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Large-N Limit As a Classical Limit: Baryon in Two-Dimensional QCD

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Large-N Limit As a Classical Limit: Baryon in Two-Dimensional QCD Large N Limit as a Classical Limit: Baryon in − Two-Dimensional QCD and Multi-Matrix Models by Govind Sudarshan Krishnaswami Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Sarada G. Rajeev arXiv:hep-th/0409279v1 27 Sep 2004 Department of Physics and Astronomy The College Arts & Sciences University of Rochester Rochester, New York 2004 ii Curriculum Vitae The author was born in Bangalore, India on June 28, 1977. He attended the Uni- versity of Rochester from 1995 to 1999 and graduated with a Bachelor of Arts in math- ematics and a Bachelor of Science in physics. He joined the graduate program at the University of Rochester as a Sproull Fellow in Fall 1999. He received the Master of Arts degree in Physics along with the Susumu Okubo prize in 2001. He pursued his research in Theoretical High Energy Physics under the direction of Professor Sarada G. Rajeev. iii Acknowledgments It is a pleasure to thank my advisor Professor S. G. Rajeev for the opportunity to be his student and learn how he thinks. I have enjoyed learning from him and working with him. I also thank him for all his support, patience and willingness to correct me when I was wrong. I am grateful to Abhishek Agarwal, Levent Akant, Leslie Baksmaty and John Vargh- ese for their friendship, our discussions and collaboration. Leslie’s comments on the introductory sections as I wrote this thesis, were very helpful. I thank Mishkat Bhat- tacharya, Alex Constandache, Subhranil De, Herbert Lee, Cosmin Macesanu, Arsen Melikyan, Alex Mitov, Jose Perillan, Subhadip Raychaudhuri and many other students for making my time in the department interesting, and for their help and friendship. Thanks are also due to the Physics and Astronomy department for all the support I have received as a student at Rochester. Prof. Arie Bodek helped with his advice and discussions on comparison of our results with experimental data. Prof. Nick Bigelow, Ashok Das, Joe Eberly, Yongli Gao, Richard Hagen, Bob Knox, Daniel Koltun, Susumu Okubo, Lynne Orr, Judy Pipher, Yonathan Shapir, Ed Thorndike, Steve Teitel, Emil Wolf, Frank Wolfs and Wenhao Wu have taught me courses and been generous with their advice and support as have all the other faculty in our department. I would also like to acknowledge Prof. Eric Blackman, Tom Ferbel, Steve Manly and Jonathan Pakianathan, who have been on my exam committees. I have also learnt from and had useful discus- sions with many of the faculty in the mathematics department, including Fred Cohen, Michael Gage, Steve Gonek, Allan Greenleaf, Naomi Jochnowitz and Douglas Ravenel. The staff at the Physics and Astronomy department has been a great help. I would like to thank Connie Jones, Barbara Warren, Judy Mack, Diane Pickersgill, Shirley Brignall, Sue Brightman, Pat Sulouff, Marjorie Chapin, Dave Munson, Mary Beth Vogel and all the others for going out of their way for me. I would also like to thank my parents Leela and Krishnaswami, my sister Maithreyi, and my wife Deepa for all their love and support. The Bartholomeusz’, Bhojs, Govenders, Rajamanis, Rajeevs, Savi and Madu have been family away from home while I was a student at Rochester. iv Abstract In this thesis, I study the limit of a large number of colors (N) in a non-abelian gauge theory. It corresponds to a classical limit where fluctuations in gauge-invariant observables vanish. The large-dimension limit for rotation-invariant variables in atomic physics is given as an example of a classical limit for vector models. The baryon is studied in Rajeev’s reformulation of two-dimensional QCD in the large-N limit: a bilocal classical field theory for color-singlet quark bilinears, whose phase space is an infinite grassmannian. In this approach, ’t Hooft’s integral equation for mesons describes small oscillations around the vacuum. Baryons are topological solitons on a disconnected phase space, labelled by baryon number. The form factor of the ground-state baryon is determined variationally on a succession of increasing- rank submanifolds of the phase space. These reduced dynamical systems are rewritten as interacting parton models, allowing us to reconcile the soliton and parton pictures. The rank-one ansatz leads to a Hartree-type approximation for colorless valence quasi- particles, which provides a relativistic two-dimensional realization of Witten’s ideas on baryon structure in the 1/N expansion. The antiquark content of the baryon is small and vanishes in the chiral limit. The valence-quark distribution is used to model parton distribution functions measured in deep inelastic scattering. A geometric adaptation of steepest descent to the grassmannian phase space is also given. Euclidean large-N multi-matrix models are reformulated as classical systems for U(N) invariants. The configuration space of gluon correlations is a space of non- commutative probability distributions. Classical equations of motion (factorized loop equations) contain an anomaly that leads to a cohomological obstruction to finding an action principle. This is circumvented by expressing the configuration space as a coset space of the automorphism group of the tensor algebra. The action principle is in- terpreted as the partial Legendre transform of the entropy of operator-valued random variables. The free energy and correlations in the N limit are determined varia- → ∞ tionally. The simplest variational ansatz is an analogue of mean-field theory. The latter compares well with exact solutions and Monte-Carlo simulations of one and two-matrix models away from phase transitions. Contents 1 Introduction 1 1.1 QCD and the Large-N ClassicalLimit .................... 1 1.2 QCD, Wilson Loop, Gluon Correlations . 14 1.3 d as a Classical Limit in Atomic Physics . 18 →∞ 1.4 The Large-N limit: Planar Diagrams and Factorization . 21 I Baryon in the Large-N Limit of 2d QCD 25 2 From QCD to Hadrondynamics in Two Dimensions 27 2.1 Large-N Limitof2dQCD........................... 27 2.2 Classical Hadron Theory on the Grassmannian . ..... 31 3 Ground-state of the Baryon 36 3.1 Steepest Descent on the Grassmannian . 37 3.2 Separable Ansatz: Formulation . 39 3.2.1 Rank-OneConfigurations . 39 3.2.2 Quantizing the Separable Ansatz: Interacting ValenceQuarks. 40 3.2.3 Hartree Approximation and the Large-N limit ........... 42 3.2.4 Pl¨ucker Embedding and Density Matrix . 43 3.3 SeparableAnsatz:Solution . 44 3.3.1 Potential Energy in Momentum space . 45 3.3.2 Integral Equation for Minimization of Baryon Mass . ...... 47 v vi 3.3.3 Small Momentum Behavior of Wave function . 48 3.3.4 Exact Solution for N = ,m = 0: Exponential Ansatz . 50 ∞ 3.3.5 EffectofFiniteCurrentQuarkMass . 53 3.3.6 Variational ground-state for Finite N ................ 55 3.4 BeyondtheSeparableAnsatz . 56 3.4.1 Fixed Rank Submanifolds of Grassmannian . 56 3.4.2 RankThreeAnsatz .......................... 58 3.4.3 FockSpaceDescription . 60 3.4.4 Valence, Sea and Antiquarks: Bogoliubov Transformation . 65 3.4.5 Variational Estimate of Antiquark Content . 66 3.5 Application to Deep Inelastic Scattering . ....... 71 II Large-N Matrix Models as Classical systems 77 4 Free Algebras and Free Probability Theory 80 4.1 Tensor Algebras and Operator-valued Random Variables . ......... 81 4.2 Non-Commutative Probability Distributions . ........ 83 4.3 Automorphisms and Derivations of Tensor Algebras . ....... 85 4.4 Free Entropy of Non-Commutative Probability Theory . ........ 89 5 Euclidean Large-N Matrix Models 91 5.1 Loop Equations, Classical Configuration Space . ....... 92 5.2 ClassicalActionPrinciple . 95 5.2.1 Anomaly as a Cohomology Class . 95 5.2.2 Configuration Space as a Coset Space / ........... 97 P G SG 5.2.3 Finite Change of Variable: Formula for χ on / .........100 G SG 5.3 χ asanEntropy ................................104 5.4 Variational Approximations . 107 5.4.1 Two Matrix Model with Interaction g (A4 + B4) 2cAB . 107 4 − 5.4.2 Two Matrix Model with A2 + A2 [A , A ]2 Interaction . 109 1 2 − 1 2 vii 6 Summary 115 1 A Definition of p2 Finite Part Integrals 118 B One Matrix Model 122 B.1 Equations of Motion and Classical Action . 122 B.2 Mean Field Theory for Quartic One Matrix Model . 126 B.3 Beyond Mean Field Theory: Non-linear Ansatz . 128 C Single and Multivariate Wigner Distribution 130 D Anomaly is a Closed 1-form 133 E Group Cohomology 134 F Formula for Entropy 136 Bibliography 138 List of Figures a(m) πm2 3.1 h(a). The solution for the exponent p is the point at which the horizontal line g˜2 intersects the curve h(a). For m = 0, a = 0 is a solution: in the limit of chiral symmetry, the wave function goes to a non-zero constant at the origin. 49 1 2 2 2 m2 m2 + 2 + 3.2 Estimate of baryon mass in units ofg ˜ N as a function of a. The curves top to bottom are for 2 = .05, .02, .005, .001. As 2 0 , both the minimum value of (mass) and the value of a that minimizes it tend to 0 , recovering the exact massless exponential solution in the chiral limit. 54 4 g˜ g˜ → ˜ a bp m2 3.3 The power a in the wave function ψ(p)= Ap e− plotted as a function of 1000 g˜2 . 54 2 2 2 m ζ− 3.4 ζ measuring departure from valence-quark approximation plotted as a percent as a function of 1000 g˜2 . The probability of finding an anti-quark in the baryon is 2 . The valence-quark approximation becomes exact in the chiral limit. 69 − 1+2ζ− a m2 3.5 The power p governing behavior of the wave functions near the origin, plotted as a function of 1000ν = 1000 2 .
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