High-statistics study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>τ</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:msub><mml:mi>ν</mml:mi><mml:mi>τ</mml:mi></mml:msub></mml:math>decay

We report a high-statistics measurement of the branching fraction for ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ and the invariant mass spectrum of the produced ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ system using $72.2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data recorded with the Belle detector at the KEKB asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider. The branching fraction obtained is $(25.24\ifmmode\pm\else\textpm\fi{}0.01\ifmmode\pm\else\textpm\fi{}0.39)%$, where the first error is statistical and the second is systematic. The unfolded ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ mass spectrum is used to determine resonance parameters for the $\ensuremath{\rho}(770)$, ${\ensuremath{\rho}}^{\ensuremath{'}}(1450)$, and ${\ensuremath{\rho}}^{\ensuremath{'}\ensuremath{'}}(1700)$ mesons. We also use this spectrum to estimate the hadronic ($2\ensuremath{\pi}$) contribution to the anomalous magnetic moment of the muon (${a}_{\ensuremath{\mu}}^{\ensuremath{\pi}\ensuremath{\pi}}$). Our result for ${a}_{\ensuremath{\mu}}^{\ensuremath{\pi}\ensuremath{\pi}}$ integrated over the mass range $\sqrt{s}=2{m}_{\ensuremath{\pi}}\ensuremath{-}1.8\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ is ${a}_{\ensuremath{\mu}}^{\ensuremath{\pi}\ensuremath{\pi}}=(523.5\ifmmode\pm\else\textpm\fi{}1.5(\mathrm{exp})\ifmmode\pm\else\textpm\fi{}2.6(\mathrm{Br})\ifmmode\pm\else\textpm\fi{}2.5(\mathrm{isospin}))\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, where the first error is due to the experimental uncertainties, the second is due to the uncertainties in the branching fractions, and the third is due to the uncertainties in the isospin-violating corrections.