The reactions $\pi \pi \to \pi \pi$ and $\gamma \gamma \to \pi \pi$ 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 $\mathcal{O}(p^4)$ 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 $\pi \pi \to \pi \pi$ and $\gamma \gamma \to \pi \pi$ 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 $\gamma \gamma \to \pi \pi$ data up to $\sqrt{s}\approx 0.5$ GeV. It also yields a $\gamma \gamma \to \pi \pi$ 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.

Comments: 17 pages, 7 figures. This version, which will be published in European Physical Journal A, contains clarification and more explanation of several points, as well as additional references