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