Instanton--anti-instanton pair induced contributions to $R_{e^+e^-\to hadrons}$ and $R_{\tau \to hadrons}$

Abstract

The instanton--anti-instanton pair induced asymptotics of perturbation theory expansion for the cross section of electron--positron pair annihilation to hadrons and hadronic width of $\tau$-lepton was found. For $N_f = N_c$ the nonperturbative instanton contribution is finite and may be calculated without phenomenological input. The instanton induced peturbative asymptotics was shown to be enhanced as $(n+10)!$ and in the intermediate region $n<15$ may exceed the renormalon contribution. Unfortunately, the analysis of $\sim 1/n$ corrections shows that for $n \sim 10$ the obtained asymptotic expressions are at best only the order of magnitude estimate. The asymptotic series for $R_{e^+ e^- \rightarrow hadrons}$ , though obtained formally for $N_f =N_c$, is valid up to energies $\sim 15$Gev. The instanton--anti-instanton pair nonperturbative contribution to $R_{\tau \rightarrow hadrons}$ blows up. On the one hand, this means that instantons could not be considered {\it ab--initio} at such energies. On the other hand, this result casts a strong doubt upon the possibility to determine the $\alpha_s$ from the $\tau$--lepton width.