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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) chiralperturbation theory ($\chi$PT) by including the $\sigma$ meson as an additionalexplicit degree of freedom, as proposed by Soto, Talavera, and Tarr\'us andothers. In this effective field theory, denoted $\chi$PT$_S$, the $\sigma$meson's well-established mass and decay width are not sufficient to properlyrenormalize its self energy. At $\mathcal{O}(p^4)$ another low-energy constantappears in the dressed $\sigma$-meson propagator; we adjust it so that theisoscalar pion-pion scattering length is also reproduced. We compare theresulting amplitudes for the $\pi\pi\rightarrow\pi\pi$ and$\gamma\gamma\rightarrow\pi\pi$ reactions to data from threshold through theenergies at which the $\sigma$-meson resonance affects observables. Theleading-order (LO) $\pi \pi$ amplitude reproduces the $\sigma$-meson poleposition, the isoscalar $\pi \pi$ scattering lengths and $\pi \pi$ scatteringand $\gamma \gamma \rightarrow \pi \pi$ data up to $\sqrt{s} \approx 0.5$ GeV.It also yields a $\gamma\gamma\rightarrow\pi\pi$ amplitude that obeys the Wardidentity. The value obtained for the $\pi^0$ polarizability is, however, onlyslightly larger than that obtained in standard $\chi$PT.

The reactions $\pi\pi\rightarrow\pi\pi$ and $\gamma\gamma\rightarrow\pi\pi$ in $\chi$PT with an isosinglet scalar resonance