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Wigner Distributions in Light-Cone Quark Models. Barbara Pasquini Pavia U. & INFN, Pavia. i n collaboration with Cédric Lorcé Mainz U. & INFN, Pavia. Outline. Generalized Transverse Momentum Dependent Parton Distributions (GTMDs). FT b . Wigner Distributions. - PowerPoint PPT Presentation
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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 ?