Light-Front Holography, Ads/QCD, and Hadronic Phenomena Stanley J

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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 confining 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 amplitude 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 momentum. The result is a single-variable light-front Schrödingerequation 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 consequences 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 Schrödinger 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 ψn (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 Schrödingerwavefunctions 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 inelastic 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 gluonic 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 |ud¯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 Schrödingerequa- 1 − x 2 × J0 ζq ψud/π¯ (x, ζ) , (3) tion whose eigensolutions ψn,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)|Φ(z)|2, (4) ternal orbital angular momentum. The result z3 is a single-variable LF Schrödingerequation 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 general 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. 2 3 dz 1 − x 2 F (Q ) = R dx 3 J0 zQ |Φ(z)| . Light-Front Holography can be derived by ob- 0 z x serving the correspondence between matrix el- (6) ements obtained in AdS/CFT with the corresponding formula using the LF representa- Comparing with the light-front QCD form factor tion. [19] The light-front electromagnetic form (3) for arbitrary values of Q [19] factor in impact space [19,20,21] can be written R3 |Φ(ζ)|2 as a sum of overlap of light-front wave functions |ψ(x, ζ)|2 = x(1 − x) , (7) of the j = 1, 2, ··· , n − 1 spectator constituents: 2π ζ4

n−1 Z where we identify the transverse LF variable ζ, 2 X Y 2 X F (q ) = dxjd b⊥j eq 0 ≤ ζ ≤ ΛQCD, with the holographic variable z. n j=1 q Matrix elements of the energy-momentum ten- µν n−1 sor Θ which define the gravitational form fac- X 2 × exp iq⊥ · xjb⊥j |ψn(xj, b⊥j)| , (1) tors play an important role in hadron physics. µν j=1 Since one can define Θ for each parton, one can identify the momentum fraction and contribution where the normalization is defined by to the orbital angular momentum of each quark n−1 X Y Z flavor and gluon of a hadron. For example, the dx d2b |ψ (x , b )|2 = 1. (2) j ⊥j n/H j ⊥j spin-flip form factor B(q2), which is the analog of n j=1 2 the Pauli form factor F2(Q ) of a nucleon, pro- The formula is exact if the sum is over all Fock vides a measure of the orbital angular momentum states n. For definiteness we shall consider a two- carried by each quark and gluon constituent of a 4

2 2 1 2 2 hadron at q = 0. Similarly, the spin-conserving where H(Q , z) = 2 Q z K2(zQ) and A(0) = 1. form factor A(q2), the analog of the Dirac form Using the integral representation of H(Q2, z) 2 factor F1(q ), allows one to measure the momen- Z 1 r ! tum fractions carried by each constituent. This 2 1 − x is the underlying physics of Ji’s sum rule [23]: H(Q , z) = 2 x dx J0 zQ , (11) 0 x z 1 hJ i = 2 [A(0)+B(0)], which has prompted much of the current interest in the generalized parton we can write the AdS gravitational form factor distributions (GPDs) measured in deeply virtual Compton scattering. An important constraint is Z 1 Z dz P A(Q2) = 2R3 x dx B(0) = i Bi(0) = 0; i.e. the anomalous grav- 3 0 z itomagnetic moment of a hadron vanishes when ! r1 − x summed over all the constituents i. This was × J zQ |Φ(z)|2 . (12) originally derived from the equivalence principle 0 x of gravity [10]. The explicit verification of these relations, Fock state by Fock state, can be ob- Comparing with the QCD gravitational form fac- tained in the LF quantization of QCD in light- tor (9) we find an identical relation between the cone gauge [3]. Physically B(0) = 0 corresponds LF wave function ψ(x, ζ) and the AdS wavefunc- to the fact that the sum of the n orbital angular tion Φ(z) given in Eq. (7) which was obtained momenta L in an n-parton Fock state must van- from the mapping of the pion electromagnetic ish since there are only n − 1 independent orbital transition amplitude. angular momenta. One can also derive light-front holography The LF expression for the helicity-conserving using a first semiclassical approximation to gravitational form factor in impact space is [24] transform the fixed light-front time bound-state Hamiltonian equation of motion in QCD to a cor- n−1 X Y Z X responding wave equation in AdS space. [15] To A(q2) = dx d2b x j ⊥j f this end we compute the invariant hadronic mass n j=1 f M2 from the hadronic matrix element n−1 X 2 × exp iq⊥ · xjb⊥j |ψn(xj, b⊥j)| , (8) 0 2 0 hψH (P )|HLF |ψH (P )i = M hψH (P )|ψH (P )i, j=1 H (13) which includes the contribution of each struck parton with longitudinal momentum xf and cor- expanding the initial and final hadronic states in responds to a change of transverse momentum terms of its Fock components. We use the frame xjq for each of the j = 1, 2, ··· , n − 1 spectators. + 2 + + − P = P ,M /P ,~0⊥ where HLF = P P . For n = 2, there are two terms which contribute The LF expression for M2 In impact space is to the f-sum in (8). Exchanging x ↔ 1−x in the second integral we find n−1 Z 2 X Y 2 ∗ Z 1 Z MH = dxj d b⊥j ψn(xj, b⊥j) 2 dx Aπ(q ) = 4π ζdζ n j=1 0 (1 − x) −∇2 + m2 ! r ! X b⊥q q 1 − x 2 × ψn(xj, b⊥j) × J0 ζq ψqq/π¯ (x, ζ) , (9) xq x q + (interactions), (14) 2 2 where ζ = x(1 − x)b⊥ and Aπ(0) = 1. We now consider the expression for the hadronic gravita- plus similar terms for antiquarks and gluons tional form factor in AdS space [25] (mg = 0). Z dz To simplify the discussion we will consider a A (Q2) = R3 H(Q2, z) |Φ (z)|2 , (10) π z3 π two-parton hadronic bound state. In the limit of 5

zero quark masses mq → 0 equation for φ

Z 1 Z d2 1 − 4L2 2 dx 2 ∗ 2 M = d b⊥ ψ (x, b⊥) − 2 − 2 + U(ζ) φ(ζ) = M φ(ζ), 0 x(1 − x) dζ 4ζ × −∇2 ψ(x, b ) + (interactions). (15) (19) b⊥ ⊥

The functional dependence for a given Fock an effective single-variable light-front Schrödinger state is given in terms of the invariant mass equation which is relativistic, covariant and an- alytically tractable. Using (15) one can readily n generalize the equations to allow for the kinetic X 2 X k2 + m2 k2 M2 = kµ = ⊥a a → ⊥ , energy of massive quarks [26]. In this case, how- n a x x(1 − x) a=1 a a ever, the longitudinal mode X(x) does not decou- (16) ple from the effective LF bound-state equations. In the hard-wall model one has U(z) = 0; con- the measure of the off-energy shell of the bound finement is introduced by requiring the wavefunc- 2 2 state, M −Mn. Similarly in impact space the tion to vanish at z = z0 ≡ 1/ΛQCD. [27] In relevant variable for a two-parton state is ζ2 = the case of the soft-wall model, [28] the potential 2 x(1 − x)b⊥. Thus, to first approximation LF dy- arises from a “dilaton” modification of the AdS namics depend only on the boost invariant vari- metric; it has the form of a harmonic oscillator 4 2 2 able Mn or ζ, and hadronic properties are en- U(z) = κ z + 2κ (L + S − 1). coded in the hadronic mode φ(ζ) from the rela- The resulting mass spectra for mesons at zero tion quark mass is M2 = 4κ2(n + L + S/2) in the soft-wall model discussed here. The spectral pre- φ(ζ) ψ(x, ζ, ϕ) = eiMϕX(x)√ , (17) dictions for both light meson and baryon states 2πζ for the hard and soft-wall holographic models discussed here are compared with experimental data thus factoring out the angular dependence ϕ and in [19,26,29,30]. The corresponding wavefunc- the longitudinal, X(x), and transverse mode φ(ζ) tions (see Fig. 1) display confinement at large in- with normalization hφ|φi = R dζ |hζ|φi|2 = 1. The terquark separation and conformal symmetry at mapping of transition matrix elements for arbi- short distances, reproducing dimensional count- trary values of the momentum transfer [19,20,24] ing rules for hard exclusive amplitudes. gives X(x) = px(1 − x). We can write the Laplacian operator in (15) in circular cylindrical coordinates (ζ, ϕ) and factor 3. Vacuum Effects and Light-Front Quan- out the angular dependence of the modes in terms tization 2 of the SO(2) Casimir representation L of orbital The LF vacuum is remarkably simple in light- angular momentum in the transverse plane. Us- cone quantization because of the restriction k+ ≥ ing (17) we find [15] 0. For example in QED, vacuum graphs such as e+e−γ associated with the zero-point energy Z d2 1 d L2 φ(ζ) do not arise. In the Higgs theory, the usual M2 = dζ φ∗(ζ)pζ − − + √ dζ2 ζ dζ ζ2 ζ Higgs vacuum expectation value is replaced with Z a k+ = 0 zero mode [6]; however, the resulting ∗ + dζ φ (ζ) U(ζ) φ(ζ), (18) phenomenology is identical to the standard analysis. where all the complexity of the interaction terms Hadronic condensates play an important role in the QCD Lagrangian is summed up in the ef- in quantum chromodynamics (QCD). Conven- fective potential U(ζ). The light-front eigenvalue tionally, these condensates are considered to be 2 equation HLF |φi = M |φi is thus a LF wave properties of the QCD vacuum and hence to 6

(a)bb (b) quark and antiquark cannot get arbitrarily great, 0 10 0 10 20 20 one cannot create a quark condensate which has 0.10 0.10 uniform extent throughout the universe. The 45 (x,b)

ψ orders of magnitude conflict of QCD with the 0.05 0.05 observed value of the cosmological condensate is 0 0 thus removed [33]. A new perspective on the na- 0 0 ture of quark and gluon condensates in quantum 0.5 0.5 chromodynamics is thus obtained: [31,32,33] the x x 1.0 1.0 2-2008 spatial support of QCD condensates is restricted 8755A1 to the interior of hadrons, since they arise due to the interactions of confined quarks and gluons. In the LF theory, the condensate physics is Figure 1. Pion light-front wavefunction ψπ(x, b⊥) replaced by the dynamics of higher non-valence for the AdS/QCD (a) hard-wall (ΛQCD = 0.32 Fock states as shown by Casher and Susskind. [4] GeV) and (b) soft-wall ( κ = 0.375 GeV) models. In particular, chiral symmetry is broken in a limited domain of size 1/mπ, in analogy to the limited physical extent of superconductor phases. This novel description of chiral symmetry break- be constant throughout spacetime. Recently a ing in terms of “in-hadron condensates” has also new perspective on the nature of QCD conden- been observed in Bethe-Salpeter studies. [34,35] µν sates hqq¯ i and hGµν G i, particularly where they This picture explains the results of recent stud- have spatial and temporal support, has been pre- ies [36,37,38] which find no significant signal for sented. [5,31,32,33] Their spatial support is re- the vacuum gluon condensate. stricted to the interior of hadrons, since these con- AdS/QCD also provides a description of chi- densates arise due to the interactions of quarks ral symmetry breaking by using the propagation and gluons which are confined within hadrons. of a scalar field X(z) to represent the dynamical For example, consider a meson consisting of a running quark mass. The AdS solution has the 3 light quark q bound to a heavy antiquark, such as form [39,40] X(z) = a1z + a2z , where a1 is pro- a B meson. One can analyze the propagation of portional to the current-quark mass. The coeffi- ¯ 3 the light q in the background field of the heavy b cient a2 scales as ΛQCD and is the analog of hqq¯ i; quark. Solving the Dyson-Schwinger equation for however, since the quark is a color nonsinglet, the the light quark one obtains a nonzero dynamical propagation of X(z), and thus the domain of the mass and, via the connection mentioned above, quark condensate, is limited to the region of color hence a nonzero value of the condensate hqq¯ i. But confinement. Furthermore the effect of the a2 this is not a true vacuum expectation value; in- term varies within the hadron, as characteristic of stead, it is the matrix element of the operator an in-hadron condensate. The AdS/QCD picture qq¯ in the background field of the ¯b quark. The of condensates with spatial support restricted to change in the (dynamical) mass of the light quark hadrons is also in general agreement with results in this bound state is somewhat reminiscent of the from chiral bag models [41,42,43], which modify energy shift of an electron in the Lamb shift, in the original MIT bag by coupling a pion field to that both are consequences of the fermion being the surface of the bag in a chirally invariant man- in a bound state rather than propagating freely. ner. Similarly, it is important to use the equations of motion for confined quarks and gluon fields 4. Hadronization at the Amplitude Level when analyzing current correlators in QCD, not free propagators, as has often been done in tra- The conversion of quark and gluon partons ditional analyses of operator products. Since af- is usually discussed in terms of on-shell hard- ter a qq¯ pair is created, the distance between the scattering cross sections convoluted with ad hoc 7 probability distributions. The LF Hamiltonian formulation of quantum field theory provides a natural formalism to compute hadronization at the amplitude level. [44] In this case one uses light-front time-ordered perturbation theory for the QCD light-front Hamiltonian to generate the off-shell quark and gluon T-matrix helicity amplitude using the LF generalization of the Lippmann-Schwinger formalism:

LF LF T = HI

LF 1 LF + HI 2 2 HI + ··· MInitial − Mintermediate + i (20)

2 PN 2 2 Here Mintermediate = i=1 (k⊥i + mi )/xi is the invariant mass squared of the intermediate state Figure 2. Illustration of an event amplitude gen- LF + − ∗ and HI is the set of interactions of the erator for e e → γ → X for hadronization QCD LF Hamiltonian in the ghost-free light-cone processes at the amplitude level. Capture oc- LF 2 2 2 gauge [45]. The T matrix element is evalu- curs if ζ = x(1 − x)b⊥ > 1/ΛQCD in the QCD AdS/QCD hard-wall model of confinement; i.e., ated between the out and in eigenstates of HLF . k2 The event amplitude generator is illustrated for 2 ⊥ 2 if M = x(1−x) < ΛQCD. e+e− → γ∗ → X in Fig. 2. The LFWFS of AdS/QCD can be used as the interpolating amplitudes between the off- shell quark and gluons and the bound-state gluonic scattering amplitudes in QCD. hadrons. Specifically, if at any stage a set of (c) Each amplitude can be renormalized us- color-singlet partons has light-front kinetic energy ing the “alternate denominator” counterterm P 2 2 i k⊥i/xi <ΛQCD, then one coalesces the virtual method [9], rendering all amplitudes UV finite. partons into a hadron state using the AdS/QCD (d) The renormalization scale in a given renor- LFWFs. This provides a specific scheme for de- malization scheme can be determined for each termining the factorization scale which matches skeleton graph even if there are multiple physi- perturbative and nonperturbative physics. cal scales. This scheme has a number of important com- (e) The T LF matrix computation allows for putational advantages: the effects of initial and final state interactions of (a) Since propagation in LF Hamiltonian the- the active and spectator partons. This allows for ory only proceeds as τ increases, all particles leading-twist phenomena such as diffractive DIS, + propagate as forward-moving partons with ki ≥ the Sivers spin asymmetry and the breakdown of 0. There are thus relatively few contributing τ- the PQCD Lam-Tung relation in Drell-Yan pro- ordered diagrams. cesses. (b) The computer implementation can be (f) ERBL and DGLAP evolution are naturally highly efficient: an amplitude of order gn for a incorporated, including the quenching of DGLAP given process only needs to be computed once. In evolution at large xi where the partons are far off- fact, each non-interacting cluster within T LF has shell. a numerator which is process independent; only (g) Color confinement can be incorporated at the LF denominators depend on the context of every stage by limiting the maximum wavelength the process. This method has recently been used of the propagating quark and gluons. by L. Motyka and A. M. Stasto [8] to compute (h) This method retains the quantum mechan- 8 ical information in hadronic production ampli- Static Dynamic tudes which underlie Bose-Einstein correlations • Square of Target LFWFs Modified by Rescattering: ISI & FSI and other aspects of the spin-statistics theo- • No Wilson Line Contains Wilson Line, Phases rem. Thus Einstein-Podolsky-Rosen QM correla- • Probability Distributions No Probabilistic Interpretation tions are maintained even between far-separated • Process-Independent Process-Dependent - From Collision • T-even Observables T-Odd (Sivers, Boer-Mulders, etc.) hadrons and clusters. • No Shadowing, Anti-Shadowing Shadowing, Anti-Shadowing, Saturation A similar off-shell T-matrix approach was used • Sum Rules: Momentum and Jz Sum Rules Not Proven to predict antihydrogen formation from virtual • DGLAP Evolution; mod. at large x DGLAP Evolution positron–antiproton states produced inpA ¯ colli- • No Diffractive DIS Hard Pomeron and Odderon Diffractive DIS

sions [46]. e– e– General remarks about orbital angulacurrentr mo- 2 quark jet mentum !*

quark final state interaction 5. Dynamical Effects of Rescattering ! S Ψn(x , k , λ ) spectator i i i system ⊥ proton 11-2001 8624A06 n ! ! ! Initial- and final-state rescattering, neglected i=1(xiR + b i) = R ⊥ ⊥ ⊥ in the parton model, have a profound effect in ! ! ! xiR + b i QCD hard-scattering reactions, predicting single- ⊥ ⊥ Figure 3. Staticn! ! vs dynamic structure functions spin asymmetries [47,48], diffractive deep lepton- i b i = 0 ⊥ ⊥ ! hadron inelastic scattering [49], the breakdown of n i xi = 1 the Lam Tung relation in Drell-Yan reactions [50], ! nor nuclear shadowing and non-universal anti- included in the nuclear light-front wavefunctions, shadowing [51]—leading-twist physics which is and a probabilistic interpretation of the nuclear not incorporated in the light-front wavefunctions DIS cross section is precluded. In addition, one of the target computed in isolation. It is thus finds that antishadowing in deep inelastic lepton- important to distinguish [52] “static” or “station- nucleus scattering is is not universal [51], but de- ary” structure functions which are computed di- pends on the flavor of each quark and antiquark rectly from the LFWFs of the target from the struck by the lepton. Evidence of this phenomena “dynamic” empirical structure functions which has been reported by Scheinbein et al. [56] take into account rescattering of the struck quark. The distinction between static structure func- Since they derive from the LF eigenfunctions of tions; i.e., the probability distributions computed the target hadron, the static structure functions from the square of the light-front wavefunctions, have a probabilistic interpretation. The wave- versus the nonuniversal dynamic structure func- function of a stable eigenstate is real; thus the tions measured in deep inelastic scattering is sum- static structure functions cannot describe diffrac- marized in Fig. 3. tive deep inelastic scattering nor the single-spin asymmetries since such phenomena involves the 6. Conclusions complex phase structure of the γ∗p amplitude. One can augment the light-front wavefunctions Light-Front Holography is one of the most re- with a gauge link corresponding to an external markable features of AdS/CFT. [57] It allows field created by the virtual photon qq¯ pair cur- one to project the functional dependence of the rent [53,54], but such a gauge link is process de- wavefunction Φ(z) computed in the AdS fifth di- pendent [48], so the resulting augmented wave- mension to the hadronic frame-independent light- functions are not universal [49,53,55]. front wavefunction ψ(xi, b⊥i) in 3 + 1 physical It should be emphasized that the shadowing space-time. The variable z maps to the LF vari- of nuclear structure functions is due to the de- able ζ(xi, b⊥i). To prove this, we have shown that structive interference between multi-nucleon am- there exists a correspondence between the ma- plitudes involving diffractive DIS and on-shell in- trix elements of the electromagnetic current and termediate states with a complex phase. The the energy-momentum tensor of the fundamen- physics of rescattering and shadowing is thus not tal hadronic constituents in QCD with the corre- 9 sponding transition amplitudes describing the in- phenomenological quantities such as heavy quark teraction of string modes in anti-de Sitter space decays, generalized parton distributions and par- with the external sources which propagate in the ton structure functions. AdS interior. The agreement of the results for We also note the distinction between between both electromagnetic and gravitational hadronic static structure functions such as the probabil- transition amplitudes provides an important con- ity distributions computed from the square of the sistency test and verification of holographic map- light-front wavefunctions versus dynamical struc- ping from AdS to physical observables defined on ture functions which include the effects of rescat- the light-front. The transverse coordinate ζ is tering. We have also shown that the LF Hamilto- closely related to the invariant mass squared of nian formulation of quantum field theory provides the constituents in the LFWF and its off-shellness a natural formalism to compute hadronization at in the LF kinetic energy, and it is thus the nat- the amplitude level. ural variable to characterize the hadronic wave- The AdS/QCD model is semiclassical, and thus function. In fact ζ is the only variable to appear it only predicts the lowest valence Fock state in the light-front Schrödingerequations predicted structure of the hadron LFWF. One can sys- from AdS/QCD in the limit of zero quark masses. tematically improve the holographic approxima- The use of the invariant coordinate ζ in light- tion by diagonalizing the QCD LF Hamiltonian front QCD allows the separation of the dynamics on the AdS/QCD basis, or by generalizing the of quark and gluon binding from the kinemat- variational and other systematic methods used in ics of constituent spin and internal orbital angu- chemistry and nuclear physics [58]. The action of lar momentum. The result is a single-variable the non-diagonal terms in the QCD interaction LF Schrödingerequation which determines the Hamiltonian also generates the form of the higher spectrum and LFWFs of hadrons for general spin Fock state structure of hadronic LFWFs. In con- and orbital angular momentum. This LF wave trast with the original AdS/CFT correspondence, equation serves as a first approximation to QCD the large NC limit is not required to connect LF and is equivalent to the equations of motion QCD to an effective dual gravity approximation. which describe the propagation of spin-J modes on AdS [15]. The AdS/LF equations correspond Acknowledgments to the kinetic energy terms of the partons inside a hadron, whereas the interaction terms build con- Presented by SJB at Light Cone 2009: Rel- finement. Since there are no interactions up to ativistic Hadronic And Particle Physics, 8-13 the confining scale in this approximation, there Jul 2009, São José dos Campos, Brazil. We are no anomalous dimensions. The eigenvalues of are grateful to Professor Tobias Frederico and these equations for both meson and baryons give his colleagues at the Instituto Tecnológico de a good representation of the observed hadronic Aeronáutica(ITA) , for their outstanding hospi- spectrum, especially in the case of the soft-wall tality. We thank Carl Carlson, Stan Glazek, Paul model. The predicted LFWFs have excellent phe- Hoyer, Dae Sung Hwang, Bo-Qiang Ma, Pieter nomenological features, including predictions for Maris, Craig Roberts, Ivan Schmidt, Robert the electromagnetic form factors and decay con- Shrock, and James Vary for helpful conversations stants. This may explain the experimental suc- and collaborations. Section 3 is based on collab- cess of power-law scaling in hard exclusive reac- orations with Robert Shrock. This research was tions where there are no indications of the effects supported by the Department of Energy contract of anomalous dimensions. DE–AC02–76SF00515. SLAC-PUB-13778. 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