Thermal dilepton rates from quenched lattice QCD [Elektronische Ressource] : a study of thermal spectral functions in the continuum limit of quenched lattice QCD, at vanishing and finite momentum / Anthony Francis. Fakultät für Physik - Elementary Particles and Quantum Fields
173 Pages
English

Thermal dilepton rates from quenched lattice QCD [Elektronische Ressource] : a study of thermal spectral functions in the continuum limit of quenched lattice QCD, at vanishing and finite momentum / Anthony Francis. Fakultät für Physik - Elementary Particles and Quantum Fields

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PhD-Thesis in Theoretical PhysicsThermal Dilepton Rates fromQuenched Lattice QCDA study of thermal spectral functions in the continuumlimit of quenched lattice QCD, at vanishing and finitemomentumDissertationzur Erlangung des Doktorgradesan der Fakula¨t fu¨r Physikder Universita¨t Bielefeldvorgelegt vonAnthony FrancisBielefeldSeptember 2011Thermal Dilepton Rates from Quenched Lattice QCDA study of thermal spectral functions in the continuum limit of quenchedlattice QCD, at vanishing and finite momentumAbstract: We study light valence quark Euclidean correlation functions in finite temperature latticeQCD. The calculations have been performed in quenched lattice QCD at T ' 1.45T for four values ofc3the lattice cut-off on lattices up to size 128 48. This allows to perform a continuum extrapolationof the correlation function in the Euclidean time interval 0.2 τT 0.5 to better than 1% accuracy.3Additionally we study the temperature dependence of our results on 128 40, 32 and 16 lattices cor-responding to the temperatures T ' 1.2, 1.45 and 3.0T , as well as the momentum dependence in thec3vector case at T ' 1.45T on a lattice sized 128 48. Subsequently we compute the first two non-cvanishing thermal moments of the vector and pseudo scalar meson spectral functions on all lattices.

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Published 01 January 2011
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PhD-Thesis in Theoretical Physics
Thermal Dilepton Rates from
Quenched Lattice QCD
A study of thermal spectral functions in the continuum
limit of quenched lattice QCD, at vanishing and finite
momentum
Dissertation
zur Erlangung des Doktorgrades
an der Fakula¨t fu¨r Physik
der Universita¨t Bielefeld
vorgelegt von
Anthony Francis
Bielefeld
September 2011Thermal Dilepton Rates from Quenched Lattice QCD
A study of thermal spectral functions in the continuum limit of quenched
lattice QCD, at vanishing and finite momentum
Abstract: We study light valence quark Euclidean correlation functions in finite temperature lattice
QCD. The calculations have been performed in quenched lattice QCD at T ' 1.45T for four values ofc
3the lattice cut-off on lattices up to size 128 48. This allows to perform a continuum extrapolation
of the correlation function in the Euclidean time interval 0.2 τT 0.5 to better than 1% accuracy.
3Additionally we study the temperature dependence of our results on 128 40, 32 and 16 lattices cor-
responding to the temperatures T ' 1.2, 1.45 and 3.0T , as well as the momentum dependence in thec
3vector case at T ' 1.45T on a lattice sized 128 48. Subsequently we compute the first two non-c
vanishing thermal moments of the vector and pseudo scalar meson spectral functions on all lattices.
Using the constraints gained byour data analysis, wethen proceed to extract information on the spectral
representation of the vector correlator and discuss resulting consequences for the electrical conductivity
and the thermal dilepton rate in the plasma phase.
Finally we discuss the spectral function of the pseudo scalar and examine renormalization errors using
the degeneracy conditions of the symmetry restored plasma phase of QCD.
Dissertation zur Erlangung des Doktorgrades.
Vorgelegt der: Universit¨at Bielefeld,
Fakulta¨t fu¨r Physik,
Theoretische Physik,
Universit¨atsstraße 25,
D-33615 Bielefeld.
Datum der Abgabe: 14. September, 2011
Betreuung und Begutachtung durch:
Prof. Dr. Edwin Laermann
Prof. Dr. Frithjof Karsch
Author: Anthony Sebastian Francis
Geboren: 26.08.1982
E-mail Addresse: afrancis@physik.uni-bielefeld.de
Gedruckt auf alterungsbesta¨ndigem Papier nach DIN-ISO 9706Contents
Preface xiii
1 Introduction and Motivation 1
1.1 Quantum Chromodynamics and Yang-Mills Theory . . . . . . . . . . . . . 2
1.2 Heavy-Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Production of Photons and Dileptons in Heavy-Ion Collisions . . . 9
1.2.2 Status of Experimental Dilepton Production Data . . . . . . . . . 11
2 Foundations of Lattice Quantum Field Theory 13
2.1 Lattice Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Lattice Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 SU(3) Pure Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Quenched QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Fermion Discretization in Lattice QCD . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Wilson-Clover Fermions . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Connecting to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Parameters and Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 The Phenomenology of Spectral Functions 41
3.1 Spectral Functions in Non-Interacting Theory . . . . . . . . . . . . . . . . 45
3.1.1 Free Continuum Spectral Functions . . . . . . . . . . . . . . . . . 45
3.1.2 Properties of the Free Continuum Spectral Function . . . . . . . . 51
3.1.3 Free Discretized Spectral Functions . . . . . . . . . . . . . . . . . . 56
3.2 Expectations for Interacting Theory . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 Linear Response and Transport Coefficients . . . . . . . . . . . . . 63
3.2.2 Heavy Quark Diffusion from the Langevin Equation . . . . . . . . 65
3.2.3 Light Quarks in a Boltzmann Gas . . . . . . . . . . . . . . . . . . 68
3.3 Hard Thermal Loops and Alternative Approaches . . . . . . . . . . . . . . 71
3.3.1 Dileptons from Hard Thermal Loops . . . . . . . . . . . . . . . . . 71
3.3.2 Spectral Functions from AdS=CFT Correspondence . . . . . . . . 73
4 Lattice Methodology 75
4.1 Lattice Correlation Functions and the Kernel . . . . . . . . . . . . . . . . 76
4.2 Thermal Moments of the Correlation Function . . . . . . . . . . . . . . . 78
vContents
4.3 An Ill Posed Problem and its Bayesian Solution . . . . . . . . . . . . . . . 81
4.3.1 Maximum Entropy Method . . . . . . . . . . . . . . . . . . . . . . 82
4.3.2 Remarks on the Bayesian solution . . . . . . . . . . . . . . . . . . 84
4.4 An Alternative Route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 The Vector Spectral Function at Vanishing Momentum 89
5.1 The Vector SPF and Dilepton Rate in the Continuum at T'1:45T . . . 90c
5.1.1 The Spectral Function of the Time-Time Vector Correlator . . . . 91
5.1.2 Spatial and Full Vector Correlation Functions . . . . . . . . . . . . 93
5.1.3 Continuum Extrapolation of the Vector Correlation Function . . . 96
5.1.4 Computation of Thermal Moments . . . . . . . . . . . . . . . . . . 97
5.1.5 Analyzing the Vector Correlator and computing the SPF . . . . . 100
5.1.6 The Thermal Dilepton Rate and Electrical Conductivity . . . . . . 109
5.2 Temperature Dependence of the Vector SPF on the Lattice . . . . . . . . 111
5.2.1 Temperature Evolution of the Thermal Moments . . . . . . . . . . 114
5.2.2 Consequences for the Spectral Functions . . . . . . . . . . . . . . . 116
6 The Vector Spectral Function at Finite Momentum 119
6.1 Analyzing the Vector Correlation Function at Finite Momentum . . . . . 121
6.1.1 Thermal Moments of the Finite Momentum Correlators . . . . . . 123
6.1.2 Toy Models of the Correlation Function . . . . . . . . . . . . . . . 125
6.2 The Time-Time Vector Channel and its Thermal Moments . . . . . . . . 130
6.2.1 On a Non-Zero Intercept in the Longitudinal Channel . . . . . . . 133
6.3 Consequences for the Spectral Functions at Finite Momentum . . . . . . . 135
7 Notes on the Pseudo Scalar and Other Spectral Functions 137
7.1 The Pseudo Scalar Correlator and its Thermal Moments . . . . . . . . . . 137
7.1.1 The Correlator Ratio at T'1:45T . . . . . . . . . . . . . . . . . 138c
7.1.2 The Continuum Extrapolation . . . . . . . . . . . . . . . . . . . . 140
7.1.3 Thermal Moments of the Pseudo Scalar Correlator . . . . . . . . . 140
7.1.4 MEM analysis of the Pseudo Scalar Channel . . . . . . . . . . . . 141
7.2 Midpoints of the Current Correlators at Finite Temperature . . . . . . . . 143
Summary and Concluding Remarks 147
Bibliography 151
Acknowledgements 157
Eigenst¨andigkeitserkl¨arung 159
viList of Tables
2.1 Tableofrenormalizationconstantsfromnon-perturbative(NP)andtadpole-
improved perturbative (TI) calculations in the case of vanishing mass at
scale =1=a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Overview of basic calculation parameters. . . . . . . . . . . . . . . . . . . 37
2.3 The AWI, RGI quark masses and their values in MS=T. . . . . . . . . . . 38
2.4 Table of quark masses in the MS-scheme in units of temperature at T'
1:45T and in [MeV]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39c
32.5 Number of configurations analyzed on lattices sized N N . . . . . . . . 39
(1) (2)
3.1 The trace operation yields the channel specific constants a , a and inH H
(3)
case of non-vanishing mass a . . . . . . . . . . . . . . . . . . . . . . . . 47
H
2 35.1 Quark number susceptibility ( =T ) calculated on lattices of size 128 N . 92q
5.2 Several values of the vector correlation functions expressed in units of
the corresponding free field values and normalized with the quark number
susceptibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 The ratio of the second thermal moment and its corresponding free value
3for our lattices 128 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Parameters for the fits shown in Fig.5.12. . . . . . . . . . . . . . . . . . . 107
5.5 Table of results for the quark number susceptibility over temperature T=T . 113c
5.6 The fit parameters of the simple Breit-Wigner+continuum Ansatz forT'
1:45T ; 1:2T and T'3:0T .. . . . . . . . . . . . . . . . . . . . . . . . . . 117c c c
36.1 Table of available momenta injpj=T atT'1:45T on lattices sized 128 c
48; 32; 24 and 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
(2;0) (2;0);free6.2 The ratioR =R for the full, spatial, transverse and longitudinal
H H
vector channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 The parameters of the toy models used to generate the data in Fig.6.4. . . 126
viiList of Figures
1.1 The eightfold way and the properties of quarks. . . . . . . . . . . . . . . . 2
+1.2 The production of hadrons by that of muons from e e -annihilation. . . . 4
1.3 Sketchofthestandardinterpretationofarelativisticheavy-ioncollision[18]. 6
1.4 Experimental spectral density and the vacuum polarization in Euclidean
theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Evolution of a HIC event and dilepton production [26]. . . . . . . . . . . . 9
1.6 a.) Emission of an on-shell photon. b.) Emission of a virtual photon
decaying into dileptons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Sketch of the dilepton spectrum, taken from [28]. . . . . . . . . . . . . . . 11
1.8 Experimental results for dilepton rate from p-p and Au-Au collisions. . . . 12
2.1 Latticediscretizationoffinitetemperaturefieldtheoryin(1+1)-Dimensions,
due to the periodic boundary conditions space-time is a torus. . . . . . . . 14
2.2 Infinitesimal parallelogram of continuum theory (left) and the elementary
plaquette on the lattice (right). . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Vacuum polarization effects are excluded in quenched QCD. . . . . . . . . 21
2.4 Doublers appear at the corners of the Brillouin zone. . . . . . . . . . . . 22
2.5 The improvement term in the plaquettes is seen to resemble a four leaf
clover, hence clover term. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 The lattice spacing a()[fm]. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
22.7 The critical hopping parameter over bare lattice coupling