AdS/QCD and Hadronic Phenomena

Transcription

1 and Hadronic Phenomena, November 16, 2007 Stan Brodsky SLAC, Stanford University Research Associate ! 1

2 QCD Lagrangian Yang-Mills Gauge Principle: Invariance under Color Rotation and Phase Change at Every Point of Space and Time Dimensionless Coupling Renormalizable Asymptotic Freedom Color Con nement November 16,

3 Although we know the QCD Lagrangian, we have only begun to understand its remarkable properties and features. Novel QCD Phenomena: hidden color, color transparency, strangeness asymmetry, intrinsic charm, anomalous heavy quark phenomena, anomalous spin e ects, single spin asymmetries, odderon, di ractive deep inelastic scattering, dangling gluons, shadowing, antishadowing, QGP, CGC,... Truth is stranger than fiction, but it is because Fiction is obliged to stick to possibilities. Mark Twain November 16,

4 Goal: Use AdS/CFT to provide an approximate, covariant, and analytic model of hadron structure with con nement at large distances, conformal behavior at short distances Analogous to the Schrodinger Theory for Atomic Physics Holographic Model 4

5 Applications of AdS/CFT to QCD Changes in physical length scale mapped to evolution in the 5th dimension z in collaboration with Guy de Teramond November 16,

6 P.A.M Dirac, Rev. Mod. Phys. 21, 392 (1949) Evolve in ordinary time Dirac s Amazing Idea: The Front Form σ = ct z Evolve in light front time! τ = t + z/c November 16, 2007 Instant Form 6 Front Form 6

7 Light Front Wavefunctions: rigorous representation of composite systems in quantum eld theory x = k+ P + = k0 + k 3 P 0 + P 3 Fixed τ = t + z/c x i P +,x ip + k i P +, P Ψ n (x i, k i,λ i ) ni x i =1 Invariant under boosts! Independent of P ni k i = 0 November 16,

8 Tis a mistake / Time flies not It only hovers on the wing Once born the moment dies not tis an immortal thing...a moment standing still for ever. James Montgomery 1833 November 16,

9 Light-Front Dynamics and Mapping AdS Modes Light-Front Wave Functions in QCD Hadronic bound state expanded in n-particle Fock eigenstates ψ h = n ψ n/h ψ h of the LF Hamiltonian H LF = P 2 = P + P P 2, H LF P = M 2 P, at fixed LF time τ = t + z/c (Dirac 49; Pauli and Pinsky, sjb Phys. Rept. 1988). Fock components ψ n/h (x i, k i )= n; x i, k i, ψh (P +, P ), frame independent and encode hadron properties in high momentum-transfer collisions. Momentum fraction x i = k + i /P + and k i are the relative coordinates of parton i in Fock-state n n x i =1 i=1 n k i =0. i=1 Define transverse position coordinates x i r i = x i R + b i n n b i =0, x i r i = R. i=1 i=1 November 16,

10 Light Front QCD Heisenberg Matrix Formulation H QCD LC Ψ h = M 2 h Ψ h DLCQ Discretized Light Cone Quantization Eigenvalues and Eigensolutions give Hadron Spectrum and Light Front wavefunctions Hans Christian Pauli & sjb 10

11 LIGHT-FRONT SCHRODINGER EQUATION A + =0 November 16, G. P. Lepage, sjb 11

12 Angular Momentum on the Light-Front J z = n i=1 n 1 si z + j=1 l z j. Conserved LF Fock state by Fock State l z j = i( k 1 j k 2 j k 2 j k 1 j ) n 1 orbital angular momenta Nonzero Anomalous Moment -->Nonzero orbital angular momentum November 16,

13 B-Decays Ψ n (x i, k i,λ i ) LFWFs GPDs Distribution Amplitudes November 16,

14 Calculation of Form Factors in Equal-Time Theory Need vacuum uctuations Calculation of Form Factors in Light-Front Theory zero for q + = 0 zero!! November 16,

15 F 2 (q 2 ) 2M = a e j 1 2 Drell, sjb [dx][d 2 k ] j [ 1 q L ψ a (x i, k i,λ i ) ψa(x i, k i,λ i )+ 1 q R ψ a (x i, k i,λ i ) ψa(x i, k i,λ i ) ] k i = k i x i q k j = k j +(1 x j )q q R,L = q x ± iq y Must have Δl z = ±1 to have nonzero F 2 (q 2 ) November 16,

16 Anomalous gravitomagnetic moment B(0) Okun, Kobzarev, Teryaev: B(0) Must vanish because of Equivalence Theorem graviton sum over constituents Hwang, Schmidt, sjb; Holstein et al B(0) = 0 Each Fock State November 16,

17 Angular Momentum on the Light-Front J z = n i=1 n 1 si z + j=1 l z j. Conserved LF Fock state by Fock State l z j = i( k 1 j k 2 j k 2 j k 1 j ) n 1 orbital angular momenta Nonzero Anomalous Moment -->Nonzero orbital angular momentum November 16,

18 p,s z >= n (x i, k i, i ) n; k i, i > n=3 sum over states with n=3, 4,...constituents The Light Front Fock State Wavefunctions n (x i, k i, i ) are boost invariant; they are independent of the hadron s energy and momentum P μ. The light-cone momentum fraction are boost invariant. x i = k+ i p + = k0 i + kz i P 0 + P z n i k + i = P +, November 16, 2007 n i x i = 1, Intrinsic heavy quarks Mueller: BFKL DYNAMICS n i k i = 0. ū(x) d(x) s(x) s(x) 18 Fixed LF time 18

19 Light Antiquark Flavor Asymmetry Naïve Assumption from gluon splitting: E866/NuSea (Drell-Yan) 19

20 uudc c > Fluctuation in Proton QCD: Probability 2 QCD MQ 2 e + e l + l > Fluctuation in Positronium QED: Probability (m e ) 4 M 4 l OPE derivation - M.Polyakov et al. <p G3 μν m 2 p >vs. <p F μν 4 Q m 4 p > l c c in Color Octet Distribution peaks at equal rapidity (velocity) Therefore heavy particles carry the largest momentum fractions High x charm! ˆx i = m i nj m j Hoyer, Peterson, Sakai, sjb November 16,

21 Hoyer, Peterson, Sakai, sjb Intrinsic Heavy-Quark Fock States Rigorous prediction of QCD, OPE Color Octet + Color Octet Fock State! Probability P Q Q 1 M 2 Q P Q QQ Q α2 s P Q Q P c c/p 1% Large E ect at high x Greatly increases kinematics of colliders such as Higgs production Kopeliovich, Schmidt, So er, sjb Severely underestimated in conventional parameterizations of heavy quark distributions Pumplin, Tung Many empirical tests November 16,

22 Measure c(x) in Deep Inelastic Lepton-Proton Scattering c Hoyer, Peterson, SJB November 16,

23 Measurement of Charm Structure Function J. J. Aubert et al. [European Muon Collaboration], Production Of Charmed Particles In 250-Gev Mu+ - Iron Interactions, Nucl. Phys. B 213, 31 (1983). First Evidence for Intrinsic Charm factor of 30! DGLAP / Photon Gluon Fusion: factor of 30 too sma November 16,

24 EMC data: c(x, Q 2 ) > 30 DGLAP Q 2 = 75 GeV 2, x =0.42 High x F pp J/ψX High x F pp J/ψJ/ψX High x F pp Λ c X High x F pp Λ b X High x F pp Ξ(ccd)X (SELEX) November 16,

25 Weak Exclusive Decay D J + (0) B Exact Formula! Hwang, SJB B W D + = n B n b g d c n + + D b B c c n+2 d g n D + Annihilation amplitude needed for Lorentz Invariance Non-perturbative complication from IC Fock states! November 16,

26 Hadronization at the Amplitude Level τ = x + e + q γ q g x, k e Event amplitude generator 1 x, k p H ψ(x, k,λ i ) Construct helicity amplitude using Light Front Perturbation theory; coalesce quarks via LFWFs November 16,

27 Light-Front Wavefunctions Fixed τ = t + z/c x i P +,x i P + k i P +, P ni x i =1 P + = P 0 + P z ni k i = 0 Ψ n (x i, k i,λ i ) Invariant under boosts! Independent of P November 16,

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29 AdS/CFT Holographic Model G. de Teramond SJB ψ(σ, b ) 0 b (GeV 1 ) σ = ct z τ = t + z/c σ 4 3-dimensional photograph: meson LFWF at fixed LF Time November 16,

30 Prediction from AdS/CFT: Meson LFWF x ψ M (x, k 2 ) Soft Wall model k GeV 1.4 φ M (x, Q 0 ) de Teramond, sjb x(1 x) 1.5 November 16,

31 Conformal Theories are invariant under the Poincare and conformal transformations with M μν, P μ, D, K μ, the generators of SO(4,2) SO 4,2 has a mathematical representation on AdS5 31

32 Scale Transformations Isomorphism of SO(4, 2) of conformal QCD with the group of isometries of AdS space SO(1, 5) ds 2 = R2 invariant measure z 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. 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. November 16,

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37 Truncated AdS/CFT (Hard-Wall) model: cut-off at z 0 =1/Λ QCD breaks conformal invariance and allows 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 can be obtained (Soft-Wall Model) Karch, Katz, Son and Stephanov (2006). We will consider both holographic models November 16,

38 AdS/CFT: Anti de Sitter Space / Conformal Field Theory Maldacena: Map AdS 5 X S 5 to conformal N=4 SUSY QCD is not conformal; however, it has manifestations of a scale invariant theory: Bjorken scaling, dimensional counting for hard exclusive processes Conformal window: α s (Q 2 ) const at small Q 2 Use mathematical mapping of the conformal group SO 4,2 to AdS5 space November 16,

39 Conformal QCD Window in Exclusive Processes Does α s develop an IR fixed point? Dyson Schwinger Equation Alkofer, Fischer, LLanes-Estrada, Deur... Recent lattice simulations: evidence that α s becomes constant and is not small in the infrared Furui and Nakajima, hep-lat/ (Green dashed curve: DSE). Αs q Log_10 q GeV November 16,

40 IR Fixed-Point for QCD? Dyson-Schwinger Analysis: QCD Coupling has IR Fixed Point Evidence from Lattice Gauge Theory De ne coupling from observable: indications of IR xed point for QCD e ective charges Con ned or massive gluons: Decoupling of QCD vacuum polarization at small Q 2 Π(Q 2 ) Serber Uehling 15π α Q 2 m 2 Q 2 << 4m 2 Justi es application of AdS/CFT in strong coupling conformal window l + l November 16,