1. Universality of T-odd effects in single spin azimuthal asymmetries P.J. Mulders Vrije Universiteit Amsterdam pjg.mulders@few.vu.nl BNL December 2003 Universality of T-odd effects in single spin and azimuthal asymmetries, D. Boer, PM and F. Pijlman, NP B667 (2003) 201-241; hep-ph/0303034

2. 4-12-2003BNL p j mulders2 Content Soft parts in hard processes twist expansion gauge link Illustrated in DIS Two or more (separated) hadrons transverse momentum dependence T-odd phenomena Illustrated in SIDIS and DY Universality Items relevant for other processes Illustrated in high p T hadroproduction

3. 4-12-2003BNL p j mulders3 Soft physics in inclusive deep inelastic leptoproduction

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

5. Lightcone dominance in DIS

6. 4-12-2003BNL p j mulders6 Leading order DIS In limit of large Q 2 the result of ‘handbag diagram’ survives … + contributions from A + gluons A+A+ A + gluons  gauge link Ellis, Furmanski, Petronzio Efremov, Radyushkin

7. Color gauge link in correlator Matrix elements  A +  produce the gauge link U(0,  ) in leading quark lightcone correlator A+A+

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

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

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

11. Lightcone correlator momentum density   = ½      Sum over lightcone wf squared

12. 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

13.  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 Related to the helicity formalism Anselmino et al. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

14. 4-12-2003BNL p j mulders14 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 f 1 q (x) = q(x), g 1 q (x) =  q(x), h 1 q (x) =  q(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 Perturbative QCD  evolution

15. 4-12-2003BNL p j mulders15 Hard processes with two or more hadrons

16. SIDIS cross section  variables  hadron tensor

17. (calculation of) cross section SIDIS Full calculation + + … + + PARTON MODEL

18. Lightfront dominance in SIDIS

19. 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

20. 4-12-2003BNL p j mulders20 Leading order SIDIS In limit of large Q 2 only result of ‘handbag diagram’ survives Isolating parts encoding soft physics ? ?

21. Lightfront correlator (distribution) Lightfront correlator (fragmentation) + 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

22. 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  ( J ) +  d  G +  Ji, Yuan, PLB 543 (2002) 66 Belitsky, Ji, Yuan, hep-ph/0208038

23. Distribution A+A+ A+A+ including the gauge link (in SIDIS or DY) SIDIS SIDIS   [-] DY DY   [+] hep-ph/0303034

24. Distribution  for plane waves T|P> = |P>  But... T U   T = U    this does affect   (x,p T )  it does not affect  (x)   appearance of T-odd functions in   (x,p T ) including the gauge link (in SIDIS or DY)

25. Parameterizations including p T Constraints from Hermiticity & Parity  Dependence on …(x, p T 2 )  Without T: h 1  and f 1T  nonzero! T-odd functions Ralston & Soper NP B 152 (1979) 109 Tangerman & Mulders PR D 51 (1995) 3357  Fragmentation f  D g  G h  H  No T-constraint: H 1  and D 1T  nonzero!

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

27. Lightcone correlator momentum density Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Remains valid for  (x,p T ) … and also after inclusion of links for   (x,p T ) Sum over lightcone wf squared Brodsky, Hoyer, Marchal, Peigne, Sannino PR D 65 (2002) 114025

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

29. Integrated distributions T-odd functions only for fragmentation

30. Weighted distributions Appear in azimuthal asymmetries in SIDIS or DY These are process-dependent (through gauge link)

31. Collinear structure of the nucleon! Matrix representation for M = [  (x)  + ] T reminder

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

33. Matrix representation for M = [  (z,k T )   ] T  p T -dependent functions  FF’s: f  D g  G h  H  No T-inv constraints H 1  and D 1T  nonzero!

34. Matrix representation for M = [  (z,k T )   ] T  p T -dependent functions  FF’s after k T -integration leaves just the ordinary D 1 (z)  R/L basis for spin 0  Also for spin 0 a T-odd function exist, H 1  (Collins function) e.g. pion

35. 4-12-2003BNL p j mulders35 Summarizing SIDIS Beyond just extending DIS by tagging quarks … Transverse momenta of partons become relevant, appearing in azimuthal asymmetries DF’s and FF’s depend on two variables,   (x,p T ) and   (z,k T ) Gauge link structure is process dependent (   p T -dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance This allows T-odd functions h 1  and f 1T  (H 1  and D 1T  ) appearing in single spin asymmetries

36. 4-12-2003BNL p j mulders36 T-odd effects in single spin asymmetries

37. T-odd  single spin asymmetry  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 Conclusion:with time reversal constraint only even-spin asymmetries But time reversal constraint cannot be applied in DY or in  1-particle inclusive DIS or e + e 

38. Single spin asymmetries  OTO  T-odd fragmentation function (Collins function) or  T-odd distribution function (Sivers function)  Both of the above also appear in SSA in pp    X  Different asymmetries in leptoproduction! Boer & Mulders PR D 57 (1998) 5780 Boglione & Mulders PR D 60 (1999) 054007 Collins NP B 396 (1993) 161 Sivers PRD 1990/91

40. 4-12-2003BNL p j mulders40 Process dependence and universality

41. Difference between  [+] and  [-] Integrate over p T

42. Difference between  [+] and  [-]  integrated quark distributions transverse moments measured in azimuthal asymmetries ±

43. Difference between  [+] and  [-] gluonic pole m.e.

44. 4-12-2003BNL p j mulders44 Time reversal constraints for distribution functions Time reversal   (x,p T )    (x,p T )  G        T-even (real) T-odd (imaginary)

45. Consequences for distribution functions    (x,p T ) =   (x,p T ) ±  G Time reversal  SIDIS  [+] DY  [-]

46. Distribution functions    (x,p T ) =   (x,p T ) ±  G Sivers effect in SIDIS and DY opposite in sign Collins hep-ph/0204004

47. 4-12-2003BNL p j mulders47 Time reversal constraints for fragmentation functions Time reversal   out (z,p T )    in (z,p T )  G        T-even (real) T-odd (imaginary)

48. 4-12-2003BNL p j mulders48 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 )

49. Fragmentation functions    (x,p T ) =   (x,p T ) ±  G Time reversal does not lead to constraints Collins effect in SIDIS and e + e  unrelated! If  G = 0 But at present this seems (to me) unlikely

50. 4-12-2003BNL p j mulders50 T-odd phenomena T-invariance does not constrain fragmentation T-odd FF’s (e.g. Collins function H 1  ) T-invariance does constrain  (x) No T-odd DF’s and thus no SSA in DIS T-invariance does not constrain  (x,p T ) T-odd DF’s and thus SSA in SIDIS (in combination with azimuthal asymmetries) are identified with gluonic poles that also appear elsewhere (Qiu-Sterman, Schaefer-Teryaev) Sign of gluonic pole contribution process dependent In fragmentation soft T-odd and (T-odd and T-even) gluonic pole effects arise No direct comparison of Collins asymmetries in SIDIS and e + e  ( unless  G = 0)

51. 4-12-2003BNL p j mulders51 What about hadroproduction?

52. 4-12-2003BNL p j mulders52 Issues in hadroproduction Weighted functions will appear in L-R asymmetries (p T now hard scale!) There are various possibilities with gluons G(x,p T ) – unpolarized gluons in unpolarized nucleon  G(x,p T ) – transversely polarized gluons in a longitudinally polarized nucleon G T (x,p T ) – unpolarized gluons in a transversely polarized nucleon (T-odd) H  (x,p T ) – longitudinally polarized gluons in an unpolarized nucleon …

53. 4-12-2003BNL p j mulders53 Issues in hadroproduction Contributions of   (x,p T ) and  G not necessarily in one combination A N ~ … G(x a )  f 1T  (1)[-] (x b )  D 1 (z c ) + … f 1 (x a )  f 1T  (1)[+] (x b )  D 1 (z c ) + … f 1 (x a )  h 1 (x b )  H 1  [-] (z c ) + … f 1 (x a )  h 1 (x b )  H 1  [+] (z c ) + … f 1 (x a )  G T (x b )  D 1 (z c ) Many issues to be sorted out

54. 4-12-2003BNL p j mulders54 Thank you for your attention

55. Relations among distribution functions 1. Equations of motion 2. Define interaction dependent functions 3. Use Lorentz invariance

56. Distribution functions    (x,p T ) =   (x,p T ) ±  G (omitting mass terms) Sivers effect in SIDIS and DY opposite in sign Collins hep-ph/0204004

57. Fragmentation functions    (x,p T ) =   (x,p T ) ±  G Collins effect in SIDIS and e + e  unrelated! including relations

58. Example of a single spin asymmetry  example of a leading azimuthal asymmetry  T-odd fragmentation function (Collins function)  involves two chiral-odd functions  Best way to get transverse spin polarization h 1 q (x) Tangerman & Mulders PL B 352 (1995) 129 Collins NP B 396 (1993) 161 example:  OTO in ep   epX