04/22/23 1

Transverse Momentum Dependent Parton Distributions at an EIC

Feng Yuan Lawrence Berkeley National Laboratory

Thank Hasch, Burton, Prokudin, and many others, INT-Write-up for the EIC

04/22/23 2

Kinematics compared to previous machines

04/22/23 3Courtesy from Sichtermann-Vogelsang for EIC-White paper

Physics programs Spin and parton imaging of nucleon

Quark/gluon helicity distributions Spin puzzle

3-D imaging of partons (OAM) Transverse coordinate space (GPDs) Transverse momentum space (TMDs)

Gluon saturation in large nucleus at small-x Electroweak physics, new physics beyond SM Cross-links among the above, and …

04/22/23 4

INT-Write-up, Boer, et al., arXive:1108.1713

04/22/23 5

Wigner DistributionW(x,r,kt)

d3 r

Transverse Momentum

Dependent PDF f(x,kt)Generalized Parton Distr.

H(x,ξ,t)

d 2kt dzF.T.

d2k t

PDF f(x)

dx

Form Factors

F1(Q),F2(Q)

GPD

Transverse momentum dependent parton distribution Straightforward extension

Spin average, helicity, and transversity distributions

Transverse momentum-spin correlationsNontrivial distributions, STXPT In quark model, depends on S- and P-

wave interference

04/22/23 6Mulders and Qiu’s talks

Deformation when nucleon is transversely polarized

04/22/23 7

Lattice Calculation of the IP density ofUp quark, QCDSF/UKQCD Coll., 2006

Quark Sivers function fit to the SIDISData, Anselmino, et al. 20009

k y

kx

-0.5

0.5

0.0

-0.5 0.0 0.5

Where can we learn TMDs Semi-inclusive hadron production in

deep inelastic scattering (SIDIS) Drell-Yan lepton pair, photon pair

productions in pp scattering Dijet correlation in DIS Relevant e+e- annihilation processes …

04/22/23 8

9 04/22/23

Novel Single Spin Asymmetries

U: unpolarized beamT: transversely polarized target

Semi-inclusive DIS

Spin-orbital correlation Leading order TMDs

04/22/23 10

All these TMDs can be probed through different azimuthalangle distributions

Two major contributions

Sivers effect in the distribution

Collins effect in the fragmentation

Other contributions…

04/22/23 11

kTST

P

(zk+pT)~pTXsT

(k,sT)

ST (PXkT)

Universality of the Collins Fragmentation

04/22/23 12

ep--> e Pi X e+e---> Pi Pi X pp--> jet(->Pi) XMetz 02, Collins-Metz 02,Yuan 07,Gamberg-Mukherjee-Mulders 08,10Meissner-Metz 0812.3783Yuan-Zhou, 0903.4680

Exps: BELLE, HERMES, STAR, COMPASS

Collins asymmetries in SIDIS

04/22/23 13

Summarized in theEIC Write-up

Collins effects in e+e-

04/22/23 14

BELLE Coll., 2008 Collins functions extracted from theData, Anselmino, et al., 2009

Sivers effect It is the final state interaction

providing the phase to a nonzero SSA Non-universality in general Only in special case, we have “Special Universality”

Brodsky,Hwang,Schmidt 02Collins, 02;Ji,Yuan,02;Belitsky,Ji,Yuan,02

Sivers asymmetries in SIDIS

04/22/23 16

Non-zero Sivers effectsObserved in SIDIS

Jlab Hall A 3He

Great Potential at the EIC High precision, detailed study Wide range of Q2 coverage, QCD

dynamics Sea quark TMDs Gluon TMDs Large transverse momentum,

connecting to collinear factorization

04/22/23 17

Electron-Ion Collider Projections: Impressive coverage on Q2, x, z, and PT

04/22/23 18√S=140,50,15 GeV

Quark Sivers function extracted from the data

04/22/23 19

Alexei Prokudin, et al.

Leading order fit, simple Gaussian assumption for the Sivers function

There are still theoretical uncertaintiesIn the fit: scale dependence, highorder corrections, …

Inner band is the impact from the planed EIC kinematics

Sea quark:

04/22/23 20

Large transverse momentum Only possible with the EIC QCD dynamics, evolution effects Q2 dependence Pt dependence Twist-three mechanism

04/22/23 21

Qiu’s talk

EIC coverage

04/22/23 22

120fb-1

A unified picture (leading pt/Q)

04/22/23 23

QΛQCD PT

Collinear/longitudinal

<<<<

Transverse momentum dependent

Ji-Qiu-Vogelsang-Yuan,2006Yuan-Zhou, 2009

PT

Collins-Soper-Sterman Resummation

Separate the singular and regular parts

TMD factorization in b-space

04/22/23 24Kang-Xiao-Yuan, 2011

Collinear divergence--splitting

Sivers function

04/22/23 25

Similar to the weighted asymmetry, and the splittingVogelsang-Yuan, 2009

Hard factor at one-loop order Same as the spin-average case

04/22/23 26

Final resum form

Sudakov the same

04/22/23 27

Coefficients at one-loop order

Together with A(2), this resum at Next-to-leading-logarithmic level

Phenomenological implementation is underway Unpolarized, Nadolsky et al, 2001 ALL, Koike-Nagashima-Vogelsang 2006

04/22/23 28

Directly working on TMDs

04/22/23 29

Aybat-Collins-Qiu-Rogers, 2011

Q2-dependence Aybat-Prokudin-Rogers, 2011

04/22/23 30Needs a cross check!

TMD gluon distributions It is not easy, because gluon does

not couple to photon directly Can be studied in two-particle

processes

04/22/23 31

Dijet In DIS

Vogelsang-Yuan, 2007Dominguez-Xiao-Yuan, 2010Boer-Brodsky-Mulders-Pisano, 2010

Di-photon In pp

Qiu-Schlegel-Vogelsang, 2011

Summary Wonderful physics associated with

TMDs: nucleon tomography; spin-orbital correlation; QCD dynamics;

We have learned something from HERMES, COMPASS, Jlab, BELLE

Electron-ion collider is the ultimate machine to study TMDs

04/22/23 32