Optimization of Re+e- and "freezing" of the QCD couplant at low energies.

Abstract

The new result for the third-order QCD corrections to ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$, unlike the old, incorrect result, is nicely compatible with the principle-of-minimal-sensitivity optimization method. Moreover, it leads to infrared fixed-point behavior: the optimized couplant ${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}$/\ensuremath{\pi} for ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ does not diverge at low energies, but ``freezes'' to a value 0.26 below about 300 MeV. This provides some direct theoretical evidence, purely from perturbation theory, for the ``freezing'' of the couplant---an idea that has long been a popular and successful phenomenological hypothesis. We use the ``smearing'' method of Poggio, Quinn, and Weinberg to compare the resulting theoretical prediction for ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ with experimental data down to the lowest energies, and find excellent agreement.