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 $\phi$ 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 $\tau^\pm \rightarrow \pi^\pm \pi^0 \nu$ can be performed. Supplemented with a few pieces of information on the $\rho^0-\omega-\phi$ 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.