Generalized Transverse-Momentum Distributions

Cédric LorcéMainz UniversityGermanyBarbara

PasquiniPavia UniversityItaly

In collaboration with:

Outline

FFsTransverse

charge densities

Wigner distributions

Parton distributions

Physical interpretation

Spin densities

Generalized Transverse-Momentum Distributions

Transverse-Momentum Distributions

Generalized Parton Distributions

Form Factors

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

Parton DistributionsW

Vector Net # of quarks

Axial Net quark longitudinal polarization

Net quark transverse polarization

Tensor

Charges

Experi

Charges

Experi

No clear interpretation !

-# of quarks changes

-Momentum transfer

Charges

Experi

Drell-Yan-West frame

-Momentum transfer

No clear interpretation in momentum space !

Proton

Neutron

Transverse charge densitiesCharges

Experi

Probabilistic interpretation in position space

Drell-Yan-West frame

2D Fourier transform

[Miller (07)]

[Carlson, Vdh (08)]

Transverse charge densities

Charges

Experi

Spin densities

FFs PDFs GPDs

[Belitsky & al. (04)]

[Burkardt (01,03)]

Hadron 3D picture !

Position

Charges

GPDsTMDs

Experi

Spin densities

Complementary hadron 3D picture !

No direct connection

Momentum space

Mean moment

Displacement

PositionMoment

um transfer

Charges

GPDsTMDs

GTMDsExperi

Longitudinal

Transverse

Wigner distribution

Spin densities

[Meißner & al. (2009)]

Charges

FFsPDFs

GPDsTMDs

GTMDsExperi

Complete PictureW

Wigner distribution

Spin densities

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)

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

(1 out of 16)

3Q light-cone model

Longitudinal

Transverse

Example: Unpol. up Quark in Unpol. Proton

Orbital angular momentum?

(1 out of 16)

3Q light-cone model

favored

unfavored

Example: Unpol. up Quark in Unpol. Proton(1 out of 16)

Longitudinal

Transverse

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

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

More u than d in central region! n

[Miller (2007)]

Hard exclusive meson leptoproduction

Handbag approximation

[Burkardt (2003)]

Bint S

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

S Diquark

AV Diquark

Cov. Parton

Quark Target

[Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)]

[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

Quark polarization

LC helicity Canonical spin

Spherical symmetry

Axial symmetry about

TMDs LCQMQSM

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

GPDs (vector & axial)LCQMQSM

GPDs (tensor)LCQMQSM

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

Unpolarized u quark in unpolarized proton

k T fixed

,k T fixed

Unpolarized u quark in unpolarized proton

Generalized Transverse-Momentum Distributions

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