Cédric Lorcé

Cédric LorcéIPN Orsay - LPT Orsay

Parton Wigner Distributions of the

nucleon

June 25 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy

The outline

• Zoo of parton distribution functions• Physical interpretation• Wigner distributions and OAM• Model calculations• Conclusions

The outline


The charges

Charges

• Polarization

Depends on :

VectorParton number

TensorParton

transversity

AxialParton helicity

DIS

The parton distribution functions (PDFs)

PDFs

Charges

• Polarization• Longitudinal momentum (fraction)

Depends on :

PDFs

Elastic scattering

The form factors (FFs)

FFsPDFs

Charges

• Polarization• Longitudinal momentum (fraction)• Momentum transfer

Depends on :

FFs

DVCS

The generalized PDFs (GPDs)

GPDs

FFsPDFs

Charges

• Polarization• Longitudinal momentum (fraction)• Momentum transfer

Depends on :

GPDs

SIDIS

The transverse momentum-dependent PDFs (TMDs)

• Polarization• Longitudinal momentum (fraction)• Momentum transfer• Transverse momentum

Depends on :

TMDs

No direct connection

TMDs

FFsPDFs

Charges

GPDs

???

The generalized TMDs (GTMDs)


Depends on :GTMDs

TMDs

FFsPDFs

Charges

GPDs

GTMDs

TMCs

TMFFs

GTMDs

TMDs

???

The complete zoo

FFsPDFs

Charges


Depends on :

GPDs

GTMDs

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

The double parton scattering

[Thürman, Master thesis (2012)]

DPDFs

DPDFs

• Polarization• Longitudinal momentum (fraction)• Momentum transfer• Transverse momentum• Inter-parton distance

Depends on :

[Diehl, Ostermeier, Schäfer (2012)]

The outline


The physical interpretation

Initial/final

Position

Momentum

Average/difference

Fourier-conjugated variables


Breit frame

Lorentz contraction

Creation/annihilation of pairs

Position w.r.t. the CM

Non-relativistic !

[Ernst, Sachs, Wali (1960)][Sachs (1962)]


Drell-Yan frame

Lorentz contraction

Creation/annihilation of pairs

Position w.r.t. the center of momentum

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


Quark Wigner operator

Canonical momentum

• Either fix the gauge such that , i.e. work with + boundary condition

Dirac matrix ~ quark

polarization

Wilson line

• Or split the Wilson line to form Dirac variables


Quark Wigner operator

Fixed light-front time

No need for time-ordering !

Non-relativistic Wigner distribution

Relativistic Wigner distribution

[Ji (2003)][Belitsky, Ji, Yuan (2004)]

[C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]

3+3D

2+3DGTMDs

The phase-space picture

GTMDs

TMDs

FFsPDFs

Charges

GPDs

2+3D

2+1D

2+0D

0+3D

0+1D

The outline


The phase-space distribution

Wigner distribution

Probabilistic interpretation

Expectation value

Heisenberg’s uncertainty

relations

Position space

Momentum space

Phase space

Galilei covariant

• Either non-relativistic• Or restricted to transverse position

[Wigner (1932)][Moyal (1949)]

The quark orbital angular momentum

GTMD correlator

[C.L., Pasquini (2011)]

Wigner distribution

Orbital angular momentum

[Meißner, Metz, Schlegel (2009)]

Parametrization

Unpolarized quark density

[Meißner, Metz, Schlegel (2009)]

The parametrization @ twist-2 and =0Parametrization :

GTMDs

TMDs GPDs

Monopole Dipole Quadrupole

Nu

cle

on

pola

riza

tion

Quark polarization

FSIISI

The path dependence

Orbital angular momentum

[C.L., Pasquini, Xiong, Yuan (2012)][Hatta (2012)]

[Ji, Xiong, Yuan (2012)][C.L. (2013)]

Drell-Yan

Reference point

SIDIS

Canonical

[Jaffe, Manohar (1990)] [Ji (1997)]Kineti

c

The proton spin decompositions

• Does not satisfy canonical relations• Incomplete decomposition

• Gauge-invariant decomposition• Accessible in DIS and DVCS

Pros:

Cons:

News:[Wakamatsu (2009,2010)]

• Complete decomposition

Pros:

Cons:

• Satisfies canonical relations• Complete decomposition

• Gauge-variant decomposition• Missing observables for the OAM

News:[Chen et al.

(2008)]

• Gauge-invariant extension

• OAM accessible via Wigner distributions [C.L., Pasquini (2012)]

[C.L., Pasquini, Xiong, Yuan(2012)]

[Hatta (2012)]

Canonical

Kinetic[Jaffe, Manohar (1990)] [Ji (1997)]

[C.L. (2013)][Leader, C.L. (in preparation)]

Reviews :

The outline


Overlap representation

Momentum Polarization

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

Light-front quark models Wigner rotation

The light-front overlap representation

Wigner distribution of unpolarized quark in unpolarized nucleon


The model results

favored

disfavored

Left-right symmetry No net quark OAM

Distortion induced by the nucleon longitudinal polarization


The model results

Proton spin

u-quark OAM

d-quark OAM

Average transverse quark momentum in a longitudinally polarized nucleon

[C.L., Pasquini, Xiong, Yuan (2012)]

The model results

« Vorticity »

Distortion induced by the quark longitudinal polarization


The model results

Quark spin

u-quark OAM

d-quark OAM

Quark spin-nucleon spin correlation


The model results

Proton spin

u-quark spin

d-quark spin


The model results

The emerging picture


[Burkardt (2005)][Barone et al.

(2008)]

Longitudinal

Transverse

The canonical and kinetic OAM

Quark canonical OAM

Quark naive canonical OAM

[Burkardt (2007)][Efremov et al.

(2008,2010)][She, Zhu, Ma (2009)][Avakian et al. (2010)][C.L., Pasquini (2011)]

[C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]

[Hatta (2012)]

Model-dependent !

Quark kinetic OAM

[Ji (1997)]

[Penttinen et al. (2000)][Kiptily, Polyakov (2004)]

[Hatta (2012)]

but

No gluons and not QCD EOM !


Pure twist-3

The conclusions

• Twist-2 parton distributions provide multidimensional pictures of the nucleon

• Relativistic phase-space distributions exist. Open question: how to access them?

• Both kinetic (Ji) and canonical (Jaffe-Manohar) are measurable (twist-2 and twist-3)

• Model calculations can test spin sum rules

Backup slides

OAM and origin dependenceRelative IntrinsicNaive

Transverse center of

momentumPhysical interpretation ?

Depends on proton

position

Equivalence

Intrinsic RelativeNaive

Momentum conservation

Momentum

Fock expansion of the proton state

Fock statesSimultaneous eigenstates

of

Light-front helicity


Light-front wave functions

Proton state

Eigenstates of parton light-front helicity

Eigenstates of total OAM

Probability associated with the N, Fock state

Normalization


gauge

Fock-state contributions



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

GTMDs

TMDs

GPDsKinetic OAM

Naive canonical OAM

Canonical OAM

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 spin

LC helicity Canonical spin

Nu

cleon

pola

riza

tion

Quark polarization Quark polarization

Nu

cleon

pola

riza

tion


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


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

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Cédric Lorcé