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Wigner Distributions and light-front quark models

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Wigner Distributionsandlight-front quark models

Barbara Pasquini

Pavia U. & INFN, Pavia

in collaboration with

Cédric Lorcé Feng Yuan XiaonuXiongIPN and LPT, U. Paris Sud LBNL, Berkeley CHEP, Peking U.

Outline

Generalized Transverse Momentum Dependent Parton Distributions (GTMDs)

FT b

Wigner DistributionsParton distributions in the Phase Space

Results in light-front quark models

Quark Orbital Angular Momentum from:

- Wigner distributions
- Pretzelosity TMD
- GPDs

Generalized TMDs and Wigner Distributions

[Meißner, Metz, Schlegel (2009)]

GTMDs

Quark polarization

4 X 4 =16 polarizations 16 complex GTMDs (at twist-2)

Nucleon polarization

»: fraction of longitudinal momentum transfer

x: average fraction of quark longitudinal momentum

Fourier transform

¢: nucleon momentum transfer

16 real Wigner distributions

[Ji (2003)]

[Belitsky, Ji, Yuan (2004)]

k?: average quark transverse momentum

TMFFs

Spin densities

GTMDs

PDFs

TMSDs

FFs

GPDs

TMDs

Charges

2D Fourier transform

Wigner distribution

Transverse charge densities

¢= 0

[ Lorce, BP, Vanderhaeghen, JHEP05 (2011)]

Transverse

Longitudinal

GTMDs

[Wigner (1932)]

[Belitsky, Ji, Yuan (04)]

[Lorce’, BP (11)]

QM

QFT (Breit frame)

QFT (light cone)

Fourier conjugate

Fourier conjugate

Heisenberg’s uncertainty relations

Quasi-probabilistic

- real functions, but in general not-positive definite

GPDs

TMDs

- quantum-mechanical analogous of classical density on the phase space

correlations of quark momentum and position in the transverse planeas function of quark and nucleon polarizations

one-body density matrix in phase-space in terms of overlap of light-cone wf (LCWF)

Third 3D picture with probabilistic interpretation !

- not directly measurable in experiments

needs phenomenological models with input from experiments on GPDs and TMDs

No restrictions from Heisenberg’s uncertainty relations

quark-quark correlator

LCWF:

invariant under boost, independent of P

internal variables:

[Brodsky, Pauli, Pinsky, ’98]

(» =0)

momentum wf

spin-flavor wf

rotation from canonical spin to light-cone spin

Bag Model, LCÂQSM, LCCQM, Quark-Diquarkand Covariant Parton Models

Common assumptions :

- No gluons
- Independent quarks

[Lorce’, BP, Vanderhaeghen (2011)]

Light-Cone Helicity and Canonical Spin

Canonical boost

Light-cone boost

modeldependent:

for k?! 0, the rotation reduced to the identity

LC helicity

Canonical spin

Light-Cone Constituent Quark Model

- momentum-space wf

[Schlumpf, Ph.D. Thesis, hep-ph/9211255]

parameters fitted to anomalous magnetic moments of the nucleon

: normalization constant

- spin-structure:

free quarks

(Melosh rotation)

- SU(6) symmetry

Applications of the model to:

GPDs and Form Factors: BP, Boffi, Traini (2003)-(2005);TMDs: BP, Cazzaniga, Boffi (2008); BP, Yuan (2010);

Azimuthal Asymmetries: Schweitzer, BP, Boffi, Efremov (2009)GTMDs:Lorce`, BP, Vanderhaeghen (2011)

typical accuracy of ¼ 30 % in comparison with exp. datain the valence region, but it

violates Lorentz symmetry

Transverse

Longitudinal

k

T

[Lorce’, BP, PRD84 (2011)]

fixed angle between k? and b? and fixed value of |k?|

Generalized Transverse Charge Density

q

b?

Transverse

Longitudinal

fixed

=

3Q light-cone model

[Lorce’, BP, PRD84 (2011)]

Transverse

Longitudinal

fixed

unfavored

=

favored

3Q light-cone model

[Lorce’, BP, PRD84 (2011)]

up quark

down quark

- left-right symmetry of distributions ! quarks are as likely to rotate clockwise as to rotate anticlockwise
- up quarks are more concentrated at the center of the proton than down quark

unfavored

- integrating over b ? transverse-momentum density

favored

Monopole

Distributions

- integrating over k ?

charge density in the transverse plane b?

[Miller (2007); Burkardt (2007)]

Unpol. quark in long. pol. proton

fixed

Proton spin

u-quark OAM

- projection to GPD and TMD is vanishing! unique information on OAM from Wigner distributions

d-quark OAM

Quark Orbital Angular Momentum

[Lorce’, BP, PRD84(2011)]

[Lorce’, BP, Xiong, Yuan:arXiv:1111.4827] [Hatta:arXiv:111.3547}

Definition of the OAM

OAM operator :

Unambiguous in absence of gauge fields

state normalization

No infinite normalization constants

No wave packets

Wigner distributionsfor unpol. quark in long. pol. proton

Quark Orbital Angular Momentum

Proton spin

u-quark OAM

d-quark OAM

[Lorce’, BP, Xiong, Yuan:arXiv:1111.4827]

Quark OAM: Partial-Wave Decomposition

eigenstate of total OAM

Lzq = ½ - Jzq

Lzq =1

Lzq =2

Lzq = -1

Lzq =0

Jzq

:probability to find the proton in a state with eigenvalue of OAM Lz

TOTAL OAM (sum over three quark)

squared of partial wave amplitudes

Quark OAM: Partial-Wave Decomposition

distribution in x of OAM

TOT

up

down

Lz=0

Lz=-1

Lz=+1

Lz=+2

Lorce,B.P., Xiang, Yuan, arXiv:1111.4827

Quark OAM from Pretzelosity

“pretzelosity”

transv. pol. quarks in transv. pol. nucleon

model-dependent relation

first derived in LC-diquark model and bag model

[She, Zhu, Ma, 2009; Avakian, Efremov, Schweitzer, Yuan, 2010]

valid in all quark models with spherical symmetry in the rest frame

chiral even and charge even

chiral odd and charge odd

[Lorce’, BP, arXiv:1111.6069]

no operator identity

relation at level of matrix elements of operators

Light-Cone Quark Models

- No gluons
- Independent quarks
- Sphericalsymmetry in the nucleon rest frame

symmetricmomentum wf

rotation from canonical spin to light-cone spin

spin-flavor wf

non-relativistic axial charge

non-relativistic tensor charge

spherical symmetry in the rest frame

the quark distribution does not depend on the direction of polarization

Quark OAM

- from Wigner distributions (Jaffe-Manohar)

- from GPDs: Ji’s sum rule

“pretzelosity”

- from TMD

transv. pol. quarks in transv. pol. nucleon

model-dependent relation

TMD

LCWF overlap representation

GTMDs Jaffe-Manohar

GPDsJi sum rule

sum over all parton contributions

Conservation of transverse momentum:

Conservation of longitudinal momentum

LCWFs are eigenstates of total OAM

0

1

For totalOAM

pretzelosity

Jaffe-Manohar

Ji

what is the origin of the differences for the contributions from the individual quarks?

OAM depends on the origin

But if

~

transverse center of momentum

~

???

Talk of Cedric Lorce’

Summary

- GTMDs $ Wigner Distributions

- the most complete information on partonic structure of the nucleon

- Results for Wigner distributions in the transverse plane

- Orbital Angular Momentum from phase-space average with Wigner distributions

- rigorous derivation for quark contribution (no gauge link)

- Orbital Angular Momentum from pretzelosity TMD

- non-trivial correlations between b? and k? due to orbital angular momentum

- model-dependent relation valid in all quark model with spherical symmetry in the rest frame

- LCWF overlap representations of quark OAM from Wigner distributions, TMD and GPDs

- they are all equivalent for the total-quark contribution to OAM, but differ forthe individual quark contribution