  The complementarity between Chiral Perturbation Theory and the Linear Sigma
Model is exploited to study $\pi^0\pi^0$ production in $\phi$ radiative decays,
where the effects of the $f_0(980)$ scalar resonance, and those of its more
controversial $\sigma(500)$ partner, should become manifest via the $\phi\to
K^+ K^- (\gamma)\to\pi^0\pi^0\gamma $ decay chain. The recently reported data
on $\phi\to\pi^0\pi^0\gamma$ coming from the VEPP-2M $e^+ e^-$ collider in
Novosibirsk and the DA$\Phi$NE $\phi$-factory in Frascati can be reasonably
described in our approach, which we propose as a promising first step towards
more detailed analyses. The $f_0(980)$ contribution, which appears as a
moderately narrow peak at the high part of the dipion mass spectrum, can be
interpreted as the isoscalar member of the scalar nonet with a large $f_0
K\bar{K}$ coupling and an $f_0\pi\pi$ coupling suppressed by almost ideal
$\sigma$-$f_0$ mixing. Indeed, the mixing angle in the flavour basis is found
to be $\phi_{S}\approx -6^\circ$, if the $f_0$-propagator is approximated by a
simple Breit-Wigner, or $\phi_{S}\approx -9^\circ$, if an improved two-channel
analysis is performed. The $\sigma(500)$ resonance, which is then strongly
coupled to pion pairs, yields a tiny contribution because, in our approach, its
coupling to kaon pairs is proportional to $m^2_\sigma-m^2_K$ and thus quite
small.
