Title: Finite energy chiral sum rules and τ spectral functions
Abstract: A combination of finite energy sum rule techniques and chiral perturbation theory (\ensuremath{\chi}PT) is used in order to exploit recent ALEPH data on the non-strange \ensuremath{\tau} vector $(V)$ and axial-vector $(A)$ spectral functions with respect to an experimental determination of the \ensuremath{\chi}PT quantity ${L}_{10}.$ A constrained fit of ${R}_{\ensuremath{\tau},V\ensuremath{-}A}^{(k,l)}$ inverse moments $(l<0)$ and positive spectral moments $(l>~0)$ adjusts simultaneously ${L}_{10}$ and the nonperturbative power terms of the operator product expansion. We give explicit formulas for the first $k=0,1$ and $l=\ensuremath{-}1$ non-strange inverse moment chiral sum rules to one-loop order generalized \ensuremath{\chi}PT. Our final result reads ${L}_{10}^{r}{(M}_{\ensuremath{\rho}})=\ensuremath{-}(5.13\ifmmode\pm\else\textpm\fi{}0.19)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3},$ where the error includes experimental and theoretical uncertainties.