1. 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

2. 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

3. 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

4. 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

5. 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/
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

6. 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
…………
……………

7. 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’

8. 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

9. 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

10. 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)

11. Light Cone Constituent Quark Modelx 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

12. 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

13. 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

14. 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

15. 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

16. Unpolarized quarks in a transversely pol. nucleonH
- 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)

17. 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

18. Transversely pol. quarks in a unpolarized nucleonH
- 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)

19. 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

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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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/
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