QCD Wehrl and entanglement entropies in a gluon spectator model at small-$x$

Abstract

Recent studies have shown that hadronic multiplicity in deep inelastic scattering can be associated with entanglement entropy. However, such definitions are intrinsically longitudinal and do not capture the full phase-space structure of the proton. In this work, we investigate the proton Wehrl entropy constructed from the gluon Husimi distribution, which provides a positive phase-space description within the present definitions and model calculations. Within this framework, we employ a gluon light-front spectator model based on soft-wall AdS/QCD-inspired wave functions, with free parameters constrained by global NNPDF fits, allowing us to compute both parton distribution functions and Wigner distributions. The Husimi distribution is obtained via Gaussian smearing of the Wigner distribution with width given by the saturation scale in the GBW model. We show that from a normalized Husimi distribution one can decompose the Wehrl entropy into an entanglement entropy term and a residual term associated with transverse degrees of freedom. Numerical results for the proton entanglement entropy are shown and compared with CMS data, while the Wehrl entropy is presented for different values of the virtuality.