International Workshop on Transversity: New Developments in Nucleon Spin Structure June 2004 Color Gauge Invariance in Hard Processes Vrije Universiteit Amsterdam Fetze Pijlman A. Bacchetta D. Boer C.J. Bomhof P.J. Mulders in collaboration with

2 Contents transversity and transverse momentum dependent correlators gauge links appearing in hard scattering processes gauge links and their consequences gauge links in more complex hard QCD processes summary

3 Transversity in semi-inclusive DIS cross-sectionleading order in M/Q transversity in the hadronic tensor: Collins, Nucl.Phys.B396:161, 1993

4 Example of a proper deconvolution concerning transversity Longitudinal polarized target (beam direction) and unpolarized electron beam: transverse moment of the Collins function: Bacchetta, Mulders, F.P., hep-ph/ Korotkov, Nowak, Oganessyan, Eur.Phys.J. C18:639, 2001 Mulders, Tangerman, Nucl.Phys. B461:197, 1996

5 Calculation of the hadronic tensor 1.assume factorization 2.make an expansion in M/Q If then gluons coming from the proton give leading contributions (leading twist) Resumming all gluons coming from the proton = Efremov, Radyushkin, Theor.Math.Phys. 44:774, 1981

6 Introduction of the gauge links Resummation of gluons in SIDIS at leading order: Resummation of gluons in Drell-Yan at leading order: Belitsky, Ji, Yuan, Nucl.Phys.B656:165,2003

7 Relevance of gauge links 1) nice gauge invariant correlators, affects transverse moments 2) the origin of the Sivers effect and Qiu-Sterman mechanism in SIDIS and DY Boer, Mulders, F.P., Nucl.Phys. B667:201,2003 3) links could spoil the Lorentz Invariance Relations such as Goeke, Metz, Pobylitsa, Polyakov, Phys.Lett.B567:27,2003 4) the link dependence might lead to additional distribution functions like Bacchetta, Mulders, F.P., hep-ph/ ) for fragmentation there are 2 sources for T-odd effects which prohibit a simple sign relation between the functions measured in SIDIS and e + e - annihilation Boer, Mulders, F.P., Nucl.Phys. B667:201,2003

8 Input from the experimental side 1) a nonzero Sivers function proves the existence of T-odd distribution functions and gluonic poles. It would stress the importance of the link structure 2) a nonzero stresses the link dependency in correlators and indicates a violation of the Lorentz Invariance Relations. asymmetry for jet production in DIS (polarized beam, unpolarized target, quark masses neglected): Bacchetta, Mulders, F.P., hep-ph/

9 Links in large transverse momentum in SIDIS 1) hard poles are cancelled and soft poles remain and give the gauge link 2) link structure is similar to ordinary SIDIS ( ) C.J. Bomhof, P.J. Mulders, F.P., hep-ph/ soft pole hard pole

10 Introduction of a (Wilson) loop link for DY at large transverse momentum: link for DY: transverse moments: ordinary DY: difference is exactly large momentum: a gluonic pole C.J. Bomhof, P.J. Mulders, F.P., hep-ph/

11 Summary and outlook 1)We found more complex link structures in hard scattering processes involving loops. The differences in link structures appear in azimuthal asymmetries as gluonic poles. 2) Nonzero gluonic poles imply: T-odd distribution functions and require a link prescription to describe the asymmetries. 3) The link dependence leads to the introduction of new functions. One of these new functions, can be measured in jet production in DIS. 4) We will study link structures and their consequences in more general scattering processes like pion production in proton proton scattering.

12