Leading Chiral Logarithms of $K_{S} \to \gamma \gamma$ and $K_{S} \to \gamma~l^+ l^-$ at two Loops

We obtain the leading divergences at two-loop order for the decays $K_{S} \to \gamma \gamma$ and $K_{S} \to \gamma~\l^+ \l^-$ using only one-loop diagrams. We then find the double chiral logarithmic corrections to the decay branching ratio of $K_{S} \to \gamma \gamma$ and to the decay rate for $K_{S} \to \gamma~\l^+ \l^-$. It turns out that these effects are numerically small and therefore make a very small enhancement on the branching ratio and decay rate. We also derive an expression for the corrections of type $\log \mu~\times$ LEC. Numerical analysis done for the process $K_{S} \to \gamma \gamma$ shows that these single logarithmic effects can be sizable but come with opposite signs with respect to the double chiral logarithms.