Lattice Quantum ChromoDynamics with approximately chiral fermions [Elektronische Ressource] / vorgelegt von Dieter Hierl
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Lattice Quantum ChromoDynamics with approximately chiral fermions [Elektronische Ressource] / vorgelegt von Dieter Hierl

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Lattice Quantum ChromoDynamicswith approximately chiral fermionsDissertationzur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.)der naturwissenschaftlichen Fakulta¨t II - Physikder Universita¨t Regensburgvorgelegt vonDieter Hierlaus HemauMai 2008Die Arbeit wurde von Prof. Dr. Andreas Scha¨fer angeleitet.Das Promotionsgesuch wurde am 22. April 2008 eingereicht.Das Promotionskolloquium fand am 27. Juni 2008 statt.Pru¨fungsausschuss:Vorsitzender: Prof. Dr. W. Wegscheider1. Gutachter: Prof. Dr. A. Scha¨fer2. Gutachter: Prof. Dr. T. Wettigweiterer Pru¨fer: Prof. Dr. J. Fabian¨FUR MEINE ELTERNContents1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I Quantum Chromodynamics 52 The QCD action and its symmetries 102.1 The QCD action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 The path integral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Local and global symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Lattice Quantum ChromoDynamics
with approximately chiral fermions
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der naturwissenschaftlichen Fakulta¨t II - Physik
der Universita¨t Regensburg
vorgelegt von
Dieter Hierl
aus Hemau
Mai 2008Die Arbeit wurde von Prof. Dr. Andreas Scha¨fer angeleitet.
Das Promotionsgesuch wurde am 22. April 2008 eingereicht.
Das Promotionskolloquium fand am 27. Juni 2008 statt.
Pru¨fungsausschuss:
Vorsitzender: Prof. Dr. W. Wegscheider
1. Gutachter: Prof. Dr. A. Scha¨fer
2. Gutachter: Prof. Dr. T. Wettig
weiterer Pru¨fer: Prof. Dr. J. Fabian¨FUR MEINE ELTERNContents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I Quantum Chromodynamics 5
2 The QCD action and its symmetries 10
2.1 The QCD action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 The path integral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Local and global symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Perturbation theories 18
3.1 Introduction to QCD perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Introduction to chiral perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Finite volume effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Random Matrix Theory 26
4.1 Introduction to RMT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Chiral RMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
II Lattice Quantum Chromodynamics 29
5 Discretizations 33
5.1 The lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 The gauge action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
vvi CONTENTS
6 Chiral fermions 41
6.1 Chiral symmetry on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Consequences of the Ginsparg-Wilson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.3 The chirally improved Dirac operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.4 The parametrized fixed-point Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7 Ensemble creation 51
7.1 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.2 Running a simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.3 Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8 Analysis 58
8.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.3 Improvements I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.4 Improvements II - Low-mode averaging for meson correlators . . . . . . . . . . . . . . . . . . . . 69
8.5 Improvements III - Correlators using covariant operators . . . . . . . . . . . . . . . . . . . . . . . 74
9 Data Modeling 79
9.1 Setting the scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.2 Two-point correlation functions in Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.3 LECs from ”unphysical” regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.4 Extrapolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9.5 Interpreting the data and its errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
III LQCD with chirally improved fermions 93
10 The baryon spectrum in the quenched approximation 95
10.1 Computing the baryon masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.2 Effective masses, eigenvectors and fit ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.3 Nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.4 Sigma and Xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10.5 Lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10.6 Delta and Omega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10.7 Chiral extrapolations for the fine lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.8 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105CONTENTS vii
11 The pentaquark 107
11.1 Quark models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.2 The Quantum Numbers of the Pentaquark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
11.3 Details of our lattice calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
11.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11.6 What is still missing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
IV LQCD with 2+1 flavors using the fixed-point action 119
12 Algorithm for dynamical fermions 121
12.1 Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12.2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
12.3 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
12.4 Relative gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
12.5 Determinant breakup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
12.6 Nested Accept/Reject steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
12.7 Matrix-vector multiplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
13 Low Energy Constants 136
13.1 The delta regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
13.2 The epsilon regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
13.3 Comparing results in the epsilon-regime with RMT predictions . . . . . . . . . . . . . . . . . . . 142
13.4 Comparing results in the epsilon-regime with ChPT predictions . . . . . . . . . . . . . . . . . . . 146
V Conclusion 151
Summary 152
Outlook 155
Acknowledgements 156
Appendix 157
A Definitions 158
A.1 Gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.2 Gell-Mann matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A.3 Grassmann Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
A.4 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161viii CONTENTS
B Path integral formalism on the lattice 163
B.1 The generating functional for fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.2 Expectation values of fermionic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
C Chiral transformations (extended) 165
C.1 Left- and right handed projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
C.2 Global covariant densities and conserved currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
C.3 Local covariant densities and conserved currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
C.4 Neglecting contact terms in the densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
C.5 The AWI mass in 2+1 flavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
D Chiral transformations with non-constant operator R 170
D.1 The general chiral transformations on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
D.2 Neglecting the contact terms in the densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
D.3 Covariant conserved currents with non-constant operator R . . . . . . . . . . . . . . . . . . . . . 172
E Group Theory 173
E.1 Young tableaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
E.2 SU(2) symmetry group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
E.3 SU(3) symmetry group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
E.4 SU(6) symmetry group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
F Quark models 180
F.1 The spin-0 and spin-1 meson nonets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
F.2 The baryon octet and decuplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
F.3 Diquarks and triquarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
G Parameters for CI fermions 183
G.1 Parameters of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
G.2 Dirac structure and quark sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
H Parameters for FP fermions 185
H.1 Parameters in the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
H.2 Parameters of the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Bibliography 189Chapter 1
Introduction
1.1 Background
The so-called standard model of elementary particle physics provides actually the description of all phenomena
in particle physics. Only gravitation, which is acting very weakly on elementary particles, is not included in it.
Research activities in the last 30 years have verified the standard model with a very high degree of accuracy.
It serves the description of both, the particle contents and the particle dynamics, i.e., the forces between the
matter particles. Those forces were represented by the exchange of particles, the gauge bosons.
According to the standard model matter consists of 12 matter particles (6 quarks and 6 leptons) and 3 forces
(electromagnetic, weak and strong force), which were described by 12 gauge bosons (photon, 8 gluons and 3
electroweak bosons). In addition to that, it is believed that the so-called Higgs particles explain the creation of
particle masses. It has not been found yet, but there is a strong hope of the whole particle physics community
to find it at the LHC starting 2008. The 12 matter particles are grouped into 3 families or generations. The
particles of higher generations are heavy copies of the particles in the first generation and are not stable. They
decay into particles of the first generation, so that nearly all matter surrounding us consists of first generation
particles.
Thelanguageofthestandardmodelisquantumfieldtheory(QFT).Allpredictionsaremathematicallyderived
from the LagrangedensityL. Due to the fact that the Lagrangedensity contains more than 27 free parameters
(at least 3 coupling constants, 12 particle masses, 6 mixing angles for quarks and leptons, 2 angles for the
1description of CP violation and the Higgs particle mass), which have to be tuned to get physical results, there
is no doubt that there has to be a more fundamental theory in physics.
A very successful part of the standard model is the quantum field theory of the strong force, the quantum
chromodynamics abbreviated as QCD. There the basic parameters are the quark and gluon fields. Like every
quantum field theory in the standard model it is a local gauge theory. The gauge fields are the 8 gluons
which assure the local gauge invariance and create the interaction among the matter fields, the quarks. From
experiments one finds that the color forces are realized by the SU(3) group.C
In contrasttothe abelian QED,wherethe mediatorparticle,the photon, is not charged,in QCDthe mediator
particles, the gluons, are also carrying color charge, which makes the color force always attractive. For QED
and QCD the strength of the interaction is scale dependent. While for QED the coupling becomes smaller
when the energy scale decreases it is vice versa for QCD, i.e., the coupling is small at high energy transfers and
becomes larger when the energy decreases. The latter behavior is known as asymptotic freedom. The energy
to separate two quarks, which are bound within a hadron, increases when the distance between the two quarks
becomes larger and larger. At a certain point the energy to separate the two quarks is high enough to create a
new quark/anti-quarkpair out of the vacuum. Therefore, quarks have never been observed as isolated particles
up to now. This fact is known as confinement.
1Cabibbo-Kobayashi-Maskawa-Matrix (CKM-Matrix)
12 Chapter 1: Introduction
In perturbation theory one performs an expansion in the coupling constant, which is expected to be smaller
thanO(1). Hence, for QCD perturbation theory can be applied for the high energy regime, where the coupling
α is small. Investigating the low energy regime of QCD perturbation theory does no longer hold and ones
has to use non-perturbative methods or different expansion parameters which are smaller than O(1) again.
There exists a effective perturbation theory, chiral perturbation theory abbreviated as χPT or ChPT, where
an expansion is done in the pion mass, the pion decay constant and hadron momenta. But this is, of course,
only an effective theory which has to be tuned via so-called low energy constants (LECs). However, if we can
determine those LECs to a high precision we can also calculate physical quantities in the low-energy regime of
QCD using perturbation theory methods.
A huge amount of evidence has been found, that QCD is the right theory of the strong interaction. However,
a complete understanding of non-perturbative QCD effects based on the fundamental equations exclusively is
still missing. This gap of knowledge limits the extraction of the free parameters in the standard model from
experiments. Calculations in Lattice QCD (LQCD) applying Monte Carlo simulations can help to fill that gap,
because lattice field theory provides a systematic way to solve QCD from first principles.
Thephysicsinthelow-energyregimeisstronglyinfluencedbythechiralsymmetryanditsbreaking. Foralong
time it was a fundamental problem to introduce chirality in lattice simulations. Naive lattice discretizations
always violate explicitly the chiral symmetry of the massless Dirac operator. One way to circumvent this
problem is to use solutions of the Ginsparg-Wilson relation. The spectral properties of Ginsparg-Wilson Dirac
operators have made it possible to study the structure of the QCD vacuum, in which the reasons of the chiral
symmetry breaking are hidden, very efficiently.
Another problem in Lattice QCD is the huge amount of computer time needed to generate independent
configurations. So, in the beginningusually the fermiondeterminant wassetto aconstant. This approximation
is equal to omitting the internal quark loops and simplifies the technical procedure tremendously. For many
observables the effects of these internal loops are indeed negligible and one has to add only a relatively small
systematical error to the results. With the progress in computer technology, however, full QCD simulations
(also called dynamical quark simulations) have become the standard and we are now even in the start-up phase
for so-called dynamical chiral QCD calculations.
1.2 Outline of this work
This work is made up of five parts. In part I, we concentrate on QCD in the continuum. There we lay
the foundations for the present work. In Chapter 2, we first introduce the QCD action and discuss several
symmetries and their breaking. Two important ways one can go to solve QCD, perturbative QCD (pQCD)
and chiral Perturbation theory (χPT), are discussed in Chapter 3. We also need predictions from the Random
Matrix Theory (RMT), especially from Chiral Random Matrix Theory (ChRMT). A short introduction into
those topics can be found in Chapter 4.
Solving QCD in the low energy regime from first principles means to apply non-perturbative methods like
lattice quantum chromodynamics(LQCD). In part II, we explain the used tools of LQCD and guide the reader
fromthestarttotheendofaLQCDsimulation. FirstweintroduceinChapter5thediscretizationswhichallow
to put a continuous field theory on a discrete space-time lattice. Because of its central importance in this work
we discuss the used fermion actions in greaterdetail in Chapter 6. Fromhere on one has to evaluate an integral
with nearly infinite degrees of freedom. This will be reduced to a Monte Carlo calculation and to the task to
create configurations which are good representatives for the physical vacuum. In Chapter 7, we explain this
calculation. From these configurations one can extract the physics of interest by evaluating expectation values
of operators, which have to have the correct symmetries. This is the analysis part of a lattice calculation and
we are dealing with that in Chapter 8. Once we have the data produced, it has to be brought in a shape such
that it can be compared with other results. Furthermore, one has to estimate the statistical and systematical
uncertainties and usually to extrapolate these results to physical regions afterwards. In Chapter 9, we will pick
up these issues.
InPartIII,weshowexampleswhichcanbecalculatedbyLQCDmethods. Wewillstartinachronologicalway
withthechirallyimprovedDiracoperator(CIDiracoperator). Herewemadeuseofthequenchedapproximation