Short range correlations in quark and nuclear matter [Elektronische Ressource] / vorgelegt von Frank Frömel

Sujets

• Physik

Informations

• Publié par: justus-liebig-universitat_giessen
• Publié le: 01 janvier 2007
• Nombre de lectures: 23
• Poids de l'ouvrage: 3 Mo

Extrait

Short-range correlations
in quark and nuclear matter
Dissertation zur Erlangung des Doktorgrades
der Naturwissenschaftlichen Fakult¨ at
der Justus-Liebig-Universit¨ at Gießen
Fachbereich 07 – Mathematik und Informatik,
Physik, Geographie
vorgelegt von
Frank Fr¨omel
aus Altenstadt
Gießen, Juni 2007Dekan: Prof. Dr. Bernd Baumann
I. Gutachter: Prof. Dr. Ulrich Mosel
II. Gutachter: PD Dr. Stefan Leupold
Tag der mundlic¨ hen Prufung:¨ 18.07.2007Contents
1 Introduction 1
2 The formalism 11
2.1 TheNJLinteraction........................... 1
2.2 1/N expansionintheNJLmodel......... 14c
2.2.1 Order counting in 1/N ........... 14c
2.2.2 Clasificationofself-energydiagrams... 15
2.2.3 TheRandomPhaseApproximation............... 17
2.3 Mean-fieldapproaches............... 18
2.3.1 TheO(1)Hartre+RPAapproach..... 18
2.3.2 Hartre–Fock,RPA,andGoldstonemodes........... 21
2.4 TheO(1/N)approach............... 24c
2.4.1 CoupledsetofDyson–Schwingerequations... 24
2.4.2 Regularizationscheme........... 25
2.4.3 Furtherremarks............... 26
2.5 1/N corectionsandchiralsymmetry.................. 27c
2.5.1 Chirally invariant 1/N extensions..... 28c
2.5.2 Consequences for theO(1/N)approach............ 29c
2.6 Quark fields and propagators ........... 32
2.6.1 Green’sfunctionsonaclosedtime-path............. 33
2.6.2 Lorentzstructure......................... 34
2.6.3 Quarkspectralfunction .......... 36
2.6.4 Real and imaginary parts of the propagators .......... 37
2.6.5 Spectralfunctionintheenergy–momentumplane....... 40
2.7 Meson propagators and spectral functions . .............. 41
2.8 Self-energy,polarizationsandwidths....... 43
2.8.1 Quarkself-energy.............. 43
2.8.2 RPAmesonpolarization.......... 45
3 Calculation of self-energies and polarizations 47
3.1 Mean-fieldself-energy........................... 47
3.1.1 Hartree self-energy ............. 47
3.1.2 Fockself-energy.... 48
3.2 Quarkcondensateandquarkdensity....... 49
3.3 Collisional self-energies and quark widths . . .............. 50
iContents
3.4 Collisional polarizations and RPA meson widths . ........... 52
3.5 Dispersionintegrals................. 54
3.6 RealpartsoftheRPAmesonpolarizations.... 56
3.6.1 Time-orderedandretardedpolarizations. 56
3.6.2 DecompositionoftheRPApolarizations............ 58
3.6.3 Thenon-dispersivepartoftheRPApolarizations....... 59
3.6.4 ThedispersivepartoftheRPApolarizations.......... 61
3.7 MasesoftheRPAmesons............. 67
3.8 Realpartsofthequarkself-energy ................... 69
3.8.1 Decompositionoftheself-energy...... 70
3.8.2 Dispersionrelation.. 71
3.8.3 Quarkwidth................. 72
4 Quark and meson scattering 75
4.1 Mean-fieldspectralfunctions............ 75
4.1.1 Quarkspectralfunction .......... 75
4.1.2 RPAmesonspectralfunctions....... 76
4.2 Thresholds...................... 7
4.2.1 Processes with bound qq¯states ................. 78
4.2.2 Quark–quarkscateringanddecayproceses.......... 84
4.3 Densitydependence...... 89
4.3.1 Saturationeffects.............. 90
4.3.2 Densitydependenceoftheindividualprocesses......... 91
4.4 On-shelwidth ................... 92
4.4.1 Relevantproceses.. 92
4.4.2 RoleoftheRPApionmas......... 94
4.4.3 Effectsofhigherordercorections..... 98
4.5 RPAmesonwidth............................. 99
5 Numerics and results 105
5.1 Detailsofthecalculation... 105
5.1.1 NJLparametersets........................ 105
5.1.2 Iterativeprocedure.. 106
5.1.3 Numericalgrid.... 109
5.1.4 Quarks,antiquarksandeffectivemasses . 110
5.2 Collisional broadening .......................... 10
5.2.1 Generalstructure... 10
5.2.2 Corespondencetoscateringanddecayproceses....... 13
5.2.3 Comparisontotheloop-expansion..... 18
5.3 RPA mesons in theO(1/N)approach ................. 119c
5.3.1 Structureofwidthandspectralfunction. 120
5.3.2 Comparisontomean-fieldresultsandchiralproperties .... 121
5.4 Chiralphasetransition............... 123
iiContents
5.4.1 TheHartre+RPAapproximation................ 123
5.4.2 Phase transition in theO(1/N)approach........... 124c
5.5 On-shelself-energy................. 128
ret5.5.1 Real part of Σ... 129s
ret5.5.2 Real part of Σ.............. 1300
5.5.3 On-shelwidth .... 132
5.6 Averagequarkwidth................ 134
5.6.1 Averagewidthofthepopulatedstates ............. 134
5.6.2 Averagewidthofalquarkstates................ 136
5.6.3 Furtherremarks.... 137
5.7 Momentumdistribution .............. 138
6 Nuclear matter at high densities and finite temperatures 141
6.1 Themodel...................... 143
6.1.1 Self-consistentapproach.......... 143
6.1.2 Short-rangeinteractions... 148
6.1.3 Mean-fieldself-energy............ 152
6.2 NumericalDetailsandResults...................... 15
6.2.1 Detailsofthecalculation.......... 15
6.2.2 Comparison to phenomenological NN potentials at T=0... 158
6.2.3 Spectralfunctionandwidth........ 160
6.2.4 On-shellwidth .......................... 161
6.2.5 Averagewidth..... 166
6.2.6 Influence of nucleon resonances . . .... 171
7 Summary and Outlook 175
A Notation and conventions 183
B RPA on the Hartree(–Fock) level 187
C Feynman rules of the real-time formalism 195
D Energy integral over the retarded quark propagator 203
E Quarks and antiquarks, density and momentum distribution 207
F Dispersive and non-dispersive terms 217
G Relations between Bose and Fermi distributions 219
H Numerical Implementation 221
Bibliography 231
iiiContents
Deutsche Zusammenfassung 241
Danksagung 249
iv1 Introduction
We know from our everyday experience that the properties of an object depend on
the surrounding environment. Consider, for example, the different weight of an item
– or the different resistance to its motion – in air and in water. Such effects, aris-
ing from interactions with the medium, play also an important role in microscopic
many-body systems. In this thesis, we will explore the influence of dynamical corre-
lations on the properties of quarks and nucleons in quark matter and nuclear matter,
respectively. We will focus on the short-range correlations here. In comparison to
the well-investigated long-range correlations that determine the bulk properties of the
quarks and nucleons – like effective masses and binding energies – they have a more
subtle influence on the in-medium properties of the particles.
To illustrate the difference between long- and short-range correlations, we can con-
sider a simple system of interacting “particles” [Buß04]: Persons that stroll along a
street in a pedestrian area. The upper panels of Fig. 1.1 show the path of a walker
in three situations with a varying number of other pedestrians (late at night, on a
regular weekday, and on the Saturday before Christmas). The lower panels indicate
the corresponding probabilities to find the walker at a certain (instantaneous) veloc-
ity|v|. The first scenario does not require much discussion. Late at night, no other
persons are in the street. The walker can move along a straight line with constant
velocity. This corresponds to the motion of free particles in the vacuum.
In the second case, a few other pedestrians will be present. This is a typical example
for long-range correlations. The spaces between the pedestrians remain large and our
walker can see them from far away. Thus, he can adjust his path long before he runs
into another person. Compared to the first case, he has to slow down a little. However,
he can still walk with a constant velocity along the smoothly curved path. Note that
we would obtain the same velocity distribution if there were no other persons but the
street had a little slope that slows down the walker. This corresponds to a so-called
mean-field approach where the interactions between the particles are absorbed into
an effective potential or an effective mass.
The third scenario shows the effect of short-range correlations. On the Saturday
before Christmas, the street will be crowded with other pedestrians – moving in non-
uniform ways and obstructing the view of the walker. Hence, he cannot plan his path
in advance. It becomes necessary for the walker to adjust his velocity (and direction)
dynamically. At some times it will be better to stop for a moment, at other times
it will be better to rush through a gap. The probability to find the walker at a
certain velocity turns from a sharp peak into a broad distribution. It should be clear
from our simple example that the magnitude of the broadening, i.e. the importance
11 Introduction
Figure 1.1: A simple example for long- and short-range correlations: The path of a
walker through a pedestrian area (upper panels) – late at night (left), at daytime
on a regular weekday (middle), and at noon on the Saturday before Christmas
(right). White dots stand for other pedestrians. The lower panels indicate the
probability to find the walker at a certain (instantaneous) velocity |v|.The
dashed lines in the middle and the right panel denote the free case.
of the short-range correlations, depends strongly on the density of the medium. A
description of the short-range effects by a simple potential is not feasible. Hence,
their investigation will be more complicated than the study of long-range effects.