Wigner distributions and light front quark models
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Wigner Distributions and light-front quark models. Barbara Pasquini Pavia U. & INFN, Pavia. i n collaboration with Cédric Lorcé Feng Yuan Xiaonu Xiong IPN and LPT, U. Paris Sud LBNL, Berkeley CHEP, Peking U. Outline.

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Wigner Distributions and light-front quark models

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Wigner distributions and light front quark models

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.


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

TMFFs

Spin densities

GTMDs

PDFs

TMSDs

FFs

GPDs

TMDs

Charges

2D Fourier transform

Wigner distribution

Transverse charge densities

¢= 0

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


W igner d istributions

Transverse

Longitudinal

GTMDs

Wigner Distributions

[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


Lcwf overlap representation

quark-quark correlator

LCWF Overlap Representation

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)]


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


U npol up q uark in u npol p roton

Transverse

Longitudinal

k

T

Unpol. up Quark in Unpol.Proton

[Lorce’, BP, PRD84 (2011)]

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

Generalized Transverse Charge Density

q

b?


U npol up q uark in u npol p roton1

Transverse

Longitudinal

Unpol. up Quark in Unpol.Proton

fixed

=

3Q light-cone model

[Lorce’, BP, PRD84 (2011)]


U npol up q uark in u npol p roton2

Transverse

Longitudinal

Unpol. up Quark in Unpol.Proton

fixed

unfavored

=

favored

3Q light-cone model

[Lorce’, BP, PRD84 (2011)]


Wigner distributions and light front quark models

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)]


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

Quark Orbital Angular Momentum

Proton spin

u-quark OAM

d-quark OAM

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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


Wigner distributions and light front quark models

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’


Wigner distributions and light front quark models

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


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