Problems with extractingmsfrom flavor breaking in hadronic τ decays

A numerical error is pointed out in the existing expression for the $\mathcal{O}\left({\alpha }_s^2\right)$ longitudinal component of the squared-mass $\mathrm{}(D=2\mathrm{})\mathrm{}$ contribution to the hadronic τ decay rate. The corrected version is found to be such that, to $\mathcal{O}\left({\alpha }_s^2\right),$ each term in the resulting series is larger than the previous one, hence ruling out the direct use of flavor breaking in hadronic τ decays as a means of extracting the strange quark mass. An alternate approach, in which one uses the model spectral functions previously developed for the strangeness-changing scalar channel (and employed in alternate sum rule analyses of $m_s)$ as input to the τ decay analysis, is shown to provide mutual consistency checks for the two different methods. The results for $m_s$ implied by τ decay data and the existing model spectral function are also presented.