Noncollinear parton structure in hard processes

1. ‘The partonic structure of hadrons’ Trento, May 9-14, 2005 Noncollinear parton structure in hard processes Piet Mulders mulders@few.vu.nl

2. Content • Intrinsic transverse momenta • Partonic structure of hadrons: correlators (distribution/fragmentation) • Color gauge invariance • Gauge link structure • Transverse moments • gluonic poles

3. k p Kh h H PH Intrinsic transverse momenta • Hard processes: Sudakov decomposition for momenta: p = xPH + pT + sn • zero: pT.PH = n2 = pT.n large: PH.n ~ s hadronic: pT2 ~ PH2 = MH2 small: s ~ (p.PH,p2,MH2)/s • Parton virtuality enters in s and is integrated out  FHq(x,pT) • Lightlike vector n enters in the treatment of distributions F(x,pT), but is irrelevant in cross sections • Similarly for • fragmentation: • k = z-1Kh + kT + s’n’ • correlator Dqh(z,kT)

4. Intrinsic transverse momenta • Aim in a hard process is to get acces to the partons (quarks and gluons). • Determine momenta from observed momenta, e.g. DIS x = xB = Q2/2P.q SIDIS z = zh = P.Kh/P.q • Also possible for transverse momenta e.g. SIDIS qT = kT – pT = q + xBP – Ph/zh = - Kh/zh

5. Partonic structure of hadrons hard process k p Ph h H PH fragmentation correlator distribution correlator Need dp-… for ‘separation’ Ph Ph PH PH

6. (calculation of) cross section in DIS Full calculation + + + … + LEADING (in 1/Q)

7. (calculation of) cross section in SIDIS Full calculation + + LEADING (in 1/Q) + … +

8. Correlators for hadrons • Describe the partonic structure of hadrons • Hadrons enter as QM states: |PH> • Partons enter as field operators: i,r(x) or Am,a(x) • Correlators are matrix elements of nonlocal combinations of quark and gluon operators: • nonlocality  momentum dependence lightfront correlators F(x,pT): nonlocality x+ = x.n = 0 lightcone correlators F(x): nonlocality x.n = xT = 0 • The fact that xT 0 leads to nontrivial gauge link structure • Related to antiquark-hadron scattering amplitude after integration over p- (T-ordering irrelevant) (Jaffe, Diehl, Gousset)

9. Lightfront correlators Collins & Soper NP B 194 (1982) 445 Jaffe & Ji, PRL 71 (1993) 2547; PRD 57 (1998) 3057

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

11. DIS  F[U] (unifies Brodsky, Schmidt, Ji, Yuang, …) SIDIS  F[U+] Color gauge invariance • Nonlocal combinations of colored fields must be joined by a gauge link: • Gauge link structure is determined by interactions of gluons between soft and hard part • Link structure for pT-dependent functions in SIDIS involves xT

12. Parametrization and interpretation • Parametrization of F(x,pT) and F(x) in terms of distribution functions f..(x,pT2) and explicit linear pTa • Parametrization of D(z,kT) and D(z) in terms of fragmentation functions D..(z,kT2) and explicit linear kTa • Constraints come from P, C and sometimes T • Time reversal: • Fields have a particular behavior: T-even or T-odd • Behavior of states: T|P0,P,…>in = |P0,-P,…>out • For plane waves: T|P0,P> = |P0,-P> (in = out) • Behavior of gauge link: T U[0,] T† = U[0,-] • F(x): no T-odd functions D(z): T-odd functions (but only subleading: H, DT and EL) F(x,pT): T-odd functions f1T and h1 D(z,kT): T-odd functions D1T and H1 • True for any link, but links may lead to certain functions

13. leading part • M/P+ parts appear as M/Q terms in cross section • T-reversal applies toF(x)  no T-odd functions Parametrization of lightcone correlator Jaffe & Ji NP B 375 (1992) 527 Jaffe & Ji PRL 71 (1993) 2547

14. 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

15. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Matrix representationfor M = [F(x)g+]T Quark production matrix, directly 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

16. 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! • Similar functions (of course) exist as fragmentation functions (no T-constraints) H1^ (Collins) and D1T^

17. 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

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

19. 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 measure T-odd quantities (such as T-odd distribution or fragmentation functions) time reversal * *

20. Lepto-production of pions H1 is T-odd and chiral-odd

21. DIS  F[U] SIDIS  F[U+] DY  F[U-] Color gauge invariance • Nonlocal combinations of colored fields must be joined by a gauge link: • Gauge link structure is determined by interactions of gluons between soft and hard part • Link structure for pT dependent functions depends on the hard process!

22. C. Bomhof, P.J. Mulders and F. Pijlman PLB 596 (2004) 277 Link structure for fields in correlator 1 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 a gauge link to lightcone infinity e.g.

23. 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 a gauge link to lightcone infinity • The correlator F(x,pT) enters for each contributing term in squared amplitude with specific link U□ = U+U-† F[U+Tr(U□)](x,pT) F[U+U□](x,pT)

24. Transverse moments • The correlators F[U](x,pT) and D[U](z,kT) depend on gauge link, which is process dependent. • This will very likely lead to severe problems for factorization including pT, s = f(x,pT2)  …  s D(z,kT2)  … • Look at transverse moments obtained by integration over transverse momentum after weighing F[U](x) =  d2pT pTaF[U](x,pT) • Beware of QCD corrections. Things only works if pTaF[U](x,pT)  as(pT2)/pT2  pT2 

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

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

27. 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[-]

28. 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[-]

29. 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 (except when pDG= 0)

30. Gluonic poles • In general one has F[U]a(x) = Fa(x) + CG[U]pFGa(x,x) • Examples: CG[U±] = ±1 CG[U+U□] = 3 • But uniquely defined functions in gluonic pole m.e. (T-odd for distributions) E.g. there is only one Sivers function f1T(1)(x) contained in pFG • One needs to consider specific hard cross sections for gluonic poles (in order to absorb the factors CG[U]) sℓqgℓq = +sℓqℓqfor SIDIS sqgq*ℓℓ* = -sqq*ℓℓ*for DY sqgqqq = SCG[D]sqqqq for pp [D] [D] In preparation: Bacchetta, Bomhof, Pijlman, Mulders

31. Conclusions • Intrinsic transverse momentum dependence is of interest for the partonic structure of hadrons • Intrinsic transverse momentum accessible in experiment via azimuthal asymmetries, 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 gluonic pole matrix element • Identification of gluonic pole matrix elements may be possible from more processes than only SIDIS and DY (sign flip) and can be made transparant by calculating the appropriate ‘gluonic pole hard cross sections’.