Scale Setting for $\alpha_s$ Beyond Leading Order

TL;DR

This paper extends the Brodsky-Lepage-Mackenzie scale-setting method to higher orders in $oldsymbol{oldsymbol{ ext{alpha}_s}}$, providing a numerical procedure for optimal scale determination with significant corrections in key processes.

Contribution

It introduces a general method for applying the BLM scale-setting prescription at higher orders, including cases with small coefficients, and provides a numerical approach within dimensional regularization.

Findings

Significant scale corrections found for $R_{e^+ e^-}$, $ ext{Gamma}(B o X_u e ar{ u})$, $ ext{Gamma}(t o b W)$.

Method applicable when series coefficients are known or anomalously small.

Provides a numerical procedure for optimal scale setting in QCD calculations.

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.