1. Transverse Momentum Dependent Parton Distributions
Feng Yuan
Lawrence Berkeley National Laboratory
12/5/2018

2. What we learn from TMDs New structure New s Dynamics New phenomena
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3. What we have learned
Unpolarized TMDs from, mainly, Drell-Yan, W/Z boson productions
Indications of polarized quark distributions (Sivers and cousins), from low energy SIDIS (HERMES, COMPASS, Jlab)
TMDlib:
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4. What we are missing
Precision, detailed, mapping of polarized quark/gluon distribution
Universality/evolution more evident
Spin correlation in momentum and coordinate space/tomography
Small-x TMDs: links to other hot fields (CGC)
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5. Establish the case for TMD physics (with Drell-Yan and e+e-)
What an EIC can offer
Establish the case for TMD physics
(with Drell-Yan and e+e-)
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6. Kinematics coverage
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7. TMD Parton Distributions
The definition contains explicitly the gauge links
QCD factorization has been proved for the hard processes in terms of TMDs
Collins-Soper 1981, Ji-Ma-Yuan 2004, Collins 2011
Collins-Soper 1981,
Collins 2002,
Belitsky-Ji-Yuan 2002
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8. Factorization
High Q2 is essential for QCD factorization, leading power contribution
Only hard processes, Q>>PT, we can unambiguously extract the TMDs

9. Transverse momentum dependent parton distribution
Straightforward extension
Spin average, helicity, and transversity distributions
PT-spin correlations
Nontrivial distributions, STXPT
In quark model, depends on S- and P-wave interference
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10. Alex Prokudin
@EIC-Whitepaper
Quark Sivers function leads to an azimuthal asymmetric distribution of quark in the transverse plane
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11. TMD at EIC: Semi-inclusive DIS
Novel Single Spin Asymmetries
Nucleon Tensor Charge
U: unpolarized beam
T: transversely polarized target
Novel Spin Phenomena
12/5/2018

12. Universality of the Collins Fragmentation
ep--> e Pi X
e+e---> Pi Pi X
pp--> jet(->Pi) X
Metz 02, Collins-Metz 02,
Yuan 07,
Gamberg-Mukherjee-Mulders 08,10
Meissner-Metz
Yuan-Zhou,
Exps: BELLE, HERMES,
STAR, COMPASS
12/5/2018

13. Collins asymmetries in SIDIS
Summarized in the
EIC Write-up
arXiv:
12/5/2018

14. Collins effects in e+e-
BELLE Coll., 2008
BARBAR Coll., 2014
Collins functions extracted from the
Data, Anselmino, et al., 2009
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15. 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 02
Collins, 02;
Ji,Yuan,02;
Belitsky,Ji,Yuan,02

16. Sivers asymmetries in SIDIS
Non-zero Sivers effects
Observed in SIDIS
Jlab Hall A 3He
12/5/2018

17. Sivers Asymmetries in DIS and Drell-Yan
Initial state vs. final state interactions
“Universality”: QCD prediction * *
DIS
Drell-Yan
HERMES/COMPASS

18. TMD predictions rely on
Non-perturbative TMDs constrained from experiments
QCD evolutions, in particular, respect to the hard momentum scale Q
Strong theory/phenomenological efforts in the last few years
Need more exp. data/lattice calculations
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19. Constraints from SIDIS with Evolution
Sun, Yuan, PRD 2013
12/5/2018

20. Predictions at RHIC √S = 500GeV Drell-Yan Q=6GeV
Sun, Yuan, PRD 2013
Additional theory uncertainties: x-dependence of the TMDs comes from a fit to fixed target drell-yan and w/z production at Tevatron ---Nadolsky et al.
√S = 500GeV
Drell-Yan Q=6GeV
12/5/2018

21. y=0 y=0 Pt(GeV) √S = 510GeV Pt(GeV) -0.06 -0.06 Rapidity of W
See also, Kang et al, 2014
y=0
y=0
Pt(GeV)
√S = 510GeV
Pt(GeV)
-0.06
-0.06
Rapidity of W
Rapidity of W
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22. QCD evolution reduces the asymmetries about
a factor of 3 for W/Z

23. Electron-Ion Collider Projections: Impressive coverage on Q2, x, z, and PT
12/5/2018

24. Quark Sivers function extracted from the data
Leading order fit, simple Gaussian
assumption for the Sivers function
There are still theoretical uncertainties
In the fit: scale dependence, high
order corrections, …
Inner band is the impact from
the planed EIC kiematics
Alexei Prokudin, et al.
Work in progress with QCD evolutions
12/5/2018

25. Sea quark:
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26. Unique at EIC: Large transverse momentum
Only Possible with the EIC
QCD dynamics, evolution effects
Q2 dependence
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27. EIC coverage
120fb-1
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28. A unified picture (leading pt/Q)
Transverse momentum dependent
Collinear/
longitudinal PT
ΛQCD PT Q
<<
<<
Ji-Qiu-Vogelsang-Yuan,2006
Yuan-Zhou, 2009
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29. The Q2 is smaller, and the Y piece is more important.
P.Sun’s talk at the INT workshop, Feb. 2014

30. TMD gluon distributions
It is not easy, because gluon does not couple to photon directly
Can be studied in two-particle processes
Di-photon
In pp
Dijet
In DIS
Vogelsang-Yuan, 2007
Dominguez-Xiao-Yuan, 2010
Boer-Brodsky-Mulders-Pisano, 2010
Qiu-Schlegel-Vogelsang, 2011
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31. Questions and comments
Why do we care?
It concerns the fundamental question, as what Rutherford asked more than 100 years ago
Are there anything exciting?
Any unknown will lead to new physics, in particular, those associated with spin
Does EIC have the capability to do so?
Yes, tons of simulations have shown that
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32. Will it attract young researchers?
Yes, it has and will continue to do so, because the physics is interesting, puzzling, challenging at the same time
TMD is a developing area, many progresses, interesting from both theory/experiment
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33. Transverse Wigner Distributions
Integrate out z in the Wigner function
Depends on x, kt, bt
Also referred as GTMD in the literature
See for example, Metz, et al., 09; Pasquini, 10,11
It has close connection to the small-x parton distributions in large nuclei
e.g., gluon number distr.
Mueller, NPB 1999
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34. Feynman Parton: one-dimension
DIS
Inclusive cross sections probe the momentum (longitudinal) distributions of partons inside nucleon
Hadronic reactions
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35. Extension to transverse direction…
Semi-inclusive measurements (in DIS or Drell-Yan processes)
Transverse momentum dependent (TMD) parton distributions
Deeply Virtual Compton Scattering and Exclusive processes
Generalized parton distributions (GPD)
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36. Nucleon tomography
EIC Focus
Current Lattice Simulations
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37. Further integrate out x
AMO Exp.
Quark model calculation: Xiong, et al.
Skovsen et al.
(Denmark) PRL91,
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38. Wigner function: Phase Space Distributions
Define as
When integrated over x (p), one gets the momentum (probability) density
Not positive definite in general, but is in classical limit
Any dynamical variable can be calculated as
Wigner 1933

39. Wigner distribution for the quark
The quark operator
Wigner distributions
After integrating over r, one gets TMD
After integrating over k, one gets Fourier transform of GPDs
Ji: PRL91,062001(2003)

40. TMD Parton Distributions
The definition contains explicitly the gauge links
The polarization and kt dependence provide rich structure in the quark and gluon distributions
Mulders-Tangerman 95, Boer-Mulders 98
Collins-Soper 1981,
Collins 2002,
Belitsky-Ji-Yuan 2002
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41. Generalized Parton Distributions
Mueller, et al. 1994; Ji, 1996, Radyushkin 1996
Off-diagonal matrix elements of the quark operator (along light-cone)
It depends on quark momentum fraction x and skewness ξ, and nucleon momentum transfer t
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42. Parton’s orbital motion through the Wigner Distributions
Phase space distribution:
Projection onto p (x) to get the
momentum (probability) density
Quark orbital angular
momentum
Wigner distribution can be interpreted as the phase space distribution, depends on the space and momentum variables. From this, we can derive the orbital angular momentum intuitively as the mean values of r\cross p. This is not only an intuitive picture for the orbital angular momentum, but also well defined in QCD. It, actually, links different formalisms of the orbital angular momentum in the literature. We have also applied a simple quark model to show how the quark Wigner distribution as shown in this plot. This Wigner distribution indeed provide nice imaging of partons inside nucleon. Of course, we hope we can obtain this picture from more solid method, such as lattice QCD, that brings me to the last slide
Well defined in QCD:
Ji, Xiong, Yuan, PRL, 2012; PRD, 2013
Lorce, Pasquini, Xiong, Yuan, PRD, 2012

43. Transverse profile for the quark distribution: kt vs bt
Quark distribution calculated from
a saturation-inspired model
A.Mueller 99, McLerran-Venugopalan 99
GPD fit to the DVCS data from HERA,
Kumerick-D.Mueller, 09,10
12/5/2018

44. Gluon distribution One of the TMD gluon distributions at small-x
GPD fit to the DVCS data from HERA,
Kumerick-Mueller, 09,10
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45. Deformation when nucleon is transversely polarized
-0.5
0.5
0.0
0.5 ky
0.0
-0.5 kx
Quark Sivers function fit to the SIDIS
Data, Anselmino, et al
Lattice Calculation of the IP density of
Up quark, QCDSF/UKQCD Coll., 2006
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46. Two major contributions
Sivers effect in the distribution
Collins effect in the fragmentation
Other contributions… kT ST
ST (PXkT) P
(zk+pT)
(k,sT)
~pTXsT
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