Angular momentum and generalized parton distributions for the proton with basis light-front quantization

We study the unpolarized and the helicity-dependent generalized parton distributions (GPDs) for the valence quarks of the proton in both momentum space and position space within the basis light-front quantization (BLFQ) framework. The GPDs for the valence quarks are computed from the eigenvectors of a light-front effective Hamiltonian in the valence Fock sector consisting of a three-dimensional confinement potential and a one-gluon exchange interaction with fixed coupling. Employing these GPDs, we obtain the spatial distributions of quark angular momentum inside the proton. In our BLFQ approach, we explore various definitions of angular momentum density and illustrate the differences between them arising from terms that integrate to zero. We also discuss the flavor contributions to the quark angular momentum densities.

Figure 1

The valence quark GPDs of the proton: (a) $H(x,0,t)$, (c) $E(x,0,t)$, and (e) $\tilde{H}(x,0,t)$ are for the valence up quark; (b), (d), and (f) are the same as (a), (c), and (e), respectively, but for the valence down quark as functions of $x$ and $-t$. All the GPDs are presented at our model scale ${\mu }_0^2=0.195{\mathrm{GeV}}^2$.

Figure 2

The valence quark GPDs of the proton in the transverse impact-parameter space: (a) $H(x,b_{\perp })$, (c) $E(x,b_{\perp })$, and (e) $\tilde{H}(x,b_{\perp })$ are for the valence up quark; (b), (d), and (f) are same as (a), (c), and (e), respectively, but for the valence down quark as functions of $x$ and $b_{\perp }$. All the GPDs are presented at our model scale ${\mu }_0^2=0.195{\mathrm{GeV}}^2$.

Figure 3

Angular momentum distributions summing over the up and the down quark contributions multiplied by $b_{\perp }$ as functions of $b_{\perp }$: (a) the kinetic TAM density $⟨J^z⟩(b_{\perp })$ (black band) as the sum of the spin $⟨S^z⟩(b_{\perp })$ in Eq. (22) (red band) and the kinetic OAM $⟨L^z⟩(b_{\perp })$ in Eq. (21) (blue band) contributions; (b) the kinetic TAM density $⟨J^z⟩(b_{\perp })$ (black band) resulting from the sum of the naive TAM density $⟨J_{\text{naive}}^z⟩(b_{\perp })$ in Eq. (33) (blue band) and the corresponding correction term $⟨J_{\mathrm{corr}}^z⟩(b_{\perp })$ in Eq. (34) (red band); (c) the kinetic TAM density $⟨J^z⟩(b_{\perp })$ (black band) expressed as the sum of the Belinfante-improved TAM $⟨J_{\text{Bel}}^z⟩(b_{\perp })$ density in Eq. (24) (blue band) and the total divergence term $⟨M^z⟩(b_{\perp })$ in Eq. (25) (red band); (d) the Belinfante-improved TAM $⟨J_{\text{Bel}}^z⟩(b_{\perp })$ density (black band) decomposed into its monopole $⟨J_{\mathrm{Bel}}^{\mathrm{mono}}⟩(b_{\perp })$ in Eq. (35) (blue band) and quadrupole $⟨J_{\mathrm{Bel}}^{\mathrm{quad}}⟩(b_{\perp })$ in Eq. (36) (red band) contributions. The bands reflect our ${\alpha }_s$ uncertainty of 10%.

Figure 4

Angular momentum densities of the up and the down quarks multiplied by $b_{\perp }$ as functions of $b_{\perp }$. (a), (c), (e), (g) are for the down quark, while (b), (d), (f), (h) are for the up quark. The legends in (a), (c), (e), (g) or (b), (d), (f), (h) are the same as described in Figs. 3 but for quarks.