Cédric Lorcé IPN Orsay - LPT Orsay Introduction to the GTMDs and the Wigner distributions June , Palace Hotel, Como, 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 Polarization Depends on : Vector Parton number Tensor Parton transversity Axial Parton helicity

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

Elastic scattering The form factors (FFs) FFsPDFs Charges Polarization Longitudinal momentum (fraction) Momentum transfer Depends on : FFs Cf. Kroll

DVCS The generalized PDFs (GPDs) GPDs FFsPDFs Charges Polarization Longitudinal momentum (fraction) Momentum transfer Depends on : FFs Cf. d’Hose, Guidal, Mueller, Murray, Pasquini, …

SIDIS The transverse momentum-dependent PDFs (TMDs) Polarization Longitudinal momentum (fraction) Momentum transfer Transverse momentum Depends on : TMDs No direct connection TMDs FFsPDFs Charges GPDs Cf. Anselmino, Aghasyan, Mulders, Scimemi, Signori, …

??? The generalized TMDs (GTMDs) Polarization Longitudinal momentum (fraction) Momentum transfer Transverse momentum Depends on : GTMDs TMDs FFsPDFs Charges GPDs GTMDs Cf. Liuti

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

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

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

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

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

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

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

The physical interpretation Quark Wigner operator Fixed light-front timeNo need for time-ordering ! Non-relativistic Wigner distribution Relativistic Wigner distribution [Ji (2003)] [Belitsky, Ji, Yuan (2004)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] 3+3D 2+3D GTMDs

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

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

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

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

[Meißner, Metz, Schlegel (2009)] The twist-2 and  =0 Parametrization : GTMDs TMDsGPDs MonopoleDipoleQuadrupole Nucleon polarization Quark polarization

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

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

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

Overlap representation MomentumPolarization [C.L., Pasquini, Vanderhaeghen (2011)] Light-front quark modelsWigner rotation The light-front overlap representation

Wigner distribution of unpolarized quark in unpolarized nucleon [C.L., Pasquini (2011)] The model results favored disfavored Left-right symmetryNo net quark OAM

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

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

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

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

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

The emerging picture [C.L., Pasquini (2011)] [Burkardt (2005)] [Barone et al. (2008)] LongitudinalTransverse Cf. Bacchetta

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

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

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

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