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Generalized TMDs of the Proton

Generalized TMDs of the Proton. 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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Generalized TMDs of the Proton

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  1. Generalized TMDsof the Proton 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 (generalized transverse charge density)

  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, arXiv:1102.4704 ]

  5. Transverse Longitudinal GTMDs GPDs TMDs Wigner Distributions [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [Lorce’, BP: arXiv:1012.0169] QM QFT (Breit frame) QFT (light cone) Heisenberg’s uncertainty relations Quasi-probabilistic Third 3D picture ! No restrictions from Heisenberg’s uncertainty relations

  6. Wigner Distributions • 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

  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, arXiv:1102.4704 ]

  8. Transverse Longitudinal k T Example:Unpol. upQuark in Unpol. Proton (1 out of 16) Generalized Transverse Charge Density fixed angle between k? and b? and fixed value of |k?| q b?

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

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

  11. 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, in preparation]

  12. 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

  13. 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)]

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

  15. 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)

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

  17. Backup

  18. 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

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

  20. 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

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

  22. 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

  23. 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)

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