Wigner DistributionsWigner Distributions

andand

light-front quark modelslight-front quark models

Barbara PasquiniPavia U. & INFN, Pavia

in collaboration with

Cédric Lorcé Feng Yuan Xiaonu Xiong

IPN and LPT, U. Paris Sud LBNL, Berkeley CHEP, Peking U.

OutlineOutline

Generalized Transverse Momentum Dependent Parton Distributions (GTMDs)

Wigner DistributionsParton distributions in the Phase Space

FT b

Results in light-front quark models

Quark Orbital Angular Momentum from: Wigner distributions Pretzelosity TMD GPDs

Generalized TMDs and Wigner DistributionsGeneralized TMDs and Wigner Distributions

GTMDsGTMDs

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

[Meißner, Metz, Schlegel (2009)]

Quark polarization

Nucleon polarization

x: average fraction of quark

longitudinal momentum

»: fraction of longitudinal momentum transfer

k?: average quark transverse momentum

¢: nucleon momentum transfer

Fourier transform

16 real Wigner distributions[Ji (2003)]

[Belitsky, Ji, Yuan (2004)]

GTMDs

Charges

PDFs

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

Wigner distribution

2D Fourier transform

GPDsTMFFs

FFs

Spin densities

Transverse charge densities

¢ = 0

TMDs

TMSDs

Longitudinal

Transverse

Wigner Distributions

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

[Lorce’, BP (11)]

QMQFT (Breit frame)QFT (light cone)

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

real functions, but in general not-positive definite

quantum-mechanical analogous of classical density on the phase space

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

not directly measurable in experiments

needs phenomenological models with input from experiments on GPDs and TMDs

GPDs

TMDs

GTMDs Third 3D picture with probabilistic interpretation !

No restrictions from Heisenberg’s uncertainty relations

Heisenberg’s uncertainty relations

Quasi-probabilistic

Fourier conjugate

Fourier conjugate

LCWF Overlap Representation

Common assumptions : No gluons Independent quarks

Bag Model, LCÂQSM, LCCQM, Quark-Diquark and Covariant Parton Models

[Lorce’, BP, Vanderhaeghen (2011)]

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

invariant under boost, independent of P

internal variables:

LCWF:

[Brodsky, Pauli, Pinsky, ’98]

quark-quark correlator(» =0)

Canonical boost

Light-cone boost

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

LC helicityCanonical spin

model dependent:

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

parameters fitted to anomalous

magnetic moments of the nucleon : normalization constant

[Schlumpf, Ph.D. Thesis,

hep-ph/9211255]

momentum-space wf

SU(6) symmetry

Light-Cone Constituent Quark ModelLight-Cone Constituent Quark Model

spin-structure:

(Melosh rotation)

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) GTMDs: Lorce`, BP, Vanderhaeghen (2011)

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

Longitudinal

Transverse

k T

b?

Generalized Transverse Charge Density

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

[Lorce’, BP, PRD84 (2011)]

Unpol. up Quark in Unpol. Proton

Longitudinal

Transverse

fixed

3Q light-cone model

=

[Lorce’, BP, PRD84 (2011)]

Unpol. up Quark in Unpol. Proton

Longitudinal

Transverse

Unpol. up Quark in Unpol. Proton

fixed

3Q light-cone model

=favored

unfavored

[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

favored

unfavored

Proton spin

u-quark OAM

d-quark OAM

Unpol. quark in long. pol. proton

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

fixed

[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 Quark Orbital Angular Momentum

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

Proton spin

u-quark OAM

d-quark OAM

Quark Orbital Angular Momentum Quark Orbital Angular Momentum

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

eigenstate of total OAM

squared of partial wave amplitudes

TOTAL OAM (sum over three quark)

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, arXiv:1111.4827

Quark OAM from PretzelosityQuark OAM from Pretzelosity

model-dependent relation

“pretzelosity”transv. pol. quarks in transv. pol. nucleon

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

first derived in LC-diquark model and bag model

valid in all quark models with spherical symmetry in the rest frame [Lorce’, BP, arXiv:1111.6069]

chiral even and charge even chiral odd and charge odd

no operator identityrelation at level of matrix elements of operators

No gluons Independent quarks Spherical symmetry in the nucleon rest frame

Light-Cone Quark Models

symmetricmomentum wf

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

non-relativistic axial charge non-relativistic tensor charge

spherical symmetry in the rest frame

Quark OAMQuark OAM

from GPDs: Ji’s sum rule

from Wigner distributions (Jaffe-Manohar)

from TMD

model-dependent relation

“pretzelosity”transv. pol. quarks in transv. pol. nucleon

GPDsJi sum rule

GTMDs Jaffe-Manohar

LCWF overlap representation

TMD

LCWFs are eigenstates of totaltotal OAM

For totaltotal OAM

Conservation of transverse momentum: Conservation of longitudinal momentum

sum over all parton contributions

0 1

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

transverse center of

momentumJaffe-Manohar

Ji

pretzelosity

???

~

~

Talk of Cedric Lorce’

OAM depends on the origin

But if

SummarySummary GTMDs $ Wigner Distributions

- the most complete information on partonic structure of the nucleon

Results for Wigner distributions in the transverse plane

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

Orbital Angular Momentum from phase-space average with Wigner distributions

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

- rigorous derivation for quark contribution (no gauge link)

Orbital Angular Momentum from pretzelosity TMD

- 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