Collins effect in the collinear factorization approach

1. Collins effect in the collinear factorization approach Jian Zhou (ShanDong University, China & LBNL, US) Collaborators: Feng Yuan (LBNL, US) Based on the paper: Phys.Rev.Lett.103:052001,2009.

2. Outline: • 1: Brief review • 2: TMDs in the collinear factorization approach • 3: Collins function in the collinear factroization approach • 4: Summary & outlook

3. L π p p R Single spin asymmetry

4. 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) A phase difference: the phase difference is needed because the structure S ·(p × k) violate the 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, 1978), It is in general suppressed byαSmq/Q See also, Ma-Sang, arXive: 0809.1791 We have to go beyond this naïve picture

6. ST (zk+pT) ~pTXsT kT Two mechanisms in QCD ST (PXkT) P • 1:Transverse momentum dependent (TMD) factorizaion Sivers distribution function f1T┴ (x,kT2)Sivers 90 Collins fragmentation function H1┴(x,kT2)Collins 93 • 2:Collinear higher-twist factorization twist-3 distribution function TF(x,x1) Qiu-Sterman 91; Efremov-Teryaev 82, 84 twist-3 fragmentation function EF(x,x1) ? Koike 02; Meissner; Metz 08

7. The unification of two mechanisms • Twist-three: QCD<< PT assuring the perturbative calculation make sense • TMD: low PT, require additional hard scale like Q2 in DIS and Drell-Yan, PT<<Q • Overlap: QCD<< PT<<Q, unifying these two Mechanisms • Crucial step: TMD distributions at large kT X. Ji, J.W. Qiu, W.Vogelsang, F. Yuan, 06

8. A unified picture (leading pt/Q) Transverse momentum dependent Collinear/ longitudinal PT QCD QT Q << << Courtesy of Feng Yuan

9. Polarized TMD Quark Distributions Nucleon Unpol. Long. Trans. Quark Unpol. Long. Trans. Boer, Mulders, Tangerman (96&98)

10. Perturbative tail is calculable Power counting, Brodsky-Farrar, 1973 Integrated Parton Distributions Twist-three functions Transverse momentum dependence

11. Generic results Large logs Splitting kernel Kt-even TMDs

12. kT-odd TMD distributions at large KT Generally speaking, TMD distributions can be calculated by using collinear approach radiated gluon lead to large kT gluon rescattering lead to asymmetry kT distribution factorized into twist-3 collinear functions accordingly, TF(x,x1), TF(σ)(x,x1) ,etc. The calculation of Collins function follows the similar procedure, but with significant difference !

13. Large kt TMDs • Color factors, CF: a1-4,b1-4,c2,c4 • 1/2Nc: c1,c3, d1-4 • CA/2: e1-4 • a1-4 • b1-4,c1,c3,e1-4 • b1-4,c1-4,d1-4,e1-4

14. X. Ji, J.W. Qiu, W.Vogelsang, F. Yuan, 06 • g1T and h1L J. Zhou, F. Yuan, Z-T Liang,2009 Sivers and Boer-Mulders

15. where ,

16. spin dependent Drell-Yan procsess Single transverse spin asymmetry AUT. Double spin(longitudinal-transverse) asymmetry ALT. S. Arnold, A. Metz and M. Schlegel • In the naive parton model, • these structure functions either are zero or power suppresed.

17. and two more twist-3 functions: higher-twist collinear factorization as long as QT >>ΛQCD

18. Structure functions at small QT In the small QT limit, ΛQCD<< QT<<Q Jian Zhou, Feng Yuan, Zuo-Tang Liang, arXiv:0909.2238

19. where,

20. On the TMD factorization side... QT<<Q, ensuring TMD factorization is valid X. Ji, J. P. Ma and F. Yuan

21. Structure functions in the TMD factorization S. Arnold, A. Metz and M. Schlegel Using the large kT TMD distributions expressions, we find the structure functions from two approaches do match in the overlap region ΛQCD<< QT<<Q

22. Collins function and its kT moment • Kt-moment defines a twist-3 fragmentation function

23. F-type fragmentation gluon pole combining with the different matrix elements E1(z, z1) + process dependent process independent twist-3 correlation function contribute to Collins function iH1(z, z1) X.Ji, PRD94; Koike, 02-06 Yuan-Zhou, 09 It is not ruled out by time reversal invariance argument ! The imaginary phase necessary for nonzero SSA comes up automatically ! correspondingly define: EF(z,z1), HF(z,z1)

24. Universality of the Collins Fragmentation pp--> jet(->Pi) X ep--> e Pi X e+e--> Pi Pi X Metz 02, Collins-Metz 02, Yuan 07, 08 Gamberg-Mukherjee-Mulders 08 Conjecture: the Collins function should be the same among the different processes, such as e^+e^- , SIDIS and pp.

25. Universality of the Collins Fragmentation • The arguments of EF(z,z1) are fixed by picking up pole contribution • soft gluon pole contribution z=z1 • hard gluon pole contribution z1=zh, z>zh fortunately... • Thanks to its support properties: • EF(z,z1)=0 when z=z1 or z>z1 S. Meißner A. Metz 08 Process dependent contribution to Collins function vanishes ! We are only left with contributions from HF \hat{H} (the moment of collins function)

26. Collins function at large kt typpical diagrams: where we changed the normalization of HF(z,z1)

27. Collins contribution in SIDIS • This result can be reproduced by the TMD factorization with Collins • function calculated, the quark transversity distribution • This demonstrate that the TMD and collinear approaches are consistent • in the intermediate transverse momentum region for the Collins effects

28. Summary • We have identified the correspondent collinear twist-three fragmentation function for the Collins effects • The Collins function calculated from this twist-three function isuniversal, does not dependent on the gauge link direction • We have shown that the TMD and collinear approaches are consistent in the intermediate transverse momentum region. outlook • cos(2φ) azimuthal asymmetry in the process e+e--> Pi Pi Xusing collinear factorization approach • SSA in the process pp--> jet(->Pi) X from fragmentation effect using collinear factroization appraoch Thank you for your attention.