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Generalized Transverse-Momentum Distributions. Cédric Lorcé Mainz University Germany. In collaboration with:. Barbara Pasquini Pavia University Italy. GTMDs. Wigner distributions. Spin densities. GPDs. TMDs. Transverse charge densities. PDFs. FFs. Outline.

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generalized transverse momentum distributions

Generalized Transverse-Momentum Distributions

Cédric Lorcé

Mainz University

Germany

In collaboration with:

Barbara Pasquini

Pavia University

Italy

outline

GTMDs

Wigner distributions

Spin densities

GPDs

TMDs

Transverse

charge densities

PDFs

FFs

Outline

Physical interpretation

Parton distributions

Generalized Transverse-Momentum Distributions

Transverse-Momentum Distributions

Generalized Parton Distributions

Parton Distribution Functions

Form Factors

3 2 d p icture of the n ucleon

Quark-quark correlator

3(+2?)DPicture of the Nucleon

Information on quark distribution

Quark models

Experiments

Wave function (often just N=3)

ES, DIS, SIDIS, DVCS, …

p arton d istributions

Charges

Parton Distributions

Wave function (often just N=3)

Quark models

Vector

Net # of quarks

Information on quark distribution

Axial

Net quark longitudinal polarization

Tensor

Net quark transverse polarization

Experiments

ES, DIS, SIDIS, DVCS, …

p arton d istributions1

PDFs

Charges

Parton Distributions

Wave function (often just N=3)

Quark models

Information on quark distribution

Experiments

ES, DIS, SIDIS, DVCS, …

p arton d istributions2

PDFs

FFs

Charges

Parton Distributions

No clear interpretation !

  • # of quarks changes
  • Momentum transfer

Wave function (often just N=3)

Quark models

Information on quark distribution

Experiments

ES, DIS, SIDIS, DVCS, …

p arton d istributions3

Drell-Yan-West frame

PDFs

FFs

Charges

Parton Distributions

No clear interpretation in momentum space !

  • Momentum transfer

Wave function (often just N=3)

Quark models

Information on quark distribution

Experiments

ES, DIS, SIDIS, DVCS, …

p arton d istributions4

Drell-Yan-West frame

Proton

Neutron

PDFs

FFs

Charges

Transverse charge densities

Parton Distributions

Probabilistic interpretation in position space

Wave function (often just N=3)

Quark models

2D Fourier transform

Information on quark distribution

NB:

Experiments

ES, DIS, SIDIS, DVCS, …

[Miller (07)]

[Carlson, Vdh (08)]

p arton d istributions5

Spin densities

PDFs

FFs

GPDs

Charges

Parton Distributions

FFs

PDFs

GPDs

Hadron 3D picture !

[Belitsky & al. (04)]

[Burkardt (01,03)]

Wave function (often just N=3)

Quark models

2D Fourier transform

Information on quark distribution

Transverse charge densities

Experiments

ES, DIS, SIDIS, DVCS, …

p arton d istributions6

Spin densities

PDFs

FFs

GPDs

TMDs

Charges

Parton Distributions

Mean momentum

Displacement

Complementary hadron 3D picture !

Momentum space

Position space

Wave function (often just N=3)

Quark models

Momentum transfer

Position

2D Fourier transform

No direct connection

Information on quark distribution

Transverse charge densities

Experiments

ES, DIS, SIDIS, DVCS, …

p arton d istributions7

Wigner distribution

Spin densities

GTMDs

PDFs

Transverse

FFs

GPDs

TMDs

Longitudinal

Charges

Parton Distributions

2D Fourier transform

[Meißner & al. (2009)]

Wave function (often just N=3)

Quark models

Information on quark distribution

Transverse charge densities

Experiments

ES, DIS, SIDIS, DVCS, …

c omplete p icture

TMFFs

Spin densities

GTMDs

PDFs

TMSDs

FFs

GPDs

TMDs

Charges

Complete Picture

2D Fourier transform

Wigner distribution

Transverse Wigner distribution

Wave function (often just N=3)

Quark models

Information on quark distribution

Transverse charge densities

Experiments

ES, DIS, SIDIS, DVCS, …

[C.L., Pasquini (submitted, 2011)]

w igner d istributions

Transverse

Longitudinal

GTMDs

GPDs

TMDs

Wigner Distributions

[Wigner (1932)]

[Belitsky, Ji, Yuan (04)]

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

QM

QFT (Breit frame)

QFT (light cone)

Heisenberg’s uncertainty relations

Quasi-probabilistic

Third 3D picture !

No restrictions from Heisenberg’s uncertainty relations

e xample u npol up q uark in u npol p roton

Transverse

Longitudinal

Example:Unpol. upQuark in Unpol. Proton

(1 out of 16)

[Wigner (1932)]

[Belitsky, Ji, Yuan (04)]

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

QM

QFT (Breit frame)

QFT (light cone)

fixed

3Q light-cone model

e xample u npol up q uark in u npol p roton1

Transverse

Longitudinal

Example:Unpol. upQuark in Unpol. Proton

(1 out of 16)

[Wigner (1932)]

[Belitsky, Ji, Yuan (04)]

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

QM

QFT (Breit frame)

QFT (light cone)

fixed

unfavored

favored

3Q light-cone model

Orbital angular momentum?

slide16

Transverse

Longitudinal

Example:Unpol. upQuark in Unpol. Proton

(1 out of 16)

[Wigner (1932)]

[Belitsky, Ji, Yuan (04)]

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

QM

QFT (Breit frame)

QFT (light cone)

0.1 GeV²

0.2 GeV²

0.3 GeV²

3Q light-cone model

0.4 GeV²

words of caution

Transverse

Longitudinal

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
summary
Summary
  • 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
slide20

Wigner Distributions

cQSM

ru/2

rd

Unpolarized u and d quarks in unpolarized proton

LCQM

n

More u than d in central region!

[Miller (2007)]

handbag approximation

2

DIS

~

Im

2

SIDIS

~

Im

Hard exclusive

meson leptoproduction

DVCS

Handbag approximation
s ome examples t ransverse c harge d ensities

Bint

S

Anomalous magnetic moment

Orbital angular momentum

u

d

Magnetic moment

d

Induced electric dipole moment

X

Some examples:Transverse Charge Densities

Neutron

[Burkardt (2003)]

Helicity flip

angular momentum

TMDs

GPDs

Angular momentum

Ji

Ji

Jaffe-Manohar

  • Each term is gauge-invariant
  • No decomposition of
  • Decomposition is gauge-dependent
  • OAM in LCWFs refers to (easy)

Ji’s sum rule

Pretzelosity

[Avakian & al. (2010)]

Model-dependent!

Trans. pol. quark in trans. pol. proton

model relations for tmds twist two
Model relations for TMDs (twist-two)

*=SU(6)

Linear relations

Quadratic relation

*

*

Flavor-dependent

*

*

*

*

*

Flavor-independent

*

*

*

*

*

Bag

cQSM

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)]

lc helicity and canonical spin
LC helicity and canonical spin

LC helicity

Canonical spin

Quark polarization

Quark polarization

Nucleon polarization

Nucleon polarization

spherical symmetry

2

2

2

=

=

+

= 0

=

=

= -

= -

Spherical symmetry

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

Axial symmetry about

Axial symmetry about

slide27
TMDs

cQSM

LCQM

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

gpds vector axial
GPDs (vector & axial)

cQSM

LCQM

~

H

H

~

E

E

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

gpds tensor
GPDs (tensor)

cQSM

LCQM

~

HT

HT

~

ET

ET

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

summary1
Summary
  • 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 structurein many models
light front and instant form wfs
Light front- and instant-form WFs

Assumption :

  • in instant form (automatic w/ spherical symmetry)

More convenient to work in canonical spin basis

wigner distributions

k

k

T

T

Wigner Distributions

q

b

,

q

fixed

Unpolarized u quark in unpolarized proton

cQSM

slide33

k

k

k

k

k

T

T

T

T

T

Wigner Distributions

Unpolarized u quark in unpolarized proton

,

q

fixed

cQSM