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Generalized Transverse-Momentum Distributions

Generalized Transverse-Momentum Distributions. Cédric Lorcé Mainz University Germany. In collaboration with:. Barbara Pasquini Pavia University Italy. GTMDs. Wigner distributions. Spin densities. GPDs. TMDs. Transverse charge densities. PDFs. FFs. Outline.

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Generalized Transverse-Momentum Distributions

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  1. Generalized Transverse-Momentum Distributions Cédric Lorcé Mainz University Germany In collaboration with: Barbara Pasquini Pavia University Italy

  2. GTMDs Wigner distributions Spin densities GPDs TMDs Transverse charge densities PDFs FFs Outline Physical interpretation Parton distributions Generalized Transverse-Momentum Distributions Transverse-Momentum Distributions Generalized Parton Distributions Parton Distribution Functions Form Factors

  3. Quark-quark correlator 3(+2?)DPicture of the Nucleon Information on quark distribution Quark models Experiments Wave function (often just N=3) ES, DIS, SIDIS, DVCS, …

  4. Charges Parton Distributions Wave function (often just N=3) Quark models Vector Net # of quarks Information on quark distribution Axial Net quark longitudinal polarization Tensor Net quark transverse polarization Experiments ES, DIS, SIDIS, DVCS, …

  5. PDFs Charges Parton Distributions Wave function (often just N=3) Quark models Information on quark distribution Experiments ES, DIS, SIDIS, DVCS, …

  6. PDFs FFs Charges Parton Distributions No clear interpretation ! • # of quarks changes • Momentum transfer Wave function (often just N=3) Quark models Information on quark distribution Experiments ES, DIS, SIDIS, DVCS, …

  7. Drell-Yan-West frame PDFs FFs Charges Parton Distributions No clear interpretation in momentum space ! • Momentum transfer Wave function (often just N=3) Quark models Information on quark distribution Experiments ES, DIS, SIDIS, DVCS, …

  8. Drell-Yan-West frame Proton Neutron PDFs FFs Charges Transverse charge densities Parton Distributions Probabilistic interpretation in position space Wave function (often just N=3) Quark models 2D Fourier transform Information on quark distribution NB: Experiments ES, DIS, SIDIS, DVCS, … [Miller (07)] [Carlson, Vdh (08)]

  9. Spin densities PDFs FFs GPDs Charges Parton Distributions FFs PDFs GPDs Hadron 3D picture ! [Belitsky & al. (04)] [Burkardt (01,03)] Wave function (often just N=3) Quark models 2D Fourier transform Information on quark distribution Transverse charge densities Experiments ES, DIS, SIDIS, DVCS, …

  10. Spin densities PDFs FFs GPDs TMDs Charges Parton Distributions Mean momentum Displacement Complementary hadron 3D picture ! Momentum space Position space Wave function (often just N=3) Quark models Momentum transfer Position 2D Fourier transform No direct connection Information on quark distribution Transverse charge densities Experiments ES, DIS, SIDIS, DVCS, …

  11. Wigner distribution Spin densities GTMDs PDFs Transverse FFs GPDs TMDs Longitudinal Charges Parton Distributions 2D Fourier transform [Meißner & al. (2009)] Wave function (often just N=3) Quark models Information on quark distribution Transverse charge densities Experiments ES, DIS, SIDIS, DVCS, …

  12. TMFFs Spin densities GTMDs PDFs TMSDs FFs GPDs TMDs Charges Complete Picture 2D Fourier transform Wigner distribution Transverse Wigner distribution Wave function (often just N=3) Quark models Information on quark distribution Transverse charge densities Experiments ES, DIS, SIDIS, DVCS, … [C.L., Pasquini (submitted, 2011)]

  13. Transverse Longitudinal GTMDs GPDs TMDs Wigner Distributions [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) Heisenberg’s uncertainty relations Quasi-probabilistic Third 3D picture ! No restrictions from Heisenberg’s uncertainty relations

  14. Transverse Longitudinal Example:Unpol. upQuark in Unpol. Proton (1 out of 16) [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) fixed 3Q light-cone model

  15. Transverse Longitudinal Example:Unpol. upQuark in Unpol. Proton (1 out of 16) [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) fixed unfavored favored 3Q light-cone model Orbital angular momentum?

  16. Transverse Longitudinal Example:Unpol. upQuark in Unpol. Proton (1 out of 16) [Wigner (1932)] [Belitsky, Ji, Yuan (04)] [C.L., Pasquini (in preparation)] QM QFT (Breit frame) QFT (light cone) 0.1 GeV² 0.2 GeV² 0.3 GeV² 3Q light-cone model 0.4 GeV²

  17. Transverse Longitudinal Words of caution • No known processes to extract GTMDs • Wigner distributions are quasi-probabilistic • Issues concerning universality of TMDs • Fragmentation functions not so well known • Extrapolations needed for Fourier transform • Scale-dependence • Twist-two picture • gauge • Problems with transverse gauge link

  18. Summary • Quark-quark correlator • Most complete information on hadron structure • GTMDs are ‘’mother’’ distributions • 2D Fourier transform on the light cone • Correct interpretation (number of partons is fixed) • GTMDs are connected to Wigner distributions • Example of Wigner distribution • Unpolarized quark in unpolarized proton • 3Q light-cone model • Distortions connected to OAM

  19. Backup

  20. Wigner Distributions cQSM ru/2 rd Unpolarized u and d quarks in unpolarized proton LCQM n More u than d in central region! [Miller (2007)]

  21. 2 DIS ~ Im 2 SIDIS ~ Im Hard exclusive meson leptoproduction DVCS Handbag approximation

  22. Bint S Anomalous magnetic moment Orbital angular momentum u d Magnetic moment d Induced electric dipole moment X Some examples:Transverse Charge Densities Neutron [Burkardt (2003)] Helicity flip

  23. TMDs GPDs Angular momentum Ji Ji Jaffe-Manohar • Each term is gauge-invariant • No decomposition of • Decomposition is gauge-dependent • OAM in LCWFs refers to (easy) Ji’s sum rule Pretzelosity [Avakian & al. (2010)] Model-dependent! Trans. pol. quark in trans. pol. proton

  24. Model relations for TMDs (twist-two) *=SU(6) Linear relations Quadratic relation * * Flavor-dependent * * * * * Flavor-independent * * * * * Bag cQSM LCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)] [C.L., 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)]

  25. LC helicity and canonical spin LC helicity Canonical spin Quark polarization Quark polarization Nucleon polarization Nucleon polarization

  26. 2 2 2 = = + = 0 = = = - = - Spherical symmetry [C.L., Pasquini (in preparation)] Axial symmetry about Axial symmetry about

  27. TMDs cQSM LCQM [C.L., Pasquini, Vdh (in preparation)]

  28. GPDs (vector & axial) cQSM LCQM ~ H H ~ E E [C.L., Pasquini, Vdh (in preparation)]

  29. GPDs (tensor) cQSM LCQM ~ HT HT ~ ET ET [C.L., Pasquini, Vdh (in preparation)]

  30. Summary • Quark-quark correlator • Most complete information on hadron structure • GTMDs are ‘’mother’’distribution • 2D Fourier transform on the light cone • Correct interpretation of FFs • GTMDs can be related to Wigner distributions • Distortions due to orbital angular momentum • TMDs • Model relations due to spherical symmetry • LC helicity and canonical spin connected by a rotation • 3Q amplitude • Same structurein many models

  31. Light front- and instant-form WFs Assumption : • in instant form (automatic w/ spherical symmetry) More convenient to work in canonical spin basis

  32. k k T T Wigner Distributions q b , q fixed Unpolarized u quark in unpolarized proton cQSM

  33. k k k k k T T T T T Wigner Distributions Unpolarized u quark in unpolarized proton , q fixed cQSM

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