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  1. Quark Helicity Distribution at large-x Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory Collaborators: H. Avakian, S. Brodsky, A. Deur, arXiv:0705.1553 [hep-ph]

  2. Outline • Introduction • Issues at large-x • Quark orbital angular momentum contribution • Outlook

  3. Physics Motivation for large-x • Global fit for the PDFs • New Physics at Tevatron and LHC • Precision Test of EW physics at Colliders • … • Itself is very interesting to study QCD effects, such as resummation; and nucleon structure, such as quark orbital angular momentum,…

  4. Importance of high x in global fit Djouadi and Ferrag, hep-ph/0310209

  5. New physics or PDF uncertainty? Inclusive Jet production at Tevatron, hep-ex/0506038 High x Partons relevant

  6. Theoretical Issues at High x • Resummation • Power counting, pQCD predictions

  7. Why Perturbative calculable • All propagators are far off-shell: ~ kT2/(1-x)>>QCD2 • Spectator power counting

  8. Power counting of Large x structure • Drell-Yan-West (1970) • Farrar-Jackson (1975) • Brodsky-Lepage (1979) • Brodsky-Burkardt-Shmidt (1995) fit the polarized structure functions.

  9. Spin-0 Spin-1 Power counting from LZ=0 • Eight propagators (1-x)8, (1-x)-4 from the scattering, (1-x)-1 from the phase space integral • (1-x)3 • Spectator two quarks with spin-1 configuration will be suppressed by (1-x)2 relative to spin-0 • q- ~(1-x)2q+ q+~ q-~

  10. Quark polarization at large-x JLab Hall A, PRL04 • Power counting rule: q+~ (1-x)3, q-~(1-x)5 • The ratio of  q/q will approach 1 in the limit of x->1 • Brodsky-Burkardt-Shmidt (1995)

  11. Quark orbital angular momentum of proton Light-front wave function decomposition: Total quark spin Lz=0 Lz=1 Lz=1

  12. OAM relevance to nucleon structure • Finite orbital angular momentum is essential for • Anomalous magnetic moment of nucleons • Helicity-flip Pauli form factor F2 • g2 structure function • Asymmetric momentum-dependent parton distribution in a transversely polarized nucleon, Sivers function • Large-x quark helicity distribution • …

  13. For example, the Sivers functions • Quark Orbital Angular Momentum e.g, Sivers function ~ the wave function amplitude with nonzero orbital angular momentum! Vanishes if quarks only in s-state! Ji-Ma-Yuan, NPB03 Brodsky-Yuan, PRD06

  14. Spin-0 Lz=1 contributions to q- • No suppression from the partonic scattering part • Intrinsic pt expansion will lead to power suppression • Total suppression factor will be

  15. Orbital angular momentum contribution • It does not change the power counting • q-~ (1-x)5 • It introduces double logarithms to q- • q-~(1-x)5 log2(1-x) • Coming from additional factor 1/y3y3’ in the intrinsic pt expansion • q-/q+~(1-x)2 log2(1-x) at x->1

  16. Large-x Partons OAM contribution to the Pauli form factor F2 Lz=0 Lz=1 Expansion~1/x3y3 It predicts that F2 goes like (ln2Q2)/Q6 and hence F2/F1 ~ (ln2Q2)/Q2 Belitsky-Ji-Yuan, PRL, 2003

  17. Quark orbital angular momentum contribution at large-x Power counting rule Brodsky-Burkardt- Schmidt 95 Leader-Sidorov-Stamenov 98 q-/q+~ (1-x)2 Quark-orbital-angular Momentum contribution Avakian-Brodsky-Deur-Yuan,07 q-/q+~(1-x)2 log2(1-x) It will be interested to see how this compares with the future data from JLab

  18. 12GeV JLab Upgrade

  19. Outlook • Soft divergence factorization • 1/kt4 for Pion distribution • Quantitative connections to the quark orbital angular momentum • Model building, light-cone wave functions of nucleon • Large logarithms resummation • Log2(1-x) terms should be resummed consistently

  20. Applications to GPDs: GPD for Pion Gauge Link

  21. Where is the t-dependence • In the leading order, there is no t-dependence • Any t-dependence is suppressed by a factor • (1-x)2 (See also, Burkardt, hep-ph/0401159)

  22. Power counting results for pion GPD • in the limit of x->1, • We can approximate the GPD with forward PDF at large x,

  23. GPDs for nucleon Helicitynonflip Helicity flip

  24. Helicity non-flip amplitude • The propagator • Power behavior Forward PDF

  25. Helicity flip amplitude • Since hard scattering conserves quark helicity, to get the • helicity flip amplitude, one needs to consider the hadron • wave function with one-unit of orbital angular momentum Lz=1 Lz=0 • In the expansion of the amplitude at small transverse • momentum l, additional suppression of • (1-x)2 will arise

  26. Two kinds of expansions • Propagator: • Wave function: • The power behavior for the helicity flip amplitude • GPDE Forward PDF • GPDH

  27. Summary for the GPDs’ power prediction • No t-dependence at leading order • Power behavior at large x ~(1-x)2 Forward PDF ~ (1-x)3 = Log(1-x) should also show up

  28. Conclusion • Quark orbital angular momentum contribution changes significantly power counting results for the quark helicity distribution at large-x • More precise determination of these contributions requires a model building and/or a NLO global fit with all experimental data

  29. Factorization (II) • Leading region

  30. Factorization (III) • Factorization formula • The power behavior of q(x) is entirely determined by the eikonal cross section Ji,Ma,Yuan,PLB610(2005)

  31. Power Counting for GPDs • No t-dependence at leading order • Power behavior at large x ~ (1-x)2 Forward PDF ~ (1-x)3 = ~ (1-x)5 Yuan, PRD69(2004)

  32. Why Resummation is Relevant • Additional scale, Q2>> (1-x)Q2>>QCD2 • Real andVirtual contributions are “imbalanced” IR cancellation leaves large logarithms (implicit)

  33. Example I Net Enhancement for the DIS Structure functions A. Vogt

  34. Large-x Partons Example II: Spin Asymmetry • Resummation effects cancel exactly in moment space, and “almost” in x-space W. Vogelsang