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Cédric LorcéIPN Orsay - LPT Orsay
Orbital Angular Momentum in QCD
June 27 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy
The outline
Dark spin
Quark spin?
~ 30 %
• The decompositions in a nutshell• Canonical formalism and Chen et al. approach• Geometrical interpretation of gauge symmetry• Path-dependence and measurability• Conclusions
Basic question
Ji (1997)Jaffe-Manohar (1990)
The decompositions in a nutshell
Sq
SgLg
Lq Sq
Jg
Lq
Noether’s theorem
Ji (1997)Jaffe-Manohar (1990)
Chen et al. (2008)
The decompositions in a nutshell
Sq
SgLg
Lq
Sq
SgLg
Lq
Sq
Jg
Lq
Gauge-invariant extension (GIE)
Noether’s theorem
Wakamatsu (2010)
Ji (1997)Jaffe-Manohar (1990)
Chen et al. (2008)
The decompositions in a nutshell
Sq
SgLg
Lq
Sq
SgLg
Lq Sq
SgLg
Lq
Sq
Jg
Lq
Gauge-invariant extension (GIE)
Noether’s theorem
Wakamatsu (2010)
Ji (1997)Jaffe-Manohar (1990)
Chen et al. (2008)
Canonical Kinetic
The decompositions in a nutshell
Sq
SgLg
Lq
Sq
SgLg
Lq Sq
SgLg
Lq
Sq
Jg
Lq
Gauge-invariant extension (GIE)
Noether’s theorem
Wakamatsu (2010)
Ji (1997)Jaffe-Manohar (1990)
Chen et al. (2008)
Canonical Kinetic
The decompositions in a nutshell
Sq
SgLg
Lq
Sq
SgLg
Lq Sq
SgLg
Lq
Sq
Jg
Lq
Gauge-invariant extension (GIE)
Noether’s theorem
The Chen et al. approach
Gauge transformation (assumed)
[Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]
The Chen et al. approach
Gauge transformation (assumed)
Pure-gauge covariant derivatives
[Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]
The Chen et al. approach
Gauge transformation (assumed)
Field strength
Pure-gauge covariant derivatives
[Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]
The canonical formalism
Textbook
Gauge covariant
Gauge invariant
Dynamical variables
Lagrangian
Dirac variables
Dressing field Gauge transformation
[Dirac (1955)][Mandelstam
(1962)]
[C.L. (2013)]
Pure gauge
Physical polarizations
The analogy with General Relativity
Degrees of freedom
[C.L. (2012,2013)]
Dual role
Pure gauge
Physical polarizations
The analogy with General Relativity
Geometrical interpretation
Parallelism Curvature
Degrees of freedom
[C.L. (2012,2013)]
Dual role
Pure gauge
Physical polarizations
Analogy with General
Relativity
The analogy with General Relativity
Geometrical interpretation
Parallelism Curvature
Inertial forces
Gravitational forces
Degrees of freedom
[C.L. (2012,2013)]
Dual role
[Wakamatsu (2010)][Chen et al. (2008)]
The Stueckelberg symmetry
Ambiguous!
[Stoilov (2010)][C.L. (2013)]
Sq
SgLg
Lq Sq
SgLg
Lq
Coulomb GIE
[Hatta (2011)][C.L. (2013)]
Sq
SgLg
Lq
Light-front GIE
Lpot
LpotSq
Sg
Lg
Lq
Infinitely many possibilities!
Gauge
GIE1
GIE2
Gauge-variant operator
« Natural » gauges
Lorentz-invariant extensions~
Rest
Center-of-mass
Infinite momentum
« Natural » frames
The gauge-invariant extension (GIE)
The geometrical interpretation[Hatta (2012)]
[C.L. (2012)]Parallel transport
Path dependent !
Stueckelberg symmetry
Non-local !
The path dependence[Ji, Xiong, Yuan (2012)]
[Hatta (2012)][C.L. (2013)]
Canonical quark OAM operator
FSIISI
SIDISDrell-Yan
The path dependence[Ji, Xiong, Yuan (2012)]
[Hatta (2012)][C.L. (2013)]
Naive T-even
Canonical quark OAM operator
Light-front
Lq
FSIISI
SIDISDrell-Yan
The path dependence[Ji, Xiong, Yuan (2012)]
[Hatta (2012)][C.L. (2013)]
Coincides locally with kinetic quark OAM
Naive T-even
Canonical quark OAM operator
x-based Fock-SchwingerLight-front
LqLq
The gauge symmetry
Quantum electrodynamics
Passive
« Physical »
[C.L. (in preparation)]
« Background »
The gauge symmetry
Quantum electrodynamics
Passive Active
« Physical »
[C.L. (in preparation)]
« Background »
The gauge symmetry
Quantum electrodynamics
Passive Active
« Physical »
[C.L. (in preparation)]
« Background »
Active x (Passive)-1
The gauge symmetry
Quantum electrodynamics
Passive Active
« Physical »
[C.L. (in preparation)]
« Background »
Active x (Passive)-1
Stueckelberg
The semantic ambiguity
Observables
« measurable »
Quid ?
« physical »
« gauge invariant »
Measurable, physical, gauge invariant (active and passive)
E.g. cross-sections
The semantic ambiguity
PathStueckelbergBackground
Observables
« measurable »
Quid ?
« physical »
« gauge invariant »
Measurable, physical, gauge invariant (active and passive)
Expansion scheme
E.g. cross-sections
dependent
E.g. collinear factorization
The semantic ambiguity
PathStueckelbergBackground
Observables
Quasi-observables
« measurable »
Quid ?
« physical »
« gauge invariant »
Measurable, physical, gauge invariant (active and passive)
« Measurable », « physical », « gauge invariant » (only passive)
Expansion scheme
E.g. cross-sections
E.g. parton distributions
dependent
E.g. collinear factorization
Canonical Kinetic
The observability
Sq
SgLg
Lq
Sq
SgLg
Lq Sq
SgLg
Lq
Sq
Jg
Lq
Not observableObservable Quasi-observable
[Wakamatsu (2010)]
[Ji (1997)][Jaffe-Manohar (1990)]
[Chen et al. (2008)]
The gluon spin
[Jaffe-Manohar (1990)] [Hatta (2011)]
Light-front GIE Light-front gauge
Gluon helicity distribution
Local fixed-gauge interpretation
Non-local gauge-invariant interpretation
« Measurable », gauge invariant but non-local
The kinetic and canonical OAM
Quark naive canonical OAM (Jaffe-Manohar)
[Burkardt (2007)][Efremov et al.
(2008,2010)][She, Zhu, Ma (2009)][Avakian et al. (2010)][C.L., Pasquini (2011)]
Model-dependent !
Kinetic OAM (Ji)
[Ji (1997)]
[Penttinen et al. (2000)][Kiptily, Polyakov (2004)]
[Hatta (2012)]
but
No gluons and not QCD EOM!
[C.L., Pasquini (2011)]
Pure twist-3
Canonical OAM (Jaffe-Manohar) [C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]
[Hatta (2012)]
Wakamatsu (2010)
Ji (1997)Jaffe-Manohar (1990)
Chen et al. (2008)
Canonical Kinetic
The conclusion
Sq
SgLg
Lq
Sq
SgLg
Lq Sq
SgLg
Lq
Sq
Jg
Lq
Not observable Observable
Quasi-observableQuasi-observable
[PRD79 (2009) 014507] [Nucl. Phys. A825 (2009) 115]
[PRL104 (2010) 112001][PRD79 (2009) 113011]
GTMDs
TMDs
Charges
PDFs
GPDs
FFsTMCs
TMFFs[PRD84 (2011) 034039]
[PLB710 (2012) 486]
[PRD84 (2011) 014015][PRD85 (2012) 114006]
[JHEP1105 (2011) 041]
[PRD74 (2006) 054019][PRD78 (2008) 034001]
[PRD79 (2009) 074027]
Phase-space densities
The parton distributions
« Vorticity »
The twist-2 OAM
Quark Wigner operator
[C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]
[Hatta (2012)]
Quark OAM operator
Exact relation
Overlap representation
Momentum Polarization
[PRD74 (2006) 054019][PRD78 (2008) 034001]
[PRD79 (2009) 074027]
Light-front quark models Wigner rotation
The light-front wave functions