Evolution of entropy at small $x$

We explore the evolution of the Deep Inelastic Scattering (DIS) entropy, defined as $S(x,μ^2) \simeq \ln[xg(x,μ^2)]$ at small Bjorken variable $x$, where $μ$ is the observable scale and the gluon distribution $xg(x,μ^2)$ is derived from the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations. We aim to evolve the DIS entropy, which is not directly observable, using a Laplace transform technique. This approach allows us to obtain an analytical solution for the DIS entropy based on known initial gluon distribution functions. We consider both leading-order (LO) and higher-order approximations for the DIS entropy, incorporating the evolved gluon distribution function at the initial scale. The DIS entropy, influenced by purely gluonic emissions, varies with higher-order corrections to the running coupling. By comparing theoretical predictions with charged hadron multiplicity data, we define the evolution. Additionally, we investigate the derivative of the scaling entropy, modeling it as a function of the running coupling, to determine the parameter $λ$, known as the Pomeron intercept. We find that the values of $λ(x,μ^2)$ decrease as the order of evolution increases, which is consistent with the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron in the LO and NLO approximations. This investigation provides insights into the dynamics of Quantum Chromodynamics (QCD) at high energies.

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