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 $χ$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 $χ$PT$_3$ with a new effective theory $χ$PT$_σ$ 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: $π, K, η$ and a QCD dilaton $σ$ which we identify with the $f_0(500)$ resonance. We discuss the construction of the $χ$PT$_σ$ Lagrangian and its implications for meson phenomenology at low-energies. Our main results include a simple explanation for the $ΔI = 1/2$ rule in $K$-decays and an estimate for the Drell-Yan ratio in the infrared limit.

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