Precise Measurement of the Mass of the $\tau$ Lepton

An optimized energy scan near the $\tau$ pair production threshold has been performed using the BESIII detector. About $24$ pb$^{-1}$ of data, distributed over four scan points, was collected. The $\tau$ mass is determined directly from the threshold behavior of the $\tau$ pair production cross section in the $e^{+}e^{-}$ collisions. The key question in the measurement is how to determine the beam energy precisely. Here the beam energy measurement system (BEMS) for BEPC-II is used to determine the beam energy. The relative systematic uncertainty of the electron and positron beam energy determination in our experiment is estimated as $2\times10^{-5}$; the relative uncertainty of the beam's energy spread is about $6\%$. This analysis is based on the combined data from the $ee$, $e\mu$, $eh$, $\mu\mu$, $\mu h$, $hh$, $e\rho$, $\mu\rho$ and $\pi\rho$ final states, where $h$ denotes a charged $\pi$ or $K$. The mass of the $\tau$ lepton is measured as \begin{eqnarray} \label{finaltaumass} m_{\tau} = 1776.91\pm0.12~^{+0.09}_{-0.12} ~\mbox{MeV/c$^2$} \end{eqnarray} which is consistent with results from any other groups included by the Particle Data Group, but has the smallest uncertainty.