$ΔI=1/2$ rule for kaon decays derived from QCD infrared fixed point

Abstract: This article gives details of our proposal to replace ordinary chiral $SU(3)L\times SU(3)_R$ perturbation theory $\chi$PT$_3$ by 3-flavor chiral-scale perturbation theory $\chi$PT$\sigma$. In $\chi$PT$\sigma$, amplitudes are expanded at low energies and small $u,d,s$ quark masses about an infrared fixed point $\alpha^{}\mathrm{IR}$ of 3-flavor QCD. At $\alpha^{}_\mathrm{IR}$, the quark condensate $\langle \bar{q}q\rangle_{\mathrm{vac}} \not= 0$ induces nine Nambu-Goldstone bosons: $\pi, K, \eta$ and a $0^{++}$ QCD dilaton $\sigma$. Physically, $\sigma$ appears as the $f_{0}(500)$ resonance, a pole at a complex mass with real part $\lesssim m_K$. The $\Delta I=1/2$ rule for nonleptonic $K$-decays is then a consequence of $\chi$PT$\sigma$, with a $K_S\sigma$ coupling fixed by data for $\gamma\gamma\rightarrow\pi\pi$ and $K{S} \to \gamma\gamma$. We estimate $R_\mathrm{IR} \approx 5$ for the nonperturbative Drell-Yan ratio $R = \sigma(e^{+}e^{-}\rightarrow\mathrm{hadrons})/ \sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})$ at $\alpha^{}_\mathrm{IR}$, and show that, in the many-color limit, $\sigma/f_0$ becomes a narrow $q\bar{q}$ state with planar-gluon corrections. Rules for the order of terms in $\chi$PT$_\sigma$ loop expansions are derived in Appendix A, and extended in Appendix B to include inverse-power Li-Pagels singularities due to external operators. This relates to an observation that, for $\gamma\gamma$ channels, partial conservation of the dilatation current is not equivalent to $\sigma$-pole dominance.