Partonic Entropy of the Proton from DGLAP Evolution

TL;DR

The paper defines partonic entropy S(t) from the collinear PDFs in the proton and proves that S(t) increases with the DGLAP evolution scale t, highlighting that unbounded growth arises from the small-x region. It provides a detailed derivation using the DGLAP kernels and proves positivity of dS/dt, then interprets S(t) as a relative entropy with respect to a uniform distribution and as a dynamical entropy with respect to the initial state. To address the unbounded growth, the authors illustrate, through a simple small-x ansatz and a reaction-diffusion saturation model, how saturation effects can bound S(t) and yield a finite entropy in the high-energy limit, with the latter model showing S(y) approaching a finite Poisson-based value as y → ∞. They further connect the partonic entropy to entanglement entropy in DIS, discussing both theoretical interpretations and phenomenological tests against data, and suggest that nonlinear QCD evolution is essential for a complete, saturation-informed picture. Overall, the work links information-theoretic measures to QCD parton dynamics and provides a framework for testing the entanglement interpretation of partonic entropy in high-energy scattering.

Abstract

We investigate the concept of partonic entropy of the proton within the Dokshitzer--Gribov--Lipatov--Altarelli--Parisi (DGLAP) evolution scheme of collinear parton distributions. We show that such entropy increases monotonically with the evolution scale. The mechanism underlying this growth is illustrated using a simplified model of DGLAP evolution, which highlights the importance of including saturation effects at small $x$ in the evolution of parton distributions. Based on existing literature, we present two simplified models of parton saturation at small $x$. In one of these models, partonic entropy is identified with entanglement entropy and proposed as an observable to be tested experimentally.

Figures (3)

• Figure 1: $P(x,t)$ given by Eq. (\ref{['eq:P']}) as a function of $x$ at $Q^2=1,100~{\rm GeV}^2$ (dashed and solid red lines, respectively). The blue line shows the entropy density $s(x,t)$ given by Eq. (\ref{['eq:sdens']}) at $Q^2=100~{\rm GeV}^2$. The sharp drop near $x\approx 0.2$ indicates that $s(x,t)$ becomes negative when $P(x,t)<1$.
• Figure 2: Partonic entropy (\ref{['eq:entropy']}) (upper blue line) and relative entropy (\ref{['eq:pentropy']}) (lower red line) as a function of the evolution parameter $t=\log_{10}(Q^2/1\,{\rm GeV}^2$).
• Figure 3: Red dotted lines: entropy (\ref{['eq:Sab']}) as a function of $\alpha$ for $\beta-1=0.2,1,5,10$ (from bottom to top). Blue dotted lines: entropy (\ref{['eq:Sbiggger']}) for $x_{min} = 10^{-8}$ (this choice is for illustration only) and the above values of $\beta$.