1. Generalized Transverse- Momentum Distributions Cédric Lorcé Mainz University Germany Barbara Pasquini Pavia University Italy In collaboration with:

2. Outline GPDsTMDsGTMDsFFs Transverse charge densities Wigner distributions Parton distributions Physical interpretation Spin densities Generalized Transverse- Momentum Distributions Transverse-Momentum Distributions Generalized Parton Distributions Form Factors PDFs Parton Distribution Functions

3. Experiments Quark models Information on quark distribution 3 (+2?) D P icture of the N ucleon Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … Quark-quark correlator

4. Charges Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … VectorNet # of quarks AxialNet quark longitudinal polarization Net quark transverse polarizationTensor

5. Charges Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … PDFs

6. Charges FFs Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … No clear interpretation ! - - # of quarks changes - - Momentum transfer PDFs

7. Charges FFs Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … Drell-Yan-West frame - - Momentum transfer No clear interpretation in momentum space ! PDFs

8. ProtonNeutron Transverse charge densities Charges FFs Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … Probabilistic interpretation in position space Drell-Yan-West frame 2D Fourier transform [Miller (07)] [Carlson, Vdh (08)] PDFs NB:

9. Transverse charge densities Charges FFs GPDs Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … 2D Fourier transform Spin densities FFsPDFsGPDs [Belitsky & al. (04)] [Burkardt (01,03)] Hadron 3D picture ! PDFs

10. Position space Charges FFs GPDsTMDs Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … Transverse charge densities 2D Fourier transform Spin densities Complementary hadron 3D picture ! No direct connection Momentum space Mean momentum Displacement Position Momentum transfer PDFs

11. Charges FFs GPDsTMDs GTMDs Experiments Quark models Information on quark distribution P arton D istributions Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … 2D Fourier transform Longitudinal Transverse Wigner distribution Transverse charge densities Spin densities PDFs [Meißner & al. (2009)]

12. Charges FFsPDFs GPDsTMDs GTMDs Experiments Quark models Information on quark distribution C omplete P icture Wave function (often just N=3) ES, DIS, SIDIS, DVCS, … Wigner distribution Transverse charge densities Spin densities 2D Fourier transform TMSDs TMFFs Transverse Wigner distribution [C.L., Pasquini (submitted, 2011)]

13. Longitudinal Transverse W igner D istributions [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) GPDs TMDs GTMDs Heisenberg’s uncertainty relations Quasi-probabilistic Third 3D picture ! No restrictions from Heisenberg’s uncertainty relations

14. Longitudinal Transverse E xample: U npol. up Q uark in U npol. P roton fixed [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) (1 out of 16) 3Q light-cone model

15. Longitudinal Transverse E xample: U npol. up Q uark in U npol. P roton fixed Orbital angular momentum? [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) (1 out of 16) 3Q light-cone model favored unfavored

16. E xample: U npol. up Q uark in U npol. P roton (1 out of 16) Longitudinal Transverse [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) 0.1 GeV² 0.2 GeV² 0.3 GeV² 0.4 GeV² 3Q light-cone model

17. Words of caution Longitudinal Transverse No known processes to extract GTMDs Wigner distributions are quasi-probabilistic Issues concerning universality of TMDs Fragmentation functions not so well known Extrapolations needed for Fourier transform Scale-dependence Twist-two picture gauge Problems with transverse gauge link

18. Quark-quark correlator  Most complete information on hadron structure  GTMDs are ‘’mother’’ distributions 2D Fourier transform on the light cone  Correct interpretation (number of partons is fixed)  GTMDs are connected to Wigner distributions Example of Wigner distribution  Unpolarized quark in unpolarized proton  3Q light-cone model  Distortions connected to OAM Summary

19. Backup

20. Unpolarized u and d quarks in unpolarized proton  u /2  d More u than d in central region! n [Miller (2007)]  QSM LCQM Wigner Distributions

21. Hard exclusive meson leptoproduction Handbag approximation DVCS DIS 2 ~ImSIDIS 2 ~Im

22. [Burkardt (2003)] B int S d u d X Neutron Anomalous magnetic moment Orbital angular momentum Induced electric dipole moment Helicity flip Magnetic moment S ome examples: T ransverse C harge D ensities

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

24. ** * * * * Model relations for TMDs (twist-two) Flavor-dependent Flavor-independent Linear relationsQuadratic relation Bag  QSM LCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)] [C.L., Pasquini (in preparation)] [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) * * * * * *

25. LC helicity and canonical spin Quark polarization Nucleon polarization Quark polarization Nucleon polarization LC helicityCanonical spin

26. = = 0 Spherical symmetry = 222 + [C.L., Pasquini (in preparation)] Axial symmetry about = = - =

27. TMDs LCQM  QSM [C.L., Pasquini, Vdh (in preparation)]

28. GPDs (vector & axial) LCQM  QSM H E H E ~ ~ [C.L., Pasquini, Vdh (in preparation)]

29. GPDs (tensor) LCQM  QSM HTHT ETET ~ ~ HTHT ETET [C.L., Pasquini, Vdh (in preparation)]

30. Quark-quark correlator  Most complete information on hadron structure  GTMDs are ‘’mother’’ distribution 2D Fourier transform on the light cone  Correct interpretation of FFs  GTMDs can be related to Wigner distributions  Distortions due to orbital angular momentum TMDs  Model relations due to spherical symmetry  LC helicity and canonical spin connected by a rotation 3Q amplitude  Same structure in many models Summary

31. Light front- and instant-form WFs Assumption :   in instant form (automatic w/ spherical symmetry) More convenient to work in canonical spin basis

32. ,  k T b Unpolarized u quark in unpolarized proton k T  fixed  QSM Wigner Distributions

33. ,k T  fixed k T k T k T k T Unpolarized u quark in unpolarized proton  QSM Wigner Distributions