Strong coupling from an improved <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi></mml:math> vector isovector spectral function

Abstract

We combine ALEPH and OPAL results for the spectral distributions measured in $\ensuremath{\tau}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\nu}}_{\ensuremath{\tau}}$, $\ensuremath{\tau}\ensuremath{\rightarrow}2{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ and $\ensuremath{\tau}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}3{\ensuremath{\pi}}^{0}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ decays with (i) recent BABAR results for the analogous $\ensuremath{\tau}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{K}^{0}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ distribution and (ii) estimates of the contributions from other hadronic $\ensuremath{\tau}$-decay modes obtained using CVC and electroproduction data, to obtain a new and more precise nonstrange, inclusive vector, isovector spectral function. The BABAR ${K}^{\ensuremath{-}}{K}^{0}$ and CVC/electroproduction results provide us with alternate, entirely data-based input for the contributions of all exclusive modes for which ALEPH and OPAL employed Monte-Carlo-based estimates. We use the resulting spectral function to determine ${\ensuremath{\alpha}}_{s}({m}_{\ensuremath{\tau}})$, the strong coupling at the $\ensuremath{\tau}$ mass scale, employing finite energy sum rules. Using the fixed-order perturbation theory (FOPT) prescription, we find ${\ensuremath{\alpha}}_{s}({m}_{\ensuremath{\tau}})=0.3077\ifmmode\pm\else\textpm\fi{}0.0075$, which corresponds to the five-flavor result ${\ensuremath{\alpha}}_{s}({M}_{Z})=0.1171\ifmmode\pm\else\textpm\fi{}0.0010$ at the $Z$ mass. While we also provide an estimate using contour-improved perturbation theory (CIPT), we point out that the FOPT prescription is to be preferred for comparison with other ${\ensuremath{\alpha}}_{s}$ determinations employing the $\overline{\mathrm{MS}}$ scheme, especially given the inconsistency between CIPT and the standard operator product expansion recently pointed out in the literature. Additional experimental input on the dominant $2\ensuremath{\pi}$ and $4\ensuremath{\pi}$ modes would allow for further improvements to the current analysis.

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