Ruben Sandapen (Acadia & Mt. A) in collaboration with M. Ahmady & F. Chishtie. September 5 th 2016

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1 Holographic Distribution Amplitudes for mesons Ruben Sandapen (Acadia & Mt. A) in collaboration with M. Ahmady & F. Chishtie Diffraction 2016 Progress in QCD session September 5 th

2 Outline Overview of light- front holography (LFH) Phenomenological improvements to LFH Holographic Distribution Amplitudes for pseudoscalar and vector mesons Comparison with data and other methods for pion 2

3 Holographic duality Strongly- coupled gauge theory in 4d = Weakly- coupled gravitational theory in 5d anti- de Sitter AdS=CFT (Maldacena, late 1990 s) Hope: AdS=QCD (Active contemporary area of research) Focus here on findings of Brodsky & de Teramond (2010 s): AdS = Light- front QCD (in semi- classical approximation) Semi- classical : zero quark masses & no quantum loops (conformal LF ChromoDynamics) Review article with all original references: 3

4 LF Schrodinger Equation for mesons LF QCD Heisenberg equation semi- classical approximation (zero quark masses and no loops) LF holographic Schrodinger Equation ζ = x(1 x) r An infinite set of coupled integral equations reduces to a single variable wave equation No truncation: higher Fock states hidden in effective potential Potential cannot be derived from first principles in QCD 4

5 Why holographic? LFH Schrodinger Equation maps exactly onto the equation of motion for spin- J string modes propagating freely in higher dimensional anti- de Sitter (AdS) space Transverse separation at equal LF time maps onto 5 th dimension of AdS ζ z Angular momentum in physical spacetime maps onto (mass x radius) in AdS Confinement in physical spacetime given by breaking of conformal symmetry in AdS 5

6 AdS/LF QCD duality Quadratic dilaton in AdS implies a harmonic oscillator potential in physical spacetime λ. = κ ζ = x(1 x) r Mapping of pion EM form factor in AdS and physical spacetime fixes longitudinal mode X x = x(1 x) Uniqueness of harmonic potential : de Alfaro, Fubini & Furlan (76) showed that it is possible to generate a mass scale and confinement in QM while retaining the conformal symmetry of the action. In semiclassical LF QCD, this uniquely determine the harmonic oscillator potential. 6

7 Solving the holographic LF Schrodinger equation Pion M 2,2,2 =0 λ = κ. Regge trajectories Measured Regge slope gives confinement scale X x = x(1 x) 7

8 Universal dilaton scale Regge slope of mesons: κ = GeV Diffractive rho production: κ = 0.56 GeV Prediction of QCD scale parameter in agreement with world average: κ = GeV 8

9 Holographic meson wavefunction No free parameter (use universal scale κ = 0.55 GeV) but Massless quarks Implicit assumption that helicity dependence decouples from dynamics Phenomenology: need to account for both quark masses and helicities 9

10 Accounting for (light) quark masses ζ = x(1 x) r Fourier conjugates Restore quark masses in momentum space λ = κ. In impact space, this results in a modified longitudinal mode This prescription is by Brodsky & de Teramond (2008) 10

11 Helicity dependence of holographic wavefunction Assumption: helicity dependence same as if meson was point- like Used in many dipole model predictions of diffractive vector meson production Vector (rho) : Pseudoscalar (pion): 11

12 Holographic LFWF for pion and ρ- meson rho pion NR limit: degenerate DAs with universal dilaton scale 12

13 Normalization condition Assumption: no higher Fock states contributions Has to be tested against data 13

14 Twist- 2 Distribution Amplitudes in QCD Perturbative evolution given by ERBL equations Not specified at low, non- perturbative starting scale μ 1 GeV 14

15 Holographic DAs We can then deduce that With NR helicities: all degenerate For phenomenology of rho DAs: Focus here on pion DA 15

16 ERBL evolution Efremov- Radyushkin (78); Brodsky- Lepage (79) Perturbative evolution known since 80 s Require DA at starting non- perturbative scale QCD Sum Rules, lattice QCD, Schwinger- Dyson predict moments of DA at starting scale 16

17 Pion twist- 2 holographic DA φ π (x,µ) Asymptotic µ=1 GeV µ=0.5 GeV µ=0.3 GeV κ = 0.55 GeV =0.330 GeV x LFH pion DA hardly evolves beyond 1 GeV 17

18 Moments of pion DA Stefanis & Pimikov, Nucl. Phys. A945(2016) 248 Asymptotic μ = 2 GeV Here Refs [6,5]: QCD Sum Rules with non- local condensates Ref [9]: Dyson- Schwinger Equations Ref [1] : QCD Sum- Rules with local condensates Ref [49]: Lattice QCD Ref [43]: Original Light Front Holography prediction with massless quarks, a lower dilaton scale and no dynamical helicity dependence κ = 0.55 GeV =0.330 GeV μ = 1 GeV 18

19 EM pion form factor F π (Q 2 ) LFH (NR) LFH (new) CEA (73) Cornell (74) Cornell (76) Cornell (78) CERN (86) CEBAF (01) Pion radius r A =0.667 fm (Prediction) PDG: 0.672±0.008 fm (Exp) κ = 0.55 GeV =0.330 GeV Q 2 [GeV 2 ] Theory curve : Prelim. Dotted: no dynamical helicity dependence Good agreement in low Q^2 region (as expected with holographic LFWF) 19

20 Pion- to- photon transition form factor Pion decay constant f A =140 MeV (Our prediction) 0.25 PDG: MeV(Experiment) 0.2 Q 2 F γ γ π Asymptotic DA LFH DA (NR) LFH DA (new) BaBar (09) Belle (12) CLEO (98) CELLO (91) Q 2 [GeV 2 ] κ = 0.55 GeV =0.330 GeV Theory curve: prelim. Dotted: non- dynamical helicity Good agreement except for BaBar (09) anomaly 20

21 Conclusions Gravity dual to semi- classical light- front QCD is known: light- front holography Universal dilaton scale We account for both quark masses and dynamical helicity effects Holographic pion DA with constituent quark mass leads to good agreement with data on form factors 21

22 Acknowledgements Funding from Canadian National Science and Engineering Research Council of Canada (NSERC) 22