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

Abstract.

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 Tarrús 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 $O(p^{4})$ 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.