1. Polarized structure functions Piet Mulders ‘Lepton scattering and the structure of nucleons and nuclei’ September 16-24, 2004 pjg.mulders@few.vu.nl

2. Content Spin structure & transversity Transverse momenta & azimuthal asymmetries T-odd phenomena & single spin asymmetries

3. DIS Known leptonic part Completeness allows reduction in hadronic tensor to commutator [J  (x),J (0)] Known structure of current in terms of quarks OPE ….

4. Deep inelastic scattering (DIS)

5. Lepton tensor Lepton tensor can also be expanded using the spacelike and timelike vectors Tensor encompasses many ‘polarization options’

6. Polarized DIS

7. Semi-inclusive deep inelastic scattering Known lepton part with much flexibility (unused in DIS) Involves two hadrons and hence a much more complex hadronic tensor

8. SIDIS

9. (calculation of) cross section in DIS Full calculation + … + + +PARTON MODEL

10. Lightcone dominance in DIS

11. Leading order DIS In limit of large Q 2 the result of ‘handbag diagram’ survives … + contributions from A + gluons ensuring color gauge invariance A + gluons  gauge link Ellis, Furmanski, Petronzio Efremov, Radyushkin A+A+

12. Parametrization of lightcone correlator Jaffe & Ji NP B 375 (1992) 527 Jaffe & Ji PRL 71 (1993) 2547 leading part M/P + parts appear as M/Q terms in  T-odd part vanishes for distributions but is important for fragmentation

13. Basis of partons  ‘Good part’ of Dirac space is 2-dimensional  Interpretation of DF’s unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity

14.  Off-diagonal elements (RL or LR) are chiral-odd functions  Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY Matrix representation for M = [  (x)  + ] T Quark production matrix, directly related to the helicity formalism Anselmino et al. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

15. Results for DIS Structure functions in (sub)leading order in 1/Q Two of three (Polarized) quark densities for each flavor: Not accessible in DIS

16. (calculation of) cross section in SIDIS “Full” calculation + + … + + PARTON MODEL

17. Lightfront dominance in SIDIS Three external momenta P P h q transverse directions relevant q T = q + x B P – P h /z h or q T = -P h  /z h

18. Leading order SIDIS In limit of large Q 2 only result of ‘handbag diagram’ survives Isolating parts encoding soft physics ? ?

19. Lightfront correlators no T-constraint T|P h,X> out = |P h,X> in Collins & Soper NP B 194 (1982) 445 Jaffe & Ji, PRL 71 (1993) 2547; PRD 57 (1998) 3057

20. Distribution From  A T (  )  m.e. including the gauge link (in SIDIS) A+A+ One needs also A T G +  =  + A T  A T  (  )= A T  ( ∞ ) +  d  G +  Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M, Pijlman, hep-ph/0303034

21. Parametrization of  (x,p T ) Link dependence allows also T-odd distribution functions since T U[0,  ] T = U[0,-  ] Functions h 1  and f 1T  (Sivers) nonzero! These functions (of course) exist as fragmentation functions (no T-symmetry) H 1  (Collins) and D 1T 

22. Interpretation unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity need p T T-odd

23.  p T -dependent functions T-odd: g 1T  g 1T – i f 1T  and h 1L   h 1L  + i h 1  (imaginary parts) Matrix representation for M = [   ±] (x,p T )  + ] T Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

24. T-odd  single spin asymmetry with time reversal constraint only even-spin asymmetries the time reversal constraint cannot be applied in DY or in  1-particle inclusive DIS or e  e  In those cases single spin asymmetries can be used to select T-odd quantities * *  W  (q;P,S;P h,S h ) =  W  (  q;P,S;P h,S h )  W  (q;P,S;P h,S h ) = W  (q;P,S;P h,S h )  W  (q;P,S;P h,S h ) = W  (q;P,  S;P h,  S h )  W  (q;P,S;P h,S h ) = W  (q;P,S;P h,S h ) _ __ _ _ ___ _ _ _ _ time reversal symmetry structure parity hermiticity * *

25. Leptoproduction of pions H 1  is T-odd and chiral-odd

27. COLLINS ASYMMETRY RESULTS OF COMPASS A coll depends on p hT, z h, x Bj with more statistics, the full analysis is foreseen from 2002 data: A coll vs x BjSign!

28. COLLINS ASYMMETRY RESULTS OF COMPASS from 2002 data: A Coll vs z h all the tests made are consistent with the fact that systematic effects, if present, are smaller than statistical errors Sign!

29. Distribution A+A+ A+A+ including the gauge link (in SIDIS or DY) SIDIS SIDIS   [-] DY DY   [+]

30. Difference between  [+] and  [-] upon integration  integrated quark distributions transverse moments measured in azimuthal asymmetries ± Back to the lightcone (theoretically clean) twist 2 twist 2 & 3

31. Difference between  [+] and  [-] upon integration gluonic pole m.e. (T-odd) In momentum space: Conclusion: T-odd parts are gluon-driven (QCD interactions)

32. Time reversal constraints for distribution functions Time reversal   (x,p T )    (x,p T )  G        T-even (real) T-odd (imaginary) Conclusion: T-odd effects in SIDIS and DY have opposite signs

33. Time reversal constraints for fragmentation functions Time reversal   out (z,p T )    in (z,p T )  G        T-even (real) T-odd (imaginary)

34. Time reversal constraints for fragmentation functions  G out    out   out    out T-even (real) T-odd (imaginary) Time reversal   out (z,p T )    in (z,p T ) Conclusion: T-odd effects in SIDIS and e  e  are not related

35. other hard processes qq-scattering as hard subprocess insertions of gluons collinear with parton 1 are possible at many places this leads for ‘external’ parton fields to gauge link to lightcone infinity e.g. C. Bomhof, P.J. Mulders and F. Pijlman PLB 596 (2004) 277

36. other hard processes qq-scattering as hard subprocess insertions of gluons collinear with parton 1 are possible at many places this leads for ‘external’ parton fields to gauge link to lightcone infinity The correlator  (x,p T ) enters for each contributing term in squared amplitude with specific link The link may enhance the effect of the (T-odd) gluonic pole contribution involving also specific color factors Finding the right observables, however is crucial

37. Conclusions Hard processes  quark and gluon structure of hadrons (quark distributions, their chirality and transverse polarization) Many new observables accessible when going beyond collinearity, often in combination with (transverse) polarization (among others the simplest access to transverse quark polarization) Going beyond collinearity gives access to gluon dynamics in hadrons, which can be done in a controlled way via weighted asymmetries (twist limited, t  3), use of chirality, and the specific time-reversal behavior of single spin asymmetries.