1. C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town Department of Physics, Stellenbosch University South Africa XII WORKSHOP ON PARTICLES & FIELDS MAZATLAN NOVEMBER 2009 ELECTROMAGNETIC AND HADRONIC FORM FACTORS IN RENORMALIZABLE QUANTUM FIELD THEORIES

2. AD-HOC MODELS FOR FORM FACTORS NO SYSTEMATIC IMPROVEMENT SYSTEMATIC IMPROVEMENT: PERTURBATIVE CALCULATIONS IN RENORMALIZABLE QUANTUM FIELD THEORIES

3. 1) Kroll – Lee – Zumino QFT 2) QCD ∞ : QCD IN THE LIMIT N c → ∞

4. Kroll – Lee – Zumino QFT RENORMALIZABLE QFT PLATFORM (  &  0 ) TO JUSTIFY VECTOR MESON (RHO) DOMINANCE (K,L,Z, 1967) ALLOWS FOR A SYSTEMATIC & MEANINGFUL PERTURBATIVE EXPANSION BEYOND THE LEADING (TREE-LEVEL) ORDER ELECTROMAGNETIC FORM FACTORS OF π & SCALAR FORM FACTOR OF π (chiral perturbation theory) (CAD, Jottar, Loewe, Willers, 2007,2008)

5.  PT  KLZ  QCD

6. VECTOR MESON DOMINANCE Abelian, TREE-LEVEL model No truly QFT platform Not subject to PERTURBATION THEORY improvement

8. CALCULATING IN KLZ Regularization using DIMENSIONAL REGULARIZATION Renormalization (fields, masses, couplings) Renormalization subtraction point for vertex diagram: q 2 = 0 Renormalization subtraction point for vacuum polarization diagram: q 2 = M 2 ρ

18. SCALAR FORM FACTOR OF THE PION  CHIRAL PERTURBATION THEORY

19. CHIRAL PERTURBATION THEORY  QCD SHARE THE SAME GLOBAL GAUGE SYMMETRIES  PT provides effective Lagrangian for QCD at low energies (near threshold) AT NLO AND BEYOND  LEC’s  l 4 NOT MEASURABLE

23. @ NLO IN KLZ = 0.4 fm 2, l 4 = 3.4, F  /F = 1.05 LATTICE QCD: l 4 = 3.6 – 5.0 NNLO IN KLZ ???

24. KLZ RENORMALIZABLE QFT PLATFORM TO EXTEND VMD BEYOND THE LEADING ORDER IN PERTURBATION THEORY A SENSIBLE PERTURBATIVE EXPANSION g   6 & (g  / 4  ) 2  0.2  PT  KLZ  QCD

25. QCD ∞ Lim N c → ∞ (N c = 3) ( t’Hooft ’74 & Witten ’79) Spectrum: ∞ number of zero width resonances Im G M2M2

26. QCD IN THE LIMIT N c → ∞ INFINITE NUMBER OF 0-WIDTH RESONANCES NEEDS A REALIZATION (MODEL) FOR HADRONIC MASSES & COUPLINGS DUAL-RESONANCE MODEL (VENEZIANO) DUAL QCD  PION, NUCLEON, DELTA FORM FACTORS CORRECTIONS DUE TO FINITE WIDTH (  /M)

27. Real Spectral Function Im G E2E2

28. CORRECTIONS to 1/N c  / M  10 %

29. Dual - QCD ∞ Dual Resonance Model Veneziano (1968) ∞ number of zero width resonances, equally spaced Masses & couplings fixed to give an Euler Beta Function

34. Nucleon Form Factors Dual-Large N c QCD F 1 (q 2 ) F 2 (q 2 ) G M (q 2 ) G E (q 2 ) G E (q 2 ) / G M (q 2 )

39. FORM FACTORS OF Δ(1236) G * M (q 2 ), G * E (q 2 ), G * C (q 2 )

42. DUAL QCD  DUAL RESONANCE MODEL (VENEZIANO) M N & g n A SINGLE FREE PARAMETER (q 2 < 0) UNITARIZATION (q 2 > 0) OTHER APPLICATIONS ???