Table Of ContentFacult´e des Sciences
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The Nucleon-Nucleon
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Interaction in a Chiral
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Constituent Quark Model
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Daniel Bartz
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Th`ese pr´esent´ee en vue de l’obtention du
Grade de Docteur en Sciences Physiques
Mars 2002
To Lorena
Acknowledgments
First and foremost, I wish to express my deep gratitude to my supervisor Fl. Stancu for
giving me inspiration and guidance in the research work contained in this thesis. I am very
grateful to her for prompt help whenever I needed it. Her professional and human support
was so fundamental that it is really hard to estimate what a chance I had meeting her. I
would also like to thank her especially for careful readings of all the manuscripts.
Many thanks to J.-P. Jeukenne and J. Cugnon, who offered me the possibility to start,
and to achieve this thesis.
I should not forget the University of Li`ege and the Institut Interuniversitaire des Sciences
Nucl´eaires (IISN) for funding this work and conference appearances. Thus, providing me
with the opportunity to interact with a wide scientific community.
I am also indebted to all the members of the group of fundamental theoretical physics for
the scientific atmosphere. I owe special acknowledgments to that crazy friend and colleague
Michel Mathot for all of its “good” advices, special thanks also to the Others...
Chapter 1
Introduction
The strong interaction between two nucleons is the basic ingredient of nuclear physics.
Since about 1980 there have been many efforts to derive the nuclear force from the under-
lying theory of strong interactions, the quantum chromodynamics (QCD). However the basic
equations of QCD are known for their difficulties to obtain a comprehensive solution at all
energies. Indeed, the QCD is often split into two parts : the high-energy and the low-energy
regimes. In the high-energy region, where the coupling constant of QCD becomes very small,
leading to the asymptotic freedom, a perturbative treatment is applicable, allowing for a
satisfactory description of many experimental data, such as the deep inelastic lepton-hadron
scattering. Ontheotherhand,duetoitsnon-perturbativecharacterinthelowenergyregime,
as forexamplethenucleonor thenucleon-nucleon interaction, QCDcannotbesolved exactly.
Moreover, in this energy region new characteristics of QCD become important such as the
confinement of quarks and gluons and the spontaneous breaking of chiral symmetry. This
renders difficult the treatment of QCD using its own degrees of freedom. A first hope could
come from lattice calculations. However, certainly more computer power and more deve-
lopment of sophisticated algorithms are still necessary. Consequently, several so-called QCD
inspired models have been used to study the nucleon properties and to derive the nucleon-
nucleon interaction, where the original degrees of freedom of QCD are replaced by effective
degrees of freedom.
Among all the available models, in this thesis we are concerned with a particular cons-
tituent quark model. Our choice goes to a new potential model : the Goldstone boson ex-
change (GBE) or the chiral constituent quark model, where the interaction between quarks
is due to a pseudoscalar meson exchange (Goldstone boson). This model incorporates the
chiral symmetry breaking effect of QCD, the importance of which becomes more and more
clear to nuclear physicists.
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2 Introduction
The GBE model reproduces well the baryon properties and it is interesting to find out
if the nucleon-nucleon problem, as a system of six quarks, can be described as successfully.
A strong motivation of using this model is that it contains all necessary ingredients for des-
cribing the nucleon-nucleon interaction. Its hyperfine interaction, which is flavor and spin
dependent,besides ashort-rangepart, essential for baryon spectroscopy, also contains a long-
range part of Yukawa type, which is necessary to describe the long-range behaviour of the
nucleon-nucleon interaction. This allows a fully consistent description of the nucleon-nucleon
interaction and avoids the extra addition of meson exchange at the nucleon level, as imposed
in the one-gluon exchange models, or of meson exchange between quarks, overlapped on the
gluon exchange, like in the so-called hybrid models.
Within the GBE model, besides one-meson exchanges, one can naturally include two or
more pseudoscalar meson exchanges. The incorporation of two-meson exchanges which gives
rise to a spin independent central component, simulated by a σ-meson exchange, could in
principleexplain the middle-range attraction of the nuclear forces. Apartfrom that, together
withtheone-pionexchange, twocorrelatedpionsbringacontributiontothetensorpartofthe
quark-quark interaction, required to explain the binding of the deuteron and the behaviour
of the 3S phase shift.
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The best known way to study the nucleon-nucleon (NN) interaction as the interaction
between two composite particles is via the resonating group method (RGM), or the closely
related, the generator coordinate method (CGM). Here we chose the RGM to treat the
nucleon-nucleon problem as a system of six interacting quarks. An important feature of this
method is that the exchange of quarks between nucleons can be treated exactly. This leads
to important non-local effects in the NN potential and allows the study of both bound and
scattering states of the NN system.
This thesis is divided in six chapters and six appendices. The work presented here begins
with a short historical overview of the nucleon-nucleon interaction studies. The review pre-
sented in Chapter 2 starts from the approach beginningwith Yukawas’s theory in the thirties
and ends with the most modern considerations where quark structure and chiral symmetry
are taken into account. In particular we shortly introduce the three best known potential
models : the one-gluon exchange model (OGE), the hybrid model and the GBE model. To
our knowledge, our work is the first study of the NN interaction within the GBE model and
based on RGM.
Chapter 3 is entirely devoted to the detailed study of the GBE model. The origin and its
foundationsarediscussedindetails. Resultsandinparticularits successinreproducingspec-
traof light baryonsis presented. We also underlinehow theGBE modelprovides apromising
basis for further investigations, such as the nucleon-nucleon interaction. We also note that
the long-range part of the quark-quark interaction which does not play an important role
in the baryon spectra, leaves room for more adjustment of the GBE parametrization when
studying the nucleon-nucleon interaction.
The application of the GBE interaction to the study of six-quark systems starts in Chap-
ter4. OurfirstgoalistofindoutiftheGBEinteraction canexplaintheshort-rangerepulsion
of a two-nucleon system. That chapter constitutes in fact a first step of our investigation
towards finding out whether or not the GBE model can give rise to a short-range repulsion
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in the nucleon-nucleon interaction. To do so we use the Born-Oppenheimer approximation
and consider two different ways of constructing six-quark states : the cluster model basis
and the molecular orbital basis. We show why the molecular basis is more appropriate to
the derivation of the nucleon-nucleon interaction. Also we show how the details of the GBE
model parametrization does not affect the results qualitatively. Finally, we propose the in-
troduction of a scalar-meson exchange interaction between quarks which can simulate the
missing middle-range attraction.
However the results obtained in Chapter 4 represent only a preliminary study, inasmuch
as we derive only a static local potential. A dynamical treatment of the nucleon-nucleon
interaction is developed in Chapter 5 where we apply the RGM to study the NN interaction.
This method allows us to calculate bound and scattering states of the NN system in taking
into account the compositeness of the nucleon. One of the concerns of Chapter 5 is to find
outtheroleplayed byGBE interaction andthequarkinterchange duetotheantisymmetriza-
tion, on theshort-rangeNNinteraction. Inthatchapter wefirstpresenttheresonating group
method in the context of the NN problem. Next we apply it to the study of the relative s-
wave scattering of two nucleons. The influence of the coupled channels NN ∆∆ CC in
− −
RGM calculations is presented. Then, searching for an agreement of the 1S phase shift with
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experimental data, we propose to explicitly include a σ-meson exchange which can provide
an important amount of middle-range attraction. In order to reproduce the experimental
data for the 3S phase shift, we have to introduce the tensor part of hyperfine interaction
1
potential which can provide the binding of the two nucleons.
Our concluding remarks are gathered together in the last chapter. Many of the analytical
details related to Chapter 5 are given in the Appendices A-F.
