Optimization of ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{-}}$ and ‘‘freezing’’ of the QCD couplant at low energies

The new result for the third-order QCD corrections to ${\mathit{R}}_{\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$, 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 ${\mathrm{\alpha }}_{\mathit{s}}$/π for ${\mathit{R}}_{\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$ 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 ${\mathit{R}}_{\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$ with experimental data down to the lowest energies, and find excellent agreement.