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Status of Chiral-Scale Perturbation Theory

Chiral-scale perturbation theory $χ$PT$_σ$ has been proposed as an alternative to chiral $SU(3)_L\times SU(3)_R$ perturbation theory which explains the $ΔI = 1/2$ rule for kaon decays. It is based on a low-energy expansion about an infrared fixed point in three-flavor QCD. In $χ$PT$_σ$, quark condensation $\langle\bar q q \rangle_\mathrm{vac} \neq 0$ induces nine Nambu-Goldstone bosons: $π, K, η$ and a QCD dilaton $σ$ which we identify with the $f_0(500)$ resonance. Partial conservation of the dilatation and chiral currents constrains low-energy constants which enter the effective Lagrangian of $χ$PT$_σ$. These constraints allow us to obtain new phenomenological bounds on the dilaton decay constant via the coupling of $σ/f_0$ to pions, whose value is known precisely from dispersive analyses of $ππ$ scattering. Improved predictions for $σ\to γγ$ and the $σNN$ coupling are also noted. To test $χ$PT$_σ$ for kaon decays, we revive a 1985 proposal for lattice methods to be applied to $K \to π$ on-shell.