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Can We Learn Quark Orbital Motion from SSAs?

Can We Learn Quark Orbital Motion from SSAs?. Feng Yuan RIKEN/BNL Research Center Brookhaven National Laboratory. Outline. Why naïve parton model fails for SSAs Two mechanisms: Sivers and twist-3 Unifying these two What we learn from SSA? Summary. Statistics: Big SSA!. Systematics

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Can We Learn Quark Orbital Motion from SSAs?

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

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

  3. 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 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 Workshop on Parton Orbital Angular Momentum

  7. 1/2 1/2−1 −1/2 1/2 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. Beyond the naïve parton model in which quarks are collinear 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 Workshop on Parton Orbital Angular Momentum

  9. 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 Where are the Phases Brodsky, Hwang, Schmidt, 02 Collins, 02 Ji, Belitsky, Yuan, 02 Efremov & Teryaev: 1982 & 1984 Qiu & Sterman: 1991 & 1999 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) 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 Workshop on Parton Orbital Angular Momentum

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

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

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

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

  16. 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 Workshop on Parton Orbital Angular Momentum

  17. 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 Workshop on Parton Orbital Angular Momentum

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

  19. Soft Facotor Workshop on Parton Orbital Angular Momentum

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

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

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

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

  26. What do we learn from SSA? 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) Workshop on Parton Orbital Angular Momentum

  28. Lz≠0 Amplitude and Sivers Function Lz=1 Lz=0 Ji, Ma, Yuan, Nucl. Phys. B (2003) Workshop on Parton Orbital Angular Momentum 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)

  29. 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 Workshop on Parton Orbital Angular Momentum

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