Dirac Spectra in Dense QCD

144 Pages·2012·2.886 MB·English

by  Takuya Kanazawa

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Takuya Kanazawa
Dirac Spectra in Dense QCD
Doctoral Thesis accepted by
The University of Tokyo, Japan
123
Author Supervisor
Dr. TakuyaKanazawa Prof.TetsuoHatsuda
Universityof Regensburg Theoretical Research Division
Regensburg NishinaCenter, RIKEN
Germany Wako
Japan
ISSN 2190-5053 ISSN 2190-5061 (electronic)
ISBN 978-4-431-54164-6 ISBN 978-4-431-54165-3 (eBook)
DOI 10.1007/978-4-431-54165-3
SpringerTokyoHeidelbergNewYorkDordrechtLondon
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(cid:2)SpringerJapan2013
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Parts of this thesis have been published in the following journal articles:
• T. Kanazawa,T. WettigandN. Yamamoto,‘‘SingularvaluesoftheDiracoper-
atorindenseQCD-liketheories,’’JHEP1112(2011)007
• G. Akemann, T. Kanazawa, M. J. Phillips and T. Wettig, ‘‘Random matrix
theoryofunquenchedtwo-colourQCDwithnonzerochemicalpotential,’’JHEP
1103 (2011) 066
• T. Kanazawa, T. Wettig and N. Yamamoto, ‘‘Exact results for two-color QCD
at low and high density,’’ PoS LAT2010 (2010) 219
• T. Kanazawa, T. Wettig and N. Yamamoto, ‘‘Chiral random matrix theory for
two-color QCD at high density,’’ Phys. Rev. D81 (2010) 081701
• T. Kanazawa, T. Wettig and N. Yamamoto, ‘‘Chiral Lagrangian and spectral
sum rules for two-color QCD at high density,’’ PoS LAT2009 (2009) 195
• T. Kanazawa, T. Wettig and N. Yamamoto, ‘‘Chiral Lagrangian and spectral
sum rules for dense two-color QCD,’’ JHEP 0908 (2009) 003
• N. YamamotoandT. Kanazawa,‘‘DenseQCDinaFiniteVolume,’’Phys.Rev.
Lett. 103 (2009) 032001
Supervisor’s Foreword
Physicsofthestronginteractionatfinitetemperatureand/orbaryondensityisone
of the most interesting topics in modern nuclear and particle physics. At tem-
peraturesgreater than 1012K,nuclei dissolveintoaplasma ofquarksandgluons,
knownasquark–gluonplasma:ouruniversestartedfromsuchanextremestateof
matter, and experimental efforts are currently being undertaken to create the
quark–gluon plasma by using the relativistic heavy-ion collisions at the RHIC
(RelativisticHeavyIonCollider)andatLHC(LargeHadronCollider).Forbaryon
densitiesgreaterthan1012kg/cm3,nucleidissolveintoadegenerateFermisystem
ofquarks,so-calledquarkmatter.Thisexoticmattermaybeformedinthecentral
core of the neutron stars.
The theoretical basis for describing the quark–gluon plasma and the quark
matter is provided by quantum chromodynamics (QCD), specifically the color
SU(3) gauge theory for the strong interaction. Due to its highly non-perturbative
nature, solving QCD analytically is one of the most challenging problems in
modern mathematical physics. On the other hand, Monte Carlo simulations of
QCDformulatedonaspace–timelattice,referredtoaslatticeQCD,havebecome
a powerful tool for solving QCD numerically. However, lattice QCD simulations
at finite baryon density are extremely difficult because of the notorious ‘‘sign
problem’’ caused by the complex QCD action at finite baryon chemical potential;
this sign problem has been preventing us from making progress in ‘‘dense QCD’’
for many years.
The present thesis by Dr. Takuya Kanazawa introduces a novel way to derive
non-perturbativeandaccurateresultsindenseQCD.Takingtwo-colorQCD(color
SU(2)gaugetheory)asaprimaryexample,Dr.Kanazawahasstudiedtheeigenvalue
distributionoftheDiracoperatorbyformulatingandsolvinganewnon-Hermitian
chiralrandommatrixtheory(ChRMT).Inparticular,heshowsthattheChRMTwith
weak non-Hermiticity can be mapped to two-color QCD with small chemical
potential,andtheChRMTwithstrongnon-HermiticitycanbemappedtotheBCS
regime of two-color QCD at high density. The spontaneous symmetry-breaking
patternofdensetwo-colorQCDplaysakeyroleinprovingsuchcorrespondence.
The approach developed by Dr. Kanazawa and his colleagues offers a theoretical
vii
viii Supervisor’sForeword
framework not only for a deeper understanding of the sign problem in QCD and
QCD-liketheoriesbutalsofornewinsights intoQCDphasesfromlowdensityto
high density. Also, the present thesis contains a transparent description of dense
QCDanditsrelationtochiralperturbationtheoryandchiralrandommatrixtheory,
making it a good introductory monograph for students and researchers who are
interestedinthisfield.
ItiswiththegreatestpleasurethatIintroduceDr.TakuyaKanazawa’sworkfor
publicationintheSpringerThesesseries.Hisworkwasnominatedasanoutstanding
Physics Ph.D. Thesis of the Fiscal Year 2010 by the Department of Physics,
GraduateSchoolofScience,theUniversityofTokyo.
Tokyo, March 2012 Tetsuo Hatsuda
Acknowledgments
IwouldliketoexpressmywholeheartedgratitudetoProf.T.Hatsudaforourfruitful
discussionsonvarioustopicsinphysics,hiscontinuousencouragementthroughout
thelastfiveyears,andhiscriticalreadingofmydoctoralthesis.Iamalsogratefulto
Prof.T.Wettigforpatientlyhelpingmetounderstandrandommatrixtheoriesand
giving me the opportunity to stay at the Physics Department of the Regensburg
UniversityinOctober2010.MygratitudealsogoestoDr.N.Yamamotoforopening
myeyestotherichphysicsofdenseQCD.Ilearnednumerousimportantconcepts
andtechniquesinquantumfieldtheoryfrommyilluminatingdiscussionswithhim.
Were it not for fruitful collaborations with Prof. Wettig, Prof. G. Akemann,
Dr.YamamotoandM.Phillips,itwouldhavebeenentirelyimpossibletocomplete
theresearchthatledtothisthesis.Iowemostofmyknowledgeoncondensedmatter
physics to S. Uchino, H. Shimada, T. Kimura, T. Sagawa and N. Sakumichi.
Mydiscussions with them have always been a great sourceofdelight andamaze-
ment for me. I would like to acknowledge tireless support of all members of the
nuclear theory group, especially Prof. S. Sasaki, Prof. T. Hirano, Dr. N. Ishii,
A.Rothkopf,Y.AkamatsuandY.Hirono.IamgratefultoProf.T.Matsuiforhaving
taught me the richness and profoundness of theoretical physics when I was an
undergraduatestudentattheKomabacampus.IthankProf.H.Fujii,N.Tanjiandthe
lateY.OhnishioftheKomabanucleartheorygroupforourvaluablediscussionsin
study meetingsduring myfirst two yearsat the graduate school. Isincerely thank
Prof. J. Verbaarschot, Prof. T. Yoneya, Prof. Y. Kikukawa, Prof. S. Ejiri, Prof.
M. Ueda, Prof. M. Nitta, Prof. M. Tachibana and Prof. K. Fukushima for our
stimulatingdiscussionsonvarioustopicsinphysics,includingquarkconfinement,
latticeQCD,coldatomicgases,topologicalsolitons,anddenseQCD.
FinallyIwouldliketoexpressmydeepappreciationtomyfamilyforsupporting
and encouragingme through the hard timesand good times alike.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 QCD with Chemical Potential and Matrix Models. . . . . . . . . . . . . 7
2.1 Phases of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Color Superconductivity and the CFL Phase . . . . . . . . . 8
2.2 Two-Color QCD at Low Density. . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Why ChPT is Worth Doing . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 ChPT for Two-Color QCD at Small l. . . . . . . . . . . . . . 20
2.3 Chiral Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Dirac Spectrum and ChPT. . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 The e-regime of ChPT. . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.3 Chiral Random Matrix Theory (ChRMT). . . . . . . . . . . . 32
2.3.4 ChRMT at l6¼0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Dirac Operator in Dense QCD. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Dense QCD in a Finite Volume . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Dense Two-Color QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Symmetry Breaking and Chiral Lagrangian . . . . . . . . . . 57
3.2.2 Finite-Volume Analysis. . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 ChRMT for Dense Two-Color QCD: Construction . . . . . . . . . . 70
3.4 ChRMT for Dense Two-Color QCD: Solution . . . . . . . . . . . . . 75
3.4.1 Spectral Density at Finite N. . . . . . . . . . . . . . . . . . . . . 76
3.4.2 The Finite-Volume Partition Function . . . . . . . . . . . . . . 79
3.4.3 Microscopic Spectral Density at Maximal
Non-Hermiticity: l^ ¼1. . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.4 Microscopic Spectral Density at Weak
Non-Hermiticity: Nl^2(cid:2)Oð1Þ. . . . . . . . . . . . . . . . . . . . 83
xi
xii Contents
3.5 Sign Problem in Dense Two-Color QCD and ChRMT. . . . . . . . 87
3.5.1 Structure of the Dirac Spectrum . . . . . . . . . . . . . . . . . . 87
3.5.2 Average Sign Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Three-fold Way at High Density. . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 QCD at Large Isospin Density (b¼2). . . . . . . . . . . . . . . . . . . 102
4.2.1 QCD Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.2 RMT Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 QCD with Real Quarks at Large Density ðb¼4Þ . . . . . . . . . . . 113
4.3.1 QCD Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 RMT Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127