Sino-German Workshop on Frontiers in QCD

1. Sino-German Workshop on Frontiers in QCD Azimuthal Asymmetries In Semi-inclusive Deep-Inelastic Scattering Liang Zuo-tang, Shandong University 山东大学　梁作堂 Collaborator: Wang Xin-Nian (LBNL) Contents • Introduction • Collinear expansion in Semi-Inclusive DIS • Results and discussions 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

2. Sino-German Workshop on Frontiers in QCD Introduction: • Intrinsic transverse momentum • Gauge link 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

3. Azimuthal asymmetry in unpolarized SiDIS 1977: Georgi & Politzer, “Clean test to pQCD”, Phys. Rev. Lett. 40,3 (77). lepton plan 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

4. Azimuthal asymmetry in unpolarized SiDIS 1978: Cahn, Intrinsic momentum effects. Phys. Lett. B78B, 269 (1978). Generalized parton model to include an intrinsic transverse momentum but no gluon radiation in the final state: “Realistic studies”: Monte-Carlo, experiments etc. small ph, moderate Q, intrinsic k contribution dominates; large ph , large Q, Georgi-Politzer mechanism dominates. In DIS at intermediate energies: intrinsic transverse momentum k effects have to be taken into account. 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

5. Azimuthal asymmetry in polarized SiDIS Motivation: Single-spin asymmetry in pp collisions azimuthal asymmetry in singly polarized pp collisions up left spin forward right 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

6. Azimuthal asymmetry in polarized SiDIS History: theoretical studies on single-spin asymmetry in p()+p+X Factorization 1978: Kane, Pumplin, Repko, pQCD aN[q()+qq+q]=0. 1991: Sivers, asymmetric quark distribution (Sivers effect) 1993: Boros, Liang, Meng, quark orbital angular momentum & “surface effect” Collins, Proof of non-existence of Sivers effect, asymmetric fragmentation function (Collins effect) 2002: Brodsky, Hwang, Schmidt, quark orbital angular momentum & “final state interaction”. Ji, Yuan, final state interaction = gauge link Collins, 1993’s proof is wrong because forgot gauge link, 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

7. Azimuthal asymmetry in polarized SiDIS Conclusion: both Sivers and Collins effects can exist when gauge link is taken into account. intrinsic transverse momentum and gauge link are important in studying azimuthal asymmetries in SiDIS 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

8. Semi-inclusive DIS with : A BASIC QUESTION The general framework to write the cross section Cross section for epehX Parton distributions Cross section for eqeq without Frag. function for qh+X =   (without transverse momentum: collinear approximation) ？ (with transverse momentum, with gauge link) 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

9. Sino-German Workshop on Frontiers in QCD Gauge link and Collinear expansion in DIS and SiDIS. 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

10. Inclusive DIS: where does the gauge link come from? DIS without QCD interaction The hard part: contracted with the leptonic tensor The matrix element: parton distributions No QCD interactions. Not (color) gauge invariant. 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

11. Inclusive DIS with QCD interaction … ... Not gauge invariant! parton correlation function 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

12. Inclusive DIS with QCD interaction Ellis, Furmanski, Petronzio, (82) Qiu, Sterman (90,91) Collinear expansion:  Expanding the hard parts around k =xp: Decomposition of the gluon field: Using the identities such as, to replace the derivatives etc. 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

13. Inclusive DIS with QCD interaction Contain QCD interactions. (Color) gauge invariant ! 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

14. Inclusive DIS with QCD interaction Definition of parton distributions 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

15. Inclusive DIS with QCD interaction Conclusion: Collinear expansion is the necessary procedure to obtain the gauge invariant parton distributions. Question: How about semi-inclusive DIS? 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

16. Semi-Inclusive DIS with QCD interaction Consider first e +pe + q + X, i.e., no fragmentation … ... 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

17. Semi-Inclusive DIS with QCD interaction The same collinear expansion technique leads to Gauge invariant parton distributions/correlations (with gauge link) 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

18. Sino-German Workshop on Frontiers in QCD Consequences and discussions 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

19. SiDIS with : Direct consequence I Consider the contribution from Cross section for epeqX Parton distributions = Cross section for eqeq without  (without transverse momentum)  (with transverse momentum) 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

20. SiDIS with : Direct consequence II A great simplification of : left cut right cut 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

21. SiDIS with : Direct consequence II This leads to Only is relevant in semi-inclusive deep-inelastic lepton-nucleon scattering. 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

22. SiDIS with : differential cross-section to 1/Q Further calculations in the standard way:  Expanding the matrices ’s in terms of ’s: The general Lorentz-structure of the coefficient functions are: un-integrated parton distribution Sivers’ function Equation of motion 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

23. SiDIS with : differential cross-section to 1/Q To the order of 1/Q: should be used to extract Sivers function and so on. where is the azimuthal angle of 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

24. SiDIS with : An observation By integrating over a single-spin asymmetry at fixed k- the terms vanish if integrated over k-from -∞ to +∞, no single-spin asymmetry for inclusive DIS, but … … 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

25. Single-spin asymmetry for the “triggered” inclusive DIS Consider a jet of hadrons It corresponds to a “triggered” inclusive DIS, i.e., a large momentum hadron in photon direction 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

26. Single-spin asymmetry for the “triggered” inclusive DIS At the hadron level, it corresponds to Measurements by e.g. HERMES at HERA? 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

27. Conclusions and Outlook Collinear expansion can be extended to SiDIS; Naïve extension of parton distribution convoluting with eq  eq cross section to include intrinsic transverse momentum is incorrect. The existence of the projection operator simplifies very much. A new single-spin asymmetry is predicted to 1/Q. Extending to include fragmentation is underway. 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)