Structure and Dynamics of the Nucleon Spin on the Light-Cone Barbara Pasquini in collaboration with Sigfrido Boffi, Sofia Cazzaniga University of Pavia and INFN Pavia, Italy

Outline Three-Quark Light-Cone Amplitudes of the Nucleon Spin-Spin and Spin-Orbit Correlations Conclusions Transverse Momentum Dependent Parton Distributions Generalized Parton Distributions Form Factors in the transverse plane Parton Distributions shape of the nucleon spin densities charge and helicities densities kinematical relations of the light-cone spin

Three Quark Light Cone Amplitudes LCWF: invariant under boost, independent of P internal variables: ‘uds’ basis total quark helicity Jq classification of LCWFs in angular momentum components Jz = Jzq + Lz (X. Ji, J.-P. Ma, F. Yuan, 03) 6 independent wave function amplitudes: isospin symmetry parity time reversal Lz = 0 Lz = 1 Lz = -1 Lz = 2

Light Cone Constituent Quark Model Instant form: x0 time; x1, x2, x3 space Light-front form: x+ time; x-, x space Instant Form (canonical) eigenvalue equation free mass operator : interaction operator Light-front eigenvalue equation generalized Melosh rotations

Light Cone Spin Jz = Jzq Jz = Jzq +Lzq Lzq = -1 Lzq =0 Lzq =1 Lzq =2 Instant-form wave function: Schlumpf, Ph.D. Thesis, hep-ph/9211255 momentum-space component: S wave spin and isospin component: SU(6) symmetric Jz = Jzq Melosh Rotations Light-cone wavefunction breaking of SU(6) symmetry non-zero quark orbital angular momentum Jz = Jzq +Lzq Six independent wave function amplitudes 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

Relevance of OAM to Nucleon Structure Transverse Momentum Dependent Parton Distributions Generalized Parton Distributions Nucleon Spin Densities Anomalous Magnetic Moment of the Nucleon Helicity-Flip Pauli Form Factor ………… ……………

Quark-Quark Distribution Correlation Function + quark-number density G = +5 quark-helicity density isx+ 5 transverse-spin density k and  dependent correlator: l, k l’, k’

Spin-Spin and Spin-Orbit correlations of quarks in the nucleon FT   b =0 TMD PDs GPDs FT   b spin densities PDs =0 Form Factors FT   b charge densities

Spin-Orbit Correlations and the Shape of the Nucleon G.A. Miller, PRC76 (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 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, -1 Jakob, et al., (1997) deformation due only to Lz=1 and Lz=-1 components x S s up quark x S K =0 K =0.25 GeV K =0.5 GeV G.A. Miller, PRC76 (2007)

Light Cone Constituent Quark Model x S x S s 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, in preparation

Angular Momentum Decomposition of h1Tu 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] x k2 SUM compensation of opposite sign contributions no deformation B.P., Cazzaniga, Boffi, in preparation

Spin dependent densities for down quark Diquark spectator model: contribution of Lz=+1 and Lz=-1 components x S x S s Light-cone CQM: contribution of Lz=+1 and Lz=-1 components plus contribution of Lz=0 and Lz=2 components x S s x S

Angular Momentum Decomposition of h1Td h1T d [GeV-3] h1T d [GeV-3] Lz=+1 and Lz=-1 Lz=0 and Lz=2 k2 x k2 x SUM partial cancellation of different angular momentum components non-spherical shape h1T d [GeV-3] x k2

Nucleon Spin densities average transverse position of the partons x b R b: transverse distance between the struck parton and the centre of momentum of the hadron xi, bi Nucleon state polarized in the X direction in IMF Impact parameter dependent GPD for the  pol. state quark density in proton state  pol. First moments of qX: probability density of unpolarized quark in a  pol. nucleon in impact parameter space Burkardt, 2003

Unpolarized quarks in a transversely pol. nucleon H - Sxby E’/M H - Sxby E’/M up up up down down down  flavor dipole moment correlation q and Sivers function and consistent with HERMES data, PRL 94 (2005)

Transverse Spin Densities Fourier transform of Tensor GPDs at  = 0: distributions in the transverse plane of transversely polarized quarks in a transversely polarized nucleon Diehl, Haegler, 2005 transversity basis projector on the transverse quark spin s monopole dipole quadrupole First moments of : transverse spin probability densities in impact parameter space

Transversely pol. quarks in a unpolarized nucleon H - sxby (E’T+2H’T) H - sxby (E’T+2H’T) up up up down down down down  spin-flavor dipole moment: correlation Tq and Boer-Mulders function sizeable Boer-Mulders effect consistent with lattice results (QCDSF/UKQCD Coll.,PRL98, 2007)

Transversely pol. quarks in a transversely pol. nucleon sx S x (HT-bHT/4M2) sx Sx (b2x-b2y) H’’T  (b, sx, Sx ) up up up down down down -0.04 up average quadrupole distortion 0.07 down

Transversely pol. quarks in a transversely pol. nucleon Sy Sy bx E’ – sxby(E’T+2 HT)/M sx Sy bx by H’’T  (b, sx, Sy ) up up up down down down

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

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  u u 2u+u 2u-u d (d+d)/2 d (d-d)/2

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

Summary Relativistic effects due to Melosh rotations in LCWF introduce a non trivial spin structure and correlations between quark spin and quark orbital angular momentum Spin-Orbit Correlations in TMD Parton Distributions shape of the nucleon interplay of different angular momentum components Transverse Spin Densities in the Impact Parameter Space Charge and helicity densities of the nucleon show unexpected distributions for up and down quark distributions Parton Distributions non-trivial relations for the valence quark contributions to f1, g1, h1 at the hadronic scale Transversely pol. quarks in an unpolarized nucleon sizeable Boer-Mulder function effect with the same sign for up and down quark, as seen by lattice results Unpolarized quarks in a transversely pol. nucleon opposite sign for Sivers function of up and down quark as seen by HERMES

Longitudinally pol. quarks in a longitudinally pol. nucleon Probability density in impact parameter space of longitudinally polarized quarks in a longitudinally polarized nucleon Burkardt, 2003 H H H + H

Momentum Space WF spin and isospin component: SU(6) symmetric momentum-space component: S wave Three free parameters: mq, ,  fitted to reproduce the magnetic moment of the proton and the axial coupling constant gA m q = 263 MeV  = 607 MeV  = 3.5 Schlumpf, PhD thesis, hep-ph/9211255 Light-cone wavefunction breaking of SU(6) symmetry non-zero quark orbital angular momentum Melosh Rotations The boost to infinite momentum frame (Melosh Rotations) introduces a non trivial spin structure and a correlation between quark spin and quark orbital angular momentum