Semiinclusive DIS at Small x Feng Yuan Lawrence

Semi-inclusive DIS at Small -x Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory Ref: Marquet, Xiao, Yuan, ar. Xiv: 0906. 1454 12/7/2020 1

Outline n We started this project with many questions ¨ TMD, kt, … After this exercise, I see some hope to understand better n More works need to be done n 12/7/2020 2

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

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

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

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

n TMDs are process dependent (Fragmentation is different) ¨ Gauge link direction changes from DIS to Drell-Yan process ¨ More complicated structure for dijetcorrelation in pp collisions, standard factorization breaks Collins-Qiu 08 n Light-cone singularity beyond Born diagram ¨ Transverse momentum resummation 12/7/2020

One-Loop Real Contribution energy dep. =v. p 2/v 2 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 (p. QCD 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 n Transverse momentum resummation ¨ Collins-Soper-Sterman, 85 ¨ C. P. Yuan, et al; Sterman, Vogelsang, et al; Qiu, Zhang; Catani, Mangano, de Florian, et al, … n Match between TMD and collinear approach on SSA and other phenomena ¨ Ji, Qiu, Vogelsang, Yuan, Koike, Zhou, … ¨ Bacchetta, Boer, Diehl, Mulders, … 12/7/2020 11

TMD at small-x: kt-factorization? What is the relevant transverse momentum n TMD distribution and/or un-integrated gluon distribution n Factorization n ¨ Is there any factorization ¨ How to express the factorization formula ¨ NLO corrections 12/7/2020 12

Inclusive DIS n Q Transverse momentum is not manifested ¨ Integral form of un-integrated gluon dis. Or dipole cross section n Power counting? ¨ in terms 1/Q or 1/Qs ¨ or 1/Log(1/x) 12/7/2020 13

Advantage of SIDIS n 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 n Can be related to the TMD factorization discussed before n 12/7/2020 14

SIDIS at small-x n Q What are the relevant scales ¨ Q, virtuality of the photon ¨ Pt, transverse momentum of hadron ¨ Qs, saturation scale n We are interested in the region of Q>>Qs, Pt ¨ TMD factorization makes sense 12/7/2020 15

Dipole picture for DIS Fragmentation function 12/7/2020 16

SIDIS Differential Cross section Unintegrated gluon dis. 12/7/2020 17

Assumptions for the hadron production n Separate the fragmentation from the dipole scattering ¨ KKT 04, DHJ 05, BUW 07, p. A scattering ¨ Nuclear effects may break down this assumption (talks by Qiu, Wang, …) n Transverse momentum in the fragmentation ¨ We can add it, it won’t change the power counting analysis 12/7/2020 18

TMD limit: Q>>p n T Keep the leading power contribution, neglect all higher power corrections 12/7/2020 19

Recall: TMD factorization 12/7/2020 20

Up to this order n Trivial factors from ¨ Soft factor ¨ Fragmentation function ¨ Hard factor 12/7/2020 21

TMD quark Mc. Lerran-Venugopalan 98 n Reproduce the SIDIS cross section with the TMD quark distribution and the TMD factorization 12/7/2020 22

Comments We don’t lose the sensitivity to the saturation physics even with Large Q n We gain the direct probe for the transverse momentum dependence of partons n Beyond the leading order? n Additional dynamics involved n ¨ Soft gluon resummation 12/7/2020 23

Transverse momentum resum Collins-Soper-Sterman 85 n Collins-Soper evolution, leading log approximation, More comprehensive studies by Nadolsky, C. P. Yuan, et al 12/7/2020 24

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

More interesting Ratio relative to that at 10 -2 26

Ready to extend to the gluon case n Mueller 94, Kovchegov-Mueller 98 TMD gluon=un-integrated gluon (? ) Small-x and transverse momentum resummation 12/7/2020 27

Perspective Will be able to calculate the hard factor at one-loop order n Check the TMD factorization including all factors n ¨ Fragmentation ¨ Soft factor ¨ Gluon distribution at one-loop 12/7/2020 28

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