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Duality in Unpolarized Structure Functions: Validation and Application

This workshop focuses on the validation and application of quark-hadron duality in the F2 structure function. It explores the limitations of the Bloom-Gilman integral and the use of Bjorken scaling in obtaining F2 structure function data. The workshop also discusses the success of duality in the resonance region and its implications for large x parton distribution modeling. Additionally, it explores the potential of duality in predicting low-energy neutrino reactions and its relevance to lattice QCD calculations.

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Duality in Unpolarized Structure Functions: Validation and Application

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

  2. Duality in the F2 Structure Function First observed ~1970 by Bloom and Gilman at SLAC by comparing resonance production data with deep inelastic scattering data • Integrated F2 strength in Nucleon Resonance region equals strength under scaling curve. Integrated strength (over all w’) is called Bloom-Gilman integral Shortcomings: • Only a single scaling curve and no Q2 evolution (Theory inadequate in pre-QCD era) • No sL/sT separation  F2 data depend on assumption of R = sL/sT • Only moderate statistics F2 Q2 = 0.5 Q2 = 0.9 F2 Q2 = 1.7 Q2 = 2.4 w’ = 1+W2/Q2

  3. Duality in the F2 Structure Function First observed ~1970 by Bloom and Gilman at SLAC Now can truly obtain F2 structure function data, and compare with DIS fits or QCD calculations/fits (CTEQ/MRST/GRV) Use Bjorken x instead of Bloom-Gilman’s w’ • Bjorken Limit: Q2, n • Empirically, DIS region is where logarithmic scaling is observed: Q2 > 5 GeV2, W2 > 4 GeV2 • Duality: Averaged over W, logarithmic scalingobserved to work also for Q2 > 0.5 GeV2, W2 < 4 GeV2, resonance regime (note: x = Q2/(W2-M2+Q2) • JLab results: Works quantitatively to better than 10% at surprisingly low Q2

  4. Bloom-Gilman Duality in F2 Today The scaling curve determines the Q2 behavior of the resonances! … or … the resonances “fix” the scaling curve!

  5. Low Q2 Too! Even Down to Photoproduction Bodek and Yang duality-based model works at all Q2 values tested – DIS to photoproduction!

  6. Quantification: Resonance Region F2 w.r.t. Alekhin NNLO Scaling Curve (Q2 ~ 1.5 GeV2)

  7. Quantification in a Fixed W2 Framework (S. Liuti et al., PRL 2002) F2exp/F2pQCD+TMC = 1 + CHT(x)/Q2 + DH(x,Q2) 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 Q2 > 0.5 (1.0) GeV2, to better than 10 (5) % for the F2 structure function in both the N-D and N-S11 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 One heavy quark, Relativistic HO N. Isgur et al : Nc ∞ qq infinitely narrow resonances qqq only resonances Q2 = 1 Q2 = 5 u Scaling occurs rapidly! • 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 • Distinction between Resonance and Scaling regions is spurious • Bloom-Gilman Duality must be invoked even in the Bjorken Scaling region  Bjorken Duality

  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 Q2 = 4 (GeV/c)2 • Higher Twistnot directly comparable with Lattice QCD • If Duality holds, comparison with Lattice QCD more robust

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

  12. 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 W2 (=4 GeV2) and Q2 (=1 GeV2). “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)

  13. Duality in FT and FL Structure Functions Duality works well for both FT and FL above Q2 ~ 1.5 (GeV/c)2

  14. Quantification Ratio Resonance Region / Scaling Curves Large x Structure Functions SLAC DIS data Q2 = 6 GeV2 x Undershoot at large x

  15. E00-116: Experiment to Push to larger Q2 (larger x) Emphasis: 1) With Duality Verified at Lower Q2, Now Can Use Data to Extend Range in x; 2) Need Large x data to Constrain Higher Moments; 3) Quantify Q2 vs. Q2(1-x) Evolution But show great numerical agreement compared to (internal) scaling curve  determines Q2 behavior of the resonances Preliminary data again do overshoot QCD calculations

  16. E00-116: Experiment to Push to larger Q2 (larger x) Same behavior at largest Q2 of E00-116 analyzed to date Additional data (4 more resonance scans) at higher Q2 for both proton and deuteron  reach x ~ 0.9. Data final in 1-2 months.

  17. Close and Isgur Approach to Duality How many states does it take to approximate closure? Proton W~1.5 o.k. Neutron W~1.7 o.k. Phys. Lett. B509, 81 (2001): Sq =Sh Relative photo/electroproduction strengths in SU(6) “The proton – neutron difference is the acid test for quark-hadron duality.” Scatter electrons off deuteron The BONuS experiment will measure neutron structure functions……. Detect 70 MeV/c spectator proton at backward angles in “BONuS” detector 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)) Hall C Experiment E00-108 Spokespersons: Hamlet Mkrtchyan (Yerevan) Gabriel Niculescu (JMU) Rolf Ent (JLab) H,D(e,e’p+/-)X (e,e’) W2 = M2 + Q2 (1/x – 1) For Mm small, pm collinear with g, and Q2/n2 << 1 (e,e’m) W’2 = M2 + Q2 (1/x – 1)(1 - z) z = Em/n

  20. Duality in Meson Electroproduction hadronic description quark-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 Q2,W2  3 GeV2(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 > 0.4 At 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 Q2 variation not corrected yet x = 0.32 W’ < 2.0 GeV W’ > 1.4 GeV x = 0.32 No visual “bumps and valleys” in pion ratios off deuterium

  23. From deuterium data: D-/D+ = (4 – Np+/Np-)/(4Np+/Np- - 1) z = 0.55 • 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! 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 – Np+/Np-)/(4Np+/Np- - 1)

  25. The Origins of Quark-Hadron Duality – Semi-Inclusive Hadroproduction 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 Destructive interference leads to factorization and duality Predictions: Duality obtained by end of second resonance region Factorization and approximate duality for Q2,W2 < 3 GeV2

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

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

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

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

  30. Duality in Unpolarized Structure Functions: Validation and Application 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. “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)

  31. QCD and the Operator-Product Expansion 1 0 • Moments of the Structure Function Mn(Q2) = dxxn-2F(x,Q2) If n = 2, this is the Bloom-Gilman duality integral! • Operator Product Expansion Mn(Q2) = (nM02/Q2)k-1Bnk(Q2) higher twistlogarithmic dependence • Duality is described in the Operator Product Expansion as higher twist effects being small or cancelingDeRujula, Georgi, Politzer (1977)  k=1

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

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

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