Pole position of the a 1 ( 1260 ) from τ -decay

We perform an analysis of the three-pion system with quantum numbers ${J}^{PC}={1}^{++}$ produced in the weak decay of $\ensuremath{\tau}$ leptons. The interaction is known to be dominated by the axial meson ${a}_{1}(1260)$. We build a model based on approximate three-body unitary and fix the free parameters by fitting it to the ALEPH data on ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ decay. We then perform the analytic continuation of the amplitude to the complex energy plane. The singularity structures related to the $\ensuremath{\pi}\ensuremath{\pi}$ subchannel resonances are carefully addressed. Finally, we extract the ${a}_{1}(1260)$ pole position ${m}_{p}^{({a}_{1}(1260))}\ensuremath{-}i{\mathrm{\ensuremath{\Gamma}}}_{p}^{({a}_{1}(1260))}/2$ with ${m}_{p}^{({a}_{1}(1260))}=(1209\ifmmode\pm\else\textpm\fi{}{4}_{\ensuremath{-}9}^{+12})\text{ }\text{ }\mathrm{MeV}$, ${\mathrm{\ensuremath{\Gamma}}}_{p}^{({a}_{1}(1260))}=(576\ifmmode\pm\else\textpm\fi{}1{1}_{\ensuremath{-}20}^{+89})\text{ }\text{ }\mathrm{MeV}$.