Strong vacuum polarization and self-damping of e+e− annihilation

If the hadronic contribution to vacuum polarization, which is proportional to the ration of e⁺e^? annihilation into hadrons to that into μ⁺μ^? rises asymptotically as a power $\text{β >}\text{1}\text{2}$ of the squared four-momentum, the ratio $\text{σ}\text{(e}{}^{\mathrm{+}}\text{e}{}^{\mathrm{?}}\text{→hadrons)}\text{σ}$ point $\text{(}{}^{\mathrm{\sigma }}\text{point =}\text{(4πα}{}^2\text{)}\text{(3s))}$ is bound to be less than $\text{?}\text{(3}\text{tg}\text{πβ)}\text{(8α)}$ and this limit is approached sooner for higher β. Other models of vacuum polarizations are also considered together with their possible origin and implications.