1. Wigner Distributions and light-front quark models Barbara Pasquini Pavia U. & INFN, Pavia in collaboration with Cédric Lorcé Feng Yuan Xiaonu Xiong IPN and LPT, U. Paris Sud LBNL, Berkeley CHEP, Peking U.

2. Outline Generalized Transverse Momentum Dependent Parton Distributions (GTMDs) Wigner Distributions Parton distributions in the Phase Space FT    b  Results in light-front quark models Quark Orbital Angular Momentum from:  Wigner distributions  Pretzelosity TMD  GPDs

3. Generalized TMDs and Wigner Distributions GTMDs 4 X 4 =16 polarizations 16 complex GTMDs (at twist-2) [ Meißner, Metz, Schlegel (2009)] Quark polarization Nucleon polarization x: average fraction of quark longitudinal momentum » : fraction of longitudinal momentum transfer k ? : average quark transverse momentum ¢ : nucleon momentum transfer Fourier transform 16 real Wigner distributions [Ji (2003)] [Belitsky, Ji, Yuan (2004)]

4. GTMDs Charges PDFs [ Lorce, BP, Vanderhaeghen, JHEP05 (2011)] Wigner distribution 2D Fourier transform GPDs TMFFs FFs Spin densities Transverse charge densities ¢ = 0 TMDs TMSDs

5. Longitudinal Transverse W igner D istributions [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [Lorce’, BP (11)] QM QFT (Breit frame) QFT (light cone) correlations of quark momentum and position in the transverse plane as function of quark and nucleon polarizations  real functions, but in general not-positive definite  quantum-mechanical analogous of classical density on the phase space one-body density matrix in phase-space in terms of overlap of light-cone wf (LCWF)  not directly measurable in experiments needs phenomenological models with input from experiments on GPDs and TMDs GPDs TMDs GTMDs Third 3D picture with probabilistic interpretation ! No restrictions from Heisenberg’s uncertainty relations Heisenberg’s uncertainty relations Quasi-probabilistic Fourier conjugate

6. LCWF Overlap Representation Common assumptions :  No gluons  Independent quarks Bag Model, LC Â QSM, LCCQM, Quark-Diquark and Covariant Parton Models [ Lorce’, BP, Vanderhaeghen (2011)] momentum wfspin-flavor wfrotation from canonical spin to light-cone spin invariant under boost, independent of P  internal variables: LCWF: [Brodsky, Pauli, Pinsky, ’98] quark-quark correlator ( » =0)

7. Canonical boost Light-cone boost Light-Cone Helicity and Canonical Spin LC helicity Canonical spin model dependent: for k ? ! 0, the rotation reduced to the identity

8. parameters fitted to anomalous magnetic moments of the nucleon : normalization constant [Schlumpf, Ph.D. Thesis, hep-ph/9211255]  momentum-space wf  SU(6) symmetry Light-Cone Constituent Quark Model  spin-structure: (Melosh rotation) free quarks Applications of the model to: GPDs and Form Factors: BP, Boffi, Traini (2003)-(2005); TMDs: BP, Cazzaniga, Boffi (2008); BP, Yuan (2010); Azimuthal Asymmetries: Schweitzer, BP, Boffi, Efremov (2009) GTMDs: Lorce`, BP, Vanderhaeghen (2011) typical accuracy of ¼ 30 % in comparison with exp. data in the valence region, but it violates Lorentz symmetry

9. Longitudinal Transverse  k T b?b? Generalized Transverse Charge Density fixed angle between k ? and b ? and fixed value of |k ? | [Lorce’, BP, PRD84 (2011)] U npol. up Q uark in U npol. P roton

10. Longitudinal Transverse fixed 3Q light-cone model = [Lorce’, BP, PRD84 (2011)] U npol. up Q uark in U npol. P roton

11. Longitudinal Transverse U npol. up Q uark in U npol. P roton fixed 3Q light-cone model = favored unfavored [Lorce’, BP, PRD84 (2011)]

12. up quarkdown quark  left-right symmetry of distributions ! quarks are as likely to rotate clockwise as to rotate anticlockwise  up quarks are more concentrated at the center of the proton than down quark  integrating over b ? transverse-momentum density  integrating over k ? charge density in the transverse plane b ? [Miller (2007); Burkardt (2007)] Monopole Distributions favored unfavored

13. Proton spin u-quark OAM d-quark OAM Unpol. quark in long. pol. proton  projection to GPD and TMD is vanishing ! unique information on OAM from Wigner distributions fixed

14. [Lorce’, BP, PRD84(2011)] [Lorce’, BP, Xiong, Yuan:arXiv:1111.4827] [Hatta:arXiv:111.3547} Definition of the OAM OAM operator : Unambiguous in absence of gauge fields state normalization No infinite normalization constants No wave packets Wigner distributions for unpol. quark in long. pol. proton Quark Orbital Angular Momentum

15. [Lorce’, BP, Xiong, Yuan:arXiv:1111.4827] Proton spin u-quark OAM d-quark OAM Quark Orbital Angular Momentum

16. L z q = ½ - J z q L z q =2L z q =1 L z q =0L z q = -1 JzqJzq Quark OAM: Partial-Wave Decomposition :probability to find the proton in a state with eigenvalue of OAM L z eigenstate of total OAM squared of partial wave amplitudes TOTAL OAM (sum over three quark)

17. Quark OAM: Partial-Wave Decomposition OAML z =0L z =-1L z =+1L z =+2TOT UP0.013-0.0460.1390.0250.131 DOWN-0.013-0.0900.0870.011-0.005 UP+DOWN0-0.1360.2260.0360.126 0.620.1360.2260.0181 updown TOT L z =0 L z =-1 L z =+2 L z =+1 distribution in x of OAM Lorce,B.P., Xiang, Yuan, arXiv:1111.4827

18. Quark OAM from Pretzelosity model-dependent relation “pretzelosity” transv. pol. quarks in transv. pol. nucleon [She, Zhu, Ma, 2009; Avakian, Efremov, Schweitzer, Yuan, 2010] first derived in LC-diquark model and bag model valid in all quark models with spherical symmetry in the rest frame [Lorce’, BP, arXiv:1111.6069] chiral even and charge evenchiral odd and charge odd no operator identity relation at level of matrix elements of operators

19.  No gluons  Independent quarks  Spherical symmetry in the nucleon rest frame the quark distribution does not depend on the direction of polarization Light-Cone Quark Models symmetric momentum wf spin-flavor wf rotation from canonical spin to light-cone spin non-relativistic axial chargenon-relativistic tensor charge spherical symmetry in the rest frame

20. Quark OAM  from GPDs: Ji’s sum rule  from Wigner distributions (Jaffe-Manohar)  from TMD model-dependent relation “pretzelosity” transv. pol. quarks in transv. pol. nucleon

21. GPDs Ji sum rule GTMDs Jaffe-Manohar LCWF overlap representation TMD total LCWFs are eigenstates of total OAM total For total OAM Conservation of transverse momentum:Conservation of longitudinal momentum sum over all parton contributions 0 1

22. what is the origin of the differences for the contributions from the individual quarks? transverse center of momentum Jaffe-Manohar Ji pretzelosity ??? ~ ~ Talk of Cedric Lorce’ OAM depends on the origin But if

23. Summary  GTMDs $ Wigner Distributions - the most complete information on partonic structure of the nucleon  Results for Wigner distributions in the transverse plane - non-trivial correlations between b ? and k ? due to orbital angular momentum  Orbital Angular Momentum from phase-space average with Wigner distributions - they are all equivalent for the total-quark contribution to OAM, but differ for the individual quark contribution - rigorous derivation for quark contribution (no gauge link)  Orbital Angular Momentum from pretzelosity TMD - model-dependent relation valid in all quark model with spherical symmetry in the rest frame  LCWF overlap representations of quark OAM from Wigner distributions, TMD and GPDs