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 SUð3ÞL × SUð3ÞR perturbation theory χPT3 by three-flavor chiral-scale perturbation theory χPTσ. In χPTσ, amplitudes are expanded at low energies and small u; d; s quark masses about an infrared fixed point αIR of three-flavor QCD. At αIR, the quark condensate h¯qqivac ≠ 0 induces nine Nambu-Goldstone bosons: π; K; η, and a 0þþ QCD dilaton σ. Physically, σ appears as the f0ð500Þ resonance, a pole at a complex mass with real part ≲mK. The ΔI ¼ 1=2 rule for nonleptonic K decays is then a consequence of χPTσ, with a KSσ coupling fixed by data for γγ → ππ and KS → γγ. We estimate RIR ≈ 5 for the nonperturbative Drell-Yan ratio R ¼ σðeþe− → hadronsÞ=σðeþe− → μþμ−Þ at αIR and show that, in the many-color limit, σ=f0 becomes a narrow q¯q state with planar-gluon corrections. Rules for the order of terms in χPTσ 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 γγ channels, partial conservation of the dilatation current is not equivalent to σ-pole dominance.