1. Duality in Unpolarized Structure Functions: Validation and Application First Workshop on Quark Hadron Duality and the Transition to Perturbative QCD Frascati June 6-8, 2005 (Italy) Rolf Ent

2. First observed ~1970 by Bloom and Gilman at SLAC by comparing resonance production data with deep inelastic scattering data Integrated F 2 strength in Nucleon Resonance region equals strength under scaling curve. Integrated strength (over all  ’) is called Bloom-Gilman integral Shortcomings: Only a single scaling curve and no Q 2 evolution (Theory inadequate in pre-QCD era) No  L /  T separation  F 2 data depend on assumption of R =  L /  T Only moderate statistics Duality in the F 2 Structure Function  ’ = 1+W 2 /Q 2 Q 2 = 0.5 Q 2 = 0.9 Q 2 = 1.7Q 2 = 2.4 F2F2 F2F2

3. First observed ~1970 by Bloom and Gilman at SLAC Now can truly obtain F 2 structure function data, and compare with DIS fits or QCD calculations/fits (CTEQ/MRST/GRV) Use Bjorken x instead of Bloom- Gilman’s  ’  Bjorken Limit: Q 2,    Empirically, DIS region is where logarithmic scaling is observed: Q 2 > 5 GeV 2, W 2 > 4 GeV 2  Duality: Averaged over W, logarithmic scaling observed to work also for Q 2 > 0.5 GeV 2, W 2 < 4 GeV 2, resonance regime (note: x = Q 2 /(W 2 -M 2 +Q 2 )  JLab results: Works quantitatively to better than 10% at surprisingly low Q 2 Duality in the F 2 Structure Function

4. Bloom-Gilman Duality in F 2 Today The scaling curve determines the Q 2 behavior of the resonances! … or … the resonances “fix” the scaling curve!

5. Low Q 2 Too! Even Down to Photoproduction Bodek and Yang duality-based model works at all Q 2 values tested – DIS to photoproduction!

6. Quantification: Resonance Region F 2 w.r.t. Alekhin NNLO Scaling Curve (Q 2 ~ 1.5 GeV 2 )

7. Quantification in a Fixed W 2 Framework (S. Liuti et al., PRL 2002) F 2 exp /F 2 pQCD+TMC = 1 + C HT (x)/Q 2 +  H(x,Q 2 ) Previous Extractions (DIS Only)  Duality not only a leading-twist phenomenon

8. If one integrates over all resonant and non-resonant states, quark-hadron duality should be shown by any model. This is simply unitarity. However, quark-hadron duality works also, for Q 2 > 0.5 (1.0) GeV 2, to better than 10 (5) % for the F 2 structure function in both the N-  and N-S 11 region! (Obviously, duality does not hold on top of a peak! -- One needs an appropriate energy range) One resonance + non- resonant background Few resonances + non- resonant background Why does local quark-hadron duality work so well, at such low energies? ~ quark-hadron transition Confinement is local ….

9. Quark-Hadron Duality – Theoretical Efforts N. Isgur et al : N c  ∞ qq infinitely narrow resonances qqq only resonances  Distinction between Resonance and Scaling regions is spurious  Bloom-Gilman Duality must be invoked even in the Bjorken Scaling region  Bjorken Duality One heavy quark, Relativistic HO Scaling occurs rapidly! Q 2 = 5 Q 2 = 1 F. Close et al : SU(6) Quark Model How many resonances does one need to average over to obtain a complete set of states to mimic a parton model?  56 and 70 states o.k. for closure  Similar arguments for e.g. DVCS and semi-inclusive reactions u

10. Quark-Hadron Duality - Applications CTEQ currently planning to use duality for large x parton distribution modeling Neutrino community using duality to predict low energy (~1 GeV) regime  Implications for exact neutrino mass  Plans to extend JLab data required and to test duality with neutrino beams Duality provides extended access to large x regime Allows for direct comparison to QCD Moments  Lattice QCD Calculations now available for u-d (valence only) moments at Q 2 = 4 (GeV/c) 2  Higher Twist not directly comparable with Lattice QCD  If Duality holds, comparison with Lattice QCD more robust

11. G0 SOS HMS E94-110/E00-116/E00-108 Experiments performed at JLab-Hall C

12. “The successful application of duality to extract known quantities suggests that it should also be possible to use it to extract quantities that are otherwise kinematically inaccessible.” (CERN Courier, December 2004) E94-110 : Precise Measurement of Separated Structure Functions in Nucleon Resonance Region  The resonance region is, on average, well described by NNLO QCD fits.  This implies that Higher-Twist (FSI) contributions cancel, and are on average small.  The result is a smooth transition from Quark Model Excitations to a Parton Model description, or a smooth quark-hadron transition.  This explains the success of the parton model at relatively low W 2 (=4 GeV 2 ) and Q 2 (=1 GeV 2 ).

13. 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

14. Quantification Ratio Resonance Region / Scaling CurvesLarge x Structure Functions SLAC DIS data Q 2 = 6 GeV 2 x Undershoot at large x

15. E00-116: Experiment to Push to larger Q 2 (larger x) Emphasis: 1) With Duality Verified at Lower Q 2, Now Can Use Data to Extend Range in x; 2) Need Large x data to Constrain Higher Moments; 3) Quantify Q 2 vs. Q 2 (1-x) Evolution Preliminary data again do overshoot QCD calculations But show great numerical agreement compared to (internal) scaling curve  determines Q 2 behavior of the resonances

16. E00-116: Experiment to Push to larger Q 2 (larger x) Same behavior at largest Q 2 of E00-116 analyzed to date Additional data (4 more resonance scans) at higher Q 2 for both proton and deuteron  reach x ~ 0.9. Data final in 1-2 months.

17. Close and Isgur Approach to Duality Phys. Lett. B509, 81 (2001):  q =   h Relative photo/electroproduction strengths in SU(6) “The proton – neutron difference is the acid test for quark-hadron duality.” How many states does it take to approximate closure? Proton W~1.5 o.k. Neutron W~1.7 o.k. Detect 70 MeV/c spectator proton at backward angles in “BONuS” detector Scatter electrons off deuteron The BONuS experiment will measure neutron structure functions……. Commissioning Run in CLAS now!

18. Neutron Quark-Hadron Duality – Projected Results (CLAS/BONUS)  Sample neutron structure function spectra  Plotted on proton structure function model for example only  Neutron resonance structure function essentially unknown  Smooth curve is NMC DIS parameterization Systematics ~5%

19. Duality in Meson Electroproduction (predicted by Afanasev, Carlson, and Wahlquist, Phys. Rev. D 62, 074011 (2000)) (e,e’) (e,e’m) W 2 = M 2 + Q 2 (1/x – 1) W’ 2 = M 2 + Q 2 (1/x – 1)(1 - z) z = E m / For M m small, p m collinear with , and Q 2 / 2 << 1 Hall C Experiment E00-108Spokespersons:Hamlet Mkrtchyan (Yerevan) Gabriel Niculescu (JMU) Rolf Ent (JLab) H,D(e,e’  +/- )X

20. Duality in Meson Electroproduction hadronic descriptionquark-gluon description Transition Form Factor Decay Amplitude Fragmentation Function Requires non-trivial cancellations of decay angular distributions If duality is not observed, factorization is questionable Duality and factorization possible for Q 2,W 2  3 GeV 2 (Close and Isgur, Phys. Lett. B509, 81 (2001))

21. Factorization P.J. Mulders, hep-ph/0010199 (EPIC Workshop, MIT, 2000) At large z-values easier to separate current and target fragmentation region  for fast hadrons factorization (Berger Criterion) “works” at lower energies At W = 2.5 GeV: z > 0.4At W = 5 GeV: z > 0.2 (Typical JLab) (Typical HERMES)

22. E00-108 data Experiment ran in August 2003  Close-to-final data analysis  Small Q 2 variation not corrected yet No visual “bumps and valleys” in pion ratios off deuterium x = 0.32 W’ > 1.4 GeVW’ < 2.0 GeV

23. From deuterium data: D - /D + = (4 – N  + /N  - )/(4N  + /N  - - 1)  D - /D + ratio should be independent of x but should depend on z  Find similar slope versus z as HERMES, but slightly larger values for D - /D +, and definitely not the values expected from Feynman- Field independent fragmentation at large z! z = 0.55 x = 0.32

24. W’ = 2 GeV ~ z = 0.35  data predominantly in “resonance region” What happened to the resonances? From deuterium data: D - /D + = (4 – N  + /N  - )/(4N  + /N  - - 1)

25. The Origins of Quark-Hadron Duality – Semi-Inclusive Hadroproduction Destructive interference leads to factorization and duality Predictions: Duality obtained by end of second resonance region Factorization and approximate duality for Q 2,W 2 < 3 GeV 2 F. Close et al : SU(6) Quark Model How many resonances does one need to average over to obtain a complete set of states to mimic a parton model?  56 and 70 states o.k. for closure

26. How Can We Verify Factorization? Neglect sea quarks and assume no p t dependence to parton distribution functions  Fragmentation function dependence drops out  [  p (  + ) +  p (  - )]/[  d (  + ) +  d (  - )] = [4u(x) + d(x)]/[5(u(x) + d(x))] =  p /  d independent of z and p t

27. Dependence on z Still to quantify (preliminary data only) … Lund MC Berger Criterion W’ > 1.4 GeV

28. Dependence on p t … but data not inconsistent with factorization Lund MC

29. Dependence on x Lund MC … trend of expected x-dependence, but slightly low?

30. Duality in Unpolarized Structure Functions: Validation and Application “It is fair to say that (short of the full solution of QCD) understanding and controlling the accuracy of quark-hadron duality is one of the most important and challenging problems for QCD practitioners today.” M. Shifman, Handbook of QCD, Volume 3, 1451 (2001) Duality was well established in the ’70s in the pre- and early-QCD era. The existence of duality is well known, based on Regge theory, and these ideas have not changed since that time. Modern studies show that duality works quantitatively better than expected, and works in all observables tested to date. - Hence, duality can be used as a tool for other studies. The origin of duality is the subject of modern theoretical research.

32. Moments of the Structure Function M n (Q 2 ) = dx x n-2 F(x,Q 2 ) If n = 2, this is the Bloom-Gilman duality integral! Operator Product Expansion M n (Q 2 ) =  (nM 0 2 / Q 2 ) k-1 B nk (Q 2 ) higher twist logarithmic dependence Duality is described in the Operator Product Expansion as higher twist effects being small or canceling DeRujula, Georgi, Politzer (1977) QCD and the Operator-Product Expansion 0 1 k=1 

33. Example: e + e - hadrons Textbook Example Only evidence of hadrons produced is narrow states oscillating around step function R =

34. Factorization Both Current and Target Fragmentation Processes Possible. At Leading Order: Berger Criterion  > 2 Rapidity gap for factorization Separates Current and Target Fragmentation Region in Rapidity