Excited Hadrons in Two-Flavor Lattice QCD Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) Eingereicht von Georg P. Engel Angefertigt am Institut f¨ur Physik der Karl-Franzens-Universit¨at Graz Betreuer und Gutachter Univ.Prof. Dr. Christian B. Lang Zweitgutachter Prof. Dr. Sasa Prelovsek Mai 2012 iii Pr¨aambel Ich best¨atige, dass es sich bei der hier vorgelegten Dissertation um eine Origi- nalarbeit handelt, die von mir selbstst¨andig angefertigt und abgefasst wurde. ............................................ (Georg P. Engel) iv To Anna v Abstract. The majority of the experimental knowledge about QCD observables is contained in the excited hadron spectrum. The present thesis provides an ab- initio, non-perturbative determination of the excited meson and baryon spectrum, using the lattice regularization of QCD. We use a Hybrid Monte Carlo algorithm to generate seven ensembles with two flavors of dynamical Chirally Improved quarks. The advantages of the improved action lie in small discretization effects and frequent tunneling of topological sectors, reducing autocorrelation. The pion masses are in the range of 250 to 600 MeV, the results are extrapolated to the physical pion mass. Three further ensembles are generated to investigate finite volume effects and to perform the infinite volume limit for specific observables. The strange hadron spectrum is accessed using partial quenching for the strange quark. The variational method is applied to access excited states and also to investigate the content of the physical states. The latter applies in particular to the approximate C-parity of strange mesons, the singlet/octet mixing of Λ baryons and the octet/decuplet mixing of Σ and Ξ baryons. The construction of interpolators is discussed for specific cases. In some baryon channels, Fierz identities force point-like interpolators to vanish exactly. We show that interpolators can be constructed nevertheless and propose two strategies, based on quark smearing and the Rarita-Schwinger condition, respectively. In general, our results compare nicely with experiment, and we even predict some new states and allow for insights concerning the content of the physical states. Part of the work has been published in [1–6], and further publications are in preparation. vi Zusammenfassung. Aus dem Experiment wissen wir, dass fast alle Hadro- nen angeregte Zust¨ande sind. Die vorliegende Dissertation liefert eine ab-initio, nicht-perturbative Bestimmung des angeregten Meson- und Baryonspektrum unter Verwendung der Gitter-Regularisierung von QCD. Mit Hilfe eines Hybrid Monte Carlo Algorithmus werden sieben Ensembles mit zwei dynamischen Chirally Im- proved Quarks erzeugt. Die Vorteile der verbesserten Wirkung liegen in kleinen Diskretisierungseffekten und h¨aufigem Tunneln der topologischen Sektoren, was eine geringe Autokorrelation zur Folge hat. Die Pionmassen in der vorgelegten Arbeit liegen im Bereich von 250 bis 600 MeV, die Resultate werden zur physikalischen Pion- masse extrapoliert. Drei weitere Ensembles f¨ur unterschiedliche Gittergr¨oßen werden erzeugt um Effekte des endlichen Volumens zu untersuchen und um den Limes zu unendlichem Volumen f¨ur bestimmte Observablen durchzuf¨uhren. Strange Quarks werden durch Partial Quenching eingef¨uhrt. Mit Hilfe der sogenannten Variations- methode studieren wir angeregte Zust¨ande und erforschen den Inhalt der physikalis- chen Zust¨ande. Insbesondere untersuchen wir die C-Parit¨at von strange Mesonen, das Mischen von Singlet und Oktett in Λ Baryonen und das Mischen von Oktett und Dekuplet in Σ und Ξ Baryonen. Die Konstruktion von Interpolatoren wird explizit f¨ur bestimmte F¨alle diskutiert. In einigen Baryon Kan¨alen erzwingen Fierz- Identit¨aten ein exaktes Verschwinden von punktf¨ormigen Interpolatoren. Wir zeigen, dass dennoch Interpolatoren konstruiert werden k¨onnen, und schlagen zwei Strate- gien daf¨ur vor, welche sich auf ausgedehnte Quarkquellen bzw. die Rarita-Schwinger Bedingung st¨utzen. Im Allgemeinen zeigen unsere Resultate gute Ubereinstimmung¨ mit dem Experiment. Dar¨uber hinaus k¨onnen wir auch einige Vorhersagen machen und finden Einblicke in den Inhalt der physikalischen Zust¨ande. Teile der Arbeit wurden in [1–6] publiziert, und weitere Publikationen sind in Vorbereitung. Contents 1 Introduction 1 2 QCD in the Continuum 5 3 QCD on the Lattice 7 3.1 TheLatticeRegulator .......................... 7 3.2 PhysicsontheLattice .......................... 9 3.2.1 ChiralSymmetry......................... 9 3.2.2 Poincar´eSymmetry . 10 3.2.3 Observables and Euclidean Correlators . 11 3.2.4 SettingtheScale ......................... 12 4 Setup of the Simulation 13 4.1 Action................................... 13 4.1.1 The Improved Gauge Action . 13 4.1.2 The Chirally Improved Dirac Operator . 14 4.2 Monte Carlo Integration of the Functional Integral . ...... 16 4.2.1 Generating Gauge Configurations . 16 4.2.2 The Hybrid Monte Carlo Algorithm . 17 4.2.3 Numerical Improvements for the HMC Algorithm . 18 4.3 MethodsinHadronSpectroscopy . 19 4.3.1 LinkSmearing .......................... 19 4.3.2 QuarkSmearing ......................... 21 4.3.3 Inverting the Dirac Operator . 24 4.3.4 Interpolator Construction . 24 4.3.5 Extracting Excited States . 28 4.4 From the Lattice to Nature: Hadron Phenomenology . 31 4.4.1 Determining and Approaching the Physical Point . 31 4.4.2 Energy Spectra in Finite and Infinite Systems . 32 5 Simulation Properties 35 5.1 Details of the Simulation . 35 5.2 MonteCarloTimeHistories . 36 vii viii Contents 6 Scale and Low Energy Parameters 41 6.1 Scale.................................... 41 6.2 SettingtheStrangeQuarkMass . 43 6.3 AxialWardIdentityQuarkMass . 43 6.4 DecayConstants ............................. 44 7 Results for the Meson Spectrum 47 7.1 AnalysisFlow............................... 47 7.2 IsovectorLightMesons. .. .. .. .. .. .. .. 48 7.2.1 Scalars............................... 48 7.2.2 Vectors .............................. 50 7.2.3 Tensors .............................. 52 7.3 MesonswithStrangeValenceQuarks. 53 7.3.1 Scalars............................... 54 7.3.2 Vectors .............................. 55 7.3.3 Tensors .............................. 58 7.4 IsoscalarLightMesons. .. .. .. .. .. .. .. 60 7.5 Summaryof the Results for the Meson Spectrum . 61 8 Results for the Baryon Spectrum 63 8.1 ResultsforLightBaryons . 63 8.1.1 Nucleons ............................. 63 8.1.2 DeltaBaryons .......................... 66 8.2 ResultsforStrangeBaryons . 68 8.2.1 LambdaBaryons ......................... 68 8.2.2 SigmaBaryons .......................... 72 8.2.3 XiBaryons ............................ 75 8.2.4 OmegaBaryons.......................... 77 8.3 Summary of the Results for the Baryon Spectrum . 78 9 Finite Volume and Other Systematic Effects 81 9.1 PossibleSourcesofSystematicErrors . 81 9.2 Finite Volume Generalities . 82 9.3 Volume Dependence of the Scale Setting . 83 9.4 Volume Dependence of Meson Energy Levels . 86 9.4.1 Strange Scalar Mesons . 87 9.4.2 Strange Vector Mesons . 88 9.5 Volume Dependence of Baryon Energy Levels . 88 9.5.1 Nucleon .............................. 89 9.5.2 DeltaBaryons .......................... 89 9.5.3 LambdaBaryons ......................... 90 9.5.4 SigmaBaryons .......................... 90 Contents ix 9.5.5 XiBaryons ............................ 91 9.5.6 OmegaBaryons.......................... 91 9.6 Summary ................................. 92 10 Summary and Conclusion 99 A The Chirally Improved Dirac Operator 101 B Meson Interpolators 103 B.1 C-Parity of Meson Interpolators . 103 B.1.1 General Considerations . 103 B.1.2 C-Parity at Zero Momentum: Gaussian Interpolators . 104 B.1.3 C-Parity at Zero Momentum: Derivatives . 106 B.1.4 C-Parity for Non-Zero Momentum . 107 B.2 Meson Interpolator Tables . 108 C Baryon Interpolators 115 C.1 Generalities ................................ 115 C.2 Baryon Interpolator Tables . 115 D Details of Fits and Results 119 x Contents List of Figures 3.1 Thelatticeregulator ........................... 8 4.1 The L¨uscher-Weisz improved gauge action . 14 4.2 The Chirally Improved Dirac operator . 15 4.3 Theconceptoflinksmearing . 20 5.1 Monte Carlo history for the dimensionless pion mass . ..... 38 5.2 Monte Carlo history and histogram for the topology . 38 5.3 SpectrumoftheDiracoperator . 39 5.4 Number of conjugate gradient steps needed . 39 6.1 Settingthescale ............................. 42 6.2 The ratio of the pseudoscalar decay constants FK/Fπ ......... 45 7.1 Energy levels for light scalar mesons: I(J)=1(0)........... 49 7.2 Eigenvectors and masses for light scalar mesons: I(J P C ) = 1(0++) . 50 7.3 Energy levels for light vector mesons: I(J)=1(1)........... 51 P C 7.4 Energy levels for light tensor mesons: I(J ) = 1(2−−) ....... 52 P C + 7.5 Energy levels for light tensor mesons: I(J ) = 1(2− ) ....... 52 7.6 Energy levels for light tensor mesons: I(J P C ) = 1(2++) ....... 53 7.7 Energy levels for strange scalar mesons: I(J) = 1/2(0) . 55 P 7.8 Energy levels for strange vector mesons: I(J ) = 1/2(1−)...... 56 7.9 Energy levels and eigenvectors for strange mesons: I(J P ) = 1/2(1+) 57 P 7.10 Eigenvectors for strange tensor mesons: I(J ) = 1/2(2−) ...... 58 7.11 Energy levels for strange tensor mesons: I(J) = 1/2(2) . 59 P C 7.12 Energy levels for isoscalar vector mesons: I(J ) = 0(1−−)..... 60 7.13 Energy levels for isoscalar tensor mesons: I(J P C ) = 0(2++)..... 61 7.14 Energylevelsformesons: Summary . 62 8.1 Energy levels for nucleon spin 1/2