1. Sep 21st 2015, INFN Frascati National Laboratories, Frascati, Italy
Quark and gluon OAM :
Where are we ?
[K.-F. Liu, C.L., arXiv: ]
Based on :
Cédric Lorcé
CPHT
Sep 21st 2015, INFN Frascati National Laboratories, Frascati, Italy

2. Outline Spin decompositions Relations to parton distributions
Myths and facts
Conclusions

3. Spin decompositions (1990-2008)
Canonical
Kinetic Sq SG LG Lq Sq Lq JG
Gauge fixed !
« Incomplete » ?
[Jaffe, Manohar (1990)]
[Ji (1997)]

4. Spin decompositions (2008-now)
Canonical
Kinetic Sq SG LG Lq Sq Lq LG SG
Gauge invariant !
« Complete »
[Chen et al. (2008)]
[Wakamatsu (2010)]

5. Spin decompositions (2008-now)
Canonical
Kinetic Sq LG Lq Lq Sq SG LG SG
[Wakamatsu (2010)]
[Leader, C.L. (2014)]

6. Ambiguity in Chen et al. approach
Background
Dynamical
Pure-gauge field
Non-abelian Stueckelberg field
Gauge covariant field
For given (i.e. gauge freedom is fixed),
is arbitrary (Stueckelberg symmetry)
~ copy of gauge symmetry
In practice : gauge symmetry is preserved
Stueckelberg symmetry is broken
[C.L. ( )]
[Leader, C.L. (2014)]
[Wakamatsu (2014)]

7. Ambiguity in Chen et al. approach
Experimental setup and theoretical framework determine Stueckelberg breaking
Example in high-energy physics
System
« Factorized » system
Wilson line
~ background field
Local
Non-local
[C.L. ( )]
[Leader, C.L. (2014)]

8. Phase-space (Wigner) distribution
Parton distribution zoo
LFWF
Phase-space (Wigner) distribution
Theoretical tools
GTMDs
2+3D
TMDs
GPDs
0+3D
2+1D
PDFs
FFs
«Physical» objects
2+0D
0+1D
Charges
[C.L., Pasquini, Vanderhaeghen (2011)]

9. Relations to parton distributions
Kinetic
Twist-2
GPDs
[Ji (1997)]
Twist-3
GPDs
[Penttinen et al. (2000)]
[Hatta, Yoshida (2012)]
See M. Burkardt’s talk
Twist-2
GTMDs
[C.L., Pasquini (2011)]
[Hatta (2011)]
[Ji, Xiong, Yuan (2012)]
[C.L. (2012)]
[Liu, C.L. (2015)]

10. Relations to parton distributions
Canonical
Twist-2
PDFs
[Ma, Schmidt (1998)]
Model-dependent !
Twist-2
TMDs
[She, Zhu, Ma (2009)]
[Avakian et al. (2010)]
[Efremov et al. (2010)]
[C.L., Pasquini (2012)]
Model-dependent !
See M. Burkardt’s talk
Twist-2
GTMDs
[C.L., Pasquini (2011)]
[Hatta (2011)]
[Ji, Xiong, Yuan (2012)]
[C.L. (2012)]

11. Myths and facts Spin densities Angular momentum densities
Trivial
Less trivial
Angular momentum densities
Ad hoc pure twist-2 generalization via Mellin moments
[Hoodbhoy, Ji, Lu (1998)]
Pure twist-2 part of transverse boost operator
in rest frame of transversely polarized target
[Ji, Xiong, Yuan (2012)]

12. Myths and facts OAM densities
involves , so twist-3 naturally contributes
Ad hoc pure twist-2 generalization via Mellin moments
-independence of divergence terms
[Hoodbhoy, Ji, Lu (1998)]
Wandzura-Wilczek approximation
[Hägler et al. (2004)]
NB :

13. Myths and facts Transverse densities Generalized Ji relation
[Burkardt ( )]
Generalized Ji relation
[Polyakov (2003)]
[Leader, C.L. (2014)]
Divergence term !

14. Myths and facts OAM phase-space densities [C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)]
[C.L. (2013)]
ISI/FSI
ISI/FSI
Density mode
Inflation mode
Orbital mode
Spiral mode
[C.L., Pasquini (in preparation)]

15. Myths and facts OAM phase-space densities [C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)]
[C.L. (2013)]
ISI/FSI
ISI/FSI
Density mode
Inflation mode
Orbital mode
Spiral mode
[C.L., Pasquini (in preparation)]

16. Myths and facts Canonical Gauge link dependence But
See M. Burkardt’s talk
Gauge link dependence
Drell-Yan
SIDIS
ISI
FSI
Canonical
[C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)]
[C.L. (2013)]
[Liu, C.L. (2015)]
[C.L., Pasquini (in preparation)]
ISI/FSI
But
[Hatta (2012)]
[Hatta, Yoshida (2012)]
[Ji, Xiong, Yuan (2013)]
[C.L. (2013)]
[C.L. (2015)]

17. Myths and facts Kinetic Gauge link dependence But
See M. Burkardt’s talk
Gauge link dependence
Kinetic
[C.L. (2013)]
But
[Ji, Xiong, Yuan (2012)]
[C.L. (2013)]
[Liu, C.L. (2015)]
NB :

18. Conclusions
Both canonical and kinetic decompositions are physical and interesting
Formal ambiguities are determined by both experimental setup and theoretical framework
Relations between integrated AM contributions and parton distributions are well established
Relations between AM densities and parton distributions are pretty confused (lots of myths)
Be careful !

19. Backup slides

20. Total EM is gauge invariant and conserved
Energy-momentum tensor
Canonical EMT
[Jaffe, Manohar (1990)]
Gauge transformation
Superpotential
Freedom of the EM distribution
satisfies
Total EM is gauge invariant and conserved

21. Energy-momentum tensor
Ji EMT
QCD EOM
Gauge invariant
[Ji (1997)]
Belinfante-Rosenfeld EMT
QCD identity
Gauge invariant
and symmetric