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Modeling the Transverse-Momentum Dependent Parton Distributions

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Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005)

Transverse-Momentum DependentParton Distributions

Barbara Pasquini (Uni Pavia & INFN Pavia, Italy)

- TMDs and Quark Model Results
- Light-Cone Constituent Quark Model
- Results for T-even and T-odd TMDs
- Leading-Twist Spin Asymmetries in SIDIS due to T-even TMDs
- Conclusions

- origin of quark-model relations among T-even TMDS

- quark-quark correlation function ! overlap representation in terms of light-cone amplitudes which are eigenstates of orbital angular momentum

l, x, k

l’, x, k

+

quark-number density

G =

+5

quark-helicity density

isx+ 5

transverse-spin density

:Wilson line which ensures the color gauge invariance

polarization of quark and nucleon in terms of light-cone helicities

nucleon polarization

Relations among T-even TMDs in quark models

(bag model, light-cone model, chiral quark soliton model, covariant parton model, scalar diquark model)

linear relations

quadratic relation

What are the common features of the models which lead to these relations?

Rotations in light-front dynamics depend on the interaction

we study the rotational symmetries for TMDs in the basis of canonical spin

- The active quark does not interact with the other quarks in the nucleon during the interaction with the external probe

we apply the free boost operator to relate the light-cone helicity and the canonical spin

rotation around an axis orthogonal to z and k? of an angle µ=µ(k)

Quark Spin Polarization

z

z

k?

k?

LC

Canonical

z

z

k?

k?

z

z

k?

k?

Rotational Symmetries in Canonical-Spin Basis

- Cilindrical symmetry around z direction

nucleon spin

quark spin

- Cilindrical symmetry around Tyk k?

- Spherical symmetry: invariance for any spin rotation

- Spherical symmetry and SU(6) spin-flavor symmetry

Spherical symmetry is a sufficient but not necessary condition for quark-TMD relations

[C. Lorce’, B.P., in preparation]

- Spectator models (Jakob, Mulders, Rodrigues, 1997; Gamberg, Goldstein, Schlegel, 2008)
- Light-Cone Chiral quark Soliton Model (C. Lorce’, BP, Vanderhaeghen, in preparation)
- Light-Cone Constituent Quark model (BP, Cazzaniga, Boffi, 2008)
- Covariant parton model (Efremov, Teryaev, Schweitzer, Zavada, 2008)
- Bag model (Yuan, Schweitzer, Avakian, 2009)

see talk of P. Schweitzer

All quark models which satisfy the TMD relations have spherical symmetry (Lz=0)

- Spherical Symmetry is a good approximation for quark models
- relations can give useful guidelines in phenomenological studies of quark TMDs

Light-Cone Fock Expansion

! talk of S. Brodsky

fixed light-cone time

frame INdependent

probability amplitude to find the N parton configuration with the complex of quantum number in the nucleon with helicity

in the light-cone gauge A+=0, the total angular momentum is conserved Fock state by Fock state ) each Fock-state component can be expanded in terms of eigenfunction of the light-front orbital angular momentum operator

Three Quark Light Cone Amplitudes

Lzq = 2

parity

time reversal

isospin symmetry

6 independent wave function amplitudes:

Lzq =1

Lzq =2

Lzq = -1

Lzq =0

Lzq = 1

- classification of LCWFs in angular momentum components

[Ji, J.P. Ma, Yuan, 03; Burkardt, Ji, Yuan, 02]

total quark LC helicity Jq

Jz = Jzq + Lzq

Lz q=0

Lzq=-1

Lzq =1

Lzq =2

Lzq = -1

Lzq =0

- Phenomenological LCWF for the valence (qqq) component:

- momentum-space component: S wave
- spin and isospin component in the instant-form: SU(6) symmetric
- )Melosh rotation to convert the canonical spin of quarks in LC helicity

Schlumpf, Ph.D. Thesis, hep-ph/9211255

parameters fitted to anomalous magnetic moments of the nucleon

: normalization constant

Jz = Jzq

Jz = Jzq+Lzq

Six independent wave function amplitudes : eigenstates of the total orbital angular momentum operator in Light-Front dynamics

The six independent wave function amplitudes obtained from the Melosh rotations satisfy the model independent classification scheme in four orbital angular momentum components

- SU(6) symmetry Nu =2 Nd =1 Pu =4/ 3 Pd = -1/3

momentum dependent wf factorized from spin-dependent effects

B.P., Cazzaniga, Boffi, PRD78, 2008

Orbital angular momentum content

f1

g1L

h1

up

up

up

TOT

TOT

S wave

S wave

P wave

P wave

D wave

D wave

f1

g1L

h1

down

down

down

- Total results obey SU(6) symmetry relations: f1u = 2 f1d, g1Lu=-4 g1Ld, h1u=-4 h1d

- The partial wave contributions do not satisfy SU(6) symmetry relations!

Orbital angular momentum content

g1T (1)

h1L?(1)

h1T?(1)

S-P int.

up

P-P int.

TOT

P-D int.

P-D int.

S-D int.

S-P int.

g1T (1)

h1L?(1)

h1T?(1)

down

S-P int.

P-P int.

P-D int.

TOT

P-D int.

S-P int.

S-D int.

ONLY TOTAL results (and not partial wave contr.) obey SU(6) symmetry relations:g1T(1) u = -4g1T(1) d, h1L?(1) u =-4 h1L?(1) d, h1T?(1) u=-4 h1T?(1) d

Light-cone quark model

up

down

up

down

Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009)

BP, Cazzaniga, Boffi, PRD78 (2008)

- Lattice calculation supports results of Light-cone quark model:

see talk of B. Musch

¸

¸’

¸2

¸3

+ h.c.

P+, ¤

P+, ¤’

- Gauge link provides the necessary phase to make non-zero T-odd distributions
- We use the one-gluon exchange approximation in the light-cone gauge A+=0

! conservation of the helicity at the quark-gluon interaction vertex: ¸2=¸3

Sivers function

Boer-Mulders function

distribution of unpolarized quark in a transversely polarized nucleon

distribution of transversely polarized quark in a unpolarized nucleon

¤ = - ¤’

¸ = ¸’

¤ = ¤’

¸ = -¸’

¢ Lz = 1

¢ Lz = 1

T-odd distributions from interference of S-P and P-D waves of LCWFs

TOT

S-P int.

P-D int.

P-D int.

S-P int.

TOT

- Burkardt sum rule is reproduced exactly:

[Burkardt, PRD69 (2004)]

LCQM at the hadronic scale Q02=0.094 GeV2

LCQM evolved to Q02= 2.5 GeV2with evolution code of unpolarized PDF

Anselmino et al., EPJA39, (2009);PRD71 (2005)

Collins et al., PRD73 (2006)

Efremov et al.,PLB612 (2005)

B.P., Yuan: arXiv: 1001.5398 [hep-ph]: Brodsky, B.P., Xiao,Yuan, PLB327 (2010)

S-P int.

P-D int.

S-P int.

P-D int.

TOT

TOT

LCQM at the hadronic scale Q02=0.094 GeV2

LCQM evolved to Q02= 2.5 GeV2with evolution code of transversity

Barone, Melis, ProkudinarXiv:0912.5194 [hep-ph]

Zhang, Lu, Ma, Schmidt PRD78 (2008)

B.P., Yuan: arXiv: 1001.5398 [hep-ph]: Brodsky, B.P., Xiao,Yuan, PLB327 (2010)

X=beam polarization Y=target polarizationweight=ang. distr. hadron

Collinear double spin-asymmetries ALL and A1

- convolution integrals between parton distributions and fragmentation functions can be solved analytically without approximation

- no complications due to k?dependence

- evolution equations and fragmentation functions are known

we can test the model under “controlled conditions”:

- in which range and with what accuracy is the model applicable?
- how stable are the results under evolution?

evolved to exp. <Q2> =3 GeV2

at Q2=2.5 GeV2 [Kretzer, PRD62, 2000]

initial scale Q20=0.079 GeV2 where q <xq>=1

SMC

HERMES

inclusive longitudinal asymmetry

- description of exp. datawithin accuracy of 20-30%in the valence region
- very weak scale dependence

[Boffi, Efremov, Pasquini, Schweitzer, PRD79, 2009]

Strategy to calculate the azimuthal spin asymmetries

- we focus on the x-dependence of the asymmetries, especially in the valence-x region

- we adopt Gaussian Ansatz

- low hadronic scale !<k2? (f1)>(MODEL) =0.08 GeV2is much smaller than phenomelogical value <k2?(f1)>(PHEN>)=0.33 GeV2

(fit to SIDIS HERMES data assuming gaussian Ansatz) [Collins, et al., PRD73, 2006]

- we rescale the model results for <k2?(TMD)> with <k2? (f1)>(MODEL)/<k2?(f1)>(PHEN)

- we do not discuss the z and Ph dependence of azimuthal asymmetries because here integrals over the x dependence extend to low x-region where the model is not applicable

gaussian ansatz

from Light-Cone CQM evolved at Q2=2.5 GeV2, from GRV at Q2=2.5 GeV2

Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005)

from HERMES & BELLE data

HERMES data: Diefenthaler, hep-ex/0507013

More recent HERMES and BELLE data not included in the fit of Collins function

COMPASS data: Alekseev et al., PLB673(2009)

x h1q

- Dashed area: extraction of transversity from BELLE, COMPASS, and HERMES data

Anselmino et al., PRD75, 2007

- Predictions from Light-Cone CQM evolved from the hadronic scale Q20to Q2= 2.5 GeV2using two different momentum-dependent wf

solution of relativistic potential model

- no free parameters
- fair description of nucleon form factors

- phenomenological wf
- three fit parameters
- , and mq fitted to the anomalous magnetic moments of the nucleon and to gA

up

down

Schlumpf, Ph.D. Thesis, hep-ph/9211255

Faccioli, et al., NPA656, 1999Ferraris et al., PLB324, 1995

BP, Pincetti, Boffi, PRD72, 2005

at Q2=2.5GeV2

evolved to Q2 =2.5 GeV2 using evolution eq. of

initial scale Q20=0.079 GeV2

[Kretzer, PRD62, 2000]

evolved to Q2 =2.5

COMPASS

[Kotzinian, et al., arXiv:0705.2402]

- from Light-Cone CQM evolved at Q2=2.5 GeV2, with the evolution equations of
- from GRV at Q2=2.5 GeV2

from HERMES & BELLE data

Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005)

HERMES Coll.

Airapetian, PRL84, 2000;Avakian, Nucl. Phys. Proc. Suppl. 79 (1999)

Model results compatible with CLAS data for ¼+ and ¼0 but cannot explain the SSA for ¼-CLAS Coll,: arXiv:1003.4549 [hep-ex]

- from Light-Cone CQM evolved at Q2=2.5 GeV2, with the evolution equations of
- from GRV at Q2=2.5 GeV2

from HERMES & BELLE data

Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005)

- smaller predictions than expected from positivity boundswith f1 and g1 from GRV at Q2=2.5 GeV2

COMPASS Coll.

Kotzinian, arXiv:0705.2402

- experiment planned at CLAS12(H. Avakian at al., LOI 12-06-108)

- preliminary HERMES data compatible with zero ! talk of L. Pappalardo

- Model relations among T-even TMDs in quark models
- Model calculation in a Light-Cone CQM
- Predictions for all the leading-twist azimuthal spin asymmetries in SIDIS due to T-even TMDs

- spherical symmetry of the models is a sufficient but not necessary condition
- useful guideline for parametrizations of quark contributions to TMDs

- representation in terms of overlap of LCWF with different orbital angular momentum
- predictions for T-odd TMDs consistent with phenomenological parametrizations

and : the model is capable to describe the data in the valence region with accuracy of 20-30%

Collins asymmetry is the only non-zero within the present day error bars

! very good agreement between model predictions and exp. data

: :available exp. data compatible with zero within error bars! model results with “approximate” evolution are compatible with data

- k? dependence of the model is not of gaussian form
- how well can it be approximated by a gaussian form?

=1 in Gauss Ansatz

results agree within 20%

with Gaussian Ansatz

Gaussian Ansatz gives uncertainty within typical accuracy of the model

exact result

from Bacchetta et al., PLB659, 2008

:orbital angular momentum decomposition

gaussian ansatz

- from Light-Cone CQM evolved at the hadronic scale Q2=0.079 GeV2
- from GRV at Q2=2.5 GeV2

from HERMES & BELLE data

giutgo

COMPASS Coll.

Kotzinian, arXiv:0705.2402

TOT

S-D wave int.

P waves int.

Pretzelosity in SIDIS: perspectives

- At small x:
- There will be data from COMPASS proton target
- There will be data from HERMES

- More favorable conditions at intermediate x (0.2-0.6)
- experiment planned at CLAS with 12 GeV(H. Avakian at al., LOI 12-06-108)

Light-Cone CQM

Error projections for 2000 hours run time at CLAS12

Schweitzer, Boffi, Efremov, BP, in preparation

- opposite sign of h1

with

with

- chiral odd, no gluons
- h1L:SP and PD interference termsh1: SS and PP diagonal terms

h1L h1

- Wandzura-Wilczek-type approximationAvakian, et al., PRD77, 2008

h1L (1)

WW approx.

Light-Cone CQM

WW approx.

- large, larger than f1 and h1
- sign opposite to transversity
- light-cone quark model and bag model peaked at smaller x
- related to chiral-odd GPD

down

Light-Cone CQMBP, Cazzaniga, BoffiPRD78, 2008

[Meissner, et al., PRD76, 2007and arXiv: 0906.5323 [hep-ph]

up

Bag Model

Avakian, et al., PRD78, 2008

down

- positivity satisfied
- helicity – transversity = pretzelosity

Chiral quark soliton ModelL. Cedric, B,.P., Vdh, in preparation

Soffer inequality

up

Relations of TMDs in Valence Quark Models

(1)

Avakian, Efremov, Yuan, Schweitzer, (2008)

(2)

(3)

- (1), (2), and (3) hold in Light-Cone CQM ModelsBP, Pincetti, Boffi, PRD72, 2005; BP, Cazzaniga, Boffi, PRD78, 034025 (2008)
- (1) and (2) hold in Bag Model Avakian, Efremov, Yuan, Schweitzer, PRD78, 114024 (2008)
- (1) and (2) hold in the diquark spectator model for the separate scalar and axial-vector contributions, (3) is valid more generally for both u and d quarks Jakob, Mulders, Rodrigues, NPA626, 937 (1997) She, Zhu, Ma, PRD 79, 054008 (2009)
- (1) and (2) are not valid in more phenomenological versions of the diquark spectator model for the axial-sector, but hold for the scalar contributionBacchetta, Conti, Radici, PRD78, 074010 (2008)
- (2) holds in covariant quark-parton model Efremov, Schweitzer, Teryaev, Zavada, arXiv: 0903.3490 [hep-ph]

- no gluon dof valid at low hadronic scale
- not restricted to S and P wave contributions
- SU(6) symmetry is not crucial in the relation withthe pretzelosity

see talks by Schweitzer,Ma,Zavada

Charge density of partons in the transverse plane

- Infinite-Momentum-Frame Parton charge density in the transverse plane

no relativistic corrections

G.A. Miller, PRL99, 2007

neutron

proton

fit to exp. form factor by Kelly, PRC70 (2004)

Quark distributions in the neutron

B.P., Boffi,PRD76 (2007)(meson cloud model)

down

up

u

Helicity density in the transverse plane

- probability to find a quark with transverse position b and light-cone helicity in the nucleon with longitudinal polarization

2u+u

2u-u

d

(d+d)/2

d

(d-d)/2

Non relativistic limit (k 0) :

consistent with Soffer bounds and (Ma, Schmidt, Soffer,’97)

up

down

B.P., Pincetti, Boffi, 05

Light cone wave function overlap representation of Parton Distributions

- Melosh rotations: relativistic effects due to the quark transverse motion

Spin-Orbit Correlations and the Shape of the Nucleon

spin-dependent charge density operator in non relativistic quantum mechanics

spin-dependent charge density operator in quantum field theory

nucleon state transversely polarized

Probability for a quark to have a momentum k and spin direction n in a nucleon polarized in the S direction

TMD parton distributions integrated over x

G.A. Miller, PRC76 (2007)

- Fix the directions of S and n the spin-orbit correlations measured with is responsible for a non-spherical distribution with respect to the spin direction

- Diquark spectator model: wave function with angular momentum components Lz = 0, +1, -1

deformation due only to Lz=1 and Lz=-1 components

Jakob, et al., (1997)

s

S

x

x

S

x

: chirally odd tensor correlations matrix element from angular momentum components with |Lz-L’z|=2

up quark

K =0

K =0.5 GeV

K =0.25 GeV

G.A. Miller, PRC76 (2007)

Light Cone Constituent Quark Model

s

S

x

x

S

x

up quark

deformation induced from the Lz=+1 and Lz=-1 components

adding the contribution from Lz=0 and Lz=2 components

B.P., Cazzaniga, Boffi, PRD78, 2008

Angular Momentum Decomposition of h1Tu

Lz=0 and Lz=2

Lz=+1 and Lz=-1

h1T u [GeV-3]

h1T u [GeV-3]

k2

x

k2

x

SUM

h1T u [GeV-3]

compensation of opposite sign contributions

k2

x

no deformation

B.P., Cazzaniga, Boffi, PRD78, 2008

Spin dependent densities for down quark

s

s

S

S

x

x

x

x

S

S

x

x

- Diquark spectator model: contribution of Lz=+1 and Lz=-1 components

- Light-cone CQM: contribution of Lz=+1 and Lz=-1 components plus contribution of Lz=0 and Lz=2 components

Angular Momentum Decomposition of h1Td

SUM

h1T d[GeV-3]

partial cancellation ofdifferent angular momentum components

k2

x

non-spherical shape

h1T d[GeV-3]

h1T d[GeV-3]

Lz=0 and Lz=2

Lz=+1 and Lz=-1

k2

k2

x

x

Evolve in light-front time¿ = t+z

¾=t-z

Front Form

Instant Form

coordinates

x0 timex1, x2, x3 space

x+=x0+x3 timex-=x0-x3, x1, x2 space

Hamiltonian

P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949)

P P

D D

P P

P P

S S

Light Cone Amplitudes Overlap Representation of TMDs

¢Lz=0

S S

P P

P P

D D

¢Lz=0

¢Lz=0

down

up

Spectator ModelJakob, et al., NPA626 (1997)

- large, larger than f1 and h1
- sign opposite to transversity
- light-cone quark model and bag model peaked at smaller x
- related to chiral-odd GPD

Light-Cone CQMBP, Cazzaniga, BoffiPRD78, 2008

[Meissner, et al., PRD76, 2007and arXiv: 0906.5323 [hep-ph]

Bag Model

Avakian, et al., PRD78, 2008

down

- positivity satisfied
- helicity – transversity = pretzelosity

Soffer inequality

up

Rotational Symmetries in Canonical-Spin Basis

- Cilindrical symmetry around z direction

nucleon spin

quark spin

- Cilindrical symmetry around Tyk k?

- Spherical symmetry: invariance for any spin rotation

- Spherical symmetry and SU(6) spin-flavor symmetry

- Spherical symmetry is a sufficient but not necessary condition for quark-TMD relations

[C. Lorce’, B.P., Vdh, in preparation]

- Cilindrical symmetry around z direction:

nucleon spin

quark spin

- Cilindrical symmetry around Tx

- Spherical symmetry: invariance for any spin rotation

- Spherical symmetry and SU(6) spin-flavor symmetry

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