A J=0 e 2 qs 0 F (t) Local J=0 fixed pole contribution. Szczepaniak, Llanes- Estrada, sjb

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1 A J=0 e 2 qs 0 F (t) Local J=0 fixed pole contribution Szczepaniak, Llanes- Estrada, sjb Light-cone wavefunction representation of deeply virtual Compton scattering! Stanley J. Brodsky a, Markus Diehl a,, Dae Sung Hwang b a March 28,

2 H ~ H q q!"#$%&'%()*%+#,%-.+/&"0%'23%+0&'2/ DIS at!=t=0 ( x,0,0) ' q( x), % q( % x) ( x,0,0) ' ( q( x), ( q( % x) % Form factors (sum rules) ~ dx H " q # q $ dx) H ( x,!, t) q ' F ( t) Dirac f.f. q $ dx) " E ( x,!, t) # ' F 2 ( t) Pauli f.f. $ q ( x,!, t) ' G A, q ( t), $ % ~ dx E q ( x,!, t) ' G P, q ( t) H q ~ q ~ q, E, H, E ( x,, t) q! Verified using LFWFs Diehl,Hwang, sjb Quark angular momentum (Ji s sum rule) q ' % J G = J xdx H 2 2 $% " ( x,!,0) & E ( x,!,0)# q q X. Ji, Phy.Rev.Lett.78,60(997) March 28,

3 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 March 28,

4 Scale Transformations Isomorphism of SO(4, 2) of conformal QCD with the group of isometries of AdS space SO(, 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. March 28,

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9 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 March 28,

10 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_0 q GeV March 28,

11 Deur, Korsch, et al: Effective Charge from Bjorken Sum Rule Γ p n bj (Q 2 ) g A 6 [ αg s (Q 2 ) π ]! s (Q)/" JLab CLAS JLab PLB ! s,g /" world data! s,f3 /" GDH limit pqcd evol. eq ! s,# /" OPAL 0 - March 28, Q (GeV) 36

12 Deur, Korsch, et al.! s /"! s,g /" JLab Fit GDH limit pqcd evol. eq. Burkert-Ioffe Bloch et al. 0 - Cornwall Godfrey-Isgur Bhagwat et al. Maris-Tandy Lattice QCD Fisher Fischer et et al. al March 28, Q (GeV) 37

13 IR Fixed-Point for QCD? Dyson-Schwinger Analysis: Fixed Point Evidence from Lattice Gauge Theory QCD Coupling has IR Define coupling from observable: indications of IR fixed point for QCD effective charges Confined gluons and quarks have maximum wavelength: Decoupling of QCD vacuum polarization at small Q 2 Π(Q 2 ) 5π α Q 2 m 2 Serber-Uehling Q 2 << 4m 2 Justifies application of AdS/CFT in strongcoupling conformal window l + l March 28,

14 Constituent Counting Rules A C dσ dt (s, t) = F (θ cm) s [n tot 2] s = E 2 cm B D F H (Q 2 ) [ Q 2]n H n tot = n A + n B + n C + n D Fixed t/s or cos θ cm Farrar & sjb; Matveev, Muradyan, Tavkhelidze Conformal symmetry and PQCD predict leading-twist scaling behavior of fixed-cm angle exclusive amplitudes Characteristic scale of QCD: 300 MeV Many new J-PARC, GSI, J-Lab, Belle, Babar tests March 28,

15 Q 4 F p (Q2 ) [GeV 4 ] F (Q 2 ) [ /Q 2] n, n = Q 2 [GeV 2 ] From: M. Diehl et al. Eur. Phys. J. C 39, (2005). Phenomenological success of dimensional scaling laws for exclusive processes dσ/dt /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 energies Farrar and sjb (973); Matveev et al. (973). Derivation of counting rules for gauge theories with mass gap dual to string theories in warped space (hard behavior instead of soft behavior characteristic of strings) Polchinski and Strassler (200). March 28,

16 Conformal behavior: Q 2 F π (Q 2 ) const Q 2 F! (GeV/c) CERN!-e scattering DESY (Ackermann) DESY (Brauel) JLab (Tadevosyan) this work QCD Sum Rules (Nesterenko, 982) pqcd (Bakulev et al, 2004) BSE-DSE (Maris and Tandy, 2000) Disp. Rel. (Geshkenbein, 2000) Q 2 (GeV/c) 2 Determination of the Charged Pion Form Factor at Q2=.60 and 2.45 (GeV/c)2. By Fpi2 Collaboration (T. Horn et al.). Jul pp. e-print Archive: nucl-ex/ March 28,

17 Test of PQCD Scaling Constituent counting rules Farrar, sjb; Muradyan, Matveev, Tavkelidze s 7 dσ/dt(γp π + n) const f ixed θ CM scaling PQCD and AdS/CFT: θ cm = 90 o s n tot 2dσ dt (A + B C + D) = F A+B C+D (θ CM ) s 7dσ dt (γp π+ n) = F(θ CM ) n tot = = 9 No sign of running coupling Conformal invariance March 28,

18 IO6 IO5 IO4 O3 O2 I I I I I I III I III IO I f I I I I I E \ \\ \\\ -cl \ \ \ \\ P \\\ q YP --(p +d, 0 yp-n-a++ 0 YP--rr P a yp-tt+n A )'p-k+fl IO0 0-l lo-* 2 s(gev 2 ) r 0.0 I I I I I I 8 IO I2 I4 I6 s(gev 2 ) Conformal Invariance: dσ dt (γp MB) = F (θ cm) s 7 March 28,

19 Quark-Counting : dσ dt (pp pp) = F (θ CM) s 0 n = = 0 ± cm 2 Best Fit n = 9.7 ± 0.5 Reflects underlying conformal scale-free interactions GeV 2 Angular distribution -- quark interchange March 28,

20 Conformal symmetry: Template for QCD Take conformal symmetry as initial approximation; then correct for non-zero beta function and quark masses Eigensolutions of ERBL evolution equation for distribution amplitudes Commensurate scale relations: relate observables at corresponding scales: Generalized Crewther Relation Fix Renormalization Scale (BLM) Use AdS/CFT V. Braun et al; Frishman, Lepage, Sachrajda, sjb March 28,

21 Polchinski & Strassler: AdS/CFT builds in conformal symmetry at short distances; counting rules for form factors and hard exclusive processes; non-perturbative derivation Goal: Use AdS/CFT to provide an approximate model of hadron structure with confinement at large distances, conformal behavior at short distances de Teramond, sjb: Holographic Model: Initial semiclassical approximation to QCD. Predict light-quark hadron spectroscopy, form factors. Karch, Katz, Son, Stephanov: Linear Confinement Mapping of AdS amplitudes to 3+ Light-Front equations, wavefunctions Use AdS/CFT wavefunctions as expansion basis for diagonalizing H LF QCD ; variational methods March 28,

22 AdS/CFT Use mapping of conformal group SO(4,2) to AdS5 Scale Transformations represented by wavefunction in 5th dimension x 2 µ λ 2 x 2 µ z λz ψ(z) Hard wall model: Confinement at large distances and conformal symmetry in interior Match solutions at small z to conformal dimension of hadron wavefunction at short distances ψ(z) z at z 0 Truncated space simulates bag boundary conditions 2 ψ(z 0 ) = 0 0 < z < z 0 z 0 = Λ QCD March 28,

23 Physical AdS modes Φ P (x, z) e ip x Φ(z) are plane waves along the Poincaré coordinates with four-momentum P µ and hadronic invariant mass states P µ P µ = M 2. For small-z Φ(z) z. The scaling dimension of a normalizable string mode, is the same dimension of the interpolating operator O which creates a hadron out of the vacuum: P O = 2 + L Twist dimension of meson Φ(z) z Confinement in the 5th dimension March 28, A z 48 z 0 = Λ QCD Identify hadron by its interpolating operator at z -- > 0 de Teramond, sjb 48

24 Bosonic Solutions: Hard Wall Model Conformal metric: ds 2 = g lm dx l dx m. x l = (x µ, z), g lm ( R 2 /z 2) η lm. Action for massive scalar modes on AdS d+ : S[Φ] = 2 d d+ x g 2 [ g lm l Φ m Φ µ 2 Φ 2], g (R/z) d+. Equation of motion ( g g lm g x l 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 )z z + z 2 M 2 (µr) 2] Φ(z) = 0. Solution: Φ(z) z as z 0, Φ(x, z) = Cz d 2 J d Φ(z) = Cz d/2 J d/2 (zm) 2 d = 4 (zm), = 2 (d + ) d 2 + 4µ 2 R 2. = 2 + L (µr) 2 = L 2 4 March 28,

25 Let Φ(z) = z 3/2 φ(z) AdS Schrodinger Equation for bound state of two scalar constituents: [ d2 dz 2 + V(z) ]φ(z) = M 2 φ(z) V(z) = 4L2 4z 2 Interpret L as orbital angular momentum Derived from variation of Action in AdS 5 Hard wall model: truncated space φ(z = z 0 = Λ c ) = 0. March 28,

26 Match fa"-off at sma" z to conformal twist-dimension at short distances twist (Φ µ = 0 gauge). = 2 + L O 2+L Pseudoscalar mesons: O 3+L = ψγ 5 D {l... D lm }ψ 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) Φ(z) 3 2 z 2 Φ(z) A7 S = 0 z z 872A8 z z 0 = Λ QCD Fig: Meson orbital and radial AdS modes for Λ QCD = 0.32 GeV. March 28,

27 Higher Spin Bosonic Modes HW Each hadronic state of integer spin S 2 is dual to a normalizable string mode Φ(x, z) µ µ 2 µ S = ɛ µ µ 2 µ S e ip x Φ S (z). with four-momentum P µ and spin polarization indices along the 3+ physical coordinates. Wave equation for spin S-mode W. S. l Yi, Phys. Lett. B 448, 28 (999) [ z 2 2 z (d+ 2S)z z + z 2 M 2 (µr) 2] Φ S (z) = 0, Solution Φ(z) S = ( z R We can identify the conformal dimension: ) S Φ(z)S = Ce ip x z d 2 J d (zm) ɛ(p ) µ µ 2 µ S, 2 = 2( d + (d 2S) 2 + 4µ 2 R 2). Normalization: R d 2S Λ QCD 0 dz z d 2S Φ2 S(z) =. March 28,

28 Φ(z) Φ(z) A z A z Fig: Orbital and radial AdS modes in the hard wall model for Λ QCD = 0.32 GeV. (a) S = 0 (b) S = f 4 (2050) a 4 (2040) (GeV 2 ) 4 2 π (40) b (235) π 2 (670) ω (782) ρ (770) a 0 (450) a 2 (320) f (285) ρ (700) ρ 3 (690) ω 3 (670) ω (650) A L f 2 (270) a (260) Fig: Light meson and vector meson orbital spectrum Λ QCD = 0.32 GeV L March 28,

29 State I J P L S O π(40) qγ 5 2 τq b (235) + 0 iqγ 5 2 τq π 2 (670) qγ 5 2 (3 i j δ ij 2 ) 2 τq ρ(770) 0 q α 2 τq ω(782) 0 0 q α q a (260) + iq ( α ) 2 τq f 2 (270) iq [ 3 2 (α i j + α j i ) α δ ij ]q f (285) 0 + iq ( α )q a 2 (320) 2 + iq [ 3 2 (α i j + α j i ) α δ ij ] 2 τq a 0 (450) 0 + iq α 2 τq Tensor decomposition of total angular momentum interpolating operators O, [O] = 2 + L March 28,

30 Let Φ(z) = z 3/2 φ(z) AdS Schrodinger Equation for bound state of two scalar constituents: [ d2 dz 2 + V(z) ]φ(z) = M 2 φ(z) Hard wall model: truncated space V(z) = 4L2 4z 2 φ(z = z 0 = /Λ 0 ) = 0 Soft wall model: Harmonic oscillator confinement V(z) = 4L2 4z 2 + κ 4 z 2 Derived from variation of Action in AdS 5 March 28,