Entanglement entropy of the proton in coordinate space

We calculate the entanglement entropy of a model proton wave function in coordinate space by integrating out degrees freedom outside small circular region $\overline{A}$ radius $L$, where $L$ is much smaller than size proton. The provides nonperturbative distribution three valence quarks. In addition, we include perturbative emission single gluon and gluons $\overline{A}$. For both quarks gluons, obtain same simple result: ${S}_{E}=\ensuremath{-}\ensuremath{\int}\frac{dx}{\mathrm{\ensuremath{\Delta}}x}{N}_{{L}^{2}}(x)\mathrm{log}[{N}_{{a}^{2}}(x)]$, $a$ UV cutoff $\mathrm{\ensuremath{\Delta}}x$ longitudinal resolution scale. Here ${N}_{S}(x)$ number partons (of appropriate species) with momentum fraction $x$ inside an area $S$. It related to standard parton ${N}_{S}(x)=\frac{S}{{A}_{p}}\mathrm{\ensuremath{\Delta}}xF(x)$, ${A}_{p}$ denotes transverse