1. C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town PION & NUCLEON FORM FACTORS IN (NON-PERTURBATIVE) QCD UTFSM 2008

2. Credits M.Loewe M.Fetea J.Rozowski C.van Gend H.Castillo R.Rontsch B.Willers J.Jonker M.Shifman, A.Vainshtein, V.Zakharov B.Ioffe, A.Radyushkin V.Braun et al. A.Lenz et al. V.Belyaev & I. Kogan

3. PION & NUCLEON FF QCD SUM RULES (π, N) DUAL-LARGE N C QCD (π, N)

4. QCD SUM RULES (SVZ)

7. QUARK-HADRON DUALITY

9. FINITE ENERGY SUM RULES

10. PION FORM FACTOR

12. VECTOR MESON DOMINANCE

15. PROTON ELECTRIC & MAGNETIC FORM FACTORS  <P f |   F 1 (q 2 ) + i   q a F 2 (q 2 )/M | P i >

16. e - p + CROSS SECTION G E (q 2 ) = F 1 (q 2 ) + (a q 2 /4M 2 ) F 2 (q 2 ) G M (q 2 ) = F 1 (q 2 ) + a F 2 (q 2 )  R = (- q 2 /4M 2 ) G 2 M (q 2 ) +  G 2 E (q 2 )  

17. ROSENBLUTH METHOD Unpolarized e - p + scattering Measure  R for constant q 2 varying  Determine G M (q 2 ) from intercept Determine G E (q 2 ) from slope Assume Scaling Law :  G E /G M = 1    G E /G M - q 2 1

18. Polarized e - p + Scattering Jefferson Lab Measure longitudinal & transverse polarizations of the recoil proton: P l, P t  G E /G M  P t / P l  G E /G M  1 + 0.13 q 2 A zero at – q 2  8 GeV 2

19. Reconciliation between Rosenbluth & Polarization Measurements Second order correction more important in Rosenbluth than in Polarization

20. QCD SUM RULES F 1,2 (-q 2 )

26. QCD Sum Rules SUMMARY F π (q 2 ) : Reasonable fit for -1 GeV 2 ≤ q 2 ≤ - 10 GeV 2  F 1 (q 2 ) : Reasonable fit for -1 GeV 2 ≤ q 2 ≤ - 30 GeV 2 F 2 (q 2 ) : → 1/(q 2 ) DISASTER !!!

27. DUAL – LARGE N c QCD QCD ∞

28. QCD  Lim N c   (N c = 3) ( t’Hooft ’74 & Witten ’79) Spectrum:  number of zero width resonances Im  M2M2

29. Real Spectral Function Im  E2E2

30. CORRECTIONS to 1/N c  / M  10 %

31. RESONANCES  - p + coupling :  -  0 - p +  0 : J PC = 1 - - M   770 MeV M  ’  1340 MeV M ,,  1720 MeV M ,,,  2034 MeV

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

33. M  = 769 MeV M  ’  1340 MeV [EXP.: 1465  25 MeV] M  ’’  1720 MeV [EXP.: 1700  20 MeV] M  ’’’  2034 MeV [EXP.: 2149  17 MeV]

34. PION FORM FACTOR Dual-Large N c QCD

43. 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 )

48. SUMMARY QCD sum rules: F π OK F 1 OK F 2 NOT DUAL – N c ∞ : F π, F 1 & F 2 + G E / G M PERFECT FITS