Measurement of the mass of the $tau$-lepton and an upper limit on the mass difference between $tau^+$ and $tau^-$

The mass of the $tau$ lepton has been measured in the decay mode $tau to 3pi u_tau$ using a pseudomass technique. The result obtained from $414 mathrm{fb}^{-1}$ of data collected with the Belle detector is $M_tau = (1776.61pm 0.13 {(stat.)} pm 0.35 {(sys.)})$ MeV/$c^2$. The upper limit on the relative mass difference between positive and negative $tau$ leptons is $|M_{tau^+}-M_{tau^-}|/M_{tau} < 2.8 times 10^{-4}$ at 90% confidence level.