Transverse Momentum Dependent Parton Distributions

1. Transverse Momentum Dependent Parton Distributions Feng Yuan Lawrence Berkeley National Laboratory

2. What we learn from TMDs New structure New s Dynamics New phenomena

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: http://arxiv.org/abs/1408.3015

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)

5. What an EIC can offer Establish the case for TMD physics (with Drell-Yan and e+e-)

6. Kinematics coverage

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

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

10. Alex Prokudin @EIC-Whitepaper Quark Sivers function leads to an azimuthal asymmetric distribution of quark in the transverse plane

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. 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 0812.3783 Yuan-Zhou, 0903.4680 Exps: BELLE, HERMES, STAR, COMPASS

13. Collins asymmetries in SIDIS Summarized in the EIC Write-up arXiv:1108.1713

14. Collins effects in e+e- BELLE Coll., 2008 BARBAR Coll., 2014 Collins functions extracted from the Data, Anselmino, et al., 2009

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

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

19. Constraints from SIDIS with Evolution Sun, Yuan, PRD 2013

20. Predictions at RHIC 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

21. 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

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

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

25. Sea quark:

26. Unique at EIC: Large transverse momentum • Only Possible with the EIC • QCD dynamics, evolution effects • Q2 dependence

27. EIC coverage 120fb-1

28. A unified picture (leading pt/Q) Transverse momentum dependent Collinear/ longitudinal PT ΛQCD PT Q << << Ji-Qiu-Vogelsang-Yuan,2006 Yuan-Zhou, 2009

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

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

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

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 partondistributions in large nuclei e.g., gluon number distr. Mueller, NPB 1999

34. Hadronic reactions Feynman Parton: one-dimension DIS • Inclusive cross sections probe the momentum (longitudinal) distributions of partons inside nucleon

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)

36. Nucleon tomography EIC Focus Current Lattice Simulations

37. Further integrate out x AMO Exp. Quark model calculation: Xiong, et al. Skovsen et al. (Denmark) PRL91, 090604

38. Wigner function: Phase Space Distributions Wigner 1933 • 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

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

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

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

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 Well defined in QCD: Ji, Xiong, Yuan, PRL, 2012; PRD, 2013 Lorce, Pasquini, Xiong, Yuan, PRD, 2012

43. Transverse profile for the quark distribution: ktvsbt 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

44. Gluon distribution One of the TMD gluon distributions at small-x GPD fit to the DVCS data from HERA, Kumerick-Mueller, 09,10

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. 20009 Lattice Calculation of the IP density of Up quark, QCDSF/UKQCD Coll., 2006

46. ST kT Two major contributions • Sivers effect in the distribution • Collins effect in the fragmentation • Other contributions… ST (PXkT) P (zk+pT) (k,sT) ~pTXsT