CORE Metadata, citation and similar papers at core.ac.uk Provided by UNT Digital Library SLAC-PUB-13778 Light-Front Holography, AdS/QCD, and Hadronic Phenomena Stanley J. Brodskya and Guy F. de T´eramondb aSLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA bUniversidad de Costa Rica, San Jos´e,Costa Rica AdS/QCD, the correspondence between theories in a modified five-dimensional anti-de Sitter space and confin- ing field theories in physical space-time, provides a remarkable semiclassical model for hadron physics. Light-front holography allows hadronic amplitudes in the AdS fifth dimension to be mapped to frame-independent light-front wavefunctions of hadrons in physical space-time, thus providing a relativistic description of hadrons at the am- plitude level. We identify the AdS coordinate z with an invariant light-front coordinate ζ which separates the dynamics of quark and gluon binding from the kinematics of constituent spin and internal orbital angular momen- tum. The result is a single-variable light-front Schr¨odingerequation with a confining potential which determines the eigenspectrum and the light-front wavefunctions of hadrons for general spin and orbital angular momentum. The mapping of electromagnetic and gravitational form factors in AdS space to their corresponding expressions in light-front theory confirms this correspondence. Some novel features of QCD are discussed, including the con- sequences of confinement for quark and gluon condensates. The distinction between static structure functions, such as the probability distributions computed from the square of the light-front wavefunctions, versus dynamical structure functions which include the effects of rescattering, is emphasized. A new method for computing the hadronization of quark and gluon jets at the amplitude level, an event amplitude generator, is outlined. 1. The Light-Front Hamiltonian Approach τ: In contrast, setting the initial condition using to QCD conventional instant time t requires simultaneous scattering of photons on each constituent. Thus One of the most important theoretical tools in it is natural to set boundary conditions at fixed τ atomic physics is the Schr¨odinger wavefunction, and then evolve the system using the light-front which describes the quantum-mechanical struc- (LF) Hamiltonian P − =P 0 −P 3 = id/dτ: The in- ture of an atomic system at the amplitude level. + − 2 variant Hamiltonian HLF = P P −P? then has Light-front wavefunctions (LFWFs) play a simi- eigenvalues M2 where M is the physical mass. lar role in quantum chromodynamics, providing Its eigenfunctions are the light-front eigenstates a fundamental description of the structure and whose Fock state projections define the light-front internal dynamics of hadrons in terms of their wavefunctions. Given the LF Fock state wave- constituent quarks and gluons. The LFWFs of H + + functions n (xi; k?i; λi); where xi = k =P , bound states in QCD are relativistic generaliza- Pn Pn i=1 xi = 1; i=1 k?i = 0, one can immedi- tions of the Schr¨odingerwavefunctions of atomic ately compute observables such as hadronic form physics, but they are determined at fixed light- factors (overlaps of LFWFs), structure functions cone time τ = t + z=c { the \front form" intro- (squares of LFWFS), as well as the generalized duced by Dirac [1] { rather than at fixed ordinary parton distributions and distribution amplitudes time t: which underly hard exclusive reactions. When a flash from a camera illuminates a A remarkable feature of LFWFs is the fact that scene, each object is illuminated along the light- they are frame independent; i.e., the form of the front of the flash; i.e., at a given τ. Similarly, LFWF is independent of the hadron's total mo- when a sample is illuminated by an x-ray source, + 0 3 mentum P = P + P and P?: The simplic- each element of the target is struck at a given ity of Lorentz boosts of LFWFs contrasts dra- 1 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2 matically with the complexity of the boost of • Amplitudes in light-front perturbation the- wavefunctions defined at fixed time t: [2] Light- ory are automatically renormalized using front quantization is thus the ideal framework the \alternate denominator" subtraction to describe the structure of hadrons in terms of method [9]. The application to QED has their quark and gluon degrees of freedom. The been checked at one and two loops. [9] constituent spin and orbital angular momentum properties of the hadrons are also encoded in the • One can easily show using LF quantization LFWFs. The total angular momentum projec- that the anomalous gravitomagnetic mo- z Pn z Pn−1 z ment B(0) of a nucleon, as defined from tion [3] J = i=1 Si + i=1 Li is conserved Fock-state by Fock-state and by every interaction the spin flip matrix element of the gravita- in the LF Hamiltonian. Other advantageous fea- tional current, vanishes Fock-state by Fock tures of light-front quantization include: state [3], as required by the equivalence principle. [10] • The simple structure of the light-front vac- • LFWFs obey the cluster decomposition the- uum allows an unambiguous definition of orem, providing the only proof of this the- the partonic content of a hadron in QCD. orem for relativistic bound states. [11] The chiral and gluonic condensates are properties of the higher Fock states [4,5], • The LF Hamiltonian can be diagonalized rather than the vacuum. In the case of the using the discretized light-cone quantiza- Higgs model, the effect of the usual Higgs tion (DLCQ) method. [12] This nonpertur- vacuum expectation value is replaced by a bative method is particularly useful for solv- constant k+ = 0 zero mode field. [6] ing low-dimension quantum field theories such as QCD(1 + 1): [13] • If one quantizes QCD in the physical light- cone gauge (LCG) A+ = 0, then gluons • LF quantization provides a distinction be- only have physical angular momentum pro- tween static (square of LFWFs) distribu- jections Sz = ±1. The orbital angular mo- tions versus non-universal dynamic struc- menta of quarks and gluons are defined un- ture functions, such as the Sivers single- ambiguously, and there are no ghosts. spin correlation and diffractive deep inelas- tic scattering which involve final state in- • The gauge-invariant distribution amplitude teractions. The origin of nuclear shadowing φ(x; Q) is the integral of the valence LFWF and process independent anti-shadowing in LCG integrated over the internal trans- also becomes explicit. This is discussed fur- 2 2 ther in Sec. 5. verse momentum k? < Q because the Wilson line is trivial in this gauge. It is • LF quantization provides a simple method also possible to quantize QCD in Feynman to implement jet hadronization at the am- gauge in the light front. [7] plitude level. This is discussed in Sec. 4. • LF Hamiltonian perturbation theory pro- • The instantaneous fermion interaction in vides a simple method for deriving analytic LF quantization provides a simple deriva- forms for the analog of Parke-Taylor am- tion of the J = 0 fixed pole contribution to plitudes [8] where each particle spin Sz is deeply virtual Compton scattering. [14] quantized in the LF z direction. The glu- onic g6 amplitude T (−1 − 1 ! +1 + 1 + 1 + • Unlike instant time quantization, the 1 + 1 + 1) requires ∆Lz = 8; it thus must Hamiltonian equation of motion in the LF vanish at tree level since each three-gluon is frame independent. This makes a direct vertex has ∆Lz = ±1: However, the order connection of QCD with AdS/CFT meth- g8 one-loop amplitude can be nonzero. ods possible. [15] 3 2. Light-Front Holography quark π+ valence Fock state jud¯i with charges 2 1 eu = 3 and ed¯ = 3 . For n = 2, there are two A key step in the analysis of an atomic sys- terms which contribute to the q-sum in (1). Ex- tem such as positronium is the introduction of changing x $ 1−x in the second integral we find the spherical coordinates r; θ; φ which separates the dynamics of Coulomb binding from the kine- Z 1 Z 2 dx matical effects of the quantized orbital angular F + (q ) = 2π ζdζ π x(1 − x) momentum L. The essential dynamics of the 0 r ! atom is specified by the radial Schr¨odingerequa- 1 − x 2 × J0 ζq ud/π¯ (x; ζ) ; (3) tion whose eigensolutions n;L(r) determine the x bound-state wavefunction and eigenspectrum. In 2 2 our recent work, we have shown that there is where ζ = x(1 − x)b? and Fπ+ (q =0) = 1. an analogous invariant light-front coordinate ζ We now compare this result with the electro- which allows one to separate the essential dy- magnetic form-factor in AdS space [22]: namics of quark and gluon binding from the Z dz kinematical physics of constituent spin and in- F (Q2) = R3 J(Q2; z)jΦ(z)j2; (4) ternal orbital angular momentum. The result z3 is a single-variable LF Schr¨odingerequation for where J(Q2; z) = zQK (zQ). Using the integral QCD which determines the eigenspectrum and 1 representation of J(Q2; z) the light-front wavefunctions of hadrons for gen- eral spin and orbital angular momentum. [15] If Z 1 r ! 2 1 − x one further chooses the constituent rest frame J(Q ; z) = dx J0 ζQ ; (5) Pn 0 x (CRF) [16,17,18] where i=1 ki = 0, then the kinetic energy in the LFWF displays the usual we write the AdS electromagnetic form-factor as 3-dimensional rotational invariance. Note that if the binding energy is nonzero, P z 6= 0; in this Z 1 Z r ! frame.