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Understanding Angular Momentum in Hadrons

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Understanding Angular Momentum in Hadrons Understanding Angular Momentum in Hadrons Kunal Kathuria Charlottesville, VA A Thesis Presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Physics UNIVERSITY OF VIRGINIA August, 2013 Declaration of Authorship I, Kunal Kathuria, declare that this thesis titled, `THESIS TITLE' and the work pre- sented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is my own work. I have acknowledged all main sources of help. Signed: Date: i "Humility means not being anxious to be honored by others." \Eloquence is truth concisely stated." [AC Bhaktivedanta Swami] If you can't explain it to a six year old, you don't understand it yourself. [Albert Einstein] At the heart of all knowledge lies simplicity of expression. UNIVERSITY OF VIRGINIA Abstract Department of Physics Doctor of Philosophy by Kunal Kathuria Charlottesville, VA The spin puzzle has been a relatively long-standing unresolved issue in hadron physics. We review and derive, using a wave-packet formalism, the first step preceding all spin sum rules: the connection of the angular momentum operator to the gravitomagnetic form factors of the energy-momentum tensor (EMT). We present our spin sum rule for spin one hadrons and comment on why the deuteron is an important system to study in an effort to understanding hadron spin and interactions better. We will visit a number of illuminating properties of spin seen from the wave packet approach (e.g. consequences/promotion of azimuthal symmetry in momentum space as the general rule for origin independence of angular momentum, merits of the different decompositions in the literature, a brand new way to express angular momentum, common \accepted" mistakes associated with the derivation etc) and will also say why this approach is the necessary starting point for all treatments. Additionally, we will attempt to clear a number of misconceptions surrounding the operators entering the sum rule (what the different relevant operators are, what the decompositions are, the merits of a partonic decomposition, the EMT in the free vs interacting theory, it being canonical vs Belinfante vs symmetric vs conserved etc) with simple and pertinent observations. Acknowledgements I would like to express my gratitude to my PhD advisor Simonetta Liuti for her very supportive attitude and very natural and open comportment during my work with her. I am grateful to her for her humble and gracious approach; for considering ideas just on merit without bias and strongly encouraging the originality she sees in her students. I am also thankful to Gary Goldstein for his collaboration and help in general. Thanks goes to all my collaborators in work and fellow students for discussions along the way, especially to J. Osvaldo Hernandez. I would like to say thank you to the physics department for supporting me as a graduate student and helping me earn a PhD in physics. I am extremely grateful for the support and presence of Amit Acara dasa and Mayapriya devi dasi, without whom my efforts would have been meaningless, and the rest of my family here, including my wife Prachi. I owe a big thanks to my parents for indulging my creative desires all along and I also thank my sister, Kamna. I thank my wife Prachi, the artist, for drawing the figures I needed for this document. Finally, thank you to Sunil Patel for providing the template for this manuscript. iv Contents Declaration of Authorshipi Abstract iii Acknowledgements iv List of Figures viii List of Tables ix Abbreviationsx Physical Constants xi 1 Introduction1 1.1 Preface: Writing Style.............................1 1.2 Introduction...................................2 1.2.1 Background...............................2 1.2.2 The Spin Crisis.............................4 1.2.3 Current Work..............................6 1.2.4 Specifics on Kinematical Regime...................7 1.2.5 Light-Cone Coordinates........................8 1.2.6 Brief Introduction to Generalized Parton Distributions.......8 1.2.7 Connection with the Energy-Momentum Tensor........... 13 2 Hadronic Spin Decomposition 14 2.1 The Two Decompositions and Operators Related to Angular Momentum. 14 2.2 Overview and Wave-Packet Results...................... 18 2.2.1 Overview................................ 18 2.2.2 The Nucleon Spin Sum Rule Developed by X. Ji.......... 19 2.2.3 Effective Zero Orbital Angular Momentum from Symmetry Con- siderations............................... 20 2.2.4 Azimuthal Momentum Symmetry is the Criterion for Zero OAM. 21 2.2.5 Necessity of Wave Packets and Comment on Separability of An- gular Momentum............................ 23 v Contents vi 2.2.6 Dark Angular Momentum....................... 24 2.2.7 Another Observable.......................... 25 2.3 A careful wave-packet treatment....................... 26 2.3.1 Relating Angular Momentum to the EMT.............. 26 2.3.2 Derivation of Sum Rule........................ 29 2.3.3 Obtaining a sum-rule with wave-packets............... 34 2.3.4 Translation-invariance for momentum and spin and translation- dependence for OAM.......................... 35 2.4 Gravitational Form Factors.......................... 37 2.4.1 The Connection with GPDs...................... 37 2.4.2 Checking Individual Terms for Conservation............. 39 3 Deuteron 42 3.1 Background................................... 42 3.2 Energy-Momentum Tensor Decomposition.................. 43 3.2.1 Trace Anomaly............................. 44 3.3 Spin One Sum Rule.............................. 44 3.4 Helicity Amplitudes: Formalism........................ 51 3.4.1 Simple Examples of Transformations................. 55 3.4.2 Accessing the Compton Form Factors................ 61 4 Simple Physical Arguments Related to Angular Momentum 62 4.1 Boost-like Operators.............................. 62 4.1.1 Commutation of Boost and Rotations................ 62 4.1.2 Degeneracy in Spin........................... 64 4.1.3 Pauli-Lubanski Vector: Contribution to Light-Cone Boost or An- gular Momentum?........................... 69 4.1.4 Good and Bad Components of the Pauli-Lubanski Vector..... 72 4.1.5 Longitudinal and Transverse Angular Momentum for the Nucleon and the Deuteron............................ 73 4.1.6 What About Orbital Angular Momentum?............. 77 4.1.7 What Does a Boost Do Classically and Quantum Mechanically to Angular Momentum?......................... 77 4.1.8 Meaning of Spin and Orbital Contributions in a Stationary State. 82 4.1.9 Why Transverse Angular Momentum Dominates.......... 84 4.1.10 Angular Momentum Contribution of the Magnetic Form Factor for the Proton and Deuteron..................... 85 4.1.11 What Does Boosting Do to Spin Individually?........... 86 4.1.12 Ji's Recent Argument for Gluon Spin................. 86 4.1.13 Spin-Orbit Coupling.......................... 87 4.1.14 Is There Origin Dependence in Angular Momentum? (On the Density Interpretation)........................ 88 4.1.15 Covariance of the Off-forward Matrix Element........... 88 4.1.16 Aside: Operator-shifting in Field Theory.............. 89 4.1.17 Comparing the Ji and JM Decompositions of the EMT....... 90 4.1.18 Further phenomenological development: Quark-Gluon Separation. 93 4.1.19 Eigenstates of the Pauli-Lubanski Spin Vector............ 93 4.1.20 Normalization of T µν ......................... 94 Contents vii 4.1.21 Burkardt's Torque Analysis...................... 95 4.1.22 On a Comment from Leader's Paper................. 95 5 Vorticity 97 5.1 Introduction and Observations........................ 97 5.1.1 The Nature of Vorticity........................ 97 5.1.2 Extension to Field Theory: Vorticity Density............ 100 5.1.3 The Circulation............................. 100 5.1.4 Vorticity is a Conserved Charge.................... 102 5.2 Calculations: Relating Vorticity and Angular Momentum......... 103 5.2.1 Involving Off-diagonal Elements of EMT............... 103 5.2.2 Writing Momentum as a Function of Vorticity............ 103 5.2.3 Evaluating the Becattini Term: Redefinition of J in terms of vorticity104 6 Conclusions and Philosophy 107 6.1 Summary of Contributions........................... 107 6.2 Brief Points of Philosophy........................... 110 6.2.1 Research Perspective.......................... 113 A Review 115 B Derivation of the Vorticity-Angular Momentum Connection 119 C Constructing the Transformation Matrices 122 Bibliography 125 List of Figures 1.1 Relative contributions of quark-spin, quark OAM, gluon spin, gluon OAM (JM) and quark-spin, quark OAM 0, gluon AM (Ji) of the two decomposi- tions of angular momentum of the proton in a simplified model, calculated by M. Burkardt. Each model sums to the same whole: total angular mo- mentum of the proton..............................5 1.2 Deeply Virtual Compton Scattering.....................9 2.1 Zero angular momentum of 2-particle system at any time in simple clas- sical example.................................. 21 z 2.2 Gaussian wave-packet centered at p0 ..................... 31 3.1 (color online) Upper panel: Total angular
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