Analytical $\pi\pi$ scattering amplitude and the light scalars-II

In the paper Phys. Rev. {\bf D83}, 054008 (2011) we constructed the $\pi\pi$ scattering amplitude $T^0_0$ with regular analytical properties in the $s$ complex plane, describing both experimental data and the results based on chiral expansion and Roy equations. Now the results obtained during development of our work are presented. We dwell on questions dealing with the low $\sigma-f_0$ mixing, inelasticity description and the kaon loop model for $\phi\to \gamma(\sigma+f_0)$ reaction, and show a number of new fits. In particular, we show that the minimization of the $\sigma-f_0$ mixing results in the four-quark scenario for light scalars: the $\sigma$(600) coupling with the $K\bar K$ channel is suppressed relatively to the coupling with the $\pi\pi$ channel, and the $f_0$(980) coupling with the $\pi\pi$ channel is suppressed relatively to the coupling with the $K\bar K$ channel. The correct analytical properties of the $\pi\pi$ scattering amplitude are reached with the help of rather complicated background function. We also suggest much more simple background parameterization, practically preserving the resonance features, which is comfortable for experimental data analysis, but allows to describe the results based on chiral expansion and Roy equations only on the real $s$ axis.