Spectrum of N=1 super Yang Mills theory on the lattice with a light gluino [Elektronische Ressource] / vorgelegt von Roland Peetz

Spectrum of N=1 super Yang Mills theory on the lattice with a light gluino [Elektronische Ressource] / vorgelegt von Roland Peetz

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Roland PeetzSpectrum ofN=1 Super Yang Mills Theoryon the Lattice with aLight Gluino2003Theoretische PhysikSpectrum ofN=1 Super Yang Mills Theoryon the Lattice with aLight GluinoInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im Fachbereich Physikder Mathematisch-Naturwissenschaftlichen Fakult¨atder Westf¨alischen Wilhelms-Universit¨at Munster¨vorgelegt vonRoland Peetzaus Wrexham2003Dekan: Prof.Dr.H.ZachariasErster Gutachter: Prof.Dr.G.Munster¨Zweiter Gutachter: Prof.Dr.M.StinglTag der mundlic¨ hen Prufungen:¨ 30.1.2004; 3.2.2004Tag der Promotion: 3.2.2004Contents1 Introduction 11.1 Why Supersymmetry? . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Why on the Lattice? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 The Physics of N=1 Super Yang Mills Theory 62.1 N=1 SYM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Non-Zero Gluino Mass . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Low Energy Effective Action . . . . . . . . . . . . . . . . . . . . . . . . 123 The Numerics of N=1 Super-Yang-Mills Theory 143.1 Simulating SUSY on the Lattice . . . . . . . . . . . . . . . . . . . . . . 143.1.1 The SU(2), N =1 SYM Lattice Action . . . . . . . . . . . . . . 153.1.2 The TSMB Algorithm . . . . . . . . . . . . . . . . . . . . . . . 183.2 Measuring Observables . . . . . . . . . . . . . . .

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Roland Peetz
Spectrum of
N=1 Super Yang Mills Theory
on the Lattice with a
Light Gluino
2003Theoretische Physik
Spectrum of
N=1 Super Yang Mills Theory
on the Lattice with a
Light Gluino
Inaugural-Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften im Fachbereich Physik
der Mathematisch-Naturwissenschaftlichen Fakult¨at
der Westf¨alischen Wilhelms-Universit¨at Munster¨
vorgelegt von
Roland Peetz
aus Wrexham
2003Dekan: Prof.Dr.H.Zacharias
Erster Gutachter: Prof.Dr.G.Munster¨
Zweiter Gutachter: Prof.Dr.M.Stingl
Tag der mundlic¨ hen Prufungen:¨ 30.1.2004; 3.2.2004
Tag der Promotion: 3.2.2004Contents
1 Introduction 1
1.1 Why Supersymmetry? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why on the Lattice? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The Physics of N=1 Super Yang Mills Theory 6
2.1 N=1 SYM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Non-Zero Gluino Mass . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Low Energy Effective Action . . . . . . . . . . . . . . . . . . . . . . . . 12
3 The Numerics of N=1 Super-Yang-Mills Theory 14
3.1 Simulating SUSY on the Lattice . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 The SU(2), N =1 SYM Lattice Action . . . . . . . . . . . . . . 15
3.1.2 The TSMB Algorithm . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Measuring Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Autocorrelation Times . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Jackknife Error Estimation . . . . . . . . . . . . . . . . . . . . . 24
3.3 String Tension σ and Sommer Scale R . . . . . . . . . . . . . . . . . . 250CONTENTS II
3.4 Obtaining Masses of States . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Fermionic Correlators . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Gluino-Glue Bound States . . . . . . . . . . . . . . . . . . . . . 31
3.4.4 Adjoint Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Matrix Inversion Methods . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1 Volume Source Technique . . . . . . . . . . . . . . . . . . . . . 34
3.5.2 Stochastic Estimators . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 The Massless Gluino Limit . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Analysis and Results 45
4.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 SET Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 VST Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Static Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.1 Details of Mass Fits . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.2 Gluino-Glue Bound States . . . . . . . . . . . . . . . . . . . . . 57
4.4.3 Adjoint Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.4 Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83CONTENTS III
4.6 The Massless Gluino Limit . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6.1 Critical Hopping Parameter from Ward Identities . . . . . . . . 86
4.6.2 Critical Hopping Parameter from OZI Arguments . . . . . . . . 87
4.7 Finite Volume Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.8 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Conclusions and Outlook 96
A Conventions and Methods 99
A.1 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.2 Majorana Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.3 Smearing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 Combining Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B Overview of Spectrum Results 103
References 104Chapter 1
Introduction
In this thesis we study the N = 1 supersymmetric Yang Mills theory on the lattice.
Supersymmetryextends the symmetries of the Standard Model of elementaryparticles
to include a symmetry between fermions and bosons.
This introductory chapter will first make the case of why one should study supersym-
metrictheoriesatall,despitethefactthattothisdatetheexistenceofsupersymmetric
partnersoftheusualparticleshasnotbeenverifiedinacceleratorexperiments. Wewill
then outline the reasons for a lattice investigation. Finally we will give an overview of
the structure of this study.
1.1 Why Supersymmetry?
There is a great amount of interest in supersymmetric theories. The reasons for the
apparent mismatch to its experimental status are numerous.
To begin with, it has to be clearly stated that the Standard Model (SM) currently
faces only two significant challenges from experiment. One, somewhat unrelated to
supersymmetry (SUSY), is the existence of dark matter. Up to now there is no clear
way how to include any such, possibly only gravitationally interacting fields, in the
Standard Model. The second, more exciting experimental challenge in the context
of SUSY are the by now famous results from the experiment E821 at the Brookhaven
NationalLaboratorymeasuringtheanomalousmagneticmomentofthemuon. Thereis
adiscrepancybetweenpredictionfromtheory,i.e.theSM,andtheexperimentalvalue.
However, there still remain uncertainties in the corresponding theoretical calculation,
which draws on other experimentally determined input parameters. On some of these
theaccuracyisnotyetsufficient. Sothejurystillhastodecide,whethertheSMfalsely
predicts the outcome of that experiment. Therefore one cannot argue the failure of the