Entropy and DIS structure functions

TL;DR

This work defines and computes entanglement entropy in DIS using observable proton structure functions, eliminating reliance on non-physical parton distributions. It leverages a momentum-space framework with a BDH/Martin1 parametrization of $F_2$ and a Laplace-transform–based method to obtain $F_L$, enabling a closed-form expression for the DIS entropy $S(x,Q^2)$. The results show good agreement with H1 DIS data for the entropy derived from $F_2$ and $F_L$, and after a $2/3$ rescaling, with charged-hadron entropy trends from HERA, including Regge-like baselines. The study reveals a characteristic $Q^2$ evolution: entropy rises for $Q^2\lesssim 20~\text{GeV}^2$, plateaus up to $\sim 100~\text{GeV}^2$, and then decreases, with at $x_{\min}=Q^2/s$ the charged-hadron entropy approaching zero while the DIS entropy remains finite, controlled by an effective intercept in the evolution.

Abstract

Entanglement entropy in Deep Inelastic Scattering (DIS) from the DIS structure functions has emerged as a novel tool for probing observable quantities. The method proposed by Kharzeev-Levin to determine entanglement entropy in DIS from parton distribution functions (PDFs) improves on the momentum-space approach proposed by Lappi et al.[Eur. Phys. J. C {\bf84}, 84 (2024)] and future developed by Boroun and Ha [Phys. Rev. D {\bf109}, 094037 (2024)] using Laplace transform techniques. The entropy of charged hadrons is obtained from the parameterization of the proton structure function and compared with H1 data and HSS and HERA PDFs. Our results for the entanglement entropy align very well with the H1 data across a wide range of $x$ and $Q^2$. Finally, the behavior of the entanglement entropy is described at fixed $\sqrt{s}$ to the minimum value of $x$ given by $Q^2/s$ which indicates that the polarization of the exchanged photon for entropy determination is transverse at this specific kinematic point.

Figures (5)

• Figure 1: The entanglement entropy results for the DIS entropy (DE, black-dashed curve, Eq.(\ref{['SF2 eq']})) and the entropy of charged hadrons (CE, red-dashed-dot-dot curve, Eq.(\ref{['SF3 eq']})) with respect to the parametrization of $F_{2}(x,Q^2)$ are compared with the H1 experimental data (blue-circles) H1A as accompanied with total errors and the results from HSS (green-dashed-dot curve) and HERAPDF (brown-solid curve) as shown in Kutak2. The error bands are a result of the parametrization of $F_{2}(x,Q^{2})$.
• Figure 2: The entanglement entropy results are compared at a fixed value of the Bjorken variable $x$ ($x=10^{-3}$) and in a wide range of $Q^2$ for the DIS entropy (DE, black-dashed curve, Eq.(\ref{['SF2 eq']})) and the entropy of charged hadrons (CE, red-dashed-dot-dot curve, Eq.(\ref{['SF3 eq']})) with respect to the parametrization of $F_{2}(x,Q^2)$ in comparison to the H1 experimental data (blue-circles) H1A along with total errors. The error bands are a result of the parametrization of $F_{2}(x,Q^{2})$.
• Figure 3: The entanglement entropy extracted is shown as a function of $Q^2$ at $x_{\mathrm{min}}=Q^2/s$ for $\sqrt{s}=319~\mathrm{GeV}$ according to the DIS entropy (DE, black-dashed curve, Eq.(\ref{['SF2 eq']})) and the entropy of charged hadrons (CE, red-dashed-dot-dot curve, Eq.(\ref{['SF3 eq']})). The error bands result from the parametrization of $F_{2}(x,Q^{2})$.
• Figure 4: The entanglement entropy extracted is shown as a function of $x_{\mathrm{min}}$ for $\sqrt{s}=319~\mathrm{GeV}$ based on the entropy of charged hadrons (red-solid curve). The error bands result from the parametrization of $F_{2}(x,Q^{2})$ (red-dashed curves).
• Figure 5: The figure displays the effective intercept behavior of the entanglement entropy extracted as a function of $x_{\mathrm{min}}$ for $\sqrt{s}=319~\mathrm{GeV}$. It is based on the entropy of charged hadrons (brown-solid curve) and is compared with the HSS HSS (black- dashed curve) and HERAPDF HZ (blue-dashed-dot curve) intercepts within the bin values of $Q^{2}= [5-10~\mathrm{GeV}^2]$ and $Q^{2}= [40-100~\mathrm{GeV}^2]$, respectively.