Problems in the derivations of the renormalization group equation for the low momentum nucleon interactions

TL;DR

This paper critically examines four derivations of the renormalization group equation for low-momentum nucleon interactions, highlighting issues and proposing modifications to ensure consistency, especially in the presence of bound states.

Contribution

It clarifies the validity of different derivations of the RGE for V_{low k} and proposes necessary modifications when bound states are involved.

Findings

Two derivations based on semi-group composition law are unjustified.

The RGE remains valid if bound state wavefunctions share low-momentum components.

Discussion of various aspects of the V_{low k} approach.

Abstract

We carefully examine all the four derivations of the renormalization group equation (RGE) for the so-called $\Vlk$ potential, given by Bogner, \textit{et. al.}[nucl-th/0111042]. Two derivations based on the ``semi-group composition law'' are shown to be unjustified, while the other two based on the completeness relation of the model space must be modified if there are bound states. It is however shown that the RGE is unchanged if the bound state wavefunctions in the reduced theory are required to have the same low-momentum components as those in the original theory. Several aspects of the $\Vlk$ approach are also discussed.