Analysis of the decay $K_L\to {\pi }^0\gamma \gamma$ and expectations for the decays $K_L\to {\pi }^0{e}^{+}{e}^{-}$ and $K_L\to {\pi }^0{\mu }^{+}{\mu }^{-}$

A two-component model of the decay $K_L\to {\pi }^0\gamma \gamma$, in which the amplitude is the sum of a pion-loop contribution and a vector-meson-mediated term, is compared with data from the NA31 and E731 experiments. An estimate is obtained for the parameter $G$ characterizing the strength of the vector-meson component. The amplitude of $K_L\to {\pi }^0\gamma \gamma$, determined in this way, is used to calculate the $\mathrm{CP}$-conserving amplitude of the decays $K_L\to {\pi }^0{e}^{+}{e}^{-}$ and $K_L\to {\pi }^0{\mu }^{+}{\mu }^{-}$, without neglect of lepton masses. This complex amplitude (containing both absorptive and dispersive parts) is allowed to interfere with the $\mathrm{CP}$-violating amplitude expected from the short-distance interaction $s\overline{d}\to l^{+}{l}^{-}$ and from the $\mathrm{CP}$ impurity of the $K_L$ wave function. Results are obtained for the branching ratios of $K_L\to {\pi }^0{l}^{+}{l}^{-}$ and for the $\mathrm{CP}$-violating $l^{+}{l}^{-}$ asymmetry in the Dalitz plot.