Appearance of single spin asymmetries in hard scattering processes

1. Appearance of single spin asymmetries in hard scattering processes Piet Mulders mulders@few.vu.nl Trento June 13, 2007

2. Content • Introduction: the partonic structure of hadrons • Transverse momentum dependence (TMD) • Gauge links and dependence on hard processes • Electroweak processes (SIDIS, Drell-Yan and annihilation) • Hadron-hadron scattering processes • Gluonic pole cross sections • Factorization breaking for TMDs made explicit • Conclusions

3. Introduction:The partonic structure of hadrons For (semi-)inclusive measurements, we want to investigate the factorization of cross sections in hard scattering processes into hard squared QCD-amplitudes and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (f y or G) lightcone TMD lightfront: x+ = 0 FF

4. Quarks • Integration over x- = x.P allows ‘twist’ expansion • Gauge link essential for color gauge invariance • Arises from all ‘leading’ matrix elements containing y A+...A+ y

5. Generic hard processes • E.g. qq-scattering as hard subprocess • Matrix elements involving parton 1 and additional gluon(s) A+ = A.n appear at same (leading) order in ‘twist’ expansion • insertions of gluons collinear with parton 1 are possible at many places • this leads for correlator involving parton 1 to gauge links to lightcone infinity Link structure for fields in correlator 1 C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 [hep-ph/0406099]; EPJ C 47 (2006) 147 [hep-ph/0601171]

6. SIDIS SIDIS  F[U+] =F[+] DY  F[U-] = F[-]

7. A 2  2 hard processes: qq  qq • E.g. qq-scattering as hard subprocess • The correlator F(x,pT) enters for each contributing term in squared amplitude with specific link U□ = U+U-† F[Tr(U□)U+](x,pT) F[U□U+](x,pT)

8. Gluons • Using 3x3 matrix representation for U, one finds in gluon correlator appearance of two links, possibly with different paths. • Note that standard field displacement involves C = C’

9. f2 - f1 K1 df K2 pp-scattering Sensitivity to intrinsic transverse momenta • In a hard process one probes partons (quarks and gluons) • Momenta fixed by kinematics (external momenta) DIS x = xB = Q2/2P.q SIDIS z = zh = P.Kh/P.q • Also possible for transverse momenta SIDIS qT = q + xBP – Kh/zh-Kh/zh = kT – pT 2-particle inclusive hadron-hadron scattering qT = K1/z1+ K2/z2- x1P1- x2P2 K1/z1+ K2/z2 = p1T + p2T – k1T – k2T • Sensitivity for transverse momenta requires  3 momenta SIDIS: g* + H  h + X DY: H1 + H2  g* + X e+e-: g*  h1 + h2 + X hadronproduction: H1 + H2  h1 + h2 + X  h + X (?) p x P + pT k z-1 K + kT Knowledge of hard process(es)!

10. In collinear cross section In weighted azimuthal asymmetries Transverse moment TMD correlation functions (unpolarized hadrons) quark correlator F(x, pT) • T-odd • Transversely • polarized quarks

11. Integrating F[±](x,pT)  F[±](x)  collinear correlator

12. transverse moment FG(p,p-p1) T-even T-odd Integrating F[±](x,pT)  Fa[±](x)

13. Gluonic poles • Thus: F[U](x) = F(x) F[U]a(x) = Fa(x) + CG[U]pFGa(x,x) • Universal gluonic pole m.e. (T-odd for distributions) • pFG(x) contains the T-odd functions h1(1)(x) [Boer-Mulders] and (for transversely polarized hadrons) the function f1T(1)(x) [Sivers] • F(x) contains the T-even functions h1L(1)(x) and g1T(1)(x) • For SIDIS/DY links: CG[U±] = ±1 • In other hard processes one encounters different factors: CG[U□U+] = 3, CG[Tr(U□)U+] = Nc ~ ~ Efremov and Teryaev 1982; Qiu and Sterman 1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201

14. A 2  2 hard processes: qq  qq • E.g. qq-scattering as hard subprocess • The correlator F(x,pT) enters for each contributing term in squared amplitude with specific link U□ = U+U-† F[Tr(U□)U+](x,pT) F[U□U+](x,pT)

15. Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268 D1 CG [D1] = CG [D2] D2 D3 CG [D3] = CG [D4] D4 examples: qqqq in pp

16. Bacchetta, Bomhof, D’Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153 D1 For Nc: CG [D1] -1 (color flow as DY) examples: qqqq in pp

17. (gluonic pole cross section) y Gluonic pole cross sections • In order to absorb the factors CG[U], one can define specific hard cross sections for gluonic poles (which will appear with the functions in transverse moments) • for pp: etc. • for SIDIS: for DY: Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]

18. examples: qgqg in pp D1 Only one factor, but more DY-like than SIDIS D2 D3 D4 Note: also etc.

19. e.g. relevant in Bomhof, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019 examples: qgqg D1 D2 D3 D4 D5

20. examples: qgqg

21. examples: qgqg

22. Pictures for qgqg part in pp (9 diagrams) ‘gluonic pole’ cross section ‘normal’ partonic cross section Both are gauge-invariant combinations (‘of course’ but ‘non-trivial’)

23. examples: qgqg It is also possible to group the TMD functions in a smart way into two! (nontrivial for nine diagrams/four color-flow possibilities) But still no factorization!

24. ‘Residual’ TMDs • We find that we can work with basic TMD functions F[±](x,pT) + ‘junk’ • The ‘junk’ constitutes process-dependent residual TMDs • The residuals satisfies Fint (x) = 0 andpFint G(x,x) = 0, i.e. cancelling kT contributions; moreover they most likely disappear for large kT no definite T-behavior definite T-behavior

25. end Conclusions • Appearance of single spin asymmetries in hard processes is calculable • For integrated and weighted functions factorization is possible • For TMDs one cannot factorize cross sections, introducing besides the normal ‘partonic cross sections’ some ‘gluonic pole cross sections’ • Opportunities: the breaking of factorization can be made explicit and be attributed to specific matrix elements Related: Qiu, Vogelsang, Yuan, hep-ph/0704.1153 Collins, Qiu, hep-ph/0705.2141 Qiu, Vogelsang, Yan, hep-ph/0706.1196 Meissner, Metz, Goeke, hep-ph/0703176