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

Evgeny Epelbaum,Jambul Gegelia,Ulf-G Meißner

doi:10.1016/j.nuclphysb.2017.10.008

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.

Highlights

More than two decades after the publication of the ground-breaking papers by Weinberg on the chiral effective field theory (EFT) approach to few-nucleon systems [1,2], the problem of renormalization and power counting within this formalism still remains a hotly debated issue
We find that the scaling of coupling constants uncovered by the Wilsonian renormalization group (RG) analysis corresponds to the choice of the renormalization scheme when all subtraction points are chosen of the order of the soft scale of the problem
By choosing the renormalization point corresponding to the coupling constant of the momentum- and energy-independent contact interaction of the order of the hard scale of the problem while taking all other renormalization points of the order of the soft scale, one recovers Weinberg’s power counting [1,2] with renormalized coupling constants being of natural size both for natural as well as unnaturally large scattering lengths

Summary

Introduction

More than two decades after the publication of the ground-breaking papers by Weinberg on the chiral effective field theory (EFT) approach to few-nucleon systems [1,2], the problem of renormalization and power counting within this formalism still remains a hotly debated issue. The above-mentioned renormalization problem can be avoided by treating the pion exchange contributions to the potential perturbatively as proposed by Kaplan, Savage and Wise [7] Their approach makes use of dimensional regularization supplemented by the power divergence subtraction scheme. In the current work we consider exactly renormalizable models of the nucleon–nucleon (NN) potential which lead to well-defined scattering amplitudes This allows us to compare subtractive renormalization to the Wilsonian RG approach. By exploring the full space of subtractive renormalization, we explicitly demonstrate that both the KSW and Weinberg approaches are consistent schemes and allow for a systematic expansion of the scattering amplitude.

On the power counting for the scattering amplitude and the potential

An exactly solvable model of contact interactions

Wilsonian renormalization group analysis

A toy model with a long-range interaction

Findings

Summary and conclusions