single spin asymmetries in hard reactions

1. Los Alamos 5 Aug. 2002 single spin asymmetries inhard reactions P.J. Mulders Vrije Universiteit Amsterdam mulders@nat.vu.nl

2. Content • Observables in (SI)DIS in field theory language  lightcone/lightfront correlations • Relation to lightcone wave functions • Single-spin asymmetries in hard reactions  T-odd correlations • T-odd in final (fragmentation) and initial state (distribution) correlations • Conclusions and further work lightcone2002 p j mulders

3. Soft physics in inclusive deep inelastic leptoproduction lightcone2002 p j mulders

4. (calculation of) cross sectionDIS Full calculation + + + … PARTON MODEL +

5. Leadingorder DIS • In limit of large Q2 only result of ‘handbag diagram’ survives • Isolate part encoding soft physics ? ? lightcone2002 p j mulders

6. Lightcone dominance in DIS

7. Distribution functions Soper Jaffe Ji NP B 375 (1992) 527 Parametrization consistent with: Hermiticity, Parity & Time-reversal

8. Distribution functions Selection via specific probing operators (e.g. appearing in leading order DIS, SIDIS or DY) Jaffe Ji NP B 375 (1992) 527

9. Bacchetta Boglione Henneman Mulders PRL 85 (2000) 712 Lightcone correlatormomentum density y+ = ½ g-g+ y Sum over lightcone wf squared

10. Kogut & Soper Lightfront quantization • Good fields are • independent, • satisfying CCR’s • Suitable to define • the partons in QCD

11. Basis for 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

12. Matrix representation Related to the helicity formalism Anselmino et al. • Off-diagonal elements (RL or LR) are chiral-odd functions • Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY

13. chiral-odd functions diagonal in transverse spin basis

14. Color gauge link in correlator • Diagrams containing correlator • <yA+y> produce the gauge • link U(0,x) in quark-quark • lightcone correlator

15. F(x) subleading • include M/P+ parts • gives M/Q terms in s • T-odd only for FF in e.g. e+e- Jaffe Ji NP B 375 (1992) 527 Jaffe Ji PRL 71 (1993) 2547

16. Summarizing DIS • Structure functions (observables) are identified with distribution functions (lightcone quark-quark correlators) • DF’s are quark densities that are directly linked to lightcone wave functions squared • There are three DF’s f1q(x) = q(x), g1q(x) =Dq(x), h1q(x) =dq(x) • Longitudinal gluons (A+, not seen in LC gauge) are absorbed in DF’s • Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators lightcone2002 p j mulders

17. Soft physics in semi-inclusive (1-particle incl) leptoproduction lightcone2002 p j mulders

18. SIDIS cross section • variables • hadron tensor

19. (calculation of) cross sectionSIDIS Full calculation + + PARTON MODEL + … +

20. Leading order SIDIS • In limit of large Q2 only result of ‘handbag diagram’ survives • Isolating parts encoding soft physics ? ? lightcone2002 p j mulders

21. Lightfront dominance in SIDIS

22. Lightfront dominance in SIDIS Three external momenta P Ph q transverse directions relevant qT = q + xB P – Ph/zh or qT = -Ph^/zh

23. Lightfront correlator(distribution) + Lightfront correlator (fragmentation) no T-constraint T|Ph,X>out =|Ph,X>in Collins Soper NP B 194 (1982) 445

24. Ralston Soper NP B 152 (1979) 109 distribution functions in SIDIS Tangerman Mulders PR D 51 (1995) 3357 Constraints from Hermiticity & Parity • Dependence on …(x, pT2) • T-invariance: • h1^ = f1T^ = 0? • T-odd functions • Fragmentation f  D g  G h  H • No T-constraint: H1^ and D1T^ nonzero!

25. Ralston Soper NP B 152 (1979) 109 Distribution functions in SIDIS Tangerman Mulders PR D 51 (1995) 3357 Selection via specific probing operators (e.g. appearing in leading order SIDIS or DY)

26. Bacchetta Boglione Henneman Mulders PRL 85 (2000) 712 Lightcone correlatormomentum density Remains valid for F(x,pT) y+ = ½ g-g+ y Sum over lightcone wf squared

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

28. Matrix representationfor M = [F(x)g+]T Collinear structure of the nucleon!

29. pT-dependent functions Matrix representationfor M = [F(x,pT) g+]T T-odd: g1T g1T – i f1T^ and h1L^  h1L^ + i h1^

30. pT-dependent functions Matrix representation for M = [D(z,kT) g-]T • FF’s: f  D g  G h  H • No T-inv constraints H1^ and D1T^ nonzero!

31. pT-dependent functions Matrix representation for M = [D(z,kT) g-]T • R/L basis for spin 0 • Also for spin 0 a T-odd function exist, H1^ (Collins function) e.g. pion • FF’s after kT- integration leaves just the ordinary D1(z)

32. pT-dependent Distribution and fragmentation functions T-odd ? • pT-integrated no T-odd Mulders Tangerman NP B 461 (1996) 197

33. Sample azimutal asymmetrysLTO • example of a leading azimuthal asymmetry • without appropriate weights this would be a convolution instead of a factorized expression Kotzinian NP B 441 (1995) 234 Tangerman Mulders PL B 352 (1995) 129

34. Sample single spin asymmetrysOTO • example of a leading azimuthal asymmetry • T-odd fragmentation function (Collins function) • T-odd  single spin asymmetry • involves two chiral-odd functions • Best way to get transverse spin polarization h1q(x) Collins NP B 396 (1993) 161 Tangerman Mulders PL B 352 (1995) 129

35. Summarizing SIDIS • Transverse momenta of partons become relevant, effects appearing in azimuthal asymmetries • DF’s and FF’s depend on two variables, F(x,pT) and D(z,kT) • Fragmentation functions are not constrained by time-reversal invariance • This allows T-odd functions H1^ and D1T^, appearing in single spin asymmetries lightcone2002 p j mulders

36. T-odd phenomena • T-invariance does not constrain fragmentation • T-odd FF’s (e.g. Collins function H1^) • T-invariance does constrain F(x) • No T-odd DF’s and thus no SSA in DIS • What about T-invariance and F(x,pT)? lightcone2002 p j mulders

37. Color gauge link in correlator Diagrams containing correlator <yA+y> produce the gauge links U(0,-J) and U(-J,x) in quark-quark matrix element Boer Mulders NP B 569 (2000) 505 hep-ph/9906223 ?

38. Distribution • for plane waves T|P> = |P> • But...T U-T = U+ • this does not affect F(x) • it does affect F(x,pT) • appearance of T-odd functions in F(x,pT) • color gauge inv requires leading transverse gluons including the gauge link ? From <y(0)AT(-J)y(x)> m.e.?

39. T-odd phenomena Related work: • T-invariance does not constrain F(x,pT) • Effects are connected to gauge link • They require ‘leading’ pT-effects • They change of sign from SIDIS to DY • They spoil density interpretation of functions and relation to lc wf • Phenomenology of T-odd functions exist • Similar T-odd effects exist at subleading (twist-3) level • Experiment can give answers via specific experiments Boer Mulders Teryaev PR D 57 (1998) 3057 hep-ph/9710223 Brodsky Hwang Schmidt hep-ph/0201296 Qiu Sterman Ji Yuan hep-ph/0206057 Schaefer Teryaev Collins hep-ph/0204004 Brodsky Hoyer PR D 65 (2002) 114025 Mulders Tangerman NP B 461 (1996) 197 lightcone2002 p j mulders

40. Single spin asymmetriessOTO • T-odd fragmentation function (Collins function) or • T-odd distribution function (Sivers function) • Both of the above can explain pp pX SSA • Different asymmetries in leptoproduction! Collins NP B 396 (1993) 161 Boer Mulders PR D 57 (1998) 5780 Boglione Mulders PR D 60 (1999) 054007

41. Conclusions • Hard reactions with two hadrons (DY, SIDIS) offer possibilities to access parton transverse momenta pT • Experimental access via azimuthal asymmetries, often also requiring polarization • T-odd phenomena occur in single-spin asymmetries • These are natural for fragmentation functions and seem possible for pT-dependent distribution functions involving necessary <yATy> pieces in gauge link lightcone2002 p j mulders