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

TL;DR

This paper presents a phenomenological approach to describe the weak two-pion form factor using scattering data, reducing modeling complexity and improving the analysis of hadronic decay processes.

Contribution

It introduces a method to directly extract the weak kernel from scattering data, minimizing modeling and enhancing the understanding of the form factor across energy ranges.

Findings

Good qualitative description with vector-meson dominance

Accurate reproduction below 0.8 GeV with chiral perturbation theory

Effective inclusion of kaon loops for full energy range

Abstract

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.