Wigner Distributions
and
light-front quark models
Barbara Pasquini
Pavia U. & INFN, Pavia
in collaboration with
Cédric Lorcé
IPN and LPT, U. Paris Sud
Feng Yuan
LBNL, Berkeley
Xiaonu Xiong
CHEP, Peking U.
Outline
Generalized Transverse Momentum Dependent Parton Distributions (GTMDs)
FT   b
Wigner Distributions
Parton 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
Nucleon
polarization
4 X 4 =16 polarizations
x: average fraction of quark
longitudinal momentum
16 complex GTMDs (at twist-2)
»: fraction
of longitudinal
Fourier
transform
momentum transfer
16 real Wigner distributions
k?: average quark transverse momentum
[Ji (2003)]
¢: nucleon momentum
[Belitsky, Ji, transfer
Yuan (2004)]
2D Fourier
transform
GTMDs
Wigner distribution
TMDs
TMSDs
TMFFs
PDFs
GPDs
FFs
Spin densities
Transverse charge
densities
¢ =0
Charges
[ Lorce, BP, Vanderhaeghen, JHEP05 (2011)]
Wigner Distributions
Transverse
[Wigner (1932)]
[Belitsky, Ji, Yuan (04)]
[Lorce’, BP (11)]
QM
QFT (Breit frame)
QFT (light cone)
Fourier conjugate
Longitudinal
Fourier conjugate
Heisenberg’s
uncertainty relations
Quasi-probabilistic
 real functions, but in general not-positive definite
correlations of quark momentum and position in the transverse plane
as function GPDs
of quark and nucleon polarizations
 quantum-mechanical analogous of classical density on the phase space
TMDs
one-body density matrix in phase-space in terms of overlap of light-cone wf (LCWF)
Third 3D picture with
probabilistic interpretation !
GTMDs
 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
LCWF:
invariant under boost, independent of P
internal variables:
[Brodsky, Pauli, Pinsky, ’98]
quark-quark correlator
(» =0)
momentum wf
spin-flavor wf
rotation from canonical
spin to light-cone spin
Bag Model, LCÂQSM, LCCQM, Quark-Diquark and Covariant Parton Models
Common assumptions :
 No gluons
 Independent quarks
[Lorce’, BP, Vanderhaeghen (2011)]
Light-Cone Helicity and Canonical Spin
Canonical
boost
Light-cone
boost
model dependent:
LC helicity
Canonical spin
for k? ! 0, the rotation reduced to the identity
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
 SU(6) symmetry
(Melosh rotation)
Applications of the model to:
typical accuracy of ¼ 30 %
in comparison with exp. data
in the valence region, but it
violates Lorentz symmetry
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)
Unpol. up Quark in Unpol. Proton
[Lorce’, BP, PRD84 (2011)]
Transverse
Longitudinal
Generalized Transverse Charge Density
fixed angle between k? and b? and fixed value of |k?|
T
k
b?
q
Unpol. up Quark in Unpol. Proton
Transverse
Longitudinal
fixed
=
3Q light-cone model
[Lorce’, BP, PRD84 (2011)]
Unpol. up Quark in Unpol. Proton
Transverse
Longitudinal
fixed
unfavored
3Q light-cone model
[Lorce’, BP, PRD84 (2011)]
favored
=
up quark
down quark
unfavored
favored
 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
Monopole
 integrating over k ?
charge density in the transverse plane b?
[Miller (2007); Burkardt (2007)]
Distributions
Unpol. quark in long. pol. proton
fixed
Proton spin
u-quark OAM
d-quark OAM
 projection to GPD and TMD is vanishing
! unique information on OAM from Wigner distributions
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 distributions
for unpol. quark in
long. pol. proton
Quark Orbital Angular Momentum
Proton spin
u-quark OAM
[Lorce’, BP, Xiong, Yuan:arXiv:1111.4827]
d-quark OAM
Quark OAM: Partial-Wave Decomposition
eigenstate of total OAM
Lzq = ½ - Jzq
Lzq = -1
Lzq =0
Lzq =1
Lzq =2
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
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
distribution in x of OAM
TOT
up
Lz=0
Lz=-1
Lz=+1
Lz=+2
Lorce,B.P., Xiang, Yuan, arXiv:1111.4827
down
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]
chiral even and charge even
chiral odd and charge odd
no operator identity
relation at level of matrix
elements of operators
valid in all quark models with spherical symmetry in the rest frame
[Lorce’, BP, arXiv:1111.6069]
Light-Cone Quark Models
 No gluons
 Independent quarks
 Spherical symmetry in the nucleon rest frame
symmetric
spin-flavor wf rotation from canonical
momentum wf
spin to light-cone spin
non-relativistic axial charge
non-relativistic tensor charge
spherical symmetry in the rest frame
Quark OAM
 from Wigner distributions (Jaffe-Manohar)
 from TMD
“pretzelosity”
transv. pol. quarks in transv. pol. nucleon
model-dependent relation
 from GPDs: Ji’s sum rule
LCWF overlap representation
GTMDs
Jaffe-Manohar
TMD
GPDs
Ji sum rule
sum over all parton contributions
Conservation of transverse momentum:
0
LCWFs are eigenstates of total OAM
For total OAM
Conservation of longitudinal momentum
1
what is the origin of the differences for the contributions from the individual quarks?
OAM depends on the origin
But if
pretzelosity
Jaffe-Manohar
Ji
~
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
- non-trivial correlations between b? and k? due to orbital angular momentum
 Orbital Angular Momentum from phase-space average with Wigner distributions
- 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
- they are all equivalent for the total-quark contribution to OAM, but differ for
the individual quark contribution