Transverse momentum dependent parton distributions in a light-cone quark model

The leading twist transverse momentum dependent parton distributions (TMDs) are studied in a light-cone description of the nucleon where the Fock expansion is truncated to consider only valence quarks. General analytic expressions are derived in terms of the six amplitudes needed to describe the three-quark sector of the nucleon light-cone wave function. Numerical calculations for the T-even TMDs are presented in a light-cone constituent quark model, and the role of the so-called pretzelosity is investigated to produce a nonspherical shape of the nucleon.

Figure 1

The TMDs $f_1$, $g_{1L}$, $h_1$ as functions of $x$ and ${\mathit{k}}_{\perp }^2$ are shown in the upper, middle and lower panels, respectively. Results for up (down) quarks are given in the left (right) panels.

Figure 2

The TMDs $h_{1L}$ and $h_{1T}^{\perp }$ as functions of $x$ and ${\mathit{k}}_{\perp }^2$ are shown in the upper and lower panels, respectively. Results for up (down) quarks are given in the left (right) panels.

Figure 3

The contribution to the TMD $h_{1T}^{u\perp }$ from the $L_z=\pm 1$ wave components (left panel) and (right panel) from the $L_z=0$ and $L_z=2$ wave components as a function of $x$ and ${\mathit{k}}_{\perp }^2$. Upper (lower) panels for up (down) quarks.

Figure 4

The parton distribution $h_{1T}^{\perp q}(x)$ (left panels) and the transverse moment $h_{1T}^{(1)\perp q}(x)$ (right panels). Solid curves: results from the light-cone CQM model with the momentum wave function of Ref. 46. Dashed curves in the upper panels: results from the light-cone CQM model with the momentum wave function in the hypercentral model of Ref. 48, 49. Dashed curves in the lower panels: results from the spectator model of Ref. 40. Dotted curves: results from the bag model.

Figure 5

The transverse moments $h_{1L}^{(1)\perp q}$ (left panels) and $g_{1T}^{(1)q}$ (right panels) as functions of $x$. Upper (lower) panels for up (down) quarks. Solid curves refer to the result obtained with the light-cone CQM model; dashed curves refer to the Wandzura-Wilczek-type approximation.

Figure 6

Transverse shape of the proton ${\widehat{\rho }}_{\mathrm{REL}T}({\mathit{k}}_{\perp }^2,\mathit{n})/{\tilde{f}}_1({\mathit{k}}_{\perp }^2)$ assuming a struck $u$ quark. The horizontal axis is the direction of ${\mathit{S}}_{\perp }$ and $\mathit{n}={\widehat{\mathit{S}}}_{\perp }$, ${\varphi }_n=0$. The shapes vary from the outer circle to the internal line as $k_{\perp }$ is increased from 0 to 2.0 GeV in steps of 0.25 GeV. The left figure is the result when only the contribution from the $P$-waves to ${\tilde{h}}_{1T}^{\perp }$ is taken into account, and the right picture shows the total results when the $S$- and $D$-waves are also included.

Figure 7

The same as in Fig. 6 except that ${\varphi }_n=\pi$. The shapes vary from the inner circle to the external line as $k_{\perp }$ is increased from 0 to 2.0 GeV in steps of 0.25 GeV.

Figure 8

Transverse shape of the proton ${\widehat{\rho }}_{\mathrm{REL}T}({\mathit{k}}_{\perp },\mathit{n})/{\tilde{f}}_1({\mathit{k}}_{\perp }^2)$ assuming a struck $d$ quark. The horizontal axis is the direction of ${\mathit{S}}_{\perp }$ and $\mathit{n}={\widehat{\mathit{S}}}_{\perp }$, ${\varphi }_n=0$. The shapes vary from the inner circle to the external line as $k_{\perp }$ is increased from 0 to 2.0 GeV in steps of 0.25 GeV. The left figure is the result when only the contribution from the $P$-waves to ${\tilde{h}}_{1T}^{\perp }$ is taken into account, and the right picture shows the total results when the $S$- and $D$-waves are also included.

Figure 9

The same as in Fig. 8 except that ${\varphi }_n=\pi$. The shapes vary from the outer circle to the internal line as $k_{\perp }$ is increased from 0 to 2.0 GeV in steps of 0.25 GeV.