Nucleon structure from basis light-front quantization

We produce the light-front wave functions (LFWFs) of the nucleon from a basis light-front approach in the leading Fock-sector representation. We solve for the mass eigenstates from a light-front effective Hamiltonian, which includes a confining potential adopted from light-front holography in the transverse direction, a longitudinal confinement, and a one-gluon exchange interaction with fixed coupling. We then employ the LFWFs to obtain the electromagnetic and axial form factors, the parton distribution functions (PDFs), and the generalized parton distribution functions for the nucleon. The electromagnetic and axial form factors of the proton agree with the experimental data, whereas the neutron form factors deviate somewhat from the experiments in the low-momentum transfer region. The unpolarized, the helicity, and the transversity valence quark PDFs, after QCD scale evolution, are fairly consistent with the global fits to the data at the relevant experimental scales. The helicity asymmetry for the down quark also agrees well with the measurements; however, the asymmetry for the up quark shows a deviation from the data, especially in the small $x$ region. We also find that the tensor charge agrees well with the extracted data and the lattice QCD predictions, while the axial charge is somewhat outside the experimental error bar. The electromagnetic radii of the protons, the magnetic radius of the neutron, and the axial radius are in excellent agreement with the measurements, while the neutron charge radius deviates from the experiment.

Figure 1

Dirac FFs (upper panel) and the Pauli FFs (lower panel) for the up and down quarks. The gray and red bands are BLFQ results for the up and down quarks, respectively, reflecting our ${\alpha }_s$ uncertainty of 10%. The experimental data are taken from Refs. [49, 50, 51].

Figure 2

Sachs FFs for the proton and the neutron. The red bands are BLFQ results. The data are taken from Refs. [8, 211, 212, 213, 214, 215] for $G_E^p(Q^2)$, Refs. [216, 217, 218, 219, 220, 221, 222] for $G_E^n(Q^2)$, Refs. [213, 223] for $G_M^p(Q^2)$, and Refs. [224, 225, 226, 227, 228] for $G_M^n(Q^2)$.

Figure 3

Ratio of Sachs form factor $R^i={\mu }_i{G}_E^i/G_M^i$. The red bands are BLFQ results. The data are taken from Refs. [7, 8, 19, 214, 215, 229, 230, 231, 232] for the proton and Refs. [216, 217, 218, 219, 220, 221, 233, 234, 235] for the neutron.

Figure 4

The ratio of the proton Pauli form factor $F_2^p$ and the Dirac form factor $F_1^p$; (a) $Q^2{F}_2^p/({\kappa }_p{F}_1^p)$ and (b) $(Q^2/{\mathrm{Log}}^2[Q^2/{\mathrm{\Lambda }}^2])F_2^p/({\kappa }_p{F}_1^p)$ for three different values of $\mathrm{\Lambda }=0.2$, 0.3 and 0.4 GeV. The data are taken from Refs. [7, 8, 10, 211, 212].

Figure 5

The axial form factors $G_{\mathrm{A}}=G_{\mathrm{A}}^u-G_A^d$ and $G_{\mathrm{A}}^u$, $G_{\mathrm{A}}^d$ as the function of $Q^2$. The gray band ($G_{\mathrm{A}}$), red band ($G_{\mathrm{A}}^u$), and orange band ($G_{\mathrm{A}}^d$) are the BLFQ results. The extracted data are taken from Refs. [52, 62] and the lattice calculations from [63]. The magenta line represents the dipole fit to the experimental data [52].

Figure 6

Quark transverse charge and anomalous magnetization densities for the unpolarized nucleon: (a, c) for the down quark; (b, d) for the up quark. The red bands correspond to the uncertainty range of the BLFQ results. The blue and black lines represent the parametrizations of Kelly [253] and Bradford et al. [254], respectively.

Figure 7

The nucleon transverse charge and anomalous magnetization densities for the unpolarized nucleon: (a, c) for the proton; (b, d) for the neutron. The red bands correspond to the uncertainty range of the BLFQ results. The blue and black lines represent the parametrizations of Kelly [253] and Bradford et al. [254], respectively.

Figure 8

The unpolarized proton’s PDFs of the proton $x{f}^q(x)$ for the up (top curves) and down (bottom curves) quarks. The light red bands correspond to the uncertainty in the BLFQ results. We compare our results with the global fits: CTEQ [75] (blue solid), NNPDF [72] (purple dashed), MMHT [74] (black dashed).

Figure 9

The helicity PDFs for the up and down quarks in the proton. We compare BLFQ results (red solid line for down quark and black dashed line for up quark with uncertainties indicated by light red bands) with measured data from the COMPASS Collaboration [87].

Figure 10

Helicity PDFs of the isovector combination in the proton. The BLFQ result (red line with uncertainty indicated by the light red band) is compared with NNPDF global fit (blue band) [90] and measured data [81, 82, 83, 84, 85, 86, 87].

Figure 11

Helicity asymmetry of $u+\overline{u}$ (upper curve/data) and $d+\overline{d}$ (lower curve/data) in the proton. The BLFQ result (light red band) is compared with the measured data [81, 82, 83, 85, 86].

Figure 12

Quark transversity PDFs in the proton: (a) up quark distribution, (b) down quark distribution. The BLFQ results (gray bands) are compared with the global fits [118, 119, 126]. The green lines correspond to the Soffer bound [263] multiplied by the momentum fraction $\frac{1}{2}|x{f}_1^f(x)+x{g}_1^f(x)|$.

Figure 13

Quark tensor charges in the proton: (a) for the up quark, (b) for the down quark at ${\mu }^2=2.4{\mathrm{GeV}}^2$. The blue data (1) represent the BLFQ predictions. The other data points correspond to the phenomenological extractions of the tensor charges from different models: (2) He and Ji (${\mu }^2\sim 1{\mathrm{GeV}}^2$) [265], (3) Barone et al. (${\mu }^2\sim 25{\mathrm{GeV}}^2$) [266], (4) Schweitzer et al. (${\mu }^2\sim 0.6{\mathrm{GeV}}^2$) [274] (5) Gamberg and Goldstein (${\mu }^2\sim 1{\mathrm{GeV}}^2$) [267], (6) Pasquini et al. (${\mu }^2=10{\mathrm{GeV}}^2$) [268], (7) Wakamatsu (${\mu }^2=2.4{\mathrm{GeV}}^2$) [269], (8) Pasquini et al. (${\mu }^2=10{\mathrm{GeV}}^2$) [270, 271], (9) Lorcé (${\mu }^2=0.36{\mathrm{GeV}}^2$) [272], (10) Pitschmann et al. (${\mu }^2=4{\mathrm{GeV}}^2$) [273], (11) Anselmino et al. (${\mu }^2=0.8{\mathrm{GeV}}^2$) [119], (12) Radici et al. (${\mu }^2=10{\mathrm{GeV}}^2$) [122], (13) Kang et al. (${\mu }^2=10{\mathrm{GeV}}^2$) [121], (14) Abdel-Rehim et al. (${\mu }^2=4{\mathrm{GeV}}^2$) [68], and (15) Bhattacharya et al. (${\mu }^2=4{\mathrm{GeV}}^2$) [70].

Figure 14

GPDs as functions of $x$ and $-t$ for the up and down quarks in the proton at the model scale: upper panels for the down quark; lower panels for the up quark. Note the sign change on the vertical axis of panel (b).

Figure 15

$x$ dependence of $⟨b_{\perp }^2⟩$ for quarks in the proton. The line corresponds to the BLFQ prediction, and the band indicates its uncertainty. The data points are taken from Ref. [277].