Light-Front Holography and Novel Collider Physics

Light-Front Holography:A New Approximation to QCDInicio Acerca de RRR en Chile Como AsExploring QCD, Cambridge, AugustRRR 20-24, 2007en ChileHigh Energy Physics in the RRR en Chile LHC > Sedes Era > UTFSMThird International WorkshopStan BrodskyUTFSMUniversidad Tecnica Federico Santa MariaValparaiso, Chile, January 4-8, 2010

H QEDQED atoms: positroniumand muonium(H 0 + H int ) |Ψ >= E |Ψ > Coupled Fock states[− ∆22m red+ V eff ( S, r)] ψ(r) = E ψ(r)Effective two-particle equationIncludes Lamb Shift, quantum corrections[− 1 d 22m red dr 2 + 1 ( + 1)2m red r 2+ V eff (r, S, )] ψ(r) = E ψ(r)Spherical Basisr, θ, φV eff → V C (r) = − α rSemiclassical first approximation to QEDCoulomb potentialBohr Spectrum

Light-Front Holography and Non-Perturbative QCDGoal:Use AdS/QCD duality to constructa first approximation to QCDHadron SpectrumLight-Front Wavefunctions,Form Factors, DVCS, etcGeneral remarks about orbital angular momentumΨ n (x i , k ⊥i , λ i )Exploring QCD, Cambridge, August 20-24, 2007 Page 9in collaboration withGuy de Teramond and Alexandre DeurCentral problem for strongly-coupled ni=1(x iR⊥ + b ⊥i ) = R gauge theories⊥Chile LHCJanuary 8, 2010LF Holography and QCD4Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

H QEDHQCDLFQCD Meson Spectrum(H 0 LF + H I LF )|Ψ >= M 2 |Ψ >Coupled Fock states[ k 2 ⊥ + m2x(1 − x) + V LFeff ] ψ LF (x, k ⊥ ) = M 2 ψ LF (x, k ⊥ )Effective two-particle equation[− d2 −1 + 4L2+dζ2 ζ 2 = x(1 − x)b 2 ⊥Azimuthal Basisζ 2 + U(ζ, S, L)] ψ LF (ζ) = M 2 ψ LF (ζ) ζ, φU(ζ, S, L) = κ 2 ζ 2 + κ 2 (L + S − 1/2)Semiclassical first approximation to QCDConfining AdS/QCDpotential

• QCD LagrangianL QCD = − 14g 2 Tr (Gµν G µν )+iψD µ γ µ ψ + mψψ• LF Momentum Generators P =(P + ,P − , P ⊥ ) in terms of dynamical fields ψ, A ⊥P − = 1 dx − d 2 x ⊥ ψγ + (i∇ ⊥) 2 + m 22i∂ + ψ + interactionsP + = dx − d 2 x ⊥ ψγ + i∂ + ψP ⊥ = 1 dx − d 2 x ⊥ ψγ + i∇ ⊥ ψ2• LF Hamiltonian P − generates LF time translationsψ(x),P− = iand the generators P + and P ⊥ are kinematical∂∂x + ψ(x)Chile LHCJanuary 8, 2010LF Holography and QCD8Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front Bound State Hamiltonian Equationτ = t + z/c• Construct light-front invariant Hamiltonian for the composite system: H LF = P µ P µ = P − P + −P 2 ⊥H LF | ψ H = M 2 H | ψ H • State |ψ H (P + , P ⊥ ,J z ) is expanded in multi-particle Fock states | n of the free LF Hamiltonian:⎧|ψ H = |uud ⎪⎨ψ n/H |n, |n = |uudgn⎪⎩ |uudqq ···where k 2 i = m2 i ,k i =(k + i ,k− i , k ⊥i), for each component i• Fock components ψ n/H (x i , k ⊥i , λ z i ) are independent of P + and P ⊥ and depend only on relativepartonic coordinates: momentum fraction x i = ki + /P + , transverse momentum k ⊥i and spin λ z innx i =1, k ⊥i =0.i=1DIS, Form Factors, DVCS, etc. measure proton LFWFPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLAC9Hadron 2009, FSU, Tallahassee, December 1, 2009 Page 9i=1HOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front Wavefunctions: rigorous representation of compositesystems in quantum P field + = Ptheory0 + P zx = k+P + = k0 + k 3P 0 + P 3Fixed τ = t + z/cGeneral remarkmentumx P + , P +i = k+A(σ, ∆ ⊥ ) = 1 i PGeneral2π dζe2 σζ + = k0 +k 3P 0 +P zM(ζ,remarks∆⊥ )x i P + about, x iorbiψ(σ, b P⊥ + kR ⊥i⊥ )⊥P + , P ⊥mentumẑx iR⊥ + b ⊥ix i P + , x iP⊥ + β = dα s(Q 2 )kd ln Q 2 < 0⊥iProcess IndependentL = R × P ni ni=1Direct b⊥i = 0(x ⊥ iP⊥ ζ = Q2 Link to QCD Lagrangian! u+ k ⊥i2p·qLΨnix i = 1n (x i , i = (x iR⊥ + b ⊥i ) × P ẑkx iP⊥ + ⊥i , λ i )k ⊥i i = b ⊥i × k ⊥i L = R × P ni k⊥i = 0 ⊥i = L i − x iR⊥ × P = b ⊥i × PInvariant under L i = (x iR⊥ + boosts! Independent of P b ⊥i ) × P nix i = 1 ni=1(x i R⊥ + b ⊥i ) = R ⊥Chile LHCJanuary 8, 2010LF Holography and QCD10Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

y conservation of the z projection ofIn general the light-cone wavefunctions satisfy consAngular Momentum on the Light-Frontngular momentum:J z =he sum over siz represents the contribution of the intrthe intrinsic spins of the n Fock stateonstituents. The sum over orbital angular momenta l zta lj =j z = −i( kj1 ∂∂ke n−1 relative momenta. 2 − kj2 ∂ )derivesj∂k 1 n-1 orbital from angular momentajThis excludes the contributionibution to the orbital angular momentumue to the motion of the center of mass, which is not an inot WeNonzero an intrinsic canAnomalous property see howMomentthe--> of angularNonzero the hadron.orbitalmomentumangular momentum!sum rulemavefunctions rule Eq. Eqs. (62)(20) is and satisfied (23) offor the the QEDPress andmodelMedia : SLAC National Accelerator LaboratorysysLF Holography and QCDChile LHCJanuary 8, 2010n∑i=1si z ∑n−1+j=1l z j .11ConservedLF Fock state by Fock State!(62)LF Spin Sum RuleStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

|p,S z >= ∑ Ψ n (x i ,k ⊥i ,λ i )|n;k ,λ ⊥i i >n=3sum over states with n=3, 4, ...constituentsThe Light Front Fock State WavefunctionsChile LHCJanuary 8, 2010Ψ n (x i ,k ⊥i ,λ i )are boost invariant; they are independent of the hadron’s energyand momentum P µ .The light-cone momentum fractionare boost invariant.n∑ik + i = P + ,x i = k+ ip + = k0 i + kz iP 0 + P zn∑ix i = 1,Intrinsic heavy quarksc(x), b(x) at high xn∑ik ⊥ i =0 ⊥ .ep → eπ + nP π/p 30%Violation of Gottfried sum rule¯s(x) = s(x)ū(x) = ¯d(x) Fixed LF timeφ M (x, Q 0 ) ∝ x(1 − x)LF Holography and QCD12Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

QCD and the LF Hadron WavefunctionsAdS/QCDLight-Front HolographyLF Schrodinger . EqnHeavy Quark Fock StatesIntrinsic CharmInitial and Final StateRescatteringDDIS, DDIS, T-OddQuark Non-Universal & Flavor StructAntishadowingBaryon ExcitationsGluonic propertiesGeneral remarks about orbital angular momentumDGLAPCoordinate spacerepresentationΨ n (x i , k ⊥i , λ i ) ni=1(x i R⊥ + b ⊥i ) = R ⊥Orbital Angular MomentumSpin, Chiral PropertiesCrewther RelationQuark & Flavor Structurex i R⊥ + b ⊥iHard Exclusive AmplitudesForm FactorsCounting RulesJ=0 Fixed PoleDVCS, GPDs. TMDsLF Overlap, incl ERBLHadronization atAmplitude Level ni b⊥i = 0 ⊥ nix i = 1Nuclear ModificationsBaryon AnomalyColor TransparencyDistribution amplitudeERBL Evolutionφ p (x 1 , x 2 , Q 2 )Baryon Decay

te between the struck and spectatorF 2 (q 2 )2M= λi ,c[dx][d 2 k ⊥ ] i ,f i i=1constituents; 2(2π) 3 namely, we i=1A(σ, ∆ ⊥ ) = 1 i12π dζe2 σζ M(ζ,e jaj2 × ∆⊥ )Drell, sjbwhere n denotes the number of constituents in Fock state a ank ⊥j = k ⊥j + (1 − x j )q ⊥ (14) possible {λ1i }, {c i }, and {f i } in−q L ψ↑∗ a (x i , k ⊥i, λ i ) ψa(x ↓ i , k ⊥i , λ i ) + 1 state a. The arguments of theq R ψ↓∗ a (x i , k ⊥i, λ i ) ψa(x ↑ i , k ⊥i , λ i ) fint j andP + wave, P ⊥ function differentiate between the struck and spectator cons,β = 0have [13, 15]xk i ⊥i P + = , kx k ⊥i iP⊥ − x+ i q k ⊥ ⊥i⊥j = k ⊥j + (1 −(15)x j B(0))q ⊥ = 0 FockF 3 (q 2 )2M for the struck[dx][d 2 constituentk ⊥ ] iejjaj2 × andre i = j. Note that because of the frame choice q + = 0, onlys of theζlight-front= Q2q R,L = q x ± iq y2p·q Fock states appear [14]. 1 −q L ψ↑∗ a (x i , k ⊥i, λ i ) ψa(x ↓ i , k ⊥i , λ i ) − 1 k ⊥i = k ⊥i − x i q ⊥q R ψ↓∗ a (x i , k ⊥i, λ i ) ψa(x ↑ ψ(x, bi , k ⊥i , λ i ) ⊥ ).ˆx, ŷforplaneeach 6 spectator i, where i = j. Note that because of the frameummationsdiagonalare over all(n xcontributing= n) overlapsFockof thestateslight-fronta and struckFockconstituentstates appearcha[Here, as Mearlier, 2 (L) we ∝ L-refrain from including the constituents’ b ⊥ (GeV) color −1 and flndence in the arguments of the light-front wave functions.6The phase-sration isIdentifyMust have ∆ z = ±1 to have nonzero F 2 (q 2 z ↔ ζ =)x] [d 2 k ⊥ ] ≡ Chile LHCJanuary 8, 2010i=1λ i ,c i ,f i n dxi d 2 k ⊥i2(2π) 3Same matrix elements appear in Sivers effect-- connection to quark anomalous moments16π 3 δLF Holography and QCD141 − nnx i δ (2) k ⊥ii=1Press and Media : SLAC National Accelerator Laboratoryi=1Stan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS,

Anomalous gravitomagnetic moment B(0)Terayev, Okun, et al: B(0) Must vanish because ofEquivalence Theoremgravitonsum over constituents-Hwang, Schmidt, sjb;Holstein et alB(0) = 0Each Fock StateChile LHCJanuary 8, 2010LF Holography and QCD15Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Calculation of Form Factors in Equal-Time TheoryInstant FormNeed vacuum-induced currentsCalculation of Form Factors in Light-Front TheoryFront FormComplete AnswerAbsent zero for forq + q + = 00 zero !!Chile LHCJanuary 8, 2010LF Holography and QCD16Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front QCDHeisenberg MatrixFormulationH QCDLFL QCD → H QCDLFPhysical gauge: A + = 0= [ m2 + k⊥2 ] i + HLFintxiLF : Matrix in Fock SpaceH intH QCDLF |Ψ h >= M 2 h|Ψ h >338Eigenvalues and Eigensolutions give HadronSpectrum and Light-Front wavefunctionsFig. 6. A few selectedChile LHCJanuary 8, 2010LF Holography and QCD17Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front QCD Features and Phenomenology• Hidden color, Intrinsic glue, sea, Color Transparency• Physics of spin, orbital angular momentum• Near Conformal Behavior of LFWFs at Short Distances;PQCD constraints• Vanishing anomalous gravitomagnetic moment• Relation between edm and anomalous magnetic moment• Cluster Decomposition Theorem for relativistic systems• OPE: DGLAP, ERBL evolution; invariant mass schemeChile LHCJanuary 8, 2010LF Holography and QCD18Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

More generally, consider a meson in SU(N). The kernel of the integral equation (3.14) isLight-Front illustrated QCDin Fig. 2 in terms of the block matrix nS.J. Brodsky et LC al. / Physics h =Reports 2 : x , k , λ Hn : x, h 301 H.C. k , λ . Pauli The structure & sjbof thismatrix depends of course H QCD on the way one has arranged the Fock space, see Eq. (3.7). Note that most(1998) h h>of the block matrix elements LF |Ψ h >= M 2vanish due to the natureh|Ψof the h >light-cone 299—486 Discretized interaction Light-Cone as defined inHeisenberg Matrix FormulationQuantizationEigenvalues and Eigensolutions give Hadron Spectrum and Light-Front wavefunctionsFig. 2. The Hamiltonian matrix for a SU(N)-meson. The matrix elements are represented by energy diagrams. Withineach block they are all of the same type: either vertex, fork or seagull diagrams. Zero matrices are denoted by a dot ( ) ).The single gluon is absent since it cannot be color neutral.Fig. 6. A few selected matrix elements of the QCD front form Hamiltonian H"P in LB-convention.DLCQ: Frame-independent, No fermion doubling; Minkowski Spaceor the DLCQ: instantaneous Periodic BC fermion in xlines − . Discrete use the factor k + ; frame-independent ¼ Fig. 5 or Fig. truncation 6, or the correspoables in Section 4. For the instantaneous boson lines use the factor ¼ .

= 1 − ζ1 ∆ 1 − i∆ 2√ 1 − ζ 2Measily checks that ∑ ni=1 x ′ Press and Media : SLAC National Accelerator Laboratoryi = 1and ∑ ni=1⃗k⊥i ′ = ⃗0 ⊥ . In Eqs. (39) and (40) onChile LHCJanuary 8, 201016π 3ζ 16π δ 1 − x j δ4 ( 1 − ζ )√ E (n→n) (x, ζ,t)Example of LFWF 2representation 1 − ζ j=1j=1n(of) (∏ dx× δ(x i d− 2 ⃗k ⊥ix 1 )ψ(n)( ↑∗ x′n∑i , ⃗k ⊥i ′ , λ ) n)GPDs (n => ↑∑)16π i ψ(n)( 3 δ 1 − x j δ (2) xi ⃗k ⊥j , ⃗k ⊥i , λ i ,1 − ζ1 − ζ H (n→n) (x, ζ,t) −2 n,λ i i=1= ( √ ) 2−n∑∫1 − ζn,λ i= ( √ ) 2−n2M ∑1 − ζE (n→n)(x, ζ,t)1 ∆ 1 − i∆∫2√ E (n→n)(x,∏ n ζ,t)1 − ζ dx i d 2 ⃗k ⊥i= ( √ ) 2−n∑ ∏ n16πdx i d 3 2 ⃗k ⊥i1 − ζ n,λ i i=1 16π 3n,λ i∫16π 3 δLF Holography and QCD20(1 −) (n∑n)∑j=1 x j δ (2) ⃗k ⊥j j=1⃗k ⊥j) (n∑n)∑x j δ (2) ⃗k ⊥ji=1× δ(x (n)( ↑∗j=1x′i , ⃗k ⊥i ′ , λ ) j=1↓ )× δ(x − x i ψ(n)(xi1 )ψ ↑∗ ) ↓ ψ , ⃗k ⊥i , λ i ,re the where arguments the arguments of theof the final-state wavefunction are are given given by byx ′ 1 = x 1 − ζi=116π 3x 1 ′ = x 1 − ζ1 − ζ , ⃗k ⊥1 ′ = ⃗k ⊥1 − 1 − x 11 − ζ , ⃗k ⊥1 ′ = ⃗k ⊥1 − 1 − x 1∆1 − ζ 1 − ζ ⃗ ⊥ for the struck quark,for the struck quark,x i ′ = x xii ′ = x i1 − ζ , 1 − ζ , ⃗k⃗k ⊥i ′ = ⊥i ′ ⃗k = ⃗k ⊥i +⊥i + x x i1i− ζ∆1 − ζ ⃗ ⃗ (41)∆ ⊥ for the spectators i = 2, ..., n.⊥ for the spectators i = 2, ..., n.over all possible combinations of helicities λ iand over all parton numbers nj=1j=1× δ(x − x 1 )ψ(n)( ↑∗ x′i , ⃗k ⊥i ′ , λ ) ↑ )i ψ(n)(xi , ⃗k ⊥i , λ i , (39)16π 3 δ(1 −(n)(x′i , ⃗k ′ ⊥i , λ iDiehl, Hwang, sjb(n)(xi , ⃗k ⊥i , λ i) , (40)One easily checks that ∑ ni=1 xi ′ = 1and ∑ ni=1⃗k⊥i ′ = ⃗0 ⊥ . In Eqs. (39) and (40) one has tosum over all possible combinations of helicities λ iand over all parton numbers n in theFock states. We also imply a sum over all possible ways of numbering Stan theBrodskypartons in then-particle Fock state so that the struck quark has the index i = 1.SLACHOME RESEARCH SCIENTIFIC PROGRAMS

H~Hqq!"#$%&'%()*%+#,%-.+/&"0%1'23%1+0&'2/DIS at !=t=0( x,0,0)' q(x),% q(% x)( x,0,0)' ( q(x),( q(% x)11% 1Form factors (sum rules)~dx H"q#q$ dx)H ( x,!,t)q' F 1 ( t) Dirac f.f.1q$ dx)" E ( x,!,t)# ' F 2 ( t) Pauli f.f.$q( x,!,t)' GA,q(t) ,1$% 1~dx Eq( x,!,t)' GP,q(t)Hq ~ q ~ q, E , H , E ( x,, t)q!Verified using LFWFsDiehl, Hwang, sjbQuark angular momentum (Ji’s sum rule)1q'1% J G =1Jxdx H2 2$% 1" ( x,!,0)& E ( x,!,0)#qqX. Ji, Phy.Rev.Lett.78,610(1997)Chile LHCJanuary 8, 2010LF Holography and QCD21Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

J=0 Fixed pole in real and virtual Compton scattering• Effective two-photon contact term• Seagull for scalar quarks• Real phaseM = s 0 e 2 qF q (t)γ ∗ (q)γDamashek, Gilman;Close, Gunion, sjbLlanes-Estrada,Szczepaniak, sjb• Independent of Q 2 at fixed tpp• Moment: Related to Feynman-Hellman Theorem• Fundamental test of local gauge theoryQ 2 -independent contribution to Real DVCS amplitudeChile LHCJanuary 8, 2010s 2 dσdt (γ∗ p → γp) = F 2 (t)LF Holography and QCD22Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

StaticDynamic• Square of Target LFWFs• No Wilson Line• Probability Distributions• Process-Independent• T-even Observables• No Shadowing, Anti-Shadowing• Sum Rules: Momentum and J zModified by Rescattering: ISI & FSIContains Wilson Line, PhasesNo Probabilistic InterpretationProcess-Dependent - From CollisionT-Odd (Sivers, Boer-Mulders, etc.)Shadowing, Anti-Shadowing, SaturationSum Rules Not Proven• DGLAP Evolution; mod. at large x DGLAP Evolution• No Diffractive DISHard Pomeron and Odderon Diffractive DISHwang,Schmidt, sjb,Mulders, BoerQiu, StermanCollins, QiuPasquini, Xiao,Yuan, sjbGeneral remarks about orbital e angular momentumcurrent2– quark jet!*e –Ψ n (x i , k ⊥i , λ i )Sprotonquarkfinal stateinteractionspectatorsystem11-20018624A06Chile LHCJanuary 8, 2010 ni=1(x i R⊥ + b ⊥i ) = R ⊥LF Holography and QCD23Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Applications of AdS/CFT to QCDChanges inphysicallength scalemapped toevolution in the5th dimension zin collaboration with Guy de TeramondChile LHCJanuary 8, 2010LF Holography and QCD24Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Application of AdS/CFT to QCDChanges inphysicallength scalemapped toevolution in the5th dimension zString TheoryBottom-UpTop-Down

Conformal Theories are invariant under thePoincare and conformal transformations withM µν , P µ , D, K µ ,the generators of Sthe generators of SO(4,2)Analytically continueSO(4,2) has a mathematical representation on AdS51s−M 2 +iMΓChile LHCJanuary 8, 2010Stan Brodskyq 2 →LF Holography q 2 + and iQCD→ q 2 + iMΓSLAC26Press and Media : SLAC National Accelerator LaboratoryHOME RESEARCH SCIENTIFIC PROGRAMS

AdS/CFT: Anti-de Sitter Space / Conformal Field TheoryMaldacena:Map AdS 5 X S 5 to conformal N=4 SUSY• QCD is not conformal; however, it hasmanifestations of a scale-invariant theory: Bjorkenscaling, dimensional counting for hard exclusiveprocesses• Conformal window:α s (Q 2 ) const at small Q 2 .• Use mathematical mapping of the conformal groupSO(4,2) to AdS5 spaceHigh Q 2 from short distancesF π (Q 2 )Chile LHCJanuary 8, 2010LF Holography and QCD27Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLAC2 2 2 1HOME RESEARCH SCIENTIFIC PROGRAMS

Conformal Behavior of QCD in InfraredConformal QCD Window in Exclusive ProcessesR fixed point? D-S Equation Alkofer, Fischer, LLanes-Es• Does α s develop an IR fixed point? Dyson–Schwinger Equation Alkofer, Fischer, LLanes-Estrada,Deur . . .ions: evidence that α s becomes constant and not small• Recent lattice simulations: evidence that α s becomes constant and is not small in the infraredep-lat/0612009Furui and Nakajima, hep-lat/0612009 (Green dashed curve: DSE). curve: DSE)• Phenomenological success of dimensional scaling laws for exclusive processesΑsqChile LHCJanuary 8, 201043.532.521.5dσ/dt ∼ 1/s n−2 , n = n A + n B + n C + n D ,implies QCD is a strongly coupled conformal theory at moderate but not asymptotic energies10.5Farrar and sjb (1973); Matveev et al. (1973).• Derivation of counting rules for gauge theories with mass gap dual to string theories in warped space-0.4-0.2 0 0.2 0.4 0.6 0.8 1Log_10qGeV(hard behavior instead of soft behavior characteristic of strings) Polchinski and Strassler (2001).LF Holography and QCD28Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Deur, Korsch, et al.! s/"1! s,g1/" JLabFitGDH limitpQCD evol. eq.Burkert-IoffeBloch et al.10 -1CornwallGodfrey-IsgurBhagwat et al.Maris-TandyLattice QCD1Fischer et al.10 -1DSE gluoncouplings10 -1 1 10 -1 1Q (GeV)Chile LHCJanuary 8, 2010Fig: Infrared conformal window ( from Deur et al., arXiv:0803.4119 )LF Holography and QCD29Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

IR Conformal Window for QCD• Dyson-Schwinger Analysis:Fixed Point• Evidence from Lattice Gauge Theory• Stability ofChile LHCJanuary 8, 2010A LF Holography and QCD30− 3QCD gluon coupling has IR• Define coupling from observable: indications of IRfixed point for QCD effective charges• Confined gluons and quarks have maximum wavelength:+ · · · +Decoupling of QCD vacuum polarization at small Q 2Π(Q 2 ) →15πα Q 2m 2Π(Q 2 Serber-Uehling) →15π• Justifies application of AdS/CFT in strong-couplingQ 2

Maximal Wavelength of Confined Fields• Colored fields confined to finite domain(x − y) 2 < Λ −2QCD• All perturbative calculations regulated in IR• High momentum calculations unaffected• Bound-state Dyson-Schwinger Equation• Analogous to Bethe’s Lamb Shift CalculationQuark and Gluon vacuum polarization insertionsdecouple: IR fixed Point Shrock, sjbJ. D. Bjorken,SLAC-PUB 1053Cargese Lectures 1989Chile LHCJanuary 8, 2010A strictly-perturbative space-time region can be defined as one whichhas the property that any straight-line segment lying entirely within the regionhas an invariant length small compared to the confinement scale(whether or not the segment is spacelike or timelike).LF Holography and QCD31Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Scale Transformations• Isomorphism of SO(4, 2) of conformal QCD with the group of isometries of AdS spaceSO(1, 5)ds 2 = R2z 2 (η µνdx µ dx ν − dz 2 ),x µ → λx µ , z → λz, maps scale transformations into the holographic coordinate z.• AdS mode in z is the extension of the hadron wf into the fifth dimension.invariant measure• Different values of z correspond to different scales at which the hadron is examined.x 2 → λ 2 x 2 ,z → λz.x 2 = x µ x µ : invariant separation between quarks• The AdS boundary at z → 0 correspond to the Q → ∞, UV zero separation limit.Chile LHCJanuary 8, 2010LF Holography and QCD32Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACaltech High Energy Seminar, Feb 6, 2006 Page 11HOME RESEARCH SCIENTIFIC PROGRAMS

Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, 2010 Cambridge, August 20-24, 2007 SLAC Page 733HOME RESEARCH SCIENTIFIC PROGRAMS

Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, 2010 Cambridge, August 20-24, 2007 SLAC Page 834HOME RESEARCH SCIENTIFIC PROGRAMS

Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, 2010 Cambridge, August 20-24, 2007 SLAC Page 935HOME RESEARCH SCIENTIFIC PROGRAMS

Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, 2010 Cambridge, August 20-24, 2007 SLAC Page 1036HOME RESEARCH SCIENTIFIC PROGRAMS

• In the “ semi-classical” approximation to QCD with massless quarks and no quantum loops the βfunction is zero, and the approximate theory is scale and conformal invariant.• Isomorphism of SO(4, 2) of conformal QCD with the group of isometries of AdS spaceds 2 = R2z 2 (η µνdx µ dx ν − dz 2 ).• Semi-classical correspondence as a first approximation to QCD (strongly coupled at all scales).• x µ → λx µ , z → λz, maps scale transformations into the holographic coordinate z.• Different values of z correspond to different scales at which the hadron is examined: AdS boundary atz → 0 corresponds to the Q → ∞, UV zero separation limit.• There is a maximum separation of quarks and a maximum value of z at the IR boundary• Truncated AdS/CFT (Hard-Wall) model: cut-off at z 0 = 1/Λ QCD breaks conformal invariance andallows the introduction of the QCD scale (Hard-Wall Model) Polchinski and Strassler (2001).• Smooth cutoff: introduction of a background dilaton field ϕ(z) – usual linear Regge dependence canbe obtained (Soft-Wall Model) Karch, Katz, Son and Stephanov (2006).Chile LHCJanuary 8, 2010LF Holography and QCD37Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front Holography:Map AdS/CFT− d2to 3+1 LF TheoryRelativistic LF radial equation−d 2− d2dζ 2 + 1 − 4L24ζ 2d 2 ζ + V (ζ) dζ 2 + V (ζ) ζ 2 = x(1 − x)b 2 ⊥ .= M 2 φ(ζ)Frame Independent= M 2 φ(ζ)+ U(ζ) φ(ζ) = M 2 φ(ζ)G. de Teramond, sjbChile LHCJanuary 8, 2010J z =xS(1 p z −= x) b ni=1Si z + ⊥n−1LF Holography and QCD39zx (1 − x) b ⊥z ∆i=1 z i = 1 2ψ(x, b ⊥ ) = ψ(ζ)x (1 − x) b ⊥ψ(x, b ⊥ ) = ψ(ζ)φ(z)U(z) = κ 4 z 2 each + 2κFock 2 (L + State S −ψ(x, 1) b ⊥ ) = ψ(ζ)φ(z)ζ = x(1 soft −wallx) b 2 ⊥φ(z)confining potentialJ z ζ =Press and Media : SLAC National Accelerator Laboratoryp = Sq z x(1 −+ Sg z x) b 2 ⊥+ L z zq + L z g = 1 ζ = x(1 − x) b 2 ⊥zStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front Quantization of QCD and AdS/CFT• Light-front (LF) quantization is the ideal framework to describe hadronicstructure in terms of quarks and gluons: simple vacuum structure gives anunambiguous definition of the partonic content of a hadron, exact formulae forform factors, physics of angular momentum of constituents ...• Frame-independent LF Hamiltonian equation: similar structure as AdS EOMP µ P µ |P>=(P − P + − P 2 ⊥)|P >= M 2 |P>• First semiclassical approximation to the bound-state LF Hamiltonianequation in QCD is equivalent to equations of motion in AdS and can besystematically improvedGdT and Sjb PRL 102, 081601 (2009)Chile LHCJanuary 8, 2010LF Holography and QCD40Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

AdS/CFT• Use mapping of conformal group SO(4,2) to AdS5Chile LHCJanuary 8, 20100 < z < z 0z 0 = 1Λ QCDψ(z 0 ) = 0LF Holography and QCD41ψ(z) ∼ z ∆ at z → 0x 2 µ → λ 2 x 2 µψ(z 0 ) = 00 < z < z 0ψ(z) ∼ z ∆ at z → 00 < z < zψ(z[C F = N 0C 2 ) 0−1 = 02N]Cz 0 = 1ψ(z 0 ) = 00 < z < z 0• Truncated space simulates “bag” boundary conditionsz 0 = 1Λ QCDx 2 µ → λ 2 x 2 µ• Scale Transformations represented by wavefunction in5th dimensionz → λz• Match solutions at small z toz →conformal λztwist ψ(z dimension0 ) = 0of hadron wavefunction at short distances• Hard wall model: Confinement at large distances andconformal symmetry in interiorΛ QCDF H (Q 2 ) × [Q 2 ] nH−1 ∼ conψ(z) ∼ z ∆ at z → 0z 0 = 1[Q 2 ] n H−1 F H (Q 2 ) ∼ constaψ(z) ∼ z ∆ at z → 0F H (Q 2 ) ∼ [ 1 Q 2]n H−1ψ(z) ∼ z ∆ at zψ(z) → 0 ∼ z ∆ at z →Press and Media : SLAC National Accelerator Laboratoryf d (Q 2 ) ≡Λ QCDStan BrodskySLACF d (Q 2 )Q 2 Q 22HOME RESEARCH SCIENTIFIC PROGRAMS

2 Bosonic ModesBosonic Solutions: Hard Wall Model• Conformal metric: ds 2 = g m dx dx m . x = (x µ , z), g m → R 2 /z 2 η m .• Action for massive scalar modes on AdS d+1 :S[Φ] = 1 2d d+1 x √ g 1 2g m ∂ Φ∂ m Φ − µ 2 Φ 2 ,√ g → (R/z) d+1 .• Equation of motion1 ∂ √√ g gm ∂g ∂x ∂x m Φ + µ 2 Φ = 0.• Factor out dependence along x µ -coordinates , Φ P (x, z) = e −iP ·x Φ(z), P µ P µ = M 2 :z 2 ∂ 2 z − (d − 1)z ∂ z + z 2 M 2 − (µR) 2 Φ(z) = 0.• Solution: Φ(z) → z ∆ as z → 0,Chile LHCJanuary 8, 2010Φ(x, z) = Cz d 2 J∆− dΦ(z) = Cz d/2 J ∆−d/2 (zM)• Normalization2d = 4R d−1 Λ−1(zM) , ∆ = 1 2d + d 2 + 4µ 2 R 2 .∆ = 2 + L (µR) 2 = L 2 − 4QCDLF Holography 0 and QCD42dzz d−1 Φ2 S=0(z) = 1.Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Let Φ(z) = z 3/2 φ(z)AdS Schrodinger Equation for bound stateof two scalar constituents:−d 2dz 2 − 1 − 4L24z 2φ(z) = M 2 φ(z)L: light-front orbital angularmomentumDerived from variation of Action in AdS 5Hard wall model: truncated spaceφ(z = z 0 = 1 Λ c) = 0.Chile LHCJanuary 8, 2010[ − d2dz 2 + V(z)]φ(z) = M2 φ(z)LF Holography and QCD43Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

AdS/QCDMatch fall-off at small z to conformal twist-dimensionat short distancesO 2+L• Pseudoscalar mesons: O 3+L = ψγ 5 D {1 . . . D m }ψ(Φ µ = 0 gauge).G. F. de Téramond• 4-d mass spectrum from boundary conditions on the normalizable string modes at z = z 0 ,Φ(x, z o ) = 0, given by the zeros of Bessel functions β α,k : M α,k = β α,k Λ QCD• Normalizable AdS modes Φ(z)twist∆ = 2 + L544Φ(z)2-20068721A7321S = 0Chile LHCJanuary 8, 2010z ∆γd → npz ∆Φ(z)00-40 1 2 3 4 0 1 2 3 42-2006z8721A8zz 0 =Λ 1QCD-2LF Holography and QCD442γγ → π + π −γd → npFig: Meson orbital and radial AdS modes for Λ QCD = 0.32 GeV.γγ → K + K − γd → npγγ → π + π −s = Ecm 2 = W 2 γγ →= Q 2 π + π −Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACaltech High Energy Seminar, Feb 6, 2006 Page 19z ∆z 0γγ → K + K −HOME RESEARCH SCIENTIFIC PROGRAMS

Φ(z)3242Φ(z)01-22-20078721A1800 1 2 3 4z2-20078721A19-40 1 2 3 4zFig: Orbital and radial AdS modes in the hard wall model for Λ QCD = 0.32 GeV .(a)S = 0(b) S = 1f 4(2050)a 4(2040)(GeV 2 )42π (140)b 1(1235)π 2(1670)ω (782)ρ (770)a 0(1450)a 2(1320)f 1(1285)ρ (1700)ρ 3(1690)ω 3(1670)ω (1650)01-20068694A160 2 4Lf 2(1270)a 1(1260)0 2 4Fig: Light meson and vector meson orbital spectrum Λ QCD = 0.32 GeVPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, Cambridge, 2010 August 20-24, 2007 SLAC45Page 23LHOME RESEARCH SCIENTIFIC PROGRAMS

Son and Stephanov (2006)] retain conformal AdS metrics but introduceon the profile of a dilaton background Soft-Wall field ϕ(z) Model =±κ 2 zd Stephanov (2006)] retain conformal AdS 2metrics but introducoft-Wall ModelS = d 4 xdz √ ge ϕ(z) L,ofile of a dilaton background field ϕ(z) =±κ 2 z 2• Soft-wall model [Karch, Katz, Son and Stephanov (2006)] retain conformal AdS metrics but intrsmooth cutoff wich depends on the profile of a dilaton background field ϕ(z) =±κ 2 z 2Retain g m conformal ∂ Φ∂ m ΦAdS − µ metrics 2 Φ 2 but introduceS = d 4 xdz √ smooth cutoffd 4 xdz √ ge ϕ(z) L,which depends on the profile of a dilaton ge ϕ(z) background L, fieldfield=L = 1 23 ∓ 2κ 2 z 2 z ∂ z + z 2 M 2 − (µR) 2 Φ(z) =0• Equation of motion for scalar field L = 1 2 g m ∂ Φ∂ m Φ − µ 2 Φ 2= 1 g2m ∂ Φ∂ m Φz 2 ∂z 2 − − µ 2 Φ 23 ∓ 2κ 2 z 2 z ∂ z + z 2 M 2 − (µR) 2 Φ(z) =0dilaton’ ϕ = +κ 2 z 2 . Lowest possible state (µR) 2 = −42 with z 2 (µR) z ∂ 2 ≥−4. z + z 2 M 2 − (µR) 2 Φ(z) =02 =0, Φ(z) ∼ z 2 e −κ2 z 2 , r 2 ∼ 1 κ 2Karch, Katz, Son and Stephanov (2006)]• LH holography requires ‘plus dilaton’ ϕ = +κ 2 z 2 . Lowest possible state (µR) 2 = −4M 2 =0, Φ(z) ∼ z 2 e −κ2 zte of two massless quarks with scaling dimension 2: 2 the , r pion2 ∼ 1 κ 2ϕ = +κ 2 z 2 . Lowest possible state (µR) 2 = −4A chiral symmetric bound state of two massless quarks with scaling dimension 2: the pionMassless pion

• Erlich, Karch, Katz, Son, Stephanov • de Teramond, sjbAdS Soft-Wall Schrodinger Equation forbound state of two scalar constituents:−d 2dz 2 − 1 − 4L24z 2+ U(z) φ(z) = M 2 φ(z)U(z) = κ 4 z 2 + 2κ 2 (L + S − 1)Derived from variation of ActionDilaton-Modified AdS 5e Φ(z) = e +κ2 z 2Positive-sign dilatonChile LHCJanuary 8, 2010LF Holography and QCD47Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Guyds 2 = e κ2 z 2 R 2z 2 (dx2 0 − dx 2 1 − dx 2 3 − dx 2 3 − dz 2 )Agrees withKlebanov andMaldacena forpositive-signexponent ofdilatonChile LHCJanuary 8, 2010LF Holography and QCD48Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Quark separationincreases with L645Φ(z)Φ(z)02-5(GeV 2 )Pion haszero mass!2-20078721A200 0 4 8z2-20078721A210 4 8zFig: Orbital and radial AdS modes in the soft wall model for κ = 0.6 GeV .42(a)π (140)S = 0 (b) S = 0b 1 (1235)π 2 (1670)π (140)π (1300) π (1800)Soft WallModelPion massautomaticallyzero!m q = 008-20078694A190 2 4L0 2 4nLight meson orbital (a) and radial (b) spectrum for κ = 0.6 GeV.Press and Media : SLAC National Accelerator LaboratoryChile LHCStan Brodskyxploring January QCD, Cambridge, LF Holography and QCD8, 2010 August 20-24, 2007 SLAC Page 2649HOME RESEARCH SCIENTIFIC PROGRAMS

Higher-Spin Hadrons• Obtain spin-J mode Φ µ1···µ Jwith all indices along 3+1 coordinates from Φ by shifting dimensionsΦ J (z) = zR −JΦ(z)• Substituting in the AdS scalar wave equation for Φz 2 ∂ 2 z − 3−2J − 2κ 2 z 2 z ∂ z + z 2 M 2 − (µR) 2 Φ J =0• Upon substitution z →ζφ J (ζ)∼ζ −3/2+J e κ2 ζ 2 /2 Φ J (ζ)we find the LF wave equation− d2dζ 2 − 1 − 4L24ζ 2 + κ 4 ζ 2 +2κ 2 (L + S − 1) φ µ1···µ J= M 2 φ µ1···µ Jwith (µR) 2 = −(2 − J) 2 + L 2Chile LHCJanuary 8, 2010LF Holography and QCD50Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Higher Spin Bosonic Modes SW• Effective LF Schrödinger wave equation− d2− d2dz 2 − 1 − 4L24z 2with eigenvaluesdζ 2 − 1 − 4L24ζ 2M 2 = 2κ 2 (2n + 2L + S).Soft-wall model+ κ 4 z 2 + + κ 4 2κ ζ 2 + 2 (L+ 2κ 2 (L+ S−1)φ (ζ) = M 2 S (z) = Mφ S (ζ)2 φ S (z)Same slope in n and L• Compare with Nambu string result (rotating flux tube): M 2 n(L) = 2πσ (n + L + 1/2) .(a)S = 1f 4(2050)a 4(2040)(b)S = 1(GeV 2 )42ω (782)ρ (770)a 2(1320)ρ 3(1690)ω 3(1670)ρ (770)ρ (1450)ρ (1700)ReggeTrajectoriesf 2(1270)05-20068694A200 2 4L0 2 4Vector mesons orbital (a) and radial (b) spectrum for κ = 0.54 GeV.• Glueballs in the bottom-up approach: (HW) Boschi-Filho, Braga and Carrion (2005); (SW) Colangelo,De Facio, Jugeau and Nicotri( 2007).Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyExploring QCD, Cambridge, August 20-24, 2007 LF Holography and QCDPage 27January 8, 2010SLAC51nHOME RESEARCH SCIENTIFIC PROGRAMS

AdS space (Fig. ??).Φ(z)Quark separation increases with L6(a)L = 34L = 2Φ(z)Φ(z)Φ(z)50n = 0n = 2(b)2L = 1L = 0-5n = 1n = 32-20078721A2000 4 8zz2-20078721A210 4 8uzzFigure 2: Meson wavefunctions f 4(2050) is AdS space f 4 (2050) in the soft-wall holographic model ofconfinement: (a) orbital modes and (b) radial modes. Constituent quark and antiquark4fly away from each otherρ 3(1690)ρas the orbital 3 (1690)and radial quantum number increases.(GeV 2 )2(a)Hadronicω (782)ω (782)spectrum. Thus AdS/CFT ρ (770) and light-front ρ (770) holography provide a quantumρ (770)fmechanical wave equation f 2(1270) formalism 2(1270)for hadron physics. The soft-wall model, in particular,appears to provide 0 a very0useful first approximation to QCD. The solutions of the light-5-20068694A20Lnfront equation 8694A20 determine the L masses of the hadrons, ngiven the total internal spin S, theorbital angular momenta L of the constituents, and the index n, the number of nodes ofChile LHCJanuary 8, 2010a 2(1320)ω 3(1670)a 4(2040)4S = 1 ρ (1700) S = 1(GeV 2 )2(a)(b)a 2 (1320)ρ (1450)ρ (1450)0 2 4 0 2 40 2 4 0 2 45-2006ρ (770)ω 3 (1670)a 4 (2040)LF Holography and QCD52Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACthe wavefunction in ζ. For example, if the total quark spin S is zero, the meson bound(b)ρ (1700)HOME RESEARCH SCIENTIFIC PROGRAMS

igenvaluesM 2 = 2κ 2 (2n + 2L + S).S = 1are with Nambu string result (rotating flux tube):M 2 n(L)(a)f 4(2050)a 4(2040)(b)(GeV 2 )42ω (782)ρ (770)ρ (1700)Vectorρ 3(1690)mesons orbital (a) and radial (b) spectrum foa 2(1320)f 2(1270)ω 3(1670)ρ (770)ρ (1450)alls in the bottom-up approach: (HW) Boschi-Filho, Bragacio, Jugeau and Nicotri( 2007).05-20068694A200 2 4L0 2 4n

1 −− 2 ++ 3 −− 4 ++ J P CM 2LParent and daughter Regge trajectories for the I = 1 ρ-meson family (red)and the I = 0 ω-meson family (black) for κ = 0.54 GeVPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDSakharov January Conference, 8, 2010Moscow, May 19, 2009 SLAC54Page 23HOME RESEARCH SCIENTIFIC PROGRAMS

Propagation of external perturbation z i ∝ m ⊥i suppressed = m 2 i + inside k2 ⊥ AdS.At large enough Q ∼ r/R 2 X , the = interaction cū ¯dū occurs in the large-r conformal region. Importacontribution to the FF integral from the boundary near z ∼ 1/Q.F (Q 2 ) I→F = dzα z 3Φ F (z)J(Q, z)Φ I (z)s (Q 2 )J(Q, z), Φ(z)1 J(Q, z)High Q 2fromsmall z ~ 1/QChile LHCJanuary 8, 20100.40.215πα z1 2 3 4 5Consider a specific AdS mode Φ (n) dual to an n partonic Fock state |n. At small z, Φ (scales as Φ (n) ∼ z ∆ n. Thus:Hadron Form Factors from AdS/CFTJ(Q, z) = zQK 1 (zQ)0.8β(Q 2 ) = dα Φ(z)s(Q 2 )0.6 d log Q 2 → 0Π(Q 2 ) →Q 2

Spacelike pion form factor from AdS/CFTF π (q 2 )10.80.6q 2 (GeV 2 )0.4Data CompilationBaldini, Kloe and Volmer0.2However J/ψ → ρπChile LHCJanuary 8, 2010F π (q 2 )is largest two-body hadron decay0-10 -8 -6 -4 -2 0q 2 (GeV 2 )Small value for ψ → ρπρSoft Wall: Harmonic Oscillator ConfinementHowever J/ψ → ρπis largest two-body hadron decayHard Wall: Truncated Space ConfinementOne parameter - set by pion decay constantSmall value for ψ → ρπLF Holography and QCD56de Teramond, sjbSee also: RadyushkinPress and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Current Matrix Elements in AdS Space (SW)• Propagation of external current inside AdS space described by the AdS wave equationz 2 ∂ 2 z − z 1 + 2κ 2 z 2 ∂ z − Q 2 z 2 J κ (Q, z) = 0.sjb and GdTGrigoryan and Radyushkin• Solution bulk-to-boundary propagatorJ κ (Q, z) = Γ1 + Q24κ 2where U(a, b, c) is the confluent hypergeometric functionΓ(a)U(a, b, z) = ∞0 Q2U4κ 2 , 0, κ2 z 2 ,e −zt t a−1 (1 + t) b−a−1 dt.Soft WallModel• Form factor in presence of the dilaton background ϕ = κ 2 z 2F (Q 2 ) = R 3 dzz 3 e−κ2 z 2 Φ(z)J κ (Q, z)Φ(z).• For large Q 2 4κ 2J κ (Q, z) → zQK 1 (zQ) = J(Q, z),the external current decouples from the dilaton field.Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACExploring QCD, Cambridge, August 20-24, 2007 57Page 34HOME RESEARCH SCIENTIFIC PROGRAMS

Spacelike pion form factor from AdS/CFTF π (q 2 )10.80.6q 2 (GeV 2 )0.4Data CompilationBaldini, Kloe and Volmer0.2However J/ψ → ρπChile LHCJanuary 8, 2010F π (q 2 )is largest two-body hadron decay0-10 -8 -6 -4 -2 0q 2 (GeV 2 )Small value for ψ → ρπρSoft Wall: Harmonic Oscillator ConfinementHowever J/ψ → ρπis largest two-body hadron decayHard Wall: Truncated Space ConfinementOne parameter - set by pion decay constantSmall value for ψ → ρπLF Holography and QCD58de Teramond, sjbSee also: RadyushkinPress and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

• Analytical continuation to time-like region q 2 → −q 2M ρ = 2κ = 750 MeV(M ρ = 4κ 2 = 750 MeV)• Strongly coupled semiclassical gauge/gravity limit hadrons have zero widths (stable).21log IF π (q 2 )I0-1-2-3-10 -5 0 5 10q 2 (GeV 2 )Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACExploring QCD, Cambridge, August 20-24, 2007 59Page 407-20078755A4Space and time-like pion form factor for κ = 0.375 GeV in the SW model.• Vector Mesons: Hong, Yoon and Strassler (2004); Grigoryan and Radyushkin (2007).HOME RESEARCH SCIENTIFIC PROGRAMS

Dressed soft-wall current bring in higherFock states and more vector meson polese +e − γ ∗ π + π −Chile LHCJanuary 8, 2010LF Holography and QCD60Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Note: Analytical Form of Hadronic Form Factor for Arbitrary Twist• Form factor for a string mode with scaling dimension τ , Φ τ in the SW modelΓ1+ Q2F (Q 2 4κ) = Γ(τ) 2 .Γ τ + Q24κ 2• For τ = N ,Γ(N + z) = (N − 1 + z)(N − 2 + z) . . . (1 + z)Γ(1 + z).• Form factor expressed as N − 1 product of poles• For large Q 2 :F (Q 2 ) =F (Q 2 ) =F (Q 2 ) =1, N = 2,1 + Q24κ 2 2 , N = 3,1 + Q24κ2 + Q22 4κ 2· · ·1 + Q24κ 2 2 + Q24κ 2 · · ·(N − 1)! , N.N −1+ Q24κ 2 4κF (Q 2 2 (N−1)) → (N − 1)!Q 2 .Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACExploring QCD, Cambridge, August 20-24, 2007 61Page 43HOME RESEARCH SCIENTIFIC PROGRAMS

V, Γ ρ = 400MeV and Γ ρ = 300 MeV. The value of |α1 |= 0.38,Spacelike and timelike pion form factorto a 15 % admission of a four-quark state.Preliminaryκ = 0.534 GeVQ 2 (GeV 2 )Q 2 (GeV 2 )|π >= ψ q¯q |q¯q > + ψ q¯qq¯q |q¯qq¯q >re 1: pion form factor Q 2 F π (Q 2 ) as a function of Q 2 = −q 2 .Figure 2: Timelike pion form factor log|F π (q 2 )| as a function of q 2 .esΓ ρ = 120 MeV, Γ ρ = 300 MeVdsky and G. F. de Teramond, Phys. Rev. D 77, 056007 (2008)07.3859 [hep-ph]].P q¯qq¯q = 15%32Q 2 (GeV 2 )

Holographic Model Light-Front for QCDRepresentationLight-Front Wavefunctionsof Two-Body Meson Form FactorSJB and GdT in preparation• Drell-Yan-West form factor in the light-cone (two-parton state)F (q 2 ) = q 1e q dx0 d 2 k⊥16π 3 ψ∗ P (x, k ⊥ − xq ⊥ ) ψ P (x, k ⊥ ).q 2 ⊥ = Q 2 = −q 2• Fourrier transform to impact parameter space b ⊥ψ(x, k ⊥ ) = √ 4πd 2 b⊥ e i b ⊥·k ⊥ ψ(x, b⊥ )• Find (b = | b ⊥ |) :F (q 2 ) = 10= 2πdx d 2 b⊥ e ix b ⊥·q ⊥ ψ(x, b) 2 10dx ∞0b db J 0 (bqx) ψ(x, b) 2 ,SoperChile LHCJanuary 8, 2010LF Holography and QCD63Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACaltech High Energy Seminar, Feb 6, 2006 Page 33HOME RESEARCH SCIENTIFIC PROGRAMS

Holographic Mapping of AdS Modes to QCD LFWFs• Integrate Soper formula over angles:F (q 2 ) = 2π 10(1 − x)dxxψ(x, b ⊥ ) = ψ(ζ)ζdζJ 0ζq 1 − xx˜ρ(x, ζ),with ρ(x, ζ) QCD effective transverse charge density.• Transversality variableζ = x1 − xn−1 x j b ⊥j .• Compare AdS and QCD expressions of FFs for arbitrary Q using identity:z 1 1 − xdxJ 0ζQ = ζQK 1 (ζQ),xthe solution for J(Q, ζ) = ζQK 1 (ζQ) !0φ(z)ζ =z ∆z 0 = 1Λ QCDx(1 − x) b 2 ⊥j=1

• Electromagnetic form-factor in AdS space:where J(Q 2 , z) = zQK 1 (zQ).F π +(Q 2 ) = R 3 dzz 3 J(Q2 , z) |Φ π +(z)| 2 ,• Use integral representation for J(Q 2 , z)J(Q 2 , z) = 10dx J 0ζQ 1 − xx• Write the AdS electromagnetic form-factor asF π +(Q 2 ) = R 3 10 dz 1 − xdxz 3 J 0 zQx• Compare with electromagnetic form-factor in light-front QCD for arbitrary Q ˜ψ qq/π (x, ζ) 2 = R32π x(1 − x)|Φ π(ζ)| 2ζ 4|Φ π +(z)| 2with ζ = z, 0 ≤ ζ ≤ Λ QCDPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLAC65From String to Things, INT, Seattle, April 10, 2008 Page 29HOME RESEARCH SCIENTIFIC PROGRAMS

ψ(x, b ⊥ ) = ψ(ζ)φ(z)ζ =zChile LHCJanuary 8, 2010x LF(3+1) (1 − x) b AdS 5⊥ψ(x, b ⊥ ) = ψ(ζ)φ(z)x(1 − x) b 2 ⊥x (1 −ζx)= b ⊥ x(1 − x) b 2 ⊥ψ(x, b ⊥ ) = ψ(ζ)zLF Holography and ∆QCD66ψ(x, b ⊥ ) = ψ(ζφ(z)ζ =x (1 − x) b ⊥x (1 − x) bz⊥ψ(x, b ⊥ ) = ψ(ζ)x (1 − x) b ⊥z ∆ψ(x, b ⊥ ) = ψ(ζ)φ(z)φ(z)ψ(x, b ⊥ ) = ψ(ζ)z ∆z ∆φ(z)ζ x(1 − x) b 2 ⊥z 0 = 1 ψ(x, φ(z)Λ x(1 − x)bζ = x(1 − x) z QCDb 2 ⊥ 0 = 1⊥ ) =φ(ζ)zζ = x(1z 0 =− x) 1b 2 2πζLight-Front Holography: Unique ⊥ mapping derived from ζ = x(1 equality − x) b 2z⊥γd → np Λz 0 = 1 ofLF and AdS formula QCD for current matrix elements z ∆zzzφ(z)ζ =zz ∆Press and Media : SLAC National Accelerator LaboratoryStan Brodskyγdz 0 →= SLAC1npΛ QCDHOME RESEARCH SCIENTIFIC PROGRAMSz ∆x(1 −(x(1 − xΛ QCDΛ QCD

Gravitational Form Factor in AdS space• Hadronic gravitational form-factor in AdS spacewhere H(Q 2 , z) = 1 2 Q2 z 2 K 2 (zQ)A π (Q 2 ) = R 3 dzz 3 H(Q2 , z) |Φ π (z)| 2 ,Abidin & Carlson• Use integral representation for H(Q 2 , z)H(Q 2 , z) = 2• Write the AdS gravitational form-factor as0 10x dx J 0zQ 1 − x 1 dz 1 − xA π (Q 2 ) = 2R 3 x dxz 3 J 0 zQ |Φ π (z)| 2x• Compare with gravitational form-factor in light-front QCD for arbitrary Q ˜ψ qq/π (x, ζ) 2 = R32π x(1 − x)|Φ π(ζ)| 2ζ 4 ,which is identical to the result obtained from the EM form-factorIdentical to LF Holography obtained from electromagnetic currentPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyFrom String to Things, INT, Seattle, April 10, LF 2008 Holography and QCDPage 31January 8, 2010SLAC67xHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front Holography:Map AdS/CFT− d2to 3+1 LF TheoryRelativistic LF radial equation!−d 2− d2dζ 2 + 1 − 4L24ζ 2d 2 ζ + V (ζ) dζ 2 + V (ζ) ζ 2 = x(1 − x)b 2 ⊥ .= M 2 φ(ζ)Frame Independent= M 2 φ(ζ)+ U(ζ) φ(ζ) = M 2 φ(ζ)G. de Teramond, sjbChile LHCJanuary 8, 2010J z =xS(1 p z −= x) b ni=1Si z + ⊥n−1LF Holography and QCD68zx (1 − x) b ⊥z ∆i=1 z i = 1 2ψ(x, b ⊥ ) = ψ(ζ)x (1 − x) b ⊥ψ(x, b ⊥ ) = ψ(ζ)φ(z)U(ζ) = each κ 4 ζ 2 Fock + 2κ 2 State (L + Sψ(x,− b ⊥1)) = ψ(ζ)φ(z)ζ = x(1 − x) b 2 ⊥φ(z) soft wallconfining potential:Jp z ζ == Sq z x(1 −+ Sg z x) b 2 ⊥+ L z zq + L z g = 1 ζ = x(1 − x) b 2 ⊥zPress and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

H QEDHQCDLFQCD Meson Spectrum(H 0 LF + H I LF )|Ψ >= M 2 |Ψ >Coupled Fock states[ k 2 ⊥ + m2x(1 − x) + V LFeff ] ψ LF (x, k ⊥ ) = M 2 ψ LF (x, k ⊥ )Effective two-particle equation[− d2 −1 + 4L2+dζ2 ζ 2 = x(1 − x)b 2 ⊥Azimuthal Basisζ 2 + U(ζ, S, L)] ψ LF (ζ) = M 2 ψ LF (ζ) ζ, φU(ζ, S, L) = κ 2 ζ 2 + κ 2 (L + S − 1/2)Semiclassical first approximation to QCDConfining AdS/QCDpotential

Derivation of the Light-Front Radial Schrodinger Equationdirectly from LF QCDM 2 =Changevariables= 10 10M 2 = d 2 k⊥dx16π 3dxx(1 − x) k2⊥x(1 − x)ψ(x, k ⊥ )( ζ, ϕ), ζ = x(1 − x) b ⊥ := 2 + interactionsd 2 b⊥ ψ ∗ (x, b ⊥ )− ∇ 2 b⊥ψ(x, b ⊥ ) + interactions.− d2∇ 2 = 1 ζddζζ d dζdζ φ ∗ (ζ) ζdζ 2 − 1 dζ dζ + L2 φ(ζ)√ζ 2 ζ+ dζ φ ∗ (ζ)U(ζ)φ(ζ)dζ φ ∗ (ζ)− d2dζ 2 − 1 − 4L24ζ 2 + U(ζ) φ(ζ)+ 1ζ 2 ∂ 2∂ϕ 2Chile LHCJanuary 8, 2010LF Holography and QCD70Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

• Find (L = |M|)M 2 =dζφ ∗ (ζ) ζ− d2dζ 2 − 1 ζ ddζ + L2 φ(ζ)ζ 2 √ + ζdζφ ∗ (ζ) U(ζ) φ(ζ)where the confining forces from the interaction terms is summed up in the effective potential U(ζ)• Ultra relativistic limit m q → 0 longitudinal modes X(x) decouple and LF eigenvalue equationH LF |φ = M 2 |φ is a LF wave equation for φ− d2dζ 2 − 1 − 4L24ζ 2 + U(ζ) φ(ζ) =M 2 φ(ζ) confinementkinetic energy of partons• Effective light-front Schrödinger equation: relativistic, frame-independent and analytically tractable• Eigenmodes φ(ζ) determine the hadronic mass spectrum and represent the probability amplitude tofind n-massless partons at transverse impact separation ζ within the hadron at equal light-front time• Semiclassical approximation to light-front QCD does not account for particle creation and absorptionbut can be implemented in the LF Hamiltonian EOM or by applying the L-S formalismPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyHadron 2009, FSU, Tallahassee, DecemberLF 1, 2009 Holography and QCDPage 12January 8, 2010SLAC71HOME RESEARCH SCIENTIFIC PROGRAMS

H QEDQED atoms: positroniumand muonium(H 0 + H int ) |Ψ >= E |Ψ > Coupled Fock states[− ∆22m red+ V eff ( S, r)] ψ(r) = E ψ(r)Effective two-particle equationIncludes Lamb Shift, quantum corrections[− 1 d 22m red dr 2 + 1 ( + 1)2m red r 2+ V eff (r, S, )] ψ(r) = E ψ(r)Spherical Basisr, θ, φV eff → V C (r) = − α rSemiclassical first approximation to QEDCoulomb potentialBohr Spectrum

Example: Pion LFWF• Two parton LFWF bound state:ψ qq/π HW (x, b ⊥) = Λ QCD x(1 − x) √πJ1+L (β L,k ) J L x(1 − x) |b⊥ |β L,k Λ QCD θ b 2 ⊥ ≤Λ−2 QCD,x(1 − x)qq/π (x, b ⊥) = κ L+1 2n!(n + L)! [x(1 − x)] 1 2 +L |b ⊥ | L e − 12 κ2 x(1−x)b 2 ⊥ LLn κ 2 x(1 − x)b 2 ⊥ .ψ SW(a)b0 10 20(b)b0 10 20ψ(x,b)0.100.050.20.10 0000.5x0.5x1.01.07-20078755A1Fig: Ground state pion LFWF in impact space. (a) HW model Λ QCD = 0.32 GeV, (b) SW model κ = 0.375 GeV.Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, Cambridge, 2010 August 20-24, 2007 SLAC Page 3773HOME RESEARCH SCIENTIFIC PROGRAMS

• Consider Orbital excitations the AdSconstructed 5 metric: by the L-th application of the raising operatords 2 = R2z 2 (η µν dx µ dx ν − dz 2 ).ds 2 invariant if x µ → λx µ , z → λz,on the ground state:Maps scale transformations to scale changes of the the holographic coordinate z.• In the light-front ζ-representationWe define light-front coordinates x ± = x 0 ± x 3 .Then η µν dx µ dx ν = dx 2 0 − dx 2 3 − dx 2 ⊥ = dx + dx − 2− dxφ ⊥ L (ζ) = ζ|L = C L ζ (−ζ)L 1 d LJ 0 (ζM)ζ dζanda † L = −iΠ La † |L = c L |L + 1.= C L ζJL (ζM) .ds 2 = − R2 (dx 2 z 2 ⊥ + dz 2 ) for x + = 0.• The• ds 2 solutions φ L are solutionsis invariant if dx 2 ⊥ → λ 2 of thedx 2 light-front equation (L = 0, ±1, ±2, · · · )⊥ , and z → λz, at equal LF time.− d2dζ 2 − 1 − L24ζ 2 φ(ζ) = M 2 φ(ζ),• Maps scale transformations in transverse LF space to scale changes of the holographic coordinate z.• Mode Holographic spectrum connection from boundary of AdSconditions 5 to the light-front. : φ (ζ = 1/Λ QCD ) = 0.Light-Front/AdS 5 Duality• The Casimir effective for the wave rotation equation groupinSO(2).the two-dim transverse LF plane has the Casimir representation L 2corresponding to the SO(2) rotation group [The Casimir for SO(N) ∼ S N−1 is L(L + N − 2) ].Exploring QCD, Cambridge, August 20-24, 2007 Page 20Press and Media : SLAC National Accelerator LaboratoryExploringChile QCD, LHC Cambridge, August 20-24, 2007 Stan Brodsky Page 3LF Holography and QCDJanuary 8, 2010SLAC74HOME RESEARCH SCIENTIFIC PROGRAMS

Prediction from AdS/CFT: Meson LFWFφ M (x, Q 0 ) ∝0.2ψ(x, k ⊥ )x(1 − x)0.8 0.60.40.2de Teramond, sjbψ M (x, k 2 ⊥ )0.150.10.05“Soft Wall”modelµ RNote couplingµ R = Qk 2 ⊥, x01.31.4ψ(x, k ⊥ ) (GeV)1.5κ = 0.375 GeVmassless quarksµ F = µ Rψ M (x, k ⊥ ) =Q/2 < µ R < 2QChile LHCJanuary 8, 20104πκ x(1 − x) e−k 2 ⊥2κ 2 x(1−x)LF Holography and QCD75φ M (x, Q 0 ) ∝Connection of Confinement to µ TMDs Rx(1 − x)Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

e get(5)dsThe 2 ¼ R2wave z 2 In this model the function gðxÞ in matching condition (8)dxfunction dx ;(13) does PHYSICAL not consider isREVIEW Dmassive fixed 80, 055014 quarks. gðxÞ ¼1 for large values of Q 2 4 2 . In this(4)(2009)case the current JWe ¼include diagð1; the 1; quark 1; Meson 1 masses 1Þ; wave function following fromtheholographic prescription ðQ 2 ;zÞ decouples from the dilaton field[22]. The examples models considered in this work correspond tosuggested by Brodsky and Téramond [24]. First one shouldwhere (6) z is the holographic Alfredocoordinate Vega, 1 Ivanand Schmidt, , that 1 Tanja represent Branz, 2 mesons withThomas Gutsche, 2 n ¼ L ¼ 0, althoughand Valery E. Lyubovitskij 2, both*for scalars andperform the Fourier transform of (13)different things in both 1 Departamento holographical de Físicamodels y Centroconsidered,vectors we findEstudios Subatómicos, Universidad Técnica Federico Santa María,in one model is Casilla 110-V, Valparaíso, Chile2 Institut für Theoretische Physik, Universität Tübingen, Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14,c ð1Þ the scale parameter4A characterizingtheq 1 q 2ðx; k ? Þ¼1k 2 pffiffiffiffiffip? dilaton field and in the other MESON one pthis ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi WAVE characterize FUNCTION D–72076exp the1 ðÞ ¼Tübingen, Germany 1 xð1 (Received xÞ5 June 2009; 2 published 2 1 xð1 10 September xÞ : (14) 1 2 expð1=2Þ 2 1 2 ffiffiffi expð1=2Þ 2 1 2 : (12)FROM HOLOGRAPHIC MODELSPHYSIanomalous dimension as you can see later when we discuss Using Eq. (12) and 2009) keeping in mind that 2 ¼ xð1 xÞb 2 ? ,the models considered. We An consider the light-front wave function for the valence quark state of mesons using the AdS/CFTInψa second correspondence, stepas has thebeen quark suggestedmasses by Brodskyare and Téramond. introduced Two kinds byc¯c (x, important step is to set up the the meson LFWF of this model iselectromagnetic current as k of wave functions, obtaineddifferent holographic ⊥ )m c =1.5 GeV,κ=0.894 GeVSoft-Wall models, are discussed.extending the kinetic energy of massless quarks with K 0 ¼JðQ 2 k;zÞ¼ Z sffiffiffiffiffiffiffiffiffiffiffiffi~c 2 1DOI: 10.1103/PhysRevD.80.055014PACS numbers: 11.25.Tq, 12.39.Ki, 14.40.Aq, 14.40.Cs?dxgðxÞJxð1 xÞto the case 0Qð1Þq 1 q 2ðx; b ? Þ¼ 1A 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1pffiffiffiffixð1 xÞ exp1 x2 2 1 xð1 xÞb2 ?:: (5)0 of massive x quarks:(13)Putting z ¼ and comparing In this work meson wave functions obtained in theI. INTRODUCTIONEqs. (1) and (3) we getThe wave function (13) doescontext of AdS/CFT ideas [21,22] are studied consideringThe hadronicK 0 !waveK ¼function K þinterms 2 12 ; of quark 2 and 12 gluon¼ m2 1two kinds of holographic soft-wall models. First we considerthe more usual model with a quadratic dilatonx þ m2 0.06 not consider massive quarks.We include 21 x : the (15) quark masses following the prescription00.04 0degrees ~ðx; of freedoms Þ ¼ xgðxÞ jðÞj 2plays an important role in QCDprocessNote,predictions. 1the changeFor example, x 250 2 : (6) suggested by Brodsky and Téramond [24]. First one shouldperform the Fourier transformproposedknowledgeinof(15) the waveis equivalent[10,14,22]. Thentowethediscuss predictions 0.02 of (13)of 250 a recent modelfunction allows to calculate distribution amplitudes and which considers a logarithmic dilaton as suggested byFinally, consideringstructure following functionsthe case change or converselywith in two500(10) thesepartonsprocessesq 01 andcanq give 2 Einstein’s c ð1Þ4Aq equations for an AdS metric. It also 5001 q 2ðx; k ? Þ¼1k 2 ?pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexp includesphenomenological restrictions on the wave functions. anomalous dimensions 0.2[13] and allows to reproduce thekIn principle the Bethe-Salpeterd approach 2 750 [1] and discrete Regge behavior even in the baryonic sector.quantization in the light-front formalism [2] allow to obtainhadronic wave functions butThe work is structured as follows. Sec. II is devoted todin 2 !d2k~practice severald 2 þ 2 12 : 0.4(16)750n¼2 ðx; Þ ¼ j ~c q1 q 2ðx; Þj 21 xð1 xÞ 2 2 1 xð1 xÞ : (14)1ð1 xÞ 2 A 2 ; (7) In a second step the quark masses are introduced byproblemspresent to realize this [3,4]. Thereforethe extraction 0.6of wave functions for scalar/pseudoscalar1000extending thexkinetic energy of massless quarksapproximate mesons using the two holographic models. In Sec. III 1000 with Kwhere 2 ¼ xð1 xÞb wesolutions Finally for hadronic we 2 0 ¼?and A is the normalization constant,we obtain the relation between the AdS modes andobtain0.8 kbound states are usually considered concentrate 2 ? on the pion wave function discussing the adjustmentof the model parameters. Distribution amplitudesxð1 xÞto the case of massive quarks:using in a first step specific quarks models to obtain thethe meson LFWF ~cvalence quark wave function.and parton distributions for the valence state are calculatedThere c ð1Þ4Aq 1 are q 2ðx; several k ? Þ¼1k 2 ? 2 q1q 2ðx; ÞFIG. 12nonperturbative p3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(color online).approaches exp Wave function cto obtain in both models. In the pion case we consider both currentproperties of distribution amplitudes 1 xð1and/or xÞhadronic wave2 2 1 xð1 Kand constituent xÞc c ðx; k ? Þ2quark 2 : according to section IV. We consider in this0 ! K ¼ K 0 þ 2894 MeV, the value suggested by the Regge slope1formasses. charmonium In Sec. states. IV we The calculate left graph correspoj ~c q1 qfunctions 2ðx; Þj 2 ¼ A 2 xð1from QCD, and Eq. xÞgðxÞ jðÞj212 ; 2 12 ¼ m2 1now (26). we have 2 possibility : (8)x þ m2 21 x : (15)to Note, decay constants the change in the simplified case when the valencePress and Media : SLAC National Accelerator LaboratoryChile include LHC techniques based on the anti–de Sitter space/con-constrained field theory by the (AdS/CFT) probability correspondence. condition LF Holography and Sec. QCD V.component is dominant.(17) proposed in (15) is equivalent to theConclusions Stan are presented Brodsky infollowing change in (10)Here A isformalJanuary 8, 2010SLACAlthoughNote,ainrigorous the caseQCD dualof massiveis unknown, B. Pion quarksa simple distribution 76the normalization amplitude andZd 2 d 2 21 and q 2(7)tion conodesand: (8)on1 (9)ence Fockf masslesse case forNext we(Model 1s.Z 1(13)HOME RESEARCH SCIENTIFIC PROGRAMS

Example: Evaluation of QCD Matrix Elements• Pion decay constant f π defined by the matrix element of EW current J + W :with0 ψ u γ + 1 2 (1 − γ 5)ψ d π − = i P + f π√2 π− = |du = √ 1 1 N C√2NCc=1• Find light-front expression (Lepage and Brodsky ’80):b † c d↓ d† c u↑ − b† c d↑ d† c u↓ 0 .f π = 2 N C 10dx d 2 k⊥16π 3 ψ qq/π(x, k ⊥ ).• Using relation between AdS modes and QCD LFWF in the ζ → 0 limitf π = 1 832 R3/2 limζ→0Φ(ζ)ζ 2 .• Holographic result (Λ QCD =0.22 GeV and κ=0.375 GeV from pion FF data): Exp: f π =92.4 MeVf HWπ =√38J 1 (β 0,k ) Λ QCD = 91.7 MeV, f SWπ =√38κ = 81.2 MeV,Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploringJanuaryQCD,8,Cambridge,2010August 20-24, 2007SLAC77Page 42HOME RESEARCH SCIENTIFIC PROGRAMS

Hadron Distribution Amplitudesφ H (x i , Q)x i = 1iP + = P 0 + P zFixed τ = t + z/cβ = dα s(Q 2 )d ln Q 2 < 0ux1 − x• Fundamental gauge invariant non-perturbativex i = k+P + = k0 +k input 3 toP 0 +Phard exclusive processes, heavy hadron decays. Definedzfor Mesons, Baryonsψ(σ, b ⊥ )• Evolution Equations fromPQCD, OPE, Conformal InvarianceLepage, sjbBraun, GardiLepage, sjbk 2 ⊥ < Q 2Efremov, RadyushkinSachrajda, Frishman Lepage, sjb• Compute from valence light-front wavefunction in lightconegauge Qφ M (x, Q) = d 2 k ψq¯q (x, k ⊥ )Chile LHCJanuary 8, 2010LF Holography and QCD78Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Second Moment of Pion Distribution Amplitude 1< ξ 2 >=−1dξ ξ 2 φ(ξ)ξ = 1 − 2x< ξ 2 > π = 1/5 = 0.20< ξ 2 > π = 1/4 = 0.25φ asympt ∝ x(1 − x)φ AdS/QCD ∝ x(1 − x)Lattice (I) < ξ 2 > π = 0.28 ± 0.03Lattice (II) < ξ 2 > π = 0.269 ± 0.039Donnellan et al.Braun et al.Chile LHCJanuary 8, 2010LF Holography and QCD79Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

ERBL Evolution of Pion Distribution AmplitudeΦx,Q 2 fΠ0.40.30.20.1ERBL evolution at Q 2 2,10,100 GeV 2Q 2 = 2 GeV 2x(1 − x)Q 2 = 100 GeV 2 x(1 − x)0.00 100 200 300 400 5000.5 1.0500 xF. Cao, GdT, sjb (preliminary)Chile LHCJanuary 8, 2010LF Holography and QCD80Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

2 Fermionic Modes• Baryons Spectrum in ”bottom-up” holographic QCDGdT and and Brodsky: Sjb hep-th/0409074, hep-th/0501022.From Nick Evans• Baryons Spectrum in ”bottom-up” holographic QCD• Conformal metric x = (x µ , z):GdT and Brodsky: hep-th/0409074, hep-th/0501022.dsBaryons in2Ads/CFT= g m dx dx m• Conformal metric x = (x µ , z):= R2z 2 (η µνdx µ dx ν − dz 2 ).ds 2 = g m dx dx m• Action for massive fermionic modes on AdS d+1 : = R2S[Ψ, Ψ] =• Action for massivefermionic modes• Equation of motion: iΓ D − µ on AdS d+1 :Ψ(x, z) = 0• Equation of motion:d d+1 x √ g Ψ(x, z 2 (η µνdx µ dx ν − dz 2 ).z) iΓ D − µ Ψ(x, z).From Nick EvansS[Ψ, Ψ] = d d+1i zη m Γ ∂ m + d x √ g Ψ(x, z)iΓ D − µ Ψ(x, z).2 Γ z + µR Ψ(x ) = 0.iΓ D − µ Ψ(x, z) = 0 i zη m Γ ∂ m + d 2 Γ z + µR Ψ(x ) = 0.Helmholtz Institut, Bonn, Oct 16, 2007 Page 20Chile LHCJanuary 8, 2010LF Holography and QCD81Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

4 Fermionic Modes• Baryons Spectrum in ”bottom-up” holographic QCDGdT and Brodsky: hep-th/0409074, hep-th/0501022.Hard-Wall ModelBaryons in Ads/CFT• Conformal metric x = (x µ , z):• Action for massive fermionic modes on AdS 5 :• Equation of motion:• Action for massive fermionic modes on AdS d+1 :From Nick EvansFrom Nick EvansS[Ψ, Ψ] = d 4 xdz √ ds 2 = g m dx dx mg Ψ(x, z) iΓ D − µ Ψ(x, z)= R2iΓ D − µ z 2 (η µνdx µ dx ν − dz 2 ).Ψ(x, z) =0 i zη m Γ ∂ m + d 2 Γ S[Ψ, Ψ] =zd d+1 +xµR √ Ψ(x g Ψ(x,)=0z) iΓ D − µ Ψ(x, z).• Solution (µR = ν +1/2) • Equation of motion: iΓ D − µ Ψ(x, z) = 0Ψ(z) =Cz 5/2 [J ν (zM)u + + J ν+1 (zM)u − ]i zη m Γ ∂ m + d • Hadronic mass spectrum determined from IR boundary conditions 2 Γ z + µR Ψ(x ) = 0.ψ ± (z =1/Λ QCD )=0M + = β ν,k Λ QCD ,M − = β ν+1,k Λ QCDHelmholtz Institut, Bonn, Oct 16, 2007with scale independent mass ratio• Obtain spin-J mode Φ µ1···µ J−1/2, J> 1 2, with all indices along 3+1 from Ψ by shifting dimensionsPress and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACHadron 2009, FSU, Tallahassee, December 1, 2009 82Page 21HOME RESEARCH SCIENTIFIC PROGRAMS

BaryonsHolographic Light-Front Integrable Form and Spectrum• In the conformal limit fermionic spin- 1 2 modes ψ(ζ) and spin- 3 2 modes ψ µ(ζ)are two-component spinor solutions of the Dirac light-front equationwhere H LF = αΠ and the operatorαΠ(ζ)ψ(ζ) = Mψ(ζ),Π L (ζ) = −iddζ − L + 1 2ζγ 5and its adjoint Π † L(ζ) satisfy the commutation relationsΠ L (ζ), Π † L (ζ) = 2L + 1ζ 2 γ 5 .• Supersymmetric QM between bosonic and fermionic modes in AdS?,Chile LHCJanuary 8, 2010LF Holography and QCD83Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Soft-Wall Model• Equivalent to Dirac equation in presence of a holographic linear confining potentializη m Γ ∂ m + d 2 Γ z+ µR + κ 2 z Ψ(x )=0.• Solution (µR = ν +1/2, d= 4)Ψ + (z) ∼ z 5 2 +ν e −κ2 z 2 /2 L ν n(κ 2 z 2 )Ψ − (z) ∼ z 7 2 +ν e −κ2 z 2 /2 L ν+1n (κ 2 z 2 )• EigenvaluesM 2 =4κ 2 (n + ν + 1)• Obtain spin-J mode Φ µ1···µ J−1/2, J> 1 2, with all indices along 3+1 from Ψ by shifting diChile LHCJanuary 8, 2010by shifting dimensionsLF Holography and QCD84Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

• Note: in the Weyl representation (iα = γ 5 β)⎛ ⎞iα = ⎝ 0 I ⎠ , β =−I 0⎛ ⎞⎝ 0 I ⎠ , γ 5 =I 0⎛ ⎞⎝ I 0 ⎠ .0 −I• Baryon: twist-dimension 3 + L (ν = L + 1)O 3+L = ψD {1 . . . D q ψD q+1 . . . D m }ψ, L =m i .i=1• Solution to Dirac eigenvalue equation with UV matching boundary conditionsψ(ζ) = C ζ [J L+1 (ζM)u + + J L+2 (ζM)u − ] .Baryonic modes propagating in AdS space have two components: orbital L and L + 1.• Hadronic mass spectrum determined from IR boundary conditionsψ ± (ζ = 1/Λ QCD ) = 0,given byM + ν,k = β ν,kΛ QCD , M − ν,k = β ν+1,kΛ QCD ,with a scale independent mass ratio.Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploringJanuary QCD, Cambridge, 8, 2010 August 20-24, 2007 SLAC Page 4685HOME RESEARCH SCIENTIFIC PROGRAMS

8(a)N (2600)I = 1/2 (b) I = 3/2! (2420)(GeV 2 )64N (1700)N (1675)N (1650)N (1535)N (1520)N (2250)N (2190)N (2220)! (1232)! (1950)! (1920)! (1910)! (1905)! (1930)201-20068694A14N (939)N (1720)N (1680)0 2L4 6! (1700)! (1620)0 2L56704 6Fig: Light baryon orbital spectrum for Λ QCD = 0.25 GeV in the HW model. The 56 trajectory corresponds to Leven P = + states, and the 70 to L odd P = − states.Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACExploring QCD, Cambridge, August 20-24, 2007 86Page 48HOME RESEARCH SCIENTIFIC PROGRAMS

Non-Conformal Extension of Algebraic Structure (Soft Wall Model)• We write the Dirac equation(αΠ(ζ) − M) ψ(ζ) = 0,in terms of the matrix-valued operator ΠΠ ν (ζ) = −iddζ − ν + 12γ 5 − κ 2 ζγ 5 ,ζand its adjoint Π † , with commutation relations• Solutions to the Dirac equationΠ ν (ζ), Π † ν(ζ)= 2ν + 1ζ 2− 2κ 2 γ 5 .• Eigenvaluesψ + (ζ) ∼ z 1 2 +ν e −κ2 ζ 2 /2 L ν n(κ 2 ζ 2 ),ψ − (ζ) ∼ z 3 2 +ν e −κ2 ζ 2 /2 L ν+1n (κ 2 ζ 2 ).M 2 = 4κ 2 (n + ν + 1).Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACExploring QCD, Cambridge, August 20-24, 2007 87Page 49HOME RESEARCH SCIENTIFIC PROGRAMS

Glazek and Schaden [Phys. Lett. B 198, 42 (1987)]: (ω B /ω M ) 2 = 5/8 4κ 2 for ∆n = 1• ∆ spectrum identical to Forkel and Klempt, Phys. Lett. B 679, 77 (2009)4κ 2 for ∆L = 12κ 2 for ∆S = 1M 2LParent and daughter 56 Regge trajectories for the N and ∆ baryon families for κ = 0.5 GeVPress and Media : SLAC National Accelerator LaboratoryChile LHCStan Brodsky2009 January JLab UsersLF Holography and QCD8, 2010 Group Meeting, June 8, 2009 SLAC Page 2688HOME RESEARCH SCIENTIFIC PROGRAMS

indebted to B. Metsch and H. Petry for numerous discussionson the quark model.ReferencesE. Klempt et al.: ∆ ∗ resonances, Light-Front quark models, QCD,” International chiral symmetry School of and Subnuclear AdS/QCD1. W.M. Yao et al., J. Phys. G 33 (2006) 1 and 2007 partial11/2 +update for the 2008 edition.M 2 (GeV 2 13/2 +2. The database can be accessed)at SAID facility at the websitehttp://gwdac.phys.gwu.ed(2002) 031601.34. J. Polchinski and M. J. Strassler,5/2 + Phys. Rev. Lett. 889/28! 15/2+(2950)3. R. A. Arndt, W.J. Briscoe, I.I. Strakovsky and R.L. Workman,35. E. Klempt, Phys. Rev. C 66 ! 7/2 (2002) +(2390) 058201. 11/2 "Phys. Rev. C 74 (2006) 045205.36. Y. Nambu, Phys. Lett. B 80 ! 9/2(1979) +(2300) 372.4. S. Capstick, N. Isgur, Phys. Rev. D 34 (1986) 2809. 37. M. Baker and R. Steinke, Phys. !N=011/2+(2420) Rev. D 65 (2002) 094042.6! 1/2+(1910) ! 5/2"(2223)5/2 +7/2 "5. U. Löring, B. C. Metsch and H. R. Petry, Eur. Phys. J. A 38. M. Baker, “From QCD to dual superconductivity to effectivestring theory,” 5th International Conference on Quark! 3/2+(1920) ! 7/2"(2200)7/2 +9/2 "10 (2001) 395, 447.9/2! 5/2+(1905)11/26. L. Y. Glozman, W. Plessas, K. Varga and R. F. Wagenbrunn,Phys. Rev. D 58 (1998) 094030.cia, Italy, 10-14 Sep 2002. Published in *Gargnano 2002,Confinement and the Hadron Spectrum, Gargnano, Bres-"! 7/2+(1950)!41/2"(1620)1/2 +3/2 "11/2 + ! 13/2"(2750)7. M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson and Quark confinement and the hadron spectrum* 204-210,D.B. Lichtenberg, Rev. Mod. Phys. 65 (1993) 1199.arXiv:hep-ph/0301032.! 3/2"(1700)3/2 + ! 5/2"(2350)8. M. Kirchbach, M. Moshinsky and Yu. F. Smirnov, Phys.7/2 N=1!! 1/2"(1900) 5/2+(2200)Rev. D 64 (2001) 114005.7/2! 3/2+(1232)! 3/2"(1940)! 9/2"(2400)9. E. Santopinto, Phys. Rev. C 72 (2005) 022201.10. L.Y. Glozman, Phys. Lett. B 475 (2000) 329.11. R.L. Jaffe, 2D. Pirjol and A. Scardicchio, ! 1/2 Phys. +(1750) Rev. D 74 ! 5/2"(1930)(2006) 057901.! 3/2+(1600)12. See, e.g., L.Y. Glozman, Phys. Rev. Lett. 99 (2007)191602, and references therein.L+N13. O. Aharony, 0 S. S. Gubser, J. M. Maldacena, H. Ooguri andY. Oz, Phys. Rept. 3230(2000) 183. 1 2 3 4 5 614. E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253.15. I.R. Klebanov, “TASI lectures: Introduction to theAdS/CFT correspondence,” arXiv:hep-th/0009139.16. A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys.Rev. D 74 (2006) 015005.17. S.J. Brodsky, Eur. Phys. J. A 31 (2007) 638.2. Regge trajectory for ∆ ∗ resonances as a function of the leading intrinsic orbital angular momentum L and thetion quantum number N (corresponding to n 1 + n 2 in quark models). The line represents a prediction of the metrdS/QCD E. Klempt model [18,19]. et al.: Resonances ∆ ∗ resonances, with N = quark 0 and N models, = 1 are listed chiral above symmetry or below theand trajectory. AdS/QCD The mass pred27, 1.64, 1.92, 2.20, 2.43, 2.64, 2.84 GeV. The two states reported in [20,21] are indicated by arrows.18. H. Forkel, M. Beyer and T. Frederico, JHEP 0707 (2007)077.19. H. Forkel, M. Beyer and T. Frederico, Int. J. Mod. Phys.[17], E 16 ∆(1930)D (2007) 2794. 35 was interpreted as L = 3, S = 1/220. I. Horn et al., preceding paper.21. 2 tion.I. HornThe new evidence for ∆(1940)D 33 – whichto (GeV be a 2 et al., “Evidence for a parity doublet ∆(1920)P 33and ∆(1930)D ) Chile natural 33LHC fromspin γp → pπ partner 0 η”, arXiv:. of ∆(1930)D 35 – sug-22. D. DrechselL = 1,January and T.S = 3/2,8, Walcher, 2010 “Hadron structure at low QN = 1 quantum numbers 2 ,”for both,arXiv:0711.3396 [hep-ph].31. G. P. Lepage and S. J. Brodsky, Phys. Rev. Lett. 43 (1979)545 [Erratum-ibid. 43 (1979) 1625].32. S. J. Brodsky and G. F. de Teramond, “AdS/CFT andPhysics: 45th Course: Searching for the “Totally Unexpected”in the LHC Era, Erice, Sicily, Italy, 29 Aug - 7Sep 2007, arXiv:0802.0514 [hep-ph].33. S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 77(2008) 056007.LF Holography and QCD89mass (1.62 GeV) 1 ; the seven states (4,5) should havGeV. The predicted masses for Press and Media L + : SLAC N National = Accelerator 3 Laboratory (6,7)9/2 +Stan BrodskySLAC9/2 +11/2 +13/2 +(8,9) are 2.20 5/2 and + 2.42 GeV, respectively. The traj9/2 " ! 15/2 + (2950)continues with ! 7/2+(2390) the calculated11/2 masses " 2.64 for L +HOME RESEARCH SCIENTIFIC PROGRAMS

SU(6) S L Baryon State56123270123212561232701232125612327012320 N 1 2+(939)0 ∆ 3 2+(1232)1 N 1 21 N 1 21 ∆ 1 22 N 3 22 ∆ 1 2−(1535) N3 −(1520)−(1650) N3 −(1700) N5 −(1675)+(1910) ∆323 N 5 23 N 3 −2N 5 −23 ∆ 5 24 N 7 +24 ∆ 5 +22−(1620) ∆3 −(1700)+(1720) N5 +(1680)222+(1920) ∆5 +(1905) ∆7 +(1950)∆ 7 +25 N 9 −25 N 7 −2−N 7 22N 7 −22−(2190) N9 −(2250)−(1930) ∆7 −2N 9 −2N 9 2+(2220)∆ 9 +2N 112N 112∆ 112−(2600)−22+(2420)N 13 −2Press and Media : SLAC National Accelerator LaboratoryChile LHCStan Brodskyxploring QCD, Cambridge, August 20-24, 2007 LF Holography and QCDPage 47January 8, 2010SLAC90HOME RESEARCH SCIENTIFIC PROGRAMS

Space-Like Dirac Proton Form Factor• Consider the spin non-flip form factorsF + (Q 2 ) = g +F − (Q 2 ) = g −dζ J(Q, ζ)|ψ + (ζ)| 2 ,dζ J(Q, ζ)|ψ − (ζ)| 2 ,where the effective charges g + and g − are determined from the spin-flavor structure of the theory.• Choose the struck quark to have S z = +1/2. The two AdS solutions ψ + (ζ) and ψ − (ζ) correspondto nucleons with J z = +1/2 and −1/2.• For SU(6) spin-flavor symmetryF p 1 (Q2 ) = dζ J(Q, ζ)|ψ + (ζ)| 2 ,F1 n (Q 2 ) = − 1 dζ J(Q, ζ) |ψ + (ζ)| 2 − |ψ − (ζ)| 2 ,3where F p 1 (0) = 1, F n 1(0) = 0.Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDExploring January QCD, 8, Cambridge, 2010 August 20-24, 2007 SLAC Page 5291HOME RESEARCH SCIENTIFIC PROGRAMS

• Scaling behavior for large Q 2 : Q 4 F p 1 (Q2 ) → constant Proton τ = 3Q 4 F p 1 (Q2 ) (GeV 4 )1.20.80.49-20078757A200 10 20 30Q 2 (GeV 2 )SW model predictions for κ = 0.424 GeV. Data analysis from: M. Diehl et al. Eur. Phys. J. C 39, 1 (2005).Press and Media : SLAC National Accelerator LaboratoryChile LHCStan BrodskyLF Holography and QCDJanuary 8, 2010SLACHelmholtz Institut, Bonn, Oct 16, 2007 92Page 29HOME RESEARCH SCIENTIFIC PROGRAMS

• Scaling behavior for large Q 2 : Q 4 F n 1 (Q2 ) → constant Neutron τ = 30Q 4 F n 1 (Q2 ) (GeV 4 )-0.1-0.2-0.3-0.4 0 10 20 309-20078757A1Q 2 (GeV 2 )SW model predictions for κ = 0.424 GeV. Data analysis from M. Diehl et al. Eur. Phys. J. C 39, 1 (2005).Chile LHCJanuary 8, 2010LF Holography and QCD93Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

21.5F p 2 (Q2 )1Spacelike Pauli Form FactorFrom overlap of L = 1 and L = 0 LFWFsHarmonic OscillatorConfinementNormalized to anomalousmomentκ = 0.49 GeVPreliminaryAdS/QCD Nochiraldivergence!0.5G. de Teramond, sjbF 2 (Q 2 ) = 1 + OQ2m π m pin chiral perturbation theory0F 2 (Q 2 )0 1 2 3 4 5 6Q 2 (GeV 2 )Chile LHCJanuary 8, 2010LF Holography and QCD94Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACJADE determination of α s (M Z )HOME RESEARCH SCIENTIFIC PROGRAMS

String TheoryGoal: First Approximant to QCDCounting rules for Hard ExclusiveScatteringRegge TrajectoriesQCD at the Amplitude LevelAdS/CFTAdS/QCDMapping of Poincare’ andConformal SO(4,2) symmetries of 3+1spaceto AdS5 spaceConformal behavior at shortdistances+ Confinement at large distanceSemi-Classical QCD / Wave EquationsHolographyBoost Invariant 3+1 Light-Front Wave EquationsJ =0,1,1/2,3/2 plus LIntegrable!Hadron Spectra, Wavefunctions, DynamicsChile LHCJanuary 8, 2010LF Holography and QCD95Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Light-Front QCDHeisenberg MatrixFormulationH QCDLFL QCD → H QCDLFPhysical gauge: A + = 0= [ m2 + k⊥2 ] i + HLFintxiLF : Matrix in Fock SpaceH intH QCDLF |Ψ h >= M 2 h|Ψ h >338Eigenvalues and Eigensolutions give HadronSpectrum and Light-Front wavefunctionsFig. 6. A few selected

More generally, consider a meson in SU(N). The kernel of the integral equation (3.14)illustrated in Fig. 2 in terms of the block matrix n : x , k , λ Hn : x, k , λ . The structure of th Light-Front matrix depends QCDof course onH QCD the wayLC|Ψ one has arrangedh = M 2 h |Ψ the Fock space, see Eq. (3.7). Note that moof the block matrix elements vanish due to the naturehof the light-cone interaction as definedHeisenberg EquationS.J. Brodsky et al. / Physics Reports 301 (1998) 299—486Fig. 6. A few selected matrix elements of the QCD front form Hamiltonian H"P in LB-convention.Use AdS/QCD basis functionsPress and Media : SLAC National Accelerator LaboratoryFig. 2. The Hamiltonian matrix for a SU(N)-meson. The matrix elements are represented by energy diagrams. WithChile each LHCblock they are all of the sameStan BrodskyLF type: Holography either vertex, and fork QCD or seagull diagrams. Zero matrices are denoted by a dot (January The 8, 2010 single gluon is absent since it cannot be color neutral.SLAC97HOME RESEARCH SCIENTIFIC PROGRAMS

Use AdS/CFT orthonormal Light Front Wavefunctionsas a basis for diagonalizing the QCD LF Hamiltonian• Good initial approximation• Better than plane wave basis• DLCQ discretization -- highly successful 1+1Pauli, Hornbostel,Hiller, McCartor, sjb• Use independent HO LFWFs, remove CM motion• Similar to Shell Model calculations• Hamiltonian light-front field theory within an AdS/QCD basis.J.P. Vary, H. Honkanen, Jun Li, P. Maris, A. Harindranath,G.F. de Teramond, P. Sternberg, E.G. Ng, C. Yang, sjbChile LHCJanuary 8, 2010LF Holography and QCD98Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

New Perspectives for QCD from AdS/CFT• LFWFs: Fundamental frame-independent description ofhadrons at amplitude level• Holographic Model from AdS/CFT : Confinement at largedistances and conformal behavior at short distances• Model for LFWFs, meson and baryon spectra: manyapplications!• New basis for diagonalizing Light-Front Hamiltonian• Physics similar to MIT bag model, but covariant. No problemwith support 0 < x < 1.• Quark Interchange dominant force at short distancesChile LHCJanuary 8, 2010LF Holography and QCD99Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Non-Perturbative QCD Coupling From LF HolographyRunning Coupling from Modified AdS/QCDWith A. Deur and S. J. BrodskyDeur, de Teramond, sjb• Consider five-dim gauge fields propagating in AdS 5 space in dilaton background ϕ(z) =κ 2 z 2S = − 1 d 4 xdz √ ge ϕ(z) 1 4g52 G 2• Flow equation1g5 2(z) = 1eϕ(z)g5 2(0) or g5(z) 2 =e −κ2 z 2 g5(0)2where the coupling g 5 (z) incorporates the non-conformal dynamics of confinement• YM coupling α s (ζ) =g 2 YM (ζ)/4π is the five dim coupling up to a factor: g 5(z) → g YM (ζ)• Coupling measured at momentum scale QαsAdS (Q) ∼ ∞0ζdζJ 0 (ζQ) αsAdS (ζ)• Solutionwhere the coupling α AdSsαsAdS (Q 2 )=αs AdS (0) e −Q2 /4κ 2 .incorporates the non-conformal dynamics of confinement

Running Coupling from Light-Front Holography and AdS/QCDAnalytic, defined at all scales, IR Fixed Pointnormalization! s(Q)/"10.8α s (Q)π0.60.40.20AdS/QCDLF HolographyAdS/QCD with! s,g1extrapolation! s,g1/" (pQCD)! s,g1/" world data! s,F3/"GDH limit! s,#/" OPALJLab CLAS PLB 665 249Hall A/CLAS PLB 650 4 244Lattice QCDα AdSs (Q)/π = e −Q2 /4κ 2κ =0.54 GeV10 -1 1 10Deur, de Teramond, sjb, (preliminary)Q (GeV)

! s,g1/" Hall A/CLAS JLab ! s,g1/" CLAS JLab! s/"1FitGDH limitpQCD evol. eq.Burkert-IoffeBloch et al.10 -1CornwallGodfrey-Isgur1Fischer et al.Bhagwat et al.Maris-TandyLattice QCDAdS/CFT(norm. to ",#=0.54),10 -110 -1 1 10 -1 1 10Deur, de Teramond, sjb, (preliminary)Q (GeV)

• β-functiond! seff/dlog(Q 2 )0-0.25-0.5-0.75β AdS (Q 2 )=!(Q)0-0.25-0.5-0.75-1-1.25dd log Q 2 αAdS s (Q 2 )= πQ2 /4κ 24κ 2 e−Q2 .PRELIMINARY-1-1.25-1.5-1.75-2-2.25" s,F3Lattice QCDAdS/QCD LFHolography" s,g1Hall A/CLAS" s,g1CLASGDH sum ruleconstraint on " s,g1" s,g1(pQCD)-1.5-1.75Hadron 2009, FSU, Tallahassee, December 1, 2009-2-2.25Q !(GeV)s,F3/"10 -1 1 10Effective coupling from LF holography for κ =0.54Hall A/CLASGeVPLB650 4 244Lattice QCDJLab CLAS PLB665 249AdS/QCD withGDH sum rule! s,g1extrapolationconstraint on ! s,g1AdS/QCD LFBjorken sum ruleHolographyconstraint on ! s,g110 -1 1 10Deur, de Teramond, sjb, (preliminary)QPage

Running Coupling for Static Potential from AdS/QCD s(r)3.53Deur, de Teramond, sjb, (preliminary)2.521.510.50Chile LHCJanuary 8, 2010AdS/QCD with s,g1extrapolation s,g1Jlab+pQCD+sum rulesAdS/QCD LF HolographyLF Holography and QCD104Lattice QCD Vfrom s,g1using CSR0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r (fm)Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Applications of NonperturbativeRunning Coupling from AdS/QCD• Sivers Effect in SIDIS, Drell-Yan• Double Boer-Mulders Effect in DY• Diffractive DIS• Heavy Quark Production at ThresholdAll involve gluon exchange at smallmomentum transferChile LHCJanuary 8, 2010LF Holography and QCD105Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Single-spinasymmetriesLeading TwistSivers Effecte –Hwang,Schmidt, sjbiS p ·q ×p qPseudo- T-Odde – !*quarkcurrentquark jetfinal stateinteractionCollins, BurkardtJi, YuanQCD S- and P-Coulomb Phases--Wilson LineSprotonLight-Front WavefunctionS and P- Wavesspectatorsystem11-20018624A06Leading-TwistRescatteringViolates pQCDFactorization!Chile LHCJanuary 8, 2010LF Holography and QCD106Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Final-State Interactions ProducePseudo T-Odd (Sivers Effect)Hwang, Schmidt, sjbCollins• Leading-Twist Bjorken Scaling!• Requires nonzero orbital angular momentum of quarkiS ·p jet ×q• Arises from the interference of Final-State QCD Coulomb phases in S- and P-waves;• Wilson line effect -- gauge independent• Relate to the quark contribution to the target protonanomalous magnetic moment and final-state QCD phases• QCD phase at soft scale!e – !*quarke –currentquark jetfinal stateinteractionS• New window to QCD coupling and running gluon mass in the IRproton• QED S and P Coulomb phases infinite -- difference of phases finite!spectatorsystem11-20018624A06• Alternate: Retarded and Advanced Gauge: Augmented LFWFs Pasquini, Xiao, Yuan, sjbMulders, Boer Qiu, StermanChile LHCJanuary 8, 2010LF Holography and QCD107Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Features of Soft-Wall AdS/QCD• Single-variable frame-independent radial Schrodinger equation• Massless pion (m q =0)• Regge Trajectories: universal slope in n and L• Valid for all integer J & S. Spectrum is independent of S• Dimensional Counting Rules for Hard Exclusive Processes• Phenomenology: Space-like and Time-like Form Factors• LF Holography: LFWFs; broad distribution amplitude• No large Nc limit• Add quark masses to LF kinetic energy• Systematically improvable -- diagonalize H LF on AdS basisChile LHCJanuary 8, 2010LF Holography and QCD108Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Formation of Relativistic Anti-HydrogenChile LHCJanuary 8, 2010Measured at CERN-LEAR and FermiLab)−¯H(¯pe + )¯H(¯pe + )¯H(¯pe + )¯p¯p¯pπq → γ ∗ qe +e +e +eγ ∗+e −e −ee −LF Holography and QCD109Munger, Schmidt, sjb¯H(¯pe + )πCoulomb fieldy¯p yZCoalescence of off-shell p co-moving positron and e antiproton+< p| G3 µν e +m 2 |p > vs.Qb ⊥ ≤ 1m red αWavefunction maximal at small impact separation and equal rapidityγe −“Hadronization” ¯ at the Amplitude Level < p| G3 µνcos 2φ m 2 |p > vs.Qq¯pτ = t + z/cbτ = ⊥ ≤t + 1 z/cγm red αPress and Media : SLAC National Accelerator LaboratoryStan BrodskySLAC4 2HOME RESEARCH SCIENTIFIC PROGRAMS

Hadronization 1 − x, − k ⊥at the Amplitude LevelChile LHCJanuary 8, 2010τ = x +e +e +e −e −γ ∗ e +e +e −γ ∗e +e −γ ∗e −γ ∗g¯qqe +e −γ ∗gψpp →gp +H (x,J/ψ k ⊥ , λ+i )pg ¯qψ H (x, k ⊥ , λ i )e −¯q x, k 1 − x, − ⊥ k ⊥g¯q q1 − x, pγ ∗qe + − k ⊥HEvent amplitudee +generator ¯qq pp → p + J/ψ + pe −ψ(x, kpp → p + J/ψ + p ⊥ , λx, ke i )−⊥gqpp → p + J/ψ + γpConstruct helicity amplitude using ∗ γ ∗Light-FrontPerturbation ¯q theory; coalesce quarkse + 1 − x, − k ⊥via LFWFspp → p + J/ψ + pe +qe −p HLF Holography and QCD110ψ H (x, k ⊥ , λ i )p Hx, k ⊥e −Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Features of LF T-Matrix Formalism“Event Amplitude Generator”• Coalesce color-singlet cluster to hadronic state ifnM 2 k⊥i 2n =+ m2 i< Λ 2 QCDx ii=1• The coalescence probability amplitude is the LFwavefunction Ψ n (x i , k ⊥i , λ i )• No IR divergences: Maximal gluon and quark wavelength fromconfinementP + , P +A(σ, ∆ ⊥ ) = 12πP + , P ⊥dζei2σx i P + , x iP⊥ + k ⊥iẑChile LHCJanuary 8, 2010L = R × PLF Holography and QCD111xP + i P + , x= P i 0 P⊥ ++ P z k ⊥ixζ =i k+ Q2Press and Media : SLAC National Accelerator LaboratoryStan Brodsky2p·q SLACP + = k0 +k 3P 0 +P zHOME RESEARCH SCIENTIFIC PROGRAMS

Features of LF T-Matrix Formalism“Event Amplitude Generator”• Same principle as antihydrogen production: off-shellcoalescence• coalescence to hadron favored at equal rapidity, smalltransverse momenta• leading heavy hadron production: D and B mesons producedat large z• hadron helicity conservation if hadron LFWF has L z =0• Baryon AdS/QCD LFWF has aligned and anti-aligned quarkspinA(σ, ∆ ⊥ ) = 1P + , P +P + , P ⊥2πdζei2σx i P + , x iP⊥ + k ⊥iẑChile LHCJanuary 8, 2010L = R × PLF Holography and QCD112xP + i P + , x= P i 0 P⊥ ++ P z k ⊥ixζ =i k+ Q2Press and Media : SLAC National Accelerator LaboratoryStan Brodsky2p·q SLACP + = k0 +k 3P 0 +P zHOME RESEARCH SCIENTIFIC PROGRAMS

Chiral Symmetry Breaking in AdS/QCD• Chiral symmetry breaking effect in AdS/QCDdepends on weighted z 2 distribution, notconstant condensateδM 2 = −2m q < ¯ψψ > ×dz φ 2 (z)z 2• z 2 weighting consistent with higher Fock statesat periphery of hadron wavefunction• AdS/QCD: confined condensate• Suggests “In-Hadron” CondensatesErlich etal.•Chile LHCJanuary 8, 2010LF Holography and QCD113de Teramond, Shrock, sjbPress and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Modern Physics Letters AVol. 23, Nos. 17–20 (2008) 1336–1345c○ World Scientific Publishing Company“One of the gravest puzzles oftheoretical physics”DARK ENERGY ANDTHE COSMOLOGICAL CONSTANT PARADOXA. ZEEDepartment of Physics, University of California, Santa Barbara, CA 93106, USAKavil Institute for Theoretical Physics, University of California,Santa Barbara, CA 93106, USAzee@kitp.ucsb.edu1.I give a brief and idiosyncratic overview of the cosmological constant paradox.(Ω Λ ) QCD ∼ 10 45Ω Λ = 0.76(expt)Gravity knows about everything, whatever its origin, luminous or dark, even theenergy contained in fluctuating quantum fields.As is well known, this leads us to one of the gravest puzzles of theoreticalphysics. Consider the Feynman diagram(Ω Λ ) EW ∼ 10 56with the graviton coupling to a matterfield (for example an electron field) loop. If we claim to understand the physicsof the electron field up to an energy scale of M, then the graviton sees an energydensity given schematically by Λ ∼ M 4 + M 2 m 2 e log( M m e) + m 4 e log( M m e) + · · · . Justabout any reasonable choice of M leads to a humongous energy density!!! In fact,even if the first two terms were to be mysteriously deleted, there is still an energydensity of order m 4 e , that is, an energy density corresponding to one electron massin a volume the size of the Compton wavelength of the electron, filling all of space,which is clearly unacceptable.Apparently, this disastrous prediction of quantum field theory has nothing towithin hadrons, not LF vacuumdo with quantum gravity. Indeed, the quantum field theory we need for the matterfield is merely free field theory: we are just adding up zero point energy of harmonicoscillators.QCD Problem Solved if Quark and Gluon condensates resideShrock, sjb

Light-Front(Front Form)Formalism

Maximum wavelength of bound quarks and gluonsk > 1Λ QCDgλ < Λ QCDq¯bB-MesonShrock, sjbUse Dyson-Schwinger Equation for bound-state quark propagator:find confined condensate< ¯b|¯qq|¯b > not < 0|¯qq|0 >Chile LHCJanuary 8, 2010LF Holography and QCD116Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

5[iE π (q; P ) + γ · P F π (q; P ) + γ · qq · P G π (q; p) + σ µ νq µ P ν H π (q; P )] (3Pion mass and decay constant.Pieter Maris, Craig D. Roberts (Argonne, PHY) , Peter C. Tandy (Kent State U.) . ANL-PHY-8753-TH-97,KSUCNR-103-97, Jul 1997. 12pp.Published in Phys.Lett.B420:267-273,1998.e-Print: nucl-th/9707003 d 4 qZ 2 (ζ, Λ)N c tr D Λ(2π) 4 γ 5γ µ S(q + P/2)Γ π (q; P )S(q − P/2), (4Pi- and K meson Bethe-Salpeter amplitudes.Pieter Maris, Craig D. Roberts Λ (Argonne, d 4 q PHY) . ANL-PHY-8788-TH-97, Aug 1997. 34pp.Published Z 4 (ζ, Λ)N in Phys.Rev.C56:3369-3383,1997.c tr De-Print: nucl-th/9708029(2π) 4 S0 (q) . (5Concerning the quark condensate.K. Langfeld (Tubingen U.) , H. Markum (Vienna, Tech. U.) , R. Pullirsch (Regensburg U.) , C.D. Roberts(Argonne, PHY & Rostock U.) , S.M. Schmidt (Tubingen U. & HGF, Bonn) . ANL-PHY-10460-TH-2002,MPG-VT-UR-239-02, Jan 2003. 7pp.97) Published 3369; in Phys.Rev.C67:065206,2003.[arXiv:nucl-th/9708029].e-Print: nucl-th/0301024Maris and C. D. Roberts, “π and K meson Bethe-Salpeter amplitudes,”. [2] defines“In-Meson the in-mesonCondensate”condensate:Valid even for m q → 0− ¯qq π ζ = f π 0|¯qγ 5 q|π . (6f π nonzeroway to proving Eq. (1) herein is a relation of the Goldberger-Treiman typePress and Media : SLAC National Accelerator LaboratoryitChile LHCJanuary 8, 2010LF Holography and QCD1170 2Stan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

de Teramond, SjbIn presence of quark masses the Holographic LF wave equation is (ζ = z)− d2dζ 2 + V (ζ) + X2 (ζ)ζ 2 φ(ζ) = M 2 φ(ζ), (1)and thusδM 2 = X2ζ 2 . (2)The parameter a is determined by the Weisberger terma = 2 √ x.ThusX(z) = m √ xz − √ x ¯ψψz 3 , (3)andδM 2 = i m2ix i− 2 im i ¯ψψz 2 + ¯ψψ 2 z 4 , (4)where we have used the sum over fractional longitudinal momentum i x i = 1.Mass shift from dynamics inside hadronic boundaryChile LHCJanuary 8, 2010LF Holography and QCD118Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Quark and Gluon condensates reside withinhadrons, not LF vacuum• Bound-State Dyson-Schwinger Equations• Spontaneous Chiral Symmetry Breaking within infinitecomponentLFWFsMaris, Roberts,TandyCasherSusskind• Finite size phase transition - infinite # Fock constituents• AdS/QCD Description -- CSB is in-hadron Effect• Analogous to finite-size superconductor!• Phase change observed at RHIC within a single-nucleus-nucleuscollisions-- quark gluon plasma!• Implications for cosmological constant -- reduction by 45 orders ofmagnitude!Shrock, sjbChile LHCJanuary 8, 2010“Confined QCD Condensates”LF Holography and QCD119Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

• Color Confinement: Maximum Wavelength of Quarkand Gluons• Conformal symmetry of QCD coupling in IR• Conformal Template (BLM, CSR, ...)• Motivation for AdS/QCD• QCD Condensates inside of hadronic LFWFs• Technicolor: confined condensates inside oftechnihadrons -- alternative to Higgs• Simple physical solution to cosmological constantconflict with Standard ModelShrock and sjbChile LHCJanuary 8, 2010LF Holography and QCD120Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Features of AdS/QCD LF Holography• Based on Conformal Scaling of Infrared QCD Fixed Point• Conformal template: Use isometries of AdS5• Interpolating operator of hadrons based on twist, superfield dimensions• Finite Nc = 3: Baryons built on 3 quarks -- Large Nc limit not required• Break Conformal symmetry with dilaton• Dilaton introduces confinement -- positive exponent• Origin of Linear and HO potentials: Stochastic arguments (Glazek);General ‘classical’ potential for Dirac Equation (Hoyer)• Effective Charge from AdS/QCD at all scales• Conformal Dimensional Counting Rules for Hard Exclusive Processes• Use CRF (LF Constituent Rest Frame) to reconstruct 3D Image ofHadrons (Glazek, de Teramond, sjb)Chile LHCJanuary 8, 2010LF Holography and QCD121Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS

Conformal Template• BLM scale-setting: Retain conformal series; nonzero β-terms set multiplerenormalization scales. No renormalization scale ambiguity. Result isscheme-independent.• Commensurate Scale Relations between observables based on conformaltemplate -- prime tests of QCD independent of theory conventions• BLM scales for three-gluon coupling; multi-scale problems• Analytic scheme for coupling unification• Direct high p T processes, baryon anomaly, intrinsic heavy quarks• IR Fixed point -- conformal symmetry motivation for AdS/CFT• Light-Front Schrödinger Equation: analytic first approximation to QCD• Dilaton-modified AdS 5 : Predict Hadron Spectra, Light-Front Wavefunctions,Form Factors, Hadronization at Amplitude Level• Non-Perturbative running QCD coupling -- new range of QCD phenomenasuch as Sivers and Diffractive DIS fraction calculable; IR fixed pointChile LHCJanuary 8, 2010LF Holography and QCD122Press and Media : SLAC National Accelerator LaboratoryStan BrodskySLACHOME RESEARCH SCIENTIFIC PROGRAMS