  The quadratic pion scalar radius, <r^2>^\pi_s, plays an important role for
present precise determinations of \pi\pi scattering. The solution of the
Muskhelishvili-Omn\`es equations for the non-strange null isospin (I) pion
scalar form factor determines that <r^2>^\pi_s=(0.61\pm 0.04) fm^2. However, by
using an Omn\`es representation of this form factor, Yndur\'ain recently
obtains <r^2>^\pi_s=(0.75\pm 0.07) fm^2. A large discrepancy between both
values, given the precision, then results. We show that Yndur\'ain's method is
indeed compatible with the determinations from the Muskhelishvili-Omn\`es
equations once a zero in the scalar form factor for some S-wave I=0 T-matrices
is considered. Once this is accounted for, the resulting value is
<r^2>^\pi=(0.63\pm 0.05) fm^2.
  On the other hand, we perform a theoretical study of the reaction
\gamma\gamma\to \pi^0\pi^0 based on dispersion relations. The large source of
uncertainty for \sqrt{s}\gtrsim 0.5 GeV, due to variations in the phase used in
the Omn\`es function above the K\bar{K} threshold, is removed by taking one
more subtraction in the dispersion relation. This allows us to make sharper
predictions for the cross section so that one could use this reaction to
distinguish between different low energy \pi\pi parameterizations, once
independent experiments are available. We also study the role played by the
\sigma or f_0(600) meson in this reaction and determine its width to two
photons.
