Electro-weak $\pi\pi$ form factor and $\pi\pi$ scattering: towards a phenomenological tool

The weak two-pion form factor $F_V^{\pi\pi}$ is described as the product of a weak kernel $\cal{K}_W$ by a strong function $\Theta_{\pi\pi}^P$, determined directly from $\pi\pi$ scattering data. As the latter accounts at once for all effects associated with resonances, intermediate $K\bar{K}$ loops, and other possible inelasticities present in $\pi\pi$ scattering, the need of modeling is restricted to $\cal{K}_W$ only. The procedure proposed allows one to asses the weak kernel directly, which has a dominant cut beginning at the $K\bar{K}$ threshold. Even the simplest vector-meson-dominance choice for $\cal{K}_W$ already yields a good qualitative description of $F_V^{\pi\pi}$. The energy sector below $0.8$ GeV is quite well reproduced when a precise theoretical chiral perturbation $\pi\pi$ amplitude is used as input, together with the single free parameter $F_V G_V/F^2=1.20$. The inclusion of kaon loops, along well established lines and using few parameters, produces a good description of the form factor in the entire energy range allowed by $\tau$ decays. This indicates that the replacement of modeling by direct empirical scattering information can also be useful in the construction of theoretical tools to be used in analysesof hadronic heavy meson decay data.