ABSTRACT Title of dissertation: Lattice QCD Simulations of Baryon Spectra and Development of Improved Interpolating Field Operators Ikuro Sato, Doctor of Philosophy, 2005 Dissertation directed by: Professor Stephen J. Wallace Department of Physics Large sets of baryon interpolating field operators are developed for use in lattice QCD studies of baryons with zero momentum. Because of the discretization of space, the continuum rotational group is broken down to a finite point group of cube. Operators are classified according to the irreducible representations of the double octahedral group. At first, three-quark quasi- local operators are constructed for each isospin and strangeness with suitable symmetry of Dirac indices. Nonlocal baryon operators are formulated in a second step as direct products of the quasi-local spinor structures together with lattice displacements. Appropriate Clebsch-Gordan coefficients of the octahedral group are used to form linear combinations of such direct products. The construction maintains maximal overlap with the continuum SU(2) group in order to provide a physically interpretable basis. Nonlocal operators provide direct couplings to states that have nonzero orbital angular momentum. Monte Carlo simulations of nucleon and delta baryon spectra are carried out with anisotropic lattices of anisotropy 3.0 with β = 6.1. Gauge configurations are generated by the Wilson gauge action in quenched approximation with space-time volumes (1.6 fm)3 2.1 fm and (2.4 fm)3 2.1 fm. × × The Wilson fermion action is used with pion mass 500 MeV. The variational method is applied ≃ to matrices of correlation functions constructed using improved operators in order to extract mass eigenstates including excited states. Stability of the obtained masses is confirmed by varying the dimensions of the matrices. The pattern of masses for the low-lying states that we compute is consistent with the pattern that is observed in nature. Ordering of masses is consistent for positive-parity excited states, but mass splittings are considerably larger than the physical values. Baryon masses for spin S 5/2 states are obtained in these simulations. Hyperfine mass splittings ≥ are studied for both parities. No significant finite volume effect is seen at the quark mass we used. Lattice QCD Simulations of Baryon Spectra and Development of Improved Interpolating Field Operators by Ikuro Sato Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2005 Advisory Committee: Professor Stephen J. Wallace, Chair/Advisor Professor Xiangdong Ji Professor Thomas Cohen Professor James J. Kelly Professor Alice Mignerey c Copyright by Ikuro Sato 2005 ACKNOWLEDGMENTS All the experiences that I have had at University of Maryland are invaluable and irreplace- able. I would like to thank my adviser, Dr. Stephen J. Wallace. He is a gentle, wise, responsible and respectable man, who has given me so many things. His door is always open for students. He always has paid great attention to me, encouraged me, and born with my stupid questions. I would like to thank all the other collaborators: Dr. David G. Richards, Dr. Colin Morn- ingstar, Dr. Robert Edwards, Dr. Rudolf Fiebig, Dr. Urs M. Heller, Dr. Subhasish Basak, and Dr. George T. Fleming. David always welcomed me when I visited the Jefferson Lab. He gave me much advice, and it has always been fun to be with him. Colin also taught me a lot. Communications with him made me feel that I went one step further. Subhasish and I have been working closely on baryon spectroscopy. It has been very comfortable to work with a person like him. Anisotropic lattice tuning and propagator computations were entirely his accomplishment. George provided me a lot of useful information. The matrix diagonalization program and the fitting program used in this work were developed by him. I owe my gratitude to all the other faculty members of the TQHN group: Dr. Xiangdong Ji, Dr. Thomas Cohen and Dr. James Griffin, and our secretary, Loretta Robinette. Communications with them have been always pleasant and kept the atmosphere in this group warm. My other colleagues, Dr. Daniel Dakin, Ahmad Idilbi, Yingchuan Li, and the postdoc of our group, Dr. Joydip Kundu have enriched my graduate experiences. I would like to thank my undergraduate adviser, Dr. Kenneth H. Hicks. I learned a lot from this extraordinary individual, who gave me direction in my life. He is a person full of insights. He always has a goal and makes efforts. Lastly, I owe my deepest thanks to my family in Japan. Their understanding and care are immeasurable. ii TABLE OF CONTENTS List of Tables v List of Figures vii 1 Introduction 1 1.1 Reviewofparticlephysics . ....... 2 1.1.1 Quarkmodel ................................... 3 1.1.2 Colorcharges.................................. 5 1.1.3 Forcesinnature ................................ 6 1.2 QuantumChromodynamics . .... 8 1.2.1 QCDlagrangian ................................. 8 1.2.2 Asymptoticfreedom .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.2.3 Euclidean path integral approach and Monte Carlo simulation ....... 10 1.2.4 Latticegaugetheory .. .. .. .. .. .. .. .. .. .. .. .. ... 13 1.2.5 Conversionintophysicalscale. ....... 17 1.2.6 Quenchedapproximation . ... 18 1.3 Hadronicresonancesonlattice . ........ 19 1.4 Overviewofthesis ................................ .... 20 2 The Invariant Groups on Cubic Lattice and Baryon Source Construction 22 2.1 Motivationandoverview. ...... 22 2.2 OctahedralGroupandLatticeOperators . ......... 24 2.2.1 Integer angular momentum : O ........................ 25 2.2.2 Half-integer angular momenta: OD ....................... 28 2.2.3 Improperpointgroupsandparity . ..... 30 2.3 Quasi-localBaryonicOperators . ......... 34 2.3.1 Quasi-localNucleonOperators . ...... 36 2.3.2 Quasi-local∆andΩOperators . .... 40 2.3.3 Quasi-localΛBaryonOperators . ..... 41 2.3.4 Quasi-localΣandΞOperators . .... 43 2.4 NonlocalBaryonicOperators . ....... 45 2.4.1 Displaced quark fields and irreps of O ..................... 45 2.4.2 Direct products and Clebsch-Gordan coefficients . .......... 47 2.4.3 One-linkoperators . .. .. .. .. .. .. .. .. .. .. .. .. ... 48 2.4.4 Two-linkoperators. .. .. .. .. .. .. .. .. .. .. .. .. ... 54 2.4.5 One-link displacements applied to two different quarks............ 58 2.5 Summary ......................................... 59 3 Computational Techniques and Lattice Setup for the Simulations 62 3.1 Overview ........................................ 62 3.2 Correlationmatrix and variationalmethod . ........... 62 3.3 Constant gauge-fields and orthogonality relation . .............. 67 3.4 Anisotropiclattices. ....... 70 3.4.1 AnisotropicWilsongaugeaction . ...... 71 3.4.2 AnisotropicWilsonfermionaction . ....... 72 3.5 Smearingmethods ................................. ... 74 3.6 Latticesetup .................................... ... 79 iii 4 Lattice QCD Simulations of Excited Baryon Masses 80 4.1 Overview ........................................ 80 4.2 Numericalcheckoforthogonality . ......... 80 4.3 Equalityofdifferentrows . ...... 83 4.4 N ∗ spectrum ....................................... 86 4.4.1 The G1 spectrum................................. 88 4.4.2 The G2 spectrum................................. 106 4.4.3 The H spectrum ................................. 115 4.4.4 The Aˆ1 linkoperators .............................. 126 4.5 Deltabaryonspectrum. .. .. .. .. .. .. .. .. .. .. .. .. ..... 129 4.5.1 The G1 and H spectra.............................. 130 4.5.2 The G2 spectrum................................. 137 4.6 Comparisonwith physicalbaryonspectra . .......... 139 4.7 Volumedependenceofbaryonmasses. ........ 149 4.8 Summary ......................................... 151 5 Summary and Future 153 A Group Theoretical Projection Operator 156 B Dirac Matrices 157 C Symmetry of Three Dirac Fields 159 D Relations of Nµ1µ2µ3 toMostCommonlyUsedNucleonOperators 164 E Clebsch-Gordan Coefficients of Cubic Group 165 F Jackknife Method 169 Bibliography 171 iv LIST OF TABLES 1.1 Fundamental particles and their properties. ............. 2 1.2 The four fundamental forces in the nature. ........... 7 2.1 Irreducible character table of O ............................. 26 2.2 The subduction of SU(2) to O for integer j ...................... 27 2.3 Bases of irreducible representations of O in terms of spherical harmonics, Yl.m for the lowest values of l. .................................. 28 2.4 Irreducible character table of OD. Only the spinorial irrep are presented. 28 2.5 The subduction of SU(2) to OD ............................. 29 2.6 Bases of irreducible representations of OD. ...................... 30 2.7 Translation of Dirac index µ to ρ- and s-spin indices. Index µ is expressed in Dirac- Paulirepresentation. .. .. .. .. .. .. .. .. .. .. .. .. .... 31 2.8 Quasi-local Nucleon operators. All operators have MA Diracindices. 38 2.9 Quasi-local ∆ operators. All operators have S Dirac indices.............. 41 2.10 Quasi-localΛbaryonoperators.. .......... 42 2.11 Quasi-localΣoperators. ........ 44 + 1 0 2 3 2.12 The T one-link baryon operators. Note that Dˆ Tˆ , Dˆ Tˆ , and Dˆ − Tˆ . 51 1 ≡ 1 ≡ 1 ≡ 1 2.13 The E one-link baryon operators. All operators have mixed Jz............ 52 2.14 Allowed combinations of Dirac indices for different one-link (A1,T1, E) baryons. The displacement is always taken on the third quark for simplicity. The third quark of the Λ and Σ baryons is chosen to be strange