Wigner Distributions in Light-Cone Quark Models

1. Wigner Distributionsin Light-Cone Quark Models Barbara Pasquini Pavia U. & INFN, Pavia in collaboration with Cédric Lorcé Mainz U. & INFN, Pavia

2. Outline Generalized Transverse Momentum Dependent Parton Distributions (GTMDs) FT b Wigner Distributions • Results in light-cone quark models • unpolarized quarks in unpolarized nucleon • Quark Orbital Angular Momentum

3. Generalized TMDs GTMDs • Complete parametrization : 16 GTMDs at twist-2 [Meißner, Metz, Schlegel (2009)] • Fourier Transform : 16 Wigner distributions [Belitsky, Ji, Yuan (2004)] »: fraction of longitudinal momentum transfer x: average fraction of quark longitudinal momentum k?: average quark transverse momentum ¢: nucleon momentum transfer

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

5. Transverse Longitudinal GTMDs GPDs TMDs Wigner Distributions [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [Lorce’, BP: PRD84 (11)] QM QFT (Breit frame) QFT (light cone) Heisenberg’s uncertainty relations Quasi-probabilistic Third 3D picture with probabilistic interpretation! No restrictions from Heisenberg’s uncertainty relations

6. Wigner Distributions • Wigner distributions in QCD: at »=0 ! diagonal in the Fock-space N N • real functions, but in general not-positive definite N=3 ! overlap of quark light-cone wave-functions not probabilistic interpretation correlations of quark momentum and position in the transverse planeas function of quark and nucleon polarizations • no known experiments can directly measure them ! needs phenomenological models

7. Light-Cone Quark Models LCWF: invariant under boost, independent of P internal variables: [Brodsky, Pauli, Pinsky, ’98] momentum wf spin-flavor wf rotation from canonical spin to light-cone spin Bag Model, ÂQSM, LCQM, Quark-Diquark and Covariant Parton Models Bag Model, ÂQSM, LCQM, Quark-Diquark and Covariant Parton Models Common assumptions : • No gluons • Independent quarks [ Lorce, BP, Vanderhaeghen, JHEP 05 (2011)Lorce, BP, arXiv:1104.5651 ]

8. Light-Cone Constituent Quark Model • momentum-space wf [Schlumpf, Ph.D. Thesis, hep-ph/9211255] parameters fitted to anomalous magnetic moments of the nucleon : normalization constant • spin-structure: free quarks (Melosh rotation) • SU(6) symmetry 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)

9. Transverse Longitudinal k T Example:Unpol. upQuark in Unpol. Proton (1 out of 16) [Lorce’, BP, PRD84 (2011)] Generalized Transverse Charge Density fixed angle between k? and b? and fixed value of |k?| q b?

10. Transverse Longitudinal Example:Unpol. upQuark in Unpol. Proton (1 out of 16) fixed = 3Q light-cone model [Lorce’, BP, PRD84 (2011)]

11. Transverse Longitudinal Example:Unpol. upQuark in Unpol. Proton (1 out of 16) fixed unfavored = favored 3Q light-cone model [Lorce’, BP, PRD84 (2011)]

12. Transverse Longitudinal Example:Unpol. upQuark in Unpol. Proton (1 out of 16) 0.1 GeV² 0.2 GeV² 0.3 GeV² 3Q light-cone model 0.4 GeV² [Lorce’, BP, PRD84 (2011)]

13. up quark down 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 Monopole Distributions • integrating over k ? charge density in the transverse plane b? [Miller (2007); Burkardt (2007)]

14. Transverse Charge Distribution unpolarized u and d quarks in unpolarized proton neutron proton charge distribution in the transverse plane [Miller (2007); Burkardt (2007)]

15. Quark Orbital Angular Momentum Wigner distribution for Unpolarized quark in a Longitudinally pol. nucleon Orbital Motion in the Transverse Space up quark: down quark: -0.6 0 0.6 -0.6 0 0.6 --0.6 0 0.6 -0.6 0 0.6 Lorce,B.P., Xiang, Yuan, in preparation

16. Quark OAM: Partial-Wave Decomposition eigenstate of OAM :probability to find the proton in a state with eigenvalue of OAM Lz Lzq = ½ - Jzq Lzq =1 Lzq =2 Lzq = -1 Lzq =0 Jzq squared of LCWFs

17. Quark OAM: Partial-Wave Decomposition distribution in x of OAM TOT up down Lz=0 Lz=-1 Lz=+1 Lz=+2 Lorce,B.P., Xiang, Yuan, in preparation

18. Quark Orbital Angular Momentum • from Wigner distributions: intrinsic OAM with respect to centre of momentum • from TMD: OAM with respect to the origin of axis in the transverse plane model-dependent relation • bag model [Avakian, Efremov, Schweitzer, Yuan, PRD81 (2010)] • light-cone diquark model [She, Zhu, Ma, PRD79 (2009)] • all quark models with spherical symmetry in the rest frame [Lorce’, BP, Xiang, Yuan, in preparation] • from GPDs: Ji’s sum rule [Ji, PRL78 (1997)] What is the relations among these three definitions?

19. All three definitions give the same results for the total-quark contribution to OAM but not for the individual flavor contribution the three definitions refer to OAM calculated with respect to different points n-th parton contribution: Total-quark contribution:

20. Summary • GTMDs $ Wigner Distributions - the most complete information on partonic structure of the nucleon • General Formalism for 3-quark contribution to GTMDs - applicable for large class of models: LCQMs, ÂQSM, Bag model • Results for Wigner distributions in the transverse plane - anisotropic distribution in k? for unpolarized quarks in unpolarized nucleon - non-trivial correlations between b? and k? due to orbital angular momentum • Orbital Angular Momentum from phase-space average with Wigner distributions - comparison with different definition from TMDs and GPDs ! they are all equivalent for the total-quark contribution to OAM

21. Integrating over b ? • Integrating over k ? charge density in the transverse plane b? neutron proton charge distribution in the transverse plane [Miller (2007); Burkardt (2007)]

22. LC helicity amplitudes nucleon ( ¤’ ¤ ) quark ( ¸’ ¸ ) Independentquarks LC helicity $canonical spin rotation around an axis orthogonal to z and k LC helicity canonical spin

23. Light-Cone Helicity and Canonical Spin rotation around an axis orthogonal to z and k LC helicity canonical spin Light-Cone CQM Chiral Quark-Soliton Model Bag Model (Melosh rotation)

24. º! quark polarization ¹! nucleon polarization ( ) 16 GTMDs Active quark : Spectator quarks : Model Independent Spin Structure

25. Backup

26. k k k T T T Unpolarized u quark in unpolarized proton µ 0 ¼/2 ¼ 3/2 ¼ k T 0.1 GeV 0.2 GeV 0.3 GeV , q fixed k 0.4 GeV T

27. k k T T Generalized Transverse Charge Densities = + µ = 0 fixed µ = ¼/2 Unpolarized u quark in unpolarized proton

28. k T Wigner function for transversely pol. quark in longitudinally pol. nucleon T-odd sx b q µ = ¼/2 µ = 0 k ,µ fixed T

29. k T Wigner function for transversely pol. quark in longitudinally pol. nucleon fixed sx Dipole Monopole Monopole + Dipole µ = 0 µ = ¼/2

30. k k k T T T u quark pol. in x direction in longitudinally pol.proton µ 0 ¼/2 ¼ 3/2 ¼ k T 0.1 GeV 0.2 GeV 0.3 GeV , q fixed k 0.4 GeV T

31. Integrating over k ? density in the transverse plane k?of transversely pol. u and d quark in longitudinally pol nucleon • Integrating over b ? sx up down BP, Cazzaniga, Boffi, PRD78 (2008)Lorce`, BP, in preparation Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009)