modeling the transverse-momentum dependent parton distributions

Modeling the Transverse-Momentum Dependent

Parton Distributions

Barbara Pasquini (Uni Pavia & INFN Pavia, Italy)

OutlineTMDs 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

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

origin of quark-model relations among T-even TMDS

l, x, k

l’, x, k

++5

G = isx+ 5

quark-number density

quark-helicity density

transverse-spin density

TMDs

:Wilson line which ensures the color gauge invariance

quark polarization

nucle

on p

olar

izatio

n

Relations among T-even TMDs in quark models

linear relations

quadratic relation

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

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

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

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

LC Canonical

z

k?

k?

k?

z

z

k?

k?

k?

z

z

z

Rotational Symmetries in Canonical-Spin Basis

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]

Cilindrical symmetry around Ty k k?

Cilindrical symmetry around z direction

nucleon spin

quark spin

TMD Relations in Quark Models

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

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

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

Light-Cone Fock Expansion

internal variables:

frame INdependent

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

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

fixed light-cone time

! talk of S. Brodsky

Three Quark Light Cone Amplitudes

6 independent wave function amplitudes: isospin symmetry

parity time reversal

total quark LC helicity Jq

classification of LCWFs in angular momentum components

Jz = Jzq + Lz

q

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

Lzq =2Lz

q =1Lzq =0Lz

q = -1

Lzq = 1 Lz q= 0 Lz

q = 2 Lzq

= -1

Light-Cone Quark Model Phenomenological LCWF for the valence (qqq) component:

Jz = Jzq

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

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

momentum components

Lzq =2Lz

q =1Lzq =0Lz

q = -1

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

dynamics

Jz = Jzq +Lz

q

TMDs in a Light-Cone CQM

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

S waveP waveD wave

TOT

Orbital angular momentum contentf1 g1L h1

S waveP waveD wave

TOT

f1 g1L h1

up up up

downdowndown

Total results obey SU(6) symmetry relations: f1u = 2 f1

d, g1Lu=-4 g1L

d, h1u=-4 h1

d

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

TOT

Orbital angular momentum content

g1T (1) h1L

?(1) h1T?(1)

S-P int.P-D int.

S-P int.P-D int.

P-P int.

S-D int.

TOT

g1T (1) h1L

?(1) h1T?(1)

S-P int.

P-D int.S-P int.

P-D int. P-P int.

S-D int.

down

up

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

up down up down

Light-cone quark model

Lattice QCD

BP, Cazzaniga, Boffi, PRD78 (2008)

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

Lattice calculation supports results of Light-cone quark model:

see talk of B. Musch

T-odd TMDs

Gauge link provides the necessary phase to make non-zero T-odd distributionsWe 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 functiondistribution 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

P+, ¤ P+, ¤’

¸ ¸’

¸2 ¸3 + h.c.

S-P int.

P-D int. S-P int.

P-D int.

Sivers function

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

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

Collins et al., PRD73 (2006)Efremov et al.,PLB612 (2005)

LCQM at the hadronic scale Q0 2=0.094 GeV2

LCQM evolved to Q0 2= 2.5 GeV2

with evolution code of unpolarized PDF

TOT

TOT

Burkardt sum rule is reproduced exactly: [Burkardt, PRD69 (2004)]

Boer-Mulders function

P-D int.S-P int.

S-P int.

P-D int.TOT TOT

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

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

Zhang, Lu, Ma, Schmidt PRD78 (2008)

LCQM at the hadronic scale Q0 2=0.094 GeV2

LCQM evolved to Q0 2= 2.5 GeV2

with evolution code of transversity

SIDIS l N ! l’ h X

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

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?

no complications due to k? dependence

ALL and A1

SMC

HERMES

at Q2=2.5 GeV2 [Kretzer, PRD62, 2000]evolved to exp. <Q2> =3 GeV2initial scale Q2

0=0.079 GeV2 where q <xq>=1

inclusive longitudinal asymmetry

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

description of exp. datawithin accuracy of 20-30%in the valence region

very weak scale dependence

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 GeV2 is 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

Collins SSA

from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005)

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

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

HERMES data: Diefenthaler, hep-ex/0507013

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

gaussian ansatz

Transversity

x h1q

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

up

down

Anselmino et al., PRD75, 2007

Predictions from Light-Cone CQM evolved from the hadronic scale Q20 to Q2= 2.5

GeV2

using two different momentum-dependent wf

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

phenomenological wf three fit parameters , and mq fitted to

the anomalous magnetic moments of the nucleon and to gA

Faccioli, et al., NPA656, 1999

Ferraris et al., PLB324, 1995

solution of relativistic potential model no free parameters fair description of

nucleon form factors

BP, Pincetti, Boffi, PRD72, 2005

gaussian ansatz

initial scale Q20=0.079 GeV2

evolved to Q2 =2.5 GeV2 using evolution eq. of

at Q2=2.5 GeV2

[Kretzer, PRD62, 2000]

COMPASS[Kotzinian, et al., arXiv:0705.2402]

evolved to Q2 =2.5


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

gaussian ansatz

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

HERMES Coll.

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

gaussian ansatz

Kotzinian, arXiv:0705.2402COMPASS Coll.

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

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

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

SummaryModel 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

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

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

representation in terms of overlap of LCWF with different orbital angular momentum

predictions for T-odd TMDs consistent with phenomenological parametrizations

spherical symmetry of the models is a sufficient but not necessary condition

useful guideline for parametrizations of quark contributions to TMDs

BACKUP SLIDES

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

=1 in Gauss Ansatz

with Gaussian Ansatz

exact result

results agree within 20% Gaussian Ansatz gives

uncertainty within typical accuracy of the model

from Bacchetta et al., PLB659, 2008


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

from GRV at Q2=2.5 GeV2

gaussian ansatz

Kotzinian, arXiv:0705.2402COMPASS Coll.

:orbital angular momentum decomposition

S-D wave int.P waves int.

TOT

giutgo

Pretzelosity in SIDIS: perspectivesAt 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

chiral odd, no gluons h1L

: SP and PD interference terms h1: SS and PP diagonal terms

opposite sign of h1

h1L h1

with with

h1L

Wandzura-Wilczek-type approximation Avakian, et al., PRD77, 2008

WW approx.

h1L (1)

WW approx.

Light-Cone CQM

Pretzelosity

down

up

Light-Cone CQM

BP, Cazzaniga, BoffiPRD78, 2008

Chiral quark soliton Model

L. Cedric, B,.P., Vdh, in preparation

Bag Model

Avakian, et al., PRD78, 2008

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

positivity satisfied

helicity – transversity = pretzelosity

Soffer inequality

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

down

up

Relations of TMDs in Valence Quark Models

(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

(1)

(2)(3)

see talks by Schweitzer,Ma,Zavada

Avakian, Efremov, Yuan, Schweitzer, (2008)

Charge density of partons in the transverse plane Infinite-Momentum-Frame Parton charge density in the transverse plane

G.A. Miller, PRL99, 2007

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

model)

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

neutron proton

Quark distributions in the neutron

down up

no relativistic corrections

Helicity density in the transverse plane

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

u

u

d

d

2u+u

2u-u

(d+d)/2

(d-d)/2

Light cone wave function overlap representation of Parton Distributions

Melosh rotations: relativistic effects due to the quark transverse motion

B.P., Pincetti, Boffi, 05

downup

Non relativistic limit (k 0) :consistent with Soffer bounds and (Ma, Schmidt, Soffer,’97)

Spin-Orbit Correlations and the Shape of the Nucleon

spin-dependent charge density operator in non relativistic quantum mechanics

G.A. Miller, PRC76 (2007)

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

Spin-dependent densities

K =0

K =0.25 GeV K =0.5 GeV

xs xSxSxs

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

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

0, +1, -1Jakob, et al., (1997)

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

up quark

G.A. Miller, PRC76 (2007)

Light Cone Constituent Quark Model

xSxs xs xS

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

up quark

Angular Momentum Decomposition of h1Tu

Lz=+1 and Lz=-1

k2 x

h1T u

[GeV-3]

Lz=0 and Lz=2

k2 x

h1T u

[GeV-3]

h1T u

[GeV-3]

xk2

SUM

compensation of opposite sign contributions

no deformationB.P., Cazzaniga, Boffi, PRD78, 2008

Spin dependent densities for down quark

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

xSxs xs xS

xSxs xs xS

Angular Momentum Decomposition of h1Td

Lz=+1 and Lz=-1

k2 x

h1T d

[GeV-3] Lz=0 and Lz=2

k2

h1T d

[GeV-3]

SUM

partial cancellation ofdifferent angular momentum components

non-spherical shape

h1T d

[GeV-3]

xk2

x

Evolve in ordinary time

Evolve in light-front time

¿ = t+z ¾=t-z

Instant Form

coordinatesx0 time

x1, x2, x3 spacex+=x0+x3 time

x-=x0-x3, x1, x2 space

Hamiltonian

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

Front Form

Light Cone Amplitudes Overlap Representation of TMDs

S S P P

P P D D

S S P P

S S P P

P P D D

¢ Lz=0

¢ Lz=0

¢ Lz=0

P S

P D

P S

P D

S P S P

P D P D

P P D S

|¢ Lz |=1

|¢ Lz |=1

|¢ Lz |=2

Pretzelosity

down

up

Light-Cone CQM

BP, Cazzaniga, BoffiPRD78, 2008

Spectator Model

Jakob, et al., NPA626 (1997)

Bag Model

Avakian, et al., PRD78, 2008

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

positivity satisfied

helicity – transversity = pretzelosity

down

up

Soffer inequality

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

Rotational Symmetries in Canonical-Spin Basis

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

Cilindrical symmetry around z direction

nucleon spin

quark spin



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

Cilindrical symmetry around Ty k k?

Rotational Symmetries

Cilindrical symmetry around z direction:

Cilindrical symmetry around Tx

nucleon spin

quark spin



modeling the transverse-momentum dependent parton distributions

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