Lattice quantum chromodynamics close to the light cone [Elektronische Ressource] / presented by Daniel Grünewald

Dissertationsubmitted to theCombined Faculties for the Natural Sciences and for Mathematicsof the Ruperto–Carola University of Heidelberg, Germanyfor the degree ofDoctor of Natural Sciencespresented by¨Dipl.-Phys. Daniel Grunewaldborn in Zell a.d. Mosel, GermanyOral examination: June 18, 2008LatticeQuantum Chromodynamicsclose to theLight ConeReferees: Prof. Dr. Hans-Ju¨rgen PirnerProf. Dr. Michael G. SchmidtQuantenchromodynamik auf dem Gitterin der N¨ahe des LichtkegelsZusammenfassungWir benutzen “Nahe dem Lichtkegel” Koordinaten um eine Gitter Formulierung von Yang-Mills The-orien a la Wilson einzufuhr¨ en die auf den Lichtkegel extrapoliert werden kann. Diese Art von For-mulierung ist predestiniert um nicht perturbative Hochenergiephysik wie z.B. Strukturfunktionenauf dem Gitter zu beschreiben. Die numerische Standardmethode der Gittereichtheorie, n¨amlichdas Monte Carlo Sampling des Euklidischen Pfad Integrals scheitert jedoch in dieser Formulierungan dem “sign” Problem genauso wie bei der numerischen Behandlung der Quantenchromodynamik(QCD) bei endlicher baryonischer Dichte. Dieses Problem kann in unserem Fall jedoch dadurch um-gangen werden, daß wir zu einer Hamilton‘schen Formulierung ub¨ ergehen.
Publié le : mardi 1 janvier 2008
Lecture(s) : 13
Tags :
Source : ARCHIV.UB.UNI-HEIDELBERG.DE/VOLLTEXTSERVER/VOLLTEXTE/2008/8601/PDF/DISSERTATION.PDF
Nombre de pages : 181
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Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto–Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
presented by
¨Dipl.-Phys. Daniel Grunewald
born in Zell a.d. Mosel, Germany
Oral examination: June 18, 2008Lattice
Quantum Chromodynamics
close to the
Light Cone
Referees: Prof. Dr. Hans-Ju¨rgen Pirner
Prof. Dr. Michael G. SchmidtQuantenchromodynamik auf dem Gitter
in der N¨ahe des Lichtkegels
Zusammenfassung
Wir benutzen “Nahe dem Lichtkegel” Koordinaten um eine Gitter Formulierung von Yang-Mills The-
orien a la Wilson einzufuhr¨ en die auf den Lichtkegel extrapoliert werden kann. Diese Art von For-
mulierung ist predestiniert um nicht perturbative Hochenergiephysik wie z.B. Strukturfunktionen
auf dem Gitter zu beschreiben. Die numerische Standardmethode der Gittereichtheorie, n¨amlich
das Monte Carlo Sampling des Euklidischen Pfad Integrals scheitert jedoch in dieser Formulierung
an dem “sign” Problem genauso wie bei der numerischen Behandlung der Quantenchromodynamik
(QCD) bei endlicher baryonischer Dichte. Dieses Problem kann in unserem Fall jedoch dadurch um-
gangen werden, daß wir zu einer Hamilton‘schen Formulierung ub¨ ergehen. Wir leiten einen effektiven
Hamilton Operator auf dem Gitter her, der die Dynamik der QCD im reinen Eichfeldsektor bestimmt
und der prinzipiell dazu benutzt werden kann, Operatorerwartungswerte im Grundzustand mittels
eines Quanten Diffusions Monte Carlo Algorithmus numerisch zu bestimmen. Wir bestimmen die
Grundzustands-Wellenfunktion analytisch im starken und im schwachen Kopplungs Limes. Diese bei-
denLimitesmotiviereneinenGrundzustands-Wellenfunktions-Ansatz,gegebendurchdasProduktvon
einzelnen Plaquette Wellenfunktionen, der in Bezug auf die Energie mittels dem Ritz‘schen Variation-
sprinzip optimiert wird und dann auf den Lichtkegel extrapoliert werden kann. Zusa¨tzlich dazu bes-
timmenwirdieGrundszustands-WellenfunktiondesHamilton OperatorsimKontinuumimLichtkegel
Limes. Wir entwickeln eine Methode die Gluonen Verteilungsfunktion von Mesonen, die durch ihre
Valenzquark Verteilungen modelliert sind, mittels der optimierten Grundzustands-Wellenfunktion im
Lichtkegellimes zu bestimmen.
Lattice Quantum Chromodynamics
close to the Light Cone
Abstract
We use near light cone coordinates in order to establish a Wilsonian lattice formulation of Yang-
Mills theories which can be extrapolated onto the light cone. Such a formulation is predestinated
for the description of non-perturbative high energy physics like structure functions on the lattice.
The numerical standard approach of lattice gauge theory namely the Monte Carlo sampling of the
Euclidean path integral fails because of a sign problem similar to Quantum Chromodynamics (QCD)
atfinitebaryonicdensity. However,wecancircumventthesignproblembyswitchingtoaHamiltonian
formulation. We develop an effective lattice Hamiltonian describing the dynamics of the pure gauge
sector of QCD which is in principle capable of determining ground state expectation values by means
of Quantum-Diffusion-Monte Carlo methods. We analytically compute the ground state in the weak
and strong coupling limit. These two analytical limits motivate a single plaquette ground state
ansatz valid over the whole coupling range which is optimized with respect to the energy by the Ritz
variational principle and which can be extrapolated onto the light cone. In addition, we compute the
continuum ground state wave functional of the near light cone Hamiltonian in the light cone limit.
We develop a method to determine gluon distribution functions of mesons modeled by their valence
quark distribution applying the optimized near light cone ground state wave functional in the light
cone limit.Contents
1 Introduction 1
2 Light cone field theory 7
2.1 Yang Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Light cone quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Transverse lattice method . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Near light cone coordinates . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Zero mode dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Lattice formulation 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Euclidean path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Hilbert space of SU(2) lattice gauge theory . . . . . . . . . . . 43
3.3.2 Transfer matrix construction of the Hamiltonian . . . . . . . . . 47
3.4 Longitudinal Momentum Operator . . . . . . . . . . . . . . . . . . . . 52
3.5 The effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 The nlc ground state on the lattice 61
4.1 Strong coupling solution . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Weak coupling solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Variational optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Quality of the effective Hamiltonian . . . . . . . . . . . . . . . . . . . . 85
5 Ground state in the light cone limit 87
6 Application: Gluon distribution functions 99
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Lattice description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
viiviii CONTENTS
6.3 Strong coupling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Summary and conclusions 115
A Notations and conventions 123
A.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.2 Minkowski Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.3 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B Color Algebra 125
B.1 The group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.2 Gauge transformation as rotation in color space . . . . . . . . . . . . . 128
C Local heat bath algorithm 133
D Hilbert space 135
D.1 Elementary commutation relations . . . . . . . . . . . . . . . . . . . . 135
D.2 Computation of expectation values . . . . . . . . . . . . . . . . . . . . 137
E Ground state calculations 141
E.1 Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
E.1.1 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . 141
E.1.2 Computation of the wave functional correlator . . . . . . . . . . 147
E.2 Alternative strong coupling . . . . . . . . . . . . . . . . . . . . . . . . 153
E.3 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
The bibliography 163Chapter 1
Introduction
Quantum Chromo Dynamics (QCD) is predominantly accepted to be the fundamental
theory of the strong interaction. It is a Lorentz invariant gauge theory with quarks
and gluons as fundamental degrees of freedom. In contrast to quantum electrodynam-
ics (QED), the gauge group of QCD is non-Abelian. As a result, the gauge bosons, i.e.
the gluons carry color charge themselves and can interact with each other. This yields
to asymptotic freedom of the QCD coupling constant [1]. Asymptotic freedom means,
thatthecouplingofquarksandgluonsislargeatlargedistancessoastoconfinequarks.
Atthesametimethecouplingispredictedtobesmallatshortdistancessothatquarks
behave as free particles at asymptotic energies. In this regime, perturbation theory is
applicable.
One of the great achievements of QCD is the successful description of the deviations
from the naive parton model seen in experiment. In the parton model the deep in-
elastic scattering (DIS) cross section is given by the incoherent sum of the scattering
off point-like constituents of the hadron. The existence of approximately point-like
constituents inside the hadrons was demonstrated by the classic electron DIS exper-
iments performed at SLAC [2, 3, 4]. The surprising result was that the measured
cross-sections did not fall off exponentially as the inelasticity of the reaction increased.
Instead, they had a scaling behavior, i.e. the structure functions became independent
2of the virtuality Q in the Bjorken limit in which the virtual photon energy ν and the
2virtuality tend to infinity at fixed Bjorken variable x = Q /(2Mν). Here M denotes
the target hadron mass. This was indicative for a point-like structure inside the target
nucleons and gave rise to the parton model. Today, we identify the partons with the
colouredquarksandgluons,i.e. thefundamentaldegreesoffreedomofQCD.Aplot[5]
of the electromagnetic structure function of the proton as a function of the virtuality
for different Bjorken x including the most recent DIS data for a fixed proton target
[6, 7, 8, 9] is shown in Fig. 1.1. Bjorken scaling is observed at high x, but is gradually
broken towards low x. These deviations can be explained by the QCD improved par-
ton model in which perturbation theory is applied to calculate corrections to the naive
12 Chapter 1. Introduction
ZEUS
x=6.32E-5
x=0.000102
ZEUS NLO QCD fit
x=0.000161
x=0.000253
tot. error
x=0.0004
x=0.0005
5 x=0.000632
x=0.0008
ZEUS 96/97
x=0.0013 BCDMS
E665
x=0.0021
NMC
4
x=0.0032
x=0.005
x=0.008
3
x=0.013
x=0.021
x=0.032
2
x=0.05
x=0.08
x=0.13
x=0.181
x=0.25
x=0.4
x=0.65
0
2 3 4 5
1 10 10 10 10 10
2 2Q (GeV )
Figure 1.1: The structure function F measured by ZEUS, H1 and various fixed target2
2experiments plotted against Q [5]. Bjorken scaling is observed at high x, but is
gradually broken towards low x. The line represents a QCD based fit to the data .
parton model predictions. In order to achieve that, one uses the QCD factorization
theorem (cf. [10] and references therein) which states that it is possible to separate the
non-perturbative long distance physics from the perturbative short distance physics in
the calculation of any physical cross section involving large momentum transfers. This
implies that the observable cross sections can be written in factorized form, namely as
a convolution of a hard partonic cross section and of the soft hadronic matrix elements
like the parton distribution functions for example. While the partonic cross section
is governed by the hard scale and can be calculated perturbatively, the parton dis-
tributions are of completely non-perturbative origin. Their most important property
is that they are independent of the particular hard process under consideration and
depend only on the hadron. Today, several collaborations exist which provide param-
eterizations of these distributions [11, 12, 13]. The parton distributions are typically
em
F -log (x)
2 10

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