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Quantum Chromodynamics and Other Field Theories on the Light Cone Physics Reports 301 (1998) 299Ð486 Quantum chromodynamics and other field theories on the light cone Stanley J. Brodsky!, Hans-Christian Pauli", Stephen S. Pinsky# ! Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA " Max-Planck-Institut fu( r Kernphysik, D-69029 Heidelberg, Germany # Ohio State University, Columbus, OH 43210, USA Received October 1997; editor: R. Petronzio Contents 1. Introduction 302 4.2. Quantum chromodynamics in 1#1 2. Hamiltonian dynamics 307 dimensions (KS) 362 2.1. Abelian gauge theory: quantum 4.3. The Hamiltonian operator in 3#1 electrodynamics 308 dimensions (BL) 366 2.2. Non-abelian gauge theory: 4.4. The Hamiltonian matrix and its Quantum chromodynamics 311 regularization 376 2.3. Parametrization of spaceÐtime 313 4.5. Further evaluation of the Hamiltonian 2.4. Forms of Hamiltonian dynamics 315 matrix elements 381 2.5. Parametrizations of the front form 317 4.6. Retrieving the continuum formulation 381 2.6. The Poincare« symmetries in the front 4.7. E¤ective interactions in 3#1 dimensions 385 form 319 4.8. Quantum electrodynamics in 3#1 2.7. The equations of motion and the dimensions 389 energyÐmomentum tensor 322 4.9. The Coulomb interaction in the front form 393 2.8. The interactions as operators acting in 5. The impact on hadronic physics 395 Fock space 327 5.1. Light-cone methods in QCD 395 3. Bound states on the light cone 329 5.2. Moments of nucleons and nuclei in the 3.1. The hadronic eigenvalue problem 330 light-cone formalism 402 3.2. The use of light-cone wavefunctions 334 5.3. Applications to nuclear systems 407 3.3. Perturbation theory in the front form 336 5.4. Exclusive nuclear processes 409 3.4. Example 1: The qqN -scattering amplitude 338 5.5. Conclusions 411 3.5. Example 2: Perturbative mass 6. Exclusive processes and light-cone renormalization in QED (KS) 341 wavefunctions 412 3.6. Example 3: The anomalous magnetic 6.1. Is PQCD factorization applicable to moment 345 exclusive processes? 414 3.7. (1#1)-dimensional: Schwinger model 6.2. Light-cone quantization and heavy (LB) 349 particle decays 415 3.8. (3#1)-dimensional: Yukawa model 352 6.3. Exclusive weak decays of heavy hadrons 416 4. Discretized light-cone quantization 358 6.4. Can light-cone wavefunctions be 4.1. Why discretized momenta? 360 measured? 418 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S0370-1573(97)00089-6 QUANTUM CHROMODYNAMICS AND OTHER FIELD THEORIES ON THE LIGHT CONE Stanley J. BRODSKY!, Hans-Christian PAULI", Stephen S. PINSKY# ! Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA " Max-Planck-Institut fu( r Kernphysik, D-69029 Heidelberg, Germany # Ohio State University, Columbus, OH 43210, USA AMSTERDAM Ð LAUSANNE Ð NEW YORK Ð OXFORD Ð SHANNON Ð TOKYO S.J. Brodsky et al. / Physics Reports 301 (1998) 299Ð486 301 7. The light-cone vacuum 419 9.2. Flavor symmetries 450 7.1. Constrained zero modes 419 9.3. Quantum chromodynamics 453 7.2. Physical picture and classification of zero 9.4. Physical multiplets 454 modes 429 10. The prospects and challenges 456 7.3. Dynamical zero modes 434 Appendix A. General conventions 460 8. Non-perturbative regularization and Appendix B. The LepageÐBrodsky convention renormalization 436 (LB) 462 8.1. TammÐDanco¤ Integral equations 437 Appendix C. The KogutÐSoper convention (KS) 463 8.2. Wilson renormalization and confinement 442 Appendix D. Comparing BD- with LB-spinors 465 9. Chiral symmetry breaking 446 Appendix E. The DiracÐBergmann method 466 9.1. Current algebra 447 References 476 Abstract In recent years light-cone quantization of quantum field theory has emerged as a promising method for solving problems in the strong coupling regime. The approach has a number of unique features that make it particularly appealing, most notably, the ground state of the free theory is also a ground state of the full theory. We discuss the light-cone quantization of gauge theories from two perspectives: as a calculational tool for representing hadrons as QCD bound states of relativistic quarks and gluons, and also as a novel method for simulating quantum field theory on a computer. The light-cone Fock state expansion of wavefunctions provides a precise definition of the parton model and a general calculus for hadronic matrix elements. We present several new applications of light-cone Fock methods, including calculations of exclusive weak decays of heavy hadrons, and intrinsic heavy-quark contributions to structure functions. A general non-perturbative method for numerically solving quantum field theories, ªdiscretized light-cone quantizationº, is outlined and applied to several gauge theories. This method is invariant under the large class of light-cone Lorentz transformations, and it can be formulated such that ultraviolet regularization is independent of the momentum space discretization. Both the bound-state spectrum and the corresponding relativistic light-cone wavefunc- tions can be obtained by matrix diagonalization and related techniques. We also discuss the construction of the light-cone Fock basis, the structure of the light-cone vacuum, and outline the renormalization techniques required for solving gauge theories within the Hamiltonian formalism on the light cone. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 11.10.Ef; 11.15.Tk; 12.38.Lg; 12.40.Yx 302 S.J. Brodsky et al. / Physics Reports 301 (1998) 299Ð486 1. Introduction One of the outstanding central problems in particle physics is the determination of the structure of hadrons such as the proton and neutron in terms of their fundamental quark and gluon degrees of freedom. Over the past 20 years, two fundamentally di¤erent pictures of hadronic matter have developed. One, the constituent quark model (CQM) [469], or the quark parton model [144,145], is closely related to experimental observation. The other, quantum chromodynamics (QCD) is based on a covariant non-abelian quantum field theory. The front form of QCD [172] appears to be the only hope of reconciling these two. This elegant approach to quantum field theory is a Hamiltonian gauge-fixed formulation that avoids many of the most di¦cult problems in the equal-time formulation of the theory. The idea of deriving a front form constituent quark model from QCD actually dates from the early 1970s, and there is a rich literature on the subject [74,119,135,30,6,120,304,305,332,350,87,88,235Ð237]. The main thrust of this review will be to discuss the complexities that are unique to this formulation of QCD, and other quantum field theories, in varying degrees of detail. The goal is to present a self-consistent framework rather than trying to cover the subject exhaustively. We will attempt to present su¦cient background material to allow the reader to see some of the advantages and complexities of light-front field theory. We will, however, not undertake to review all of the successes or applications of this approach. Along the way we clarify some obscure or little-known aspects, and o¤er some recent results. The light-cone wavefunctions encode the hadronic properties in terms of their quark and gluon degrees of freedom, and thus all hadronic properties can be derived from them. In the CQM, hadrons are relativistic bound states of a few confined quark and gluon quanta. The momentum distributions of quarks making up the nucleons in the CQM are well-determined experimentally from deep inelastic lepton scattering measurements, but there has been relatively little progress in computing the basic wavefunctions of hadrons from first principles. The bound-state structure of hadrons plays a critical role in virtually every area of particle physics phenomenology. For example, in the case of the nucleon form factors and open charm photoproduction the cross sections depend not only on the nature of the quark currents, but also on the coupling of the quarks to the initial and final hadronic states. Exclusive decay processes will be studied intensively at B-meson factories. They depend not only on the underlying weak transitions between the quark flavors, but also the wavefunctions which describe how B-mesons and light hadrons are assembled in terms of their quark and gluon constituents. Unlike the leading twist structure functions measured in deep inelastic scattering, such exclusive channels are sensitive to the structure of the hadrons at the amplitude level and to the coherence between the contributions of the various quark currents and multi-parton amplitudes. In electro-weak theory, the central unknown required for reliable calculations of weak decay amplitudes are the hadronic matrix elements. The coe¦cient functions in the operator product expansion needed to compute many types of experimental quantities are essentially unknown and can only be estimated at this point. The calculation of form factors and exclusive scattering processes, in general, depend in detail on the basic amplitude structure of the scattering hadrons in a general Lorentz frame. Even the calculation of the magnetic moment of a proton requires wavefunctions in a boosted frame. One thus needs a practical computational method for QCD which not only determines its spectrum, but which can provide also the non-perturbative hadronic matrix elements needed for general calculations in hadron physics. S.J. Brodsky et al. / Physics Reports 301 (1998) 299Ð486 303 An intuitive approach for solving relativistic bound-state problems would be to solve the gauge-fixed Hamiltonian eigenvalue problem. The natural gauge for light-cone Hamiltonian theories is the light-cone gauge A`"0. In this physical gauge the gluons have only two physical transverse degrees of freedom. One imagines that there is an expansion in multi-particle occupation number Fock states. The solution of this problem is clearly a formidable task, and if successful, would allow one to calculate the structure of hadrons in terms of their fundamental degrees of freedom.
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