PoS(QCD-TNT-II)008 n http://pos.sissa.it/ 1 space at fixed light-front + and radial quantum number L in 3 ζ -function with an infrared fixed point which β extracted from measurements of the Bjorken sum rule and its ) 2 ) Q Q ( ( 1 g are sublimated into a color-confining potential for quarks. We α AdS s 2 α 1 GeV < 2 Q . This is consistent with a flux-tube interpretation of QCD where soft glu- 2 ∗ to an invariant impact separation variable z 1 GeV < 2 Q [email protected] [email protected] Speaker. discuss a number of phenomenological hadronic properties which support this picture. time. A key feature ishadrons – the the determination relativistic analogs of of the the Schrödingerone frame-independent wavefunctions of to light-front atomic compute wavefunctions physics form which of allow factors, transversityhadronization, distributions, and spin other properties hadronic of observables. the One thusmodel valence obtains quarks, for a jet hadron one-parameter color-confining physics atperturbative the effective amplitude coupling level. AdS/QCD also predicts the form ofbelow the non- ons with virtualities Gauge/gravity duality leads toperturbative a approximation simple, analytical, to and the“Light-Front phenomenologically full Holography”, compelling light-front successfully non- QCD describesbaryons, the Hamiltonian. their elastic spectroscopy and of transition This form light-quarkSchrödinger approach, factors, and meson and called other Dirac and hadronic equations properties. oftories The the which bound-state soft-wall have AdS/QCD the model same predict slope linear in orbital Regge angular trajec- momentum agrees with the effective coupling for both mesons and baryons.of Light-front AdS holography connects space the fifth-dimensional coordinate ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

International Workshop on QCD Green’s Functions,5-9 Confinement September and 2011 Phenomenology Trento, Italy Guy F. de Téramond Universidad de Costa Rica, San José, Costa Rica E-mail: Stanley J. Brodsky AdS/QCD, Light-Front Holography, and Sublimated Gluons SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA E-mail: PoS(QCD-TNT-II)008 ] 5 ] It , the 7 τ , ] 6 9 Stanley J. Brodsky ] between higher- 1 maps to a LF variable z However, this traditional . t ] leads to an analytic and phenomeno- at fixed time 2 i wavefunctions away from the hadron’s rest Ψ | t E = i 2 ] is plagued by complex vacuum and relativistic ef- Ψ | 10 , along the front of a light-wave as in a flash picture. H τ . This leads to single-variable light-front Schrödinger 3 x + ] Worse, evaluating a current matrix element in the instant 0 1 physical space-time. The variable x 11 + [ = . t ] by perturbing the AdS metric around its static solution. [ + x 8 in 3 ) = i ⊥ c , requiring the exact synchrony in time of many simultaneous probes. ], is a powerful non-perturbative alternative, providing a boost-invariant t b / ~ , z i 10 x + ( t ψ computed in the AdS fifth dimension to the hadronic frame-independent light- = ) z τ ( ] by matching the electromagnetic (EM) current matrix elements in AdS space [ 4 Φ , 3 which measures the invariant separation of the constituents within a hadron at equal ) i ⊥ b ~ Light front (LF) holographic methods allow one to project the functional dependence of the In atomic physics, one computes the properties of electrons within an atom from its Schrödinger A central problem of hadron physics is how to compute the properties and spectroscopy of , i x ( ζ “front-form” of Dirac [ In contrast, the constituents ofat a the bound same state instant inCalculating time an a instant-form hadronic wavefunction form must factorwavefunction be from in the measured the initial to traditional final instantThe state, boost a form dynamical invariance requires of problem boosting as LFWFs difficult the contrastsfunctions as dramatically hadron’s defined solving with QCD at the itself. fixed complexity ofform time boosting requires the computing wave- the interaction of the probe with all of connected currents fluctuating in front wavefunction frame is an intractable dynamical problem. In contrast, quantization at fixedframework light-front for time describing the structure ofture hadrons of in the terms light-front vacuum of allows their anin constituents. unambiguous QCD definition The in of simple the terms struc- partonic of content thelarge of distance hadronic a hadronic light-front hadron states wavefunctions (LFWFs), and the thements underlying constituent of link degrees the between of constituents freedom at ofare short determined a distances. at bound the Measure- state same such light-front as time in form factors or in deep inelastic scattering wavefunction light-front time wavefunction, the eigenfunction of thenian. nonrelativistic In approximation principle, to one thefor could exact the also eigenstates QED of calculate Hamilto- the hadronic QCD Hamiltonian: spectroscopy and wavefunctions by solving with the corresponding expression usinghas light-front also quantization been shown in that physicalthe one space energy-momentum obtains time. tensor identical [ [ holographic mapping using the matrix elements of and Dirac equations whose eigensolutions determineorbital the angular spectrum of momentum. hadrons for Thefactors, general transversity resulting spin distributions, spin and light-front properties wavefunctions of the allowhadronic valence quarks, observables. one jet One to hadronization, thus and compute obtains other a form at one-parameter the color-confining amplitude model level. for hadron physics method – called the “instant form” byfects, Dirac, as [ well as by the fact that the boost of such fixed- hadrons in terms ofAs the we confined shall quark discuss, and gluon AdS/QCD, degrees which of uses freedom the of AdS/CFT the correspondence QCD [ Lagrangian. AdS/QCD, Light-Front Holography, and Sublimated Gluons 1. Introduction dimensional anti–de Sitter (AdS) spaceuseful guide, and combined conformal with field “Light-Front theorieslogically Holography”, compelling in [ first physical approximation to space-time QCD. asintroduced Light-front holographic [ a methods were originally PoS(QCD-TNT-II)008 = µ (2.1) P 0 and 0, and µ needed P = i = i H = L ⊥ Ψ k | LF ~ 1 n H h = n i 0 is particularly contributes only Stanley J. Brodsky ∑ = z = S H / + and then evolves the n A ψ τ ] Its eigenfunctions are the 13 highlights the importance of quark are functions of LF momentum frac- z ) J , i i λ , H ⊥ . The invariant Hamiltonian Ψ k | ~ τ , 2 i H d x / ( M H id 3 / -parton LF wavefunctions = n n = i ] by applying anti-periodic boundary conditions in ψ 3 H P is the physical hadron mass. The Heisenberg equation 14 Ψ | − relative transverse momenta satisfying 0 M LF , is conserved, Fock-state by Fock-state, and by every inter- P 1 H z i = L = − 1 1 i P − x where = i n 1 2 ∑ = n i + ∑ M z i S 1 = with n i 3 3 i ∑ Remarkably, the LFWFs are independent of the hadron’s total momentum P k . + + i = 0 0 i λ k z P J ] ] = , so that once they are known in one frame, they are known in all frames; Wigner 3 15 This method has been used successfully to solve many lower dimension quantum field ] has eigenvalues + is determined canonically from the QCD Lagrangian. [ + i 12 . P k P 3 2 13 ⊥ x + LF P ~ = 0 − H i P − 0 x x In general, hadron eigenstates have components with different orbital angular momentum; In the light-front formalism, one sets boundary conditions at fixed The light-front Fock state wavefunctions Light-front quantization is thus the ideal framework to describe the structure of hadrons in − 1 light-front Fock components with equal probability. Higher Fock states with extra quark-anti = P = . = -function. [ + + z for phenomenology. Heisenberg’s problem on thecretized light-front light-front can be quantization solved (DLCQ) numerically using [ dis- for example, the proton eigenstate in LF holographic QCD with massless quarks has tions useful since there are no ghostsmentum. and They one also has allows a direct onethe physical amplitude to interpretation level. formulate of hadronization orbital in angular inclusive mo- and exclusive reactions at L quark pairs also arise. Theincluding resulting the LFWFs possibility then to leadlevel. to compute The a the soft-wall new model hadronization range also of of predictsβ hadron quark the phenomenology, form and of gluon the non-perturbative jets effective at coupling and the its amplitude P transformations and Melosh rotations are not required. Light-cone gauge 2. Light-Front Quantization and Light-Front Wavefunctions a small fraction of theorbital nucleon’s angular total momentum. angular momentum Insingle-spin fact, asymmetries, such the as nucleon the anomalousL Sivers moment, effect, the would vanish Pauli unless form the factor, quarks and carry nonzero system using the LF Hamiltonian for QCD on the light-front thus takes the form where light-front eigenstates which define the frame-independent light-frontvalues wavefunctions, yield and the its hadronic eigen- spectrum, the boundeigensolutions states on as well the as free the Fock continuum. basis The gives projection the of the spin projections action in the LF Hamiltonian. The empirical observation that the quark spin σ theories. [ P terms of their quark andmentum gluon properties degrees of of the freedom. hadronsprojection The are [ constituent also spin encoded and in orbital the angular LFWFs. mo- The total angular momentum AdS/QCD, Light-Front Holography, and Sublimated Gluons the acausal instant form vacuum, and each contributing diagram is frame-dependent. PoS(QCD-TNT-II)008 2 Q ] for the 7 , 6 Stanley J. Brodsky ] ] measured in deeply 23 , one can compute vir- ) 19 i λ , i ⊥ k ~ where the struck quark is far- , i , defined from the integral over x x ) ( Q H ERBL evolution of the distribution , / i . n x 2 ) Fock state, provides a fundamental ( ψ n H W φ . Form factors, exclusive weak transition n ] they are the nonperturbative inputs to the 17 , 2. The resulting distributions obey the DGLAP 4 16 + 0 ] so that the fall-off of the DIS cross sections in n 21 = n ] This theorem can be proven in the light-front formalism are (assuming the “handbag" approximation) overlaps of the 22 and p 0 of the valence (smallest γ n 0. [ , and the generalized parton distributions [ 2 ] between string theories defined in 5-dimensional AdS space and → = Q 1 νπ p n ] In each case, the derived quantities satisfy the appropriate operator ) = ] The LF vacuum is trivial up to zero modes in the front form, thus ` ≤ ∗ 0 γ i ( 20 12 ] The DLCQ approach can in principle be applied to QED(3+1) and → 2 ⊥ B k ~ B 24 , 14 ] such as 18 One of the most significant theoretical advances in recent years has been the application of Given the frame-independent light-front wavefunctions One can solve the LF Hamiltonian problem for theories in one-space and one-time by Heisen- A key example of the utility of the light-front is the Drell-Yan West formula [ A fundamental theorem for gravity can be derived from the equivalence principle: the anoma- conformal field theories in physical space-time, to study the dynamics of strongly coupled quan- and ERBL evolution equations as aical function factorization of the scheme. maximal [ invariant mass, thus providing a phys- Fock state by Fock state. [ QCD(3+1). This nonperturbative methodframe-independent; has however, no in fermion practice, doubling, thefound is to resulting in be large computational Minkowski matrix challenging, space, diagonalizationbased so on and problem alternative AdS/QCD has are methods necessary. and approximations such as those 3. AdS/QCD the AdS/CFT correspondence [ eliminating contributions to the cosmological constant from QED or QCD. [ berg matrix diagonalization. Forlutions example, of the mesons complete and seteral baryons of color, in discrete multiple QCD(1+1) and flavors, can and continuum(DLCQ) be general eigenso- method. obtained quark [ to masses using any the desired discretized precision light-cone for quantized gen- factorized form of hard exclusive amplitudes andmomentum exclusive where heavy one hadron can decays iterate in the pQCD. hardfactorization At scattering theorems, high kernel, and this ERBL yields evolution dimensional of counting the rules, distribution amplitudes. tually all exclusive and inclusive hadrongluon observables. distributions For which example, are the measured valencethe in and squares deep sea of inelastic quark the lepton and LFWFs scattering summedamplitudes (DIS) [ over are all defined Fock from states virtual Compton scattering initial and final LFWFs with product expansions, sum rules, andoff evolution shell, equations. DGLAP At evolution large is quenched, [ AdS/QCD, Light-Front Holography, and Sublimated Gluons spacelike form factors offinal electromagnetic LFWFs. currents, The gauge-invariant given distribution amplitude the as transverse simple momenta convolutions ofmeasure initial of and the hadron at the amplitude level; [ satisfies Bloom-Gilman inclusive-exclusive duality at fixed lous gravitomagnetic moment defined from thetensor spin-flip is matrix identically element zero of the energy-momentum amplitudes and DGLAP evolution ofobservables structure can functions also are be computed automatically from satisfied.ables, the one LFWFs; Transversity must however, include in the the lensing case effect of of pseudo-T-odd observ- final state or initial state interactions. PoS(QCD-TNT-II)008 ] - ) 0 n → → β , 32 ] or, → µ L (3.1) (3.2) ϕ to the , x ] with z 30 S ) ( 2 32 at 2 , or [ , ) and τ 4 ) z z ( z ( 2 M z λ λ represents the 2 → SO 1 ) κ . In the case of 2 − ) z → z d Stanley J. Brodsky z ( z / ( (+ 4 Φ ] that the QCD Φ / , where → 2 ) z 27 ) 2 2 z κ / ( 3 S -function have a maximal ϕ ± 0. Since there is a window β + Re → L 2 1 physical space-time to scale 2 + . The resulting spectrum repro- . + Q mc and the principal radial quantum z ], one can relate the solutions by n
( 2 , L 2 = 33 ) κ 2 dz 00 4 g − dz ν √ − 2 ) = ν dx n geometrical representation of the conformal , µ mc 5 dx L , µ dx S ( ) = dx µν 5 2 z η ( ] We assume a dilaton profile exp µν V M ) η 31 z ( ( 2 2 λ z e R ] and empirical effective charges, [ 2 2 . In the AdS/CFT duality, the amplitude z R = z 26 ] The QCD infrared fixed point arises since the propagators of 2 = ds 28 2 ]; however as shown in Ref. [ ds in the fifth dimension coordinate of the hadron at short distances. 31 ) τ 1 − S + in the fifth dimension: L ( m 2 κ ] The decoupling of quantum loops in the infrared is analogous to QED where vac- 2 29 + . The modified metric can be interpreted in AdS space as a gravitational potential for 5 2 z 4 . n ] Dyson Schwinger theory, [ κ 25 QCD is not conformal but there is in fact much empirical evidence from lattice gauge the- The conformal invariance can also be broken by the introduction of an additional warp factor We thus begin with a conformal approximation to QCD to model an effective dual gravity . Thus one can match scale transformations of the theory in 3 ) = for AdS z µ ( x λ 3 2 ory, [ matches the twist-dimension extension of the hadron into the additional fifth dimension. The behavior of transformations in the fifth dimension in the AdS metric to include confinement forces: which is invariant under scaleλ changes of the coordinate in the fifth dimension wavelength. [ opposite sign to that of Ref. [ duces linear Regge trajectories foris mesons: proportional to the internalnumber spin, orbital angular momentum The results are equivalent to introducing a dilaton profile, provided that an object of mass group represents scale transformations withinduced the with a conformal sharp window. cut-off in Confinement the infrared can region be of intro- AdS space, as in the “hard-wall” model, [ description in AdS space. The five-dimensional AdS group of isometries of Anti-de Sitter space. The AdS metric is function vanishes in the infrared. [ uum polarization corrections to the photon propagatorwhere decouple the at QCD coupling is largefor and massless approximately quarks. constant, Thus, QCD even resembles thoughmatical a QCD representation conformal of is theory the not conformal conformally group invariant,an in one analytic five-dimensional can Anti-de first use Sitter approximation the space to mathe- to the construct theory. 3.1 AdS/QCD Models more successfully, using a dilaton backgroundlarge in distances the as fifth in dimension the to “soft-wall” produce model. a [ smooth cutoff at a canonical transformation. The positive dilaton background leads to the harmonic potential [ the confined quarks and gluons in the loop integrals contributing to the AdS/QCD, Light-Front Holography, and Sublimated Gluons tum field theories. The essentialtheories principle is the underlying isomorphism the of AdS/CFT the group approach of to Poincaré conformal and conformal gauge transformations U PoS(QCD-TNT-II)008 . ] = 3 35 ζ = (4.1) is the C modes N ⊥ J b ~ ] Stanley J. Brodsky 2 describing the quark ) ζ ] We thus will choose the ( the gravitational potential , φ ) z 32 ζ ( .[ in the soft-wall model with a φ κ 2 ) / 1 1 M represents the off-light-front energy − ζ S i ∼ z ) = bound states at fixed light-front time has h + ζ ¯ ] where axial and vector currents become q ( L q ( φ 2 37 in the Schrödinger equation. Here  , κ ) 2 GeV, which enters the confinement potential. r 2 ζ / . In effect 36 ( 1 + 6 ⊥ k 2 U ] This additional warp factor leads to a well-defined ~ ] Its eigenvalues determine the hadronic spectra and ∼ ] has argued that the Coulomb and linear potentials ζ + 2, and one can study baryons with finite color 4 34 κ 32 2 , 39 κ ≤ L 2 32 4 J ζ 4 ) = − ζ . This correspondence provides a direct connection between 1 ] provides a remarkable connection between the equations of ( τ 4 . Furthermore, for large values of − U , κ 2 3 / 2 in AdS space with LF wavefunctions ζ , 1 d . For the positive solution, the potential is non-monotonic and has d 2 ) ] have shown that a harmonic oscillator confining potential naturally z z ∼ ( − 38 0 Φ  z ] The resulting equation for the mesonic is a Lorentz invariant coordinate which measures the distance between the quark and 2 2 ⊥ b ~ ) x − The identification of orbital angular momentum of the constituents is a key element in the Light-front holography [ Glazek and Schaden [ 1 ( x where the confining potentialpositive-sign dilaton-modified is AdS metric. [ shell and invariant mass dependenceof of quark the and bound gluon state;momentum. binding it [ allows from the the separation kinematics of of the constituent dynamics spin and internal orbital angular the form of a single-variable relativistic Lorentz invariant Schrödinger equation [ Fourier conjugate of the transverse momentum in AdS space. Thestructure resulting and light-front internal dynamics wavefunctions of provideThere hadronic a is states only fundamental in one description parameter, terms of the of mass the their scale constituent quark and gluons. antiquark; it is analogous to the radial coordinate its eigenfunctions are related to the light-frontangular wavefunctions of momentum. hadrons for This general LF spin wave and equationand orbital serves it as is a semiclassical equivalent first to approximation the to QCD, equations of motion which describe the propagation of spin- in contrast with the usualthe AdS/QCD primary framework entities as [ in effective chiralnot theory. limited Unlike to the hadrons top-down of string maximum theory spin approach, one is motion in AdS space andon the the Hamiltonian light formulation front of atthe QCD fixed in hadronic LF physical amplitudes time space-time quantized q description of the internal structure of hadronsand using gluon holographic degrees principles. of In freedom our approach are quark explicitly introduced in the gauge/gravity correspondence, [ arises as an effectivehigher potential gluonic between Fock states. heavy quark Also, states Hoyer [ when one stochastically eliminates scale-dependent effective coupling. are uniquely allowed in theharmonic Dirac oscillator potential equation in at the the corresponding classical Klein-Gordon equation. level. The linear4. potential becomes Light-Front a Holography AdS/QCD, Light-Front Holography, and Sublimated Gluons the negative solution, the potentialto decreases infinitely monotonically, large and values thus of an an absolute object minimum in at AdSincreases will exponentially, confining fall any object toconfining distances positive sign dilaton solution. [ and gluon constituent structure of hadrons in physical space-time. In the case of a meson, PoS(QCD-TNT-II)008 . , z n S + for L 2 + and κ n L ( 2 κ 4 which is com- = ++ ) (2040) 2 (2050) 4 4 4 6 GeV; and (b) the Stanley J. Brodsky x f a . 0 − . The spectrum also M ζ 1 = ( is their relative orbital κ x -- L PC 3 L (1670) (1690) n=2 n=1 n=0 J 3 3 p ρ ω ∼ 0) is massless for zero quark ) state, x = ( 1 2 3 4 ++ (1320) q (2300) (1950) (1270) 2 n=3 L 2 2 2 π 2 f f f a , ] thus verifying the consistency of φ for a change of one unit of spin 0 9 2 , and ¯ for the soft-wall (a) and hard-wall (b) = 8 κ 2 q n -- 0 , (782) (770) 1 (1650) (1420) (1700) (1450) -meson family with ρ 0 ω ρ ρ and the fifth-dimension AdS coordinate ω ω π and 2 = ζ 0 6 4 2 L 2 S 9-2009 8796A1 M 54 GeV. . 7 0 has a string-theory Regge form = κ -+ 4.1 4 for a unit change in the radial quantum number, 4 ]. The pion ( 2 κ 40 +- provide the light-front wavefunctions of the valence Fock state 3 4.1 1 is the combined spin of the , -+ L PC 0 (1880) (1670) 2 2 2 J n=2 n=1 n=0 ] as illustrated for the pion in Fig. π π 3 = -meson families with [ ] It has also been shown that one obtains the identical holographic mapping S ) ω 3 ⊥ 0 b +- ~ 1234 , = 1 n=3 x I b(1235) ( ψ -+ π 0 (1800) (1300) 0 Parent and daughter Regge trajectories for (a) the , the square of the eigenmasses are linear in both the orbital angular momentum π π counts the number of nodes of the wavefunction in the radial variable n -meson and 4 2 6 0 i.e. , where the data are from Ref. [ 2 ρ ; 1 The eigensolutions of Eq. The mapping between the LF invariant variable The meson spectrum predicted by Eq. ) 2-2011 8796A5 M 1 2 / = patible with moments determined fromobservables lattice such gauge as theory. hadronic One form can factors then (overlaps immediately of compute LFWFs), structure functions (squares of was originally obtained by matching theAdS expression for space electromagnetic with current the matrix elements correspondingphysical in expression space time. for [ the current matrix element, using LF theory in mass, consistent with the chiral invariancehadron of spectrum massless by quarks simply in QCD. adding Thus 4 one can compute the Remarkably, the same rule holds for three-quark baryons as we shall show below. of the hadrons models. The resulting distribution amplitude has a broad form Figure 1: angular momentum as determined by the hadronic light-front wavefunctions. S a change in one unit in the orbital quantum number I using the matrix elements of the energy-momentum tensor, [ AdS/QCD, Light-Front Holography, and Sublimated Gluons In the case of mesons the holographic mapping from AdS to physical observables defined on the light front. 5. The Hadron Spectrum in Light-Front AdS/QCD where depends on the internal spin S.Fig. This is illustrated for the pseudoscalar and vector meson spectra in PoS(QCD-TNT-II)008 ]. 64 , 63 , 62 , 61 , Stanley J. Brodsky 60 , ) is derived from the in the LF Dirac equa- 59 ζ 4.1 2 , κ 7-2007 58 8755A1 0.2 0.1 0 , ∼ 0 20 ) 57 ζ x , ( b 0.5 U 56 10 ] It is possible to extend the model , , predicting the same multiplicity of 2 spinor component representation is 42 0 55 L 1.0 , × to the LF wavefunctions in the physical ] The duality between these two methods + z 2 54 n , of its interpolating operator at short distances ∼ 53 τ 2 , , in a 2 b) ( 2 LF 52 8 M , D 51 0.10 0.05 0 = 0 , 20 x in the LF equation of motion ( ] For fermionic modes the light-front matrix Hamiltonian LF ) for the AdS/QCD (a) hard-wall and (b) soft-wall models. 50 to the twist 2 b , ⊥ H 0.5 z ζ 65 b Casimir term in Schrödinger theory. One thus establishes 10 ~ , / , 49 i x 2 2 , ( r L ψ 0 π | / 1.0 48 ) ψ , 1 M 47 + = , ` i ] thus the twist of a hadron equals the number of constituents plus the (

46 `

ψ

(x,b) , ψ | 41 ] The term in the AdS equations of motion. The interaction terms build confinement (a) 45 2 LF . , L D ζ 44 ) was derived by taking the LF bound-state Hamiltonian equation of motion , 4.1 43 , Pion LF wavefunction 33 . , in analogy to the ) τ Individual hadrons in AdS/QCD are identified by matching the power behavior of the hadronic For baryons, the light-front wave equation is a linear equation determined by the LF transfor- Higher spin modes follow from shifting dimensions in the AdS wave equations. In the soft- Equation ( ϕ Figure 2: , ζ eigenvalue equation corresponding to the dilaton modification of AdS space. [ the interpretation of amplitude at the AdS boundary at small mation properties of spin 1/2 states. A linear confining potential reduction of the LF kinetic energy when( one transforms to two-dimensional cylindrical coordinates states for mesons and baryonsto as hadrons observed experimentally. with [ heavyCoulomb quark corrections. constituents For by other introducing recentsee nonzero calculations Refs. of quark the masses [ hadronic and spectrum based short-range on AdS/QCD, wall model the usual Regge behavior is found as required by the AdS/CFT dictionary.chiral The super-multiplets; twist [ also equals the dimension of fields appearing in provides a direct connectionHamiltonian between formulation the of description QCD in oftime physical hadronic space-time modes quantized in on the AdS light-front space at and fixed the LF as the starting point. [ relative orbital angular momentum.plitude One eigensolutions then in can the apply fifth-dimensionspace-time coordinate light-front variable holography to relate the am- tion leads to linear Regge trajectories. [ AdS/QCD, Light-Front Holography, and Sublimated Gluons LFWFs), as well as the generalizedhard parton exclusive distributions reactions. and For distribution example, amplitudes hadronic whichLFWFs underly form in factors the can Drell-Yan West be formula. predicted from the overlap of PoS(QCD-TNT-II)008 (5.1) (5.2) 4 , is a twist (2420) i Δ ` family. The 5 GeV. 1 . ∆ = 0 m i ∑ = Stanley J. Brodsky 3 . We thus require κ = ] states are shown. L L 40 , + ψ } L m 2 ` 2 state. (1950) (1905) (1920) (1910) Δ Δ Δ Δ D , , = + − ... , L ψ ψ ) 1 2 + , 1, q ) ζ 1 ` M M 2 2 = n=3 n=2 n=1 n=0 baryon families for D ζ κ = = n 2 ( ∆ ψ 1 q κ ` − + ( + ν ν n n D and (b) L L 0 ζψ ζψ (1600) (1232) 2 2 N 2 2 Δ Δ ... / / relative to the lowest ground state, the proton. , only confirmed PDG [ 2 2 1 κ κ , 1 ` ζ ζ 2 { 2 2 − − κ κ κ D 9 2 − + − − ψ e e ψ ψ N(2200) ν ν = are well accounted for in this model as the first and 1 2 1 2 = + + S corresponds to the first radial state of the ) 1 3 2 2 L ζ ζ ∆ + + ) + ζ ζ 3 ν ν . O ∼ ∼ 1710 and ) − − ( 1660 1 ) ) 2 ( + − N ζ ζ κ + ∆ ψ ψ ( ( 4 ν + − ζ ζ L d d = d d ψ ψ + N(1680) N(1720) L N(1900) n − ( ∆ and the 2 , 2 ) κ κ 4 with scaling behavior given by its twist-dimension 3 4 = n=3 n=2 n=1 n=0 L = 1440 2 ( n + N ∆ M 2 Positive-parity Regge trajectories for the / (a) 0 1 2 3 4 N(940) N(1710) N(1440) is explained in this framework as a doubly excited 0 2 6 4 1 to match the short distance scaling behavior. Higher spin modes are obtained by shifting ] in the LF Hamiltonian equations. By contrast, if we start with a five-dimensional action ) 2 . As for the predictions for mesons in Fig. Figure 3: + 66 3 12-2011 8796A3 M The resulting predictions for the spectroscopy of positive-parity light baryons are shown in The baryon interpolating operator L 1900 = ( N with eigenfunctions ν dimensions for the fields. Thus, asbaryonic in state the is meson sector, theSince increase our in starting the point mass to squared find forfix the higher bound the state overall equation energy of scale motionpion identical for baryons [ for is mesons the and light-front, baryons we by imposing chiral symmetry to the The Roper state second radial states. Likewise the for a scalar field in presence of a positive sign dilaton, theFig. pion is automatically massless. and eigenvalues 3, dimension 9 AdS/QCD, Light-Front Holography, and Sublimated Gluons equivalent to the system of coupled linear equations PoS(QCD-TNT-II)008 . , , ) q ∼ µ ]. x µ z ) = ) = γ z z (6.2) (6.3) (6.1) ¯ 43 , , q 1905 q µ 0 ( e x ( ∆ . = ) P = 2 2 2, Φ q µ Q ( J ( = – the hard wall M V F L Stanley J. Brodsky , ] µ 2 5 ) 0

QCD ) P Λ ζ / is the four-momentum , + 1 x q ( P π ( ∼ / µ ¯ z d ε u
ψ are the Minkowski coordinates q

5 , − ! i ] ) ¯ x z P 5 d ( u x 2 − | − 0 for a photon propagating in AdS space 1 Φ is the determinant of the metric tensor. P ) . The expression on the right-hand side ) degenerate pair and the z z µ g r respectively and , is the solution of the AdS wave equation 4 , ] ε ) q x 0 2 5 δ ( ) 0 (no physical polarizations along the AdS ζ P 4 z Q M ) , (

], but further insight is required to understand = 2 A 0 π 1720 V z q J , with boundary conditions [ ( 5, and from momentum conservation at the vertex from 2 48 and ( ) 3 ( A ζ ) N 10 z dz V P q d ··· ∼ and plane waves along Minkowski coordinates zQ − , ζ ( Z ) ) 1 − ) we find [ 2 1 z 3 Z , P 1. We now compare this result with the electromagnetic = R valence Fock state x 6.1 ) M ( ] − x ] The coordinates of AdS 1680 + 0 zQK N P 4 ( , = P π , ) = ) = − Φ 30 ( N 2 dx 3 0 M µ 1 4 M ( ) = Q in the gauge P = δ 2, z ( x ∂ µ 4 , ) . It is the EM matrix element of the quark current q 1 ←→ F ) z 2 P describing a meson in AdS is [ ) ( = 0 ) , 2 z π + Z ) Q 2 L , q , with states which are degenerate within error bars. Our results for the π z 2 ( q ( ) π x , ( F ( z ) ( V x 2 0 M , ( V ∗ P F µ x P · 0) x µ Φ and iq ) = Φ ) ) 1950 0 2 z − > 2 ⊥ ( , q P e = ( b ∆ 2 ( x ) ) ( q + + q , M x ( π ) M x states agree with those of Ref. [ − , in order to avoid coupling to Fock states with different numbers of constituents. P F µ − 1. + ε ∆ 1 gA = J ( = ( 1920 2 x √ i ) = ( with invariant mass ) = Q 0 P ∆ z = ). The bulk-to-boundary propagator | ) labeled , and also sets the gap scale. [ z ) xdz , z 2 = µ z 4 ) ( 0 ζ x ) represents the space-like QCD electromagnetic transition amplitude in physical space-time z ( d ( Φ , µ x The propagation of the pion in AdS space is described by a normalizable mode The form factor is computed in the light front from the matrix elements of the plus-component In the higher dimensional gravity theory, the hadronic transition matrix element corresponds The model is successful in explaining the important parity degeneracy observed in the light 2 µ QCD · J Z 6.1 and | A Q iP 1910 Λ 0 ( ( / µ − P integration over Minkowski variables in ( variable e Extracting the overall factor where form factor in AdS spaceto time. The incoming electromagnetic field propagates in AdS according of the current For definiteness we shall consider the given by ( to the coupling of an external electromagnetic field h better the complex structure of thealso of a the property of negative parity the baryons. hard wall Parity model, degeneracy but of radial baryons states is are6. not well Hadronic described Form in Factors this in model Light-Front [ AdS/QCD with the extended field To simplify the discussion we will first describe a model with a wall at x The pion has initial andtransferred final to four the momentum pion byof the ( photon with polarization and represents a local coupling to pointlikeamplitudes constituents. look Although very the different, expressions for one the can transition carried show out that at a fixed precise light-front mapping time. of [ the matrix elements can be baryon spectrum, such as the AdS/QCD, Light-Front Holography, and Sublimated Gluons ∆ positive parity model – which limits the propagation of the1 string modes in AdS space beyond the IR separation V PoS(QCD-TNT-II)008 (6.4) and the smaller  κ ] x x − 69 1 q Stanley J. Brodsky zQ . We find the result  0 Q 0 diverge term-by-term; dxJ 1 0 = R 2 . The factorization given in | 0.0 Q ⊥ ) = 2 z b ~ , | 2 ) ] boxes are JLAB data. [ x GeV Q 2 ( , 68 0.5 q − - V ) 1 πζ ζ ( . Continuous line: confined current, dashed 2 ( ) x 2 φ q p p ( ) for arbitrary values of 1.0 ] ) . The pion form factor and the vector meson π - x 4 = F ( ] where we identify the transverse impact LF 67 6.2 are kinematical generators which commute with 3 ζ X components. The confined EM current also leads ⊥ 11 ¯ ,[ q P ~ ) → q 1.5 x ) = - z ϕ , − , and z 1 ζ ( , the EM current becomes modified as it propagates in an + , x x P ( 2.0 i.e. - p M ψ 2 q I . Π ) = − F x P ( 2.5 - X 1.0 0.8 0.6 0.4 0.2 0, as illustrated on Fig. ] for the pion form factor. The DYW formula is an exact expression for = and 7 2 ) , Q 6 ζ 1. We use the integral representation of ( Φ 2 ) = / 0 3 ) = R with the holographic variable 2 / Space-like electromagnetic pion form factor ζ Q ζ ( F The results for the elastic form factor described above correspond to a “free” current prop- Alternatively, one can use a truncated basis of states in the LF Fock expansion with a limited ) = ( ) is a natural factorization in the light front since the corresponding canonical generators, the ζ ( 6.4 poles residing in the dressed current in the soft wall model require choosing a value of Figure 4: agating on AdS space.light-front It formula is [ dual to the electromagnetic point-like current in the Drell-Yan-West number of constituents and the non-perturbative polecurrent structure as can in be the generated with Heisenberg a picture, dressed EM factorization of the LFWF to compare with the light-front QCD form factor ( the LF Hamiltonian generator 7. Elastic Form Factor with a Dressed Current the form factor. It is writtentrary as number an of infinite constituents. sum This ofin allows an AdS one overlap space. of to LF map However, Fock this state-by-statetime-like components to mapping form with the has factor an effective the cannot arbi- gravity be shortcoming theory obtainedstates that in the is the multiple included. time-like pole region Furthermore, structure unlessfor the of an example moments the infinite one of obtains number the an of form infinite Fock charge factor radius. at [ line free current. Triangles are the data compilation from Baldini, [ IR deformed AdS space tocurrent which simulate includes confinement. any number The ofto dressed virtual finite current moments is at dual to a hadronic EM longitudinal and transverse generators variable AdS/QCD, Light-Front Holography, and Sublimated Gluons where φ ( PoS(QCD-TNT-II)008 ) ) τ ), z 2 , Φ Q (the 2 7.5 ( (7.2) (7.6) (7.4) (7.5) (7.1) (7.3) 0 and Q τ τ ( F = V creates a ] n ¯ q q 73 → and the bulk-to- Stanley J. Brodsky − e τ ]. + e 72 has the form of a har- , ) Fock component 71 2 z , , . τ 2 , )  x 70 κ 2  − − 2 2 ± ) with twist τ 1 , ( 2 z ( ρ Q , , / 2 τ 2 2 M x / | P 7.1 2 κ 2 ) z , z 2 + 2 ζ = 0 κ κ ( 1 , 2 defined at the asymptotic boundary of − − 2  + ) 2 e 3 hadron, we compute the spin non-flip e z τ κ 2 ψ τ 2 ( Q | κ 4 z ··· Q τ 4 = ) 1 ] The remarkable analytical form of ( . In the case of soft-wall potential the EM x   Φ τ ζ − τ 0 4 τ 2 ). This agrees with the fact that the field 2 , 2 τ 2 ρ z P U ) 2 2 Q κ M is x κ Q  7.1 which determines the mass scale of the hadronic ) − ( τ dx 2 + − 1 ] The proton and neutron Dirac form factors are 2 composition of the vector mesons in the spectrum. e 12 κ τ V Φ κ 1 1 ¯ − Q q 3 P ( 65 4 ζ z dz q τ 2 1 d (  0 0 + 2 2 ρ mode ( Γ Z Z 1 Z Q M . The normalizable solution for a meson of twist 2 = τ 2 z  = s z z 2 + i 4 L Γ ) = κ ] τ 1 2 κ ) =  Φ Q 73 z | ( ) = ( , τ ) = p z τ z 1 4 , Φ , ) = F 2 Φ h 2 2 Q 1 product of poles along the vector meson Regge radial trajectory. Q Q ( vector meson mass and its radial excitations, incorporates the correct ( ( − τ V V ρ ) we find the elastic form factor for a twist F τ ) the expression for the hadronic state ( 7.4 1 instead of the 6.3 ± ] 2 than the canonical value of = ) can be conveniently written in terms of the integral representation [ 4 z is the Tricomi confluent hypergeometric function. The modified current √ L ) / 7.3 c 0) [ , b 0 state is given by > , 2 is the probability for the twist a = ( q . Eq. ( τ L P − U ∞ As an example of the scaling behavior of a twist Substituting in ( The effective potential corresponding to a dilaton profile exp = → 2 2 Q where AdS space, and thus thebulk-to-boundary scaling propagator is dimension [ of ( given by couples to a local hadronic interpolating operator of twist boundary propagator ( which is expressed asFor a a pion, forthe example, form the factor lowest is Fockexpressed the state in well – terms known the of monopole the valencescaling form. state behavior [ – from is the aconfinement. constituent’s twist-2 hard state, scattering and with thus the photon and thenucleon mass form factor gap in from the soft wall model. [ Other recent computations of the pion form factor are given in Refs. [ with normalization where has the same boundaryQ conditions as the free current, and reduces to a free current in the limit AdS/QCD, Light-Front Holography, and Sublimated Gluons by a factor of 1 quark pair with monic oscillator confining potential spectra. This shift is apparently due to the fact that the transverse current in number of constituents for a givenorbital Fock component) corresponding to the lowest radial PoS(QCD-TNT-II)008 τ 35 . ) 2 (7.8) (7.7) (7.9) 2 (7.11) (7.10) / 1 30 GeV ] + 2 4 n Q ( 2 25 κ 4 Stanley J. Brodsky = " 20 2 2 . Q 2 ! M , / n 1 2  F ζ 2 2 15 | 4 , ) κ 2 Q | − ζ ) ( e 2 b − / , 10 5 x ψ . ( ζ 2 2 3 κ ψ Q − | 4 . To compare with physical data we | κ 2 5 x , 5 | )) ) 2 ) x − )= ζ Q τ ( , − ζ ) x + ( x 1 ( 0 ( − ψ ρ | 0.1 0.2 0.3 0.4 0.0 − ! ! ! ! in the soft wall model. Data compilation from  Ψ 1 bQ ) dx ) ( ζ 1 2 , 0 )( , 0 13 2 1 Q 2 / Z ( 2 − n Q 1 ζ 35 ( 2 τ F bdbJ κ 4 ) = V are shown in Fig. 2 ∞ − 2 Q 0 ζ e ) ) to their physical values located at Q Z 2 2 d ) = ( ( / 30 π GeV 3 Q Q F 7.5 and Z ( 2 2 , ζ n x 3 1 1 ) Q 2 ( 2 F κ − 4 ρ ) = Q 25 2 ( ). Q Q | p 1 √ , ⊥ ) = F x 0. Plus and minus components of the twist 3 nucleon LFWF are 2 b 4 ( " | and 2 )= Q 20 Q ρ ( ) ζ Q = n ! ) = 2 ( 1 p 1 b 0 F + Q ( F ( n ψ 4 1 p 15 1 in a modified AdS space. For the soft-wall model the expression is [ F Q F τ 4 1, Φ Q 10 ) = ] Predictions for 0 ( ] 75 p 5 1 74 F The form factor in light-front QCD can be expressed in terms of an effective single-particle It is also possible to find a precise mapping of a confined EM current propagating in a warped We can also compute an effective density on the gravity side corresponding to a twist Electromagnetic Current 0 0.0 0.6 0.4 0.2 1.4 1.2 1.0 0.8 density [ The results for 7.1 Effective Light-Front Wave Function From Holographic Mapping of a Confined AdS space to the light-frontwe QCD find Drell-Yan-West expression an for effective the LFWF, formstates which factor. generated corresponds by In to the this a case “dressed”done superposition we confined analytically. of current. an For infinite the number soft-wall of model Fock this mapping can be where have shifted the poles in expression ( where Figure 5: AdS/QCD, Light-Front Holography, and Sublimated Gluons Diehl. [ for a two-parton state ( hadronic mode PoS(QCD-TNT-II)008 ] ) x 0, = 85 − 2 = u , 7.12 ] over β (7.12) (7.13) + 84 − 88 A e and 1 ) u x ] This result 2) appearing at α ( ) 89 = 0 . The resulting x ]. i ( τ s Stanley J. Brodsky ψ 80 we discuss further | n uduJ h 10 ∞ 0 R = n ] as shown by Chang and ψ ]. Nucleon transition form . . Such distributions [ ) 86 77 ) x x 1 x 1 ( ] distribution ( ln ln / 83 2 2 ) ) x x ] and [ is the primary origin of the sea quark − − 1 i 1 76 ( ( as in QCD(1+1). Thus in holographic ¯ 2 Q 2 ⊥ ] where soft gluons interact so strongly κ pairs, but surprisingly no dynamical glu- ¯ b q 2 2 ¯ q / 90 κ ¯ qq 2 ⊥ q 2 1 k uudQ − ) we use the integral − | e → e ) is a matrix in Fock space which represents an ) ) q
x 1 ) x x 1 7.10 ( 14 x 2.1 ], and structure functions is Ref. [ − ln and ¯ − 1 ln 79 π ( 1 ¯ insertions connected to the valence quarks in the pro- q ( q κ q qq gg π κ . We find the effective two-parton LFWF ( 4 , an intrinsic strangeness [ ) → x ) ] and [ → ( x q ) = ln ¯ ( 78 ) = ¯ Q γ , d ⊥ ] Experiments show that the sea quarks have remarkable nonper- ¯ ⊥ e q Q b k q , 6= 82 , = x , x ) → ( γ ( x → x ( ψ 81 ψ ¯ q gg u q , qq → ) with the QCD expression ( qq 7.11 and the relation , ] β 4 / 87 2 1, as well as intrinsic charm and bottom distributions at large . α To illustrate the relevance of higher Fock states and the absence of dynamical gluons at the The LF Hamiltonian eigenvalue equation ( The five-quark Fock state of the proton’s LFWF 0 − e > β 1 2 ons. This unusual property of AdS/QCD may explain the dominance of quark interchange [ QCD, higher Fock states can have any number of extra which leads to potential in quantum field theoryaction can be in considered LF as time, an similar instantaneous four-point to effective the inter- instantaneous gluon exchange in the light-cone gauge hadronic scale, we discuss a simplepion semi-phenomenological model where of we the include elastic form the factor first of the two components in a Fock expansion of the pion wave function for the active and spectatorto quarks, obtain respectively. generalized Holographic parton QCD distributions methods (GPDs) have in also Refs. been [ used x will arise rigorously from ton self-energy; in fact, they fit a universal intrinsic quark model, [ infinite number of coupled integral equations for the Fock components quark annihilation or gluon exchange contributions inis large consistent angle elastic with scattering. the [ flux-tube interpretation of QCD [ experimental results in hadron physics which support this picture. in impact space. Theand momentum it is space given expression by follows from the Fourier transform of ( The effective LFWF encodes non-perturbativeterm-by-term holographic dynamical mapping, aspects unless that onehas cannot adds the be an right infinite learned analytical number from properties ofEM to a terms. form reproduce factor. the Furthermore, Unlike bound it the statean “true” vector infinite valence meson LFWF, number pole the of in effective Fock the LFWF, components, which time-like is represents not a symmetric sum in of the longitudinal variables factors have been studied in Refs. [ turbative features, such as ¯ Peng. [ that they are sublimated into a color confinement potential for quarks. In Sec. AdS/QCD, Light-Front Holography, and Sublimated Gluons To compare ( 8. Higher Fock Components in Light-Front Holographic QCD distributions of the proton. [ PoS(QCD-TNT-II)008 i ¯ q → q | 2 5 q 2 . The 6 (and the ] squares ¯ q 68 GeV ¯ qq 2 ,[ q 4 Q P 120 MeV. The et al. Stanley J. Brodsky = 00 ρ Γ 3 respectively. " 2 ] The potential between Q ¯ ! qq 92 Π mass and the masses of the q F 5 γ ρ + 2 γ log twist-two and twist-4 states q state. The value of + i − ) to introduce a finite width: ¯ q 0 360 MeV and and ¯ ¯ 7.5 q qq = 1 = 5 q 0 | γ ρ PC + J Γ γ q . The time-like structure of the pion form factor 0 ) in QCD is usually restricted to the perturbative 1 2 2 1 0 containing a dynamical gluon as the interference ! ! 2 15 ) i / 2 is determined from the 1 ¯ qg Q 7 κ 140 MeV, ( q + , where the s | n 2 α = ( ··· 2 ρ 6 κ Γ + 4 GeV 4 2 = Q τ → 5 i 2 ¯ q ] an effective coupling or charge can be defined directly from phys- M ¯ qq 91 q " 4 | = 13 %, the admixture of the 2 ¯ q ¯ q Q ! ¯ qq ¯ Π qq q q F 3 ψ P 2 Q + 2 ] = 2 τ 69 i ¯ q . We choose the values q | are created by the interpolating operators ¯ Γ 1 π i / Structure of the space- and time-like pion form factor in light-front holography for a truncation of ¯ ¯ q q iM q 2 ¯ qq ψ 0 √ q The definition of the running coupling in perturbative quantum field theory is scheme-dependent. The concept of a running coupling Since the charge form factor is a diagonal operator, the final expression for the form factor | 0.2 0.1 0.0 0.5 0.4 0.3 0.6 = + i 2 π As discussed by Grunberg, [ domain. However, as in QED, itthe is full useful space-like to and define time-like themomentum domains. transfer coupling is as a The an complex study analytic problem function of because valid of the over gluonic non-Abelian self-coupling QCD and color coupling confinement. at small becomes opposite in sign. 9. Nonperturbative Running Coupling from Light-Front Holography and displays a rich pole structure with constructivewith and the destructive interferences, admixture which of is the incompatible twist three state ical observables. Effective charges defined fromin different the observables leading-twist can domain using be commensurate relatedinfinitely scale to relations heavy each (CSR). quarks other [ can be defined analytically in momentum transfer space as the product of corresponding to the truncation uppole to twist term. four is In theand the sum are strongly of stable. two coupled terms, semiclassical One a gauge/gravity monopole can and limit, nonetheless a hadrons modify three- the have formula zero ( widths the pion wave function up to twist-four. Triangles are the data compilation from Baldini | q results for the pionresults form correspond factor to with twistwidths) two are input and in the four model. Fockradial The excitations value states follow of are from shown in Fig. Figure 6: AdS/QCD, Light-Front Holography, and Sublimated Gluons are JLAB data. [ PoS(QCD-TNT-II)008 ] 5 is g 93 has 2 (9.3) (9.1) (9.2) ) Q 2 and the Q ( ) -function -function 1 φ g β β , α ζ at large ( consistent with ) . The coupling of the confining . This effective z 2 2 . Thus 2 Stanley J. Brodsky ) Q q Q scheme, where the ( ζ YM / as a function of the ( 1 ) g ) g q with the invariant im- AdS 2 s space in presence of a ( YM z α V Q 5 g ( α F → ] The coupling AdS s C ) z π α 94 ( 4 5 g and the square of the coupling − . 2 has been measured at intermediate 2 in the five-dimensional action. At ζ , 1 z 2 2 g 2 κ κ ) = F α κ q − ( 2 5 e 1 , = g V 2 ) ∝ z − 5 ϕ ( propagating in AdS π g , ϕ ) 5 4 ] Since z ) F ( / z ge ) ϕ 95 scheme of Appelquist, Dine, and Muzinich. [ / e ζ √ R . However, the fall-off of ( V 16 x 2 α 5 ) = ]. We can also make use of the Q 2 YM d = ( z can be considered to be the nonperturbative extension g ( g 93 Z 2 ) 2 4 1 − 5 . This coupling incorporates confinement and agrees well √ 2 ) = g Q − q is ( ζ ( 5 − = AdS s ] how the LF holographic mapping of effective classical gravity S = AdS s α , rather than the pQCD logarithmic fall-off. It agrees with hadron 2 2 α 15 κ Q / ]. 2 Q 96 − which appears in the LF Hamiltonian: , which introduces the energy scale e is determined from the Bjorken sum rule. [ ζ ) 29 ) ∼ z 2 ) as the effective coupling of the theory at the length scale ( ) 2 Q ϕ consistent with asymptotic freedom, and it vanishes at small ( 9.1 Q 1 2 ( g Q α AdS s α and asymptotic freedom at large is mapped, modulo a constant, into the Yang-Mills (YM) coupling 2 effective charge defined in Ref. [ ) z Q , where it reaches a minimum, signaling the transition region from the infrared (IR) confor- ( V 2 5 α then incorporates the non-conformal dynamics of confinement. The five-dimensional cou- κ g In contrast with the 3-dimensional radial coordinates of the non-relativistic Schrö-dinger the- We have shown with Deur [ Let us consider a five-dimensional gauge field ) ∼ z ( 2 5 ory, the natural light-front variables are the two-dimensional cylindrical coordinates has dimensions of length. We can identify the prefactor theory in physical space-time usingpact light-front separation holography. variable One identifies where the metric determinant of AdS in the AdS action ( g pling mal region, characterized by hadronic degreesregime where of the freedom, relevant degrees to of a freedom pQCD are conformal the quark ultraviolet and (UV) gluon constituents. The In fact, the holographic coupling of the energies, it is particularly useful for studying the transition from small to large distances. physics data extracted phenomenologically from differenttions observables, of models as with well built-in as confinement with andlogical the lattice extended simulations. predic- coupling We can also show be thatderived defined a from which phenomeno- light-front implements holography the becomes significantly pQCDQ negative behavior. in The the non-perturbative regime vanishes at large dilaton background in AdS space, modifiedlytically by simple a color-confining non-perturbative positive-sign effective dilaton coupling background, can bewith used effective to charge identify observables an andat lattice ana- small simulations. It also exhibits an infrared fixed point AdS/QCD, Light-Front Holography, and Sublimated Gluons the running coupling times thecharge defines Born a gluon renormalization propagator: scheme – the strong coupling the advantage that it is the best-measurednition effective charge, of and the it effective coupling can to be large used distances. to [ extrapolate the defi- space-like momentum transfer exponential: an infrared fixed point [ quadratic order in the field strength the action is PoS(QCD-TNT-II)008 . , ) = µ 0 ζ A
. In 9.4 2 (9.5) (9.4) 0 gives 0 / 1 φ = L − 3 2 2 Z − Q ) 1 GeV. The = 0 also displays g ∼ 1 / 7 µ ren AdS g Q A α Stanley J. Brodsky phys ]. ] The lattice results g 27 32 = ( with experimental and 7 scheme: The renormalization of the 0 1 QCD . g g L 1 is the two-dimensional Fourier 1 , in the QCD Lagrangian density − ) − 3 Q λ ζ Z µν ( . 2 G ] which is also shown in the figure. = → κ 0 one has the constraint on the slope λ AdS 4 s 0 is the square of the space-like four- g / 97 α 2 = → ren > ) Q 2 QCD which is computed from the Bjorken sum − Q Q 2 ⊥ 1 L e µν g ζ q ~ ) ( G α leads to an integral which diverges at large 0 ). Integration over the azimuthal angle − 0 normalized to the ( ] from several experiments. Fig. J 2 z = ζ 9.3 2 27 ( AdS s 2 d AdS 17 s κ ) q α ζ α ζ − ∞ − sets the mass scale of the resulting running coupling. ( 0 = Z ) = L serves as the UV regulator, and the renormalized QCD = 2 AdS s ϕ 2 ∼ a Q α (solid line) is compared in Fig. Q ) ( 2 ) 2 Q AdS and field strength s ( Q is the bare coupling in the Lagrangian in the UV-regulated theory, α µ ( ] have been scaled to match the perturbative UV domain. The 0 A AdS s g λ AdS 25 α (a) agree well in the strong coupling regime up to α ) from the scale dependent measure for the gauge fields can be un- 7 → µ 9.2 ] where A , coupling. This defines with the physical QCD running coupling in its nonperturbative domain. 0 15 1 g ) g 2 has been determined in Ref. [ . The physical coupling measured at the scale 2 / x 1 with a rescaled Lagrangian density 1 from Ref. [ 3 Q g ( Z α 7 µν 0 = AdS s G 54 GeV is determined from the vector meson Regge trajectory. [ 0 light-front frame where 2 α . / phys 1 0 = g − 3 = ] over a large range of momentum transfer. At + Z from the Gerasimov-Drell-Hearn (GDH) sum rule [ by a compensating rescaling of the coupling strength q κ 94 1 = The hadronic model obtained from the dilaton-modified AdS space provides a semi-classical The effective coupling The flow equation ( The couplings in Fig. g Details on the comparison with other effective charges are given in Ref. [ α QCD µν ren . momentum transferred to the hadronic bound state. Using this ansatz we then have from Eq. ( π value shown in Fig. first approximation to QCD. Color confinement is introduced by the harmonic oscillator potential, As is the casedynamics in at lattice large gauge distances. theory,wall The holographic color model QCD are confinement couplings thus in defined similarproperties in from AdS/QCD in concept, lattice the reflects and nonperturbative gauge both nonperturbative domain, theory schemes up are and to expected a the to rescaling have soft of similar their respective momentum scales. effective charge rule [ the Bessel transform In contrast, the negative dilaton solution We identify L coupling is thus equivalent to the renormalization of the vector potential and field strength: lattice gauge theory, the latticecoupling spacing is determined from the normalizationpropagator. of the The gluon inverse field of strength the as lattice it size appears in the gluon lattice data. For this comparison to beAdS meaningful, coupling we as have to the impose the same normalization on the other couplings from different observables as well as of The results show no sign ofallows a us phase transition, to cusp, extend orbelow, the other the non-analytical functional smooth behavior, behavior a dependence of fact of the which AdSperturbative the strong domain. coupling coupling [ also to allows us large to distances. extrapolate its form As to discussed the derstood as a consequence ofnon-Abelian field-strength gluon renormalization. field In physical QCD we can rescale the AdS/QCD, Light-Front Holography, and Sublimated Gluons light-cone fraction transform of the LF transverse coupling in the G PoS(QCD-TNT-II)008 0 1 (9.6) (9.7) ) with g1 1 GeV, " . We use 9.5 ∼ CLAS Q (GeV) 7 (pQCD) g1 g1 g1 " " )), and thus Q " 9.5 From Hall A/CLAS From From GDH sum From ) is the analytical Stanley J. Brodsky rule constraint on -function compared β 9.6 1 , ) 2 . ]. The smoothly extrap- ) is given by Eq. ( Q 2 ( κ 15 in Eq. ( 4 − ( 1 AdS 1 g / g fit 2 g ) α 2 Q is also shown on Fig. α − F3 Q s " e ( α 2 2 1 fit . In order to have a fully analytical AdS Modified AdS g -1 54 GeV compared with effective QCD Q κ . Lattice QCD (2007) From Lattice QCD (2004) 7 – and 10 α 0 4 π 7 = − ) + 2 κ 0

-1 -2 Q

) = -0.5 -1.5 ! (Q) ( -1.75 -2.25 -0.25 -0.75 -1.25 2 + . Q g 2 ( ) 2 18 AdS 10 Q ( α 2 3 GeV 1 Q (GeV) . AdS g Q 0 α are smeared step functions which match the two regimes. d = is also shown on Fig. ) log 2 2 s d τ ) = ]. The couplings are chosen to have the same normalization at τ / 2 α ) 27 2 0 Q [ ( Q ) = 1 and 1 2 – the plain black line in Fig. − g g , 2 2 Q π α Q ( ( 1 (2007) ± e ) = AdS by matching to a perturbative coupling at the transition scale 0 / β 8 GeV ( AdS + Modi fied F3 . 0 1 α AdS 1 s ( AdS g / α = α 1 2 0 represents the width of the transition region. Here Q τ (pQCD) world data JLab CLAS Hall A/CLAS OPAL ) = / / / / / 2 g1 g1 g1 g1 Modified AdS AdS GDH limit Lattice QCD (2004) -function for the nonperturbative effective coupling obtained from the LF holographic -Function from AdS/QCD Q (a) Effective coupling from LF holography for ( β β The smoothly extrapolated result (dot-dashed line) for ± -1 . g 10 0 1

0

0.2 0.6 0.4 0.8

s (Q)/ The = 2 The parameter where 9.1 The mapping in a positive dilaton modified AdS background is where pQCD contributions become important,olated as result described (dot-dashed in line) Ref. for model, we write [ the normalization fit to the measured coupling the parameters Q couplings extracted from different observables and lattice results. (b) Prediction for the the logarithmic fall-off from pQCD quantumcoupling loops is will an analytical dominate function in of thisof the regime. applicability momentum of Since transfer the at strong all scales, we can extend the range Figure 7: but effects from gluon creation and absorptionturbative are confining not included effects in vanish this exponentially effective theory. at The large nonper- momentum transfer (Eq. ( AdS/QCD, Light-Front Holography, and Sublimated Gluons to lattice simulations, JLab and CCFR results for the Bjorken sum rule effective charge. PoS(QCD-TNT-II)008 ) ] ] 25 102 (9.8) (9.9) 9.10 (9.11) (9.10) ). Near ] since the 9.7 96 1 GeV. We also Stanley J. Brodsky -function in QCD at short distances, > β large, expresses the τ Q z 2 Q ] Burkert and Ioffe [ ] For example, in deuteron ) for 101 . 99 0 ,[ , Q 9.9 ). The conditions given by Eqs. 98 > et al. , Q 0 9.10 0 because the theory develops a fixed - -function of QCD will extrapolate be- , for 0 β ) = → , 9.8 -function for an effective QCD coupling 0 ∞ > , Q β ] Condition ( 0 > Q → 29 = β Q -scheme. d dQ 0 for ( Q V 0. [ , 19 β ] Bloch, Fisher , -function has a minimum. Since there is only one = 0 0 Q β 96

→ Q < ) = 1 GeV and the pQCD results for β 2 ) 0 d dQ < -function extracted from LF holography, as well as the Q < Q -functions obtained from phenomenology and lattice cal- β Q ( → β Q β Q for ( where the , β 0 0 ). The are governed by their twist scaling behavior Q < ) 9.6 z β ] are seen to have a similar shape and magnitude. ( -function obtained from LF holography. d dQ (b) are the Φ β (b) corresponds to the light-front holographic result Eq. ( 25 7 7 ) follow immediately from Eqs. ( is compatible with the momentum transfer for the onset of scaling behavior ] Dimensional counting is built into the AdS/QCD soft and hard wall models 0 9.11 Q -function is zero and the approximate theory is scale invariant in the limit of mass- ] When quantum corrections are included, the conformal behavior is preserved at 100 β because of asymptotic freedom and near 103 ] given by Eq. ( ) expresses the fact that QCD approaches a conformal theory in both the far ultraviolet 1 GeV, we can interpret the results as a transition from the nonperturbative IR do- Q ] 15 ' 9.8 ) describe the essential behavior of the full 30 κ 2 9.11 Eq. ( Judging from these results, we infer that the actual Also shown on Fig. 0. [ - ' 0 → 9.8 ( whose scheme/definition is similar to that of the observe that the general conditions are satisfied by our model basic anti-screening behavior of QCD where theis strong essentially coupling negative, vanishes. thus The thereaches coupling increases its monotonically maximum from value: thedefines UV it the to has transition the a region IR finite at wherehadronic-partonic value it transition, for the minimum a is theory anexpressed with absolute in minimum; a Eq thus mass the ( gap. additional conditions Equation ( very large and deep infrared regions. In theabsorption, the semiclassical approximation to QCD withoutless particle quarks. creation [ or Q main to the quarkmomentum and scale gluon degrees of freedom in the perturbativephoto-disintegration UV the regime. onset of The scaling corresponds transition cleon to involved. momentum [ transfer of 1.0 GeV to the nu- z culations. For clarity, we present onlyand the the LF holographic experimental predictions, data the supplemented latticesponds results by to from, the the [ extrapolated relevant approximation sumcoupling obtained rules. by [ matching The to dot-dashed AdS curveforms results obtained corre- to from the the perturbative works of Cornwall,and [ Furui and Nakajima, [ tween the non-perturbative results for point. An infrared fixed pointpropagators is of in the fact colored a fields naturaltion consequence have of of a the color maximum gluon confinement wavelength, self-energy [ all decouple loop at integrals in the computa- in exclusive reactions where quark counting rules are observed. [ since the AdS amplitudes AdS/QCD, Light-Front Holography, and Sublimated Gluons The solid line in Fig. PoS(QCD-TNT-II)008 ¯ ] c ' ¯ c qg 2 q 106 κ ] two 4 9. The 89 / > t 2 ,[ Q almost never 0 et al. ψ Stanley J. Brodsky of the quarkonium 1. Gluon jets are 0 light-cone gauge, ¯ c ± c = = + z ] ] The Gaussian fall-off of A S 15 107 , .[ 9 is incorrect. However, this 2 / κ 88 ] where the 2 4 ψ / 2 M Q 108 − ' e 2 q ∝ ; in fact measurements are consistent with ) 8 1 2 t ] We can understand the breakdown of the Q ∝ ]. Similarly, there is no clear evidence for ( 27 ) are physically absent. Thus the physical effects .[ puzzle shows that the usual Zweig picture of ) 105 2 QCD pp 20 2 / are dominated by quark exchange and interchange , Q amplitudes each with small gluon virtuality ρπ gluonium bound states has been difficult., even in the is the largest two-body mode where → ( 42 cm AdS 1 s [ , the DGLAP evolution of structure functions, and the g s θ α 1GeV → qq pp 2 α ρπ ggg gg ( 2 QCD Q ' γ ψ is clearly incorrect in the short-distance domain Λ σ → dt / d → 2 J and → Q qq ψ log QCD / o ψ / Fock state of one of the final-state mesons. The suppression of gg J β / i J / ¯ domain. However, gluons with smaller virtuality are sublimated in c AdS s π 2 ¯ α , as predicted by quark-interchange. [ 4 qc 4 q t | 4 ' 1 u GeV 2 ) effective charge , as predicted at leading order in QCD. The existence of asymptotic free- s ] 2 9 4 1 > Q ] rather than gluon exchange contributions. As shown by Landshoff, [ g 42 ∝ ( 2 = . The infamous s ) 88 ] Empirically, the rapidity plateau of a gluon jet is higher than that of a quark Q A F α C C pp ρπ : is then due to the node in the excited quarkonium radial wavefunction. 2 104 → Q .[ ρπ ¯ qg pp → q ( where the asymptotic freedom property of QCD becomes evident, as seen from the fall- 0 σ dt 2 d Landshoff mechanism predicts annihilation to three gluons,puzzling each decay pattern with is consistent virtuality with a mechanism [ hybrid states. [ gluon exchange implies thatnated large-angle by a proton-proton sequence elastic of three scattering would be domi- decays to state flows into the trajectory. ψ “gluon factory” reaction exclusive hadron-hadron scattering reactions. In fact, as shownamplitudes by [ Carroll body scattering amplitudes at fixed → These striking phenomenological features are consisted with the AdS/QCD prediction that As we have discussed above, an important feature of AdS/QCD is its prediction for the running An essential prediction of quantum chromodynamics is the existence of color-octet spin-1 − 3. There is no solid evidence for the existence4. of Experiments the find Odderon that the – the three-gluon exchange 2. One would normally expect to see gluon exchange dominate large-angle elastic scattering 1. Finding clear evidence for e + gluon quanta with virtuality below dom at large jet by the factor ERBL evolution of distribution amplitudesever, empirical are evidence confirming all gluonic based degrees of onexample, freedom the at existence small virtualities of is hard lacking. gluons. For How- coupling constant in the infrared domain: gluon quanta. If one quantizes QCD using light-front quantization and AdS/QCD, Light-Front Holography, and Sublimated Gluons 10. Sublimated Gluons the AdS/QCD prediction for 1 GeV off of the measured AdS/QCD prediction at hardof scales freedom as in evidence the forterms the of appearance the effective of confining dynamical potential. gluon degrees then the gluon quanta have positive metric and physical polarization clearly seen in hard QCDe processes, such as the three-jet events in electron-positron annihilation PoS(QCD-TNT-II)008 c / z + t = function is τ β Stanley J. Brodsky at all spatial points t in the QCD f The purpose of the running n ]. A single global PMC scale, . ) 2 µ function; in fact, when the renor- 115 ( , s β α 114 -matrix elements by the disconnected , S 0 terms in a perturbative expansion arising 113 ] between observables, and the scale-setting 6= β 111 21 which is instantaneous in light-front time 2 1 + k ] See also Refs. [ of the running coupling , replaced by an effective potential which confines quarks. The µ 109 ] and Brodsky have shown how to determine the PMC/BLM scale 112 , sublimated 109 ] We also note that in (1+1) QCD quantized in light-cone gauge – gluon quanta ] commensurate scale relations [ 90 110 ] are thus independent of the choice of renormalization scheme – a key requirement of 0. The resulting scale-fixed predictions using the “principle of maximum conformality” 109 = β A key problem in making precise perturbative QCD predictions is the uncertainty in deter- When one makes a measurement in hadron physics, such as deep inelastic lepton-proton scat- Physical eigenstates are built from operators acting on the vacuum. It is thus important to DiGiustino, Wu, [ coupling in any gauge theory ismalization to scale sum is all set terms properly, involving all the non-conformal from renormalization are summedturbative into series the are running then coupling.with identical to The that remaining of(PMC) terms [ a in conformal the theory; per- i.e., the corresponding theory mining the renormalization scale tering, one probes hadron’s constituents consistentat with instant causality time. – at a Similarly,consistent given with when light the one front finite speed makes time, of not observations light. in cosmology, information isdistinguish two obtained very different concepts ofQCD. The the conventional vacuum instant-form vacuum in is quantum a field state defined theories at such the same as time QED and remains. 11. The Principle of Maximal Conformality valid at leading order, can beThe derived elimination from basic of properties the of renormalizationthe the precision scheme perturbative QCD of ambiguity cross QCD using section. tests, thephysics but beyond PMC it the will will Standard not also Model. increase only the increase sensitivity of collider experiments to12. new Vacuum Condensates and the Cosmological Constant in the universe. Inconnected; contrast, i.e., the or front-form withinlowest vacuum energy the only eigenstate observer’s of senses the light-cone. instant-form phenomena Hamiltonian.QED which The For is example, are saturated instant-form the with instant-form causally vacuum quantum vacuum loops in isone of defined leptons must and as photons. then the In normal-order calculationsvacuum of the loops. physical vacuum processes and divide the renormalization group invariance. The results avoidscale-setting renormalon resummation in and the agree Abelian with QED limit.procedure, The [ PMC is alsomethods the used theoretical in principle lattice underlying gauge thealso theory. BLM correctly determined. The number of active flavors for QCD processes up to NNLO. [ AdS/QCD, Light-Front Holography, and Sublimated Gluons of gluons are evidently AdS/QCD picture is consistenttube with model. string [ descriptions ofare confinement absent; and only the the Isgur-Paton confining flux potential PoS(QCD-TNT-II)008 , n µν T to the z Stanley J. Brodsky Moreover, the same . ζ -function which reflects the β However, in the Bethe-Salpeter . 2 π f and its / and the principal quantum number i ) 2 0 L | Q ( ¯ qq s | represents a pion decay constant via an an . The LF vacuum is remarkably simple be- α 0 c h i q / π z bound-state, the GMOR relation is replaced by | determined in the AdS fifth dimension m ¯ q 2 q − ) 5 22 q z t γ − ( ¯ q | = Φ = 0 τ ] The usual argument for a quark vacuum condensate is 2 π m 118 ] In particular, chiral symmetry is broken in a limited do- ≡ −h , π 116 ρ 117 The LF vacuum thus coincides with the vacuum of the free LF ] the 45 orders of magnitude conflict of QCD with the observed . 0 119 where , ≥ , π + 29 f k , / i 23 π | q , in analogy to the limited physical extent of superconductor phases. This novel 5 π γ ¯ m q | / 0 h q m 2 − The combination of anti-de Sitter space methods with light-front holography provides an accu- The AdS/QCD correspondence is now providing important insight into the properties of QCD The cosmological constant measures the matrix element of the energy momentum tensor In contrast, the front-form (light-front) vacuum is defined as the lowest mass eigenstate of In fact, in the LF analysis, one finds that the spatial support of QCD condensates is restricted = 2 π rate first approximation for the spectra and wavefunctions of meson and baryon light-quark bound dynamics of confinement. as well as dynamical form factors.other fixed The CM theory angle predicts exclusive dimensional reactions.allows counting Moreover, one for as form to we factors have map and discussed, thevalence light-front LFWFs hadronic holography of amplitudes each hadron astechniques a provide function a of prediction a for covariant the impact QCD variable coupling needed to compute exclusiveby reactions. a positive sign In dilaton particular,parameter metric, description the which of nonperturbative represents soft-wall hadron color dynamics, AdS/QCD including confinement,meson successful model leads predictions and to for modified baryon the a spectrum remarkable forof one- zero linear trajectories quark with mass, the including same a slope zero-mass in orbital pion, angular a Regge spectrum the Gell-Mann–Oakes–Renner formula: in the background universe. Ita corresponds to probe the of measurement finite of mass;observer. the it gravitational If interactions only the of senses universe the iscausal. causally empty, the One connected automatically appropriate domain obtains vacuum within a stateas the vanishing is argued light-cone cosmological thus in of constant the Refs. from the LF thevalue vacuum LF of since [ the vacuum. it cosmological Thus, is condensate is removed,gluon and condensates a in new QCD perspective on is the thus nature obtained. of quark and 13. Conclusions description of chiral symmetryserved breaking in in Bethe-Salpeter studies. terms [ of “in-hadron condensates” has also been ob- main of size 1 m elementary pseudoscalar current. light-front Hamiltonian quantized at fixed AdS/QCD, Light-Front Holography, and Sublimated Gluons and light-front formalisms, where the pion is a to the interior of hadrons, physicsons. which The arises condensate due physics to normally thedynamics associated interactions with of of the confined higher instant-form quarks non-valence vacuum andmethod is Fock glu- replaced by states by Casher the as and shown Susskind. in [ the context of the infinite momentum cause of the restriction Hamiltonian (up to possible zerocausal modes and set Lorentz invariant; by whereas boundaryserver’s the conditions). Lorentz instant frame. form The vacuum front-form is vacuum acausal is and depends on the ob- PoS(QCD-TNT-II)008 ). 4.1 , Eq. ( z Stanley J. Brodsky contributions to the ] The confining cou- φ ] and the Sommerfeld- 123 puzzle. 122 ρπ → ψ / J extracted from the Bjorken sum rule. The 1 g α -function. The AdS/QCD running coupling is in β 23 which measures the separation of the quark and gluonic ζ and its ) Q ( AdS s α gives a successful accounting of the spectra of the light-quark meson and ) -function of QCD and the essential dynamics of confinement, thus giving 2 ]. z β 2 1 GeV. Our analysis indicates that light-front holography captures the char- κ 124 ( ∼ Q ] the Boer-Mulders functions which lead to anomalous cos2 -function displays a transition from nonperturbative to perturbative regimes at a mo- β ]. 121 32 , 120 Invited talk presented by SJB at the International Workshop on QCD Green’s Functions, Con- As we have seen, the gauge/gravity duality leads to a simple analytical and phenomenolog- We have also shown that the light-front holographic mapping of effective classical grav- There are many phenomenological applications where detailed knowledge of the QCD cou- finement and Phenomenology, 5-9 September 2011,Xing-Gang Trento, Italy. Wu, We Mike thank Cornwall, Leonardo Sadataka di Giustino, Furui, Craig Roberts, Robert Shrock, and Peter Tandy Sakharov-Schwinger correction to heavy-quark production at threshold. [ Acknowledgments ically compelling nonperturbative approximationexample, to the soft-wall the holographic full model leads light-frontperturbative to hadron QCD a dynamics remarkable Hamiltonian. despite one-parameter the description fact ofof For that non- explicit explicit gluonic gluonic degrees of quanta freedom are atQCD absent. small models virtuality The which is absence are a based feature on ofter the color-confining gauge/gravity holographic spaces duality following on the warped five-dimensional original anti-deexplain Maldacena Sit- several correspondence. outstanding The anomalies sublimation ofmechanism of hadron in soft large-angle physics hadron-hadron gluons including scattering can the and the absence of the Landshoff baryons [ The eigenvalues give the hadronic spectrum, andtions the of eigenmodes the represent hadronic the constituents probability distribu- atdilaton a profile given exp scale. In particular, the soft-wall model with a positive very good agreement with theholographic effective coupling acteristics of the full lepton pair angular distribution in the unpolarized Drell-Yan reaction, [ pling from light-front holography thus can provide abreaking quantitative physics. understanding of [ this factorization- ity in AdS space,perturbative modified effective coupling by the positive-sign dilaton background predicts the form of a non- mentum scale pling and the renormalizedThis gluon includes propagator exclusive and at semi-exclusive processes relatively ascreate soft well momentum the as the transfer leading-twist rescattering are interactions Siversing, which essential. single-spin [ correlations in semi-inclusive deep inelastic scatter- further support to the application of thecoupled gauge/gravity QCD. duality to the confining dynamics of strongly AdS/QCD, Light-Front Holography, and Sublimated Gluons states. One also obtains an elegant connectionquantized between on a semiclassical the first light-front, approximation to withresulting QCD, bound-state hadronic Hamiltonian modes equation propagating of motion onequations in a in QCD the fixed leads invariant impact AdS to variable relativistic background. light-frontconstituents wave The within the hadron atvariable equal relativistic light-front Schrödinger-like time. equation in This the corresponds AdS to fifth the dimension effective coordinate single- PoS(QCD-TNT-II)008 Stanley J. 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