Bottom Quark Mass from Upsilon Mesons

The bottom quark pole mass $M_b$ is determined using a sum rule which relates the masses and the electronic decay widths of the $\Upsilon$ mesons to large $n$ moments of the vacuum polarization function calculated from nonrelativistic quantum chromodynamics. The complete set of next-to-next-to-leading order (i.e. ${\cal{O}}(\alpha_s^2, \alpha_s v, v^2)$ where $v$ is the bottom quark c.m. velocity) corrections is calculated and leads to a considerable reduction of theoretical uncertainties compared to a pure next-to-leading order analysis. However, the theoretical uncertainties remain much larger than the experimental ones. For a two parameter fit for $M_b$, and the strong $\bar{{MS}}$ coupling $\alpha_s$, and using the scanning method to estimate theoretical uncertainties, the next-to-next-to-leading order analysis yields 4.74 GeV $\le M_b\le 4.87$ GeV and $0.096 \le \alpha_s(M_z) \le 0.124$ if experimental uncertainties are included at the 95% confidence level and if two-loop running for $\alpha_s$ is employed. $M_b$ and $\alpha_s$ have a sizeable positive correlation. For the running $\bar{{MS}}$ bottom quark mass this leads to 4.09 GeV $\le m_b(M_{\Upsilon(1S)}/2)\le 4.32$ GeV. If $\alpha_s$ is taken as an input, the result for the bottom quark pole mass reads 4.78 GeV $\le M_b\le 4.98$ GeV (4.08 GeV $\le m_b(M_{\Upsilon(1S)}/2)\le 4.28$ GeV) for $0.114\lsim \alpha_s(M_z)\le 0.122$. The discrepancies between the results of three previous analyses on the same subject by Voloshin, Jamin and Pich, and K\"uhn et al. are clarified. A comprehensive review on the calculation of the heavy quark-antiquark pair production cross section through a vector current at next-to-next-to leading order in the nonrelativistic expansion is presented.