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PION & NUCLEON FORM FACTORS IN (NON-PERTURBATIVE) QCD. C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town. UTFSM 2008. M.Loewe M.Fetea J.Rozowski C.van Gend H.Castillo R.Rontsch B.Willers J.Jonker. M.Shifman, A.Vainshtein, V.Zakharov
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PION & NUCLEON FORM FACTORS IN (NON-PERTURBATIVE) QCD C.A. Dominguez Centre for Theoretical Physics & Astrophysics University of Cape Town UTFSM 2008
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 Credits
PION & NUCLEON FF • QCD SUM RULES (π, N) • DUAL-LARGE NC QCD (π, N)
PROTONELECTRIC & MAGNETIC FORM FACTORS • <Pf | J(p+) | Pi> <Pf | F1(q2) • + i q a F2 (q2)/M | Pi>
e- p+ CROSS SECTION • GE (q2) = F1(q2) + (a q2/4M2) F2(q2) • GM (q2) = F1(q2) + a F2(q2) • R = (- q2/4M2) G2M (q2) + G2E (q2)
ROSENBLUTH METHOD • Unpolarized e- p+ scattering • Measure R for constant q2 varying • Determine GM (q2) from intercept • Determine GE (q2) from slope • Assume ScalingLaw : GE /GM = 1 GE/GM 1 - q2
Polarized e- p+ ScatteringJefferson Lab • Measure longitudinal & transverse polarizations of the recoil proton: Pl, Pt • GE /GM Pt / Pl • GE /GM1 + 0.13 q2 • A zero at – q2 8 GeV2
Reconciliation between Rosenbluth & PolarizationMeasurementsSecond order correction more important in Rosenbluth than in Polarization
QCD Sum Rules SUMMARY • Fπ(q2) : Reasonable fit for -1 GeV2 ≤ q2 ≤ - 10 GeV2 • F1 (q2) : Reasonable fit for -1 GeV2 ≤ q2 ≤ - 30 GeV2 • F2 (q2) : → 1/(q2) DISASTER !!!
QCD • Lim Nc (Nc = 3) ( t’Hooft ’74 & Witten ’79) • Spectrum: number of zero width resonances Im M2
Real Spectral Function Im E2
CORRECTIONS to 1/Nc / M 10 %
RESONANCES • - p+ coupling : - 0 - p+ • 0 : JPC = 1- - • M 770 MeV • M ’ 1340 MeV • M ,, 1720 MeV • M ,,, 2034 MeV
Dual - QCD • Dual Resonance Model Veneziano (1968) • number of zero width resonances, equally spaced • Masses & couplings fixed to give an Euler Beta Function
M = 769 MeV M’ 1340 MeV [EXP.: 1465 25 MeV] M’’ 1720 MeV [EXP.: 1700 20 MeV] M’’’ 2034 MeV [EXP.: 2149 17 MeV]
PION FORM FACTOR Dual-Large Nc QCD
Nucleon Form FactorsDual-Large Nc QCD • F1 (q2) • F2 (q2) • GM (q2) • GE (q2) • GE (q2) / GM (q2)
SUMMARY • QCD sum rules: FπOK F1OK F2NOT • DUAL – Nc ∞ : Fπ, F1 & F2 + GE / GM PERFECT FITS