Instanton–anti-instanton pair induced contributions to Re+e−→hadrons and Rτ→hadrons

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 τ -lepton was found. For Nf = Nc 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 ∼ 1/n corrections shows that for n ∼ 10 the obtained asymptotic expressions are at best only the order of magnitude estimate. The asymptotic series for Re+e−→hadrons , though obtained formally for Nf = Nc, is valid up to energies ∼ 15Gev. The instanton–anti-instanton pair nonperturbative contribution to Rτ→hadrons blows up. On the one hand, this means that instantons could not be considered ab–initio at such energies. On the other hand, this result casts a strong doubt upon the possibility to determine the αs from the τ–lepton width.