1. Moments and Structure Functions at Low Q 2 Rolf Ent, DIS2004 - Formalism - F 2 Moments: Old Analysis (R “Guess”…) - E94-110 L/T Separation  F 2, F 1, F L Moments - Low Q 2 Intermezzo - Spin-dependent Moments - Summary - (Skipped Extended GDH and Baldin Sum Rules)

2. Thomas Jefferson National Accelerator Facility Unpolarized structure functions F 1 (x,Q 2 ) and F 2 (x,Q 2 ), or F T (x,Q 2 ) [=2xF 1 (x,Q 2 )] and F L (x,Q 2 ), to separate by measuring R =  L /  T Polarized structure functions g 1 (x,Q 2 ) and g 2 (x,Q 2 ) Q 2 : Four-momentum transfer x : Bjorken variable (=Q 2 /2 M ) : Energy transfer M : Nucleon mass W : Final state hadronic mass L T U Inclusive Electron Scattering – Formalism

3. Thomas Jefferson National Accelerator Facility. Moments of the Structure Function M n (Q 2 ) = dx x n-2 F(x,Q 2 ). Operator Product Expansion M n (Q 2 ) =  (nM 0 2 / Q 2 ) k-1 B nk (Q 2 ) higher logarithmic twist dependence QCD and the Operator-Product Expansion 0 1 k=1  At High Q 2 : ln(Q 2 ) dependence of moments one of first “proofs” of QCD   (QCD) At Low Q 2 : Unique regime for JLab to determine (1/Q 2 ) m  Higher Twist effects Lowest moments are calculable in Lattice QCD (LQCD): least computer intensive!

4. Thomas Jefferson National Accelerator Facility Moments of F 2 p @ Low Q 2 n = 2 n = 4 n = 6 n = 8 50% of momentum carried by quarks (Momentum Sum Rule) Uses old R parameterization

5. Thomas Jefferson National Accelerator Facility Moments of F 2 p @ Low Q 2 Proton Charge (Coulomb Sum Rule) Elastic contribution n = 2 n = 4 n = 6 n = 8 50% of momentum carried by quarks (Momentum Sum Rule) (A combination of Hall C + CLAS data has been used to constrain the Constituent Quark radius; Here only Hall C data are shown)

6. Thomas Jefferson National Accelerator Facility Moments of F 2 p @ Low Q 2 Proton Charge (Coulomb Sum Rule) Elastic contribution n = 2 n = 4 n = 6 n = 8 50% of momentum carried by quarks (Momentum Sum Rule) total n = 2 elastic  -region W 2 > 4 GeV 2 (“DIS”) S 11 -region @ Q 2 = 2 (GeV/c) 2 30% of M 2 comes from the resonance region

7. Thomas Jefferson National Accelerator Facility Rosenbluth Separations Rosenbluth Separations Hall C E94-110: a global survey of longitudinal strength in the resonance region…... TT  L +  T =  L  T / (polarization of virtual photon)

8. Thomas Jefferson National Accelerator Facility World's L/T Separated Resonance Data (All data for Q 2 < 9 (GeV/c) 2 ) R =  L /  T < <

9. Thomas Jefferson National Accelerator Facility World's L/T Separated Resonance Data Now able to study the Q 2 dependence of individual resonance regions! Clear resonant behaviour can be observed! Use R to extract F 2, F 1, F L (All data for Q 2 < 9 (GeV/c) 2 ) R =  L /  T <

10. Thomas Jefferson National Accelerator Facility E94-110 L-T Separated Structure Functions  Good agreement between results from (180!) direct L/T separations and global fit of all data

11. Thomas Jefferson National Accelerator Facility Duality in F T and F L Structure Functions Duality works well for both F T and F L above Q 2 ~ 1.5 (GeV/c) 2

12. MRST (NNLO) + TM MRST (NNLO) Alekhin SLAC E94-110 Also good agreement with SLAC L/T Data DIS data provide lower x data for moment extraction

13. F 2 ~ Q 2 F 2 p @ Q 2 < 0.5 Resonance region “slides” to lower x at low Q 2, but does not disappear like Q 2 yet Low Q 2 Intermezzo

14. Thomas Jefferson National Accelerator Facility E99118 Preliminary: Still  R = 0.1 point-to-point (mainly due to bin centering assumptions to x = 0.1) R @ Low Q 2 < 0.5 R shows surprisingly flat Q 2 behaviour down to very low Q 2 (also shown by SLAC R1998 parameterization)  does not appear to follow current conservation prescription yet: R stays rather constant (The x < 0.2, Low Q 2 (~0.1 GeV 2 ) region pushed the envelope of the E94-110 (and E99-118) experiments. Dedicated data for this region were taken last year.) Low Q 2 Intermezzo

15. Thomas Jefferson National Accelerator Facility n = 2 Moments of F 2, F 1 and F L : n = 2 Moments of F 2, F 1 and F L : M n (Q 2 ) = dx x n-2 F(x,Q 2 ) Elastic Contributions Even flatter Q 2 dependence (  smaller higher twist!) with “correct” F 2 - smaller Q 2 dependence in F 2 moment than in F 1, F L moments DIS: SLAC fit to F 2 and R Resonances: E94-110 (Hall C) fit F 1 EL = G M 2  (x-1) F 2 EL = (G E 2 +  G M 2 )  (x-1) F L EL = G E 2  (x-1) 1 +   = Q 2 /4M p 2 F2F2 F1F1 FLFL 0 1

16. Thomas Jefferson National Accelerator Facility n = 4 Moments of F 2, F 1 and F L Neglecting elastics, n = 4 Cornwall-Norton moments have only a small Q 2 dependence as well. Momentum sum rule This is only at leading twist and for zero proton mass ⇒ Must remove non-zero proton mass effects from data to extract moment of xG(x,Q 2 ) ⇒ Work in progress M L (n) =  s (Q 2 ){ 4M 2 (n) + 2c∫dx xG(x,Q 2 )} 3(n+1)(n+1)(n+2) Gluon distributions! F2F2 F1F1 FLFL

17. Nachtmann Moments n = 4n = 2 F2F2 F2F2 F1F1 FLFL F1F1 FLFL

18. Thomas Jefferson National Accelerator Facility Moments of g 1 p (=  1 p ) 30% of Spin carried by quarks (Ellis-Jaffe Sum Rule) ~  of proton (GDH Sum Rule) Elastic not included in Moment as shown  With Elastic included no zero crossing, and Q 2 dependence far smoother CLAS EG1 Data

19. Thomas Jefferson National Accelerator Facility Moments of g 1 p (=  1 p ) 30% of Spin carried by quarks (Ellis-Jaffe Sum Rule) ~  of proton (GDH Sum Rule) Zero crossing mainly due to cancellation of  (negative) and S 11 Resonances Elastic not included in Moment as shown  With Elastic included no zero crossing, and Q 2 dependence far smoother CLAS EG1 Data

20. Thomas Jefferson National Accelerator Facility Moments of g 1 p (=  1 p ) 30% of Spin carried by quarks (Ellis-Jaffe Sum Rule) ~  of proton (GDH Sum Rule) Zero crossing mainly due to cancellation of  (negative) and S 11 Resonances Elastic not included in Moment as shown  With Elastic included no zero crossing, and Q 2 dependence far smoother =  1/2 –  3/2 SU(6) unbroken: M p = M   ground state of  3/2 In this scenario the  is a  function and all higher twist  it plays the same role as the elastic in F 2 CLAS EG1 Data

21. Thomas Jefferson National Accelerator Facility Moments of g 1 n and g 1 p -g 1 n 1n1n Hall A 3 He(e,e’) to extract g 1 n and its moment  1 n (E94-010) Similar ideas as with proton Here, whole region negative

22. Thomas Jefferson National Accelerator Facility Moments of g 1 n and g 1 p -g 1 n 1n1n Hall A 3 He(e,e’) to extract g 1 n and its moment  1 n Similar ideas as with proton Here, whole region negative Combine Hall A g 1 n with Hall B g 1 p data

23. Thomas Jefferson National Accelerator Facility Moments of g 1 n and g 1 p -g 1 n 1n1n  1 p -  1 n Bjorken Sum Rule (Verification of QCD) Hall A 3 He(e,e’) to extract g 1 n and its moment  1 n Similar ideas as with proton Here, whole region negative Combine Hall A g 1 n with Hall B g 1 p data Chiral Perturbation Theory (  PT)

24. Thomas Jefferson National Accelerator Facility Moments of g 2 n (=  2 n ) Burkhardt-Cottingham Sum Rule Dispersion relation for forward spin-flip Compton amplitude (similar assumption as GDH). Doesn’t follow from OPE and valid at all Q 2 Many scenarios of g 2 low x behavior would invalidate the sum rule. Here, the part of the integral that comes from the nucleon resonance region is shown. It’s ~0, mostly from a cancellation of the large elastic and  contributions Contribution beyond resonance region as calculated from g 1 Hall A 3 He(e,e’) (E94-010)

25. Thomas Jefferson National Accelerator Facility Summary – Moments at Low Q 2 Resonances are an integral part of the Structure Function Moments at Low Q 2 (Note: QCD deals with the Moments and does not care what contributes to the moments!) 2xF 1, F L, F T Structure Functions well determined in the Resonance Region (for Q 2 < 5 (GeV/c) 2 ) The Structure Function Moments have a smooth behaviour as a function of Q 2, and in fact pick up almost uniquely the quark-quark interactions in the SU(6) ground states. This seems to indicate that the parton-hadron transition occurs in a local and small region, with only few resonances  “Quark-Hadron Duality” Revised Nachtmann Moments (using measured R) at low Q 2 show very limited Q 2 dependence. F 2 Moments have less Q 2 dependence than F 1 and F L moments.

26. Why is I GDH (Q 2 ) interesting?

27. Extended Baldin Sum Rule Q 2 = 0, photoproduction GDH Sum Rule Q 2 > 0, electroproduction Extended GDH Sum Rule Baldin Sum Rule Extended Baldin Sum Rule * Where κ : anomalous magnetic moment of the nucleon. α, β: electric and magnetic polarizabilities respectively α, β: electric and magnetic polarizabilities respectively ν 0 : pion photoproduction threshold ν 0 : pion photoproduction threshold * D. Drechsel, B. Pasquini, M. Vanderhaeghen hep-ph/0212124 Dec 2002 Need L/T separated data!

28. Measurement of the extended Baldin Sum Rule for the proton Extended Baldin Integral goes smoothly from Q 2 > 5 to real photon point (Q 2 = 0). Extended Baldin Integral goes smoothly from Q 2 > 5 to real photon point (Q 2 = 0). (PDG) (D. Babusci et al. 1998)

29. Multiply with Q 4 /2M to emphasize transition of Baldin Sum Rule to Perturbative DIS Region ~ 2xF 1 Moment