Asymptotic estimate of then-loop QCD contribution to the totale+e−annihilation cross section

A technique for estimating the large-order QCD perturbative contributions to the total ${\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$ total cross section is presented which is based upon the renormalization group and momentum analyticity. Our estimate for the coefficient of (${\mathrm{\alpha }}_{\mathit{s}}$/π${)}^3$ has recently been confirmed by two exact calculations. We find that the effective expansion parameter is, in fact, not ${\mathrm{\alpha }}_{\mathit{s}}$/π but rather 4${\mathrm{\pi }}^2$ ${\mathit{eb}}_1$(${\mathrm{\alpha }}_{\mathit{s}}$/π), where ${\mathit{b}}_1$ is the first coefficient in the expansion of the β function.