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
 General formalism for the 3-quark contribution to GTMDs
 Results for Wigner distributions in light-cone quark models
 unpolarized quarks in unpolarized nucleon (generalized transverse charge density)
 Relations between GPDs and TMDs
 Relations among TMDs in various 3-quark models
 Orbital Angular Momentum in terms of LCWF: Ji’s vs. Jaffe-Manohar’s definition
Generalized TMDs
GTMDs
 Complete parametrization : 16 GTMDs
[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
Wigner-Ds
FT   b
FT
TMDs
GPDs
spin densities
FT
PDs
Form Factors
charge densities
Wigner Distributions
 Quantum phase-space distribution: most complete information on wave function
quantum molecular dynamics, signal analysis, quantum info, optics, image processing,…
[Wigner, (1932)]
 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
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
Light-cone gauge A+=0
Wilson line r reduces to the identity
Twist-2 operators
º ! quark polarization
¹! nucleon polarization
(
16 GTMDs
)
3-Quark Overlap representation
Quark line :
Model Independent Spin Structure
Active quark :
Spectator quarks :
3-Quark Overlap representation
16 GTMDs
Assumption :

symmetry
[C.Lorce’, B. Pasquini, M. Vanderhaeghen (in preparation)]
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)
Wigner function
for unpolarized quark in unpolarized nucleon
T
k
q
b
k , µ fixed
µ = ¼/2
T
µ=0
[C.Lorce’, B. P. (in preparation)]
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
µ = ¼/2
µ=0
Generalized Transverse Charge Densities
T
k
=
+
µ=0
T
k
fixed
Unpolarized u quark in
unpolarized proton
µ = ¼/2
 Integrating over b ?
TMD
 Integrating over k ?
charge density in the transverse plane b?
unpolarized u and d quarks in
unpolarized proton
neutron
proton
charge distribution in the
transverse plane
[Miller (2007); Burkardt (2007)]
GTMDs
TMDs
GPDs
GPDs and TMDs probe the same overlap of quark LCWFs in different kinematics
nucleon
quark
at »=0
UU
UT
LL
TU
TT
TT
LT
0
TL
0
Relations between GPDs and TMDs in Quark Models
GPDs and TMDs probe the same overlap of LCWFs in different kinematics
there exist relations in particular kinematical limits?
Trivial Relations
UU
LL
TT
Non-Trivial Relations
TT
valid also in spectator model : Meissner, Metz, Goeke, PRD76(2007)
(with a factor 3 instead of 2)
Model dependent relations
[Burkardt, Hwang, 2003; Burkardt, 2005]
SSA= GPDFSI
valid for both Sivers and Boer-Mulders functions in spectator model, but breaks down at higher-orders
LT
0
TL
0
GPDs at »=0 vanish because of time-reversal invariance
sx
up
down
BP, Cazzaniga, Boffi, PRD78 (2008); Lorce`, BP, in preparation
Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009)
Light-cone quark model:
! consistent with lattice calculations
Relations among TMDs in Quark Models
Linear relations
Quadratic relation
Flavor-dependent
**
*
Flavor-independent
**
*
**
*
Bag
**
*
[Jaffe & Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)]
ÂQSM
[Lorcé & Pasquini (in preparation)]
LCQM
[Pasquini & al. (2005-2008)]
S Diquark
[Ma & al. (1996-2009), Jakob & al. (1997), Bacchetta & al. (2008)]
AV Diquark
[Ma & al. (1996-2009), Jakob & al. (1997)] [Bacchetta & al. (2008)]
Cov. Parton
[Efremov & al. (2009)]
Quark Target
[Meißner & al. (2007)]
Common assumptions :
 No gluons
 Independent quarks
Light-cone Helicity and Canonical Spin
Quark polarization
Nucleon polarization
LC helicity
 Rotations in light-front dynamics depend on the interaction, while are kinematical in canonical
quantization
we study the rotational symmetries for TMDs in the basis of canonical spin
rotation around an axis
orthogonal to z and k? of
an angle µ=µ(k)
Nucleon polarization
Canonical spin
Quark polarization
Rotational Symmetries in Canonical-Spin Basis
 Cilindrical symmetry around z direction
nucleon spin
quark spin
 Cilindrical symmetry around Ty k k?
 Spherical symmetry: invariance for any spin rotation
 Spherical symmetry and SU(6) spin-flavor symmetry
[C. Lorce’, B.P., in preparation]
Orbital Angular Momentum
not unique decomposition
gauge invariant,
but contains interactions through
the gauge covariant derivative
[ X. Ji, PRL 78, (1997) ]
not gauge invariant,
but diagonal in the LCWFs basis
[ R.L. Jaffe, NPB 337, (1990) ]
Ji’s sum rule
quark orbital angular momentum:
What is the difference between the two definitions in a quark model without gauge fields?
scalar diquark model: M. Burkardt, PRD79, 071501 (2009); LCCQM: BP, F. Yuan, in preparation
Three Quark Light Cone Amplitudes
 classification of LCWFs in angular momentum components
total quark helicity Jq
J z = J z q + Lz q
Lzq = -1
Lzq =0
[Ji, J.P. Ma, Yuan, 03;
Burkardt, Ji, Yuan, 02]
Lzq =1
parity
time reversal
isospin symmetry
6 independent wave function amplitudes:
0
1
2
LLzzqqq== -1
Lzq =2
Quark Orbital Angular Momentum
Jaffe-Manohar and Ji OAM should coincide when A=0
! no-gluons, only quark contribution
 Jaffe-Manohar definition:
overlap of LCWFs with ¢Lz=0
 Ji’s definition:
¢Lz=0
¢Lz=
1
¢Lz=0
interference between LCWFs with different Lz
it is not trivial to have the same orbital angular momentum for the quark contribution
Distribution in x of Orbital Angular Momentum
Definition of Jaffe and Manohar: contribution from different partial waves
TOT
up
down
Lz=0
Lz=-1
Lz=-1
Lz=+2
total result, sum of up and down contributions: Jaffe-Manohar’s vs. Ji’s definition
Jaffe-Manohar
Ji
even in a model without gauge fields the two definitions give different distributions in x
Orbital Angular Momentum
 Definition of Jaffe and Manohar: contribution from different partial waves
= 0 ¢ 0.62 + (-1) ¢ 0.14 + (+1) ¢ 0.23 + (+2) ¢ 0.018 = 0.126
 Definition of Ji:
[BP, F. Yuan, in preparation]
[scalar diquark model: M. Burkardt, PRD79, 071501 (2009)]
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? even for unpolarized quarks in unpolarized nucleon
 GPDs and TMDs probe the same overlap of 3-quark LCWF in different kinematics
- give complementary information useful to reconstruct the nucleon wf
 Relations of TMDs in a large class of models due to rotational symmetries in the
quark-spin space
- useful to test them with experimental observables in the valence region
 Orbital Angular Momentum in terms of LCWFs:
- in quark models, the total OAM (but not the distributions in x) is the same from JI
and Jaffe-Manohar definitions
Backup
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
º ! quark polarization
¹! nucleon polarization
(
16 GTMDs
Active quark :
Spectator quarks :
Model Independent Spin Structure
)
Orbital Angular Momentum in the Light-Front
 Light-cone Gauge A+=0 and advanced boundary condition for A
 generalization of the relation for the anomalous magnetic moment:
[Brodsky, Drell, PRD22, 1980]
complex LCWFs due to FSI/ISI
[Brodsky, Gardner, PLB 643, 2006]
[Brodsky, BP, Yuan, Xiao, PLB 667, 2010