Hadron multiplicity as the limit of jet multiplicity at high resolution

Recently exact numerical results from the evolution equation for parton multiplicities in QCD jets have been obtained. A comparison with various approximate results is presented. A good description is obtained not only of the jet multiplicities measured at LEP-1 but also of the hadron multiplicities for $cms$ energies above 1.6 GeV in \epem annihilation. The solution suggests that a final state hadron can be represented by a jet in the limit of small (nonperturbative) $k_{\perp }$ cut-off $Q_0$. In this description using as adjustable parameters only the QCD scale $\mathrm{\Lambda }$ and the cut-off $Q_0$, the coupling ${\alpha }_s$ can be seen to rise towards large values above unity at low energies.