Wigner distributions and quark orbital

angular momentum

Cédric Lorcé

and

May 14 2012, JLab, Newport News, VA, USA

Outline Parton distribution zoo

PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation

Proton spin puzzle Decompositions Relations with observables

Model results Overlap representation Wigner distributions

Outline Parton distribution zoo

PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation

Proton spin puzzle Decompositions Relations with observables

Model results Overlap representation Wigner distributions

Parton distribution zooCharges

Charges

Partonic interpretation (twist-2)

Vector

Tensor

Axial

0D Picture

Parton distribution zoo

Partonic interpretation (twist-2)

DIS

Parton Distribution Functions

Charges

PDFs

1D Picture

Parton distribution zoo

Partonic interpretation (twist-2)

ES

Form Factors

PDFs

Charges

FFs

2D Picture

Impact parameter

Transverse center of

momentum

DVCS

Parton distribution zoo

Partonic interpretation (twist-2)

Generalized PDFs

GPDs

Charges

PDFs FFs

3D Picture

[Soper (1977)][Burkardt (2000,2003)]

[Diehl, Hägler (2005)]

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

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

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)

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

Twist-2 structureGTMDs

Nuc

leon

pol

ariz

atio

n

Quark polarization

Monopole Dipole Quadrupole

Naive OAM

[Meißner, Metz, Schlegel (2009)]

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

Outline Parton distribution zoo

PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation

Proton spin puzzle Decompositions Relations with observables

Model results Overlap representation Wigner distributions

• 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)]

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

OAM and origin dependenceRelative IntrinsicNaive

Transverse center of

momentumPhysical interpretation ?

Depends on proton

position

Equivalence

Intrinsic RelativeNaive

Momentum conservation

Outline Parton distribution zoo

PDFs, FFs, GPDs, TMDs, GTMDs Partonic interpretation

Proton spin puzzle Decompositions Relations with observables

Model results Overlap representation Wigner distributions

Momentum

Fock expansion of the proton state

Fock statesSimultaneous eigenstates

of

Light-front helicity

Overlap representation

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

Fock-state contributions

Overlap representation[C.L., Pasquini

(2011)][C.L. et al. (2012)]

GTMDs

TMDs

GPDsKinetic OAM

Naive canonical OAM

Canonical OAM

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

Model calculationsUnpold quark in unpold

nucleon [C.L., Pasquini (2011)]

favored

disfavored

Left-right symmetry No net quark OAM

Model calculationsUnpold quark in longitudinally pold

nucleon

Proton spinu-quark OAMd-quark OAM

[C.L., Pasquini (2011)]

Model calculations

Proton spinu-quark OAMd-quark OAM

[C.L. et al. (2012)]Unpold quark in longitudinally pold

nucleon

Quark spinu-quark OAMd-quark OAM

Model calculationsLongitudinally pold quark in unpold

nucleon [C.L., Pasquini (2011)]

Proton spinu-quark spind-quark spin

Model calculationsLongitudinally pold quark in longitudinally pold

nucleon [C.L., Pasquini (2011)]

Emerging pictureLongitudinal

[C.L., Pasquini (2011)]

Transverse

[Burkardt (2005)][Barone et al.

(2008)]

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

Incoherent scatteringDVCS vs. SIDIS

DVCS SIDIS

GPDs TMDs

FFs

Factorization

Compton form factor

Cross section • process dependent• perturbative

• « universal »• non-perturbative

hard soft

GPDs vs. TMDsGPDs TMDs

Correlator

Correlator

Dirac matrix

Wilson line

Off-forward! Forward!

FSIISI

e.g. SIDISe.g. DY

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)]

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)

*

*

*

*

*

*

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)

Geometrical explanationAxial symmetry about

z

Geometrical explanationAxial symmetry about

z