Scale Setting for $\alpha_s$ Beyond Leading Order

Abstract

We present a general procedure for applying the scale-setting prescription of Brodsky, Lepage and Mackenzie to higher orders in the strong coupling constant $\alphas$. In particular, we show how to apply this prescription when the leading coefficient or coefficients in a series in $\alphas$ are anomalously small. We give a general method for computing an optimum scale numerically, within dimensional regularization, and in cases when the coefficients of a series are known. We find significant corrections to the scales for $R_{e^+ e^-}$, $\Gamma(B \to X_u e \bar{\nu})$, $\Gamma(t \to b W)$, and the ratios of the quark pole to $\MSbar$ and lattice bare masses.