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Hadron tomography through Wigner distributions

This paper delves into the concept of hadron tomography via Wigner distributions, focusing on the intricacies of parton distributions in phase space. We explore the light-cone wave function approach, model predictions, and the significance of orbital angular momentum in quark dynamics. A comparison of different definitions and the implications of quark polarizations is presented. Our findings shed light on various topics, including transverse and longitudinal parton distribution functions, generalized parton distributions, and their applications in nuclear and quantum physics.

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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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