This article gives details of our proposal to replace ordinary chiral $SU(3{)}_L\times SU(3{)}_R$ perturbation theory $\chi {\mathrm{PT}}_3$ by three-flavor chiral-scale perturbation theory $\chi {\mathrm{PT}}_{\sigma }$. In $\chi {\mathrm{PT}}_{\sigma }$, amplitudes are expanded at low energies and small $u,d,s$ quark masses about an infrared fixed point ${\alpha }_{\mathrm{IR}}$ of three-flavor QCD. At ${\alpha }_{\mathrm{IR}}$, the quark condensate $⟨\overline{q}q{⟩}_{\text{vac}}\ne 0$ 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 $\lesssim \hspace{0.17em}{m}_K$. The $\mathrm{\Delta }I=1/2$ rule for nonleptonic $K$ decays is then a consequence of $\chi {\mathrm{PT}}_{\sigma }$, with a $K_S\sigma$ coupling fixed by data for $\gamma \gamma \to \pi \pi$ and $K_S\to \gamma \gamma$. We estimate $R_{\mathrm{IR}}\approx 5$ for the nonperturbative Drell-Yan ratio $R=\sigma (e^{+}{e}^{-}\to \text{hadrons})/\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})$ at ${\alpha }_{\mathrm{IR}}$ and show that, in the many-color limit, $\sigma /f_0$ becomes a narrow $q\overline{q}$ state with planar-gluon corrections. Rules for the order of terms in $\chi {\mathrm{PT}}_{\sigma }$ 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.

• Received 5 March 2012

DOI:https://doi.org/10.1103/PhysRevD.91.034016