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Accessing the quark orbital angular momentum with Wigner distributions

This presentation discusses the use of Wigner distributions to access the orbital angular momentum of quarks and its relevance in understanding the proton spin puzzle. It explores the issues related to relativity and provides model results and comparisons between different approaches.

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Accessing the quark orbital angular momentum with Wigner distributions

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  1. Accessing the quark orbital angular momentum with Wigner distributions Cédric Lorcé and September 12, 2012, Puerto del Carmen, Lanzarote (Canary Islands), Spain

  2. Outline • Phase-space distributions • Wigner distributions • Issues due to relativity • Light-front perspective • Proton spin puzzle • Decompositions • Relations with observables • Model results • 3Q light-front models • Longitudinal angular momentum structure • Comparison between different OAM

  3. Outline • Phase-space distributions • Wigner distributions • Issues due to relativity • Light-front perspective • Proton spin puzzle • Decompositions • Relations with observables • Model results • 3Q light-front models • Longitudinal angularmomentum structure • Comparisonbetweendifferent OAM

  4. Phase-space distribution Quantum Mechanics [Wigner (1932)] [Moyal (1949)] Wigner distribution Position-space density Momentum-space density Quantum average

  5. Phase-space distribution Wigner distribution Numerous applications in • Nuclear physics • Quantum chemistry • Quantum molecular dynamics • Quantum information • Quantum optics • Classical optics • Signal analysis • Image processing • Heavy ion collisions • … [Antonov et al. (1980-1989)] Heisenberg’s uncertainty relation Quasi-probabilistic interpretation

  6. Phase-space distribution Difficulties in Quantum Field Theory Lorentz contraction No relativistic concept of center of mass No separation of intrinsic and extrinsic coordinates Pair creation/annihilation

  7. Change of perspective Instant and front forms of dynamics Space-time foliations Instant form Front form Kinematics maps a sheet onto itself Dynamics connects points on different sheets [Dirac (1949)]

  8. Change of perspective Advantages and inconvenients Extreme Lorentz contraction « pancake » Longitudinal momentum Transverse momentum ~ NR mass for transverse motion Transverse plane has Galilean symmetry Impact parameter Looks non-relativistic ! Transverse center of momentum No pair creation/annihilation in frame with (Quasi) probabilistic interpretation ! [Soper (1977)] [Burkardt (2000,2003)]

  9. Change of perspective Wigner distribution in Quantum Field Theory Wigner operator Wilson line Dirac matrix Wigner distribution in Breit frame Non-relativistic ! [Belitsky, Ji, Yuan (2004)] Wigner distribution in Drell-Yan frame [C.L., Pasquini (2011)] Generalized Transverse Momentum-dependent parton Distributions [Meißner, Metz, Schlegel (2009)]

  10. Outline • Phase-space distributions • Wigner distributions • Issues due to relativity • Light-front perspective • Proton spin puzzle • Decompositions • Relations with observables • Model results • 3Q light-front models • Longitudinal angularmomentum structure • Comparisonbetweendifferent OAM

  11. Proton spin puzzle [C.L. (2012)] Ji Jaffe-Manohar [Ji (1997)] [Jaffe, Manohar (1990)] Kinetic Canonical Pros: Pros: • Gauge-invariant decomposition • Accessible in DIS and DVCS • Satisfies canonical relations • Complete decomposition Cons: Cons: • Does not satisfy canonical relations • Incomplete decomposition • Gauge-variant decomposition • Missing observables for the OAM News: News: • Complete decomposition • Gauge-invariant extension [Wakamatsu (2009,2010)] [Chen et al. (2008)] • OAM accessible via Wigner distributions [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan(2011)] [Hatta (2011)]

  12. Proton spin puzzle Quark orbital angular momentum [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan(2011)] Kinetic Canonical ISI FSI e.g. DY e.g. SIDIS [Ji, Xiong, Yuan (2012)] [Burkardt (2012)]

  13. GTMDs TMDs PDFs FFs GPDs Parton distribution zoo Projections DVCS ES DIS SIDIS

  14. Outline • Phase-space distributions • Wigner distributions • Issues due to relativity • Light-front perspective • Proton spin puzzle • Decompositions • Relations with observables • Model results • 3Q light-front models • Longitudinal angularmomentum structure • Comparisonbetweendifferent OAM

  15. Model calculations Twist-2 ~ LO in U L T 3Q light-front wave functions ~ Light-Cone Constituent Quark Model Chiral Quark-Soliton Model Symmetric momentum WF SU(6) WF Melosh rotation Valence WF Canonical spin Light-front helicity

  16. Model calculations Twist-2 ~ LO in U L T Unpold quark in unpold nucleon [C.L., Pasquini (2011)] favored disfavored Left-right symmetry No net quark OAM

  17. Model calculations Twist-2 ~ LO in U L T Unpold quark in longitudinally pold nucleon [C.L., Pasquini (2011)] Proton spin u-quark OAM d-quark OAM

  18. Model calculations Twist-2 ~ LO in U L T Unpold quark in longitudinally pold nucleon [C.L. et al. (2012)] Proton spin u-quark OAM d-quark OAM

  19. Model calculations Twist-2 ~ LO in U L T Longitudinally pold quark in unpold nucleon [C.L., Pasquini (2011)] Quark spin u-quark OAM d-quark OAM

  20. Model calculations Twist-2 ~ LO in U L T Longitudinally pold quark in longitudinally pold nucleon [C.L., Pasquini (2011)] Proton spin u-quark spin d-quark spin

  21. Emerging picture Longitudinal Transverse [Burkardt (2005)] [Barone et al. (2008)] [C.L., Pasquini (2011)]

  22. GTMDs TMDs GPDs Comparison of different OAM Fock-state contributions [C.L., Pasquini (2011)] [C.L. et al. (2012)] Kinetic OAM Naive canonical OAM Canonical OAM

  23. GTMDs TMDs GPDs Comparison of different OAM Light-front 3Q models [C.L., Pasquini (2011)] Models are notQCD Truncation of Fock space can spoil Lorentz covariance [Carbonell, Desplanques, Karmanov, Mathiot (1998)] In model calculations, one should expect but

  24. Summary • Phase-space distributions • Wigner distributions • Issues due to relativity • Light-front perspective • Proton spin puzzle • Decompositions • Relations with observables • Model results • 3Q light-front models • Longitudinal angular momentum structure • Comparison between different OAM

  25. Phase-space distribution Wigner distribution They can even be « measured » in some cases [Hosseini et al. (2011)] a,b, Photon-number distribution for input and output pulses, respectively. The blue solid lines show the fitted Poissonian distribution. The green dotted line represents the no-cloning limit and the red dashed line shows the boundary for the quantum limit. The error bars are calculated by dividing the data into 100 equal subsets and finding the error in the mean. c,d, Reconstructed Wigner functions of input and output states for N =3.4. x and p represent the amplitude and phase of the coherent state, respectively.

  26. Charges Parton distribution zoo Charges Partonic interpretation (twist-2) Vector Axial Tensor 0D Picture

  27. PDFs Charges Parton distribution zoo Parton Distribution Functions DIS Partonic interpretation (twist-2) 1D Picture

  28. PDFs FFs Charges Parton distribution zoo Form Factors ES Partonic interpretation (twist-2) Transverse center of momentum Impact parameter 2D Picture

  29. PDFs FFs GPDs Charges Parton distribution zoo Generalized PDFs DVCS Partonic interpretation (twist-2) [Soper (1977)] [Burkardt (2000,2003)] [Diehl, Hägler (2005)] 3D Picture

  30. TMDs PDFs FFs GPDs Charges Parton distribution zoo Transverse-Momentum dependent PDFs SIDIS No direct connection Partonic interpretation (twist-2) Mean momentum Displacement Momentum space Position space Momentum transfer Mean position 3D Picture gauge

  31. GTMDs TMDs PDFs FFs GPDs Charges Parton distribution zoo Generalized TMDs ??? ??? Partonic interpretation (twist-2) Quasi-probabilistic interpretation [Wigner (1932)] [Belitsky , Ji, Yuan(2004)] [C.L., Pasquini (2011)] 5D Picture

  32. GTMDs TMDs TMFFs TMCs PDFs TMFFs TMFFs TMCs TMCs FFs GPDs Charges Parton distribution zoo Complete set [C.L., Pasquini, Vanderhaeghen (2011)] Partonic interpretation (twist-2) 4D Picture 2D Picture

  33. Twist-2 structure Monopole Dipole GPDs TMDs Quadrupole Quark polarization Quark polarization Nucleon polarization Nucleon polarization -odd Naive T-odd

  34. Twist-2 structure Monopole Dipole GTMDs [Meißner, Metz, Schlegel (2009)] Quadrupole Quark polarization Nucleon polarization Naive OAM

  35. Twist-3 structure Monopole Dipole GPDs TMDs Quadrupole Quark polarization Quark polarization Nucleon polarization Nucleon polarization -odd Naive T-odd

  36. Quark spin and OAM GPDs TMDs GTMDs Quark spin ALL Quark spin ALL Quark spin ALL Quark kinetic OAM Quark canonical OAM Quark canonical OAM AUU+AUT ATT AUL Twist-2 [Burkardt (2007)] [Efremov et al. (2008,2010)] [She, Zhu, Ma (2009)] [Avakian et al. (2010)] [C.L., Pasquini (2011)] [Ji (1997)] [C.L., Pasquini (2011)] [Hatta (2011)] [C.L. et al. (2012)] AUL Twist-3 • Model-dependent • Not intrinsic! Pure twist-3! [Penttinen et al. (2000)]

  37. OAM and origin dependence Naive Relative Intrinsic Depends on proton position Momentum conservation Transverse center of momentum Physical interpretation ? Equivalence Intrinsic Naive Relative

  38. Overlap representation Fock expansion of the proton state Fock states Simultaneous eigenstates of Momentum Light-front helicity

  39. Overlap representation Light-front wave functions Eigenstates of parton light-front helicity Eigenstates of total OAM gauge Proton state Probabilityassociated with the N,b Fock state Normalization

  40. DVCS vs. SIDIS Incoherent scattering DVCS SIDIS FFs GPDs TMDs Factorization Compton form factor Cross section hard soft • process dependent • perturbative • « universal » • non-perturbative

  41. GPDs vs. TMDs GPDs TMDs Dirac matrix Correlator Correlator Off-forward! Forward! Wilson line ISI FSI e.g. DY e.g. SIDIS

  42. Quark polarization Quark polarization Nucleon polarization Nucleon polarization LC helicity and canonical spin [C.L., Pasquini (2011)] LC helicity Canonical spin

  43. Interesting relations *=SU(6) Model relations Linear relations Quadratic relation Flavor-dependent * * * * * Flavor-independent * * * * * * * Bag LFcQSM LFCQM 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)]

  44. Geometrical explanation [C.L., Pasquini (2011)] Preliminaries Conditions: • Quasi-free quarks • Spherical symmetry Wigner rotation (reduces to Melosh rotation in case of FREE quarks) Canonical spin Light-front helicity

  45. Geometrical explanation Axial symmetry about z

  46. Geometrical explanation Axial symmetry about z

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