Ads/QCD and Light Front Holography: a New Approximation to QCD*

CPC(HEP & NP), 2010, 34(9): 1229–1235 Chinese Physics C Vol. 34, No. 9, Sep., 2010

AdS/QCD and light front holography: A new approximation to QCD *

Stanley J. Brodsky1) Guy F. de T´eramond2)

1 SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA 2 Universidad de Costa Rica, San Jos´e, Costa Rica

Abstract The combination of Anti-de Sitter space (AdS) methods with light-front holography leads to a semi-classical ﬁrst approximation to the spectrum and wavefunctions of meson and baryon light-quark bound states. Starting from the bound-state Hamiltonian equation of motion in QCD, we derive relativistic light- front wave equations in terms of an invariant impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron at equal light-front time. These equations of motion in physical space-time are equivalent to the equations of motion which describe the propagation of spin-J modes in anti–de Sitter (AdS) space. Its eigenvalues give the hadronic spectrum, and its eigenmodes represent the probability distributions of the hadronic constituents at a given scale. Applications to the light meson and baryon spectra are presented. The predicted meson spectrum has a string-theory Regge form M2 = 4κ2(n+L+S/2); i.e., the square of the eigenmass is linear in both L and n, where n counts the number of nodes of the wavefunction in the radial variable ζ. The space-like pion and nucleon form factors are also well reproduced. One thus obtains a remarkable connection between the description of hadronic modes in AdS space and the Hamiltonian formulation of QCD in physical space-time quantized on the light-front at ﬁxed light-front time τ. The model can be systematically improved by using its complete orthonormal solutions to diagonalize the full QCD light- front Hamiltonian or by applying the Lippmann-Schwinger method in order to systematically include the QCD interaction terms. Key words gauge/gravity correspondence, light-front dynamics, strongly coupled QCD, meson and baryon spectrum, meson and nucleon form factors PACS 11.25Tq, 12.38.A, 12.38.Lg

1 Introduction by the front of a light wave [1], has the form of a relativistic Lorentz invariant Schrödinger equation A long-sought goal in hadron physics is to find d2 1 4L2 a simple analytic first approximation to QCD analo- − +U(ζ) φ(ζ) = 2φ(ζ), (1) −dζ2 − 4ζ2 M gous to the Schrödinger-Coulomb equation of atomic physics. This problem is particularly challenging since where the confining potential is U(ζ) = κ4ζ2+2κ2(L+ the formalism must be relativistic, color-confining, S 1) in a soft dilaton modified background. There − and consistent with chiral symmetry. is only one parameter, the mass scale κ 1/2 GeV, ∼ We have recently shown that the combination of which enters the confinement potential. Here S = 0,1 Anti-de Sitter space (AdS) methods with light-front is the spin of the q and q,¯ L is their relative orbital (LF) holography leads to a remarkably accurate first angular momentum as determined in the light-front approximation for the spectra and wavefunctions of formalism and ζ = x(1 x)b2 is a Lorentz invari- − ⊥ meson and baryon light-quark bound states. The re- ant coordinate whicph measures the distance between sulting equation for the mesonic qq¯ bound states at the quark and antiquark; it is analogous to the radial fixed light-front time τ = t + z/c , the time marked coordinate r in the Schrödinger equation. In effect

Received 19 January 2010 * Supported by Department of Energy Department of Energy contract DE-AC02-76SF00515 1) E-mail: [email protected] 2) E-mail: [email protected] ©2010 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 1230 Chinese Physics C (HEP & NP) Vol. 34

ζ represents the off-light-front energy shell or invariant mass dependence of the bound state; it allows the separation of the dynamics of quark and gluon binding from the kinematics of constituent spin and internal orbital angular momentum [2]. We thus obtain a single-variable LF relativistic Schrödinger equation which determines the spectra and light-front wavefunctions (LFWFs) of hadrons for general spin and orbital angular momentum. This LF wave equation serves as a semiclassical first approximation to QCD, and it is equivalent to the equations of motion which describe the propagation of spin-J modes in AdS space. Light-front holography thus provides a remarkable connection between the description of hadronic modes in AdS space and the Fig. 2. Parent and daughter Regge Regge tra- Hamiltonian formulation of QCD in physical space- jectories for the I = 1 ρ-meson and the I = 0 time quantized on the light-front at fixed LF time τ. ω-meson families for κ = 0.54 GeV. The LFWFs of bound states in QCD are relativis- 2 Mesons in light-front AdS/QCD tic generalizations of the Schrödinger wavefunctions of atomic physics, but they are determined at fixed The meson spectrum predicted by Eq. (1) has a light-cone time τ = t + z/c – the “front form” intro- string-theory Regge form 2 = 4κ2(n+L+S/2); i.e., M duced by Dirac [1] – rather than at fixed ordinary the square of the eigenmasses are linear in both L time t. It is natural to set boundary conditions at and n, where n counts the number of nodes of the fixed τ and then evolve the system using the light- wavefunction in the radial variable ζ. This is illus- front (LF) Hamiltonian P − =P 0 P 3 = id/dτ. The in- trated for the pseudoscalar and vector meson spectra − variant Hamiltonian H = P +P − P 2 then has eigen- in Figs. 1 and 2, where the data are from Ref. [3] The LF − ⊥ values 2 where is the physical mass. Its eigen- pion (S = 0,n = 0,L = 0) is massless for zero quark M M functions are the light-front eigenstates whose Fock mass, consistent with the chiral invariance of massless state projections define the light-front wavefunctions. QCD. Thus one can compute the hadron spectrum The eigensolutions of Eq. (1) provide the light-front by simply adding 4κ2 for a unit change in the radial wavefunctions of the valence Fock state of the hadrons quantum number, 4κ2 for a change in one unit in the ψ(x,b ) as illustrated for the pion in Fig. 3 for the orbital quantum number L and 2κ2 for a change of ⊥ soft and hard wall models. The resulting distribution one unit of spin S. Remarkably, the same rule holds amplitude has a broad form φπ(x) x(1 x) which for three-quark baryons as we shall show below. ∼ − is compatible with moments determinedp from lattice gauge theory. One can then immediately compute observables such as hadronic form factors (overlaps of LFWFs), structure functions (squares of LFWFs), as well as the generalized parton distributions and distribution amplitudes which underly hard exclusive reactions. For example, hadronic form factors can be predicted from the overlap of LFWFs, as in the Drell- Yan West formula. The prediction for the space-like pion form factor is shown in Fig. 4. The pion form factor and the vector meson poles residing in the dressed current in the soft wall model require choosing a value of κ smaller by a factor of 1/√2 than the canonical value of κ which determines the mass scale of the hadronic spectra. This shift is apparently due to the fact that the transverse current in e+e− qq¯ creates Fig. 1. Parent and daughter Regge trajectories → a quark pair with Lz = 1 instead of the Lz = 0 qq¯ for the π-meson family for κ = 0.6 GeV. composition of the vector mesons in the spectrum. No. 9 Stanley J. Brodsky et al: AdS/QCD and light front holography: A new approximation to QCD 1231

Fig. 3. Pion LF wavefunction ψπ(x,b⊥) for the AdS/QCD (a) hard-wall (ΛQCD = 0.32 GeV) and (b) soft-wall (κ = 0.375 GeV) models.

and gluons in the loop integrals contributing to the β function have a maximal wavelength. [9] The decou- pling of quantum loops in the infrared is analogous to QED where vacuum polarization corrections to the photon propagator decouple at Q2 0. → We thus begin with a conformal approximation to QCD to model an eﬀective dual gravity description

in AdS space. One uses the five-dimensional AdS5 geometrical representation of the conformal group to represent scale transformations within the conformal window. Confinement can be effectively introduced with a sharp cut-off in the infrared region of AdS space, the “hard-wall” model [10], or, more successfully, using a dilaton background in the fifth Fig. 4. Space-like scaling behavior for 2 2 dimension which produces a smooth cutoff and lin- Q Fπ(Q ). The continuous line is the predic- ear Regge trajectories, the “soft-wall” model [11]. tion of the soft-wall model for κ = 0.375 GeV. The dashed line is the prediction of the The soft-wall AdS/CFT model with a positive-sign

hard-wall model for ΛQCD = 0.22 GeV. The dilaton-modified AdS space leads to the potential U(z) = κ4z2+2κ2(L+S 1) [12], We assume a dilaton triangles are the data compilation of Baldini − et al. [4], the filled boxes are JLAB 1 data [5] profile exp(+κ2z2) [12–14], with opposite sign to that and the empty boxes are JLAB 2 data [6]. of Ref. [11]. Glazek and Schaden [15] have shown that a har- In the standard applications of AdS/CFT meth- monic oscillator confining potential naturally arises ods, one begins with Maldacena’s duality between the as an effective potential between heavy quark states conformal supersymmetric SO(4,2) gauge theory and when one stochastically eliminates higher gluonic a semiclassical supergravity string theory defined in a Fock states. Also, Hoyer [16] has argued that the 10 dimension AdS S5 space-time [7]. The essential Coulomb and a linear potentials are uniquely allowed 5 × mathematical tool underlying Maldacena’s observa- in the Dirac equation at the classical level. The linear tion is the fact that the effects of scale transforma- potential becomes a harmonic oscillator potential in tions in a conformal theory can be mapped to the the corresponding Klein-Gordon equation. z dependence of amplitudes in AdS5 space. QCD Individual hadrons in AdS/QCD are identified by is not conformal but there is in fact much empirical matching the power behavior of the hadronic ampli- evidence from lattice, Dyson Schwinger theory and tude at the AdS boundary at small z to the twist of effective charges that the QCD β function vanishes its interpolating operator at short distances x2 0, → in the infrared [8]. The QCD infrared fixed point as required by the AdS/CFT dictionary. The twist arises since the propagators of the confined quarks corresponds to the dimension of fields appearing in 1232 Chinese Physics C (HEP & NP) Vol. 34 chiral super-multiplets [17]; thus the twist of a hadron tions of the pion form factor are given in Refs. [40, 41]. equals the number of constituents plus the relative orbital angular momentum. We then apply light-front 3 Baryons in light-front AdS/QCD holography to relate the amplitude eigensolutions in the fifth dimension coordinate z to the LF wavefunc- For baryons, the light-front wave equation is a lin- tions in the physical spacetime variable ζ. Light- ear equation determined by the LF transformation Front Holography can be derived by observing the properties of spin 1/2 states. A linear confining po- correspondence between matrix elements obtained in tential U(ζ) κ2ζ in the LF Dirac equation leads to ∼ AdS/CFT [18] with the corresponding formula using linear Regge trajectories [44]. For fermionic modes the LF representation [19, 20]. Identical results are the light-front matrix Hamiltonian eigenvalue equa- 2 obtained from the mapping of the QCD gravitational tion DLF ψ = ψ , HLF = D , in a 2 2 spinor | i M| i LF × form factor with the expression for the hadronic gravi- component representation is equivalent to the system tational form factor in AdS space [21, 22], a nontrivial of coupled linear equations test of consistency. 1 ν + Equation (1) was derived by taking the LF bound- d 2 2 ψ− ψ− κ ζψ− = ψ+, state Hamiltonian equation of motion as the starting −dζ − ζ − M 2 2 1 point [2]. The term L /ζ in the LF equation of mo- ν + d 2 2 tion (1) is derived from the reduction of the LF ki- ψ+ ψ+ κ ζψ+ = ψ− . (2) netic energy when one transforms to the radial ζ and dζ − ζ − M angular coordinate ϕ, in analogy to the `(` + 1)/r2 with eigenfunctions

1 2 2 Casimir term in Schrödinger theory. One thus estab- 2 +ν −κ ζ /2 ν 2 2 ψ+(ζ) z e Ln(κ ζ ), lishes the interpretation of L in the AdS equations of ∼ 3 2 2 ψ (ζ) z 2 +ν e−κ ζ /2Lν+1(κ2ζ2), (3) motion. The interaction terms build confinement and − ∼ n correspond to truncation of AdS space [2] in an effec- and eigenvalues tive dual gravity approximation. The duality between 2 = 4κ2(n+ν +1). (4) these two methods provides a direct connection be- M

tween the description of hadronic modes in AdS space The baryon interpolating operator 3+L = m O and the Hamiltonian formulation of QCD in physical ψD{`1 ...D`q ψD`q+1 ...D`m}ψ, L = i=1 ì is a twist space-time quantized on the light-front at fixed LF 3, dimension 9/2 + L with scaling bPehavior given by time τ. its twist-dimension 3 + L. We thus require ν = L + 1 The identification of orbital angular momentum to match the short distance scaling behavior. Higher of the constituents is a key element in the descrip- spin fermionic modes are obtained by shifting dimen- tion of the internal structure of hadrons using holo- sions for the fields as in the bosonic case. Thus, as graphic principles. In our approach quark and gluon in the meson sector, the increase in the mass squared degrees of freedom are explicitly introduced in the for higher baryonic state is ∆n = 4κ2, ∆L = 4κ2 and gauge/gravity correspondence [23], in contrast with ∆S = 2κ2, relative to the lowest ground state, the the usual AdS/QCD framework [24, 25] where axial proton. and vector currents become the primary entities as in The predictions for the 56-plet of light baryons effective chiral theory. under the SU(6) flavor group are shown in Fig. 5. Unlike the top-down string theory approach, one As for the predictions for mesons in Fig. 2, only con- is not limited to hadrons of maximum spin J 6 2, firmed PDG [3] states are shown. The Roper state

and one can study baryons with finite color NC = 3. N(1440) and the N(1710) are well accounted for in Higher spin modes follow from shifting dimensions in this model as the first and second radial states. Like- the AdS wave equations. In the soft-wall model the wise the ∆(1660) corresponds to the first radial state usual Regge behavior is found 2 n + L, predict- of the ∆ family. The model is successful in explaining M ∼ ing the same multiplicity of states for mesons and the important parity degeneracy observed in the light baryons as observed experimentally [26]. It is possi- baryon spectrum, such as the L=2, N(1680) N(1720) − ble to extend the model to hadrons with heavy quark degenerate pair and the L = 2, ∆(1905), ∆(1910), constituents by introducing nonzero quark masses ∆(1920), ∆(1950) states which are degenerate within and short-range Coulomb corrections. For other re- error bars [42]. Parity degeneracy of baryons is also cent calculations of the hadronic spectrum based on a property of the hard wall model, but radial states AdS/QCD, see Refs. [27–39]. Other recent computa- are not well described in this model [43]. No. 9 Stanley J. Brodsky et al: AdS/QCD and light front holography: A new approximation to QCD 1233

As an example of the scaling behavior of a twist τ = 3 hadron, we compute the spin non-ﬂip nucleon form factor in the soft wall model [44]. The proton and neutron Dirac form factors are given by

F p(Q2) = dζ J(Q,ζ) ψ (ζ) 2, (5) 1 Z | + | 1 F n(Q2) = dζ J(Q,ζ)[ ψ (ζ) 2 ψ (ζ) 2], (6) 1 −3 Z | + | −| − | p n where F1 (0) = 1, F1 (0) = 0. The non-normalizable mode J(Q,z) is the solution of the AdS wave equation for the external electromagnetic current in presence of a dilaton background and the exp( κ2z2) [20, 45]. Plus and minus components of the twist 3 nucleon LFWF are

2 3/2 −κ2ζ2/2 3 5/2 −κ2ζ2/2 ψ+(ζ)=√2κ ζ e , Ψ−(ζ)=κ ζ e . (7) 4 p 2 4 n 2 The results for Q F1 (Q ) and Q F1 (Q ) and are shown in Fig. 6.

4 Novel applications of light-front holography Fig. 5. 56 Regge trajectories for the N and ∆ baryon families for κ = 0.5 GeV. 1) The AdS/QCD model does not account for particle creation and absorption, and thus it will break down at short distances where hard gluon exchange and quantum corrections become important. This semi-classical first approximation to QCD can be systematically improved by using the complete orthonormal solutions of Eq. (1) to diagonalize the QCD light- front Hamiltonian [47] or by applying the Lippmann- Schwinger method to systematically include the QCD interaction terms, together with a variational princi- ple. In either case, the result is the full Fock state structure of the hadron eigensolution. 2) One can model heavy-light and heavy hadrons by including non-zero quark masses in the LF kinetic k2 2 energy i( ⊥i +mi )/xi as well as the effects of the one-gluonP exchange potential. 3) Given the LFWFs one can compute jet hadronization at the amplitude level from first principles [48]. LF quantization also provides a distinction between static (the absolute square of LFWFs) structure functions versus non-universal dynamic structure functions, such as the Sivers single-spin correlation and distributions measured in diffractive deep inelas- tic scattering which involve final state interactions. The origin of nuclear shadowing and process inde- pendent anti-shadowing also becomes explicit. 4 p 2 4 n Fig. 6. Predictions for Q F1 (Q ) and Q F1 4) Light-front holographic mapping of effective (Q2) in the soft wall model for κ = 0.424 GeV. classical gravity in AdS space, modified by a positive- The data compilation is from Diehl [46]. sign dilaton background exp(+κ2z2), can be used 1234 Chinese Physics C (HEP & NP) Vol. 34 to identify [49] a non-perturbative effective coupling ture also explains the results of recent studies [60–62] AdS 2 αs (Q ) and β-functions which are in agreement which find no significant signal for the vacuum gluon with available data extracted from different observ- condensate. The 45 orders of magnitude conflict of ables and with the predictions of a class of mod- QCD with the observed value of the cosmological con- els with built-in confinement and lattice simulations. densate can thus be explained [56]. The AdS/QCD β-function shows the transition from perturbative to non-perturbative conformal regimes 5 Conclusions at a momentum scale Q 1 GeV, thus giving fur- ∼ ther support to the application of the gauge/gravity We have derived a connection between a semi- duality to the confining dynamics of strongly cou- classical first approximation to QCD, quantized on pled QCD. There are many phenomenological appli- the light-front, and hadronic modes propagating on cations where detailed knowledge of the QCD cou- a fixed AdS background. This leads to an effective pling and the renormalized gluon propagator at rel- relativistic Schrödinger-like equation in the AdS fifth atively soft momentum transfer is essential. This in- dimension coordinate z (1). We have shown how this cludes the rescattering (final-state and initial-state AdS wave equation can be derived in physical space- interactions) which create the leading-twist Sivers time as an effective equation for valence quarks in single-spin correlations in semi-inclusive deep in- light-front quantized theory, where one identifies the elastic scattering [50, 51], the Boer-Mulders func- AdS fifth dimension coordinate z with the LF coor- tions which lead to anomalous cos2φ contributions dinate ζ. to the lepton pair angular distribution in the unpo- We originally derived the light-front holographic larized Drell-Yan reaction [52] and the Sommerfeld- correspondence from the identity of electromagnetic Sakharov-Schwinger correction to heavy quark pro- and gravitational form factors computed in AdS and duction at threshold [53]. The confining AdS/QCD LF theory [19–21]. Our derivation shows that the coupling can thus lead to a quantitative understand- fifth-dimensional mass µ in the AdS equation of ing of this factorization-breaking physics [54]. motion is directly related to orbital angular mo- 5) AdS/QCD provides a description of chiral sym- mentum L in physical space-time. The result is metry breaking by using the propagation of a scalar physically compelling and phenomenologically suc- field X(z) to represent the dynamical running quark cessful. The resulting Schrödinger-like light-front mass. The AdS solution has the form [24, 25] X(z) = AdS/QCD equation provides successful predictions a z + a z3, where a is proportional to the current- 1 2 1 for the light-quark meson and baryon spectra as func- quark mass. The coefficient a scales as Λ3 and 2 QCD tion of hadron spin, quark angular momentum, and is the analog of qq¯ ; however, since the quark is a h i radial quantum number . The pion is massless for color nonsinglet, the propagation of X(z), and thus zero mass quarks in agreement with chiral invari- the domain of the quark condensate, is limited to the ance arguments. The predictions for form factors are region of color confinement. Furthermore the effect also successful. The predicted power law fall-off of of the a term varies within the hadron, as charac- 2 hard exclusive amplitudes agrees with dimensional teristic of an “in-hadron” condensate. A similar so- counting rules as required by conformal invariance at lution is found in the soft wall model in presence of a small z [20, 44]. The AdS/QCD holographic model positive sign dilaton [14]. A new perspective on the model can be systematically improved by using its nature of quark and gluon condensates in quantum complete orthonormal solutions to diagonalize the chromodynamics is thus obtained [9, 55, 56]: the spa- full QCD light-front Hamiltonian or by applying the tial support of QCD condensates is restricted to the Lippmann-Schwinger method in order to systemat- interior of hadrons, since they arise due to the inter- ically include the QCD interaction terms. We have actions of confined quarks and gluons. In LF theory, also outlined many new applications of this formal- the condensate physics is replaced by the dynamics ism. of higher non-valence Fock states as shown by Casher and Susskind [57]. In particular, chiral symmetry is Presented by SJB at the Fifth International Con- broken in a limited domain of size 1/mπ, in analogy to ference On Quarks and Nuclear Physics (QNP09), the limited physical extent of superconductor phases. 21-26 Sep 2009, Beijing, China. We thank Carl This novel description of chiral symmetry breaking Carlson, Alexandre Deur, Josh Erlich, Stan Glazek, in terms of in-hadron condensates has also been ob- Hans-Guenter Dosch, Paul Hoyer, Dae Sung Hwang, served in Bethe-Salpeter studies [58, 59]. This pic- Eberhard Klempt, Mariana Kirchbach, Craig Roberts, No. 9 Stanley J. Brodsky et al: AdS/QCD and light front holography: A new approximation to QCD 1235

Ivan Schmidt, Robert Shrock, James Vary, and Fen Zuo for helpful conversations and collaborations.

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