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Exploring Generalized TMDs and GTMDs: Physical Interpretation and Constraints

Delve into the world of Generalized TMDs and GTMDs, understanding their physical interpretations and constraints. Learn about phase space, spatial distributions, momentum-space density, and their implications in quantum mechanics and quantum field theory.

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Exploring Generalized TMDs and GTMDs: Physical Interpretation and Constraints

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  1. SummerSchool 2017 GPDs and GTMDs CPHT Cédric Lorcé June 28, Temple University, Philadelphia, USA

  2. Outline Generalized TMDs Physical interpretation Phase space Physical content How to constrain GTMDs Multidimensional Universe no. 2 by rogerhitchcock

  3. 1. Generalized TMDs

  4. Where we are TMDs PDFs Charges Nonlocal quark operator Gauge link See T. Rodgers’ lectures

  5. Through the Looking-Glass TMDs PDFs FFs Charges Off-forward amplitudes Form factors

  6. Through the Looking-Glass TMDs PDFs FFs GPDs Charges Generalized PDFs

  7. Through the Looking-Glass GTMDs TMDs PDFs FFs GPDs Charges Generalized TMDs

  8. Parton distribution zoo GTMDs TMDs PDFs FFs GPDs Charges [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini, Vanderhaeghen (2011)]

  9. 2. Physical interpretation

  10. Spatial distributions Localized state in momentum space in position space

  11. Spatial distributions Localized state in momentum space in position space Phase-space compromise

  12. Spatial distributions Localized state in momentum space in position space Phase-space compromise Density in the Breit frame Breit frame Exercise 1: Derive these Breit-frame expressions

  13. Galilean symmetry All is fine as long as space-time symmetry is Galilean Position operator can be defined CoM position

  14. Lorentz symmetry But in Special Relativity, space-time symmetry is Lorentzian No separation of CoM and internal coordinates Position operator is ill-defined ! Further issues : Creation/annihilation of pairs Lorentz contraction Spoils (quasi-) probabilistic interpretation

  15. Light-front operators Transverse space-time symmetry is Galilean Transverse position operator can be defined ! « CoM » position Longitudinal momentum plays the role of mass in the transverse plane [Kogut, Soper (1970)]

  16. Quasi-probabilistic interpretation What about the further issues with Special Relativity ? Transverse boosts are Galilean No transverse Lorentz contraction ! No sensitivity to longitudinal Lorentz contraction ! Particle number is conserved in Drell-Yan frame Drell-Yan frame is conserved and positive

  17. Relativistic densities Localized state in momentum space in 2D position space [Soper (1977)] [Burkardt (2000)] [Burkardt (2003)]

  18. Relativistic densities Localized state in momentum space in 2D position space Phase-space compromise [Soper (1977)] [Burkardt (2000)] [Burkardt (2003)]

  19. Relativistic densities Localized state in momentum space in 2D position space Phase-space compromise Density in the symmetric Drell-Yan frame [Soper (1977)] [Burkardt (2000)] [Burkardt (2003)]

  20. Partonic picture GTMDs TMDs PDFs FFs GPDs Charges

  21. Partonic picture GTMDs TMDs PDFs FFs GPDs Charges 2+3D 0+3D 2+1D 0+1D 2+0D See P. Nadolsky’s lectures

  22. 3. Phase space

  23. Classical Mechanics State of the system Momentum Particles follow well-defined trajectories Position [Gibbs (1901)]

  24. Statistical Mechanics Phase-space density Position-space density Momentum Momentum-space density Phase-space average Position [Gibbs (1902)]

  25. Quantum Mechanics Wigner distribution Position-space density Momentum Momentum-space density Phase-space average Position [Wigner (1932)] [Moyal (1949)]

  26. Quantum Mechanics Wigner distribution Momentum Exercise 2: Show that Position Symmetric derivative [Wigner (1932)] [Moyal (1949)]

  27. Applications Wigner distributions have applications in: Harmonic oscillator • Nuclear physics • Quantum chemistry • Quantum molecular dynamics • Quantum information • Quantum optics • Classical optics • Signal analysis • Image processing • Quark-gluon plasma • … Quasi-probabilistic Heisenberg’s uncertainty relations

  28. Quantum Field Theory Covariant Wigner operator Time ordering ? Field operators [Carruthers, Zachariasen (1976)] [Carruthers, Zachariasen (1983)] [Ochs, Heinz(1997)]

  29. Quantum Field Theory Covariant Wigner operator Time ordering ? Field operators Equal-time Wigner operator [Carruthers, Zachariasen (1976)] [Carruthers, Zachariasen (1983)] [Ochs, Heinz(1997)]

  30. Quantum Field Theory Covariant Wigner operator Time ordering ? Field operators Equal-time Wigner operator Phase-space/Wigner distribution [Carruthers, Zachariasen (1976)] [Carruthers, Zachariasen (1983)] [Ochs, Heinz(1997)]

  31. Quantum Field Theory Equal light-front time Wigner operator Connection with partonic physics

  32. Quantum Field Theory Equal light-front time Wigner operator Connection with partonic physics Non-relativistic 3+3D Wigner distribution [Ji (2003)] [Belitsky, Ji, Yuan (2004)]

  33. Quantum Field Theory Equal light-front time Wigner operator Connection with partonic physics Non-relativistic 3+3D Wigner distribution [Ji (2003)] [Belitsky, Ji, Yuan (2004)] Relativistic 2+3D Wigner distribution GTMDs [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)]

  34. Instant form vs light-front form Our intuition is instant form and not light-front form NB : is invariant under light-front boosts At leading twist, it can be thought of as instant form phase-space (Wigner) distribution in IMF ! See M. Constantinou and M. Engelhardt’s lectures Transverse momentum Longitudinal momentum Transverse position 2+3D In IMF, the nucleon looks like a pancake

  35. 4. Physical content

  36. Parametrization GTMDs TMDs GPDs Twist-2 Monopole Dipole Quadrupole Quark polarization Nucleon polarization Complete parametrizations : Quarks [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini (2013)] Quarks & gluons

  37. Parametrization GTMDs TMDs GPDs Twist-2 Monopole Dipole Quadrupole Quark polarization Nucleon polarization Complete parametrizations : Quarks [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini (2013)] Quarks & gluons

  38. Parametrization GTMDs TMDs GPDs Twist-2 Monopole Dipole Quadrupole Quark polarization Nucleon polarization Complete parametrizations : Quarks [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini (2013)] Quarks & gluons

  39. Parametrization GTMDs TMDs GPDs Twist-2 Monopole Dipole Quadrupole Quark polarization Nucleon polarization Complete parametrizations : Quarks [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini (2013)] Quarks & gluons

  40. Parametrization GTMDs TMDs GPDs Twist-2 Monopole Dipole Quadrupole Quark polarization Nucleon polarization New ! Complete parametrizations : Quarks [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini (2013)] Quarks & gluons

  41. Light-front wave functions (LFWFs) Fock expansion of the nucleon state

  42. Light-front wave functions (LFWFs) Fock expansion of the nucleon state Probability associated with the Fock states

  43. Light-front wave functions (LFWFs) Fock expansion of the nucleon state Probability associated with the Fock states Linear and angular momentum conservation gauge

  44. Light-front wave functions (LFWFs) Overlap representation GTMDs Momentum Polarization [C.L., Pasquini, Vanderhaeghen (2011)]

  45. Model results Wigner distribution of unpolarized quark in unpolarized nucleon 2+2D [C.L., Pasquini (2011)]

  46. Model results Wigner distribution of unpolarized quark in unpolarized nucleon 2+2D Left-right symmetry [C.L., Pasquini (2011)]

  47. Model results Wigner distribution of unpolarized quark in unpolarized nucleon 2+2D favored disfavored [C.L., Pasquini (2011)] favored disfavored

  48. Model results Quark spin-nucleon spin correlation 2+2D Proton spin u-quark spin d-quark spin [C.L., Pasquini (2011)]

  49. Model results Distortion correlated to nucleon spin 2+2D Proton spin u-quark OAM d-quark OAM [C.L., Pasquini (2011)]

  50. Model results Average transverse quark momentum correlated to nucleon spin [C.L., Pasquini, Xiong, Yuan (2012)]

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