The phase diagram of neutral quark matter [Elektronische Ressource] / von Stefan Bernhard Rüster

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• Physik

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• Publié par: goethe_universitat_frankfurt_am_main
• Publié le: 01 janvier 2007
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

The Phase Diagram of Neutral
Quark Matter
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
der Johann Wolfgang Goethe - Universitat¨
in Frankfurt am Main
von
Stefan Bernhard Ru¨ster
aus Alzenau in Ufr.
Frankfurt 2006
(D 30)2
vom Fachbereich Physik der
Johann Wolfgang Goethe - Universita¨t als Dissertation angenommen.
Dekan: Prof. Dr. Aßmus
Gutachter: Prof. Dr. Rischke und HD PD Dr. Schaffner-Bielich
Datum der Disputation: 14. Dezember 2006Abstract
In this thesis, I study the phase diagram of dense, locally neutral three-flavor quark matter as a
function ofthe strangequarkmass, the quarkchemicalpotential, andthe temperature, employing
a general nine-parameter ansatz for the gap matrix. At zero temperature and small values of
the strange quark mass, the ground state of quark matter corresponds to the color–flavor-locked
(CFL) phase. At some critical value of the strange quark mass, this is replaced by the recently
proposed gapless CFL (gCFL) phase. I also find several other phases, for instance, a metallic
CFL (mCFL) phase, a so-calleduSC phase where all colors of up quarks are paired, as well as the
standard two-flavor color-superconducting (2SC) phase and the gapless 2SC (g2SC) phase.
I also study the phase diagram of dense, locally neutral three-flavor quark matter within the
framework of a Nambu–Jona-Lasinio (NJL) model. In the analysis, dynamically generated quark
masses are taken into account self-consistently. The phase diagram in the plane of temperature
and quark chemical potential is presented. The results for two qualitatively different regimes,
intermediate and strong diquark coupling strength, are presented. It is shown that the role of
gapless phases diminishes with increasing diquark coupling strength.
In addition, I study the effect of neutrino trapping on the phase diagram of dense, locally
neutral three-flavor quark matter within the same NJL model. The phase diagrams in the plane
of temperature and quark chemical potential, as well as in the plane of temperature and lepton-
number chemical potential are presented. I show that neutrino trapping favors two-flavor color
superconductivity and disfavors the color–flavor-lockedphase at intermediate densities of matter.
At the same time, the location of the critical line separating the two-flavor color-superconducting
phase and the normal phase of quark matter is little affected by the presence of neutrinos. The
implications of these results for the evolution of protoneutron stars are briefly discussed.4Acknowledgments
I amvery gratefulto my advisorProf.Dr. Dirk Rischkewho suggestedthe topicfor my thesis. He
introduced me to quantum field theory and color superconductivity. I learnt a lot in his lectures
and in private communication. I thank him for his suggestions and advices. I am very thankful to
Prof. Dr. Igor Shovkovy. I thank him for the excellent cooperation, the discussions, suggestions,
and advices. I am grateful to our colleagues Verena Werth and PD Dr. Michael Buballa from
the Institut fu¨r Kernphysik at the Technische Universit¨at Darmstadt for the teamwork. I thank
HosseinMalekzadehforthecooperationconcerningthespin-zeroA-phaseofcolor-superconducting
quark matter.
I am grateful to HD PD Dr. Ju¨rgen Schaffner-Bielich. I learnt a lot in his lectures, seminars,
and in our astro group meetings. I also thank him and Matthias Hempel for the cooperation and
discussions concerning the outer crust of nonaccreting cold neutron stars.
I am grateful to the computer trouble team for removing computer problems. I am thankful
forusingtheCenterforScientificComputing(CSC)oftheJohannWolfgangGoethe- Universita¨t.
I am very grateful to my parents who supported me during the whole time of my study.6Contents
Abstract 3
Acknowledgments 5
Contents 7
List of Figures 9
List of Tables 11
1 Introduction 13
1.1 The phase diagram of strongly interacting matter . . . . . . . . . . . . . . . . . . . 13
1.2 Color superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1 The 2SC phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.2 The CFL phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.3 Spin-one color superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 The formation of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.2 Main sequence stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.3 Red giants and red super giants . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.4 Compact stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.1 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.2 Structure of neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.3 Properties of neutron star matter . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.4 Toy models of neutral normal quark matter . . . . . . . . . . . . . . . . . . 33
1.5 Color superconductivity in neutron stars . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.1 Toy models of neutral color-superconducting quark matter . . . . . . . . . . 38
2 The phase diagram of neutral quark matter 41
2.1 The phase diagram of massless quarks . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.1 Quantum chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.2 The effective action of quarks . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.3 Propagators and self-energies in projector representation . . . . . . . . . . . 49
2.1.4 The potential part of the effective action of quarks . . . . . . . . . . . . . . 49
2.1.5 The kinetic part of the effective action of quarks . . . . . . . . . . . . . . . 51
2.1.6 The pressure of color-superconducting quark matter . . . . . . . . . . . . . 54
2.1.7 Results at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.1.8 Results at nonzero temperature . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2 The phase diagram with a self-consistent treatment of quark masses . . . . . . . . 66
2.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 The phase diagram with the effect of neutrino trapping . . . . . . . . . . . . . . . 838 CONTENTS
2.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3.2 Simplified considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3 Conclusions 101
3.1 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Open questions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A Definitions of matrices 109
A.1 The Pauli matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.1.1 Spin projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.2 Matrices in Dirac space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.2.1 Projectors in Dirac space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.3 The generators of the SU(3) group . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B Useful formulae 113
B.1 Non-interacting massless fermions and antifermions at nonzero temperature . . . . 113
B.2 The inverse Dirac propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.3 The tree-level quark propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.4 The Feynman gauged gluon propagator . . . . . . . . . . . . . . . . . . . . . . . . 115
B.5 The determinant of the inverse quark propagator . . . . . . . . . . . . . . . . . . . 116
B.6 The logarithm of the determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.7 The Dirac trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.8 The trace of the logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.9 Cubic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
B.10 Summation over the fermionic Matsubara frequencies . . . . . . . . . . . . . . . . . 123
C Zusammenfassung 127
Bibliography 133
Lebenslauf 141List of Figures
1.1 The knowledge about the phase diagram of strongly interacting matter in 2003. . . 14
1.2 The one-gluon exchange interaction between two quarks in QCD. . . . . . . . . . . 16
1.3 The Hertzsprung-Russell diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 The proton-proton cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 The carbon-nitrogen-oxygen cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 The evolution of neutron stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7 A schematic representation of a pulsar. . . . . . . . . . . . . . . .