The low-energy effective theory of QCD at small quark masses in a finite volume [Elektronische Ressource] / vorgelegt von Christoph Lehner
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The low-energy effective theory of QCD at small quark masses in a finite volume [Elektronische Ressource] / vorgelegt von Christoph Lehner

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136 Pages
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The low-energy effective theory ofQCD at small quark masses in afinite volumeD I S S E R T A T I O Nzur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.)der Naturwissenschaftlichen Fakultat¨ II - Physikder Universitat¨ Regensburgvorgelegt vonChristoph Lehneraus RegensburgJanuar 2010Promotionsgesuch eingereicht am: 30.11.2009Die Arbeit wurde angeleitet von: Prof. Dr. Tilo Wettig¨PRUFUNGSAUSSCHUSS:Vorsitzender: Prof. Dr. Christian Back1. Gutachter: Prof. Dr. Tilo Wettig2. Prof. Dr. Andreas Schafer¨weiterer Prufer:¨ Prof. Dr. Jaroslav FabianAbstractAt low energies the theory of quantum chromodynamics (QCD) can be described effectively in termsof the lightest particles of the theory, the pions. This approximation is valid for temperatures wellbelow the mass difference of the pions to the next heavier particles.We study the low-energy effective theory at very small quark masses in a finite volumeV . Thepcorresponding perturbative expansion in 1= V is called" expansion. At each order of this expansiona finite number of low-energy constants completely determine the effective theory. These low-energyconstants are of great phenomenological importance.In the leading order of the" expansion, called" regime, the theory becomes zero-dimensional andis therefore described by random matrix theory (RMT).

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The low-energy effective theory of
QCD at small quark masses in a
finite volume
D I S S E R T A T I O N
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der Naturwissenschaftlichen Fakultat¨ II - Physik
der Universitat¨ Regensburg
vorgelegt von
Christoph Lehner
aus Regensburg
Januar 2010Promotionsgesuch eingereicht am: 30.11.2009
Die Arbeit wurde angeleitet von: Prof. Dr. Tilo Wettig
¨PRUFUNGSAUSSCHUSS:
Vorsitzender: Prof. Dr. Christian Back
1. Gutachter: Prof. Dr. Tilo Wettig
2. Prof. Dr. Andreas Schafer¨
weiterer Prufer:¨ Prof. Dr. Jaroslav FabianAbstract
At low energies the theory of quantum chromodynamics (QCD) can be described effectively in terms
of the lightest particles of the theory, the pions. This approximation is valid for temperatures well
below the mass difference of the pions to the next heavier particles.
We study the low-energy effective theory at very small quark masses in a finite volumeV . Thep
corresponding perturbative expansion in 1= V is called" expansion. At each order of this expansion
a finite number of low-energy constants completely determine the effective theory. These low-energy
constants are of great phenomenological importance.
In the leading order of the" expansion, called" regime, the theory becomes zero-dimensional and
is therefore described by random matrix theory (RMT). The dimensionless quantities of RMT are
mapped to dimensionful quantities of the low-energy effective theory using the leading-order low-
energy constants andF . In this way andF can be obtained from lattice QCD simulations in the
" regime by a fit to RMT predictions.
For typical volumes of state-of-the-art lattice QCD simulations, finite-volume corrections to the
RMT prediction cannot be neglected. These corrections can be calculated in higher orders of the"
expansion. We calculate the finite-volume corrections to andF at next-to-next-to-leading order in
the" expansion. We also discuss non-universal modifications of the theory due to the finite volume.
These results are then applied to lattice QCD simulations, and we extract andF from eigenvalue
correlation functions of the Dirac operator.
As a side result, we provide a proof of equivalence between the parametrization of the partially
quenched low-energy effective theory without singlet particle and that of the super-Riemannian mani-
fold used earlier in the literature. Furthermore, we calculate a special version of the massless sunset
diagram at finite volume without constant mode which was not known before.
Apart from the universal regime of QCD, random matrix models can be used as schematic models
that describe certain features of QCD such as the chiral phase transition. These models are
defined at fixed topological charge instead of fixed vacuum angle. Therefore special care has to be
taken when different topological sectors are combined. We classify different schematic random ma-
trix models in terms of the topological domain of Dirac eigenvalues, i.e., the part of eigenvalues that
is affected by topology. If the topological domain extends beyond the microscopic eigenvalues, ad-
ditional normalization factors need to be included to allow for finite topological fluctuations. This is
important since the mass of the pseudoscalar singlet particle eta’ is related to topological fluctuations,
and the factors thus solve the corresponding U(1) problem.A
iiiivAcknowledgments
I would like to thank Tilo Wettig for his guidance and support throughout the course of this work
and for initiating prolific collaborations with Stony Brook University and KEK. I am grateful to Jac
Verbaarschot for his hospitality and many interesting discussions during my stay in Stony Brook and
his visits in Regensburg. I would also like to thank Shoji Hashimoto for inviting me to Tsukuba and
for a very fruitful and interesting collaboration. I am obliged to Jacques Bloch for countless interest-
ing discussions and for sharing his insight in many topics inside and outside of physics. I would also
like to thank Robert Lohmayer for many interesting discussions and especially for proof-reading this
thesis. I am grateful to Volodya Braun, Falk Bruckmann, Hidenori Fukaya, Tetsuo Hatsuda, Thomas
Hemmert, Sasha Manashov, and Munehisa Ohtani for very beneficial and stimulating discussions. I
am obliged to Andreas Schafer¨ for his guidance and support as head of the Elitestudiengang Physik
mit integriertem Doktorandenkolleg. This work was supported by BayEFG.
I would like to thank my dear grandparents for their support and for providing a warm and wel-
coming atmosphere. A special thank goes to my girlfriend Natascha who supported me also in lean
periods of my work.
vviContents
I Introduction 1
1 Construction of QCD 3
1.1 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Lagrangian of spin 1=2 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Euclidean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Construction of the low-energy effective theory of QCD 21
2.1 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Chiral symmetry of supersymmetric QCD . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 The effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 The effective theory in a finite volume . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Invariant integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
II The epsilon expansion 45
3 The universal limit 47
3.1 The effective theory to leading order in" . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 The partition function of chiral random matrix theory . . . . . . . . . . . . . . . . . 48
3.3 Proof of equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Dirac eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Leading-order corrections 57
4.1 The partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Finite-volume corrections to . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4olume toF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 The optimal lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Next-to-leading-order corrections 63
5.1 The partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 The two-loop propagator at finite volume . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Two quark flavors in an asymmetric box . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Results from lattice QCD 77
6.1 The low-energy constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 The logyF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
viiIII Schematic models of QCD 81
7 The Dirac spectrum at nonzero temperature and topology 83
7.1 A schematic random matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 The partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3 Compact Hubbard-Stratonovich transformation . . . . . . . . . . . . . . . . . . . . 84
7.4 Non-compact Hubbard-Stratonovich transformation . . . . . . . . . . . . . . . . . . 85
7.5 The limit of large matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8 The axial anomaly at nonzero temperature 91
8.1 Topology and the microscopic domain of QCD . . . . . . . . . . . . . . . . . . . . 91
8.2 Chiral random matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.3 Normalization factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.4 Chiral condensate and topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.5 Eigenvalue fluctuations and microscopic universality . . . . . . . . . . . . . . . . . 100
8.6 Topological and pseudoscalar susceptibility . . . . . . . . . . . . . . . . . . . . . . 102
8.7 The topological domain and lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . 105
IV Epilogue 107
9 Conclusions and outlook 109
A One-loop propagators at finite volume 111
A.1 Poisson’s sum over momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.2 The spectrum of the harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . 112
A.3 Massive propagators at finite volume . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.4 Masslessators at finite volume . . . . . . . . . . . . . . . . . . . . . . . . . 120
Bibliography 123
viiiPart I
Introduction
1