1. Single spin asymmetries in pp scattering Piet Mulders Trento July 2-6, 2006 mulders@few.vu.nl _

2. Content Single Spin Asymmetries (SSA) in pp scattering Introduction: what are we after? SSA and time reversal invariance Transverse momentum dependence (TMD) Through TMD distribution and fragmentation functions to transverse moments and gluonic poles Electroweak processes (SIDIS, Drell-Yan and annihilation) Hadron-hadron scattering processes Gluonic pole cross sections What can pp add? Conclusions _ _

3. Introduction: what are we after? The partonic structure of hadrons For (semi-)inclusive measurements, cross sections in hard scattering processes factorize into a hard squared amplitude and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (    or G) lightcone lightfrontTMD FF

4. The partonic structure of hadrons Quark distribution functions (DF) and fragmentation functions (FF) –unpolarized q(x) = f 1 q (x) and D(z) = D 1 (z) –Polarization/polarimetry  q(x) = g 1 q (x) and  q(x) = h 1 q (x) –Azimuthal asymmetries g 1T (x,p T ) and h 1L (x,p T ) –Single spin asymmetries h 1  (x,p T ) and f 1T (x,p T ); H 1  (z,k T ) and D 1T (z,k T ) Form factors Generalized parton distributions FORWARD matrix elements x section one hadron in inclusive or semi- inclusive scattering NONLOCAL lightcone OFF- FORWARD Amplitude Exclusive NONLOCAL lightfront LOCAL NONLOCAL lightcone pictures? appendix

5. SSA and time reversal invariance QCD is invariant under time reversal (T) Single spin asymmetries (SSA) are T-odd observables, but they are not forbidden! For distribution functions a simple distinction between T-even and T-odd DF’s can be made –Plane wave states (DF) are T-invariant –Operator combinations can be classified according to their T-behavior (T-even or T-odd) Single spin asymmetries involve an odd number (i.e. at least one) of T-odd function(s) The hard process at tree-level is T-even; higher order  s is required to get T-odd contributions Leading T-odd distribution functions are TMD functions

6. Intrinsic transverse momenta In a hard process one probes partons (quarks and gluons) Momenta fixed by kinematics (external momenta) DISx = x B = Q 2 /2P.q SIDIS z = z h = P.K h /P.q Also possible for transverse momenta SIDIS q T = k T – p T = q + x B P – K h /z h   K h  /z h 2-particle inclusive hadron-hadron scattering q T = p 1T + p 2T – k 1T – k 2T = K 1 /z 1 + K 2 /z 2  x 1 P 1  x 2 P 2  K 1  /z 1 + K 2  /z 2 Sensitivity for transverse momenta requires  3 momenta SIDIS:  * + H  h + X DY: H 1 + H 2   * + X e+e-:  *  h 1 + h 2 + X hadronproduction: H 1 + H 2  h + X  h 1 + h 2 + X p  x P + p T k  z -1 K + k T K2K2 K1K1      pp-scattering

7. TMD correlation functions (unpolarized hadrons)  (x, p T ) Transverse moment T-odd Transversely polarized quarks quark correlator In collinear cross section In azimuthal asymmetries pictures?

8. Color gauge invariance Nonlocal combinations of colored fields must be joined by a gauge link: Gauge link structure is calculated from collinear A.n gluons exchanged between soft and hard part Link structure for TMD functions depends on the hard process! SIDIS   [U + ] =  [+] DY   [U - ] =  [  ] DIS   [U]

9. Integrating  [±] (x,p T )   [±] (x) collinear correlator 

10. Integrating  [±] (x,p T )     [±] (x) transverse moment  G (p,p  p 1 ) T-evenT-odd

11. Gluonic poles Thus   [±]  (x) =    (x) + C G [±]  G  (x,x) C G [±] = ±1 with universal functions in gluonic pole m.e. (T-odd for distributions) There is only one function h 1  (1) (x) [Boer-Mulders] and (for transversely polarized hadrons) only one function f 1T  (1) (x) [Sivers] contained in  G These functions appear with a process-dependent sign Situation for FF is (maybe) more complicated because there are no T-constraints Efremov and Teryaev 1982; Qiu and Sterman 1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201 Metz and Collins 2005 What about other hard processes (e.g. pp and pp scattering)? _

12. Other hard processes C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 Link structure for fields in correlator 1 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

13. 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  (x,p T ) enters for each contributing term in squared amplitude with specific link  [Tr(U □ )U + ] (x,p T ) U □ = U + U  †  [U □ U + ] (x,p T ) C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277

14. Gluonic pole cross sections Thus   [U]  (x) =    (x) + C G [U]  G  (x,x) C G [U ± ] = ±1 C G [U □ U + ] = 3, C G [Tr(U □ )U + ] = N c with the same uniquely defined functions in gluonic pole matrix elements (T-odd for distributions)

15. examples: qq  qq C G [D 1 ] D1D1 = C G [D 2 ] = C G [D 4 ] C G [D 3 ] D2D2 D3D3 D4D4 Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

16. Gluonic pole cross sections In order to absorb the factors C G [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: Similarly for gluon processes Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171 (gluonic pole cross section) y

17. examples: qq  qq D1D1 For N c  : C G [D 1 ]   1 (color flow as DY)

18. Conclusions Single spin asymmetries in hard processes can exist They are T-odd observables, which can be described in terms of T-odd distribution and fragmentation functions For distribution functions the T-odd functions appear in gluonic pole matrix elements Gluonic pole matrix elements are part of the transverse moments appearing in azimuthal asymmetries Their strength is related to path of color gauge link in TMD DFs which may differ per term contributing to the hard process The gluonic pole contributions can be written as a folding of universal (soft) DF/FF and gluonic pole cross sections Belitsky, Ji, Yuan, NPB 656 (2003) 165 Boer, Mulders, Pijlman, NPB 667 (2003) 201 Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030 Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171 Eguchi, Koike, Tanaka, hep-ph/0604003 Ji, Qiu, Vogelsang, Yuan, hep-ph/0604023 end

19. Local – forward and off-forward Local operators (coordinate space densities): PP’  Static properties: Examples: (axial) charge mass spin magnetic moment angular momentum Form factors

20. Nonlocal - forward Nonlocal forward operators (correlators): Specifically useful: ‘squares’ Momentum space densities of  -ons: Selectivity at high energies: q = p Sum rules  form factors

21. Nonlocal – off-forward Nonlocal off-forward operators (correlators AND densities): Sum rules  form factors Forward limit  correlators GPD’s b Selectivity q = p

22. Caveat We study forward matrix elements, including transverse momentum dependence (TMD), i.e. f(p ||,p T ) with enhanced nonlocal sensitivity! This is not a measurement of orbital angular momentum (OAM). Direct measurement of OAM requires off-forward matrix elements, i.e. GPD’s. One may at best make statements like: linear p T dependence  nonzero OAM no linear p T dependence  no OAM back

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