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Wigner Distributions and light-front quark models

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

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Wigner Distributions and light-front quark models

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

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

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

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

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

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

  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)GTMDs:Lorce`, BP, Vanderhaeghen (2011) typical accuracy of ¼ 30 % in comparison with exp. datain the valence region, but it violates Lorentz symmetry

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

  10. Transverse Longitudinal Unpol. up Quark in Unpol.Proton fixed = 3Q light-cone model [Lorce’, BP, PRD84 (2011)]

  11. Transverse Longitudinal Unpol. up Quark in Unpol.Proton fixed unfavored = favored 3Q light-cone model [Lorce’, BP, PRD84 (2011)]

  12. 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 unfavored • integrating over b ? transverse-momentum density favored Monopole Distributions • integrating over k ? charge density in the transverse plane b? [Miller (2007); Burkardt (2007)]

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

  14. Quark Orbital Angular Momentum [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 distributionsfor unpol. quark in long. pol. proton

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

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

  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, arXiv:1111.4827

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

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

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

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

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

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

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