Test of optimization procedures

On the basis of a recent O(${\alpha }_s^3$) calculation by Gorishny, Kataev, and Larin of the ratio &=${\mathrm{\sigma }}_{\mathrm{tot}({\mathrm{e}}^{\mathrm{-}}{\mathrm{e}}^{+}}$ →hadrons)/σ($e^{\mathrm{-}}$ $e^{+}$→${\mu }^{\mathrm{-}}$ ${\mu }^{+}$) we carry out the following test of two optimization procedures [principle of minimal sensitivity (PMS) and fastest apparent convergence (FAC)]: First, using O(${\alpha }_s^2$) calculations, we determine the optimal ${R_e^{\mathrm{-}}{e}^{+}}^{\mathrm{}\left(2\right)\mathrm{}}$; then, using the O(${\alpha }_s^3$) calculation, we determine the optimal ${R_e^{\mathrm{-}}{e}^{+}}^{\mathrm{}\left(3\right)\mathrm{}}$. We form the fractional difference ${(R_e^{\mathrm{-}}{e}^{+}}^{\mathrm{}\left(3\right)\mathrm{}}$ ${\mathrm{-}{R}_e^{\mathrm{-}}{e}^{+}}^{\mathrm{}\left(2\right)\mathrm{}}$) ${/R_e^{\mathrm{-}}{e}^{+}}^{\mathrm{}\left(2\right)\mathrm{}}$, and compare it with the same quantity in the usual schemes (minimal subtraction and modified minimal subtraction). We find that this difference is largest in the FAC scheme and next to largest in the PMS scheme. Our results cast doubt on the usefulness of optimization.