Validity of Chiral Perturbation Theory for $K^0-{\bar{K}}^0$ Mixing

Chiral perturbation theory relates the $\left|\Delta S\right|=2\mathrm{}$ matrix element $〈{\bar{K}}^0|{\left(\bar{s}d\right)}_{V-A}\times {\left(\bar{s}d\right)}_{V-A}|K^0〉$ to the $\left|\Delta S\right|=1\mathrm{}$ matrix element $〈{\pi }^{+}{\pi }^0|{\left(\bar{s}d\right)}_{V-A}{\left(\bar{u}u\right)}_{V-A}+{\left(\bar{s}u\right)}_{V-A}\times {\left(\bar{u}d\right)}_{V-A}-{\left(\bar{s}d\right)}_{V-A}{\left(\bar{d}d\right)}_{V-A}|K^{+}〉$. The latter matrix element is measured in $K^{+}\to {\pi }^{+}{\pi }^0$ decay and the former matrix element is relevant for the predictions that the standard model makes for $\mathrm{CP}$ nonconservation in $K^0-{\bar{K}}^0$ mixing. In this paper the corrections of order $m_K^4\ln m_K^2$ to lowest-order chiral perturbation theory (i.e., order $m_K^2$) for these matrix elements are computed. The correction is large for the $\left|\Delta S\right|=2\mathrm{}$ matrix element indicating a breakdown of chiral perturbation theory.