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

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’ RSC Meeting, RIKEN

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

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

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

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 RSC Meeting, RIKEN

Connecting these Two Relations at the matrix elements level TF(x,x)=s d2k |k|2 qT(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

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

Recall the TMD Factorization RSC Meeting, RIKEN

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

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

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

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

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

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

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

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

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

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 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 RSC Meeting, RIKEN

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

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