Search

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.

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 Tarrus and others. In this effective field theory, denoted $ \chi$ PTS, 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.

We present procedures based on Bayesian statistics for estimating, from data, the parameters of effective field theories (EFTs). The extraction of low-energy constants (LECs) is guided by theoretical expectations in a quantifiable way through the specification of Bayesian priors. A prior for natural-sized LECs reduces the possibility of overfitting, and leads to a consistent accounting of different sources of uncertainty. A set of diagnostic tools are developed that analyze the fit and ensure that the priors do not bias the EFT parameter estimation. The procedures are illustrated using representative model problems, including the extraction of LECs for the nucleon mass expansion in SU(2) chiral perturbation theory from synthetic lattice data., Comment: 40 pages, 40 figures. Final version to appear in JPhysG