15-Gen-091 Riassunto della lezione precedente scomposizione gauge-invariante di operatore momento angolare: definizione di momento angolare orbitale regola di somma di spin e sua evoluzione con scala 2 GPD a twist=2: proprietà, interpretazione intuitiva, trasformata di Fourier nel parametro di impatto b e localizzazione dei partoni nel piano generalizzazione della momentum sum rule per operatori tower di tipo vettoriale ( ), assiale-vettoriale ( 5 ), etc.. polinomialità delle GPD non polarizzate H,E def. gauge-invariante di operatori momento (da tensore energia-impulso T ) e momento angolare (da tensore momento angolare M ) polinomialità : fattori di forma generalizzati regola di somma di momento e di spin

15-Gen-092 q 2 0 s channel crossing q 2 0 t channel 4 vectors (k 1 +k 2 p 1 +p 2 ) 8 elements in J : (, k 1, k 2, p ) (1, q) p=(p 1 +p 2 )/2 q=(p 1 -p 2 ) q J = 0 6 elements 3 complex form factors a possible parametrization: exclusive Drell-Yan N.B. eliminate p and q p

15-Gen-093 Dispersion Relation F(t) analytic function in t 2 C with cut [t 0 = 4m 2, 1) C R Im F(t) 0 only in [t 0 =4m 2, 1 ) vector-meson poles and multi-hadron continuum G E (q 2 ), G M (q 2 ) A(q 2 ) q 2 0 complex analytic continuation G E (Q 2 =q 2 e -i ), G M (Q 2 =q 2 e -i ) 0 q 2 0 real hermitean Hamiltonian

15-Gen-094 elastic scattering cross section Rosenbluth separation fixed Q 2, vary G E slope, G M intercept large errors in G E question in space-like domain

15-Gen-095 polarization transfer method Rosenbluth method

15-Gen-096 exps repeated and checked for consistency; see also JLab – Hall A Qattan et al. P.R.L. 94 (05) Blunden et al. P.R. C72 (05) ? space-like form factor puzzle 1.Rosenbluth method 2. after correction for nonBorn (2 ) 3. polarization transfer method

15-Gen-097 pQCD JLab polar. transfer data space-like form factor puzzle: where is onset of QCD scaling? Q 2 10 no scaling!

15-Gen-098 exclusive Drell-Yan cross section parametrizzazione dati sperimentali (analisi di Fourier) twist 2 annichilazione collineare ~ QPM R T 2 spin-flip R TT ~ 1 >> ~ violazione regola di Lam-Tung R angular asymmetry 1 spin-flip R LT

15-Gen-099 need good coverage over whole range in q 2 0 Rosenbluth plot change E, e at fixed q 2 linear plot in q 2 0 measure at fixed q 2 get R r cos 2 typical of Born diagram non cos 2 explore 2 mechanisms? (continua) poor exp. angular coverage; data mainly from tot with |G E |=|G M | true only at threshold |G E | is unknown! threshold =1 |G E |=|G M |

15-Gen-0910 polarized exclusive Drell-Yan cross section A x, A z require polarization of the electron: P e+ 0 phases of FF from residual interactions (FSI) of baryon system : interference of channels with different phases ( Im [G E * G M ] ) ambiguity $ - solved by FSI ! T-odd mechanisms are allowed : generates A y 0

15-Gen-0911 IJL CQM pQCD improved Brodsky et al. P.R. D69 (04) Interesting model selectivity polarized data would discriminate among models giving the same space-like results but no polarization data available phases of G E/M unknown ! 7 threshold q o2 =4M 2 |G E |=|G M | Im [G E * G M ] = 0

15-Gen-0912 but fit to pQCD T-L fit to pQCD S-L puzzles in the time-like domain space like analytic continuation time-like pQCD + analyticity but fit to |G M p | / 2 |G M | only 1 measurement for neutron (ADONE-1998) again with

15-Gen-0913 space-liketime-like continued

15-Gen-0914 ADONE Q 2 = 4.4 GeV (1973) CERN Q 2 ~ 3.6 (1977) Orsay-DM1 Q 2 ~ (1979) Orsay-DM2 Q 2 =4-5 (1983) LEAR Q 2 ~ (1994) E760 Q 2 ~ (1993) FENICE Q 2 ~3.7-6 (1994) E835 Q 2 ~ (1999) (2003) CLEO Q 2 ~11-12 (2005) BES Q 2 ~4-9 (2005) BaBar Q 2 ~2-20 (2005) World data

15-Gen-0915 world data for |G M | show steep rise at threshold tail of resonance at q 2 4M 2 : 1900) or baryonium or..? should show up as a dip in some hadronic cross section E687 diffractive photoproduction of 6 W » 1.9 GeV below threshold ?