Chiral-Scale Perturbation Theory About an Infrared Fixed Point

We review the failure of lowest order chiral $SU(3)_L \times SU(3)_R$ perturbation theory $\chi$PT$_3$ to account for amplitudes involving the $f_0(500)$ resonance and $O(m_K)$ extrapolations in momenta. We summarize our proposal to replace $\chi$PT$_3$ with a new effective theory $\chi$PT$_\sigma$ based on a low-energy expansion about an infrared fixed point in 3-flavour QCD. At the fixed point, the quark condensate $\langle\bar{q}q\rangle_\mathrm{vac}\neq 0$ induces nine Nambu-Goldstone bosons: $\pi, K, \eta$ and a QCD dilaton $\sigma$ which we identify with the $f_0(500)$ resonance. We discuss the construction of the $\chi$PT$_\sigma$ Lagrangian and its implications for meson phenomenology at low-energies. Our main results include a simple explanation for the $\Delta I = 1/2$ rule in $K$-decays and an estimate for the Drell-Yan ratio in the infrared limit.