General aspects of the nucleon-nucleon interaction and nuclear matter properties [Elektronische Ressource] / vorgelegt von Oliver Plohl

General aspects ofthe nucleon-nucleon interactionand nuclear matter propertiesDissertationZur Erlangung des Grades eines Doktorsder Naturwissenschaftender Fakult¨at fur¨ Mathematik und Physikder Eberhard-Karls-Universit¨at zu Tubingen¨vorgelegt vonOliver Plohlaus Ostfildern2008Tag der mundlic¨ hen Prufung:¨ 25. Juli 2008Dekan: Prof. Dr. Nils Schopohl1. Berichterstatter: Prof. Dr. Christian Fuchs2. Berich Prof. Dr. Dr. h.c. mult. Amand F¨aßlerZusammenfassungDas Thema dieser Doktorarbeit ist, zunac¨ hst grundlegende modelunabh¨angigeEigenschaftenderNukleon-Nukleon(NN)WechselwirkungimVakuumhinsicht-lichihrerrelativistischenStrukturunddieKonsequenzendarausfur¨ Eigenschaf-ten von Kernmaterie zu untersuchen.Hierfur¨ werden relativistische und nicht-relativistische Meson-Austausch Po-tentiale,ph¨anomenologischePotentialeundaufeffektiverFeldtheorie(EFT)be-ruhende Potentiale auf eine relativistische Operator Basis der Clifford-Algebraabgebildet. Der Vergleich der unterschiedlichen Potentiale auf der Ebene von¨kovarianten Amplituden zeigt eine bemerkenswerte Ubereinstimmung.Weiterhin wird die relativistische Selbstenergie in der Hartree-Fock (HF)Naherungbestimmt.DasAuftreteneinesskalarenundvektoriellenFeldesinder¨Gr¨oßenordnung von mehreren hundert MeV ist eine universelle Eigenschaft vonrelativistischen Beschreibungen von Kernmaterie. Im Rahmen von QCD Sum-menRegelnsinddieseFelderengmitderDichteabhangigk¨ eitchiralerKondensa-te verknupft.
Publié le : mardi 1 janvier 2008
Lecture(s) : 28
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Source : TOBIAS-LIB.UB.UNI-TUEBINGEN.DE/VOLLTEXTE/2008/3552/PDF/THESIS_PLOHL.PDF
Nombre de pages : 169
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General aspects of
the nucleon-nucleon interaction
and nuclear matter properties
Dissertation
Zur Erlangung des Grades eines Doktors
der Naturwissenschaften
der Fakult¨at fur¨ Mathematik und Physik
der Eberhard-Karls-Universit¨at zu Tubingen¨
vorgelegt von
Oliver Plohl
aus Ostfildern
2008Tag der mundlic¨ hen Prufung:¨ 25. Juli 2008
Dekan: Prof. Dr. Nils Schopohl
1. Berichterstatter: Prof. Dr. Christian Fuchs
2. Berich Prof. Dr. Dr. h.c. mult. Amand F¨aßlerZusammenfassung
Das Thema dieser Doktorarbeit ist, zunac¨ hst grundlegende modelunabh¨angige
EigenschaftenderNukleon-Nukleon(NN)WechselwirkungimVakuumhinsicht-
lichihrerrelativistischenStrukturunddieKonsequenzendarausfur¨ Eigenschaf-
ten von Kernmaterie zu untersuchen.
Hierfur¨ werden relativistische und nicht-relativistische Meson-Austausch Po-
tentiale,ph¨anomenologischePotentialeundaufeffektiverFeldtheorie(EFT)be-
ruhende Potentiale auf eine relativistische Operator Basis der Clifford-Algebra
abgebildet. Der Vergleich der unterschiedlichen Potentiale auf der Ebene von
¨kovarianten Amplituden zeigt eine bemerkenswerte Ubereinstimmung.
Weiterhin wird die relativistische Selbstenergie in der Hartree-Fock (HF)
Naherungbestimmt.DasAuftreteneinesskalarenundvektoriellenFeldesinder¨
Gr¨oßenordnung von mehreren hundert MeV ist eine universelle Eigenschaft von
relativistischen Beschreibungen von Kernmaterie. Im Rahmen von QCD Sum-
menRegelnsinddieseFelderengmitderDichteabhangigk¨ eitchiralerKondensa-
te verknupft.¨ Es zeigt sich, dass unabhangig¨ von der Wahl der NN Wechselwir-
kung große skalare und vektorielle Felder auftreten, sobald die Symmetrien der
Lorentz Gruppe wiederhergestellt sind. Im Rahmen der chiralen EFT (chEFT)
wird gezeigt, dass kurzreichweitige Kontakt-Terme in nachst zu fuhrenderOrd-¨ ¨
nung, die mit der Spin-Bahn Wechselwirkung verknupft¨ sind, diese Felder er-
zeugen.
Um Auswirkungen von NN Korrelationen abzuschatzen,¨ wird die Zustands-
gleichung mit dem chiralen EFT Potential fur Kern- bzw. Neutronenmaterie¨
in der Bruckner-HF (BHF) Naherung¨ bestimmt. Wahrend¨ erwartungsgem¨aß ei-
¨ne deutliche Uberbindung eintritt (in nac¨ hst zu fuhrender¨ Ordnung wird S¨atti-
gungsverhaltenbeobachtet),zeigtdieSymmetrieenergieimVergleichmitphano-¨
menologischen Potentialen (in der gleichen N¨aherung) bzw. anderen Zug¨angen
ein realistisches Verhalten. Bei der Untersuchung der Pionmassenabhangigkeit¨
im Rahmen der chEFT in n¨achst zu fuhrender¨ Ordnung zeigt sich, dass die
GroßenordnungderskalarenundvektoriellenFelderimchiralenLimesbestehen¨
bleibt und nukleare Materie gebunden ist. Im Gegensatz zum Fall einer gr¨oße-
renPionmassealsdiephysikalischeverringernsichimchiralenLimessowohldie
Bindungsenergie als auch die S¨attigungsdichte.
Der vorliegende Formalismus erlaubt nun im Rahmen der chEFT einen kon-
sistenten Vergleich der In-Medium Nukleonmasse und der Dichteabhangigkeit¨
des skalaren Kondensates, welches unter Anwendung des Hellmann-Feynman
Theorems (in HF und BHF Naherung) bestimmt wird. Es zeigt sich, dass¨
die In-Medium Nukleonmasse und das skalare Kondensat entkoppeln. Im Ge-
gensatz zu QCD Summen Regeln bestimmen kurzreichweitige Kontakt-Terme
die In-Medium Nukleonmasse, wohingegen virtuelle Pionen niedriger Impulse
hauptsachlich zur Reduzierung des chiralen In-Medium Kondensates beitragen.¨Abstract
The subject of the present thesis is at first the investigation of model indepen-
dent properties of the nucleon-nucleon (NN) interaction in the vacuum concern-
ing the relativistic structure and the implications for nuclear matter properties.
Relativisticandnon-relativisticmeson-exchangepotentials,phenomenological
potentialsswellaspotentialsbasedoneffectivefieldtheory(EFT)aretherefore
mapped on a relativistic operator basis given by the Clifford Algebra. This
allows to compare the various approaches at the level of covariant amplitudes
where a remarkable agreement is found.
Furthermore, the relativistic self-energy is determined in the Hartree-Fock
(HF)approximation. Theappearanceofascalarandvectorfieldofseveralhun-
dred MeV magnitude is a general feature of relativistic descriptions of nuclear
matter. Within QCD sum rules these fields arise due to the density dependence
of chiral condensates. We find that independent of the applied NN interaction
large scalar and vector fields are generated when the symmetries of the Lorentz
group are restored. In the framework of chiral EFT (chEFT) it is shown, that
these fields are generated by short-range next-to-leading order (NLO) contact
terms, which are connected to the spin-orbit interaction.
To estimate the effect arising from NN correlations the equation of state of
nuclear and neutron matter is calculated in the Brueckner-HF (BHF) approx-
imation applying chEFT. Although, as expected, a clear over-binding is found
(at NLO a saturating behavior is observed), the symmetry energy shows realis-
tic properties when compared to phenomenological potentials (within the same
approximation) and other approaches. The investigation of the pion mass de-
pendence within chEFT at NLO shows that the magnitude of the scalar and
vector fields persists in the chiral limit – nuclear matter is still bound. In con-
trast to the case of a pion mass larger than the physical one the binding energy
and saturation density are decreased in the chiral limit.
The present formalism allows within chEFT to perform a consistent compar-
ison of the in-medium nucleon mass and the density dependence of the scalar
condensate derived from the Hellmann-Feynman theorem (in HF and BHF ap-
proximation). A decoupling of the in-medium nucleon mass and the scalar con-
densateisobserved. ItturnsoutthatincontrasttoQCDsumrulestheeffective
nucleonmassinmatterismainlydeterminedbyshort-rangecontacttermswhile
virtual low-momentum pions provide the essential contributions responsible for
the reduction of the in-medium scalar condensate.Contents
1. Introduction 9
2. Mean fields in nuclear matter 23
2.1. Density functional approach to QHD . . . . . . . . . . . . . . . 24
2.1.1. The σω-model . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.2. Lagrange density and field equations . . . . . . . . . . . 27
2.1.3. Mean field theory . . . . . . . . . . . . . . . . . . . . . . 29
2.2. The Dirac-Brueckner theory . . . . . . . . . . . . . . . . . . . . 33
2.2.1. T-matrix approximation . . . . . . . . . . . . . . . . . . 35
2.2.2. Self-energy in nuclear matter. . . . . . . . . . . . . . . . 37
2.3. In-medium QCD sum rules . . . . . . . . . . . . . . . . . . . . . 40
3. The NN interaction 47
3.1. Boson exchange potentials . . . . . . . . . . . . . . . . . . . . . 48
3.2. Non-relativistic potentials . . . . . . . . . . . . . . . . . . . . . 51
3.2.1. Non-relativistic reduction . . . . . . . . . . . . . . . . . 51
3.2.2. Meson-theoretical potentials . . . . . . . . . . . . . . . . 52
3.2.3. Phenomenological potentials . . . . . . . . . . . . . . . . 53
3.3. EFT interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1. Chiral EFT potentials . . . . . . . . . . . . . . . . . . . 54
3.3.2. Quark mass dependence of the chiral EFT interaction . . 58
3.3.3. Renormalization Group approach to the NN in . 60
4. Dirac structure of the NN interaction 63
4.1. Covariant operators in Dirac space . . . . . . . . . . . . . . . . 64
4.2. Projection onto covariant operators . . . . . . . . . . . . . . . . 66
4.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.1. Effective low momentum potentials . . . . . . . . . . . . 72
5. Nuclear matter properties 75
5.1. Self-energy in Hartree-Fock approximation . . . . . . . . . . . . 78
5.1.1. Results for various NN interactions . . . . . . . . . . . . 79
5.1.2. Reliability of tree level calculations . . . . . . . . . . . . 80
5.1.3. Comparison with non-relativistic single particle potential 81
7Contents
5.2. Self-energy from chiral EFT . . . . . . . . . . . . . . . . . . . . 85
5.3. Self-consistency at Hartree-Fock . . . . . . . . . . . . . . . . . . 90
5.4. EOS in chiral EFT . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4.1. Symmetry energy from chiral EFT . . . . . . . . . . . . 106
5.5. Self-energy and EOS in the chiral limit . . . . . . . . . . . . . . 108
6. Nucleon mass in nuclear matter 117
6.1. HF self-energy vs. QCD in-medium sum rules . . . . . . . . . . 120
6.2. Chiral condensate in nuclear matter . . . . . . . . . . . . . . . . 122
6.2.1. Results: HF and BHF approximation . . . . . . . . . . . 125
⁄6.3. The effective nucleon mass M . . . . . . . . . . . . . . . . . . . 132
7. Summary and conclusions 133
A. Notation and conventions 139
B. Momentum space OBEP 141
C. Scalar quark condensate in matter 145
Bibliography 147
81. Introduction
The field of nuclear physics was born with the discovery of the neutron by
Chadwick in 1932 [1]. Since then the nuclear force, i.e., the interaction between
twonucleons,hadbeentheheartofnuclearphysicsandhasbeeninvestigatedall
overtheworldforthepast75years. Thereasonforthatoutstandingimportance
of the nuclear force is that traditionally the main intention in nuclear physics
is to determine the properties of nuclei and nuclear matter in terms of a bare
nucleon-nucleon interaction.
Going back in history, the idea of meson exchange is based on Yukawa’s
fundamental hypothesis from 1935, namely, that the nuclear force is mediated
by the exchange of massive particles [2]. Yukawa’s original concept of a scalar
field which interacts with the nucleons was modified consequently soon after
to vector [3] and then to pseudo-scalar and pseudo-vector fields [4] and also
concepts of mixed meson exchange theories came up [5]. The empirically found
sign of the quadrupole moment of the deuteron could be explained correctly
by the inclusion of a pseudo-scalar field. Therefore an isovector, pseudo-scalar
boson has been predicted by Pauli [6] long before a massive particle with this
properties was found experimentally in 1947/48.
AftertheexperimentalfindingofthepionTaketani, NakamuraandSasaki[7]
proposed a subdivision of the nuclear force, an attractive long-range part for a
relative distance of the two nucleons larger than 2 fm dominated by one-pion
exchange (OPE), an intermediate range (1 fm ‚ r • 2 fm) and a short-range
(r• 1 fm) or core region. The short-range part is is the mathematically most
complicated part. From a nowadays perspective it is clear that besides multi-
pion exchange (or the exchange of heavy mesons) quark-gluon exchange plays
also a crucial role. It therefore becomes evident that the main difference among
all theories of the nuclear force arise due to the different description of the
interaction at high momenta in the short-range region.
Inordertoderivethenucleon-nucleon(NN)interactionthefirstfield-theoretic
attempts were based on pion exchange. OPE was well established describing
the long-range part of the nuclear interaction since it proved to be suitable to
describe NN scattering data and the properties of the deuteron. However, it
turned out that it was not possible to get a sufficiently strong spin-orbit force
by incorporating two-pion exchange (TPE). Moreover serious ambiguities arose
from multi-pion exchange which led to the conclusion that the ”pion theories”
developedatthattimewerenotfeasibleinordertodescribetheNNinteraction.
91. Introduction
The situation changed when heavy mesons were discovered experimentally in
the early 1960s (due to the experimental investigation of a strong short-ranged
spin-orbit force and from the electromagnetic structure of the nucleon the exis-
tence of vector mesons had been already suggested before [8, 9, 10, 11]). Now
with the inclusion of vector bosons one-boson-exchange (OBE) models were
developed which described NN scattering data fairly well. These models are
based on the assumption that multi-pion exchange can be represented by the
exchange of adequate multi-pion resonances. However, a main problem is the
inclusion of the scalar-isoscalar sigma meson since its experimental evidence
is still controversial. The sigma boson describing the intermediate range at-
traction is connected to correlated TPE. Therefore a lot of effort was spent
in order to derive the contribution from TPE to the NN interaction. One ap-
proach of deriving the OBE potentials was based on dispersion relations which
led, e.g., to a model of the nuclear force constructed from using one-pion ex-
change, dispersion-theoretical TPE and ω meson exchange and additionally a
phenomenological short-range part.
Anotherpromisingwaywastoconstructrelativisticmeson-theoreticalnucleon-
nucleon potentials in the framework of field theory. The advantage of a field-
theoretical approach is that it can account for a well-defined off-shell behavior
and medium effects when applied to the many-body problem. These properties
are natural consequences of meson exchange. In summary, the field-theoretical
approach led to more and more complex meson exchange potentials, which go
far beyond the traditional OBE models. They are constructed from OPE, TPE
including also virtual isobar excitation and finally all relevant diagrams of 3π-
and 4π-exchange. Modern one-boson-exchange potentials (OBEP) as e.g. the
Bonn potentials [12] are based on the exchange of these mesons and provide
high precision fits to nucleon-nucleon scattering data. Meson-nucleon coupling
constants and form factors are empirically fixed from the data. Thus OBEPs
are the result of relativistic phenomenology at the level of the elementary NN
interaction.
With the onset of quantum chromo-dynamics (QCD) which is the fundamen-
tal theory of strong interactions it became obvious that the nucleon-nucleon in-
teractionisnotfundamental. Althoughitisawellknownfactthatthenucleon-
nucleon interaction is entirely determined by the underlying quark-gluon dy-
namics a quantitative understanding of the NN interaction in the language of
QCD is far from being realized due to the non-perturbative character of QCD
in the low-energy regime leading to formidable mathematical problems. Even
lattice-gauge theory which is a promising tool for the treatment of low-energy
QCD nowadays, fails to be appropriate concerning the NN force due to compu-
tational restrictions which appear to persist even in the near future.
Since a direct solution of QCD in the low-energy regime is not possible quark
clustermodelsweredevelopedinspiredbyQCD.Someofthesemodelswereable
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