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

We obtain the leading divergences at two-loop order for the decays $K_{S} \to γγ$ and $K_{S} \to γ~ł^+ ł^-$ using only one-loop diagrams. We then find the double chiral logarithmic corrections to the decay branching ratio of $K_{S} \to γγ$ and to the decay rate for $K_{S} \to γ~ł^+ ł^-$. 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 $\log μ~\times$ LEC. Numerical analysis done for the process $K_{S} \to γγ$ shows that these single logarithmic effects can be sizable but come with opposite signs with respect to the double chiral logarithms.