Understanding Spin Dependence of SSA: Nonpert. to Pert. Transition

1. P? Dependence of SSA-- from nonpert. to pert. Feng Yuan RBRC , Brookhaven National Laboratory References: Ji, Ma, Yuan, Phys. Rev. D70, 074021; Phys. Lett. B597, 299; hep-ph/0503015; to be published; SIR05 JLab

2. Outline • Introduction • Low PT region: TMD parton distributions and QCD factorization • High PT region: quark-gluon-quark correlation (ETQS mechanism) • Summary SIR05 JLab

3. Inclusive and Semi-inclusive DIS Inclusive DIS: Partonic Distribution depending on the longitudinal momentum fraction Semi-inclusive DIS: Probe additional information for partons’ transverse distribution in nucleon SIR05 JLab

4. Different PT Region • Integrate out PT -- similar to inclusive DIS, probe int. PDF • Large PT (>>QCD) -- hard gluon radiation, can be calculated from perturbative QCD, Polarized ->q-g-q correlations • Low PT (~QCD) -- nonperturbative information (TMD): new factorization formula SIR05 JLab

5. TMD Physics • A way to measure Transversity Distribution, the last unkownleading twist distribution Collins 1993 • The Novel Single Spin Asymmetries • Connections with GPDs, and Quantum Phase Space Wigner distributions • Quark Orbital Angular Momentum and Many others … SIR05 JLab

6. TMD Distribution: the definition Gauge Invariance requires the Gauge Link Brodsky,Hwang,Schmidt 02’ Collins 02’ Belitsky,Ji,Yuan 02’ SIR05 JLab

7. This definition is also consistent with the QCD factorization SIR05 JLab

8. Illustrate the Factorization(one-loop order) • Take an on-shell quark target • Calculate the TMD Dis. and F.F. • Separate the cross section into different pieces • Show the soft divergence is cancelled out SIR05 JLab

9. TMD Dis. At One-loop SIR05 JLab

10. No soft divergence • Collinear divergence : ln(m2) • Double Logarithms: ln2(2 b2) --- Collins-Soper equation SIR05 JLab

11. Factorization Soft Factor SIR05 JLab

12. The Factorization Applies to • Semi-inclusive DIS (polarized and unpolarized) • Drell-Yan at Low transverse momentum • Di-hadron production in e+e- annihilation (extract the Collins function) • Di-jet and/or di-hadron correlation at hadron collider (work in progress) • Many others, … SIR05 JLab

13. Large Logarithms Resummation • At low PT, PT<< Q, we must resum the large logarithms snln2n-1(Q2/PT2) -- Dokshitzer, Diakonov, Troian, 1978 -- Parisi, Petronzio, 1979 • These large logarithms can be resummed by solving the energy evolution equation for the TMD parton dis. -- Collins-Soper 1981 SIR05 JLab

14. TMD: the Energy Dependence • The TMD distributions depend on the energy of the hadron! (or Q in DIS) • One can write down an evolution equation in ζ=2(P+)2v-/v+ • K and G obey an RG equation, Collins and Soper (81’) μ independent! SIR05 JLab

15. Spin-dep. Collins-Soper Eq. • The evolution kennel is the same for all the leading-twist TMDquark distributions • Kt-even ones: q(x, k┴), ΔqL, δqT • Kt-odd ones: qT(x, k┴), δq, ΔqT, δqL Idilbi, Ji, Ma, Yuan: PRD70 (2004) SIR05 JLab

16. Large Logarithms Resumm. (II) • After resummation, large logarithms associated with Q2 can be factorized into the Sudakovform factors, e.g. • And the Sudakov form factor SIR05 JLab

17. Double Logarithmic (DL) Approx. • If Q2 is not too large, DL approx. applies. The Sudakov suppression form factor then only depends on Q2 • The Q2 dependence of the structure functions can be factorized out • We can predict the PT distribution at higher Q2 from that of lower Q2 The PT spectrum of the polarization asymmetry will be the same for different Q2 at fixed B and

18. Phenomenogical applications of the QCD resummation to the PT spectrum of EW bosons production have been very successful Yuan, Nadolsky, Ladinsky, Landry, Qiu, Zhang, Berger, Li, Laenen, Sterman, Vogelsang, Kulesza, Bozzi, Catani, deFlorian, Kulesza, Stirling, and many others, … working even at NNLL level for some • Our interest however, is in the region of not so large Q2, where DLA (LLA) should apply SIR05 JLab

19. Drell-Yan at Fixed Target QT spectrum from E288, PRD23,604(81) Normalized to the same spectrum! Consequence of DLA SIR05 JLab

20. At very large Q2 (e.g., Z0 and W boson), DL Approx. breaks down SIR05 JLab

21. Power Counting at Large PT • kt-even distributions have the same dependence on kt • kt-odd distributions are suppressed at large kt • Power Counting Rule kt-even: 1/kt2 kt-odd: 1/kt3 SIR05 JLab

22. SIDIS: at Large PT • Large PT SIDIS can be calculated from perturbative QCD (same for TMDs) • Polarized scattering depends on the quark-gluon correlations in the nucleon • Single Spin Asymmetry (Sivers and Collins) at large P? is not suppressed by 1/Q, but by 1/PT SIR05 JLab

23. An example: SSA at Large PT • ETQS mechanism • It is suppressed by 1/PT Efremov & Teryaev: 1982 & 1984 Qiu & Sterman: 1991 & 1999 Quark-gluon correlations Ji,Ma,Yuan, work in progress SIR05 JLab

24. Transition from Perturbative region to Nonperturbative region? • Compare different region of PT Nonperturbative TMD Perturbative region SIR05 JLab

25. Summary • Q2 evolution of the PT spectrum is calculable from perturbative QCD, by solving the associated Collins-Soper eqs • Large PT SSA depends on the quark-gluon-quark correlation of nucleon (ETQS), which is suppressed by 1/PT, not 1/Q • Detailed study of the PT dependence will provide us information for the perturbative to nonperturbative transition of QCD SIR05 JLab