1. “Time-reversal-odd” distribution functions in chiral models with vector mesons Alessandro Drago University of Ferrara

2. Outline T-odd distributions in QCD Chiral models with vector mesons as dynamical gauge bosons Large N c expansion and the relation

3. Brodsky, Hwang and Schmidt mechanism

4. Gauge link and factorization in Drell-Yan and in DIS (Collins 2002)

5. From QCD to chiral lagrangians In QCD the two main ingredients are: Gauge theory  Wilson lines Factorization  violation of universality How to compute “T-odd” distributions in chiral models? If we are not using a gauge theory we have no necessity to introduce Wilson lines…

6. Chiral lagrangians with a hidden local symmetry Bando, Kugo, Uehara, Yamawaki, Yanogida 1985 A theorem: “Any nonlinear sigma model based on the manifold G/H is gauge equivalent to a linear model with a G global H local symmetry and the gauge bosons corresponding to the hidden local symmetry H local are composite gauge bosons” For instance: G = SU(2) L SU(2) R H = SU(2) V

7. What is a hidden symmetry? Kinetic term of a nonlinear sigma model: Under the global SU(2) L X SU(2) R symmetry: We rewrite U(x) in terms of two auxiliary variables: The transformation rules under [SU(2) L X SU(2) R ] global X [SU(2) V ] local are:

8. L V and L A are both invariant under [SU(2) L X SU(2) R ] global X [SU(2) V ] local Any linear combination of L V and L A is equivalent to the original kinetic term.

9. About the  mesons So far we have given no dynamics to V  “For simplicity” we add a kinetic term: The  meson acquires a mass via spontaneous breaking of the hidden local SU(2) V symmetry g  = g universality M  2 = 2 g  2 f  2 KSRF relation

10. Introducing matter fields linear representation of H local singlet of G global … is the simplest term After the gauge fixing matter fields transform non-linearly because h(x) has to be restricted to the  dependent form.

11. Again about matter fields (quarks) Take a representation  of G whose restriction to H contains  0 Defining    so that    g)  we have two  types of quarks: “constituent quarks”  (singlet of G global ) “current quarks”  (singlet of H local ) Vector mesons are non-singlet of H local and therefore they are “composed” of constituent quarks .

12. Sivers function in  models     constituent quarks Almost identical to the QCD diagram.  has a physical mass. The exchange of a  meson can also be included

13. The 1/N c expansion At leading order (1/N c ) 0  hedgehog solution (the problem of time reversal has already been solved by the link operators…)

14. Conclusions “T-odd” distributions can be computed also in chiral models, at least if vector mesons are introduced as gauge bosons. At leading order in 1/N c : Work to be done: to compute explicitely these distributions. Open problem: if one does not introduce a dynamics associated with the vector mesons, T-odd distributions can still be computed?