Three-dimensional imaging of the nucleon in momentum space

1. [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] Three-dimensional imaging of the nucleon in momentum space Cédric Lorcé Mainz University Germany

2. Outline • Semi-inclusive DIS and TMDs • Model relations and spherical symmetry • Orbital angular momentum

3. Deep-Inelastic Scattering DIS 2 ~ Im PDFs E.g. :

4. Semi-inclusive DIS [Collins (1993)] [Bacchetta & al. (2007)] SIDIS 2 FFs ~ Im TMDs TMDs FFs E.g. :

5. Semi-inclusive DIS

6. Transverse-Momentum Distributions Dirac matrix selects quark polarization TMDs TMDs parametrize the quark-quark correlator Quark density in momentum space

7. Quark polarization Nucleon polarization Transverse-Momentum Distributions Convenient to think in terms of net polarization Light-cone helicity

8. Model calculations [C.L., Pasquini, Vanderhaeghen (2011)] ~ 3Q light-cone wave functions cQSM LCCQM

9. Model relations Linear relations Quadratic relation * * Flavor-dependent * Flavor-independent * * * * * * * * * Bag cQSM LCCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe & Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)] [C.L., Pasquini & Vanderhaeghen (2011)] [Pasquini & al. (2005-2008)] [Ma & al. (1996-2009), Jakob & al. (1997), Bacchetta & al. (2008)] [Ma & al. (1996-2009), Jakob & al. (1997)][Bacchetta & al. (2008)] [Efremov & al. (2009)] [Meißner & al. (2007)] *=SU(6)

10. LC helicity and canonical spin Bag Model, cQSM, LCCQM, Quark-Diquark Model (Ma) and Covariant Parton Model Common assumption : Quasi-free quarks Wigner rotation (reduces to Melosh rotation in case of FREE quarks) LC helicity Canonical spin

11. Quark polarization Quark polarization Nucleon polarization Nucleon polarization LC helicity and canonical spin [C.L., Pasquini (2011)] LC helicity Canonical spin

12. 2 2 2 = = + = 0 = = - Spherical symmetry [C.L., Pasquini (2011)] Axial symmetry about Axial symmetry about

13. TMDs GPDs Angular momentum Ji Ji Jaffe-Manohar • Each term is gauge-invariant • No decomposition of • Decomposition is gauge-dependent • OAM in LCWFs refers to (easy) Ji’s sum rule Pretzelosity [Avakian & al. (2010)] Model-dependent! Trans. pol. quark in trans. pol. proton

14. Summary • SIDIS sensitive to quark 3-momentum • Complementary to DVCS • TMDs describe spin-spin and spin-orbit correlations • Interesting relation to quark OAM • Stronger contraints on nucleon LCWFs • Phenomenological relations due to spherical symmetry [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)]

15. Backup

16. LC helicity and canonical spin Canonical boost Light-cone boost

17. Spherical symmetry [C.L., Pasquini (2011)] Bag Model, cQSM, LCCQM, Quark-Diquark Model (Ma) and Covariant Parton Model Common assumption : Explicit or implicit rotational symmetry The probability does not depend on the direction of canonical polarization

18. Spherical symmetry [C.L., Pasquini (2011)] Axial symmetry about = = 0 2 2 2 2 + = +

19. Spherical symmetry [C.L., Pasquini (2011)] Axial symmetry about = = 0 2 2 2 = +

20. Spherical symmetry [C.L., Pasquini (2011)] Axial symmetry about = = -

21. Why do relations appear in models? Bag Model, cQSM, LCCQM, Quark-Diquark (Ma) and Covariant Parton Models Spherical symmetry Axial symmetries Not independent!

22. LCWFs without explicit dependence satisfy flavor-independent relations! Why do relations appear in models? What about Quark-Diquark models of Jakob & al. and Bacchetta & al.? • Quark and diquark (a priori) not independent • WF defined directly in Front Form Scalar diquark (Yukawa) Axial-vector diquark Jakob & al. Bacchetta & al. ~ Independent constituents? Spherical symmetry? Instant Form WFs?

23. Formalism Independent quarks LC helicity Canonical spin Light-Cone Quark Model (Melosh rotation) Chiral Quark-Soliton Model S-wave P-wave Bag Model S-wave P-wave

24. Formalism Assumption : • in instant form (automatic w/ spherical symmetry) More convenient to work in canonical spin basis