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 $\text{alphas}$. In particular, we show how to apply this prescription when the leading coefficient or coefficients in a series in $\text{alphas}$ 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 $R_{e^{+}{e}^{-}}$, $\Gamma (B\to X_{u}e\bar{\nu })$, $\Gamma (t\to bW)$, and the ratios of the quark pole to $\text{MSbar}$ and lattice bare masses.

Comments: Lattice 2000 (Perturbation Theory), 5 pages, 7 figures, LaTeX