1. The transverse spin structure of the nucleon
Mauro Anselmino, Torino University and INFN, Vancouver, July 31, 2007

2. The longitudinal structure of nucleons is “simple” It has been studied for almost 40 yearsxP P Q2

3. essentially x and Q2 degrees of freedom ….

4. Very good or good knowledge of q(x,Q2), g(x,Q2) and Δq(x,Q2) Poor knowledge of Δg(x,Q2)
(Research Plan for Spin Physics at RHIC)
Talk by L. De Nardo

5. The transverse structure is much more interesting and less studied
spin-k┴ correlations?
orbiting quarks? ?
Transverse Momentum Dependent distribution functions
Space dependent distribution functions

6. The mother of all functions M. Diehl, Trento workshop, June 07
Wigner function
(Belitsky, Ji, Yuan)
GPD’s
TMD’s

7. SSA in SIDIS: Sivers functions
TMDs in SIDIS
SSA in SIDIS: Sivers functions
Collins function from e+e- unpolarized processes (Belle) and first extraction of transversity
SSA in hadronic processes
Future measurements and transversity
(Trento workshop on “Transverse momentum, spin, and position distributions of partons in hadrons”, June 07)

8. Main source of information on transverse nucleon structure
SIDIS kinematics according to Trento conventions (2004)

9. Polarized SIDIS cross section, up to subleading order in 1/Q
Kotzinian, NP B441 (1995) 234
Mulders and Tangermann, NP B461 (1996) 197
Boer and Mulders, PR D57 (1998) 5780
Bacchetta et al., PL B595 (2004) 309
Bacchetta et al., JHEP 0702 (2007) 093
SIDISLAND

10. SIDIS in parton model with intrinsic k┴
factorization holds at large Q2, and
Ji, Ma, Yuan

11. Azimuthal dependence induced by quark intrinsic motion
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

12. Boer-Mulders function
Sivers function
Boer-Mulders function

13. 8 leading-twist spin-k┴ dependent distribution functions
Courtesy of Aram Kotzinian

14. Polarizing fragmentation function
Collins function
Polarizing fragmentation function

16. Large K+ asymmetry!

18. Talk by G. Schnell

19. COMPASS measured Collins and Sivers asymmetries for positive (●) and negative (○) hadrons
small values due to deuteron target: cancellation between u and d contributions
talk by T. Iwata

20. (Kretzer fragmentation functions)
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin
A fit of HERMES + COMPASS pion data, information on u and d Sivers funtions
no sea contribution
sea contribution?
(Kretzer fragmentation functions)

21. Fragmentation functions
DSS = de Florian, Sassot, Stratmann KRE = Kretzer HKNS = Hirai, Kumano, Nagai, Sudoh

22. M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin
fit of HERMES + COMPASS pion and kaon data, with new set of fragmentation functions (de Florian, Sassot, Stratmann)

25. Predictions for JLab

26. Present knowledge of Sivers function (u,d)
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
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.

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

28. Total amount of intrinsic momentum carried by partons of flavour a
for a proton moving along the +z-axis and polarization vector S

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

30. Sivers function and orbital angular momentum
D. Sivers
Sivers mechanism originates from
then it is related to the quark orbital angular momentum
For a proton moving along z and polarized along y

31. Sivers function and proton anomalous magnetic moment
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:
related result (M. Burkardt)

32. Collins function from e+e– processes (spin effects without polarization, D. Boer)j1
j2-p e-
thrust-axis
e+e- CMS frame:
KEK

34. Fit of BELLE data

35. Extraction of Collins functions and transversity distributions from fitting HERMES + COMPASS + BELLE data
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin C. Türk

36. h1u and h1d have opposite signs
What do we learn (if anything yet) from the transversity distributions and the Collins functions ?
h1u and h1d have opposite signs
They are both smaller than Soffer bound
Still large uncertainties
Tensor charges appear to be smaller than results from lattice QCD calculations M. Wakamatsu, hep-ph/
Collins functions well below the positivity bound. Favoured and unfavoured ones have opposite signs, comparable magnitudes
Only the very beginning …

37. TMDs and SSAs in hadronic collisions (abandoning safe grounds ….)
(collinear configurations)
factorization theorem (?) a b c X
PDF FF
pQCD elementary interactions

38. RHIC data
excellent agreement with data for unpolarized cross section, but no SSA

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

40. talk by L. Bland STAR-RHIC √s = 200 GeV 1.2 < pT < 2.8
and AN stays at high energies ….
talk by L. Bland

41. SSA in hadronic processes: intrinsic k┴, factorization?
Two main different (?) approaches
Generalization of collinear scheme (M. A., M. Boglione, U. D’Alesio, E. Leader, F. Murgia, S. Melis) a b c X

42. main remaining contribution to SSA from Sivers effect
It generalizes to polarized case
plenty of phases
main remaining contribution to SSA from Sivers effect

43. U. D’Alesio, F. Murgia
E704 data
STAR data
fit
prediction

44. Higher-twist partonic correlations (Efremov, Teryaev; Qiu, Sterman; Kouvaris, Vogelsang, Yuan)
contribution to SSA
twist-3 functions
hard interactions
“collinear expansion” at order ki┴

45. fits of E704 and STAR data
Kouvaris, Qiu, Vogelsang, Yuan

46. Gluonic pole cross sections and SSA in
Bacchetta, Bomhof, Mulders, Pijlman; Vogelsang, Yuan
factorization ? (Collins)
Sivers contribution to SSA
gluonic pole cross sections take into account gauge links
Diagram dependent Gauge link Colour factors
(breaking of factorization?)

47. Gluonic pole cross sections and SSA in
to be compared with the usual cross section

48. Non-universality of Sivers Asymmetries:
Unique Prediction of Gauge Theory !
Simple QED
example:
DIS: attractive
Drell-Yan: repulsive
Same in QCD:
As a result:

49. factorization holds, two scales, M2, and
TMDs and SSAs in Drell-Yan processes (returning to safer grounds and looking at future ….) p
Q2 = M2 qT qL l+ l– γ*
factorization holds, two scales, M2, and

50. Unpolarized cross section already very interesting
Collins-Soper frame

51. ! ! ! (Polarized) Drell-Yan cross sections allow to
access many TMDs (Boer-Mulders, …) D-YLAND
verify whether
and offer the golden channel to measure the transversity distribution
! ! !
Talks by R. Kaiser, M. Contalbrigo

52. Conclusions
Increasing knowledge about quark transverse motion, and spin-k┴ correlations (Sivers function)
First experimental information on h1
Work in progress on GPDs and quark space distribution (Lq)
Towards resolving the orbital motion of quarks in a proton …
More data from HERMES, COMPASS, JLab and RHIC
Thanks