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![Page 1: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/1.jpg)
Wigner distributions and quark orbital
angular momentum
Cédric Lorcé
and
May 14 2012, JLab, Newport News, VA, USA
![Page 2: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/2.jpg)
Outline Parton distribution zoo
PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation
Proton spin puzzle Decompositions Relations with observables
Model results Overlap representation Wigner distributions
![Page 3: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/3.jpg)
Outline Parton distribution zoo
PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation
Proton spin puzzle Decompositions Relations with observables
Model results Overlap representation Wigner distributions
![Page 4: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/4.jpg)
Parton distribution zooCharges
Charges
Partonic interpretation (twist-2)
Vector
Tensor
Axial
0D Picture
![Page 5: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/5.jpg)
Parton distribution zoo
Partonic interpretation (twist-2)
DIS
Parton Distribution Functions
Charges
PDFs
1D Picture
![Page 6: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/6.jpg)
Parton distribution zoo
Partonic interpretation (twist-2)
ES
Form Factors
PDFs
Charges
FFs
2D Picture
Impact parameter
Transverse center of
momentum
![Page 7: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/7.jpg)
DVCS
Parton distribution zoo
Partonic interpretation (twist-2)
Generalized PDFs
GPDs
Charges
PDFs FFs
3D Picture
[Soper (1977)][Burkardt (2000,2003)]
[Diehl, Hägler (2005)]
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Parton distribution zoo
Partonic interpretation (twist-2)
SIDIS
Transverse-Momentum dependent PDFs
No direct connection
TMDs
Position space
Momentum space
Mean momentu
mDisplacement
Mean position
Momentum
transfergauge Charges
PDFs FFs
GPDs
3D Picture
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Parton distribution zoo
Partonic interpretation (twist-2)
Generalized TMDsGTMDs
TMDs
[Wigner (1932)][Belitsky , Ji, Yuan (2004)]
[C.L., Pasquini (2011)]
Quasi-probabilistic interpretation
??????
Charges
PDFs
GPDs
FFs
5D Picture
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Parton distribution zooComplete set
GTMDs
TMDs[C.L., Pasquini, Vanderhaeghen
(2011)]
Charges
PDFs
GPDs
FFsTMCs
TMFFs
TMFFs
TMFFs
TMCs
TMCs
4D Picture 2D Picture
Partonic interpretation (twist-2)
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Twist-2 structureGPDs TMDs
Nuc
leon
pol
ariz
atio
n
Quark polarization Quark polarization
Nuc
leon
pol
ariz
atio
n
Monopole Dipole Quadrupole
Naive T-odd
-odd
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Twist-2 structureGTMDs
Nuc
leon
pol
ariz
atio
n
Quark polarization
Monopole Dipole Quadrupole
Naive OAM
[Meißner, Metz, Schlegel (2009)]
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Twist-3 structureGPDs TMDs
Nuc
leon
pol
ariz
atio
n
Quark polarization Quark polarization
Nuc
leon
pol
ariz
atio
n
Naive T-odd
-odd
Monopole Dipole Quadrupole
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Outline Parton distribution zoo
PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation
Proton spin puzzle Decompositions Relations with observables
Model results Overlap representation Wigner distributions
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• Does not satisfy canonical relations• Incomplete decomposition
Proton spin puzzleJi Jaffe-Manohar
[Ji (1997)] [Jaffe, Manohar (1990)]
• Gauge-invariant decomposition• Accessible in DIS and DVCS
Kinetic CanonicalPros:Cons:
Pros:Cons:
• Satisfies canonical relations• Complete decomposition
• Gauge-variant decomposition• Missing observables for the OAM
Improvements:
[Wakamatsu (2009,2010)]• Complete decomposition
Improvements:
[Chen et al. (2008)]
• Gauge-invariant extension
• OAM accessible via Wigner distributions [C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan(2011)]
[Hatta (2011)]
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Quark spin and OAMGPDs TMDs GTMDs
Quark spin
Quark kinetic OAM
ALL
AUU+AU
T
[Penttinen et al. (2000)]Pure twist-3!
[Ji (1997)]
AUL
ALL ALLQuark spin Quark spin
Quark canonical OAM
Quark canonical OAMATT
• Model-dependent• Not intrinsic!
[C.L., Pasquini (2011)][Hatta (2011)]
[C.L. et al. (2012)]
[Burkardt (2007)][Efremov et al.
(2008,2010)][She, Zhu, Ma (2009)][Avakian et al. (2010)][C.L., Pasquini (2011)]
AULTwist-2
Twist-3
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OAM and origin dependenceRelative IntrinsicNaive
Transverse center of
momentumPhysical interpretation ?
Depends on proton
position
Equivalence
Intrinsic RelativeNaive
Momentum conservation
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Outline Parton distribution zoo
PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation
Proton spin puzzle Decompositions Relations with observables
Model results Overlap representation Wigner distributions
![Page 19: Wigner distributions and quark orbital angular momentum Cdric Lorc and May 14 2012, JLab, Newport News, VA, USA.](https://reader035.fdocuments.us/reader035/viewer/2022062402/5a4d1b937f8b9ab0599c24f8/html5/thumbnails/19.jpg)
Momentum
Fock expansion of the proton state
Fock statesSimultaneous eigenstates
of
Light-front helicity
Overlap representation
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Light-front wave functions
Proton state
Eigenstates of parton light-front helicity
Eigenstates of total OAM
Probability associated with the N, Fock state
Normalization
Overlap representation
gauge
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Fock-state contributions
Overlap representation[C.L., Pasquini
(2011)][C.L. et al. (2012)]
GTMDs
TMDs
GPDsKinetic OAM
Naive canonical OAM
Canonical OAM
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Model calculationsLight-front 3Q
models [C.L., Pasquini (2011)]
Models are not QCD
Truncation of Fock space spoils Lorentz covariance
In model calculations, one would expect but
[Carbonell, Desplanques, Karmanov, Mathiot (1998)]
GTMDs
TMDsGPDs
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Model calculationsUnpold quark in unpold
nucleon [C.L., Pasquini (2011)]
favored
disfavored
Left-right symmetry No net quark OAM
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Model calculationsUnpold quark in longitudinally pold
nucleon
Proton spinu-quark OAMd-quark OAM
[C.L., Pasquini (2011)]
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Model calculations
Proton spinu-quark OAMd-quark OAM
[C.L. et al. (2012)]Unpold quark in longitudinally pold
nucleon
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Quark spinu-quark OAMd-quark OAM
Model calculationsLongitudinally pold quark in unpold
nucleon [C.L., Pasquini (2011)]
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Proton spinu-quark spind-quark spin
Model calculationsLongitudinally pold quark in longitudinally pold
nucleon [C.L., Pasquini (2011)]
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Emerging pictureLongitudinal
[C.L., Pasquini (2011)]
Transverse
[Burkardt (2005)][Barone et al.
(2008)]
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Summary Parton distribution zoo
PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation
Twist-2 phase-space distributions
Proton spin puzzle Decompositions
Kinetic versus canonical Relations with observables
Model results Overlap representation Wigner distributions
Consistent emerging picture
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Incoherent scatteringDVCS vs. SIDIS
DVCS SIDIS
GPDs TMDs
FFs
Factorization
Compton form factor
Cross section • process dependent• perturbative
• « universal »• non-perturbative
hard soft
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GPDs vs. TMDsGPDs TMDs
Correlator
Correlator
Dirac matrix
Wilson line
Off-forward! Forward!
FSIISI
e.g. SIDISe.g. DY
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LC helicity and canonical spinLC helicity Canonical spin
Nucle
on p
olar
izatio
n
Quark polarization Quark polarization
Nucle
on p
olar
izatio
n
[C.L., Pasquini (2011)]
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Interesting relationsModel
relations
* **
***
Flavor-dependent
Flavor-independent
Linear relations Quadratic relation
BagLFQSMLFCQM
S DiquarkAV DiquarkCov. Parton
Quark Target
[Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)][C.L., Pasquini, Vanderhaeghen (2011)][Pasquini & al. (2005-2008)][Ma & al. (1996-2009), Jakob & al. (1997), Bacchetta & al. (2008)][Ma & al. (1996-2009), Jakob & al. (1997)] [Bacchetta & al. (2008)][Efremov & al. (2009)][Meißner & al. (2007)]
*=SU(6)
*
*
*
*
*
*
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Geometrical explanationPreliminaries
Quasi-free quarks Spherical symmetry
[C.L., Pasquini (2011)]
Conditions:
Light-front helicity
Canonical spin
Wigner rotation
(reduces to Melosh rotation in case of FREE
quarks)
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Geometrical explanationAxial symmetry about
z
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Geometrical explanationAxial symmetry about
z