1. Can We Learn Quark Orbital Motion from SSAs?
Feng Yuan
RIKEN/BNL Research Center
Brookhaven National Laboratory
Feb , 2006
Workshop on Parton Orbital Angular Momentum

2. Workshop on Parton Orbital Angular Momentum
Outline
Why naïve parton model fails for SSAs
Two mechanisms: Sivers and twist-3
Unifying these two
What we learn from SSA?
Summary
Feb , 2006
Workshop on Parton Orbital Angular Momentum

3. Workshop on Parton Orbital Angular Momentum
Statistics: Big SSA!
Systematics
AN is significant in the fragmentation region of the polarized beam: Valence feature
AN and its sign show a strong dependence on the type of polarized beam and produced particles: Flavor dependence
Feb , 2006
Workshop on Parton Orbital Angular Momentum

4. Why Does SSA Exist? Single Spin Asymmetry is proportional to
Im (MN * MF)
where MN is the normal helicity amplitude
and MF is a spin flip amplitude
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) formally violate naïve time-reversal invariance

5. Naïve Parton Model Fails
If the underlying scattering mechanism is hard, the naïve parton model generates a very small SSA: (G. Kane et al, PRL41, 1978)
The only way to generate the hadron helicity-flip is through quark helicity flip, which is proportional to current quark mass mq
To generate a phase difference, one has to have pQCD loop diagrams, proportional to αS
Therefore a generic pQCD prediction goes like
AN ~ αS mq/Q
Every factor suppresses the SSA!

6. Beyond the Naïve Parton Model
Transverse Momentum Dependent Parton Distributions
Sivers function, Sivers 90
Collins function, Collins 93
Brodsky, Hwang, Schmidt 02
Collins 02
Belitsky, Ji, Yuan 02
Twist-three Correlations
Efremov-Teryaev, 82, 84
Qiu-Sterman, 91,98
Feb , 2006
Workshop on Parton Orbital Angular Momentum

7. Parton Orbital Angular Momentum and Gluon Spin
The hadron helicity flip can be generated by other mechanism in QCD
Quark orbital angular momentum (OAM): Therefore, the hadron helicity flip can occur without requiring the quark helicity flip.
1/2
−1/2
1/2−1
Beyond the naïve parton model in which quarks are collinear
Feb , 2006
Workshop on Parton Orbital Angular Momentum

8. Parton OAM and Gluons (cont.)
A collinear gluon carries one unit of angular momentum because of its spin. Therefore, one can have a coherent gluon interaction -1
1/2
1/2
1/2
−1/2
Quark-gluon quark correlation function!
Efremov & Teryaev: 1982 & 1984
Qiu & Sterman: 1991 & 1999
Feb , 2006
Workshop on Parton Orbital Angular Momentum

9. Workshop on Parton Orbital Angular Momentum
Where are the Phases
TMD: the factorizable final state interactions --- the gauge link in the definition of the TMDs
Twist-three quark-gluon correlation: poles from the hard scattering amplitudes
Brodsky, Hwang, Schmidt, 02
Collins, 02
Ji, Belitsky, Yuan, 02
Efremov & Teryaev: 1982 & 1984
Qiu & Sterman: 1991 & 1999
Feb , 2006
Workshop on Parton Orbital Angular Momentum

10. Unifying the Two Mechanisms (P? dependence of DY)
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 emerge, showing consistence of the theory
(Ji, Qiu, Vogelsang, Yuan, to appear)
Feb , 2006
Workshop on Parton Orbital Angular Momentum

11. A General Diagram in Twist-3
Antiquark distribution:
\bar q(x’)
Twist-3 quark-gluon
Correlation: TF(x1,x2)
Collinear Factorization:
Qiu,Sterman, 91
Feb , 2006
Workshop on Parton Orbital Angular Momentum

12. Workshop on Parton Orbital Angular Momentum
Soft and Hard Poles
Soft: xg=0
Hard: xg= 0
Feb , 2006
Workshop on Parton Orbital Angular Momentum

13. Diagrams from Soft Poles
Feb , 2006
Workshop on Parton Orbital Angular Momentum

14. Diagrams from Hard Poles
Feb , 2006
Workshop on Parton Orbital Angular Momentum

15. Workshop on Parton Orbital Angular Momentum
Cross sections
Unpolarized cross section
Polarized cross section,
Feb , 2006
Workshop on Parton Orbital Angular Momentum

16. Workshop on Parton Orbital Angular Momentum
Low q? limit
Keeping the leading order of q?/Q,
Which should be reproduced by the Sivers function at the same kinematical limit, by the factorization
Feb , 2006
Workshop on Parton Orbital Angular Momentum

17. Workshop on Parton Orbital Angular Momentum
TMD Factorization
When q?¿ Q, a TMD factorization holds,
When q?ÀQCD, all distributions and soft factor can be calculated from pQCD, by radiating a hard gluon
Feb , 2006
Workshop on Parton Orbital Angular Momentum

18. TMD Antiquark at k?ÀQCD
See, e.g., Ji, Ma, Yuan, 04
Feb , 2006
Workshop on Parton Orbital Angular Momentum

19. Workshop on Parton Orbital Angular Momentum
Soft Facotor
Feb , 2006
Workshop on Parton Orbital Angular Momentum

20. Sivers Function from twist-3: soft poles
Feb , 2006
Workshop on Parton Orbital Angular Momentum

21. Hard Poles for Sivers Function
Feb , 2006
Workshop on Parton Orbital Angular Momentum

22. Sivers Function at Large k?
1/k?4 follows a power counting
Plugging this into the factorization formula, we indeed reproduce the polarized cross section calculated from twist-3 correlation
Feb , 2006
Workshop on Parton Orbital Angular Momentum

23. Factorization Arguments
Reduced diagrams for different regions of the gluon momentum:
along P direction, P’, and soft Collins-Soper 81
Feb , 2006
Workshop on Parton Orbital Angular Momentum

24. Workshop on Parton Orbital Angular Momentum
Final Results
P? dependence
Which is valid for all P? range
Sivers function at low P?
Qiu-Sterman Twist-three
Feb , 2006
Workshop on Parton Orbital Angular Momentum

25. Transition from Perturbative region to Nonperturbative region?
Compare different region of P?
Nonperturbative TMD
Perturbative region
Feb , 2006
Workshop on Parton Orbital Angular Momentum

26. What do we learn from SSA?
Feb , 2006
Workshop on Parton Orbital Angular Momentum

27. Nonzero Sivers function implies
Nonzero Quark Orbital Angular Momentum
e.g, Siver’s function ~ the wave function amplitude with orbital angular momentum! Vanishes if quarks only in s-state!
Friends:
Pauli Form Factor F2(t)
Spin-dependent structure function g2(x)
Generalized Parton Distribution E(x, ξ, t)
Feb , 2006
Workshop on Parton Orbital Angular Momentum

28. Lz≠0 Amplitude and Sivers Function
All distributions can be calculated using the wave function. The amplitudes are not real because of FSI. Siver’s function:
Similar expressions for F2(Q), g2(x) and E(x,t)
Lz=0
Lz=1
Ji, Ma, Yuan, Nucl. Phys. B (2003)
Feb , 2006
Workshop on Parton Orbital Angular Momentum

29. Workshop on Parton Orbital Angular Momentum
Concluding Remarks
Nonzero Sivers function indeed indicates the existence of the Quark Orbital Angular Momentum
However, there is no definite relation between these two so far
We, as theorists, need to work hard for that goal, as asked by experimentalists
Feb , 2006
Workshop on Parton Orbital Angular Momentum