Partially Quenched Chiral Perturbation Theory and a Massless Up Quark: A Lattice Calculation of the Light-Quark-Mass Ratio DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Daniel R. Nelson, B.A. * * * * * The Ohio State University 2002 Dissertation Committee: Approved by Professor Junko Shigemitsu, Adviser Professor Gregory Kilcup Adviser Professor Richard Furnstahl Department of Physics Professor Richard Hughes ABSTRACT The nontrivial topological structure of the QCD gauge vacuum generates a CP breaking term in the QCD Lagrangian. However, measurements of the neutron elec- tric dipole moment have demonstrated that the term's coefficient is unnaturally small, a dilemma known as the strong CP problem. A massless up quark has long been seen as a potential solution, as the term could then be absorbed through the resulting free- dom to perform arbitrary chiral rotations on the up quark field. Through the light-quark-mass ratio mu=md, leading order Chiral Perturbation Theory appears to rule this scenario out. However, the Kaplan-Manohar ambigu- ity demonstrates that certain strong next-to-leading order corrections are indistin- guishable from the effects of an up quark mass. Only a direct calculation of the Gasser-Leutwyler coefficient combination 2L L can resolve the issue. 8 − 5 New theoretical insights into partial quenched Chiral Perturbation Theory have revealed that a calculation of the low-energy constants of the partially quenched chiral Lagrangian is equivalent to a determination of the physical Gasser-Leutwyler coeffi- cients. The coefficient combination in question is directly accessible through the pion mass's dependence on the valence quark mass, a dependence ripe for determination via Lattice Quantum Chromodynamics. ii We carry out such a partially quenched lattice calculation using Nf = 3 staggered fermions and the recently developed smearing technique known as hypercubic block- ing. Through the use of several ensembles, we make a quantitative assessment of our 3 systematic error. We find 2L L = 0:22 0:14 10− , which corresponds to 8 − 5 × a light-quark-mass ratio of m =m = 0:408 0:035. Thus, our study rules out the u d massless-up-quark solution to the strong CP problem. This is the first calculation of its type to use a physical number of light quarks, N = 3, and the first determination of 2L L to include a comprehensive study of f 8 − 5 statistical error. iii to Mom and Dad for their unconditional love iv ACKNOWLEDGMENTS Special thanks go to George Fleming, whose enthusiasm, discipline, and leadership made this work possible. I offer thanks to Gregory Kilcup for the use of his matrix inversion code, as well as for his procurement of the necessary computer resources. I thank the Ohio Supercomputer Center, which provided a majority of the com- puter time consumed by this calculation. Finally, thanks to Angie Linn for her boundless patience. v VITA September 19, 1975 . Born Pittsburgh, Pennsylvania 1997 . .B.A. Physics / Computer Science Juniata College 1997 - 2001 . Teaching Assistant Department of Physics The Ohio State University 1999 . .Programmer Ohio Supercomputer Center 1999 - 2002 . Physics Researcher Department of Physics The Ohio State University PUBLICATIONS Research Publications D.R. Nelson, G.T. Fleming and G.W. Kilcup, Nucl. Phys. Proc. Suppl. 106 (2002) 221, hep-lat/0110112. D.R. Nelson, G.T. Fleming and G.W. Kilcup, (2001), hep-lat/0112029. G.T. Fleming, D.R. Nelson and G.W. Kilcup, (2002), hep-lat/0209141. FIELDS OF STUDY Major Field: Physics Studies in Lattice Quantum Chromodynamics: Professor Gregory Kilcup vi TABLE OF CONTENTS Page Abstract . ii Dedication . iv Acknowledgments . v Vita . vi List of Tables . xii List of Figures . xiv Chapters: 1. Introduction . 1 2. Quantum Chromodynamics . 5 2.1 Symmetries of Quantum Chromodynamics . 6 2.1.1 Color Symmetry . 7 2.1.2 Flavor Symmetry . 7 2.1.3 Spontaneous Symmetry Breaking . 9 2.1.4 Chiral Condensate . 10 2.2 U(1)A Problem . 12 2.2.1 Adler-Bardeen-Jackiw Anomaly . 13 2.2.2 Gauge-Variant Axial Vector Current . 14 2.3 θ Vacua . 16 2.4 Strong CP . 20 vii 3. Chiral Perturbation Theory . 23 3.1 Leading Order Chiral Perturbation Theory . 25 3.2 Leading Order Quark Mass Ratios . 27 3.3 Next-To-Leading Order Chiral Perturbation Theory . 29 3.4 Next-To-Leading Order Quark Mass Ratios . 31 3.5 Phenomenological Results . 35 3.5.1 Meson Decay Constants . 36 3.5.2 Gell-Mann-Okubo Relation . 37 3.6 Kaplan-Manohar Ambiguity . 37 3.7 Theoretical Estimates . 42 3.7.1 Resonance Saturation . 42 3.7.2 Large Nc . 43 3.8 Indirect Phenomenological Results . 45 3.8.1 0 Branching Ratios . 45 3.8.2 ρ0-! Mixing . 46 3.8.3 Baryon Masses . 46 3.8.4 Accepted Values . 47 3.9 First Principles Calculation . 48 3.10 Degenerate Quark Masses . 49 4. Lattice Quantum Field Theory . 50 4.1 Discretization . 50 4.2 Ensemble Creation . 53 4.2.1 Markov Chains . 54 4.2.2 Ergodicity . 54 4.2.3 Detailed Balance . 55 4.2.4 Autocorrelation Length . 57 4.2.5 Acceptance Rate . 58 4.2.6 Thermalization . 58 4.3 Two-Point Correlation Functions . 59 5. Lattice Quantum Chromodynamics . 62 5.1 Gluon Fields . 62 5.1.1 Gauge Discretization . 62 5.1.2 Gauge Action . 64 5.1.3 Gauge Coupling . 65 5.1.4 Gauge Updates . 66 5.2 Quark Fields . 67 viii 5.2.1 Fermion Discretization . 67 5.2.2 Fermion Action . 68 5.2.3 Color Symmetry . 70 5.2.4 Quark Propagators . 70 5.2.5 Quenched Approximation . 72 5.2.6 Partially Quenched Approximation . 73 5.2.7 Fermion Doubling Problem . 74 5.3 Staggered Fermions . 76 5.3.1 Staggered Action . 77 5.3.2 Shift Symmetry . 78 5.3.3 Even-Odd Symmetry . 79 5.3.4 Quarks . 80 5.3.5 Quark Bilinears . 82 5.3.6 Meson States . 86 5.3.7 Quark Action . 91 5.3.8 Flavor Symmetry . 92 5.3.9 N = 4 . 93 f 6 5.4 Conjugate Gradient . 94 5.5 R Algorithm . 96 5.5.1 Hybrid Molecular Dynamics . 97 5.5.2 Ensemble Equilibrium . 99 5.5.3 Finite Step-Size Error . 101 5.5.4 N = 4 . 102 f 6 5.6 Hypercubic Blocking . 103 5.7 Sommer Scale . 107 5.7.1 Static Quark Potential . 108 5.7.2 Wilson Loops . 108 5.7.3 Corrected Cornell Potential . 110 5.8 Meson Propagators . 113 5.8.1 Bilinear Correlators . 114 5.8.2 Meson Masses . 119 5.8.3 Meson Decay Constants . 120 6. Partially Quenched Chiral Perturbation Theory . 124 6.1 Partially Quenched Quantum Chromodynamics . 125 6.2 Partially Quenched Flavor Symmetry . 127 6.3 Graded Groups . 128 6.4 Partially Quenched Chiral Lagrangian . 129 6.5 Next-to-Leading Order Meson Properties . 132 6.6 Physical Results from Partially Quenched Calculations . 133 6.7 Calculation of the Gasser-Leutwyler Coefficients . 135 ix 6.8 Constant Dynamical Quark Mass . 136 7. Data Modeling . 138 7.1 Lattice Measurements . 138 7.2 Data Blocking . 139 7.3 Data Correlation . 140 7.4 Theoretical Models . 142 7.5 Jackknife Error . 144 7.6 Specific Applications ..