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

Abstract

We obtain the leading divergences at two-loop order for the decays $K_S\to \gamma \gamma$ and $K_S\to \gamma {\text{l}}^{+}{\text{l}}^{-}$ 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 $K_S\to \gamma {\text{l}}^{+}{\text{l}}^{-}$. 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.

Comments: v3: more references, typos corrected, version accepted for publication