Improvement of Renormalization-Scale Uncertainties Within Empirical Determinations of the b-Quark Mass

Abstract

Accurate determinations of the MS-bar b-quark mass $m_b(m_b)$ from $σ(e^+e^-\to{\rm hadrons})$ experimental data currently contain three comparable sources of uncertainty; the experimental uncertainty from moments of this cross-section, the uncertainty associated with $α_s(M_z)$, and the theoretical uncertainty associated with the renormalization scale. Through resummation of all logarithmic terms explicitly determined in the perturbative series by the renormalization-group (RG) equation, it is shown that the renormalization-scale dependence is virtually eliminated as a source of theoretical uncertainty in $m_b(m_b)$. This resummation also reduces the estimated effect of higher-loop perturbative contributions, further reducing the theoretical uncertainties in $m_b(m_b)$. Furthermore, such resummation techniques improve the agreement between the values of the MS-bar b-quark mass extracted from the various moments of $R(s)=σ(e^+e^-\to{\rm hadrons})/σ_{pt}$ [$σ_{pt}=4πα^2/(3s)$], obviating the need to choose an optimummoment for determining $m_b(m_b)$. Resummation techniques are also shown to reduce renormalization-scale dependence in the relation between b-quark MS-bar and pole mass and in the relation between the pole and $1S$ mass.