SLAC–PUB–15972 Light-Front Holographic QCD and Emerging Confinement Stanley J. Brodsky SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA Guy F. de T´eramond Universidad de Costa Rica, San Jos´e, Costa Rica Hans G¨unter Dosch Institut f¨ur Theoretische Physik, Philosophenweg 16, D-6900 Heidelberg, Germany Joshua Erlich College of William and Mary, Williamsburg, VA 23187, USA arXiv:1407.8131v2 [hep-ph] 13 Feb 2015 [email protected], [email protected], [email protected], [email protected] (Invited review article to appear in Physics Reports) This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-76SF00515 and HEP. Abstract In this report we explore the remarkable connections between light-front dynamics, its holographic mapping to gravity in a higher-dimensional anti-de Sitter (AdS) space, and conformal quantum mechanics. This approach provides new insights into the origin of a fundamental mass scale and the physics underlying confinement dynamics in QCD in the limit of massless quarks. The result is a relativistic light-front wave equation for arbitrary spin with an effective confinement potential derived from a conformal action and its embedding in AdS space. This equation allows for the computation of essen- tial features of hadron spectra in terms of a single scale. The light-front holographic methods described here gives a precise interpretation of holographic variables and quan- tities in AdS space in terms of light-front variables and quantum numbers. This leads to a relation between the AdS wave functions and the boost-invariant light-front wave functions describing the internal structure of hadronic bound-states in physical space- time. The pion is massless in the chiral limit and the excitation spectra of relativistic light-quark meson and baryon bound states lie on linear Regge trajectories with iden- tical slopes in the radial and orbital quantum numbers. In the light-front holographic approach described here currents are expressed as an infinite sum of poles, and form factors as a product of poles. At large q2 the form factor incorporates the correct power- law fall-off for hard scattering independent of the specific dynamics and is dictated by the twist. At low q2 the form factor leads to vector dominance. The approach is also extended to include small quark masses. We briefly review in this report other holo- graphic approaches to QCD, in particular top-down and bottom-up models based on chiral symmetry breaking. We also include a discussion of open problems and future applications. Contents 1 Introduction 4 1.1 Motivation................................... 4 1.2 The AdS/CFT correspondence and holographic QCD . 6 1.3 Light-frontholographicQCD. 12 1.4 Confinement and conformal algebraic structures . ... 14 1.5 Otherapproachesandapplications . 16 1.6 Contentsofthisreview ............................ 17 2 A Semiclassical Approximation to Light-Front Quantized QCD 19 2.1 The Dirac forms of relativistic dynamics . 22 2.2 Light-frontdynamics ............................. 23 2.3 Light-frontquantizationofQCD . 25 2.3.1 Representation of hadrons in the light-front Fock basis . 28 2.4 Semiclassical approximation to QCD in the light front . 31 2.4.1 Inclusion of light quark masses . 36 3 Conformal Quantum Mechanics and Light-Front Dynamics 39 3.1 One-dimensionalconformalfieldtheory . 40 3.2 Connection to light-front dynamics . 45 3.3 Conformal quantum mechanics, SO(2, 1) and AdS2 ............ 47 4 Higher-Spin Wave Equations and AdS Kinematics and Dynamics 49 4.1 Scalarandvectorfields ........................... 50 4.2 Arbitraryintegerspin ............................ 52 4.2.1 Confining interaction and warped metrics . 57 4.2.2 Higher spin in a gauge invariant AdS model . 58 4.3 Arbitrary half-integer spin . 59 1 5 Light-Front Holographic Mapping and Hadronic Spectrum 63 5.1 Integerspin .................................. 65 5.1.1 A light-front holographic model for mesons . 66 5.1.2 Meson spectroscopy in a gauge invariant AdS model . 72 5.1.3 Lightquarkmassesandmesonspectrum . 73 5.2 Half-integer spin . 78 5.2.1 A light-front holographic model for baryons . 80 6 Light-Front Holographic Mapping and Transition Amplitudes 87 6.1 Mesonelectromagneticformfactor . 88 6.1.1 MesonformfactorinAdSspace . 88 6.1.2 Meson form factor in light-front QCD . 90 6.1.3 Light-front holographic mapping . 91 6.1.4 Soft-wallformfactormodel . 96 6.1.5 Time-like form factors in holographic QCD . 101 6.2 Nucleonelectromagneticformfactors . 105 6.2.1 Computing nucleon form factors in light-front holographic QCD . 107 7 Other Bottom-Up and Top-Down Holographic Models 112 7.1 Bottom-upmodels .............................. 113 7.2 A bottom-up model with chiral symmetry breaking . 121 7.3 Top-downmodels ............................... 124 7.4 TheSakai-Sugimotomodel. 128 8 Summary and Conclusion 132 8.1 Open problems and future applications . 136 A Riemannian Geometry and Anti-de Sitter Space 144 A.1 Basics of non-Euclidean geometry . 144 A.1.1 Covariant derivative and parallel transport . 146 A.1.2 Space-timecurvature . .. .. 147 A.1.3 Spinors in non-Euclidean geometry . 149 A.2 Maximally symmetric spaces . 151 A.2.1 Definition . 151 A.2.2 Anti-de Sitter space-time AdSd+1 .................. 152 1 A.2.3 Relation between Conf(R ), SO(2, 1) and the isometries of AdS2 155 2 B Light-Front Metric Conventions and Spinors 158 C Notes on Conformal Quantum Mechanics 160 D Useful Formulas for Higher Spin Equations in Anti-de Sitter Space 164 D.1 Arbitraryintegerspin ............................ 164 D.1.1 The action in the local Lorentz frame . 164 D.1.2 Warpedmetric ............................ 166 D.2 Arbitrary half integer spin . 167 D.2.1 Generaltreatment .......................... 167 3 D.2.2 Spin- 2 Rarita-Schwinger field in AdS space . 169 E Light-Front Holographic Mapping and the Energy-Momentum Tensor 171 E.1 Gravitationalformfactorofcomposite hadrons . .. 171 E.1.1 Meson gravitational form factor in AdS space . 172 E.1.2 Meson gravitational form factor in light-front QCD . 175 E.1.3 Light-front holographic mapping . 177 F Propagators in the Limiting Theory of AdS5 179 F.1 AdS boundary conditions and gauge/gravity correspondence . ..... 179 F.2 Two-point functions for arbitrary spin and Migdal procedure . .... 181 G Some Useful Formulæ 186 G.1 Solutions of the equations of motion in AdS space . 186 G.1.1 Ausefulintegral ........................... 189 H Integrability and Light-Front Effective Hamiltonians 191 H.1 Light-front effective bosonic Hamiltonians . 192 H.1.1 Light-fronthard-wallmodel . 192 H.1.2 Light-frontsoft-wallmodel . 194 H.2 Light-front effective fermionic Hamiltonians . 196 H.2.1 Light-fronthard-wallmodel . 197 H.2.2 Light-frontsoft-wallmodel. 199 I Equations of Motion for p-Form Fields in AdS 202 3 Chapter 1 Introduction 1.1 Motivation Quantum Chromodynamics (QCD), the SU(3) color gauge field theory of quarks and gluons, is the standard theory of strong interactions. High energy experiments, such as the deep inelastic electron-proton scattering pioneered at SLAC [1], which revealed the quark structure of the proton, and continued at DESY [2] to extremely short distances, have shown that the basic elementary interactions of quarks and gluons are remarkably well described by QCD [3]. Yet, because of its strong-coupling nature, it has been difficult to make precise predictions outside of its short-distance perturbative domain where it has been tested to high precision. Unlike Quantum Electrodynamics (QED), the fundamental theory of electrons and photons, the strong couplings of quarks and gluons at large-distances makes the calculation of hadronic properties, such as hadron masses, a very difficult problem to solve, notwithstanding that the fundamental QCD Lagrangian is well established. In particular, one has no analytical understanding of how quarks and gluons are permanently confined and how hadrons emerge as asymptotic states in this theory [4]. In fact, in the limit of massless quarks no scale appears in the QCD Lagrangian. The classical Lagrangian of QCD is thus invariant under conformal transformations [5, 6]. Nonetheless, the quantum theory built upon this conformal theory displays color confinement, a mass gap, and asymptotic freedom. One then confronts a fundamental question: how does the mass scale which determines the masses of the light- quark hadrons, the range of color confinement, as well as the running of the coupling appear in QCD? Euclidean lattice methods [7] provide an important first-principle numerical simula- 4 tion of nonperturbative QCD. However, the excitation spectrum of hadrons represents a difficult challenge to lattice QCD due to the enormous computational complexity beyond ground-state configurations and the unavoidable presence of multi-hadron thresholds [8]. Furthermore, dynamical observables in Minkowski space-time are not obtained directly from Euclidean space lattice computations. Other methods, as for example the Dyson- Schwinger equations, have also led to many important insights, such as the infrared fixed-point behavior of the strong coupling constant and the pattern of dynamical quark mass generation [9, 10, 11, 12]. In practice, however, these analyses have been limited to ladder approximation in Landau gauge. A problem, common to all realistic relativistic quantum field theories, is especially flagrant in QCD: the only known analytically tractable treatment is perturbation the- ory, which obviously is not the most appropriate tool for solving a strongly interacting theory with permanently confined constituents. In fact, according to the Kinoshita-Lee- Nauenberg theorem, which applies to any order of perturbation theory, a description of confinement using perturbative QCD is not possible in a simple way [13, 14]. Thus, an important theoretical goal is to find an initial approximation to QCD in its strongly coupled regime relevant at large distances, which is both analytically tractable and can be systematically improved.