$\sigma(e^+e^-\to{hadrons})$ and the Heavy Quark Masses

The precise data for the total cross section $\sigma(e^+e^-\to{hadrons})$ from the charm threshold region, when combined with the evaluation of moments with three loop accuracy, lead to a direct determination of the short distance $\bar{\rm MS}$ charm quark mass $m_c(m_c)=1.304(27)$. Applying the same approach to the bottom quark we obtain $m_b(m_b)=4.191(51)$ GeV. A complementary method for the determination of $m_b$ is based of the analysis of the $\Upsilon(1S)$ system which is confronted with a next-to-next-to-next-to-leading order calculation of the corresponding energy level. This leads to $m_b(m_b)=4.346(70)$ GeV.