Mueller’s dipole wave function in QCD: Emergent Koba-Nielsen-Olesen scaling in the double logarithm limit
Yizhuang Liu, Maciej A. Nowak, Ismaïl Zahed
Abstract
We analyze Mueller's QCD dipole wave function evolution in the double logarithm approximation (DLA). Using complex analytical methods, we show that the distribution of the dipole in the wave function (gluon multiplicity distribution) asymptotically satisfies the Koba-Nielsen-Olesen (KNO) scaling, with a nontrivial scaling function $f(z)$ with $z=\frac{n}{\overline{n}}$. The scaling function decays exponentially as $2(2.55{)}^{2}z{e}^{\ensuremath{-}\frac{z}{0.3917}}$ at large $z$, while its growth is log-normal as ${e}^{\ensuremath{-}\frac{1}{2}{\mathrm{ln}}^{2}z}$ for small $z$. A detailed analysis of the Fourier-Laplace transform of $f(z)$ allows for performing the inverse Fourier transform and accessing the nonasymptotic bulk region around the peak. The bulk and asymptotic results are shown 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 ${Q}^{2}$. A numerical tabulation of $f(z)$ is included. Remarkably, the same scaling function is found to emerge in the resummation of double logarithms in the evolution of jets. Using the generating function approach, we show why this is the case. The absence of KNO scaling in noncritical and super-renormalizable theories is briefly discussed. We also discuss the universal character of the entanglement entropy in the KNO scaling limit and its measurement using the emitted multiplicities in DIS and ${e}^{+}{e}^{\ensuremath{-}}$ annihilation.