The ground state of bottomium to two loops and higher
  We consider the properties of the ground state of bottomium. The $\Upsilon$ mass is evaluated to two loops, and including leading higher order [$O(\alpha_s^5\log\alpha_s)$] and $m_c^2/m_b^2$ corrections. This allows us to present updated values for the pole mass and $\bar{MS}$ mass of the $b$ quark: $m_b=5022\pm58$ MeV, for the pole mass, and $\bar{m}_b(\bar{m}_b)=4286\pm36$ MeV for the $\bar{MS}$ one. The value for the \msbar mass is accurate including and $O(\alpha_s^3)$ corrections and leading orders in the ratio $m_c^2/m_b^2$. We then consider the wave function for the ground state of $\bar{b}b$, which is calculated to two loops in the nonrelativistic approximation. Taking into account the evaluation of the matching coefficients by Beneke and Signer one can calculate, in principle, the width for the decay $\Upsilon\to e^+e^-$ to order $\alpha_s^5$. Unfortunately, given the size of the corrections it is impossible to produce reliable numbers. The situation is slightly better for the ground state of toponium, where a decay width into $e^+e^-$ of 11 -- 14 keV is predicted. 
F. J. Yndurain