Generalized Transverse-Momentum Distributions

Cédric LorcéMainz UniversityGermanyBarbara

PasquiniPavia UniversityItaly

In collaboration with:

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

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

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

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

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

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

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:

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

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

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)]

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)]

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

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

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

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

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

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

Backup

Unpolarized u and d quarks in unpolarized

proton

u/2d

More u than d in central region! n

[Miller (2007)]

QSM

LCQM

Wigner Distributions

Hard exclusive meson leptoproduction

Handbag approximation

DVCS

DIS

2

~ Im

SIDIS

2

~ Im

[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

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)]

* **

**

*

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)

*

*

*

*

*

*

LC helicity and canonical spinQuark polarization

Nu

cleon

pola

riza

tion

Quark polarization

Nu

cleon

pola

riza

tion

LC helicity Canonical spin

=

= 0

Spherical symmetry

=

2 22

+

[C.L., Pasquini (in preparation)]

Axial symmetry about

Axial symmetry about

=

= -

=

= -

TMDs LCQMQSM

[C.L., Pasquini, Vdh (in preparation)]

GPDs (vector & axial)LCQMQSM

H

E

H

E~

~

[C.L., Pasquini, Vdh (in preparation)]

GPDs (tensor)LCQMQSM

HT

ET

~

~HT

ET

[C.L., Pasquini, Vdh (in preparation)]

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

Light front- and instant-form WFs

Assumption : in instant form (automatic w/ spherical symmetry)

More convenient to work in canonical spin basis

,

k T

b

Unpolarized u quark in unpolarized proton

k T fixed

QSM

Wigner Distributions

,k T fixed

k T

k T

k T

k T

Unpolarized u quark in unpolarized proton

QSM

Wigner Distributions