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Accessing transversity via single spin (azimuthal) asymmetries

COMPASS workshop Paris, March 2004. Accessing transversity via single spin (azimuthal) asymmetries. 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. P.J. Mulders Vrije Universiteit Amsterdam

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Accessing transversity via single spin (azimuthal) asymmetries

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  1. COMPASS workshop Paris, March 2004 Accessing transversity via single spin (azimuthal) asymmetries 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 P.J. Mulders Vrije Universiteit Amsterdam pjg.mulders@few.vu.nl

  2. 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 pT hadroproduction COMPASS p j mulders

  3. Soft physics in hard processes (e.g. inclusive deep inelastic leptoproduction) COMPASS p j mulders

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

  5. Lightcone dominance in DIS

  6. Leadingorder DIS • In limit of large Q2 the result of ‘handbag diagram’ survives • … + contributions from A+ gluons A+ Ellis, Furmanski, Petronzio Efremov, Radyushkin A+ gluons  gauge link COMPASS p j mulders

  7. Matrix elements <yA+y> produce the gauge link U(0,x) in leading quark lightcone correlator Color gauge link in correlator A+

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

  9. Distribution functions • M/P+ parts appear as M/Q terms in s • 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 Selection via specific probing operators (e.g. appearing in leading order DIS, SIDIS or DY) Jaffe & Ji NP B 375 (1992) 527

  11. Lightcone correlatormomentum density Production matrix: y+ = ½ g-g+ y 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. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Matrix representationfor M = [F(x)g+]T 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

  14. 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 • Perturbative QCD  evolution COMPASS p j mulders

  15. Hard processes with two or more hadrons COMPASS p j mulders

  16. SIDIS cross section • variables • hadron tensor

  17. (calculation of) cross sectionSIDIS “Full” calculation + + PARTON MODEL + … +

  18. Lightfront dominance in SIDIS

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

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

  21. Lightfront correlator(distribution) + Lightfront correlator (fragmentation) Collins & Soper NP B 194 (1982) 445 no T-constraint T|Ph,X>out =|Ph,X>in Jaffe & Ji, PRL 71 (1993) 2547; PRD 57 (1998) 3057

  22. Distribution A+ including the gauge link (in SIDIS) One needs also AT G+a = +ATa ATa(x)= ATa(∞) +dh G+a Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M, Pijlman, hep-ph/0303034 From <y(0)AT()y(x)> m.e.

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

  24. Distribution • for plane waves T|P> = |P> • But... T U[0, ]T = U[0,- ] • this does affect F[](x,pT) • appearance of T-odd functions in F[](x,pT) including the gauge link (in SIDIS or DY)

  25. Ralston & Soper NP B 152 (1979) 109 Parameterizations including pT Tangerman & Mulders PR D 51 (1995) 3357 Constraints from Hermiticity & Parity • Dependence on …(x, pT2) • Without T: • h1^ and f1T^ • nonzero! • T-odd functions • Fragmentation f  D g  G h  H • No T-constraint: H1^ and D1T^ nonzero!

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

  27. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Lightcone correlatormomentum density Remains valid for F(x,pT) … and also after inclusion of links forF[](x,pT) Sum over lightcone wf squared Brodsky, Hoyer, Marchal, Peigne, Sannino PR D 65 (2002) 114025

  28. 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

  29. Difference between F[+] and F[-] Integrate over pT

  30. Integrated distributions T-odd functions only for fragmentation

  31. Weighted distributions Appear in azimuthal asymmetries in SIDIS or DY These are process-dependent (through gauge link) and thus need in fact [±] superscript!

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

  33. pT-dependent functions Matrix representationfor M = [F[±](x,pT)g+]T T-odd: g1T g1T – i f1T^ and h1L^  h1L^ + i h1^ Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

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

  35. pT-dependent functions Matrix representationfor 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)

  36. 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, F[](x,pT) and D[](z,kT) • Gauge link structure is process dependent ( []) • pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance • This allows T-odd functions h1^ and f1T^ (H1^ and D1T^) appearing in single spin asymmetries COMPASS p j mulders

  37. T-odd effects in single spin asymmetries COMPASS p j mulders

  38. T-odd  single spin asymmetry • Wmn(q;P,S;Ph,Sh) = -Wnm(-q;P,S;Ph,Sh) • Wmn(q;P,S;Ph,Sh) = Wnm(q;P,S;Ph,Sh) • Wmn(q;P,S;Ph,Sh) = Wmn(q;P, -S;Ph, -Sh) • Wmn(q;P,S;Ph,Sh) = Wmn(q;P,S;Ph,Sh) symmetry structure hermiticity * _ _ _ _ _ _ parity _ _ _ _ _ _ time reversal * 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-

  39. Example of a single spin asymmetry example:sOTO in ep epX • example of a leading azimuthal asymmetry • T-odd fragmentation function (Collins function) • 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

  40. Single spin asymmetriessOTO • T-odd fragmentation function (Collins function) or • T-odd distribution function (Sivers function) • Both of the above also appear in SSA in pp  pX • Different asymmetries in leptoproduction! • But be aware now of [±] dependence Collins NP B 396 (1993) 161 Sivers PRD 1990/91 Boer & Mulders PR D 57 (1998) 5780 Boglione & Mulders PR D 60 (1999) 054007

  41. Process dependence and universality COMPASS p j mulders

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

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

  44. Time reversal constraints for distribution functions T-odd (imaginary) Time reversal: F[+](x,pT)  F[-](x,pT) pFG F[+] F T-even (real) F[-] COMPASS p j mulders

  45. Consequences for distribution functions SIDIS F[+] DY F[-] F[](x,pT) = F(x,pT) ± pFG Time reversal 

  46. Distribution functions F[](x,pT) = F(x,pT) ± pFG Sivers effect in SIDIS and DY opposite in sign Collins hep-ph/0204004

  47. Time reversal constraints for fragmentation functions T-odd (imaginary) Time reversal: D[+]out(z,pT)  D[-]in(z,pT) pDG D[+] D T-even (real) D[-] COMPASS p j mulders

  48. Time reversal constraints for fragmentation functions T-odd (imaginary) Time reversal: D[+]out(z,pT)  D[-]in(z,pT) D[+]out pDG out D out T-even (real) D[-]out COMPASS p j mulders

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