$\Delta I=1/2$ 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 $⟨\bar{q}q{⟩}_{\mathrm{vac}}\ne 0$ 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 $\lesssim m_K$. 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 $\gamma \gamma \to \pi \pi$ and $K_S\to \gamma \gamma$. We estimate $R_{\mathrm{IR}}\approx 5$ for the nonperturbative Drell-Yan ratio $R=\sigma (e^{+}{e}^{-}\to \mathrm{hadrons})/\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})$ at ${\alpha }_{\mathrm{IR}}^{}$, and show that, in the many-color limit, $\sigma /f_0$ becomes a narrow $q\bar{q}$ 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.

Comments: 20 pages, 11 figures. This article is an expanded version of the letter arXiv:1203.1321 (2012). v4: Fig. 1 moved to second page to match PRD formatting, minor changes to text and references