June 25 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy Parton Wigner Distributions of the nucleon Cédric Lorcé IPN Orsay - LPT Orsay June 25 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy

The outline Zoo of parton distribution functions Physical interpretation Wigner distributions and OAM Model calculations Conclusions

The outline Zoo of parton distribution functions Physical interpretation Wigner distributions and OAM Model calculations Conclusions

The charges Charges Depends on : Polarization Vector Axial Tensor Parton number Axial Parton helicity Tensor Parton transversity Charges

The parton distribution functions (PDFs) Depends on : Polarization Longitudinal momentum (fraction) DIS PDFs PDFs Charges

The form factors (FFs) PDFs FFs Charges Depends on : Polarization Longitudinal momentum (fraction) Momentum transfer Elastic scattering PDFs FFs FFs Charges

The generalized PDFs (GPDs) Depends on : Polarization Longitudinal momentum (fraction) Momentum transfer GPDs DVCS PDFs FFs GPDs Charges

The transverse momentum-dependent PDFs (TMDs) Depends on : Polarization Longitudinal momentum (fraction) Momentum transfer Transverse momentum No direct connection TMDs FFs PDFs Charges GPDs SIDIS TMDs

The generalized TMDs (GTMDs) Depends on : GTMDs Polarization Longitudinal momentum (fraction) Momentum transfer Transverse momentum TMDs GPDs ??? PDFs FFs GTMDs Charges

The complete zoo GTMDs TMDs TMFFs GPDs TMCs PDFs FFs Charges Depends on : GTMDs Polarization Longitudinal momentum (fraction) Momentum transfer Transverse momentum TMDs TMFFs GPDs ??? TMCs PDFs FFs GTMDs Charges [C.L., Pasquini, Vanderhaeghen (2011)]

The double parton scattering Depends on : Polarization Longitudinal momentum (fraction) Momentum transfer Transverse momentum Inter-parton distance DPDFs DPDFs [Diehl, Ostermeier, Schäfer (2012)] [Thürman, Master thesis (2012)]

The outline Zoo of parton distribution functions Physical interpretation Wigner distributions and OAM Model calculations Conclusions

The physical interpretation Initial/final Average/difference Position Momentum Fourier-conjugated variables

The physical interpretation [Ernst, Sachs, Wali (1960)] [Sachs (1962)] Breit frame Non-relativistic ! Position w.r.t. the CM Lorentz contraction Creation/annihilation of pairs

The physical interpretation [Soper (1977)] [Burkardt (2000)] Drell-Yan frame Position w.r.t. the center of momentum Lorentz contraction Creation/annihilation of pairs

The physical interpretation Dirac matrix ~ quark polarization Quark Wigner operator Wilson line Canonical momentum Either fix the gauge such that , i.e. work with + boundary condition Or split the Wilson line to form Dirac variables

The physical interpretation Quark Wigner operator Fixed light-front time No need for time-ordering ! Non-relativistic Wigner distribution [Ji (2003)] [Belitsky, Ji, Yuan (2004)] 3+3D Relativistic Wigner distribution [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] 2+3D GTMDs

The phase-space picture GTMDs 2+3D TMDs GPDs 0+3D 2+1D PDFs FFs 0+1D 2+0D Charges

The outline Zoo of parton distribution functions Physical interpretation Wigner distributions and OAM Model calculations Conclusions

Heisenberg’s uncertainty relations The phase-space distribution [Wigner (1932)] [Moyal (1949)] Wigner distribution Galilei covariant Either non-relativistic Or restricted to transverse position Probabilistic interpretation Heisenberg’s uncertainty relations Expectation value Position space Momentum space Phase space

Unpolarized quark density The quark orbital angular momentum [C.L., Pasquini (2011)] GTMD correlator Wigner distribution Orbital angular momentum Unpolarized quark density Parametrization [Meißner, Metz, Schlegel (2009)]

The parametrization @ twist-2 and x=0 [Meißner, Metz, Schlegel (2009)] GTMDs Quark polarization Nucleon polarization TMDs GPDs Monopole Dipole Quadrupole

The path dependence Orbital angular momentum Canonical Kinetic [C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] [Ji, Xiong, Yuan (2012)] [C.L. (2013)] Orbital angular momentum Reference point [Jaffe, Manohar (1990)] [Ji (1997)] Canonical Kinetic FSI ISI Drell-Yan SIDIS

The proton spin decompositions [C.L. (2013)] [Leader, C.L. (in preparation)] Reviews : Canonical Kinetic [Jaffe, Manohar (1990)] [Ji (1997)] Pros: Satisfies canonical relations Complete decomposition Pros: Gauge-invariant decomposition Accessible in DIS and DVCS Cons: Gauge-variant decomposition Missing observables for the OAM Cons: Does not satisfy canonical relations Incomplete decomposition News: Gauge-invariant extension News: Complete decomposition [Chen et al. (2008)] [Wakamatsu (2009,2010)] OAM accessible via Wigner distributions [C.L., Pasquini (2012)] [C.L., Pasquini, Xiong, Yuan(2012)] [Hatta (2012)]

The outline Zoo of parton distribution functions Physical interpretation Wigner distributions and OAM Model calculations Conclusions

The light-front overlap representation [C.L., Pasquini, Vanderhaeghen (2011)] Overlap representation Momentum Polarization Light-front quark models Wigner rotation

The model results [C.L., Pasquini (2011)] Wigner distribution of unpolarized quark in unpolarized nucleon favored disfavored Left-right symmetry No net quark OAM

The model results [C.L., Pasquini (2011)] Distortion induced by the nucleon longitudinal polarization Proton spin u-quark OAM d-quark OAM

The model results [C.L., Pasquini, Xiong, Yuan (2012)] Average transverse quark momentum in a longitudinally polarized nucleon « Vorticity »

The model results [C.L., Pasquini (2011)] Distortion induced by the quark longitudinal polarization Quark spin u-quark OAM d-quark OAM

The model results Quark spin-nucleon spin correlation Proton spin [C.L., Pasquini (2011)] Quark spin-nucleon spin correlation Proton spin u-quark spin d-quark spin

The model results [C.L., Pasquini (2011)]

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

No gluons and not QCD EOM ! The canonical and kinetic OAM Quark canonical OAM [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] Quark naive canonical OAM [Burkardt (2007)] [Efremov et al. (2008,2010)] [She, Zhu, Ma (2009)] [Avakian et al. (2010)] [C.L., Pasquini (2011)] Model-dependent ! Quark kinetic OAM [Ji (1997)] [Penttinen et al. (2000)] [Kiptily, Polyakov (2004)] [Hatta (2012)] Pure twist-3 No gluons and not QCD EOM ! but [C.L., Pasquini (2011)]

The conclusions Twist-2 parton distributions provide multidimensional pictures of the nucleon Relativistic phase-space distributions exist. Open question: how to access them? Both kinetic (Ji) and canonical (Jaffe-Manohar) are measurable (twist-2 and twist-3) Model calculations can test spin sum rules

Backup slides

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

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

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

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

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

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

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

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)]

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

Geometrical explanation Axial symmetry about z

Geometrical explanation Axial symmetry about z