Transcript Document

Wigner Distributions
in
Light-Cone Quark Models
Barbara Pasquini
Pavia U. & INFN, Pavia
in collaboration with
Cédric Lorcé
Mainz U. & INFN, Pavia
Outline
Generalized Transverse Momentum Dependent Parton Distributions (GTMDs)
FT   b
Wigner Distributions
 Results in light-cone quark models
 unpolarized quarks in unpolarized nucleon
 Quark Orbital Angular Momentum
Generalized TMDs
GTMDs
 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
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: PRD84 (11)]
QM
QFT (Breit frame)
QFT (light cone)
Longitudinal
Heisenberg’s
uncertainty relations
Quasi-probabilistic
GPDs
TMDs
GTMDs
Third 3D picture
with probabilistic
interpretation!
No restrictions from Heisenberg’s
uncertainty relations
Wigner Distributions
 Wigner distributions in QCD:
at »=0 ! diagonal in the Fock-space
N
N
N=3 ! overlap of quark light-cone wave-functions
 real functions, but in general not-positive definite
not probabilistic interpretation
correlations of quark momentum and position in the transverse plane
as function of quark and nucleon polarizations
 no known experiments can directly measure them ! needs phenomenological models
Light-Cone Quark Models
LCWF:
invariant under boost, independent of P
internal variables:
[Brodsky, Pauli, Pinsky, ’98]
momentum wf
spin-flavor wf
rotation from canonical
spin to light-cone spin
Bag Model, ÂQSM, LCQM, Quark-Diquark and Covariant Parton Models
Common assumptions :
 No gluons
 Independent quarks
[ Lorce, BP, Vanderhaeghen, JHEP 05 (2011)
Lorce, BP, arXiv:1104.5651 ]
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
(Melosh rotation)
 SU(6) symmetry
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
(1 out of 16)
Transverse
[Lorce’, BP, PRD84 (2011)]
Longitudinal
Generalized Transverse Charge Density
fixed angle between k? and b? and fixed value of |k?|
T
k
b?
q
Example: Unpol. up Quark in Unpol. Proton
(1 out of 16)
Transverse
Longitudinal
fixed
=
3Q light-cone model
[Lorce’, BP, PRD84 (2011)]
Example: Unpol. up Quark in Unpol. Proton
(1 out of 16)
Transverse
Longitudinal
fixed
unfavored
3Q light-cone model
[Lorce’, BP, PRD84 (2011)]
favored
=
Example: Unpol. up Quark in Unpol. Proton
(1 out of 16)
Transverse
Longitudinal
0.1 GeV²
0.2 GeV²
0.3 GeV²
3Q light-cone model
[Lorce’, BP, PRD84 (2011)]
0.4 GeV²
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
Monopole
 integrating over k ?
charge density in the transverse plane b?
[Miller (2007); Burkardt (2007)]
Distributions
Transverse Charge Distribution
unpolarized u and d quarks in
unpolarized proton
neutron
proton
charge distribution in the
transverse plane
[Miller (2007); Burkardt (2007)]
Quark Orbital Angular Momentum
Wigner distribution
for Unpolarized quark in a Longitudinally pol. nucleon
Orbital Motion in the Transverse Space
0.6
down quark:
-0.6
-0.6
0
0
0.6
up quark:
--0.6
0
0.6
-0.6
Lorce,B.P., Xiang, Yuan, in preparation
0
0.6
Quark OAM: Partial-Wave Decomposition
eigenstate of 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
squared of LCWFs
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, in preparation
down
Quark 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
model-dependent relation
 bag model
 light-cone diquark model
[Avakian, Efremov, Schweitzer, Yuan, PRD81 (2010)]
[She, Zhu, Ma, PRD79 (2009)]
 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?
All three definitions give the same results
for the total-quark contribution to OAM
but not for the individual flavor contribution
the three definitions refer to OAM calculated with respect to different points
n-th parton contribution:
Total-quark contribution:
OAM
UP
DOWN
TOT
0.131
-0.005
0.126
0.169
-0.042
0.126
0.071
0.055
0.126
Summary
 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
- anisotropic distribution in k? for unpolarized quarks in unpolarized nucleon
- non-trivial correlations between b? and k? due to orbital angular momentum
 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
 Integrating over b ?
 Integrating over k ?
charge density in the transverse plane b?
neutron
proton
charge distribution in the
transverse plane
[Miller (2007); Burkardt (2007)]
LC helicity amplitudes
nucleon ( ¤’ ¤ )
quark
( ¸’ ¸ )
LC helicity $ canonical spin
Independent quarks
LC helicity
canonical spin
rotation around an axis
orthogonal to z and k
Light-Cone Helicity and Canonical Spin
LC helicity
Light-Cone CQM
(Melosh rotation)
canonical spin
rotation around an axis
orthogonal to z and k
Chiral Quark-Soliton Model
Bag Model
º ! quark polarization
¹! nucleon polarization
(
16 GTMDs
Active quark :
Spectator quarks :
Model Independent Spin Structure
)
Backup
3/2 ¼
k
k
0.1 GeV
k
0.2 GeV
0.3 GeV
0.4 GeV
k
T
¼
T
¼/2
Unpolarized u quark in
unpolarized proton
T
0
, q fixed
T
µ
T
k
Generalized Transverse Charge Densities
T
k
=
+
µ=0
T
k
fixed
Unpolarized u quark in
unpolarized proton
µ = ¼/2
Wigner function
for transversely pol. quark in longitudinally pol. nucleon
b
k ,µ fixed
k
T
sx
T-odd
q
µ = ¼/2
µ=0
T
Wigner function
for transversely pol. quark in longitudinally pol. nucleon
Dipole
T
k
sx
fixed
Monopole
µ=0
µ = ¼/2
Monopole + Dipole
3/2 ¼
k
k
k
0.1 GeV
0.2 GeV
0.3 GeV
k
T
¼
T
¼/2
u quark pol. in x direction
in longitudinally pol.proton
T
0
, q fixed
T
µ
T
k
0.4 GeV
 Integrating over k ?
 Integrating over b ?
sx
density in the transverse plane k?
of transversely pol. u and d quark in longitudinally pol nucleon
up
BP, Cazzaniga, Boffi, PRD78 (2008)
Lorce`, BP, in preparation
down
Haegler, Musch, Negele, Schaefer,
Europhys. Lett. 88 (2009)