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

2. 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’
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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
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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
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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
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6. 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
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7. 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
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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, , 2006)
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9. Recall the TMD Factorization
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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?
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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
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12. Factorization guidelines
Reduced diagrams for different regions of the gluon momentum:
along P direction, P’, and soft Collins-Soper 81
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13. Final Results P? dependence Which is valid for all P? range
Sivers function at low P?
Qiu-Sterman Twist-three
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14. Transition from Perturbative region to Nonperturbative region?
Compare different region of P?
Nonperturbative TMD
Perturbative region
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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
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16. 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/
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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
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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)
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19. 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
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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
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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
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