1. Semi-inclusive DIS at Small-x
Feng Yuan
Lawrence Berkeley National Laboratory
RBRC, Brookhaven National Laboratory
Ref: Marquet, Xiao, Yuan, arXiv:
11/18/2018

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
11/18/2018

3. Inclusive and Semi-inclusive DISQ
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

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 v is not n to avoid l.c. singularity !!
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
11/18/2018

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, …
11/18/2018

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
11/18/2018

13. Inclusive DIS Transverse momentum is not manifested Power counting?Q
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)
11/18/2018

14. Advantage of SIDIS Q
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
11/18/2018

15. SIDIS at small-x What are the relevant scalesQ
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
11/18/2018

16. Dipole picture for DIS
Fragmentation function
11/18/2018

17. SIDIS Differential Cross section
Unintegrated gluon dis.
11/18/2018

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
11/18/2018

19. TMD limit: Q>>pT
Keep the leading power contribution, neglect all higher power corrections
11/18/2018

20. Recall: TMD factorization
11/18/2018

21. Up to this order Trivial factors from Soft factor
Fragmentation function
Hard factor
11/18/2018

22. TMD quark
McLerran-Venugopalan 98
Reproduce the SIDIS cross section with the TMD quark distribution and the TMD factorization
11/18/2018

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
11/18/2018

24. Transverse momentum resum Collins-Soper-Sterman 85
Collins-Soper evolution, leading log approximation,
More comprehensive studies by Nadolsky, C.P. Yuan, et al
11/18/2018

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
11/18/2018

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
11/18/2018

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