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 is associated with an 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 Wehrl entropy of the proton constructed from Husimi distribution obtained from the Gaussian smearing of the Wigner distribution. We show that the entanglement entropy naturally emerges from the normalization condition of the Husimi distribution within this framework. In addition, the Wehrl entropy contains a contribution associated with transverse degrees of freedom. Numerical results for the proton Wehrl entropy are presented for different values of the virtuality.

Figures (4)

• Figure 1: Unpolarized gluon PDF $f^{1}_g(x)$ as a function of the longitudinal momentum fraction $x$, in the range $0.0001 < x < 1$ for $Q^{2} = 4$ GeV$^{2}$. The solid blue line and the blue shadow represents the NNPDF 4.0 at NNLO data set for the unpolarized gluon PDF and its uncertainty band with 1$\sigma$, respectively. The dashed red line and the red shadow represents the fit of the Eq. (\ref{['softwall']}) with the free parameters $N_g$, $a$, and $b$ and the uncertainty band respectively.
• Figure 2: The panels displays the Kharzeev-Levin entanglement entropy as a function of Bjorken-$x$ for the CMS pp data CMS:2010qvf, and the theoretical prediction of the model studied in this work in blue solid line. The left panel shows the entropy from the CMS data in red dot for $\sqrt{s} = 7$ TeV and $\sqrt{s} = 2.36$ TeV with pseudo-rapidity $|\eta| < 0.5$, in blue solid line is shown this work model. The middle and right panel show the entropy from the CMS data in red dot for $\sqrt{s} = 7$ TeV, $\sqrt{s} = 2.36$ TeV and $\sqrt{s} = 0.9$ TeV with pseudo-rapidity $|\eta| < 1.0$ and $|\eta| < 2.0$ respectively, with the blue solid line the theoretical prediction.
• Figure 3: This figure shows the plots for the first moment of the Wigner (left) and Husimi (right) distributions for the unpolarized gluon in a unpolarized proton for a fixed transverse momentum, $k_x = 0.4$ GeV. This plots are obtained using the fit parameters from the Table \ref{['params_table']} for $Q^{2} = 4$ GeV$^{2}$.
• Figure 4: Plots of the Wehrl entropy as a function of the longitudinal momentum fraction as defined in Eq. (\ref{['wehrldecomposition']}). The full Wehrl entropy is shown by the solid blue line, the transverse entropy is shown by the dotted black line, the entanglement entropy is shown by the dashed red line. The curves are presented for three different virtuality scales: $Q = 2$ GeV (top--left panel), $Q = 5$ GeV (top--right panel), and $Q = 10$ GeV (bottom panel). The shaded bands represent the 1$\sigma$ uncertainties associated with the PDF parametrization.