$m_{b}(m_{Z})$ from jet production at the $Z$ peak in the Cambridge algorithm

Two ratios of two- and three-jet heavy quark production rates in $e^+ e^-$ annihilation at the $Z$ peak are presented at next-to-leading order for the Cambridge jet-clustering algorithm. We study the uncertainties due to the unknowledge of higher order corrections. We show that for one of the ratios, when expressed in terms of the running mass at the $m_Z$ scale, the results are particularly stable with respect to the change in the renormalization scale. The experimental study of this observable opens the possibility for an improvement in the measurement of $m_b(m_Z)$ made by DELPHI, thus confirming the running of the $b$-quark mass.