Scattering Vector Mesons in D4/D8 model

Scattering Vector Mesons in D4/D8 model Marcus A. C. Torres C. A. Ballon Bayona H. Boschi-Filho N. Braga Instituto de Física Universidade Federal do Rio de Janeiro Light-Cone 2009: Relativistic Hadronic and Particle Physics

Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

Gauge/String duality vs. the Real QCD Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

Gauge/String duality vs. the Real QCD From Gauge/String duality towards Real QCD AdS/CFT provides means to reach strong coupling gauge theory, but it is not a QCD. To do: Break SUSY, add quarks, masses, χ, confinement D4-D8 brane model scale.

D4/D8 pion form factors Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

D4/D8 pion form factors Q 2 F Π Q 2 0.8 0.6 0.4 0.2 0 2 4 6 8 10 Q2 GeV 2 (a) Q 2 F π(q 2 ) plot from D4/D8 (b) Q 2 F π(q 2 ) prediction from model hardwall (dashed line), softwall (continuous line) [5] models and data

Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

N c (large) D4 branes fills M 1,3, x 4 (τ), with x 4 compactified in a circle. 4d SU(N c) YM on the brane N f probe branes D8, D8 fills M 1,3, {x 5, x 8 } and radial U(τ) χ symm SU(N f ) SU(N f ) dynamically broken.

Holography: 4d from 5d quarks confined D8-D8 gauge fields independent of {x 5, x 8 } directions. KK modes: vector ( v n µ) and axial vector mesons (a n µ). A µ(x µ, z) = P n=1 v (n) µ (x µ )ψ 2n 1 (z) + a (n) µ (x µ )ψ 2n (z) + A L, A R (x) ψ n: eigenfunctions of e.o.m. (orthonormal). λ n: eigenvalues m 2 n of (axial) vector mesons. S D8 d 9 x e φ det(g MN + 2πα F MN ) d 4 xl kin+mass 2g v n tr ( v n µv µ) +g v l v m v n ( µ v lν ν v lµ)[ v m µ, v n ν ] ( +g v l a m a µ v lν ν v lµ)[ a m n µ,aν] n +gv l a m a n( µ a nν ν a nµ ) ([ ] [ ]) vµ,a l ν m v l ν,aµ m

Scattering Vector Mesons Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

Scattering Vector Mesons v nν g v m v n v p g v p γ µ v pµ v mρ Figure: Vector meson EM scattering. VMD F v n v m(q2 ) = p=1 v nν (k) J (V)µ (0) v mρ (k ) = f µνρ (k,k ) F v n v m(q2 ), g v p g m 2 v n v m v p v p Q 2 p=1 + 1 = 1 g v pg v n v m v p Q 2 m 2 v p p=1 g v pg v n v m v p Q 2 + m 2 v p = 1 + m2 v p Q 2.

Scattering Vector Mesons Sum Rules n m 2 v n MKK 2 Table: Masses and coupling constants = λ 2n 1 κ 1/2 g v n M 2 KK κ 1/2 g v n v 1 v 1 1 0.66931 2.10936 0.44658 2 2.87432 9.10785 0.14654 3 6.59118 20.7957 1.8434 10 2 4 11.79669 37.1502 3.6885 10 4 5 18.48972 58.1701 2.6953 10 4 6 26.67017 83.834 3.0775 10 5 7 36.33796 114.152 1.8572 10 5 8 47.49318 148.103 6.9961 10 6 9 53.6285 188.695 3.5081 10 6 F v m v n(q2 = 0) = 9 g v p g v n v m v p p=1 m 2 v p δ mn

Scattering Vector Mesons v 1 Elastic form factor Fv1 Q 2 1.0 Q 4 Fv1 Q 2 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 2 4 6 8 10 Q2 GeV 2 (a) Elastic form factor F v 1 vs Q 2 0 5 10 15 20 Q2 GeV 2 (b) Q 4 F v 1 vs Q 2 Superconvergence (Grigoryan and Radyushkin): 9 n=1 g v ng v n v 1 v 1 0.000945(M KK) 2 9 n=1 g v ng v n a 1 a 1 0.000903(M KK) 2

Scattering Vector Mesons a 1 Elastic form factor 1.0 0.8 0.6 0.4 Q 4 Fa 1 Q2 2.0 1.5 1.0 0.2 0.5 5 10 15 20 25 (c) Elastic form factor F a 1versus Q 2 5 10 15 20 25 Q2 GeV 2 (d) Q 4 F a 1 versus Q 2 Figure: a 1 elastic form factor

Wave functions Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

Wave functions Vanishing of higher 3-vertex coupling g v n = m 2 v nκ dz(1+z 2 ) 1/3 ψ 2n 1, g v 1 v 1 v p = κ dz(1+z 2 ) 1/3 ψ 2 1 ψ 2p 1 7 6 5 4 4 2 6 4 2 3 2 1 100 50 50 100 100 50 50 100 2 4 100 50 50 100 2 4 6 (a) ψ 1 (b) ψ 2 (c) ψ 60 ψ 2 1 selects small z region. In this region ψ 2p 1 cos(zλ 1/2 2p 1 )

Wave functions Regge Trajectory Λn 40 30 20 10 0 2 4 6 8 10 12 λ n(*) and 0.25n 2 (+) vs n n 1000 100 10 3 1 0.3 λ n(*) n 1.9 n 1.27 0.1 1 3 10 30 60 Loglog n

Quadratic radius Outline 1 Motivation Gauge/String duality vs. the Real QCD D4/D8 pion form factors 2 D4/D8 Brane Model 3 Work Scattering Vector Mesons Wave functions Quadratic radius 4 Summary

Quadratic radius Using the ρ (vector) meson form factor expression in the low Q 2 limit, in terms of the electric form factor: ( G ρ c (Q 2 ) = 1 Q2 6m 2 v 1 F v 1(Q 2 ) ) We found r 2 ρ = 6 d dq 2 Gρ c (Q 2 ) Q 2 =0 r 2 ρ = 0.57fm 2 While results from Dyson-Schwinger equations : r 2 ρ = 0.54fm 2

Summary Besides some weaknesses (mass spectrum, UV behavior), D4-D8 model shows adequate results, concerning vector meson EM scattering. Outlook Would our results improve if quarks were massive? What are the general features that makes D4-D8 model interesting? D4-D8 is not really a Top-down model, since M KK and κ are fixed by experimental values. What can be done?

Appendix For Further Reading For Further Reading I T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113 (2005) 843 [arxiv:hep-th/0412141]. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 114 (2005) 1083 [arxiv:hep-th/0507073]. H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D 76, 095007 (2007) [arxiv:0706.1543 [hep-ph]]. H. R. Grigoryan and A. V. Radyushkin, Phys. Lett. B 650, 421 (2007) [arxiv:hep-ph/0703069]. S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 77, 056007 (2008) [arxiv:0707.3859 [hep-ph]].