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 behaviour of the true QCD couplant. 
M.R. Ahmady, F.A. Chishtie, V. Elias, A.H. Fariborz, D.G.C. McKeon,   T.N. Sherry, A. Squires, T.G. Steele