Transverse structure of electron in momentum space in basis light-front quantization

We investigate the leading-twist transverse-momentum-dependent distribution functions (TMDs) for a physical electron, a spin-$1/2$ composite system consisting of a bare electron and a photon, using the basis light-front quantization (BLFQ) framework. The light-front wave functions of the physical electron are obtained from the eigenvectors of the light-front QED Hamiltonian. We evaluate the TMDs using the overlaps of the light-front wave functions. The BLFQ results are found to be in excellent agreement with those TMDs calculated using lowest-order perturbation theory.

Figure 1

The mass counterterm $\mathrm{\Delta }{m}_e=m_e-M_e$ as a function of the basis truncation parameter in BLFQ. All the results are evaluated at $K=N_{\max }$ and $b=M_e$. The solid (blue) line with dots represents the BLFQ results, while the dashed (red) line corresponds to the linear fitting of the results: $\mathrm{\Delta }{m}_e=-0.339+0.0218N_{\max }$.

Figure 2

The electron wave function renormalization factor $Z_2$, as a function of the basis truncation parameters in BLFQ. All the results are evaluated at $K=N_{\max }$ and $b=M_e$. The dots connected by solid (blue) line segments represent the BLFQ results, while the dashed (red) line corresponds to the logarithm fitting of the results: $1/Z_2=0.9279+0.03392\ln (N_{\max })$.

Figure 3

The BLFQ computations at different truncation parameters and the perturbative results (black dashed lines) [55, 56] for $f_1^e$ in the transverse direction (left) and longitudinal direction (right). We use $f_1^e$ at $x\approx 0.5$ and at $(k^{\perp }{)}^2=1.1({\mathrm{MeV}}^2)$ as examples. Curves with different dash styles (colors) are the BLFQ results at different $N_{\max }$ with fixed $K=50$ and $b=M_e$.

Figure 4

The BLFQ results after averaging and the perturbative results (black dashed lines) [55, 56] for $f_1^e$ in the transverse direction (left) and longitudinal direction (right). Calculations are performed at $x_e\approx 0.5$ (left) and at $k_e^{\perp }=1.25({\mathrm{MeV}}^2)$ (right). Curves with different dash styles (colors) are results averaged from the BLFQ computations at different $N_{\max }$ sets with fixed $K=50$ and $b=M_e$.

Figure 5

The BLFQ results after averaging and the perturbative results (black dashed lines) [55, 56] for $f_1^e$ in the transverse direction (left) and longitudinal direction (right). Calculations are performed at $x_e\approx 0.5$ (left) and at $k_e^{\perp }=1.25({\mathrm{MeV}}^2)$ (right). Curves with different dash styles (colors) are results averaged from the BLFQ computations at $N_{\max }=\{50,52,54\}$, $b=M_e$ but different $K$ as specified in the legends.

Figure 6

$f_1^e$ after averaging over the BLFQ computations at $N_{\max }=\{n,n+2,n+4\}$, $K=n$ and $b=M_e$. (Here $n$ is the lowest $N_{\max }$ we use for averaging.) We also plot the perturbative results (black dashed lines) [55, 56] for comparison. In the left panel we plot in the IR region. In the right panel we show the UV region. In the legend of the right panel, we list the square of the UV cutoff at $x=0.5$ introduced by the $N_{\max }$ truncation, i.e., $(b·\sqrt{N_{\max }/2}{)}^2$. Curves with different dash styles (colors) are results averaged from the BLFQ computations at different $N_{\max }$ and $K$, with fixed $b=M_e$.

Figure 7

The root mean square deviation $\mathrm{\Delta }\mathrm{TMD}$ of the BLFQ results from the perturbation results as a function of $n$. All the BLFQ results are averaged over $N_{\max }=\{n,n+2,n+4\}$, $K=n$. The blue and the orange lines represent the average deviations for $f_1^e$ and $h_{1L}^{\perp e}$ at $x\approx 0.5$, respectively. We only include the deviation within the UV and IR cutoff, i.e., $b/\sqrt{N_{\max }}<|k^{\perp }|<b\sqrt{N_{\max }/2}$.

Figure 8

The BLFQ results (solid lines with different color) and the perturbative results (black lines with different dash styles) [55, 56] for five nonzero leading-twist TMDs neglecting the gauge link. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$. The left panels show $k^{\perp }$ dependence at selected $x$ and also the envelope of the oscillation using shadowing as our numerical uncertainty. The right ones show $x$ dependence at selected $k^{\perp }$. Curves with different dash styles or colors are results at different $x$ ($k^{\perp }$) for left (right) panels.

Figure 9

3D plots of the BLFQ results for five nonzero TMDs neglecting the gauge link. Results are all obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$.

Figure 10

Plots for the electron PDFs $f_1^e(x)$, $g_1^e(x)$ and $h_1^e(x)$. The BLFQ results (solid lines with different markers) are compared with the perturbative results (black lines with different dash styles) [55, 56]. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$. We also show an enlarged version of those curves in the range $0.2\le x\le 0.6$ for a clearer comparison.

Figure 11

Density plots in the transverse-momentum plane at different $x$ for the unpolarized distribution, i.e., $f_1^e$. The BLFQ results are shown in the lower row and the perturbative results [55, 56] are shown in the upper row. From left to right, the longitudinal momentum fraction of the bare electron is 0.1, 0.5 and 0.8, respectively. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$.

Figure 12

Density plots in the transverse-momentum plane at different $x$ for distribution with both physical and bare electron polarized in the same longitudinal direction, i.e., $(f_i^e+g_{1L}^e)/2$. The BLFQ results are shown in the lower row and the perturbative results [55, 56] are shown in the upper row. From left to right, the longitudinal momentum fraction of the bare electron is 0.1, 0.5 and 0.8, respectively. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$.

Figure 13

Density plots in the transverse-momentum plane at different $x$ for distribution with both physical and bare electron polarized in the same transverse direction, i.e., $(f_i^e+h_1^e)/2$. The BLFQ results are shown in the lower row and the perturbative results [55, 56] are shown in the upper row. From left to right, the longitudinal momentum fraction of the bare electron is 0.1, 0.5 and 0.8, respectively. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$.

Figure 14

Density plots in the transverse-momentum plane at different $x$ for distribution of a longitudinally polarized bare electron in a $y$-axis polarized physical electron, i.e., $(f_i^e+\frac{k^y}{M_e}{g}_{1T}^e)/2$. The BLFQ results are shown in the lower row and the perturbative results [55, 56] are shown in the upper row. From left to right, the longitudinal momentum fraction of the bare electron is 0.1, 0.5 and 0.8, respectively. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$.

Figure 15

Density plots in the transverse-momentum plane at different $x$ for distribution of a $y$-axis polarized bare electron in a longitudinally polarized physical electron, i.e., $(f_i^e+\frac{k^y}{M_e}{h}_{1L}^{\perp e})/2$. The BLFQ results are shown in the lower row and the perturbative results [55, 56] are shown in the upper row. From left to right, the longitudinal momentum fraction of the bare electron is 0.1, 0.5 and 0.8, respectively. The BLFQ results are obtained by averaging over the BLFQ computations at $N_{\max }=\{100,102,104\}$, $K=100$ and $b=M_e$.