Problems With Extracting $m_s$ from Flavor Breaking in Hadronic $τ$ Decays

A numerical error is pointed out in the existing expression for the order $\alpha_s^2$ longitudinal component of the squared-mass (D=2) contribution to the hadronic $\tau$ decay rate. The corrected version is found to be such that, to order $\alpha_s^2$, each term in the resulting series is larger than the previous one, ruling out the direct use of flavor breaking in hadronic $\tau$ decays as a means of extracting the strange quark mass. An alternate approach, using 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 $\tau$ decay analysis, is shown to provide mutual consistency checks for the two different methods. The results for $m_s$ implied by $\tau$ decay data, and the existing model spectral function are also presented.