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Exploring Semi-Inclusive Deep Inelastic Scattering and Transverse Momentum Distributions

This project delves into the intricacies of Semi-Inclusive Deep Inelastic Scattering (SIDIS) and Transverse Momentum Distributions (TMDs). It aims to clarify the underlying mechanisms of parton distribution in nucleons and nuclei, using various factorization approaches. The analysis presents the nuances between Inclusive and Semi-Inclusive DIS, assesses the role of transverse momentum in scattering processes, and discusses the impact of saturation effects. Continued research is encouraged to enhance our understanding of TMDs and their applications in high-energy physics.

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Exploring Semi-Inclusive Deep Inelastic Scattering and Transverse Momentum Distributions

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  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

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