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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.
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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
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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)]
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GTMDs
Charges
PDFs
[ Lorce, BP, Vanderhaeghen, JHEP05 (2011)]
Wigner distribution
2D Fourier transform
GPDsTMFFs
FFs
Spin densities
Transverse charge densities
¢ = 0
TMDs
TMSDs
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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
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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)
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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
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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
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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
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Longitudinal
Transverse
fixed
3Q light-cone model
=
[Lorce’, BP, PRD84 (2011)]
Unpol. up Quark in Unpol. Proton
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Longitudinal
Transverse
Unpol. up Quark in Unpol. Proton
fixed
3Q light-cone model
=favored
unfavored
[Lorce’, BP, PRD84 (2011)]
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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
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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
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[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
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[Lorce’, BP, Xiong, Yuan:arXiv:1111.4827]
Proton spin
u-quark OAM
d-quark OAM
Quark Orbital Angular Momentum Quark Orbital Angular Momentum
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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)
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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
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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
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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
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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
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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
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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
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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