Transcript Generalized TMDs

Generalized TransverseMomentum Distributions
Cédric Lorcé
In collaboration with:
Barbara Pasquini
Pavia University
Italy
Mainz University
Germany
Outline
Parton distributions
Physical
interpretation
GTMDs
Wigner
distributions
Generalized TransverseMomentum Distributions
TMDs
GPDs
Transverse-Momentum
Distributions
Generalized Parton
Distributions
PDFs
Parton Distribution
Functions
FFs
Form Factors
Spin densities
Transverse
charge densities
3(+2?)D Picture of the
Nucleon
Quark-quark correlator
Information on quark distribution
Experiments
ES, DIS, SIDIS, DVCS, …
Quark models
Wave function
(often just N=3)
(often just N=3)
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
Parton Distributions
Vector
Net # of quarks
Axial
Net quark longitudinal polarization
Tensor
Net quark transverse polarization
Charges
ES, DIS, SIDIS, DVCS, …
Experiments
Wave function
(often just N=3)
Quark models
Information on quark distribution
Parton Distributions
PDFs
Charges
ES, DIS, SIDIS, DVCS, …
Experiments
Wave function
(often just N=3)
Quark models
Information on quark distribution
Parton Distributions
No clear interpretation !
-#
of quarks changes
-Momentum
transfer
PDFs
Charges
FFs
(often just N=3)
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
Parton Distributions
Drell-Yan-West frame
No clear interpretation
in momentum space !
-Momentum
transfer
PDFs
Charges
FFs
(often just N=3)
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
Parton Distributions
Drell-Yan-West frame
Probabilistic interpretation
in position space
2D Fourier transform
NB:
Proton
PDFs
Charges
Neutron
FFs
[Miller (07)]
[Carlson, Vdh (08)]
Transverse charge
densities
(often just N=3)
FFs
PDFs
GPDs
Hadron 3D picture !
[Belitsky & al. (04)]
[Burkardt (01,03)]
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
Parton Distributions
2D Fourier
transform
PDFs
Charges
GPDs
Spin densities
FFs
Transverse charge
densities
(often just N=3)
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
Parton Distributions
Mean
momentum
Displacement
Momentum
space
Position
space
Momentum
transfer
TMDs
Complementary
hadron 3D picture !
Position
No direct
connection
PDFs
Charges
2D Fourier
transform
GPDs
Spin densities
FFs
Transverse charge
densities
Parton Distributions
(often just N=3)
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
[Meißner & al. (2009)]
2D Fourier
transform
GTMDs
Transverse
Longitudinal
Wigner distribution
TMDs
PDFs
Charges
GPDs
Spin densities
FFs
Transverse charge
densities
(often just N=3)
ES, DIS, SIDIS, DVCS, …
Wave function
Experiments
Quark models
Information on quark distribution
Complete Picture
2D Fourier
transform
GTMDs
Wigner distribution
Transverse Wigner
distribution
TMDs
TMSDs
TMFFs
PDFs
Charges
GPDs
Spin densities
FFs
Transverse charge
densities
[C.L., Pasquini (submitted, 2011)]
Wigner Distributions
Transverse
[Wigner (1932)]
[Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
QM
QFT (Breit frame)
QFT (light cone)
Longitudinal
Heisenberg’s
uncertainty relations
Quasi-probabilistic
GPDs
TMDs
GTMDs
Third 3D picture !
No restrictions from Heisenberg’s
uncertainty relations
Example: Unpol. up Quark in Unpol. Proton
Transverse
[Wigner (1932)]
[Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
Longitudinal
fixed
3Q light-cone model
(1 out of 16)
QM
QFT (Breit frame)
QFT (light cone)
Example: Unpol. up Quark in Unpol. Proton
Transverse
[Wigner (1932)]
[Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
Longitudinal
fixed
unfavored
favored
3Q light-cone model
Orbital angular momentum?
(1 out of 16)
QM
QFT (Breit frame)
QFT (light cone)
Example: Unpol. up Quark in Unpol. Proton
Transverse
[Wigner (1932)]
[Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
(1 out of 16)
QM
QFT (Breit frame)
QFT (light cone)
Longitudinal
0.1 GeV²
0.2 GeV²
0.3 GeV²
3Q light-cone model
0.4 GeV²
Words of caution

No known processes to extract GTMDs

Wigner distributions are quasi-probabilistic

Issues concerning universality of TMDs

Fragmentation functions not so well known

Extrapolations needed for Fourier transform

Scale-dependence

Twist-two picture


gauge
Problems with transverse gauge link
Transverse
Longitudinal
Summary

Quark-quark correlator



2D Fourier transform on the light cone



Most complete information on hadron structure
GTMDs are ‘’mother’’ distributions
Correct interpretation (number of partons is fixed)
GTMDs are connected to Wigner distributions
Example of Wigner distribution



Unpolarized quark in unpolarized proton
3Q light-cone model
Distortions connected to OAM
Backup
Wigner Distributions
ru/2
rd
Unpolarized u and d quarks
in unpolarized proton
More u than d in central region!
n
[Miller (2007)]
cQSM
LCQM
Handbag approximation
2
DIS
~
Im
~
Im
2
SIDIS
DVCS
Hard exclusive
meson leptoproduction
Some examples: Transverse Charge Densities
Neutron
d
d
u
Bint
S X
[Burkardt (2003)]
Orbital angular
momentum
Anomalous magnetic
moment
Helicity flip
Magnetic moment
Induced electric
dipole moment
Angular momentum
Ji
Jaffe-Manohar

Each term is gauge-invariant

Decomposition is gauge-dependent

No decomposition of

OAM in LCWFs refers to
Ji’s sum rule
(easy)
[Avakian & al. (2010)]
Pretzelosity
Modeldependent!
Trans. pol. quark in
trans. pol. proton
GPDs
TMDs
Model relations for TMDs
*=SU(6)
Linear relations
Flavor-dependent
Flavor-independent
Bag
Quadratic relation
**
*
**
*
**
*
**
*
[Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)]
cQSM
[C.L., Pasquini (in preparation)]
LCQM
[Pasquini & al. (2005-2008)]
S Diquark
(twist-two)
[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)]
LC helicity and canonical spin
Quark polarization
Canonical spin
Nucleon polarization
Nucleon polarization
LC helicity
Quark polarization
Spherical symmetry
[C.L., Pasquini (in preparation)]
Axial symmetry about
2
=
=
= 0
Axial symmetry about
=
=
=-
=-
2
+
2
TMDs
cQSM
LCQM
[C.L., Pasquini, Vdh (in preparation)]
GPDs
(vector & axial)
cQSM
H
~
H
E
~
E
LCQM
[C.L., Pasquini, Vdh (in preparation)]
GPDs
(tensor)
cQSM
HT
~
HT
ET
~
ET
LCQM
[C.L., Pasquini, Vdh (in preparation)]
Summary

Quark-quark correlator
Most complete information on hadron structure
 GTMDs are ‘’mother’’ distribution


2D Fourier transform on the light cone




TMDs



Correct interpretation of FFs
GTMDs can be related to Wigner distributions
Distortions due to orbital angular momentum
Model relations due to spherical symmetry
LC helicity and canonical spin connected by a
rotation
3Q amplitude

Same structure in many models
Light front- and instant-form WFs
Assumption :

in instant form
(automatic w/ spherical symmetry)
More convenient to work in canonical spin basis
Wigner Distributions
q
T
k
b
cQSM
T
k
, q fixed
Unpolarized u quark in
unpolarized proton
Wigner Distributions
T
k
T
k
T
cQSM
T
k
Unpolarized u quark in
unpolarized proton
k
T
k
, q fixed