Semi-inclusive DIS at Small-x Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory Ref: Marquet, Xiao, Yuan, arXiv:0906.1454 11/18/2018

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

Inclusive and Semi-inclusive DIS Q 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

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

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

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 !!

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

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

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)

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)

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

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

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

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

SIDIS at small-x What are the relevant scales Q 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

Dipole picture for DIS Fragmentation function 11/18/2018

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

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

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

Recall: TMD factorization 11/18/2018

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

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

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

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

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

More interesting Ratio relative to that at 10-2

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

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

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