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Hadron tomography through Wigner distributions

Hadron tomography through Wigner distributions. Cédric Lorcé. Old. Germany. [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (in preparation)]. New. France. Outline. Wigner distributions Parton distributions in phase space Model predictions

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Hadron tomography through Wigner distributions

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  1. Hadron tomography throughWigner distributions Cédric Lorcé Old Germany [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (in preparation)] New France

  2. Outline • Wigner distributions • Parton distributions in phase space • Model predictions • Light-cone wave function approach • Orbital angular momentum • Comparison of different definitions

  3. Outline • Wigner distributions • Parton distributions in phase space • Model predictions • Light-cone wave function approach • Orbital angular momentum • Comparison of different definitions

  4. 2D Fourier transform Parton distributions cf. Bacchetta’s talk Transverse Longitudinal PDFs TMDs DIS e.g. SIDIS Alessandro’s « elephant » FFs GPDs GTMDs elastic scattering e.g. DVCS ??? TDs cf. Hoyer’s talk IPDs Wigner distributions [Miller (2007)] [Carlson, Vanderhaeghen (2008)] [Alexandrou et al. (2009,2010)] [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] [Burkardt (2000,2003)]

  5. Wigner distributions Definition of the GTMD correlator cf. Schlegel’s talk [Meißner, Metz, Schlegel (2009)] Quark polarization Gauge link Nucleon polarization 4 X 4 = 16 polarization configurations 16 complex GTMDs (at leading twist) Definition of the Wigner distributions Non-relativistic : Relativistic : [Ji (2003)] [Belitsky, Ji, Yuan (2004)] [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] 16 real Wigner distributions (at leading twist)

  6. Wigner distributions [Wigner (1932)] [Moyal (1949)] Wigner distributions have applications in: • Nuclear physics • Quantum chemistry • Quantum molecular dynamics • Quantum information • Quantum optics • Classical optics • Signal analysis • Image processing • Heavy ion collisions • … E.g. [Antonov et al. (1980-1989)] They can even be « measured » in some cases E.g. [Lvovsky et al. (2001)] Heisenberg’s uncertainty relations Quasi-probability distribution

  7. Outline • Wigner distributions • Parton distributions in phase space • Model predictions • Light-cone wave function approach • Orbital angular momentum • Comparison of different definitions

  8. … … Quark-quark correlator Overlap representation [C.L., Pasquini, Vanderhaeghen (2011)] Drell-Yan-West frame ~ • Light-cone constituent quark model Melosh rotation Symmetric momentum WF SU(6) WF • Chiral quark-soliton model Valence WF Canonical spin LC helicity

  9. Unpol. quark in unpol. proton [C.L., Pasquini (2011)] disfavored favored Left-right symmetry no net quark OAM

  10. Unpol. quark in long. pol. proton [C.L., Pasquini (2011)] Proton spin u-quark OAM d-quark OAM

  11. Unpol. quark in long. pol. proton [C.L., Pasquini, Xiong, Yuan (in preparation)] Proton spin u-quark OAM d-quark OAM

  12. Long. pol. quark in unpol. proton [C.L., Pasquini (2011)] Quark spin u-quark OAM d-quark OAM

  13. Long. pol. quark in long. pol. proton [C.L., Pasquini (2011)] Proton spin u-quark spin d-quark spin

  14. Long. pol. quark in long. pol. proton [C.L., Pasquini (2011)]

  15. Outline • Wigner distributions • Parton distributions in phase space • Model predictions • Light-cone wave function approach • Orbital angular momentum • Comparison of different definitions

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

  17. Orbital angular momentum [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (in preparation)] Definition of the OAM OAM operator : Unambiguous in absence of gauge fields State normalization No infinite normalization constants No wave packets Unpol. quark in long. pol. proton Wigner distributions

  18. Naive Jaffe-Manohar Ji Orbital angular momentum [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (in preparation)] Overlap representation Spectators are involved! NB : LCWFs are eigenstates of total OAM For totalOAM we find

  19. Naive Jaffe-Manohar Ji Comparison using quark models Scalar quark-diquark model : [Burkardt, Hikmat (2009)] (after regularization) [Avakian & al. (2010)] MIT Bag model : (no spectators) [C.L., Pasquini (2011)] Wigner distributions GPDs TMDs Spherically symmetric quark models : [C.L., Pasquini, Xiong, Yuan (in preparation)]

  20. Naive Jaffe-Manohar Ji Possible origin of the differences I [C.L., Pasquini, Xiong, Yuan (in preparation)] OAM depends on the origin But if ~ ~ Transverse center of momentum ???

  21. Possible origin of the differences II Jaffe-Manohar Ji Canonical energy-momentum tensor Belifante energy-momentum tensor OAM density operator Total AM density operator Total AM density operator ? [Bakker, Leader, Trueman (2004)]

  22. Summary [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (in preparation)] • Wigner distributions • Parton phase-space distributions • Model calculations • 3Q LCWFs (cQSM, LCCQM) • Correlations between quark polarization/motion and nucleon polarization • Comparison of different definitions of OAM • Overlap representation in non-gauge theories • Total OAM is the same but not individual contributions • Origin dependence, EOM?

  23. Backup

  24. TMFFs GTMDs PDFs Transverse TMSDs FFs TMDs GPDs Longitudinal Charges Completepicture @ Momentum space Position space Transverse density in momentum space Transverse density in position space

  25. Model relations Linear relations Quadratic relation * * Flavor-dependent * Flavor-independent * * * * * * * * * Bag cQSM LCCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe & Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)] [C.L., Pasquini & Vanderhaeghen (2011)] [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)] *=SU(6)

  26. 2 2 2 = = + = 0 = = - Spherical symmetry [C.L., Pasquini (2011)] Axial symmetry about Axial symmetry about

  27. LC helicity and canonical spin Canonical boost Light-cone boost

  28. Spherical symmetry [C.L., Pasquini (2011)] Bag Model, cQSM, LCCQM, Quark-Diquark Model (Ma) and Covariant Parton Model Common assumption : Explicit or implicit rotational symmetry The probability does not depend on the direction of canonical polarization

  29. Formalism Independent quarks LC helicity Canonical spin Light-Cone Quark Model (Melosh rotation) Chiral Quark-Soliton Model S-wave P-wave Bag Model S-wave P-wave

  30. Formalism Assumption : • in instant form (automatic w/ spherical symmetry) More convenient to work in canonical spin basis

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