1. June 28, Temple University, Philadelphia, USA
Summer School 2017
GPDs and GTMDs
Cédric Lorcé
CPHT
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. See T. Rodgers’ lectures
Where we are
Nonlocal quark operator
Gauge link
See T. Rodgers’ lectures
TMDs
PDFs
Charges

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

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

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

8. Parton distribution zoo
GTMDs
TMDs
GPDs
PDFs
FFs
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. Exercise 1: Derive these Breit-frame expressions
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
Position operator is ill-defined !
No separation of CoM and internal coordinates
Further issues :
Lorentz contraction
Creation/annihilation of pairs
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
GPDs
PDFs
FFs
Charges

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

22. 3. Phase space

23. Particles follow well-defined trajectories
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 Exercise 2: Show that
Momentum
Exercise 2: Show that
Position
Symmetric derivative
[Wigner (1932)]
[Moyal (1949)]

27. Heisenberg’s uncertainty relations
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. Connection with partonic physics
Quantum Field Theory
Equal light-front time Wigner operator
Connection with partonic physics

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

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

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

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

40. Complete parametrizations : Quarks
Twist-2
Monopole
Dipole
Quadrupole
GTMDs
Quark polarization
Nucleon polarization
New !
TMDs
GPDs
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
favored
disfavored
[C.L., Pasquini (2011)]

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

49. Model results Distortion correlated to nucleon spin 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)]

51. Model results Distortion correlated to quark spin Quark spin
u-quark OAM
d-quark OAM
[C.L., Pasquini (2011)]

52. Model results
[C.L., Pasquini (2011)]

53. Phase-space transverse modes
[C.L., Pasquini (2016)]

54. Phase-space transverse modes
[C.L., Pasquini (2016)]

55. parity and time-reversal
Phase-space transverse modes
Properties under
parity and time-reversal
[C.L., Pasquini (2016)]

56. Phase-space transverse modesUU

57. Phase-space transverse modesUU

58. Phase-space transverse modesLL

59. Phase-space transverse modesLU
OAM !
[C.L., Pasquini (2011)]

60. Phase-space transverse modesLU
OAM !
[C.L., Pasquini (2011)]

61. Phase-space transverse modesLU
OAM !
[C.L., Pasquini (2011)]

62. Phase-space transverse modesUL
Spin-orbit !
[C.L., Pasquini (2011)]

63. Angular correlations TMDs GPDs Quark polarization Nucleon polarization
[C.L., Pasquini (2016)]

64. 5. How to constrain GTMDs

65. Tomography
In quantum optics, Wigner distributions are « measured » using homodyne tomography
[Lvovski et al. (2001)]
[Bimbard et al. (2014)]
Idea : measuring projections of Wigner distributions from different directions
Exercise 3: Find the 3D hidden picture 3D 2D 2D
Binocular vision in phase space !

66. TMDs and GPDs GTMDs TMDs GPDs PDFs FFs Charges
Partial constraints from TMDs and GPDs
Straight gauge link
Lorentz invariance relations
TMDs
GPDs
Twist-2 TMDs and twist-3 PDFs
[Mulders, Tangerman ( )]
[Boer, Mulders (1998)]
[Goeke et al. (2003)]
[Metz, Schweitzer, Teckentrup (2009)]
[Kanazawa et al. (2016)]
PDFs
FFs
Twist-2 GTMDs and twist-3 GPDs
[Rajan et al. (2016)]
[Courtoy, Miramontes (2017)]
Charges

67. Possible physical processes
Recently, several observables sensitive to GTMDs have been proposed
eA scattering
Dijet production
Longitudinal SSA
pA scattering
Double parton scattering (DPS)
Ultra-peripheral collions (UPCs)
pN scattering
Exclusive double Drell-Yan
[Hatta, Xiao, Yuan (2016)]
[Hatta, Nakagawa, Yuan, Zhao (2016)]
[Ji, Yuan , Zhao (2016)]
[Hagiwara, Hatta, Xiao, Yuan (2017)]
[Hagiwara et al. (2017)]
[Bhattacharya, Metz, Zhou (2017)]

68. Lattice calculations
Lattice simulation with « large » nucleon momentum
Concept of quasi-distributions
Pioneer calculation of quark OAM
[Hägler et al. ( )]
[Ji (2013)]
[Engelhardt (2017)]
Jaffe-Manohar OAM
Ji OAM
Jaffe-Manohar OAM
Nucleon rapidity

69. Summary

70. Summary GTMDs add information about parton position to TMDs
Concept of phase space exists in QFT
Wigner functions have simple relation to OAM
Spin structure is richer than nucleon spin budget
GTMDs (partially) constrained by experiments and Lattice QCD
GTMDs
LFWFs
GPDs
TMDs
FFs
PDFs