June 28, Temple University, Philadelphia, USA Summer School 2017 GPDs and GTMDs Cédric Lorcé CPHT June 28, Temple University, Philadelphia, USA

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

1. Generalized TMDs

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

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

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

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

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

2. Physical interpretation

Spatial distributions Localized state in momentum space in position space

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

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

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

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

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)]

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

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

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

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)]

Partonic picture GTMDs TMDs GPDs PDFs FFs Charges

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

3. Phase space

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

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

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

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

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

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

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

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)]

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

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)]

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)]

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

4. Physical content

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Phase-space transverse modes UU

Phase-space transverse modes UU

Phase-space transverse modes LL

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

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

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

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

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

5. How to constrain GTMDs

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 !

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 (1994-97)] [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

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)]

Lattice calculations Lattice simulation with « large » nucleon momentum Concept of quasi-distributions Pioneer calculation of quark OAM [Hägler et al. (2009-16)] [Ji (2013)] [Engelhardt (2017)] Jaffe-Manohar OAM Ji OAM Jaffe-Manohar OAM Nucleon rapidity

Summary

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