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Three-Flavor Color Superconductivity

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Three-Flavor Color Superconductivity Three-Flavor Color Superconductivity Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe-Universit¨at in Frankfurt am Main von Hossein Malekzadeh aus dem Iran Frankfurt am Main, Dezember 2007 (D 30) 2 vom Fachbereich Physik der Johann Wolfgang Goethe–Universit¨at als Dissertation angenommen. Dekan: Prof. Dr. W. Aßmus Gutachter: Prof. Dr. D.-H. Rischke, Prof. Dr. Adrian Dumitru Datum der Disputation: Dezember 7, 2007. Acknowledgements I would like to express my special thanks to Prof. Dirk Rischke for all the generous supports that during my Ph.D thesis he gave mentally and practically to me. I learned a lot of things from him which will be certainly very useful for me in future as well. Without his helps and encouragements it would be difficult to come to the end of this thesis. Thank you Dirk! I am also very grateful to Prof. Igor Shovkovy for very instructive discussions we had. He was the one who taught this stimulating field to me. Also, I had a lot of fun with him during lunch breaks and in other occasions. Since I wanted to take ten watermelons with my tow hands, I needed to work harder. On other words, along my Ph.D thesis, I wanted to work on other fields as well, although the project were not published. I had great times when I was collaborating with Prof. Adrian Dumitru and Prof. Michael Strickland. I thank both of you for teaching me those fascinating topics. I also had a wonderful chance to talk to Prof. Robert Pisarski. I thank him as the person who encouraged me to continue working on this field and also as one of my referees. His kindness is very special. I also thank Stefan Ruester, Andreas Schmitt, and Qun Wang for very fruitful discussions and collaborations. I always had great times with Andreas in many conferences, lots of joke and fun. Looking forward for other conferences. I thank all my dear friends with whom I had always fun. Phillip Reuter for the time we shared the office, which were full of jokes and sometimes discussions. I thank him that gave me good times during lunches break, without his unique humor, which worked as a tasty sauce, it was impossible to enjoy German food in the Mensa. Thank you man! Additionally, my thanks go to Amir Haghighirad for his truly friendship, Jacquelyn Hostler for her kindness, Benjamin Koch for his unfinishable nice humor, Gregor Kaczor for repeating the timeless jokes which were sometimes very funny, Sophie Nahrwold with whom I hanged out anytime I wanted to, Jorge Noronha for funny times, imitating everybody starting from me and ending to Prof. St¨ocker, Basil Sa’d for showing the cool side of life, and Nan Su for exhibiting the exciting part of life. My thanks also go to the scientific administrator Dr. Joachim Reinhardt, to the system administrators Alexander Achenbach, and Gebhard Zeeb, to the secretaries Mrs. Walburga Bergmann, Mrs. Veronika Palade, Mrs. Laura Quist, and Mrs. Daniela Radulescu and to Joe Laperal-Gomez for their support. My innumerable thanks of course are for my family. Maman, Baba, Roya, Lida, va Vida mamnun be khatere hame chiz. I thank Frankfurt International Graduate School for Sciences for the financial supports. 3 4 Abstract I investigate some of the inert phases in three-flavor, spin-zero color-superconducting quark matter: the CFL phase (the analogue of the B phase in superfluid 3He), the A and A* phases, and the 2SC and sSC phases. I compute the pressure of these phases with and without the neutrality condition. Without the neutrality condition, after the CFL phase the sSC phase is the dominant phase. However, including the neutrality condition, the CFL phase is again the energetically favored phase except for a small region of intermediate densities where the 2SC/A* phase is favored. It is shown that the 2SC phase is identical to the A* phase up to a color rotation. In addition, I calculate the self-energies and the spectral densities of longitudinal and transverse gluons at zero temperature in color-superconducting quark matter in the CFL phase. I find a collective excitation, a plasmon, at energies smaller than two times the gap parameter and momenta smaller than about eight times the gap. The dispersion relation of this mode exhibits a minimum at some nonzero value of momentum, indicating a van Hove singularity. 5 6 Kurzfassung In dieser Arbeit werden verschiedene inerte Phasen von Quarkmaterie mit drei verschiedenen Quarkarten (flavors) die sich in einem farbsupraleitenden Zustand mit Spin null befinden un- tersucht. Insbesondere werden die CFL Phase, die A, die A*, die 2SC und die sSC Phasen studiert. F¨ur diese Phasen wird der Druck sowohl ohne eine Neutralit¨atsbedingung als auch mit einer Neutralit¨atsbedingung errechnet. Es zeigt sich, dass ohne die eine Neutralit¨atsbedingung die CFL und die sSC Phasen dominieren. Ber¨ucksichtigt man jedoch die Neutralit¨at der Quarkmaterie, wird meist die CFL Phase energetisch favorisiert. Lediglich f¨ur eine kleine Teilmenge des Parameterraumes bei mittelgroßen Dichten sind die 2SC und die A* Phasen energetisch favorisiert. Es wird gezeigt, dass sich die 2SC und die A* Phasen abgesehen von einem konstanten Farbfaktor gleichen. Anschließend werden f¨ur die CFL Phase die Selbsten- ergien und spektralen Energiedichten longitudinaler und transversaler Gluonen bei einer Tem- peratur T = 0 in farbsupraleitender Quarkmaterie berechnet. Bei Energien, die kleiner sind als der doppelte L¨uckenparameter (gap) und bei Impulsen die kleiner sind als das Achtfache des L¨uckenparameters zeigt es sich, dass es eine kollektive Anregung (Plasmon) m¨oglich ist. Die Dispersionsrelation dieser Anregung hat ein Minimum bei endlichem Impuls, was auf eine van Hove Singularit¨at hinweist. 7 8 Contents 1 Introduction 15 1.1 Superconductivity ............................... 15 1.1.1 Magnetic flux quantisation . 16 1.1.2 Meissner-Ochsenfeldeffect. 17 1.1.3 Londontheory ................................. 18 1.1.4 Ginzburg-Landautheory. 18 1.1.5 Cooperinstability ............................. 19 1.1.6 BCStheory................................... 21 1.2 Superfluidity .................................... 23 1.2.1 Quasiparticleconcept . 25 1.2.2 Fermiliquids .................................. 26 1.2.3 Orderparameter ................................ 29 1.3 Spontaneoussymmetrybreaking . ...... 30 1.3.1 SSBincondensedmatter . 30 1.3.2 SSB in relativistic quantum field theory . ....... 33 1.3.3 Goldstonemodes ................................ 34 1.3.4 Higgsmechanism................................ 36 1.4 Stronginteractions .............................. ..... 37 1.5 PhasediagramofQCD ............................... 41 1.6 NJLmodel ....................................... 43 1.7 Nuclearastrophysics . ..... 45 1.7.1 Compactstars ................................. 46 1.7.2 Compact stars and dense-matter physics . ...... 46 1.7.3 Electricalneutralityofstars . ....... 47 1.7.4 Nuclear matter versus neutron star . ...... 48 1.7.5 Pionandkaoncondensation. 48 1.7.6 Quarkstars................................... 49 1.8 Colorsuperconductivity . ...... 50 1.8.1 Introduction .................................. 50 1.8.2 Gapequation.................................. 52 9 10 CONTENTS 2 Three-flavor, spin-zero color superconductivity 57 2.1 Introduction.................................... 57 2.2 Pressure without neutrality condition . .......... 59 2.2.1 Derivationofgapequation . 59 2.2.2 Solutionofgapequation. 64 2.2.3 Pressure..................................... 67 2.2.4 Aphase..................................... 69 2.2.5 A*phase .................................... 70 2.2.6 PlanarorsSCphase .............................. 71 2.2.7 Polaror2SCphase............................... 72 2.2.8 CFLphase ................................... 73 2.3 Pressure including neutrality condition . ............ 73 2.3.1 NJLmodelforCSC .............................. 74 2.3.2 Tadpoles .................................... 76 2.3.3 Calculationoftadpoles . 77 2.3.4 TadpolesinAphase .............................. 77 2.3.5 TadpolesinA*phase ............................. 78 2.3.6 Pressure..................................... 78 2.4 Patternofsymmetrybreaking. ...... 83 3 Gluon self-energy in CFL phase 85 3.1 Introduction.................................... 85 3.2 Generatingfunctional . ..... 87 3.3 Gluonself-energyinnormalphase . ...... 88 3.3.1 Nambu-Gorkov, flavor, and color spaces . ..... 90 3.3.2 MixedRepresentation . 91 3.3.3 HDLlimit.................................... 94 3.4 Gluon self-energy in the superconducting phase . ........... 94 3.4.1 Nambu–Gorkovspace ............................. 95 3.4.2 Colorandflavorspace ............................ 95 3.4.3 Mixed representations for quark propagators . ......... 98 3.4.4 NG bosons and longitudinal gluon modes . 102 3.5 Explicit calculation of the gluon self-energy . ............ 109 3.5.1 Imaginaryparts ................................ 113 3.5.2 Realparts.................................... 118 3.5.3 Spectraldensities. 120 4 Summary and Outlook 123 5 Zusammenfassung 127 List of Figures 1.1 Meissnereffect ................................... 17 1.2 Gapenergy ....................................... 22 1.3 Ground state of fermionic dilute gas . ........ 28 1.4 Spontaneously symmetry breaking . ....... 35 1.5 One-loop contributions to QCD coupling constant . .......... 39 1.6 Asymptotictheory ................................ 40 1.7 PhasediagramofQCD ............................... 42 1.8 Dyson equation for quark propagator . ........ 44 1.9 Piondecay ......................................
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