1. 0 Simonetta Liuti University of Virginia Structure of Nucleons and Nuclei Workshop Como, June 10 th- 14 th, 2013 N&N-StructureSimonetta Liuti Generalized TMDs presented by Osvaldo Gonzalez Hernandez - Torino

2. N&N-StructureSimonetta Liuti 1 In collaboration with: Aurore Courtoy (Liege U.) Gary Goldstein (Tufts U.) Osvaldo Gonzalez Hernandez (INFN Torino since Fall 2012) Graduate Students Kunal Kathuria (U.Va.) Evan Askanazi (U.Va.) Abha Rajan (U.Va.)

3. Question of what components do L q and measure (Ji, Xiong, Yuan, 2011, 2012, Burkardt 2012) N&N-StructureSimonetta Liuti 2

4. A Sum Rule was constructed which identified components of the Energy Momentum Tensor (EMT) with the Angular Momentum carried by quarks and gluons. (Jaffe&Manohar (JM)) A Saga in Several Episodes: Developing a Sum Rule for Angular Momentum 1990 N&N-StructureSimonetta Liuti 3

5. N&N-StructureSimonetta Liuti 4 Partonic picture:  work directly in A + =0 gauge  quark and gluon spin components are identified with the n=1 moments of spin dependent structure functions from DIS, ΔΣ and ΔG. ΔG ΔΣ

6. New Relation (X.Ji) 1997 q p’ + = p –Δ q'=q+Δ p+q P’P’ e p+p+ New processes (DVCS …) were thought of, whose structure functions – the GPDs - admit n=2 moments that were identified with the (spin+OAM) quark and gluon components of the SR P H,E N&N-StructureSimonetta Liuti 5

7. N&N-StructureSimonetta Liuti 6 Lots of work followed in a series of papers by Wakamatsu, Leader and collaborators, Chen et al., Hatta, and Burkardt to understand the origin of this discrepancy.

8. N&N-StructureSimonetta Liuti 7 Conceptual issues: “What type of information can one obtain?” The goal is to give a partonic interpretation of the observables Once a scheme, whichever, is defined to connect with experiment, we stick to it. Corollary: If L can is not observable, does it exist? Practical issues: “how can one extract information from experiment”? Example: GPDs are hard to extract because they need to be “deconvoluted” from Compton Form Factors… Both points pose important theoretical problems. We would like to do some phenomenology

9. N&N-StructureSimonetta Liuti 8 p’p’p x,k T ’ b in GPD b out x,k T

10. N&N-StructureSimonetta Liuti 9 zTzT p’p’p b out b in TMD x,k T p’=p, k T ’=k T

11. N&N-StructureSimonetta Liuti 10 (using constraints from Jlab flavor separated form factor data, G. Cates et al, 2011) Transverse coordinate density distributions PauliDirac d d u u GPD What do we obtain (or wish to obtain) from experiment

12. N&N-StructureSimonetta Liuti 11 TMDs Calculations done using reggeized diquark model, O. Gonzalez et al., arXiv:1206.1876 What do we obtain (or wish to obtain) from experiment Bacchetta, Conti Radici Gonzalez et al. Q o 2 ≈ 0.3 GeV 2 d u This needs to be evolved to the scale of the data (Collins, Rogers, Boglione, Prokudin, …)

13. N&N-StructureSimonetta Liuti 12 Unintegrated GPDs  GTMDs What partonic configurations do they correspond to? Fourier Transform wrt Δ T  Wigner Distributions Notice! Two transverse momenta, simultaneously present

14. N&N-StructureSimonetta Liuti 13 zTzT p’p’p GTMD b GTMDs correlate partonic configurations with both: a shift in transverse position from the initial to final state  z T an average transverse position  b. average shift When can these configurations exist in the “impulse approximation”, and when do we need to introduce FSI?

15. N&N-StructureSimonetta Liuti 14 How would we observe GTMDs? q, Λ γ k’= k –Δ. λ’ q’=q+Δ, Λ γ ’ k, λ p’= k-Δ, Λ’ p, Λ u-channel two body scattering k’= k –Δ. λ’ k, λ ϑ

16. N&N-StructureSimonetta Liuti 15 k’ CM = k CM –Δ. λ’ k CM, λ p’ CM = k CM -Δ, Λ’ p CM, Λ ϑ This angle determined by Δ T z y x CoM motion determined by Transverse Plane

17. N&N-StructureSimonetta Liuti 16 In CoM the scattering happens in one plane therefore there is only one transverse direction defined by Δ T In CoM frame Δ T and individual partons k T are parallel to one another Average k T describes the motion of the CoM

18. N&N-StructureSimonetta Liuti 17 λ'λ same as GPDs “New” terms can exist GTMDs in Cartesian Basis Meissner, Metz, Schlegel, JHEP 2009

19. N&N-StructureSimonetta Liuti 18 Spin(Helicity/Transversity) Basis Quark-proton helicity amplitude

20. N&N-Structure Simonetta Liuti 19 Chiral Even Sector G 14 F 11 2F 13 -F 11 2G 13 -G 14 UsHelicity amps contentMetz

21. N&N-StructureSimonetta Liuti 20 Can the new GTMDs F 14 and G 11 be defined at leading twist? In two-body scattering these terms are Parity violating (valid in CM frame, demonstration in general frame hinges on appropriate Lorentz Transformation) In a quark-target model only H, H are present ~

22. N&N-StructureSimonetta Liuti 21 Why are they important: the idea has been emerging that canonical OAM can be related to F 14 Ji, Xiong and Yuan, 2012 Lorce and Pasquini, 2011

23. N&N-StructureSimonetta Liuti 22 However F 14 drops out of the observables Not because it contains “even more information on the structure of the proton than what can be measured”, but for a good reason…

24. N&N-StructureSimonetta Liuti 23 The demonstration is rather technical Out of all the functions that parameterize the generalized correlator (MMS’09)… … the matrix elements for the “type 3” ones transform the opposite way under Parity. These define F 14

25. N&N-StructureSimonetta Liuti 24 Summary so far… We confirm that canonical AM cannot be measured if defined as because it corresponds – in two body scattering – to a combination of helicity amplitudes that violates Parity

26. N&N-StructureSimonetta Liuti 25 Consider now the angular momentum sum rule including twist 3 contributions Belitsky and Mueller (2000), Polyakov et al. (2000), Hatta (2011) -J q SqSq -L q Twist 3 decomposition of hadronic tensor OAM is a twist three contribution

27. N&N-StructureSimonetta Liuti 26 In order to describe GTMDs including partonic interactions, i.e. moving out of a 2 body scattering picture, we need a situation where even in the CoM, one can define two independent transverse vectors as if there were a third particle, so that the additional combinations giving rise to “LU”-type contributions can exist

28. N&N-StructureSimonetta Liuti 27 Twist 3 GTMDs spin flip spin non flip MMS’09

29. N&N-StructureSimonetta Liuti 28

30. N&N-StructureSimonetta Liuti 29 Finally…a prerequisite for all these calculations is that we use a model whose parameters are constrained by a quantitative fit to all data which are relevant for GPDs (Form Factors, DIS and DVCS)

31. N&N-StructureSimonetta Liuti 30 Sections of Wigner distributions: Completely Unpolarized Case kTkT u quark

32. N&N-StructureSimonetta Liuti 31 All together: integrated over x and k T (this is a fit to real data: DVCS+G.Cates et al) d quarku quark

33. N&N-StructureSimonetta Liuti 32APS Denver 4/12/13Simonetta Liuti32 Sections of Wigner distributions: Transversely Polarized Case kTkT u quark

34. N&N-StructureSimonetta Liuti 33 All together: integrated over x and k T Δ 1 E u - Δ 1 E d

35. N&N-StructureSimonetta Liuti 34 Twist 3 component  G 2

36. N&N-StructureSimonetta Liuti 35 Summary of this part and Open Questions G 2 can be measured directly, and it does have a partonic interpretation, if one views it as a twist 3 quantity. We can distinguish in models a WW part defining OAM, and a genuine twist 3 part. Canonical angular momentum cannot be measured directly, but it can be linked to G 2 through M. Burkardt’s “force-type” correction We need to devise experiments that are sensitive to this contributions (they might be already in the available data!) Question of L vs. T interpretation We focused mainly on quarks, question of gluon component One way to shed light on these questions is to look at deuteron

37. N&N-StructureSimonetta Liuti 36 Spin 1 systems, due to 1)The presence of additional L components (D-waves) 2)Isoscalarity provide a crucial test the working of the angular momentum sum rules

38. Nucleon DVCS Deuteron H ISO =H u +H d E ISO =E u +E d λNλN λqλq λ z=p N + /P + D What are the quark and gluon angular momenta in the deuteron? N&N-StructureSimonetta Liuti 37

39. Sum Rules in Deuteron (K.Kathuria) OAM Momentum Spin 1/2 Form factors from the respective Energy Momentum tensors, T μν N&N-StructureSimonetta Liuti 38

40. If f ++ (z) = f +0 (z)=δ(1-z) then H 2 =H+E F 1 +F 2 = G M GMGM Differently from the nucleon, in the transverse case, we are not finding the same relation (other GPDs describing charge and tensor component enter…). More details later… Longitudinal Transverse Deuteron DeuteronNucleon N&N-StructureSimonetta Liuti 39

41. How does Ji sum rule differ from JM in the deuteron? Effect of evolution Models are important Model calculations of L with w.f.s’ seem to lead to similar conclusions as M.Burkardt, more to explore here… avenue to compare different schemes? Using GPDs from Goldstein, Gonzalez, SL, PRD84 N&N-StructureSimonetta Liuti 40

42. Nuclear effect much larger than in unpolarized scattering Needs to be treated systematically… N&N-StructureSimonetta Liuti 41

43. Observables: DVCS from deuteron subleading Can the deuteron help us understand the role of gluon OAM? (Brodsky, Gardner, 2006) By connecting L g to SSA in ≈ 0 Both L q and L g contribute! Since L q disappears because of isospin symmetry, if A UT π is 0 then L g is 0 N&N-StructureSimonetta Liuti 42

44. Two ways to go beyond the two body scattering scenario: 3 1 2 1 2 3

45. Our model for fracture functions (work in progress…) scattered quark outgoing qqqq-bar Initial proton h q-bar frag. B B A A

46. N&N-StructureSimonetta Liuti 45 GG, S. Liuti and O. Gonzalez, GPDS, THEIR RELATIONSHIPS WITH TMDS & RELATED TOPICS Proceedings of QCD-N12, Bilbao, Spain - 2012

47. N&N-StructureSimonetta Liuti 46 Conclusions We conclude that the distribution of an unpolarized quark in a longitudinally polarized nucleon does indeed measure OAM We provide a theoretical basis for the “simple moment” of Wigner distributions, connecting to twist 3 GPDs Belitsky, Mueller, Polyakov, Hatta,… Now one can think seriously of observables (DV processes including twist 3 GPDs) Our argument stresses the fact that in order to measure OAM one has to go beyond a two body scattering picture Transverse vs. Longitudinal sum rule? Gluon components?  we should look into Spin 1 systems also

48. N&N-StructureSimonetta Liuti 47 G. Goldstein, O. Gonzalez Hernandez, S.Liuti, J.Phys. G, 39 (2012) 11500 G. Goldstein, O. Gonzalez Hernandez, S.Liuti, Phys. Rev. D84, 034007 (2011) S.K. Taneja, K. Kathuria, S.Liuti, G. Goldstein, Phys. Rev. D86 (2012) G. Goldstein, O. Gonzalez Hernandez, S.Liuti, K. Kathuria, arXiv:1206.1876, subm. PRC G. Goldstein, O. Gonzalez Hernandez, S.Liuti, K. Kathuria, “Proceedings of 4 th QCD Evolution Workshop”, Jefferson Lab, May (2012)

49. N&N-StructureSimonetta Liuti 48 Backup

50. N&N-StructureSimonetta Liuti 49 Sections of Wigner distributions: Mixed Transverse/Longitudinal Basis