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Wigner Distributions in Light-Cone Quark Models

Wigner Distributions in Light-Cone Quark Models. Barbara Pasquini Pavia U. & INFN, Pavia. i n collaboration with Cédric Lorcé Mainz U. & INFN, Pavia. Outline. Generalized Transverse Momentum Dependent Parton Distributions (GTMDs). FT    b . Wigner Distributions.

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Wigner Distributions in Light-Cone Quark Models

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  1. Wigner Distributions in 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 • General formalism for the 3-quark contribution to GTMDs • Results for Wigner distributions in light-cone quark models • unpolarized quarks in unpolarized nucleon (generalized transverse charge density) • Relations between GPDs and TMDs • Relations among TMDs in various 3-quark models • Orbital Angular Momentum in terms of LCWF: Ji’s vs. Jaffe-Manohar’s definition

  3. Generalized TMDs GTMDs • Complete parametrization : 16 GTMDs [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 ¢: nucleon momentum transfer k?: average quark transverse momentum

  4. GTMDs Wigner-Ds FT b FT TMDs GPDs spin densities FT PDs Form Factors charge densities

  5. Wigner Distributions • Quantum phase-space distribution: most complete information on wave function quantum molecular dynamics, signal analysis, quantum info, optics, image processing,… [Wigner, (1932)] • Wigner distributions in QCD: at »=0 ! diagonal in the Fock-space N N N=3 ! overlap of quark light-cone wave-functions • real functions, but in general not-positive definite 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

  6. 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 Common assumptions : • No gluons • Independent quarks

  7. 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)

  8. ( ) Light-cone gauge A+=0 Wilson line r reduces to the identity Twist-2 operators º! quark polarization ¹! nucleon polarization 16 GTMDs

  9. 3-Quark Overlap representation Quark line : Model Independent Spin Structure Active quark : Spectator quarks :

  10. 3-Quark Overlap representation 16 GTMDs Assumption : • symmetry [C.Lorce’, B. Pasquini, M. Vanderhaeghen (in preparation)]

  11. 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)

  12. k T Wigner function for unpolarized quark in unpolarized nucleon q b k , µ fixed T µ = ¼/2 µ = 0 [C.Lorce’, B. P. (in preparation)]

  13. 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 µ = ¼/2 0.4 GeV T µ = 0

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

  15. Integrating over b ? TMD • Integrating over k ? charge density in the transverse plane b? unpolarized u and d quarks in unpolarized proton neutron proton charge distribution in the transverse plane [Miller (2007); Burkardt (2007)]

  16. GTMDs TMDs GPDs GPDs and TMDs probe the same overlap of quark LCWFs in different kinematics nucleon quark at »=0 UU UT LL TU TT TT 0 TL 0 LT

  17. Relations between GPDs and TMDs in Quark Models GPDs and TMDs probe the same overlap of LCWFs in different kinematics there exist relations in particular kinematical limits? Trivial Relations UU LL TT Non-Trivial Relations TT valid also in spectator model : Meissner, Metz, Goeke, PRD76(2007)(with a factor 3 instead of 2) Model dependent relations [Burkardt, Hwang, 2003; Burkardt, 2005] SSA= GPD­FSI valid for both Sivers and Boer-Mulders functions in spectator model, but breaks down at higher-orders

  18. 0 TL 0 LT GPDs at »=0 vanish because of time-reversal invariance sx up down Light-cone quark model: ! consistent with lattice calculations BP, Cazzaniga, Boffi, PRD78 (2008); Lorce`, BP, in preparation Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009)

  19. Relations among TMDs in Quark Models Linear relations Quadratic relation * * Flavor-dependent * * * * * * * Flavor-independent * * * Bag ÂQSM LCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe & Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)] [Lorcé & 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)] Common assumptions : • No gluons • Independent quarks

  20. Light-cone Helicity and Canonical Spin LC helicity Canonical spin Quark polarization Quark polarization Nucleon polarization Nucleon polarization • Rotations in light-front dynamics depend on the interaction, while are kinematical in canonical quantization we study the rotational symmetries for TMDs in the basis of canonical spin rotation around an axis orthogonal to z and k? of an angle µ=µ(k)

  21. Rotational Symmetries in Canonical-Spin Basis nucleon spin quark spin • Cilindrical symmetry around z direction • Cilindrical symmetry around Tyk k? • Spherical symmetry: invariance for any spin rotation • Spherical symmetry and SU(6) spin-flavor symmetry [C. Lorce’, B.P., in preparation]

  22. Orbital Angular Momentum not unique decomposition gauge invariant, but contains interactions through the gauge covariant derivative not gauge invariant, but diagonal in the LCWFs basis [ R.L. Jaffe, NPB 337, (1990) ] [ X. Ji, PRL 78, (1997) ] Ji’s sum rule quark orbital angular momentum: What is the difference between the two definitions in a quark model without gauge fields? scalar diquark model: M. Burkardt, PRD79, 071501 (2009); LCCQM: BP, F. Yuan, in preparation

  23. Three Quark Light Cone Amplitudes Lzq = 2 6 independent wave function amplitudes: Lzq =1 Lzq =2 Lzq = -1 Lzq =0 Lzq = 1 • classification of LCWFs in angular momentum components [Ji, J.P. Ma, Yuan, 03; Burkardt, Ji, Yuan, 02] Jz = Jzq + Lzq total quark helicity Jq parity time reversal Lz q=0 isospin symmetry Lzq=-1

  24. Quark Orbital Angular Momentum Jaffe-Manohar and Ji OAM should coincide when A=0 ! no-gluons, only quark contribution • Jaffe-Manohar definition: overlap of LCWFs with ¢Lz=0 • Ji’s definition: ¢Lz= 1 ¢Lz=0 ¢Lz=0 interference between LCWFs with different Lz it is not trivial to have the same orbital angular momentum for the quark contribution

  25. Distribution in x of Orbital Angular Momentum Definition of Jaffe and Manohar: contribution from different partial waves TOT up down Lz=0 Lz=-1 Lz=-1 Lz=+2 total result, sum of up and down contributions: Jaffe-Manohar’s vs. Ji’s definition Jaffe-Manohar Ji even in a model without gauge fields the two definitions give different distributions in x

  26. Orbital Angular Momentum • Definition of Jaffe and Manohar: contribution from different partial waves = 0 ¢ 0.62 + (-1) ¢ 0.14 + (+1) ¢ 0.23 + (+2) ¢ 0.018 = 0.126 • Definition of Ji: [BP, F. Yuan, in preparation][scalar diquark model: M. Burkardt, PRD79, 071501 (2009)]

  27. 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? even for unpolarized quarks in unpolarized nucleon • GPDs and TMDs probe the same overlap of 3-quark LCWF in different kinematics - give complementary information useful to reconstruct the nucleon wf • Relations of TMDs in a large class of models due to rotational symmetries in the quark-spin space - useful to test them with experimental observables in the valence region • Orbital Angular Momentum in terms of LCWFs: - in quark models, the total OAM (but not the distributions in x) is the same from JI and Jaffe-Manohar definitions

  28. Backup

  29. 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

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

  31. 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

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

  33. Orbital Angular Momentum in the Light-Front • Light-cone Gauge A+=0 and advanced boundary condition for A • generalization of the relation for the anomalous magnetic moment: [Brodsky, Drell, PRD22, 1980] complex LCWFs due to FSI/ISI [Brodsky, Gardner, PLB 643, 2006] [Brodsky, BP, Yuan, Xiao, PLB 667, 2010

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