Entropy and DIS structure functions

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 further 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, 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. The effect of adding a simple higher twist term to the description of entropy at low-$x$ and low-$Q^2$ values for comparison with HERA data is investigated.

💡 Research Summary

The manuscript presents a comprehensive study of entanglement entropy in deep‑inelastic scattering (DIS) by expressing it directly in terms of observable structure functions rather than the non‑observable parton distribution functions (PDFs). Building on the Kharzeev‑Levin proposal, the authors replace the momentum‑space PDF‑based approach of Lappi et al. and Boroun‑Ha with a formulation that uses the proton’s inclusive structure functions F₂(x,Q²) and the longitudinal structure function F_L(x,Q²).

The theoretical framework starts from the von Neumann entropy S = −Tr ρ_A ln ρ_A, where the reduced density matrix ρ_A describes the part of the proton wave‑function probed by the virtual photon. By invoking the Schmidt decomposition, the number of effective partons N is related to the gluon and singlet PDFs via N = x g(x,Q²)+x Σ(x,Q²). The authors then invert the standard collinear factorization relations (F_i = C_ij ⊗ f_j) to express the singlet density x Σ and the gluon density x g directly through derivatives of the measured structure functions. Explicitly, they obtain

x Σ(x,Q²) = F₂(x,Q²)/⟨e²⟩,

x g(x,Q²) = η