Wigner distributions and quark orbital angular momentum Cédric Lorcé and May , JLab, Newport News, VA, USA

Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

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

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

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

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

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

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

Parton distribution zoo Complete set GTMDs TMDs [C.L., Pasquini, Vanderhaeghen (2011)] Charges PDFs GPDs FFsTMCs TMFFs TMCs 4D Picture2D Picture Partonic interpretation (twist-2)

Twist-2 structure GPDsTMDs Nucleon polarization Quark polarization Nucleon polarization MonopoleDipoleQuadrupole Naive T-odd-odd

Twist-2 structure GTMDs Nucleon polarization Quark polarization MonopoleDipoleQuadrupole Naive OAM [Meißner, Metz, Schlegel (2009)]

Twist-3 structure GPDsTMDs Nucleon polarization Quark polarization Nucleon polarization Naive T-odd-odd MonopoleDipoleQuadrupole

Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

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

Quark spin and OAM GPDsTMDsGTMDs Quark spin Quark kinetic OAM A LL A UU +A UT [Penttinen et al. (2000)] Pure twist-3! [Ji (1997)] A UL A LL Quark spin Quark canonical OAM A TT Model-dependent Not intrinsic! [C.L., Pasquini (2011)] [Hatta (2011)] [C.L. et al. (2012)] [Burkardt (2007)] [Efremov et al. (2008,2010)] [She, Zhu, Ma (2009)] [Avakian et al. (2010)] [C.L., Pasquini (2011)] A UL Twist-2 Twist-3

OAM and origin dependence RelativeIntrinsicNaive Transverse center of momentum Physical interpretation ? Depends on proton position Equivalence IntrinsicRelativeNaive Momentum conservation

Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

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

Light-front wave functions Proton state Eigenstates of parton light-front helicity Eigenstates of total OAM Probability associated with the N,  Fock state Normalization Overlap representation gauge

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

Model calculations Light-front 3Q models [C.L., Pasquini (2011)] Models are not QCD Truncation of Fock space spoils Lorentz covariance In model calculations, one would expectbut [Carbonell, Desplanques, Karmanov, Mathiot (1998)] GTMDs TMDs GPDs

Model calculations Unpol d quark in unpol d nucleon [C.L., Pasquini (2011)] favored disfavored Left-right symmetryNo net quark OAM

Model calculations Unpol d quark in longitudinally pol d nucleon Proton spin u-quark OAM d-quark OAM [C.L., Pasquini (2011)]

Model calculations Proton spin u-quark OAM d-quark OAM [C.L. et al. (2012)] Unpol d quark in longitudinally pol d nucleon

Quark spin u-quark OAM d-quark OAM Model calculations Longitudinally pol d quark in unpol d nucleon [C.L., Pasquini (2011)]

Proton spin u-quark spin d-quark spin Model calculations Longitudinally pol d quark in longitudinally pol d nucleon [C.L., Pasquini (2011)]

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

Summary Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Twist-2 phase-space distributions Proton spin puzzle  Decompositions Kinetic versus canonical  Relations with observables Model results  Overlap representation  Wigner distributions Consistent emerging picture

Incoherent scattering DVCS vs. SIDIS DVCSSIDIS GPDs TMDs FFs Factorization Compton form factor Cross section process dependent perturbative « universal » non-perturbative hardsoft

GPDs vs. TMDs GPDsTMDs Correlator Dirac matrix Wilson line Off-forward!Forward! FSIISI e.g. SIDISe.g. DY

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

Interesting relations Model relations ** * * * * Flavor-dependent Flavor-independent Linear relationsQuadratic relation Bag LF  QSM LFCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. ( )] [C.L., Pasquini, Vanderhaeghen (2011)] [Pasquini & al. ( )] [Ma & al. ( ), Jakob & al. (1997), Bacchetta & al. (2008)] [Ma & al. ( ), Jakob & al. (1997)] [Bacchetta & al. (2008)] [Efremov & al. (2009)] [Meißner & al. (2007)] * =SU(6) * * * * * *

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

Geometrical explanation Axial symmetry about z

Geometrical explanation Axial symmetry about z