The HLS approach to $(g-2{)}_{\mu }$~: A Solution to the "$\tau$ versus $e^{+}{e}^{-}$" Puzzle

The Hidden Local Symmetry (HLS) Model provides a framework able to encompass several physical processes and gives a unfied description of these in an energy range extending up to the $\varphi$ mass. Supplied with appropriate symmetry breaking schemes, the HLS Model gives a broken Effective Lagrangian (BHLS). The BHLS Lagrangian gives rise to a fit procedure in which a simultaneous description of the $e^{+}{e}^{-}$ annihilations to ${\pi }^{+}{\pi }^{-}$, ${\pi }^0\gamma$, $\eta \gamma$, ${\pi }^{+}{\pi }^{-}{\pi }^0$, $K^{+}{K}^{-}$, $K_L{K}_S$ and of the dipion spectrum in the decay ${\tau }^{\pm }\to {\pi }^{\pm }{\pi }^0\nu$ can be performed. Supplemented with a few pieces of information on the ${\rho }^0-\omega -\varphi$ system, the $\tau$ dipion spectrum is shown to predict accurately the pion form factor in $e^{+}{e}^{-}$ annihilations. Physics results derived from global fits involving or excluding the $\tau$ dipion spectra are found consistent with each others. Therefore, no obvious mismatch between the $\tau$ and $e^{+}{e}^{-}$ physics properties arises and the $\tau -e^{+}{e}^{-}$ puzzle vanishes within the broken HLS Model.