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Cédric Lorcé

Cédric Lorcé. Introduction to the GTMDs and the Wigner distributions. IPN Orsay - LPT Orsay. June 10 2013, Palace Hotel, Como, Italy. The outline. Zoo of parton distribution functions Physical interpretation Wigner distributions and OAM Model calculations Conclusions. The outline.

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Cédric Lorcé

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  1. Cédric Lorcé Introduction to the GTMDs and the Wigner distributions IPN Orsay - LPT Orsay June 10 2013, Palace Hotel, Como, Italy

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

  3. The outline • Zoo of parton distribution functions • Physicalinterpretation • Wigner distributions and OAM • Model calculations • Conclusions

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

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

  6. The form factors (FFs) PDFs FFs Charges Depends on : • Polarization • Longitudinal momentum (fraction) • Momentumtransfer Elastic scattering FFs Cf. Kroll

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

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

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

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

  11. 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)]

  12. The outline • Zoo of parton distribution functions • Physicalinterpretation • Wigner distributions and OAM • Model calculations • Conclusions

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

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

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

  16. 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

  17. 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

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

  19. The outline • Zoo of parton distribution functions • Physicalinterpretation • Wigner distributions and OAM • Model calculations • Conclusions

  20. 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

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

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

  23. The path dependence [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 ISI FSI Drell-Yan SIDIS

  24. The proton spin decompositions [C.L. (2013)] [Leader, C.L. (in preparation)] Reviews : Cf. Burkardt, Zhang Canonical Kinetic [Jaffe, Manohar (1990)] [Ji (1997)] Pros: Pros: • Satisfies canonical relations • Complete decomposition • Gauge-invariant decomposition • Accessible in DIS and DVCS Cons: Cons: • Does not satisfy canonical relations • Incomplete decomposition • Gauge-variant decomposition • Missing observables for the OAM News: News: • Gauge-invariant extension • 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)]

  25. The outline • Zoo of parton distribution functions • Physicalinterpretation • Wigner distributions and OAM • Model calculations • Conclusions

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

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

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

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

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

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

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

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

  34. 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 Cf. Liuti No gluons and not QCD EOM ! but [C.L., Pasquini (2011)]

  35. 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

  36. Backup slides

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

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

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

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

  44. 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)]

  45. 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

  46. Geometrical explanation Axial symmetry about z

  47. Geometrical explanation Axial symmetry about z

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