# Investigations of the renormalization group approach to the nucleon-nucleon interaction

Abstract (Summary)

This thesis work has investigated the Renormalization Group theory for the nucleonnucleon
interaction. Conventional nuclear many-body calculations have the following
sources of non-perturbative physics: 1. a strongly repulsive short-range interaction,
2. a tensor force, e.g. from pion exchange, which is highly singular at short-distances,
3. the presence of low-energy bound states or nearly bound states (in the S waves).
The RG approach exploits the insensitivity of low-energy processes to the details of
the high-energy physics. Using any of the high-precision potentials as input, the highmomentum
intermediate states in the Lippmann-Schwinger equation for the T matrix
in a particular partial-wave are cut-off at ?. The details of the physics beyond this
limit of resolution are integrated out and included in the potential by requiring that
the half-off shell T matrix elements be independent of the cut-off ?. This requirement
leads to a low-momentum potential “Vlow k”, which is energy independent.
The choice of the regulator which cuts off the high momentum intermediate states
is investigated. Sharp cut-offs, though straight forward, lead to convergence issues in
few-body calculations that are eliminated using smooth regulators. The construction
of low-momentum potentials using a smooth regulator is explored in detail. In the
course of this study, a three-step process to calculate Vlow k requiring the cut-off independence
of the fully-off shell T matrix elements has been established and this yields
better numerical stability than the energy-independent RG.
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The complex eigenvalues (Weinberg eigenvalues) of the operator G0(z)V , which
appears in the Lippmann-Schwinger equation, are a useful tool for investigating the
convergence of the Born series. Weinberg eigenvalues for Vlow k potentials, including
chiral effective theory potentials, have been investigated as a function of cut-off. The
studies reveal the density and/or scale dependence of the sources of non-perturbative
physics. The in-medium eigenvalues near the Fermi surface give a good estimate
of the pairing gaps. Using two-particle Nambu-Gorkov propagators, the eigenvalue
equation at E = 2?F is the gap equation and the eigenvectors corresponding to the
largest eigenvalue gives the first approximation to the gap function ?(k), which can
be further iterated using the BCS gap equation to give self-consistent gaps.
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This work is dedicated to my parents
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Bibliographical Information:

Advisor:

School:The Ohio State University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:nucleon interactions renormalization group nuclear physics

ISBN:

Date of Publication: