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Theoretical developments

Transversity workshop Trento, 17/06/2004. Theoretical developments. piet mulders. pjg.mulders@few.vu.nl. Content. Spin structure & transversity Transverse momenta & azimuthal asymmetries T-odd phenomena & single spin asymmetries. Collinear hard processes, e.g. DIS.

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Theoretical developments

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  1. Transversity workshop Trento, 17/06/2004 Theoretical developments piet mulders pjg.mulders@few.vu.nl

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

  3. Collinear hard processes, e.g. DIS • Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) • Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. • There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) • Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators • DF’s are quark densities that are directly linked to lightcone wave functions squared • Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

  4. A+ Ellis, Furmanski, Petronzio Efremov, Radyushkin A+ gluons  gauge link Leadingorder DIS • In limit of large Q2 the result of ‘handbag diagram’ survives • … + contributions from A+ gluons ensuring color gauge invariance

  5. Collinear hard processes, e.g. DIS • Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) • Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. • There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) • Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators • DF’s are quark densities that are directly linked to lightcone wave functions squared • Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

  6. leading part Parametrization of lightcone correlator • 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

  7. Collinear hard processes, e.g. DIS • Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) • Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. • There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) • Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators • DF’s are quark densities that are directly linked to lightcone wave functions squared • Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

  8. 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

  9. Collinear hard processes, e.g. DIS • Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone) • Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv. • There are three ‘leading’ DF’s f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x) • Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators • DF’s are quark densities that are directly linked to lightcone wave functions squared • Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

  10. Non-collinear processes, e.g. SIDIS • Relevant in electroweak processes with two hadrons (SIDIS, DY) • 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

  11. Leading order SIDIS • In limit of large Q2 only result of ‘handbag diagram’ survives • Isolating parts encoding soft physics ? ?

  12. Lightfront correlators 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

  13. Non-collinear processes, e.g. SIDIS • Relevant in electroweak processes with two hadrons (SIDIS, DY) • 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

  14. Distribution including the gauge link (in SIDIS) A+ 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.

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

  16. Non-collinear processes, e.g. SIDIS • Relevant in electroweak processes with two hadrons (SIDIS, DY) • 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

  17. Parametrization of F(x,pT) • Link dependence allows also T-odd distribution functions since T U[0,] T = U[0,-] • Functions h1^ and f1T^ (Sivers) nonzero! • These functions (of course) exist as fragmentation functions (no T-symmetry) H1^ (Collins) and D1T^

  18. 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

  19. 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

  20. pT-dependent DF’stwist structure • For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE • For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist  t • Transverse momentsFa(x,pT)   d2pTpTa F(x,pT) project out the parts in F[](x,pT) proportional to pT. They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon) • Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum • The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd • The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link • Factorization of explicit pT-dependent functions requires ‘soft factors’ • Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTaf at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]

  21. Difference between F[+] and F[-] upon integration Back to the lightcone  integrated quark distributions transverse moments measured in azimuthal asymmetries ±

  22. pT-dependent DF’stwist structure • For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE • For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist  t • Transverse momentsFa(x,pT)   d2pTpTa F(x,pT) project out the parts in F[](x,pT) proportional to pT. They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon) • Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum • The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd • The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link • Factorization of explicit pT-dependent functions requires ‘soft factors’ • Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTaf at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]

  23. Difference between F[+] and F[-] upon integration In momentum space: gluonic pole m.e. (T-odd)

  24. pT-dependent DF’stwist structure • For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE • For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist  t • Transverse momentsFa(x,pT)   d2pTpTa F(x,pT) project out the parts in F[](x,pT) proportional to pT. They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon) • Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum • The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd • The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link • Factorization of explicit pT-dependent functions requires ‘soft factors’ • Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTaf at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]

  25. T-odd phenomena • T-odd phenomena appear in single spin asymmetries • T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs • T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links • Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

  26. 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) _ _ _ _ _ _ _ _ _ _ _ _ T-oddsingle spin asymmetry symmetry structure hermiticity * parity • 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 time reversal *

  27. T-odd phenomena • T-odd phenomena appear in single spin asymmetries • T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs • T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links • Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

  28. Time reversal constraints for distribution functions T-odd (imaginary) Time reversal: F[+](x,pT)  F[-](x,pT) pFG F[+] F T-even (real) Conclusion: T-odd effects in SIDIS and DY have opposite signs F[-]

  29. T-odd phenomena • T-odd phenomena appear in single spin asymmetries • T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs • T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links • Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

  30. 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[-]

  31. 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 Conclusion: T-odd effects in SIDIS and e+e- are not related

  32. T-odd phenomena • T-odd phenomena appear in single spin asymmetries • T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs • T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links • Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

  33. 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

  34. 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 F(x,pT) enters for each contributing term in squared amplitude with specific link • The link may enhance the effect of the gluonic pole contribution involving also specific color factors

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