1. Collins effect in the collinear factorization approach Jian Zhou (ShanDong University, China & LBNL, US) Collaborators: Feng Yuan (LBNL, US) Based on the paper: e-Print: arXiv:0903.4680

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

3. Single spin asymmetry pp π L R Naive parton model: 1978, Kane, Pumplin, Repko

4. Two mechanisms in QCD  1: Transverse momentum dependent (TMD) factorizaion Sivers distribution function f 1T ┴ (x,k T 2 ) Sivers 90 Collins fragmentation function H 1 ┴ (x,k T 2 ) Collins 93  2: Collinear higher-twist factorization twist-3 distribution function T F (x,x1) Qiu-Sterman 91; Efremov-Teryaev 82, 84 t wist-3 fragmentation function E F (x,x1) ? Koike 02; Meissner; Metz 08 kTkT STST P S T (P X k T ) (zk+p T ) ~pTXsT~pTXsT

5. The unification of two mechanisms  Twist-three:  QCD << P T assuring the perturbative calculation make sense  TMD: low P T, require additional hard scale like Q 2 in DIS and Drell-Yan, P T <<Q  Overlap:  QCD << P T <<Q, unifying these two Mechanisms  Crucial step: TMD distributions at large k T X. Ji, J.W. Qiu, W.Vogelsang, F. Yuan, 06

6. k T -odd TMD distributions at large K T Generally speaking, TMD distributions can be calculated by using collinear approach radiated gluon lead to large k T gluon rescattering lead to asymmetry k T distribution factorized into twist-3 collinear functions accordingly, T F (x,x 1 ), T F (σ) (x,x 1 ),etc. The calculation of Collins function follows the similar procedure, but with significant difference !

7. Collins function and its k T moment  Kt-moment defines a twist-3 fragmentation function

8. Yuan-Zhou, 09 twist-3 correlation function contribute to Collins function X.Ji, PRD94; Koike, 02-06 i H 1 (z, z 1 ) It is not ruled out by time reversal invariance argument ! The imaginary phase necessary for nonzero SSA comes up automatically ! gluon pole process dependent process independent combining with the different matrix elements F-type fragmentation correspondingly define: E F (z,z 1 ), H F (z,z 1 ) E 1 (z, z 1 )+

9. Universality of the Collins Fragmentation ep--> e Pi Xe+e--> Pi Pi X pp--> jet(->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.

10. Universality of the Collins Fragmentation The arguments of E F (z,z 1 ) are fixed by picking up pole contribution  soft gluon pole contribution z=z 1  hard gluon pole contribution z 1 =z h, z>z h fortunately... Thanks to its support properties: E F (z,z 1 )=0 when z=z 1 or z>z 1 S. Mei ß ner A. Metz 08 Process dependent contribution to Collins function vanishes ! We are only left with contributions from H F \hat{H} (the moment of collins function)

11. Collins function at large kt typpical diagrams: where we changed the normalizationof H F (z,z 1 )

12. 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

13. Summary  We have identified the correspondent collinear twist-three fragmentation function for the Collins effects  The Collins function calculated from this twist-three function is universal, 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 X using collinear factorization approach  SSA in the process pp--> jet(->Pi) X from fragmentation effect using collinear factroization appraoch Thank you for your attention.