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 $\overline{A}$ of radius L, where L is much smaller than the size of the proton. Here, 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 $\overline{A}$. For both quarks and gluons, we obtain the same simple result: SE = –∫ $\frac{dx}{Δx}$ NL2(x) log [Na2(x)], where a is the UV cutoff in coordinate space and Δx is the longitudinal resolution scale. Here NS(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 by NS(x) = $\frac{S}{Ap}$ ΔxF(x), where Ap denotes the transverse area of the proton.