The chiral phase transition of QCD with 2+1 flavors [Elektronische Ressource] : a lattice study on Goldstone modes and universal scaling / vorgelegt von Wolfgang Unger

Sujets

• Physik

Informations

• Publié par: universitat_bielefeld
• Publié le: 01 janvier 2010
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

The Chiral Phase Transition of QCD with 2+1 Flavors
A lattice study on
Goldstone modes and universal scaling
Dissertation
zur Erlangung des Doktorgrades
an der Fakultat fur Physik¨ ¨
der Universitat Bielefeld¨
vorgelegt von
Wolfgang Unger
August 2010◦◦Gedruckt auf alterungsbestandigem Papier ISO 9706¨Abstract
ThisPhDthesisisconcernedwiththechiralphasetransitionofQCDwithtwodegeneratelightquark
masses and a strange quark mass close to its physical value. We analyze the quark mass dependence
of the chiral condensate and chiral susceptibilities close to the transition temperature. The analysis
is twofold:
FirstweprovideevidencefortheinfluenceofthermalfluctuationsofGoldstonemodesonthechiral
condensate at finite temperature. We show that at temperatures below but close to the chiral phase
transition at vanishing quark mass this leads to a characteristic dependence of the light quark chiral
condensate on the square root of the light quark massm. As a consequence the chiral susceptibilityl
−1/2
shows a strong quark mass dependence for all temperatures belowT and diverges likem in thec l
chiral limit. We separately examine the divergence of disconnected and connected parts of the light
quarksusceptibilityanddiscussthevolumeaswellascut-offdependenceofsusceptibilitiesandchiral
condensates.
Second we analyze the critical behavior of the chiral transition with a scaling analysis based
on the O(N) scaling functions. We find strong evidence for 2nd order O(N) scaling in the chiral
limit of the light quark mass and with physical strange quark mass. Z(2) scaling is disfavored for
cfinite values of the m , which indicates that the physical strange quark mass is above the tricriticall
phys tricmass, m > m . The scaling fits are based on the magnetic equation of state for the chirals s
condensate. We compare these fit results also with the corresponding scaling functions for the chiral
susceptibilities and identify the Goldstone contributions and attempt to identify the connected and
disconnectedsusceptibilitycontributions. Wediscussthedeviationsfromscalingandcompareresults
for two different lattice spacings. Finally we present the result on the pseudocritical line for zero
chemical potential and the curvature of the critical line for non-zero chemical potential to lowest
order.
iiiContents
1 Introduction: The Phase Diagram of Quantum Chromodynamics 1
1.1 The Confinement-Deconfinement Transition . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Chiral Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Lattice QCD at Finite Temperature 7
2.1 Discretization of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 The Lattice QCD Partition Function . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 The Wilson Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Staggered Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Improved Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Gauge Part: Tree-level Improved Symanzik Action . . . . . . . . . . . . . . . . 12
2.2.2 Fermionic Part: p4fat3 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 The RHMC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Hybrid Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Non-zero Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Taylor Expansion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 Chiral Condensate at Non-zero Density . . . . . . . . . . . . . . . . . . . . . . 18
3 Critical Phenomena 19
3.1 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Classification of Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.2 Second order Transitions and Landau Theory . . . . . . . . . . . . . . . . . . . 21
3.1.3 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 O(N) Symmetric Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 General Properties of O(N) Models . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Goldstone Modes in O(N) Models . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 O(N) Scaling Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Magnetic Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.2 Scaling Function for the Susceptibilities and the Cumulant . . . . . . . . . . . 35
3.3.3 Finite Size Scaling and Binder Cumulant . . . . . . . . . . . . . . . . . . . . . 35
3.4 Finite Size Effects in the O(N) Model below T . . . . . . . . . . . . . . . . . . . . . . 37c
3.4.1 Higher Order Effective Lagrangian at Infinite Volume . . . . . . . . . . . . . . 37
3.4.2 Expansion Schemes in Chiral Perturbation Theory . . . . . . . . . . . . . . . . 37
4 Spontaneous Chiral Symmetry Breaking and the Chiral Transition 41
4.1 Chiral Symmetry and its Spontaneous Breaking . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 QCD Lagrangian in the Chiral Limit . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Spontaneous Chiral Symmetry Breaking and Goldstone Theorem . . . . . . . . 43
iii4.1.3 Explicit Chiral Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Chiral Perturbation Theory for 2+1 Flavors . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Chiral Perturbation Theory at Zero Temperature . . . . . . . . . . . . . . . . . 46
4.2.2 Chiral Perturbation Theory at Finite Temperature . . . . . . . . . . . . . . . . 52
4.3 The QCD Phase Diagram Revisited: The Rˆole of the Chiral Anomaly . . . . . . . . . 55
4.3.1 Linear σ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.2 Order of the Chiral Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 Chiral Susceptibilities and the Anomaly . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Staggered Chiral Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1 Staggered Fermions and Taste Breaking . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 Elements of Staggered Chiral Perturbation Theory . . . . . . . . . . . . . . . . 59
4.4.3 Scalar Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.4 Chiral Condensate from SχPT . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.5 Chiral Susceptibilities from Scalar Correlators . . . . . . . . . . . . . . . . . . . 63
5 Analysis of 2+1 Flavor Lattice Data 69
5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 Random Noise Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.2 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.3 Ferrenberg-Swendsen Reweighting . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Setup of Lattice Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.2 Setting the Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.3 Chiral Observables on the Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.4 Determination of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87c
5.3 Goldstone Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1 Goldstone Fits for the Chiral Condensate . . . . . . . . . . . . . . . . . . . . . 94
5.3.2 Goldstone Fits for the Chiral Susceptibilities . . . . . . . . . . . . . . . . . . . 98
5.3.3 Estimate of Taste Breaking Effects in Chiral Observables . . . . . . . . . . . . 101
5.3.4 SχPT Fits of Chiral Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 The Universality Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4.1 Binder Cumulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4.2 Magnetic Equation of State for the Chiral Condensates . . . . . . . . . . . . . 106
5.4.3 The pseudo-critical line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4.4 The critical line at finite density . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.5 Comparison with Literature and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5.1 Universal scaling in N =2 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 120f
5.5.2 Universal scaling in N =2+1 QCD . . . . . . . . . . . . . . . . . . . . . . . . 121f
5.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A O(N) Non-linear Sigma Model 125
A.1 Angular Distribution Integrals for Binder Cumulant . . . . . . . . . . . . . . . . . . . 125
B Chiral Perturbation Theory 1