UNIVERSITY OF CINCINNATI _____________ , 20 _____ I,______________________________________________, hereby submit this as part of the requirements for the degree of: ________________________________________________ in: ________________________________________________ It is entitled: ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ Approved by: ________________________ ________________________ ________________________ ________________________ ________________________ CURVATURE DEPENDENCE OF CLASSICAL SOLUTIONS EXTENDED TO HIGHER DIMENSIONS A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTORATE OF PHILOSOPHY (PH.D.) in the Department of Physics of the College of Arts and Sciences 2003 by Athula Herat B.Sc., University of Colombo, Sri Lanka, 1994 M.S., University of Cincinnati, 1996 Committee Chair: Dr. Peter Surnayi Abstract I study the curvature dependence of extended classical solutions of interacting field theories. The main motivation behind my study of classical solutions is confinement. It has become apparent that classical objects (monopoles and vortices) are responsible for color confinement in QCD. Therefore, studying the properties of these objects is extremely important. The main goal of this investigation is to accurately determine the energy of extended classical solutions. In most applications it is assumed that the energy depends simply on the length of the object, irrespective of the shape. I show that the curvature effects are highly non-trivial. I start with the kink solutions of the (2+1) dimensional linear sigma model. In particular, I show that the curvature energy of a kink in two spatial dimensions, as a prototype of extended classical solutions, is always negative. Assuming that the deviations of the kink from the straight line are small, I derive a closed form for the curvature energy. This investigation clearly demonstrates that the energy of the (2+1) dimensional kink has a positive length term and a negative curvature term. Next I look at extended vortex solutions. Vortices are (2+1) dimensional soliton solutions of gauge theories coupled to complex scalar Higgs field. In the trivially extended vortices the core of the vortex forms a straight line. Recent lattice studies show that vortices appear not be straight. Therefore, it is important to study the curvature effects of vortex solutions. However, unlike in the case of the kink, no analytical vortex solutions exist, which makes the study of curved vortices an ex- i tremely complex one. This problem can be made less daunting if circular vortices are considered. Therefore, I study the circular vortex of the U(1) gauge theory. I use numerical techniques to obtain the circular vortex configurations that minimize the Hamiltonian. I show that the curvature energy of the circular vortex is negative, which clearly demonstrates that the total energy of the (3+1) dimensional circular vortex is less than the length energy. I.e. the extended abelian vortex prefers the curved state to the straight one. ii iii Acknowledgments Throughout my graduate student life at the Department of Physics, University of Cincinnati, I have have been extremely lucky to have had the guidance, support, and the friendship of many people. I would like to take this opportunity to extend my gratitude to all of them. First and foremost, I would like to thank my advisor Dr. Peter Suranyi for giving me the opportunity to work with him. His underlying support through the years, academically, professionally, as well as personally, has made it possible for me to achieve this goal. It has been a pleasure to work under his guidance and I am indebted him for his patient help throughout the course of my research. I am extremely grateful to Dr. Rohana Wijewardhana for being both a mentor and a friend. His guidance and assistance from the very first day I set foot in Cincinnati, especially during times when the going was tough, has been invaluable to me. I would like to extend a special thanks to Dr. Frank Pinski for his invaluable advice and assistance with the numerical aspects of the research. I thank Dr. Brian Meadows and Dr. Mark Jarrell as well as all of the above for serving in my dissertation committee. It has been a great privilege to work and interact with my talented and gifted colleagues and friends. I thank them all for their cooperation and their entertaining company. I am grateful to David Will for helping me solve every single computer issue that came up. I would also like to thank Dr. Mike Sokoloff for his help in various aspects of my stay at the Department of Physics. iv Without the dedication and the support of my family, I would not have had the courage to undertake this endeavor. I owe a special debt of gratitude to my wonderful wife, Manori, for her loving support and for the countless sacrifices she has made over the years so that I could achieve my dream. To my beautiful daughter Hiruni, I am so glad that you came into my life during this entire process. I could not have completed this effort without the both of you. I am also grateful to my sister and brother-in-law (Tamara and Sampath Jayawardane) and my parents-in-law (Walter and Indrani Singhabahu) for their love and support. I would like to dedicate this work to my parents (Samson and Ramani Herat) who have been a constant source of encouragement and inspiration to me throughout my life. I am grateful to them for their warmth, support and guidance. v Contents 1 Introduction 5 1.1 Confinement . 6 1.2 Models of Confinement . 7 1.2.1 Dual superconductor model of confinement . 8 1.2.2 Abelian Projection . 11 1.2.3 Abelian Projection on a Lattice . 14 1.2.4 Center vortex model of confinement . 18 1.2.5 Monopoles vs. Vortices . 23 2 The Undulating Kink 25 2.1 Introduction . 25 2.2 Kink solution . 27 2.3 Extension to 2+1 dimensions . 29 2.4 Exact solutions for small deviations from the straight line . 32 2.5 Non-infinitesimal undulation . 36 1 3 The Circular Vortex 44 3.1 Introduction . 44 3.2 The Straight Vortex . 46 3.2.1 Solutions of the straight Vortex . 52 3.2.2 Results . 55 3.3 The Circular Vortex . 57 3.4 Numerical Treatment . 62 3.4.1 Variational Method . 63 3.4.2 Jacobian Iteration Method . 64 3.5 Results . 65 4 Conclusions 68 4.1 Classical Objects and Confinement . 69 4.1.1 't Hooft's dual superconductivity model . 69 4.1.2 Abelian Projection . 70 4.1.3 Center Vortex Model of Confinement . 72 4.2 The Undulating Kink . 75 4.3 The Circular Vortex . 77 2 List of Figures 1.1 (a) The Abrikosov string between the monopoles in the superconductor and (b) an analogue of the Abrikosov string between the chromo-electrically charged particles in the dual superconductor. 9 1.2 Abelian and nonabelian potentials (with self energy V0 subtracted), Ref. [17]. 16 1.3 The abelian potential (diamonds) in comparison with the photon contribu- tion (squares), the monopole contribution (crosses) and the sum of these two parts (triangles), Ref. [17]. 17 1.4 (a) The "Spaghetti vaccum" of condensed vortices (b) The "monopole vacumm" appears as a result of abelian projection [25] . 24 2.1 (a) A plot of the static kink(anti-kink) solution (2.4) (b) The energy density of the kink (2.5). It is localized. 29 3.1 Plot of the straight vortex field configurations for different values of the coupling constant. (a) β = βc = 1=4, critical coupling. (b) β = 3βc. (c) β = 2βc. (d) β = βc=2. (e) β = βc=3. 56 3 3.2 Plot of the variation method approximate circular vortex field configurations for critical coupling, β = βc = 1=4 . 64 3.3 Plot of the circular vortex field configurations for different values of the coupling constant. (a) β = βc = 1=4, critical coupling. (b) β = 3βc. (c) β = 2βc. (d) β = βc=2. (e) β = βc=3. 66 4 Chapter 1 Introduction One of the major developments of twentieth century physics has been recognition that all the known interactions can be described in terms of gauge theories [1]. Gauge the- ories in general exhibit a rich spectrum of finite energy classical solutions. Of these vortices, monopoles, and instantons are the best known topological solutions in 2, 3, and 4 dimensions. These classical objects play a major role in describing a wide vari- ety of phenomena such as magnetic field lines in type II superconductors [3], cosmic strings in grand unified theories [4], and objects responsible for confinement in dy- namical models of nonabelian gauge theories [5][6], to name a few. Our investigations of classical solutions extended to higher dimensions have been manily motivated by the recent success of these dynamical models of confinement. 5 1.1 Confinement Ample experimental evidence exists to support the fact that the strongly interacting particles, hadrons (i.e. protons, neutrons,etc.), are made of more elementary con- stituents called quarks. There is further evidence that, in addition to their electric charge, quarks carry an additional quantum number called color. In fact, quarks come in three colors. A non-abelian gauge theory, Quantum ChromoDynamics(QCD) ex- plains the interactions between quarks as due to the exchange of a set of particles called the gluons. This theory is similar to Quantum Electrodynamics (QED) that explains the Electromagnetic interactions between electrically charged particles as due to the exchange of a particle, the photon. The main difference is that gluons interact among themselves while the photons do not. The self interactions between gluons play an important role in determining the nature of the color force. The strength of the strong force between quarks is given by the gauge coupling strength.