The static quark potential and scaling behavior of SU(3) lattice Yang-Mills theory [Elektronische Ressource] / von Silvia Necco

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• Physik
• Astronomie

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2003
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

The static quark potential and scaling behavior of
SU(3) lattice Yang-Mills theory
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–Universitat zu Berlin¨
von
Frau Dipl.–Phys. Silvia Necco
geboren am 26.05.1974 in Torino, Italien
Prasident der Humboldt-Universitat zu Berlin:¨ ¨
Prof. Dr. Ju¨rgen Mlynek
Dekan der Mathematisch–Naturwissenschaftlichen Fakultat I:¨
Prof. Dr. Michael Linscheid
Gutachter:
1. Prof. Dr. Volkard Linke
2. Dr. Rainer Sommer
3. Prof. Dr. Ulrich Wolff
eingereicht am: 27. Februar 2003
Tag der mundlichen Prufung: 15. Mai 2003¨ ¨2Contents
Introduction 1
1 Lattice gauge fields 9
1.1 Wilson action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Continuum limit and improvement . . . . . . . . . . . . . . . . . . . 11
1.2.1 The Symanzik program and the Lu¨scher-Weisz action . . . . . 12
1.3 Renormalization group improved actions . . . . . . . . . . . . . . . . 13
1.3.1 RG gauge improved actions in 2-parameter space . . . . . . . 16
1.3.2 FP action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 The static quark potential 19
2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The static quark potential in perturbation theory . . . . . . . . . . . 21
2.3 Quark confinement and the bosonic string picture . . . . . . . . . . . 28
2.4 Evaluation of the potential on the lattice . . . . . . . . . . . . . . . . 29
2.4.1 Setting the scale in Yang-Mills lattice theories . . . . . . . . . 30
2.4.2 Definition of the force . . . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Computation of the potential and force . . . . . . . . . . . . . 32
2.4.4 Analysis details . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.6 The ratio r /r and parametrization of r /a . . . . . . . . . . 36c 0 0
2.5 Continuum force and potential . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Comparison with perturbation theory . . . . . . . . . . . . . . . . . . 42
2.7 Comparison with the bosonic string model . . . . . . . . . . . . . . . 48
3 Scaling properties of RG actions: the critical temperature 51
3.1 The critical temperature T . . . . . . . . . . . . . . . . . . . . . . . 52c
3.2 Evaluation of r /a for RG actions . . . . . . . . . . . . . . . . . . . . 550
3.2.1 Effective potential . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 The force at tree level . . . . . . . . . . . . . . . . . . . . . . 57
3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.4 Parametrization of r /a . . . . . . . . . . . . . . . . . . . . . 600
3.2.5 Scaling of T r . . . . . . . . . . . . . . . . . . . . . . . . . . . 61c 0
3.3 Scaling of α (μ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63qq
iii CONTENTS
4 Scaling properties of RG actions: the glueball masses 67
4.1 Glueball states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Irreducible representations of the cubic group on Wilson loops . . . . 70
4.3 Wave functions of glueball operators . . . . . . . . . . . . . . . . . . 71
4.4 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Analysis details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Conclusions 89
Appendix 93
A Numerical results 93
A.1 Wilson action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.2 RG actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B Transfer matrix and Hamiltonian formalism 99
B.1 Wilson action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B.2 Improved actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2.1 Unphysical poles in the propagator . . . . . . . . . . . . . . . 103
C 3-dimensional lattice propagator in coordinate space 107
C.1 Lattice 3-dimensional propagator. . . . . . . . . . . . . . . . . . . . . 107
C.2 Recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
C.3 Numerical computation of G(~x) . . . . . . . . . . . . . . . . . . . . . 109
Acknowledgments 117List of Figures
1.1 Paths contributing to the Lu¨scher-Weisz action . . . . . . . . . . . . 13
2.1 A Wilson loopC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20r,t
2.2 Running coupling obtained by integration of the RG with truncation
of the β-functions at 2- and 3-loop. . . . . . . . . . . . . . . . . . . . 27
2.3 The potential evaluated at r∼r /2 for different couplings as a func-0
tion of exp(−t Δ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 35min
2.4 The ratio r /r for 5.95≤β ≤ 6.57 including the continuum extrap-c 0
olation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Interpolation of r /a. . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
22.6 Continuum extrapolation of r F(xr ). . . . . . . . . . . . . . . . . . . 40cc
2.7 The force in the continuum limit and for finite resolution, where the
discretization errors are estimated to be smaller than the statistical
errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Continuum extrapolation of V (r). . . . . . . . . . . . . . . . . . . . . 41I
2.9 The static potential in the continuum limit . . . . . . . . . . . . . . . 43
2.10 Running coupling in the qq scheme. . . . . . . . . . . . . . . . . . . . 44
2.11 Perturbative matching between the qq and the SF-scheme. . . . . . . 46
2.12 The potential compared to different perturbative expressions. . . . . . 47
2.13 The potential compared with the parameter-free prediction of the
bosonic string model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.14 Theforcecomparedwiththeparameter-freepredictionofthebosonic
string model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.15 The force compared with the bosonic string model at large distances. 50
√
23.1 T / σ as function of 1/N for different actions. . . . . . . . . . . . . 54c t
3.2 The effective potential as function of t for r ≈ r for the Wilson0
plaquette action and for the RG-improved Iwasaki and DBW2 action. 56
3.3 The force at tree level for several actions. . . . . . . . . . . . . . . . . 58
3.4 Parametrization of r /a for the Iwasaki action. . . . . . . . . . . . . . 600
3.5 Parametrization of r /a for the DBW2 action. . . . . . . . . . . . . . 610
3.6 T r for different actions. . . . . . . . . . . . . . . . . . . . . . . . . . 62c 0
3.7 Continuum extrapolation of T r for the Iwasaki and Wilson action. . 63c 0
3.8 α at finite lattice spacing for Iwasaki and DBW2 action comparedqq
with the continuum result. . . . . . . . . . . . . . . . . . . . . . . . . 65
3.9 α at finite lattice spacing for Iwasaki and DBW2 action comparedqq
with the continuum result in the short distance region. . . . . . . . . 65
iiiiv LIST OF FIGURES
4.1 Wilson loops un to length 8. . . . . . . . . . . . . . . . . . . . . . . . 72
++4.2 The effective masses for the A channel, evaluated with Iwasaki1
action, at different lattice spacings. . . . . . . . . . . . . . . . . . . . 79
++4.3 The effective masses for the A channel, evaluated with DBW2 ac-1
tion, at different lattice spacings. . . . . . . . . . . . . . . . . . . . . 80
++4.4 The effective masses for the T channel, evaluated with Iwasaki2
action, at different lattice spacings. . . . . . . . . . . . . . . . . . . . 80
++4.5 The effective masses for the T channel, evaluated with DBW2 ac-2
tion, at different lattice spacings. . . . . . . . . . . . . . . . . . . . . 81
++ 24.6 The 0 glueball mass normalized with r as function of (a/r ) for0 0
different actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
++ 24.7 The 2 glueball mass normalized with r as function of (a/r ) for0 0
different actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.8 ThephasediagramofSU(3)gaugetheoriesinthefundamental-adjoint
coupling space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
++4.9 Plotforthevalueofmassofthe0 glueballstatefromquencheddata
and for the lightest scalar meson with N = 2 flavours of dynamicalf
quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
ˆB.1 The curves defined by the condition Im(w) = 0 in the k = 0 plane.3
Inside the curve Im(w)=0, while outside Im(w)=0. . . . . . . . . . 105
B.2 Distribution of the poles for different actions, scanning the Brillouin
zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6List of Tables
2.1 Coefficients e in b for several schemes . . . . . . . . . . . . . . . . . 24i 2
2.2 Matching coefficients for different schemes. . . . . . . . . . . . . . . . 24
2.3 Ratio of Λ-parameters and 2-loop coefficient of the β-function for
various schemes for N =0. . . . . . . . . . . . . . . . . . . . . . . . 26f
02.4 2-loop coefficients for s =s =Λ /Λ and N =0. . . . . . . . . . . 270 S S f
2.5 Simulation parameters . . .