A general effective action for quark matter and its application to color superconductivity [Elektronische Ressource] / von Philipp Tim Reuter

Sujets

• Physik

Informations

• Publié par: goethe_universitat_frankfurt_am_main
• Publié le: 01 janvier 2006
• Nombre de lectures: 40
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

A General Effective Action for
Quark Matterand its Application to
Color Superconductivity
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
der Johann Wolfgang Goethe-Universit¨at
in Frankfurt am Main
von
Philipp Tim Reuter
aus Bonn
Frankfurt am Main, 2005
(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, HD Dr. J. Schaffner-Bielich
Datum der Disputation: 30.11.2005Contents
1 Introduction 9
1.1 Quarkmatterandstronginteractions ........................ 9
1.2 Color superconductivity . . . . . . . . ............ 16
1.3 Search for an effectiveapproach............................ 23
2 A general effective action for quark matter 29
2.1 Derivingtheeffectiveaction.............................. 29
2.1.1 Settingthestage.................... 29
2.1.2 Integratingoutirelevantquarkmodes.................... 31
2.1.3 Integratingouthardgluonmodes............... 34
2.1.4 Tre-leveleffectiveaction....................... 37
2.2 Recoveryofknowneffectivetheories................. 42
2.2.1 HTL/HDLeffectiveaction....................... 42
2.2.2 High-densityeffectivetheory ................. 43
2.3 Towards a general effective Theory ...................... 49
2.3.1 Power counting quark loops at large µ and small T ......... 49
3 Application to color superconductivity 61
3.1 CalculationoftheQCDgapparameter........................ 61
3.1.1 CJTformalismfortheeffectivetheory............ 61
3.1.2 Dyson-Schwinger equations for relevant quarks and soft gluons . . . . . . 67
3.1.3 Solutionofthegapequationtosub-leadingorder.............. 70
3.2 Theimaginarypartofthegapfunction ................... 82
3.2.1 Solvingthecomplexgapequation .................. 86
3.2.2 Estimating the order of magnitude ofA andB ........... 92
3.2.3 EstimatingH[A]andH[B]andφ ......................106
0
3.2.4 Reproducing Reφ( + iη,k)tosubleadingorder..........10
k
3.2.5 Calculating Im φ( + iη,k)neartheFermisurface.............11
k
4 Summary and Outlook 113
A Matsubara sums in quark loops 117
B Zusammenfassung 121
34 CONTENTSList of Figures
1.1 The dependence of α on the considered energy scale Q............... 12s
1.2 Thephasediagramofstronglyinteractingmatter.............. 15
1.3 Feynmandiagramforonegluonexchangeamongtwoquarks............ 17
1.4 Theseparationofscales............................. 24
2.1 The full propagator for irrelevant quarks. . . . . . . ................ 3
−1
2.2 The diagrammatic symbol for the factor (1 + gAG ) .......... 3
0,22
¯2.3 The term Ψ gB Ψ . .................................. 33
1 1
−1
2.4 The graphical representation of the term Tr lnG inEq.(2.24)...... 34q
22
2.5 The term A J . .................................... 36
2 B
2.6 The fermionic contribution to the term A J ............ 36
2 loop
2.7 The term A J . .................................... 37
2 V
2.8 The term A Π A acordingtoEq.(2.40). ............ 37
2 22 2
2.9 The term A Π A .................................... 38
2 B 2
2.10 The fermionic contribution to the term A Π A .......... 38
2 loop 2
2.11 The term A Π A .................................... 39
2 V 2
2.12 The three- and four-gluon vertices in S [A ]............. 39A 1
¯2.13 The term Ψ gB[A ]Ψ intheeffectiveaction(2.46)................. 40
1 1 1
−12.14 The term Tr lnG [A ]intheeffectiveaction(2.46)......... 40q 1
22
−1 ¯2.15 The term Tr ln [A ,Ψ ,Ψ ]intheeffectiveaction(2.46)............ 41g 1 1 1
22
2.16 The termJ ∆ J intheeffectiveaction(2.46)............... 41
B 22 B
2.17AparticularpatchcoveringtheFermisurface................. 45
2.18 The integration region in the quark-quarkhole contribution in Eq. (2.99) . . . . 57
3.1 Diagrammatic representation of Γ ,Eq.(3.20).................... 65
2
QCD
3.2 Diagrammatic representation of Γ ................. 66
2
3.3 The contourCinEq.(3.35)........................... 71
3.4 SameasinFig.3.3,butformagnetichardgluonexchange......... 72
3.5 Evaluating the Matsubara sum for HDL-resummed gluon propagators. . . . . . . 74
3.6 A contour circumventing all non-analyticities of φ(K)................ 83
3.7 The contourCinEq.(3.94)........................... 8
3.8 Deforming the contourC........................ 89
3.9 Hard gluon exchange with momentum p>Λ ∼ µ.............. 97
gl
3.10 The integration regions of ξ and ω in Eq. (3.127). ............. 98t
5
∆6 LIST OF FIGURES
4.1 Momentum regime of quarks near the Fermi surface and soft gluons. . . . . . . . 115
B.1 Trennung der Skalen EundΛ.............................12
¯B.2 Der Term Ψ gB Ψ .......................124
1 1
−1
B.3 Die graphische Darstellung des Terms Tr lnG inEq.(B.6)............124q
22List of Tables
t
3.1 Estimates forA andA at different energy scales and ζ M......... 96
cut
cut
t <3.2 Estimates forA andA at different energy scales and ζ M......... 96
cut
cut
∼
t
3.3 Estimates forA andA at different energy scales and ζ M........ 9
pole
pole
t <3.4 Estimates forA andA at different energy scales and ζ M. ....... 9
pole
pole ∼
t
3.5 Estimates forB andB at different energy scales and ζ M..........103
cut
cut
t <3.6 Estimates forB andB at different energy scales and ζ M......103
cut
cut
∼
t
3.7 Estimates forB andB at different energy scales and ζ M.........106
pole
pole
t <3.8 Estimates forB andB at different energy scales and ζ M.........106
pole
pole
∼
78 LIST OF TABLESChapter 1
Introduction
1.1 Quark matter and strong interactions
In contrast to philosophers who search for meaning in nature, physicists investigate the properties
of nature and search for the laws that inanimate nature obeys. Apparently, the properties of
any given piece of (known) matter depend on its temperature and its density. It is therefore
not at all a naive question to ask: “What happens to some piece of matter if I heat and squeeze
it. . . further and further?” One correct answer could be: “You may use QCD to describe it.”
At least this applies to “normal” matter made of atoms. Atoms have a nucleus, which is
composed of neutrons and protons, which in turn consist of quarks. Quarks are fermions and
interact with each other by exchanging gluons, which mediate the strong interaction. The charge
corresponding to the strong interaction is called color and the quantum field theory describing
this interaction Quantum Chromodynamics (QCD). Generally, all particles that interact strongly
are composed of quarks and called hadrons. Those hadrons, which are composed of a quark and
an antiquark, are called mesons and those composed of three quarks, as the neutrons and protons
mentioned above, baryons. While all quarks carry color charge, all hadrons are in total color
neutral.
As long as the temperature and density of the considered system are not too large, i.e.
below the scale Λ 200 MeV, quarks are strongly coupled and always confined into these
QCD
color neutral hadrons. As a consequence of this strong coupling perturbative approaches are
impossible. The quarks are deconfined at temperatures or densities above Λ .Welabove
QCD
this scale the strong interaction exhibits the phenomenon of asymptotic freedom [1], where QCD
becomes weakly coupled. In this regime all hadrons vanish and quarks and gluons form a state
called the quark-gluon plasma or simply quark matter [2].
Quarks also feel the electroweak and the gravitational force. The first one is described by a
quantum field theory, which comprises electromagnetic and weak interactions. The theory for
the latter, general relativity, describes gravitation in terms of the curvature of space-time and
is a classical field theory. Due to their relative weakness as compared to the strong force it is
often justified to neglect the gravitional and the electroweak against the strong interaction and
describe the considered matter by QCD only. As a counter example related to this work, where
QCD alone is not sufficient, one may allude to neutron stars. These are the compact remnants
910 Chapter 1. Introduction
of supernova explosions of type II [3, 4, 5]. They have extremely large masses comparable to the
mass of our sun, but only very small radii of several kilometers. The density in their inner cores
are possibly sufficient to deconfine the quarks. Since these stellar objects have to be electrically
neutral and β−equilibrated, electroweak processes have to be considered. Furthermore, neutron
stars are held together gravitationally. In order to describe their bulk properties adequately, one
therefore has to account for gravity as well.
In the present work, however, only the strong interaction will be accounted for and, more-
over, only its weak coupling regime will be considered. The aim is to derive an effective action
for strongly interacting matter in the deconfined phase, i.e. for quark matter [6]. This moti-
vated by the occurence of several well seperated momentum scales at high temperatures and/or
baryonic densities, cf. Sec. 1.3. The derivation is performed in Sec. 2.1. The field of application
comprises low energy phen