Unique Description for SSAs in DIS and Hadronic Collisions Feng Yuan RBRC, Brookhaven National Laboratory Collaborators: Kouvaris, Ji, Qiu, Vogelsang SPIN2006, Kyoto

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? z x y S? P P’ SPIN 2006, Kyoto

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 SPIN 2006, Kyoto

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 SPIN 2006, Kyoto

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, Pijlman, 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, 00-04 SPIN 2006, Kyoto

How Do They Contribute? 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 SPIN 2006, Kyoto

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 SPIN 2006, Kyoto

Territories and unification Twist-three: the single inclusive hadron production in pp, require large P? TMD: low P?, require additional hard scale like Q2 in DIS and Drell-Yan, P?¿ Q Connecting these two, at the matrix elements level TF(x,x)=s d2k |k|2 qT(x,k) Boer, Mulders, Pijlman 03 Qiu-Sterman Sivers SPIN 2006, Kyoto

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, PRL97, 082002;PRD73,094017; PLB638,178, 2006) SPIN 2006, Kyoto

Recall the TMD Factorization SPIN 2006, Kyoto

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? SPIN 2006, Kyoto

SSA in the Twist-3 approach Fragmentation function: \hat q(x’) Twist-3 quark-gluon Correlation: TF(x1,x2) Collinear Factorization: Qiu,Sterman, 91 SPIN 2006, Kyoto

Factorization guidelines Reduced diagrams for different regions of the gluon momentum: along P direction, P’, and soft Collins-Soper 81 SPIN 2006, Kyoto

Final Results P? dependence Which is valid for all P? range Sivers function at low P? Qiu-Sterman Twist-three SPIN 2006, Kyoto

Transition from Perturbative region to Nonperturbative region? Compare different region of P? Nonperturbative TMD Perturbative region SPIN 2006, Kyoto

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 SPIN 2006, Kyoto

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 Indicate the consistency of SSAs in DIS and hadron collisions not corrected for acceptance and smearing See also, Efremov, et al., PLB612,233 (20005) SPIN 2006, Kyoto

Dijet-correlation at RHIC Proposed by Boer-Vogelsang Initial state and/or final state interactions? Bacchetta-Bomhof-Mulders-Pijlman,04-06 We can illustrate the initial and final state interactions in a model-independent way, at nonzero leading order, for example SPIN 2006, Kyoto

At this order, the asymmetry can be related to that in DIS, in leading power of q?/P? qTDIS--- Sivers function from DIS q?--- imbalance of the dijet Hsivers depends on subprocess (see Bomhof’s talk) SPIN 2006, Kyoto

Predictions for dijet-correlation AN d-quark Modified Initial/final VY05 u-quark VY05: Vogelsang-Yuan, Phys.Rev.D72:054028,2005 Sivers functions fit to the HERMES data Model II: uT(1/2)/u=-0.75x(1-x),dT(1/2)/d=2.76x(1-x) Cautions: Scale difference, HERMES <Q2>~2GeV2 Sudakov effects may affect the asymmetry, Boer-Vogelsang 04 SPIN 2006, Kyoto

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 SPIN 2006, Kyoto

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 SPIN 2006, Kyoto