Optimized perturbation theory

Conventional perturbation theory gives different results in different renormalization schemes, a problem which is especially serious in quantum chromodynamics (QCD). I propose a theoretical resolution of this ambiguity which uses the full renormalization-group invariance of the theory. The idea is that, in any kind of approximation scheme which does not respect the known invariances of the exact result, the "optimum" approximant is the one that is "most invariant," i.e., least sensitive to variations in the unphysical parameters. I discuss this principle in several examples, including the Halliday-Suranyi expansion for the anharmonic oscillator. Turning to massless field theories, I identify the unphysical variables which label a particular renormalization scheme as the renormalization point $\mu$, and the $\beta$-function coefficients. I describe how perturbative approximations depend on these unphysical variables, and show how to find the stationary point which represents the "optimum" result. Certain renormalization-scheme invariants, in one-to-one correspondence with the perturbation-series coefficients, arise naturally in the analysis. An application to the $e$,$\mu$ magnetic moments in QED provides a partial test of these ideas, with encouraging results. I suggest possible further theoretical developments, and advocate the method as a sound basis for quantitative QCD phenomenology.