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Quark phase-space distributions and orbital angular momentum

This presentation by Cédric Lorcé at École Polytechnique on April 19, 2012, delves into the intricate relationship between quark phase space distributions and orbital angular momentum (OAM). Topics include parton distribution functions, Wigner distributions, and various models linking OAM to generalized parton distributions (GPDs) and transverse-momentum distributions (TMDs). The discussion extends to both unpolarized and polarized quarks in nucleons, addressing the complexities of OAM decomposition and its implications for our understanding of proton structure.

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Quark phase-space distributions and orbital angular momentum

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  1. Quark phase-space distributionsand orbital angular momentum Cédric Lorcé and 19 Apr 2012, Ecole Polytechnique, Palaiseau, France

  2. Outline [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2011)] • Introduction • Parton distribution functions • Model results and discussion • Wigner distributions • Link with OAM

  3. Charges Charges Forward, local Vector Axial Tensor

  4. PDFs Charges Parton Distribution Functions Forward, non-local DIS

  5. PDFs FFs Charges Form Factors Non-forward, local ES 2D Fourier transform!

  6. PDFs FFs GPDs Charges Generalized PDFs Hadron 3D picture ! Non-forward, non-local DVCS [Burkardt (2000,2003)] [Belitsky et al. (2004)]

  7. TMDs PDFs FFs GPDs Charges Transverse-Momentum PDFs Complementary hadron 3D picture ! Forward, non-local SIDIS No direct connection Mean momentum Displacement Momentum space Position space Momentum transfer Mean position

  8. GTMDs TMDs PDFs FFs GPDs Charges Generalized TMDs Hadron 5D picture ! Non-forward, non-local Quasi-probabilistic interpretation in phase-space [Wigner (1932)] [Belitsky , Ji, Yuan(2004)] [C.L., Pasquini, Vanderhaeghen (2011)]

  9. Unpol. quark in unpol. proton [C.L., Pasquini (2011)] Based on LFCQM disfavored favored Left-right symmetry no net quark OAM

  10. Unpol. quark in long. pol. proton [C.L., Pasquini (2011)] Based on LFCQM Proton spin u-quark OAM d-quark OAM

  11. Unpol. quark in long. pol. proton [C.L., Pasquini, Xiong, Yuan (2011)] Based on LFCQM Proton spin u-quark OAM d-quark OAM

  12. Long. pol. quark in unpol. proton [C.L., Pasquini (2011)] Based on LFCQM Quark spin u-quark OAM d-quark OAM

  13. Long. pol. quark in long. pol. proton [C.L., Pasquini (2011)] Based on LFCQM Proton spin u-quark spin d-quark spin

  14. Angular momentum decompositions 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 Improvements: Improvements: • 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)]

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

  16. Summary [C.L., Pasquini, Vanderhaeghen (2011)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2011)] • Introduction • Parton distribution functions • 3D and 5D partonic pictures • Model results and discussion • Wigner distributions • Unpolarized/polarized quark in unpolarized/polarized nucleon • Link with OAM • Kinematic and canonical decompositions • Relations to parton distributions

  17. Backup

  18. Decompositions of total OAM Fock expansion of the proton state Fock states Simultaneous eigenstates of Momentum Light-front helicity

  19. How can one access to OAM? Overlap representation [Hägler, Mukherjee, Schäfer (2004)] [C.L., Pasquini, Xiong, Yuan (2011)] [C.L., Pasquini (2011)] Flavor contribution TMDs TMDs GTMDs GPDs Pure quark system [C.L., Pasquini (2011)] Conservation of transverse momentum NB: also valid for N,b Fock states Conservation of longitudinal momentum Anomalous gravitomagnetic sum rule! [Brodsky, Hwang, Ma, Schmidt (2001)]

  20. Physical interpretation Models Chiral quark-soliton model [Wakamatsu, Tsujimoto (2005)] [Wakamatsu (2010)] Non-perturbative sea contribution Scalar quark-diquark [Burkardt, Hikmat (2009)] Regularization-dependent 3Q light-front wave functions [C.L., Pasquini (2011)] Artifacts?

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