Multiplicity distribution and entropy of produced gluons in deep inelastic scattering at high energies

In this paper we found the multiplicity distribution of the produced gluons in deep inelastic scattering at large $z=\ln\LbQ^2_s/Q^2\Rb\,\,\gg\,\,1$ where $ Q_s $ is the saturation momentum and $Q^2$ is the photon virtuality. It turns out that this distribution at large $n > \bar{n}$ almost reproduces the KNO scaling behaviour with the average number of gluons $ \bar{n} \propto \exp\Lb z^2/2 \kappa\Rb$, where $\kappa = 4.88 $ in the leading order of perturbative QCD. TheKNO function $\Psi\Lb \frac{n}{\bar{n}}\Rb = \exp\Lb -\,n/\bar{n}\Rb$. For $n < \bar{n}$ we found that $\sigma_n \propto\Big( z - \sqrt{2 \,\kappa\,\ln (n-1)}\Big)/(n-1)$. Such small $n$ determine the value of entropy of produced gluons $S_E = 0.3\, z^2/(2\,\kappa)$ at large $z$. The factor $0.3$ stems from the non-perturbative corrections that provide the correct behaviour of the saturation momentum at large $b$.