Hadronic Form Factors in Ads/QCD

Transcription

1 Hadronic in Ads/QCD A.V. Old Dominion University and Jefferson Lab in collaboration with H.R. Grigoryan (JLab & LSU) May 15, 28

2 Quark counting rules Hadronic form factors: (1/Q 2 ) nq 1 counting rules Expectation: some fundamental/easily visible reason Most natural suspect: scale invariance Implementation: hard exchange in a theory with dimensionless coupling constant QCD: (α s /Q 2 ) nq 1 Suppression: Fπ as (Q 2 ) = 2αs π s /Q 2 ] [s = 4π 2 f 2π.67 GeV 2 Looks like O(α s ) correction to VMD s F VMD π (Q 2 ) 1/(1 + Q 2 /m 2 ρ)

3 Quark counting rules Known: α s /π.1 is penalty for an extra loop Growing consensus: pqcd gives small correction dominant contribution comes from soft terms ed by GPDs F(x, Q 2 ) with exponential fall-off e Q2 g(x) for fixed x: F n (Q 2 ) 1 e (1 x)2 Q 2 /Λ 2 (1 x) 2n 1 }{{} dx f n (x) ( Λ 2 Q 2 ) n 1

4 AdS/QCD & Form factor (from Brodsky & de Teramond) F (Q 2 ) = 1/Λ Nonnormalizable mode (for large Q) dζ ζ 3 Φ P (ζ)j(q, ζ)φ P (ζ) J(Q, ζ) = ζqk 1 (ζq) K(ζQ) Normalizable modes: Φ(ζ) = Cζ 2 J L (β L,k ζλ) ζ 2 φ(ζ) F (Q 2 ) = 1/Λ dζ ζ φ P (ζ)j(q, ζ)φ P (ζ) φ(ζ) satisfies Dirichlet b.c. for z = 1/Λ φ(ζ) is nonzero for z = if L = [ φ(ζ) J (β,k )]

5 for vector mesons (Erlich et al., Da Rold & Pomarol) AdS 5 metric with hard-wall ds 2 = 1 z 2 ( ηµν dx µ dx ν dz 2), z z = 1/Λ, 5D gauge action for vector field S AdS = 1 4g5 2 d 4 x dz g Tr ( F MN F MN ) Field-strength tensor F MN = M A N N A M i[a M, A N ] A M = t a A a M, ta SU(2), a = 1, 2, 3; M, N =, 1, 2, 3, z AdS/QCD correspondence with 4D field õ(p) à µ (p, z) = õ(p) V (p, z) V (p, ɛ)

6 Two-point Function Bulk-to-boundary propagator V (p, z) satisfies ( ) 1 z z z zv (p, z) + p 2 V (p, z) = with Neumann b.c. z V (p, z ) = gauge invariant condition F µz (x, z ) = Bilinear term of the action (after integration by parts) S (2) AdS = 1 d 4 [ ] p 1 z V (p, z) 2g5 2 (2π) 4 õ (p)ãµ(p) z V (p, ɛ) 2-point function J µ J ν δ 2 S (2) AdS /δaµ δa ν z=ɛ Tensor structure d 4 x e ip x J a µ(x)j b ν() = δ ab ( η µν p µ p ν /p 2) Σ(p 2 )

7 Bound state expansion Solution for V (p, z) with Neumann b.c. V (p, z) = P z [Y (P z )J 1 (P z) J (P z )Y 1 (P z)] Two-point function Σ(p 2 ) = πp2 2g 2 5 Kneser-Sommerfeld expansion [ Y (P z) J (P z) Y ] (P z ) J (P z ) z=ɛ Y (P z )J (P z) J (P z )Y (P z) J (P z ) = 4 π n=1 J (γ,n z/z ) [J 1 (γ,n )] 2 (P 2 z 2 γ2,n )

8 Bound states Two-point function is given by sum of poles Σ(p 2 ) = 2p2 g 2 5 z2 Masses: M n = γ,n /z n=1 [J 1 (γ,n )] 2 p 2 M 2 n Positive residues f 2 n = lim p2 M 2 n f 2 n = p2 2g 2 5 ln p 2 { (p 2 M 2 n) Σ(p 2 ) } 2M 2 n g 2 5 z2 J 2 1 (γ,n) Agrees with the usual definition J a µ ρ b n = δ ab f n ɛ µ Matching with QCD result Σ QCD (p 2 ) (N c /24π 2 ) ln p 2 fixes g 5 g 2 5 = 12π 2 /N c

9 Three-Point Function Trilinear term of action calculated on V (q, z) solution: S (3) AdS = ɛ abc 2g5 2 z d 4 dz x ɛ z 3-point correlator J α a (p 1 )J β b ( p 2)J µ c (q) Dynamical part has Mercedes-Benz form W (p 1, p 2, q) Bound state expansion z ɛ dz z V (p 1, z) V (p 1, ɛ) ( µa a ν) A µ,b A ν,c V (p 2, z) V (p 2, ɛ) V (q, z) V (q, ɛ) V (p, z) V (p, ) V(p, z) = n=1 g 5 f n p 2 Mn 2 ψ n (z)

10 Wave functions of ψ type Expansion over functions (Hong, Yoon and Strassler) 2 ψ n (z) = z J 1 (γ,n ) zj 1(M n z) Obeying equation of motion with p 2 = M 2 n Satisfying ψ n () = and z ψ n (z ) = at IR boundary Normalized according to z dz z ψ n(z) 2 = 1

11 Shape of ψ-type wave functions Ψ z z z Do not look like bound state w.f. in quantum mechanics

12 EM current channel For spacelike q (with q 2 = Q 2 ) [ J (Q, z) = Qz K 1 (Qz) + I 1 (Qz) K ] (Qz ) I (Qz ) Bound-state expansion J (Q, z) = g 5 f m Q 2 + Mm 2 ψ m (z) m=1 Infinite tower of vector mesons (Son, HJS) Transition form factors F nk (Q 2 ) = z dz z J (Q, z) ψ n(z) ψ k (z)

13 Green s function formalism Green s function for equation of motion Two-point function Σ(P 2 ) = 1 g 2 5 Coupling constants G(p; z, z ) = n=1 ψ n (z)ψ n (z ) p 2 M 2 n [ [ ]] 1 1 z z z zg(p; z, z ) z,z =ɛ g 5 f n = [ 1 z zψ n (z) ] z=

14 Wave functions of φ type Introducing φ wave functions φ n (z) 1 2 M n z zψ n (z) = z J 1 (γ,n ) J (M n z) Give couplings g 5 f n /M n as their values at the origin Satisfy Dirichlet b. c. φ n (z ) = at confinement radius Are normalized by z dz z φ n (z) 2 = 1

15 Shape of φ-type wave functions φ z z z Are analogous to bound state wave functions in quantum mechanics Is it possible to write form factors in terms of φ functions!?

16 Form factors in terms of φ functions Elastic form factor F nn (Q 2 ) = z dz z J (Q, z) ψ n(z) 2 Use e.o.m. for J(Q, z) and ψ/φ connection φ n (z) = 1 M n z zψ n (z), ψ n (z) = z M n z φ n (z) F nn (Q 2 ) = z dz z J (Q, z) φ n (z) 2 Q2 z dz 2Mn 2 z J (Q, z) ψ n(z) 2 1 = 1 + Q 2 /2Mn 2 z dz z J (Q, z) φ n (z) 2

17 Three form factors for vector mesons ρ + (p 2, ɛ ) J µ EM () ρ+ (p 1, ɛ) = ɛ βɛ α [ η αβ (p µ 1 + pµ 2 ) G 1(Q 2 ) +(η µα q β η µβ q α )(G 1 (Q 2 ) + G 2 (Q 2 )) 1 M 2 qα q β (p µ 1 + pµ 2 ) G 3(Q 2 ) ] gives (also Son & Stephanov, HJS) ɛ βɛ α [ ηαβ (p 1 + p 2 ) µ + 2(η αµ q β η βµ q α ) ] F nn (Q 2 ) Prediction: G 1 (Q 2 ) = G 2 (Q 2 ) = F nn (Q 2 ); G 3 (Q 2 ) = Moments: magnetic µ = 2, quadrupole t D = 1/M 2, same result as for pointlike meson (Brodsky & Hiller)

18 +++ Form +++ component of 3-point correlator gives combination ( ) F(Q 2 ) = G 1 (Q 2 ) + Q2 Q 2M 2 G 2(Q ) 2M 2 G 3 (Q 2 ) prediction F nn (Q 2 ) = z dz z J (Q, z) φ n (z) 2 Direct analogue of diagonal bound state form factors in quantum mechanics For ρ-meson, F(Q 2 ) coincides with () helicity transition that has leading 1/Q 2 behavior in pqcd

19 Low-Q 2 behavior Free-field version K(Qz) zqk 1 (Qz) vs. full propagator [ J (Q, z) = Qz K 1 (Qz) + I 1 (Qz) K ] (Qz ) I (Qz ) K(Qz) has logarithmic branch singularity K(Qz) = 1 z2 Q 2 [ ] 1 2γ E ln(q 2 z 2 /4) + O(Q 4 ) 4 leading to incorrect infinite slope at Q 2 = J (Q, z) is analytic in Q 2 J (Q, z) = 1 z2 Q 2 ] Qz 1 [1 ln z2 + O(Q 4 ) 4 z 2 Low-Q 2 expansion for form factor F 11 (Q 2 ) Q2 Q M 2 M 4 + O(Q6 )

20 Comparison with constituent quark CQM uses plane wave approximation similar to using K(Qz) Massless quarks infinite slope at Q 2 = Constituent masses m q 3 MeV: place lowest Q-channel singularity at 2m q m ρ AdS/QCD lesson: use massless quarks and current operator with Q-channel bound states

21 Electric Radius Electric G C, magnetic G M and quadrupole G Q G C = G 1 + Q 2 /6M 2 G Q, G M = G 1 + G 2, G Q = ( 1 + Q 2 /4M 2) G 3 G 2. prediction for electric form factor ) G (n) C (Q2 ) = (1 Q2 6M 2 F nn (Q 2 ) = 1 Q2 /6M Q 2 /2Mn 2 F nn (Q 2 ) G (1) C (Q2 ) Q2 Q M 2 M 4 + O(Q6 ) Electric radius of ρ-meson r 2 ρ C =.53 fm 2 Close to (.54 fm 2 ) obtained within DSE (Bhagwat & Maris) and m 2 π limit of lattice calculations (Lasscock et al.)

22 Vector meson dominance patterns Generalized VMD representation (Son, HYS) F 11 (Q 2 ) = m=1 Overlap integrals F m, Q 2 /M 2 m F m,11 = 4 F m,11 = 4 1 1, F 11 (Q 2 ) = Relation between two VMD patterns m=1 F m, Q 2 /M 2 m dξ ξ 2 J 1(γ,m ξ) J 2 1 (γ,1 ξ) γ,m J 2 1 (γ,m)j 2 1 (γ,1) dξ ξ 2 J 1(γ,m ξ) J 2 (γ,1 ξ) γ,m J 2 1 (γ,m)j 2 1 (γ,1) F m,11 F m,11 = 1 Mm/2M 2 1 2

23 Low-Q 2 Sum Rules Charge Sum Rules F m,11 = 1, F m,11 = 1 m=1 m=1 For F 11 : 1 = For F 11 : 1 = Slopes are given by sums F m,11 /Mm 2 or F m,11 /Mm 2 m=1 m=1 For F 11 : = For F 11 :.692 = Two-resonance dominance (also HYS)

24 Large-Q 2 Sum Rule Asymptotically F 11 (Q 2 ) is suppressed by 1/Q 2 compared to F 11 (Q 2 ) since F 11 (Q 2 ) = F 11(Q 2 ) 1 + Q 2 /2M 2 1 Known: F 11 (Q 2 ) M 2 /Q 2 (SJB/GdT, A.R.) Superconvergence/conspiracy (HYS) relation m=1 M 2 m M 2 1 F m,11 = Numerically: = (agrees with HYS) Again, sum rule is saturated by first two states

25 Large-Q 2 behavior of F 11 (Q 2 ) Asymptotic normalization of F 11 (Q 2 ) given by 1 Q 2 m=1 M 2 mf m,11 A M 2 1 Q 2 with A = (A.R.) strongly exceeds naïve VMD expectation A = 1 Numerically: = , result is dominated by second bound state Sum rule MmF 2 m,11 = φ 1 () 2 dχ χ 2 K 1 (χ) = 2 φ 1 () 2 m=1

26 Take with z 2 barrier (Karch et al.) Equation for bulk-to-boundary propagator V (p, z) [ ] 1 z 2 z z e κ2 z V + p 2 1 z 2 z e κ2 V = Solution normalized to 1 for z = : 1 [ V(p, z) = a dx x a 1 exp x ] 1 x κ2 z 2, where a = p 2 /4κ 2. Integrating by parts: 1 [ V(p, z) = κ 2 z 2 dx (1 x) 2 xa exp x ] 1 x κ2 z 2

27 ψ Wave Functions Generating function for Laguerre polynomials L 1 n(κ 2 z 2 ) [ 1 (1 x) 2 exp x ] 1 x κ2 z 2 = L 1 n(κ 2 z 2 ) x n n= Representation analytically continuable for a < V(p, z) = κ 2 z 2 n= L 1 n(κ 2 z 2 ) a + n + 1 = g 5 f n Mn 2 p 2 ψ n(z) n= has poles at locations p 2 = 4(n + 1)κ 2 M 2 n ψ wave functions ψ n (z) = z 2 2 n + 1 L1 n(κ 2 z 2 )

28 φ Wave Functions Coupling constants g 5 f n = 1 z 2 z e κ2 z ψ n (z) = 8(n + 1)κ 2 z=ɛ φ wave functions φ n (z) = 1 2 M n z e κ z 2 z ψ n (z) = 2 e κ 2 z 2 L M n(κ 2 z 2 ) n φ (z) = 2e κ2 z 2, φ 1 (z) = 2e κ2 z 2 (1 κ 2 z 2 ) Inverse relation between ψ and φ wave functions ψ n (z) = 1 ze κ 2 z 2 z φ n (z) M n

29 Normalization conditions (with κ = 1) Elastic form factors F nn (Q 2 ) = dz z e z2 ψ m (z) ψ n (z) = δ mn dz z e z2 φ m (z) φ n (z) = δ mn dz 2J z e z (Q, z) ψ n (z) 2 1 = 1 + Q 2 /2Mn 2 dz z e z2 J (Q, z) φ n (z) Q 2 /2Mn 2 F nn (Q 2 )

30 ρ-meson Dominance Form factor of the lowest state F (Q 2 ) = 2 dz z e z2 J (Q, z) Using representation for J (Q, z) with a = Q 2 /4 gives F (Q 2 ) = a = Q 2 /M 2 Exact vector dominance is due to overlap integral F m, 2 dz z 3 e z2 L 1 m(z 2 ) = δ m For F (Q 2 ) two states contribute F (Q 2 ) = 1 (1 + a)(1 + a/2) = Q 2 /M Q 2 /M1 2

31 Electric Radius in Model Small-Q 2 expansion for electric form factor [ G C(Q 2 ) = 1 5 Q 2 ] 3 M Q4 M 4 + O(Q 6 ) Electric radius of ρ-meson r 2 ρ C =.655 fm 2 Larger than in hard-wall (.53 fm 2 ) For higher excitations, the slope d dq 2 F nn(q 2 ) increases logarithmically Q2 = S n M 2 S n ln (n + 1) (n + 1) While size increases linearly r 2 n n/m 2 ρ

32 Large-Q 2 behavior Large-Q 2 behavior of F form factor F nn (Q 2 ) = dz z J (Q, z) Φ n (z)φ n (z) follows from J (Q, z) zqk 1 (zq) K(zQ) as Q and K(zQ) e zq, so that only z 1/Q work F nn (Q 2 ) Φ2 n() Q 2 dχ χ 2 K 1 (χ) = 2 Φ2 n() Q 2 In hard-wall : Φ H 2Mρ () = γ,1 J 1 (γ,1 ) M ρ Fρ H (Q 2 ) 2.56m2 ρ Q 2 In soft-wall : Φ S () = M ρ 2.77 M ρ F S ρ (Q 2 ) m2 ρ Q 2

33 Action including χsb Full action of hard-wall z [ 1 SAdS B = Tr d 4 x dz z 3 (DM X) (D M X) + 3 z 5 X X 1 ] 8g5 2z (BMN (L) B (L)MN + B(R) MN B (R)MN ) DX = X ib (L) X + ixb (R), B (L,R) = V ± A, X(x, z) = v(z)u(x, z)/2, Chiral field: U(x, z) = exp [2it a π a (x, z)], t a = σ a /2 Pion field: π a (x, z) v(z) = (m q z + σz 3 ) with m q quark mass, σ condensate Longitudinal component of axial field A a M (x, z) = M ψ a (x, z) gives another pion field ψ a (x, z)

34 Pion wave function Ψ Model satisfies Gell-Mann Oakes Renner relation m 2 π m q Chiral limit m q =, then π(z) = 1 Analytic result for Ψ(z) ψ(z) π(z) (Da Rold & Pomarol) [ ( α ) 1/3 ( Ψ(z) = z Γ (2/3) I 1/3 αz 3 ) ( I 1/3 αz 3 ) I ( ) ] 2/3 αz 3 2 I 2/3 (αz 3) where α = g 5 σ/3 Ψ(z) satisfies Ψ() = 1, Neumann b.c. Ψ (z ) = and fπ 2 = 1 ( ) 1 g5 2 z zψ(z) z=ɛ Ψ(z) ψ(ζ, a) ζ z/z a αz 3 Ψ Ζ, a Ζ a = a = 1 a = 2.26 a = 5 a = 1

35 Pion wave function Φ Conjugate wave function Φ(z) = 1 g 2 5 f 2 π ( 1 z zψ(z) As usual, s = 4π 2 f 2 π.67 GeV 2 ) = 2 ( ) 1 s z zψ(z) Analytic result for Φ(z) φ(ζ, a) φ(ζ, a) = 3 [ ] [ ζ2 2 a 4/3 ( g5 2f π 2 Γ I 3 2 1/3 2/3 aζ 3 ) ( + I 2/3 aζ 3 ) I ] 2/3 (a) I 2/3 (a) Φ(z) satisfies Φ() = 1 and Dirichlet b.c. Φ (z ) = Φ(z) φ(ζ, a) ζ z/z a αz 3 φ Ζ, a Ζ a = a = 1 a = 2.26 a = 5 a = 1

36 Parameters of z is fixed through ρ-meson mass: z = z ρ = (323 MeV) 1 From Φ() = 1, it follows that g5f 2 π 2 1/3 Γ(2/3) = 3 2 Γ(1/3) I 2/3 ( αz 3 ) I 2/3 (αz 3 ) α2/3 Experimental f π is obtained for α = (424 MeV) 3 Then a αz 3 equals 2.26 a Note: I 2/3 (a)/i 2/3 (a) 1 for a 1 value of f π is basically determined by α alone

37 In terms of Ψ(z): F π (Q 2 ) = 1 [ z ( z ) ] 2 Ψ g5 2f π 2 dz z J (Q, z) + g2 5v 2 z z 4 Ψ2 (z) Normalization can be checked from z ) F π (Q 2 ) = dz J (Q, z) z (Ψ(z) Φ(z) that gives F π () = z ) dz z (Ψ(z) Φ(z) = Ψ() Φ() = 1

38 Pion Charge Radius Pion charge radius for small α is determined by z rπ 2 = 4 } 3 z2 {1 a2 4 + O(a4 ) Pion charge radius for large a αz 3 rπ 2 a 2 Γ(1/3) 2 4/3 Γ(2/3) ( ) 2/3 [ ( a ) ] α 3 ln.566 a-dependence in fm 2 for z = z ρ 2 r Π a Experimentally r 2 π.45 fm 2 Model value r 2 π a= fm 2 is too small

39 Pion Charge Radius In terms of f π : rπ 2 = 3 a 2 2π 2 fπ ( ) αz 3 2π 2 fπ 2 ln 2.54 Compare to Nambu-Jona-Lasinio rπ 2 NJL = 3 2π 2 fπ }{{ ( ) m 2 8π } 2 fπ 2 ln σ m 2 π }{{}.34fm 2.11fm 2

40 at Large Q 2 Form factor in terms of Ψ(z) and Φ(z): z F π (Q 2 ) = dz z J (Q, z) [g 25f 2πΦ ] 2 (z) + 9α2 g5 2f π 2 z 2 Ψ 2 (z) Q 2 F Π Q Q2 Total (in GeV 2 ) Φ 2 term Ψ 2 term For large Q, only z 1/Q work: F π (Q 2 ) 2 g2 5f 2 πφ 2 () Q 2 = 4π2 f 2 π Q 2 s Q 2

41 Monopole interpolation: F mono π (Q 2 ) = Comparison with experiment F Π Q 2 F mono Π Q Q 2 /s Q pqcd: Fπ pqcd (Q 2 ) 2α s π s Q 2.2 F AdS/QCD π (Q 2 )

42 π γ γ form factor π, p T {J µ EM (x) J EM()} ν e iq1x d 4 x = ɛ µναβ ( q 1 α q 2 β F γ γ π Q 2 1, Q 2 ) 2 p = q 1 + q 2 and q 2 1,2 = Q 2 1,2 For real photons in QCD is fixed by axial anomaly c F γ γ π(, ) = N 12π 2 f π

43 Extending AdS/QCD Model Need to have isoscalar fields gauging U(2) L U(2) R B µ = t a B a µ + 1 4D currents correspond to following 5D gauge fields Need Chern-Simons term S (3) CS [B] = k N c 48π 2 ɛµνρσ Tr ˆB µ 2 J A,a µ (x) A a µ(x, z) J {I=} µ (x) ˆV µ (x, z) J {I=1},a µ (x) V a µ (x, z) ] d 4 x dz ( z B µ ) [F νρ B σ + B ν F ρσ (axial gauge B z = )

44 Chern-Simons term Take SCS AdS[B L, B R ] = S (3) CS [B L] S (3) CS [B R] with B L,R = V ± A, and keep only A a σ = σψ a S AdS CS = k N c π 2 ɛµνρσ z { [ d 4 x ψ a ( ρ Vµ a ) ( σ ˆVν )] dz ( z ψ a ) ( ρ Vµ a ) ( ) } σ ˆVν z=z After adding compensating surface term S anom = k N z c 8π 2 ɛµνρσ d 4 x dz ( z ψ a ) ( ρ Vµ a ) ( ) σ ˆVν Structure similar to πωρ interaction term L πωρ = N cg 2 8π 2 f π ɛ µναβ π a ( µ ρ a ν) ( α ω β ), (1) in hidden local symmetries approach (cf. Fujiwara et al., Meissner)

45 Conforming to QCD Anomaly Bare form factor K AdS bare(q 2 1, Q 2 2) = k 2 z J (Q 1, z)j (Q 2, z) z ψ(z) dz QCD axial anomaly corresponds to K QCD (, ) = 1 Calculation for bare form factor gives K bare (, ) = k 2 z On IR boundary z = z : z ψ(z) dz = k 2 ψ(z ) = k 2 [ ] 1 Ψ(z ) Ψ(z, a) = 3 Γ (2/3) πi 2/3 (a) Artifact of hard-wall b.c. ( ) 1/3 1 2a 2 Ψ(z, a = 2.26) =.14

46 Model for Form Take k = 2 and add surface term K(Q 2 1, Q 2 2) = Ψ(z )J (Q 1, z )J (Q 2, z ) z J (Q 1, z)j (Q 2, z) z Ψ(z) dz Extra term rapidly decreases with Q 1 and/or Q 2 J (Q, z ) = 1 I (Qz ) Effects of fixing Ψ(z ) artifact are wiped out for large Q 1,2

47 One real and one highly virtual photon For large Q 1 and/or Q 2 K(Q 2 1, Q 2 2) s 2 One real photon: In pqcd: z K(, Q 2 ) Φ()s 2Q 2 J (Q 1, z)j (Q 2, z) Φ(z) z dz dχ χ 2 K 1 (χ) = s Q 2 K pqcd (, Q 2 ) = s 1 ϕ π (x) 3Q 2 dx s x 3Q 2 Iϕ Coincides with AdS/QCD if I ϕ = 3, e.g., for ϕ π (x) = 6x(1 x) (asymptotic DA)

48 Equal large photon virtualities For large Q 1 and/or Q 2 K(Q 2 1, Q 2 2) s 2 z Equal photon virtualities: K(Q 2, Q 2 ) Φ()s Q 2 J (Q 1, z)j (Q 2, z) Φ(z) z dz pqcd result does not depend on pion DA dχ χ 3 [K 1 (χ)] 2 = s 3Q 2 K pqcd (Q 2, Q 2 ) = s 3 1 and coincides with AdS/QCD! ϕ π (x) dx xq 2 + (1 x)q 2 = s 3Q 2

49 Non-equal large photon virtualities Take Q 2 1 = (1 + ω)q 2 and Q 2 2 = (1 ω)q 2 Leading-order pqcd gives in this case K pqcd (Q 2 1, Q 2 2) = s 1 ϕ π (x) dx 3Q ω(2x 1) s 3Q 2 Iϕ (ω) AdS/QCD gives Φ()s 1 ω 2Q 2 2 = ( s 3Q 2 ) { 3 4ω 3 dχ χ 3 K 1 (χ 1 + ω)k 1 (χ 1 ω) [ ( )]} 1 ω 2ω (1 ω 2 ) ln 1 + ω {...} coincides with pqcd I ϕ (ω) for ϕ(x) = 6x(1 x)

50 Comparison with data Brodsky-Lepage interpolation K BL (, Q 2 ) = Q 2 /s (2) Our (red) is very close to BL interpolation (blue) Q 2 K,Q 2 GeV Q2 GeV 2 CLEO data represented by black dash-dotted line NLO pqcd fits data. Fits give DA s with I ϕ 3

51 Form factors of vector mesons: 1 Charge radius agrees with DSE and lattice calculations 2 Lessons for constituent quark s 3 () light-front helicity form factor F(Q 2 ) indeed behaves like 1/Q 2 for large Q 2 4 Existence of GVMD representation has nothing to do with asymptotic large-q 2 behavior 5 Exact ρ-dominance for F(Q 2 ) in soft-wall Pion form factor 1 Charge radius too small compared to experimental 2 Large-Q 2 asymptotics is s /Q 2 vs. pqcd (2α s /π)s /Q 2 3 Overshoots data: AdS/QCD pion is too small again amplitude 1 Extension to U(2) L U(2) R and Chern-Simons term 2 Fixing normalization by conforming to QCD anomaly 3 Large-Q 2 behavior coincides with pqcd calculations for asymptotic pion DA