Three-Quark Light-Cone Wave function of the Nucleon. Spin-Spin and Spin-Orbit Correlations in T-even TMDs

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1 TMDs and Azimuthal Spin Asymmetries in Light-Cone Quark Models Barbara Pasquini (Uni Pavia & INFN Pavia, Italy) in collaboration with: S. Boffi (Uni Pavia & INFN Pavia) A.V. Efremov (JINR, Dubna) P. Schweitzer (Uni Connecticut)

2 Outline Three-Quark Light-Cone Wave function of the Nucleon Spin-Spin and Spin-Orbit Correlations in T-even TMDs overlap representation in terms of three-quark light-cone amplitudes which are eigenstates of orbital angular momentum results in a light-cone CQM Leading-Twist Single Spin Asymmetries in SIDIS due to T-even TMDs Comparison with available experimental data from CLAS, COMPASS, HERMES Conclusions

3 Light-Cone Fock Expansion fixed light-cone time internal variables: ables frame INdependent probability amplitude to find the N parton configuration with the complex of quantum number β in the nucleon with helicity λ 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

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

5 Light-Cone Quark Model Phenomenological LCWF for the valence (qqq) component: momentum-space component: S wave Schlumpf, Ph.D. Thesis, hep-ph/ ph/ spin and isospin component in the rest frame: SU(6) symmetric fitted to anomalous magnetic moments of the nucleon : normalization constant J z = J q z Melosh rotation to convert the rest-frame spins of quarks in LF spins J q z = J zq +L q z Six independent wave function amplitudes : eigenstates of the total orbital angular momentum operator in Light-Front dynamics L zq = -1 L zq =0 L zq =1 L zq =2 The six independent wave function amplitudes obtained from the Melosh rotations satisfy the model independent classification scheme in four orbital angular momentum components

6 Time-Even TMDs λ, x, k λ, x, k Light-cone Gauge A + =0 and advanced boundary condition for A no gauge link Γ = γ + γ + γ 5 iσ x+ γ 5 quark-number density quark-helicity density transverse-spin density

7 Light Cone Amplitudes Overlap Representation of TMDs L z =0 S S P P P P D D L z =0 S S P P P P D D L z =0 S S P P

8 L z =1 P S P S P D P D L z =1 S P S P P D P D P P L z =2 2 D S

9 TMDs in a Light-Cone CQM SU(6) symmetry N u =2 N d =1 P u =4/ 3 P d = -1/3 momentum m dependent n wf factorized from spin-dependent p n n effects 3 relations between the TMDs B.P., Cazzaniga, Boffi, PRD78, 2008

10 Orbital angular momentum content TOT S wave Pwave Dwave f 1 g 1L h 1 up up up f 1 g 1L h 1 TOT S wave Pwave Dwave down down down Total results obey SU(6) symmetry relations: f 1u = 2 f 1d, g 1Lu =-4 g 1Ld, h 1u =-4 h 1 d The partial wave contributions do not satisfy SU(6) symmetry relations!

11 Orbital angular momentum content g 1T (1) h 1L (1) h 1T (1) up TOT S-P int. P-D Dint. P-D int. S-P int. P-P int. S-D int. down TOT g 1T (1) h 1L (1) h 1T (1) S-P int. P-P int. P-D int. P-D Dint. S-P int. S-D int. ONLY TOTAL results (and not partial wave contr.) obey SU(6) symmetry relations: g 1T (1) u = -4g 1T (1) d, h 1L (1) u =-4 h 1L (1) d, h 1T (1) u =-4 h 1T (1) d

12 Relations of TMDs in Valence Quark Models (1) (2) Avakian, Efremov, Yuan, Schweitzer, (2008) (3) (1), (2), and (3) hold in Light-Cone CQM Models BP, Pincetti, Boffi, PRD72, 2005; BP, Cazzaniga, Boffi, PRD78, (2008) (1) and (2) hold in Bag Model Avakian, Efremov, Yuan, Schweitzer, PRD78, (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) Gamberg, Goldstein, Schlegel, PRD77, (2008) She, Zhu, Ma, PRD 79, (2009) (1) and (2) holds in covariant quark-parton model Efremov, Schweitzer, Teryaev, Zavada, PRD80, (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 contribution Bacchetta, Conti, Radici, PRD78, (2008) no gluon dof valid at low hadronic scale there are NO EXACT relations between TMDs in QCD, but having well-motivated approximations is valuable!

13 helicity transversity = pretzelosity Positivity constraint: Bacchetta, Boglione, Henneman, Mulders PRL85, (2000) Soffer inequality scale of the model: Q 2 0= GeV 2 where 2 2 q <x q >=1 after evolution to Q =2.5 GeV up up h 1T (1) g 1 h 1 g 1 h 1 approximate evolution for h 1T (1), using evolution equations of transversity

14 Light-cone quark model Lattice QCD up down up down BP, Cazzaniga, Boffi, PRD78 (2008) <k x >=55.81 MeV (up) g 1T = - h 1L <k x >= MeV (down) Haegler, Musch, Negele, Schaefer, arxiv: [hep-lat] g 1T : <k x >=67(5) MeV; h 1L = <k x >=-60(5) MeV (up) g 1T : <k x >=-30(5) MeV ; h 1L = 16(5) MeV (down) Light-cone quark model: Lattice calculation: supports predictions from light-cone QM

15 correlations in k, S correlations in b, S correlations in k, s correlations in b, s correlations in k,s,s correlations in b,s,s correlations in k, Λ, s time-reversal odd GPD=0 correlations in k, S, λ time-reversal odd GPD=0 Diehl, Haegler, 2005; Meissner, Metz, Schlegel, 2009

16 correlations in k, S, s correlations in b, S, s trivial relations: nontrivial, model-dependent relation: up up down no Fourier transform down

17 SIDIS l N l h X X=beam polarization Y=target t polarization weight=ang. distr. hadron Bacchetta, et al., JHEP0702, 2007

18 Collinear double spin-asymmetries A LL and A 1 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?

19 A LL and A 1 at Q 2 =2.5 GeV 2 [Kretzer, PRD62, 2000] evolved to exp. <Q 2 > =3 GeV 2 initial scale Q 2 0=0.079 GeV 2 where q <x q >=1 SMC HERMES inclusive i longitudinal l asymmetry [Boffi, Efremov, Pasquini, Schweitzer, PRD79, 2009] description of exp. data within accuracy of 20-30% in the valence region very weak scale dependence

20 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 from Bacchetta et al., PLB659, 2008 results agree within 20% Gaussian Ansatz gives uncertainty within typical accuracy of the model

21 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 <k 2 (f 1 )>(MODEL) =0.08 GeV 2 is much smaller than phenomelogical value <k 2 (f 1 )>(PHEN>)=0.33 GeV 2 (fit to SIDIS HERMES data assuming gaussian Ansatz) [Collins, et al., PRD73, 2006] we rescale the model results for <k 2 (TMD)> with <k 2 (f 1 )>(MODEL)/ <k 2 (f 1 )>(PHEN) we do not discuss the z and P h dependence of azimuthal asymmetries because here integrals over the x dependence extend to low x-region where the model is not applicable

22 Collins SSA gaussian ansatz from Light-Cone CQM evolved at Q 2 =2.5 GeV 2, from GRV at Q 2 =2.5 GeV 2 from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005) HERMES data: Diefenthaler, hep-ex/ More recent HERMES and BELLE data not included in the fit of Collins function COMPASS data: Alekseev et al., PLB673, (2009)

23 Transversity 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 Q 2 0 to Q 2 = 2.5 GeV 2 using two different momentum-dependent wf x h q 1 phenomenological wf three fit parameters β, γ and m q fitted to the anomalous magnetic moments of the nucleon and to g A Schlumpf, Ph.D. Thesis, hep-ph/ up down BP, Pincetti, Boffi, PRD72, 2005 solution of relativistic potential model no free parameters fair description of nucleon form factors Faccioli, et al., NPA656, 1999 Ferraris et al., PLB324, 1995

24 h 1L chiral odd, no gluons opposite sign of h 1 h 1L : SP and PD interference terms h 1 : SS and PP diagonal terms h 1L h 1 with with Wandzura-Wilczek-type approximation Avakian, et al., PRD77, 2008 h (1) 1L WW approx. WW approx. Light-Cone CQM

25 gaussian ansatz from Light-Cone CQM evolved at Q 2 =2.5 GeV 2, with the evolution equations of from GRV at Q 2 =2.5 GeV 2 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 preliminary CLAS data for π + and π 0 but cannot explain the SSA for π - more precise data expected

26 gaussian ansatz initial scale Q 2 0=0.079 GeV 2 evolved to Q 2 =2.5 GeV 2 using evolution eq. of evolved to Q 2 =2.5 at Q 2 =2.5 GeV 2 [Kretzer, PRD62, 2000] COMPASS [Kotzinian, et al., arxiv: ]

27 gaussian ansatz from Light-Cone CQM evolved at Q 2 =2.5 GeV 2, with the evolution equations of from GRV at Q 2 =2.5 GeV 2 from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD73 (2006); Anselmino et al., PRD75 (2007); Vogelsang, Yuan, PRD72 (2005) smaller predictions than expected from positivity bounds with f 1 and g 1 from GRV at Q 2 =2.5 GeV 2 experiment planned at CLAS12 (H. Avakian at al., LOI ) analysis in progress for HERMES data COMPASS Coll. Kotzinian, i arxiv:

28 Summary Model independent classification of TMDs in terms of three-quark Light-Cone amplitudes with different orbital angular momentum Model calculation in a Light-Cone CQM relativistic effects due to Melosh rotations in LCWF introduce a non trivial spin structure and correlations between quark spin and quark orbital angular momentum three non-trivial relations among T-even TMDs valid at low scale in a large class of relativistic quark models one nontrivial model-dependent relation between pretzelosity and chiral-odd GPD Predictions for all Azimuthal Spin Asymmetries in SIDIS due to T-even TMDs and : the model is capable to describe the data in the valence region with accuracy of 20-30% TMDs in the light-cone CQM have not gaussian shape, but we checked that, t within the accuracy of the model, the k dependence in the azimuthal asymmetries can be approximated with gaussian Ansatz 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