Multiplicity distribution of dipoles in QCD from the Le-Mueller-Munier equation

In this paper we derived in QCD the Balitsky-Fadin-Kuraev-Lipatov (BFKL) linear, inhomogeneous equation for the factorial moments of multiplicity distribution (${M}_{k}$) from Le-Mueller-Munier equation. In particular, the equation for the average multiplicity of the color-singlet dipoles ($N$) turns out to be the homogeneous BFKL while ${M}_{k}\ensuremath{\propto}{N}^{k}$ at small $x$. Second, using the diffusion approximation for the BFKL kernel we show that the factorial moments are equal to ${M}_{k}=k!N{(N\ensuremath{-}1)}^{k\ensuremath{-}1}$ which leads to the multiplicity distribution $\frac{{\ensuremath{\sigma}}_{n}}{{\ensuremath{\sigma}}_{\text{in}}}=\frac{1}{N}{(\frac{N\ensuremath{-}1}{N})}^{n\ensuremath{-}1}$. We also suggest a procedure for finding corrections to this multiplicity distribution which will be useful for descriptions of the experimental data.