1. The transverse structure of the nucleon (resolving the quark motion inside a nucleon) Mauro Anselmino, Torino University and INFN, JLab, December 15, 2006  the need for parton intrinsic motion  transverse momentum dependent distribution and fragmentation functions (TMD) spin and k ┴ : transverse Single Spin Asymmetries SSA in SIDIS Spin effects in unpolarized e + e – → h 1 h 2 X at BELLE what do we learn from TMD’s?

2. Plenty of theoretical and experimental evidence for transverse motion of partons within nucleons and of hadrons within fragmentation jets uncertainty principle gluon radiation ±1 ± ± k┴k┴ q T distribution of lepton pairs in D-Y processes Partonic intrinsic motion p p Q 2 = M 2 qTqT qLqL l+l+ l–l–

3. p T distribution of hadrons in SIDIS Hadron distribution in jets in e + e – processes Large p T particle production in Transverse motion is usually integrated, but there might be important spin- k ┴ correlations p xp k┴k┴

4. M. Arneodo et al (EMC): Z. Phys. C 34 (1987) 277 Intrinsic motion in unpolarized SIDIS, in collinear parton model thus, no dependence on azimuthal angle Ф h at zero-th order in pQCD the experimental data reveal that

5. Cahn: the observed azimuthal dependence is related to the intrinsic k ┴ of quarks (at least for small P T values) These modulations of the cross section with azimuthal angle are denoted as “Cahn effect”. assuming collinear fragmentation, φ = Φ h

6. SIDIS with intrinsic k ┴ kinematics according to Trento conventions (2004) factorization holds at large Q 2, and Ji, Ma, Yuan

7. The full kinematics is complicated as the produced hadron has also intrinsic transverse momentum with respect to the fragmenting parton neglecting terms of order one has

8. assuming: one finds: with clear dependence on(assumed to be constant) and Find best values by fitting data on Φ h and P T dependences

9. EMC data, µp and µd, E between 100 and 280 GeV M.A., M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia and A. Prokudin

10. Large P T data explained by NLO QCD corrections

11. dashed line: parton model with unintegrated distribution and fragmentation functions solid line: pQCD contributions at LO and a K factor (K = 1.5) to account for NLO effects

12. S p p'p' – p PTPT θ – p ' 5 independent helicity amplitudes z y x Example: Transverse single spin asymmetries: elastic scattering

13. needs helicity flip + relative phase – + + + x + + + + at quark level but large SSA observed at hadron level! QED and QCD interactions conserve helicity, up to corrections Single spin asymmetries at partonic level. Example:

14. BNL-AGS √s = 6.6 GeV 0.6 < p T < 1.2 E704 √s = 20 GeV 0.7 < p T < 2.0 experimental data on SSA observed transverse Single Spin Asymmetries

15. and A N stays at high energies …. STAR-RHIC √s = 200 GeV 1.2 < p T < 2.8

16. PDF FF pQCD elementary interactions a b c X X (collinear configurations) factorization theorem

17. RHIC data

18. Transverse Λ polarization in unpolarized p-Be scattering at Fermilab

20. “Sivers moment”

21. “Collins moment”

22. and now ….? Polarization data has often been the graveyard of fashionable theories. If theorists had their way, they might just ban such measurements altogether out of self-protection. J.D. Bjorken St. Croix, 1987

23. needs k ┴ dependent quark distribution in p ↑ and/or p ┴ dependent fragmentation of polarized quark z y x ΦSΦS ΦπΦπ X p S PTPT Transverse single spin asymmetries in SIDIS in collinear configurations there cannot be (at LO) any P T

24. spin-k ┴ correlations? orbiting quarks? Transverse Momentum Dependent distribution functions Space dependent distribution functions (GPD) ?

25. p S k┴k┴ φ q pqpq φ SqSq p┴p┴ Amsterdam group notations spin-k ┴ correlations Sivers functionCollins function

26. p SqSq k┴k┴ φ q pqpq φ SΛSΛ p┴p┴ Amsterdam group notations spin-k ┴ correlations Boer-Mulders function polarizing F.F.

27. 8 leading-twist spin-k ┴ dependent distribution functions Courtesy of Aram Kotzinian

28. Sivers function dependence on S T in cross section The Sivers mechanism for SSA in SIDIS processes

29. Sivers angle SSA: after integration over k ┴ one obtains:

30. data are presented for the sinΦ moment of the analyzing power

31. Parameterization of the Sivers function q = u, d. The Sivers function for sea quarks and antiquarks is assumed to be zero.

32. M.A., U.D’Alesio, M.Boglione, F. Murgia, A.Kotzinian, A Prokudin from Sivers mechanism

33. Deuteron target

34. First p ┴ moments of extracted Sivers functions, compared with models M.A, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin data from HERMES and COMPASS hep-ph/0511017

35. The first and 1/2-transverse moments of the Sivers quark distribution functions. The fits were constrained mainly (or solely) by the preliminary HERMES data in the indicated x-range. The curves indicate the 1-σ regions of the various parameterizations. M. Anselmino, M. Boglione, J.C. Collins, U. D’Alesio, A.V. Efremov, K. Goeke, A. Kotzinian, S. Menze, A. Metz, F. Murgia, A. Prokudin, P. Schweitzer, W. Vogelsang, F. Yuan

36. predictions for JLab, proton target, 6 GeV 0.4 ≤ z h ≤ 0.7 0.02 ≤ P T ≤ 1 GeV/c 0.1 ≤ x B ≤ 0.6 0.4 ≤ y ≤ 0.85 Q 2 ≥ 1 (GeV/c) 2 W 2 ≥ 4 GeV 2 1 ≤ E h ≤ 4 GeV

37. predictions for JLab, proton target, 12 GeV 0.4 ≤ z h ≤ 0.7 0.02 ≤ P T ≤ 1.4 GeV/c 0.05 ≤ x B ≤ 0.7 0.2 ≤ y ≤ 0.85 Q 2 ≥ 1 (GeV/c) 2 W 2 ≥ 4 GeV 2 1 ≤ E h ≤ 7 GeV

38. S number density of partons with longitudinal momentum fraction x and transverse momentum k ┴, inside a proton with spin S M. Burkardt, PR D69, 091501 (2004) What do we learn from the Sivers distribution?

39. Then we can compute the total amount of intrinsic momentum carried by partons of flavour a for a proton moving along the +z-axis and polarization vector S

40. Sivers functions extracted from A N data in give also opposite results, with Numerical estimates from SIDIS data U. D’Alesio

41. Collins mechanism for SSA q q’ initial quark transverse spin is transmitted to the final quark (in collinear configuration, here ↑,↓ means perpendicular to the leptonic plane)

42. The polarization of the fragmenting quark q' can be computed in QED depolarization factor

43. Collins effect in SIDIS transversity distribution

44. fit to HERMES data on assuming W. Vogelsang and F. Yuan Phys. Rev. D72, 054028 (2005)

45. A. V. Efremov, K. Goeke and P. Schweitzer (h 1 from quark-soliton model)

47. Collins function from e + e – processes (spin effects without polarization) e + e - CMS frame: BELLE @ KEK e+e+     e-e- e+e+ thrust-axis

48. single quark or antiquark are not polarized, but there is a strong correlation between their spins cross section for detecting the final hadrons inside the jets contains the product of two Collins functions

49. Cosine modulations clearly visible M. Grosse Perdekamp, A. Ogawa, R. Seidl Talk at SPIN2006 BELLE data

50. U = unlike charged pions L = like charged pions

51. fit of HERMES data on Collins functions and transversity distributions from a global best fit of HERMES, COMPASS and BELLE data M.A., M. Boglione, U.D’Alesio, A.Kotzinian, F. Murgia, A Prokudin, C. Türk, in preparation

52. fit of COMPASS data on

53. fit of BELLE data on

54. Extracted favoured and unfavoured Collins functions positivity bound Efremov et al. Vogelsang, Yuan

55. Extracted transversity distributions Soffer bound

56. Predictions for Collins asymmetry at JLab, proton target, 6 GeV

57. Predictions for Collins asymmetry at JLab, proton target, 12 GeV

58. Sivers function and orbital angular momentum Sivers mechanism originates from then it is related to the quark orbital angular momentum D. Sivers For a proton moving along z and polarized along y Open issues and wild ideas …

59. Sivers function and proton anomalous magnetic momentum M. Burkardt, S. Brodsky, Z. Lu, I. Schmidt Both the Sivers function and the proton anomalous magnetic moment are related to correlations of proton wave functions with opposite helicities in qualitative agreement with large z data:

60. Conclusions Towards the transverse spin and momentum structure of the nucleon via azimuthal and P hT dependence in SIDIS Information on quark intrinsic motion Spin-k ┴ correlation from Sivers function Orbiting quarks? Extracting Collins function and accessing transversity Much more data needed …