![Page 1: Generalized Transverse- Momentum Distributions Cédric Lorcé Mainz University Germany Barbara Pasquini Pavia University Italy In collaboration with:](https://reader035.vdocument.in/reader035/viewer/2022081603/56649eab5503460f94bb1b54/html5/thumbnails/1.jpg)
Generalized Transverse-Momentum Distributions
Cédric LorcéMainz UniversityGermanyBarbara
PasquiniPavia UniversityItaly
In collaboration with:
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Outline
GPDs
TMDs
GTMDs
FFsTransverse
charge densities
Wigner distributions
Parton distributions
Physical interpretation
Spin densities
Generalized Transverse-Momentum Distributions
Transverse-Momentum Distributions
Generalized Parton Distributions
Form Factors
PDFs
Parton Distribution Functions
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Experiments
Quark models
Information on quark distribution
3(+2?)D Picture of the Nucleon
Wave function (often just N=3)ES, DIS, SIDIS, DVCS, …
Quark-quark correlator
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Charges
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
Vector Net # of quarks
Axial Net quark longitudinal polarization
Net quark transverse polarization
Tensor
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Charges
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
PDFs
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Charges
FFs
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
No clear interpretation !
-# of quarks changes
-Momentum transfer
PDFs
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Charges
FFs
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
Drell-Yan-West frame
-Momentum transfer
No clear interpretation in momentum space !
PDFs
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Proton
Neutron
Transverse charge densitiesCharges
FFs
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
Probabilistic interpretation in position space
Drell-Yan-West frame
2D Fourier transform
[Miller (07)]
[Carlson, Vdh (08)]
PDFs
NB:
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Transverse charge densities
Charges
FFs
GPDs
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
2D Fourier transform
Spin densities
FFs PDFs GPDs
[Belitsky & al. (04)]
[Burkardt (01,03)]
Hadron 3D picture !
PDFs
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Position
space
Charges
FFs
GPDsTMDs
Experi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
Transverse charge densities
2D Fourier transform
Spin densities
Complementary hadron 3D picture !
No direct connection
Momentum space
Mean moment
um
Displacement
PositionMoment
um transfer
PDFs
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Charges
FFs
GPDsTMDs
GTMDsExperi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Parton DistributionsW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
2D Fourier transform
Longitudinal
Transverse
Wigner distribution
Transverse charge densities
Spin densities
PDFs
[Meißner & al. (2009)]
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Charges
FFsPDFs
GPDsTMDs
GTMDsExperi
ments
Quark
m
odels
Info
rmati
on o
n q
uark
dis
trib
uti
on
Complete PictureW
ave f
un
ctio
n (
oft
en
ju
st
N=
3)
ES
, D
IS,
SID
IS,
DV
CS
, …
Wigner distribution
Transverse charge densities
Spin densities
2D Fourier transform
TMSDs
TMFFs
Transverse Wigner distribution
[C.L., Pasquini (submitted, 2011)]
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Longitudinal
Transverse
Wigner Distributions
[Wigner (1932)][Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
QMQFT (Breit frame)QFT (light cone)
GPDs
TMDs
GTMDs
Heisenberg’s uncertainty
relations
Quasi-probabilistic
Third 3D picture !
No restrictions from Heisenberg’s uncertainty relations
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Longitudinal
Transverse
Example: Unpol. up Quark in Unpol. Proton
fixed
[Wigner (1932)][Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
QMQFT (Breit frame)QFT (light cone)
(1 out of 16)
3Q light-cone model
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Longitudinal
Transverse
Example: Unpol. up Quark in Unpol. Proton
fixed
Orbital angular momentum?
[Wigner (1932)][Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
QMQFT (Breit frame)QFT (light cone)
(1 out of 16)
3Q light-cone model
favored
unfavored
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Example: Unpol. up Quark in Unpol. Proton(1 out of 16)
Longitudinal
Transverse
[Wigner (1932)][Belitsky, Ji, Yuan (04)]
[C.L., Pasquini (in preparation)]
QMQFT (Breit frame)QFT (light cone)
0.1 GeV²
0.2 GeV²
0.3 GeV²
0.4 GeV²
3Q light-cone model
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Words of caution
Longitudinal
Transverse 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
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Quark-quark correlator Most complete information on hadron structure GTMDs are ‘’mother’’ distributions
2D Fourier transform on the light cone 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
Summary
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Backup
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Unpolarized u and d quarks in unpolarized
proton
u/2d
More u than d in central region! n
[Miller (2007)]
QSM
LCQM
Wigner Distributions
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Hard exclusive meson leptoproduction
Handbag approximation
DVCS
DIS
2
~ Im
SIDIS
2
~ Im
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[Burkardt (2003)]
Bint S
d
u
d
X
Neutron
Anomalous magnetic moment
Orbital angular momentum
Induced electric dipole moment
Helicity flip
Magnetic moment
Some examples: Transverse Charge
Densities
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Angular momentum
Ji’s sum rule
Each term is gauge-invariant
No decomposition of
Ji Jaffe-ManoharJi
Decomposition is gauge-dependent
OAM in LCWFs refers to (easy)
TMDsGPDs
Pretzelosity
Trans. pol. quark in trans. pol.
proton
Model-dependent
!
[Avakian & al. (2010)]
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* **
**
*
Model relations for TMDs (twist-two)
Flavor-dependent
Flavor-independent
Linear relations Quadratic relation
Bag
QSM
LCQM
S Diquark
AV Diquark
Cov. Parton
Quark Target
[Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)]
[C.L., Pasquini (in preparation)]
[Pasquini & al. (2005-2008)]
[Ma & al. (1996-2009), Jakob & al. (1997), Bacchetta & al. (2008)]
[Ma & al. (1996-2009), Jakob & al. (1997)] [Bacchetta & al. (2008)]
[Efremov & al. (2009)]
[Meißner & al. (2007)]
*=SU(6)
*
*
*
*
*
*
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LC helicity and canonical spinQuark polarization
Nu
cleon
pola
riza
tion
Quark polarization
Nu
cleon
pola
riza
tion
LC helicity Canonical spin
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=
= 0
Spherical symmetry
=
2 22
+
[C.L., Pasquini (in preparation)]
Axial symmetry about
Axial symmetry about
=
= -
=
= -
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TMDs LCQMQSM
[C.L., Pasquini, Vdh (in preparation)]
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GPDs (vector & axial)LCQMQSM
H
E
H
E~
~
[C.L., Pasquini, Vdh (in preparation)]
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GPDs (tensor)LCQMQSM
HT
ET
~
~HT
ET
[C.L., Pasquini, Vdh (in preparation)]
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Quark-quark correlator Most complete information on hadron structure GTMDs are ‘’mother’’ distribution
2D Fourier transform on the light cone Correct interpretation of FFs GTMDs can be related to Wigner distributions Distortions due to orbital angular momentum
TMDs Model relations due to spherical symmetry LC helicity and canonical spin connected by a
rotation
3Q amplitude Same structure in many models
Summary
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Light front- and instant-form WFs
Assumption : in instant form (automatic w/ spherical symmetry)
More convenient to work in canonical spin basis
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,
k T
b
Unpolarized u quark in unpolarized proton
k T fixed
QSM
Wigner Distributions
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,k T fixed
k T
k T
k T
k T
Unpolarized u quark in unpolarized proton
QSM
Wigner Distributions