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The Rho Meson Spectrum and Kaon-Pion Scattering with Partial

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The Rho Meson Spectrum and Kaon-Pion Scattering with Partial THE ρ MESON SPECTRUM AND Kπ SCATTERING WITH PARTIAL WAVE MIXING IN LATTICE QCD by Andrew D. Hanlon B.Sc. in Physics and Computer Science, Michigan State University, 2013 Submitted to the Graduate Faculty of the Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2017 UNIVERSITY OF PITTSBURGH DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Andrew D. Hanlon It was defended on September 28, 2017 and approved by Adam Leibovich, Department of Physics and Astronomy, University of Pittsburgh Colin Morningstar, Department of Physics, Carnegie Mellon University Eric Swanson, Department of Physics and Astronomy, University of Pittsburgh Daniel Boyanovsky, Department of Physics and Astronomy, University of Pittsburgh Vladimir Savinov, Department of Physics and Astronomy, University of Pittsburgh Dissertation Advisors: Adam Leibovich, Department of Physics and Astronomy, University of Pittsburgh, Colin Morningstar, Department of Physics, Carnegie Mellon University ii Copyright c by Andrew D. Hanlon 2017 iii THE ρ MESON SPECTRUM AND Kπ SCATTERING WITH PARTIAL WAVE MIXING IN LATTICE QCD Andrew D. Hanlon, PhD University of Pittsburgh, 2017 The finite-volume QCD spectrum in the I = 1, S = 0, parity-odd, G-parity-even channels for zero total momentum is studied using lattice QCD, and the K-matrix for Kπ scattering is investigated to determine the mass and decay width of the K∗(892) from first principles. The recently developed stochastic LapH method has proven to be a valuable tool in lattice QCD calculations when all-to-all quark propagators are needed, as is the case for isoscalar mesons and two-hadron operators. This method is especially important for large volumes where other methods do not scale well. These calculations were done with 412 gauge configurations using clover-improved Wilson fermions on a large anisotropic 323 × 256 lattice with a pion mass near 240 MeV. The stationary states determined to be single-particle dominated are compared with the experimental resonances and are found to be in reasonable agreement. Additionally, the initial development of tetraquark operators is described. iv TABLE OF CONTENTS PREFACE ......................................... xiv 1.0 INTRODUCTION .................................1 2.0 LATTICE QCD ...................................6 2.1 Continuum QCD................................6 2.1.1 Euclidean Spacetime...........................9 2.2 Discretization of the QCD Action....................... 11 2.2.1 Fermionic action............................. 12 2.2.2 The Fermion Doubling Problem..................... 15 2.2.3 The Wilson Gauge Action........................ 16 2.2.4 Improved Actions............................ 18 2.2.5 Tuning the Lattice and Setting the Scale................ 20 2.3 Monte Carlo Integration............................ 22 2.3.1 The Hybrid Monte Carlo Algorithm.................. 24 2.3.2 The Rational Hybrid Monte Carlo Algorithm............. 26 2.4 Energies from Temporal Correlation Matrices................ 26 2.4.1 Hermiticity................................ 28 3.0 CONSTRUCTION OF HADRONIC OPERATORS ............ 31 3.1 Symmetry Channel Transformations..................... 32 3.2 The Basic Building Blocks........................... 35 3.2.1 Gauge-Link Smearing.......................... 36 3.2.2 LapH Smearing of the Quark Fields.................. 37 3.2.3 Gauge-Covariant Displacements..................... 42 v 3.2.4 Charge Conjugation and G-Parity................... 43 3.3 Elemental Operators.............................. 44 3.3.1 Flavor Structure............................. 44 3.3.2 Baryons.................................. 47 3.3.3 Mesons.................................. 49 3.4 The Lattice Symmetry Group......................... 52 3.4.1 The Octahedral Group O ........................ 56 3.4.2 The Point Group Oh ........................... 62 3.4.3 The Little Groups............................ 64 3.4.3.1 The Subductions onto the Little Group Irreps........ 64 3.4.4 Transformation of the Building Blocks................. 66 3.5 Group-Theoretical Projections onto Symmetry Channels.......... 67 3.6 Two-Hadron Operators............................. 70 3.6.1 Comparison with Local Two-Hadron Operators............ 71 4.0 THE ESTIMATION OF TEMPORAL CORRELATORS USING STOCHAS- TIC LAPH ...................................... 74 4.1 Distillation and The LapH Subspace..................... 78 4.2 Stochastic Estimate of Matrix Inverses.................... 79 4.2.1 Variance Reduction through Noise Dilution.............. 81 4.3 Stochastic Estimate of Quark Lines in the LapH Subspace......... 83 4.4 Temporal Correlators.............................. 86 4.4.1 Meson Temporal Correlators...................... 87 4.4.2 Multi-Hadron Correlators........................ 89 4.5 Dilution Schemes................................ 90 5.0 TETRAQUARK OPERATORS ........................ 91 5.1 Color Structure................................. 91 5.1.1 More Complicated Color Structure................... 93 5.1.2 Elemental Operators........................... 100 5.2 Computational Details............................. 101 6.0 ENERGIES FROM TEMPORAL CORRELATORS ............ 103 vi 6.1 Euclidean Space and Thermal Effects..................... 103 6.2 Temporal Correlator Matrix.......................... 105 6.2.1 The Generalized Eigenvalue Problem.................. 106 6.2.2 The Correlator Matrix Pivot...................... 108 6.2.2.1 Alternative Pivot Methods................... 110 6.3 Fitting to Temporal Correlators........................ 111 7.0 THE LUSCHER¨ QUANTIZATION CONDITION ............. 113 7.1 The Quantization Condition.......................... 115 7.1.1 Deriving The Quantization Condition................. 117 7.2 The K-matrix.................................. 121 7.2.1 The S-matrix............................... 122 7.2.2 Parameterizing the K-matrix...................... 125 7.3 The Box Matrix................................. 126 7.4 Block Diagonalization............................. 127 7.5 Fitting Strategies................................ 128 7.5.1 The Spectrum Method.......................... 128 7.5.2 The Determinant Residual Method................... 129 8.0 RESULTS ...................................... 131 8.1 Computational Details............................. 131 8.2 Operator Selection............................... 132 8.3 The I = 1;S = 0;P = −1;G = +1 Bosonic Spectra............. 133 + 8.3.1 A1u .................................... 135 + 8.3.2 A2u .................................... 142 8.3.2.1 Single Hadron Operators.................... 142 8.3.2.2 All operators, Single Hadron Improved Operators...... 142 + 8.3.3 Eu .................................... 150 + 8.3.4 T1u .................................... 156 8.3.4.1 Single Hadron Operators.................... 156 8.3.4.2 All operators, Single Hadron Improved Operators...... 158 + 8.3.5 T2u .................................... 169 vii 8.3.5.1 Single Hadron Operators.................... 169 8.3.5.2 All operators, Single Hadron Improved Operators...... 169 8.3.6 Finite-volume Spectrum Conclusions.................. 180 8.4 The K∗(892) Resonance............................ 181 9.0 CONCLUSIONS AND OUTLOOKS ..................... 187 APPENDIX A. RESAMPLING ........................... 189 A.1 Jackknife.................................... 190 A.2 Bootstrap.................................... 191 APPENDIX B. FITTING ............................... 193 APPENDIX C. OPERATOR CHOICES ...................... 196 + C.1 A1u ........................................ 197 + C.2 A2u ........................................ 201 + C.3 Eu ........................................ 204 + C.4 T1u ........................................ 208 + C.5 T2u ........................................ 213 BIBLIOGRAPHY .................................... 218 viii LIST OF TABLES 3.1 Elemental baryon operators............................ 48 3.2 Elemental meson operators............................ 51 3.3 Reference momentum and rotation choices................... 57 3.4 The conjugacy classes for the octahedral group O ............... 58 3.5 The choice of matrices used for the representation of O ............ 59 3.6 The occurrence of the octahedral irreps in the subduced representations of SO(3)....................................... 61 3.7 The conjugacy classes for the double octahedral group OD .......... 61 3.8 The choice of matrices used for the double-valued representations of O .... 62 3.9 The occurrence of the double octahedral irreps in the subduced representations of SU(3)...................................... 63 3.10 The subduced representations of the octahedral group onto the little groups. 65 5.1 The SU(3) Clebsch-Gordan coefficients used for constructing different color structures for tetraquarks............................. 95 7.1 The B-matrix irrep relationship to the lattice symmetry irrep......... 128 + 8.1 The lowest 33 fitted energies in the A1u channel................ 136 + 8.2 The A2u fitted energies using only single-hadron operators........... 142 + 8.3 The lowest 18 fitted energies in the A2u channel................ 143 + 8.4 The lowest 36 fitted energies in the Eu channel................. 150 + 8.5 The T1u fitted energies using only single-hadron operators........... 156 + 8.6 The lowest 64 fitted energies in the T1u channel................. 159 + 8.7 The T2u fitted energies using only single-hadron operators........... 169 ix + 8.8 The lowest 47 fitted energies
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