  The lowest-lying resonance in the QCD spectrum is the $0^{++}$ isoscalar
$\sigma$ meson, also known as the $f_0(500)$. We augment SU(2) chiral
perturbation theory ($\chi$PT) by including the $\sigma$ meson as an additional
explicit degree of freedom, as proposed by Soto, Talavera, and Tarr\'us and
others. In this effective field theory, denoted $\chi$PT$_S$, the $\sigma$
meson's well-established mass and decay width are not sufficient to properly
renormalize its self energy. At $\mathcal{O}(p^4)$ another low-energy constant
appears in the dressed $\sigma$-meson propagator; we adjust it so that the
isoscalar pion-pion scattering length is also reproduced. We compare the
resulting amplitudes for the $\pi\pi\rightarrow\pi\pi$ and
$\gamma\gamma\rightarrow\pi\pi$ reactions to data from threshold through the
energies at which the $\sigma$-meson resonance affects observables. The
leading-order (LO) $\pi \pi$ amplitude reproduces the $\sigma$-meson pole
position, the isoscalar $\pi \pi$ scattering lengths and $\pi \pi$ scattering
and $\gamma \gamma \rightarrow \pi \pi$ data up to $\sqrt{s} \approx 0.5$ GeV.
It also yields a $\gamma\gamma\rightarrow\pi\pi$ amplitude that obeys the Ward
identity. The value obtained for the $\pi^0$ polarizability is, however, only
slightly larger than that obtained in standard $\chi$PT.
