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NEW PERSPECTIVES in QUANTUM CHROMODYNAMICS * Invited SLAC-PUB-i July 1993 T/E NEW PERSPECTIVES IN QUANTUM CHROMODYNAMICS STANLEY J. BRODSKY NEW PERSPECTIVES IN QUANTUM CHROMODYNAMICS * Stanford Linear Accelerator Center Stanford University, Stanford, California 94309 Stanley J. Brodsky ABSTRACT In these lectures I will discuss three central topics in quantum chromodynamics: (1) the use of light cone Stanford Linear Accelerator Center quantization and Fock space methods to determine the long and Stanford University, Stanford, California 94309 short-distance structure of quark and gluon distributions within hadrons; (2) the role of spin, heavy quarks, and nuclei in unrav• eling fundamental phenomenological features of QCD; and (3) a new approach to understanding the scale and scheme dependence of perturbative QCD predictions. 1. INTRODUCTION Invited Lectures presented at the Symposium/Workshop "Particle Physics at the Fermi Scale" One of the most challenging problems in theoretical high energy physics Beijing, People's Republic of China is to compute the bound-state structure of the proton and other hadrons from quantum chromodynamics (QCD), the field theory of quarks and May 27-June 4, 1993 gluons. The goal is to not only calculate the spectrum of hadron masses from first principles, but also to derive the momentum and spin distri• butions of the quarks and gluons which control high energy hadron interactions. In the first chapter, I will discuss new methods based on "light-cone" quantization which have been proposed as alternatives to lattice theory for solving non-perturbative problems in QCD and • Work supported by the Department of Energy, contract DE-AC03-76SF00515. other field theories. The basic idea is a generalization ol Heisenberg's challenging numerical computational problem. In the DLCQ .r.ciho.j. pioneering matrix formulation of quantum mechanics: if one could nu• developed by H. C. Pauli and myself, the size of the quark and gluon merically diagonalize the matrix of the Hamiltonian representing the basis and the discretization of the transverse momenta quickly leads underlying QCD interaction, then the resulting eigenvalues would give to very large matrices. In addition, the Hamiltonian must be supple• the hadron spectrum, while the corresponding eigenstates would de• mented by renormalization terms. Approximate methods have been scribe each hadron in terms of its quark and gluon degrees of freedom. developed which use effective light-cone Hamiltonians and a truncation The new ingredient which appears to make this method tractable of the quark and gluon basis states (the "Light-Front Tamm-DancofT is quantization on the light-cone—which sets boundary conditions as method") or a combination of light-cone quantization with traditional if the observer were travelling at the speed of light! For example, if lattice gauge theory in the transverse dimensions. In the case of quan• one shines a laser directed along the 2-axis on an atom, the scattered tum electrodynamics in 3+1 dimensions, the positronium spectrum has photons determine the coordinates of each electron at a fixed value of been obtained at large coupling strength (a = 0.3) by solving an inte• t — z/c. One then can use the equations of motion of quantum elec• gral equation derived from the truncated QED light-cone Hamiltonian. trodynamics QED to predict the coordinates of the electrons at later The most subtle problem now confronting light-cone quantization values of / — z/c. methods is how to understand the spontaneous symmetry breaking The foundations of light-cone quantization date back to Dirac, who normally associated with the structure of the vacuum. In light-cone in 1949 showed that there are remarkable advantages in quantizing quantization the momentum-independent "zero modes" of the quan• relativistic field theories at fixed "iight-cone time" t — z/c rather than tum fields are determined from constraint equations derived from the ordinary time. In the traditional formulation, handling a moving bound equations of motion of the theory. Although the vacuum is simple in state is as complicated as diagonalizing the Hamiltonian itself. On the light-cone quantization, the values of the zero modes determine the other hand, quantization on the light-cone can be formulated without phase and physics of the theory. There are other fundamental renor• having to choose a specific frame of reference. Thus a light-cone QCD malization and gauge invariance issues that still have to be completely Hamiltonian describes bound states of confined relativistic quarks and understood, such as how symmetries lost in the truncation to a finite gluons of arbitrary four-momentum. It also provides a precise defini• basis of free quark and gluon states can be restored, how to deal con• tion of hadron structure in terms of quarks and gluons and a general sistently with massless particles, and how to control singularities. All calculus for computing relativistic scattering amplitudes, form factors, of these problems make the field quite exciting and challenging. electroweak transitions, and other hadronic phenomena. In addition to its potential for solving QCD problems, light-cone The problem of computing the hadronic spectrum and the corre• quantization has already led to many new insights into the quanti• sponding light-cone wavefunctions of QCD can thus be reduced to the zation of gauge theories. Light-cone quantization not only provides diagonalization of a matrix representation of the light- cone Hamilto• a consistent language for representing hadrons as QCD bound-STATES nian. This method, called '"discretized light-cone quantization" (DLCQ), of relativistic quarks and gluons, but it also provides a novel method has now been successfully applied to a number of quantum field theories for simulating quantum field theory on a computer and understanding in one-space and one-time dimension, including QCD. quantum elec• non-perturbative features of QCD. trodynamics, Yukawa models, and the two-dimensional matrix models In the second chapter, I will discuss a number of interesting spin, of superstring theory. For QCD(l-rl), complete numerical solutions heavy quark, and nuclear effects which test fundamental features OF for the spectrum and light-cone wavefunctions can be obtained as a perturbative and non-perturbative QCD. These include constraints on function of the coupling strength, the quark masses, and the number the shape and normalization of the polarized quark and gluon struc• of flavors and colors. ture functions of the proton; the principle of hadron helicity retention Light-cone quantization of QCD in physical space-time is a highly in high Xf inclusive reactions; predictions based on total hadron he- 2 3 licity conservation in high momentum transfer exclusive reactions; the example, in electroweak theory, the central unknown required for re• dependence of nuclear structure functions and shadowing on virtual liable calculations of weak decay amplitudes are the hadronic matrix photon polarization; and general constraints on the magnetic moment elements: the computation of the B meson decay into particular hadron of hadrons. I also will discuss the implications of several measurements channels requires detailed knowledge of both the light and heavy hadron which are in striking conflict with leading-twist perturbative QCD pre• wavefunctions. The coefficient functions in the operator product expan• dictions, such as the extraordinarily large spin correlation ANN ob• sion needed to compute leading and higher twist structure functions served in large angle proton-proton scattering, the anomalously large and other inclusive cross sections are also essentially unknown. Form p7r branching ratio of the J/v\ and the rapidly changing polarization factors and exclusive scattering processes depend in detail on the ba• dependence of both J/^ and continuum lepton pair hadroproduction sic amplitude structure of the scattering hadrons in a general Lorentz observed at large if. frame. Even the calculation of the proton magnetic moment requires In the third chapter, which is based on recent work with Hung Jung an understanding of hadron wavefunctions in a boosted frame. Lu, I will discuss a crucial problem in perturbative quantum chromody• In this chapter I will discuss the light-cone quantization of gauge namics: the setting of the renormalization scale in perturbative QCD theories from two perspectives: as a language for representing hadrons predictions. First I discuss the use of generalized renormalization group as QCD bound-states of relativistic quarks and gluons, and also as equations which allow a careful analysis of the theoretical error when a novel method for simulating quantum field theory on a computer. the scale is unspecified. I also discuss difficulties encountered when one The light-cone Fock state expansion of wavefunctions at fixed light- uses various scale setting procedures. I then discuss the validity and cone time in fact provides a precise definition of the parton model use of the automatic scale fixing procedure developed by Lepage and and a general calculus for hadronic matrix elements. The Hamilto• Mackenzie and myself. nian formulation of quantum field theory quantized at fixed light-cone time has led to new non-perturbative calculational tools for numerically solving quantum field theories. In particular, the "discretized light- 2. LIGHT-CONE QUANTIZATION AND QCD 2 cone quantization," method (DLCQ)" has been successfully applied to several gauge theories, including QCD in one-space and one-time di• A primary goal of particle physics is to understand the structure of mension, and quantum electrodynamics in physical space-time at large hadrons in terms of their fundamental quark and gluon degrees of free• coupling strength. Other non-perturbative methods based on light- dom. It is important to predict not only the spectrum of the hadrons, 3 but also to derive from first principles the hadron structure functions cone quantization, such as the transverse lattice and the Light-Front that control inclusive reactions, the form of the hadron distribution Tamm-Dancoff method are also being developed as new alternatives amplitudes that control exclusive processes, and the behavior of the to conventional lattice gauge theory.
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