Renormalization of three-quark operators for the nucleon distribution amplitude [Elektronische Ressource] / vorgelegt von Thomas Kaltenbrunner
216 Pages
English

Renormalization of three-quark operators for the nucleon distribution amplitude [Elektronische Ressource] / vorgelegt von Thomas Kaltenbrunner

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer

Description

Renormalization ofThree-Quark Operators for theNucleon Distribution AmplitudeDissertationzur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.)der Naturwissenschaftlichen Fakult¨at II - Physikder Universit¨at Regensburgvorgelegt vonThomas Kaltenbrunneraus RegensburgSeptember 2008Promotionsgesuch eingereicht am: 26.08.2008Die Arbeit wurde angeleitet von: Prof. Dr. Andreas Sch¨afer¨PRUFUNGSAUSSCHUSS:Vorsitzender: Prof. Dr. Josef Zweck1. Gutachter: Prof. Dr. Andreas Sch¨afer2. Gutachter: Prof. Dr. Vladmir Braunweiterer Pruf¨ er: Prof. Dr. Ingo MorgensternContentsPreface 51 A Phenomenological Introduction to QCD 71.1 Quarks, Baryons and Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 The Nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Elastic Scattering and Form Factors . . . . . . . . . . . . . . . . . . . . . . . 10Deep Inelastic Scattering and Structure Functions . . . . . . . . . . . . . . . 121.3 The Nucleon Distribution Amplitude . . . . . . . . . . . . . . . . . . . . . . . 17Reinvestigating the Elastic Form Factors . . . . . . . . . . . . . . . . . . . . . 17Introducing the Nucleon Distribution Amplitude . . . . . . . . . . . . . . . . 18An Ab Initio Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Continuum QCD 232.1 The Euclidean Action of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 232.

Subjects

Informations

Published by
Published 01 January 2009
Reads 14
Language English
Document size 3 MB

Renormalization of
Three-Quark Operators for the
Nucleon Distribution Amplitude
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der Naturwissenschaftlichen Fakult¨at II - Physik
der Universit¨at Regensburg
vorgelegt von
Thomas Kaltenbrunner
aus Regensburg
September 2008Promotionsgesuch eingereicht am: 26.08.2008
Die Arbeit wurde angeleitet von: Prof. Dr. Andreas Sch¨afer
¨PRUFUNGSAUSSCHUSS:
Vorsitzender: Prof. Dr. Josef Zweck
1. Gutachter: Prof. Dr. Andreas Sch¨afer
2. Gutachter: Prof. Dr. Vladmir Braun
weiterer Pruf¨ er: Prof. Dr. Ingo MorgensternContents
Preface 5
1 A Phenomenological Introduction to QCD 7
1.1 Quarks, Baryons and Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 The Nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Elastic Scattering and Form Factors . . . . . . . . . . . . . . . . . . . . . . . 10
Deep Inelastic Scattering and Structure Functions . . . . . . . . . . . . . . . 12
1.3 The Nucleon Distribution Amplitude . . . . . . . . . . . . . . . . . . . . . . . 17
Reinvestigating the Elastic Form Factors . . . . . . . . . . . . . . . . . . . . . 17
Introducing the Nucleon Distribution Amplitude . . . . . . . . . . . . . . . . 18
An Ab Initio Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Continuum QCD 23
2.1 The Euclidean Action of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Perturbation Theory in the Path Integral Approach . . . . . . . . . . . . . . 25
Introducing the Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
The Generating Functional and Free Propagators . . . . . . . . . . . . . . . . 25
Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Loop Divergences and Need for Regularization . . . . . . . . . . . . . . . . . 28
2.4 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Ren of the Action. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Ren of Composite Operators . . . . . . . . . . . . . . . . . . . . 36
Renormalization Group Equation and Running Coupling . . . . . . . . . . . . 37
3 Lattice QCD 39
3.1 Naive Discretization of the Free Action . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Introducing Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Wilson Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 The Gauge Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Order a Improved Wilson Fermions . . . . . . . . . . . . . . . . . . . . . . . . 43
3.6 The Generating Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7 (Hybrid) Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Performing the Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.9 Chiral Symmetry Breaking and Chiral Actions . . . . . . . . . . . . . . . . . 48
12 CONTENTS
4 Irreducible Multiplets of Three-Quark Operators 51
4.1 The Symmetry of the Hypercubic Lattice . . . . . . . . . . . . . . . . . . . . 52
The Hypercubic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
The Spinorial Hypercubic Group . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Construction of Irreducible Three-Quark Operators . . . . . . . . . . . . . . . 55
Irreducibility in SO and O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 4
Irreducibility in H(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Three-Quark Operators and Renormalization . . . . . . . . . . . . . . . . . . 58
4.4 Isospin Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Identities due to Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Operators without Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Operators with Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Consequences for Renormalization . . . . . . . . . . . . . . . . . . . . . . . . 64
5 The RI-MOM Renormalization Scheme 67
5.1 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
An Appropriate Matrix Element on the Lattice . . . . . . . . . . . . . . . . . 68
Relation to Calculable Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 69
The Three-Quark Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Setup of the RI-MOM Renormalization Scheme . . . . . . . . . . . . . . . . . 70
Continuum and Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
The Renormalization Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Definition of the Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
The Quark Field Renormalization . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Scheme Matching and RG Behavior 75
6.1 The Scheme Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
General One-Loop Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
MS←mRIDetermination of Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Renormalization Group Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 79
The Renormalization Group Equation . . . . . . . . . . . . . . . . . . . . . . 79
The Scaling Function ΔZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
The Anomalous Dimension of the Three-Quark Vertex . . . . . . . . . . . . . 81
6.3 Input from Continuum Perturbation Theory . . . . . . . . . . . . . . . . . . . 86
General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Operators without Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Operators with One Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Operators with Two Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 92
Details on the Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 The Results 105
7.1 Technical Details of the Lattice Calculation . . . . . . . . . . . . . . . . . . . 105
Fixed Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Available Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Implementation of the Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 107
Even-Odd Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Choice of the Quark Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . 108CONTENTS 3
Chiral Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2 Data Analysis and Error Estimation . . . . . . . . . . . . . . . . . . . . . . . 109
MSExtracting Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Renormalization Group Behavior and Estimation of Systematic Errors . . . . 111
Influence of the Chosen Quark Momenta . . . . . . . . . . . . . . . . . . . . . 115
MS7.3 Results for Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.4 Renormalization of Moments of the NDA . . . . . . . . . . . . . . . . . . . . 122
Relating Moments of the NDA to Three-Quark Operators . . . . . . . . . . . 122
The Zeroth Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
The Next-to-Leading Twist Constants λ and λ . . . . . . . . . . . . . . . . 1251 2
The Proton Decay Constants α and β . . . . . . . . . . . . . . . . . . . . . . 126
The First Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
The Second Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.5 The Nucleon Distribution Amplitude . . . . . . . . . . . . . . . . . . . . . . . 131
Advanced Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Discussion and Comparison with Other Approaches. . . . . . . . . . . . . . . 133
A Model for the Nucleon Distribution Amplitude . . . . . . . . . . . . . . . . 134
Beyond the Distribution Amplitude: An Outlook . . . . . . . . . . . . . . . . 136
8 Summary and Conclusion 141
Acknowledgements 143
A Conventions and Formulas for Perturbation Theory 145
A.1 The Weyl Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.2 Scheme Matching for the Quark Field Renormalization . . . . . . . . . . . . . 146
B Irreducibly Transforming Three-Quark Operators 147
B.1 Operators without Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.2 Operators with One Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.3 Operators with Two Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 151
C Isospin induced Identities 157
C.1 Operators without Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.2 Operators with One Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 158
C.3 Operators with Two Derivatives - Preparations . . . . . . . . . . . . . . . . . 160
C.4 Operators with Two Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 161
D The Renormalization Matrices 165
4
D.1 Operators without Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . . 1651
12
D.2 Operators without Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . . 1671
8D.3 Operators with One Derivative in τ . . . . . . . . . . . . . . . . . . . . . . . 168
12
D.4 Operators with One Derivative in τ . . . . . . . . . . . . . . . . . . . . . . 1691
12
D.5 Operators with One Derivative in τ . . . . . . . . . . . . . . . . . . . . . . 1712
4
D.6 Operators with Two Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . 1741
4
D.7 Operators with Two Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . 1802
8D.8 Operators with Two Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . 1844 CONTENTS
12
D.9 Operators with Two Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . 1871
12
D.10Operators with Two Derivatives in τ . . . . . . . . . . . . . . . . . . . . . . 1952Preface
Fascinated by nature scientists all along have tried to figure out the mechanisms that govern
the world around them. In the twentieth century four major theories were developed that
have been very successfull in describing the fundamental physical interactions on the smallest
and largest length scales. In the 1900s and 1910s, Einstein introduced his special and general
theories of relativity, which marked the major break-through in the description of gravita-
tion. He succeeded in explaining the interaction between masses and energies by means of
a revolutionary concept of space and time and thus provided a solid theoretical framework
that predicts the dominating long-distance interaction in our universe with highest accuracy.
However, the fundamental interactions between the elementary particles happen on tiny dis-
tancesandaregovernedbyaninherentlydifferentsetoftheories,whosemajorcommunityare
quantum effects. Quantum mechanics, developed in the 1920s and 1930s mainly by Heisen-
berg and Schr¨odinger, can be seen as a foundation for the three relativistic quantum field
theories that describe the electromagnetism, the weak interaction and the strong interaction.
Many amenities of the daily routine are based on the electromagnetism, whereas the weak
interaction is commonly known for the decay of radioactive atoms and governs processes on
much smaller length scales. The strong interaction finally reigns on even smaller distances
and keeps together the subatomic building-blocks of nature, such as neutrons and protons.
Searching for truly fundamental particles and trying to understand the mass spectrum of
the mid-1960s’ particle zoo, Zweig and Gell-Mann postulated the existence of quarks as ele-
mentary particles. These particles bear not only an electric charge, but also a so-called color
charge. In experiments quarks appear exclusively in color-neutral bound states, leading to a
systematic and very successfull explanation of the observed hadrons. Introducing gluons that
mediate the strong force and interact with the quarks, quantum chromodynamics emerged
as the underlying physical theory. In this non-abelian gauge theory the color-neutrality of
observableparticlesisknownasconfinementandisinherentlylinkedtothepropertyofasymp-
totic freedom: While at small energy scales all fundamental particles are tied closely together
by the strong force, they behave like quasi-free particles at very high energies.
Duetothisbehaviorquantumchromodynamics(QCD)canbetreatedperturbativelyatlarge
energy scales. Even though the non-abelian character of the theory leads to a self-interaction
of the gluons that makes the approach calculational demanding, an expansion in the strong
coupling constant results in a very precise description of experimental high-energy processes.
However, many other processes cannot be described by this perturbative approach, because
they also involve substantial contributions from the low energy regime. In order to access
these interesting and fundamentally non-perturbative aspects of quantum chromodynamics,
lattice QCD was developed in the 1970s. This method is based on Monte-Carlo integra-
tions in association with a statistical interpretation of QCD on a discretized four-dimensional
space-time lattice. On the one hand this method facilitates the exploration of perturbatively
56 PREFACE
inaccessible regions and thus allows to discover entirely new features of quantum chromody-
namics. On the other hand the approach is computationally very challenging so that CPU
time and computer architecture define the limiting factors for most lattice studies.
Likeinmostquantumfieldtheories, alsotheradiativecorrectionsinquantumchromodynam-
ics suffer at first sight from infinities. In order to extract finite physical information from the
theory, one first has to regularize the theory by introducing appropriate cutoffs in the mo-
mentum region. While this has to be done explicitly in the perturbative method, the lattice
approachprovidesinfraredandultravioletcutoffsimplicitlyduetothediscretizedspace-time.
In a second step one has to renormalize the theory, i.e., link the – in general in an arbitrary
way – regularized theory to physical observables and experiment. It was demonstrated by
’t Hooft and Veltman in 1972 that quantum chromodynamics is in fact a renormalizable
field theory. Therefore any at first sight occurring divergence can be rendered finite in a
well-defined way such that the theory is capable of predicting real physics.
The main focus of this thesis will be on the internal structure of nucleons. Our main
goal is to learn more about the momentum distribution of the three valence quarks inside
the nucleon. This information is encoded in the so-called nucleon distribution amplitude
(NDA), which is closely linked to the nucleon wave function and enters the description of any
exclusive scattering process at high energies that involves nucleons. The nucleon distribution
amplitude can be inferred from matrix elements of local three-quark operators, and a central
issue in our approach is the renormalization these operators. Once the NDA is renormalized,
the momentum fractions carried by each valence quark of definite spin and flavor can be
read off. This knowledge is essential for calculating amplitudes of exclusive processes at
high energies, like electron-proton scattering. Besides, also the scope of recent and future
particle accelerators promotes the interest in a better understanding of the internal structure
of nucleons.
The nucleon distribution amplitude is a purely non-perturbative quantity that is domi-
nated by soft contributions. Therefore it must be studied in the framework of lattice QCD.
The thesis is organized as follows: We start with a phenomenological discussion of scattering
experiments with nucleons and give some insight into the relation between form factors and
the NDA. After introducing the basics of continuum and lattice QCD we focus on the renor-
malizationpropertiesofthethree-quarkoperatorsfromwhichlowmomentsofthedistribution
amplitude can be calculated. In a first step we reduce the operator mixing under renormal-
ization. This is accomplished by constructing irreducible multiplets of three-quark operators
with respect to the spinorial hypercubic group H(4), which represents the space-time symme-
try of the lattice. After isospin-symmetrization identities between these operators are derived
and an independent subset of operators is chosen as a basis for the renormalization. Then
we introduce an RI-MOM like renormalization scheme that is applicable on the lattice and in
the continuum. It is this step that facilitates the main goal of this thesis, namely to convert
the lattice-regularized operators into operators that are renormalized in the MS continuum
scheme. Hence the following chapter is dedicated to this matching to the MS renormaliza-
tion scheme, whereby we first renormalize the operators in the RI-MOM like scheme on the
lattice and then derive a matching to the MS scheme in continuum perturbation theory. The
thus derived renormalization matrices for the three-quark operators represent the main result
and are finally applied to renormalize moments of the nucleon distribution amplitude. We
conclude by comparing this first rigorous determination of the low moments of the nucleon
distribution amplitude in the MS scheme with previous models and sum rule calculations.