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We analyze Mueller's QCD dipole wave function evolution in the doublelogarithm approximation (DLA). Using complex analytical methods, we show thatthe distribution of dipole in the wave function (gluon multiplicitydistribution) asymptotically satisfies the Koba-Nielsen-Olesen (KNO) scaling,with a non-trivial scaling function $f(z)$ with $z=\frac n{\bar n}$. Thescaling function decays exponentially as $2(2.55)^2ze^{-\frac{z}{0.3917}}$ atlarge $z$, while its growth is log-normal as $e^{-\frac{1}{2}\ln^2 z}$ forsmall-$z$. A detailed analysis of the Fourier-Laplace transform of $f(z)$,allows for performing the inverse Fourier transform, and access thenon-asymptotic bulk-region around the peak. The bulk and asymptotic results areshown to be in good agreement with the measured hadronic multiplicities in DIS,as reported by the H1 collaboration at HERA in the region of large $Q^2$. Anumerical tabulation of $f(z)$ is included. Remarkably, the same scalingfunction is found to emerge in the resummation of double logarithms in theevolution of jets. Using the generating function approach, we show why this isthe case. The absence of KNO scaling in non-critical and super-renormalizabletheories is briefly discussed. We also discuss the universal character of theentanglement entropy in the KNO scaling limit, and its measurement using theemitted multiplicities in DIS and $e^+e^-$ annihilation.

Mueller's dipole wave function in QCD: emergent KNO scaling in the double logarithm limit