Optimal renormalization-group improvement of the perturbative series for the e(+)e(-) annihilation cross section

Using renormalization-group methods, we derive differential equations for the all-orders summation of logarithmic corrections to the QCD series for R(s)=sigma(e(+)e(-)--hadrons) sigma(e(+)e(-)--mu(+)mu(-)), as obtained from the imaginary part of the purely perturbative vector-current correlation function. We present explicit solutions for the summation of leading and up to three subsequent subleading orders of logarithms. The summations accessible from the four-loop vector correlator not only lead to a substantial reduction in sensitivity to the renormalization scale, but necessarily impose a common infrared bound on perturbative approximations to R(s), regardless of the infrared behavior of the true QCD couplant.