Semi-inclusive DIS at Small-x

1. Semi-inclusive DIS at Small-x Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory Ref: Marquet, Xiao, Yuan, arXiv:0906.1454

2. Outline • We started this project with many questions • TMD, kt, … • After this exercise, I see some hope to understand better • More works need to be done

3. Inclusive and Semi-inclusive DIS Inclusive DIS: Partonic Distribution depending on the longitudinal momentum fraction Q Semi-inclusive DIS: Probe additional information for partons’ transverse distribution in nucleon/nucleus Metz and Gamberg’s talks Q

4. TMD: Naïve Factorization • SIDIS Cross section • Naïve factorization (unpolarized structure function) Hadron tensor TMD distr. TMD frag. Mulders, Tangelman, Boer (96 & 98)

5. TMD Factorization Collins-Soper, 81 Collins-Soper-Sterman,85 Ji-Ma-Yuan, 04 Collins-Metz 04 Scherednikov-Stefanis, 07 • Leading order in pt/Q • Additional soft factor

6. TMD: the definition In Feynman Gauge, the gauge link v is not n to avoid l.c. singularity !!

7. TMDs are process dependent (Fragmentation is different) • Gauge link direction changes from DIS to Drell-Yan process • More complicated structure for dijet-correlation in pp collisions, standard factorization breaks Collins-Qiu08 • Light-cone singularity beyond Born diagram • Transverse momentum resummation

8. One-Loop Real Contribution energy dep. =v.p2/v2 Evolution to resummation: Collins-Soper 1981, Collins-Soper-Sterman 1985

9. How Factorization works: gluon radiation • Vertex corrections (single quark target) q p′ k p Four possible regions for the gluon momentum k: 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line) 4) k is hard (pQCD correction)

10. One-Loop Factorization (real gluon) • Gluon Radiation (single quark target) q p′ k p Three possible regions for the gluon momentum k: 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line)

11. Applications • Transverse momentum resummation • Collins-Soper-Sterman, 85 • C.P. Yuan, et al; Sterman, Vogelsang, et al; Qiu, Zhang; Catani, Mangano, de Florian, et al,… • Match between TMD and collinear approach on SSA and other phenomena • Ji, Qiu,Vogelsang, Yuan, Koike, Zhou, … • Bacchetta, Boer, Diehl, Mulders, …

12. TMD at small-x:kt-factorization? • What is the relevant transverse momentum • TMD distribution and/or un-integrated gluon distribution • Factorization • Is there any factorization • How to express the factorization formula • NLO corrections

13. Inclusive DIS • Transverse momentum is not manifested • Integral form of un-integrated gluon dis. Or dipole cross section • Power counting? • in terms 1/Q or 1/Qs • or 1/Log(1/x) Q

14. Advantage of SIDIS • Direct probe for the transverse momentum dependence of partons • Saturation effects explicitly show up in the transverse momentum distribution • Factorization can be argued for large Q • Can be related to the TMD factorization discussed before Q

15. SIDIS at small-x • What are the relevant scales • Q, virtuality of the photon • Pt, transverse momentum of hadron • Qs, saturation scale • We are interested in the region of Q>>Qs, Pt • TMD factorization makes sense Q

16. Dipole picture for DIS Fragmentation function

17. SIDIS Differential Cross section Unintegrated gluon dis.

18. Assumptions for the hadron production • Separate the fragmentation from the dipole scattering • KKT 04, DHJ 05, BUW 07, pA scattering • Nuclear effects may break down this assumption (talks by Qiu, Wang, …) • Transverse momentum in the fragmentation • We can add it, it won’t change the power counting analysis

19. TMD limit: Q>>pT • Keep the leading power contribution, neglect all higher power corrections

20. Recall: TMD factorization

21. Up to this order • Trivial factors from • Soft factor • Fragmentation function • Hard factor

22. TMD quark • Reproduce the SIDIS cross section with the TMD quark distribution and the TMD factorization McLerran-Venugopalan 98

23. Comments • We don’t lose the sensitivity to the saturation physics even with Large Q • We gain the direct probe for the transverse momentum dependence of partons • Beyond the leading order? • Additional dynamics involved • Soft gluon resummation

24. Transverse momentum resumCollins-Soper-Sterman 85 • Collins-Soper evolution, leading log approximation, More comprehensive studies by Nadolsky, C.P. Yuan, et al

25. Phenomenology: quark distributions ratios Transverse Mometum Broading with Q GBW model for dipole Cross section

26. More interesting Ratio relative to that at 10-2

27. Ready to extend to the gluon case • Mueller 94, Kovchegov-Mueller 98 TMD gluon=un-integrated gluon (?) Small-x and transverse momentum resummation

28. Perspective • Will be able to calculate the hard factor at one-loop order • Check the TMD factorization including all factors • Fragmentation • Soft factor • Gluon distribution at one-loop

29. Summary • Semi-inclusive DIS provides additional probe to saturation physics at small-x, with advantage to directly probe the transverse momentum dependence • Nuclear effects • Factorizable or non-factorizable • Gluon TMDs can be also studied similarly