Quark spin and orbital angular momentum from proton generalized parton distributions

We calculate the leading-twist helicity-dependent generalized parton distributions (GPDs) of the proton at finite skewness in the Nambu–Jona-Lasinio (NJL) model of quantum chromodynamics (QCD). From these (and previously calculated helicity-independent GPDs) we obtain the spin decomposition of the proton, including predictions for quark intrinsic spin and orbital angular momentum. The inclusion of multiple species of diquarks is found to have a significant effect on the flavor decomposition, and resolving the internal structure of these dynamical diquark correlations proves essential for the mechanical stability of the proton. At a scale of $Q^2=4$ ${\mathrm{GeV}}^2$ we find that the up and down quarks carry an intrinsic spin and orbital angular momentum of $S_u=0.534$, $S_d=-0.214$, $L_u=-0.189$, and $L_d=0.210$, whereas the gluons have a total angular momentum of $J_g=0.151$. The down quark is therefore found to carry almost no total angular momentum due to cancellations between spin and orbital contributions. Comparisons are made between these spin decomposition results and lattice QCD calculations.

Figure 1

Diagrams contributing to the leading-twist proton GPDs. On the left is the quark diagram and on the right is the diquark diagram. The single line is the dressed quark propagator, the double line is the diquark propagator, the shaded oval is the Faddeev vertex, and the shaded circles represent the dressed quark and diquark GPDs.

Figure 2

The helicity-dependent proton GPDs at the model scale of $Q^2=0.16$ ${\mathrm{GeV}}^2$, where the GPD ${\tilde{E}}^q(x,\xi ,t)$ has been scaled by a factor of $\tau =-t/\left(4M_N^2\right)$. The top row is for $\xi =0$ and the bottom row has $\xi =0.5$. The transparent (orange) surface is up quarks and the opaque (blue) surface is down quarks.

Figure 3

The helicity-dependent proton GPDs at the scale of $Q^2=4.0$ ${\mathrm{GeV}}^2$, where the GPD ${\tilde{E}}^q(x,\xi ,t)$ has been scaled by a factor of $\tau =-t/\left(4M_N^2\right)$. The top row is for $\xi =0$ and the bottom row has $\xi =0.5$. The transparent (orange) surface is up quarks and the opaque (blue) surface is down quarks.

Figure 4

Spin decomposition at the model scale. The scalar-diquark-only model variant is contrasted with the full model containing both scalar and axial vector diquarks.

Figure 5

Comparison of NJL model spin decomposition to the lattice QCD results of Ref. [43] at $Q^2=4$ ${\mathrm{GeV}}^2$. The NJL results contain contributions from both scalar and axial vector diquarks.