1. Yu-kun Song (USTC) 2013.10.28 Jinan YKS, Jian-hua Gao, Zuo-tang Liang, Xin-Nian Wang, arXiv:1308.1159 Collinear expansion and SIDIS at twist-3

2. Outline Introduction Collinear expansion extended to SIDIS Azimuthal asymmetries & nuclear effects Discussions and outlook

3. A large scale (Q) and a small scale (k  ) Large scale → factorization Small scale → structure information (intrinsic k  ) An ideal probe of nucleon/nuclear structure ! TMD factorization works at leading twist Higher twist Gauge invariant parton correlation functions Factorization proof/arguments NLO calculations Experiments : Compass, Jlab, EIC,… Semi-inclusive DIS

4. X Sterman-Libby power counting Leading twist Gauge invariant parton distribution functions Finite, perturbatively calculable partonic cross section Leading twist: collinear approximation

5. Higher twist: collinear expansion Gauge invariant parton correlation functions Finite, perturbatively calculable partonic cross section X Higher twist (1/Q power corrections)

6. Systematic applications to SIDIS Collinear expansion in DIS Ellis, Furmanski, Petronzio, 1982, 1983 Qiu, 1990 Collinear expansion applied to SIDIS e+N →e+q+X Liang, Wang, 2006 Nuclear medium effects of azimuthal asymmetries Gauge link → nuclear modification of PDFs Liang, Wang, Zhou, 2008 SIDIS e+N →e+q+X at twist-4 YKS, Gao, Liang, Wang, 2011 Doublely polarized SIDIS e+N →e+q+X at twist-3 YKS, Gao, Liang, Wang, Arxiv: 1308.1159

7. [Ellis, Furmanski, Petronzio, 1982,1983 ;Qiu,1990] Collinear Expansion: 1. Taylor expand at, and decompose 2. Apply Ward Identities 3. Sum up and rearrange all terms, Collinear expansion in DIS

8. In the low region, we consider the case when final state is a quark(jet) Compared to DIS, the only difference is the kinematical factor Collinear expansion is naturally extended to SIDIS Parton distribution/correlation functions are -dependent Collinear expansion in SIDIS e+N →e+q+X [Liang, Wang, 2006]

9. : color gauge invariant Hadronic tensor for SIDIS

10. Structure of correlation matrices Time reversal invariance relate and Lorentz covariance + Parity invariance, SIDIS DY

11. Structure of correlation matrices QCD equation of motion,, induce relations

12. Relations from QCD EOM Sum up and, one has (up to twist-3) Explicit color gauge invariance for and. Explicit EM gauge invariance

13. Unpolarized SIDIS at twist-4 level Cross section for Twist-4 parton correlation functions 13 [YKS, Gao, Liang, Wang,2011]

14. Doubly polarized e+N →e+q+X at twist-3 [YKS, Gao, Liang, Wang, 2013] Leading twist Twist-3

15. k T - broadening of PDF in a nucleus Two facts about the gauge link Generated by QCD multiple gluon scattering between struck quark and medium It induce physical effects, cannot be removed by a wise choice of gauge → Different interaction induce different PDFs: f N q & f A q More FSI !

16. k T - broadening of PDF in a nucleus Relations between nucleon and nuclear PDFs simplify under “maximal two-gluon exchange ” approximation. It is just Gaussian broadening. More FSI diffuse the scattered parton!

17. Nuclear modification of ˂cosφ>˂cos2φ> Nuclear twist-3/4 parton correlation function Gaussian ansatz for distribution Take identical Gaussian parameter for parton distribution/correlation functions ˂cosφ>˂cos2φ> are Suppressed!

18. Nuclear modification for depend on dependence Nuclear modification of ˂sinφ> LU

19. k T - dependence Nuclear modification of ˂sinφ> LU Sensitive to the ratio of γ/α !

20. Discussions and outlook Collinear expansion is the systematic and essential method to calculate higher twist effects to SIDIS. Gauge invariance of correlation functions are automatically fulfilled as a result of collinear expansion. Azimuthal asymmetries for doubly polarized e+N →e+q+X are obtained up to twist-3, and for unpolarized case up to twist-4. Much more abundant azimuthal asymmetries at high twist, and their gauge invariant expressions are obtained. Extension to hadronic production process are interesting and are underway. Thanks for your attention!