Probing the Onset of Maximal Entanglement inside the Proton in Diffractive Deep Inelastic Scattering

It has been proposed that at small Bjorken $x$, or equivalently at high energy, hadrons represent maximally entangled states of quarks and gluons. This conjecture is in accord with experimental data from the electron-proton collider HERA at the smallest accessible $x$. In this Letter, we propose to study the onset of the maximal entanglement inside the proton using diffractive deep inelastic scattering. It is shown that the data collected by the H1 Collaboration at HERA allow one to probe the transition to the maximal entanglement regime. By relating the entanglement entropy to the entropy of final-state hadrons, we find a good agreement with the H1 data using both the exact entropy formula as well as its asymptotic expansion which indicates the presence of a nearly maximally entangled state. Finally, future opportunities at the Electron Ion Collider are discussed.

Figure 1

Kinematics of the neutral current diffractive DIS process $ep\to epX$.

Figure 2

Probabilities $p_n(y_X)$ with $y_X=\ln (1/\beta )$ as extracted from leading order diffractive PDFs for $n=1,\dots ,50$ for the charged hadron multiplicities. The shaded region indicates the region in $\beta$ probed by the H1 dataset.

Figure 3

Exact and asymptotic entropy as a function of $\beta$. H1 data [60] extracted from the multiplicity distributions are shown, where statistical and systematic uncertainty are added in quadrature and presented as error bars. The theoretical uncertainty bands correspond to PDF and its scale uncertainty added in quadrature, where the scale uncertainty is obtained from the variation of the factorization scale of the leading order diffractive PDFs in the range $Q\to [Q/2,2Q]$.