Deep inelastic scattering as a probe of entanglement: confronting experimental data

Parton distributions can be defined in terms of the entropy of entanglement between the spatial region probed by deep inelastic scattering (DIS) and the rest of the proton. For very small $x$, the proton becomes a maximally entangled state. This approach leads to a simple relation $S = \ln N$ between the average number $N$ of color-singlet dipoles in the proton wave function and the entropy of the produced hadronic state $S$. At small $x$, the multiplicity of dipoles is given by the gluon structure function, $N = x G(x,Q^2)$. Recently, the H1 Collaboration analyzed the entropy of the produced hadronic state in DIS, and studied its relation to the gluon structure function; poor agreement with the predicted relation was found. In this letter we argue that a more accurate account of the number of color-singlet dipoles in the kinematics of H1 experiment (where hadrons are detected in the current fragmentation region) is given not by $xG(x,Q^2)$ but by the sea quark structure function $x\Sigma(x,Q^2)$. Sea quarks originate from the splitting of gluons, so at small $x$ $x\Sigma(x,Q^2)\,\sim\, xG(x,Q^2)$, but in the current fragmentation region this proportionality is distorted by the contribution of the quark-antiquark pair produced by the virtual photon splitting. In addition, the multiplicity of color-singlet dipoles in the current fragmentation region is quite small, and one needs to include $\sim 1/N$ corrections to $S= \ln N$ asymptotic formula. Taking both of these modifications into account, we find that the data from the H1 Collaboration in fact agree well with the prediction based on entanglement.