1. Overview on Transverse Momentum Dependent Distribution and Fragmentation Functions M. Boglione

2. Glasgow HERMES Symposium2M. Boglione  The nucleon has an internal structure  x is the fraction of proton momentum carried by the parton  The cross section is the incoherent sum of all partonic contributions convoluted with the parton distribution function, which only depends on x at LO q (E,k) (E,k’)  xP P

3. Glasgow HERMES SymposiumM. Boglione3  QCD corrections induce Q 2 dependence  f(x) → f(x,Q 2 )  DGLAP evolution equations exactly predict this Q 2 dependence NOTE : As we will see later, the evolution of some TMD distribution and fragmentation functions is yet unknown

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5. Glasgow HERMES SymposiumM. Boglione5 q Δq ΔTqΔTq D.E. Soper, Phys. Rev. D 15 (1977) 1141; Phys. Rev. Lett. 43 (1979) 1847; J.C. Collins and D.E. Soper, Nucl. Phys. B194 (1982) 445; R.L. Jaffe, Nucl. Phys. B 229 (1983) 205.

6. Glasgow HERMES SymposiumM. Boglione6  Very good knowledge of unpolarized distribution functions, q(x,Q 2 ) and g(x,Q 2 )  Fairly good knowledge of longitudinally polarized, partonic distributions, Δq(x,Q 2 ); poor knowledge of longitudinally polarized gluons Δg(x,Q 2 )  NO direct information on transversely polarized partonic distributions, Δ T q(x,Q 2 ), from DIS

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8. Glasgow HERMES SymposiumM. Boglione8  There is no gluon transversity distribution function  Transversity cannot be studied in deep inelastic scattering because it is chirally odd  Transversity can only appear in a cross-section convoluted to another chirally odd function

9. Glasgow HERMES SymposiumM. Boglione9 ℓ ℓ’ℓ’  P k k’ f q (x) D h/q (z) distribution function X Incoming proton elementary cross- section fragmentation function h final detected hadron

10. Glasgow HERMES SymposiumM. Boglione10 Intrinsic Transverse Momentum Intrinsic Transverse Momentum

11. Glasgow HERMES SymposiumM. Boglione11 P xP kk Plenty of theoretical and experimental evidence for transverse motion of partons within nucleons, and of hadrons within fragmentation jets

12. Glasgow HERMES SymposiumM. Boglione12 q T distribution of lepton pairs in D-Y processes p T distribution of hadrons in SIDIS

13. Glasgow HERMES SymposiumM. Boglione13 Distribution and fragmentation functions now depend  on the lightcone momentum fraction (x for the distributions and z for the fragmentations)  on Q 2 ( → pQCD evolution),  on the intrinsic transverse momentum of the partons, (k  for the distributions and p  for the fragmentations) OPEN QUESTIONS:  How do TMD’s depend on the intrinsic transverse momentum ? Gaussian behaviour in the central region … Power law decrease at large transverse momentum…  Does the partonic intrinsic transverse momentum k  (p  ) depend on x (z) ?

14. Glasgow HERMES SymposiumM. Boglione14 Mulders and Tangermann, NP B461 (1996) 197, Boer and Mulders, PR D57 (1998) 5780

15. Glasgow HERMES SymposiumM. Boglione15 Courtesy of Aram Kotzinian

16. Glasgow HERMES SymposiumM. Boglione16 Functions in bold face survive k  integration Functions in shaded cells are naïve T-odd Functions in red box are chirally odd Courtesy of A. Bacchetta

17. M. Boglione 17 M. Anselmino, M. Boglione, U. D-Alesio, E. Leader, S. Melis, F. Murgia, PR D71, 014002 (2005), PR D73, 014020 (2006) X A a M. Anselmino, M. Boglione, U. D-Alesio, S. Melis, F. Murgia, A. Prokudin (2010) in preparation The distribution function f a/A is the number density of partons of type a inside hadron A

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21. Glasgow HERMES SymposiumM. Boglione21 Functions in bold face survive k  integration Functions in shaded cells are naïve T-odd Functions in red box are chirally odd

22. Glasgow HERMES SymposiumM. Boglione22 General SIDIS Cross Section General SIDIS Cross Section

23. Glasgow HERMES SymposiumM. Boglione23 TMD-PDF hard scattering TMD-FF All three fundamental blocks contain phases. The most general expression for the cross-section is obtained when all of them are kept into account.

24. Glasgow HERMES SymposiumM. Boglione24 SIDIS in parton model with intrinsic k ┴ Factorization holds at large Q 2, and (Ji, Ma, Yuan)

25. Glasgow HERMES SymposiumM. Boglione25 EMC data, µp and µd, E between 100 and 280 GeV CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb, AIP Conf.Proc.1149:423-426,2009. W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara

26. Glasgow HERMES SymposiumM. Boglione26 EMC data, µp and µd, E between 100 and 280 GeV At further dependence on is generated A cos φ dependence is also generated by Boer-Mulders ⊗ Collins term, via a kinematical effect in d Δσ ̂, not included in this fit.  Assume a simple, factorized form for the TMD distribution and fragmentation functions, with a gaussian dependence on the intrinsic transverse momentum  Determine the free parameters by fitting experimental data M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, Phys. Rev. D71:074006,2005.

27. Glasgow HERMES SymposiumM. Boglione27 W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara Comparison with M. Anselmino, M. Boglione, A. Prokudin, C. Türk Eur. Phys. J. A 31, 373-381 (2007) does not include the Boer – Mulders contribution

28. Glasgow HERMES SymposiumM. Boglione28 P T dependence of data in agreement with a Gaussian k ⊥ dependence of unpolarized TMDs Hint of a z-dependence at small z values No hint of x dependence in the explored region solid line → Gaussian TMD’s with CLAS data, arXiv:0809.1153 [hep-ex]

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30. 30 Unpolarized proton and lepton Longitudinally polarized proton, polarized lepton Transversely polarized proton, unpolarized lepton Transversely polarized proton, polarized lepton  Studying Sivers, Collins and other mechanisms is complicated by the fact that all these effects mix and overlap in the polarized SIDIS cross section and azimuthal asymmetries  Way out : build appropriately “weighted” azimuthal asymmetries ! The F structure functions contain all the TMD’s Kotzinian, NP B441 (1995) 234, Mulders and Tangerman, 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, M. Anselmino, M. Boglione, U.D’Alesio, S. Melis, F. Murgia, A. Prokudin, (in preparation).

31. Glasgow HERMES SymposiumM. Boglione31 (Avakian, Efremov, Schweitzer, Metz, Teckentrup, arXiv:0902.0689) chiral-even TMDs chiral-odd TMDs Cahn kinematical effects

32. Glasgow HERMES SymposiumM. Boglione32 TMD’s in SIDIS one by one … TMD’s in SIDIS one by one …

33. Glasgow HERMES SymposiumM. Boglione33 The Sivers function is related to the probability of finding an unpolarized quark inside a transversely polarized proton X The Sivers function inbeds the correlation between the proton spin and the quark transverse momentum The Sivers function is T-odd

34. Glasgow HERMES SymposiumM. Boglione34 HERMES Collaboration, Phys.Rev.Lett.103:152002,2009.

35. Glasgow HERMES SymposiumM. Boglione35 Courtesy of F. Bradamante COMPASS vs HERMES, problems? COMPASS proton data S. Levorato for the COMPASS Collaboration, Transversity 2008 COMPASS Collaboration, Phys. Lett. B673: 127-135, 2009 COMPASS deuteron data Compass finds Sivers = 0. Up to now, only HERMES data reveal non-zero Sivers effect. We need further check s!

36. Glasgow HERMES Symposium36 Determining the Sivers function Models → Fits → 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. M. Boglione Bacchetta, Gamberg, Goldstain, Mukherjee, Metz, Amrath, Shaefer, Yang, Brodsky, Schmidt, Hwang, Scopetta, Courtoy, Frattini, Vento, Radici, Pasquini, Yuan …

37. Glasgow HERMES SymposiumM. Boglione37 M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk, Eur.Phys.J.A39:89-100,2009. New data, old fit !

38. Glasgow HERMES SymposiumM. Boglione38 Update the fit using new analysis from HERMES and COMPASS and, possibly, more modern fragmentation functions Study evolution equations Test new parametrizations with non-gaussian intrinsic transverse momentum dependence Comparison SIDIS vs. Drell-Yan

39. Glasgow HERMES SymposiumM. Boglione39 Spin-orbit correlations A non-zero Sivers function requires non-zero orbital angular momentum ! Lattice [P. Haegler et al.] L u >0 L d <0 [Matthias Burkardt]

40. Glasgow HERMES SymposiumM. Boglione40 Courtesy of Alexei Prokudin

41. Glasgow HERMES SymposiumM. Boglione41 Courtesy of Alexei Prokudin

42. Glasgow HERMES SymposiumM. Boglione42 The Boer-Mulders function is related to the probability of finding a polarized quark inside an unpolarized proton X The Boer-Mulders function inbeds the correlation between the quark spin and its transverse momentum The Boer- Mulders function is chirally odd and T-odd

43. Glasgow HERMES SymposiumM. Boglione43 W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara F. Giordano and R. Lamb, AIP Conf.Proc.1149:423-426,2009. L. P. Gamberg and G. R. Goldstein, Phys.Rev.D77:094016,2008. Diquark spectator model

44. Glasgow HERMES SymposiumM. Boglione44 V. Barone, S. Melis, A. Prokudin ArXiv: 0912.5194 A. Prokudin, talk at Workshop on Transverse Spin Physics, Beijing (2008 ) There are two contributions to : Boer-Mulders + Collins (no suppression in 1/Q) Cahn effect at O(k  2 /Q 2 ), which turns out to be large

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46. Glasgow HERMES SymposiumM. Boglione46 The Collins function is related to the probability that a transversely polarized struck quark will fragment into a spinless hadron The Collins function is chirally odd X The Collins function inbeds the correlation between the fragmenting quark spin and the transverse momentum of the produced hadron

47. 47 The Collins effect in SIDIS couples to transversity COMPASS 2002-2004 S. Levorato for the COMPASS Collaboration, Transversity 2008 M. Boglione

48. Glasgow HERMES SymposiumM. Boglione48 We need to determine two convoluted unknown functions – Fix one of the two functions according to some theoretical model and use SIDIS data to determine the other (see for example Efremov, Goeke, Schweitzer) – Perform a simultaneous fit of SIDIS and e+e - →h 1 h 2 X BELLE data. BELLE Thrust axis method BELLE Hadronic plane method

49. Glasgow HERMES SymposiumM. Boglione49 M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk

50. Glasgow HERMES SymposiumM. Boglione50 what about the last 3 TMDs? any relation with the others? neglecting twist-3 contributions similar to the Wandzura-Wilczek relation supported by experiment for a recent model of all twist-2 TMDs see Bacchetta et al., arXiv:0807.0323

51. Glasgow HERMES SymposiumM. Boglione51 HERMES data, PRL 84 (2000) 4047; PL B562 (2003) 182 COMPASS data, arXiv:0705.2402 H. Avakian., A.V. Efremov, P. Schweitzer, F. Yuan → Bag Model predictions arXiv:0805.3355 [hep-ph]

52. Glasgow HERMES Symposium52 Spin-orbit correlations P. Schweitzer, talk given at INT Program 09-3, Seattle

53. Glasgow HERMES Symposium53 The Collins fragmentation function is universal (no initial/final state interactions, no effects induced by requiring color gauge invariance) J. Collins and A. Metz, Phys. Rev. Lett. 93,252001 (2004), F. Yuan, arXiv:0903.4680 The Sivers distribution function (naively time reversal odd) is subject to initial/final state interaction – color gauge invariance requirements induce color factors (process dependence). C. J. Bomhof, P. J. Mulders,F. Pijlman, Eur. Phys. J. C 47, 147 (2006) Example: Mark Schlegel, talk given at INT Program 09-3, Seattle

54. Glasgow HERMES SymposiumM. Boglione54 A. Prokudin, talk at Workshop on Transverse Spin Physics, Beijing (2008 )

55. Glasgow HERMES SymposiumM. Boglione55  Universality  Factorization  Evolution  Orbital Angular Momentum TMD’s

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