Tomographic image of the proton

We have carried out a phenomenological analysis of the latest deep virtual Compton scattering experimental data based on the generalized parton distribution formalism. This allows us to extract the dependence of the spatial size of the proton on the quark’s longitudinal momentum. This results in the first continuous two-dimensional momentum-space image and tomography of the proton based on experimental data.

Figure 1

$t$-dependence of the CFF ${\mathcal{H}}_{\mathrm{Im}}$ for 20 CLAS ($\xi$, $Q^2$) bins ($Q^2$ in ${\mathrm{GeV}}^2$). Open squares: Results of the CLAS $\sigma$ and $\mathrm{\Delta }\sigma$ fit. Solid circles: Results of the fit to CLAS $\sigma$ and $\mathrm{\Delta }\sigma$ data, as well as to the $lTSA$ and $DSA$ data. Solid triangles: Results of the Hall A $\sigma$ and $\mathrm{\Delta }\sigma$ fit. Stars: VGG reference CFFs. Solid curve: Exponential fit of the open squares.

Figure 2

Upper: Amplitude $A$ of ${\mathcal{H}}_{\mathrm{Im}}$, multiplied by $\xi$, as a function of $\xi$. Lower: $t$-slope $B$ of ${\mathcal{H}}_{\mathrm{Im}}$ as a function of $\xi$. Data points: eight-CFF fit of JLab (circles) and HERMES data (square), the latter as extracted in Refs. [9, 13]. For the JLab points, we kept only the eight lowest $\xi$ bins of Fig. 1 which provide the lowest uncertainty results. The fits to these data points according to Eqs. (4) and (5) are shown by the bands, corresponding with a $1\sigma$ variation of $a_A$ and $a_B$, given by Eq. (6). The theory curves correspond with the dual model and the double distribution (DD) model for three choices of the valence (sea) profile parameters $b_v$ ($b_s$).

Figure 3

$x$-dependence of the ratio $B_0(x)/B(x)$, where $B_0$ is the exponential $t$-slopes of $H_{-}^p(x,0,t)$ according to Eq. (14), and $B$ is the exponential $t$-slope of $H_{+}^p(x,0,t)$ according to Eq. (3). The theory curves are as described in the caption of Fig. 2.

Figure 4

$x$-dependence of $⟨b_{\perp }^2⟩$ for quarks in the proton. The data points correspond with the results obtained in this work for $B(x)$, as displayed in Fig. 2. They have been multiplied by the correction factor $B_0/B\simeq 0.925\pm 0.025$ in the $x$ range of the data. The resulting model uncertainty is shown by the outer error bars. The band shows the empirical result using the logarithmic ansatz for $B_0(x)$ of Eqs. (17), (18) with the parameter $a_B$ determined from the proton Dirac radius. The inset is the corresponding three-dimensional representation.