Unique Description for Single Transverse Spin Asymmetries

1. Unique Description for Single Transverse Spin Asymmetries Feng Yuan RBRC, Brookhaven National Laboratory Collaborators: Kouvaris, Ji, Qiu, Vogelsang RSC Meeting, RIKEN

2. x z y What is Single Spin Asymmetry? • Scattering a transverse spin polarized proton on unpolarized target (another hadron or a photon) • the cross section contains a term K? S? P P’ RSC Meeting, RIKEN

3. Why Does SSA Exist? • Single Spin Asymmetry requires • Helicity flip: one must have a reaction mechanism for the hadron to change its helicity (in a cut diagram) • Final State Interactions (FSI): to generate a phase difference between two amplitudes The phase difference is needed because the structure S ·(p × k) violate the naïve time-reversal invariance RSC Meeting, RIKEN

4. Naïve Parton Model Fails to Explain Large SSAs • If the underlying scattering mechanism is hard, the naïve parton model generates a very small SSA: (G. Kane et al, PRL41, 1978) • It is in general suppressed by αS mq/Q • We have to go beyond the naïve parton model to understand the large SSAs observed in hadronic reactions RSC Meeting, RIKEN

5. Two Mechanisms in QCD • Transverse Momentum Dependent (TMD) Parton Distributions and Fragmentations • Sivers function, Sivers 90 • Collins function, Collins 93 • Gauge invariant definition of the TMDs: Brodsky, Hwang, Schmidt 02; Collins 02 ; Belitsky, Ji, Yuan 02; Boer, Mulders, Pijman, 03 • The QCD factorization: Ji, Ma, Yuan, 04; Collins, Metz, 04 • Twist-three Correlations (collinear factorization) • Efremov-Teryaev, 82, 84 • Qiu-Sterman, 91,98 • Kanazawa-Koike, 99-02 RSC Meeting, RIKEN

6. TMD: the quark orbital angular momentum leads to hadron helicity flip The factorizable final state interactions --- the gauge link provides the phase Twist-three: the gluon carries spin, flipping hadron helicity The phase comes from the poles in the hard scattering amplitudes How Do They Contribute? RSC Meeting, RIKEN

7. Connecting these Two • Relations at the matrix elements level TF(x,x)=s d2k |k|2qT(x,k) • Boer, Mulders, Pijman 03 • Attempts to connect these two in some processes, no definite conclusion: • Ma, Wang, 03 • Bacchetta, 05 Qiu-Sterman Sivers RSC Meeting, RIKEN

8. Unifying the Two Mechanisms (P? dependence of SSAs) • At low P?, the non-perturbative TMD Sivers function will be responsible for its SSA • When P?» Q, purely twist-3 contributions • For intermediate P?, QCD¿ P?¿ Q, we should see the transition between these two • An important issue, at P?¿ Q, these two should merge, showing consistence of the theory (Ji, Qiu, Vogelsang, Yuan, PLB638,178;PRD73,094017; PRL97, 082002, 2006) RSC Meeting, RIKEN

9. Recall the TMD Factorization RSC Meeting, RIKEN

10. SIDIS: at Large P? • When q?ÀQCD, the Pt dependence of the TMD parton distribution and fragmentation functions can be calculated from pQCD, because of hard gluon radiation • Single Spin Asymmetry at large P? is not suppressed by 1/Q, but by 1/P? RSC Meeting, RIKEN

11. SSA in the Twist-3 approach Fragmentation function: \hat q(x’) Twist-3 quark-gluon Correlation: TF(x1,x2) Collinear Factorization: Qiu,Sterman, 91 RSC Meeting, RIKEN

12. Factorization guidelines Reduced diagrams for different regions of the gluon momentum: along P direction, P’, and soft Collins-Soper 81 RSC Meeting, RIKEN

13. Final Results • P? dependence • Which is valid for all P? range Sivers function at low P? Qiu-Sterman Twist-three RSC Meeting, RIKEN

14. Transition from Perturbative region to Nonperturbative region? • Compare different region of P? Nonperturbative TMD Perturbative region RSC Meeting, RIKEN

15. More over • At low P?¿ Q, the second term vanishes, use the Sivers function only • P?-moment of the asymmetry can be calculated from twist-three matrix element • In SIDIS, for the Sivers asymmetry Boer-Mulders-Tangelmann, 96,98 RSC Meeting, RIKEN

16. Compare to the HERMES data • TF from the fit to single inclusive hadron asymmetry data (fixed target & RHIC) • See Vogelsang’s talk, Kouvaris-Qiu-Vogelsang-Yuan, hep-ph/0609238 not corrected for acceptance and smearing RSC Meeting, RIKEN

17. Dijet-correlation at RHIC • Proposed by Boer-Vogelsang • Initial state and/or final state interactions? • Bacchetta-Bomhof-Mulders-Pijlman,04-06 • Nonzero leading order in a model-independent way, for example RSC Meeting, RIKEN

18. At this order, the asymmetry can be related to that in DIS • qTDIS--- Sivers function from DIS q?--- imbalance of the dijet Hsivers depends on subprocess (see Bomhof’s talk) RSC Meeting, RIKEN

19. Predictions for dijet-correlation AN • VY05: Vogelsang-Yuan, Phys.Rev.D72:054028,2005 • Sivers functions fit to the HERMES data • Model I: uT(1/2)/u=-0.81x(1-x),dT(1/2)/u=1.86x(1-x) • Model II: uT(1/2)/u=-0.75x(1-x),dT(1/2)/d=2.76x(1-x) • Caution: Sudakov effects may reduce the asymmetry, Boer-Vogelsang 04 d-quark Modified Initial/final VY05 u-quark RSC Meeting, RIKEN

20. Global Picture for SSAs SIDIS Large P? SIDIS Drell-Yan TMD Gluon Sivers Charmonium Large q? Drell-Yan Quark-gluon correlation Dijet-Corr. Dijet at RHIC Single Inclusive at RHIC RSC Meeting, RIKEN

21. Summary • We are in the early stages of a very exciting era of transverse spin physics studies • Many data are coming out, and new experiments are proposed to provide a detailed understanding of the spin degrees of freedom, especially for the quark orbital motion • We will learn more about nucleon structure from these studies RSC Meeting, RIKEN