  The effect of isoscalar S-wave multi-channel pion-pion scattering
($\pi\pi\to\pi\pi,K\bar{K},\eta\eta$) is considered in the analysis of data on
decays of the $\Upsilon$-meson family -- $\Upsilon(2S)\to\Upsilon(1S)\pi\pi$,
$\Upsilon(3S)\to\Upsilon(1S)\pi\pi$ and $\Upsilon(3S)\to\Upsilon(2S)\pi\pi$.
The analysis, which aims at studying the scalar mesons, is performed jointly
considering the multi-channel pion-pion scattering, which is described in our
model-independent approach based on analyticity and unitarity and using an
uniformizing variable method, and the charmonium decay processes
$J/\psi\to\phi(\pi\pi, K\bar{K})$, $\psi(2S)\to J/\psi(\pi\pi)$. Results of the
analysis confirm all our earlier conclusions on the scalar mesons. It is also
shown that in the final states of the $\Upsilon$-meson family decays (except
for the $\pi\pi$ scattering) the contribution of the coupled processes, e.g.,
$K\bar{K}\to\pi\pi$, is important even if these processes are energetically
forbidden. This is in accordance with our previous conclusions on the wide
resonances: If a wide resonance cannot decay into a channel which opens above
its mass but the resonance is strongly connected with this channel (e.g. the
$f_0(500)$ and the $K\bar}$ channel), one should consider this resonance as a
multi-channel state with allowing for the indicated channel taking into account
the Riemann-surface sheets related to the threshold branch-point of this
channel and performing the combined analysis of the considered and coupled
channels.

High Energy Physics - Phenomenology