Nucleon wave function from lattice QCD [Elektronische Ressource] / vorgelegt von Nikolaus Warkentin

Nikolaus Warkentin

Sujets

• Physik

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2008
• Nombre de lectures: 13
• Langue: English
• Poids de l'ouvrage: 3 Mo

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Nucleon Wave Function from Lattice QCD
DISSERTATION
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat)
der naturwissenschaftlichen Fakultät II - Physik
der Universität Regensburg
vorgelegt von
Nikolaus Warkentin
aus Regensburg
April 2008Promotionsgesuch eingereicht am 22. April 2008
Promotionskolloquium am 28. Mai 2008
Die Arbeit wurde angeleitet von Prof. Dr. Andreas Schäfer
Prüfungsauschuß:
Vorsitzender: Prof. Dr. Ch. Back
1. Gutachter: Prof. Dr. A. Schäfer
2. Prof. Dr. V. Braun
Weiterer Prüfer: Prof. Dr. I. MorgensternABSTRACT
In this work we develop a systematic approach to calculate moments of leading-
twist and next-to-leading twist baryon distribution amplitudes within lattice QCD.
Using two flavours of dynamical clover fermions we determine low moments of
nucleon distribution amplitudes as well as constants relevant for proton decay cal-
culations in grand unified theories. The deviations of the leading-twist nucleon
distribution amplitude from its asymptotic form, which we obtain, are less pro-
nounced than sometimes claimed in the literature. The results are applied within
the light cone sum rule approach to calculate nucleon form factors that are com-
pared with recent experimental data.Contents
1 The Global Frame 4
1.1 Standard Model ... . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 ... and a Glimpse Beyond . . . . . . . . . . . . . . . . . . . . . . 8
2 Continuum QCD 11
2.1 Non-Abelian Gauge Theories . . . . . . . . . . . . . . . . . . . . 12
2.2 The Theory of Strong Interaction . . . . . . . . . . . . . . . . . . 14
2.3 QCD Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Asymptotic Freedom & Confinement . . . . . . . . . . . 17
2.3.2 QCD Scale and the Origin of Hadron Masses. . . . . . . . 19
2.3.3 Nucleon Form Factors . . . . . . . . . . . . . . . . . . . 20
2.4 Operator Product Expansion . . . . . . . . . . . . . . . . . . . . 22
2.5 Distribution Amplitudes . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 In a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.2 Leading-Twist Nucleon Distribution Amplitudes . . . . . 28
2.5.3 Moments of Leading-Twist Distribution . . . 32
2.5.4 Modelling thewist NDA . . . . . . . . . . . . 34
2.5.5 Moments of NLTW Nucleon Distribution Amplitudes . . 36
2.6 Detour to Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . 37
2.6.1 The Axial Anomaly . . . . . . . . . . . . . . . . . . . . . 38
2.6.2 Spontaneous Chiral Symmetry Breaking . . . . . . . . . . 39
2.6.3 Low-Energy Effective Theory . . . . . . . . . . . . . . . 40
2.7 GUT Decay Constants . . . . . . . . . . . . . . . . . . . . . . . 40
1CONTENTS
3 Lattice QCD 43
3.1 Path Integral and Correlation Functions . . . . . . . . . . . . . . 44
3.2 Two-Point Correlation Functions . . . . . . . . . . . . . . . . . . 45
3.3 Euclidisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Lattice QCD Action . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.1 Gauge Action . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Fermion Action . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . 50
3.5.2 The Green’s Function . . . . . . . . . . . . . . . . . . . . 52
3.5.3 The APEmille Machine . . . . . . . . . . . . . . . . . . . 53
3.6 Two-Point Correlators on the Lattice . . . . . . . . . . . . . . . . 54
3.7 The Transfer Matrix on the Lattice . . . . . . . . . . . . . . . . . 56
3.8 Operator Overlap Improvement . . . . . . . . . . . . . . . . . . . 57
3.9 Setting the Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.10 Operator Choice on the Lattice . . . . . . . . . . . . . . . . . . . 61
3.11 Details of the Lattice Calculation . . . . . . . . . . . . . . . . . 64
3.11.1 Common Properties . . . . . . . . . . . . . . . . . . . . 64
3.12 Moments of Distribution Amplitudes . . . . . . . . . . . . . . . . 65
3.12.1 Leading Twist . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Renormalisation 69
5 Main Results 72
5.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Unconstrained Analysis . . . . . . . . . . . . . . . . . . . . . . 76
5.2.1 Normalisation constants . . . . . . . . . . . . . . . . . . 76
5.2.2 Higher Moments . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Constrained Analysis of Higher Moments . . . . . . . . . . . . . 82
5.3.1 Partially Constrained Analysis . . . . . . . . . . . . . . . 82
5.3.2 Fully . . . . . . . . . . . . . . . . 85
5.3.3 Modelling the Nucleon Distribution Amplitude . . . . . . 87
5.4 Phenomenological Results . . . . . . . . . . . . . . . . . . . . . 89
5.4.1 Comparison to Other Estimates . . . . . . . . . . . . . . 89
5.4.2 Light Cone Sum Rule Results . . . . . . . . . . . . . . . 89
6 Discussions and Outlook 94
A Definitions and Relations 98
A.1 Weyl representation . . . . . . . . . . . . . . . . . . . . . . . . 98
A.2 Operator Relations . . . . . . . . . . . . . . . . . . . . . . . . . 99
2CONTENTS
B Lattice Setup 101
C Raw Lattice Results 103
Acknowledgements 112
Bibliography 124
3CHAPTER 1
The Global Frame
Quantum field theories are the state-of-the-art in modern physics. The develop-
ment of quantum mechanics and the aim to include properties of fields in this
framework resulted finally in the formulation of the first quantum field theory, the
quantum electrodynamics. This theory demonstrates the successful unification of mechanics and electrodynamics allowing highly precise calculations of
matter properties at the atomic scale. Many effects, like anomalous magnetic mo-
ment of the electron, the Lamb shift of the energy levels of hydrogen, could be
predicted and are tested to a precision, which can only rarely be reached within
physics. Quantum electrodynamics was not only the first physical relevant quan-
tum field theory, it served also as a prototype for other quantum field theories.
Although it seems that quantum electrodynamics is driven to its limits, we are
still detecting new properties and effects within this theory, like in the field of
cavity-quantum electrodynamics.
From the theoretical point of view, the next step was to describe not only the
electromagnetic force by a quantum field theory but also the other fundamental
forces which act at nucleonic scale, namely the weak and the strong interaction.
Up to now, only the gravitation resists to be formulated as a quantum field theory.
The present knowledge of the interplay and some partial connections between
the different quantum field theories is condensed in the standard model of particle
physics. It is the essence of what is known by the physicists about the fundamental
forces in the nature up to our day.
Therefore the aim of todays and tomorrows experiments is a better understand-
ing of the complete standard model and, may be even more important, the search
for new physics to answer the unresolved secrets of nature. Hence it is not only
4THEGLOBALFRAME
important to understand each known force separately but also the interplay and
the hidden connections of the different sectors of the standard model are of the
key importance for the future. In quantum mechanics probably the most impor-
tant breakthrough was achieved by calculating the different properties of the most
simplest object, the hydrogen wave function. Within the standard model we have
still not reached the point to be able to calculate the wave functions of the most
simple objects, the hadrons like mesons and baryons. As the hadrons are built up
from more elementary particles which interact through weak and strong forces,
the calculation would involve obviously both of these forces. However, as the
name may implicate, the weak force is less important in this cases and is not taken
into account within this work .
The theory of the strong interaction is Quantum Chromodynamics (QCD)
which will be the basis of the calculations in this work. However, as already
pointed out, QCD cannot be studied isolated but the connections to other parts of
the standard model are also of crucial importance. Any prediction and also any
description of tomorrows and todays experiments involves all parts of the standard
model. Thus, to approve or to falsify the standard model we need highly precise
theoretical descriptions of all ingredients in standard model. The understanding
of the nucleon properties is of particular importance for experiments. To inspect
the nature at the femtometer scales we need microscopes with very high resolu-
tion. Thus we need very high energies which are at the moment only reachable
if we use nucleons as probe. But as long as there is a lack of the true theoretical
understanding of the nucleon properties, all experimental results and
predictions are limited by our present knowledge. Thus in full analogy to the
hydrogen wave function, we would like to have an analogous description of the
nucleon. The knwoledge of the full nucleon wave function would be an enormous
improvement, but the calculation of that seems to be almost impossible due to the
intricacy of the quantum chromodynamics. However, as long as we can not ac-
cess the full nucleon wave