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. 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: $S_E=-\int \frac{dx}{\mathrm{\Delta }x}{N}_{L^2}(x)\log [N_{a^2}(x)]$, where $a$ is the UV cutoff in coordinate space and $\mathrm{\Delta }x$ 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 by $N_S(x)=\frac{S}{A_p}\mathrm{\Delta }xF(x)$, where $A_p$ denotes the transverse area of the proton.

• Received 24 April 2023
• Accepted 26 June 2023

DOI:https://doi.org/10.1103/PhysRevD.108.014014

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Published by the American Physical Society