Wigner Distributions in Light-Cone Quark Models Barbara Pasquini

Pavia U. & INFN, Pavia in collaboration with

Cédric Lorcé

Mainz U. & INFN, Pavia

Outline

Generalized Transverse Momentum Dependent Parton Distributions (GTMDs) FT    b  Wigner Distributions  General formalism for the 3-quark contribution to GTMDs  Results for Wigner distributions in light-cone quark models  unpolarized quarks in unpolarized nucleon (generalized transverse charge density)  Relations between GPDs and TMDs  Relations among TMDs in various 3-quark models  Orbital Angular Momentum in terms of LCWF: Ji’s vs. Jaffe-Manohar’s definition

Generalized TMDs

GTMDs  Complete parametrization : 16 GTMDs [Meißner, Metz, Schlegel (2009)]  Fourier Transform : 16 Wigner distributions [Belitsky, Ji, Yuan (2004)] x: average fraction of quark k ?

longitudinal momentum : average quark transverse momentum » : fraction of longitudinal momentum transfer ¢ : nucleon momentum transfer

GTMDs TMDs

FT    b 

Wigner-Ds GPDs

FT

spin densities PDs Form Factors

FT

charge densities

Wigner Distributions

 Quantum phase-space distribution: most complete information on wave function quantum molecular dynamics, signal analysis, quantum info, optics, image processing,… [Wigner, (1932)]  Wigner distributions in QCD: at » =0 !

diagonal in the Fock-space N N N=3 !

overlap of quark light-cone wave-functions  real functions, but in general not-positive definite not probabilistic interpretation correlations of quark momentum and position in the transverse plane as function of quark and nucleon polarizations  no known experiments can directly measure them !

needs phenomenological models

Light-Cone Quark Models

LCWF: invariant under boost, independent of P  internal variables: [Brodsky, Pauli, Pinsky, ’98] momentum wf spin-flavor wf rotation from canonical spin to light-cone spin Bag Model, Â QSM, LCQM, Quark-Diquark and Covariant Parton Models Common assumptions :   No gluons Independent quarks

Light-Cone Helicity and Canonical Spin

LC helicity canonical spin rotation around an axis orthogonal to z and k ?

Light-Cone CQM Chiral Quark-Soliton Model Bag Model (Melosh rotation)

Light-cone gauge A + =0 Wilson line r reduces to the identity Twist-2 operators º !

quark polarization ¹ !

nucleon polarization ( ) 16 GTMDs

3-Quark Overlap representation

Quark line : Model Independent Spin Structure Active quark : Spectator quarks :

3-Quark Overlap representation

16 GTMDs Assumption :  symmetry [C.Lorce’, B. Pasquini, M. Vanderhaeghen (in preparation)]

Light-Cone Constituent Quark Model

 momentum-space wf [Schlumpf, Ph.D. Thesis, hep-ph/9211255] parameters fitted to anomalous magnetic moments of the nucleon : normalization constant  spin-structure: free quarks (Melosh rotation)  SU(6) symmetry Applications of the model to: GPDs and form factors: BP, Boffi, Traini (2003)-(2005); TMDs : BP, Cazzaniga, Boffi (2008); BP, Yuan (2010); Azimuthal asymmetries: Schweitzer, BP, Boffi, Efremov (2009)

k

T

b

Wigner function for unpolarized quark in unpolarized nucleon

q k

T

, µ fixed [C.Lorce’, B. P. (in preparation)] µ = ¼ /2 µ = 0

µ

0

¼

/2

¼

3/2

¼ k

T

, q fixed k

T

Unpolarized u quark in unpolarized proton k

T

k

T

k

T

0.1 GeV 0.2 GeV 0.3 GeV 0.4 GeV µ = ¼ /2 µ = 0

k

T

Generalized Transverse Charge Densities

= + k

T

fixed Unpolarized u quark in unpolarized proton µ = 0 µ = ¼ /2

 Integrating over b ? TMD  Integrating over k ?

charge density in the transverse plane b ?

proton unpolarized u and d quarks in unpolarized proton neutron charge distribution in the transverse plane [Miller (2007); Burkardt (2007)]

GTMDs GPDs TMDs

GPDs and TMDs probe the same overlap of quark LCWFs in different kinematics nucleon quark UU LL TT LT 0 at » =0 UT TU TT TL 0

Relations between GPDs and TMDs in Quark Models

GPDs and TMDs probe the same overlap of LCWFs in different kinematics there exist relations in particular kinematical limits?

Trivial Relations UU LL TT Non-Trivial Relations TT valid also in spectator model : Meissner, Metz, Goeke, PRD76(2007) (with a factor 3 instead of 2) Model dependent relations [Burkardt, Hwang, 2003; Burkardt, 2005] SSA= GPDFSI valid for both Sivers and Boer-Mulders functions in spectator model, but breaks down at higher-orders

s x LT 0 TL 0 GPDs at » =0 vanish because of time-reversal invariance

up down

BP, Cazzaniga, Boffi, PRD78 (2008); Lorce`, BP, in preparation Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009) Light-cone quark model: !

consistent with lattice calculations

Relations among TMDs in Quark Models

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)] [Lorcé & 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)] Common assumptions :   No gluons Independent quarks

Light-cone Helicity and Canonical Spin

LC helicity Quark polarization  Rotations in light-front dynamics depend on the interaction, while are kinematical in canonical quantization we study the rotational symmetries for TMDs in the basis of canonical spin rotation around an axis orthogonal to z and k an angle µ = µ (k) ?

of Canonical spin Quark polarization

Rotational Symmetries in Canonical-Spin Basis

 Cilindrical symmetry around z direction nucleon spin quark spin  Cilindrical symmetry around T y k k ?

 Spherical symmetry: invariance for any spin rotation  Spherical symmetry and SU(6) spin-flavor symmetry [C. Lorce’, B.P., in preparation]

Orbital Angular Momentum

not unique decomposition gauge invariant, but contains interactions through the gauge covariant derivative [ X. Ji, PRL 78, (1997) ] Ji’s sum rule quark orbital angular momentum: not gauge invariant, but diagonal in the LCWFs basis [ R.L. Jaffe, NPB 337, (1990) ] What is the difference between the two definitions in a quark model without gauge fields?

scalar diquark model: M. Burkardt, PRD79, 071501 (2009); LCCQM: BP, F. Yuan, in preparation

Three Quark Light Cone Amplitudes  classification of LCWFs in angular momentum components [Ji, J.P. Ma, Yuan, 03; Burkardt, Ji, Yuan, 02] L z q = -1 L z q =0 J z = J z q + L z q L z q =1 total quark helicity J q L z q =2 parity time reversal isospin symmetry 6 independent wave function amplitudes: L z z q q = 2

Quark Orbital Angular Momentum

Jaffe-Manohar and Ji OAM should coincide when A=0 !

no-gluons, only quark contribution  Jaffe-Manohar definition: overlap of LCWFs with ¢ L z =0  Ji’s definition: ¢ L z =0 ¢ L z = 1 ¢ L z =0 interference between LCWFs with different L z it is not trivial to have the same orbital angular momentum for the quark contribution

Distribution in x of Orbital Angular Momentum Definition of Jaffe and Manohar: contribution from different partial waves TOT L z =0 L z =-1 L z =-1 L z =+2 up down total result, sum of up and down contributions: Jaffe Manohar’s vs. Ji’s definition Jaffe-Manohar Ji even in a model without gauge fields the two definitions give different distributions in x

Orbital Angular Momentum  Definition of Jaffe and Manohar: contribution from different partial waves = 0 ¢ 0.62 + (-1) ¢ 0.14 + (+1) ¢ 0.23 + (+2) ¢ 0.018 = 0.126

 Definition of Ji: [BP, F. Yuan, in preparation] [scalar diquark model: M. Burkardt, PRD79, 071501 (2009)]

 GTMDs $ Wigner Distributions

Summary

- the most complete information on partonic structure of the nucleon  General Formalism for 3-quark contribution to GTMDs - applicable for large class of models: LCQMs, Â QSM, Bag model  Results for Wigner distributions in the transverse plane - anisotropic distribution in k ?

even for unpolarized quarks in unpolarized nucleon  GPDs and TMDs probe the same overlap of 3-quark LCWF in different kinematics - give complementary information useful to reconstruct the nucleon wf  Relations of TMDs in a large class of models due to rotational symmetries in the quark-spin space - useful to test them with experimental observables in the valence region  Orbital Angular Momentum in terms of LCWFs: - in quark models, the total OAM (but not the distributions in x) is the same from JI and Jaffe-Manohar definitions

Backup

Wigner function for transversely pol. quark in longitudinally pol. nucleon

T-odd b s x k

T

q k

T

, µ fixed µ = ¼ /2 µ = 0

Wigner function for transversely pol. quark in longitudinally pol. nucleon

s x k

T

fixed Dipole µ = 0 µ = ¼ /2 Monopole Monopole + Dipole

µ

0

¼

/2

¼

3/2

¼ k

T

, q fixed u quark pol. in x direction in longitudinally pol.proton k

T

k

T

k

T

k

T

0.1 GeV 0.2 GeV 0.3 GeV 0.4 GeV

º !

quark polarization ¹ !

nucleon polarization 16 GTMDs ( ) Active quark : Spectator quarks : Model Independent Spin Structure

Orbital Angular Momentum in the Light-Front

 Light-cone Gauge A + =0 and advanced boundary condition for A   generalization of the relation for the anomalous magnetic moment: [Brodsky, Drell, PRD22, 1980] complex LCWFs due to FSI/ISI [Brodsky, Gardner, PLB 643, 2006] [Brodsky, BP, Yuan, Xiao, PLB 667, 2010