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Information to Users INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the twct directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zed) Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Lattice Studies of the 77 Meson DISSERTATION Presented in Partial Fnlfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Lakshmi Venkataraman. M.S. * * y * * The Ohio State University 1997 Dissertation Committee: Approved by Dr. Gregory Kilcup, .A.dviser Dr. Junko Shigemitsu C o-A dviser Dr. Richard Kass Co-Adviser Dr. Richard Furnstahl Department of Physics DMI Number: 9813367 Copyright 1997 by Venkataraman, Lakshmi All rights reserved. UMI Microform 9813367 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 © Copyright by Lakshmi Venkataraman 1997 ABSTRACT This thesis presents some studies of the r}' meson on the lattice. The eta-prime meson is the flavor singlet cousin of the pseudoscalar octet mesons with a much heavier mass. In the continuum, a qualitative explanation exists to account for its heavy mass within the framework of quantum chromodynamics (QCD) in the limit of large number of colors. However, a quantitative model independent calculation is possible using lattice regularization of QCD which demands a vast amount of supercomputer time. In this thesis, we present such a numerical calculation to obtain the mass of 77' using the staggered fermion formulation both in the quenched and full QCD. The quenched calculation gives a value of 891(101 )MeV while the dynamical calculation yields 780(I87)MeV. Both numbers are consistent with the experimental value within the shown statistical errors. In a closely related study, we compute the lowest 32 eigenmodes of the staggered Dirac operator. The modes are then used to study the validity of the Atiyah-Singer index theorem on the lattice and to construct hadronic correlators. We find that the shifted zero modes of the lattice Dirac operator do allow some of the features of the index theorem to be preserved on the lattice. As a consequence, we show that the disconnected 77' correlator is almost completely determined by these topological zero modes verifying the conventional notion that the f/(l) anomaly gives mass to 77'. To my parents. Ill ACKNOWLEDGMENTS First of all, I would like to thank Prof. Kilcup for his patient guidance of my work over the last few years. His constructive criticism helped an eager but inexperienced graduate student to leam about the methodology of research. I would also like to thank my dissertation committee members Prof. Shigemitsu. Prof. Kass and Prof. Furnstahl for their helpful suggestions which improved the presentation of this dissertation. Finally, none of these would have been possible without the love, encouragement and moral support of my parents and my life partner. IV VITA A pril 19. 1967 ........................................................... Born - Madras. India 1987 ..............................................................................B.Sc. Physics 1989 ..............................................................................M.Sc. Physics 1991 ..............................................................................M.Tech. Computer Science and Data Processing 1991-present .............................................................Graduate Teaching and Research Associate, The Ohio State University. PUBLICATIONS Research Publications L. Venkataraman and G. Kilcup “Applications of the Eigenmodes of the Dirac Oper­ a to r” Proceedings of Lattice '91, July 1997. L. Venkataraman, G. Kilcup and J. G randy “The Staggered rf with Smeared Opera­ tors” Nucl. Phys. B proc. SuppL, 53, 1997 pp. 259-261. G. Kilcup, D. Pekurovsky and L. Venkataraman “On the N f and a dependence of Bfc” Nucl. Phys. B proc. SuppL, 53, 1997 pp. 345-348. G. Kilcup, J. Grandy and L. Venkataraman “r/' and Cooling with Staggered Fermions” Nucl. Phys. B proc. Suppl., 53, 1996 pp. 358-361. Instructional Publications L. Venkataraman and W. D. Ploughe “Magnetic Field of a Long Straight Wire’’ A A P T Sum mer Meeting, University of Notre Dame, Notre Dame, IN, Aug. 1994. L. Venkataraman, R. Padma and P.C. Deshmukh “Classroom Demonstration of an .Atomic Hartree-Fock Program” IV Annual Convention of Indian Association of Physics Teachers, Raipur, India, Oct, 1989. FIELDS OF STUDY Major Field: Physics Studies in Lattice QCD: Prof. G. Kilcup VI TABLE OF CONTENTS Page A b s tr a c t.............................................................................................................................................. ii D e d ic a tio n .......................................................................................................................................... iii Acknowledgments ........................................................................................................................... iv V i t a ...................................................................................................................................................... V List of T ables ................................................................................................................................... ix List of Figures ................................................................................................................................ x C hapters: 1. Introduction ............................................................................................................................ I 1.1 O v e r v ie w .................................................................................................................. 1 1.2 Relation of Our Work to Previous W o rk .................................................... 3 1.3 Contributions of This Thesis .......................................................................... -5 1.4 Organization of This T hesis .............................................................................. 6 9 Lattice Background ......................................................................................................... S 2.1 Overview ................................................................................................................. S 2.2 Staggered Fermions ............................................................................................... 12 2.3 Hadron Correlators ............................................................................................... 16 2.4 Sources of Errors ................................................................................................... 20 VI1 3. Numerical Techniques.................................................................................................. 24 3.1 Computational Requirements for Lattice Q CD ......................................... 24 3.2 Fermion Matrix Inversion ................................................................................ 27 3.2.1 Description of the CO A lgorithm .................................................... 29 3.2.2 Implementation issues .......................................................................... 33 3.3 Eigenvalues and Eigenvectors of the Staggered P .................................. 36 3.3.1 Description of the subspace iteration algorithm ....................... 38 3.3.2 Implementation of the A lgorithm .................................................... 41 3.4 The Lanczos A lgorithm ................................................................................... 47 3.4.1 Cullum and Willoughby’s Lanczos M ethod .................................. 48 3.4.2 Numerical Implementation and T ests ............................................. 51 3.5 Summary and Conclusions ............................................................................... 54 4. T]' meson with Staggered Ferm ions ......................................................................... 55 4.1 Continuum t)' L o r e .............................................................................................. 55 4.2 Lattice calculation of m o ..................................................................................
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