Speaker
Description
We discuss recent results leading to a new understanding of one of the two
main approaches to determining $\alpha_s$ from finite-energy sum rule analyses of non-strange hadronic $\tau$ decay data. This approach involves fits to sets of differently weighted spectral integrals with a single common upper endpoint, $s=s_0$, in which several in-principle-present, higher-dimension condensate contributions are unavoidably neglected. We show that, contrary to current understanding, $\alpha_s$ in this approach is obtained from a purely perturbative analysis of a smaller subset of FESRs, with the retained lower-dimension condensates, $C_D$, playing no role whatsoever in this determination. Results for the retained $C_D$ are, moreover, shown to be produced by redundant, one-spectral-integral-in, one-fit-parameter-outmatchings susceptible to generically large theoretical systematic uncertainties. Internal self-consistency checks in the restricted subspace of the FESRs responsible for determining $\alpha_s$ are shown to reveal discrepancies comparable in size to currently quoted errors. The utility of alternate analyses employing, not just multiple weights, but also multiple $s_0$, in dealing with these issues is highlighted. We also discuss the relative roles of theory-side non-perturbative condensate and duality violating contributions, as well as further improvements to the analyses and internal self-consistency tests improved experimental results for the two $4\pi$ distributions would make possible.
