian renormalization group versus subtractive renormalization in effective field theories for nucleon--nucleon scattering

## TL;DR

This paper compares subtractive and Wilsonian renormalization group methods in effective field theories for nucleon-nucleon scattering, highlighting limitations of the Wilsonian approach and confirming the consistency of Weinberg's power counting.

## Contribution

It demonstrates that Weinberg's power counting scheme is consistent and systematically expandable within pionless EFT, and shows the limitations of Wilsonian RG in certain parameter regimes.

## Key findings

- Wilsonian RG with a single cutoff does not cover all subtractive scheme parameters.
- Weinberg's power counting is valid for systematic expansion of scattering amplitudes.
- Limitations of Wilsonian RG in cases of large scattering length.

## Abstract

We compare the subtractive renormalization and the Wilsonian renormalization group approaches in the context of an effective field theory for the two-nucleon system. Based on an exactly solvable model of contact interactions, we observe that the standard Wilsonian renormalization group approach with a single cutoff parameter does not cover the whole space spanned by the renormalization scale parameters of the subtractive formalism. In particular, renormalization schemes corresponding to Weinberg's power counting in the case of an unnaturally large scattering length are beyond the region covered by the Wilsonian renormalization group approach. In the framework of pionless effective field theory, also extended by the inclusion of a long-range interaction of separable type, we demonstrate that Weinberg's power counting scheme is consistent in the sense that it leads to a systematic order-by-order expansion of the scattering amplitude.