1. Transversity and Transverse-Momentum-Dependent Partonic Functions Alessandro Bacchetta

2. Alessandro Bacchetta - Transverse-momentum- dependent functions 2/58 Outline Theoretical framework Transversity Sivers function Boer-Mulders function

3. Theoretical framework

4. Alessandro Bacchetta - Transverse-momentum- dependent functions 4/58 Semi-inclusive Deep Inelastic Scattering proton pion lepton

5. Alessandro Bacchetta - Transverse-momentum- dependent functions 5/58 Feynman diagrams & Factorization proton lepton pion SIDIS Partonic scattering amplitude Fragmentation amplitude Distribution amplitude

6. Alessandro Bacchetta - Transverse-momentum- dependent functions 6/58 Feynman diagrams & Factorization Partonic scattering amplitude Fragmentation amplitude Distribution amplitude proton lepton pion SIDIS electron positron pion e – e + to pions proton lepton antilepton Drell-Yan

7. Alessandro Bacchetta - Transverse-momentum- dependent functions 7/58 Feynman diagrams & Factorization proton lepton pion SIDIS proton lepton antilepton Drell-Yan proton pion p-p to pions electron positron pion e – e + to pions

8. Alessandro Bacchetta - Transverse-momentum- dependent functions 8/58 Factorization proofs Factorization proofs for involving TMD functions LEADING TWIST (1/ Q 0 ), NOT for pp to pions – Collins, Soper, NPB 193 (81) – Ji, Ma, Yuan, PRD 71 (04) – Ji, Ma, Yuan, PLB 597 (04) – Collins, Metz, PRL 93 (04) See talk by A. Metz

9. Alessandro Bacchetta - Transverse-momentum- dependent functions 9/58 Nonperturbative elements 2 =

10. Alessandro Bacchetta - Transverse-momentum- dependent functions 10/58 Origin of the gauge link...

11. Alessandro Bacchetta - Transverse-momentum- dependent functions 11/58 Different processes SIDIS Drell-Yan pp to hadrons Bacchetta, Bomhof, Mulders, Pijlman, hep-ph/0505268 See talk by C. Bomhof

12. Alessandro Bacchetta - Transverse-momentum- dependent functions 12/58 The correlation function integrated over k T   axis 0a SIDIS

13. Alessandro Bacchetta - Transverse-momentum- dependent functions 13/58 The correlation function integrated over k T   axis 0a Drell-Yan

14. Alessandro Bacchetta - Transverse-momentum- dependent functions 14/58 The correlation function UN integrated over k T   axis 0 a  Belitsky, Ji, Yuan, NPB656 (03) SIDIS

15. Alessandro Bacchetta - Transverse-momentum- dependent functions 15/58 The correlation function UN integrated over k T 0 a   axis  Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03) Drell-Yan

16. Alessandro Bacchetta - Transverse-momentum- dependent functions 16/58 Practical consequences Some TMD functions will not be sensitive to the differences in the structure of the gauge link (i.e. they will be connected to standard partonic cross-sections) Some TMD functions will be sensitive to it and will be multiplied by prefactors to be computed for each partonic subprocess (i.e. they will be connected to gluonic-pole cross-sections) See talk by C. Bomhof

17. Alessandro Bacchetta - Transverse-momentum- dependent functions 17/58 Decomposition of the correlation function integrated over k T Leading twist only Unpolarized distr. func. Helicity distr. func. Transversity distr. func.

18. Alessandro Bacchetta - Transverse-momentum- dependent functions 18/58 Decomposition of the correlation function UN integrated over k T Boer-Mulders Sivers Sivers, PRD 43 (91) Boer, Mulders, PRD 57 (98) Mulders, Tangerman, NPB 461 (96) Goeke, Metz, Schlegel, PLB 618 (05)

19. Alessandro Bacchetta - Transverse-momentum- dependent functions 19/58 Parton distribution functions with transverse spin - Sivers - Boer-Mulders - Transversity Photon moves into the screen/ proton moves out of the screen

20. Transversity

21. Alessandro Bacchetta - Transverse-momentum- dependent functions 21/58 Definition of transversity distribution function Probability to find a quark with momentum xP + and transverse spin s q in a proton with transverse spin S - NOTE: transverse momentum is integrated over! Notation of Anselmino et al.

22. Alessandro Bacchetta - Transverse-momentum- dependent functions 22/58 Transversity: birth and growth Ralston, Soper, NPB 152 (79)

23. Alessandro Bacchetta - Transverse-momentum- dependent functions 23/58 Helicity and transversity        

24. Alessandro Bacchetta - Transverse-momentum- dependent functions 24/58 Transversity vs helicity Different evolution h 1 @ 0.079 GeV 2 g1g1 h 1 @ 0.75 GeV 2 g1g1 S. Scopetta, V. Vento,PLB 424 (1997)

25. Alessandro Bacchetta - Transverse-momentum- dependent functions 25/58 Transversity vs helicity Different evolution Different integrals (axial and tensor charge of the nucleon). E.g. from lattice QCD: S. Aoki et al., PRD 56 (1997) M. Göckeler et al. [QCDSF/UKQCD], PLB (05)

26. Alessandro Bacchetta - Transverse-momentum- dependent functions 26/58 Transversity vs helicity Different evolution Different integrals (axial and tensor charge of the nucleon). E.g. from lattice QCD: Different sum-rules Bakker, Leader, Trueman, PRD 70 (04) S. Aoki et al., PRD 56 (1997) M. Göckeler et al. [QCDSF/UKQCD], PLB (05)

27. Alessandro Bacchetta - Transverse-momentum- dependent functions 27/58 Comparison of tensor charges Barone, Drago, Ratcliffe, PR 359 (2002) lattice  quark soliton 1 spectator bag  quark soliton 2  quark non-relativistic

28. Alessandro Bacchetta - Transverse-momentum- dependent functions 28/58 First data: Collins asymmetry hadron plane lepton plane Airapetian et al. [HERMES], PRL 94 (05) Alexakhin et al. [COMPASS], PRL 94 (05) See talk by G. Schnell, A. Martin

29. Alessandro Bacchetta - Transverse-momentum- dependent functions 29/58 First attempt to combine BELLE and HERMES data Efremov, Goeke, Schweitzer, PRD 73 (06) 1. Get Collins function from BELLE Talk by R. Seidl Talks by P. Schweitzer and M. Boglione

30. Alessandro Bacchetta - Transverse-momentum- dependent functions 30/58 First attempt to combine BELLE and HERMES data 2. Check size of transversity at HERMES See talk by P. Schweitzer

31. Alessandro Bacchetta - Transverse-momentum- dependent functions 31/58 Where to observe it Schnell,Martin, Avakian, Elschenbroich Martin, Radici, Giordano Heppelman, Dalpiaz Talks by Koch, Fischer Heppelman, Aidala

32. The Sivers function

33. Alessandro Bacchetta - Transverse-momentum- dependent functions 33/58 Introducing the Sivers function Probability to find a quark with momentum xP + and k T in a proton with transverse spin S see e.g. Bacchetta, D’Alesio, Diehl, Miller, PRD 70 (04) - T-odd

34. Alessandro Bacchetta - Transverse-momentum- dependent functions 34/58 Sivers function: birth and growth D. Sivers, PRD41 (90)

35. Alessandro Bacchetta - Transverse-momentum- dependent functions 35/58 Two ingredients Final-state interactions (included in the gauge link) Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03) Transverse-spin dependent distribution of quarks in transverse space Burkardt, PRD 66 (02); Diehl, Hägler, EPJ C44 (05)

36. Alessandro Bacchetta - Transverse-momentum- dependent functions 36/58 down up Final-state interactions proton quarks Side viewFront view

37. Alessandro Bacchetta - Transverse-momentum- dependent functions 37/58 down up Final-state interactions photon Side viewFront view NOTE: QCD tells us that the FSI has to be attractive, since quark and remnants form a color antisymmetric state

38. Alessandro Bacchetta - Transverse-momentum- dependent functions 38/58 down up Final-state interactions photon Burkardt, PRD 66 (02) Chromodynamic lensing Side viewFront view See talk by M. Burkardt

39. Alessandro Bacchetta - Transverse-momentum- dependent functions 39/58 down up Change of sign in Drell-Yan proton photon quarks Side viewFront view antiquark Clear-cut prediction of QCD Collins, PLB 536 (02)

40. Alessandro Bacchetta - Transverse-momentum- dependent functions 40/58 down up Distortions in transverse space proton quarks Side viewFront view

41. Alessandro Bacchetta - Transverse-momentum- dependent functions 41/58 down up Distortions in transverse space proton quarks Side viewFront view  A distortion in the distribution of quarks in transverse space can give rise to a nonzero Sivers function The presence of spin can distort the distribution of quarks in transverse space (orbital angular momentum of quarks is required)

42. Alessandro Bacchetta - Transverse-momentum- dependent functions 42/58 Relation to GPDs The transverse-space deformation of unpolarized quarks in a transversely polarized nucleon is described by the generalized parton distribution function E Work of Burkardt

43. Alessandro Bacchetta - Transverse-momentum- dependent functions 43/58 Lattice-QCD studies Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032) up quarks Sivers function for up quarks expected to be NEGATIVE See talk by P. Hägler

44. Alessandro Bacchetta - Transverse-momentum- dependent functions 44/58 Model calculations spectator proton up down Bacchetta, Schäfer, Yang, PLB 578 (04) Brodsky, Hwang, Schmidt PLB530 (02) Gamberg, Goldstein, Oganessyan, PRD 67 (03) See talk by L. Gamberg

45. Alessandro Bacchetta - Transverse-momentum- dependent functions 45/58 Fits to HERMES and COMPASS data Airapetian et al. [HERMES], PRL 94 (05) Anselmino et al., PRD72 (05)

46. Alessandro Bacchetta - Transverse-momentum- dependent functions 46/58 Comparison with COMPASS data Alexakhin et al. [COMPASS], PRL94 (05), Anselmino et al., PRD72 (05) deuteron

47. Alessandro Bacchetta - Transverse-momentum- dependent functions 47/58 Different fits Anselmino et al., hep-ph/0511017 [20] Anselmino et al., PRD72 (05) [21] Vogelsang, Yuan, PRD72 (05) [23] Collins et al., hep-ph/0510342

48. Alessandro Bacchetta - Transverse-momentum- dependent functions 48/58 New preliminary kaon data See talk by U. Elschenbroich

49. Alessandro Bacchetta - Transverse-momentum- dependent functions 49/58 Future experimental data HERMES@DESY:  0 and weighted asymmetries (this year?) COMPASS@CERN: proton data (this year?) CLAS@Jlab STAR,PHENIX@RHIC: jet-production in pp collisions EIC (DIS with polarized protons) FAIR@GSI (Drell-Yan with polarized protons) See talk by U. D’Alesio

50. The Boer-Mulders function

51. Alessandro Bacchetta - Transverse-momentum- dependent functions 51/58 Definition of the Boer-Mulders function Probability to find a quark with momentum xP + and k T and spin s q T-odd -

52. Alessandro Bacchetta - Transverse-momentum- dependent functions 52/58 Boer-Mulders function: birth and growth Boer, Mulders, PRD57 (98)

53. Alessandro Bacchetta - Transverse-momentum- dependent functions 53/58 down up Relation to transverse space proton quarks Side viewFront view

54. Alessandro Bacchetta - Transverse-momentum- dependent functions 54/58 down up      Relation to transverse space proton quarks Side viewFront view A distortion in the distribution of quark spin in transverse space can give rise to a Boer-Mulders function Burkardt, hep-ph/0510408 Quark spin can be unevenly distributed in transverse space (orbital angular momentum of quarks is required)  

55. Alessandro Bacchetta - Transverse-momentum- dependent functions 55/58 Lattice-QCD studies Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032) up quarks Boer-Mulders function for up quarks expected to be NEGATIVE

56. Alessandro Bacchetta - Transverse-momentum- dependent functions 56/58 Model calculations up down Bacchetta, Schäfer, Yang, PLB 578 (04) See talk by L. Gamberg, G. Goldstein

57. Alessandro Bacchetta - Transverse-momentum- dependent functions 57/58 Where to look for it Unpolarized semi-inclusive DIS (HERMES, COMPASS, H1, ZEUS, EIC, LHEC) Unpolarized Drell-Yan (GSI) Jet-production in unpolarized pp collisions (RHIC, LHC)

58. Alessandro Bacchetta - Transverse-momentum- dependent functions 58/58 Conclusions Many progresses from the theoretical, experimental, phenomenological sides…...Pay attention to the rest of the workshop!

59. Alessandro Bacchetta - Transverse-momentum- dependent functions 59/58 downup Semi-inclusive Deep Inelastic Scattering proton pion  lepton