The quark orbital angular momentum from Wigner distributions and light-cone wave functions

We investigate the quark orbital angular momentum of the nucleon in the absence of gauge-field degrees of freedom, by using the concept of the Wigner distribution and the light-cone wave functions of the Fock state expansion of the nucleon. The quark orbital angular momentum is obtained from the phase-space average of the orbital angular momentum operator weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon. We also derive the light-cone wave function representation of the orbital angular momentum. In particular, we perform an expansion in the nucleon Fock state space and decompose the orbital angular momentum into the $N$-parton state contributions. Explicit expressions are presented in terms of the light-cone wave functions of the three-quark Fock state. Numerical results for the up and down quark orbital angular momenta of the proton are shown in the light-cone constituent quark model and the light-cone chiral quark-soliton model.