Testing the running of the QCD coupling in particle energy spectra

The spectra of charged particles in the energy variable $\xi = \log 1/x$ (``the hump backed plateau'') are analyzed in terms of moments in order to investigate the sensitivity to the running of $\alpha_s$ in particular towards small scales, and to further test thereby the local parton hadron duality picture. The modified leading log approximation (MLLA) of QCD provides a very satisfying description of the $e^+e^-$ data in the full range of c.m. energies from about 1 up to 91 GeV. The same calculation but with fixed coupling $\alpha_s$ yields moments which are inconsistent with the data already close to threshold. In particular the soft particles ($E\le 1$~GeV) reflect directly the running of $\alpha_s$.