Modeling the TransverseMomentum Dependent Parton Distributions Barbara Pasquini

Modeling the Transverse-Momentum Dependent Parton Distributions Barbara Pasquini (Uni Pavia & INFN Pavia, Italy)

Outline TMDs and Quark Model Results Ø origin of quark-model relations among T-even TMDS Light-Cone Constituent Quark Model Ø quark-quark correlation function ! overlap representation in terms of light-cone amplitudes which are eigenstates of orbital angular momentum Results for T-even and T-odd TMDs Leading-Twist Spin Asymmetries in SIDIS due to T-even TMDs Conclusions

TMDs , x, k G= ’, x, k + quark-number density + 5 quark-helicity density isx+ 5 transverse-spin density : Wilson line which ensures the color gauge invariance

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

v Rotations in light-front dynamics depend on the interaction we study the rotational symmetries for TMDs in the basis of canonical spin v 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? LC Canonical z k?

Rotational Symmetries in Canonical-Spin Basis v Cilindrical symmetry around z direction nucleon spin quark spin v Cilindrical symmetry around Ty k k? v Spherical symmetry: invariance for any spin rotation v 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]

TMD Relations in Quark Models v. Spectator models (Jakob, Mulders, Rodrigues, 1997; Gamberg, Goldstein, Schlegel, 2008) v Light-Cone Chiral quark Soliton Model (C. Lorce’, BP, Vanderhaeghen, in preparation) v Light-Cone Constituent Quark model (BP, Cazzaniga, Boffi, 2008) v Covariant parton model (Efremov, Teryaev, Schweitzer, Zavada, 2008) v Bag model (Yuan, Schweitzer, Avakian, 2009) see talk of P. Schweitzer All quark models which satisfy the TMD relations have spherical symmetry (Lz=0) v 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 Ø internal variables: Ø 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 v classification of LCWFs in angular momentum components total quark LC helicity Jq Jz = Jzq + Lzq = -1 Lzq =0 Lzq =1 parity time reversal isospin symmetry 6 independent wave function amplitudes: 1 2 0 LLzzqqq== -1 [Ji, J. P. Ma, Yuan, 03; Burkardt, Ji, Yuan, 02] Lzq =2

Light-Cone Quark Model v Phenomenological LCWF for the valence (qqq) component: ü momentum-space component: S wave parameters fitted to anomalous magnetic moments of the nucleon : normalization constant ü spin and isospin component in the instant-form: SU(6) symmetric Schlumpf, Ph. D. Thesis, hep-ph/9211255 Jz = Jzq ) Melosh rotation to convert the canonical spin of quarks in LC helicity Jz = Jzq +Lzq Six independent wave function amplitudes : eigenstates of the total orbital angular momentum operator in Light-Front dynamics Lzq = -1 Lzq =0 Lzq =1 Lzq =2 The six independent wave function amplitudes obtained from the Melosh rotations satisfy the model independent classification scheme in four orbital angular momentum components

TMDs in a Light-Cone CQM Ø 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, PRD 78, 2008

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

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

Lattice QCD Light-cone quark model up down BP, Cazzaniga, Boffi, PRD 78 (2008) up Haegler, Musch, Negele, Schaefer, Europhys. Lett. 88 (2009) v Lattice calculation supports results of Light-cone quark model: see talk of B. Musch down

T-odd TMDs ¸ ¸’ ¸ 2 P+ , ¤ ¸ 3 P+, ¤’ + h. c. v Gauge link provides the necessary phase to make non-zero T-odd distributions v. 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 distribution of unpolarized quark in a transversely polarized nucleon ¤ = - ¤’ ¸ = ¸’ ¢ Lz = 1 Boer-Mulders function distribution of transversely polarized quark in a unpolarized nucleon ¤ = ¤’ ¸ = -¸’ ¢ Lz = 1 T-odd distributions from interference of S-P and P-D waves of LCWFs

Sivers function TOT P-D int. S-P int. TOT v Burkardt sum rule is reproduced exactly: LCQM at the hadronic scale Q 0 2=0. 094 Ge. V 2 S-P int. P-D int. [Burkardt, PRD 69 (2004)] LCQM evolved to Q 0 2= 2. 5 Ge. V 2 with evolution code of unpolarized PDF Anselmino et al. , EPJA 39, (2009); PRD 71 (2005) Collins et al. , PRD 73 (2006) Efremov et al. , PLB 612 (2005) B. P. , Yuan: ar. Xiv: 1001. 5398 [hep-ph]: Brodsky, B. P. , Xiao, Yuan, PLB 327 (2010)

Boer-Mulders function S-P int. P-D int. S-P int. TOT LCQM at the hadronic scale Q 0 2=0. 094 Ge. V 2 P-D int. TOT LCQM evolved to Q 0 2= 2. 5 Ge. V 2 with evolution code of transversity Barone, Melis, Prokudin ar. Xiv: 0912. 5194 [hep-ph] Zhang, Lu, Ma, Schmidt PRD 78 (2008) B. P. , Yuan: ar. Xiv: 1001. 5398 [hep-ph]: Brodsky, B. P. , Xiao, Yuan, PLB 327 (2010)

SIDIS l N ! l’ h X X=beam polarization Y=target polarization weight=ang. distr. hadron

Collinear double spin-asymmetries ALL and A 1 Ø convolution integrals between parton distributions and fragmentation functions can be solved analytically without approximation Ø no complications due to k? dependence Ø evolution equations and fragmentation functions are known we can test the model under “controlled conditions”: ü in which range and with what accuracy is the model applicable? ü how stable are the results under evolution?

ALL and A 1 at Q 2=2. 5 Ge. V 2 [Kretzer, PRD 62, 2000] evolved to exp. <Q 2> =3 Ge. V 2 initial scale Q 20=0. 079 Ge. V 2 where q <xq>=1 SMC HERMES inclusive longitudinal asymmetry Ødescription of exp. data within accuracy of 20 -30% in the valence region Øvery weak scale dependence [Boffi, Efremov, Pasquini, Schweitzer, PRD 79, 2009]

Strategy to calculate the azimuthal spin asymmetries Ø we focus on the x-dependence of the asymmetries, especially in the valence-x region Ø we adopt Gaussian Ansatz Ø low hadronic scale ! <k 2? (f 1)>(MODEL) =0. 08 Ge. V 2 is much smaller than phenomelogical value <k 2? (f 1)>(PHEN>)=0. 33 Ge. V 2 (fit to SIDIS HERMES data assuming gaussian Ansatz) [Collins, et al. , PRD 73, 2006] Ø we rescale the model results for <k 2? (TMD)> with <k 2? (f 1)>(MODEL)/ <k 2? (f 1)>(PHEN) Ø 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 gaussian ansatz Ø Ø from Light-Cone CQM evolved at Q 2=2. 5 Ge. V 2, from GRV at Q 2=2. 5 Ge. V 2 from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD 73 (2006); Anselmino et al. , PRD 75 (2007); Vogelsang, Yuan, PRD 72 (2005) 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. , PLB 673 (2009)

Transversity v Dashed area: extraction of transversity from BELLE, COMPASS, and HERMES data Anselmino et al. , PRD 75, 2007 v Predictions from Light-Cone CQM evolved from the hadronic scale Q 20 to Q 2= 2. 5 Ge. V 2 using two different momentum-dependent wf x h 1 q ü phenomenological wf ü three fit parameters solution of relativistic potential model up ü , and mq fitted to the anomalous magnetic moments of the nucleon and to g. A Schlumpf, Ph. D. Thesis, hep-ph/9211255 down ü no free parameters ü fair description of nucleon form factors Faccioli, et al. , NPA 656, 1999 Ferraris et al. , PLB 324, 1995 BP, Pincetti, Boffi, PRD 72, 2005

gaussian ansatz initial scale Q 20=0. 079 Ge. V 2 evolved to Q 2 =2. 5 COMPASS [Kotzinian, et al. , ar. Xiv: 0705. 2402] using evolution eq. of at Q 2=2. 5 Ge. V 2 [Kretzer, PRD 62, 2000]

gaussian ansatz Ø from Light-Cone CQM evolved at Q 2=2. 5 Ge. V 2, with the evolution equations of Ø from GRV at Q 2=2. 5 Ge. V 2 Ø from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD 73 (2006); Anselmino et al. , PRD 75 (2007); Vogelsang, Yuan, PRD 72 (2005) HERMES Coll. Airapetian, PRL 84, 2000; Avakian, Nucl. Phys. Proc. Suppl. 79 (1999) Model results compatible with CLAS data for ¼+ and ¼ 0 but cannot explain the SSA for ¼CLAS Coll, : ar. Xiv: 1003. 4549 [hep-ex]

gaussian ansatz Ø from Light-Cone CQM evolved at Q 2=2. 5 Ge. V 2, with the evolution equations of Ø from GRV at Q 2=2. 5 Ge. V 2 Ø from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD 73 (2006); Ø smaller predictions than expected from positivity bounds with f 1 and g 1 from GRV at Q 2=2. 5 Ge. V 2 COMPASS Coll. Kotzinian, ar. Xiv: 0705. 2402 ü experiment planned at CLAS 12 (H. Avakian at al. , LOI 12 -06 -108) ü preliminary HERMES data compatible with zero ! talk of L. Pappalardo Anselmino et al. , PRD 75 (2007); Vogelsang, Yuan, PRD 72 (2005)

Summary Model relations among T-even TMDs in quark models Ø spherical symmetry of the models is a sufficient but not necessary condition Ø useful guideline for parametrizations of quark contributions to TMDs Model calculation in a Light-Cone CQM Ø representation in terms of overlap of LCWF with different orbital angular momentum Ø predictions for T-odd TMDs consistent with phenomenological parametrizations Predictions for all the leading-twist 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% 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

BACKUP SLIDES

Gaussian Ansatz Ø k? dependence of the model is not of gaussian form Ø how well can it be approximated by a gaussian form? =1 in Gauss Ansatz with Gaussian Ansatz exact result from Bacchetta et al. , PLB 659, 2008 results agree within 20% Gaussian Ansatz gives uncertainty within typical accuracy of the model

: orbital angular momentum decomposition gaussian ansatz Ø from Light-Cone CQM evolved at the hadronic scale Q 2=0. 079 Ge. V 2 Ø from GRV at Q 2=2. 5 Ge. V 2 from HERMES & BELLE data Efremov, Goeke, Schweitzer, PRD 73 (2006); Ø Anselmino et al. , PRD 75 (2007); Vogelsang, Yuan, PRD 72 (2005) giutgo COMPASS Coll. Kotzinian, ar. Xiv: 0705. 2402 TOT S-D wave int. P waves int.

Pretzelosity in SIDIS: perspectives At small x: ü There will be data from COMPASS proton target ü There will be data from HERMES More favorable conditions at intermediate x (0. 2 -0. 6) ü experiment planned at CLAS with 12 Ge. V (H. Avakian at al. , LOI 12 -06 -108) Light-Cone CQM Error projections for 2000 hours run time at CLAS 12 Schweitzer, Boffi, Efremov, BP, in preparation

h 1 L v chiral odd, no gluons v opposite sign of h 1 v h 1 L : SP and PD interference terms h 1: SS and PP diagonal terms v h 1 L h 1 with v Wandzura-Wilczek-type approximation Avakian, et al. , PRD 77, 2008 with h 1 L (1) WW approx. Light-Cone CQM

Pretzelosity v large, larger than f 1 and h 1 down v sign opposite to transversity Light-Cone CQM v light-cone quark model and bag model peaked at smaller x BP, Cazzaniga, Boffi PRD 78, 2008 v related to chiral-odd GPD [Meissner, et al. , PRD 76, 2007 and ar. Xiv: 0906. 5323 [hep-ph] Bag Model up Avakian, et al. , PRD 78, 2008 v positivity satisfied Chiral quark soliton Model Soffer inequality v helicity – transversity = pretzelosity down L. Cedric, B, . P. , Vdh, in preparation up

Relations of TMDs in Valence Quark Models (1) (2) Avakian, Efremov, Yuan, Schweitzer, (2008) (3) Ø (1), (2), and (3) hold in Light-Cone CQM Models BP, Pincetti, Boffi, PRD 72, 2005; BP, Cazzaniga, Boffi, PRD 78, 034025 (2008) Ø (1) and (2) hold in Bag Model Avakian, Efremov, Yuan, Schweitzer, PRD 78, 114024 (2008) Ø (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, NPA 626, 937 (1997) She, Zhu, Ma, PRD 79, 054008 (2009) Ø (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, PRD 78, 074010 (2008) Ø (2) holds in covariant quark-parton model Efremov, Schweitzer, Teryaev, Zavada, ar. Xiv: 0903. 3490 [hep-ph] ü no gluon dof valid at low hadronic scale ü not restricted to S and P wave contributions ü SU(6) symmetry is not crucial in the relation with the pretzelosity see talks by Schweitzer, Ma, Zavada

Charge density of partons in the transverse plane v Infinite-Momentum-Frame Parton charge density in the transverse plane no relativistic corrections G. A. Miller, PRL 99, 2007 fit to exp. form factor by Kelly, PRC 70 (2004) B. P. , Boffi, PRD 76 (2007) (meson cloud model) neutron proton Quark distributions in the neutron down up

Helicity density in the transverse plane v probability to find a quark with transverse position b and light-cone helicity in the nucleon with longitudinal polarization u 2 u+ u u 2 u- u d (d+ d)/2 d (d- d)/2

Light cone wave function overlap representation of Parton Distributions v Melosh rotations: relativistic effects due to the quark transverse motion consistent with Soffer bounds and (Ma, Schmidt, Soffer, ’ 97) v Non relativistic limit (k 0) : up B. P. , Pincetti, Boffi, 05 down

Spin-Orbit Correlations and the Shape of the Nucleon G. A. Miller, PRC 76 (2007) 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

Spin-dependent densities Ø Fix the directions of S and n the spin-orbit correlations measured with responsible for a non-spherical distribution with respect to the spin direction is : chirally odd tensor correlations matrix element from angular momentum components with |L z-L’z|=2 v Diquark spectator model: wave function with angular momentum components L z = 0, +1, -1 deformation due only to Lz=1 and Lz=-1 components Jakob, et al. , (1997) sx up quark Sx Sx K =0 K K =0. 5 =0. 25 Ge. V G. A. Miller, PRC 76 (2007)

Light Cone Constituent Quark Model sx Sx up quark Sx deformation induced from the Lz=+1 and Lz=-1 components adding the contribution from Lz=0 and Lz=2 components B. P. , Cazzaniga, Boffi, PRD 78, 2008

Angular Momentum Decomposition 1 T of uh Lz=0 and Lz=2 Lz=+1 and Lz=-1 h 1 T u [Ge. V-3] k 2 x x SUM h 1 T u [Ge. V-3] compensation of opposite sign contributions no deformation k 2 x B. P. , Cazzaniga, Boffi, PRD 78, 2008

Spin dependent densities for down quark v Diquark spectator model: contribution of Lz=+1 and Lz=-1 components sx Sx Sx v Light-cone CQM: contribution of Lz=+1 and Lz=-1 components plus contribution of L z=0 and Lz=2 components sx Sx Sx

Angular Momentum Decomposition 1 T of dh h 1 T d [Ge. V-3] Lz=+1 and Lz=-1 k 2 x SUM h 1 T d [Ge. V-3] partial cancellation of different angular momentum components non-spherical shape Lz=0 and Lz=2 k 2 x x

Evolve in light-front time ¿ = t+z Evolve in ordinary time ¾=t-z Front Form Instant Form coordinates x 0 time x 1, x 2, x 3 space x+=x 0+x 3 time x-=x 0 -x 3, x 1, x 2 space Hamiltonian P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949)

Light Cone Amplitudes Overlap Representation of TMDs ¢ Lz=0 S S P P P P D D ¢ Lz=0 S S P P

|¢ Lz |=1 P S P D |¢ Lz |=1 S P P D P P |¢ Lz |=2 D S

Pretzelosity down v large, larger than f 1 and h 1 v sign opposite to transversity v light-cone quark model and bag model peaked at smaller x v related to chiral-odd GPD [Meissner, et al. , PRD 76, 2007 and ar. Xiv: 0906. 5323 [hep-ph] Light-Cone CQM BP, Cazzaniga, Boffi PRD 78, 2008 up Bag Model Avakian, et al. , PRD 78, 2008 v positivity satisfied Soffer inequality Spectator Model down Jakob, et al. , NPA 626 (1997) up v helicity – transversity = pretzelosity

Rotational Symmetries in Canonical-Spin Basis v Cilindrical symmetry around z direction nucleon spin quark spin v Cilindrical symmetry around Ty k k? v Spherical symmetry: invariance for any spin rotation v Spherical symmetry and SU(6) spin-flavor symmetry üSpherical symmetry is a sufficient but not necessary condition for quark-TMD relations [C. Lorce’, B. P. , Vdh, in preparation]

Rotational Symmetries v Cilindrical symmetry around z direction: nucleon spin quark spin v Cilindrical symmetry around Tx v Spherical symmetry: invariance for any spin rotation v Spherical symmetry and SU(6) spin-flavor symmetry