Instanton Correlations in the Finite Temperature QCD Vacuum by Fernando P´erez B.S. Physics, Universidad de Antioquia, 1995. M.Sc. Physics, Universidad de Antioquia, 1999. A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics 2002 This thesis entitled: Instanton Correlations in the Finite Temperature QCD Vacuum written by Fernando P´erez has been approved for the Department of Physics Anna Hasenfratz Thomas A. DeGrand Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii P´erez, Fernando (Ph.D., Physics) Instanton Correlations in the Finite Temperature QCD Vacuum Thesis directed by Professor Anna Hasenfratz Instantons, discovered in 1975, are solutions to the Euclidean-space classi- cal equations of Quantum Chromodynamics (QCD). They have been found to play a crucial role in many low energy phenomena driven by the strong interactions. Instan- tons are the leading candidate for explaining spontaneous chiral symmetry breaking, and they provide a mechanism for fermion propagation in the QCD vacuum. At finite temperature, QCD is expected to undergo a transition into a chirally sym- metric phase known as a quark-gluon plasma. In lattice calculations with dynamical fermions, the instantons are expected to form ‘molecules’, polarized in the time di- rection. In this work, we study finite temperature lattice configurations and find evidence for this structure by analyzing the two-point charge correlator. We also present tools that we developed for visualizing numerical data from lattice calcula- tions. Dedicated to Irene Gonz´alez. For a lifetime of friendship which has changed me like no other. v Acknowledgements This project has been a challenging road to travel for me. I would never have been able to do it without the help and support of many people, whom I want to express my sincere thanks to. First and foremost, I would like to thank my advisor, Anna Hasenfratz. It was the elegance of her approach to physics what brought me to this field. It was her incredible patience and good nature what allowed me to complete this project. A very special thanks goes to professor Thomas DeGrand, whose talent and knowledge of physics will always inspire me. I would also like to thank him for his frequent technical assistance and for the configurations and fermionic data used for visualization. I would like to thank Andrei Alexandru for many useful discussions about the topology of instantons, and their interpretation as tunneling events between inequivalent vacua. To my former professors Manuel P´aez, Alonso Sep´ulveda, Lorenzo De La Torre and Jorge Mahecha, for their passion for physics. To professor Allan Franklin, for his support in a difficult period. To the MILC collaboration, especially Doug Toussaint, for the lattice code and configurations which this project is based upon. To Prabhu Ramachandran from the MayaVi project and Eric Jones from the SciPy project for contributing such valuable tools and their generous assistance on their internals. vi To my fellow graduate students Chet Nieter and Archie Paulson, for their friendship in and out of physics. I would also like to thank Chet for the code used for generating artificial instanton configurations. To Germ´an V´elez and Lucas Monz´on, for they are proof that exceptional friendships make all the difference in life. To my parents, for feeding my dreams with the best of examples, and for supporting those dreams in all situations. And finally to my incredible wife Adriana, who has never stopped being by my side or believing in me. This work was partially supported by the U.S. Department of Energy. It would not have been possible without the generous contribution of computing time by the Colorado High Energy Physics Experimental Group. Contents Chapter 1 Introduction 1 2 QCD - The Theory of Strong Interactions 4 2.1 Quarks.................................... 4 2.2 MathematicalOutlineofQCD . 6 3 QCD on the Lattice 9 3.1 ThePathIntegralFormulation . 9 3.2 GaugeSymmetry.............................. 11 3.3 TheLatticeAction............................. 14 3.4 TheLatticeasaRegulator . 19 3.5 Simulations ................................. 20 3.5.1 MarkovChainsandMonteCarloMethods . 20 3.5.2 OverviewofMethods . 23 3.5.3 Monte Carlo Methods for Systems with Fermions . 29 3.6 MeasurementsontheLattice . 32 3.6.1 ComparingwiththeContinuum . 32 3.6.2 Spectroscopy ............................ 33 4 Instantons in QCD 36 4.1 TopologyandtheQCDVacuum . 37 viii 4.2 Quantum MechanicalTunneling Revisited . ... 43 4.3 InstantonsandChiralSymmetry . 48 4.4 Instantons in the SU(3)Vacuum ..................... 51 5 Topology on the Lattice 54 5.1 Smoothing.................................. 54 5.2 Operators .................................. 58 6 Finite Temperature 61 6.1 HighTemperatureQCDintheContinuum . 61 6.2 Finite T ontheLattice........................... 63 6.3 Finite T EffectsonInstantons. 64 6.4 PreviousWorkonPairFormation . 71 7 TopologicalChargeCorrelationAnalysis 74 7.1 LatticesProcessed ............................. 75 7.2 Finite Temperature Quenched Configurations . .... 76 7.3 Directly Looking at Dynamical Configurations . .... 78 7.4 ComparisonofAPEandHYPblocking . 83 7.5 TheprocessofAPEblocking . 85 7.6 CorrelationMeasurementsat24APESteps . .. 90 7.7 AddressingtheEffectsofPeriodicity . ... 94 7.8 SimplePairingModel ........................... 97 7.9 FittingtheCorrelationData . 100 8 Visualization of Lattice Quantities 107 8.1 AMixedLanguageApproach . .107 8.2 ArtificialInstantons ... .... ... .... .... ... .... ...112 8.3 The Relation Between Topological and Fermionic Observables. .114 ix 9 Summary and Outlook 122 Bibliography 124 Figures Figure 3.1 Dispersion relation for the continuum action and the na¨ıve discretiza- tion, along a momentum axis. The extra zeros at the edges of the Brillouinzoneappearasextrafermions. 16 3.2 Dispersion relation for the continuum action and the Wilson dis- cretization, along a momentum axis. The Wilson action shows no extrazerosintheBrillouinzone. 18 4.1 Doublewellpotential. ........................... 45 4.2 DoublewellpotentialinEuclideanspace. .... 46 4.3 One-dimensional instanton for the potential (4.33), centered at the origin. .................................... 47 4.4 Instanton distribution in SU(3), taken from [41]. The solid line rep- resents the prediction from the phenomenological Instanton Liquid Model..................................... 52 5.1 PathsusedinbuildingtheAPElinks. 56 5.2 Schematic structure of HYP links in three dimensions, from Ref. [38]. In a), the fat link is built from the four double-lined staples; in b) each double-lined link is built from two staples which extend only into the hypercubesattachedtotheoriginallink. .. 57 xi 5.3 Comparison of topological charge operators for a background with a single SU(2)instanton,fromRef.[25]. 59 6.1 QCDphasediagramatfirstapproximation. .. 62 6.2 At T = 0 the net topological charge vanishes as m 0. Likely q → mechanisms for this are the formation of tight I-A pairs or screening incloudsofnetvanishingcharge. 66 6.3 Fermion propagators at finite temperature. .... 68 6.4 The quark-induced interaction picture for the instanton polarization effectatthephasetransition. 70 6.5 Around the phase transition, we expect a difference between quenched and dynamical simulations, induced by the virtual quarks. ...... 72 7.1 Plots of c(r, t) for the pure gauge configurations. All lattices were blocked 24 APE steps. The column on the right is a detail of the x = [5, 12]region............................... 77 7.2 Distribution of topological charge density for lattice 704, after 24 steps ofAPEblocking. .............................. 79 7.3 Topological charge density isosurfaces (lattice 704). mq =0.008, β = 5.85, 24 APE steps. Three-dimensional slices taken at x =0, 4, 8, 12, 16,20..................................... 80 7.4 Topological charge density isosurfaces (lattice 1304). mq =0.008, β = 5.85, 24 APE steps. Three-dimensional slices taken at x =0, 4, 8, 12, 16,20..................................... 81 7.5 Topological charge density isosurfaces (lattice 1600). mq =0.008, β = 5.85, 24 APE steps. Three-dimensional slices taken at x =0, 4, 8, 12, 16,20..................................... 82 xii 7.6 Comparison of APE and HYP blocking. The left column shows the effect of 6, 12 and 24 APE steps (top to bottom). The right column shows2,3and4HYPsteps. .. ... .... .... ... .... ... 84 7.7 Comparison of APE and HYP blocking for mq = 0.008, β = 5.85 lattices. Only the x = [5, 12]regionisshown. 86 7.8 Comparison of APE and HYP blocking, continued. ... 87 7.9 APE processshown for the x = 0 slice of lattice 1600 (mq =0.008, β = 5.85). From left to right and top to bottom, 6, 9, 12, 18, 24 and 36 APEstepsshown. ............................. 88 7.10 Effects of APE blocking on c(r, t), shown for the full mq =0.008, β = 5.85 set of configurations. The plots corresponding to 6, 9 and 12 APE steps are taken from Fig. 7.7. Those for 18 and 24 steps are from Fig. 7.8. They are reproduced here in one figure to show the wholeAPEprocess. ............................ 89 7.11 Topological correlator for all β values with mq = 0.008 and 24 APE steps. The second column shows a detail of the x = [5, 12] region. 91 7.12 Topological correlator for all β values with mq = 0.016 and