Simulations of lattice fermions with chiral symmetry in quantum chromodynamics [Elektronische Ressource] / von Stanislav Shcheredin

Stanislav Shcheredin

Sujets

• Physik

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2004
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Simulations of Lattice Fermions with Chiral
Symmetry in Quantum Chromodynamics
D I S S E R T A T I O N
zur Erlangung des akademischen Grades
doctor rerum naturalium
(dr. rer. nat.)
im Fach Physik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultat¨ I
Humboldt-Universit¨at zu Berlin
von
Herr Dipl.-Phys. Stanislav Shcheredin
geboren am 17.12.1978 im Moskauer Gebiet
Pr¨asident der Humboldt-Universit¨at zu Berlin:
Prof. Dr. Jurg¨ en Mlynek
Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at I:
Prof. Thomas Buckhout, Ph.D.
Gutachter:
1. Prof. Dr. P. Damgaard
2. Prof. Dr. M. Muller-Preußk¨ er
3. Prof. Dr. P. Weisz
eingereicht am: 1. November 2004
Tag der mundlic¨ hen Prufung:¨ 17. Januar 2005Abstract
This thesis is dedicated to explore the feasibility of extraction of the low
energy constants of the chiral Lagrangian in the epsilon–regime of quenched
QCD. We apply two formulations of the Ginsparg-Wilson fermions, namely,
the Neuberger operator and the hypercube overlap operator to compute the
observables of interest. As a main result we present the comparison of the
distributions of the leading individual eigenvalues of the Neuberger operator
in QCD and the analytical predictions of chiral random matrix theory. We
observeagoodagreementaslongaseachsideofthephysicalvolumeexceeds
about 1.12 fm. At the same time the chiral condensate Sigma can also be
estimated. It turns out that this bound for L is generic and sets the size of
the physical volume where the axial correlator behaves according to chiral
perturbation theory. This allows us to compute a value for the pion decay
constant. The simulations also show that due to the high probability of the
near-zero modes it is prohibitively difficult to sample the axial correlator
in the neutral topological sector. In the higher sectors, however, we observe
that the sensitivity of the analytical predictions for the axial correlator to
extract Sigma is lost to a large extent. As an alternative procedure we only
considerthecontributionfromthezeromodes.Hereweareabletoobtainan
estimate for the pion decay constant and alpha, where alpha is a low energy
constant peculiar to quenching. We calculate the topological susceptibility,
both for the Neuberger operator and for the overlap hypercube operator.
It turns out that the result with the overlap hypercube operator is closer
to the continuum limit. Also the locality properties are superior to those of
the Neuberger fermions. As a theoretical development the Lusc¨ her topology
conserving gauge action is investigated. This enables us to sample the ob-
servables of interest in the epsilon–regime without recomputing the index.
Keywords:
lattice QCD, chiral perturbation theory, random matrix theory, epsilon
regimeZusammenfassung
Das Ziel dieser Dissertation besteht darin, die Realisierbarkeit der Berech-
nungen der Niederenergie-Konstanten der chiralen Lagrangedichte zur Ge-
winnungphysikalischerInformationenimepsilon–RegimederquenchedQCD
zu erforschen. Wir haben der Neuberger Operator und Overlap Hyperkubus
Operator eingesetzt. Ein Hauptergebniss dieser Arbeit ist der Vergleich der
Wahrscheinlichkeitsverteilungen einzelner Eigenwerte des Neuberger Opera-
tors in der QCD mit den analytischen Vorhersagen der Theorie der Zufalls-
¨matritzen.WirbeobachteneineguteUbereinstimmungsolangejedeSeitedes
physikalischen Volumens groߨ er als etwa 1.12 fm ist. Dabei kann auch das
chirale Kondensat Sigma abgeschatzt werden. Es ergab sich, daßdiese untere¨
Schranke von L allgemein gilt und die Gr¨oße des physikalischen Volumens,
auf dem der Axialkorrelator den Vorhersagen des chiralen St¨orungstheorie
folgt, festlegt. Damit koennen wir die Pionzerfallskonstante bestimmen. Un-
sere Simulationen zeigen, daßwegen der großen Wahrscheinlichkeit niedriger
EigenwertedieMessungdesAxialkorrelatorsimtopologischenneutralenSek-
tor extrem aufwandig¨ ist. Doch reicht die Empfindlichkeit der Vorhersagen
der chiralen Storungstheorie in hoheren topologischen Sektoren bei der ge-¨ ¨
gebenen Statistik nicht zur Bestimmung von Sigma aus. Als alternative Me-
thode,gehenwirdazuuber,alleindenBeitragderNullmodenzubetrachten.¨
Hier koennen wir Abschat¨ zungen fur¨ die Pionzerfallskonstante und alpha ge-
winnen. Wir berechnen die topologische Suszeptibilit¨at fur¨ den Neuberger
und Overlap Hyperkubus Operator. Im letzten Fall ist der berechnete Wert
naher¨ beim Kontinuumslimes. Die Lokalisierung fur¨ den Overlap Hyperku-
bus Operator ist auch besser als fur den Neuberger Operator. Unser anderes¨
Ziel ist die Erforschung einer topologieerhaltenden Eichwirkung.
Schlagworter:¨
Gitter QCD, chirale Storungstheorie, Theorie der Zufallsmatritzen, Epsilon-¨
RegimeAcknowledgments
First of all I would like to express my gratitude to Prof. Michael Muller-¨
Preußker for having made it possible for me to make my Ph.D. in his the-
oretical group. In particular I received a great scientific supervision as well
as the all possible support. I would also like to thank Dr. Karl Jansen for
letting me join the χLF collaboration and for many interesting ideas which
stimulated me as a scientist. Especially I am indebted to Dr. Wolfgang
Bietenholz who became my direct supervisor during my Ph.D. His careful
and kind supervision as well as the fascinating ideas filled me all these years
and contributed a great deal to my world view as well as to my scientific
education. Further I would like to acknowledge Prof. J. J. M. Verbaarschot,
Prof. P. Damgaard and Prof. G. Akemann for useful discussions concerning
the issues addressed in the thesis. I would also like to acknowledge all my
colleagues: Dr. F. Hofheinz, Dipl. Phys. A. Sternbeck and Dipl. Phys.
D. Peschka for creating a wonderful working atmosphere and helping me to
figure out a number of small things which occur in everyday life. And last
but not least I would like to thank my mother and grandmother for the kind
support.
iiiContents
1 Motivation 1
2 Theoretical background 5
2.1 QCD at first glance. Gluons and quarks . . . . . . . . . . . . 5
2.2 Chiral symmetry breaking and eigenvalues of the Dirac operator 7
2.2.1 Implications of the chiral symmetry for the structure
of the Dirac operator . . . . . . . . . . . . . . . . . . . 7
2.2.2 Spectral density of the Dirac operator. . . . . . . . . . 11
2.3 χPT as a low energy effective theory of QCD. . . . . . . . . . 12
2.3.1 The chiral Lagrangian and its low energy constants . . 12
2.3.2 The p– and –expansion of χPT . . . . . . . . . . . . . 15
2.3.3 Quenched χPT: first order expressions for the axial-
vector correlation function . . . . . . . . . . . . . . . . 18
2.3.4 Zero mode contributions to the pseudo-scalar correla-
tion function . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Chiral random matrix theory . . . . . . . . . . . . . . . . . . 25
2.4.1 Microscopic spectral properties . . . . . . . . . . . . . 29
2.4.2 Bulk eigenvalues. Unfolding of the spectra . . . . . . . 33
2.5 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1 The Wilson gauge action and the Wilson fermions . . . 34
2.5.2 Staggered fermions . . . . . . . . . . . . . . . . . . . . 37
2.5.3 TheGinsparg–WilsonrelationandtheNeubergerover-
lap operator . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.4 The hypercube Dirac operator . . . . . . . . . . . . . . 40
2.5.5 The overlap hypercube Dirac operator . . . . . . . . . 47
3 Lattice simulations 48
3.1 Quenched simulations of the gauge fields . . . . . . . . . . . . 48
3.2 A numerical treatment of the overlap Dirac operator . . . . . 49
4 Numerical treatment of the hypercube fermions 57
ivCONTENTS v
5 Results on the pion dispersion relation 63
6 Eigenvalue distributions of the overlap Dirac operator 67
6.1 Microscopic regime . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.1 Distributions of individual eigenvalues . . . . . . . . . 69
6.1.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Bulk eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 The unfolded spectrum . . . . . . . . . . . . . . . . . . 78
7 Topological susceptibility 80
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2 Results with overlap fermions . . . . . . . . . . . . . . . . . . 81
8 Mesonic two-point functions 85
8.1 The axial-vector correlator . . . . . . . . . . . . . . . . . . . . 85
8.2 Subtleties of numerical simulations in the –regime . . . . . . 88
8.3 Zero mode contributions to the pseudo-scalar correlator . . . . 91
9 Topology conserving Lusc¨ her gauge action 102
9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
9.2 Results on the topological histories . . . . . . . . . . . . . . . 104
10 Conclusions 109
A A local Hybrid Monte Carlo algorithm 124List of Figures
2.1 The distributions for the lowest eigenvalue at|ν| = 0, 1 and
2, as predicted by χRMT in Eq. (2.101). . . . . . . . . . . . . 32
2.2 1-space, 2-space and 3-space hyper-links. . . . . . . . . . . . 44
2 4 43.1 The condition number of the Q operator in V = 8 and 10
volumes at β = 5.85 a