Bulk viscosity of spin-one color superconductors [Elektronische Ressource] / von Basil A. Saʻd

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Bulk Viscosity of Spin-OneColor SuperconductorsDissertationzur Erlangung des Doktorgradesder Naturwissenschaftenvorgelegt beim Fachbereich Physikder Johann-Wolfgang-Goethe-Universit¨atin Frankfurt am MainvonBasil A. Sa’daus JordanienFrankfurt am Main, August 27, 2009(D 30)vom Fachbereich Physik der Johann Wolfgang Goethe–Universit¨atals Dissertation angenommen.Dekan: Prof. Dr. D. H. RischkeGutachter: Prof. Dr. D. H. RischkeDatum der Disputation: May 26, 2008Contents1 Introduction 11.1 QCD Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Properties of Neutron Stars . . . . . . . . . . . . . . . . . . . 61.2.2 Structure of Neutron Stars . . . . . . . . . . . . . . . . . . . . 101.3 Color Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 CFL Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 2SC Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Spin-1 Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.4 The Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . 172 Weak Interaction Rates in Quark Matter 212.1 Quark Matter in β-Equilibrium . . . . . . . . . . . . . . . . . . . . . 222.2 Weak Interaction rates . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Down-Quarkβ Decay . . . . . . . . . . . . . . . . . . . . . . . 252.2.

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Bulk Viscosity of Spin-One
Color Superconductors
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
der Johann-Wolfgang-Goethe-Universit¨at
in Frankfurt am Main
von
Basil A. Sa’d
aus Jordanien
Frankfurt am Main, August 27, 2009
(D 30)vom Fachbereich Physik der Johann Wolfgang Goethe–Universit¨at
als Dissertation angenommen.
Dekan: Prof. Dr. D. H. Rischke
Gutachter: Prof. Dr. D. H. Rischke
Datum der Disputation: May 26, 2008Contents
1 Introduction 1
1.1 QCD Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Properties of Neutron Stars . . . . . . . . . . . . . . . . . . . 6
1.2.2 Structure of Neutron Stars . . . . . . . . . . . . . . . . . . . . 10
1.3 Color Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 CFL Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 2SC Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Spin-1 Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 The Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Weak Interaction Rates in Quark Matter 21
2.1 Quark Matter in β-Equilibrium . . . . . . . . . . . . . . . . . . . . . 22
2.2 Weak Interaction rates . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Down-Quarkβ Decay . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Strange-Quarkβ Decay . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Non-Leptonic Weak Interaction . . . . . . . . . . . . . . . . . 32
3 Two-Flavor Quark Matter 37
3.1 Bulk Viscosity Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Bulk Viscosity in the Normal Phase . . . . . . . . . . . . . . . . . . . 41
3.3 Bulk Viscosity in Spin-One Color-Superconducting Phases . . . . . . 45
4 Three-Flavor Quark Matter 53
4.1 Contributing Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Bulk Viscosity in Strange Quark Matter . . . . . . . . . . . . . . . . 55
4.3 Bulk Viscosity in Normal Phase . . . . . . . . . . . . . . . . . . . . . 62
iii CONTENTS
5 R-mode Instabilities 71
5.1 R-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Dissipative Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 Gravitational Radiation Timescale . . . . . . . . . . . . . . . 75
5.2.2 Shear Viscosity Timescale . . . . . . . . . . . . . . . . . . . . 76
5.2.3 Bulk Viscosity Timescale . . . . . . . . . . . . . . . . . . . . . 76
5.3 R-mode Instability Window for npe Stars . . . . . . . . . . . . . . . . 77
5.4 The R-mode Instability Window for Quark Stars. . . . . . . . . . . . 78
5.4.1 R-mode Instability Window for the Normal Phase . . . . . . . 79
5.4.2 R-mode Instability Window for the CFL Phase . . . . . . . . 82
5.4.3 R-mode Instability Window for the 2SC Phase . . . . . . . . . 84
5.4.4 R-mode Instability Window for the CSL Phase . . . . . . . . . 87
6 Summary and Outlook 91List of Figures
1.1 A sketch showing the general features of the QCD phase diagram
from Ref. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Distribution of observed neutron star masses [2]. . . . . . . . . . . . . 8
1.3 The spins of almost all known pulsars vs. their ages [3]. . . . . . . . . 9
1.4 Possible phases and structures of a neutron star [4]. . . . . . . . . . . 11
2.1 The values of = (thick lines) and 10 (thin lines) vs. thed s e
strange quark mass for different values of the coupling constant α . . 24s
3.1 Diagrammatic representation of the weak (Urca) processes that con-
tribute to the bulk viscosity of non-strange quark matter in stellar
cores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 The bulk viscosity for the normal phase of two-flavor quark matter
as a function of the period of the density oscillations. . . . . . . . . . 44
3.3 The reduction factor as a function of ϕ≡ φ/T for the CSL, planar,
polar, and A phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 The reduction factor as a function of the temperature for the CSL,
planar, polar, and A phases. . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Thebulkviscosity asafunctionofthetemperatureforspin-onecolor-
3 −1superconductingquarkmatter. Theoscillationfrequencyisω =10 s . 50
3.6 The bulk viscosity as a function of the temperature for a toy model
of a spin-one color superconductor in which all quasiparticle modes
3 −1are gapped. The oscillation frequency is ω = 10 s . . . . . . . . . . 51
4.1 Diagrammatic representation of the weak processes that contribute
to the bulk viscosity of quark matter in stellar cores. . . . . . . . . . 54
iiiiv LIST OF FIGURES
4.2 (Color online) The bulk viscosity for the normal phase of three-flavor
quark matter as a function of the period of density oscillations. Re-
sults forset Bareshown intheupper panel andthoseforset Ainthe
lower panel. For each temperature, the dots on the lines correspond
to the values of the frequency defined in Eq. (4.25). . . . . . . . . . . 68
4.3 (Color online) The bulk viscosity for the normal phase of three-flavor
quark matter as a function ofthe temperature forseveral fixed values
of the frequency of density oscillations. Results for set B are shown
in the upper panel and those for set A in the lower panel. . . . . . . . 69
4.4 (Coloronline)Theratioζ/ζ asafunctionoftemperatureforsetA.non
The results forthree fixed values ofthedensity oscillation frequencies
1 1 1areshown: = 1Hz(solidline), =10Hz(dashedline), = 100Hz
τ τ τ
1(dotted line), and = 1000 Hz (dashed-dotted line). . . . . . . . . . . 70
τ
5.1 The r-mode instability window for normal quark stars (thick lines)
for strange quark masses of 100, 200, and 300 MeV, and neutron
stars (thin line). the shaded box represents the region in which most
LMXB’s are observed. . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 The quark contribution to the bulk viscosity as a function of temper-
atureat constant frequency “1000Hz” andr-mode instabilities ofthe
CFL phase for different values of the strange quark mass. The thin
lines show the results for unpaired quark matter.. . . . . . . . . . . . 83
5.3 Contributions to the process u +s ↔ u +d in the 2SC phase. A
gapped fermion is marked with the gap Δ at the respective line [5]. 852SC
5.4 Same as Fig. 5.2, but for the 2SC phase. . . . . . . . . . . . . . . . . 86
5.5 Same as Fig. 5.2, but for the CSL phase. . . . . . . . . . . . . . . . . 88List of Tables
1.1 Functions ω and λ for four spin-one color superconductors. . . . . 16ii ξ,i
1.2 The values of a , λ , and d for the 2SC, CFL, and the CSL phases. . 19r r
4.1 Values of the chemical potentials and the coefficient functions for the
parameter sets A, and B. All numbers are in MeV. . . . . . . . . . . 64
vvi LIST OF TABLESAcknowledgements
First of all, I would like to thank my advisor and boss Dirk Rischke for guiding me
through this work, providing me with advice and help throughout my Ph.D.. He
did it with enthusiasm and patience that made my work much more exciting and
rewarding.
Second, thanks to Igor Shovkovy, without whom this thesis would not have been
possible. He closely worked with me throughout the better part of my Ph.D., pro-
vided me with insights, and trying to convince him of my results only made my
work more concrete, it was simply great working with you, Igor.
Throughout my work, I found discussions with Ju¨rgen Schaffner-Bielich and Igor
Mishustin to be very fruitful and insightful.
Therearemanyfriendsandcolleagueswho,throughtheirsupportandfruitfuldis-
cussions, made this work more pleasant to go through. Of them I mention Veronica
Dexheimer, Ben Koch, Irina Sagert, and Hussein Malekzadeh.
I must mention Jorge and Jaki Noronha-Hostler, they were the best of colleagues
and friends a person could ask for, they have both supported me and helped me
throughout my time in Frankfurt, and sometimes, outside Frankfurt. One of their
many ways of helping me was to proof-read this thesis.
The German translation of the summary and the abstract were done by the dear
Sophie Nahrwold and checked by J¨org Ruppert, thank you both very much.
Many thanks go to the Frankfurt Institute for Advanced Studies (FIAS) and
the Frankfurt International Graduate School for Science (FIGSS) under the wise
leadership of Horst St¨ocker.
vii