Wigner DistributionsWigner Distributionsin in

Light-Cone Quark ModelsLight-Cone Quark Models

Barbara PasquiniPavia U. & INFN, Pavia

in collaboration with

Cédric LorcéMainz U. & INFN, Pavia

OutlineOutline

Generalized Transverse Momentum Dependent Parton Distributions (GTMDs)

Wigner Distributions

FT b

Results in light-cone quark models

unpolarized quarks in unpolarized nucleon

Quark Orbital Angular Momentum

Generalized TMDsGeneralized TMDs

GTMDsGTMDs

Complete parametrization : 16 GTMDs at twist-2

[Meißner, Metz, Schlegel (2009)]

Fourier Transform : 16 Wigner distributions

[Belitsky, Ji, Yuan (2004)]

x: average fraction of quark

longitudinal momentum

»: fraction of longitudinal momentum transfer

k?: average quark transverse momentum

¢: nucleon momentum transfer

GTMDs

Charges

PDFs

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

Wigner distribution

2D Fourier transform

GPDsTMFFs

TMSDs FFs

¢ = 0

TMDs Spin densities

Transverse charge densities

Longitudinal

Transverse

Wigner Distributions

[Wigner (1932)][Belitsky, Ji, Yuan (04)]

[Lorce’, BP: PRD84 (11)]

QMQFT (Breit frame)QFT (light cone)

GPDs

TMDs

GTMDs

Heisenberg’s uncertainty relations

Quasi-probabilistic

Third 3D picture with probabilistic

interpretation!No restrictions from Heisenberg’s

uncertainty relations

Wigner DistributionsWigner Distributions

real functions, but in general not-positive definite

Wigner distributions in QCD: at »=0 ! diagonal in the Fock-space

N N

N=3 ! overlap of quark light-cone wave-functions

not probabilistic interpretation

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

no known experiments can directly measure them ! needs phenomenological models

Light-Cone Quark ModelsLight-Cone Quark Models

invariant under boost, independent of P

internal variables:

LCWF:

[Brodsky, Pauli, Pinsky, ’98]

Bag Model, ÂQSM, LCQM, Quark-Diquark and Covariant Parton Models

momentum wf spin-flavor wf rotation from canonical spin to light-cone spin

Common assumptions : No gluons Independent quarks

[ Lorce, BP, Vanderhaeghen, JHEP 05 (2011)Lorce, BP, arXiv:1104.5651 ]

Bag Model, ÂQSM, LCQM, Quark-Diquark and Covariant Parton Models

parameters fitted to anomalous

magnetic moments of the nucleon : normalization constant

[Schlumpf, Ph.D. Thesis,

hep-ph/9211255]

momentum-space wf

spin-structure:

(Melosh rotation)

SU(6) symmetry

Light-Cone Constituent Quark ModelLight-Cone Constituent Quark Model

free quarks

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)

Example: Unpol. up Quark in Unpol. Proton

Longitudinal

Transverse

(1 out of 16)

k T

b?

Generalized Transverse Charge Density

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

[Lorce’, BP, PRD84 (2011)]

Longitudinal

Transverse

Example: Unpol. up Quark in Unpol. Proton

fixed

3Q light-cone model

=

(1 out of 16)

[Lorce’, BP, PRD84 (2011)]

Longitudinal

Transverse

Example: Unpol. up Quark in Unpol. Proton

fixed

3Q light-cone model

=

(1 out of 16)

favored

unfavored

[Lorce’, BP, PRD84 (2011)]

Example: Unpol. up Quark in Unpol. Proton (1 out of 16)

Longitudinal

Transverse

0.1 GeV²

0.2 GeV²

0.3 GeV²

0.4 GeV²

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

integrating over b ? transverse-momentum density

integrating over k ? charge density in the transverse plane b? [Miller (2007); Burkardt (2007)]

Monopole

Distributions

unpolarized u and d quarks in unpolarized proton

proton neutron

[Miller (2007); Burkardt (2007)]

charge distribution in the transverse plane

Transverse Charge DistributionTransverse Charge Distribution

Quark Orbital Angular Momentum Quark Orbital Angular Momentum

Wigner distribution for Unpolarized quark in a Longitudinally pol. nucleon

Orbital Motion in the Transverse Space

--0.6 0 0.6 -0

.6

0

0.6

-0.6

0

0

.6

-0.6 0 0.6

up quark: down quark:

Lorce,B.P., Xiang, Yuan, in preparation

Lzq = ½ - Jz

q Lzq =2Lz

q =1Lzq =0Lz

q = -1

Jzq

Quark OAM: Partial-Wave DecompositionQuark OAM: Partial-Wave Decomposition

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

squared of LCWFs

eigenstate of OAM

Quark OAM: Partial-Wave DecompositionQuark OAM: Partial-Wave Decomposition

OAM Lz=0 Lz=-1 Lz=+1 Lz=+2 TOT

UP 0.013 -0.046 0.139 0.025 0.131

DOWN -0.013 -0.090 0.087 0.011 -0.005

UP+DOWN 0 -0.136 0.226 0.036 0.126

<P" |P"> 0.62 0.136 0.226 0.018 1

up downTOT

Lz=0

Lz=-1

Lz=+2

Lz=+1

distribution in x of OAM

Lorce,B.P., Xiang, Yuan, in preparation

Quark Orbital Angular MomentumQuark Orbital Angular Momentum

from Wigner distributions: intrinsic OAM with respect to centre of momentum

from TMD: OAM with respect to the origin of axis in the transverse plane

light-cone diquark model [She, Zhu, Ma, PRD79 (2009)]

bag model [Avakian, Efremov, Schweitzer, Yuan, PRD81 (2010)]

model-dependent relation

all quark models with spherical symmetry in the rest frame [Lorce’, BP, Xiang, Yuan, in preparation]

from GPDs: Ji’s sum rule [Ji, PRL78 (1997)]

What is the relations among these three definitions?

the three definitions refer to OAM calculated with respect to different points

OAM UP DOWN TOT

0.131 -0.005 0.126

0.169 -0.042 0.126

0.071 0.055 0.126

n-th parton contribution:

Total-quark contribution:

SummarySummary

GTMDs $ Wigner Distributions

- the most complete information on partonic structure of the nucleon

General Formalism for 3-quark contribution to GTMDs

- applicable for large class of models: LCQMs, ÂQSM, Bag model

Results for Wigner distributions in the transverse plane

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

- anisotropic distribution in k? for unpolarized quarks in unpolarized nucleon

Orbital Angular Momentum from phase-space average with Wigner distributions

- comparison with different definition from TMDs and GPDs ! they are all equivalent for the total-quark contribution to OAM

proton neutron

[Miller (2007); Burkardt (2007)]

charge distribution in the transverse plane

Integrating over b ?

Integrating over k ? charge density in the transverse plane b?

LC helicity amplitudes

( ¸’ ¸ )

Independent quarks

LC helicity canonical spin rotation around an axis orthogonal to z and k

LC helicity $ canonical spin

quark

nucleon ( ¤’ ¤ )

LC helicity canonical spin rotation around an axis orthogonal to z and k

Light-Cone CQM Chiral Quark-Soliton Model Bag Model

(Melosh rotation)

Light-Cone Helicity and Canonical SpinLight-Cone Helicity and Canonical Spin

16 GTMDs

º ! quark polarization

¹! nucleon polarization

Active quark :

Spectator quarks :

Model Independent Spin Structure

( )

Backup

µ 0 ¼/2 ¼ 3/2 ¼

k T

k T

k T

Unpolarized u quark in unpolarized proton

,k T fixed

k T

0.1 GeV

0.2 GeV

0.3 GeV

0.4 GeV

= +

k T

k T fixed

Unpolarized u quark in unpolarized proton

Generalized Transverse Charge DensitiesGeneralized Transverse Charge Densities

µ = ¼/2

µ = 0

Wigner functionWigner function for transversely pol. quark in longitudinally pol. nucleon for transversely pol. quark in longitudinally pol. nucleon

µ = ¼/2

µ = 0

T-odd

k Tb

sx

k T,µ fixed

Wigner functionWigner function for transversely pol. quark in longitudinally pol. nucleon for transversely pol. quark in longitudinally pol. nucleon

Dipole

µ = 0

µ = ¼/2

Monopole + Dipole

k T fixed sx

Monopole

¼ 3/2 ¼

u quark pol. in x direction in longitudinally pol.proton

,k T fixed

0.1 GeV

0.2 GeV

0.3 GeV

0.4 GeV

k T

k T

k T

k T

µ 0 ¼/2

Integrating over b ? density in the transverse plane k? of transversely pol. u and d quark in longitudinally pol nucleon

sxupup downdown

BP, Cazzaniga, Boffi, PRD78 (2008)Lorce`, BP, in preparation

Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009)

Integrating over k ?